JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 25, 2018

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Volume 25, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

July 15, 2018

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (sixteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $800, Electronic OPEN ACCESS. Individual:Print $400. For any other part of the world add $150 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2018 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

Fractional Differential Equations Nonlinear Analysis, Fractional Dynamics Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis.

Ravi P. Agarwal Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 tel: 361-593-2600 [email protected] Differential Equations, Difference Equations, Inequalities

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Jerry L. Bona Department of Mathematics The University of Illinois at Chicago 851 S. Morgan St. CS 249 Chicago, IL 60601 e-mail:[email protected] Partial Differential Equations, Fluid Dynamics

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Computational Mathematics Ram N. Mohapatra Department of Mathematics University of Central Florida Orlando, FL 32816-1364 tel.407-823-5080 [email protected] Real and Complex Analysis, Approximation Th., Fourier Analysis, Fuzzy Sets and Systems

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Choonkil Park Department of Mathematics Hanyang University Seoul 133-791 S. Korea, [email protected] Functional Equations

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Locally and globally small Riemann sums and Henstock integral of fuzzy-number-valued functions Muawya Elsheikh Hamida ∗, Luoshan Xu a , Zengtai Gong b School of Mathematical Science, Yangzhou University, Yangzhou 225002, China College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R.China a

b

Abstract In this paper, we first define and discuss the locally small Riemann sums (LSRS) for fuzzy-numbervalued functions. In addition the necessary and sufficient conditions have been obtained for a fuzzy-number-valued function which has (LSRS), i.e., if a fuzzy-number-valued function is Henstock (H) integrable on [a, b] then it has (LSRS) and the converse is always true. Secondly, the globally small Riemann sums (GSRS) for fuzzynumber-valued functions is defined and discussed, and the necessary and sufficient conditions have been given for a fuzzy-number-valued function which has (GSRS), i.e., if a fuzzy-number-valued function is (H) integrable on [a, b] then it has (GSRS) and the converse is always true. Finally, by Egorov, s Theorem, we obtain the dominated convergence theorem for globally small Riemann sums (GSRS) of fuzzy-number-valued functions. Keywords: Fuzzy numbers; fuzzy integrals; (H) integral; (LSRS); (GSRS).

1

Introduction

Since the concept of fuzzy sets was firstly introduced by Zadeh in 1965 [22], it has been studied extensively from many different aspects of the theory and applications, such as fuzzy topology, fuzzy analysis, fuzzy decision making and fuzzy logic, information science and so on. fuzzy integrals of fuzzy-number-valued functions have been studied by many authors from different points of views, including Goetschel [9], Nanda [15], Kaleva [12], Wu [18, 19] and other authors [1, 3, 4, 5, 6, 8]. The locally and globally small Riemann sums have been introduced by many authors from different points of views. In 1986, Schurle characterized the Lebesgue integral in (LSRS) (locally small Riemann sums) property [16]. The (LSRS) property has been used to characterized the Perron (P ) integral on [a, b] [17]. By considering the equivalency between the (P ) integral and the Henstock-Kurzweil (HK) integral, the (LSRS) property has been used to characterized the (HK) integral on [a, b] [13]. The (LSRS) property brought a research to have global characterization on the Riemann sums of an (HK) integrable function on [a, b]. This research has been done by considering the following fact: Every (HK) integrable function on [a, b] is measurable, however, there is no guarantee the boundedness of the function. A measurable function f is (HK) integrable on [a, b] depends on it behaves on the set of x in which |f (x)| is large, i.e. |f (x)| ≥ N for some N . This fact has been characterized in (GSRS) (globally small Riemann sums) property [13]. The (GSRS) property involves one characteristic of the primitive of an (HK) integrable function. That is the primitive of the (HK) integral on [a, b] is ACG∗ (generalized strongly absolutely continuous) on [a, b]. This is not a simple concept. In 2015, Indrati [11] introduced a countably Lipschitz condition of a function which is simpler than the ACG∗ , and proved that the (HK) integrable function or it, s primitive could be characterized in countably Lipschitz condition. Also, by considering the characterization of the (HK) integral in the (GSRS) property, it showed that the relationship between (GSRS) property and countably Lipschitz condition of an (HK) integrable function on [a, b]. In this paper, we first define and discuss the locally small Riemann sums (LSRS) for fuzzy-number-valued functions. In addition the necessary and sufficient conditions have been obtained for a fuzzy-number-valued ∗ Corresponding author. Tel.: +8613218977118. E-mail address: [email protected], [email protected] (M.E. Hamid), [email protected] (L.S. Xu) and [email protected] (Z.T. Gong).

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Elsheikh Hamid et al 11-18

J. COMPUTATIONAL APPLICATIONS, VOL. 25,small NO.1, Riemann 2018, COPYRIGHT EUDOXUS PRESS, LLC M.E. Hamid, L.S.ANALYSIS Xu, Z.T. AND Gong: Locally and globally sums and2018 Henstock integral...

function which has (LSRS), i.e., if a fuzzy-number-valued function is (H) integrable on [a, b] then it has (LSRS) and the converse is always true. Secondly, the globally small Riemann sums (GSRS) for fuzzy-number-valued functions is defined and discussed, and the necessary and sufficient conditions have been given for a fuzzy-numbervalued function which has (GSRS), i.e., if a fuzzy-number-valued function is (H) integrable on [a, b] then it has (GSRS) and the converse is always true. Finally, by Egorov, s Theorem, we obtain the dominated convergence theorem for globally small Riemann sums (GSRS) of fuzzy-number-valued functions. The paper is organized as follows, in Section 2 we shall review the relevant concepts and properties of fuzzy sets and the definition of (H) integrals for fuzzy-number-valued functions. Section 3 is devoted to discussing the locally small Riemann sums (LSRS) of fuzzy-number-valued functions. In section 4 we shall investigate the globally small Riemann sums (GSRS) of fuzzy-number-valued functions by Egorov, s Theorem, we obtain the dominated convergence theorem for globally small Riemann sums (GSRS) of fuzzy-number-valued functions. The last section provides Conclusions.

2

Preliminaries

Definition 2.1 [10, 13] Let δ : [a, b] → R+ be a positive real-valued function. P = {[xi−1 , xi ]; ξi } is said to be a δ-fine division, if the following conditions are satisfied: (1) a = x0 < x1 < x2 < ... < xn = b; (2) ξi ∈ [xi−1 , xi ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi ))(i = 1, 2, · · · , n). For brevity, we write P = {[u, v]; ξ}, where [u, v] denotes a typical interval in P and ξ is the associated point of [u, v]. Definition 2.2 [10, 13] A real-valued function f (x) is said to be (H) integrable to G on [a, b] if for every ε > 0 there is a function δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} we have X f (ξ)(v − u) − G < ε (2.1) (P )

Rb

As usual, we write (RH) a f (x)dx = G and f (x) ∈ RH[a, b]. For the results about fuzzy number space E 1 . we recall that E 1 = {u : R → [0, 1] : u satisfies (1)-(4) below}: (1) u is normal, i.e., there exists a x0 ∈ R such that u(x0 ) = 1; (2) u is a convex fuzzy set, i.e., u(rx + (1 − r)y) > min(u(x), u(y)), x, y ∈ R, r ∈ [0, 1]; (3) u is upper semi-continuous; (4) cl{x ∈ R : u(x) > 0} is compact, where clA denotes the closure of A. For 0 < r 6 1, denote [u]r = {x : u(x) > r}. Then from (1)-(4), it follows that the r−level set [u]r is a close interval for all r ∈ [0, 1] (refer to [2, 7, 9, 12, 14, 20, 21]). We write ur = [u]r = [ur− , ur+ ] or [u− (r), u+ (r)]. For u, v ∈ E 1 , k ∈ R, the addition and scalar multiplication are defined by the equations: r r [u + v]r = [u]r + [v]r , i.e., ur− + v− = [u + v]r− and ur+ + v+ = [u + v]r+ ;

[ku]r = k[u]r , i.e., [ku]r− = min{kur− , kur+ } and [ku]r+ = max{kur− , kur+ }, respectively. r r Define D(u, v) = sup d([u]r , [v]r ) = sup max{|ur− − v− |, |ur+ − v+ |}, where d is Hausdorff metric. Furtherr∈[0,1]

r∈[0,1]

more, we write + k˜ ukE 1 = D(˜ u, ˜ 0) = sup max{|u− λ |, |uλ |}. λ∈[0,1]

Notice that k · kE 1 = D(·, ˜ 0) doesn’t stands for the norm of E 1 . r r For u, v ∈ E 1 , u 6 v means ur− 6 v− , ur+ 6 v+ (see [2, 7, 9, 12, 14, 20, 21]). Using the results of [2, 7, 9, 12, 14, 20, 21], we recall that: (1) (E 1 , D) is a complete metric space, (2) D(u + w, v + w) = D(u, v), (3) D(u + v, w + e) 6 D(u, w) + D(v, e), (4) D(ku, kv) = |k|D(u, v), k ∈ R, 12

Elsheikh Hamid et al 11-18

J. COMPUTATIONAL APPLICATIONS, VOL. 25,small NO.1, Riemann 2018, COPYRIGHT EUDOXUS PRESS, LLC M.E. Hamid, L.S.ANALYSIS Xu, Z.T. AND Gong: Locally and globally sums and2018 Henstock integral...

(5) D(u + v, ˜ 0) 6 D(u, ˜ 0) + D(v, ˜ 0), (6) D(u + v, w) 6 D(u, w) + D(v, ˜ 0), where ˜ 0 = χ{0} and u, v, w, e ∈ E 1 . Definition 2.3 [18] A fuzzy-number-valued function f˜(x) is said to be (H) integrable to A˜ ∈ E 1 if for every ε > 0 there is a function δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X ˜ 0 there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have

X

f˜(ξ)(v − u) E 1 < ε, (3.1) whenever P = {[u, v]; ξ} is a δ-fine division of an interval [r, s] ⊂ (t − δ(t), t + δ(t)), t ∈ [r, s] and Σ sums over P . If there exists a z ∈ E 1 such that x = y + z, then we call z the H− difference of x and y, denoted by x − y. Lemma 3.1 [18] Let f˜ ∈ F H[a, b] and F˜ be the primitive of f˜(x) then F˜ satisfies the H− difference. Lemma 3.2 (Henstock Lemma). If a fuzzy-number-valued function f˜ : [a, b] → E 1 is (H) integrable on [a, b] with primitive F˜ , i.e., for every ε > 0 there is a positive function δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X X D( f˜(ξ)(v − u), F˜ (u, v)) < ε. (3.2) P P Then for any sum of parts from , we have 1

X X D( f˜(ξ)(v − u), F˜ (u, v)) < ε. 1

(3.3)

1

The proof is similar to the Theorem 3.7 [13]. Theorem 3.1 If f˜(x) is (H) integrable on [a, b] then it has LSRS. Proof Let F˜ be the primitive of f˜(x). Given ε > 0 there is a δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X X
D f˜(ξ)(v − u), F˜ (u, v) < ε. (3.4) Where F˜ (u, v) = F˜ (v) − F˜ (u). By the continuity of F˜ at ξ,
D F˜ (u), F˜ (v) < ε whenever [u, v] ⊂ (ξ − δ(ξ), ξ + δ(ξ)). Therefore for t ∈ [a, b] and any δ-fine division P = {[u, v]; ξ} of [r, s] ⊂ (t − δ(t), t + δ(t)), we have X X

X





f˜(ξ)(v − u) E 1 ≤ D f˜(ξ)(v − u), F˜ (u, v) + D F˜ (r), F˜ (s)
0 there is a positive function δ(ξ) > 0 such that whenever P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } are two δ-fine divisions, we have X X
D f˜(ξ1 )(v1 − u1 ), f˜(ξ2 )(v2 − u2 ) < ε. (3.5) (P1 )

(P2 )

Theorem 3.2 If a fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS on [a, b] then f˜(x) is (H) integrable on any closed sub-interval C ⊂ (a, b). (Where C = [r, s]). Proof A fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS means that for every ε > 0 there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have

X

f˜(ξ)(v − u) E 1 < ε, (3.6) whenever P = {[u, v]; ξ} is a δ-fine division of an interval C ⊂ (t − δ(t), t + δ(t)), t ∈ C and Σ sums over P . (i) If there t ∈ [a, b] with C ⊂ (t − δ(t), t + δ(t)) we have the following discussion: (1) If t ∈ C then for every ε > 0 there is a two δ-fine divisions P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } on C, such that X X
D f˜(ξ1 )(v1 − u1 ), f˜(ξ2 )(v2 − u2 ) < ε. (3.7) (P1 )

(P2 )

According to the Cauchy criterion, then f˜(x) is (H) integrable on C. (2) If t ∈ / C then there is a closed interval E ⊂ (t − δ(t), t + δ(t)), with the result that t ∈ E and C ⊂ E (where E = [g, h] ). As a result, for every ε > 0 there is a two δ-fine divisions P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } on E, such that X X
D f˜(ξ1 )(v1 − u1 ), f˜(ξ2 )(v2 − u2 ) < ε. (3.8) (P1 )

(P2 )

According to the Cauchy criterion, then f˜(x) is (H) integrable on E. Because C ⊂ E and f˜(x) is (H) integrable on E then f˜(x) is (H) integrable on C. (ii) If C * (t − δ(t), t + δ(t)) then there is a positive function δ on [a, b] which resulted in the presence that P = {(Ci , ti ) : i = 1, 2, · · · , k} is a δ-fine division of the interval C. It follows that f˜(x) is (H) integrable on Ci for i = 1, 2, · · · , k. Then f˜(x) is (H) integrable on C. This completes the proof.  Corollary 3.1 If a fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS on [a, b] then f˜(x) is (H) integrable on C for any simple set C ⊂ (a, b). Notice that a simple set C means that there exists finite closed sub-interval Ci which belongs to (a, b) such k S that C = Ci . i=1

Theorem 3.3 If a fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS on [a, b] then f˜(x) is (H) integrable on [a, b]. Proof A fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS then for every ε > 0 there is δ ∗ (ξ) > 0 such that for every t ∈ [a, b], we have

X

f˜(ξ)(v − u) E 1 < ε, (3.9) whenever P = {[u, v]; ξ} is a δ ∗ -fine division of an interval C ⊂ (t − δ(t), t + δ(t)), t ∈ C and Σ sums over P . According to the Corollary 3.1, f˜(x) is (H) integrable on C for any simple set C ⊂ (a, b). T S Rows set {Ei }, Ei Ej = φ, ∀i 6= j with property (a, b) = Ei , Ei is a closed interval. Thus for above ε > 0, there is a positive numbers n0 with property [ µ{[a, b] − Ei } < ε, (3.10) i≤n0

where µ is Lebesgue measure. 14

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J. COMPUTATIONAL APPLICATIONS, VOL. 25,small NO.1, Riemann 2018, COPYRIGHT EUDOXUS PRESS, LLC M.E. Hamid, L.S.ANALYSIS Xu, Z.T. AND Gong: Locally and globally sums and2018 Henstock integral...

For any i, there is a positive function δi such that for any δi -fine division on Ei , we have Z X D( f˜(ξ)(v − u), (H) f˜(x)dx) < ε.

(3.11)

Ei

Define a positive function δ by the formula:  ∗ 1   min{δ (ξ), 2 d(ξ, ∂[a, b])} δ(ξ) = ∗   min{δ (ξ), δi (ξ)},

S

if ξ ∈

Ei ,

i>n0

S

if ξ ∈

Ei .

i≤n0

T For each C = {C} = {C1 , C2 , · · · , Ck } with Cj = Ei Q (where Q = [u, v]), for one i ≤ n0 and one Q with {[u, v]; ξ} is a δ-fine division and ξ ∈ (a, b), we have (i) If Cj = Ei for i ≤ n0 . Because f˜(x) is (H) integrable on Ei and f˜(x) is (H) integrable on Cj consequently k S f˜(x) is (H) integrable on Cj . Selected a positive function δ∗ with δ∗ (ξ) = min{δj (ξ) : j = 1, 2, · · · , k}, then j=1

for each δ∗ -fine division P = {[u, v]; ξ} on

k S

Cj , we have

j=1

Z D (H)

k S

f˜(x)dx,

X


f˜(ξ)(v − u) < ε.

(3.12)

Cj

j=1

Thus obtained:

X

C (H)

Z C

f˜(x)dx E 1

Z ≤

D (H)

k S

f˜(x)dx,

X

Cj

k

X


X

f˜(ξ)(v − u) + f˜(ξ)(v − u) E 1 j=1

j=1


N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X

f˜(ξ)(v − u) E 1 < ε, (4.1) kf˜(ξ)kE 1 >n

where the

P

is taken over P and for which kf˜(ξ)kE 1 > n.

Theorem 4.1 Let f˜(x) be (H) integrable to F˜ (a, b) on F H[a, b] and F˜n (a, b) the integral of f˜n (x) on F H[a, b], where f˜n (x) = f˜(x) when ||f˜(x)||E 1 6 n and ˜ 0 otherwise. If F˜n (a, b) → F˜ (a, b) as n → ∞ then f˜(x) has GSRS. Proof Given ε > 0 there is a δn (ξ) > 0 such that for every δn -fine division P = {[u, v]; ξ} of [a, b], we have X D( f˜n (ξ)(v − u), F˜n (a, b)) < ε, (4.2) where (F H)

Rb a

f˜n (x)dx = F˜n (a, b). D(

X

f˜(ξ)(v − u), F˜ (a, b)) < ε,

(4.3)

Rb

where (F H) a f˜(x)dx = F˜ (a, b). Choose N so that whenever n > N D(F˜n (a, b), F˜ (a, b)) < ε.

(4.4)

Therefore for n > N and δn -fine division P = {[u, v]; ξ} of [a, b], we have X X X

f˜n (ξ)(v − u), f˜(ξ)(v − u)) f˜(ξ)(v − u) E 1 = D( kf˜(ξ)kE 1 >n

6

X X D( f˜n (ξ)(v − u), F˜n (a, b)) + D(F˜n (a, b), F˜ (a, b)) + D(F˜ (a, b), f˜(ξ)(v − u))


N and a suitably chosen δ-fine division P = {[u, v]; ξ}, we have

6

D(F˜n (a, b), F˜m (a, b)) X D(F˜n (a, b), f˜(ξ)(v − u)) + D(

+



kf˜(ξ)kE 1 6n

X

f˜(ξ)(v − u)

f˜(ξ)(v − u), F˜m (a, b))

kf˜(ξ)kE 1 6m E1

kf˜(ξ)kE 1 >n


m

4ε.

That is, F˜n (a, b) converge to a fuzzy number, say F˜ (a, b), as n → ∞. Again, for suitably chosen N and δ(ξ) and for every δ-fine division P = {[u, v]; ξ}, we have X D( f˜(ξ)(v − u), F˜ (a, b)) 6 D(F˜ (a, b), F˜N (a, b)) X X

+ D(F˜N (a, b), f˜(ξ)(v − u)) + f˜(ξ)(v − u) E 1 kf˜(ξ)kE 1 6N


N

3ε.

That is, f˜(x) is (F H) integrable on [a, b]. This completes the proof.

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Elsheikh Hamid et al 11-18

J. COMPUTATIONAL APPLICATIONS, VOL. 25,small NO.1, Riemann 2018, COPYRIGHT EUDOXUS PRESS, LLC M.E. Hamid, L.S.ANALYSIS Xu, Z.T. AND Gong: Locally and globally sums and2018 Henstock integral...

Theorem 4.3 Let f˜n (x) ∈ F H[a, b], n = 1, 2, 3 · · · and satisfy: (1) lim f˜n (x) = f˜(x) almost everywhere in [a, b]; n→∞ (2) there exists a Lebesgue (L) integrable (H integrable) function h(x) on [a, b] such that D(f˜n (x), f˜m (x)) < h(x).

(4.5)

Then, f˜n (x) has GSRS on [a, b] uniformly for any n. Naturally, f˜ is (SF H) integrable on [a, b]. Furthermore, Z b Z b ˜ lim (SF H) fn (x)dx = (SF H) f˜(x)dx. (4.6) n→∞

a

a

Rx Proof Let ε > 0. Since H(x) = (L) a h(t)dt is absolutely continuous on [a, b], there exists a positive number P η > 0 such that |H(bi ) − H(ai )| < ε whenever {[ai , bi ]} is a finite collection of non-overlapping intervals in [a, b] P that satisfy (bi − ai ) < η. Since lim f˜n (x) = f˜(x) almost everywhere in [a, b], and n→∞

D(f˜n , f˜)

= sup max{|(fn (x))r− − (f (x))r− |, |(fn (x))r+ − (f (x))r+ |} r∈[0,1] r

r

r

r

= sup max{|(fn (x))−k − (f (x))−k |, |(fn (x))+k − (f (x))+k |} rk ∈[0,1]

is a sequence of Lebesgue (L) measurable functions, where rk ∈ [0, 1] is the set of rational numbers, by Egorov’s Theorem, there exists an open set G with L(G) < η such that lim f˜n (x) = f˜(x) uniformly for x ∈ [a, b]\G. Then, n→∞ there is an natural number N , such that for any n, m > N , and for any x ∈ [a, b]\G, we have D(f˜n (x), f˜m (x)) < ε. Since h(x) is (H) integrable on [a, b], there is a δh (ξ) > 0 such that for any δh -fine division P = {[u, v]; ξ} of [a, b], we have Z b X h(ξ)(v − u) − (L) h(t)dt < ε. (4.7) a

Define

 δh (ξ), δ(ξ) = δ(ξ), satisfying (ξi − δ(ξi ), ξi + δ(ξi )) ⊂ G,

if ξ ∈ [a, b] \ G, if ξ ∈ [a, b].

Then, it follows that for a δ-fine division P0 = {[xi−1 , xi ]; ξi } of [a, b], X X D( f˜n (ξi )(xi − xi−1 ), f˜m (ξi )(xi − xi−1 )) X X ≤ D( f˜n (ξi )(xi − xi−1 ), f˜m (ξi )(xi − xi−1 )) ξi ∈[a,b]\G

+

D(

X

ξi ∈[a,b]\G

f˜n (ξi )(xi − xi−1 ),

ξi ∈G

X

≤

X

f˜m (ξi )(xi − xi−1 ))

ξi ∈G

D(f˜n (ξi ), f˜m (ξi ))(xi − xi−1 ) +

X

D(f˜n (ξi ), f˜m (ξi ))(xi − xi−1 )

ξi ∈G

ξi ∈[a,b]\G

Z Z X ε(b − a) + h(ξi )(xi − xi−1 ) − h(t)dt + h(t)dt


0 such that for any δ -fine D(F˜n [a, b], A) N1 N1 division P = {[u, v]; ξ} of [a, b], for any n > NN1 , we have X D( f˜n (ξ)(v − u), F˜n [a, b]) X ≤ D(F˜n [a, b], F˜N1 [a, b]) + D( f˜N1 (ξ)(v − u), F˜N1 [a, b]) X X + D( f˜n (ξ)(v − u), f˜N1 (ξ)(v − u))
0 such that ⟨T x − ty, x − y⟩ ≥ α∥x − y∥2 , ∀x, y ∈ H1 . (iii) T is said to be β-inverse strongly monotone(or, β-ism), if there exists a constant β > 0 such that ⟨T x − ty, x − y⟩ ≥ β∥T x − T y∥2 , ∀x, y ∈ H1 . (iv) T is said to be firmly nonexpansive, if ⟨T x − ty, x − y⟩ ≥ ∥T x − T y∥2 , ∀x, y ∈ H1 . Next, let M : H1 → 2H1 be a multi-valued mappings. We recall the following definitions: • M is called monotone if for all x, y ∈ H1 , u ∈ M x and v ∈ M y such that ⟨x − y, u − v⟩ ≥ 0. ∗ Corresponding

author: Email address: [email protected] (R. Wangkeeree) and [email protected] (K. Rattanaseeha) and [email protected] (R. Wangkeeree). 1

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2

R. WANGKEEREE, K. RATTANASEEHA AND R. WANGKEEREE

• A monotone mappings M : H1 → 2H1 is maximal if the Graph(M ) is not properly contained in the graph of any other monotone mapping. For more precisely, a monotone mappings M is maximal if and only if for (x, u) ∈ H1 ×H1 , ⟨x−y, u−v⟩ ≥ 0, for every (y, v) ∈ Graph(M ) implies that u ∈ M x, where Graph(M ) := {(x, y) ∈ H1 ×H1 : y ∈ M x}. Let M : H1 → 2H1 be a multi-valued mappings. Then, the resolvent mapping associated with M , is defined by JλM (x) := (I + λM )−1 (x), ∀x ∈ H1 for some λ > 0, where I stands identity operator on H1 . We note that for all λ > 0 the resolvent operator JλM is single-valued, nonexpansive and firmly nonexpansive. For a given single-valued operator F : H1 → H1 , Hartman and Stampacchia [12] introduced the variational inequality problem (in short, VIP) : { Find x∗ ∈ C such that (VIP) ⟨F (x∗ ), x − x∗ ⟩ ≥ 0, ∀x ∈ C. The VIP is a powerful tool to investigate and study a wide class of unrelated problems arising in industrial, regional, physical, pure and applied sciences in a unified and general framwork. Variational inequalities have been extended and generallized in several direction using novel and new techniques. The following existence result of solutions for VIP can be found in [12]. Let H1 be a real Hilbert space, C a nonempty, compact and convex subset H1 . Then, if F : C → H1 is continuous, there exists x∗ ∈ C such that ⟨F (x∗ ), x − x∗ ⟩ ≥ 0, ∀x ∈ C. Recently, in 2011, Moudafi [24] introduced the following split monotone variational inclusion problem (in short, SMVIP): { Find x∗ ∈ H1 such that 0 ∈ f1 (x∗ ) + B1 (x∗ ), (SMVIP) y ∗ = Ax∗ ∈ H2 solves 0 ∈ f2 (y ∗ ) + B2 (y ∗ ). where B1 : H1 → 2H1 is a multi-valued mappings on a Hilbert space H1 , B2 : H2 → 2H2 is a multivalued mappings on a Hilbert space H2 , A : H1 → H2 is a bounded linear operator, f1 : H1 → H1 and f2 : H2 → H2 are two given single-valued operators. If f1 ≡ 0 and f2 ≡ 0, then SMVIP reduces to the following split variational inclusion problem (in short, SVIP) : Find x∗ ∈ H1 such that 0 ∈ B1 (x∗ ),

(1.2)

y ∗ = Ax∗ ∈ H2 solves 0 ∈ B2 (y ∗ ).

(1.3)

and When looked separately, (1.2) is the variational inclusion problem and we denoted its solution set by SOLVIP(B1 ). The SVIP (1.2)-(1.3) constitutes a pair of variational inclusion problems which have to be solved so that the image y ∗ = Ax∗ under a given bounded linear operator A, of the solution x∗ of SVIP (1.2) in H1 is the solution of another SVIP (1.3) in another space H2 , we denote the solution set of SVIP (1.3) by SOLVIP(B2 ). The solution set of The SVIP (1.2)-(1.3) is denoted by Γ : {x∗ ∈ H1 : x∗ ∈ SOLVIP(B1 ) and Ax∗ ∈ SOLVIP(B2 )}. l,Recently, Byrne et al. [3] studied the weak and strong convergence of the following iterative method for SVIP. For given x0 ∈ H1 , compute iterative sequence {xn } generated by the following scheme. ( ) xn+1 = JλB1 (x) xn + γA∗ (JλB1 − I)Axn , (1.4) for λ > 0 and A∗ is the adjoint of A, L = ∥A∗ A∥ and γ ∈ (0, L2 ). It is proved, in [3], that the sequence {xn } generated by (1.4) converges strongly to x∗ which is the solution of SVIP. Very recently, Kazami and Rizvi [13] studied and analyzed the strong convergence of the iterative method for approximating a common solution of SVIP and FPP for a nonexpansive mapping in a real Hilbert space. Let g : H1 → H1 be a contraction mapping with constant α ∈ (0, 1) and S : H1 → H1 be a nonexpansive mapping. For a given x0 ∈ H1 arbitrarily, let {un } and {xn } be generated by ( ) un = JλB1 xn + γA∗ (JλB2 − I)Axn ; xn+1

= αn g(xn ) + (1 − αn )Sun ,

20

(1.5)

WANGKEEREE et al 19-31

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

THE GENERAL ITERATIVE METHODS FOR SPLIT VARIATIONAL INCLUSION PROBLEM

3

where λ > 0 and γ ∈ (0, L1 ), L is the spectral radius of the operator A∗ A and A∗ is the adjoint of A and {αn } is a sequence in (0, 1). They proved that, under some certain conditions imposed on the parameters {αn }, the sequences {un } and {xn } both converge strongly to z ∈ Fix(S) ∩ Γ, where z = PFix(S)∩Γ g(z). On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, e.g., [11, 27, 28, 29] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space: 1 θ(x) = min ⟨Gx, x⟩ − ⟨x, b⟩, (1.6) x∈C 2 where G is a linear bounded operator, C is the fixed point set of a nonexpansive mapping T and b is a given point in H. Let H be a real Hilbert space. Recall that a linear bounded operator B is strongly positive if there is a constant γ¯ > 0 with property ⟨Gx, x⟩ ≥ γ¯ ∥x∥2 for all x ∈ H.

(1.7)

Marino and Xu [25] introduced the following general iterative scheme basing on the viscosity approximation method introduced by Moudafi [14]: xn+1 = (I − αn G)T xn + αn βf (xn ), n ≥ 0.

(1.8)

where G is a strongly positive bounded linear operator on H. They proved that if the sequence {αn } of parameters satisfies appropriate conditions, then the sequence {xn } generated by (1.8) converges strongly to the unique solution of the variational inequality ⟨(G − βf )x∗ , x − x∗ ⟩ ≥ 0, x ∈ C

(1.9)

which is the optimality condition for the minimization problem 1 min ⟨Gx, x⟩ − h(x), x∈C 2 where h is a potential function for βf (i.e., h′ (x) = βf (x) for x ∈ H). Motivated by the work of Kazmi and Rizvi [13] and Moudafi [24] and Marino and Xu [25] and by the ongoing research in this direction, we suggest and analyze a general iterative method for approximating a common solution of SVIP and FPP which solves the variational inequality (1.9). More precisely, let g : H1 → H1 be a contraction mapping with constant α ∈ (0, 1), S : H1 → H1 be a nonexpansive mapping and G : H1 → H1 be a strongly positive, bounded linear operator with constant µ and µ 0 0 and γ ∈ (0, L is the spectral radius of the operator A A and A is the adjoint of A and {αn } is a sequence in (0, 1) and B1 : H1 → 2H1 , B2 : H2 → 2H2 two multi-valued mappings on H1 , and H2 , respectively. We prove that the iterative method (1.10) converges strongly to a common element of SVIP and FPP for a nonexpansive mapping, which is a solution of a certain optimization problem related to a strongly positive linear operator. The result presented in this paper generalize the corresponding results of Kazmi and Rizvi [13] and Moudafi [24], Marino and Xu [25] and many others. 1 L ),

2. Preliminaries For a real Hilbert space H1 with the norm ∥ · ∥ and the inner product ⟨·, ·⟩, it is well known that for any λ ∈ (0, 1), ∥λx + (1 − λ)y∥2 = λ∥x∥2 + (1 − λ)∥y∥2 − λ(1 − λ)∥x − y∥2 , ∀x, y ∈ H1 .

(2.1)

Further, every nonexpansive operator T : H1 → H1 satisfies, for all (x, y) ∈ H1 × H1 , the inequality ⟨(x − T (x)) − (y − T (y)), T (y) − T (x)⟩ ≤

21

1 ∥(T (x) − x) − (T (y) − y)∥2 2

(2.2)

WANGKEEREE et al 19-31

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

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R. WANGKEEREE, K. RATTANASEEHA AND R. WANGKEEREE

and therefore, we get, for all (x, y) ∈ H1 × Fix(T ), 1 ∥T (x) − x∥2 . (2.3) 2 A mapping T : H1 → H1 is said to be averaged if and only if it can be written as the average of the identity mapping and a nonexpansive mapping, i.e., ⟨(x − T (x), (y − T (x)⟩ ≤

T := (1 − α)I + αS where α ∈ (0, 1) and S : H1 → H1 is nonexpansive and I is the identity operator on H1 Proposition 2.1. (i) If T = (1 − α)S + αV , where S : H1 → H1 is averaged, V : H1 → H1 is nonexpansive and α ∈ (0, 1), then T is averaged. (ii) The composite of finitely many averaged mapping is averaged. (iii) If the mapping are averaged and have a nonempty common fixed point, then N ∩

F ix(Ti ) = F ix(T1 , T2 , . . . , TN ).

i=1

(iv) If T is τ − ism, then for γ > 0, γT is

τ − ism γ

1 2 For every point x ∈ H1 , there exists a unique nearest point in C, denoted by PC x, such that

(v) T is averaged if and only if , its complement I − T is τ − ism for some τ >

∥x − PC x∥ ≤ ∥x − y∥, ∀y ∈ C. PC is called the (nearest point or metric) projection of H1 onto C. In addition, PC x is characterized by the following properties: PC x ∈ C and ⟨x − PC x, y − PC x⟩ ≤ 0,

(2.4)

∥x − y∥2 ≥ ∥x − PC x∥2 + ∥y − PC x∥2 , ∀x ∈ H1 , y ∈ C. Recall that a mapping T : H1 → H1 is said to be firmly nonexpansive mapping if

(2.5)

∥T x − T y∥2 ≤ ⟨T x − T y, x − y⟩, ∀x, y ∈ H1 . It is well known that PC is a firmly nonexpansive mapping of H1 onto C and satisfies ∥PC x − PC y∥2 ≤ ⟨x − y, PC x − PC y⟩, ∀x, y ∈ H1 .

(2.6)

If G an α−inverse-strongly monotone mapping of C into H1 , then it is obvious that G is continuous. We also have that for all x, y ∈ C and λ > 0, ∥(I − λG)x − (I − λG)y∥2

1 α −Lipschitz

=

∥x − y − λ(Gx − Gy)∥2

=

∥x − y∥2 − 2λ⟨Gx − Gy, x − y⟩ + λ2 ∥Gx − Gy∥2

≤

∥x − y∥2 + λ(λ − 2α)∥Gx − Gy∥2

(2.7)

So, if λ ≤ 2α, then I − λG is a nonexpansive mapping of C into H1 . Next, we denote weak convergence and strong convergence by notations ⇀ and →, respectively. A space X is said to satisfy Opials condition [31] if for each sequence {xn } in X which converges weakly to a point x ∈ X, we have lim inf ∥xn − x∥ < lim inf ∥xn − y∥, ∀y ∈ X, y ̸= x. n→∞

n→∞

Lemma 2.2. [25] Let H1 be a Hilbert space, C be a nonempty closed convex subset of H, and f : H1 → H1 be a contraction with coefficient 0 < α < 1, and G be a strongly positive linear bounded operator with coefficient γ¯ > 0. Then, for 0 < γ < αγ¯ , ⟨x − y, (G − γf )x − (G − γf )y⟩ ≥ (¯ γ − γα)∥x − y∥2 , x, y ∈ H1 . That is, G − γf is strongly monotone with coefficient γ¯ − γα. Lemma 2.3. [25] Assume G is a strongly positive linear bounded operator on a Hilbert space H1 with coefficient γ¯ > 0 and 0 < ρ ≤ ∥G∥−1 . Then ∥I − ρG∥ ≤ 1 − ρ¯ γ.

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Lemma 2.4. [23] Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn } be a sequence in [0, 1] with 0 < lim inf n→∞ βn ≤ lim supn→∞ βn < 1. Suppose that xn+1 = (1−βn )yn +βn xn for all integers n ≥ 0 and lim supn→∞ (∥yn+1 − yn ∥ − ∥xn+1 − xn ∥) ≤ 0. Then limn→∞ ∥yn − xn ∥ = 0. Lemma 2.5. [31] Let H1 be a Hilbert space, C a closed convex subset of H1 , and S : C → C a nonexpansive mapping with F (S) ̸= ∅. If {xn } is a sequence in C weakly converging to x ∈ C and if {(I − S)xn } converges strongly to y, then (I − S)x = y; in particular, if y = 0, then x ∈ F ix(S). Lemma 2.6. [26] Assume {an } is a sequence of nonnegative real numbers such that an+1 ≤ (1 − σn )an + δn , where {σn } is a sequence in (0, 1) and {δn } is a sequence in R such that ∑∞ (1) n=1 σn = ∞; ∑∞ δn (2) lim supn→∞ ≤ 0 or n=1 |δn σn | < ∞. σn Then limn→∞ αn = 0. 3. Main Results In this section, we prove a strong convergence theorem for the general iterative methods for approximating the common element of SVIP and FPP which is the unique solution for the variational inequality (1.9). First, we have the following technical lemma, which is immediately consequence of the definition of resolvent mapping: Lemma 3.1. SVIP is equivalent to find x∗ ∈ H1 such that y ∗ = Ax∗ ∈ H2 , x∗ = JλB1 (x∗ ) and y ∗ = JλB2 (y ∗ ) for some λ > 0. Theorem 3.2. Let H1 and H2 be two real Hilbert spaces. Let A : H1 → H2 be a bounded linear operator. Assume that B1 : H1 → 2H1 and B2 : H2 → 2H2 are maximal monotone mappings. Let S : H1 → H1 be a nonexpansive mapping such that Fix(S) ∩ Γ ̸= ∅. Let f : H1 → H1 be a contraction mapping with constant α ∈ (0, 1) and G : H1 → H1 a strongly positive, bounded linear operator with µ constant µ such that ∥G∥ = 1, and 0 < β < α . For given ∀x0 ∈ H1 , let the sequences {un } and {xn } be generated by (1.10), where {αn } is a sequence in (0, 1) satisfying the following conditions : (i) lim ∑∞n→∞ αn = 0; (ii) ∑n=1 αn = ∞ and ∞ (iii) n=1 |αn − αn−1 | < ∞. Then the sequences {un } and {xn } both converge strongly to z ∈ Fix(S) ∩ Γ, where z = PFix(S)∩Γ (I − G + βf )(z). Moreover, z is a unique solution of the variational inequality (1.9). Proof We observe that PFix(S)∩Γ (I − G + βf ) is a contraction. Indeed, applying Lemma 2.3 with ∥G∥ = 1, we have, ∥PFix(S)∩Γ (I − G + βf )(x) − PFix(S)∩Γ (I − G + βf )(y)∥

≤

∥(βf + (I − G))(x) − (βf + (I − G))(y)∥

≤ β∥f (x) − f (y)∥ + ∥I − G∥∥x − y∥ ≤ γα∥x − y∥ + (1 − γ)∥x − y∥ ≤ (1 − (γ − αβ)) ∥x − y∥, for all x, y ∈ H1 . Therefore, Banach’s Contraction Mapping Principle guarantees that PFix(S)∩Γ (I − G + βf ) has a unique fixed point, say z ∈ H1 . That is, z = QF (γf + (I − G))(z). Next, we devide the proof into five steps as follows. Step 1. We first show that the sequences {xn } is bounded. Let p ∈ F ix(S) ∩ Γ, then we have that p = JλB1 p, Ap = JλB2 (Ap) and Sp = p. So, we have

2 ( ) ( )

∥un − p∥2 = JλB1 xn + γA∗ JλB2 − I Axn − p

2 ( ) ( )

= JλB1 xn + γA∗ JλB2 − I Axn − JλB1 p

2 ( )

≤ xn + γA∗ JλB2 − I Axn − p

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≤

(

2 ) ⟨ ( ) ⟩

∥xn − p∥2 + γ 2 A∗ JλB2 − I Axn + 2γ xn − p, A∗ JλB2 − I Axn .

(3.1)

⟨( ) ( ) ⟩ ∥xn − p∥2 + γ 2 JλB2 − I Axn , AA∗ JλB2 − I Axn ⟨ ( ) ⟩ + 2γ xn − p, A∗ JλB2 − I Axn .

(3.2)

It follow that ∥un − p∥2

Since γ2

≤

⟨( ) ( ) ⟩ ⟨( ) ( ) ⟩ JλB2 − I Axn , AA∗ JλB2 − I Axn ≤ Lγ 2 JλB2 − I Axn , JλB2 − I Axn

(

2 )

= Lγ 2 JλB2 − I Axn

and using (2.3), we have ⟨ ( ) ⟩ 2γ xn − p, A∗ JλB2 − I Axn

(3.3)

⟨ ( ) ⟩ = 2γ A(xn − p), JλB2 − I Axn ⟨ ( ) ( ) ( ) ⟩ = 2γ A(xn − p) + JλB2 − I Axn − JλB2 − I Axn , JλB2 − I Axn { }

2 ⟨ ( ) ⟩ ( )

B2 B2 B2 = 2γ Jλ Axn − Ap, Jλ − I Axn − Jλ − I Axn {

2 (

2 ( ) ) 1

B2

≤ 2γ

Jλ − I Axn − JλB2 − I Axn 2

(

2 )

≤ −γ JλB2 − I Axn .

}

From the inequalities (3.2), (3.3) and (3.4), we can conclude that

(

2 )

∥un − p∥2 ≤ ∥xn − p∥2 + γ(Lγ − 1) JλB2 − I Axn .

(3.4)

(3.5)

Since γ ∈ (0, L1 ), we obtain ∥un − p∥2 ≤ ∥xn − p∥2 , which implies that ∥un − p∥ ≤ ∥xn − p∥.

(3.6)

Therefore ∥xn+1 − p∥ = ∥αn βf (xn ) + (I − αn G)Sun − p∥ ≤ βαn ∥f (xn ) − f (p)∥ + ∥(I − αn G)∥∥Sun − p∥ + αn ∥βf (p) − Gp∥ ≤ βαn α∥xn − p∥ + (1 − αn µ)∥un − p∥ + αn ∥βf (p) − Gp∥ ≤ βαn α∥xn − p∥ + (1 − αn µ)∥xn − p∥ + αn ∥βf (p) − Gp∥ ( ) = βαn α + (1 − αn µ) ∥xn − p∥ + αn ∥βf (p) − Gp∥ ( ) ∥βf (p) − Gp∥ = 1 − αn (µ − βα) ∥xn − p∥ + αn (µ − βα) (µ − βα) { } ∥βf (p) − Gp∥ ≤ max ∥xn − p∥, . µ − βα By induction, we have

} ∥βf (p) − Gp∥ ∥xn − p∥ ≤ max ∥x1 − p∥, , ∀n ≥ 1. µ − βα {

(3.7)

Hence {xn } is bounded and consequently, we deduce that {un }, {f (xn) } and {Sun } are bounded. Step 2. We show that the sequences {xn } is asymptotically regular, i.e., ∥xn+1 − xn ∥ → 0 as n → ∞.

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For each n ∈ N, we notice that

( ) ∥xn+1 − xn ∥ = ∥αn βf (xn ) + (I − αn G)Sun − αn−1 βf (xn−1 ) + (I − αn−1 G)Sun−1 ∥ ≤ ∥(I − αn G)(Sun − Sun−1 ) − (αn − αn−1 )GSun−1 ( ) +βαn f (xn ) − f (xn−1 ) + β(αn − αn−1 )f (xn−1 )∥ ≤ (1 − αn µ)∥Sun − Sun−1 ∥ + |αn − αn−1 |∥GSun−1 ∥ +βαn ∥(f (xn ) − f (xn−1 )∥ + β|αn − αn−1 |∥f (xn−1 )∥ ≤ (1 − αn µ)∥un − un−1 ∥ + |αn − αn−1 |∥GSun−1 ∥ +βααn ∥xn − xn−1 ∥ + β|αn − αn−1 |∥f (xn−1 )∥ =

(1 − αn µ)∥un − un−1 ∥ + βααn ∥xn − xn−1 ∥ ( ) +|αn − αn−1 | ∥Gun−1 ∥ + β∥f (xn−1 )∥

≤ (1 − αn µ)∥un − un−1 ∥ + βααn ∥xn − xn−1 ∥ + |αn − αn−1 |K (3.8) } B1 1 where K = sup ∥Gun−1 ∥ + β∥f (xn−1 )∥ : n ∈ N . Since, for γ ∈ (0, L ), the mapping Jλ (I + γA∗ (JλB2 − I)A) is averaged and hence nonexpansive, therefore

( ) ( ) ( ( ) )

∥un − un−1 ∥ = JλB1 xn + γA∗ JλB2 − I Axn − JλB1 xn−1 + γA∗ JλB2 − I Axn−1

( ) ) ( ) ) ( (

≤ JλB1 I + γA∗ JλB2 − I A xn − JλB1 I + γA∗ JλB2 − I A xn−1 {

≤ ∥xn − xn−1 ∥.

(3.9)

It follows from (3.8) and (3.9) that

( ) ∥xn+1 − xn ∥ ≤ 1 − αn (µ − βα) ∥xn − xn−1 ∥ + |αn − αn−1 |K.

By applying Lemma 2.6 with βn = αn (µ − βα) and δn = |αn − αn−1 |K, we obtain lim ∥xn+1 − xn ∥ = 0.

(3.10)

n→∞

Step 3. We show that ∥xn+1 − p∥ → 0 as n → ∞. For each n ∈ N, ∥xn+1 − p∥2

= ∥αn βf (xn ) + (I − αn G)Sun − p∥2 = ∥αn (βf (xn ) − Gp) + (I − αn G)(Sun − p)∥2 ≤

∥(I − αn G)(Sun − p)∥2 + 2αn ⟨βf (xn ) − Gp, xn+1 − p⟩

≤ (1 − αn µ)2 ∥Sun − p∥2 + 2αn β⟨f (xn ) − f (p), xn+1 − p⟩ +2αn ⟨βf (xn ) − Gp, xn+1 − p⟩ ≤ (1 − αn µ)2 ∥un − p∥2 + 2αn β⟨f (xn ) − f (p), xn+1 − p⟩ +2αn ⟨βf (p) − Gp, xn+1 − p⟩ ≤ (1 − αn µ)2 ∥un − p∥2 + 2αn βα∥xn − p∥∥xn+1 − p∥ +2αn ∥βf (p) − Gp∥∥xn+1 − p∥. Thus, from (3.5), we obtain ∥xn+1 − p∥2

≤ (1 − αn µ)2

{

(

2 )

∥xn − p∥2 + γ(Lγ − 1) JλB2 − I Axn

(3.11) }

+2αn βα∥xn − p∥∥xn+1 − p∥ +2αn ∥βf (p) − Gp∥∥xn+1 − p∥

(

2 ) ( ) ( )

= 1 − 2αn µ + (αn µ)2 ∥xn − p∥2 + (1 − αn µ)2 γ(Lγ − 1) JλB2 − I Axn +2αn βα∥xn − p∥∥xn+1 − p∥ ≤

+2αn ∥βf (p) − Gp∥∥xn+1 − p∥

(

2 ) ( )

∥xn − p∥2 + αn µ2 ∥xn − p∥2 − 1 − αn µ)2 (γ(1 − Lγ) JλB2 − I Axn

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+2αn βα∥xn − p∥∥xn+1 − p∥ +2αn ∥βf (p) − Gp∥∥xn+1 − p∥.

(3.12)

Therefore,

(

2 )

(1 − αn µ)2 (γ(1 − Lγ) JλB2 − I Axn ≤ ∥xn − p∥2 − ∥xn+1 − p∥2 + αn µ2 ∥xn − p∥2 + 2αn βα∥xn − p∥∥xn+1 − p∥ + 2αn ∥βf (p) − Gp∥∥xn+1 − p∥ ( ) ≤ ∥xn+1 − xn ∥ ∥xn − p∥ + ∥xn+1 − p∥ + αn µ2 ∥xn − p∥2 + 2αn βα∥xn − p∥∥xn+1 − p∥ + 2αn ∥βf (p) − Gp∥∥xn+1 − p∥.

Since γ(1 − Lγ) > 0, and αn → 0 and ∥xn+1 − xn ∥ → 0 as n → ∞, we have

(

)

lim JλB2 − I Axn = 0.

(3.13)

n→∞

Furthermore, using (3.7), (3.11) and γ ∈ (0, L1 ), we notice that

2 ( ) ( )

∥un − p∥2 = JλB1 xn + γA∗ JλB2 − I Axn − p

2 ( ) ( )

= JλB1 xn + γA∗ JλB2 − I Axn − JλB1 p ⟨ ( ) ⟩ ) ≤ un − p, xn + γA∗ JλB2 − I Axn − p ( ) 1{ ∥un − p∥2 + ∥xn + γA∗ JλB2 − I Axn − p∥2 = 2 ( ( ) )} − ∥(un − p) − xn + γA∗ JλB2 − I Axn − p

(

2 ) 1{

= ∥un − p∥2 + ∥xn − p∥2 + γ(Lγ − 1) JλB2 − I Axn 2 ( ) } − ∥(un − xn ) − γA∗ JλB2 − I Axn ∥2 [ 1{ ∥un − p∥2 + ∥xn − p∥2 − ∥un − xn ∥2 ≤ 2 ( ) ( ) ⟨ ⟩]} + γ 2 ∥A∗ JλB2 − I Axn ∥2 − 2γ un − xn , A∗ JλB2 − I Axn 1{ ≤ ∥un − p∥2 + ∥xn − p∥2 − ∥un − xn ∥2 2 } + 2γ∥A(un − xn )∥∥(JλB2 − I)Axn ∥ . Thus, we obtain ∥un − p∥2 ≤ ∥xn − p∥2 − ∥un − xn ∥2 + 2γ∥A(un − xn )∥∥(JλB2 − I)Axn ∥.

(3.14)

It follows from (3.11) and (3.14) that [ ] ∥xn+1 − p∥2 ≤ (1 − αn µ)2 ∥xn − p∥2 − ∥un − xn ∥2 + 2γ∥A(un − xn )∥∥(JλB2 − I)Axn ∥ + 2αn βα∥xn − p∥∥xn+1 − p∥ + 2αn ∥βf (p) − Gp∥∥xn+1 − p∥ = (1 − 2αn µ + (αn µ)2 ∥xn − p∥2 − (1 − αn µ)2 ∥un − xn ∥2 + (1 − αn µ)2 2γ∥A(un − xn )∥∥(JλB2 − I)Axn ∥ + 2αn βα∥xn − p∥∥xn+1 − p∥ + 2αn ∥βf (p) − Gp∥∥xn+1 − p∥ ≤ ∥xn − p∥2 + αn µ2 ∥xn − p∥2 − (1 − αn µ)2 ∥un − xn ∥2 + (1 − αn µ)2 2γ∥A(un − xn )∥∥(JλB2 − I)Axn ∥

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+ 2αn βα∥xn − p∥∥xn+1 − p∥ + 2αn ∥βf (p) − Gp∥∥xn+1 − p∥. Therefore, (1 − αn µ)2 ∥un − xn ∥2 ≤ ∥xn − p∥2 − ∥xn+1 − p∥2 + αn µ2 ∥xn − p∥2 + (1 − αn µ)2 2γ∥A(un − xn )∥∥(JλB2 − I)Axn ∥ + 2αn βα∥xn − p∥∥xn+1 − p∥ + 2αn ∥βf (p) − Gp∥∥xn+1 − p∥ ( ) ≤ ∥xn+1 − xn ∥ ∥xn − p∥ + ∥xn+1 − p∥ + αn µ2 ∥xn − p∥2 + (1 − αn µ)2 2γ∥A(un − xn )∥∥(JλB2 − I)Axn ∥ + 2αn βα∥xn − p∥∥xn+1 − p∥ + 2αn ∥βf (p) − Gp∥∥xn+1 − p∥. Since αn → 0 as n → ∞, and from (3.10) and (3.13), we obtain lim ∥un − xn ∥ = 0.

(3.15)

n→∞

Since ∥Sun − un ∥ ≤ ∥Sun − xn ∥ + ∥xn − un ∥, it follows that ∥Sun − un ∥ → 0 as n → ∞.

(3.16)

Step 4. We will show that lim sup⟨(G − βf )z, xn − z⟩ ≤ 0, where z = PFix(S)∩Γ (I − G + βf )(z). n→∞ Since {un } is bounded, we consider a weak cluster point w of {un }. Hence, there exists a subsequence {uni } of {un }, which converges weakly to w. Now, S being ( nonexpansive, by (3.16) ) and Lemma 2.5, we B1 B2 ∗ obtain that w ∈ Fix(S). On the other hand, uni = Jλ xni + γA (Jλ − I)Axni can be rewritten as (xni − uni ) + A∗ (JλB2 − I)Axni ∈ B1 uni . (3.17) λ By passing to limit i → ∞ in (3.17) and by taking into account (3.13), (3.15) and the fact that the graph of a maximal monotone operator is weakly-trongly closed, we obtain 0 ∈ B1 (w). Furthermore, since {un } and {xn } have the same asymptoical behavior, {Axni } weakly converges to Aw. Again, by (3.13) and the fact that the resolvent JλB2 is nonexpansive and Lemma 3.1, we obtain that Aw ∈ B2 (Aw). Thus w ∈ Fix(S) ∩ Γ. Since z = PFix(S)∩Γ (I − G + βf )(z). Indeed, we have lim sup⟨(G − βf )z, z − xn ⟩ = n→∞

=

lim ⟨(G − βf )z, z − xni ⟩

i→∞

⟨(G − βf )z, z − w⟩ ≤ 0.

(3.18)

Step 5. Finally, we will show that xn → z as n → ∞. We have ∥xn+1 − z∥2 = ∥αn βf (xn ) + (I − αn G)Sun − z∥2 = ∥αn (βf (xn ) − Gz) + (I − αn G)(Sun − z)∥2 ≤ ∥(I − αn G)(Sun − z)∥2 + 2αn ⟨βf (xn ) − Gz, xn+1 − z⟩ ≤ (1 − αn µ)2 ∥un − z∥2 + 2αn β⟨f (xn ) − f (z), xn+1 − z⟩ + 2αn ⟨βf (xn ) − Gz, xn+1 − z⟩ ≤ (1 − αn µ)2 ∥xn − z∥2 + 2αn β⟨f (xn ) − f (z), xn+1 − z⟩ + 2αn ⟨βf (z) − Gz, xn+1 − z⟩ ≤ (1 − αn µ)2 ∥xn − z∥2 + 2αn βα∥xn − z∥∥xn+1 − z∥ + 2αn ∥βf (z) − Gz∥∥xn+1 − z∥ [ ] ≤ (1 − αn µ)2 ∥xn − z∥2 + αn βα ∥xn − z∥2 + ∥xn+1 − z∥2 + 2αn ∥βf (z) − Gz∥∥xn+1 − z∥

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( ) ≤ (1 − αn µ)2 + αn βα ∥xn − z∥2 + αn βα∥xn − z∥2 + ∥xn+1 − z∥2 + 2αn ∥βf (z) − Gz∥∥xn+1 − z∥, which implies that 1 − 2αn µ + (αn µ)2 + αn βα ∥xn − z∥2 1 − αn βα 2αn + ⟨βf (z) − Gz, xn+1 − z⟩ 1 − αn βα [ ] 2(µ − βα)αn (αn µ)2 = 1− ∥xn − z∥2 + ∥xn − z∥2 1 − αn βα 1 − αn βα

∥xn+1 − z∥2 ≤

2αn + ⟨βf (z) − Gz, xn+1 − z⟩ 1 − αn βα ] [ 2(µ − βα)αn ∥xn − z∥2 ≤ 1− 1 − αn βα [ ] 2(µ − βα)αn (αn µ2 )M 1 + + ⟨βf (z) − Gz, xn+1 − z⟩ 1 − αn βα 2(µ − βα) µ − βα = (1 − σn )∥xn − z∥2 + σn δn , 2(µ − βα)αn (αn µ2 )M 1 and δn = + ⟨βf (z) − 1 − αn βα 2(µ − βα) µ − βα ∑∞ δn Gz, xn+1 − z⟩. It is easily to see that σn → 0, n=1 σn = ∞ and lim supn→∞ ≤ 0 by (3.18). Thus, σn by Lemma 2.6, we deduce that xn → z as n → ∞. Further it follows from ∥un − xn ∥ → 0, un ⇀ w ∈ Fix(S) ∩ Γ and xn → z as n → ∞, that is z = w. This completes the proof. □ where M = sup{∥xn − z∥2 : n ∈ N}, σn =

Remark 3.3. In general case, if G is any strongly positive bounded linear operator with coefficient γ¯ µ and 0 < γ < . We define a bounded linear operator G on E by α G = ∥G∥−1 G. It is easy to see that G is a strongly positive with coefficient ∥G∥−1 µ > 0 such that ∥G∥ = 1 and 0 < ∥G∥−1 γ
0 and γ ∈ (0, and {αn } is a sequence in (0, 1) such that (i) lim ∑∞n→∞ αn = 0; (ii) ∑n=1 αn = ∞ and ∞ (iii) n=1 |αn − αn−1 | < ∞. Then the sequences {un } and {xn } both converge strongly to z ∈ Fix(S) ∩ Γ, where z = PFix(S)∩Γ f (z). 1 L ),

Applying Theorem 3.2, we can establish the strong convergence for new iterative method as the following theorem. Theorem 3.6. Let H1 and H2 be two real Hilbert spaces. Let A : H1 → H2 be a bounded linear operator. Assume that B1 : H1 → 2H1 and B2 : H2 → 2H2 are maximal monotone mappings. Let S : H1 → H1 be a nonexpansive mapping such that Fix(S) ∩ Γ ̸= ∅. Let f : H1 → H1 be a contraction mapping with constant α ∈ (0, 1) and G : H1 → H1 a strongly positive, bounded linear operator with µ constant µ and 0 < β < α . For any given y1 ∈ H1 , let the sequences {u′n } and {yn } generated by ( ) u′n = JλB1 yn + γA∗ (JλB2 − I)Ayn ; yn+1

= αn βf (Su′n ) + (I − αn G)Su′n , n ≥ 1, ∗

(3.22) ∗

where λ > 0 and γ ∈ (0, L is the spectral radius of the operator A A and A is the adjoint of A and {αn } is a sequence in (0, 1) such that (i) lim ∑∞n→∞ αn = 0; (ii) ∑n=1 αn = ∞ and ∞ (iii) n=1 |αn − αn−1 | < ∞. Then the sequences {u′n } and {yn } both converge strongly to z obtained in Theorem 3.2. 1 L ),

Proof. Let {xn } be the sequence given by x1 = y1 and ( ) un = JλB1 xn + γA∗ (JλB2 − I)Axn ; xn+1

= αn βf (xn ) + (I − αn G)Sun , n ≥ 1.

(3.23)

From Theorem 3.2, xn → z. Next, we claim that yn → z. Since JλB1 and JλB2 both are firmly nonexpansive, they are averaged. For γ ∈ (0, L1 ), L, the mapping (I + γA∗ (JλB2 I)A) is averaged, see [15] . It follows from Proposition 2.1 (ii) that the mapping JλB1 (I + γA∗ (JλB2 I)A) is averaged and hence nonexpansive. For each n ≥ 1, we can estimate the following

∥xn+1 − yn+1 ∥ = ∥αn βf (xn ) + (I − αn G)Sun − αn βf (Su′n ) − (I − αn G)Su′n ∥ ≤ ∥αn βf (xn ) − αn βf (Su′n )∥ + ∥(I − αn G)Sun − (I − αn G)Su′n ∥ ≤ αn βα∥Su′n − xn ∥ + (1 − αn µ)∥un − u′n ∥

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R. WANGKEEREE, K. RATTANASEEHA AND R. WANGKEEREE

≤ αn βα∥Su′n − Sz∥ + αn βα∥Sz − xn ∥ + (1 − αn µ)∥xn − yn ∥ ≤ αn βα∥u′n − z∥ + αn βα∥z − xn ∥ + (1 − αn µ)∥xn − yn ∥ ≤ αn βα∥yn − z∥ + αn βα∥z − xn ∥ + (1 − αn µ)∥xn − yn ∥ ≤ αn βα∥yn − xn ∥ + αn βα∥xn − z∥ + αn βα∥z − xn ∥ + (1 − αn µ)∥xn − yn ∥ = (1 − αn (µ − βα))∥xn − yn ∥ + αn (µ − βα)

2βα ∥xn − z∥. µ − βα

∑∞ It follows from n=1 αn = ∞, limn→∞ ∥xn −z∥ = 0 and Lemma 2.6 that ∥xn −yn ∥ → 0. Consequently, yn → z as required. □ References 1. F.E. Browder, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24(1967), 82-90. 2. F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20(1967), 197-228. 3. C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18, (2002), 441-453. 4. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759775 (2012) 5. Y. Censor, A. Gibali, S. Reich, The split variational inequality problem. The Technion-Israel Institute of Technology, Haifa(2010). arXiv:1009.3708. 6. L.C. Ceng, Q.H. Ansari, J.C. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Analysis: Theory, Methods Applications,In press. 7. L.C. Ceng, J.C. Yao, Strong convergence theorems by a relaxed extrgradient method for a general system of variational inequalities, Math. Methods Oper.Res. 67 (2008), 375-390. 8. P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert space. J. Nonlinear Convex Anal. 6 (2005), 117-136. 9. F. Cianciaruso, G. Marino, L. Muglia, Y. Hong, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, Fixed Point Theory and Applications, vol.10.1155, 383-745, 2010. 10. S.S. Chang, Y.J. Cho, J.K. Kim, L. Yang, Multiple-set split feasibility problems for asymptotically strict pseudocontractions, Vol. 2012, Article ID 491760, 12 pp, 2012. 11. F. Deutsch, I. Yamada, Minimizing certain convex functions over the intersection of the fixed point set of nonexpansive mappings, Numer. Funct. Anal. Optim. 19 (1998) 33-56. 12. Hartman, P. and G. Stampacchia, On some nonlinear elliptic dirential functional equations, Acta. Math. 115, 153188.1966. 13. K. R. Kazmi, S.H.Rizvi, An iterative method for split variational inclusion problem and fixed point problem for nonexpansive mapping, Optim Lett, DOI 10.1007/s11590-013-0629-2, 2013. 14. A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55. 15. A. Moudafi, Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275-283 (2011). 16. K. Goebel, W.A.Kirk, Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge, 1990. 17. B. Qu and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21,(2005), 55-65. 18. X. Qin, Y.J. Cho, S.M. Kang, Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. 72, (2010), 99-112. 19. S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mappings in a Hilbert spaces, Nonlinear Anal. 69 (2008), 1025-1033. 20. S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2008), 548-558. 21. S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), 455-469. 22. S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506-515. 23. T. Suzuki, Strong convergence of krasnoselskii and manns type sequences for one-parameter nonexpansive semigroups without bochner integrals. J. Math. Anal. Appl. 305 (2005), 227-239. 24. A. Moudafi, Split monotone variational inclusions, Journal of Optimization Theory and Applications, 150 (2011), 275-283. 25. G. Marino, H.K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), 43-52. 26. H.K. Xu, Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298 (2004), 279-291. 27. H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002) 240-256. 28. H.K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003) 659-678. 29. I. Yamada, The hybrid steepest descent method for the variational inequality problem of the intersection of fixed point sets of nonexpansive mappings, in: D. Butnariu, Y. Censor, S. Reich (Eds.), Inherently Parallel Algorithm for Feasibility and Optimization, Elsevier, 2001, pp. 473504.

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13

30. K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mapping and applications. Taiwanese J. Math. 5, (2001), 387-404. 31. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 , 595-597, 1967. 32. F.Wang, H.K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. doi:10.1016/j.na.2011.03.044, 2011. 33. Y.H. Yao, R.D. Chen, G. Marino, and Y.C. Liou, Applications of Fixed-Point and Optimization Methods to the Multiple-Set Split Feasibility Problem, Journal of Applied Mathematics Volume 2012, Article ID 927530, (2012), 21 PP.

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A FIXED POINT ALTERNATIVE TO THE STABILITY OF AN ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY BANACH SPACES CHOONKIL PARK AND SUN YOUNG JANG∗ Abstract. In this paper, we solve the following additive ρ-functional inequalities       t x+y − f (x) − f (y) , t ≥ (0.1) N f (x + y) − f (x) − f (y) − ρ 2f 2 t + ϕ(x, y) and     x+y t N 2f − f (x) − f (y) − ρ (f (x + y) − f (x) − f (y)) , t ≥ (0.2) 2 t + ϕ(x, y) in fuzzy normed spaces, where ρ is a fixed real number with ρ 6= 1. Using the fixed point method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [23] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [14, 27, 52]. In particular, Bag and Samanta [3], following Cheng and Mordeson [10], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [26]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [4]. We use the definition of fuzzy normed spaces given in [3, 31, 32] to investigate the Hyers-Ulam stability of additive ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [3, 31, 32, 33] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [30, 31]. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; additive ρ-functional inequality; fixed point method; HyersUlam stability. ∗ Corresponding author: Sun Young Jang (email: [email protected]).

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Definition 1.2. [3, 31, 32, 33] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn −x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N limn→∞ xn = x. Definition 1.3. [3, 31, 32, 33] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [4]). The stability problem of functional equations originated from a question of Ulam [51] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [19] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Th.M. Rassias [43] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [15] by replacing the unbounded Cauchy difference by a general control function in the spiritof Th.M. Rassias’ approach.  x+y 1 The functional equation f 2 = 2 f (x)+ 12 f (y) is called the Jensen equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [9, 20, 22, 24, 25, 28, 39, 40, 41, 45, 46, 47, 48, 49, 50]). Gil´anyi [17] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [44]. Fechner [13] and Gil´anyi [18] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [38] investigated the Cauchy additive functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k

(1.2)

and the Cauchy-Jensen additive functional inequality



x+y +z (1.3)

2 and proved the Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces. Park [36, 37] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. 



kf (x) + f (y) + 2f (z)k ≤

2f

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We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.4. [6, 11] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; n (2) the sequence {J x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [21] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [1, 5, 7, 8, 12, 16, 30, 34, 35, 41, 42]). In Section 2, we solve the additive ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we solve the additive ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. Throughout this paper, assume that X is a real vector space and (Y, N ) is a fuzzy Banach space. 2. Additive ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in fuzzy Banach spaces. Let ρ be a real number with ρ 6= 1. We need the following lemma to prove the main results. Lemma 2.1. Let f : X → Y be a mapping satisfying     x+y f (x + y) − f (x) − f (y) = ρ 2f − f (x) − f (y) 2 for all x, y ∈ X. Then f : X → Y is additive.

(2.1)

Proof. Letting x = y = 0 in (2.1), we get −f (0) = 0 and so f (0) = 0. Replacing y by x in (2.1), we get f (2x) − 2f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Thus     x+y f (x + y) − f (x) − f (y) = ρ 2f − f (x) − f (y) = ρ(f (x + y) − f (x) − f (y)) 2 and so f (x + y) = f (x) + f (y) for all x, y ∈ X. 

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Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ(x, y) ≤ ϕ (2x, 2y) 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying       t x+y N f (x + y) − f (x) − f (y) − ρ 2f − f (x) − f (y) , t ≥ 2 t + ϕ(x, y)

(2.2)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f 2xn exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2L)t (2.3) N (f (x) − A(x), t) ≥ (2 − 2L)t + Lϕ(x, x) for all x ∈ X and all t > 0.


Proof. Letting y = x in (2.2), we get N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x)

(2.4)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: 

t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, x) 

d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [29, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, x) for all x ∈ X and all t > 0. Hence       x x N (Jg(x) − Jh(x), Lεt) = N 2g − 2h , Lεt 2 2       x x L = N g −h , εt 2 2 2 ≥ =

Lt 2 Lt 2


≥ x

+ ϕ x2 , 2 t t + ϕ(x, x)

Lt 2

+

Lt 2 L 2 ϕ(x, x)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S.

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It follows from (2.4) that L x t , t ≥ 2 2 t + ϕ(x, x)

 



N f (x) − 2f



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L2 . By Theorem 1.4, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., 1 x = A(x) (2.5) 2 2 for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set  

A

M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.5) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality n



N - lim 2 f n→∞

for all x ∈ X; (3) d(f, A) ≤

1 1−L d(f, Jf ),

x 2n



= A(x)

which implies the inequality d(f, A) ≤

L . 2 − 2L

This implies that the inequality (2.3) holds. By (2.2), x y N 2 f −f −f n 2 2n         x+y x y t n+1 n n n
−ρ 2 f −2 f −2 f ,2 t ≥ n+1 n n 2 2 2 t + ϕ 2xn , 2yn 

n

 

x+y 2n











for all x, y ∈ X, all t > 0 and all n ∈ N. So 

x y −f 2n 2n         x+y x y n n −ρ 2n+1 f − 2 f − 2 f ,t ≥ n+1 n 2 2 2n

 

N 2n f

x+y 2n











−f

for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 2n t Ln + 2n ϕ(x,y) 2n

t 2n

+

t 2n Ln 2n ϕ (x, y)

= 1 for all x, y ∈ X and all

x+y − A(x) − A(y) 2 for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is Cauchy additive, as desired. 







A(x + y) − A(x) − A(y) = ρ 2A

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Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm k · k. Let f : X → Y be amapping satisfying       x+y N f (x + y) − f (x) − f (y) − ρ 2f − f (x) − f (y) , t 2 t ≥ (2.6) t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2p − 2)t N (f (x) − A(x), t) ≥ p (2 − 2)t + 2θkxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 21−p , and we get the desired result.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y ϕ(x, y) ≤ 2Lϕ , 2 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying (2.2). Then A(x) := N -limn→∞ exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2L)t N (f (x) − A(x), t) ≥ (2 − 2L)t + ϕ(x, x) for all x ∈ X and all t > 0.

1 2n f

(2n x)

(2.7)

Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2 for all x ∈ X. It follows from (2.4) that   1 1 t N f (x) − f (2x), t ≥ 2 2 t + ϕ(x, x) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 12 . Hence 1 d(f, A) ≤ , 2 − 2L which implies that the inequality (2.7) holds. The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm k · k. Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2p )t N (f (x) − A(x), t) ≥ (2 − 2p )t + 2θkxkp for all x ∈ X and all t > 0.

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Proof. The proof follows from Theorem 2.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−1 , and we get the desired result.  3. Additive ρ-functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in fuzzy Banach spaces. Let ρ be a fuzzy number with ρ 6= 1. Lemma 3.1. Let f : X → Y be a mapping satisfying f (0) = 0 and   x+y 2f − f (x) − f (y) = ρ (f (x + y) − f (x) − f (y)) 2 for all x, y ∈ X. Then f : X → Y is additive.

(3.1)

Proof. Letting y = 0 in (3.1), we get 2f x2 − f (x) = 0 and so f (2x) = 2f (x) for all x ∈ X. Thus   x+y f (x + y) − f (x) − f (y) = 2f − f (x) − f (y) = ρ(f (x + y) − f (x) − f (y)) 2 and so f (x + y) = f (x) + f (y) for all x, y ∈ X. 


Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ(x, y) ≤ ϕ (2x, 2y) 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and     x+y t N 2f − f (x) − f (y) − ρ (f (x + y) − f (x) − f (y)) , t ≥ (3.2) 2 t + ϕ(x, y) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f 2xn exists for each x ∈ X and defines an additive mapping A : X → Y such that (1 − L)t N (f (x) − A(x), t) ≥ (3.3) (1 − L)t + ϕ(x, 0) for all x ∈ X and all t > 0.


Proof. Letting y = 0 in (3.2), we get         x x t N f (x) − 2f , t = N 2f − f (x), t ≥ 2 2 t + ϕ(x, 0) for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S:   t d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥ , ∀x ∈ X, ∀t > 0 , t + ϕ(x, 0)

(3.4)

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [29, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X.

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C. PARK, S. Y. JANG

Let g, h ∈ S be given such that d(g, h) = ε. Then N (g(x) − h(x), εt) ≥

t t + ϕ(x, 0)

for all x ∈ X and all t > 0. Hence x x − 2h , Lεt 2 2       x L x −h , εt = N g 2 2 2 

 

 



N (Jg(x) − Jh(x), Lεt) = N 2g

Lt 2

≥


≥

+ ϕ x2 , 0 t t + ϕ(x, 0) Lt 2

=

Lt 2

+

Lt 2 L 2 ϕ(x, 0)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (3.4) that t x ,t ≥ 2 t + ϕ(x, 0)

 



N f (x) − 2f



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 1. By Theorem 1.4, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., x 2

 

A

1 = A(x) 2

(3.5)

for all x ∈ X. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (3.5) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + ϕ(x, 0) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality n

N - lim 2 f n→∞

for all x ∈ X; (3) d(f, A) ≤

1 1−L d(f, Jf ),



x 2n



= A(x)

which implies the inequality d(f, A) ≤

1 . 1−L

This implies that the inequality (3.3) holds.

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY BANACH SPACES

By (3.2), x+y 2n+1

y N 2 f −2 f −2 f 2n          x+y x y t n n
−ρ 2 f −f −f ,2 t ≥ n n n 2 2 2 t + ϕ 2xn , 2yn 

n+1





n



x 2n



n





for all x, y ∈ X, all t > 0 and all n ∈ N. So        x+y x y n+1 n n N 2 f −2 f −2 f n+1 n 2 2 2n          x+y x y − ρ 2n f −f −f ,t ≥ n n 2 2 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

t 2n t Ln + 2n ϕ(x,y) 2n

t 2n

+

t 2n Ln 2n ϕ (x, y)

= 1 for all x, y ∈ X and all

x+y − A(x) − A(y) = ρ (A(x + y) − A(x) − A(y)) 2 for all x, y ∈ X. By Lemma 3.1, the mapping A : X → Y is Cauchy additive, as desired. 



2A



Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and     t x+y − f (x) − f (y) − ρ (f (x + y) − f (x) − f (y)) , t ≥ (3.6) N 2f 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2p − 2)t (2p − 2)t + 2p θkxkp

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 21−p , and we get the desired result.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y ϕ(x, y) ≤ 2Lϕ , 2 2 for all x, y ∈ X. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.3). Then A(x) := N limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (1 − L)t N (f (x) − A(x), t) ≥ (3.7) (1 − L)t + Lϕ(x, 0) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 2

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C. PARK, S. Y. JANG

for all x ∈ X. It follows from (3.4) that 1 t N f (x) − f (2x), Lt ≥ 2 t + ϕ(x, 0) 



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L. Hence d(f, A) ≤

L , 1−L

which implies that the inequality (3.7) holds. The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with the norm k · k. Let f : X → Y be a mapping satisfying f (0) = 0 and (3.6). Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2p )t N (f (x) − A(x), t) ≥ (2 − 2p )t + 2p θkxkp for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−1 , and we get the desired result.  Acknowledgments S. Y. Jang was supported by Research Program of University of Ulsan, 2016. The authors would like to thank the useful comments of referees. References [1] L. Aiemsomboon and W. Sintunavaat, On new stability results for generalized Cauchy functional equations on groups by using Brzdek’s fixed point theorem, J. Fixed Point Theory Appl. (in press). [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687– 705. [4] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [5] A. Bahyrycz and J. Brzdek, On functions that are approximate fixed points almost everywhere and Ulam’s type stability, J. Fixed Point Theory Appl. (in press). [6] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [7] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [8] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [9] I. Chang and Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. [10] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [11] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [12] M. Eshaghi Gordji and Th.M. Rassias, Fixed points and generalized stability for quadratic and quartic mappings in C ∗ -algebras, J. Fixed Point Theory Appl. (in press).

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[13] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [14] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [15] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [16] A. G. Ghazanfari and Z. Alizadeh, On approximate ternary m-derivations and σ-homomorphisms, J. Fixed Point Theory Appl. (in press). [17] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [18] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [19] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [20] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [21] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [22] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [23] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [24] H. Kim, M. Eshaghi Gordji, A. Javadian and I. Chang, Homomorphisms and derivations on unital C ∗ algebras related to Cauchy-Jensen functional inequality, J. Math. Inequal. 6 (2012), 557–565. [25] H. Kim, J. Lee and E. Son, Approximate functional inequalities by additive mappings, J. Math. Inequal. 6 (2012), 461–471. [26] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [27] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [28] J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. [29] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [30] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [31] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [32] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [33] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [34] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007, Art. ID 50175 (2007). [35] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008, Art. ID 493751 (2008). [36] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [37] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [38] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007). [39] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [40] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [41] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [42] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96.

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[43] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [44] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ ankov´ [45] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [46] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [47] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [48] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [49] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [50] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [51] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [52] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, South Korea E-mail address: [email protected] Sun Young Jang Department of Mathematics, University of Ulsan, Ulsan 44610, South Korea E-mail address: [email protected]

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FOURIER SERIES OF HIGHER-ORDER GENOCCHI FUNCTIONS AND THEIR APPLICATIONS TAEKYUN KIM, DAE SAN KIM, LEE CHAE JANG, AND DMITRY V. DOLGY

Abstract. In this paper, we derive some identities for higher-order Genocchi functions arising from Fourier series for them . In addition, we give some application of these identities related to Bernoulli function.

1. Introduction The numbers Gk , (k ≥ 0), in the Taylor expansion ∞ ∑ tn 2t = G , n et + 1 n=0 n!

(see [2 − 11]),

(1.1)

are known as the Genocchi numbers. These numbers arise in the series expansion of trigonometric functions, and are extremely important in the number theory and analysis. The Genocchi polynomials Gn (x), (n ≥ 0), are defined by the generating function ( ) ∞ ∑ 2t tn xt e = Gn (x) , (see [5, 12, 13]). (1.2) t e +1 n! n=0 Note that Gn (x) ∈ Z[x] with deg Gn (x) = n − 1, for n ≥ 1. Let f (x) be a square integrable function defined on [−p, p]. Then the Fourier series of f (x) is given by ) ∞ ( a0 ∑ nπ nπ + x + bn sin x , an cos (1.3) 2 p p n=1 where a0 =

1 p

∫

p

f (x)dx, −p

and bn =

1 p

∫

an =

p

f (x) sin −p

1 p

∫

p

f (x) cos −p

nπ x dx, p

nπ x dx, p

(see [6, 7]).

(1.4)

(1.5)

The Fourier series in (1.3) can be alternatively given as follows: ∞ ∑

Cn e

nπi p x

,

(i =

√

−1),

(1.6)

(see [6, 7]).

(1.7)

n=−∞

where 1 Cn = 2p

∫

p

f (x)e−

nπi p x

dx, ,

−p

2010 Mathematics Subject Classification. 11B83, 42A16. Key words and phrases. Fourier series, Genocchi polynomials, Genocchi functions. 1

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2

Fourier series of higher-order Genocchi functions and their applications

For r ∈ N, the higher-order Genocchi polynomials are defined by the generating function to be ( )r ∞ ∑ 2t tn xt , (see [5]). e = G(r) n (x0 t e +1 n! n=0 (r)

(1.8)

(r)

When x = 0, Gn = Gn (0) are called the higher-order Genocchi numbers. For any real number x, we define < x >= x − [x] ∈ [0, 1),

(1.9) (r)

where [·] is the Gauss symbol. Note that < x > is the fractional part of x. Thus, Gm (< x >) are functions defined on (−∞, ∞) and periodic with period 1, which are called Genocchi functions of order r. In this paper, we derive some identities of Genocchi functions of order r arising from Fourier series for them. In addition, we give some application of these identities related to Bernoulli functions. 2. Fourier series of higher-order Genocchi functions and their applications From (1.8), we note that (r) G(r) m (x) = 0, for 0 ≤ m ≤ r − 1, and Gr (x) = r!.

(2.1)

Now, we assume that m ≥ r + 1 ≥ 2. We first observe that r−1)

(r) G(r) m (x + 1) = 2mGm−1 (x) − Gm (x), (m ≥ 0).

(2.2)

Indeed, ∞ ∑

G(r) m (x

m=0

(

(

)r 2t = e ext (et + 1 − 1) et + 1 ( ( )r−1 ( ) )r 2t 2t 2t xt t 1) − = e (e + ext et + 1 et + 1 et + 1 ∞ ∞ ∑ ∑ tm tm (x) = 2t G(r−1) − G(r) m m (x) m! m=0 m! m=0

tm + 1) = m!

=2

2t t e +1

∞ ∑

)r

(x+1)t

(r−1)

mGm−1 (x)

m=0

=

∞ ( ∑

(2.3)

∞ ∑ tm tm − G(r) (x) m! m=0 m m!

(r−1)

2mGm−1 (x) − G(r) m (x)

m=0

) tm . m!

For x = 0 in , we have (r−1)

(r) G(r) m (1) = 2mGm−1 (0) − Gm (0).

(2.4)

By (2.4), we get (r−1)

(r) (r) G(r) m (1) = Gm (0) ⇔ Gm (0) = mGm−1 (0).

(2.5)

Gm (< x >) is piecewise C ∞ . Moreover, Gm (< x >) is continuous for those (r, m) with Gm (0) = (r−1) (r) mGm−1 (0), and discontinuous with jump discontinuities at integers for those (r, m) with Gm (0) ̸= (r)

(r)

(r)

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T. Kim, D. S.Kim, L. C. Jang, D. V. Dolgy (r−1)

3

(r)

mGm−1 (0). The Fourier series of Gm (< x >) is ∞ ∑

Cn(r,m) e2πinx , (i =

√

−1),

(2.6)

n=−∞

where

∫

1

Cn(r,m) =

−2πinx G(r) dx m (< x >)e

0

∫ =

1

(2.7) −2πinx G(r) dx. m (x)e

0

Now, we observe that

∫

1

Cn(r,m) =

−2πinx G(r) dx m (x)e

0

∫ 1 1 [ (r) −2πinx ]1 2πin (r) = + (x)e−2πinx dx Gm+1 e G m+1 m + 1 0 m+1 0 ) 1 ( (r) 2πin (r,m+1) (r) = Gm+1 (0) − Gm+1 (0) + C m+1 m+1 n ) 2πin (r,m+1) 2 ( (r) = (0) − Gm+1 (0) + (m + 1)G(r−1) C . m m+1 m+1 n Replacing m by m − 1 in (2.8), we have ) 2 ( (r) 2πin (r,m) (r−1) Cn = Cn(r,m−1) + Gm (0) − mGm−1 (0) . m m Assume first that n ̸= 0. Then we have ) m (r,m−1) 1 ( (r) (r−1) Cn(r,m) = + Cn Gm (0) − mGm−1 (0) 2πin ( πin )) m 1 ( (r) m − 1 (r,,m−2) (r−1) = + Cn Gm−1 (0) − (m − 1)Gm−2 (0) 2πin 2πin πin ) 1 ( (r) (r−1) + Gm (0) − mGm−1 (0) πin ) m(m − 1) (r,m−2) m 1 ( (r) (r−1) = (0) + (0) − (m − 1)G C G m−1 m−2 (2πin)2 n 2 (πin)2 ( ) 1 (r−1) + G(r) m (0) − mGm−1 (0) πin { )} m(m − 1) m − 2 (r,m−3) 1 ( (r) (r−1) = C + G (0) − (m − 2)G (0) m−2 m−3 (2πin)2 2πin n πin ) ( ( ) m 1 1 (r) (r−1) (r−1) (r) + (0) − (m − 1)G (0) + G G (0) − mG (0) m m−2 m−1 m−1 2 (πin)2 πin ) m(m − 1)(m − 2) (r,m−3) m(m − 1) 1 ( (r) (r−1) = C + G (0) − (m − 2)G (0) n m−2 m−3 (2πin)3 22 (πin)3 ) ) 1 ( (r) m 1 ( (r) (r−1) (r−1) (0) − (m − 1)G G (0) + G (0) − mG (0) + m m−1 m−2 m−1 2 (πin)2 πin = ··· =

(2.8)

(2.9)

(2.10)

m−1 ) ∑ (m)k−1 1 ( (r) m! (r−1) (r,1) C + G (0) − (m + 1 − k)G (0) . n m+1−k m−k (2πin)m−1 2k−1 (πin)k k=1

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4

Fourier series of higher-order Genocchi functions and their applications

Note that ∫ Cn(r,1)

1

=

G1 (x)e−2πinx dx = 0, (r)

(2.11)

0

(r)

(1)

since G1 (x) = 0, for r ≥ 2 and G1 (x) = 1. From (2.10) and (2.11), we have

Cn(r,m) =

m−1 ∑ k=1

) (m)k−1 1 ( (r) (r−1) Gm−k+1 (0) − (m − k + 1)Gm−k (0) k−1 k 2 (πin) ∑

min{m+1−r,m−1}

=

k=1

) (m)k−1 1 ( (r) (r−1) G (0) − (m − k + 1)G (0) m−k+1 m−k 2k−1 (πin)k

(2.12)

(r)

Here we used the fact that Gm = 0, for 0 ≤ m ≤ r − 1. Assume next that n = 0. Then we have ∫ (r,m)

C0

1

]1 ) 1 ( (r) 1 [ (r) (r) Gm+1 (x) = Gm+1 (1) − Gm+1 (0) m+1 m+1 0 ) ( (r) (0) − Gm+1 (0) (m + 1)G(r−1) m

G(r) m (x)dx =

= 0

2 = m+1

(2.13)

Before proceeding further, we recall the following facts about Bernoulli functions Bn (< x >) :

Bm (< x >) = −m!

∞ ∑

e2πinx , for m ≥ 2, (2πin)m n=−∞

(see [1]),

(2.14)

n̸=0

and ∞ ∑ e2πinx − = 2πin n=−∞

{

B1 (< x >), for x ∈ /Z 0, for x ∈ Z.

(2.15)

n̸=0

The series in (2.14) converges uniformly, but that in (2.15) converges only pointwise. Assume first that (r) (r−1) (r) (r) (r) Gm (0) = mGm−1 (0). Then Gm (1) = Gm (0). Gm (< x >) is piecewise C ∞ , and continuous. Hence (r) (r) the Fourier series of Gm (< x >) converges uniformly to Gm (< x >), and we have ) 2 ( (r) (m + 1)G(r−1) (0) − Gm+1 (0) G(r) m m (< x >) = m+1   min{m+1−r,m−1} ∞ ( ) ∑ ∑ 1 (m) k−1 (r) (r−1)  Gm−k+1 (0) − (m − k + 1)Gm−k (0)  e2πinx , + k−1 (πin)k 2 n=−∞ n̸=0

(2.16)

k=1

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5

for all x ∈ (−∞, ∞). In addition, we can express this in terms of Bernoulli functions Bm (< x >). G(r) m (< x >) =

) 2 ( (r) (m + 1)G(r−1) (0) − G (0) m m+1 m+1 ∑

min{m+1−r,m−1}

−

k=1

2(m)k−1 k!

(

 ∞ ) ∑  (r) (r−1) Gm−k+1 (0) − (m − k + 1)Gm−k (0) −k!

 2πinx

e  k (2πin) n=−∞ n̸=0

) 2 ( (r) = (m + 1)G(r−1) (0) − Gm+1 (0) m m+1 min{m+1−r,m−1} ) ∑ 2(m)k−1 ( (r−1) (r) + (m + 1 − k)Gm−k (0) − Gm+1−k (0) Bk (< x >) k! k=2 { ( ) B1 (< x >), for x ∈ /Z (r−1) (r) + 2 mGm−1 (0) − Gm (0) × 0, for x ∈ Z. (2.17) Therefore, we obtain the following theorem. Theorem 2.1. Let m ≥ r + 1 ≥ 2. Assume that (r−1)

G(r) m (0) = mGm−1 (0). Then we have ) 2 ( (r) (r) (0) − G (0) (a) Gm (< x >) = (m + 1)G(r−1) m m+1 m+1   min{m+1−r,m−1} ∞ ( ) ∑ ∑ (m)k−1 1 (r) (r−1)  + Gm−k+1 (0) − (m − k + 1)Gm−k (0)  e2πinx , k−1 (πin)k 2 n=−∞ k=1

n̸=0

for all x ∈ (−∞, ∞), where the convergence is uniform. ) 2 ( (r) (m + 1)G(r−1) (0) − Gm+1 (0) m m+1 min{m+1−r,m−1} ) ∑ 2(m)k−1 ( (r−1) (r) + (m + 1 − k)Gm−k (0) − Gm+1−k (0) Bk (< x >), k!

(b) G(r) m (< x >) =

k=2

for all x ∈ (−∞, ∞), where Bk (< x >) is the Bernoulli function. (r)

(r−1)

(r)

(r)

(r)

Assume next that Gm (0) ̸= mGm (0). Then Gm (1) ̸= Gm (0), and hence Gm (< x >) is piecewise (r) ∞ C and discontinuous with jump discontinuities at integers. Thus,(the Fourier series)of Gm (< x >) (r) (r) (r−1) (r) converges pointwise to Gm (< x >), for x ∈ / Z, and converges to 21 Gm (1) + Gm (0) = mGm−1 (0), for x ∈ Z. Therefore, we obtain the following theorem.

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Fourier series of higher-order Genocchi functions and their applications (r)

(r)

Theorem 2.2. Let m ≥ r + 1 ≥ 2. Assume that Gm (0) ̸= mGm−1 (0). Then we have ∞ ) ∑ 2 ( (r) (r−1) (m + 1)Gm (0) − Gm+1 (0) + (a) m+1 n=−∞

( min{m+1−r,m−1} ( )k ∑ 1 (m)k−1 2k−1 πin

n̸=0

)

( ) (r−1) (r) × Gm+1−k (0) − (m + 1 − k)Gm−k (0) e2πinx

k=1

{ (r) Gm (< x >), = (r−1) mGm−1 (0),

for x ∈ / Z, for x ∈ Z.

Here the convergence is pointwise. (b)

) 2 ( (r) (m + 1)G(r−1) (0) − G (0) m m+1 m+1 min{m+1−r,m−1} ) ∑ 2(m)k−1 ( (r−1) (r) + (m + 1 − k)Gm−k (0) − Gm+1−k (0) Bk (< x >) k! k=1

= G(r) m (< x >),

for x ∈ / Z,

and ) 2 ( (r) (0) − Gm+1 (0) (m + 1)G(r−1) m m+1 min{m+1−r,m−1} ) ∑ 2(m)k−1 ( (r) (r−1) + (m + 1 − k)Gm−k (0) − Gm+1−k (0) Bk (< x >) k! =

k=2 (r−1) mGm−1 (0),

for x ∈ Z.

Here Bk (< x >) is the Bernoulli function.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1970. L. Carlitz, Some unusual congruences for the Bernoulli and Genocchi numbers, Duke Math. J., 35 (1968), 563–566. J. M. Gandhi, Some integrals for Genocchi numbers, Math. Mag., 33(1959/1960), 21–23. J. M. Gandhi, Congruences for Genocchi numbers, Duke Math. J., 31 (1964), 519–527. D. S. Kim and T. Kim, Some identities involving Genocchi polynomials and numbers, Ars combin., 121 (2015), 403–412. T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 17(2008), no. 2, 131–136. T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Apol. Anal., 2008 Art. ID 581582. 11 pp. T. Kim, S.-H. Rim, D. V. Dolgy and S.-H. Lee, Some identities of Genocchi polynomials arising from Genocchi bases, J. Inequal. Appl., 2013 2013:43, 6 pp. T. Kim, J. Choi and Y. H. Kim, A note on the values of Euler zeta functions at positive integers, Adv. Stud. Contemp. Math. (Kyungshang), 22(2012), no. 1, 27–34. T. Kim, Some identities for the Bernoulli the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20(2010), no. 1, 23–28. T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys. 15(2008), no. 4, 481–486. D. H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly., 95(1988), 905–911. J. Riordan and P. R. Stein, Proof of a conjecture on Genocchi numbers, Discrete Math, 5(1973), 381–388.

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7

Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Graduate School of Education, Kon-Kuk University, Seoul 143-701, republic of Korea E-mail address: [email protected] Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea, Institute of Mathematics and Computer Science, Far Eastern State University, 690950 Vladivostok, Russia E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

g; 'h;m CONVEX AND (g; log ') CONVEX DOMINATED FUNCTIONS AND HADAMARD TYPE INEQUALITIES RELATED TO THEM MUSTAFA GÜRBÜZ

Abstract. In this paper, we present the notion of g; 'h;m convex and (g; log ') convex dominated function and present some properties of them. Besides, we attain some Hermite-Hadamard-type inequalities for g; 'h;m convex and (g; log ') convex dominated functions. Our results generalize some …ndings about Hermite-Hadamard-type inequalities in the literature.

1. Introduction The inequality (1.1)

f

a+b 2

1 b

a

Z

b

f (x) dx

a

f (a) + f (b) 2

which holds for every convex function f; from a closed set [a; b] to R, is known in the literature as Hermite-Hadamard’s inequality (see [13]). In [1], Dragomir and Ionescu introduced the following class of functions. De…nition 1. Let g : I ! R be a convex function on the interval I: The function f : I ! R is called g convex dominated on I if the following condition is satis…ed: (1.2)

j f (x) + (1 g (x) + (1

for all x; y 2 I and

) f (y) ) g (y)

f ( x + (1 g ( x + (1

) y)j ) y)

2 [0; 1] :

In [1] and [2], the authors connect together some disparate threads through a Hermite-Hadamard motif. The …rst of these threads is the unifying concept of a g convex-dominated function. In [3], Hwang et al. established some inequalities of Fejér type for g convex-dominated functions. Finally, in [4], [5] and [6] authors introduced several new di¤erent kinds of convex -dominated functions and then gave Hermite-Hadamard-type inequalities for this classes of functions. In [7], S. Varošanec introduced the following class of functions. I and J are intervals in R; (0; 1) J and functions h and f are real non-negative functions de…ned on J and I, respectively. Date : October 12, 2016. 2000 Mathematics Subject Classi…cation. Primary 26D15, Secondary 26D10, 05C38. Key words and phrases. Convex dominated functions, Hermite-Hadamard Inequality, 'h;m convex functions, (g; 'h ) convex dominated functions, (g; log ') convex dominated functions. 1

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De…nition 2. Let h be a non-negative function from J which is a subset of R to R, h 6 0. f : I ! R is called an h convex function, or that f belongs to the class SX (h; I) ; if f is non-negative and for all x; y 2 I and 2 (0; 1]; we get (1.3)

f ( x + (1

) y)

h ( ) f (x) + h (1

) f (y):

If the inequality (1.3) is reversed, then f is said to be h concave, i.e. f 2 SV (h; I) : Youness have de…ned the ' convex functions in [9]. A function ' : [a; b] ! [c; d] where [a; b] R: De…nition 3. A function f : [a; b] ! R is said to be ' convex on [a; b] if for every two points x 2 [a; b] ; y 2 [a; b] and t 2 [0; 1] the following inequality holds: (1.4)

f (t' (x) + (1

t) ' (y))

tf (' (x)) + (1

t) f (' (y)) :

In [8], Sar¬kaya de…ned a new kind of ' convexity using h convexity as following: De…nition 4. Let I be an interval in R and h : (0; 1) ! (0; 1) be a given function. We say that a function f : I ! [0; 1) is 'h convex if (1.5)

f (t' (x) + (1

t) ' (y))

h (t) f (' (x)) + h (1

t) f (' (y))

for all x; y 2 I and t 2 (0; 1) : If inequality (1.5) is reversed, then f is said to be 'h concave. In particular, if f satis…es (1.5) with h (t) = t; h (t) = ts (s 2 (0; 1)) ; h (t) = 1t ; and h (t) = 1; then f is said to be ' convex, 's convex, ' Godunova-Levin function and ' P function, respectively. In [10], Özdemir et al. de…ned (h m) convexity and obtained Hermite-Hadamardtype inequalities as following. De…nition 5. Let h : J R ! R be a non-negative function. We say that f : [0; b] ! R is a (h m) convex function, if f is non-negative and for all x; y 2 [0; b] ; m 2 [0; 1] and 2 (0; 1) ; we have (1.6)

f ( x + m (1

) y)

h ( ) f (x) + mh (1

If the inequality is reversed, then f is said to be (h

) f (y) :

m) concave function on [0; b] :

In [2], Dragomir et al. proved the following theorem for g convex dominated functions related to (1.1): De…nition 6. A function f : I ! [0; 1) is said to be log convex or multiplicatively convex if log t is convex, or, equivalently, if for all x; y 2 I and t 2 [0; 1] one has the inequality (1.7)

f (tx + (1

t

t) y)

1 t

[f (x)] [f (y)]

:

We note that if f and g are convex and g is increasing, then g f is convex; moreover, since f = exp (log f ) ; it follows that a log convex function is convex, but the converse may not necessarily be true [12]. This follows directly from (1.7) because, by the aritmetic-geometric mean inequality, we have t

1 t

[f (x)] [f (y)]

tf (x) + (1

t) f (y)

for all x; y 2 I and t 2 [0; 1] :

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3

For some results related to this classical results, (see [13], [14], [15], [16], [17]) and the references therein. In [17], Sar¬kaya has de…ned the log ' convex function as following. De…nition 7. Let us consider a ' : [a; b] ! [a; b] where [a; b] R and I stands for a convex subset of R: We say that a function f : I ! R+ is a log ' convex if (1.8)

f (t' (x) + (1

t

t) ' (y))

1 t

[f (' (x))] [f (' (y))]

for all x; y 2 I and t 2 [0; 1] : Theorem 1. Let g : I ! R be a convex function and f : I ! R a g convex dominated mapping. Then for all a; b 2 I with a < b; Z b Z b a+b 1 1 a+b f (1.9) f (x) dx g (x) dx g 2 b a a b a a 2 and (1.10)

f (a) + f (b) 2

Z

1 b

a

b

f (x) dx

a

g (a) + g (b) 2

1 b

a

Z

b

g (x) dx:

a

In [4], Kavurmac¬et al. proved the next theorem: Theorem 2. Let h : J ! R be a non-negative function, h 6= 0, g : I ! R be an h convex function and the real function f : I ! R be a (g; h) convex dominated on I. Then one has the inequalities: (1.11) Z b Z b 1 a+b 1 a+b 1 1 f (x) dx f g (x) dx g b a a 2 b a a 2 2h 12 2h 12 and (1.12) [f (a) + f (b)]

Z

1

1

h( )d

b

0

for all x; y 2 I and

a

Z

b

f (x) dx

[g (a) + g (b)]

a

Z

1

h( )d

0

1 b

a

Z

b

g (x) dx

a

2 [0; 1] :

In [6], Özdemir et al. proved the following theorem: Theorem 3. Let a nonnegative function g : I R ! R belong to the class of P (I) : The real function f : I R ! R is (g; P (I)) convex dominated on I. If a; b 2 I with a < b and f; g 2 L1 [a; b], then one has the inequalities: Z b Z b 2 a+b 2 a+b (1.13) f (x) dx f g (x) dx g b a a 2 b a a 2 and (1.14)

[f (a) + f (b)]

1 b

a

Z

b

f (x) dx

[g (a) + g (b)]

a

1 b

a

Z

b

g (x) dx

a

for all x; y 2 I: In [11], Özdemir et al. proved the following theorem:

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Theorem 4. Let h : (0; 1) ! (0; 1) be a given function, g : I ! [0; 1) be a given 'h convex function. If f : I ! [0; 1) is Lebesgue integrable and (g; 'h ) convex dominated on I for linear continious, non-constant function ' : [a; b] ! [a; b], then the following inequalities hold: Z '(b) 1 1 ' (a) + ' (b) f (x) dx f ' (b) ' (a) '(a) 2 2h 12 (1.15) 1 ' (b)

' (a)

Z

'(b)

g (x) dx

'(a)

1 2h

' (a) + ' (b) 2

g

1 2

and (1.16)

[f (' (a)) + f (' (b))]

Z

1

1

h (t) dt

' (b)

0

[g (' (a)) + g (' (b))]

Z

1

h (t) dt

0

1 ' (b)

Z

' (a) Z

' (a)

'(b)

f (x) dx

'(a)

'(b)

g (x) dx

'(a)

for all x; y; a 2 [0; b], t 2 (0; 1) and m 2 (0; 1] : In the following sections our main results are given: We introduce the notion of g; 'h;m convex and (g; log ') convex dominated function and present some properties of them. Besides, we present some Hermite-Hadamard-type inequalities for g; 'h;m convex and (g; log ') convex dominated functions. Our results generalize the Hermite-Hadamard-type inequalities in [2], [4], [6] and [11]. 2. g; 'h;m

convex dominated functions

De…nition 8. Let h : (0; 1) ! R be a non-negative function, h 6= 0; g : [0; b] [0; 1) ! R+ be a given 'h;m convex function. The real function f : [0; b] ! R+ is called g; 'h;m convex dominated on [0; b] if the following condition is satis…ed (2.1)

jh (t) f (' (x)) + mh (1 h (t) g (' (x)) + mh (1

t) f (' (y)) t) g (' (y))

f (t' (x) + m (1 g (t' (x) + m (1

t) ' (y))j t) ' (y))

for all x; y 2 [0; b], t 2 (0; 1) and m 2 [0; 1]. In particular, if f satis…es (2.1) with m = 1; then f is said to be (g; 'h ) convex dominated function. If the inequality (2.1) is reversed, then f is said to be 'h;m concave dominated function on [0; b] : The next simple characterisation of g; 'h;m convex dominated functions holds. Lemma 1. Let h : (0; 1) ! (0; 1) be a given function, g : [0; b] [0; 1) ! R+ be a given 'h;m convex function and f : [0; b] ! R+ be a real function. The following statements are equivalent: (1) f is g; 'h;m convex dominated on [0; b] : (2) The mappings g f and g + f are 'h;m convex on [0; b] : (3) There exist two 'h;m convex mappings l; k de…ned on [0; b] such that f=

1 2

(l

k) and g =

54

1 2

(l + k) :

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Proof. 1()2 The condition (2.1) is equivalent to g (t' (x) + m (1

t) ' (y))

h (t) g(' (x))

mh(1

t)g(' (y))

h (t) f (' (x)) + mh(1

t)f (' (y))

f (t' (x) + m (1

t) ' (y))

h (t) g(' (x)) + mh(1

t)g(' (y))

g (t' (x) + m (1

t) ' (y))

for all x; y 2 [0; b] and t 2 (0; 1) : The two inequalities may be rearranged as (g + f ) (t' (x) + m (1

t) ' (y))

h (t) (g + f ) (' (x)) + mh(1

t) (g + f ) (' (y))

and (g

f ) (t' (x) + m (1

h (t) (g

t) ' (y))

f ) (' (x)) + mh(1

t) (g

f ) (' (y))

which are eqivalent to the 'h;m convexity of g + f and g f; respectively. 2()3 Let we de…ne the mappings f; g as f = 21 (l k) and g = 12 (l + k). Then if we sum and subtract f and g; respectively, we have g + f = l and g f = k: By the condition 2 in Lemma 1, the mappings g f and g + f are 'h;m convex on [0; b] ; so l; k are 'h;m convex mappings on [0; b] too. Theorem 5. Let h : (0; 1) ! (0; 1) be a given function, g : [0; b] [0; 1) ! R+ be a given 'h;m convex function. If f is de…ned from [0; b] to [0; 1) and it is Lebesgue integrable with g; 'h;m convex dominated on [0; b] for linear continuous, nonconstant function ' : [0; b] ! [0; b] ; then the following inequalities hold: 1 ' (b)

' (a)

Z

'(b)

'(a)

m2 f (x) dx + ' (b) ' (a)

Z

'(b) m

f (x) dx

'(a) m

1 h

' (a) + ' (b) 2

f

1 2

(2.2) 1 ' (b)

' (a)

Z

'(b)

g (x) dx +

'(a)

m2 ' (b) ' (a)

Z

'(b) m '(a) m

g (x) dx

and f (' (a)) + m f ' (2.3) 1 m'(

b m

)

'(a)

a m

R m'( mb ) '(a)

+f ' f (x) dx +

b m

+ mf '

m'(

h R1 0

1 2

g

' (a) + ' (b) 2

h (t) dt

R m'( mb2 ) f (x) dx a ) '( ) '( m )

m b m2

b m2

1

a m

Z 1 a b b h (t) dt g (' (a)) + m g ' +g ' + mg ' m m m2 0 " # Z m'( mb ) Z m'( b2 ) m 1 m g (x) dx + g (x) dx b a a m' m ' (a) '(a) m' mb2 ' m '( m )

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for all x; y; a 2 [0; b], t 2 (0; 1) and m 2 (0; 1] : Proof. By the De…nition 8 with t = 12 ; x = a + (1 [0; 1] and m 2 (0; 1] ; as the mapping f is g; 'h;m we have that 1 2

h

f (' ( a + (1 f

1 2

h g

'( a+(1

a b )b; y = (1 )m + m ; 2 convex dominated function,

)b)) + mf ' (1 a )b)+m'((1 )m +

b m

a )m + )

b m

2

g (' ( a + (1

)b)) + mg ' (1 a )m +

)b) + m' (1 2

' ( a + (1

) b m

a b + m m ! :

Then using the linearity of ' function, we have h

h

1 2 1 2

f ( ' (a) + (1

g ( ' (a) + (1

1

) ' (b)) + mf

m 1

) ' (b)) + mg

m

' (a) +

' (a) +

m

m

' (b)

' (a) + ' (b) 2

f

' (b)

' (a) + ' (b) 2

g

:

If we integrate the above inequality with respect to over [0; 1] ; the inequality (2.2) is proved. Since f is g; 'h;m convex dominated on [0; b] ; we have jh (t) f (' (x)) + mh (1 h (t) g (' (x)) + mh (1

t) f (' (y)) t) g (' (y))

f (t' (x) + m (1 g (t' (x) + m (1

t) ' (y))j t) ' (y))

for all x; y > 0 which gives for x = a and y = b h (t) f (' (a)) + mh (1

h (t) g (' (a)) + mh (1 and for x =

a m;

y=

mh (t) f '

mh (t) g '

b m2 ;

a m a m

t) f

b m

'

t) g '

b m

f

t' (a) + m (1

g t' (a) + m (1

t) '

t) '

b m b m

multiplying with m; + m2 h (1

+ m2 h (1

t) f

'

t) g '

56

b m2 b m2

mf

t'

mg t'

a + m (1 m a + m (1 m

t) '

t) '

b m2 b m2

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for all t 2 (0; 1) : By properties of modulus and adding the above inequalities side by side we get, a h (t) f (' (a)) + mf ' m f t' (a) + m (1 t) '

h h (t) g (' (a)) + mg ' g t' (a) + m (1

a m

t) '

b + mh (1 t) f ' m + mf ' mb2 a + mf t' m + m (1 t) ' mb2

b m

i

+ mh (1

b m

t) g '

+ mg t'

b m

+ mg '

a + m (1 m

t) '

b m2

b m2

Thus, integrating over t on [0; 1] we obtain the inequality (2.3). Remark 1. Under the assupmtions of Theorem 5, if we choose m = 1, the inequalities (2.2) and (2.3) reduce to Hermite-Hadamard type inequalities for (g; 'h ) convex dominated functions given as (1.15) and (1.16) in [11]. Remark 2. Under the assupmtions of Theorem 5, if we choose m = 1, h (t) = t, t 2 (0; 1) and the function ' is the identity, then the inequalities (2.2) and (2.3) reduce to Hermite-Hadamard type inequalities for convex-dominated functions given as (1.9) and (1.10) in [2]. Remark 3. Under the assupmtions of Theorem 5, if we choose m = 1, h (t) = ts , t; s 2 (0; 1) and the function ' is the identity, then the inequalities (2.2) and (2.3) reduce to Hermite-Hadamard type inequalities for (g; s) convex-dominated functions given as (1.9) and (1.10) in [4]. Remark 4. Under the assupmtions of Theorem 5, if we choose m = 1, h (t) = 1t , t 2 (0; 1) and the function ' is the identity, then the inequalities (2.2) and (2.3) reduce to Hermite-Hadamard type inequalities for (g; Q (I)) convex-dominated functions given as (1.9) and (1.10) in [6].

3. (g; log ') convex dominated functions De…nition 9. Let g : [a; b] R ! (0; 1) be a given log ' convex mapping where ' : [a; b] ! [a; b] . The real function f : [a; b] ! (0; 1) is called (g; log ') convex dominated on [a; b] if it holds (3.1)

t

1 t

[f (' (x))] [f (' (y))] t

1 t

[g (' (x))] [g (' (y))]

f (t' (x) + (1 g (t' (x) + (1

t) ' (y)) t) ' (y))

for all x; y 2 [a; b] and t 2 [0; 1]. Proposition 1. Let g : [a; b] R ! (0; 1) be a given log ' convex mapping where ' : [a; b] ! [a; b] and f : [a; b] ! (0; 1) be a (g; log ') convex dominated function on [a; b] :Then the mapping g + f is ' convex on [a; b] :

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Proof. The condition (3.1) is equivalent to g (t' (x) + (1

t

t) ' (y))

1 t

[g (' (x))] [g (' (y))]

t

1 t

f (t' (x) + (1

t) ' (y))

t

1 t

g (t' (x) + (1

t) ' (y))

[f (' (x))] [f (' (y))] [g (' (x))] [g (' (y))]

for all x; y 2 [a; b] and t 2 [0; 1] : The left side of the inequality may be rearranged as (g + f ) (t' (x) + (1 t

t) ' (y)) 1 t

[f (' (x))] [f (' (y))] tf (' (x)) + (1

t

1 t

+ [g (' (x))] [g (' (y))]

t) f (' (y)) + tg (' (x)) + (1

= t (f + g) (' (x)) + (1

t) g (' (y))

t) (f + g) (' (y))

which is eqivalent to the ' convexity of f + g. Theorem 6. Let g : [a; b] R ! (0; 1) be a given log ' convex mapping and f : [a; b] ! (0; 1) is Lebesgue integrable and (g; log ') convex dominated function on [a; b] for linear continuous function ' : [a; b] ! [a; b] ; then the following inequalities hold:

1 ' (b)

' (a)

(3.2) 1 ' (b) (3.3)

' (a)

' (b)

' (a)

G (f (x) ; f (' (a) + ' (b)

x)) dx

f

'(b)

G (g (x) ; g (' (a) + ' (b)

x)) dx

Z

'(b)

G (f (x) ; f (' (a) + ' (b)

x)) dx

f

'(a)

Z

'(b)

g (x) dx

g

'(a)

' (a) + ' (b) 2 ' (a) + ' (b) 2

g

'(a)

' (a) 1

'(b)

'(a)

Z

1 ' (b)

Z

' (a) + ' (b) 2

' (a) + ' (b) 2

and 1

L (f (' (b)) ; f (' (a)))

' (b)

' (a)

(3.4) L (g (' (b)) ; g (' (a)))

1 ' (b)

' (a)

Z

'(b)

f (x) dx

'(a)

Z

'(b)

g (x) dx

'(a)

for all x; y 2 [a; b] :

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9

Proof. By the De…nition 3.1 with t = 21 ; x = a+(1 )b; y = (1 ) a+ b and 2 [0; 1] ; as the mapping f is (g; log ') convex dominated function, we have that [f (' ( a + (1

1

1

)b))] 2 [f (' ((1

) a + b))] 2

1

[g (' ( a + (1

1

)b))] 2 [g (' ((1

' ( a + (1

f

) a + b))] 2

g

)b) + ' ((1 2

' ( a + (1

)b) + ' ((1 2

) a + b) ) a + b)

Then using the linearity of ' function we have

[g ( ' (a) + (1

1

1

)' (b))] 2 [f ((1

(3.5) [f ( ' (a) + (1

) ' (a) + ' (b))] 2

1

1

)' (b))] 2 [g ((1

) ' (a) + ' (b))] 2

If we integrate the above inequality with respect to (3.2) is proved. p On the other hand, if we use the inequality ab we have 1

[f ( ' (a) + (1 g ( ' (a) + (1

:

over [0; 1] ; the inequality in 1 2

(a + b) for a; b > 0 on (3.5)

) ' (a) + ' (b))] 2

) ' (b)) + g ((1 2

' (a) + ' (b) 2

g

1

)' (b))] 2 [f ((1

' (a) + ' (b) 2

f

) ' (a) + ' (b))

' (a) + ' (b) 2

f

g

' (a) + ' (b) 2

:

If we integrate the above inequality with respect to over [0; 1] ; the inequality in (3.3) is proved. To prove the inequality in (3.4), …rstly we use the De…nition 3.1 for x = a and y = b, we have t

1 t

[f (' (a))] [f (' (b))] t

1 t

[g (' (a))] [g (' (b))]

f (t' (a) + (1 g (t' (a) + (1

t) ' (b)) t) ' (b))

Then, we integrate the above inequality with respect to t over [0; 1] ; we get Z 1 Z 1 t f (' (a)) f (' (b)) dt f (t' (a) + (1 t) ' (b)) dt f (' (b)) 0 0 g (' (b))

Z

1

0

= g (' (b))

g (' (a)) g (' (b))

g (' (a)) g (' (b))

1

t

dt

Z

1

g (t' (a) + (1

t) ' (b)) dt

0

1 log g (' (a))

log g (' (b)) ' (b) Z '(b) 1 g (' (b)) g (' (a)) g (x) dx = log f (' (b)) log f (' (a)) ' (b) ' (a) '(a) Z '(b) 1 = L (g (' (b)) ; g (' (a))) g (x) dx: ' (b) ' (a) '(a)

59

1 ' (a)

Z

'(b)

g (x) dx

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If similar calculation is made for the function f; the proof is completed. Corollary 1. If function ' is the identity in (3.2), (3.3) and (3.4), then we have Z b 1 a+b G (f (x) ; f (a + b x)) dx f b a a 2 (3.6) 1 b (3.7)

a

b

b

a

a

b

G (g (x) ; g (a + b

x)) dx

a+b 2

g

a

1

1 and

Z

Z

b

G (f (x) ; f (a + b

x)) dx

f

a

Z

b

g (x) dx

a+b 2

a+b 2

g

a

1

L (f (b) ; f (a))

b

a

(3.8) L (g (b) ; g (a))

1 b

a

for all x; y 2 [a; b] :

Z

b

f (x) dx

a

Z

b

g (x) dx

a

References [1] Dragomir S.S. and Ionescu N.M. On some inequalities for convex-dominated functions, Anal. Num. Theor. Approx., 19 (1990), 21-28. MR 936: 26014 ZBL No.733: 26010. [2] Dragomir S.S., Pearce C.E.M. and Peµcari´c J.E. Means, g Convex Dominated & HadamardType Inequalities; Tamsui Oxford Journal of Mathematical Sciences 18(2) 2002, 161-173.161173. [3] Hwang S. and Ho M. Inequalities of Fejér Type for G convex Dominated Functions, Tamsui Oxford Journal of Mathematical Sciences, 25 (1) (2009) 55-69. [4] Kavurmac¬H., Özdemir M.E. and Sar¬kaya M.Z. New De…nitions and Theorems via Di¤erent Kinds of Convex Dominated Functions, submitted. [5] Özdemir M.E., Kavurmac¬ H. and Tunç M. Hermite-Hadamard-type inequalities for new di¤erent kinds of convex dominated functions, submitted. [6] Özdemir M.E.,Tunç M. and Kavurmac¬ H. Two new di¤erent kinds of convex dominated functions and inequalities via Hermite-Hadamard type, submitted. [7] S. Varošanec, On h convexity, J. Math. Anal. Appl. 326 (2007) 303-311. [8] Sar¬kaya M.Z. On Hermite-Hadamard-type inequalities for 'h convex functions, RGMIA Ress. Rep. Coll., 15(37) 2012. [9] Youness E. A. E-Convex Sets, E-Convex Functions and E - Convex Programming, Journal of Optimization Theory and Applications, 102, 2(1999), 439-450. [10] Özdemir M.E. Akdemir A.O. and Set E. On (h m) Convexity and Hadamard-Type Inequalities, Transylvanian Jour. of Math. and Mechanics, 8, 1(2016), 51-58. [11] Özdemir M.E., Gürbüz M. and Kavurmac¬ H. Hermite-Hadamard-Type Inequalities for (g 'h ) Convex Dominated Functions, Journal of Ineq. and Appl., 2013:184. [12] Pearce C.E.M., Peµcari´c J.E. and Simic V. Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl. 220 (1998) 99-109. [13] Dragomir S.S. and Pearce C.E.M. Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.

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11

[14] Dragomir S.S. and Agarwal R.P. Two inequalities for di¤erentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91-95. [15] Set E., Özdemir M.E. and Dragomir S.S. On the Hermite-Hadamard inequality and other integral inequalities involving two functions, Journal of Ineq. and Appl., Art. ID 148102, 9 pages, 2010. [16] Set E., Özdemir M.E. and Dragomir S.S. On Hadamard-Type inequalities involving several kinds of convexity, Journal of Ineq. and Appl., Art. ID 286845, 12 pagesi 2010. [17] Sar¬kaya M.Z. On Hermite-Hadamard Inequalities for Product of Two log ' Convex Functions, arXiv:1203.5495v1 [math.FA] 25 Mar 2012. Agri Ibrahim Cecen University, Faculty of Education, Department of Mathematics, 04100, AG¼ r¬, Turkey E-mail address : [email protected]

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FIXED POINT THEOREM AND A UNIQUENESS THEOREM CONCERNING THE STABILITY OF FUNCTIONAL QUATIONS IN MODULAR SPACES CHANGIL KIM In this paper, we investigate a xed point theorem in modular spaces, whose induced modular is lower semi-continunous, for a mapping with some conditions in place of the condition of bounded orbit. Using this xed point theorem, we prove the generalized Hyers-Ulam stability for the following additive-quadratic functional equation f (2x + y ) + f (2x y ) 2f (x + y ) 2f (x y ) 2f (2x) + 4f (x) = 0 in modular spaces. Abstract.

1. Introduction and preliminaries A problem that mathematicians has dealt with, for almost fty years, is how to generalize the classical function space Lp ". A rst attempt was made by Birnhaum and Orlicz in 1931. This generalization found many applications in dierential and integral equations with kernels of nonpower types. The more abstract generalization was given by Nakano [13] based on replacing the particular integral form of the functional by an abstract one that satisfes some good properties. This functional was called modular. This idea was re ned and generalized by Musielak and Orlicz [11] in 1959. Modular spaces have been studied for almost forty years and there is a large set of known applications of them in various parts of analysis([6], [7], [9], [10], [12], [14], [17], [20]). It is well known that xed point theories are one of powerful tools in solving mathematical problems. Banach's contraction principle is one of the pivotal results in xed point theories and they have a board set of applications. Khamsi, Kozowski and Reich [4] investigated the xed point theorem in modular spaces. In [5], Khamsi proved a series of xed point theorems in modular spaces, where the modulars do not satisfy 42 -conditions. Lemma 1.1. [5] Let X be a modular space whose induced modular is lower semicontinuous and let C  X be a -complete subset. If T : C ! C is a contraction, that is, there is a constant L 2 [0; 1) such that (T x T y)  L(x y); 8x; y 2 C and T has a bounded orbit at a point x0 convergent to a point w 2 C .

2 C,

then the sequence fT n x0 g is -

2010 Mathematics Subject Classi cation. 39B52, 39B72, 47H09. Key words and phrases. Fixed point theorem, Hyers-Ulam stability, modular spaces. * Corresponding author.

1

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The stability problem for functional equations rst was planed in 1940 by Ulam [18]. \Let G1 be a group and G2 a metric group with the metric d. Given a constant  > 0, does there exist a constant c > 0 such that if a mapping f : G1 ! G2 satis es d(f (xy); f (x)f (y)) < c for all x; y 2 G1 , then there exists a unique homomorphism h : G1 ! G2 with d(f (x); h(x)) <  for all x 2 G1 ?" In the next year, Hyers [3] gave the rst armative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [1] for additive mappings and by Rassias [15] for linear mappings by considering an unbounded Cauchy dierence, the latter of which has in uenced many developments in the stability theory. This area is then referred to as the generalized Hyers-Ulam stability. In 1994, P. Gavruta [2] generalized these theorems for approximate additive mappings controlled by the unbounded Cauchy dierence with regular conditions. Recently, Sadeghi [16] presented a xed point method to prove the generalized Hyers-Ulam stability of functional equations in modular spaces with the 42 condition and using the xed point theorem Lemma 1.1, Wongkum, Chaipunya, and Kumam [19] proved the generalized Hyers-Ulam stability for quadratic mappings in a modular space whose modular is convex, lower semi-continuous but do not satisfy the 42 -condition. Lee and Jung [8] proved a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations in Banach spaces. In this paper, we investigate a xed point theorem in modular spaces, whose induced modular is lower semi-continuous, for a mapping with some conditions in place of the condition of a bounded orbit. Using this xed point theorem, we will prove a general uniqueness theorem that can be applied to the generalized HyersUlam stability of additive-quadratic functional equations in modular spaces without 42 -conditions. 2. Fixed point Theorems in modular spaces In this section, we will prove a xed point theorem in modular spaces, whose induced modular is (convex) lower semi-continuous, for a mapping with some conditions in place of the condition of a bounded orbit. De nition 2.1. Let X be a vector space over a eld K(=R or C). (1) A generalized functional  : X ! [0; 1] is called a modular if (M1) (x) = 0 if and only if x = 0, (M2) (x) = (x) for every scalar  with jj = 1, and (M3) (z )  (x) + (y) whenever z is a convex combination of x and y. (2) If (M3) is replaced by (M4) (x + y)  (x) + (y) for all x; y 2 V and for all nonnegative real numbers , with  + = 1, then we say that  is convex. Remark 2.2. Let  be a modualr on a vector space X . Then by (M1) and (M3), we can easily show that for any positive real number  with  < 1, (x)  (x)

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for all x 2 X and hence we have

3

(x)  (2x)

for all x 2 X . For any modular  on X , the modular space X is de ned by X := fx 2 X j (x) ! 0 as  ! 0g: Let X be a modular space and let fxn g be a sequence in X . Then (i) fxn g is called -convergent to a point x 2 X , denoted by lim x = x or x = nlim n!1 n  !1 xn ; if (xn x) ! 0 as n ! 1, (ii) fxn g is called -Cauchy if for any  > 0, one has (xn xm ) <  for suciently large m; n 2 N, and (iii) a subset K of X is called -complete if each -Cauchy sequence in K is -convergent to a ponit in K . Proposition 2.3. Let  be a modualr on a vector space X and S : X ! X an one-to-one linear map. De ne a map S : X ! [0; 1] by S (x) = (S (x)); 8x 2 X:

Then we have

(1) S is a modular on X , (2) if  is a convex modular on X , then S is a convex modular on X , and (3) if  is lower semi-continuous, then S is lower semi-continuous. Suppose that S is an isomorphism. Then we have

(4) S (XS ) = X and (5) if C is a -complete subset of X and S (C ) = C , then C is a S -complete subset of XS .

(1) Suppose that S (x) = 0. Then by (M1), S (x) = 0 and since S is oneto-one, x = 0. If x = 0, then S (0) = (S (0)) = (0) = 0. Hence S satis es (M1). Since S is a linear map, S satis es (M2). Let z = x + y for x; y 2 X and non-negative real numbers ; with  + = 1. Since S is a linear map and  is a modular, by (M3), we have S (x + y) = (S (x) + S (y))  (S (x)) + (S (y))  S (x) + S (y) and thus S satis es (M3). (2) is trivial. (3) Suppose that fxn g is a sequence in XS such that fxn g is S -convergent to x in XS . Then fS (xn )g is -convergent to S (x). Since  is lower semi-continuous and fS (xn )g is -convergent to S (x), S (x) = (S (x))  lim inf (S (xn ))  lim inf  (x ): (2.1) n!1 n!1 S n and hence S is lower semi-continuous. (4) Let x 2 X . Then

Proof.

lim (x) = lim  (S 1 (x)) = 0 !0 S and so S 1 (x) 2 XS . Hence X  S (XS ). For the converse, let x 2 XS . Then lim S (x) = lim (S (x)) = 0 !0 !0 !0

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and so S (x) 2 X . Hence S (XS )  X . (5) Suppose that C is a -complete subset of X with S (C ) = C . By (4), C  XS . Let fxn g be a S -Cauchy sequence in C . For any n; m 2 N, (S (xn ) S (xm )) = S (xn xm ); fS (xn )g is a -Cauchy sequence in C . Since C is a -complete subset of X , there is an y 2 C such that fS (xn )g is -convergent to y. Then clearly, fxn g is S convergent to S 1 (y) 2 C and so C is a S -complete subset of XS .  A modular space X is said to satisfy the 42 -condition if there exists k  2 such that (2x)  k(x) for all x 2 X . Now, we will prove a xed point theorem in modular spaces where the map T do not assume to be the boundedness of an orbit. Our results exploit one unifying hypothesis in which some conditions are assumed. Lemma 2.4. Let X be a modular space whose induced modular is lower semicontinuous and let C  X be a -complete subset. Let S : X ! X be an isomorphism and T : C ! C a mapping such that S (C ) = C and ST x = T Sx for all x 2 C . Suppose that there is a constant L 2 [0; 1) and xo 2 C such that (T xo xo ) < 1 and (2.2) (x + y)  S (x) + S (y); S (T x T y)  L(x y); 8x; y 2 C: Then there is a unique xed point w 2 C of T such that 2 (2.3) (S 2 x0 w)  1 L (T x0 x0 ):

Further, we have

lim (T n S 2 x0 ) = w:

n!1

By Proposition 2.3, S is a modular, C is a S -complete subset of XS , and S (XS ) = X . By (M1) and (2.2), we have (x)  S (x); S (T x T y)  L(x y)  LS (x y) for all x; y 2 C and so T is a S -contraction. By (M1) and (2.2), we have S (S 2 T 2 x0 S 2 x0 ) = (S 1 T 2 x0 S 1 x0 )  S (S 1 T 2 x0 S 1 T x0 ) + S (S 1 T x0 S 1 x0 )  L(S 1 T x0 S 1 x0 ) + (T x0 x0 )  (L + 1)(T x0 x0 ):

Proof.

and

S (S 2 T n x0 S 2 x0 )  S (S 1 T n x0 S 1 T x0 ) + S (S 1 T x0 S 1 x0 )  L(S 1 T n 1 x0 S 1 x0 ) + (T x0 x0 ) = LS (S 2 T n 1 x0 S 2 x0 ) + (T x0 x0 ) for all n 2 N. By induction, we have 1 (T x x ) S (S 2 T n x0 S 2 x0 )  nk=01 Lk (T x0 x0 )  0 0 1 L

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5

for all n 2 N. For any non-negative integers m; n with m > n, (2.4) S (S 3 T n x0 S 3 T m x0 ) = (S 2 T n x0 S 2 T m x0 )  S (S 2 T n x0 S 2 x0 ) + S (S 2 T m x0 S 2 x0 )  1 2 L (T x0 x0 ):

Since ST x = T Sx for all x 2 C , T has a bounded orbit at a point S 3 x0 in C  XS and thus by Lemma 1.1, fT n S 3 x0 g is S -convergent to a point !0 2 C . Let ! = S!0 . Then lim (T n S 2 x0 ) = w n!1 and since S is lower semi-continuous, by (2.4), we have (2.3). Now, we claim that w is a unique xed point of T . Since S is a lower semicontinuous, we have (w T w) = S (w0 T w0 )  lim inf  (T n+1 S 3 x0 T w0 ) n!1 S  lim inf L(T n S 3 x0 w0 ) = 0 n!1 and hence w is a xed point of T . Suppose that v is another xed point of T . Since ST x = T Sx for all x 2 C , by (2.2) and (2.3), we have (S 1 w S 1 v)  S (T n w T n S 2 x0 ) + S (T n S 2 x0 T n v)  Ln 1 4 L (T x0 x0 )

for all n 2 N and since 0  L < 1, w = v.



Replacing (2.2) by (2.5), we have the following lemma. The proof is similar to the proof of Lemma 2.4. Lemma 2.5. Let X be a modular space whose induced modular is lower semicontinuous and let C  X be a -complete subset. Let S : X ! X be an isomorphism and T : C ! C a mapping such that S (C ) = C and ST x = T Sx for all x 2 C . Suppose that there are real numbers r; L and xo 2 C such that 0 < r < 1, L 2 [0; 1r ), (T xo xo ) < 1, and

(2.5)

(x + y)  rS (x) + rS (y); S (T x T y)  L(x y); 8x; y 2 C:

Then there is a unique xed point w 2 C of T such that

lim (T n S 2 x0 ) = w

n!1

and

2r2 (T x x ): (S 2 x0 w)  0 0 1 rL Proof. By Proposition 2.3, S is a modular, C is a S -complete subset of XS , and S (XS ) = X . By (M1) and (2.5), we have

(2.6)

(x)  S (x); S (T x T y)  L(x y)  rLS (x y)

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for all x; y 2 C and so T is a S -contraction. By (M1) and (2.5), we have S (S 2 T 2 x0 S 2 x0 ) = S (S 1 T 2 x0 S 1 x0 )  rS (S 1 T 2 x0 S 1 T x0 ) + rS (S 1 T x0 S  rL(S 1 T x0 S 1 x0 ) + r(T x0 x0 )  r(rL + 1)(T x0 x0 ): and S (S 2 T n x0 S 2 x0 )  rS (S 1 T n x0 S 1 T x0 ) + rS (S 1 T x0 S  rL(S 1 T n 1 x0 S 1 x0 ) + r(T x0 x0 ) = rLS (S 2 T n 1 x0 S 2 x0 ) + r(T x0 x0 ) for all n 2 N. By induction, we have

1 x0 )

1 x0 )

r S (S 2 T n x0 S 2 x0 )  nk=01 rk+1 Lk (T x0 x0 )  1 rL (T x0 x0 ) for all n 2 N. For any non-negative integers m; n with m > n, S (S 3 T n x0 S 3 T m x0 ) = (S 2 T n x0 S 2 T m x0 )  rS (S 2 T n x0 S 2 x0 ) + rS (S 2 T m x0 S 2 x0 ) 2  1 2rrL (T x0 x0 ):

The rest of the proof is similar to the proof of Lemma 2.4 .



3. Uniquness theorem for the stability of functional equations and its applications Throughout this section, we assume that V is a linear space and X is a complete modular space whose induced modular is lower semi-continuous. In this section, we prove that, if for given map f : V ! X , there is a mapping F : V ! X , which is near f in X , with some properties possessed by additive-quadratic mappings, then F is uniquely determined. De ne a set M by M := fg : V ! X j g(0) = 0g and a generalized function e on M by for each g 2 M, e(g) := inf fc > 0 j (g(x))  c (x); 8x 2 V g; where : V ! [0; 1) is a mapping. Similar to Lemma 10 in [19], we have the following lemma : Lemma 3.1. We have the following :

(1) (2) (3) (4) (5)

M is a linear space,

e is a modular on M, and if  is convex, then e is convex, Me = M and Me is e-complete, and e is lower semi-continuous.

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(1), (2), and (3) are trivial. (4) By the de nition of Me, Me = M. Take any e-Cauchy sequence fgn g in Me. Then fgn (x)g is a -Cauchy sequence in X for all x 2 X . Since X is -complete, there is a mapping g : V ! X such that (gn (x) g(x)) ! 0 as n ! 1 for all x 2 X . For any  > 0, there is an m 2 N such that (gm (0) g(0)) = (g(0))   and hence g 2 Me = M. Let  > 0 be given. Since fgn g is a e-Cauchy sequence in Me, there is a k 2 N such that for any n; m 2 N with n; m  k, (gn (x) gm (x))   (x); 8x 2 V and since  is a lower semi-continuous, we have (gn (x) g(x))  lim inf (gn (x) gm (x))   (x) m!1 Proof.

for all x 2 X . Hence fgn g is e-convergent and thus Me is e-complete. (5) Suppose that fgn g is a sequence in Me which is e-convergent to g 2 Me. Then fgn (x)g is -convergent to g(x) for all x 2 V . Let  > 0 be given. Then for any n 2 N, there is a positive real number cn such that e(gn )  cn  e(gn ) + 

and so





(g(x))  lim inf (gn (x))  lim inf c (x)  lim inf e(gn ) +  (x) n!1 n!1 n n!1 for all x 2 X . Hence e is lower semi-continuous.



Now, with Lemma 2.4 and Lemma 3.1, we will show the following uniquness concerning the stability of additive-quadratic mappings and using these, we prove the generalized Hyers-Ulam stability for additive-quadratic mappings Theorem 3.2. Let  : V ! [0; 1) be a mapping and L a positive real number sucht that 0  L < 21 and (3.1) (2x)  L(x) for all x 2 V . Let f; F : V ! X be mappings such that (3.2) (f (x) F (x))  M [(x) + ( x)] for all x 2 V and some non-negative real number M and

(3.3)

F (2x) = 3F (x) + F ( x)

for all x 2 X . Then F is determined by

(3.4)

1 F (x) = lim h 1 + 1 f (2n x) +  1  n!1 8 22n+4 2n+4 22n+4

for all x 2 V .

1 f ( 2n x)i 2n+4

Proof. Let (x) = (x) + ( x) for all x 2 V . By Lemma 3.1, Me = M is e-complete and e is lower semi-continuous. De ne T : Me ! Me by T g(x) = 3 g(2x) 1 g( 2x) for all g 2 Me and all x 2 V and S : Me ! Me by Sg = 2g for 8 8

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all g 2 Me. Then S is an isomorphism. Suppose that g; h 2 Me and e(g h)  c for some positive real number c. By (M3) and (3.1), we have 3 1 3 1 S (T g(x) T h(x)) = ( g(2x) 4 4 g( 2x) 4 h(2x) + 4 h( 2x))  (g(2x) h(2x)) + (g( 2x) h( 2x))  c( (2x) + ( 2x))  2cL (x) for all x 2 V and so eS (T g T h)  2Le(g h): By (M3), we have e(g h)  eS (g) + eS (h) for all g; h 2 Me. By (3.3), F is a xed point of T and since eS = fS , by (3.2), we get (S 1 f (x) T S 1 f (x))  (f (x) F (x)) + (T F (x) T f (x))  M (x) + S (T F (x) T f (x))  M (x) + 2L(F (x) f (x)) for all x 2 V and thus e(S 1 f T S 1 f )  (1 + 2L)M < 1: By Lemma 2.4, there is a unique xed point G 2 Me of T such that lim e(T n S 3 f G) = 0 n!1 and 2 1 1 e(S 3 f G)  1 2L e(S f T S f ): Since F is a xed point of T , S 3 F is a xed point of T and by (3.2), we have 2 e(S 1 f T S 1 f ) e(S 3 f S 3 F )  e(S 1 f S 1 F )  1 2L 2 because 1 < 1 2L . Hence by the uniquness of G in Lemma 2.4, S 3 F = G and 



(3.5) lim e T n S 3 f S 3 F = 0: n!1 By induction, there are sequences fan g and fbn g such that T n f (x) = an f (2n x) + bn f ( 2n x) for all x 2 V and all n 2 N. By the de nition of T , 3 1 b f (2n+1 x) +  3 b 1 a f ( 2n+1 x) T n+1 f (x) = an 8 8n 8n 8 n and so 8 1 3 >

:b

n+1

= 83 bn 18 an

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8 1 >

:a 1 n+1 bn+1 = 2 (an bn )

for all n 2 N and so we get

for all n 2 N. Thus

8 1 >

: bn = 22n1+1

+ 2n1+1 1 2n+1

1 + 1 f (2n x) +  1 1 f ( 2n x)i 22n+4 2n+4 22n+4 2n+4 for all x 2 V and hence we have (3.4).  For any mapping f : V ! X , let f (x) f ( x) f (x) + f ( x) fo (x) = ; fe (x) = : 2 2 Then fo is odd and fe is even. By the fact that f (x) = fo (x) + fo (x) for all x 2 V , we can easily show the following corollary : S 3 F (x) = nlim !1

h

Corollary 3.3. All conditions in Theorem 3.2 are assumed. Then F is determined by

1 F (x) = lim 1 f (2n x);  n!1 n+3 o 8 o 2 1 1 n !1 22n+3 fe (2 x); 8 Fe (x) = nlim

(3.6) (3.7) and

(3.8)

for all x 2 V . Proof.

1 F (x) = lim  1 f (2n x) + 1 f (2n x)  n!1 2n+4 e 16 2 2n+4 o

Note that h1

 Fo (x)

i

1 n 2n+3 fo (2 x)

8 1 F (x) 1 F ( x) 1 f (2n x) + 1 f ( 2n x)i =  16 16 2n+4 2n+4  1   i h1   8 F (x) 22n+4 + 2n1+4 f (2n x) 22n1+4 2n1+4 f ( 2n x) h     i +  18 F ( x) 22n1+4 + 2n1+4 f ( 2n x) 22n1+4 2n1+4 f (2n x) for all x 2 V and for all n 2 N. By (3.4), we have (3.6) and similarly, we have (3.7). Thus we get (3.8).  If F is additive or quadratic or additive-quadratic, then F satis es (3.3) and hence we have the following corollary : h

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Corollary 3.4. All conditions in Theorem 3.2 are assumed. If F is addiive(quadratic, addiitive-quadratic, resp.) then F is determind by

1 F (x) = lim 1 f (2n x)  1 F (x) = lim 1 f (2n x);  n!1 n+3 o  n!1 2n+3 e 8 2 8 2 1 f (2n x) + 1 f (2n x); resp: 1 F (x) = lim  n!1 2n+4 e 16 2 2n+4 o

for all x 2 V .

Similar to the proof of Theorem 3.2, we can show the following theorem for modular spaces with convex modulars. Theorem 3.5. All conditions in Theorem 3.2 are assumed. Suppose that  is convex and L is a positive real number sucht that 0  L < 2. Then F is determined by

(3.9)

1 F (x) = lim h 1 + 1 f (2n x) +  1  n!1 8 22n+4 2n+4 22n+4

for all x 2 V .

1 f ( 2n x)i

2n+4

Proof. Let (x) = (x) + ( x) for all x 2 V . By Lemma 3.1, Me = M is e-complete and e is lower semi-continuous. De ne T : Me ! Me by T g(x) = 3 g(2x) 1 g( 2x) for all g 2 Me and all x 2 V and S : Me ! Me by Sg = 2g for 8 8 all g 2 Me. Then S is an isomorphism. Suppose that g; h 2 Me and e(g h)  c for some positive real number c. By (M3) and (3.1), we have

3 1 g( 2x) 3 h(2x) + 1 h( 2x)) 4 4 4 4 3 1  4 (g(2x) h(2x)) + 4 (g(2x) h( 2x))  cL (x)

S (T g(x) T h(x)) = ( g(2x)

for all x 2 V and so Further, clearly we have

eS (T g T h)  Le(g h):

1 1 2 2 for all g; h 2 Me. By (3.3), F is a xed point of T and by (3.2), we get 1 1 (S 1 f (x) T S 1 f (x))  (f (x) F (x)) + (T F (x) T f (x)) 2 2  12 M (x) + 12 S (T F (x) T f (x))  12 (1 + L)M (x) for all x 2 V and thus 1 e(S 1 f T S 1 f )  (1 + L)M < 1: 2 By Lemma 2.5, there is a unique xed point G 2 Me of T such that 1 e(S 1 f T S 1 f ): e(S 3 f G)  2 L e(g h)  eS (g) + eS (h)

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and further, we have

lim e(T n S 3 f G) = 0: Since F is a xed point of T , S 3 F is a xed point of T and 1 1 e(S 1 f T S 1 f ) e(S 3 f S 3 F )  e(S 1 f S 1 F )  4 2 L n!1

because 21  2 1L . Hence by the uniquness of G in Lemma 2.4, S 3 F = G. The rest proof is similar to the proof of Theorem 3.2.  Using Lemma 2.5 and Theorem 3.5, we can show the generalized Hyers-Ulam stability for aditive-quadratic mappings. Corollary 3.6. Let V be a linear space and X a -complete modular space whose induced modular is convex lower semi-continuous. Suppose that f : V ! X is a mapping such that

(f (x + y) + f (x y) 2f (x) f (y) f ( y))  (x; y) for all x; y 2 V and let  : V 2 ! [0; 1) be a mapping satisfying (3.11) (2x; 2y)  L(x; y); 8x; y 2 V for some real number L with 0  L < 2. Then there is a unique additive-quadratic mapping G : V ! X such that

(3.10)

(3.12) for all x 2 V .

1 4

( f (x) G(x)) 

3 8(2 L) [(x; x) + ( x; x)]

Let (x) = (x; x) and (x) = (x) + ( x) for all x 2 V . By Lemma 3.1, Me = M is e-complete and e is lower semi-continuous. De ne T : Me ! Me by T g(x) = 83 g(2x) 18 g( 2x) for all g 2 Me and all x 2 V and S : Me ! Me by Sg = 2g for all g 2 Me. Then S is an isomorphism and (2.5) in Lemma 2.5 holds for r = 12 . Letting y = x in (3.10), we get Proof.

(3.13) (f (2x) 3f (x) f ( x))  (x; x) for all x 2 V and by (3.13), we have 3 1 3 (T f (x) f (x))  (x; x) + ( x; x)  (x) 8 8 8 for all x 2 V . Hence we get 3 (3.14) e(T f f )  8 and by Lemma 2.5, there is a unique xed point G 2 Me of T such that 3 e(S 2 f G)  8(2 L) : For any n 2 N, let 1 1 1 1 an = 2n+4 + n+4 ; bn = 2n+4 2 2 2 2n+4 :

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Since G is a xed point of T , G satis es (3.3) and by Theorem 3.5, we have 1 G(x) = lim [a f (2n x) + b f ( 2n x)]  n!1 n n 8 for all x 2 V . By (M3), we get (3.15)  1 1 1 1 1  9 G(x + y) + 9 G(x y) 8 G(x) 9 G(y) 9 G( y) 2 2 2 2 2 1 1 n n  26  8 G(x + y) an f (2 (x + y)) bn f ( 2 (x + y))   + 216  18 G(x y) an f (2n (x y)) bn f ( 2n (x y))   + 216  14 G(x) 2an f (2n x) 2bn f ( 2n x)   + 216  18 G(y) an f (2n y) bn f ( 2n y)   + 216  18 G( y) an f (2n ( y)) bn f ( 2n ( y))   + a2n6  f (2n (x + y)) + f (2n (x y)) 2f (2n x) f (2n y) f (2n ( y)) 







+ jb2n6 j  f ( 2n (x + y)) + f ( 2n (x y)) 2f ( 2n x) f ( 2n y) f ( 2n ( y)) and by (3.11), we have (3.16)   an n (x + y )) + f (2n (x y )) 2f (2n x) f (2n y ) f (2n ( y ))  f (2 26 + jb2n6 j  f ( 2n (x + y)) + f ( 2n (x y)) 2f ( 2n x) f ( 2n y) f ( 2n ( y))

 a2n6 (2n x; 2n y) + jb2n6 j ( 2n x; 2n y) i h  Ln an (x; y) + jbn j ( x; y) 26

26 for all x; y 2 V and for all n 2 N. Since 0 < an ; jbn j < 2 n , by (3.15) and (3.16), we can show that G is an additive-quadratic mapping. Since every additive-quadratic mapping satisifes (3.3), G is a unique additive-quadratic mapping with (3.12).  References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] P. Gavruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [3] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [4] M. A. Khamsi, W. M. Kozowski and S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal. 14(1990), 935-953. [5] M. A. Khamsi, Quasicontraction mappings in modular spaces without 2-condition, Fixed Point Theory and Applications, 2008(2008), 1-6. [6] S. Koshi and T. Shimogaki, On F-norms of quasi-modular spaces , J. Fac. Sci. Hokkaido Univ. Ser. 15(1961), 202-218. [7] M. Krbec, Modular interpolation spaces, Z. Anal. Anwendungen 1 (1982), 25-40.

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[8] Y. H. Lee and S. M. Jung, General uniqueness theorem concerning the stability of additive and quadratic functional equations journal of function spaces 2015(2015), 1-8. [9] A. Luxemburg, Banach function spaces, Ph. D. thesis, Delft Univrsity of technology, Delft, The Netherlands, 1959. [10] L. Maligranda, Orlicz Spaces and Interpolation, in: Seminars in Math., Vol. 5, Univ. of Campinas, Brazil, 1989. [11] J. Musielak and W. Orlicz ,On modular spaces, Studia Math. 18 (1959), 49-65. [12] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math., Vol. 1034, Springerverlag, Berlin, 1983. [13] H. Nakano, Modular semi-ordered spaces, Tokyo, Japan, 1959. [14] W. Orlicz, Collected Papers, Vols. I, II, PWN, Warszawa, 1988. [15] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [16] G. Sadeghi, A xed point approach to stability of functional equations in modular spaces, Bulletin of the Malaysian Mathematical Sciences Society. Second Series, 37(2014), 333-344. [17] Ph. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math., Tomus specialis in honorem Ladislai Orlicz I (1978), 331-353. [18] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York; 1964. [19] K. Wongkum, P. Chaipunya, and P. Kumam, On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without 42 -conditions, Journal of function spaces 2015(2015), 1-6. [20] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc. 90(1959), 291-311. Department of Mathematics Education, Dankook University, 152 Jukjeon-ro, Suji-gu, Yongin-si, Gyeonggi-do 448-701, Republic of Korea

E-mail address : [email protected]

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Rough fuzzy ideals in BCK/BCI-algebras Sun Shin Ahn1 and Jung Mi Ko2,∗ 1 2

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Department of Mathematics, Gangneung-Wonju National University, Gangneung, 25457, Korea

Abstract. The notions of rough ideals and rough fuzzy ideals in BCK/BCI-algebras are introduced and some properties of such ideals are investigated. The relations between the upper(lower) rough ideals and the upper (lower) approximations of their homomorphic images are discussed.

1. Introduction The notion of rough sets was introduced by Pawlark ([11]). The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. It is turning out to be methodologically significant to the domains of artificial intelligence and cognitive sciences, especially in the representation of reasoning with vague and/or imprecise knowledge, data analysis, machine learning, and knowledge discovery ([11,12]). The algebraic approach to rough sets was studied in [8]. Biswas and Nanda ([1]) introduced the notion of rough subgrougs, and Kuroki and Morderson ([6]) discussed the structure of rough sets and rough groups. Kuroki and Wang ([7]) gave some properties of lower and upper approximations with respect to the normal subgroups and the fuzzy normal subgroups, and Kuroki ([5]) introduced the notion of rough ideals in semigroup, which is an extended notion of ideals in semigroups, and gave some properties of such ideals. Xiao and Zhang ([13]) established the notion of rough prime ideals and rough fuzzy prime ideals in a semigroup. Imai and Is´eki ([2]) introduced two classes of abstract algebras : BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. C. R. Lim and H. S. Kim ([8]) introduced the notion of a rough set in BCK/BCI-algebras. By introducing the notion of a quick ideal in BCK/BCI-algebras, they obtained some relations between quick ideals and upper(lower) rough quick ideals in BCK/BCI-algebras. In this paper, we introduce the notion of rough ideals and rough fuzzy ideals in BCK/BCIalgebras, and we give some properties of such ideals. Also, we discuss the relations between the upper(lower) rough ideals and the upper (lower) approximations of their homomorphic images. 0

2010 Mathematics Subject Classification: 06F35; 03G25. Keywords: rough sets; rough ideals; rough fuzzy ideals. The corresponding author. 0 E-mail: [email protected] (S. S. Ahn); [email protected] (J. M. Ko) 0 This study was supported by Gangneung-Wonju National University. 0

∗

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2. Preliminaries A BCI-algebra ([9]) is a non-empty set X with a constant 0 and a binary operation “ ∗ ” satisfying the axioms, for all x, y, z ∈ X: (i) (ii) (iii) (iv)

((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (x ∗ (x ∗ y)) ∗ y = 0, x ∗ x = 0, x ∗ y = 0 and y ∗ x = 0 imply x = y.

A BCK-algebra is a BCI-algebra satisfying the axiom: (v) 0 ∗ x = 0 for all x ∈ X. We can define a partial ordering ≤ on X by x ≤ y if and only x ∗ y = 0. In any BCI-algebra X, the following hold: for any x, y, z ∈ X, (1) (2) (3) (4)

x ∗ 0 = x, (x ∗ y) ∗ z = (x ∗ z) ∗ y, x ≤ y implies x ∗ z ≤ y ∗ z and z ∗ y ≤ z ∗ x, (x ∗ z) ∗ (y ∗ z) ≤ (x ∗ z).

Let X be a BCK/BCI-algebra and let 0 ∈ I ⊆ X. A set I is called an ideal of X if for all x, y ∈ X, x ∗ y ∈ I and y ∈ I imply x ∈ I. An ideal I is said to be closed if 0 ∗ x ∈ I whenever x ∈ I. Let S be a non-empty subset of X. Then S is called a subalgebra of X if, for any x, y ∈ S, x∗y ∈ S. A closed ideal of a BCK/BCI-algebra X is a subalgebra of X. An equivalence relation ρ on X is called a congruence relation on X if (x ∗ u, y ∗ v) ∈ ρ for any (x, y), (u, v) ∈ ρ. We denote by [a]ρ the ρ-congruence class containing the element a ∈ X. Let X/ρ be the set of all ρ-equivalence classes on X, i.e., X/ρ := {[a]ρ |a ∈ X}. For any [x]ρ , [y]ρ ∈ X/ρ, if we define [x]ρ ∗ [y]ρ := [x ∗ y]ρ = {z ∈ X|(z, x ∗ y) ∈ ρ}, then it is well defined, since ρ is a congruence relation. A congruence relation ρ on a BCK/BCIalgebra X is said to be regular if [x]ρ ∗ [y]ρ = [0]ρ = [y]ρ ∗ [x]ρ implies [x]ρ = [y]ρ for any [x]ρ , [y]ρ ∈ X/ρ. Theorem 2.1. ([9]) Let X be a BCK-algebra and let ρ be a congruence relation on X. Then ρ is regular if and only if X/ρ is a BCK-algebra. Let I be an ideal of X. We define a relation ρI on X as follows: ρI := {(x, y)|x ∗ y, y ∗ x ∈ I}.

(*)

Then ρI is a regular congruence relation ([4]). Let Con(X) be the set of all congruences on X. We define a subset Iρ of X from ρ ∈ Con(X) by Iρ := {x ∗ y|(x, y) ∈ ρ}.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Rough fuzzy ideals in BCK/BCI-algebras

Proposition 2.2. ([4]) Let A be an ideal. If A is closed, then A = IρA . Let X be a BCK/BCI-algebra and let ρ be a congruence relation on X. Let P(X) denote the power set of X and P ∗ (X) = P(X) \ {∅}. Define the functions ρ, ρ− : P(X) → P(X) as follows: for any ∅ ̸= A ∈ P(X), ρ− (A) := {x ∈ X|[x]ρ ⊆ A} and ρ− (A) := {x ∈ X|[x]ρ ∩ A ̸= ∅}. The set ρ− (A) is called the ρ-lower approximation of A, while ρ− (A) is called the ρ-upper approximation of A. For a non-empty subset A of X, ρ(A) = (ρ− (A), ρ− (A)) is called a rough set with respect to ρ of P(X) × P(X) if ρ− (A) ̸= ρ− (A). A subset A of X is said to be definable if ρ− (A) = ρ− (A). The pair (X, ρ) is called an approximation space. A congruence relation ρ on a set X is called complete if [x]ρ ∗ [y]ρ = [x ∗ y]ρ for any x, y ∈ X. 3. Rough ideals in BCK/BCI-algebras Let X be a BCK/BCI-algebra and let ∅ ̸= A ⊆ X. Let ρ be a congruence relation on X. Then A is called an upper (a lower, respectively) rough ideal of X if ρ− (A) (ρ− (A), respectively) is an ideal of X. Theorem 3.1. Let ρI be a congruences relation on a BCK/BCI-algebra X as in (∗). If A is a closed ideal of X, then it is an upper rough ideal of X. Proof. Since A is an ideal of X, 0 ∈ A. Hence A ∩ [0]ρI ̸= ∅. Therefore 0 ∈ ρI − (A). Let x, y ∈ X with x∗y, y ∈ ρI − (A). Then ([x]ρI ∗[y]ρI )∩A = ([x∗y]ρI )∩A ̸= ∅ and [y]ρI ∩A ̸= ∅. Hence there exist α, β ∈ A such that α ∈ [x]ρI ∗ [y]ρI = [x ∗ y]ρI and β ∈ [y]ρI . Therefore α = p ∗ q for some p ∈ [x]ρI , q ∈ [y]ρI . Since β, q ∈ [y]ρI , we have (β, y), (y, q) ∈ ρI and so (β, q) ∈ ρI . Hence [β]ρI = [q]ρI . Since (q ∗ β, q ∗ q) = (q ∗ β, 0) ∈ ρI , we have (q ∗ β) ∗ 0 = q ∗ β ∈ A by Proposition 2.2. Using β ∈ A, we have q ∈ A. Since p ∗ q, q ∈ A and A is an ideal of X, we obtain p ∈ A. Therefore p ∈ [x]ρI ∩ A ̸= ∅. Thus x ∈ ρI − (A), completing the proof. □ Theorem 3.1 shows that the notion of an upper rough ideal is an extended notion of a closed ideal in BCK/BCI-algebras. Example 3.2. Let X := {0, 1, 2, 3} be a BCK-algebra with the following Cayley table: ∗ 0 1 2 3

0 0 1 2 3

1 0 0 1 3

2 0 0 0 3

3 0 0 1 0

If we take A := {0, 2}, then it is not an ideal of X, since 1 ∗ 2 = 0 ∈ A, but 1 ∈ / A. On the while, let ρ be a congruence relation on X such that {0, 1, 2}, {3} are all ρ-congruence classes of X. Then ρ− (A) = {0, 1, 2} is an ideal of X.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Sun Shin Ahn and Jung Mi Ko

Theorem 3.3. Let X be a BCK/BCI-algebra and let A be a closed ideal of X. Then ρ− (A), if it is non-empty, is an ideal of X. Proof. Since A is a closed ideal of X, it is a subalgebra of X. Since ρ− (A) ̸= ∅, ρ− (A) is a subalgebra of X. Hence 0 ∈ ρ− (A). Let x, y ∈ X with x ∗ y, y ∈ ρ− (A). Then [x]ρ ∗ [y]ρ = [x ∗ y]ρ ⊆ A, [y]ρ ⊆ A. If α ∈ [x]ρ , then (α, x) ∈ ρ. Since ρ is a congruence relation on X, we have (α ∗ y, x ∗ y) ∈ ρ and so α ∗ y ∈ [x ∗ y]ρ ⊆ A. Since A is an ideal of X and y ∈ A, we get α ∈ A, i.e., [x]ρ ⊆ A, proving that x ∈ ρ− (A). □ Let ρ be a regular congruence relation on a BCK-algebra X and let ∅ ̸= A ⊆ X. The lower and upper approximations can be presented in an equivalent form as shown below: ρ− (A)/ρ ={[x]ρ ∈ X/ρ | [x]ρ ⊆ A} ρ− (A)/ρ ={[x]ρ ∈ X/ρ | [x]ρ ∩ A ̸= ∅}. Proposition 3.4. Let ρ be a regular congruence relation on a BCK-algebra X. If A is a subalgebra of X, then ρ− (A)/ρ is a subalgebra of the quotient BCK-algebra X/ρ. Proof. Since A is a subalgebra of X, there exists an element x ∈ A such that [x]ρ ∩ A ̸= ∅, i.e., ρ− (A)/ρ ̸= ∅. Let [x]ρ and [y]ρ be any elements of ρ− (A)/ρ. Then [x]ρ ∩ A ̸= ∅ and [y]ρ ∩ A ̸= ∅. This means that there exist a, b ∈ X such that a ∈ [x]ρ ∩A and b ∈ [y]ρ ∩A. Then a∗b ∈ [x]ρ ∗[y]ρ . Since A is a subalgebra of X, a ∗ b ∈ A. This means that [x]ρ ∗ [y]ρ ∈ ρ− (A)/ρ, completing the proof. □ Proposition 3.5. Let ρ be a regular congruence relation on a BCK-algebra X. If A is a subalgebra of X, then ρ− (A)/ρ is, if it is non-empty, a subalgebra of the quotient BCK-algebra X/ρ. □

Proof. Straightforward.

Theorem 3.6. Let ρI be a regular congruence relation on a BCK-algebra X as in (∗). If A is an ideal of X, then ρI − (A)/ρI is an ideal of the quotient BCK-algebra X/ρI . Proof. Since 0 ∈ ρI − (A), we have [0]ρI ∩A ̸= ∅ and hence [0]ρI ∈ ρI − (A)/ρI . Let [x]ρI ∗[y]ρI , [y]ρI ∈ ρI − (A)/ρI . Then ([x]ρI ∗ [y]ρI ) ∩ A = [x ∗ y]ρI ∩ A ̸= ∅ and [y]ρI ∩ A ̸= ∅. Hence there exist α ∈ A with α ∈ [x]ρI ∗ [y]ρI = [x ∗ y]ρI and β ∈ A for some β ∈ [y]ρI . Therefore α = p ∗ q for some p ∈ [x]ρI , q ∈ [y]ρI . Since β, q ∈ [y]ρI , we have (β, y), (y, q) ∈ ρI and so (β, q) ∈ ρI . Hence [β]ρI = [q]ρI . Since (q ∗ β, q ∗ q) = (q ∗ β, 0) ∈ ρI , we have (q ∗ β) ∗ 0 = q ∗ β ∈ A by Proposition 2.2. Using β ∈ A, we have q ∈ A. Since A is an ideal of X and q ∈ A, we have p ∈ A. Thus p ∈ [x]ρI ∩ A, proving [x]ρI ∈ ρI − (A)/ρI . □ Theorem 3.7. Let ρ be a regular congruence relation on a BCK-algebra X. If A is an ideal of X, then ρ− (A)/ρ, if it is non-empty, an ideal of the quotient BCK-algebra X/ρ.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Rough fuzzy ideals in BCK/BCI-algebras

Proof. Since ρ− (A)/ρ ̸= ∅, ρ− (A)/ρ is a subalgebra of X/ρ. Hence [0]ρ ∈ ρ− (A)/ρ. Let [x]ρ ∗ [y]ρ , [y]ρ ∈ ρ− (A)/ρ for some [x]ρ ∈ X/ρ. Hence [x ∗ y]ρ ⊆ A and [y]ρ ⊆ A. Therefore x ∗ y ∈ ρ− (A), y ∈ ρ− (A). If α ∈ [x]ρ , then (α, x) ∈ ρ. Since ρ is a congruence relation on X, we have (α ∗ y, x ∗ y) ∈ ρ. Hence α ∗ y ∈ [x ∗ y]ρ ⊆ A. Hence α ∈ A, because A is an ideal of X and y ∈ A. Therefore [x]ρ ⊆ A, proving that [x]ρ ∈ ρ− (A)/ρ. □ Theorem 3.8. Let ρ be a regular congruence relation on a BCK-algebra X. If A is an upper rough ideal of X, then ρ− (A)/ρ is an ideal of the quotient algebra X/ρ. Proof. Since 0 ∈ ρ− (A), we have [0]ρ ∩ A ̸= ∅ and hence [0]ρ ∈ ρ− (A)/ρ. Let [x]ρ ∗ [y]ρ = [x ∗ y]ρ , [y]ρ ∈ ρ− (A)/ρ for some [x]ρ ∈ X/ρ. Then ([x]ρ ∗ [y]ρ ) ∩ A = [x ∗ y]ρ ∩ A ̸= ∅ and [y]ρ ∩ A ̸= ∅. Hence x ∗ y, y ∈ ρ− (A). Since ρ− (A) is an ideal of X, we have x ∈ A. Thus x ∈ [x]ρ ∩ A ̸= ∅, proving [x]ρ ∈ ρ− (A)/ρ. □ Theorem 3.9. Let ρ be a regular congruence relation on a BCK-algebra X. If A is a lower rough ideal of X, then ρ− (A)/ρ is, if it is non-empty, an ideal of the quotient BCK-algebra X/ρ. Proof. Since ρ− (A)/ρ ̸= ∅, ρ− (A)/ρ is a subalgebra of X/ρ and hence [0]ρ ∈ ρ− (A)/ρ. Let [x]ρ ∗ [y]ρ , [y]ρ ∈ ρ− (A)/ρ for some [x]ρ ∈ X/ρ. Hence [x ∗ y]ρ ⊆ A and [y]ρ ⊆ A. Therefore x ∗ y ∈ ρ− (A), y ∈ ρ− (A). Since ρ− (A) is an ideal of X, we have x ∈ ρ− (A). Thus [x]ρ ⊆ A. □ 4. Approximations of fuzzy sets Let µ and λ be two fuzzy subsets of X. The inclusion λ ⊆ µ is denoted by λ(x) ≤ µ(x) for all x ∈ X, and µ ∩ λ is defined by (µ ∩ λ)(x) = µ(x) ∧ λ(x) for all x ∈ X. Definition 4.1. Let ρ be a congruence relation on a BCK/BCI-algebra X and µ a fuzzy subset of X. Then we define the fuzzy sets ρ− (µ) and ρ− (µ) as follows: ρ− (µ)(x) := ∧a∈[x]ρ µ(a) and ρ− (µ)(x) := ∨a∈[x]ρ µ(a). The fuzzy sets ρ− (µ) and ρ− (µ) are called the ρ-lower approximations and ρ-upper approximations of the fuzzy set µ, respectively. A set ρ(µ) = (ρ− (µ), ρ− (µ)) is called a rough fuzzy set with respect to ρ if ρ− (µ) ̸= ρ− (µ). Definition 4.2. ([3]) A fuzzy set µ of a BCK/BCI-algebra X is called a fuzzy ideal of X if (F1 ) µ(0) ≥ µ(x) for all x ∈ X, (F2 ) µ(x) ≥ min{µ(x ∗ y), µ(y)} for all x, y ∈ X. Let µ and ν be fuzzy ideals of a BCK/BCI-algebra X. Then µ ∩ ν is also a fuzzy ideal of X. A fuzzy subset µ of a BCK/BCI-algebra X is called an upper (a lower, respectively) rough fuzzy ideal of X if ρ− (µ) (ρ− (µ), respectively) is a fuzzy ideal of X.

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Sun Shin Ahn et al 75-84

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Sun Shin Ahn and Jung Mi Ko

Theorem 4.3. Let ρ be a congruence relation on a BCK/BCI-algebra X. If µ is a fuzzy ideal of X, then ρ− (µ) is a fuzzy ideal of X. Proof. Since µ is a fuzzy ideal of X, µ(0) ≥ µ(x) for all x ∈ X. Hence we obtain ρ− (µ)(0) = ∨z∈[0]ρ µ(z) ≥ ∨x′ ∈[x]ρ µ(x′ ) = ρ− (µ)(x). For any x, y ∈ X, we have ρ− (µ)(x) = ∨x′ ∈[x]ρ µ(x′ ) ≥ ∨x′ ∗y′ ∈[x]ρ ∗[y]ρ ,y′ ∈[y]ρ {min{µ(x′ ∗ y ′ ), µ(y ′ )}} = ∨x′ ∗y′ ∈[x∗y]ρ ,y′ ∈[y]ρ {min{µ(x′ ∗ y ′ ), µ(y ′ )}} ≥ min{∨x′ ∗y′ ∈[x∗y]ρ µ(x′ ∗ y ′ ), ∨y′ ∈[y]ρ µ(y ′ )} = min{ρ− (µ)(x ∗ y), ρ− (µ)(y)}. Thus ρ− (µ) is a fuzzy ideal of X.

□

Theorem 4.4. Let ρ be a congruence relation on a BCK/BCI-algebra X. If µ is a fuzzy ideal of X, then ρ− (µ) is, if it is non-empty, a fuzzy ideal of X. Proof. Since µ is a fuzzy ideal of X, µ(0) ≥ µ(x) for all x ∈ X. Hence for all x ∈ X, we have ρ− (µ)(0) = ∧z∈[0]ρ µ(z) ≥ ∧z′ ∈[x]ρ µ(z ′ ) = ρ− (µ)(x). For any x, y ∈ X, we obtain ρ− (µ)(x) = ∧x′ ∈[x]ρ µ(x′ ) ≥ ∧x′ ∗y′ ∈[x]ρ ∗[y]ρ ,y′ ∈[y]ρ {min{µ(x′ ∗ y ′ ), µ(y ′ )}} = ∧x′ ∗y′ ∈[x∗y]ρ ,y′ ∈[y]ρ {min{µ(x′ ∗ y ′ ), µ(y ′ )}} = min{∧x′ ∗y′ ∈[x∗y]ρ µ(x′ ∗ y ′ ), ∧y′ ∈[y]ρ µ(y ′ )} = min{ρ− (µ)(x ∗ y), ρ− (µ)(y)}. □

Thus ρ− (µ) is a fuzzy ideal of X.

Let µ be a fuzzy subset of a BCK/BCI-algebra X and let (ρ− (µ), ρ− (µ)) be a rough fuzzy set. If ρ− (µ) and ρ− (µ) are fuzzy ideals of a BCK/BCI-algebra X, then we call (ρ− (µ), ρ− (µ)) a rough fuzzy ideal of X. Therefore we have: Corollary 4.5. If µ is a fuzzy ideal of a BCK/BCI-algebra X, then (ρ− (µ), ρ− (µ)) is a rough fuzzy ideal of X. If µ, λ are fuzzy ideals of a BCK/BCI-algebra X, then (ρ− (µ ∩ λ), ρ− (µ ∩ λ)) is a rough fuzzy ideal of X. Let µ be a fuzzy subset of a BCK/BCI-algebra X. Then the sets µt := {x ∈ X|µ(x) ≥ t}, µX t := {x ∈ X|µ(x) > t}, where t ∈ [0, 1], are called respectively, a t-level subset and a t-strong level subset of µ. Theorem 4.6. ([3]) Let µ be a fuzzy subset of a BCK/BCI-algebra X. Then µ is a fuzzy ideal of X if and only if µt and µX t are, if they are non-empty, ideals of X for every t ∈ [0, 1].

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Rough fuzzy ideals in BCK/BCI-algebras

Lemma 4.7. Let ρ be a congruence relation on a BCK/BCI-algebra X. If µ is a fuzzy subset of X and t ∈ [0, 1], then (1) (ρ− (µ))t = ρ− (µt ), − X (2) (ρ− (µ))X t = ρ (µt ). Proof. (1) We have x ∈ (ρ− (µ))t ⇔ ρ− (µ)(x) ≥ t ⇔ ∧a∈[x]ρ µ(a) ≥ t ⇔ ∀a ∈ [x]ρ , µ(a) ≥ t ⇔ [x]ρ ⊆ µt ⇔ x ∈ ρ− (µt ). (2) Also we have − x ∈ (ρ− (µ))X t ⇔ ρ (µ)(x) > t ⇔ ∨a∈[x]ρ µ(a) > t − X ⇔ ∃a ∈ [x]ρ , µ(a) > t ⇔ [x]ρ ∩ µX t ̸= ∅ ⇔ x ∈ ρ (µt ).

□ Theorem 4.8. Let ρ be a congruence relation on a BCK/BCI-algebra X. Then µ is a lower (an upper) rough fuzzy ideal of X if and only if µt , µX t are, if they are non-empty, lower (upper) rough ideals of X for every t ∈ [0, 1]. □

Proof. By Theorem 4.6 and Lemma 4.7, we can obtain the conclusion easily. 5. Problems of Homomorphism

Lemma 5.1. Let f be a surjective homomorphism of a BCK/BCI-algebra X to a BCK/BCIalgebra Y and let A be any subset of X. Let ρ2 be a congruence relation on Y , and ρ1 := {(x1 , x2 ) ∈ X × X|(f (x1 ), f (x2 )) ∈ ρ2 }. Then (1) ρ1 is a congruence relation on X, (2) If ρ2 is complete and f is single-valued, then ρ1 is complete, − (3) f (ρ− 1 (A)) = ρ2 (f (A)), (4) f (ρ1 − (A)) ⊆ ρ2 − (f (A)). If f is single-valued, then f (ρ1 − (A)) = ρ2 − (f (A)). Proof. (1) It is clear that ρ1 is a congruence relation on X. (2) Let x′ be any element of [x1 ∗ x2 ]ρ1 . Since ρ2 is complete, by the definition of ρ1 , we know that f (x′ ) ∈ [f (x1 ∗ x2 )]ρ2 = [f (x1 )]ρ2 ∗ [f (x2 )]ρ2 . Since f is surjective, there exist x′1 , x′2 ∈ X such that f (x′1 ) ∈ [f (x1 )]ρ2 , f (x′2 ) ∈ [f (x2 )]ρ2 , and f (x′ ) = f (x′1 ) ∗ f (x′2 ) = f (x′1 ∗ x′2 ). Since f is single-valued, by the definition of ρ1 , we have x′1 ∈ [x1 ]ρ1 , x′2 ∈ [x2 ]ρ1 , such that x′ = x′1 ∗ x′2 . Thus x′ ∈ [x1 ]ρ1 ∗ [x2 ]ρ1 . This means that [x1 ∗ x2 ]ρ1 ⊆ [x1 ]ρ1 ∗ [x2 ]ρ1 . On the other hand, we have [x1 ]ρ1 ∗ [x2 ]ρ1 ⊆ [x1 ∗ x2 ]ρ1 . Therefore ρ1 is complete. (3) Let y be any element of f (ρ1 − (A)). Then there exists x ∈ ρ1 − (A) such that f (x) = y. Hence [x]ρ1 ∩ A ̸= ∅. Then there exists x′ ∈ [x]ρ1 ∩ A. Then f (x′ ) ∈ f (A) and by the definition of ρ1 ,

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Sun Shin Ahn and Jung Mi Ko

we have f (x′ ) ∈ [f (x)]ρ2 . So [f (x)]ρ2 ∩ f (A) ̸= ∅, which implies y = f (x) ∈ ρ2 − (f (A)). Thus f (ρ1 − (A)) ⊆ ρ2 − (A)). Conversely, let y ∈ ρ2 − (f (A)). Then there exists x ∈ X such that f (x) = y. Hence [f (x)]ρ2 ∩ f (A) ̸= ∅. So there exists x′ ∈ A such that f (x′ ) ∈ f (A) and f (x′ ) ∈ [f (x)]ρ2 . Then by the definition of ρ1 , we have x′ ∈ [x]ρ1 . Thus [x]ρ1 ∩ A ̸= ∅ which implies x ∈ ρ1 − (A). So y = f (x) ∈ f (ρ1 − (A)). It means that ρ2 − (f (A)) ⊆ f (ρ1 − (A)). From the above, we have f (ρ1 − (A)) = ρ2 − (f (A)). (4) Let y be any element of f (ρ1 − (A)). Then there exists x ∈ ρ1 − (A) such that f (x) = y, so we have [x]ρ1 ⊆ A. Let y ′ ∈ [y]ρ2 . Then there exists x′ ∈ X such that f (x′ ) = y ′ and f (x′ ) ∈ [f (x)]ρ2 . Hence x′ ∈ [x]ρ1 ⊆ A, and so y ′ = f (x′ ) ∈ f (A). Thus [y]ρ2 ⊆ f (A) which yields that y ∈ ρ2 − (f (A)). So we have f (ρ1 − (A)) ⊆ ρ2 − (f (A)). Assume that f is single-valued and suppose y ∈ ρ2 − (f (A)). Then there exist x ∈ X such that f (x) = y and [f (x)]ρ2 ⊆ f (A). Let x′ ∈ [x]ρ1 . Then f (x′ ) ∈ [f (x)]ρ2 ⊆ f (A), and so x′ ∈ A. Thus [x]ρ1 ⊆ A which yields x ∈ ρ1 − (A). Then y = f (x) ∈ f (ρ1 (A)), and so ρ2 − (f (A)) ⊆ f (ρ1 − (A)). From the above, we have f (ρ1 − (A)) = ρ2 − (f (A)). □ Theorem 5.2. Let f be a surjective homomorphism of a BCK/BCI-algebra X to a BCK/BCIalgebra Y . Let ρ2 be a congruence relation on Y and A be a subset of X. If ρ1 := {(x1 , x2 ) ∈ X × X|(f (x1 ), f (x2 )) ∈ ρ2 }, then ρ1 − (A) is an ideal of X if and only if ρ2 − (f (A)) is an ideal of Y. Proof. Assume that ρ1 − (A) is an ideal of X. Since 0 ∈ ρ1 − (A), [0]ρ1 ∩ A ̸= ∅. Hence there exists x′ ∈ [0]ρ1 ∩ A. Then f (x′ ) ∈ f (A), and by the definition of ρ1 , we have f (x′ ) ∈ [f (0)]ρ2 . So [f (0)]ρ2 ∩ f (A) ̸= ∅ which means f (0) ∈ ρ2 − (f (A)). Let x′ , y ′ ∈ Y with x′ , y ′ ∗ x′ ∈ ρ2 − (f (A)). Then there exist x, z ∈ A such that f (x) = x′ and f (z) = y ′ ∗ x′ . Hence [f (x)]ρ2 ∩ f (A) ̸= ∅ and [f (z)]ρ2 ∩ f (A) ̸= ∅. Therefore there exists b ∈ A such that f (b) ∈ [f (x)]ρ2 . By the definition of ρ1 , b ∈ [x]ρ1 and so b ∈ [x]ρ1 ∩ A. Hence [x]ρ1 ∩ A ̸= ∅. Thus x ∈ ρ1 − (A). Since f is surjective, there exists y ∈ X such that f (y) = y ′ . Put u := y ∗ ((y ∗ x) ∗ z). Then u ∈ X. Since f ((y ∗ x) ∗ z) =f (y ∗ x) ∗ f (z) =f (y ∗ x) ∗ y ′ ∗ x′ (∵ f (z) = y ′ ∗ x′ ) =(f (y) ∗ f (x)) ∗ (y ′ ∗ x′ ) =(y ′ ∗ x′ ) ∗ (y ′ ∗ x′ ) = 0′ , we have f (u) = f (y ∗ ((y ∗ x) ∗ z)) = f (y) ∗ f ((y ∗ x) ∗ z) = f (y) ∗ 0′ = f (y) = y ′ . Since [f (z)]ρ2 ∩ f (A) ̸= ∅, we obtain [y ′ ∗ x′ ]ρ2 ∩ f (A) =([y ′ ]ρ2 ∗ [x′ ]ρ2 ) ∩ f (A) =([f (u)]ρ2 ∗ [f (x)]ρ2 ) ∩ f (A) =[f (u ∗ x)]ρ2 ∩ f (A) ̸= ∅.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Rough fuzzy ideals in BCK/BCI-algebras

Then there exists a ∈ A such that f (a) ∈ f (A) and f (a) ∈ [f (u ∗ x)]ρ2 . By the definition of ρ1 , we have a ∈ [u ∗ x]ρ1 . Hence [u ∗ x]ρ1 ∩ A ̸= ∅ and so u ∗ x ∈ ρ1 − (A). Since ρ1 − (A) is an ideal of X and x ∈ ρ1 − (A), we get u ∈ ρ1 − (A). Therefore f (u) = y ′ ∈ f (ρ1 − (A)) = ρ2 − (f (A)). Thus ρ2 − (f (A)) is an ideal of Y . Conversely, suppose that ρ2 − (f (A)) is an ideal of Y . Since f (0) = 0′ ∈ ρ2 − (f (A)), [f (0)]ρ2 ∩ f (A) ̸= ∅. Hence there exists y ′ ∈ [f (0)]ρ2 ∩ f (A). Since f is surjective, there exists x′ ∈ X such that f (x′ ) = y ′ . Hence f (x′ ) ∈ [f (0)]ρ2 ∩ f (A). Therefore f (x′ ) ∈ f (A). By the definition of ρ1 , x′ ∈ [0]ρ1 and x′ ∈ A. Hence [0]ρ1 ∩ A ̸= ∅, which means 0 ∈ ρ1 − (A). Let x1 , x2 ∈ X with x1 ∗ x2 , x2 ∈ ρ1 − (A). By Lemma 5.1, we obtain that f (x1 ∗ x2 ) = f (x1 ) ∗ f (x2 ), f (x2 ) ∈ f (ρ1 − (A)) = ρ2 − (f (A)). Since ρ2 − (f (A)) is an ideal of Y , we have f (x1 ) ∈ ρ2 − (f (A)). Hence [f (x1 )]ρ2 ∩ f (A) ̸= ∅. Therefore y ′ ∈ [f (x1 )]ρ2 ∩ f (A). Since f is surjective, there exists x′ ∈ X such that f (x′ ) = y ′ . Hence f (x′ ) = y ′ ∈ [f (x1 )]ρ2 ∩ f (A). Therefore f (x′ ) ∈ f (A). By the definition of ρ1 , there exists x′ ∈ [x1 ]ρ1 and x′ ∈ A. Therefore [x1 ]ρ1 ∩ A ̸= ∅, which means x1 ∈ ρ1 − (A). Thus ρ1 − (A) is an ideal of X. □ Theorem 5.3. Let f be an isomorphism of a BCK/BCI-algebra X to a BCK/BCI-algebra Y . Let ρ2 be a complete congruence relation on Y and let A be a subset of X. If ρ1 := {(x1 , x2 ) ∈ X × X|(f (x1 ), f (x2 )) ∈ ρ2 }, then ρ1 − (A) is an ideal of X if and only if ρ2 − (f (A)) is an ideal of Y. Proof. By Lemma 5.1, we have f (ρ1 − (A)) = ρ2 − (f (A)). The proof is similar to the proof of Theorem 5.2. □ By Theorem 5.2 and Theorem 5.3, we can obtain the following conclusion easily in quotient BCK/BCI-algebras. Corollary 5.4. Let f be an isomorphism of a BCK/BCI-algebra X to a BCK/BCI-algebra Y . Let ρ2 be a complete congruence relation on Y and let A be a subset of X. If ρ1 := {(x1 , x2 ) ∈ X × X|(f (x1 ), f (x2 )) ∈ ρ2 }, then ρ1 − (A)/ρ1 ( resp. ρ1 − (A)/ρ1 ) is an ideal of X/ρ1 if and only if ρ2 − (f (A))/ρ2 (resp. ρ2 − (f (A))/ρ2 ) is an ideal of Y /ρ2 . Theorem 5.5. Let f be a surjective homomorphism of a BCK/BCI-algebra X to a BCK/BCIalgebra Y . Let ρ2 be a complete congruence relation on Y and let A be a fuzzy subset of X. If ρ1 := {(x1 , x2 ) ∈ X × X|(f (x1 ), f (x2 )) ∈ ρ2 }, then (1) ρ1 − (A) is a fuzzy ideal of X if and only if ρ2 − (f (A)) is a fuzzy ideal of Y. (2) If f is single-valued, then ρ1 − (A) is a fuzzy ideal of X if and only if ρ2 − (f (A)) is a fuzzy ideal of X. Proof. (1) By Theorem 4.6, we obtain that ρ1 − (A) is a fuzzy ideal of X if and only if (ρ1 − (A))X t is, − − X if it is non-empty, an ideal of X for every t ∈ [0, 1]. By Lemma 4.7, we have (ρ1 (A))t = ρ1 (AX t ).

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Sun Shin Ahn and Jung Mi Ko − X By Theorem 5.2, we obtain that ρ1 − (AX t ) is an ideal of X if and only if ρ2 (f (At )) is an ideal X of Y . It is clear that f (AX t ) = (f (A))t . From this and Lemma 4.7, we have − X − X ρ2 − (f (AX t )) = ρ2 (f (A)t ) = (ρ2 (f (A)))t .

By Theorem 4.6, we obtain that (ρ2 − (f (A)))X t is an ideal of Y for every t ∈ [0, 1] if and only if − ρ2 (f (A)) is a fuzzy ideal of Y . Thus the conclusion holds. (2) Since f is single valued, by Lemma 5.1, we have f (ρ1 − (A)) = ρ2 − (f (A)). The proof is similar to that of (1). □ Corollary 5.6. Let f be an isomorphism of a BCK/BCI-algebra X to a BCK/BCI-algebra Y . Let ρ2 be a complete congruence relation on Y and A a fuzzy subset of X. If ρ1 := {(x1 , x2 ) ∈ X × X|(f (x1 ), f (x2 )) ∈ ρ2 }, then ρ1 − (At )/ρ1 (resp. ρ1 − (AX t )/ρ1 ) is an ideal of X/ρ1 if and only − X if ρ2 − (f (At ))/ρ2 (resp. ρ2 (f (At ))/ρ2 ) is an ideal of Y /ρ2 . References [1] R. Biswas and S. Nanda, Rough groups and rough subgroups, Bull. Pol. Ac. Math. 42 (1994), 251-254. [2] K. Is´eki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japo. 23 (1978), 1-26. [3] Y. B. Jun, Chracterization of fuzzy ideals by their level ideals in BCK(BCI)-algebras, Math. Japop. 38 (1993), 67-71. [4] M. Kondo, Congruences and closed ideals in BCI-algebras, Math. Japo. 48 (1997), 491-496. [5] N. Kuroki, Rough ideals in semigroups, Inform. Sci., 100 (1995), 139-163. [6] N. Kuroki and J. N. Mordeson, Structure of rough sets and rough groups, J. Fuzzy Math. 5 (1997), 183-191. [7] N. Kuroki and P. P. Wang, The lower and upper approximations in a fuzzy group, Infrom. Sci. 90 (1996), 203-220. [8] C. R. Lim and H. S. Kim, Rough ideals in BCK/BCI-algebras, Bull. Pol. Ac. Math. 51(2003), 59-67. [9] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa, Seoul, 1994. [10] J. N. Mordeson, Rough set theory applied to (fuzzy) ideals theory, Fuzzy Sets and Systems 121 (2001), 315-324. [11] Z. Pawlak, Rough sets, Int. J. Inform. Comp. Sci. 11 (1982), 341-356. [12] Z. Pawlak, Rough sets and fuzzy sets, Fuzzy Sets and Systems 17 (1985), 99-102. [13] Q. M. Xiao and Z. L. Zhang, Rough prime ideals and rough fuzzy prime ideals in semigroups, Inform. Sci. 176 (2006), 725-733.

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A Lebesgue integrable space of Boehmians for a class of Dκ transformations Shrideh Al-Omari

1

and Dumitru Baleanu

2

October 28, 2016 Abstract Boehmians are objects obtained by an abstract algebraic construction similar to that of field of quotients and it in some cases just gives the field of quotients. As Boehmian spaces are represented by convolution quotients, integral transforms have a natural extension onto appropriately defined spaces of Boehmians. In this paper, we have defined convolution products and a class of delta sequences and have examined the axioms necessary for generating the Dκ spaces of Boehmians. The extended Dκ transformation has therefore been defined as a one-to-one onto mapping continuous with respect to ∆ and δ convergences. Over and above, it has been asserted that the necessary and sufficient conditions for an integrable sequence to be in the range of the Dκ transformation is that the class of quotients belongs to the range of the representative. Further results related to the inverse problem are also discussed. keywords: Integral transform; analogue system; generalized integral; discrete system; Boehmian. ∗

Correspondence : Email: [email protected]

1

Introduction

As some physical situations were determined by initial value problems which are not smoothly enough but are generalized functions, numerous integral transforms were defined in a context of distributions, ultradistributions, tempered distributions, tempered ultradistributions and Boehmian spaces. The Laplace transform method of right-side distributions was treated in [17] and [18] to solve various types of ordinary differential equations. In [19] Loonker and Banerji have given a solution of Volterra-Abel integral equations by aid of a distributional wavelet transform. Indeed, if the differential equation u ´ = w , w being the heaviside step function, is considered then no classical conclusion can be drawn at this point. But, on generalized sense, if S denotes a space of rapid descents 0 (rapidly decreasing functions) and S be its dual of slow growth, then for every

1

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v ∈ S we write u ´ (v)

= −u (´ v) Z ∞ = − u (x) v´ (x) dx −∞ 0

∞

Z

Z

α´ v (x) dx −

= − −∞

(x + α) v´ (x) dx 0

Z = −αv (0) + αv (0) − Z ∞ = v (x) dx Z0 ∞ w (x) v (x) dx =

∞

x´ v (x) dx 0

−∞

= w (v) where α is some suitable constant. Let κ be a sampling period and vα be an analogue function. In some engineering applications, the classical D transform was presented as an equivalence between discrete and analogue systems as [8] Z
r −1 1 Dvα (r, κ) := Dvα (r) := vα (t) e−tκ tκ−1 dt (1) r! R+

where D (vα ∗ yα ) (r, κ) =

r P

Dyα (r − k, κ) Dvα (r, κ) , ∗ being the Fourier con-

0

volution product defined by [7] Z (vα ∗ yα ) (t) :=

vα (τ ) yα (t − τ ) dt.

(2)

R+

Let xα be an analogue function and κ be a sampling period. Then, treating r as a positive real number, say ξ, then the existed integral, denoted by Dκ , is given as Z
ξ −1 1 Dκ vα (ξ, κ) = vα (t) e−tκ tκ−1 dt, (3) ξ! R+

where ξ ∈ R+ ; R+ := (0, ∞) . In this paper, without reading the efficiency of this integral in discrete and analogue systems, we attempt to investigate the extension of this integral to a class of Boehmians, being recent in the space of generalized functions. We derive virtuous products, give definitions and derive some properties of the existence of the given integral in the class of generalized functions. Boehmian spaces were inaugurated by the idea of regular operators which is a 2

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subalgebra of Mikusi´ nski operators. According to literature, we briefly recall the general construction of Boehmian spaces. Let G be a group and S be a subgroup of G. We assume to each pair of elements f ∈ G and ω ∈ S, is assigned the product f ∗ g such that : (1) ω, ψ ∈ S implies ω ∗ ψ ∈ S and ω ∗ ψ = ψ ∗ ω. (2) f ∈ G and ω, ψ ∈ S implies (f ∗ ω) ∗ ψ = f ∗ (ω ? ψ) . (3) f, g ∈ G, ω ∈ S and λ ∈ R, implies (f + g)∗ω = f ∗ω +g ∗ω, λ (f ∗ ω) = (λf ) ∗ ω.Let ∆ be a family of sequences from S such that: (1) f, g ∈ G, (δn ) ∈ ∆ and f ∗ δn = g ∗ δn (n = 1, 2, ...) implies f = g, n ∈ N. (2) (ωn ) , (δn ) ∈ ∆ implies (ωn ∗ ψn ) ∈ ∆. Members of ∆ are called delta sequences.  Let A be a pair of sequences defined by A = ((fn ) , (ωn )) : (fn ) ⊆ G N , (ωn ) ∈ ∆ , where n ∈ N, then members of ((fn ) , (ωn )) ∈ A are called quotient of sequences, denoted by [fn /ωn ] , if fn ∗ ωm = fm ∗ ωn , ∀n, m ∈ N. Two quotients of sequences fn /ωn and gn ψn are equivalent, fn /ωn ∼ gn ψn , if fn ∗ ψm = gm ∗ ωn , ∀n, m ∈ N. The relation ∼ is an equivalent relation on A. The equivalence class containing fn /ωn is denoted by [fn /ωn ] . These equivalence classes are called Boehmians. The space of all Boehmians is denoted by β1 . The sum of two Boehmians and multiplication by a scalar can be defined in a natural way [fn /ωn ] + [gn /ψn ] = [(fn ∗ ψn + gn ∗ ωn ) / (ωn ∗ ψn )] , α [fn /ωn ] = [αfn /ωn ] ,α ∈ C, space of complex numbers. The operations ∗ and Dα are given by [fn /ωn ]∗[gn /ψn ] = [(fn ∗ gn ) / (ωn ∗ ψn )] and Dα [fn /ωn ] = [Dα fn /ωn ] whereas, ∗ can be extended to β × S in the form that If [fn /ωn ] ∈ β1 and ω ∈ S, then [fn /ωn ] ∗ ω = [fn ∗ ω/ωn ] . However, soon after the topic has been initiated, numerous integral transforms were extended to Boehmian spaces by many authors in [1] , [2] , [6], [9 − 16] , [20 − 23] and many others. Definition 1 The Mellin type convolution product between two signals xα and yα is defined by the integral equation ( see [4]) Z
(vα yα ) (x) = vα y −1 x yα (x) y −1 dy (4) R+

when the integral exists.
2 2 The Lebesgue space of integrable functions defined on R+ is denoted by l1 R+ and the set of smooth functions of bounded supports over R+ is denoted by ϑ (R+ ) (see [3] for definition, properties and convergence in ϑ (R+ )).

2

Convolution products and Boehmians

In this section, we
establish the prerequisite axioms of the Boehmian space
2 B l1 R+ , ϑ, , • with the operations and • where • is a convolution product defined as follows.

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2 Definition 2 Let the casual analogue signals vα , yα ∈ l1 R+ be given. Then, between vα and yα , we define a product • given as Z
(vα • yα ) (ξ, κ) = vα ξ, y −1 κ yα (y) dy (5) R+

provided the above integral exists.

2 Proving axioms of the space B l1 R+ , ϑ, , • begins with the following theorem.
2 Theorem 3 Given vα ∈ l1 R+ and yα ∈ ϑ (R+ ) . Then we get vα • yα ∈
2 l1 R+ .

R 2 vα ξ, y −1 κ dξdκ < Proof The hypothesis that vα ∈ l1 R+ implies 2 R+

M1 (y > 0) . Hence, with the aid of the Fubini’s theorem together with the hypothesis that yα ∈ ϑ (R+ ) we confirm Z Z Z
−1 |(vα • yα ) (ξ, κ)| dξdκ = vα ξ, y κ yα (y) dy dξdκ 2 2 R R+ R+ + Z Z
vα ξ, y −1 κ |yα (y)| dydξdκ ≤ 2 P R+

Z ≤ M1

|yα (y)| dy < ∞ P

where P is an interval in R+ including the support of yα . Hence the theorem is finished.
2 and that yα , zα ∈ ϑ (R+ ) be analogue signals. Theorem 4 Let vα ∈ l1 R+ Then vα • (yα zα ) = (vα • yα ) • zα . Proof On account of (4) and (5) we are permitted to write Z

(vα • (yα zα )) (ξ, κ) = vα ξ, y −1 κ yα t−1 y dyzα (t) t−1 dt.

(6)

2 R+

The substitution u = yt−1 implies dy = tdu. Therefore, (6) can be expressed as Z

(vα • (yα zα )) (ξ, κ) = vα ξ, u−1 t−1 κ yα (u) zα (t) dudt 2 R+

Z =


(vα • yα ) ξ, t−1 κ zα (t) dt.

R+

4

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The proof is therefore finished.
2 Theorem 5 Given vα ∈ l1 R+ . For every yα ∈ ϑ (R+ ) , we get Dκ (vα yα ) (ξ, κ) = ((Dκ vα ) • yα ) (ξ, κ) Proof Applying (3) to (4) gives Z  

ξ −1 1 vα y −1 x yα (y) y −1 dy e−xκ Dκ (vα • yα ) (ξ, κ) = xκ−1 dx ξ! 2 R+

=

1 ξ!

Z

vα y −1 x




−1

e−xκ



xκ−1


ξ

yα (y) y −1 dxdy.(7)

2 R+

Let zy = x, then dx = ydz. Therefore, on account of (7) we obtain that Z  
ξ −1 1 Dκ (vα yα ) (ξ, κ) = vα (x) e−yzκ xκ−1 yα (y) dzdy. ξ!

(8)

2 R+

Hence, by the Fubini’s theorem, we finish the proof of the theorem.
2 and yα ∈ ϑ (R+ ) . We get Theorem 6 Given r˘ ∈ C, vα ∈ l1 R+ (˘ rvα ) • yα = r˘ (vα • yα ) . Proof of this theorem is straightforward follows from definitions. Hence it is omitted.
2 Theorem 7 Given vα , zα ∈ l1 R+ . For every yα ∈ ϑ (R+ ) , we get (vα + zα ) • yα = vα • yα + zα • yα . Proof of above theorem follows from simple integration. Details are therefore omitted.
2 Theorem 8 Given vα,n → vα as n → ∞ in l1 R+ . For every yα ∈ ϑ (R+ ) , we get vα,n • yα → vα • yα as n → ∞. Proof of above theorem is a direct conclusion of Theorem 4. Hence it is avoided. ∞ By ∆ we mean the subset of ϑ (R+ ) such that for every sequence (µα,n )0 ∈ ϑ (R+ ), n ∈ N , we have. R R 0 00 000 i : µα,n dy = 1; i : |µα,n | dy < ∞; i : suppµα,n ⊆ (0, an ) , limn→∞ an = R+

R+

0. Elements of ∆ are said to be delta sequences or approximating identities.
2 Theorem 9 Given vα ∈ l1 R+ . For every (µα,n ) ∈ ϑ (R+ ) , we get limn→∞ vα • µα,n = vα .

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2 2 Proof Let vα ∈ l1 R+ and ϑ
R+ be the set of
smooth functions of bounded 2 2 . Hence, for a given  > 0, we is dense in l1 R+ supports over R2 , then ϑ R +
2 can find ψα ∈ ϑ R+ such that kvα − ψα k < . (9)
Define gα (y) = ψα ξ, y −1 κ , then gα (y) is uniformly continuous mapping in ϑ (R+ ) for every ξ, κ > 0. Therefore, for each  > 0 we find δ > 0 so that − < gα (y1 ) − gα (y2 ) <  wherever −δ < y − y −1 < δ. Since y and y −1 belong to R+ and that gα ∈ ϑ (R+ ) , we get
− < gα (y) − gα y −1 <  (10) when −δ < y − y −1 < δ.

2 Also, since ψα is of bounded support in ϑ R+ it follows suppψα ξ, y −1 κ ⊆ [a1 , a2 ] × P for some compact subset P of R+ . Hence

ψα ξ, y −1 κ = 0, ξ, y −1 κ ∈ / [a1 − δ, a2 + δ] × P. (11) The hypothesis that suppµα,n → 0, n → ∞ asserts that we can find N ∈ N such that suppµα,n (y) ⊂ [0, δ] (12) R for every n ≥ N. By the property µα,n dy = 1 we write R+

  Z Z

  −1 ψα ξ, y κ − ψα (ξ, κ) µα,n (y) dy  dξdκ k(ψα • µα,n ) − ψα k =  2 R+ R+ Z

gα (y) − gα y −1 |µα,n (y)| dydξdκ ≤ 3 R+

Z

aZ2 +δZδ

P

a1

=



gα (y) − gα y −1 |µα,n (y)| dydξdκ.

0

By virtue of (10) and (12) and the assumption that

R

|µα,n | dy < M we have

R+ aZ2 +δ

kψα • µα,n − ψα k
0 such that d(f (x), h(x)) ≤ δ for all x ∈ G1 ? This problem was solved affirmatively by D. H. Hyers under the assumption that G2 is a Banach space (see Hyers [19], Hyers-Isac-Rassias [20]). Since then Ulam problems of many other functional 2010 Mathematics Subject Classification. 46F99, 39B82. Key words and phrases. convolution, distribution, hyperfunction, heat kernel, sine addition formula, Ulam problem. * Corresponding author. 1

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2

CHANG-KWON CHOI and JEONGWOOK CHANG∗

equations have been investigated [13, 14, 15, 21, 23, 24, 25, 26, 27, 28]. Among the results, Sz´ekelyhidi has developed his idea of using invariant subspaces of functions defined on a group or semigroup he prove the Ulam-Hyers stability problem for functional equation (1.2)

f (x + y) = f (x)g(y) + g(x)f (y),

x, y ∈ Rn ,

which arises from the sine addition formula [30, 31]. Using his elegant idea, Chung and Chang [7] prove the parallel Ulam-Hyers stability problem for functional equation (1.3)

f (x − y) = f (x)g(y) − g(x)f (y),

x, y ∈ Rn ,

which arises from the sine subtraction formula. As a result it was proved that if f, g : Rn → C satisfy (1.4)

|f (x − y) − f (x)g(y) + g(x)f (y)| ≤ M,

x, y ∈ Rn

for some M > 0, then either there exist λ1 , λ2 ∈ C, not both zero, and L > 0 such that (1.5)

|λ1 f (x) − λ2 g(x)| ≤ L

for all x ∈ Rn , or else (1.6)

f (x − y) = f (x)g(y) − g(x)f (y)

for all x, y ∈ Rn . Also in the sequel, the functions f and g satisfying both (1.4) and (1.5) were investigated. Schwartz introduced the theory of distributions in his monograph Th´eorie des distributions [29] in which Schwartz systematizes the theory of generalized functions, basing it on the theory of linear topological spaces, relates all the earlier approaches, and obtains many important results. After his elegant theory appeared, many important concepts and results on the classical spaces of functions have been generalized to the space of distributions. For example, the space L∞ (Rn ) of bounded n 0 measurable functions on Rn has been generalized to the space DL ∞ (R ) of bounded distributions as a n 0 subspace of distributions and later the space DL∞ (R ) is further generalized to the space A0L∞ (Rn ) of bounded hyperfunctions. It is very natural to consider the following stability problem for the functional equation in distributions and hyperfunctions u, v with respect to bounded distributions and bounded hyperfunctions (1.7)

0 2n u ◦ T − u ⊗ v + v ⊗ u ∈ DL ) [resp. A0L∞ (R2n )], ∞ (R

0 2n where DL ) and A0L∞ (R2n ) are the spaces of bounded distributions and bounded hyperfunc∞ (R tions, T : R2n → Rn such that T (x, y) = x − y for all x, y ∈ Rn , and ◦, ⊗ denote pullback and tensor product of generalized functions respectively. In [10] the distributional version of the stability of (1.2) was proved. In this paper, as a parallel result we prove the stability of (1.7). As in [10] the main tool is the heat kernel method initiated by T. Matsuzawa [22] which represents the generalized functions in some class as the initial values of solutions of the heat equation with appropriate growth

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STABILITY OF THE SINE-COSINE FUNCTIONAL EQUATION IN HYPERFUNCTIONS

3

conditions [12, 22]. Making use of the heat kernel method we can convert (1.7) to the classical UlamHyers stability problem of the functional inequality; there exist C > 0 and N ≥ 0[ resp. for every  > 0 there exists C > 0 ] such that  N 1 1 (1.8) |˜ u(x − y, t + s) − u ˜(x, t)˜ v (y, s) + v˜(x, t)˜ u(y, s)| ≤ C + [ resp. C e(1/t+1/s) ] t s for all x, y ∈ Rn , t, s > 0, where u ˜, v˜, w, ˜ k˜ : Rn × (0, ∞) → C are solutions of the heat equation whose initial values are u, v, w, k respectively. In Section 3, we consider the stability problem (1.8) with a more general setting, which will be used, combined with the heat kernel method [12, 22], to prove the stability problem of (1.7). 2. Distributions and hyperfunctions We first introduce the spaces S 0 of Schwartz tempered distributions and G 0 of Gelfand hyperfunctions(see [16, 17, 18, 22, 29] for more details of these spaces). We use the notations: p α1 αn α αn 1 |α| = α1 + · · · + αn , α! = α1 ! · · · αn !, |x| = x21 + · · · x2n , xα = xα 1 · · · xn and ∂ = ∂1 · · · ∂n , for x = (x1 , · · · , xn ) ∈ Rn , α = (α1 , · · · , αn ) ∈ Nn0 , where N0 is the set of non-negative integers and ∂ ∂j = ∂x . j Definition 2.1. [29] We denote by S or S(Rn ) the Schwartz space of all infinitely differentiable functions ϕ in Rn such that kϕkα,β = sup |xα ∂ β ϕ(x)| < ∞

(2.1)

x

Nn0 ,

for all α, β ∈ equipped with the topology defined by the seminorms k · kα,β . The elements of S are called rapidly decreasing functions and the elements of the dual space S 0 are called tempered distributions. Definition 2.2. [16, 17] We denote by G or G(Rn ) the Gelfand space of all infinitely differentiable functions ϕ in Rn such that kϕkh,k =

sup x∈Rn , α, β∈Nn 0

|xα ∂ β ϕ(x)| 0. We say that ϕj −→ 0 as j → ∞ if ||ϕj ||h,k −→ 0 as j → ∞ for some h, k, and denote by G 0 the strong dual space of G and call its elements Gelfand hyperfunctions. As a generalization of the space L∞ of bounded measurable functions, L. Schwartz introduced 0 the space DL ∞ of bounded distributions as a subspace of tempered distributions. Definition 2.3. [29] We denote by DL1 (Rn ) the space of smooth functions on Rn such that ∂ α ϕ ∈ L1 (Rn ) for all α ∈ Nn0 equipped with the topology defined by the countable family of seminorms X kϕkm = k∂ α ϕkL1 , m ∈ N0 . |α|≤m 0 We denote by DL ∞ the strong dual space of DL1 and call its elements bounded distributions.

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As a generalization of bounded distributions, the space A0L∞ of bounded hyperfunctions has been introduced as a subspace of G 0 . Definition 2.4. [12] We denote by AL1 the space of smooth functions on Rn satisfying kϕkh = sup α

k∂ α ϕkL1 0. We say that ϕj → 0 in AL1 as j → ∞ if there is a positive constant h such that k∂ α ϕj kL1 sup → 0 as j → ∞. h|α| α! α We denote by A0L∞ the strong dual space of AL1 . It is well known that the following topological inclusions hold: G ,→ S ,→ DL1 ,

0 0 0 DL ∞ ,→ S ,→ G ,

G ,→ AL1 ,→ DL1 ,

0 0 0 DL ∞ ,→ AL∞ ,→ G .

It is known that the space G(Rn ) consists of all infinitely differentiable functions ϕ(x) on Rn which can be extended to an entire function on Cn satisfying (2.2)

|ϕ(x + iy)| ≤ C exp(−a|x|2 + b|y|2 ),

x, y ∈ Rn

for some a, b, C > 0(see [16]). Definition 2.5. Let uj ∈ G 0 (Rnj ) for j = 1, 2. Then the tensor product u1 ⊗ u2 of u1 and u2 , defined by hu1 ⊗ u2 , ϕ(x1 , x2 )i = hu1 , hu2 , ϕ(x1 , x2 )i i for ϕ(x1 , x2 ) ∈ G(Rn1 × Rn2 ), belongs to G 0 (Rn1 × Rn2 ).

3. Stability of (1.8) Throughout this paper hG, +i is a 2-divisible commutative group, f, g : G × (0, ∞) → C and N denotes a fixed nonnegative real number. We consider the stability problems of each of the following functional inequalities;

(3.1)

there exist C > 0 and d > 0 such that  N 1 1 |f (x − y, t + s) − f (x, t)g(y, s) + g(x, t)f (y, s)| ≤ C + + d, t s

∀ x, y ∈ G, t, s > 0;

f or every  > 0, there exists C > 0 which depends on  such that (3.2)

|f (x − y, t + s) − f (x, t)g(y, s) + g(x, t)f (y, s)| ≤ C e(1/t+1/s) ,

∀ x, y ∈ G, t, s > 0.

From now on, a function a from a semigroup hS, +i to the field C of complex numbers is said to be an additive function provided a(x + y) = a(x) + a(y) for all x, y ∈ S and m : S → C is said to be an exponential function provided m(x + y) = m(x)m(y) for all x, y ∈ S.

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We introduce the following conditions (3.3) and (3.4) on f : G × (0, ∞) → C and N ; there exist C > 0 and d > 0 such that (3.3)

|f (x, t)| ≤ Ct−N + d,

∀ x ∈ G, t > 0;

f or every  > 0, there exists C > 0 which depends on  such that (3.4)

|f (x, t)| ≤ C e/t ,

∀ x ∈ G, t > 0.

Using the idea in [20, p. 104] we obtain the following (See [10] for the proofs). Lemma 3.1. Let f, g : G × (0, ∞) → C satisfy the inequality; for each y ∈ G and s > 0 there exist positive constants C = C(y, s) and d = d(y, s) [resp. for each y ∈ G, s > 0 and  > 0 there exists a positive constant C = C (y, s) ] such that |f (x − y, t + s) − f (x, t)g(y, s)| ≤ Ct−N + d [resp. C e/t ]

(3.5)

for all x ∈ G, t > 0. Then either f satisfies (3.3) [resp.(3.4)] or g is an exponential function. Lemma 3.2. Let m : G × (0, ∞) → C be a nonzero exponential function satisfying (3.3) [resp.(3.4)]. Then m can be written in the form m(x, t) = m1 (x)m2 (t), where m1 : G → C, m2 : (0, ∞) → C is exponential functions satisfying |m1 (x)| = 1 for all x ∈ G. Lemma 3.3. Let m be a nonzero exponential function satisfying (3.3) [resp.(3.4)]. Suppose that f : G × (0, ∞) → C satisfies the inequality; there exist positive constants C and d [resp. for each  > 0, there exists a positive constant C ] such that  N 1 1 + + d [resp. C e/t ] (3.6) |f (x + y, t + s) − f (x, t)m(y, s) − f (y, s)m(x, t)| ≤ C t s for all x, y ∈ G, t, s > 0. Then we have  f (x, t) = a(x)m1 (x)m2 (t) + 2f

t 0, 2



  t m1 (x)m2 + R(x, t), 2

where a : G → C is an additive function, m : (0, ∞) → C is an exponential function, λ ∈ C and R : G × (0, ∞) → C satisfies |R(x, t)| ≤ Ct−N + d [ resp.(3.4)] for all x ∈ G, t > 0. Theorem 3.4. Suppose that f, g : G × (0, ∞) → C satisfy the inequality (3.1) [resp.(3.2)]. Then either (3.7)

f (x − y, t + s) − f (x, t)g(y, s) + g(x, t)f (y, s) = 0

for all x, y ∈ G, t, s > 0, or else there exist λ1 , λ2 ∈ C, not both zero, such that λ1 f (x, t) − λ2 g(x, t) satisfies (3.1) [resp.(3.2)].

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Proof. It suffices to prove that f, g satisfies (3.7) when λ1 f (x, t) − λ2 g(x, t) satisfies (3.3) [resp.(3.4)] only for λ1 = λ2 = 0. Let F (x, y, t, s) = f (x − y, t + s) − f (x, t)g(y, s) + g(x, t)f (y, s).

(3.8)

Choosing y1 and s1 with f (y1 , s1 ) 6= 0 we have g(x, t) = k1 f (x, t) + k2 f (x − y1 , t + s1 ) − k2 F (x, y1 , t, s1 ),

(3.9) where k1 = (3.10)

g(y1 ,s1 ) f (y1 ,s1 )

and k2 =

−1 f (y1 ,s1 ) .

f (x − y) − z, (t + s) + r)

From (3.8) and (3.9) we have




=f (x − y, t + s)g(z, r) − g(x − y, t + s)f (z, r) + F (x − y, z, t + s, r) =f (x − y, t + s)g(z, r)   − k1 f (x − y, t + s) + k2 f (x − y − y1 , t + s + s1 ) − k2 F (x − y, y1 , t + s, s1 ) f (z, r) + F (x − y, z, t + s, r)   = f (x, t)g(y, s) − g(x, t)f (y, s) + F (x, y, t, s) g(z, r)   − k1 f (x, t)g(y, s) − g(x, t)f (y, s) + F (x, y, t, s) f (z, r)  + k2 f (x, t)g(y + y1 , s + s1 ) − g(x, t)f (y + y1 , s + s1 )  + F (x, y + y1 , t, s + s1 ) − F (x − y, y1 , t + s, s1 ) f (z, r) + F (x − y, z, t + s, r), and also we have
(3.11) f x − (y + z), t + (s + r) = f (x, t)g(y + z, s + r) − g(x, t)f (y + z, s + r) + F (x, y + z, t, s + r). From (3.10) and (3.11) we have   (3.12) f (x, t) g(y, s)g(z, r) − k1 g(y, s)f (z, r) + k2 g(y + y1 , s + s1 )f (z, r) − g(y + z, s + r)   + g(x, t) − f (y, s)g(z, r) + k1 f (y, s)f (z, r) − k2 f (y + y1 , s + s1 )f (z, r) + f (y + z, s + r) =F (x, y + z, t, s + r) − F (x − y, z, t + s, r) − F (x, y, t, s)g(z, r) + k1 F (x, y, t, s)f (z, r)   − k2 F (x, y + y1 , t, s + s1 ) − F (x − y, y1 , t + s, s1 ) f (z, r). Fixing y, z, s, r in (3.12), using (3.1) and (3.8) we have F (x, y + z, t, s + r) − F (x − y, z, t + s, r) − F (x, y, t, s)g(z, r) + k1 F (x, y, t, s)f (z, r)   − k2 F (x, y + y1 , t, s + s1 ) − F (x − y, y1 , t + s, s1 ) f (z, r)  ≤ 2C

1 1 + t r

N

 + 2d + C1

1 1 + t s

N

 + d1 + C2

1 1 + t s1

N + d2

≤ C 0 t−N + d0 , where C 0 = 2N (2C + C1 + C2 ), d0 = 2N (2Cr−N + C1 s−N + C2 s−N 1 ) + 2d + d1 + d2 .

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Similarly, using (3.2) we obtain that for every  > 0 there exists C > 0 such that F (x, y + z, t, s + r) − F (x − y, z, t + s, r) − F (x, y, t, s)g(z, r) + k1 F (x, y, t, s)f (z, r)   − k2 F (x, y + y1 , t, s + s1 ) − F (x − y, y1 , t + s, s1 ) f (z, r) ≤ 2C e(1/t+1/r) + C1 C e(1/t+1/s) + C2 C e(1/t+1/s1 ) ≤ C0 e/t , where C0 = C (2e/r + C1 e/s + C2 e/s1 ). Thus, by the assumption that λ1 f (x, t) − λ2 g(x, t) satisfies (3.3) [resp.(3.4)] only for λ1 = λ2 = 0 we have g(y, s)g(z, r) − k1 g(y, s)f (z, r) + k2 g(y + y1 , s + s1 )f (z, r) − g(y + z, s + r) = f (y, s)g(z, r) − k1 f (y, s)f (z, r) + k2 f (y + y1 , s + s1 )f (z, r) − f (y + z, s + r) = 0. Thus, it follows that (3.13)

F (x, y + z, t, s + r) − F (x − y, z, t + s, r)   = − k1 F (x, y, t, s) + k2 F (x, y + y1 , t, s + s1 ) − k2 F (x − y, y1 , t + s, s1 ) f (z, r) + F (x, y, t, s)g(z, r).

Now, if we fix x, y, t, s, the left hand side of (3.13) satisfies (3.3) [resp. (3.4)] as a function of z and r. From the right hand side of (3.13), using the assumption that λ1 f (x, t) − λ2 g(x, t) satisfies (3.3) [resp.(3.4)] only for λ1 = λ2 = 0 it follows that F ≡ 0. This completes the proof.  Theorem 3.5. Let f, g : G × (0, ∞) → C satisfy (3.1) [resp. (3.2)]. Then (f, g) satisfies one of the following : (i) both f and g satisfy (3.3) [resp.(3.4)], (ii) f (x, t) = a(x)m(t) + R(x, t), g(x, t) = λf (x, t) + m(t) for all x ∈ G, t > 0, where a : G → C is an additive function, m : (0, ∞) → C is an exponential function, λ ∈ C and R : G × (0, ∞) → C satisfies |R(x, t)| ≤ Ct−2N + d [ resp.(3.4)] for all x ∈ G, t > 0 and for some C, d > 0, (iii) f (x − y, t + s) − f (x, t)g(y, s) + g(x, t)f (y, s) = 0 for all x, y ∈ G, t, s > 0. Proof. Assume that (f, g) does not satisfy (iii). Then by Lemma 3.4 there exist λ1 , λ2 ∈ C, not both zero, such that λ1 f (x, t) − λ2 g(x, t) satisfies (3.3) [resp.(3.4)]. (Case 1) f (6= 0) satisfies (3.3) [resp.(3.4)]. Assume that f (6= 0) satisfies (3.3). Choosing y0 ∈ G, s0 > 0 such that f (y0 , s0 ) 6= 0, dividing |f (y0 , s0 )| in both sides of (3.1) and using the triangle inequality we have !  N 1 1 1 |g(x, t)| ≤ |f (x − y0 , t + s0 )| + |f (x, t)g(y0 , s0 )| + C + +d |f (y0 , s0 )| t s0  N 1 1 ≤ C1 (t + s0 )−N + d1 + C2 t−N + d2 + C3 + + d3 t s0 ≤ C 0 t−N + d0

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for all x ∈ G, t > 0 and for some positive constants C1 , C2 , C3 , d1 , d2 , d3 , C 0 and d0 . Similarly, if f satisfies (3.4) we can show that for every  > 0 there exists C0 > 0 such that |g(x, t)| ≤ C0 e/t for all x ∈ G, t > 0. Thus, we obtain the case (ii). (Case 2) f does not satisfy (3.3) [resp. (3.4)]. Assume that f does not satisfy (3.3). In this case we must have λ2 6= 0 and we can write (3.14)

g(x, t) = −

λ1 f (x, t) + B(x, t) := λf (x, t) + B(x, t) λ2

for all x ∈ G, t > 0, where R satisfies (3.3) [resp. (3.4)]. Putting (3.14) in (3.1) we have  N 1 1 +d (3.15) |f (x − y, t + s) − f (x, t)B(y, s) + B(x, t)f (y, s)| ≤ C + t s for all x, y ∈ G, t, s > 0. Using the triangle inequality and fixing y and s in (3.15) we have  N 1 1 |f (x − y, t + s) − f (x, t)B(y, s)| ≤ |B(x, t)f (y, s)| + C + + d ≤ C 0 t−N + d0 t s for all x, y ∈ G, t, s > 0 and for some positive constants C 0 and d0 . Applying Lemma 3.1 we have (3.16)

B(x, t) = m(x, t)

for all x ∈ G, t > 0, where m is an exponential function on G × (0, ∞). Now, applying Lemma 3.2 we have (3.17)

R(x, t) = m(x, t) = m1 (x)m2 (t)

for all x ∈ G, t > 0, where m1 : G → C, m2 : (0, ∞) → C are exponential functions. Replacing (x, t) by (y, s) in (3.15) we have N  1 1 + +d (3.18) |f (−x + y, t + s) − f (y, s)B(x, t) + B(y, s)f (x, t)| ≤ C t s for all x, y ∈ G, t, s > 0. From (3.15) and (3.18), using the triangle inequality, putting y = 0 and replacing t, s by 2t we have (3.19)

|f (x, t) + f (−x, t)| ≤ C22N +1 t−N + 2d

for all x ∈ G, t > 0. Replacing x by −x, y by −y in (3.15), we have  (3.20)

|f (−x + y, t + s) − f (−x, t)B(−y, s) + B(−x, t)f (−y, s)| ≤ C

1 1 + t s

N +d

for all x, y ∈ G, t, s > 0. From (3.20) and using (3.19) with fixing y and s we have (3.21)

|f (−x + y, t + s) − f (x, t)B(−y, s) + B(−x, t)f (y, s)| ≤ C1 t−N + d1

for all x ∈ G, t > 0. From (3.20) and (3.21) with fixing y and s we have (3.22)

|f (x, t) (B(y, s) − B(−y, s)) − f (y, s) (B(x, t) − B(−x, t)) | ≤ C2 t−N + d2

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for all x, y ∈ G, t, s > 0. Since f does not satisfy (3.3), it follows from (3.22) that B(y, s) = B(−y, s) for all y ∈ G, s > 0 and hence m1 (y) = 1 for all y ∈ G. Thus, we have (3.23)

g(x, t) = λf (x, t) + m2 (t)

for all x ∈ G, t > 0. From (3.15), (3.17) and (3.19) we have (3.24) N 1 1 +d + |f (x + y, t + s) − f (x, t)m2 (s) − f (y, s)m2 (t)| ≤ |f (y, s) + f (−y, s)||m2 (t)| + C t s  N 1 1 2N +1 −N −N ≤ (C2 t + 2d)Ct +C + +d t s 2N  1 1 + d0 + ≤ C0 t s 

for all x, y ∈ G, t, s > 0 and for some C 0 > 0, d0 > 0. Similarly, if f satisfies (3.4) we can show that for every  > 0 there exists C0 > 0 such that (3.25)

|f (x + y, t + s) − f (x, t)m2 (s) − f (y, s)m2 (t)| ≤ C0 e/t

for all x ∈ G, t > 0. Applying Lemma 3.3 with (3.23) and (3.24) we have     t t m2 + R(x, t) (3.26) f (x, t) = a(x)m2 (t) + 2f 0, 2 2 for all x ∈ G, t > 0, where a is an additive mapping and R satisfies (3.3)[resp. (3.4)]. Replacing

(y, s) by (x, t) in (3.1) we see that f (0, t) satisfies (3.3). Thus, 2f 0, 2t m2 2t + R(x, t) satisfies

(3.3)[resp. (3.4)]. Replacing 2f 0, 2t m2 2t + R(x, t) by R(x, t) and m2 by m we get the case (iii). This completes the proof. 

4. Main results In this section as a main result of the paper we consider the stability of (1.6). The main tools of our proof are based on structure theorems for generalized functions and the heat kernel method initiated by T. Matsuzawa [22] which represents the generalized functions as initial values of solutions of the heat equation with appropriate growth conditions [8, 9, 11, 12, 22]. For the proof of our theorem we employ the n-dimensional heat kernel Et (x) given by Et (x) = (4πt)−n/2 exp(−|x|2 /4t), t > 0. In view of (2.2), we can see that the heat kernel Et belongs to the Gelfand space G(Rn ) for each t > 0. Thus, for each u ∈ G 0 (Rn ), the convolution (u ∗ Et )(x) := huy , Et (x − y)i is well defined. We call (u ∗ Et )(x) the Gauss transform of u. From now on we denote by u ˜(x, t) the Gauss transform of

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u. It is well known that the Gauss transform u ˜(x, t) is a smooth solution of the heat equation such that u ˜(x, t) → u in weak star topology as t → 0+ , i.e., Z hu, ϕi = lim+ u ˜(x, t)ϕ(x)dx t→0

for all ϕ ∈ G. We first discuss the solutions of the corresponding trigonometric functional equations in the space G of Gelfand generalized functions. Lemma 4.1. The solutions u, v ∈ G 0 of the equation (4.1)

u◦T −u⊗v+v⊗u=0

are either (4.2)

u = λ(ec·x − e−c·x ),

v = γec·x + (1 − γ)e−c·x

or else (4.3)

u = c · x,

v = 1 + λc · x.

Proof. As a consequence of the results in [4, 15] the solutions (u, v) of (4.1) are equal to the smooth solutions (f, g) of the equation (4.4)

f (x − y) − f (x)g(y) + f (y)g(x) = 0

for all x, y ∈ Rn . By [2, Theorem 11] all solutions of (4.4) are given by (4.5)

f (x) = λ(m(x) − m(−x)),

g(x) = γm(x) + (1 − γ)m(−x)

or else (4.6)

f (x) = a(x),

g(x) = 1 + λa(x),

where m is an exponential function and a is an additive function. From (4.5) and (4.6) m and a are smooth functions and hence m(x) = ec·x and a(x) = c · x for some c ∈ Cn . Thus, we get (4.2) and (4.3). This completes the proof.  The proof of Theorem 2.3 of [11] works even when p = ∞, i.e., we obtain the following. 0 n Lemma 4.2. [11] The Gauss transform u ˜(x, t) := (u ∗ E)(x, t) of u ∈ DL ∞ (R ) is a smooth solution of the heat equation (∆ − ∂/∂t )˜ u = 0 satisfying: (i)There exist constants C > 0, N ≥ 0 such that

(4.7)

|˜ u(x, t)| ≤ Ct−N

for all x ∈ Rn , t > 0.

(ii) u ˜(x, t) → u as t → 0+ in the sense that for every ϕ ∈ DL1 , Z hu, ϕi = lim+ u ˜(x, t)ϕ(x) dx. t→0

Conversely, every smooth solution u ˜(x, t) of the heat equation satisfying the estimate (4.7) can be 0 n uniquely expressed as u ˜(x, t) = (u ∗ E)(x, t) for some u ∈ DL ∞ (R ).

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The following lemma is a special case of Theorem 3.5 of [12] when p = ∞ where the space is denoted by BL∞ (Rn ).

A0L∞ (Rn )

Lemma 4.3. [12] The Gauss transform u ˜(x, t) := (u ∗ E)(x, t) of u ∈ A0L∞ (Rn ) is a smooth solution of the heat equation (∆ − ∂/∂t )˜ u = 0 satisfying : (i) For every  > 0 there exists a constant C > 0 such that (4.8)

|˜ u(x, t)| ≤ C e/t

for all x ∈ Rn , t > 0.

(ii) u ˜(x, t) → u as t → 0+ in the sense that for every ϕ ∈ AL1 , Z ˜(x, t)ϕ(x) dx. hu, ϕi = lim+ u t→0

Conversely, every smooth solution u ˜(x, t) of the heat equation satisfying the estimate (4.8) can be 0 n uniquely expressed as u ˜(x, t) = (u ∗ E)(x, t) for some u ∈ DL ∞ (R ). The following structure theorem for bounded distributions is well known. n 0 Lemma 4.4. [29] Every u ∈ DL ∞ (R ) can be expressed as X (4.9) u= ∂ α fα |α|≤p

for some p ∈ N0 where fα are bounded continuous functions on Rn . The equality (4.9) implies that Z X hu, ϕi = (−1)|α| fα (x)∂ α ϕ(x)dx |α|≤p

for all ϕ ∈ DL1 . As a special case of Theorem 3.4 of [12] when p = ∞ where the space A0L∞ (Rn ) is denoted by BL∞ (Rn ) we obtain the following. Lemma 4.5. [12] Every u ∈ A0L∞ (Rn ) can be expressed by ! ∞ X k (4.10) u= ak ∆ g + h k=0

where ∆ denotes the Laplacian, g, h are bounded continuous functions on Rn and ak , k = 0, 1, 2, . . . satisfy the following estimates; for every L > 0 there exists C > 0 such that |ak | ≤ CLk /k!2 for all k = 0, 1, 2, . . .. The following properties of the heat kernel will be useful, which can be found in [22]. Proposition 4.6. [22] For each t > 0, Et (·) is an entire function and the following estimate holds; there exists C > 0 such that (4.11)

|∂xα Et (x)| ≤ C |α| t−(n+|α|)/2 α!1/2 exp(−|x|2 /8t).

Also for each t, s > 0 we have Z (4.12)

(Et ∗ Es )(x) :=

Et (x − y)Es (y)dy = Et+s (x).

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Now, we state and prove the main theorem. Theorem 4.7. Let u, v ∈ G 0 (Rn ). Then (u, v) satisfies (4.1) if and only if (u, v) satisfies one of the followings: 0 n 0 n (i) u, v ∈ DL ∞ (R ) [resp. AL∞ (R )], 0 n 0 n (ii) u = c · x + r, v = 1 + λc · x for some c ∈ Cn , λ ∈ C and r ∈ DL ∞ (R ) [resp. AL∞ (R )], (iii) u = λ(ec·x − e−c·x ), v = γec·x + (1 − γ)e−c·x for some c ∈ Cn , λ, γ ∈ C. Proof. We use the same method as in the proof of [10, Theorem 4.6]. Here we give the proof for the reader. Convolving the tensor product Et (x)Es (y) of n-dimensional heat kernels in the left hand side of (4.1), in view of the semigroup property (Et ∗ Es )(x) = Et+s (x) of the heat kernel we have Z (4.13)

[(u ◦ T ) ∗ (Et (ξ)Es (η))](x, y) = huξ ,

Et (x − ξ − η)Es (y − η) dηi

= huξ , (Et ∗ Es )(x − y − ξ)i = huξ , Et+s (x − y − ξ)i =u ˜(x − y, t + s).

Similarly we have

(4.14)

[(u ⊗ v) ∗ (Et (ξ)Es (η))](x, y) = u ˜(x, t)˜ v (y, s), [(v ⊗ u) ∗ (Et (ξ)Es (η))](x, y) = v˜(x, t)˜ u(y, s),

where u ˜(x, t), v˜(x, t) are the Gauss transforms of u, v, respectively. Let w := u ◦ T − u ⊗ v + v ⊗ u. 2n 0 2n 0 ). Using (4.9) and ) [resp. A0L∞ (R2n )]. First, we suppose that w ∈ DL Then w ∈ DL ∞ (R ∞ (R (4.11) we have

|[w ∗ (Et (ξ)Es (η))](x, y)| ≤

X

|[∂ α fα ∗ (Et (ξ)Es (η))](x, y)|

|α|≤p

≤

X

α |[fα ∗ ∂ξ,η (Et (ξ)Es (η))](x, y)|

|α|≤p

≤

X

α kfα kL∞ k∂ξ,η (Et (ξ)Es (η))kL1

|α|≤p

X

≤ C1

k∂ξβ Et (ξ)kL1 k∂ηγ Es (η)kL1

|β|+|γ|≤p

X

≤ C2

t−(n+|β|)/2 s−(n+|γ|)/2

|β|+|γ|≤p

 ≤C

1 1 + t s

107

N + d,

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where N = n + p/2 and the constants C and d depend only on p. Secondly we suppose that w ∈ A0L∞ (R2n ). Then, using (4.11) we have X k! k∆k (Et (ξ)Es (η))kL1 ≤ k∂ 2α (Et (ξ)Es (η))kL1 α! |α|=k

X

≤

|β|+|γ|=k

X

≤

|β|+|γ|=k

X

≤

k! k∂ 2β Et (ξ)kL1 k∂η2γ Es (η)kL1 β!γ! ξ k!(2β)!1/2 (2γ)!1/2 M 2k −n/2−|β| −n/2−|γ| t s β! γ! k!(2M )2k t−n/2−|β| s−n/2−|γ|

|β|+|γ|=k

√ n+k ≤ k!(2 nM )2k (1/t + 1/s) . Now, by the structure (4.10) of bounded hyperfunctions together with the growth condition of ak , k = 0, 1, 2, . . . we have |[w ∗ (Et (ξ)Es (η))](x, y)| ≤

∞ X

kak (∆k g) ∗ (Et (ξ)Es (η))kL∞ + kh ∗ (Et (ξ)Es (η))kL∞

k=0

≤ kgkL∞

∞ X

kak ∆k (Et (ξ)Es (η))kL1 + khkL∞ kEt (ξ)Es (η)kL1

k=0 ∞ X 1 n+k (4nM 2 L)k (1/t + 1/s) + khkL∞ ≤ C1 k!

≤ C2

k=0 ∞ X

k=0

1 k n+k  (1/t + 1/s) + khkL∞ k!

≤ C e(1/t+1/s) , where L is taken so that 4nM 2 L <  and the constant C depends only on w and . Thus, we have the inequality; there exist C > 0 and d > 0 [resp. for every  > 0 there exists C > 0 ] such that N  1 1 + + d [resp. C e(1/t+1/s) ] (4.15) |˜ u(x − y, t + s) − u ˜(x, t)˜ v (y, s) + v˜(x, t)˜ u(y, s)| ≤ C t s where u ˜, v˜ are the Gauss transforms of u, v, respectively, given in Lemma 4.2. Replacing f by u ˜, g by v˜ in Theorem 3.5 and using the continuity of u ˜ and v˜ we obtain one of the followings (I) ∼ (III): (I) both u ˜ and v˜ satisfy (3.3) [resp.(3.4)], (II) u ˜(x, t) = c · xebt + R(x, t), v˜(x, t) = λ˜ u(x, t) + ebt , n n where c ∈ C , b, λ ∈ C and R : R × (0, ∞) → C satisfies |R(x, t)| ≤ Ct−2N + d [ resp.(3.4)] for all x ∈ Rn , t > 0 and for some C, d > 0, (III) u ˜(x − y, t + s) − u ˜(x, t)˜ v (y, s) + v˜(x, t)˜ u(y, s) = 0 for all x, y ∈ Rn , t, s > 0. By Lemma 4.2, case (I) implies (i). For the case (II), since u ˜, v˜ are solutions of the heat equation we must have b = 0 and so is R(x, t) = u ˜(x, t) − c · x. Letting t → 0+ in (II) we obtain case (ii).

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Finally, letting t → 0+ in (III) we have (4.16)

u ◦ T − u ⊗ v + v ⊗ u = 0.

The nontrivial solutions of the equation (4.16) are given by (iii) or u = c · x, v = 1 + λc · x which is included in the case (ii). This completes the proof.  Acknowledgment The present research was conducted by the research fund of Dankook University in 2015. References [1] J. Acz´ el, Lectures on functional equations in several variables, Academic Press, New York-London, 1966 [2] J. Acz´ el and J. Dhombres, Functional equations in several variables, Cambridge University Press, New YorkSydney, 1989. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950), 64–66. [4] J. A. Baker, Distributional methods for functional equations, Aequationes Math. 62(2001), 136–142. [5] J. A. Baker, On a functional equation of Acz´ el and Chung, Aequationes Math. 46(1993) 99–111. [6] J. Chung and J. Chang, On the stability of trigonometric functional equations in distributions and hyperfunctions, Abstract and Applied Analysis Volume 2013, Article ID 275915, 12 pages. [7] J. Chang and J. Chung, On a generalized Hyers-Ulam stability of trigonometric functional equations, Journal of Applied Mathematics, Volume 2012, Article ID 610714, 14 pages. [8] J. Chung, S.-Y. Chung and D. Kim, A characterization for Fourier hyperfunctions, Publ. Res. Inst. Math. Sci. 30(1994), 203–208. [9] J. Chung, S.-Y. Chung and D. Kim, Une caract´ erisation de l’espace de Schwartz C. R. Acad. Sci. Paris S´ er. I Math. 316(1993), 23–25. [10] J. Chung and D. Kim, Ulam problem for the sine addition formula in hyperfunctions, Publ. RIMS Kyoto Univ. 50 (2014), 227–250. [11] S.-Y. Chung, An heat equation approach to distributions with Lp growth, Comm. Kor. math. Soc. 9(1994), No. 4, 897–903. [12] S. Y. Chung, D. Kim and E. G. Lee, Periodic hyperfunctions and Fourier series, Proc. Amer. Math. Soc. 128(2000), 2421–2430. [13] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Inc., Palm Harbor, Florida, Florida,2003. [14] I. Feny¨ o, On an inequality of P. W. Cholewa, Internat Schriftenreihe Numer. math., Vol 80(1987) 277-280. [15] G. L. Forti, Hyer-Ulam stability of functional equation in several variables, Aeq. Math. 50(1995), 143–190. [16] I. M. Gelfand and G.E. Shilov, Generalized functions II, Academic Press, New York, 1968. [17] I. M. Gelfand and G. E. Shilov, Generalized functions IV, Academic, Press, New York, 1968. [18] L. H¨ ormander, The analysis of linear partial differential operators I,Springer-Verlag, Berlin-New York, 1983. [19] D. H. Hyers, On the stability of the linear functional equations, Proc. Nat. Acad. Sci. USA 27(1941), 222-224. [20] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Birkhauser, 1998. [21] S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011). [22] T. Matsuzawa, A calculus approach to hyperfunctions II, Trans. Amer. Math. Soc. 313(1989), 619–654. [23] J. M. Rassias, On Approximation of Approximately Linear Mappings by Linear Mappings, J. Funct. Anal. 46(1982), 126–130. [24] J. M. Rassias, On Approximation of Approximately Linear Mappings by Linear Mappings, Bull. Sci. Math. 108(1984), 445–446. [25] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Th. 57(1989), 268–273.

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[26] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251(2000), 264–284. [27] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 7(1978), 297-300. [28] K. Ravi and M. Arunkumar, On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation, Intern. J. Appl. Math. Stat, 7(2007), 143–156. [29] L. Schwartz, Th´ eorie des distributions, Hermann, Paris, 1966. [30] L. Sz´ ekelyhidi, The stability of sine and cosine functional equations, Proc. Amer. Math. Soc. 110(1990), 109–115. [31] L. Sz´ ekelyhidi, The stability of d’Alembert type functional equations, Acta Sci. Math. Szeged. 44(1982c), 313–320. [32] S. M. Ulam, Problems in modern mathematics, Chapter VI, Wiley, New York, 1964. [33] D.V. Widder, The heat equation, Academic Press, New York, 1975. Chang-Kwon Choi Department of Mathematics, Jeonbuk National University, Jeonju, 561-756, Republic of Korea E-mail address: [email protected] Jeongwook Chang Department of Mathematics Education, Dankook University, Yongin 448-701, Republic of Korea E-mail address: [email protected]

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Effect of cytotoxic T lymphocytes on HIV-1 dynamics Shaimaa A. Azoza and Abdelmonem M. Ibrahimb a Department of Mathematics and Institute of Applied Mathematics, University of British Columbia, Canada. b Electrical and Computer Engineering, University of British Columbia, Canada. Email: [email protected] Abstract The purpose of this paper is to study the effect of the cytotoxic T lymphocytes (CTLs) on an HIV-1 dynamics. The model considers that the virus infects the macrophages in addition to the CD4+ T cells. The role of the CTLs is to kill the infected macrophages and CD4+ T cells. The time delay which accounts the time of infection and the time of producing new active HIV-1 is modeled. The HIV-1 dynamics is modeled as a 6-dimensional nonlinear delay differential equations. The incidence rate of infection and killer rate of infected cells are given by general nonlinear functions. We study the qualitative behavior of the system. The global stability analysis has been established using Lyapunov method and LaSalle invariance principle. We present an example and perform numerical simulations to emphasize our theoretical results. Keywords: Global stability; HIV infection; time delay; Immune response; Direct Lyapunov method.

1

Introduction

Recently, the study of Human Immunodeficiency Virus type-1 (HIV-1) and Acquired Immunodeficiency Syndrome (AIDS) has become a topic of interest in the mathematical literature. In the pursuit of understanding the interaction between the HIV-1 and immune system, several mathematical models have been proposed. The following is the basic model of HIV-1 infection dynamics that has been described and studied in [1]: x˙ = λ − dx − βxv, y˙ = βxv − δy, v˙ = ky − rv. Here, the concentrations of uninfected CD4+ T cells, infected CD4+ T cells and virus are represented by x, y and v, respectively. The production rate of CD4+ T cells is represented by λ, while the infection rate, and thus the infected CD4+ T cell production rate, is represented by βxv, where β is the infection rate constant. The uninfected cells and infected cells are die with rate dx and δy, respectively. k represents the rate constant of virion generation by CD4+ T cells while r represents the rate constant of viral particle emptying from the plasma. Replication models assume cytotoxic T lymphocyte cells (CTLs) to be the main host defence restricting viral replication in vivo and thus the main determinant of viral load. Nowak and Bangham [2] constructed the first model of HIV taking into account CTLs as: x˙ = λ − dx − βxv, y˙ = βxv − δy − pyz, v˙ = ky − rv, z˙ = cyz − bz.

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where z represent the concentration of CTLs which multiply at a rate cyz when stimulated by infected cells, while bz represents the death rate of this population of CTLs. Delay differential equations are used to introduce delays into the infection equations and/or equations for virus production to account for the intracellular phase of the viral life cycle. This delay is defined as period between infection of a CD4+ T-cell and the point at which the infected cell begins to produce viral particles (see, e.g. [3], [5], [6], [7], [8], [9], [10]). Complications have been shown to occur ([11], [12], [13]) when time delays are introduced into infection models with immune responses. Such complications include stable periodic solutions and chaos. The use of general kernel function to represent distributed intracellular delays has been motivated by the argument that constant delays may not biologically realistic ([22],[23],[24]). In contrast to Nakata’s [17] investigation of the stability of an immunity mediated HIV-1 model with two finite distributed intracellular delays, Wang et al. [16] and Li and Shu [14] examined the stability of an infection model with infinite distributed intracellular delays by constructing Lyapunov functionals. Yuan and Zou [15] proposed and developed an appropriate mathematical model for HIV-1 infection by incorporating distributed delay into the cell infection equation and another virus production equation and nonlinear incidence rate and a nonlinear removal rate for the infected cells. However, the presence of the macrophages has been neglected. Our aim in this paper is to study the effect of the CTL immune response of the global dynamics of a distributed delayed HIV-1 model which describe the interaction between the virus and two target cells, CD4+ T cells and macrophages. The motivation for considering the two target cell model is the observation that the rate of viral load decline was considerably lower after the rapid first phase of decay during the 1-2 weeks after antiretroviral treatment ([3],[4],[18]). The model is a 6-dimensional nonlinear ODES that takes into account cytotoxic T lymphocyte cells (CTLs) with nonlinear incidence rate and distributed delays using distributed kernels reflecting the variance in time required for viral entry into cells and the variability in time required for intracellular virion reproduction. The positive invariance properties and the boundedness of the solutions for the model are studied. By constructing explicit Lyapunov functionals and using the LaSalle invariance principle, which are extensions and modified forms of the Lyapunov functionals given in [15], we prove that the steady states of the model are globally asymptotically stable (GAS) and the dynamics of the system is fully determined by the basic reproduction number R0 .

2

Mathematical model

We shall examine a deterministic model of HIV infection, which represents the interaction of HIV with two co-circulation populations of target cells, representing CD4+ T and macrophages cells. The system takes into consideration the distributed invasion and production delays and (i) We assume that the incidence rate is given by a nonlinear form. (ii) The model takes into consideration cytotoxic T lymphocyte cells (CTLs) immune response: x˙ 1 (t) = µ1 − k1 x1 (t) − α1 x1 (t)f1 (v(t)), Z∞ y˙ 1 (t) = α1 e−m1 τ G1 (τ )x1 (t − τ )f1 (v(t − τ ))dτ − ry1 (t) − βy1 (t)h1 (z(t)),

(1) (2)

0

x˙ 2 (t) = µ2 − k2 x2 (t) − α2 x2 (t)f2 (v(t)), Z∞ y˙ 2 (t) = α2 e−m2 τ G2 (τ )x2 (t − τ )f1 (v(t − τ ))dτ − ry2 (t) − βy2 (t)h2 (z(t)),

(3) (4)

0

∞  Z Z∞ v(t) ˙ = N r  e−n1 τ Ψ1 (τ )y1 (t − τ )dτ + e−n2 τ Ψ2 (τ )y2 (t − τ )dτ  − dv(t), 0

(5)

0

z(t) ˙ = λ (y1 (t) + y2 (t)) − qz(t).

(6)

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The state variables describes the plasma concentrations of: x1 , y1 , represent the uninfected and infected CD4+ T cells; x2 , y2 , represent the uninfected and infected macrophages. Eq. (1) and (3) describe the populations of target cells, where µ1 and µ2 perform the rates of new generations of CD4+ T cell and macrophages from sources within the body, k1 , k2 are the death rate constants, and α1 , α2 are the infection rate constants. Equation (2) and (4) represent the population dynamics of the infected target cells, where r represent the clearance rate and it killed at rate βy1 (t)h1 (z(t)) and βy2 (t)h2 (z(t)), respectively. The CTL cells are produced at a rate λ(y1 + y2 )z and are decayed at a rate qz . Assume the kernel functions Gi and Ψi , i = 1, 2 satisfy Gi (τi ) > 0, R∞ R∞ iτ Ψi (τi ) > 0. Let us denote ai = e−mi τ Gi (τ )dτ, bi = e−n Ψi (τ )dτ , i = 1, 2. Thus 0 < ai ≤ 1, 0 < bi ≤ 1 i 0

0

All parameters are assumed to be positive. The function fi (v) and hi (z) are continuously differentiable and guarantee this conditions are met:  0 (C1): fi (0) = 0, fi0 (ξi ) exists and satisfies fi0 (ξi ) ≥ 0 and fiξ(ξi i ) 6 0 in (0, ∞), (C2): hi (0) = 0, hi (ζi ) is strictly increasing in (0, ∞),

2.1

Positively and Boundedness

To prove the positively and the boundedness of the solutions, it is biologically reasonable to consider the following non-negative initial conditions for the system (1-6), define the Banach space of fading memory type Cα = {ϕ ∈ C ((−∞, 0], R) : ϕ(θ)eαθ is uniformly continuous for θ ∈ (−∞, 0] and kϕk < ∞} where α is a positive constant and kϕk = supθ≤0 |ϕ(θ)| eαθ . Let C+ α = {ϕ ∈ Cα : ϕ(θ) ≥ 0 for θ ∈ (−∞, 0]}. The initial conditions for system (1-6) are given as: x1 (θ) = ϕ1 (θ), y1 (θ) = ϕ2 (θ), x2 (θ) = ϕ3 (θ), y2 (θ) = ϕ4 (θ), v(θ) = ϕ5 (θ), z(θ) = ϕ6 (θ) for θ ∈ [−∞, 0] , ϕi ∈ Cα+ , i = 1, 2, ..., 6.

(7)

By the fundamental theory of functional differential equations (see [20] and [21]), model (1-6) with initial conditions (7) has a unique solution and the following lemma establishes the positivity and boundedness of the solutions. Lemma 1. Let (x1 (t), y1 (t), x2 (t), y2 (t), v(t), z(t)) be the solution of system (1-6) with the initial conditions (7), then x1 (t), y1 (t), x2 (t), y2 (t), v(t) and z(t) are all positive and bounded for all t > 0. Proof. First, we will prove that xi (t) > 0, i = 1, 2, for all t ≥ 0. Assume that xi (t) loses its nonnegativity on some local existence interval [0, υ] for some constant υ and let t∗ ∈ [0, υ] be such that xi (t∗ ) = 0. From (1) and (3) we have xi (t∗ ) = µi > 0. Hence xi (t∗ ) > 0 for some t ∈ (t∗ , t∗ + ), where  > 0 is sufficiently small. This leads to a contradiction and hence xi (t) > 0, for all t ≥ 0. Further by using the variation of parameters method and Eq. (2), (4) and (5) we have Rt

yi (t) = yi (0)e− 0 (r+βh(z(s)))ds Z t R Z t + αi e− s (r+h(z(η)))dη 0

∞

e−mi η Gi (η)x(s − η)fi (v(s − η))dηds;

i = 1, 2.

0 −dt

v(t) = v(0)e

Z + Nr

t

e 0

−d(t−s)

∞

Z

2 X

0

e−ni η Ψi (η)yi (s − η)dηds,

i=1

confirming that yi (t) ≥ 0, i = 1, 2, and v(t) ≥ 0 for all t ≥ 0. Now from (6) we get z(t) = z(0)e−qt + λ

Z 0

t

e−q(t−s)

2 X

yi (s)ds.

i=1

Then z(t) ≥ 0, for all t ≥ 0, and this prove the positively of the solution. Now we shall prove that the solution are bounded, from Eq.(1) and (3), we have x˙ i (t) 6 µi − ki xi (t), this implies lim supt→∞ xi (t) ≤ µkii , i = 1, 2, let

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Ui (t) =

R∞ 0

e−mi τ Gi (τ )xi (t − τ )dτ + yi (t), then U˙ i (t) =

R∞

=

R∞

0 0

e−mi τ Gi (τ )x˙ i (t − τ )dτ + y˙ i (t), e−mi τ Gi (τ )(µi − ki xi (t − τ ) − αi xi (t − τ )fi (v(t − τ )))dτ

Z∞ + αi e−mi τ Gi (τ )xi (t − τ )fi (v(t − τ ))dτ − ryi (t) − βyi (t)hi (z(t)), 0

= µi ai − αi

R∞ 0

e−mi τ Gi (τ )xi (t − τ )dτ − ryi (t) − βyi (t)hi (z(t)),

≤ µi ai − pi Ui (t). It follows that lim supt→∞ Ui (t) ≤ µpi ai i , where pi = min{α1 , α2 , r}. Since xi (t) > 0, yi (t) ≥ 0 and z(t) ≥ 0 P2 then lim sup i=1 yi (t)t→∞ ≤ Li , lim sup yi (t)t→∞ ≤ Li , i = 1, 2. On the other hand, v(t) ˙ ≤ N r(b1 L1 + b2 L2 ) − dv(t) ≤ N rbL − dv(t). R∞

where bi = 0 e−ni τ Ψi (τ )dτ. From Eq.(6) we get z(t) ˙ ≤ λL − qz(t). Then lim supt→∞ v(t) ≤ N rbL and lim d λL supt→∞ z(t) ≤ q . Therefore, x1 (t), y1 (t), x2 (t), y2 (t), v(t), and z(t) are ultimately bounded and this complete the proof of boundedness of solutions.

2.2

Basic reproduction number and steady state

To obtain the equilibrium points of model (1)-(6), we consider the following equations: µ1 − k1 x1 − α1 x1 f1 (v) = 0, α1 a1 x1 f1 (v) − ry1 − βy1 h1 (z) = 0, µ2 − k2 x2 − α2 x2 f2 (v) = 0, α2 a2 x2 f2 (v) − ry2 − βy2 h2 (z) = 0, N r (b1 y1 + b2 y2 ) − dv = 0, λ (y1 + y2 ) − qz = 0. We find that if z 6= 0, there is two steady states E0 = (x01 , 0, x02 , 0, 0, 0) where x01 = E ∗ = (x∗1 , y1∗ , x∗2 , y2∗ , v ∗ , z ∗ ) satisfies the equations:

µ1 k1 ,

x02 =

µ2 k2 ,

and

P2 ∗ µi d ∗ , v , i=1 yi = ∗ ki + αi fi (v ) N rb λ P2 ∗ λd ∗ αi ai fi (v ∗ ) ∗ z∗ = v , yi∗ = x . i=1 yi = q N rbq r + βhi (z ∗ ) i

x∗i =

The basic reproduction number, R0 , for system (1)-(6) is given by: R0 =

µ1 α1 a1 N b1 f10 (0) µ2 α2 a2 N b2 f20 (0) + = R1 + R2 , k1 d k2 d

where, R1 and R2 are the basic reproduction numbers for CD4+ T cells and macrophages cells, severally. Now, we shall prove that R0 > 1 is a sufficient condition to ensure the existence of an infected steady state E ∗ = (x∗1 , y1∗ , x∗2 , y2∗ , v ∗ , z ∗ ). Using the above calculations the existence of an infected equilibrium is equivalent to the existence of a positive root of the equation L(v) = 0, where   µ1 µ2 rd ∗ βd ∗ λdv ∗ ∗ L(v ∗ ) = α1 a1 f1 (v ∗ ) + α a f (v ) − v − v h , 2 2 2 k1 + α1 f1 (v ∗ ) k2 + α2 f2 (v ∗ ) N rb N rb N rbq

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and it satisfies L(0) = 0, L(+∞) = −∞, and µ1 µ2 rd + α2 a2 f20 (0) − k1 k2 N rb d α1 a1 µ1 N bf10 (0) α2 a2 µ2 N bf20 (0) d = + −1= (R0 − 1) > 0. Nb k1 d k2 d Nb

L0 (0) = α1 a1 f10 (0)

It follows from the continuity of the function L(v) in [0, ∞) that L(v) = 0 has at least one positive root. Hence, we see that the condition at least has one infected equilibrium E ∗ when R0 > 1. We can rewrite the model as: x˙ 1 (t) = µ1 − k1 x1 (t) − α1 x1 (t)f1 (v(t)), Z∞ y˙ 1 (t) = ϑ1 e−m1 ζ g1 (ζ)x1 (t − ζ)f1 (v(t − ζ))dζ − ry1 (t) − βy1 (t)h(z(t)),

(8) (9)

0

x˙ 2 (t) = µ2 − k2 x2 (t) − α2 x2 (t)f2 (v(t)), Z∞ y˙ 1 (t) = ϑ2 e−m2 ζ g2 (ζ)x2 (t − ζ)f1 (v(t − ζ))dζ − ry2 (t) − βy2 (t)h(z(t)),

(10) (11)

0

Z∞ Z∞ −n1 ζ ψ1 (ζ)y1 (t − ζ)dζ + γ2 e−n2 ζ ψ2 (ζ)y2 (t − ζ)dζ − dv(t), v(t) ˙ = γ1 e z(t) ˙ = λ (y1 (t) + y2 (t)) − qz(t). For simplify, we taked h1 = h2 = h, ϑi = αi ai , γi = N rbi , gi (ζ) =

3

(12)

0

0

(13) Gi (ζ) ai ,

ψi (ζ) =

Ψi (ζ) bi .

Global stability

In this section, we going to show that the steady states satisfy the global stability condition: Theorem 1. Let Conditions C1 and C2 hold true and R0 ≤ 1, then the infection-free equilibrium E0 is globally asymptotically stable. R∞ R∞ P3 Proof. Define Hi (t) = t gi (ζ)dζi , Pi (t) = t ψi (ζ)dζ, and consider laypunov function W (t) = i=1 Wi (t), where,  2 2 X µi 1 αi µi r αi µi β R z(t) αi µi xi (t) − yi (t) + v(t) + W1 (t) = + h(ζ)dζ, 2 ki ϑi ki ϑ1 ϑ2 ki ϑi ki λ 0 i=1 ∞

W2 (t) =

Z 2 X αi µi i=1

W3 (t) =

2 X i=1

Hi (ζi )xi (t − ζi )fi (v(t − ζi ))dζi ,

ki 0

αi µi r ki ϑi

Z∞ Pi (ζi )yi (t − ζi )dζi , 0

It clear that, W (t) ≥ 0 and W (t) = 0 if and only if xi (t) = µkii and yi (t) = v(t) = z(t) = 0. The derivative Wi (t) of along the solution is:    Z∞ 2 X ˙ 1 (t) =  xi (t) − µi (µi − ki xi (t) − αi xi (t)f (v(t))) + αi µi gi (ζi )xi (t − ζi )fi (v(t − ζi ))dζi W ki ki i=1 0

Z∞ αi µi r αi µi β αi µi r αi µi rd − yi (t) − yi (t)h(z(t)) + ψi (ζi )yi (t − ζi )dζi − v(t) ki ϑ i ki ϑi ki ϑi ki ϑi γi 0  αi µi β + h(z(t)) [λyi (t) − qz(t)] , ki λϑi

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Note that Hi (0) = 1, Hi (∞) = 0 and dHi (t) = −gi (t)dt. Using integration by parts, we calculate the derivative of W2 : ∞

∞

Z 2 X αi µi

Z 2 X d (xi (t − ζi )fi (v(t − ζi ))) d (xi (t − ζi )fi (v(t − ζi ))) αi µi ˙ 2 (t) = W Hi (ζi ) dζi = − Hi (ζi ) dζi , ki dt ki dζi i=1 i=1 0 0   ∞ Z 2 X αi µi αi µi  xi (t − ζi )fi (v(t − ζi ))dHi (ζi ) , =− Hi (ζi )xi (t − ζi )fi (v(t − ζi )) p∞ ζ=0 + k ki i i=1 0   ∞ Z 2 X αi µi αi µi  gi (ζi )xi (t − ζi )fi (v(t − ζi ))dζi  . xi (t)fi (v(t)) − = k ki i i=1 0

Similarly  Z∞ ˙ 3 (t) =  αi µi r yi (t) − αi µi r ψi (ζi )yi (t − ζi )dζi  , W ki ϑ i ki ϑi i=1 2 X



0

Therefore ˙ (t) = W

2 X

"

i=1

αi µi + ki

2  αi µi µi − αi x2i (t)fi (v(t)) + xi (t)fi (v(t)) −ki xi (t) − ki ki

Z∞ αi µi r αi µi β gi (ζi )xi (t − ζi )fi (v(t − ζi ))dζi − yi (t) − yi (t)h(z(t)) ki ϑi ki ηi 0

αi µi r + ki ϑi

Z∞ αi µi rd αi µi β ψi (ζi )yi (t − ζi )dζi − v(t) + yi (t)h(z(t)) ki ϑi γi ki ϑi 0

Z∞ αi µi αi µi αi µi βq z(t)h(z(t)) + xi (t)fi (v(t)) − − gi (ζi )xi (t − ζi )fi (v(t − ζi ))dζi ki λϑi ki ki 0  Z∞ αi µi r αi µi r + yi (t) − ψi (ζi )yi (t − ζi )dζi  . ki ϑi ki ϑi 0

2 X

"

 2   µi 2µi µ2 −ki xi (t) − − αi fi (v(t)) x2i (t) − xi (t) + 2i ki ki ki i=1  αi µ2 αi µi rd αi µi βq + 2 i fi (v(t)) − v(t) − z(t)h(z(t)) . ki ki ηi γi ki ληi

˙ (t) = W

Hence 2 X

"

 2  2 µi µi αi µi βq −ki xi (t) − − αi fi (v(t)) xi (t) − − z(t)h(z(t)) k k ki λϑi i i i=1   αi µi rd µi ϑi γi fi (v(t)) + v(t) −1 . ki ϑi γi ki rd v(t)

˙ (t) = W

But from Condition (C1), we have ˙ (t) ≤ W

2 X i=1

fi (v(t)) v(t)

6 fi0 (0). Hence

"

#  2  2 µi µi αi µi rd αi µi βq − αi fi (v(t)) xi (t) − + (R0 − 1) v(t) − z(t)h(z(t)) . −ki xi (t) − ki ki ki ϑi γi ki λϑi

˙ 1 ≤ 0. To prove the global stability of the infected equilibrium, we need to use this lemma: If R0 ≤ 1, then, W

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Lemma 2. If satisfies Condition (C1), then g (F (σ)) ≤ g(σ), σ > 0 with the equality holding only at ∗ σ = 1, where F (σ) = ff(v(v∗σ) ) , and g(u) = u − 1 − ln u and with g : (0, ∞) → [0, ∞) has the global minimum g(1) = 0 and positive elsewhere for ζi ∈ (0, ∞). Proof. Since F (1) = 1 and the derivative of g(σ) has the same sign as σ − 1 for σ > 0, we need only to prove that σ ≤ F (σ) ≤ 1 for σ ∈ (0, 1) and 1 ≤ F (σ) ≤ σ for σ ∈ [1, ∞) . The proof of case σ ∈ [1, ∞) is similar to that case of σ ∈ (0, 1) , so we will only consider the case when σ ∈ (0, 1) . Note that σ ≤ F (σ) ≤ 1 is ∗ ) f (v ∗ σ) f (v ∗ ) equivalent to f (v v ∗ ≤ v ∗ σ ≤ v ∗ σ for σ ∈ (0, 1) , from Condition (C1) we completed the proof. Theorem 2. Let conditions C1 and C2 hold true and R0 > 1, then the chronic infection equilibrium E ∗ is globally asymptotically stable for all positive solution. Proof. Define V1 =

2 X

 gi

i=1

V3 =

2 X i=1

 gi

xi (t) x∗i



yi (t) yi∗



∞

,

V2 =

2 Z X



Hi (ζi )gi

i=1 0

,

V4 =

2 X

 gi

i=1

z(t) Z V5 = [h (ζi ) − h (z ∗ )] dζi , ,

v (t) v∗

V6 =

i=1 0

z∗

 dζi ,

 ,

∞

2 Z X

xi (t − ζi )fi (v(t − ζi )) x∗i fi (v ∗ )



Ψi (ζi ) gi

yi (s) yi∗

 ds.

with the infected steady state conditions: µi = ki x∗i + αi x∗i fi (v ∗ ) ,

γi yi∗ = dv ∗ ,

ϑi x∗i fi (v ∗ ) = ryi∗ + βyi∗ h (z ∗ ) ,

λyi∗ = qz ∗ .

(14)

we will let the function V (t) and study the derivative of the Lyapunov functional as: V (t) = x∗i V1i (t) + αi x∗i fi (v ∗ ) V2i (t) +

αi yi∗ αi rv ∗ αi β αi βyi∗ h (z ∗ ) V3i (t) + V4i (t) + V5i (t) + V4i (t) ϑi ϑi γi λϑi dϑi

+ αi x∗i fi (v ∗ ) V6i (t) . satisfies V (t) ≥ 0 with the equality holding if and only xi (t) = x∗i , yi (t) = yi∗ , v (t) = v ∗ , z (t) = z ∗ and xi (t − ζi )fi (v(t − ζi )) = x∗i fi (v ∗ ) , yi (t − ζi ) = yi∗ . We get   x∗i 1 ˙ (ki x∗i + αi x∗i fi (v ∗ ) − ki xi (t) − αi xi (t)fi (v(t))) , V1i (t) = ∗ 1 − xi xi (t)   2 −ki (xi (t) − x∗i ) x∗i xi (t)f (v(t)) fi (v(t)) ∗ = + α f (v ) 1 − − + . (15) i i x∗i xi (t) xi (t) x∗i fi (v ∗ ) fi (v ∗ ) xi (t)fi (v(t)) V˙ 2i (t) = − ln x∗i fi (v ∗ )



xi (t)fi (v(t)) x∗i fi (v ∗ )

 −

Z∞ xi (t − ζi )fi (v(t − ζi )) gi (ζi ) dζi x∗i fi (v ∗ ) 0

  Z∞ xi (t − ζi )fi (v(t − ζi )) dζi , + gi (ζi ) ln x∗i fi (v ∗ ) 0

where Hi (0) = 1, Hi (∞) = 0, dHi (t) = −gi (t) dt.  ∞    Z ∗ 1 y ηi gi (ζi )xi (t − ζi )fi (v(t − ζi ))dζi − ryi (t) − βyi (t)h(z(t)) . V˙ 3i (t) = ∗ 1 − i yi yi (t) 0

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Using Eq. (14) Z∞ Z∞ xi (t − ζi )fi (v(t − ζi )) ϑi x∗i fi (v ∗ ) xi (t − ζi )yi∗ fi (v(t − ζi )) gi (ζi ) dζ − g (ζ ) dζi , i i i x∗i fi (v ∗ ) yi∗ x∗i yi (t) fi (v ∗ ) 0 0     ϑi x∗i fi (v ∗ ) yi (t) h (z (t)) h (z (t)) yi (t) yi (t) h (z (t)) h (z (t)) − + − − + r 1 − . (16) yi∗ yi∗ h (z ∗ ) h (z ∗ ) yi∗ yi∗ h (z ∗ ) h (z ∗ )  ∞    Z Z∞ ∗ v 1 γ1 ψ1 (ζ1 )y1 (t − ζ1 )dζ1 + γ2 ψ2 (ζ2 )y2 (t − ζ2 )dζ2 − dv(t) , V˙ 4i (t) = ∗ 1 − yi v (t)

ϑi x∗i fi (v ∗ ) V˙ 3i (t) = yi∗

0

0

Using Eq. (14)  v (t) V˙ 4i (t) = d 1 − ∗ + v

 Z∞ Z∞ yi (t − ζi ) v ∗ yi (t − ζi )  ψi (ζi ) dζi − ψi (ζi ) dζi , yi∗ v (t) yi∗ 0

0

V˙ 5i (t) = [h (z (t)) − h (z ∗ )] [λyi (t) − qz (t)] . Using Eq. (14)   yi (t) yi (t) h (z (t)) h (z (t)) . V˙ 5i (t) = −q [h (z (t)) − h (z ∗ )] [z (t) − z ∗ ] + λyi∗ h (z ∗ ) 1 − ∗ + − yi yi∗ h (z ∗ ) h (z ∗ ) Similar to V˙ 2i (t) the derivative of V˙ 6i (t),differentiating gives yi (t) V˙ 6i (t) = ∗ − yi

  Z∞ Z∞ yi (t − ζi ) yi (t − ζi ) dζi , ψi (ζi ) dζi + ψi (ζi ) ln yi∗ yi (t) 0

0

It follows that " V˙ (t) =

P2

i=1

  2 −ki (xi (t) − x∗i ) x∗i fi (v(t)) + αi x∗i fi (v ∗ ) 1 − + xi (t) xi (t) fi (v ∗ )

  Z∞ Z∞ xi (t − ζi )fi (v(t − ζi )) xi (t − ζi )yi∗ fi (v(t − ζi )) ∗ ∗ dζi − αi xi fi (v ) gi (ζi ) dζi (v ) gi (ζi ) ln ∗ ∗ xi fi (v ) x∗i yi (t) fi (v ∗ ) 0 0     αi ryi∗ yi (t) yi (t) h (z (t)) h (z (t)) y (t) h (z (t)) h (z (t)) i ∗ ∗ + 1− ∗ + − − αi xi fi (v ) − ϑi yi yi∗ h (z ∗ ) h (z ∗ ) yi∗ h (z ∗ ) h (z ∗ )   ∞ ∞ Z Z αi rv ∗ d  v (t) yi (t − ζi ) v ∗ yi (t − ζi )  + 1 − ∗ + ψi (ζi ) dζ − ψi (ζi ) dζi i ∗ ϑ i γi v yi v (t) yi∗ 0 0   ∞ Z Z∞ ∗ ∗ ∗ v (t) yi (t − ζi ) v yi (t − ζi )  αi βyi h (z )  1 − ∗ + ψi (ζi ) dζi − ψi (ζi ) dζi + ϑi v yi∗ v (t) yi∗ 0 0   ∗ ∗ αi βq α βy h (z ) yi (t) yi (t) h (z (t)) h (z (t)) i i ∗ ∗ − [h (z (t)) − h (z )] [z (t) − z ] + 1− ∗ + − λϑi ϑi yi yi∗ h (z ∗ ) h (z ∗ )    Z∞ Z∞ yi (t − ζi ) yi (t − ζi ) yi (t) +αi x∗i fi (v ∗ ) ∗ − αi x∗i fi (v ∗ ) ψi (ζi ) dζi + αi x∗i fi (v ∗ ) ψi (ζi ) ln dζi  , yi yi∗ yi (t) +αi x∗i fi

∗

0

0

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We have

rdv ∗ γi

= ryi∗ = ϑi x∗i fi (v ∗ ) − βyi∗ h (z ∗ ), then

  P2 −ki (xi (t) − x∗i )2 x∗i fi (v(t)) v (t) V˙ (t) = i=1 + αi x∗i fi (v ∗ ) 3 − + − αi x∗i fi (v ∗ ) ∗ ∗ xi (t) xi (t) fi (v ) v ∞   Z Z∞ xi (t − ζi )fi (v(t − ζi )) xi (t − ζi )yi∗ fi (v(t − ζi )) ∗ ∗ dζ + α x f (v ) g (ζ ) ln dζi − αi x∗i fi (v ∗ ) gi (ζi ) i i i i i i x∗i yi (t) fi (v ∗ ) x∗i fi (v ∗ ) 0

0

Z∞ yi (t) h (z (t)) h (z (t)) yi (t − ζi ) ∗ ∗ ∗ ∗ − αi xi fi (v ) − dζi − αi xi fi (v ) ψi (ζi ) ∗ ∗ ∗ yi h (z ) h (z ) yi∗ 



0

  Z∞ yi (t − ζi ) αi βq ∗ ∗ ∗ ∗ [h (z (t)) − h (z )] [z (t) − z ] + αi xi fi (v ) ψi (ζi ) ln dζi , − λϑi yi (t) 0

We can write

Z∞ Z∞ 3 = 2 gi (ζi ) dζi + ψi (ζi ) dζi . 0

0

Hence " 2 P2 −ki (xi (t) − x∗i ) αi βq ˙ V (t) = i=1 − [h (z (t)) − h (z ∗ )] [z (t) − z ∗ ] xi (t) λϑi ∞     ∗  Z xi (t − ζi )yi∗ fi (v(t − ζi )) x∗ xi ∗ ∗  − ln −g +αi xi fi (v ) gi (ζi ) −g xi (t) x∗i yi (t) fi (v ∗ ) x(t) 0

   ∗  Z∞ v yi (t − ζi ) x(t − ζi )y fi (v(t − ζi )) xi (t − ζi )f (v(t − ζi )) − ln dζi + ψi (ζi ) −g + ln x∗i yi (t)fi (v ∗ ) xi (t)fi (v(t)) v (t) yi∗ 0   v ∗ yi (t − ζi ) yi (t − ζi ) fi (v(t)) v(t) − − ln + ln dζ − , i v(t)yi∗ yi (t) fi (v ∗ ) v∗       P2 fi (v(t)) v(t) v (t) fi (v(t)) − ln − + ln ≤ i=1 αi x∗i fi (v ∗ ) , fi (v ∗ ) fi (v ∗ ) v∗ v∗ P2 = i=1 [αi x∗i fi (v ∗ ) (g(F (σ)) − g(σ))] , ∗

∗ ˙ ˙ where σ = v(t) v ∗ and using Lemma 2, we get V (t) ≤ 0 and V (t) = 0 if and only if xi (t) = xi , z(t) = z ∗ , yi∗ fi (v(t − ζi )) = yi (t)fi (v ∗ ), v ∗ yi (t − ζi ) = v(t)yi∗ and v(t) = v ∗ for ζi ∈ [0, ∞). Then the solutions converge to Γ, which is the largest invariant subset of {V˙ (t) = 0} and by conforming LaSalle’s invariance principle, we get that E ∗ is GAS in Γ.

4

Numerical simulations

In this section, we present an instance to explain the main results given in Theorem 1 and 2 by using the Lyapunov direct method. We have determined a set of conditions which guarantee that the steady states of model (1)-(6) are GAS. Table 1 have the estimate values of model (1)-(6) parameters. The effects of two main factors on the qualitative behavior of the system which include therapy efficacy ε and time delay τ will be studied below in details. Using MATLAB we have implemented all computations. This example is obtained from the model (1)-(6) by choosing particular template of the functions fi (v(t)) and hi (z(t)) as follow: f1 (v(t)) =

v v , f2 (v(t)) = , h1 (z(t)) = z(t), h2 (z(t)) = z(t) , 1 + ω1 v 1 + ω2 v

where ω1 , ω2 ≥ 0 are constants. Further more, we are going to choose a particular form of the probability distribution functions Gi (τ ) and Ψi (τ ) as Gi (τ ) = δ(τ − τi ), Ψi (τ ) = δ(τ − τi ), i = 1, 2, where δ(.) is the

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Dirac delta function, τ1 and τ2 are constants where τi ∈ [0, ∞], i = 1, 2. are constants where R∞ Ψi (τ )dτ = 1. Using Dirac delta function properties we get: 0

R∞ 0

Gi (τ )dτ =

Z∞ Z∞ −mi τ −mi τi gi = e δ(τ − τi )dτ = e−mi τi δ(τ − τi )dτ = e 0

0

Z∞ Z∞ ψi = e−ni τ δ(τ − τi )dτ = e−ni τi δ(τ − τi )dτ = e−ni τi , 0

0

Z∞ δ(τ − τi )e−mi τ xi (t − τ )fi (v(t − τ ))dτ = e−mi τi xi (t − τi )fi (t − τi ), 0

Z∞ δ(τ − τi )yi (t − τi )dτ = e−ni τi yi (t − τi ). 0

Referring to the previous relations, we can rewrite model (1)-(6) as follows v(t) , 1 + ω1 v(t) v(t − τ1 ) y˙ 1 (t) = α1 e−m1 τ1 x1 (t − τ1 ) − ry1 (t) − βy1 (t)z(t), 1 + ω1 v(t − τ1 ) v(t) x˙ 2 (t) = µ2 − k2 x2 (t) − α2 x2 (t) 1 + ω2 v(t) v(t − τ2 ) − ry2 (t) − βy2 (t)z(t), y˙ 1 (t) = α2 e−m2 τ2 F2 x2 (t − τ2 ) 1 + ω2 v(t − τ2 )
v(t) ˙ = N r e−n1 τ1 y1 (t − τ1 ) + e−n2 τ2 y2 (t − τ2 ) − dv(t), x˙ 1 (t) = µ1 − k1 x1 (t) − α1 x1 (t)

z(t) ˙ = λ (y1 (t) + y2 (t)) − qz(t).

(17) (18) (19) (20) (21) (22)

To study the effect of drug efficacy, we choose α1 = (1 − ε)a0 and α2 = (1 − ε)b0 . We have chosen the initial conditions: IC: ϕ1 (u) = 600, ϕ2 (u) = 1, ϕ3 (u) = 500, ϕ4 (u) = 1, ϕ5 (u) = 10 and ϕ6 (u) = 40, u ∈ [−∞, 0]. Table 1: We define the parameter values of model (17-22) as follow: Parameter µ1 k1 a0 ω1 r β N d λ q

Value 10 cells mm−3 day−1 0.01 day−1 0.004 day−1 0.05 virus−1 mm3 0.3 day−1 0.001 5 virus cells−1 3 day−1 3 day−1 0.1 day−1

Parameter µ2 k2 b0 ω2 m1 m2 n1 n2 τ1 = τ2 ε

Value 6 cells mm−3 day−1 0.01 day−1 0.001 day−1 0.05 cells−1 mm3 1 day−1 1 day−1 1 day−1 1 day−1 varied varied

Case I: Effect of drug efficacy on the dynamical behavior of the system: In this case, we fix the delay parameter τ1 = τ2 = τ = 0.5. Figures 1-6 show the effect of drug efficacy on the stability of the steady states and the evolution of the uninfected and infected for each CD4+ T cells and macrophages, free virus particles and immune response. We observe that, as the drug efficacy is increased from

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ε = 0 to ε = 0.8, E1 still exists and is a globally asymptomatically stable. Moreover the concentrations of uninfected CD4+ T cells and macrophages are increasing and converging to their normal values µk11 = 1000 cells mm−3 , µk22 = 600 cells mm−3 , respectively. While the concentrations of CD4+ T, macrophages infected cells and free viruses are decaying and tend to zero when ε = 0.8. The concentration of cytotoxic T lymphocytes (CTLs) immune response is increasing for the values of equal to 0, 0.2, 0.5 and tend to zero when ε equal to 0.8. It means that the numerical results are consistent with the theoretical results that are given in theorem 1,2. We can see from the simulation results that the treatment with such drug efficacy succeeded to eliminate the HIV virus from the blood. Case II: Effect of time delay on the dynamical behavior of the system: In this case, we confirm the effect of delay parameter in pre-treatment case where ε = 0.0. Figures 7-12 show the effect of time delay on the stability of the steady states and the evolution of the uninfected and infected for each CD4+ T cells and macrophages, free virus particles and immune response. We observe that, as time delay is increased from τ = 0.1 to 0.9, E1 still exists and is a globally asymptomatically stable. Moreover the concentrations of uninfected CD4+ T cells and macrophages are increasing for the values of τ except τ = 0.1. The concentrations of CD4+ T, macrophages infected cells and free viruses are decaying with the increasing of time delay values and tend to zero when τ = 0.9. While the concentration of cytotoxic T lymphocytes (CTLs) immune response is increasing for the values of equal to 0.1, 0.3, 0.5 and it tend to zero when τ equal to 0.9. It means that the numerical results are consistent with the theoretical results that are given in theorem 1,2. Moreover from a biological point of view, the intracellular delay plays a similar role as an antiviral treatment in eliminating the virus. Where, sufficiently large delay repress viral replication and works on virus clearance. This awaken us to the significance of medications running on the prolong of intracellular delay period. 9

1000 eps = 0 eps = 0.2 eps = 0.5 eps = 0.8

8

900

7

850

6

Infected CD4 T cells

Uninfected CD4 T cells

950

800 750 700

eps = 0 eps = 0.2 eps = 0.5 eps = 0.8

5 4 3

650

2

600

1 0

550 0

50

100

150

200

250

300

350

400

450

0

500

50

100

150

200

250

300

350

400

450

500

Time(days)

Time(days)

Figure 1: The evolution of uninfected CD4+T cells against time with constant time delay τ = 0.5.

121

Figure 2: The evolution of infected CD4+T cells against time with constant time delay τ = 0.5.

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2.5

600

eps = 0 eps = 0.2 eps = 0.5 eps = 0.8

2 eps = 0 eps = 0.2 eps = 0.5 eps = 0.8

560

Infected macrophages cells

Uninfected macrophages cells

580

540

520

1.5

1

0.5 500

0

480 0

50

100

150

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Figure 3: The evolution of uninfected macrophages cells against time with constant time delay τ = 0.5.

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Figure 4: The evolution of infected macrophages cells against time with constant time delay τ = 0.5.

4

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eps = 0 eps = 0.2 eps = 0.5 eps = 0.8

eps = 0 eps = 0.2 eps = 0.5 eps = 0.8

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3

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Figure 5: The evolution of free viruses against time with constant time delay τ = 0.5.

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Figure 6: The evolution of immune response against time with constant time delay τ = 0.5.

12

1000 950

tau = 0.1 tau = 0.3 tau = 0.5 tau = 0.9

900

tau = 0.1 tau = 0.3 tau = 0.5 tau = 0.9

10

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Uninfected CD4 T cells

250

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Figure 7: The pre-treatment evolution of uninfected CD4+T cells against time.

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Figure 8: The pre-treatment evolution of infected CD4+T cells against time.

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600

4.5 tau = 0.1 tau = 0.3 tau = 0.5 tau = 0.9

4 3.5

Infected macrophages cells

Uninfected macrophages cells

tau = 0.1 tau = 0.3 tau = 0.5 tau = 0.9

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Figure 9: The pre-treatment evolution of uninfected macrophages cells against time.

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Figure 10: The pre-treatment evolution of infected macrophages cells against time.

7

350 tau = 0.1 tau = 0.3 tau = 0.5 tau = 0.9

6

tau = 0.1 tau = 0.3 tau = 0.5 tau = 0.9

300

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Free virus

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Figure 11: The pre-treatment evolution of free viruses against time.

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Figure 12: The pre-treatment evolution of immune response against time.

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5

Conclusion

In this paper, we suggested a distributed delayed human immunodeficiency virus (HIV) models with CTL and two target cells as a system of nonlinear ODES. We demonstrated the positively and boundedness of the solutions and calculate the steady states of the model. Besides we have used suitable Lyapunov functions to set the global asymptotic stability of the steady states. We have derived the basic reproduction number R0 and established that the global dynamics are completely established by the value of the related reproduction number.

References [1] M.A. Nowak, R. Anderson, M. Boerlijst, S. Bonhoeffer, R. May and A. McMichael, HIV-1 evolution and disease progression, Science 274 (1996) 1008-1010. [2] M.A. Nowak and C.R.M. Bangham, Population dynamics of immune responses to persistent viruses, Science 272 (1996) 74-79. [3] A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global stability of HIV infection models with intracellular delays, Journal of the Korean Mathematical Society, 49(4) (2012), 779-794. [4] A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences, 36 (2013), 383-394. [5] A. M. Elaiw, N. H. AlShamrani, Mathematical analysis of a cell mediated immunity in a virus dynamics model with nonlinear infection rate and removal, Journal of Computational Analysis and Applications, 21 (3) (2016), 578-586. [6] A. M. Elaiw, N. H. AlShamrani and M. A. Alghamdi, Global stability analysis of a delayed viral infection model with antibodies and general nonlinear incidence rate, 20 (2) (2016), 277-295. [7] A.V. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May and M.A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proceedings of the National Academy of sciences USA, 93 (1996), 7247-7251. [8] J.E. Mittler, B. Markowitz, D.D. Ho, and A.S. Perelson, Improved estimates for HIV-1 clearance rate and intracellular delay,AIDS, 13 (1999), 1415-1417. [9] P.W. Nelson, J.D. Murray and A.S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Mathematical Bioscience, 163 (2000), 201-215. [10] R.V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell HIV-1 that includes a time delay, Journal of Mathematical Biology 46 (2003) 425-444. [11] J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Mathematical Medicine and Biology, IMA. [12] K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune responses, Physica D 226 (2007) 197-208. [13] H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of HIV infection model with CTL-response delay, Comput. Mathematical Applied, 62 (2011), 3091-3102. [14] M. Y. Li and H. Shu, Global dynamics of a mathematical model for HTLV-1 infection of CD4+ T cells with CTL response, Nonlinear Anal. Real World Applied, 13 (2012), 1080-1092.

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[15] Z. Yuan and X. Zou, Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays, Math. Biosciences and engineering, 10 (2013), 483-498. [16] S. Liu and L. Wang, Global stability of an HTLV-1 model with distributed intracellular delays and combination therapy, Math. Biosciences and engineering, 7 (2010), 675-685. [17] Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, Math. Biosciences and engineering, 375 (2011), 14-27. [18] A.S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz, D.D. Ho, Decay characteristics of HIV-1- infected compartments during combination therapy, Nature 387 (1997) 188-191. [19] A.S. Perelson, P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review 41(1) (1999) 3-44. [20] T.A. Burton, Volterra integral and differential equations,in Mathematics, Science and Engineering, Elsevier, Amsterdam-Boston 202(2) (2005). [21] J. Hale, S.M. Verduyn Lunel, Introduction to Functional differential equations, Springer-Verlag, (1993). [22] P. Nelson, and A.S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infected, Math. Biosciences 179 (2002) 73-94. [23] H. Zhu, and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math Medicine and Biology, IMA 25 (2008) 99-112. [24] H. Zhu, and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Continuous Dynam. Systems-B, 12 (2009) 511-524.

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The pseudo-T -direction and pseudo-Nevanlinna direction of K-quasi-meromorphic mapping ∗ Hong Yan Xua Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

Abstract By applying Ahlfors’ theory of covering surfaces, we prove that for quasi-meromorphic T (r,f ) mapping f satisfying lim supr→∞ (log = +∞, there exists at least one pseudo-T -direction r)2 of f . We also prove that there exists at least one pseudo-Nevanlinna direction of f which is also pseudo-T -direction of f under the same condition. Key words: K-quasi-meomrophic mapping; pseudo-T-direction; pseudo-Nevanlinna direction 2000 Mathematics Subject Classification : 30D 60.

1

Introduction, definitions and results

It is very interesting topic on singular directions of meromorphic functions in the fields of complex analysis([3, 6, 8, 13, 10, 14]), such as Julia direction, Borel direction, T-direction, Hayman direction, and so on. In 1997, Sun and Yang [7] extended the value distribution theory of meromorphic functions (see [3, 13] for standard references) to the corresponding theory of quasi-meromorphic mappings [1, 7]. In fact, for value distribution of quasi-meromorphic mappings f , the singular direction for f is also one of the main research objects. In [7], Sun and Yang obtained an existence theorem of the Borel direction by using the filling disc theorem of quasi-meromorphic mappings. Later, there were some important results about singular directions for quasi-meromorphic mappings. In 1999, Chen and Sun [1] gave the definition of Nevanlinna directions of quasi-meromorphic mappings on the complex plane and proved that there exists at least one Nevanlinna direction for quasi-meromorphic mappings of infinite order by using type function, and they also obtained that the Nevanlinna direction for quasi-meromorphic mappings of infinite order is also one Borel direction with respect to the type function. In 2004, Liu and Yang [4] studied the relationship between the Julia direction and the Nevanlinna direction of quasi-meromorphic mappings by applying a fundamental inequality of quasi-meromorphic mappings on an angular domain. For a meromorphic function f , Zheng [14] introduced a new singular direction called a T direction conjectured that a transcendental meromorphic function f must have at least one T T (r,f ) direction and proved that lim sup (log r)2 = +∞. Later, H. Guo, J. H. Zheng and T. W. Ng [2] r→∞

proved that the conjecture is true by using Ahlfors-Shimizu character T (r, Ω) of a meromorphic function in an angular domain Ω. Xuan [12] studied the existence of T -direction of algebroid function dealing with multiple values. In 2006, Li and Gu [5] proved that there exists at least one T (r,f ) Nevanlinna direction for a K-quasi-meromorphic mapping f under the condition lim sup (log r)2 = r→∞

∗ The

author was supported by the NSF of China(11561033,11561031), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008), and the Foundation of Education Department of Jiangxi of China (GJJ150902).

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+∞. In this paper we will further investigate some new singular direction of K-quasi-meromorphic mapping f . Before stating our main results, we will introduce some definitions and notations, which can be found in [7, 11]. Definition 1.1 (see [7]). Let f be a complex and continuous functions in a region D. If for any rectangle R = {x + iy; a < x < b, c < y < d} in D, f (x + iy) is an absolutely continuous function of y for almost every x ∈ (a, b), and f (x + iy) is an absolutely continuous function of x for almost every y ∈ (c, d), then f is said to be absolutely continuous on lines in the region D. We also call that f is ACL in D. Definition 1.2 (see [7, Definition 1.1]). Let f be a homemorphism from D to D0 . If (i) f is ACL in D, (ii) there exists K ≥ 1 such that f (z) = u(x, y) + iv(x, y) satisfies |fz | + |fz¯| ≤ K(|fz | − |fz¯|) a. e. in D, then f is called an univalent K-quasiconformal mapping in D. If D0 is a region on Riemann sphere V , then f is named an univalent K-quasi-meromorphic mapping in D. Definition 1.3 (see [7, Definition 1.2]) Let f be a complex and continuous function in the region D. For every point z0 in D, if there is a neighborhood U (⊂ D) and a positive integer n depending on z0 , such that  1 f (z0 ) = ∞, (f (z)) n , F (z) = 1 (f (z) − f (z0 )) n + f (z0 ), f (z0 ) 6= ∞. is an univalent K-quasi-meromorphic mapping, then f is named n-valent K-quasi-meromorphic mapping at point z0 . If f is n-valent K-quasi-meromorphic at every point of D, then f is called a K-quasi-meromorphic mapping in D. Let V be the Riemann sphere whose diameter is 1. For any complex number a, let n(r, a) be the number of zero points of f (z) − a in disc |z| < r, counted according to their multiplicities, nl) (r, a) be the number of zeros of f (z) − a with multiplicity ≤ l in disc |z| < r, counted according to their multiplicities. Let Fr be the covering surface f (z) = u(x, y) + iv(x, y) on sphere V and S(r, f ) be the average covering times of Fr to V , S(r, f ) =

|Fr | 1 = |V | π

r

Z

2π

Z

0

0

|fz |2 − |fz¯|2 rdϕdr, (1 + |f |2 )2

where |Fr | and |V | are the areas of Fr and V respectively, Z r S(r, f ) T (r, f ) = dr, r 0 Z r n(t, a) − n(0, a) N (r, a) = dt + n(0, a) log r, t 0 Z r l) n (t, a) − nl) (0, a) l) N (r, a) = dt + nl) (0, a) log r. t 0 Let Ω(ϕ1 , ϕ2 ) = {z ∈ C : ϕ1 < arg z < ϕ2 }(0 ≤ ϕ1 < ϕ2 ≤ 2π), we denote Z Z |Fr | 1 r ϕ2 |fz |2 − |fz¯|2 S(r, ϕ1 , ϕ2 ; f ) = = rdϕdr, |V | π 0 ϕ1 (1 + |f |2 )2 Z T (r, ϕ1 , ϕ2 ; f ) = 0

r

S(r, ϕ1 , ϕ2 ; f ) dr, r

when ϕ1 = 0, ϕ2 = 2π, we note S(r, 0, 2π; f ) = S(r, f ), T (r, 0, 2π; f ) = T (r, f ).

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For any complex number a, let n(r, ϕ1 , ϕ2 ; a) be the number of zero points of f (z) − a in sector Ω(ϕ1 , ϕ2 ) ∩ {z : |z| < r}, counted according to their multiplicities, nl) (r, ϕ1 , ϕ2 ; a) be the number of zeros of f (z) − a with multiplicity ≤ l in sector Ω(ϕ1 , ϕ2 ) ∩ {z : |z| < r}, counted according to their multiplicities. We define Z r n(t, ϕ1 , ϕ2 ; a) − n(0, ϕ1 , ϕ2 ; a) N (r, ϕ1 , ϕ2 ; a) = dt + n(0, ϕ1 , ϕ2 ; a) log r, t 0 Z r l) n (t, ϕ1 , ϕ2 ; a) − nl) (0, ϕ1 , ϕ2 ; a) l) N (r, ϕ1 , ϕ2 ; a) = dt + nl) (0, ϕ1 , ϕ2 ; a) log r. t 0 Next we give the definitions concerning the Nevanlinna direction of K-quasi-meromorphic mappings dealing with multiple values . Definition 1.4 Let f be a K-quasi-meromorphic mapping and l be a positive integer. Then we call δ l) (a, ϕ0 ) the deficiency of the value a in the direction ∆(ϕ0 ): arg z = ϕ0 , 0 ≤ ϕ0 < 2π. We call a the deficiency value of f in the direction ∆(ϕ0 ) if δ l) (a, ϕ0 ) > 0, where δ l) (a, ϕ0 ) = 1 − lim sup lim sup ε→+0

r→∞

N l) (r, ϕ0 − ε, ϕ0 + ε; a) . T (r, ϕ0 − ε, ϕ0 + ε; f )

Definition 1.5 We call ∆(ϕ0 ) : arg z = ϕ0 the pseudo-Nevanlinna direction of f if, for any system aj ∈ C ∪ {∞}(j = 1, 2, . . . , q) of distinct values and any system kj (j = 1, 2, . . . , q) such that kj is a positive integer or +∞ such that  q  X 1 1− > 2, (1) kj + 1 j=1 and

q X j=1

kj δ kj (aj , ϕ0 ) ≤ 2. kj + 1

Similarly, we give the pseudo-T-direction of K-quasi-meromorphic mapping as follows. Definition 1.6 Let f be the K-quasi-meromorphic mapping. A direction B : arg z = ϕ0 (0 ≤ ϕ0 ≤ 2π) is called a T -direction of f if, for any ε(0 < ε < π2 ), and any system aj ∈ C ∪ {∞}(j = 1, 2, . . . , q) of distinct values and any system kj (j = 1, 2, . . . , q) such that kj is a positive integer or +∞ satisfying (2), there exists at least one integer j(1 ≤ j ≤ q) such that lim sup r→∞

N kj ) (r, ϕ0 − ε, ϕ0 + ε, aj ) > 0. T (r, f )

Now, we will give an existence theorem of pseudo-T -direction of K-quasi-meromorphic mapping f as follows. Theorem 1.1 Let f be the K-quasi-meromorphic mapping satisfying lim sup r→∞

T (r, f ) = +∞, (log r)2

(2)

then there exists at least one pseudo-T -direction of f . We also investigate the problem on the relationship between pseudo-Nevanlinna direction and T (r,f ) pseudo-T -direction of f under the condition lim sup (log r)2 = +∞, and obtain the following result: r→∞

Theorem 1.2 Let f be the K-quasi-meromorphic mapping satisfying (2). Then there exists at least one direction which is both one pseudo-Nevanlinna direction of f and one pseudo-T -direction of f .

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

2

Some Lemmas

Let F be a finite covering surface of F1 , F is bounded by a finite number of analytic closed Jordan curves, its boundary is denoted by ∂F . We call the part of ∂F , which lies the interior of F1 , the relative boundary of F , and denote its length by L. Let D be a domain of F1 , its boundary consists of finite number of points or analytic closed Jordan curves, and F (D) be the part of F , which lies above D. We denote the area of F, F1 , F (D) and D by |F |, |F1 |, |F (D)| and |D|, respectively. We call |F (D)| |F | , S(D) = S= |F1 | |D| the mean covering numbering of F relative to F1 , D, respectively. Lemma 2.1 (see [9, Theorem 3]) Let F be a simply connected finite covering surface on the unit sphere V , and let kj (j = 1, 2, . . . , q) be q positive integers. Let Dj (j = 1, 2, . . . , q) be q(≥ 2) disjoint spherical disks with radius δ/3(> 0) on V and without a pair of Dj such that their spherical distance k ) is less than δ and let nj j be the number of simply connected islands in F (Dj ), which consist of not more than kj sheets, then    q  q X X 1 C + 9πh kj kj ) nj ≥  1− − 2 S − L, k +1 kj + 1 δ3 j=1 j=1 j where L is the length of the relative boundary of F . By applying Lemma 2.1, we can get an important inequality of K-quasi-meromorphic mapping in an angular domain as follows. Lemma 2.2 Suppose that f (z) is a K-quasi-meromorphic mapping, and let kj (j = 1, 2, . . . , q) be q positive integers, and {aj } are q(≥ 3) distinct points on V and without a pair of {aj } such that k ) their spherical distance is less than δ + 2δ/3, nj j be the number of zeros of f (z) − aj , which are consisted of not more than kj multiplicities, then    q q  X X 1 kj C + 9πh kj ) 1− nj ≥  − 2 S − L. k + 1 k + 1 δ3 j j=1 j j=1 Lemma 2.3 (see [5, Lemma 2.2]). Let f (z) be a K-quasi-meromorphic mapping on the angular domain Ω(ϕ0 − δ, ϕ0 + δ), a1 , . . . , aq (q ≥ 3) are distinct points on the unit sphere V and the spherical distance of any two points is no smaller than γ ∈ (0, 21 ). Let F0 = V \ {a1 , a2 , . . . , aq }, D = Ω(r, ϕ0 − ϕ, ϕ0 + ϕ) ∩ {z : |z| > 1} \ {f −1 (a1 ), f −1 (a2 ), . . . , f −1 (a1 )} and Dr = D ∩ {z : |z| < r}(r > 1), Fr = f (Dr ) ⊂ V , then for any positive number ϕ satisfying 0 < ϕ < δ, we have  1 √ 1 d(S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )) 2 L(∂f (Dr )) ≤ 2Kπ (log r) 2 dϕ √ √ 1 1 2 + 2Kδrµ (r, ϕ0 − δ, ϕ0 + δ) + 2Kδµ 2 (1, ϕ0 − δ, ϕ0 + δ).

(3)

where Fr is the covering surface of F0 and L(∂f (Dr )) is the length of the relative boundary of Fr relative to F0 , and Z ϕ0 +δ |fz |2 − |fz¯|2 µ(r, ϕ0 − δ, ϕ0 + δ) = rdϕ. iϕ 2 2 ϕ0 −δ (1 + |f (re | )

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Lemma 2.4 Let f (z) be a K-quasi-meromorphic mapping on the angular domain Ω(ϕ0 −δ, ϕ0 +δ), and kj (j = 1, 2, . . . , q) q positive integers. If a1 , . . . , aq (q ≥ 3) are distinct points on the unit sphere V and the spherical distance of any two points is no small than γ ∈ (0, 21 ). Then    q  X 1  1− − 2 S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) k + 1 j j=1 ≤

q X

2C 2 γ −6 π 2 K kj    log r nkj ) (r, ϕ0 − δ, ϕ0 + δ; aj ) + P q 1 k +1 j=1 j j=1 1 − kj +1 − 2 (δ − ϕ)    q  X 1 1 1 1 1 1− + − 2 S(1, ϕ0 − ϕ, ϕ0 + ϕ; f ) + 2Cγ −3 δ 2 K 2 r 2 µ 2 (r, ϕ0 − δ, ϕ0 + δ) kj + 1 j=1 1

1

1

+ 2Cγ −3 δ 2 K 2 µ 2 (1, ϕ0 − δ, ϕ0 + δ)

(4)

and  q  X  1− j=1

≤

1 kj + 1





− 2 T (r, ϕ0 − ϕ, ϕ0 + ϕ; f )

q X

2C 2 γ −6 π 2 K kj    (log r)2 N kj ) (r, ϕ0 − δ, ϕ0 + δ; aj ) + P q 1 k + 1 j j=1 j=1 1 − kj +1 − 2 (δ − ϕ)    q  X 1  + 1− − 2 T (1, ϕ0 − ϕ, ϕ0 + ϕ; f ) k + 1 j j=1    q  X 1 + 1− − 2 S(1, ϕ0 − ϕ, ϕ0 + ϕ; f ) log r k + 1 j j=1 1

1

1

+ 2Cγ −3 δ 2 K 2 µ 2 (1, ϕ0 − δ, ϕ0 + δ) log r + λ(r, ϕ0 − δ, ϕ0 + δ)

(5)

for any ϕ, 0 < ϕ < δ, where C is a constant depending only on {a1 , a2 , . . . , aq }. λ(r, ϕ0 −δ, ϕ0 +δ) = R ϕ +δ |fz |2 −|fz¯|2 1 1 Rr 0 +δ) 1 ) 2 dr, (µ(r, ϕ0 − δ, ϕ0 + δ) = ϕ00−δ (1+|f 2Cγ −3 δ 2 K 2 1 ( µ(r,ϕ0 −δ,ϕ r (reiϕ |2 )2 rdϕ) 1

1

1

1

λ(r, ϕ0 − δ, ϕ0 + δ) ≤ 2Cγ −3 δ 2 π 2 K 2 (T (r, ϕ0 − δ, ϕ0 + δ; f )) 2 log T (r, ϕ0 − δ, ϕ0 + δ; f ) (6) R outside a set Eδ of r at most, where Eδ consists of a series of intervals and satisfies Eδ (r log r)−1 dr < +∞. Proof: Under the condition of Lemma 2.3 and Lemma 2.2, we have S(Dr ) = S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f ).

(7)

Using Lemma 2.1, we easily obtain  q  X  1− j=1

≤

q X j=1

1 kj + 1



 − 2 [S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )]

kj nkj ) (r, ϕ0 − δ, ϕ0 + δ; aj ) + Cγ −3 L(∂(Dr )). kj + 1

130

(8)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

where C is a constant depending only on {a1 , a2 , . . . , aq }. Taking (3) into (8), we have    q  X 1  1− − 2 [S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )] k + 1 j j=1 q X

√ 1 kj nkj ) (r, ϕ0 − δ, ϕ0 + δ; aj ) − Cγ −3 2Kδrµ 2 (r, ϕ0 − δ, ϕ0 + δ) k +1 j=1 j √ 1 − Cγ −3 2Kδµ 2 (1, ϕ0 − δ, ϕ0 + δ)  1 √ 1 d(S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )) 2 ≤Cγ −3 2Kπ (log r) 2 . dϕ −

(9)

We denote  A(r, ϕ) = 

q  X j=1

1 1− kj + 1



 − 2 [S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )]

q X

√ 1 kj nkj ) (r, ϕ0 − δ, ϕ0 + δ; aj ) − Cγ −3 2Kδrµ 2 (r, ϕ0 − δ, ϕ0 + δ) k +1 j=1 j √ 1 − Cγ −3 2Kδµ 2 (1, ϕ0 − δ, ϕ0 + δ). −

(10)

By (9) and (10), we have A(r, ϕ) ≤ Cγ

−3

1  √ 1 d(S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )) 2 2Kπ (log r) 2 . dϕ

(11)

And from (10), it follows that A(r, ϕ) is an increasing function of ϕ. Thus, there exists δ0 > 0, such that A(r, ϕ) ≤ 0 for 0 < ϕ ≤ δ0 and A(r, ϕ) > 0 for ϕ > δ0 . Now, two following cases will be considered: Case 1. For ϕ > δ0 , by (11) we have [A(r, ϕ)]2 ≤ 2C 2 γ −6 Kπ 2

d(S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )) log r. dϕ

(12)

By (10) we have    q  X dA(r, ϕ)  1 d(S(r, ϕ0 − ϕ, ϕ0 + ϕ; f ) − S(1, ϕ0 − ϕ, ϕ0 + ϕ; f )) = 1− − 2 . (13) dϕ k + 1 dϕ j j=1 From (12) and (13) we have 2C 2 γ −6 Kπ 2 log r dA(r, ϕ)   [A(r, ϕ)]2 ≤ P · , q 1 dϕ 1 − − 2 j=1 kj +1 i.e., 2C 2 γ −6 Kπ 2 log r dA(r, ϕ)   dϕ ≤ P · . q 1 [A(r, ϕ)]2 j=1 1 − kj +1 − 2 For the above inequality, by integrating its two sides, we have Z δ Z δ 2C 2 γ −6 Kπ 2 log r dA(r, ϕ) 2C 2 γ −6 Kπ 2 log r 1     ≤ · . δ−ϕ= dϕ ≤ P P 2 q q 1 1 [A(r, ϕ)] A(r, ϕ) ϕ ϕ j=1 1 − kj +1 − 2 j=1 1 − kj +1 − 2

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Thus

2C 2 γ −6 Kπ 2 log r    . 1 − 2 (δ − ϕ) 1 − j=1 kj +1

A(r, ϕ) ≤ P q

(14)

Case 2. Because A(r, ϕ) ≤ 0 when 0 < ϕ ≤ δ0 , the above inequality also holds. From Case 1 and Case 2, we can easily get 2C 2 γ −6 Kπ 2 log r    , 1 1 − − 2 (δ − ϕ) j=1 kj +1

A(r, ϕ) ≤ P q

for any ϕ, 0 < ϕ < δ. Thus, from (10) we can get (4) easily. Then, by dividing r and integrating from 1 to r on each sides of (4) we get    q  X 1  1− − 2 T (r, ϕ0 − ϕ, ϕ0 + ϕ; f ) k + 1 j j=1 ≤

q X

2C 2 γ −6 π 2 K kj    N kj ) (r, ϕ0 − δ, ϕ0 + δ; aj ) + P (log r)2 q 1 k + 1 j 1 − − 2 (δ − ϕ) j=1 j=1 kj +1    q  X 1 + 1− − 2 T (1, ϕ0 − ϕ, ϕ0 + ϕ; f ) kj + 1 j=1    q  X 1 + 1− − 2 S(1, ϕ0 − ϕ, ϕ0 + ϕ; f ) log r kj + 1 j=1 1

1

1

1

1

+ 2Cγ −3 δ 2 K 2 µ 2 (1, ϕ0 − δ, ϕ0 + δ) log r + 2Cγ −3 δ 2 K 2

Z

r

1



µ(r, ϕ0 − δ, ϕ0 + δ) r

 12 dr.

From the definitions of S(r, ϕ1 , ϕ2 ; f ), µ(r, ϕ0 − δ, ϕ0 + δ) and λ(r, ϕ0 − δ, ϕ0 + δ), and Schwarz’s inequality we get "Z   1 #2 r µ(r, ϕ0 − δ, ϕ0 + δ) 2 2 2 −6 dr (λ(r, ϕ0 − δ, ϕ0 + δ)) = 4C γ δK r 1 Z r Z r ≤ 4C 2 γ −6 δK µ(r, ϕ0 − δ, ϕ0 + δ)dr r−1 dr 1 1 Z r 2 −6 ≤ 4C γ πδK log r dS(r, ϕ0 − δ, ϕ0 + δ; f ) 1 2 −6

≤ 4C γ

πδKS(r, ϕ0 − δ, ϕ0 + δ; f ) log r

= 4C 2 γ −6 πδK

dT (r, ϕ0 − δ, ϕ0 + δ; f ) r log r. dr

(15)

Choosing r0 , r0 > 0 such that T (r0 , ϕ0 − δ, ϕ0 + δ; f ) > 1, and setting Eδ = {r0 < r < ∞ : (λ(r, ϕ0 − δ, ϕ0 + δ))2 > 4C 2 γ −6 πδKT (r, ϕ0 − δ, ϕ0 + δ; f )(log T (r, ϕ0 − δ, ϕ0 + δ; f ))2 }, thus we have Z Z dr dT (r, ϕ0 − δ, ϕ0 + δ; f ) ≤ 2 Eδ r log r Eδ T (r, ϕ0 − δ, ϕ0 + δ; f )[log T (r, ϕ0 − δ, ϕ0 + δ; f )] ≤ [log T (r0 , ϕ0 − δ, ϕ0 + δ; f )]−1 < +∞. Then for r > r0 and r 6∈ Eδ , we have (5). Thus, the proof of Lemma 2.4 is completed.

(16) 2

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Lemma 2.5 (see [11, Lemma 2.4] or [12]). Let F (r) be a positive nondecreasing function defined for 1 < r < +∞ and satisfy F (r) = +∞. (17) lim sup 2 r→∞ (log r) R dr 1 Then, for any subset E ⊂ (1, +∞) satisfying E r log r < ℘ (℘ ≥ 2), lim sup r→∞,r∈(1,+∞)\E

F (r) = +∞. (log r)2

Lemma 2.6 Let f (z) be the K-quasi-meromorphic mapping and m(m > 1) be a positive integer. 2π Put ϕ0 = 0, ϕ1 = 2π m , . . . , ϕm−1 = (m − 1) m . Let   2π ∆(ϕi ) = z | | arg z − ϕi | < (0 ≤ i ≤ m − 1). m Then among these m angular domains {∆(ϕi )}, there exists at least an angular domain ∆(ϕi ) such that for any system aj (j = 1, 2, . . . , q) of distinct values and any system kj (j = 1, 2, . . . , q such that kj is a positive integer or +∞ and that q  X 1− j=1

1 kj + 1

 > 2,

there exists at least one integer j(1 ≤ j ≤ q) such that lim sup r→∞

N kj ) (r, ∆(ϕi ), a) > 0. T (r, f )

Proof: Suppose that the conclusion is false. Then for any ∆(ϕi )(i = 0, 1, . . . , m − 1), there is a system aij (j = 1, 2, . . . , q) of distinct values and a system kji (j = 1, 2, . . . , q) such that kji is a positive integer or +∞ and that q X 1 (1 − i ) > 2, k j +1 j=1 for any j(1 ≤ j ≤ q), we have i

N kj ) (r, ∆(ϕi ), aij ) = 0. lim sup T (r, f ) r→∞

(18)

2kπ Let β be any positive integer. Put ϕi,k = 2π m i + βm , 0 ≤ i ≤ m − 1, 0 ≤ k ≤ β − 1. For any given number r > 1, writing ∆i,k (r) = {z||z| < r, ϕi,k < ϕi,k+1 } .

Then {|z| < r} =

β−1 X m−1 X

∆i,k (r).

k=0 i=0

Put

  ϕi+1,0 + ϕi+1,1 ϕi,0 + ϕi,1 ∆i = z| ≤ argz ≤ , 2 2 ∆0i = {z|ϕi,0 < argz < ϕi+1,1 }, 0 ≤ i ≤ m − 1,  !  q  q X  X 1 1 > 2. 1− = min 1− i 1≤i≤m  kj + 1 kj + 1  j=1

j=1

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From Lemma 2.4 we have    q q  X X kji i 1 1 1  nkj ) (r, ∆0i , aij )+O(log r)+hi r 2 µ 2 (r, ϕi,0 , ϕi+1,1 ). 1− − 2 S(r, ∆i , f ) ≤ i kj + 1 k +1 j=1 j j=1 Add from i = 0 to m − 1 andP divide both sides of this inequality by r and integrate both sides m−1 from 1 to r, and since T (r, f ) = i=0 T (r, ∆i , f ), then the following inequality can be obtained    q  X 1  1− − 2 T (r, f ) k + 1 j j=1 ≤

q m−1 XX i=0 j=1

m−1 X kji kji ) 0 i 2 N (r, ∆ , a ) + O((log r) ) + λ(r, ϕi,0 , ϕi+1,1 ), i j i kj + 1 i=0

(19)

where 1 1 2π 1 2 (1 + ) (T (r, ϕi,0 , ϕi+1,1 ; f )) 2 log T (r, ϕi,0 , ϕi+1,1 ; f ) m β R at most outside a set Ei of r, where Ei satisfies Ei (r log r)−1 dr < +∞(i = 0, 1, . . . , m − 1). For any i ∈ {0, 1, . . . , m−1} and ℘ ≥ 2, there exists ri > 0 such that T (ri , ϕi,0 , ϕi+1,1 ; f ) > e℘m for r > ri . Then it follows from (16) that Z 1 1 1 1 dr ≤ < < . r log r log T (r, ϕ , ϕ ; f ) ℘m ℘ i,0 i+1,1 Ei 

λ(r, ϕi,0 , ϕi+1,1 ) ≤ hi

Put E = ∪m−1 i=0 Ei , then Z E

Z m−1 XZ 1 1 1 1 1 dr ≤ dr ≤ m max dr < m · < . 0≤i≤m−1 r log r r log r ℘m ℘ Ei r log r i=0 Ei

By applying Lemma 2.5 to this set E and T (r, f ), we obtain that lim sup r→∞,r∈(1,∞)\E

T (r, f ) = +∞. (log r)2

There exists {rn } ∈ (r, +∞)\E, lim

n→∞

T (rn , f ) = +∞. (log rn )2

For this sequence {rn }, by (19) we have    q  X 1  1− − 2 T (rn , f ) k + 1 j j=1 ≤

q m−1 XX i=0 j=1

m−1 X kji kji ) 0 i 2 N (r , ∆ , a ) + O((log r ) ) + λ(rn , ϕi,0 , ϕi+1,1 ). n n i j i kj + 1 i=0

From (18), by dividing both sides of the above inequality by T (rn , f ) and letting n → ∞, we obtain   Pq  Pq  1 1 j=1 1 − kj +1 − 2 ≤ 0 that is, j=1 1 − kj +1 ≤ 2, a contradict. Thus, this completes the proof of Lemma 2.6. 2

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3

The Proof of Theorem 1.1

Proof: By Lemma 2.6, we can choose subsequence of {θm }, assume that θm → θ0 when m → ∞. Then B : arg z = θ0 is a pseudo-T-direction of f . In fact, for any ε(0 < ε < π2 ), when m is sufficiently large, we have ∆(θm ) ⊂ Ω(θ0 , ε). By Lemma 2.6, we have lim sup r→∞

N kj ) (r, θ0 , ε, aj ) N kj ) (r, ∆(θm ), aj ) ≥ lim sup > 0. T (r, f ) T (r, f ) r→∞ 2

Thus, we complete the proof of Theorem 1.1.

4

The Proof of Theorem 1.2

Proof: Suppose δ ∈ (0, 2π), we can choose r0 > 0 such that T (r0 , ϕ0 − δ, ϕ + δ; f ) > e℘ . Then it follows from (16) that Z 1 1 1 dr ≤ < . r log r log T (r , ϕ − δ, ϕ + δ; f ) ℘ 0 0 Eδ By applying Lemma 2.4 for the set Eδ and T (r, f ), it follows that lim sup r→∞,r∈(1,∞)\Eδ

T (r, f ) = +∞. (log r)2

So, there exists a sequence {rn } ∈ (r, +∞)\Eδ , lim

n→∞

T (rn , f ) = +∞. (log rn )2

(20)

By applying the finite covering theorem at [0, 2π], there exists some ϕ0 such that ϕ0 ∈ [0, 2π] and lim sup n→∞

T (rn , ϕ0 − ϕ, ϕ0 + ϕ; f ) >0 T (rn , f )

(21)

for an arbitrary ϕ, 0 < ϕ < ϕ0 . Thus, we will prove that the direction ∆(ϕ0 ) : arg z = ϕ0 is one pseudo-Nevanlinna direction of f (z) which is also the pseudo-T -direction of f (z). Step one. We firstly prove that the direction ∆(ϕ0 ) : arg z = ϕ0 is one pseudo-Nevanlinna direction of f (z). Otherwise, for an arbitrary positive number ε0 > 0, there exists a system aj ∈ C ∪ {∞}(j = 1, 2, . . . , q) of distinct values and a system kj (j = 1, 2, . . . , q) such that kj is a positive integer or +∞ and that  q  X 1 1− > 2, (22) kj + 1 j=1 the following inequality holds q X j=1

kj δ kj (aj , ϕ0 ) > 2 + ε0 . kj + 1

From the definition of δ kj (aj , ϕ0 ), we get lim sup lim sup ϕ→+0

r→+∞

q X j=1

q

kj N kj ) (r, ϕ0 − ϕ, ϕ0 + ϕ; aj ) X 1 < (1 − ) − 2 − ε0 . kj + 1 T (r, ϕ0 − ϕ, ϕ0 + ϕ; f ) k +1 j j=1

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Thus, there exists some ϕ0 (> 0), and for any ϕ, 0 < ϕ < ϕ0 , we have lim sup r→+∞

q X j=1

q

kj N kj ) (r, ϕ0 − ϕ, ϕ0 + ϕ; aj ) X 1 < (1 − ) − 2 − ε0 . kj + 1 T (r, ϕ0 − ϕ, ϕ0 + ϕ; f ) kj + 1 j=1

(23)

Then for any ϕ, 0 < ϕ < ϕ0 , set T (ϕ) = lim sup n→+∞

T (rn , ϕ0 − ϕ, ϕ0 + ϕ; f ) . T (rn , f )

(24)

Obviously, T (ϕ) is an increasing function in interval [0, ϕ0 ]. From (21) we have T (ϕ) > 0. So, 0 < T (ϕ) ≤ 1. Since the increasing of T (ϕ) in interval [0, ϕ0 ] and the continuous theorem for monotonous functions, we can see that all discontinuous points of T (ϕ) constitute a countable set at most. Then, by Lemma 2.4, we can get   q X 1  (1 − ) − 2 T (rn , ϕ0 − ϕ, ϕ0 + ϕ; f ) k + 1 j j=1 ≤

q X j=1

kj N kj ) (rn , ϕ0 − δ, ϕ0 + δ; aj ) + O(log rn )2 kj + 1 1

+ O((T (rn , ϕ0 − δ, ϕ0 + δ; f )) 2 log T (rn , ϕ0 − δ, ϕ0 + δ; f )) for 0 < ϕ < δ < ϕ0 and rn 6∈ Eδ . Thus, it follows from (23)-(25) that     q q X X  (1 − 1 ) − 2 T (ϕ) <  (1 − 1 ) − 2 − ε0  T (δ). kj + 1 kj + 1 j=1 j=1

(25)

(26)

Then, we get from (26) T (ϕ) → T (δ),

ϕ → δ.

(27)

By combining (26) with (27), we can obtain T (δ) = 0, which is a contradiction to T (δ) > 0. Then ∆(ϕ0 ) : argz = ϕ0 is the pseudo-Nevanlinna direction of f (z). Step two. We will prove that ∆(ϕ0 ) : argz = ϕ0 is the pseudo-T -direction of f (z). Otherwise, there exists ε0 > 0 and there is a system aj (j = 1, 2, . . . , q) of distinct values and a system kj (j = 1, 2, . . . , q) such that kj is a positive integer or +∞ and that q X (1 − j=1

1 ) > 2, kj + 1

for any j(1 ≤ j ≤ q), we have lim sup r→∞

N kj ) (r, ϕ0 − ε0 , ϕ0 + ε0 , aj ) = 0. T (r, f )

Then there exists a sequence {rn } such that N kj ) (rn , ϕ0 − ε0 , ϕ0 + ε0 , aj ) = 0. r→∞ T (rn , f ) lim

(28)

For ϕ ∈ (0, ε0 ), similar to (24), we define T (ϕ), then 0 < T (ϕ) ≤ 1. By Lemma 2.4, for the above sequence {rn } ⊂ (1, +∞)\Eδ and 0 < ϕ < ϕ0 < δ, we have   q q X X kj 1  (1 −  T (rn , ϕ0 −ϕ, ϕ0 +ϕ; f ) ≤ ) − 2 N kj ) (rn , ϕ0 −ε0 , ϕ0 +ε0 ; aji )+O((log rn )2 ) i k + 1 k + 1 j j j=1 j=1

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1

(29) +O((T (rn , ϕ0 − ε0 , ϕ0 + ε0 ; f )) 2 log T (rn , ϕ0 − ε0 , ϕ0 + ε0 ; f )). Pq By (21), (28), (29) and j=1 (1 − kj1+1 ) > 2, we can obtain T (ϕ) ≤ 0 which is a contradiction with T (ϕ) > 0. Therefore ∆(ϕ0 ) : argz = ϕ0 is the pseudo-T -direction of f (z). Thus, this completes the proof of Theorem 1.2. 2

References [1] T. W. Chen, D. C. Sun, Singular directions of quasi-meromorphic mappings, Acta Math. Sci. Ser. A Chin. Ed. 19(4) (1999): 472-478. [2] H. Guo, J. H. Zheng, T. W. Ng, On a new singular direction of meromorphic functions, Bull. Austral. Math. Soc. 69 (2004): 277-287. [3] W. Hayman, Meromorphic Functions, Clarendon, Oxford,1964. [4] M. S. Liu, Y. Yang, The Nevanlinna direction and Julia direction of quasi-meromorphic mappings, Acta Math. Sci. Ser. A Chin. Ed. 24(5) (2004): 578-582. [5] C. H. Li, Y. X. Gu, A fundamental inequality for K-quasi-meromorphic mappings in an angular domain and its application, Acta Math. Sinica Ser. A 49(6) (2006): 1279-1287. [6] W. C. Lu, On the λ∗ -Logarithmic Type of Analytic Functions epresented by Laplace-Stieltjes Transformation, J. Jiangxi Norm. Univ., Nat. Sci. 40 (6) (2016), 591-594. [7] D. C. Sun, L. Yang, Value distribution of K-quasi-meromorphic mappings, Sci. China Ser. A 27(2) (1997): 132-139. [8] J. Wang, K. Xia, F. Long, The Poles of Meromorphic Solutions of Fermat Type Differential-Difference Equations, J. Jiangxi Norm. Univ., Nat. Sci. 40 (5) (2016), 497-499. [9] Z. J. Wu, Y. Q. Chen, Z. X. Xuan, An inequality of meromorphic functions and its application, The Scientific World Journal 2014(2014) Art. 242851, 9 pages. [10] Z. J. Wu, B. Wang, The Characteristic Function of E-Valued Meromorphic Functions, J. Jiangxi Norm. Univ., Nat. Sci. 40 (5) (2016), 500-504. [11] H. Y. Xu, T. S. Zhan, On the existence of T -direction and Nevanlinna direction of K-quasimeromorphic mapping dealing with multiple values, Bull. Malays. Math. Sci. Soc. 33 (2) (2010): 281-294. [12] Z. X. Xuan, On the existence of T -direction of algebroid functions: A problem of J.H. Zheng, J. Math. Anal. Appl. 341 (1) (2008): 540-547. [13] L. Yang, Value distribution theory, Springer-Verlag, Belin,1993. [14] J. H. Zheng, On transcendental meromorphic functions with radially distributed values, Sci. China 47 (2004): 401-416.

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Some inequalities on small functions and derivatives of meromorphic functions on annuli ∗ Hua Wanga and Hong-Yan Xub† a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, P.R. China

b

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China email: [email protected]

Abstract In this paper, we firstly establish the second main theorem about meromorphic functions on annuli concerning small functions. Then, by using this theorem, we deal with the uniqueness of meromophic functions sharing some small functions on annuli and obtain the results of meromporphic functions sharing five small functions on annuli, which is an answer to the question of Cao and Yi. In addition, we investigate the properties of meromorphic functions on annuli, and obtain a form of Yang’s inequality on annuli by reducing the coeffcients of Hayman’s inequality. Moreover, we also study Hayman’s inequality in different forms, and obtain accurate estimates of sums of deficiencies. Key words: Small function, Nevanlinna theory, the annulus. Mathematical Subject Classification (2010): 30D30, 30D35.

1

Introduction

Firstly, we always assumed that the reader is familiar with the notations of the Nevanlinna theory such as T (r, f ), m(r, f ), N (r, f ) and so on (see [6, 22, 23]). In 1920s, R. Nevanlinna (see [17]) first established the famous Nevanlinna characteristic of meromorphic functions. It is well known that the Nevanlinna characteristic is powerful, and Nevanlinna theory of value distribution play an important role in the research of complex analysis, which has been used to deal with various complex problems, such as: complex differential equation, complex difference equation, uniqueness of meromorphic functions, complex dynamic systems, etc. Among many basic theorems in Nevanlinna theory, the second main theorem is very important to study the value distribution, uniqueness, singular direction, which is listed as follows. Theorem 1.1 (see [6, 23]). Let f (z) be a non-identically-constant meromorphic function, let a1 , . . . , aq be distinct complex numbers, one of which can be equal to ∞. Then q X j=1

m(r,

1 ) < 2T (r, f ) − N1 (r, f ) + S(r, f ), f − aj

∗ The

authors were supported by the NSF of China (11561033,11561031), the Natural Science Foundation of Jiangxi Province in China (20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ150902) of China. † Corresponding author.

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(q − 2)T (r, f )
0 (q − 2 − ε)T (r, f )
0. Then for p = 1, 2, . . . ., we have   1 (p − 1)N 0 (r, f ) ≤ (1 + ε)N0 r, (p) + (1 + ε)N01 (r, f ) + S1 (r, f ), (1) f where N01 (r, f ) = N0 (r, f ) − N 0 (r, f ). Proof: For any given ε > 0 and positive integer n, we choose a positive integer n(> pε ), and consider for all z ∈ A. Let W (z) = W (1, z, z 2 , · · · , z p+n−1 , f, zf, · · · , z n f ) as the Wronskian determinant of 1, z, z 2 , · · · , z p+n−1 , f, zf, · · · , z n f . Since f is a transcendental meromorphic function, we can suppose that W (z) 6= 0. It is easy to see that W (z) is a homogeneous differential polynomial of degree p + 1 in f with polynomial coefficients of z and without f (j) (z)(j < p) in each term of W (z). Let B(z) = W (z) · (f (p) (z))−n−1 , from Lemma 3.1, it follows m0 (r, B) = S1 (r, f ). From the first fundamental theorem for meromorphic function on annuli, we have N0 (r,

1 ) ≤ T0 (r, B) + O(1) = N0 (r, B) + m0 (r, B) + O(1) B ≤ N0 (r, B) + S1 (r, f ).

(2)

Next, we will estimate the number of zeros and poles of B on A. From the definition of W (z), we have W (z) = f p+2n+1 W (f −1 , zf −1 , · · · , z p+n−1 f −1 , 1, z, · · · , z n ). If z0 is a pole of f of order t, then W (z) = O((z − z0 )−t(p+2n+1) ),

z → z0 .

Hence   B(z) = O (z − z0 )(n+1)(p+t)−t(p+2n+1)   = O (z − z0 )n(p−1)−(p+n)(t−1) ,

143

(3)

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as z → z0 . 0 ∞ ∗ Let N t (r), N t (r) and N t (r) be the counting functions for those poles of f of order t on A, where B(z) has a zero, pole or finite nonzero value, respectively, each pole being counted only once. From (2) and (3), we get ∞ X 1 0 (n(p − 1) − (p + n)(t − 1))N p (r) ≤ N0 (r, ) ≤ N0 (r, B) + S1 (r, f ) B t=1

≤

∞ X ∞ ((p + n)(t − 1) − n(p − 1))N t (r) t=1

  1 + (n + 1)N0 r, (p) + S1 (r, f ). f

(4)

∗

If a pole of f contributes to N t (r), then from (3) it follows n(p − 1) − (p + n)(t − 1) ≤ 0 and ∗

∗

n(p − 1)N t (r) ≤ (p + n)(t − 1)N t (r). Summing for t = 1, 2, · · · in above and substituting to (4), we obtain n(p − 1)

∞ X

N t (r) ≤(p + n)

t=1 ∞

0

  ∞ X 1 (t − 1)N t (r) + (n + 1)N0 r, (p) + S1 (r, f ), f t=1

(5)

∗

where N t (r) = N t (r) + N t (r) + N t (r). Noting ∞ ∞ X X   (t − 1)N t (r) = tN t (r) − N t (r) t=1

t=1 ∞ X   = Nt (r) − N t (r) = N0 (r, f ) − N 0 (r, f ), t=1

since n > pε and (5), we have proved Lemma 3.3. 2 By using the same argument as in the proof of Lemma 3.3, we can get the following lemma. Lemma 3.4 Let f (z) and aj (z)(j = 1, 2, . . . , p; p ≥ 3) be stated as in Theorem 2.1. Set W (f ) = W (a1 (z), a2 (z), . . . , ap (z), f (z)). If aj (z)(j = 1, 2, . . . , p; p ≥ 3) are linearly independent, then for ε > 0, we have   1 pN 0 (r, f ) ≤ N0 r, + (1 + ε)N0 (r, f ) + S1 (r, f ). W (f ) Lemma 3.5 Let f be a transcendental or admissible meromorphic function on the annulus A = {z : 0 < |z| < ∞}. Then for any ε > 0 and positive integer k, we have   1 1 1 (6) N 0 (r, f ) < N0 r, (k) + N0 (r, f ) + εT0 (r, f ) + S1 (r, f ). k k f Proof: Replacing ε with

ε 3

in Lemma 3.3, it follows     ε 1 1 1 1 N 0 (r, f ) < N0 r, (k) + N0 (r, f ) + N0 r, (k) k k 3k f f ε + N0 (r, f ) + S1 (r, f ). 3k

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Since   1 ≤ T0 (r, f (k) ) + O(1) N0 r, (k) f   f (k) + m0 (r, f ) + N0 (r, f (k) ) + O(1) ≤ m0 r, f ≤ m0 (r, f ) + N0 (r, f ) + kN 0 (r, f ) + S1 (r, f ) ≤ (k + 1)T0 (r, f ) + S1 (r, f ), from (7), we have ε ε k+2 1
N0 r, (k) + N0 (r, f ) ≤ εT0 (r, f ) + S1 (r, f ) 3k 3k 3k f ≤ εT0 (r, f ) + S1 (r, f ).

(8)

2

From (7) and (8), we get (6) easily.

Lemma 3.6 (see [7, Theorem 2]). Let f be a transcendental or admissible meromorphic function on the annulus A = {z : 0 < |z| < ∞}, and k be a positive integers. Then T0 (r, f ) < N 0 (r, f ) + N0 (r,

1 1 1 ) + N0 (r, (k) ) − N0 (r, (k+1) ) + S1 (r, f ). f f −1 f

Lemma 3.7 Let f be a transcendental or admissible meromorphic function on the annulus A = {z : 0 < |z| < ∞}. Then for any ε > 0 and positive integer k, we have     1 1 1 1 N 0 (r, f ) < N0 r, + N0 r, (k) + εT0 (r, f ) + S1 (r, f ). (9) k f k f −1 Proof: From Lemma 3.5 we have   k+1 1 N0 r, (k+1) > (k + 1)N 0 (r, f ) − N0 (r, f ) − εT0 (r, f ) − S1 (r, f ). 2 f Substituting the above inequality back into Lemma 3.6, we obtain     1 1 kN 0 (r, f ) < N0 r, + N0 r, (k) + (N0 (r, f ) − T0 (r, f )) f f −1 k+1 + εT0 (r, f ) + S1 (r, f ). 2 Therefore

N 0 (r, f ) <
0 and positive integer k, we have     1 1 1 1 N 0 (r, f ) < N0 r, (k) N0 r, (k) + + εT0 (r, f ) + S1 (r, f ). 2k 2k f −a f −b Proof: By using Lemma 3.8 for f (k) and three distinct complex numbers a, b, ∞, we have         1 1 (k) (k) + N0 r, (k) ≤N0 r, f T0 r, f + N0 r, (k) f −a f −b   − N00 (r) + S1 r, f (k) ,  

1 where N00 (r) = 2N0 r, f (k) − N0 r, f (k+1) + N0 r, f (k+1) . Thus, we get       1 1 T0 r, f (k) ≤N 0 (r, f ) + N0 r, (k) + N0 r, (k) f −a f −b     1 − N0 r, (k+1) + S1 r, f (k) . f

(10)

Since T0 (r, f (k) ) = m0 (r, f (k) ) + N0 (r, f ) + kN 0 (r, f ), then by applying Lemma 3.7 for f (k+1) , it follows   1 N0 r, (k+1) >(k + 1)N 0 (r, f ) − N0 (r, f ) − (k + 1)εT0 (r, f ) − (k + 1)S1 (r, f ). f Substituting the two above inequalities back into (10) , we get     1 1 k+1 1 1 N 0 (r, f ) < + + N0 r, (k) N0 r, (k) εT0 (r, f ) 2k 2k 2k f −a f −b k+2 + S1 (r, f (k) ) 2k     1 1 1 1 N0 r, (k) N0 r, (k) < + + ε0 T0 (r, f ) 2k 2k f −a f −b +S1 (r, f (k) ). From the definition of S1 (r, ∗) and T0 (r, f ) ≤ T0 (r, f (k) ) ≤ (k + 1)T0 (r, f ) + S1 (r, f ), where S1 (r, f ) is as stated in Lemma 3.1, we can get the conclusion of Lemma 3.9. Thus, we can complete the proof of Lemma 3.9. 2

4 4.1

Proofs of Theorem 2.1 and Theorem 2.2 The proof of Theorem 2.1

Without any loss of generalities, suppose that {a1 (z), a2 (z), . . . , ap (z)} is a maximum linearly independent subset of aj (z)(j = 1, 2, . . . , q), then p ≤ q and each aj (z)(j = 1, 2, . . . , q) can be linearly expressed in terms of aj (z)(j = 1, 2, . . . , p). Set W (f ) = W (a1 , a2 , . . . , ap (z), f ), then W (f ) = bp f (p) + bp−1 f (p−1) + · · · + b1 f 0 + b0 f,

(11)

where bj (j = 1, 2, . . . , p) are small functions with respect to f . It follows from (11) that N0 (r, W (f )) = pN 0 (r, f ) + N0 (r, f ) + S1 (r, f )

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and m0 (r, W (f )) ≤ m0 (r, f ) + m0 (r,

W (f ) ) = m0 (r, f ) + S1 (r, f ). f

(13)

Thus from (12), (13) it follows that T0 (r, W (f )) ≤ pN 0 (r, f ) + T0 (r, f ) + S1 (r, f ).

(14)

From the definition of W (f ), we have W (f − aj ) = W (f ) for j = 1, 2, . . . , q. Thus, it follows by Lemma 3.1 that W (f ) W (f − aj ) m0 (r, ) = m0 (r, ) = S1 (r, f ). (15) f − aj f − aj Set q X 1 F (z) = . f (z) − aj (z) j=1 Then it follows from (14),(15) and by Lemma 3.4 that 1 ) + m0 (r, F W (f )) W (f ) 1 ) + S1 (r, f ) ≤ T0 (r, W (f )) − N0 (r, W (f ) 1 ≤ pN 0 (r, f ) + T0 (r, f ) − N0 (r, ) + S1 (r, f ) W (f ) ≤ T0 (r, f ) + (1 + ε)N0 (r, f ) + S1 (r, f ).

m0 (r, F ) ≤ m0 (r,

Then it follows by Lemma 3.2 that qT0 (r, f ) = T0 (r, F ) + S1 (r, f ) ≤

q X

N0 (r,

1 ) + T0 (r, f ) + (1 + ε)N0 (r, f ) + S1 (r, f ) f − aj

N0 (r,

1 ) + (2 + ε)T0 (r, f ) + S1 (r, f ). f − aj

j=1

≤

q X j=1

Hence, this completes the proof of Theorem 2.1.

4.2

The proof of Theorem 2.2

Suppose f (z) 6≡ g(z). By applying Theorem 2.1, since f and g share a1 , . . . , a5 CM , we have (3 − ε)T0 (r, f ) ≤

5 X j=1

N0 (r,

1 1 ) + S1 (r, f ) ≤ N0 (r, ) + S1 (r, f ) f − aj f −g

≤ T0 (r, f ) + T0 (r, g) + S1 (r, f ), that is, (2 − ε)T0 (r, f ) ≤ T0 (r, g) + S1 (r, f ).

(16)

(2 − ε)T0 (r, g) ≤ T0 (r, f ) + S1 (r, g).

(17)

Similarly, we have Thus for any small number ε(> 0), it follows from (16) and (17) that (1 − ε) [T0 (r, f ) + T0 (r, g)] ≤ S1 (r, f ) + S1 (r, g), which is a contradiction with the assumption that f, g are transcendental or admission. Therefore, we have f ≡ g.

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5

Proofs of Theorem 2.3 and Theorem 3.4

5.1

The proof of Theorem 2.3

From Lemma 3.4 and Lemma 3.7, we get         1 1 1 1 T0 (r, f ) < 1+ N0 r, + 1+ N0 r, (k) k f k f −1   1 −N0 r, (k+1) + εT0 (r, f ) + S1 (r, f ). f Now, we will prove the inequality of the sum of deficiencies as follows. First, by using the above inequality for the function f −a b , then it follows           1 1 1 1 1 N0 r, + 1+ N0 r, (k) − N0 r, (k+1) T0 (r, f ) < 1 + k f −a k f −b f + εT0 (r, f ) + S1 (r, f )         1 1 1 1 < 1+ N0 r, + 1+ N0 r, (k) + εT0 (r, f ) + S1 (r, f ). k f −a k f −b Dividing the both sides of the above inequality by T0 (r, f ), we have      1 1   N N r, r, 0 0 (k) f −a 1  f −b  < 1 + 2 + ε + S1 (r, f ) . 1+ 1− +1− k T0 (r, f ) T0 (r, f ) k T0 (r, f )

(18)

From the definitions of δ0 (a, f ), δ0k (a, f (k) ), then it follows from (18) that    1  δ0 (a, f ) + δ0k (b, f (k) ) 1+ k      1 1   N N r, r, 0 0 (k) f −a 1 f −b  ≤ 1+ +1− lim inf 1 − k r→∞ T0 (r, f ) T0 (r, f )   S1 (r, f ) 2 . ≤ lim sup 1 + + ε + lim inf r→∞ k T0 (r, f ) r→∞ Since f is a transcendental or admission on annuli, we have lim

r→∞

S1 (r, f ) = 0. T0 (r, f )

(19)

Hence, δ0 (a, f ) + δ0k (b, f (k) ) ≤

k+2 . k+1

Thus, this completes the proof of Theorem 2.3.

5.2

The proof of Theorem 2.4

By Lemma 3.9, it follows from (10) that           1 1 1 1 (k) T0 r, f ≤ 1+ N0 r, (k) + 1+ N0 r, (k) 2k 2k f −a f −b     1 −N0 r, (k+1) + εT0 (r, f ) + S1 r, f (k) . f

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The above inequality implies      1 1   N r, N r, (k) 0 0 (k) (k) 1  f −a f −b  < 1 + 1 + ε + S1 (r, f ) . − 1+ 2− (k) (k) (k) 2k k T0 (r, f ) T0 (r, f ) T0 (r, f )

(20)

Thus, it follows from (20) and the definition of δα,β (a, f ) that    1  δ0 (a, f (k) ) + δ0 (b, f (k) ) 1+ 2k      1 1   N r, N r, 0 0 1 f (k) −a f (k) −b  ≤ 1+ lim inf 1 − +1− (k) 2k r→∞ T0 (r, f ) T0 (r, f (k) )   S1 (r, f (k) ) 1 . ≤ lim sup 1 + + ε + lim inf r→∞ T0 (r, f (k) ) k r→∞ Since f is a transcendental or admission on annuli, we have the following equalities easily lim inf r→∞

S1 (r, f (k) ) = 0. T0 (r, f (k) )

(21)

Since ε is arbitrary, it follows from (21) that δ0 (a, f (k) ) + δ0 (b, f (k) ) ≤ 1 +

1 . 2k + 1

Therefore, we complete the proof of Theorem 2.4.

References [1] S. Axler, Harmomic functions from a complex analysis viewpoit, Amer. Math. Monthly 93 (1986), 246-258. [2] T. B. Cao and H. X. Yi, Uniqueness theorems of meromorphic functions sharing sets IM on annuli, Acta Mathematica Sinica (Chinese Series), 54 (4) (2011), 623-632. (in Chinese). [3] T. B. Cao, H. X. Yi and H. Y. Xu, On the multiple values and uniqueness of meromorphic functions on annuli, Comput. Math. Appl. 58 (2009), 1457-1465. [4] H. H. Chen, Singular directions related to the Hayman inequality, Progress in Math., 16(1) (1987), 73-79, (in Chinese). [5] C.T. Chuang, Une g´en´eralisation d’une in´egalit´e de Nevanlinna, Sci. Sinica 13 (1964), 887-895. [6] W. Hayman, Meromorphic functions, Oxford: Clarendon Press, 1964. [7] Y. Hu, H. Y. Xu, The Milloux’s inequality of meromorphic functions on annulis, J. Jiangxi Norm. Univ., Nat. Sci. 33 (3) (2009), 293-296. [8] A. Ya. Khrystiyanyn and A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. I, Mat. Stud. 23(2005), No.1, 19-30. [9] A. Ya. Khrystiyanyn and A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. II, Mat. Stud. 24 (2005), No.2, 57-68. [10] A. A. Kondratyuk and I. Laine, Meromorphic functions in multiply connected domains, Laine, Ilpo (ed.), fourier series methods in complex analysis. Proceedings of the workshop, Mekrij¨ arvi, Finland, July 24-29, 2005. Joensuu: University of Joensuu, Department of Mathematics (ISBN 952-458-8889/pbk). Report series. Department of mathematics, University of Joensuu 10, 9-111 (2006).

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[11] R. Korhonen, Nevanlinna theory in an annulus, value distribution theory and related topics, Adv. Complex Anal. Appl. 3 (2004), 167-179. [12] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993. [13] X. M. Li, H. X. Li, X. Zhang, Normal Families of Meromorphic Functions Concerning Composite Meromorphic Functions and Fixed Points, J. Jiangxi Norm. Univ. Nat. Sci. 40(6) (2016), 578-594. [14] L. W. Liao, The new developments in the research of nonlinear complex differential equations. J. Jiangxi Norm. Univ. Nat. Sci. 39(4) (2015), 331-339. [15] M. Lund and Z. Ye, Logarithmic derivatives in annuli, J. Math. Anal. Appl. 356 (2009), 441-452 [16] M. Lund and Z. Ye, Nevanlinna theory of meromorphic functions on annuli, Sci. China. Math. 53 (2010), 547-554. [17] R. Nevanlinna, Eindentig keitss¨ atze in der theorie der meromorphen funktionen, Acta. Math. 48 (1926), 367-391. [18] N. Steinmetz, Eine Varallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes, J. Rein Angew. Math. 368 (1986), 131-141. [19] Z. J. Wu and B. Wang, The Characteristic Function of E-Valued Meromorphic Functions, J. Jiangxi Norm. Univ., Nat. Sci. 40 (5) (2016), 500-504. [20] H. Y. Xu and Z. X. Xuan, The uniqueness of analytic functions on annuli sharing some values, Abstract and Applied Analysis 2012 (2012), Art. 896596, 1-13. [21] K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), 225-294. [22] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Science Press 1995/Kluwer 2003. [23] L. Yang, Value distribution theory, Berlin: Springer-Verlag/ Beijing: Science Press, 1993. [24] J. H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad J. Math. 47 (2004), 152-160. [25] J. H. Zheng, On uniqueness of meromorphic functions with shared values in one angular domains, Complex Var. Elliptic Equ. 48 (2003), 777-785. [26] J. H. Zheng, Value Distribution of Meromorphic Functions, Tsinghua University Press, Beijing, Springer, Heidelberg, 2010. [27] J. H. Zheng, N. Wu, Hayman T directions of meromorphic functions, Taiwanese J Math. 14 (6) (2010), 2219-2228.

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WEIGHTED COMPOSITION OPERATORS BETWEEN WEIGHTED HILBERTIAN BERGMAN SPACES IN THE UNIT POLYDISK NING CAO, GANG WANG AND CEZHONG TONG∗

Abstract. In this paper, we prove that the topology spaces of nonzero weighted composition operators acting on some Hilbert spaces of holomorphic functions in the unit polydisk are path connected, which generalized Hosokawa, Izuchi and Ohno’s results in single complex variables’ case [9]. Keywords: Weighted Hilbertian Bergman spaces, weighted composition operator, polydisk, norm topology, Hilbert-Schmidt topology.

1. Introduction Let H(DN ) be the space of analytic functions on the open unit polydisk DN := {z = (z1 , . . . , zN ) ∈ CN : |zi | < 1, i = 1, 2, . . . , N } and H ∞ the space of bounded analytic functions on DN with the supremum norm k · k∞ . When N = 1, the unit polydisk reduces to the unit open disc D in the complex plane C. Let S(DN ) be the set of analytic self-maps of DN . Every ϕ = (ϕ1 , · · · , ϕN ) ∈ S(DN ) induces the composition operator Cϕ
defined by Cϕ f = f ◦ ϕ for f ∈ H DN . If u ∈ H(DN ), the multiplication, Mu : H(DN ) → H(DN ), is defined by Mu (f )(z) = u(z) · f (z) for any f ∈ H(DN ) and z ∈ DN . If u ∈ H(DN ) and ϕ ∈ S(DN ), we call the operator Mu Cϕ to be the weighted composition operator. Much effort has been expended on characterizing those analytic maps which induce bounded or compact composition operators between those classic spaces of analytic functions. Readers interested in this topic can refer to the books [16] by Shapiro, [7] by Cowen and MacCluer, and [23, 24] by Zhu. An active topic is the topological structure of the space of composition operators acting on function spaces. If X is a Banach space of analytic functions, 2010 Mathematics Subject Classification. Primary: 47B33; Secondary: 47B38. ∗ Corresponding author. Tong was supported in part by the National Natural Science Foundation of China (Grant Nos. 11301132). 1

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N. CAO, G. WANG, C. TONG

we employ the symbol C(X) to represent the space of composition operators on X equipped with the operator norm topology. In 1981, Berkson [3] firstly studied the topological structure of C(H 2 (D)). Central problem focuses on both the structure of C(H 2 (D)) and the compact differences of its members. In 1989, MacCluer [12] showed that, on the weighted Bergman space A2s (D) for s ≥ −1, all the compact composition operators can be connected by paths, and she gave necessary conditions for two composition operators to have compact difference. At about the same time, Shapiro and Sundberg [17] gave further results on compact difference and isolation and they believe that the compact composition operators should form a connected component of the set C(H 2 (D)). In 2008, Gallardo-Guti¨ errez and co-workers [8] showed that there exists noncompact composition operators in the component generated by all compact composition operators. In 2005, Moorhouse [13] characterized compact difference for composition operators acting on A2λ (D), λ > −1, and gave a partial answer to the component structure of C(A2λ (D)). Later, Kriete and Moorhouse [11] extended that results to linear combinations. In 2012, Saukko [14, 15] obtained a complete characterization of bounded and compact differences between standard weighted Bergman spaces. Recently, Choe, Koo and Park [5, 6] extend Moorhouses characterization to the Bergman spaces in unit polydisk and unit ball. In 2015, Hosokawa, Izuchi and Ohno [9] investigate the topology space of weighted composition operators acting between some Hilbert spaces on D in general, and they also consider the Hilbert-Schmidt norm topology. Readers interested in those related topic can refer recent papers [19, 20, 21, 22] and the references therein. Generally speaking, theory of composition operators on the spaces of holomorphic functions in the unit polydisk are far from complete. To completely characterize the boundedness and compactness of composition operators on Hardy spaces and weighted Bergman spaces is still open. In [2], Bayart showed that the study of boundedness of composition operators on the polydisk is a difficult problem, and many obstacles are caused by differences between the torus of DN (distinguishing boundary of DN ) and the whole boundary. Stessin and Zhu [18] characterized the boundedness of composition operators between different weighted Bergman spaces in the polydisk. Inspired by [2, 9, 18], we continue to investigate the topology spaces of weighted composition operators between different weighted Bergman spaces in the unit polydisk. On those spaces, we will also consider the Hilbert Schmidt topology spaces of weighed composition operators. 2. Preliminaries Let dA(z) = dxdy/π denote the normalized area measure of D. For s > −1 the weighted Hilbertian Bergman space A2s (DN ) consists of all functions

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3

f ∈ H(DN ) such that kf k2s

Z

|f (z)|2 dvs (z) < ∞,

= DN

where dvs (z) = dAs (z1 ) · · · dAs (zN ) and dAs (zj ) = (s + 1)(1 − |zj |2 )s dA(zj ). The inner product of A2s (DN ) is given by Z hf, gis =

f (z)g(z)dvs (z), DN

where f, g ∈ A2s . And the reproducing kernel of A2s (DN ) is given by kA2s ,w (z) =

N Y

1 . (1 − w¯j zj )2+s j=1

When s = 0, A20 (DN ) is the classical Bergman space. It is well known that kz α k2s

Z

α 2

|z | dvs (z) =

= DN

N Z Y j=1

α

|zj j |2 dAs (zj ) = ΓN (s + 1)

D

N Y

Γ(αj + 1) Γ(αj + s + 2) j=1

αN where α = (α1 , . . . , αN ) and z α = z1α1 · · · zN . We believe the following pointwise esitmate is well known, and we list it with a simple proof for the completeness.

Lemma 2.1. Let p > 0 and s > −1. If f ∈ Aps (DN ), then |f (w)| ≤

kf ks N Q

(1 − |wi |2 )

2+α p

i=1

for each w = (w1 , . . . , wN ) ∈ DN . Proof. Fixing w2 , . . . , wN , function f (ζ1 , w2 , . . . , wN ) is analytic with respect to ζ1 ∈ D. It is well known that kf (ζ1 , w2 , . . . , wN )kps |f (w1 , w2 , . . . , wN )|p ≤ (1 − |w1 |2 )2+s R |f (ζ1 , w2 , . . . , wN )|p dAs (ζ1 ) D = . (1 − |w1 |2 )2+s

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We can also estimate |f (ζ1 , w2 , . . . , wN )|p by the similar inequality with respect to w2 , and then up to wN , hence we have

≤ ≤ ... ≤

|f (w1 , w2 , . . . , wN )|p R |f (ζ1 , w2 , . . . , wN )|p dAs (ζ1 ) D (1 − |w1 |2 )2+s R R |f (ζ1 , ζ2 , . . . , wN )|p dAs (ζ2 )dAs (ζ1 ) D D (1 − |w1 |2 )2+s (1 − |w2 |2 )2+s R R · · · D |f (ζ1 , ζ2 , . . . , ζN )|p dAs (ζN ) . . . dAs (ζ2 )dAs (ζ1 ) D (1 − |w1 |2 )2+s . . . (1 − |wN |2 )2+s

That is |f (w)| ≤

kf ks N Q

(1 − |wi |2 )

. 2+α p

i=1

 It is an obvious consequence that the point evaluation w : f 7→ f (w) = f (w1 , . . . , wN ) is a bounded linear functional on Aps (DN ), and max

sup kw kAps < ∞

i=1,...,N |wi |≤r

for every 0 < r < 1. If −1 < s0 < s, we can immediately have that A2s0 ⊂ A2s by a direct computation kf ks ≤ Ckf ks0 where the constant C depends only on s and s0 . Let Cw (A2s0 , A2s ) be the space of nonzero bounded weighted composition operators from a weighted Hilbertian Bergman space A2s0 to another A2s with the operator norm topology, that is, Cw (A2s0 , A2s ) = {Mu Cϕ : Mu Cϕ : A2s0 → A2s is bounded, u 6= 0}. And Cw (A2s ) = Cw (A2s , A2s ). For a bounded linear operator T : X 0 → X, we write kT kX 0 ,X its operator norm. If −1 < s0 < s, for Mu Cϕ ∈ Cw (A2s ), we have kMu Cϕ f kA2s ≤ kMu Cϕ kA2s kf ks ≤ C · kMu Cϕ kA2s kf ks0 for every f ∈ A2s0 . Hence Mu Cϕ : A2s0 → A2s is bounded and (2.1)

kMu Cϕ kA20 ,A2s ≤ kMu Cϕ kA2s s

for every Mu Cϕ ∈ Cw (A2s ).

Restricting Mu Cϕ ∈ Cw (A2s ) on A2s0 , we may consider that Mu Cϕ is also a bounded linear operator from A2s0 to A2s . We note that Cw (A2s0 , A2s ) = Cw (A2s ) as sets, and if Mu Cϕ ∈ Cw (A2s0 , A2s ), then u ∈ A2s .

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5

3. Operator norm topology spaces Now we discuss the operator norm topology spaces Cw (A2s0 , A2s ). Lemma 3.1. If ϕ = (ϕ1 , . . . , ϕN ) ∈ S(DN ) and kϕk∞ := max kϕi k∞ < 1, i=1,...,N

then Cϕ f ∈ H ∞ for every f ∈ A2s and kCϕ f k∞ ≤ kf ks max

sup

i=1,...,N |wi |≤kϕk∞

kw kA2s .

Proof. For f ∈ H and z ∈ DN , we have |(Cϕ f )(z)| = |f (ϕ(z))| ≤ kf ks kϕ(z) kA2s ≤ kf ks max

sup

i=1,...,N |wi |≤kϕk∞

kw kA2s ,

so we get the assertion.



The main result is following. Theorem 3.2. If −1 < s0 < s, then the space Cw (A2s0 , A2s ) is path connected. Proof. Let Mu Cϕ ∈ Cw (A2s0 , A2s ). Since Cw (A2s0 , A2s ) = Cw (A2s ) as sets, we have u ∈ A2s and kMu Cϕ kA2s < ∞. Let 0 ≤ r < 1. For f ∈ A2s , by Lemma 3.1 we have f ◦ rϕ = f (rϕ1 , . . . , rϕN ) ∈ H ∞ and by Lemma 2.1, kMu Crϕ f ks

= ku(f ◦ rϕ)ks ≤ kf ◦ rϕk∞ kuks ≤ kuks kf ks max sup kw kA2s . i=1,...,N |wi |≤r

Hence Mu Crϕ ∈ Cw (A2s ), so Mu Crϕ ∈ Cw (A2s0 , A2s ). Fixing 0 ≤ t0 ≤ 1, we apply the similar method P in [9] to show that kMu Ct0 ϕ − Mu Ctϕ kA20 ,A2s → 0 as t → t0 . Let g(z) = α cα z α ∈ A2s0 . For s each 0 ≤ t ≤ 1, let X |α| gt (z) = cα (t0 − t|α| )z α . α

Since

A2s0

kgt k2s

⊂

A2s ,

A2s .

we have g ∈ Note that Z X X |α| |α| cα (t0 − t|α| )z α dvs (z) = hgt , gt is = cα (t0 − t|α| )z α DN

2

α

|cα | |z| dvs (z)

DN

α

≤ 4

α

α

Z X |α| = (t0 − t|α| )2 XZ α

DN

|cα |2 |z|α dvs (z) = 4kgk2s ,

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we have gt ∈ A2s . Hence k(Mu Ct0 ϕ − Mu Ctϕ )gk2s

2

X

|α| cα (t0 − t|α| )ϕα = u

α s

= kMu Cϕ gt k2s ≤ kMu Cϕ k2A2s kgt k2s X |α| = kMu Cϕ k2A2s |cα |2 |t0 − t|α| |2 kz α k2s α

 kz α k2s X ≤ sup −t | |cα |2 kz α k2s0 kz α k2s0 |α|>0 α 2  α kz k s |α| kgk2s0 . ≤ kMu Cϕ k2A2s sup |t0 − t|α| | α kz ks0 |α|>0 

kMu Cϕ k2A2s

|α| |t0

|α| 2

Then   kz α ks |α| |t0 − t|α| | α . kz ks0 |α|>0

kMu Ct0 ϕ − Mu Ctϕ kA20 ,A2s ≤ kMu Cϕ kA2s sup s

For any positive integer n, we have  α  α  X  |α| kz α ks |α| |α| kz ks |α| kz ks sup t0 − t α ≤ + 2 sup . t0 − t α α kz ks0 kz ks0 |α|>0 |α|≥n2 kz ks0 2 |α| −1. For each 0 ≤ r ≤ 1, if the operator Mu Cϕ : A2s0 → A2s is a Hilbert-Schmidt operator for some nonzero u ∈ A2s and ϕ ∈ S(DN ), then Mu Crϕ is also a Hilbert-Schmidt operator from A2s0 to A2s . Proof. The Lemma follows immediately from the computations that X kMu Crϕ (z α )k2 s kMu Crϕ k2A20 ,A2 ,HS = s s kz α k2s0 α≥0

=

X α≥0

kuϕα k2 r2|α| α 2 s ≤ kMu Cϕ k2A20 ,A2 ,HS < ∞. s s kz ks0  0

Theorem 4.2. If −1 < s < s, then the topology space connected.

157

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Proof. If Mu Cϕ ∈ Cw,HS (A2s0 , A2s ), recall that {z α /kz α k : α} is an orhonomal basis, and we have X kuϕα k2 s (4.1) kMu Cϕ k2A20 ,A2 ,HS = < ∞. s s kz α k2s0 |α|≥0

Cw,HS (A2s0 , A2s )

Besides, Mu Ctϕ ∈ for every 0 ≤ t ≤ 1 by Lemma 4.1. If we fix 0 ≤ t0 ≤ 1. we can also prove kMu Ct0 ϕ − Mu Ctϕ kA20 ,A2s ,HS → 0 s as t → t0 as following statements. For any positive integer N , we have kMu Ct0 ϕ − Mu Ctϕ k2A20 ,A2 ,HS = s

≤

s

X ku(t|α| − t|α| )ϕα k2 s 0 kz α k2s0

|α|≥0

2 kuϕα k2 X |α| X kuϕα k2 s s + . t0 − t|α| kz α k2s0 kz α k2s0

|α|≤N

|α|>N

Take ε > 0 arbitrary. Then by (4.1), we may take N large enough so that X kuϕα k2 s < ε. kz α k2s0 |α|>N

Hence kMu Ct0 ϕ − Mu Ctϕ k2A20 ,A2 ,HS < ε + s

s

2 kuϕα k2 X |α| s t0 − t|α| kz α k2s0

|α|≤N

By letting t → t0 , we have lim sup kMu Ct0 ϕ − Mu Ctϕ k2A20 ,A2 ,HS < ε. s

t→t0

s

Let ε → 0 then, we have the topology space Cw,HS (A2s0 , A2s ) is path connected.  5. Final remarks Lemma 5.1. Let s > −1. If kϕk∞ < 1 and u ∈ A2s , then Mu Cϕ ∈ Cw (A2s ) and is compact. Proof. By the first paragraph of the proof in Theorem 3.2, we have Mu Cϕ ∈ Cw (A2s ). To show that Mu Cϕ is compact, let {fn } be a sequence in A2s such that there is a positive constant K satisfying kfn ks < K for every n. By the pointwise estimate (Lemma 2.1), we may assume that fn converges to some f ∈ H(DN ) uniformly on compact subsets of DN . By the assumption, fn ◦ ϕ → f ◦ ϕ in H ∞ . Hence both u(fn ◦ ϕ) and u(f ◦ ϕ) are in A2s , and kMu Cϕ fn − u(f ◦ ϕ)ks ≤ kuks kfn ◦ ϕ − f ◦ ϕk∞ → 0, Thus Mu Cϕ ∈

Cw (A2s )

is compact.

n → ∞. 

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WEIGHTED COMPOSITIONS IN THE POLYDISK

9

Proposition 5.2. If −1 < s0 < s, then Cw (A2s0 , A2s ) consists of all compact weighted composition operators. Proof. For 0 < r < 1, by Lemma 5.1 Mu Crϕ ∈ Cw (A2s ) is compact. Hence Mu Crϕ : A2s → A2s , which can be regarded as the composition of id : A2s0 → A2s and Mu Crϕ : A2s → A2s , is compact. Since the algebra of compact operators is closed in norm topology, we get Mu Cϕ is compact since it can be approximated by compact operators Mu Crϕ by Theorem 3.2.  We note that on many spaces, the compact (weighted) composition operators form a path connected subset in the topology space of bounded (weighted) composition operators. The compactness has played an important role in the proof of the main result. References [1] R. F. Allen, K. C. Heller and M. A. Pons, Compact differences of composition operators on weighted Dirichlet spaces, Cent. Eur. J. Math., 12(7), 1040-1051(2014). [2] F. Bayart, Composition oeprators on the polydisk induced by affine maps, J. Funct. Anal., 260, 1969-2003(2011). [3] E. Berkson, Composition opertors isolated in the uniform operator topology, Proc. Amer. Math. Soc., 81, 230-232(1981). [4] B. Berndtsson, Interpolating sequences for H ∞ in the ball, Indag. Math. (Proc.), 88(1), 1-10(1985). [5] B. R. Choe, H. Koo and I. Park, Compact differences of composition operators over polydisk. Integr. Equ. Oper. Theory, 73, 57-91(2012). [6] B. R. Choe, H. Koo and I. Park, Compact differences of composition operators on the Bergman spaces over the Ball, Potential Anal., 40, 81-102(2014). [7] C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. [8] E. A. Gallardo-Guti´ errez, M. J. Gonz´ alez, P. J. Nieminen and E. Saksman, On the connected component of compact composition operators on the Hardy space, Advances in Mathematics, 219, 986-1001(2008). [9] T. Hosokawa, K. J. Izuchi, and S. Ohno, Weighted composition operators between Hilbert spaces of analytic functions in the operator norm and Hilbert-Schmidt norm topologies, J. Math. Anal. Appl., 421, 1546-1558(2015). [10] K. Heller, B. MacCluer, R. Weir, Compact differences of composition operators in several variables. Integr. Equ. Oper. Theory, 69, 247-268(2011). [11] T. Kriete and J. Moorhouse, Linear relations in the Calkin algebra for composition operators, Trans. Amer. Math. Soc., 359, 2915-2944(2007). [12] B. D. MacCluer, Components in the space of composition operators, Integral Equations Operator Theory, 12, 725-738(1989). [13] J. Moorhouse, Compact differences of composition operators, Journal of Functional Analysis, 219, 70-92(2005). [14] E. Saukko, Difference of composition operators between standard weighted Bergman spaces, J. Math. Anal. Appl., 381, 789-798(2011). [15] E. Saukko, An application of atomic decomposition in Bergman spaces to the study of differences of composition operators, 262, 3872-3890(2012). [16] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer Verlag, New York, 1993.

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[17] J. H. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacific J. Math., 145, 117-152(1990). [18] M. Stessin and K. Zhu, Composition operators induced by symbols defined on a polydisk, J. Math. Anal. Appl., 319, 815-829(2006). [19] C. Tong, and Z. Zhou, The compactness of the sum of weighted composition operators on the ball algebra, J. Ineq. Appl., 2011(45), 1-10(2011). [20] C. Z. Tong and Z. H. Zhou, Intertwining relations for Volterra operators on the Bergman space, Illinois J. Math. 57(1), 195-211(2013). [21] C. Z. Tong and Z. H. Zhou, Compact intertwining relations for composition operators between the weighted Bergman spaces and the weighted Bloch spaces, J. Korean Math. Soc., 51(1), 125-135(2014). [22] C. Tong, C. Yuan and Z. Zhou, Topological structures of derivative weighted composition operators on the Bergman space, J. Function Spaces, 2015(8), 1-8(2015). [23] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker. Inc, New York, 1990. [24] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball. Grad. Texts in Math, Springer, 2005.

College of Information Engineering Qingdao Binhai University Qingdao 266555 P.R. China. E-mail address: [email protected]

College of Information Engineering Qingdao Binhai University Qingdao 266555 P.R. China. E-mail address: [email protected]

Department of Mathematics Hebei University of technology Tianjin 300401 P.R. China. E-mail address: [email protected]

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On the L∞ convergence of a nonlinear difference scheme for Schr¨odinger equations Xiaoman Liu1 Yongmin Liu2 1 Department of Mathematics, Southeast University Nanjing, 210096, P.R.China 2 School of Mathematics and Statistics, Jiangsu Normal University Xuzhou, 221116, P.R.China

Abstract In this article, a nonlinear difference scheme for Schr¨ odinger equations is studied. The existence of the difference solution is proved by Brouwer fixed point theorem. With the aid of the fact that the difference solution satisfies two conservation laws, the difference solution is proved to be bounded in the L∞ norm. Then, the difference solution is shown to be unique and second order convergent in the L∞ norm. Finally, a convergent iterative algorithm is presented.

MSC(2010): 35A35, 35K55, 65M12, 65M15 Keywords: Schr¨ odinger equations, Nonlinear difference scheme, Solvability, Convergence

1

Introduction

The Schr¨ odinger equation is one of the most important equations in quantum mechanics. This model equation also arises in many other branches of science and technology, e.g. optics, seismology and plasma physics. Recently, a growing interest is on the numerical solution to the nonlinear Schr¨odinger equations. Many authors investigated the finite difference methods for solving this kind of equations, including the conservation, solvability, stability, convergence and the symplectic geometry (see [1] − [8]). Consider nonlinear Schr¨ odinger equations i

∂u ∂ 2 u + + q|u|2 u = 0, 0 < x < 1, 0 < t ≤ T, ∂t ∂x2 u(x, 0) = φ(x), 0 ≤ x ≤ 1, u(0, t) = 0, u(1, t) = 0, 0 < t ≤ T,

1 Corresponding

(1.1) (1.2) (1.3)

author: X.M. Liu, email: [email protected]

1

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On the L∞ convergence of a nonlinear difference scheme

2

where q is a real constant, φ(x) is a known function and φ(0) = φ(1) = 0, u(x, t) is an unknown function. Take two positive integers m and n. Denote h = 1/m, τ = T /n, so we have xj = jh, 0 ≤ j ≤ m, tk = kτ, 0 ≤ k ≤ n. Denote Ωhτ = {(xj , tk ) | 0 ≤ j ≤ m, 0 ≤ k ≤ n}. Suppose u = {ukj | 0 ≤ j ≤ m, 0 ≤ k ≤ n} be a discrete grid function on Ωhτ . Introduce the following notations: ) ) ) 1 ( k+1 1 ( k+1 1 ( k+1 k+ 1 uj + ukj , δt uj 2 = uj − ukj , Dt ukj = uj − uk−1 , j 2 τ 2τ ) ) 1( k 1( uj+1 − ukj , δx2 ukj = δx ukj+ 1 − δx ukj− 1 . δx ukj+ 1 = 2 2 2 h h The author of [9] developed the following nonlinear difference scheme for (1.1)-(1.3) ( 2 ) k+ 1 2 k+ 1 k+ 1 u 2 = 0, iδt uj 2 + δx2 uj 2 + 2q ukj + uk+1 j j (1.4) 1 ≤ j ≤ m − 1, 0 < k ≤ n − 1, k+ 12

uj

=

u0j = φ(xj ),

1 ≤ j ≤ m − 1,

(1.5)

0 < k ≤ n.

(1.6)

uk0 = 0, ukm = 0,

The contents in [9] pointed out that the difference scheme preserves the densities and the energy of the solution, and the author also proved that the difference scheme is uniquely solvable and convergent with the convergence order of (τ 2 + h2 ) in L2 norm under some constraints on the stepsizes. On this basis, we proof further that this difference scheme is convergent with the convergence order of (τ 2 + h2 ) in L∞ norm. In this paper, we will analyze the difference scheme (1.4)-(1.6). The remainder of the paper is arranged as follows. In Section 2, the existence of the difference solution is shown by the Brouwer fixed point theorem. Then with the aid of the conversations of the difference solution, the boundedness and uniqueness of difference solution are proved. In Section 3, the convergence of the difference scheme is discussed. The difference scheme is proved to be convergent with the convergence order of O(τ 2 + h2 ) in L∞ norm. In Section 4, an iterative algorithm for the difference scheme with the proof of the convergence is given. A short conclusion section ends the paper.

2

The existence of the difference solution

In this section, we will prove that the finite difference scheme (1.4)-(1.6) exists a solution.

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On the L∞ convergence of a nonlinear difference scheme

3

Let H = {v | v = (v0 , v1 , . . . , vm ), vj ∈ C, 0 ≤ j ≤ m, v0 = vm = 0} be the space of complex grid functions on Ωh . Given any complex grid functions v, w ∈ H, denote the inner product (v, w) = h

m−1 ∑

vj w ¯j .

j=1

The discrete Lp norm ∥ · ∥p , maximum-norm ∥ · ∥∞ and H01 norm | · |1 are defined respectively as follows v u m−1 u ∑ p ∥v∥p = t h |vj |p , p ≥ 1, ∥v∥∞ = max |vi |, 0≤i≤m

j=1

v u m−1 u ∑ vj+1 − vj 2 . |v|1 = th h j=0 For abbreviation, we write ∥ · ∥2 as ∥ · ∥. In order to illustrate the existence of the difference solution, we need the following lemma. Lemma 2.1. (Brouwer Fixed Point Theorem [10]) Let (H, (·, ·)) be a finite dimensional inner product space, ∥ · ∥ the associated norm, and Π : H → H be continuous. Assume moreover that ∃α > 0, ∀z ∈ H, ∥z∥ = α, Re(Π(z), z) ≥ 0. Then, there exists an element z ∗ ∈ H such that Π(z ∗ ) = 0 and ∥z ∗ ∥ ≤ α. Theorem 2.2. The solution of difference scheme (1.4) − (1.6) exists. } { Proof. Suppose ukj | 0 ≤ j ≤ m be the numerical solution. Using the notation introduced before, we rewrite the difference scheme (1.4)-(1.6) in the following form ( 2 ) ) ( 2 k+ 1 k+ 1 k+ 1 k+ 1 i τ2 uj 2 − ukj + δx2 uj 2 + 2q ukj + 2uj 2 − ukj uj 2 = 0, (2.1) 1 ≤ j ≤ m − 1, k+ 12

u0

k+ 12

= 0, um

Let

k+ 12

vj = uj

= 0.

(2.2)

, 0 ≤ j ≤ m,

then (2.1)-(2.2) can be written as

i

) ) 2( q( k2 vj − ukj + δx2 vj + |uj | + |2vj − ukj |2 vj = 0, 1 ≤ j ≤ m − 1, τ 2

163

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4

v0 = 0, vm = 0.

(2.4)

Define the mapping Π : H → H, { 0, ] j = 0, m, [ (Π(v))j = vj − ukj − i τ2 δx2 vj + 2q (|ukj |2 + |2vj − ukj |2 )vj , 1 ≤ j ≤ m − 1. Computing the inner product of Π(v) and v, we obtain (Π(v), v) =

h

m−1 ∑ [ j=1

] vj − ukj − i τ2 δx2 vj − i τ4q (|ukj |2 + |2vj − ukj |2 )vj v¯j

) δx2 vj )¯ vj − i τ4q (|ukj |2 + |2vj − ukj |2 |vj |2 j=1 2 m ( ) ∑ τ 2 k = ∥v∥ − (u , v) − i 2 h δx vj− 12 − i τ4q |ukj |2 + |2vj − ukj |2 |vj |2 .

= ∥v∥2 − (uk , v) − i τ2 h

m−1 ∑ (

j=1

So taking the real part of the inner product Re(Π(v), v)

k ∥v∥2 − Re(u ( , v) ) m−1 ∑ k 2 = ∥v∥ − Re h uj v¯j j=1  ( )2 ( )2  m−1 m−1 ∑ k ∑ ≥ ∥v∥2 − Re  h2 uj + v¯j 

=

j=1

( ) 2 = ∥v∥ − 12 ∥uk ∥2 )+ ∥v∥2 ( = 12 ∥v∥2 − ∥uk ∥2

j=1

When ∥v∥ = ∥uk ∥, Re(Π(v), v) ≥ 0. Using Lemma 2.1, we have ∀v ∈ H, ∥v∥ = 1 ∥uk+ 2 ∥ = ∥uk ∥ > 0, Re(Π(v), v) ≥ 0. Then there exists an element v ∗ ∈ H such that Π(v ∗ ) = 0 and ∥v ∗ ∥ ≤ ∥uk ∥. Hence, it is easily seen that the solution {vj | 0 ≤ j ≤ m} satisfies the difference scheme (2.3)-(2.4). The proof is complete.

3

The uniqueness of the difference solution

Theorem 3.1. ([9])The solution { of difference scheme (1.4) } − (1.6) is conservative. In more precisely, let ukj | 0 ≤ j ≤ m, 0 ≤ k ≤ n be the solution of (1.4) − (1.6), we have ∥uk ∥2 = ∥u0 ∥2 , E k = E 0 , 1 ≤ k ≤ n, where Ek = h

m−1 ∑ j=0

2 q m−1 ∑ 4 uk . δx ukj+ 1 − h 2 2 j=1 j

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5

Theorem 3.2. ([9])The difference solution{ of (1.4) − (1.6) is bounded } in the discrete L∞ norm. In more precisely, let ukj | 0 ≤ j ≤ m, 0 ≤ k ≤ n be the solution of (1.4) − (1.6), we have ( ) q 0 4 1 1 2 0 6 0 2 k 2 |q| ∥u ∥ + |u |1 − ∥u ∥4 , 1 ≤ k ≤ n, ∥u ∥∞ ≤ 2 8 2 Using Theorems 3.1 and 3.2, we can obtain Theorem 3.3. The difference solution of (1.4) − (1.6) is unique. Proof. To proof this theorem, we can prove the solution of difference scheme (2.3)-(2.4) is unique. Let {vj | 0 ≤ j ≤ m} and {wj | 0 ≤ j ≤ m} be the solutions of (2.3)-(2.4). Then we have ) q( k2 2 |uj | + |2wj − ukj |2 wj = 0, 1 ≤ j ≤ m − 1, (3.1) i (wj − ukj ) + δx2 wj + τ 2 w0 = 0, wm = 0.

(3.2)

Denote θj = vj − wj , 0 ≤ j ≤ m. Subtracting (3.1)-(3.2) from (2.3)-(2.4) respectively, we obtain the following equations i τ2 θj + δx2 θj + 2q |ukj |2 θj +

q 2

) ( |2vj − ukj |2 vj − |2wj − ukj |2 wj = 0, 1 ≤ j ≤ m − 1,

(3.3)

θ0 = 0, θm = 0.

(3.4)

Multiplying (3.3) by −ihθ¯j , summing up for j from 1 to m−1, we can obtain 2 τh

m−1 ∑

|θj |2 − ih

j=1 m−1 ∑ (

−i 2q h

m−1 ∑ ( j=1

|2vj −

j=1

m−1 ) ∑ k2 2 δx2 θj θ¯j − i 2q h |uj | |θj | j=1

ukj |2 vj

− |2wj −

ukj |2 wj

)

(3.5) θ¯j = 0.

Adding the term −|2vj − ukj |2 wj to the part of the forth term in (3.5). Meanwhile, noticing the equality |a|2 − |b|2 = (a − b)¯ a + b(a − b) where both a and b are complex functions, we have |2vj − ukj |2 vj − |2wj − ukj |2 wj = |2vj − ukj |2 (vj −[wj ) + (|2vj − ukj |2 − |2wj − ukj |2 )wj

] = |2vj − ukj |2 θj + 2(vj − wj )2vj − ukj + 2(vj − wj )(2wj − ukj ) wj [ ] = |2vj − ukj |2 θj + 2 θj 2vj − ukj + θ¯j (2wj − ukj ) wj .

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6

Thus, taking the real part of (3.5) with the rewritten forth term, we obtain   m−1   [ ] ∑ 2 q ∥θ∥2 − Im h 2 |θj |2 (2vj − ukj ) + (2wj − ukj )wj = 0. 2  τ j=1 According to Theorem 3.2 which illustrates that the solution v, w are boundedness, we easily get that |v|, |w| ≤ ∥uk ∥∞ . So using Theorem 3.2 and CauchySchwarz inequality, we have 2 2 τ ∥θ∥

≤ ≤

|q|h |q|h

m−1 ∑ ( j=1 m−1 ∑ ( j=1

) |2vj − ukj | + |2wj − ukj | |θj |2 |wj | ) 2|vj | + |ukj | + 2|wj | + |ukj | ∥θ∥2 |wj |

≤ 6|q| · ∥uk ∥2∞ ∥θ∥2 Denote the right term of the inequality in Theorem 3.2 be a constant c1 , we have 2 ∥θ∥2 ≤ 6c21 |q| · ∥θ∥2 . τ When τ < 3c21|q| , we get ∥θ∥2 = 0. Hence, vj = wj , 0 ≤ j ≤ m. The proof is 1 complete.

4

The convergence of the finite difference scheme

Suppose that the continuous problem (1.1)-(1.3) has a smooth solution u, and Ujk = {u(xj , tk ) | 0 ≤ j ≤ m, 0 ≤ k ≤ n} is the solution u under the mapping Ωhτ . In this section, we will illustrate that the solution ukj of the difference scheme (1.4)-(1.6) is convergent to the solution Ujk with the convergence order of O(τ 2 + h2 ) in the L∞ norm. Denote c0 = max ∥u(·, t)∥∞ , (4.1) 0≤t≤T

ekj

=

Ujk

−

ukj ,

0 ≤ j ≤ m, 0 ≤ k ≤ n.

Lemma 4.1. (Gronwall Inequality [9]) Assume {Gn | n ≥ 0} is a nonnegative sequence, and satisfies Gn+1 ≤ (1 + cτ )Gn + τ g, n = 0, 1, 2, . . . , where c and g are nonnegative constants. Then G satisfies ( g) Gn ≤ ecnτ G0 + , n = 0, 1, 2, . . . . c Lemma 4.2. ([11]) For any complex functions U, V, u, v, one has | |U |2 V − |u|2 v| ≤ (max {|U |, |V |, |u|, |v|})2 · (2|U − u| + |V − v|).

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On the L∞ convergence of a nonlinear difference scheme

7

Lemma 4.3. ([9]) Denote Vh = {v | v = {vi | 0 ≤ i ≤ m} is the grid f unction on Ωh } , V˙ h = {v | v = {vi | 0 ≤ i ≤ m} ∈ Vh , v0 = vm = 0}. (1) Suppose v ∈ V˙ h , so there is ∥v∥∞ ≤

1 |v|1 . 2

(2) Suppose v ∈ V˙ h . For any ε > 0, there is ∥v∥2∞ ≤ ε|v|21 +

1 ∥v∥2 . 4ε

Theorem 4.4. Suppose that the difference scheme (1.4) − (1.6) has the solution ukj and the equations (1.1) − (1.3) has the solution Ujk . When τ is small enough, there exists a constant C independent of h, τ such that ∥ek ∥∞ ≤ C(τ 2 + h2 ), 0 ≤ k ≤ n.

(4.2)

Proof. Subtracting(1.4)-(1.6) from (1.1)-(1.3) respectively, we obtain the error equations [ ] k+ 1 k+ 1 k+ 1 k+ 1 iδt ei 2 + δx2 ei 2 + 2q (|Ujk |2 + |Ujk+1 |2 )Uj 2 − (|ukj |2 + |uk+1 |2 )uj 2 j = Rjk , 1 ≤ j ≤ m − 1, 0 ≤ k ≤ n − 1, (4.3) e0j = 0, 1 ≤ j ≤ m − 1, ek0

=

0, ekm

(4.4)

0 ≤ k ≤ n.

= 0,

(4.5)

In using the Taylor expansion with Lagrange remainder, we can get [ ] 2 τ 2 ∂3u h2 ∂ 4 u ∂4u ′ τ ∂4u k Rj = (x , η )+ (ξ , t ) + (ξ , t ) + (xj , ζjk ), j jk jk k k+1 j,k+1 24 ∂t3 24 ∂x4 ∂x4 8 ∂x2 ∂t2 where

′ ηjk , ζjk ∈ (tk , tk+1 ), ξjk , ξj,k+1 ∈ (xj−1 , xj+1 ).

Therefore there exists a constant c2 such that |Rjk | ≤ c2 (τ 2 + h2 ), 1 ≤ j ≤ m − 1, 0 < k ≤ n − 1. k+ 12

k+ 12

= (|Ujk |2 + |Ujk+1 |2 )Uj

Let Gj term

−(|Ujk |2

k+ 12

Gj

+

k+ 1 |Ujk+1 |2 )uj 2

to the k+ 12

k+ 12

− (|ukj |2 + |uk+1 |2 )uj j

k+ 1 Gj 2 ,

) k+ 1 ( + |Ujk |2 + |Ujk+1 |2 − |ukj |2 − |uk+1 |2 uj 2 j ¯ k + uk (U k − uk ) + [(U k − uk )U

(|Ujk |2 + |Ujk+1 |2 )ej

=

(|Ujk |2 + |Ujk+1 |2 )ej 2 j j j j j 1 ¯ k+1 + uk+1 (U k+1 − uk+1 )]uk+ 2 +(U k+1 − uk+1 )U

=

(|Ujk |2 + |Ujk+1 |2 )ej

j

j

j k+ 12

, and add the

we obtain

=

k+ 1

(4.6)

j

j

j

j

j

k+ 12

¯ k + uk e¯k + ek+1 U ¯ k+1 + uk+1 e¯k+1 )u + (ekj U j j j j j j j j

167

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On the L∞ convergence of a nonlinear difference scheme

8

Noticing the initial-boundary value conditions (1.2)-(1.3) and (1.5)-(1.6), we have ( ) k+ 1 ( ) k+ 1 k+ 1 G0 2 = |U0k |2 + |U0k+1 |2 U0 2 − |uk0 |2 + |uk+1 |2 u0 2 = 0, 0 ( k 2 ) k+ 1 ( ) k+ 21 k+ 1 k+1 2 2 Gm 2 = |Um | + |Um | Um 2 − |ukm |2 + |uk+1 um = 0. m | According to Lemma 4.2, we get ( { })2 ( ) k+ 1 k+ 1 k+ 1 k+ 1 |Gj 2 | ≤ max |Ujk |, |Uj 2 |, |ukj |, |uj 2 | · 2|ekj | + |ej 2 | , or we can say there exists a positive constant c3 such that ( ) 1 ∥Gk+ 2 ∥2 ≤ c3 ∥ek ∥2 + ∥ek+1 ∥2 , 0 < k ≤ n − 1, ) ( 1 |Gk+ 2 |21 ≤ c3 ∥ek ∥2 + ∥ek+1 ∥2 + |ek |21 + |ek+1 |21 , 0 < k ≤ n − 1. k+ 12

Multiplying the (4.3) by h¯ ej ih

m−1 ∑ j=1

k+ 12

(δt ej

k+ 12

)¯ ej

+h

m−1 ∑

1 i 2τ (∥ek+1 ∥2 − ∥ek ∥2 ) − h

+ 2q h

m−1 ∑ ( j=1

¯ k + uk e¯k + ekj U j j j

j=1

(4.8)

, summing j from 1 to m − 1, we have

k+ 21

k+ 12

(δx2 ej

j=1 m ∑

(4.7)

)¯ ej

+ 2q h

k+ 1

|δx ej− 12 |2 + 2q h

j=1 m−1 ∑ (

2

¯ k+1 ek+1 U j j

+

m−1 ∑

j=1

uk+1 e¯k+1 j j

)

k+ 12 k+ 12 e¯j

Gj

=h

m−1 ∑ j=1

k+ 21

Rjk e¯j

,

) k+ 1 |Ujk |2 + |Ujk+1 |2 |ej 2 |2 k+ 12 k+ 12 e¯j

uj

=h

m−1 ∑ j=1

k+ 12

Rjk e¯j

.

Taking the imaginary part and then using (4.1), (4.6) and Theorem 3.2, we can get 1 2τ

(

∥ek+1 ∥2 − ∥ek ∥2

)

≤

|q| 2 h

m−1 ∑ j=1

( ) ¯ k + uk e¯k + ek+1 U ¯ k+1 + uk+1 e¯k+1 | ekj U j j j j j j j

k+ 1 k+ 1 ·uj 2 e¯j 2 |

≤

|q| 2 h

m−1 ∑ ( j=1

k+ 1 ·c1 |ej 2 |

≤

|q| 2 h(c0

m−1 ∑ j=1

k+ 12

|Rjk e¯j

|

|ekj |c0 + c1 |ekj | + |ek+1 |c0 + c1 |ek+1 | j j

+h

m−1 ∑

k+ 21

|Rjk ||¯ ej

j=1 m−1 ∑

+ c1 )c1

+ 21 c22 (τ 2 ≤

+h

j=1 2 2

+h ) +

[

|q| 1 2 (c0 + c1 )c1 + 4 1 2 2 2 2 + 2 c2 (τ + h ) , 0

)

|

(|ekj | + |ek+1 |) · 21 (|ekj | + |ek+1 |) j j

h 2

]

m−1 ∑ [

1 k 2 2 (|ej |

j=1 k 2

] + |ek+1 |2 ) j

(∥e ∥ + ∥ek+1 ∥2 ) ≤ k ≤ n − 1.

Thus, ( ( )) k+1 2 1 1 (− τ |q|(c ( 0 + c1 )c1 + 2 ∥e)) ∥ ≤ 1 + τ |q|(c0 + c1 )c1 + |q| ∥ek ∥2 + τ c22 (τ 2 + h2 )2 , 0 ≤ k ≤ n − 1. 2

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On the L∞ convergence of a nonlinear difference scheme

9

( ) Let β = τ |q|(c0 + c1 )c1 + 12 . When β ≤ 13 , we have ∥ek+1 ∥2 ≤ (1 + 3β)∥ek ∥2 +

1 τ c2 (τ 2 + h2 )2 , 1−β 2

or we can say [ ( )] 1 3 k+1 2 ∥e ∥ ≤ 1 + 3τ |q|(c0 + c1 )c1 + ∥ek ∥2 + τ c22 (τ 2 +h2 )2 , 0 ≤ k ≤ n−1. 2 2 According to Gronwall Inequality in Lemma 4.1, we obtain [ ( )] [ ] 1 c22 (τ 2 + h2 )2 k 2 0 2 ∥e ∥ ≤ exp 3kτ |q|(c0 + c1 )c1 + · ∥e ∥ + , 2 2|q|(c0 + c1 )c1 + 1 1 ≤ k ≤ n. By the initial-boundary value conditions, we could easily know ∥e0 ∥ = 0, so [ ] c2 ∥ek ∥ ≤ exp 23 T (|q|(c0 + c1 )c1 + 21 ) √ (τ 2 + h2 ) 2|q|(c0 +c1 )c1 +1 (4.9) = c4 (τ 2 + h2 ), 0 ≤ k ≤ n. k+ 1

Multiplying the (4.3) by −hδt e¯j 2 , summing j from 1 to m − 1 and taking the real part, we have { } m−1 ∑ ( 2 k+ 12 ) ( k+ 12 ) −Re h δ x ej δt e¯j { j=1 } { } (4.10) ) m−1 m−1 ∑ k( ∑ k+ 21 ( k+ 12 ) q k+ 12 = 2 Re h δt e¯j − Re h R j δ t ej . Gj j=1

j=1

Now, we estimate each term of (4.10). Firstly, simplifying the left of (4.10), we obtain { } m−1 ∑ ( 2 k+ 12 ) ( k+ 12 ) −Re h δ x ej δt e¯j j=1 ) ( ) m ( ∑ k+ 1 k+ 1 = h δx ej− 12 · δx δt e¯j− 12 2 2 j=1 ) ( ) m ( ∑ k+ 12 k+ 21 = h δx ej− 1 · δt δx e¯j− 1 2 2 j=1 ( ) ( ) m ∑ k+1 1 1 k k = h ¯k+1 ¯ 1 1 − δx e 2 δx ej− 12 + δx ej− 12 · τ δx e j− 2 j− 2 j=1 ) m ( ∑ 1 = 2τ h |δx ek+1 |2 − |δx ekj− 1 |2 , j− 1 j=1

that is

2

2

   m−1 ∑ ( k+ 1 ) ( k+ 1 ) ) 1 ( k+1 2 −Re h δx2 ej 2 δt e¯j 2 = |e |1 − |ek |21 .   2τ j=1

169

(4.11)

Xiaoman Liu et al 161-175

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On the L∞ convergence of a nonlinear difference scheme

10

Let the right term of (4.10) be A1 , A2 separately. By the error equation (4.3), we have k+ 12

δt e¯j

k+ 12

= −iδx2 e¯j

q ¯ k+ 12 ¯ jk . − iG + iR 2 j

(4.12)

Substituting (4.12) into A1 , we obtain { } m−1 ∑ k+ 21 ( k+ 12 ) Re h Gj δt e¯j { j=1 } ) m−1 1 1 ∑ k+ 21 ( q ¯ k+ 2 2 k+ 2 k ¯ = Re h −iδx e¯j Gj − 2 iGj + iRj } { j=1 m−1 m−1 1 ∑ 2 k+ 21 k+ 12 ∑ 1 k+ q ¯k G 2 − i 2 ∥Gk+ 2 ∥2 + ih δx e¯j Gj R = Re −ih j j j=1 j=1 { } m−1 m−1 ∑ 2 k+ 12 k+ 12 ∑ ¯ k k+ 12 = Im −h δ e¯ G +h R G x j

j=1

j

j

j=1

j

= B1 + B2 , where B1 ≤ |h

m−1 ∑

k+ 12

δx2 e¯j

k+ 12

Gj

j=1

B2 ≤ |h

m−1 ∑

 k+ 12

¯ jk G R j

j=1

|≤|−h

|≤

m ∑

k+ 1

k+ 1

2

2

δx e¯j− 12 δx Gj− 12 | ≤

j=1 m−1 ∑

h ¯ k |2 + |R 2 j=1 j

m−1 ∑ j=1

 k+ 12

|Gj

|2  ≤

) 1 1 ( k+ 1 2 |e 2 |1 + |Gk+ 2 |21 , 2 ) 1 1 ( ¯k 2 ∥R ∥ + ∥Gk+ 2 ∥2 . 2

Then according to (4.6)-(4.8), we can estimate the first right term A1 as follow ( ) 1 1 ¯ k ∥2 + ∥Gk+ 12 ∥2 A1 ≤ 4q |ek+ 2 |21 + |Gk+ 2 |21 + ∥R ) ( 1 ≤ 4q |ek+ 2 |21 + 4q c3 ∥ek ∥2 (+ ∥ek+1 ∥2 + |ek |21)+ |ek+1 |21 +[4q c22 (τ 2 + h2 )2 + 4q c3 ∥e]k ∥2 + ∥ek+1 ∥2 ( ) 1 = 4q |ek+ 2 |21 + c22 (τ 2 + h2 )2 + 2q c3 ∥ek ∥2 + ∥ek+1 ∥2 ( ) +[4q c3 |ek |21 + |ek+1 |21 ] ( ) 1 ≤ 4q |ek+ 2 |21 + c22 (τ 2 + h2 )2 + qc3 c24 (τ 2 + h2 )2 + 4q c3 |ek |21 + |ek+1 |21 ( ) ( ) 1 = 4q c3 |ek |21 + |ek+1 |21 + 4q |ek+ 2 |21 + qc3 c24 + 4q c22 (τ 2 + h2 )2 . According to (4.6), we can also estimate the second right term A2 as follow A2 ≤

) 1( ) 1 1 1( k 2 ∥R ∥ + |ek+ 2 |21 ≤ c21 (τ 2 + h2 )2 + |ek+ 2 |21 . 2 2

Now, substituting the three estimations just represented before into (4.10),

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On the L∞ convergence of a nonlinear difference scheme we obtain ≤ ≤ ≤ =

(

11

)

1 |ek+1 |21 − |ek |21 2τ ( ( ) ) 1 q k 2 k+1 2 |1 + 4q |ek+ 2 |21 + qc3 c24 + 4q c22 (τ 2 + h2 )2 4 c3 |e |1 + |e 1 + 21 c22 (τ 2 + h2 )2 + 21 |ek+ 2 |21 ( ( ) 2 ) q k+ 12 2 2 k 2 k+1 2 2 2 |1 + qc3 c24 + q+2 |1 ) + q+2 4 c3 (|e |1 + |e 4 |e 4 c2 (τ + h )) ( q+2 2 q+2 1 q k 2 k+1 2 k 2 k+1 2 2 |1 + )4 2 ((|e |1 + |e |1)) + qc3 c4 + 4 c2 (τ 2 4 c3 |e |1(+ |e 2qc3 +q+2 q+2 2 k 2 k+1 2 2 |e |1 + |e |1 + qc3 c4 + 4 c2 (τ 2 + h2 )2 , 8

+ h2 )2

that is ( ) k+1 2 ( ) k2 1 − τ 2qc3 +q+2 |e |1 ≤ 1(+ τ 2qc3 +q+2 |e) |1 4 4 2 2 2 2 c + 2qc3 c24 + q+2 2 (τ + h ) , 0 ≤ k ≤ n − 1. 2 . When β ≤ 13 , we have Let β = τ 2qc3 +q+2 4 ) ( ) ( 2qc3 + q + 2 3 q+2 2 k+1 2 k 2 2 |e |1 ≤ 1 + 3τ |e |1 + τ 2qc3 c4 + c (τ 2 + h2 )2 , 4 2 2 2 0 ≤ k ≤ n − 1. Denote 2qc3 + q + 2 q+2 2 , c6 = 2qc3 c24 + c , 4 2 2 then we rewrite the inequality as follow c5 =

3 |ek+1 |21 ≤ (1 + 3τ c5 )|ek |21 + τ c6 (τ 2 + h2 )2 , 0 ≤ k ≤ n − 1. 2 Using Gronwall inequality, we get ( ) c6 (τ 2 + h2 )2 |ek |21 ≤ exp(3kτ c5 ) · |e0 |21 + , 1 ≤ k ≤ n. 2c5 By the initial-boundary value conditions, we also know |e0 |21 = 0, so |ek |21

≤ ≤

2

2 2

) exp(3kτ c5 ) · c6 (τ 2c+h 5 c6 (τ 2 + h2 )2 , 0 ≤ k ≤ n. exp(3c5 T ) 2c 5

(4.13)

According to (1) in Lemma 4.3, we have ∥ek ∥2∞

≤ ≤

1 k 2 4 |e |1 c6 2 8c5 exp(3c5 T )(τ

Denote

√ C=

+ h2 )2 , 0 ≤ k ≤ n.

c6 exp(3c5 T ). 8c5

Therefore, when τ is small enough, there exists a constant C independent of h, τ such that ∥ek ∥∞ ≤ C(τ 2 + h2 ), 0 ≤ k ≤ n. (4.14) This completes the proof.

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On the L∞ convergence of a nonlinear difference scheme

5

12

Iterative algorithm

There are some discrete methods about the nonlinear Schr¨odinger equations [12]-[13]. In this section, we use an iterative method [9] to compute the solution of the difference scheme (2.3)-(2.4). Define the following iterative method ( ( ) ) (l) (l−1) (l) (l) i τ2 vj − ukj + δx2 vj + 2q |ukj |2 + |2vj − ukj |2 vj = 0, (5.1) 1 ≤ j ≤ m − 1, 0 ≤ k ≤ n − 1, (l)

v0 = 0, vm (l) = 0, (0)

where vj

(5.2)

= ukj , 0 ≤ j ≤ m, l = 1, 2, . . .. (l)

Multiplying the (5.1) by h¯ vj , summing j from 1 to m − 1 and taking the imaginary part, we have m−1 ∑ 2 (l) 2 (l) ∥v ∥ − Reh ukj v¯j = 0, τ j=1

that is 2 (l) 2 τ ∥v ∥

= Reh ( ≤ h

j=1 m−1 ∑ j=1

( h

=

m−1 ∑

m−1 ∑ j=1

(l)

ukj v¯j

(ukj )2 ·

m−1 ∑ j=1

) 12 (l) (¯ vj )2

) 12 ( (ukj )2

·

h

m−1 ∑ j=1

) 12 (l) (¯ vj ) 2

= ∥uk ∥ · ∥v (l) ∥. Thus, ∥v (l) ∥ ≤ ∥uk ∥, l = 1, 2, . . . .

(5.3)

Denote (l)

(l)

εj = vj − vj , 0 ≤ j ≤ m. Theorem 5.1. Suppose that the solution is {ukj | 0 ≤ j ≤ m}, τ is sufficiently small enough, then the iterative method (5.1) − (5.2) is convergent. Proof. Subtracting (5.1)-(5.2) from (2.3)-(2.4), we obtain [ ] (l) (l) (l−1) (l) i τ2 εj + δx2 εj + 2q (|ukj |2 + |2vj − ukj |2 )vj − (|ukj |2 + |2vj − ukj |2 )vj (5.4) = 0, 1 ≤ j ≤ m − 1, (l)

ε0 = ε(l) m = 0.

172

(5.5)

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On the L∞ convergence of a nonlinear difference scheme

13

Multiplying the (5.4) by hε¯j (l) , summing j from 1 to m − 1, we have i τ2 h

m−1 ∑

(l)

|εj |2 + h

j=1 m−1 ∑ [

+ 2q h

|2vj −

j=1

m−1 ∑ j=1

(l)

(δx2 εj )ε¯j (l) + 2q h

ukj |2 vj

−

(l−1) |2vj

−

j=1

(l) ukj |2 vj

(l−1)

Noticing the term in brackets, we add |2vj (l−1)

m−1 ∑

(l)

|ukj |2 |εj |2 ]

(5.6) ε¯j (l) = 0.

− ukj |2 vj to this term as follow

(l)

|2vj − ukj |2 vj −(|2vj −) ukj |(2 vj ) (l−1) (l) (l−1) = |2vj − ukj |2 vj − vj + |2vj − ukj |2 − |2vj − ukj |2 vj ( ) (l−1) (l) (l−1) (l−1) − ukj |2 εj + 2εj = |2vj − ukj )2ε¯j (l−1) vj . (2vj − ukj ) + (2vj Then, substituting this rewritten term into (5.6) and taking the imaginary part, we have

≤

2 (l) 2 τ ∥ε ∥ m−1 ∑ ( (l−1) |q| (2vj 2εj 2 h j=1

(l−1)

− ukj ) + (2vj

) − ukj )2ε¯j (l−1) vj ε¯j (l)

) m−1 ∑ ( (l−1) |q| · ∥ε(l−1) ∥∞ · ∥ε(l) ∥∞ · h |2vj − ukj | + |2vj − ukj | |vj | ( j=1 ) ≤ |q| · ∥ε(l−1) ∥∞ · ∥ε(l) ∥∞ · ∥2v∥ + ∥uk ∥ + ∥2v (l−1) ∥ + ∥uk ∥ ∥v∥ ≤ |q| · ∥ε(l−1) ∥∞ · ∥ε(l) ∥∞ · 6∥v∥2 . ≤

According to (5.3) and Theorem 3.1, we obtain ∥ε(l) ∥2

≤ =

3τ |q| · ∥ε(l−1) ∥∞ · ∥ε(l) ∥∞ · ∥uk ∥2 3τ |q| · ∥ε(l−1) ∥∞ · ∥ε(l) ∥∞ · ∥u0 ∥2 .

(5.7)

Similarly, taking the real part of (5.6), we have |ε(l) |21 ≤

|q| k 2 2 ∥u ∥

· ∥ε(l) ∥2∞ +

|q| (l) 2 2 ∥ε ∥∞

·h

m−1 ∑ [ j=1

(l−1) 2

4(vj

(l−1) k uj

) + (ukj )2 − 4vj

]

+|q| · ∥ε(l−1) ∥∞ · ∥ε(l) ∥∞ · 6∥v∥2 ( ) |q| k 2 (l) 2 (l) 2 k 2 k 2 k 2 ≤ |q| 2 ∥u ∥ ∥ε ∥∞ + 2 ∥ε ∥∞ · 4∥u ∥ + ∥u ∥ + 4∥u ∥ +6|q| · ∥uk ∥2 ∥ε(l−1) ∥∞ ∥ε(l) ∥∞ ≤ 5|q| · ∥uk ∥2 ∥ε(l) ∥2∞ + 6|q| · ∥uk ∥2 ∥ε(l−1) ∥∞ ∥ε(l) ∥∞ ≤ 5|q| · ∥u0 ∥2 ∥ε(l) ∥2∞ + 6|q| · ∥u0 ∥2 ∥ε(l−1) ∥∞ ∥ε(l) ∥∞ . According to (2) in Lemma 4.3, for any α > 0, there is ∥ε(l) ∥2∞

≤ ≤

1 (l) 2 α|ε ∥ε(l) ∥2 ( |1 + 4α ) 0 2 (l) 2 α 5|q| · ∥u ∥ ∥ε ∥∞ + 6|q| · ∥u0 ∥2 ∥ε(l−1) ∥∞ ∥ε(l) ∥∞ 1 · 3τ |q| · ∥u0 ∥2 ∥ε(l−1) ∥∞ ∥ε(l) ∥∞ . + 4α

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On the L∞ convergence of a nonlinear difference scheme That is ∥ε(l) ∥∞

14

( ) ≤ α 5|q| · ∥u0 ∥2 ∥ε(l) ∥∞ + 6|q| · ∥u0 ∥2 ∥ε(l−1) ∥∞ 1 + 4α · 3τ |q| · ∥u0 ∥2 ∥ε(l−1) ∥∞ .

Taking α = 1/(12|q| · ∥u0 ∥2 ), we get ∥ε(l) ∥∞ ≤

5 (l) 1 ∥ε ∥∞ + ∥ε(l−1) ∥∞ + 9τ q 2 ∥u0 ∥4 ∥ε(l−1) ∥∞ , 12 2

that is ∥ε(l) ∥∞ ≤ When τ q 2 ∥u0 ∥4 ≤ ∥ε(l) ∥∞

12 7

(

) 1 + 9τ q 2 ∥u0 ∥4 ∥ε(l−1) ∥∞ . 2

1 144 ,

we have ( ) 12 1 9 27 (l−1) ≤ + ∥ε(l−1) ∥∞ = ∥ε ∥∞ . 7 2 144 28

This completes the proof.

6

Conclusion

In this paper, we consider a nonlinear finite difference scheme for the Schr¨odinger equations. We prove that the difference scheme has a unique and bounded solution and the finite difference solution is convergent with the convergence order of O(τ 2 + h2 ) in L∞ norm. Finally we give a convergent iterative method to compute the solution of the difference scheme. Acknowledgement: This work is supported by the Natural Science Foundation of China (11171285) and the Foundation Research Project of Jiangsu Province of China (BK20161158).

References [1] T. Kato, On nonlinear Schr¨odinger equations, Ann. Inst. H. Poincat´e Phys. Th´eor. 46, 113-129(1987). [2] J. Zhang, On the finite-time behaviour for nonlinear Schr¨odinger equations, Commun. Math. Phys. 162, 249-260(1994). [3] Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear Schr¨ odinger equation, J. Comput. Phys. 148, 397-415(1999). [4] A. Kurtinaitis and F. Ivanauska, Finite difference solution methods for a system of the nonlinear Schr¨odinger equations, Nonlinear Anal. Mode. Control. 9, 247-258(2004).

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On the L∞ convergence of a nonlinear difference scheme

15

[5] W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schr¨ odinger equation with wave operator, SIAM J. Numer. Anal. 50, 492-521(2012). [6] L. Ignat and E. Zuazua, Convergence rates for dispersive approximation schemes to nonlinear Schr¨ odinger equations, J. Math. Pures Appl. 98, 479517(2012). [7] Z. Sun, X. Wu, J. Zhang and D. Wang, A linearized difference scheme for semilinear parabolic equations with nonlinear absorbing boundary conditions, Appl. Math. Comput. 218, 5187-5201(2012). [8] N. Saito and T. Sasaki, Finite difference approximation for nonlinear Schr¨ odinger equations with application to blow-up computation, Japan J. Indust. Appl. Math. 33, 427-470(2016). [9] Z. Sun, Numerical Methods of the Partial Differential Equations, second ed., Science Press, Beijing, 2005. [10] G. Akrivis, Finite difference discretization of the cubic Schr¨odinger equation, IMA J. Numer. Anal. 13, 115-124(1993). [11] Z. Sun and D. Zhao, On the L∞ convergence of a difference scheme for coupled nonlinear Schr¨ odinger equations, Comput. Math. Appl. 59, 32863300(2010). [12] A. Mai and Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schr¨ odinger equations with superlinear nonlinearities, Abstr. Appl. Anal. 2013, 1-11(2013). [13] G. Sun and A. Mai, Ground state solutions for discrete nonlinear Schr¨ odinger equations with potentials, J. Comput. Anal. Appl. 19, 3951(2015).

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SINGLE POINT V.S. TOTAL BLOW-UP FOR A REACTION DIFFUSION EQUATION WITH NONLOCAL SOURCE DENGMING LIU

Abstract. In this paper, we consider the following initial-boundary value problem of a semilinear parabolic equation with local and nonlocal sources Z ut = ∆u + up + uq (x, t) dx, (x, t) ∈ B × (0, T ) , B

 where p, q > 0, B = x ∈ RN : |x| < R . We completely classify blow-up solutions into total blow-up case and single point blow-up case according to the different values of the nonlinear parameters, and give the blow-up rates of solutions near the blow-up time.

1. Introduction In this paper, we deal with the property of the blow-up solution of the following reaction-diffusion equation with local and nonlocal sources  R   ut = ∆u + up + B uq (x, t) dx, (x, t) ∈ B × (0, T ) ,    (1.1) u (x, t) = 0, (x, t) ∈ ∂B × (0, T ) ,      u (x, 0) = u0 (x) , x ∈ B,  where p, q > 0, B = x ∈ RN : |x| < R . Throughout this paper, we assume that the initial data u0 ∈ C 2 (B) ∩ C(B), u0 (x) = u (r) ≥ 0 with r = |x|, and u00 (r) < 0 for r ∈ (0,RR]. Moreover, we assume that there exists a positive constant δ such that ∆u0 +up0 + B uq0 dx ≥ δ. When min {p, q} ≥ 1, we can easily show the local existence and uniqueness of classical solution of problem (1.1). If min {p, q} < 1, the existence of maximal solution can be proved. Moreover, if max {p, q} > 1, we can prove that the solution of (1.1) blows up in finite time for large initial data. In this paper, we consider the blow-up set of problem (1.1) and denote the blow-up time by T . We now begin with the definition of the blow-up point for a blow-up solution. Definition 1.1. A point x ∈ B is called a blow-up point if there exists a sequence (xn , tn ) such that xn → x, tn % T and u (xn , tn ) → ∞ as n → ∞. The set of all blow-up points is called the blow-up set. For simplification, we denote the blow-up set by S. When S = B, we call this phenomenon “total blowup” and when the blow-up set include only one point, we call this “single point blow-up”. 2000 Mathematics Subject Classification. 35K55, 35K65. Key words and phrases. Nonlocal source; Single point blow-up; Total blow-up. 1

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D. LIU

In 1984, Weissler(see [1]) considered the property of the blow-up solution for the following one-dimensional initial-boundary value problem    ut = ∆u + up , (x, t) ∈ (−R, R) × (0, T ) ,    (1.2) u (x, t) = 0, (x, t) ∈ {−R, R} × (0, T ) ,      u (x, 0) = u0 (x) , x ∈ [−R, R] , where p > 1, and obtained the single point blow-up phenomenon under some suitable conditions. In [2], Friedman and McLeod generalized Weissler’s results to N -dimensional case, and showed that the blow-up point is only the origin, namely, S = {0}. Chadam et al. in [3] studied the following problem with localized reaction term     ut = ∆u + uq (x∗ , t) , (x, t) ∈ B × (0, T ) ,   (1.3) u (x, t) = 0, (x, t) ∈ ∂B × (0, T ) ,      u (x, 0) = u0 (x) , x ∈ B, and proved that total blow-up occurs whenever a solution blows up, that is, S = B. Souplet [4,5] extended the results in [3] to the case for the moving source x∗ (t) and obtained the precise blow-up profiles of the total blow-up solution. Recently, Okada and Fukuda in [7] dealt with the single point and total blow-up for the following problem    ut = ∆u + up + uq (x∗ , t) , (x, t) ∈ B × (0, T ) ,    (1.4) u (x, t) = 0, (x, t) ∈ ∂B × (0, T ) ,      u (x, 0) = u0 (x) , x ∈ B. They showed that p = q + 1 is a cut off between the single point blow-up and the total blow-up for x∗ = 0, and p = q is the critical exponent of the single point blow-up and the total blow-up for x∗ 6= 0. Motivated by above works, we investigate problem (1.1). Similar to [7], the main purpose of this article is to evaluate the effect of the competition between up and R q u dx on the single blow-up and total blow-up. Motivated by the idea of Souplet B in [6], through modifying the construction of auxiliary functions used in [7], we completely classify blow-up solutions into total blow-up case and single point blowup case according to the different values of p and q, and give the blow-up rates of solutions near the blow-up time. In order to state our results, we first let ϕ be a solution of    ϕt = ∆ϕ, (x, t) ∈ B × (0, T ) ,    (1.5) ϕ (x, t) = 0, (x, t) ∈ ∂B × (0, T ) ,      ϕ (x, 0) = ϕ0 (x) ≥ 0, x ∈ B, where ϕ0 ∈ C 2 (B) ∩ C(B), ϕ0 (x) = ϕ (r) with r = |x|, and ϕ00 (r) < 0 for r ∈ (0, R]. The main results of this article are stated as follows.

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Theorem 1.2. Suppose q > p and q > 1, and let u be a solution of (1.1) with u0 = λϕ0 (λ > 0), then there exists a positive constant λ0 (ϕ0 ) such that if λ > λ0 (ϕ0 ), then u blows up on the whole domain, that is, S = B; Moreover, the following estimate − 1 − 1 C1 (T − t) q−1 ≤ u (x, t) ≤ C2 (T − t) q−1 , t → T (1.6) holds for any compact subset of B, here C1 , C2 are positive constants. Theorem 1.3. Suppose p ≥ q and p > 1, then all blow-up solutions of problem (1.1) blow up only at the origin, namely, S = {0}; Moreover, there exist positive constants C3 and C4 such that 1 − p−1

C3 (T − t)

1 − p−1

≤ u (0, t) ≤ C4 (T − t)

,

t → T.

(1.7)

Remark 1.4. From Theorems 1.1 and 1.2, we know that p = q is the critical exponent for single point blow-up and total blow-up. This paper is organized as follows. In the next section, we will give some lemmas. In section 3, we concern with the single point blow-up and the total blow-up, and give the proofs of Theorems1.1 and 1.2, respectively. 2. Preliminary In this section, we will state two important lemmas, which will be used in the sequel. Lemma 2.1. Suppose q > 1 and q > p, let u (x, t) be a solution of (1.1) with u0 (x) = λϕ0 (x), then there exists a positive constant λ0 (ϕ0 ) such that u (x, t) ≥

ϕ (x, t) u (0, t) ≡ ψ (x, t) u (0, t) , 2ϕ0 (0)

(x, t) ∈ B × [0, T ),

(2.1)

holds if λ > λ0 (ϕ0 ). Proof. Using maximum principle (see [4]), we have 0 ≤ ϕ (x, t) ≤ ϕ0 (x) ≤ ϕ0 (0) . Because of q > p and u0 (0) = λϕ0 (0), we can choose λ large enough such that Z p−q 2ψ (x, t) up−q (0) ≤ {λϕ (0)} ≤ ψ q (x, t) dx. (2.2) 0 0 B

Now, letting U = u (x, t) − ψ (x, t) u (0, t) , after a series of simple computation, we have   Z Z p q p q Ut − ∆U = u + u dx − ψ (x, t) ∆u (0, t) + u (0, t) + u dx B (2.3) Z B 1 ≥ uq dx − ψ (x, t) up (0, t) . 2 B R On the other hand, ∆u0 + up0 + B uq0 dx ≥ δ means ut ≥ 0, thus, for any t ∈ [0, T ), we see u (0, t) ≥ u0 (0) . (2.4)

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Combining (2.2), (2.3) with (2.4), we know Z  1 p−q q q Ut − ∆U ≥ u dx − 2ψ (x, t) u0 (0) u (0, t) 2 ZB  Z 1 q q q ≥ u dx − ψ u (0, t) dx 2 B Z B q = U Φdx, 2 B where Z 1

q−1

[θu + (1 − θ) ψu (0, t)]

Φ=

(2.5)

dθ.

0

In addition, for any (x, t) ∈ ∂B × (0, T ), we have U (x, t) = 0,

(2.6)

and U (x, 0) = u0 (x) − ψ (x, 0) u0 (0) = λϕ0 (x) −

ϕ0 (x) λϕ0 (x) λϕ0 (0) = ≥ 0. (2.7) 2ϕ0 (0) 2

From (2.5), (2.6), (2.7) and maximum principle, it follows that U (x, t) ≥ 0,

(x, t) ∈ B × [0, T ),

which leads to (2.1). The proof of Lemma 2.1 is complete.



Lemma 2.2. Suppose p > 1 and q ≥ 0, let u (x, t) be a solution of (1.1), then there exists a positive constant η such that   Z ut ≥ ηϕ (x, t) up + uq dx , (x, t) ∈ B × [0, T ). (2.8) B

Proof. Putting 

p

Z

J (x, t) = ut (x, t) − ηϕ (x, t) u +

 u dx , q

B

where η will be chosen later. Computing directly, we obtain   Z p−1 p−1 p q Jt − ∆J − pu J ≥ ηpϕu u + u dx − ut + ∆u  B Z  Z p q − η (ϕt − ∆ϕ) u + u dx + q (1 − ηϕ) uq−1 ut dx B

B 2

+ 2ηpup−1 ∇u · ∇ϕ + ηp (p − 1) ϕup−2 |∇u| Z ≥ 2ηpup−1 ∇u · ∇ϕ + q (1 − ηϕ) uq−1 ut dx. B

(2.9) Since u and ϕ are radially symmetric and monotone decreasing with respect to r, we have x x x x xn  xn  1 2 1 2 ∇u · ∇ϕ = ur , ,··· , · ϕr , ,··· , = ur ϕr ≥ 0. r r r r r r R On the other hand, by maximum principle, the assumption ∆u0 + up0 + B uq0 dx ≥ δ implies that ut ≥ 0. Choosing η small enough such that 1 − ηϕ (x, t) ≥ 0,

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(2.10)

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we then have Jt − ∆J − pup−1 J ≥ 0.

(2.11)

Moreover, we can verify that  (x) + (x) dx J (x, 0) = ut (x, 0) − ηϕ0 (x) B   Z Z p q p q = ∆u0 (x) + u0 (x) + u0 (x) dx − ηϕ0 (x) u0 (x) + u0 (x) dx B B   Z ≥ µ − ηϕ0 (0) up0 (0) + uq0 (0) dx 

Z

up0

uq0

B

≥ 0, (2.12) holds for sufficiently small η. In addition, for any (x, t) ∈ ∂B × (0, T ), we have J (x, t) = 0.

(2.13)

From (2.11), (2.12), (2.13) and maximum principle, it follows that J (x, t) ≥ 0,

(x, t) B × [0, T ),

which yields (2.8). The proof of Lemma 2.2 is complete.



3. Proof of Theorems 1.1 and 1.2 In this section, we will discuss the single point and total blow-up phenomena according to the different values of p and q. Firstly, we give the proof of Theorem 1.1. Proof of Theorem 1.1. Since lim u (0, t) = +∞,

t→T

we can easily show the total blow-up result under the condition q > p from Lemma 2.1. Moreover, noticing the fact that max u (x, t) = u (0, t) , x∈B

we have ∆u (0, t) ≤ 0. On the other hand, thanks to q > p and q > 1, there exists t1 ∈ (0, T ) such that Z ut (0, t) = ∆u (0, t) + up (0, t) + uq (0, t) dx B Z (3.1) p q ≤ u (0, t) + u (0, t) dx B

≤ (|B| + 1) uq (0, t) ,

t ∈ (t1 , T ) .

Combining (3.1) with Lemma 2.1, we obtain 1 − q−1

C1 (T − t)

≤ Cu (0, t) ≤ u (x, t) ,

(x, t) ∈ K × (t1 , T ) ,

(3.2)

where K is any compact subset of B.

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Furthermore, it follows that, from (2.1) and (2.8), Z ut (0, t) ≥ ηϕ (0, t) uq dx ≥ η |B| ϕ (0, t) ψ q (0, t) uq (0, t) .

(3.3)

B

Integrating (3.3) from 0 to t, we conclude 1 − q−1

u (x, t) ≤ u (0, t) ≤ C2 (T − t)

,

(x, t) ∈ B × (0, T ) .

(3.4)

Combining (3.2) with (3.3), we get (1.6). The proof of Theorem 1.1 is complete.  Now, by the method of contradiction, we give the proof of Theorem 1.2. Proof of Theorem 1.2. Using mean value theorem, and noticing the fact that u (x, t) is radially symmetric and monotone decreasing with respect to r, we know that there exists a unique point x∗ such that Z uq dx = |B| uq (x∗ , t) , x∗ 6= 0 and x∗ 6∈ ∂B. B

We suppose that, on the contrary, there is a blow-up point x0 6= 0 (|x0 | ≤ |x∗ | = r0 ). Since u (x, t) is radially symmetric and monotone decreasing on r, then for any r1 ∈ [0, r0 ], we see lim u (r1 , t) = ∞. t→T

Letting 0 < µ1 < µ < µ2 < r0 , and  S0 (µ, γ) = x = (x1 , x2 , · · · , xN ) ∈ RN : µ < x1 < µ + γ, 0 < xj < γ (j = 2, · · · N ) , here γ is a sufficiently small constant such that S0 (µ, γ) ∈ B (µ2 ) \B (µ1 ). Defining an auxiliary function as the form F (x, t) = ux1 (x, t) + b (x) um (x, t) ,

(x, t) ∈ S0 × [0, T ),

(3.5)

where , m will be determined later, and b (x) = sin

N π (x1 − µ) Y πxj sin . γ γ j=2

Calculating directly, we obtain   2m∇b · ∇u m−1 p−1 Ft − ∆F − pu − u F ux1 Z π 2 N m =  (m − p) bup+m−1 + mbum−1 uq dx + bu γ2 B π 2 N m =  (m − p) bup+m−1 + mb |B| um−1 uq (x∗ , t) + bu γ2 2m2 ∇b · ∇u 2m−1 2 − m (m − 1) bum−2 |∇u| + bu ux1 π 2 N m 2m2 ∇b · ∇u 2m−1 ≤  (m − p) bup+m−1 + mb |B| um+q−1 + bu + bu . γ2 ux1 On the other hand, it is easy to verify that 2m∇b · ∇u 2mπN µ2 0< < . ux1 γµ1

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p For the case p = q, we can choose m < 1+|B| and τ1 large enough such that   2m∇b · ∇u m−1 p−1 Ft − ∆F − pu u − F ux1   2mπN µ2 m π2 N (3.6) m−1 p ≤ −bu [p − m (1 + |B|)] u − 2 u − u γ γµ1 ≤ 0,

holds for every (x, t) ∈ S0 × [τ1 , T ). For the case p > q, we can take m < p and τ2 large enough such that, for any (x, t) ∈ S0 × [τ2 , T ), the following inequality holds   2m∇b · ∇u m−1 u F Ft − ∆F − pup−1 − ux1   π2 N 2mπN µ2 m (3.7) ≤ −bum−1 (p − m) up − m |B| uq − 2 u − u γ γµ1 ≤ 0. Next, putting τ = max {τ1 , τ2 }, and taking  small enough, such that F (x, τ ) = ux1 (x, τ ) + b (x) um (x, τ ) (3.8)

≤ max ux1 (x, τ ) +  max um (x, τ ) < 0. x∈S0

x∈S0

In addition, we can easily check that, F (x, t) = ux1 (x, t) < 0,

(x, t) ∈ ∂S0 × [τ, T ).

(3.9)

Combining (3.6), (3.7) and (3.8) with (3.9), we conclude immediately that F (x, t) ≤ 0,

for any (x, t) ∈ S0 × [τ, T ),

which implies − u−m ux1 ≥ b (x) .

(3.10)

Fixing a0 = (a2 , · · · aN ) ∈ RN −1 , and denoting a1 = (µ + γ, a2 , · · · aN ) . Integrating (3.10) by x1 from µ to µ + γ, we have Z µ+γ N 1 2γ Y πxj ≤ . 0< b (x) dx1 = sin π j=2 γ (m − 1) um−1 (a1 , t) µ

(3.11)

Since lim u (a1 , t) = +∞,

t→T

from (3.11), we have a contradiction. Hence, u (x, t) blows up only at the origin. In light of max u (x, t) = u (0, t) , x∈B

then similar to the process of the derivation of (3.1) and (3.2), we find that 1 − p−1

C3 (T − t)

≤ u (0, t) ,

182

t → T.

(3.12)

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Now, using Lemma 2.2, we get ut (0, t) ≥ ηϕ (0, t) up (0, t) .

(3.13)

From (3.13), it follows immediately that 1 − p−1

u (0, t) ≤ C4 (T − t)

,

t → T.

(3.14)

Combining (3.12) with (3.14), we arrive at (1.7). The proof of Theorem 1.2 is complete.  Acknowledgements The author would like to thank professor Chunlai Mu of Chongqing University for his encouragements and discussions. This work is supported by the Scientific Research Fund of Hunan Provincial Education Department (16A071, 15C0537). References [1] F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Diff. Equns., (55)1984, 204-224. [2] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., (34)1985, 425-447. [3] J. M. Chadam, A. Pierce and H. M. Yin, The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl., (169)1992, 313-328. [4] P. Souplet, Blow-up in nonlocal reaction diffusion equations, SIAM J. Math. Anal., (29)1998, 1301-1334. [5] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Diff. Equns., (153)1999, 374-406. [6] P. Souplet, Single-point blow-up for a semilinear parabolic system, J. Eur. Math. Soc., 11(2009), 169-188. [7] A. Okada and I. Fukuda, Total versus single point blow-up of solutions of a semilinear parabolic equation with localized reaction, J. Math. Anal. Appl., (281)2003, 485-500. Dengming Liu1 School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, PR China

1Corresponding author: [email protected]

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COMMON FIXED POINT RESULTS FOR WEAKLY COMPATIBLE MAPPINGS USING C-CLASS FUNCTIONS GENO KADWIN JACOB1 , ARSLAN HOJAT ANSARI2 , CHOONKIL PARK∗3 , N. ANNAMALAI4 Abstract. In this paper, using the concept of C -class fuction, we prove the existence of common fixed point for generalized Zamfirescu-type mappings and generalized weakly Zamfirescutype mappings. Our results generalize so many results in the literature.

1. Introduction In 1922, Banach proved the existence of fixed point on complete metric space (X, d). A mapping f has been considered to be a contraction and a self-mapping. Definition 1.1. Let (X, d) be a metric space. A mapping f : X → X is said to be a contraction mapping if there exists k ∈ [0, 1) such that d(f (x), f (y)) ≤ kd(x, y). Later many authours have proved fixed point existence on several type of generalized contractions. Kannan type and Chatterjea type mappings were significant type of mappings since they provided existence of fixed point for non-continuous mappings in literature (see [4, 6]). In 1972, Zamfirescu [7] generalized functions of Banach, Kannan and Chatterjea by introducing a new kind of mapping and proved the existence of fixed points for mappings. Definition 1.2. Let (X, d) be a metric space. A mapping f : D → X is said to be a Zamfirescu mapping if for all x, y ∈ X it satisfies the condition d(f (x), f (y)) ≤ kMf (x, y) for some k ∈ [0, 1), where n  1 o 1 Mf (x, y) := max d(x, y), d(x, f (x)) + d(y, f (y)) , d(x, f (y)) + d(y, f (x)) . 2 2 Apart all these generalizations, Dugundji and Granas [5] in 1978 generalized the contraction mapping as follows. Definition 1.3. [5] Let (X, d) be a metric space and D ⊂ X. A mapping f : D → X is said to be a weakly contraction mapping if there exists α : D × D → [0, 1] such that Θ(a, b) := sup α(x, y) : a ≤ d(x, y) ≤ b < 1 for every 0 < a ≤ b and for all x, y ∈ D, d(f (x), f (y)) ≤ α(x, y)d(x, y). 2010 Mathematics Subject Classification. Primary 47H10; 54H25. Key words and phrases. cyclic coupled contraction; best proximity point; multivalued mapping; fixed point; C-class function. ∗ Corresponding author.

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In 2014, the concept of C-class functions was introduced by Ansari [2]. By using this concept, we can generalize many fixed point theorems in the literature. Definition 1.4. [2] A function F : [0, ∞)2 → R is called a C-class function (also denoted as C ) if it is continuous and satisfies the following: (1) F (s, t) ≤ s; (2) F (s, t) = s implies that either s = 0 or t = 0 for all s, t ∈ [0, ∞). Example 1.5. [2] The following functions f : [0, ∞)2 → R are elements of C, for all s, t ∈ [0, ∞): (1) f (s, t) = s − t, f (s, t) = s ⇒ t = 0; (2) f (s, t) = ms, 0 e, f (s, 1) = s ⇒ s = 0. Definition 1.6. Let Ψ denote all the functions ψ : [0, ∞) → [0, ∞) which satisfy (i) ψ(t) = 0 if and only if t = 0, (ii) ψ is continuous, (iii) ψ(s) ≤ s, ∀s > 0. Definition 1.7. Let (X, d) be a metric space. Then f, g : X → X are said to be weakly compatible if f g(x) = gf (x) for x ∈ X whenever f (x) = g(x). Lemma 1.8. ( [3]) Suppose that (X, d) is a metric space. Let {xn } be a sequence in X such that d(xn , xn+1 ) → 0 as n → ∞. If {xn } is not a Cauchy sequence, then there exist an ε > 0 and sequences of positive integers {m(k)} and {n(k)} with m(k) > n(k) > k such that d(xm(k) , xn(k) ) ≥ ε, d(xm(k)−1 , xn(k) ) < ε and (i) limk→∞ d(xm(k)−1 , xn(k)+1 ) = ε; (ii) limk→∞ d(xm(k) , xn(k) ) = ε; (iii) limk→∞ d(xm(k)−1 , xn(k) ) = ε. In this paper, we prove the existence of common fixed point for two weakly compatible mappings on a complete metric space. 2. Main results Definition 2.1. Let (X, d) be a metric space. Consider two self-mappings f and g on X and let α : X × X → [0, 1] be a function. Then g is an f -weakly generalized Zamfirescu type mapping if, for all F ∈ C, ψ ∈ Ψ and for all x, y ∈ X,  n 
1 d(g(x), g(y)) ≤F α f (x), f (y) max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) , 2  n
 1 ψ α f (x), f (y) max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) . 2

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Theorem 2.2. Let (X, d) be a complete metric space and f, g : X → X mappings such that g is an f -weakly generalized Zamfirescu type mapping. Then f and g have a unique common fixed point on X if the following conditions are satisfied: (1) g(X) ⊂ f (X); (2) f (X) is complete; (3) f, g are weakly compatible. Proof. Choose x0 ∈ Y arbitrarily and xn ∈ X such that f (xn ) = g(xn−1 ). Then d(f (xn ), f (xn+1 )) = d(g(xn−1 ), g(xn )) (2.1)  n
  1 ≤ F α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )), d(f (xn−1 ), g(xn−1 )) + d(f (xn ), g(xn )) , 2 o 1 d(f (xn ), g(xn−1 )) + d(f (xn−1 ), g(xn )) , 2  n
 1 ψ α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )), d(f (xn−1 ), g(xn−1 )) + d(f (xn ), g(xn )) , 2 o 1 d(f (xn ), g(xn−1 )) + d(f (xn−1 ), g(xn )) 2 n
 1 ≤ α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )), d(f (xn−1 ), g(xn−1 )) + d(f (xn ), g(xn )) , 2 o 1 d(f (xn ), g(xn−1 )) + d(f (xn−1 ), g(xn )) 2 n
 1 ≤ α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )), d(f (xn−1 ), f (xn )) + d(f (xn ), f (xn+1 )) , 2 o 1 d(f (xn ), f (xn )) + d(f (xn−1 ), f (xn+1 )) 2 n
 1 ≤ α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )), d(f (xn−1 ), f (xn )) + d(f (xn ), f (xn+1 )) , 2 o 1 d(f (xn−1 ), f (xn )) + d(f (xn ), f (xn+1 )) 2 n
o 1 ≤ α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )), d(f (xn−1 ), f (xn )) + d(f (xn ), f (xn+1 )) . 2 Claim: d(f (xn ), f (xn+1 )) ≤ α(f (xn−1 ), f (xn ))d(f (xn−1 ), f (xn )). Suppose that d(f (xn ), f (xn+1 )) ≤ α(f (xn−12 ),f (xn )) [d(f (xn−1 ), f (xn )) + d(f (xn ), f (xn+1 ))] α(f (xn−1 ), f (xn )) d(f (xn−1 ), f (xn )) 2 − α(f (xn−1 ), f (xn )) ≤ α(f (xn−1 ), f (xn ))d(f (xn−1 ), f (xn )).

=⇒ d(f (xn ), f (xn+1 )) ≤

Then {d(f (xn ), f (xn+1 ))} is positive, decreasing and converges to some d ∈ [0, ∞). Now, letting n → ∞ in (2.1), we get n
 1 d ≤ lim α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )), d(f (xn−1 ), g(xn−1 )) + d(f (xn ), g(xn )) , n→∞ 2 o 1 d(f (xn ), g(xn−1 )) + d(f (xn−1 ), g(xn )) 2 ≤d,

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which implies that n
d = lim α f (xn−1 ), f (xn ) max d(f (xn−1 ), f (xn )),

(2.2)  1 o 1 d(f (xn−1 ), g(xn−1 )) + d(f (xn ), g(xn )) , d(f (xn ), g(xn−1 )) + d(f (xn−1 ), g(xn )) . 2 2 Again, letting n → ∞ in (2.1) and using (2.2), we get n→∞

d ≤ F (d, ψ(d)) ≤ d. Thus F (d, ψ(d)) = d, which implies that d = 0. Next, we prove that {f (xn )} is Cauchy. Suppose not. Then by Lemma 1.8, there exist sequences of positive integers {m(k)} and {n(k)} with mk > nk ≥ k such that d(f (xmk−1 ), f (xnk )) and d(f (xmk ), f (xnk )) converge to some δ > 0. So d(f (xmk ), f (xnk )) = d(g(xmk −1 ), g(xnk−1 ))  n
≤ F α f (xmk −1 ), f (xnk −1 ) max d(f (xmk −1 ), f (xnk −1 )),

(2.3)

 1 d(f (xmk −1 ), g(xmk −1 )) + d(f (xnk −1 ), g(xnk −1 )) , 2 o 1 d(f (xmk −1 ), g(xnk −1 )) + d(f (xnk −1 ), g(xmk −1 )) , 2 n


ψ α f (xmk −1 ), f (xnk −1 ) max d(f (xmk −1 ), f (xnk −1 )),

 1 d(f (xmk −1 ), g(xmk −1 )) + d(f (xnk −1 ), g(xnk −1 )) , 2 o 1 d(f (xmk −1 ), g(xnk −1 )) + d(f (xnk −1 ), g(xmk −1 )) 2 n


≤ α f (xmk −1 ), f (xnk −1 ) max d(f (xmk −1 ), f (xnk −1 )),

 1 d(f (xmk −1 ), g(xmk −1 )) + d(f (xnk −1 ), g(xnk −1 )) , 2 o 1 d(f (xmk −1 ), g(xnk −1 )) + d(f (xnk −1 ), g(xmk −1 )) 2 n  1 ≤ max d(f (xmk −1 ), f (xnk −1 )), d(f (xmk −1 ), g(xmk −1 )) + d(f (xnk −1 ), g(xnk −1 )) , 2 o 1 d(f (xmk −1 ), g(xnk −1 )) + d(f (xnk −1 ), g(xmk −1 )) 2 nh i ≤ max

d(f (xmk −1 ), f (xnk )) + d(f (xnk ), f (xnk−1 )) , d(f (xnk−1 ), f (xnk )),

o 1 d(f (xmk −1 ), f (xnk )) + d(f (xnk−1 ), f (xnk )) + d(f (xnk ), f (xmk )) . 2 Letting n → ∞ in (2.3), we get n
δ = lim α f (xmk −1 ), f (xnk −1 ) max d(f (xmk −1 ), f (xnk )), k→∞

 1 d(f (xmk −1 ), g(xmk −1 )) + d(f (xnk ), g(xnk )) , 2 o 1 d(f (xmk −1 ), g(xnk )) + d(f (xnk ), g(xmk −1 )) . 2

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COMMON FIXED POINT RESULTS FOR WEAKLY COMPATIBLE MAPPINGS

Again, letting n → ∞ in (2.3) and using (2.4), δ ≤ F (δ, ψ(δ)) ≤ δ. Thus F (δ, ψ(δ)) = δ, which implies that δ = 0, which is a contradiction. Therefore, {f (xm )} is Cauchy and converges to x = f (u) for some x ∈ X. Next, we prove that d(f (u), g(u)) = 0. d(g(u), x) = lim d(g(u), g(xn )) n→∞  n  1 ≤ lim F α(f (u), f (xn ))max d(f (u), f (xn )), d(f (u), g(u)) + d(f (xn ), g(xn )) , n→∞ 2 o 1 d(f (xn ), g(u)) + d(f (u), g(xn )) , 2  n  1 ψ α(f (u), f (xn ))max d(f (u), f (xn )), d(f (u), g(u)) + d(f (xn ), g(xn )) , 2 o 1 d(f (xn ), g(u)) + d(f (u), g(xn )) 2 n  1 ≤ lim α(f (u), f (xn ))max d(f (u), f (xn )), d(f (u), g(u)) + d(f (xn ), g(xn )) , n→∞ 2 o 1 d(f (xn ), g(u)) + d(f (u), g(xn )) 2 n  1 = lim α(f (u), f (xn ))max d(f (u), f (xn )), d(f (u), g(u)) + d(f (xn ), f (xn+1 )) , n→∞ 2 o 1 d(f (xn ), g(u)) + d(f (u), f (xn+1 ) 2 1 1 ≤ d(f (u), g(u)) ≤ d(x, g(u)). 2 2 So x = g(u) = f (u) on X. Therefore, by the weak compatiblity of f and g, we have f (x) = f g(u) = gf (u) = g(x). Claim: x is a common fixed point of f and g. d(x, g(x)) = d(g(u), g(x))  n 1 ≤ F α(f (u), f (x))max d(f (u), f (x)), d(f (u), g(u)) + d(f (x), g(x)), 2 o 1 d(f (u), g(x)) + d(f (x), g(u)) , 2  n 1 ψ α(f (u), f (x))max d(f (u), f (x)), d(f (u), g(u)) + d(f (x), g(x)), 2 o 1 d(f (u), g(x)) + d(f (x), g(u)) 2 n 1 ≤ α(f (u), f (x))max d(f (u), f (x)), d(f (u), g(u)) + d(f (x), g(x)), 2 o 1 d(f (u), g(x)) + d(f (x), g(u)) 2 ≤ d(x, g(x)). Thus F (d(x, g(x)), ψ(d(x, g(x)))) = d(x, g(x)), which implies d(x, g(x)) = 0. So x is a common fixed point of f and g on X. Uniqueness of common fixed point:

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Suppose that x and x0 are common fixed points of f and g. Then d(x, x0 ) = (f (x), f (x0 )) = d(g(x), g(x0 ))  n  1 ≤ F α(f (x), f (x0 ))max d(f (x), f (x0 )), d(f (x), g(x)) + d(f (x0 ), g(x0 )) , 2 o 1 d(f (x), g(x0 )) + d(f (x0 ), g(x)) , 2  n  1 0 ψ α(f (x), f (x ))max d(f (x), f (x0 )), d(f (x), g(x)) + d(f (x0 ), g(x0 )) , 2 o 1 d(f (x), g(x0 )) + d(f (x0 ), g(x)) 2 n  1 0 0 ≤ α(f (x), f (x ))max d(f (x), f (x )), d(f (x), g(x)) + d(f (x0 ), g(x0 )) , 2 o 1 d(f (x), g(x0 )) + d(f (x0 ), g(x)) 2 n  o 0 0 1 ≤ α(f (x), f (x )) max d(x, x ), d(x, x0 ) + d(x0 , x) 2 0 ≤ d(x, x ). Therefore, F (d(x, x0 ), ψ(d(x, x0 )) = d(x, x0 ), which implies d(x, x0 ) = 0. So x is the unique common fixed point of f and g on X.  Definition 2.3. Let (X, d) be a metric space. Consider two self-mappings f and g on X and let α : X × X → [0, 1] be a function. Then g is said to satisfy condition (A) on f if, for all F ∈ C, ψ ∈ Ψ, k ∈ [0, 1) and for all x, y ∈ X, n 
1 d(g(x), g(y)) ≤ k α f (x), f (y) max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) . 2 Corollary 2.4. Let (X, d) be a complete metric space and f, g : X → X mappings such that g satisfies the condition (A) on f . Then f and g have a unique common fixed point on X if the following conditions are satisfied: (1) g(X) ⊂ f (X); (2) f (X) is complete; (3) f, g are weakly compatible. Proof. Choose x0 ∈ Y arbitrarily. Let xn ∈ X be the element such that f (xn ) = g(xn−1 ) and define a function F1 : [0, ∞)2 → R as F1 (s, t) = ks for all k ∈ [0, 1) which is a C-class function. Since f and g satisfy the condition (A), n
 1 d(g(x), g(y)) ≤ k α f (x), f (y) max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) 2

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COMMON FIXED POINT RESULTS FOR WEAKLY COMPATIBLE MAPPINGS

 n
 1 = F1 α f (x), f (y) max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) 2  n
 1 ψ α f (x), f (y) max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) . 2 Hence by Theorem 2.2, f and g have a unique common fixed point in X.



Definition 2.5. Let (X, d) be a metric space. Let f and g be two self-mappings on X. Then g is said to satisfy condition (B) on f if, for all F ∈ C, ψ ∈ Ψ and for all x, y ∈ X, d(g(x), g(y))  n  1 o 1 ≤ F max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , d(f (x), g(y)) + d(f (y), g(x)) , 2 2  n  1 o 1 . ψ max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , d(f (x), g(y)) + d(f (y), g(x)) 2 2 Corollary 2.6. Let (X, d) be a complete metric space and f, g : X → X mappings such that g satisfies the condition (B) on f . Then f and g have a unique common fixed point on X if the following conditions are satisfied: (1) g(X) ⊂ f (X); (2) f (X) is complete; (3) f, g are weakly compatible. Proof. By Theorem 2.2, if α(x, y) = 1 for all x, y ∈ X, then the mappings f and g have a unique common fixed point on X .  Definition 2.7. Let (X, d) be a metric space and f, g be two self-mappings on X. Then g is said to satisfy condition (C) on f if, for all x, y ∈ X and a, b, c ∈ [0, 1), n  b d(g(x), g(y)) ≤ max a(d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o c d(f (x), g(y)) + d(f (y), g(x)) . 2 Remark 2.8. If we choose f = I X (I X is the identity mapping in the condition (C), then we obtain the definition of Zamfierscu mapping [7]. Corollary 2.9. Let (X, d) be a complete metric space and f, g : X → X mappings such that g satisfies the condition (C) on f . Then f and g have a unique common fixed point on X if the following conditions are satisfied: (1) g(X) ⊂ f (X); (2) f (X) is complete; (3) f, g are weakly compatible.

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G. K. JACOB, A. H. ANSARI, C. PARK, N. ANNAMALAI

Proof. Choose x0 ∈ Y arbitrarily. Let xn ∈ X be elements such that f (xn ) = g(xn−1 ) and define a function F1 : [0, ∞)2 → R as F1 (s, t) = ks for all k ∈ [0, 1) which is a C-class function. Since f and g satisfy the condition (C), n  b d(g(x), g(y)) ≤ max a(d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o c d(f (x), g(y)) + d(f (y), g(x)) 2 n  1 ≤ k max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) 2  n  1 = F1 max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) , 2  n  1 ψ max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) . 2 Hence by Corollary 2.6, f and g have a unique common fixed point in X.  Definition 2.10. Let (X, d) be a metric space and f and g two self-mappings on X. Then g is said to satisfy condition (D) on f if, for all x, y ∈ X, n  1 d(g(x), g(y)) ≤ max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) 2  n  1 − Ψ max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) 2 Corollary 2.11. Let (X, d) be a complete metric space and f, g : X → X mappings such that g satisfies the condition (D) on f . Then f and g have a unique common fixed point on X if the following conditions are satisfied: (1) g(X) ⊂ f (X); (2) f (X) is complete; (3) f, g are weakly compatible. Proof. Choose x0 ∈ Y arbitrarily. Let xn ∈ X be elements such that f (xn ) = g(xn−1 ) and define a function F2 : [0, ∞)2 → R as F2 (s, t) = s − t which is a C-class function. Since f and g satisfies the condition (D), n  1 d(g(x), g(y)) ≤ max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) 2

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 n  1 − Ψ max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) 2  n  1 = F2 max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) , 2  n  1 ψ max (d(f (x), f (y)), d(f (x), g(x)) + d(f (y), g(y)) , 2 o 1 d(f (x), g(y)) + d(f (y), g(x)) . 2 Hence by Corollary 2.6, f and g have a unique common fixed point in X.



Definition 2.12. [1] Let X be a normed linear space. Then a set Y ∈ X is called q-starshaped with q ∈ Y if the segment [q, x] = {(1 − k)q + kx : 0 ≤ k ≤ 1} joining q to x is contained in Y for all x ∈ Y . Definition 2.13. Let (X, d) be a metric space, f, T two self-mappings on X and let α : X × X → [0, 1] be a function. Then T is said to be a f -weakly generalized almost Zamfirescu mapping if, for all x, y ∈ X and a, b, c ∈ (0, 1), kT (x) − T (y)k  n
 b ≤F α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) , 2  n
 b Ψ α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c . dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) 2 Theorem 2.14. Let f and T be self-mappings on a nonempty q-starshaped subset Y of a Banach space X, where T is a f -weakly generalized almost Zamfirescu mapping and satisfies the following conditions: (1) f is linear and q = f (q); (2) T (X) ⊂ f (X); (3) f (X) is complete; (4) f, T are weakly compatible. Define a mapping Tn on Y by Tn (x) = (1 − βn )q + βn T (x), where {βn } is a sequence of numbers in (0, 1). Then for each n, Tn and f have exactly one common fixed point xn in Y such that f (xn ) = xn = (1 − βn )q + βn T (xn ). Also T and f have a common fixed point x ∈ Y . Moreover, if {xn } is Cauchy and lim βn = 1, then xn → x. n→∞

Proof. By definition, kT (x) − T (y)k  n
 b ≤F α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2

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o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) , 2  n 
b Ψ α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) 2 n
 b ≤α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) , 2 n  b ≤ max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) , 2 n  b ≤ max akf (x) − f (y)k, kf (x) − T (x)k + kf (y) − T (y)k , 2 o c kf (x) − T (y)k + kf (y) − T (x)k . 2 Therefore, by Corollary 2.9, T and f have a common fixed point x ∈ Y . By definition, kTn (x) − Tn (y)k = βn kT (x) − T (y)k  n
 b ≤βn F α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) , 2  n
 b Ψ α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) 2 n
 b ≤βn α f (x) − f (y) max akf (x) − f (y)k, dist(f (x), [q, T (x)]) + dist(f (y), [q, T (y)]) , 2 o c dist(f (x), [q, T (y)]) + dist(f (y), [q, T (x)]) , 2 n
 1 ≤βn α f (x) − f (y) max kf (x) − f (y)k, kf (x) − T (x)k + kf (y) − T (y)k , 2 o 1 kf (x) − T (y)k + kf (y) − T (x)k . 2 Therefore, by Corollary 2.4, Tn and f have a common fixed point x ∈ Y . By the assumption that {xn } is Cauchy, let us consider {xn } → y. If lim βn = 1, then n→∞

kxn − xk = kTn (xn ) − T (x)k = k(1 − βn )p + βn T (xn ) − T (x)k ≤ k(1 − βn )pk + βn kT (xn ) − T (x)k

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KADWIN JACOB et al 184-194

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.1, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

COMMON FIXED POINT RESULTS FOR WEAKLY COMPATIBLE MAPPINGS

 n
≤ k(1 − βn )pk + βn F α f (x) − f (y) max akf (xn ) − f (x)k,  b dist(f (xn ), [q, T (xn )]) + dist(f (x), [q, T (x)]) , 2 o c dist(f (xn ), [q, T (x)]) + dist(f (x), [q, T (xn )]) ,  2 n
Ψ α f (x) − f (y) max akf (xn ) − f (x)k,  b dist(f (xn ), [q, T (xn )]) + dist(f (x), [q, T (x)]) , 2 o c dist(f (xn ), [q, T (x)]) + dist(f (x), [q, T (xn )]) 2 n ≤ k(1 − βn )pk + βn max akf (xn ) − f (x)k,  b kf (xn ) − T (xn )k + kf (x) − T (x)k , 2 o c kf (xn ) − T (x)k + kf (x) − T (xn )k 2 n b  1 − βn 

xn − pk, ≤ k(1 − βn )pk + βn max akxn − xk, 2 βn  1 − β  o c 1 n kxn − xk + kx − xn + pk . 2 βn βn Letting n → ∞, we obtain n o c ky − xk ≤ max aky − xk, 0, ky − xk + kx − yk ≤ kky − xk, 2 where k = max{a, c}. Therefore, x = y and so xn → x.



References [1] M. Abbas, D. Ilic, Common fixed points of generalised almost nonexpasive mappings, Filomat 24:3 (2010), 11-18. [2] A.H. Ansari, Note on ϕ-ψ-contractive type mappings and related fixed point, The 2nd Regional Conference on Mathematics and Applications. Payame Noor University. 2014, pp. 377–380. [3] G.V.R. Babu, P.D. Sailaja, A fixed point theorem of generalized weakly contractive maps in orbitally complete metric spaces, Thai J. Math. 9 (2011), 1–10. [4] S.K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727–730. [5] J. Dugundji. A. Granas, Weakly contractive maps and elementary domain invariance theorem, Bull. Soc. Math. Cr`ece 19 (1978), 141–151. [6] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71–76. [7] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23 (1972), 292–298. 1

Department of Mathematics, Bharathidasan University, Trichy- 620 024, Tamil Nadu, India E-mail address: [email protected] 2

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran E-mail address: [email protected] 3 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail address: [email protected] 4

Department of Mathematics, Bharathidasan University, Trichy- 620 024, Tamil Nadu, India E-mail address: [email protected]

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KADWIN JACOB et al 184-194

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 1, 2018

Locally and globally small Riemann sums and Henstock integral of fuzzy-number-valued functions, Muawya Elsheikh Hamid, Luoshan Xu, and Zengtai Gong,……………………11 The general iterative methods for split variational inclusion problem and fixed point problem in Hilbert spaces, Rattanaporn Wangkeeree, Kiattisak Rattanaseeha, and Rabian Wangkeeree,19 A fixed point alternative to the stability of an additive 𝜌-functional inequalities in fuzzy Banach spaces, Choonkil Park and Sun Young Jang,…………………………………………………32 Fourier series of higher-order Genocchi functions and their applications, Taekyun Kim, Dae San Kim, Lee Chae Jang, and Dmitry V. Dolgy,…………………………………………………44 �𝑔, 𝜑ℎ,𝑚 � −convex and (𝑔, log 𝜑) −convex dominated functions and Hadamard type inequalities related to them, Mustafa Gürbüz,……………………………………………………………51 Fixed point theorem and a uniqueness theorem concerning the stability of functional equations in modular spaces, Changil Kim,……………………………………………………………….62 Rough fuzzy ideals in BCK/BCI-algebras, Sun Shin Ahn and Jung Mi Ko,…………………75 A Lebesgue integrable space of Boehmians for a class of 𝐷𝐾 transformations, Shrideh Al-Omari and Dumitru Baleanu,…………………………………………………………………………85 Stability of the sine-cosine functional equation in hyperfunctions, Chang-Kwon Choi and Jeongwook Chang,………………………………………………………………………………96 Effect of cytotoxic T lymphocytes on HIV-1 dynamics, Shaimaa A. Azoz and Abdelmonem M. Ibrahim,…………………………………………………………………………………………111 The pseudo-T-direction and pseudo-Nevanlinna direction of K-quasi-meromorphic mapping, Hong Yan Xu,………………………………………………………………………………….126 Some inequalities on small functions and derivatives of meromorphic functions on annuli, Hua Wang and Hong-Yan Xu,………………………………………………………………………138 Weighted composition operators between weighted Hilbertian Bergman spaces in the unit polydisk, Ning Cao, Gang Wang, and Cezhong Tong,…………………………………………151

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 1, 2018 (continued) On the 𝐿∞ convergence of a nonlinear difference scheme for Schrodinger equations, Xiaoman Liu and Yongmin Liu,…………………………………………………………………………161 Single point v.s. total blow-up for a reaction diffusion equation with nonlocal source, Dengming Liu,……………………………………………………………………………………………176 Common fixed point results for weakly compatible mappings using C-class functions, Geno Kadwin Jacob, Arslan Hojat Ansari, Choonkil Park, and N. Annamalai,…………………184

Volume 25, Number 2 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

August 2018

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (sixteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

COUPLED FIXED POINT THEOREM IN PARTIALLY ORDERED MODULAR METRIC SPACES AND ITS AN APPLICATION ¨ ¨ ¨ ALI˙ MUTLU, KUBRA OZKAN, AND UTKU GURDAL

Abstract. In this article, we extend certain coupled fixed theorem which was introduced for mappings having the mixed monotone property in various metric spaces to partially ordered modular metric spaces. In addition to this, we express some results about this theorem. Finally, we show using our main theorem that there exists a unique solution for a given nonlinear integral equation.

1. Introduction The first time in literature, Guo and Lakshmikantham [12] introduced the concept of coupled fixed point in 1987. After that, Bhaskar and Lakshmikantham [5] introduced the concept of the mixed monotone property and expressed certain coupled fixed point theorems which are considered as the most interesting fixed point theorems for mappings having this property in ordered metric spaces. They showed the existence of a unique solution for a periodic boundary value problem. Since the coupled fixed point theorems in the study of nonlinear integral equations and differential equations are important tools, many researcher have studied them in various partially ordered metric spaces, e.g. [3, 4, 6, 13, 16, 18, 20, 21, 22, 23, 24]. Lately, a lot of significant results related to fixed point theorems have been extended to modular metric spaces which was introduced by Chistyakov via Fmodular [7] in 2008 and developed the theory of this spaces in 2010 [8]. And then, many authors made various studies on these structures, e.g. [1, 2, 9, 10, 11, 14, 15, 17]. In this article, we extend certain coupled fixed theorem which was introduced for mappings having the mixed monotone property in various metric spaces to partially ordered modular metric spaces. In addition to this, we investigate some results about this theorem. Finally, we show using our main theorem that there exists a unique solution for a given nonlinear integral equation. 2. Modular Metric Spaces Here, we express a series of definitions of some fundamental notions related to modular metric spaces. Definition 2.1. [19] Let X be a vector space on R and ρ : X → [0, ∞] be a function. If ρ satisfies the following conditions, we call that ρ is a modular on X: (1) ρ(0) = 0; (2) If a ∈ X and ρ(γa) = 0 for all numbers γ > 0, then a = 0; 2010 Mathematics Subject Classification. 46A80, 47H10, 54H25. Key words and phrases. Modular metric space, coupled fixed point, complete modular metric. 1

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(3) ρ(−a) = ρ(a), for all a ∈ X; (4) ρ(γa + θb) ≤ ρ(a) + ρ(b) for all γ, θ ≥ 0 with γ + θ = 1 and a, b ∈ X. Let X 6= ∅ and ω : (0, ∞) × X × X → [0, ∞] be a function where λ ∈ (0, ∞). Throughout this article, the value ω(λ, a, b) is denoted by ωλ (a, b) for all a, b ∈ X and λ > 0. Definition 2.2. [8] Let X 6= ∅ and ω : (0, ∞) × X × X → [0, ∞] be a function. If ω satisfies the following conditions for all a, b, c ∈ X, we call that ω is a metric modular on X: (m1) ωλ (a, b) = 0 for all λ > 0 ⇔ a = b; (m2) ωλ (a, b) = ωλ (b, a) for all λ > 0; (m3) ωλ+µ (a, b) ≤ ωλ (a, c) + ωµ (c, b) for all λ, µ > 0. From [8, 9], we know that as fix a0 ∈ X, the two sets Xω = Xω (a0 ) = {a ∈ X : ωλ (a, a0 ) → 0 as λ → ∞} and Xω∗ = Xω∗ (a0 ) = {a ∈ X : ∃λ = λ(a) > 0 such that ωλ (a, a0 ) < ∞} are said to be modular spaces. From [8, 9], the modular space Xω can be equipped by a metric dω (a, b) = inf{λ > 0 : ωλ (a, b) ≤ λ} which is generated by ω for any a, b ∈ Xω where ω is a modular on X. Definition 2.3. [15] Let Xω be a modular metric space, {an }n∈N be a sequence in Xω and C ⊆ Xω . Then, (1) {an }n∈N is called a modular convergent sequence such that an → a, a ∈ Xω if ωλ (an , a) → 0 as n → ∞ for all λ > 0. (2) {an }n∈N is called a modular Cauchy sequence if and only if for all  > 0 there exists n() ∈ N such that ωλ (an , am ) <  for each m, n ≥ n() and λ > 0. (3) C is called complete modular if every modular Cauchy sequence {an } in C is a modular convergent in C. 3. Main Results Let ≤ be a ordered relation and Xω be a modular metric space. Throughout this article, (Xω , ≤) denotes partially ordered modular metric space. Definition 3.1. Let (Xω , ≤) be a partially ordered modular metric space. The mapping F : Xω × Xω → Xω has the mixed monotone property if F holds the following conditions for any a, b ∈ Xω a1 ≤ a2 ⇒ F (a1 , b) ≤ F (a2 , b), a1 , a2 ∈ Xω and b1 ≥ b2 ⇒ F (a, b1 ) ≤ F (a, b2 ), b1 , b2 ∈ Xω . These imply that F is monotone non-decreasing in a and monotone non-increasing in b.

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Coupled Fixed Point Theorem in Partially Ordered Modular Metric Spaces and Its an Application 3

Definition 3.2. Let Xω be a modular metric space, (a, b) ∈ Xω × Xω and F : Xω × Xω → Xω be a mapping. (a, b) is called a coupled fixed point of the mapping if F (a, b) = a and F (b, a) = b. Theorem 3.3. Let (Xω , ≤) be a partially ordered complete modular metric space, the mapping F : Xω × Xω → Xω has the mixed monotone property in Xω and k, l be non-negative constants such that k + l < 1. Suppose that we have the following condition for all a, b, p, q ∈ Xω and λ > 0 ωλ (F (a, b), F (p, q)) ≤ kωλ (a, p) + lωλ (b, q)

(3.1)

where a ≥ p, q ≥ b. If there exist a0 , b0 ∈ Xω with a0 ≤ F (a0 , b0 ) and b0 ≥ F (b0 , a0 ), then F has a unique coupled fixed point. Proof. Let a0 , b0 ∈ Xω with a0 ≤ F (a0 , b0 ) and b0 ≥ F (b0 , a0 ). We take a1 , b1 ∈ Xω with a1 = F (a0 , b0 ) and b1 = F (b0 , a0 ). Again we take a2 , b2 ∈ Xω with a2 = F (a1 , b1 ) and b2 = F (b1 , a1 ). Repeating this way, we obtain sequences {an } and {bn } in Xω with an+1 = F (an , bn ) and bn+1 = F (bn , an ) +

for all n ∈ N . In view of mixed monotone property of F , we get a0 ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ an+1 ≤ · · · and b0 ≥ b1 ≥ b2 ≥ · · · ≥ bn ≥ bn+1 ≥ · · · . Then, by (3.1), we get (3.2)

ωλ (an , an+1 )

= ωλ (F (an−1 , bn−1 ), F (an , bn )), ≤ kωλ (an−1 , an ) + lωλ (bn−1 , bn )

for all n ∈ N+ , λ > 0 and k + l < 1. Similarly, (3.3)

ωλ (bn , bn+1 )

= ωλ (F (bn−1 , an−1 ), F (bn , an )), ≤ kωλ (bn−1 , bn ) + lωλ (an−1 , an )

for all n ∈ N+ , λ > 0 and k + l < 1. Therefore, letting en = ωλ (an , an+1 ) + ωλ (bn , bn+1 ) for all n ∈ N+ , λ > 0. Using equations (3.2) and (3.3), we get en

= ≤ = =

ωλ (an , an+1 ) + ωλ (bn , bn+1 ) kωλ (an−1 , an ) + lωλ (bn−1 , bn ) + kωλ (bn−1 , bn ) + lωλ (an−1 , an ) (k + l)(ωλ (an−1 , an ) + ωλ (bn−1 , bn )) (k + l)en−1 .

Then, we obtain that (3.4)

0 ≤ en ≤ (k + l)en−1 ≤ (k + l)2 en−2 ≤ · · · ≤ (k + l)n e0 .

If e0 = 0, then e0 = ωλ (a0 , a1 ) + ωλ (b0 , b1 ) = 0. Therefore, we get ωλ (a0 , a1 ) = 0 and ωλ (b0 , b1 ) = 0. So, from condition (m2) of modular metric spaces, we get a0 ≤ a1 = F (a0 , b0 )

and b0 ≥ b1 = F (b0 , a0 ).

This implies that (a0 , b0 ) is a coupled fixed point of F .

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Now, let e0 > 0. Preserving the generality, suppose that for m, n ∈ N and n < m, there exist n λ ∈ N satisfying m−n

ω

λ m−n

(an , an+1 ) + ω

λ m−n

(bn , bn+1 ) = en

λ for all m−n > 0 and n ≥ n λ . We get m−n (3.5) ωλ (an , am ) ≤ ω λ (an , an+1 ) + ω λ (an+1 , an+2 ) + · · · + ω λ (am−1 , am ), m−n m−n m−n ωλ (bn , bm ) ≤ ω λ (bn , bn+1 ) + ω λ (bn+1 , bn+2 ) + · · · + ω λ (bm−1 , bm ) m−n

m−n

m−n

for each n < m. Thus, from (3.4) and (3.5), we get ωλ (an , am ) + ωλ (bn , bm ) ≤ (ω

λ m−n

+(ω

(an , an+1 ) + ω

λ m−n

λ m−n

(am−1 , am ) + ω

(bn , bn+1 )) + · · ·

λ m−n

(bm−1 , bm )),

= en + en+1 + · · · + em−1 , ≤ (k + l)n e0 + (k + l)n+1 e0 + · · · + (k + l)m−1 e0 , = ≤

(3.6)

((k + l)n + (k + l)n+1 + · · · + (k + l)m−1 )e0 , (k + l)n e0 1 − (k + l)

for n < m and λ > 0. Let k + l = δ. Since there exists n0 such that from (3.6), we have for each n, m ≥ n0 that

δ n0 1−δ e0

< ,

ωλ (an , am ) + ωλ (bn , bm ) <  for an arbitrary  > 0. Then, {an } and {bn } are Cauchy sequences in Xω . Using completeness of Xω , we can talk about existence of a, b ∈ Xω with lim an = a and lim bn = b.

n→∞

n→∞

There exists n0 ∈ N with ω λ (an , a) < 2 and every  > 0. So, from (3.1), we get

 2

and ω λ (bn , b) < 2

 2

for all n ≥ n0 , λ > 0

ωλ (F (a, b), a) ≤ ω λ (F (a, b), an+1 ) + ω λ (an+1 , a) 2

2

= ω λ (F (a, b), F (an , bn )) + ω λ (an+1 , a) 2

2

≤ kω λ (an , a) + lω λ (bn , b) + ω λ (an+1 , a) 2 2 2    < k +l + 2 2 2   = (k + l) + <  2 2 for all λ > 0 and each n ∈ N. Then, ωλ (F (a, b), a) = 0. So, F (a, b) = a. In similar way, we get F (b, a) = b. These imply that (a, b) is a coupled fixed point of F . We assume that F has an another coupled fixed point (a∗ , b∗ ). Then, for λ > 0 we get ωλ (a∗ , a) = ωλ (F (a∗ , b∗ ), F (a, b)) ≤ kωλ (a∗ , a) + lωλ (b∗ , b) and ωλ (b∗ , b) = ωλ (F (b∗ , a∗ ), F (b, a)) ≤ kωλ (b∗ , b) + lωλ (a∗ , a). Therefore, we get (3.7)

ωλ (a∗ , a) + ωλ (b∗ , b) ≤ (k + l)(ωλ (a∗ , a) + ωλ (b∗ , b)).

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Coupled Fixed Point Theorem in Partially Ordered Modular Metric Spaces and Its an Application 5

Since k + l < 1, from (3.7) we get ωλ (a∗ , a) + ωλ (b∗ , b) = 0 for all λ > 0. Hence, we obtain that ωλ (a∗ , a) = 0 ⇔ a∗ = a and ωλ (b∗ , b) = 0 ⇔ b∗ = b. Therefore, (a, b) is a unique coupled fixed point of F .



If we take equal the constants k, l in Theorem (3.3), the following corollary is obtained. Corollary 3.4. Let (Xω , ≤) be a partially ordered complete modular metric space, the mapping F : Xω × Xω → Xω has the mixed monotone property in Xω and k ∈ [0, 1). Suppose that we have the following condition for all a, b, p, q ∈ Xω and λ>0 k (3.8) ωλ (F (a, b), F (p, q)) ≤ (ωλ (a, p) + ωλ (b, q)) 2 where a ≥ p, q ≥ b. If there exist a0 , b0 ∈ Xω with a0 ≤ F (a0 , b0 ) and b0 ≥ F (b0 , a0 ), then F has a unique coupled fixed point. Theorem 3.5. Let (Xω , ≤) be a partially ordered complete modular metric space. Suppose that Xω satisfies the following conditions (i) if a non-decreasing sequence an → a, then an ≤ a for all n, (ii) if a non-increasing sequence bn → b, then bn ≥ b for all n. Let the mapping F : Xω ×Xω → Xω has the mixed monotone property in Xω and k, l be non-negative constants such that k + l < 1. Suppose that we have the following condition for all a, b, p, q ∈ Xω and λ > 0 ωλ (F (a, b), (p, q)) ≤ kωλ (a, p) + lωλ (b, q) where a ≥ p, q ≥ b. If there exist a0 , b0 ∈ Xω with a0 ≤ F (a0 , b0 ) and b0 ≥ F (b0 , a0 ), then F has a unique coupled fixed point. Proof. This proof can be made in analogy to the proof of Theorem (3.3). Here, it will be enough only to show that F (a, b) = a and F (b, a) = b for proof. Let  > 0. Since F (an−1 , bn−1 ) = an → a and F (bn−1 , an−1 ) = bn → b, there exists n0 ∈ N with ωλ (an , a) = ωλ (F (an−1 , bn−1 ), a) < 3 (3.9) ωλ (bn , b) = ωλ (F (bn−1 , an−1 ), b) < 3 for all n ≥ n0 . Letting n ≥ n0 and using the equations (3.1) and (3.9), we get ωλ (F (a, b), a) ≤ ω λ (F (a, b), an+1 ) + ω λ (an+1 , a) 2

2

= ω λ (F (a, b), F (an , bn )) + ω λ (an+1 , a) 2

2

≤ kω λ (an , a) + lω λ (bn , b) + ω λ (an+1 , a) 2 2 2    < k +l + 3 3 3   = (k + l) + < . 3 3

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¨ A. Mutlu, K. Ozkan, U. G¨ urdal

Then, we get F (a, b) = a. In a similar way, we obtain that ωλ (F (a, b), a) <  implies F (b, a) = b. On the other hand, uniqueness of the coupled fixed point of F can be shown in a similar way with the proof of Theorem (3.3).  Corollary 3.6. Let (Xω , ≤) be a partially ordered complete modular metric space. Suppose that Xω satisfies the following conditions; (i) if a non-decreasing sequence an → a, then an ≤ a for all n, (ii) if a non-increasing sequence bn → b, then bn ≥ b for all n. Let the mapping F : Xω × Xω → Xω has the mixed monotone property in Xω and k ∈ [0, 1). Suppose that we have the following condition for all a, b, p, q ∈ Xω and λ>0 k ωλ (F (a, b), (p, q)) ≤ (ωλ (a, p) + ωλ (b, q)) 2 where a ≥ p, q ≥ b. If there exist a0 , b0 ∈ Xω with a0 ≤ F (a0 , b0 ) and b0 ≥ F (b0 , a0 ), then F has a unique coupled fixed point. Example 3.7. Let Xω = R. If we take the usual partial order ≤ in R, then (R, ≤) is a partially ordered set. We define a mapping ω : (0, ∞) × R × R → [0, ∞) by for all a, b ∈ R and λ > 0. It can be said that Xω is a complete ωλ (a, b) = |a−b| λ modular metric space. We take a mapping F : R × R → R such that F (a, b) = a+b 6 . We easily see that F has the mixed monotone property. Then, we have a+b p+q , ) ωλ (F (a, b), F (p, q)) = ωλ ( 6 6 p+q | a+b 6 − 6 )| = λ 1 |a − p + b − q| = 6 λ 1 |a − p| |b − q| ( + ) ≤ 6 λ λ 1 = (ωλ (a, p) + ωλ (b, q)) 6 for any a, b, p, q ∈ Xω So, the equation (3.8) is satisfied for k = 31 . Therefore, from Corollary (3.4), F has a unique coupled fixed point. Also, there are a0 = 0 ≤ F (0, 0) = F (a0 , b0 ) and b0 = 0 ≥ F (0, 0) = F (b0 , a0 ). It is obvious that (0, 0) is the coupled fixed point of F . On the other hand, if F : Xω × Xω → Xω is defined by F (a, b) = a+b 2 , then F satisfies the condition (3.8) for k = 1. Then, we get a+b p+q , ) ωλ (F (a, b), F (p, q)) = ωλ ( 2 2 p+q | a+b 2 − 2 )| = λ 1 |a − p + b − q| = 2 λ 1 |a − p| |b − q| ≤ ( + ) 2 λ λ 1 = (ωλ (a, p) + ωλ (b, q)). 2

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Coupled Fixed Point Theorem in Partially Ordered Modular Metric Spaces and Its an Application 7

Therefore, coupled fixed points of F are both (0, 0) and (1, 1). Namely, F has not a unique the coupled fixed point. Then, the conditions k < 1 in Corollary (3.4) and k + l < 1 in Theorem (3.3) are the most appropriate conditions for satisfying the uniqueness of coupled fixed point. 4. Application Here, we show that there exists a unique solution of a nonlinear integral equation using the Theorem (3.3). We consider the following nonlinear integral equations: Z S (4.1) a(s) = f (s, a(t), b(t))dt, s ∈ [0, S] = I 0

Z b(s)

S

f (s, b(t), a(t))dt, s ∈ [0, S] = I

= 0

where S ∈ R+ (S > 0) and f : I × R × R → R. Let Xω = C(I, R) and Xω be a partially ordered set. We define the order relation as follows: a ≤ b ⇔ a(s) ≤ b(s) for a, b ∈ C(I, R) and all s ∈ I. We can easily see that Xω is a complete modular metric space such that ωλ (a, b) = sup s∈I

|a(s) − b(s)| , λ

for all a, b ∈ X and λ > 0. Assumption 4.1. There exist two non-negative constants k and l with k + l < 1 such that 1 k(a − p) + l(q − b) (4.2) 0 ≤ f (s, a, b) − f (s, p, q) ≤ ( ) S λ for all s ∈ I, a, b, p, q ∈ Xω and λ > 0 where a ≥ p, q ≥ b. Definition 4.2. (α, β) ∈ C(I, R) × C(I, R) is called a coupled lower and upper solution of the integral equations (4.1) if α(s) ≤ β(s) and Z S α(s) ≤ f (s, α(t), β(t))dt 0

Z β(s) ≤

S

f (s, β(t), α(t))dt 0

for all s ∈ I. Theorem 4.3. We suppose that the Assumption (4.1) is satisfied. The integral equations (4.1) have a unique solution in C(I, R) if there exists a coupled lower and upper solution for equations (4.1). Proof. Let Xω = C(I, R). Xω is a partially ordered set if we define the order relation such that for a, b ∈ C(I, R) and all s ∈ I a ≤ b ⇔ a(s) ≤ b(s).

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It is obvious that Xω is a complete modular metric space such that

(4.3)

ωλ (a, b) = sup s∈I

|a(s) − b(s)| λ

for a, b ∈ C(I, R) and all λ > 0. Also, we define a partial order on Xω × Xω = C(I, R) × C(I, R) such that (a, b) ≤ (p, q) ⇒ a(s) ≤ p(s) and q(s) ≤ b(s) for (a, b), (p, q) ∈ Xω × Xω and all s ∈ I. Now, we define F : Xω × Xω → Xω with Z (4.4)

S

f (s, a(t), b(t))dt

F (a, b)(s) = 0

for a, b ∈ C(I, R) and s ∈ I. We need to show that F has the mixed monotone property. If a1 ≤ a2 , that is, a1 (s) ≤ a2 (s) for all s ∈ I, by Assumption (4.1) we get Z F (a1 , b)(s) − F (a2 , b)(s)

S

Z

S

f (s, a1 (t), b(t))dt −

= 0

Z

f (s, a2 (t), b(t))dt 0

S

(f (s, a1 (t), b(t)) − f (s, a2 (t), b(t)))dt

= 0

≤ 0. Then, F (a1 , b)(s) ≤ F (a2 , b)(s) for all s ∈ I. That is, F (a1 , b) ≤ F (a2 , b). Similarly, if b1 ≥ b2 , that is, b1 (s) ≥ b2 (s) for all s ∈ I, by Assumption (4.1), we get Z F (a, b1 )(s) − F (a, b2 )(s)

S

Z f (s, a(t), b1 (t))dt −

= 0

Z

S

f (s, a(t), b2 (t))dt 0

S

(f (s, a(t), b1 (t)) − f (s, a(t), b2 (t)))dt

= 0

≤ 0. Then, F (a, b1 )(s) ≤ F (a, b2 )(s) for all s ∈ I. That is, F (a, b1 ) ≤ F (a, b2 ). Therefore, F (a, b) is monotone nondecreasing in a and monotone nonincreasing in b. Now, we show that F has a coupled fixed point. Let a ≥ p and q ≥ b. Then, a(s) ≥ p(s) and q(s) ≥ b(s) for all s ∈ I. From equation (4.2), (4.3) and (4.4), we

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obtain that |F (a, b)(s) − F (p, q)(s)| λ

= = ≤ ≤ ≤ ≤ = =

RS f (s, a(t), b(t))dt − 0 f (s, p(t), q(t))dt| λ RS | 0 (f (s, a(t), b(t)) − f (s, p(t), q(t)))dt| λ RS |f (s, a(t), b(t)) − f (s, p(t), q(t))|dt 0 λ Z 1 S |k(a(t) − p(t)) + l(q(t) − b(t))| ( )dt S 0 λ Z 1 S |a(t) − p(t)| |b(t) − q(t)| (k +l )dt S 0 λ λ Z 1 S |a(z) − p(z)| |b(z) − q(z)| (k sup + l sup )dt S 0 λ λ z∈I z∈I 1 |a(z) − p(z)| |b(z) − q(z)| · S · (k sup + l sup ) S λ λ z∈I z∈I |b(z) − q(z)| |a(z) − p(z)| + l sup k sup λ λ z∈I z∈I |

RS 0

for all λ > 0 which implies that |F (a, b)(s) − F (p, q)(s)| |a(z) − p(z)| |b(z) − q(z)| ≤ k sup + l sup . λ λ λ z∈I z∈I Therefore, we obtain that ωλ (F (a, b), F (p, q)) ≤ kωλ (a, p) + lωλ (b, q) for a, b, p, q ∈ Xω and all λ > 0. On the other hand, let (α, β) be a coupled lower and upper solution of the integral equations (4.1), then we get α(s) ≤ F (α, β)(s) and β(s) ≥ F (β, α)(s) for all s ∈ I where α, β ∈ I. Then, α ≤ F (α, β) and β ≥ F (β, α). We obtain that Theorem (3.3) is satisfied. Therefore, from Theorem (3.3), we get a unique solution (a, b) ∈ Xω × Xω of integral equations (4.1).  References 1. A.A.N. Abdou and M.A. Khamsi, Fixed point results of pointwise contractions in modular metric spaces, Fixed Point Theory Appl. 2013 (2013), Article ID 163. 2. A.A.N. Abdou and M.A. Khamsi, Fixed points of multivalued contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2014 (2014), Article ID 249 . 3. I. Altun and G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo 58 (2009), 319-325. 4. I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. 2010 (2010), 17 pages, Article ID 621492. 5. T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379-1393. 6. L.j. C´ır´ıc and V. Lakshmikantham, Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces, Stoch. Anal. Appl. 27 (2009), 1246-1259.

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7. V.V. Chistyakov, Modular metric spaces generated by F-modulars, Folia Math. 14 (2008), 3-25. 8. V.V. Chistyakov, Modular metric spaces I basic concepts, Nonlinear Anal. 72 (2010), 1-14. 9. V.V. Chistyakov, Fixed points of modular contractive maps, Doklady Mathematics 86 (2012), no.1, 515-518. 10. Y.J. Cho, R. Saadati and G. Sadeghi, Quasi-contractive mappings in modular metric spaces, J. Appl. Math. 2012 (2012), 5 pages Article ID 907951. 11. M.E. Ege and C. Alaca, Some properties of modular S-metric spaces and its fixed point results, Journal of Computational Analysis and Applications. 20 (2016), no.1, 24–33. 12. D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (1987), 623-632. 13. E. Karapinar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59 (2010), 3656-3668. 14. E. Kilinc and C. Alaca, A Fixed point theorem in modular metric spaces, Adv. Fixed Point Theory 4 (2014), 199–206. 15. C. Mongkolkeha, W. Sintunavarat and P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory Appl. 2011 (2011), 2011:93. 16. V. Lakshmikantham and L.j. C´ır´ıc, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009), 4341-4349. 17. P. Kumam, Fixed point theorems for nonexpansive mapping in modular spaces, Arch Math. (Brno) 40 (2004), 345–353. 18. N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74 (2011), 983992. 19. J. Musielak and W. Orlicz, On modular spaces, Studia Math. 18 (1959), 49–65. 20. A. Mutlu, N. Yolcu, B. Mutlu and N. Bildik, On Common Coupled Fixed Point Theorems for Comparable Mappings in Ordered Partially Metric Spaces, Abstract and Applied Analysis 2014 (2014), 6 pages Article ID 486384. 21. A. Mutlu, N. Yolcu and B. Mutlu, Coupled Fixed Point Theorem for Mixed Monotone Mappings on Partially Ordered Dislocated Quasi Metric Spaces, Global Journal of Mathematics 1 (2015), no.1, 12-17. 22. A. Mutlu, N. Yolcu and B. Mutlu, Fixed Point Theorems in Partially Ordered Rectangular Metric Spaces, British Journal of Mathematics & Computer Science 15 (2016), no.2, 1-9. 23. F. Sabetghadam, H.P. Masiha and A.H. Sanatpour, Some coupled fixed point theorems in cone metric spaces, Fixed Point Theory Appl. (2009) 8 pages Article ID 125426 doi:10.1155/2009/125426. 24. B. Samet, Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517. Manisa Celal Bayar University , Faculty of Science and Arts, Department of Mathematics Muradiye Campus, 45047, Manisa/TURKEY E-mail address: [email protected], [email protected] Manisa Celal Bayar University , Faculty of Science and Arts, Department of Mathematics Muradiye Campus, 45047, Manisa/TURKEY E-mail address: [email protected] Manisa Celal Bayar University , Faculty of Science and Arts, Department of Mathematics Muradiye Campus, 45047, Manisa/TURKEY E-mail address: [email protected]

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Connected m − Kn−residual graph and its application in cryptology Huiming Duan1,2 , Xinping Xiao1 ,∗Congjun Rao1 1.College of Science,Wuhan University of Technology, Wuhan, 430070, China 2.College of Science, Chongqing University of Posts and Telecommunications, Nanan, Chongqing, 400065, China

Abstract The minimum order and extremal graph of a connected m−Kn −residual graph were first proposed by Erd¨ os, Harary and Klawe, in addition to two important conjectures related to the graph. They addressed said conjectures solely from the perspective of Kn −residual graphs, and did not study the m − Kn −residual graphs m > 1. In this study, we found that with n > 2m + 2, the minimum order of connected m − Kn −residual graph was (m + n)(m + 1) and the unique extremal graph was Km+n × Km+1 . This conclusion agreed with the conjectures proposed by Erd¨ os, Harary and Klawe. Moreover, with n = 2m + 2, we identified at least two non isomorphic m − Kn −residual graphs; this did not correspond with the conjecture. The m − Kn −residual graph was determined only by m, n and the maximum connected branch r. The relationship among m, n and r was similar to that among the parameters of Hill cryptosystem implementation steps. According to these observations and principle knowledge regarding Hill cryptosystem implementation, the novel binary cryptosystem related m − Kn −residual graphs was established. We also built a Hill password encryption algorithm that ensures the binary cryptosystem is effective. The complexity of the minimum order and extremal graphs of connected m−Kn −residual graphs make the ciphertext, plaintext, and relationship between the keys highly complex and give the binary cryptosystem favorable performance. Keywords: Residual graph; Minimum order; Extremal graph; Isomorphic.

1

Introduction

The residual graph, which represents an important branch of graph theory, was first proposed by Erd¨os, Harary and Klawe[1], and has since been widely applied throughout information science, networking, computer science, and other fields[2-7]. By definition, a residual graph is built by removing points in the closed neighborhood N (u) that are isomorphic with the original graph; each removed point the closed neighborhood N (u) in the graph has the same nature as its counterpart the original graph. For example, K3 is a highly stable triangle that retains its original shape but with even higher stability after the adjacent edges and vertices of m − K3 −residual graph are removed for m times. In cryptology, an important component of information security[8-9], known information is denoted by graphs and kept in cipher form. Corresponding residual graphs are constructed according to their relevant definitions. Many residual graphs can be constructed, but it is difficult to select only one as the ciphertext: Only a sufficiently complex(i.e.,unique)representative residual graph can ensure information security. The complete graph Kn is often utilized in the computer networking and computer aided design fields[10-12], making it a useful research object in regards to the minimum order and extremal graph of connected m − Kn −residual graph. Erd¨os, Harary and Klawe originally defined this concept[1] and they concluded that as n > 1, n 6= 2, the minimum order of connected m − Kn−residual graph is 2(n + 1); as n 6= 2, 3, 4, Kn+1 × K2 is the unique connected Kn −residual graph with the minimum order. They also proposed the following two conjectures for connected m − Kn −residual graphs. Conjecture 1: If n 6= 2, then every connected m − Kn −residual graph has at least min{2n(m + 1), (n + m)(m + 1)} vertices. Conjecture 2: If n is large, there is a unique smallest connected m − Kn −residual graph. ∗ Corresponding

author. Tel:+86-13507164166; E-mail:[email protected] (X.P. Xiao).

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It is difficult to study the minimum order and residua graph of connected m − Kn −residual graph. As shown in Fig.1, for example, we successfully constructed a residual graph with order of 6 from m = 2 and n = 1, but it was very difficult to prove that the minimum order was 6 or even that the graph as-drawn was the unique extremal graph.

Fig.1 2 − K1 −residual graph.

There has been notable progress towards resolving problems related to the minimum order and extremal graph of connected m − Kn −residual graphs. Author [13], for example, investigated complete residual graphs of odd order to find that there is no Kn −residual graph of odd order for any odd n. Author [14] studied the Kn −residual graph to determine several other important properties and ultimately obtain the minimum order and extremal graph with n = 2, 3, 4 . Author [15] explored the 2−Kn −residual graph to find that when n > 5, n 6= 6 , the results are in accordance with the two conjectures listed above. Other researchers [16] found that when n ≥ 9 , the 3 − Kn −residual graph is also in accordance with the conjectures. Author [17] examined the nature of a connected residual graph when n = 3, 4. In this study, we explored a connected m − Kn −residual graph to find that when n > 2m + 2, the minimum order was (m + n)(m + 1) and the unique extremal graph was Km+n × Km+1 . This also agreed well with the conjectures described above. In addition, when n = 2m + 2, there were two isomorphic connected m − Kn −residual graphs as least−this does not align with the conjectures. Recent years have seen rapid advancements in modern cryptography [18-23]. Information security has become especially important with the advent of the internet. Protecting information via secure, efficient cryptosystems is a popular research subject, to this effect. Hill cryptosystems [23-24] are widely used in encryption and digital signature applications, but do contain loopholes. Researchers are still looking for a more effective cryptosystem. At present, the relationship between graph theory and cryptography is understood primarily as a relationship between the block cipher and DNA algorithm components of graph theory [25-27]. In other words, graph theory is an important theoretical basis for cryptography. Again, there are some loopholes in the security of the traditional Hill cryptosystem [28-30]. Despite notable achievements in improving cryptosystem security, there is much room for further improvement. In this paper, plaintext is denoted by m, n, and the maximum connected branch r. The parameters of the Hill cryptosystem and minimum order and extremal graph of the connected residual graph were carefully assessed, and a novel Hill password encryption algorithm was established. The binary cryptosystem was found to be secure due to the complexity of the minimum order and extremal graphs of connected m−Kn −residual graphs, which makes the ciphertext, plaintext, and relationship between the keys highly complex. The security of this binary cryptosystem was also found to be adjustable. The remainder of this paper is organized as follows. Section 2 provides some background information on the residual graph concept, and Section 3 introduces the minimum order and extremal graph of the connected m − Kn −residual graph. In Section 4, the binary cipher is proposed according to several parameters of the connected m−Kn −residual graph and the security of the resulting system is discussed. Section 5 concludes the paper.

2

Preliminaries The following concepts and results for solving the m − Kn −residual graphs.

Definition 2.1 V (G) is the number of vertices in a graph G, N (u) is the closed neighborhood of vertex u ∈ V (G), and N ∗ (u) are the original neighborhood and closed neighborhood of u in G. Definition 2.2 For u ∈ V (G), define Gu = G − N [u]. For convenience, we use notation hSi to mean G[S](the subgraph induced by S in G).

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Definition 2.3 Let F ⊂ G, then the degree of F in G is the cardinality of its boundary dG (F ) = P dG (x) − dF (x), x ∈ F . Definition 2.4 A graph G is said to be a F −residual graph if for every vertex v in G, the graph obtained from G by removing the closed neighborhood of v is isomorphic to F. We inductively define a multiply −F −residual graph by saying that G is an m−F −residual graph if the removal of the closed neighborhood of any vertex of G results in a (m − 1) − F −residual graph, where of course a 1 − F −residual graph is simply said to be a F −residual graph. Definition 2.5 Let X, Y ⊂ V (G), X is said to be adjacent to Y , and viceversa, if there exist x ∈ X and y ∈ Y , then xy ∈ E(G). If xy ∈ E(G) for all x ∈ X and y ∈ Y , then X is said to be complete adjacent to Y , and viceversa, for example, X and Y are said to be nonadjacent if there are no edges between them. Definition 2.6 Let G1 and G2 are two disjoint graphs the join, G = G1 +G2 , of G1 and G2 is defined as follows V (G) = V (G1 ) ∪ V (G2 ) two vertex u and v are adjacent to each other, if and only if, u ∈ V (G1 ) and v ∈ V (G2 ), or uv ∈ E(G1 ) or uv ∈ E(G2 ). The known supporting results are summarized in the following Lemma. Lemma 2.1 [1] If G is a connected F-residual graph, then for any vertex u in G, the degree d(u) = ν(G) − ν(F ) − 1. Lemma 2.2 [17] If G = G1 + G2 , then G is m − F −residual graph, if and only if, both G1 and G2 are m − F −residual graphs. Lemma 2.3 [15] If G is a 2 − Kn −residual graph, when ν(G) = 3n + t, 1 ≤ 2n, then it hasn’t u ∈ G, which makes d(u) = n + t − 1, when n ≥ 5, n 6= 6, then ν(G) = 3n + 6, G ∼ = Kn+2 × K3 . Lemma 2.4 [16] If G is a 3 − Kn −residual graph, when n ≥ 11, then ν(G) = 4n + 12, G ∼ = Kn+3 × K4 . Lemma 2.5 [17] Assume G is an mKn −residual graph, G 6= (m + 1)Kn , then ν(G) ≥ 2(m + 1)n, and Km+1,m+1 [Kn ] is the unique extremal graphs.

3

On connected m − Kn −residual graphs

In order to obtain the minimum order and extremal graph of the connected m − Kn −residual graph, we need the following Lemma. Lemma 3.1 Assume G is an m − Kn −residual graph, m ≥ 2, u ∈ G, and F ⊂ G, F is the maximum connected subgraph, and F is a r − Kn −residual graph, (0 ≤ r ≤ m − 2), if ν(G) < ν(F ) + 2(m − r)n, then there certainly a v ∈ Gu , which makes Gv = G − N [v] connected. Proof. In the following, by reductio absurdum, we suppose there don’t exist vertexes v ∈ Gu , which make Gv = G − N [v] connected, then we need to prove ν(G) ≥ ν(F1 ) + 2(m − r)n. Set H = {F |F ⊂ Gv , v ∈ Gu , u ∈ F }. Let F1 ∈ H, ν(F1 ) = max{ν(F )|F ∈ H}, and Gv = F1 ∪ G1 , then F1 is a r − Kn −residual graph. When r = 0, F1 ∼ = Kn , then G1 is an (m − r − 2) − Kn −residual graph. For every w ∈ (G − N (F1 )) ⊂ Gu , let Gw = F2 ∪ G2 , where F1 ⊂ Gw , hence F1 ⊂ F2 , because F1 is the biggest connected components, then F2 = F1 , Gw = F1 ∪ G2 , N (F ) ∩ G2 = ∅, G2 ⊂ [G − N (F1 )], G2 is an (m − r − 2) − Kn −residual graph too, G − N (F1 ) is a (m − r − 1) − Kn −residual graph, and ν(G − N (F1 )) ≥ (m − r)n. Let N (F1 ) − F1 = X, then G − N (F1 ) is complete adjacent to X, in the following, we have discussed F1 according to three conditions. Case 1. F1 is complete adjacent to X, then G = hN (F1 ) ∪ (G − N (F1 ))i = hX ∪ F1 ∪ (G − N (F1 ))i = hXi + hF1 ∪ (G − N (F1 ))i = hXi + hG − Xi, because |X| ≥ (m + 1)n, then ν(G) ≥ (m + 1)n + ν(F1 ) + (m − r)n ≥ ν(F1 ) + 2(m − r)n.

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Case 2. F1 is a (r + 1) independent, let {u0 , u1 , · · · , ur } ⊂ F1 , let X − N (u0 , u1 , · · · , ur ) = X1 , then X1 = 6 ∅, hence we have 0

G = G − N (u0 , u1 , · · · , ur ) = hXi + hG − N (F1 )i, 0

by Lemma 2.2, then G is a (m − r − 1) − Kn −residual graph, and |X1 | ≥ (m − r)n, according to the proving in Case 1, we know that ν(G) ≥ ν(F1 ) + 2(m − r)n. Case 3. There is a l independent set and l < r+1. Set u1 , u2 , · · · , ul } ⊂ F1 , let X−N (u1 ,u2 , · · · , ul ) = X2 , then X2 6= ∅. Let 00

00

G = G − N (u1 , u2 , · · · , ul ) = hX1 i + hG ∪ (G − N (F1 ))i, because|X| ≥ |X2 | ≥ (m − l + 1)n > (m − r)n, according to the proving in Case 1, we know that ν(G) ≥ ν(F1 ) + 2(m − r)n. From the proves above, we know that if ν(G) < ν(F ) + 2(m − r)n, then there certainly exist a vertices v ∈ Gu , which makes Gv = G − N [v] connected. Theorem 3.1 Assume G is an m−Kn −residual graph, m ≥ 2, if δ(G) = n, then ν(G) ≥ (m+3)n+m−1, and the Fig.3 is the only extremal graph. Proof. We take induction to m, let G1 be a 2 − Kn −residual graph, if exist w ∈ G1u , which makes Gw = G1 − N (w) connected, then dG1 (w) = n + t − 2, d(w) = n + t − 1, Gw = H1 ∪ H2 ∼ = 2Kn and u ∈ H1 ∼ = Kn , i = 1, 2. By Lemma 3.1, ν(G1 ) ≥ δ(G) + ν(Gw ) + 1 ≥ n + 4n + 1 = 5n + 1, Gw is a Kn −residual graph, when ν(G1 ) = 5n+1, we can specify the Fig.1 is the minimum 2−Kn −residual graph. Suppose that Theorem is true for m ≥ 2, and assume G is an (m + 1) − Kn −residual graph, and δ(G) = n, let u ∈ G, d(u) = n, and assume ν(G) ≤ (m + 4)n + m. In the following, we prove that it must exist a vertex v ∈ Gu with the case of the above assumptions, which makes Gv connected. By contrary, by Lemma 3.1, we have ν(G) ≥ ν(F ) + 2(m − r + 1)n, and F is a connected r − Kn −residual graph, if F ∼ = Kn . We have ν(G) ≥ n + 2(m + 1)n = (m + 4)n + (m − 1)n = (m + 4)n + m + (m − 1)n − 1) − 1 > (m + 4)n + m, this is a contradiction. Since u ∈ F , d(w) = n, we have r 6= 1. If r ≥ 2, by induction hypothesis of F is a connected r − Kn −residual graph with δ(F ) = n, hence ν(F ) ≥ (r + 3)n + r − 1 and ν(G) ≥ (r + 3)n + r − 1 + 2(m + 1 − r)n = (m + 4)n + (m + 1 − r)n + r − 1 > (m + 4)n + m, which contradicts that ν(G) ≤ (m + 4)n + m. So we have there must be a vertex v ∈ Gu , which makes Gv connected. Let u ∈ Gv , then n ≤ δ(Gv ) ≤ dGv (u) ≤ d(u) = n, so we have δ(Gv ) = n, and by induction hypothesis, we have ν(Gu ) ≤ (m + 3)n + m − 1, and by Lemma 2.1, ν(G) = ν(Gu ) + d(v) + 1 ≥ (m + 3)n + m − 1 + n + 1 = (m + 4)n + m. So we have ν(G) = (m + 4)n + m, and ν(G) < (m + 4)n + m is not true, then ν(G) ≥ (m + 4)n + m. When ν(G) = (m + 4)n + m, we can construct (m − 1) − Kn −residual graph and m − Kn −residual graph, the Fig.2 is the minimum (m − 1) − Kn −residual graph, and the Fig.3 is the minimum m − Kn −residual graph. In the following, we have proved the Fig.3 is the only extremal graph. Let v ∈ Gu , Gv = X1 ∪ X2 ∪ Y1 ∪ Y2 ∪ Vj , j = 3, 4, · · · , m + 1, Xi ∼ = Yi , i = 1, 2, Vj ∼ = Kn+1 . Let Vj = {xj , yj } ∪ Cj , j = 3, 4, · · · , m + 1, and let X = X1 ∪ X2 ∪ {x3 , x4 , · · · , xm+1 }, Y = Y1 ∪ Y2 ∪ {y3 , y4 , · · · , ym+1 }, then X is complete adjacent to Y . Suppose u ∈ C3 , Gu is the Fig.2, we have G = hN (v) ∪ Gv i, Gu = h(Gv − N (u)) ∪ N (v)i = h((Gu ) − V3 ) ∪ Vm+2 i, where Vm+2 = N (u) = {xm+2 , ym+2 } ⊂ Cm+2 , xm+2 is complete adjacent to Y − {y3 }, ym+2 is complete adjacent to X − {x3 }, hence we have let X ∗ = X ∪ {xm+2 }, Y ∗ = Y ∪ {ym+2 }, then X ∗ is complete adjacent to Y ∗ , so that the Fig.3 of G is the only extremal graph. 2

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Fig.1 2 − Kn −residual graph with order 5n + 1

Fig.2 (m − 1) − Kn -residual graph with order (m + 2)n + m − 2

Fig.3 m − Kn -residual graph with order (m + 3)n + m − 1

On the basis of the conclusion from literature [15] and the definition of m − Kn −residual graph, we have Theorem 3.2 Let G be a connected m − Kn −residual graph, ν(G) = (m + 1)n + t, 1 ≤ t ≤ 2n, m > 2, n > 4, then G has no vertex of degree n + t − 1. Proof. We take induction to m, when m = 2, by Lemma 2.3, G has no vertex of degree n + t − 1. When m = 3, G is a connected 3 − Kn −residual graph ν(G) = 4n + t, 1 ≤ t ≤ 2n. Supposing there exist such vertices like u ∈ G, which make d(u) = n + t − 1 and Gu = H1 ∪ H2 ∪ H3 = 3Kn , because ν(G) = 4n + t, 1 ≤ t ≤ 2n, then ν(G) ≤ 6n ≤ 6n + 2. By Lemma 3.1, there exists at least v ∈ Gu , and we might as well suppose v ∈ H3 , which makes Gv connected, then Gv − N (u) = H1 ∪ H2 ∼ = 2Kn , and by Lemma 2.1, ν(Gv ) = ν(G) − d(v) − 1, let d(v) = n − r, ν(Gv ) = 4n + t − n − r − 1 = 3n + t1 , let t1 = n − r − 1, ν(Gv ) = 3n + t1 , 1 ≤ t1 ≤ 2n, because Gv is 2 − Kn −residual graph, it is contradictory to the conclusion of literature [15] that there are no vertexes makes d(u) = n + t − 1. We suppose let G be a connected m − Kn −residual graph, and there doesn’t exist such vertices like u ∈ G, which make d(u) = n + t − 1. For G be a connected (m + 1) − Kn −residual graph, ν(G) = (m + 2)n + t, 1 ≤ t ≤ 2n, m > 2, n > 4. Supposing there exists u ∈ G, which makes d(u) = n + t − 1, then Gu = H1 ∪ H2 ∪ · · · ∪ Hm+1 = (m + 1)Kn , because ν(G) = (m + 2)n + t,1 ≤ t ≤ 2n, then ν(G) ≤ (m + 2)n + 2n= mn + 4n < mn + 4n + m= (m + 4)n + m, by Lemma 2.3 and Lemma 3.1, there exists v ∈ Gu , let v ∈ Hm+1 , which makes Gv is connected then 5

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Gv − N (u) = H1 ∪ H2 ∪ · · · ∪ Hm ∼ = mKn , and by Lemma2.1, ν(Gv ) = ν(G)−d(v)−1, let d(v) = n−r, ν(Gv ) = (m+2)n+t−n−r−1 = (m+1)n+t1 , t1 = n − r − 1 , it is contradictory to the induction and assumption that Gv is an m − Kn −residual graph, then there don’t exist vertices, which make d(u) = n + t − 1, u ∈ G.2 Theorem 3.3 There are at least two non isomorphic minimum m−Kn1 −residual graph, One is Km+n1 × Km+1 , and the other is Gm [K n21 ], where Gm ∼ = m − K2 −residual graph, n1 = 2m + 2, and this example doesn’t meet the conclusion Erd¨ os and Harary and Klawe made in [1]. Proof. Let G ∼ = Km+n1 × Km+1 , ν(G) = (m + n1 )(m + 1), and Gm ∼ = m − K2 −residual graph, by literature [1], ν(Gm ) ≥ 3m + 2, ν(Gm [K n21 ]) = n21 (3m + 2), and n1 = 2m + 2, so when n = n1 , it doesn’t meet the conclusion Erd¨ os and Harary and Klawe made that only one Km+n1 × Km+1 is an m − Kn1 −residual graph. 2 Theorem 3.4 Let G be a connected m − Kn −residual graph, if n > 2m + 2, m ≥ 3, then ν(G) ≥ (m + n)(m + 1), and when ν(G) = (m + n)(m + 1), G ∼ = Km+n × Km+1 is a connected m − Kn −residual graph of minimum order, it is only such graph, so we show that the conjectures are true, when n > 2m+2. Proof. At first, we prove ν(G) ≥ (m + n)(m + 1) and construct m − Kn −residual graph, and show the conjecture [1] is true. Let G1 be 2 − Kn −residual graph, when n > 6, by Lemma 2.3, G1 ∼ = Kn+2 × K3 , and G1 is the Fig.4, all points in the same square are mutually adjacent, and two vertices, which are joined by a line, are adjacent. Let u ∈ G, we have Gu is (m − 1) − Kn −residual graph, suppose ν(Gu ) = (m + n − 1)m, and Gu ∼ = Km+n−1 × Km . Let ,n+m−1 }, (3.1) Gu = hH1 ∪ H2 · · · ∪ Hm i = {xjr |j=1,2,··· r=1,2,··· ,m j i i i, and if i = j, then x is adjacent to x , if i = 6 j, x is nonadjacent to where Hr = hx1r , x2r , · · · , xn+m−1 r l l k xjm , l 6= k, k, l = 1, 2, · · · , m, adjacent to the Fig.5, all points in the same square are mutually adjacent, and two vertices, which are joined by a line, are adjacent. According to the induction and assumption that ν(Gu ) ≥ (m + n − 1)m. When Gu is disconnected, by Lemma 2.5, let Gu = (m − 1)nKn , and ν(Gu ) ≥ 2(m − 1)n. Because n ≥ 2m + 2, m ≥ 3, then ν(Gu ) ≥ 2(m − 1)n > (m + n − 1)m, by definition of residual graph, if and only if, when Gu is connected, ν(G) is minimum. According to the Fig.5, we can construct m − Kn −residual graph in the Fig.6, when ν(G) = (m + n)(m + 1). The Fig.6, all points in the same square are mutually adjacent, and two vertices, which are joined by a line, are adjacent. By Gu is connected and Lemma 2.1, ν(G) = ν(Gu ) + d(v) + 1, according to the induction and assumption that ν(Gu ) ≥ (m + n − 1)m, and by the Fig.6, d(u) ≥ n + 2m − 1, and by Lemma 2.1, hence ν(G) = ν(Gu ) + d(v) + 1 ≥ (m + n − 1)m + n + 2m − 1 + 1 = (m + n)(m + 1). In the following, exist a vertex u ∈ G, which makes Gu connected. Because ν(G) = (m + n)(m + 1), by Lemma 3.1, when F is r − Kn −residual graph, 0 ≤ r ≤ (m − 2), ν(G) = ν(F ) + 2(m − r)n. In the following, we prove ν(G) = ν(F ) + 2(m − r)n ≥ (m + n)(m + 1), when F ∼ = Kn , and n > 2m + 2, we have ν(G) ≥ n + 2mn > (m + n)(m + 1), if r ≥ 1 by induction hypothesis F is a connected r − Kn −residual graph. So ν(F ) ≥ (r + n)(r + 1), and ν(G) = ν(F ) + 2(m − r)n= (r + n)(r + 1) + 2(m − r)n, because of (r + n)(r + 1) + 2(m − r)n − (m + n)(m + 1)> (m − r)2 + m − r > 0, then ν(G) = ν(F ) + 2(m − r)n = (r + n)(r + 1) + 2(m − r)n > (m + n)(m + 1). So it must exist a vertex u ∈ G, which makes Gu connected. From the proves above, we know ν(G) ≥ (m + n)(m + 1), when n > 2m + 2, because 2n(m + 1) > (m + n)(m + 1), so we show the conjecture [1] is true.

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Fig.4 2 − Kn −residual graph with minimum order 3n + 6

Fig.5 (m − 1) − Kn −residual graph with minimum order (m + n − 1)m

Fig.6 m − Kn −residual graph with minimum order (m + n)(m + 1)

In the following, we prove uniqueness, we have a discussion according to five conditions, so we show that the conjecture [2] is true. Fact 1. Let F = H1 ∪ H2 ∪ · · · ∪ Hm ∼ = Km+n−1 × Km , where Hi ∼ = Kn+m−1 , i = 1, 2, · · · , m, then Hi and Hj have bijection θ : V (Hi ) → V (Hj ), i, j = 1, 2, · · · , m, and ui ∈ Hi is adjacent to θ(ui ) ∈ Hj , where i 6= j, i, j = 1, 2, · · · , m. If H ⊂ F , and H ⊂ Ks , 3 ≤ s ≤ n + m − 1, then H ⊂ Hw , 1 ≤ w ≤ n. Fact 2. Let u ∈ G, ν(Gu ) = (m + n − 1)(m − 1), hence Gu ∼ = Km+n−1 × Km . Adjacent to (3.1), let 7

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∗ ∼ G2 = G − N [xn+m−1 ] = hH0∗ ∪ H1∗ ∪ H3∗ ∪ · · · ∪ Hm i = Km+n−1 × Km , 2 we have = hx11 , x21 , · · · , xn+m−2 i ⊂ G2 Km+n−2 ∼ = H1 − xn+m−1 1 1

by Fact 1, without loss of generality, we may assume that hx11 , x11 , · · · , xn+m−2 i ⊂ H1∗ = hx00 , x11 , · · · , xn+m−2 i 1 1 n+m−2 n+m−2 1 2 ∗ 0 1 hx2 , x2 , · · · , x2 i ⊂ H2 = hx2 , x2 , · · · , x2 i ······ ∗ hx1m , x2m , · · · , xn+m−2 i ⊂ Hm = hx0m , x1m , · · · , xn+m−2 i (3.2) m m j j If x0 is adjacent to x1 , where j = 0, 1, · · · , n + m − 1. Obvious x00 = u, then H0∗ = hx00 , x10 , · · · , xn+m−2 i. 0 n+m−1 0 We now prove x01 is adjacent to xn+m−1 . Suppose the contrary, let G = G − N [x ], x ∈ G , 3 3 1 1 1 n+m−1 by (3.1), (3.2), we have x01 is adjacent to {x11 , x21 , · · · , xn+m−1 } ⊂ N [x ]. Thus 1 1 d(x01 ) ≥ dG3 (x01 ) + n + m − 2 =n+m−1+n+m−2 > n + m − 1 − m = n + 2m − 1. 0 So x01 is adjacent to xn+m−1 , x is adjacent to H1 . Set 1 1 H1 = hx01 , x11 , · · · , xn+m−1 i∼ = Km+n . 1 Similarly, x0w is adjacent to Hw , set Hw = hx0w , x1w , · · · , xn+m−1 i∼ = Km+n , w = 1, 2, · · · , m. w Similarly, in ∗ G − N ∗ [xn+m−1 ] = hH0∗ ∪ H1∗ ∪ H3∗ ∪ · · · ∪ Hm i, 2 ∗ i complete adjacent to H0∗ . Obvious, we have xn+m−1 ∈ N ∗ [xn+m−1 ] = hH0∗ ∪ H1∗ ∪ H3∗ ∪ · · · ∪ Hm 0 2

x0n+m−1 ∈ (H2 ∪ xn+m−1 ∪ xn+m−1 ∪ · · · ∪ xn+m−1 ) ⊂ N ∗ [xn+m−1 ], m 1 3 2 hence H0 = hx00 , x10 , · · · , xn+m−1 i∼ = Km+n . 0 Fact 3. Any vertex in Hr is adjacent to single vertex in Hs , r 6= s. Suppose the contrary, let xj0 ∈ H0 be nonadjacent to Hm , then G∗ = G − N [xj0 ] ∼ = Kn+m−1 × Km , j ∗ but Hm ∼ ⊂ G∗ , contrary to G∗ ∼ = Km+n , Hm = Kn+m−1 × Km , hence x0 is adjacent to Hm . If Hm has j two vertices adjacent to x0 , by

dH0 (xj0 ) = n + m + m − 1 = n + 2m − 1, d(xj0 ) = n + m + m + 1 = n + 2m + 1, we have this is nonadjacent to one of Hw , which is a contradiction. Similarly, xji−1 is adjacent to Hi , i = 1, 2, · · · , m, and is adjacent to only one vertex in Hi . Fact 4. By Fact 3, we have x01 adjacent to H2 . If x01 is adjacent to xj2 , j 6= 0, by xj2 , j 6= 0 is adjacent to xj1 , thus H1 has two vertices adjacent to xj2 , j 6= 0, contrary to Fact 3, so x01 adjacent to x02 . Similar, so x0i adjacent to x0j , and xn+m−1 adjacent to xn+m−1 , i = 1, 2, · · · , m. 0 i j Fact 5. Since x0 is adjacent to xjm for j = 0, n+m−1, let xj0 be nonadjacent to xjm for j 6= 0, n+m−1, by Fact 3, we have xj0 is adjacent to xim , i 6= j. Since xjl is adjacent to xjk , l 6= k, l, k = 1, 2, · · · , m, set xj0

G − N [xj0 ] = h(H1 − xj1 ) ∪ (H2 − xj2 ) ∪ · · · ∪ (Hm − xjm )i ∼ = Kn+m−1 × Km

(3.3)

By Fact 4, we have xt1 adjacent to xtm , t 6= i, j, by (3.3), we have xi1 adjacent to xjm , contrary (3.1), hence xj0 is adjacent to xjm . Similarly, xj0 is adjacent to xjm . Hence ,n+m−1 G = hXi = hxjr |j=0,1,2,··· i, r=0,1,2,··· ,m

where xir is adjacent to xjs , if and only if r = s,i 6= j, or i = j, r 6= s, thus G ∼ = Km+n × Km+1 .2

8

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4

Binary cryptosystem of connected m − Kn −residual graphs

The implementation principle and steps of the traditional Hill cryptosystem were followed to establish the novel, binary cryptosystem proposed here. The Hill cryptosystem is a symmetrical cipher that is effectively resistant to frequency analysis that was first proposed by Prof. Lester S. Hill in 1929. It is implemented in the following steps. (1) The digitization of plaintext M = [m1 , m2 , · · · , mk ] (41 characters, including 26 letters, the figures from 0 to 9 and punctuation corresponded to the figures from 0 to 40, respectively), with t components taken as a row vector (if the amount could not reach t in the last row, the space is required for supplementation.) that constitute a nt matrix, written as M . (2) Matrix A of t · t is constructed in Z41 where gcd(det(A), 41) = 1 is required as the encryption key. (3) Encryption operation C = E(M ) ≡ M A(mod41) is carried out to obtain the ciphertext. (4) Decryption operation M = D(C) ≡ CA−1 (mod41) is carried out to recover the plaintext. It is difficult for the Hill cryptosystem to withstand a plaintext attack. After attackers intercept the ciphertext C, they are able to guess certain words used in the plaintext to attempt to ascertain the key K, then can calculate M K to determine whether ciphertext C can be generated. A large amount of information is stored in a computer system in a form of figure and transmitted through a public signal channel. Unfortunately, these computer systems and signal channels are very susceptible to attack in an open environment. The complexities of the minimum order and extremal graph of the connected m − Kn −residual graph can be used to denote the plaintext with different Kn , thus forming the proposed binary cryptosystem of the connected m − Kn −residual graph. The corresponding encryption algorithm is as follows. (1) Each character in plaintext M is translated into the corresponding figure ni ; 41 characters, (26 letters, figures from 0 to 9, and punctuation corresponded to the figures from 0 to 40, respectively, where ni ∈ Z41 .) (2)For encryption operation, each figure ni corresponds to a Kni . The sequence [m1 , m2 , · · · , mk ] represents the multiples, where mk ∈ Z. Finally, the largest connected branch r can be identified and assigned to an extremal graph, i.e., the ciphertextC. (3) For decryption operation, each ciphertext C equals an extremal graph and the complete graph Knj is determined according to the multiple sequences [m1 , m2 , · · · , mk ] and the largest connected branch r. The corresponding nj can also be identified, nj ∈ Z41 (i.e., j(j = 1, 2, · · · , 41)); the resulting graph is the plaintext M . These implementation steps are depicted in Fig. 8. The binary cryptosystem of the connected m − Kn −residual graph makes full use of all available complexity in constructing the minimum order and extremal graph of the connected m − Kn −residual graph, which ensures optimal binary cryptosystem security. Even if attackers intercept the ciphertext C and knew that the minimum order and extremal graph of the connected m − Kn −residual graph have been used for encryption, they are unable to solve the decryption operation and thus cannot obtain the plaintext.

5

Conclusions

(1) The minimum order and extremal graphs of connected m − Kn −residual graphs form an open question put forward by Erd¨ os, Harary, and Klawe which was addressed in this study. (2) It is difficult to determine the minimum order and extremal graphs of connected m − Kn −residual graphs. (3) We confirmed that the m and n values identified are in accordance with the conjectures of Erd¨os, Harary, and Klawe; namely, that the minimum order and extremal graphs of connected m − Kn −residual graphs do exist. This had important practical significance for establishing the proposed binary cryptosystem. (4) A new binary password system comprised of an image-encryption-based password system and encryption algorithm was proposed here. This system represents an enhanced relationship between graph theory and cryptography. (5) According to the complexity of the minimum order and extremal graph of the connected m − Kn −residual graph, the ciphertext, plaintext, and relationship between the keys is highly complex and the binary cryptosystem performs well. The security of the binary cryptosystem can be effectively adjusted according to these factors. 9

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Fig.8 Encryption and decryption operation

6

Acknowledgements

The authors are grateful to the editor for their valuable comments. This work was supported by National Natural Science Foundation of China (Grant no. 11671001,51479151, 71671135, 71540027,61472056) ,Chongqing Frontier and Applied Basic Research Project ( cstc2015jcyjA00034, cstc2015jcyjA00015), and The Science and Technology Research Program of Chongqing Municipal Educational Committee( KJ1600425,KJ150 12024).

References [1] P. Erd¨ os, F. Harary and M. Klawe, Residually-Complete Graphs, Annals of Discrete Mathematics 6 (1980) 117-123. [2] A.Matthias, S. Friedl, Residual properties of graph manifold groups, Topology and its Applications 158(10) (2011) 1179-1191. [3] M.S. Lin, C.C. Ting, A polynomial-time algorithm for computing K-terminal residual reliability of dtrapezoid graphs, Information Processing Letters 115(2)(2015)371-376. [4] A.A.Chernyak, Residual reliability of P-threshold graphs, Discrete Applied Mathematics 135(1) (2004) 83–95. [5] B. Trotta, Residual Properties of Simple Graphs, Bulletin of the Australian Mathematical Society 82(3) (2008) 488-504. [6] M.S.Lin, C.C.Ting, A polynomial-time algorithm for computing K-terminal residual reliability of d-trapezoid graphs, Information processing letters 115 (2015) 371-376. [7] P. Luksic, T. Pisanski, Distance-residual graphs, Mathematics 9(3) (2006) 104-111. [8] K. Patra, B. Nemade, D. P. Mishra and P. P. Satapathy, Cued-Click Point Graphical Password Using Circular Tolerance to Increase Password Space and Persuasive Features, Procedia Computer Science 79 (2016) 490498.

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[9] T.Y. Chang, C.J. Tsai and J.H. Lin, A graphical-based password keystroke dynamic authentication system for touch screen handheld mobile devices, Journal of Systems and Software 85(5) (2012) 1157-1165. [10] F. Septyanto, K. A. Sugeng, On the Rainbow and Strong Rainbow Connection Numbers of the m-Splitting of the Complete Graph Kn , Procedia Computer Science 74(2015)155-161. [11] R. El-Shanawany, H. O.F. Gronau, M. Grttmller, Orthogonal double covers of Kn ,n by small graphs, Discrete Applied Mathematics 138(1-2) (2004) 47-63. [12] A. Aguado, Structures, relative stabilities, and electronic properties of potassium clusters Kn (13 ≤ n ≤ 80) 1021(1) (2013) 135-143. [13] S. Yang, H. Duan, Residually Complete Graph with odd order, Acta Mathematics Applicatae Sinica 34(5) 2011 778-785. [14] H. Duan, Y. Li, On connected Kn − Residual Graphs, Operations Research Transactions 20(2) (2016) 38-48. [15] J. Liao, S. Yang, Y. deng, On connected 2 − Kn −residual graphs, Mediterranean Journal of Mathematics 6 (2012) 12–27. [16] H. Duan, B. Zeng, Z. Dou, On connected three multiply Kn − Residual Graphs, Operations Research Transactions 18(2) (2014) 59-68. [17] J. Liao, S. Yang, Two improvement on the Erd¨ os, Harary and Klawe conjecture, Mediterranean Journal of Mathematics 3 (2014) 1-16. [18] A. Thomas, E. M. Manuel, Embedment of Montgomery Algorithm on Elliptic Curve Cryptography over RSA Public Key Cryptography, Procedia Technology 24 (2016) 911-917. [19] E. Bellini, N. Murru, An efficient and secure RSA-like cryptosystem exploiting Rdei rational functions over conics, Finite Fields and Their Applications 39 (2016) 179-194. [20] R. Rizk, Y. Alkady, Two-phase hybrid cryptography algorithm for wireless sensor networks, Journal of Electrical Systems and Information Technology 2(3) (2015) 296-313. [21] P. Singh, V. Neema, S. Daulatabad and A. P. Shah, Subthreshold Circuit Designing and Implementation of Finite Field Multiplier for Cryptography Application, Procedia Computer Science 79 (2016) 597-602. [22] S. Chandra, B. Mandal, Sk. S. Alam and S. Bhattacharyya, Content Based Double Encryption Algorithm Using Symmetric Key Cryptography, Procedia Computer Science 57 (2015) 1228-1234. [23] K. Adinarayana Reddy, B. Vishnuvardhan, Madhuviswanatham, A.V.N. Krishna, A Modified Hill Cipher Based on Circulant Matrices,Procedia Technology 4 (2012) 114-118. [24] B. Acharya, M. Dhar Sharma, S. Tiwari, V. K. Minz, Privacy protection of biometric traits using modified hill cipher with involutory key and robust cryptosystem, Procedia Computer Science 2 (2010) 242-247. [25] Bakhtiari, M. Aizaini, Serious Security Weakness in RSA Cryptosystem, International Journal of Computer Science Issues 9(1) (2012) 175-178. [26] Bassam J. Mohd, Thaier Hayajneh, Athanasios V. Vasilakos, A survey on lightweight block ciphers for lowresource devices: Comparative study and open issues, Journal of Network and Computer Applications 58 (2015) 73-93. [27] X.J. Tong, Z. Wang, Y. Liu, M. Zhang, L.J. Xu, A novel compound chaotic block cipher for wireless sensor networks,Communications in Nonlinear Science and Numerical Simulation. 22 (2015) 120-133. [28] X. Xu, L. He, A. Shimada, R. Taniguchi, H.M. Lu, Learning unified binary codes for cross-modal retrieval via latent semantic hashing, Neurocomputing 213(12) (2016) 191-203. [29] Y.C. Wan, M.W. Wang, Z.W. Ye, X.D. Lai, A feature selection method based on modified binary coded ant colony optimization algorithm, Applied Soft Computing 49 (2016) 248-258. [30] V. Thayananthan, A. Albeshri, Big Data Security Issues Based on Quantum Cryptography and Privacy with Authentication for Mobile Data Center, Procedia Computer Science 50 (2015) 149-156.

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A comparative analysis of the Harry Dym model with and without singular kernel. E.F. Doungmo Goufo∗ , P. Tchepmo, Z. Ali and A. Kubeka Department of Mathematical Sciences, University of South Africa, Florida, 0003 South Africa ∗

Corresponding author email: [email protected]

Abstract The question that raised after the recent introduction of the derivative with no singular kernel was: Does this non-singularity have a comprehensive impact on real life phenomena like wave motion or other related motions? We comprehensively analyze the Harry Dym model generalized with two types of derivatives, namely a derivative with singular kernel, the Caputo derivative and the other one without singular kernel called the CaputoFabrizio derivative. Using Picard L-stability combined with the fixed-point theorem, the well-posedness of both models are proved together with their existence and uniqueness results existence. Techniques to approximate numerical solutions are provided for each of the two models with graphical representations performed and compared for several values given to the derivative order α. Similar behaviors are noticed for soliton waves related to close values of α and are compared to the soliton wave of the standard first order (α = 1) Harry Dym model. Keywords: Harry Dym equation; existence, uniqueness, derivative with and without singular kernel, approximated solutions AMS Mathematics Subject Classification: 35F10, 26A33, 35D05. 1. Introduction It is well known that one of the most popularly used derivative with fractional order is the one introduced in 1967 by Michele Caputo and called the Caputo derivative [1] given by c

Dtα r(x,t)

1 = Γ (1 − α)

Z t

(t − τ)−α

0

d r (x, τ) dτ, dτ

(1.1)

0 < α ≤ 1, with its associated anti-derivative well known to be defined by I0−α r(x,t)

1 = Γ (α)

Z t

r (x, τ)

0

(t − τ)1−α

dτ.

(1.2)

However in April 2015, Caputo and Fabrizio [5] observed that the Caputo fractional derivative is unable to properly described some features related to some behavior happening in the fields of classical thermal media, classical viscoelastic materials or electromagnetic. Hence they proposed a new definition, the fractional derivative with no singular kernel in order to address the noticed unsolved 1 228

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issues. The new version called the Caputo-Fabrizio derivative differentiate from the Caputo derivative by having no singular kernel in the integral part. It reads as   Z (2 − α)M(α) t d α(t − τ) cf α Dt r(x,t) = r (x, τ) exp − dτ. (1.3) 2 (1 − α) 1−α 0 dτ 0 < α ≤ 1, where M(α) is a normalization function such that M(0) = M(1) = 1. Its associated fractional integral (anti-derivative) is given by [8]: cf α It r(x,t)

=

2α 2(1 − α) r(x,t) + (2 − α)M(α) (2 − α)M(α)

Z t

r (x, τ) dτ,

(1.4)

0

α ∈ [0, 1] t ≥ 0. This expression shows the integral as a sort of average between the function r and its integral. Since then, many authors have improved the concepts. Starting with Losada and Nieto [8] who developed the associated fractional integral (1.3) and Doungmo Goufo [11] or Doungmo Goufo and Atangana [12] who proposed the related Riemann-Liouville version. Nevertheless, the literature is full of several other definitions of fractional derivative. General formulation is done in [9] and many useful properties are intensively analyzed in [1–3, 10], especially in the analysis of the spread of diseases [14–17]. The main goal of this work is to apply both Caputo and Caputo-Fabrizio derivatives to the same Harry Dym equation and see their impact on the output behavior of the solutions. In other words, does the non-singularity have a significant influence on a real life process like wave motion or other motion? Recall that unlike Caputo derivative, there is no singularity at t = τ for Caputo-Fabrizio derivative and the following equality for Caputo-Fabrizio fractional derivative is satisfied: lim c f Dtα r(x,t) =

α→1

∂ r (x,t) ∂t

(1.5)

and lim c f Dtα r(x,t) = r (x,t) − r(x, 0).

(1.6)

α→0

This paper investigates the non-linear third-order Harry Dym differential equation within the scope of the two types of derivatives mentioned above. Note that the traditional Harry Dym model can be solved using the Lax operator [18–20] and is associated with the Sturm-Liouville operator. 2. Solvability with Caputo fractional derivative Let Ω = (a, b), R 3 T > 0 R 3 b > a ∈ R and r ∈ C0 [Ω × [0, T ]] . Let α ∈ [0; 1], β ∈ (0, +∞) then, the non-linear Dym equation expressed with the Caputo time fractional derivative is investigated in this section. Existence and uniqueness of the exact solution are shown for the model under investigation that reads as C

Dtα r(x,t) = r3 rxxx (x,t),

(2.1)

r(x, 0) = g(x)

(2.2)

subject to the initial condition

with C Dtα r(x,t) the Caputo derivative as defined in (1.1) and g : Ω 7−→ R+ . We start by transforming (2.1) into an integral form by applying the anti-derivative integral (1.2) on both sides to get 2 229

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r(x,t) − r(x, 0) =

1 Γ(α)

Z t 0

(t − y)α−1 (r3 rxxx (x,t))dy

(2.3)

Let us proceed by simplicity and consider the operator with three variables Ξ(x,t, r) = r3 rxxx (x,t).

(2.4)

The next goal is to show that operator Ξ with respect to variable r verifies the Lipschitz condition. For that,

kΞ(x,t, r) − Ξ(x,t, v)k = r3 rxxx (x,t) − v3 vxxx (x,t) C0 [Ω×[0,T ]]

(2.5)

Assuming that r and v are bounded functions, there is a positive constants k1 > 0 and k2 > 0 such that krkC3 0 [Ω×[0,T ]] ≤ k1

and

kvkC3 0 [Ω×[0,T ]] ≤ k2 .

(2.6)

Furthermore, using the properties of the norm and the Lipschitz condition for the first order derivative function ∂x there is a positive constant ϑ such that (2.5) becomes kΞ(x,t, r) − Ξ(x,t, v)kC0 [Ω×[0,T ]] ≤ k1 k2 ϑ 3 kr − vkC0 [Ω×[0,T ]] .

(2.7)

Putting K = k1 k2 ϑ 3 , we finally get kΞ(x,t, r) − Ξ(x,t, v)kC0 [Ω×[0,T ]] ≤ K kr − vkC0 [Ω×[0,T ]] , which therefore proves the desired Lipschitz condition. This enables us to evaluate the following norm kr(x,t)kC0 [Ω×[0,T ]] , t ∈ [0, T ]. Assuming the existence and the boundedness of the initial condition g, there is a positive constant C such that kg(x)kC0 [Ω×[0,T ]] ≤ C for any x ∈ Ω. Whence, 1 kr(x,t)kC0 [Ω×[0,T ]] ≤ kr(x, 0)kC0 [Ω×[0,T ]] + Γ(α) R t K α−1 dy ≤ kg(x)kC0 [Ω×[0,T ]] + Γ(α) 0 (t − y) α KT ≤ C + αΓ(α)

Rt

α−1 Ξ(x, y, r(x, y))dy 0 (t − y)

,

(2.8)

which yields the following propositions: Proposition 2.1. Assuming that g given in (2.2) is bounded, let 0 < α < 1 and Ξ[x,t, r(x,t)] : [Ω × [0, T ]] × B −→ A (with A ⊃ B) be a continuous function with respect to t for any fixed x ∈ Ω, r ∈ B. If r(x,t) ∈ C0 [Ω × [0, T ]] , then the function r(x,t) verifies the model (2.1)-(2.2) if and only if r(x,t) verifies the corresponding Volterra integral equation (2.3). 3 230

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Proof. To prove the necessity condition we assume that r(x,t) ∈ C0 [Ω × [0, T ]] satisfies the equations (2.1)-(2.2). Because Ξ(x, y, r(x, y) ∈ C [Ω × [0, T ] × B] for any r ∈ B¯ then (2.1) means there exists the Caputo fractional derivative of r in C [Ω × [0, T ]] . However C α 0 Dt r

=

∂ −α
I [r(x,t) − r(x, 0)]. ∂t 0

(2.9)

Exploiting I0−α [r(x,t) − r(x, 0)] ∈ C0 [Ω × [0, T ]] and applying the results in [13] for γ = 0 to V (x,t) = r(x,t) − r(x, o), yields I0α C0 Dtα r(x,t) = I0α C0 Dα0 [r(x,t) − r(x, 0)] i− j r1−α (x, 0)α− j j=1 Γ(α − j + 1) 1

(2.10)

= r(x,t) − r(x, 0) − ∑

with r1−α (x,t) = I01−α [r(x,t) − r(x, 0)]. Using the integration by parts in (2.10) and differentiating the resulting expression give (1− j)

r1−α (x,t) =


∂ 2−α I0 [∂t r(x,t) − r(x, 0)] ∂t

(2.11)

Changing of variable t = β + ρ(y − β ) leads to (1− j) r1−α (x,t)

(y − β )1−α = Γ(1 − α)

Z1


(1 − ρ)−α r1− j [β + ρ(y − β )] .

(2.12)

0

Recalling 0 < α < 1 and r1− j (x,t) ∈ C [Ω × [0, T ]] , equation (2.12) and (2.10) take the form I0α C0 Dtα r(x,t) = r(x,t) − r(x, 0)

(2.13)

Since I0α Ξ(x, τ, r(x, τ)) ∈ C0 [Ω × [0, T ]] and using the Lipschitz condition of Ξ, we obtain kI0α Ξ(x, τ, r(x, τ)kC0 [Ω×[0,T ]] ≤

KT α . αΓ(α)

Nonetheless, applying I0α on both sides of (2.1) and making use of the initial condition we prove the necessity condition by recovering the Volterra version (2.3). Conversely for the sufficient condition, assume r(x,t) ∈ C [Ω × [0, T ]] verifies the Volterra version (2.3) of equations (2.1)-(2.2). It suffices to show that r(x,t) satisfies the initial condition (2.2). 4 231

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The differentiation of the two sides of the Volterra version yields 1 ∂t r(x,t) = Γ(α − 1)

Zt

Ξ(x, τ, r(x, τ))(t − τ)α dτ.

0

Changing again the variable t = β + ρ(y − β ) in the Volterra expression for k = 1, leads to rk (x,t) =

(y − β ) Γ(α − k)

Zt 0

Ξ(x, β + ρ(y − β ), r(x, β + ρ(y − β ))) dy (1 − β )1−α+k

Passing to the limit as y −→ β + and making use of the continuity of K show that the sought initial condition is verified, and the sufficient condition is proved. Proposition 2.2. Considering 0 < α < 1 and the Lipschitz condition for Ξ then, there is a unique solution for equation (2.3) in the space C0,α [0, T ] × Ω Proof. From the above analysis in Theorem 2.1, it is sufficient to show the existence of the unique solution r(x,t) ∈ C0 [Ω × [0, T ]] of the Volterra equation (2.3). Indeed, the model (2.3) holds in any interval [0, τ] ⊆ [0, T ]. Hence, we select the adequate t1 ∈ [0, T ] so that Kt1α αΓ(α) < 1 and then, prove the desired existence result of a unique r(x,t) ∈ C0 [Ω × [0,t1 ]] . We can exploit the technique of successive approximation and set

r0 (x,t) = r(x, 0) 1 rn (x,t) = rn−1 (x, 0) + Γ(α)

Zt

Ξ(x, τ, rn−1 (x, τ)) dτ, (t − τ)1−α

0

n ∈ N.

(2.14)

Obviously, r(x, 0) ∈ C0 [0, T ] and the differentiation of (2.14) with respect to t yields 1 ∂t rn (x,t) = Γ(α − 1)

Zt 0

Ξ(x, τ, rn−1 (x, τ)) dτ. (t − τ)−α

Using the differentiability of r(x, 0) with respect to t leads to rn (x,t) ∈ C0 [0, T ]. The quantity krn (x,t) − rn−1 (x,t)kC0 [Ω×[0,t1 ]] can be evaluated for n ∈ N. Whence, krn (x,t) − rn−1 (x,t)kC0 [Ω×[0,t1 ]] ≤

Kt1α . αΓ(α)

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Furthermore, kr2 (x,t) − r1 (x,t)kC0 [Ω×[0,t1 ]] ≤ kI0α Ξ(x, τ, r1 (x, τ)) − Ξ(x, τ, r0 (x, τ))kC0 [Ω×[0,t1 ]] t1α kr1 (x,t) − r0 (x,t)kC0 [Ω×[0,t1 ]] αΓ(α) Kt1α tα ≤ · 1 αΓ(α) αΓ(α) ≤

The above iteration is repeated n-times and yields  kr2 (x,t) − r1 (x,t)kC0 [Ω×[0,t1 ]] ≤

Kt1α αΓ(α)

n−1 ·

t1α αΓ(α)

Hence, the sequence {rn (x,t)}n∈N has r(x,t) ∈ C0 [Ω × [0,t1 ]] as its limit function. Moreover, the assumption α t1 αΓ(α) < 1 gives lim krn (x,t) − rn−1 (x,t)kC0 [Ω×[0,t1 ]] = 0. However by considering t1 = T, we can estimate n−→∞

Ξ(x,t, rn (x, τ)) − Ξ(x,t, r(x,t)). Taking into account the Lipschitz condition of Ξ leads to



t Zt

1 Z Ξ(x, τ, rn (x, τ)) 1 Ξ(x, τ, r(x, τ))

dτ − dτ

Γ(α)

1−α 1−α (t − τ) Γ(α) (t − τ)

0

0

 ≤

C0 [Ω×[0,T ]]

 Lt1α krn (x,t) − r(x,t)kC0 [Ω×[0,t1 ]] αΓ(α)

and



t t Z Z

1 Ξ(x, τ, rn (x, τ)) 1 Ξ(x, τ, r(x, τ))

lim dτ − dτ

1−α 1−α n−→∞ Γ(α) (t − τ) Γ(α) (t − τ)

0 0

= 0,

C0 [Ω×[0,T ]]

which prove that r(x,t) satisfies (2.3) in the space C0 [Ω × [0, T ]]. Uniqueness result Considering now that there are two separate solutions r1 (x,t) and r2 (x,t) verifying (2.3) on [0,t1 ] then, kr1 (x,t) − r2 (x,t)kC0 [Ω×[0,t1 ]] ≤

Kt1α kr1 (x,t) − r2 (x,t)kC0 [Ω×[0,t1 ]] . αΓ(α)

Kt α

1 This gives 1 ≤ αΓ(α) which is a contradiction. The solution is unique in C0 [Ω × [0,t1 ]] We consider now the closed interval [t1 ,t2 ] with t2 = t1 + h1 where h1 > 0 and t2 < T. For t ∈ [t1 ,t2 ],

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we have 1 r(x,t) = Γ(α)

Zt t1

Ξ(x, τ, rn (x, τ)) 1 dτ + r(x, 0) + (t − τ)1−α Γ(α)

Zt1 0

Ξ(x, τ, rn (x, τ)) dτ. (t − τ)1−α

Taking into account the uniqueness result on [0,t1 ], then

r(x,t) =

1 where r1∗ (x,t) = r(x, 0) + Γ(α)

r1∗ (x,t) +

Rt1 Ξ(x,τ,rn (x,τ)) 0

(t−τ)1−α

1 Γ(α)

Zt t1

Ξ(x, τ, rn (x, τ)) dτ (t − τ)1−α

dτ is a given function.

we repeat the same analysis presented above and there is an unique solution r(x,t) ∈ C0 [Ω × [t1 ,t2 ]] for (2.3). Taking another interval [t2 ,t3 ] so that t3 = t2 + h2 with h2 > 0 and t3 < T, the same analysis is performed to finally obtain the existence of an unique solution r(x,t) ∈ C0 [Ω × [0, T ]] of equation (2.3) and therefore, proves the existence of an unique solution of equation (2.1) in the space C0,α [Ω × [0, T ]]. Corollary 2.3. Assume Ξ satisfies Theorem 2.1. If the inequality KT α αΓ(α) ≤ 1 holds, then the sequence rn (x,t), (n ∈ N) tends to the exact solution r(x,t). Moreover, we have for any n ∈ N, kr(x,t) − rn (x,t)kC0 [Ω×[0,t1 ]] ≤

Tα Kn Tα αΓ(α) 1 − αΓ(α)

Proof. The proof is done by the mathematical induction on n. For kr1 (x,t) − r0 (x,t)kC0 [Ω×[0,t1 ]] ≤

KT α αΓ(α)

 kr2 (x,t) − r1 (x,t)kC0 [Ω×[0,t1 ]] ≤ K

T αΓ(α)

2

Thus by induction, Tα krn (x,t) − rn−1 (x,t)kC0 [Ω×[0,t1 ]] ≤ αΓ(α)



KT α αΓ(α)

n−1

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but, kr(x,t) − rn (x,t)kC0 [Ω×[0,t1 ]] = lim krn+ j (x,t) − rn (x,t)kC0 [Ω×[0,t1 ]] j−→∞

= krn+1 (x,t) − rn (x,t)kC0 [Ω×[0,t1 ]] + krn+2 (x,t) − rn+1 (x,t)kC0 [Ω×[0,t1 ]] + · · · n+1  n+2  Tα Tα n+1 n +K +··· ≤K αΓ(α) αΓ(α)  n+1 " ∞  k # α α T KT = Kn ∑ αΓ(α) k=0 αΓ(α)

(2.15)

and the corollary is proved. This implies the following existence and uniqueness results for our model (2.1)-(2.2): Corollary 2.4. The function r(x,t) is the strong solution of the sequence rn (x,t) given by (2.14). Proof. This result is an immediate consequence of Propositions 2.2 and 2.1 3. Analyzis with Caputo-Fabrizio fractional derivative 3.1. Introduction and formulation Definition 3.1 (Piccard’s L-stability). Consider the Banach space (B, kk), the self-map L of B and the recursive technique σn+1 = g(L, σn ). Let us assume that B(L), containing all the fixed points of L has at least one element and σn converges to an element b of B(L). Let {un } ⊆ B and set dn = kun+1 − g(L, un )k. Therefore lim dn = 0 n−→∞

leads to lim un = b. In this case, we say that the reccurence formula σn+1 = g(L, σn ) is L-stable. n−→∞

Remark 3.1. Assuming that {un } has a upper boundary, σn+1 = Lσn is called Piccard’s iteration if the conditions of Definition 3.1 are satisfied and therefore, will be L-stable. Lemma 3.1. Let (B, kk) be a Banach space and L a self-map of B verifying kLx − Lyk ≤ Ckx − Lxk +Ckx − yk for all x, y ∈ B, where 0 ≤ C, 0 ≤ α ≤ 1. If L admits a fixed point f then, L is Picard L-stable. Proof. [4, Theorem 3.1] Proposition 3.2. The self-map L expressed as L(rn (x,t)) = rn+1 (x,t) = rn (x,t) +c f Itα [r3 rxxx (x,t))] is L-stable in L2 (a, b) 8 235

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Proof. The first step is to prove that L has a fixed-point. For that set i, j ∈ N then,

    kLri (x,t)−Lr j (x,t)k = kri+1 (x,t)−r j+1 (x,t)k = ri (x,t) +c f Itα ri3 ∂x33 ri − r j (x,t) −c f Itα r3j ∂x33 r j with cf α It r(t)

=

2α 2(1 − α) r(t) + (2 − α)M(α) (2 − α)M(α)

Z t

r (τ) dτ, 0

the anti-derivative associated to the Caputo-Fabrizio derivative as given in (1.4). Then, making use of the boundedness the function u, the Lipschitz condition for the first order operator ∂x with the same constants k1 , k2 , ϑ as in the previous section, we have kLri (x,t) − Lr j (x,t)k ≤ kri (x,t) − r j (x,t)k + kc f Itα [ri3 ∂x33 ri − r3j ∂x33 r j ]k ≤

2α 2(1 − α) k1 k2 ϑ 3 kri − r j k + k1 k2 ϑ 3 kri − r j k. (2 − α)M(α) (2 − α)M(α)

Thus, kLri (x,t) − Lr j (x,t)k ≤ K kri (x,t) − r j (x,t)k With K =

2(1 − α) 2α k1 k2 ϑ 3 kri − r j k + k1 k2 ϑ 3 . (2 − α)M(α) (2 − α)M(α)

Consequently L is Lipschitz continuous with respect to r and this means the non-linear operator L has a fixed point. To conclude the proof, it is necessary to remark that taking C = 0 and C = P, the conditions of Lemma 3.1 are verified for L and then, L is Picard L-stable. 4. Numerical Solvability This section deals with some numerical schemes associated with both models. So a technique to determine the solution for each model using integral iterative methods is presented. We The model with Caputo derivative is iteratively solved making use of Laplace transform while the model with the Caputo-Fabrizio derivative exploit the Sumudu transform. Similar results are obtained as shown below. 4.1. Numerical Approximations with Caputo derivative Applying the Laplace transform L on both sides of the model (2.1)-(2.2) iteratively yields   pr(x, p) − g(x) = L r3 rxxx (x,t) (p), equivalently r(x, p)

=

 g(x) 1  3 + L r rxxx (x,t) (p). p p 9 236

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Taking the inverse Laplace transform L −1 yields r(x,t) = g(x) + L

−1



  1  3 L r rxxx (x,t) (t). p

Now, we can introduce the following iterative formula r0 (x,t) = g(x) rn+1 (x,t) = rn (x,t) + L

−1



  1  3 L r rxxx (x,t) (t). p

the above formula leads to a numerical approximation with Caputo derivative and the approximate solution reads as r(x,t) = lim rn (x,t) n−→∞

4.2. Numerical Approximations Caputo-Fabrizio derivative Here we first recall the following important relation   pF(p) − f (o) S c0 f Dtα f (t) = M(α) 1−α −αp where F(p) = S ( f (t)) is the Sumudu transform of f (t). Applying the Sumudu transform S on both sides of equation (2.1) yields   pr(x, p) − g(x) = S r3 rxxx (x,t) (p), 1−α −αp equivalently r(x, p) =

  g(x) + (1 − α − α p)S r3 rxxx (x,t) (p). p

The inverse Sumudu transform S −1 yields r(x,t)  
= g(x) + S −1 (1 − α − α p)S r3 rxxx (x,t) (t). repeating as above, the following iterative formula introduced r0 (x,t) = g(x)  
rn+1 (x,t) = rn (x,t) + S −1 (1 − α − α p)S r3 rxxx (x,t) (t). which leads to a numerical approximation with Caputo-Fabrizio derivative and the approximate solution reads as r(x,t) = lim rn (x,t). n−→∞

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Those recurrence schemes are used and numerical representations of both models can be depicted in Fig.1 to Fig.5 for different values of the order α. The graphics in Fig.1, Fig.2, Fig.3, Fig.4 and Fig.5, performed respectively for α = 0.2, α = 0.3, α = 0.4, α = 0.8 and α = 1.0 using the Caputo derivative and compared to one using the Caputo-Fabrizio derivative show the standard well-known wave solution of the Harry Dym equation. It is clear that the figures show similar behavior for solutions of both models. 5. Concluding remarks We have proved the existence and uniqueness of the solution to for the nonlinear Harry Dym equation modelled with both the classical Caputo derivative and the newly introcuced derivative of fractional order with no singular kernel. It is the first time that the same model of Harry Dym is analyzed using both derivatives in the same work. This proves that there is a possible way to extent the nonlinear Harry Dym model to the scope of fractional calculus.Two numerical methods suitable to approximate the solutions of model with both derivatives have been presented with numerical simulations performed for α = 0.2, α = 0.3, α = 0.4, α = 0.8 and α = 1.0. Each figure exhibits solution with similar behavior for the involved wave associated to the Dym equation. This paper innovates by pointing out another concrete application of the Caputo-Fabrizio derivative, new in the literature and till under investigation. More complex investigation will certainly follow.

Fig. 1: Representation of the solution r(x,t) when α = 0.2 with both derivatives.

Fig. 2: Representation of the solution r(x,t) when α = 0.3 with both derivatives. 11 238

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Fig. 3: Representation of the solution r(x,t) when α = 0.4 with both derivatives.

Fig. 4: Representation of the solution r(x,t) when α = 0.8 with both derivatives.

Fig. 5: Representation of the solution r(x,t) when α = 1.0 with both derivatives. References [1] Caputo M., Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Ast. Soc. 13 (5), 529-539, 1967; Reprinted in: Fract. Calc. Appl. Anal. 11 (1), 3-14, (2008) 12 239

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[2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. [3] Atangana A. and Doungmo Goufo E.F., ”Extension of Matched Asymptotic Method to Fractional Boundary Layers Problems,” Mathematical Problems in Engineering, vol. 2014, Article ID 107535, 7 pages, 2014. doi:10.1155/2014/107535 [4] D. Eberly, Stability Analysis for Systems of Differential Equations, Geometric Tools, LLC, (2008). http://www.geometrictools.com. [5] Caputo M, Fabrizio M, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2), (2015) 1–13 [6] Caputo Michele , Carcione Jos´e M., Botelho Marco A. B., Modeling Extreme-Event Precursors with the Fractional Difusion Equation, Fractional Calculus and Applied Analysis, 18 (1) 208–222, (2015). [7] Doungmo Goufo EF, A biomathematical view on the fractional dynamics of cellulose degradation, Fractional Calculus and Applied Analysis, Vol. 18, No 3, 554–564, (2015). DOI: 10.1515/fca-20150034. [8] Losada J, Nieto JJ, Properties of the new fractional derivative without singular Kernel. Progr. Fract. Differ. Appl, 1(2), (2015) 87–92 [9] Tenreiro Machado J., Mainardi F., and Kiryakova V. Fractional calculus: quo vadimus? (Where are we going?). Fract. Calc. Appl. Anal., 18(2):495–526, 2015. [10] Doungmo Goufo EF, A biomathematical view on the fractional dynamics of cellulose degradation, Fractional Calculus and Applied Analysis, Vol. 18, No 3, 554–564, (2015). DOI: 10.1515/fca-20150034 [11] Doungmo Goufo EF, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg–de Vries–Bergers equation, Mathematical Modelling and Analysis, Vol. 21, Issue 2, 188198, 2016, http://dx.doi.org/10.3846/13926292.2016.1145607. [12] Doungmo Goufo EF, Atangana A, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, Vol. 131, Issue 8, 2016. DOI: 10.1140/epjp/i2016-16269-1 [13] Luchko Y. and Gorenflo R., The initial value problem for some fractional differential equations with the Caputo derivative. Preprint Series A0898, Freic Universitat Berlin, 1998, Fachbreich Mathematik and Informatik, 1998. [14] Area I, Batarfi H, Losada J, Nieto JJ, Shammakh W, Torres A. On a Fractional Order Ebola Epidemic Model. Adv Difference Equ. 2015; Art. ID 278, 12 pp. [15] Rachah A, Torres DFM. Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa. Discrete Dyn Nat Soc. 2015; Art. ID 842792, 9 pp. [16] Area I, Losada J, Nda¨ırou F, Nieto JJ, Tcheutia DD. Mathematical modeling of 2014 Ebola outbreak. Math Method Appl Sci.; in press. [17] Du, M, Wang, Z, Hu, H: Measuringmemorywith the order of fractional derivative. Sci. Rep. 3, 3431 (2013) [18] Beals R. and Coifman R., Scattering and Inverse Scattering for First Order Systems, Commun. Pure and Appl. Math. 37 (1984) 39-90 [19] Gerdjikov V. and Yanovski A., Completeness of the Eigenfunctions for the Caudrey-Beals-Coifman System, J. Math. Phys. 35 (1994) 3687-3725 [20] Z. Yan, The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equation, MMRC, AMSS, Academis Sinica, Beijing 22(2003), 275-284. [21] Khan Y., Sayevand K., Fardi M., Ghasemi M., A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations, Applied Mathematics and Computation 2014, 249, 229–236.

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Product-type Operators from Weighted Bergman Spaces to Bloch-Orlicz Spaces Zhi-jie Jiang

(School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, P. R. China) Abstract: Let D be the open unit disk in the complex plane C and H(D) the class of all analytic functions on D. Let ϕ be an analytic self-map of D and u ∈ H(D). By constructing some suitable test functions in weighted Bergman space, in this paper the boundedness and compactness of the product-type operators Dn Mu Cϕ , Dn Cϕ Mu , Cϕ Dn Mu , Mu Dn Cϕ , Mu Cϕ Dn and Cϕ Mu Dn from weighted Bergman space to Bloch-Orlicz space are characterized in terms of the symbol functions u and ϕ. Keywords: Weighted Bergman-type space; Bloch-Orlicz space; product-type operator; boundedness; compactness MR (2010) Subject Classification: 47B38; 47B33, 47B37 Chinese Library Classification: O177.2

1 Introduction Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C and H(D) the class of all analytic functions on D. Let ϕ be an analytic self-map of D and u ∈ H(D). The weighted composition operator Wϕ,u on H(D) is defined by Wϕ,u f (z) = u(z)f (ϕ(z)), z ∈ D. If u ≡ 1, it becomes the composition operator, usually denoted by Cϕ . If ϕ(z) = z, it becomes the multiplication operator, usually denoted by Mu . Since Wϕ,u = Mu Cϕ , it is a producttype operator. A standard problem is to provide function theoretic characterizations when ϕ and u induce a bounded or compact weighted composition operator (see, for example, [2, 4, 8, 13, 15, 23, 26, 27] and the references therein). Let n ∈ N0 = N ∪ {0}. The nth differentiation operator Dn on H(D) is defined by Dn f (z) = f (n) (z), z ∈ D, Foundation item: Supported by the Sichuan Province University Key Laboratory of Bridge Nondestruction Detecting and Engineering Computing (No.2016QZJ01), the Key Fund Project of Sichuan Provincial Department of Education (No.15ZA0221) and the Cultivation Project of Sichuan University of Science and Engineering (No.2015PY04). E-mail: [email protected]

1

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where f (0) = f . If n = 1, it is the differentiation operator D. A systematic study of some other product-type operators started by Stevi´c et al. since the publication of papers [11] and [12]. Before that there were a few papers in the topic, e.g., [5]. The publication of paper [11] first attracted some attention in product-type operators DCϕ and Cϕ D (see, e.g., [14, 18, 20] and the references therein). The publication of paper [12] attracted some attention in product-type operators involving integral-type ones (see, e.g., [9, 19, 21] and the references therein). Now there is a great interest in various product-type operators (see, e.g., [6, 7, 10, 16, 30, 31] and the references therein). By using multiplication, composition and the nth differentiation operators, we define the product-type operators in the following six ways Dn Mu Cϕ , Dn Cϕ Mu , Cϕ Dn Mu , Mu Dn Cϕ , Mu Cϕ Dn , Cϕ Mu Dn .

(1.1)

When n = 1, they were studied by Sharma in [17]. They were also studied on the weighted Bergman space in a unified manner by Stevi´c et al. in [24] and [25]. By constructing some test functions in weighted Bergman space, here we characterize the boundedness and compactness of the product-type operators in (1.1) from weighted Bergman space to Bloch-Orlicz space. Because some more suitable test functions were not found in weighted Bergman space, before this work we didn’t find any result on these operators from weighted Bergman space to Bloch-Orlicz space. Let dA(z) =

1 dxdy be π 2 α

the normalized Lebesgue measure on D. For α > −1, let

dAα (z) = (α + 1)(1 − |z| ) dA(z) be the weighted Lebesgue measure on D. For p ≥ 1, the famous weighted Bergman space Apα consists of all f ∈ H(D) such that Z p kf kApα = |f (z)|p dAα (z) < ∞. D

It is well known that the weighted Bergman space Apα with the norm k · kApα is a Banach space. For some results of the weighted Bergman space, see, for example, [28, 29]. Let Ψ be a strictly increasing convex function on [0, +∞) such that Ψ(0) = 0. The BlochOrlicz space B Ψ was introduced in [15] by Ramos Fern´andez, is the class of all f ∈ H(D) such that sup(1 − |z|2 )Ψ(λ|f 0 (z)|) < ∞ z∈D

for some λ > 0 depending on f . Ramos Fern´andez in [15] proved that B Ψ is isometrically equal to µΨ -Bloch space, where µΨ (z) =

1 Ψ−1 (

, 1 ) 1−|z|2

z ∈ D.

Hence, B Ψ is a Banach space with the norm given by kf kBΨ = |f (0)| + sup µΨ (z)|f 0 (z)|. z∈D

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This space generalizes some other spaces. For example, if Ψ(t) = tp with p > 0, then the space B Ψ coincides with the weighted Bloch space B α , where α = 1/p. Also, if Ψ(t) = t log(1 + t), then B Ψ coincides with the Log-Bloch space (see [1]). Let X and Y be Banach spaces. A linear operator L : X → Y is bounded if there exists a positive constant K such that kLf kY ≤ Kkf kX for all f ∈ X. The operator L : X → Y is compact if it maps bounded sets into relatively compact sets. In this paper, the letter C denotes a positive constant which may differ from one occurrence to the other. The notation a . b means that there exists a positive constant C such that a ≤ Cb. When a . b and b . a, we write a  b.

2 Prerequisites The first result is a alternative to Proposition 3.11 in [3], which characterizes the compactness in terms of sequential convergence. So the proof is omitted. Lemma 2.1. Let T be one of the operators in (1.1). Then the bounded operator T : Apα → B Ψ is compact if and only if for every bounded sequence {fj }j∈N in Apα such that fj → 0 uniformly on every compact subset of D as j → ∞, it follows that lim kT fj kBΨ = 0.

j→∞

For k = 0, the following lemma was proved in [29], while for k ≥ 1 it essentially follows from the Jensen’s inequality (see [6]). Lemma 2.2. Let α > −1 and p ≥ 1. Then for each k ∈ N0 , there exists a positive constant Ck = C(α, p, k) independent of f ∈ Apα and z ∈ D such that |f (k) (z)| ≤

Ck kf kApα (1 − |z|2 )k+

α+2 p

.

In order to construct some test functions in weighted Bergman space, for a fixed w ∈ D and i ∈ N0 we define the following function kw,i (z) =

(1 − |w|2 )i+ (1 − wz)i+

α+2 p

2α+4 p

, z ∈ D.

Then from [6], we know that fw,i ∈ Apα and sup kkw,i kApα . 1.

(2.1)

w∈D

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By using some suitable linear combinations of the functions kw,i , we obtain the test function in Apα in the following result, which will be used in the proofs of our main results. Lemma 2.3. Let w ∈ D and n ∈ N. Then for each fixed k ∈ {0, 1, . . . , n + 1}, there exist constants a0,k , a1,k , . . . , an+1,k such that the function fw,k (z) =

n+1 X

ai,k kw,i (z)

i=0

satisfies wk

(k)

fw,k (w) =

(1 −

|w|2 )k+

(j)

α+2 p

and fw,k (w) = 0

(2.2)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. Moreover, sup kfw,k kApα . 1.

(2.3)

w∈D

Proof. We write a = (2α + 4)/p. From a direct calculation, it follows that the system (2.2) is equivalent to the following system  n+1 P   (a + i)ai,k = 0    i=0   n+1 P    (a + i)(α + i + 1)ai,k = 0    i=0    ······· k−1 n+1 P Q   (a + i + j)ai,k = 1    i=0 j=0     ·······    n n+1  Q P   (a + i + j)ai,k = 0. 

(2.4)

i=0 j=0

Hence we only need to prove that there exist constants a0,k , a1,k , . . . , an+1,k such that the system (2.4) holds. By Lemma 3 in [22], the determinant of the system (2.4) equals Qn+1 j=1 j!, which is different from zero. So there exist constants a0,k , a1,k , . . . , an+1,k such that the system (2.4) holds. From (2.1) the asymptotic expression of supw∈D kfw,k kApα . 1 is obvious. Remark 2.1. It is not hard to see that fw,k → 0 uniformly on every compact subset of D as |w| → 1− . Stevi´c in [22] used the Fa`a di Bruno’s formula of the following version (f ◦ ϕ)(n) (z) =

n X

f (k) (ϕ(z))Bn,k (ϕ0 (z), . . . , ϕ(n−k+1) (z)),

(2.5)

k=0

244

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where Bn,k (x1 , ..., xn−k+1 ) is the Bell polynomial. For n ∈ N the sum can go from k = 1 since Bn,0 (ϕ0 (z), ..., ϕ(n−k+1) (z)) = 0, however we will keep the summation since for n = 0 the only existing term B0,0 is equal to 1. From (2.5) and the Leibnitz formula the next Lemma 2.4 follows. Lemma 2.4. Let f , u ∈ H(D) and ϕ be an analytic self-map of D. Then n+1 n+1 X
(n+1) X
j = f (k) (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) . u(z)f (ϕ(z)) k=0

j=k

3 Boundedness of the product-type operators First we characterize the boundedness of the operator Dn Mu Cϕ : Apα → B Ψ . Theorem 3.1. Let α > −1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Dn Mu Cϕ : Apα → B Ψ is bounded. (ii) The functions u and ϕ satisfy the following conditions: n+1
P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) j=k Ik := sup k, we get k n+1 X X
0 (j) i Dn Mu Cϕ hk (z) = hk (ϕ(z)) Cn+1 u(n+1−i) (z)Bi,j (ϕ0 (z), . . . , ϕ(i−j+1) (z)) j=0

=

k X j=0

k · · · (k − j + 1)(ϕ(z))k−j

i=j n+1 X

i Cn+1 u(n+1−i) (z)Bi,j (ϕ0 (z), . . . , ϕ(i−j+1) (z)).

(3.3)

i=j

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

From (3.3), the boundedness of function ϕ and the triangle inequality, by noticing that the coefficient at n+1 X

j Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z))

j=k

is independent of z and finally using hypothesis (3.2) we easily obtain n+1 X j Lk := sup µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) ≤ CkDn Mu Cϕ k. (3.4) z∈D

j=k

By induction we see that (3.4) holds for each k ∈ {0, 1, . . . , n + 1}. For a fixed w ∈ D and k ∈ {0, 1, . . . , n + 1}, by Lemma 2.3 there exist constants a0,k , a1,k , . . . , an+1,k such that the function fϕ(w),k (z) =

n+1 X

ai,k kϕ(w),i (z),

i=0

satisfies k

(k) fϕ(w),k (ϕ(w))

=

ϕ(w) (1 −

(j)

α+2 |ϕ(w)|2 )k+ p

and fϕ(w),k (ϕ(w)) = 0

(3.5)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. Moreover, sup kfϕ(w),k kApα ≤ C.

(3.6)

w∈D

Then from (3.5), (3.6) and the boundedness of Dn Mu Cϕ : Apα → B Ψ , we have

n+1
P j µΨ (w)|ϕ(w)|k Cn+1 u(n+1−j) (w)Bj,k ϕ0 (w), . . . , ϕ(j−k+1) (w) j=k

Ik (w) :=

(1 − |ϕ(w)|2 )k+

α+2 p

≤ kDn Mu Cϕ fϕ(w),k kBΨ ≤ CkDn Mu Cϕ k.

(3.7)

From (3.7) we see that

sup Ik (z) ≤ C Dn Mu Cϕ , z∈D

from which we obtain

n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) sup |ϕ(z)|>1/2

j=k

(1 −

|ϕ(z)|2 )k+

≤ CkDn Mu Cϕ k.

α+2 p

246

(3.8)

Zhi-jie Jiang 241-254

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

On the other hand, from (3.4) we get

n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) j=k

sup

(1 − |ϕ(z)|2 )k+

|ϕ(z)|≤1/2

≤ CLk ≤ CkDn Mu Cϕ k.

α+2 p

(3.9) Hence from (3.8) and (3.9) we see that Ik < ∞ for each k ∈ {0, 1, ..., n + 1}. (ii) ⇒ (i). From Lemma 2.2 and Lemma 2.4, for all f ∈ Apα we have


0 sup µΨ (z) (Dn Mu Cϕ f (z) z∈D

n+1 n+1 X X
j (k) = sup µΨ (z) f (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) z∈D

k=0

≤ sup µΨ (z) z∈D

≤

n+1 X

n+1 X

j=k

n+1 (k) X
j f (ϕ(z)) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z)

k=0

j=k

Ck Ik kf kApα .

(3.10)

k=0

It is clear that |(Dn Mu Cϕ f )(0)| ≤ Ckf kApα .

(3.11)

Hence from (3.10) and (3.11) it follows that Dn Mu Cϕ : Apα → B Ψ is bounded. Remark 3.1. If Dn Cϕ Mu : Apα → B Ψ is a zero operator, then it is obvious that kDn Cϕ Mu k = 0. Hence, the case is usually excluded from such considerations. Remark 3.2. Since Dn Cϕ Mu = Dn Mu◦ϕ Cϕ , the characterization of the boundedness of Dn Cϕ Mu : Apα → B Ψ can be directly obtained from Theorem 3.1. So we omit here. Noticing that (Cϕ Dn Mu f )0 (z) =

n+1 X

k Cn+1 u(n+1−k) (ϕ(z))ϕ0 (z)f (k) (ϕ(z)),

k=0

we can obtain the following result whose proof is similar to that of Theorem 3.1. So we also omit. Theorem 3.2. Let α > −1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Cϕ Dn Mu : Apα → B Ψ is bounded.

247

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

(ii) The functions u and ϕ satisfy the following conditions: Jk := sup

µΨ (z)|u(n+1−k) (ϕ(z))||ϕ0 (z)| (1 − |ϕ(z)|2 )k+

z∈D

α+2 p

−1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Mu Dn Cϕ : Apα → B Ψ is bounded. (ii) The functions u and ϕ satisfy the following conditions: µΨ (z) u0 (z)Bn,k (ϕ0 (z), . . . , ϕ(n−k+1) (z)) + u(z)Bn+1,k (ϕ0 (z), . . . , ϕ(n−k+2) (z)) Mk := sup −1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Mu Cϕ Dn : Apα → B Ψ is bounded. (ii) The functions u and ϕ satisfy the following conditions: R: = sup z∈D

µΨ (z)|u0 (z)| (1 − |ϕ(z)|2 )n+

α+2 p

< ∞,

and S := sup z∈D

µΨ (z)|u(z)||ϕ0 (z)| (1 − |ϕ(z)|2 )n+1+

248

α+2 p

< ∞.

Zhi-jie Jiang 241-254

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Remark 3.3. Noticing that Cϕ Mu Dn = Mu◦ϕ Cϕ Dn , we can obtain the characterization of the boundedness of Cϕ Mu Dn : Apα → B Ψ from Theorem 3.4. Here we omit.

4 Compactness of the product-type operators We first characterize the compactness of the operator Dn Mu Cϕ : Apα → B Ψ . Theorem 4.1. Let α > −1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Dn Mu Cϕ : Apα → B Ψ is compact. (ii) The functions u and ϕ satisfy Lk < ∞ and

n+1
P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k ϕ0 (z), . . . , ϕ(j−k+1) (z) j=k

lim

(1 − |ϕ(z)|2 )k+

|ϕ(z)|→1−

=0

α+2 p

for each k ∈ {0, 1, . . . , n + 1}. Proof. (i) ⇒ (ii). Suppose that the operator Dn Mu Cϕ : Apα → B Ψ is compact. Then it is clear that the operator Dn Mu Cϕ : Apα → B Ψ is bounded. Hence from the proof of Theorem 3.1 it follows that Lk < ∞ for each k ∈ {0, 1, . . . , n + 1}. Consider a sequence {ϕ(zi )}i∈N in D such that |ϕ(zi )| → 1− as i → ∞. If such a sequence does not exist, then the last condition in (ii) obviously holds. Without loss of generality, we may suppose that |ϕ(zi )| > 1/2 for all i ∈ N. For each fixed k ∈ {0, 1, . . . , n + 1}, by using this sequence we define the function sequence fi,k (z) = fϕ(zi ),k (z), i ∈ N. Then from Lemma 2.3 and Remark 2.1, we see that supi∈N kfi,k kApα ≤ C and fi,k → 0 uniformly on every compact subset of D as i → ∞, moreover k

(k) fi,k (ϕ(zi ))

=

ϕ(zi ) (1 − |ϕ(zi

(j)

and fi,k (ϕ(zi )) = 0

α+2 )|2 )k+ p

(4.1)

for each j ∈ {0, 1, . . . , n + 1} \ {k}. Then from Lemma 2.1 we have lim kDn Mu Cϕ fi,k kBΨ = 0.

(4.2)

i→∞

Combing (4.1) and (4.2), for each fixed k ∈ {0, 1, . . . , n + 1} we get

n+1 P j µΨ (zi ) Cn+1 u(n+1−j) (zi )Bj,k (ϕ0 (zi ), . . . , ϕ(j−k+1) (zi )) lim

i→∞

j=k

(1 − |ϕ(zi )|2 )k+

α+2 p

249

= 0.

(4.3)

Zhi-jie Jiang 241-254

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

(ii) ⇒ (i). We first prove that Dn Mu Cϕ : Apα → BΨ is bounded. We observe that the last condition in (ii) implies that for every ε > 0, there is an η ∈ (0, 1), such that for any z ∈ K = {z ∈ D : |ϕ(z)| > η}

n+1 P j µΨ (z) Cn+1 u(n+1−j) (z)Bj,k (ϕ0 (z), . . . , ϕ(j−k+1) (z)) Ik (z) =

j=k

(1 − |ϕ(z)|2 )k+

−1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Cϕ Dn Mu : Apα → B Ψ is compact. (ii) The functions u and ϕ satisfy the following conditions: sup µΨ (z)|u(n+1−k) (ϕ(z))||ϕ0 (z)| < ∞ z∈D

and lim

µΨ (z)|u(n+1−k) (ϕ(z))||ϕ0 (z)|

|ϕ(z)|→1−

(1 − |ϕ(z)|2 )k+

α+2 p

=0

for each k ∈ {0, 1, . . . , n + 1}. Theorem 4.3. Let α > −1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Mu Dn Cϕ : Apα → B Ψ is compact. (ii) The functions u and ϕ satisfy the following conditions:

sup µΨ (z) u0 (z)Bn,k (ϕ0 (z), . . . , ϕ(n−k+1) (z)) + u(z)Bn+1,k (ϕ0 (z), . . . , ϕ(n−k+2) (z)) < ∞, z∈D

lim

µΨ (z) u0 (z)Bn,k (ϕ0 (z), . . . , ϕ(n−k+1) (z)) + u(z)Bn+1,k (ϕ0 (z), . . . , ϕ(n−k+2) (z)) (1 − |ϕ(z)|2 )k+

|ϕ(z)|→1−

α+2 p

=0

for each k ∈ {0, 1, . . . , n}, sup µΨ (z)|u(z)||ϕ0 (z)|n+1 < ∞, z∈D

and lim

|ϕ(z)|→1−

µΨ (z)|u(z)||ϕ0 (z)|n+1 (1 − |ϕ(z)|2 )n+1+

251

α+2 p

= 0.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.2, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Theorem 4.4. Let α > −1, p ≥ 1, ϕ be an analytic self-map of D and u ∈ H(D). Then the following statements hold. (i) The operator Mu Cϕ Dn : Apα → B Ψ is compact. (ii) The functions u and ϕ are such that u ∈ B Ψ , sup µΨ (z)|u(z)||ϕ0 (z)| < ∞, z∈D

lim

µΨ (z)|u0 (z)|

|ϕ(z)|→1−

(1 − |ϕ(z)|2 )n+

α+2 p

= 0,

and lim

|ϕ(z)|→1−

µΨ (z)|u(z)||ϕ0 (z)| (1 − |ϕ(z)|2 )n+1+

α+2 p

= 0.

Remark 4.2. Noticing that Cϕ Mu Dn = Mu◦ϕ Cϕ Dn , we can obtain the characterization of the compactness of Cϕ Mu Dn : Apα → B Ψ from Theorem 4.4. Here we omit.

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[25] Stevi´c, S., Sharma, A. K. and Bhat, A., Essential norm of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 2011, 218:2386-2397. [26] Zhang, X.J., Guan, Y. and Li, M., Composition operators between logarithmic weighted Bloch type spaces on the unit ball, Adv. Math. (China), 2015, 44(3):553-561 (in Chinese). [27] Zhao, L. K., Fredholm weighted composition operators on the weighted Dirichlet space, Adv. Math. (China), 2014, 43(3):419-424 (in Chinese). [28] Zhu, K., Operator theory in function space, New York: Marcel Dekker, 1990. [29] Zhu, K., Spaces of holomorphic functions in the unit ball, New York: Springer-Verlag, 2005. [30] Zhu, X., Products of differentiation, composition and multiplication operator from Bergman type spaces to Bers spaces, Integral Transforms Spec. Funct., 2007, 18:223-231. [31] Zhu, X., Generalized weighted composition operators from Bloch spaces into Bers-type spaces, Filomat., 2012, 26:1163-1169.

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On some recent fixed point results for α-admissible mappings in b-metric spaces Huaping Huang1 , Guantie Deng1,∗, Zhanmei Chen1 , Stojan Radenović2 1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, PR China 2. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Beograd, Serbia

Abstract: The purpose of this paper is to present some fixed point theorems for weak α-admissible mappings type in the setting of b-metric spaces. The results greatly optimize and improve some fixed point results in the existing literature. Moreover, we highlight our assertions by utilizing an example. In addition, we use our results to obtain the existence of solution for a class of nonlinear integral equations. Keywords: α-admissible mapping, α-contraction mapping, rational α-Geraghty contraction of type, fixed point, integral equation

1

Introduction Since Polish mathematician Banach proved the well-known Banach contraction map-

ping principle in metric spaces in 1922 (see [1]), fixed point theory occupies a prominent place in strong research activity. Due to its applications in finding the existence of solutions for the nonlinear Volterra integral equations, nonlinear integro-differential equations and existence of equilibria in game theory as well, it has become the most celebrated tool in nonlinear analysis. During the past decades, scholars extend this principle towards different spaces, such as G-metric spaces, 2-metric spaces, fuzzy metric spaces, probabilistic metric spaces, cone metric spaces, partial metric spaces, modular metric spaces, b-metric spaces, etc (see [2-10]). Whereas, the most influential spaces among them, i.e., b-metric spaces, or metric type spaces called by some authors, introduced by Bakhtin [9] or Czerwik [10], have a rapid development. Compared with other spaces, people are willing to deal with fixed point problems or the variational principle for single-valued or multi-valued operators in b-metric spaces, based on the fact that b-metrics have no continuity in general. On the other hand, people fascinate fixed point results by substituting the Banach contractive mapping, such as Kannan contraction mapping, Chatterjea contraction mapping, α-ψ-contractive type mapping, cyclic contractive mapping, multivalued contraction ∗

Correspondence: [email protected] (G. Deng) 255

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mapping, and so on (see [6-14]). Recently, Samet et al. [12] introduced the notion of αadmissible mapping in the framework of metric spaces, and very recently, Sintunavarat [15] introduced the concepts of α-admissible mapping type S, weak α-admissible mapping, weak α-admissible mapping type S, as some generalizations of α-admissible mapping. Moreover, [15] proved fixed point theorems based on his new types of α-admissibility in the setup of b-metric spaces. In this paper, inspired by [15], we introduce the notion of α-admissibility mapping, and obtain some fixed point theorems, as compared to the main results of [15], with much simpler conditions and more straightforward proofs. Furthermore, we cope with some fixed point results for the mappings on rational α-Geraghty contraction of type in terms of α-admissibility in b-metric spaces. In addition, we give an application in the existence of a solution for a class of nonlinear integral equations. Our conditions are weak and applicable compared to the applications from [15]. For the sake of reader, the following definitions and results will be needed in the sequel. Definition 1.1([16]). A mapping ϕ : [0, ∞) → [0, ∞) is said to be an altering distance function if it holds: (1) ϕ is nondecreasing and continuous; (2) ϕ(t) = 0 if and only if t = 0. Definition 1.2([15]). Let X be a nonempty set and s ≥ 1 a given real number. Let α : X × X → [0, ∞) and f : X → X be mappings. We say f is an α-admissible mapping type S if for all x, y ∈ X, α(x, y) ≥ s leads to α(f x, f y) ≥ s. In particular, f is called α-admissible mapping if s = 1. Remark 1.3 Usually, use A(X, α) and As (X, α) to denote the collection of all α-admissible mappings on X and the collection of all α-admissible mappings type S on X. It is worth reminding that the class of α-admissible mappings and the class of α-admissible mappings type S are independent, in other words, A(X, α) 6= As (X, α) in general case. Definition 1.4([15]). Let X be a nonempty set and s ≥ 1 a given real number. Let α : X × X → [0, ∞) and f : X → X be mappings. We say f is a weak α-admissible mapping type S if for all x ∈ X, α(x, f x) ≥ s leads to α(f x, f f x) ≥ s. In particular, f is called weak α-admissible mapping if s = 1. Remark 1.5. Customarily, utilize WA(X, α) and WAs (X, α) to denote the collection of all weak α-admissible mappings on X and the collection of all weak α-admissible mappings type S on X. Clearly, A(X, α) ⊆ WA(X, α) and As (X, α) ⊆ WAs (X, α). Definition 1.6([10]). Let X be a nonempty set and s ≥ 1 a real number. A mapping d : X × X → [0, ∞) is called a b-metric if for all x, y, z ∈ X, the following conditions are satisfied: (b1) d(x, y) = 0 if and only if x = y; (d2) d(x, y) = d(y, x); (d3) d(x, z) ≤ s[d(x, y) + d(y, z)]. In this case, (X, d) is called a b-metric space. Definition 1.7([17]). Let (X, d) be a b-metric space, x ∈ X and {xn } a sequence in X. 256

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Then we say (i) {xn } b-converges to x if d(xn , x) → 0 as n → ∞. In this case, we write lim xn = x. n→∞

(ii) {xn } is a b-Cauchy sequence if d(xn , xm ) → 0 as n, m → ∞. (iii) (X, d) is b-complete if every b-Cauchy sequence is b-convergent in X. (iv) a function f : X → Y is b-continuous at a point x ∈ X if {xn } ⊂ X b-converges to x, then {f xn } b-converges to f x, where (Y, ρ) is a b-metric space. Throughout this paper, unless otherwise specified, X is a nonempty set, f : X → X is a mapping, Fix(f ) denotes the set of all fixed points of f on X, that is, Fix(f ) := {x ∈ X|f x = x}. Also, for each elements x and y in a b-metric space (X, d) with coefficient s ≥ 1, let   d(x, f y) + d(y, f x) Ms (x, y) := max d(x, y), d(x, f x), d(y, f y), . 2s Lemma 1.8([18]). Let (X, d) be a b-metric space with coefficient s ≥ 1 and let {xn } and {yn } be b-convergent to points x, y ∈ X, respectively. Then we have 1 d(x, y) ≤ lim inf d(xn , yn ) ≤ lim sup d(xn , yn ) ≤ s2 d(x, y). 2 n→∞ s n→∞ In particular, if x = y, then we have limn→∞ d(xn , yn ) = 0. Moreover, for each z ∈ X, we have

1 d(x, z) ≤ lim inf d(xn , z) ≤ lim sup d(xn , z) ≤ sd(x, z). n→∞ s n→∞ Definition 1.9([15]). Let (X, d) be a b-metric space with coefficient s ≥ 1, let α : X × X → [0, ∞) be a mapping and let ψ, ϕ : [0, ∞) → [0, ∞) be two altering distance functions. A mapping f : X → X is said to be an (α, ψ, ϕ)s -contraction mapping if x, y ∈ X with α(x, y) ≥ s =⇒ ψ(s3 d(f x, f y)) ≤ ψ(Ms (x, y)) − ϕ(Ms (x, y)).

(1.1)

In this case, write Ωs (X, α, ψ, ϕ) as the collection of all (α, ψ, ϕ)s -contraction mappings. Theorem 1.10([15]). Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1, let ψ, ϕ : [0, ∞) → [0, ∞) be two altering distance functions and let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: (S1 ) f ∈ Ωs (X, α, ψ, ϕ) ∩ WAs (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (S3 ) α has a transitive property type S, that is, for x, y, z ∈ X, α(x, y) ≥ s and α(y, z) ≥ s ⇒ α(x, z) ≥ s; (S4 ) f is b-continuous. Then Fix(f ) 6= ∅. Theorem 1.11([15]). Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1, let ψ, ϕ : [0, ∞) → [0, ∞) be two altering distance functions and let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: 257

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(S1 ) f ∈ Ωs (X, α, ψ, ϕ) ∩ WAs (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (S3 ) α has a transitive property type S; (Se4 ) X is αs -regular, that is, if {xn } is a sequence in X such that α(xn , xn+1 ) ≥ s for all n ∈ N and xn → x ∈ X as n → ∞, then α(xn , x) ≥ s for all n ∈ N. Then Fix(f ) 6= ∅. Corollary 1.12([15]). Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1, let ψ, ϕ : [0, ∞) → [0, ∞) be two altering distance functions and let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: (Se1 ) f ∈ Ωs (X, α, ψ, ϕ) ∩ As (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (S3 ) α has a transitive property type S; (S4 ) f is b-continuous. Then Fix(f ) 6= ∅. Corollary 1.13([15]). Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1, let ψ, ϕ : [0, ∞) → [0, ∞) be two altering distance functions and let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: (Se1 ) f ∈ Ωs (X, α, ψ, ϕ) ∩ As (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (S3 ) α has a transitive property type S; (Se4 ) X is αs -regular. Then Fix(f ) 6= ∅.

2

Main results

Definition 2.1. Let (X, d) be a b-metric space with coefficient s ≥ 1, let α : X × X → [0, ∞) be a mapping and ε > 1 be a constant. A mapping f : X → X is said to be an α-contraction mapping if x, y ∈ X with α(x, y) ≥ s =⇒ sε d(f x, f y) ≤ Ms (x, y).

(2.1)

In this case, write Ωs (X, α) as the collection of all α-contraction mappings. Theorem 2.2. Let (X, d) be a b-complete b-metric space with coefficient s > 1. Let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: (S1 ) f ∈ Ωs (X, α) ∩ WAs (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (S3 ) f is b-continuous. Then Fix(f ) 6= ∅. 258

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Proof. By (S2 ), for x0 ∈ X, construct a Picard iteration sequence {xn } satisfying xn+1 = f xn , n ∈ N. Assume that xn0 = xn0 +1 for some n0 , then Fix(f ) = {xn0 } 6= ∅, in this case, the conclusion is satisfied. So set xn 6= xn+1 for all n, that is, d(xn , xn+1 ) > 0 for all n. Let us prove the following inequality: d(xn+1 , xn+2 ) ≤ λd(xn , xn+1 ),

(2.2)

where λ ∈ [0, 1s ) is a constant. Indeed, in view of f ∈ WAs (X, α) and α(x0 , f x0 ) ≥ s, it implies that α(x1 , x2 ) = α(f x0 , f f x0 ) ≥ s. Repeating this process, we make a conclusion that α(xn , xn+1 ) ≥ s for all n. Making the most of (2.1), we have sε d(xn+1 , xn+2 ) = sε d(f xn , f xn+1 ) ≤ Ms (xn , xn+1 )  = max d(xn , xn+1 ), d(xn , f xn ), d(xn+1 , f xn+1 ),  d(xn , f xn+1 ) + d(xn+1 , f xn ) 2s  = max d(xn , xn+1 ), d(xn , xn+1 ), d(xn+1 , xn+2 ),  d(xn , xn+2 ) + d(xn+1 , xn+1 ) 2s   d(xn , xn+1 ) + d(xn+1 , xn+2 ) ≤ max d(xn , xn+1 ), d(xn+1 , xn+2 ), 2 = max{d(xn , xn+1 ), d(xn+1 , xn+2 )}. (2.3) If d(xn , xn+1 ) ≤ d(xn+1 , xn+2 ), then by (2.3), it follows that sε d(xn+1 , xn+2 ) ≤ d(xn+1 , xn+2 ). Hence, d(xn+1 , xn+2 ) = 0, it is a contradiction. If d(xn+1 , xn+2 ) ≤ d(xn , xn+1 ), then by (2.3), it establishes that sε d(xn+1 , xn+2 ) ≤ d(xn , xn+1 ). As a result, (2.2) holds, where λ =

1 sε

∈ [0, 1s ).

Now by [11, Lemma 3.1], taking advantage of (2.2), we claim that {xn } is a b-Cauchy sequence. Since (X, d) is b-complete, we know that {xn } b-converges to some point x ∈ X. Finally, we show x ∈ Fix(f ). Actually, by using (S3 ), it is not hard to verify that d(f x, x) ≤ s[d(f x, f xn ) + d(f xn , x)] = s[d(f x, f xn ) + d(xn+1 , x)] → 0 as n → ∞. 259

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Therefore, d(f x, x) = 0, that is to say, x ∈ Fix(f ). Theorem 2.3. Let (X, d) be a b-complete b-metric space with coefficient s > 1. Let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: (S1 ) f ∈ Ωs (X, α) ∩ WAs (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (Se3 ) X is αs -regular. Then Fix(f ) 6= ∅. Proof. Making full use of the proof of Theorem 2.2, we obtain a sequence {xn } satisfying xn+1 = f xn → x ∈ X as n → ∞. Then by (Se3 ), we get α(xn , x) ≥ s for all n ∈ N. By virtue of (S1 ), we have sε d(f xn , f x) ≤ Ms (xn , x)  = max d(xn , x), d(xn , f xn ), d(x, f x),  d(xn , f x) + d(x, f xn ) 2s  ≤ max d(xn , x), s[d(xn , x) + d(xn+1 , x)], d(x, f x),  d(xn , x) + d(x, f x) d(x, xn+1 ) + 2 2s   d(x, f x) → max 0, 0, d(x, f x), = d(x, f x) (n → ∞), 2 which implies that lim d(f xn , f x) ≤

n→∞

1 d(x, f x). sε

(2.4)

Note that 1 d(x, f x) ≤ d(x, f xn ) + d(f xn , f x) = d(x, xn+1 ) + d(f xn , f x). s

(2.5)

Taking the limit as the above inequality (2.5) and utilizing (2.4), we speculate that 1 1 d(x, f x) ≤ ε d(x, f x), s s which follows that d(x, f x) = 0, that is, x ∈ Fix(f ). Corollary 2.4. Let (X, d) be a b-complete b-metric space with coefficient s > 1. Let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: (Se1 ) f ∈ Ωs (X, α) ∩ As (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (S3 ) f is b-continuous. Then Fix(f ) 6= ∅. 260

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Corollary 2.5. Let (X, d) be a b-complete b-metric space with coefficient s > 1. Let α : X × X → [0, ∞) and f : X → X be given mappings. Suppose that the following conditions hold: (Se1 ) f ∈ Ωs (X, α) ∩ As (X, α); (S2 ) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (Se3 ) X is αs -regular. Then Fix(f ) 6= ∅. Remark 2.6. Theorem 2.2, Theorem 2.3, Corollary 2.4 and Corollary 2.5 greatly optimize and improve Sintunavarat’s theorems, i.e., Theorem 1.10, Theorem 1.11, Corollary 1.12 and Corollary 1.13, respectively. Actually, on the one hand, compared with (1.1), (2.1) not only deletes the limitation of the altering distance functions ψ and ϕ, but also it dispenses with an item ϕ(Ms (x, y)) which makes the condition become much wider. These are some great improvements. Moreover, our index ε > 1 is arbitrary, and it clearly contains ε = 3. Hence, our range ε > 1 is much larger and more applicable. On the other hand, our theorems dismiss the condition of transitive property type S for the mapping α. That is to say, the conditions of our theorems are weaker than Sintunavarat’s theorems. Therefore, our conclusions may be more convenient than Sintunavarat’s in applications. Remark 2.7. From the proofs of our theorems, it is easy to see that we do not use Lemma 1.8. Our proofs are much simpler since we do not refer to b-discontinuity of bmetric. Whereas, in order to overcome the difficulty of the b-discontinuity of b-metric, the proofs of Sintunavarat’s theorems are very comprehensive based on the fact of depending on Lemma 1.8 strongly. Example 2.8. Let X = R and define d(x, y) = |x − y|2 for all x, y ∈ X. Then (X, d) is a b-complete b-metric space with coefficient s = 2. Define mappings f : X → X and α : X × X → [0, ∞) by  x , x ∈ [0, 16 ], 4 3 fx = 1 2 16 x + 3 , x ∈ ( 3 , ∞), 8 and

 α(x, y) =

5 4

+

1 , |x−y|

0,

x, y ∈ [0, 16 ], 3 otherwise.

Let us prove f ∈ Ωs (X, α). Actually, assume that x, y ∈ X with α(x, y) ≥ s = 2 and hence x, y ∈ [0, 16 ] with |x − y| ≤ 43 . Let 1 < ε ≤ 4 be a constant. Then 3 2ε d(f x, f y) = 2ε |f x − f y|2 x y 2 = 2ε − 4 4 ε−4 = 2 |x − y|2 ≤ Ms (x, y). 261

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As a consequence, f ∈ Ωs (X, α). We also can verify f ∈ WAs (X, α). Indeed, if x ∈ X such that α(x, f x) ≥ s = 2, ] and |x − f x| ≤ 43 . Thus x ∈ [0, 16 ]. This indicates that f f x ∈ [0, 19 ] then x, f x ∈ [0, 16 3 9 and hence α(f x, f f x) =

5 16 17 + ≥ > s = 2. 4 3|x| 4

Otherwise, it is obvious that f is b-continuous and there exists x0 = 1 such that α(x0 , f x0 ) = α(1, f 1) =

1 31 5 + = ≥ 2 = s. 4 |1 − f 1| 12

Consequently, all the conditions of Theorem 2.2 hold. Thus Fix(f ) = {0} 6= ∅. However, we cannot use Theorem 1.10 to get Fix(f ) 6= ∅, since α is unsuitable for the condition (S3 ) of this theorem. Indeed, put x = 4, y = 3, z = 2. Though α(x, y) = 9 4

9 4

>2

7 4

and α(y, z) = > 2, whereas, α(x, z) = < 2. So (S3 ) does not hold in this example. In other words, Theorem 2.2 is more superior than Theorem 1.10. In the sequel, let s ≥ 1 be a constant and let Fs denote the class of all functions β : [0, ∞) → [0, 1s ) satisfying the following condition: lim sup β(tn ) = n→∞

1 implies that tn → 0 as n → ∞. s

Definition 2.9. Let (X, d) be a b-metric space with coefficient s ≥ 1, and let α : X ×X → [0, ∞) be a function. A mapping f : X → X is called a rational α-Geraghty contraction of type Iε,β if there exist ε > 0 and β ∈ Fs such that x, y ∈ X with α(x, y) ≥ s =⇒ α(x, y)sε d(f x, f y) ≤ β(MI (x, y))MI (x, y), where



d (x, f x) d (y, f y) d (x, f x) d (y, f y) MI (x, y) = max d (x, y) , , 1 + d (x, y) 1 + d (f x, f y)

(2.6)

 .

Definition 2.10. Let (X, d) be a b-metric space with coefficient s ≥ 1, and let α : X × X → [0, ∞) be a function. A mapping f : X → X is called a rational α-Geraghty contraction of type IIε,β if there exist ε > 0 and β ∈ Fs such that x, y ∈ X with α(x, y) ≥ s =⇒ α(x, y)sε d(f x, f y) ≤ β(MII (x, y))MII (x, y),

(2.7)

where 

d (x, f x) d (x, f y) + d (y, f y) d (y, f x) , 1 + s [d (x, f x) + d (y, f y)]  d (x, f x) d (x, f y) + d (y, f y) d (y, f x) . 1 + s [d (x, f y) + d (y, f x)]

MII (x, y) = max d (x, y) ,

262

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Definition 2.11. Let (X, d) be a b-metric space with coefficient s ≥ 1, and let α : X × X → [0, ∞) be a function. A mapping f : X → X is called a rational α-Geraghty contraction of type IIIε,β if there exist ε > 0 and β ∈ Fs such that x, y ∈ X with α(x, y) ≥ s =⇒ α(x, y)sε d(f x, f y) ≤ β(MIII (x, y))MIII (x, y),

(2.8)

where  MIII (x, y) = max d (x, y) ,

d (x, f x) d (y, f y) , 1 + s [d (x, y) + d (x, f y) + d (y, f x)]  d (x, f y) d (x, y) . 1 + sd (x, f x) + s3 [d (y, f x) + d (y, f y)]

Theorem 2.12. Let (X, d) be a b-complete b-metric space with coefficient s > 1, and let α : X × X → [0, ∞) be a function and f : X → X be a mapping. Suppose that the following conditions hold: (i) f is a rational α-Geraghty contraction of type Iε,β (resp. type IIε,β or type IIIε,β ); (ii) f ∈ As (X, α) and there exists x0 ∈ X such that α(x0 , f x0 ) ≥ s; (iii) f is b-continuous or X is αs -regular. Then Fix(f ) 6= ∅. Proof. By (ii) and the proof of Theorem 2.2, we can construct a Picard iteration sequence {xn } satisfying xn+1 = f xn and α(xn , xn+1 ) ≥ s for all n ∈ N. Let us prove that d(xn+1 , xn+2 ) ≤ λd(xn , xn+1 )

(2.9)

for all n ∈ N, where λ ∈ [0, 1s ). First of all, let f be a rational α-Geraghty contraction of type Iε,β . Then by (2.6), we have sε d(xn+1 , xn+2 ) = sε d(f xn , f xn+1 ) ≤ α(xn , xn+1 )sε d(f xn , f xn+1 ) ≤ β(MI (xn , xn+1 ))MI (xn , xn+1 ) 1 ≤ MI (xn , xn+1 ), s

(2.10)

which follows that sε+1 d(xn+1 , xn+2 ) ≤ MI (xn , xn+1 )   d (xn , xn+1 ) d (xn+1 , xn+2 ) d (xn , xn+1 ) d (xn+1 , xn+2 ) = max d (xn , xn+1 ) , , 1 + d (xn , xn+1 ) 1 + d (xn+1 , xn+2 )   d (xn , xn+1 ) d (xn+1 , xn+2 ) d (xn , xn+1 ) d (xn+1 , xn+2 ) ≤ max d (xn , xn+1 ) , , d (xn , xn+1 ) d (xn+1 , xn+2 ) = max {d (xn , xn+1 ) , d (xn+1 , xn+2 )} . (2.11) 263

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If d (xn , xn+1 ) ≤ d(xn+1 , xn+2 ), then from (2.11), it leads to d (xn+1 , xn+2 ) ≤

1 sε+1

d (xn+1 , xn+2 ) < d (xn+1 , xn+2 ) .

This is a contradiction. So d(xn+1 , xn+2 ) ≤ In this case, (2.9) is satisfied, where λ =

1 sε+1

1 sε+1

d (xn , xn+1 ) .

∈ [0, 1s ).

Secondly, let f be a rational α-Geraghty contraction of type IIε,β . Then similarly by (2.10), we have 1 sε d(xn+1 , xn+2 ) ≤ MII (xn , xn+1 ), s which establishes that sε+1 d(xn+1 , xn+2 ) ≤ MII (xn , xn+1 )  d (xn , xn+1 ) d (xn , xn+2 ) + d (xn+1 , xn+2 ) d (xn+1 , xn+1 ) = max d (xn , xn+1 ) , , 1 + s [d (xn , xn+1 ) + d (xn+1 , xn+2 )]  d (xn , xn+1 ) d (xn , xn+2 ) + d (xn+1 , xn+2 ) d (xn+1 , xn+1 ) 1 + s [d (xn , xn+2 ) + d (xn+1 , xn+1 )]   d (xn , xn+1 ) d (xn , xn+2 ) d (xn , xn+1 ) d (xn , xn+2 ) , = max d (xn , xn+1 ) , 1 + s [d (xn , xn+1 ) + d (xn+1 , xn+2 )] 1 + sd (xn , xn+2 )   sd (xn , xn+1 ) [d (xn , xn+1 ) + d (xn+1 , xn+2 )] d(xn , xn+1 )d(xn , xn+2 ) , ≤ max d (xn , xn+1 ) , 1 + s [d (xn , xn+1 ) + d (xn+1 , xn+2 )] sd(xn , xn+2 )   sd (xn , xn+1 ) [d (xn , xn+1 ) + d (xn+1 , xn+2 )] d(xn , xn+1 ) ≤ max d (xn , xn+1 ) , , s [d (xn , xn+1 ) + d (xn+1 , xn+2 )] s ≤ d (xn , xn+1 ) . Accordingly, (2.9) is also satisfied, where λ =

1 sε+1

∈ [0, 1s ).

Thirdly, let f be a rational α-Geraghty contraction of type IIIε,β . Then similarly by (2.13), we have 1 sε d(xn+1 , xn+2 ) ≤ MIII (xn , xn+1 ), s which implies that sε+1 d(xn+1 , xn+2 ) ≤ MIII (xn , xn+1 )  d (xn , xn+1 ) d (xn+1 , xn+2 ) = max d (xn , xn+1 ) , , 1 + s [d (xn , xn+1 ) + d (xn , xn+2 ) + d (xn+1 , xn+1 )]  d (xn , xn+2 ) d (xn , xn+1 ) 1 + sd (xn , xn+1 ) + s3 [d (xn+1 , xn+1 ) + d (xn+1 , xn+2 )]  d (xn , xn+1 ) d (xn+1 , xn+2 ) ≤ max d (xn , xn+1 ) , , 1 + s [d (xn , xn+1 ) + d (xn , xn+2 )]  s[d(xn , xn+1 ) + d(xn+1 , xn+2 )]d(xn , xn+1 ) 1 + sd(xn , xn+1 ) + s3 d(xn+1 , xn+2 )   d (xn , xn+1 ) d (xn+1 , xn+2 ) s[d(xn , xn+1 ) + d(xn+1 , xn+2 )]d(xn , xn+1 ) ≤ max d (xn , xn+1 ) , , d (xn , xn+1 ) s[d(xn , xn+1 ) + d(xn+1 , xn+2 )] = max {d (xn , xn+1 ) , d (xn+1 , xn+2 )} . 264

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A similar discussion as to (2.11), we can get (2.9), too. In a word, under the conditions of (i) and (ii), we always acquire (2.9). As a consequence, by using [11, Lemma 3.1] and the b-completeness of (X, d), there exists a point x ∈ X such that xn → x as n → ∞. Now by (iii), if f is b-continuous, then   x = lim xn+1 = lim f (xn ) = f lim xn = f x, n→∞

n→∞

n→∞

that is, x ∈ Fix(f ). If X is αs -regular, then α(xn , x) ≥ s. Put M (xn , x) ∈ {MI (xn , x), MII (xn , x), MIII (xn , x)}, it follows immediately from (2.6), (2.7) and (2.8) that α(xn , x)sε d(f xn , f x) ≤ β(M (xn , x))M (xn , x).

(2.12)

lim M (xn , x) = 0.

(2.13)

We show that n→∞

Indeed, for one thing,   d (xn , xn+1 ) d (x, f x) d (xn , xn+1 ) d (x, f x) , MI (xn , x) = max d (xn , x) , 1 + d (xn , x) 1 + d (xn+1 , f x) → max {0, 0, 0} = 0, as n → ∞. For another thing, 

d (xn , xn+1 ) d (xn , f x) + d (x, f x) d (x, xn+1 ) , 1 + s [d (xn , xn+1 ) + d (x, f x)]  d (xn , xn+1 ) d (xn , f x) + d (x, f x) d (x, xn+1 ) 1 + s [d (xn , f x) + d (x, xn+1 )] ≤ max {d (xn , x) , d (xn , xn+1 ) s [d (xn , x) + d (x, f x)] + d (x, f x) d (x, xn+1 ) ,

MII (xn , x) = max d (xn , x) ,

d (xn , xn+1 ) s [d (xn , x) + d (x, f x)] + d (x, f x) d (x, xn+1 )} → max {0, 0, 0} = 0, as n → ∞. For the third thing,  MIII (xn , x) = max d(xn , x),

d(xn , xn+1 )d(x, f x) , 1 + s[d(xn , x) + d(xn , f x) + d(x, xn+1 )]  d(xn , f x)d(xn , x) 1 + sd(xn , xn+1 ) + s3 [d(x, xn+1 ) + d(x, f x)] → max {0, 0, 0} = 0, as n → ∞.

Thus (2.13) holds. Using (2.12) and (2.13), we speculate that sε d(x, f x) ≤ sε+1 [d(x, f xn ) + d(f xn , f x)] ≤ sε+1 d(x, xn+1 ) + sε+1 α(xn , x)d(f xn , f x) ≤ sε+1 d(x, xn+1 ) + sβ(M (xn , x))M (xn , x) ≤ sε+1 d(x, xn+1 ) + M (xn , x) → 0, as n → ∞, which establishes that d(x, f x) = 0, that is to say, x ∈ Fix(f ). 265

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3

Application In this section, we prove an existence theorem for a solution of the following nonlinear

integral equation by using our results in the previous section: Z b x(c) = φ(c) + K(c, r, x(r))dr,

(3.1)

a

where a, b ∈ R, x ∈ C[a, b] (the set of all continuous functions from [a, b] into R, φ : [a, b] → R and K : [a, b] × [a, b] × R → R are given mappings. The following theorem greatly improves Theorem 3.1 of [15] with simpler conditions, which illustrates the superiority of our results. Theorem 3.1. Consider the nonlinear integral equation (3.1). Suppose that the following conditions hold: (i) K : [a, b] × [a, b] × R → R is continuous and nondecreasing in the third order; (ii) there exists p > 1 satisfying the following condition: for each r, c ∈ [a, b] and x, y ∈ C[a, b] with x(w) ≤ y(w) for all w ∈ [a, b], we have |K(c, r, x(r)) − K(c, r, y(r))| ≤ ζ(c, r)|x(r) − y(r)|,

(3.2)

where ζ : [a, b] × [a, b] → [0, ∞) is a continuous function satisfying  Z b 1 p sup ζ(c, r) dr ≤ εp−ε 2 (b − a)p−1 c∈[a,b] a and ε > 1 is a constant. (iii) there exists x0 ∈ C[a, b] such that x0 (c) ≤ φ(c)+

Rb a

K(c, r, x0 (r))dr for all c ∈ [a, b].

Then the nonlinear integral equation (3.1) has a solution. Proof. Put X = C[a, b] and define a mapping f : X → X by Z

b

K(c, r, x(r))dr

(f x)(c) = φ(c) + a

for all x ∈ X and c ∈ [a, b]. Define a mapping d : X × X → [0, ∞) by d(x, y) = sup |x(c) − y(c)|p

(p > 1)

c∈[a,b]

for all x, y ∈ X. Then (X, d) is a b-complete b-metric space with coefficient s = 2p−1 . Define a mapping α : X × X → [0, ∞) by  p−1 2 , x(c) ≤ y(c) for all c ∈ [a, b], α(x, y) = τ, otherwise, where 0 < τ < 2p−1 . Since K is nondecreasing in the third order, we get f ∈ As (X, α) ⊂ WAs (X, α). By (iii), it infers (S2 ) in Theorem 2.3 is satisfied. Also, we get that condition (Se3 ) in Theorem 2.3 also holds (see [21]).

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Finally, we show f ∈ Ωs (X, α). To prove this fact, we first choose q ∈ R such that 1 p

+

1 q

= 1. Assume that x, y ∈ X such that α(x, y) ≥ s = 2p−1 , that is, x(c) ≤ y(c) for all

c ∈ [a, b]. From (ii) and the Hölder inequality, for each c ∈ [a, b] we get 2εp−ε |(f x)(c) − (f y)(c)|p Z b p εp−ε |K(c, r, x(r)) − K(c, r, y(r))|dr ≤2 a "Z  1 Z b

1q dr

≤ 2εp−ε

|K(c, r, x(r)) − K(c, r, y(r))|p dr

a εp−ε

≤2

b

q

(b − a)

 p1 #p

a p q

Z

b p

p



ζ(c, r) |x(r) − y(r)| dr Z b  p εp−ε p ≤2 (b − a) q ζ(c, r) d(x, y)dr a Z b  εp−ε p−1 p ≤2 (b − a) Ms (x, y) ζ(c, r) dr a

a

≤ Ms (x, y). This implies that sε d(f x, f y) ≤ Ms (x, y). Hence f ∈ Ωs (X, α). Thus all the conditions of Theorem 2.3 are satisfied and hence f has a fixed point in X. It follows that the nonlinear integral equation (3.1) has a solution. Remark 3.2. Compared with [15, Theorem 3.1], our Theorem 3.1 has many superiorities. First, our condition (ii) is much simpler than (ii) from [15, Theorem 3.1]. Indeed, our condition (3.2) is weaker than the corresponding condition of [15, Theorem 3.1]. Moreover, we delete the function Υ(t). Whereas, Υ(t) is a complex function with very strong conditions. Otherwise, our function ζ(c, r) satisfies the wider condition since ε is arbitrary. Further, even if ε = 3, our condition for ζ(c, r) is also much weaker. Remark 3.3. In [15, Theorem 3.1], there exist some mistakes. For instance, the incorrect equality from the proof of [15, Theorem 3.1] appears as follows: !p
p s3 d(f x, f y) =

23p−3 sup |(f x)(t) − (f y)(t)|

.

t∈[a,b]

In fact, it should be the following: !p s3 d(f x, f y)


p

=

23p−3 sup |(f x)(t) − (f y)(t)|p

.

t∈[a,b]

Due to such mistake, the conditions from [15, Theorem 3.1] need some revisions. Similar revisions should be done in Corollary 3.2 and Corollary 3.3 from [15, Theorem 3.1].

Acknowledgements The research was partially supported by the National Natural Science Foundation of China (11271045). 267

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References [1] S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales., Fund. Math., 3 (1922), 133-181. [2] Z. Mustafa, F. Awawdeh, W. Shatanawi, Fixed point theorem for expansive mappings in G-metric Spaces, Int. J. Contemp. Math. Sciences, 5(50) (2010), 2463-2472. [3] Mona. S. Bakry, H. M. Abu-Donia, Fixed-point theorems for a probabilistic 2-metric spaces, J. King Saud University (Science), 22 (2010), 217-221. [4] C. Ionescu, S. Rezapour, M.-E. Samei, Fixed points of some new contractions on intuitionistic fuzzy metric spaces, Fixed Point Theory Appl., 2013, 2013: 168. [5] G. Cain, R. Kasriel, Fixed and periodic points of local contraction mappings on probabilistic metric spaces, Theory Comput. Systems, 9(3) (1975), 289-297. [6] L. -G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332(2) (2007), 1468-1476. [7] T. Abdeljawad, J. Alzabut, A. Mukheimer, Y Zaidan, Banach contraction principle for cyclical mappings on partial metric spaces, Fixed Point Theory Appl., 2012, 2012: 154. [8] A.-A. Abdou, M.-A Khamsi, Fixed points of multivalued contraction mappings in modular metric spaces, Fixed Point Theory Appl., 2014, 2014: 249. [9] I. A. Bakhtin, The contraction principle in quasi-metric spaces, Funct. Anal., 30 (1989), 26-37. [10] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform., Univ. Ostrav., 1 (1993), 5-11. [11] M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed point results in metric type spaces, Fixed Point Theory Appl., 2010, Article ID 978121, 15 pages. [12] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-ψ-contractive type mappings., Nonlinear Anal., 75 (2012), 2154-2165. [13] R. Kannan, Some results on fixed points. II. Amer. Math. Monthly, 76 (1969), 405408. [14] M. Imdad, A. Erduran, Suzuki-type generalization of Chatterjea contraction mappings on complete partial metric spaces, J. Oper., 2013 (2013), Article ID 923843, 5 pages.

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[15] W. Sintunavarat, Nonlinear integral equations with new admissibility types in bmetric spaces, J. Fixed Point Theory Appl., DOI 10.1007/s11784-015-0276-6, 20 pages. [16] H. Huang, S. Radenović, J. Vujaković, On some recent coincidence and immediate consequences in partially ordered b-metric spaces, Fixed Point Theory and Applications, 2015, 2015: 63. [17] M. Boriceanu, M. Bota and A. Petrusel, Mutivalued fractals in b-metric spaces, Cent. Eur. J. Math., 8 (2010), 367-377. [18] A. Aghajani, M. Abbas, J. R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64(4) (2014), 941-960. [19] H. Huang, L. Paunović, S. Radenović, On some new fixed point results for rational Geraghty contractive mappings in ordered b-metric spaces, J. Nonlinear Science Appl., 8(5) (2015), 800-807. [20] N. Hussain, E. Karapınar, P. Salimi, F. Akbar, α-admissible mappings and related fixed point theorems, Fixed Point Theory Appl., 2013, 2013: 114. [21] J. J. Nieto, R. Rodíguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. (Engl. Ser.) 23 (2007), 2205-2212.

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Anti Implicative IF-Ideals in BCK/BCI-algebras Servet Kutukcu* and Adnan Tuna (Dedicated to the 79th birthday of Ivo G. Rosenberg.) Department of Mathematics, Faculty of Science and Arts Ondokuz May¬s University, 55139, Samsun, Turkey Department of Mathematics, Faculty of Science and Arts University of Ni¼ gde, 51240, Ni¼ gde, Turkey Abstract:Using triangular norms, we present a new classi…cation of fuzzy subalgebras, ideals and implicative ideals in BCK/BCI-algebras. Keywords: t-norm; anti if-ideal; anti implicative if-ideal; BCK/BCI-algebra.

1

Introduction

BCK/BCI-algebras are an important class of logical algebras introduced by Imai and Iseki [7], and was extensively investigated by several researches. BCK/BCI-algebras generalize, on the one hand, the notion of the algebra of sets with the set subtraction as the only fundemental non-nullary operation and, on the other hand, the notion of the implication algebra (see [7]). In 1965, Zadeh [16] introduced the notion of fuzzy sets and in 1991, Xi [15] applied this notion to BCK/BCI-algebras. In 1990, Biswas [4] introduced the notion of anti fuzzy subgroups of groups and in 2008, modifying Biswas’ idea, Kutukcu and Sharma [10] introduced the notion of anti fuzzy ideals in BCC-algebras. In the present paper, we introduce the notions of anti if-subalgebras, anti if-ideals and anti implicative if-ideals of BCK/BCI-algebras with respect to arbitrary t-conorms and t-norms. Illustrating with examples, we prove that our de…nitions are more general than the classical ones. We also prove that an if-subset of a BCK/BCI-algebra is an anti if-ideal if and only if the complement of this if-subset is an anti if-ideal. We also discuss some relationships between such notions. Let us recall [7,8] that a BCI-algebra is an algebra (X; ; 0) of type (2; 0) which satis…es the following conditions, for all x; y; z 2 X: (i) ((x y) (x z)) (z y) = 0; (ii) (x (x y)) y = 0; (iii) x x = 0; (iv) x y = 0 and y x = 0 imply x = y. A BCI-algebra X satisfying the additional condition (v) for all x 2 X; 0 x = 0 is called a BCK-algebra. We can de…ne a partial ordering on X by x y if and only if x y = 0. Furthermore, in any BCK/BCI-algebra X, the following properties hold, for all x; y; z 2 X: (i) (x y) z = (x z) y; (ii) *Corresponding Author E-mail: [email protected] (S. Kutukcu), [email protected] (A. Tuna) 2010 MR Subject Classi…cation: 06F35, 03G25, 03E72, 94D05

1

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x (x (x y)) = x y; (iii) x y x; (iv) x 0 = x; (v) (x z) (y z) x y; (vi) x y implies x z y z and z y z x. A non-empty subset A of a BCK/BCI-algebra X is called an ideal of X if 0 2 A; and x y 2 A and y 2 A imply x 2 A for all x; y 2 X. A non-empty subset A of a BCK/BCI-algebra X is called an implicative ideal of X if 0 2 A; and (x (y x)) z 2 A and z 2 A imply x 2 A for all x; y; z 2 X. Any implicative ideal is an ideal, but not conversely. A mapping f of a BCK/BCI-algebra X into a BCK/BCI-algebra Y is called a homomorphism if f (x y) = f (x) f (y) for all x; y 2 X. By a triangular conorm (shortly t-conorm) S [14], we mean a binary operation on the unit interval [0; 1] which satis…es the following conditions, for all x; y; z 2 [0; 1]: (i) S(x; 0) = x; (ii) S(x; y) S(x; z) if y z; (iii) S(x; y) = S(y; x); (iv) S(x; S(y; z)) = S(S(x; y); z). Some important examples of t-conorms are SL (x; y) = min fx + y; 1g, SP (x; y) = x + y xy and SM (x; y) = max fx; yg : By a triangular norm (shortly t-norm) T [14], we mean a binary operation on the unit interval [0; 1] which satis…es the following conditions, for all x; y; z 2 [0; 1]: (i) T (x; 1) = x; (ii) T (x; y) T (x; z) if y z; (iii) T (x; y) = T (y; x); (iv) T (x; T (y; z)) = T (T (x; y); z). Some important examples of t-norms are TL (x; y) = max fx + y 1; 0g, TP (x; y) = xy and TM (x; y) = min fx; yg : A t-conorm S and a t-norm T are called associated [11], i.e. S(x; y) = 1 T (1 x; 1 y) for all x; y 2 [0; 1]. For example, t-conorm SM and t-norm TM are associated [6,9-11]. Also, it is well known [6,9] that if S is a t-conorm and T is a t-norm, then max fx; yg S(x; y) and min fx; yg T (x; y) for all x; y 2 [0; 1], respectively. Note that, the concepts of t-conorms and t-norms are known as the axiomatic skeletons that we use for characterizing fuzzy unions and intersections, respectively. These concepts were originally introduced by Menger [13] and several properties and examples for these concepts were proposed by many authors (see [6,9-11,13,14]). A fuzzy subset A in an arbitrary non-empty set X is a function A : X ! [0; 1]. The complement of A , denoted by cA , is the fuzzy subset in X given by cA (x) = 1 A (x) for all x 2 X. De…nition 1.1 ([15]) A fuzzy subset A in a BCK/BCI-algebra X is called a fuzzy BCK/BCI-subalgebra of X if min f A (x); A (y)g A (x y) for all x; y 2 X.

De…nition 1.2 ([15]) A fuzzy subset A in a BCK/BCI-algebra X is called a fuzzy ideal of X if A (0)

A (x)

min f

A (x

y);

A (y)g

for all x; y 2 X. De…nition 1.3 ([10,12]) A fuzzy subset A in a BCK/BCI-algebra X is called a implicative fuzzy ideal of X if min f A ((x (y x)) z); A (z)g A (0) A (x) for all x; y; z 2 X.

De…nition 1.4 ([12]) A fuzzy subset A in a BCK/BCI-algebra X is called an anti fuzzy BCK/BCIsubalgebra of X if max f A (x); A (y)g A (x y)

for all x; y 2 X.

2

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De…nition 1.5 ([12]) A fuzzy subset A in a BCK/BCI-algebra X is called an anti fuzzy ideal of X if max f A (x y); A (y)g A (0) A (x) for all x; y 2 X. As a generalization of the notion of fuzzy subsets in X, Atanassov [2] introduced the concept of intuitionistic fuzzy subsets (or simply if-sets) de…ned on X as objects having the form A = f(x;

A (x);

A (x))

: x 2 Xg

where the functions A : X ! [0; 1] and A : X ! [0; 1] denote the degree of membership (namely A (x)) and the degree of non-membership (namely A (x)) of each element x in X to the set A, respectively, and 0 1 for all x in X. A (x) + A (x) In [3], for every two if-subsets A and B in X, we have (i) A (ii) (iii)

B i¤

A (x)

B (x)

and

A (x)

A = f(x;

A (x);

c A (x))

: x 2 Xg ;

A = f(x;

c A (x);

A (x))

: x 2 Xg :

B (x)

for all x 2 X,

For the sake of simplicity, we shall use the symbol A = (

2

A;

A)

for the if-subset A = f(x;

A (x);

A (x))

Anti IF-Ideals

De…nition 2.1 An if-subset A = ( A ; A ) in a BCK/BCI-algebra X is said to be an anti intuitionistic fuzzy BCK/BCI-subalgebra of X (or simply, an anti if-BCK/BCI-subalgebra of X) if (i)

A (x

y)

max f

(ii)

A (x

y)

min f

A (x); A (x);

A (y)g ; A (y)g

for all x; y 2 X. De…nition 2.2 An if-subset A = ( A ; A ) in a BCK/BCI-algebra X is said to be an anti intuitionistic fuzzy BCK/BCI-subalgebra of X with respect to a t-conorm S and a t-norm T (or simply, an (S; T )anti if-BCK/BCI-subalgebra of X) if (i)

A (x

y)

S(

A (x);

A (y))

(ii)

A (x

y)

T(

A (x);

A (y))

for all x; y 2 X. Remark 2.3 Every anti if-BCK/BCI-subalgebra of a BCK/BCI-algebra is an (S; T )-anti if-BCK/BCIsubalgebra of X such that S = SM and T = TM , but it is clear that the converse is not true. If A (x) = 1 A (x) for all x 2 X, then every anti if-BCK/BCI-subalgebra of a BCK/BCI-algebra X is an anti fuzzy BCK/BCI-subalgebra of X. Also, if A (x) = 1 A (x) for all x 2 X, S = SM and T = TM , then every (S; T )-anti if-BCK/BCI-subalgebra of a BCK/BCI-algebra X is an anti fuzzy BCK/BCI-subalgebra of X. 3

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De…nition 2.4 An if-subset A = ( X if (i)

A (0)

A (x)

(ii)

A (x)

max f

(iii)

A (x)

min f

and

A (0)

A (x

y);

A (x

y);

A;

A)

in a BCK/BCI-algebra X is said to be an anti if-ideal of

A (x); A (y)g ; A (y)g

for all x; y 2 X. De…nition 2.5 An if-subset A = ( A ; A ) in a BCK/BCI-algebra X is said to be an anti if-ideal of X with respect to a t-conorm S and a t-norm T (or simply, an (S; T )-anti if-ideal of X) if (i)

A (0)

A (x)

and

A (0)

(ii)

A (x)

S(

A (x

y);

A (y));

(iii)

A (x)

T(

A (x

y);

A (y))

A (x);

for all x; y 2 X. Remark 2.6 Every anti if-ideal of a BCK/BCI-algebra is an (S; T )-anti if-ideal of X such that S = SM and T = TM , but it is clear that the converse is not true. If A (x) = 1 A (x) for all x 2 X, then every anti if- ideal of a BCK/BCI-algebra X is an anti fuzzy ideal of X. Also, if A (x) = 1 A (x) for all x 2 X, S = SM and T = TM , then every (S; T )-anti if-ideal of a BCK/BCI-algebra X is an anti fuzzy ideal of X. Example 2.7 Let X = f0; 1; 2; 3g be a BCK-algebra with the Cayley table as follows j 0 1 2 3 0 0 0 0 0 1 0 0 1 1 2 1 0 2 2 3 3 3 0 3 De…ne an if-set 8 8 A = ( A ; A ) in X by x=0 x=0 < 1, < 0, 1=2, x = 1 or 2 and A (x) = 1=3, x = 1 or 2 A (x) = : : 1, x=3 0, x=3 It is easy to check that 0 1, A (0) A (x). Also, A (x) + A (x) A (x) and A (0) SM ( A (x y); A (y)) and A (x) TL ( A (x y); A (y)) for all x; y 2 X. Hence A = ( A ; A ) A (x) is an (SM ; TL )-anti if-ideal of X. Also note that t-conorm SM and t-norm TL are not associated. Remark 2.8 Note that, the above example holds even with the t-conorm SM and t-norm TM , and hence A = ( A ; A ) is also an (SM ; TM )-anti if-ideal of X. Therefore, every anti if-ideal of X is an (S; T )-anti if-ideal but the converse is not true. Example 2.9 Let X = f0; a; b; c; dg be a BCK-algebra with the Cayley table as follows

4

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j 0 a b c d 0 0 0 0 0 0 a 0 a 0 0 a b b 0 0 0 b c b a 0 0 c d d d d 0 d De…ne an if-set A = ( A ; A ) in X by 1=2, x 2 f0; a; bg 1=3, x 2 f0; a; bg and A (x) = A (x) = 3=4, otherwise 1=4, otherwise. It is easy to check that 0 1, A (0) A (x). Also, A (x) + A (x) A (x) and A (0) SL ( A (x y); A (y)) and A (x) TP ( A (x y); A (y)) for all x; y 2 X. Hence A = ( A ; A ) A (x) is an (SL ; TP )-anti if-ideal of X. But A = ( A ; A ) is not an anti if-ideal of X. Lemma 2.10 If A = ( A ; A ) is an (S; T )-anti if-ideal of a BCK/BCI-algebra X, then so is ( A ; cA ) such that t-conorm S and t-norm T are associated.

A=

Proof. Since A = ( A ; A ) is an (S; T )-anti if-ideal of X, then A (0) A (x) for all x 2 X and so c c c c 1 1 A (x). Also, for all x; y 2 X, we have A (x), hence A (0) A (0) A (x)

S(

A (x

y);

A (y))

and so 1

c A (x)

S(1

c A (x

y); 1

c A (y))

S(1

c A (x

y); 1

c A (y)):

which implies c A (x)

1

Since S and T are associated, we have c A (x)

Thus,

A=(

A;

c A)

T(

c A (x

y);

c A (y)):

is an (S; T )-anti if-ideal of X.

Lemma 2.11 If A = ( A ; A ) is an (S; T )-anti if-ideal of a BCK/BCI-algebra X, then so is ( cA ; A ) such that t-conorm S and t-norm T are associated.

A=

Proof. The proof is similar to the proof of Lemma 2.10. Combining the above two lemmas, it is easy to see that the following theorem is valid. Theorem 2.12 A = ( A ; A ) is an (S; T )-anti if-ideal of a BCK/BCI-algebra X if and only if and A are (S; T )-anti if-ideals of X such that t-conorm S and t-norm T are associated. Corollary 2.13 A = ( A ; A ) is an (S; T )-anti if-ideal of a BCK/BCI-algebra X if and only if and cA are anti fuzzy ideals of X such that t-conorm S and t-norm T are associated. Lemma 2.14 Let A = ( A ; ordering on X then A (x)

A

A

A)

be an (S; T )-anti if-ideal of a BCK/BCI-algebra X. If is a partial (y) and A (y) y. A (x) for all x; y 2 X such that x A

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Proof. Let X be a BCK/BCI-algebra. It is known [8] that x y if and only if x y = 0 for all x; y 2 X. Let A = ( A ;

A)

is a partial ordering on X de…ned by be an (S; T )-anti if-ideal of X. Then

A (x)

S(

A (x

y);

A (y))

= S(

A (0);

A (y))

A (y)

A (x)

T(

A (x

y);

A (y))

= T(

A (0);

A (y))

A (y):

and These complete the proof. Theorem 2.15 Let A = ( A ; A ) be an (S; T )-anti if-ideal of a BCK/BCI-algebra X. A is an (S; T )anti if-BCK/BCI-subalgebra of X: Proof. Let A = ( A ; from Lemma 2.14 that

A)

be an (S; T )-anti if-ideal of X. Since x y x for all x; y 2 X, it follows (x y) A (x y). Then A A (x) and A (x)

A (x

y)

A (x)

S(

A (x

y);

A (y))

S(

A (x);

A (y))

A (x

y)

A (x)

T(

A (x

y);

A (y))

T(

A (x);

A (y))

and and so A is an (S; T )-anti if-BCK/BCI-subalgebra of X. Remark 2.16 The converse of the above theorem does not hold in general. In fact, suppose that X be the BCK-algebra in Example 2.7. De…ne an if-set A = ( A ; A ) in X by 8 8 x=0 x=0 < 0, < 1, 1=2, x = 1 1=3, x=1 (x) = and (x) = A A : : 1, x = 2 or 3 0, x = 2 or 3

By routine calculations, we know that A = ( not an (SM ; TM )-anti if-ideal of X because minf A (2 1); A (1)g.

A;

A ) is an (SM ; TM )-anti if-BCK-subalgebra of X but (2) = 1 > maxf A (2 1); A (1)g and A (2) = 0 < A

If A = ( A ; A ) is an if-subset in a BCK/BCI-algebra X and f is a self mapping of X, we de…ne mappings A [f ] : X ! [0; 1] by A [f ](x) = A (f (x)) and A [f ] : X ! [0; 1] by A [f ](x) = A (f (x)) for all x 2 X, respectively. Proposition 2.17 If A = ( A ; A ) is an (S; T )-anti if-ideal of a BCK/BCI-algebra X and f is an increasing endomorphism of X, then ( A [f ]; A [f ]) is an (S; T )-anti if-ideal of X. Proof. For any given x; y 2 X, we have A [f ](x)

= S( A (f (x) f (y)); A (f (x)) = S( A (f (x y)); A (f (y))) = S( A [f ](x y); A [f ](y));

A (f (y)))

A [f ](x)

= T ( A (f (x) f (y)); A (f (x)) = T ( A (f (x y)); A (f (y))) = T ( A [f ](x y); A [f ](y)):

A (f (y)))

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Also, since X is a BCK/BCI-algebra, we have 0 x = 0 and so 0 x for all x 2 X. Since f is increasing, we have f (0) f (x) for all x 2 X and, from Lemma 2.14, A (f (0)) A (f (x)) and A (f (x)) A (f (0)) i.e., A [f ](0) A [f ](0) for all x 2 X. This completes A [f ](x) and A [f ](x) the proof. If f is a self mapping of a BCK/BCI-algebra X and B = ( B ; B ) is an if-subset in f (X), then the if-subset A = ( A ; A ) in X de…ned by A = B f and A = B f (i.e., A (x) = B (f (x)) and A (x) = B (f (x)) for all x 2 X) is called the preimage of B under f . Theorem 2.18 An onto increasing homomorphic preimage of an (S; T )-anti if-ideal is an (S; T )-anti if-ideal. Proof. Let f : X ! Y be an onto homomorphism of BCK/BCI-algebras, B = ( (S; T )-anti if-ideal of Y , and A = ( A ; A ) be preimage of B under f . Then, we have A (x)

= S( B (f (x) f (y)); B (f (x)) = S( B (f (x y)); B (f (y))) = S( A (x y); A (y));

B (f (y)))

A (x)

= T ( B (f (x) f (y)); B (f (x)) = T ( B (f (x y)); B (f (y))) = T ( A (x y); A (y))

B (f (y)))

for all x; y 2 X. Also, A (0) = B (f (0)) B (f (x)) = A (x) and A (0) = (x) for all x 2 X. Hence, A = ( ; ) is an (S; T )-anti if-ideal of X. A A A

B (f (0))

B;

B)

be an

B (f (x))

=

Lemma 2.19 ([9]) Let S and T be a t-conorm and a t-norm, respectively. Then S(S(x; y); S(z; t)) = S(S(x; z); S(y; t)); T (T (x; y); T (z; t)) = T (T (x; z); T (y; t)) for all x; y; z; t 2 [0; 1]: Theorem 2.20 Let S be a t-conorm, T be a t-norm and X = X1 X2 be the direct product BCK/BCIalgebra of BCK/BCI-algebras X1 and X2 . If A1 = ( A1 ; A1 ) (resp. A2 = ( A2 ; A2 )) is an (S; T )-anti if-ideal of X1 (resp. X2 ), then A = ( A ; A ) is an (S; T )-anti if-ideal of X de…ned by A = A1 A2 and A = A1 A2 such that A (x1 ; x2 )

=(

A1

A2 )(x1 ; x2 )

= S(

A1 (x1 );

A2 (x2 ));

A (x1 ; x2 )

=(

A1

A2 )(x1 ; x2 )

= T(

A1 (x1 );

A2 (x2 ))

for all (x1 ; x2 ) 2 X.

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Proof. Let x = (x1 ; x2 ) and y = (y1 ; y2 ) be any elements of X. Since X is a BCK/BCI-algebra, we have A (x)

= ( A1 A2 )(x1 ; x2 ) = S( A1 (x1 ); A2 (x2 )) S(S( A1 (x1 y1 ); A1 (y1 )); S( A2 (x2 y2 ); A2 (y2 ))) = S(S( A1 (x1 y1 ); A2 (x2 y2 )); S( A1 (y1 ); A2 (y2 ))) = S(( A1 A2 )(y1 ; y2 )) A2 )(x1 y1 ; x2 y2 ); ( A1 = S(( A1 A2 )(y1 ; y2 )) A2 )((x1 ; x2 ) (y1 ; y2 )); ( A1 = S( A (x y); A (y));

A (x)

= ( A1 A2 )(x1 ; x2 ) = T ( A1 (x1 ); A2 (x2 )) T (T ( A1 (x1 y1 ); A1 (y1 )); T ( A2 (x2 y2 ); A2 (y2 ))) = T (T ( A1 (x1 y1 ); A2 (x2 y2 )); T ( A1 (y1 ); A2 (y2 ))) = T (( A1 A2 )(x1 y1 ; x2 y2 ); ( A1 A2 )(y1 ; y2 )) = T (( A1 A2 )((x1 ; x2 ) (y1 ; y2 )); ( A1 A2 )(y1 ; y2 )) = T ( A (x y); A (y))

Also, A (0)

A (0)

= ( A1 A2 )(0; 0) = S( A1 (0); S( A1 (x1 ); A2 (x2 )) = ( A1 = A (x);

A2 (0))

= ( A1 A2 )(0; 0) = T ( A1 (0); T ( A1 (x1 ); A2 (x2 )) = ( A1 = A (x):

A2 (0))

A2 )(x1 ; x2 )

A2 )(x1 ; x2 )

This completes the proof. De…nition 2.21 Let S be a t-conorm and T be a t-norm, and let A = ( A ; A ) and B = ( B ; be if-sets in a BCK/BCI-algebra X. Then S-product of A and B , and T -product of A and written [ A : B ]S and [ A : B ]T , are de…ned by [

A : B ]S (x)

= S(

A (x);

B (x));

[

A : B ]T (x)

= T(

A (x);

B (x))

B) B,

for all x 2 X, respectively. Theorem 2.22 Let S be a t-conorm and T be a t-norm, and let A = ( A ; A ) and B = ( B ; (S; T )-anti if-ideals of a BCK/BCI-algebra X. If S1 is a t-conorm which dominates S, that is, S1 (S(x; y); S(z; t))

B)

be

S(S1 (x; z); S1 (y; t)) 8

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and T1 is a t-norm which dominates T , that is, T1 (T (x; y); T (z; t)) for all x; y; z; t 2 [0; 1], then ([

T (T1 (x; z); T1 (y; t))

A : B ]S1 ; [ A : B ]T1 )

is an (S; T )-anti if-ideal of X.

Proof. For any x; y 2 X, we have [

[

A : B ]S1 (x)

A : B ]T1 (x)

= S1 ( A (x); B (x)) S1 (S( A (x y); A (y)); S( B (x y); S(S1 ( A (x y); B (x y)); S1 ( A (y); = S([ A : B ]S1 (x y); [ A : B ]S1 (y)); = T1 ( A (x); B (x)) T1 (T ( A (x y); A (y)); T ( B (x y); T (T1 ( A (x y); B (x y)); T1 ( A (y); = T ([ A : B ]T1 (x y); [ A : B ]T1 (y)):

B (y))) B (y)))

B (y))) B (y)))

Also, [ [

A : B ]S1 (0)

= S1 (

A (0);

B (0))

S1 (

A (x);

B (x))

=[

A : B ]S1 (x);

A : B ]T1 (0)

= T1 (

A (0);

B (0))

T1 (

A (x);

B (x))

=[

A : B ]T1 (x)

This completes the proof.

3

Anti Implicative IF-Ideals

De…nition 3.1 A fuzzy subset A in a BCK/BCI-algebra X is said to be an anti implicative fuzzy ideal of X if (i)

A (0)

(ii)

A (x)

A (x);

max f

A ((x

(y x)) z);

A (z)g

for all x; y; z 2 X. De…nition 3.2 An if-subset A = ( if-ideal of X if (i)

A (0)

A (x)

(ii)

A (x)

max f

(iii)

A (x)

min f

and A ((x A (x

A (0)

A;

A)

in a BCK/BCI-algebra X is said to be an anti implicative

A (x);

(y x)) z); (y x)) z);

A (z)g ; A (z)g

for all x; y; z 2 X.

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De…nition 3.3 An if-subset A = ( A ; A ) in a BCK/BCI-algebra X is said to be an anti implicative if-ideal of X with respect to a t-conorm S and a t-norm T (or simply, an (S; T )-anti implicative if-ideal of X) if (i)

A (0)

A (x)

and

(ii)

A (x)

S(

A ((x

(iii)

A (x)

T(

A (x

A (0)

A (x);

(y x)) z); (y x)) z);

A (z)); A (z))

for all x; y; z 2 X. Remark 3.4 Every anti implicative if-ideal of a BCK/BCI-algebra is an (S; T )-anti implicative ifideal of X such that S = SM and T = TM , but it is clear that the converse is not true. If A (x) = 1 A (x) for all x 2 X, then every anti implicative if-ideal of a BCK/BCI-algebra X is an anti implicative fuzzy ideal of X. Also, if A (x) = 1 A (x) for all x 2 X, S = SM and T = TM , then every (S; T )-anti implicative if-ideal of a BCK/BCI-algebra X is an anti implicative fuzzy ideal of X. Example 3.5 In Example 2.9, it is easy to see that A = ( if-ideal of X.

A;

A)

is also an (SL ; TP )-anti implicative

Remark 3.6 An (S; T )-anti if-ideal of a BCK/BCI-algebra X need not to be (S; T )-anti implicative if-ideal. For instance, in Example 2.7, we know that A = ( A ; A ) is an (SM ; TL )-anti if-ideal of X but it is not an (SM ; TL )-anti implicative if-ideal of X, because A (1) > SM ( A ((1 (2 1)) 0); A (0)) and A (1) < TL ( A ((1 (2 1)) 0); A (0)). Theorem 3.7 Any (S; T )-anti implicative if-ideal of a BCK/BCI-algebra X is an (S; T )-anti if-ideal of X. Proof. In De…nition 3.3, let z = y and y = x. Hence A (x) S( A ((x (x x)) y); A (y)) and T ( A ((x (x x)) y); A (y)). Since x x = 0 and x 0 = x, we obtain (ii) and (iii) in A (x) De…nition 2.5. This completes the proof. Theorem 3.8 Let A = ( A ; A ) be an (S; T )-anti if-ideal of a BCK/BCI-algebra X. Then A = ( A ; A ) is an (S; T )-anti implicative if-ideal of X i¤ A (x) A (x (y x)) A (x (y x)) and A (x) for all x; y 2 X. Proof. Assume that A = ( A ; A ) is an (S; T )-anti implicative if-ideal. Taking z = 0 in (ii) and (iii), and using (i) in De…nition 3.3, we get the inequalities. Conversely, since A = ( A ; A ) is an (S; T )-anti if-ideal, hence S( A ((x (y x)) z); A (z)); A (x) A (x (y x)) A (x)

A (x

(y x))

T(

A ((x

(y x)) z);

A (z)):

This completes the proof. Lemma 3.9 Let A = ( in X, then A (x) S(

A;

A)

A (y);

be an (S; T )-anti if-ideal of a BCK/BCI-algebra X. If x y T ( A (y); A (z)) for all x; y 2 X. A (z)) and A (x)

z holds

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Proof. Since x y

z holds for all x; y 2 X, we have A (x

y)

S( S( = S(

A ((x

y) z); A (z)) A (z z); A (z)) A (0); A (z)) A (z)

it follows that A (x)

S(

A (x

A (x

y)

y);

A (y))

T( T( = T(

S(

A (y);

A (z));

A ((x

y) z); A (z)) (z z); A A (z)) A (0); A (z)) A (z)

hence A (x)

T(

A (x

y);

A (y))

T(

A (y);

A (z)):

This completes the proof. Theorem 3.10 ([12]) A BCK-algebra is implicative i¤ it is both commutative and positive implicative. Theorem 3.11 ([12]) If X is an implicative BCK-algebra, then x x; y; z 2 X.

((x

(y

x))

z)

z for all

Theorem 3.12 In an implicative BCK/BCI-algebra, every (S; T )-anti if-ideal is an (S; T )-anti implicative if-ideal. Proof. The proof is easily follows from Lemma 3.9 and Theorem 3.11. Theorem 3.13 The intersection of any set of (S; T )-anti implicative if-ideals of a BCK/BCI-algebra X is also an (S; T )-anti implicative if-ideal whenever S and T are continuous norms. Proof. Let Ai = ( x; y; z 2 X,

Ai ;

\

Ai ) i2I Ai

(0) = inf

(\

Ai ) (0)

Ai

(x)

Also, \

be a family of (S; T )-anti implicative if-ideals of X. Then, for any

=

= sup f inf

= S(inf = S( \

(\

Ai ) (x)

Ai (0)

inf

Ai (x)

Ai (0)g

sup f

Ai (x)g

Ai (x)

inf S(

Ai ((x Ai

= \

= (\

Ai ((x

y) z) ; inf

((x y) z); \

(x);

Ai ) (x):

y) z); Ai (z)

Ai

Ai

Ai (z))

)

(z));

= sup Ai (x) sup fT ( Ai ((x y) z); = T (sup f Ai ((x y) z)g ; sup f Ai (z)g) = T ((\ Ai ) ((x y) z); (\ Ai ) (z)):

Ai (z))g

This completes the proof. 11

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4

Conclusions

In this work, we introduce the notions of anti intuitionistic fuzzy BCK/BCI- subalgebras, anti intuitionistic fuzzy ideals and anti implicative intuitionistic fuzzy ideals with the help of arbitrary t-conorms and t-norms, and discuss some properties such as product, direct product and relations between them. But there are still some open problems. How can we de…ne the notions of anti intuitionistic fuzzy …lters and anti intuitionistic fuzzy congruences with recpect to arbitrary t-conorms and t-norms on a BCK/BCI-algebra? What are the relations between such notions, between the cosets of an anti intuitionistic fuzzy …lter and anti intuitionistic fuzzy congruences? These could be a topic of further research. Furthermore, using that generalizations, one could de…ne the notion anti intuitionistic fuzzy subgroups in BCK/BCI-algebras with respect to arbitrary t-conorms and t-norms in the sense of [1] and [7]. Using the idea of Dudek et al. [5], one could also generalize the notion of fuzzy topological anti BCK/BCI-algebras to intuitionistic fuzzy structures. The notions given in this paper can be fundamental to other sciences. For instance, in the last decade, most of researchers are focused on Content Based Image Retrieval, shortly CBIR, and managing uncertainty becomes a fundamental topic in image database. Intuitionistic fuzzy set theory can be ideally suited to deal with this kind of uncertainty. This fuzziness is mainly due to similarity of media features, imperfection in the feature extraction algorithms, esc. Using the concept of this paper, one could develop an anti intuitionistic fuzzy model for image data and provide an anti intuitionistic fuzzy subalgebra for dealing with such data. Moreover, new anti intuitionistic fuzzy algebraic operators could be de…ned in order to capture the fuzziness related to the semantic descriptors of an image, and built thematic categorizations of multimedia documents using ontological information and anti intuitionistic fuzzy subalgebra in triangular norm systems. Problem. Can we replace in the statement of Theorem 3.13, the condition "S and T are continuous" with "inf a>0 S(a; a) = 0 and supb B ≥ 0 and p > 1. Hence, we can get the desired result (2.18). We give, now, some special cases of Theorem 2.4. Corollary 2.10 In Theorem 2.4, if we take p = q = 2, then we get i  2a + η(b, a)  Γ(α + 1) h α α J f (a + η(b, a)) + J f (a) − f + − (a+η(b,a)) 2η α (b, a) a 2 (  1 Z Z i  12  1  12  η 2 (b, a) h 00 1  2 00 2 2 √ ≤ |f (a)| + |f (b)| h(t )dt + h(t )dt 1 2 2 6 0 2 ) Z  12 1  2α + 1  12  1 2 + , h(t )dt α + 1 2α + 3 0 where h is super-multiplicative. Corollary 2.11 Under assumptions of Theorem 2.4, letting h(t) = t, then the inequality (2.18) becomes the following inequality for the preinvex function i  2a + η(b, a)  Γ(α + 1) h α α J f (a + η(b, a)) + J f (a) − f (a+η(b,a))− 2η α (b, a) a+ 2 i η 2 (b, a)  1  q1 h 00 ≤ |f (a)| + |f 00 (b)| 2 q+1 ) (   1  p1  1 1+ p1  1 1+ q1  1  q1 1  pα + p − 1  p1 + 1 − q+1 + , × p+1 2 2 2 α + 1 pα + p + 1 specially for α = 1, we get the following inequality for preinvex functions Z a+η(b,a)  2a + η(b, a)  1 f (t)dt − f 2η(b, a) 2 a 1h   i 2 η (b, a) 1 q ≤ |f 00 (a)| + |f 00 (b)| 4 q+1 ( )    1  p1  1 1+ q1  1  q1 2p − 1  p1 × + 1 − q+1 + . 2p + 2 2 2 2p + 1

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In Theorem 2.4, if we choose h(t) = 1, then we get i  2a + η(b, a)  Γ(α + 1) h α α J f (a + η(b, a)) + J f (a) − f + − (a+η(b,a)) 2η α (b, a) a 2 ( ) i 1  1  p1 η 2 (b, a) h 00 1  pα + p − 1  p1 00 ≤ |f (a)| + |f (b)| + , 2 2 p+1 α + 1 pα + p + 1 for P -preinvex functions. Corollary 2.12 In Theorem 2.4, if we take h(t) = ts , then the inequality (2.18) becomes the following inequality for s-preinvex functions i  2a + η(b, a)  Γ(α + 1) h α α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2 1h   i 2 1 η (b, a) q ≤ |f 00 (a)| + |f 00 (b)| 2 sq + 1 ( )   1  p1  1 1+ p1  1 s+ q1  1  q1 1  pα + p − 1  p1 × + 1 − sq+1 + . p+1 2 2 2 α + 1 pα + p + 1 Theorem 2.5 Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative and supermultiplicative function, h(t) ≥ t for 0 ≤ t ≤ 1. Assume that f : [a, a + η(b, a)] ⊆ [0, ∞) → R be a twice differentiable mapping on (a, a + η(b, a)) with η(b, a) > 0 such that f 00 ∈ L1 [a, a + η(b, a)]. If |f 00 |q is h-preinvex on [a, a + η(b, a)], q ≥ 1 and |f 00 (x)| ≤ M , x ∈ [a, a+η(b, a)], then the following inequalities for fractional integrals with α > 0 hold: i Γ(α + 1) h α 2a + η(b, a)
α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2  1 Z Z  q1  1  q1 i M η 2 (b, a)  1 1− q1 h 2 t[h(t) + h(1 − t)]dt + ≤ (1 − t)[h(t) + h(1 − t)]dt 1 2 8 0 2  Z 1 1   qα  2 −1 q + qα h(t) + h(1 − t) dt (2.21) 2 (α + 1) 0  Z 1  q1  Z 1 h  i  q1 i M η 2 (b, a)  1 1− q1 h 2 2 2 ≤ [h(t ) + h(t − t )]dt + h(t − t2 ) + h (1 − t)2 dt 1 2 8 0 2  Z   q1 2qα − 1  1  + qα . h(t) + h(1 − t) dt 2 (α + 1) 0 (2.22) Proof. Continuing from inequality (2.16) in the proof of Theorem 2.3, using properties of absolute value again, recurring to definition of λ(t) and power

14

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mean inequality for q ≥ 1, we have i Γ(α + 1) h α 2a + η(b, a)
α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2  1 1 Z Z 1− q1  2
q  q1 η 2 (b, a)  2 ≤ tdt t f 00 a + tη(b, a) dt 2 0 0 Z 1 1− q1  Z 1
q  q1 + (1 − t)dt (1 − t) f 00 a + tη(b, a) dt 1 2

1 2

+

1 α+1

Z

1

1− q1  Z 1dt

0

1

 q1 
q
q 1 − tα+1 − (1 − t)α+1 f 00 a + tη(b, a) dt .

0

According to the h-preinvexity of |f 00 |q and |f 00 | ≤ M , we get Z

1 2


q t f 00 a + tη(b, a) dt ≤

1 2

Z

0

h i t h(1 − t)|f 00 (a)|q + h(t)|f 00 (b)|q dt

0

≤ Mq

Z

1 2

t[h(t) + h(1 − t)]dt. 0

Similarly, we also have Z 1 Z 1
q (1 − t) f 00 a + tη(b, a) dt ≤ M q (1 − t)[h(t) + h(1 − t)]dt 1 2

1 2

and Z

1



1 − tα+1 − (1 − t)α+1

q
f 00 a + tη(b, a) q dt

0

Z 1  1  [h(t) + h(1 − t)]dt, ≤ 1 − qα M q 2 0 where we use the fact that 1 − (1 − t)α+1 − tα+1


q

 q 1 ≤ 1 − (1 − t)α+1 + tα+1 ≤ 1 − (2−α )q = 1 − qα 2

for any t ∈ [0, 1] with q ≥ 1. Also Z 12

Z

1

(1 − t)dt =

tdt = 0

1 2

1 . 8

Using these results, we see that the inequality (2.21) is proved. To prove (2.22), and using the additional properties of h in hypothetical conditions, we further have Z 12 Z 21 t[h(t) + h(1 − t)]dt ≤ [h(t2 ) + h(t − t2 )]dt 0

0

15

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and Z

1

1

Z (1 − t)[h(t) + h(1 − t)]dt ≤

1 2

i h  h(t − t2 ) + h (1 − t)2 dt.

1 2

Hence, the proof of (2.22) is completed. Elementary calculation yields the following result. Corollary 2.13 In Theorem 2.5, if we choose h(t) = t, we can obtain i Γ(α + 1) h α 2a + η(b, a)
α f (a) − f f (a + η(b, a)) + J J − + (a+η(b,a)) 2η α (b, a) a 2 M η 2 (b, a)  1 2qα − 1  ≤ + qα , 2 4 2 (α + 1) specially for h(t) = 1, we get i Γ(α + 1) h α 2a + η(b, a)
α J f (a + η(b, a)) + J f (a) − f + − (a+η(b,a)) 2η α (b, a) a 2 2qα − 1  M η 2 (b, a)  1 + qα . ≤ 1− q1 4 2 (α + 1) 2 Theorem 2.6 Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative and superadditive function, h(t) ≥ t for 0 ≤ t ≤ 1. Suppose that f : [a, a+η(b, a)] ⊆ [0, ∞) → R be a twice differentiable mapping on (a, a + η(b, a)) with η(b, a) > 0 such that f 00 ∈ L1 [a, a + η(b, a)]. If |f 00 |q is h-preinvex on [a, a + η(b, a)], p, q > 1, 1 1 00 p + q = 1 and |f (x)| ≤ M , x ∈ [a, a + η(b, a)], then the following inequalities for fractional integrals with α > 0 hold: i  2a + η(b, a)  Γ(α + 1) h α α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2  1 Z Z  q1  1  q1 i M η 2 (b, a)  1 1+ p1  1  p1 h 2 ≤ [h(t) + h(1 − t)]dt + [h(t) + h(1 − t)]dt 1 2 2 p+1 0 2  Z 1 1 1     1 pα + p − 1 p q + [h(t) + h(1 − t)]dt (2.23) α + 1 pα + p + 1 0   M η 2 (b, a) 1  1  p1 q1 1  pα + p − 1  p1 q1 ≤ h (1) + h (1) . 2 2 p+1 α + 1 pα + p + 1 (2.24) Proof. Continuing from inequality (2.16) in the proof of Theorem 2.3, using properties of absolute value again, recurring to definition of λ(t) and H¨older’s

16

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inequality for q > 1, we have i  2a + η(b, a)  Γ(α + 1) h α α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2  1 1 Z Z
q i q1 η 2 (b, a)  2 p  p1 h 2 00 ≤ t dt f a + tη(b, a) dt 2 0 0 Z 1  p1 h Z 1
i 1 f 00 a + tη(b, a) q dt q + (1 − t)p dt 1 2

1 2

1 + α+1

1

hZ



1−t

α+1

α+1

− (1 − t)

p i p1 h Z dt

0

1

 00
i 1 f a + tη(b, a) q dt q .

0 00 q

00

According to the h-preinvexity of |f | and |f | ≤ M , we can get Z 21 Z Z 12 00
f a + tη(b, a) q dt ≤ |f 00 (a)|q h(1 − t)dt + |f 00 (b)|q 0

0

≤M

q

Z

1 2

h(t)dt

0

1 2

[h(t) + h(1 − t)]dt. 0

Similarly we also have Z 1 Z 1 00
f a + tη(b, a) q dt ≤ M q [h(t) + h(1 − t)]dt 1 2

1 2

and Z

1

00
f a + tη(b, a) q dt ≤ M q

0

Z

1

[h(t) + h(1 − t)]dt. 0

By virtue of the above results and the fact (2.19) and the inequality (2.20), we complete the proof of (2.23). Using the supper-additive property of h in the assumptions, we further have h(t) + h(1 − t) ≤ h(1). Hence, the proof of (2.24) is completed. Finally we shall obtain estimate of Riemann-Liouville fractional HermiteHadamard inequality for for h-preincave functions. Theorem 2.7 Let h : J ⊆ R → R ([0, 1] ⊆ J) be a non-negative function, h(t) ≥ t for 0 ≤ t ≤ 1 and f : [a, a + η(b, a)] ⊆ [0, ∞) → R be a twice differentiable mapping on (a, a+η(b, a)) with η(b, a) > 0 such that f 00 ∈ L1 [a, a+ η(b, a)]. If |f 00 |q is h-preincave on [a, a + η(b, a)], p, q > 1, p−1 + q −1 = 1, then the following inequality for fractional integrals with α > 0 hold: i  2a + η(b, a)  Γ(α + 1) h α α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2 1 1 ih 1 h    i  2 η (b, a) 1 1 pα + p − 1 p 1 2a + η(b, a)  p q 00 ≤ + f 2 2p + 2 α + 1 pα + p + 1 2 2h( 12 ) 17

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Proof. Continuing from inequality (2.16) in the proof of Theorem 2.3, using properties of absolute value again, recurring to definition of λ(t) and H¨older’s inequality, we have i  2a + η(b, a)  Γ(α + 1) h α α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2  1 1 Z Z 1 1
q i q η 2 (b, a)  2 p  p h 2 00 ≤ t dt f a + tη(b, a) dt 2 0 0 Z 1  p1 h Z 1
i 1 f 00 a + tη(b, a) q dt q + (1 − t)p dt 1 2

1 2

1 α+1  η 2 (b, a)  +

≤

hZ

2

1



1 − tα+1 − (1 − t)α+1

p i p1 h Z dt

0

1

 00
i 1 f a + tη(b, a) q dt q

0

 p1 h Z

1 2p+1 (p + 1)  pα + p − 1  p1 h Z

1 2

Z 00
 1  f a + tη(b, a) q dt q +

0

1 1 2

00
 1 i f a + tη(b, a) q dt q

1 1 f 00 α + 1 pα + p + 1 0 To prove the second inequality above, we here use the fact (2.19) and the inequality(2.20) again. Also, |f 00 |q is h-preincave on [a, a + η(b, a)], by inequalities (1.9) we have Z 1 Z 12   q 00 00

f a+tη(b, a) q dt ≤ 1 f 00 2a + η(b, a) . f a+tη(b, a) q dt ≤ 2 2h( 12 ) 0 0

+

1 q


q i . a + tη(b, a) dt

Similarly, we also have Z 1 00
f a + tη(b, a) q dt ≤ 1 2

and Z 0

1

00
f a + tη(b, a) q dt ≤

1 00  2a + η(b, a)  q f 2 2h( 12 ) 1 00  2a + η(b, a)  q f . 2 2h( 21 )

Therefore, we can get i  2a + η(b, a)  Γ(α + 1) h α α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2 1 1 1 h    ih i  η 2 (b, a) 1 1 pα + p − 1 p 1 2a + η(b, a)  p q 00 ≤ + f . 2 2p + 2 α + 1 pα + p + 1 2 2h( 21 ) Direct computation provides the following corollary. Corollary 2.14 In given conditions of Theorem 2.7, if we take h(t) = t, we obtain the following inequality for the preincave functions i  2a + η(b, a)  Γ(α + 1) h α α 2η α (b, a) Ja+ f (a + η(b, a)) + J(a+η(b,a))− f (a) − f 2 1 1 i h      2 1 pα + p − 1 p 00 2a + η(b, a) η (b, a) 1 p ≤ + f . 2 2p + 2 α + 1 pα + p + 1 2 18

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Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11601267) and the Natural Science Foundation of Fujian Province of China(No. 2016J01023).

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 2, 2018

Coupled fixed point theorem in partially ordered modular metric spaces and its an application, Ali Mutlu, Kubra Ozkan, and Utku Gurdal,…………………………………………………207 Connected 𝑚 − 𝐾𝑛 −residual graph and its application in Cryptology, Huiming Duan, Xinping Xiao, and Congjun Rao,………………………………………………………………………217 A comparative analysis of the Harry Dym model with and without singular kernel, E.F. Doungmo Goufo, P. Tchepmo, Z. Ali, and A. Kubeka,…………………………………….228 Product-type Operators from Weighted Bergman Spaces to Bloch-Orlicz Spaces, Zhi-jie Jiang, …………………………………………………………………………………………………241 On some recent fixed point results for 𝛼-admissible mappings in b-metric spaces, Huaping Huang, Guantie Deng, Zhanmei Chen, and Stojan Radenovic,…………………………….255 Anti Implicative IF-Ideals in BCK/BCI-algebras, Servet Kutukcu and Adnan Tuna,………270 On (IT)-commutativity condition in fixed point consideration of set-valued and single-valued mappings, Servet Kutukcu, Sushil Sharma, and Bhavana Deshpande,…………………….283 Some Generalizations in Intuitionistic Fuzzy Metric Spaces, Servet Kutukcu and Sunny Chauhan,………………………………………………………………………………………292 Existence and uniqueness of positive solutions for singular higher order fractional differential equations with in finite-point boundary value conditions, Qianqian Leng, Jiandong Yin, and Pinghua Yan,………………………………………………………………………………….302 On the (h,q)-Euler polynomials and numbers, Lee-Chae Jang and Jeong Gon Lee,……….311 Hermite-Hadamard type inequalities for n-times differentiable and 𝛼-logarithmically preinvex functions, Shuhong Wang,……………………………………………………………………319 Existence of solutions for boundary value problems of fractional differential equation in Banach spaces, Yabing Gao and Pengyu Chen,………………………………………………………329 On the Higher Order Difference equation 𝑥𝑛+1 = 𝐴𝑥𝑛 + 𝐵𝑥𝑛−𝑙 + 𝐶𝑥𝑛−𝑘 + 𝐷𝑥

𝛾𝑥𝑛−𝑘

𝑛−𝑠 +𝐸𝑥𝑛−𝑡

, M. M.

El-Dessoky and M. A. El-Moneam,…………………………………………………………342

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 2, 2018 (continued) An additive (𝛼, 𝛽)-functional equation and linear mappings in Banach spaces, Choonkil Park, Sun Young Jang, and Young Cho,……………………………………………………………355 Riemann-Liouville fractional Hermite-Hadamard inequalities for h-preinvex functions, Tingsong Du, Shanhe Wu, Shoujiang Zhao, and Muhammad Uzair Awan,……………………………364

Volume 25, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

September 2018

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (sixteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

SOME HERMITE−HADAMARD AND SIMPSON TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA FRACTIONAL INTEGRALS WITH APPLICATIONS MUHAMMAD IQBAL, MUSTAFA HABIB, NASIR SIDDIQUI, AND MUHAMMAD MUDDASSAR

Abstract. In this paper, a new general identity for Riemann−Liouville fractional integrals is established. Then by making use of the established identity, we establish some new inequalities of the Simpson and the Hermite−Hadamard type for functions whose absolute values of derivatives are convex. Our results have some relationships with the results, proved in [3, 6, 10], and the analysis used in the proofs is simple.

1. Introduction Definition 1. Let I ⊂ R be an interval. The function f : I → R is said to be convex on I, if for all a, b ∈ I with a ≤ b and λ ∈ [0, 1] , satisfies the inequality f (λa + (1 − λ) b) ≤ λf (a) + (1 − λ)f (b) The inequalities discovered by Hermite and Hadamard for convex functions are very important in the literature (see, e.g.,[[12], p. 137], ).These inequalities state that if f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b. Then   Z b 1 f (a) + f (b) a+b . (1) ≤ f (x) dx ≤ f 2 b−a a 2 Both inequalities hold in the reversed direction for f to be concave. Hadamard’s inequality for convex functions has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found; see, for example, ([3],[5]−[6], [9]−[10], [12]) and the references cited therein. In [10], a variant of Hermite−Hadamard type inequalities was obtained, which follows as: Theorem 1. Let f : I ◦ ⊆ R → R be a differentiable function on I ◦ and let a, b ∈ I ◦ with a < b. If |f 0 | is convex function on [a, b], then the following inequality holds:     1 Z b a + b b − a |f 0 (a)| + |f 0 (b)| f (x)dx − f ≤ (2) b − a a 2 4 2 In [6] authors proved the following version of Hermite−Hadamard type inequalities: Date: December 5, 2016. 2000 Mathematics Subject Classification. 26D15, 26A51, 26A33. Key words and phrases. Hermite−Hadamard’s Inequality, Simpson’s Inequality, Convex Functions, Riemann−Liouville fractional integral. 1

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2

M. IQBAL, M. HABIB, N. SIDDIQUI, AND M. MUDDASSAR

Theorem 2. Let f : I ◦ ⊆ R → R be a differentiable function on I ◦ and let a, b ∈ I ◦ with a < b. If |f 0 | is convex function on [a, b], then the following inequality holds: Z b f (a) + f (b) b − a  |f 0 (a)| + |f 0 (b)|  1 f (x)dx ≤ − (3) 2 b−a a 4 2 The Simpson’s inequality is very important and well known in the literature. For recent refinements, counterparts, generalizations and new Simpson’s type inequalities, see ([7],[13], [16], [17]). In [16], Sarikaya et al. obtained inequality for differentiable convex mappings which is connected with Simpson’s inequality, is as follow: Theorem 3. Let f : I ◦ ⊆ R → R be a differentiable function such that f 0 ∈ L[a, b] where a, b ∈ I ◦ with a < b. If |f 0 | is convex function on [a, b], then the following inequality holds:      0  Z b 0 1 f (a) + f (b) a+b 1 ≤ 5(b − a) |f (a)| + |f (b)| f (x)dx 2f + − (4) 3 2 2 b−a a 36 2

In [3], the authors generalize some inequalities related to Hermite−Hadamard and Simpsons inequality for functions whose derivatives in absolute value are convex functions as: Theorem 4. Let f : I ◦ ⊆ R → R be a differentiable function on I ◦ and let 1 a, b ∈ I ◦ with a < b. If 0 ≤ λ ≤ 1 and |f 0 | q for q ≥ 1 is a convex on [a, b], then the following inequality holds:   Z b a+b 1 f (a) + f (b) − +λ f (x)dx (1 − λ)f 2 2 b−a a   q1 1 b−a 1 ≤ (1 − 2λ + 2λ2 )1− q 8 6 h  
1 × 2 − 3λ + 2λ3 |f 0 (a)|q + 4 − 9λ + 12λ2 − 2λ3 |f 0 (b)|q q  
1 i (5) + 4 − 9λ + 12λ2 − 2λ3 |f 0 (a)|q + 2 − 3λ + 2λ3 |f 0 (b)|q q Remark 1. On letting λ =, 0, 1, 13 with q = 1 , inequality (5) reduces to inequalities (2), (3) and (4), respectively.. It is well known that the integral inequalities play an important role in nonlinear analysis. In the recent years, these inequalities have been improved and generalized in a number of ways and a large number of research papers have been written on these inequalities, (see, [1]−[2], [4], [10], [15]) and the references therein. In recent paper, [10] Sarikaya et. al. proved a variant of Hermite−Hadamard’s inequalities in fractional integral forms as follows: Theorem 5. Let f : [a, b] → R be a positive function with 0 ≤ a < b and f ∈ L[a, b]. If f is convex function on [a, b], then the following inequalities for fractional integrals hold:   Γ(α + 1) α f (a) + f (b) a+b ≤ [J + f (b) + Jbα− f (a)] ≤ (6) f 2 2(b − a)α a 2 Remark 2. For α = 1, inequality (6) reduces to inequality (1). In the following, we will give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used further in this paper.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

SOME INTEGRAL INEQUALITIES

3

Definition 2. Let f ∈ L[a, b], the Reimann−Liouville integrals Jaα+ and Jbα− of order α > 0 with a ≥ 0 are defined by Z x 1 Jaα+ = (x − t)α−1 f (t)dt, x>α Γ(α) a and Z b 1 (t − x)α−1 f (t)dt, x 0}. 0 The closure of suppe u defines the 0-level of u e, i.e. [e u] = cl(suppe u). Here cl(M ) denotes the closure of set M. Fuzzy set u e ∈ F (Rn ) is called a fuzzy number if (1) u e is a normal fuzzy set, i.e. there exists an x0 ∈ Rn such that u e(x0 ) = 1, (2) u e is a convex fuzzy set, i.e. u e(λx + (1 − λ)y) ≥ min{e u(x), u e(y)} for any x, y ∈ Rn and λ ∈ [0, 1], (3) u e is upper semi-continuous, S (4) [e u]0 = cl(suppe u) = cl( r∈(0,1] [e u]r ) is compact. We will denote E n the set of fuzzy numbers [9, 10, 11, 12]. It is clear that any u ∈ Rn can be regarded as a fuzzy number u e defined by ( 1, x = u, u e(x) = 0, otherwise. In particular, the fuzzy number e 0 is defined as e 0(x) = 1 if x = 0, and e 0(x) = 0 otherwise. n r Definition 2.1. [13] If u e ∈ E , and [e u] is a cell, i.e., for any r ∈ [0, 1], [e u]r =

n Y

− + − + + − + [u− i (r), ui (r)] = [u1 (r), u1 (r)] × [u2 (r), u2 (r)] × · · · × [un (r), un (r)],

i=1 + u− i (r), ui (r)

+ where ∈ R with u− e a fuzzy n-cell number. Denote i (r) ≤ ui (r) (i = 1, 2, · · · , n), then we call u n the collection of all fuzzy n-cell numbers by L(E ). − For any r ∈ [0, 1], li [e u]r = u+ i (r) − ui (r) (i = 1, 2, · · · , n) is called the r-level length of a fuzzy n-cell number u e with respect to the ith component. + Theorem 2.1. [13] (Representation theorem). If u e ∈ L(E n ), then for i = 1, 2, · · · , n, u− i (r), ui (r) are real-valued functions on [0, 1], and satisfy (1) u− i (r) are non-decreasing, left continuous at r ∈ (0, 1] and right continuous at r = 0, (2) u+ i (r) are non-increasing, left continuous at r ∈ (0, 1] and right continuous at r = 0, + − + (3) u− i (r) ≤ ui (r) (it is equivalent to ui (1) ≤ ui (1)). Conversely if ai (r), bi (r) (i = 1, 2, · · · , n) are real-valued functions on [0, 1] which satisfy conditions Qn (1)-(3), then there exists a unique u e ∈ L(E n ) such that [e u]r = i=1 [ai (r), bi (r)] for any r ∈ [0, 1]. Theorem 2.2. [13] Let u e, ve ∈ L(E n ) and k ∈ R. Then for any r ∈ [0, 1], Qn + − + r r r (1) [e u + ve] = [e u] + [e v ] = i=1 [u− i (r) + vi (r), ui (r) + vi (r)],

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Shexiang Hai and Zengtai Gong: The differentiability for fuzzy n-cell mappings and the KKT optimality ...

( Q n + [ku− i (r), kui (r)], k ≥ 0, Qi=1 (2) [ke u] = k[e u] = n + − i=1 [kui (r), kui (r)], k < 0, Q n − − − + + − + + (3) [e uve]r = i=1 [min{ui (r)vi (r), ui (r)vi (r), ui (r)vi (r), ui (r)vi (r)}, r

r

− − + + − + + max{u− i (r)vi (r), ui (r)vi (r), ui (r)vi (r), ui (r)vi (r)}]. Given u e, ve ∈ L(E n ), the distance D : L(E n ) × L(E n ) → [0, +∞) between u e and ve is defined by the equation − + + D(e u, ve) = sup d([e u]r , [e v ]r ) = sup max {| u− i (r) − vi (r) |, | ui (r) − vi (r) |}. r∈[0,1] 1≤i≤n

r∈[0,1]

Then (L(E n ), D) is a complete metric space, and satisfies D(e u + w, e ve + w) e = D(e u, ve), D(ke u, ke v ) = |k|D(e u, ve) n for any u e, ve, w e ∈ L(E ), k ∈ R. In recent years, several authors have discussed different ordering relation of fuzzy numbers [14]. To the best of our knowledge, very few investigations have been appeared to study ordering relation of fuzzy n-cell numbers. For this reason, an ordering c of fuzzy n-cell numbers will be introduced. Definition 2.2. Let τ : L(E n ) → Rn be a vector-valued function defined by τ (e u)

R 1 R ··· R e r x1 dx1 dx2 ···dxn R 1 R ··· R[u] R 1 R ··· R[u] r x2 dx1 dx2 ···dxn r xn dx1 dx2 ···dxn R Re (2 0 r R ··· R[u] dr, 2 r dr, · · · , 2 r R ··· Re r 1dx1 dx2 ···dxn dr) 0 0 1dx1 dx2 ···dxn 1dx1 dx2 ···dxn ··· [u] [u] e r e r [u] e R1 R1 R1 − − − ( 0 r(e u+ u+ u+ n (r) + un (r))dr), 1 (r) + u1 (r))dr, 0 r(e 2 (r) + u2 (r))dr, · · · , 0 r(e

= =

where

R1 0

R

r

R ··· e r xi dx1 dx2 ···dxn R R[u] dr ··· [u] 1dx1 dx2 ···dxn e r

R

(i = 1, 2, · · · , n) are the Lebesgue integral of r

R ··· e r xi dx1 dx2 ···dxn R R[u] ··· [u] 1dx1 dx2 ···dxn e r n

(i =

1, 2, · · · , n) on [0, 1]. The vector-valued function τ is called a ranking value function defined on L(E ). In this case τ (e u) represents a centroid of the fuzzy n-cell number u e. From the ranking value function n τ (e u), we consider the following ordering relation c on L(E ). Definition 2.3. Let u e, ve ∈ L(E n ), C ⊆ Rn be a closed convex cone with 0 ∈ C and C 6= Rn . We say that u e c ve (e u precedes ve) if τ (e v ) ∈ τ (e u) + C (τ (e v ) − τ (e u) ∈ C). Obviously the order relation c is reflexive and transitive, and c is a partially ordered relation on L(E n ). For u e, ve ∈ L(E n ), if either u e c ve or ve c u e, then we say that u e and ve are comparable, otherwise 1 non-comparable. If u e, ve ∈ E , C = [0, +∞) ⊆ R, then Definition 2.3 coincides with Definition 2.5 from [14]. We say that u e ≺c ve if u e c ve and τ (e u) 6= τ (e v ). Sometimes we may write ve c u e (resp. ve c u e) instead of u e c ve (resp. u e ≺c ve). Remark 2.1. Let u e, ve ∈ L(E n ), k1 , k2 ∈ R. According to Theorem 2.2 and Definition 2.2, it is easy to verify that τ (k1 u e + k2 ve) = k1 τ (e u) + k2 τ (e v ). Theorem 2.3. Let u e1 , u e2 , ve1 , ve2 , ∈ L(E n ), k1 , k2 ∈ [0, +∞], C ⊆ Rn be a closed convex cone with 0 ∈ C n and C 6= R . If u e1 c ve1 and u e2 c ve2 , then k1 u e1 + k2 u e2 c k1 ve1 + k2 ve2 . The proof is similar to the proof of Theorem 2.3 in [15] 3. Generalized difference for fuzzy n-cell numbers Definition 3.1. [16] Let u e, ve ∈ L(E n ). The generalized difference (g-difference for short) of u e and ve is given by its level sets as [e u g ve]r =

n Y

− + + − − + + [ inf min{u− i (β) − vi (β), ui (β) − vi (β)}, sup max{ui (β) − vi (β), ui (β) − vi (β)}],

i=1

β≥r

β≥r

where β ∈ [r, 1]. Remark 3.1. If u e, ve ∈ E 1 , we have [e u g ve]r = [ inf min{u− (β) − v − (β), u+ (β) − v + (β)}, sup max{u− (β) − v − (β), u+ (β) − v + (β)}], β≥r

β≥r

which coincides with Definition 7 of reference [8].

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According to Proposition 13 in [7], we have the following conclusion. Theorem 3.1. Let u e, ve ∈ L(E n ). If li [e u]r ≤ li [e v ]r or li [e u]r ≥ li [e v ]r for any r ∈ [0, 1] and i = 1, 2, · · · , n, then n the g-difference u e g ve exists and u e g ve ∈ L(E ). From now on, throughout this paper, we will assume that the g-difference u e g ve for any fuzzy n-cell numbers u e and ve exists. Theorem 3.2. For any u e, ve, w, e ∈ L(E n ), we have (1) u e g u e=e 0, u e g e 0=u e, e 0 g u e = −e u, (2) u e g ve = −(e v g u e), (3) k(e u g ve) = ke u g ke v , for any k ∈ R, (4) k1 u e g k2 u e = (k1 − k2 )e u, for any k1 , k2 ∈ R and k1 · k2 ≥ 0, (5) u e g (−e v ) = ve g (−e u), (−e u) g ve = (−e v ) g u e, (6) (e u + ve) g ve = u e, (7) e 0 g (e u g ve) = ve g u e = (−e u) g (−e v ), (8) u e g ve = ve g u e=w e if and only if w e = −w. e Proof. The proof of (1), (3) are immediate. (2) According to Definition 3.1, for any r ∈ [0, 1], we have −[e v g u e]r Qn + + − − + + = − i=1 [inf β≥r min{vi− (β) − u− i (β), vi (β) − ui (β)}, supβ≥r max{vi (β) − ui (β), vi (β) − ui (β)}] Qn − − + + = i=1 [− supβ≥r max{vi (β) − ui (β), vi (β) − ui (β)}, + + − inf β≥r min{vi− (β) − u− i (β), vi (β) − ui (β)}]

=

Qn

− i=1 [− supβ≥r (− min{ui (β)

+ − vi− (β), u+ i (β) − vi (β)}),

− + + − inf β≥r (− max{u− i (β) − vi (β), ui (β) − vi (β)})]

=

Qn

=

[e u g ve]r .

i=1 [inf β≥r

− + + − − + + min{u− i (β) − vi (β), ui (β) − vi (β)}, supβ≥r max{ui (β) − vi (β), ui (β) − vi (β)}]

It follows from Theorem 2.2 that u e g ve = −(e v g u e). (4) For any r ∈ [0, 1], it follows from Definition 3.1 that [k1 u e g k2 u e]r Qn − + − + = i=1 [inf β≥r min{(k1 − k2 )ui (β), (k1 − k2 )ui (β)}, supβ≥r max{(k1 − k2 )ui (β), (k1 − k2 )ui (β)}]. If k1 − k2 ≥ 0, for any r ∈ [0, 1], it is obvious that [k1 u e g k2 u e]r =

n Y

+ [(k1 − k2 )u− u]r . i (r), (k1 − k2 )ui (r)] = [(k1 − k2 )e

i=1

On the other hand, if k1 − k2 < 0, for any r ∈ [0, 1], we have from Theorem 2.2 that [k1 u e g k2 u e]r Qn − + − + = i=1 [inf β≥r min{(k1 − k2 )ui (β), (k1 − k2 )ui (β)}, supβ≥r max{(k1 − k2 )ui (β), (k1 − k2 )ui (β)}] Qn − + − + = i=1 [(k1 − k2 ) supβ≥r max{ui (β), ui (β)}, (k1 − k2 ) inf β≥r min{ui (β), ui (β)}] Qn + − = i=1 [(k1 − k2 )ui (r), (k1 − k2 )ui (r)] =

[(k1 − k2 )e u]r .

Then k1 u e g k2 u e = (k1 − k2 )e u.

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(5) According to Definition 3.1 and Theorem 2.2, for any r ∈ [0, 1], we have [e u g (−e v )]r Qn − + + − − + + − = i=1 [inf β≥r min{ui (β) + vi (β), ui (β) + vi (β)}, supβ≥r max{ui (β) + vi (β), ui (β) + vi (β)}] Qn − + + − − + + − = i=1 [inf β≥r min{vi (β) + ui (β), vi (β) + ui (β)}, supβ≥r max{vi (β) + ui (β), vi (β) + ui (β)}] =

[e v g (−e u)]r .

Then u e g (−e v ) = ve g (−e u). It follows from (3) that (−e u) g ve = (−e v ) g u e. (6) For any r ∈ [0, 1], we have from Theorem 2.2 that Qn − − − + + + [(e u + ve) g ve]r = i=1 [inf β≥r min{(ui (β) + vi (β)) − vi (β), (ui (β) + vi (β)) − vi (β)}, − − + + + supβ≥r max{(u− i (β) + vi (β)) − vi (β), (ui (β) + vi (β)) − vi (β)}]

=

Qn

+ − + min{u− i (β), ui (β)}, supβ≥r max{ui (β), ui (β)}]

=

Qn

+ u− i (β), supβ≥r ui (β)]

=

Qn

=

[e u]r .

i=1 [inf β≥r

i=1 [inf β≥r − i=1 [ui (r),

u+ i (r)]

Then (e u + ve) g ve = u e. (7) It follows from (1), (2) and (3) that the proof of (7) is immediate. (8) We have from (2) that the proof of (8) is immediate. For any u e, ve ∈ L(E n ), using the method with that Bede proved Proposition 15 in [8], we can show that D(e u, ve) = D(e u g ve, e 0). 4. The differentiability and convexity for fuzzy n-cell mappings In this work, let M be a convex set of m-dimensional Euclidean space Rm . We consider mapping Fe from M into L(E n ), such a mapping is called a fuzzy n-cell mapping. For the sake of brevity, Fe is called a fuzzy Qn mapping. For any r ∈ [0, 1], we denote [Fe(t)]r by Fr (t) = i=1 [Fi− (r, t), Fi+ (r, t)]. Let Fe : M → L(E n ) be a fuzzy mapping and u e ∈ L(E n ). For t0 ∈ intM, we write limt→t0 Fe(t) = u e, if, e for every ε > 0, there exists a δ > 0 such that, for 0 < kt − t0 k < δ, we have D(F (t), u e) < ε. We say that Fe is continuous at t0 ∈ intM if limt→t0 Fe(t) = Fe(t0 ). Theorem 4.1. Let Fe : M → L(E n ) be a fuzzy mapping such that Fe(t) = f (t) · u e, where f (t) : M → R be n a real-valued function on M, u e ∈ L(E ) and u e 6= 0. If f is continuous at t0 , then Fe is continuous at t0 and lim Fe(t) = u e · lim f (t).

t→t0

t→t0

Proof. Assume that f is continuous at t0 . Then for every ε > 0, there exists a δ > 0 such that, for 0 < kt − t0 k < δ, we have |f (t) − f (t0 )| < D(euε ,e0) . According to the sign-preserving theorem of limit, we have D(Fe(t), Fe(t0 ))

= D(f (t) · u e, f (t0 ) · u e) =

+ supr∈[0,1] max1≤i≤n {|f (t) − f (t0 )| · |u− i (r)|, |f (t) − f (t0 )| · |ui (r)|}

+ = |f (t) − f (t0 )| supr∈[0,1] max1≤i≤n {|u− i (r)|, |ui (r)|}

= |f (t) − f (t0 )| · D(e u, e 0)
0 is a constant, T is a Volterra integral operator, 0 < t1 < t2 < · · · < tm < a, Ik ∈ C(E, E), k = 1, 2, · · · , m and x0 ∈ E. Under wide monotone conditions and the noncompactness measure condition of nonlinearity f , we obtain the existence of extremal mild solutions and unique mild solution between lower and upper solutions. The results obtained generalize the recent conclusions on this topic. An example is also given to illustrate that our results are valuable. Key Words: Impulsive fractional order integro-differential evolution equation; lower and upper solution; equicontinuous semigroup; measure of noncompactness; monotone iterative technique MR(2010) Subject Classification: 34A37; 34G20; 35R11; 45J05 ∗

Research supported by NNSF of China (11661071, 11501455), Key Project of Gansu Provincial National Science Foundation (1606RJZA015), The Science Research Project for Colleges and Universities of Gansu Province (2015A-213) and Project of NWNU-LKQN-14-6. † Corresponding author. E-mail address: [email protected].

1

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1

Introduction

Fractional order models are found to be more adequate than integer order models in some real world problems. Fractional order derivatives describe the property of memory and heredity of materials, and this is the major advantage of fractional order derivatives compared with integer order derivatives. In recent years, fractional order differential calculus has attracted many physicists, mathematicians and engineers, and notable contributions have been made to both theory and applications of fractional differential equations. It has been found that the differential equations involving fractional order derivatives in time are more realistic to describe many phenomena in practical cases than those of integer order in time. For instance, fractional calculus concepts have been used in the modelling of neurons [1], viscoelastic materials [2]. Other examples from fractional order differential equations can be found in [3-7] and the references therein. One of the branches of fractional differential equations and dynamics is the theory of time fractional order evolution equations. Since time fractional order semilinear evolution equations are abstract formulations for many problems arising in engineering and physics, time fractional evolution equations have attracted increasing attention in recent years, see [8-16] and the references therein. In this article, we use a monotone iterative technique based on the presence of lower and upper solutions to discuss the existence of mild solutions for the initial value problem (IVP) of impulsive time fractional order partial differential equation of Volterra type in an ordered Banach space E   D0 q u(t) + Au(t) = f (t, u(t), T u(t)), t ∈ J, t 6= tk ,    (1.1) ∆u|t=tk = Ik (u(tk )), k = 1, 2, · · · , m,     u(0) = x0 , where D0 q is the Caputo fractional derivative of order q, 0 < q < 1, A : D(A) ⊂ E → E is a closed linear operator, −A generates a C0 -semigroup S(t)(t ≥ 0) in E, J = [0, a], a > 0 is a constant, f ∈ C(J × E × E, E), x0 ∈ E, 0 < t1 < t2 < · · · < tm < a, Ik ∈ C(E, E), k = 1, 2, · · · , m, and Z t

T u (t) :=

K(t, s)u(s)ds

(1.2)

0

is a Volterra integral operator with integral kernel K ∈ C(4, R+ ), 4 = {(t, s) ∈ R2 | 0 ≤ s ≤ − + t ≤ a}, ∆u|t=tk stands the jump of u(t) at t = tk , i.e., ∆u|t=tk = u(t+ k ) − u(tk ), where u(tk ) and u(t− k ) represent the right and left limits of u(t) at t = tk , respectively. The monotone iterative technique based on lower and upper solutions is an effective and 2

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flexible method, which yields monotone sequences of lower and upper approximate solutions that converge to the minimal and maximal solutions between the lower and upper solutions. In 1982, Du and Lakshmikantham [17] established a monotone iterative method for an initial value problem of ordinary differential equation in an ordered banach space. Later, Li [18,19], Chen and Li [20,21] developed the monotone iterative method for the abstract evolution equations in abstract space. The theory of impulsive differential equations is a new and important branch of differential equation theory, which has an extensive physical background and realistic mathematical model, and hence has been emerging as an important area of investigation in recent years, see [22]. Correspondingly, the existence of solutions to impulsive integro-differential equations in Banach spaces has also been studied by several authors, see for example [23,24] and the references therein. But all of the results mentioned above are for the differential equations of integer order. To the best of the author’s knowledge, no results yet exist for the initial value problem of the impulsive time fractional order integro-differential evolution equation (1.1) by using the monotone iterative technique. The purpose of this paper is to establish the monotone iterative method for IVP (1.1) in an ordered Banach space E. Under the positivity assumption for the C0 semigroup S(t) and some monotone conditions combined with the noncompactness measure condition of nonlinearity f , we obtain the results on the existence and uniqueness of mild solutions for IVP(1.1).

2

Preliminaries

Let (E, k · k) be an ordered Banach space with the partial order “ ≤ ”. Then the positive cone P = {x ∈ E | x ≥ θ} is normal with normal constant N . Denote C(J, E) the Banach space of all E-value continuous functions with the supremum norm kukC = sup ku(t)k. Clearly, t∈J

C(J, E) is also an ordered Banach space, which partial order “ ≤ ” is reduced by the positive cone PC = {u ∈ C(J, E) | u(t) ≥ θ, t ∈ J}. Set P C(J, E) = { u : J → E | u(t) is continuous at t 6= tk , left continuous at t = tk , and u(t+ k ) exists, k = 1, 2, · · · , m }. Evidently, P C(J, E) is also an ordered Banach space with the supremum norm kukP C = supt∈J ku(t)k, its partial order “ ≤ ” is reduced by the positive function cone PP C = {u ∈ P C(J, E) | u(t) ≥ θ, t ∈ J}. PP C is also a normal with the same normal constant N . For v, w ∈ P C(J, E) with v ≤ w, we use [v, w] to denote the order interval {u ∈ P C(J, E) | v ≤ u ≤ w}, and for every t ∈ J, we use [v(t), w(t)] to represent the order interval {x ∈ E | v(t) ≤ 0 x ≤ w(t)} in E. Let J := J\{t1 , t2 , · · · , tm }. We denote by P C 1 (J, E) = {u ∈ P C(J, E) ∩ 0 0 0 − C 1 (J , E) | u (t+ k ) and u (tk ) exist}. Set L(J, E) be the RBanach space of all E-valued Bochner t integrable functions defined on J with the norm kuk1 = 0 ku(t)kdt. 3

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Let 0 < q < 1. The Caputo fractional order derivative of order q with the lower limit 0 for a function g ∈ C 1 (J) is defined as D0 q f (t) =

1 Γ(1 − q)

t

Z 0

g 0 (s) ds, (t − s)q

t > 0,

(2.1)

where Γ(·) is the Gamma function. See [7]. If g is an abstract function with values in E, the definition of its Caputo fractional order derivative is same. In that case then the integrals appeared in (2.1) is taken in Bochner’s sense.

Let A : D(A) ⊂ E → E be a closed linear operator and E1 denote the Banach space D(A) with the graphic norm kxk1 = kxk + kAxk. We assume that −A generates a C0 -semigroup S(t) (t ≥ 0) of linear bounded operators in E. Denote by L (E) the Banach space of all linear bounded operators in E. By the exponential boundedness of C0 -semigroup, there exist M > 0 and ω ∈ R such that kS(t)kL (E) ≤ M eωt , t ≥ 0. (2.2) If ω = 0, we call S(t) a uniformly bounded semigroup. Let h ∈ L(J, E) and consider the initial value problem of the linear time fractional order evolution equation (LIVP)  D0 q u(t) + Au(t) = h(t), t ∈ J, (2.3) u(0) = x . 0 By [11], we have the following existence and uniqueness result Lemma 2.1 Assume that the C0 -semigroup S(t) (t ≥ 0) generated by −A is a uniformly bounded and analytic semigroup. If h ∈ C(J, E) is uniformly H¨ older continuous on J, then the linear initial value problem (2.3) has a unique solution expressed by Z u(t) = U (t)x0 +

t

(t − s)q−1 V (t − s)h(s)ds

(2.4)

0

where U (t), V (t) : [0, ∞) → L (E) are strongly continuous functions of linear bounded operator value given by Z ∞

ζq (ϑ)S(tq ϑ)xdϑ,

U (t)x =

x ∈ E, t ≥ 0,

0

Z V (t)x = q

(2.5)

∞

ϑζq (ϑ)S(tq ϑ)xdϑ, x ∈ E, t ≥ 0,

0

where ζq (ϑ) =

1 −1−(1/q) ϑ ρq (ϑ−1/q ) q

(2.6)

4

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is a probability density function on (0, +∞), in which ∞ 1X Γ(nq + 1) ρq (ϑ) = (−1)n−1 ϑ−qn−1 sin(nπq), π n!

ϑ ∈ (0, +∞).

n=0

By [11], the probability density function ζq (ϑ) given by (2.6) satisfies the following condition Z ∞ Z ∞ 1 . (2.7) ζq (ϑ)dϑ = 1, ϑζq (ϑ)dϑ = Γ(1 + q) 0 0 When S(t)(t ≥ 0) is a uniformly bounded C0 -semigroup, we can also define two operator value functions U (t) and V (t) by (2.5). From [11, 12], we have the following result Lemma 2.2 Assume that S(t)(t ≥ 0) is a uniformly bounded C0 -semigroup. Then operators U (t) and V (t) defined by (2.5) have the following properties: (1). For evrey t ≥ 0, U (t) and V (t) are linear bounded operators, and kU (t)xk ≤ M kxk,

kV (t)xk ≤

M kxk, Γ(q)

x ∈ E.

(2.8)

(2). U (t) and V (t) are strongly continuous on [0, +∞). (3). When S(t)(t ≥ 0) is equicontinuous, U (t) and V (t) are continuous in [0, +∞) by the operator norm. In this case, by means of Lemma 2.2 (2), the function u gived by (2.5) belongs to C(J, E), we call it a mild solution of the linear fractional order evolution equation (2.3). That is: Definition 2.1 Let S(t)(t ≥ 0) be a uniformly bounded C0 -semigroup and h ∈ L(J, E). By the mild solution of the LIVP (2.3), we mean that the function u ∈ C(J, E) satisfying the integral equation Z t

(t − s)q−1 V (t − s)h(s)ds.

u(t) = U (t)x0 + 0

Let h ∈ P C(J, E), yk ∈ E, k = 1, 2, · · · , m. We consider the initial value problem of the linear impulsive time fractional order evolution equation (LIVP)   D0 q u(t) + Au(t) = h(t), t ∈ J, t 6= tk ,    (2.9) ∆u|t=tk = yk , k = 1, 2, · · · , m,     u(0) = x0 ∈ E. Let J1 = [0, t1 ], Jk = (tk−1 , tk ], k = 2, 3, · · · , m + 1, where tm+1 = a. Using Definition 2.1, from J1 to Jm+1 interval by interval, we can easily obtain the following result. 5

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Lemma 2.3 For every h ∈ P C(J, E) and yk ∈ E, k = 1, 2, · · · , m, the LIVP (2.9) has a unique mild solution u ∈ P C(J, E) given by  Z t   U (t)x + (t − s)α−1 V (t)(t − s)h(s)ds, t ∈ J1 ,  0   0   Z t     U (t − t1 )(u(t1 ) + y1 ) + (t − s)q−1 V (t − s)h(s)ds, t ∈ J2 , t1 u(t) = (2.10)    ··· ··· ···     Z t     U (t − tm )(u(tm ) + ym ) + (t − s)q−1 V (t − s)h(s)ds, t ∈ Jm+1 . tm

Remark 2.1 Note that the operator value functions U (t) and V (t) do not possess the properties of semigroup. The mild solution of the LIVP (2.9) can be expressed only by using piecewise function. We consider the nonlinear impulsive time fractional order evolution equation (1.1). By Lemma 2.3, a function u ∈ P C(J, E) is called a mild solution of IVP (1.1) if u satisfies the piecewise integral equation  Z t   U (t)x0 + (t − s)α−1 V (t)(t − s)f (s, u(s), T u(s))ds, t ∈ J1 ,    0   Z t     U (t − t1 )(u(t1 ) + I1 (u(t1 ))) + (t − s)q−1 V (t − s)f (s, u(s), T u(s))ds, t ∈ J2 , t 1 u(t) =    ··· ··· ···     Z t     U (t − tm )(u(tm ) + Im (u(tm ))) + (t − s)q−1 V (t − s)f (s, u(s), T u(s))ds, t ∈ Jm+1 . tm

We will use the monotone iterative method based on lower and upper solutions to discuss the existence of extremal mild solutions for IVP (1.1). Next, we introduce the concepts of lower and upper solutions for IVP (1.1). Definition 2.2 If a function v0 ∈ P C 1 (J, E) ∩ P C(J, E1 ) and satisfies inequalities   D0 q v0 (t) + Av0 (t) ≤ f (t, v0 (t), T v0 (t)), t ∈ J, t 6= tk ,    ∆v0 |t=tk ≤ Ik (v0 (tk )), k = 1, 2, · · · , m,     v0 (0) ≤ x0 ,

(2.11)

we called it a lower solution of IVP (1.1). If all the inequalities of (2.11) are inverse, we call it an upper solution of IVP (1.1). 6

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Our discussion needs that the S(t)(t ≥ 0) is a positive C0 -semigroup, that is, S(t)x ≥ θ for any x ≥ θ and t ≥ 0. For more details of the properties of the positive C0 -semigroup, see [19,25]. Clearly, by the (2.5) we obtain that: Lemma 2.4 If S(t)(t ≥ 0) is a uniformly bounded positive C0 -semigroup in E, then U (t) and V (t) are positive operators in E for every t ∈ [0, +∞). Let C ≥ 0 is a constant, I denote the identity operator in E. It is easy to see that −(A + CI) generates a C0 -semigroup S1 (t) = e−Ct S(t) (t ≥ 0) in E, and S1 (t) is a positive C0 semigroup if S(t) is a positive C0 -semigroup. If C ≥ ω, then by (2.2), S1 (t) is a a uniformly bounded C0 -semigroup, for more details please see [26]. Hence we can define the corresponding operator value functions U1 (t) and V1 (t) as follows ∞

Z

ζq (ϑ)S1 (tq ϑ)xdϑ,

U1 (t)x = 0 Z

x ∈ E, t ≥ 0, (2.12)

∞ q

ϑζq (ϑ)S1 (t ϑ)xdϑ, x ∈ E, t ≥ 0.

V1 (t)x = q 0

U1 (t) and V1 (t) have completely same properties with U (t) and V (t). If the semigroup S(t) is not uniformly bounded, we choose C ≥ ω such that S1 (t) uniformly bounded. In this case, the mild solution of IVP (1.1) can be expressed by U1 (t) and V1 (t). Next, we recall some properties about the measure of noncompactness that will be used in the proof of our main results. Let α(·) denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definitions and properties of the measure of noncompactness, see [27]. For ∀ B ⊂ P C(J, E) and t ∈ J, set B(t) = {u(t) : u ∈ B} ⊂ E. If B is bounded in P C(J, E), then B(t) is bounded in E and α(B(t)) ≤ α(B). Lemma 2.5 on J, and

[27]

Let B ⊂ C(J, E) be bounded and equicontinuous. Then α(B(t)) is continuous α

n Z

u(t) | u ∈ B

o

Z ≤

α(B(t))dt.

J

J

Lemma 2.6 [28] Assume that B = {un } ⊂ P C(J, E) is a countable set and there exists a function m ∈ L1 (J, R+ ) such that for every n ∈ N kun (t)k ≤ m(t),

a.e. t ∈ J.

Then α(B(t)) is Lebesgue integral on J, and Z o n Z un (t)dt ≤ 2 α(B(t))dt. α J

J

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Our discussion also need the following generalized Gronwall inequality which can be fund in [29]. Lemma 2.7 Let C0 ≥ 0 be a constant and a ∈ L(J) be a nonnegative function. If ϕ ∈ L(J) is nonnegative and satisfies Z t (t − s)q−1 ϕ(s)ds, t ∈ J, ϕ ≤ a(t) + C0 0

then ϕ(t) ≤ a(t) +

Z thX ∞ i (C0 Γ(q))n (t − s)nq−1 a(s) ds, Γ(nq) 0

t ∈ J.

n=1

3

Main Results

In this section, we use the monotone iterative method based on lower and upper solutions to discuss the existence of mild solution for IVP (1.1). We assume that the operator A : D(A) ⊂ E → E satisfies (H0) A : D(A) ⊂ E → E be a closed linear operator, −A generates a positive and equicontinuous C0 -semigroup S(t) (t ≥ 0). Our main results as follows: Theorem 3.1 Let E be an ordered Banach space, and let the positive cone P be normal. Assume that A : D(A) ⊂ E → E satisfies the assumption (H0), f ∈ C(J × E × E, E), Ik ∈ C(E, E), k = 1, 2, · · · , m, and IVP (1.1) has a lower solution v0 and an upper solution w0 with v0 ≤ w0 . If the following conditions are satisfied: (H1) There exists a constant C > 0 such that f (t, x2 , y2 ) − f (t, x1 , y1 ) ≥ −C(x2 − x1 ), for ∀ t ∈ J, v0 (t) ≤ x1 ≤ x2 ≤ w0 (t), and T v0 (t) ≤ y1 ≤ y2 ≤ T w0 (t). (H2) Ik (x) is increasing on the order interval [v0 (t), w0 (t)] for t ∈ J, k = 1, 2, · · · , m. (H3) There exists a constant L > 0 such that α({f (t, xn , yn )}) ≤ L(α({xn }) + α({yn })), for ∀ t ∈ J, and increasing or decreasing monotonic sequences {xn } ⊂ [v0 (t), w0 (t)] and {yn } ⊂ [T v0 (t), T w0 (t)], 8

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then IVP (1.1) has minimal and maximal mild solutions between v0 and w0 , and they can be obtained by a monotone iterative procedure starting from v0 and w0 , respectively. Proof. Without losing the generality, in the assumption (H1) we may assume that C ≥ ω, where ω is the growth exponent of S(t) given by (2.2). Then the C0 -semigroup S1 (t) = e−Ct S(t),

t≥0

(3.1)

generated by −(A + CI) is uniformly bounded, positive and equicontinuous. Let U1 (t)(t ≥ 0) and V1 (t)(t ≥ 0) be the operator value function defined by (2.12), then they have the properties in Lemma 2.2, specially they satisfy kU1 (t) xk ≤ M kxk,

kV1 (t) xk ≤

M kxk, Γ(q)

t ≥ 0, x ∈ E.

(3.2)

For every u ∈ P C(J, E), set G(u)(t) = f (t, u(t), T u(t)) + Cu(t),

t ∈ J.

(3.3)

Then G : P C(J, E) → P C(J, E) is a continuous mapping. We rewrite the equation (1.1) to the form of   D0 q u(t) + (A + CI) u(t) = G(u) (t), t ∈ J, t 6= tk ,    (3.4) ∆u|t=tk = Ik (u(tk )), k = 1, 2, · · · , m,     u(0) = x0 , then by Lemma 2.3, the mild solution of this equation, equivalently IVP (1.1), which means that u ∈ P C(J, E) satisfies the piecewise integral equation  Z t    U1 (t)x0 + (t − s)α−1 V1 (t)(t − s) G(u)(s) ds, t ∈ J1 ,    0   Z t     U1 (t − t1 )(u(t1 ) + I1 (u(t1 ))) + (t − s)q−1 V1 (t − s) G(u)(s) ds, t ∈ J2 , t u(t) = 1    ··· ··· ···     Z t      (t − s)q−1 V1 (t − s)G(u)(s)ds, t ∈ Jm+1 . U1 (t − tm )(u(tm ) + Im (u(tm ))) + tm

We define the mapping Q : [v0 , w0 ] → P C(J, E) by

9

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Qu(t) =

 Z t    (t − s)α−1 V1 (t)(t − s) G(u)(s) ds, t ∈ J1 , U1 (t)x0 +    0   Z t     U1 (t − t1 )(u(t1 ) + I1 (u(t1 ))) + (t − s)q−1 V1 (t − s) G(u)(s) ds, t ∈ J2 , t1

   ··· ··· ···     Z t      U (t − t )(u(t ) + I (u(t ))) + (t − s)q−1 V1 (t − s)G(u)(s)ds, t ∈ Jm+1 , m m m m  1 tm

(3.5) then the mild solution of IVP (1.1) is equivalent to the fixed point of Q. Clearly, Q : [v0 , w0 ] → P C(J, E) is continuous. Since operators U1 (t) and V1 (t) are positive, by the assumptions (H1) and (H2), Q is increasing in [v0 , w0 ]. We use monotone iterative method of increasing operator to find the fixed point of Q. Firstly, we show that v0 ≤ Qv0 and Qw0 ≤ w0 . Let h(t) = D0 q v0 (t)+Av0 (t)+Cv0 (t). Then h ∈ P C(J, E) and v0 is the unique mild solution of the the linear impulsive time fractional evolution equation (LIVP)   D0 q u(t) + (A + CI) u(t) = h(t), t ∈ J, t 6= tk ,    (3.6) ∆u|t=tk = ∆v0 |t=tk , k = 1, 2, · · · , m,     u(0) = v0 (0) ∈ E. By the definition of lower solution v0 (0) ≤ x0 ,

h(t) ≤ G(v0 )(t),

∆v0 |t=tk ≤ Ik (v0 (tk )),

t ∈ J 0,

k = 1, 2, · · · , m.

Hence by Lemma 2.3 and the positivity of operators U1 (t) and V1 (t), we have  Z t    U1 (t)v0 (0) + (t − s)α−1 V1 (t)(t − s)h(s)ds, t ∈ J1 ,    0   Z t     U1 (t − t1 )(v0 (t1 ) + ∆v0 |t=t1 ) + (t − s)q−1 V1 (t − s)h(s)ds, t ∈ J2 , t1 v0 (t) =    ··· ··· ···     Z t      U (t − t )(v (t ) + ∆v | ) + (t − s)q−1 V1 (t − s)h(s)ds, t ∈ Jm+1 m 0 m 0 t=tm  1 tm

10

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≤

 Z t    (t − s)α−1 V1 (t)(t − s)G(v0 )(s) ds, t ∈ J1 , U (t)v (0) + 1 0    0   Z t     U1 (t − t1 )(v0 (t1 ) + I1 (v0 (t1 ))) + (t − s)q−1 V1 (t − s)G(v0 )(s) ds, t ∈ J2 , t1

   ··· ··· ···     Z t      (t − s)q−1 V1 (t − s)G(v0 )(s) ds, t ∈ Jm+1 U1 (t − tm )(v0 (tm ) + Im (v0 (tm ))) + tm

= Q v0 (t). This means that v0 ≤ Qv0 . Using a similar method, we can prove that Qw0 ≤ w0 . Combining these facts and the increasing property of Q in [v0 , w0 ], we see that Q maps [v0 , w0 ] into itself. Hence, Q : [v0 , w0 ] → [v0 , w0 ] is a continuously increasing operator. Secondly, we prove that the image set Q([v0 , w0 ]) is equicontinuous in every interval Jk , k = 1, 2,· · · , m + 1. For ∀ u ∈ [v0 , w0 ], by the assumptions (H1) and (H2), we have G(v0 ) ≤ G(u)(t) ≤ G(w0 ),

t ∈ J,

and v0 (tk ) + Ik (v0 (tk )) ≤ u(tk ) + Ik (u(tk )) ≤ w0 (tk ) + Ik (w0 (tk )), k = 1, 2, · · · , m. Hence by the normality of the cone P , there exist positive constant M ∗ and Lk , k = 1, 2, · · · , m, such that kG(u)(t)k ≤ M ∗ , t ∈ J, (3.7) ku(tk ) + Ik (u(tk ))k ≤ Lk , k = 1, 2, · · · , m. 0

00

0

00

00

0

Consider the case of J1 . Let t , t ∈ J1 and 0 < t < t . We show that kQu(t )−Qu(t )k → 0 00 0 independently of u as t − t → 0. By the definition of Q, we have 00

0

00

0

Qu (t ) − Qu (t ) = U1 (t )x0 − U1 (t )x0 00

t

Z +

00

00

00

0

(t − s)q−1 V1 (t − s)G(u)(s)ds

0

t

0

Z

t

00

[(t − s)q−1 − (t − s)q−1 ]V1 (t − s)G(u)(s)ds

+ 0 0

Z +

t

0

00

0

(t − s)q−1 [V1 (t − s) − V1 (t − s)]G(u)(s)ds

0

11

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= S11 + S12 + S13 + S14 , where 00

0

S11 = U1 (t )x0 − U1 (t )x0 , 00

t

Z S12 =

00

00

00

0

(t − s)q−1 V1 (t − s)G(u)(s)ds,

t0 0

t

Z

00

[(t − s)q−1 − (t − s)q−1 ]V1 (t − s)G(u)(s)ds,

S13 = 0 0

t

Z

0

00

0

(t − s)q−1 [V1 (t − s) − V1 (t − s)]G(u)(s)ds.

S14 = 0

Since 00

0

kQu (t ) − Qu (t )k ≤ kS11 k + kS12 k + kS13 k + kS14 k, 00

0

we only need to check that kS1i k → 0 independently of u ∈ [v0 , w0 ] as t − t → 0, i = 1, 2, 3, 4. For S11 , by Lemma 2.2 (2), U1 (t)x0 is continuous on J, hence it is uniformly continuous on J and we have 00 0 00 0 kS11 k = kU1 (t ) x0 − U1 (t ) x0 k → 0 (t − t → 0). For S12 and S13 , by (3.2) and (3.7) we have 00

t

Z kS12 k ≤

00

00

(t − s)q−1 kV1 (t − s)G(u)(s)kds

t0

M M ∗ 00 0 (t − t )q → 0 q Γ(q)

≤

00

0

(t − t → 0).

0

Z kS13 k ≤

t

0

00

00

[(t − s)q−1 − (t − s)q−1 ] kV1 (t − s)G(u)(s)kds

0

≤

MM∗ Γ(q)

0

Z

t

0

00

[(t − s)q−1 − (t − s)q−1 ]ds

0

=

MM∗ 0 q 00 00 0 [(t ) − (t )q + (t − t )q ] q Γ(q)

≤

M M ∗ 00 0 (t − t )q → 0 q Γ(q)

00

0

(t − t → 0).

For S14 , using (3.2), (3.7), Lemma 2.2(3) and the Lebesgue bounded convergence theorem of integration, we have Z t0 0 00 0 kS14 k ≤ (t − s)q−1 kV1 (t − s) − V1 (t − s)k · kG(u)(s)k ds. 0

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0

≤ M

∗

t

Z

0

00

0

(t − s)q−1 kV1 (t − s) − V1 (t − s)k ds

0 0

= M

∗

t

Z

00

0

00

0

rq−1 kV1 (t − t + r) − V1 (r)k dr

0

≤ M∗

a

Z

00

rq−1 kV1 (t − t + r) − V1 (r)k dr → 0

0

(t − t → 0).

0 00

0

00

0

As a result, kQ(u)(t ) − Q(u)(t )k tends to 0 independently of u ∈ [v0 , w0 ] as t − t → 0, which means that Q([v0 , w0 ]) is equicontinuous in the interval J1 . 0

00

0

00

Consider the case of J2 . For t , t ∈ J2 with t < t , we have 00

0

00

0

Qu (t ) − Qu (t ) = ( U1 (t − t1 ) − U1 (t − t1 ) ) (u(t1 ) + I1 (u(t1 )) ) 00

t

Z +

00

00

00

0

(t − s)q−1 V1 (t − s)G(u)(s) ds

0

t

0

Z

t

00

[(t − s)q−1 − (t − s)q−1 ]V1 (t − s)G(u)(s) ds

+ t1 0

Z

t

0

00

0

(t − s)q−1 [V1 (t − s) − V1 (t − s)]G(u)(s) ds

+ t1

= S21 + S22 + S23 + S24 , where 00

0

00

00

00

0

S21 = ( U1 (t − t1 ) − U1 (t − t1 ) ) (u(t1 ) + I1 (u(t1 )) ), 00

t

Z S22 =

(t − s)q−1 V1 (t − s)G(u)(s) ds,

0

t

0

Z

t

00

[(t − s)q−1 − (t − s)q−1 ]V1 (t − s)G(u)(s) ds,

S23 = t1 0

Z

t

S24 =

0

00

0

(t − s)q−1 [V1 (t − s) − V1 (t − s)]G(u)(s) ds.

t1

It is obvious that 00

0

kQu (t ) − Qu (t )k ≤ kS21 k + kS22 k + kS23 k + kS24 k. 00

0

Therefore, we only need to check that kS2i k → 0 independently of u ∈ [v0 , w0 ] as t − t → 0, i = 1, 2, 3, 4. For S21 , by Lemma 2.2 (3) and (3.7), we have that 00

0

kS21 k = k( U1 (t − t1 ) − U1 (t − t1 ) ) (u(t1 ) + I1 (u(t1 )) )k 13

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00

0

≤ kU1 (t − t1 ) − U1 (t − t1 )k · ku(t1 ) + I1 (u(t1 ))k 00

0

≤ L1 kU1 (t − t1 ) − U1 (t − t1 )k → 0

00

0

(t − t → 0).

For S22 , similarly to S12 , we have M M ∗ 00 0 (t − t )q → 0 q Γ(q)

kS22 k ≤

00

0

(t − t → 0).

For S23 , by (3.2) and (3.7) we have 0

t

Z

0

00

00

[(t − s)q−1 − (t − s)q−1 ] kV1 (t − s)G(u)(s)kds

kS23 k ≤ t1

MM∗ Γ(q)

≤

MM∗

=

q Γ(q)

0

Z

t

0

00

[(t − s)q−1 − (t − s)q−1 ]ds

t1 0

00

M M ∗ 00 0 (t − t )q → 0 q Γ(q)

≤

00

0

[(t − t1 )q − (t − t1 )q + (t − t )q ] 00

0

(t − t → 0).

For S24 , by (3.7) and lemma 2.2(3), we have 0

Z

t

0

00

0

(t − s)q−1 kV1 (t − s) − V1 (t − s)k · kG(u)(s)k ds.

kS24 k ≤ t1

0

≤ M

∗

Z

t

0

00

0

(t − s)q−1 kV1 (t − s) − V1 (t − s)k ds

t1 0

= M

∗

Z

t1 −t

00

0

rq−1 kV1 (t − t + r) − V1 (r)k dr

0

≤ M∗

Z

a

00

0

rq−1 kV1 (t − t + r) − V1 (r)k dr → 0

00

0

(t − t → 0).

0 00

0

00

0

Consequently, kQu (t ) − Qu (t )k tends to 0 independently of u ∈ [v0 , w0 ] as t − t → 0. This means that Q([v0 , w0 ]) is equicontinuous in the interval J2 . Continuing such a process interval by interval up to Jm+1 , we can prove that Q([v0 , w0 ] is equicontinuous in every interval Jk , k = 1, 2, · · · , m + 1. Now, we define two sequences {vn } and {wn } in [v0 , w0 ] by the iterative schemes vn = Q vn−1 ,

wn = Q wn−1 ,

n = 1, 2, · · · .

(3.8)

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Then from the monotonicity of Q, one can easy to prove that v 0 ≤ v 1 ≤ · · · v n ≤ · · · ≤ wn ≤ · · · ≤ w1 ≤ w0 .

(3.9)

We prove that {vn } and {wn } are uniformly convergent in J. For convenience, let B = {vn | n ∈ N} and B0 = {vn−1 | n ∈ N}. Since B = Q(B0 ) ⊂ Q([v0 , w0 ]), so that B is equicontinuous in every interval Jk , k = 2, 3, · · · , m. From B0 = B ∪ {v0 } it is follows that α(B0 (t)) = α(B(t)) for t ∈ J. Denote t ∈ J.

ϕ(t) = α(B(t)) = α(B0 (t)),

(3.10)

By Lemma 2.5, ϕ ∈ P C(J, R+ ). We from J1 to Jm+1 interval by interval show that ϕ(t) ≡ 0 in J. For every t ∈ J, by Lemma 2.6 we get that n Z t o α(T (B0 )(t)) = α K(t, s)vn−1 (s)ds n ∈ N 0

Z

t

≤ 2

α({K(t, s)vn−1 (s) | n ∈ N})ds 0

Z

t

K(t, s) α({vn−1 (s) | n ∈ N})ds

= 2 0

t

Z ≤ 2K0

ϕ(s)ds, 0

where K0 = max(t,s)∈∆ K(t, s). Therefore Z

t q−1

(t − s)

t

Z

q−1

α(T (B0 )(s)) ds ≤ 2K0

(t − s)

0

0

= ≤

2K0 q

Z

2aK0 q

hZ

s

i ϕ(r)dr ds

0 t

(t − r)q ϕ(r)dr,

0

Z

t

(t − s)q−1 ϕ(s)ds,

t ∈ J.

(3.11)

0

For ∀ t ∈ J1 , by (3.5), using Lemma 2.6, the assumption (H3), (3.2) and (3.11), we have ϕ(t) = α(B(t)) = α( Q(B0 )(t) ) = α

n

Z U1 (t)x0 +

t

(t − s)q−1 V1 (t − s)G(vn−1 )(s) ds

o

0

= α

n Z

t

(t − s)q−1 V1 (t − s)G(vn−1 )(s) ds

o

0

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t

Z

α



2M Γ(q)

Z

t

2M Γ(q)

Z

2M Γ(q)

Z

≤ 2

(t − s)q−1 V1 (t − s) G(vn−1 )(s)




ds

0

≤ = ≤

(t − s)q−1 α ({ G(vn−1 )(s) ds }) ds

0 t

(t − s)q−1 α ({ f (s, vn−1 , T vn−1 (s)) + Cvn−1 (s) }) ds

0 t

(t − s)q−1 [L( α(B0 (s)) + α( T (B0 )(s)) ) + Cα(B0 (s)) ] ds

0

Z t Z t 2M 2M q−1 (t − s)q−1 α(T (B0 )(s)) ds (L + C) (t − s) ϕ(s) ds + Γ(q) Γ(q) 0 0 Z 2M  2aK0 L  t (t − s)q−1 ϕ(s)ds. L+C + Γ(q) q 0

= ≤

Hence by Lemma 2.7, ϕ(t) ≡ 0 in J1 . In particular, α(B(t1 )) = α(B0 (t1 )) = ϕ(t1 ) = 0, this means that B(t1 ) and B0 (t1 ) are precompact in E. Hence, from the continuity of I1 we obtain that I1 (B0 (t1 )) is precompact in E, and α(I1 (B0 (t1 ))) = 0. For ∀ t ∈ J2 , since α({U1 (t − t1 )[vn−1 (t1 ) + I1 (vn−1 (t1 ))]}) ≤ α(U1 (t − t1 )(B0 (t1 ) + I1 (B0 (t1 )) ) ≤ M (α(B0 (t1 )) + α(I1 (B0 (t1 ))) ) = 0, using (3.5) and a similar argument above, we have ϕ(t) = α(B(t)) = α( Q(B0 )(t) ) = α

n

Z

t

U1 (t − t1 )[vn−1 (t1 ) + I1 (vn−1 (t1 ))] +

o (t − s)q−1 V1 (t − s)G(vn−1 )(s) ds

t1

≤ α({U1 (t − t1 )[vn−1 (t1 ) + I1 (vn−1 (t1 ))]}) +α

t

n Z

o (t − s)q−1 V1 (t − s)G(vn−1 )(s) ds

t1

= α

t

n Z

(t − s)q−1 V1 (t − s)G(vn−1 )(s) ds

o

0

Z

t

≤ 2

α



Z

t

(t − s)q−1 V1 (t − s) G(vn−1 )(s)




ds

t1

≤

2M Γ(q)

(t − s)q−1 α ({ G(vn−1 )(s) }) ds

t1

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= ≤ = ≤ ≤

2M Γ(q)

Z

2M Γ(q)

Z

t

(t − s)q−1 α ({ f (s, vn−1 , T vn−1 (s)) + Cvn−1 (s) }) ds

t1 t

(t − s)q−1 [ L(α(B0 (s)) + α(T (B0 )(s))) + Cα(B0 (s)) ] ds

t1

2M (L + C) Γ(q)

Z

t q−1

(t − s) t1

2M L ϕ(s) ds + Γ(q)

Z

t

(t − s)q−1 α(T (B0 )(s)) ds

t1

Z t Z 2M 2M L t q−1 (L + C) (t − s) ϕ(s) ds + (t − s)q−1 α(T (B0 )(s)) ds Γ(q) Γ(q) 0 0 Z 2M  2aK0 L  t (t − s)q−1 ϕ(s)ds. L+C + Γ(q) q 0

Again by Lemma 2.7, ϕ(t) ≡ 0 on J2 , from which we obtain that α(B0 (t2 )) = 0, and therefore α(I2 (B0 (t2 ))) = 0. Continuing such a process interval by interval up to Jm+1 , we can prove that ϕ ≡ 0 in every Jk . Therefore, for every Jk , {vn } is equicontinuous on Jk and {vn (t)} is precompact in E for every t ∈ Jk . By the Arzela-Ascoli theorem, {vn } has a subsequence which is uniformly convergent in Jk . Combining this with the monotonicity (3.9), we easily prove that {vn } itself is uniformly convergent in Jk , k = 1, 2, · · · , m + 1. Consequently, {vn (t)} is uniformly convergent over the whole of J. Using a similar argument to that for {vn (t)}, we can prove that {wn (t)} is also uniformly convergent on J. Hence, {vn } and {wn } are convergent in the Banach space P C(J, E). Set u = lim vn n→∞

and u = lim wn n→∞

in P C(J, E).

(3.12)

Letting n → ∞ in (3.8) and (3.9), we see that v0 ≤ u ≤ u ≤ w0 and u = Q u and u = Q u.

(3.13)

By the monotonicity of Q, it is easy to prove that u and u are the minimal and maximal fixed points of Q in [v0 , w0 ], and therefore, they are the minimal and maximal mild solutions of IVP (1.1) in [v0 , w0 ], respectively. 2

This completes the proof of Theorem 3.1.

In Theorem 3.1, if E is weakly sequentially complete, the condition (H3) holds automatically. In fact, when E is an ordered and weakly sequentially complete Banach space, by Theorem 2.2 in paper [30], we know that any monotonic and order-bounded sequence is precompact. Let {xn } and {yn } be two increasing or decreasing sequences in condition (H3), then by condition (H1), { f (t, xn , yn ) + Cxn } is monotonic and order-bounded sequence. By the property of measure of 17

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noncompactness, we have α({ f (t, xn , yn )}) ≤ α({ f (t, xn , yn ) + Cxn }) + C α({xn }) = 0. Hence, condition (H3) holds for any L > 0. From Theorem 3.1, we obtain that Corollary 3.1 Let E be an ordered and weakly sequentially complete Banach space, and let the positive cone P be normal. Assume that A : D(A) ⊂ E → E satisfies the assumption (H0), f ∈ C(J × E × E, E), and Ik ∈ C(E, E), k = 1, 2, · · · , m. If IVP (1.1) has a lower solution v0 and an upper solution w0 with v0 ≤ w0 and the assumptions (H1) and (H2) are satisfied, then IVP (1.1) has minimal and maximal mild solutions between v0 and w0 , which can be obtained by a monotone iterative procedure starting from v0 and w0 , respectively. Now we discuss the uniqueness of the mild solution for IVP (1.1) in [v0 , w0 ]. In theorem 3.1, if replacing the assumption (H3) by the condition: (H4) There exist positive constants C1 and C2 such that f (t, x2 , y2 ) − f (t, x1 , y1 ) ≤ C1 (x2 − x1 ) + C2 (y2 − y1 ), for ∀ t ∈ J, and v0 (t) ≤ x1 ≤ x2 ≤ w0 (t), T v0 (t) ≤ y1 ≤ y2 ≤ T w0 (t), we have the following uniqueness result. Theorem 3.2 Let E be an ordered Banach space, and let the positive cone P be normal. Assume that A : D(A) ⊂ E → E satisfies the assumption (H0), f ∈ C(J × E × E, E), and Ik ∈ C(E, E), k = 1, 2, · · · , m. If IVP(1.1) has a lower solution v0 and an upper solution w0 with v0 ≤ w0 such that conditions (H1), (H2) and (H4) hold, then IVP (1.1) has a unique mild solution between v0 and w0 , which can be obtained by a monotone iterative procedure starting from v0 or w0 . Proof. We firstly prove that (H1) and (H4) can deduce (H3). For t ∈ J, let {xn } ⊂ [v0 (t), w0 (t)] and {yn } ⊂ [T v0 (t), T w0 (t)] be two increasing sequences. For m, n ∈ N with m > n, by (H1) and (H4), θ ≤ ( f (t, xm , ym ) − f (t, xn , yn ) ) + C(xm − xn ) ≤ (C + C1 )(xm − xn ) + C2 (ym − yn ). By this and the normality of cone P , we have kf (t, xm , ym ) − f (t, xn , yn )k 18

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≤ N k(C + C1 )(xm − xn ) + C2 (ym − yn )k + C kxm − xn k ≤ (C + N C + N C1 )kxm − xn k + N C2 kym − yn k. From this and the definition of the measure of noncompactness, it follows that α({ f (t, xn , yn )}) ≤ (C + N C + N C1 ) α({xn }) + N C2 α({yn }) ≤ L (α({xn }) + α({yn })), where L = C + N C + N C1 + N C2 . If {xn } and {yn } are two decreasing sequences, the above inequality is also valid. Hence (H3) holds. Therefore, by Theorem 3.1, IVP (1.1) has a minimal mild solution u and a maximal mild solution u in [v0 , w0 ]. Let {vn } and {wn } be the sequences defined by the iterative scheme (3.8). Then by the proof of Theorem 3.1, we know that (3.9), (3.12) and (3.13) are valid. We show that u(t) ≡ u(t) on J. Set ψ(t) = ku(t) − u(t)k,

t ∈ J,

(3.14)

then ψ ∈ P C(J, R+ ). We need to show that ψ(t) ≡ 0 on J. We from J1 to Jm+1 interval by interval show that ψ(t) ≡ 0 on J. For every t ∈ J1 , using (3.5), (3.8), (3.9) and assumption (H1) and (H4), we obtain that θ ≤ u(t) − u(t) = Qu (t) − Qu (t) Z

t

(t − s)q−1 V1 (t − s) ( G(u)(s) − G(u)(s) ) ds

= 0

Z ≤

t


(t − s)q−1 V1 (t − s) (C + C1 )(u(s) − u(s)) + C2 (T u(s) − T u(s)) ds.

0

Hence, by the the normality of cone P and (3.2), we have ψ(t) = ku(t) − u(t)k

Z t


q−1 ≤ N (t − s) V1 (t − s) (C + C1 )(u(s) − u(s)) + C2 (T u(s) − T u(s)) ds 0

≤ ≤ ≤

Z t Z t i MN h q−1 (C + C1 ) (t − s) ψ(s) ds + C2 (t − s)q−1 kT u(s) − T u(s)k ds Γ(q) 0 0 Z t Z t Z s h i MN q−1 q−1 (C + C1 ) (t − s) ψ(s) ds + C2 K0 (t − s) ψ(r)drds Γ(q) 0 0 0 Z Z i t MN h C2 K0 t (C + C1 ) (t − s)q−1 ψ(s) ds + (t − r)q ψ(r)dr Γ(q) q 0 0 19

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≤ =

Z t Z i MN h aC2 K0 t q−1 (t − s)q−1 ψ(s)ds (C + C1 ) (t − s) ψ(s) ds + Γ(q) q 0 0 Z   t MN aC2 K0 (t − s)q−1 ψ(s) ds. C + C1 + Γ(q) q 0

So we obtain that ψ(t) ≡ 0 on J1 by Lemma 2.7. For t ∈ J2 , noting that u(t1 ) = u(t1 ) and I1 (u(t1 )) = I1 (u(t1 )), using (3.5) and the same argument as above for t ∈ J1 , by (3.5) we can prove that Z NM  aC2 K0  t (t − s)q−1 ψ(s)ds ψ(t) ≤ C + C1 + Γ(q) q t1 Z NM  aC2 K0  t ≤ (t − s)q−1 ψ(s)ds, t ∈ J2 . C + C1 + Γ(q) q 0 Again by Lemma 2.7, we have ψ(t) ≡ 0 on J2 . Continuing such a process interval by interval up to Jm+1 , we obtain that ψ(t) ≡ 0 over the whole of J. Hence, u(t) ≡ u(t) on J and u e := u = u is a unique mild solution of IVP (1.1) in [v0 , w0 ], which can be obtained by the monotone iterative procedure (3.8) starting from v0 or w0 . 2

This completes the proof of Theorem 3.2.

Remark 3.1 Since the condition (H4) can be more easily verified than (H3), the applications of Theorem 3.2 are convenient.

4

Application

In order to illustrate the applicability of our main results, we consider the initial-boundary value problem of time fractional order parabolic partial differential equation with impulses and integral term  q ∂t u − ∇2 u = g(t, x, u(t, x), T u(t, x)), (t, x) ∈ J × Ω, t 6= tk ,       ∆u|t=tk = ck u(tk , x), x ∈ Ω; k = 1, 2, · · · , m, (4.1)   u| = 0, ∂Ω      u(0, x) = ϕ0 (x), x ∈ Ω, where ∂t q is the Caputo fractional order partial derivative of order q, 0 < q < 1, ∇2 is the Laplace operator, J = [0, a], a > 0, 0 < t1 < t2 < · · · < tm < a, Ω ⊂ RN is a bounded domain with a sufficiently smooth boundary ∂Ω, g ∈ C(J × Ω × R × R) and satisfies the growth condition |g(t, x, ξ, η)| ≤ b0 + b1 |ξ| + b2 |η|,

(t, x, ξ, η) ∈ J × Ω × R × R,

(4.2)

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with positive constants b0 , b1 , b2 , and t

Z

K(t, s)u(s, x)ds

T u(t, x) :=

(4.3)

0

is a Volterra integral operator with integral kernel K ∈ C(4, R+ ), 4 := {(t, s) ∈ R2 | 0 ≤ s ≤ t ≤ a}; c1 , c2 , · · · , cm are positive constants, and the initial value function ϕ0 ∈ L2 (Ω). Let E = L2 (Ω), P = {u ∈ L2 (Ω) | u(x) ≥ 0, a. e. x ∈ Ω}. Then E is a Banach space, P is a normal cone of E. We define an operator A in L2 (Ω) by D(A) = H 2 (Ω) ∩ H01 (Ω),

Au = −∇2 u,

(4.4)

From [26, Chapter 7, Theorem 3.2], we know that −A generates a positive and analytic semigroup S(t) (t ≥ 0) in E. Let f (t, v, w) := g(t, x, v(x), w(x)),

t ∈ J,

v, w ∈ E.

(4.5)

Then by the condition (4.2), f : J × E × E → E is continuous. Let Ik = ck I, k = 1, 2, · · · , m, where I is the identity operator in L2 (Ω). For the fuction u : J × Ω → R, let u(t) = u(t, ·). Then the equation (4.1) be transformed into the following abstract form of IVP (1.1) in L2 (Ω)   D0 q u(t) + Au(t) = f (t, u(t), T u(t)), t ∈ J, t 6= tk ,    (4.6) ∆u|t=tk = Ik (u(tk )), k = 1, 2, · · · , m,     u(0) = ϕ0 . Let λ1 be the first eigenvalue of A. It is well known that λ1 > 0 and it has a unique positive eigenfunction φ1 ∈ C 2 (Ω) ∩ C0 (Ω) satisfied maxx∈Ω φ1 (x) = 1. Let µ(t) = 1 +

P

t>tk ck ,

t ∈ J,

(4.7)

then µ ∈ P C(J) and ∆µ|t=tk = ck , k = 1, 2, · · · , m. In order to solve the problem (4.1), we make the following assumptions: (A1) ϕ0 ∈ L2 (Ω) and 0 ≤ ϕ0 (x) ≤ φ1 (x) for a. e. x ∈ Ω. (A2) g(t, x, 0, 0) ≥ 0 and g(t, x, µ(t)φ1 (x), φ1 (x) T µ(t)) ≤ λ1 µ(t) φ1 (x) for every (x, t) ∈ J × Ω. (A3) In J × Ω × R × R, gη 0 (t, x, ξ, η).

g(t, x, ξ, η) has continuous partial derivative gξ 0 (t, x, ξ, η) and

Theorem 4.1 If the assumptions (A1)-(A3) are satisfied, then the equation (4.1) has a unique L2 -mild solution between 0 and µ(t)φ1 (x). 21

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Proof. Consider IVP (4.6). From Definition 2.2 and the assumptions (A1) and (A2) we see that v0 (t) ≡ 0 is a lower solution of IVP (4.6) and w0 (t) = µ(t)φ1 is an upper solution of IVP (4.6). From (A3) it is easy to verify that f satisfies the assumption (H1) and (H4). Clearly, for Ik = ck I, k = 1, 2, · · · , m, (H2) holds. Therefore, by Theorem 3.2, IVP (4.6) has a unique mild solution between v0 and w0 , that is, the equation (4.1) has a unique L2 -mild solution between 0 and µ(t)φ1 (x). 2

References [1] B. Lundstrom, M. Higgs, W. Spain, A. Fairhall, Fractional by neocortial pyramidal neurons. Nat Neurosci. 11(2008), 1335-1342. [2] Y. Rossikhin, M. Shitikova, Application of fractional derivatives to the analysis of damped vibrationsof viscoelaticsingle mass system. Acta Mech. 120(1997), 109-125. [3] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett. 37(2014), 26-33. [4] X. Zhang, Positive solutions for a class of singular fractional differential equation with infinite points boundary value conditions, Appl. Math. Lett. 39(2015), 22-27. [5] P. Chen, Y. Li, Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness, Abst. Appl. Anal. Vol. 2013, Article ID 784816, 12 pages. [6] Z. Ouyang, Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl. 61(2011), 860-870. [7] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. [8] C. Lizama, Solutions of two-term time fractional order differential equations with nonlocal initial conditions, Electron. J. Qual. Theo. Diff. Equat. 82(2012), 1-9. [9] C. Lizama, An operator theoretical approach to a class of fractional order differential equations. Appl. Math. Lett. 24(2)(2011), 184-190. [10] J. Wang, Y. Zhou, W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Syst. Contr. Lett. 61(2012), 472-476. [11] M.M. EI-Borai, Some probability densities and funddamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, 14(2002), 433-440. [12] M.M. El-Borai, K.E. El-Nadi, E.G. El-Akabawy, On some fractional evolution equations, Comput. Math. Appl. 59(2010), 1352-1355. [13] P. Chen, Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys. 65(2014), 711-728. [14] P. Chen, Y. Li, Q. Li, Existence of mild solutions for fractional evolution equations with nonlocal initial conditions, Ann. Polon. Math. 110(2014), 13-24. [15] R. Wang, T.J. Xiao, J. Liang, A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lette. 24(2011), 1435-1442. [16] K. Balachandran, J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal. 72(2010), 4587-4593. [17] S. Du, V. Lakshmikantham, Monotone iterative technique for differential equtions in Banach spaces, J. Math. Anal. Appl. 87(1982), 454-459.

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[18] Y. Li, Existence of solutions to initial value problems for abstract semilinear evolution equations, Acta Mathematica Sinica, 48(2005), 1089-1094. [19] Y. Li, The positive solutions of abstract semilinear evolution equations and their applications, Acta Mathematica Sinica, 39(1996), 666-672. [20] P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math. 63(2013), 731-744. [21] P. Chen, Y. Li, Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces, Nonlinear Anal. 74(2011), 3578-3588. [22] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific,Singapore, 1989. [23] J. Liang, J. H. Liu and T.J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Model. 49(2009), 798-804. [24] P. Chen, Y. Li, H. Yang, Perturbation method for nonlocal impulsive evolution equations, Nonlinear Anal. Hybrid Syst. (8)2013, 22-30. [25] J. Banasiak, L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer-verlag, London, 2006. [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-verlag, Berlin, 1983. [27] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. [28] H.P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of rector-valued functions, Nonlinear Anal. 7(1983), 1351-1371. [29] H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328(2007), 1075-1081. [30] Y. Du, Fixed pionts of increasing operators in Banach spaces and applications, Appl. Anal. 38 (1990),1-20.

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EVP, MINIMAX THEOREMS AND EXISTENCE OF NONCONVEX EQUILIBRIA IN COMPLETE G-METRIC SPACES E. HASHEMI AND R. SAADATI*

Abstract. We prove generalized EVP (Ekeland’s variational principle) and generalized Takahashi’s nonconvex minimization theorem by Ω-distance on G-metric spaces. As a result of last theorems, we get generalized flower petal theorem.

1. introduction EVP, studied first one in 1972, has found a multitude of applications in different fields of analysis. It has also served to provide simple and elegant proofs of known results. The best references for those are by Ekeland himself: his survey article [2], his book with Aubin [1] and [4].

2. EVP Definition 2.1. [3] Let X 6= ∅. The function G : X × X × X → [0, ∞) is said to be G-metric when (i) G(u, v, w) = 0 if u = v = w (coincidence), (ii) G(u, u, v) > 0 for all u, v ∈ X, where u 6= v, (iii) G(u, u, w) ≤ G(u, v, w) for all u, v, w ∈ X, with w 6= v, (iv) G(u, v, w) = G(P {u, v, w}), where p is a permutation of u, v, w (symmetry), (v) G(u, v, w) ≤ G(u, a, a) + G(a, v, w) for all u, v, w, a ∈ X (rectangle inequality). In this paper, ϕ : (−∞, ∞) → (0, ∞) is a nondecreasing function. We say the function h : X → (−∞, ∞] is lower semicontinuous from above (shortly lsca) at w0 ∈ X when for every sequence {wn } in X with wn → w0 and h(w1 ) ≥ h(w2 ) ≥ · · · ≥ h(wn ) ≥ · · · , we have Key words and phrases. Ω-distance; Generalized EVP; Lower semicontinuous from above function; Generalized Caristi’s (common) fixed point theorem; Nonconvex minimax theorem; Nonconvex equilibrium theorem; Generalized flower petal theorem. *The corresponding author. 1

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E. HASHEMI AND R. SAADATI

h(w0 ) ≤ limn→∞ h(wn ). We say h is lsca on X when h is lsca at every point of X. We say h is proper if h 6= ∞. Definition 2.2. [3] Let (X, G) be a G-metric space. (1) A sequence {un } in X is a G-Cauchy sequence when, for every ε > 0, there exists a positive integer n0 such that if m, n, l ≥ n0 , then G(um , un , ul ) < ε. (2) A sequence {un } in X is G-convergent to a point u ∈ X when for every ε > 0, there exists a positive integer n0 such that for all m, n ≥ n0 we have G(um , un , u) < ε. Definition 2.3. [5] Let (X, G) be a G-metric space. A function Ω : X × X × X → [0, ∞) is said to be an Ω-distance on X when (a) Ω(u, v, w) ≤ Ω(u, a, a) + Ω(a, v, w) for all u, v, w ∈ X; (b) For any u ∈ X, Ω(u, ., .) : X → [0, ∞) is lower semicontinuous; (c) For each ε > 0, there exists a δ > 0 such that Ω(u, a, a) ≤ δ and Ω(a, v, w) ≤ δ imply G(u, v, w) ≤ ε. Example 2.4. [5] Let (X, d) be a metric space and G : X 3 → [0, ∞) defined by G(u, v, w) = max{d(u, v), d(u, w), d(v, w)} for all u, v, w ∈ X. Then Ω = G is an Ω-distance on X. Lemma 2.5. [5] Let (X, G) be a G-metric space and Ω an Ω-distance on X.

Let also

{un }, {vn } be sequences in X, {αn } and {βn } sequences in [0, ∞) converging to zero and let u, v, w, a ∈ X. Then we have (1) if Ω(v, un , un ) ≤ αn and Ω(un , v, w) ≤ βn for n ∈ N, then G(v, v, w) < ε and hence w = v; (2) if Ω(vn , un , un ) ≤ αn and Ω(un , um , w) ≤ βn for any m > n ∈ N, Then G(vn , vm , w) → 0 and hence vn → w; (3) if Ω(un , um , ul ) ≤ αn for any l, m, n ∈ N with n ≤ m ≤ l, then {un } is a G-Cauchy sequence; (4) if Ω(un , a, a) ≤ αn for any n ∈ N, then {un } is a G-Cauchy sequence. Lemma 2.6. Let Ω be an Ω-distance on X × X × X. If {un } is a sequence in X with lim supn→∞ {Ω(un , um , ul ) : n ≤ m ≤ l} = 0, then {un } is a G-Cauchy sequence in X. Proof. Suppose αn = sup{Ω(un , um , ul )}. We have limn→∞ αn = 0. By Lemma 2.5, we obtain that {un } is a G-Cauchy sequence in X.



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3

Lemma 2.7. Let f : X → (−∞, ∞] be a function and Ω a Ω-distance on X × X × X. We define the set P (u) by P (u) = {v ∈ X : v 6= u, Ω(u, v, v) ≤ ϕ(f (u))(f (u) − f (v))}. If v ∈ P (u), then we have f (v) ≤ f (u) and P (v) ⊆ P (u). Proof. Let v ∈ P (u). Then v 6= u and Ω(u, v, v) ≤ ϕ(f (u))(f (u)−f (v)). Since Ω(u, v, v) ≥ 0 for any u, v ∈ X and ϕ is nondecreasing in (0, ∞), we have f (u) ≥ f (v). If P (v) = ∅, then P (v) ⊆ P (u). If P (v) 6= ∅, then let w ∈ P (v). We have w 6= v and Ω(v, w, w) ≤ ϕ(f (v))(f (v) − f (w)). Then, we have f (v) ≥ f (w). Also we have

Ω(u, w, w) ≤ Ω(u, v, v) + Ω(v, w, w) ≤ ϕ f (u) f (u) − f (w) . We claim that w 6= u. Let w = u; then Ω(u, w, w) = 0. On the other hand



Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) ≤ ϕ f (u) f (u) − f (w) = 0. Then Ω(u, v, v) = 0; for each ε > 0, we have Ω(u, w, w) = 0 < δ, Ω(w, v, v) = 0 < δ =⇒ G(w, v, v) < ε. Then G(w, v, v) = 0 and w = v, which is a contradiction. Therefore w ∈ P (u) and hence P (v) ⊆ P (u).



Proposition 2.8. Let f : X → (−∞, ∞] be a proper lsca and bounded from below function. Let also Ω be an Ω-distance on X × X × X. For each u ∈ X, let 

P (u) = v ∈ X : v 6= u, Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) . If {un } is a sequence in X such that P (un ) is a nonempty set and un+1 ∈ P (un ) for all n ∈ N, T then there exists u0 ∈ X such that un → u0 and u0 ∈ ∞ n=1 P (un ). T Also, if f (un+1 ) ≤ inf w∈P (un ) f (w) + 1/n for all n ∈ N, then ∞ n=1 P (un ) has only one point. Proof. First we show that {un } is a Cauchy sequence. Whereas un+1 ∈ P (un ), by Lemma 2.7, f (un ) ≥ f (un+1 ) for all n ∈ N, so {f (un )} is nonincreasing. On the other hand, f is bounded from below; then r = limn→∞ f (un ), so f (un ) ≥ r for all n ∈ N. We show that lim supn→∞ {Ω(un , um , um ) : m > n} = 0. We have Ω(un , um , um ) ≤ Ω(un , un+1 , un+1 ) + Ω(un+1 , um , um ) ≤ Ω(un , un+1 , un+1 ) + Ω(un+1 , un+2 , un+2 ) + Ω(un+2 , um , um )

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Pm−1



Ω(uj , uj+1 , uj+1 ) ≤ ϕ f (u1 ) f (un ) − r , for all m, n ∈ N

with m > n. Put αn = ϕ f (u1 ) f (un ) − r for all n ∈ N. We have sup{Ω(un , um , um ) : m >

Therefore Ω(un , um , um ) ≤

j=n

n} ≤ αn for all n ∈ N. So limn→∞ f (un ) = r. We have limn→∞ αn = 0 and lim sup{Ω(un , um , um ) : m > n} = 0.

n→∞

By Lemma 2.6, {un } is a G-Cauchy sequence, and X is a G-complete metric space, so there T exists u0 ∈ X such that un → u0 . We show that u0 ∈ ∞ n=1 P (un ). Since f is lsca, then f (u0 ) ≤ limn→∞ f (un ) = r ≤ f (uk ). Suppose that n ∈ N is fixed for all m ∈ N with

P m > n. We have Ω(un , um , um ) ≤ m−1 j=n Ω(uj , uj+1 , uj+1 ) ≤ ϕ f (un ) (f (un ) − f (u0 ) . Since Ω(u, ., .) : X → (0, ∞) is lower semi continuous, then

Ω(un , u0 , u0 ) ≤ ϕ f (un ) f (un ) − f (u0 ) .

(2.1)

Also u0 6= un for all n ∈ N. Suppose contrary, that there exists j ∈ N such that u0 = uj .



Since Ω(uj , uj+1 , uj+1 ) ≤ ϕ f (uj ) f (uj ) − f (uj+1 ) ≤ ϕ f (uj ) f (uj ) − f (u0 ) = 0, so we would have Ω(uj , uj+1 , uj+1 ) = 0. Similarly, we would have Ω(uj+1 , uj+2 , uj+2 ) = 0. Now, for ε > 0, we would have Ω(uj , uj+1 , uj+1 ) = 0 < δ and Ω(uj+1 , uj+2 , uj+2 ) = 0 < δ. Then G(uj , uj+2 , uj+2 ) < ε, and by Definition 2.2, we would have uj = uj+2 , which is contradiction. Since uj+1 ∈ P (uj ), then P (uj+1 ) ⊆ P (uj ) and uj+2 ∈ P (uj+1 ), so uj+2 ∈ P (uj ) and therefore uj+2 6= uj . We conclude that u0 6= un for all n ∈ N. By (2.1) we have u0 ∈ T∞ T∞ n=1 P (un ) 6= ∅. Now we assume that f (un+1 ) ≤ inf w∈P (un ) f (w) + 1/n for n=1 P (un ), thus T∞ T all n ∈ N. We show that ∞ n=1 P (un ); then n=1 P (un ) = {u0 }. Let t ∈

Ω(un , t, t) ≤ ϕ f (un ) f (un ) − f (t)
≤ ϕ f (u1 ) f (un ) − inf


f (w)

w∈S(un )




≤ ϕ f (u1 ) f (un ) − f (un+1 ) + 1/n).
Let βn = ϕ f (u1 ) f (un ) − f (un+1 ) + 1/n), for all n ∈ N. Then limn→∞ βn = 0, thus limn→∞ Ω(un , t, t) = 0; also {um } is G-Cauchy. Then limn→∞ Ω(um , um , un ) = 0, and we T obtain un → t. By uniqueness, we have t = u0 . Then ∞  n=1 P (un ) = {u0 }. Theorem 2.9 (Generalized EVP). Let f : X → (−∞, ∞] be a proper lsca and bounded from below function. Let also Ω be an Ω-distance on X × X × X. Then there exists t ∈ X such that

Ω(t, u, u) > ϕ f (t) f (t) − f (u) for all u ∈ X with u 6= t.

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Proof. Suppose contrary, that for each u ∈ X, there exists v ∈ X with v 6= u such that
Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v)). That would mean that P (u) 6= ∅ for each u ∈ X. Since f is proper, there would exist u ∈ X such that f (u) 6= ∞. We define a sequence {un } as follows: let u1 = x, and choose u2 ∈ P (u1 ) such that f (u2 ) ≤ inf u∈P (u1 ) f (u) + 1. Suppose un ∈ X is so defined, and choose un+1 ∈ P (un ) such that f (un+1 ) ≤ inf u∈P (un ) f (u) + 1/n. T By Proposition 2.8, there would exist u0 ∈ X such that ∞ n=1 P (un ) = {u0 }. By Lemma 2.7, T∞ P (u0 ) ⊆ n=1 P (un ) = {u0 }, so P (u0 ) = {u0 }, which is a contradiction. Therefore, there exists t ∈ X such that

Ω(t, u, u) > ϕ f (t) f (t) − f (u) for all u ∈ X with u 6= t.



Theorem 2.10 (Generalized Caristi’s common fixed point theorem for a family of multivalued maps). Let f : X → (−∞, ∞] be a proper lsca and bounded from below function. Let also Ω be an Ω-distance on X × X × X. Let J be any index set and for each j ∈ J, suppose Pj : X → 2X is a multivalued map with nonempty values such that for each u ∈ X, there exists v = v(u, j) ∈ Pj (u) with

Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) . T Then there exists t ∈ X such that t ∈ j∈J Pj (t), and Ω(t, t, t) = 0.

(2.2)


Proof. By Theorem 2.9, there exists t ∈ X such that Ω(t, u, u) > ϕ f (t) f (t) − f (u)) for all T u ∈ X with u 6= t. Now we show that t ∈ j∈J Pj (t) and Ω(t, t, t) = 0. According to the

assumption, there exists w(t, j) ∈ Pj (t) such that Ω(t, w, w) ≤ ϕ f (t) f (t) − f (w(t, j)) . We claim that w(t, j) = t, for all j ∈ J. If, on the contrary, w(t, j0 ) 6= t for some j0 ∈ J, then

Ω(t, w, w) ≤ ϕ f (t) f (t) − f (w) < Ω(t, w, w). which is a contradiction. Therefore t = w(t, j) ∈ Pj (t) for all j ∈ P .
Since Ω(t, t, t) ≤ ϕ f (t) f (t) − f (t)) = 0, we obtain Ω(t, t, t) = 0.



Corollary 2.11 (Generalized Caristi’s common fixed point theorem for a family of single-valued maps). Let f : X → (−∞, ∞] be a proper lsca and bounded from below function. Let also Ω be an Ω-distance on X × X × X. Let J be any index set and for each j ∈ J, let gj : X → X be a single-valued map so that


Ω u, gj (u), gj (u) ≤ ϕ f (u) f (u) − f (gj (u))

(2.3)

is established for each u ∈ X. Then there exists t ∈ X such that gj (t) = t for each j ∈ J and Ω(t, t, t) = 0.

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Proof. Let Pj : X → X and Pj (x) = {gj (u)}, for all u ∈ X and j ∈ J. Then by Theorem 2.10, gj (t) = t for each j ∈ J and Ω(t, t, t) = 0.



Remark 2.12. (a) Corollary 2.11 implies Theorem 2.10. Suppose that for each u ∈ X, there exists v(u, j) ∈ Pj (u) such that


Ω u, v(u, j), v(u, j)) ≤ ϕ f (u) f (u) − f (v(u, j)) for each j ∈ J, and let gj (u) = v(u, j). Then gj is single-valued map and


Ω u, gj (u), gj (u)) ≤ ϕ f (u) f (u) − f (gj (u)) for all u ∈ X. By Corollary 2.11, there exists t ∈ X such that t = gj (t) ∈ Pj (t) for each j ∈ J and Ω(t, t, t) = 0 (b) Theorem 2.10 implies Theorem 2.9. Suppose contrary, that for each u ∈ X, there exists v ∈ X with v 6= u such that

Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) . Define P : X → 2X \ {∅} by

P (u) = {v ∈ X : v 6= u, Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) }. By Theorem 2.10, P has a fixed point t ∈ X; this means t ∈ P (t). This is a contradiction, because t ∈ / P (t). Theorem 2.13 (Nonconvex maximal element theorem for a family of multivalued maps). Let f : X → (−∞, ∞] be a proper lsca and bounded from below function. Let also Ω be an Ωdistance on X ×X ×X and J be any index set. For each j ∈ J, let Pj : X → 2X be a multivalued map. Suppose that for each (u, j) ∈ X × J with Pj (u) 6= ∅, there exists v = v(u, j) ∈ X with v 6= u such that (2.2) holds. Then there exists t ∈ X such that Pj (t) = ∅ for each j ∈ J.

Proof. By Theorem 2.9, there exists t ∈ X, such that Ω(t, u, u) > ϕ f (t) f (t) − f (u) for all u ∈ X with u 6= t. We prove that Pj (t) = ∅ for each j ∈ J. Indeed, if Pj0 (t) 6= ∅, for some j0 ∈ J, according to the assumption, there would exist w = w(t, j0 ) ∈ X with w 6= t such



that Ω(t, w, w) ≤ ϕ f (t) f (t) − f (w) . Also Ω(t, w, w) > ϕ f (t) f (t) − f (w) , which is a contradiction.



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Remark 2.14. Theorem 2.13 implies Theorem 2.9. Suppose contrary, that for each u ∈ X, there exists v ∈ X with v 6= u such that

Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) . For each u ∈ X, we define

P (u) = {v ∈ X : v 6= u, Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) }. Then P (u) 6= ∅ for all u ∈ X. But by Theorem 2.13, there would exist t ∈ X such that P (t) = ∅, which is a contradiction. 3. Nonconvex optimization and minimax theorems Theorem 3.1 (Generalized Takahashi’s nonconvex minimization theorem). Let f : X → (−∞, ∞] be a proper lsca and bounded from below function. Also, let Ω be an Ω-distance on X × X × X. Suppose that for any u ∈ X with f (u) > inf w∈X f (w) there exists v ∈ X with v 6= u such that (2.2) holds. Then there exists t ∈ X such that f (t) = inf w∈X f (w).

Proof. By Theorem 2.9, there exists t ∈ X such that Ω(t, u, u) > ϕ f (t) f (t) − f (u) , for all u ∈ X, u 6= t. Now we prove that f (t) = inf w∈X f (w). On the contrary, let f (t) > inf w∈X f (w). According to the assumption, there would exist v = v(t) ∈ X, with v 6= t such

that Ω(t, v, v) ≤ ϕ f (t) f (t) − f (v) . Then we would have

Ω(t, v, v) ≤ ϕ f (t) f (t) − f (v) < Ω(t, v, v) which is a contradiction.



Remark 3.2. Using Theorem 3.1, we can infer Theorem 2.9. If we could not, then for each u ∈ X, there would exist v ∈ X with v 6= u such that

Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) . By Theorem 3.1, there would exist t ∈ X such that f (t) = inf w∈X f (w). According to the assumption, there would exist z ∈ X with z 6= u, such

that Ω(t, z, z) ≤ ϕ f (t) f (t) − f (z) ≤ 0. Then Ω(t, z, z) = 0 and f (t) = f (z) = inf w∈X f (w).

There would exist w ∈ X with w 6= z such that Ω(z, w, w) ≤ ϕ f (z) f (z) − f (w) ≤ 0. Then we would have Ω(z, w, w) = 0 and f (t) = f (z) = f (w) = inf u∈X f (u). Since Ω(t, w, w) ≤ Ω(t, z, z) + Ω(z, w, w), then Ω(t, w, w) = 0. For ε > 0 we would have Ω(t, z, z) = 0 < δ, Ω(z, w, w) = 0 < δ; then G(t, w, w) < ε, that is, t = w. Also for ε > 0 we would have Ω(z, t, t) = 0 < δ, Ω(t, w, w) = 0 < δ; then G(z, w, w) < ε that is, z = w, which is a contradiction.

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Theorem 3.3 (Nonconvex minimax theorem). Let G : X × X → (−∞, ∞] be a proper lsca and bounded from below function in the first argument. Suppose that for each u ∈ X with {x ∈ X : G(u, x) > inf a∈X G(a, x)} 6= ∅, there exists v = v(u) ∈ X with v 6= u such that

Ω(u, v, v) ≤ ϕ G(u, t) G(u, t) − G(v, t)

(3.1)

for all t ∈ {x ∈ X : G(u, x) > inf a∈X G(a, x)}. Then inf u∈X supv∈X G(u, v) = supv∈X inf u∈X G(u, v). Proof. By Theorem 3.1, for every v ∈ X, there exists u(v) ∈ X such that G u(v), v
inf u∈X G(u, v). Then, supv∈X G u(v), v = supv∈X inf u∈X G(u, v).




=

Replacing u(v) by an arbitrary u ∈ X, we obtain inf sup G(u, v) = sup inf G(u, v).

u∈X v∈X



v∈X u∈X

Theorem 3.4 (Nonconvex equilibrium theorem). Let G and ϕ be the same as in Theorem 3.3. Let for each u ∈ X with {x ∈ X : G(u, x) < 0} = 6 ∅, there exist v = v(u) ∈ X with v 6= u such that (3.1) holds for all t ∈ X. Then there exists y ∈ X such that G(y, v) > 0 for all v ∈ X. Proof. By Theorem 2.9, for each w ∈ X, there exists y(w) ∈ X such that Ω(y(w), u, u) >

ϕ G(y(w), w) G(y(w), w) − G(u, w) for all u ∈ X with u 6= y(w). We show that there exists y ∈ X such that G(y, v) ≥ 0 for all v ∈ X. Suppose contrary, that for each u ∈ X there exists v ∈ X such that G(u, v) < 0. Then for each u ∈ X, {x ∈ X : G(u, x) < 0} 6= ∅. According to the assumption, there would exist v = v(y(w)), y 6= y(w) such that

Ω(y(w), v, v) ≤ ϕ G(y(w), w) G(y(w), w) − G(v, w) , which is a contradiction.  Example 3.5. Let X = [0, 1] and G(u, v, w) = max{|u − v|, |u − w|, |v − w|}. Then (X, G) is a complete G-metric space. Suppose that a, b are positive real numbers with a > b. Let H : X × X → R with H(u, v) = au − bv. Therefore, the function u 7→ H(u, v) is proper, lower semicontinuous and bounded from below, and H(1, v) ≥ 0 for every v ∈ X. Also H(u, v) ≥ 0 for every u ∈ [ ab , 1] and every v ∈ X. In fact, for every u ∈ [0, ab ], H(u, v) = au − bv < 0 when v ∈ [ ab u, 1]. Then set {x ∈ X : H(u, x) < 0} 6= ∅ for every u ∈ [0, ab ]. Let u, v ∈ X, u ≥ v; we have u − v = a1 {(au − bx) − (av − bx)}, for every x ∈ X. Define ϕ : [0, ∞) → [0, ∞) by

ϕ(t) = a1 . Then G(u, v, v) ≤ ϕ H(u, x) H(u, x) − H(v, x) , for every u > v, and u, v, x ∈ X. By Theorem 3.4, there exists y ∈ X such that H(y, v) > 0 for every v ∈ X.

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Theorem 3.6. Let Ω, ϕ be the same as in Theorem 2.9. For each j ∈ J, let Pj : X → X be multivalued maps with nonempty values, gj , hj : X × X → R be functions and {aj } and {bj } families of real numbers. Suppose that: (i) For each (u, j) ∈ X × J, there exists v = v(u, j) ∈ Pj (u) such that gj (u, v) ≥ aj and

Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) ; (ii) For each (x, j) ∈ X × J, there exists w = w(x, j) ∈ Pj (x) such that hj (x, w) ≤ bj and

Ω(x, w, w) ≤ ϕ f (x) f (x) − f (w) . Then there exists u0 ∈ Pj (uo ) such that gj (u0 , u0 ) ≥ aj and hj (u0 , u0 ) ≤ bj for all j ∈ J and Ω(u0 , u0 , u0 ) = 0.

Proof. By Theorem 2.9, there exists t ∈ X such that Ω(t, u, u) > ϕ f (t) f (t) − f (u) , for all u ∈ X with u 6= t. For each j ∈ J, by (i) there exists w1 = w1 (t, j) ∈ Pj (t) such that

gj (t, w1 ) ≥ aj and Ω(t, , w1 , w1 ) ≤ ϕ f (t) f (t) − f (w1 ) . Also according to (ii), there exists

w2 = w2 (t, j) ∈ Pj (t) such that hj (t, w2 ) ≤ bj and Ω(t, , w2 , w2 ) ≤ ϕ f (t) f (t) − f (w2 ) . If

w1 6= t, then Ω(t, , w1 , w1 ) ≤ ϕ f (t) f (t) − f (w1 ) < Ω(t, w1 , w1 ), which is a contradiction.

Therefore w1 = t. Similarly, we have w2 = t. Since Ω(t, , t, t) ≤ ϕ f (t) f (t) − f (t) = 0, hence Ω(t, t, t) = 0.



Remark 3.7. (a) In Theorem 3.6, put gj = hj = Fj and aj = bj = cj ; then there exists u0 ∈ Pj (u0 ) such that Fj (u0 , u0 ) = cj for all j ∈ J and Ω(u0 , u0 , u0 ) = 0. (b) In (a), put Pj (u) = X for all u ∈ X; then there exists u0 ∈ X such that Fj (u0 , u0 ) = cj for all j ∈ J and Ω(u0 , u0 , u0 ) = 0. Remark 3.8. From Theorem 3.5, we can infer Theorem 2.9. Suppose contrary, that for each u ∈ X, there exists v ∈ X with v 6= u such that

Ω(u, v, v) ≤ ϕ f (u) f (u) − f (v) . Define P : X → X \ {∅} by P (u) = {v ∈ X : v 6= u} and a function F : X × X → R by F (u, v) = χP (u) (v), where χA is the characteristic function for an arbitrary set A. We would have v ∈ P (u) ⇐⇒ F (u, v) = 1. Then for each u ∈ X, there would exist v ∈ X such that

F (u, v) = 1 and Ω(t, , u, u) ≤ ϕ f (t) f (t) − f (u) . According to Remark 3.7(a) with c = 1, there would exist u0 ∈ X such that F (u0 , u0 ) = 1 and Ω(u0 , u0 , u0 ) = 0. Then u0 ∈ P (u0 ). This is a contradiction.

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4. Applications Let (X, G) be a G-metric space and a, b ∈ X. Suppose that κ : X → (0, ∞) is a function and Ω a Ω-distance on X. Define
Ωε (a, b, κ) = {u ∈ X : εΩ(a, u, u) ≤ κ(a) Ω(b, a, a) − Ω(b, u, u) } such that ε ∈ (0, ∞) and a, b ∈ X. Lemma 4.1. Let Ω, f,and ϕ be the same as in Theorem 2.9. Let ε > 0 and Ω be an Ω-distance on X. Suppose that there exists x ∈ X such that f (x) < ∞ and Ω(x, x, x) = 0. Then there exists t ∈ X such that

(i) εΩ(x, t, t) ≤ ϕ f (x) f (x) − f (t) ;

(ii) Ω(t, u, u) > ϕ f (t) f (t) − f (u) for all u ∈ X with u 6= t. Proof. Let x ∈ X, f (x) < +∞ and Ω(x, x, x) = 0. Put

V = {u ∈ X : εΩ(x, u, u) ≤ ϕ f (x) f (x) − f (u) }. The space (V, G) is a nonempty complete G-metric space. By Theorem 2.9, there exists t ∈ V

such that εΩ(t, u, u) > ϕ f (t) f (t) − f (u) for all u ∈ V with u 6= t. For any u ∈ X \ V , since

ε[Ω(x, t, t) + Ω(t, u, u)] ≥ εΩ(x, u, u) > ϕ f (x) f (x) − f (u)

≥ εΩ(x, t, t) + ϕ f (t) f (t) − f (u) ,


we have εΩ(t, u, u) > ϕ f (t) f (t) − f (u) for all u ∈ X \ V . Then εΩ(t, u, u) > ϕ f (t) f (t) −
f (t) for all u ∈ X with u 6= t.  Theorem 4.2 (Generalized flower petal theorem). Suppose that N is a proper complete subset of a G-metric space X and a ∈ N . Let Ω be an Ω-distance on X with Ω(a, a, a) = 0. Let b ∈ X \ N , Ω(b, N, N ) = inf u∈N Ω(b, u, u) ≥ r, Ω(b, a, a) = s > 0, and let there exist a
function κ : X → (0, ∞) satisfying κ(u) = ϕ Ω(b, u, u) for some nondecreasing function ϕ : (−∞, ∞] → (0, ∞). Then for each ε > 0, there exists t ∈ N ∩ Ωε (a, b, κ) such that Ωε (t, b, κ) ∩ (N \ {t}) = ∅ and Ω(a, t, t) ≤ ε−1 κ(a)(s − r). Proof. The space (N, G) is a complete G-metric space. Consider the function f : N → (−∞, ∞] defined by f (u) = Ω(b, u, u). Since f (a) = Ω(b, a, a) = s < ∞ and Ω(b, N, N ) = inf u∈N Ω(b, u, u) ≥ r, f is a proper lower semicontinuous and bounded from below function. By Lemma 4.2, there exists t ∈ N such that

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(i) εΩ(a, t, t) ≤ κ(a) f (a) − f (t) ;
(ii) εΩ(t, u, u) > κ(t) f (t) − f (t) for all u ∈ N with u 6= t. T Applying (i), we get t ∈ N Ωε (a, b, κ). Applying (i) again, we get Ω(a, t, t) ≤ ε−1 κ(a) Ω(b, a, a)−

Ω(b, t, t) ≤ ε−1 κ(a)(s − r). By (ii), we obtain εΩ(t, u, u) > κ(t) Ω(b, t, t) − Ω(b, u, u) for all u ∈ N with u 6= t. Therefore u ∈ / Ωε (t, b, κ) for all u ∈ N \ {t} or Ωε (t, b, κ) ∩ (N \ {t}) = ∅.  Acknowledgement The authors are grateful to the reviewer for their valuable comments and suggestions. References [1] J.-P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Wiley (1984). [2] I. Ekeland, Remarques sur les problmes variationnels, I, C. R. Acad. Sci. Paris S´er. A–B, 275 (1972), 1057–1059. [3] Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289–392. [4] S. Plubtieng, Th. Seangwattana, The Borwein-Preiss variational principle for nonconvex countable systems of equilibrium problems, J. Nonlinear Sci. Appl. 9 (2016), no. 5, 2224–2232. [5] R. Saadati, S. M. Vaezpour, P. Vetro, B. E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Math. Comput. Modelling, 52 (2010), 797–801.

(Eshagh Hashemi) Department of Mathematics, College of Basic Sciences, Karaj Branch, Islamic Azad University, Alborz, Iran. E-mail address: eshagh [email protected]

(Reza Saadati) Department of Mathematics, Iran University of Science and Technology, Tehran, Iran E-mail address: [email protected]

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GEOMETRIC PROPERTIES OF BESSEL FUNCTIONS FOR THE CLASSES OF JANOWSKI STARLIKE AND CONVEX FUNCTIONS V.RADHIKA1 , S.SIVASUBRAMANIAN2 , N.E.CHO AND G.MURUGUSUNDARAMOORTHY4

3,∗

1 Department

of Mathematics Easwari Engineering College Chennai-600089, India E-Mail:[email protected] 2 Department

of Mathematics University College of Engineering Anna University Tindivanam-604001, India E-Mail:[email protected] 3,∗ Department

of Applied Mathematics Pukyong National University Busan 608-737, Republic of Korea Email:[email protected] 4 Department

of Mathematics School of Advanced Sciences VIT University Vellore-632014, India Email:[email protected] Abstract. Applications of Bessel differential equations have attracted the univalent function theorists in recent years. In the present investigation, we establish certain sufficient conditions for Bessel function to be in the class of Janowski starlike and Janowski convex functions. Further, certain sufficient condition for an integral operator defined using Bessel function to be in the class of Janowski starlike and Janowski convex functions are determined.

2010 Mathematics Subject Classification: 30C45 Key words and phrases: Analytic functions, Bessel functions, Univalent functions, Starlike functions, Convex functions * Corresponding author

1

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1. Introduction Let A be the class of functions f normalized by f (z) = z +

∞ X

an z n ,

(1.1)

n=2

which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1} and S the subclass of A consisting of functions which are also univalent in U. A function f ∈ A is said to be starlike of order α (0 ≤ α < 1), if and only if  0  zf (z) < > α (z ∈ U). f (z) This function class is denoted by S ∗ (α). We also write S ∗ (0) =: S ∗ , where S ∗ denotes the class of functions f ∈ A that are starlike in U with respect to the origin. A function f ∈ A is said to be convex of order α (0 ≤ α < 1) if and only if   zf 00 (z) < 1+ 0 > α (z ∈ U). f (z) This class is denoted by K(α). Further, K = K(0), the well-known standard class of convex functions. It is an established fact that f ∈ K(α) ⇐⇒ zf 0 ∈ S ∗ (α). There has been a continuous interest shown on the Geometric and other related properties of Bessel functions (like hypergeometric functions) after many papers have been published by Baricz [2](see also the other works of Baricz) in recent times. One such problem of Baricz [3] was to find conditions on the triplet p, b and c such that the function up,b,c is starlike and convex of order α. In earlier investigations, finding conditions on the parameters for which the Gaussian hypergeometric function belong to the various classes of functions have been discussed in detail by Shanmugam [20], Sivasubramanian et al. [21] and Sivasubramanian and Sokol [22] (See also [6, 7, 10, 11, 12, 13, 16, 17] ). Let us consider the following second-order linear homogenous differential equation (see, for details, [3]): z 2 ω 00 (z) + bzω 0 (z) + [cz 2 − p2 + (1 − b)p]ω(z) = 0 (b, c, p ∈ C). (1.2) The function ωp,b,c (z), which is called the generalized Bessel function of the first kind of order p, is defined as a particular solution of (1.2). Further, the function ωp,b,c (z) has the familiar representation ∞  z 2n+p X (−c)n
(z ∈ C), (1.3) ωp,b,c (z) = 2 n!Γ p + n + b+1 2 n=0 where Γ stands for the Euler gamma function. The series (1.3) permits the study of Bessel, modified Bessel and spherical Bessel functions all together. Solutions of (1.2) are referred as the generalized Bessel function of order p. The particular solution given by (1.3) is called the generalized Bessel function of the first kind order of p. Although the series defined above is convergent everywhere, the function ωp,b,c is generally not univalent in U. By ratio test, the radius of convergence for the series in (1.3) is infinity and hence ωp,b,c (z) converges everywhere for all b, c, p ∈ C and for all z ∈ U.

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It is worth mentioning that, in particular, for b = c = 1 in (1.3), we obtain the familiar Bessel function of the first kind of order p defined by ∞  z 2n+p X (−1)n Jp (z) = (z ∈ C). (1.4) n!Γ(p + n + 1) 2 n=0

Further, for the choices c = 1 and b = 2 in (1.3), we obtain the familiar spherical Bessel function of the first kind of order p defined by ∞  z 2n+p X (−1)n Sp (z) = (z ∈ C). (1.5) n!Γ(p + n + 3/2) 2 n=0

For the choices of b = 1 and c = −1 in (1.3), we obtain the modified Bessel function of the first kind of order p defined by ∞  z 2n+p X 1 (z ∈ C). (1.6) Ip (z) = n!Γ(p + n + 1) 2 n=0

From (1.3), it is clear that ω(0) = 0. Therefore, it follows from (1.3) that    ∞ b + 1 −1 p X (−c/4)n Γ (p + (b + 1)/2) 2n p ωp,b,c (z) = 2 Γ p + z z (z ∈ C). 2 n!Γ (p + n + (b + 1)/2)

(1.7)

n=0

Let us set up,b,c (z) =

∞ X

bn z n ,

n=0

where bn =

(−c/4)n Γ (p + (b + 1)/2) . n!Γ (p + n + (b + 1)/2)

Hence, (1.7) becomes    b + 1 −1 p p z up,b,c (z 2 ). (1.8) ωp,b,c (z) = 2 Γ p + 2 By using the well-known Pochhammer symbol (or the shifted factorial) (λ)µ defined, for λ, µ ∈ C and in terms of the Euler Γ function, by ( 1 (µ = 0; λ ∈ C \ {0}) Γ(λ + µ) = (λ)µ := Γ(λ) λ(λ + 1) · · · (λ + n − 1) (µ = n ∈ N; λ ∈ C). In view of the fact that (0)0 := 1, the series representation for the function up,b,c is given by up,b,c (z) =

∞ X (−c/4)n z n n=0

κ := p + (b + 1)/2 6∈ Z− 0

(κ)n (n)!




(1.9)

and therefore, zup,b,c (z) = z +

∞ X (−c/4)n−1 z n (κ)n−1 (n − 1)!

κ := p + (b + 1)/2 6∈ Z− 0




(1.10)

n=2

Z− 0

where N := {1, 2, ...} and := {0, −1, −2, ...}. The function up,b,c is called the generalized and normalized Bessel function of the first kind of order p. We note that by the ratio test, the radius of

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convergence of the series up,b,c is infinity. Moreover, the function up,b,c is analytic in C and satisfies the differential equation 4z 2 u00 (z) + 4κzu0 (z) + czu(z) = 0. Also, if b, p, c ∈ C and κ 6∈ Z− 0 , then the function up,b,c satisfies the recursive relation 4ku0p,b,c (z) = −cup+1,b,c (z) (z ∈ C).

(1.11)

Further, for z = 1, we denote up,b,c (z) simply by up (1). For f ∈ A, we define the operator Ip,b,c f (z) by ∞ X (−c/4)n−1 an z n Ip,b,c f (z) = zup,b,c (z) ∗ f (z) = z + , (1.12) (κ)n−1 (n − 1)! n=2

where κ = p + (b + 1)/2 6∈ Z− 0 . In fact, the function Ip,b,c f (z) given by (1.12) is an elementary transform of the generalized hypergeometric function. Thus, it is easy to see that Ip,b,c f (z) = z0 F1 (κ; −c/4z) ∗ f (z). For the special choices of b = c = 1 in (1.12), Ip,b,c f reduces to Jp : A → A related with Bessel function, defined by ∞ X (−1/4)n−1 an z n . (1.13) Jp f (z) = zup,1,1 (z) ∗ f (z) = z + (p + 1)n−1 (n − 1)! n=2

For the special choices of b = 1 and c = −1 in (1.12), Ip,b,c f reduces to Mp : A → A related with modified Bessel function, defined by ∞ X zn an (1.14) Mp f (z) = zup,1,−1 (z) ∗ f (z) = z + (4)n−1 (p + 1)n−1 (n − 1)! n=2

where ∗ denotes the usual Hadamard product or convolution of power series. If f and g are analytic in U, then we say that the function f is subordinate to g, if there exists a Schwarz function w(z), analytic in U with w(0) = 0 and |w(z)| < 1 (z ∈ U), such that f (z) = g(w(z)) (z ∈ U). We denote this subordination by f ≺ g or f (z) ≺ g(z) (z ∈ U). For −1 ≤ F < E ≤ 1, let 

∗

S [E, F ] = and

zf 0 (z) 1 + Ez f ∈A: ≺ (z ∈ U) f (z) 1 + Fz



 1 + Ez zf 00 (z) ≺ (z ∈ U) . f 0 (z) 1 + Fz It is fairly straightforward to see that S ∗ [1, −1] is the familiar class of starlike functions S ∗ , S ∗ [1 − 2γ, −1] (0 ≤ γ < 1) is the class of starlike functions of order γ and also the class S ∗ [λ, 0] is denoted by Sλ∗ . Further, K [1, −1] is the familiar class of convex functions K, K [1 − 2γ, −1] (0 ≤ γ < 1) is the class of convex functions of order γ and also the class K [λ, 0] is denoted by Kλ . These two classes have been investigated in several works, for example, see [18, 19]. The connection between the Janowski starlike, Janowski convex functions and the Bessel functions is not considered so far. In the present paper, we obtain mapping properties between various subclasses of S motivated by the works of Anbudurai and Parvatham [1] (see also [5, 10, 11, 12, 16, 17, 18, 24]). 

K [E, F ] =

f ∈A:1+

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2. Sufficient conditions for Bessel functions to be in S ∗ [E, F ] and K[E, F ] involving Jack’s Lemma In the present section, we determine certain sufficient conditions involving Jack’s Lemma for up,b,c and z up,b,c to be in the class of Janowski starlike and Janowski convex functions. To prove the main theorems we need the following lemma. Lemma 2.1. [9] Let ω be regular in the unit disk U with ω(0) = 0. If |ω(z)| attains a maximum value on the circle |z| = r (0 ≤ r < 1) at a point z, then z1 ω 0 (z1 ) = mω(z1 ) where m is real and m ≥ 1. Lemma 2.2. [18] Let a function f of the form (1.1) satisfy 0 1−α 00 α 2 α zf (z) zf (z) < (E − F )(2 + E + E ) − 1 f (z) f 0 (z) (1 + |F |)(1 + E)2α

(2.1)

for fixed constants E, F and α such that −1 ≤ F < E ≤ 1, α ≥ 0 and z ∈ U. Then f ∈ S ∗ [E, F ]. Theorem 2.1. Let f ∈ A. If 00 β (Ip,b,c f (z))0 − 1 1−β z(Ip,b,c f (z)) < 1 (β ≥ 0), (Ip,b,c f (z))0 2β

(2.2)

then Ip,b,c f is univalent in U. Proof. We know that Ip,b,c f (z) = z +

∞ X n=2

(−c/4)n−1 an z n (κ)n−1 (n − 1)!

in A. Define ω by ω(z) = (Ip,b,c f (z))0 − 1 for z ∈ U. Then it follows that ω is analytic in U with ω(0) = 0. In view of (2.2), we have β β 0 0 zω (z) 1 1 1−β zω (z) (2.3) |ω(z)| 1 + ω(z) = |ω(z)| ω(z) 1 + ω(z) < 2β . Suppose that there exists a point z1 ∈ U such that max |ω(z)| = |ω(z1 )| = 1. Then, by Lemma |z|≤|z1 |

3.1, we can put

z1 ω 0 (z1 ) = m ≥ 1. ω(z1 )

Therefore, we obtain β   β z1 ω 0 (z1 ) 1 1 ≥ m |ω(z1 )| ≥ β ω(z1 ) 1 + ω(z) 2 2 which contradicts the condition (2.3). This shows that |ω(z)| = |(Ip,b,c f (z))0 − 1| < 1 which implies that 0 for z ∈ U. Therefore, by the Noshiro-Warschawski theorem [8], Ip,b,c f is univalent in U.  Theorem 2.2. Let f ∈ A, c ∈ C and κ > 0. If up,b,c defined by (1.9) satisfies the inequality zu0 (z) E−F p,b,c , (2.4) < up,b,c (z) 1 + |F | where −1 ≤ F < E < 1, −1 ≤ F ≤ 0 and z ∈ U, then zup,b,c ∈ S ∗ [E, F ].

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Proof. Let us define a function F by F (z) = zup,b,c (z) (z ∈ U). In view of (2.4), we have 0 zF (z) < E−F . − 1 F (z) 1 + |F | An application of Lemma 2.2 with α = 0 proves Theorem 2.2.

(2.5) 

Theorem 2.3. Let f ∈ A, c ∈ C and κ > 0. If up,b,c defined by (1.9) satisfies the inequality zu00 (z) (E − F )(2 + E + E 2 ) p,b,c , (2.6) 0 < up,b,c (z) (1 + |F |)(1 + E)2 where −1 ≤ F < E < 1 and −1 ≤ F ≤ 0, then up,b,c is starlike of order (E + F )/2F and type |F | with respect to 1. Proof. Let h : U → C be defined by h(z) =

up,b,c (z) − b0 . b1

Then h ∈ A and h satisfies 00 zu00 (z) zh (z) p,b,c h0 (z) = u0 (z) p,b,c
0. If up,b,c defined by (1.9) satisfies the inequality zu0 E−F p+1,b,c (z) , (2.7) < up+1,b,c (z) 1 + |F | where −1 ≤ F < E < 1, −1 ≤ F ≤ 0 and c 6= 0, then up,b,c (z) ∈ K[E, F ]. Proof. By virtue of Theorem 2.2, zup+1,b,c ∈ S ∗ [E, F ]. In view of (1.11), zu0p,b,c (z) = b1 up+1,b,c (z), where b1 = −c/4κ 6= 0. Therefore, we have zu0p,b,c ∈ S ∗ [E, F ], which implies up,b,c ∈ K[E, F ].  Remark 2.1. Note that, the conclusions of Theorem 2.2, Theorem 2.3 and Theorem 2.4 hold in the disk |z| < 4/|c| where 0 < |c| < 4 which is larger than the unit disk. By applying as in Theorems 2,3, and 5 of Owa and Srivastava [15] to the function F (z) = 0 F1 (κ, z) and using the transformation F (z) = up,b,c (−4z/c) and replacing z by −cz/4, we obtain that Theorem 2.2, Theorem 2.3 and Theorem 2.4 hold in the disk |z| < 4/|c|. Theorem 2.5. Let c ∈ C, −1 ≤ F < E < 1, −1 ≤ F ≤ 0 and κ > 0. If the Bessel’s inequality |c| (1 − F ) up+1,b,|c| (1) + (E − F )up,b,|c| (1) ≤ 2(E − F ) (2.8) 4κ is satisfied, then zup,b,c ∈ S ∗ [E, F ].

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Proof. A special case of Theorem 3 in [1] gives a sufficient condition for a function f ∈ S ∗ [E, F ] and is given by ∞ X [n(1 − F ) − (1 − E)] |An | ≤ E − F, n=2

where An =

(−c/4)n−1 . (κ)n−1 (n − 1)!

To prove the theorem, we need to show that ∞ X T : = [n(1 − F ) − (1 − E)] |An | n=2 ∞ X

(−c/4)n−1 = [n(1 − F ) − (1 − E)] (κ)n−1 (n − 1)! n=2 ∞ ∞ (−c/4)n−1 n−1 X X (−c/4) = (1 − F ) n − (1 − E) (κ)n−1 (n − 1)! (κ)n−1 (n − 1)! ≤ (1 − F )

n=2 ∞ X

n=2

= (1 − F )

n=2

n−1

(|c|/4) + (E − F ) (κ)n−1 (n − 2)!

∞ X n=2

(|c|/4)n−1 (κ)n−1 (n − 1)!


|c| up+1,b,|c| (1) + (E − F ) up,b,|c| (1) − 1 , 4κ

which is bounded above by E − F if (2.8) is satisfied.



For the choices of E = λ and F = 0, we get the following corollary. Corollary 2.1. Let c ∈ C, −1 ≤ F < E < 1, −1 ≤ F ≤ 0 and κ > 0. If the Bessel’s inequality |c| u (1) + λup,b,c (1) ≤ 2λ 4κ p+1,b,|c|

(2.9)

is satisfied, then zup,b,|c| ∈ Sλ∗ . Theorem 2.6. Let c ∈ C and κ > 0. If the Bessel’s inequality (1 − F )

(|c|/4)2 |c| up+2,b,|c| (1) + (2 + E − 3F ) up+1,b,|c| (1) + (E − F ) up,b,|c| (1) ≤ 2(E − F ) (2.10) κ(κ + 1) 4κ

is satisfied, then the operator zup,b,c ∈ K [E, F ]. Proof. By an analogous similar result [1] mentioned as in the earlier theorem, a sufficient condition for f ∈ K [E, F ] is that ∞ X n [n(1 − F ) − (1 − E)] |An | ≤ E − F, n=2

where An =

(−c/4)n−1 . (κ)n−1 (n − 1)!

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Then we have to show that T1 :=

∞ X

n [n(1 − F ) − (1 − E)] |An | ≤ E − F.

(2.11)

n=2

Writing n = n − 1 + 1, and proceeding with the calculation as in the previous theorem, we get ∞ (−c/4)n−1 X T1 = n (n(1 − F ) − (1 − E)) (κ)n−1 (n − 1)! ≤

n=2 ∞ X

n=2

∞

X (|c|/4)n−1 (|c|/4)n−1 (n(1 − F ) − (1 − E)) + (n(1 − F ) − (1 − E)) . (κ)n−1 (n − 2)! (κ)n−1 (n − 1)! n=2

Breaking the above inequality into two parts and simplifying, we observe that the summation is bounded above by E − F if (2.10) is satisfied.  For the choices of E = λ and F = 0, we get the following corollary. Corollary 2.2. Let c ∈ C and κ > 0. If the Bessel’s inequality (|c|/4)2 |c| up+2,b,|c| (1) + (2 + λ) up+1,b,|c| (1) + λup,b,|c| (1) ≤ 2λ κ(κ + 1) 4κ

(2.12)

is satisfied, then the operator zup,b,c ∈ Kλ . Remark 2.2. For the choices of E = 1 − 2α (0 ≤ α < 1) and F = −1, each of the above theorems reduces to the results obtained by Baricz [3]. 3. Inclusion properties involving the class of Janowski starlike and convex functions Let a function f ∈ A is said to be in the class Rτ (A, B) if f 0 (z) − 1 τ (A − B) − B(f 0 (z) − 1) < 1 (−1 ≤ B < A ≤ 1; τ ∈ C \ {0}; z ∈ U).

(3.1)

Clearly, a function f belongs to Rτ (A, B) if and only if there exists a function w regular in U satisfying w(0) = 0 and |w(z)| < 1 z ∈ U such that 1 + Aw(z) 1 (z ∈ U). 1 + (f 0 (z) − 1) = τ 1 + Bw(z) The class Rτ (A, B) was introduced by Dixit and Pal [6]. For τ = 1, A = β, B = −β, (0 < β ≤ 1), Rτ (A, B) reduces to the class of functions f ∈ A satisfying the inequality 0 f (z) − 1 f 0 (z) + 1 < β (z ∈ U; 0 < β ≤ 1), which was studied by Caplinger and Cauchy [4] and Padmanaban [14]. Now we aim at investigating various mapping and inclusion properties involving the class of Janowski starlike and Janowski convex functions. To prove the main results we need the following lemmas.

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Lemma 3.1. [6] Let a function f of the form (1.1) be in Rτ (A, B). Then |an | ≤

(A − B)|τ | . n

The result is sharp for the function  Z z (A − B)|τ |z n−1 dz (n ≥ 2; z ∈ U) f (z) = 1+ 1 + Bz n−1 0 Lemma 3.2. [6] Let a function f of the form (1.1) satisfy the inequality ∞ X

(1 + |B|)n|an | ≤ (A − B)|τ | (−1 ≤ B < A ≤ 1; τ ∈ C).

n=2

Then f ∈ Rτ (A, B). The result is sharp for the function f (z) = z +

(A − B)τ n z (n ≥ 2; z ∈ U). (1 + |B|)n

Theorem 3.1. Let c ∈ C, κ > 0. Suppose that f ∈ Rτ (A, B). If the Bessel’s inequality
(κ − 1) 1 2up,b,|c| (1) − 4 up−1,b,|c| (1) − 1 ≤ +1 c (1 + B)

(3.2)

is satisfied, then zup,b,c (z 2 ) ∗ f (z) ∈ Rτ (A, B) Proof. Suppose that f ∈ Rτ (A, B). We note that zup,b,c (z 2 ) = z +

∞ X n=2

n−1 ( −c 4 ) z 2n−1 . (κ)n−1 (n − 1)!

By Lemma 3.2, it is enough to show that ∞ X

−c n−1 ) ( 4 an ≤ (A − B)|τ |. (1 + |B|) (2n − 1) (κ)n−1 (n − 1)!

n=2

Then by a similar proof as in the earlier theorem, we get  
(κ − 1) (A − B) |τ | (1 + |B|) 2up,b,|c| (1) − 4 up−1,b,|c| (1) − 1 − 1 ≤ (A − B) |τ |, |c| which completes the proof of Theorem 3.1.



Theorem 3.2. Let c ∈ C and κ > 0. Suppose that f ∈ Rτ (A, B) and satisfy the condition up,b,|c| (1) ≤

1 + 1. 1 + |B|

(3.3)

Then Ip,b,c f ∈ Rτ (A, B). Proof. Let f be of the form (1.1) belong to the class Rτ (A, B). By Lemma 3.2, it suffices to show that ∞ X n(1 + |B|)|An | ≤ (A − B)|τ |, (3.4) n=2

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where

(−c/4)n−1 an . (κ)n−1 (n − 1)! By virtue of Lemma 3.1 and making use of the fact that |−c/4|n ≤ (|c|/4)n , we obtain ∞ ∞ X X (−c/4)n−1 (|c|/4)n−1 n(1 + |B|) an ≤ (1 + |B|) |τ | (A − B) (κ)n−1 (n − 1)! (κ)n−1 (n − 1)! An =

n=2

n=2

(1 + |B|) |τ | (A − B) [up,b,|c| (1) − 1],

=

which is bounded above by (A − B)|τ | in view of (3.3). This completes the proof of Theorem 3.2.  Theorem 3.3. Let c ∈ C and κ > 0. Suppose that f ∈ Rτ (A, B). If the Bessel’s inequality (1 − F )up,b,|c| (1) − (1 − E)

4(κ − 1) E−F (1 − E)4(κ − 1) up−1,b,|c| (1) ≤ +E−F − |c| (A − B)|τ | |c|

(3.5)

is satisfied, then the operator Ip,b,c f ∈ S ∗ [E, F ]. Proof. Let f be of the form (1.1) belong to the class Rτ (A, B). A special case of Theorem 3 [1] gives a sufficient condition that ∞ X [n(1 − F ) − (1 − E)] |An | ≤ E − F, n=2

where An =

(−c/4)n−1 an . (κ)n−1 (n − 1)!

Then we have to show that T :=

∞ X

[n(1 − F ) − (1 − E)] |An | ≤ E − F.

(3.6)

n=2

Since, f ∈ Rτ (A, B), in virtue of Lemma 3.1, ∞ (−c/4)n−1 (A − B)|τ | X T ≤ [n(1 − F ) − (1 − E)] (κ)n−1 (n − 1)! n n=2 # " ∞ ∞ n−1 n−1 X X (−c/4) (−c4) = (A − B)|τ | (1 − F ) − (1 − E) (κ)n−1 (n − 1)! (κ)n−1 (n!) n=2 n=2 " # ∞ ∞ n−1 X X (|c|/4)n−1 (|c|/4) ≤ (A − B)|τ | (1 − F ) − (1 − E) (κ)n−1 (n − 1)! (κ)n−1 (n!) n=2 n=2    4(κ − 1) |c|/4 = (A − B)|τ | (1 − F )(up,b,|c| (1) − 1) − (1 − E) up−1,b,|c| (1) − 1 − , |c| (κ − 1) which is bounded above by E − F if (3.5) is satisfied.



For the choices of E = λ and F = 0, we get the following corollary.

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Corollary 3.1. Let c ∈ C, κ > 0 and λ ∈ [0, 1]. Suppose that f ∈ Rτ (A, B) and satisfy the condition   4(κ − 1) 1 up,b,|c| (1) + (λ − 1) [up−1,b,|c| (1) − 1] ≤ + 1 λ. c (A − B) |τ | Then the operator Ip,b,c f ∈ Sλ∗ . Theorem 3.4. Let c ∈ C and κ > 0. Suppose that f ∈ Rτ (A, B). If the Bessel’s inequality   |c| 1 (1 − F ) up+1,b,|c| (1) + (E − F )up,b,|c| (1) ≤ (E − F ) +1 4κ (A − B)|τ |

(3.7)

is satisfied, then the operator Ip,b,c f ∈ K [E, F ]. Proof. Let f be of the form (1.1) belong to the class Rτ (A, B). We need to show (see [1]) that ∞ X

n [(n(1 − F ) − (1 − E))] |An | ≤ E − F,

n=2

where An =

(−c/4)n−1 an . (κ)n−1 (n − 1)!

Since, f ∈ Rτ (A, B), in virtue of Lemma 3.1, T : =

T

≤

= ≤ =

∞ X

n [n(1 − F ) − (1 − E)] |An | ≤ E − F

n=2 ∞ X

(−c/4)n−1 (n(1 − F ) − (1 − E)) (A − B)|τ | (κ)n−1 (n − 1)! n=2 # " ∞ ∞ (−c/4)n−1 n−1 X X (−c/4) (A − B)|τ | (1 − F ) n − (1 − E) (κ)n−1 (n − 1)! (κ)n−1 (n − 1)! n=2 n=2 # " ∞ ∞ X X (|c|/4)n−1 (|c|/4)n−1 + (E − F ) (A − B)|τ | (1 − F ) (κ)n−1 (n − 2)! (κ)n−1 (n − 1)! n=2 n=2  
|c| (A − B)|τ | (1 − F ) up+1,b,|c| (1) + (E − F ) up,b,|c| (1) − 1 , 4κ

which is bounded above by E − F if (3.7) is satisfied. This completes the proof of Theorem 3.4.



For the choices of E = λ and F = 0, we get the following corollary. Corollary 3.2. Let c ∈ C and κ > 0. Suppose that f ∈ Rτ (A, B). If the Bessel’s inequality   1 |c| u (1) + λ up,b,|c| (1) ≤ λ +1 (3.8) 4κ p+1,b,|c| (A − B)|τ | is satisfied, then the operator Ip,b,c f ∈ Kλ .

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4. Sufficient conditions for Bessel’s integral operator to be in the class of S ∗ [E, F ] and K[E, F ] As in the work of Baricz [3], one can look at other linear operators acting on up,b,c to obtain similar results. In this section, we make use of this idea in the case of a particular integral operator. We continue our earlier work that was done in the earlier section. That is, we determine sufficient conditions for the integral operator g defined by (4.1) to be in the class of Janowski starlike and Janowski convex functions as follows: Z z g(z) = up (t)dt 0

= z+ = z+

∞ X bn−1 n=2 ∞ X n=2

n

zn

(−c/4)n−1 n z . (κ)n−1 (n!)

(4.1)

Theorem 4.1. Let c ∈ C and κ > 0. Further, let −1 ≤ F < E < 1 and −1 ≤ F ≤ 0. If the Bessel’s inequality 4(κ − 1) 4(κ − 1) (1 − F )up,b,|c| (1) − (1 − E) up−1,b,|c| (1) ≤ 2(E − F ) − (1 − E) (4.2) |c| |c| is satisfied, then the function g ∈ S ∗ [E, F ] where g is defined by (4.1). Proof. To prove the theorem, we have to show that ∞ X [n(1 − F ) − (1 − E)] |Bn | ≤ E − F, T :=

(4.3)

n=2

where Bn =

(−c/4)n−1 . (κ)n−1 (n)!

Then T

=

∞ X n=2

(−c/4)n−1 [n(1 − F ) − (1 − E)] (κ)n−1 (n)! ∞ X

∞

X (|c|/4)n−1 (|c|/4)n−1 − (1 − E) (κ)n−1 (n − 1)! (κ)n−1 (n)! n=2 n=2   
4(κ − 1) |c|/4 = (1 − F ) up,b,|c| (1) − 1 − (1 − E) up−1,b,|c| (1) − 1 − , |c| (κ − 1)

≤ (1 − F )

which is bounded above by E − F if (4.2) is satisfied.



For the choices of E = λ and F = 0, we get the following corollary. Corollary 4.1. Let c ∈ C and κ > 0. Further, let λ ≥ 0. If the Bessel’s inequality 4(κ − 1) 4(κ − 1) up,b,|c| (1) − (1 − λ) up−1,b,|c| (1) ≤ 2λ + (1 − λ) |c| |c|

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(4.4)

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is satisfied, then the function g ∈ Sλ∗ where g is defined by (4.1). Theorem 4.2. Let c ∈ C and κ > 0. Further, let −1 ≤ F < E < 1 and −1 ≤ F ≤ 0 and g be defined as in (4.1). If the Bessel’s inequality |c| u (1) + (E − F )up,b,|c| (1) ≤ 2(E − F ) 4κ p+1,b,|c| is satisfied, then the integral operator g ∈ K [E, F ]. (1 − F )

(4.5)

Proof. We have to show that T4 :=

∞ X

n (n(1 − F ) − (1 − E)) |Bn | ≤ E − F,

(4.6)

n=2

where Bn =

(−c/4)n−1 . (κ)n−1 (n)!

Then T4 = ≤

∞ X n=2 ∞ X

(−c/4)n−1 n [n(1 − F ) − (1 − E)] (κ)n−1 (n)! [n(1 − F ) − (1 − E)]

n=2

= (1 − F )

∞ X n=2

(|−c/4|)n−1 (κ)n−1 (n − 1)! ∞

X (|c|/4)n−1 (|c|/4)n−1 + (E − F ) (κ)n−1 (n − 2)! (κ)n−1 (n − 1)! n=2


|c| = (1 − F ) up+1,b,|c| (1) + (E − F ) up,b,|c| (1) − 1 , 4κ which is bounded above by E − F if (4.5) is satisfied.



For the choices of E = λ and F = 0, we get the following corollary. Corollary 4.2. Let c ∈ C and κ > 0. Further, let −1 ≤ F < E < 1 and −1 ≤ F ≤ 0 and g be defined as in (4.1). If the Bessel’s inequality |c| u (1) + λup,b,|c| (1) ≤ 2λ 4κ p+1,b,|c|

(4.7)

is satisfied, then g ∈ Kλ . 5. Consequences and observations Since the study generalized Bessel function permits the study of Bessel, modifed Bessel and spherical Bessel functions all together, each of these Theorems can also be stated for the Bessel, modified Bessel and spherical Bessel functions for special choices of the parameters b and c. However, we leave all these results for the interested readers.

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Acknowledgements The first and the third authors are supported by a grant from Department of Science and Technology, Government of India vide ref: SR/FTP/MS-022/2012 under fast track scheme and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2016R1D1A1A09916450), respectively. References [1] M. Anbu Durai and R. Parvatham, Convolutions with Hypergeometric functions, Bull. Malay. Math. Soc. (Second Series) 23 (2000), 153–161. ´ Baricz, Geometric properties of generalized Bessel function, publ. Math. Debrecan 73 (2008), 155– [2] A. 178. ´ Baricz, Generalized Bessel functions of the first kind, PhD thesis, Babes-Bolyai University, Cluj[3] A. Napoca, 2008. [4] T. R. Caplinger and W. M. Causey, A class of univalent functions, Proc. Amer. Math. Soc. 39 (1973), 357–361. [5] E. Deniz, H. Orhan and H.M. Srivastava, Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions, Taiwansese J. Math. 15(2) (2011), 883–917. [6] K. K. Dixit And S. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math. 26(9)(1995), 889–896. [7] A. Gangadharan, T. N. Shanmugam and H. M. Srivastava, Generalized hypergeometric functions associated with k-uniformly convex functions, Comput. Math. Appl. 44 (2002), 1515–1526. [8] A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), 364–370. [9] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 3 (1971), 469–474. [10] T. Janani and G. Murugusundaramoorthy, Inclusion results on subclasses of starlike and convex functions associated with struve functions, Ital. J. Pure Appl. Math. 32 (2014), 467–476. [11] S. R. Mondal and A. Swaminathan, Geometric properties of generalized Bessel functions, Bull. Malay. Math. Soc. 35(1) (2012), 179–194. [12] G. Murugusundaramoorthy, K. Vijaya and M. Kasturi, A note on subclasses of starlike and convex functions associated with Bessel functions, J. Nonlinear Funct. Anal. 2014, 1–11. [13] G. Murugusundaramoorthy and N. Magesh, On certain subclasses of analytic functions associated with hypergeometric functions, Appl. Math. Lett. 24 (2011), 494–500. [14] K. S. Padmanabhan, On a certain class of functions whose derivatives have a positive real part in the unit disc, Ann. Polon. Math. 23 (1970/71), 73–81. [15] S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987), 1057-1077 . [16] S. Porwal and K. K. Dixit, An application of generalized Bessel functions on certain analytic functions, Acta Univ. M. Belii. Ser. Math. 2013, 51–57,. [17] S. Porwal and K. K. Dixit, An application of certain convolution operator involving hypergeometric functions, J. Rajasthan Acad. Phys. Sci. 9(2) (2010), 173–186. [18] R. K. Raina, On Univalent and starlike Wright’s hypergeometric function, Rend. Sem. Mat. Univ. Padova 95 (1996), 11–22.

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[19] T. N. Shanmugam, Convolution and differential subordination, Internat. J. Math. Math. Sci. 12 (1989), 333–340 [20] T. N. Shanmugam, Hypergeometric functions in the geometric function theory, Appl. Math. Comput. 187 (2007), 433–444. [21] S. Sivasubramanian, T. Rosy and K. Muthunagai, Certain sufficient conditions for a subclass of analytic functions involving Hohlov operator, Comput. Math. Appl. 62 (2011), 4479–4485. [22] S. Sivasubramanian and J. Sok´ ol, Hypergeomtric transforms in certain classes of functions, Math. Comp. Modelling, 54 (2011), 3076–3082. [23] A. Swaminathan, Certain suffienct conditions on Gaussian hypergeometric functions, J. Inequal. Pure Appl. Math. 5(4), Art.83 (2004), 10 pp. [24] N. Yagmur and H. Orhan, Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal. 2013 (2013), Article ID 954513, 6 pp.

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ON THE JENSEN-TYPE INEQUALITY FOR THE g¯-INTEGRAL. JEONG GON LEE, LEE-CHAE JANG

Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 54538, Republic of Korea E-mail : [email protected] Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea E-mail : [email protected]

Abstract. We consider the pseudo-integral with respect to a σ − ⊕-measure of set-valued functions which was defined by Grb´ıc et al. Rom´ an-Flores et al.(2007) proved the Jensen type inequality for fuzzy integral with respect to a fuzzy measure. In this paper, we prove the Jensen type inequality for the g¯-integral with respect to a σ − ⊕g -measure under some sufficient conditions.

1. Introduction Benvenuti-Mesiar [2], Deschrijver [3], J. Fang [4], Mesiar-Pap [9], Ralescu-Adams [10], and Wu-Wang-Ma [12] provided the properties and applications of the generalized fuzzy integral which is a generalization of fuzzy integrals. The integrals of set-valued functions was introduced by Aumanm [1], and Jang [6,7] and Zhang-Guo [13] investigated some properties of the generalized fuzzy integral of set-valued functions. Not long ago, authors in [8,11] proved the Jensen type inequality for the fuzzy integral and for the generalized Sugeno integral. We consider the pseudo-integral with respect to a σ −⊕-measure of set-valued functions which was defined by Grb´ıc et al [5]. Rom´an-Flores et al. [11] proved the Jensen type inequality for fuzzy integral with respect to a fuzzy measure. In this paper, we prove the Jensen type inequality for g¯-integral with respect to a σ − ⊕g measure under some sufficient conditions.

2. Jensen type inequality for the g-integral Let [a, b] be a closed (in some cases can be considered semiclosed) subinterval of R = [−∞, ∞] and let ≼ be a total order on [a, b]. We introduce a semiring which is a structure ([a, b], ⊕, ⊙) as follows.

Key words and phrases. Sugeno integral, σ − ⊕-measure, g¯-integral, Jensen inequality. 1

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Definition 2.1. ([2,7,9]) (1) A function ⊕ : [a, b] × [a, b] −→ [a, b] is called a pseudo-addition if it is commutative, non-decreasing with respect to ≼, associative, and with a zero (natural) element denoted by 0, that is, for each x ∈ [a, b], 0 ⊕ x = x holds (usually 0 is either a or b). (2) A function ⊙ : [a, b] × [a, b] −→ [a, b] is called a pseudo-multiplication if it is commutative, positively non-decreasing, that is, x ≼ y implies x ⊙ z ≼ y ⊙ z for all z ∈ [a, b]+ = {x|x ∈ [a, b], 0 ≼ x}, associative and there exists a unit element 1 ∈ [a, b], that is, for each x ∈ [a, b], 1 ⊙ x = x. (3) The structure ([a, b], ⊕, ⊙) is called a semiring if 0 ⊙ x = 0 and ⊙ is a distributive pseudo-multiplication with respect to ⊕, that is, x ⊙ (y ⊕ z) = (x ⊙ y) ⊕ (x ⊙ z). (4) A set function µ : Σ −→ [a, b] is a σ − ⊕-measure if it satisfies the following two conditions: (i) µ(∅) = 0 (if ⊕ is not idempotent); ∞ (ii) µ(∪∞ i=1 Ai ) = ⊕i=1 µ(Ai ) holds for any sequence (Ai )i∈N of disjoint sets from Σ. We note that for a real interval [a, b] = [0, ∞], a pseudo-addition ⊕ and a pseudomultiplication ⊙ are generated by a strictly monotone bijective function g : [0, ∞] −→ [0, ∞], that is, pseudo-operations are given by x ⊕g y = g −1 (g(x) + g(y)) and x ⊙g y = g −1 (g(x)g(y)). Now, the pseudo-integral, known as the g-integral, of some measurable function f : X −→ [0, ∞] is (∫ ) ∫ ∫ ⊕g (g) f dµ = f ⊙g dµ = g −1 (g ◦ f ) d(g ◦ µ) (1) X

X

X

where g ◦ µ is the Lebesgue measure and the integral on the right-hand side of (A) is the Lebesgue integral (see [7,9]). Let (X, Σ, µ) be a σ − ⊕-measure ∑ space. Grb´ıc et al. [5] defined the pseudo-integral of an interval-valued function F on A ∈ as follows; ∫ ⊕ ∫ ⊕ F ⊙ dµ = { f ⊙ dµ|f ∈ S(F )} (2) A

A

1 where µ is a σ − ⊕-measure and S(F ) is the set of all selections of F ∑. Let L (η) be 1the set of all Lebesgue integrable functions on the Lebesgue space ([0, ∞), , η) and f ∈ L⊕ (µ) if and only if g ◦ f ∈ L1 (g ◦ µ). We introduce the definition of g-integrable boundedness of a set-valued function F as follows:

Definition 2.2. ([5]) Let g be a strictly monotone bijective function. A set-valued function F is g -integrable bounded if there is a function h ∈ L1⊕ (µ) such that (i) ⊕α∈F (x) α ≼ h(x), for the idempotent pseudo-addition, (ii) supα∈F (x) α ≼ h(x), for the pseudo-addition given by an increasing generator g, (iii) inf α∈F (x) α ≼ h(x), for the pseudo-addition given by a decreasing generator g. From Proposition 11 in [5], we note that if F is a pseudo-integrable bounded set-valued ∫⊕ function, then F is pseudo-integrable, that is, X F ⊙ dµ ̸= ∅.

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Theorem 2.1. (Theorem 2.4 [5]) Let F be a pseudo-integrable bounded interval-valued function with border functions fl and fr . Then we have [∫ ⊕ ] ∫ ⊕ ∫ ⊕ F ⊙ dµ = fl ⊙ dµ, fr ⊙ dµ , (3) X

X

X

Now, we obtain the following Jensen type inequality for the g-integral with respect to a σ − ⊕g -measure. ∑ Theorem 2.2. Let g be∫ a decreasing function and (X, , g ◦µ) be the Lebesgue measure space and f ∈ L1⊕ µ with (g) X f dµ = m. If Φ : [o, ∞) −→ [0, ∞) is strictly increasing function such that Φ(x) ≤ x, for every x ∈ [0, m] and Φ(f ) ∈ L1⊕ (µ), then we have ( ∫ ) ∫ Φ (g) f dµ ≤ (g) Φ(f )dµ. (4) X

X

Proof. Since Φ(f ) ≤ f and g is decrasing, g ◦ Φ(f ) ≥ g ◦ f.

(5)

By (5) and monotonicity of the Lebesgue integral with respect to g ◦ µ, we have ∫ ∫ g ◦ Φ(f ) dg ◦ µ ≥ g ◦ f dg ◦ µ. X

(6)

X

Since g −1 is decreasing, by (6), we have g

−1

∫ g ◦ Φ(f ) dg ◦ µ ≤ g

−1

∫ g ◦ f dg ◦ µ.

X

By (7),

(7)

X

( ∫ ) Φ (g) f dµ =

( ) ∫ −1 Φ g g ◦ f dg ◦ µ X ∫ ≤ g −1 g ◦ f dg ◦ µ ∫X ≤ g −1 g ◦ Φ(f ) dg ◦ µ ∫X = (g) Φ(f )dµ.

X

X

□ ∑ Theorem 2.3. Let g be an increasing function and (X, , g ◦ µ) be the Lebesgue measure ∫ space and f ∈ L1⊕ (µ) with (g) X f dµ = m. If Φ : [0, ∞) −→ [0, ∞) is strictly increasing function such that Φ(x) ≥ x, for every x ∈ [0, m], and Φ(f ) ∈ L1⊕ (µ), then ( ∫ ) ∫ Φ (g) f dµ ≥ (g) Φ(f )dµ. (8) X

X

Proof. Since Φ(f ) ≤ f and g is increasing, g ◦ Φ(f ) ≤ g ◦ f.

(9)

By (9) and monotonicity of the Lebesgue integral with respect to g ◦ µ, we have

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∫

∫ g ◦ Φ(f ) dg ◦ µ ≤

X

g ◦ f dg ◦ µ.

(10)

X

Since g −1 is increasing, by (10), we have g −1

∫

g ◦ Φ(f ) dg ◦ µ ≤ g −1

∫

X

g ◦ f dg ◦ µ.

(11)

X

By (11), ( ∫ ) f dµ = Φ (g) X

( ) ∫ Φ g −1 g ◦ f dg ◦ µ X ∫ −1 ≥ g (g ◦ f ) dg ◦ µ ∫X ≥ g −1 g ◦ Φ(f ) dg ◦ µ ∫X Φ(f )dµ. = (g) X

□

3. Jensen type inequality for the g¯-integral Let I([0, ∞]) be the set of all bounded closed intervals in [0, ∞] as follows : I([0, ∞]) = {¯ a = [al , ar ]|al , ar ∈ [0, ∞] and al ≤ ar } For these intervals, we define the order, the strictly order, and strong strictly order of intervals as follows: Definition 3.1. ([5]) If a ¯ = [al , ar ], ¯b = [bl , br ] ∈ I([0, ∞]), then we define order (≤), strictly order ( 0 is said to be m-convex, where m ∈ [0, 1], if for every x, y ∈ [0, b] and t ∈ [0, 1], we have f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y).

(1.2)

The class of (α, m)-convex functions was also first introduced in [16] and it is defined as follows: Definition 1.2 ([16]). The function f : [0, b] → R, b > 0 is said to be (α, m)-convex, where (α, m) ∈ [0, 1]2 , if we have f (tx + m(1 − t)y) ≤ tα f (x) + m(1 − tα )f (y)

(1.3)

for all x, y ∈ [0, b] and t ∈ [0, 1]. Also, the m-convex and (α, m)-convex functions on the co-ordinates defined in a rectangle from the plane were introduced as follows. Definition 1.3 ([17]). Let 4 := [0, b] × [0, d] be the bidimensional interval in R20 with b > 0 and d > 0. For some m ∈ [0, 1], the function f : 4 → R is said to be m-convex if the following inequality f (λx + (1 − λ)z, λy + m(1 − λ)w) ≤ λf (x, y) + m(1 − λ)f (z, w) (1.4) 1991 Mathematics Subject Classification. 15A45, 15A46, 15A47, 26A51, 26D15. Key words and phrases. integral inequality; operator m-convex function on the co-ordinates; operator (α, m)convex function on the co-ordinates. This paper was typeset using AMS-LATEX. 1

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holds for all (x, y), (z, w) ∈ 4 and λ ∈ (0, 1). Definition 1.4 ([17]). For some m ∈ [0, 1], a function f : 4 := [0, b] × [0, d] ⊆ R20 → R which is m-convex on 4 will be called m-convex on the co-ordinates with b > 0 and d > 0 if the partial mappings fy : [a, b] → R, fy (u) := f (u, y) and fx : [c, d] → R, fx (v) := f (x, v) are m-convex for all y ∈ [c, d] and x ∈ [a, b]. Definition 1.5 ([17]). Let 4 := [0, b] × [0, d] be the bidimensional interval in R20 with b > 0 and d > 0. For some (α, m) ∈ [0, 1]2 , the function f : 4 → R is said to be (α, m)-convex if the following inequality f (λx + (1 − λ)z, λy + m(1 − λ)w) ≤ λα f (x, y) + m(1 − λα )f (z, w)

(1.5)

holds for all (x, y), (z, w) ∈ 4 and λ ∈ (0, 1). Definition 1.6 ([17]). For some (α, m) ∈ [0, 12 ], a function f : 4 := [0, b] × [0, d] ⊆ R20 → R which is (α, m)-convex on 4 will be called (α, m)-convex on the co-ordinates with b > 0 and d > 0 if the partial mappings fy : [a, b] → R, fy (u) := f (u, y) and fx : [c, d] → R, fx (v) := f (x, v) are (α, m)-convex for all y ∈ [c, d] and x ∈ [a, b]. In recent years several extensions and generalizations have been considered for classical convexity. A significant generalization of convex functions is that of operator functions introduced by S. S. Dragomir in [6]. We review the operator order in B(H) and the continuous functional calculus for a bounded self-adjoint operator. For self-adjoint operators A, B ∈ B(H), we write A ≤ B if hAx, xi ≤ hBx, xi for every vector x ∈ H, we call it the operator order. Let A be a bounded self-adjoint linear operator on a complex Hilbert space (H; h., .i). The Gelfand map establishes a ∗-isometrically isomorphism Φ between the set C(Sp(A)) of all continuous complex-valued functions defined on the spectrum of A, denoted Sp(A), and the C ∗ -algebra C ∗ (A) generated by A and the identity operator 1H on H as follows (see for instance [8], p.3). For any f, g ∈ C(Sp(A)) and any α, β ∈ C, we have (i) (ii)

Φ(αf + βg) = αΦ(f ) + βΦ(g); Φ(f g) = Φ(f )Φ(g)

(iii) kΦ(f ) k=k f k :=

and sup

Φ(f ∗ ) = Φ(f )∗ ; | f |;

t∈Sp(A)

(iv)

Φ(f0 ) = 1H

and Φ(f1 ) = A,

where

f0 (t) = 1

and f1 (t) = t

for t ∈ Sp(A).

With this notation, we define f (A) := Φ(f )

for all f ∈ C(Sp(A))

(1.6)

and we call it the continuous functional calculus for a bounded self-adjoint operator A. A real valued continuous function f on an interval I ⊆ R is said to be operator convex (operator concave) if the operator inequality f ((1 − λ)A + λB) ≤ (≥)(1 − λ)f (A) + λf (B)

475

(1.7)

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3

holds in the operator order in B(H), for all λ ∈ [0, 1] and for every bounded self-adjoint operators A and B in B(H) whose spectra are contained in I. In [20], Wang defined operator m-convex and (α, m)-convex functions in the following way:. Definition 1.7. Let [0, b] ⊆ R0 with b > 0 and K be a convex set of B(H)+ . A continuous function f : [0, b] → R is said to be operator m-convex on [0, b] for operators in K, if f (tA + m(1 − t)B) ≤ tf (A) + m(1 − t)f (B)

(1.8)

in the operator order in B(H), for all t ∈ [0, 1] and every positive operators A and B in K whose spectra are contained in [0, b] and for some fixed m ∈ [0, 1]. Definition 1.8. Let [0, b] ⊆ R0 with b > 0 and K be a convex set of B(H)+ . A continuous function f : [0, b] → R is said to be operator (α, m)-convex on [0, b] for operators in K, if f (tA + m(1 − t)B) ≤ tα f (A) + m(1 − tα )f (B)

(1.9)

in the operator order in B(H), for all t ∈ [0, 1] and every positive operators A and B in K whose spectra are contained in [0, b] and for some fixed (α, m) ∈ [0, 1]2 . Also, author proved the following inequalities in [20]: Theorem 1.1 ([20]). Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ (0, 1]2 . Then for all positive operator A, B ∈ K with spectra in R0 , the following inequality holds: (

) Z 1 A B f (B) + αmf m f (A) + αmf m , . (1.10) f (tA + (1 − t)B) dt ≤ min α+1 α+1 0 Theorem 1.2 ([20]). Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ (0, 1]2 . Then for all positive operator A, B ∈ K with spectra in R0 , the following inequalities hold:     Z 1 A+B 1 (1 − t)A + tB α f ≤ α f (tA + (1 − t)B) + m(2 − 1)f dt 2 2 0 m       A B 1 f (A) + f (B) + m(α + 2α − 1) f +f ≤ α+1 2 (α + 1) m m      A B 2 α + αm (2 − 1) f +f . (1.11) m2 m2 Theorem 1.3 ([20]). Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ (0, 1]2 . Then for all positive operator A, B ∈ K with spectra in R0 , the following inequality holds:
  A
Z 1 B f (A) + f (B) + αm f m +f m . (1.12) f (tA + (1 − t)B) dt ≤ 2(α + 1) 0 Theorem 1.4 ([20]). Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ (0, 1]2 . Then for all positive operator A, B ∈ K with spectra in R0 , the following inequality holds: Z 1   (1 + mα)[f (A) + f (B)] f (tA + m(1 − t)B) + f (tB + m(1 − t)A) dt ≤ . (1.13) α+1 0

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Theorem 1.5 ([20]). Let the continuous function f : R0 → R be operator (α, m)-convex for operators in K ⊆ B(H)+ with (α, m) ∈ (0, 1]2 . Then for all positive operator A, B ∈ K with spectra in R0 , the following inequalities hold:   m 2−m B + (mA) f 2 2   Z 1
1 (1 − t)(2 − m)B + tm2 A 2 α f t(2 − m)B + (1 − t)m A + m(2 − 1)f ≤ α dt 2 0 m 
1 ≤ α f ((2 − m)B) + m α + 2α − 1 f (mA) 2 (α + 1)   (2 − m)B 2 α . (1.14) + m α(2 − 1)f m2 For recent results related to Hermite-Hadamard type inequalities are given in [1], [4], [5], [7], [8], [9], [10], [13], [14], and plenty of references therein. The main purpose of this paper is to establish some new Hadamard type inequalities for operator m)-convex and (α, m)-convex functions on the co-ordinates. 2. operator co-ordinated m-convex and (α, m)-convex functions Let I1 , I2 be real intervals and let f : I1 × I2 → R be a Borel measurable and essentially bounded function. Let X = (X1 , X2 ) be a 2-tuple of bounded self-adjoint operators on Hilbert spaces H1 , H2 such that the spectrum of Xi is contained in Ii for i = 1, 2. We say that such a 2-tuple is in the domain of f . If Z λi Ei (dλi ), i = 1, 2 Xi = Ii

is the spectral decomposition of Xi where Ei is a bounded positive measure on Ii , we define Z f (X) = f (λ1 , λ2 )E1 (dλ1 ) ⊗ E2 (dλ2 ) I1 ×I2

as a bounded self-adjoint operator on the tensor product H1 ⊗ H2 . If the Hilbert spaces are of finite dimension, then the above integrals become finite sums, and we may consider the functional calculus for arbitrary real functions. This construction have the property that f (X1 , X2 ) = f1 (X1 ) ⊗ f2 (X2 ), whenever f can be separated as a product f (t1 , t2 ) = f1 (t1 )f2 (t2 ) of 2 functions each depending on only one variable. With above functional calculus, we say that a function f : I1 × I2 → R is operator convex if f is continuous and the operator inequality f (tX + (1 − t)Y ) ≤ tf (X) + (1 − t)f (Y )

(2.1)

holds for all 2-tuples of self-adjoint operators X = (X1 , X2 ) and Y = (Y1 , Y2 ) in the domain of f acting on any Hilbert spaces H1 , H2 and for all t ∈ [0, 1]. In [21], Hermite-Hadamard type inequality for the co-ordinated operator convex functions is given. Theorem 2.1. Suppose that a continuous function f : I1 × I2 ⊆ R2 → R is operator convex on the co-ordinates for all 2-tuples of self-adjoint operators in the domain of f acting on any Hilbert spaces H1 , H2 . Then we have the inequalities   A+C B+D f , 2 2

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1 ≤ 2 Z ≤

0 1

Z

1 4



B+D f tA + (1 − t)C, 2



Z dt + 0

1



  A+C f , λB + (1 − λ)D dλ 2

1

f (tA + (1 − t)C, λB + (1 − λ)D) dt dλ

0

≤

1

Z

5

Z

0 1

1

Z f (tA + (1 − t)C, B) dt +

Z

0 1

f (tA + (1 − t)C, D) dt 0 1

Z f (A, λB + (1 − λ)D) dλ +

+ 0

 f (C, λB + (1 − λ)D) dλ

0

f (A, B) + f (A, D) + f (C, B) + f (C, D) , ≤ 4 where (A, B), (C, D) ∈ B(H1 ) ⊗ B(H2 ) with spectra in I1 × I2 .

(2.2)

For some fundamental results on operator convex and operator monotone functions of several variables, see [11], [12], [15], and the references therein Now we give the concepts of operator m-convex and (α, m)-convex functions on the coordinates. Definition 2.1. A continuous function f : [0, b]×[0, d] ⊆ R20 → R is said to be operator m-convex with b > 0 and d > 0 for some fixed m ∈ [0, 1] if the operator inequality f (tX1 + (1 − t)Y1 , tX2 + m(1 − t)Y2 ) ≤ tf (X1 , X2 ) + m(1 − t)f (Y1 , Y2 )

(2.3)

holds for all 2-tuples of self-adjoint operators X = (X1 , X2 ) and Y = (Y1 , Y2 ) in the domain of f acting on any Hilbert spaces H1 , H2 and for all t ∈ (0, 1). Definition 2.2. A continuous function f : [0, b] × [0, d] ⊆ R20 → R which is operator m-convex on [0, b] × [0, d] with b > 0 and d > 0 is said to be operator m-convex on the co-ordinates for some fixed m ∈ [0, 1] if the partial mapping fX2 : I1 → R, fX2 (u) := f (u, X2 ) and fX1 : I2 → R, fX1 (v) := f (X1 , v) are operator m-convex for all operators X2 ∈ B(H2 ) and X1 ∈ B(H1 ) whose spectra are contained in [0, d] and [0, b], respectively. Definition 2.3. A continuous function f : [0, b] × [0, d] ⊆ R20 → R is said to be operator (α, m)-convex with b > 0 and d > 0 for some fixed (α, m) ∈ [0, 1]2 if the operator inequality f (tX1 + (1 − t)Y1 , tX2 + m(1 − t)Y2 ) ≤ tα f (X1 , X2 ) + m(1 − tα )f (Y1 , Y2 )

(2.4)

holds for all 2-tuples of self-adjoint operators X = (X1 , X2 ) and Y = (Y1 , Y2 ) in the domain of f acting on any Hilbert spaces H1 , H2 and for all t ∈ (0, 1). Definition 2.4. A continuous function f : [0, b] × [0, d] ⊆ R20 → R which is operator (α, m)convex on [0, b] × [0, d] with b > 0 and d > 0 is said to be operator (α, m)-convex on the co-ordinates for some fixed (α, m) ∈ [0, 1]2 if the partial mapping fX2 : I1 → R, fX2 (u) := f (u, X2 ) and fX1 : I2 → R, fX1 (v) := f (X1 , v) are operator (α, m)-convex for all operators X2 ∈ B(H2 ) and X1 ∈ B(H1 ) whose spectra are contained in [0, d] and [0, b], respectively.

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Remark 2.1. It can be easily seen that for (α, m) ∈ {(1, 1), (1, m)} one obtains the classes of operator convex and operator m-convex functions of two variables, respectively. The following lemmas hold: Lemma 2.1. For b > 0, d > 0, and some fixed m ∈ [0, 1], every operator m-convex mapping f : [0, b] × [0, d] ⊆ R20 → R is operator m-convex on the co-ordinates, but the converse is not generally true. Proof. Suppose that f is operator m-convex mapping on [0, b] × [0, d]. Consider fX1 : [0, d] → R, fX1 (v) := f (X1 , v). Then for all t ∈ (0, 1) and operators A, C ∈ B(H2 ) with spectra in [0, d], one has fX1 (tA + m(1 − t)C) = f (tX1 + (1 − t)X1 , tA + m(1 − t)C) ≤ tf (X1 , A) + m(1 − t)f (X1 , C) = tfX1 (A) + m(1 − t)fX1 (C), where X1 ∈ B(H1 ) with spectra in [0, b]. It shows the operator m-convexity of fX1 . The fact that fX2 : [0, b] → R, fX2 (u) := f (u, X2 ) is also operator m-convex on [0, b] for all operators X2 ∈ B(H2 ) with spectra in [0, d] goes likewise and we shall omit the details. In [21], authors gave a mapping f : [0, 1]2 → R0 defined by f (r1 , r2 ) = r1 × r2 which is operator convex on the co-ordinates but is not operator convex. We consider the same function with m = 1 to prove that the operator m-convexity on the co-ordinates does not imply the operator m-convexity. The Lemma 2.1 is thus proved.  Similarly, we state the following elementary results without proof. Lemma 2.2. For b > 0, d > 0, and some fixed (α, m) ∈ [0, 1]2 , every operator (α, m)-convex mapping f : [0, b] × [0, d] ⊆ R20 → R is operator (α, m)-convex on the co-ordinates, but the converse is not generally true. 3. Hermite-Hadamard type inequalities for operator m-convex and (α, m)-convex functions on the co-ordinates We will now point out some new inequalities of the Hermite-Hadamard type. Theorem 3.1. Let some fixed (α, m) ∈ (0, 1]2 and a continuous function f : R20 → R be operator (α, m)-convex on the co-ordinates for all 2-tuples of positive self-adjoint operators in the domain of f acting on any Hilbert spaces H1 , H2 . Then one has Z 1Z 1 min{v1 , v2 } + min{v3 , v4 } , (3.1) f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt ≤ 2(α + 1) 0 0 where (A, B), (C, D) ∈ B(H1 ) × B(H2 ) with spectra in R20 , and  Z 1 Z 1  D dt, v1 = f (tA + (1 − t)C, B) dt + αm f tA + (1 − t)C, m 0 0  Z 1 Z 1  B v2 = f (tA + (1 − t)C, D) dt + αm f tA + (1 − t)C, dt, m 0 0  Z 1 Z 1  C v3 = f (A, λB + (1 − λ)D) dλ + αm f , λB + (1 − λ)D dλ, m 0 0  Z 1 Z 1  A v4 = f (C, λB + (1 − λ)D) dλ + αm f , λB + (1 − λ)D dλ. m 0 0

479

(3.2)

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Proof. Since the spectrum of tA + (1 − t)C and λB + (1 − λ)D are contained in R0 , and f is R1 R1 continuous, the operator valued integrals 0 f (tA + (1 − t)C) dt, 0 f (λB + (1 − λ)D) dλ and R1R1 f (tA + (1 − t)C, λB + (1 − λ)D) dt dλ exist. 0 0 From the operator co-ordinated (α, m)-convexity of f and the inequality (1.10) it is easy to see that Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dt 0 (
C , λB + (1 − λ)D) f (A, λB + (1 − λ)D)) + αmf m ≤min , α+1
) A f (C, λB + (1 − λ)D)) + αmf m , λB + (1 − λ)D) . α+1 Integrating this inequality on [0, 1] over λ, we deduce Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dt dλ 0 0 (Z  Z 1  1 1 C ≤ f (A, λB + (1 − λ)D) dλ + αm f min , λB + (1 − λ)D dλ, α+1 m 0 0  ) Z 1 Z 1  A f (C, λB + (1 − λ)D) dλ + αm , λB + (1 − λ)D dλ . f m 0 0 By a similar argument we get Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt 0 0 (Z  Z 1  1 1 D ≤ min f (tA + (1 − t)C, B) dt + αm f tA + (1 − t)C, dt, α+1 m 0 0  ) Z 1 Z 1  B f (tA + (1 − t)C, D) dt + αm dt . f tA + (1 − t)C, m 0 0 Summing the inequalities (3.3) and (3.4) and dividing by 2, we get the inequality (3.1). The proof thus is complete.

(3.3)

(3.4)



Corollary 3.1.1. Under the assumptions of Theorem 3.1, choosing α = 1, we get the inequality for operator m-convex: Z 1Z 1 min{u1 , u2 } + min{u3 , u4 } f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt ≤ , (3.5) 4 0 0 where 1

1

  D f tA + (1 − t)C, dt, m 0 0  Z 1 Z 1  B u2 = f (tA + (1 − t)C, D) dt + m f tA + (1 − t)C, dt, m 0 0  Z 1 Z 1  C , λB + (1 − λ)D dλ, u3 = f (A, λB + (1 − λ)D) dλ + m f m 0 0 Z

u1 =

Z

f (tA + (1 − t)C, B) dt + m

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Z

1

Z f (C, λB + (1 − λ)D) dλ + m

u4 =

0

0

1



 A f , λB + (1 − λ)D dλ. m

Furthermore, for α, m = 1 we have Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt 0 0 Z 1 Z 1 1 ≤ f (tA + (1 − t)C, B) dt + f (tA + (1 − t)C, D) dt 4 0 0  Z 1 Z 1 + f (A, λB + (1 − λ)D) dλ + f (C, λB + (1 − λ)D) dλ . 0

(3.6)

(3.7)

0

Theorem 3.2. Let some fixed (α, m) ∈ (0, 1]2 and a continuous function f : R20 → R be operator (α, m)-convex on the co-ordinates for all 2-tuples of positive self-adjoint operators in the domain of f acting on any Hilbert spaces H1 , H2 . Then one has   A+C B+D f , 2 2 Z 1      A+C A + C (1 − λ)B + λD 1 α f , λB + (1 − λ)D + m(2 − 1)f , dλ ≤ α+1 2 2 2 m 0     Z 1  B+D (1 − t)A + tC B + D + f tA + (1 − t)C, + m(2α − 1)f , dt 2 m 2 0          A+C B+D 1 A+C B+D f ,B + f , D + f A, ≤ α+2 + f C, 2 (α + 1) 2 2 2 2          A+C B A+C D A B+D C B+D + m(α + 2α − 1) f , +f , +f , +f , 2 m 2 m m 2 m 2      A + C B A + C D + αm2 (2α − 1) f , 2 +f , 2 2 m 2 m     A B+D C B+D +f , , +f , (3.8) m2 2 m2 2 where (A, B), (C, D) ∈ B(H1 ) × B(H2 ) with spectra in R20 . Proof. By operator co-ordinated (α, m)-convexity of f and and the inequality (1.11), we can give   A+C B+D f , 2 2      Z 1 (1 − t)A + tC B + D 1 B+D α ≤ α f tA + (1 − t)C, + m(2 − 1)f , dt 2 0 2 m 2      1 B+D B+D ≤ α+1 f A, + f C, 2 (α + 1) 2 2      A B + D C B+D α + m(α + 2 − 1) f , +f , m 2 m 2      A C B + D B + D 2 α + αm (2 − 1) f +f (3.9) , , m2 2 m2 2 and 

A+C B+D f , 2 2



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9

   Z 1  A+C A + C (1 − λ)B + λD α f , λB + (1 − λ)D + m(2 − 1)f , dλ 2 2 m 0      A+C 1 A+C ≤ α+1 f ,B + f ,D 2 (α + 1) 2 2      A+C B A+C D α + m(α + 2 − 1) f , +f , 2 m 2 m      A+C D A+C B 2 α (3.10) + αm (2 − 1) f , 2 +f , 2 2 m 2 m

1 ≤ α 2

Summing the inequalities (3.9) and (3.10) and dividing by 2, we get the inequality (3.8). The proof is completed.



Corollary 3.2.1. Under the assumptions of Theorem 3.2, choosing α = 1, we get the inequality for operator m-convex:   A+C B+D , f 2 2 Z 1      A+C 1 A + C (1 − λ)B + λD f ≤ , λB + (1 − λ)D + mf , dλ 4 0 2 2 m     Z 1  B+D (1 − t)A + tC B + D + , f tA + (1 − t)C, + mf dt 2 m 2 0          1 A+C A+C B+D B+D ≤ f ,B + f , D + f A, + f C, 16 2 2 2 2          A+C B A+C D A B+D C B+D + 2m f , +f , +f , +f , 2 m 2 m m 2 m 2          A+C B C B+D A+C D A B+D + m2 f , 2 +f , 2 +f , + f , . (3.11) 2 m 2 m m2 2 m2 2 Furthermore, for α, m = 1 we have   A+C B+D , f 2 2 Z 1     Z 1  1 A+C B+D ≤ , λB + (1 − λ)D dλ + dt f f tA + (1 − t)C, 2 0 2 2 0          1 A+C A+C B+D B+D ≤ f ,B + f , D + f A, + f C, . (3.12) 4 2 2 2 2 Theorem 3.3. Let some fixed (α, m) ∈ (0, 1]2 and a continuous function f : R20 → R be operator (α, m)-convex on the co-ordinates for all 2-tuples of positive self-adjoint operators in the domain of f acting on any Hilbert spaces H1 , H2 . Then one has Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt 0 0 Z 1 Z 1 1 ≤ f (tA + (1 − t)C, B) dt + f (tA + (1 − t)C, D) dt 4(α + 1) 0 0 Z 1     Z 1  B D + αm f tA + (1 − t)C, dt + f tA + (1 − t)C, dt m m 0 0 Z 1 Z 1 + f (A, λB + (1 − λ)D) dλ + f (C, λB + (1 − λ)D) dλ 0

0

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Z

1

+ αm 0

   Z 1  A C f f , λB + (1 − λ)D dλ + , λB + (1 − λ)D dλ , m m 0 

(3.13)

where (A, B), (C, D) ∈ B(H1 ) × B(H2 ) with spectra in R20 . Proof. Using operator co-ordinated (α, m)-convexity of f and the inequality (1.12), we can write Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dt 0  1 f (A, λB + (1 − λ)D) + f (C, λB + (1 − λ)D) ≤ 2(α + 1)      A C + αm f , λB + (1 − λ)D + f , λB + (1 − λ)D . m m Integrating this inequality on [0, 1] over λ, we deduce Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dt dλ 0 0 Z 1 Z 1 1 ≤ f (C, λB + (1 − λ)D) dλ f (A, λB + (1 − λ)D) dλ + 2(α + 1) 0 0 Z 1     Z 1  A C + αm f , λB + (1 − λ)D dλ + f , λB + (1 − λ)D dλ . m m 0 0 By a similar argument we get Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt 0 0 Z 1 Z 1 1 ≤ f (tA + (1 − t)C, B) dt + f (tA + (1 − t)C, D) dt 2(α + 1) 0 0    Z 1  Z 1  D B dt + f tA + (1 − t)C, dt . + αm f tA + (1 − t)C, m m 0 0

(3.14)

(3.15)

Summing the inequalities (3.14) and (3.15) and dividing by 2, we get the inequality (3.13). The proof thus is complete.



Corollary 3.3.1. Under the assumptions of Theorem 3.3, choosing α = 1, we get the inequality for operator m-convex: Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt 0 0 Z 1  Z 1  Z 1 1 B ≤ dt f (tA + (1 − t)C, B) dt + f (tA + (1 − t)C, D) dt + m f tA + (1 − t)C, 8 0 m 0 0   Z 1 Z 1  Z 1 D + f tA + (1 − t)C, dt + f (A, λB + (1 − λ)D) dλ + f (C, λB + (1 − λ)D) dλ m 0 0 0 Z 1     Z 1  A C f f +m , λB + (1 − λ)D dλ + , λB + (1 − λ)D dλ . (3.16) m m 0 0 Furthermore, for α, m = 1 we have Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt 0

0

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1 ≤ 4

Z

1

1

Z

f (tA + (1 − t)C, D) dt

f (tA + (1 − t)C, B) dt + 0

0

Z

1

Z f (A, λB + (1 − λ)D) dλ +

+

11

0

1

 f (C, λB + (1 − λ)D) dλ .

(3.17)

0

Theorem 3.4. Let some fixed (α, m) ∈ (0, 1]2 and a continuous function f : R20 → R be operator (α, m)-convex on the co-ordinates for all 2-tuples of positive self-adjoint operators in the domain of f acting on any Hilbert spaces H1 , H2 . Then one has Z 1Z 1   f (tA + m(1 − t)C, λB + m(1 − λ)D) + f (tC + m(1 − t)A, λB + m(1 − λ)D) dt dλ 0 0 Z 1 Z 1 1 + mα f (tA + m(1 − t)C, D) dt ≤ f (tA + m(1 − t)C, B) dt + 2(α + 1) 0 0  Z 1 Z 1 + f (A, λB + m(1 − λ)D) dλ + f (C, λB + m(1 − λ)D) dλ , (3.18) 0

0

where (A, B), (C, D) ∈ B(H1 ) × B(H2 ) with spectra in R20 . Proof. Using operator co-ordinated (α, m)-convexity of f and the inequality (1.13), we can write Z 1   f (tA + m(1 − t)C, λB + m(1 − λ)D) + f (tC + m(1 − t)A, λB + m(1 − λ)D) dt 0

(1 + mα)[f (A, λB + m(1 − λ)D) + f (C, λB + m(1 − λ)D] ≤ . α+1 Integrating this inequality on [0, 1] over λ, we deduce Z 1Z 1   f (tA + m(1 − t)C, λB + m(1 − λ)D) + f (tC + m(1 − t)A, λB + m(1 − λ)D) dt dλ 0 0 Z 1  Z 1 1 + mα f (A, λB + m(1 − λ)D) dλ + f (C, λB + m(1 − λ)D) dλ . (3.19) ≤ α+1 0 0 By a similar argument we get Z 1Z 1   f (tA + m(1 − t)C, λB + m(1 − λ)D) + f (tC + m(1 − t)A, λB + m(1 − λ)D) dt dλ 0 0 Z 1  Z 1 1 + mα ≤ f (tA + m(1 − t)C, B) dt + f (tA + m(1 − t)C, D) dt . (3.20) α+1 0 0 Summing the inequalities (3.19) and (3.20) and dividing by 2, we get the inequality (3.18). The proof thus is complete.



Corollary 3.4.1. Under the assumptions of Theorem 3.3, choosing α = 1, we get the inequality for operator m-convex: Z 1Z 1   f (tA + m(1 − t)C, λB + m(1 − λ)D) + f (tC + m(1 − t)A, λB + m(1 − λ)D) dt dλ 0 0 Z 1 Z 1 1+m f (tA + m(1 − t)C, B) dt + f (tA + m(1 − t)C, D) dt ≤ 4 0 0  Z 1 Z 1 + f (A, λB + m(1 − λ)D) dλ + f (C, λB + m(1 − λ)D) dλ . (3.21) 0

0

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Furthermore, for α, m = 1 we have Z 1Z 1 f (tA + (1 − t)C, λB + (1 − λ)D) dλ dt 0 0 Z 1 Z 1 1 ≤ f (tA + (1 − t)C, D) dt f (tA + (1 − t)C, B) dt + 4 0 0  Z 1 Z 1 + f (A, λB + (1 − λ)D) dλ + f (C, λB + (1 − λ)D) dλ . 0

(3.22)

0

Theorem 3.5. Let some fixed (α, m) ∈ (0, 1]2 and a continuous function f : R20 → R be operator (α, m)-convex on the co-ordinates for all 2-tuples of positive self-adjoint operators in the domain of f acting on any Hilbert spaces H1 , H2 . Then one has   2−m m 2−m m f C + (mA), D + (mB) 2 2 2 2 Z 1   2−m 1 m 2 f ≤ α+1 C + (mA), λ(2 − m)D + (1 − λ)m B dλ 2 2 2 0  Z 1  2−m m (1 − λ)(2 − m)D + λm2 B α f + m(2 − 1) C + (mA), dλ 2 2 m 0  Z 1  2−m m + f t(2 − m)C + (1 − t)m2 A, D + (mB) dt 2 2 0   Z 1  2 m (1 − t)(2 − m)C + tm A 2 − m + m(2α − 1) , D + (mB) dt f m 2 2  0   
2−m 1 2−m m m ≤ α+2 f C + (mA), (2 − m)D + m α + 2α − 1 f C + (mA), mB 2 (α + 1) 2 2 2 2     2−m m (2 − m)D m 2−m + m2 α(2α − 1)f C + (mA), D + (mB) + f (2 − m)C, 2 2 m2 2 2  
2−m m + m α + 2α − 1 f mA, D + (mB) 2 2   (2 − m)C 2 − m m + m2 α(2α − 1)f , D + (mB) , (3.23) m2 2 2 where (A, B), (C, D) ∈ B(H1 ) × B(H2 ) with spectra in R20 . Proof. From operator co-ordinated (α, m)-convexity of f and the inequality (1.14), we can deduce   m 2−m m 2−m C + (mA), D + (mB) f 2 2 2 2  Z 1  1 2−m m 2 ≤ α f t(2 − m)C + (1 − t)m A, D + (mB) 2 0 2 2   2 (1 − t)(2 − m)C + tm A 2 − m m α + m(2 − 1)f , D + (mB) dt m 2 2     
1 2−m m 2−m m α ≤ α+1 f (2 − m)C, D + (mB) + m α + 2 − 1 f mA, D + (mB) 2 (α + 1) 2 2 2 2   (2 − m)C 2 − m m + m2 α(2α − 1)f , D + (mB) (3.24) m2 2 2

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13

and   2−m m 2−m m f C + (mA), D + (mB) 2 2 2 2  Z 1  2−m 1 m f ≤ α C + (mA), λ(2 − m)D + (1 − λ)m2 B 2 0 2 2   2−m m (1 − λ)(2 − m)D + λm2 B + m(2α − 1)f C + (mA), dλ 2 2 m     
2−m 1 2−m m m ≤ α+1 f C + (mA), (2 − m)D + m α + 2α − 1 f C + (mA), mB 2 (α + 1) 2 2 2 2   2−m m (2 − m)D . (3.25) + m2 α(2α − 1)f C + (mA), 2 2 m2 Summing the inequalities (3.24) and (3.25) and dividing by 2, we get the inequality (3.23). The proof is completed.



Corollary 3.5.1. Under the assumptions of Theorem 3.5, choosing α = 1, we get the inequality for operator m-convex:   2−m m 2−m m f C + (mA), D + (mB) 2 2 2 2 Z 1   1 m 2−m ≤ C + (mA), λ(2 − m)D + (1 − λ)m2 B dλ f 4 0 2 2  Z 1  2−m m (1 − λ)(2 − m)D + λm2 B +m f C + (mA), dλ 2 2 m 0  Z 1  m 2−m 2 + D + (mB) dt f t(2 − m)C + (1 − t)m A, 2 2 0    Z 1 2 (1 − t)(2 − m)C + tm A 2 − m m +m f , D + (mB) dt m 2 2     0 2−m 1 2−m m m ≤ C + (mA), (2 − m)D + 2mf C + (mA), mB f 16 2 2 2 2     2 − m m (2 − m)D m 2 − m 2 +m f C + (mA), + f (2 − m)C, D + (mB) 2 2 m2 2 2   2−m m + 2mf mA, D + (mB) 2 2   2 − m m (2 − m)C 2 +m f , D + (mB) , (3.26) m2 2 2 Furthermore, for α, m = 1 we have   A+C B+D f , 2 2 Z 1     Z 1  A+C B+D 1 f , λB + (1 − λ)D dλ + f tA + (1 − t)C, dt ≤ 2 0 2 2      0    1 A+C A+C B+D B+D ≤ f ,B + f , D + f A, + f C, . 4 2 2 2 2

486

(3.27)

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Acknowledgements. This work was supported by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZZ16175. Competing interests. The authors declare that they have no competing interests. References [1] M. Alomari and M. Darus, The Hadamard’s inequality for s-convex function of 2-variables on the coordinates, Int. Journal of Math. Analysis, 2:13 (2008), 629–638. [2] S. S. Dragomir and G. Toader, Some inequalities for m-convex functions, Studia Univ. Babes-Bolyai, Mathematica, 38:1 (1993), 21–28. [3] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math., 3:1 (2002), 45–55. [4] S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math., 3:2 (2002), 775–788; Available online at http://jipam.vu.edu.cn. [5] S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math., 3:3 (2002), Article 35; Available online at http://jipam.vu.edu.cn. [6] S. S. Dragomir, Hermite-Hadamard’s type inequalities for operator convex functions, Applied Mathematics and Computation, 218 (2011), 766–772; Available online at http://www.elsevier.com/locate/amc. [7] S. S. Dragomir and S. Fitzpatrick, The Hadamards inequality for s-convex functions in the second sense, Demonstratio Math., 32:4 (1999), 687–696. [8] T. Furuta, J. M. Hot, J. Peˇ cari´ c, and Y. Seo, Mond-Peˇ cari´ c method in operator inequalities, Inequalities for Bounded Selfadjoint Operators on a Hilbert Space. Element, Zagreb, 2005. [9] A. G. Ghazanfari,Some new Hermite-Hadamard type inequalities for two operator convex functions; Available online at http://arXiv:1207.0928v1[math.FA]4Jul2012. [10] A. G. Ghazanfari,The Hermite-Hadamard type inequalities for operator s-convex functions, Journal of Advanced Research in Pure Mathematics, 6:3 (2014), 52–61; Available online at http://dx.doi.org/10.5373/ jarpm.1876.110613. [11] F. Hansen, Operator convex functions of several variables, Publ. RIMS, Kyoto Univ., 33 (1997), 443–463; Available online at http://dx.doi.org/10.2977/prims/1195145324. [12] F. Hansen, Operator monotone functions of several variables; Available online at http://arXiv:math/ 0205147v1[math.OA]14May2002. [13] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100–111. [14] E. Kikianty, Hermite-Hadamard inequality in the geometry of Banach spaces, PhD thesis, Victoria University, 2010. [15] A. Kor´ ayi, On some classes of analytic functions of several variables, Transactions of the American Mathematical Society, 101:3 (1961), 520–554; Available online at http://www.jstor.org/stable/1993476. [16] V. G. A. Mihesan, A generalization of the convexity , Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca, Romania (1993). [17] M. E. Ozdemir, E. Set, and M. Z. Sardari, Some new Hadamard type inequalities for co-ordinated m-convex and (α, m) -convex functions, Hacettepe Journal of Mathematics and Statistics, 40 (2011), 219–229. [18] E. Set, M. Z. Sardari, M. E. Ozdemir, and J. Rooin, On generalizations of the Hadamard inequality for (α, m)-convex functions, RGMIA Res. Rep. Coll., 12:4, Article 4 (2009). [19] G. Toader,Some generalizations of the convexity, Proc. Colloq. Approx. Opt. ClujNapoca, (1984), 329–338; [20] S. H. Wang, New integral inequalities of Hermite-Hadamard type for operator m−convex and (α; m)−convex functions, J. Computational analysis and applications, 22:4 (2017), 744–753, in press. [21] S. H. Wang, Hermite-Hadamard type inequalities for operator convex functions on the co-ordinates, J. Nonlinear Sci. Appl., (2016), submit. (Wang) College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China. E-mail address: [email protected] (Wu) Department of Mathematics, Longyan College, Longyan 364012, Fujian, China. E-mail address: [email protected]

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN FUZZY BANACH SPACES: A FIXED POINT APPROACH CHOONKIL PARK1 , JUNG RYE LEE2∗ , AND DONG YUN SHIN3 Abstract. Let 3 1 1 1 f (x + y) − f (−x − y) + f (x − y) + f (y − x) − f (x) − f (y), 4 ( 4 4 4 ) ( ) ( ) x+y x−y y−x +f +f − f (x) − f (y). M2 f (x, y) : = 2f 2 2 2 Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities t N (M1 f (x, y) − ρM2 f (x, y), t) ≥ (0.1) t + φ(x, y) and t N (M2 f (x, y) − ρM1 f (x, y), t) ≥ (0.2) t + φ(x, y) in fuzzy Banach spaces, where ρ is a fixed real number with ρ ̸= 1. M1 f (x, y) :

=

1. Introduction and preliminaries Katsaras [19] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [15, 21, 48]. In particular, Bag and Samanta [3], following Cheng and Mordeson [11], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [20]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [4]. We use the definition of fuzzy normed spaces given in [3, 25, 26] to investigate the Hyers-Ulam stability of additive ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [3, 25, 26, 27] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c ̸= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x ̸= 0, N (x, ·) is continuous on R. 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; fixed point method; additive-quadratic ρ-functional inequality; Hyers-Ulam stability. ∗ Corresponding author.

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The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [25, 28]. Definition 1.2. [3, 28, 26, 27] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn −x, t) = 1 for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N limn→∞ xn = x. Definition 1.3. [3, 28, 26, 27] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [4]). The stability problem of functional equations originated from a question of Ulam [47] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [17] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [39] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [16] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation f (x+y)+f (x−y) = 2f (x)+2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [46] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [12] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian group. The stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 5, 9, 10, 14, 22, 24, 29, 34, 35, 36, 40, 41, 42, 43, 44, 45, 49, 50]). We recall a fundamental result in fixed point theory. Theorem 1.4. [6, 13] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J;

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(3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−α d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [18] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [7, 8, 30, 31, 38]). Park [32, 33] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. In Section 2, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. Throughout this paper, assume that X is a real vector space and (Y, N ) is a fuzzy Banach space. Let ρ be a real number with ρ ̸= 1. 2. Additive-quadratic ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.1) in fuzzy Banach spaces. Theorem 2.1. Let φ : X 2 → [0, ∞) be a function such that there exists an L < 1 with φ(x, y) ≤

L L φ (2x, 2y) ≤ φ (2x, 2y) 4 2

(2.1)

for all x, y ∈ X. (i) Let f : X → Y be an odd mapping satisfying N (M1 f (x, y) − ρM2 f (x, y), t) ≥

t t + φ(x, y)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(

x 2n

)

(2 − 2L)t (2 − 2L)t + Lφ(x, x)

(2.2) exists for each x ∈ X and

(2.3)

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.2). Then Q(x) := N ( ) limn→∞ 4n f 2xn exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2 − 2L)t N (f (x) − Q(x), t) ≥ (2.4) (2 − 2L)t + Lφ(x, x) for all x ∈ X and all t > 0. Proof. (i) Letting y = x in (2.2), we get N (f (2x) − 2f (x), t) ≥

490

t t + φ(x, x)

(2.5)

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and so

(

( )

N f (x) − 2f

)

x t ( ) ,t ≥ 2 t + φ x2 , x2

(2.6)

for all x ∈ X. Consider the set S := {g : X → Y } and introduce the generalized metric on S: {

}

d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

t , ∀x ∈ X, ∀t > 0 , t + φ(x, x)

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [23, Lemma 2.1]). Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + φ(x, x) for all x ∈ X and all t > 0. Hence

(

( )

( )

x N (Jg(x) − Jh(x), Lεt) = N 2g 2 ≥

Lt 2

)

( ( )

( )

x x x L − 2h , Lεt = N g −h , εt 2 2 2 2 Lt Lt t 2 2 = ( x x ) ≥ Lt L t + φ(x, x) + φ 2, 2 2 + 2 φ(x, x)

)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. ( ) ( ) It follows from (2.6) that N f (x) − 2f x2 , L2 t ≥

t t+φ(x,x)

for all x ∈ X and all t > 0. So

d(f, Jf ) ≤ By Theorem 1.4, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e., L 2.

( )

x 1 = A(x) (2.7) 2 2 for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set A

M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.7) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + φ(x, x) for all x ∈ X;

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(2) d(J n f, A) → 0 as n → ∞. This implies the equality ( ) x n N - lim 2 f = A(x) n→∞ 2n for all x ∈ X; 1 (3) d(f, A) ≤ 1−L d(f, Jf ), which implies the inequality d(f, A) ≤

L . 2 − 2L

This implies that the inequality (2.3) holds. By (2.2), ( ( ( ) ( ) ( )) ) x+y x y t n n (x y) N 2 f −f −f ,2 t ≥ n n n 2 2 2 t + φ 2n , 2n and so

(

N 4

( ( n

f

x+y 2n

)

(

+f

x−y 2n

)

(

− 2f

x 2n

)

(

− 2f

for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

y 2n

))

)

,t ≥

t 2n Ln t + 2n φ(x,y) 2n

t 2n

+

t 2n n L 2n φ (x, y)

= 1 for all x, y ∈ X and all

N (A(x + y) − A(x) − A(y), t) = 1 for all x, y ∈ X and all t > 0. So the mapping A : X → Y is additive. (ii) Letting y = x in (2.2), we get ( ) 1 t N f (2x) − 2f (x), t ≥ 2 t + φ(x, x) and so ( ( ) ) t x t 2( ( ) ) = N f (x) − 4f ,t ≥ t x x 2 t + 2φ x2 , x2 2 + φ 2, 2

(2.8)

(2.9)

for all x ∈ X. Now we consider the linear mapping J : S → S such that ( ) x Jg(x) := 4g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + φ(x, x) for all x ∈ X and all t > 0. Hence ( ( ) ( ) ) ( ( ) ( ) ) x x x x L N (Jg(x) − Jh(x), Lεt) = N 4g − 4h , Lεt = N g −h , εt 2 2 2 2 4 Lt Lt t 4 4 ≥ Lt = ( x x ) ≥ Lt L t + φ(x, x) 4 + φ 2, 2 4 + 4 φ(x, x) for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h)

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for all g, h ∈ S. ( ) ( ) It follows from (2.9) that N f (x) − 4f x2 , L2 t ≥

t t+φ(x,x)

for all x ∈ X and all t > 0. So

d(f, Jf ) ≤ By Theorem 1.4, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., L 2.

( )

1 x = Q(x) (2.10) 2 4 for all x ∈ X. Since f : X → Y is even, Q : X → Y is a even mapping. The mapping Q is a unique fixed point of J in the set Q

M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (2.10) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − Q(x), µt) ≥ t + φ(x, x) for all x ∈ X; (2) d(J n f, Q) → 0 as n → ∞. This implies the equality (

n

N - lim 4 f n→∞

for all x ∈ X; (3) d(f, Q) ≤

1 1−L d(f, Jf ),

x 2n

This implies that the inequality (2.4) holds. By (2.2), (

(

and so

(

N 4

( n

1 f 2 1 f 2

(

(

x+y 2n

)

x+y 2n

)

= Q(x)

which implies the inequality d(f, Q) ≤

N 4n

)

1 + f 2 1 + f 2

(

(

x−y 2n x−y 2n

)

(

−f )

(

−f

L . 2 − 2L

x 2n x 2n

)

(

−f )

(

−f

for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,

y 2n y 2n

))

)

, 4n t ≥ ))

)

,t ≥

t 4n t Ln + φ(x,y) 4n 4n

(

t

t+φ

t 4n

+

(

) x y 2n , 2n

t 4n n L 4n φ (x, y)

= 1 for all x, y ∈ X and all )

1 1 N Q(x + y) + Q(x − y) − Q(x) − Q(y), t = 1 2 2 for all x, y ∈ X and all t > 0. So the mapping Q : X → Y is quadratic.

□

Corollary 2.2. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm ∥ · ∥. (i) Let f : X → Y be an odd mapping satisfying N (M1 f (x, y) − ρM2 f (x, y), t) ≥

493

t t + θ(∥x∥p + ∥y∥p )

(2.11)

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for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2p − 2)t (2p − 2)t + 2θ∥x∥p

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.11). Then Q(x) := N limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2p − 4)t N (f (x) − Q(x), t) ≥ p (2 − 4)t + 4θ∥x∥p for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.1 by taking φ(x, y) := θ(∥x∥p + ∥y∥p ) for all x, y ∈ X. Choosing L = 21−p for an odd mapping case and L = 22−p for an even mapping case, then we obtain the desired results. □ Theorem 2.3. Let φ : X 2 → [0, ∞) be a function such that there exists an L < 1 with (

x y φ (x, y) ≤ 2Lφ , 2 2

)

(

x y , ≤ 4Lφ 2 2

)

(2.12)

for all x, y ∈ X.. (i) Let f : X → Y be an odd mapping satisfying (2.2). Then A(x) := N -limn→∞ exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

1 2n f

(2n x)

(2 − 2L)t (2 − 2L)t + φ(x, x)

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.2). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (2 − 2L)t N (f (x) − Q(x), t) ≥ (2 − 2L)t + φ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.1. (i) It follows from (2.5) that (

)

1 1 t N f (x) − f (2x), t ≥ 2 2 t + φ(x, x) for all x ∈ X and all t > 0. (ii) It follows from (2.8) that

)

(

1 1 t N f (x) − f (2x), t ≥ 4 2 t + φ(x, x) for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 2.1.

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Corollary 2.4. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm ∥ · ∥. (i) Let f : X → Y be an odd mapping satisfying (2.11). Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2 − 2p )t (2 − 2p )t + 2θ∥x∥p

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (2.11). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (4 − 2p )t N (f (x) − Q(x), t) ≥ (4 − 2p )t + 4θ∥x∥p for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.3 by taking φ(x, y) := θ(∥x∥p + ∥y∥p ) for all x, y ∈ X. Choosing L = 2p−1 for an odd mapping case and L = 2p−2 for an even mapping case, then we obtain the desired results. □ 3. Additive-quadratic ρ-functional inequality (0.2) In this section, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (0.2) in fuzzy Banach spaces. Theorem 3.1. Let φ : X 2 → [0, ∞) be a function satisfying (2.1). (i) Let f : X → Y be an odd mapping satisfying t N (M2 f (x, y) − ρM1 f (x, y), t) ≥ t + φ(x, y) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(

x 2n

)

(3.1)

exists for each x ∈ X and

(1 − L)t (1 − L)t + φ(x, x)

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.1). Then Q(x) := N ( ) limn→∞ 4n f 2xn exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that (1 − L)t N (f (x) − Q(x), t) ≥ (1 − L)t + φ(x, x) for all x ∈ X and all t > 0. Proof. (i) Letting y = 0 in (3.1), we get (

N f (x) − 2f

( )

)

(

x , t = N 2f 2

( )

x 2

)

− f (x), t ≥

t t + φ(x, 0)

(3.2)

for all x ∈ X.

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Consider the set S := {g : X → Y } and introduce the generalized metric on S: {

}

d(g, h) = inf µ ∈ R+ : N (g(x) − h(x), µt) ≥

t , ∀x ∈ X, ∀t > 0 , t + φ(x, 0)

where, as usual, inf ϕ = +∞. It is easy to show that (S, d) is complete (see [23, Lemma 2.1]). The rest of the proof is similar to the proof of Theorem 2.1 (i). (ii) Letting y = 0 in (3.1), we get (

( )

N f (x) − 4f

)

(

x , t = N 4f 2

( )

x 2

)

− f (x), t ≥

t t + φ(x, 0)

(3.3)

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.1 (ii).

□

Corollary 3.2. Let θ ≥ 0 and let p be a real number with p > 2. Let X be a normed vector space with norm ∥ · ∥. (i) Let f : X → Y be an odd mapping satisfying N (M2 f (x, y) − ρM1 f (x, y), t) ≥

t t + θ(∥x∥p + ∥y∥p )

(3.4)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2p − 2)t (2p − 2)t + 2p θ∥x∥p

for all x ∈ X and all t > 0. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.4). Then Q(x) := N limn→∞ 4n f ( 2xn ) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(2p − 4)t (2p − 4)t + 2p θ∥x∥p

for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 3.1 by taking φ(x, y) := θ(∥x∥p + ∥y∥p ) for all x, y ∈ X. Choosing L = 21−p for an odd mapping case and L = 22−p for an even mapping case, then we obtain the desired results. □ Theorem 3.3. Let φ : X 2 → [0, ∞) be a function satisfying (2.12). (i) Let f : X → Y be an odd mapping satisfying (3.1). Then A(x) := N -limn→∞ exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

1 2n f

(2n x)

(1 − L)t (1 − L)t + Lφ(x, x)

for all x ∈ X and all t > 0.

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(ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.1). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(1 − L)t (1 − L)t + Lφ(x, x)

for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.1. (i) It follows from (3.2) that (

1 t N f (x) − f (2x), 2 2 and so

(

)

≥

t t + φ(2x, 0)

)

1 2Lt t N f (x) − f (2x), Lt ≥ = 2 2Lt + φ(2x, 0) t + φ(x, 0) for all x ∈ X and all t > 0. (ii) It follows from (3.3) that (

1 t N f (x) − f (2x), 4 4 and so

(

)

≥

t t + φ(2x, 0)

)

4Lt t 1 = N f (x) − f (2x), Lt ≥ 4 4Lt + φ(2x, 0) t + φ(x, 0) for all x ∈ X and all t > 0. The rest of the proof is similar to the proof of Theorem 2.1.

□

Corollary 3.4. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with norm ∥ · ∥. (i) Let f : X → Y be an odd mapping satisfying (3.4). Then A(x) := N -limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

(2 − 2p )t (2 − 2p )t + 2p θ∥x∥p

for all x ∈ X. (ii) Let f : X → Y be an even mapping satisfying f (0) = 0 and (3.4). Then Q(x) := N limn→∞ 41n f (2n x) exists for each x ∈ X and defines a quadratic mapping Q : X → Y such that N (f (x) − Q(x), t) ≥

(4 − 2p )t (4 − 2p )t + 2p θ∥x∥p

for all x ∈ X. Proof. The proof follows from Theorem 3.3 by taking φ(x, y) := θ(∥x∥p + ∥y∥p ) for all x, y ∈ X. Choosing L = 2p−1 for an odd mapping case and L = 2p−2 for an even mapping case, then we obtain the desired results. □

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Department of Mathematics, Daejin University, Kyunggi 11159, Republic of Korea E-mail address: [email protected] 4

Department of Mathematics, University of Seoul, Seoul 02504, Republic of Korea E-mail address: [email protected]

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Optimal control of a special predator-prey system with functional response and toxicant Jiangbi Liu1

Hongwei Luo2

∗

1

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, PR China 2 School of Information Engineering, Gansu Forestry Technological College, Tianshui, 741020, PR China

Abstract This paper is devoted to the optimal harvesting problem for a diffusive population dynamics with functional response in a polluted environment . C0 -semigroup theory is used to obtain the existence and uniqueness of the positive strong solution for the controlled system. The first order necessary optimality condition is derived by means the technique of tangent-normal cones and adjoint system of the state. The second-order necessary and sufficient optimality conditions are established by making use of the second order Fr´echet derivative of the associated Lagrange function. Keywords: Optimal harvesting; optimal conditions; functional response; toxicant

1

Introduction

The optimal control problems of population dynamics have been widely studied, such as N.C. Apreutesei [1] studied for a Lotka-Volterra system of three differential equations, some necessary conditions of optimality were founded in order to maximize the total number of individuals. W.Ko [2-3] considered a diffusive two-competing-prey and one-predator system with functional response (Beddington-DeAngelis and ratio-dependent), showed the properties for the positive steady-state solutions of the corresponding elliptic system with Robin boundary. Then N.C. Apreutesei [4] studied for a reaction-diffusion system as follows  ∂y1    ∂t = α1 ∆y1 + y1 g1 (y1 ) + u1 y1 − y1 y2 f (y1 ),     ∂y2   = α2 ∆y2 − ay2 + by1 y2 f (y1 ) + cy2 y3 h(y3 ),    ∂t ∂y3 (1.1) = α3 ∆y3 + y3 g3 (y3 ) + u3 y3 − y3 y2 h(y3 ),   ∂t    ∂yi    (t, x) = 0, on Σ = [0, T ] × ∂Ω, i = 1, 2, 3,   ∂v   yi (0, x) = yi0 (x), x ∈ Ω, i = 1, 2, 3. the author considered the general functional response yi f (yi ), which contains the classical various Holling type, the existence of an optimal solution and first and second order optimality conditions ∗ Corresponding

author. E-mail: [email protected]

1

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were proved. E.Casas [5] investigated an abstract formulation for optimization problems in some Lp spaces, devoted to reduce the classical gap between the necessary and sufficient conditions for optimization problems in Banach spaces. Other models from population dynamics and optimal control problems can be found in [6-9]. However, those papers were not take into account toxicant factor. Among the practical problems, it determines the real rate of the biological individual and the behavior of individual. To this end, Luo [10-12] first formulated a new age-dependent toxicant population model in an environment with small toxicant capacity, effectively bridge the research between age-structure and polluted environment. Inspired his works, this paper propose a more realistic models with toxicant-population in a small content of the environment. The aim of this paper is to seek the maximum of the following functional, which gives the profit from harvesting less the cost of harvesting: J(u, ν) =

3 Z X i=1

0

T

Z

  1 1 Ki ui (t, x)yi (t, x) − Ci u2i (t, x) dxdt − 2 2 Ω

Z

T

C4 [ν(t)]2 dt.

(OH)

0

where Ki are selling price factors, positive constants Ci and C4 represents the cost factors of harvesting and the cost factor of administering pollution of environment, respectively; u = (u1 , u2 , u3 ) are the proportions of the populations to be harvested, ν(t) is the exogenous toxicant input rate the moment t, and the state y = (y1 , y2 , y3 ) is the solution of the following system corresponding to (u1 , u2 , u3 ):  ∂y1  = α1 ∆y1 + y1 [g1 (y1 ) − r1 c10 ] − y1 y2 f (y1 ) − u1 y1 ,    ∂t    ∂y2   = α2 ∆y2 − (a − r2 c20 )y2 + by1 y2 f (y1 ) + cy2 y3 h(y3 ) − u2 y2 ,    ∂t    ∂y3 = α3 ∆y3 + y3 [g3 (y3 ) − r3 c30 ] − y3 y2 h(y3 ) − u3 y3 , (1.2) ∂t   dc  i0   = kce (t) − gci0 (t) − mci0 (t), i = 1, 2, 3,   dt    3  X  dce   = −k c (t)[y (t) + y (t) + y (t)] + g ci0 (t)yi (t) − hce (t) + ν(t) 1 e 1 2 3 1  dt i=1 for (t, x) ∈ Q, subject to some Neumann boundary conditions ∂yi (t, x) = 0, on Σ = [0, T ] × ∂Ω, i = 1, 2, 3 ∂ν and to the initial conditions yi (0, x) = yi0 (x), x ∈ Ω, i = 1, 2, 3. which descried a diffusive one-predator and two-competing-prey system in a spatially inhomogeneous environment, where Q = (0, T ) × Ω, Ω is a bounded domain in Rd (d ≥ 1) with the boundary ∂Ω of class C 2+σ (σ > 0), we denote by yi (t, x) the density of individuals of ith population at the moment t and in the location x ∈ Ω. c0 (t) is the concentration of the toxicant in an organism at the moment t, ce (t) is the concentration of the toxicant in the environment at the moment t. The function ui is the harvesting rate of population yi , and the coefficients α1 , α2 , α3 , a, b, c are all positive constants. For the simplicity, we have assumed that f and h depend only on y1 and on y3 respectively, but the reasoning and the main results remain true also in the case when f and h depend on y2 too, parameter a is the per capita death rate of the predator.

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Optimal control of a special predator-prey system with functional response and toxicant

3

The admissible control set is defined as  Uad = (u, ν) ∈ [L2 (Q)]3 × L∞ (0, T )| 0 ≤ ui (t, x) ≤ 1, a.e in Q, 0 ≤ ν(t) ≤ h a.e in (0, T ) . Throughout this paper, we always assume that: (H1 ) g1 , g3 are continuous and bounded on (0, ∞); (H2 ) f, h are continuous and positive on (0, ∞) and bounded on bounded sets; (H3 ) yi0 ∈ H 2 (Ω), yi0 > 0 on Ω and ∂yi0 /∂ν = 0 a.e. on ∂Ω, i = 1, 2, 3; (H4 ) ν(·) ∈ L2 [0, T ], 0 ≤ ν(t) ≤ ν1 < +∞; (H5 ) 0 ≤ ci0 (0) ≤ 1, 0 ≤ ce (0) ≤ 1; (H6 ) g ≤ k ≤ g + m, ν ≤ h. The paper is organized as follows: In section 2, we use results from the semigroup theory and some well-known existence theorems from [13-14] to derive the global existence and uniqueness of a positive strong solution of the controlled system (1.2), Section 3 is devoted to first order necessary optimality conditions for (OH). Necessary and sufficient second order optimality conditions are given in Section 4.

2

Basic properties of the solution

This section concerns the most important properties of the dynamics system with diffusion. Existence, uniqueness and positivity of the solution will be proved. Thus formally, system (1.1) can be written as an infinite dimensional Cauchy problem of the form   dy (t) = Ay(t) + F (t, y(t)), t ∈ [0, T ], (2.1) dt  y(0) = y0 , where A : D(A) ⊂ X → X is the infinitesimal generator of a C0 -semigroup of contractions {S(t)}t≥0 on the Banach space X, if X is a Hilbert space, A is called dissipative if (Ax, x) ≤ 0, ∀x ∈ D(A), and F : [0, T ] × X → X is measurable in t and Lipschtiz in x ∈ X uniformly with respect to t. We shall employ a general existence result which we use in the sequel (Proposition 1.2, p.175,[14]). Theorem 2.1 For each y0 ∈ X, the initial value problem (2.1) has a unique mild solution y ∈ C([0, T ]; X), and Z t y(t) = S(t)y0 + S(t − s)F (s, y(s))ds, t ∈ [0, T ]. 0

In addition, if X is a Hilbert space, A is self-adjoint and dissipative on X, and y0 ∈ D(A), then the mild solution is in fact a strong solution and y ∈ W 1,2 ([δ, T ]; X), ∀δ ∈ [0, T ]. Thus, we work in the Hilbert space H = (L2 (Ω))3 , where the operator A : D(A) ⊂ H → H,     F1 (t, y(t)) α1 ∆ 0 0 A= 0 F (t, y(t)) =  F2 (t, y(t))  , α2 ∆ 0 , F3 (t, y(t)) 0 0 α3 ∆ o n ∂yi = 0 on ∂Ω, i = 1, 2, 3 , for y = (y1 , y2 , y3 ) ∈ D(A), D(A) = y = (y1 , y2 , y3 ) ∈ (H 2 (Ω))3 , ∂ν y 0 = (y10 , y20 , y30 ) is the initial value of y, and F = (F1 , F2 , F3 ) is the nonlinear term in (2.1), that is  F1 (t, y(t)) = y1 g1 (y1 ) − y1 y2 f (y1 ) − u1 y1 ,    (2.2) F2 (t, y(t)) = −ay2 + by1 y2 f (y1 ) + cy2 y3 h(y3 ) − u2 y2 ,    F3 (t, y(t)) = y3 g3 (y3 ) − y3 y2 h(y3 ) − u3 y3 ,

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Theorem 2.2 Suppose that y 0 = (y10 , y20 , y30 ) ∈ D(A), and yi0 > 0, i = 1, 2, 3. Then for each u ∈ Uad , the system (2.1) has a unique nonnegative solution (y(t, x), c0 (t), ce (t)), such that (i) (yi (t, x), ci0 (t), ce (t)) ∈ L∞ (Q) ∩ L2 (0, T ; H 2 (Ω) ∩ L∞ (0, T ; H 1 (Ω))) × L∞ (0, T ) × L∞ (0, T ), (ii) 0 ≤ ci0 (t) ≤ 1, 0 ≤ ce (t) ≤ 1, ∀ t ∈ (0, T ). Proof Since F is not satisfy Lipschtiz conditions, we cannot apply the theorem 2.1 directly for our problem, usually we use a truncation procedure for F , consider the truncated initial value problem  N   ∂y (t, x) = Ay N (t) + F N (t, y(t)), t ∈ [0, T ], ∂t (2.3)   y N (0) = y , 0 where F N = (F1N , F2N , F3N ) is obtained from F = (F1 , F2 , F3 ), a fixed large number N > 0. If |yi | ≤ N , then yi in F (t, y1 , y2 , y3 ) remains unchanged, if yi > N , then yi from (2.2) is replaced by N , if yi < −N , then yi from (2.2) is replaced by −N . Thus function F N becomes Lipschitz continuous with respect to t, according theorem 2.1, the problem (2.3) admits a unique strong solution y N = (y1N , y2N , y3N ) ∈ W 1,2 ([δ, T ]; H) ∩ L2 (0, T ; D(A)), ∀δ ∈ [0, T ]. To begin with, we shall that y ∈ L2 (0, T ; H 2 (Ω)∩L∞ (0, T ; H 1 (Ω)). On one hand, from theorem 2.1 we know y ∈ L2 (0, T ; H 2 (Ω), on the other hand, from (2.3) we derive that Z Z Z N 2 Z ∂y1N ∂y1 ∆y1N dsdx + α2 |∆y1N |2 dsdx = |F1 (t, y(t))|2 dsdx, dsdx − 2α1 ∂t Q Q Q Q ∂t Using the regularity of y1N and the Green’s formula, we have Z N 2 Z Z Z Z ∂y1 |∇y1N |2 dx + α2 |∆y1N |2 dsdx = |F1 (t, y(t))|2 dsdx + 2α1 |∇y10 |2 dx. dsdx + 2α1 ∂t Q Ω Q Q Ω Since y1N ∈ W 1,2 (0, T ; H) and y10 ∈ H 2 (Ω), by the Lipschitz property of F1N we deduce that Z Z Z 2α1 |∇y1N |2 dx ≤ |y N |dxds + 2α1 |∇y10 |2 dx < +∞. Ω ∞

Q

Ω

1

Thus, we have y1 ∈ L (0, T ; H (Ω)), analogously y2 , y3 are proved. Further more, it remains to prove that y N ∈ L∞ (Q), (ci0 , ce ) ∈ L∞ (0, T ). Indeed, consider the following auxiliary initial value problems  N   ∂ρ1 (t, x) = ∆ρN (t) + F N (t, y(t)) − M , t ∈ [0, T ], N 1 1 ∂t (2.4)   ρN (0) = y 0 − ky 0 k ∞ 1 1 1 L (Ω) and

where MN

 N   ∂ω1 (t, x) = ∆ω N (t) + F N (t, y(t)) + M , t ∈ [0, T ], N 1 1 ∂t   ω N (0) = y 0 + ky 0 k ∞ , 1 1 1 L (Ω) n o N = max kFi (·, y(t))kL∞ (Q) , kyi0 kL∞ (Q) , i = 1, 2, 3 .

(2.5)

N By theorem 2.1 the function ρN 1 and ω1 in C([0, T ]; X) is a mild solution to problem (2.4) and (2.5), the solution of these can be written as Z t 0 0 ρ1 (t) = S(t)(y1 − ky1 kL∞ (Ω) ) + S(t − s)(F1N (s, y1 , y2 , y3 ) − MN )ds, 0

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Optimal control of a special predator-prey system with functional response and toxicant

ω(t) =

S(t)(y10

+

ky10 kL∞ (Ω) )

5

t

Z

S(t − s)(F1N (s, y1 , y2 , y3 ) + MN )ds.

+ 0

Remark that their solutions are ρ1 (t, x) = y1N (t, x) − MN t − ky10 kL∞ (Ω) , ω1 (t, x) = y1N (t, x) + MN t + ky10 kL∞ (Ω) , since |F1N (t, y N )| ≤ MN , from the comparison principle of linear parabolic equation, we deduce N that ρN 1 (0) ≤ 0, ω1 (0) ≥ 0, that is |y1N (t, x)| ≤ MN t + ky10 kL∞ (Ω) , and in the same manner to prove that y2N , y3N hold for (t, x) ∈ Q. Therefor, yiN ∈ L∞ (Q). To prove (ci0 , ce ) ∈ L∞ (0, T ), we define G : X → X, from (1.2) we can deduce that Z

t

Gi (t) = ci0 (t) = ci0 (0) exp{−(g + m)t} + k

 ce (s) exp (s − t)(g + m) ds, i = 1, 2, 3,

(2.6)

0

G4 (t) = ce (t) = ce (0) exp

n

t

Z −

3 X

0

+

Z th

g1

0

3 X

 o k1 yi (τ ) + h dτ

i=1

i nZ
ci0 (s)yi (s) + ν(s) exp t

i=1

s

3 X

(2.7)  o k1 yi (τ ) + h dτ ds,

i=1

if the hypothesis (H6 ) hold, it is clear that 0 ≤ ci0 (t) ≤ 1, 0 ≤ ce (t) ≤ 1 and (ci0 , ce ) ∈ L∞ (0, T ), i = 1, 2, 3. the specific process can refer to [15]. Moreover, we shall that yiN are positive on Q, to this end, let yiN = (yiN )+ − (yiN )− , where (yiN )+ (t, x) = sup{yiN (t, x), 0}, (yiN )− (t, x) = − inf{yiN (t, x), 0}, i = 1, 2, 3.

(2.8)

Multiplying the first equation from (2.1) by y1N we have   1 ∂ |(y1N )− |2 = α1 (y1N )− ∆(y1N )− + |(y1N )− |2 g1 (y1N ) − y2N f (y1N ) − u1 . 2 ∂t

(2.9)

Integrating (2.9) on Ω and using Greens formula we get Z Z Z   1 ∂ |(y1N )− |2 dx = −α1 |∇(y1N )− |2 dx + |(y1N )− |2 g1 (y1N ) − y2N f (y1N ) − u1 dx. 2 Ω ∂t Ω Ω By integrating over [0, t], for t ∈ [0, T ], and take into consideration of the uniformly boundedness of yiN , it is not difficult to see that there exists a constant C N > 0 depending on N such that 1 2

Z

|(y1N )− |2 dx + α1

Z tZ

Ω

0

|∇(y1N )− |2 dxds ≤ C N

Z tZ

Ω

0

|y1N (s)|2 dxds.

Ω

Gronwalls inequality lead to Z

|(y1N )− |2 dx ≤ 0, ∀t ∈ [0, T ],

Ω

(y1N )−

that is = 0, by the definition of (2.8) we conclude that y1N (t, x) > 0, analogously we get N y2 (t, x) > 0 and y3N (t, x) > 0.

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In addition, we prove the uniqueness of the solution. For any (y 1 , c10 , c1e ) and (y 2 , c20 , c2e ) are two solutions of problem (1.2), where y 1 = (y11 , y21 , y31 ), c10 = (c110 , c120 , c130 ), y 2 = (y12 , y22 , y32 ), c20 = (c210 , c220 , c230 ), we denote by ϕ = y 1 − y 2 , then ϕ is the solution of                         

∂ϕ1 = α1 ∆ϕ1 + F1 (t, y11 , y21 , y31 ) − F1 (t, y12 , y22 , y32 ), ∂t ∂ϕ2 = α2 ∆ϕ2 + F2 (t, y11 , y21 , y31 ) − F2 (t, y12 , y22 , y32 ), ∂t ∂ϕ3 = α3 ∆ϕ3 + F3 (t, y11 , y21 , y31 ) − F3 (t, y12 , y22 , y32 ), ∂t ∂ϕ1 ∂ϕ2 ∂ϕ3 = = = 0, on Σ, ∂ν ∂ν ∂ν ϕ1 (0, x) = ϕ2 (0, x) = ϕ3 (0, x) = 0.

(2.10)

Suppose g1 , g3 , f, h ∈ C 1 [0, ∞), g1 , g3 are bounded and f, h are positive and have at most polynomial growth, then from (2.2) we obtain |Fi (t, y11 , y21 , y31 ) − Fi (t, y12 , y22 , y32 )| ≤ c(|ϕ1 | + |ϕ2 | + |ϕ3 |), where c is a positive constant. Multiplying (2.10) by ϕ1 , ϕ2 , ϕ3 respectively, and integrating on ΩT = Ω × (0, t) we get 3

1X 2 i=1

Z

2

|ϕi (t)| dx + Ω

3 Z X

2

αi |∇ϕi | dsdx =

3 Z X

ΩT

i=1

ϕi (Fi (t, y11 , y21 , y31 ) − Fi (t, y12 , y22 , y32 ))dsdx

ΩT

i=1

Z tZ

(|ϕ1 (s)|2 + |ϕ2 (s)|2 + |ϕ3 (s)|2 )dsdx.

≤C 0

Ω

(2.11) From (2.11) and Gronwall’s lemma we have Z

(|ϕ1 (s)|2 + |ϕ2 (s)|2 + |ϕ3 (s)|2 ) ≤ 0,

Ω

which yields that ϕ1 = ϕ2 = ϕ3 = 0, thus we have proved the uniqueness of the yi . However, we can follow by (2.6) and (2.7) |c10 (t)

−

c20 (t)|

=

3 X

|c1i0 (t)

−

c2i0 (t)|

Z ≤ 3k

−

|c1e (s) − c2e (s)|ds, i = 1, 2, 3.

(2.12)

0

i=1

|c1e (t)

t

c2e (t)|

≤ M1

3 Z X i=1

t

|c1i0 (s) − c2i0 (s)|ds,

(2.13)

0

where M1 is constant. We define an equivalent norm in X as follows: k(ci0 , ce )k∗ = Ess

sup e−λt t∈(0,T )

505

3 nX

o |ci0 (t)| + |ce (t)| ,

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7

by (2.11) and (2.12) we obtain kG(x1 ) − G(x)2 k∗ = kGi (x1 ) − Gi (x2 ), G4 (x1 ) − G4 (x2 )k∗ Z tnX 3 o ≤ M2 Ess sup e−λt |c1i0 (s) − c2i0 (s)| + |c1e (s) − c2e (s)| ds t∈(0,T )

≤ M2 Ess

sup e−λt t∈(0,T )

≤ M2 kx1 − x2 k∗ Ess

0

Z

i=1 t

3 n o X eλs e−λs [ |c1i0 (s) − c2i0 (s)| + |c1e (s) − c2e (s)|] ds

0

sup {e−λt t∈(0,T )

Z

i=1 t λs

e ds}

0

M2 1 kx − x2 k∗ , λ where M2 is constant, choose λ > M2 yields that G is a strict contraction on (X, k · k∗ ) and consequently has a unique fixed point. Thus, the system (1.2) has a unique solution (yi , ci0 , ce ). the proof is completed. ≤

3

Necessary optimality conditions

In this section, we find some necessary optimality conditions in order to maximize the profit from harvesting less the cost of harvesting. Theorem 3.1 If (u∗ , ν ∗ ) is an optimal control and (y ∗ , c∗i0 , c∗e ) is the corresponding optimal state, then  (K − q )y ∗  i i i , i = 1, 2, 3, a.e. in Q, u∗i (t, x) = Li Ci (3.1)  q (t)  7 ∗ ν (t) = L4 , a.e. in (0, T ), C4 where  x Hj and q = (q1 , q2 , . . . , q7 ) is the solution of following adjoint system corresponding to (u∗ , ν ∗ ).  ∂q1  = −α1 ∆q1 + [g1 (y1∗ ) − r1 c∗10 + y1∗ g10 (y1∗ ) − u∗1 − y2∗ f (y1∗ ) − y1∗ y2∗ f 0 (y1∗ )]q1    ∂t     − [by2∗ f (y1∗ ) + by1∗ y2∗ f 0 (y1∗ )]q2 + [k1 c∗e − g1 c∗10 ]q7 + K1 u∗1 ,      ∂q2   = −α2 ∆q2 + y1∗ f (y1∗ )q1 − [−a − r2 c∗20 + by1∗ f (y1∗ ) + cy3∗ h(y3∗ )]q2    ∂t    + y3∗ h(y3∗ )q3 + [k1 c∗e − g1 c∗20 ]q7 + K2 u∗2 ,      ∂q 3 = −α3 ∆q3 − [g3 (y3∗ ) − r3 c∗30 + y3∗ g30 (y1∗ ) − u∗3 − y2∗ h(y3∗ ) − y3∗ y2∗ h0 (y3∗ )]q3 (3.2)  ∂t  ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗  − cy2 [h(y3 ) + y3 h (y3 )]q2 + [k1 ce − g1 c30 ]q7 + K3 u3 ,       ∂qj   = (g + m)qj − g1 yi∗ q7 , j = i + 3, i = 1, 2, 3,   ∂t    6 3  X X  ∂q7   = −k q + k yi q7 + hq7 ,  j 1  ∂t   j=4 i=1   qi (T, x) = 0, x ∈ Ω,

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∂qi = 0 a.e. on Σ, i = 1, 2, 3. ∂ν Proof Existence and uniqueness of the solution q to system (3.2) follows by theorem 2.2. Denote by NUad (u∗ , ν ∗ ) the normal cone at Uad in (u∗ , ν ∗ ), NUad (u∗ , ν ∗ ) = {v1 ∈ L2 (Q), v2 ∈ L2 (0, T ) satisfying the following formula},    v1 (t, x) ≤ 0, when u(t, x) = 0,  v2 (t) ≤ 0, when ν(t) = 0, v1 (t, x) = 0, when 0 ≤ u(t, x) ≤ 1, v (t) = 0, when 0 ≤ ν(t) ≤ h,   2 v1 (t, x) ≥ 0, when u(t, x) = 1, v2 (t) ≥ 0, when ν(t) = h, for any given (ϑ1 , ϑ2 ) ∈ TUad (u∗ , ν ∗ ) ϑ1 = (ϑ11 , ϑ21 , ϑ31 ), as ε > 0 small enough, (u∗ + εϑ1 , ν ∗ + εϑ2 ) ∈ Uad , we get J(u∗ + εϑ1 , ν ∗ + εϑ2 ) ≤ J(u∗ , ν ∗ ). (3.3) Substituting (2.1) into (3.3) gives that 3 Z X i=1

0

T

Z

3

Ki (u∗i + εϑi1 )yiε dxdt −

Ω

≤

3 Z X i=1

0

T

Z

1X 2 i=1

T

Z

Z

0

Ω 3

Ki u∗i yi∗ dxdt

Ω

Ci (u∗i + εϑi1 )2 dxdt −

1X − 2 i=1

Z

T

0

Z

Ci [u∗i ]2 dxdt −

Ω

1 2

1 2 Z

T

Z

C4 (ν ∗ + εϑ2 )2 dt

0 T

C4 [ν ∗ ]2 dt,

0

that is 3 Z X i=1

0

T

Z Ω

Ki u∗i zi∗ dxdt

+

3 Z X i=1

0

T

Z

(Ki yi∗

−

Ci u∗i )ϑi1 dxdt

Z −

Ω

T

ν ∗ ϑ2 dt ≤ 0,

(3.4)

0

where zi (t, x) = lim

ε→0+

cε (t) − c∗i0 (t) cε (t) − c∗e (t) yiε (t, x) − yi∗ (t, x) , zi+3 (t) = lim i0 , z7 (t) = lim e , i = 1, 2, 3, ε ε ε ε→0+ ε→0+

(y ε , cε0 , cεe ) is the state corresponding to (u∗ + εϑ1 , ν ∗ + εϑ2 ), it follows from the state system (1.2) that z = (z1 , z2 , . . . , z7 ) is the solution of                                                 

∂z1 = α1 ∆z1 + z1 [g1 (y1∗ ) + y1∗ g10 (y1∗ ) − u∗1 − y2∗ f (y1∗ ) − y1∗ y2∗ f 0 (y1∗ )] ∂t − y1∗ z2 f (y1∗ ) − ϑ11 y1∗ , ∂z2 = α2 ∆z2 + +bz1 [y2∗ f (y1∗ ) + y1∗ y2∗ f 0 (y1∗ )] + z2 [−a − u∗2 + by1∗ f (y1∗ ) + cy3∗ h(y3∗ )] ∂t + cz3 [y2∗ h(y3∗ ) + y3∗ y2∗ h0 (y3∗ )] − ϑ21 y2∗ , ∂z3 = α3 ∆z3 + z3 [g3 (y3∗ ) + y3∗ g30 (y1∗ ) − u∗3 − y2∗ h(y3∗ ) − y3∗ y2∗ h0 (y3∗ )] ∂t − z2 y3∗ h(y3∗ ) − ϑ31 y3∗ , ∂zj = kz7 (t) − gzj (t) − mzj (t), j = i + 3, i = 1, 2, 3, ∂t 3 3 3 3 X X X X ∂z7 = −k1 c∗e (t) zi + g1 ci0 (t)zi (t) + g1 yi∗ zj (t) − [k1 yi∗ (t) + h]z7 (t) + ϑ2 (t), ∂t i=1 i=1 i=1 i=1 (3.5)

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for all (t, x) ∈ Q, subject to the boundary and initial conditions ∂qi = 0 a.e. on Σ, i = 1, 2, 3. ∂ν zi (0, x) = zj (0) = 0, x ∈ Ω, j = i + 3, i = 1, 2, 3. Multiplying the (3.5) by q1 , q2 , . . . , q7 respectively, integrating on Q and (0, T ) and using (3.2) yield 3 Z X i=1

0

T

Z

Ki u∗i (t, x)zi (t, x)dxdt = −

Ω

3 Z X i=1

T

0

Z

yi∗ (t, x)qi (t, x)ϑ1i dxdt +

Z

T

ϑ2 (t)q7 (t)dt. (3.6) 0

Ω

Substituting (3.6) into (3.4) we obtain that 3 Z X

T

0

i=1

Z

[(Ki − qi )yi∗ − Ci u∗i ]ϑ1i dxdt +

Ω

Z

T

(−C4 ν ∗ + q7 )ϑ2 dt ≤ 0.

(3.7)

0

By using the concept of normal cone Uad at (u∗ , ν ∗ ) [16], we get   (Ki − qi )yi∗ − Ci u∗i , −C4 ν ∗ + q7 ∈ NUad (u∗ , ν ∗ ), the proof is completed by the characteristics properties of the normal vector [14].

4

Second order optimality conditions

In this section, we discuss the second order sufficient conditions for the controlled system, since the second order optimality conditions can be solved by using the second order Fr´echet derivative of the associated Lagrange function, so we introduce the Lagrange function firstly, Z Z  T ∂y T dtdx, (4.1) L(y, u, ν, q) = J(u, ν) − q(yt − Ay − F ) dtdx − q ∂ν Q Σ  ∂y ∂y ∂y  ∂y 1 2 3 = , , , we employ ∂ν ∂ν ∂ν ∂ν the method from [17-18], let X = (y, c0 , ce ), U = (u, v), Q = (q1 , q2 , · · · , q7 ), then (4.1) can be written in detail as Z Z TZ   1 T 1 C4 ν 2 dt L(X, U, Q) = K1 u1 y1 + K2 u2 y2 + K3 u3 y3 − (C1 u21 + C2 u22 + C3 u23 ) dxdt − 2 2 0 0 Ω Z  Z ∂q2 ∂q3  ∂q1 y1 + y2 + y3 dtdx + (α1 y1 ∆q1 + α2 y2 ∆q2 + α3 y3 ∆q3 )dtdx + ∂t ∂t ∂t Q Q Z + [y1 g1 (y1 )q1 + y3 g3 (y3 )q3 − u1 y1 q1 − u2 y2 q2 − u3 y3 q3 + y1 y2 f (y1 )(bq2 − q1 ) here the upper index T is the transposed of any matrix and

Q

Z − ay2 q2 + y2 y3 h(y3 )(cq2 − q3 )]dtdx + 0

Z −

T

6 X

qj [kce − (g + m)ci0 ]dt

j=4

T

q7 [k1 ce (y1 + y2 + y3 ) + g1 (c10 y1 + c20 y2 + c30 y3 ) − hce + ν]dt, 0

(4.2)

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we introduce the Hamiltonian function secondly H(X, U, Q) =K1 u1 y1 + K2 u2 y2 + K3 u3 y3 + y1 g1 (y1 )q1 + y3 g3 (y3 )q3 − u1 y1 q1 − u2 y2 q2 − u3 y3 q3 + y1 y2 f (y1 )(bq2 − q1 ) − ay2 q2 + y2 y3 h(y3 )(cq2 − q3 ) +

6 X

qj [kce − (g + m)ci0 ]

j=4

− q7 [k1 ce (y1 + y2 + y3 ) + g1 (c10 y1 + c20 y2 + c30 y3 ) − hce + ν], Assume that g1 , g3 , f and h are functions of class C 2 . If u ¯, ν¯ is an admissible control and y, q are the corresponding state and adjoint state, then the associated Hessian matrix at (y, u, q) is      D2 H(y, u, ν, q) =      and  H11     H  12 H23    H33   H14

H11 H21 0 H41 0 0 0

H12 0 H32 0 H52 0 0

0 H23 H33 0 0 H63 0

H14 0 0 −C1 0 0 0

0 H25 0 0 −C2 0 0

0 0 H36 0 0 −C3 0

0 0 0 0 0 0 −C4

     ,    

= q¯1 [2g10 (¯ y1 ) + y¯1 g100 (¯ y1 )] − y¯2 (¯ q1 − b¯ q2 )[2f 0 (¯ y1 ) + y¯1 f 00 (¯ y1 )], = −(¯ q1 − b¯ q2 )[f (¯ y1 ) + y¯1 f 0 (¯ y1 )], = −(¯ q3 − c¯ q2 )[h(¯ y3 ) + y¯3 f 0 (¯ y3 )], 0 = q¯3 [2g3 (¯ y3 ) + y¯3 g300 (¯ y3 )] − y¯2 (¯ q3 − c¯ q2 )[2h0 (¯ y3 ) + y¯3 h00 (¯ y3 )] = H41 = K1 − q¯1 , H25 = H52 = K2 − q¯2 , H36 = H63 = K3 − q¯3 .

Then, we have Z

00

(y, u, ν)D2 H(y, u, q)(y, u, ν)T dtdx,

L (¯ y, u ¯, ν¯, q¯)[(y, u, ν), (y, u, ν)] = Q

that is L00 (¯ y, u ¯, ν¯, q¯)[(y, u, ν), (y, u, ν)] Z = (y1 )2 [¯ q1 (2g10 (¯ y1 ) + y¯1 g100 (¯ y1 )) − y¯2 (¯ q1 − b¯ q2 )(2f 0 (¯ y1 ) + y¯1 f 00 (¯ y1 ))]dtdx Q Z + (y3 )2 [¯ q3 (2g30 (¯ y3 ) + y¯3 g300 (¯ y3 )) − y¯2 (¯ q3 − c¯ q2 )(2h0 (¯ y3 ) + y¯3 h00 (¯ y3 ))]dtdx Q Z + 2[−y1 y2 (¯ q1 − b¯ q2 )(f (¯ y1 ) + y¯1 f 0 (¯ y1 )) − y2 y3 (¯ q3 − c¯ q2 )(h(¯ y3 ) + y¯3 f 0 (¯ y3 ))]dtdx Q Z + [2u1 y1 (K1 − q¯1 ) + 2u2 y2 (K2 − q¯2 ) + 2u3 y3 (K3 − q¯3 )]dtdx Q

Z − Q

(C1 u21 + C2 u22 + C3 u23 )dtdx −

Z

T

C4 ν 2 dt.

0

Now we can formulate the second order optimality conditions for our problem. Theorem 4.1 (i) (Second order necessary optimality conditions.) Under the hypotheses of Theorem 3.1, if (u∗ , ν ∗ ) is an optimal pair and q is the corresponding adjoint variable, then L00 (y ∗ , u∗ , ν ∗ , q)[(y, u, ν), (y, u, ν)] ≤ 0, ∀ (u, ν) ∈ NUad (u∗ , ν ∗ ).

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(ii) (Second order sufficient optimality conditions.) ∀ (u∗ , ν ∗ ) ∈ Uad , together with its corresponding state y ∗ and adjoint state q, if (y ∗ , u∗ , ν ∗ , q) satisfies the first order necessary condition (3.1) and the condition L00 (y ∗ , u∗ , ν ∗ , q)[(y, u, ν), (y, u, ν)] < κ(kv1 k2L2 (Q) + kv2 k2L2 (0,T ) ), ∀ (v1 , v2 ) ∈ NUad , for some κ > 0, then (y ∗ , u∗ , ν ∗ ) is an optimal local solution of the controlled system (1.2).

Acknowledgements The authors would like to thank the anonymous referees for their valuable comments and suggestions. The work has been supported by the Natural Science Foundation of China (11561041) and the Nature Science Foundation of Gansu Province of China (1506RJZA071).

References [1] N.C. Apreutesei, Necessary optimality conditions for a Lotka-Volterra three species system, Math. Model. Nat. Phenom. 1 (2006) 123-135. [2] W. Ko, K. Ryu, Analysis of diffusive two-competing-prey and one-predator systems with Beddington-DeAngelis functional response, Nonlinear Anal. 71 (9) (2009) 4185-4202. [3] W. Ko, I. Ahn, A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: II stationary pattern formation, J. Math. Anal. Appl. 397 (1) (2013) 29-45. [4] N. Apreutesei, G. Dimitriu, R. Strugariu, An optimal control problem for a two-prey and one-predator model with diffusion, Comput. Math. Appl. 67 (2014) 2127-2143. [5] E. Casas, M. Mateos, F. Tr¨ oltzsch, Necessary and sufficient optimality conditions for optimization problems in function spaces and applications to control theory, ESAIM: Proceedings. 13 (2003). [6] N. Apreutesei, G. Dimitriu, On a prey-predator reaction-diffusion system with Holling type III functional response, J. Comput. Appl. Math. 235 (2) (2010) 366-379. [7] N.C. Apreutesei, An optimal control problem for a pest, predator, and plant system, Nonlinear Anal.RWA. 13 (3) (2012) 1391-1400. [8] W. Chen, M. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling 42 (2005) 31-44. [9] Y. Du, Y. Lou, Qualitative behavior of positive solutions of a predatorCprey model: effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 321-349. [10] Z.X. Luo, Z.R. He, Optimal control for age-dependent population hybrid system in a polluted environment, Appl. Math. Comput. 228 (2014) 68-76. [11] Z.X. Luo, X.L. Fan, Optimal control for an age-dependent competitive species model in a polluted environment. Comput. Math. Appl. 228 (2014) 91-101. [12] Z.X. Luo, Optimal control for an age-dependent predator-prey system in a polluted environment, J. Appl. Math. Comput. 44 (1-2) (2014) 491-509.

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[13] I. Vrabie, C0 -Semigroups and Applications, in: Math. Stud., vol. 191, North-Holland, 2003. [14] V. Barbu, Mathematical Methods in Optimization of Differential Systems, Kluwer Academic Publishers, Dordrecht, 1994. [15] Z.E. Ma, G.R. Cui , W.D. Wang, Persistence and extinction of a population in a polluted environment, Math. Biosci. 101 (1) (1990) 75-97. [16] V. Barbu, M. Iannelli, Optimal control of population dynamics. Optim. Theory. Appl. 102 (1) ( 1999) 1-14. [17] E. Casas, F. Tr¨ oltzsch, Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim. 13 (2) (2002) 406-431. [18] J.-P. Raymond, F. Tr¨ oltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints, Discrete and continuous dynamical systems, 6 (2) (2000) 431-450.

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Existence results for new extended vector variational-like inequality Kasamsuk Ungchittrakoola,b,∗, Boonyarit Ngeonkama a Department b Research

of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Center for Academic Excellence in Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Abstract In this paper, we establish and study some new existence theorems for a new extended vector variational-like inequality in a Banach space. The results are proved by using the new definition of g − f − η − ϕ − µ−quasimonotone of Stampacchia and of Minty type mappings. The obtained results in this article can be viewed as some new and generalized forms which can be applied to several problems. Keywords: New extended vector variational-like inequality; Existence result; C-convex; KKM-mapping; g − f − η − ϕ − µ−quasimonotonicity; g − f − η − ϕ − µ−pseudomonotoncity.

1. Introduction In 1980, Giannessi introduced a generalization of variational inequality is the vector variational inequality (for short, VVI) in a finite-dimensional Euclidean space, see [8]. For the past years, vector variational inequalities and their generalizations have been studied and applied in various directions. The vector variational-like inequalities is a generalized form of a vector variational inequalities related to the class of η-connected sets which is much more general than the class of convex sets. It well Known that monotonicity plays an important role to proving existence of solutions of vector variational inequalities and vector variational-like inequalities. Some important generalizations of monotonicity, such as quasimonotonicity, proper quasimonotonicity, pseu-domonotonicity, dense pseudomonotonicity, semimonotonicity, have been introduced and considered to study various variational inequalities and other related problems. In [9] Ahmad and Irfan obtained existence results for extended vector variational-like inequality and equilibrium problems by using g-h-η-quasimonotone of Stampacchia and Minty types. In this paper, we introduce a new definition for a new extended vector variational-like inequality and we define a new and general form of definitions for quasimonotone of Stampacchia and Minty type mappings. We have some ideas to establish some sufficient conditions to guarantee the existence of solutions. The new problems can be viewed as some unified forms of the previous problem, that is, extended vector variational-like inequalities considered and studied by Ahmad and Irfan [9]. Let X and Y be two real Banach spaces, K ⊂ X be a nonempty, closed and convex subset, C ⊂ Y be a pointed, closed and convex cone in Y such that intC ̸= ∅ where intC denote the ∗ Corresponding

author. Tel.:+66 55963250; fax:+66 55963201. Email addresses: [email protected] (Kasamsuk Ungchittrakool), [email protected] (Boonyarit Ngeonkam)

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interior of C. Then for x, y ∈ Y , a partial order ≥C in Y is defined as x ≥C y ⇔ x − y ∈ C. Let L(X, Y ) be the space of all continuous linear mappings from X to Y . Let T1 , T2 , · · · , TN : K → L(X, Y ), g, f : K → K, η : K × K → X and ϕ, µ : K × K → Y are mappings. We consider the following new extended vector variational-like inequalities:   F ind x ∈ Ksuch that,   ⟩ ⟨ ∑ N (N EV V LI − I) Ti (x), η(g(y), g(x)) + ϕ (f (y), f (x)) − µ (f (x), f (y)) ≥C 0,  i=1    ∀y ∈ K. and

  F ind x ∈ Ksuch that,   ⟩ ⟨ ∑ N (N EV V LI − II) Ti (y), η(g(x), g(y)) + ϕ (f (x), f (y)) − µ (f (y), f (x)) ≤C 0,  i=1    ∀y ∈ K.

Special cases: (i) If T3 , T4 , · · · , TN ≡ 0, T1 = S, T2 = T , ϕ = h, µ ≡ 0 and f = g then (NEVVLI-I) and (NEVVLI-II) reduces to the following extended vector variational-like inequalities considered and studied by Ahmad and Irfan [9]   F ind x ∈ Ksuch that, (EV V LI − I) ⟨S(x) + T (x), η(g(y), g(x))⟩ + h(g(y), g(x)) ≥C 0,   ∀y ∈ K, and   F ind x ∈ Ksuch that, (EV V LI − II) ⟨S(y) + T (y), η(g(x), g(y))⟩ + h(g(x), g(y)) ≤C 0,   ∀y ∈ K, (ii) If T2 , T3 , · · · , TN ≡ 0, T1 = T , ϕ = h, µ ≡ 0 and f = g = I then (NEVVLI-I) and (NEVVLI-II) reduces to the following vector variational-like inequalities considered and studied by Ahmad [1] { F ind x ∈ Ksuch that, (V V LI − I) ⟨T (x), η(y, x)⟩ + h(y, x) ≥C 0, ∀y ∈ K, and

{ (V V LI − I)

F ind x ∈ Ksuch that, ⟨T (y), η(x, y)⟩ + h(x, y) ≤C 0, ∀y ∈ K,

(iii) If T2 , T3 , · · · , TN ≡ 0, T1 = T , ϕ ≡ 0, µ ≡ 0 and g = I then (NEVVLI-I) and (NEVVLI-II) reduces to the following vector variational-like inequalities considered and studied by Zhao and Xia [12] { F ind x ∈ Ksuch that, (V V LI − I) ⟨T (x), η(y, x)⟩ ≥C 0, ∀y ∈ K, 2

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and { (V V LI − I)

F ind x ∈ Ksuch that, ⟨T (y), η(x, y)⟩ ≤C 0, ∀y ∈ K,

The following concepts and results are needed for the results. Definition 1.1. A mapping f : K → Y is said to be hemicontinuous if, for any fixed x, y ∈ K, the mapping t 7→ f (x + t(y − x)) is continuous at 0+ . Definition 1.2. Let C : K → 2Y be a set-valued mapping, h : K × K → Y and g : K × K → X are the single-valued mappings. Then (i) h(·, v) is said to be C-convex in the first argument if h(tu1 + (1 − t)u2 , v) ∈ th(u1 , v) + (1 − t)h(u2 , v) − C, ∀u1 , u2 ∈ K, t ∈ [0, 1], (ii) If K is an affine set, the η(x, y) is said to be affine with respect to u if for any given v ∈ K η(tu1 + (1 − t)u2 , v) = tη(u1 , v) + (1 − t)η(u2 , v), ∀u1 , u2 ∈ K, t ∈ R, with u = (tu1 + (1 − t)u2 ) ∈ K. Definition 1.3. Let T1 , T2 , · · · , TN : K → L(X, Y ), g, f : K → K, η : K × K → X and ϕ, µ : K × K → Y are mappings. Then T1 , T2 , · · · , TN are said to be g-f -η-ϕ-µ-pseudomonotone if for any x, y ∈ K, ⟨N ⟩ ∑ Ti (x), η(g(y), g(x)) + ϕ (f (y), f (x)) − µ (f (x), f (y)) ≥C 0, i=1

⇒

⟨N ∑

⟩ Ti (y), η(g(x), g(y))

+ ϕ (f (x), f (y)) − µ (f (y), f (x)) ≤C 0.

i=1

Example 1.4. Let X = R, K = R+ , Y = R2 , C = R2+ and ( ) ( x) ( x) ( x) 1 2 3 N T1 (x) = , T2 (x) = , T3 (x) = , · · · , TN (x) = 1 2−x 3−x N −x ( ) 8y − 12x g(x) = 3x, f (x) = 2x, η(y, x) = 4y − 5x, ϕ(y, x) = , 6y 2 − 5xy − 11x2 ) ( 3x − 4y , ∀x, y ∈ K. µ(x, y) = 4x2 − 2xy − 6y 2 Thus η(g(y), g(x)) = η(3y, 3x) = 4(3y) − 5(3x) = 12y − 15x,

3

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ϕ(f (y), f (x)) = ϕ(2y, 2x)   8(2y) − 12(2x) = 6(2y)2 − 5(2x)(2y) − 11(2x)2  ( =

) 16y − 24x , 24y 2 − 20xy − 44x2

and µ(f (x), f (y)) = µ(2y, 2x)   3(2x) − 4(2y) = 4(2x)2 − 2(2x)(2y) − 2(2y)2  ( =

) 6x − 8y . 16x2 − 8xy − 24y 2

Then ∀x, y ∈ K ⟨N ⟩ ∑ Ti (x), η(g(y), g(x)) + ϕ (f (y), f (x)) − µ (f (x), f (y)) i=1

    ) 16y − 24x 6x − 8y 1 + 2x + 3 x + · · · + N x (12y − 15x) + 24x2 − 20xy − 44y 2  − 16y 2 − 8xy − 24x2  1 + 2−x + 3−x + · · · + N −x

( =

  ) 16y − 24x − 6x + 8y 1 + 2x + 3 x + · · · + N x (12y − 15x) + 24y 2 − 20xy − 44x2 − 16x2 + 8xy + 24y 2  1 + 2−x + 3−x + · · · + N −x

( =

( =

1 + 2 + 3 + ··· + N 1 + 2−x + 3−x + · · · + N −x x

x

x

)



 24y − 30x (12y − 15x) + 48y 2 − 12xy − 60x2 

(

) ( ) 1 + 2x + 3 x + · · · + N x 2(12y − 15x) 15x) + (12y − 1 + 2−x + 3−x + · · · + N −x 2(12y − 15x)(y + x) ( ) ( ) x x x 1 + 2 + 3 + ··· + N 2 = (12y − 15x) + (12y − 15x) 1 + 2−x + 3−x + · · · + N −x 4(y + x) [( ) ( )] 1 + 2x + 3 x + · · · + N x 2 = (12y − 15x) + 1 + 2−x + 3−x + · · · + N −x 4(y + x) ( ) 1 + 2x + 3 x + · · · + N x + 2 = (12y − 15x) ≥C 0, 1 + 2−x + 3−x + · · · + N −x + 4y + 4x

=

implies that 12y ≥ 15x. Thus, 12x ≤ 15x ≤ 12y ≤ 15y. Therefore, 12x − 15y ≤ 0.

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So it follows that ⟨N ⟩ ∑ Ti (y), η(g(x), g(y)) + ϕ (f (x), f (y)) − µ (f (y), f (x)) i=1

( =

1 + 2 + 3 + ··· + N 1 + 2−y + 3−y + · · · + N −y y

y

y

)



   16x − 24y 6y − 8x (12x − 15y) + 24y 2 − 20xy − 44x2  − 16x2 − 8xy − 24y 2 

  ) 16x − 24y − 6y + 8x 1 + 2y + 3 y + · · · + N y (12x − 15y) + 24x2 − 20xy − 44y 2 − 16y 2 + 8xy + 24x2  1 + 2−y + 3−y + · · · + N −y

( =

( =

1 + 2 + 3 + ··· + N 1 + 2−y + 3−y + · · · + N −y y

y

y

)



 24x − 30y (12x − 15y) + 48x2 − 12xy − 60y 2 

(

) ( ) 1 + 2y + 3 y + · · · + N y 2(12x − 15y) = (12x − 15y) + 1 + 2−y + 3−y + · · · + N −y 2(12x − 15y)(x + y) ) ( ) ( y y y 1 + 2 + 3 + ··· + N 2 = (12x − 15y) + (12x − 15y) 1 + 2−y + 3−y + · · · + N −y 4(x + y) [( ) ( )] 1 + 2y + 3y + · · · + N y 2 = (12x − 15y) + 1 + 2−y + 3−y + · · · + N −y 4(x + y) ( ) y y y 1 + 2 + 3 + ··· + N + 2 = (12x − 15y) ≤C 0. 1 + 2−y + 3−y + · · · + N −y + 4x + 4y ⇒ T1 , T2 , · · · , TN are g-f -η-ϕ-µ-pseudomonotone. ∗

Definition 1.5. A multi-valued operator S : X → 2X is called quasimonotone if for all x, y ∈ X the following implications hold: ∃x∗ ∈ S(x) : ⟨x∗ , y − x⟩ > 0 ⇒ ∃y ∗ ∈ S(y) : ⟨y ∗ , y − x⟩ ≥ 0. ∗

Definition 1.6. A multi-valued operator S : X → 2X is called properly quasimonotone if for all x1 , x2 , ..., xn ∈ X and every y ∈ Conv{x1 , x2 , ..., xn } there exist i such that ∀x∗i ∈ S(xi ) : ⟨x∗i , y − xi ⟩ ≥ 0. Definition 1.7. A mapping T : K → L(X, Y ) is said to be properly quasimonotone of Stampacchia type if for all n ∈ N for all vectors v1 , v2 , ..., vn ∈ K and scalars λi ≥ 0, i = 1, 2, ..., n with ∑n ∑n i=1 λi = 1 and u := i=1 λi vi , ⟨T u, vi − u⟩ ≥C 0 holds for some i. T is said to be properly quasimonotone of Minty type if for all vectors v1 , v2 , ..., vn ∈ K and scalars λi ≥ 0, i = 1, 2, ..., n ∑n ∑n with i=1 λi = 1 and u := i=1 λi vi , ⟨T vi , vi − u⟩ ≤C 0 holds for some i. Definition 1.8. A mapping T : K → L(X, Y ) is said to be properly g-η-quasimonotone of Stampacchia type if for all n ∈ N for all vectors v1 , v2 , ..., vn ∈ K and scalars λi ≥ 0, i = 1, 2, ..., n with ∑n ∑n i=1 λi = 1 and u := i=1 λi vi , ⟨T u, η(g(vi ), g(u))⟩ ≥C 0 holds for some i. T is said to be properly g-η-quasimonotone of Minty type if for all vectors v1 , v2 , ..., vn ∈ K and scalars λi ≥ 0, i = 1, 2, ..., n ∑n ∑n with i=1 λi = 1, and u := i=1 λi vi , ⟨T vi , η(g(vi ), g(u))⟩ ≤C 0 holds for some i. Definition 1.9. A mapping T : K → L(X, Y ) is said to be properly g-h-η-quasimonotone of Stampacchia type if for all n ∈ N for all vectors v1 , v2 , ..., vn ∈ K and scalars λi ≥ 0, i = 1, 2, ..., n ∑n ∑n with i=1 λi = 1 and u := i=1 λi vi , ⟨T u, η(g(vi ), g(u))⟩+h(g(vi ), g(u))≥C 0 holds for some i. T is said to be properly g-h-η-quasimonotone of Minty type if for all vectors v1 , v2 , ..., vn ∈ K and scalars ∑n ∑n λi ≥ 0, i = 1, 2, ..., n with i=1 λi = 1, and u := i=1 λi vi , ⟨T vi , η(g(u), g(vi ))⟩+h(g(u), g(vi ))≤C 0 holds for some i. 5

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Definition 1.10. A mapping T : K → L(X, Y ) is said to be properly g-f -η-ϕ-µ-quasimonotone of Stampacchia type if for all n ∈ N for all vectors v1 , v2 , ..., vn ∈ K and scalars λi ≥ 0, i = 1, 2, ..., n ∑n ∑n with i=1 λi = 1 and u := i=1 λi vi , ⟨T (u), η(g(vi ), g(u))⟩ + ϕ(f (vi ), f (u)) − µ(f (u), f (vi ))≥C 0 holds for some i. T is said to be properly g-f -η-ϕ-µ-quasimonotone of Minty type if for all vec∑n ∑n tors v1 , v2 , ..., vn ∈ K and scalars λi ≥ 0, i = 1, 2, ..., n with i=1 λi = 1, and u := i=1 λi vi , ⟨T (vi ), η(g(u), g(vi ))⟩ + ϕ(f (u), f (vi )) − µ(f (vi ), f (u))≤C 0 holds for some i. Example 1.11. Let X = R, K = R+ , Y = R2 , C = R2+ and ( ) ( ) ( ) ( ) 1 2 3 N T1 (x) = , T2 (x) = , T3 (x) = , · · · , TN (x) = x x2 x3 xN ( ) 5y + 3x g(x) = 2x, f (x) = 3x, η(y, x) = 7y − 5x, ϕ(y, x) = , 5y 2 + 3x2 ( ) 2x + 3y µ(x, y) = , ∀x, y ∈ K. 2x2 + 3y 2 Thus η(g(y), g(x)) = η(2y, 2x) = 7(2y) − 5(2x) = 14y − 10x, ϕ(f (y), f (x)) = ϕ(3y, 3x)   5(3y) + 3(3x) = 5(3y)2 + 3(3x)2  ( =

) 15y + 9x , 45y 2 + 27x2

and µ(f (x), f (y)) = µ(3x, 3y)   2(3x) + 3(3y) = 2(3x)2 + 3(3y)2  ) 6x + 9y . 18x2 + 27y 2

( =

We claim that T1 , T2 , · · · , TN are properly g-f -η-ϕ-µ-quasimonotone of Stampacchia type. Suppose ∑n to the contrary that there exists xi ∈ K, ti ≥ 0, i = 1, 2, ..., n with i=1 ti = 1 such that ⟨N ∑

⟩ Ti (x), η(g(xi ), g(x))

+ ϕ (f (xi ), f (x)) − µ (f (x), f (xi )) C 0, i = 1, 2, ..., n

i=1

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where xi =

∑n i=1

λi xi , it follows that ⟨N ∑

⟩ + ϕ (f (xi ), f (x)) − µ (f (x), f (xi ))

Ti (x), η(g(xi ), g(x))

i=1

(

) ( ) 15xi + 9x 6x + 9xi − 45xi 2 + 27x2 18x2 + 27xi 2   ( ) 6xi + 3x (1 + 2 + 3 + · · · + N )(14xi − 10x)   C 0 = + 18x i + 9x (x + x2 + x3 + · · · + xN )(14xi − 10x)

=

1+2+3+···+N x + x2 + x3 + · · · + xN

)

(

(14xi − 10x) +

( =

(1 + 2 + 3 + · · · + N )(14xi − 10x) + 6xi + 3x (x + x2 + x3 + · · · + xN )(14xi − 10x) + 18xi + 9x

) C 0

i = 1, 2, ..., n,

which is a contradiction, since (1 + 2 + 3 + · · · + N )(14xi − 10x) + 6xi + 3x ≥C 0, and (x + x2 + x3 + · · · + xN )(14xi − 10x) + 18xi + 9x ≥C 0, for atleast one i. Thus T1 , T2 , · · · , TN are properly g-f -η-ϕ-µ-quasimonotone of Stampacchia type. Lemma 1.12. Let T1 , T2 , · · · , TN : K → L(X, Y ), η : K × K → X, ϕ, µ : K × K → Y and g : K → K be mappings. If T1 , T2 , · · · , TN are g-f -η-ϕ-µ-pseudomonotone and properly g-f -η-ϕ-µquasimonotone of Stampacchia type, then T1 , T2 , · · · , TN are properly g-f -η-ϕ-µ-quasimonotone of Minty type. Proof. The fact directly follows from Definitions 1.3 and 1.9. Definition 1.13. Let D be a nonempty subset of a topological Hausdorff space E. A mapping G : D → 2E (the family of nonempty subset of E) is said to be a KKM mapping if for any finite subset {x1 , x2 , ..., xn } ⊂ D, ∪n we have Co {x1 , ..., xn } ⊂ i=1 G(xi ). where Co denotes the convex hull operator. Lemma 1.14 ([6]). Let D be a nonempty subset of a topological Hausdorff vector space E and G : D → 2E be a KKM mapping. If G(x) is closed for any x ∈ D, and compact for some x ∈ D, ∩ then G(x) ̸= ∅. x∈D

Lemma 1.15. Let Y be a topological vector space with a pointed, closed and convex cone C such that intC ̸= ∅. If u, v ∈ Y and u ∈ / C and v ∈ −C, then tv + (1 − t)u ∈ / C, ∀t ∈ (0, 1). Proof. Assume that u, v ∈ Y and u ∈ / C and v ∈ −C. We must to show that tv + (1 − t)u ∈ / C ∀t ∈ (0, 1). Suppose to the contrary that there exists some t ∈ (0, 1) such that tv + (1 − t)u ∈ C. Since C is cone and v ∈ −C, we have −tv ∈ C. Thus tv + (1 − t)u + (−tv) ∈ C + C ⊂ C and hence 1 (1 − t)u ∈ C. By (1 − t) > 0 and C is cone, it follows that (1−t) (1 − t)u ∈ C. So u ∈ C. Which is a contradiction. Hence tv + (1 − t)u ∈ / C, ∀t ∈ (0, 1). This completes the proof.

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2. Existence results In this section, we establish some existence results for (NEVVLI-I) and (NEVVLI-II) by using Lemma 1.14. Lemma 2.1. Let T1 , T2 , · · · , TN : K → L(X, Y ), η : K × K → X, ϕ, µ : K × K → Y be mappings and g, f : K → K is affine mapping satisfying the following conditions: (a) T1 , T2 , · · · , TN are g-f -η-ϕ-µ-pseudomonotone; ⟨N ⟩ ∑ (b) for any fixed x ∈ X, the mapping y 7→ Ti (y), η(g(x), g(y)) is hemicontinuous and i=1

ϕ(f (x), f (y)) and µ(f (y), f (x)) are continuous with {zt } → x0 ∈ K, zt ∈ K; (c) ϕ(·, f (y)) is C-convex in the first variable and ϕ(f (x), f (x)) = 0, ∀x ∈ K; (d) µ(f (y), ·) is C-concave in the second variable and µ(f (x), f (x)) = 0, ∀x ∈ K; (e) η(·, g(y)) is affine in the first variable and η(g(x), g(x)) = 0, ∀x ∈ K. Then for any x0 ∈ K, the following statements are equivalent ⟨N ⟩ ∑ (I) Ti (x0 ), η(g(x), g(x0 )) + ϕ (f (x), f (x0 )) − µ (f (x0 ), f (x)) ≥C 0, i=1

(II)

⟨N ∑

⟩ Ti (x), η(g(x0 ), g(x)) + ϕ (f (x0 ), f (x)) − µ (f (x), f (x0 )) ≤C 0.

i=1

Proof. T1 , T2 , · · · , TN are g-f -η-ϕ-µ-pseudomonotone, it follows that (I) ⇒ (II). (II) ⇒ (I). Suppose that (II) holds for any x0 ∈ K ⟨N ⟩ ∑ Ti (y), η(g(x), g(y)) + ϕ (f (x), f (y)) − µ (f (y), f (x)) ≤C 0. i=1

For arbitrary z ∈ K, letting zt = (1 − t)x0 + tz, t ∈ (0, 1), we have zt ∈ K by convexity of K. Hence we have ⟨N ⟩ ∑ Ti (zt ), η(g(x0 ), g(zt )) + ϕ (f (x0 ), f (zt )) − µ (f (zt ), f (x0 )) ≤C 0. i=1

Now we show that ⟩ ⟨N ∑ Ti (zt ), η(g(z), g(zt )) + ϕ (f (z), f (zt )) − µ (f (zt ), f (z)) ≥C 0. i=1

Suppose to the contrary that there exists some t ∈ (0, 1) such that ⟨N ⟩ ∑ Ti (zt ), η(g(z), g(zt )) + ϕ (f (z), f (zt )) − µ (f (zt ), f (z)) C 0. i=1

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As C is a convex cone and in veiw of (c), (d) and (e) we get ⟨N ⟩ ∑ 0= Ti (zt ), η(g(zt ), g(zt )) + ϕ (f (zt ), f (zt )) − µ (f (zt ), f (zt )) i=1

=

⟨N ∑

⟩

Ti (zt ), η(g((1 − t)x0 + tz), g(zt ))

+ ϕ (f ((1 − t)x0 + tz), f (zt ))

i=1

− µ (f (zt ), f ((1 − t)x0 + tz)) ⟨N ⟩ ∑ = Ti (zt ), η((1 − t)g(x0 ) + tg(z), g(zt )) + ϕ ((1 − t)f (x0 ) + tf (z), f (zt )) i=1

− µ (f (zt ), (1 − t)f (x0 ) + tf (z)) ⟨N ⟩ ∑ = Ti (zt ), (1 − t)η(g(x0 )g(zt )) + tη(g(z), g(zt )) + ϕ ((1 − t)f (x0 ) + tf (z), f (zt )) i=1

− µ (f (zt ), (1 − t)f (x0 ) + tf (z)) ⟨N ⟩ ∑ ≤C Ti (zt ), (1 − t)η(g(x0 )g(zt )) + tη(g(z), g(zt )) + (1 − t)ϕ (f (x0 ), f (zt )) i=1

+ tϕ(f (z), f (zt )) − [(1 − t)µ (f (zt ), f (x0 )) + tµ(f (zt ), f (z))] ⟩ ⟩ ⟨N ⟨N ∑ ∑ Ti (zt ), tη(g(z), g(zt )) Ti (zt ), (1 − t)η(g(x0 )g(zt )) + = i=1

i=1

+ (1 − t)ϕ (f (x0 ), f (zt )) + tϕ(f (z), f (zt )) − (1 − t)µ (f (zt ), f (x0 )) − tµ(f (zt ), f (z)) ⟨N ⟩ ⟨N ⟩ ∑ ∑ = (1 − t) Ti (zt ), η(g(x0 )g(zt )) + t Ti (zt ), η(g(z), g(zt )) i=1

i=1

+ (1 − t)ϕ (f (x0 ), f (zt )) + tϕ(f (z), f (zt )) − (1 − t)µ (f (zt ), f (x0 )) − tµ(f (zt ), f (z)) {⟨ N ⟩} ∑ =t Ti (zt ), η(g(z), g(zt )) + ϕ(f (z), f (zt )) − tµ(f (zt ), f (z)) i=1

+ (1 − t) ∈t

{⟨ N ∑

{⟨ N ∑

⟩} Ti (zt ), η(g(x0 )g(zt )) + ϕ (f (x0 ), f (zt )) − µ (f (zt ), f (x0 ))

i=1

⟩}

Ti (zt ), η(g(z), g(zt )) + ϕ(f (z), f (zt )) − tµ(f (zt ), f (z))

i=1

+ (1 − t)

{⟨ N ∑

⟩} Ti (zt ), η(g(x0 )g(zt )) + ϕ (f (x0 ), f (zt )) − µ (f (zt ), f (x0 ))

− C,

i=1

which implies that {⟨ N ⟩} ∑ t Ti (zt ), η(g(z), g(zt )) + ϕ(f (z), f (zt )) − tµ(f (zt ), f (z)) i=1

+ (1 − t)

{⟨ N ∑

⟩} Ti (zt ), η(g(x0 )g(zt )) + ϕ (f (x0 ), f (zt )) − µ (f (zt ), f (x0 ))

i=1

∈ C.

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Which is a contradiction. Hence ⟨N ⟩ ∑ Ti (zt ), η(g(z), g(zt )) + ϕ (f (z), f (zt )) − µ (f (zt ), f (z)) ≥C 0. i=1

Condition (b) implies that ⟨N ⟩ ∑ Ti (x0 ), η(g(z), g(x0 )) + ϕ (f (z), f (x0 )) − µ (f (x0 ), f (z)) ≥C 0, ∀x ∈ K. i=1

This completes the proof. Theorem 2.2. Let X and Y be two real Banach spaces and K ⊂ X a nonempty, compact and convex set. Let T1 , T2 , · · · , TN : K → L(X, Y ), η : K × K → X, ϕ, µ : K × K → Y and g, f : K → K are the mappings satisfying the following conditions: ⟨N ⟩ ∑ (i) for any fixed y ∈ X, the mapping x 7→ Ti (x), η(g(y), g(x)) , ϕ(f (y), f (x)) and µ(f (x), f (y)) i=1

are continuous; (ii) T1 , T2 , · · · , TN are properly g-f -η-ϕ-µ-quasimonotone of Stampacchia type; (iii) for all x ∈ K, η(g(x), g(x)) = 0 and ϕ(f (x), f (x)) = 0 = µ(f (x), f (x)). Then there exists x ∈ K such that ⟨N ⟩ ∑ Ti (x), η(g(y), g(x)) + ϕ (f (y), f (x)) − µ (f (x), f (y)) ≥C 0, ∀y ∈ K. i=1

Proof. Define a multivalued mapping M1 : K → 2K by {⟨ N ⟩ } ∑ M1 (z) = Ti (x), η(g(z), g(x)) + ϕ (f (z), f (x)) − µ (f (x), f (z)) ≥C 0 , ∀z ∈ K, i=1

then M1 (z) is nonempty for each z ∈ K. We claim that M1 is a KKM mapping. In fact if it is ∑n not the case then there exists {x1 , x2 , ..., xn } ⊂ K, x = i=1 ti xi with ti > 0, i = 1, 2, ..., n and ∪m ∑n / i=1 M1 (xi ). i=1 ti = 1 such that x ∈ This implies that ⟩ ⟨N ∑ Ti (x), η(g(xi ), g(x)) + ϕ (f (xi ), f (x)) − µ (f (x), f (xi )) ≥C 0. i=1

This contradicts condition (ii). Therefore M1 is a KKM mapping; Now we prove that for any z ∈ K, M1 (z) is closed. In veiw of (i), let there exists a net {xn } ⊂ M1 (z) such that xn → x ∈ K. Because ⟨N ⟩ ∑ Ti (xn ), η(g(z), g(xn )) + ϕ (f (z), f (xn )) − µ (f (xn ), f (z)) ≥C 0, ∀n, i=1

we have ⟨N ∑

⟩ Ti (x), η(g(z), g(x))

+ ϕ (f (z), f (x)) − µ (f (x), f (z)) ≥C 0.

i=1

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Hence x ∈ M1 (z) and so M1 (z) is closed. It follows from the compactness of K and closedness of ∩ M1 (z) ⊂ K, that M1 (z) is compact. Thus by Lemma 1.14, we have z∈K M1 (z) ̸= ∅. Hence there exist x ∈ K such that ⟨N ⟩ ∑ Ti (x), η(g(y), g(x)) + ϕ (f (y), f (x)) − µ (f (x), f (y)) ≥C 0 ∀y ∈ K. i=1

This completes the proof. Theorem 2.3. Let K be a nonempty, bounded, closed and convex subset a real reflexive Banach space X and Y a real Banach space. Let T1 , T2 , · · · , TN : K → L(X, Y ), η : K × K → X, ϕ, µ : K × K → Y and g, f : K → K are the mappings satisfying the following conditions: ⟨N ⟩ ∑ (i) for any fixed y ∈ X, the mapping Ti (y), η(g(·), g(y)) , ϕ(f (·), f (y)) and µ(f (y), f (·)) are i=1

lower semicontinuous; (ii) T1 , T2 , · · · , TN are properly g-f -η-ϕ-µ-quasimonotone of Minty type; (iii) forall x ∈ K, η(g(x), g(x)) = 0 and ϕ(f (x), f (x)) = 0 = µ(f (x), f (x)). Then there exists x ∈ K such that ⟨N ⟩ ∑ Ti (y), η(g(x), g(y)) + ϕ (f (x), f (y)) − µ (f (y), f (x)) ≤C 0, ∀y ∈ K. i=1

Proof. Define a multivalued mapping M2 : K → 2K by { ⟨N ⟩ } ∑ M2 (z) = x ∈ K : Ti (y), η(g(x), g(y)) + ϕ (f (x), f (y)) − µ (f (y), f (x)) ≤C 0 , ∀z ∈ K. i=1

then M2 (z) is nonempty for each z ∈ K. We claim that M2 is not KKM mapping, then there ∑n ∑n exists {x1 , x2 , ..., xn } ⊂ K, x = ti xi with ti > 0, i = 1, 2, ..., n and i=1 ti = 1 such that i=1 ∪m x∈ / i=1 M2 (xi ). This implies that ⟨N ⟩ ∑ Ti (xi ), η(g(x), g(xi )) + ϕ (f (x), f (xi )) − µ (f (xi ), f (x)) C 0, i = 1, 2, ..., n. i=1

This contradicts condition (ii). Therefore M2 is a KKM mapping. Since K is bounded, M2 (z) is bounded. From (ii), we have M2 (z) is convex. Next, we will show that M2 (z) closed. In veiw of (i), let there exists a net {xn } ⊂ M2 (z) such that xn → x ∈ K. Because ⟨N ⟩ ∑ Ti (y), η(g(xn ), g(y)) + ϕ (f (xn ), f (y)) − µ (f (y), f (xn )) ≤C 0, ∀n, i=1

we have ⟨N ∑

⟩ Ti (y), η(g(x), g(y))

+ ϕ (f (x), f (y)) − µ (f (y), f (x)) ≤C 0.

i=1

Hence x ∈ M2 (z) and so M2 (z) is closed. Since X is reflexive, M2 (z) is weakly compact for all z ∈ K. It follows from Lemma 1.14, that z∈K M2 (z) ̸= ∅. Hence there exist x ∈ K such that ⟨N ⟩ ∑ Ti (y), η(g(x), g(y)) + ϕ (f (x), f (y)) − µ (f (y), f (x)) ≤C 0, ∀y ∈ K.

∩

i=1

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It is useful to mention that the result of Theorem 2.2 can be viewed as an improvement of the following corollary. Corollary 2.4 ([9, Theorem 2.1]). Let X and Y be two real Banach spaces and K ⊂ X a nonempty, compact and convex set. Let S, T : K → L(X, Y ), η : K × K → X, h : K × K → Y and g : K → K are the mappings satisfying the following conditions: (a) for any fixed y ∈ X, the mapping continuous;

x 7→ ⟨S(y) + T (y), η(g(x), g(y))⟩ and h(g(x), g(y)) are

(b) S and T are properly g-h-η-quasimonotone of Stampacchia type; (c) for all x ∈ K, η(g(x), g(x)) = 0 = h(g(x), g(x)). Then there exists x ∈ K such that ⟨S(x) + T (x), η(g(y), g(x))⟩ + h(g(y), g(x)) ≥C 0, ∀y ∈ K. Proof. By taking T3 , T4 , · · · , TN ≡ 0, T1 = S, T2 = T , ϕ = h, µ ≡ 0 and f = g in Theorem 2.2, we obtain the desired results. Acknowledgements The authors would like to express their thanks to the referees for their constructive and helpful suggestions the improvement of this paper. Moreover, The first author would like to thank Naresuan University for financial support. References [1] R. Ahmad, Existence results for vector variational-like inequalities, Thai J. Math, 9(3), 553561(2011). [2] R. Ahmad, S.S. Irfan, On generalized nonlinear variational-like inequality problem, Appl. Math. Lett, 19, 294-297(2006). [3] Q.H. Ansari, J.C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optim. Theory Appl, 108, 527-541(2001). [4] Q.H. Ansari, J.C. Yao, On nondifferentiable and nonconvex vector optimization problems, J. Optim. Theory Appl, 106 (3), 487-500(2000). [5] A. Deniilidis, N. Hadjisavvas, Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions, J. Optm. Theory Appl, 102, 525-536(1999). [6] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Anal, 266, 519547(1984). [7] A.P. Farajzadeh, B.S. Lee, Vector variational-like inequality problem and vector optimization problem, Appl. Math. Lett, 23, 48-52(2010). [8] F. Giannessi, Theorems of alternative quadratic programs and complementarity problems, in: R. Cottle, F. Giannessi, J.L. Lions (Eds.), Variational Inequalities and Complementarity Problems, John Wiley and Sons, 1980. [9] S.S. Irfan and R. Ahmad, Existence results for extended vector variational-like inequality. Journal of the Egyptian Mathematical Society, 23, 144-148(2015).

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[10] S.S. Irfan, R. Ahmad, Generalized multivalued vector variational inequalities, J. Glob. Optim, 46 (1), 25-30(2010). [11] S.K. Mishraa, S.Y. Wang, Vector variational-like inequalities and non-smooth vector optimization problems, Nonlinear Anal, 64, 1939-1945(2006). [12] Y. Zhao, Z. Xia, Existence results for system of variational-like inequalities, Nonlinear Anal. Real World Appl, 8, 1370-1378(2007).

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Existence of solutions for a new semi-linear evolution equations with impulses Huanhuan Zhang

1,2†

, Yongxiang Li

∗

2

1. School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Lanzhou, 730030, Peoples’s Republic of China, 2. Department of Mathematics, Northwest Normal University, Lanzhou 730070, Peoples’s Republic of China

Abstract By using monotone iterative technique and operator semigroup theorem, we consider the existence of mild solutions for a class of nonlocal semi-linear evolution equation with not instantaneous impulses in ordered Banach spaces. Finally, an example is given to show the existence results. Key Words: evolution equation; not instantaneous impulses; operator semigroup; upper and lower solutions; monotone iterative technique; mild solutions MR(2010) Subject Classification: 34K30; 34K45; 47H05.

1

Introduction

The impulsive differential equations are used to describe mathematical models of many real processes and phenomena studied by physical, chemical, biological, population dynamics, industrial robotics, economics, engineering and so on, see [1]. Applied impulsive mathematical models have become an active research subject in nonlinear science and have attracted more attention in many fields , see[2-4] and references therein. For more details on differential equations with “abrupt and instantaneous” impulses, one can see for instance the monographs[5-7] and the references therein. By means of monotone iterative method coupled with lower and upper solutions, some sufficient ∗

Research supported by NNSF of China (11261053, 11401473, 11501455), NSF of Gansu Province (1208RJZA129), the Fundamental Research Funds for the Central Universities(31920130010). † Corresponding author. E-mail address: [email protected] (H. Zhang)

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conditions for the existence of solutions of impulsive integro-differential equations were established in [8]. Recently, the existence results to impulsive differential equations with nonlocal conditions was studied in [9-15]. Moreover, Chen, Li and Yang[16] used the perturbation method and monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions for the nonlocal impulsive evolution equation in ordered Banach spaces. However, it seems that the models with instantaneous impulses could not explain the certain dynamics of evolution processes in pharmacotherapy. For example, one considers the hemodynamic equilibrium of a person, the introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous process. Hernandez and O’Regan[17] and Pierri et al.[18] initially studied on Cauchy problems for a new type first order evolution equations with not instantaneous impulses of the form:  0 u (t) = Au(t) + f (t, u(t)), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     u(t) = hi (t, u(t)), t ∈ (ti , si ], i = 1, 2, · · · , m.     u(0) = u0 . Wang and Li[19], Yu et al.[20] considered periodic boundary value problems for nonlinear evolution equations with non instantaneous impulses. Wang et al.[21] discussed a class of new fractional differential equations with not instantaneous impulses. However, to the best of our knowledge, the existence mild solutions for nonlocal evolution equations with not instantaneous impulses by means of monotone iterative technique has not been investigated yet. Motivated by this consideration, in this paper, we discuss the existence of mild solutions for the nonlocal evolution equation with not instantaneous impulses in an ordered Banach space X  0 u (t) + Au(t) = f (t, u(t)), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     u(t) = hi (t, u(t)), t ∈ (ti , si ], i = 1, 2, · · · , m, (1.1)     u(0) = g(u), where A : D(A) ⊂ X → X is a closed linear operator and −A generates a C0 -semigroup T (t)(t ≥ 0) in X; 0 = s0 < t1 ≤ s1 < t2 ≤ s2 < t3 ≤ s3 < · · · < tm−1 ≤ sm−1 < tm ≤ sm < tm+1 = a are pre-fixed numbers, J = [0, a], a > 0 is a constant; f ∈ C([0, a], X). hi ∈ C([ti , si ] × X, X) for all i = 1, 2, · · · , m.

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2

Preliminaries

Throughout this paper, Let X be a Banach space, A : D(A) ⊂ X → X be a closed linear operator and −A generate a C0 -semigroup T (t)(t ≥ 0) in X. Denote M ≡ sup kT (t)k, t∈J

which is a finite number. For more details of the properties of the operator semigroups and positive C0 -semigroup, we refer to the monographs[22, 23] and [24]. Let X be an ordered Banach space with the norm k·k and partial order ” ≤ ”, whose positive cone K = {x ∈ X|x ≥ θ} is normal with normal constant N . Let C(J, X) with the norm kukC = max ku(t)k, then C(J, X) is an ordered Banach space induced by the t∈J

convex cone KC = {u ∈ C(J, X) | u(t) ≥ 0, t ∈ J}, and KC is also a normal cone. Let J 0 = J \ {t1 , t2 , · · · , tm }, J 00 = J \ {0, t1 , t2 , · · · , tm }. Evidently, P C(J, X) = {u : J → X | u(t) is continuous in J 0 , and left continuous at tk , and u(t+ k ) exists, k = 1, 2, · · · , m}. P C(J, X) is a Banach space with the norm k · kP C = sup ku(t)k. t∈J

Evidently, P C(J, X) is also an order Banach space with the partial order ” ≤ ” induced by the positive cone KP C = {u ∈ P C(J, X)|u(t) ≥ θ, t ∈ J}. KP C is normal with the same normal constant N . For v, w ∈ P C(J, X) with v ≤ w, we use [v, w] to denote the order interval {u ∈ P C(J, X) | v ≤ u ≤ w} in P C(J, X), and [v(t), w(t)] to denote the order interval {u ∈ X | v(t) ≤ u(t) ≤ w(t), t ∈ J} in X. We use X1 to denote the Banach space D(A) with the graph norm k · k1 = k · k + kA · k. For more details and definitions of the partial and cone, we refer to the monographs [25, 26]. Definition 2.1. If functions v0 ∈ P C(J, X) ∩ C 1 (J 00 , X) ∩ C(J 0 , X1 ) satisfy  0 v (t) + Av0 (t) ≤ f (t, v0 (t)), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,    0 v0 (t) ≤ hi (t, v0 (t)), t ∈ (ti , si ], i = 1, 2, · · · , m.    v0 (0) ≤ g(v0 ),

(2.1)

we call v0 a lower solution of problem (1.1); if all the inequalities of (2.1) are inverse, we call it an upper solution of problem (1.1). Next, we recall some properties of measure of noncompactness that will be used in the proof of our main results. Let α(·) denotes the Kuratowski measure of noncompactness of the bounded set.For the details of the definition and properties of the measure of noncompactness, see [25]. The following lemmas are needed in our arguments. Lemma 2.3.([27]) Let B ⊂ C(J, X) be bounded and equicontinuous. Then α(B(t)) is continuous on J, and α(B) = max α(B(t)) = α(B(J)). t∈J

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Lemma 2.4. ([28]) Let B = {un } ⊂ C(J, X)(n = 1, 2, · · ·) be a bounded and countable set. Then α(B(t)) is Lebesgue integral on J, and Z n Z o α un (t)dt | n ∈ N ≤ 2 α(B(t))dt. (2.2) J

3

J

Linear nonlocal problem

Let I = [t0 , t], t0 ≥ 0. It is well-known ([22])that for any x0 ∈ D(A) and h ∈ C (I, X), the initial value problem of linear evolution equation ( 0 u (t) + Au(t) = h(t), t ∈ I, (3.1) u(t0 ) = x0 , 1

has a unique classical solution u ∈ C 1 (I, X) ∩ C(I, X1 ) expressed by Z t u(t) = T (t − t0 )x0 + T (t − s)h(s)ds, t ∈ I

(3.2)

t0

If x0 ∈ X and h ∈ C(I, X), the function u given by (3.2) belongs to C(I, X), which is known as a mild solution of IVP(3.1). To prove our main results, for any h ∈ P C(J, X) and yi ∈ P C(J, X), i = 1, 2, · · · , m, we consider the linear nonlocal evolution equation with not instantaneous impulses in X  0 u (t) + Au(t) = h(t), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     u(t) = yi (t), t ∈ (ti , si ], i = 1, 2, · · · , m. (3.3)     u(0) = g(u). Theorem 3.1. Let X be a Banach space, A : D(A) ⊂ X → X be a closed linear operator and −A generate a C0 -semigroup T (t)(t ≥ 0) in X. For any h ∈ P C(J, X), yi ∈ P C(J, X), i = 1, 2, · · · , m, g : P C(J, X) → X, problem(3.3) has a unique mild solution u ∈ P C(J, X) given by  Rt  u(t) = T (t)g(u) + T (t − τ )h(τ )dτ, t ∈ [0, t1 ];  0   u(t) = yi (t), t ∈ (ti , si ], i = 1, 2, · · · , m;     u(t) = T (t − s )y (s ) + R t T (t − τ )h(τ )dτ, t ∈ (s , t ], i = 1, 2, · · · , m. i i i i i+1 si (3.4) 4 528

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Proof Let t ∈ [0, t1 ], problem(3.3) is equivalent to the linear nonlocal evolution equation without impulse ( 0 u (t) + Au(t) = h(t), t ∈ [0, t1 ], (3.5) u(0) = g(u). Then (3.5) has a unique mild solution u ∈ C([0, t1 ], X) given by Z t T (t − τ )h(τ )dτ. u(t) = T (t)g(u) + 0

Let t ∈ (ti , si ], then u(t) = yi (t), i = 1, 2, · · · , m. Let t ∈ (si , ti+1 ], problem(3.3) is equivalent to the initial value problem of linear evolution equation ( 0 u (t) + Au(t) = h(t), t ∈ (si , ti+1 ], , i = 1, 2, · · · , m, (3.6) u(si ) = yi (si ). Then (3.6) has a unique mild solution u ∈ C([si , ti+1 ], X) given by Z t T (t − τ )h(τ )dτ. u(t) = T (t − si )yi (si ) + si

Inversely, we can verify directly that the function u ∈ P C(J, X) defined by(3.4) is a mild solution of problem(3.3). Hence problem(3.3) has a unique mild solution u ∈ P C(J, X) given by (3.4). This completes the proof. Remark 3.2. In Theorem 3.1, let X be an ordered Banach space,−A generate a positive C0 -semigroup T (t)(t ≥ 0) in X. For any h ≥ θ,g ≥ θ and yi ≥ θ, i = 1, 2, · · · , m, then the mild solution of problem(3.3) is a positive solution.

4

The main results Now, we are in a position to state and prove our main results of this section.

Theorem 4.1. Let X be an ordered Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate a compact and positive C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). Assume that problem(1.1) has lower and upper solutions v0 and w0 with v0 (t) ≤ w0 (t)(t ∈ J). Suppose that the following conditions are satisfied: (H1) There exists a constant C ≥ 0 such that f (t, x2 ) − f (t, x1 ) ≥ −C(x2 − x1 ),

t ∈ J,

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for any t ∈ J, and v0 (t) ≤ x1 ≤ x2 ≤ w0 (t). (H2) The impulsive functions hi (i = 1, 2, · · · , m) are satisfy the conditions hi (t, x2 ) ≥ hi (t, x1 ), i = 1, 2, · · · , m, for ∀t ∈ J, v0 (t) ≤ x1 ≤ x2 ≤ w0 (t). (H3) The nonlocal function g(u) is increasing in u for u ∈ [v0 , w0 ]. (H4) hi ∈ C(J × X, X)(i = 1, 2, · · · , m) are compact operators. (H5) g : P C(J, X) → X is compact operator. Then the problem (1.1) has minimal and maximal mild solutions u and u between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 . Proof It is easy to see that −(A + CI) generates a positive compact semigroup S(t) = e−Ct T (t). Define D = [v0 , w0 ]. Let M = sup kS(t)k, we define an operator t∈J

Q : D → P C(J, X) by  Rt   S(t)g(u) + 0 S(t − τ )(f (τ, u(τ )) + Cu(τ ))dτ, t ∈ [0, t1 ];      hi (t, u(t)), t ∈ (ti , si ], i = 1, 2, · · · , m; (Qu)(t) = Rt   S(t − si )hi (si , u(si )) + si S(t − τ )(f (τ, u(τ )) + Cu(τ ))dτ,      t ∈ (s , t ], i = 1, 2, · · · , m. i

(4.1)

i+1

Since f , hi and g are continuous, so Q : D → P C(J, X) is continuous. Clearly, from Theorem 3.1, the mild solutions of problem (1.1) are equivalent to the fixed point of operator Q. (i) We show Q : D → P C(J, X) is an increasing operator. For ∀x1 , x2 ∈ D and x1 ≤ x2 , from the assumptions (H1) and (H2), we have f (t, x1 (t)) + Cx1 (t) ≤ f (t, x2 (t)) + Cx2 (t), t ∈ J.

(4.2)

hi (t, x1 (t)) ≤ hi (t, x2 (t)), i = 1, 2, · · · , m.

(4.3)

and Combining the positive of C0 -semigroup S(t) with (4.2), (4.2) and (H3), we have Z t S(t)g(x1 ) + S(t − τ )(f (τ, x1 (τ )) + Cx1 (τ ))dτ 0 Z t ≤ S(t)g(x2 ) + S(t − τ )(f (τ, x2 (τ )) + Cx2 (τ ))dτ, t ∈ [0, t1 ]; 0

hi (t, x1 (t)) ≤ hi (t, x2 (t)), t ∈ (ti , si ],

i = 1, 2, · · · , m;

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t

Z

S(t − τ )(f (τ, x1 (τ )) + Cx1 (τ ))dτ

S(t − si )hi (si , x1 (si )) + s Z it

≤ S(t − si )hi (si , x2 (si )) +

S(t − τ )(f (τ, x2 (τ )) + Cx2 (τ ))dτ, si

t ∈ (si , ti+1 ],

i = 1, 2, · · · , m.

Namely, Q : D → P C(J, X) is an increasing operator. (ii) We show v0 ≤ Q(v0 ), Q(w0 ) ≤ w0 . Let  0 v0 (t) + Av0 (t) + Cv0 (t) = f¯(t) t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     ¯ i (t), t ∈ (ti , si ], i = 1, 2, · · · , m. v0 (t) = h     v0 (0) = g¯(v0 ), by the definition of v0 , we have  ¯ f (t) ≤ f (t, v0 (t)) + Cv0 (t), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     ¯ i (t) ≤ hi (t, v0 (t)), t ∈ (ti , si ], i = 1, 2, · · · , m. h     g¯(v0 ) ≤ g(v0 ), By Theorem 3.1, (4.5)and (4.6), we have  Rt  S(t)¯ g (v ) + S(t − τ )f¯(τ )dτ, t ∈ [0, t1 ]; 0  0     ¯ i (t), t ∈ (ti , si ], i = 1, 2, · · · , m;  h v0 (t) = R  ¯ i (si ) + t S(t − τ )f¯(τ )dτ,  S(t − si )h  si     t ∈ (s , t ], i = 1, 2, · · · , m. i i+1

(4.4)

(4.5)

(4.6)

and Z

t

S(t − τ )f¯(τ )dτ

S(t)¯ g (v0 ) + Z0 t ≤ S(t)g(v0 ) +

S(t − τ )(f (τ, v0 (τ )) + Cv0 (τ ))dτ,

t ∈ [0, t1 ];

0

¯ i (t) ≤ hi (t, v0 (t)), t ∈ (ti , si ], i = 1, 2, · · · , m; h Z t ¯ i (si ) + S(t − si )h S(t − τ )f¯(τ )dτ si

Z

t

≤ S(t − si )hi (si , v0 (si )) +

S(t − τ )(f (τ, v0 (τ )) + Cv0 (τ ))dτ, si

t ∈ (si , ti+1 ],

i = 1, 2, · · · , m. 7 531

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Namely, v0 (t) ≤ Q(v0 )(t). Similarly, it can be shown that Q(w0 )(t) ≤ w0 (t). Therefore, Q : [v0 , w0 ] → [v0 , w0 ] is a continuously increasing operator. (iii) We prove that the operator Q has fixed points on [v0 , w0 ]. Now, we define two sequences {vn } and {wn } by the iterative scheme vn = Q(vn−1 ),

wn = Q(wn−1 ),

n = 1, 2, · · ·

(4.7)

Then from the monotonicity of operator Q it follows that v0 ≤ v1 ≤ v2 ≤ · · · ≤ vn ≤ · · · ≤ wn ≤ · · · ≤ w2 ≤ w1 ≤ w0 .

(4.8)

Next, we prove that {vn } and {wn } are convergent in J. Let B = {vn | n ∈ N}, B0 = {vn−1 | n ∈ N}, then B0 = {v0 } ∪ B and B = Q(B0 ). For any vn−1 ∈ B0 , let Z t S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ, t ∈ [0, t1 ]; (Q1 vn−1 )(t) = S(t)g(u) + 0

t ∈ (ti , si ], i = 1, 2, · · · , m; Z t (Q3 vn−1 )(t) = S(t − si )hi (si , vn−1 (si )) + S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ, (Q2 vn−1 )(t) = hi (t, vn−1 (t)),

si

t ∈ (si , ti+1 ],

i = 1, 2, · · · , m.

For 0 < t ≤ a, by the assumption (H1), we know that f (t, v0 (t)) + Cv0 (t) ≤ f (t, vn−1 (t)) + Cvn−1 (t) ≤ f (t, w0 (t)) + Cw0 (t). Since f (t, v0 (t)) and f (t, w0 (t)) are continuous in compact set [0, a], so their image sets are compact sets in X, namely image sets are bounded. Combining this fact with the normality of cone K in X, we have ∃M1 > 0, ∀vn−1 ∈ B0 , kf (t, vn−1 (t)) + Cvn−1 (t)k ≤ kf (t, v0 (t)) + Cv0 (t)k + N0 kf (t, w0 (t)) + Cw0 (t) − f (t, v0 (t)) − Cv0 (t)k

(4.9)

≤ M1 . Case 1. For interval [0, t1 ] and any 0 <  < t1 , let Z t (W1 vn−1 )(t) := S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ 0

and (W1 vn−1 )(t)

Z

t−

S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ,

:= 0

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then k(W1 vn−1 )(t)) − (W1 vn−1 )(t)k Z t S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ = k 0 Z t− − S(t − s)τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ k 0 Z t ≤ kS(t − τ )kkf (τ, vn−1 (τ )) + Cvn−1 (τ )kdτ t−

≤ M M1 . Therefore, Y1 (t) , {(W1 vn−1 )(t) | vn−1 ∈ B0 } is precompact in X by using the total boundedness. On the other hand, by the assumption (H5), {S(t)g(vn−1 ) | vn−1 ∈ B0 } is precompact in X due to the compactness of S(t). Therefore, {(Q1 vn−1 )(t) | vn−1 ∈ B0 } is precompact in X for t ∈ [0, t1 ]. Case 2. For t ∈ (ti , si ], i = 1, 2, · · · , m, the set {(Q2 vn−1 )(t) | vn−1 ∈ B0 } is precompact in X by the assumption (H4). Case 3. For t ∈ (si , ti+1 ], i = 1, 2, · · · , m, similar to the case 1, {(Q3 vn−1 )(t) | vn−1 ∈ B0 } is precompact in X by (4.9)and the assumption (H4). Hence, {vn (t)} = {Q(vn−1 )(t) | vn−1 ∈ B0 } is precompact in X for t ∈ J, combining this fact with the monotonicity of {vn }, we easily prove that {vn (t)} is convergent. Let {vn (t)} → u(t) in t ∈ J. Similarly, we prove that {wn (t)} → u(t) in t ∈ J. Evidently {vn (t)}, {wn (t)} ∈ P C(J, X), so u(t) and u(t) is bounded integrable in J). Since for any t ∈ J, vn (t) = Q(vn−1 )(t), wn (t) = Q(wn−1 )(t), letting n → ∞, by the Lebesgue dominated convergence theorem, we have u(t) = Q(u)(t), u(t) = Q(u)(t) and u(t), u(t) ∈ P C(J, X). Combining this with monotonicity (4.8), we have v0 (t) ≤ u(t) ≤ u(t) ≤ w0 (t). Next, we prove that u(t) and u(t) are the minimal and maximal fixed points of Q in [v0 , w0 ], respectively. In fact, for any u∗ ∈ [v0 , w0 ], Q(u∗ ) = u∗ , we have v0 ≤ u∗ ≤ w0 and v1 = Q(v0 ) ≤ Q(u∗ ) = u∗ ≤ Q(w0 ) = w1 . Continuing such progress, we get vn ≤ u∗ ≤ wn . Letting n → ∞, we get u(t) ≤ u∗ ≤ u(t). Therefor, u(t) and u(t) are the minimal and maximal mild solutions of the problem (1.1) between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 , respectively. From the Theorem4.1, we obtain the following result. Theorem4.2. Let X be an ordered Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator 9 533

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and −A generate a compact and positive C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). Assume that problem(1.1) has lower and upper solutions v0 and w0 with v0 (t) ≤ w0 (t)(t ∈ J). Suppose that conditions (H1), (H2), (H3) and the following conditions are satisfied: (H6) {hi (·, xn )}(i = 1, 2, · · · , m) are precompact in X, for any increasing or decreasing monotonic sequence {xn } ⊂ [v0 , w0 ]. (H7) {g(xn )} is a precompact set in X, for any increasing or decreasing monotonic sequence {xn } ⊂ [v0 , w0 ]. Then problem (1.1) has minimal and maximal mild solutions u and u between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 . Next, we discuss the existence of the mild solutions for problem (1.1) under the function g is continuous in PC(J,X) and noncompactness measure conditions. Theorem 4.3. Let X be an ordered Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate an equicontinuous and positive C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). hi ∈ C(J × X, X)(i = 1, 2, · · · , m). g : P C(J, X) → X be a continuous function. Assume that problem(1.1) has lower and upper solutions v0 and w0 with v0 (t) ≤ w0 (t)(t ∈ J). Suppose that conditions (H1), (H2) and (H3) hold, and satisfy: (H8) There exist a constant L > 0 such that α({f (t, xn )}) ≤ Lα({xn }), for all t ∈ J, and increasing or decreasing sequence {xn } ⊂ [v0 (t), w0 (t)]. (H9) There exist constants 0 < Li < 1(i = 1, 2, · · · , m) such that α({hi (t, xn )}) ≤ Li α({xn }), (i = 1, 2, · · · , m), for all t ∈ J, and increasing or decreasing sequence {xn } ⊂ [v0 (t), w0 (t)]. 0

(H10) There exist a constant L > 0 such that 0

α({g(xn )}) ≤ L α({xn }), for all t ∈ J, and increasing or decreasing sequence {xn } ⊂ [v0 (t), w0 (t)]. 0

(H11) M [Li + L + 2(L + C)a] < 1(i = 1, 2, · · · , m). Then the problem (1.1) has minimal and maximal mild solutions u and u between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 . Proof From Theorem 4.1, we know that Q : [v0 , w0 ] → [v0 , w0 ] is continuous. Furthermore, if conditions (H1), (H2) and (H3) are satisfied, the iterative sequences {vn } and 10 534

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{wn } defined by (4.7) satisfying (4.8). Therefore, for any t ∈ J, {vn (t)} and {wn (t)} are monotone and order-bounded sequences in X. Next, we prove that {vn } and {wn } are convergent in J. Since T (t)(t ≥ 0) is an equicontinuous C0 -semigroup, so S(t)(t ≥ 0) also is an equicontinuous C0 -semigroup. Let B = {vn | n ∈ N} and B0 = {vn−1 | n ∈ N}, by (4.8) and the normality of the positive cone K, then B and B0 are bounded. (i) We prove that Q(B0 ) is equicontinuous in P C(J, X). Combining (H2) and (H3) with the normality of cone K in X, we have ∃M2 > 0, M3 > 0, ∀vn−1 ∈ B0 , kg(vn−1 )k ≤ kg(v0 )k + N0 kg(w0 ) − g(v0 )k ≤ M2 . khi (t, vn−1 (t))k ≤ khi (t, v(t))k + N0 khi (t, w0 (t)) − g(hi (t, v0 (t))k ≤ M3 .

(4.10) (4.11)

Case 1. For ∀t0 , t00 ∈ [0, t1 ] and t0 < t00 , by (4.9) and (4.10) we have that k(Qvn−1 )(t00 ) − (Qvn−1 )(t0 )k Z t00 00 S(t00 − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ = kS(t )g(vn−1 ) + 0 Z t0 −S(t0 )g(vn−1 ) − S(t0 − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ k 0

≤ kS(t00 ) − S(t0 )kkg(vn−1 )k Z t0 + kS(t00 − τ ) − S(t0 − τ )kkf (τ, vn−1 (τ )) + Cvn−1 (τ )kdτ 0 Z t00 + kS(t00 − τ )kkf (τ, vn−1 (τ )) + Cvn−1 (τ )kdτ t0 Z t0 0 00 0 ≤ M2 kS(t )kkS(t − t ) − Ik + M1 kS(t00 − τ ) − S(t0 − τ )kdτ + M M1 (t00 − t0 ) 0 Z t0 ≤ M2 M kS(t00 − t0 ) − Ik + M1 kS(t00 − t0 + τ ) − S(τ )kdτ + M M1 (t00 − t0 ) 0

→ 0(t00 − t0 → 0). Case 2. For ∀t0 , t00 ∈ (ti , si ](i = 1, 2, · · · , m) and t0 < t00 , we have that k(Qvn−1 )(t00 ) − (Qvn−1 )(t0 )k = khi (t00 , vn−1 (t00 )) − hi (t0 , vn−1 (t0 ))k → 0(t00 − t0 → 0).

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Case 3. For ∀t0 , t00 ∈ (si , ti+1 ](i = 1, 2, · · · , m) and t0 < t00 , by (4.9) and (4.11) we have that k(Qvn−1 )(t00 ) − (Qvn−1 )(t0 )k Z t00 00 S(t00 − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ = kS(t )hi (si , vn−1 (si )) + si t0

−S(t0 )hi (si , vn−1 (si )) −

Z

S(t0 − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ k

si

≤ kS(t00 ) − S(t0 )kkhi (si , vn−1 (si ))k Z t0 kS(t00 − τ ) − S(t0 − τ )kkf (τ, vn−1 (τ )) + Cvn−1 (τ )kdτ + si t00

Z

kS(t00 − τ )kkf (τ, vn−1 (τ )) + Cvn−1 (τ )kdτ Z t0 0 00 0 ≤ M3 kS(t )kkS(t − t ) − Ik + M1 kS(t00 − τ ) − S(t0 − τ )kdτ + M M1 (t00 − t0 ) +

t0

≤ M3 M kS(t00 − t0 ) − Ik + M1

Z

si t0 −si

kS(t00 − t0 + τ ) − S(τ )kdτ + M M1 (t00 − t0 )

0

→ 0(t00 − t0 → 0). Therefore, Q(B0 ) is equicontinuous in P C(J, X). (ii) We prove that α(B(t)) = 0 for t ∈ J. It follows from B0 = {v0 } ∪ B that α(B(t)) = α(B0 (t)) for t ∈ J. Case 1. For t ∈ [0, t1 ], by Lemma 2.3 and Lemma 2.4, we have that α(B(t)) = α((QB0 )(t)) Z t n o = α S(t)g(vn−1 ) + S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ | n ∈ N 0

≤ α({S(t)g(vn−1 ) | n ∈ N}) n Z t o S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ | n ∈ N +α 0

Z

t

≤ M α({g(vn−1 ) | n ∈ N}) + 2 α({S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ | n ∈ N}) 0 Z t 0 ≤ M L α(B0 ) + 2M (L + C)α(B0 (τ ))dτ 0

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0

≤ M L max α(B(t)) + 2M (L + C)a max α(B(t)) t∈J

t∈J

0

≤ M [L + 2(L + C)a] max α(B(t)). t∈J

Case 2. For t ∈ (ti , si ], i = 1, 2, · · · , m, by (H9) we have n o α(B(t)) = α((QB0 )(t)) = α hi (t, vn−1 (t)) | n ∈ N ≤ Li α(B0 (t)) ≤ Li max α(B(t)) < max α(B(t)). t∈J

t∈J

Case 3. For t ∈ (si , ti+1 ](i = 1, 2, · · · , m), we have α(B(t)) = α((QB0 )(t)) = α

n

Z

t

S(t − si )hi (si , vn−1 (si )) +

S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ | n ∈ N

o

si

≤ α({S(t − si )hi (si , vn−1 (si )) | n ∈ N}) o n Z t S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ | n ∈ N +α si

≤ M α({hi (si , vn−1 (si )) | n ∈ N}) Z t α({S(t − τ )(f (τ, vn−1 (τ )) + Cvn−1 (τ ))dτ | n ∈ N}) +2 si Z t ≤ M Li α(B0 ) + 2M (L + C)α(B0 (τ ))dτ si

≤ M Li max α(B(t)) + 2M (L + C)a max α(B(t)) t∈J

t∈J

≤ M [Li + 2(L + C)a] max α(B(t)). t∈J

By (H11), we have α(B(t)) < max α(B(t)), then α(B(t)) = 0 in t ∈ J. Therefore, t∈J

{vn (t)} is precompact in X for t ∈ J, combining this fact with the monotonicity of {vn }, we easily prove that {vn (t)} is convergent. Let {vn (t)} → u(t) in t ∈ J. The same idea can be used to prove that {wn (t)} → u(t) in t ∈ J. Similar to the proof of Theorem 4.1, we know that u(t) and u(t) are the problem(1.1) between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 , respectively. This completes the proof of Theorem 4.3.  Remark 4.4. Analytic semigroup and differentiable semigroup are equicontinuous semigroup ([22]). In the application of partial differential equations, such as parabolic 13 537

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and strongly damped wave equations, the corresponding solution semigroup are analytic semigroup. Therefore, Theorem 4.3. has extensive applicability. we discuss the existence of the mild solutions for problem (1.1) under the positive cone is regular. Theorem 4.5. Let X be an ordered Banach space, whose positive cone K is regular. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate a positive C0 semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). hi ∈ C(J × X, X)(i = 1, 2, · · · , m). g : P C(J, X) → X be a continuous function. Assume that problem(1.1) has lower and upper solutions v0 and w0 with v0 (t) ≤ w0 (t)(t ∈ J). Suppose that conditions (H1), (H2) and (H3) are satisfied. Then the problem (1.1) has minimal and maximal mild solutions u and u between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 . Proof From Theorem 4.1 we know that Q : [v0 , w0 ] → [v0 , w0 ] is a continuously increasing operator. Similarly, the two sequences {vn (t)} and {wn (t)} are defined in [v0 , w0 ] by the iterative scheme (4.7). By conditions (H1), (H2) and (H3), then {vn (t)} and {wn (t)} are ordered-monotonic and ordered-bounded sequences in X. Using the regularity of the cone K, any ordered-monotonic and ordered-bounded sequence in X is convergent. Similar to the proof of Theorem 4.1, we know that u(t) and u(t) are the problem(1.1) between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 , respectively. This completes the proof of Theorem 4.5.  Corollary 4.6. Let X be an ordered and weakly sequentially complete Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate a positive C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). hi ∈ C(J × X, X)(i = 1, 2, · · · , m). g : P C(J, X) → X be a continuous function. Assume that problem(1.1) has lower and upper solutions v0 and w0 with v0 (t) ≤ w0 (t)(t ∈ J). Suppose that conditions (H1), (H2) and (H3) are satisfied. Then the problem (1.1) has minimal and maximal mild solutions u and u between v0 and w0 , which can be obtained by monotone iterative sequences starting from v0 and w0 . Proof In an ordered and weakly sequentially complete Banach space, the normal cone K is regular. Then the proof is complete.  Next, we discuss the existence of mild solutions of problem (1.1), when we don’t assume the lower and upper solutions of problem (1.1) be exist. Theorem 4.7. Let X be an ordered Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator and

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−A generate a positive and compact C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). hi ∈ C(J × X, X)(i = 1, 2, · · · , m). g : P C(J, X) → X be a continuous function. Suppose that conditions (H1)–(H5) hold and the following condition is satisfied: (H12) ∃b ≥ 0, h ∈ P C(J, X), h ≥ θ, yi (si ) ∈ D(A), yi ≥ θ, i = 1, 2, · · · , m and g(u) ∈ D(A), g(u) ≥ θ for any u ∈ P C(J, X), such that f (t, u) ≤ bu + h(t), bu − h(t) ≤ f (t, u),

hi (t, u) ≤ yi (t), −yi (t) ≤ hi (t, u),

u ≥ 0; u ≤ 0.

Then the problem (1.1) has minimal and maximal mild solutions , which can be obtained by monotone iterative procedure. Proof For h(t) ≥ θ, yi (t) ≥ θ, we consider the linear nonlocal evolution equation with not instantaneous impulses in X  0 u (t) + Au(t) − bu(t) = h(t), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     u(t) = yi (t), t ∈ (ti , si ], i = 1, 2, · · · , m. (4.12)     u(0) = g(u), Since −(A − bI) generate a positive C0 -semigroup S(t) = ebt T (t)(t ≥ 0) in X. By Theorem 3.1 and assumption (H12), we know that the problem (4.12) has a unique positive solution u∗ ≥ θ. Let v0 = −u∗ , w0 = u∗ , by the conditions (H1)–(H3) and (H12), we get  0 v0 (t) + Av0 (t) = bv0 (t) − h(t) ≤ f (t, v0 (t)), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     v0 (t) = −yi (t) ≤ hi (t, v0 (t)), t ∈ (ti , si ], i = 1, 2, · · · , m.     v0 (0) = −g(−v0 ) ≤ g(v0 ), (4.13) and  0 w0 (t) + Aw0 (t) = bw0 (t) + h(t) ≥ f (t, w0 (t)), t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,     w0 (t) = yi (t) ≥ hi (t, w0 (t)), t ∈ (ti , si ], i = 1, 2, · · · , m.     w0 (0) ≥ g(w0 ), (4.14) So, we inferred that v0 and w0 are a lower solution and an upper solution of the problem (1.1), respectively. Therefore by Theorem 4.1., the conclusion holds. Then the proof is complete. 15 539

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Meanwhile, we can obtain the following results from Theorem 4.2, 4.3, 4.5 and Corollary 4.6, respectively. Corollary 4.8. Let X be an ordered Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate a positive and compact C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). g, hi (i = 1, 2, · · · , m) are continuous and map a monotonic set into a precompact set and conditions (H1)–(H3) and (H12) hold, then the problem (1.1) has minimal and maximal mild solutions , which can be obtained by monotone iterative procedure. Corollary 4.9. Let X be an ordered Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate a positive C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). g, hi (i = 1, 2, · · · , m) are continuous and for any monotonic sequence {xn } satisfy conditions (H8)–(H11) and conditions (H1)–(H3) as well as (H12) hold, then the problem (1.1) has minimal and maximal mild solutions , which can be obtained by monotone iterative procedure. Corollary 4.10. Let X be an ordered Banach space, whose positive cone K is regular. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate a positive C0 semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). g, hi (i = 1, 2, · · · , m) are continuous and conditions (H1)–(H3) as well as (H12) hold, then the problem (1.1) has minimal and maximal mild solutions , which can be obtained by monotone iterative procedure. Corollary 4.11. Let X be an ordered and weakly sequentially complete Banach space, whose positive cone K is normal, and N0 be the normal constant. Let A : D(A) ⊂ X → X be a closed linear operator and −A generate a positive C0 -semigroup T (t)(t ≥ 0) in X. f ∈ C(J × X, X). g, hi (i = 1, 2, · · · , m) are continuous and conditions (H1)–(H3) as well as (H12) hold, then the problem (1.1) has minimal and maximal mild solutions, which can be obtained by monotone iterative procedure.

5

Application

In this section, we present one example, which indicates how our abstract results can be applied to concrete problems. Example 5.1. Consider the following nonlocal parabolic partial differential equa-

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tion with not instantaneous impulses:  ∂ u(x, t) + A(x, D)u(x, t) = f (x, t, u(x, t)), x ∈ Ω,  ∂t    t ∈ J, t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,    u(x, t) = hi (x, t, u(x, t)), x ∈ Ω, t ∈ (ti , si ], i = 1, 2, · · · , m,    Bu = 0, (x, t) ∈ ∂Ω × J,     u(x, 0) = g(u), x ∈ Ω,

(5.1)

where J = [0, a], a > 0 is a constant, 0 = s0 < t1 ≤ s1 < t2 ≤ s2 < t3 ≤ s3 < · · · < tm−1 ≤ sm−1 < tm ≤ sm < tm+1 = a are pre-fixed numbers, integer n ≥ 1, Ω ⊂ Rn is a bounded domain with a sufficiently smooth boundary ∂Ω, A(x, D) = −

n X n X i=1 j=1

n

aij (x)

X ∂ ∂2 + + a0 (x) ai (x) ∂xi ∂yj ∂x i i=1

is a strongly elliptic operator of second order, coefficient functions aij (x), ai (x) and ∂u a0 (x) are H¨ older continuous in Ω, Bu = b0 (x)u + δ ∂n is a regular boundary operator on ∂Ω, f : Ω × J × R → R is continuous, hi : Ω × J × R → R are also continuous, i = 1, 2, · · · , m, g is a continuous function. Let X = Lp (Ω) with p > n + 2, K = {u ∈ Lp (Ω) | u(x) ≥ 0 a.e. x ∈ Ω}, and define the operator A as follows: D(A) = {u ∈ W 2,p (Ω) | Bu = 0}, Au = A(x, D)u. We know that X is a Banach space, K is a regular cone of X, and −A generates a positive and analytic C0 -semigroup T (t)(t ≥ 0) in X (see [22]). Define u(t) = u(·, t), f (t, u(t)) = f (·, t, u(·, t)), hi (t, u(t)) = hi (·, t, u(·, t)), then system (5.1) can be reformulated as problem (1.1) in X. We assume that the following conditions hold: (i) Let f (x, t, 0) ≥ 0, hi (x, t, 0) ≥ 0, g(0) ≥ 0, x ∈ Ω. (ii) There exist w = w(x, t) ∈ P C(Ω×J)∩C 2,1 (Ω×J 00 ), and w(x, t) ≥ 0, x ∈ Ω, t ∈ J such that  ∂ w(x, t) + A(x, D)w(x, t) ≥ f (x, t, w(x, t)), x ∈ Ω,  ∂t    t ∈ J, t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , m,    w(x, t) ≥ hi (x, t, w(x, t)), x ∈ Ω, t ∈ (ti , si ], i = 1, 2, · · · , m,    Bw = 0, (x, t) ∈ ∂Ω × J,     w(x, 0) ≥ g(w), x ∈ Ω, (iii) The partial derivative fu0 (x, t, u) is continuous on any bounded domain. (iv) For any u1 , u2 ∈ [0, w(x, t)] with u1 ≤ u2 , for any x ∈ Ω, i = 1, 2, · · · , m,we have hi (x, t, u1 (x, t)) ≤ hi (x, t, u2 (x, t)), g(u1 ) ≤ g(u1 ) 17 541

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Theorem 5.2. If assumptions (i), (ii), (iii) and (iv) are satisfied, then the impulsive parabolic partial differential equation (5.1) has minimal and maximal mild solutions between 0 and w(x, t), which can be obtained by a monotone iterative procedure starting from 0 and w(x, t), respectively. Proof From assumptions (i) and(ii) we know that 0 and w(x, t) are lower and upper solutions of problem (5.1), respectively. (iii) implies that condition (H1) is satisfied. (iv) implies that conditions (H2) and (H2) are satisfied. So, by Theorem 4.5., we have the result. Then the proof is complete.

6

Conclusions

In this paper, we consider the existence of mild solutions for the new nonlocal evolution equation with impulses. We initially use the monotone iterative technique to the problem under new impulsive conditions. Hence the results are new.

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[12] Y.K. Chang, A. Anguraj, K. Karthikeyan, Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Nonlinear Anal. 71(2009). 4377-4386. [13] Y.K. Chang , V. Kavitha , M. Mallika Arjunan, Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators, Nonlinear Analysis: Hybrid Systems 4(2010). 32-43 [14] N. Abada, M. Benchohra, H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differ. Equations 246(2009). 3834-3863. [15] T. Cardinali, P. Rubbioni, Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces, Nonlinear Anal.75(2012). 871-879. [16] P.Y. Chen, Y.X. Li, H. Yang, Perturbation method for nonlocal impulsive evolution equations, Nonlinear Analysis: Hybrid Systems 8(2013). 22-30. [17] E. Hernandez, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc. 141(2013). 1641-1649. [18] M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput. 219(2013). 6743-6749. [19] J.R. Wang, X.Z. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput. 46(2014). 321-334. [20] X.L. Yu, J.R. Wang, Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces, Commun. Nonlinear Sci. Numer. Simul. 22(2015). 980-989. [21] J.R. Wang, Y. Zhou, Z. Lin, On a new class of impulsive fractional differential equations, Appl. Math. Comput. 242(2014). 649-657. [22] A. Pazy, Semigroup of linear operators and applications to partial differential equations, Springer-Verlag, Berlin, 1983. [23] J. Banasiak, L. Arlotti, Perturbations of Positive Semigroups with Applications, SpringerCverlag, London, 2006. [24] Y.X. Li, The positive solutions of abstract semilinear evolution equations and their applications, Acta Math. Sinica 39(5)(1996). 666-672 (in Chinese). [25] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. [26] D.Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. [27] D.J. Guo, J.X. Sun, Ordinary Differential Equations in Abstract Spaces, Shandong Science and Technology, Jinan, 1989. [28] H. P. Heinz; On the behaviour of measure of noncompactness with respect to differentiation and integration of rector-valued functions, Nonlinear Anal. 7(1983). 1351-1371.

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Eigenvalue for a system of Caputo fractional differential equations ∗ Xiaofeng Zhang, Hanying Feng† Department of Mathematics, Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, Hebei, P. R. China

Abstract: In this article, we study the existence of positive solutions for a system of nonlinear differential equations of mixed Caputo fractional orders 8 c α D0+ u(t) + λf (t, u(t), v(t)) = 0, 0 < t < 1, > > < c β D0+ v(t) + µg(t, u(t), v(t)) = 0, 0 < t < 1, u(0) = u0 (0) = u00 (1) = u000 (0) = 0, > > : v(0) = v 0 (0) = v 00 (1) = v 000 (0) = 0, β α c where 3 < α, β ≤ 4 are real numbers, c D0+ , D0+ are the Caputo fractional derivatives, and f, g : [0, 1] × [0, +∞) × [0, +∞) → [0, +∞) are given continuous functions. By using Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained. Keywords: Fractional order differential equation, Positive solution, Existence, Krasnoselskii’s fixed point theorem, Eigenvalue. 2010 Mathematics Subject Classification: 34B15, 34B16, 34B18

1

Introduction

Fractional differential equations describe many phenomena in various fields of engineering and scientific disciplines such as physics, biophysics, chemistry, economics, control theory, see [4, 8]. Recently, fractional differential equations have been of great interest, there are a large number of papers dealing with the existence of positive solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis (such as upper and lower solution method, Leray-Schauder theory, etc.), see [1, 2, 5, 7, 9, 10]. In this paper, we consider the system of Caputo fractional differential equations  c α D u (t) + λf (t, u (t) , v (t)) = 0, 0 < t < 1,    c 0+ β D0+ v (t) + µg (t, u (t) , v (t)) = 0, 0 < t < 1, (1.1)  u (0) = u0 (0) = u00 (1) = u000 (0) = 0,   v (0) = v 0 (0) = v 00 (1) = v 000 (0) = 0, β α c where 3 < α, β ≤ 4 are real numbers, c D0+ , D0+ are the Caputo fractional derivatives, and f, g : [0, 1] × [0, +∞) × [0, +∞) → [0, +∞) are given continuous functions. By using Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained. This paper is organized as follows. In Section 2, we present some basic definitions and properties from the fractional calculus theory. In Section 3, based on the Krasnoselskii’s fixed point theorem, we prove two existence theorems of the positive solutions for BVP (1.1). In section 4, an example is presented to illustrate the main results.

2

Preliminaries Let us start with the necessary definitions which are used throughout this paper. ∗ Supported

by NNSF of China (11371368) and HEBNSF of China (A2014506016). author. E-mail address: [email protected] (H. Feng)

† Corresponding

1

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Definition 2.1 ([2]). The Riemann-Liouville fractional integral of order α > 0 of a function f : (0, +∞) → R is given by α I0+ f (t)

1 = Γ(α)

Z

t

(t − s)α−1 f (s)ds,

t > 0,

0

provided the right-hand side is pointwise defined on (0, +∞). Definition 2.2 ([3, 8]).For a function f : (0, +∞) → R, the Caputo derivative of fractional order is defined as Z t f (n) (s) 1 c α ds, t > 0, D0+ f (t) = Γ(n − α) 0 (t − s)α−n+1 where n = [α] + 1, [α] denotes the integer part of the number α. α Lemma 2.1 ([3, 9]). Let α > 0, then fractional differential equation c D0+ u(t) = 0 has solutions u(t) = C1 + C2 t + · · · + Cn tn−1 , Ci ∈ R, i = 1, 2, · · ·, n, n = [α] + 1. Lemma 2.2 ([3, 9]). Let α > 0, then α c α I0+ D0+ u(t) = u(t) + C1 + C2 t + · · · + Cn tn−1 , Ci ∈ R, i = 1, 2, · · ·, n, n = [α] + 1.

In the following, we present Green’s function of BVP (1.1). Lemma 2.3 . Let h1 ∈ C[0, 1] and 3 < α ≤ 4, the unique solution of problem c

is

α D0+ u(t) + h1 (t) = 0, 0 < t < 1,

(2.1)

u(0) = u0 (0) = u00 (1) = u000 (0) = 0, Z 1 u(t) = G1 (t, s)h1 (s)ds,

(2.2)

0

where

 (α − 1)(α − 2)t2 (1 − s)α−3 − 2(t − s)α−1   , 0 ≤ s ≤ t ≤ 1,  2Γ(α) G1 (t, s) = 2 α−3   t (1 − s) , 0 ≤ t ≤ s ≤ 1.  2Γ(α − 2)

(2.3)

Here G1 (t, s) is called the Green’s function of BVP (2.1) and (2.2). Proof. We may apply Lemma 2.2 to reduce (2.1) to an equivalent integral equation α u(t) = −I0+ h1 (t) + C1 + C2 t + C3 t2 + C4 t3 ,

for some C1 , C2 , C3 , C4 ∈ R. Consequently, the general solution of (2.1) is u(t) = −

1 Γ(α)

Z

t

(t − s)α−1 h1 (s)ds + C1 + C2 t + C3 t2 + C4 t3 .

0

1 2Γ(α − 2) Therefore, the unique solution of problem (2.1) and (2.2) is

Z

By (2.2), there are C1 = C2 = C4 = 0, and C3 =

1

(1 − s)α−3 h1 (s)ds.

0

Z t Z 1 1 1 (t − s)α−1 h1 (s)ds + t2 (1 − s)α−3 h1 (s)ds Γ(α) 0 2Γ(α − 2) 0 ¸ Z t· 2 Z 1 2 t (1 − s)α−3 (t − s)α−1 t (1 − s)α−3 = − h1 (s)ds + h1 (s)ds 2Γ(α − 2) Γ(α) 2Γ(α − 2) 0 t Z t Z 1 2 (α − 1)(α − 2)t2 (1 − s)α−3 − 2(t − s)α−1 t (1 − s)α−3 = h1 (s)ds + h1 (s)ds 2Γ(α) 2Γ(α − 2) 0 t Z 1 = G1 (t, s)h1 (s)ds.

u(t) = −

0

2

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The proof is finished. Lemma 2.4 . The function G1 (t, s) defined by (2.3) possesses the following properties: (1) G1 (t, s) > 0, for t, s ∈ (0, 1) ; 1 1 max G1 (t, s) = G1 (1, s), for s ∈ (0, 1) . (2) 1 min 3 G1 (t, s) ≥ 0≤t≤1 16 16 ≤t≤ 4 4 (α − 1)(α − 2)t2 (1 − s)α−3 − 2(t − s)α−1 t2 (1 − s)α−3 Proof. Let g1 (t, s) = , g2 (t, s) = . 2Γ(α) 2Γ(α − 2) (1) Since 3 < α ≤ 4, 0 < s ≤ t < 1, so (α − 1)(α − 2)t2 (1 − s)α−3 > 2t2 (1 − s)α−3 > 2t2 (t − s)α−3 ≥ 2(t − s)α−1 , therefore, g1 (t, s) > 0, obviously, g2 (t, s) > 0, thus G1 (t, s) > 0, for t, s ∈ (0, 1). (2) Since α−3

α−2

∂g1 (t, s) (α − 1) (α − 2) t (1 − s) − (α − 1) (t − s) = ∂t Γ (α)

∂g2 (t, s) t(1 − s)α−3 = > 0, ∂t Γ(α − 2)

> 0,

so G1 (t, s) is monotone increasing function for t. Thus, 0 ≤ G1 (t, s) ≤ max G1 (t, s) = G1 (1, s),

t, s ∈ [0, 1].

0≤t≤1

Noticing that ¢α−1 ¡  α−3 (α − 1) (α − 2) (1 − s) − 32 41 − s 1    , s ∈ (0, ], 1 32Γ (α) 4 min 3 G1 (t, s) = G1 ( , s) = α−3 1  4 (1 − s) 1  4 ≤t≤ 4  , s ∈ [ , 1). 32Γ (α − 2) 4 α−3

max G1 (t, s) = G1 (1, s) =

0≤t≤1

(α − 1) (α − 2) (1 − s) 2Γ (α)

Next we proof min G1 (t, s) ≥

1 3 4 ≤t≤ 4

α−1

− 2 (1 − s)

,

s ∈ (0, 1).

1 1 max G1 (t, s) = G1 (1, s). 16 0≤t≤1 16

When 0 < s ≤ 41 , since 3 < α ≤ 4, we have 1 1 1 1 (1 − s)α−1 , ( − s)α−1 = ( )α−1 (1 − 4s)α−1 ≤ ( )2 (1 − 4s)α−1 < 4 4 4 16 so

min G1 (t, s) ≥

When

1 4

1 3 4 ≤t≤ 4

1 1 max G1 (t, s) = G1 (1, s). 16 0≤t≤1 16

≤ s < 1, we obtain α−3

min 3 G1 (t, s) = 1 4 ≤t≤ 4

α−3

(1 − s) (α − 1)(α − 2) (1 − s) = 32Γ (α − 2) 32Γ (α)

, α−1

1 1 (α − 1)(α − 2)(1 − s)α−3 (1 − s) max G1 (t, s) = G1 (1, s) = − . 16 0≤t≤1 16 32Γ(α) 16Γ (α) Obvious that, min G1 (t, s) ≥

1 3 4 ≤t≤ 4

1 1 max G1 (t, s) = G1 (1, s), 16 0≤t≤1 16

s ∈ (0, 1) .

The proof is finished. Lemma 2.5 . If the function f ∈ C([0, 1] × [0, +∞) × [0, +∞), [0, +∞)), then the unique solution of BVP (1.1) satisfied 1 kuk . min 3 u(t) ≥ 1 16 4 ≤t≤ 4 3

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Proof. From lemma Z 1 2.3, we known Z u (t) = G1 (t, s) f (s, u(s), v(s)) ds ≤ Z kuk = max |u(t)| = max 0≤t≤1

max G1 (t, s) f (s, u(s), v(s)) ds,

0 0≤t≤1

0

and

1

0≤t≤1

Z

1

G1 (t, s) f (s, u(s), v(s)) ds ≤ 0

1

max G1 (t, s) f (s, u(s), v(s)) ds.

0 0≤t≤1

From lemma 2.4, we have Z 1 G1 (t, s) f (s, u(s), v(s)) ds min 3 u(t) = 1 min 3 1 4 ≤t≤ 4

4 ≤t≤ 4

≥

1 16

Z

0

1

max G1 (t, s) f (s, u(s), v(s)) ds ≥

0 0≤t≤1

1 max 16 0≤t≤1

Z

1

G1 (t, s) f (s, u(s), v(s)) ds = 0

The proof is finished. Similarly, we can obtain G2 (t, s) if α is replaced by β,  β−1 (β − 1)(β − 2)t2 (1 − s)β−3 − 2 (t − s)    , 0 ≤ s ≤ t ≤ 1, 2Γ (β) G2 (t, s) = 2 β−3  t (1 − s)   , 0 ≤ t ≤ s ≤ 1. 2Γ(β − 2)

1 kuk . 16

(2.4)

The function G2 (t, s) defined by (2.4) have the same properties with G1 (t, s), so 1 1 max G2 (t, s) = G2 (1, s), s ∈ (0, 1) . min 3 G2 (t, s) ≥ 1 16 0≤t≤1 16 4 ≤t≤ 4 Lemma 2.6 ([6]). Let E be a Banach space, and let P ⊂ E be a cone in E. Assume Ω1 , Ω2 be two open subsets of E with θ ∈ Ω1 ⊂ Ω1 ⊂ Ω2 , and let T : P → P be a completely continuous operator such that either (i)kT wk ≤ kwk, w ∈ P ∩ ∂Ω1 , kT wk ≥ kwk, w ∈ P ∩ ∂Ω2 , or (ii)kT wk ≥ kwk, w ∈ P ∩ ∂Ω1 , kT wk ≤ kwk, w ∈ P ∩ ∂Ω2 holds. Then T has a fixed point in P ∩ Ω2 \Ω1 .

3

Main results and proof

In this section, we establish the existence of positive solutions for BVP (1.1). For convenience, we introduce the following notations f (t, u, v) , f0 = lim inf+ min u+v u+v→0 t∈[ 1 , 3 ] 4 4 f (t, u, v) f 0 = lim sup max , u+v u+v→0+ t∈[0,1]

g (t, u, v) g0 = lim inf+ min , u+v→0 t∈[ 1 , 3 ] u + v 4 4 g (t, u, v) g 0 = lim sup max , u+v→0+ t∈[0,1] u + v

f (t, u, v) f∞ = lim inf min , u+v→∞ t∈[ 1 , 3 ] u+v 4 4 f (t, u, v) f ∞ = lim sup max , u+v u+v→∞ t∈[0,1]

g (t, u, v) g∞ = lim inf min , u+v→∞ t∈[ 1 , 3 ] u + v 4 4 g (t, u, v) g ∞ = lim sup max . u+v→∞ t∈[0,1] u + v

By using the Green’s functions Gi (t, s) (i = 1, 2) , from Section 2, the problem (1.1) can be written equivalently as the following nonlinear system of integral equations  Z 1   G1 (t, s)f (s, u(s), v(s))ds, 0 ≤ t ≤ 1,  u(t) = λ Z0 1    v (t) = µ G2 (t, s)g(s, u(s), v(s))ds, 0 ≤ t ≤ 1. 0

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We consider the Banach space X = C [0, 1] with the norm kuk = max |u (t)|, and the Banach space 0≤t≤1

Y = X × X with the norm k(u, v)kY = kuk + kvk. We define the ( cone P ⊂ Y by

) 1 k(u, v)kY , t ∈ [0, 1] . P = (u, v) ∈ Y |u (t) ≥ 0, v (t) ≥ 0, 1 min 3 (u(t) + v(t)) ≥ 16 4 ≤t≤ 4 For λ, µ > 0, we define the operators T1 , T2 : Y → X and T : Y → Y respectively by  Z 1   G1 (t, s)f (s, u(s), v(s))ds, 0 ≤ t ≤ 1,  T1 (u, v)(t) = λ Z0 1    T2 (u, v)(t) = µ G2 (t, s)g(s, u(s), v(s))ds, 0 ≤ t ≤ 1, 0

and T (u, v) = (T1 (u, v) , T2 (u, v)) , (u, v) ∈ Y. Thus, the solutions of BVP (1.1) are the fixed points of the operator T . Lemma 3.1. T : P → P is a completely continuous operator. Proof. Let (u, v) ∈ P be an arbitrary element. From the definition T1 (u, v) and Lemma 2.4, we get Z 1 kT1 (u, v)k = max |T1 (u, v)(t)| ≤ max G1 (t, s)f (s, u(s), v(s))ds, 0≤t≤1

0 0≤t≤1

Z

G1 (t, s)f (s, u(s), v(s))ds ≥

min 3 T1 (u, v)(t) = 1 min 3 1

4 ≤t≤ 4

1

4 ≤t≤ 4

0

1 max ≥ 16 0≤t≤1

Z

1 16

1

G1 (t, s) f (s, u(s), v(s)) ds = 0

In the similar manner, we deduce 1 min 3 T2 (u, v)(t) ≥ 4 ≤t≤ 4

Z

1

max G1 (t, s) f (s, u(s), v(s)) ds

0 0≤t≤1

1 kT1 (u, v)k . 16

1 kT2 (u, v)k . 16

Thus we have 1 1 min 3 (T1 (u, v)(t) + T2 (u, v)(t)) ≥ (kT1 (u, v)k + kT2 (u, v)k) ≥ T k(u, v)kY . 1 16 16 4 ≤t≤ 4 Hence T (u, v) ∈ P, that is T (P ) ⊂ P. According to the Arzela-Ascoli theorem, we can easily get that T : P → P is a completely continuous operator. The proof is completed. Next, for α1 , α2 , α1 , α2 > 0 such that α1 + α2 = 1, α1 + α2 = 1, we define the numbers L1 , L2 , L3 , L4 by R 34 R 34 R1 R1 1 G1 (1, s)ds 1 G2 (1, s)ds G1 (1, s)ds G2 (1, s)ds 4 4 0 L1 = , L2 = , L3 = , L4 = 0 . 256α1 α1 256α2 α2 ³

Theorem 3.1. If f 0 , g 0 , f∞ , g∞ ∈ (0, ∞), L11f∞ < L21f 0 and L31g∞ < L41g0 hold, then for any λ ∈ ´ 1 1 and µ ∈ ( L31g∞ , L41g0 ), BVP (1.1) has at least one positive solution (u(t), v(t)), t ∈ [0, 1]. L1 f∞ , L2 f 0 ³ ´ ³ ´ Proof. When λ ∈ L11f∞ , L21f 0 and µ ∈ L31g∞ , L41g0 , choosing ε > 0, such that 1 L1 (f∞ − ε)

≤λ≤

1 L2

(f 0

+ ε)

1

,

L3 (g∞ − ε)

≤µ≤

1 L4 (g 0 + ε)

(3.1)

By the definition of f 0 , g 0 , there exists R1 > 0, such that for all t ∈ [0, 1], u, v ∈ R+ , with 0 ≤ u + v ≤ R1 , we have (3.2) f (t, u, v) ≤ (f 0 + ε)(u + v), g(t, u, v) ≤ (g 0 + ε)(u + v). Now define the set Ω1 = {(u, v) ∈ Y, k(u, v)kY < R1 }. Let (u, v) ∈ P ∩ ∂Ω1 , that is (u, v) ∈ P with k(u, v)kY = R1 , so u(t) + v(t) ≤ R1 for all t ∈ [0, 1], thus Z 1 Z 1 G1 (1, s)(f 0 + ε)(u(s) + v(s))ds G1 (t, s)f (s, u(s), v(s))ds ≤ λ T1 (u, v)(t) =λ 0

Z

≤λ(f 0 + ε)

0 1

G1 (1, s)dsk(u, v)kY ≤ 0

1 L2

Z

1

G1 (1, s)dsk(u, v)kY 0

≤α1 k(u, v)kY . 5

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Therefore, kT1 (u, v)k ≤ α1 k(u, v)kY . In the similar manner, we deduce Z

1

0

T2 (u, v)(t) ≤ µ(g + ε) 0

G2 (1, s)ds k(u, v)kY ≤ α2 k(u, v)kY .

So, kT2 (u, v)k ≤ α2 k(u, v)kY . Then for (u, v) ∈ P ∩ ∂Ω1 , we deduce kT (u, v)kY = kT1 (u, v)k + kT2 (u, v)k ≤ (α1 + α2 ) k(u, v)kY = k(u, v)kY . By the definition of f∞ , g∞ , there exists R2 > 0, such that for all t ∈ [0, 1], u, v ∈ R+ , with u + v ≥ R2 , we have f (t, u, v) ≥ (f∞ − ε)(u + v), g(t, u, v) ≥ (g∞ − ε)(u + v). (3.3) Now define the set Ω2 = {(u, v) ∈ Y, k(u, v)kY < R2 }. Let (u, v) ∈ P ∩ ∂Ω2 , that is (u, v) ∈ P with 1 k(u, v)kY for all t ∈ [ 41 , 34 ], thus, it follows from Lemma 2.4 that k(u, v)kY = R2 , so u(t) + v(t) ≥ 16 Z 1 Z 34 T1 (u, v)(t) =λ G1 (t, s)f (s, u(s), v(s))ds ≥ λ G1 (t, s)f (s, u(s), v(s))ds 1 4

0

Z

Z 3 1 λ(f∞ − ε) 4 ≥λ G1 (1, s)(u(s) + v(s))ds G1 (1, s)(f∞ − ε)(u(s) + v(s))ds = 1 1 16 16 4 4 Z 3 Z 34 λ(f∞ − ε) 4 1 ≥ G1 (1, s)ds k(u, v)kY ≥ G1 (1, s)ds k(u, v)kY ≥ α1 k(u, v)kY . 1 256 256L1 41 4 3 4

Therefore, kT1 (u, v)k ≥ α1 k(u, v)kY . Similarly, we have T2 (u, v)(t) ≥

µ(g∞ − ε) 256

Z

3 4 1 4

G2 (1, s)ds k(u, v)kY ≥ α2 k(u, v)kY .

So, kT2 (u, v)k ≥ α2 k(u, v)kY . Then for (u, v) ∈ P ∩ ∂Ω2 , we deduce kT (u, v)kY = kT1 (u, v)k + kT2 (u, v)k ≥ (α1 + α2 ) k(u, v)kY = k(u, v)kY . By using Lemma 2.6, we conclude that T has a fixed point (u, v) ∈ P ∩ (Ω2 \ Ω1 ) such that R1 ≤ kuk + kvk ≤ R2 . Theorem 3.2. If f0 , g0 , f ∞ , g ∞ ∈ (0, ∞), L11f0 < L21f ∞ and L31g0 < L41g∞ hold, then for any λ ∈ ( L11f0 , L21f ∞ ) and µ ∈ ( L31g0 , L41g∞ ), BVP (1.1) has at least one positive solution (u(t), v(t)), t ∈ [0, 1]. ³ ´ Proof. When λ ∈ L11f0 , L21f ∞ and µ ∈ ( L31g0 , L41g∞ ), choosing ε > 0, such that 1 L1 (f0 − ε)

≤λ≤

1 L2

(f ∞

+ ε)

1

,

L3 (g0 − ε)

≤µ≤

1 L4

(g ∞

+ ε)

.

(3.4)

By the definition of f0 , g0 , there exists R3 > 0, such that for all t ∈ [0, 1], u, v ∈ R+ , with 0 ≤ u + v ≤ R3 , we have f (t, u, v) ≥ (f0 − ε)(u + v), g(t, u, v) ≥ (g0 − ε)(u + v). (3.5) Now define the set Ω3 = {(u, v) ∈ Y, k(u, v)kY < R3 }. Let (u, v) ∈ P ∩ ∂Ω3 , that is (u, v) ∈ P with 1 k(u, v)kY for all t ∈ [ 14 , 34 ], thus, it follows from Lemma 2.4 that k(u, v)kY = R3 , so u(t) + v(t) ≥ 16 Z 1 Z 34 T1 (u, v)(t) =λ G1 (t, s)f (s, u(s), v(s))ds ≥ λ G1 (t, s)f (s, u(s), v(s))ds 1 4

0

Z

Z 3 1 λ(f0 − ε) 4 G1 (1, s)(f0 − ε)(u(s) + v(s))ds = G1 (1, s)(u(s) + v(s))ds 1 1 16 16 4 4 Z 3 Z 43 λ(f0 − ε) 4 1 ≥ G1 (1, s)dsk(u, v)kY ≥ G1 (1, s)dsk(u, v)kY ≥ α1 k(u, v)kY . 1 256 256L1 14 4 ≥λ

3 4

6

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Therefore, kT1 (u, v)k ≥ α1 k(u, v)kY . In the similar manner, we deduce Z

µ(g0 − ε) T2 (u, v)(t) ≥ 256

3 4 1 4

G2 (1, s)ds k(u, v)kY ≥ α2 k(u, v)kY .

So, kT2 (u, v)k ≥ α2 k(u, v)kY . Then for (u, v) ∈ P ∩ ∂Ω3 , we deduce kT (u, v)kY = kT1 (u, v)k + kT2 (u, v)k ≥ (α1 + α2 ) k(u, v)kY = k(u, v)kY . Next, we define the functions f ∗ , g ∗ : [0, 1] × R+ → R+ , f ∗ (t, x) = max

0≤u+v≤x

max

0≤u+v≤x ∗

f (t, u, v), g ∗ (t, x) =

g(t, u, v), t ∈ [0, 1], x ∈ R+ . Then f (t, u, v) ≤ f ∗ (t, x), g(t, u, v) ≤ g (t, x) for all t ∈ [0, 1] , u ≥

0, v ≥ 0 and u + v ≤ x. The functions f ∗ (t, ·), g ∗ (t, ·) are nondecreasing for every t ∈ [0, 1], and satisfy the conditions f ∗ (t, x) g ∗ (t, x) lim sup max ≤ f ∞, lim sup max ≤ g∞ . x x x→∞ t∈[0,1] x→∞ t∈[0,1] Therefore, for ε > 0, there exists R4 > 0, such that for all x ≥ R4 and t ∈ [0, 1], we can get f ∗ (t, x) f ∗ (t, x) ≤ lim sup max + ε ≤ f ∞ + ε, x x x→∞ t∈[0,1] g ∗ (t, x) g ∗ (t, x) ≤ lim sup max + ε ≤ g ∞ + ε, x x x→∞ t∈[0,1]

so f ∗ (t, x) ≤ (f ∞ + ε)x, g ∗ (t, x) ≤ (g ∞ + ε)x. We consider R4 ≥ R4 + R3 and define the set Ω4 = {(u, v) ∈ Y, k(u, v)kY < R4 } . Let (u, v) ∈ P ∩ ∂Ω4 , that is (u, v) ∈ P with k(u, v)kY = R4 or equivalently kuk + kvk = R4 . By the definition of f ∗ , g ∗ , we can get for all t ∈ [0, 1], f (t, u(t), v(t)) ≤ f ∗ (t, k(u, v)kY ),

g(t, u(t), v(t)) ≤ g ∗ (t, k(u, v)kY ).

(3.6)

Thus Z

Z

1

T1 (u, v)(t) =λ 0

Z

G1 (1, s)(f

∞

+ ε)R4 ds = λ(f

0

1 L2

G1 (1, s)f ∗ (t, k(u, v)kY )ds

0 1

≤λ ≤

1

G1 (t, s)f (s, u(s), v(s))ds ≤ λ

Z 0

∞

Z

+ ε)

1

G1 (1, s)dsk(u, v)kY 0

1

G1 (1, s)dsk(u, v)kY ≤ α1 k(u, v)kY .

Therefore, kT1 (u, v)k ≤ α1 k(u, v)kY . Similarly, we have Z T2 (u, v)(t) ≤ µ(g

∞

+ ε) 0

1

G2 (1, s)dsk(u, v)kY ≤α2 k(u, v)kY .

So, kT2 (u, v)k ≤ α2 k(u, v)kY . Then for(u, v) ∈ P ∩ ∂Ω4 , we deduce kT (u, v)kY = kT1 (u, v)k + kT2 (u, v)k ≤ (α1 + α2 )k(u, v)kY = k(u, v)kY . By using Lemma 2.6, we conclude that T has a fixed point (u, v) ∈ P ∩ (Ω4 \ Ω3 ) such that R3 ≤ kuk + kvk ≤ R4 .

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4

Example Example 4.1. Consider the following system of fractional differential equations  7 c 2  D0+ u(t) + λf (t, u(t), v(t)) = 0, 0 < t < 1,    c 10 3 v(t) + µg(t, u(t), v(t)) = 0, 0 < t < 1, D0+  u(0) = u0 (0) = u00 (1) = u000 (0) = 0,    v(0) = v 0 (0) = v 00 (1) = v 000 (0) = 0, In the system (4.1), α = 27 , β =

10 3

(4.1)

and

(t + 1)[p1 (u + v) + q1 e−(u+v) ](u + v) , u+v+1 (t + 1)2 [p2 (u + v) + q2 e−(u+v) ](u + v) g(t, u, v) = , u+v+1

f (t, u, v) =

for t ∈ [0, 1], u, v ≥ 0, where p1 , p2 , q1 , q2 > 0. 0.2902 0.0007 0.3120 0 0 We deduce L1 ≈ 0.0007 α1 , L2 ≈ α1 , L3 ≈ α2 , L4 ≈ α2 . We have f = 2q1 , g = 4q2 , f∞ = p1 , g∞ = p2 . For α1 , α2 > 0 with α1 + α2 = 1, we consider α1 = α1 , α2 = α2 . Then, the conditions L11f∞ < L21f 0 and L31g∞ < L41g0 become L1 p1 > 2L2 q1 ,

L3 p2 > 4L4 q2 .

p1 p2 ≥ 830 and ≥ 1783, then the above conditions are satisfied. Therefore, by Theorem q1 q2 3.1, there exists one positive solution (u(t), v(t)), t ∈ [0, 1].

For example, if

References [1] C. Bai, J. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput. 150 (2004) 611-621. [2] Z. Bai, H. L¨ u, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005) 495-505. [3] Z. Bai, T. Qiu. Existence of positive solutions for singular fractional differential equation. Appl. Math. Comput. 215 (2009) 2761-2767 [4] S. Das, Functional Fractional Calculus for System Identification andControls, Springer, New York (2008). [5] W. Feng, S. Sun, Z. Han, Y. Zhao, Existence of solutions for a singular system of nonlinear fractional differential equations, Comput. Math. Appl. 62 (2011) 1370-1378. [6] D. Guo, J. Sun, Z. Liu, Functional Methods in Nonlinear Ordinary Differential Equations, Shandong Science and Technology Press, Jinan, 1995 (in Chinese). [7] J. Henderson, R. Luca, Positive solutions for a system of nonlocal fractional boundary value problems, Fract. Calc. Appl. Anal. 16 (2013) 985-1008. [8] I. Podlubny, Fractional Differential Equations, Academic Press, SanDiego (1999). [9] S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Elect. J. Diff. Equ. 36 (2006) 1-12. [10] Y. Zhao, S. Sun, Z. Han, W. Feng, Positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders, Adv. Diff. Equ. 2011 (2011) 1-13.

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A COMMON FIXED POINT THEOREM FOR A PAIR OF GENERALIZED CONTRACTION MAPPINGS WITH APPLICATIONS MUHAMMAD NAZAM, MUHAMMAD ARSHAD, AND CHOONKIL PARK∗ Abstract. In this article, we introduce abscissa dominating function F : [0, ∞)2 → R and define a generalized (α, F, ψ, ϕ)-contraction mapping which retrieves Banach’s contraction, Geraghty type contraction and weak contraction as particular cases. We establish a common fixed points theorem for a pair of generalized (α, F, ψ, ϕ)-contraction mappings in complete partial metric spaces and apply this theorem to show the existence of solution of system of integral equations. This result and its consequences generalize many existing results both in partial metric spaces and metric spaces. We give examples to illustrate our results and to express the usefulness of these results in the literature.

1. Introduction A partial metric on a nonempty set X is a function p : X × X → [0, ∞) such that (p1 ) (p2 ) (p3 ) (p4 )

x = y ⇔ p(x, x) = p(x, y) = p(y, y), p(x, x) ≤ p(x, y), p(x, y) = p(y, x), p(x, y) ≤ p(x, z) + p(z, y) − p(z, z).

Partial metrics were introduced in [12] as a generalization of the notion of metric to allow non-zero self distance for the purpose of modeling partial objects in reasoning about data flow networks. The self distance p(x, x) is to be understood as a quantification of the extent to which x is unknown. A partial metric space is a pair (X, p) such that X is a nonempty set and p is a partial metric on X. Matthews [12] proved an analogue of Banach’s fixed point theorem in partial metric spaces. After this remarkable fixed point theorem, many authors took interest in partial metric spaces and its topological properties and established many well known fixed point results successfully (see [1, 2, 3, 4, 5, 6, 7, 8, 11]). In this paper, continuing the study of fixed point theorems in partial metric spaces, we shall establish a common fixed points theorem for a pair of generalized (α, F, ψ, ϕ)-contraction mappings and shall discuss its consequences. The result proved in this paper generalizes many existing results in the literature (see [5, 7, 8, 14]). We explain hypotheses of our result through an example. In the last section of this paper, we apply this theorem to show the existence of solution of system of integral equations.

2010 Mathematics Subject Classification. Primary: 47H09; 47H10; 54H25. Key words and phrases. common fixed points; generalized contraction mapping; partial metric space. ∗ Corresponding author.

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2. Preliminaries Throughout this paper, we denote (0, ∞) by R+ , [0, ∞) by R+ 0 , (−∞, +∞) by R and the set of natural numbers by N. Following concepts and results will be required for the proofs of main results. Matthews [12] proved that every partial metric p on X induces a metric dp : X × X → R+ 0 by dp (x, y) = 2p (x, y) − p (x, x) − p (y, y) ,

(2.1)

for all x, y ∈ X. Notice that a metric on a set X is a partial metric p such that p(x, x) = 0 for all x ∈ X. Following [12], each partial metric p on X generates a T0 topology τ (p) on X. The base of the topology τ (p) is the family of open p-balls {Bp (x, ) : x ∈ X,  > 0}, where Bp (x, ) = {y ∈ X : p (x, y) < p (x, x) + } for all x ∈ X and  > 0. A sequence {xn }n∈N in (X, p) converges to a point x ∈ X if and only if p(x, x) = limn→∞ p(x, xn ). Definition 1. [12] Let (X, p) be a partial metric space. (1) A sequence {xn }n∈N in (X, p) is called a Cauchy sequence if limn,m→∞ p(xn , xm ) exists and is finite. (2) A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn }n∈N in X converges, with respect to τ (p), to a point x ∈ X such that p(x, x) = limn,m→∞ p(xn , xm ). Definition 2. [15] Let S : X → X and α : X × X → R+ 0 be two functions. Then S is said to be α-admissible if α(x, y) ≥ 1 implies α(S(x), S(y)) ≥ 1 ∀ x, y ∈ X. Definition 3. [10] Let S : X → X and α : X × X → R+ 0 be two functions. Then S is said to be a triangular α-admissible mapping if (1) α(x, y) ≥ 1 implies α(S(x), S(y)) ≥ 1, (2) α(x, z) ≥ 1 and α(z, y) ≥ 1 imply α(x, y) ≥ 1 for all x, y, z ∈ X. Definition 4. [1] Let S, T : X → X and α : X × X → R+ 0 be two functions. The pair (S, T ) is said to be triangular α-admissible if (1) α(x, y) ≥ 1 implies α(S(x), T (y)) ≥ 1 and α(T (x), S(y)) ≥ 1, (2) α(x, z) ≥ 1 and α(z, y) ≥ 1 imply α(x, y) ≥ 1 for all x, y, z ∈ X. The following lemma will be helpful in the sequel. Lemma 1. [12] (1) A partial metric space (X, p) is complete if and only if the metric space (X, dp ) is complete. (2) A sequence {xn }n∈N in X converges to a point x ∈ X, with respect to τ (dp ) if and only if limn→∞ p(x, xn ) = p(x, x) = limn,m→∞ p(xn , xm ). (3) If limn→∞ xn = υ such that p(υ, υ) = 0 then limn→∞ p(xn , y) = p(υ, y) for every y ∈ X.

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GENERALIZED CONTRACTION MAPPINGS

Lemma 2. [5] Let S : X → X be a triangular α-admissible mapping. Assume that there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1. Define a sequence {xn } by xn+1 = S(xn ). Then we have α(xn , xm ) ≥ 1 for all m, n ∈ N ∪ {0} with n < m. Lemma 3. [1] Let S, T : X → X be triangular α-admissible mappings. Assume that there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1. Define sequence x2i+1 = S(x2i ), and x2i+2 = T (x2i+1 ), where i = 0, 1, 2, . . . .. Then we have α(xn , xm ) ≥ 1 for all m, n ∈ N ∪ {0} with n < m. Definition 5. A continuous function F : [0, ∞)2 → R is called an abscissa dominating function if for any u, v ∈ R+ 0 , the following conditions hold: (1) F(u, v) < u, (2) If F(u, v) = u, then either u = 0 or v = 0. An extra condition F(0, 0) = 0 could be imposed in some cases if required. Let ∆c denote the class of all abscissa dominating functions. Example 1. (1) F(u, v) = u − v. (2) F(u, v) = ru, for some r ∈ (0, 1). u (3) F(u, v) = for some r ∈ (0, ∞). (1 + v)r u log(t + a ) , for some a > 1. (4) F(u, v) = (1 + v) 1 (1 + v)r − l, l > 1, for r ∈ (0, ∞). (5) F(u, v) = (u + l) (6) F(u, v) = uβ(u), where β : R+ 0 → [0, 1). and continuous. −j R e ∞ (7) F(u, v) = uπ −1/2 0 √ dj. j+v + Let Φ denote the class of the functions ϕ : R+ 0 → R0 which satisfy the following conditions:

(a) ϕ is continuous; (b) ϕ(t) > 0, t > 0 and ϕ(0) ≥ 0, + and Ψ denote the class of all the functions ψ : R+ 0 → R0 which satisfy the following conditions:

(1) ψ is increasing; (2) ψ(t) > 0, t > 0 and ψ(t) = 0 imply t = 0. 3. Main results This section contains definitions, a common fixed point result for a pair of generalized (α, F, ψ, ϕ)contraction mappings in the setting of partial metric spaces and examples to support this result. We begin with following definitions. Definition 6. Let (X,p) be a partial metric space and α : X × X → [0, ∞) be a function. Mappings S, T : X → X are called a pair of generalized (α, F, ψ, ϕ)-contraction mapping if for all x, y ∈ X, the contractive condition α(x, y)ψ (p(S(x), T (y))) ≤ F(ψ(M (x, y)), ϕ(M (x, y)))

554

(3.1)

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holds, where F ∈ ∆c , ψ ∈ Ψ, ϕ ∈ Φ and   p(x, S(x))p(y, T (y)) p(x, S(x))p(y, T (y)) M (x, y) = max p(x, y), . , 1 + p(x, y) 1 + p(S(x), T (y)) If we set S = T in (3.1), then we obtain the following contractive condition α(x, y)ψ (p(T (x), T (y))) ≤ F(ψ(N (x, y)), ϕ(N (x, y))), where 

p(x, T (x))p(y, T (y)) p(x, T (x))p(y, T (y)) N (x, y) = max p(x, y), , 1 + p(x, y) 1 + p(T (x), T (y))

 .

The following theorem is one of the main results. Theorem 1. Let (X, p) be a complete partial metric space, α : X × X → R+ 0 be a function. Suppose that S, T : X → X are continuous mappings satisfying the following conditions: (1) (2) (3) (4)

(S, T ) is a pair of (α, F, ψ, ϕ)-contraction mappings, (S, T ) is triangular α-admissible, there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1, α(x, y) ≥ 1 for all x, y ∈ F ix(T, S).

Then (S, T ) have a unique common fixed point. Proof. We begin with the following observation. M (x, y) = 0 if and only if x = y is a common fixed point of (S, T ). Indeed, if x = y is a common fixed point of (S, T ), then T (y) = T (x) = x = y = S(y) = S(x) and   p(x, x)p(x, x) p(x, x)p(x, x) M (x, y) = max p(x, x), = p(x, x). , 1 + p(x, x) 1 + p(x, x) From the contractive condition (3.1), we get ψ (p(x, x)) = ψ (p(S(x), T (y))) ≤ α(x, y)ψ (p(S(x), T (y))) ≤ F (ψ (M (x, y)) , ϕ (M (x, y))) , which is only possible if p(x, x) = 0. So M (x, y) = 0. Conversely, if M (x, y) = 0, then by (P1 ) and (P2 ) it is easy to check that x = y is a fixed point of S and T . On the other hand, if M (x, y) > 0, we construct an iterative sequence xn of points in X such a way that x2i+1 = S(x2i ) and x2i+2 = T (x2i+1 ) where i = 0, 1, 2, . . . . We observe that if xn = xn+1 , then xn is a common fixed point of S and T . Suppose that xn 6= xn+1 for all n ≥ 0. Since α(x0 , x1 ) ≥ 1 and the pair (S, T ) is α-admissible, by Lemma 3, we have α(xn , xn+1 ) ≥ 1 for all n ∈ N ∪ {0}.

(3.2)

Thus, for F ∈ ∆c , we have ψ (p(x2i+1 , x2i+2 )) = ψ (p(S(x2i ), T (x2i+1 ))) ≤ α(x2i , x2i+1 )ψ (p(S(x2i ), T (x2i+1 ))) ≤ F (ψ (M (x2i , x2i+1 )) , ϕ (M (x2i , x2i+1 ))) for all i ∈ N ∪ {0}.

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GENERALIZED CONTRACTION MAPPINGS

Now   p(x2i , S(x2i ))p(x2i+1 , T (x2i+1 ))    ,   p(x2i , x2i+1 ),  1 + p(x , x ) 2i 2i+1 M (x2i , x2i+1 ) = max p(x2i , S(x2i ))p(x2i+1 , T (x2i+1 ))       1 + p(S(x2i ), T (x2i+1 ))   p(x2i , x2i+1 )p(x2i+1 , x2i+2 ) p(x2i , x2i+1 )p(x2i+1 , x2i+2 ) = max p(x2i , x2i+1 ), , 1 + p(x2i , x2i+1 ) 1 + p(x2i+1 , x2i+2 ) ≤ max {p(x2i , x2i+1 ), p(x2i+1 , x2i+2 )} . From the definition of F, the case M (x2i , x2i+1 ) = p(x2i+1 , x2i+2 ) is impossible. Indeed, if x2i+1 6= x2i+2 , then ψ (p(x2i+1 , x2i+2 )) ≤ F (ψ (M (x2i , x2i+1 )) , ϕ (M (x2i , x2i+1 ))) < ψ (M (x2i , x2i+1 )) = ψ (p(x2i+1 , x2i+2 )) , which is a contradiction. Therefore, M (x2i , x2i+1 ) = p(x2i , x2i+1 ). Thus ψ (p(x2i+1 , x2i+2 )) ≤ F (ψ (M (x2i , x2i+1 )) , ϕ (M (x2i , x2i+1 ))) ≤ F (ψ (p(x2i , x2i+1 )) , ϕ (p((x2i , x2i+1 )) < ψ (p(x2i , x2i+1 ))) and so ψ (p(x2i+1 , x2i+2 )) < ψ (p(x2i , x2i+1 )) . The definition of ψ implies that p(x2i+1 , x2i+2 ) < p(x2i , x2i+1 ). Thus p(xn+1 , xn+2 ) < p(xn , xn+1 ), for all n ∈ N ∪ {0}.

(3.3)

Hence we deduce that the sequence {p(xn , xn+1 )}n∈N is nonnegative and nonincreasing. Consequently, there exists r ≥ 0 such that limn→∞ p(xn , xn+1 ) = r. We assert that r = 0. Suppose, on contrary, that r > 0. If r > 0, then letting n → +∞ in the following inequality ψ (p(xn+1 , xn+2 )) ≤ F (ψ (p(xn , xn+1 )) , ϕ (p((xn , xn+1 ))) ≤ ψ (p(xn , xn+1 )) ,

(3.4)

we get ψ(r) ≤ F(ψ(r), ϕ(r)) < ψ(r), which is a contradiction. Thus r = 0. Hence lim p(xn−1 , xn ) = 0.

n→+∞

(3.5)

Now, we claim that the sequence {xn } is a Cauchy sequence in (X, p). Suppose, on contrary, that {xn } is not a Cauchy sequence. Then limn,m→∞ p(xn , xm ) 6= 0 and there exists  > 0 for which we can find two subsequences {xmk } and {xnk } of {xn } such that mk is the smallest index for mk > nk > k satisfying p(xmk , xnk ) ≥ . (3.6) This means that p(xmk , xnk−1 ) < .

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(3.7)

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By the triangle inequality, we have  ≤ p(xmk , xnk ) ≤ p(xmk , xnk−1 ) + p(xnk−1 , xnk ) − p(xnk−1 , xnk−1 ) ≤ p(xmk , xnk−1 ) + p(xnk−1 , xnk ) <  + p(xnk−1 , xnk ). That is,  <  + p(xnk−1 , xnk )

(3.8)

for all k ∈ N. In the view of (3.8), (3.5), we have lim p(xmk , xnk ) = .

(3.9)

k→∞

Again using the triangle inequality, we have p(xmk , xnk ) ≤ p(xmk , xmk+1 ) + p(xmk+1 , xnk ) − p(xmk+1 , xmk+1 ) ≤ p(xmk , xmk+1 ) + p(xmk+1 , xnk ) ≤ p(xmk , xmk+1 ) + p(xmk+1 , xnk+1 ) + p(xnk+1 , xnk ) − p(xnk+1 , xnk+1 ) ≤ p(xmk , xmk+1 ) + p(xmk+1 , xnk+1 ) + p(xnk+1 , xnk ) and p(xmk+1 , xnk+1 ) ≤ p(xmk+1 , xmk ) + p(xmk , xnk+1 ) − p(xmk , xmk ) ≤ p(xmk+1 , xmk ) + p(xmk , xnk+1 ) ≤ p(xmk+1 , xmk ) + p(xmk , xnk ) + p(xnk , xnk+1 ) − p(xnk , xnk ) ≤ p(xmk+1 , xmk ) + p(xmk , xnk ) + p(xnk , xnk+1 ). Taking the limit as k → +∞ and using (3.5) and (3.9), we obtain lim p(xmk+1 , xnk+1 ) = .

(3.10)

k→+∞

By Lemma 3 and α(xnk , xmk+1 ) ≥ 1, we have


ψ p(xnk+1 , xmk+2 ) = ψ p(S(xnk ), T (xmk+1 )) ≤ α(xnk , xmk+1 )ψ p(S(xnk ), T (xmk+1 ))


≤ F ψ M (xnk , xmk+1 ) , ϕ M (xnk , xmk+1 ) . This implies that limk→∞ p(xnk , xmk+1 ) = 0 < , which is a contradiction. So limn,m→∞ p(xn , xm ) = 0, which implies that {xn } is a Cauchy sequence in (X, p). From (2.1), we obtain that dp (xn , xm ) ≤ 2p(xn , xm ). Therefore, limn,m→∞ dp (xn , xm ) = 0 and thus by Lemma 1, {xn } is a Cauchy sequence in both (X, p) and (X, dp ). Since (X, p) is a complete partial metric space, by Lemma 1, (X, dp ) is also a complete metric space. Thus there exists υ ∈ X such that xn → υ, that is, limn→∞ dp (xn , υ) = 0. Then again from Lemma 1, we get lim p(υ, xn ) = p(υ, υ) =

n→∞

lim p(xn , xm ).

(3.11)

n,m→∞

Due to limn,m→∞ p(xn , xm ) = 0, it follows from (3.11) that p(υ, υ) = 0 and {xn } converges to υ with respect to τ (p). Moreover, x2n+1 → υ and x2n+2 → υ. Now the continuity of T implies υ = lim xn = lim x2n+1 = lim x2n+2 = lim T (x2n+1 ) = T ( lim x2n+1 ) = T (υ). n→∞

n→∞

n→∞

n→∞

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Analogously, υ = S(υ). Thus we have S(υ) = T (υ) = υ. Hence (S, T ) have a common fixed point. Now we show that υ is the unique common fixed point of S and T . Assume the contrary, that is, there exists ω ∈ X such that υ 6= ω and ω = T (ω). From the contractive condition (3.1), we have ψ (p(υ, ω)) ≤ F (ψ (M (υ, ω)) , ϕ (M (υ, ω))) < ψ (M (υ, ω)) , but



p(υ, S(υ))p(ω, T (ω)) p(υ, S(υ))p(ω, T (ω)) M (υ, ω) = max p(υ, ω), , 1 + p(υ, ω) 1 + p(S(υ), T (ω)) This implies that M (υ, ω) = p(υ, ω).

 .

This means that p(υ, ω) < p(υ, ω), which is a contradiction and so p(υ, ω) = 0. Consequently, υ is a unique common fixed point of the pair (S, T ).  It is also possible to remove the continuity of the mappings S and T by replacing a weaker condition: (C) If {xn } is a sequence in X such that α(xn , xn+1 ) ≥ 1 for all n ∈ N ∪ {0} and xn → υ ∈ X as n → +∞, then there exists a subsequence {xnk } of {xn } such that α(xnk , υ) ≥ 1 for all k. Theorem 2. Let (X, p) be a complete partial metric space and α : X × X → R+ 0 be a function. Suppose that S, T : X → X are mappings such that (1) (2) (3) (4) (5)

(S, T ) is a pair of (α, F, ψ, ϕ)-contraction mappings, (S, T ) is triangular α-admissible, there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1, α(x, y) ≥ 1 for all x, y ∈ F ix(T, S), (C) holds.

Then (S, T ) have a unique common fixed point. Proof. Following the proof of Theorem 1, we know that x2n+1 → υ and x2n+2 → υ as n → +∞. We only have to show that υ is a common fixed point of the pair (S, T ). Due to the hypothesis (4), there exists a subsequence {xnk } of {xn } such that α(x2nk , υ) ≥ 1 for all k. Now by using (3.1) for all k, we have
ψ (p(x2nk +1 , T (υ))) = ψ p(S(x2nk ) , T (υ)) ≤ α(x2nk , υ)ψ (p(S(x2nk ), T (υ))) ≤ F (ψ (M (x2nk , υ)) , ϕ (M (x2nk , υ))) and so ψ (p(x2nk +1 , T (υ))) ≤ F (ψ (M (x2nk , υ)) , ϕ (M (x2nk , υ))) , which implies that p(x2nk +1 , T (υ)) ≤ M (x2nk , υ).

(3.12)

On the other hand, we obtain   p(x2nk , S(x2nk ))p(υ, T (υ)) p(x2nk , S(x2nk ))p(υ, T (υ)) , . M (x2nk , υ) = max p(x2nk , υ), 1 + p(x2nk , υ) 1 + p(S(x2nk ), T (υ)) Letting k → ∞, we have lim M (x2nk , υ) ≤ max {p(υ, S(υ)), p(υ, T (υ))} .

k→∞

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(3.13)

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Case I. Assume that limk→∞ M (x2nk , υ) = p(υ, T (υ)). Suppose that p(υ, T (υ)) > 0. Otherwise, the result is obvious. Letting k → ∞ in (3.12), we obtain that p(υ, T (υ)) < p(υ, T (υ)), which is a contradiction. Thus we obtain that p(υ, T (υ)) = 0. Due to (P M 1) and (P M 2), we have υ = T (υ). Case II. Assume that limk→∞ M (x2nk , υ) = p(υ, S(υ)). Then arguing as above, we get υ = S(υ). Thus υ = T (υ) = S(υ).    p(x, S(x))p(y, S(y)) p(x, S(x))p(y, S(y)) If we set T = S and M (x, y) = max p(x, y), in Theo, 1 + p(x, y) 1 + p(S(x), S(y)) rems 1 and 2, then we obtain the following results. Corollary 1. Let (X, p) be a complete partial metric space and α : X × X → R+ 0 be a function. Suppose that S : X → X is a continuous mapping such that (1) (2) (3) (4)

S is a (α, F, ψ, ϕ)-contraction mapping, S is triangular α-admissible, there exists x0 ∈ X such that α(x0 , S(x0 ) ≥ 1, α(x, y) ≥ 1 for all x, y ∈ F ix(S).

Then S has a unique fixed point υ ∈ X and {S n (x)} converges to υ for every x ∈ X. Corollary 2. Let (X, p) be a complete partial metric space and α : X × X → R+ 0 be a function. Suppose that S satisfies the following conditions: (1) (2) (3) (4) (5)

S is a (α, F, ψ, ϕ)-contraction mapping, S is triangular α-admissible, there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1, α(x, y) ≥ 1 for all x, y ∈ F ix(S), (C) holds.

Then S has a unique fixed point υ ∈ X and {S n (x)} converges to υ for every x ∈ X. Remark 1. For a partial metric space (X, p), we have the following observations: (1) If we set p(x, x) = 0 and F(x, y) = β(x)x for all x, y ∈ X in Corollaries 1 and 2, then we obtain the results presented by Chandok [4]. (2) If we set M (x, y) = max {p(x, y), p(x, S(x)), p(y, S(y))}, p(x, x) = 0 and F(x, y) = β(x)x for all x, y ∈ X in Theorems 1 and 2, then the results presented by Cho et al. [5] can be viewed as particular cases of Theorems 1 and 2. 4. Consequences The following corollaries shall support our claim that Theorem 1 is a generalized version of many corresponding results and shorten the proofs of many results presented in the literature. The results established in [14] can be viewed as particular cases of Corollary 3. Corollary 3. ([14]) Let (X, p) be a complete partial metric space and α : X × X → R+ 0 be a function. Let S, T : X → X be a pair of self-mappings such that (1) (S, T ) is a pair of Geraghty type contraction mappings, (2) (S, T ) is triangular α-admissible,

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GENERALIZED CONTRACTION MAPPINGS

(3) there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1, (4) α(x, y) ≥ 1 for all x, y ∈ F ix(T, S), (5) either S, T are continuous or the condition (C) holds. Then (S, T ) have a unique common fixed point υ in X with p(υ, υ) = 0. Proof. Setting F(x, y) = xβ(x), ψ(t) = t, ϕ(t) = t in Theorem 1, we obtain the required result.



Corollary 4. ([14]) Let (X, p) be a complete partial metric space and α : X × X → R+ 0 be a function. Let S, T : X → X be a pair of self-mappings such that (1) the pair (S, T ) satisfies α(x, y)p(S(x), T (y)) ≤ κM (x, y) where κ ∈ (0, 1), (2) (3) (4) (5)

(S, T ) is triangular α-admissible, there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1, α(x, y) ≥ 1 for all x, y ∈ F ix(T, S), either S, T are continuous or the condition (C) holds.

Then (S, T ) have a unique common fixed point υ in X with p(υ, υ) = 0. Proof. Setting F(x, y) = κx, ψ(t) = t, ϕ(t) = t in Theorem 1, we obtain the required result.



Corollary 5 generalizes the results proved in [13]. Corollary 5. ([13]) Let (X, p) be a complete partial metric space and α : X × X → R+ 0 be a function. Let S, T : X → X be a pair of self-mappings such that (1) the pair (S, T ) satisfies α(x, y)ψ(p(S(x), T (y))) ≤ ψ(M (x, y)) − ϕ(M (x, y)), (2) (3) (4) (5)

(S, T ) is triangular α-admissible, there exists x0 ∈ X such that α(x0 , S(x0 )) ≥ 1, α(x, y) ≥ 1 for all x, y ∈ F ix(T, S), either S, T are continuous or thecondition (C) holds.

Then (S, T ) have a unique common fixed point υ in X with p(υ, υ) = 0. Proof. Setting F(x, y) = x − y in Theorem 1, we obtain the required result.



To illustrate the results proved in this paper and to show the superiority of a pair of (α, F, ψ, ϕ)contraction mappings than the contractions used in [4, 5], we present the following example. Example 2. Let X = {1, 2, 3}. Define p : X × X → R+ 0 by 5 1 2 3 p(1, 3) = p(3, 1) = , p(1, 1) = , p(2, 2) = , p(3, 3) = , 7 10 10 10 3 4 p(1, 2) = p(2, 1) = , p(2, 3) = p(3, 2) = . 7 7 It is easy to check that p is a partial metric and define α : X × X → R+ 0 by  1 if x, y ∈ X; α(x, y) = 0 if otherwise.

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Define the mappings S, T : X → X as follows: S(x) = 1 for each x ∈ X, T (1) = T (3) = 1, T (2) = 3. In addition, define F(x, y) = β(x)x for all x ∈ X, where β : R+ 0 → [0, 1) defined by β (M (x, y)) =

9 for all x, y ∈ X. Note that S(x) and T (x) belong to X and are continuous. The pair (S, T ) is 10 α-admissible. Indeed, α(x, y) = 1 implies α(S(x), T (y)) = 1. We shall show that the condition (3.1) in Theorem 2 is satisfied. If x = 2, y = 3, then α(2, 3) = 1 and   p(2, S(2))p(3, T (3)) p(2, S(2))p(3, T (3)) M (2, 3) = max p(2, 3), , 1 + p(2, 3) 1 + p(S(2), T (3))   9 4 9 9 = , = max , , 7 20 14 14 p (S(2), T (3)) = p (1, 1) =

1 . Now 10

81 1 = α(2, 3)p (S(2), T (3)) ≤ β(M (2, 3))M (2, 3) = 10 140 holds. Similarly, for other cases (x = 1, y = 3 and x = 2, y = 1), it is easy to check that the contractive condition (3.1) in Theorem 1 is satisfied. Consequently, all the conditions (1-4) of Theorem 1 are satisfied. Hence (S, T ) have a unique common fixed point (x = 1). Nevertheless, the contractive condition (3) in [5] does not hold for this particular case. Indeed, for x = 2, y = 3, M (2, 3) = max {d(2, 3), d(2, T (2)), d(3, T (3))}   4 4 5 5 = max , , = , 7 7 7 7 α(2, 3)d(T (2), T (3)) =

5 9  = β(M (2, 3))M (2, 3). 7 14

Similarly, the contractive condition (2.1) in [4] does not hold for this particular case. Indeed, for x = 2, y = 3 and ψ(t) = t,   d(2, T (2))d(3, T (3)) d(2, T (2))d(3, T (3)) M (2, 3) = max d(2, 3), d(2, T (2)), d(3, T (3)), , 1 + d(2, 3) 1 + d(T 2, T 3)   4 4 5 20 5 5 = max , , , , = , 7 7 7 77 21 7

α(2, 3)ψ (d (T (2), T (3))) =

5 9  = β(ψ (M (2, 3))) ψ (M (2, 3)) . 7 14

Here we have assumed that p(x, y) = d(x, y) for all x, y ∈ X such that x 6= y.

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5. Application to system of integral equations In this section, we shall apply Theorem 1 to show the existence of solution of a pair of simultaneous Volterra-Hammerstein integral equations Z 1 x(t) = f (t) + λ K(t, s)Fn (s, x(s)) ds, (5.1) 0 1

Z y(t) = f (t) + λ

K(t, s)Gn (s, y(s)) ds

(5.2)

0

for all t ∈ [0, 1], where f (t) is known, K(t, s), Fn (s, x(s)) and Gn (s, y(s)) are real-valued functions that are measurable both in t and s on [0, 1], and λ is a real number. Let X = L1 ([0, 1], R) and p(x, y) = d(x, y) + cn for all x, y ∈ X, where Z 1 |x(s) − y(s)| ds d(x, y) = kx(s) − y(s)kX = 0

and {cn } is a sequence of positive real numbers satisfying cn → 0 as n → ∞. It is easy to verify that (X, p) is a complete partial metric space. We define F : [0, ∞)2 → R by F(x, y) = β(x)x for all x ∈ X and ψ(t) = t. + Let Θ represent the class of functions φ : R+ 0 → R0 with the following properties (1) φ is increasing, (2) For each t > 0, φ(t)

1 π 1 1 π − + A1 ≤ − a − 2y0 1 − cos πa y 2a 2(a − 2y0 ) a π | πa y − a−2y (y − y0 )| 0 + 2(a − 2y0 )(1 − cos πa y) π | πa y − a−2y (y − y0 )| 0 + π π 2(a − 2y0 )(1 − cos a y)(1 − cos a−2y (y − y0 )) 0

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(2.2)

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and |Py− (x)

−

− P˜y−y (x)| 0

1 π 1 π 1 − + A2 − ≤ a − 2y0 1 + cos πa y 2a 2(a − 2y0 ) a π | πa y − a−2y (y − y0 )| 0 + 2(a − 2y0 )(1 + cos πa y) π | πa y − a−2y (y − y0 )| 0 + π π 2(a − 2y0 )(1 + cos a y)(1 + cos a−2y (y − y0 )) 0

(2.3)

hold for every y0 < y < a − y0 . Proof Firstly, we have Py+ (x)

+ − P˜y−y (x) = 0

π (y − y0 ) sin a−2y sin πa y 0 − π π 2a(cosh πa x − cos πa y) 2(a − 2y0 )(cosh a−2y x − cos a−2y (y − y0 )) 0 0

=I1 + I2 + I3 + I4 , where  sin πa y 1 1 − , I1 = 2a 2(a − 2y0 ) cosh πa x − cos πa y π sin πa y − sin a−2y (y − y0 ) 0 I2 = , 2(a − 2y0 )(cosh πa x − cos πa y) π π sin a−2y (y − y0 ) cos πa y − cos a−2y (y − y0 ) 0 0 I3 = , π π 2(a − 2y0 ) (cosh πa x − cos πa y)(cosh a−2y x − cos a−2y (y − y0 )) 0 0 

I4 =

π (y − y0 ) sin a−2y 0

π cosh a−2y x − cosh πa x 0

2(a − 2y0 )

π π x − cos a−2y (y − y0 )) (cosh πa x − cos πa y)(cosh a−2y 0 0

.

Obviously, 1 |I1 | ≤ 1 − cos πa y

1 1 − 2a 2(a − 2y0 ) .

By mean value theorem of differentials, it is easy to see that |I2 | ≤ |I3 | ≤ I4 =

| πa y −

π (y a−2y0

− y0 )|

, 2(a − 2y0 )(1 − cos πa y) π | πa y − a−2y (y − y0 )| 0 π 2(a − 2y0 )(1 − cos πa y)(1 − cos a−2y (y − y0 )) 0

,

π sin a−2y (y − y0 ) 0

π ( a−2y − πa )x · sinh ξ 0

2(a − 2y0 )

π π (cosh πa x − cos πa y)(cosh a−2y x − cos a−2y (y − y0 )) 0 0

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π

π where ξ is between πa x and a−2y x. Note that there exists M > 0 such that |x|e− a |x| ≤ 0 π 1 , cosh πa x ≥ 2 and cosh a−2y x ≥ 2 for all |x| > M . Therefore, we have 4 0

cosh

|x| ≤ − cos πa y

π x a

1 2

and | sinh ξ| ≤ π π cosh a−2y0 x − cos a−2y (y − y ) 0 0 Hence

2 |I4 | ≤ a − 2y0

4|x| |x| ≤ π |x| ≤ 1 π cosh a x ea | sinh ξ| 2e|ξ| ≤ ≤ 4. π π 1 cosh x cosh a−2y x a−2y 2 0 0

π π a − 2y0 − a .

If |x| ≤ M , then π π π π | sinh ξ| ≤ sinh |x| + sinh |x| ≤ sinh M + sinh M, a a − 2y0 a a − 2y0 which follows that π M (sinh πa M + sinh a−2y M) π π 1 0 |I4 | ≤ π a − 2y0 − a . 2(a − 2y0 ) (1 − cos πa y)(1 − cos a−2y (y − y )) 0 0 Let

( A1 = max

then

π M (sinh πa M + sinh a−2y M) 2 0 , π π a − 2y0 2(a − 2y0 )(1 − cos a y)(1 − cos a−2y0 (y − y0 ))

π π |I4 | ≤ A1 − , a − 2y0 a

) ,

x ∈ R,

(2.2) is thus proved. Similarly, we can prove (2.3). Lemma 2.4 (Harmonic Majorant) Let v(z) be a nonnegative subharmonic function in Sa satisfying (2.1), then Z Mv (y) = v(x + iy)dx R

is convex in (0, a) and there exist two positive measures µ1 , µ2 ∈ M (R) with kµ1 k, kµ2 k ≤ C such that Z ∞ Z ∞ + u(x + iy) = Py (x − t)dµ1 (t) + Py− (x − t)dµ2 (t). −∞

−∞

Moreover, v(z) ≤ u(z) for all z ∈ Sa . Proof There exists a sequence {yk } such that limk→∞ yk = 0. By (2.1), {vyk }, {va−yk } are bounded linear functionals on C0 (R) and they are uniformly bounded, where vy (x) = v(x + iy). Based on Banach-Alaoglu theorem, there exist µ1 , µ2 ∈ M (R) and a subsequence

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{ykj } such that {vykj } converges weakly to µ1 as j → ∞ and {va−ykj } converges weakly to µ2 as j → ∞. That is, for each ϕ ∈ C0 (R), Z ∞ Z ∞ lim v(t + iykj )ϕ(t)dt = ϕ(t)dµ1 (t), j→∞

Z

−∞

−∞

∞

Z

∞

v(t + i(a − ykj ))ϕ(t)dt =

lim

j→∞

ϕ(t)dµ2 (t). −∞

−∞

Accordingly, we obtain that  Z  kµ1 k = sup ϕ(t)dµ1 (t) : ϕ ∈ C0 (R), kϕk∞ = 1 ZR ≤ lim inf v(t + iykj )dt ≤ C, j→∞

R

 Z  kµ2 k = sup ϕ(t)dµ2 (t) : ϕ ∈ C0 (R), kϕk∞ = 1 ZR ≤ lim inf v(t + i(a − ykj )dt ≤ C. j→∞

R

Because of ϕ(t) = Py+ (x − t)(or Py− (x − t)) ∈ C0 (R), in particular, we have Z ∞ Z ∞ + v(t + i(a − ykj ))Py− (x − t)dt lim v(t + iykj )Py (x − t)dt + lim j→∞ −∞ j→∞ −∞ Z ∞ Z ∞ + Py (x − t)dµ1 (t) + Py− (x − t)dµ2 (t) , u(x + iy). = −∞

−∞

0 For any fixed 0 < y0 < y1 < a, let r = y1 −y , then the function a Z Z + u˜(z) = v(t + iy0 )Py (x − t)dt + v(t + iy1 )Py− (x − t)dt

R

R

is harmonic in Sa (see [15, Theorem 1]). Assume that ε > 0 and A > exp{ πε min{yC0 ,a−y1 } } + 1. 0 Since v(rz + iy0 ) is subharmonic in the set {z = x + iy : x ∈ R, − yr0 < y < a−y }, then there r (1) (2) exist two sequences of continuous functions {un (t)} and {un (t)} decreasing to v(rt + iy0 ) and v(rt + iy1 ) on [−A, A], respectively. Let Z A Z A + (1) Un (z) = Py (x − t)un (t)dt + Py− (x − t)u(2) n (t)dt, −A

−A

then by Lemma 2.2, Un (z) is harmonic in Sa and (2) |Un (z)| ≤ max{max u(1) n (t), max un (t)} = An . |t|≤A

|t|≤A

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Therefore, the function Vn (z) = v(rz + iy0 ) − 2ε log |z + i| − Un (z) is subharmonic in Sa , and by Lemma 2.1, we speculate that Vn (z) ≤

2C − ε log |x2 + (y + 1)2 | + An → −∞ (z → ∞, 0 < y < a). π min{y0 , a − y1 }

It follows that lim sup Vn (z) ≤ z→t,0 0 such that ∫ ∥h(t, x1 (t), y) − h(t, x2 (t), y)∥2 µ(dy) ≤ L3 ∥x1 (t) − x2 (t)∥2 , Z

582

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∫ J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC ∥h(t, x(t), y) − h(t, x(t), y)∥4 µ(dy) ≤ L4 ∥x1 (t) − x2 (t)∥4 ,

Z

for all x1 , x2 , x ∈ X and a.e. t ∈ J. (H4 ) There are some constants L5 , L6 > 0 such that ∫ ∥h(t, x(t), y)∥2 µ(dy) ≤ L5 (1 + ∥x(t)∥2 ), ∫Z ∥h(t, x(t), y)∥4 µ(dy) ≤ L6 (1 + ∥x(t)∥4 ) Z

for all x1 , x2 , x ∈ X and a.e. t ∈ J. ∫T (H5 ) The function σ : [0, ∞) → L02 (K, X) satisfies 0 ∥σ(s)∥2L0 ds < ∞. 2 Now, we consider the existence result for system (1.1). Theorem 3.1. Assume that hypotheses (H1 ) − (H5 ) hold. Then for any u ∈ L2 (J, U ) the stochastic system (1.1) has a unique mild solution on J, if √ 2T 2 [M 2 L1 + l(L3 + L4 )] < 1. Proof. We define an operator F : C(J, L2 (Ω, X)) → C(J, L2 (Ω, X)) by ∫ t ∫ t (F x)(t) = S(t)x0 + S(t − s)Bu(s)ds + S(t − s)f (s, x(s))ds 0 0 ∫ t ∫ t∫ H + S(t − s)σ(s)dBQ (s) + S(t − s)h(s, x(s), y)N (ds, dy). 0

0

Z

Using the contraction mapping principle, we will show that the operator F has a fixed point. To prove this, we subdivide the proof into four steps. Step 1. For any x ∈ C(J, L2 (Ω, X)), we show that F maps C(J, L2 (Ω, X)) into itself. For all x ∈ C(J, L2 (Ω, X)), we have E∥(F x)(t)∥2

∫ t

2

∫ t

2



+ 5E

≤ 5E∥S(t)x0 ∥2 + 5E S(t − s)Bu(s)ds S(t − s)f (s, x(s))ds



0

0

∫ t

2

∫ t∫

2



H

+5E S(t − s)σ(s)dBQ (s) + 5E S(t − s)h(t, x(t), y)N (ds, dy)

0 0 Z [ ] ∫ T ≤ 5M 2 E∥x0 ∥2 + E∥Bu∥2 T 2 + T L2 (1 + E∥x∥2C ) + cH(2H − 1)T 2H−1 ∥σ(s)∥2L0 (V,U ) ds Q

0

{ (∫ t∫ ) (∫ t∫ ) 21 } 2 4 +l E ∥h(t, x(t), y)∥H µ(dy)ds + E ∥h(t, x(t), y)∥H µ(dy)ds 0

Z

0

Z

[ ∫ ≤ 5M 2 E∥x0 ∥2 + E∥Bu∥2 T 2 + T L2 (1 + E∥x∥2C ) + cH(2H − 1)T 2H−1 [

∫

(∫

) 12 ] E(1 + ∥x(s)∥4 )ds

T 0

] ∥σ(s)∥2L0 (V,U ) ds Q

√ +l L5 E(1 + ∥x(s)∥2 )ds + L6 0 [ 0 2 2 2 2 ≤ 5M E∥x0 ∥ + E∥Bu∥ T + T L2 (1 + E∥x∥2C ) t

∫ +cH(2H − 1)T

t

]

T

2H−1 0

∥σ(s)∥2L0 (V,U ) ds Q

+ l(L5 T +

√ L6 T )E(1 + ∥x(s)∥2 )

(3.1)

for all t ∈ J. From the inequality (3.1) and the assumptions, one can see that there exists M1 > 0 such that E∥(F x)(t)∥2

≤ M1 (1 + T sup E∥x(s)∥2 ) s∈J

for all t ∈ J. Thus, F maps C(J, L2 (Ω, X)) into itself. Step 2. We prove that F is a contraction mapping. Let x1 , x2 ∈ C(J, L2 (Ω, X)), for t ∈ J we have E∥(F x1 )(t) − (F x2 )(t)∥2

∫ t

≤ E S(t − s)[f (s, x1 (s)) − f (s, x2 (s))]ds

0

583

Liang Zhao 579-589

2 COPYRIGHT 2018 EUDOXUS PRESS, LLC ∫ t∫ J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.3, 2018,

S(t − s)[h(t, x1 (t), y) − h(t, x2 (t), y)]N (ds, dy)

0 Z { (∫ t∫ ) ≤ 2M 2 T 2 L1 sup E∥x1 (t) − x2 (t)∥X 2 + 2l E ∥h(t, x1 (t), y) − h(t, x2 (t), y)∥2C µ(dy)ds +

t∈[0,T ]

0

(∫ t∫

∥h(t, x1 (t), y) −

+E 0

Z

h(t, x2 (t), y)∥4C µ(dy)ds

) 21 }

Z

∫ t √ ≤ 2M 2 T 2 L1 sup E∥x1 (s) − x2 (s)∥2C + 2l(L3 + L4 ) E∥x1 (s) − x2 (s)∥2 ds s∈J 0 √ ≤ 2T 2 [M 2 L1 + l(L3 + L4 )] sup E∥x1 (s) − x2 (s)∥2C . s∈J

√ Since 2T [M L1 + l(L3 + L4 )] < 1, then F is a contraction mapping and hence there exists a unique fixed point x(·) in C(J, L2 (Ω, X)) which is the mild solution of problem (1.1). 2

4

2

Controllability results

In this section, we discuss the controllability results for System (1.1). Before starting, we consider the following assumption: (H5 ) The linear operator LT0 ∈ L2 (U, X) is defined by ∫ T T L0 u = S(T − s)Bu(s)ds. 0

has an inverse operator (LT0 )−1 which takes values in L2 (J, U ) \ ker LT0 , where ker LT0 = {x ∈ L2 (J, U ), LT0 x = 0}, and there are positive constants Mb , ML such that ∥B∥2 ≤ Mb , ∥(LT0 )−1 ∥2 ≤ ML . To the readers’ convenience, we give the definitions of controllability as follows. Definition 4.1. System (1.1) is said to be completely controllable on the interval J if Rt (x0 ) = C(J, L2 (Ω, X)), that is, all the points in C(J, L2 (Ω, X)) can be exactly reached from arbitrary initial condition x(0) = x0 and xT at time T . Theorem 4.2. Assume that hypotheses (H1 ) − (H5 ) hold. Then the stochastic system (1.1) is completely controllable on J, if { ( ) } √ √ 3 T M 2 L1 T + 2M 2 Mb2 ML [M 2 L1 T + (L3 + L4 )]T 2 + l(L3 + L4 ) < 1. Proof. Fix T > 0 and let ZT = C(J, L2 (Ω, X)) be the Banach space of all functions from J into L2 (Ω, X), endowed with the supremum norm ( ∥µ∥ZT =

sup E∥µ(t)∥2

) 12 .

t∈[0,T ]

Let’s consider the set GT = {x ∈ ZT : x(0) = x0 }. We easily know that GT is a closed subset of ZT equipped with norm ∥ · ∥ZT . By assumption (H5 ), one can choose the feedback control function ux (t) as { ∫ T ux (t) = B ∗ S ∗ (T − t)E (LT0 )−1 (xT − S(T )x0 − S(T − s)f (s, x(s))ds [∫

∫

T

−

S(T −

H s)σ(s)dBQ (s)

T

0

] } S(T − s)h(s, x(s), y)N (ds, dy) |Ft .

∫

+

0

0

Z

We will prove that if we use this control ux (t), the operator Φ define on ∥ · ∥ZT by ( [ ∫ t ∗ ∗ T −1 xb − S(T )x0 Φ(x)(t) = S(t)x0 + S(t − s)BB S (T − s)E (L0 ) ∫

0 T

−

[∫

T

S(T − η)f (s, x(η))dη − 0

∫

T

∫

+ 0

H S(T − η)σ(η)dBQ (η) 0

] ] ∫ t S(t − s)f (s, x(s))ds S(T − η)h(η, x(η), y)N (dη, dy) |Ft ds + 0

Z

584

Liang Zhao 579-589

∫ tANALYSIS AND APPLICATIONS, ∫ t ∫ VOL. 25, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC J. COMPUTATIONAL H S(t − s)σ(s)dBQ (s) +

+

S(t − s)h(s, x(s), y)N (ds, dy).

0

0

Z

has a fixed point on J. To prove that the operator Φ has a fixed point on J, we divide the subsequent proof into the following two steps. Step 1. For any x ∈ GT , let’s prove that t → Φ(x)(t) is continuous on J in the L2 (Ω, X)-sense. Let 0 < t < t + δ < T , here t, t + δ are belong to J, and δ > 0 is sufficiently small. Then we have E∥Φ(x)(t + δ) − Φ(x)(t)∥2

∫ t

2

≤ 9E∥S(t + δ)x0 − S(t)x0 ]∥ + 9E [S(t + δ − s) − S(t − s)]f (s, x(s))ds

0

∫ t+δ

2

∫ t



+9E S(t + δ − s)f (s, x(s))ds [S(t + δ − s) − S(t − s)]B

+ 9E 2

t

0

( ∫ ∗ ∗ T −1 ×B S (T − s)(L0 ) xT − S(T )x0 − ∫

∫

T

−

T

T

S(T − η)f (η, x(η))dη 0

∫

H S(T − η)σ(η)dBQ (η) − 0

0

Z

0

0

Z

) 2

S(T − η)h(η, x(η), y)N (dη, dy) ds

∫ t+δ ( ∫ T

∗ ∗ T −1

+9E S(t + δ − s)BB S (T − s)(L0 ) xT − S(T )x0 − S(T − η)f (η, x(η))dη t 0 ) 2 ∫ T ∫ T∫

H − S(T − η)σ(η)dBQ (η) − S(T − η)h(η, x(η), y)N (dη, dy) ds

∫ t

2

∫ t+δ

2



H H

+9E [S(t + δ − s) − S(t − s)]σ(s)dBQ (s) + 9E S(t − s)σ(s)dBQ (s)

. 0 t

∫ t∫

2

+9E [S(t + δ − s) − S(t − s)]h(t, x(t), y)N (ds, dy)

0

∫

+9E

Z t+δ ∫

t

≤

9

9 ∑

2

S(t + δ − s)h(t, x(t), y)N (ds, dy)

Z

I9 .

i=1

We can easily know that I1 ≤ ∥S(t + δ) − S(t)∥2 E∥x0 ∥2 → 0

as δ → 0.

By using the well-known H¨older’s inequality, we get ∫ t I2 ≤ t ∥S(t + δ − s) − S(t − s)∥2 sup E(1 + ∥x(s)∥2 )ds → 0

as δ → 0.

s∈J

0

∫

t+δ

I3 ≤ M 2 t

sup E(1 + ∥x(s)∥2 )ds → 0

as δ → 0.

s∈J

t

By H¨older’s inequality again, Lemma 2.4 and the condition (H3 ), we get ( ∫ t 2 4 T 2 I4 ≤ 5t ∥S(t + δ − s) − S(t − s)∥ ∥B∥ ∥L0 ∥ E∥xT ∥2 + M 2 E∥x0 ∥2 0

∫ T

2

∫ T

2



H

+E S(T − η)f (η, x(η))dη + E S(T − η)σ(η)dBQ (η)

0 0

∫ T ∫

2 )

+E S(T − η)h(η, x(η), y)N (dη, dy)

ds 0 Z ( ∫ t 2 2 ≤ 5Mb ML ∥S(t + δ − s) − S(t − s)∥ E∥xT ∥2 + M 2 E∥x0 ∥2 0 T

∫ { ( +l E

∫

T

sup E(1 + ∥x(η)∥2 )dη + M 2 cH(2H − 1)T 2H−1

+M 2 T ∫

η∈J

0 T

∫

2

)

(

∫

T

∫

0

∥σ(η)∥2L0 dη 2

4

E∥h(η, x(η), y)∥ µ(dy)dη E∥h(η, x(η), y)∥ µ(dy)dη + E 0 Z ( t ∥S(t + δ − s) − S(t − s)∥2 E∥xT ∥2 + M 2 E∥x0 ∥2

0

) ) 21 }

ds

Z

∫

≤ 5Mb2 ML 0

585

Liang Zhao 579-589

∫ T ∫ TANALYSIS AND APPLICATIONS, VOL. 25, NO.3, 2018, COPYRIGHT J. COMPUTATIONAL 2018 EUDOXUS PRESS, LLC sup E(1 + ∥x(η)∥2 )dη + M 2 cH(2H − 1)T 2H−1

+M 2 T

η∈J

0

0

) √ +l(L5 T + L6 T )E(1 + ∥x(s)∥2 ) ds.

∥σ(η)∥2L0 dη 2

Hence, by Lebesgue’s dominated convergence, one can know that I4 → 0 as δ → 0. For a similar way, we obtain ∫ t+δ ( ∫ T I5 ≤ 5M 2 Mb2 ML E∥xT ∥2 + M 2 E∥x0 ∥2 + M 2 T sup E(1 + ∥x(η)∥2 )dη t

∫

η∈J

0 T

+M 2 cH(2H − 1)T 2H−1

∥σ(η)∥2L0 dη 2 ) sup E(1 + ∥x(η)∥2 )dη ds → 0 as δ → 0. 0

∫

T

+L4 M 2

η∈J

0

∫

t

≤ cH(2H − 1)t2H−1

I6

∥S(t + δ − s) − S(t − s)∥2 ∥σ(s)∥2L0 ds as δ → 0.

0

∫

2

t+δ

I8

≤ cH(2H − 1)M 2 δ 2H−1 ∥σ(s)∥2L0 ds as δ → 0. 2 t ∫ t √ ≤ (L5 T + L6 T ) ∥S(t + δ − s) − S(t − s)∥2 sup E(1 + ∥x(s)∥2 )ds as δ → 0.

I9

∫ √ ≤ l(L5 T + L6 T )

I7

s∈J

0 t+δ

sup E(1 + ∥x(s)∥2 )ds as δ → 0. s∈J

t

Then, by the strong continuous of S(t) and the Lebesgue’s dominated convergence theorem, we know that the right hand of Ii (i = 1, · · · , 9) tends to 0 as δ → 0. Hence, Φ(x)(t) is continuous on J in the L2 (Ω, X)-sense. Next, we prove that Φ is a contraction mapping. Let x, z ∈ C(J, L2 (Ω, X)) are two mild solution of (1.1), then E∥Φ(x)(t) − Φ(z)(t)∥2H

∫ t

2

≤ 3E S(t − s)[f (s, x(s)) − f (s, z(s))]ds

0

∫ t

2

+3E S(t − s)B(s)[ux (s) − uz (s)]ds

0

∫ t∫

2

+3E S(t − s)[h(s, x(s), y) − h(s, z(s), y)]N (ds, dy)

0

Z

≤ 3J1 + 3J2 + 3J3 . We can easily show that ≤ T M 2 L1 T sup E∥x(t) − z(t)∥2H , t∈J √ ≤ l(L3 + L4 ) sup E∥x(t) − z(t)∥2 .

J1 J3

(4.1) (4.2)

t∈J

Since E∥ux (t) − uz (t)∥2

(∫ T

∗ ∗ T −1 S(T − s)(f (s, x(s)) − f (s, z(s)))ds ≤ E B S (T − t)(L ) 0

0 ) 2 ∫ T∫

− S(T − s)[h(s, x(s), y) − h(s, z(s), y)]N (ds, dy)

0

Z

(

∫

T

≤ 2M Mb ML M 2 T

E∥f (s, x(s)) − f (s, z(s))∥2 ds

2

{ (∫ +l E 0

T

∫

0

) ( E∥h(η, x(η), y)∥2 µ(dy)dη + E

∫

0

∫ E∥h(η, x(η), y)∥4 µ(dy)dη

0

Z

∫ √ 2 2 ≤ 2M Mb ML [M L1 T + (L3 + L4 )]

T

) 21 }

)

Z

T

sup E∥x(s) − z(s)∥2 ds, s∈[0,t]

we have sup E∥ux (t) − uz (t)∥2 ≤ 2M 2 Mb ML [M 2 L1 T + (L3 + t∈J

√

L4 )]T sup E∥x(t) − z(t)∥2 . t∈J

586

Liang Zhao 579-589

Hence

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

J2

≤ 2M 4 Mb2 ML [M 2 L1 T + (L3 +

√ L4 )]T 2 sup E∥x(t) − z(t)∥2 ,

(4.3)

t∈J

By inequalities (4.1)-(4.3), we get E∥Ψ(x)(t) − Ψ(z)(t)∥2 { ( ) } √ √ 2 2 2 2 2 + l(L3 + L4 ) ∥x − z∥2C . ≤ 3 T M L1 T + 2M Mb ML [M L1 T + (L3 + L4 )]T ( { ) } √ √ 2 2 2 2 2 Since 3 T M L1 T + 2M Mb ML [M L1 T + (L3 + L4 )]T + l(L3 + L4 ) < 1. Therefore, Φ is a contraction mapping and hence there exists a unique fixed point x(·) in C(J, L2 (Ω, X)) which is the mild solution of system (1.1). Thus, system (1.1) is complete controllable on J.

5

An example

Let’s consider the following stochastic partial differential equations driven by fractional Brownian motion and Poisson noise process:  ] [ 2 ∂ x(θ, t)    + F (θ, t, x(θ, t)) + g(θ, t) dt + σ(t)dB H (t) dx(θ, t) =  2  ∂θ  ∫  e (dt, dµ), in Ω × [0, τ ], + cos tx(θ, t)µN (5.1)  Z    x(θ, t) = 0, on ∂Ω × [0, τ ],    x(θ, 0) = x (θ), θ ∈ Ω, 0

where B H is a fractional Brownian motion, Ω is a bounded open set in R, F : Ω × J × R → R is nonlinear function, measurable with respect to θ and almost everywhere continuous with respect to t. Let {q(t), t ∈ [0, τ ]} be the Poisson jump process taking values in the space H = [0, ∞) with a σ-finite intensity measure λ(dµ) on the completely probability space (Σ, F, P ). We denote the Poisson counting measure as N (dt, dµ), which is induced by q(·), and compensating martingale measure given by e (dt, dµ) = N (dt, dµ) − λ(dµ)dt. N Take X = Y = U = L2 ([0, τ ]) and the operator A : D(A) ⊂ X → X is defined by Ax = x′′ , D(A) = {x ∈ X : x, x′ are absolutely continuous, x′′ ∈ X, x(0) = x(π) = 0}. Then, A can be written as Ax = −

∞ ∑

n2 (x, xn )xn ,

x ∈ D(A),

n=1

√ where xn (x) = 2/π sin ny(n = 1, 2, · · · ) is an orthonormal basis of X. It is well known that A is the infinitesimal generator of a differentiable semigroup T (t)(t > 0) in X given by ∞ ∑

T (t)x =

exp−n t (x, xn )xn , 2

x ∈ X,

and ∥T (t)∥ ≤ e−1 < 1 = M.

n=1

In order to define the operator Q : Y → R, we choose a sequence {ln }n∈N ⊂ R+ , let Qen = ln en , and assume that tr(Q) =

∞ √ ∑ ln < ∞. n=1

Thus, we define the fractional Brownian motion in Y as B H (t) =

∞ √ ∑ ln γnH (t)en , n=1

where H ∈ ( 12 , 1) and {γnH }n∈n is a sequence of one-dimensional fractional Brownian motion mutually independent. Let x(t)(·) = x(·, t), f (t, x)(·) = F (·, t, x(·)). Define the bounded operator B : U → X by Bu(t)(θ) = g(θ, t), θ ∈ Ω, u ∈ U . Hence, by the above choice, it’s easily known that the system (5.1) can be written into (1.1) and all the conditions of Theorem 4.2 are satisfied. Then by the Theorem 4.2, the stochastic partial differential equations driven by fractional Brownian motion and Poisson noise process is completely controllable on [0, τ ]. 587

Liang Zhao 579-589

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.3, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC References

[1] D. Applebaum, F. Tang, Stochastic flows of diffeomorphisms on manifolds driven by infinite-dimensional semimartingales with jumps, Stoch. Proc. Appl., 92 (2001) 219-236 [2] L.H. Bai, J. Ma, Stochastic differential equations driven by fractional Brownian motion and Poisson point process, Bernoulli, 21(1) (2015) 303-334. [3] K. Balachandran, S. Karthikeyan, J.H. Kim, Controllability of semilinear stochastic integrodifferential systems, Kybernetika, 43(1) (2007) 31-44. [4] D. Barraez, H. Leiva, N. Merentes, M. Narv´aez, Exact Controllability of Semilinear Stochastic Evolution Equations, African Diaspora J. Math, 11(1) (2011) 124-139. [5] M. Besal´ u, C. Rovira, Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion, Bernoulli, 18(1) (2012) 24-45. [6] W. Bian, Controllability of nonlinear evolution systems with preassigned responses, J. Optim. Theory Appl., 100 (1999) 265-285. [7] B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics and Probability Letters, 82(8) (2012) 1549-1558. [8] T. Caraballo, M.J. Garrido-Atienza, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis, 74 (2011) 3671-3684. [9] N.T. Dung, Mackey-Glass equation driven by fractional Brownian motion, Physica A, 391(2012) 5465-5472. [10] S. Hajji, E. Lakhel, Neutral stochastic functional differential equation driven by fractional Brownian motion and Poisson point processes, (2013) 18 pages. arXiv admin note: text overlap with arXiv:1312.6147 [11] Hamdy M. Ahmed, Controllability of impulsive neutral stochastic differential equations with fractional Brownian motion, IMA Journal of Mathematical Control and Information, (2014) 1-14, doi:10.1093/imamci/dnu019 [12] E. Hausenblas, SPDEs driven by Poisson random measure with non-Lipschitz coefficients: Existence results, Probab. Th. Relat. Fields 137 (2007) 161-200. [13] H. Leiva, Exact controllability of the suspension bridge model proposed by Lazer and McKenna, J. Math. Anal. Appl. 309 (2005) 404-419. [14] H. Leiva, Exact controllability of semilinear evolution equation and applications, Int. J. Systems Control Communications, 1 (1) (2008) 1-12. [15] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981. [16] K. Itˆo, Stochastic Analysis, North-Holland Publishing Company Amsterdam, New York, Oxford, 1982. [17] R.E. Kalman, Controllablity of linear dynamical systems, Contributions Differ. Equat., 1 (1963) 190-213. [18] S. Karthikeyan, K. Balachandran, Controllability of nonlinear stochastic neutral impulsive systems, Nonlinear Analysis: HS, 3(3)(2009) 266-276. [19] J. Klamka, L. Socha, Some remarks about stochastic controllability, IEEE Trans. Automat. Control, 22 (1977) 880-881. [20] C. Knoche, SPDE’s in infinite dimensions with Poisson noise, Comptes Rendus Math´ematique, Acad´emie des Sciences. Paris, Serie I, 339 (2004) 647-652. [21] C. Knoche, Mild solutions of SPDEs driven by Poisson noise in infinite dimensions and their dependence on initial conditions, Preprint no. 05-003 of the Collaborative Research Centre 701, Faculty of Mathematics, University of Bielefeld, Bielefeld, Germany, 2005. [22] W. Laukajtys, L. Slomi´ nski, Penalization methods for reflecting stochastic differential equations with jumps, Stoch. Stoch. Rep. 75 (5) (2003) 275-293. [23] J. Le´on, S. Tindel, Malliavin calculus for fractional delay equations, J. Theoretical Probability, 25(3) (2012) 854-889. [24] B. Liu, Stability of solutions for stochastic impulsive systems via comparison approach, IEEE Trans. Auto. Control, 53 (9) (2008) 2128-2133. [25] Z.H. Liu, X.W. Li, On the controllability of impulsive fractional evolution inclusions in Banach spaces, J. Optim. Theory Appl., 156 (2013) 167-182. 588

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[26] J. Luo, T. Taniguchi, The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps, Stochastics Dyna., 9 (1) (2009) 135-152. [27] N.I. Mahmudov, Controllability of linear stochastic systems in Hilbert spaces, J. Math. Anal. Appl. 259 (2001) 64-82. [28] N.I. Mahmudov, S. Zorlu, Controllability of non-linear stochastic systems. Inter. J. Control, 76 (2) (2003) 95-104. [29] X.R. Mao, Stochastic Differential Equations and Their Applications, Horwood, Chichester, UK, 1997. [30] Y. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Springer, Berlin, 2008. [31] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. [32] P.E. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 2004. [33] G.D. Prato, J. Zabczyk, Stochastic Equations In Infinite Dimensions. Encyclopedia Math. Appl. vol 44, Cambridge University Press, Cambridge, UK, 1992. [34] J. Ren, J. Wu, Multivalued stochastic differential equations driven by Poisson point processes, in: Proc. of Workshop on Stochastic Analysis and Finance, July 2009. [35] M. Royer, Backward stochastic differential equations with jumps and related non-linear expectations, Stochastic Process. Appl. 116 (2006) 1358-1376. [36] L.J. Shen, J.P. Shi, J.T. Sun, Complete controllability of impulsive stochastic integro-differential systems, Automatica 46 (2010) 1068-1073. [37] R. Subalakshmi, K. Balachandran, Approximate controllability of nonlinear stochastic impulsive integrodifferential systems in hilbert spaces, Chaos Solitons Fractals, 42(4) (2009) 2035-2046. [38] S. Tindel, C. Tudor, F. Viens, Stochastic evolution equations with fractional brownian motion, Probab. Theory Related Fields, 127 (2) (2003) 186-204. [39] J. Unterberger, Stochastic calculus for fractinal Brownian montion with Hurst Exponent H > path method by analytic extension, The Annals of Probability, 27 (2) (2009) 565-614.

1 4

: A rough

[40] H.X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIMA J. Control Optim. 22 (1983) 405-422.

589

Liang Zhao 579-589

 

590

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 3, 2018

Some Hermite-Hadamard and Simpson type inequalities for convex functions via fractional integrals with applications, Muhammad Iqbal, Mustafa Habib, Nasir Siddiqui, and Muhammad Muddassar,…………………………………………………………………………………397 The differentiability for fuzzy n-cell mappings and the KKT optimality conditions for a class of fuzzy constrained minimization problem, She-Xiang Hai and Zeng-Tai Gong,……………407 Monotone iterative technique for fractional partial differential equations with impulses, Xuping Zhang and Yongxiang Li,……………………………………………………………………418 EVP, minimax theorems and existence of nonconvex equilibria in complete G-metric spaces, E. Hashemi and R. Saadati,……………………………………………………………………441 Geometric properties of Bessel functions for the classes of Janowski starlike and convex functions, V. Radhika, S. Sivasubramanian, N. E. Cho, and G. Murugusundaramoorthy,…452 On the Jensen-type inequality for the 𝑔̅ -integral, Jeong Gon Lee and Lee-Chae Jang,………467

Some new Hermite-Hadamard type inequalities for operator 𝑚-convex and (𝛼, 𝑚)-convex functions on the co-ordinates, Shu-Hong Wang and Shan-He Wu,…………………………474 Additive-quadratic 𝜌-functional inequalities in fuzzy Banach spaces: a fixed point approach, Choonkil Park, Jung Rye Lee, and Dong Yun Shin,…………………………………………488 Optimal control of a special predator-prey system with functional response and toxicant, Jiangbi Liu and Hongwei Luo,…………………………………………………………………………500 Existence results for new extended vector variational-like inequality, Kasamsuk Ungchittrakool and Boonyarit Ngeonkam,……………………………………………………………………512 Existence of solutions for a new semi-linear evolution equations with impulses, Huanhuan Zhang and Yongxiang Li,…………………………………………………………………………….525 Eigenvalue for a system of Caputo fractional differential equations, Xiaofeng Zhang and Hanying Feng,…………………………………………………………………………………544 A common fixed point theorem for a pair of generalized contraction mappings with applications, Muhammad Nazam, Muhammad Arshad, and Choonkil Park,………………………………552

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 3, 2018 (continued) Inner-outer factorization on the Nevanlinna space in a strip, Cuiqiao Wang, Guantie Deng, and Huaping Huang,………………………………………………………………………………565 Controllability of Stochastic Evolution Differential Equations Driven by Fractional Brownian Motion and Poisson Jumping Processes, Liang Zhao,………………………………………579

Volume 25, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

October 15, 2018

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (sixteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

A note on non-instantaneous impulsive fractional neutral integro-differential systems with state-dependent delay in Banach spaces Selvaraj Suganya,∗ Dumitru Baleanu†, Palaniyappan Kalamani‡ and Mani Mallika Arjunan§

Abstract In this research, we establish the existence results for non-instantaneous impulsive fractional neutral integro-differential systems with state-dependent delay in Banach space. By utilizing the Banach contraction principle and condensing fixed point theorem coupled with semigroup theory, we build up the desired results. To acquire the main results, our working concepts are that the functions deciding the equation fulfill certain Lipschitz conditions of local type which is similar to the hypotheses [5]. In the end, an example is given to show the abstract theory.

Keywords:

Fractional order differential systems, Caputo fractional integral operator, non-

instantaneous impulses , state-dependent delay, fixed point theorem, semigroup theory. MSC 2010: Primary 34K30, 26A33; Secondary 35R10, 47D06.

1

Introduction Fractional calculus may be considered an old and yet novel topic. In fact, the concepts are almost as

old as their more familiar integer-order counterparts. In 1965 Leibniz and L’Hopital had correspondence where they discussed the meaning of the derivative of order one half.

Since then, many famous

mathematicians have worked on this and related questions, creating the field which is known today as fractional calculus. The fractional calculus is also considered a novel topic, since it is only during the last three decades that it has been the subject of specialised conferences and treatises. This was stimulated by the fact that many important applications of fractional calculus have been found in numerous diverse and widespread ∗

Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641 042, Tamil Nadu, India.

E. Mail:

[email protected] † Corresponding author: Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey and Institute of Space Sciences, Magurele-Bucharest, Romania, E. Mail: [email protected] ‡ Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641 042, Tamil Nadu, India.

E. Mail:

[email protected] § Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641 042, Tamil Nadu, India. E. Mail: [email protected]

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fields in science, engineering and finance. Many authors have pointed out that fractional derivatives and integrals are very suitable for modelling the memory and hereditary properties of various materials and processes that are governed by anomalous diffusion. This represents the main advantage of using the fractional derivatives in comparison with classical integer-order models, in which such effects are not taken into account. For more details, we suggest the reader to refer the monographs [6, 24, 47], and the papers [1, 3, 10, 23, 28, 34, 44], and the references cited therein. Due to the diverse applications in science and technology, functional differential equations turn out to be the most essential branch of research in mathematical sciences. The work on non-integer order functional differential equations with state-dependent delay (abbreviate, SDD), is going on last few years. Furthermore, the study for such kind of the differential equations with SDD, we refer the papers [2, 4, 9, 11–13, 15, 16, 21, 41, 42, 45, 46]. An important feature of real-world dynamic processes that has attracted considerable interest by scientists is the effect of abrupt changes. Hereby, “abrupt” is meant in the sense of a multi-scale problem, i.e. the state of a system changes only slowly for a long time interval, and then undergoes a drastic change within a very short time interval. For example, a football may be flying through the air for several seconds before it changes its flight direction within milliseconds during a collision with a goal post. For the mathematical description of this system, the specification of two sets of equations is appropriate: one for the flight phase, and one for the collision phase. Several mathematical models can be developed for the football example. In a simplified setting, the motion of the football could be described by the position and velocity of its center of mass, and the encounter with the goal post could be treated as an inelastic collision (i.e. by an immediate change of the football’s velocity). For the description of the collision of the ball with the goal post leads to differential equations in which the velocity experiences, at the time of the collision, a so-called impulse. For additional information on this concept and pertinent advancements of impulsive differential equations (abbreviated, IDEs), for instance [7, 8, 26, 39]. However, in [22, 32], the authors suggested a new class of abstract IDEs for which the impulses are not instantaneous. In particular, in [22], the authors investigate the new type of IDEs with NII of the form u′ (t) = Au(t) + f (t, u(t)), u(t) = gi (t, u(t)),

t ∈ (si , ti+1 ], i = 0, 1, 2, . . . , N,

t ∈ (ti , si ], i = 1, 2, · · · , N,

(1.1) (1.2)

u(0) = x0 ,

(1.3)

and set up the existence and uniqueness solutions of mild and classical solutions by applying well-known fixed point theorems. From the above system (1.1)-(1.3), we observe that the impulses start instantly at the points ti and their action continue on a finite time interval [ti , si ]. As indicated in [22], there are actually several distinct aspirations for the research of this kind of model. For more details on this theory and on its applications, we suggest the reader to refer [14, 22, 30, 32]. Moreover, Pierri et al. [32] have generalized the results of [22], by employing the theory of analytic 2 604

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semigroup and fractional power of closed operators and proven the existence results of solutions for a class of semilinear IDEs with NII in Banach space. Furthermore, in [17, 19, 20, 25], the authors analyzed the different types of IFDEs with NII in Banach spaces under appropriate fixed point theorem. Recently, Suganya et al.[40] researched the existence results for fractional neutral integro-differential system with SDD and NII in Banach space through the utilization of the Hausdorff’s measures of non-compactness and Darbo-Sadovskii fixed point theorem. On the other hand, the existence results for impulsive fractional neutral integro-differential systems(abbreviated, IFNIDS) with SDD and NII in Bh phase space axioms have not yet been completely examined. This persuade us to explore the existence results of these types of structures with NII in Banach spaces. Motivated by the effort of the aforementioned papers [5, 17, 21, 22], the principle motivation behind this manuscript is to research the existence of mild solutions for an IFNIDS with SDD of the model C



Dtα z(t) − Q1 t, zζ(t,zt ) , Czζ(t,zt ) = A z(t) + Q2 t, zζ(t,zt ) , Czζ(t,zt )
+ Q3 t, zζ(t,zt ) , Czζ(t,zt ) , t ∈ (si , ti+1 ], i = 0, 1, 2, . . . , N, z(t) = gi (t, zζ(t,zt ) ), t ∈ (ti , si ], i = 1, 2, . . . , N, z0 = ϕ(t) ∈ Bh ,

t ∈ (−∞, 0],

(1.4) (1.5) (1.6)

where A denotes the infinitesimal generator of an analytic semigroup {T(t)}t≥0 in a Banach space X;

C Dα t

is the Caputo fractional derivative operator of order α with 0 < α ≤ 1; I = [0, T ] is an operational

interval; Q1 , Q2 , Q3 : I × Bh × Bh → X, ζ : I × Bh → R are appropriate Z functions, and Bh is a phase t

space outlined in next section. The term Czζ(t,zt ) is given by Czζ(t,zt ) =

0

K(t, s)(zζ(s,zs ) )ds, where K ∈

C(D, R+ ) is the set of all positive functions which are continuous on D = {(t, s) ∈ I ×I : 0 ≤ s ≤ t ≤ T } Z t and C ∗ = sup K(t, s)ds. Here 0 = t0 = s0 < t1 ≤ s1 < t2 ≤ s2 < · · · < tN ≤ sN < tN +1 = T, are t∈[0,T ] 0

prefixed numbers, and gi ∈ C((ti , si ] × Bh , X) for all i = 1, 2, . . . , N, is stand for impulsive conditions.

For almost any continuous function z characterized on (−∞, T ] and for almost any t ≥ 0, we

designate by zt the part of Bh characterized by zt (θ) = z(t + θ) for θ ≤ 0. Now zt (·) speaks to the historical backdrop of the state from every θ ∈ (−∞, 0] likely the current time t.

Contrary to the recent results, this paper has some useful features including the integral term in

the involved functions Q1 , Q2 , Q3 and define a suitable mild solution of the model (1.4)-(1.6) with the help of probability density function. Then, based on local Lipschitz conditions of the involved functions, we establish the existence results for IFNIDS with SDD and NII of the problem (1.4)-(1.6) under appropriate fixed point theorem, and the outcomes in [17, 21] might be viewed as the particular situations. We organize the paper as follows. We provide some basis definitions, lemmas and theorems in Section 2 as these are useful for establish our results. Section 3 focuses on the existence of mild solutions for the model (1.4)-(1.6) with the help of the fixed point theorem. Section 4 provides an example to illustrate the acquired abstract concept.

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2

Preliminaries From now on, X represents Banach space with norm k · k, C(I , X) denotes the space of all X-valued

continuous functions on I and L (X) is the Banach space of all linear and bounded operators on X. Furthermore, the notation Br (z, X) stands for the closed ball with center at z and the radius r > 0 in X. Let A : D(A ) ⊂ X → X is the infinitesimal generator of an analytic semigroup of uniformly bounded

linear operator on X. Let 0 ∈ ̺(A ), then it is possible to describe the fractional power A µ , 0 < µ ≤ 1,

as a closed linear operator on its domain D(A µ ). Moreover, the subspace D(A µ ) is dense in X and the expression kzkµ = kA µ zk, z ∈ D(A µ ), defines a norm on D(A µ ). For 0 < ν ≤ µ ≤ 1, Xµ → Xν and the imbedding is compact whenever the resolvent operator of A is compact. Also for every 0 < µ ≤ 1, there exists a positive constant Mµ such that

kA µ T(t)k ≤

Mµ , tµ

0 < t ≤ T.

For additional information about the above preliminaries, we refer to [31, 35]. To portray properly our system, we claim that a function z : [σ, τ ] → X is a normalized piecewise

continuous function on [σ, τ ] if z is piecewise continuous and left continuous on (σ, τ ]. By the symbol PC([σ, τ ]; X), we mean the space of normalized piecewise continuous functions from [σ, τ ] into X.

Specifically, we signify the space PC established by all functions z : [0, T ] → X in ways that z is

+ continuous at t 6= ti , z(t− i ) = z(ti ) and z(ti ) exists, for all i = 1, 2, · · · , N . It is not difficult to find out

that PC is a Banach space having the norm kzkPC = sup kz(s)k. s∈[0,T ]

Once the delay is infinite, then we should talk about the theoretical phase space Bh in a beneficial way. Thus, in this manuscript, we deliberate phase spaces Bh which are same as it was described in [21]. As a result, we bypass the details. We assume that the phase space (Bh , k · kBh ) is a semi-normed linear space of functions mapping

(−∞, 0] into X, and fulfilling the subsequent elementary adages as a result of Fu et al. [18] and Ganga Ram Gautam et al. [21].

If z : (−∞, T ] → X, T > 0, is continuous on I and z0 ∈ Bh , then for every t ∈ I the accompanying

conditions hold:

(P1 ) zt is in Bh ; (P2 ) kz(t)kX ≤ Hkzt kBh ; (P3 ) kzt kBh ≤ E1 (t) sup{kz(s)kX : 0 ≤ s ≤ t} + E2 (t)kz0 kBh , where H > 0 is a constant and

E1 (·) : [0, +∞) → [0, +∞) is continuous, E2 (·) : [0, +∞) → [0, +∞) is locally bounded, and E1 , E2 are independent of z(·).

E2∗ = sup E2 (s).

For our convenience, denote E1∗ = sup E1 (s), s∈I

s∈I

Define the space BT = {z : (−∞, T ] → X such that

z0 ∈ Bh

and the constraint

z|I ∈ PC} ,

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where z|I is the constraint of z to the real compact interval on I . to be a seminorm in BT , it is described by

The function k · kBT

kzkBT = kϕkBh + sup{kz(s)kX : s ∈ [0, T ]}, z ∈ BT . To stay away from the reiterations of a few definitions utilized as a part of this paper we refer the readers: such as for the definition of the fractional integral, Riemann-Liouville fractional integral operator, the generalized Mittag-Leffler special function, Wright-type function and the Caputo’s derivative one can see the papers [17, 35, 40] and the monographs [24, 33, 47]. Currently, we are have the ability to define the mild solution for the problem (1.4)-(1.6). For this, initially we treat the following model C

Dtα z(t) = A z(t) + Q2 (t),

(2.1)

z(0) = z0 ,

(2.2)

where C Dtα and A are just like described in (1.4)-(1.6). By thinking the proofs as in [35, Lemma 6 and Lemma 9], we directly define the mild solution for the model (2.1)-(2.2). Definition 2.1. A function z : I → X is considered to be a mild solution of problem (2.1)-(2.2) if z ∈ C(I , X) fulfills the accompanying integral equation: z(t) = Tα (t)z0 +

Z

t 0

Sα (t − s)Q2 (s)ds,

t ∈ I.

For additional reference about this concept, we suggest the reader to refer[35, 38, 40]. Before we characterize the mild solution for the structure (1.4)-(1.6), finally, we treat the following system C

  Dtα z(t) − Q1 (t, z(t)) = A z(t) + Q2 (t, z(t)) + Q3 (t, z(t)), t ∈ (si , ti+1 ], i = 0, 1, . . . , N,

where

(2.7)

z(t) = gi (t, z(t)), t ∈ (ti , si ], i = 1, 2, · · · , N,

(2.8)

z(0) = z0 ,

(2.9)

C D α , g (t, z(t)) i t

appropriate functions.

and A are same as defined in (1.4)-(1.6) and z0 ∈ X, Q1 , Q2 , Q3 are

We remark that, the impulses in problem (2.7)-(2.9) start abruptly at the points ti and their action continues on the interval [ti , si ]. In addition, the function z takes an abrupt impulse at ti and follows different rules in the two subintervals (ti , si ] and (si , ti+1 ] of the interval (ti , ti+1 ]. At the point si , the function z is continuous. On the results received in the papers [35–37, 43, 48], first we define the mild solution for the system

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(2.7)-(2.9) is given by                               x(t) =                             

Z t A Sα (t − s)Q1 (s, z(s))ds Tα (t)[z0 − Q1 (0, z0 )] + Q1 (t, z(t)) + 0   Z t + Sα (t − s) Q2 (s, z(s)) + Q3 (s, z(s)) ds, t ∈ [0, t1 ], 0

g1 (t, z(t)),

t ∈ (t1 , s1 ],

Z

t

Tα (t − s1 )d1 + Q1 (t, z(t)) + A Sα (t − s)Q1 (s, z(s))ds 0   Z t + Sα (t − s) Q2 (s, z(s)) + Q3 (s, z(s)) ds, t ∈ (s1 , t2 ], 0

··· ,

gi (t, z(t)),

t ∈ (ti , si ], i = 1, 2, · · · , N, Z t Tα (t − si )di + Q1 (t, z(t)) + A Sα (t − s)Q1 (s, z(s))ds 0   Z t + Sα (t − s) Q2 (s, z(s)) + Q3 (s, z(s)) ds, t ∈ (si , ti+1 ], 0

where

Z

si

di = gi (s, z(si )) − Q1 (si , z(si )) − A Sα (si − s)Q1 (s, z(s))ds 0   Z si − Sα (si − s) Q2 (s, z(s)) + Q3 (s, z(s)) ds, i = 1, 2, · · · , N. 0

Remark 2.1. From the discussion in [40], we clearly see that our definition of mild solution fulfills the given model (2.7)-(2.9). In accordance with the above discussion, we determine the mild solution of the model (1.4)-(1.6). Definition 2.2. [40, Definition 2.8] A function z : (−∞, T ] → X is called a mild solution of the

model (1.1)-(1.3) if z0 = ϕ ∈ Bh , z(·)|I ∈ PC and for each s ∈ [0, t) the function A Sα (t −

s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) ) is integrable and

z(t) = Tα (t)[ϕ(0) − Q1 (0, ϕ(0), 0)] + Q1 (t, zζ(t,zt ) , Czζ(t,zt ) ) Z t + A Sα (t − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )ds 0 Z t
+ Sα (t − s)Q2 s, zζ(s,zs ) , Czζ(s,zs ) ds 0 Z t
+ Sα (t − s)Q3 s, zζ(s,zs ) , Czζ(s,zs ) ds, t ∈ [0, t1 ], 0

gi (t, zζ(t,zt ) ),

t ∈ (ti , si ], i = 1, 2, · · · , N,

Tα (t − si )di + Q1 (t, zζ(t,zt ) , Czζ(t,zt ) ) Z t + A Sα (t − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )ds 0

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(2.16)

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+

Z

t

0

+

Z

t

0

where


Sα (t − s)Q2 s, zζ(s,zs ) , Czζ(s,zs ) ds


Sα (t − s)Q3 s, zζ(s,zs ) , Czζ(s,zs ) ds, t ∈ (si , ti+1 ], Z

si

di = gi (si , zζ(si ,zsi ) ) − Q1 (si , zζ(si ,zsi ) , Czζ(si ,zsi ) ) − A Sα (si − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )ds Z si Z 0si

− Sα (si − s)Q2 s, zζ(s,zs ) , Czζ(s,zs ) ds − Sα (si − s)Q3 s, zζ(s,zs ) , Czζ(s,zs ) ds, 0

0

i = 1, 2, · · · , N.

(2.17)

Now, we turn to the statement of Condensing fixed point theorem [21, Theorem 2.9]. Theorem 2.1. Let B be a convex, bounded and closed subset of Banach space X and let P : B → B be

a condensing map. Then P has a fixed point.

3

Existence results In this section, we show and demonstrate the existence of solutions for the model (1.4)-(1.6) under

different fixed point theorems and we consider ϕ ∈ Bh a fixed function, I = [0, T ]. To simplify writing

of the text, in what follows, we assume that 0 ≤ ζ(t, ψ) ≤ t for all ψ ∈ Bh . Presently, we itemizing the subsequent suppositions:

(H1) The function Q1 : I × Bh × Bh → X is continuous and we can find constants β ∈ (0, 1), LQ1 > 0, e Q > 0 and L∗ > 0 in ways that Q1 is Xβ -valued and fulfills the subsequent assumptions: L 1 Q1 e Q k ψ1 − ψ2 kB , k (A )β Q1 (t, ϕ1 , ψ1 ) − (A )β Q1 (t, ϕ2 , ψ2 ) kX ≤ LQ1 kϕ1 − ϕ2 kBh + L 1 h k (A )β Q1 (t, ϕ, ψ) kX ≤ LQ1 k ψ kBh +L∗g ,

where L∗Q1 = max kQ1 (t, 0, 0)kX , t∈I

for all

t∈I

and ψ, ϕ1 , ϕ2 , ψ1 , ψ2 ∈ Bh .

eQ > 0 (H2) The function Q2 : I ×Bh ×Bh → X is continuous and we can find positive constants LQ2 , L 2 and L∗Q2 > 0 in ways that

and

e Q kψ1 − ψ2 kB , kQ2 (t, ϕ1 , ψ1 ) − Q2 (t, ϕ2 , ψ2 )kX ≤ LQ2 kϕ1 − ϕ2 kBh + L 2 h L∗Q2 = max kQ2 (t, 0, 0)kX , t∈I

for all

t∈I

and ϕ1 , ϕ2 , ψ1 , ψ2 ∈ Bh .

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eQ > 0 (H3) The function Q3 : I ×Bh ×Bh → X is continuous and we can find positive constants LQ3 , L 3 and L∗Q3 > 0 in ways that

e Q kψ1 − ψ2 kB , kQ3 (t, ϕ1 , ψ1 ) − Q3 (t, ϕ2 , ψ2 )kX ≤ LQ3 kϕ1 − ϕ2 kBh + L 3 h

and

L∗Q3 = max kQ3 (t, 0, 0)kX , t∈I

for all

t∈I

and ϕ1 , ϕ2 , ψ1 , ψ2 ∈ Bh .

(H4) The function gi : (ti , si ]×Bh → X, i = 1, 2, · · · , N are continuous and there exist positive constants Lgi > 0, L∗gi > 0 such that

kgi (t, ϕ1 ) − gi (t, ϕ2 )kX ≤ Lgi kϕ1 − ϕ2 kBh , kgi (t, ϕ)kX ≤ Lgi kϕkBh + L∗gi ,

where L∗gi = max kgi (t, 0)kX , t∈(ti ,si ]

for all

t ∈ (ti , si ] and ϕ, ϕ1 , ϕ2 ∈ Bh .

(H5) For every r > 0, there exist constants LQ1 (r) > 0, LQ2 (r) > 0, LQ3 (r) > 0 and Lgi (r) > 0 such that k (A )β Q1 (t, ϕt2 , Cψt2 ) − (A )β Q1 (t, ϕt1 , Cψt1 ) k ≤ LQ1 (r)(1 + C ∗ )|t2 − t1 |, k Q2 (t, ϕt2 , Cψt2 ) − Q2 (t, ϕt1 , Cψt1 ) k ≤ LQ2 (r)(1 + C ∗ )|t2 − t1 |, k Q3 (t, ϕt2 , Cψt2 ) − Q3 (t, ϕt1 , Cψt1 ) k ≤ LQ3 (r)(1 + C ∗ )|t2 − t1 |, and kgi (t, ϕt2 ) − gi (t, ϕt1 )k ≤ Lgi (r)|t2 − t1 |,

t, t1 , t2 ∈ I ,

for all function z : (−∞, T ] → X such that z0 = ψ ∈ Bh , z : I → X is continuous and max kz(s)k ≤ r.

0≤s≤T

(H6) The function ζ : I × Bh → [0, ∞) satisfies: (i) For every ψ ∈ Bh , the function t 7→ ζ(t, ψ) is continuous. (ii) There exists a constant Lζ > 0 such that |ζ(t, ϕ2 ) − ζ(t, ϕ1 )| ≤ Lζ kϕ2 − ϕ1 kBh , ϕ1 , ϕ2 ∈ Bh for all t ∈ I . (H7) The following inequalities holds: (i) Let max

1≤i≤N

(

MM0 LQ1 kϕkBh +

ML∗gi



M1−β Γ(β + 1) T αβ · + (M + 1) M0 + Γ(αβ + 1) β "



L∗Q1

M(M + 1)T α ∗ {LQ2 + L∗Q3 } + (E1∗ r + cn ) MLgi Γ(α + 1)   M1−β Γ(β + 1) T αβ eQ ) + (M + 1) M0 + · (LQ1 + C ∗ L 1 Γ(αβ + 1) β #) M(M + 1)T α eQ + L e Q )} ≤ r, for some r > 0. + {(LQ2 + LQ3 ) + C ∗ (L 2 3 Γ(α + 1) +

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(ii) Let "

  M1−β Γ(β + 1)T αβ eQ M(Lgi + 2Lgi (r)Lζ ) + (M + 1) M0 + (LQ1 + C ∗ L 1 1≤i≤N βΓ(αβ + 1) # M(M + 1)T α ∗ e ∗ ∗ e Q ) + (LQ (r) + LQ (r))L } < 1 + LQ1 (r)L ) + {(LQ2 + LQ3 ) + C (LQ2 + L 3 2 3 Γ(α + 1)

Λ=

E1∗

max

be such that 0 ≤ Λ < 1, where 2Lζ (1 + C ∗ ) = L∗ . (H8) The functions Q1 , Q2 , Q3

and gi , i

=

1, 2, · · · , N

are continuous and there exist

µQ1 (t), µ eQ1 (t), µQ2 (t), µ eQ2 (t), µQ3 (t), µ eQ3 (t), µgi ∈ C(I , [0, ∞)) in a way that k(A )β Q1 (t, ϕ1 , ϕ2 )kX ≤ µQ1 (t)kϕ1 kBh + µ eQ1 (t)kϕ2 kBh , kQ2 (t, ϕ1 , ϕ2 )kX ≤ µQ2 (t)kϕ1 kBh + µ eQ2 (t)kϕ2 kBh ,

kQ3 (t, ϕ1 , ϕ2 )kX ≤ µQ3 (t)kϕ1 kBh + µ eQ3 (t)kϕ2 kBh , kgi (t, ϕ)kX ≤ µgi (t)kϕkBh , i = 1, 2, · · · , N,

for all t ∈ I

and

ϕ, ϕj ∈ Bh , j = 1, 2.

Theorem 3.1. Assume that the conditions (H1)-(H7) hold. Then the structure (1.4)-(1.6) has a unique mild solution on (−∞, T ]. Proof. We will transform the model (1.4)-(1.6) into a fixed-point problem. Recognize the operator Υ : BT → BT specified by

(Υx)(t) =

                                                        

Tα (t)[ϕ(0) − Q1 (0, ϕ(0), 0)] + Q1 (t, zζ(t,zt ) , Czζ(t,zt ) ) Z t + A Sα (t − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )ds Z0 t
+ Sα (t − s)Q2 s, zζ(s,zs ) , Czζ(s,zs ) ds Z0 t
+ Sα (t − s)Q3 s, zζ(s,zs ) , Czζ(s,zs ) ds, t ∈ [0, t1 ], 0

gi (t, zζ(t,zt ) ),

t ∈ (ti , si ], i = 1, 2, · · · , N,

Tα (t − si )di + Q1 (t, zζ(t,zt ) , Czζ(t,zt ) ) Z t + A Sα (t − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )ds Z0 t
+ Sα (t − s)Q2 s, zζ(s,zs ) , Czζ(s,zs ) ds Z0 t
+ Sα (t − s)Q3 s, zζ(s,zs ) , Czζ(s,zs ) ds, t ∈ (si , ti+1 ], 0

with di , i = 1, 2, 3, · · · , N , defined by (2.17).

In perspective of [40, Theorem 2.1] and for any z ∈ X and β ∈ (0, 1), we obtain kA Sα (t − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )kX

= kA 1−β Sα (t − s)A β Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )kX 9 611

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(3.1)

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h Z

≤

α

rφα (r)(t − s) A T((t − s) r)dr A Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )

0 X Z ∞  ≤ αM1−β (t − s)αβ−1 r β φα (r)dr kA β Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )kX . ∞

α−1

1−β

i

α

β

(3.2)

0

On the other hand, from

Z

Γ(1 + αq ) , for all q ∈ [0, 1] (see [48, Lemma 3.2]), we have Γ(1 + q) 0 Z ∞ Z ∞ Γ(1 + β) 1 β r φα (r)dr = ψα (r)dr = . (3.3) βα r Γ(1 + αβ) 0 0 ∞

r −q ψα (r)dr =

From (3.2) and (3.3), we conclude that kA Sα (t − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )kX ≤

αM1−β Γ(1 + β) kA β Q1 (s, zζ(s,zs ) , Czζ(s,zs ) )kX . Γ(1 + αβ)(t − s)1−αβ

(3.4)

It is obvious that the function s → A Tα (t − s)Q1 (s, zζ(s,zs ) , Czζ(s,zs ) ) is integrable on [0, t) for every t > 0.

It is clear that the fixed points of the operator Υ are mild solutions of the model (1.4)-(1.6). We express the function ye(·) : (−∞, T ] → X as ye(t) =

 ϕ(t),

t ≤ 0;

T (t)ϕ(0), α

t ∈ I,

then ye0 = ϕ. For every function x ∈ C(I , R) with x(0) = 0, we allocate as x e is characterized by  0, t ≤ 0; x e(t) = x(t), t ∈ I .

If z(·) obeys (2.16), we are able to split it as z(t) = ye(t) + x(t), t ∈ I , which suggests zt = yet + xt , for each t ∈ I and also the function x(·) obeys 
 −Tα (t)Q1 (0, ϕ(0), 0) + Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )    Z t     + A Sα (t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds     Z0 t  
  + Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds     Z0 t  
  Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds, t ∈ [0, t1 ],   +  0  x(t) = gi (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) ), t ∈ (ti , si ], i = 1, 2, · · · , N,      Tα (t − si )dei + Q1 (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) )   Z t      + A Sα (t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds    Z0 t  
   + Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds     Z0 t 
   + Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds, t ∈ (si , ti+1 ], 0

10

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where dei = gi (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )

− Q1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) ) Z si − A Sα (si − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds 0 Z si
− Sα (si − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds Z 0 si
− Sα (si − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds, i = 1, 2, · · · , N. (3.5) 0

Let

BT0

= {x ∈ BT : x0 = 0 ∈ Bh }. Let k · kB0 be the seminorm in BT0 described by T

x ∈ BT0 ,

kxkB0 = sup kx(t)kX + kx0 kBh = sup kx(t)kX , T

t∈I

t∈I

as a result (BT0 , k · kB0 ) is a Banach space. Set Br = {x ∈ BT0 : kxkX ≤ r} for some r ≥ 0; then for T

each r, Br ⊂ BT0 is clearly a bounded closed convex set. For x ∈ Br (0, BT0 ), from phase space axioms

(P1 ) − (P3 ) and along with the above discussion, we receive kxζ(s,xs +eys ) + yeζ(s,xs +eys ) kBh

≤ kxζ(s,xs +eys ) kBh + ke yζ(s,xs +eys ) kBh

≤ E1∗

sup

0≤τ ≤ζ(s,xs +e ys )

kx(τ )kX + E2∗ kx0 kBh + E1∗

sup 0≤τ ≤ζ(s,xs +e ys )

ke y (τ )k + E2∗ ke y0 kBh

≤ E1∗ sup kx(τ )kX + E1∗ kTα (t)kL (X) kϕ(0)k + E2∗ kϕkBh 0≤τ ≤s

≤ E1∗ sup kx(τ )kX + (E1∗ MH + E2∗ )kϕkBh . 0≤τ ≤s

In the event that kxkX < r, r > 0, then kxζ(s,xs +eys ) + yeζ(s,xs+eys ) kBh ≤ E1∗ r + cn ,

s ∈ I,

(3.6)

where cn = (E1∗ MH + E2∗ )kϕkBh . We delimit the operator Υ : BT0 → BT0 by 
 −Tα(t)Q1 (0, ϕ(0), 0) + Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )    Z t     + A Sα (t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds     Z0 t  
  + Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds     Z0 t  
  Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds, t ∈ [0, t1 ],   +  0  (Υz)(t) = gi (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) ), t ∈ (ti , si ], i = 1, 2, · · · , N,      Tα (t − si )dei + Q1 (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) )   Z t      + A Sα (t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds    Z0 t  
   + Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds     Z0 t 
   + Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds, t ∈ (si , ti+1 ], 0

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with dei , i = 1, 2, · · · , N , defined by (3.5).

It is vindicated that the operator Υ has a fixed point if and only if Υ admits a fixed point.

Remark 3.1. As a result, we have the following estimations: I1 = kTα (t)Q1 (0, ϕ(0), 0)kX

Z ∞

α

≤ φα (r)T(t r)drQ1 (0, ϕ, 0)

0

−β

I2

I3

I4

I5

X

β

= Mk(A ) kk(A ) Q1 (0, ϕ, 0)kX h i ≤ MM0 LQ1 kϕkBh + L∗Q1 , where M0 = k(A )−β k.


= Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) X 
−β ≤ k(A ) k k(A )β Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) − (A )β Q1 (t, 0, 0)kX  β + k(A ) Q1 (t, 0, 0)kX   ∗ ∗ ∗ e ≤ M0 (LQ1 + LQ1 C )(E1 r + cn ) + LQ1 .

Z t



= A S (t − s)Q (s, x + y e , Cx + y e )ds α 1 ζ(s,xs +e ys ) ζ(s,xs+e ys ) ζ(s,xs +e ys ) ζ(s,xs+e ys )

0 X  αβ  M1−β Γ(β + 1) t1 e Q C ∗ )(E1∗ r + cn ) + L∗ , t ∈ [0, t1 ]. · (LQ1 + L ≤ Q1 1 Γ(αβ + 1) β

Z t




= S (t − s)Q s, x + y e , Cx + y e ds α 2 ζ(s,xs +e ys ) ζ(s,xs +e ys ) ζ(s,xs +e ys ) ζ(s,xs +e ys )

0 X

Z t Z ∞

≤ α rφα (r)(t − s)α−1 T((t − s)α r)dr

0  0 

× Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) ds

X Z ∞ Z t ≤α rφα (r)dr (t − s)α−1 kT((t − s)α r)kL (X) 0 0

 

×

Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds X   M(t1 )α e Q C ∗ )(E1∗ r + cn ) + L∗ , t ∈ [0, t1 ]. ≤ (LQ2 + L Q2 2 Γ(α + 1)

Z t




= Sα(t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0 X   α M(t1 ) e Q C ∗ )(E ∗ r + cn ) + L∗ , t ∈ [0, t1 ]. ≤ (LQ3 + L 1 Q3 3 Γ(α + 1)

I6 = kgi (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) )kX

≤ kgi (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) ) − gi (t, 0)kX + kgi (t, 0)kX

≤ Lgi (E1∗ r + cn ) + L∗gi , t ∈ (ti , si ].

I7 = kgi (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )kX

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≤ Lgi (E1∗ r + cn ) + L∗gi , t ∈ (si , ti+1 ]. I8 = kQ1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )kX   ∗ ∗ ∗ e ≤ M0 (LQ1 + LQ1 C )(E1 r + cn ) + LQ1 , t ∈ (si , ti+1 ].

Z si

I9 = A Sα (si − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds

0 X   M1−β Γ(β + 1) (si )αβ e Q C ∗ )(E ∗ r + cn ) + L∗ , t ∈ (si , ti+1 ]. ≤ · (LQ1 + L 1 Q1 1 Γ(αβ + 1) β

Z si




I10 = Sα (si − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0 X   M(si )α ∗ ∗ ∗ e Q C )(E r + cn ) + L ≤ (LQ2 + L 1 Q2 , t ∈ (si , ti+1 ]. 2 Γ(α + 1)

Z si




I11 = Sα (si − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0 X   M(si )α e Q C ∗ )(E ∗ r + cn ) + L∗ , t ∈ (si , ti+1 ]. ≤ (LQ3 + L 1 Q3 3 Γ(α + 1)

Z t



I12 = A Sα(t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds

0 X   αβ M1−β Γ(β + 1) (ti+1 ) e Q C ∗ )(E1∗ r + cn ) + L∗ , t ∈ (si , ti+1 ]. ≤ · (LQ1 + L Q1 1 Γ(αβ + 1) β

Z t




I13 =

Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds 0 X   M(ti+1 )α e Q C ∗ )(E1∗ r + cn ) + L∗ , t ∈ (si , ti+1 ]. (LQ2 + L ≤ Q2 2 Γ(α + 1)

Z t




I14 = Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0 X   α M(ti+1 ) e Q C ∗ )(E ∗ r + cn ) + L∗ , t ∈ (si , ti+1 ]. ≤ (LQ3 + L 1 Q3 3 Γ(α + 1)


I15 =

Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )


− Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )

X


≤

Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) − Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) + Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) − Q1





t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )

X

e Q C + LQ (r)L )kx − xk 0 , +L 1 1 BT  = A Sα (t − s) Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )

0 

− Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ) ds

≤ I16




M0 E1∗ (LQ1

Z t

∗

∗

X

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M1−β Γ(β + 1) tαβ e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ [0, t1 ]. · 1 E1∗ (LQ1 + L 1 1 BT Γ(αβ + 1) β

Z t 


= Sα (t − s) Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys )

0 


− Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

≤

I17

X

I18

M(t1 )α ∗ e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ [0, t1 ]. ≤ E (LQ2 + L BT 2 2 Γ(α + 1) 1

Z t 


= Sα (t − s) Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys )

0 


− Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

X

M(t1 )α ∗ e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ [0, t1 ]. E (LQ3 + L ≤ BT 3 3 Γ(α + 1) 1

I19 = kgi (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) ) − gi (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) )kX ≤ E1∗ [Lgi + 2Lgi (r)Lζ ]kx − xkB0 , t ∈ (ti , si ]. T

I20 = kgi (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) ) − gi (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )kX ≤ E1∗ [Lgi + 2Lgi (r)Lζ ]kx − xkB0 , t ∈ (si , ti+1 ]. T

I21 = kQ1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )

− Q1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )kX

I22

e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ (si , ti+1 ]. ≤ M0 E1∗ (LQ1 + L BT 1 1

Z si 

= A Sα (si − s) Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )

0 

− Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ) ds

X

I23

M1−β Γ(β + 1) (si )αβ ∗ e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ (si , ti+1 ]. ≤ · E1 (LQ1 + L 1 1 BT Γ(αβ + 1) β

Z si 


= Sα (si − s) Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys )

0 


− Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

X

)α

M(si e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ (si , ti+1 ]. E ∗ (LQ2 + L 2 BT 2 Γ(α + 1) 1

Z si 


= Sα (si − s) Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys )

0 


− Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

≤ I24

X

I25

M(si )α ∗ e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ (si , ti+1 ]. ≤ E (LQ3 + L 3 3 BT Γ(α + 1) 1

Z t 

= A Sα (t − s) Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )

0

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− Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ) ds

X

M1−β Γ(β + 1) (ti+1 e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ (si , ti+1 ]. · E1∗ (LQ1 + L BT 1 1 Γ(αβ + 1) β

Z t 


= Sα (t − s) Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys )

0 


− Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

≤ I26

)αβ

X

I27

M(ti+1 )α ∗ e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ (si , ti+1 ]. E (LQ2 + L ≤ BT 2 2 Γ(α + 1) 1

Z t 


= Sα (t − s) Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys )

0 


− Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

X

M(ti+1 )α ∗ e Q C ∗ + LQ (r)L∗ )kx − xk 0 , t ∈ (si , ti+1 ]. E (LQ3 + L ≤ 3 3 BT Γ(α + 1) 1

Now, we start proving the main proof of this Theorem. We demonstrate that Υ maps Br (0, BT0 ) into Br (0, BT0 ). For any x(·) ∈ BT0 , by employing Remark 3.1, we sustain k(Υx)(t)kX =

5 X

Ii

i=1

! M1−β Γ(β + 1) tαβ 1 L∗Q1 ≤ MM0 LQ1 kϕkBh + M0 (M + 1) + · Γ(αβ + 1) β " ! αβ M Γ(β + 1) M(t1 )α t 1−β eQ ) + {L∗ + L∗Q3 } + (E1∗ r + cn ) M0 + · 1 (LQ1 + C ∗ L 1 Γ(α + 1) Q2 Γ(αβ + 1) β # M(t1 )α eQ + L eQ )} + {(LQ2 + LQ3 ) + C ∗ (L 2 3 Γ(α + 1) ≤ r,

t ∈ [0, t1 ].

k(Υx)(t)kX = I6 ≤ Lgi (E1∗ r + cn ) + L∗gi , t ∈ (ti , si ], k(Υx)(t)kX =

14 X

i = 1, 2, · · · , N.

Ii

i=7

≤ max

1≤i≤N

(

  M1−β Γ(β + 1)  αβ αβ + M0 (M + 1) + · M(si ) + (ti+1 ) L∗Q1 βΓ(αβ + 1) "   {L∗Q2 + L∗Q3 } M(si )α + (ti+1 )α + (E1∗ r + cn ) MLgi

ML∗gi



M Γ(α + 1)   M1−β Γ(β + 1) αβ αβ eQ ) + M0 (M + 1) + · {M(si ) + (ti+1 ) } (LQ1 + C ∗ L 1 βΓ(αβ + 1) #) M eQ + L eQ )}{M(si )α + (ti+1 )α } ≤ r, t ∈ (si , ti+1 ], + {(LQ2 + LQ3 ) + C ∗ (L 2 3 Γ(α + 1) +

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Then, for all t ∈ I , we conclude that k(Υx)(t)kX (

  M1−β Γ(β + 1) T αβ MM0 LQ1 kϕkBh + ML∗gi + (M + 1) M0 + · L∗Q1 1≤i≤N Γ(αβ + 1) β " M(M + 1)T α ∗ {LQ2 + L∗Q3 } + (E1∗ r + cn ) MLgi + Γ(α + 1)   M1−β Γ(β + 1) T αβ eQ ) · (LQ1 + C ∗ L + (M + 1) M0 + 1 Γ(αβ + 1) β #) M(M + 1)T α eQ + L eQ )} {(LQ2 + LQ3 ) + C ∗ (L + 2 3 Γ(α + 1)

≤ max

≤ r.

Thus, Υ maps the ball Br (0, BT0 ) into itself. Now, we prove that Υ is a contraction on Br (0, BT0 ). Let us consider x, x ∈ Br (0, BT0 ), then from estimations Ij , j = 15, · · · , 27, we sustain " ! αβ α M Γ(β + 1) t 1−β e Q C ∗ + LQ (r)L∗ ) + M(t1 ) k(Υx)(t) − (Υx)(t)kX ≤ E1∗ M0 + (LQ1 + L · 1 1 1 Γ(αβ + 1) β Γ(α + 1) # eQ + L e Q ) + L∗ (LQ (r) + LQ (r))} kx − xk 0 , t ∈ [0, t1 ]. {(LQ2 + LQ3 ) + C ∗ (L B 2 3 2 3 T

k(Υx)(t) − (Υx)(t)kX ≤ E1∗ [Lgi + 2Lgi (r)Lζ ]kx − xkB0 , t ∈ (ti , si ]. T "   M1−β Γ(β + 1) k(Υx)(t) − (Υx)(t)kX ≤ E1∗ max MLgi + M0 (M + 1) + · {M(si )αβ + (ti+1 )αβ } 1≤i≤N βΓ(αβ + 1) e Q + LQ (r)L∗ ) + (LQ1 + C ∗ L 1 1

M eQ + L eQ ) {(LQ2 + LQ3 ) + C ∗ (L 2 3 Γ(α + 1) #

+ L∗ (LQ2 (r) + LQ3 (r))}{M(si )α + (ti+1 )α } kx − xkB0 , t ∈ (si , ti+1 ]. T

As a result, for all t ∈ I , we conclude that k(Υx)(t) − (Υx)(t)kX " ≤

E1∗

max

1≤i≤N



M1−β Γ(β + 1)T αβ MLgi + (M + 1) M0 + βΓ(αβ + 1)



e Q + LQ (r)L∗ ) (LQ1 + C ∗ L 1 1

# M(M + 1)T α ∗ ∗ e e Q ) + L (LQ (r) + LQ (r))} kx − xk 0 + {(LQ2 + LQ3 ) + C (LQ2 + L 2 3 BT 3 Γ(α + 1)

≤ Λkx − xkB0 . T

From the assumption (H7) and in the perspective of the contraction mapping principle, we understand that Υ includes a unique fixed point x ∈ BT0 which is a mild solution of the model (1.4)-(1.6) on (−∞, T ]. The proof is now completed.

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Theorem 3.2. Let the assumption (H8) hold and ( )   M1−β Γ(β + 1)T αβ ∗ ∗ (kµQ1 k∞ + C ke µQ1 k∞ ) < 1. Λ1 = E1 max Mkµgi k∞ + (M + 1) M0 + 1≤i≤N βΓ(αβ + 1) (3.7) Then the system (1.4)-(1.6) has a mild solution on I . Proof. Let Υ : BT0 → BT0 be the operator same as defined in Theorem 3.1. Now, we demonstrate that Υ has a fixed point.

Remark 3.2. From the hypothesis (H8) along with the above discussion, we sustain I28 = kTα (t)Q1 (0, ϕ(0), 0)kX I29

I30

≤ MM0 kµQ1 k∞ kϕkBh .


= Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) X   ∗ ∗ ≤ M0 (E1 r + cn ) kµQ1 k∞ + ke µQ1 k∞ C .

Z t



= A S (t − s)Q (s, x + y e , Cx + y e )ds α 1 ζ(s,xs +e ys ) ζ(s,xs +e ys ) ζ(s,xs +e ys ) ζ(s,xs +e ys )

0

tαβ 1

I32

X

M1−β Γ(β + 1) · Γ(αβ + 1) β

Z t




= S (t − s)Q s, x + y e , Cx + y e ds α 2 ζ(s,xs +e ys ) ζ(s,xs +e ys ) ζ(s,xs +e ys ) ζ(s,xs +e ys )

0 X   M(t1 )α ≤ (E ∗ r + cn ) kµQ2 k∞ + ke µQ2 k∞ C ∗ , t ∈ [0, t1 ]. Γ(α + 1) 1

Z t




= Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0 X   M(t1 )α ∗ ∗ ≤ (E r + cn ) kµQ3 k∞ + ke µQ3 k∞ C , t ∈ [0, t1 ]. Γ(α + 1) 1 ≤

I31

  ∗ ∗ (E1 r + cn ) kµQ1 k∞ + ke µQ1 k∞ C , t ∈ [0, t1 ].

I33 = kgi (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) )kX ≤ kµgi k∞ (E1∗ r + cn ), t ∈ (ti , si ].

I34 = kgi (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )kX ≤ kµgi k∞ (E1∗ r + cn ), t ∈ (si , ti+1 ].

I35 = kQ1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )kX   ∗ ∗ ≤ M0 (E1 r + cn ) kµQ1 k∞ + ke µQ1 k∞ C , t ∈ (si , ti+1 ].

Z si

I36 = A Sα (si − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds

0 X   M1−β Γ(β + 1) (si )αβ ∗ ≤ · (E1 r + cn ) kµQ1 k∞ + ke µQ1 k∞ C ∗ , t ∈ (si , ti+1 ]. Γ(αβ + 1) β

Z si




I37 = Sα (si − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0

X

17

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I38

I39

I40

I41

  M(si )α ∗ ∗ (E r + cn ) kµQ2 k∞ + ke ≤ µQ2 k∞ C , t ∈ (si , ti+1 ]. Γ(α + 1) 1

Z si


= Sα (si − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0 X   M(si )α ≤ (E ∗ r + cn ) kµQ3 k∞ + ke µQ3 k∞ C ∗ , t ∈ (si , ti+1 ]. Γ(α + 1) 1

Z t



= A Sα (t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds

0 X   αβ M1−β Γ(β + 1) (ti+1 ) · (E1∗ r + cn ) kµQ1 k∞ + ke µQ1 k∞ C ∗ , t ∈ (si , ti+1 ]. ≤ Γ(αβ + 1) β

Z t




=

Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds 0 X   M(ti+1 )α ∗ ≤ (E r + cn ) kµQ2 k∞ + ke µQ2 k∞ C ∗ , t ∈ (si , ti+1 ]. Γ(α + 1) 1

Z t




= Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds

0 X   M(ti+1 )α ∗ ≤ (E r + cn ) kµQ3 k∞ + ke µQ3 k∞ C ∗ , t ∈ (si , ti+1 ]. Γ(α + 1) 1

Now, we will give the main proof of this theorem. We remark that (Υx)(t) ⊂ BT0 . Let Br be the set

same as defined in Theorem 3.1, where ( r ≥ max

1≤i≤N

"

MM0 kµQ1 k∞ kϕkBh + (E1∗ r + cn ) Mkµgi k∞

  M1−β Γ(β + 1) T αβ + (M + 1) M0 + · (kµQ1 k∞ + C ∗ ke µQ1 k∞ ) Γ(αβ + 1) β #) M(M + 1)T α {(kµQ2 k∞ + kµQ3 k∞ ) + C ∗ (ke µQ2 k∞ + ke µQ3 k∞ )} . + Γ(α + 1)

It is obvious that Br is closed bounded and convex subset of BT0 . Let x ∈ Br (0, BT0 ) then for t ∈ [0, t1 ], we receive

k(Υx)kB0 ≤ MM0 kµQ1 k∞ kϕkBh + T

+ C ∗ ke µQ1 k∞ ) +

"

(E1∗ r

+ cn )

M1−β Γ(β + 1) (t1 )αβ · M0 + Γ(αβ + 1) β #



(kµQ1 k∞

M(t1 )α {(kµQ2 k∞ + kµQ3 k∞ ) + C ∗ (ke µQ2 k∞ + ke µQ3 k∞ )} . Γ(α + 1)

In the similar manner for t ∈ (ti , si ], we sustain k(Υx)kB0 ≤ kµQ1 k∞ kϕkBh (E1∗ r + cn ), i = 1, 2, · · · , N. T

Similarly, for t ∈ (si , ti+1 ], we find that

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k(Υx)kB0 ≤ max T

1≤i≤N



(E1∗ r

(

+ cn ) Mkµgi k∞

 M1−β Γ(β + 1) αβ αβ · {M(si ) + (ti+1 ) } (kµQ1 k∞ + C ∗ ke µQ1 k∞ ) + M0 (M + 1) + βΓ(αβ + 1) ) M + {(kµQ2 k∞ + kµQ3 k∞ ) + C ∗ (ke µQ2 k∞ + ke µQ3 k∞ )}{M(si )α + (ti+1 )α } . Γ(α + 1)

From this, we notice that kΥxkB0 ≤ r for every t ∈ I . Therefore, Υ(Br ) ⊆ Br . In order to utilizing T

the Theorem 2.2, we have to prove that the operator Υ is a condensing operator. For this, we split Υ by Υ = Υ1 + Υ2 , where 
 −Tα (t)Q1 (0, ϕ(0), 0) + Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )    Z t     + A Sα (t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs+eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) )ds, t ∈ [0, t1 ],    0      g (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) ), t ∈ (ti , si ], i = 1, 2, · · · , N,   i      T (t − s ) eζ(si ,xsi +eysi ) )  α i gi (si , xζ(si ,xsi +e ysi ) + y (Υ1 x)(t) =   −Q1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )    Z si      − A Sα (si − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) )ds    0     +Q1 (t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) )   Z t      + A Sα (t − s)Q1 (s, xζ(s,xs +eys ) + yeζ(s,xs+eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) )ds, t ∈ (si , ti+1 ], 0

and

(Υ2 x)(t) =

                                            

Z

t


Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs+eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) ds 0Z t
+ Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds, t ∈ [0, t1 ], 0

0,

t ∈ (ti , si ], i = 1, 2, · · · , N,  Z si
−Tα (t − si ) Sα (si − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds 0  Z si
− Sα (si − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds Z t0
+ Sα (t − s)Q2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds Z0 t
+ Sα (t − s)Q3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) ds, t ∈ (si , ti+1 ]. 0

Firstly, we show that Υ1 is continuous, so we consider a sequence xn → x ∈ Br . In perspective of (3.1),

we notice that

kxnζ(t,xn +eyt ) + yeζ(t,xnt+eyt ) kBh ≤ E1∗ r + cn . t

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Remark 3.3. By utilizing the hypothesis (H8) and Definition 2.2, we receive: (i) For every t ∈ [0, t1 ], we obtain   n n Q1 t, xζ(t,xn +eyt ) + yeζ(t,xnt +eyt ) , Cxζ(t,xn +eyt ) + yeζ(t,xnt+eyt ) t t   → Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) and since

 

n

Q1 t, xn n n n eζ(t,xt +eyt ) , Cxζ(t,xn +eyt ) + yeζ(t,xt +eyt ) ζ(t,xt +e yt ) + y

t  

−Q1 t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )

≤ 2I29 . X

(ii) For every t ∈ [0, t1 ], we obtain   n n A Sα (t − s)Q1 s, xζ(s,xns+eys ) + yeζ(s,xns+eys ) , Cxζ(s,xns +eys ) + yeζ(s,xns+eys )   → A Sα (t − s)Q1 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) and since

Z t   

n n

, Cx + y e n A S (t − s) Q s, x + y e n α 1 ys ) ys ) ζ(s,xs +e ζ(s,xs +e ζ(s,xn ys ) ζ(s,xn ys )

s +e s +e 0  

−Q1 s, xζ(s,xs +eys ) + yeζ(s,xs+eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) ds

≤ 2I30 . X

(iii) For each t ∈ (ti , si ], we sustain   n n gi t, xζ(t,xn +eyt ) + yeζ(t,xnt +eyt ) , Cxζ(t,xn +eyt ) + yeζ(t,xnt+eyt ) t t   → gi t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt ) and since

 

n n

gi t, x eζ(t,xnt +eyt ) , Cxζ(t,xn +eyt ) + yeζ(t,xnt +eyt ) ζ(t,xn yt ) + y

t +e t  

−gi t, xζ(t,xt +eyt ) + yeζ(t,xt +eyt ) , Cxζ(t,xt +eyt ) + yeζ(t,xt +eyt )

≤ 2I33 . X

(iv) For every t ∈ (si , ti+1 ], we receive

gi (si , xnζ(si ,xns +eys ) + yeζ(si ,xns +eysi ) ) → gi (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) ) i

i

i

and since



gi (si , xn

≤ 2I34 . n +e + y e ) − g (s , x + y e ) n i i ζ(s ,x y ) ζ(s ,x +e y ) ζ(s ,x +e y ) s s s s s ζ(s ,x +e y ) i i i

si si i si i i i i i X

20

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(v) For all t ∈ (si , ti+1 ], we get Q1 (si , xnζ(si ,xns +eys ) + yeζ(si ,xns +eysi ) , Cxnζ(si ,xns +eys ) + yeζ(si ,xns +eysi ) ) i

and since

i

i

i

i

i

→ Q1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )



Q1 (si , xn eζ(si ,xns +eysi ) , Cxnζ(si ,xns +eys ) + yeζ(si ,xns +eysi ) ) ζ(si ,xn ysi ) + y

si +e i i i i

−Q1 (si , xζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) , Cxζ(si ,xsi +eysi ) + yeζ(si ,xsi +eysi ) )

≤ 2I35 . X

(vi) For every t ∈ (si , ti+1 ], we obtain   n n A Sα (si − s)Q1 s, xζ(s,xns+eys ) + yeζ(s,xns+eys ) , Cxζ(s,xns +eys ) + yeζ(s,xns+eys )   → A Sα (si − s)Q1 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) and since

Z



   n n A Sα (si − s) Q1 s, xζ(s,xns +eys ) + yeζ(s,xns+eys ) , Cxζ(s,xns +eys ) + yeζ(s,xns+eys ) 0  

−Q1 s, xζ(s,xs +eys ) + yeζ(s,xs+eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) ds

≤ 2I36 . si

X

(vii) For every t ∈ (si , ti+1 ], we find that   n n A Sα (t − s)Q1 s, xζ(s,xns+eys ) + yeζ(s,xns+eys ) , Cxζ(s,xns +eys ) + yeζ(s,xns+eys )   → A Sα (t − s)Q1 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) and since

Z t   

n n

n n A S (t − s) Q s, x + y e , Cx + y e α 1 ζ(s,xs +e ys ) ζ(s,xs +e ys ) ζ(s,xn ys ) ζ(s,xn ys )

s +e s +e 0  

−Q1 s, xζ(s,xs +eys ) + yeζ(s,xs+eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) ds

≤ 2I39 . X

From the Remark 3.3, for all [0, t1 ], we have

k(Υ1 xn ) − (Υ1 x)kB0 ≤ 2(I29 + I30 ), T

and for every t ∈ (si , ti+1 ], we receive n

k(Υ1 x ) − (Υ1 x)kB0

T

  ≤ 2 M(I34 + I35 + I36 ) + I29 + I39 .

Since the functions Q1 and gi , i = 1, 2, · · · , N are continuous, so we conclude that k(Υ1 xn ) − (Υ1 x)kB0 → 0 T

as

n → ∞.

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Hence Υ1 is continuous. Next, we prove that the operator Υ1 is contraction on Br (0, BT0 ). Indeed, let x, x ∈ Br (0, BT0 ), for

[0, t1 ], we sustain n

k(Υ1 x ) − (Υ1 x)kB0 ≤ T

E1∗ (kµQ1 k∞



M1−β Γ(β + 1) (t1 )αβ + C ke µQ1 k∞ ) M0 + · Γ(αβ + 1) β ∗



kx − xkB0 , T

and for t ∈ (ti , si ], we get k(Υ1 xn ) − (Υ1 x)kB0 ≤ E1∗ kµgi k∞ kx − xkB0 , T

T

and for every t ∈ (si , ti+1 ], we sustain k(Υ1 xn ) − (Υ1 x)kB0 T ( ≤ E1∗ max

1≤i≤N

Mkµgi k∞

)  M1−β Γ(β + 1) αβ αβ ∗ + M0 (M + 1) + · {M(si ) + (ti+1 ) } (kµQ1 k∞ + C ke µQ1 k∞ ) kx − xkB0 . T βΓ(αβ + 1) 

Then for all t ∈ I , we find that k(Υ1 xn ) − (Υ1 x)kB0 T ( ≤

E1∗

max

1≤i≤N



M1−β Γ(β + 1)T αβ Mkµgi k∞ + (M + 1) M0 + βΓ(αβ + 1)



)

∗

(kµQ1 k∞ + C ke µQ1 k∞ ) kx − xkB0

T

≤ Λ1 kx − xkB0 . T

Since Λ1 < 1, which implies that Υ1 is a contraction. Next, we prove that the operator Υ2 is completely continuous on Br (0, BT0 ). First, we prove Υ2 is continuous, so consider a sequence xn → x ∈ Br . By applying the condition (H8), I31 , I32 , I37 , I38 , I40 , I41

and in perspective of Remark 3.3, for all t ∈ [0, t1 ], we get

k(Υ2 xn ) − (Υ2 x)kB0 ≤ 2(I31 + I32 ), T

and for all t ∈ (si , ti+1 ], we receive

  k(Υ2 xn ) − (Υ2 x)kB0 ≤ 2 M(I37 + I38 ) + I40 + I41 . T

Since the functions Q2 and Q3 are continuous, so we conclude that k(Υ2 xn ) − (Υ2 x)kB0 → 0 T

as

n → ∞.

Hence Υ2 is continuous. Next, we show that the operator Υ2 maps bounded sets into bounded set in Br . It is enough to show that there exists a positive constant Λ2 such that for each x ∈ Br one has kΥ2 xkB0 ≤ Λ2 . For all T

t ∈ I , we obtain kΥ2 xkB0

T

  M(M + 1)T α ∗ ∗ ≤ (E1 r + cn ) (kµQ2 k∞ + kµQ3 k∞ ) + C (ke µQ2 k∞ + ke µQ3 k∞ ) ≤ Λ2 . Γ(α + 1) 22 624

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Finally, we show that Υ2 is a family of equi-continuous functions. Let τ1 , τ2 ∈ [0, t1 ] be such that

0 ≤ τ1 < τ2 ≤ t1 . Then

k(Υ2 x)(τ2 ) − (Υ2 x)(τ1 )kX Z τ1
≤ kSα (τ2 − s) − Sα (τ1 − s)kL (X) kQ2 s, xζ(s,xs +eys ) + yeζ(s,xs+eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs+eys ) kX ds 0 Z τ2
+ kSα (τ2 − s)kL (X) kQ2 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) kX ds Zτ1τ1
+ kSα (τ2 − s) − Sα (τ1 − s)kL (X) kQ3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) kX ds Z0 τ2
+ kSα (τ2 − s)kL (X) kQ3 s, xζ(s,xs +eys ) + yeζ(s,xs +eys ) , Cxζ(s,xs +eys ) + yeζ(s,xs +eys ) kX ds τ1

≤

(E1∗ r

  Z ∗ + cn ) (kµQ2 k∞ + kµQ3 k∞ ) + C (ke µQ2 k∞ + ke µQ3 k∞ )

τ1

kSα (τ2 − s) − Sα (τ1 − s)kL (X) ds

0

! M(τ2 − τ1 )α + , Γ(α + 1)

and for all τ1 , τ2 ∈ (si , ti+1 ], we have k(Υ2 x)(τ2 ) − (Υ2 x)(τ1 )kX   Z ∗ ∗ ≤ (E1 r + cn ) (kµQ2 k∞ + kµQ3 k∞ ) + C (ke µQ2 k∞ + ke µQ3 k∞ ) ! M(τ2 − τ1 )α M(si )α + + kTα (τ2 − si ) − Tα (τ1 − si )kL (X) . Γ(α + 1) Γ(α + 1)

0

τ1

kSα (τ2 − s) − Sα (τ1 − s)kL (X) ds

Since Tα and Sα are strongly continuous, so lim kSα (τ2 − s) − Sα (τ1 − s)kL (X) = 0, lim kTα (τ2 − τ2 →τ1

τ2 →τ1

si ) − Tα (τ1 − si )kL (X) = 0. From this, we conclude that k(Υ2 x)(τ2 ) − (Υ2 x)(τ1 )kX → 0 as τ2 → τ1 . This proves that Υ2 is a family of equi-continuous functions. Hence, the operator Υ2 is completely continuous. Therefore the operator Υ = Υ1 + Υ2 is a condensing operator from BT0 into BT0 , where Υ1 is contraction and Υ2 is completely continuous. Finally, from Theorem 2.2, we infer that there exists a mild solution of the structure (1.4)-(1.6). This completes the proof.

4

Example To prove our theoretical results, we treat the IFNIDS with SDD of the model " Z t u(s − ζ1 (s)ζ2 (ku(s)k), z) α Dt u(t, z) + e2(s−t) ds 49 −∞ # Z t Z s u(τ − ζ (τ )ζ (ku(τ )k), z) ∂2 1 2 + sin(t − s) e2(τ −s) dτ ds = 2 u(t, z) 36 ∂z 0 −∞ Z t u(s − ζ1 (s)ζ2 (ku(s)k), z) + e2(s−t) ds 9 −∞

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+

Z Z

t

sin(t − s)

0 t

Z

s

e2(τ −s)

−∞

u(τ − ζ1 (τ )ζ2 (ku(τ )k), z) dτ ds 25

u(s − ζ1 (s)ζ2 (ku(s)k), z) ds 64 −∞ Z t Z s N [ u(τ − ζ1 (τ )ζ2 (ku(τ )k), z) + sin(t − s) e2(τ −s) dτ ds, (t, z) ∈ (si , ti+1 ] × [0, π], (4.1) 16 0 −∞

+

e2(s−t)

i=1

u(t, 0) = 0 = u(t, π),

t ∈ [0, T ],

(4.2)

u(t, z) = ϕ(t, z), t ≤ 0, z ∈ [0, π], Z t u(s − ζ1 (s)ζ2 (ku(s)k), z) u(t, z) = e2(s−t) ds, (t, z) ∈ (ti , si ] × [0, π], i = 1, 2, . . . , N, 81 −∞ where

C Dq t

(4.3) (4.4)

is Caputo’s fractional derivative of order 0 < q < 1, 0 = t0 = s0 < t1 < t2 < · · ·
0. 1

We can find a constant M > 0 in a way that k T(t) k≤ M. If we fix β = 12 , then the operator (A ) 2

is given by

1

(A ) 2 w = 1 2

in which (D(A ) ) =

(

∞ X

n=1

ω(·) ∈ X :

∞ X

n=1

1

−n2 hw, wn iwn , w ∈ (D(A ) 2 ), )

2

n hω, wn iwn ∈ X . Then

Sα (t)w = α =

Z

∞ 0

∞ X

n=1

For the phase space, we choose h =

rφα (r)tα−1 T(tα r)dr,

Eα,α (−n2 tα )hw, wn iwn , w ∈ X.

e2s ,

s < 0, then l =

determine kϕkBh =

Z

Z

0

h(s)ds = −∞

1 < ∞, for t ≤ 0 and 2

0 −∞

h(s) sup kϕ(θ)kL2 ds. θ∈[s,0]

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Hence, for (t, ϕ) ∈ [0, T ] × Bh , where ϕ(θ)(z) = ϕ(θ, z), (θ, z) ∈ (−∞, 0] × [0, π]. Set u(t)(z) = u(t, z),

ζ(t, ϕ) = ζ1 (t)ζ2 (kϕ(0)k),

we have Q1 (t, ϕ, H ϕ)(z) =

Z

0

e2(s)

−∞ 0

Z

ϕ ds + (H ϕ)(z), 49

ϕ fϕ)(z), e2(s) ds + (H 9 −∞ Z 0 ϕ c Q3 (t, ϕ, H ϕ)(z) = e2(s) ds + (Hcϕ)(z), 64 −∞ Z 0 ϕ gi (t, ϕ)(z) = e2(s) ds, i = 1, 2, · · · , N, 81 −∞

Q2 (t, ϕ, Hfϕ)(z) =

where (H ϕ)(z) =

Z

t

sin(t − s)

0

(Hfϕ)(z) = (Hcϕ)(z) =

Z

t

sin(t − s)

0

Z

t

0

sin(t − s)

Z Z Z

0 −∞ 0 −∞ 0 −∞

e2(τ )

ϕ dτ ds, 36

e2(τ )

ϕ dτ ds, 25

e2(τ )

ϕ dτ ds, 16

then using these configurations, the system (4.1)-(4.4) is usually written in the theoretical form of design (1.4)-(1.6). To treat this system we assume that ζi : [0, ∞) → [0, ∞), i = 1, 2 are continuous. Now, we can see

that for t ∈ [0, 1], ϕ, ϕ ∈ Bh , we have

2 Z t Z 0



ϕ

2(τ ) ϕ ds + − s)k e dτ ds dz k(A ) Q1 (t, ϕ, H ϕ)kX ≤ e k sin(t



49 36 0 0 −∞ −∞ 2 ! 12 Z π Z 0 Z 0 1 1 ≤ e2(s) sup kϕkds + e2(s) sup kϕkds dz 49 −∞ 36 −∞ 0 √ √ π π ≤ kϕkBh + kϕkBh 49 36 eQ kϕkB , ≤ LQ1 kϕkBh + L 1 h Z

1 2

eQ = where LQ1 + L 1 1

√ 85 π 1764 ,

π

Z

0

2(s)

and

1

k(A ) 2 Q1 (t, ϕ, H ϕ) − (A ) 2 Q1 (t, ϕ, H ϕ)kX



2 ! 21 Z t Z π Z 0 Z 0



ϕ ϕ ϕ ϕ

≤ e2(s) k sin(t − s)k e2(τ )

49 − 49 ds +

36 − 36 dτ ds dz 0 −∞ 0 −∞ 2 ! 12 Z π Z 0 Z 0 1 1 ≤ e2(s) sup kϕ − ϕkds + e2(s) sup kϕ − ϕkds dz 49 −∞ 36 −∞ 0 25 627

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√ √ π π ≤ kϕ − ϕkBh + kϕ − ϕkBh 49 36 e Q kϕ − ϕkB . ≤ LQ1 kϕ − ϕkBh + L 1 h

Similarly, we conclude kQ2 (t, ϕ, Hfϕ)kL2

eQ = where LQ2 + L 2

Z

2 Z 0 Z t



2(s) ϕ 2(τ ) ϕ ≤ e k sin(t − s)k e

ds +

dτ ds dz 9 25 0 −∞ −∞ 0 2 ! 12 Z π Z 0 Z 0 1 1 2(s) 2(s) ≤ e sup kϕkds + e sup kϕkds dz 9 −∞ 25 −∞ 0 √ √ π π ≤ kϕkBh + kϕkBh 9 25 e Q kϕkB , ≤ LQ2 kϕkBh + L 2 h Z

√ 34 π 225 ,

π

0

!1 2

and

fϕ)kL2 kQ2 (t, ϕ, Hfϕ) − Q2 (t, ϕ, H



2 ! 12 Z π Z 0 Z 0 Z t



ϕ ϕ ϕ ϕ

e2(τ ) ≤ e2(s) k sin(t − s)k

25 − 25 dτ ds dz

9 − 9 ds + −∞ 0 −∞ 0 2 ! 21 Z π Z 0 Z 0 1 1 ≤ e2(s) sup kϕ − ϕkds + e2(s) sup kϕ − ϕkds dz 9 25 0 −∞ −∞ √ √ π π ≤ kϕ − ϕkBh + kϕ − ϕkBh 9 25 eQ kϕ − ϕkB . ≤ LQ2 kϕ − ϕkBh + L 2 h

Correspondingly, we have kQ3 (t, ϕ, Hcϕ)kL2

eQ = where LQ3 + L 3

2 Z t Z 0



2(s) ϕ 2(τ ) ϕ ≤ e k sin(t − s)k e

ds +

dτ ds dz 64 16 0 −∞ −∞ 0 2 ! 12 Z π Z 0 Z 0 1 1 ≤ e2(s) sup kϕkds + e2(s) sup kϕkds dz 64 16 0 −∞ −∞ √ √ π π ≤ kϕkBh + kϕkBh 64 16 e Q kϕkB , ≤ LQ3 kϕkBh + L 3 h

√ 80 π 1024 ,

Z

π

Z

0

!1

and

kQ3 (t, ϕ, Hcϕ) − Q3 (t, ϕ, Hcϕ)kL2



2 ! 21 Z t Z π Z 0 Z 0



ϕ ϕ ϕ ϕ

≤ e2(s) k sin(t − s)k e2(τ )

64 − 64 ds +

16 − 16 dτ ds dz 0 −∞ 0 −∞ 2 ! 12 Z π Z 0 Z 0 1 1 ≤ e2(s) sup kϕ − ϕkds + e2(s) sup kϕ − ϕkds dz 64 −∞ 16 −∞ 0 26 628

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√ √ π π ≤ kϕ − ϕkBh + kϕ − ϕkBh 64 16 e Q kϕ − ϕkB . ≤ LQ3 kϕ − ϕkBh + L 3 h

Finally,

kgi (t, ϕ)kX = ≤

Z

π

0

Z

π

0

Z 

2 2(s) ϕ e

ds dz 81 −∞ 0

1 81

Z

0

−∞

e2(s) sup kϕkds

!1 2

2

!1 2

dz

≤ Lgi kϕkBh , i = 1, 2, · · · , N, where Lgi =

√

π 81 ,

and kgi (t, ϕ) − gi (t, ϕ)kX

2 ! 21

Z π Z 0

ϕ 2(s) ϕ

= e

81 − 81 ds dz 0 −∞ ≤

Z

0

π



1 81

Z

0

2(s)

e

−∞

sup kϕ − ϕkds

2

dz

!1 2

≤ Lgi kϕ − ϕkBh . Therefore the conditions (H1)-(H6) are all fulfilled. Furthermore, we assume that E1 ∗ = 1, M =

1, M0 = 1, M 1 = 1, α = 12 , T = 1, C ∗ = 1 and Lζ = 1. Moreover, the appropriate values of the constants 2

Lgi (r), LQ1 (r), LQ2 (r) and LQ3 (r), obtain " Λ=

E1∗

max

1≤i≤N



M1−β Γ(β + 1)T αβ M(Lgi + 2Lgi (r)Lζ ) + (M + 1) M0 + βΓ(αβ + 1)



eQ (LQ1 + C ∗ L 1

# α M(M + 1)T eQ + L e Q ) + (LQ (r) + LQ (r))L∗ } < 1 + LQ1 (r)L∗ ) + {(LQ2 + LQ3 ) + C ∗ (L 2 3 2 3 Γ(α + 1)

be such that 0 ≤ Λ < 1, where 2Lζ (1 + C ∗ ) = L∗ . Thus the condition (H7) holds. Hence by Theorem

3.1, we realize that the system (4.1)-(4.4) has a unique mild solution on [0,1].

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Characterizations of positive implicative superior ideals induced by superior mappings Seok Zun Song1,† , Young Bae Jun2 and Hee Sik Kim3,∗ 1

2 3

Department of Mathematics, Jeju National University, Jeju 690-756, Korea Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea

Research Institute for Natural Sci., Department of Mathematics, Hanyang University, Seoul, 04763, Korea

Abstract. The notion of positive implicative superior ideals of BCK-algebras is introduced, and their properties are investigated. Relations between a superior ideal and a positive implicative superior ideal in BCK-algebras are studied, and conditions for a superior ideal to be a positive implicative superior ideal are provided. Characterizations of positive implicative superior ideals induced by superior mappings are discussed.

1. Introduction Algebras have played an important role in pure and applied mathematics and have its comprehensive applications in many aspects including dynamical systems and genetic code of biology (see [1], [2], [7], and [12]). Starting from the four DNA bases order in the Boolean lattice, S´aanchez et al. [11] proposed a novel Lie Algebra of the genetic code which shows strong connections among algebraic relationship, codon assignments and physicochemical properties of amino acids. A BCK/BCI-algebra (see [3, 4, 10]) is an important class of logical algebras introduced by Is´eki and was extensively investigated by several researchers. Jun and Song [5] introduced the notion of BCK-valued functions and investigated several properties. They established block-codes by using the notion of BCK-valued functions, and shown that every finite BCK-algebra determines a block-code. In [6], Jun and Song introduced the notion of superior mapping by using partially ordered sets. Using the superior mapping, they introduced the concept of superior subalgebras and (commutative) superior ideals in BCK/BCI-algebras, and investigated related properties. They also discussed relations among a superior subalgebra, a superior ideal and a commutative superior ideal. In this paper, we introduce the notion of positive implicative superior ideals of BCK-algebras, and investigate properties. We investigate relations between a superior ideal and a positive implicative superior ideal in BCK-algebras. We provide conditions for a superior ideal to be a positive implicative superior ideal, and discuss characterizations of positive implicative superior ideals. 0

2010 Mathematics Subject Classification: 06F35, 03G25, 06A11. Keywords: superior mapping, superior subalgebra, superior ideal, positive implicative superior ideal. ∗ Correspondence: +82 2 2220 0897 (Phone) 0 E-mail: [email protected]; [email protected]; [email protected] 0

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Seok Zun Song1,† , Young Bae Jun2 and Hee Sik Kim3,∗ 2. Preliminaries We display basic definitions and properties of BCK/BCI-algebras that will be used in this paper. For more details of BCK/BCI-algebras, we refer the reader to [3], [8], [9] and [10]. An algebra L := (L; ∗, 0) is called a BCI-algebra if it satisfies the following conditions: (I) (∀x, y, z ∈ L) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (II) (∀x, y ∈ L) ((x ∗ (x ∗ y)) ∗ y = 0), (III) (∀x ∈ L) (x ∗ x = 0), (IV) (∀x, y ∈ L) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). If a BCI-algebra L satisfies the following identity: (V) (∀x ∈ L) (0 ∗ x = 0), then L is called a BCK-algebra. A BCK-algebra L is said to be positive implicative if it satisfies: (∀x, y, z ∈ L) ((x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z)) .

(2.1)

Any BCK/BCI-algebra L satisfies the following conditions: (∀x ∈ L) (x ∗ 0 = x) ,

(2.2)

(∀x, y, z ∈ L) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) ,

(2.3)

(∀x, y, z ∈ L) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) ,

(2.4)

(∀x, y, z ∈ L) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y)

(2.5)

where x ≤ y if and only if x ∗ y = 0. A subset A of a BCK/BCI-algebra L is called an ideal of L if it satisfies: 0 ∈ A,

(2.6)

(∀x, y ∈ L) (x ∗ y ∈ A, y ∈ A ⇒ x ∈ A) .

(2.7)

A subset A of a BCK-algebra L is called a positive implicative ideal of L if it satisfies (2.6) and (∀x, y, z ∈ L) ((x ∗ y) ∗ z ∈ A, y ∗ z ∈ A ⇒ x ∗ z ∈ A) .

(2.8)

Let L be a set of parameters and let U be a partially ordered set with the partial ordering  and the first element e. For a mapping f˜ : L → P(U ), we consider the mapping ( ||f˜|| : L → U, x 7→

sup f˜(x) if ∃ sup f˜(x), e otherwise,

(2.9)

    which is called the superior mapping of L with respect to f˜, L . In this case, we say that f˜, L is a pair on (U, ) (see [6]).

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Characterizations of positive implicative superior ideals Definition 2.1 ([6]). Let L := (L, ∗, 0) be a BCK/BCI-algebra. By a superior ideal on (L, f˜), we mean   the superior mapping ||f˜|| of L with respect to f˜, L which satisfies the following conditions:   (∀x ∈ L) ||f˜||(0)  ||f˜||(x) ,   (∀x, y ∈ L) ||f˜||(x)  sup{||f˜||(x ∗ y), ||f˜||(y)} .

(2.10) (2.11)

Proposition 2.2 ([6]). If ||f˜|| is a superior ideal on (L, f˜), then ||f˜||(x)  ||f˜||(y) for all x, y ∈ L with x ≤ y.

3. Positive implicative superior ideals In what follows, let L := (L, ∗, 0) be a BCK-algebra unless otherwise specified, where L is a set of parameters. Definition 3.1. By a positive implicative superior ideal on (L, f˜), we mean the superior mapping ||f˜||  of L with respect to f˜, L which satisfies the condition (2.10) and   (∀x, y, z ∈ L) ||f˜||(x ∗ z)  sup{||f˜||((x ∗ y) ∗ z), ||f˜||(y ∗ z)} .

(3.1)

Example 3.2. Let L = {0, 1, 2, 3} be a set with a binary operation ‘∗’ shown in Table 1.

Table 1. Cayley table for the binary operation ‘∗’ ∗ 0 1 2 3

0 0 1 2 3

1 0 0 2 3

2 0 0 0 3

3 0 1 2 0

Then L := (L, ∗, 0) is a BCK-algebra (see [10]). Let U = {a, b, c, d, e, f } be ordered as pictured in Figure A3. e r rf HH H r Hr d c  @ b @r r a Figure A3

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Seok Zun Song1,† , Young Bae Jun2 and Hee Sik Kim3,∗   (1) Let f˜, L be a pair on (U, ) where f˜ is given as follows:  {a, b}     {a, d, f } f˜ : L → P(U ), x 7→  {b, c, d, f }    {a, b, c}

if if if if

x = 0, x = 1, x = 2, x = 3.

  Then the superior mapping ||f˜|| of L with respect to f˜, L is described as follows: ||f˜||(0) = b, ||f˜||(1) = ||f˜||(2) = f and ||f˜||(3) = c. By routine calculations, we know that ||f˜|| is a positive implicative superior ideal on (L, f˜). (2) Let (˜ g , L) be a pair on (U, ) where g˜ is given as follows:   if x = 0,  {a, b} g˜ : L → P(U ), x 7→ {b, c, d, e} if x = 3,   {b, c, d, f } if x ∈ {1, 2}. Then the superior mapping ||˜ g || of L with respect to (˜ g , L) is described as follows: ||˜ g ||(0) = b, ||f˜||(1) = ||f˜||(2) = f and ||f˜||(3) = e. It is not a positive implicative superior ideal on (L, f˜) since there does not exist sup{||f˜||((3 ∗ 2) ∗ 1), ||f˜||(2 ∗ 1)} because ||f˜||((3 ∗ 2) ∗ 1) = e and ||f˜||(2 ∗ 1) = f are noncomparable. Example 3.3. Let U = {a, b, c, d, e, f } be ordered as pictured in Figure B3. rd a r @

@r b @

re @

@r f

@r c

Figure B3 Let L = {0, 1, 2, 3, 4} be a set with a binary operation ‘∗’ shown in Table 2.

Table 2. Cayley table for the binary operation ‘∗’ ∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 1 4

2 0 1 0 3 4

637

3 0 0 2 0 4

4 0 1 0 3 0

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Characterizations of positive implicative superior ideals   Then L := (L, ∗, 0) is a BCK-algebra (see [10]). Let f˜, L be a pair on (U, ) where f˜ is defined by   {b, c} if x = 0,      if x = 1,  {a, b, e} f˜ : L → P(U ), x 7→ {b, e, f } if x = 2,    {a, c, e, f } if x = 3,     {d, e} if x = 4.   Then the superior mapping of L with respect to f˜, L is described as follows: ||f˜||(0) = b, ||f˜||(1) = c, ||f˜||(2) = e, ||f˜||(3) = c and ||f˜||(4) = d, and it is neither a superior ideal nor a positive implicative superior ideal on (L, f˜).   Theorem 3.4. Let f˜, L be a pair on (U, ). If ||f˜|| is a positive implicative superior ideal on (L, f˜), then the nonempty set ||f˜||α := {x ∈ L | ||f˜||(x)  α} is a positive implicative ideal of L for all α ∈ U . Proof. Let α ∈ U be such that ||f˜||α 6= ∅. Clearly 0 ∈ ||f˜||α . Let x, y, z ∈ L be such that (x∗y)∗z ∈ ||f˜||α and y ∗ z ∈ ||f˜||α . Then ||f˜||((x ∗ y) ∗ z)  α and ||f˜||(y ∗ z)  α. It follows from (3.1) that ||f˜||(x ∗ z)  sup{||f˜||((x ∗ y) ∗ z), ||f˜||(y ∗ z)}  α. Thus x ∗ z ∈ ||f˜||α , and therefore ||f˜||α is a positive implicative ideal of L.



  Corollary 3.5. Let f˜, L be a pair on (U, ). If ||f˜|| is a positive implicative superior ideal on (L, f˜), then the set A := {x ∈ L | ||f˜||(x) = ||f˜||(0)} is a positive implicative ideal of L. Theorem 3.6. Every positive implicative superior ideal is a superior ideal. Proof. Let ||f˜|| be a positive implicative superior ideal on (L, f˜). If we take z = 0 in (3.1) and use (2.2), then ||f˜||(x) = ||f˜||(x ∗ 0)  sup{||f˜||((x ∗ y) ∗ 0), ||f˜||(y ∗ 0)} = sup{||f˜||(x ∗ y), ||f˜||(y)} for all x, y ∈ L. Hence ||f˜|| is a superior ideal on (L, f˜).



The converse of Theorem 3.6 is not true in general as seen in the following example. Example 3.7. Let L = {0, a, b, c} be a set with a binary operation ‘∗’ shown in Table 3. Then L := (L, ∗, 0) is a BCK-algebra (see [10]). Let U = {1, 2, 3, · · · , 8} be ordered as pictured in Figure 1.

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Seok Zun Song1,† , Young Bae Jun2 and Hee Sik Kim3,∗

Table 3. Cayley table for the binary operation ‘∗’ ∗ 0 a b c

0 0 a b c

a 0 0 a c

b 0 0 0 c

c 0 a b 0

1 rY HH

r * 2  3 HH r  *  YH H  HHr 5 r  4 H YH * H  YH H  H r H Hr Y H * 7 6 H  HH r  8 Figure 1 



Consider a pair f˜, L in which f˜ is given as follows:   if x ∈ {0},  {6, 8} ˜ f : L → P(U ), x 7→ {4, 6, 7} if x ∈ {a, b},   {2, 3, 5, 6, 7} if x = c. Then the superior mapping ||f˜|| on (L, f˜) is described as follows: ||f˜||(0) = 6, ||f˜||(a) = ||f˜||(b) = 3 and ||f˜||(c) = 2. Routine calculations show that ||f˜|| is a superior ideal on (L, f˜). But it is not a positive implicative superior ideal on (L, f˜) since ||f˜||(b ∗ a) = 3  6 = sup{||f˜||((b ∗ a) ∗ a), ||f˜||(a ∗ a)}. We provide conditions for a superior ideal to be a positive implicative superior ideal. Theorem 3.8. For a superior ideal ||f˜|| on (L, f˜), the following are equivalent. (i) ||f˜|| is a positive implicative superior ideal on (L, f˜).  (ii) (∀x, y ∈ L) ||f˜||(x ∗ y)  ||f˜||((x ∗ y) ∗ y) . Proof. Assume that ||f˜|| is a positive implicative superior ideal on (L, f˜). If we put z = y in (3.1), then ||f˜||(x ∗ y)  sup{||f˜||((x ∗ y) ∗ y), ||f˜||(y ∗ y)} = sup{||f˜||((x ∗ y) ∗ y), ||f˜||(0)} = ||f˜||((x ∗ y) ∗ y) for all x, y ∈ L. Conversely, let ||f˜|| be a superior ideal on (L, f˜) which satisfies the condition (ii). Note that ((x ∗ z) ∗ z) ∗ (y ∗ z) ≤ (x ∗ z) ∗ y = (x ∗ y) ∗ z

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Characterizations of positive implicative superior ideals for all x, y, z ∈ L. It follows from (ii), (2.11) and Proposition 2.2 that ||f˜||(x ∗ z)  ||f˜||((x ∗ z) ∗ z)  sup{||f˜||(((x ∗ z) ∗ z) ∗ (y ∗ z)), ||f˜||(y ∗ z)}  sup{||f˜||((x ∗ y) ∗ z), ||f˜||(y ∗ z)} for all x, y, z ∈ L. Therefore ||f˜|| is a positive implicative superior ideal on (L, f˜).



Theorem 3.9. For a superior ideal ||f˜|| on (L, f˜), the following are equivalent. ˜ (i) ||f˜|| is a positive  implicative superior ideal on (L, f ).  (ii) (∀x, y, z ∈ L) ||f˜||((x ∗ z) ∗ (y ∗ z))  ||f˜||((x ∗ y) ∗ z) . Proof. Suppose that ||f˜|| is a positive implicative superior ideal on (L, f˜). Then ||f˜|| is a superior ideal on (L, f˜) by Theorem 3.6. Note that ((x ∗ (y ∗ z)) ∗ z) ∗ z = ((x ∗ z) ∗ (y ∗ z)) ∗ z ≤ (x ∗ y) ∗ z for all x, y, z ∈ L. It follows from (2.4), Theorem 3.8 and Proposition 2.2 that ||f˜||((x ∗ z) ∗ (y ∗ z)) = ||f˜||((x ∗ (y ∗ z)) ∗ z)  ||f˜||(((x ∗ (y ∗ z)) ∗ z) ∗ z)  ||f˜||((x ∗ y) ∗ z) for all x, y, z ∈ L. Conversely, let ||f˜|| be a superior ideal on (L, f˜) which satisfies the second condition. Using (2.11) and the second condition, we have ||f˜||(x ∗ z) = sup{||f˜||((x ∗ z) ∗ (y ∗ z)), ||f˜||(y ∗ z)}  sup{||f˜||((x ∗ y) ∗ z), ||f˜||(y ∗ z)} for all x, y, z ∈ L. Therefore ||f˜|| is a positive implicative superior ideal on (L, f˜).    Theorem 3.10. Let ||f˜|| be the superior mapping of L with respect to f˜, L . Then ||f˜|| is a positive implicative superior ideal on (L, f˜) if and only if it satisfies the condition (2.10) and   (∀x, y, z ∈ L) ||f˜||(x ∗ y)  sup{||f˜||(((x ∗ y) ∗ y) ∗ z), ||f˜||(z)} . (3.2) Proof. Assume that ||f˜|| is a positive implicative superior ideal on (L, f˜). Then ||f˜|| is a superior ideal on (L, f˜) by Theorem 3.6, and so ||f˜|| satisfies the condition (2.10). Using (2.11), (III), (2.2), (2.4) and Theorem 3.9, we have ||f˜||(x ∗ y)  sup{||f˜||((x ∗ y) ∗ z), ||f˜||(z)} = sup{||f˜||(((x ∗ z) ∗ y) ∗ (y ∗ y)), ||f˜||(z)}  sup{||f˜||(((x ∗ z) ∗ y) ∗ y), ||f˜||(z)} = sup{||f˜||(((x ∗ y) ∗ y) ∗ z), ||f˜||(z)} for all x, y, z ∈ L.

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Seok Zun Song1,† , Young Bae Jun2 and Hee Sik Kim3,∗ Conversely, suppose that ||f˜|| satisfies two conditions (2.10) and (3.2). Then ||f˜||(x) = ||f˜||(x ∗ 0)  sup{||f˜||(((x ∗ 0) ∗ 0) ∗ z), ||f˜||(z)} = sup{||f˜||(x ∗ z), ||f˜||(z)} for all x, z ∈ L, and so ||f˜|| is a superior ideal on (L, f˜). If we take z = 0 in (3.2) and use (2.2) and (2.10), then ||f˜||(x ∗ y)  sup{||f˜||(((x ∗ y) ∗ y) ∗ 0), ||f˜||(0)} = sup{||f˜||((x ∗ y) ∗ y), ||f˜||(0)} = ||f˜||((x ∗ y) ∗ y) for all x, y ∈ L. Therefore ||f˜|| is a positive implicative superior ideal on (L, f˜) by Theorem 3.8.    Lemma 3.11. Let ||f˜|| be the superior mapping of L with respect to a pair f˜, L on (U, ). Then ||f˜|| is a superior ideal on (L, f˜) if and only if it satisfies the following assertion:   (∀x, y, z ∈ L) (x ∗ y) ∗ z = 0 ⇒ ||f˜||(x)  sup{||f˜||(y), ||f˜||(z)} .

(3.3)

Proof. Assume that ||f˜|| is a superior ideal on (L, f˜). Let x, y, z ∈ L be such that x ∗ y ≤ z. Then (x ∗ y) ∗ z = 0, and so ||f˜||(x ∗ y)  sup{||f˜||((x ∗ y) ∗ z), ||f˜||(z)} = sup{||f˜||(0), ||f˜||(z)} = ||f˜||(z) by (2.11) and (2.10). It follows that ||f˜||(x)  sup{||f˜||(x ∗ y), ||f˜||(y)}  sup{||f˜||(z), ||f˜||(y)}. Conversely, suppose that the assertion (3.3) is valid. Since (0 ∗ x) ∗ x = 0 and (x ∗ (x ∗ y)) ∗ y = 0 for all x, y ∈ L, it follows from (3.3) that ||f˜||(0)  sup{||f˜||(x), ||f˜||(x)} = ||f˜||(x) and ||f˜||(x)  sup{||f˜||(x ∗ y), ||f˜||(y)} for all x, y ∈ L. Therefore ||f˜|| is a superior ideal on (L, f˜).

 



Corollary 3.12. Let ||f˜|| be the superior mapping of L with respect to a pair f˜, L on (U, ). Then ||f˜|| is a superior ideal on (L, f˜) if and only if it satisfies the following assertion: ||f˜||(x)  sup{||f˜||(a1 ), ||f˜||(a2 ), · · · , ||f˜||(an )}

(3.4)

for all x, a1 , a2 , · · · , an ∈ L with (· · · ((x ∗ a1 ) ∗ a2 ) ∗ · · · ) ∗ an = 0.

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Characterizations of positive implicative superior ideals   Theorem 3.13. Let ||f˜|| be the superior mapping of L with respect to a pair f˜, L on (U, ). Then ||f˜|| is a positive implicative superior ideal on (L, f˜) if and only if it satisfies the following assertion: ||f˜||(x ∗ y)  sup{||f˜||(a), ||f˜||(b)}

(3.5)

for all x, y, a, b ∈ L with (((x ∗ y) ∗ y) ∗ a) ∗ b = 0. Proof. Assume that ||f˜|| is a positive implicative superior ideal on (L, f˜). Then ||f˜|| is a superior ideal on (L, f˜) by Theorem 3.6. Let x, y, a, b ∈ L be such that (((x ∗ y) ∗ y) ∗ a) ∗ b = 0. Using Theorem 3.8(ii) and (3.3), we have ||f˜||(x ∗ y)  ||f˜||((x ∗ y) ∗ y)  sup{||f˜||(a), ||f˜||(b)}. Conversely, suppose that ||f˜|| satisfies the condition (3.5) for all x, y, a, b ∈ L with (((x∗y)∗y)∗a)∗b = 0. Assume that (x ∗ u) ∗ v = 0 for all x, u, v ∈ L. Then (((x ∗ 0) ∗ 0) ∗ u) ∗ v = 0, and so ||f˜||(x) = ||f˜||(x ∗ 0)  sup{||f˜||(u), ||f˜||(v)} by (3.5). It follows from Lemma 3.11 that ||f˜|| is a superior ideal on (L, f˜). Note that (((x ∗ y) ∗ y) ∗ ((x ∗ y) ∗ y)) ∗ 0 = 0 for all x, y ∈ L. Using (3.5) and (2.10), we have ||f˜||(x ∗ y)  sup{||f˜||((x ∗ y) ∗ y), ||f˜||(0)} = ||f˜||((x ∗ y) ∗ y) for all x, y ∈ L. Therefore ||f˜|| is a positive implicative superior ideal on (L, f˜) by Theorem 3.8.    Corollary 3.14. Let ||f˜|| be the superior mapping of L with respect to a pair f˜, L on (U, ). Then ||f˜|| is a positive implicative superior ideal on (L, f˜) if and only if it satisfies the following assertion: ||f˜||(x ∗ y)  sup{||f˜||(a1 ), ||f˜||(a2 ), · · · , ||f˜||(an )}

(3.6)

for all x, y, a1 , a2 , · · · , an ∈ L with (· · · ((((x ∗ y) ∗ y) ∗ a1 ) ∗ a2 ) ∗ · · · ) ∗ an = 0.   Theorem 3.15. Let ||f˜|| be the superior mapping of L with respect to a pair f˜, L on (U, ). Then ||f˜|| is a positive implicative superior ideal on (L, f˜) if and only if it satisfies the following assertion: ||f˜||((x ∗ z) ∗ (y ∗ z))  sup{||f˜||(a), ||f˜||(b)}

(3.7)

for all x, y, z, a, b ∈ L with (((x ∗ y) ∗ z) ∗ a) ∗ b = 0. Proof. Assume that ||f˜|| is a positive implicative superior ideal on (L, f˜). Then ||f˜|| is a superior ideal on (L, f˜) by Theorem 3.6. Suppose that (((x ∗ y) ∗ z) ∗ a) ∗ b = 0 for all x, y, z, a, b ∈ L. Then ||f˜||((x ∗ z) ∗ (y ∗ z))  ||f˜||((x ∗ y) ∗ z)  sup{||f˜||(a), ||f˜||(b)} by Theorem 3.9 and Lemma 3.11. Conversely, suppose that ||f˜|| satisfies the condition (3.7) for all x, y, z, a, b ∈ L with (((x∗y)∗z)∗a)∗b = 0. Let x, y, a, b ∈ L be such that (((x ∗ y) ∗ y) ∗ a) ∗ b = 0. Then ||f˜||(x ∗ y) = ||f˜||((x ∗ y) ∗ (y ∗ y))  sup{||f˜||(a), ||f˜||(b)} by (3.7). It follows from Theorem 3.13 that ||f˜|| is a positive implicative superior ideal on (L, f˜).

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Seok Zun Song1,† , Young Bae Jun2 and Hee Sik Kim3,∗   Corollary 3.16. Let ||f˜|| be the superior mapping of L with respect to a pair f˜, L on (U, ). Then ||f˜|| is a positive implicative superior ideal on (L, f˜) if and only if it satisfies the following assertion: ||f˜||((x ∗ z) ∗ (y ∗ z))  sup{||f˜||(a1 ), ||f˜||(a2 ), · · · , ||f˜||(an )}

(3.8)

for all x, y, z, a1 , a2 , · · · , an ∈ L with (· · · ((((x ∗ y) ∗ z) ∗ a1 ) ∗ a2 ) ∗ · · · ) ∗ an = 0. Theorem 3.17. Let ||f˜|| and ||˜ g || be superior ideals on (L, f˜) and (L, g˜), respectively, such that ||f˜||(0) = ||˜ g ||(0) and ||˜ g ||(x)  ||f˜||(x) for all x(6= 0) ∈ L. If ||f˜|| is a positive implicative superior ideal on (L, f˜), then ||˜ g || is a positive implicative superior ideal on (L, g˜). Proof. For any x, y, z ∈ L, let u := (x ∗ y) ∗ z. Then ||˜ g ||(((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)) = ||˜ g ||(((x ∗ z) ∗ (y ∗ z)) ∗ u) = ||˜ g ||(((x ∗ u) ∗ z) ∗ (y ∗ z))  ||f˜||(((x ∗ u) ∗ z) ∗ (y ∗ z))  ||f˜||(((x ∗ u) ∗ y) ∗ z) = ||f˜||(((x ∗ y) ∗ z) ∗ u) = ||f˜||(0) = ||˜ g ||(0), and so ||˜ g ||(((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)) = ||˜ g ||(0). It follows from (2.11) that ||˜ g ||((x ∗ z) ∗ (y ∗ z))  sup{||˜ g ||(((x ∗ z) ∗ (y ∗ z)) ∗ ((x ∗ y) ∗ z)), ||˜ g ||((x ∗ y) ∗ z)} = sup{||˜ g ||(0), ||˜ g ||((x ∗ y) ∗ z)} = ||˜ g ||((x ∗ y) ∗ z). Therefore ||˜ g || is a positive implicative superior ideal on (L, g˜) by Theorem 3.9.



References [1] J. D. Bashford and P. D. Jarvis, The genetic code as a peridic table: algebraic aspects, BioSystems 57 (2000), 147–161. [2] L. Frappat, A. Sciarrino and P. Sorba, Crystalizing the genetic code, J. Biological Physics 27 (2001), 1–34. [3] Y. Huang, BCI-algebra, Science Press, Beijing, 2006. [4] K. Is´eki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japonica 23 (1978), 1–26. [5] Y. B. Jun and S. Z. Song, Codes based on BCK-algebras, Inform. Sci. 181 (2011), 5102–5109. [6] Y. B. Jun and S. Z. Song, Superior subalgebras and ideals of BCK/BCI-algerbas, Discuss. Math. Gen. Algebra Appl. 36 (2016), 85–99. [7] M. K. Kinyon and A. A. Sagle, Quadratic dynamical systems and algebras, J. Differential Equations 117 (1995), 67–126. [8] J. Meng, Commutative ideals in BCK-algebras, Pure Appl. Math. (in China) 9 (1991), 49–53. [9] J. Meng, On ideals in BCK-algebras, Math. Japonica 40 (1994), 143–154. [10] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa, Seoul, 1994. [11] R. S´aanchez, R. Grau and E. Morgado, A novel Lie algebra of the genetic code over the Galois field of four DNA bases, Mathematical Biosciences 202 (2006), 156–174. [12] J. J. Tian and B. L. Li, Coalgebraic structure of genetics inheritance, Mathematical Biosciences and Engineering 1 (2004), 243–266.

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The uniqueness of meromorphic functions sharing sets in an angular domain ∗ Yong Zheng Zhoua , Hong-Yan Xub† and Zu-Xing Xuanc a

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

b

Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

c

Basic Department, Beijing Union University,

No.97 Bei Si Huan Dong Road Chaoyang District Beijing, 100101, China

Abstract By using the Tsuji’s characteristic, we deal with the uniqueness problem of meromorphic functions sharing sets in an angular domain and obtain some theorems which improve and extend the results given by Zheng, Xuan. Key words: Meromorphic function; Angular domain; Uniqueness; Tsuji’s characteristic. Mathematical Subject Classification (2010): 30D30 30D35.

1

Introduction and main results

The purpose of this paper is to investigate the uniqueness of meromorphic functions sharing sets in an angular domain by using the Tsuji’s characteristic functions of angular domain. It is assumed that the readers are familiar with the notations of the Nevanlinna theory such as T (r, f ), m(r, f ), N (r, f ) and so on, that can be found, for instance, in [5, 17]. S b We use C to denote the open complex plane, C(= C {∞}) to denote the extended complex plane, and Ω(⊂ C) to denote an angular domain. Let S be a set of distinct b and Ω ⊆ C. Define elements in C [ E(S, Ω, f ) = {z ∈ Ω|fa (z) = 0, counting multiplicities}, a∈S ∗ This

project was supported by the NSF of China(11561033), the Natural Science foundation of Jiangxi Province in China (20151BAB201008). The last author is supported by the project of Beijing Municipal Science and Technology(D161100003516003). † Corresponding author.

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E(S, Ω, f ) =

[

{z ∈ Ω|fa (z) = 0,

ignoring

multiplicities},

a∈S

where fa (z) = f (z) − a if a ∈ C and f∞ (z) = 1/f (z). Let f and g be two non-constant meromorphic functions in C. If E(S, Ω, f ) = E(S, Ω, g), we say that f and g share the set S CM (counting multiplicities) in Ω. If E(S, Ω, f ) = E(S, Ω, g), we say f and g share the set S IM (ignoring multiplicities) b we say f and g share the value a in Ω. In particular, when S = {a}, where a ∈ C, CM in Ω if E(S, Ω, f )) = E(S, Ω, g), and we say f and g share the value a IM in Ω if E(S, Ω, f ) = E(S, Ω, g). When Ω = C, we give the simple notation as before, E(S, f ), E(S, f ) and so on(see [13]). Let l be a nonnegative integer or infinity. For a ∈ C ∪ {∞}, we denote by E l (a, Ω, f ) the set of all a-points of f in Ω, where an a-point of multiplicity k is counted one times if k ≤ l and zero times if k > l. R.Nevanlinna(see [9]) proved the following well-known theorem. Theorem 1.1 (see [9].) If f and g are two non-constant meromorphic functions that share five distinct values a1 , a2 , a3 , a4 , a5 IM in Ω = C, then f (z) ≡ g(z). After his theorems, the uniqueness problems of meromorphic functions sharing values in the whole complex plane attracted many investigations (see [15]). In 2004, Zheng [19] studied the uniqueness problem under the condition that five values are shared in some angular domain in C. It is an interesting topic to investigate the uniqueness with shared values in the remaining part of the complex plane removing an unbounded closed set, see [3, 4, 7, 8, 10, 13, 18, 19, 20]. Zheng [20], Cao and Yi [2], Xu and Yi [13] continued to investigate the uniqueness of meromorphic functions sharing five values and four values, Lin, Mori and Tohge [7] and Lin, Mori and Yi [8] investigated the uniqueness of meromorphic and entire functions sharing sets in an angular domain. To state theirs results, we need the following basic notations and definitions of meromorphic functions in an angular domain(see [5, 19, 20]). In 2009, the present author [14] investigated the uniqueness of meromorphic functions with finite order sharing some values in an angular domain and obtained the following theorem Theorem 1.2 (see [14]). Let f (z) and g(z) be both transcendental meromorphic funcb and tions, and let f (z) be of finite order λ (lower order µ) and such that for some a ∈ C (p) an integer p ≥ 0, δ = δ(a, f ) > 0. For m pair of real numbers {αj , βj } satisfying −π ≤ α1 ≤ β1 ≤ α2 ≤ β2 ≤ · · · ≤ αm ≤ βm ≤ 2π and

m X 4 (αj+1 − βj ) < arcsin σ j=1

r

δ , 2

π π where σ = max{ω, µ}, ω = max{ β1 −α , . . . , βm −α }, assume that aj (j = 1, 2, . . . , q) be q 1 m distinct complex numbers, and let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying

k1 ≥ k2 ≥ · · · ≥ kp ,

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E kj (aj , Ω, f ) = E kj (aj , Ω, g), q X j=3

where Ω =

Sm

j=1 {z

kj > 2, kj + 1

(2)

: αj ≤ arg z ≤ βj }. If ω < λ(f ), then f (z) ≡ g(z).

In 2009, Cao and Yi [2] investigated the uniqueness problem of two transcendental meromorphic functions f, g sharing five values IM in an angular domain and obtained the following result which extended Theorem 1.1 to an angular domain. Theorem 1.3 (see [2, Theorem 1.3].) Let f and g be two transcendental meromorphic functions. Given one angular domain Ω = {z : α < arg z < β} with 0 < β − α ≤ 2π, we assume that f and g share five distinct values aj (j = 1, 2, 3, 4, 5) IM in Ω. Then f (z) ≡ g(z), provided that lim

r→∞

Sα,β (r, f ) = ∞, log(rT (r, f ))

(r 6∈ E),

where Sα,β (r, f ) is called the Nevanlinna’s angular characteristic. Moreover, Cao and Yi [2] also investigated the two uniqueness problems of two transcendental meromorphic functions f, g sharing four distinct values CM in an angular domain X and f, g sharng two distinct values CM in an angular domain X and the other two distinct values IM in an angular domain X, and they obtained two interesting results which extended the analogous results as in the whole complex plane to an angular domain. In 2011, Xu and Cao [11, 12] improve the results given by Cao and Yi[1, 2] to some extent. Most recently, Zheng [21] prove the following theorem by using the Tsuji’s characteristic to extend the five IM theorem of Nevanlinna’s to an angular domain. The Tsuji’s characteristic will be introduced in Section 2. Theorem 1.4 (see [21]). Let f (z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and f (z) be transcendental in the Tsuji’s sense. Assume that aj (j = 1, 2, . . . , 5) be 5 distinct complex numbers. If E(aj , Ω, f ) = E(aj , Ω, g), then f (z) ≡ g(z). In this paper, we will deal with the uniqueness of meromorphic functions sharing sets in an angular domain by using the Tsuji’s characteristic and obtain the following results which are improvement of Theorem 1.4. Theorem 1.5 Let f (z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and f (z) be transcendental in the Tsuji’s sense. Suppose that Sj = {aj , aj + b, . . . , aj + (l − 1)b},

j = 1, 2, . . . , q,

with b 6= 0, Si ∩Sj = ∅, (i 6= j). Let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1) and E kj ) (Sj , Ω, f ) = E kj ) (Sj , Ω, g), (j = 1, 2, . . . , q). (3)

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Furthermore, let ΘT (f ) =

X

ΘT (0, f − a) −

a

j=1

=

s=0 δT (0, f

− (aj + sb))

km + 1 +

ΘT (0, f − (aj + sb)),

j=1 s=0

Pm−1 Pl−1 A1

q X l−1 X

+

q X l−1 X kj + δT (0, f − (aj + sb)) kj + 1 j=m s=0

(2l − 1)kn (lm − 3l + 1)km − + Θ0 (f ) − 2 km + 1 kn + 1

and Pn−1 Pl−1 A2

j=1

=

s=0 δT (0, g

− (aj + sb))

kn + 1 +

+

q X l−1 X kj + δT (0, g − (aj + sb)) kj + 1 j=n s=0

(ln − 3l + 1)kn (2l − 1)km − + Θ0 (g) − 2, kn + 1 km + 1

where m and n are positive integers in {1, 2, . . . , q} and a is an arbitrary complex number or ∞. If min{A1 , A2 } ≥ 0, and max{A1 , A2 } > 0. (4) Then f1 (z) ≡ f2 (z). From Theorem 1.5, we can get the following corollaries. Corollary 1.1 Let f (z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and f (z) be transcendental in the Tsuji’s sense. Suppose that Sj = {aj , aj + b, . . . , aj + (l − 1)b},

j = 1, 2, . . . , q,

with b 6= 0, Si ∩Sj = ∅, (i 6= j). Let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1) and E kj ) (Sj , Ω, f ) = E kj ) (Sj , Ω, g), (j = 1, 2, . . . , q). If q X l−1 X j=3 s=0

kj (2 − 2l)k3 + > 2. kj + 1 k3 + 1

Then f (z) ≡ g(z). Proof: Let m = n = 3. Since ΘT (f ) ≥ 0, ΘT (g) ≥ 0, δT (0, f − (aj + sb)) ≥ 0 and δT (0, g − (aj + sb)) ≥ 0 for j = 1, 2, . . . , q, one can deduce from Theorem 1.5 that Corollary 1.1 follows. 2 The following corollary is an analog of a result due to Yi (Theorem 10.7 in [15], see also [16]) on C.

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Corollary 1.2 Let f (z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and f (z) be transcendental in the Tsuji’s sense. Suppose that Sj = {aj , aj + b, . . . , aj + (l − 1)b},

j = 1, 2, . . . , q,

with b 6= 0, q > 4, Si ∩ Sj = ∅, (i 6= j). If E(Sj , Ω, f ) = E(Sj , Ω, g), (j = 1, 2, . . . , q). Then f (z) ≡ g(z). Proof: Let k1 = k2 = . . . = kq = ∞. One can deduce from Corollary 1.1 that Corollary 1.2 follows immediately. 2 Let l = 1. Then it is easily derived the following corollary from Corollary 1.1, which is an analog of the Corollary of Theorem 3.15 in [15]. Corollary 1.3 Let f (z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and f (z) be transcendenb tal in the Tsuji’s sense. Let aj (j = 1, 2, . . . , q) be q distinct complex numbers in C, and kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1) and E kj ) (aj , Ω, f ) = E kj ) (aj , Ω, g), (j = 1, 2, . . . , q). Then (i) if q = 7, then f (z) ≡ g(z). (ii) if q = 6 and k3 ≥ 2, then f (z) ≡ g(z). (iii) if q = 5, k3 ≥ 3 and k5 ≥ 2, then f (z) ≡ g(z). (iv) if q = 5 and k4 ≥ 4, then f (z) ≡ g(z). (v) if q = 5, k3 ≥ 5 and k4 ≥ 3, then f (z) ≡ g(z). (vi) if q = 5, k3 ≥ 6 and k4 ≥ 2, then f (z) ≡ g(z). Another main theorem of this paper is listed as follows. Theorem 1.6 Let f (z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and f (z) be transcendental in the Tsuji’s sense. Suppose that Sj = {c + aj , c + aj w, . . . , c + aj wl−1 },

j = 1, 2, . . . , q,

with aj 6= 0, (j = 1, 2, . . . , q), w = exp( 2πi l ), Si ∩ Sj = ∅, (i 6= j). Let kj (j = 1, 2, . . . , q) be positive integers or ∞ satisfying (1) and E kj ) (Sj , Ω, f ) = E kj ) (Sj , Ω, g),

(j = 1, 2, . . . , q).

(5)

Furthermore, let ΘT (f ) =

X

ΘT (0, f − a) −

a

Pm−1 Pl−1 A1

j=1

=

ΘT (0, f − (c + aj ws )),

j=1 s=0

s=0 δT (0, f

− (c + aj ws ))

km + 1 +

q X l−1 X

+

q X l−1 X kj + δT (0, f − (c + aj ws )) kj + 1 j=m s=0

l(m − 2)km lkn − + ΘT (f ) − 2 km + 1 kn + 1

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and Pn−1 Pl−1 A2

j=1

=

s=0 δT (0, g

− (c + aj ws ))

kn + 1 +

+

q X l−1 X kj + δT (0, g − (c + aj ws )) kj + 1 j=n s=0

l(n − 2)kn lkm − + ΘT (g) − 2, kn + 1 km + 1

where m and n are positive integers in {1, 2, . . . , q} and a is an arbitrary complex number or ∞. If min{A1 , A2 } ≥ 0, and max{A1 , A2 } > 0. (6) Then (f (z) − c)l ≡ (g(z) − c)l . From Theorem 1.6, we can get the following corollary immediately. Corollary 1.4 Let f (z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg z < β} with 0 ≤ α < β ≤ 2π and f (z) be transcendental in the Tsuji’s sense. Suppose that Sj = {c + aj , c + aj w, . . . , c + aj wl−1 },

j = 1, 2, . . . , q,

with aj 6= 0, (j = 1, 2, . . . , q), q > 2 + 2l , w = exp( 2πi l ), Si ∩ Sj = ∅, (i 6= j). If E(Sj , Ω, f ) = E(Sj , Ω, g) for j = 1, 2, . . . , q, then (f (z) − c)l ≡ (g(z) − c)l . Proof: Set m = n = 1 and k1 = k2 = . . . = ∞. Since ΘT (f ) ≥ 0, ΘT (g) ≥ 0, δT (0, f − (aj + sb)) ≥ 0 and δT (0, g − (aj + sb)) ≥ 0 for j = 1, 2, . . . , q. Then Corollary 1.4 follows immediately from Theorem 1.6. 2

2

Preliminaries

In this section, we will introduce some notations of Tsuji’s characteristic in an angular π domain (see [6, 21]). For meromorphic function f in an angular domain Ω and ω = β−α , we define Z π−arcsin(r−ω ) −1 −1 1 1 Mα,β (r, f ) = log+ f (rei(α+ω θ) sinω θ) ω 2 dθ, 2π arcsin(r−ω ) r sin θ   X sin ω(βn − α) 1 Nα,β (r, f ) = − ω , |bn |ω r −1 1 0). From (8) we have (ql + p − 2)T(r, f )
ΘT (f ) − ε holds for any given ε(> 0). Using a similar discussion as in the proof of Theorem 1.5, we obtain     l(n − 1)kn l(m − 1)km + B1 − ε T(r, f ) + + B2 − ε T(r, g) km + 1 kn + 1 q X l−1 X


0, Ωε = Ω(α + ε, β − ε). Then for ε > 0, we have Z r N (t, Ω, f ) N (r, Ω, f ) 2 +ω dt, N(r, f ) ≤ ω rω tω+1 1 and N(r, f ) ≥ ωcω

N (cr, Ωε , f ) + ω 2 cω rω

Z 1

cr

N (t, Ωε , f ) dt tω+1

π where 0 < c < 1 is a constant depending on ε, ω = β−α and N (t, Ω, f ) = n(t, Ω, f ) is the number of poles of f (z) in Ω ∩ {z : 1 < |z| ≤ t}.

Rr 1

n(t,Ω,f ) dt, t

From Lemma 5.1, we can get that f is transcendental in Tsuji sense if f satisfies condition (9). Thus, we can get the following results Theorem 5.2 Let the assumptions of Theorems 1.5-1.6 and Corollaries 1.1-1.4 be given with the exception of that f (z) is transcendental in Tsuji sense. Assume that for some b and ε > 0, a∈C N (r, Ωε , f = a) = ∞, (9) lim sup rω log r r→∞ where ω, N (t, Ω, f ) are stated as in Lemma 5.1. Then f (z) ≡ g(z).

References [1] T. B. Cao, H. X. Yi, Analytic functions sharing three values DM in one angular domain, J.Korean Math. Soc. 45(6)(2008), 1523-1534. [2] T. B. Cao, H. X. Yi, On the uniqueness of meromorphic functions that share four values in one angular domain, J. Math. Anal. Appl. 358(2009), 81-97. [3] A. A. Gol’dberg, Nevanlinna’s lemma on the logarithmic derivative of a meromorphic function, Mathematical Notes 17(4) (1975), 310-312. [4] A. A. Gol’dberg, I. V. Ostrovskii, The Distribution of Values of Meromorphic Function, Nauka, Moscow, 1970 (in Russian).

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[5] W. K. Hayman, Meromorphic Functions, Oxford Univ. Press, London, 1964. [6] J. K. Langley, An application of the Tsuji characteristic, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 38 (1991), 299-318. [7] W. C. Lin, S. Mori, K. Tohge, Uniqueness theorems in an angular domain, Tohoku Math. J. 58 (2006), 509-527. [8] W. C. Lin, S. Mori, H. X. Yi, Uniqueness theorems of entire functions with shared-set in an angular domain, Acta Mathematica Sinica 24(2008), 1925-1934. [9] R. Nevanlinna, Le th´eor`eme de Picard-Borel et la th´eorie des fonctions m´eromorphes, Repringting of the 1929 original, Chelsea Publishing Co., New York, 1974(in Frech). [10] Z. J. Wu, A remark on uniqueness theorems in an angular domain, Proc, Japan Acad. Ser. A 64(6)(2008), 73-77. [11] H. Y. Xu, T. B. Cao, Uniqueness of two analyticfunctions sharing four values in an angular domain, Ann. Polon.Math. 99 (2010), 55-65. [12] H. Y. Xu, T. B. Cao, Uniqueness of meromorphicfunctions sharing four values IM and one set in an angular domain,Bulletin of the Belgian Mathematical Society -Simon Stevin, 17(5) (2010), 937-948. [13] J. F. Xu, H. X. Yi, On uniqueness of meromorphic functions with shared four values in some angular domains, Bull. Malays. Math. Sci. Soc. 31(1) (2008), 57-65. [14] Z. X. Xuan, On uniqueness of meromorphic functions with multiple values in some angular domains, Journal of Inequalities and Applications, Volume 2009 (2009), Article ID 208516, 10 pages. [15] H. X. Yi, C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press/Kluwer., Beijing, 2003. [16] H. X. Yi , On the uniqueness of meromorphic functions, Acta Math. Sinica (Chin. Ser.) 31(4) (1988), 570-576. [17] L. Yang, Value Distribution Theory, Springer/Science Press, Berlin/Beijing, 1993/1982. [18] Q. C. Zhang, Meromorphic functions sharing values in an angular domain, J. Math. Anal. Appl. 349(2009), 100-112. [19] J. H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad J. Math. 47(2004), 152-160. [20] J. H. Zheng, On uniqueness of meromorphic functions with shared values in one angular domains, Complex Var. Elliptic Equ. 48(2003), 777-785. [21] J. H. Zheng, Value distrbution of meromorphic functions, Springer, Beijing: Tsinghua Univ. Press. 2010.

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A new generalization of Fibonacci and Lucas p−numbers a ¨ Yasin YAZLIKa,∗, Cahit KOME , Vinay MADHUSUDANANb a Department b Department

of Mathematics, Nev¸sehir Hacı Bekta¸s Veli University, Nev¸sehir, Turkey of Mathematics, Manipal Institute of Technology, Manipal University, Manipal, India

Abstract In this paper, we define a new generalization of the Fibonacci and Lucas p−numbers. Further, we build up the tree diagrams for generalized Fibonacci and Lucas p−sequence and derive the recurrence relations of these sequences by using these diagrams. Also, we show that the generalized Fibonacci and Lucas p−sequences can be reduced into the various number sequences. Finally, we develop Binet formulas for the generalized Fibonacci and Lucas p−numbers and present the numerical and graphical results, which obtained by means of the Binet formulas, for specific values of a, b and p. Keywords: The generalized Fibonacci p−numbers, The generalized Lucas p−numbers, Binet formula. 2010 MSC: 11B39

1. Introduction Fibonacci and Lucas sequences are one of the most popular and fascinating sequences that arise in various situations, especially in mathematics, physics and related fields. The classical Fibonacci and Lucas sequences are defined by Fn+2 = Fn+1 + Fn and Ln+2 = Ln+1 + Ln , for n ∈ N, with initial conditions F0 = 0,

F1 = 1 and L0 = 2,

L1 = 1, respectively. One of the most important sources of this area is [1],

which was written by Thomas Koshy, and contains numerous applications, generalizations and recurrence relations of Fibonacci and Lucas numbers. In recent years, many authors have studied generalizations of the Fibonacci and Lucas sequences [2–12]. For instance, in [8, 10] the authors defined the generalized Fibonacci {qn }n∈N0 sequence as

q0 = 0, q1 = 1, qn+2 =

  aqn+1 + qn , if n ≡ 0 (mod 2)

(1)

 bqn+1 + qn , if n ≡ 1 (mod 2),

∗ Corresponding

Author ¨ Email addresses: [email protected] (Yasin YAZLIK), [email protected] (Cahit KOME), [email protected] (Vinay MADHUSUDANAN)

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and the generalized Lucas {ln }n∈N0 sequence as in the form   bln+1 + ln , if n ≡ 0 (mod 2) l0 = 2, l1 = a, ln+2 =  aln+1 + ln , if n ≡ 1 (mod 2).

(2)

Stakhov and Rozin introduced Fibonacci and Lucas p−numbers, one of the most significant mathematical discoveries of the modern Fibonacci numbers theory, and they presented some properties of this sequence, Fp (n) = Fp (n − 1) + Fp (n − p − 1)

(3)

Lp (n) = Lp (n − 1) + Lp (n − p − 1),

(4)

and

in [13], with the initial conditions Fp (0) = 0, Fp (1) = 1, Fp (2) = 1, . . . , Fp (p) = 1 and Lp (0) = p + 1, Lp (1) = 1, Lp (2) = 1, . . . , Lp (p) = 1, respectively. After that, Kocer et al. defined the m−extension of the Fibonacci and Lucas p−numbers, Fp,m (n + p + 1) = mFp,m (n + p) + Fp,m (n)

(5)

Lp,m (n + p + 1) = mLp,m (n + p) + Lp,m (n),

(6)

and

with initial conditions Fp,m (0) = 0, Fp,m (1) = 1, Fp,m (2) = m, Fp,m (3) = m2 , . . . , Fp,m (p + 1) = mp and Lp,m (0) = p + 1, Lp,m (1) = m, Lp,m (2) = m2 , Lp,m (3) = m3 , . . . , Lp,m (p + 1) = mp+1 , where p and n are nonnegative integers and m is a positive real number [14]. The main purpose of the present article is to give a wider generalization of the generalized Fibonacci and Lucas sequence given by (1) and (2), the Fibonacci and Lucas p-sequences given by (3) and (4) and the m−extension of the Fibonacci and Lucas p-sequences given by (5) and (6) to introduce a new class of the recurrence numerical sequences called the generalization of Fibonacci and Lucas p−numbers.

2. Generalized Fibonacci and Lucas p−numbers Definition 2.1. For any positive real numbers a, b and positive integer p, the generalized Fibonacci ∞ p−sequence {fn }∞ n=0 and Lucas p−sequence {`n }n=0 are defined recursively by     b`n−1 + `n−p−1 , if n ≡ 0 (mod 2) afn−1 + fn−p−1 , if n ≡ 0 (mod 2) fn = and `n =   bfn−1 + fn−p−1 , if n ≡ 1 (mod 2), a`n−1 + `n−p−1 , if n ≡ 1 (mod 2),

where n ≥ p + 1 and the initial conditions of fn and `n are p

f0 = 0, f1 = 1, f2 = a, . . . , fp = ab 2 c bb

p−1 2 c

(7)

2

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and `0 = p + 1, `1 = a, `2 = ab, . . . , `p = ab

p+1 2 c

p

bb 2 c ,

(8)

respectively. Note that, these sequences can be reduced to different sequences for specific values of p, a and b. It is not difficult to see from the following table that Fibonacci, Lucas, Pell, Pell−Lucas, k−Fibonacci, k−Lucas, Fibonacci p, Lucas p, Pell p, Pell−Lucas p, m−extension of Fibonacci p and m−extension of Lucas psequences are special cases of generalized Fibonacci and Lucas p−sequence. p

a

b

fn

`n

1

1

1

Classical Fibonacci sequence Fn

Classical Lucas sequence Ln

1

2

2

Classical Pell sequence Pn

Classical Pell-Lucas sequence Qn

1

k

k

k−Fibonacci numbers {Fk,n }∞ n=0

k−Lucas numbers {Lk,n }∞ n=0

p

1

1

Fibonacci p−sequence Fp,n

Lucas p−sequence Lp,n

p

2

2

Pell p−sequence Fp,n

Pell-Lucas p−sequence Lp,n

p

m

m

m−extension of Fibonacci p−numbers Fp,m,n

m−extension of Lucas p−numbers Lp,m,n

Let a and b be positive real numbers, p be a positive integer and ξ(n) = n − 2b n2 c. We can construct the tree diagrams for the generalized Fibonacci and Lucas p−numbers as: fn a

1−ξ(n) ξ(n)

b

fn−2

a

1 fn−p−1

fn−1 aξ(n) b1−ξ(n)

`n ξ(n) 1−ξ(n)

1 fn−p−2

aξ(n+p) b1−ξ(n+p) fn−p−2

b

1 `n−p−1

`n−1 a1−ξ(n) bξ(n)

1 fn−2p−2

Figure 1: Tree diagram for generalized Fibonacci p−numbers

`n−2

1

a1−ξ(n+p) bξ(n+p)

`n−p−2

`n−p−2

1 `n−2p−2

Figure 2: Tree diagram for generalized Lucas p−numbers.

By considering Figures 1 and 2, we will derive the recurrence relations for fn and `n . First we suppose that p is even. Then, {fn } satisfies the recurrence relation fn

= a1−ξ(n) bξ(n) fn−1 + fn−p−1   = a1−ξ(n) bξ(n) aξ(n) b1−ξ(n) fn−2 + fn−p−2 + aξ(n+p) b1−ξ(n+p) fn−p−2 + fn−2p−2 = abfn−2 + a1−ξ(n) bξ(n) fn−p−2 + aξ(n) b1−ξ(n) fn−p−2 + fn−2p−2   = abfn−2 + a1−ξ(n) bξ(n) + aξ(n) b1−ξ(n) fn−p−2 + fn−2p−2 = abfn−2 + (a + b) fn−p−2 + fn−2p−2 . 3

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Next, we suppose that p is odd. Then, fn also satisfies the recurrence relation fn

= a1−ξ(n) bξ(n) fn−1 + fn−p−1   = a1−ξ(n) bξ(n) aξ(n) b1−ξ(n) fn−2 + fn−p−2 + aξ(n+p) b1−ξ(n+p) fn−p−2 + fn−2p−2 = abfn−2 + a1−ξ(n) bξ(n) fn−p−2 + a1−ξ(n) bξ(n) fn−p−2 + fn−2p−2 = abfn−2 + 2a1−ξ(n) bξ(n) fn−p−2 + fn−2p−2 = abfn−2 + 2 (fn−p−1 − fn−2p−2 ) + fn−2p−2 = abfn−2 + 2fn−p−1 − fn−2p−2 .

In a similar way, we can easily obtain the same recurrence relation for `n . Let   fn , if α0 = 0, α1 = 1, α2 = a, . . . , αp = ab p2 c bb p−1 2 c αn = p  `n , if α0 = p + 1, α1 = a, α2 = ab, . . . , αp = ab p+1 2 c bb 2 c be a sequence that satisfies both fn and `n . Thereby, αn satisfies the recurrence relation   abαn−2 + (a + b) αn−p−2 + αn−2p−2 , if p is even, αn =  abαn−2 + 2αn−p−1 − αn−2p−2 , if p is odd. By considering eq. (9), the characteristic polynomial of αn is   x2p+2 − abx2p − (a + b) xp − 1, if p is even, αp (x) =  x2p+2 − abx2p − 2xp+1 + 1, if p is odd. By taking r = x2 , we can express the characteristic equation (10) as   rp+1 − abrp − (a + b) r p2 − 1, if p is even, βp (r) =  rp+1 − abrp − 2r p+1 2 + 1, if p is odd.

(9)

(10)

(11)

Lemma 2.1. Assume that p is odd. Then the characteristic equation of the generalized Fibonacci and Lucas p−numbers αp (x) does not have multiple roots. For the other case, it can easily seen that there are no multiple real roots. However, whether there exists complex multiple roots or not is an open problem, and we suggest that interested readers study it with us. Proof of Lemma. The characteristic equation of the generalized Fibonacci and Lucas p−numbers for odd p can be written in the form αp (x) = (xp+1 − 1)2 − abx2p and its derivative is αp0 (x) = 2(p + 1)xp (xp+1 − 1) − 2pabx2p−1 . 4

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Then, αp (x) = 0 if and only if ab =

(xp+1 − 1)2 x2p

and αp0 (x) = 0 if and only if (p + 1)x xp+1 − 1 (p + 1)xp (xp+1 − 1) = . 2p−1 px p xp   √ p+1 Upon simplifying, we obtain ab = x ab. Therefore, αp (x) and αp0 (x) vanish for the same x if and p only if for some root x of αp (x), √   √ p+1 p ab ab = x ab or equivalently, x = . p p+1 ab =

So, for every p and ab, if such an x is a root, it is a multiple root. Let t be a multiple root. Then,  2 p+1 t2 . Since αp (t) = 0, we have ab = p t

2p+2

2 p+1 − t2p+2 − 2tp+1 + 1 = 0 p (2p + 1) 2p+2 − t − 2tp+1 + 1 = 0 p2 

(2p + 1)t2(p+1) + 2p2 tp+1 − p2 = 0. When treated as a quadratic equation in tp+1 , the discriminant is 4p4 + 4p2 (2p + 1) = 4p2 (p + 1)2 , and therefore, the solutions are tp+1 = −p,

p . 2p + 1

But substituting the same ab in αp0 (t) = 0, we get p

p+1

2(p + 1)t (t

 − 1) − 2p

p+1 p

2

t2p+1 = 0

p(tp+1 − 1) − (p + 1)tp+1 = 0 tp+1 = −p  Then, by ab =

p+1 p

2

t2 , we have ab =

(p + 1)2 2p

.

(−p) p+1 The equation has multiple roots exactly when ab and p are related as above. Note that for odd values of p, ab will be a real number, and then, ab < 0. This is a contradiction. Therefore the characteristic equation αp (x) has distinct roots. The proof is complete. 5

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We will describe the terms of the sequence {αn } clearly by using the Binet formula. So, we can give the generalized Binet formula for the generalized Fibonacci and Lucas p−numbers with the following theorem. Theorem 2.1. Suppose that the characteristic equation (11) has (p+1) distinct roots, r1 , r2 , . . . , rp+1 . Then αn satisfies the relation 

αn

X



rk1 rk2 . . . rkp+1−j

 1≤k1 j . Therefore (i, j)-th element of V−1 can be written as  X   rk1 rk2 . . . rkp+1−j     1≤k

1 4.

They proved the regularities of the solution to

the linear stochastic problem corresponding to the stochastic Burgers’ equation and then obtained the global existence and uniqueness results of the stochastic Burgers’ equation. For more contributions on stochastic calculus with fractional noise, we refer the reader to [4, 16–18, 20, 22, 26, 29, 30] and reference therein. It should be noted that most of the papers and books on stochastic partial differential equations with fractional noise are devoted to the case of additive noise. However, there are few papers that consider the case of multiplicative fractional noise [2, 9, 14, 18]. Enlightened by the above contributions, in this paper we will consider a stochastic elastic equation driven by multiplicative multi-parameter white noise. Let H = (h0 , h1 , . . . , hd ) with

1 2

< hi < 1, i = 0, 1, . . . , d. Consider the following

stochastic elastic equation dut (t, x) + ∆2 u(t, x) = u(t, x) ⋄ W H (t, x), dt

(1.1)

where t > 0, x ∈ D = (0, 1)d with initial and boundary conditions u(0, x) = u0 (x),

∂u (0, x) = υ0 (x), x ∈ D, ∂t

u(t, x) = ∆u(t, x) = 0, on ∂D. We assume that u0 and υ0 are deterministic functions defined on [0, 1]d , u0 and ∆u0 both vanish at ∂D, W H is a (d + 1)-dimensional fractional white noise and u ⋄ W H is Wick product which will be defined in section 2. Stochastic elastic equation driven by Brownian motion has been studied by many authors, (see [5, 6, 12, 21, 31]). However, there are few papers consider stochastic elastic equation driven by fractional Brownian motion. Such equation is a fourth order partial differential equation and has very wide applications in structural engineering. As an engineering problem, it has its applications in beams, bridges and other structures, see [5, 26]. 2 696

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Our aim in this paper is to obtain the existence, uniqueness, the asymptotic behavior and the H¨older continuity of the mild solution of problem (1.1) in a distribution space. The keys to the proof are the Wiener chaos expansion of the solution and the undetermined coefficient method. Throughout the paper, we use the letter C denotes a constant that may not be the same form one occurrence to anther, even in the same line. We express the dependence on some parameters by writing the parameters as arguments, e.g. C = C(H). The remaining of this paper is organized as follows. In section 2, we give preliminaries of the stochastic integral with respect to multi-parameter fractional white noise. In section 3, we prove the the existence, uniqueness, asymptotic behavior and H¨older continuity in a distribution space of the solutions of problem (1.1).

2

Stochastic integral with respect to multi-parameter fractional white noise In this section, we introduce the definition of stochastic integral with respect to

the d−parameter fractional Brownian fields for Hurst index H = (h1 , · · · , hd ), ( 21 < hi < 1, i = 1, · · · , d) by using the fractional white noise analysis method. For more contributions about white noise analysis, we refer the reader to [3, 13]. Let h ∈ (0, 1). A fractional Brownian motion Bth , t ≥ 0, of Hurst index h is a continuous Gaussian stochastic process, such that for all s, t ∈ R+ , 1 B0h = 0, E(Bth ) = 0, E(Bth Bsh ) = (|t|2h + |s|2h − |t − s|2h ). 2

(2.1)

Since we are concerned with the fractional Brownian motions of multi-parameter, some notations must be introduced. Let u = (u1 , · · · , ud ) ∈ Rd , denote du = du1 · · · dud . Fix h with

1 2

< h < 1. We put ϕh (s, t) = h(2h − 1)|s − t|2h−2 , s, t ∈ R,

(2.2)

and ϕH (u, υ) =

d ∏

ϕhi (ui , υi ), u = (u1 , · · · , ud ), υ = (υ1 , · · · , υd ) ∈ Rd

(2.3)

i=1

for H = (h1 , · · · , hd ), ( 21 < hi < 1, i = 1, · · · , d). Let S(Rd ) be the Schwartz space of rapidly decreasing smooth functions on Rd and S ′ (Rd ) be the dual of S(Rd ), i.e., S ′ (Rd ) is the space of tempered distributions on Rd . 3 697

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∫ The action of ω ∈ S ′ (Rd ) on f ∈ S(Rd ) is given by ⟨ω, f ⟩ = Rd ω(x)f (x)dx. Denote ∫ ⟨f, g⟩H = f (u)g(υ)ϕH (u, υ)dudυ, f, g ∈ S(Rd ). (2.4) R2d

If we equip S(Rd ) with the inner product ⟨·, ·⟩H and the norm ∥f ∥H = ⟨f, f ⟩H , then the completion of S(Rd ), denote by L2ϕH (Rd ), becomes a separable Hilbert space. Now let Ω = S ′ (Rd ). The map f → e− 2 ∥f ∥H is positive definite on S(Rd ), by the Bochner1

2

Minlos theorem [13], there exists a probability measure PH on the Borel subset B(Ω) of Ω such that

∫

ei⟨ω,f ⟩ dPH (ω) = e− 2 ∥f ∥H , ∀f ∈ S(Rd ). 1

2

(2.5)

Ω

Let E denotes the expectation under the probability measure PH , then E[⟨·, f ⟩] = 0, E[⟨·, f ⟩2 ] = ∥f ∥2H .

(2.6)

Now define a square integrable stochastic field B H (x), x ∈ Rd as B H (x) = B H (x, ω) = ⟨ω, I[0,x] (·)⟩,

(2.7)

where x = (x1 , · · · , xd ) ∈ Rd , I[0,x] = I[0,x1 ] · · · I[0,xd ] and for every i ∈ {1, 2, · · · , d},  0 ≤ yi ≤ xi ,  1 −1 xi ≤ yi ≤ 0, excetp xi = yi = 0, I[0,xi ] (yi ) = (2.8)  0 otherwise. From (2.7), we see that B H (x) is a Gaussian field and for every x, y ∈ Rd , E[B H (x)] = 0, E[B H (x)B H (y)] =

d 1 ∏ (|xi |2hi + |yi |2hi − |xi − yi |2hi ), 2d

(2.9)

i=1

where we have used the well known identity ∫ xi ∫ yi 1 ϕh (s, t)dsdt = (|xi |2hi + |yi |2hi − |xi − yi |2hi ). 2 0 0

(2.10)

By Kolmogorov’s continuity theorem, B H (x) has a continuous version. The fractional Brownian field is defined as the continuous version of B H (x), which we sill denote it by B H (x). Similarly to the case of stochastic integral with respect to fractional Brownian motion for deterministic function [3, 7, 10], we have the following lemma. Lemma 2.1 If f ∈ L2H (Rd ), then variable and [∫ ] [∫ H E f (x)dB (x) = 0, E Rd

∫ Rd

f (x)dB H (x) is well-defined Gaussian random ]2 H

f (x)dB (x)

Rd

∫ = R2d

f (u)f (υ)ϕH (u, υ)dudυ. (2.11)

4 698

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

The Hermite polynomials hn (x) are defined by 1 2

hn (x) = (−1)n e 2 x

dn − 1 x2 (e 2 ), n = 0, 1, 2, . . . . dxn

(2.12)

For example, the first Hermite polynomials are h0 (x) = 1, h1 (x) = x, h2 (x) = x2 − 1, h3 (x) = x3 − 3x, · · · .

(2.13)

Let ξn (x) be the Hermite functions defined by √ 1 1 x2 ξn (x) = π − 4 ((n − 1)!)− 2 hn−1 ( 2x)e− 2 , n = 1, 2, . . . .

(2.14)

It is proved in [13] that ξn ∈ S(R) and the collection {ξn }∞ n=1 constitutes an orthonormal basis of L2 (R). The most important property of ξn used in this paper is { √ 1 Cn− 12 , |x| ≤ 2 n, |ξn (x)| ≤ n = 1, 2, . . . , √ 2 Ce−γx , |x| > 2 n,

(2.15)

where constants C and γ are independent of n. Lemma 2.2 [3] Let

1 2

h− 12

< h < 1. The fractional integral I− h− 12

I− where ch =

√

∫

∞

f (u) = ch

is defined by

3

(t − u)h− 2 f (t)dt,

(2.16)

u

h(2h − 1)Γ(3/2 − h)/(Γ(h − 1/2)Γ(2 − 2h)) and Γ denotes the gamma h− 12

function. Then I−

is an isometry from L2ϕh (R) to L2 (R). h− 21 −1 ) (ξn )(u).

Now we define ηnh (u) = (I−

Then by Lemma 2.2 and the properties of

2 ξn , {ηnh }∞ n=1 is an orthonormal basis of Lϕh (R).

Let δ = (δ1 , . . . , δd ) denote d-dimensional multi-indices with δi ∈ N, i = 1, . . . , d, where N is the set of natural numbers. Then the family of tensor products ηδ (x1 , x2 , . . . , xd ) := ηδh11 (x1 )ηδh22 (x2 ) · · · ηδhdd (xd ), δ = (δ1 , . . . , δd ) ∈ Nd

(2.17)

forms an orthonormal basis of L2ϕH (Rd ). Let {δ i = (δ1i , . . . , δdi )}∞ i=1 be a fixed ordering of Nd . From a detailed proof in [13], we can assume that the ordering has the properties that i < j ⇒ δ1i + δ2i + · · · + δdi ≤ δ1j + δ2j + · · · + δdj .

(2.18)

and 1

j d ≤ δ1j · δ2j · · · δdj ≤ j.

(2.19)

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Yinghan Zhang 695-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

( ) Let J = NN 0 c denote the set of all multi-indices α = (α1 , α2 , . . .) with elements αi ∈ N0 = N ∪ {0} and with compact support, i.e., with only finitely many αi ̸= 0. ∑ Denote |α| = ∞ i=1 αi . Define ej (x) = ηδj (x) = η hj1 (x1 )η hj2 (x2 ) · · · η hjd (xd ), x = (x1 , . . . , xd ), j = 1, 2, . . . . δ1

δ2

δd

(2.20)

Then {ej } forms an orthonormal basis of L2ϕH (Rd ). If α = (α1 , α2 , . . . , αm ) ∈ J , define Hα (ω) =

m ∏

hαi (⟨ω, ei ⟩).

(2.21)

i=1

Then we have the following fractional Wiener Ito chaos expansion theorem. Theorem 2.1 [3] The family {Hα }α∈J constitutes an orthogonal basis for L2 (PH ) and for α = (α1 , α2 , . . .) ∈ J , ∥Hα ∥2L2 (PH ) = E[Hα2 ] = α! = α1 !α2 ! · · · .

(2.22)

Moreover, if F ∈ L2 (PH ), then there exist constants cα ∈ R, α ∈ J , such that F (ω) =

∑

cα Hα (ω),

(2.23)

α∈J

where the convergence holds in L2 (PH ) and ∥F ∥2L2 (PH ) =

∑

α!c2α .

(2.24)

α∈J

Now we compute the Wiener Ito chaos expansion of the fractional Brownian field B H (x).

By (2.7), ∞ ∞ ∑ ∑ B (x) = ⟨ω, I[0,x] ⟩ = ⟨ω, ⟨I[0,x] , ei ⟩H ei ⟩ = ⟨I[0,x] , ei ⟩H ⟨ω, ei ⟩ H

=

∞ [∫ ∑ i=1

0

i=1

x∫ Rd

i=1

] ei (υ)ϕH (u, υ)dυdu Hε(i) (ω),

(2.25)

where ε(i) = (0, . . . , 0, 1, 0, . . .) denote the ith unit vector. The fractional Hida test function and distribution spaces are defined as follows. Definition 2.1 The fractional Hida test function space (S)H = ∑ (S)H,k is the set of all ψ(ω) = α∈J aα Hα (ω) ∈ L2 (PH ) such that ∥ψ∥2H,k =

∑

α!a2α (2N)kα < ∞, k ∈ N,

∩∞

k=0 (S)H,k ,

where

(2.26)

α∈J

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∏ γj if γ = (γ , γ , . . .) ∈ J . The fractional Hida distribution where (2N)γ = ∞ 1 2 j=1 (2j) ∪∞ ∗ ∗ space (S)H = q=0 (S)H,−q , where (S)∗H,−q is the set of all formal expansions G(ω) = ∑ α∈J bα Hα (ω) such that ∑

∥G∥2H,−q =

α!a2α (2N)−qα < ∞, q ∈ N.

(2.27)

α∈J

The family of seminorms ∥ · ∥H,k , k ∈ N gives rise to a topology on (S)H and (S)∗H can be identified with the dual of (S)H by the action ∑

≪ G, ψ ≫=

α!aα bα .

(2.28)

α∈J

Definition 2.2 G : Rd → (S)∗H is dx-integrable in (S)∗H if ≪ G(x), ψ ≫∈ L1 (Rd ), for all ψ ∈ (S)H . If G : Rd → (S)∗H is dx-integrable in (S)∗H , We define element of (S)∗H such that ∫ ∫ ≪ G(x)dx, ψ ≫= Rd

Rd

∫ Rd

(2.29)

G(x)dx to be the unique

≪ G(x), ψ ≫ dx, for all ψ ∈ (S)H .

(2.30)

The fractional noise W H (x) is defined by the formal derivative of B H (x) in (S)∗H , ] ∞ [∫ ∑ H W (x) = (2.31) ei (υ)ϕH (x, υ)dυ Hε(i) (ω). Rd

i=1

Then, W H (x) ∈ (S)∗H and

d H dx B (x)

= W H (x) in (S)∗H .

∑ ∗ Definition 2.3 The Wick product F ⋄ G of F (ω) = α∈J aα Hα (ω) ∈ (S)H and ∑ G(ω) = α∈J bα Hα (ω) ∈ (S)∗H is defined by   ∑ ∑ ∑  F ⋄ G(ω) = aα bβ Hα+β (ω) = aα bβ  Hγ (ω). (2.32) γ∈J

α,β∈J

α+β=γ

Based on the preparations above, now we define the fractional Wick Ito Skorohod integral as follows. Definition 2.4 Suppose G : Rd → (S)∗H is a given function and G(x) ⋄ W H (x) is dx∫ integrable in (S)∗H . Then the fractional Wick Ito Skorohod integral Rd G(x)dB H (x) is defined by

∫

∫ H

G(x)dB (x) = Rd

Rd

G(x) ⋄ W H (x)dx.

For a interval in Rd , the integral can be defined as ∫ x ∫ H G(y)dB (y) = G(y)I[0,x] (y)dB H (y). 0

(2.33)

(2.34)

Rd

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3

Main results Let H = (h0 , h1 , . . . , hd ) with

1 2

< hi < 1, i = 0, 1, . . . , d. In this section, we consider

the following stochastic elastic equation dut (t, x) + ∆2 u(t, x) = u(t, x) ⋄ W H (t, x), dt

(3.1)

where t > 0, x ∈ D = (0, 1)d , with initial and boundary conditions u(0, x) = u0 (x),

∂u (0, x) = υ0 (x), x ∈ D, ∂t

u(t, x) = ∆u(t, x) = 0, on ∂D. We assume that u0 and υ0 are deterministic functions defined on [0, 1]d , u0 and ∆u0 both vanish at x = 0 and x = 1, and that W H is a (d + 1)-dimensional fractional white noise has been defined in section 2. Let r = (r1 , . . . , rd ) ∈ Nd , x = (x1 , . . . , xd ) ∈ Rd . Define φr (x) =

√ ∏d 2d i=1 sin(ri πxi ).

Then {φr }r∈Nd satisfy the boundary conditions of (3.1) and compose of a complete orthonormal system on L2 (D) which diagonalize ∆ with λr = π |r| = π 2

2

2

d ∑

ri2 ,

(3.2)

i=1

the corresponding eigenvalues. For a given function g : D → R and ρ ∈ R, define  ∥g∥ρ,2

:= 

∑

1 2

(1 + |r| ) |⟨g, φr ⟩| 2 ρ

2

,

r∈Nd

where ⟨·, ·⟩ stands for the usual scalar product in L2 (D), and denote by H ρ,2 (D) the set of functions g : D → R such that ∥g∥ρ,2 < ∞. Notice that H ρ,2 (D) is a subspace of the fractional Sobolev space of fractional differential order α and integrability order p = 2 (see [27]). For a special case ρ = 0, it is clear that H ρ,2 (D) = L2 (D) and we will denote ∥ · ∥0,2 by ∥ · ∥. By Parseval’s identity, we have ∥g∥2 =

∑

|⟨g, φr ⟩|2 , ∀g ∈ L2 (D).

(3.3)

r∈Nd

Since the fundamental solution of υtt + ∆2 υ = 0, υ = ∆u = 0 on ∂D, υ|t=0 = ϕ(x), υt |t=0 = ψ(x) on D 8 702

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is given by υ(t, x) =

∫ ∫ ∑ ∑ sin(λr t) φr (x) ψ(y)φr (y)dy + cos(λr t)φr (x) ϕ(y)φr (y)dy, λr D D d d r∈N

r∈N

(3.4) we can define the solution of (3.1) as follows. Definition 3.1 A random field u = u(t, x) : R+ × D × Ω → R is said to be a solution of (3.1), if (i) u = u(t, x) : R+ × D × Ω → R is jointly measurable. (ii) There exists constant q ∈ N, such that for almost all x ∈ D and t ≥ 0, ∑ ∫ t ∫ sin(λr (t − s)) φr (x)φr (y)u(s, y)dB H (s, y) λ r 0 D d

r∈N

is well defined as an element of (S)∗H,−q , and that

2

∫ ∑ ∫ t∫

sin(λ (t − s)) r H

φr (x)φr (y)u(s, y)dB (s, y)

λr D 0 D

r∈Nd

dx < ∞, ∀t ≥ 0.

H,−q

(iii) It holds in (S)∗H,−q that ∫ ∑ sin(λr t) φr (x) υ0 (y)φr (y)dy u(t, x) = λr D r∈Nd ∫ ∑ + cos(λr t)φr (x) u0 (y)φr (y)dy D

r∈Nd

∑ ∫ t ∫ sin(λr (t − s)) φr (x)φr (y)u(s, y)dB H (s, y). (3.5) + λ r 0 D d r∈N

Let ∫ ∑ sin(λr t) U0 (t, x) = φr (x) υ0 (y)φr (y)dy λr D r∈Nd ∫ ∑ + cos(λr t)φr (x) u0 (y)φr (y)dy. D

r∈Nd

Then by the definition of the stochastic integral, the solution of (3.1), if it exists, can be written as ∑ ∫ t ∫ sin(λr (t − s)) u(t, x) = U0 (t, x) + φr (x)φr (y)u(s, y) ⋄ W H (s, y)dsdy. (3.6) λ r D d 0 r∈N

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If we assume that the solution of (3.1) exists in (S)∗H and it has a formal expansions u(t, x) =

∑

cα (t, x)Hα ,

(3.7)

α∈J

where cα (t, x), α ∈ J are coefficients of Wiener chaos expansion of u, which are undetermined. Let cα−ε(i) = 0 if αi = 0.

(3.8)

Then by the formal expansion of W H , we obtain that [∞ (∫ )] ∑ ∑ u(t, x) ⋄ W H (t, x) = cα−ε(i) (t, x) ei (υ)ϕH (t, x; υ)dυ Hα (ω). (3.9) α∈J

Rd+1

i=1

Brought (3.9) into (3.6), we derive that ∑ ∫ t ∫ sin(λr (t − s)) φr (x)φr (y)u(s, y) ⋄ W H (s, y)dsdy. u(t, x) − U0 (t, x) = λ r 0 D r∈Nd  ∞ ∑ ∑ ∫ t ∫ sin(λr (t − s)) ∑  = φr (x)φr (y) cα−ε(i) (s, y) λr D i=1 |α|≥1 r∈Nd 0 (∫ ) ] × ei (υ)ϕH (s, y; υ)dυ dsdy Hα . (3.10) Rd+1

Therefore, by (3.10) and (3.7), we get cα (t, x) = U0 (t, x) if α = 0 and for |α| ≥ 1, ∞ ∑ ∫ t ∫ sin(λr (t − s)) ∑ cα (t, x) = φr (x)φr (y) cα−ε(i) (s, y) λr 0 D i=1 r∈Nd (∫ ) × ei (υ)ϕH (s, y; υ)dυ dsdy.

(3.11)

Rd+1

We will need the following preliminaries and lemmas to estimate cα . The boundedness and H¨older continuity of U0 are given by the following lemma. Lemma 3.1 Assume that υ0 ∈ H ϱ,2 (D) for some ϱ ≥ −2 and u0 ∈ H ρ,2 (D) for some ρ ≥ 0, then U0 (t, ·) ∈ L2 (D), and sup ∥U0 (t, ·)∥ < +∞.

(3.12)

t∈R+

Moreover, if υ0 ∈ H ϱ,2 (D) for some ϱ ≥ 0 and u0 ∈ H ρ,2 (D) for some ρ ≥ 2, then, for any t, s ∈ R+ with |t − s| < 1, ∥U0 (t, ·) − U0 (s, ·)∥2 ≤ C|t − s|2 .

(3.13)

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Proof. It is clear that ∥U0 (t, ·)∥ ≤ ∥I(t, ·)∥ + ∥J(t, ·)∥, with ∫ ∑ sin(λr t) I(t, x) = φr (x) υ0 (y)φr (y)dy, λr D d r∈N ∫ ∑ J(t, x) = cos(λr t)φr (x) u0 (y)φr (y)dy. D

r∈Nd

By (3.2), (3.3) and the assumptions of υ0 , we have ∥I(t, ·)∥2 =

∑

| < I(t, ·), φr > |2

r∈Nd

=

∑ sin2 (λr t) ∑ | < φr , υ0 > |2 2 < +∞. | < φ , υ > | ≤ C r 0 λ2r |r|4 d d

r∈N

r∈N

Similarly, we have ∥J(t, ·)∥2 ≤ C

∑

| < φr , u0 > |2 < +∞.

r∈Nd

Thus, the first part of the lemma if proved. For the second part, we have ∥U0 (t, ·) − U0 (s, ·)∥2 ≤ 2(∥I1 (t, s; ·)∥2 + ∥I2 (t, s; ·)∥2 ), where

∫ ∑ sin(λr t) − sin(λr s) I1 (t, s; x) = φr (x) υ0 (y)φr (y)dy, λr D d r∈N ∫ ∑ (cos(λr t) − cos(λr s))φr (x) u0 (y)φr (y)dy. I2 (t, s; x) = D

r∈Nd

By the assumptions on u0 , υ0 , we get ∥I1 (t, s; ·)∥2 =

∑

| < I1 (t, s; ·), φr > |2

r∈Nd

∑ | sin(λr t) − sin(λr s)|2 | < φr , υ0 > |2 2 λ r r∈Nd ∑ | < φr , υ0 > |2 ≤ C(t − s)2 , ≤ C(t − s)2 =

r∈Nd

and ∥I2 (t, s; ·)∥2 =

∑

| < I2 (t, s; ·), φr > |2

r∈Nd

=

∑

| cos(λr t) − cos(λr s)|2 | < φr , υ0 > |2

r∈Nd

≤ C(t − s)2

∑

|λr |2 | < φr , υ0 > |2

r∈Nd

≤ C(t − s)

2

∑

(1 + |r|2 )2 | < φr , υ0 > |2 ≤ C(t − s)2 .

r∈Nd

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Bringing together the above estimates, we obtain ∥U0 (t, ·) − U0 (s, ·)∥2 ≤ C(t − s)2 , with a constant C independent of t. Thus the lemma is proved. ¶ ∫ Let Φi (t, x) = Rd+1 ei (s, y)ϕH (t, x; s, y)dsdy, then we have the following lemma. Lemma 3.2 There exists constant C = C(h), such that 1

sup |Φi (t, x)| ≤ C(h)iσ(h) (1 ∨ th− 2 ), i ∈ N,

(3.14)

x∈D 1

1

where h = max{h0 , h1 , . . . , hd }, 1 ∨ th− 2 = max{1, th− 2 } and { 1 2 2 4 (h − 3 ), if h < 3 , σ(h) = 1 2 2 2 (h − 3 ), if h ≥ 3 .

(3.15)

Proof. By the definitions of ei and ϕH , we have ∫ ei (s, y)ϕH (t, x; s, y)dsdy Rd+1

∫ =

R

Let

1 2

ηδhi0 (s)ϕh0 (t, s)ds 0

d ∫ ∏ k=1 R

ηδhik (y)ϕhk (x, y)dy.

(3.16)

k

< h < 1 and n ∈ N. It is proved in [? ] that ∫ t∧s 3 3 2 ϕh (t, s) = ch (s − u)h− 2 (t − u)h− 2 du.

where ch =

√

(3.17)

−∞

h(2h − 1)Γ(3/2 − h)/(Γ(h − 1/2)Γ(2 − 2h)). Therefore, by (2.15) and

Lemma 3.2, we have ∫ ∫ ∫ t∧s 3 3 h h− 2 h− h ηn (s)ϕh (t, s)ds = c ηn (s) (s − u) 2 (t − u) 2 duds h R −∞ R ∫ t ∫ +∞ 3 h− 23 h 2 h− 2 du (s − u) ηn (s)ds = ch (t − u) −∞ u ∫ t ∫ t 1 3 1 h− h− 23 2 h− 2 2 h I− ηn (u)du = ch (t − u) ξn (u)du = ch (t − u) ch −∞ [∫−∞ ] ∫ ≤ ch

(t − u)h− 2 n− 12 du + 3

√ |u|≤2 n

(t − u)h− 2 e−γu du . 3

1

√ |u|>2 n

2

√ If 0 ≤ t ≤ 2 n, it follows from (3.18) that ∫ ηnh (s)ϕh (t, s)ds R [∫ √ [ ] ] ∫ t −2 n 3 1 3 1 2 (t − u)h− 2 ≤ ch ue−γu du + (t − u)h− 2 n− 12 du √ u −∞ −2 n [ ] √ h− 3 √ 1 1 e−4γn − 1 1 ≤ ch n 2 (t + 2 n) 2 + n− 12 (t + 2 n)h− 2 4γ h − 12 √ 1 1 1 2 1 ≤ C(h)n− 12 (t + 2 n)h− 2 ≤ C(h)n 2 (h− 3 ) (1 ∨ th− 2 ).

(3.18)

(3.19)

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√ For the case of t > 2 n, we have [∫ ∫ R

ηnh (s)ϕh (t, s)ds

≤ ch

√ 2 n

√ −2 n

(t − u)h− 2 n− 12 du 3

1

] ] ∫ t 1 −γu2 h− 32 −γu2 e du + (t − u) ue du + √ (t − u) u −∞ 2 n [ ] √ √ √ 1 3 1 1 1 e−4γn (t + 2 n)h− 2 n− 12 (t + 2 n)h− 2 − (n− 12 − e−4γn )(t − 2 n)h− 2 ≤ ch + 1 h − 12 4γn 2 √ 1 1 1 2 1 ≤ C(h)n− 12 (t + 2 n)h− 2 ≤ C(h)n 2 (h− 3 ) (1 ∨ th− 2 ). (3.20) ∫

√ −2 n [

h− 32

With the estimates of (3.19), (3.20), we obtain from (3.16) that ∫

Rd+1

d ∏ 2 1 1 ei (s, y)ϕH (t, x; s, y) ≤ C(h0 , h1 , · · · , hd ) (δki ) 2 (hk − 3 ) (1 ∨ th1 − 2 ) k=0

≤ C(h0 , h1 , · · · , hd )(

d ∏

1

2

1

1

δki ) 2 (h− 3 ) (1 ∨ th− 2 ) ≤ C(h)iσ(h) (1 ∨ th− 2 ),

(3.21)

k=0

where h = max{h0 , h1 , · · · , hd } and σ(h) is given by (3.15). This proves the lemma. ¶ Using the above lemmas, now we can estimate cα (t, x). Lemma 3.3 Assume that all the assumptions of Lemma 3.1 are satisfied, then for every α ∈ J and t ≥ 0, cα (t, ·) ∈ L2 (D). Moreover, there exists constant C = C(h), such that ∥cα (t, ·)∥2 ≤ C(h)2|α| Proof.

(N)(2σ(h)+1)α 2 (t ∨ t2h+1 )|α| . α!

(3.22)

Let α = (α1 , α2 , . . . , αm ) and Cα (t) = ∥cα (t, ·)∥2 . By Cauchy-Schwartz

inequality, we obtain Cα (t) =

∑

| < cα (t, ·), φr > |2

r∈Nd

= ≤ ≤ ≤

2 m ∑ ∫ t ∫ sin(λr (t − s)) ∑ φr (y) cα−ε(i) (s, y)Φi (s, y)dsdy 0 D λr i=1 r∈Nd 2 ∫ m ∫ t ∑∑ sin(λr (t − s)) φr (y)cα−ε(i) (s, y)Φi (s, y)dyds m λr D 0 r∈Nd i=1 2 ∫ ∫ t ∫ m t ∑∑ sin(λr (t − s)) 2 m ds φr (y)cα−ε(i) (s, y)Φi (s, y)dy ds λr 0 0 D r∈Nd i=1 m ∫ t ∑ ∑ Ctm | < φr , cα−ε(i) (s, ·)Φi (s, ·) > |2 ds

≤ Ctm

i=1 0 r∈Nd m ∫ t ∑ i=1

0

∥cα−ε(i) (s, ·)Φi (s, ·)∥2 ds.

(3.23)

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1

Owing to Lemma 3.2, supx∈D |Φi (t, x)| ≤ C(h)iσ(h) (1 ∨ th− 2 ). Hence, by (3.23), Cα (t) ≤ Ctm

m ∫ ∑ 0

i=1

≤ C(h) tm 2

t

∥cα−ε(i) (s, ·)Φi (s, ·)∥2 ds

m ∑

∫ i

0

i=1

≤ C(h) (t ∨ t )m 2

t

2σ(h)

2h

m ∑

i

1

Cα−ε(i) (s)(1 ∨ (sh− 2 ))2 ds ∫

t

2σ(h) 0

i=1

Cα−ε(i) (s)ds.

(3.24)

Now let n = |α| = α1 + · · · + αm . By iterating the above equation, we get Cα (t) ≤ C(h)2 (t ∨ t2h )m

m ∑

∫ i2σ(h) 0

i=1

s ∫ s1

0

0

(i1 i2 · · · in )2σ(h) ×

i1 ,i2 ,...,in =1

∫

sn−2

···

0

Cα−ε(i) (s)ds m ∑

≤ C(h)2n (t ∨ t2h )n [1α1 2α−2 · · · mαm ] ∫ t∫

t

0

Cα−ε(i1 ) −ε(i2 ) ···−ε(in ) (sn−1 )dsn−1 · · · ds1 ds.

(3.25)

Since for some β ∈ J with βj = 0 we have Cβ−ε(j) = 0, and by Lemma 3.1, C0 (t) = ∥U0 (t, ·)∥ is bounded on R+ , thus we derive from (3.25) that n! Cα (t) ≤ C(h)2n (t ∨ t2h )n [1α1 2α−2 · · · mαm ] [1α1 2α2 · · · mαm ]2σ(h) × α1 ! · · · αm ! ∫ sn−2 ∫ t ∫ s ∫ s1 ··· C0 (sn−1 )dsn−1 · · · ds1 ds 0

0

0

0

≤ C(h)2n (t ∨ t2h )n = C(h)2|α|

n! tn [1α1 2α2 · · · mαm ]2σ(h)+1 α1 ! · · · αm ! n!

(N)(2σ(h)+1)α 2 (t ∨ t2h+1 )|α| . α!

(3.26)

This proves the lemma. ¶ The following lemma will be used in the proof of our main results. Lemma 3.4 Let constants a > 0, b ∈ R, q ∈ R. Then the sufficient and necessary condition for

∑

(aN)bα (2N)−qα < ∞

(3.27)

α∈J

is q > max{b + 1, b log2 a}. Proof. For α = (α1 , α2 , . . . , ) ∈ J , let α0 = max{j; αj ̸= 0}. Then ∑ α∈J

bα

(aN) (2N)

−qα

=

∞ ∑

∑

bα

(aN) (2N)

n=0 α∈J ,α0 =n

−qα

:=

∞ ∑

an .

n=0

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We assume q > b log2 a and q > b + 1, then a0 = 1 and for n ≥ 1, ∑ ∑ an = (aN)bα (2N)−qα = (aN)bα (2N)−qα α1 ,...,αn−1 ≥0,αn ≥1

α∈J ,α0 =n

( =

 ) n−1  ∞ ∞ ∑ ∏ ∑  (an)bαn (2n)−qαn  (aj)bαj (2j)−qαj 

αn =1

( =

j=1

αn =1

=

αj =0

   ]α ) n−1 ]α ∞ [ ∞ [ ∑ (an)b n  ∏  ∑ (aj)b j  (2n)q (2j)q αj =0

j=1

(an)b (2n)q − (an)b

n−1 ∏ j=1

n (2j)q (an)b ∏ (2j)q = . (2j)2 − (aj)b (2n)q (2j)2 − (aj)b

(3.28)

j=1

By (3.28), we have an (an)b (2n + 2)q (2n + 2)q − (an + a)b = × an+1 (an + a)b (2n)q (2n + 2)q ( )b [( )q ] ( ) n n+1 (an + a)b 1 q−b (an)b = = 1+ . − − n+1 n (2n)q n (2n)q

(3.29)

This gives q − b (an)b an −1≥ − . an+1 n (2n)q Hence

(

an −1 an+1

(3.30)

)

≥ q − b > 1. (3.31) ∑∞ Therefore, by Abel’s criterion for convergence, n=0 an < ∞. Conversely, if q ≤ ( ) an b log2 a, then, by (3.28), an = ∞. If q = b+1, then, by (3.29), lim inf n→∞ n an+1 −1 = ∑∞ 1 − 21 ( a2 )b < 1, by Abel’s criterion, n=0 an = ∞. Hence, condition q > max{b + lim inf n n→∞

1, b log2 a} is sufficient and necessary for the convergence of (3.27). ¶ We now present our main results. Theorem 3.1 Assume that all the conditions in Lemma 3.1 are satisfied, then there exists a unique solution of (3.1) in the sense of Definition 3.1. Proof.

From the above analyses and lemmas, we know that the stochastic field

u(t, x) with a formal expansions u(t, x) =

∑

cα (t, x)Hα ,

(3.32)

α∈J

where cα (t, x) = U0 (t, x) if α = 0 and for |α| ≥ 1, ∞ ∑ ∑ ∫ t ∫ sin(λr (t − s)) φr (x)φr (y) cα (t, x) = cα−ε(i) (s, y) λr 0 D d i=1 r∈N (∫ ) × ei (υ)ϕH (s, y; υ)dυ dsdy,

(3.33)

Rd+1

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is a solution of (3.1) in the sense of Definition 3.1, if for almost every x ∈ D and t ≥ 0, ∫ it belongs to (S)∗H,−q for some q ∈ N and D ∥u(t, x)∥2H,−q dx < +∞, (∀t ≥ 0). Now we ∫ compute D ∥u(t, x)∥2H,−q dx. By Lemma 3.3, we have ∫ ∑ ∥u(t, x)∥2H,−q dx = α!∥cα (t, ·)∥2 (2N)−qα D

α∈J

∑ C(h)2|α| (N)(2σ(h)+1)α (t2 ∨ t2h+1 )|α| (2N)−qα . α!

≤

(3.34)

α∈J

It is prove in [13] that for α ∈ J , |α|! ≤ α!(2N)2α . Therefore, ∫

∑ C(h)2|α| (N)(2σ(h)+1)α (t2 ∨ t2h+1 )|α| (2N)−qα (2N)2α |α|!

∥u(t, x)∥2H,−q dx ≤ D

=

(3.35)

α∈J

∞ ∑

(t2

∨

n!

n=0

)(2σ(h)+1)α ∑ ( 2 C(h) 2σ(h)+1 N (2N)−(q−2)α .

t2h+1 )n

(3.36)

|α|=n

If we choose a natural number q > max{4 + 2σ(h), 2 + 2 log2 C(h)}, then by Lemma )(2σ(h)+1)α ( 1 ∑ 3.4, |α|=n C(h) 2σ(h)+1 N (2N)−(q−2)α < ∞. Thus ∫ ∥u(t, x)∥2H,−q dx ≤ C(h) D

∞ ∑ (t2 ∨ t2h+1 )n n=0

n!

= C(h)et

2 ∨t2h+1

.

(3.37)

Now we prove the uniqueness. Assume that stochastic fields u(t, x), υ(t, x) with formal expansions u(t, x) =

∑

aα (t, x)Hα , υ(t, x) =

α∈J

∑

bα (t, x)Hα

(3.38)

α∈J

are two solutions of (3.1) in the sense of Definition 3.1. Then u(t, x) − υ(t, x) is a solution of of (3.1) in the sense of Definition 3.1 with zero initial conditions, that is, a0 (t, x) − b0 (t, x) = 0. Therefore, similarly to the proof of Lemma 3.3, we can derive that ∥aα (t, ·) − bα (t, ·)∥2 n! ≤ C(h)2n (t ∨ t2h )n [1α1 2α−2 · · · mαm ] [1α1 2α2 · · · mαm ]2σ(h) × α1 ! · · · αm ! ∫ t ∫ s ∫ s1 ∫ sn−2 ··· ∥a0 (sn−1 , ·) − b0 (sn−1 , ·)∥2 dsn−1 · · · ds1 ds = 0. (3.39) 0

0

0

0

Thus the theorem follows. ¶ The following corollary deals with the asymptotic properties of u(t, x) in (S)∗H,−q . 16 710

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Corollary 3.1 Let u(t, x) be the solution of (3.1) and q > max{4+2σ(h), 2+2 log2 C(h)}, where C(h) is given by (3.37), then u(t, x) ∈ (S)∗H,−q for almost all (t, x) ∈ R+ × [0, 1], and lim sup

∫ ln( D ∥u(t, x)∥2H,−q dx) t2h+1

t→+∞

< ∞.

(3.40)

Proof. By (3.37), we know that for sufficiently large t, ∫ 2h+1 ∥u(t, x)∥2H,−q dx ≤ C(h)et .

(3.41)

D

Therefore

∫ ln( D ∥u(t, x)∥2H,−q dx)

lim sup

t2h+1

t→+∞

ln(C(h)) + t2h+1 = 1. t2h+1 t→+∞

≤ lim sup

(3.42)

This shows the corollary. ¶ We now study the H¨older property of the trajectories of the solution of equation (3.1) in the distribution space (S)∗H,−q . Theorem 3.2 Fix T > 0 and let u(t, x) be the solution of (3.1) in time interval [0, T ] with initial conditions υ0 ∈ H β1 ,2 (D) for some β1 ≥ 0 and u0 ∈ H β2 ,2 for some β2 ≥ 2, then there exists q ∈ N, such that for any t, τ ∈ [0, T ] with |t − τ | < 1, ∫ ∥u(t, x) − u(τ, x)∥2H,−q dx ≤ C|t − τ |.

(3.43)

D

Proof. Let 0 ≤ τ < t ≤ T. First we estimate the term ∥cα (t, ·) − cα (τ, ·)∥. Since

=

cα (t, x) − cα (τ, x) ∑ ∫ t ∫ sin(λr (t − s)) r∈Nd

−

0

∑∫

r∈Nd

λr

D τ

0

∫

φr (x)φr (y)

∞ ∑

cα−ε(i) (s, y)Φi (s, y)dsdy

i=1 ∞

D

∑ sin(λr (τ − s)) φr (x)φr (y) cα−ε(i) (s, y)Φi (s, y)dsdy λr i=1

∞ ∑ ∫ t ∫ sin(λr (t − s)) ∑ = φr (x)φr (y) cα−ε(i) (s, y)Φi (s, y)dsdy λr τ D i=1 r∈Nd ∑ ∫ τ ∫ sin(λr (t − s)) − sin(λr (τ − s)) φr (x)φr (y) + λr D d 0

×

r∈N ∞ ∑

cα−ε(i) (s, y)Φi (s, y)dsdy

i=1

. = I(t, τ ; x) + J(t, τ ; x). By using the methods used in the proof of Lemma 3.3, we can derive that (N)(2σ(h)+1)α 2 (T ∨ T 2h+1 )|α| , α! (N)(2σ(h)+1)α 2 ≤ (t − τ )C(h)2|α| (T ∨ T 2h+1 )|α| . α!

∥I(t, τ ; ·)∥2 ≤ (t − τ )2 C(h)2|α| ∥J(t, τ ; ·)∥2

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(2σ(h)+1)α

Thus we have ∥cα (t, ·) − cα (τ, ·)∥2 ≤ C(t − τ )C(h)2|α| (N)

α!

definition of the norm ∥ · ∥2H,−q , we can divide the term

∫ D

(T 2 ∨ T 2h+1 )|α| . By the

∥u(t, x) − u(τ, x)∥2H,−q dx

into two terms, that is ∫ ∥u(t, x) − u(τ, x)∥2H,−q dx 



D

≤ 2 ∥U0 (t, ·) − U0 (τ, ·)∥2 +

∑

α!∥cα (t, ·) − cα (τ, ·)∥2 (2N)−qα 

α∈J ,|α|≥1

](2σ(h)+1)α ∑[ 2 1 ≤ C(t − τ ) (C(h)(T ∨ T h+ 2 )) 2σ(h)+1 N (2N)−qα . α∈J

Bringing together the above estimates and by Lemma 3.1, Lemma 3.3, we obtain that ∫ ∥u(t, x) − u(τ, x)∥2H,−q dx ≤ C(h, T )(t − τ ), D 1

if q > max{2σ(h)+2, 2 log2 C(h)(T ∨ T h+ 2 )}, where C(h) is given by (3.37). The proof is finished. ¶ Acknowledge This research is supported by the Fundamental Research Funds for the Central Universities under grant FRF-TP-15-100A1.

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[10] T.E. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I. Theory, SIAM J. Control Optim. 38 (2000) 582–612. [11] M. Erraoui, D. Nualart, Y. Ouknine, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet, Stoch. Dyn. 3 (2003) 121–139. [12] O.H. Galal, M.A. El-Tawil, A.A. Mahmoud, Stochastic beam equations under random dynamic loads, Inter. J. Solid. Struct. 39 (2002) 1031–1040. [13] H. Holden, B. ∅ksendal, J. Uboe, T. Zhang, Stochastic Partial Differential Equations, a Modeling, White Noise Functional Approach, Second Edition published by Springer Science+Business Media, LLC, 2010. [14] Y. Hu, Heat equation with fractional white noise potential, Appl. Math. Optim. 43 (2001) 221–243. [15] Y. Hu, D. Nualart, Stochastic heat equation driven by fractional noise and local time, Probab. Theory Related Fields 143 (2009) 285–328. [16] Y. Hu, J. Huang, D. Nualart, S. Tindel, Stochastic heat equations with general multiplicative Gaussian noise: H¨older continuity and intermittency, Electron. J. Probab. 20 (2015), 1–50. [17] J.W. Jeon, Y.T. Kim, Stochastic Green’s theorem for fractional Brownian sheet and its application, J. Korean Statistical Society, 41 (2012) 225–234. [18] Y. Jiang, K. Shi, Y. Wang, Stochastic fractional Anderson models with fractional noises, Chin. Ann. Math. 31B(1) (2010) 101–118. [19] Y. Jiang, X. Wang, Y. Wang, On a stochastic heat equation with first order fractional noises and applications to finance, J. Math. Anal. Appl. 396 (2012) 656–669. [20] Y. Jiang, T. Wei, X. Zhou, Stochastic generalized Burgers equation driven by fractional noise, J. Diff. Eqn. 252 (2012) 1934–1961. [21] J.U. Kim, On a stochastic plate equation, Appl. Math. Optim. 44 (2001) 33–48. [22] J. Liu, L. Yan, Solving a nonlinear fractional stochastic partial differential equation with fractional noise, J. Theor. Probab. 29 (2016), 307–347. [23] Y.S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin Heidelberg, 2008. [24] D. Nualart, Malliavin Calculus and Related Topics, Springer-Verlag, 1995. [25] B. Pasik-Duncan, T.E. Duncan, B. Maslowski, Linear stochastic equations in a Hilbert space with fractional Brownian motion, in: Control Theory Applications in Financial Engineering and Manufacturing, Springer-Verlag, 2006, pp. 201–222 (Chapter 11). [26] G.D. Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University press, Cambridge, UK, 1992. [27] L. Quer-Sardanyons, S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stochastic Process. Appl. 117 (2007) 1448–1472. [28] S. Tindel, C.A. Tudor, F. Viens, Stochastic evolution equations with fractional Brownian motion, Probab. Theory Related Fields 127 (2003) 186–204. [29] J.B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Mathematics, 1180, Springer-Verlag, Berlin, (1986) 265–439. [30] G. Wang, M. Zeng, B. Guo, Stochastic Burger’s equation driven by fractional Brownian motion, J. Math. Anal. Appl. 371 (2010) 210–222. [31] T. Zhang, Large deviations for stochastic nonlinear beam equations, J. Funct. Anal. 248 (2007) 175–201. 19 713

Yinghan Zhang 695-713

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

FOURIER SERIES OF SUMS OF PRODUCTS OF EULER FUNCTIONS AND THEIR APPLICATIONS TAEKYUN KIM, DAE SAN KIM, GWAN-WOO JANG, AND LEE CHAE JANG

Abstract. We consider three types of sums of products of Euler functions and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions.

1. Introduction Let Em (x) be the Euler polynomials given by the generating function ∞ ∑ 2 tm xt e = Em (x) , t e +1 m! m=0

(see [1, 2, 14]).

(1.1)

For any real number x, we let < x >= x − [x] ∈ [0, 1)

(1.2)

denote the fractional part of x. Here we will consider the following three types of sums of products of Euler functions and derive their Fourier series expansions. Further, we will express each of them in terms of Bernoulli functions Bm (< x >). ∑m (1) αm (< x >) = ∑ k=0 Ek (< x >)Em−k (< x >), (m ≥ 1); m 1 (2) βm (< x >) = k=0 k!(m−k)! Ek (< x >)Em−k (< x >), (m ≥ 1); ∑m−1 1 (3) γm (< x >) = k=1 k(m−k) Ek (< x >)Em−k (< x >), (m ≥ 2). For elementary facts about Fourier analysis, the reader may refer to any book (for example, see [8,13,15,18]). As to γm (< x >), we note that the polynomial identity (1.3 ) follows immediately from (4.16), which is in turn derived from the Fourier series expansion of γm (< x >). m−1 ∑

1 Ek (x)Em−k (x) k(m − k) k=1 ( ) m 4 ∑ m Em−s+1 =− (Hm−1 − Hm−s )Bs (x), m s m−s+1

∑m

(1.3)

s=0,s̸=1

1 j=1 j

where Hm = are the harmonic numbers. The obvious polynomial identities can be derived also for αm (< x >) and βm (< x >) from (2.21) and (2.24), and (3.15) and (3.18), respectively. It is remarkable that from the Fourier series expansion of the 2010 Mathematics Subject Classification. 11B68, 42A16. Key words and phrases. Fourier series, Euler polynomials, Euler functions. 1

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Fourier series of sums of products of Bernoulli functions and their applications

function identity:

∑m−1

1 k=1 k(m−k) Bk (
)Bm−k (< x >) we can derive the following corresponding polynomial

m−1 ∑

1 Bk (x) Bm−k (x) k (m − k) k=1 ( ) ( ) m−2 2 1 2 ∑ m 2 1 = 2 Bm + + Bm−k Bk (x) + Hm−1 Bm (x) , m 2 m m−k k m

(1.4) (m ≥ 2).

k=1

Simple modification of (1.3) yields m−1 ∑

1 2 B2k (x) B2m−2k (x) + B1 (x) B2m−1 (x) 2k (2m − 2k) 2m − 1 k=1 ( ) m 1 ∑ 1 2m 1 = B2k B2m−2k (x) + H2m−1 B2m (x) m 2k 2k m

(1.5)

k=1

+

2 B1 (x) B2m−1 , 2m − 1

(m ≥ 2) .

Letting x = 0 in (1.4) gives a slightly different version of the well-known Miki’s identity (see [3,6,16,17]): m−1 ∑

Setting x =

1 2

1 B2k B2m−2k 2k (2m − 2k) k=1 ( ) m 1 ∑ 1 2m 1 = B2k B2m−2k + H2m−1 B2m , (m ≥ 2) . m 2k 2k m k=1 ( ) ( ) ( ) m−1 in (1.5) with B m = 1−2 Bm = 21−m − 1 Bm = Bm 12 , we have m−1 2 m−1 ∑

1 B 2k B 2m−2k 2k (2m − 2k) k=1 ( ) m 1 ∑ 1 2m 1 = B2k B 2m−2k + H2m−1 B 2m , m 2k 2k m

(1.6)

(1.7) (m ≥ 2) ,

k=1

which is the Faber-Pandharipande-Zagier identity (see [4]). Some related works can be found in [9-12].

2. Fourier series of functions of the first type In this section, we consider the function αm (< x >) =

m ∑

Ek (< x >)Em−k (< x >), (m ≥ 1)

(2.1)

k=0

defined on (−∞, −∞) which is periodic of period 1. The Fourier series of αm (< x >) is ∞ ∑

2πinx A(m) , n e

(2.2)

n=−∞

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where

∫ A(m) n

1

=

3

αm (< x >)e−2πinx dx

0

∫

(2.3)

1

=

αm (x)e

−2πinx

dx.

0

Before proceeding further, we observe the following. ′ αm (x) =

m ∑

(kEk−1 (x)Em−k (x) + (m − k)Ek (x)Em−k−1 (x))

k=0

=

m ∑

(kEk−1 (x)Em−k (x) +

k=1

=

m−1 ∑

(m − k)Ek (x)Em−k−1 (x))

k=0

m−1 ∑

(k + 1)Ek (x)Em−k−1 (x) +

k=0

m−1 ∑

(m − k)Ek (x)Em−k−1 (x))

(2.4)

k=0

= (m + 1)

m−1 ∑

Ek (x)Em−1−k (x)

k=0

( From this, we have

= (m + 1)αm−1 (x). )′ αm+1 (x) = αm (x).Then we have m+2 ∫

1

αm (x)dx = 0

1 (αm+1 (1) − αm+1 (0)). m+2

(2.5)

Noting that Em (x + 1) + Em (x) = 2xm , we see that Em (1) + Em (0) = 2δm,0 . So, we have αm (1) − αm (0) = = =

m ∑ k=0 m ∑ k=0 m ∑

(Ek (1)Em−k (1) − Ek (0)Em−k (0)) ((−Ek (0) + 2δk,0 )(−Em−k (0) + 2δm−k,0 ) − Ek (0)Em−k (0))

(2.6)

(−2δm−k,0 Ek (0) − 2Em−k (0)δk,0 + 4δk,0 δm−k,0 )

k=0

= 4δm,0 − 4Em (0), (m ≥ 0). Thus, for m ≥ 1, αm (1) − αm (0) = −4Em . Also,

∫

1

1 (αm+1 (1) − αm+1 (0)) m+2 4 =− Em+1 . m+2

(2.7)

αm (x)dx = 0

(2.8)

(m)

Now, we would like to determine the Fourier coefficients An .

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Fourier series of sums of products of Bernoulli functions and their applications

Case 1 : n ̸= 0. ∫ 1 A(m) = αm (x)e−2πinx dx n 0

=−

]1 1 [ 1 αm (x)e−2πinx 0 + 2πin 2πin

∫

1

′ αm (x)e−2πinx dx

0

∫ 1 m+1 1 =− (αm (1) − αm (0)) + αm−1 (x)e−2πinx dx 2πin 2πin 0 m + 1 (m−1) 2 = An + Em 2πin ( πin ) m+1 m (m−2) 2 2 = An + Em−1 + En 2πin 2πin πin πin (m + 1)m (m−2) m + 1 2 2 = + A Em−1 + En (2πin)2 n 2πin πin πin ( ) 2 (m + 1)m m − 1 (m−3) 2 m+1 2 = A + E Em−1 + Em m−2 + (2πin)2 2πin n πin 2πin πin πin m+1 2 2 (m + 1)m(m − 1) (m−3) (m + 1)m 2 An + Em−2 + Em−1 + Em = (2πin)3 (2πin)2 πin 2πin πin πin = ··· =

(2.9)

m−1 (m + 1)m−1 (1) ∑ (m + 1)k−1 2 A + Em−k+1 , n (2πin)m−1 (2πin)k−1 πin k=1

where

∫

1

A(1) n =

α1 (x)e−2πinx dx =

0

∫

1

(2x − 1) e−2πinx dx = −

0

1 . πin

(2.10)

Hence =− A(m) n

m−1 ∑ (m + 1)k−1 2(m + 1)m−1 4 + Em−k+1 m (2πin) (2πin)k k=1

4 = m+2 Case 2: n = 0.

m ∑ (m + 2)k k=1

∫ (m)

A0

(2πin)k

Em−k+1 .

1

αm (x)dx = −

=

(2.11)

0

4 Em+1 . m+2

(2.12)

We recall here that 1 B1 = − , B2n+1 = 0, for n ≥ 1, (−1)n+1 B2n > 0, 2 (see [5], Proposition 15.1.1), and En = −

1 (2n+2 − 2)Bn+1 (n ≥ 0), n+1

(2.13)

(see [3]).

(2.14)

E2n = 0 (n ≥ 1), E2n−1 ̸= 0 (n ≥ 1), and E0 = 1.

(2.15)

From these, we see that

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From these and (2.7), we observe that αm (1) = αm (0)(αm (1) ̸= αm (0)) ⇐⇒ Em = 0(Em ̸= 0) ⇐⇒ m is an even positive integer (m is an odd positive integer).

(2.16)

Here αm (< x >) is piecewise C ∞ . In addition, αm (< x >) is continuous for all even positive integers m and discontinuous with jump discontinuities at integers for all odd positive integers m. We now recall the following facts about Bernoulli functions Bm (< x >): (a) for m ≥ 2, Bm (< x >) = −m!

∞ ∑ n=−∞,n̸=0

e2πinx . (2πin)m

(2.17)

(b) for m = 1, −

∞ ∑ n=−∞,n̸=0

e2πinx = 2πin

{

B1 (< x >), for x ∈ Zc , 0, for x ∈ Z,

(2.18)

where Zc = R − Z. Assume first that m is an even positive integer. Then αm (1) = αm (0). Thus αm (< x >) is piecewise C ∞ , and continuous. Hence the Fourier series of αm (< x >) converges uniformly to αm (< x >), and αm (< x >)

) m 4 ∑ (m + 2)k Em−k+1 e2πinx m+2 (2πin)k k=1 n=−∞,n̸=0   ( ) m ∞ 2πinx ∑ ∑ 4 4 m+2 e  =− Em+1 − Em−k+1 −k! m+2 m+2 k (2πin)k

4 =− Em+1 + m+2

(

∞ ∑

k=1

m ( ∑

n=−∞,n̸=0

) m+2 Em−k+1 Bk (< x >) k

(2.19)

4 4 Em+1 − m+2 m+2 k=2 ( ) { m+2 4 B1 (< x >), for x ∈ Zc , Em × − 0, for x ∈ Z m+2 1 ( ) m ∑ 4 m+2 =− Em−k+1 Bk (< x >), m+2 k =−

k=0,k̸=1

for all x ∈ (−∞, ∞). Hence we get the following theorem. Theorem ∑m 2.1. Let m be an even positive integer. Then we have the following. (a) k=0 Ek (< x >)Em−k (< x >) has the Fourier series expansion m ∑

Ek (< x >)Em−k (< x >)

k=0

4 =− Em+1 + m+2

∞ ∑ n=−∞,n̸=0

(

4 ∑ (m + 2)k Em−k+1 m+2 (2πin)k m

)

(2.20) e

2πinx

,

k=1

for all x ∈ (−∞, ∞), where the convergence is uniform.

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Fourier series of sums of products of Bernoulli functions and their applications

(b) m ∑

Ek (< x >)Em−k (< x >)

k=0

4 =− m+2

m ∑ k=0,k̸=1

( ) m+2 Em−k+1 Bk (< x >) k

(2.21)

for all x ∈ (−∞, ∞), where Bk (< x >) is the Bernoulli function. Assume next that m is an odd positive integer. Then αm (1) ̸= αm (0), and hence αm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers. Thus the Fourier series of αm (< x >) converges pointwise to αm (< x >), for x ∈ Zc , and converges to 1 (αm (0) + αm (1)) = αm (0) − 2Em , 2 for x ∈ Z., Thus we get the following theorem.

(2.22)

Theorem 2.2. Let m be an odd positive integer. Then we have the following. (a) ( ) ∞ m ∑ 4 ∑ (m + 2)k Em−k+1 e2πinx m+2 (2πin)k n=−∞,n̸=0 k=1 { ∑m Ek (< x >)Em−k (< x >), for x ∈ Zc , ∑k=0 = m for x ∈ Z. k=0 Ek Em−k − 2Em ,

(2.23)

(b) −

) m ( 4 ∑ m+2 Em−k+1 Bk (< x >) m+2 k k=1

=

m ∑

(2.24)

Ek (< x >)Em−k (< x >),

k=0

for x ∈ Zc ;

) m ( 4 ∑ m+2 − Em−k+1 Ek (< x >) m+2 k k=2

=

m ∑

(2.25)

Ek Em−k − 2Em ,

k=0

for x ∈ Z.

3. Fourier series of functions of the second type In this section, we consider the function βm (< x >) =

m ∑ k=0

1 Ek (< x >)Em−k (< x >), (m ≥ 1) k!(m − k)!

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(3.1)

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defined on (−∞, −∞) which is periodic of period 1. The Fourier series of βm (< x >) is ∞ ∑

Bn(m) e2πinx ,

(3.2)

n=−∞

where ∫

1

Bn(m) =

βm (< x >)e−2πinx dx

0

∫

1

=

(3.3) −2πinx

βm (x)e

dx.

0

Before proceeding further, we observe the following. } m { ∑ m−k k ′ βm Ek−1 (x)Em−k (x) + Ek (x)Em−k−1 (x) (x) = k!(m − k)! k!(m − k)! k=0

=

m ∑ k=1

=

m−1 ∑ k=0

m−1 ∑ 1 1 Ek−1 (x)Em−k (x) + Ek (x)Em−k−1 (x) (k − 1)!(m − k)! k!(m − k − 1)! k=0

(3.4)

m−1 ∑ 1 1 Ek (x)Em−1−k (x) + Ek (x)Em−1−k (x) k!(m − 1 − k)! k!(m − 1 − k)! k=0

= 2βm−1 (x). ′ (x) = 2βm−1 (x). From this, we have So, βm ( )′ βm+1 (x) = βm (x) 2

(3.5)

and ∫

1

βm (x)dx = 0

1 (βm+1 (1) − βm+1 (0)). 2

(3.6)

Using Em (1) + Em (0) = 2δm,0 , we observe that βm (1) − βm (0) =

m ∑ k=0

= =

m ∑ k=0 m ∑ k=0

=−

1 (Ek (1)Em−k (1) − Ek (0)Em−k (0)) k!(m − k)!

1 {(−Ek (0) + 2δk,0 )(−Em−k (0) + 2δm−k,0 ) − Ek (0)Em−k (0)} k!(m − k)!

(3.7)

1 {−2δk,0 Em−k (0) − 2Ek (0)δm−k,0 + 4δk,0 δm−k,0 } k!(m − k)!

4 (Em (0) − δm,0 ). m!

So, for m ≥ 1, βm (1) − βm (0) = −

720

4 Em . m!

(3.8)

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Fourier series of sums of products of Bernoulli functions and their applications

Also, we have

∫

1

1 βm (x)dx = (βm+1 (1) − βm+1 (0)) 2 0 ( ) 1 4 = − (Em+1 (0) − δm+1,0 ) 2 (m + 1)! 2 Em+1 . =− (m + 1)!

(3.9)

(m)

Now, we are ready to determine the Fourier coefficients Bn . Case 1: n ̸= 0. ∫ 1 Bn(m) = βm (x)e−2πinx dx 0

=−

1 1 [βm (x)e−2πinx ]10 + 2πin 2πin

∫

1

′ βm (x)e−2πinx dx

0

∫ 1 2 1 (βm (1) − βm (0)) + =− βm−1 (x)e−2πinx dx 2πin 2πin 0 1 (m−1) 2 1 = Bn + Em πin ( m! πin ) 1 1 (m−2) 2 1 2 1 = Bn + Em−1 + Em πin πin (m − 1)! πin m! πin 1 2 1 2 1 = Bn(m−2) + Em−1 + Em 2 2 (πin) (m − 1)! (πin) m! πin ) ( 2 2 1 1 1 (m−3) 2 1 1 = Bn + Em−2 + Em−1 + Em 2 2 (πin) πin (m − 2)! πin (m − 1)! (πin) m! πin 2 1 2 1 2 1 1 B (m−3) + Em−2 + Em−1 + Em = (πin)3 n (m − 2)! (πin)3 (m − 1)! (πin)2 m! πin = ··· =

(3.10)

m−1 ∑ 1 2 1 (1) + B Em−k+1 n m−1 (πin) (m − k + 1)! (πin)k k=1

=− =2

1 + (πin)m

m ∑ k=1

m−1 ∑ k=1

2 1 Em−k+1 (m − k + 1)! (πin)k

1 Em−k+1 . (m − k + 1)! (πin)k

Case 2: n = 0. By (3.9), we see that (m) B0

∫

1

βm (x)dx = −

= 0

2 Em+1 . (m + 1)!

(3.11)

From (3.8), we observe that βm (1) = βm (0)(βm (1) ̸= βm (0)) ⇐⇒ Em = 0(Em ̸= 0) ⇐⇒ m is an even positive integer (m is an odd positive integer).

(3.12)

Here βm (< x >) is piecewise C ∞ . In addition, βm (< x >) is continuous for all even positive integers m and discontinuous with jump discontinuities at integers for all odd positive integers m.

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Assume first that m is an even positive integer. Then βm (1) = βm (0). So βm (< x >) is piecewise C ∞ , and continuous. Thus the Fourier series of βm (< x >) converges uniformly to βm (< x >), and βm (< x >)

(

) Em−k+1 1 2 e2πinx (m − k + 1)! (πin)k n=−∞,n̸=0 k=1  ( ) m ∞ ∑ ∑ 1 2 m + 1 Em+1 − =− 2k+1 Em−k+1 −k! (m + 1)! (m + 1)! k

2 =− Em+1 + (m + 1)!

∞ ∑

k=1 m ∑

m ∑

n=−∞,n̸=0

 e2πinx  (2πin)k

( ) k+1 m + 1 2 Em−k+1 Bk (< x >) k

(3.13)

2 1 =− Em+1 − (m + 1)! (m + 1)! k=2 ( ) { 4 m+1 B1 (< x >), for x ∈ Zc , − Em × 0, for x ∈ Z (m + 1)! 1 ( ) m ∑ 1 m+1 =− 2k+1 Em−k+1 Bk (< x >), k (m + 1)! k=0,k̸=1

for all x ∈ (−∞, ∞). Hence we get the following theorem. Theorem m be an even positive integer. Then we have the following. ∑m 3.1. Let 1 (a) k=0 k!(m−k)! Ek (< x >)Em−k (< x >) has the Fourier series expansion m ∑

1 Ek (< x >)Em−k (< x >) k!(m − k)! k=0 ( m ) ∞ ∑ ∑ Em−k+1 2 1 =− Em+1 + 2 e2πinx , (m + 1)! (m − k + 1)! (πin)k n=−∞,n̸=0

(3.14)

k=1

for all x ∈ (−∞, ∞), where the convergence is uniform. (b) m ∑

1 Ek (< x >)Em−k (< x >) k!(m − k)! k=0 ( ) m ∑ 1 m+1 =− 2k+1 Em−k+1 Bk (< x >), (m + 1)! k

(3.15)

k=0,k̸=1

for all x ∈ (−∞, ∞), where Bk (< x >) is the Bernoulli function. Assume next that m is an odd positive integer. Then βm (1) ̸= βm (0), and hence βm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers. Thus the Fourier series of βm (< x >) converges pointwise to βm (< x >), for x ∈ Zc , and converges to 2 1 (βm (0) + βm (1)) = βm (0) − Em , 2 m! for x ∈ Z., Thus we get the following theorem.

(3.16)

Theorem 3.2. Let m be an odd positive integer. Then we have the following.

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Fourier series of sums of products of Bernoulli functions and their applications

(a) (

∞ ∑

2

n=−∞,n̸=0

{ ∑m ∑k=0 = m

m ∑ k=1

Em−k+1 1 (m − k + 1)! (πin)k

) e2πinx

1 k!(m−k)! Ek (< x >)Em−k (< 1 2 k=0 k!(m−k)! Ek Em−k − m! Em ,

(3.17)

x >), for x ∈ Zc , for x ∈ Z.

Here the convergence is pointwise. (b) −

( ) m ∑ 1 m+2 2k+1 Em−k+1 Bk (< x >) (m + 1)! k k=1

=

m ∑ k=0

(3.18)

1 Ek (< x >)Em−k (< x >), k!(m − k)!

for x ∈ Zc ; ( ) m ∑ 1 k+1 m + 1 − 2 Em−k+1 Bk (< x >) (m + 1)! k k=2

=

m ∑ k=0

(3.19)

1 2 Ek Em−k − Em , k!(m − k)! m!

for x ∈ Z.

4. Fourier series of functions of the third type In this section, we consider the function γm (< x >) =

m−1 ∑ k=1

1 Ek (< x >)Em−k (< x >) k(m − k)

(4.1)

defined on (−∞, −∞) which is periodic of period 1. The Fourier series of γm (< x >) is ∞ ∑

Cn(m) e2πinx ,

(4.2)

n=−∞

where ∫ Cn(m)

=

1

γm (< x >)e

−2πinx

∫ dx =

0

1

γm (x)e−2πinx dx.

(4.3)

0

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11

To proceed further, we note the following. ′ γm (x) =

m−1 ∑ k=1

=

m−2 ∑ k=0

=

m−1 ∑ 1 1 Ek−1 (x)Em−k (x) + Ek (x)Em−k−1 (x) m−k k k=1

m−1 ∑ 1 1 Ek (x)Em−1−k (x) + Ek (x)Em−1−k (x) m−1−k k k=1

2 Em−1 (x) + (m − 1) m−1

m−2 ∑ k=1

(4.4)

1 Ek (x)Em−1−k (x) k(m − 1 − k)

2 = (m − 1)γm−1 (x) + Em−1 (x). m−1 So, ′ γm (x) = (m − 1)γm (x) +

2 Em−1 (x). m−1

(4.5)

From this, we note that 1 m ∫

( γm+1 (x) −

2 Em+1 (x) m(m + 1)

)′ (4.6)

= γm (x).

1

γm (x)dx 0

]1 2 1 (γm+1 (x) − Em+1 (x)) m m(m + 1) 0 1 2 = (γm+1 (1) − γm+1 (0)) − 2 (Em+1 (1) − Em+1 (0)) m m (m + 1) 1 2 = (γm+1 (1) − γm+1 (0)) − 2 (−2Em+1 (0) + 2δm+1,0 ) m m (m + 1) 1 4 = (γm+1 (1) − γm+1 (0)) + 2 Em+1 . m m (m + 1) [

=

(4.7)

Observe that γm (1) − γm (0) =

m−1 ∑ k=1

=

m−1 ∑ k=1

=

m−1 ∑ k=1

1 (Ek (1)Em−k (1) − Ek (0)Em−k (0)) k(m − k) 1 ((−Ek (0) + 2δk,0 )(−Em−k (0) + 2δm−k,0 ) − Ek (0)Em−k (0)) k(m − k)

(4.8)

1 (−2δk,0 Em−k (0) − 2Ek (0)δm−k,0 + 4δk,0 δm−k,0 ) k(m − k)

= 0. Thus, for m ≥ 2, γm (1) − γm (0) = 0. Also, ∫ 1 1 4 4 γm (x)dx (γm+1 (x) − γm+1 (0)) + 2 Em+1 (0) = 2 Em+1 . m m (m + 1) m (m + 1) 0

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(4.9)

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We can show that ∫

1

Em−1 (x)e−2πinx dx =

0

m−1 ∑ k=1

(m − 1)k−1 Em−k . (2πin)k

(4.10)

(m)

Now, we are ready to determine the Fourier coefficients Cn . Case 1: n ̸= 0 ∫

1

Cn(m) =

γm (x)e−2πinx dx

0

=−

1 1 [γm (x)e−2πinx ]10 + 2πin 2πin

∫

1

′ γm (x)e−2πinx dx

0

∫ 1 1 1 2 =− (γm (1) − γm (0)) + (m − 1)γm−1 (x) + Em−1 (x))e−2πinx dx 2πin 2πin 0 m−1 { } ∫ 1 ∫ 1 2 1 −2πinx −2πinx dx + = (m − 1) γm−1 (x)e Em−1 (x)e dx 2πin m−1 0 0 m−1 m − 1 (m−1) 1 4 ∑ (m − 1)k−1 Cn + Em−k 2πin 2πin m − 1 (2πin)k k=1 ( ) m−2 m−1 4 1 ∑ (m − 2)k−1 1 4 ∑ (m − 1)k−1 m − 1 m − 2 (m−2) Cn + E + Em−k = m−k−1 2πin 2πin m − 2 2πin (2πin)k 2πin m − 1 (2πin)k

=

k=1

k=1

=

(m − 1)(m − 2) (m−2) m−1 4 Cn + 2 2 (2πin) (2πin) m − 2 +

1 4 2πin m − 1

m−1 ∑

m−2 ∑ k=1

(m − 2)k−1 Em−k−1 (2πin)k

(m − 1)k−1 Em−k (2πin)k

k=1

= ··· =

m−2 m−l (m − 1)! (2) ∑ (m − 1)l−1 4 ∑ (m − l)k−1 + C Em−k−l+1 (2πin)m−2 n (2πin)l m − l (2πin)k l=1

=− =

2(m − 1)! + (2πin)m

m−1 ∑ l=1

=

4 m

m−2 ∑ l=1

(m − 1)l−1 4 (2πin)l m − l

(m − 1)l−1 4 (2πin)l m − l

m−1 ∑ l=1

1 m−l

m−l ∑ k=1

k=1

m−l ∑ k=1

m−l ∑ k=1

(m − l)k−1 Em−k−l+1 (2πin)k

(m − l)k−1 Em−k−l+1 (2πin)k

(m)k+l−1 Em−k−l+1 (2πin)k+l

m s−1 ∑ 4 ∑ (m)s−1 1 = E m−s+1 m s=2 (2πin)s m−l l=1

m 4 ∑ (m)s Em−s+1 = (Hm−1 − Hm−s ), m s=2 (2πin)s m − s + 1

(4.11)

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where

∫ c(2) n =

1

γ2 (x)e−2πinx dx =

0

∫ 0

Case 2: n = 0.

∫ (m)

C0

1

( ) 1 −2πinx 2 x2 − x + e dx = − . 4 (2πin)2

1

=

13

γm (x)dx = 0

4 Em+1 . m2 (m + 1)

(4.12)

(4.13)

As γm (1) = γm (0), for all m ≥ 2, γm (< x >) is piecewise C ∞ , and continuous. Hence the Fourier series of γm (< x >) converges uniformly to γm (< x >), and γm (< x >)

(

) m 4 ∑ (m)s Em−s+1 (Hm−1 − Hm−s ) e2πinx m s=2 (2πin)s m − s + 1 n=−∞,n̸=0   ∞ m ( ) 2πinx ∑ 4 4 ∑ m Em−s+1 e  = 2 Em+1 − (Hm−1 − Hm−s ) −s! m (m + 1) m s=2 s m − s + 1 (2πin)s

4 = 2 Em+1 + m (m + 1)

∞ ∑

n=−∞,n̸=0

(4.14)

m ( ) 4 4 ∑ m Em−s+1 = 2 Em+1 − (Hm−1 − Hm−s )Bs (< x >) m (m + 1) m s=2 s m − s + 1 ( ) m 4 ∑ m Em−s+1 =− (Hm−1 − Hm−s )Bs (< x >) m s m−s+1 s=0,s̸=1

Finally, we obtain the following theorem. Theorem ∑m 4.1. 1Let m be an integer ≥ 2. Then we have the following. (a) k=0 k(m−k) Ek (< x >)Em−k (< x >) has the Fourier expansion m−1 ∑

1 Ek (< x >)Em−k (< x >) k(m − k) k=1 ( ) m ∞ ∑ 4 4 ∑ (m)s Em−s+1 = 2 Em+1 + (Hm−1 − Hm−s ) e2πinx , m (m + 1) m s=2 (2πin)s m − s + 1

(4.15)

n=−∞,n̸=0

for all x ∈ (−∞, ∞), where the convergence is uniform. (b) m−1 ∑

1 Ek (< x >)Em−k (< x >) k(m − k) k=1 ( ) m 4 ∑ m Em−s+1 =− (Hm−1 − Hm−s )Bs (< x >), s m−s+1 m

(4.16)

s=0,s̸=1

for all x ∈ (−∞, ∞), where Bs (< x >) is the Bernoulli function. References 1. A. Bayad, T. Kim, Higher recurrences for Apostol-Bernoulli-Euler numbers, Russ. J. Math. Phys., 19(1) (2012), 1-10. 2. D. Ding, J. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20(1)(2010), 7-21. 3. G. V. Dunne, C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys., 7(2)(2013), 225-249. 4. C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139(1)(2000), 173-199.

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5. G. J. Fox, Congruence relating rational values of Bernoulli and Euler polynomials, Fibonacci Quart. 39(1)(2001), 50-57. 6. I. M. Gessel, On Miki’s identities for Bernoulli numbers, J. Number Theory 110(1)(2005), 75-82. 7. K. Ireland, M. I. Rosen, A classical introduction to modern number theory, 2nd ed., Springer, 1990. 8. L. C. Jang, T. Kim, D. J. Kang, A note on the Fourier transform of fermionic p -adic integral on Zp , J. Comput. Anal. Appl., 11(3) (2009), 571-575. 9. D.S. Kim, T. Kim, Identities arising from higher-order Daehee polynomial bases, Open Math. 13(2015), 196-208. 10. D.S. Kim, T. Kim, Euler basis, identities, and their applications, Int. J. Math. Math. Sci. 2012, Art. ID 343981. 11. D.S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct., 24(9) (2013), 734-738. 12. D.S. Kim, T. Kim, S.-H. Lee, D.V. Dolgy, S.-H. Rim, Some new identities on the Bernoulli and Euler numbers, Discrete Dyn. Nat. Soc. 2011, Art. ID 856132. 13. T. Kim, A note on the Fourier transform of p-adic q-integrals on Zp , J. Comput. Anal. Appl., 11(1) (2009), 81-85. 14. T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20(1)(2010), 23-28. 15. J. E. Marsden, Elementary classical analysis, W. H. Freeman and Company, 1974. 16. H. Miki, A relation between Bernoulli numbers, J. Number Theory 10(3)(1978), 297-302. 17. K. Shiratani, S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A 36(1)(1982), 7383. 18. D. G. Zill, M. R. Cullen, Advanced Engineering Mathematics, Jones and Bartlett Publishers 2006. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China, Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Graduate School of Education, Konkuk University, Seoul 143-701, republic of Korea E-mail address: [email protected]

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Quotient subtraction algebras by an int-soft ideal Sun Shin Ahn1 and Young Hee Kim 1 2

2,∗

Department of Mathematics Education, Dongguk University, Seoul 04620, Korea

Department of Mathematics, Chungbuk National University, Cheongju, 28644, Korea

Abstract. The aim of this article is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of an intersectional soft subalgebra and an intersectional soft ideal of a subtraction algebra are introduced, and related properties are investigated. A quotient structure of a subtraction algebra using an intersectional soft ideal is constructed.

1. Introduction The real world is inherently uncertain, imprecise and vague. Various problems in system identification involve characteristics which are essentially non-probabilistic in nature [16]. In response to this situation Zadeh [17] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [18]. To solve complicated problem in economics, engineering, and environment, we can’t successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties can’t be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [14]. Maji et al. [13] and Molodtsov [14] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [14] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number 0

2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: γ-inclusive set; int-soft subalgebra; int-soft ideal; subtraction algebra. The corresponding author. 0 E-mail: [email protected] (S. S. Ahn); [email protected] (Y. H. Kim). 0

∗

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of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [13] described the application of soft set theory to a decision making problem. Maji et al. [12] also studied several operations on the theory of soft sets. Akta¸s and C ¸ a˘gman [4] studied the basic concepts of soft set theory, and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. Jun [9] discussed the union soft sets with applications in BCK/BCI-algebras. We refer the reader to the papers [1, 5, 7, 8, 15] for further information regarding algebraic structures/properties of soft set theory. In this paper, we discuss applications of the an intersectional soft sets in a subalgebra (an ideal) of a subtraction algebra. We introduce the notion of an intersectional soft subalgebra (ideal) of a subtraction algebra, and investigate some related properties. We consider a new construction of a quotient subtraction algebra induced by an int-soft ideal. Also we investigated some related properties. 2. Preliminaries

We review some definitions and properties that will be useful in our results (see [10]). By a subtraction algebra we mean an algebra (X, ∗, 0) with a single binary operation “ − ” that satisfies the following conditions: for any x, y, z ∈ S, (S1) x − (y − x) = x, (S2) x − (x − y) = y − (y − x), (S3) (x − y) − z = (x − z) − y. The subtraction determines an order relation on X: a ≤ b if and only if a−b = 0, where 0 = a−a ia an element that does not depend on the choice of a ∈ X. The ordered set (X; ≤) is a semiBoolean algebras in the sense of [2], that is, it is a meet semilattice with zero 0 in which every interval [0, a] is a Boolean algebra with respect to the induced order. Hence a ∧ b = a − (a − b); the complement of an element b ∈ [0, a] is a − b; and if b, c ∈ [0, a], then b ∨ c = (b′ ∧ c′ )′ = a − ((a − b) ∧ (a − c)) = a − ((a − b) − ((a − b) − (a − c))). In a subtraction algebra, the following are true: (a1) (x − y) − y = x − y, (a2) x − 0 = x and 0 − x = 0, (a3) (x − y) − x = 0,

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Quotient subtraction algebras by an int-soft ideal

(a4) (a5) (a6) (a7) (a8) (a9) (a10) (a11)

x − (x − y) ≤ y, (x − y) − (y − x) = x − y, x − (x − (x − y)) = x − y, (x − y) − (z − y) ≤ x − z, x ≤ y if and only if x = y − w for some w ∈ X, x ≤ y implies x − z ≤ y − z and z − y ≤ z − x for all z ∈ X, x, y ≤ z implies x − y = x ∧ (z − y), (x ∧ y) − (x ∧ z) ≤ x ∧ (y − z).

A non-empty subset A of a subtraction algebra X is called a subalgebra ([10]) of X if x − y ∈ A for any x, y ∈ A. A non-empty subset I of a subtraction algebra X is called an ideal ([10]) of X if (I1) 0 ∈ I, (I2) x − y, y ∈ I imply x ∈ I for any x, y, z ∈ X. A mapping f : X → Y of subtraction algebras is called a homomorphism if f (x−y) = f (x)−f (y) for all x, y ∈ X. Molodtsov [12] defined the soft set in the following way: Let U be an initial universe set and let E be a set of parameters. We say that the pair (U, E) is a soft universe. Let P(U ) denotes the power set of U and A, B, C, · · · ⊆ E. A fair (f˜, A) is called a soft set over U , where f˜ is a mapping given by f˜ : X → P(U ). In other words, a soft set over U is parameterized family of subsets of the universe U . For ε ∈ A, f˜(ε) may be considered as the set of ε-approximate elements of the set (f˜, A). A soft set over U can bd represented by the set of ordered pairs: (f˜, A) = {(x, f˜(x))|x ∈ A, f˜(x) ∈ P(U )}, where f˜ : X → P(U ) such that f˜(x) = ∅ if x ∈ / A. Clearly, a soft set is not a set. ˜ For a soft set (f , A) of X and a subset γ of U , the γ-inclusive set of (f˜, A), defined to be the set iA (f˜; γ) := {x ∈ A|γ ⊆ f˜(x)}. 3. Intersectional soft subalgebras In what follows let X denote a subtraction algebra unless otherwise specified. Definition 3.1. A soft set (f˜, X) over U is called an intersectional soft subalgebra (briefly, int-soft subalgebra of X if it satisfies: (3.1) f˜(x) ∩ f˜(y) ⊆ f˜(x − y) for all x, y ∈ X.

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Proposition 3.2. Every int-soft subalgebra (f˜, X) of a subtraction algebra X satisfies the following inclusion: (3.2) f˜(x) ⊆ f˜(0) for all x ∈ X. Proof. Using (3.1), we have f˜(x) = f˜(x) ∩ f˜(x) ⊆ f˜(x − x) = f˜(0) for all x ∈ X.

□

Example 3.3. Let (U = Z, X) where X = {0, 1, 2, 3} is a subtraction algebra ([11]) with the following Cayley table: ∗ 0 1 2 3

0 0 1 2 3

1 0 0 2 2

2 0 1 0 1

3 0 0 0 0

Let (f˜, X) be a soft set over U defined as follows:   Z if x = 0 ˜ f : X → P(U ), x 7→ 2Z if x ∈ {1, 2}  4Z if x = 3. It is easy to check that (f˜, X) is an int-soft subalgebra over U . Theorem 3.4. A soft set (f˜, X) of a subtraction algebra X over U is an int-soft subalgebra of X over U if and only if the γ-inclusive set iX (f˜; γ) is a subalgebra of X for all γ ∈ P(U ) with iX (f˜; γ) ̸= ∅. Proof. Assume that (f˜, X) is an int-soft subalgebra over U . Let x, y ∈ X and γ ∈ P(U ) be such that x, y ∈ iX (f˜; γ). Then γ ⊆ f˜(x) and γ ⊆ f˜(y). It follows from (3.1) that γ ⊆ f˜(x) ∩ f˜(y) ⊆ f˜(x − y) Hence x − y ∈ iX (f˜; γ). Thus iX (f˜, X) is a subalgebra of X. Conversely, suppose that iX (f˜; γ) is a subalgebra X for all γ ∈ P(U ) with iX (f˜; γ) ̸= ∅. Let x, y ∈ X, be such that f˜(x) = γx and f˜(y) = γy . Take γ = γx ∩ γy . Then x, y ∈ iX (f˜; γ) and so x − y ∈ iX (f˜; γ) by assumption. Hence f˜(x) ∩ f˜(y) = γx ∩ γy = γ ⊆ f˜(x − y). Thus (f˜, X) is an int-soft subalgebra over U . □ The subalgebra iX (f˜; γ) in Theorem 3.4 is called the inclusive subalgebra of X. Theorem 3.5. Every subalgebra of a subtraction algebra can be represented as a γ-inclusive set of an int-soft subalgebra.

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Quotient subtraction algebras by an int-soft ideal

Proof. Let A be a subalgebra of a subtraction algebra X. For a subset γ of U , define a soft set (f˜, X) over U by { γ if x ∈ A ˜ f : X → P(U ), x 7→ ∅ if x ∈ /A Obviously, A = iX (f˜; γ). We now prove that (f˜, X) is an int-soft subalgebra over U . Let x, y ∈ X. If x, y ∈ A, then x − y ∈ A because A is a subalgebra of X. Hence f˜(x) = f˜(y) = f˜(x − y) = γ, and so f˜(x) ∩ f˜(y) ⊆ f˜(x − y). If x ∈ A and y ∈ / A, then f˜(x) = γ and f˜(y) = ∅ which imply that f˜(x) ∩ f˜(y) = γ ∩ ∅ = ∅ ⊆ f˜(x − y). Similarly, if x ∈ / A and y ∈ A, then f˜(x) ∩ f˜(y) ⊆ f˜(x − y). Obviously, if x ∈ / A and y ∈ / A, then f˜(x) ∩ f˜(y) ⊆ f˜(x − y). Therefore (f˜, X) is an int-soft subalgebra over U . □ Any subalgebra of a subtraction algebra X may not be represented as a γ-inclusive set of an int-soft subalgebra (f˜, X) over U in general (see the following example). Example 3.6. Consider a subtraction algebra X = {0, 1, 2, 3} which is given Example 3.3. Consider a soft set (f˜, X) which is given by { {0, 1} if x = 0 f˜ : X → P(U ), x 7→ {1} if x ∈ {1, 2, 3} Then (f˜, X) is an int-soft subalgebra over U . follows:    X iX (f˜; γ) = {0}   ∅

The γ-inclusive set of (f˜, X) are described as if γ ∈ {∅, {1}} if γ ∈ {{0}, {0, 1}} otherwise.

The subalgebra {0, 2} cannot be a γ-inclusive set iX (f˜; γ) since there is no γ ⊆ U such that iX (f˜; γ) = {0, 2}. We make a new int-soft subalgebra from old one. Theorem 3.7. Let (f˜, X) be a soft set of a subtraction algebra X over U . Define a soft set (f˜∗ , X) of X over U by { f˜(x) if x ∈ iX (f˜; γ) ∗ ˜ f : X → P(U ), x 7→ ∅ otherwise where γ is a non-empty subset subset of U . If (f˜, X) is an int-soft subalgebra of X, then so is (f˜∗ , X).

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Proof. If (f˜, X) is an int-soft subalgebra over U , then iX (f˜; γ) is a subalgebra of X for all γ ⊆ U by Theorem 3.6. Let x, y ∈ X. If x, y ∈ iX (f˜; γ), then x − y ∈ iX (f˜; γ). Hence we have f˜∗ (x) ∩ f˜∗ (y) = f˜(x) ∩ f˜(y) ⊆ f˜(x − y) = f˜∗ (x − y). If x ∈ / iX (f˜; γ) or y ∈ / iX (f˜; γ), then f˜∗ (x) = ∅ or f˜∗ (y) = ∅. Thus f˜∗ (x) ∩ f˜∗ (y) = ∅ ⊆ f˜∗ (x − y). Therefore (f˜∗ , X) is an int-soft subalgebra over U .

□

Definition 3.8. A soft set (f˜, X) over U is called an intersectional ideal (briefly, int-soft ideal) of X if it satisfies (3.2) and (3.3) f˜(x − y) ∩ f˜(y) ⊆ f˜(x) for all x, y ∈ X. Example 3.9. (1) Let E = X be the set of parameters and let U = X be the universe set where X = {0, a, b, c} is a subtraction algebra ([3]) with the following Cayley table: ∗ 0 a b c

0 0 a b c

a 0 0 b c

b 0 a 0 c

c 0 a b 0

Let (f˜, X) be a soft set over U defined as follows: { f˜ : X → P(U ), x 7→

γ2 if x ∈ {0, a} γ1 if x ∈ {b, c}

where γ1 and γ2 are subsets of U with γ1 ⊊ γ2 It is easy to check that (f˜, X) is an int-soft ideal of X. (2) In Example 3.3, (f˜, X) is an int-soft subalgebra of X. But it is not an int-soft ideal of X, since f˜(3 − 1) ∩ f˜(2) = 2Z ⊈ 4Z = f˜(3). Proposition 3.10. Every int-soft ideal (f˜, X) of a subtraction algebra X satisfies the following inclusion: (i) (∀x, y ∈ X)(x ≤ y ⇒ f˜(y) ⊆ f˜(x)), (ii) (∀x, y, z ∈ X)(f˜((x − y) − z) ∩ f˜(y) ⊆ f˜(x − z)). Proof. (i) Let x, y ∈ X be such that x ≤ y. Then x − y = 0. Hence f˜(y) = f˜(0) ∩ f˜(y) = f˜(x − y) ∩ f˜(y) ⊆ f˜(x). (ii) Let x, y, z ∈ X. Using (S3) and (3.3), we have f˜((x − y) − z)) ∩ f˜(y) = f˜((x − z) − y)) ∩ f˜(y) ⊆ f˜(x − z). □

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Quotient subtraction algebras by an int-soft ideal

Theorem 3.11. Let (f˜, X) be a soft set of X over U . Then (f˜, X) is an int-soft ideal of X over U if and only if (3.4) (∀x, y, z ∈ X)(x − y ≤ z ⇒ f˜(z) ∩ f˜(y) ⊆ f˜(x)). Proof. Assume that (f˜, X) is an int-soft ideal of X over U . Let x, y and z ∈ X be such that x − y ≤ z. By Proposition 3.10(i) and (3.3), we have f˜(z) ∩ f˜(y) ⊆ f˜(x − y) ∩ f˜(y) ⊆ f˜(x). Conversely, suppose that (f˜, X) satisfies (3.4). By (a2), we get 0 ≤ x for any x ∈ X. Using Proposition 3.10(i), we have f˜(x) ⊆ f˜(0) for any x ∈ X. By (a4), we have x − (x − y) ≤ y for any x, y ∈ X. It follows from (3.4) that f˜(y) ∩ f˜(x − y) ⊆ f˜(x). Hence (3.3) hold. Therefore (f˜, X) is an int-soft ideal of X over U . □ Theorem 3.12. A soft set (f˜, X) of X over U is an int-soft ideal of a subtraction algebra X over U if and only if the γ-inclusive set iX (f˜; γ) is an ideal of X for all γ ∈ P(U ) with iX (f˜; γ) ̸= ∅. □

Proof. Similar to Theorem 3.4. The ideal iX (f˜; γ) in Theorem 3.12 is called the inclusive ideal of X.

Proposition 3.13. Let (f˜, X) be a soft set of a subtraction algebra X over U . Then the set Xf˜ := {x ∈ X|f˜(x) = f˜(0)} is an ideal of X. Proof. Obviously, 0 ∈ Xf˜. Let x, y ∈ X be such that x−y ∈ Xf˜ and y ∈ Xf˜. Then f˜(x−y) = f˜(0) and f˜(y) = f˜(0). By (3.3), we have f˜(0) = f˜(x − y) ∩ f˜(y) ⊆ f˜(x). It follows from (3.2) that f˜(x) = f˜(0). Hence x ∈ Xf˜. Therefore Xf˜ is an ideal of X. □ 4. Quotient subtraction algebras induced by an int-soft ideal Let (f˜, X) be an int-soft ideal of a subtraction algebra X. For any x, y ∈ X, we define a binary ˜ operation “ ∼f ” on X as follows: ˜ x ∼f y ⇔ f˜(x − y) = f˜(y − x) = f˜(0). ˜

Lemma 4.1. The operation ∼f is an equivalence relation on a subtraction algebra X. ˜

Proof. Obviously, it is reflexive and symmetric. Let x, y and z ∈ X be such that x ∼f y and ˜ y ∼f z. Then f˜(x − y) = f˜(y − x) = f˜(0) and f˜(y − z) = f˜(z − y) = f˜(0). By (a7), we have (x − z) − (y − z) ≤ x − y and (z − x) − (y − x) ≤ z − y. Using (3.4) and (3.2), we have f˜(0) = f˜(x − y) ∩ f˜(y − z) ⊆ f˜(x − z) ⊆ f˜(0) and f˜(0) = f˜(z − y) ∩ f˜(y − x) ⊆ f˜(z − x) ⊆ f˜(0). ˜ ˜ ˜ Hence f˜(x − z) = f˜(z − x) = f˜(0). Thus x ∼f z, that is, ∼f is transitive. Therefore ∼f is an equivalence relation. □ ˜

˜

˜

Lemma 4.2. For any x, y, u, v ∈ X, if x ∼f y and u ∼f v, then x − u ∼f y − v.

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Sun Shin Ahn and Young Hee Kim ˜ ˜ Proof. Let x, y, u, v ∈ X be such that x ∼f y and u ∼f v. Then f˜(x − y) = f˜(y − x) = f˜(0) and f˜(u − v) = f˜(v − u) = f˜(0). Since (x − u) − (y − u) ≤ x − y and (y − u) − (x − u) ≤ y − x, it follows from Proposition 3.10(i) that f˜(0) = f˜(x − y) ≤ f˜((x − u) − (y − u)) and f˜(0) = f˜(y − x) ≤ f˜((y − u) − (x − u)). By (3.2), we have f˜((x − u) − (y − u)) = f˜(0) and f˜((y − u) − (x − u)) = f˜(0). ˜ Hence x − u ∼f y − u. By (a4), (a9) and (S3), we have (y − (y − v)) − u = (y − u) − (y − v) ≤ v − u. Using Proposition 3.10(i), we obtain f˜(0) = f˜(v − u) ≤ f˜((y − u) − (y − v)). It follows from (3.2) that f˜((y − u) − (y − v)) = f˜(0). By a similar way, we get f˜((y − v) − (y − u)) = f˜(0). Hence ˜ ˜ ˜ ˜ ˜ y−v ∼f y−u. Since ∼f is symmetric, we have y−u ∼f y−v. Since ∼f is transitive, x−u ∼f y−v. ˜ Therefore ∼f is a congruence relation on X. □

Denote f˜x and X/f˜ the equivalence class containing x and the set of all equivalence classes of X, respectively, i.e., ˜ f˜x := {y ∈ X|y ∼f x} and X/f˜ := {f˜x |x ∈ X}. Define a binary relation − on X/f˜ as follows: f˜x − f˜y = f˜x−y for all f˜x , f˜y ∈ X/f˜. Then this operation is well-defined by Lemma 4.2. Theorem 4.3. If (f˜, X) is an int-soft ideal of a subtraction algebra X, then the quotient X/f˜ := (X/f˜; −) is a subtraction algebra. □

Proof. Straightforward.

Proposition 4.4. Let µ : X → Y be an epimorphism of subtraction algebras. If (f˜, Y ) is an int-soft ideal of Y , then (f˜ ◦ µ, X) is an int-soft ideal of X. Proof. For any x ∈ X, we have (f˜ ◦ µ)(x) = f˜(µ(x)) ⊆ f˜(0Y ) = f˜(µ(0X )) = (f˜ ◦ µ)(0X ) and (f˜ ◦ µ)(x) = f˜(µ(x)) ⊇ f˜(µ(x) −Y a) ∩ f˜(a) for any a ∈ Y . Let y be any preimage of a under µ. Then we have (f˜ ◦ µ)(x) ⊇ f˜(µ(x) −Y a)) ∩ f˜(a) = f˜(µ(x) − µ(y)) ∩ f˜(µ(y)) = f˜(µ(x −X y) ∩ f˜(µ(y)) = (f˜ ◦ µ)(x −X y) ∩ (f˜ ◦ µ)(y). Hence f˜ ◦ µ is an int-soft ideal of X.

□

Proposition 4.5. Let (f˜, X) be an int-soft ideal of a subtraction algebra X. The mapping γ : X → X/f˜, given by γ(x) := f˜x , is a surjective homomorphism, and Kerγ = {x ∈ X|γ(x) = f˜0 } = Xf˜.

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Quotient subtraction algebras by an int-soft ideal

Proof. Let f˜x ∈ X/f˜. Then there exists an element x ∈ X such that γ(x) = f˜x . Hence γ is surjective. For any x, y ∈ X, we have γ(x − y) = f˜x−y = f˜x − f˜y = γ(x) − γ(y). Thus γ is a homomorphism. Moreover, Ker γ = {x ∈ X|γ(x) = f˜0 } = {x ∈ X|x ∈ f˜0 } = {x ∈ ˜ X|x ∼f 0} = {x ∈ X|f˜(x) = f˜(0)} = Xf˜. It completes the proof. □ Proposition 4.6. Let a soft set (f˜, X) over U of a subtraction algebra X be an int-soft ideal of X. If J is an ideal of X, then J/f˜ is an ideal of X/f˜. Proof. Let a soft set (f˜, X) over U of a subtraction algebras X be an int-soft ideal of X and let J be an ideal of X. Then 0 ∈ J. Hence f˜0 ∈ J/f˜. Let f˜x , f˜y ∈ J/f˜ such that f˜x − f˜y ∈ J/f˜ and f˜y ∈ J/f˜. Since f˜x−y = f˜x − f˜y , we have x − y, y ∈ J. Since J is an ideal of X, we have x ∈ J. Hence f˜x ∈ J/f˜. □ Theorem 4.7. If J ∗ is an ideal of a quotient subtraction algebra X/f˜, then there exists an ideal J = {x ∈ X|f˜x ∈ J ∗ } in X such that J/f˜ = J ∗ . Proof. Since J ∗ is an ideal of X/f˜, f˜0 ∈ J ∗ . Hence 0 ∈ J. Let f˜x , f˜y ∈ J/f˜ be such that f˜x − f˜y , f˜y ∈ J ∗ . Since f˜x−y = f˜x − f˜y , we have x − y, y ∈ J. Since J ∗ is an ideal of J/f˜, f˜x ∈ J/f˜ and so x ∈ J. Therefore J is an ideal of X. By Proposition 4.6, we have J/f˜ = {f˜j |j ∈ J} ˜ = {f˜j |∃f˜x ∈ J ∗ such that j ∼f x} = {f˜j |∃f˜x ∈ J ∗ such that f˜x = f˜j } = {f˜j |f˜j ∈ J ∗ } = J ∗ .

□ Theorem 4.8. Let a soft set (f˜, X) over U be an int-soft ideal of a subtraction algebra X. If X/f˜ ∼ J is an ideal of X, then = X/J. J/f˜ X/f˜ X/f˜ = {[f˜x ]J/f˜|f˜ ∈ X/f˜}. If we define φ : → X/J by φ([f˜x ]J/f˜) = J/f˜ J/f˜ [x]J = {y ∈ X|x ∼J y}, then it is well defined. In fact, suppose that [f˜x ]J/f˜ = [f˜y ]J/f˜. Then ˜ f˜x ∼J/f f˜y and so f˜x−y = f˜x − f˜y , f˜y−x = f˜y − f˜x ∈ J/f˜. Hence y − x, x − y ∈ J. Therefore Proof. Note that

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X/f˜ , we have J/f˜ φ([f˜x ]J/f˜ − [f˜y ]J/f˜) = φ([f˜x − f˜y ]J/f˜) = [x − y]J

x ∼J y, i.e., [x]J = [y]J . Given [f˜x ]J/f˜, [f˜y ]J/f˜ ∈

= [x]J − [y]J = φ([f˜x ]J/f˜) − φ([f˜y ]J/f˜). Hence φ is a homomorphism. Obviously, φ is onto. Finally, we show that φ is one-to-one. If φ([f˜x ]J/f˜) = φ([f˜y ]J/f˜), then ˜ [x]J = [y]J , i.e., x ∼J y. If f˜a ∈ [f˜x ] ˜, then f˜a ∼J/f f˜x and hence f˜a−x , f˜x−a ∈ J/f˜. It follows J/f

that a − x, x − a ∈ J, i.e., a ∼J x. Since ∼J is an equivalence relation, a ∼J y and so Ja = Jy . ˜ Hence a − y, y − a ∈ J and so f˜a−y , f˜y−a ∈ J/f˜. Therefore f˜a ∼J/f f˜y . Hence f˜a ∈ [f˜y ]J/f˜. Thus [f˜x ]J/f˜ ⊆ [f˜y ]J/f˜. Similarly, we obtain [f˜y ]J/f˜ ⊆ [f˜x ]J/f˜. Therefore [f˜x ]J/f˜ = [f˜y ]J/f˜. It is completes the proof. □ References [1] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl. 59(2010) 3458-3463. [2] J. C. Abbott, Sets, Lattices and Boolean Algebras, Allyn and Bacon, Boston 1969. [3] S. S. Ahn, Y. H. Kim and K. J. Lee, A relation on subtraction algebras, Sci. Math. Jpn. Online (e-2006), 51-55. [4] H. Akta¸s and N. C ¸ a˘gman, Soft sets and soft groups, Inform. Sci. 177(2007) 2726-2735. [5] A. O. Atag¨ un and A. Sezgin, Soft substructures of rings, fields and modules, Comput. Math. Appl. 61 (2011) 592-601. [6] Y. C ¸ even and S. K¨ uc¸u ¨kko¸c, Quotient subtraction algebras, Inter. Math. Forum 6(2011) 1241-1247. [7] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621-2628. [8] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408-1413. [9] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Bull. Korean Math. Soc. 50 (2013), 1937-1956. [10] Y. B. Jun, H. S. Kim and E. H. Roh, Ideal theory of subtraction algebras, Sci. Math. Jpn. Online (e-2004), 397-402. [11] Y. B. Jun, Y. H. Kim and K. A. Oh, Subtraction algebras with additional conditions, Commun. Korrean Math. Soc. 22 (2007), 1-7. [12] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562. [13] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077-1083. [14] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19-31. [15] K. S. Yang and S. S. Ahn, Union soft q-ideals in BCI-algebras, Applied Mathematical Scineces 8(2014), 2859-2869. [16] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng. 50 (1962) 856-865. [17] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353. [18] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) - an outline, Inform. Sci. 172 (2005) 1-40.

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Characterization of weak sharp solutions for generalized variational inequalities in Banach spaces Natthaphon Artsawangb,1 , Ali Farajzadeha , Kasamsuk Ungchittrakoolb,c,∗ a Department of Mathematics, Razi University, Kermanshah, 67149, Iran of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand c Research Center for Academic Excellence in Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand b Department

Abstract In this paper, we study the solution set of variational like-inequalities (in this sense we are called η-variational inequalities) and introduce the notion of a weak sharp set of solutions to η-variational inequality problem in reflexive, strictly convex and smooth Banach space. We also present sufficient conditions for the relevant mapping to be constant on the solutions. Moreover, we characterize the weak sharpness of the solutions of η-variational inequality by primal gap function. Keywords: η-variational inequality, Gap function, Weakly sharp solution

1. Introduction Burke and Ferris [2] introduced the concept of a weak sharp minimum to present sufficient conditions for the finite identification, by iterative algorithm, of local minima associated with mathematical programming in space Rn . Patriksson [7] has generalized the concept of the weak sharpness of the solution set of a variational inequality problem (in short, VIP). Their concepts have been extended by Marcotte and Zhu [6] to introduce another the notion of weak sharp solutions for variational inequalities. They also characterized the weak sharp solutions in terms of a dual gap function for variational inequalities. The relevant results have been obtained by Zhang et al. [12]. It is further study by Wu and Wu [9–11]. Hu and Song [4] have extended the results of weak sharpness for the solutions of VIP under some continuity and monotonicity assumptions in Banach space. They also introduce the notion of weak sharp set of solutions to a variational inequality problem in a reflexive, strictly convex and smooth Banach space and present its several equivalent conditions. Liu and Wu [5] studied weak sharp solutions for the variational inequality in terms of its primal gap function. They also characterized the weak sharpness of the solution set of VIP in terms of primal gap function. Recently, AL-Hamidan et al. [1] give some characterization of weak sharp solutions for the VIP without considering the primal or dual gap function. In this paper, we provide some general two concepts of Liu and Wu [5] and Hu and Song [4] to study the weak sharpness of solution set of η-variational inequality problem in Banach space. We also give some characterizations of weak sharp solutions for the η-VIP and also present its several equivalent conditions. Our purpose in this paper is to develop the weak sharpness result in space Rn . ∗ Corresponding

author. Tel.:+66 55963250; fax:+66 55963201. Email addresses: [email protected] (Kasamsuk Ungchittrakool), [email protected] (Natthaphon Artsawang), [email protected] (Ali Farajzadeh) 1 Supported by The Royal Golden Jubilee Project Grant no. PHD/0158/2557, Thailand.

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The paper is organized as follows. Section 2 discuss the new concepts of the Gateaux differentiable and Lipschitz continuity of the primal gap function and we also introduce the main definitions. Several equivalent conditions for F to be constant are discuss and present some relationship among C η , Cη , Γ(x∗ ), and Λ(x∗ ) in Section 3. Finally, section 4 addresses the weak sharpness of C η in terms of the primal gap function is characterized. 2. Preliminaries and formulations Let E be a real Banach space with is topological dual space E ∗ and ⟨· , · ⟩ denote the pairing between E and E ∗ respectively. For a mapping from η : E × E to E . Let g be a mapping from E into Banach space Y . The mapping g is called directionally differentiable at a point x ∈ E in a direction v ∈ E if the limit g(x + tv) − g(x) g ′ (x, v) := lim t↓0 t exists. We say that g is directionally differentiable at x, if g is directionally differentiable at x in every direction v ∈ E. The mapping g is called Gateaux differentiable at x if g is directionally differentiable at x and the directional derivative g ′ (x, v) is linear and continuous in v and we denote this operator by ∇g(x), i.e. ⟨∇g(x), v⟩ = g ′ (x, v). Definition 2.1. Let g be a mapping from E into Banach space Y . The mapping g is called ηGateaux differentiable at x if g is Gateaux differentiable at x and there exists a unique ξ ∈ E ∗ such that ⟨ξ, η(v, 0)⟩ = ⟨∇g(x), v⟩ , ∀v ∈ E. We denote this operator by ∇η g(x) i.e. ⟨∇η g(x), η(v, 0)⟩ = g ′ (x, v). We defined η-subdifferential of a proper convex function f at x ∈ E is given by ∂η f (x) := {x∗ ∈ E ∗ : ⟨x∗ , η(y, x)⟩ ≤ f (y) − f (x), ∀y ∈ E}. Let C be a closed convex subset of E. The mapping PC : E → 2C defined by PC (x) := {y ∈ C : ∥x − y∥ = d(x, C)}, is called the metric projection operator. We known that if E is a reflexive and strictly convex Banach space, PC is a single-valued mapping. ∗ The duality mappings J : E → 2E and J ∗ : E ∗ → 2E are defined by J(x) = {x∗ ∈ E ∗ : ⟨x∗ , x⟩ = ∥x∗ ∥2∗ = ∥x∥2 }, ∀x ∈ E and

J ∗ (x∗ ) = {x ∈ E : ⟨x, x∗ ⟩ = ∥x∥2 = ∥x∗ ∥2∗ }, ∀x∗ ∈ E ∗ .

We know the following (see [8]) (i) (ii) (iii) (iv)

if if if if

E E E E

is is is is

smooth, then J is single-valued; reflexive, then J is onto; strictly convex, then J is one-to-one; strictly convex, then J is strictly monotone.

Thought out this paper, we let η : E × E to E be satisfy the following condition; (i) η is continuous on E × E; (ii) for any x, y ∈ E, η(x, y) = −η(y, x); 2

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(iii) for any x, y ∈ E and α, β are scalars, η(αx + βy, 0) = αη(x, 0) + βη(y, 0); (iv) there exists k > 0 such that ∥η(x, y)∥ = k∥x − y∥ for all x, y ∈ E; (v) η(E × {0}) = E. For a mapping g from a Banach space E into Banach space Y , we say that g is η-locally Lipschitz on E if for any x ∈ E there exist δ > 0 and L ≥ 0 such that ∥g(x) − g(y)∥ ≤ L∥η(x, y)∥, for all x, y ∈ B(x, δ). The following results are importance: E∗

Lemma 2.2 ([3]). Let E be a Banach space, J : E → 2

∫ a duality mapping and Φ(∥x∥) =

∥x∥

ds, 0

0 ̸= x ∈ X. Then J(x) = ∂Φ(∥x∥).

Lemma 2.3. Assume that E is a reflexive, strictly convex and smooth Banach space. Let C be a closed convex subset of E and x ˆ ∈ C. Then the following are equivalent: (i) x ˆ is a best approximation to x :∥η(x, x ˆ)∥ = inf ∥η(x, y)∥. y∈C

(ii) the inequality ⟨J(η(x, x ˆ)), η(y, x ˆ)⟩ ≤ 0, ∀y ∈ C holds. Proof. (ii) ⇒ (i) For each x ∈ E. Let x ˆ ∈ C such that ⟨J(η(x, x ˆ)), η(y, x ˆ)⟩ ≤ 0 ∀y ∈ C. Then ∥η(x, x ˆ)∥∥J(η(x, x ˆ))∥∗

= ⟨J(η(x, x ˆ)), η(x, x ˆ)⟩ ≤

⟨J(η(x, x ˆ)), η(x, x ˆ)⟩ + ⟨J(η(x, x ˆ)), η(ˆ x, y)⟩, ∀y ∈ C

= ⟨J(η(x, x ˆ)), η(x, y)⟩, ∀y ∈ C ≤

∥J(η(x, x ˆ))∥∗ ∥η(x, y)∥, ∀y ∈ C.

Hence, ∥η(x, x ˆ)∥ = inf ∥η(x, y)∥. y∈C

(i) ⇒ (ii) For each x ∈ E. Suppose that x ˆ ∈ C such that ∥η(x, x ˆ)∥ = inf ∥η(x, y)∥. y∈C

Since C is convex, we obtain that ∥η(x, x ˆ)∥ ≤ ∥η(x, (1 − t)ˆ x + ty)∥, ∀y ∈ C and t ∈ [0, 1], which implies that Φ(∥η(x, x ˆ)∥) − Φ(∥η(x, (1 − t)ˆ x + ty)∥), ∀y ∈ C and t ∈ [0, 1], ∫ x where Φ : R+ → R+ give by Φ(x) = ds, for all x ∈ R+ . 0

By Lemma 2.2, J(z) = ∂Φ(∥z∥). It follows that ⟨J(η(x, (1 − t)ˆ x + ty)), η(x, x ˆ) − η(x, (1 − t)ˆ x + ty)⟩

≤

Φ(∥η(x, x ˆ)∥) − Φ(∥η(x, (1 − t)ˆ x + ty)∥)

≤ 0, ∀y ∈ C and t ∈ [0, 1], that is, t⟨J(η(x, (1 − t)ˆ x + ty)), η(y, x ˆ)⟩ ≤ 0, ∀y ∈ C and t ∈ [0, 1]

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Therefore, ⟨J(η(x, (1 − t)ˆ x + ty)), η(y, x ˆ)⟩ ≤ 0, ∀y ∈ C and t ∈ [0, 1]. Taking t → 0, we have

⟨J(η(x, x ˆ)), η(y, x ˆ)⟩ ≤ 0, ∀y ∈ C.

Remark 2.4. By definition of η for each x ∈ E if x ˆ = PC (x) then ∥η(x, x ˆ)∥ = inf ∥η(x, y)∥. y∈C

If C is a closed convex subset of E and x ∈ C, then the η-tangent cone to C at x has the form TCη (x) =

{d ∈ E : there exists a bounded sequence {dk } ⊆ X with η(dk , 0) → d, tk ↓ 0 such that x + tk dk ∈ C, ∀k ∈ N}.

In the above, denote xk = x + tk dk ∈ C. Taking the limit as k → +∞, tk → 0, which implies that tk dk → 0, thereby leading to xk → x. Also from construction, η(xk , x) = η(dk , 0) → d. tk Thus, the η-tangent cone can be equivalently expressed as TCη (x) =

{d ∈ E : there exists sequence {xk } ⊆ C with xk → x, tk ↓ 0 η(xk , x) such that → d}. tk

Proposition 2.5. Consider a set C ⊆ E and x ∈ C. Then the following hold: (i) TCη (x) is closed; (ii) If C is convex, TCη (x) is the closure of the cone generated by η(C × {x}), that is, TCη (x) = cone(η(C × {x})) Proof. (i) Suppose that {dk } ⊆ TCη (x) such that dk → d. Since dk ∈ TCη (x), there exist {xrk } ⊆ C with xrk → x and {trk } ⊆ R+ with trk → 0 such that η(xrk , x) → dk , ∀k ∈ N. trk For a fixed k, there exists r such that ∥

η(xrk , x) 1 − dk ∥ < , ∀r ≥ r. trk k

Taking k → +∞, one can generate a sequence {xk } ⊆ C with xk → x and tk ↓ 0 such that η(xk , x) → d. tk Thus, d ∈ TCη (x). Hence, TCη (x) is closed. (ii) Suppose that d ∈ TCη (x), which implies that there exist {xk } ⊆ C with xk → x and {tk } ⊆ R+ with tk → 0 such that η(xk , x) → d. tk Observe that η(xk , x) ∈ η(C × {x}). Since tk > 0,

1 tk

> 0. Therefore,

η(xk , x) ∈ cone η(C × {x}). tk 4

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Thus, d ∈ cone(η(C × {x})). Hence, TCη (x) ⊆ cone(η(C × {x})). Conversely, for each x ∈ C. Define a sequence 1 (x − x), ∀k ∈ N. k By the convexity of C, it is obvious that {xk } ⊆ C. Taking k → +∞, xk → x, by construction, we obtain that xk = x +

kη(xk , x) = η(x, x). Set tk = k1 > 0, tk → 0 such that η(xtkk,x) → η(x, x), which implies that η(x, x) ∈ TCη (x). Since x ∈ C is arbitrary, η(C×{x}) ⊆ TCη (x). Because TCη (x) is cone, we have cone(η(C×{x})) ⊆ TCη (x). By (i), TCη (x) is closed, which implies that cone(η(C × {x})) ⊆ TCη (x). The η-normal cone to C at x is defined by NCη (x) := [TCη (x)]◦ , where A◦ := {x∗ ∈ E ∗ : ⟨x∗ , x⟩ ≤ 0, ∀x ∈ A}. If C is convex, then { {x∗ ∈ E ∗ : ⟨x∗ , η(c, x)⟩ ≤ 0 for all c ∈ C} η NC (x) := ∅,

if x ∈ C, if x ∈ / C.

Let C be a nonempty closed convex subset of reflexive, strictly convex and smooth Banach space E. For a mapping F from E into E ∗ , the η-variational inequality problem [η-VIP] is to find a vector x∗ ∈ C such that ⟨F (x∗ ), η(x, x∗ )⟩ ≥ 0 for all x ∈ C.

(2.1)

We denote the solution set of the η-VIP by C η The η-dual variational inequality problem [η-DVIP] is to find a vector x∗ ∈ C such that ⟨F (x), η(x, x∗ )⟩ ≥ 0 for all x ∈ C.

(2.2)

We denote the solution set of the η-DVIP by Cη Definition 2.6. The mapping F : E → E ∗ is said to be: (i) η-monotone on C if ⟨F (x) − F (y), η(y, x)⟩ ≥ 0 for all x, y ∈ C; (ii) η-pseudomonotone at x ∈ C if for each y ∈ C there holds ⟨F (x), η(y, x)⟩ ≥ 0 ⇒ ⟨F (y), η(y, x)⟩ ≥ 0; (iii) η-pseudomonotone + on C if it is η-pseudomonotone at each point in C and, for all x, y ∈ C, } ⟨F (y), η(x, y)⟩ ≥ 0 ⇒ F (x) = F (y). ⟨F (x), η(x, y)⟩ = 0 Now, we define the primal gap function g(x) associated with η-VIP (2.1) as g(x) := sup {⟨F (x), η(x, y)⟩}, for all x ∈ E, y∈C

and we setting Γ(x) := {y ∈ C : ⟨F (x, η(x, y))⟩ = g(x)}. Similarly, we define the dual gap function G(x) associated with η-DVIP (2.2) as G(x) := sup {⟨F (y), η(x, y)⟩}, for all x ∈ E, y∈C

and we setting Λ(x) := {y ∈ C : ⟨F (y, η(x, y))⟩ = G(x)}. 5

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3. Sufficient condition for constancy of F on C η and some properties of the primal gap function In this section, we discuss about relations among C η , Cη , Γ(x∗ ), and Λ(x∗ ). We study sufficient condition for F to be constant on C η and also study the η-Lipschitz continuity and ηsubdifferentiability of the primal gap function g in terms of the mapping F . Proposition 3.1. Let x ˆ ∈ C. Then (i) x ˆ ∈ C η ⇔ g(ˆ x) = 0 ⇔ x ˆ ∈ Γ(ˆ x); (ii) x ˆ ∈ Cη ⇔ G(ˆ x) = 0 ⇔ x ˆ ∈ Λ(ˆ x). Proof. (i) Consider x ˆ ∈ Cη

⇔

⟨F (ˆ x), η(y, x ˆ)⟩ ≥ 0, ∀y ∈ C

⇔

⟨F (ˆ x), η(ˆ x, y)⟩ ≤ 0, ∀y ∈ C

⇔

g(ˆ x) = 0.

And we also consider x ˆ ∈ Γ(ˆ x)

⇔ ⟨F (ˆ x), η(ˆ x, x ˆ)⟩ = g(ˆ x) ⇔ 0 = g(ˆ x).

Similarly, we can obtain (ii). Proposition 3.2. If F is η-pseudomonotone on C, C η ⊆ Cη . Proof. Immediate from the definitions. The following proposition we present a sufficient condition for F to be constant on C η . Proposition 3.3. Let F be η-pseudomonotone+ on C η . Then F is constant on C η Proof. Let x1 , x2 ∈ C η . Since F is η-pseudomonotone+ on C η , we have ⟨F (x1 ), η(x2 , x1 )⟩ ≥ 0 and ⟨F (x2 ), η(x1 , x2 )⟩ ≥ 0. By pseudomonotonicity of F on C η , we have ⟨F (x1 ), η(x1 , x2 )⟩ ≥ 0 it follows that ⟨F (x1 ), η(x1 , x2 )⟩ = 0. Since F is η-pseudomonotone+ on C η and ⟨F (x2 ), η(x1 , x2 )⟩ ≥ 0, implies that F (x1 ) = F (x2 ). Hence, F is constant on C η . Proposition 3.4. Let F be η-pseudomonotone+ on C and x∗ ∈ C η . Then C η = Λ(x∗ ) and F is constant on Λ(x∗ ). Proof. First, we prove that F is constant on Λ(x∗ ). For x∗ ∈ C η and c ∈ C, we have ⟨F (x∗ ), η(c, x∗ )⟩ ≥ 0. Since F is pseudomonotone, we get that ⟨F (c), η(c, x∗ )⟩ ≥ 0, ∀c ∈ C. It follows that G(x∗ ) = 0. For c ∈ Λ(x∗ ), we have ⟨F (c), η(c, x∗ )⟩ = −G(x∗ ) = 0 and hence F (c) = F (x∗ ).

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It sufficient to show that C η = Λ(x∗ ). (⊆): Let y ∗ ∈ C η . Then ⟨F (y ∗ ), η(x∗ , y ∗ )⟩ ≥ 0. Since x∗ ∈ C η ⊆ Cη , we have ⟨F (z), η(z, x∗ )⟩ ≥ 0, ∀z ∈ C. It follows that G(x∗ ) = 0, and ⟨F (y ∗ ), η(y ∗ , x∗ )⟩ ≥ 0. Therefore, ⟨F (y ∗ ), η(x∗ , y ∗ )⟩ = 0 = G(x∗ ), that is, y ∗ ∈ Λ(x∗ ). Thus C η ⊆ Λ(x∗ ). (⊇): Let y ∗ ∈ Λ(x∗ ). Then ⟨F (y ∗ ), η(x∗ , y ∗ )⟩ = G(x∗ ) ≥ 0. Since x∗ ∈ C η , we have ⟨F (x∗ ), η(y, x∗ )⟩ ≥ 0, ∀y ∈ C. Note that x∗ ∈ Λ(x∗ ), we have F (x∗ ) = F (y ∗ ). Consider, for all y ∈ C, 0 ≤ ⟨F (y ∗ ), η(y, x∗ )⟩ = ⟨F (y ∗ ), η(y, y ∗ )⟩ + ⟨F (y ∗ ), η(y ∗ , x∗ )⟩ implies 0 ≤ ⟨F (y ∗ ), η(x∗ , y ∗ )⟩ ≤ ⟨F (y ∗ ), η(y, y ∗ )⟩, ∀y ∈ C. Therefore, C η = Λ(x∗ ). Proposition 3.5. Suppose that F be η-pseudomonotone on C and x∗ ∈ C η . If F is constant on Γ(x∗ ) then F is constant on C η . And hence C η = Cη = Γ(x∗ ) = Λ(x∗ ). Proof. Since F is η-pseudomonotone on C, we have C η ⊆ Cη . Let y ∗ ∈ Cη . Then ⟨F (x∗ ), η(x∗ , y ∗ )⟩ ≥ 0. By assumption, we obtain that g(x∗ ) = 0 and hence ⟨F (x∗ ), η(y ∗ , x∗ )⟩ ≥ 0. It follows that ⟨F (x∗ ), η(x∗ , y ∗ )⟩ = 0 = g(x∗ ). Thus y ∗ ∈ Γ(x∗ ). Therefore C η ⊆ Cη ⊆ Γ(x∗ ). Let z ∗ ∈ Γ(x∗ ). Then ⟨F (x∗ ), η(x∗ , z ∗ )⟩ = g(x∗ ) = 0. From above x∗ ∈ C η ⊆ Γ(x∗ ) and F is constant on Γ(x∗ ), we obtain that F (x∗ ) = F (z ∗ ). Since x∗ ∈ C η , we have ⟨F (x∗ ), η(z, x∗ )⟩ ≥ 0, ∀z ∈ C. It follows that, for all z ∈ C, 0 ≤ ⟨F (z ∗ ), η(z, x∗ )⟩ = ⟨F (z ∗ ), η(z, z ∗ )⟩ + ⟨F (z ∗ ), η(z ∗ , x∗ )⟩ = ⟨F (z ∗ ), η(z, z ∗ )⟩ + ⟨F (x∗ ), η(z ∗ , x∗ )⟩ = ⟨F (z ∗ ), η(z, z ∗ )⟩. This implies that z ∗ ∈ C η . Thus Γ(x∗ ) ⊆ C η and hence C η = Cη = Γ(x∗ ). It sufficient to prove that Γ(x∗ ) = Λ(x∗ ). For c ∈ Γ(x∗ ), we have ⟨F (x∗ ), η(x∗ , c)⟩ = g(x∗ ) = 0, so ⟨F (c), η(x∗ , c)⟩ = 0 = G(x∗ ). Therefore, c ∈ Λ(x∗ ), which implies that Γ(x∗ ) ⊆ Λ(x∗ ). Now let c ∈ Λ(x∗ ). Then

⟨F (c), η(x∗ , c)⟩ = G(x∗ ) = 0. 7

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The pseudomonotonicity of F on C implies that ⟨F (x∗ ), η(x∗ , c)⟩ ≥ 0. In this case, ⟨F (x∗ ), η(x∗ , c)⟩ = 0 = g(x∗ ) since x∗ ∈ C η . Thus c ∈ Γ(x∗ ) and hence Λ(x∗ ) ⊆ Γ(x∗ ). Therefore C η = Cη = Γ(x∗ ) = Λ(x∗ ).

Proposition 3.6. Let F be η-pseudomonotone+ on C. Then, for x∗ ∈ C η , F is constant on Γ(x∗ ) if and only if C η = Cη = Γ(x∗ ) = Λ(x∗ ). Proof. (⇒)Suppose that F is constant on Γ(x∗ ). By Proposition 3.5, we obtain that C η = Cη = Γ(x∗ ) = Λ(x∗ ). (⇐)Assume that C η = Cη = Γ(x∗ ) = Λ(x∗ ). Let x1 , x2 ∈ Γ(x∗ ). Then ⟨F (x1 ), η(x2 , x1 )⟩ ≥ 0 and ⟨F (x2 ), η(x1 , x2 )⟩ ≥ 0 , because x1 , x2 ∈ C η . Since F is η-pseudomonotone and ⟨F (x2 ), η(x1 , x2 )⟩ ≥ 0, we obtain that ⟨F (x1 ), η(x1 , x2 )⟩ ≥ 0 , that is , ⟨F (x1 ), η(x2 , x1 )⟩ ≤ 0. Thus ⟨F (x1 ), η(x2 , x1 )⟩ = 0. Since F is η-pseudomonotone+ on C, we have F (x1 ) = F (x2 ). Proposition 3.7. Let F be η-pseudomonotone+ on C. Then the following are equivalent: (i) (ii) (iii) (iv)

F is constant on Γ(x∗ ) for each x∗ ∈ C η . C η = Cη = Γ(x∗ ) = Λ(x∗ ) for each x∗ ∈ C η . C η = Γ(x∗ ) = Λ(x∗ ) for each x∗ ∈ C η . C η = Γ(x∗ ) for each x∗ ∈ C η .

Proof. (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) are immediate. It suffices to show (iv) ⇒ (i). Suppose that C η = Γ(x∗ ) for each x∗ ∈ C η . Let x∗ ∈ C η and x1 , x2 ∈ Γ(x∗ ). Then x1 , x2 ∈ C η and ⟨F (x1 ), η(x2 , x1 )⟩ ≥ 0 and ⟨F (x2 ), η(x1 , x2 )⟩ ≥ 0. Since F is η-pseudomonotone and ⟨F (x2 ), η(x1 , x2 )⟩ ≥ 0, we obtain that ⟨F (x1 ), η(x1 , x2 )⟩ ≥ 0 , that is , ⟨F (x1 ), η(x2 , x1 )⟩ ≤ 0. Thus ⟨F (x1 ), η(x2 , x1 )⟩ = 0. Since F is η-pseudomonotone+ on C, we have F (x1 ) = F (x2 ). Next we prove that if F is η-locally Lipschitz on C η , then so is g. Lemma 3.8. Let C be compact. If F is η-locally Lipschitz on C η , then g is also η-locally Lipschitz on C η . Proof. Suppose that F is η-locally Lipschitz on C η . Let x∗ be any element in C η . Then there exist δ > 0 and L0 ≥ 0 such that ∥F (x) − F (y)∥ ≤ L0 ∥η(x, y)∥ and ∥F (x)∥ ≤ L0 for all x, y ∈ B(x∗ , δ).

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Let c ∈ Γ(x) with x ∈ B(x∗ , δ). Then g(x) − g(y)

≤ ⟨F (x), η(x, c)⟩ − ⟨F (y), η(y, c)⟩ = ⟨F (x), η(x, y)⟩ + ⟨F (x), η(y, c)⟩ − ⟨F (y), η(y, c)⟩ = ⟨η(x, y)⟩ + ⟨F (x) − F (y), η(y, c)⟩ ≤ ∥F (x)∥∥η(x, y)∥ + ∥F (x) − F (y)∥∥η(y, c)∥ ≤ L0 ∥η(x, y)∥ + L0 ∥η(x, y)∥∥η(y, c)∥.

By the compactness of C and definition of η implies that there exists a constant M ≥ 0 such that ∥η(y, c)∥ ≤ M for all y ∈ B(x∗ , δ) and c ∈ C. We set L = L0 + L0 M, we obtain that g(x) − g(y) ≤ L∥η(x, y)∥. We can conclude that g is η-locally Lipschitz on C η . The following Proposition 3.2 we present the η-subdifferential of g at x∗ ∈ C η is a singleton under sufficient condition. Proposition 3.9. Let F be η-monotone on X and x∗ ∈ C η . Suppose that g is finite on X and η-Gateaux differentiable at x∗ . Then ∂η g(x∗ ) = {F (x∗ )}. Proof. Since x∗ ∈ C η , we have g(x∗ ) = 0. For each y ∈ X and F is η-monotone, we obtain that g(y) − g(x∗ ) ≥ ⟨F (y), η(y, x∗ )⟩ ≥ ⟨F (x∗ ), η(y, x∗ )⟩. Hence F (x∗ ) ∈ ∂η g(x∗ ). Let z ∈ ∂η g(x∗ ). Then for each v ∈ X and t > 0, we get that g(x∗ + tv) − g(x∗ ) ≥ ⟨z, η(x∗ + tv, x∗ )⟩ = t⟨z, η(v, 0)⟩, that is,

g(x∗ + tv) − g(x∗ ) ≥ ⟨z, η(v, 0)⟩. t By the η-Gateaux differentiability of g at x∗ implies that g(x∗ + tv) − g(x∗ ) ≥ ⟨z, η(v, 0)⟩. t→0 t

⟨∇η g(x∗ ), η(v, 0)⟩ = lim

Therefore, ⟨z − ∇η g(x∗ ), η(v, 0)⟩ ≤ 0, for all v ∈ X. By definition of η we can set η(v, 0) = z − ∇η g(x∗ ), we have ∥z − ∇η g(x∗ )∥2 ≤ 0. This implies that z = ∇η g(x∗ ), and hence ∂η g(x∗ ) = {∇η g(x∗ )} = {F (x∗ )}. 4. Weak sharpness of C η Throughout this paper, we assume that C η and Cη are nonempty and that E is a reflexive, strictly convex, and smooth Banach space. We introduce the notion of weak sharpness solution for generalized variational inequality(η-VIP). Definition 4.1. The solution set C η is said to be weakly sharp, if F satisfies ∩ −F (x∗ ) ∈ int [TC (x) ∩ J ∗ NC η (x)]◦ for all x∗ ∈ C η . x∈C η

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Theorem 4.2. Let F be η-monotone on E and constant on Γ(x∗ ) for some x∗ ∈ C η . Suppose that g is η-Gateaux differentiable, η-locally Lipschitz on C η , and g(x) < +∞ for all x ∈ E. Then C η is weakly sharp if and only if there exists a positive number α such that αdηC η (x) ≤ g(x) for all x ∈ C,

(4.1)

where dηC η (x) := inf η ∥η(x, y)∥. y∈C

Proof. On the given assumption and by Proposition 3.5, we obtain that C η = Cη = Γ(x∗ ) = Λ(x∗ ). If C η is weakly sharp, then for any x∗ ∈ C η there exists α > 0 such that ∩ αBE ∗ ⊆ F (x∗ ) + [TC (x) ∩ J ∗ NC η (x)]◦ ,

(4.2)

x∈C η

where BE ∗ is the open unit ball in E ∗ . Since F is constant on Γ(x∗ ), α satisfies (4.2) for all x∗ ∈ C η . Therefore, for every y ∈ BE ∗ , we have ∩ [TC (x) ∩ J ∗ NC η (x)]◦ ⊆ [TC (x∗ ) ∩ J ∗ NC η (x∗ )]◦ . αy − F (x∗ ) ∈ x∈C η ∗

Thus, for every z ∈ [TC (x ) ∩ J ∗ NC η (x∗ )]. It follows that ⟨αy − F (x∗ ), z⟩ ≤ 0. Taking y =

Jz ∥Jz∥∗

(4.3)

in (4.3), we get that, for each x∗ ∈ C η , α∥Jz∥∗ =

α ⟨Jz, z⟩ ≤ ⟨F (x∗ ), z⟩. ∥Jz∥∗

This implies that for every z ∈ [TC (x∗ ) ∩ J ∗ NC η (x∗ )], we have α∥z∥ ≤ ⟨F (x∗ ), z⟩. For any x ∈ C, set x = PC η (x), we have η(x, x) ∈ TC (x) ∩ J ∗ NC η (x) by Proposition 2.5 and lemma 2.3. Therefore, ⟨F (x∗ ), η(x, x)⟩ ≥ α∥η(x, x)∥ = αdηC η (x). Conversely, suppose that there exists α > 0 such that αdηC η (x) ≤ g(x) for each x ∈ C. We claim that αBE ∗ ⊆ F (x∗ ) + [TC (x∗ ) ∩ J ∗ NC η (x∗ )]◦ for each x∗ ∈ C η .

(4.4)

If TC (x∗ ) ∩ J ∗ NC η (x∗ ) = {0} for x∗ ∈ C η , then [TC (x∗ ) ∩ J ∗ NC η (x∗ )]η = E and αBE ∗ ⊆ F (x∗ ) + [TC (x∗ ) ∩ J ∗ NC η (x∗ )]η , trivially. So it suffices to prove (4.4) to hold if TC (x∗ ) ∩ J ∗ NC η (x∗ ) ̸= {0} for x∗ ∈ C η . Now let 0 ̸= z ∈ TC (x∗ ) ∩ J ∗ NC η (x∗ ). By definition of η there exists a unique v ∈ E such that z = η(v, 0). Then ⟨J(η(v, 0)), η(v, 0)⟩ > 0 and ⟨J(η(v, 0)), η(y ∗ , x∗ )⟩ ≤ 0 for each y ∗ ∈ C η , which implies that C η is separated from x∗ + v by the hyperplane Hv = {x ∈ E : ⟨J(η(v, 0)), η(x, x∗ )⟩ = 0} = {x ∈ E : ⟨J(η(v, 0)), η(x, 0)⟩ = ⟨J(η(v, 0)), η(x∗ , 0)⟩}. 10

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Thus we can write Hv = {x ∈ E : ⟨J(η(v, 0)), η(x, 0)⟩ = β}, where β = ⟨J(η(v, 0)), η(x∗ , 0)⟩. Since η(v, 0) ∈ TC (x∗ ), for each positive sequence {ti } decreasing to 0, there exists a sequence {vi } such that {η(vi , 0)} converging to η(v, 0) and x∗ + ti vi ∈ C for sufficiently large i . By definition of η, we obtain that vi converging to v. Thus ⟨J(η(v, 0)), η(vi , 0)⟩ > 0 holds for sufficiently large i, and hence we suppose that x∗ + ti vi lies in the open set {x ∈ E : ⟨J(η(v, 0)), η(x, x∗ )⟩ > 0}. Therefore, dηC η (x∗ + ti vi ) ≥ dηHv (x∗ + ti vi ). For each x ∈ E. We set

(4.5) [⟨

y := x −

] ⟩ J(η(v, 0)), η(x, 0) − β v.

J(η(v, 0)) 2 ∗

A straightforward computation show that ⟨J(η(v, 0)), η(y, 0)⟩ = β, i.e., y ∈ Hv . Furthermore, for any z ∈ Hv , we have [⟨ ] ⟩ ⟩ ⟨ ⟩) ⟨ ⟩ J(η(v, 0)), η(x, 0) − β (⟨ J(η(x, y)), η(z, y) = J(η(v, 0)), η(z, 0) − J(η(v, 0)), η(y, 0)

J(η(v, 0)) 2 ∗

=

0.

By Lemma 2.3, we have dηHv (x∗ + ti vi )

∥η(x∗ + ti vi , y)∥ ⟨ ⟩ ti J(η(v, 0)), η(vi , 0) = ∥η(v, 0)∥

J(η(v, 0)) 2 ∗ ⟨ ⟩ ti J(η(v, 0)), η(vi , 0)

= ,

η(v, 0)

=

and hence, by (4.1), g(x∗ + ti vi ) ≥ αdC η (x∗ + ti vi ) ≥ αti

⟨Jη(v, 0), η(vi , 0)⟩ . ∥η(v, 0)∥

By Proposition 3.1, g(x∗ ) = 0 for any x∗ ∈ C η , so g(x∗ + ti vi ) − g(x∗ ) = g(x∗ + ti vi ) ≥ αti

⟨Jη(v, 0), η(vi , 0)⟩ . ∥η(v, 0)∥

Since g is η-locally Lipschitz and η-Gateaux differentiable on C η , there hold ∥g(x∗ + ti vi ) − g(x∗ + ti v)∥ ≤ Lti ∥η(vi , v)∥ for some L > 0 and all sufficiently large i and ⟨∇η g(x∗ ), η(v, 0)⟩

g(x∗ + ti v) − g(x∗ ) i→∞ ti g(x∗ + ti vi ) − g(x∗ ) = lim ≥ α∥η(v, 0)∥. i→∞ ti

=

lim

By Proposition 3.9, ∇η g(x∗ ) = F (x∗ ). Thus ⟨F (x∗ ), η(v, 0)⟩ ≥ α∥η(v, 0)∥.

11

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This implies that for each w ∈ BE ∗ , ⟨αw − F (x∗ ), η(v, 0)⟩ = ⟨αw, η(v, 0)⟩ − ⟨F (x∗ ), η(v, 0)⟩ ≤ α∥η(v, 0)∥ − α∥η(v, 0)∥ = 0. Hence αBE ∗ − F (x∗ ) ⊆ [TC (x∗ ) ∩ J ∗ NC η (x∗ )]◦ , that is, αBE ∗ ⊆ F (x∗ ) + [TC (x∗ ) ∩ J ∗ NC η (x∗ )]◦ . This shows that C η is weakly sharp since F is constant on C η . Corollary 4.3 ([5]). Let F be monotone on Rn and constant on Γ(x∗ ) for some x∗ ∈ C ∗ . Suppose that g is Gateaux differentiable, locally Lipschitz on C ∗ , and g(x) < +∞ for all x ∈ Rn . Then C ∗ is weakly sharp if and only if there exists a positive number α such that αdC ∗ (x) ≤ g(x) for all x ∈ C. Proof. By applying above Theorem 4.2, if we define η(x, y) = x − y, for all x, y ∈ E and space E = Rn , then C η can be reduce to C ∗ , where C ∗ is the solution set of variational inequalities. Moreover, the mapping g is Gateaux differentiable and locally Lipschitz on C ∗ .

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgements The third author would like to thank Naresuan University and The Thailand Research Fund for financial support. Moreover, N. Artsawang is also supported by The Royal Golden Jubilee Program under Grant PHD/0158/2557, Thailand. References [1] AI-Homidan, S., Ansari, Q.H., Nguyen, L.V.: Finite convergence analysis and weak sharp solutions for variational inequalities. Optim. Lett. 10, 805-819 (2016) [2] Burke, J.V., Ferris, M.C.:Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340-1359 (1993) [3] Dhara, A., Dutta, J.:Optimality conditions in convex optimization a finite-dimensional view. CRC press Taylor and Francis Group, New York(2012) [4] Hu, Y.H., Song,W.:Weak sharp solutions for variational inequalities in Banach spaces. J.Math. Anal. Appl. 374, 118-132 (2011) [5] Liu, Y., Wu, Z.: Characterization of weakly sharp solutions of a variational inequality by its primal gap function. Optim. Lett. 10, 563-576 (2016) [6] Marcotte, P., Zhu, D.L.: Weak sharp solutions of variational inequalities. SIAM J. Optim. 9, 179-189 (1998) [7] Patriksson, M.: A Unified Framework of Descent Algorithms for Nonlinear Programs and Variational Inequalities, Ph.D. thesis, Department of Mathematics, Linkping Institute of Technology, Linkping, Sweden (1993) 12

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[8] Takahashi, W. :Nonlinear functional analysis, Fixed point theory and its applications, Yokohama Publ., Yokohama,(2000). 2 [9] Wu, Z.L.,Wu, S.Y.:Weak sharp solutions of variational inequalities in Hilbert spaces. SIAM J. Optim. 14, 1011-1027 (2004) [10] Wu, Z.L., Wu, S.Y.: Gateaux differentiability of the dual gap function of a variational inequality. Eur. J. Oper. Res. 190, 328-344 (2008) [11] Wu, Z.L.,Wu, S.Y.: Characterizations of the solution sets of convex programs and variational inequality problems. J. Optim. Theory Appl. 130, 339-358 (2006) [12] Zhang, J.Z., Wan, C.Y., Xiu, N.H.: The dual gap functions for variational inequalities. Appl. Math. Optim. 48, 129-148 (2003)

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INEQUALITIES OF HERMITE-HADAMARD TYPE FOR n-TIMES DIFFERENTIABLE ( ; m)-LOGARITHMICALLY CONVEX FUNCTIONS M. A. LATIF, S. S. DRAGOMIR 1;2 , AND E. MOMONIAT Abstract. In this paper, some new integral inequalities of Hermite-Hadamard type are presented for functions whose nth derivatives in absolute value are ( ; m)-logarithmically convex. From our results, several inequalities of Hermite-Hadamard type can be derived in terms of functions whose …rst and second derivatives in absolute value are ( ; m)-logarithmically convex functions as special cases. Our results may provide re…nements of some results for ( ; m)-logarithmically convex functions already exist in the most recent concerned literature of inequalities.

1. Introduction Let us …rst refresh our knowledge how the following de…nition of classical convex functions is generalized. De…nition 1. A function f : I ! R, ; 6= I (1.1)

f (tx + (1

t)y)

tf (x) + (1

R, is said to be convex on I if the inequality

t)f (y)

holds for all x; y 2 I and t 2 [0; 1]. The inequalities in (1.1) are swapped if f is a concave function. The de…nition of convex functions plays an important role in the theory of convex analysis and in many other branches of pure and applied mathematics. A number of remarkable and signi…cant results in the theory of inequality hinge on this de…nition. One of the momentous results which uses the notion of convexity is stated as follows: Z b a+b 1 f (a) + f (b) (1.2) f f (x) dx ; 2 b a a 2 where f : ; 6= I R ! R, is a convex function of single variable, a, b 2 I with a < b. The inequalities in (1.2) are celebrated as Hermite-Hadamard inequality and are overturned if f is a concave function. The inequalities (1.2) have been target of extensive research because of its usefulness and usages in the theory of inequalities and in various other branches of mathematics. A vast literature is reported on the HermiteHadamard type inequalities during the past few years which generalize, improve and extend the inequalities (1.2), see for example [6, 12, 13, 14, 15, 17, 19, 23, 28] and closely related references therein. The classical convexity has been generalized in diverse ways such as s-convexity, m-convexity, ( ; m)convexity, h-convexity, logarithmic-convexity, s- logarithmic convexity, ( ; m)- logarithmic convexity and hlogarithmic-convexity but we will focus on the following generalizations of the classical convexity to prove our results. De…nition 2. [2, 33, 34] If a function f : I (1.3)

f ( x + (1

) y)

R ! (0; 1) satis…es

1

[f (x)] [f (y)]

for all x, y 2 I, 2 [0; 1], the function f is called logarithmically convex on I. If the inequality (1.3) reverses, the function f is called logarithmically concave on I. The above stated concept logarithmically convex functions is further generalized as in the de…nitions below. De…nition 3. [9] A function f : [0; b] ! (0; 1) is said to be m-logarithmically convex if f (tx + m (1

t) y)

t

m(1 t)

[f (x)] [f (y)]

holds for all x, y 2 [0; b], t 2 [0; 1] and m 2 (0; 1]. De…nition 4. [9] A function f : [0; b] ! (0; 1) is said to be ( ; m)-logarithmically convex if f (tx + m (1

t) y)

t

[f (x)]

m(1 t )

[f (y)]

holds for all x, y 2 [0; b], t 2 [0; 1] and ( ; m) 2 (0; 1]

(0; 1].

Date : Today. 2000 Mathematics Subject Classi…cation. Primary 26D15, 26D20, 26E60; Secondary 41A55. Key words and phrases. Hermite-Hadamard’s inequality, ( ; m)-logarithmically convex function, Hölder integral inequality. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1

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2

It is also obvious that if m = 1 in De…nition 3 and if ( ; m) = (1; 1) in De…nition 4, the notion of m-logarithmic convexity and ( ; m)-logarithmic convexity recapture the notion of usual logarithmic convexity. Many papers have been written by a number of mathematicians concerning Hermite-Hadamard type inequalities for di¤erent classes of convex functions see for instance the recent papers [2, 3, 4, 7, 8, 9, 16, 18, 24, 25, 27, 29, 31, 32, 33, 35] and the references within these papers. The main purpose of the present paper is to establish new Hermite-Hadamard type integral inequalities by using the notion of m- and ( ; m)-logarithmically convex functions and a new identity for n-times di¤erentiable functions from [19]in Section 2. 2. Main Results We will use the following Lemmas to establish our main results in this section. Lemma 1. [19] Let f : I R ! R be a function such that f (n) exists on I and f (n) 2 L ([a; b]) for n 2 N, where a, b 2 I with a < b, we have the identity h i k Z b n 1 k 1 + ( 1)k (b a) X 1 a+b f (a) + f (b) f (x) dx f (k) (2.1) 2 b a a 2k+1 (k + 1)! 2 k=1 n Z 1 1 t (b a) 1+t n 1 (1 t) (n 1 + t) f (n) = n+1 a+ b dt 2 n! 0 2 2 n n Z 1 1 t 1+t ( 1) (b a) n 1 (1 t) (n 1 + t) f (n) b+ a dt; + 2n+1 n! 2 2 0 where an empty sum is understood to be nil. Lemma 2. [20] If (2.2)

Z

> 0 and n 2 N [ f0g, then 8 Pn ( 1)n+1 n! > < (ln )n+1 + n! k=0 (n

1

tn

t

dt =

> :

0

Lemma 3. If (2.3)

(2.4)

G ( ; ) :=

> 0 and

1

t)

t

dt =

0

1 n+1 ;

Pn

1 k=0 (n k)!(ln )k+1 ;

6= 1 = 1:

u in Lemma 2, we get (2.3).

1 k X (ln ) ( )k

1

< 1;

k=1

where ( )k =

n!

> 0, we have

1

(1

= 1:

n! (ln )n+1

Proof. By making the substitution t = 1 Z

6= 1

1 n+1 ;

> 0 and N [ f0g, then 8 > Z 1 < n E (n; ) := (1 t) t dt = > 0 :

Lemma 4. [7] For

( 1)k ; k)!(ln )k+1

( + 1) ( + 2) ::: ( + k

1) :

From Lemma 3 and Lemma 4, by simple computations we get the following results. Lemma 5. If (2.5)

> 0 and n 2 N, then

F (n; ) := nE (n

Lemma 6. For (2.6)

1; )

> 0 and

E (n; ) =

> 0, we have

H ( ; ) := nG ( ; )

G ( + 1; ) =

8 > < > :

n! (ln 1) (ln )n+1

+

1 ln

n n+1 ;

( )k+1 =

(2.7)

s

ln 1 k=1 (n k)!(ln )k+1 ;

6= 1 = 1:

1

< 1;

( + 1) ( + 2) ::: ( + k) :

Lemma 7. [35] Let 0 < s

Pn

1 k X (n + nk ) (ln ) ( )k+1

k=1

where

n!

and

1 s

,0

s +1 s

1 and 0 < s

1. Then

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3

Theorem 1. Let I [0; 1) be an open interval and let f : I ! (0; 1) be a function such that f (n) exists on q b I. If f (n) 2 L ([a; b]) for n 2 N, where 0 a < b < 1 and f (n) is ( ; m)-logarithmically convex on 0; m for ( ; m) 2 (0; 1] (0; 1], q 2 [1; 1). Then h i k Z b n 1 k 1 + ( 1)k (b a) X f (a) + f (b) 1 a+b f (x) dx (2.8) f (k) 2 b a a 2k+1 (k + 1)! 2 k=1

n

(b a) 2n+1 n!

where F (n; ) is de…ned in Lemma 5,

=

8 < :

2;

0
1, by using Lemma 5 and Lemma 7, we have Z

1 + t)

q ( 1+t 2 )

0

0

1

n 1

(1

t)

(n

1 + t)

q( 1 2 t )

1 q

dt

+

0

Z

1

n 1

(1

t)

+

Z

(n

(n

1 + t) i1=q

q 2

F n;

1 + t)

qt 2

q ( 1+t 2 )

1 q

dt

h + F n;

i1=q

q 2

:

1 q

dt

0

Z

1

n 1

(1

t)

(n

1 + t)

q ( 1 2 t )+q

q

1 q

dt

0

=

t)

q ( 1+t 2 )

n : n+1

0

(2.12)

n 1

.

0

When

1=q

1

1, by using Lemma 5 and Lemma 7, we obtain

Z

(2.11)

f (x) dx

a

0

0

When 0
1:

f (a) + f (b) 2

=

i1=q

q 2

1

(b a) b f (n) n+1 2 n! m ( Z 1 n (1 t) where

h + F n;

and

m

jf (n) ( mb )j

Proof. From Lemma 1, the Hölder inequality and using the fact that f (n) b 0; m , we have (2.9)

i1=q

q 2

F n;

1

1

n 1

(1

t)

(n

1 + t)

1 q

q ( 1+t 2 )+q

q

dt

0

2

Z

1

(1

n 1

t)

(n

1 + t)

qt 2

1 q

dt

+

1

2

0

Z

0

= A combination of (2.9)-(2.12) gives the desired result. 753

1

2

1

(1 h

n 1

t)

F n;

(n q 2

1 + t) i1=q

qt 2

1 q

dt

h + F n;

q 2

i1=q

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4

Corollary 1. Suppose the assumptions of Theorem 1 are satis…ed and if q = 1, we have h i k k Z b n X1 k 1 + ( 1) (b a) f (a) + f (b) 1 a+b f (x) dx (2.13) f (k) k+1 2 b a a 2 (k + 1)! 2 k=1

n

(b a) f (n) 2n+1 n! where F (n; ) is de…ned in Lemma 5,

=

jf (n) (a)j m and jf (n) ( mb )j

m

b m

F n;

a)

1 2

4

1 q

1

f

h

m

b m

0

0

where

=

f (a)

, 8 >
:

1 2;

q 2

F 1;

i1=q

h + F 1;

q 2

F 1;

+ F 1;

2

(b

m

jf 00 ( mb )j (

F (2; ) =

2

2 3

a) 16

1

1 q

f

00

h

m

b m

,

q 2

F 2;

i1=q

h + F 2;

q 2

;

is as de…ned in Theorem 1 and 2 ln 2 3;

(ln )2 2 +2 ; (ln )3

6= 1; = 1:

Corollary 5. If q = 1 in Corollary 4, we have Z b 2 00 f (a) + f (b) 1 (b a) f (x) dx f (2.17) 2 b a a 16 where

i1=q

is as de…ned in Theorem 1.

00

=

;

;

2

Corollary 4. Suppose the assumptions of Theorem 1 are ful…lled and if n = 2, we have Z b 1 f (a) + f (b) f (x) dx (2.16) 2 b a a

where

i1=q

= 1:

are de…ned as in Corollary 2 and

f (a)

;

is de…ned in Theorem 1 and h i 1 1 + ; 6= 1 ln ln

Corollary 3. Corollary 2 with q = 1 gives the following result Z b m 1 (b a) 0 b f (a) + f (b) f (2.15) f (x) dx 2 b a a 4 m where F (1; ) and

2

is de…ned in Theorem 1.

Corollary 2. Under the assumptions of Theorem 1, if n = 1, we have the inequality Z b f (a) + f (b) 1 f (x) dx (2.14) 2 b a a (b

+ F n;

2

m

b m

F 2;

2

+ F 2;

2

;

is de…ned in Theorem 1 and , F (2; ) are de…ned in Corollary 4.

Theorem 2. Let I [0; 1) be an open interval and let f : I ! (0; 1) be a function such that f (n) exists on q b I. If f (n) 2 L ([a; b]) for n 2 N, where 0 a < b < 1 and f (n) is ( ; m)-logarithmically convex on 0; m for ( ; m) 2 (0; 1] (0; 1], q 2 (1; 1), we have h i k Z b n 1 k 1 + ( 1)k (b a) X 1 a+b f (a) + f (b) (2.18) f (x) dx f (k) 2 b a a 2k+1 (k + 1)! 2 k=1

(b

h n a) n(2q

1)=(q 1)

(n

2n+1 n!

(2q 1)=(q 1)

1)

i1

1 q

q 2q

h G nq

where

=

jf (n) (a)j m , G ( ; ) is de…ned in Lemma 4 and jf (n) ( mb )j 754

1 1

q + 1;

1=q

m

b m i q1 h + G nq

f (n) q 2

q + 1;

q 2

i q1

is de…ned in Theorem 1. M. A. LATIF et al 751-759

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5

Proof. Using Lemma 1, the Hölder inequality and the ( ; m)-logarithmic convexity of f (n) h i k Z b n 1 k 1 + ( 1)k (b a) X f (a) + f (b) 1 a+b (2.19) f (x) dx f (k) 2 b a a 2k+1 (k + 1)! 2

q

b on 0; m , we have

k=1

n

(b a) f (n) 2n+1 n!

m

b m

Z

1

1

q

(n

1 + t) q

1 q

dt

1

0

( Z

Z

1=q

1

q(n 1)

(1

t)

q( 1 2 t )

dt

+

0

1

(1

q(n 1)

t)

q ( 1+t 2 )

dt

0

)1=q

:

i q1

;

i q1

;

The proof follows by using similar arguments as in proving Theorem 1, using Lemma 4 and Lemma 7. Corollary 6. Under the assumptions of Theorem 2, if n = 1, we have the inequality Z b f (a) + f (b) 1 (2.20) f (x) dx 2 b a a (b

a)

q 2q

4

1 1

1=q

f

0

h

m

b m

0

where

=

f (a) m

00

jf ( mb )j

q 2

1

0

0

is the incomplete Beta function and

is de…ned in Theorem 1. 755

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6

Proof. Using Lemma 1, the Hölder inequality and the ( ; m)-logarithmic convexity of f (n) h i k k Z b n X1 k 1 + ( 1) (b a) f (a) + f (b) 1 a+b (2.23) f (x) dx f (k) k+1 2 b a a 2 (k + 1)! 2

q

b on 0; m , we have

k=1

n

(b a) f (n) 2n+1 n!

Z

m

b m

1 1=q

1

(1

q(n 1)=(q 1)

t)

0

q=(q 1)

(n 1 + t) ( Z 1

dt 1=q

q( 1 2 t )

dt

+

0

By using Lemma 7 and the fact that Z 1 q(n 1)=(q 1) q=(q (1 t) (n 1 + t) 0

=n

Z

nq+q 1 q 1

1 n

t

(n 1)q q 1

1)

q

(1

t) q

1

dt = n

Z

1

1=q q ( 1+t 2 )

dt

0

)

:

dt

nq+q 1 q 1

1 nq 1 2q ; ; n q 1 q

B

0

1 1

;

we get the required inequality (2.22) from (2.23). Corollary 8. Suppose the assumptions of Theorem 3 are satis…ed and if n = 1, we have the inequality Z b 1 q1 h i q1 h i q1 q q (b a) q 1 f (a) + f (b) 1 2 2 f (x) dx + F3 F3 ; (2.24) 2 b a a 4 2q 1 0

where

=

f (a) m

jf 0 ( mb )j

, 1 ln

F3 ( ) = and

;

6= 1 =1

1;

are de…ned as in Theorem 1.

Corollary 9. Suppose the assumptions of Theorem 3 are satis…ed and if n = 2, we have the inequality (2.25)

1

f (a) + f (b) 2

b

a

Z

b

f (x) dx

a 2

(b

a) 1+ q1

2

B

1 2q ; 2 q

1 2q ; 1 q

1 1

1

h F3

1 q

00

where

=

f (a) 00

m

jf ( mb )j

i q1

h + F3

q 2

i q1

;

, 1 ln

F3 ( ) =

q 2

1;

;

6= 1 ; =1

B (z; ; ) is the incomplete Beta function as de…ned in Theorem 3 and

is de…ned as in Theorem 1.

Theorem 4. Let I [0; 1) be an open interval and let f : I [0; 1) ! (0; 1) be a function such that f (n) q (n) exists on I. If f 2 L ([a; b]) for n 2 N, where 0 a < b < 1 and f (n) is ( ; m)-logarithmically convex on b 0; m for ( ; m) 2 (0; 1] (0; 1], q 2 (1; 1) for 0 r (n 1) q. Then i h k Z b n 1 k 1 + ( 1)k (b a) X f (a) + f (b) 1 a+b (2.26) f (x) dx f (k) 2 b a a 2k+1 (k + 1)! 2 k=1

n

(b a) 2n+1 n!

=

jf (n) (a)j m, jf (n) ( mb )j

"

(q 1) n2 q nr 2n + r + 1 (nq r 1) (nq + q r 2)

#1

1 q

f (n) h

b m

m

H r + 1;

q 2

i1=q

h + H r + 1;

q 2

i1=q

is de…ned in Theorem 1 and H ( ; ) is de…ned in Lemma 6.

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7

q

Proof. From Lemma 1, the Hölder inequality and using the fact that f (n) is ( ; m)-logarithmically convex on b 0; m , we have h i k k Z b n X1 k 1 + ( 1) (b a) f (a) + f (b) 1 a+b f (x) dx (2.27) f (k) k+1 2 b a a 2 (k + 1)! 2 k=1

m

n

(b a) b f (n) 2n+1 n! m ( Z 1

Z

1

1

(nq q r)=(q 1)

(1

t)

(n

1 + t) dt

0

(1

1=q

r

t) (n

1 + t)

q( 1 2 t )

dt

1 q

+

0

Z

1=q

1

(1

r

t) (n

1 + t)

q ( 1+t 2 )

dt

0

)

;

The rest of the proof is similar to that of the proof of Theorem 2 by using Lemma 6 and Lemma 7. Corollary 10. Suppose the assumptions of Theorem 4 are ful…lled and if r = 0, we have h i k Z b n 1 k 1 + ( 1)k (b a) X f (a) + f (b) 1 a+b f (x) dx f (k) (2.28) 2 b a a 2k+1 (k + 1)! 2 k=1

n

(b a) 2n+1 n!

"

(q 1) n2 q 2n + 1 (nq 1) (nq + q 2)

#1

1 q

b m

f (n)

m

h

H 1;

q 2

i1=q

h + H 1;

q 2

i1=q

:

i1=q

:

Corollary 11. Suppose the assumptions of Theorem 4 are ful…lled and if r = (n 1) q, we have h i k Z b n 1 k 1 + ( 1)k (b a) X 1 a+b f (a) + f (b) f (x) dx f (k) (2.29) 2 b a a 2k+1 (k + 1)! 2 k=1

n

(b a) 2n+1 n!

2nq

2n q + 1 2 (q 1)

1

1 q

f (n) h

b m

H (n

m

1) q + 1;

q 2

i1=q

h + H (n

1) q + 1;

q 2

Remark 1. Several interesting inequalities for m-logarithmically convex functions can be obtained by setting = 1 in the results presented in this section. However, we leave the details to the interested reader. Remark 2. We can get several interesting inequalities for logarithmically convex functions by setting and m = 1 in the results proved above. However, the details are left to the interested reader.

=1

3. Applications to Special Means For positive real numbers a > 0, b > 0, we consider the following means p a+b 2ab A (a; b) = , G (a; b) = ab, H (a; b) = ; 2 a+b 8 1=(b a) b > < 1e aba ; a 6= b; I (a; b) = > : a a = b; and 8 h i1=p bp+1 ap+1 > > ; p 6= 0; 1 and a 6= b; > (p+1)(b a) > > > > > > < b a ; p = 1 and a 6= b; ln b ln a Lp (a; b) = > > > > I (a; b) ; p = 0 and a 6= b; > > > > > : a; a = b:

It is well known that A, G, H, L = L 1 , I = L0 and Lp are called the arithmetic, geometric, harmonic, identric, exponential and generalized logarithmic means of positive real numbers a and b. In what follows we will use the above means and the established results of the previous section to obtain some interesting inequalities involving means. Theorem 5. Let 0 < a < b

1, r < 0, r 6=

1 and q

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8

(1) If r 6=

2, then

r+1

A a (b

Lr+1 r+1 (a; b)

; br+1

a)

2 q

3

1 2

jr + 1j

h +ar=2 L aqr=2 ; bqr=2

(2) If r =

2, then

1 H (a; b) (b +a

q

L a

1 q

3

1 2

Proof. Let f (x) =

xr+1 r+1

for x, y 2 (0; 1],

q

1

b

q

L a

q

;b

q

1=q

0

0

0

and b m

m

and

= xrq is logarithmically convex function on (0; 1]

= 12 .

1, hence

r

h F 1;

q h = f 0 (b) f 0 (a) F 1; h p ar br F 1;

0

f (a) f 0 (b)

=

Since f (a) > f (b) = br 0

q

0

1. This shows that f (x)

2 [0; 1] and q

a f (a) = f 0 (b) b

q

0

) ln f (y)

so that we have ( ; m) = (1; 1),

q 2

1=q

2 qr (ln b ln a) h +ar=2 L aqr=2 ; bqr=2

=

n b

q

0

=

i1=q

) y)

ln f (x) + (1

0

L aqr=2 ; bqr=2

1. Then f (x) = xr and

for 0 < x

0

f

1=q

1 q (ln a ln b) o 1=q a q :

q

;b

ln f ( x + (1

=

h br=2 bqr=2

1=q

1 L (a; b)

a) 1

1 qr (ln b ln a) i1=q : aqr=2

q 2

q 2

i1=q

i1=q

i1=q

h + F 1;

h + F 1;

h + F 1;

q 2

h br=2 bqr=2 aqr=2

i1=q

i1=q

q 2

q 2

i1=q

i1=q

L aqr=2 ; bqr=2

i1=q

:

Substituting the above quantities in Corollary 2, we get the required results. Remark 3. Many interesting inequalities of means can be obtained from the other results of Section 2, however, the details are left to the readers. References [1] M. W. Alomari, M. Darus and U. S. Kirmaci, Some inequalities of Hadamard type inequalities for s-convex functions, Acts Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 4, 1643-1652. [2] A. O. Akdemir and M. Tunç, On some inegral inequalities for s-logarithmically convex functions and their applications, arXiv:1212.1584 [math.FA]. [3] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45–55. [4] S. S. Dragomir and S. Fitzpatric, The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math. 32 (1999), no. 4 687-696. [5] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. [6] S. S. Dragomir, R. P. Agarwal, Two inequalities for di¤erentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998) 91–95. [7] J. Deng and J. R. Wang, Fractional Hermite-Hadamard inequalities for ( ; m)-logarithmically convex functions, J Inequal Appl 2013, 2013:364.

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9

[8] W.-D. Jiang, D.-W. Niu, Y. Hua, and F. Qi, Generalizations of Hermite-Hadamard inequality to n-time di¤erentiable functions which are s-convex in the second sense, Analysis (Munich) 32 (2012), 1001–1012. [9] R.- F. Bai, F. Qi and B. -Y. Xi, Hermite-Hadamard type inequalities for the m- and ( ; m)-logarithmically convex functions, Filomat 27:1 (2013), 1-7. [10] J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d’une fonction considerée par Riemann, J. Math Pures Appl., 58 (1893), 171–215. [11] Ch. Hermite, Sur deux limites d’une intégrale dé…nie, Mathesis 3 (1883), 82. [12] D.-Y. Hwang, Some inequalities for n-times di¤erentiable mappings and applications, Kyugpook Math. J. 43(2003), 335-343. [13] U.S. K¬rmac¬, Inequalities for di¤erentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146. [14] U.S. K¬rmac¬ and M.E. Özdemir, On some inequalities for di¤erentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 153 (2004), 361-368. [15] U.S. K¬rmac¬, Improvement and further generalization of inequalities for di¤erentiable mappings and applications, Comp and Math. with Appl., 55 (2008), 485-493. [16] H. Kavurmaci and M. Tunç, On some inequalities for s-logarithmically convex functions in the second sense via fractional inegrals, arXiv:1212.1604 [math.FA]. [17] M. A. Latif and S. S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose derivatives in absolute value are convex ’with applications to special means and to general quadrature formula, Acta Univ. M. Belii, ser. Math. (2013), 24–39. [18] M. A. Latif and S. S. Dragomir, On Hermite-Hadamard type integral inequalities for n-times di¤erentiable ( ; m)logarithmically convex functions, RGMIA Research Report Collection, 17(2014), Article 14, 16 pp. [19] M. A. Latif and S. S. Dragomir, New inequalities of Hermite-Hadamard type for n-times di¤erentiable convex and concave functions with applications. (Accepted) [20] M. A. Latif and S. S. Dragomir, On Hermite-Hadamard type integral inequalities for n-times di¤erentiable log-preinvex functions. (Accepted) [21] M. A. Latif and S. S. Dragomir,G eneralization of Hermite-Hadamard type inequalities for n-times di¤erentiable functions which are s-preinvex in the second sense with applications. (Accepted) [22] D. S. Mitrinovi´c and I. B. Lackovi´c, Hermite and convexity, Aequationes Math. 28 (1985), 229–232. [23] C.E.M. Pearce and J. Peµcari´c, Inequalities for di¤erentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51–55. [24] M. A. Noor, F. Qi and M. U. Awan, Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis 12/2013, 33(4):367–375. DOI:10.1524/anly.2013.1223 [25] M. Z. Sarikaya, A. Saglam and H. Y¬ld¬r¬m, On some Hadamard-type inequalities for h-convex functions, J Math Inequal, Vol. 2, No. 3 (2008), 335-341. [26] M. Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications Mathematical and Computer Modelling, Volume 54, Issues 9-10, November 2011, Pages 2175-2182. [27] M. Z. Sarikaya, E. Set and M. E. Ozdemir, On some new inequalities of Hadamard type involving h-convex functions, Acta Mathematica Universitatis Comenianae, Vol. LXXIX, 2(2010), pp. 265-272. [28] A. Saglam, M. Z. Sarikaya and H. Yildirim, Some new inequalities of Hermite-Hadamard’s type, Kyungpook Mathematical Journal, 50(2010), 399-410. [29] M. Tunç and A. O. Akdemir, Ostrowski type inequalities for s-logarithmically convex functions in the second sense with applications, arXiv:1301.3041 [math.CA]. [30] J. Wang, J. Deng, M. Feµckan, Exploring s-e-condition and applications to some Ostrowski type inequalities via Hadamard fractional integrals. Math. Slovaca (2013, in press). [31] S.-H. Wang, B.-Y. Xi, F. Qi, Some new inequalities of Hermite-Hadamard type for n-time di¤erentiable functions which are m-convex, Analysis (Munich) 32 (2012) 247–262. [32] B.-Y. Xi, R.-F. Bai and F. Qi, Hermite-Hadamard type inequalities for the m- and ( ; m)-geometrically convex functions, Aequationes Math. 39 (2012), in press. [33] B.-Y. Xi and F. Qi, Some integral inequalities of Hermite-Hadamard type for s-logarithmically convex functions. (Accepted) [34] T.-Y. Zhang, A.-P. Ji, F. Qi, On integral inequalities of Hermite-Hadamard type for s-geometrically convex functions, Abstr. Appl. Anal., 2012 (2012). [35] T.-Y. Zhang, M. Tunç, A.-P. Ji and B.-Y. Xi, Erratum to “On integral inequalities of Hermite-Hadamard type for s-geometrically convex functions”, Abstr. Appl. Anal., Volume 2014, Article ID 294739, 5 pageshttp://dx.doi.org/10.1155/2014/294739. School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa E-mail address : [email protected] 1

School of Engineering and Science, Victoria University, P. O. Box 14428, Melbourne City, MC8001, Australia, of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa E-mail address : [email protected] 2 School

Center for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa E-mail address : [email protected]

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Positive solutions for p-Laplacian fractional difference equation with a parameter ∗ Yongshun Zhao and Shurong Sun School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China e-mail: [email protected],e-mail: [email protected]

Abstract: In this paper, we consider a boundary value problem for fractional difference equation with p-Laplacian operator involving a parameter ( ∆[φp (∆νC u)](t) + λp−1 f (t + ν − 1, u(t + ν − 1)) = 0, t ∈ [0, b − 1]N0 , ∆u(ν − 2) = ∆νC u(0) = 0, u(ν + b) = γu(η), where 1 < ν ≤ 2 is a real number, φp (s) = |s|p−2 s, p > 1, γ ∈ (0, 1), η ∈ (ν, ν+b), ∆νC denotes the discrete Caputo fractional difference of order ν, f : [ν − 1, ν + b − 2]Nν−1 × [0, +∞) → (0, +∞) is a continuous function, b ≥ 3 is an integer, λ > 0 is a parameter. We study the existence of positive solutions to this problem by the properties of the Green function and Guo-Krasnosel’skii fixed point theorem in cones. Keywords: boundary value problem; discrete fractional calculus; existence of solutions; p-Laplacian operator; Guo-Krasnosel’skii theorem. Mathematics Subject Classification 2010: 39A12; 26A33; 34B15

1

Introduction

In recent years, fractional differential equations have received increasing attention. With the development of computer, it is well known that discrete analogues of differential equations can be very useful, especially for using computer to simulate the behavior of solutions for certain dynamic equations. More recent works to the discrete fractional calculus can be find in [1–7] and references contained therein. For example, Y. Pan and Z. Han et al. [5] studied the the existence and nonexistence of positive solutions to a boundary value problem for fractional difference equation with a parameter −∆ν y(t) = λf (t + ν − 1, y(t + ν − 1)), t ∈ [0, b + 1]N0 , y(ν − 2) = y(ν + b + 1) = 0, where 1 < ν ≤ 2 is a real number, f : [ν − 1, ν + b]Nν−1 × R → (0, +∞) is a continuous function, b ≥ 2 is an integer, λ is a parameter. The eigenvalue intervals of boundary value problem to a nonlinear fractional difference equation are considered by the properties of the Green function and Guo-Krasnosel’skii fixed point theorem in cones, some sufficient conditions to the nonexistence of positive solutions for the boundary value problem are established. Differential equations with p-Laplacian operator are applied in real life, especially in physics and engineering [8]. Some theories of fractional difference equations with p-Laplacian operator are just beginning to be investigated. W. Lv [9] investigated the following boundary value problem for fractional difference equation involving a p-Laplacian operator ∆[φp (∆α C u)](t) = f (t + α − 1, u(t + α − 1)), t ∈ [0, b]N0 , u(α − 2) = β1 u(α + b + 1), ∗ Corresponding

author: Shurong Sun, e-mail: [email protected]. This research is supported by the Natural Science Foundation of China(11571202, 61374074), and supported by Shandong Provincial Natural Science Foundation(ZR2016AM17).

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∆u(α − 2) = ∆u(α − 1) = β2 ∆u(α + b), where 1 < α ≤ 2, b ∈ N1 , β1 6= 1, β2 6= 1, ∆ is the forward difference operator with step size 1, ∆α C denotes the discrete Caputo fractional difference of order α, f : [α − 1, α + b − 1]Nα−1 × R → R is a continuous function, and φp is the p-Laplacian operator. Some existence and uniqueness results are obtained by using the Banach contraction mapping theorem. In this paper, we discuss the following boundary value problem for fractional difference equation with p-Laplacian operator ∆[φp (∆νC u)](t) + λp−1 f (t + ν − 1, u(t + ν − 1)) = 0, t ∈ [0, b − 1]N0 , ∆u(ν − 2) = ∆νC u(0) = 0, u(ν + b) = γu(η),

(1.1) (1.2)

where 1 < ν ≤ 2 is a real number, γ ∈ (0, 1), η ∈ (ν, ν + b), ∆νC denotes the discrete Caputo fractional difference of order ν, f : [ν − 1, ν + b − 2]Nν−1 × [0, +∞) → (0, +∞) is a continuous function, b ≥ 3 is an integer, λ > 0 is a parameter. φp is the p–Laplacian operator, that is, φp (s) = |s|p−2 s, p > 1. Obviously, φp is invertible and its inverse operator is φq , where q > 1 is a constant with p1 + 1q = 1. Our work presented in this article has the following features which are worth emphasizing. (i ) As far as we know, there are not many results available concerning with three–point boundary value problem of fractional difference equation which ∆νC is the standard Caputo fractional difference. (ii ) We consider the boundary value problem with p-Laplacian which arises in the modeling of different physical and natural phenomena. (iii ) We investigate the intervals of parameter λ for boundary value problem to a nonlinear fractional difference equation with p-Laplacian. The plan of the paper is as follows. In Section 2, we shall present some definitions and lemmas in order to prove our main results, the corresponding Green function and some properties of the Green function. In Section 3, we shall deduce the existence of positive solutions to problem (1.1)– (1.2) by the properties of the Green function and Guo-Krasnosel’skii fixed point theorem in cones. In Section 4, we give some examples to illustrate the theorems.

2

Preliminaries

For the convenience of the reader, we give some necessary basic definitions and lemmas that will be important to us in what follows. Γ(t+1) Definition 2.1 ([6]) We define tν := Γ(t+1−ν) for any t and ν, for which the right-hand side is defined. We also appeal to the convention that if t + 1 − ν is a pole of the Gamma function and t + 1 is not a pole, then tν = 0.

Definition 2.2 ([7]) Assume f : Na → R and ν > 0. Then the ν-th fractional sum of f (based at a) at the point t ∈ Na+ν is defined by t−ν

∆−ν a f (t) :=

1 X (t − s − 1)ν−1 f (s). Γ(ν) s=a

Note that by our convention on delta sums we can extend the domain of ∆−ν a f to Na+ν−N , where N is the unique positive integer satisfying N − 1 < ν ≤ N , by noting that a+ν−1 ∆−ν a f (t) = 0, t ∈ Na+ν−N .

Definition 2.3 ([10]) The ν-th Caputo fractional difference of a function f : Na → R, for ν > 0, ν ∈ / N, is defined by ∆νC f (t)

= ∆−(n−ν) ∆n f (t) 1 = Γ(n−ν) Σt−n+ν (t − s − 1)n−ν−1 ∆n f (s), s=a

for t ∈ Na+n−ν , where n is the smallest integer greater than or equal to ν and ∆n is the n-th forward difference operator. If ν = n, then ∆νC f (t) = ∆n f (t). 2 761

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Lemma 2.1 ([11]) Assume that ν > 0 and f is defined on Na . Then ∆−ν ∆νC f (t) = f (t) + C0 + C1 t + · · · + Cn−1 tn−1 , for some Ci ∈ R, i = 1, 2, . . . , n − 1, and n is the smallest integer greater than or equal to ν. Lemma 2.2 ([6]) Let t and ν be any numbers for which tν and tν−1 are defined. Then ∆tν = νtν−1 . Lemma 2.3 ([6]) For t and s, for which both (t − s − 1)ν and (t − s − 2)ν are defined, we find that ∆s [(t − s − 1)ν ] = −ν(t − s − 2)ν−1 . Lemma 2.4 Let f : [ν − 1, ν + b − 2]Nν−1 × [0, +∞) → (0, +∞) be given. A function u is a solution of the (1.1)–(1.2), if and only if it has the form u(t) = λ

Pb

s=0

G(t, s)φq

Ps−1

τ =0


f (τ + ν − 1, u(τ + ν − 1)) (2.2)

λγ + 1−γ

Pb

s=0 G(η, s)φq

Ps−1


τ =0 f (τ + ν − 1, u(τ + ν − 1)) , t ∈ [ν − 2, ν + b]Nν−2 ,

where G(t, s) is given by ( (ν + b − s − 1)ν−1 − (t − s − 1)ν−1 , 0 ≤ s < t − ν + 1 ≤ b, 1 G(t, s) = Γ(ν) (ν + b − s − 1)ν−1 , 0 ≤ t − ν + 1 ≤ s ≤ b.

(2.3)

Proof. If u(t) is a solution to (1.1)–(1.2). Then from (1.1), together with condition ∆νC u(0) = 0, we find that Pt−1 [φp (∆νC u)](t) = φp (∆νC u(0)) − λp−1 s=0 f (s + ν − 1, u(s + ν − 1)) = −λp−1 so ∆νC u(t)

= −λφq

Pt−1

s=0

X t−1

f (s + ν − 1, u(s + ν − 1)), t ∈ [0, b]N0 ,

 f (s + ν − 1, u(s + ν − 1)) , t ∈ [0, b]N0 ,

s=0

in view of Lemma 2.1, we have X  t−ν s−1 λ X (t − s − 1)ν−1 φq f (τ + ν − 1, u(τ + ν − 1)) + C0 + C1 t, t ∈ [ν − 2, ν + b]Nν−2 . Γ(ν) s=0 τ =0 (2.6) Furthermore, (2.3) implies that u(t) = −

λ ∆u(t) = − Γ(ν − 1)

t−(ν−1)

X

ν−2

(t−s−1)

φq

X s−1

 f (τ +ν−1, u(τ +ν−1)) +C1 , t ∈ [ν−2, ν+b−1]Nν−2 .

τ =0

s=0

By condition ∆u(ν − 2) = 0, we can get that C1 = 0. Then we obtain ! t−ν s−1 X λ X ν−1 u(t) = − (t−s−1) φq f (τ + ν − 1, u(τ + ν − 1)) +C0 , t ∈ [ν −2, ν +b]Nν−2 . (2.8) Γ(ν) s=0 τ =0 Now b

λ X u(ν + b) = − (ν + b − s − 1)ν−1 φq Γ(ν) s=0

s−1 X

! f (τ + ν − 1, u(τ + ν − 1))

+ C0 ,

τ =0

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η−ν λγ X γu(η) = − (η − s − 1)ν−1 φq Γ(ν) s=0

s−1 X

! f (τ + ν − 1, u(τ + ν − 1))

+ γC0 ,

τ =0

by condition u(ν + b) = γu(η), we obtain that λ (1−γ)Γ(ν)

C0 =

Pb

ν−1 φq s=0 (ν + b − s − 1)

λγ − (1−γ)Γ(ν)

Pη−ν

s=0 (η

P

s−1 τ =0

P

s−1 τ =0

− s − 1)ν−1 φq

 f (τ + ν − 1, u(τ + ν − 1))

 f (τ + ν − 1, u(τ + ν − 1)) .

Now, substitution of C0 and C1 into (2.6) gives P  Pt−ν s−1 λ ν−1 (t − s − 1) u(t) = − Γ(ν) φ f (τ + ν − 1, u(τ + ν − 1)) q s=0 τ =0 λ + (1−γ)Γ(ν)

Pb

λγ − (1−γ)Γ(ν)

Pη−ν

s=0 (ν

+ b − s − 1)ν−1 φq

ν−1 φq s=0 (η − s − 1)

P

s−1 τ =0

P

 f (τ + ν − 1, u(τ + ν − 1))

 f (τ + ν − 1, u(τ + ν − 1)) , t ∈ [ν − 2, ν + b]Nν−2 ,

s−1 τ =0

splitting the second sum in two parts on the basis of the following equality 1 γ 1 + = , Γ(ν) (1 − γ)Γ(ν) (1 − γ)Γ(ν) therefore, λ u(t) = − Γ(ν) λ + Γ(ν)

Pt−ν

ν−1 φq s=0 (t − s − 1)

Pb

s=0 (ν

P

s−1 τ =0

P

s−1 τ =0

+ b − s − 1)ν−1 φq

λγ + (1−γ)Γ(ν)

Pb

λγ − (1−γ)Γ(ν)

Pη−ν

 f (τ + ν − 1, u(τ + ν − 1))

ν−1 φq s=0 (ν + b − s − 1)

s=0 (η

− s − 1)ν−1 φq

 f (τ + ν − 1, u(τ + ν − 1))

P

s−1 τ =0

P

s−1 τ =0

 f (τ + ν − 1, u(τ + ν − 1))

 f (τ + ν − 1, u(τ + ν − 1)) , t ∈ [ν − 2, ν + b]Nν−2 ,

which is equivalent to (2.1) that u(t) = λ

Pb

s=0

λγ + 1−γ

G(t, s)φq

Pb

s=0

Ps−1

τ =0

G(η, s)φq


f (τ + ν − 1, u(τ + ν − 1))

Ps−1

τ =0


f (τ + ν − 1, u(τ + ν − 1)) , t ∈ [ν − 2, ν + b]Nν−2 .

On the other hand, if the function u(t) satisfies to (2.1), then u(ν + b) = γu(η). What’s more, function u(t) defined by (2.2) can transform to (2.8) that ! t−ν s−1 X λ X ν−1 (t − s − 1) u(t) = − φq f (τ + ν − 1, u(τ + ν − 1)) + C0 , t ∈ [ν − 2, ν + b]Nν−2 . Γ(ν) s=0 τ =0 Then we find that ∆u(t) = −

λ Γ(ν − 1)

t−(ν−1)

X

(t − s − 1)ν−2 φq

X s−1

s=0

 f (τ + ν − 1, u(τ + ν − 1)) , t ∈ [ν − 2, ν + b − 1]Nν−2 ,

τ =0

(2.16) and ∆νC u(t)

n h P io t−1 = −∆νC ∆−ν λφq s=0 f (s + ν − 1, u(s + ν − 1)) (2.17) = −φq

P

t−1 s=0

λ

p−1

 f (s + ν − 1, u(s + ν − 1)) , t ∈ [0, b]N0 . 4 763

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From (2.16) and (2.17), we see that ∆u(ν − 2) = ∆νC u(0) = 0. From the above proofs, it is to say function u(t) meets the boundary condition (1.2). Then we will proof that u(t) satisfies the fractional difference equation (1.1). Taking p-Laplacian operators on sides of (2.17), we find that [φp (∆νC u)](t)

=−

t−1 X

λp−1 f (s + ν − 1, u(s + ν − 1)), t ∈ [0, b]N0 .

(2.18)

s=0

By equation (2.18), function ∆[φp (∆νC u)](t) has the form ∆[φp (∆νC u)](t) = −λp−1 f (t + ν − 1, u(t + ν − 1)), t ∈ [0, b − 1]N0 , which shows that if (1.1)–(1.2) has a solution, then it can be represented by (2.2) and that every function of the form (2.2) is a solution of (1.1)–(1.2), which completes the proof. Lemma 2.5 ([12]) Let ν be any positive real number and a, b be two real numbers such that ν < a ≤ b. Then the following are valid. (i) x1ν is a decreasing function for x ∈ (0, +∞)N . (ii)

(a−x)ν (b−x)ν

is a decreasing function for x ∈ [0, a − ν)N .

Lemma 2.6 The function G(t, s) defined by (2.3) has the following properties: 1. 0 ≤ G(t, s) ≤ G(s + ν − 1, s), for t ∈ [ν − 2, ν + b]Nν−2 and s ∈ [0, b]N0 ; 2. there exists a positive number κ ∈ (0, 1) such that G(t, s) ≥ κ

min t∈[

(b+ν) 3(b+ν) , 4 ] 4

G(t, s) = κG(s + ν − 1, s),

max t∈[ν−2,ν+b]Nν−2

for s ∈ [0, b]. Proof. 1. For 0 ≤ s < t − ν + 1 ≤ b, we get (ν + b − s − 1)ν−1 ≥ (t − s − 1)ν−1 , which implies G(t, s) ≥ 0. For 0 ≤ t − ν + 1 ≤ s ≤ b, clear G(t, s) ≥ 0. On the other hand, case (1): 0 ≤ s < t − ν + 1 ≤ b, ∆t G(t, s) = −

(ν − 1)(t − s − 1)ν−2 < 0; Γ(ν)

case (2): 0 ≤ t − ν + 1 ≤ s ≤ b, ∆t G(t, s) = 0. Combining the above two cases, we have the function G(t, s) is non-increasing of t, thus G(t, s) ≤ G(s + ν − 1, s), s ∈ [0, b]N0 . 2. For s ≥ t − ν + 1 and

(b+ν) 4

≤t≤

3(b+ν) , 4

we have

G(t, s) = 1. G(s + ν − 1, s) For s < t − ν + 1 and

(b+ν) 4

≤t≤

3(b+ν) , 4

G(t,s) G(s+ν−1,s)

we have that =

(ν+b−s−1)ν−1 −(t−s−1)ν−1 (ν+b−s−1)ν−1

=1− ≥1−

(t−s−1)ν−1 (ν+b−s−1)ν−1 (

3(b+ν) −s−1)ν−1 4 ν−1

(ν+b−s−1)

.

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By Lemma 2.5 (ii), ( 3(b+ν) − s − 1)ν−1 4 (ν + b − s − 1)ν−1 is decreasing for 0 ≤ s
α1 such that φp (δ1 α1 ) ≤ ϕ(α1 ), ψ(α2 ) ≤ φp (δ2 α2 ) hold, where δ1 , δ2 are positive constants satisfying δ2 A < κδ1 B, 1−γ then for each  λ∈

−1  −1 ! δ2 KA κδ1 KB) , , 1−γ

the problem (1.1)–(1.2) has at least one positive solution. Proof. Let Ω1 = {u ∈ B : kuk ≤ α1 }. For any u ∈ P with kuk = α1 , we have φp (δ1 α1 ) ≤ ϕ(α1 ) ≤ f (t + ν − 1, u(t + ν − 1)) for (t + ν − 1, u(t + ν − 1)) ∈ [0, b − 1]N0 × [0, α1 ]. We have 7 766

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|T u(t)| = λ

Pb

λγ + 1−γ

≥λ

Ps−1

G(t, s)φq

s=0

Pb

s=0

Pl1

s=l0

τ =0

Ps−1

G(η, s)φq

G(t, s)φq


f (τ + ν − 1, u(τ + ν − 1)) τ =0

Ps−1

τ =0


f (τ + ν − 1, u(τ + ν − 1))


f (τ + ν − 1, u(τ + ν − 1))

≥ κλK

Pl1

φq

Ps−1


ϕ(α1 )

≥ κλK

Pl1

φq

Ps−1


φp (δ1 α1 )

s=l0

s=l0

τ =0

τ =0

= α1 δ1 λκKB ≥ α1 . So, kT uk ≥ kuk, u ∈ P ∩ ∂Ω1 .

(3.2)

On the other hand, let Ω2 = {u ∈ B : kuk ≤ α2 }. For any u ∈ P with kuk = α2 , we have f (t + ν − 1, u(t + ν − 1)) ≤ ψ(α2 ) ≤ φp (δ2 α2 ) for (t + ν − 1, u(t + ν − 1)) ∈ [0, b − 1]N0 × [0, α2 ]. We have
Pb Ps−1 |T u(t)| = λ s=0 G(t, s)φq τ =0 f (τ + ν − 1, u(τ + ν − 1)) λγ + 1−γ

≤ Kλ

Pb

s=0

Pb

λ ≤ K 1−γ

G(η, s)φq

s=0 φq

P

s−1 τ =0

Pb

s=0 φq

P

Ps−1

τ =0


f (τ + ν − 1, u(τ + ν − 1))

 ψ(α2 ) +

s−1 τ =0

λγ 1−γ K

Pb

s=0 φq

P

s−1 τ =0

 ψ(α2 )

 φp (δ2 α2 )

K A = δ2 α2 λ 1−γ

≤ α2 . Hence, kT uk ≤ kuk, u ∈ P ∩ ∂Ω2 .

(3.4)

Consequently, from (3.2) and (3.4), we may invoke Lemma 2.6 to deduce that T has a fixed point in the set P ∩ (Ω2 \ Ω1 ). Then the theorem is proved. Theorem 3.2 Suppose that conditions (H1)–(H2) hold. If there exist a sufficient small positive constant δ3 and sufficient large δ4 such that δ3

A < κδ4 B 1−γ

holds, then for each  λ∈

−1  −1 ! δ3 KA δ4 κKB , , 1−γ

then problem (1.1)–(1.2) has at least one positive solution. Proof. Because of condition (H1), there exist β1 > 0 and a sufficient small constant δ3 > 0 such that f (t, u) ≤ (δ3 u)p−1 , 0 < u ≤ β1 . (3.5)

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So for u ∈ P with kuk = β1 , by (2.22) and (3.5), we have for all t ∈ [ν − 2, ν + b]Nν−2 ,
Pb Ps−1 kT uk = max λ s=0 G(t, s)φq τ =0 f (τ + ν − 1, u(τ + ν − 1)) t∈[ν−2,ν+b]Nν−2

λγ + 1−γ

Pb

s=0

G(η, s)φq

Ps−1

τ =0


f (τ + ν − 1, u(τ + ν − 1)) (3.6)

λγ ≤ λkukδ3 KA + kuk 1−γ δ3 KA 1 A = λkukδ3 K 1−γ

≤ kuk. Thus, if we choose Ω1 = {u ∈ B : kuk ≤ β1 }, then (3.6) implies that kT uk ≤ kuk, u ∈ P ∩ ∂Ω1 .

(3.7)

On the other hand, condition (H2) implies that there exist a number 0 < β1 < β2 and a sufficient large constant δ4 such that f (t, u) ≥ (δ4 u)p−1 , u ≥ β2 . And then we set β2∗ =

β2 σ

(3.8)

> β2 . Then, u ∈ P and kuk = β2∗ implies

min t∈[

(b+ν) 3(b+ν) , 4 ] 4

u(t) ≥ σkuk = β2 ,

3(b+ν) ]. Therefore, for all t ∈ [ν − 2, ν + b]Nν−2 , by (2.11) and thus u(t) ≥ β2 , for all t ∈ [ (b+ν) 4 , 4 (3.8), we have that
Pb Ps−1 T u(t) = λ s=0 G(t, s)φq τ =0 f (τ + ν − 1, u(τ + ν − 1))

Pb

λγ + 1−γ

≥λ

s=0

Pl1

≥ λκ

s=l0

G(η, s)φq

G(t, s)φq

Pl1

s=l0

Kφq

Ps−1

τ =0

Ps−1

τ =0


f (τ + ν − 1, u(τ + ν − 1))


f (τ + ν − 1, u(τ + ν − 1)) (3.9)

Ps−1


τ =0 f (τ + ν − 1, u(τ + ν − 1))

≥ λkukδ4 κKB ≥ kuk. Hence, if we choose Ω2 = {u ∈ B : kuk ≤ β2∗ }, from (3.9) we have that kT uk ≥ kuk, u ∈ P ∩ ∂Ω2 .

(3.10)

Consequently, from (3.7) and (3.10), we may invoke Lemma 2.6 to deduce that T has a fixed point in the set P ∩ (Ω2 \ Ω1 ). Then the theorem is proved.

4

Examples

In this section, we will present some examples to illustrate main results. Example 4.1 Suppose that ν = 32 , b = 9, p = 32 . Take γ = 0.1, α1 = 2 and α2 = 80000. Then f (t, u) = t2 + sin u + 6 and problem (1.1)–(1.2) becomes 3

∆[φp (∆C2 u)](t) + λp−1 f (t, u) = 0, t ∈ [0, 8]N0 , 3 2

∆u(ν − 2) = ∆C u(0) = 0, u(ν + b) = 0.1u(η).

(4.1) (4.2)

1 Make δ1 = 15 and δ2 = 10 . By calculation, we have K = max(t,s)∈[− 12 , 21 G(t, s) ≈ 3.524, 2 ]N0 ×[0,9]N0 √ Pb 1 30 = φp (δ2 α1 ) < ϕ(α1 ) = 6 and 88 = ψ(α2 ) < φp (δ1 α2 ) ≈ 89. Then δ1 K 1−γ s=0 φq (s) ≈

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3.524 ×

1 9

× 285 ≈ 111.6 and δ2 κK

Pb 3(b+ν) −ν+1c 4 s=d

(b+ν) −ν+1e 4

φq (s) ≈ 15 × 0.0871 × 3.524 × 135 ≈ 690. So,

the conditions of Theorem 3.1 are satisfied. Then the boundary value problem (4.1)–(4.2) has at least one positive solution for each λ ∈ (0.001, 0.009). 1 Example 4.2 Suppose that ν = 32 , b = 9, p = 32 . Take γ = 0.1, δ3 = 1000 and δ4 = 2. Then 3 f (t, u) = (t + 1)u 2 , and problem (1.1)–(1.2) becomes 3

∆[φp (∆C2 u)](t) + λp−1 f (t, u) = 0, t ∈ [0, 8]N0 ,

(4.3)

3 2

∆u(ν − 2) = ∆C u(0) = 0, u(ν + b) = 0.1u(η). In addition, we have K = max(t,s)∈[− 21 , 21 G(t, s) ≈ 3.524, lim+ 2 ]N0 ×[0,9]N0

(4.4) maxt∈[ν−1,ν+b−2]N up−1

u→0

0 and lim

mint∈[ν−1,ν+b−2]N

1.116 and δ4 κK

ν−1

f (t,u)

up−1

u→+∞

Pb 3(b+ν) −ν+1c 4 s=d

(b+ν) −ν+1e 4

1 = +∞. Then δ3 K 1−γ

Pb

s=0

φq (s) ≈

1 1000

ν−1

f (t,u)

=

×3.524× 10 9 ×285 ≈

φq (s) ≈ 2 × 0.0871 × 3.524 × 135× ≈ 90.90. So, the conditions of

Theorem 3.2 are satisfied. Then the boundary value problem (4.3)–(4.4) has at least one positive solution for each λ ∈ (0.012, 0.896).

References [1] G. Wu, D. Baleanu, W. Luo, Mittag-Leffler function for discrete fractional modelling, Journal of King Saud University–Science, 28 (2016) 99–102. [2] I. Area, J. Losada, J. Nieto, On quasi–periodic properties of fractional sums and fractional differences of periodic functions, Applied Mathematics and Computation, 273 (2016) 190–200. [3] V. Tarasov, Discrete model of dislocations in fractional nonlocal elasticity, Journal of King Saud University–Science, 28 (2016) 33–36. [4] Y. Chen, X. Tang, The difference between a class of discrete fractional and integer order boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014) 4025–4067. [5] Y. Pan, Z. Han, D. Yang, The existence and nonexistence of positive solutions to a discrete fractional boundary value problem with a parameter, Applied Mathematics Letters, 36 (2014) 1–6. [6] C. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Computers and Mathematics with Applications, 61 (2011) 191–201. [7] C. Goodrich, A. C. Peterson, Discrete fractional calculus, Springer International Publishing, Switzerland, 2015. [8] H. Zhang, Q. Peng, Y. Wu, Wavelet inpainting based on p-Laplace operator, Acta Automatica Sinica, 33 (2003) 546–549. [9] W. Lv, Solvability for discrete fractional boundary value problems with a p-Laplacian operator, Discrete Dynamics in Nature and Society, 2013 (2013) 1–8. [10] T. Abdeljawad, On Riemann and Caputo fractional differences, Computers and Mathematics with Applications, 62 (2011) 1602–1611. [11] W. Lv, Existence of solutions for discrete fractional boundary value problems with a pLaplacian operator, Advances in Difference Equations, 2012 (2012) 1–10. [12] F. Atici, P. Eloe, Two-point boundary value problems for finite fractional difference equations, Journal of Difference Equations and Applications, 17 (2011) 445–456.

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Weighted Composition Operators from analytic Morrey spaces into Zygmund spaces∗ Shanli Ye† (School of Sciences, Zhejiang University of Science and Technology, Hangzhou 310023, China)

Abstract In this paper we characterize the boundedness and compactness of the weighted composition operator from the analytic Morrey spaces L2,λ to the Zygmund space Z, and the little analytic Morrey spaces L2,λ to the little Zygmund space Z0 , respectively. 0 Keywords Analytic Morrey space, Zygmund space; Weighted composition operator; Boundedness; Compactness 2010 MR Subject Classification 47B38, 30D99, 30H05

1

Introduction

Let D = {z : |z| < 1} be the open unit disk in the complex plane and H(D) denote the set of all analytic functions on D. Let u, φ ∈ H(D), where φ is an analytic self-map of D. Then the well-known weighted composition operator uCφ on H(D) is defined by uCφ (f )(z) = u(z)·(f ◦φ(z)) for f ∈ H(D) and z ∈ D. Weighted composition operators can be regarded as a generalization of multiplication operators Mu and composition operators Cφ . In 2001, Ohno and Zhao studied the weighted composition operators on the classical Bloch space β in [18], which has led many researchers to study this operator on other Banach spaces of analytic functions. The boundedness and compactness of it have been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type spaces, see, e.g. [4, 6, 11, 29]. ∫ 1 For an arc I ⊂ ∂D, let |I| = 2π |dζ| be the normalized arc length of I, I ∫ 1 |dζ| fI = f (ζ) , f ∈ H(D), |I| I 2π and S(I) be the Carleson box based on I with S(I) = {z ∈ D : 1 − |I| ≤ |z| < 1,

z ∈ I}. |z|

Clearly, if I = ∂D, then S(I) = D. Let L2,λ (D) represent the analytic M orrey spaces of all analytic functions f ∈ H 2 on D such that ∫ ( 1 |dζ| )1/2 sup |f (ζ) − fI |2 < ∞, λ 2π I⊂∂D |I| I ∗ The

research was supported by the National Natural Science Foundation of China (Grant No. 11671357, 11571217) and the Natural Science Foundation of Fujian Province, China(Grant No. 2015J01005). † E-mail: ye [email protected]; [email protected]

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where 0 < λ ≤ 1 and the Hardy space H 2 consists of analytic functions f in D satisfying ∫ 2π 1 sup |f (reiθ )|2 dθ < ∞. 0 0 dependent only on indexes p, λ... such that F ≤ CG, then we say that F . G. Furthermore, denote that F ≈ G (F is comparable with G) whenever F . G . F .

2

Auxiliary results

In order to prove the main results of this paper. we need some auxiliary results. Lemma 2.1 Let 0 < λ < 1. If f ∈ L2,λ , then ∥f ∥L2,λ

1−λ for every z ∈ D; (1 − |z|2 ) 2 ∥f ∥L2,λ (ii) |f ′ (z)| . 3−λ for every z ∈ D; (1 − |z|2 ) 2 ∥f ∥L2,λ (iii) |f ′′ (z)| . 5−λ for every z ∈ D. (1 − |z|2 ) 2

(i) |f (z)| .

Proof (i) and (ii) are from Lemma 2.5 in [14]. For any f ∈ L2,λ . Fix z ∈ D and let ρ = by the Cauchy integral formula, we obtain that

|f ′′ (z)| = |

1 2πi

∫ |ξ|=ρ

∥f ∥L2,λ f ′ (ξ) 1 dξ| ≤ 3−λ 2 (ξ − z) (1 − ρ2 ) 2 2π

∫

2π 0

1 + |z| , 2

∥f ∥L2,λ ∥f ∥L2,λ ρ dθ ρ = . 3−λ 5−λ . 2 2 2 − z| (1 − ρ2 ) 2 ρ − |z| (1 − |z|2 ) 2

|ρeiθ

Hence (iii) holds.

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Lemma 2.2 Let 0 < λ < 1. If f ∈ L2,λ 0 , then (i) lim (1 − |z|2 )

3−λ 2

|z|→1

(ii) lim (1 − |z|2 )

|f ′ (z)| = 0;

1−λ 2

|z|→1

(iii) lim (1 − |z|2 )

|f (z)| = 0;

5−λ 2

|z|→1

|f ′′ (z)| = 0.

The proof of (i) is similar to that of Lemma 2.5 in [14], and we easily obtain (ii) and (iii) by (i). These details are omitted here. 2,λ Lemma 2.3 Suppose uCφ : L2,λ → Z is a bounded 0 → Z0 is a bounded operator, then uCφ : L operator.

The proof is similar to that of Lemma 2.3 in [33]. The details are omitted.

3

Boundedness of uCφ

In this section we characterize the boundedness of the weighted composition operator uCφ from the analytic Morrey spaces L2,λ to the Zygmund space Z, and the little analytic Morrey spaces to the little Zygmund space Z0 , respectively. L2,λ 0 Theorem 3.1 Let u be an analytic function on the unit disc D, and φ an analytic self-map of D. Then uCφ is a bounded operator from the analytic Morrey spaces L2,λ to the Zygmund space Z if and only if the following are satisfied:

sup z∈D

sup

(1 − |z|2 )|u′′ (z)| (1 − |φ(z)|2 )

1−λ 2

< ∞;

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| (1 − |φ(z)|2 )

z∈D

sup z∈D

3−λ 2

(1 − |z|2 )|u(z)(φ′ (z))2 | (1 − |φ(z)|2 )

5−λ 2

(3.1) < ∞;

(3.2)

< ∞.

(3.3)

Proof Suppose uCφ is bounded from the analytic Morrey spaces L2,λ to the Zygmund space Z. Using functions f (z) = 1, f (z) = z and and f (z) = z 2 in L2,λ , we have u ∈ Z,

(3.4)

sup (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z) + φ(z)u′′ (z)| < +∞,

(3.5)

z∈D

and sup (1 − |z|2 )|4φ(z)φ′ (z)u′ (z) + φ2 (z)u′′ (z) + 2u(z)(φ(z)φ′′ (z) + (φ′ (z))2 )| < ∞. z∈D

Since φ(z) is a self-map, we get K1 = sup (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| < +∞

(3.6)

z∈D

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and K2 = sup (1 − |z|2 )|(φ′ (z))2 u(z)| < +∞.

(3.7)

z∈D

Fix a ∈ D with |a| > 12 , we take the test functions: fa (z) =

1 − |a|2 (1 − a ¯z)

3−λ 2

−2

(1 − |a|2 )2 (1 − a ¯z)

5−λ 2

+

(1 − |a|2 )3 (1 − a ¯z)

7−λ 2

(3.8)

for z ∈ D. Then, arguing as the proof of Lemma 3.2 in [14] we obtain that fa ∈ L2,λ and 2¯ a supa ∥fa ∥L2,λ . 1. Since fa (a) = 0, fa′ (a) = 0, fa′′ (a) = 5−λ , it follows that for all (1 − |a|2 ) 2 λ ∈ D with |φ(λ)| > 12 , we have ∥fa ∥L2,λ

& ∥uCφ fa ∥Z ≥ sup (1 − |z|2 )|(uCφ fa )′′ (z)| z∈D ( ) 2 = sup (1 − |z| )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) fa′ (φ(z)) +

z∈D fa′′ (φ(z))(φ′ (z))2 u(z)

+ u′′ (z)fa (φ(z))|.

Let a = φ(λ), it follows that ∥fa ∥L2,λ

( ) ′ & (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) fφ(λ) (φ(λ)) ′′ + fφ(λ) (φ(λ))(φ′ (λ))2 u(λ) + u′′ (λ)fφ(λ) (φ(λ))|

= (1 − |λ|2 )|(φ′ (λ))2 u(λ) ≥

(1 − |λ|2 )|φ′ (λ))2 u(λ)| (1 − |φ(λ)|2 )

5−λ 2

2φ(λ) (1 − |φ(λ)|2 )

5−λ 2

|

.

For ∀λ ∈ D with |φ(λ)| ≤ 12 , by (3.7), we have sup λ∈D

(1 − |λ|2 )|φ′ (λ))2 u(λ)| (1 −

|φ(λ)|2 )

5−λ 2

4 5−λ ≤ ( ) 2 sup (1 − |λ|2 )|φ′ (λ))2 u(λ)| < +∞. 3 λ∈D

Hence (3.3) holds. Next, we will show that (3.2) holds. Fix a ∈ D with |a| > 21 , we take another test functions: ga (z) =

1 − |a|2 (1 − a ¯z)

3−λ 2

−

12 − 2λ (1 − |a|2 )2 5 − λ (1 − |a|2 )3 5−λ + 7−λ 7 − λ (1 − a 7 − λ (1 − a ¯z) 2 ¯z) 2

(3.9)

for z ∈ D. Then ga ∈ L2,λ and supa ∥ga ∥L2,λ . 1(see [14]). Since ga (a) = 0, ga′′ (a) = 0, ga′ (a) = −2¯ a 1 3−λ , it follows that for all λ ∈ D with |φ(λ)| > 2 , we have (7 − λ)(1 − |a|2 ) 2 ∥ga ∥L2,λ

& ∥uCφ ga ∥Z ≥ sup (1 − |z|2 )|(uCφ ga )′′ (z)| z∈D ( ) = sup (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) ga′ (φ(z)) +

z∈D ga′′ (φ(z))(φ′ (z))2 u(z)

+ u′′ (z)ga (φ(z))|.

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Let a = φ(λ), it follows that ∥ga ∥L2,λ

( ) ′ & (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) gφ(λ) (φ(λ)) ′′ + gφ(λ) (φ(λ))(φ′ (λ))2 u(λ) + u′′ (λ)gφ(λ) (φ(λ))|

( ) = (1 − |λ|2 )| 2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ) ≥

−2φ(λ)

(7 − λ)(1 − |φ(λ)|2 ) 1 (1 − |λ|2 )|2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ)| . 3−λ 7−λ (1 − |φ(λ)|2 ) 2

3−λ 2

|

For ∀λ ∈ D with |φ(λ)| ≤ 12 , by (3.6), we have sup

(1 − |λ|2 )|2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ)| (1 − |φ(λ)|2 )

λ∈D

3−λ 2

4 3−λ ≤ ( ) 2 sup (1 − |λ|2 )|2φ′ (λ)u′ (λ) + φ′′ (λ)u(λ)| < +∞. 3 λ∈D

Hence (3.2) holds. Finally we will show (3.1) holds. Let ha (z) =

1 − |a|2 (1 − a ¯z)

(3.10)

3−λ 2

for z ∈ D. It is easily proved that sup 21 δ, it follows that 3−λ

(1 − |z|2 )|(uCφ f )′′ (z)| =

( ) (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) f ′ (φ(z))

+ f ′′ (φ(z))(φ′ (z))2 u(z) + u′′ (z)f (φ(z))| ( ) ≤ (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) f ′ (φ(z))| +

(1 − |z|2 )|f ′′ (φ(z))(φ′ (z))2 u(z)| + (1 − |z|2 )|u′′ (z)f (φ(z))|


0, there is a constant δ, 0 < δ < 1, such that δ < |φ(z)| < 1 implies (1 − |z|2 )|u(z)(φ′ (z))2 | (1 − |φ(z)|2 ) and

5−λ 2

(1 − |z|2 )|u′′ (z)|

< ϵ,

(1 − |φ(z)|2 )

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| (1 − |φ(z)|2 )

3−λ 2

1−λ 2

< ϵ,

< ϵ.

Let K = {w ∈ D : |w| ≤ δ}. Noting that K is a compact subset of D, we get that

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

z(uCφ fn )

=

sup (1 − |z|2 )|(uCφ fn )′′ (z)| z∈D

≤

( ) sup (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) fn′ (φ(z))|

z∈D

+

sup (1 − |z|2 )|fn′′ (φ(z))(φ′ (z))2 u(z)| + sup (1 − |z|2 )|u′′ (z)fn (φ(z))|

z∈D

z∈D

( ) . 3ϵ + sup (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) fn′ (φ(z))| |φ(z)|≤δ

+

sup (1 − |z|2 )|fn′′ (φ(z))(φ′ (z))2 u(z)| + sup (1 − |z|2 )|u′′ (z)fn (φ(z))|

|φ(z)|≤δ

|φ(z)|≤δ

≤ 3ϵ + M1 sup |fn′ (w)| + M2 sup |fn′′ (w)| + M3 sup |fn (w)|. w∈K

w∈K

w∈K

As n → ∞, ∥uCφ fn ∥Z → 0. Hence uCφ is compact. This completes the proof of Theorem 4.1. Theorem 4.2 Let u be an analytic function on the unit disc D, and φ an analytic self-map of D. Then uCφ is compact from L2,λ to Z0 if and only if the following are satisfied: 0 lim

|z|→1

lim

(1 − |z|2 )|u′′ (z)| (1 − |φ(z)|2 )

1−λ 2

= 0;

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| (1 − |φ(z)|2 )

|z|→1

lim

|z|→1

3−λ 2

(1 − |z|2 )|u(z)(φ′ (z))2 | (1 − |φ(z)|2 )

5−λ 2

(4.7)

= 0;

(4.8)

= 0.

(4.9)

Proof Assume (4.7), (4.8), and (4.9) hold. From Theorem 4.2, we know that uCφ is bounded from L2,λ to Z0 . Suppose that f ∈ L2,λ with ∥f ∥L2,λ ≤ 1. We obtain that 0 0 ( ) (1 − |z|2 )|(uCφ f )′′ (z)| = (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) f ′ (φ(z)) + f ′′ (φ(z))(φ′ (z))2 u(z) + u′′ (z)f (φ(z))| ( ) ≤ (1 − |z|2 )| 2φ′ (z)u′ (z) + φ′′ (z)u(z) f ′ (φ(z))| + (1 − |z|2 )|f ′′ (φ(z))(φ′ (z))2 u(z)| + (1 − |z|2 )|u′′ (z)f (φ(z))| .

+

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)| (1 − |φ(z)|2 )

3−λ 2

(1 − |z|2 )|(φ′ (z))2 u(z)| (1 − |φ(z)|2 )

5−λ 2

781

∥f ∥L2,λ +

∥f ∥L2,λ

(1 − |z|2 )|u′′ (z)| (1 − |φ(z)|2 )

1−λ 2

∥f ∥L2,λ ,

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

thus

sup{|(1 − |z|2 )(uCφ f )′′ (z)| : f ∈ L2,λ 0 , ∥f ∥L2,λ ≤ 1} (1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)|

.

+

(1 − |φ(z)|2 )

3−λ 2

(1 − |z|2 )|(φ′ (z))2 u(z)| (1 − |φ(z)|2 )

5−λ 2

+

(1 − |z|2 )|u′′ (z)| (1 − |φ(z)|2 )

1−λ 2

,

and it follows that lim sup{|(1 − |z|2 )(uCφ f )′′ (z)| : f ∈ L2,λ 0 , ∥f ∥L2,λ ≤ 1} = 0,

|z|→1

hence uCφ : L2,λ 0 → Z0 is compact by Lemma 4.2. Conversely, suppose that uCφ : L2,λ 0 → Z0 is compact. First, it is obvious that uCφ : L2,λ → Z0 is bounded, then by Theorem 3.2, we have u ∈ Z0 0 and that (3.11) and (3.12) hold. On the other hand, by Lemma 4.2 we have lim sup{|(1 − |z|2 )(uCφ f )′′ (z)| : f ∈ L2,λ 0 , ∥f ∥L2,λ ≤ M } = 0,

|z|→1

for some M > 0. Next, noting that the proof of Theorem 3.1 and the fact that the functions given in (3.8) are in L2,λ and have norms bounded independently of a, we obtain that 0 (1 − |z|2 )|u(z)(φ′ (z))2 |

lim

(1 − |φ(z)|2 )

|z|→1

5−λ 2

=0

for |φ(z)| > 12 . However, if |φ(z)| ≤ 12 , by (3.11), we easily have lim

(1 − |z|2 )|u(z)(φ′ (z))2 | (1 − |φ(z)|2 )

|z|→1

5−λ 2

4 5−λ ≤ ( ) 2 lim (1 − |z|2 )|u(z)(φ′ (z))2 | = 0. 3 |z|→1 Thus (4.9) holds. Also, the second statement, that (4.8), is proved similarly. We omitted it here. Similarly, noting that the functions given in (3.10) are in L2,λ and have norms bounded inde0 pendently of a, we obtain that lim

|z|→1

(1 − |z|2 )|u′′ (z)| (1 − |φ(z)|2 )

1−λ 2

(1 − |z|2 )|2φ′ (z)u′ (z) + φ′′ (z)u(z)|

. lim

(1 − |φ(z)|2 )

|z|→1

+ lim

3−λ 2

(1 − |z|2 )|u(z)(φ′ (z))2 | (1 −

|z|→1

|φ(z)|2 )

5−λ 2

+ lim (1 − |z|2 )|(uCφ ha )′′ (z)|, |z|→1

for |φ(z)| > 12 . So by (4.8) and (4.9), it follows that lim

|z|→1

(1 − |z|2 )|u′′ (z)| (1 − |φ(z)|2 )

1−λ 2

=0

for |φ(z)| > 12 . However, if |φ(z)| ≤ 12 , by u ∈ Z0 , we easily have lim

|z|→1

(1 − |z|2 )|u′′ (z)| (1 − |φ(z)|2 )

1−λ 2

4 1−λ = lim ( ) 2 (1 − |z|2 )|u′′ (z)| = 0. |z|→1 3

This completes the proof of Theorem 4.2.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Corollary 4.1 Let φ be an analytic self-map of D. Then Cφ is a compact operator from the analytic Morrey space L2,λ to the Zygmund space Z if and only if Cφ is bounded,

lim

|φ(z)|→1

(1 − |z|2 )|(φ′ (z))2 | (1 − |φ(z)|2 )

5−λ 2

= 0 and

lim

|φ(z)|→1

(1 − |z|2 )|φ′′ (z)| (1 − |φ(z)|2 )

3−λ 2

= 0.

Corollary 4.2 Let φ be an analytic self-map of D. Then Cφ is a compact operator from L2,λ to 0 Z0 if and only if (1 − |z|2 )|(φ′ (z))2 | =0 lim 5−λ |z|→1 (1 − |φ(z)|2 ) 2 and lim

|z|→1

(1 − |z|2 )|φ′′ (z)| (1 − |φ(z)|2 )

3−λ 2

= 0.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 4, 2018

A note on non-instantaneous impulsive fractional neutral integro-differential systems with statedependent delay in Banach spaces, Selvaraj Suganya, Dumitru Baleanu, Palaniyappan Kalamani and Mani Mallika Arjunan,………………………………………………………………603 Characterizations of positive implicative superior ideals induced by superior mappings, Seok Zun Song, Young Bae Jun, and Hee Sik Kim,……………………………………………634 The uniqueness of meromorphic functions sharing sets in an angular domain, Yong Zheng Zhou, Hong-Yan Xu, and Zu-Xing Xuan,……………………………………………………….644 A new generalization of Fibonacci and Lucas p-numbers, Yasin Yazlik, Cahit Kome, and Vinay Madhusudanan,……………………………………………………………………………657 Some families of generating functions for the generalized Cesàro polynomials, Nejla Özmen and Esra Erkus-Duman,………………………………………………………………………670 On uniqueness of meromorphic functions sharing one small function, Qiu Ling, Xuan Zuxing, and Qi Ping,…………………………………………………………………………………684 Stochastic elastic equation in d-dimensional space driven by multiplicative multi-parameter fractional white noise, Yinghan Zhang,……………………………………………………695 Fourier series of sums of products of Euler functions and their applications, Taekyun Kim, Dae San Kim, Gwan-Woo Jang, and Lee Chae Jang,……………………………………………714 Quotient subtraction algebras by an int-soft ideal, Sun Shin Ahn and Young Hee Kim,…728 Characterization of weak sharp solutions for generalized variational inequalities in Banach spaces, Natthaphon Artsawang, Ali Farajzadeh, and Kasamsuk Ungchittrakool,…………738 Inequalities of Hermite-Hadamard type for n-times differentiable (𝛼,m)-logarithmically convex functions, M. A. Latif, S. S. Dragomir, and E. Momoniat,………………………………….751 Positive solutions for p-Laplacian fractional difference equation with a parameter, Yongshun Zhao and Shurong Sun,……………………………………………………………………….760 Weighted Composition Operators from analytic Morrey spaces into Zygmund spaces, Shanli Ye,………………………………………………………………………………………………770

Volume 25, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

October 30, 2018

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

Journal of Computational Analysis and Applications ISSNno.’s:1521-1398 PRINT,1572-9206 ONLINE SCOPE OF THE JOURNAL An international publication of Eudoxus Press, LLC (sixteen times annually) Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240, U.S.A [email protected] http://www.msci.memphis.edu/~ganastss/jocaaa The main purpose of "J.Computational Analysis and Applications" is to publish high quality research articles from all subareas of Computational Mathematical Analysis and its many potential applications and connections to other areas of Mathematical Sciences. Any paper whose approach and proofs are computational,using methods from Mathematical Analysis in the broadest sense is suitable and welcome for consideration in our journal, except from Applied Numerical Analysis articles. Also plain word articles without formulas and proofs are excluded. The list of possibly connected mathematical areas with this publication includes, but is not restricted to: Applied Analysis, Applied Functional Analysis, Approximation Theory, Asymptotic Analysis, Difference Equations, Differential Equations, Partial Differential Equations, Fourier Analysis, Fractals, Fuzzy Sets, Harmonic Analysis, Inequalities, Integral Equations, Measure Theory, Moment Theory, Neural Networks, Numerical Functional Analysis, Potential Theory, Probability Theory, Real and Complex Analysis, Signal Analysis, Special Functions, Splines, Stochastic Analysis, Stochastic Processes, Summability, Tomography, Wavelets, any combination of the above, e.t.c. "J.Computational Analysis and Applications" is a peer-reviewed Journal. See the instructions for preparation and submission of articles to JoCAAA. Assistant to the Editor: Dr.Razvan Mezei,[email protected], Madison,WI,USA.

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Editorial Board Associate Editors of Journal of Computational Analysis and Applications Francesco Altomare Dipartimento di Matematica Universita' di Bari Via E.Orabona, 4 70125 Bari, ITALY Tel+39-080-5442690 office +39-080-3944046 home +39-080-5963612 Fax [email protected] Approximation Theory, Functional Analysis, Semigroups and Partial Differential Equations, Positive Operators.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

FOURIER SERIES OF FUNCTIONS INVOLVING EULER POLYNOMIALS TAEKYUN KIM, DAE SAN KIM, GWAN-WOO JANG, AND JONGKYUM KWON

Abstract. Recently, T. Kim introduced Fourier series expansions of certain special polynomials and investigated some interesting identities and properties of these polynomials by using those Fourier series. In this paper, we consider three types of functions involving Euler polynomials and derive their Fourier series expansions. Moreover, we express each of them in terms of Benoulli functions.

1. Introduction Let Em (x) be the Euler polynomials given by the generating function ∞ X 2 tm xt e = Em (x) , (see [1,2,5,7-11,16]). t e +1 m! m=0

(1.1)

From this equation, we can derive the following relation. ( 2, if n = 0, n E0 = 1, (E + 1) + En = 0, if n 6= 0. The Bernoulli polynomials Bm (x) are defined by the generating function ∞ X tm t xt e = B (x) , (see [1,2,5,9]). m et − 1 m! m=0

(1.2)

For any real number x, we let < x >= x − [x] ∈ [0, 1)

(1.3)

denote the fractional part of x. Here we will consider the following three types of functions involving Euler polynomials and derive their Fourier series expansions. Further, we will express each of them in terms of Bernoulli functions Bm (< x >). Pm (1) αm (< x >) = P k=0 Ek (x)xm−k , (m ≥ 1); m 1 (2) βm (< x >) = k=0 k!(m−k)! Ek (x)xm−k , (m ≥ 1); Pm−1 1 (3) γm (< x >) = k=1 k(m−k) Ek (x)xm−k , (m ≥ 2). The reader may refer to any book (for example, see [13-15,17]), for elementary facts about Fourier analysis. 2010 Mathematics Subject Classification. 11B68, 42A16. Key words and phrases. Fourier series, Euler polynomials. 1

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As to γm (< x >), we note that the polynomial identity (1.4) follows immediately from Theorems 4.2 and 4.3, which is in turn derived from the Fourier series expansion of γm (< x >). m−1 X k=1

1 Ek (x)xm−k k(m − k)

m  1 X Ek 1 2 + − Em+1 m k(m − k + 1) m(m + 1) m(m + 1) k=1     m−1 m  1 X m Hm−1 − Hm−s m X El−s+1 + Bs (x), (1 − 2Em−s+1 ) − m s=1 s m−s+1 s (l − s + 1)(m − l) l=s (1.4) Pm where Hm = j=1 1j are the harmonic numbers. The obvious polynomial identities can be derived also for αm (< x >) and βm (< x >) from Theorems 2.1 and 2.2, and Theorems 3.1 and 3.2 , respectively.

=−

2. The function αm (< x >) Pm m−k Let αm (x) = , (m ≥ 1). Then we consider the function Pm k=0 Ek (x)x αm (< x >) = k=0 Ek (< x >) < x >m−k , defined on (−∞, ∞), which is periodic with period 1. P∞ (m) The Fourier series of βm (< x >) is n=−∞ An e2πinx , where Z 1 A(m) = αm (< x >)e−2πinx dx n 0

Z

(2.1)

1

=

−2πinx

αm (x)e

dx.

0

To proceed further, we note the following.

0 αm (x) =

m X

kEk−1 (x)xm−k + (m − k)Ek (x)xm−k−1




k=0

=

m X

kEk−1 (x)xm−k +

k=1

=

m−1 X

(m − k)Ek (x)xm−k−1

k=0

m−1 X

(k + 1)Ek (x)xm−k−1 +

k=0

= (m + 1)

m−1 X

(m − k)Ek (x)xm−k−1

(2.2)

k=0 m−1 X

Ek (x)xm−1−k

k=0

= (m + 1)αm−1 (x).

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FOURIER SERIES OF FUNCTIONS INVOLVING EULER POLYNOMIALS 0 So, αm (x) = (m + 1)αm−1 (x). From this, 1

Z

αm (x)dx = 0

αm (1) − αm (0) = =



αm+1 (x) m+2

0

= αm (x).

1 (αm+1 (1) − αm+1 (0)) . m+2

m X k=0 m X

3

(2.3)

(Ek (1) − Ek δm,k ) ((−Ek + 2δk,0 )) −

k=0 m X

=−

m X

Ek δm,k

(2.4)

k=0

Ek + 2 − Em

k=0

Thus m X

αm (1) = αm (0) ⇐⇒

Ek = 2 − Em .

(2.5)

k=0

Also, Z 0

1

1 αm (x)dx = m+2

−

m+1 X

! Ek + 2 − Em+1

.

(2.6)

k=0

(m)

Now, we would like to determine the Fourier coefficients An . Case1 : n 6= 0. A(m) n

Z

1

=

αm (x)e−2πinx dx

0

i1 1 h 1 =− αm (x)e−2πinx + 2πin 2πin 0

Z

1 0 αm (x)e−2πinx dx

0

Z 1 m+1 1 (αm (1) − αm (0)) + αm−1 (x)e−2πinx dx 2πin 2πin 0 ! m X m + 1 (m−1) 1 = A + Ek − 2 + Em 2πin n 2πin k=0 !! ! m−1 m X X m+1 m (m−2) 1 1 = A + Ek − 2 + Em−1 + Ek − 2 + Em 2πin 2πin n 2πin 2πin k=0 k=0 ! ! m−1 m X X (m + 1)m (m−2) m+1 1 = Ek − 2 + Em A + Ek − 2 + Em−1 + (2πin)2 n (2πin)2 2πin =−

k=0

k=0

= ···

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TAEKYUN KIM, DAE SAN KIM, GWAN-WOO JANG, AND JONGKYUM KWON m−j+1 X

m−1 (m + 1)m−1 (1) X (m + 1)j−1 = A + (2πin)m−1 n (2πin)j j=1

=

m−j+1 X

m X (m + 1)j−1 j=1

(2πin)j

Ek − 2 + Em−j+1

k=0

m−j+1 X

m−1 (m + 1)! X (m + 1)j−1 + =− (2πin)m (2πin)j j=1

!

! Ek − 2 + Em−j+1

k=0

(2.7)

! Ek − 2 + Em−j+1

k=0

m

1 X (m + 2)j = m + 2 j=1 (2πin)j

m−j+1 X

! Ek − 2 + Em−j+1

,

k=0

R1 R1 (1) 2 where An = 0 α1 (x)e−2πinx dx = 0 (2x − 12 )e−2πinx dx = − 2πin . Case2 : n = 0.

(m) A0

1

Z = 0

1 αm (x)dx = m+2

−

m+1 X

! Ek + 2 − Em+1

.

(2.8)

k=0

∞ αm (< x >), (m ≥ 1) is piecewise Pm C . Moreover, αm (< x >) is continuous for those positive integers m with k=0 Ek = 2 − Em and discontinuous with jump Pm discontinuities at integers for those positive integers m with k=0 Ek 6= 2 − Em .

We recall the following facts about Bernoulli functions Bn (< x >) : (a) for m ≥ 2,

Bm (< x >) = −m!

∞ X n=−∞,n6=0

e2πinx . (2πin)m

(2.9)

(b) for m = 1,

−

∞ X n=−∞,n6=0

e2πinx = 2πin

( B1 (< x >), 0,

for x ∈ / Z, for x ∈ Z.

(2.10)

Pm Assume first that m is a positive integer with k=0 Ek = 2−Em . Then αm (1) = αm (0). αm (< x >) is piecewise C ∞ , and continuous. So the Fourier series of αm (< x >) converges uniformly to αm (< x >), and

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FOURIER SERIES OF FUNCTIONS INVOLVING EULER POLYNOMIALS

1 αm (< x >) = − m+2

m+1 X

5

! Ek − 2 + Em+1

k=0

 ! m−j+1 m X X 1 (m + 2)j  Ek − 2 + Em−j+1  e2πinx + j m+2 (2πin) j=1 k=0 n=−∞,n6=0 ! m+1 X 1 =− Ek − 2 + Em+1 m+2 k=0 !  m−j+1 m  X 1 X m+2 − Ek − 2 + Em−j+1 m + 2 j=1 j k=0   ∞ 2πinx X e  × −j! (2πin)j ∞ X

n=−∞,n6=0

(2.11) 1 =− m+2

m+1 X

! Ek − 2 + Em+1

k=0

!  m−j+1 m  X 1 X m+2 Ek − 2 + Em−j+1 Bj (< x >) − j m + 2 j=2 k=0 ! (   X m B1 (< x >), for x ∈ / Z, 1 m+2 − Ek − 2 + Em · , 1 m+2 0, for x ∈ Z

(2.12)

k=0

for all x ∈ (−∞, ∞). Hence we obtain the following theorem. Pm Theorem 2.1. Let m be a positive integer with k=0 Ek = 2 − Em . Then we have the P following. m (a) k=0 Ek (< x >) < x >m−k has the Fourier series expansion m X

Ek (< x >) < x >m−k

k=0

1 =− m+2 1 + m+2

m+1 X

! Ek − 2 + Em+1

k=0 ∞ X n=−∞,n6=0

 m X (m + 2)j  (2πin)j j=1

m−j+1 X

! Ek − 2 + Em−j+1  e2πinx ,

k=0

for all x ∈ (−∞, ∞), where the convergence is uniform.   m m X X 1 m+2 m−k (b) Ek (< x >) < x > =− m+2 j k=0

j=0,j6=1

m−j+1 X

! Ek − 2 + Em−j+1

Bj (< x >),

k=0

for all x ∈ (−∞, ∞), where Bk (< x >) is the Bernoulli function.

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Pm Assume next that m is a positive integer with k=0 Ek 6= 2 − Em . Then αm (1) 6= αm (0). Hence αm (< x >) is piecewise C ∞ , and discontinuous with jump discontinuities at integers. The Fourier series of αm (< x >) converges pointwise to αm (< x >) , for x ∈ / Z, and converges to m

1 1X 1 (αm (0) + αm (1)) = αm (0) − Ek + 1 − Em , 2 2 2 k=0

=1−

1 2

m−1 X

Ek ,

k=0

for x ∈ Z. Thus, we get the following theorem. Pm Theorem 2.2. Let m be a positive integer with k=0 Ek 6= 2 − Em . Then we have the following. ! m+1 X 1 Ek − 2 + Em+1 (a) − m+2 k=0  ! m−j+1 ∞ m X X X 1 (m + 2) j  + Ek − 2 + Em−j+1  e2πinx j m+2 (2πin) j=1 n=−∞,n6=0 k=0 (P m m−k , for x ∈ / Z, k=0 Ek (< x >) < x > = Pm−1 1 1 − 2 k=0 Ek , for x ∈ Z.  m  1 X m+2 (b) − j m + 2 j=0 =

m X

m−j+1 X

! Ek − 2 + Em−j+1

Bj (< x >)

k=0

Ek (< x >) < x >m−k , f or x ∈ / Z,

k=0

1 − m+2 =1−

  m X m+2 j

m−j+1 X

j=0,j6=1

! Ek − 2 + Em−j+1

Bj (< x >)

k=0

m−1 1 X Ek , f or x ∈ Z. 2 k=0

Question: For what values of m ≥ 1, does (m)

Remark 2.3. Another expression for A0 (see [3,4,6,12]) and is m−1 X m−l X l=0 j=1

m−l+1 j

Pm

=

k=0

R1 0

Ek = 2 − Em hold ?

αm (x)dx was obtained previously


El+j 4(−1)m+1 Em+1 .
+ l+j m+2 (m − l + 1) l

(−1)j

802

(2.13)

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7

So, we obtain the following identity. 1 m+2 =

m+1 X

−

! Ek + 2 − Em+1

k=0

m−1 X m−l X l=0 j=1

m−l+1 j


El+j 4(−1)m+1 + Em+1 .
m+2 (m − l + 1) l+j l

(−1)j

3. The fuction βm (< x >) Let βm (x) = function

Pm

1 m−k , k=0 k!(m−k)! Ek (x)x

βm (< x >) =

m X k=0

(m ≥ 1). Then we will consider the

1 Ek (< x >) < x >m−k , k!(m − k)!

defined on (−∞, ∞), which is periodic with period 1. The Fourier series of βm (< x >) is ∞ X

Bn(m) e2πinx ,

n=−∞

where Bn(m)

1

Z

−2πinx

=

βm (< x >)e

Z dx =

0

1

βm (x)e−2πinx dx.

0

Before proceeding further, we observe the following: m  X k 0 Ek−1 (x)xm−k βm (x) = k!(m − k)! k=0  m−k + Ek (x)xm−k−1 k!(m − k)! m X 1 Ek−1 (x)xm−k = (k − 1)!(m − k)! k=1

+

m−1 X k=0

=

m−1 X k=0

+

m−1 X k=0

1 Ek (x)xm−1−k k!(m − 1 − k)!

(3.1)

1 Ek (x)xm−1−k k!(m − 1 − k)! 1 Ek (x)xm−1−k k!(m − 1 − k)!

= 2βm−1 (x). 0 So, βm (x) = 2βm−1 (x). This implies that

Z

1

βm (x)dx = 0



βm+1 (x) 2

0

= βm (x).

 1 βm+1 (1) − βm+1 (0) . 2

803

(3.2)

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TAEKYUN KIM, DAE SAN KIM, GWAN-WOO JANG, AND JONGKYUM KWON

βm (1) − βm (0) = = −

m X k=0 m X k=0 m X


1 Ek (1) − Ek (0)δm,k k!(m − k)! 1 {(−Ek + 2δk,0 )} . k!(m − k)!

(3.3)

1 Ek δm,k k!(m − k)!

k=0 m X

=−

k=0

2 Em Ek + − . k!(m − k)! m! m!

 P m+1 Ek 2 So, 0 βm (x)dx = 12 − k=0 k!(m+1−k)! + (m+1)! − Pm Ek Em 2 Also, βm (1) = βm (0) ⇔ k=0 k!(m−k)! = m! − m! . R1

Em+1 (m+1)!



.

(m)

Now, we are going to determine the Fourier coefficients Bn . Case 1:n 6= 0.

Bn(m) =

Z

1

βm (x)e−2πinx dx

0

Z 1 i1 1 h 1 βm (x)e−2πinx + β 0 (x)e−2πinx dx 2πin 2πin 0 m 0 Z 1  1  1 =− βm (1) − βm (0) + βm−1 (x)e−2πinx dx 2πin πin 0 1 1 (m−1) Bn − (βm (1) − βm (0)) = πin 2πin   1 1 1 1 (m−2) = Bn − (βm−1 (1) − βm−1 (0)) − (βm (1) − βm (0)) πin πin 2πin 2πin 1 2 1 = Bn(m−2) − (βm−1 (1) − βm−1 (0)) − (βm (1) − βm (0)) 2 2 (πin) (2πin) 2πin = ··· =−

m−1 X 2j−1 1 (1) = B − (βm−j+1 (1) − βm−j+1 (0)) n (πin)m−1 (2πin)j j=1 m−1 X 2j−1 1 =− + (πin)m (2πin)j j=1

=

m−j+1 X

Ek 2 Em−j+1 − + k!(m − j − k + 1)! (m − j + 1)! (m − j + 1)! k=0 ! m−j+1 X 2 Em−j+1 Ek − + , k!(m − j − k + 1)! (m − j + 1)! (m − j + 1)!

m X 2j−1 (2πin)j j=1

!

k=0

(3.4) where

(1) Bn

=

R1 0

β1 (x)e

−2πinx

dx =

R1 0

(2x −

804

1 −2πinx dx 2 )e

=

1 − πin .

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9

Case 2: n = 0.

(m) B0

Z = 0

1

1 βm (x)dx = 2

−

m+1 X k=0

Ek 2 Em+1 + − k!(m + 1 − k)! (m + 1)! (m + 1)!

! . (3.5)

Let Ωm = βm (1) − βm (0) = −

m X k=0

Ek 2 Em + − , k!(m − k)! m! m!

for m ≥ 1. βm (< x >), (m ≥ 1) is piecewise C ∞ . Moreover, βm (< x >) is continuous for those positive integers m with Ωm = 0 and discontinuous with jump discontinuities at integers for those positive integers m with Ωm 6= 0. Assume first that m is a positive integer with Ωm = 0. Then βm (1) = βm (0). βm (< x >) is piecewise C ∞ , and continuous. So the Fourier series of βm (< x >) converges uniformly to βm (< x >), and βm (< x >) =

m X k=0

=

=

1 Ek (< x >) < x >m−k k!(m − k)!

1 Ωm+1 − 2

∞ X

m X  2j−1 Ω e2πinx m−j+1 j (2πin) j=1

n=−∞,n6=0

∞ X

m X 2j−1

 1 Ωm+1 + Ωm−j+1 × −j! 2 j! j=1 m X

n=−∞,n6=0

e2πinx  (2πin)j

j−1

1 2 Ωm+1 + Ωm−j+1 Bk (< x >) 2 j! j=2 ( B1 (< x >), for x ∈ / Z, + Ωm × 0, for x ∈ Z, =

for all x ∈ (−∞, ∞). (3.6) Now, we obtain the following theorem. Theorem 3.1. For each positive integer l, let Ωl = −

l X k=0

Ek 2 El + − . k!(l − k)! l! l!

Assume m = 0, for a positive integer m. Then we have the following. Pm that Ω 1 (a) k=0 k!(m−k)! Ek (< x >) < x >m−k has the Fourier series expansion m X k=0

1 1 Ek (< x >) < x >m−k = Ωm+1 − k!(m − k)! 2

∞ X

m X  2j−1 Ωm−j+1 e2πinx , j (2πin) j=1

n=−∞,n6=0

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for all x ∈ (−∞, ∞). Here the convergence is uniform. (b)

m X k=0

1 Ek (< x >) < x >m−k = k!(m − k)!

m X j=0,j6=1

2j−1 Ωm−j+1 Bk (< x >), j!

for all x ∈ (−∞, ∞). Here Bk (< x >) is the Bernoulli function. Assume next that m is a positive integer with Ωm 6= 0. Then, βm (1) 6= βm (0). βm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers. Thus the Fourier series of βm (< x >) converges pointwise to βm (< x >), for x ∈ / Z, and converges to 1 1 (βm (0) + βm (1)) = βm (0) + Ωm 2 2 ! m X Em 1 2 Em Ek = + − + − m! 2 k!(m − k)! m! m! k=0 ! m−1 X 2 1 Ek − . = 2 m! k!(m − k)!

(3.7)

k=0

for x ∈ Z. So, we obtain the following theorem. Theorem 3.2. For each positive integer l, let Ωl = −

l X k=0

Ek 2 El + − . k!(l − k)! l! l!

Assume that Ωm 6= 0, for a positive integer m. Then we have the following.

∞ m X  X 1 2j−1 (a) Ωm+1 − Ωm−j+1 e2πinx j 2 (2πin) n=−∞,n6=0 j=1 (Pm 1 Ek (< x >) < x >m−k , for x ∈ / Z, = Emk=0 1k!(m−k)! + Ω , for x ∈ Z. m! 2 m

Here the convergence is pointwise. (b) m X 2j−1 Ωm−j+1 Bj (< x >) j! j=0 =

m X k=0

1 Ek (< x >) < x >m−k , k!(m − k)! m X

j=0,j6=1

for x ∈ / Z,

2j−1 Ωm−j+1 Bj (< x >) j!

1 Em + Ωm , m! 2 Here Bk (< x >) is the Bernoulli function. =

806

for x ∈ Z.

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FOURIER SERIES OF FUNCTIONS INVOLVING EULER POLYNOMIALS

Question: For what values of m ≥ 1, does

Pm

Ek k=0 k!(m−k)!

=

2 m!

−

Em m!

11

hold ?

Remark 3.3. In a previous paper (see [3,4,6,12]), it was shown that

Z

1

βm (x)dx = 0

m−1 X m−l X

(−1)j

m+1 l+j


El+j

(m + 1)!

l=0 j=1

+

2(−1)m+1 Em+1 . (m + 1)!

(3.8)

Hence, we have the following identity. m+1 X

1 2

2 Em+1 Ek − + − k!(m + 1 − k)! (m + 1)! (m + 1)! k=0
m−1 X m−l X (−1)j m+1 2(−1)m+1 Em+1 l+j El+j + . = (m + 1)! (m + 1)! j=1

!

l=0

4. The fuction γm (< x >) Let γm (x) = function

Pm−1

1 m−k , (m k=1 k(m−k) Ek (x)x

γm (< x >) =

m−1 X k=1

≥ 2). Then we will consider the

1 Ek (< x >) < x >m−k , k(m − k)

(4.1)

defined on (−∞, ∞), which is periodic with period 1. The Fourier series of γm (< x >) is ∞ X

Cn(m) e2πinx ,

(4.2)

n=−∞

where Cn(m)

Z =

1 −2πinx

γm (< x >)e 0

Z dx =

1

γm (x)e−2πinx dx.

(4.3)

0

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To proceed further, we observe the following. m−1   X 1 0 γm (x) = kEk−1 (x)xm−k + (m − k)Ek (x)xm−k−1 k(m − k) k=1

=

m−1 X k=1

=

m−2 X k=0

=

m−1 X 1 1 Ek−1 (x)xm−k + Ek (x)xm−k−1 m−k k k=1

m−1 X 1 1 Ek (x)xm−k−1 + Ek (x)xm−k−1 m−k−1 k k=0

1 xm−1 + m−1 +

m−2 X k=1

m−2 X k=1

1 1 Ek (x)xm−k−1 + Em−1 (x) m−k−1 m−1

(4.4)

1 Ek (x)xm−k−1 k

m−2 X

 1  m−1 1 x + Em−1 (x) Ek (x)xm−1−k + k(m − 1 − k) m−1 k=1   1 = (m − 1)γm−1 (x) + xm−1 + Em−1 (x) . m−1 Thus,  1  m−1 0 γm (x) = (m − 1)γm−1 (x) + x + Em−1 (x) . m−1 From this, we have    0 1 1 1 γm+1 (x) − xm+1 − Em+1 (x) = γm (x). m m(m + 1) m(m + 1) = (m − 1)

Z

1

γm (x)dx 0

i1 1 1 1h xm+1 − Em+1 (x) γm+1 (x) − m m(m + 1) m(m + 1) 0   1 1 1 = − γm+1 (1) − γm+1 (0) − Em+1 (1) − Em+1 (0) m m(m + 1) m(m + 1)  2 1 1 + Em+1 . = γm+1 (1) − γm+1 (0) − m m(m + 1) m(m + 1) (4.5) =

γm (1) − γm (0) =

m−1 X k=1

=

m−1 X k=1

=−


1 Ek (1) − Ek (0)δm,k k(m − k)   m−1 X 1 1 −Ek (0) + 2δk,0 − Ek (0)δm,k k(m − k) k(m − k)

m−1 X k=1

(4.6)

k=1

Ek . k(m − k)

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13

Thus, γm (1) = γm (0) ⇔

m−1 X

Ek = 0. k(m − k)

k=1

(4.7)

In addition, Z 0

1

1 γm (x)dx = − m

m X k=1

Ek 1 2 + − Em+1 k(m − k + 1) m(m + 1) m(m + 1)

! . (4.8)

(m)

Now, we would like to determine the Fourier coefficients Cn . Case 1:n 6= 0. Cn(m) =

1

Z

γm (x)e−2πinx dx

0

Z 1 i1 1 1 h −2πinx γm (x)e + γ 0 (x)e−2πinx dx =− 2πin 2πin 0 m 0  m−1Z 1 1  (4.9) =− γm (1) − γm (0) + γm−1 (x)e−2πinx dx 2πin 2πin 0 Z 1 Z 1 1 1 xm−1 e−2πinx dx + Em−1 (x)e−2πinx dx + 2πin(m − 1) 0 2πin(m − 1) 0 1 1 2 m − 1 (m−1) C − Λm − Θm + Φm , = 2πin n 2πin 2πin(m − 1) 2πin(m − 1) where , for l ≥ 1, Z

1

El (x)e

−2πinx

dx =

( Pl 2 k=1

0

Z

1

xl e−2πinx dx =

0

(l)k−1 E , (2πin)k l−k+1

2 − l+1 El+1 ,

( Pl − k=1

for n 6= 0, for n = 0.

(l)k−1 , (2πin)k

1 l+1 ,

for n 6= 0, for n = 0.

Here, for m ≥ 2,

Λm = γm (1) − γm (0) = −

m−1 X k=1

Θm =

m−1 X k=1

Φm =

m−1 X k=1

(m − 1)k−1 , (2πin)k

Ek , k(m − k) (4.10)

(m − 1)k−1 Em−k . (2πin)k

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m − 1 (m−1) 1 1 2 C − Λm − Θm + Φm 2πin n 2πin 2πin(m − 1) 2πin(m − 1)  1 1 2 m − 1  m − 2 (m−2) Cn − Λm−1 − Θm−1 + Φm−1 = 2πin 2πin 2πin 2πin(m − 2) 2πin(m − 1) 1 2 1 − Λm − Θm + Φm 2πin 2πin(m − 1) 2πin(m − 1) m−1 1 m−1 (m − 1)(m − 2) (m−2) Cn − Λm−1 − Λm − Θm−1 = (2πin)2 (2πin)2 2πin (2πin)2 (m − 2) 1 2(m − 1) 2 − Θm + Φm−1 + Φm (2πin)(m − 1) (2πin)2 (m − 2) 2πin(m − 1) = ···

Cn(m) =

=

+

m−2 m−2 X (m − 1)j−1 (m − 1)! (2) X (m − 1)j−1 C − Λ − Θm−j+1 m−j+1 n m−2 j (2πin) (2πin) (2πin)j (m − j) j=1 j=1 m−2 X j=1

=−

−

m−2 1 (m − 1)! 2(m − 1)! X (m − 1)j−1 − − Λm−j+1 2 (2πin)m−1 (2πin)m (2πin)j j=1

m−2 X j=1

=−

2(m − 1)j−1 Φm−j+1 (2πin)j (m − j)

m−2 X 2(m − 1)j−1 (m − 1)j−1 Θ + Φm−j+1 m−j+1 j (2πin) (m − j) (2πin)j (m − j) j=1

m−1 X j=1

+

m−1 X j=1

m−1 X (m − 1)j−1 (m − 1)j−1 Θm−j+1 Λ − m−j+1 j (2πin) (2πin)j (m − j) j=1

2(m − 1)j−1 Φm−j+1 , (2πin)j (m − j) (4.11)

where

Z

1

Z

1

1 1 2 1 − (x2 − x)e−2πinx dx = − 2 2 2 2πin (2πin) 0 0 1 1 1 1 Λ2 = , Θ2 = , Φ2 = × (− ). 2 2πin 2πin 2 (4.12) Before proceeding further, we note the following. Cn(2)

=

−2πinx

γ2 (x)e

dx =

810

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m−1 X j=1

=−

(m − 1)j−1 Λm−j+1 (2πin)j

m−1 X j=1

1 m

=−

15

m−j (m − 1)j−1 X Ek (2πin)j k(m − j − k + 1) k=1

m−1 X m−j X j=1 k=1

(m)j Ek (2πin)j k(m − j − k + 1)

(4.13)

m−1 m−s 1 X X (m)s Ek m s=1 (2πin)s k(m − s − k + 1)

=−

k=1

1 m

=−

m−1 X j=1

=

m−1 X j=1

=

=

2 m 2 m

m m−1 X X s=1 l=s

(m)s El−s+1 . − s + 1)(m − l)

(2πin)s (l

2(m − 1)j−1 Φm−j+1 (2πin)j (m − j) m−j 2(m − 1)j−1 X (m − j)k−1 Em−j−k+1 (2πin)j (m − j) (2πin)k k=1

m−1 X m−j X j=1 k=1 m−1 X j=1

(m)j+k−1 Em−j−k+1 (2πin)j+k (m − j)

m X 1 (m)s−1 Em−s+1 m − j s=j+1 (2πin)s

=

m s−1 X 2 X (m)s−1 1 E m−s+1 m s=2 (2πin)s m −j j=1

=

2 X (m)s−1 Em−s+1 (Hm−1 − Hm−s ) m s=2 (2πin)s

=

2 X (m)s Em−s+1 (Hm−1 − Hm−s ) . m s=1 (2πin)s m − s + 1

(4.14)

m

m

m−1 X

(m − 1)j−1 Θm−j+1 (2πin)j (m − j)

j=1 m X

1 = m

s=1

(4.15) (m)s Hm−1 − Hm−s . (2πin)s m − s + 1

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Putting everything together, we have

Cn(m) =

m m−1 1 XX (m)s El−s+1 s m s=1 (2πin) (l − s + 1)(m − l) l=s

m m 1 X (m)s Hm−1 − Hm−s 2 X (m)s Em−s+1 − + (Hm−1 − Hm−s ) m s=1 (2πin)s m − s + 1 m s=1 (2πin)s m − s + 1 m

m−1 X (m)s Hm−1 − Hm−s (m)s El−s+1 (1 − 2E ) − m−s+1 s s (2πin) m−s+1 (2πin) (l − s + 1)(m − l) l=s (4.16) Case 2: n = 0.

1 X =− m s=1

! .

! Ek 1 2 = + − Em+1 . k(m − k + 1) m(m + 1) m(m + 1) 0 k=1 (4.17) Pm Question: For what values of m ≥ 1, does k=0 Ek = 2 − Em hold ?

(m) C0

1

Z

1 γm (x)dx = − m

m X

Remark 4.1. In a previous paper (see [3,4,6,12]), it was shown that

Z 0

1

m−1 X m−l X (−1)j 1 γm (x)dx = m(m2 − 1) j=1 l=1

m+1 l+j
m−2 l−1


El+j

m−1 2(−1)m+1 Em+1 X (−1)l
+ m−2 . m(m2 − 1) l−1 l=1

(4.18) So, we obtain the following identity.

! Ek 1 2 + − Em+1 k(m − k + 1) m(m + 1) m(m + 1) k=1
m−1 m−1 X (−1)j m+1 X m−l 2(−1)m+1 Em+1 X (−1)l 1 l+j El+j

+ = m−2 m−2 , m(m2 − 1) m(m2 − 1) l−1 l−1 j=1

1 − m

m X

l=1

(4.19)

l=1

for m ≥ 2. γm (< x >), (m ≥ 2) is piecewise C ∞ . Moreover, γm (< x >) is continuous for those integers m ≥ 2 with and Λm = 0, and discontinuous with jump discontinuities at integers for those integers ≥ 2 with Λm 6= 0. Assume first that Λm = 0. Then γm (1) = γm (0). γm (< x >) is piecewise C ∞ and continuous. So the Fourier series of γm (< x >) converges uniformly to γm (< x >), and

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17

γm (< x >) m X

1 =− m

k=1

∞ X

−

Ek 1 2 + − Em+1 k(m − k + 1) m(m + 1) m(m + 1)

!

m  1 X (m)s Hm−1 − Hm−s (1 − 2Em−s+1 ) m s=1 (2πin)s m − s + 1

n=−∞,n6=0

−

m−1 X l=s

 (m)s El−s+1 e2πinx (2πin)s (l − s + 1)(m − l)

! Ek 1 2 + − Em+1 k(m − k + 1) m(m + 1) m(m + 1) k=1   m−1 m    m X El−s+1 1 X m Hm−1 − Hm−s (1 − 2Em−s+1 ) − + s m s=1 s m−s+1 (l − s + 1)(m − l) 1 =− m

m X

l=s



× −s!

∞ X n=−∞,n6=0

2πinx

e (2πin)s



! Ek 1 2 + − Em+1 k(m − k + 1) m(m + 1) m(m + 1) k=1   m−1 m    m X El−s+1 1 X m Hm−1 − Hm−s (1 − 2Em−s+1 ) − + Bs (< x >) m s=2 m−s+1 s (l − s + 1)(m − l) s l=s (  m−1 X B1 (< x >), for x ∈ / Z, El  × + − l(m − l) 0, for x ∈ Z, l=1 (4.20) Pm where Hm = k=1 k1 . 1 =− m

m X

Now, we get the following theorem.

Theorem 4.2. Let m be an integer ≥ 2, with

Λm = −

m−1 X k=1

Ek = 0. k(m − k)

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Then have the following. Pwe m−1 1 (a) k=1 k(m−k) Ek (< x >) < x >m−k has the Fourier series expansion m−1 X k=1

1 Ek (< x >) < x >m−k k(m − k) m

=−

 1 X Ek 1 2 + − Em+1 m k(m − k + 1) m(m + 1) m(m + 1) k=1

−

∞ X

m  1 X (m)s Hm−1 − Hm−s (1 − 2Em−s+1 ) m s=1 (2πin)s m − s + 1

n=−∞,n6=0

−

m−1 X l=s

 (m)s El−s+1 e2πinx , (2πin)s (l − s + 1)(m − l)

for all x ∈ (−∞, ∞), where the convergence is uniform. (b) m−1 X k=1

1 Ek (< x >) < x >m−k k(m − k)

m  Ek 1 2 1 X + − Em+1 m k(m − k + 1) m(m + 1) m(m + 1) k=1   m−1   m  1 X m Hm−1 − Hm−s m X El−s+1 + (1 − 2Em−s+1 ) − Bs (< x >), m s=2 m−s+1 s (l − s + 1)(m − l) s

=−

l=s

for all x ∈ (−∞, ∞). Here Bk (< x >) is the Bernoulli function. Assume next that m is an integer ≥ 2 with Λm 6= 0. Then, γm (1) 6= γm (0). Hence γm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers. Thus the Fourier series of γm (< x >) converges pointwise to γm (< x >), for x ∈ / Z, and converges to m−1 1 1 1 X Ek (γm (0) + γm (1)) = γm (0) + Λm = − , 2 2 2 k(m − k) k=1

for x ∈ Z. Hence we obtain the following theorem. Theorem 4.3. Let m be an integer ≥ 2, with Λm = −

m−1 X k=1

Ek 6= 0. k(m − k)

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19

Then, we have the following. (a) m  1 2 1 X Ek − + − Em+1 m k(m − k + 1) m(m + 1) m(m + 1) k=1

−

∞ X

m  1 X (m)s Hm−1 − Hm−s (1 − 2Em−s+1 ) m s=1 (2πin)s m − s + 1

n=−∞,n6=0

−

m−1 X l=s

 (m)s El−s+1 e2πinx (2πin)s (l − s + 1)(m − l)

(Pm−1 =

1 k=1 k(m−k) Ek (< Pm−1 Ek 1 − 2 k=1 k(m−k) ,

x >) < x >m−k ,

for x ∈ / Z, for x ∈ Z.

Here the convergence is pointwise. (b) m  1 2 1 X Ek + − Em+1 − m k(m − k + 1) m(m + 1) m(m + 1) k=1   m−1 m    m X El−s+1 1 X m Hm−1 − Hm−s (1 − 2Em−s+1 ) − Bs (< x >) + m s=1 m−s+1 s (l − s + 1)(m − l) s l=s

=

m−1 X k=1

1 Ek (< x >) < x >m−k , for x ∈ / Z, k(m − k)

m  Ek 1 X 1 2 + − Em+1 m k(m − k + 1) m(m + 1) m(m + 1) k=1   m−1   m  1 X m Hm−1 − Hm−s m X El−s+1 + (1 − 2Em−s+1 ) − Bs (< x >) m s=2 m−s+1 s (l − s + 1)(m − l) s

−

l=s

=−

1 2

m−1 X k=1

Ek , for x ∈ Z. k(m − k)

Question: For what values of m ≥ 2, does

Pm−1

Ek k=1 k(m−k)

= 0 hold ?

References 1. A. Bayad, T. Kim, Higher recurrences for Apostol-Bernoulli-Euler numbers, Russ. J. Math. Phys., 16(2012), no.1, 1-10. 2. A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russian J. Math. Phys., 18(2011), no. 2, 133-143. 3. D.S. Kim, D. V. Dolgy, T. Kim, S.-H. Rim Some Formulae for the Product of Two Bernoulli and Euler Polynomials, Abstr. Appl. Anal. 2012, Art. ID 784307. 4. D.S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct., 24(9) (2013), 734-738. 5. D.S. Kim, T. Kim, Y.H. Lee, Some arithmetic properties of Bernoulli and Euler nembers, Adv. Stud. Contemp. Math., 22(2010), no.4, 467-480. 6. D.S. Kim, T. Kim, T. Mansour, Euler basis and the product of several Bernoulli and Euler polynomials, Adv. Stud. Contemp. Math., 24(2014), no.4, 535-547.

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7. T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 17(2008), 131–136. 8. T. Kim, On the weighted q-Euler numbers and q-Berstein polynomials , Adv. Stud. Contemp. Math., 22(2012), no.1, 7-12. 9. T. Kim, Some identities for the Bernoulli, the Euler and Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20(2015), no.1, 23-28. 10. T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal. 2008, Art. ID 581582, 11 pp. 11. T. Kim, J. Choi, Y. H. Kim, A note on the values of Euler zeta functions at positive integers, Adv. Stud. Contemp. Math. (Kyungshang), 22(2012), 27–34. 12. T. Kim, D. S. Kim, D. V. Dolgy, S.-H. Rim, Some identities on the Euler numbers arising from Euler basis polynomials, ARS Combinatoria 109(2013), 433–446. 13. T. Kim, D. S. Kim, S.-H. Rim, D. V. Dolgy, Fourier series of higher-order Bernoulli functions and their applications, J. Inequal. Appl. 2016, to appear. 14. J. E. Marsden, Elementary classical analysis, W. H. Freeman and Company, 1974. 15. B.H. Yadav, Absolute convergence of Fourier series, Thesis (Ph.D.)-Maharaja Sayajirao University of Baroda (India), 1964. 16. Y. Simsek, Interpolation functions of the Eulerian type polynomials and numbers, Adv. Stud. Contemp. Math. (Kyungshang), 23(2013), no. 2, 301-307. 17. D. G. Zill, M. R. Cullen, Advanced Engineering Mathematics, Jones and Bartlett Publishers 2006. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China, Department of Mathematics, Kwangwoon University, Seoul, 139701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea E-mail address: [email protected] Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Higher order generalization of Bernstein type operators defined by (p, q)-integers M. Mursaleen1 , Md. Nasiruzzaman1 , Nurgali Ashirbayev2 , Azimkhan Abzhapbarov 2 1 Department of Mathematics, Aligarh Muslim University, Aligarh–202002, India 2 Science-Pedagogical Faculty, M. Auezov South Kazakhstan State University, Shymkent, 160012, Kazakhstan [email protected]; [email protected]; ank [email protected]; azeke [email protected] Abstract In this paper, we introduce the higher order generalization of Bernstein type operators defined by (p, q)-integers. We establish some approximation results for these new operators by using the modulus of continuity. Keywords and phrases: (p, q)-integers; (p, q)-Bernstein operators; modulus of continuity; approximation theorems. AMS Subject Classification (2010): 41A10, 41A36.

1. Introduction and preliminaries In 1912, S.N Bernstein [4] introduced the following sequence of operators Bn : C[0, 1] → C[0, 1] defined for any n ∈ N and for any f ∈ C[0, 1] such as   n   X n k k n−k Bn (f ; x) = x (1 − x) f , x ∈ [0, 1]. (1.1) k n k=0 In approximation theory, q-type generalization of Bernstein polynomials was introduced by Lupa¸s [7]. For f ∈ C[0, 1], the generalized Bernstein polynomial based on the q-integers is defined by Phillips [15] as follows    n  n−k−1 X Y [k]q n k s Bn,q (f ; x) = x (1 − q x) f , x ∈ [0, 1]. (1.2) k q [n]q s=0 k=0 Recently, Mursaleen et al. [10] applied (p, q)-calculus in approximation theory and introduced first (p, q)-analogue of Bernstein operators and defined as:

Bn,p,q (f ; x) =

1 p

n(n−1) 2

 n X f k=0

[k] pk−n [n]

 Pn,k (p, q; x), 0 < q < p ≤ 1, x ∈ [0, 1]

where Pn,k (p, q; x) = p

k(k−1) 2



n k

 x p,q

k

n−k−1 Y

(ps − q s x).

(1.3)

s=0

1

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They have also introduced and studied approximation properties based on (p, q)integers given as: (p, q)-Bernstein-Stancu operators [11], (p, q)-Bernstein-Shurer operators [14] and (p, q)-Bleimann-Butzer-Hahn operators [13]. In the sequel, some more articles on (p, q)-approximation have also been appeared, e.g. [1], [2], [3], [6], [9], [12] and [13]. We recall some basic properties of (p, q)-integers. The (p, q)-integer [n]p,q is defined by [n]p,q =

pn − q n , n = 0, 1, 2, · · · , 0 < q < p ≤ 1. p−q

The (p, q)-Binomial expansion is (x + y)np,q := (x + y)(px + qy)(p2 x + q 2 y) · · · (pn−1 x + q n−1 y) and the (p, q)-binomial coefficients are defined by   [n]p,q ! n := . k p,q [k]p,q ![n − k]p,q ! For p = 1, all the notions of (p, q)-calculus are reduced to q-calculus. For details on (p, q)-calculus and q-calculus, one can refer [5, 7]. In this paper we use the notation [n] in place of [n]p,q . In [5], (p, q)-derivative of a function f (x) is defined by Dp,q f (x) =

f (px) − f (qx) , x 6= 0, (p − q)x

(1.4)

and the formulae for the (p, q)-derivative for the product of two functions is given as Dp,q (f g)(x) = f (px).Dp,q g(x) + {Dp,q f (x)}.g(qx),

(1.5)

Dp,q (f g)(x) = f (qx).Dp,q g(x) + {Dp,q f (x)}.g(px).

(1.6)

also Let r ∈ N ∪ {0} be a fixed number. For f ∈ C r [0, 1] and m ∈ N, we define rth order (p, q)-Bernstein type operators as follows:   i n r X 1 X 1 (i) [k] [k] [r] Bn,p,q (f ; x) = n(n−1) Pn,k (p, q; x) f x − k−n (1.7) i! pk−n [n] p [n] p 2 k=0 i=0 In this paper, using the moment estimates from [8], we give the estimates of the central moments for these operators. We also study some approximation properties of an rth order generalization of the operators defined by (1.7) using the techniques of the work on the higher order generalization of q-analogue [16]. Further, we study approximation properties and prove Voronovskaja type theorem for these operators.

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If we put p = 1, then we get the moments for q-Bernstein operators [8] and the usual generalization higher order q-Bernstein operators [16], respectively. 2. Main results We have the following elementary result. Proposition 2.1. For n ≥ 1, 0 < q < p ≤ 1 Dp,q (1 + x)np,q = [n](1 + qx)n−1 p,q .

(2.1)

Proof. By applying simple calculation on (p, q)-analogue, we have n n−1 (1 + px)np,q = pn−1 (1 + px)(1 + qx)n−1 + q n x)(1 + qx)n−1 p,q , (1 + qx)p,q = (p p,q . (2.2)

Applying (p, q)-derivative and result (2.2) we get the desired result.



Lemma 2.2. Let Bn,p,q (f ; x) be given by (1.7). Then for any m ∈ N, x ∈ [0, 1] and 0 < q < p ≤ 1 we have   
pm+n x(1 − x) x m x m+1 Bn,p,q (t − x)p,q ; x = Dp,q Bn,p,q (t − )p,q ; [n] p p   m+n−1 p [m]x(1 − x) qx m−1 qx + Bn,p,q (t − )p,q ; [n] p p n n
[m](p − q )x Bn,p,q (t − x)m + p,q ; x . [n] Proof. using (1.5) and Proposition 2.1, we have   First of all by   m Pn x 1 Pn,k (p, q; xp ) Dp,q n(n−1) k=0 t − p p

p,q

2

n X

1

= p

n(n−1) 2

(t −

k=0

x)m p,q

! m−1 n  x [m] X qx qx Dp,q {Pn,k (p, q; )} − t− Pn,k (p, q; ) . p p k=0 p p,q p (2.3)

Nowin the same way by using  (1.5) and  Proposition    2.1, we have    k k k(k−1) n x x x Dp,q Pn,k p, q; p = Dp,q p 2 [k] 1− p k p,q p =p

k(k−1) 2

 [k]

n k

 p,q

 n−k    n−k−1 ! 1 k−1 qx 1 k qx n − [n − k] . x 1− x 1− k p,q p pk p p,q p p,q (2.4)

Now by a simple calculation, we have  n−k qx 1 1 1 1− = n−k (p − qx)n−k+1 = n−k (pn−k − q n−k x)(1 − x)n−k p,q p,q p p,q p p (1 − x) (2.5)

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 n−k−1 qx 1 1 1− = n−k−1 (1 − x)n−k p,q . p p,q p (1 − x)

(2.6)

From (2.4),(2.5) and (2.6), we get   
Pn,k (p, q; x) x = n [k](pn−k − q n−k x) − pk [n − k]x , Dp,q Pn,k p, q; p p x(1 − x) which implies that   
x Pn,k (p, q; x) n−k p [k] − [n]x . (2.7) Dp,q Pn,k p, q; = n p p x(1 − x) From (2.3),  (2.7), we have  m Pn x x Dp,q Pn,k (p, q; p ) k=0 t − p p,q

1

= − p

n(n−1) 2

m−1 n  [m] X qx qx t− Pn,k (p, q; ) p k=0 p p,q p

1

n

X 1 n−k (t − x)m [k] − [n]x) + n(n−1) p,q Pn,k (p, q; x)(p n p 2 p x(1 − x) k=0 m−1 n  1 [m] X qx qx = − n(n−1) Pn,k (p, q; ) t− p k=0 p p,q p p 2 n

X 1 (t − x)m + n(n−1) p,q Pn,k (p, q; x) n x(1 − x) p p 2 k=0   [n] m [n] m m m × (p t − q x) − m (p − q )x . pm p 1

Hence n we have  o x m x Dp,q Bn,p,q (t − p )p,q ; p  
[m] qx m−1 qx [n] = − Bn,p,q (t − )p,q ; + m+n Bn,p,q (t − x)m+1 p,q ; x p p p p x(1 − x)
[m](pn − q n ) − m+n Bn,p,q (t − x)m p,q ; x . p (1 − x) This complete the proof of Lemma 2.2. 
Lemma 2.3. Let Bn,p,q (t − x)m p,q ; x be a polynomial in x of degree less than or 1 equal to m and the minimum degree of [n] is b m+1 c. Then for any fixed m ∈ N 2 and x ∈ [0, 1], 0 < q < p ≤ 1 we have Bn,p,q (t −

x)m p,q ; x




=

m−2 x(1 − x) X

[n]

b m+1 c 2

bk,m,n (p, q)xk ,

(2.8)

k=0

such that the coefficients bk,m,n (p, q) satisfy | bk,m,n (p, q) |≤ bm , k = 1, 2, · · · , m−2 and bm does not depend on x, t, p, q; where bac is an integer part of a ≥ 0.

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Proof. Clearly by Lemma 2.2 it is true for m = 2. Assuming it is true for m, then from the recurrence of Lemma 2.2 and equation (2.8) we easily get Bn,p,q (t − x)m+1 p,q ; x




=

m−1 x(1 − x) X

[n]

b m+2 c 2

bk,m+1,n (p, q)xk ,

k=0

where
1 pm+n−k [k] + pm+n−k−1 q k bk,m,n (p, q) α [n]
1 m+n+1−k m+n−k−1 k−1 − p [k − 1] + [2]p q bk−1,m,n (p, q) [n]α 1 [m](pn − q n )bk−1,m,n (p, q) + [m]pm+n−k−1 q k bk−1,m−1,n (p, q) + [n]α

bk,m+1,n (p, q) =

− [m]pm+n−k−1 q k bk−2,m−1,n (p, q). Clearly m+1 m+2 c−b c, 0 ≤ k ≤ m − 1, 2 2 which lead us that either α = 0 or α = 1. Since | bk,m,n (p, q) |≤ bm , for k = m − 1, clearly we have α=1+b

| bk,m+1,n (p, q) | ≤ + + = + =



1 1 n+1 n m−1 n+2 n m−2 p [m − 1] + p q p [m − 2] + [2]p q b + bm m [n]α [n]α 1 [m](pn − q n )bm + [m]pn q m−1 bm−1 [n]α [m]pn q m−1 bm−1

1 1 p[m − 1] + q m−1 bm + α p2 [m − 2] + [2]q m−2 bm α [n] [n] 1 [m]bm + [m]q m−1 bm−1 + [m]q m−1 bm−1 [n]α bm+1 , k = 1, 2, · · · m − 1,

and bm does not depend on x, t, p, q. This complete the proof. Remark 2.4. From the Lemma 2.3 we have

m m Bn,p,q (t − x)p,q ; x = x(1 − x)Qm−2 , Bn,p,q (t − x)p,q ; x



= 0,

(2.9)

x=0,1

where Qm−2 is a polynomial of highest degree m − 2. From the Lemma 2.2 and Lemma 2.3 we have the following theorem.

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Theorem 2.5. Let m ∈ N and 0 < q < p ≤ 1. Then there exits a constant Cm > 0 such that for any x ∈ [0, 1], we have
x(1 − x) | Bn,p,q (t − x)m m+1 . p,q ; x |≤ Cm [n]b 2 c Lemma 2.6. For any fixed m ∈ N and x ∈ [0, 1], 0 < q < p ≤ 1 we have m−k  n m m X X p − qn m m−k m−k k xm−k (t − x)kp,q (t − x) = γm,k (p − q) x (t − x)p,q = γm,k [n] k=1 k=1 (2.10) where ( γm,k =

γm−1,k−1 pk−1

−

1,

[k]γm−1,k , pk

k = 1, · · · , m − 1, γm,0 = 0 k = m,

the coefficients γm,k satisfy | γm,k |≤ γm , k = 1, · · · , m and γm does not depend on x, t, p, q. Proof. Inductively, for m = 1, it is obvious. For m ≥ 1 the relation (2.10) holds. For k = 1, · · · , m, we have m X m+1 γm,k (p − q)m−k xm−k (t − x)kp,q (t − x)m , (2.11) (t − x) = k=1

We can write t−x=


1 k k p t − q x − (p − q)[k] x p,q pk

(2.12)

(2.11),(2.12) imply that, m+1

(t − x)

=

m X

γm,k (p − q)m−k xm−k (t − x)k+1 p,q

k=1

−

m X

γm,k (p − q)m−k xm−k (t − x)kp,q

k=1

1 pk

1 (p − q)[k]p,q x pk

m X γm,m (t − x)m+1 1 p,q = + γm,k−1 (p − q)m+1−k xm+1−k (t − x)kp,q m k−1 p p k=2 m

γm,1 (p − q)m xm (t − x) X 1 − [k]γm,k (p − q)m+1−k xm+1−k (t − x)kp,q − k p p k=2 γm,m (t − x)m+1 γm,1 (p − q)m xm (t − x) p,q − pm p   m X γm,k−1 [k]γm,k + − (p − q)m+1−k xm+1−k (t − x)kp,q k−1 k p p k=2 =

=

m+1 X

γm+1,k (p − q)m+1−k xm+1−k (t − x)kp,q ,

k=1

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where ( γm+1,k =

γm,k−1 pk−1

γm,m

[k]γm,k , pk

− = 1,

k = 1, · · · , m, γm,0 = 0 k = m + 1. 

Theorem 2.7. Let m ∈ N and 0 < q < p ≤ 1. Then there exits a constant Em > 0 such that for any x ∈ [0, 1], we have | Bn,p,q ((t − x)m ; x) |≤ Em

x(1 − x) [n]b

m+1 c 2

.

Proof. From Lemma 2.6 we have

m

| Bn,p,q ((t − x) , x) | ≤

m X

 | γm,k |

k=1

≤ γm | Bn,p,q

pn − q n [n]

m−k


| Bn,p,q (t − x)kp,q , x |

m−1 X
m (t − x)p,q , x | + k=1

!
1 | Bn,p,q (t − x)kp,q , x | [n]m−k

By using Theorem 2.5 we have | Bn,p,q ((t − x)m , x) | ≤ ≤ ≤ ≤

m−1 X
m (t − x)p,q , x | +

1 x(1 − x) γm | Bn,p,q Ck k+1 m−k [n] [n]b 2 c k=1 ! m−1 X
x(1 − x) Ck γm | Bn,p,q (t − x)m m p,q , x | + [n]1+b 2 c k=1 ! m−1 x(1 − x) x(1 − x) X γm Cm b m+1 c + Ck m+1 [n] 2 [n]b 2 c k=1 ! m−1 X x(1 − x) γm C m + Ck m+1 [n]b 2 c k=1

= Em

!

x(1 − x) [n]b

m+1 c 2

 Corollary 2.8. Let m ∈ N and 0 < q < p ≤ 1. Then there exits a constant Km > 0 such that for any x ∈ [0, 1], we have Bn,p,q ((| t − x |)m ; x) ≤ Km

823

x(1 − x) . m [n] 2

(2.13)

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Proof. For an even m, clearly we have Bn,p,q ((| t − x |)m ; x) = Bn,p,q ((t − x)m ; x) x(1 − x) ≤ Em b m+1 c [n] 2 x(1 − x) = Km m [n] 2 In case if m is odd, say m = 2u + 1, we have Bn,p,q ((| t − x |)2u+1 ; x) q
q ≤ Bn,p,q (| t − x |)4u ; x Bn,p,q ((| t − x |)2 ; x) p,q s s x(1 − x) x(1 − x) ≤ E4u b 4u+1 c E2 3 [n]b 2 c [n] 2 s s x(1 − x) x(1 − x) E4u E2 = 2u [n] [n] 2 = K2u+1

x(1 − x) [n]

2u+1 2

.

This complete the proof.



[r]

Theorem 2.9. Let Bn,p,q (f ; x) be an operator from C r [0, 1] → C r [0, 1]. Then for 0 < q < p ≤ 1 there exits a constant M (r) such that for every f ∈ C r [0, 1], we have r X [r] k Bn,p,q (f ; x) kC[0,1] ≤ M (r) k f (i) k= M (r) k f kC r [0,1] . (2.14) i=0 [r]

Proof. Clearly Bn,p,q (f ; x) is continuous on [0, 1]. From (1.7) we have r X
(−1)i [r] Bn,p,q (t − x)i f (i) (t); x . Bn,p,q (f ; x) = i! i=0 From the Corollary 2.8, we have

| Bn,p,q (t − x)i f (i) (t); x | ≤ k f (i) k Bn,p,q | (t − x) |i ; x i

≤ Ki k f (i) k [n]− 2 . Therefore k

[r] Bn,p,q (f ; x)

k ≤

r X (−1)i i=0

≤ M (r)

i! r X


k Bn,p,q (t − x)i f (i) (t); x k k f (i) k .

i=0

This complete the proof.



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3. Convergence properties of Bn,p,q (f ; x) The modulus of continuity of the derivative f (r) is given by  
(r) (r) (r) ω f ; t = sup | f (x) − f (y) |:| x − y |≤ t, x, y ∈ [0, 1] .

(3.1)

Theorem 3.1. Let 0 < q < p ≤ 1 and r ∈ N ∪ {0} be a fixed number. Then for x ∈ [0, 1], n ∈ N there exits Dr > 0 such that for every f ∈ C r [0, 1] the following inequality holds ! 1 1 [r] (f ; x) − f (x) |≤ Dr r ω f (r) ; p | Bn,p,q . (3.2) [n] 2 [n] Proof. Let r ∈ N. Then for f ∈ C r [0, 1] at a given point t ∈ [0, 1], we have from the Taylor formula that

f (x) =

r X f (i) (t) i=0 Z 1

×

i!

(x − t)i +

(x − t)r ((r − 1)!)


(1 − u)r−1 f (r) (t + u(x − t)) − f (r) (t) du.

0

On applying

[r] Bn,p,q (f ; x),

f (x) −

we get

[r] Bn,p,q (f ; x)

=

n X (x − k=0

 × f

(r)



[k] )r pk−n [n]

(r − 1)!

Z

1

(1 − u)r−1 Pn,k (p, q; x)

0

    [k] [k] [k] (r) + u x − k−n −f du. pk−n [n] p [n] pk−n [n]

(3.3)

Now from the definition and properties of modulus of continuity, we have        (r) [k] [k] [k] [k] (r) (r) f + u x − k−n −f ≤ ω f ; u x − k−n pk−n [n] p [n] pk−n [n] p [n]  (r) ω f ; u x −

!    p [k] [k] 1 . ≤ [n] x − k−n + 1 ω f (r) ; p pk−n [n] p [n] [n]

(3.4) 

Now for every 0 ≤ x ≤ 1, 0 < q < p ≤ 1, k ∈ N ∪ {0}, n ∈ N and from (3.3) and (3.4), we get [r] | Bn,p,q (f ; x) − f (x) | ! n r   X 1 1 [k] p [k] (r) ≤ ω f ;p x − k−n [n] x − k−n + 1 Pn,k (p, q; x) r! p [n] p [n] [n] k=0

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1 1 = ω f (r) ; p r! [n]

!

p

r+1

[n]Bn,p,q | x − t |

r






; x + Bn,p,q (| x − t | ; x) . (3.5)

Using (3.9) and (3.5) for x ∈ [0, 1], we have !r ! 1 1 1 [r] | Bn,p,q (f ; x) − f (x) | ≤ (Kr+1 + Kr ) p ω f (r) ; p r! [n] [n] !r ! 1 1 = Dr p . ω f (r) ; p [n] [n] [r]

In order to obtain the uniform convergence of Bn,pn ,qn (f ; x) to a continuous function f , we take q = qn , p = pn where qn ∈ (0, 1) and pn ∈ (qn , 1] satisfying, lim pn = 1, lim qn = 1. n

(3.6)

n

Corollary 3.2. Let p = pn , q = qn , 0 < qn < pn ≤ 1 satisfy (3.6) and f ∈ C r [0, 1] for a fixed number r ∈ N ∪ {0}. Then r

[r]

lim [n] 2 k Bn,k (f ) − f k= 0.

(3.7)

n→∞

We say that (cf. [16]) a function f ∈ C[0, 1] belongs to LipM (α), 0 < α ≤ 1, provided | f (x) − f (y) |≤ M | x − y |α , (x, y ∈ [0, 1] and M > 0).

(3.8)

Corollary 3.3. Let p = pn , q = qn , 0 < qn < pn ≤ 1 satisfy (3.6) and f ∈ C r [0, 1] for a fixed number r ∈ N ∪ {0}. If f (r) ∈ LipM (α) then   [r] − r+α 2 k Bn,p,q (f ) − f k= O [n] . (3.9) Proof. From (3.2) and (3.8), we have [r] k Bn,p,q (f ) − f k≤ Dr M

1 1 r α . [n] 2 [n] 2 

Theorem 3.4. Let 0 < q < p ≤ 1. Suppose that f ∈ C r+2 [0, 1], where r ∈ N∪{0} is fixed then we have r (r+1) [r] (x)Bn,p,q ((t − x)r+1 ; x) Bn,p,q (f ; x) − f (x) − (−1) f (r + 1)! (−1)r f (r+2) (x)Bn,p,q ((t − x)r+2 ; x) − (r + 2)! r   x(1 − x) X 1 (r+2−i) − 21 . ≤ (Kr+2 + Kr+4 ) ω f , [n] r [n] 2 +1 i=0 i!(r + 2 − i)!

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Proof. Let f ∈ C r+2 [0, 1] and x ∈ [0, 1] for a fixed number r ∈ N ∪ {0} we have f (i) ∈ C r+2−i [0, 1], 0 ≤ i ≤ r. Then by Taylor formula we can write (i)

f (t) =

r+2−i X i=0

f (i+j) (x) (t − x)j + Rr+2−j (f ; t; x), j!

(3.10)

where f (r+2−i) (ζpn−k−1 t ) − f (r+2−i) (x) Rr+2−i (f ; t; x) = (t − x)r+2−i , (r + 2 − i)! and | ζt − x |= x − [x] ∈ [0, 1) denote the fractional part of x. In this paper, we will study the Fourier series of the following three types of functions involving Genocchi polynomials Gm (< x >). Pm (1) αm (< x >) = P k=1 Gk (< x >) < x >m−k , (m ≥ 2); m 1 Gk (< x >) < x >m−k , (m ≥ 2); (2) βm (< x >) = k=1 k!(m−k)! Pm−1 1 (3) γm (< x >) = k=1 k(m−k) Gk (< x >) < x >m−k , (m ≥ 2). The reader may refer to any book (for example, see [6,18,22]) for elementary facts about Fourier analysis. As to γm (< x >), we note that the polynomial identity (1.7) follows immediately from Theorems 4.1 and 4.2, which can be derived in turn from the Fourier series expansion of γm (< x >). m−1 X k=1

1 Gk (x)xm−k k(m − k)

! Gk 2 2 − − Gm+1 k(m − k + 1) m m(m + 1) k=1 ! m−1   m−1 X 1 X m 2Gm−s+1 Gl−s+1 2 + − (Hm−1 − Hm−s ) − Bs (x). m s=1 s m−s m−s+1 (l − s + 1)(m − l) m X

1 =− m

(1.7)

l=s

The obvious polynomial identities can be derived also for αm (< x >) and βm (< x >) from Theorems 2.1 and 2.2, and TheoremsP3.1 and 3.2 , respectively. It is noteworthy that from the Fourier series m−1 1 expansion of the function k=1 k(m−k) Bk (< x >)Bm−k (< x >) we can derive a slightly different version of the well-known Miki’s identity (see [3,5,19,20]) m−1 X

1 B2k B2m−2k 2k (2m − 2k) k=1   m 1 X 1 2m 1 = B2k B2m−2k + H2m−1 B2m , m 2k 2k m

(1.8) (m ≥ 2) .

k=1

In addition, we can derive the Faber-Pandharipande-Zagier identity (see [4]) m−1 X

1 B 2k B 2m−2k 2k (2m − 2k) k=1   m 1 1 X 1 2m = B2k B 2m−2k + H2m−1 B 2m , m 2k 2k m

(1.9) (m ≥ 2) ,

k=1

where B m =



1−2m−1 2m−1




Bm = 21−m − 1 Bm = Bm

1 2


, Some related works can be found in [1,7-11].

2. Fourier series of the first type of functions In this section, Pm we will study the Fourier series of first type of functions involving Genocchi polynomials. Let αm (x) = k=1 Gk (x)xm−k , (m ≥ 2). Note here that deg αm (x) = m − 1. Then we will consider the

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function m X

αm (< x >) =

Gk (< x >) < x >m−k , (m ≥ 2).

(2.1)

k=1

defined on (−∞, −∞) which is periodic of period 1. The Fourier series of αm (< x >) is ∞ X

2πinx A(m) , n e

(2.2)

n=−∞

where A(m) n

1

Z

αm (< x >)e−2πinx dx

= 0

(2.3)

1

Z

−2πinx

=

αm (x)e

dx.

0

Before proceeding further, we first observe the following. 0 αm (x) =

m X

(kGk−1 (x)xm−k + (m − k)Gk (x)xm−k−1 )

k=1

=

m X

(kGk−1 (x)xm−k +

k=2

=

m−1 X

(m − k)Gk (x)xm−k−1

k=1

m−1 X

(k + 1)Gk (x)x

m−k−1

m−1 X

(m − k)Gk (x)xm−k−1

+

k=1

= (m + 1)

(2.4)

k=1 m−1 X

Gk (x)xm−1−k

k=1

= (m + 1)αm−1 (x). From this, we have



αm+1 (x) m+2

0

= αm (x). Then we have

Z

1

αm (x)dx = 0

αm (1) − αm (0) =

m X

1 (αm+1 (1) − αm+1 (0)), m+2

(2.5)

(Gk (1) − Gk (0)δm,k )

k=1

=

m X

(−Gk (0) + 2δk,1 − Gk (0)δm,k ) = −

k=1

m X

(2.6) Gk + 2 − Gm ,

k=1

αm (0) = αm (1) ⇐⇒

m X

Gk = 2 − Gm ,

(2.7)

k=1

Z 0

1

1 αm (x)dx = m+2

−

m+1 X

! Gk + 2 − Gm+1

.

(2.8)

k=1

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Fourier series of functions involving Genocchi polynomials (m)

We are now ready to determine the Fourier coefficients An . Case 1 : n 6= 0. Z 1 αm (x)e−2πinx dx A(m) = n 0

=−

1 1 1  αm (x)e−2πinx 0 + 2πin 2πin

1

Z

0 αm (x)e−2πinx dx

0

Z 1 m+1 1 αm−1 (x)e−2πinx dx (αm (1) − αm (0)) + 2πin 2πin 0 ! m X m + 1 (m−1) 1 = A + Gk − 2 + Gm 2πin n 2πin k=1 ! m−1 m (m−2) 1 X m+1 A + ( Gk − 2 + Gm−1 ) = 2πin 2πin n 2πin k=1 ! m X 1 + Gk − 2 + Gm 2πin k=1 ! m−1 X (m + 1)m (m−2) m+1 Gk − 2 + Gm−1 = A + (2πin)2 n (2πin)2 k=1 ! m X 1 + Gk − 2 + Gm 2πin =−

(2.9)

k=1

= ··· m−j+1 X

m−2 (m + 1)m−2 (2) X (m + 1)j−1 A + = (2πin)m−2 n (2πin)j j=1 m−2 X (m + 1)j−1 3(m + 1)m−2 =− + m−1 (2πin) (2πin)j j=1

=

m−1 X j=1

m−j+1 X

(m + 1)j−1 (2πin)j

Gk − 2 + Gm−j+1

k=1 m−j+1 X

! Gk − 2 + Gm−j+1

k=1

! Gk − 2 + Gm−j+1

k=1

m−1 1 X (m + 1)j = m + 2 j=1 (2πin)j

where A(2) n

!

Z =

m−j+1 X

! Gk − 2 + Gm−j+1

,

k=1

1

(3x − 1)e−2πinx dx = −

0

3 . 2πin

(2.10)

Case 2: n = 0. (m)

A0

Z

m+1

1

αm (x)dx = −

= 0

X 1 ( Gk − 2 + Gm+1 ). m+2

(2.11)

k=1

αmP (< x >), (m ≥ 2) is piecewise C ∞ . Moreover, αm (< x >) is continuous for those integers m ≥ 2 m with Pk=1 Gk = 2 − Gm and discontinuous with jump discontinuities at integers for those integers m ≥ 2 m with k=1 Gk 6= 2 − Gm . We need the following facts about Bernoulli functions Bm (< x >):

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(a) for m ≥ 2, ∞ X

Bm (< x >) = −m!

n=−∞,n6=0

e2πinx . (2πin)m

(2.12)

(b) for m = 1, ∞ X

−

n=−∞,n6=0

e2πinx = 2πin



B1 (< x >), for x ∈ Zc , 0, for x ∈ Z,

(2.13)

Pm where Zc = R − Z. Assume first that m ≥ 2 is an integer with k=1 Gk = 2 − Gm . Then αm (1) = αm (0). Thus αm (< x >) is piecewise C ∞ , and continuous. So the Fourier series of αm (< x >) converges uniformly to αm (< x >), and αm (< x >)

=−

∞ X

k=1

n=−∞,n6=0

m+1 X

1 ( m+2

m−1 X

 j=1

 m−j+1 (m + 2)j X ( Gk − 2 + Gm−j+1 ) e2πinx (2πin)j k=1

Gk − 2 + Gm+1 )

k=1

1 =− m+2 m X

m+1 X

! Gk − 2 + Gm−j+1 −j!

m−j+1 X

 m−1  1 X m+2 − m + 2 j=1 j

−



m+1

X 1 1 ( Gk − 2 + Gm+1 ) + =− m+2 m+2

k=1

n=−∞,n6=0

! Gk − 2 + Gm+1

k=1

! Gk − 2 + Gm )

 ×

k=1

∞ X

1 − m+2

m−1 X j=2

m+2 j



m−j+1 X

 e2πin  (2πin)j

(2.14)

! Gk − 2 + Gm−j+1

Bj (< x >)

k=1

B1 (< x >), for x ∈ Zc , 0, for x ∈ Z,

for all x ∈ (−∞, ∞). Hence we obtain the following theorem. Pm Theorem integer with k=1 Gk = 2 − Gm . Then we have the following. Pm 2.1. Let m ≥ 2 be an (a) k=1 Gk (< x >) < x >m−k has the Fourier series expansion m X

Gk (< x >) < x >m−k

k=1

1 =− m+2 1 + m+2

m+1 X

! Gk − 2 + Gm+1

(2.15)

k=1 ∞ X



m−1 X

 n=−∞,n6=0

j=1

 m−j+1 (m + 2)j X ( Gk − 2 + Gm−j+1 ) e2πinx , (2πin)j k=1

for all x ∈ (−∞, ∞). Here the convergence is uniform.

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Fourier series of functions involving Genocchi polynomials

(b) m X

Gk (< x >) < x >m−k

k=1

1 =− m+2

m−1 X



j=0,j6=1

m+2 j

m−j+1 X



(2.16)

! Gk − 2 + Gm−j+1

Bj (< x >),

k=1

for all x ∈ (−∞, ∞), where Bj (< x >) is the Bernoulli function. Pm Next, we assume that m ≥ 2 is an integer with k=1 Gk 6= 2 − Gm . Then αm (1) 6= αm (0). Hence αm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers. The Fourier series of αm (< x >) converges pointwise to αm (< x >), for x ∈ Zc , and converges to m

1X 1 1 (αm (0) + αm (1)) = αm (0) − Gk + 1 − Gm 2 2 2 k=1

1 =1− 2

m−1 X

(2.17)

Gk .

k=1

Thus we get the following theorem.

Theorem 2.2. Let m ≥ 2 be an integer with (a) 1 m+2  Pm

−

=

1

m+1 X

! Gk − 2 + Gm+1

+

k=1

Gk (< x >) k=1P m−1 1 − 2 k=1 Gk ,

m−k

1 m+2

Pm

k=1

Gk 6= 2 − Gm . Then we have the following. 

∞ X

m−1 X

 n=−∞,n6=0

j=1

(m + 2)j (2πin)j

m−j+1 X k=1

! Gk − 2 + Gm−j+1  e2πinx (2.18)

c

, for x ∈ Z , for x ∈ Z.

Here the convergence is pointwise. (b) −

 m−j+1 m−1  X 1 X m+2 ( Gk − 2 + Gm−j+1 )Bj (< x >) m + 2 j=0 j k=1

=

m X

Gk (< x >) < x >m−k , for x ∈ Zc ;

k=1

1 − m+2 =1−

1 2

m−1 X j=0,j6=1 m−1 X



 m−j+1 X m+2 ( Gk − 2 + Gm−j+1 )Bj (< x >) j

(2.19)

k=1

Gk , x ∈ Z.

k=1

Question: For what values of m ≥ 2, does

Pm

k=1

Gk = 2 − Gm hold?

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3. Fourier series of the second type of functions Let βm (x) =

Pm

1 m−k , k=1 k!(m−k)! Gk (x)x

βm (< x >) =

(m ≥ 2). Then, we consider the function

m X k=1

1 Gk (< x >) < x >m−k , k!(m − k)!

(3.1)

defined on (−∞, −∞) which is periodic with period 1. The Fourier series of βm (< x >) is ∞ X

Bn(m) e2πinx ,

(3.2)

n=−∞

where Bn(m) =

Z

1

βm (< x >)e−2πinx dx

0

Z

(3.3)

1 −2πinx

=

βm (x)e

dx.

0

Before proceeding further, we need the following.  m  X m−k k m−k m−k−1 0 Gk−1 (x)x + Gk (x)x βm (x) = k!(m − k)! k!(m − k)! k=1

=

m X

m−1 X 1 1 Gk−1 (x)xm−k + Gk (x)xm−k−1 (k − 1)!(m − k)! k!(m − k − 1)!

k=2

=

k=1

m−1 X k=1

(3.4)

m−1 X 1 1 Gk (x)xm−1−k + Gk (x)xm−1−k k!(m − 1 − k)! k!(m − 1 − k)! k=1

= 2βm−1 (x). 0 (x) = 2βm−1 (x). From this, we see that So, βm  0 βm+1 (x) = βm (x) 2

(3.5)

and 1

Z

βm (x)dx = 0

1 (βm+1 (1) − βm+1 (0)). 2

(3.6)

We also observe that βm (1) − βm (0) =

m X k=1

=

m X

1 (Gk (1) − Gk (0)δm,k ) k!(m − k)! m

X Gk (0)δm,k 1 (−Gk (0) + 2δk,1 ) − k!(m − k)! k!(m − k)!

k=1 m X

=−

k=1

(3.7)

k=1

Gk 2 Gm + − . k!(m − k)! (m − 1)! m!

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We put Ωm = βm (1) − βm (0) = −

m X k=1

Gk 2 Gm + − , k!(m − k)! (m − 1)! m!

(3.8)

for m ≥ 2. Then βm (0) = βm (1) ⇐⇒ Ωm = 0.

(3.9)

1 βm (x)dx = Ωm+1 2 0 (m+1 ) X 1 Gk 2 Gm+1 =− − + . 2 k!(m − k + 1)! m! (m + 1)!

(3.10)

Moreover, Z

1

k=1

(m)

Now, we are going to determine the Fourier coefficients Bn . Case 1: n 6= 0. Z 1 Bn(m) = βm (x)e−2πinx dx 0

Z 1 1 1 1  βm (x)e−2πinx 0 + β 0 (x)e−2πinx dx 2πin 2πin 0 m Z 1 1 1 =− (βm (1) − βm (0)) + βm−1 (x)e−2πinx dx 2πin πin 0 1 1 (m−1) Bn − Ωm = πin  2πin  1 (m−2) 1 1 1 Bn − Ωm−1 − Ωm = πin πin 2πin 2πin 1 2 1 = Bn(m−2) − Ωm−1 − Ωm 2 2 (πin) (2πin) 2πin = ··· =−

=

(3.11)

m−2 X 2j−1 1 (2) B − Ωm−j+1 , n m−2 (πin) (2πin)j j=1

where Bn(2)

1

Z



= 0

1 2x − 2



e−2πinx dx = −

1 . πin

(3.12)

By (3.11) and (3.12), we get Bn(m) = − =−

m−2 X 2j−1 1 − Ωm−j+1 m−1 (πin) (2πin)j j=1 m−1 X j=1

(3.13)

2j−1 Ωm−j+1 . (2πin)j

Case 2: n = 0. (m) B0

Z =

1

βm (x)dx = 0

837

1 Ωm+1 . 2

(3.14)

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Here βm (< x >), (m ≥ 2) is piecewise C ∞ . Moreover, βm (< x >) is continuous for those integers m ≥ 2 with Ωm = 0 and discontinuous with jump discontinuities at integers for those integers m ≥ 2 with Ωm 6= 0. Assume first that m ≥ 2 is an integer with Ωm = 0 . Then βm (0) = βm (1). So βm (< x >) is piecewise C ∞ , and continuous. Hence the Fourier series of βm (< x >) converges uniformly to βm (< x >), and m X

1 Gk (< x >) < x >m−k k!(m − k)! k=1   ∞ m−1 X X 2j−1 1  e2πinx  = Ωm+1 − Ω j m−j+1 2 (2πin) j=1 n=−∞,n6=0   m−1 ∞ 2πinx X 2j−1 X 1 e  = Ωm+1 + Ωm−j+1 −j! j 2 j! (2πin) j=1

βm (< x >) =

(3.15)

n=−∞,n6=0

=

m−1 X

2j−1 1 Ωm+1 + Ωm−j+1 Bj (< x >) 2 j! j=2  B1 (< x >), for x ∈ Zc , + Ωm × 0, for x ∈ Z,

for all x ∈ (−∞, ∞). Thus we have the following theorem. Theorem 3.1. For each integer l ≥ 2, let Ωl = −

l X k=1

Gk 2 Gl + − . k!(l − k)! (l − 1)! l!

(3.16)

AssumePthat Ωm = 0, for an integer m ≥ 2. Then we have the following. m 1 (a) k=1 k!(m−k)! Gk (< x >) < x >m−k has the Fourier series expansion m X

1 Gk (< x >) < x >m−k k!(m − k)! k=1   ∞ m−1 X X 2j−1 1   e2πinx , = Ωm+1 − Ω j m−j+1 2 (2πin) j=1

(3.17)

n=−∞,n6=0

for all x ∈ (−∞, ∞). Here the convergence is uniform. (b) m X k=1

=

1 Gk (< x >) < x >m−k k!(m − k)! m−1 X

j=0,j6=1

2j−1 Ωm−j+1 Bj (< x >), j!

(3.18)

for all x ∈ (−∞, ∞). Here Bk (< x >) is the Bernoulli function. Assume next that m ≥ 2 is an integer with Ωm 6= 0. Then βm (1) 6= βm (0), and hence βm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers. Thus the Fourier series of

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βm (< x >) converges pointwise to βm (< x >), for x ∈ Zc , and converges to 1 1 (βm (0) + βm (1)) = βm (0) + Ωm 2 2 ! m X 1 2 Gm Gm Gk + − + − = m! 2 k!(m − k)! (m − 1)! m! k=1 ! m−1 X 1 2 Gk = − , 2 (m − 1)! k!(m − k)!

(3.19)

k=1

for x ∈ Z. Hence we obtain the following theorem. Theorem 3.2. For each integer l ≥ 2, let Ωl = −

l X k=1

Gk 2 Gl + − . k!(l − k)! (l − 1)! l!

(3.20)

Assume that Ωm 6= 0, for an integer m ≥ 2. Then we have the following. (a)   ∞ m−1 j−1 X X 2 1   e2πinx Ωm+1 − Ω j m−j+1 2 (2πin) j=1 n=−∞,n6=0  Pm 1 m−k , for x ∈ Zc , k=1 k!(m−k)! Gk (< x >) < x > = Gm 1 for x ∈ Z. m! + 2 Ωm ,

(3.21)

Here the convergence is pointwise. (b) m−1 X

2j−1 Ωm−j+1 Bj (< x >) j!

j=0 m X

=

k=1

(3.22) 1 Gk (< x >) < x >m−k , k!(m − k)!

for x ∈ Zc ; m−1 X j=0,j6=1

2j−1 Ωm−j+1 Bk (< x >) j!

Gm 1 = + Ωm , m! 2 for x ∈ Z. Here Bj (< x >) is the Bernoulli function. Pm Gm Remark: For what values of m ≥ 2, does k=1 k!(m−k)! =

2 (m−1)!

(3.23)

−

Gm m!

hold?

4. Fourier series of the third type of functions Let γm (x) =

Pm−1

1 m−k , k=1 k(m−k) Gk (x)x

γm (< x >) =

(m ≥ 2). Then we will consider the function

m−1 X k=1

1 Gk (< x >) < x >m−k , k(m − k)

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(4.1)

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defined on (−∞, −∞) which is periodic of period 1. The Fourier series of γm (< x >) is ∞ X

Cn(m) e2πinx ,

(4.2)

n=−∞

where Cn(m)

1

Z

−2πinx

=

γm (< x >)e

Z

1

dx =

0

γm (x)e−2πinx dx.

(4.3)

0

To proceed further, we need to observe the following. 0 γm (x) =

m−1 X

 1 kGk−1 (x)xm−k + (m − k)Gk (x)xm−k−1 k(m − k)

k=1

=

m−2 X

m−1 X 1 1 Gk (x)xm−k−1 + Gk (x)xm−k−1 m−k−1 k

k=1

(4.4)

k=1

= (m − 1)

m−2 X k=1

1 1 Gk (x)xm−1−k + Gm−1 (x) k(m − 1 − k) m−1

= (m − 1)γm−1 (x) +

1 Gm−1 (x). m−1

Thus, 0 γm (x) = (m − 1)γm−1 (x) +

1 Gm−1 (x). m−1

(4.5)

From this, we have 

1 1 (γm+1 (x) − Gm+1 (x)) m m(m + 1)

0 (4.6)

= γm (x)

and Z

1

γm (x)dx 0

1 1 1 (γm+1 (x) − Gm+1 (x)) m m(m + 1) 0   1 1 = (γm+1 (1) − γm+1 (0)) − (Gm+1 (1) − Gm+1 (0)) m m(m + 1)   1 2 = γm+1 (1) − γm+1 (0) + Gm+1 (0) . m m(m + 1) 

=

(4.7)

Observe that γm (1) − γm (0) =

m−1 X k=1

=

m−1 X k=1

=−

1 (Gk (1) − Gk (0)δm,k ) k(m − k) m−1 X 1 1 (−Gk (0) + 2δk,1 ) − Gk (0)δm,k k(m − k) k(m − k)

m−1 X k=1

(4.8)

k=1

Gk 2 + . k(m − k) m − 1

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Fourier series of functions involving Genocchi polynomials

So, we have

γm (1) = γm (0) ⇐⇒

m−1 X k=1

Gk 2 = . k(m − k) m−1

(4.9)

Also,

Z 0

1

1 γm (x)dx = m

−

m X k=1

Gk 2 2 + + Gm+1 k(m − k + 1) m m(m + 1)

! .

(4.10)

(m)

Now, we will determine the Fourier coefficients Cn . Case 1: n 6= 0.

Cn(m) =

Z

1

γm (x)e−2πinx dx

0

=− =−

1 1 [γm (x)e−2πinx ]10 + 2πin 2πin 1 1 (γm (1) − γm (0)) + 2πin 2πin

1

Z

0 γm (x)e−2πinx dx Z 1 (m − 1)γm−1 (x) + 0

0

 1 Gm−1 (x) e−2πinx dx m−1

Z 1 m−1 1 =− (γm (1) − γm (0)) + γm−1 (x)e−2πinx dx 2πin 2πin 0 Z 1 1 Gm−1 (x)e−2πinx dx + 2πin(m − 1) 0 m − 1 (m−1) 1 2 = C − Λm + Φm , 2πin n 2πin 2πin(m − 1)

(4.11)

where

Φm =

m−2 X k=1

(m − 1)k−1 Gm−k , (2πin)k

(4.12)

and one can show

Z 0

1

Gm (x)e−2πinx dx =



2Ωm+1 , m+1 −2 Gm+1 ,

841

for n 6= 0, for n = 0.

(4.13)

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m − 1 (m−1) 2 1 Cn Λm + Φm − 2πin 2πin 2πin(m − 1)   m − 1 m − 2 (m−2) 1 2 = C − Λm−1 + Φm−1 2πin 2πin n 2πin 2πin(m − 2) 1 2 − Λm + Φm 2πin 2πin(m − 1) 1 m−1 (m − 1)(m − 2) (m−2) Cn Λm−1 − − Λm = (2πin)2 (2πin)2 2πin 2 2(m − 1) Φm−1 + Φm + (2πin)2 (m − 2) 2πin(m − 1) = ···

Cn(m) =

=

(4.14)

m−2 m−2 X 2(m − 1)j−1 (m − 1)! (2) X (m − 1)j−1 C − Λ + Φm−j+1 m−j+1 n (2πin)m−2 (2πin)j (2πin)j (m − j) j=1 j=1

=−

m−1 X j=1

=−

m−2 X 2(m − 1)j−1 (m − 1)j−1 Φm−j+1 Λ + m−j+1 (2πin)j (2πin)j (m − j) j=1

m−1 m−2 2(m)j 1 X 1 X (m)j Λ + Φm−j+1 , m−j+1 m j=1 (2πin)j m j=1 (2πin)j (m − j)

where Cn(2) =

Z

1

xe−2πinx dx = −

0

1 . 2πin

(4.15)

(m)

In order to get a final expression for Cn , we need to observe the following. m−2 X j=1

=

2(m)j Φm−j+1 (2πin)j (m − j)

m−2 X j=1

=

m−j−1 X (m − j)k−1 2(m)j Gm−j−k+1 (2πin)j (m − j) (2πin)k k=1

m−2 X m−j−1 X j=1

=2

m−2 X j=1

=2

m−1 X s=2

=2

m−1 X s=1

k=1

2(m)j+k−1 Gm−j−k+1 (2πin)j+k (m − j)

m−1 X (m)s−1 1 Gm−s+1 m − j s=j+1 (2πin)s

(4.16)

s−1 X (m)s−1 1 G m−s+1 s (2πin) m−j j=1

(m)s Gm−s+1 (Hm−1 − Hm−s ), (2πin)s m − s + 1

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Fourier series of functions involving Genocchi polynomials

and m−1 X

(m)j Λ j m−j+1 (2πin) j=1 ( m−j ) m−1 X X (m)j Gk 2 = − + (2πin)j k(m − j − k + 1) m − j j=1 k=1

=−

m−1 X m−j X j=1 k=1

=−

m−1 X m−1 X s=1 l=s

(4.17)

m−1 X (m)j (m)j Gk + 2 j (m − j) (2πin)j k(m − j − k + 1) (2πin) j=1 m−1 X (m)s Gl−s+1 (m)s + 2 . s (m − s) (2πin)s (l − s + 1)(m − l) (2πin) s=1

Putting everything altogether,

Cn(m) =

m−1 m−1 (m)s Gl−s+1 1 X X m s=1 (2πin)s (l − s + 1)(m − l) l=s

−

+

2 m 2 m

m−1 X s=1 m−1 X s=1

(m)s (2πin)s (m − s) (m)s Gm−s+1 (Hm−1 − Hm−s ) (2πin)s m − s + 1

(4.18)

m−1 1 X (m)s m s=1 (2πin)s ( ) m−1 X 2 2Gm−s+1 Gl−s+1 × − (Hm−1 − Hm−s ) − . m−s m−s+1 (l − s + 1)(m − l)

=−

l=s

Case 2: n = 0.

(m)

C0 = =

1 m 1 m

1

Z = 

γm (x)dx 0

Λm+1 + −

m X k=1

2 Gm+1 m(m + 1)



Gk 2 2 + + Gm+1 k(m − k + 1) m m(m + 1)

(4.19) ! .

γm (< x >), (m ≥ 2) is piecewise C ∞ . Moreover, γm (< x >) is continuous for those integers m ≥ 2 with Λm = 0 and discontinuous with jump discontinuities at integers for those integers Λm 6= 0.

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15

Assume first that Λm = 0 . Then γm (0) = γm (1). So γm (< x >) is piecewise C ∞ , and continuous. So the Fourier series of γm (< x >) converges uniformly to γm (< x >), and γm (< x >) ! Gk 2 2 − − Gm+1 k(m − k + 1) m m(m + 1) k=1 (m−1 !) ∞ m−1 X X X (m)s 1 2 2Gm−s+1 Gl−s+1 e2πinx − − (Hm−1 − Hm−s ) − s m (2πin) m − s m − s + 1 (l − s + 1)(m − l) s=1 n=−∞,n6=0 l=s ! m 1 X Gk 2 2 =− − − Gm+1 m k(m − k + 1) m m(m + 1) k=1 ! m−1   m−1 X 2Gm−s+1 Gl−s+1 1 X m 2 − (Hm−1 − Hm−s ) − + m s=1 s m−s m−s+1 (l − s + 1)(m − l) (4.20) l=s   ∞ X e2πinx  × −s! (2πin)s n=−∞,n6=0 ! m Gk 2 2 1 X − − Gm+1 =− m k(m − k + 1) m m(m + 1) k=1 ! m−1   m−1 X 2Gm−s+1 Gl−s+1 1 X m 2 − (Hm−1 − Hm−s ) − Bs (< x >) + m s=2 s m−s m−s+1 (l − s + 1)(m − l) l=s !  m−1 X 2 Gl B1 (< x >), for x ∈ Zc , + − × 0, for x ∈ Z, m−1 l(m − l) 1 =− m

m X

l=1

for all x ∈ (−∞, ∞). Now, we obtain the following theorem. Pm−1 Gk Theorem 4.1. Let m ≥ 2 be an integer with Λm = − k=1 k(m−k) + following. Pm−1 1 Gk (< x >) < x >m−k has the Fourier expansion (a) k=1 k(m−k) m−1 X k=1

= 0. Then we have the

1 Gk (< x >) < x >m−k k(m − k)

1 =− m −

2 m−1

1 m

! Gk 2 2 − − Gm+1 k(m − k + 1) m m(m + 1) (4.21) k=1 (m−1 !) ∞ m−1 X X X (m)s 2Gm−s+1 2 Gl−s+1 − (Hm−1 − Hm−s ) − e2πinx s (2πin) m − s m − s + 1 (l − s + 1)(m − l) s=1 m X

n=−∞,n6=0

l=s

, for all x ∈ (−∞, ∞). Here the convergence is uniform.

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Fourier series of functions involving Genocchi polynomials

(b) m−1 X k=1

1 Gk (< x >) < x >m−k k(m − k)

! Gk 2 2 − − Gm+1 (4.22) k(m − k + 1) m m(m + 1) k=1 ! m−1   m−1 X 1 X m 2 2Gm−s+1 Gl−s+1 Bs (< x >) + − (Hm−1 − Hm−s ) − m s=2 s m−s m−s+1 (l − s + 1)(m − l)

1 =− m

m X

l=s

for all x ∈ (−∞, ∞), where Bs (< x >) is the Bernoulli function. Assume next that m ≥ 2 is an integer with Λm 6= 0.Then γm (0) 6= γm (1). γm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers.Thus the Fourier series of γm (< x >) converges pointwise to γm (< x >), for x ∈ Zc , and converges to 1 1 (γm (0) + γm (1)) = γm (0) + Λm 2 2 ! m−1 X 1 Gk 2 = − + , 2 k(m − k) m − 1

(4.23)

k=1

for x ∈ Z. Hence we have the following theorem. Pm−1 Gk 2 Theorem 4.2. Let m ≥ 2 be an integer with Λm = − k=1 k(m−k) + m−1 6= 0. Then we have the following. (a) ! m 1 X Gk 2 2 − − − Gm+1 m k(m − k + 1) m m(m + 1) k=1 (m−1 )  ∞ m−1 X X (m)s  2 X 2Gm−s+1 Gl−s+1 1 − (Hm−1 − Hm−s ) − e2πinx − (4.24) s m (2πin) m − s m − s + 1 (l − s + 1)(m − 1) s=1 n=−∞,n6=0 l=s ( Pm−1 1 G (< x >) < x >m−k , for x ∈ Zc , k=1Pk(m−k) k  = m−1 Gk 1 2 − + for x ∈ Z. k=1 k(m−k) 2 m−1 , Here the convergence is uniform. (b) ! Gk 2 2 − − Gm+1 k(m − k + 1) m m(m + 1) k=1 ! m−1   m−1 X 1 X m 2 2Gm−s+1 Gl−s+1 + − (Hm−1 − Hm−s ) − Bs (< x >) (4.25) m s=1 s m−s m−s+1 (l − s + 1)(m − l)

1 − m

m X

l=s

=

m−1 X k=1

1 Gk (< x >) < x >m−k , k(m − k)

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17

for x ∈ Zc and ! Gk 2 2 − − Gm+1 k(m − k + 1) m m(m + 1) k=1 ! m−1   m−1 X 1 X m 2 2Gm−s+1 Gl−s+1 + − (Hm−1 − Hm−s ) − Bs (< x >) (4.26) m s=2 s m−s m−s+1 (l − s + 1)(m − l) l=s ! m−1 X 2 1 Gk − + , = 2 k(m − k) m − 1

1 − m

m X

k=1

for x ∈ Z. Pm−1 Gk 2 = m−1 hold? Question For what values of m ≥ 2, does k=1 k(m−k) Acknowledgements. This paper is supported by grant NO 14-11-00022 of Russian Scientific Fund.

References 1. S. Araci, E. Sen, and M. Acikgoz, Theorems on Genocchi polynomials of higher order arising from Genocchi basis, Taiwanese J. of Math., 18(2014), no.2, 473-482. 2. M. Cenkci, M. Can, and V. Kurt, q-Extensions of Genocchi Numbers, J. Korean Math. Soc., 43(2006), 183-198. 3. G. V. Dunne, C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys., 7(2)(2013), 225-249. 4. C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139(1)(2000), 173-199. 5. I. M. Gessel, On Miki’s identities for Bernoulli numbers, J. Number Theory 110(1)(2005), 75-82. 6. L. C. Jang, T. Kim, D. J. Kang, A note on the Fourier transform of fermionic p -adic integral on Zp , J. Comput. Anal. Appl., 11(3) (2009), 571-575. 7. D.S. Kim, T. Kim, Identities arising from higher-order Daehee polynomial bases, Open Math. 13(2015), 196-208. 8. D.S. Kim, T. Kim, Euler basis, identities, and their applications, Int. J. Math. Math. Sci. 2012, Art. ID 343981. 9. D.S. Kim, T. Kim, A note on higher-order Bernoulli polynomials, J. Inequal. Appl. 2013, 2013:111. 10. D.S. Kim, T. Kim, Bernoulli basis and the product of several Bernoulli polynomials, Int. J. Math. Math. Sci. 2012, Art. ID 463659. 11. D.S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct., 24(9) (2013), 734-738. 12. T. Kim, On the Multiple q-Genocchi and Euler Numbers, Russ. J. Math. Phys., 15(2008), 481-486. 13. T. Kim, On the q-Extension of Euler and Genocchi Numbers, J. Math. Anal. Appl. 326(2007), 1458-1465. 14. T. Kim, A note on the q-Genocchi Numbers and polynomials, J. Inequalities and applications, 2007 Article ID 71452, 8pages,(2007). 15. T. Kim, Some identities for the Bernoulli, the Euler and Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20(2015), no.1, 23-28. 16. H. Liu, and W. Wang, Some identities on the the Bernoulli, Euler and Genocchi poloynomials via power sums and alternate power sums, Disc. Math., 309(2009), 3346-3363. 17. Q. M. Luo, Fouier expansions and integral representations for Genocchi poloynomials, J. Integer Seq.,12 (2009), Article 09.1.4. 18. J. E. Marsden, Elementary classical analysis, W. H. Freeman and Company, 1974. 19. H. Miki, A relation between Bernoulli numbers, J. Number Theory 10(3)(1978), 297-302. 20. K. Shiratani, S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A 36(1)(1982), 7383. 21. H. M. Srivastava, Some generalizations and basic extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. and Inf. Sci., 5(2011), no. 3, 390-414. 22. D. G. Zill, M. R. Cullen, Advanced Engineering Mathematics, Jones and Bartlett Publishers 2006.

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Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China, Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Graduate School of Education, Konkuk University, Seoul 143-701, republic of Korea E-mail address: [email protected] Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea, School of Natural Sciences, Far Eastern Federal University, 690950 Vladivostok , Russia E-mail address: [email protected]

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Lyapunov inequalities of quasi-Hamiltonian systems on time scales Taixiang Sun

Fanping Zeng

Guangwang Su

Bin Qin∗

College of Information and Statistics, Guangxi Univresity of Finance and Economics Nanning, Guangxi 530003, China

Abstract In this paper, we obtain several new Lyapunov-type inequalities for the following quasi-Hamiltonian systems x∆ (t) = −W (t)x(σ(t))−U (t)|y(t)|p−2 y(t), y ∆ (t) = V (t)|x(σ(t))|q−2 x(σ(t))+W T (t)y(t) on the time scale interval [a, b]T ≡ [a, b] ∩ T for some a, b ∈ T (σ(a) < b), where U and V are real n × n symmetric matrix-valued functions on [a, b]T with U being positive definite, W is real n × n matrix-valued function on [a, b]T with I + µ(t)W being invertible, and x, y are real vector-valued functions on [a, b]T . AMS Subject Classification: 34K11, 34N05, 39A10. Keywords: Lyapunov inequality; Quasi-Hamiltonian system; Time scale

1. Introduction In 1990, Hilger [1] initiated the theory of time scales as a theory capable of treating continuous and discrete analysis in a consistent way, based on which some authors have studied some Lyapunov inequalities for dynamic equations on time scales (see [2-4]) during the last few years. A time scale T is an arbitrary nonempty closed subset of real axis R. On a time scale T, the forward jump operator and the graininess function are defined σ(t) = inf{s ∈ T : s > t}

and

µ(t) = σ(t) − t,

respectively. For the notions used below we refer to [5,6] that provide some basic facts on time scales. In this paper, we continue this line of investigation and study Lyapunov-type inequalities for the following quasi-Hamiltonian systems x∆ (t) = −W (t)x(σ(t))−U (t)|y(t)|p−2 y(t), y ∆ (t) = V (t)|x(σ(t))|q−2 x(σ(t))+W T (t)y(t), (1.1) ? Project Supported by NNSF of China (11461003) and SF of Guangxi Univresity of Finance and Eco- nomics( 2016KY15; 2016ZDKT06; 2016TJYB06) ∗ Corresponding author: E-mail address: [email protected]

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on the time scale interval [a, b]T ≡ [a, b] ∩ T for some a, b ∈ T (σ(a) < b), where p, q ∈ (0, +∞) and 1/p + 1/q = 1, U and V are real n × n symmetric matrix-valued functions on [a, b]T with U being positive definite, W is real n × n matrix-valued function on [a, b]T with I + µ(t)W being invertible, and x, y are real vector-valued functions on [a, b]T . When n = 1, (1.1) reduces to x∆ (t) = α(t)x(σ(t)) + β(t)|y(t)|p−2 y(t), y ∆ (t) = −γ(t)|x(σ(t))|q−2 x(σ(t)) − α(t)y(t).

(1.2)

In 2011, Zhang et al. [7] obtained the following theorem. Theorem 1.1[7] Suppose that 1 − µ(t)α(t) > 0 and β(t) ≥ 0 for any t ∈ T and a, b ∈ Tk with σ(a) ≤ b. If (1.2) has a real solution (x(t), y(t)) satisfying x(a) = 0 or x(a)x(σ(a)) < 0, x(b) = 0 or x(b)x(σ(b)) < 0, max |x(t)| > 0, t∈[a,b]T

then the following inequality holds: Z

³Z

b

σ(b)

|α(t)| 4 (t) + a

´1 ´1 ³ Z b p q max{γ(t), 0} 4 (t) ≥ 2. β(t) 4 (t)

(1.3)

a

a

When n = 1 and T = R, Tiryaki et al. [8] obtained the following theorem. Theorem 1.2[8] Suppose that β(t) > 0 for any t ∈ R and a, b ∈ R with a < b. If (1.2) has a real solution (x(t), y(t)) satisfying x(a) = x(b) = 0 and maxt∈[a,b] |x(t)| > 0, then the following inequalities hold: Z

b

max{γ(t), 0}

1−q a ha (t)

and

Z

+ h1−q b (t)

dt ≥ 1

(1.4)

³ 1 1 ´1−q + dt ≥ 1, ha (t) hb (t) a Rs Rs Rt Rb where ha (t) = a β(s)e−p t α(τ )dτ ds and hb (t) = t β(s)e−p t α(τ )dτ ds. b

max{γ(t), 0}2q−2

(1.5)

For some other related results on Lyapunov-type inequalities, see, e.g. [9-16] and the related references therein.

2. Preliminaries and some lemmas For any u ∈ Rn and any U ∈ Rn×n (the space of real n × n matrices), write |u| =

√

uT u

and

|U | =

max

y∈Rn −{O}

|U y| , |y|

which are called the Euclidean norm of u and the matrix norm of U respectively, where QT is the transpose of a n × m matrix Q. It follows from the definition that for any y ∈ Rn and any U, V ∈ Rn×n , |U y| ≤ |U ||y|,

|U V | ≤ |U ||V |.

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Write Rn×n = {U ∈ Rn×n : U T = U }. It is easy to show that for any U ∈ Rn×n , s s |U | =

max |λ| and |U 2 | = |U |2 , det|λI−U |=0

where det|λI − U | denotes determinant of the matrix λI − U . An U ∈ Rn×n is said to be s positive definite (resp. semi-positive definite), written as U > 0 (resp. U ≥ 0), if y T U y > 0 (resp. y T U y ≥ 0) for all y ∈ Rn with y 6= 0. If U is positive definite (resp. semi-positive definite), then there exists a unique positive definite matrix (resp. semi-positive definite matrix), written √ √ as U , such that [ U ]2 = U . In this paper, we study Lyapunov-type inequalities of (1.1) which has some solution (x(t), y(t)) satisfying x(a) = x(b) = 0 and max |x(t)| > 0.

(2.1)

t∈[a,b]T

We first introduce the following notions and lemmas. The point t ∈ T is said to be left-dense (resp. left-scattered) if ρ(t) = t (resp. ρ(t) < t). The point t ∈ T is said to be right-dense (resp. right-scattered) if σ(t) = t (resp. σ(t) > t). If T has a left-scattered maximum M , then we define Tk = T − {M }, otherwise Tk = T. A function f : T −→ R is said to be rd-continuous provided that f is continuous at rightdense points and has finite left-sided limits at left-dense points in T. The set of all rd-continuous functions from T to R is denoted by Crd (T, R). For a function f : T −→ R, the (delta) derivative f ∆ (t) at t ∈ T is defined to be the number ( if it exists), such that for given any ε > 0, there is a neighborhood U of t with |f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s)| ≤ ε|σ(t) − s| for all s ∈ U . If the (delta) derivative f ∆ (t) exists for every t ∈ Tk , then we say that f is ∆− differentiable on T. Definition 2.1[5] Let F, f ∈ Crd (T, R). If F 4 (t) = f (t) for all t ∈ Tk , then we definite the Cauchy integral of f by Z

b

f (t) 4 t = F (b) − F (a) for any a, b ∈ T. a

Lemma 2.2[5] (Holder’s inequality) Let a, b ∈ T with a ≤ b and f1 , f2 ∈ Crd (T, R). Then Z b ´1 ³ Z b ³Z b ´1 p q |f1 (t)|p 4 t |f1 (t)f2 (t)| 4 t ≤ |f2 (t)|q 4 t , a

a

a

where p > 1 and q = p/(p − 1). Lemma 2.3[5] Suppose that W ∈ Crd (T, Rn×n ) with I + µ(t)W (t) being invertible and g ∈ Crd (T, Rn ). Let t0 ∈ T and x0 ∈ Rn . Then the initial value problem x∆ (t) = −W (t)x(σ(t)) + g(t), x(t0 ) = x0 has a unique solution

Z x(t) = eΘW (t, t0 )x0 +

850

t

t0

eΘW (t, τ )g(τ )∆τ,

Taixiang Sun et al 848-859

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

where (ΘA)(t) = −[I +µ(t)A(t)]−1 A(t) for any t ∈ Tk and eΘA (t, t0 ) is the unique matrix-valued solution of the initial value problem (

Y ∆ (t) = (ΘA)(t)Y (t), Y (t0 ) = I.

Lemma 2.4[5] Suppose that A(t) and B(t) are differentiable n × n matrix-valued functions. Then (A(t)B(t))4 = A4 (t)B(σ(t)) + A(t)B 4 (t) = A(σ(t))B 4 (t) + A4 (t)B(t). Lemma 2.5[12] Let a, b ∈ T with a ≥ b and x1 (t), x2 (t), · · · , xn (t) be ∆-integrable on [a, b]T . Write x(t) = (x1 (t), x2 (t), · · · , xn (t)). Then v n ³Z b ¯Z b ¯ u ´2 Z X ¯ ¯ u t x(t)∆t¯ = xi (t)∆t ≤ ¯ a

i=1

a

a

v u t

n b uX

Z x2i (t)∆t

i=1

b

|x(t)|∆t.

= a

Lemma 2.6[12] Let V, V1 ∈ Rn×n with V1 ≥ V (i.e., V1 − V ≥ 0) and x ∈ Rn . Then xT V x ≤ s |V1 ||x|2 .

3. Main results and proofs Write ( R p q p(p−2) t ( a |eΘW (t, s)|p |U (s)| 2 +1 |[ U (s)]−1 |p(p−2) ∆s) p , if 1 < q < 2, ξ(t) = q Rt ( a |eΘW (t, s)|p |U (s)|∆s) p , if q ≥ 2,

(3.1)

( R p q p(p−2) b ( t |eΘW (t, s)|p |U (s)| 2 +1 |[ U (s)]−1 |p(p−2) ∆s) p , if 1 < q < 2, η(t) = q Rb ( t |eΘW (t, s)|p |U (s)|∆s) p , if q ≥ 2.

(3.2)

and

Theorem 3.1 Let a, b ∈ T with σ(a) < b and V1 ∈ Rn×n with V1 (t) ≥ V (t). If (1.1) has a s solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then the following inequality holds: Z b ξ(σ(t))η(σ(t)) |V1 (t)| 4 t ≥ 1. (3.3) ξ(σ(t)) + η(σ(t)) a Proof We claim that y(t) 6≡ 0 (t ∈ [a, b]T ). Indeed, if y(t) ≡ 0 (t ∈ [a, b]T ), then the first equation of (1.1) reduces to x∆ (t) = −W (t)x(σ(t)), x(a) = 0. By Lemma 2.3, it follows x(t) = eΘW (t, a) · x(a) = 0, which is a contradiction with (2.1). Moreover, we have y T (t)U (t)y(t) ≥ 0 (6≡ 0) for t ∈ [a, b]T since U (t) > 0. Since (x(t), y(t)) satisfies the following equality (y T (t)x(t))∆ = (xσ (t))T V (t)|xσ (t)|q−2 xσ (t) − y T (t)U (t)|y(t)|p−2 y(t),

851

(3.4)

Taixiang Sun et al 848-859

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

where xσ (t) = x(σ(t)). By integrating (3.4) from a to b and taking into account that x(a) = x(b) = 0, we see Z

Z

b

σ

q−2

|x (t)|

σ

T

σ

b

(x (t)) V (t)x (t) 4 t =

a

|y(t)|p−2 y T (t)U (t)y(t) 4 t > 0.

(3.5)

a

For t ∈ [a, b]T , let t0 = a and t0 = b respectively, we obtain from Lemma 2.3 that Z x(t) = − a

Z

t

p−2

eΘW (t, τ )U (τ )|y(τ )|

t

y(τ )∆τ = − b

eΘW (t, τ )U (τ )|y(τ )|p−2 y(τ )∆τ.

Which follows that for t ∈ [a, b)T , Z σ

Z

σ(t)

x (t) = −

p−2

eΘW (σ(t), τ )U (τ )|y(τ )|

a

b

y(τ )∆τ = σ(t)

eΘW (σ(t), τ )U (τ )|y(τ )|p−2 y(τ )∆τ.

Case I: Assume that q ≥ 2. Then we have that for a ≤ τ ≤ σ(t) ≤ b , |eΘW (σ(t), τ )U (τ )|y(τ )|p−2 y(τ )| ≤ |eΘW (σ(t), τ )||y(τ )|p−2 |U (τ )y(τ )| 1

= |eΘW (σ(t), τ )||y(τ )|p−2 {y T (τ )U T (τ )U (τ )y(τ )} 2 p p 1 ≤ |eΘW (σ(t), τ )||y(τ )|p−2 {| U (τ )y(τ )||U (τ )|| U (τ )y(τ )|} 2 1

1

1

1

= |eΘW (σ(t), τ )||y(τ )|p−2 |U (τ )| 2 (y T (τ )U (τ )y(τ )) 2 1

−1

= |eΘW (σ(t), τ )||y(τ )|p−2 |U (τ )| 2 (y T (τ )U (τ )y(τ )) q (y T (τ )U (τ )y(τ )) 2 q 1 p 1 2( 1 − 1 ) = |eΘW (σ(t), τ )||y(τ )|p−2 |U (τ )| 2 (y T (τ )U (τ )y(τ )) q | U (τ )y(τ )| 2 q 1 p 1 1− 2 p−1− 2q ≤ |eΘW (σ(t), τ )||U (τ )| 2 (y T (τ )U (τ )y(τ )) q | U (τ )| q |y(τ )| 1− 1q

≤ |eΘW (σ(t), τ )||U (τ )|

1

p−1− 2q

(y T (τ )U (τ )y(τ )) q |y(τ )|

.

Combining Lemma 2.2 and Lemma 2.5 we obtain ¯ Z σ(t) ¯q ¯ ¯ |xσ (t)|q = ¯ eΘW (σ(t), τ )U (τ )|y(τ )|p−2 y(τ )∆τ ¯ a ³ Z σ(t) ´q ≤ |eΘW (σ(t), τ )U (τ )|y(τ )|p−2 y(τ )|∆τ a

³Z

σ(t)

a

³Z

1− 1q

|eΘW (σ(t), τ )||U (τ )|

≤ σ(t)

≤

´q ³ Z

p

|eΘW (σ(t), τ )| |U (τ )|4τ

a

that is

Z σ

1

σ(t)

q

|x (t)| ≤ ξ(σ(t))

p−1− 2q

(y T (τ )U (τ )y(τ )) q |y(τ )| p

σ(t)

∆τ

´q

´ y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ ,

a

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

(3.6)

a

Similarly, by letting η(t) be as in (3.2), for a ≤ σ(t) ≤ τ ≤ b, we have Z |xσ (t)|q ≤ η(σ(t))

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

(3.7)

σ(t)

852

Taixiang Sun et al 848-859

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

It follows from (3.6) and (3.7) that Z

σ(t)

η(σ(t))ξ(σ(t))

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ ≥ |xσ (t)|q η(σ(t))

a

and

Z

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ ≥ |xσ (t)|q ξ(σ(t)).

η(σ(t))ξ(σ(t)) σ(t)

Thus

ξ(σ(t))η(σ(t)) |x (t)| ≤ ξ(σ(t)) + η(σ(t)) σ

Z

q

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

a

By Lemma 2.6 and (3.5) we see Z a

b

Z |V1 (t)||xσ (t)|q ∆t ≤

b

(|V1 (t)|

a

Z

a

ξ(σ(t))η(σ(t)) |V1 (t)| ∆t ξ(σ(t)) + η(σ(t))

b

ξ(σ(t))η(σ(t)) ∆t |V1 (t)| ξ(σ(t)) + η(σ(t))

a

= a

Z

Z

b

= Z

ξ(σ(t))η(σ(t)) ξ(σ(t)) + η(σ(t))

b

≤ a

ξ(σ(t))η(σ(t)) |V1 (t)| ∆t ξ(σ(t)) + η(σ(t))

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ )∆t

Z

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ

a

Z

b

|xσ (t)|q−2 (xσ (t))T V (t)xσ (t) 4 t

a

Z

a

b

|V1 (t)||xσ (t)|q ∆t.

Since Z b Z b Z b |V1 (t)||xσ (t)|q ∆t ≥ |xσ (t)|q−2 (xσ (t))T V (t)xσ (t) 4 t = |y(t)|p−2 y T (t)U (t)y(t)∆t > 0, a

a

we get

a

Z a

b

ξ(σ(t))η(σ(t)) |V1 (t)| 4 t ≥ 1. ξ(σ(t)) + η(σ(t))

This completes the proof of Case I. Case II: Assume that 1 < q < 2. Then p > 2. Note that for a ≤ τ ≤ σ(t) ≤ b , |eΘW (σ(t), τ )U (τ )|y(τ )|p−2 y(τ )| ≤ |eΘW (σ(t), τ )||y(τ )|p−2 |U (τ )y(τ )| 1

= |eΘW (σ(t), τ )||y(τ )|p−2 {y T (τ )U T (τ )U (τ )y(τ )} 2 p p 1 = |eΘW (σ(t), τ )||y(τ )|p−2 {( U (τ )y(τ ))T U (τ ) U (τ )y(τ )} 2 p p p p 1 ≤ |eΘW (σ(t), τ )||( U (τ ))−1 U (τ )y(τ )|p−2 {| U (τ )y(τ )||U (τ )|| U (τ )y(τ )|} 2 p p 1 p ≤ |eΘW (σ(t), τ )||( U (τ ))−1 |p−2 | U (τ )y(τ )|p−2 |U (τ )| 2 | U (τ )y(τ )| p 1 p = |eΘW (σ(t), τ )||( U (τ ))−1 |p−2 |U (τ )| 2 | U (τ )y(τ )|p−1 p 2 p 1 p p−1− 2q = |eΘW (σ(t), τ )||( U (τ ))−1 |p−2 |U (τ )| 2 | U (τ )y(τ )| q | U (τ )y(τ )| p (p−1)(p−2) 2 p 1 p p−1− 2q p |y(τ )| ≤ |eΘW (σ(t), τ )||( U (τ ))−1 |p−2 |U (τ )| 2 | U (τ )y(τ )| q | U (τ )| p (p−1)(p−2) 1 p 1 p−1− 2q p = |eΘW (σ(t), τ )||( U (τ ))−1 |p−2 |U (τ )| 2 (y T (τ )U (τ )y(τ )) q | U (τ )| |y(τ )| .

853

Taixiang Sun et al 848-859

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Then we obtain σ

q

|x (t)|

¯ Z σ(t) ¯q ¯ ¯ = ¯ eΘW (σ(t), τ )U (τ )|y(τ )|p−2 y(τ )∆τ ¯ a ³ Z σ(t) ´q ≤ |eΘA (σ(t), τ )U (τ )|y(τ )|p−2 y(τ )|∆τ a

³Z

p 1 |eΘW (σ(t), τ )||( U (τ ))−1 |p−2 |U (τ )| 2

σ(t)

≤ a

´q p (p−1)(p−2) p−1− 2q p ×(y (τ )U (τ )y(τ )) | U (τ )| |y(τ )| ∆τ Z ³ σ(t) p p (p−1)(p−2) +1 p |eΘW (σ(t), τ )||( U (τ ))−1 |p−2 | U (τ )| ≤ a ´q 1 p−1− 2q ∆τ ×(y T (τ )U (τ )y(τ )) q |y(τ )| ³ Z σ(t) ´q p p(p−2) p p −1 p(p−2) +1 2 |eΘW (σ(t), τ )| |( U (τ )) | ≤ |U (τ )| 4τ 1 q

T

a

³Z ×

σ(t)

´ y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ .

a

That is

Z σ

σ(t)

q

|x (t)| ≤ ξ(σ(t))

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

(3.8)

a

Similarly, by letting η(t) be as in (3.2), for a ≤ σ(t) ≤ τ ≤ b, we have Z σ

q

b

|x (t)| ≤ η(σ(t))

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

(3.9)

σ(t)

The rest of the proof is similar to the Case I, we have Z b ξ(σ(t))η(σ(t)) |V1 (t)| 4 t ≥ 1. ξ(σ(t)) + η(σ(t)) a This completes the proof of Theorem 3.1 Corollary 3.2 Let a, b ∈ T with σ(a) < b and V1 ∈ Rn×n with V1 (t) ≥ V (t). If (1.1) has a s solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then the following inequality holds: Z

b

(ξ(σ(t)) + η(σ(t)))|V1 (t)| 4 t ≥ 4.

a

Proof Note

It follows from (3.3) that

ξ(σ(t))η(σ(t)) ξ(σ(t)) + η(σ(t)) ≤ . ξ(σ(t)) + η(σ(t)) 4 Z

b

a

That is

(3.10)

Z a

ξ(σ(t)) + η(σ(t)) |V1 (t)| 4 t ≥ 1. 4

b

(ξ(σ(t)) + η(σ(t)))|V1 (t)| 4 t ≥ 4.

This completes the proof of Corollary 3.2.

854

Taixiang Sun et al 848-859

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Corollary 3.3 Let a, b ∈ T with σ(a) < b and V1 ∈ Rn×n with V1 (t) ≥ V (t). If (1.1) has a s solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then the following inequality holds: Z

b

1

(ξ(σ(t))η(σ(t))) 2 |V1 (t)| 4 t ≥ 2.

a

(3.11)

Proof Note 1

ξ(σ(t)) + η(σ(t)) ≥ 2(ξ(σ(t))η(σ(t))) 2 . It follows from (3.3) that Z

1

b

(ξ(σ(t))η(σ(t))) 2 |V1 (t)| 4 t ≥ 1. 2

a

That is

Z

b

1

(ξ(σ(t))η(σ(t))) 2 |V1 (t)| 4 t ≥ 2.

a

This completes the proof of Corollary 3.3. Theorem 3.4 Let a, b ∈ T with σ(a) < b and V1 ∈ Rn×n with V1 (t) ≥ V (t). If (1.1) has a s solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then there exists an c ∈ (a, b) such that

Z

Z

σ(c)

ξ(σ(t))|V1 (t)| 4 t ≥ 1 and

a

Proof Let

Z a

η(σ(t))|V1 (t)| 4 t ≥ 1.

c

Z

t

F (t) =

b

ξ(σ(s))|V1 (s)| 4 s −

(3.12)

b

η(σ(s))|V1 (s)| 4 s.

t

Then we have F (a) < 0 and F (b) > 0. Hence we can choose an c ∈ (a, b) such that F (c) ≤ 0 and F (σ(c)) ≥ 0, that is Z a

and

Z

Z

c

ξ(σ(s))|V1 (s)| 4 s ≤

ξ(σ(s))|V1 (s)| 4 s ≥

η(σ(s))|V1 (s)| 4 s

c

Z

σ(c)

a

b

(3.13)

b

σ(c)

η(σ(s))|V1 (s)| 4 s.

(3.14)

From (3.6) and (3.8), we have Z σ

q

|V1 (t)||x (t)| ≤ ξ(σ(t))|V1 (t)|

σ(t)

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

(3.15)

a

Note that for a ≤ τ ≤ σ(t) ≤ σ(c) ≤ b . Integrating (3.15) from a to σ(c), we obtain Z a

σ(c)

Z σ

σ(c)

q

|V1 (t)||x (t)| 4 t ≤

³Z ξ(σ(t))|V1 (t)|

σ(t)

´ y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ 4 t

a

a

Z ≤ a

Z

c

ξ(σ(t))|V1 (t)| 4 t

σ(c)

a

Z

+ξ(σ(c))|V1 (c)|(σ(c) − c)

855

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

σ(c)

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

a

Taixiang Sun et al 848-859

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Z

Z

σ(c)

ξ(σ(t))|V1 (t)| 4 t

= a

σ(c)

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

a

Similarly, for a ≤ σ(c) ≤ σ(t) ≤ τ ≤ b, we can obtain from (3.7),(3.9) and (3.14) that Z

Z

b

|V1 (t)||xσ (t)|q 4 t ≤

σ(c)

Z

b

η(σ(t))|V1 (t)| 4 t

σ(c)

Z

ξ(σ(t))|V1 (t)| 4 t

a

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ

σ(c)

Z

σ(c)

≤

b

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

σ(c)

These yield Z

Z

b

|V1 (t)||xσ (t)|q 4 t ≤

a

Z

σ(c)

ξ(σ(t))|V1 (t)| 4 t

a

Z

Z ξ(σ(t))|V1 (t)| 4 t

a

Z

|xσ (t)|q−2 (xσ (t))T V (t)xσ (t) 4 t

a

ξ(σ(t))|V1 (t)| 4 t

a

b

Z

σ(c)

≤

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ

a

σ(c)

=

b

a

b

|V1 (t)||xσ (t)|q 4 t.

Since Z b Z b Z b σ q σ q−2 σ T σ |V1 (t)||x (t)| ∆t ≥ |x (t)| (x (t)) V (t)x (t) 4 t = |y(t)|p−2 y T (t)U (t)y(t)∆t > 0, a

a

R σ(c)

we obtain

a

a

ξ(σ(t))|V1 (t)| 4 t ≥ 1.

Next, we have from (3.7) and (3.9) that Z σ

q

|x (t)| |V1 (t)| ≤ η(σ(t))|V1 (t)|

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

(3.16)

σ(t)

Integrating (3.16) from c to b, we obtain that for a ≤ c ≤ t ≤ σ(t) ≤ τ ≤ b, Z

Z

b

σ

|V1 (t)||x (t)| 4 t ≤

c

³Z η(σ(t))|V1 (t)|

b

q

c

Z

´ y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ 4 t

b

σ(t)

Z

b

≤

η(σ(t))|V1 (t)| 4 t

c

b

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

σ(c)

Similarly, for a ≤ τ ≤ σ(t) ≤ σ(c) ≤ b, we can obtain Z

Z

c

|V1 (t)||xσ (t)|q 4 t ≤

a

Z

c

ξ(σ(t))|V1 (t)| 4 t

a

Z

σ(c)

a

Z

b

η(σ(t))|V1 (t)| 4 t

≤ c

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ

σ(c)

y T (τ )U (τ )y(τ )|y(τ )|p−2 ∆τ.

a

These yield Z a

b

Z σ

q

|V1 (t)||x (t)| 4 t ≤

η(σ(t))|V1 (t)| 4 t

c

Z = c

Z

b

η(σ(t))|V1 (t)| 4 t

856

y T (t)U (t)y(t)|y(t)|p−2 ∆t

a

Z

b

b

b

|xσ (t)|q−2 (xσ (t))T V (t)xσ (t) 4 t

a

Taixiang Sun et al 848-859

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Z ≤ Thus, we also obtain

Rb c

c

Z

b

η(σ(t))|V1 (t)| 4 t

b

a

|V1 (t)||xσ (t)|q 4 t.

η(σ(t))|V1 (t)| 4 t ≥ 1. This completes the proof of Theorem 3.4.

Theorem 3.5 Let a, b ∈ T with σ(a) < b and V1 ∈ Rn×n with V1 (t) ≥ V (t). If (1.1) has a s solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then the following inequalities hold: Z

³Z

b

|W (t)| 4 t + Z

³Z

b

a b

|U (t)|

|W (t)| 4 t +

|U (t)| 4 t

p(p−2) +1 2

p

a

a

a

a

´1 ³ Z

b

´1

b

|V1 (t)| 4 t

q

≥ 2,

if q ≥ 2,

´1 ´1 ³ Z b p p q −1 p(p−2) |V1 (t)| 4 t ≥ 2, if 1 < q < 2. |( U (t)) | 4t a

Proof From the proof of Theorem 3.1, we have Z

Z

b

|y(t)|p−2 y T (t)U (t)y(t) 4 t =

a

b

|xσ (t)|q−2 (xσ (t))T V (t)xσ (t) 4 t.

a

It follows from the first equation of (1.1) that for all a ≤ t ≤ b, Z

t

x(t) =

(−W (τ )xσ (τ ) − U (τ )|y(τ )|p−2 y(τ )) 4 τ,

a

Z x(t) =

b

(W (τ )xσ (τ ) + U (τ )|y(τ )|p−2 y(τ )) 4 τ.

t

Case I: Assume that q ≥ 2. We have ¯Z t ¯ ¯ ¯ |x(t)| = ¯ (−W (τ )xσ (τ ) − U (τ )|y(τ )|p−2 y(τ )) 4 τ ¯ a Z t ≤ |W (τ )xσ (τ ) + U (τ )|y(τ )|p−2 y(τ )| 4 τ a Z t Z t σ ≤ |W (τ )x (τ )| 4 τ + |U (τ )|y(τ )|p−2 y(τ )| 4 τ a a Z t Z t 1 p−1− 2q 1− 1 4 τ. ≤ |W (τ )||xσ (τ )| 4 τ + |U (τ )| q (y T (τ )U (τ )y(τ )) q |y(τ )| a

a

Similarly, we have Z |x(t)| ≤ t

b

Z |W (τ )||xσ (τ )| 4 τ +

b

1− 1q

|U (τ )|

1

p−1− 2q

(y T (τ )U (τ )y(τ )) q |y(τ )|

4 τ.

t

Then from Lemma 2.2 and Lemma 2.6, we obtain |x(t)| ≤ ≤ = ≤

Z Z b i 1 1h b 1− 1q T p−1− 2q σ q (y (t)U (t)y(t)) |y(t)| 4t |W (t)||x (t)| 4 t + |U (t)| 2 a a Z ³Z b ´1 i ´1 ³ Z b 1h b p q |W (t)||xσ (t)| 4 t + |U (t)| 4 t y T (t)U (t)y(t)|y(t)|p−2 4 t 2 a a a Z ³Z b ´1 i ´1 ³ Z b 1h b p q |W (t)||xσ (t)| 4 t + |U (t)| 4 t |xσ (t)|q−2 (xσ (t))T V (t)xσ (t) 4 t 2 a a a Z ³Z b ´1 ³ Z b ´1 i 1h b p q |W (t)||xσ (t)| 4 t + |V1 (t)||xσ (t)|q 4 t . |U (t)| 4 t 2 a a a

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Denote M = maxa≤t≤b |x(t)| > 0, then Z ´1 i ´1 ³ Z b ³Z b 1h b p q . M≤ |V1 (t)|M q 4 t |U (t)| 4 t |W (t)|M 4 t + 2 a a a Thus

Z

³Z

b

|W (t)| 4 t + a

´1 ³ Z

b

|U (t)| 4 t

p

a

a

´1

b

q

|V1 (t)| 4 t

≥ 2.

Case II: Assume that 1 < q < 2. Then p ≥ 2 and Z

t

|W (τ )xσ (τ ) + U (τ )|y(τ )|p−2 y(τ )| 4 τ a Z t Z t ≤ |W (τ )xσ (τ )| 4 τ + |U (τ )|y(τ )|p−2 y(τ )| 4 τ a a Z t Z t p 1 σ ≤ |W (τ )||x (τ )| 4 τ + |( U (τ ))−1 |p−2 |U (τ )| 2

|x(t)| ≤

a

a

p (p−1)(p−2) p−1− 2q p ×(y T (τ )U (τ )y(τ )) | U (τ )| |y(τ )| 4 τ. 1 q

and Z |x(t)| ≤

b

Z |W (τ )||xσ (τ )| 4 τ +

t

b

p 1 |( U (τ ))−1 |p−2 |U (τ )| 2

t

p (p−1)(p−2) p−1− 2q p ×(y (τ )U (τ )y(τ )) | U (τ )| |y(τ )| 4 τ. 1 q

T

Thus we obtain 1h 2

Z

Z

p 1 |( U (t))−1 |p−2 |U (t)| 2 a a i (p−1)(p−2) 1 p p−1− 2q p |y(t)| 4t ×(y T (t)U (t)y(t)) q | U (t)| Z ³Z b p ´1 p (p−1)(p−2) 1h b p +1 p σ p ≤ |W (t)||x (t)| 4 t + (|( U (t))−1 |p−2 | U (t)| ) 4t 2 a a ³Z b ´1 i 1 q p−1− 2q q × ((y T (t)U (t)y(t)) q |y(t)| ) 4t

|x(t)| ≤

b

σ

b

|W (t)||x (t)| 4 t +

a

Z ³Z b ´1 p p(p−2) 1h b p σ +1 −1 p(p−2) 2 |W (t)||x (t)| 4 t + |U (t)| = |( U (t)) | 4t 2 a a ³Z b ´1 i q × |xσ (t)|q−2 (xσ (t))T V (t)xσ (t) 4 t a Z ³Z b ´1 p p(p−2) 1h b p σ |W (t)||x (t)| 4 t + |U (t)| 2 +1 |( U (t))−1 |p(p−2) 4 t ≤ 2 a a ³Z b ´1 i q × |V1 (t)||xσ (t)|q 4 t . a

Similarly, we also have Z b ³Z b ´1 ³ Z b ´1 p p(p−2) p q |U (t)| 2 +1 |( U (t))−1 |p(p−2) 4 t |W (t)| 4 t + |V1 (t)| 4 t ≥ 2. a

a

a

This completes the proof of Theorem 3.5.

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REFERENCES [1] S. Hilger, Analysis on measure chains − a unified approach to continuous and discrete calculus, Results Math., 18(1990): 18-56. [2] L. Jiang, Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. 310(2005)579-593. [3] F. Wong, S. Yu, C. Yeh, W. Lian, Lyapunov’s inequality on time scales, Appl. Math. Lett. 19(2006)1293-1299. [4] X. Liu, M. Tang, Lyapunov-type inequality for higher order difference equations, Appl. Math. Comput. 232(2014)666-669. [5] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001. [6] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. [7] Q. Zhang, X. He, J. Jiang, On Lyapunov-type inequalities for nonlinear dynamic systems on time scales, Comput. Math. Appl. 62(2011): 4028-4038. [8] A. Tiryaki, D. Cakmak, M. F. Aktas, Lyapunov-type inequalities for a certain class of nonlinear systems, Comput. Math. Appl. 64(2012): 1804-1811. [9] G. Sh. Guseinov, B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl. 45(2003): 1399-1416. [10] X. He, Q. Zhang, X. Tang, On inequalities of Lyapunov for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. 381(2011): 695-705. [11] X. Tang, M. Zhang, Lyapunov inequalities and stability for linear Hamiltonian systems, J. Differential Equations. 252(2012): 358-381. [12] J. Liu, T. Sun, X. Kong, Q. He, Lyapunov inequalities of linear Hamiltonian systems on time scales, J. Comput. Anal. Appl. 21(2016)1160-1169. [13] R. P. Agarwal, M. Bohner, P. Rehak, Half-linear dynamic equations, Nonlinear Anal. Appl. (2003)1-56. [14] R. P. Agarwal, A. Ozbekler, Lyapunov type inequalities for even order differential equations with mixed nonlinearities, J. Inequal. Appl. 2015(2015)142, 10 pages. [15] R. P. Agarwal, A. Ozbekler, Disconjugacy via Lyapunov and Val´ee–Poussin type inequalities for forced differential equations, Appl. Math. Comput., 265(2015)456-468. [16] R. P. Agarwal, A. Ozbekler, Lyapunov type inequalities for second order sub and super– half–linear differential equations, Dynam. Sys. Appl. 24(2015)211-220.

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A new three-step iterative method for a countable family of pseudo-contractive mappings in Hilbert spaces Qin Chen, Li Li, Nan Lin, Baoguo Chen∗ Research Center for Science Technology and Society, Fuzhou University of International Studies and Trade, Fuzhou 350202, P.R. China

Abstract: In this paper, we propose a new three-step iterative method for a countable family of pseudo-contractive mappings in a real Hilbert space. We also prove the strong convergence of the proposed iterative algorithm under appropriate conditions. Key words: pseudo-contractive mapping; iterative method; fixed point; strong convergence AMS subject classification (2000): 47H09

1

Introduction

In this paper, we assume that H is a real Hilbert space with the inner product h·, ·i and the induced norm k · k, C is a nonempty closed convex subset of H and T : C → C is a self-mapping of C. F(T ) denotes the fixed point set of the mapping T . Recall that T is called a k-strictly pseudo-contractive mapping if there exists a constant k ∈ [0, 1) such that kT x − T yk2 ≤ kx − yk2 + kk(I − T )x − (I − T )yk2 , ∀ x, y ∈ C,

(1.1)

and T is called a pseudo-contractive mapping if kT x − T yk2 ≤ kx − yk2 + k(I − T )x − (I − T )yk2 , ∀ x, y ∈ C.

(1.2)

It is obvious that k = 0, then the mapping T is nonexpansive, that is kT x − T yk ≤ kx − yk, ∀ x, y ∈ C.

(1.3)

Finding the fixed points of nonexpansive mappings is an important topic in the theory of nonexpansive mappings and has wide applications in a number of applied areas, such as the convex feasibility problem [1, 2], the split feasibility problem [3], image recovery and signal ∗

Corresponding author. Email: [email protected].

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processing [4]. After that, as an important generalization of nonexpansive mappings, strictly pseudo-contractive mappings become one of the most interesting studied class of nonexpansive mappings. In fact, strictly pseudo-contractive mappings have more powerful application than nonexpansive mappings do such as in solving inverse problem [5]. Iterative methods for nonexpansive mappings have been extensively investigated (see e.g., [6–16, 31–33] and the references contained therein). However, iterative methods for strictly pseudo-contractive mappings are far less developed than those for nonexpansive mappings and the reason is probably that the second term appearing on the right hand side of (1.1) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strictly pseudocontractive mapping T . The most general iterative algorithm for nonexpansive mappings studied by many authors is Mann’s iteration algorithm [18] which is as following: xn+1 = αn xn + (1 − αn )T xn , n ≥ 0,

(1.4)

where x0 ∈ C is chosen arbitrarily and {αn } is a real sequence in (0, 1). Under the following P additional assumptions: (i) lim αn = 0 and (ii) ∞ n=0 αn = ∞, the sequence {xn } generated n→∞

by (1.4) is generally referred to as Mann’s iteration algorithm in the light of [18]. The Mann’s iteration algorithm dose not generally converge to a fixed point of T even the fixed point exists. For example, C is a nonempty closed convex and bounded subset of a real Hilbert space, T : C → C is nonexpansive, one can only prove that the sequence generated by Mann’s iteration algorithm (1.4) with the assumptions (i) and (ii) is an approximate fixed point sequence, that is, kxn − T xn k → 0 as n → ∞. In [19], Reich proved that if X is a uniformly convex Banach space P with a Fr´ echet differentiable norm and if {αn } is chosen such that ∞ n=0 αn (1 − αn ) = ∞, then the sequence {xn } defined by (1.4) converges weakly to a fixed point of T . To get the sequence {xn } to converge strongly to a fixed point of T (when such a fixed point exists), some type of compactness condition must be additionally imposed either on C (e.g., C is compact) or on T (e.g., T is demicompact or T is semicompact, see [20, 21]). The first convergence result for k-strictly pseudo-contractive mappings was proposed by Browder and Petryshyn [22] in 1967. They proved that if the sequence {xn } is generated by the following: xn+1 = αxn + (1 − α)T xn , n ≥ 0,

(1.5)

for any starting point x0 ∈ C and α is a constant such that k < α < 1, then the sequence {xn } converges weakly to a fixed point of k-strictly pseudo-contractive mapping T . In [23], Marino and Xu extended the result of Browder and Petryshyn [22] to Mann’s iteration algorithm (1.4), they proved that the sequence {xn } generated by (1.4) converges weakly to a fixed point of k-strictly pseudo-contractive mapping T for the conditions that k < αn < 1 for all n and P∞ n=0 (αn − k)(1 − αn ) = ∞. 2 861

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However, the well known strong convergence result for pseudo-contractive mappings is Ishikawa’s iteration algorithm which was proved by Ishikawa [24] in 1974 and it is more general than that of Mann’s iteration algorithm (1.4) in some sense. More precisely, he got the following theorem.

Theorem 1.1 ([24]) Let C be a convex compact subset of a Hilbert space H and let T : C → C be a Lipschitz pseudo-contractive mapping. For any x1 ∈ C, suppose the sequence {xn } is defined iteratively for each n ≥ 1 by (

yn = (1 − βn )xn + βn T xn , xn+1 = (1 − αn )xn + αn T yn ,

(1.6)

where {αn }, {βn } are sequences of positive number that satisfy the following there conditions: P (i) 0 ≤ αn ≤ βn ≤ 1; (ii) lim βn = 0; (iii) ∞ n=1 αn βn = ∞. Then the sequence {xn } converges n→∞

strongly to a fixed point of T. In 2001, Chidume and Mutangadura [25] gave an example to show that the Mann’s iteration algorithm (1.4) failed to be convergent to a fixed point of Lipschitz pseudo-contractive mappings. In order to obtain a strong convergence result for pseudo-contractive mappings without the compactness assumption, Zhou [26] established the hybrid Ishikawa algorithm for Lipschitz pseudo-contractive mappings as following: Theorem 1.2 ([26]) Let C be a closed convex subset of a real Hilbert space H and let T : C → C be a Lipschitz pseudo-contraction such that F(T ) 6= φ. Suppose that {αn } and {βn } are two real sequences in (0, 1) satisfying the conditions: (i) αn ≤ βn , ∀n ≥ 0; (ii) lim inf αn > 0; (iii) n→∞ 1 lim sup αn ≤ α < √ , n ≥ 0, where L ≥ 1 is the Lipschitzian constant of T . Let a n→∞ 1 + L2 + 1 sequence {xn } be generated by    x0 ∈ C,      yn = (1 − αn )xn + αn T xn ,     z = (1 − β )x + β T y , n n n n n (1.7) 2 ≤ kx − zk2 − α β (1 − 2α − L2 α2 )kx − T x k2 },  C = {z ∈ C : kz − zk  n n n n n n n n n      Qn = {z ∈ C : hxn − z, x0 − xn i ≥ 0},     x n+1 = PCn ∩Qn x0 , n ≥ 0. Then, {xn } converges strongly to a fixed point v of T , where v = PF (T ) (x0 ). We observe that the iterative algorithm (1.7) generates a sequence {xn } by projecting x0 on to the intersection of the suitably constructed closed convex sets Cn and Qn . Recently, Yao et al. [27] introduced the hybrid iterative algorithm which just involved one closed convex set for pseudo-contractive mappings in Hilbert spaces as follows: 3 862

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Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a pseudo-contractive mapping. Let {αn } be a sequence in (0, 1). Let x0 ∈ H. For C1 = C and x1 = PC1 x0 , define a sequence {xn } of C as follows:     yn = (1 − αn )xn + αn T xn ,

Cn+1 = {z ∈ Cn : kαn (I − T )yn k2 ≤ 2αn hxn − z, (I − T )yn i},    x n+1 = PCn+1 x0 , n ∈ N.

(1.8)

Theorem 1.3 ( [27]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a L-Lipschitz pseudo-contractive mapping such that F(T ) 6= φ. Assume the 1 sequence αn ∈ [a, b] for some a, b ∈ (0, ). Then the sequence {xn } generated by (1.8) L+1 converges strongly PF (T ) (x0 ). In [28], Tang et al. proposed the hybrid algorithm (1.8) to the Ishikawa’s iteration algorithm (1.6) and got the following result. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a pseudo-contractive mapping. Let {αn }, {βn } be two sequences in [0, 1]. Let x0 ∈ H. For C1 = C and x1 = PC1 x0 , define a sequence {xn } of C as follows:   yn = (1 − αn )xn + αn T zn ,        zn = (1 − βn )xn + βn T xn ,

Cn+1 = {z ∈ Cn : kαn (I − T )yn k2 ≤ 2αn hxn − z, (I − T )yn i     + 2αn βn Lkxn − T xn k · kyn − xn + αn (I − T )yn k},     x n+1 = PCn+1 x0 , n ≥ 1.

(1.9)

Theorem 1.4 ( [28]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a L-Lipschitz pseudo-contractive mapping with L ≥ 1 such that F(T ) 6= φ. Assume the sequences {αn } and {βn } in (0, 1) satisfying: (i) b ≤ αn < αn (L + 1)(1 + βn L) < a < 1, for some a, b ∈ (0, 1); (ii) lim βn = 0. Then the sequence {xn } generated by (1.9) converges n→∞

strongly PF (T ) (x0 ). Recently, Zegeye et al. [29] generalized Ishikawa’s iteration algorithm (1.6) to a common fixed point of a finite family of Lipschitz pseudo-contractive mappings and obtained the following theorem. Theorem 1.5 ( [29]) Let C be a nonempty, closed convex subset of a real Hilbert space H. Let Ti : C → C, i = 1, 2, · · · , N, be a finite family of Lipschitz pseudo-contractive mappings with Lipschitzian constants Li , for i = 1, 2, · · · , N, respectively. Assume that the interior of T F := N i=1 F(Ti ) is nonempty. Let {xn } be a sequence generated from an arbitrary x0 ∈ C by ( yn = (1 − βn )xn + βn Tn xn , (1.10) xn+1 = (1 − αn )xn + αn Tn yn , 4 863

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and {αn }, {βn } ⊂ (0, 1) satisfying the following conditions: (i) αn ≤ 1 for L := max{Li : i = βn , ∀ n ≥ 0; (ii) lim inf αn = α > 0; (iii) supn≥1 βn ≤ β < √ n→∞ 1 + L2 + 1 1, 2, · · · , N }. Then, {xn } converges strongly to a common fixed point of {T1 , T2 , · · · , TN }. where Tn := Tn(mod

N)

In [30], Cheng et al.

extended the algorithm (1.10) to a countable family of pseudo-

contractive mappings and gave a three-step iterative method, which is as follows: Theorem 1.6 ( [30]) Let C be a nonempty, closed convex subset of a real Hilbert space H, let {Tn }∞ n=1 : C → C be a countable family of uniformly closed and uniformly Lipschitz pseudocontractive mappings with Lipschitzian constants Ln , let L := supn≥1 Ln . Assume that the T interior of F := ∞ n=1 F(Tn ) is nonempty. Let {xn } be a sequence generated from an arbitrary x0 ∈ C by the following algorithm:     zn = (1 − γn )xn + γn Tn xn ,

(1.11)

yn = (1 − βn )xn + βn Tn zn ,    x n+1 = (1 − αn )xn + αn Tn yn ,

where {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions: (i) αn ≤ βn ≤ γn , ∀ n ≥ 0; (ii) lim inf αn = α > 0; (iii) supn≥1 γn ≤ γ with γ 3 L4 + 2γ 2 L3 + γ 2 L2 + γL2 + 2γ < 1. Then, n→∞

{xn } converges strongly to x∗ ∈ F. Remark 1.1 The condition (iii) of the Theorem 1.6 is not correct, it is replaced by supn≥1 γn ≤ γ with γ 3 L4 + 2γ 2 L3 + γ 2 L2 + 2γL2 + 2γ < 1. Motivated and inspired by the above works, in this paper, we propose a new three-step iterative method for a countable family of pseudo-contractive mappings in Hilbert spaces and prove its strong convergence theorem under appropriate conditions.

2

Preliminaries

In this section, we recall some definitions and useful results which will be used in the next section. Definition 2.1 Let C be a subset of a real Hilbert space H. (1) A mapping T : C → H is said to be L-Lipschitz if there exists L > 0 such that kT x − T yk ≤ Lkx − yk, ∀ x, y ∈ C. When L = 1, T is nonexpansive. If L < 1, T is called a contraction. It is easy to see that every contractive mapping is nonexpansive and every nonexpansive mapping is Lipschitz. 5 864

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(2) A countable family of mappings {Tn }∞ n=1 : C → H is said to be uniformly Lipschitz with Lipschitzian constants Ln > 0, n ≥ 1, if there exists 0 < L := supn≥1 Ln such that kTn x − Tn yk ≤ Lkx − yk, ∀ x, y ∈ C, n ≥ 1. (3) A countable family of mappings {Tn }∞ n=1 : C → H is said to be uniformly closed if T∞ ∗ ∗ xn → x and kxn − Tn xx k → 0 imply x ∈ n=1 F(Tn ). Definition 2.2 A mapping T with domain D(T ) and range R(T ) in a real Hilbert space H is said to be monotone if the inequality kx − yk ≤ kx − y + s(T x − T y)k holds for every x, y ∈ D(T ) and for all s > 0. We observe that T is monotone ⇔ hT x − T y, x − yi ≥ 0 ⇔ k(I − T )x − (I − T )yk2 ≤ kx − yk2 + kT x − T yk2 ⇔ kAx − Ayk2 ≤ kx − yk2 + k(I − A)x − (I − A)yk2 , A := I − T ⇔ A is pseudo-contractive. Furthermore, a zero of T is a fixed point of A, that is, x ∈ N (T ) := {x ∈ D(T ) : T x = 0} ⇔ x ∈ F(A) := {x ∈ D(A) : Ax = x}. Lemma 2.1 Let H be a real Hilbert space. Then for α ∈ [0, 1] the following equality kαx + (1 − α)yk2 = αkxk2 + (1 − α)kyk2 − α(1 − α)kx − yk2 holds for all x, y ∈ H. Lemma 2.2 If the sequences {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions: (i) βn ≤ γn , ∀ n ≥ 1, (ii) (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0, where α = lim inf αn , β = lim inf βn , γ ≥ supn≥1 γn , and L > 0 is a constant. Then, we have n→∞

n→∞

α > 0, β > 0 and (1 − αn )γn + αn βn (γn2 L2 + 2γn − 1) < 0. Proof. On one hand, it is obvious that α > 0, β > 0 and γ 2 L2 + 2γ − 1 < 0 because of (1−α)γ +αβ(γ 2 L2 +2γ −1) < 0. And we get that (1−αn )γ ≤ (1−α)γ and αn βn (γ 2 L2 +2γ −1) ≤ αβ(γ 2 L2 + 2γ − 1). Then (1 − αn )γ + αn βn (γ 2 L2 + 2γ − 1) ≤ (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0. 6 865

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On the other hand, it is easy to know that (1 − αn )γn ≤ (1 − αn )γ and αn βn (γn2 + 2γn − 1) ≤ αn βn (γ 2 L2 + 2γ − 1). We can obtain (1 − αn )γn + αn βn (γn2 + 2γn − 1) ≤ (1 − αn )γ + αn βn (γ 2 L2 + 2γ − 1) < 0. Hence (1 − αn )γn + αn βn (γn2 L2 + 2γn − 1) < 0.

3



The main result

Theorem 3.1 Let C be a nonempty, closed convex subset of a real Hilbert space H, let {Tn }∞ n=1 : C → C be a countable family of uniformly closed and uniformly Lipschitz pseudo-contractive mappings with Lipschitzian constants Ln , let L := supn≥1 Ln . Assume that the interior of T F := ∞ n=1 F(Tn ) is nonempty. Let {xn } be a sequence generated from an arbitrary x1 ∈ C by     zn = (1 − γn )xn + γn Tn xn , (3.1) yn = (1 − βn )xn + βn Tn zn ,    x = (1 − α )z + α y , n+1

n

n

n n

where {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions: (i) βn ≤ γn , ∀ n ≥ 1, (ii) (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0, where α = lim inf αn , β = lim inf βn and γ ≥ supn≥1 γn . Then, {xn } converges strongly to x∗ ∈ F.

n→∞

n→∞

Proof. Take p ∈ F arbitrarily. By (3.1) and Lemma 2.1, we have kxn+1 − pk2 = k(1 − αn )zn + αn yn − pk2 = k(1 − αn )(zn − p) + αn (yn − p)k2 = (1 − αn )kzn − pk2 + αn kyn − pk2 − αn (1 − αn )kzn − yn k2 ≤ (1 − αn )kzn − pk2 + αn kyn − pk2 ,

(3.2)

and kzn − pk2 = k(1 − γn )xn + γn Tn xn − pk2 = k(1 − γn )(xn − p) + γn (Tn xn − p)k2 = (1 − γn )kxn − pk2 + γn kTn xn − pk2 − γn (1 − γn )kxn − Tn xn k2 ≤ (1 − γn )kxn − pk2 + γn (kxn − pk2 + kxn − Tn xn k2 ) − γn (1 − γn )kxn − Tn xn k2 = kxn − pk2 + γn2 kxn − Tn xn k2 ,

(3.3)

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where the inequality is based on that {Tn }∞ n=1 is a countable family of pseudo-contractive mappings. Similarly, we can get kyn − pk2 = k(1 − βn )xn + βn Tn zn − pk2 = k(1 − βn )(xn − p) + βn (Tn zn − p)k2 = (1 − βn )kxn − pk2 + βn kTn zn − pk2 − βn (1 − βn )kxn − Tn zn k2 ≤ (1 − βn )kxn − pk2 + βn (kzn − pk2 + kzn − Tn zn k2 ) − βn (1 − βn )kxn − Tn zn k2 .

(3.4)

In addition, using (3.1), we have that kzn − Tn zn k2 = k(1 − γn )xn + γn Tn xn − Tn zn k2 = k(1 − γn )(xn − Tn zn ) + γn (Tn xn − Tn zn )k2 = (1 − γn )kxn − Tn zn k2 + γn kTn xn − Tn zn k2 − γn (1 − γn )kxn − Tn xn k2 ≤ (1 − γn )kxn − Tn zn k2 + γn L2 kxn − zn k2 − γn (1 − γn )kxn − Tn xn k2 = (1 − γn )kxn − Tn zn k2 + γn L2 kγn (xn − Tn xn )k2 − γn (1 − γn )kxn − Tn xn k2 = (1 − γn )kxn − Tn zn k2 + γn (γn2 L2 + γn − 1)kxn − Tn xn k2 ,

(3.5)

where the inequality is based on that {Tn }∞ n=1 is a countable family of uniformly Lipschitz mappings. Substituting (3.3) and (3.5) into (3.4), we obtain that   kyn − pk2 ≤ (1 − βn )kxn − pk2 + βn kxn − pk2 + γn2 kxn − Tn xn k2   + βn (1 − γn )kxn − Tn zn k2 + γn (γn2 L2 + γn − 1)kxn − Tn xn k2 − βn (1 − βn )kxn − Tn zn k2 = kxn − pk2 + βn γn (γn2 L2 + 2γn − 1)kxn − Tn xn k2 + βn (βn − γn )kxn − Tn zn k2 ≤ kxn − pk2 + βn γn (γn2 L2 + 2γn − 1)kxn − Tn xn k2 ,

(3.6)

where the last inequality is based on the condition (i). Therefore, substituting (3.3) and (3.6) into (3.2), we get   kxn+1 − pk2 ≤ (1 − αn ) kxn − pk2 + γn2 kxn − Tn xn k2   + αn kxn − pk2 + βn γn (γn2 L2 + 2γn − 1)kxn − Tn xn k2   = kxn − pk2 + (1 − αn )γn2 + αn βn γn (γn2 L2 + 2γn − 1) kxn − Tn xn k2 .(3.7) According to the conditions and Lemma 2.2, inequality (3.7) implies that kxn+1 − pk2 ≤ kxn − pk2 .

(3.8)

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It is obvious that lim kxn − pk exists, then {kxn − pk} is bounded. This implies that {xn }, n→∞

{Tn xn }, {zn }, {Tn zn } and {yn } are also bounded. Furthermore, we have that kxn − pk2 = kxn − xn+1 k2 + kxn+1 − pk2 + 2hxn+1 − p, xn − xn+1 i. This implies 1 1 hxn+1 − p, xn − xn+1 i + kxn − xn+1 k2 = (kxn − pk2 − kxn+1 − pk2 ). 2 2

(3.9)

Moreover, since the interior of F is nonempty, then there exists p∗ ∈ F and r > 0 such that p∗ + rh ∈ F whenever khk ≤ 1. Thus, from (3.8), we have 1 0 ≤ hxn+1 − (p∗ + rh), xn − xn+1 i + kxn − xn+1 k2 2 1 (kxn − (p∗ + rh)k2 − kxn+1 − (p∗ + rh)k2 ). = 2

(3.10)

From (3.9) and (3.10), we obtain that 1 rhh, xn − xn+1 i ≤ hxn+1 − p∗ , xn − xn+1 i + kxn − xn+1 k2 2 1 = (kxn − p∗ k2 − kxn+1 − p∗ k2 ). 2 Since h with khk ≤ 1 is arbitrary, we can take h = kxn − xn+1 k ≤

(3.11)

xn − xn+1 with khk = 1, then kxn − xn+1 k

1 (kxn − p∗ k2 − kxn+1 − p∗ k2 ). 2r

So, for n > m, we can get kxm − xn k = k(xm − xm+1 ) + (xm+1 − xm+2 ) + · · · + (xn−1 − xn )k n−1 X ≤ kxi − xi+1 k ≤

i=m n−1 X i=m

=

1 (kxi − p∗ k2 − kxi+1 − p∗ k2 ) 2r

1 (kxm − p∗ k2 − kxn − p∗ k2 ). 2r

From (3.8), we know that {kxn − p∗ k2 } converges. Therefore, {xn } is a Cauchy sequence. Since C is closed subset of Hilbert space H, then there exists x∗ ∈ C such that xn → x∗ ∈ C.

(3.12)

Furthermore, from the conditions and Lemma 2.2, we have   0 < β (α − 1)γ + αβ(1 − 2γ − γ 2 L2 ) 9 868

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  ≤ γn (αn − 1)γn + αn βn (1 − 2γn − γn2 L2 ) = (αn − 1)γn2 + αn βn γn (1 − 2γn − γn2 L2 ).

(3.13)

Then, by (3.7) and (3.13), we conclude that 

(α − 1)γβ + αβ 2 (1 − 2γ − γ 2 L2 )

∞ X

kxn − Tn xn k2

n=1

≤ ≤

∞  X

 (αn − 1)γn2 + αn βn γn (1 − 2γn − γn2 L2 ) kxn − Tn xn k2

n=1 ∞ X

(kxn − pk2 − kxn+1 − pk2 ) < ∞,

n=1

from which it follows that lim kxn − Tn xn k = 0.

n→∞

(3.14)

Since {Tn }∞ n=1 are uniformly closed mappings, then from (3.12) and (3.14), we can obtain ∗

x ∈

∞ \

F(Tn ) = F.

n=1

The proof is complete.



Remark 3.1 We now give an example of a countable family of uniformly closed and uniformly Lipschitz pseudo-contractive mappings with the interior of the common fixed points nonempty. This example comes from [30]. Suppose that H := R and C := [−1, 1] ∈ H. Let {Tn }∞ n=1 : C → C be defined by   x, x ∈ [−1, 0), Tn x := 1 1  ( + )x, x ∈ [0, 1]. 2n 2 T∞ Then F := n=1 F(Tn ) = [−1, 0], and hence the interior of the common fixed points is nonempty. Moreover, it is easy to show that {Tn }∞ n=1 is a countable family of uniformly closed and uniformly Lipschitz pseudo-contractive mappings with Lipschitz constant L := supn≥1 Ln = 2. 1 1 1 3 1 3 For this example, we can let αn = + , βn = + and γn = − for 4 n+4 10 n + 40 20 n + 40 3 n ≥ 1. Then {αn }, {βn }, {γn } ⊂ (0, 1) and βn ≤ γn , ∀ n ≥ 1. Furthermore, α = lim inf αn = , n→∞ 4 1 3 β = lim inf βn = , supn≥1 γn ≤ , and n→∞ 10 20  3 3 3 1  3 3 33 (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) = (1 − ) × + × × ( )2 × 22 + 2 × − 1 = − < 0. 4 20 4 10 20 20 4000 It satisfies all conditions in Theorem 3.1. Hence, from Theorem 3.1, we can obtain the sequence {xn } generated by (3.1) and staring with an arbitrary x1 ∈ C will converges strongly to a common fixed point of {Tn }∞ n=1 . 10 869

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4

Applications

If in Theorem 3.1, we consider a finite family of Lipschitz pseudo-contractive mappings, then we have the following result. Theorem 4.1 Let C be a nonempty, closed convex subset of a real Hilbert space H, let {Ti }N i=1 : C → C be a finite family of uniformly closed and Lipschitz pseudo-contractive mappings with Lipschitzian constants Li , for i = 1, 2, · · · , N , respectively. Assume that the interior of F := TN i=1 F(Ti ) is nonempty. Let {xn } be a sequence generated from an arbitrary x1 ∈ C by     zn = (1 − γn )xn + γn Tn xn , (4.1) yn = (1 − βn )xn + βn Tn zn ,    x = (1 − α )z + α y , n+1

where Tn := Tn(mod

N)

n

n

n n

and {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions:

(i) βn ≤ γn , ∀ n ≥ 1, (ii) (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0, where α = lim inf αn , β = lim inf βn and γ ≥ supn≥1 γn , for L := max{Li : i = 1, 2, · · · , N }. n→∞

n→∞

Then, {xn } converges strongly to a common fixed point of {T1 , T2 , · · · , TN }. If in Theorem 3.1, we consider a single Lipschitz pseudo-contractive mapping, then we may P add a condition that is ∞ n=1 γn = ∞. Theorem 4.2 Let C be a nonempty, closed convex subset of a real Hilbert space H, let T : C → C be a Lipschitz pseudo-contractive mapping with Lipschitzian constant L. Assume that the interior of F(T ) is nonempty. Let {xn } be a sequence generated from an arbitrary x1 ∈ C by    

zn = (1 − γn )xn + γn T xn , (4.2)

yn = (1 − βn )xn + βn T zn ,    x n+1 = (1 − αn )zn + αn yn , where {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions: (i) βn ≤ γn , ∀ n ≥ 1, P (ii) ∞ n=1 γn = ∞, (iii) (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0,

where α = lim inf αn , β = lim inf βn and γ ≥ supn≥1 γn . Then, {xn } converges strongly to a n→∞

n→∞

fixed point of T . Proof. Following the method of the proof of Theorem 3.1, we also obtain that   kxn+1 − pk2 ≤ kxn − pk2 + (1 − αn )γn2 + αn βn γn (γn2 L2 + 2γn − 1) kxn − T xn k2 , 11 870

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and xn → x∗ ∈ C. Now, from Lemma 2.2, we get 

2 2

(α − 1)γ + αβ(1 − 2γ − γ L )

∞ X

γn kxn − T xn k2

n=1

≤ = ≤

∞ X



 γn (αn − 1)γn + αn βn (1 − 2γn − γn2 L2 ) kxn − T xn k2

n=1 ∞  X

 (αn − 1)γn2 + αn βn γn (1 − 2γn − γn2 L2 ) kxn − T xn k2

n=1 ∞ X

(kxn − pk2 − kxn+1 − pk2 ) < ∞,

n=1

from which it follows that lim inf kxn − T xn k = 0, n→∞

and hence there exists a subsequence {xnk } of {xn } such that lim kxnk − T xnk k = 0.

n→∞

Thus, xnk → x∗ and the continuity of T imply that x∗ = T x∗ and hence x∗ ∈ F(T ).



Now, we prove a convergence theorem for a countable family of monotone mappings. Theorem 4.3 Let H be a real Hilbert space, let {Tn }∞ n=1 : H → H be a countable family of uniformly Lipschitz monotone mappings with Lipschitzian constants Ln , let L := supn≥1 Ln . T And if xn → x∗ and kTn xn k → 0, then x∗ ∈ ∞ n=1 N (Tn ). Assume that the interior of N := T∞ n=1 N (Tn ) is nonempty. Let {xn } be a sequence generated from an arbitrary x1 ∈ C by     zn = xn − γn Tn xn , (4.3)

yn = xn − βn (xn − zn ) − βn Tn zn ,    x n+1 = (1 − αn )zn + αn yn ,

where {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions: (i) βn ≤ γn , ∀ n ≥ 1, (ii) (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0, where α = lim inf αn , β = lim inf βn and γ ≥ supn≥1 γn . Then, {xn } converges strongly to x∗ ∈ N .

n→∞

n→∞

Proof. Since Tn is monotone if and only if An := I −Tn is pseudo-contractive and T∞ n=1 N (Tn ) 6= ∅, then the conclusion follows from Theorem 3.1.

T∞

n=1 F(An )

= 

If in Theorem 4.3, we consider a finite family of monotone mappings and a single monotone mapping, respectively, then we get the following corollaries. 12 871

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Corollary 4.1 Let H be a real Hilbert space, let {Ti }N i=1 : H → H be a finite family of Lipschitz monotone mappings with Lipschitzian constants Li , for i = 1, 2, · · · , N , respectively. And if TN T xn → x∗ and kTn xn k → 0, then x∗ ∈ N i=1 N (Ti ) i=1 N (Ti ). Assume that the interior of N := is nonempty. Let {xn } be a sequence generated from an arbitrary x1 ∈ C by     zn = xn − γn Tn xn ,

(4.4)

yn = xn − βn (xn − zn ) − βn Tn zn ,    x n+1 = (1 − αn )zn + αn yn , where Tn := Tn(mod

N)

and {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions:

(i) βn ≤ γn , ∀ n ≥ 1, (ii) (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0, where α = lim inf αn , β = lim inf βn and γ ≥ supn≥1 γn . for L := max{Li : i = 1, 2, · · · , N }. n→∞

n→∞

Then, {xn } converges strongly to a common zero point of {T1 , T2 , · · · , TN }. Corollary 4.2 Let H be a real Hilbert space, let T : H → H be a Lipschitz monotone mapping with Lipschitzian constant L. Assume that the interior of N (T ) is nonempty. Let {xn } be a sequence generated from an arbitrary x1 ∈ C by     zn = xn − γn T xn ,

yn = xn − βn (xn − zn ) − βn T zn ,    x n+1 = (1 − αn )zn + αn yn ,

(4.5)

where {αn }, {βn }, {γn } ⊂ (0, 1) satisfying the following conditions: (i) βn ≤ γn , ∀ n ≥ 1, P (ii) ∞ n=1 γn = ∞, (iii) (1 − α)γ + αβ(γ 2 L2 + 2γ − 1) < 0, where α = lim inf αn , β = lim inf βn and γ ≥ supn≥1 γn . Then, {xn } converges strongly to a n→∞

n→∞

zero point of T .

Competing interests The authors declare that they have no competing interests regarding the publication of this article.

Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. 13 872

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References [1] D. Butnariu, Y. Censor, P. Gurfil, E. Hadar, On the behavior of subgradient projections methods for convex feasibility problem in Euclidean spaces, SIAM J. Optim. 19 (2) (2008) 786-807. [2] S. Maruster, C. Popirlan, On the Mann-type iteration and the convex feasibility problem, J. Comput. Appl. Math. 212 (2008) 390-396. [3] H.K. Xu, A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems 22 (2006) 2021-2034. [4] D. Youla, Mathematical theory of image restoration by the method of convex projections, in: H. Stark (Ed.), Image Recovery Theory and Applications, Academic Press, Orlando, 1987, pp. 29-77. [5] O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl. 194 (1991) 911-933. [6] H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 202 (1996) 150-159. [7] D.P. Wu, S.S. Chang, G.X. Yuan, Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space, Nonlinear Anal. 63 (2005) 987-999. [8] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957-961. [9] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641-3645. [10] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-491. [11] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360. [12] S.Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266. [13] S.S. Chang, Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 323 (2006) 1402-1416. 14 873

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[29] H. Zegeye, N. Shahzad, M.A. Alghamdi, Convergence of Ishikawas iteration method for pseudocontractive mappings, Nonlinear Anal. 74 (2011) 7304-7311. [30] Q.Q. Cheng, Y.F. Su, J.L. Zhang, Convergence theorems of a three-step iteration method for a countable family of pseudocontractive mappings, Fixed Point Theory and Applications 2013, 2013: 100. [31] N. Huang, C. Ma, A new extragradient-like method for solving variational inequality problems, Fixed Point Theory and Applications 2012, 2012: 223. [32] Y. Ke, C. Ma, A new relaxed extragradient-like algorithm for approaching common solutions of generalized mixed equilibrium problems, a more general system of variational inequalities and a fixed point problem, Fixed Point Theory and Applications 2013, 2013: 126. [33] Y. Ke, C. Ma, The convergence analysis of the projection methods for a system of generalized relaxed cocoercive variational inequalities in Hilbert spaces, Fixed Point Theory and Applications 2013, 2013: 182.

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Harmonic analysis in the product of commutative hypercomplex systems Hossam A. Ghany Department of mathematics, Helwan University, Sawah street (11282), Cairo, Egypt. Department of mathematics, Taif University, Hawea (888), Taif, KSA. [email protected]

Abstract The main aim of this paper is to give integral representations for strongly negative definite functions defined on the product hypercomplex systems. Harmonic properties for strongly negative definite functions are investigated. We construct a Lèvy measure on the product hypercomplex systems, then we study the conditions that guarantee the existence of some integrations having an integrand parts as a function of the constructed kernel. Finally, we give a Lèvy - Khinchin type formula for strongly negative definite functions defined on the product hypercomplex systems.

Keywords. Lèvy – Khinchin; Hypercomplex; Negative definite. 2010 Mathematics subject classification. 43A35; 43A65; 43A25.

1. Introduction. The integral representation of negative definite functions is defined as Lèvy-Khinchin formula. This was established by Lèvy-Khinchin in1930’s for 𝐺 = ℝ. Many author’s paid attention to generalize this result in different spaces. It had been extended by Hunt [4] to Lie groups, Parthasarathy et al [8] to locally compact commutative groups, Berg et al [3] to comutative semigroups with identical involution and by Lasser [6] for commutative hypergroups. The main aim of this paper is devoted to find the integral representations for strongly negative definite functions defined on the product dual hypercomplex system. Let 𝑄 be a commutative separable locally compact metric space of points 𝑝, 𝑞, 𝑟, … ; 𝐵(𝑄) is the 𝜎 –algebra of Borel subsets on 𝑄 and 𝐵0 (𝑄) is the subring of 𝐵(𝑄), which consists of sets with compact closure. We denote by 𝐶(𝑄) the space of continuous functions on 𝑄; 𝐶𝑏 (𝑄), 𝐶∞ (𝑄) and 𝐶0 (𝑄) consists of bounded, tending to zero at infinity and compactly supported functions from 𝐶(𝑄), respectivly. For a fixed 𝑟 ∈ 𝑄, 𝐵 ∈ 𝐵(𝑄), we will denote by 𝑐(𝐴, 𝐵; 𝑟) a commutative Borel structure measure in 𝐴 ∈ 𝐵(𝑄) . The hypercomplex system 𝐿1 (𝑄, 𝑑𝑚) is the Banach algebra of functions on 𝑄 with respect to the multiplicative measure 𝑚 and convolution " ∗ " defined for any 𝜙 ∗ 𝜓 ∈ 𝐿1 (𝑄, 𝑑𝑚) by:

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(𝜙 ∗ 𝜓)(𝑟) = ∫ 𝜙(𝑝)𝑑𝑝 ∫ 𝜓(𝑞)𝑑𝑞 𝑐(𝐸𝑝 , 𝐸𝑞 ; 𝑟) 𝑄

𝑄

= ∫ ∫ 𝜙(𝑝)𝜓(𝑞) 𝑐(𝑝, 𝑞; 𝑟) 𝑑𝑚(𝑝)𝑑𝑚(𝑞) 𝑄 𝑄

= ∫ ∫ 𝜙(𝑝)𝜓(𝑞) 𝑑𝑚𝑟 (𝑝, 𝑞) 𝑄 𝑄

The space 𝐶∞ (𝑄) is a Banach space with norm ||. ||∞ = supr∈Q |(. )(𝑟)| We will denote by ℳ(𝑄), the space of Radon measure on 𝑄, i.e. the space of continuous linear functionals defined on 𝐶0 (𝑄) . Let ℳ𝑏 (𝑄) = ( 𝐶∞ (𝑄))′ denote the space of bounded Radon measures with norm ||𝜇||∞ = sup{|𝜇(𝑓)|; f ∈ C∞ , |𝑓| ≤ 1} The topology of simple convergence on functions from in the space of Radon measures, is called vague topology.

2. Strongly Negative Definite Functions. A hypercomplex system 𝐿1 (𝑄, 𝑑𝑚) may or may not have a unity. In this paper we will concern our efforts on hypercomplex system with unity. A normal hypercomplex system contain a basis unity if there exists 𝑒 ∈ 𝑄 such that 𝑒 ∗ = 𝑒 and 𝑐(𝐴, 𝐵; 𝑒) = 𝑚(𝐴∗ ∩ 𝐵),

𝐴, 𝐵 ∈ 𝐵(𝑄).

A nonzero measurable and bounded almost everywhere function 𝑄 ∋ 𝑟 → 𝜒(𝑟) ∈ ℂ is said to be a character of the hypercomplex system 𝐿1 (𝑄, 𝑑𝑚)if for all 𝐴, 𝐵 ∈ 𝐵0 (𝑄) we have ∫ 𝑐(𝐴, 𝐵; 𝑟) 𝜒(𝑟)𝑑𝑚(𝑟) = 𝜒(𝐴)𝜒(𝐵) 𝑄

and ∫ 𝜒(𝑟) 𝑑𝑚(𝑟) = 𝜒(𝐶), 𝐶 ∈ 𝐵0 (𝑄). 𝐶

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

We will denote by 𝑋ℎ the set of all bounded Hermitian characters, i.e. ̅̅̅̅̅̅ = 𝜒(𝑟 ∗ ) } 𝑋ℎ ≔ {𝜒 ∈ 𝐶𝑏 (𝑄); ∫ 𝑐(𝐴, 𝐵; 𝑟) 𝜒(𝑟)𝑑𝑚(𝑟) = 𝜒(𝐴)𝜒(𝐵), 𝜒(𝑟) 𝑄

A continuous bounded function 𝜓: 𝑄 → ℂ is called negative definite if for any 𝑟1 , 𝑟2 , … , 𝑟𝑛 ∈ 𝑄; 𝑐1 , 𝑐2 , … , 𝑐𝑛 ∈ ℂ and 𝑛 ∈ ℕ we have: ∑𝑛𝑖,𝑗=1[𝜓(𝑟𝑖 ) + ̅̅̅̅̅̅̅ 𝜓(𝑟𝑗 ) − (𝑅𝑟 ∗𝑗 𝜓)(𝑟𝑖 )] 𝑐𝑖 𝑐̅𝑗 ≥ 0, and a continuous bounded function 𝜑: 𝑄 → ℂ is called positive definite if for any 𝑟1 , 𝑟2 , … , 𝑟𝑛 ∈ 𝑄; 𝑐1 , 𝑐2 , … , 𝑐𝑛 ∈ ℂ and 𝑛 ∈ ℕ we have: ∑𝑛𝑖,𝑗=1(𝑅𝑟 ∗ 𝑗 𝜑)(𝑟𝑖 ) 𝑐𝑖 𝑐̅𝑗 ≥ 0, where 𝑅𝑟 ( 𝑟 ∈ 𝑄), denote the generalized translation operators on𝐿1 (𝑄, 𝑑𝑚). As pointed out of [1], every positive definite function 𝜑 ∈ 𝑃(𝑄) admits a unique representation in the integral form (2.1)

𝜑(𝑟) = 𝜇̂ (𝜒) = ∫𝑋 𝜒(𝑟)𝑑 𝜇(𝜒), 𝜒 ∈ 𝑋ℎ , ℎ

where 𝜇 is a finite nonnegative regular measure on the space 𝑋ℎ . Conversely, each function have the integral form (1.1) belongs to the set of all positive definite function 𝑃(𝑄). Let 𝑄1 and 𝑄2 be two commutative separable locally compact metric spaces, with identities 𝑒1 and 𝑒2 respectively, and suppose 𝐴 be a non empty subset of 𝐿1 (𝑄1 ) × 𝐿1 (𝑄2 ), then the strongly positive definite function will be defined as follows: Definition 2.1. A locally bounded continuous measurable function 𝛷 ∈ 𝐴 is called strongly positive definite, if there exists two positive definite functions 𝜑1 ∈ 𝑃(𝑄1 ) and 𝜑2 ∈ 𝑃(𝑄2 ) and a Radon measure 𝜇 ∈ ℳ+ (𝑄1 × 𝑄2 ), such that (2.2)

𝜇̂ (𝜒, 𝜏) = { 0,

𝜑1 (𝜒)+𝜑2 (𝜏), (𝜒, 𝜏) ∈ 𝐴 (𝜒, 𝜏) ∉ 𝐴 .

A locally bounded continuous measurable function 𝛹 ∈ 𝐴 is called strongly negative definite, if 𝛹(𝑒1 , 𝑒2 ) ≥ 0and exp(−𝑡𝛹) is strongly positive definite in 𝐴 for each 𝑡 > 0. Clearly each strongly positive (negative) definite function is positive (negative) definite but the converse implication does not hold. Negative definiteness is an analogue of one half of Schoenberg’s duality result, It is not known for which hypercomplex system, negative definiteness implies strong negative definiteness. The following Lemma is in fact, an adaption of

878

Hossam A. Ghany 876-888

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

whatever done for hypergroups [7], we will not repeat the proof, wherever the proof for hypergroups can be applied to the hypercomplex with necessary modification. Lemma 2.2. The sum and the point-wise limit of strongly negative definite functions on hypercomplex are also strongly negative definite. Theorem 2.3. A function 𝛹: 𝑄 → ℂ is strongly negative definite if and only if the following conditions are satisfied: (i) (ii) (iii)

𝛹(𝑒1 , 𝑒2 ) ≥ 0, 𝛹 is continuous bounded function ; ̅̅̅̅̅̅̅ = 𝛹(𝒓∗ ) for each 𝒓 ∈ 𝑄1 × 𝑄2 ; 𝛹(𝒓) if for any 𝒓1 , 𝒓2 , … , 𝒓𝑛 ∈ 𝑄1 × 𝑄2 𝑎𝑛𝑑 𝑐1 , 𝑐2 , … , 𝑐𝑛 ∈ ℂ with ∑𝑛𝑖=1 𝑐𝑖 = 0 and 𝒓𝑖 = (𝑟1𝑖 , 𝑟1𝑖 ) ∈ 𝑄1 × 𝑄2 , we have ∑𝑛𝑖,𝑗=1(𝑅𝒓∗ 𝑗 𝛹)(𝒓𝑖 ) 𝑐𝑖 𝑐̅𝑗 ≤ 0.

Proof. Suppose that the function 𝛹 is strongly negative definite. From the above definition of strongly negative definite functions, it is clear that 𝛹 satisfies (i) and (ii). Let 𝒓1 , 𝒓2 , … , 𝒓𝑛 ∈ 𝑄1 × 𝑄2 𝑎𝑛𝑑 𝑐1 , 𝑐2 , … , 𝑐𝑛 ∈ ℂ with ∑𝑛𝑖=1 𝑐𝑖 = 0 . Since, every strongly negative definite function is negative definite, so ̅̅̅̅̅̅̅̅) − (𝑅 ∗ 𝛹)(𝒓 )] 𝑐 𝑐̅ 0 ≤ ∑𝑛𝑖,𝑗=1[𝛹(𝒓𝑖 ) + 𝛹(𝒓 𝑗 𝒓 𝑗 𝑖 𝑖 𝑗 𝑛 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 𝑛 𝑛 𝑛 ̅̅̅̅̅̅̅̅̅̅̅̅̅ = (∑𝑗=1 𝑐𝑗 ) ∑𝑖=1[𝛹(𝒓𝑖 )] 𝑐𝑖 + (∑𝑖=1 𝑐𝑖 ) ∑𝑗=1[𝛹(𝒓𝑗 )] − ∑𝑛𝑖,𝑗=1[(𝑅𝒓∗𝑗 𝛹)(𝒓𝑖 )] 𝑐𝑖 𝑐̅𝑗 = − ∑𝑛𝑖,𝑗=1[(𝑅𝒓∗𝑗 𝛹)(𝒓𝑖 )] 𝑐𝑖 𝑐̅𝑗 Conversely, suppose that 𝛹 satisfies the above conditions. Let 𝑒, 𝒓1 , 𝒓2 , … , 𝒓𝑛 ∈ 𝑄1 × 𝑄2 𝑎𝑛𝑑 𝑐1 , 𝑐2 , … , 𝑐𝑛 ∈ ℂ with ∑𝑛𝑖=1 𝑐𝑖 = 0. From (iii) we have 𝑛

0 ≥ ∑ [(𝑅𝒓∗ 𝑗 𝛹)(𝒓𝑖 )] 𝑐𝑖 𝑐̅𝑗 𝑖,𝑗=0 𝑛

𝑛 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = ∑ [(𝑅𝒓∗𝑗 𝛹)(𝒓𝑖 )] 𝑐𝑖 𝑐̅𝑗 + 𝑐̅0 ∑[𝛹(𝒓𝑖 )] 𝑐𝑖 + 𝑐0 ∑[𝛹(𝒓𝑗 )] 𝑐𝑗 + 𝛹(𝒆)|𝑐0 |2

=

𝑛

𝑖,𝑗=1 ∑𝑛𝑖,𝑗=1[𝛹(𝒓𝑖 )

This implies

𝑖=1

𝑗=1

̅̅̅̅̅̅̅̅) − (𝑅 ∗ 𝛹)(𝒓 )] 𝑐 𝑐̅ + 𝛹(𝒆)|𝑐 |2 + 𝛹(𝒓 𝑗 𝒓 𝑗 𝑖 𝑖 𝑗 0

𝑛

̅̅̅̅̅̅̅̅) − (𝑅 ∗ 𝛹)(𝒓 )] 𝑐 𝑐̅ ≥ 𝛹(𝒆)|𝑐 |2 ≥ 0 ∑ [𝛹(𝒓𝑖 ) + 𝛹(𝒓 𝑗 𝒓 𝑗 𝑖 𝑖 𝑗 0 𝑖,𝑗=1

879

Hossam A. Ghany 876-888

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Corollary 2.4. For any functions 𝛷, 𝛹 on the product 𝑄1 × 𝑄2 we have: (i) If 𝛹 belongs to the set of strongly negative definite function on 𝑄1 × 𝑄2 , then the function 𝒓 → 𝛹(𝒓) − 𝛹(𝑒1 , 𝑒2 ) is also strongly negative definite function. (ii) If 𝛷 belongs to the set of strongly positive definite function on 𝑄1 × 𝑄2 , then the function 𝒓 → 𝛷(𝒓) − 𝛷(𝑒1 , 𝑒2 ) is also strongly positive definite function. Proof. Let 𝒓1 , 𝒓2 , … , 𝒓𝑛 ∈ 𝑄1 × 𝑄2 𝑎𝑛𝑑 𝑐1 , 𝑐2 , … , 𝑐𝑛 ∈ ℂ with ∑𝑛𝑖=1 𝑐𝑖 = 0. Then we have 𝑛

𝑛

𝑛

∑ [𝑅𝒓∗𝑗 (𝛹(𝒓𝑖 ) − 𝛹(𝑒1 , 𝑒2 ))] 𝑐𝑖 𝑐̅𝑗 = ∑ (𝑅𝒓∗𝑗 𝛹)(𝒓𝑖 )𝑐𝑖 𝑐̅𝑗 − 𝛹(𝑒1 , 𝑒2 ) | ∑ 𝑐𝑖 |2 𝑖,𝑗=0

𝑖,𝑗=0

𝑖=1

𝑛

∑ (𝑅𝒓∗𝑗 𝛹)(𝒓𝑖 )𝑐𝑖 𝑐̅𝑗 ≤ 0

=

𝑖,𝑗=0

This proves the strongly negative definiteness of 𝛹(𝒓) − 𝛹(𝑒1 , 𝑒2 ). Similarly, let 𝒓1 , 𝒓2 , … , 𝒓𝑛 ∈ 𝑄1 × 𝑄2 𝑎𝑛𝑑 𝑐1 , 𝑐2 , … , 𝑐𝑛 ∈ ℂ with ∑𝑛𝑖=1 𝑐𝑖 = 0. Then we find 𝑛

𝑛

𝑛

∑ [𝑅𝒓∗ 𝑗 (𝛷(𝑒1 , 𝑒2 ) − 𝛷(𝒓𝑖 ))] 𝑐𝑖 𝑐̅𝑗 = − ∑ (𝑅𝒓∗𝑗 𝛷)(𝒓𝑖 )𝑐𝑖 𝑐̅𝑗 − 𝛷(𝑒1 , 𝑒2 ) | ∑ 𝑐𝑖 |2 𝑖,𝑗=0

𝑖,𝑗=0

𝑖=1

𝑛

= − ∑ (𝑅𝒓∗𝑗 𝛷)(𝒓𝑖 )𝑐𝑖 𝑐̅𝑗 ≤ 0 𝑖,𝑗=0

Because 𝛷 belongs to the set of strongly positive definite functions, hence (ii). Theorem 2.5. For every strongly negative definite function 𝛹 on the product 𝑄1 × 𝑄2 with 1 𝛹(𝑒1 , 𝑒2 ) ≥ 0, the function 𝛹 is strongly positive definite function on the product 𝑄1 × 𝑄2 . Proof. Suppose 𝛹 strongly negative definite function on the product 𝑄1 × 𝑄2 , so exp(−𝑡𝛹) is strongly positive definite on the product 𝑄1 × 𝑄2 . This implies for all 𝑡 > 0.

|exp(−𝑡𝛹)| ≤ |exp(−𝑡𝛹(𝑒1 , 𝑒2 ))| It follows, for all (𝜒, 𝜏) ∈ 𝑄1̂ × 𝑄2 we have ∞

∞

1 = ∫ exp(−𝑡𝛹(𝜒, 𝜏)) 𝑑𝑡 = ∫ 𝜇̂𝑡 (𝜒, 𝜏)𝑑𝑡 𝛹(𝜒, 𝜏) 0

0

Where 𝜇𝑡 is the corresponding measure for exp(−𝑡𝛹). Moreover, applying Lèvy continuity Theorem, there exists a measure 𝜐 ∈ ℳ+ (𝑄1 × 𝑄2 )such that

880

Hossam A. Ghany 876-888

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

∞

υ(𝜒, 𝜏) ≔ υ̂(𝜒, 𝜏) = ∫ 𝜇̂𝑡 (𝜒, 𝜏)𝑑𝑡 0

and

1 0 will be called a convolution semigroup on 𝑄1 × 𝑄2 if it satisfies the following items: (i) (ii) (iii)

𝜇𝑡 (𝑄1 × 𝑄2 ) ≤ 1, for each 𝑡 > 0; 𝜇𝑡1 ∗ 𝜇𝑡2 = 𝜇𝑡1 +𝑡2 for each 𝑡1 , 𝑡2 > 0; lim 𝜇𝑡 = ∈𝒆 , with respect to the vague topology on 𝜇 ∈ ℳ 𝑏 (𝑄1 × 𝑄2 ). 𝑡→0

Theorem 3.1. For any strongly negative definite function 𝛹 on 𝑄1 × 𝑄2 , there exists a unique convolution semigroup on 𝑄1 × 𝑄2 such that 𝛹 is associated to (𝜇𝑡 )𝑡>0 . Proof. Firstly, we will prove that, for (𝜒, 𝜏) ∈ 𝑄1̂ × 𝑄2 , the function 𝑡 → 𝜇̂ 𝑡 (𝜒, 𝜏) is continuous. As pointed out of Ursohn’s lemma [9], there exists 𝑓 ∈ 𝐶𝑐 (𝑄1 × 𝑄2 ) that satisfies 𝑓(𝒆) = 1 𝑎𝑛𝑑 0 ≤ 𝑓 < 1. Applying the above conditions for the convolution semigroup on 𝑄1 × 𝑄2 , we have: 1 = 𝑓(𝒆) = lim < 𝜇𝑡 , 𝑓 > ≤ lim𝑖𝑛𝑓 𝜇𝑡 (𝑄1 × 𝑄2 ) ≤ lim𝑠𝑢𝑝 𝜇𝑡 (𝑄1 × 𝑄2 ) ≤ 1 𝑡→0

𝑡→0

and this shows that lim 𝜇𝑡 = ∈𝒆 𝑡→0

𝑡→0

( in the Bernolli topology).

As pointed out of [2], for each 𝑡1 , 𝑡2 > 0, we have |𝜇̂ 𝑡 (𝜒, 𝜏) − 𝜇̂ 𝑡0 (𝜒, 𝜏)| ≤ |𝜇̂ |𝑡−𝑡0 | (𝜒, 𝜏) − 1| the right hand side tends to zero uniformally on compact subset of 𝑄1̂ × 𝑄2 , so lim 𝜇𝑡 = 𝜇𝑡0 𝑡→0

( in the Bernolli topology).

Secondly, from the definition of strongly negative definite function, there exists a unique determined measures 𝜇𝑡 ∈ ℳ 𝑏 (𝑄1 × 𝑄2 ), 𝑡 > 0, such that 𝜇̂ 𝑡 (𝜒) = exp(−𝑡𝛹) It is clear that,

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Hossam A. Ghany 876-888

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

the family (𝜇𝑡 )𝑡>0 satisfies conditions (i) and (ii). The boundedness of the function 𝛹on compact subsets of 𝑄1̂ × 𝑄2 implies that lim 𝜇̂ 𝑡 (𝜒) = lim exp(−𝑡𝛹) = 1. 𝑡→0

𝑡→0

From [5], there exists a multiplicative measure 𝑚 ̂ on the dual 𝑄1̂ × 𝑄2 , such that for every 𝑓 ∈ ̂ 𝐶0 (𝑄1 × 𝑄2 ) and 𝜀 > 0, there exits 𝑔 ∈ 𝐶0 (𝑄1 × 𝑄2 )such that 𝑄1̂ × 𝑄2 ||𝑓 − 𝑔̃|| < 𝜀 and |𝜇𝑡 (𝑓) − 𝜀𝒆 (𝑓)| ≤ 2𝜀 +

∫ |𝑔(𝜒, 𝜏)| |𝜇̂ 𝑡 (𝜒, 𝜏) − 1|𝑑𝑚 ̂ (𝜒, 𝜏) 𝑄1̂ ×𝑄2

this implies (iii).

Let 𝑆 denote the set of probability and symmetric measures on 𝑄1 × 𝑄2 with compact support, i.e. 𝑆 = {𝜎; 𝜎 ∈ ℳ 1 (𝑄1 × 𝑄2 ) ∩ ℳ 𝑐 (𝑄1 × 𝑄2 ), 𝜎(𝜒, 𝜏) = 𝜎̃ (𝜒, 𝜏)} Let (𝜇𝑡 )𝑡>0 be a convolution semigroup on 𝑄1 × 𝑄2 and 𝛹: 𝑄1̂ × 𝑄2 → ℂ the strongly negative definite function associated to (𝜇𝑡 )𝑡>0 . Applying the same technique of [2] for the 1 hypercomplex system instead of semigroups, we can see that, the net ( 𝑡 𝜇𝑡 |𝑄1 ̂ × 𝑄2 \{𝒆})𝑡>0 of × 𝑄2 \{𝒆}, positive measures on 𝑄1 ̂ × 𝑄2 \{𝒆}converges vaguely as 𝑡 → 0 to a measure 𝜇 on 𝑄1 ̂ and for every 𝜎 ∈ 𝑆 , the function 𝛹 ∗ 𝜎 − 𝛹 is continuous strongly positive definite on 𝑄1̂ × 𝑄2 and the positive bounded measure 𝜇𝜎 on 𝑄1 × 𝑄2 whose Fourier transform is 𝛹 ∗ 𝜎 − 𝛹 satisfies (3.1)

(1 − 𝜎̃ )𝜇 = 𝜇𝜎 |𝑄1 × 𝑄2 \{𝒆}.

The positive measure 𝜇 on 𝑄1 × 𝑄2 \{𝒆} defined by (3.1) is called the strong Lèvy measure for the convolution semigroup (𝜇𝑡 )𝑡>0 on 𝑄1 × 𝑄2 . Theorem 3.2. Let 𝜇 denote the Lèvy measure for the convolution semigroup (𝜇𝑡 )𝑡>0 on 𝑄1 × 𝑄2 . Then (3.2)

∫𝑄

(1 − 𝑅𝑒(𝜒, 𝜏)(𝑟))𝑑 𝜇(𝜒, 𝜏) < ∞, (𝜒, 𝜏) ∈ 𝑄1̂ × 𝑄2 .

1 ×𝑄2 \{𝒆}

1 Proof. For (𝜒, 𝜏) ∈ 𝑄1̂ × 𝑄2 , let 𝜎 = 2 (𝜖(𝜒,𝜏) + 𝜖̅̅̅̅̅̅̅ ̃ = 𝑅𝑒(𝜒, 𝜏)(𝑟), substituting (𝜒,𝜏) ) ∈ 𝑆; then 𝜎 in (3.2) we get

∫

(1 − 𝑅𝑒(𝜒, 𝜏)(𝑟))𝑑 𝜇(𝜒, 𝜏) =

𝑄1 ×𝑄2 \{𝒆}

∫

(1 − 𝜎̃ (𝑟))𝑑 𝜇(𝜒, 𝜏) = 𝜇𝜎 |𝑄1 × 𝑄2 \{𝒆} < ∞.

𝑄1 ×𝑄2 \{𝒆}

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.5, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

4. Integral representation theorem. A continuous function ℎ: 𝑄1 × 𝑄2 → ℝ is called homomorphism if it satisfies ℎ(𝒓∗ ) = −ℎ(𝒓) and 𝑅𝒓 ℎ(𝒔) = ℎ(𝒓) + ℎ(𝒔), 𝒓, 𝒔 ∈ 𝑄1 × 𝑄2 . Clearly, if ℎ: 𝑄1 × 𝑄2 → ℝ is a homomorphism, then the function 𝛹 = 𝑖ℎ is strongly negative definite. A continuous function 𝑞: 𝑄1 × 𝑄2 → ℝ is called a quadratic form, if it satisfies (4.1)

𝑅𝒓 𝑞(𝒔) + 𝑅𝒓∗ 𝑞(𝒔) = 2𝑞(𝒓) + 2𝑞(𝒔), 𝒓, 𝒔 ∈ 𝑄1 × 𝑄2 .

Theorem 4.1. Let 𝛹 be a strongly negative definite function associated the convolution semigroup (𝜇𝑡 )𝑡>0 on 𝑄1 × 𝑄2 . If the Lèvy measure 𝜇 of (𝜇𝑡 )𝑡>0 is symmetric, then 𝐼𝑚𝛹 is a homomorphism. Proof. As remarked in [2], a continuous function 𝑓: 𝑄1 ×̂ 𝑄2 → ℝ which satisfies 𝑓(𝑒1 , 𝑒2 ) = 0 is a homomorphism if and only if 𝑓 ∗ 𝜐 − 𝑓 = 0 for all 𝜐 ∈ 𝑆. Since, 𝜇̌ = 𝜇 is equivalent to 𝜇̌ 𝜎 = 𝜇𝜎 for each 𝜎 ∈ 𝑆 . So, 𝐼𝑚𝛹 ∗ 𝜐 − 𝐼𝑚𝛹 = 0 for each 𝜎 ∈ 𝑆 , hence, then 𝐼𝑚𝛹 is a homomorphism. In particular, we have 𝑖 𝐼𝑚𝛹 is strongly negative definite. Lemma 4.2. For every positive definite symmetric measure 𝜇 on the product 𝑄1 × 𝑄2 \{𝒆} such that × 𝑄2 . (4.2) ∫𝑄 ×𝑄 \{𝒆}(1 − 𝑅𝑒(𝜒, 𝜏)(𝒓))𝑑 𝜇(𝒓) < ∞, (𝜒, 𝜏) ∈ 𝑄1̂ 1

2

̂ The function 𝛹𝜇 : 𝑄 1 × 𝑄2 → ℂ defined by (4.3)

(1 − 𝑅𝑒(𝜒, 𝜏)(𝒓))𝑑 𝜇(𝒓) < ∞, (𝜒, 𝜏) ∈ 𝑄1̂ × 𝑄2 ,

𝛹𝜇 ≔ ∫𝑄

1 ×𝑄2 \{𝒆}

is strongly negative definite function. Proof. To prove the function 𝛹𝜇 is strongly negative definite, we will sufficiently prove that the measure 𝜇 is strong Lèvy measure for 𝛹𝜇 . For 𝑓 ∈ 𝐶𝐶+ (𝑄1̂ × 𝑄2 ) such that 𝑓(𝜒̅ ) = 𝑓(𝜒) and ∫ 𝑓(𝜒) 𝑑𝑥 = 1, Applying Fubini’s Theorem we get (4.4)

(𝛹𝜇 ∗ 𝑓)(𝜒) = ∫𝑄 ̂ (𝑅𝜌 𝑓)(𝜒)𝛹𝜇 (𝜌)𝑑𝜌 ×𝑄 1

2

=∫𝑄 ̂ 𝑓(𝜌) ∫𝑄 ×𝑄 1

=∫𝑄

[1 − 𝑅𝑒𝜒(𝑟)𝑓̃(𝑟)]𝑑 𝜇(𝑟)

1 ×𝑄2 \{𝒆}

Specially, for 𝜒 = 1, we have ∫

[1 − 𝑅𝑒𝜒(𝑟)𝜌(𝑟)]𝑑 𝜇(𝑟)

1 ×𝑄2 \{(𝑒1 ,𝑒2 )}

2

[1 − 𝑓̃(𝑟)]𝑑 𝜇(𝑟) = ∫ 𝑓(𝜌) 𝛹𝜇 (𝜌)𝑑𝜌

𝑄1 ×𝑄2 \{𝒆}

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Clearly, 𝑑𝜐(𝑟) = [1 − 𝑓̃(𝑟)]𝑑𝜇(𝑟) is positive definite measure on 𝑄1 × 𝑄2 \{𝒆}, so can be considered as positive definite measure on 𝑄1 × 𝑄2. This implies 𝜐̂(𝜒) = 𝑅𝑒𝜐̂(𝜒) = ∫𝑄

1 ×𝑄2 \{𝒆}

𝑅𝑒𝜒(𝑟)[1 − 𝑓̃(𝑟)]𝑑 𝜇(𝑟) for 𝜒 ∈ 𝑄1̂ × 𝑄2 .

Putting 𝑓 = 𝜎 in (4.4) implies that 𝛹𝜇 ∗ 𝜎(𝜒) − 𝛹𝜇 (𝜒) = ∫

𝑅𝑒𝜒(𝑟)[1 − 𝜎̃(𝑟)]𝑑 𝜇(𝑟)

𝑄1 ×𝑄2 \{𝒆}

So, 𝛹𝜇 ∗ 𝜎 − 𝛹𝜇 is the Fourier transform of the measure [1 − 𝜎̃(𝑟)]|𝜇, this implies 𝜇 is the Lèvy measure of 𝛹𝜇 . Theorem 4.3.(Main Result) Let 𝛹: 𝑄1 ̂ × 𝑄2 → ℂ be a strongly negative definite function associated the convolution semigroup (𝜇𝑡 )𝑡>0 with a symmetric positive Lèvy measure 𝜇 such that ∫

× 𝑄2 , (1 − 𝑅𝑒(𝜒, 𝜏)(𝑟))𝑑 𝜇(𝑟) < ∞, (𝜒, 𝜏) ∈ 𝑄1̂

𝑄1 ×𝑄2 \{𝒆}

Then 𝛹 admits the integral representation 𝛹(𝜒, 𝜏) = 𝛹(𝑒) + 𝑖𝐼𝑚𝛹 + 𝑞(𝜒, 𝜏)

+

∫

(1 − 𝑅𝑒(𝜒, 𝜏)(𝑟))𝑑 𝜇(𝑟) < ∞, (𝜒, 𝜏) ∈ 𝑄1̂ × 𝑄2 ,

𝑄1 ×𝑄2 \{𝒆}

where 𝑛 (𝑅(𝜒,𝜏) 𝛹)(𝜒,𝜏)

𝑞(𝜒, 𝜏) = lim [

4𝑛2

𝑛→∞

+

𝑛 (𝑅(𝜒,𝜏) ̅̅̅̅̅̅̅ 𝛹)(𝜒,𝜏)

2𝑛

].

Proof. Regarding Theorem 4.1, the symmetries of the measure 𝜇 implies ℎ = 𝐼𝑚𝛹 is a homomorphism and 𝑖ℎ belongs to the space of strongly negative definite functions on 𝑄1̂ × 𝑄2 . Hence, the function 𝛹 − 𝐶𝐼 belongs to the space of strongly negative definite functions on 𝑄1̂ × 𝑄2 associated Lèvy measure 𝜇 , where 𝐶 = 𝛹(𝒆). This implies the function 𝛹 # = 𝛹 − 𝐶𝐼 − 𝑖ℎ belongs to the space of strongly negative definite functions on 𝑄1̂ × 𝑄2 associated Lèvy measure 𝜇. By virtue of the argument of Theorem 3.2, the integral 𝛹𝜇 ≔

∫

(1 − 𝑅𝑒(𝜒, 𝜏)(𝒓))𝑑 𝜇(𝒓)

𝑄1 ×𝑄2 \{𝒆}

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is finite for all (𝜒, 𝜏) ∈ 𝑄1̂ × 𝑄2 . Observing Lemma 4.2, we get that, the function 𝑞 = 𝛹 # − 𝛹𝜇 is a real valued symmetric function with 𝑞(𝒆) = 0. As remarked in [3], for 𝜎 ∈ 𝑆 we have 𝛹# ∗ 𝜎 − 𝛹# = 𝛹 ∗ 𝜎 − 𝛹

and (4.5)

𝛹𝜇 ∗ 𝜎 − 𝛹𝜇 = ∫𝑄

1 ×𝑄2 \{𝒆}

𝑅𝑒𝜒(𝑟)[1 − 𝜎̃(𝑟)]𝑑 𝜇(𝑟)

Applying (3.1) and (4.5), we get (4.6)

𝑞 ∗ 𝜎 − 𝑞 = (𝛹 # − 𝛹𝜇 ) ∗ 𝜎 − (𝛹 # − 𝛹𝜇 ) = 𝜇̂ 𝜎 ({𝒆}) ≥ 0

As pointed in [2], (4.6) implies that the function q is a nonnegative quadratic form on 𝑄1̂ × 𝑄2 . Recalling the integral 𝛹𝜇 ≔

∫

(1 − 𝑅𝑒(𝜒, 𝜏)(𝒓))𝑑 𝜇(𝒓)

𝑄1 ×𝑄2 \{𝒆}

By Lemma 4.2 the function 𝛹𝜇 is strongly negative definite. Since every quadratic form satisfies the following relation[2] 𝑛 (𝑅(𝜒,𝜏) 𝑞)(𝜒, 𝜏) 1 lim [ ] = 𝑞(𝜒, 𝜏) − (𝑅̅̅̅̅̅̅̅ 𝑞)(𝜒, 𝜏) 𝑛→∞ 4𝑛2 2 (𝜒,𝜏)

So (4.7)

1

𝑞(𝜒, 𝜏) − 2 (𝑅̅̅̅̅̅̅̅ (𝜒,𝜏) 𝑞)(𝜒, 𝜏) 𝑛 𝑛 (𝑅(𝜒,𝜏) 𝛹)(𝜒, 𝜏) (𝑅(𝜒,𝜏) 𝛹𝜇 )(𝜒, 𝜏) = lim [ ] − lim [ ] 𝑛→∞ 𝑛→∞ 4𝑛2 4𝑛2

= lim [ 𝑛→∞

𝑛 (𝑅(𝜒,𝜏) 𝛹)(𝜒, 𝜏)

4𝑛2

1 𝑛→∞ 4𝑛2

] − lim

∫

(1 − 𝑅𝑒((𝜒, 𝜏)(𝒓))2𝑛 )𝑑 𝜇(𝒓)

𝑄1 ×𝑄2 \{𝒆}

Since the product 𝑄1 × 𝑄2 is locally compact, then for every compact K of 𝑄1̂ × 𝑄2 , there exists a constant 𝑀𝐾 ≥ 0, a nieghbourhood 𝑁𝐾 of e and a finite subset 𝑆𝐾 of K such that for every element 𝑟 ∈ 𝑁𝐾 we have supr {1 − 𝑅𝑒(𝜒, 𝜏)(𝒓); (𝜒, 𝜏) ∈ 𝐾} ≤ 𝑀𝐾 supr {1 − 𝑅𝑒(𝜒, 𝜏)(𝒓); (𝜒, 𝜏) ∈ 𝑆𝐾 }. If (𝜒, 𝜏)(𝒓) ≠ 0, let (𝜒, 𝜏)(𝒓) = 𝜌exp(𝑖𝜗) for some 0 < 𝜌 ≤ 1 and −𝜋 ≤ 𝜗 ≤ 𝜋. Then for

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sin(𝑛𝜗)

𝜋

𝑛 ∈ ℕ the ratio 𝑛𝜗 is bounded a way from 𝑄1 × 𝑄2 on [2 , 𝜋], this implies the existence of a positive constant 𝐶 ≥ 0 such that 2

2 1 1 sin(𝑛𝜗) 𝜗 1 − cos(2𝜗) (1 − cos(2𝑛𝜗)) = [ ] [ ] [ ] 4𝑛2 2 𝑛𝜗 sin(𝑛𝜗) 2

≤ 𝐶(1 − cos(2𝜗)) Also, we have

1 − 𝜌2𝑛 1 − 𝜌 1 − 𝜌2 ≤ ≤ 4𝑛2 2𝑛 2

These gives 1 1 𝜌2𝑛 2𝑛 2𝑛 ) (1 (1 − cos(2𝑛𝜗)) (1 − 𝑅𝑒((𝜒, 𝜏)(𝒓)) ) = − 𝜌 + 4𝑛2 4𝑛2 4𝑛2 ≤

1−𝜌2 2

+ 𝐶𝜌2𝑛 (1 − cos(2𝜗))

1 − 𝜌2 ≤ + 𝐶𝜌2 (1 − cos(2𝜗)) 2 ≤

1 − 𝜌2 + 𝐶(1 − 𝑅𝑒((𝜒, 𝜏)(𝒓))2 ) 2

Applying the dominated convergence theorem gives 1 4𝑛2 Substituting in (4.7) gives (4.8)

(1 − 𝑅𝑒((𝜒, 𝜏)(𝒓))2𝑛 )𝑑 𝜇(𝒓) = 0

∫

𝑄1 ×𝑄2 \{𝒆}

1

𝑞(𝜒, 𝜏) = 2 (𝑅(𝜒,𝜏) ̅̅̅̅̅̅̅ 𝑞)(𝜒, 𝜏) + lim [ 𝑛→∞

886

𝑛 (𝑅(𝜒,𝜏) 𝛹)(𝜒,𝜏)

4𝑛2

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Observing that (𝑅̅̅̅̅̅̅̅ (𝜒,𝜏) 𝑞)(𝜒, 𝜏) = lim [ 𝑛→∞

𝑛 (𝑅(𝜒,𝜏) ̅̅̅̅̅̅̅ 𝑞)(𝜒,𝜏)

2𝑛

]

𝑛 (𝑅̅̅̅̅̅̅̅ 1 (𝜒,𝜏) 𝛹)(𝜒, 𝜏) = lim [ ] − lim 𝑛→∞ 𝑛→∞ 2𝑛 2𝑛

∫

(1 − |(𝜒, 𝜏)(𝒓)|2𝑛 )𝑑 𝜇(𝒓)

𝑄1 ×𝑄2 \{𝒆}

But 1 (1 − |(𝜒, 𝜏)(𝒓)|2𝑛 ) ≤ 1 − |(𝜒, 𝜏)(𝒓)|2 2𝑛 Applying the dominated convergence theorem again gives 1 𝑛→∞ 2𝑛 lim

and so

∫

(1 − |(𝜒, 𝜏)(𝒓)|2𝑛 )𝑑 𝜇(𝒓) = 0

𝑄1 ×𝑄2 \{𝒆}

𝑛 (𝑅(𝜒,𝜏) ̅̅̅̅̅̅̅ 𝛹)(𝜒, 𝜏) (𝑅̅̅̅̅̅̅̅ lim [ ] (𝜒,𝜏) 𝑞)(𝜒, 𝜏) = 𝑛→∞ 2𝑛

This complete the proof of the Theorem.

5.

6.

Conclusion

In this paper integral representations for strongly negative definite functions defined on the product hypercomplex systems is given. Harmonic properties for strongly negative definite functions are investigated. We construct a Lèvy measure on the product hypercomplex systems, then we study the conditions that guarantee the existence of some integrations having an integrand parts as a function of the constructed kernel. Finally, we give a Lèvy - Khinchin type formula for strongly negative definite functions defined on the product hypercomplex systems.

Competing Interests “The authors declare that they have no competing interests.”

7.

Acknowledgements I greatly thanks Prof. Dr. Ahmed Zable for his valuable discussion throughout the preparing of this paper. References

[1] Ju. M. Berezanskii and A. A. Kalyuhnyi, Harmonic analysis in hypercomplex systems, Kive, Naukova Dumka, (1992).

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[2] C. Berg and G. Forst, Potential theory on locally compact abelian groups, Springer-Verlag, Berlin-Heidelberg-New York (1975). [3] C. Berg, J.P.R. Christensen and P. Ressel, Harmonic analysis on semigroups.Theory of positive definite and related functions, Graduated texts inMath., 100, Springer-Verlag, BerlinHeidelberg-New-York (1984). [4] G. A. Hunt, Semigroups of measures on Lie groups, Trans. Amer. Math. Soc, 81(1956), 264293. [5] R.I. Jewett, Spaces with an abstract convolution of measures, Adv. in Math., 18 (1975), 1-101. [6] R. Lasser, On the Lèvy Hincin formula for commutative hypergroups, Lecture notes in Math, 1064(1984), 298-308. [7] A. S. Okb El Bab and H. A. Ghany, Harmonic analysis on hypergroups, American Institute of Physics Conf. Proc. 1309(2010), 312.

[8] K.R. Parthasarathy, Probability measures on metric spaces, Academic Press, New YorkLondon, (1967). [9] W. Rudin, Real and complex analysis, McGraw-Hill Book Co, New York, (1974).

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Nonlinear delay fractional difference equations with applications on discrete fractional Lotka–Volterra competition model J. Alzabuta, , T. Abdeljawada , D. Baleanub,c1 a

Department of Mathematics and Physical Sciences, Prince Sultan University P. O. Box 66833, 11586 Riyadh, Saudi Arabia b Department of Mathematics, C ¸ ankaya University 06530 Ankara, Turkey c

Institute of Space Sciences, Magurele–Bucharest, Romania

Abstract. The existence and uniqueness of solutions for nonlinear delay fractional difference equations are investigated in this paper. We prove the main results by employing the theorems of Krasnoselskii’s Fixed Point and Arzela–Ascoli. As an application of the main theorem, we provide an existence result on the discrete fractional Lotka–Volterra model. Keywords. Existence and uniqueness; Fractional difference equations; Krasnoselskii Fixed Point Theorem; Arzela-Ascoli’s Theorem; Discrete fractional Lotka–Volterra model. AMS subject classification: 34A08, 34A12, 39A12.

1

Introduction

Fractional differential equations have received a special attention during the last decades since it has been found that these type of equations provide an excellent instruments for the description of memory and hereditary properties of various materials and processes [1, 2, 3]. The problem of the existence of solutions for fractional differential equations, in particular, has been considered in several recent papers; ( see Refs. [4, 5, 6, 7, 8] and the references therein). For the development of the theory of fractional difference equations, which is the discrete counterpart of fractional differential equations, still there exists less interest among researchers. In fact the progress of the theory of fractional difference equations is still in its early stages. Indeed, some mathematicians have recently taken the lead to develop the qualitative properties of fractional difference equations. We name here for instance Atici et. al. [9, 10, 11, 12, 13] who developed the transform methods, properties of initial value problems and studied applications of these equations on the tumor growth, Abdeljawad et. al. [14, 15, 16, 17, 18] who investigated the properties of Riemann and Caputo’s fractional sum and difference operators, Anastassiou [19, 20] who defined a Caputo like discrete fractional difference and studied some discrete fractional inequalities, Goodrich [21, 22, 23] who established sufficient conditions for the existence of solutions for initial and boundary value problems of discrete fractional equations and Chen et. al. [24, 25, 26] who studied the stability of certain fractional difference equations. In [27, 28], Wu and Baleanu provided some applied results concerning with certain real life problems described by discrete fractional equations. For further details on these achievements, we recommend the reader to consult the new publications [29, 30]. 1

Corresponding Author E-Mail Address: [email protected]

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Obviously, the existence and uniqueness of solutions are essentially significant concept for differential equations. To the best of authors’ knowledge, there are no results concerning with the existence and uniqueness of solutions for nonlinear delay fractional difference equations. The objective of this paper is to cover this gap and study the existence and uniqueness problem for equations of the form  c α ∇0 x(t) = f (t, x(t), x(t − τ )), t ∈ N0 = {0, 1, 2, . . .}, τ ≥ 0, (1) x(t) = φ(t), t ∈ [−τ, −τ + 1, . . . , 0],

where f : N0 × R × R → R and c ∇α0 denotes the Caputo’s fractional difference of order α ∈ (0, 1). To prove our main results, we employ the Krasnoselskii Fixed Point Theorem and the Arzela-Ascoli’s Theorem. As an application of the main theorem, we provide an existence result on the discrete fractional Lotka–Volterra model.

2

Preliminaries

Throughout this paper, we will make use of the following notations, definitions and known results of discrete fractional calculus [29]. For any α, t ∈ R, the α rising function is defined by Γ(t + α) , t ∈ R\{. . . , −2, −1, 0}, 0α = 0, (2) tα = Γ(t) where Γ is the well known Gamma function satisfying Γ(α + 1) = αΓ(α). Definition 1. Let x : N0 → R, ρ(s) = s − 1, α ∈ R+ and µ > −1. Then 1. The nabla difference of x is defined by ∇x(t) = x(t) − x(t − 1), t ∈ N1 = {1, 2, . . .}. 2. The Riemann–Liouville’s sum operator of x of order α > 0 is defined by t

∇−α 0 x(t) =

1 X (t − ρ(s))α−1 x(s), t ∈ N1 . Γ(α)

(3)

s=1

3. The Riemann–Liouville’s difference operator of x of order 0 < α < 1 is defined by c

−(1−α)

∇α0 x(t) = ∇0

∇x(t) =

t X
−α 1 t − ρ(s) ∇x(s), t ∈ N1 . Γ(1 − α) s=1

(4)

4. The power rule is defined by µ ∇−α 0 t =

Γ(µ + 1) (t)α+µ , t ∈ N0 . Γ(µ + α + 1)

(5)

Lemma 1. [40] x(t) denotes a solution of equation (1) if and only if it admits the following representation t

x(t) = φ(0) +

1 X (t − ρ(s))α−1 f (s, x(s), x(s − τ )), t ∈ N0 , Γ(α)

(6)

s=1

and x(t) = φ(t), t ∈ [−τ, −τ + 1, . . . , 0].

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The space l∞ denotes the set of real bounded sequences with respect to the usual supremum norm. We recall that l∞ is a Banach space. Definition 2. A set D of sequences in l∞ is uniformly Cauchy if for every ε > 0, there exists an integer N such that |x(t) − x(s)| < ε whenever t, s > N for any x = {x(n)} in D. The following discrete version of Arzela–Ascoli’s Theorem has a crucial role in the proof of our main theorem. Theorem 1. (Arzela–Ascoli’s Theorem) A bounded, uniformly Cauchy subset D of l∞ is relatively compact. The proof of the main theorem is achieved by employing the following fixed point theorem. Theorem 2. [31] (Krasnoselskii Fixed Point Theorem) Let D be a nonempty, closed, convex and bounded subset of a Banach space (X, kxk). Suppose that A : X → X and B : D → X are two operators such that (i) A is a contraction. (ii) B is continuous and B(D) resides in a compact subset of X, (iii) for any x, y ∈ D, Ax + By ∈ D. Then the operator equation Ax + Bx = x has a solution x ∈ D.

3

Main results

We prove our main results under the following assumptions: (I) f (t, x(t), y(t)) = f1 (t, x(t)) + f2 (t, x(t), y(t)), where fi are Lipschitz functions with Lipschitz constants Lfi , i = 1, 2. (II) |f1 (t, x(t))| ≤ M1 |x(t)|, |f2 (t, x(t), y(t))| ≤ M2 |x(t)| × |y(t)| for any positive numbers M1 and M2 . Let B(N−τ , R) denote the set of all bounded functions (sequences). Define the set D = {x : x ∈ B(N−τ , R), |x| ≤ r, t ∈ N0 }, where r satisfies |φ(0)| +

M1 r + M2 r 2 ≤ r. Γ(α)

Define the operators F1 and F2 by t

F1 x(t) = φ(0) + and

1 X (t − ρ(s))α−1 f1 (s, x(s)), Γ(α) s=1

t

F2 x(t) =

1 X (t − ρ(s))α−1 f2 (s, x(s), x(s − τ )). Γ(α) s=1

It is clear that x(t) is a solution of (1) it it is a fixed point of the operator F x = F1 x+F2 x.

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Theorem 3. Let conditions (I)–(II) hold. Then, equation (1) has a solution in the set
M1 r+M2 r 2 C(α) Lf1 C(α) ≤ r. D provided that Γ(α) < 1 and |φ(0)| + Γ(α) Proof. From the assumptions on the set D, one can easily see that D is a nonempty, closed, convex and bounded set. Step.1: We prove that F1 is contractive. We can easily see that F1 x(t) − F1 y(t) = ≤

t 1 X (t − ρ(s))α−1 f1 (s, x(s)) − f1 (s, y(s)) Γ(α) s=1 t

L f1 X (t − ρ(s))α−1 |x(s) − y(s)| Γ(α) s=1

t

≤

X L f1 kx − yk (t − ρ(s))α−1 . Γ(α)

(7)

s=1

By virtue of (2), (3), (5) and since (t − 0)0 = 1, one can see that t X Γ(t + α) 0 . (t − ρ(s))α−1 (t − 0)0 = Γ(α)∇−α 0 (t − 0) = αΓ(t) s=1

Therefore, (7) becomes

Γ(T1 +α) αΓ(T1 ) is Lf1 C(α) < 1, we Γ(α)

where C(α) = sumption x∈D

F1 x(t) − F1 y(t) ≤ Lf1 C(α) kx − yk, t < T1 , Γ(α)

a positive constant depending on the order α. By the as-

conclude that F1 is contractive. Furthermore, we obtain for

F1 x(t) + F2 x(t) ≤ |φ(0)| +

t 1 X (t − ρ(s))α−1 f1 (s, x(s)) + f2 (s, x(s), x(s − τ )) Γ(α) s=1 t

M1 kxk + M2 kxk2 X (t − ρ(s))α−1 ≤ |φ(0)| + Γ(α) s=1
M1 r + M2 r 2 C(α) ≤ |φ(0)| + , Γ(α)

which implies that F1 x + F2 x ∈ D. For x ∈ D, we also get
t 2 C(α) M r 1 X 2 α−1 (t − ρ(s)) f2 (s, x(s), x(s − τ )) ≤ ≤ r, |F2 x(t)| ≤ Γ(α) Γ(α) s=1

which implies that F2 (D) ⊂ D.

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Step.2: We prove that F2 is continuous. Let a sequence xn converge to x. Taking the norm of F2 xn (t) − F2 x(t), we have F2 xn (t) − F2 x(t) ≤

≤ ≤

t 1 X (t − ρ(s))α−1 f2 (s, xn (s), xn (s − τ )) − f2 (s, x(s), x(s − τ )) Γ(α)

L f2 Γ(α)

s=1 t X s=1

  (t − ρ(s))α−1 |xn (s) − x(s)| − xn (s − τ )) − x(s − τ )


t X 2Lf2 C(α) 2Lf2 α−1 kxn − xk (t − ρ(s)) = kxn − xk. Γ(α) Γ(α) s=1

From the above discussion, we conclude that whenever xn → x, F xn → F x. This proves the continuity of F2 . To prove that F2 (D) resides in a relatively compact subset of l∞ , we let t1 ≤ t2 ≤ H to get F2 x(t2 ) − F2 x(t1 ) ≤ − ≤ +

2 1 X (t2 − ρ(s))α−1 f2 (s, x(s), x(s − τ )) Γ(α) s=1

t

t1 X s=1

(t1 − ρ(s))α−1 f2 (s, x(s), x(s − τ ))

t1 1 X (t2 − ρ(s))α−1 − (t1 − ρ(s))α−1 |f2 (s, x(s), x(s − τ ))| Γ(α) s=1 t2 X 1 (t2 − ρ(s))α−1 |f2 (s, x(s), x(s − τ ))|. Γ(α) s=t +1 1

Upon employing condition (II), we obtain

t1 t1 h X 1 X α−1 F2 x(t2 ) − F2 x(t1 ) ≤ M2 r 2 1 (t2 − ρ(s)) (t1 − ρ(s))α−1 − Γ(α) s=1 Γ(α) s=1

+

t2 i X 1 (t2 − ρ(s))α−1 . Γ(α) s=t +1 1

By using (3), we get i h F2 x(t2 ) − F2 x(t1 ) ≤ M2 r 2 ∇−α (t2 − 0)0 − ∇−α (t1 − 0)0 + ∇−α (t2 − t1 )0 . t1 0 0

From (5), it follows that

F2 x(t2 ) − F2 x(t1 ) ≤

i M2 r 2 h 0 t2 − t01 + (t2 − t1 )0 . Γ(α + 1)

This implies that F2 is bounded and uniformly subset of l∞ . Thus, by virtue of the Discrete Arzela Ascoli’s Theorem 1, we conclude that F2 is relatively compact.

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Step.3: It remains to show that for any x, y ∈ D, we have F1 x(t) + F2 y(t) ∈ D. If x = F1 x(t) + F2 y(t), then we have |x(t)| ≤ F1 x(t) + F2 y(t) ≤ |φ(0)| +

t

t 1 X (t − ρ(s))α−1 f1 (s, x(s)) + f2 (s, y(s), y(s − τ )) Γ(α) s=1

M1 kxk + M2 kxk2 X (t − ρ(s))α−1 Γ(α) s=1
2 M1 r + M2 r C(α) , ≤ |φ(0)| + Γ(α)

≤ |φ(0)| +

which implies that x(t) ∈ D. By employing the Krasnoselskii Fixed Point Theorem, we conclude that there exists x ∈ D such that x = F x = F1 x + F2 x which is a fixed point of F . Hence, equation (1) has at least one solution in D.

4

Applications

The Lotka–Volterra model has been extensively investigated through different approaches [32, 33, 34, 35, 36, 37]. However, all the above mentioned papers studied the integer order Lotka–Volterra model. In spite of the fact that the study of population and medical models of fractional order has been initiated in [12, 38, 39], there is no literature achieved in the direction of discrete fractional Lotka–Volterra model. Therefore, in this section, we employ Theorem 3 to prove an existence and uniqueness result for the solutions of this model. For a bounded sequence g on N, we define g + and g − as follows g + = sup g(t) and g − = inf g(t). t∈N

t∈N


Let f (t, x(t), x(t − τ )) = x(t) γ(t) − β(t)x(t − τ ) in equation (1), then we have the following discrete fractional Lotka–Volterra model:
 c α ∇0 x(t) = x(t) γ(t) − β(t)x(t − τ ) , t ∈ N0 (8) x(t) = φ(t), t ∈ [−τ, −τ + 1, . . . , 0], 0 < α < 1, where the coefficients γ and β satisfy the boundedness relations γ − ≤ γ(t) ≤ γ + , β − ≤ β(t) ≤ β + , which are medically and biologically feasible. Model (8) represents the interspecific competition in single species with τ denotes the maturity time period. Denote f 1 (t, x(t)) = x(t)γ(t),

f 2 (t, x(t), x(t − τ )) = −β(t)x(t)x(t − τ ).

It follows that the functions f 1 and f 2 satisfy the conditions

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(III) |f 1 (t, x(t))| ≤ γ + |x(t)|, |f 2 (t, x(t), x(t − τ ))| ≤ β + |x(t)| × |x(t − τ )|. (IV) f i are Lipschitz functions with Lipschitz constants Lfi , i = 1, 2. The solution of model (8) has the form x(t) = φ(0) +

t   1 X (t − ρ(s))α−1 x(s) γ(s) − β(s)x(s − τ ) , t ∈ N0 , Γ(α)

(9)

s=1

and x(t) = φ(t), t ∈ [−τ, −τ + 1, . . . , 0]. Define a function G by Gx(t) = G1 x(t) + G2 x(t), where t

G1 x(t) = φ(0) +

1 X (t − ρ(s))α−1 x(s)γ(s), Γ(α) s=1

and t

G2 x(t) = −

1 X (t − ρ(s))α−1 x(s)β(s)x(s − τ ). Γ(α) s=1

One can easily employ the same arguments used in the proof of Theorem 3 to complete the proof of the following theorem for equation (8). Theorem 4. Let conditions (III)–(IV) hold. Then, the
model (8) has a solution in the γ + r+β + r 2 C(α) L 1 C(α) < 1 and |φ(0)| + ≤ r. set D provided that fΓ(α) Γ(α) Remark 1. The above result can be extended to n species competitive Lotka–Volterra system of the form 


P ∇α0 xi (t) = xi (t) γi (t) − nj=1 βij (t)xj (t − τij ) , t ∈ N0 , i = 1, 2, . . . , n. xi (t) = φi (t), t ∈ [−τi , −τi + 1, . . . , 0], 0 < α < 1, τi = max1≤j≤n τij ,

(10)

where γ − ≤ γi (t) ≤ γ + , β − ≤ βij (t) ≤ β + . Remark 2. Results of this paper can be carried out for the equation 

∇α0 x(t) = f (t, x(t), x(t − τ )), t ∈ N2 = {2, 3, . . .}, τ ≥ 0, x(t) = φ(t), t ∈ [−τ, −τ + 1, . . . , 1],

(11)

where f : N0 × R × R → R and ∇α0 denotes the Riemann–Liouville’s fractional difference of order α ∈ (0, 1). The solution of equation (11) has the form t

1 X tα−1 (t − ρ(s))α−1 f (s, x(s), x(s − τ )). φ(1) + x(t) = Γ(α) Γ(α)

(12)

s=2

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5

Conclusion

A comprehensive literature survey on the predator–prey type Lotka–Volterra model reveals that a considerable amount of work has already been done by many esteemed researchers during the last century. However the concept of the model related to fractional time derivatives is an original one. The fractional Lotka–Volterra equation is obtained from the classical equations by replacing the first order time derivative by fractional derivative of order α ∈ (0, 1). One of the most significant outcomes of this evolution equation is the generation of fractional Brownian motions. It has been discernible that the discrete analogue of ordinary differential equations has tremendous applications in computational analysis and computer simulations. Motivated by this reality, the study of the discrete analogue of fractional differential equations has become pressing and compulsory. In this paper, we studied the existence and uniqueness of solutions for nonlinear delay fractional difference equations. The main theorem is proved with the help the Krasnoselskii fixed point theorem and the Arzela–Ascoli’s Theorem. Prior to the main result, we set forth some notations and definitions which enriched the knowledge of discrete fractional calculus. To demonstrate the applicability of the main theorem, we provide an existence result for the discrete fractional Lotka–Volterra model. It is to be noted that the analysis carried out in this paper is based on the use of nabla rather than delta operators. Indeed, unlike the delta operator the range of nabla fractional sum and difference operators depends only of the starting point and independent of the order α. This provides exceptional ability to treat skilfully different circumstances throughout the proofs. The delta approach can be obtained from nabla operator through the implementation of the dual identities discussed in [41].

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[35] J. X. Li, J. R. Yan: Permanence and extinction for a non linear diffusive predator–prey system, Nonlinear Anal. 71 (1?2) (2009), 399–417. [36] S. S. Chen, J. P. Shi, J. J. Wei: A note on Hopf bifurcations in a delayed diffusive Lotka–Volterra predator–prey system, Comput. Math. Appl. 62 (5) (2011), 2240–2245. [37] Z. J. Liu, L. S. Chen: Periodic solution of neutral Lotka–volterra system with periodic delays, J. Math. Anal. Appl. 324 (1) (2006), 435-?451. [38] C. N. Angstmann, B. I. Henry, A.V. McGann: A fractional–order infectivity SIR model, Physica A, 452 (15) (2016), 86–93. [39] R. Khoshsiar Ghaziani, J. Alidousti, A. Bayati Eshkaftaki: Stability and dynamics of a fractional order Leslie–Gower prey–predator model, Appl. Math. Model. 40 (3) (2016), 2075–2086. [40] T. Abdeljawad: On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc. Volume 2013 (2013), Article ID 406910, 12 page. [41] T. Abdeljawad: Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ. 2013: 36.

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Some sharp results on NLC-operators in Gp-metric spaces Huaping Huang1 , Ljiljana Gajić2 , Stojan Radenović3 , Guantie Deng1,∗ 1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, PR China 2. Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Serbia 3. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Beograd, Serbia

Abstract: In this paper we generalize, complement and improve some recent results on NLC-operators established in Gp -metric spaces. Several examples are given to support our theoretical approach. Keywords: Gp -metric space, NLC-operator, supporting sequence, Gp -complete, fixed point

1

Introduction and preliminaries

Partial metric space and G-metric space are two different generalized metric spaces. In 1994 Matthews [13] introduced partial metric space as follows: Definition 1.1. Let X be a nonempty set. A partial metric is a mapping p : X 2 → [0, +∞) which satisfies that (p1) x = y ⇔ p (x, x) = p (x, y) = p (y, y) , for all x, y ∈ X; (p2) p (x, x) ≤ p (x, y) , for all x, y ∈ X; (p3) p (x, y) = p (y, x) , for all x, y ∈ X; (p4) p (x, z) ≤ p (x, y) + p (y, z) − p (y, y) , for all x, y, z ∈ X. Then the pair (X, p) is called a partial metric space. It is clear that each (standard) metric space is a partial metric space, while on the contrary it does not hold, in general. In recent years, many authors have obtained lots of fixed point results in partial metric spaces, for example, see [12], [13], [15], [17], [21] and the references therein. On the other hand, in 2006 Mustafa and Sims [14] introduced another kind of generalized metric space, so-called G-metric space as follows: Definition 1.2. Let X be a nonempty set. A mapping G : X 3 → [0, +∞) is called G-metric if it satisfies the following conditions: ∗

Correspondence: [email protected] (G. Deng) 899

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(G1) x = y = z ⇔ G (x, y, z) = 0 for all x, y, z ∈ X; (G2) 0 < G (x, x, y), for all x, y ∈ X with x 6= y; (G3) G (x, x, y) ≤ G (x, y, z), for all x, y, z ∈ X with z 6= y; (G4) G (x, y, z) = G (P {x, y, z}), where P is a permutation of x, y, z ∈ X (symmetry in all three variables); (G5) G (x, y, z) ≤ G (x, a, a) + G (a, y, z) , for all x, y, z, a ∈ X (rectangle inequality). Then the pair (X, G) is called a G-metric space. Based on this notion, many fixed point results under different contractive conditions have been obtained (see [1], [7]-[10], [14], and the references therein). In 2011 Zand and Nezhad [23] introduced a concept as a generalization of both partial metric space and G-metric space as follows: Definition 1.3. Let X be a nonempty set. A mapping Gp : X 3 → [0, +∞) is called a Gp -metric if the following conditions are satisfied: (Gp 1) x = y = z if Gp (x, y, z) = Gp (x, x, x) = Gp (y, y, y) = Gp (z, z, z) for all x, y, z ∈ X; (Gp 2) Gp (x, x, x) ≤ Gp (x, x, y) ≤ Gp (x, y, z) for all x, y, z ∈ X; (Gp 3) Gp (x, y, z) = Gp (P {x, y, z}) , where P is a permutation of x, y, z ∈ X (symmetry in all three variables); (Gp 4) Gp (x, y, z) ≤ Gp (x, a, a)+Gp (a, y, z)−Gp (a, a, a), for all x, y, z, a ∈ X (rectangle inequality). Then the pair (X, Gp ) is called a Gp -metric space. Remark 1.4. It is worth mentioning that authors in [2], [3], [5], [19] and [23] used (Gp 2) while in [6], [18] and [20] authors used the following condition: (Gp 20 ) Gp (x, x, x) ≤ Gp (x, x, y) ≤ Gp (x, y, z) for all x, y, z ∈ X with z 6= y. In the former case (X, Gp ) is a symmetric Gp -metric space, that is., Gp (x, x, y) = Gp (x, y, y) for all x, y ∈ X. However, in the latter case this does not hold. Otherwise, each symmetric G-metric space is symmetric Gp -metric space, but the converse is not true (see Example 1 from [23]) as well as each G-metric space is Gp -metric space in the sense of [18]. However, the claim from [23] (page 87, lines 6-,7-) that each G-metric space is also Gp -metric space is false (see [18], page 79). In addition, It is noteworthy that Example 3 in [23] is symmetric G-metric space, and hence it is Gp -metric space. It is also clear that Definition 6 (because (Gp 2)) in [23] is superfluous. First our important result in this section is the following: Proposition 1.5. Every Gp -metric space (X, Gp ) in the sense of [18] defines a metric
space X, dGp as follows: dGp (x, y) = Gp (x, y, y) + Gp (x, x, y) − Gp (x, x, x) − Gp (y, y, y) , for all x, y ∈ X. Proof. Using (Gp 2), we have dGp (x, y) ≥ 0 for all x, y ∈ X. Also, if x = y, then dGp (x, y) = 0. Conversely, let dGp (x, y) = 0, then Gp (x, y, y) + Gp (x, x, y) − Gp (x, x, x) − Gp (y, y, y) = 0, 900

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that is., [Gp (x, x, y) − Gp (x, x, x)] + [Gp (x, y, y) − Gp (y, y, y)] = 0, or equivalently, Gp (x, x, y) = Gp (x, x, x) and Gp (x, y, y) = Gp (y, y, y) . Further, on account of (Gp 4) it implies that Gp (x, y, y) ≤ 2Gp (x, x, y) − Gp (x, x, x) = Gp (x, x, y). Similarly it follows that Gp (x, x, y) ≤ Gp (x, y, y) for all x, y ∈ X. Then Gp (x, y, x) = Gp (x, x, x) = Gp (y, y, y) , thus by (Gp 1) it gives x = y. It is obvious that dGp (x, y) = dGp (y, x) for all x, y ∈ X. Finally, we shall prove that dGp (x, z) ≤ dGp (x, y) + dGp (y, z) , for all x, y, z ∈ X, or equivalently, Gp (x, x, z) + Gp (x, z, z) − Gp (x, x, x) − Gp (z, z, z) ≤ Gp (x, x, y) + Gp (x, y, y) − Gp (x, x, x) − Gp (y, y, y) + Gp (y, y, z) + Gp (y, z, z) − Gp (y, y, y) − Gp (z, z, z) , that is., Gp (x, x, z) + Gp (x, z, z) ≤ Gp (x, x, y) + Gp (x, y, y) − Gp (y, y, y) + Gp (y, y, z) + Gp (y, z, z) − Gp (y, y, y) . Notice that Gp (x, x, z) = Gp (z, x, x) ≤ Gp (z, y, y) + Gp (y, x, x) − Gp (y, y, y) and Gp (x, z, z) ≤ Gp (x, y, y) + Gp (y, z, z) − Gp (y, y, y) , so the proof is completed. Remark 1.6. Our proof of this proposition is more detailed than one of [23]. Further, we announce the following definition with valid approaches which complements Definition 1.9 from [18]. Definition 1.7. Let (X, Gp ) be a Gp -metric space and {xn } a sequence in X. Then (1) {xn }n∈N is called Gp -convergent to a point x ∈ X if lim Gp (x, xn , xm ) = Gp (x, x, x). n,m→∞

In this case, we write xn → x as n → ∞; (2) {xn } is called a Gp -Cauchy sequence if lim Gp (xn , xm , xm ) = r ∈ R. Particularly, n,m→∞

{xn } is called 0-Cauchy sequence if r = 0; (3) (X, Gp ) is called Gp -complete if for every Gp -Cauchy sequence {xn } in X is Gp convergent to x ∈ X. 901

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Now, we give the following conclusion which corrects Proposition 4 of [23]: Proposition 1.8. Let (X, Gp ) be a symmetric Gp -metric space. Then for a sequence {xn } ⊆ X and a point x ∈ X the following are equivalent: (1) {xn } is Gp -convergent to x; (2) Gp (xn , xn, x) → Gp (x, x, x) as n → ∞; (3) Gp (xn , x, x) → Gp (x, x, x) as n → ∞. Proof. Since (X, Gp ) is symmetric Gp -metric space, then (2) is equivalent to (3). Taking m = n in (1), we speculate that (1) implies (2), thus, (1) implies (3). For the converse we have that Gp (x, xn , xm ) − Gp (x, x, x) = Gp (xn , xm , x) − Gp (x, x, x) ≤ Gp (xn , x, x) + Gp (x, xm , x) − Gp (x, x, x) − Gp (x, x, x) = [Gp (xn , x, x) − Gp (x, x, x)] + [Gp (xm , x, x) − Gp (x, x, x)] → 0 + 0 = 0, as n, m → ∞, then (3) implies (1). We complete the proof. Next we generalize Lemma 1.10 from [2] (see also [3], [5], [6], [18], [20]), that is., we announce the following assertion: Proposition 1.9. Let (X, Gp ) be a Gp -metric space in the sense of [18]. Then (A) if Gp (x, y, z) = 0, then x = y = z; (B) if x 6= y, then Gp (x, y, y) > 0. Proof. (A) If x 6= y 6= z 6= x, then (A) is an immediate consequence of (Gp 20 ) and (Gp 1). If for instance, x 6= y = z, then Gp (x, y, z) = Gp (x, y, y) = 0. In this case, we get Gp (x, x, x) = Gp (x, x, y) = Gp (y, y, y) = 0. Indeed, by (Gp 4) it follows that Gp (x, x, y) ≤ Gp (x, y, y) + Gp (y, x, y) − Gp (y, y, y) ≤ 2Gp (x, y, y) = 0. Since Gp (x, x, x) ≤ Gp (x, x, y) and Gp (y, y, y) ≤ Gp (x, y, y) hold for all x, y ∈ X, then we arrive at Gp (x, y, y) = Gp (x, x, y) = Gp (x, x, x) = Gp (y, y, y) = 0, so by (Gp 1), we obtain the desired result. (B) Let Gp (x, y, y) = 0. Now, based on the proof of (A) when x 6= y = z, we claim that x = y. A contradiction.

2

Auxiliary results In the sequel, let (X, Gp ) be a Gp -metric space in the sense of [18]. First of all, we

introduce the following notion: Definition 2.1. Let (X, Gp ) be a Gp -metric space, α ∈ (0, 1) a constant and T : X → X a mapping. We say that T is an NLC-operator on X if for each x ∈ X there is some 902

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n (x) ∈ N such that for each y ∈ X it holds
Gp T n(x) x, T n(x) x, T n(x) y ≤ max {αGp (x, x, y) , Gp (x, x, x)} .

(2.1)

For an NLC-operator T and x ∈ X we define supporting sequence at x as a sequence {sk }k∈N∪{0} where s0 = 0 and sk+1 = sk + n (T sk x) , k ∈ N∪ {0}. Also set JT (X) = {x ∈ X : T m x = T m+1 x for some m ∈ N} . Remark 2.2. (i) Condition (2.1) implies that for any i ≥ sk , it is valid that

 Gp T sk x, T sk x, T i x ≤ max αGp T sk−1 x, T sk−1 x, T j x , Gp (T sk−1 x, T sk−1 x, T sk−1 x) , (2.2) where j = i − sk + sk−1 ≥ sk−1 , and specially that Gp (T sk x, T sk x, T sk x) ≤ Gp (T sk−1 x, T sk−1 x, T sk−1 x) .

(2.3)

Now, fix x ∈ X\JT (X) . For k ∈ N and i ≥ sk use (2.2), repeatedly fix integers lj ≥ sj , 0 ≤ j < k and t1 , t2 , ..., tk ∈ {0, 1} such that lk := i, then

Gp T sj x, T sj x, T lj x ≤ αtj · Gp T sj−1 x, T sj−1 x, T lj−1 x for all 0 ≤ j ≤ k, where  tj =

1, 0,

if sj−1 < lj−1 , if sj−1 = lj−1 .

Let us recall (l0 , l1 , ..., lk−1 ) and (t1 , t2 , ..., tk ) as the (k, l)-descent and (k, i)-signature at x, respectively. Further put rk,i =: k − hk,i , where hk,i is a number of zeroes in (k, i)-signature at x. We shall say that x is Type 1 if there are sequences of positive integers {km }m∈N∪{0} and {im }m∈N∪{0} , one of them is strictly increasing such that for all m ∈ N∪ {0} we have that im ≥ sm and rkm ,im < rkm+1 ,im+1 . We shall say that x is Type 2 if x is not Type 1, i.e., there are k0 , B ∈ N such that for all k ≥ k0 and i ≥ sk it holds rk,i < B. (ii) In the framework of G-metric spaces, condition (2.1) becomes
Gp T n(x) x, T n(x) x, T n(x) y ≤ αGp (x, x, y) ,

(2.3’)

hence, it is iterate contractive condition of Sehgal-Guseman type in this framework (see [11], [16]). Lemma 2.3. Let T be an NLC-operator on Gp -metric space (X, Gp ) , x ∈ / JT (X), and let {sk }k∈N∪{0} be a supporting sequence at x. Then (a) if (l0 , l1 , ..., lk−1 ) is (k, i0 )-descent at x, then

Gp T sk x, T sk x, T i0 x ≤ αrk,i0 · Gp x, x, T l0 x ,

Gp T sk x, T sk x, T i0 x ≤ Gp T sj x, T sj x, T lj x for all 0 ≤ j ≤ k, where lk := i0 ; 903

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(b) if P ⊆ {0, 1, ..., k − 1} and rk,i0 < cardP (cardP is the number of elements of P ), then for some j0 ∈ P it holds
Gp T sk x, T sk x, T i0 x ≤ Gp (T sj0 x, T sj0 x, T sj0 x) . Proof. Using the definition of rk,i , (a) is obvious. To prove (b), under the hypothesis, the set {j + 1 : j ∈ P } is subset of {1, 2, ..., k} with card(P ) > rk,i0 , so there is some j0 ∈ P with tj0 +1 = 0. Then     lj +1 sj +1 Gp T sj0 +1 x, T 0 x, T i0 x ≤ Gp T lj0 +1 x, T 0 x, T i0 +1 x
≤ αtj0 +1 Gp T sj0 x, T sj0 x, T lj0 x = Gp (T sj0 x, T sj0 x, T sj0 x) , whereof (a) and sj0 = lj0 have been used. Lemma 2.4. Let T be an NLC-operator on Gp -metric space (X, Gp ) and x ∈ X, then there is some Mx > 0 such that for all i ≥ 0 it satisfies that
Gp x, x, T i x ≤ Mx ,

(2.4)

and so Gp (T j x, T j x, T i x) ≤ 3Mx , for each i, j ∈ N∪ {0}. Proof. If x ∈ JT (X), then this is obvious. Thus, let x ∈ / JT (X) and set
b (x) = Gp (x, x, x) + Gp (x, x, T x) + · · · + Gp x, x, T n(x) x . Let us prove by induction that
Gp x, x, T i x ≤

1 b (x) , for all i ∈ N. 1−α

Obviously (2.4) is true for 0 ≤ k ≤ n (x). Now assume that the same is valid for some k ≥ n (x). Then


Gp x, x, T k+1 x ≤ Gp x, x, T n(x) x + Gp T n(x) x, T n(x) x, T k+1 x

≤ Gp x, x, T n(x) x + max αGp x, x, T k+1−n(x) x , Gp (x, x, x)
α ≤ Gp x, x, T n(x) x + Gp (x, x, x) + b (x) 1−α α ≤ b (x) + b (x) 1−α 1 = b (x) , 1−α so (2.4) is proved with Mx =

1 b (x) . 1−α

Further, we have


Gp T i x, T j x, T j x ≤ Gp T i x, x, x + 2Gp T j x, x, x ≤ 3Mx , for all i, j ∈ N∪ {0} . 904

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Lemma 2.5. Let T be an NLC-operator on Gp -metric space (X, Gp ) and x ∈ X\JT (X) . If x is Type 1, then limi,j→∞ Gp (T i x, T i x, T j x) = 0. Proof. Fix m ∈ N∪ {0}. If (l0 , ..., lkm −1 ) is (skm , im )-descent, then by (a) of Lemma 2.3 we have

Gp T skm x, T skm x, T im x ≤ αrkm ,im · Gp x, x, T l0 x ≤ αrkm ,im Mx . In view of limm→∞ rkm ,im = ∞, it follows that
lim Gp T skm x, T skm x, T im x = lim Gp (T skm x, T skm x, T skm x) = 0. m→∞

m→∞

For given ε > 0, choose m0 ∈ N such that αm0 Mx < ε and Gp (T skm x, T skm x, T skm x) < ε for all m ≥ m0 . Let rk2m0 ,i ≥ m0 . Then Gp T

sk2m

0

x, T

sk2m

0


r x, T i x ≤ α k2m0 ,i · Mx ≤ αm0 Mx < ε.

Now suppose that rk2m0 ,i < m0 . For Pi = {km0 , ..., k2m0 −1 } ⊆ {0, 1, ..., k2m0 − 1}, we have card (P ) > rk2m0 ,i , so by Lemma 2.3, there exists some m0 ≤ j ≤ 2m0 − 1 such that Gp T

sk2m

0

x, T

sk2m

0


x, T i x ≤ Gp (T skj x, T skj x, T skj x, ) < ε

for each i ≥ sk2m0 . Accordingly, if i, j ≥ sk2m0 , then


s s s Gp T i x, T i x, T j x ≤ Gp T k2m0 x, T k2m0 x, T j x + Gp T i x, T i x, T k2m0

s s s s ≤ Gp T k2m0 x, T k2m0 x, T j x + 2Gp T i x, T k2m0 x, T k2m0 x < 3ε. Therefore, we prove that limi,j→∞ Gp (T i x, T i x, T j x) = 0. Lemma 2.6. Let T be an NLC-operator and x ∈ X\JT (X). If x is Type 2, then the sequence {T i x}i∈N∪{0} is Gp -Cauchy. Proof. By (2.3), it is easy to see that {Gp (T sk x, T sk x, T sk x)}k∈N∪{0} is a nonincreasing sequence, where {sk }k∈N∪{0} is a supporting sequence at x. Then there exists rx := lim Gp (T sk x, T sk x, T sk x) = inf {Gp (T sk x, T sk x, T sk x)} k

k

such that it is finite. At first let us prove that for any ε > 0, there exists m0 ∈ N such that for all m ≥ m0 and i ≥ sm , one has
Gp T sm x, T sm x, T i x ∈ (rx − ε, rx + ε) .

(2.5)

Since x is Type 2 there are k0 , B ∈ N such that for all k ≥ k0 and all i ≥ sk , there holds rk,i < B. Let ε > 0, take m1 ≥ k0 such that for all m ≥ m1 , Gp (T sm x, T sm x, T sm x) ∈ (rx − ε, rx + ε) .

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Let m ≥ m1 +B and i ≥ sm be arbitrary. For P = {m1 , ..., m1 + B − 1} ⊆ {0, 1, ..., m − 1}, we have cardP ≥ B > rm,i , then there exists m1 ≤ j ≤ m1 + B − 1 such that
rx − ε < Gp (T sm x, T sm x, T sm x) ≤ Gp T sm x, T sm x, T i x ≤ Gp (T sj x, T sj x, T sj x) < rx + ε. So we get (2.5). Now let us prove that for any ε > 0, there is k ∗ ∈ N such that for all i, j ≥ k ∗ ,
Gp T i x, T i x, T j x < rx + ε.

(2.7)

Indeed, for any ε > 0, consider m0 as in (2.5) and let i, j ≥ sm0 be arbitrary. Then


Gp T i x, T i x, T j x ≤ Gp T sm0 x, T sm0 x, T j x + Gp T i x, T i x, T sm0 x − Gp (T sm0 x, T sm0 x, T sm0 x)
< rx + ε + 2Gp T sm0 x, T sm0 x, T i x − 2Gp (T sm0 x, T sm0 x, T sm0 x) < rx + ε + 2 (rx + ε) − 2 (rx − ε) = rx + 5ε. To prove limi,j→∞ Gp (T i x, T i x, T j x) = rx , we only need to show that for any ε > 0, there exists e k ∈ N such that for all i ≥ e k, ones always have
rx − ε < Gp T i x, T i x, T j x .

(2.8)

Suppose on the contrary, that for any k there is some i0 ≥ k satisfying
Gp T i0 x, T i0 x, T i0 x ≤ rx − ε. Put z := T i0 x. Obviously, x ∈ / JT (X) implies that z ∈ / JT (X). If z is Type 1, then by Lemma 2.5 it follows that

0 = lim Gp T i z, T i z, T j z = lim Gp T i x, T i x, T j x = rx , i,j

i,j

so {T i x}i∈N∪{0} is 0-Cauchy sequence. Now suppose that z is Type 2, and let {qm }m∈N∪{0} be a supporting sequence at z. Then, for each m ∈ N∪ {0}, Gp (T qm z, T qm z, T qm z) ≤ Gp (z, z, z) ≤ rx − ε, so rz = lim Gp (T qm z, T qm z, T qm z) ≤ rx − ε. m

Note that rz
2 which is impossible, so (2.8) is satisfied. Now, for sm − i0 ≥ m − i0 ≥ j0 , we claim that
rx + rz rx + rz < Gp (T sm x, T sm x, T sm x) = Gp T sm −i0 x, T sm −i0 x, T sm −i0 x < . 2 2 In the end, from

rx − ε < Gp T i x, T i x, T i x ≤ Gp T i x, T i x, T j x , it follows that {T i x}i∈N∪{0} is a Gp -Cauchy sequence. Lemma 2.7. Let T : X → X be an operator on Gp -metric space (X, Gp ). Suppose that x ∈ X is a point such that T k x = x holds for some positive integer k, and there is y ∈ X such that

Gp (y, y, y) = lim Gp y, T i x, T i x = lim Gp T i x, T i x, T j x , (2.9) i

i,j

then T x = x. Proof. Since T k·i x = x, then for any i ∈ N∪ {0}, we have that
Gp (y, y, y) = lim Gp y, T ki x, T ki x = Gp (y, x, x) i

and
Gp (y, y, y) = lim Gp T ki x, T ki x, T ki x = Gp (x, x, x) , i

so y = x. Now (2.9) implies that
Gp (x, x, x) = lim Gp x, T ki+1 x, T ki+1 x = Gp (x, T x, T x) i

and
Gp (x, x, x) = lim Gp T ki+1 x, T ki+1 x, T ki+1 x = Gp (T x, T x, T x) . i

Thus, T x = x.

3

Main results

Both results in this section generalize many existing results in the literature (see [12, Theorem 3.1] and [4, Lemmas 3.-5, Theorems 1 and 2]). Firstly, we announce our first result for NLC-operator in Gp -complete Gp -metric space as follows. Proposition 3.1. Let T be an NLC-operator on Gp -complete Gp -metric space (X, Gp ), then (1) for each x ∈ X, the sequence {T i x}i∈N∪{0} Gp -converges to some vx ∈ X; (2) for all x, y ∈ X, one has Gp (vy , vy , vx ) = max {Gp (vx , vx , vx ) , Gp (vy , vy , vy )} . 907

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Proof. Since (X, Gp ) is Gp -complete, then for each x ∈ X, the existence of vx is assured by Lemma 2.5 and Lemma 2.6. Let us prove (2). Let x, y ∈ X and Gp (vy , vy , vy ) ≥ Gp (vx , vx , vx ). If Gp (vy , vy , vy ) = 0, then vx = vy and the claim is clear. Thus, assume that Gp (vy , vy , vy ) > 0. 1−α For any 0 < ε < 2(1+α) Gp (vy , vy , vy ), there is some m0 ∈ N such that for all i, j ≥ m0 , we have max{Gp (T i y, vy , vy ) − Gp (T i y, T i y, T i y), |Gp (T i y, T i y, T i y) − Gp (vy , vy , vy )|, Gp (T i y, T i y, vy ) − Gp (vy , vy , vy )} < ε and 


max Gp vx , vx , T j x − Gp (vx , vx , vx ) , Gp vx , T j x, T j x − Gp T j x, T j x, T j x < ε. For i, j ≥ m0 , we have

Gp T i y, T i y, T i x ≤ Gp T i y, T i y, vy − Gp (vy , vy , vy )
+ Gp (vy , vy , vx ) + Gp vx , vx , T i x − Gp (vx , vx , vx ) < 2ε + Gp (vy , vy , vx ) and


Gp (vy , vy , vx ) ≤ Gp vy , vy , T i y − Gp T i y, T i y, T i y + Gp vx , T j x, T j x

− Gp T j y, T j y, T j y + Gp T i y, T i y, T j y
< 2ε + Gp T i y, T i y, T j x . For any i0 ≥ m0 and i1 := n (T i0 y), we get
Gp (vy , vy , vx ) − 2ε ≤ Gp T i0 +i1 y, T i0 +i1 y, T i0 +i1 x

≤ max αGp T i0 y, T i0 y, T i0 x , Gp T i0 y, T i0 y, T i0 y < max {α (2ε + Gp (vy , vy , vx ) , ε + Gp (vy , vy , vy ))} . If Gp (vy , vy , vx ) − 2ε < 2αε + αGp (vy , vy , vx ) , then Gp (vy , vy , vx ) < 2ε (1 + α) < Gp (vy , vy , vx ) . This is a contradiction. As a consequence, we deduce that Gp (vy , vy , vx ) < 3ε + Gp (vy , vy , vy ) , so Gp (vy , vy , vx ) ≤ Gp (vy , vy , vy ) . 908

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Finally, by (Gp 2), we speculate that Gp (vy , vy , vx ) = Gp (vy , vy , vx ) = max {Gp (vx , vx , vx ) , Gp (vy , vy , vy )} .

Now, we announce our second result in the framework of Gp -complete Gp -metric spaces. Theorem 3.2. Let T be an NLC-opertor on Gp -complete Gp -metric space (X, Gp ), then there is a fixed point z ∈ X of T such that Gp (z, z, z) = inf {Gp (vx , vx , vx ) : x ∈ X}. Proof. For x ∈ X, put rx := Gp (vx , vx , vx ) = limk→∞ Gp (T sk x, T sk x, T sk x) for {sk }k∈N∪{0} which is the supporting sequence at x. Let I := inf {rx : x ∈ X}. For m ≥ 1, take xm ∈ X such that for all i, j ∈ N∪ {0}, it holds Gp T xim , T xim , T xjm




  1 1 ∈ I − ,I + . m m

(3.1)

At first we shall prove that limm,k→∞ Gp (xm , xm , xk ) = I. For m, k ≥ 2, let Cm,k > 0 and
Gp T j xm , T j xm , T i xk < Cm,k , i, j ∈ N∪ {0} . Fix m, k ≥ 2 and let {sq }q∈N∪{0} be the supporting sequence at xm and let l ≥ 1 be an integer such that αl · Cm,k
Gp T i xm , T i xm , T i xm for all i ∈ {0, 1, ..., sl } . Am,k : = Gp (xm , xm , T sl xm ) − Gp (T sl xm , T sl xm , T sl xm )
1, β ≥ 0, γ ≥ 0, was obtained in [31]. On the other hand, as pointed in [2], Levine’s method as presented in [30, 31] cannot be used here due to the term −∆ut + (|u|k u)t , so a different differential inequality was used to prove Theorem 3 in [3] for negative initial energy case (i.e. in the case of (1.5)) φφ00 − α(φ0 )2 + βφ2 + γφ2+q1 ≥ 0, α > 1, β ≥ 0, γ ≥ 0, q1 ≥ 0. But the above inequality cannot be used to the positive initial energy case. In this paper, we will establish a new differential inequality and prove a finite-time blow-up result under arbitrary positive initial energy. This article is organized as follows. In Section 2, we are concerned with some notations and state our main results. Following the potential well theory introduced by [28], we get global existence in Section 3. Section 3 gives also an asymptotic stability results of the problem (1.1)-(1.3). In section 4, it is shown that the weak solution of the problem (1.1) -(1.3) blow-up in the case of positive initial energy E(0) > 0 and q > k.

2

Preliminaries

In this section we present some notations and state our main results. We use the standard Lebesgue space Lp (Ω)(1 ≤ p ≤ ∞) and Sobolev space H01 (Ω). We denote by ||u||p the Lp (Ω) 3

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norm and by ||∇ · || the norm in H01 (Ω). Moreover, for later use we denote by (·, ·) the duality paining between H01 (Ω) and H −1 (Ω). In this paper, we will always assume that 0 < q < ∞, if n = 1, 2;

0 0, J(u) < d} and one can easily see that the stable set can also be defined by W = {(λ, E) ∈ [0, +∞) × R : 0 < g(λ) ≤ E < d, 0 < λ < λ0 }, − q+2

q+2

where g(λ) = 12 λ2 − C∗q+2 λq+2 , λ0 = C∗ q is the absolute maximum point of g, and finally 1 d = g(λ0 ) = ( 12 − q+2 )λ20 > 0. In order to get the energy decay of the solution, we introduce the following set W1 = {u ∈ H01 |I(u) > 0, J(u) < E1 }, −1

1 where λ1 = ((q + 2)C∗q+2 ) q , E1 = ( 21 − q+2 )λ21 . Obviously, d > E1 and W1 ⊂ W. Our main results read as follows. The first result is concerned with the global existence of weak solutions to the problem (1.1)-(1.3). Namely, we have the following theorem. 2n , if n ≥ 3; u0 , u1 ∈ H01 (Ω), assuming that Theorem 2.1 Let k > 1, if n = 1, 2; k < n−2 k > q, E(0) < d and u0 ∈ W , then the problem (1.1)-(1.3) admits a global weak solution u and u(·) ∈ W for t ≥ 0. The second result is about the asymptotic stability results of the weak solutions. Theorem 2.2 Under the assumptions of Theorem 2.1, k > q and u0 ∈ W1 , and ||∇u0 || < λ1 , E(0) < E1 , then there exist positive constant α such that the energy E(t) satisfies the energy estimates E(t) ≤ E(0) exp{−α[t − 1]+ } f or large t,

where [t − 1]+ = max{t − 1, 0}. Remark 2.2 Let us mention that the special polynomial form of the dissipation and source terms in (1.1)-(1.3) is not essential. The results can be extend to the case of more general nonlinearities under suitable assumptions. Our final result provides a finite time blowup property of the weak solutions to problem (1.1)-(1.3). Theorem 2.3 Assume that k < q. If the initial data are such that E(0) > 0,

(2.7)

(∇u0 , ∇u1 ) + (u0 , u1 ) > 0,

(2.8)

and (φ0 (0))2 −

β0 2 α0 −1 φ (0)

+

0 2γ0 φ2α0 +δ1 (1−α0 ) (0) (α0 −1)δ10

0

0 + (α02δ φ2α0 +δ2 (1−α0 ) (0) = B0 > 0, −1)δ 0

(2.9)

2

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2(k+1)2 C

2(k+1)

2 0 where α0 = 2 + k2 , β0 = q−k , γ0 = , δ0 = (q + 2)E(0), δ10 = q−k and C0 is embedding constant from H01 to L2(k+1) , then there exists

T∞ ≤ φ1−α0 (0)A−1 such that

2α0 −2−k α0 −1 ,

δ20 =

2α0 −1 α0 −1

lim φ(t) = ∞,

− t→T∞

where A2 = (α0 − 1)2 φ−2α0 (0)B0 and φ(t) = 21 ||∇u||2 + 12 ||u||2 . Finally, we state the local existence result of problem (1.1)-(1.3). Theorem2.4 Let u0 , u1 ∈ H01 (Ω), then problem (1.1)-(1.3) has a unique weak solution on [0, T0 ) for some T0 > 0, and we have either T0 = +∞ or T0 < +∞ and lim sup[||ut ||2H 1 (Ω) + ||u||2H 1 (Ω) ] = +∞.

t→T0 +

0

0

This lemma can be established by combining the arguments of Theorem 8.1 (or Theorem 6.8) and Example 9.5 in [3], Theorem 2 in [2] and [35], so we omit it.

3

Global existence and asymptotic stability of the solutions

In this section we study the existence and asymptotic stability of global solutions for problem (1.1)-(1.3). We start by the following lemma. Lemma 3.1 Suppose that u is the solution of problem (1.1)-(1.3), and u0 , u1 ∈ H01 , if u0 ∈ W and E(0) < d, then u(t) remains inside the set ∈ W for any t ≥ 0. The proof is similar to that of Lemma 2.2 in [33], so we omit it. Proof of Theorem 2.1. By Lemma 3.1, we have u(t) ∈ W for all t ∈ [0, T0 ), then I(u) > 0, J(u) < d for all t ∈ [0, T0 ). Therefore, 1 1 1 1 1 2 )||um ||q+2 ||u||q+2 ( − q+2 = ||∇u|| − q+2 − I(u) ≤ J(u) < d, 2 q+2 2 q+2 2

(3.1)

||u||q+2 q+2 < d.

(3.2)

then

By the energy equation (2.6), definition of J(u) and (3.1), we arrive 1 1 1 1 ||ut ||2 + ||∇ut ||2 + ||∇u||2 ≤ E(0) + ||u||q+2 q+2 ≤ Cd, 2 2 2 q+2

f or 0 ≤ t < T0 ,

(3.3)

It follows from (3.3) and from a standard continuous argument that the local solution u furnished by Theorem 2.4 can be extended to the whole interval [0, +∞), that is, u is a global solution. Finally from Lemma 3.1 we get u ∈ W for t ∈ [0, ∞). In order to get the energy decay of the solution, we prepare the following lemma. Lemma 3.2[29] Let ϕ(x) be a nonnegative and non-increasing function defined on [0, ∞), satisfying ϕ1+r (t) ≤ k0 (ϕ(t) − ϕ(t + 1)), t ∈ [0, T ], 6

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for k0 > 1, and r ≥ 0. Then we have, for each t ∈ [0, T ], ϕ(t) ≤ ϕ(0) exp(−α[t − 1]+ ),

if r = 0, − r1

ϕ(t) ≤ (ϕ(0)−r + k0 r[t − 1]+ )

,

if r > 0,

0 where [t − 1]+ = max{t − 1, 0}, and α = ln( k0k−1 ). Adapting the idea of Vitillaro[34], we have the following lemma. Lemma 3.3 Suppose that u is the solution of problem (1.1)-(1.3), and u0 , u1 ∈ H01 , if u0 ∈ W1 and ||∇u0 || < λ1 , E(0) < E1 , then u(t) remains inside the set W1 and ||∇u|| < λ1 , E(t) < E1 for any t ≥ 0. Lemma 3.4 Under the condition of Theorem 2.2. and q > 0, then, for t ≥ 0,

||∇u||2 ≥ 2||u||q+2 q+2 . E(t) ≥

(3.4)

q+1 q+1 ||∇u||2 ≥ ||u||q+2 q+2 , 2(q + 2) q+2

(3.5)

Proof By the definition E(t) and embedding theorem, we have E(t) ≥ 21 ||∇u||2 − ≥ 12 ||∇u||2 −

q+2 q+2 1 1 2 q+2 ||u||q+2 ≥ 2 ||∇u|| − ||u||q+2 C∗q+2 ||∇u||q+2 = g1 (||∇u||).

(3.6) −1

where g1 (λ) = 12 λ2 − C∗q+2 λq+2 . Note that g1 (λ) has the maximum at λ1 = ((q + 2)C∗q+2 ) q and the maximum value g1 (λ1 ) = E1 . We see that g1 (λ) is increasing in (0, λ1 ), decreasing in (λ1 , +∞) and g1 (λ) → −∞ as λ → ∞. Since ||∇u0 ||2 < λ1 , E(0) < E1 ,then ||∇u||2 < λ1 , for any t ≥ 0, so g1 (||∇u||) ≥ 0. By (3.6), we have q+2 1 1 2 2 ||∇u||2 − ||u||q+2 q+2 = 2 ||∇u|| + ( 2 ||∇u|| − ||u||q+2 )

≥ 12 ||∇u||2 + g1 (||∇u||), then (3.4) holds since g1 (||∇u||) > 0. Furthermore, we have q+1 1 1 E(t) ≥ ||∇u||2 − ||u||q+2 ||∇u||2 . q+2 ≥ 2 q+2 2(q + 2) So (3.5) hold. Proof of Theorem 2.2 From (2.5), we know that and E(t) is nonincreasing. Setting p F (t) = E(t) − E(t + 1), then we have 2

Z

t+1

Z

2

k

2

[||∇us || + (k + 1) |u| |us | ]ds ≥ t Ω Z t+1 Z F 2 (t) ≥ (k + 1) |u|k |us |2 ds. F (t) =

t

Z

t+1

||∇us ||2 ds,

(3.7)

t

(3.8)

Ω

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Applying the mean value theorem in (3.7), there exists t1 ∈ [t, t + 14 ] and t2 ∈ [t + 34 , t + 1] such that ||∇ut (ti )||2 ≤ 2F 2 (t),

i = 1, 2.

(3.9)

Multiplying u in (1.1), intergrating over [t1 , t2 ] × Ω, we have Z

t2

t1

[||∇u||2 − ||u||q+2 q+2 ]dt Z

2 ut u|t=t t=t1 dx −

=− Z

Ω t2 Z

Ω

2 ∇ut ∇u|t=t t=t1 dx +

Z

t2

Z

t1

t1

Ω

5 X

Z

t2

(||ut ||2 + ||∇ut ||2 )dt

t1

(3.10)

|u|k uut dxdt

∇u∇ut dxdt − (k + 1)

− =

Z

Ω

Mi .

i=1

Now we estimate the terms of the right-hand side of (3.10). By H¨older inequality, Poincar´e inequality, (3.9), Lemma 3.4, the fact that the E(t) is non-increasing, and Young inequality with ε > 0, we have Z |M1 | = | − Ω

≤

2 X

2 ut u|t=t t=t1 dx| ≤

2 X

||ut (ti )||||u(ti )||

i=1 1

C2 ||ut (ti )||||∇u(ti )|| ≤ C3 E 2 (t)F (t)

i=1

≤ C1 (ε)F 2 (t) + εE(t), Z 2 ∇ut ∇u|t=t |M2 | = | − t=t1 dx|

(3.11)

Ω

≤

2 X

1

||∇ut (ti )||||∇u(ti )|| ≤ C4 E 2 (t)F (t)

i=1

≤ C2 (ε)F 2 (t) + εE(t).

(3.12)

By Poincar´e inequality and (3.7), we have Z

t2

|M3 | = |

2

Z

2

(||ut || + ||∇ut || )dt| ≤ C5 t1

t+1

||∇us ||2 ds ≤ C5 F 2 (t).

(3.13)

t

From Lemma 3.4 and the fact that the E(t) is non-increasing, we arrive |M4 | = |

R t2 t1

∇u∇us ds| ≤

R t+1 t

2 [C3 (ε)||∇us ||2 + ε 2(q+2) q+1 ||∇u|| ]ds

≤ C3 (ε)F 2 (t) + εE(t).

(3.14)

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According to H¨older inequality, embedding theorem, the assumption k > q and the Lemma3.2, the fact that the E(t) is non-increasing and Young inequality, we have Z t2 Z |u|k+1 |ut |dxdt |M5 | ≤ (k + 1) t1 t2

Ω

Z

Z

k

|u| 2 |ut ||u|

= (k + 1) t1

k+2 2

dxdt

Ω t2

Z

Z

≤ (k + 1)( t1 t2

Z ≤ C7 F (t)(

t1 t2

Z ≤ C9 F (t)(

Z |u| |ut | dxdt) ( k

2

1 2

Ω

t2

t1

Z

1

|u|k+2 dxdt) 2

Ω

1

2 ||u||k+2 k+2 dt)

1

||∇u||k ||∇u||2 dt) 2

t1

Z

t2

≤ C10 F (t)(

1

E(t)dt) 2 t1

2

≤ C4 (ε)F (t) + εE(t).

(3.15)

Substituting (3.11)-(3.15) into (3.10), we get the estimate R t2 q+2 2 2 t1 [||∇u|| − ||u||q+2 ]dt ≤ C5 (ε)F (t) + 4εE(t).

(3.16)

On the other hand, it follows from the definition of E(t) and (3.5) in Lemma 3.4 that 1 E(t) = (||∇u||2 − ||u||q+2 q+2 ) + 2 1 ≤ (||∇u||2 − ||u||q+2 q+2 ) + 2

q 1 1 2 2 ||u||q+2 q+2 + ||ut || + ||∇ut || 2(q + 2) 2 2 1 1 q ||ut ||2 + ||∇ut ||2 + E(t). 2 2 2(q + 1)

(3.17)

Then we have 1 1 1 2 2 ≤ (||∇u||2 − ||u||q+2 q+2 ) + ||ut || + ||∇ut || . 2 2 2 Therefore, by (3.18), (3.16) and (3.13), we arrive that R t2 2(q+1) R t2 q+2 q+2 R t2 2 2 2 t1 (||∇u|| − ||u||q+2 )ds + 2(q+1) t1 (||ut || + ||∇ut || )ds t1 E(s)ds ≤ q+2 q+2 2(q+1) E(t)

≤ C6 (ε)F 2 (t) + 4ε 2(q+1) q+2 E(t). Since E(t) is non-increasing, we can choose t3 ∈ [t1 , t2 ] such that Z t2 E(t3 ) ≤ C E(s)ds.

(3.18)

(3.19)

(3.20)

t1

Then, using (2.5), t3 < t + 1, it follows from (3.20) and the fact that E(t) is non-increasing that R t+1 R t+1 R k 2 E(t) = E(t + 1) + t ||∇us ||2 ds + (k + 1) t Ω |u| |us | ds R t+1 R R t+1 k 2 ≤ E(t3 ) + t ||∇us ||2 ds + (k + 1) t Ω |u| |us | ds R t2 R t+1 R R t+1 k 2 ≤ C t1 E(s)ds + t ||∇us ||2 ds + (k + 1) t (3.21) Ω |u| |us | ds. 9

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Combining (3.21) with (3.19), (3.7) and (3.8), we have E(t) ≤ C7 (ε)F 2 (t) + 4ε

2(q + 1) E(t). q+2

(3.22)

Choosing ε sufficiently small, (3.22) leads to E(t) ≤ C8 F 2 (t), C2

where C8 = 2(1 + 2C6 + 27 ). Since E(t) is nonincreasing, using Lemma 3.2, we conclude that for each t ∈ [0, ∞), E(t) ≤ E(0) exp(−α[t − 1]+ ) 8 ). Then the exponential decay of the energy where [t − 1]+ = max{t − 1, 0}, and α = ln( CC8 −1 is obtained. The proof of Theorem 2.2 is completed.

4

Blowup of the solutions

In this section our aim is to establish sufficient condition for blow-up of solutions to problem (1.1)-(1.3). We assume that k < q and u be a weak solution to the problem (1.1)-(1.3) on the interval [0, T ]. We note that the Levine energy method [30] is one of basic methods for studying the blow-up phenomenon. The role of the differential inequality [30] φφ00 − α(φ0 )2 + βφ2 ≥ 0, α > 1, β ≥ 0, in the standard Levine method is known. The generalization of this inequality φφ00 − α(φ0 )2 + βφ2 + γφφ0 ≥ 0, α > 1, β ≥ 0, γ ≥ 0, was obtained in [31]. On the other hand, a somewhat different differential inequality was used to prove Theorem 3 in [3] φφ00 − α(φ0 )2 + βφ2 + γφ2+q1 ≥ 0, α > 1, β ≥ 0, γ ≥ 0, q1 ≥ 0. Now, we consider our main differential inequality φφ00 − α(φ0 )2 + βφ2 + γφ2+l + δφ ≥ 0,

(4.1)

where φ(t) ∈ C 2 ([0, T ]) and α > 1, β ≥ 0, γ ≥ 0, δ ≥ 0, l ≥ 0.

(4.2)

Lemma 4.1 Suppose φ(t) ∈ C 2 ([0, T ]). Let conditions (4.1) and (4.2) be satisfied and moreover the following conditions hold: 2α > 2 + l,

(4.3)

φ(t) ≥ 0, φ0 (0) > 0, φ(0) > 0, (φ0 (0))2

−

β 2 α−1 φ (0)

+

2γ 2α+δ1 (1−α) (0) (α−1)δ1 φ

+

(4.4)

2δ 2α+δ2 (1−α) (0) (α−1)δ2 φ

= B > 0,

(4.5)

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where δ1 = 2α−2−l α−1 and δ2 = es the condition

2α−1 α−1 ,

then any solution to the differential inequality (4.1) satisfi

T∞ ≤ φ1−α (0)A−1 , lim φ(t) = ∞ − t→T∞

where A2 = (α − 1)2 φ−2α (0)B. Proof Condition (4.4) imply the existence of a time t1 > 0 for which the inequality φ0 (t) > 0 for t ∈ [0, t1 ) holds. Hence, φ(t) > φ(0) > 0 for t ∈ [0, t1 ). Dividing both sides of (4.1) by φ1+α we obtain φ−α φ00 − αφ−(1+α) (φ0 )2 + βφ1−α + γφ1+l−α + δφ−α ≥ 0.

(4.6)

2

d 1−α = (1 − α)φ−α φ00 − α(1 − α)φ−(1+α) (φ0 )2 , from (4.6) it is easy to derive the Noting dt 2φ inequality 1 d2 1−α 1−α dt2 φ

+ βφ1−α + γφ1+l−α + δφ−α ≥ 0.

(4.7)

We introduce the new function ψ = φ1−α ,

(4.8)

then we obtain 1 00 1−α ψ

+ βψ + γψ

−α

1+l−α 1−α

+ δψ 1−α ≥ 0.

(4.9)

Note now that by (4.8), we have ψ 0 (t) = (1 − α)φ0 φ−α ,

(4.10)

so in view of φ0 (t) > 0 and α > 1, it follows that ψ 0 (t) ≤ 0.

(4.11)

Now multiplying (4.9) by ψ 0 (t) we obtain 1 0 00 1−α ψ (t)ψ

+ βψ 0 (t)ψ + γψ 0 (t)ψ

α−1−l α−1

α

+ δψ 0 (t)ψ α−1 ≤ 0,

since α > 1, which gives us ψ 0 (t)ψ 00 ≥ β(α − 1)ψ 0 (t)ψ + γ(α − 1)ψ 0 (t)ψ

α−1−l α−1

α

+ δ(α − 1)ψ 0 (t)ψ α−1 .

Hence 1 d 0 2 2 dt (ψ (t))

≥

β(α−1) d 2 2 dt ψ

2α−2−l where δ1 = 1 + α−1−l α−1 = α−1 , δ2 = 1 + that δ1 > 0 and δ2 > 0. Integrating (4.12), we obtain that 1 0 2 2 (ψ (t))

≥ 12 (ψ 0 (0))2 +

β(α−1) (ψ 2 2

+

α α−1

γ(α−1) d δ1 δ1 dt ψ

=

− ψ 2 (0)) +

2α−1 α−1 .

+

δ(α−1) d δ2 δ2 dt ψ ,

(4.12)

Since 2α > 2 + l and α > 1, it follows

γ(α−1) δ1 δ1 (ψ

− ψ δ1 (0)) +

δ(α−1) δ2 δ2 (ψ

− ψ δ2 (0)),

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then, we obtain the following equivalent inequality (ψ 0 (t))2 ≥ A2 + β(α − 1)ψ 2 +

2γ(α−1) δ1 ψ δ1

+

2δ(α−1) δ2 ψ δ2

≥ A2 ,

(4.13)

where A2 = (ψ 0 (0))2 − β(α − 1)ψ 2 (0) −

2γ(α−1) δ1 ψ (0) δ1

−

2δ(α−1) δ2 ψ (0). δ2

From (4.8), (4.10) and the initial condition (4.5) we get A2 = (1 − α)2 φ−2α (0)(φ0 (0))2 − β(α − 1)φ2(1−α) (0) − 2γ(α−1) φδ1 (1−α) (0) − δ1 = (α −

2δ(α−1) δ2 (1−α) φ (0) δ2 1)2 φ−2α (0)B > 0.

A further analysis of inequality (4.13) yields |ψ 0 | ≥ A > 0, ∀t ∈ [0, t0 ),

(4.14)

where t0 is the life time of the solution. Since ψ 0 < 0 by (4.11), hence, ψ 0 ≤ −A < 0.

(4.15)

Integrating the inequality (4.15) we obtain ψ(t) ≤ ψ(0) − At, in view of (4.8) and α > 1, therefore φ1−α (t) ≤ φ1−α (0) − At. As a result, we obtain the lower estimate −1

φ(t) ≥ (φ1−α (0) − At) α−1 , which implies that T 6= +∞, since otherwise there exists T∞ < φ1−α (0)A−1 such that lim φ(t) = ∞. Then the proof is completed.

− t→T∞

Proof of Theorem 2.3 We introduce the following notations 1 1 φ(t) = ||∇u||2 + ||u||2 , G(t) = ||∇ut ||2 + ||ut ||2 . 2 2 If we multiply the equation (1.1) by u, we obtain the following equality: φ00 − G + (∇ut , ∇u) + ((|u|k u)t , u) + ||∇u||2 = ||u||q+2 q+2 .

(4.16)

The third and fourth terms in (4.16) can be estimated as follows: |(∇ut , ∇u)| ≤ 2 ||∇ut ||2 +

1 2 2 ||∇u||

≤ 2 G + 1 φ

(4.17)

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and |((|u|k u)t , u)| ≤ (k + 1)||ut ||||u||k+1 2(k+1) ≤ 2 ||ut ||2 +

2(k+1)

(k+1)2 C0 2

||∇u||2(k+1) ≤ 2 G +

2(k+1)

(k+1)2 C0 

φk+1

(4.18)

for any  > 0. Hence, by the estimate (4.17) and (4.18), equality (4.16) yields φ00 − G + G + 1 φ +

2(k+1)

(k+1)2 C0 

φk+1 + ||∇u||2 ≥ ||u||q+2 q+2 .

(4.19)

Integrating (2.5) with respect to t gives 1 1 1 1 ||u||q+2 E(t) = ||ut ||2 + ||∇ut ||2 + ||∇u||2 − q+2 ≤ E(0). 2 2 2 q+2 Then we get ||u||q+2 q+2 ≥

q+2 q+2 G+ ||∇u||2 − (q + 2)E(0). 2 2

(4.20)

Then, (4.20) and (4.19) yield φ00 −

q+4 2 G

+ G + 1 φ +

Now, we choose  = 0 =

q−k 2

α0 =

2(k+1)

(k+1)2 C0 

φk+1 − 2q ||∇u||2 + (q + 2)E(0) ≥ 0.

(4.21)

> 0 so that

q+4 2

−

0 2

=2+

k 2

> 1, 2α0 = 4 + k > k + 2,

(4.22)

where we have used the fact q > k. Then (4.21) becomes φ00 − α0 G +

1 0 φ

2(k+1)

+

(k+1)2 C0 0

φk+1 + (q + 2)E(0) ≥ 2q ||∇u||2 ≥ 0.

(4.23)

From the Cauchy-Schwarz inequality, we have (φ0 )2 ≤ φG.

(4.24)

Then using (4.24) and (4.23) we arrive at the following second order differential inequality φφ00 − α0 (φ0 )2 +

1 2 0 φ

2(k+1)

+

(k+1)2 C0 0

φk+2 + (q + 2)E(0)φ ≥ 0.

(4.25)

Comparing this differential inequality with inequality (4.1), we find that α = α0 , β =

1 0 , γ

2(k+1)

=

(k+1)2 C0 0

, δ = (q + 2)E(0), l = k.

(4.26)

By (4.26) and (4.22), we know that (4.2) and (4.3) are satisfied. By (2.8) and (2.9), we know that (4.4) and (4.5) are satisfied. Then by Lemma 4.1, we see that φ(t) blows up in finite time. This theorem is proved. ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (No.11526077,11601122). 13

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References [1] S.A.Gabov. Mathematical fundamentals of linear theory of ionic sound waves in unmagnetized plasma. Matematicheskoe Modelirovanie, 1 (12) (1989), 133-148. [2] M.O.Korpusov. Blow-up of ion acoustic waves in a plasma, Sbornik: Mathematics, 202 (1) (2011), 37-64. [3] M.O.Korpusov, A.A.Panin. Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity. Izvestiya: Mathematics, 78(5)(2014), 937-985. [4] Y.D.Shang. Initial-boundary value problem for a class of fourth order nonlinear evolution equations. Mathematica Applicata, 13(1)(2000), 7-11.(in Chinese) [5] Y.D.Shang. Initial boundary value problem of equation utt − ∆u − ∆ut − ∆utt = f (u). Acta Mathematicae Applicatae Sinica, 23A(2000), 385-393. (in Chinese) [6] H.W.Zhang, Q.Y.Hu. Existence of global weak solution and stability of a class nonlinear evolution equation. Acta Mathematiea Scientia, 24A(3) (2004), 329-336.(in Chinese) [7] Y.Q.Xie, C.K.Zhong. The existence of global attractors for a class nonlinear evolution equation. Journal of Mathematical Analysis and Application, 336(2007), 54-69. [8] R.Z.Xu, X.R.Zhao, J.H.Shen. Asymptotic behavior of solution for fourth order wave equation with dispersive and dissipative terms. Applied Mathematics and Mechanics, 29 (2008), 259-262. [9] Y.Q.Xie, C.K.Zhong. Asymptotic behavior of a class of nonlinear evolution equations. Nonlinear Analysis, 71(11) (2009), 5095-5105. [10] A.N.Carvalho, J.W.Cholewa. Local well posedness, asymptotic behavior and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time. Transactions of the American Mathematical Society, 361(5) (2009), 2567-2586. [11] C.Y.Sun, L.Yang, J.Q.Duan. Asymptotic behavior for a semilinear second order evolution equation. Transactions of the American Mathematical Society, 363(11) (2011), 6085-6109 [12] C.S.Chen, H.Wang, S.L.Zhu. Global attractor and decay estimates of solutions to a class of nonlinear evolution equations. Mathematical Methods in the Applied Sciences, 34(5)(2011), 497-508. [13] R.Z.Xu, Y.B.Yang. Finite time blow up for the nonlinear fourth-order dispersive-dissipative wave equation at high energy level. International Journal of Mathematics, 23(5) (2012), 1250060 (10 pages). [14] H.F.Di, Y.D.Shang. Global existence and asymptotic behavior of solutions for the double dispersive-dissipative wave equation with nonlinear damping and source terms. Boundary Value Problems, 2015(2015) 29, DOI 10.1186/s13661-015-0288-6. [15] V.Pata, M.Squassina. On the strongly damped wave equation. Communications in Mathematical Physics, 253(3) (2005), 511-533. [16] A.N.Carvalho, J.W.Cholewa, T.Dlotko. Strongly damped wave problems: bootstrapping and regularity of solutions. Journal of Differential Equations, 244(2008), 2310-2333. [17] V.Barbu, I.Lasiecka, M.A.Rammaha. On nonlinear wave equations with degenerate damping and source terms. Transactions of the American Mathematical Society, 357(7),(2005), 2571-2611. [18] V.Barbu, I.Lasiecka, M.A.Rammaha. Blow-up of generalizedsolutions to wave equations with degenerate damping and source terms. Indiana University Mathematics Journal, 56(3),(2007), 995-1022.

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[19] D.R.Pitts, M.A.Rammaha. Global existence and nonexistence theorems for nonlinear wave equations. Indiana University Mathematics Journal, 51(6)(2002), 1479-1509. [20] M.A.Rammaha, T.A.Strei. Global existence and nonexistence for nonlinear wave equations with damping and source terms. Transactions of the American Mathematical Society, 354(9)(2002), 3621-3637. [21] V.Barbu, I.Lasiecka, M.A.Rammaha. Existence and uniqueness of solutions to wave equations with degenerate damping and source terms. Control and Cybernetics, 34(3) (2005), 665-687. [22] Q.Y. Hu, H.W. Zhang. Blowup and asymptotic stabiity of weak solutions to wave equations with nonlinear degenerate damping and source terms. Electronic Journal of Differential Equations. 2007(2007), No. 76, 1-10. [23] M.A.Rammaha, S.Sakuntasathien. Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms. Nonlinear Analysis Theory Methods and Applications, 2010, 72(5), 2658-2683. [24] M.A. Rammaha, S.Sakuntasathien. Critically and degenerately damped systems of nonlinear wave equations with source terms. Applicable Analysis, 89(8)(2010), 1201-1227. [25] X.S.Han, M.X.Wang. Global existence and blow-up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source. Mathematische Nachrichten, 284(5-6)(2011), 703-716. [26] S.T. Wu. General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms. Journal of Mathematical Analysis and Application, 406 (2013), 34-48. [27] E.Piskin. Blow up of positive initial-energy solutions for coupled nonlinear wave equations with degenerate damping and source terms. Boundary Value Problems, 2015(2015),no.43, 1-11. [28] L.E.Payne, D.Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel Journal of Mathematics, 22(3-4) (1975), 273-303. [29] M.Nakao. Asymptotic stability of the bounded or almost periodic solution of the wave equation. Journal of Mathematical Analysis and Application, 56 (1977), 336-343. [30] H.A.Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form P utt = Au + F (u). Transactions of the American Mathematical Society, 192 (1974), 1-21. [31] V.K.Kalantarov, O.A.Ladyzhenskaya. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. Journal of Soviet Mathematics, 10(1)(1978), 77-102. [32] E.Vitillaro. A potential well theory for the wave equation with nonlinear source and boundary damping terms. Glasgow Mathematical Journal, 44(2)(2002), 375-395. [33] G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms. Journal of Mathematical Analysis and Application, 239(1999), 213-226 [34] E.Vitillaro. Some new results on global nonexistence and blow-up for evolution problems with positive initial energy. Rendiconti Dellistituto Di Matematica Delluniversita Di Trieste, 31(2)(2000), 245-275. [35] J.L.Lions. Quelques m´ ethodes de r´ esolution des probl´ emes aux limites non lin´ eaires, DunodGauthier Villars, 1969.

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COMPACT AND MATRIX OPERATORS ON THE SPACE jC; 1jk G. CANAN HAZAR GÜLEÇ AND M. ALI SARIGÖL

Abstract. According to Hardy [5], Cesàro summability is usually considered for the range 1: In a more recent paper [14], the space jC jk is studied for > 1: In this paper we de…ne jC 1 jk using the Cesàro mean (C; 1) of Thorpe [26] , compute its -, - and - duals, give some algebraic and topological properties, and characterize related matrix operators, and also obtain some identities or estimates for the their operator norms and the Hausdor¤ measure of noncompactness. Further, by applying the Hausdor¤ measure of noncompactness, we establish the necessary and su¢ cient conditions for such operators to be compact. So some results in [14] is also extended to the range 1:

1. Introduction Let ! be the set of all complex sequences, c; `1 w be the set of convergent and bounded sequences. For cs ; bs and `k (k 1; `1 = `) ; we write the sets of all convergent, bounded, k-absolutely convergent series, respectively. Let A = (anj ) be an arbitrary in…nite matrix of complex numbers. By P A(x) = (An (x)) ; we 1 denote the A-transform of the sequence x, i.e., An (x) = j=0 anj xj ; provided that the series converges for n 0: We say that A de…nes a matrix transformation from U into V , and it denote by A 2 (U; V ) or A : U ! V if sequence Ax = (An (x)) 2 V for every sequence x 2 U , where U and V are subspace of w and also the sets U = f" 2 w : ("v xv ) 2 ` for all x 2 U g ; U = f" 2 w : ("v xv ) 2 cs for all x 2 U g ; U = f" 2 w : ("v xv ) 2 bs for all x 2 U g and UA = fx 2 w : A(x) 2 U g

(1.1)

are said to be the -, -, - duals of U and the domain of the matrix A in U; respectively. Further, U is said to be an BK-space if it is a complete normed space with continuous coordinates pn : U ! C de…ned by pn (x) = xn for n 0: The sequence (en ) is called a Schauder base (or brie‡y base) for a normed sequence space U if for each x 2 U there exist unique sequence of 1991 Mathematics Subject Classi…cation. 40C05, 40D25, 40F05, 46A45. Key words and phrases. Sequence spaces; Absolute Cesàro summability; Dual spaces; matrix operators; Bounded linear operator; BK spaces; Norms. 1

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HAZAR AND SARIGÖL

Pm v coe¢ cients (xv ) such that kx v=0 xv e k ! 0 (m ! 1) ; and in this case P1 we write x = v=0 xv ev : For example, the sequence e(n) is a base of lk with 1=k P1 k respect to the norm kxk`k = ; k 1; where e(n) is the sequence v=0 jxv j whose only non-zero term is 1 in the nth place for each n:Throughout k denotes the conjugate of k > 1, i.e., 1=k + 1=k = 1; and 1=k = 0 for k = 1: Let an be an in…nite series with partial sum sn . Let ( n ) be nth Cesàro Pthe n 1 1 mean (C; ) of order > 1 of the sequence (sn ) ; e:i:; n = A v=0 An v sv :The n summability jC; jk was de…ned by Flett [4] as follows. The series an is said to be summable jC; jk with index k 1 if 1 P

nk

k

1 n

n=1

n 1

< 1:

More recently the series space jC jk has been studied by the second author for > 1, [14] : The Cesàro summability (C; ) is studied usually for range 1 (see [5]). Since the above de…nition does not work for = 1, so it was separately de…ned by Thorpe [26] as follows. If the series to sequence transformation n X1 Tn = a + (n + 1) an (1.2) =0

tends to s as n tends to in…nity, then the series an is summable by Cesàro summability (C; 1) [26] : Now we de…ne the space jC 1 jk ; k 1; as the set of all series summable by the method jC; 1jk : Then, it can be written that 1 P k jC 1 jk = a = (av ) : nk 1 jTn Tn 1 j < 1 ; where (Tn ) is de…ned by n=1

(1:2), or

jC

1 jk

=

(

a = (a ) :

1 X

n=1

nk

1

j(n + 1) an

(n

1) an

k

1j

)

0 then S is called an "-net of H , if, for every h 2 H; there exists an s 2 S such that d (h; s) < "; if S is …nite, then the "-net S of H is called a …nite "-net of H: Let X and Y be Banach spaces. A linear operator L : X ! Y is called compact its domain is all of X and, for every bounded sequence (xn ) in X; the sequence (L(xn )) has a convergent subsequence in Y: We denote the class of such operators by C (X; Y ). If Q is a bounded subset of the metric space X; then the Hausdor¤ measure of noncompactness of Q is de…ned by (Q) = f" > 0 : Q has a …nite "-net in Xg ;and is called the Hausdor¤ measure of noncompactness. The following result is an important tool to compute the Hausdor¤ measure of noncompactness of a bounded subset of the BK space `k ; k 1:

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HAZAR AND SARIGÖL

Lemma 2.1. Let Q be a bounded subset of the normed space X where X = `k , for 1 k < 1 or X = c0 :If Pn : X ! X is the operator de…ned by Pr (x) = (x0 ; x1 ; :::; xr ; 0; :::) for all x 2 X; then (Q) = limr!1 supx2Q k(I Pr ) (x)k ;where I is the identity operator on X; [13]. If X and Y be Banach spaces and 1 and 2 be Hausdor¤ measures on X and Y; then, the linear operator L : X ! Y is said to be ( 1 ; 2 )-bounded if L (Q) is bounded subset of Y for every bounded subset Q of X and there exists a positive constant M such that 2 (L (Q)) M 1 (Q) for every bounded Q of X: If an operator L is ( 1 ; 2 )-bounded then the number kLk( 1 ; 2 ) = inf fM > 0 : 2 (L (Q)) M 1 (Q) for all bounded Q Xg is called the ( 1 ; 2 )measure of noncompactness of L: In particular, we write kLk( ; ) = kLk for 1 = 2 = : Lemma 2.2. Let X and Y be Banach spaces, L 2 B (X; Y ) and SX = fx 2 X : kxk 1g denote the unit sphere in X: Then, kLk = (L (SX )) ; and L 2 C (X; Y ) if and only if kLk = 0; [8]. Lemma 2.3. Let X be normed sequence space and T and denote the Hausdor¤ measures of noncompactness on MX and MX , the collections of all T bounded sets in XT and X, respectively. Then, T (Q) = (T (Q) for all Q 2 MX ; where T = (tnv ) is a triangular in…nite matrix, [9]. T

3. Continuous and compact matrix operators on the space jC

1 jk

In this section by giving some topological properties, -, -, - duals, base of jC 1 jk ;we characterize matrix operators on that space, determine their norms, and also establish the necessary and su¢ cient conditions for such operators to be compact by applying the Hausdor¤ measure of noncompactness, which also extends some results of Sar¬göl [14] to 1: First of all, we de…ne the matrix (k) (k) T (k) = (tnv ) by t00 = 1; 8 < n1=k (n 1) ; v = n 1 (k) tnv = (3.1) n1=k (n + 1) ; v = n : 0; otherwise, for 1 k < 1. Then it is clear that jC 1 jk = (`k )T (k) according to (1:1) : Further, since every triangular matrix has a unique inverse which is also triangle [27, p. 9]. there exists the inverse of T (k) and denote this inverse by S (k) : Now 1 Pn 1=k it follows from (1:2) that a0 = y0 and an = [n (n + 1)] y ;n 1; v=1 v (k) and so s00 = 1; 8 1=k < ;1 v n (k) (3.2) snv = : n (n + 1) 0; v > n:

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jC;

1jk SUM M ABILITY

5

Finally, we de…ne the following notations: 1 P D1 = " = ("v ) 2 w : s(1) vr "v converges, r

1 ;

v=r

D2 = D3 =

(

D5 =

m;r v=r

" = ("v ) 2 w : sup

D4 = (

m P

" = ("v ) 2 w : sup

m m P P

s(1) vr "v < 1 ; k

m r=1 v=r

" = ("v ) 2 w : sup

" = ("v ) 2 w :

1 P

=1

1 P

n=

1 P

n=

s(k) vr "v

)

r

Then each matrix in the class A 2 (jC 1 jk ; jC 1 j) de…nes a bounded linear operator LA : jC 1 jk ! jC 1 j such that LA (x) = A(x) for all x 2 jC 1 jk ; and A 2 (jC 1 jk ; jC 1 j) if and only if (3:4) sup

m P

r1=k

m r=1

Morever, if A 2 (jC kLA k(jC

1 jk ;jC

1 jk

; jC

= 1 j)

1

m P

an ( + 1)

v=r

1 P

1 P

k

< 1;

(3.11)

k

< 1:

jfn j

(3.12)

=1

n=1

1 j) ;

then, there exists 1

4 such that

1

0

kF k(`k ;`) and kLA k =

lim

r!1

F (r)

0

(lk ;l)

:

(3.13)

Proof. The …rst part is immediate by Theorem 4.2.8 of Wilansky [27; p:57], since jC 1 j and jC 1 jk are a BK-spaces by Theorem 3.1. For the second part, 1 let A 2 (jC 1 jk ; jC 1 j) : Then, (an ) =0 2 (jC 1 jk ) and A (x) 2 jC 1 j for every 1 1 x 2 jC 1 jk : Now by Theorem 3.1 (b); (an ) =0 2 (jC 1 jk ) i¤ (an ) =0 2 D1 \ 1 D3 for each n: This also means that (an ) =0 2 (jC 1 jk ) i¤ conditions (3:4) and (3:11) hold: Also to obtain (3:12), we consider the operators T (k) : jC 1 jk ! `k same as Theorem 3.1 (a). Then, since the space jC 1 jk is izomorf to `k ; it can be written that x 2 jC 1 jk i¤ y 2 `k ; where T (k) (x) = y ; i:e:; y0 = x0 and (k) 1 ; x 1 = 0. So by yn = Tn (x) = n1=k [(n + 1) xn (n 1) xn 1 ] for n (3:2) ; as in the proof of Theorem 3.2 we get m P

anv xv =

v=1 (n)

where, f mr = r

1=k

m P

r

1=k

r=1

(n)

`mr for 1

`(n) mr yr =

r=1

(n)

r

m P

(n)

f mr yr (n)

m; and f mr = 0 for r > m; and L

(n) `mr

=

is de…ned as in Theorem 3.2. Moreover, if any matrix H = (hnv ) 2 (`k ; c) ; then by Lemma1.4 and using Hölder’s inequality, we get 1 P

hnv xv

sup n

v=m

1 P

v=0

1=k k

jhnv j

1 P

v=m

Also, since x 2 `k ; we obtain that the series Hn (x) = formly in n, which implies lim Hn (x) = n

1 P

lim hnv xv :

v=0 n

1022

1=k k

jxv j

v hnv xv

converges in-

(3.14)

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HAZAR AND SARIGÖL (n)

Therefore, since (3:4) and (3:11) hold, F (n) = f mr 2 (`k ; c), then by (3:14) ; we get 1 1 1 P P P (n) An (x) = lim f mr yr = r 1=k `nr yr = f nr yr = F n (y); n 1 r=1

m

r=1

r=1

(n)

where, f nr = limm f mr : This shows that A(x) 2 jC 1 j for every x 2 jC 1 jk i¤ F (y) 2 jC 1 j for every y 2 `k or, equivalently T (1) F (y) 2 `; since jC 1 j = (`)T (1) ; so we obtain that F 2 (`k ; `), where F = T (1) F : Hence, it is clear that A 2 (jC 1 jk ; jC 1 j) i¤ (3:4) ; (3:11) hold, and F 2 (`k ; `) :With a few calculations, it can be easily seen that F is the same as (3:10) ; and so it follows from applying Lemma 1.2 to the matrix F that F 2 (`k ; `) i¤ (3:12) is satis…ed, and this proves the second part. Also, considering that T (k) : jC 1 jk ! `k and T (1) : jC 1 j ! ` are norm 1 isomorphism, as in Theorem 3.2, it follows that A = T (1) oF oT (k) and so, by Lemma 1.2, 1 0 kLA k(jC 1 j ;jC 1 j) = kF k(`k ;`) = kF k(`k ;`) k Finally, S = fx 2 jC 1 jk : kxk 1g : Then, by considering Lemma 2.1-Lemma 2.3, and Lemma 1.2, there exists 1 4 such that kLA k = lim

sup k(I

r!1 y2T (k) S

Pr ) F (y)kl = lim

r!1

F (r)

(lk ;l)

=

1

lim

r!1

F (r)

0

(lk ;l)

where Pr : l ! l is de…ned by Pr (y) = (y0 ; y1 ; :::; yr ; 0; :::) and the matrix (n) (n) F (r) = f~n is de…ned as: f~n = 0 for 1 n r; and fn for n > r; which proves the theorem together with Lemma 1.2. The compact operators in this class are obtained from Theorem 3.5 as follows. Corollary 3.6. Under hypotheses of Theorem 3.5, LA 2 (jC

1 jk ; jC

1 j) is compact if and only if lim

r!1

F (r)

0

(lk ;l)

= 0:

References [1] Bor, H. and Thorpe, B., On some absolute summability methods, Analysis 7 (2) (1987), 145-152. [2] Borwein, D. and Cass, F.P., Strong Nörlund summability, Math. Zeitschr. 103 (1968), 94– 111. [3] Bosanquet L.S. and Das G., Absolute summability factors for Nörlund means, Proc. London Math. Soc. (3) 38 (1979), 1-52. [4] Flett, T.M., On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957), 113-141. [5] Hardy, G. H., Divergent Series Oxford, 1949.

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1jk SUM M ABILITY

11

[6] Kuttner, B., Some remarks on summability factors, Indian J. Pure Appl. Math. 16 (9) (1985), 1017-1027. [7] Maddox, I.J., Elements of functinal analysis, Cambridge University Press, London,New York, (1970). [8] Malkowsky, E. and Rakoµcevi´c, V., An introduction into the theory of sequence space and measures of noncompactness, Zb. Rad. (Beogr) 9, (17),(2000), 143-234. [9] Malkowsky, E. and Rakoµcevi´c, V., On matrix domain of triangles, Appl. Math. Comp. 189(2), (2007), 1146-1163. [10] Mazhar, S.M., On the absolute summability factors of in…nite series, Tohoku Math. J. 23 (1971), 433-451. [11] Mohapatra, R.N. and Das, G., Summability factors of lower-semi matrix transformations, Monatshefte für Mathematika, 79 (1975), 307-3015. [12] Orhan, C. and Sar¬göl, M.A., On absolute weighted mean summability, Rocky Moun. J. Math. 23 (3) (1993), 1091-1097. [13] Rakoµcevi´c, V., Measures of noncompactness and some applications, Filomat, 12, (1998), 87-120. [14] Sar¬göl, M.A., Spaces of Series Summable by Absolute Cesàro and Matrix Operators, Comm. Math Appl. 7 (1) (2016) 11-22. [15] _____ Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci. 42 (3) (2015), 28-35. [16] _____ Matrix operators on A k , Math. Comp. Model. 55 (2012), 1763-1769. [17] _____ Matrix transformatins on …elds of absolute weighted mean summability, Studia Sci. Math. Hungar. 48 (3) (2011), 331-341. [18] _____ and Bor, H., Characterization of absolute summability factors, J. Math. Anal. Appl. 195 (1995), 537-545. [19] _____ On two absolute Riesz summability factors of in…nite series, Proc. Amer. Math. Soc. 118 (1993), 485-488. [20] _____ A note on summability, Studia Sci. Math. Hungar. 28 (1993), 395-400. [21] _____ On absolute weighted mean summability methods, Proc. Amer. Math. Soc. 115 (1) (1992), 157-160. [22] _____ Necessary and su¢ cient conditions for the equivalence of the summability methods N ; pn k and jC; 1jk , Indian J. Pure Appl. Math. 22(6) (1991), 483-489. [23] Stieglitz, M. and Tietz, H., Matrixtransformationen von folgenraumen eine ergebnisüberischt, Math Z., 154 (1977), 1-16. [24] Sulaiman, W.T., On summability factors of in…nite series, Proc. Amer. Math. Soc. 115 (1992), 313-317. [25] Sunouchi, G., Notes on Fourier Analysis, 18, absolute summability of a series with constant terms, Tohoku Math. J., 1, (1949), 57-65. [26] Thorpe, B., Matrix Transformations of Cesàro Summable Series, Acta Math. Hung., 48(34), (1986), 255-265. [27] Wilansky, A., Summability Through Functional Analysis, North-Holland Mathematical Studies, vol. 85, Elsevier Science Publisher, 1984. Department of Mathematics, Pamukkale University, E-mail address : [email protected], [email protected]

1024

TR-20007 Denizli,

TURKEY

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On value distribution of meromorphic solutions of a certain second order difference equations

∗

Yun Fei Du, Min Feng Chen†and Zong Sheng Gao LMIB & School of Mathematics and Systems Science, Beihang University, Beijing, 100191, P. R. China

Abstract. In this paper, we consider difference equation f (z + 1) + f (z − 1) = m(z) A(z)f (z)+C 1−f 2 (z) , where C is a non-zero constant, A(z) = n(z) , m(z) and n(z) are irreducible polynomials, we obtain the forms of rational solutions. (z)+C(z) To the difference equation f (z + 1) + f (z − 1) = A(z)f , where A(z), C(z) 1−f 2 (z) are non-constant small functions with respect to f (z), the Borel exceptional value, the exponents of convergence of zeros, poles and fixed points of finite order transcendental meromorphic solution f (z), and the exponents of convergence of poles of differences ∆f (z) = f (z + 1) − f (z), ∆2 f (z) = ∆f (z + 1) − ∆f (z) and divided (z) ∆2 f (z) differences ∆f f (z) , f (z) are estimated. Mathematics Subject Classification (2010). 39B32, 34M05, 30D35. Keywords. Difference equation, Rational solution, Transcendental meromorphic solution.

1

Introduction and Results

Halburd and Korhonen [6, 7] used value distribution theory and a reasoning related to the singularity confinement to single out the difference Painlev´ e I and II equations from difference equation f (z + 1) + f (z − 1) = R(z, f ), (1.1) where R is rational in both of its arguments. They obtained that if (1.1) has an admissible meromorphic solutions of finite order, then either f satisfies a difference Riccati equation, or (1.1) may be transformed into some classical difference equations, which include difference Painlev´ e I equations az + b f (z + 1) + f (z − 1) = + c, (1.2) f (z) f (z + 1) + f (z − 1) =

az + b c + 2 , f (z) f (z)

(1.3)

az + b + c, f (z)

(1.4)

(az + b)f (z) + c , 1 − f 2 (z)

(1.5)

f (z + 1) + f (z) + f (z − 1) = and difference Painlev´ e II equation f (z + 1) + f (z − 1) = ∗ This

research was supported by the National Natural Science Foundation of China (No: 11371225) and by the Fundamental Research Funds for the Central Universities. † Corresponding author. E-mail: [email protected].

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where a, b and c are constants. Recently, as the difference analogues of Nevanlinna’s theory are investigated [2, 5], many results on the complex difference equations are rapidly obtained, such as [1, 3, 10−12]. However, there are a few papers concerning with the existence of rational solution of difference Painlev´ e equations. In this paper, we will consider the forms of rational solutions, and investigate the value distribution of meromorphic solution of finite order of a certain type of difference equation which originates from the difference Painlev´ e II equation. We assume that the reader is familiar with the basic Nevanlinna’s value distribution theory of meromorphic functions (see[4, 9]). In addition, we use the notation σ(f ) to denote the order of growth of the meromorphic function f (z), λ(f ) and λ( f1 ) to denote, respectively, the exponent of convergence of zeros and poles of f (z). We also use the notation τ (f ) to denote the exponent of convergence of fixed points of f (z) which is defined as   1 log N r, f (z)−z τ (f ) = lim . r→∞ log r We denote by S(r, f ) any quantify satisfying S(r, f ) = o(T (r, f )), as r → ∞, possibly outside a set with finite measure. For every n ∈ N∗ , the forward difference ∆n f (z) are defined in the standard way [14] by ∆f (z) = f (z + 1) − f (z), ∆n+1 f (z) = ∆n f (z + 1) − ∆n f (z). Chen and Shon [3] considered the difference Painlev´ e II equation (1.5) and proved the following result. Theorem A. (See [3]) Let a, b, c be constants, ac 6= 0. Suppose that a rational function f (z) =

P (z) pz m + pm−1 z m−1 + · · · + p0 = Q(z) qz n + qn−1 z n−1 + · · · + q0

is a solution of (1.5), where P (z) and Q(z) are relatively prime polynomials, p, pm−1 , . . . , p0 and q, qn−1 , . . . , q0 are constants. Then n=m+1

and

c p = − q. a

In equation (1.5), if we replace az + b with A(z) = m(z) n(z) , where m(z) and n(z) are mutually prime polynomials, we still consider the rational solutions of equation (1.5), what will happen? Here, we obtain the following result. Theorem 1.1. Let C be non-zero constant, and A(z) = m(z) n(z) be a rational function, where m(z) and n(z) are mutually prime polynomials with deg m(z) = m, deg n(z) = n. If difference equation A(z)f (z) + C f (z + 1) + f (z − 1) = (1.6) 1 − f 2 (z) P (z) has a rational solution f (z) = Q(z) , where P (z) and Q(z) are relatively prime polynomials with deg P (z) = p, deg Q(z) = q, then (i) Suppose that m > n, then p − q = m−n and m − n must be even or q − p = m − n ≥ 1; 2 (ii) Suppose that m = n, then p − q = 0 and

m(z) = A ∈ C \ {0}, z→∞ n(z) lim

P (z) = B ∈ C \ {0}, z→∞ Q(z) lim

C = B[2(1 − B 2 ) − A] or C = ±A; (iii) Suppose that m < n, then p − q = 0 and lim

z→∞

P (z) = B ∈ C \ {0, ±1}, Q(z) 1026

C = 2B(1 − B 2 ).

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The following examples show that the difference equation (1.6) has rational solutions satisfying Theorem 1.1 (i), (ii) and (iii). Example 1.1. The difference equation f (z + 1) + f (z − 1) =

− 2z

6

+2z 4 +Cz 3 +2z 2 −Cz f (z) z 4 −1 2 1 − f (z)

+C

has a rational solution f (z) = z + z1 , where C 6= 0, m = 6, n = 4, p = 2, q = 1 and p − q = m−n = 1. 2 Example 1.2. The difference equation f (z + 1) + f (z − 1) =

(−Cz + 2)f (z) + C 1 − f 2 (z)

has a rational solution f (z) = z1 , where C 6= 0, m = 1, n = 0, p = 0, q = 1 and q − p = m − n = 1. Example 1.3. The difference equation f (z + 1) + f (z − 1) = has a rational solution f (z) =

z−1 z+1 ,

−2(z 3 +5z+10) z 3 +2z 2 −z−2 f (z) 1 − f 2 (z)

+2

where m = n = 3, p = q = 1,

−2(z 3 + 5z + 10) = −2, z→∞ z 3 + 2z 2 − z − 2 lim

lim

z→∞

z−1 = 1, z+1

and C = 2 = B[2(1 − B 2 ) − A] = 1 · [2 · (1 − 1) − (−2)]. Example 1.4. The difference equation f (z + 1) + f (z − 1) = has a rational solution f (z) = lim

z→∞

2(z+1) z−1 ,

2(z + 1) = 2, z−1

−44z 3 +36z 2 +28z+12 f (z) z 4 −2z 3 −z 2 +2z 2 1 − f (z)

− 12

where m = 3, n = 4, p = q = 1 and

C = −12 = 2B(1 − B 2 ) = 2 · 2 · (1 − 22 ).

In [3], Chen and Shon also investigated some properties of meromorphic solutions of finite order of difference Painlev´ e II equation (1.5) and obtained the following result. Theorem B. (See [3]) Let a, b, c be constants with ac 6= 0. If f (z) is a finite-order transcendental meromorphic solution of the difference Painlev´ e II equation (1.5), then: (i) f (z) has at most one non-zero finite Borel exceptional value;   1 (ii) λ f = λ(f ) = σ(f ); (iii) f (z) has infinitely many fixed points and satisfies τ (f ) = σ(f ). In this paper, we investigate the properties of a transcendental meromorphic solution of the difference equation A(z)f (z) + C(z) f (z + 1) + f (z − 1) = , (1.7) 1 − f 2 (z) where A(z), C(z) are nonconstant small functions with respect to f (z). And we obtain the following result. Theorem 1.2. Suppose that the difference equation (1.7) admits a transcendental meromorphic  solution of finite order, then (i) λ f1 = λ(f ) = σ(f );         (ii) λ ∆f1(z) = λ ∆f1(z) = σ(f ), λ ∆2 f1 (z) = λ ∆2 1f (z) = σ(f ); f (z)

f (z)

(iii) If 2z(z 2 − 1) + zA(z) + C(z) 6≡ 0, then τ (f ) = σ(f ); (iv) In particular, if A(z)±C(z) ≡ 0, then f (z) has at most one non-zero finite Borel exceptional value. 1027

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2

Lemmas for the Proof of Theorems

Lemma 2.1. (See [10, Theorem 2.4],[5]) Let f (z) be a transcendental meromorphic solution of finite order σ of the difference equation P (z, f ) = 0, where P (z, f ) is a difference polynomial in f (z) and its shifts. If P (z, a) 6≡ 0 for a slowly moving target meromorphic function a, that is, T (r, a) = S(r, f ), then   1 m r, = O(rσ−1+ε ) + S(r, f ), f −a outside of a possible exceptional set of finite logarithmic measure. Lemma 2.2. (See [10, T heorem 2.3]) Let f (z) be a transcendental meromorphic solution of finite order σ of a difference equation of the form U (z, f )P (z, f ) = Q(z, f ), where U (z, f ), P (z, f ) and Q(z, f ) are difference polynomials such that the total degree degf U (z, f ) = n in f (z) and its shifts, and degf Q(z, f ) ≤ n. Moreover, we assume that U (z, f ) contains just one term of maximal total degree in f (z) and its shifts. Then for each ε > 0, m(r, P (z, f )) = O(rσ−1+ε ) + S(r, f ), possibly outside of an exceptional set of finite logarithmic measure. Lemma 2.3. (See [13]) functions in f (z),

Let f (z) be a meromorphic function. Then for all irreducible rational Pm ai (z)f i (z) , R(z, f (z)) = Pni=0 j j=0 bj (z)f (z)

with meromorphic coefficients ai (z), bj (z)(am (z)bn (z) 6≡ 0) being small with respect to f (z), the characteristic function of R(z, f (z)) satisfies T (r, R(z, f (z))) = max{m, n}T (r, f ) + S(r, f ). Lemma 2.4. (See [2, Corollary 2.5]) Let f (z) be a meromorphic function of finite order σ and let η be a non-zero complex number. Then for each ε > 0, we have     f (z + η) f (z) m r, + m r, = O(rσ−1+ε ). f (z) f (z + η) Lemma 2.5. (See [2, T heorem 2.1]) Let f (z) be a meromorphic function with order σ = σ(f ), σ < +∞, and let η be a fixed non-zero complex number, then for each ε > 0, we have T (r, f (z + η)) = T (r, f (z)) + O(rσ−1+ε ) + O(log r). Lemma 2.6. (See [2, T heorem 2.2]) Let f (z) be a meromorphic function with exponent of  1 convergence of poles λ f = λ < ∞, η 6= 0 be fixed, then for each ε > 0, N (r, f (z + η)) = N (r, f (z)) + O(rλ−1+ε ) + O(log r). Lemma 2.7. (See [8, Remark 1]) Let f (z) be a transcendental meromorphic function. If P (z, f ) and Q(z, f ) are mutually prime polynomials in f (z), there exist polynomials of f (z), U (z, f ) and V (z, f ) such that U (z, f )P (z, f ) + V (z, f )Q(z, f ) = s(z), where s(z) and coefficients of U (z, f ) and V (z, f ) are small functions with respect to f (z). 1028

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Lemma 2.8. Let f (z) be a transcendental meromorphic function and A(z) ± C(z) 6≡ 0, then A(z)f (z) + C(z) and f (z)(1 − f 2 (z)) are mutually prime polynomials in f (z), where A(z), C(z) are nonzero small functions with respect to f (z). Proof. Since A(z), C(z) are non-zero small functions with respect to f (z) and A(z)±C(z) 6≡ 0, then C(z)(C 2 (z) − A2 (z)) 6≡ 0, T (r, C(z)(C 2 (z) − A2 (z))) = S(r, f ). There exist polynomials of f (z), U (z, f ) = A2 (z)f 2 (z) − A(z)C(z)f (z) + C 2 (z) − A2 (z) and V (z, f ) = A3 (z) such that U (z, f )(A(z)f (z) + C(z)) + V (z, f )f (z)(1 − f 2 (z)) = C(z)(C 2 (z) − A2 (z)). By Lemma 2.7, we see that A(z)f (z) + C(z) and f (z)(1 − f 2 (z)) are mutually prime. Lemma 2.9. Let f (z) be a transcendental meromorphic function and A(z) ± C(z) 6≡ 0, then 2f 3 (z) + (A(z) − 2)f (z) + C(z) and 1 − f 2 (z) are mutually prime polynomials in f (z), where A(z), C(z) are non-zero small functions with respect to f (z). Proof. Since A(z), C(z) are non-zero small functions with respect to f (z) and A(z)±C(z) 6≡ 0, then A2 (z) − C 2 (z) 6≡ 0, T (r, A2 (z) − C 2 (z)) = S(r, f ). There exist polynomials of f (z), U (z, f ) = A(z)f (z) − C(z) and V (z, f ) = 2A(z)f 2 (z) − 2C(z)f (z) + A2 (z) such that U (z, f )(2f 3 (z) + (A(z) − 2)f (z) + C(z)) + V (z, f )(1 − f 2 (z)) = A2 (z) − C 2 (z). By Lemma 2.7, we see that 2f 3 (z) + (A(z) − 2)f (z) + C(z) and 1 − f 2 (z) are mutually prime. Lemma 2.10. (See [15, T heorem 1.51]) Suppose that f1 (z), f2 (z), . . . , fn (z)(n ≥ 2) are meromorphic functions and g1 (z), g2 (z), . . . , gn (z) are entire functions satisfying the following conditions, Pn (1) j=1 fj (z)egj (z) ≡ 0; (2) gj (z) − gk (z) are not constants for 1 ≤ j < k ≤ n; (3) For 1 ≤ j ≤ n, 1 ≤ h < k ≤ n, T (r, fj (z)) = o(T (r, egh (z)−gk (z) ))(r → ∞, r 6∈ E), where E ⊂ [1, ∞) is finite linear measure or finite logarithmic measure. Then fj (z) ≡ 0 (j = 1, . . . , n). Lemma 2.11. Suppose that f (z) is a transcendental meromorphic solution of finite order of the difference equation δA(z) f (z + 1) + f (z − 1) = , (2.1) 1 − δf (z) whereδ  = ±1, A(z) is a non-constant small function with respect to f (z), then 1 (i) λ f = λ(f ) = σ(f ); (ii) f (z) has at most one non-zero finite Borel exceptional value. Proof. (i) Since f (z) is a finite order transcendental meromorphic solution of the equation (2.1), then we have P (z, f ) := δf (z)(f (z + 1) + f (z − 1)) − (f (z + 1) + f (z − 1)) + δA(z) ≡ 0.

(2.2)

By (2.2), we obtain P (z, 0) = δA(z) 6≡ 0.

(2.3)

It follows from (2.3) and Lemma 2.1 that   1 m r, = S(r, f ), f outside of a finite exceptional set of logarithmic measure. Then   1 N r, = T (r, f ) + S(r, f ), f 1029

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that is, λ(f ) = σ(f ).   Next, we will prove λ f1 = σ(f ). It follows from (2.1) that δf (z)(f (z + 1) + f (z − 1)) = f (z + 1) + f (z − 1) − δA(z).

(2.4)

Set σ(f ) = σ < ∞, by Lemma 2.2, we have m(r, f (z + 1) + f (z − 1)) = O(rσ−1+ε ) + S(r, f ),

(2.5)

possibly outside of an exceptional set of finite logarithmic measure. By (2.1) and Lemma 2.3, we obtain   δA(z) T (r, f (z + 1) + f (z − 1)) = T r, = T (r, f ) + S(r, f ). (2.6) 1 − δf (z)   Since λ f1 ≤ σ(f ) < ∞, it follows from (2.5), (2.6) and Lemma 2.6 that T (r, f ) + O(rσ−1+ε ) + S(r, f ) = N (r, f (z + 1) + f (z − 1)) (2.7)

≤ 2N (r, f ) + O(rλ( f )−1+ε ) + O(log r), 1

  then λ f1 = σ(f ). (ii) By (i), we see that 0, ∞ are not the Borel exceptional values of f (z). Suppose that f (z) has two distinct finite Borel exceptional values a(6= 0) and b(6= 0, a). Set f (z) − a . (2.8) g(z) = f (z) − b Then

  1 σ(g) = σ(f ), λ(g) = λ(f − a) < σ(g), λ = λ(f − b) < σ(g). g

That is, 0 and ∞ are the Borel exceptional values of g(z). By [15, Theorem 2.11], we see that g(z) is of regular growth, then g(z) can be rewritten as σ

g(z) = P (z)edz ,

(2.9)

where d(6= 0) is a constant, σ(g) = σ(≥ 1) is a positive integer, P (z) is a meromorphic function with σ(P ) < σ(g) = σ. By (2.8) and (2.9), we have f (z) = b +

b−a b−a =b+ g(z) − 1 P (z)edzσ − 1

(2.10)

and b−a b−a =b+ , P (z + 1)P+1 (z)edzσ − 1 P (z + 1)ed(z+1)σ − 1 b−a b−a f (z − 1) = b + =b+ , P (z − 1)P−1 (z)edzσ − 1 P (z − 1)ed(z−1)σ − 1 f (z + 1) = b +

(2.11)

where ( P+1 (z) = exp d

σ   X σ

i

i=1

) z

σ−i

(

,

  σ X σ σ−i P−1 (z) = exp d (−1)i z i i=1

) .

Substituting (2.10) and (2.11) into (2.1) yields σ

σ

σ

C1 (z)e3dz + C2 (z)e2dz + C3 (z)edz + C4 (z) = 0,

1030

(2.12)

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where  C1 (z) = [δA(z) − 2b(1 − δb)]P (z)P (z + 1)P+1 (z)P (z − 1)P−1 (z);      C2 (z) = [(a + b)(1 − δb) − δA(z)]P (z)[P (z + 1)P+1 (z) + P (z − 1)P−1 (z)]     +[2b(1 − δa) − δA(z)]P (z + 1)P+1 (z)P (z − 1)P−1 (z);  C3 (z) = [δA(z) − (a + b)(1 − δa)][P (z + 1)P+1 (z) + P (z − 1)P−1 (z)]      +[δA(z) − 2a(1 − δb)]P (z);    C4 (z) = 2a(1 − δa) − δA(z).

(2.13)

It follows from (2.13) and Lemma 2.10 that C1 (z) ≡ C2 (z) ≡ C3 (z) ≡ C4 (z) ≡ 0. Since P (z)P (z + 1)P+1 (z)P (z − 1)P−1 (z) 6≡ 0 and C1 (z) ≡ C4 (z) ≡ 0, we obtain δA(z) ≡ 2b(1 − δb) and

δA(z) ≡ 2a(1 − δa).

(2.14)

Note that A(z) is a non-constant function, which shows that (2.14) is a contradiction.

3

Proof of Theorems

Proof of Theorem 1.1 P (z) is a rational solution of (1.6). Then (1.6) can be rewritten as Suppose that f (z) = Q(z) 

P (z + 1) P (z − 1) + Q(z + 1) Q(z − 1)

  P 2 (z) m(z) P (z) 1− 2 = · +C Q (z) n(z) Q(z)

(3.1)

or n(z)[Q2 (z) − P 2 (z)][P (z + 1)Q(z − 1) + P (z − 1)Q(z + 1)] = m(z)P (z)Q(z)Q(z + 1)Q(z − 1) + Cn(z)Q2 (z)Q(z + 1)Q(z − 1). (i) Suppose that m > n. If p > q, (3.2) yields  2 2   deg(n(z)[Q (z) − P (z)][P (z + 1)Q(z − 1) + P (z − 1)Q(z + 1)]) = n + 3p + q, deg(m(z)P (z)Q(z)Q(z + 1)Q(z − 1)) = m + 3q + p,   deg(Cn(z)Q2 (z)Q(z + 1)Q(z − 1)) = n + 4q.

(3.2)

(3.3)

By n + 3p + q > n + 4q and m + 3q + p > n + 4q, then we must have n + 3p + q = m + 3q + p, that is p − q = m−n 2 , m − n must be even. P (z) P (z+1) P (z−1) If p = q, then Q(z) → B, Q(z+1) → B, Q(z−1) → B as z → ∞, where B is a nonzero constant, while 

m(z) n(z)

→ ∞ as z → ∞. If B = 1 or B = −1, then

P (z + 1) P (z − 1) + Q(z + 1) Q(z − 1)

  P 2 (z) 1− 2 → 0, Q (z)

m(z) P (z) · + C → ∞ as z → ∞. n(z) Q(z)

If B 6= ±1, then    P (z + 1) P (z − 1) P 2 (z) + 1− 2 → 2B(1 − B 2 ), Q(z + 1) Q(z − 1) Q (z)

m(z) P (z) · + C → ∞ as z → ∞. n(z) Q(z)

These show that (3.1) is a contradiction. If p < q, (3.2) yields  2 2   deg(n(z)[Q (z) − P (z)][P (z + 1)Q(z − 1) + P (z − 1)Q(z + 1)]) = n + 3q + p, deg(m(z)P (z)Q(z)Q(z + 1)Q(z − 1)) = m + 3q + p,   deg(Cn(z)Q2 (z)Q(z + 1)Q(z − 1)) = n + 4q. 1031

(3.4)

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By m + 3q + p > n + 3q + p and n + 4q > n + 3q + p, then we must have m + 3q + p = n + 4q, that is q − p = m − n ≥ 1. (ii) Suppose that m = n. If p > q, by (3.3), we see that n + 3p + q > m + 3q + p > n + 4q. This shows that there is only one term in (3.2) which has the highest degree, a contradiction. P (z) P (z+1) P (z−1) If p = q, then Q(z) → B, Q(z+1) → B, Q(z−1) → B as z → ∞, where B is a nonzero constant, while m(z) n(z) → A as z → ∞, where A is a nonzero constant. If B = 1 or B = −1, then    P 2 (z) m(z) P (z) P (z + 1) P (z − 1) + 1− 2 → 0, · + C → AB + C as z → ∞. Q(z + 1) Q(z − 1) Q (z) n(z) Q(z) So we have C ± A = 0. If B 6= ±1, then    P (z + 1) P (z − 1) P 2 (z) + 1− 2 → 2B(1 − B 2 ), Q(z + 1) Q(z − 1) Q (z)

m(z) P (z) · + C → AB + C as z → ∞. n(z) Q(z)

So we have C = B[2(1 − B 2 ) − A]. If p < q, by (3.4), we see that n + 4q > n + 3q + p = m + 3q + p. This also shows that there is only one term in (3.2) which has the highest degree, a contradiction. (iii) Suppose that m < n. If p > q, by (3.3), we see that n + 3p + q > m + 3q + p, n + 3p + q > n + 4q, that is, there is only one term in (3.2) which has the highest degree, a contradiction. P (z+1) P (z−1) P (z) → B, Q(z+1) → B, Q(z−1) → B as z → ∞, where B is a nonzero If p = q, then Q(z) constant, while m(z) n(z) → 0 as z → ∞. If B = 1 or B = −1, then    P (z + 1) P (z − 1) P 2 (z) m(z) P (z) + · + C → C as z → ∞. 1− 2 → 0, Q(z + 1) Q(z − 1) Q (z) n(z) Q(z) So we have C = 0, a contradiction. If B 6= ±1, then    P (z + 1) P (z − 1) P 2 (z) → 2B(1 − B 2 ), + 1− 2 Q(z + 1) Q(z − 1) Q (z)

m(z) P (z) · + C → C as z → ∞. n(z) Q(z)

So we have C = 2B(1 − B 2 ). If p < q, by (3.4), we see that n + 4q > n + 3q + p > m + 3q + p. That is, there is only one term in (3.2) which has the highest degree, a contradiction. This completes the proof of Theorem 1.1. Proof of Theorem 1.2 In what follows, we consider two cases: A(z) ± C(z) 6≡ 0 and A(z) ± C(z) ≡ 0. Case 1, A(z) ± C(z) 6≡ 0.   (i) Using the same method as in the proof of Lemma 2.11 (i), we can obtain that λ f1 = λ(f ) = σ(f ).   1

(ii) First, we will prove λ

= σ(f ). By equation (1.7), Lemmas 2.3, 2.5 and 2.8, we

∆f (z) f (z)

have 

3T (r, f (z))

 A(z)f (z) + C(z) + S(r, f ) f (z)(1 − f 2 (z))   f (z + 1) + f (z − 1) + S(r, f ) T r, f (z)     f (z + 1) f (z) T r, + T r, + S(r, f ) f (z) f (z − 1)     f (z + 1) f (z + 1) 2T r, + S r, + S(r, f ) f (z) f (z)   f (z + 1) 2T r, + S(r, f ) f (z)   ∆f (z) 2T r, + S(r, f ), f (z)

= T = ≤ = ≤ =

r,

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that is, 3 T (r, f (z)) ≤ T 2

 r,

∆f (z) f (z)

 + S(r, w).

(3.5)

It follows from (3.5) and Lemma 2.4 that       ∆f (z) ∆f (z) ∆f (z) N r, = T r, − m r, f (z) f (z) f (z) 3 ≥ T (r, f (z)) + S(r, f ). 2   1 Thus, λ ∆f (z) = σ(f ). f (z)   Next, we will prove λ ∆f1(z) = σ(f ). By equation (1.7), ∆f (z) − ∆f (z − 1) = f (z + 1) + f (z − 1) − 2f (z) A(z)f (z) + C(z) − 2f (z) 1 − f 2 (z) 2f 3 (z) + (A(z) − 2)f (z) + C(z) . = 1 − f 2 (z) =

(3.6)

From (3.6), Lemmas 2.3, 2.5 and 2.9, we have   2f 3 (z) + (A(z) − 2)f (z) + C(z) 3T (r, f (z)) = T r, + S(r, f ) 1 − f 2 (z) = T (r, ∆f (z) − ∆f (z − 1)) + S(r, f ) ≤ 2T (r, ∆f (z)) + S(r, ∆f (z)) + S(r, f ) ≤ 2T (r, ∆f (z)) + S(r, f ), that is, 3 T (r, f (z)) ≤ T (r, ∆f (z)) + S(r, f ). 2 It follows from (3.7) and Lemma 2.4 that N (r, ∆f (z))

Hence, λ



1 ∆f (z)



(3.7)

= T (r, ∆f (z)) − m(r, ∆f (z))     ∆f (z) ≥ T (r, ∆f (z) − m r, + m(r, f (z)) f (z) 3 ≥ T (r, f (z)) − T (r, f (z)) + S(r, f ) 2 1 = T (r, f (z)) + S(r, f ). 2

= σ(f ).

Furthermore, we will prove λ



1 ∆2 f (z)



= σ(f ). By (3.6), we have

∆2 f (z − 1) = ∆f (z) − ∆f (z − 1) =

2f 3 (z) + (A(z) − 2)f (z) + C(z) . 1 − f 2 (z)

(3.8)

From (3.8), Lemmas 2.3, 2.5 and 2.9, we have   2f 3 (z) + (A(z) − 2)f (z) + C(z) 3T (r, f (z)) = T r, + S(r, f ) 1 − f 2 (z) = T (r, ∆2 f (z − 1)) + S(r, f )

(3.9)

= T (r, ∆2 f (z)) + S(r, f ).

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It follows from (3.9) and Lemma 2.4 that N (r, ∆2 f (z))

= T (r, ∆2 f (z)) − m(r, ∆2 f (z))     ∆f (z) + m(r, f (z)) ≥ 3T (r, f (z)) − m r, f (z) ≥ 3T (r, f (z)) − T (r, f (z)) + S(r, f ) =

Thus, λ



1 ∆2 f (z)



2T (r, f (z)) + S(r, f ).

= σ(f ). 

Finally, we will prove λ  N

r,

1

 = σ(f ). It follows from (3.9) and Lemma 2.4 that

∆2 f (z) f (z)

∆2 f f

 = T

    ∆2 f ∆2 f r, − m r, f f

≥ T (r, ∆2 f (z)) − T (r, f (z)) + S(r, f )

 Then, λ

1

=

3T (r, f (z)) − T (r, f (z)) + S(r, f )

=

2T (r, f (z)) + S(r, f ).



∆2 f (z) f (z)

= σ(f ).

(iii) Suppose that f (z) is a finite order transcendental meromorphic solution of equation (1.7). Set g(z) = f (z) − z. Then g(z) is a transcendental meromorphic function with σ(g) = σ(f ) < ∞ and τ (f ) = λ(g). Substituting f (z) = g(z) + z into (1.7) yields P (z, g) := [g(z + 1) + g(z − 1) + 2z][(g(z) + z)2 − 1] + A(z)g(z) + zA(z) + C(z) ≡ 0. (3.10) Since P (z, 0) = 2z(z 2 − 1) + zA(z) + C(z) 6≡ 0, it follows from (3.10) and Lemma 2.1 that   1 = T (r, g) + S(r, g) = T (r, f ) + S(r, f ). N r, g Then, τ (f ) = λ(g) = σ(g) = σ(f ). Case 2, A(z) ± C(z) ≡ 0. Rewriting equation (1.7) as f (z + 1) + f (z − 1) =

δA(z) , 1 − δf (z)

(3.11)

where δ = ±1. By Lemma 2.11, we see that (i) and (iv) hold. Using the same method as in the proof of Case 1 (ii) and (iii), we can also obtain (ii) and (iii). If 2z(1−δz)−δA(z) 6≡ 0, then τ (f ) = σ(f ). This completes the proof of Theorem 1.2. Acknowledgements. The authors would like to thank the referee for his/her thorough reviewing with constructive suggestions and comments to the paper.

References [1] W. Bergweiler and J. K. Langlely., Zeros of difference of meromorphic functions, Math. Proc. Cambridge. Philos. Soc. 142(2007), 133-147. [2] Y. M. Chiang and S. J. Feng., On the Nevanlinna characteristic of f (z + η) and difference equations in the complex plane, Ramanujan J. 16 (2008), 105-129. [3] Z. X. Chen and K. H. Shon., Value distribution of meromorphic solutions of certain difference Painlev´ e equations, J. Math. Anal. Appl. 364 (2010), 556-566. [4] W. K. Hayman., Meromorphic Function, Clarendon Press, Oxford, 1964. [5] R. G. Halburd and R. Korhonen., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314(2006), 477-487.

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[6] R. G. Halburd and R. Korhonen., Meromorphic solution of difference equation, integrability and discrete Painlev´ e equations, J. Phys. A:Math. Theory. 40(2007), 1-38. [7] R. G. Halburd and R. Korhonen., Finite-order meromorphic solutions and the discrete Painlev´ e equations, Proc. Lond. Math. Soc. 94(2007), 443-474. [8] K. Ishizaki., Admissible solutions of Schwarizan differential equation, J. Austral. Math. Soc. 50(1991), 258-278. [9] I. Laine, Nevanlinna Theory and Complex Differential Equations. W. de Gruyter, Berlin, 1993. [10] I. Laine and C. C. Yang., Clunie theorems for difference and q-difference polynomials, J. Lond. Math. Soc. 76(2007), 556-566. [11] S. Li and Z. S. Gao., Finite order meromorphic solutions of linear difference equations, Proc. Japan. Acad. Ser A. Math. Sci. 86(2010), 10-14. [12] Q. Li and Z. B. Huang., Some results on a certain type of difference equation originated from difference Painlev´ e I equation, Adv. Differ. Equ. 2015(2015):276, 1-11. [13] A. Z. Mokhon’ko., On the Nevanlinna characteristics of some meromorphic functions, in ”Theory of functions, functional analysis and their applicatins”, Izd-vo Khar’kovsk. Un-ta, 14(1971), 83-87. [14] J. M. Whillaker., Interpolatory Funtion Theory, Cambridge Tracts in Mathematics and Mathematical Physics. no. 33. Cambridge University Press, Cambridge, 1935. [15] C. C. Yang and H. X. Yi., Uniqueness theory of meromorphic functions, Science Press, Beijing, 1995. Kluwer Academic, Dordrecht, 2003.

Yun-Fei Du LMIB & School of Mathematics and Systems Science Beihang University Beijing 100191 P. R. China E-mail: [email protected] Min-Feng Chen LMIB & School of Mathematics and Systems Science Beihang University Beijing 100191 P. R. China Email: [email protected] Zong-Sheng Gao LMIB & School of Mathematics and Systems Science Beihang University Beijing 100191 P. R. China E-mail: [email protected]

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Catalan Numbers, k-Gamma and k-Beta Functions, and Parametric Integrals Feng Qi1,2,3,† 1

Abdullah Akkurt4

H¨ useyin Yildirim4

Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China

2

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China 3

Department of Mathematics, College of Science,

Tianjin Polytechnic University, Tianjin City, 300387, China 4

Department of Mathematics, Faculty of Science and Arts, University of ˙ Kahramanmara¸s S¨ ut¸cu ¨ Imam, 46100, Kahramanmara¸s, Turkey

†

Corresponding author: [email protected], [email protected]

Abstract In the paper, the authors establish some new explicit formulas and integral representations of the Catalan numbers and a class of parametric integrals in terms of the k-gamma and k-beta functions. 2010 Mathematics Subject Classification: Primary 26A42; Secondary 11B75, 11B83, 26A06, 26A51, 33B15. Key words and phrases: explicit formula; integral representation; Catalan number; parametric integral; k-gamma function; k-beta function.

1

Introduction and main results

The Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers that occur in various counting problems in combinatorial mathematics. The nth Catalan number can be expressed in
terms of the central binomial coefficients 2n by n
  4n Γ n + 12 1 2n (2n)! Cn = = =√ , (1) n+1 n n!(n + 1)! π Γ(n + 2) R∞ where Γ(x) = 0 tx−1 e−t d t for x > 0 is the classical Euler gamma function, or say, the Euler integral of the second kind. The Catalan numbers Cn were described in the 18th century by Leonhard Euler and are named after the Belgian mathematician Eug´ene Charles Catalan. In 1988, it came to light that the Catalan numbers Cn had been used in China by the Mongolian mathematician Ming Antu by 1730, see [11, 12, 14, 16, 17, 18, 19, 20, 35]. In recent years, the Catalan numbers Cn has been

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analytically generalized and studied in [15, 21, 23, 25, 26, 27, 28, 30, 29, 31, 33, 36] and closelyrelated references therein. For more information on the Catalan numbers Cn , please refer to the monographs [2, 3, 8, 9, 10, 32, 34] and closely-related references therein. It is common knowledge [1, p. 4] that the beta function, or say, the Euler integral of the first kind R1 B(x, y) can be defined by B(x, y) = 0 tx−1 (1−t)y−1 d t and satisfies B(x, y) = Γ(x)Γ(y) Γ(x+y) for x, y > 0. The rising factorial, denoted by (x)n , is defined [13, p. 13] by (x)n = x(x+1) · · · (x+n−1) = which is frequently called the Pochhammer symbol in mathematics. Diaz and Pariguan [5, p. 180] introduced the Pochhammer k-symbol as

Γ(x+n) Γ(x)

(x)n,k = x(x + k)(x + 2k) · · · [x + (n − 1)k]. It is clear that (x)n,1 = (x)n . Diaz et al. [4, 5, 6] introduced the k-gamma and k-beta functions and proved a number of their properties. They also studied the k-zeta function and the k-hypergeometric functions based on the Pochhammer k-symbols. The k-gamma function is defined in [5, p. 180] by n!k n (kn)x/k−1 , n→∞ (x)n,α

Γk (x) = lim

k > 0. k

It was showed [5, p. 180] that the Mellin transform of the exponential function e−t function, that is, Z ∞

k

tx−1 e−t

Γk (x) =

/k

/k

is the k-gamma

d t.

0

It is easy to see that Γ(x) = Γ1 (x),

Γk (x) = k

x/k−1

  x Γ , k

Γk (x + k) = xΓk (x).

(2)

This gives rise to the k-beta function 1 Bk (x, y) = k which satisfies

Z

1

tx/k−1 (1 − t)y/k−1 d t

(3)

0

  x y Γk (x)Γk (y) 1 , and Bk (x, y) = . Bk (x, y) = B k k k Γk (x + y)

(4)

The aim of this paper is to establish some new explicit formulas and integral representations of the Catalan numbers Cn and parametric integrals Z a
β Iα,β;k (a) = x(α+1)k−1 a2k − x2k d x 0

for a, k > 0 and some α, β > −1 in terms of the k-gamma function Γk (x) and the k-beta function Bk (x, y). Our main results in this paper can be stated as the following theorems. Theorem 1.1. For k > 0 and n ∈ N, we have
Z 2n+1 n k21+2n(1−k) 2 x(2n+1)k−1 3/2 4 Γk 2 k √ Cn = k √ = d x. π(n + 1) π Γk ((n + 2)k) 22k − x2k 0

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Theorem 1.2. For a, k > 0 and n ≥ 0, we have √ k 1/2 π (n+2)k Γk In,1/2;k (a) = a 4 Γk and r In,−1/2;k (a) =

π ank Γk k 2 Γk

n+1 2 k
n+4 2 k


=

n+1 2 k
n+2 2 k




  k 1/2 (n+2)k n+1 3 a Bk k, k 2 2 2

(6)

  ank n+1 1 = B k, k . 2 2 2

(7)

For a, k > 0 and α, β > −1, we have Iα,β;k (a) =

2

  α+1 aα+2βk+1 Bk , (β + 1)k . 2 2

(8)

Proofs of main results

We are now in a position to prove our main results stated in Theorems 1.1 and 1.2. From the second relation in (2), we have       √ (2n + 1)k 1 2n + 1 (2n+1)k/2k−1 (2n−1)/2 (2n−1)/2 (2n)! π k =k Γ . Γk =k Γ n+ =k 2 2k 2 4n n! Accordingly, it follows that √
√ √ (2n−1)/2 (2n)! π Γk 2n+1 k k n −3/2 (2n)! π −3/2 π Cn 2 4 n! = =k =k . Γk ((n + 2)k) k n+1 (n + 1)! 4n n!(n + 1)! 4n

The explicit formulas in (5) thus follow. By changing variables x = at1/2k for t ∈ [0, 1], we have Z a
(n+1)k−1 h 2k
2k i1/2 a 1/2k−1 t dt In,1/2;k (a) = at1/2k a − at1/2k 2k 0 Z a(n+2)k 1 (n+1)k/2k−1 = t (1 − t)1/2+1−1 d t 2k 0   Z n+1 3 a(n+2)k a(n+2)k 1 (n+1)k/2k−1 Bk k, k . = t (1 − t)3k/2k−1 d t = 2k 2 2 2 0 Utilizing the second relation in (4) gives

3 a(n+2)k Γk n+1 2 k Γk
2 k In,1/2;k (a) = . 2 Γk n+4 2 k Further using  Γk

3 k 2

 =k

1/2

  √ 3 kπ Γ = 2 2

and

  √ 1 Γ = π, 2

we obtain the formulas in (6).

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By the Leibniz rule for differentiation d dt

Z

h(t)

F (x, t) d x = h0 (t)F (h(t), t) − g 0 (t)F (g(t), t) +

Z

h(t)

g(t)

g(t)

∂ F (x, t) d x ∂t

in [7, p. 615], differentiating with respect to a on both sides of (6) gives 0 In,1/2;k (a)

2k−1

a

Z

x(n+1)k−1
1/2 d x = a2k − x2k

= ka

0

√ Γk kπ (n + 2)ka(n+2)k−1 4 Γk

n+1 2 k . (n+4)k
2




Therefore, it follows that √ Γk kπ In,−1/2;k (a) = (n + 2)ank 4 Γk

n+1 2 k
. n+4 2 k




(9)

The formulas in (7) are thus acquired. Letting a = 2 and replacing n by 2n in (9) derive k21+2n(1−k) π(n + 1)

2

Z 0


2n+1 n x(2n+1)k−1 3/2 4 Γk 2 k
√ = Cn . dx = k √ π Γk (n + 2)k 22k − x2k

The integral representation in (5) is thus proved.   By changing variables x = a sin1/k θ for θ ∈ 0, π2 , we have Z

π/2

a sin1/k θ

In,k (a) =


(α+1)k−1

a2k − a2k sin2 θ

0

a(α+2β+1)k = k a(α+2β+1)k = k

Z

π/2

sinα θ cos2β+1 θ d θ

0

π/2

Z

2k(α+1)/2k−1

sin

θ cos


β a sin1/k−1 θ cos θ d θ k

2k(β+1)/k−1

0

  a(α+2β+1)k α+1 θdθ = Bk k, (β + 1)k , 2 2

where we used in the last step the formula Z

π/2

sin2x/k−1 θ cos2y/k−1 θ d θ =

0

k Bk (x, y), 2

0, which is a generalization of [22, Remark 6.5]. Remark 3.2. This paper is a slightly revised version of the preprint [24].

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Generalized von Neumann-Jordan and James Constants for Quasi-Banach Spaces Waqas Nazeer1, Qaisar Mehmood2 , Shin Min Kang3,4,∗ and Absar Ul Haq1 1

Division of Science and Technology, University of Education, Lahore 54000, Pakistan e-mail: [email protected] (W. Nazeer) e-mail: [email protected] (A. U. Haq) 2

3

Government Science College, Wahdat Road, Lahore 54000, Pakistan e-mail: [email protected]

Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea e-mail: [email protected] 4

Center for General Education, China Medical University, Taichung 40402, Taiwan e-mail: [email protected]

Abstract (p) The generalized von Neumann-Jordan constant CNJ (B) and the James constant J(B) for a quasi-Banach space were introduced in [7]. In this note, it is shown that (p) (p) CNJ (B) ≤ 2 for any quasi-Banach space B and CNJ (B) < 2 if and only if B is uniformly (p) non-square. Along with relationship between J(B) and CNJ (B), the criterion for the uniformly smooth quasi-Banach space is also established. 2010 Mathematics Subject Classification: 46B20, 46E30 Key words and phrases: James constant, generalized von Neumann-Jordan constant, uniform non-square

1

Introduction

Among various geometric constants of a Banach space B, the von Neumann-Jordan constant CN J (B) for a Banach space B introduced by Clarkson [2] as the smallest constant C, for which the estimates 1 kx1 + x2 k2 + kx1 − x2 k2 ≤ ≤C C 2(kx1 k2 + kx2 k2 ) hold for any x1 , x2 ∈ B with (x1 , x2 ) 6= (0, 0). Equivalently   kx1 + x2 k2 + kx1 − x2 k2 CN J (B) = sup : x1 , x2 ∈ B with (x1 , x2) 6= (0, 0) . 2(kx1 k2 + kx2 k2 ) ∗

Corresponding author

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This idea was further enhanced by many authors in [2, 4, 5, 8, 9]. The James constant J(B) of a Banach space B is defined by  J(B) = sup min(kx1 + x2 k, kx1 − x2 k) : x1 , x2 ∈ SB ,

where SB is unit sphere. (p) In [3], the authors introduced the generalized von Neumann-Jordan constant CN J (B) which is defined as   kx1 + x2 kp + kx1 − x2 kp (p) CN J (B) = sup : x1 , x2 ∈ B with (x1 , x2 ) 6= (0, 0) 2p−1 (kx1 kp + kx2 kp ) (p)

and obtained the relationship between CN J (B) and J(B). This has an analog in the quasi-Banach space, that was considered in [7]. In this note, (p) (p) It is shown that CN J (B) ≤ 2 for any quasi-Banach space B and CN J (B) < 2 if and only if (p) B is uniformly non-square. A relationship between J(B) and CN J (B) is established. We also give the criterion for the uniformly smooth quasi-Banach space.

2

Preliminaries

Recall [1], that a quasi-norm on k·k on vector space B over a field K (R or C) is a mapping B → [0, ∞) with properties • kxk = 0 ⇐⇒ x = 0. • kαxk = |α|kxk for all α ∈ K and x ∈ B. • There exists a constant C ≥ 1 such that ∀x1 , x2 ∈ B we have kx1 + x2 k ≤ C(kx1 k + kx2 k). (p)

Definition 2.1. The generalized von Neumann-Jordan constant CN J (B) for a quasiBanach space is defined by   kx1 + x2 kp + kx1 − x2 kp (p) CN J (B) = sup : x1 , x2 ∈ B with (x1 , x2 ) 6= (0, 0) , 2p−1 C p (kx1 kp + kx2 kp ) where 1 ≤ p < ∞. (p)

The parametrized formula for the constant CN J (B) is given as   kx1 + tx2 kp + kx1 − tx2 kp (p) CN J (B) = sup : x , x ∈ S , 0 ≤ t ≤ 1 , 1 2 B C p 2p−1 (1 + tp ) By taking t = 1 and x1 = x2 , we obtain the estimate (p)

CN J (B) ≥

k2x1 kp 2p 1 ≥ = . p p−1 C2 (1 + 1) C2 (1 + 1) C 2 1044

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Definition 2.2. In a quasi-Banach space B the James constant is defined as   1 min(kx1 + x2 k, kx1 − x2 k) : x1 , x2 ∈ SB . J(B) = sup C Definition 2.3. A quasi-Banach space B is said to be uniformly non-square if there exists a positive number δ < 2 such that for any x1 , x2 ∈ SB , we have



 

x1 + x2 x1 + x2



≤ δ. min , C C

Remark 2.4. As in classical case, the quasi-Banach space B is uniformly non-square if and only if J(B) < 2 Definition 2.5. The modulus of uniform smoothness of a quasi-Banach space B is defined as   kx1 + tx2 k + kx1 − tx2 k 1 ρB (t) = sup − : x1 , x 2 ∈ S B , t ≥ 0 . 2C C Definition 2.6. A quasi-Banach space B is said to be uniformly smooth if (ρB )0+ (0) = limt→0+ ρBt(t) = 0. Definition 2.7. For any quasi-Banach space B and a real number p ∈ [0, ∞), JB,p (t) is defined by JB,p (t) = sup

3



kx1 + tx2 kp + kx1 − tx2 kp 2C p

1 p

 : x1 , x 2 ∈ S B .

Main results

Theorem 3.1. Let B be a non-trivial quasi-Banach space and p ∈ [1, ∞). Then JB,p (t) ≥ ρB (t) +

1 . C

Proof. By using convexity of the function f (u) = up on (0, ∞), one can easily obtained  p kx1 + tx2 k + kx1 − tx2 k kx1 + tx2 kp + kx1 − tx2 kp ≤ , 2 2 therefore



kx1 + tx2 k + kx1 − tx2 k 2C

which implies that

For p = 1, we have JB,1 (t) =

1 C

p

≤

kx1 + tx2 kp + kx1 − tx2 kp , 2C p

1 + ρB (t) ≤ JB,p(t). C 2 (t) = E(t, B), where + ρB (t) and for p = 2, we have 2C 2 JB,2

 E(t, B) = sup (kx1 + tx2 k2 + kx1 − tx2 k2 ) : x1 , x2 ∈ SB .

This completes the proof.

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Theorem 3.2. For any quasi-Banach space B, we have (a) JB,p (t) is a non-decreasing function. (b) JB,p (t) is convex. (c) JB,p(t) is continuous function. J (t)−1 (d) B,p t is a non-decreasing function. Proof. We only prove (a) and remaining are analogous to it. Let g(t) = kx1 + tx2 kp + kx1 − tx2 kp be a convex and even function. Let 0 < t1 ≤ t2 and x1 , x2 ∈ SB . Then we have   t2 + t1 t2 − t1 p p kx1 + t1 x2 k + kx1 − t1 x2 k = g(t1 ) = g t2 + (−t2 ) 2t2 2t2 t2 + t1 t2 − t1 ≤ g(t2 ) + g(t2 ) 2t2 2t2 = g(t2 ) = kx1 + t2 x2 kp + kx1 − t2 x2 kp p ≤ 2C p JB,p (t2 ),

which implies that

kx1 + t1 x2 kp + kx1 − t1 x2 kp p ≤ JB,p (t2 ). 2C p Hence JB,p (t1 ) ≤ JB,p (t2 ). Theorem 3.3. For any quasi-Banach space B and 1 ≤ p < ∞, the generalized von (p) (p) Neumann-Jordan constant CN J (B) satisfy the inequality CN J (B) ≤ 2. Proof. As we have already defined   kx1 + tx2 kp + kx1 − tx2 kp (p) CN J (B) = sup : x1 , x2 ∈ SB with (x1 , x2 ) 6= (0, 0) , C p 2p−1 (1 + tp ) where 0 ≤ t ≤ 1. By using definition of a quasi-Banach space, we have the following inequality kx1 + tx2 kp + kx1 − tx2 kp ≤ C p (kx1 k + tkx2 k)p + C p (kx1 k + tkx2 k)p = 2C p (kx1 k + tkx2 k)p = 2C p (1 + t)p, therefore

kx1 + tx2 kp + kx1 − tx2 kp 2(1 + t)p ≤ . C p 2p−1 (1 + tp ) 2p−1 (1 + tp )

(3.1)

The function f (u) = up is convex, which leads     1+t p 1 + tp (1 + t)p = 2 · ≤ 2p = 2p−1 (1 + tp ). 2 2 Using above inequality (3.1) become kx1 + tx2 kp + kx1 − tx2 kp 1 ≤ p−2 2p−1 = 2. p p−1 p C 2 (1 + t ) 2 4 1046

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Hence (p) CN J (B)

= sup



kx1 + tx2 kp + kx1 − tx2 kp : x1 , x2 ∈ S B , 0 ≤ t ≤ 1 C p 2p−1 (1 + tp )



≤ 2. This completes the proof. (p)

Next theorem presents the relationship between CN J (B) and J(B). Theorem 3.4. Let B be a non-trivial qausi-Banach space and p ∈ (1, ∞). Then the following inequality hold: q p−1 p (p) p J(B) ≤ 2 CN J (B). Proof. For any x1 , x2 ∈ SB , we have  kx1 + x2 k + kx1 − x2 k p 2(min{kx1 + x2 k, kx1 − x2 k}) ≤ 2 2   p kx1 + x2 k + kx1 − x2 kp ≤2 2 p kx1 + x2 k + kx1 − x2 kp p p−1 = p p−1 (kx1 kp + kx2 kp) p p .C 2 C 2 (kx1 k + kx2 k ) p



(p)

≤ C p CN J (B)2p−1 (kx1 kp + kx2 kp ) (p)

= 2 · 2p−1 C p CN J (B), (p)

({min{kx1 + x2 k, kx1 − x2 k})p ≤ 2p−1 C p CN J (B), q p−1 p 1 (p) p {min{kx1 + x2 k, kx1 − x2 k)} ≤ 2 CN J (B). C Taking supremum both side, sup



1 min{kx1 + x2 k, kx1 − x2 k} C



≤2

q

p−1 p p

(p)

CN J (B).

Therefore J(B) ≤ 2

q

p−1 p p

(p)

CN J (B).

This completes the proof. Theorem 3.5. For p ∈ (1, ∞), a quasi-Banach space B is uniformly non-square if and only if there exists δ ∈ (0, 1) such that for any x1 , x2 ∈ B, we have



p p

x1 + x2 p x1 − x2 p

+

≤ (2 − δ) kx1 k + kx2 k .

2C

2C 2

(3.2)

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Proof. Let B be an uniformly non-square quasi-Banach space and on contrary assume that (3.2) is not hold. Therefore for every positive integer n, there exists xn and yn in B such that



  p p

xn + yn p xn − yn p

+

> 2 − 1 kxn k + kyn k .

2C

2C n 2

Let xn ∈ SB and yn ∈ BB = {x ∈ B : kxk ≤ 1} for all n. With out loss of generality we assume that {kyn k} converges to some γ, where 0 ≤ γ ≤ 1 (by Bolzano-Wiestrass) we have



 

xn + yn p xn − yn p 1 1 + kyn kp

< 2−

2C + 2C n 2   C(kxn kp + kyn kp ) p ≤2 2C   1 + kyn k p =2 2   1 + kyn kp ≤2 , 2

letting n → ∞, we obtain (1 + γ)p = 2p−1 1 + γp therefore

=⇒

γ = 1,



xn + yn p xn − yn p

+

2C

2C −→ 2,

which implies that



xn + yn p

2C −→ 1



xn − yn p

−→ 1. and 2C

This contradiction to the fact that B is uniformly non-square. Conversely, suppose that



p p

x1 + x2 p x1 − x2 p

+

≤ (2 − δ) kx1 k + kx2 k .

2C

2C 2 In particularly, we have

which implies that



x1 + x2 p x1 − x2 p

+

2C

2C ≤ (2 − δ),



   1 p

x1 + x2 p x1 − x2 p δ



min

2C , 2C ≤ 1 − 2 .

Hence B is uniformly non-square. This completes the proof.

Theorem 3.6. For p ∈ (1, ∞), a quasi-Banach space B is uniformly non-square if and (p) only if CN J (B) < 2. 6 1048

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Proof. From the Theorem 3.5, B is uniformly non-square if and only if there exists 0 < δ < 1 such that



p p

x1 + x2 p x1 − x2 p

+

≤ (2 − δ) kx1 k + kx2 k .

2C

2C 2

Therefore

kx1 + x2 kp + kx1 − x2 kp ≤ 2, C p 2p−1 (kx1 kp + kx2 kp )

∀(x1 , x2 ) 6= (0, 0).

(p)

Hence CN J (B) < 2. (p)

Theorem 3.7. For any quasi-Banach space B and p ∈ (1, ∞), the inequalities CN J (B) < 2 and J(B) < 2 are equivalent. Proof. From the Remark 2.4, J(B) < 2 if and only if B is uniformly non-square. Therefore (p) by using Theorem 3.6, we have CN J (B) < 2. (p) Suppose that CN J (B) < 2. Then by using the Theorem 3.4, we have J(B) < 2

1

p−1 p

2 p = 2.

This completes the proof. Theorem 3.8. Let B be a quasi-Banach space, p ∈ [1, ∞) and t > 0. Then the following conditions are equivalent: (a) JB,p (t) < 1 + t. (b) J(t, x1 ) < 1 + t. Proof. (a) ⇒ (b) Suppose on contrary that J(t, x1 ) ≥ 1 + t. Then it is enough to take J(t, x1 ) = 1 + t. Since   1 J(t, B) = sup min(kx1 + x2 k, kx1 − x2 k) : x1 , x2 ∈ SB , C by using the definition of supremum, for any  > 0, there exist x1 , x2 ∈ SB such that kx1 + tx2 k + kx1 − tx2 k ≥ min{kx1 + tx2 k, kx1 − tx2 k} 2 ≥ c(1 + t − ).

(3.3)

Applying convexity of the function f (u) = up , we get 

kx1 + tx2 k + kx1 − tx2 k 2

p  1 p

≤



kx1 + tx2 kp + kx1 − tx2 k p 2

1

p

,

therefore from (3.3) 

kx1 + tx2 kp + kx1 − tx2 k 2

p  p1

≥ min{kx1 + tx2 k, kx1 − tx2 k} ≥ c(1 + t − ).

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Since  is any arbitrary so JB,p (t) ≥ 1 + t, which leads a contradiction. (b) ⇒ (a) Suppose on contrary that JB,p (t) ≥ 1 + t. Then it is enough to take JB,p (t) = 1 + t. Again using the definition of supremum, for any  > 0 there exists x1 , x2 ∈ SB such that kx1 + tx2 kp + kx1 − tx2 kp ≥ 2C p (1 + t − )p , also using kx1 + tx2 kp + kx1 − tx2 kp ≤ 2C p (1 + t)p , 2C p (1 + t)p ≥ kx1 + tx2 kp + kx1 − tx2 kp ≥ 2C p (1 + t − )p since  is arbitrary, so kx1 + tx2 kp + kx1 − tx2 kp = 2C p (1 + t)p , which implies that kx1 + tx2 k = kx1 − tx2 k = C(1 + t). So using the definition of J(t, x1 ), we get J(t, x1 ) ≥ 1+t, which lead to a contradiction. Corollary 3.9. Let B be a quasi-Banach space, p ∈ [1, ∞) and t > 0. Then the following conditions are equivalent: (a) B is uniformly non-square. (b) JB,p (t) < 1 + t. (c) J(t, x1 ) < 1 + t. Theorem 3.10. A quasi-Banach apace B is uniformly smooth if   JB,p (t) − C1 lim = 0. t→0 t Proof. Suppose that lim

t→0



JB,p (t) − t

1 C



= 0.

From Theorem 3.1 we know that JB,p (t) ≥ ρB (t) +

1 , C

which implies that

1 ≥ ρB (t), C dividing both side by t and applying the limt→0   JB,p (t) − C1 ρB (t) lim ≤ lim = 0. t→0 t→0 t t JB,p (t) −

So by definition B is uniformly smooth. 8 1050

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4

Quasi-Banach fixed point theorem (Contraction theorem)

˜ be a quasi-metric space. A mapping T : B → T is Definition 4.1. ([6]) Let B = (B, d) called a contraction on B if there exists a positive real number α < 1 such that for all x1 , x 2 ∈ B ˜ x1 , T x2 ) ≤ αd(x ˜ 1 , x2 ). d(T ˜ where B 6= ∅. Suppose B is Theorem 4.2. Consider a quasi-metric space B = (B, d), α complete and T : B → T be a contraction on B and suppose that C n β m → 0, where β = C . Then T has exactly one fixed point. Proof. Let any x0 ∈ B and define the iterative sequence by x0 ,

x 1 = T x0 ,

x 2 = T 2 x0 ,

x 3 = T 3 x0 ,

··· ,

x n = T n x0 ,

··· .

First, we show that this iterative sequence (xn ) is cauchy. For this we take ˜ m+1 , xm) = d(T ˜ xm , T xm−1 ) d(x ˜ m, xm−1 ) ≤ αd(x ˜ xm−1 , T xm−2 ) ≤ αd(T ˜ m−1 , xm−2 ) ≤ α2 d(x · · ·· · · ˜ 1 , x0 ). ≤ αm d(x Suppose that n > m and using the definition of quasi-metric
˜ m , xn) ≤ C d(x ˜ m , xn−1 ) + d(x ˜ n−1 , xn) d(x
˜ m , xn−2 ) + d(x ˜ n−2 , xn−1 ) + C d(x ˜ n−1 , xn ) ≤ C 2 d(x
˜ m , xn−3 ) + d(x ˜ n−3 , xn−2 ) + C 2 d(x ˜ n−2 , xn−1 ) + C d(x ˜ n−1 , xn ) ≤ C 3 d(x · · ·· · ·

˜ m , xm+1 ) + · · · + C d(x ˜ n−1 , xn ) ≤ C n−m d(x
˜ m , xm+1 ) + · · · + d(x ˜ n−1 , xn ) ≤ C n−m d(x   ˜ 1 , x0 ) ≤ C n−m αm + αm+1 + · · · + αn−1 d(x m α ˜ 1 , x0 ) ≤ C n−m d(x 1−α  α m 1 ˜ 1 , x0 ). = Cn d(x C 1−α Since ˜ m , xn ) ≤ C n d(x

 α m

1 ˜ d(x1 , x0 ) 1−α

C ˜ from our supposition d(xm , xn ) → 0 as m → ∞. Since B is complete so xm → x ∈ B. Next we show that this limit x is the fixed point of T . For this using the definition of quasi-metric we have ˜ T x) ≤ C(d(x, ˜ xm) + d(x ˜ m , T x)) d(x, ˜ xm) + αd(x ˜ m−1 , x)) ≤ C(d(x, 9 1051

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˜ T x) = 0. This implies T x = x. Now because (xm) converges to x, therefore we have d(x, we prove that x is the unique fixed point of T . For this let T x = x and T y = y, therefore we have ˜ y) = d(T ˜ x, T y) ≤ αd(x, ˜ y), d(x, ˜ y) = 0 because α < 1. Hence x = y. This completes the proof. which implies d(x,

5

Conclusion

Generalized von Neumann-Jordan and James constants studied by many researcher for Banach space for example in [2, 4, 5, 8, 9] and the references therein. In this paper, we introduce the generalized von Neumann-Jordan constant and the James constant for a quasi-Banach space. Relationships between James constant and generalized von NeumannJordan constant are also presented.

References [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston, 1988. [2] J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue spaces, Ann. Math., 38 (1937), 114–115. [3] Y. Cui, W. Huang, H. Hudzik and R. Kaczmarek, Generalized von Nuemann-Jordan constant and its relationship to the fixed point property, Fixed Point Theory Appl., 2015 (2015), Article ID 40, 11 pages. [4] M. Kato and L. Maligranda, On James and Jordan-von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl., 258 (2001), 457–465. [5] M. Kato, L. Maligranda and Y. Takahashi, On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces, Stud. Math., 144 (2001), 275–295. [6] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York-London-Sydney, 1978. [7] Y. C. Kwun, Q. Mehmood, W. Nazeer, A. U. Haq and S. M. Kang, Relations between generalized von Neumann-Jordan and James constants for quasi-Banach spaces, J. Inequal. Appl., 2016 (2016), Article ID 171, 10 pages. [8] Y. Takahashi and M. Kato, Von Neumann-Jordan constant and uniformly non-square Banach spaces, Nihonkai Math. J., 9 (1998), 155–169. [9] C. Yang, Jordan-von Neumann constant for Bana´s-Fr¸aczek space, Banach J. Math. Anal., 8 (2014), 185–192.

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Shared-values of meromorphic functions on annuli

∗

Si Jun Tao1 , Hong-Yan Xu2† and Zhao-Jun Wu3 1. School of Mathematics and Computer, Xinyu University, Xinyu, Jiangxi 338004, China email: [email protected] 2. Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China email: [email protected] 3. School of Mathematics and Statistics, Hubei University of Science and Technology, Xianning 437100, China email: [email protected]

Abstract In this paper, we study the shared values and uniqueness of meromorphic functions on annulus, and obtain one theorem about meromorphic functions on annulus sharing some distinct values, and this result is an improvement of some theorems given by Cao, Yi [4, 5], Kondratyuk and Laine[9]. Key words: Meromorphic function, Nevanlinna theory, the annulus. Mathematical Subject Classification (2010): 30D30, 30D35.

1

Introduction and main resut

In 1929, R.Nevanlinna(see [14]) first investigated the uniqueness of meromorphic functions in the whole complex plane and obtained the well-known theorem—5 IM theorem of two meromorphic functions sharing five distinct values. After his theorem, there are vast references on the uniqueness of meromorphic functions sharing values and sets in the whole complex plane(see [2, 16, 18]). In recent, the uniqueness problem of meromorphic functions with shared values in some angular domain attracted many investigations (see [3, 11, 19, 20]). Thus, we always assumed that the reader is familiar with the notations of the Nevanlinna theory such as T (r, f ), m(r, f ), N (r, f ) and so on (see [6, 16, 17]). We use C to denote the open complex plane, C to denote the extended complex plane, b and X ⊆ C. and X to denote the subset of C. Let S be a set of distinct elements in C ∗ This work was supported by the NSF of China(11561033, 11201395), the Natural Science Foundation of Jiangxi Province (20151BAB201008), and the Foundation of Education Department of Jiangxi of China (GJJ150902, GJJ151222). † Corresponding author.

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Define EX (S, f ) =

[

{z ∈ X|fa (z) = 0,

counting

multiplicities},

{z ∈ X|fa (z) = 0,

ignoring

multiplicities},

a∈S

E X (S, f ) =

[ a∈S

where fa (z) = f (z) − a if a ∈ C and f∞ (z) = 1/f (z). For a ∈ C, we say that two meromorphic functions f and g share the value a CM (IM ) in X (or C), if f (z) − a and g(z) − a have the same zeros with the same multiplicities(ignoring multiplicities) in X (or C). In addition, we also use f = a g = a in X (or C) to express that f and g share the value a CM in X (or C), f = a ⇐⇒ g = a in X (or C) to express that f and g share the value a IM in X (or C), and f = a =⇒ g = a in X (or C) to express that f = a implies g = a in X (or C). As we know, the whole complex plane C and angular domain all can be regarded as simply-connected regions, many results about the uniqueness of shared values and sets in the complex plane and angular domain can also be regarded as the uniqueness of meromorphic functions in simply-connected regions. Thus, it arises naturally an interesting subject on the uniqueness for the meromorphic functions in the multiply connected region? The main purpose of this paper is to study the uniqueness of meromorphic functions in doubly connected domains of complex plane C. From the Doubly Connected Mapping Theorem [1], we can get that each doubly connected domain is conformally equivalent to the annulus {z : r < |z| < R}, 0 ≤ r < R ≤ +∞. For two cases: r = 0, R = +∞ simultaneously and 0 < r < R < +∞, the latter case the homothety z 7→ √zrR reduces q R the given domain to the annulus {z : R10 < |z| < R0 }, where R0 = r . Thus, every annulus is invariant with respect to the inversion z 7→ z1 in two cases. The basic notions of the Nevanlinna theory on annuli will be showed in the next section. In recent, there have some results on the Nevannlina Theory of meromorphic functions on the annulus (see [7, 8, 10, 12, 13, 15]). In 2005, Khrystiyanyn and Kondratyuk [7, 8] proposed the Nevanlinna theory for meromorphic functions on annuli (see also [9]). Lund and Ye [12] in 2009 studied functions meromorphic on the annuli with the form {z : R1 < |z| < R2 }, where R1 ≥ 0 and R2 ≤ ∞. However, there are few results about the uniqueness of meromorphic functions on the annulus. In 2009 and 2011, Cao [4, 5] investigated the uniqueness of meromorphic functions on annuli sharing some values and some sets, and obtained an analog of Nevanlinna’s famous five-value theorem as follows: Theorem 1.1 ([5, Thereom 3.2] or [4, Corollary 3.3]). Let f1 and f2 be two transcendental or admissible meromorphic functions on the annulus A = {z : R10 < |z| < R0 }, where 1 < R0 ≤ +∞. Let aj (j = 1, 2, 3, 4, 5) be five distinct complex numbers in C. If f1 , f2 share aj IM for j = 1, 2, 3, 4, 5, then f1 (z) ≡ f2 (z). Remark 1.1 For the case R0 = +∞, the assertion was proved by Kondratyuk and Laine [9]. In this paper, we will focus on the uniqueness problem of meromorphic functions in the field of complex analysis and obtain the main result below which improve Theorem 1.1.

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Theorem 1.2 Let f and g be two transcendental or admissible meromorphic functions on the annulus A = {z : R10 < |z| < R0 }, where 1 < R0 ≤ +∞, aj ∈ C(j = 1, 2, 3, 4) be four distinct values. We assume that f and g share four distinct values aj (j = 1, 2, 3, 4) IM on A and E A (S, f ) ⊂ E A (S, g), where S = {b1 , . . . , bm }, m ≥ 1 and b1 , . . . , bm ∈ b \ {a1 , a2 , a3 , a4 }. Then f and g share all values CM on A, thus it follows that either C f ≡ g or f is a M¨ obius transformation of g. Furthermore, if the number of the values in S is odd, then f ≡ g. Remark 1.2 The special case m = 1 of Theorem 1.2 immediately yields Theorem 1.1. In fact, when m = 1, set S = {a5 }. If f, g share a5 IM on A, which implies E A (S, f ) ⊂ E A (S, g), then by Theorem 1.2, we can get f ≡ g.

2

Basic notions in the Nevanlinna theory on annuli

For a meromorphic function f on whole plane C, the classical notations of Nevanlinna theory are denoted as follows R

n(t, f ) − n(0, f ) dt + n(0, f ) log R, t 0 Z 2π 1 log+ |f (Reiθ )|dθ, T (R, f ) = N (R, f ) + m(R, f ), m(R, f ) = 2π 0 Z

N (R, f ) =

where log+ x = max{log x, 0}, and n(t, f ) is the counting function of poles of the function f in {z : |z| ≤ t}. Let f be a meromorphic function on the annulus A = {z : R10 < |z| < R0 }, where 1 < R < R0 ≤ +∞, the notations of the Nevanlinna theory on annuli will be introduced as follows, let Z

1

N1 (R, f ) = 1 R

n1 (t, f ) dt, t

1 m0 (R, f ) = m(R, f ) + m( , f ), R

Z N2 (R, f ) = 1

R

n2 (t, f ) dt, t

N0 (R, f ) = N1 (R, f ) + N2 (R, f ),

where n1 (t, f ) and n2 (t, f ) are the counting functions of poles of the function f in {z : t < |z| ≤ 1} and {z : 1 < |z| ≤ t}, respectively. Similarly, for a ∈ C, we have N 0 (r,

1 ) f −a

1 1 ) + N 2 (R, ) f −a f −a Z R n (t, 1 ) 1 ) n1 (t, f −a 2 f −a dt + dt t t 1

= N 1 (R, Z

1

= 1 R

in which each zero of the function f − a is counted only once. In addition, we use k) (k 1 1 n1 (t, f −a ) (or n1 (t, f −a )) to denote the counting function of poles of the function 1 with multiplicities ≤ k (or > k) in {z : t < |z| ≤ 1}, each point counted only f −a

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k)

(k

k)

(k

k)

once. Similarly, we have the notations N 1 (t, f ), N 1 (t, f ), N 2 (t, f ), N 2 (t, f ), N 0 (t, f ), (k

N 0 (t, f ). The Nevanlinna characteristic of f on the annulus A is defined by T0 (R, f ) = m0 (R, f ) − 2m(1, f ) + N0 (R, f ). For a nonconstant meromorphic function f on the annulus A = {z : R10 < |z| < R0 }, where 1 < R < R0 ≤ +∞, the following properties will be used in this paper (see [7])   1 (i) T0 (R, f ) = T0 R, , f f1 (ii) max{T0 (R, f1 · f2 ), T0 (R, ), T0 (R, f1 + f2 )} ≤ T0 (R, f1 ) + T0 (R, f2 ) + O(1), f2 1 (iii) T0 (R, ) = T0 (R, f ) + O(1), f or every f ixed a ∈ C. f −a In 2005, the lemma on the logarithmic derivative on the the annulus A was obtained by Khrystiyanyn and Kondratyuk [8]. Theorem 2.1 ([8]) (Lemma on the logarithmic derivative) Let f be a nonconstant meromorphic function on the annulus A = {z : R10 < |z| < R0 }, where R0 ≤ +∞, and let λ > 0. Then (i) in the case R0 = +∞,   f0 m0 R, = O(log(RT0 (R, f ))) f R for R ∈ (1, +∞) except for the set 4R such that 4R Rλ−1 dR < +∞; (ii) if R0 < +∞, then   T0 (R, f ) f0 = O(log( m0 R, )) f R0 − R R 0 dR for R ∈ (1, R0 ) except for the set 4R such that 40 (R0 −R) λ−1 < +∞. R

In 2005, the second fundamental theorem on the the annulus A was first obtained by Khrystiyanyn and Kondratyuk [8]. Later, the other forms of the second fundamental theorem on annuli were given by Cao, Yi and Xu [5]. Theorem 2.2 ([5, Theorem 2.3]) (The second fundamental theorem) Let f be a nonconstant meromorphic function on the annulus A = {z : R10 < |z| < R0 }, where 1 < R0 ≤ +∞. Let a1 , a2 , . . . , aq be q distinct complex numbers in the extended complex plane C. Let k1 , k2 , . . . , kq be q positive integers, and let λ ≥ 0. Then (i)

(q − 2)T0 (R, f )
4) is a positive integer, that is, on A, thus, we can get a contradiction.

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4 k

(10)

< 1, from f is transcendental or admissible

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Case 2. Suppose that ϕpq := q1m ϕ1 − p1m ϕ2 ≡ 0, for some positive integers p, q. Thus, we have  0  m m f (g − a1 ) · · · (g − a4 ) p (g − b1 )2 · · · (g − bm )2 ≡ · . (11) q (f − b1 )2 · · · (f − bm )2 g 0 (f − a1 ) · · · (f − a4 ) We will consider the two following subcases: Subcase 2.1. p 6= q. Without loss of generality, we may assume that p < q. For p1 q1 some two positive integers p1 and q1 , if z1 ∈ ΓA (aj ) for some j ∈ {1, 2, 3, 4}, then (11) p p1 implies that q = q1 . Hence q1 > p1 ≥ 1, and q1 ≥ 2 which means that any aj -points (j = 1, 2, 3, 4) of g on A are multiple. By Lemma 3.2 and f, g are transcendental or admissible on A, we can get 2T0 (R, g)

=

4 X

 N0

j=1

≤

1 R, g − aj

 + S(R, g)

  4 1X 1 N0 R, + S(R, g) 2 j=1 g − aj

≤ 2T0 (R, g) + S(R, g). Thus, we can get the following equalities easily   1 T0 (R, g) = N0 R, + S(R), g − aj and

 N0 R,

1 g − aj



 = 2N 0

1 R, g − aj

j = 1, 2, 3, 4

(12)

 + S(R),

j = 1, 2, 3, 4.

(13)

From (12) and (13), we can see that ”almost all” of aj -points of g have multiplicity 2, and ”almost all” of aj -points of f are simple on the annulus A. Without loss of generality, we may assume that f and g attain the values a3 and a4 on the annulus A. Set φ1 :=

2f 0 (f − a4 ) g 0 (g − a4 ) − (f − a1 )(f − a2 )(f − a3 ) (g − a1 )(g − a2 )(g − a3 )

φ2 :=

g 0 (g − a3 ) 2f 0 (f − a3 ) − . (f − a1 )(f − a2 )(f − a4 ) (g − a1 )(g − a2 )(g − a4 )

and

Since φi (i = 1, 2) is analytic at the poles of f and of g and also at those common aj -points of f and g which have multiplicity 1 with respect to f and multiplicity 2 with respect to g, 3.1, we have T0 (R, φi ) = S(R, f ), i = 1, 2. If φi 6≡ 0, then   by Lemma   1 1 N0 r, f −a4 ≤ N0 R, φ1 = S(R, f ), which contradicts to equation (13). Then φ1 ≡ 0. Similarly, we have φ2 ≡ 0. Therefore, from the definitions of φ1 and φ2 , we have 

f − a4 f − a3

2

1063

 ≡

g − a4 g − a3

2 .

(14)

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Since f 6≡ g, and from (14), we have g − a4 f − a4 ≡− , f − a3 g − a3 which implies that f and g share a3 , a4 CM on the annulus A. Since f and g assume the value a3 there exist positive integers p1 , q1 such that ΓpA1 q1 (a3 ) 6= ∅. From the considerations above we get q1 > p1 , contradicting the fact that f and g share a3 CM . Subcase 2.2. p = q. In this subcase, (10) becomes  0 m f (g − a1 ) · · · (g − a4 ) (g − b1 )2 · · · (g − bm )2 ≡ . (f − b1 )2 · · · (f − bm )2 g 0 (f − a1 ) · · · (f − a4 ) which implies that f and g share the four values aj (j = 1, 2, 3, 4) CM on the annulus A. From the conditions of this lemma and applying Lemma 3.3, g is a M¨obius transformation of f on A. Furthermore, two of the four values, say a1 , a2 are Picard exceptional values of f and g on the annulus A. Set ∆1 :=

f 0 (f − a4 ) g 0 (g − a4 ) − (f − a1 )(f − a2 )(f − a3 ) (g − a1 )(g − a2 )(g − a3 )

∆2 :=

f 0 (f − a3 ) g 0 (g − a3 ) − . (f − a1 )(f − a2 )(f − a4 ) (g − a1 )(g − a2 )(g − a4 )

and

Using the same argument as in Subcase 2.1 for ∆1 , ∆2 , we can get f − a3 g − a3 ≡− . f − a4 g − a4 We take the M¨ obius transformations T, M and L satisfying T (ω) :=

w − a3 , w − a4

M (w) := −w

L := T −1 ◦ M ◦ T.

and

Then we have T ◦ f = −T ◦ g,

hence

g = L ◦ f.

Thus, we can see that a3 and a4 are the fixed points of L. Therefore, there exist no fixed points of L in the set S. If some b ∈ S is given. Then from b 6= a1 , a2 , there exists a z0 ∈ C such that b = f (z0 ), and from E A (S, f ) ⊆ E A (S, g) we obtain L(b) = L(f (z0 )) = g(z0 ) ∈ S. So S is invariant under L. Furthermore, we have L ◦ L = I where I denotes the identical transformation. Thus, we can get that S must contain an even number of values. Thus, we complete the proof of Theorem 1.2.

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References [1] S. Axler, Harmomic functions from a complex analysis viewpoit, Amer. Math. Monthly 93 (1986), 246-258. [2] A. Banerjee, Weighted sharing of a small function by a meromorphic function and its derivativer, Comput. Math. Appl. 53 (2007), 1750-1761. [3] T. B. Cao and H. X. Yi, On the uniqueness of meromorphic functions that share four values in one angular domain, J. Math. Anal. Appl. 358 (2009), 81-97. [4] T. B. Cao and H. X. Yi, Uniqueness theorems of meromorphic functions sharing sets IM on annuli, Acta Mathematica Sinica (Chinese Series), 54 (4) (2011), 623-632. (in Chinese). [5] T. B. Cao, H. X. Yi and H. Y. Xu, On the multiple values and uniqueness of meromorphic functions on annuli, Comput. Math. Appl. 58 (2009), 1457-1465. [6] W. Hayman, Meromorphic functions, Oxford: Clarendon Press, 1964. [7] A. Ya. Khrystiyanyn and A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. I, Mat. Stud. 23(2005), No.1, 19-30. [8] A. Ya. Khrystiyanyn and A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. II, Mat. Stud. 24 (2005), No.2, 57-68. [9] A. A. Kondratyuk and I. Laine, Meromorphic functions in multiply connected domains, Laine, Ilpo (ed.), fourier series methods in complex analysis. Proceedings of the workshop, Mekrij¨ arvi, Finland, July 24-29, 2005. Joensuu: University of Joensuu, Department of Mathematics (ISBN 952-458-888-9/pbk). Report series. Department of mathematics, University of Joensuu 10, 9-111 (2006). [10] R. Korhonen, Nevanlinna theory in an annulus, value distribution theory and related topics, Adv. Complex Anal. Appl. 3 (2004), 167-179. [11] W. C. Lin, S. Mori and K. Tohge, Uniqueness theorems in an angular domain, Tohoku Math. J. 58(2006), 509-527. [12] M. Lund and Z. Ye, Logarithmic derivatives in annuli, J. Math. Anal. Appl. 356 (2009), 441-452 [13] M. Lund and Z. Ye, Nevanlinna theory of meromorphic functions on annuli, Sci. China. Math. 53 (2010), 547-554. [14] R. Nevanlinna, Eindentig keitss¨ atze in der theorie der meromorphen funktionen, Acta. Math. 48 (1926), 367-391. [15] H. Y. Xu and Z. X. Xuan, The uniqueness of analytic functions on annuli sharing some values, Abstract and Applied Analysis 2012 (2012), Art. 896596, 1-13. [16] H. X. Yi and C. C. Yang, Uniqueness theory of meromorphic functions, Science Press 1995/Kluwer 2003. [17] L. Yang, Value distribution theory, Berlin: Springer-Verlag/ Beijing: Science Press, 1993.

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[18] X. Y. Zhang, J. F. Chen and W. C. Lin, Entire or meromorphic functions sharing one value, Comput. Math. Appl. 56 (2008), 1876-1883. [19] J. H. Zheng, On uniqueness of meromorphic functions with shared values in some angular domains, Canad J. Math. 47 (2004), 152-160. [20] J. H. Zheng, On uniqueness of meromorphic functions with shared values in one angular domains, Complex Var. Elliptic Equ. 48 (2003), 777-785.

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On the Henstock-Pettis Integral for Fuzzy Number Valued Functions Yabin Shao1 , Zengtai Gong2 and Yuping Lian3 1 School of Science Chongqing University of Posts and Telecommunications, Nanan, 400065, Chongqing, P. R. China 2 College of Mathematics and Statistics Northwest Normal University, Lanzhou 730070, Gansu, P. R. China 3 Department of Mathematics Dingxi Teachers College, Dingxi 743000, Gansu, P. R. China

Abstract. In this paper, we show that the fuzzy Henstock-Pettis integration of n−dimensional fuzzy-number-valued function could be translated into the sum of the fuzzy Henstock-Pettis integration of a n−dimensional fuzzy-numbervalued function and the Henstock-Pettis integral of a vector-valued function which valued in the kernel sets. In addition, we give the Koml´ os-type convergence theorems for such integrals. Keywords. Fuzzy number, fuzzy Henstock-Pettis integral, representation theorems, convergence theorems.

AMS (MOS) subject classification: Primary 26E50; Secondary 28B20.

1

Introduction

It is well-known that the Henstock integral includes the Riemann, improper Riemann, Lebesgue and Newton integrals [9, 12]. It is also equal to the Denjoy and Perron integrals [14]. In the theory of integrals, there are some integrals based on the Banach space-valued functions such as Pettis and Bochner integrals [3, 14, 21]. In particular, Ziat [28, 29] and Amri and Hess [1] presented a characterization of Pettis integral having as their values convex weakly compact subsets of a Banach space. Bochner and Pettis integrals are all defined by using the Lebesgue integrability of the support functions. The integrals of fuzzy-number-valued functions, as a natural generalization of set-valued functions, have been discussed by Puri and Ralescu [19], Kaleva [10], and other authors [7, 24, 25, 27]. Recently, Wu and Gong [6, 8] discussed the fuzzy Henstock integrals of fuzzy-number-valued functions which extended Kaleva [10] integration. However, for a fuzzy valued function in the n−dimensional fuzzy number space E n , the integral and its characteristic theorems have not defined or discussed. In [2], the authors shown that a fuzzy-number valued function is fuzzy Henstock integrable if and only if it can be represented by a sum of a fuzzy McShane integrable fuzzy-number valued function and a fuzzy Henstock integrable fuzzy number valued function generated by a Henstock integrable function. In 2014, K. Musial [18] established the following decomposition theorem for fuzzy mappings with values in a Banach space: a fuzzy mapping is fuzzy Henstock integrable if and only if it can be represented as a sum of a fuzzy McShane integrable fuzzy mapping and of a fuzzy Henstock integrable fuzzy mapping generated by a Henstock integrable function. As a continuation of our previous work [16, 17, 22], in this paper, we continue to develop the theory of Henstock-Pettis integrals in fuzzy number spaces. By means of replacing the Lebesgue integrability of support functions with their Henstock integrability, we give the definitions of Henstock-Pettis integral and Aumann-Henstock-Pettis integral for compact convex set-valued functions. In addition, the relationships among Henstock-Pettis integral, AumannHenstock integral and Pettis integral are investigated. Furthermore, we present the Henstock-Pettis integral, fuzzy Henstock-Pettis integral and Aumann-Henstock-Pettis integral of n−dimensional fuzzynumber-valued functions, and the relationships among them are studied. At the same time, the representation theorems and the calculations of fuzzy Henstock-Pettis integral are given. It shows that the fuzzy Henstock-Pettis integration of a n−dimensional fuzzy-number-valued function equals the sum of the Henstock-Pettis integration of an n−dimensional fuzzy-number-valued function and the Henstock-Pettis integration of a vector-valued function which valued in the kernel sets. The rest of the paper is organized as follows. In section 2, the definitions of Pettis integral, Henstock-Pettis integral and Aumann-Henstock integral for compact convex set-valued functions are

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given. In section 3, we disscus the characterization of Henstock-Pettis integral, fuzzy HenstockPettis integral and Aumann-Henstock integral for fuzzy-number-valued functions, and we give the representation theorems of fuzzy Henstock-Pettis integral. In section 4, we present a Koml´os-type convergence theorem to the fuzzy Henstock-Pettis integral and give an existence theorem for a kind of fuzzy integral inclusion. And in section 5, we present some concluding remarks.

2

Preliminaries

Let T be the closed interval on the real line R, i.e., T = [a, b] (a, b ∈ R). |T | denotes the length of T . Throughout this paper, we use Pk (Rn ) to denote the family of all nonempty compact convex subsets of Rn . For A, B ∈ Pk (Rn ), k ∈ R, the addition and scalar multiplication are defined by the equations as follows respectively:   A + B = x + y | x ∈ A, y ∈ B , aA = ax | x ∈ A . In addition, for A, B ∈ Pk (Rn ), the Hausdorff metric between them defined by:  d(A, B) = max sup inf ka − bk, sup inf kb − ak . b∈B a∈A

a∈A b∈B

A compact convex set-valued function F : T → Pk (Rn ) is said to be measurable if {t ∈ T |F (t)∩O 6= φ} is a measurable set for any open subset O ⊂ Rn . F is said to be scalarly measurable if the map σ(x, F (·)) is measurable for every x ∈ S n−1 . Certainly, a compact convex set-valued function F : T → Pk (Rn ) is measurable if it is scalarly measurable. A function f : T → Rn is called a selection of F if f (t) ∈ F (t) for any t ∈ T. A selection f is said to be measurable if the function f is strongly measurable, i.e., f is a limit of an almost everywhere convergent sequence of measurable simple functions. A compact convex set-valued function F : T → Pk (Rn ) is said to be graph measurable if the set {(t, x) ∈ T × Rn |x ∈ F (t)} is a member of the product σ−algebra generated by L and the Borel subsets of Rn in the norm topology. Here L denotes the family of all Lebesgue measurable subsets of T. Definition 2.1 ([27]). For A ∈ Pk (Rn ), x ∈ S n−1 , the support function of A is defined by σ(x, A) = sup hy, xi, y∈A

where S n−1 denotes the unit sphere of Rn , h·, ·i is the inner product in Rn . Next, we shall give the definitions of Pettis integral, Henstock-Pettis integral and AumannHenstock-Pettis integral for compact convex set-valued functions. Definition 2.2. A set-valued function F : T → Pk (Rn ) is said to be Henstock integrable to I ∈ Pk (Rn ) if for every ε > 0 there is a function δ(x) > 0 such that for any δ−fine division Π = {ξi , [xi−1 , xi ]} of T , we have X d(I, F (ξi )(xi − xi−1 )) < ε, i

and write (H)

R T

F (x)dx = I.

 Lemma 2.3 ([27]). If A ∈ Pk (Rn ), x ∈ S n−1 , then A = y ∈ Rn |hy, xi ≤ σ(x, A), x ∈ S n−1 . Lemma 2.4 ([26]). If Ar ∈ Pk (Rn ), {Arm } ⊂ Pk (Rn ), where rm is converging nondecreasingly to r ∞ T and Arm ⊃ Arm+1 ⊃ Ar (m = 1, 2, · · · ) for any x ∈ S n−1 , then Ar = Arm if σ(x, Arm ) converge m=1

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Theorem 2.5. A
set-valued function F : T → Pk (Rn ) is Henstock integrable on T iff the real-valued function σ x, F (t) is Henstock integrable uniformly on T for any x ∈ S n−1 , and Z Z

σ x, (H) F (t)dt = (H) σ x, F (t) dt. (2.1) T

T

Definition 2.6. Let F : T → Pk (Rn ) be a measurable set-valued function. F is said to be Pettis (Henstock-Pettis) integrable on T if there is a nonempty set A ∈ Pk (Rn ) such that for any x ∈ S n−1 we have Z σ(x, A) = (L) σ(x, F (t))dt T

Z (σ(x, A) = (H)

σ(x, F (t))dt), T

and write A = (P )

R T

F (t)dt (A = (wH)

R T

F (t)dt).

In particular, if the set-valued function above is defined by F : T → Rn , then the set A will become a vector in Rn , and for any x ∈ S n−1 we have Z < x, A >= (L) < x, F (t) > dt T

Z (< x, A >= (H)

< x, F (t) > dt). T

In this case, F is also said to be Pettis (Henstock-Pettis) integrable on T . Theorem 2.7. If F : T → Pk (Rn ) is a measurable set-valued function, then the family of measurable selections of F is not empty. Proof. Since Rn is a separable space, we can prove the theorem easily. Now, we use sH (F ) to denote the family of Henstock-Pettis integrable selections and sP (F ) to denote the family of Pettis integrable selections of F . Definition 2.8. The Aumann-Henstock integral of a measurable set-valued function F : T → Pk (Rn ) defined by Z Z F (t)dt = {(HP )

(AH) T

f (t)dt|f ∈ sH (F )}. T

Definition 2.9. Pick a set-valued function F : T → Pk (Rn ) and let I ⊂ T . The function f : A → Pk (Rn ) is the weak derivative of F on T if the Banach valued function (σ(x, F ))0 is differentiable almost everywhere on I and (σ(x, F ))0 = σ(x, f )) almost everywhere on T . Example. Let F : T → Pk (Rn ) be weakly differentiable. Then its weak derivative F 0 is Henstock-Pettis integrable and Z s

F 0 (t)dt = F (s) − F (a), s ∈ T.

(HP ) a

Indeed, F has the weak derivative at a point t means that there is a point F 0 (t) ∈ Rn such that for any x ∈ S n−1 , we have lim

4t→0

< x, F (t + 4t) > − < x, F (t) > =< x, F 0 (t) > . 4t

Since < x, F > is differentiable, so we have Z < x, F (s) > − < x, F (a) >= (H)

s

< x, F >0 dt, s ∈ T.

a

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On the other hand, < x, F >0 =< x, F 0 >, it implies that Z < x, F (s) > − < x, F (a) >= (H)

s

< x, F 0 (t) > dt.

a

That is

Z < x, F (s) − F (a) >= (H)

s

< x, F 0 (t) > dt.

a

Hence

Z F (s) − F (a) = (HP )

s

F 0 (t)dt.



a

Theorem 2.10. If all measurable selectionsRof F : T → Pk (Rn ) are Henstock-Pettis integrable and σ(x, F (t)) is Henstock integrable, then (AH) T F (t)dt is a compact convex set. Proof. Since σ(x, F (t)) is Henstock integrable, so σ(x, F (t)) is measurable. Now fix a measurable selection f of F and let G(t) = F (t) − f (t). Since f is Henstock-Pettis integrable, so G is AumannHenstock integrable. R Let IT = (AH) T G(t)dt and D be a countable dense subset of S n−1 . We prove the convexity of IT first. It can be proved that Z Z (AH) G(t)dt = {(wH) g(t)dt|g ∈ sH (F − f )} T

T

is a convex set. In fact, for any A, B ∈ IT , there exist g1 (t), g2 (t) ∈ sH (F − f ) such that Z Z A = (HP ) g1 (t)dt, B = (HP ) g2 (t)dt. T

(2.2)

T

In addition, for any λ ∈ [0, 1], Z λA + (1 − λ)B

Z

= λ(HP ) g1 (t)dt + (1 − λ) g2 (t)dt T Z T = (HP ) (λg1 (t) + (1 − λ)g2 (t))dt. T

That is, λA + (1 − λ)B ∈ IT . In order to prove the compactness of IT , we take a sequence of points xn ∈ IT , and then there exists R gn ∈ sH (G) with xn = (HP ) T gn (t)dt. For each n ∈ N, t ∈ T, x ∈ S n−1 , we have the inequalities −σ(−x, G(t)) ≤< x, gn (t) >≤ σ(x, G(t)).

(2.3)

Since f ∈ sH (F ) and the null function is included in G(t), σ(x, G(t)) is nonnegative Henstock integrable. It implies that the support function σ(x, G(t)) is Lebesgue integrable. Thus, each < x, gn > is Lebesgue integrable and Z Z Z (L) | < x, gn (t) > |dt ≤ (L) σ(x, G(t))dt + (L) σ(−x, G(t))dt. T

T

T

Furthermore, due to the countability of D and L1 −boundeness of each < x, gn > we can find there exist hn ∈ conv{gn , gn+1 , · · · }, such that for each x ∈ D the sequence < x, hn > is almost everywhere convergent to a measurable function hx . As for each t and n we have hn (t) ∈ G(t) and G(t) is compact, there is a cluster point h(t) ∈ G(t). It follows that there is a set N of Lebesgue measure zero such that for any x ∈ D and t ∈ / N we have < x, h(t) >= lim < x, hn (t) >= hx (t). n→∞

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Taking into formula Eq. (2.3) and the Lebesgue dominated convergence theorem, we have for any x ∈ S n−1 , Z Z lim < x, (HP ) hn (t)dt > = lim (L) < x, hn (t) > dt n→∞ n→∞ T T Z = (L) < x, h(t) > dt T Z = < x, (HP ) h(t)dt > . (2.4) T

R

We write yn = (HP ) T hRn (t)dt, then yn ∈ IT , yn ∈ conv{xn , xn+1 , · · · } and the sequence {yn } convergent to y0 = (HP ) T h(t)dt. Thus, given an arbitrary sequence {xn }, xn ∈ IT , there is a convex combination of points yn ∈ conv{xn , xn+1 , · · · } and y0 ∈ IT Rsuch that yn converge to y0 . Consequently, the set IT is compact, i.e., there exists y0 = (HP ) T h(t)dt ∈ IT such that lim yn = y0 . n→∞

3

The Henstock-Pettis integral for fuzzy number valued functions

In this section, we give the definition of Henstock-Pettis integral of fuzzy-number-valued functions and its representation theorems.  Definition 3.1 ([4, 23]). Let E n = u|u : Rn → [0, 1]}. For any u ∈ E n , u is said to be a n−dimensional fuzzy number if the following conditions are satisfied: (1) u is a normal fuzzy set, i.e., there exists an
x0 ∈ Rn, such that u(x0 ) = 1; (2) u is a convex fuzzy set, i.e., u tx + (1 − t)y ≥ min u(x), u(y) for any x, y ∈ Rn , t ∈ [0, 1]; (3) u is upper semi-continuous; (4) suppu = {x ∈ Rn | u(x) > 0} is compact, here A denotes the closure of A. For rS∈ (0, 1], denote [u]r = {x ∈ Rn | u(x) ≥ r} and we call it the r−level set of u, and [u] = [u]r . 0

r∈(0,1]

E n denotes the n−dimensional fuzzy number space. If u ∈ E n , then [u]r is a nonempty compact convex subset of Rn for each r ∈ [0, 1]. Theorem 3.2 ([4, 23]). Define D : E n × E n → [0, ∞) by the equation D(u, v) = sup d([u]r , [v]r ), u, v ∈ E n , r∈[0,1]

then (1) (E n , D) is a complete metric space; (2) D(λu, λv) = |λ|D(u, v), λ ∈ R; (3) D(u + w, v + w) = D(u, v); (4) D(u + v, w + e) ≤ D(u, w) + D(v, e); (5) D(u + v, e 0) = D(u, e 0) + D(v, e 0); (6) D(u + v, w) ≤ D(u, w) + D(v, e 0). where u, v, w, e, e 0 ∈ En, e 0 = χ({0}) . The metric space (E n , D) has a linear structure, it can be imbedded isomorphically as a convex cone with vertex θ into the Banach space of functions u∗ : I × S n−1 −→ R, where S n−1 is the unit sphere in Rn , with an imbedding function u∗ = j(u) defined by u∗ (r, x) = sup < α, x > α∈[u]α

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for all < r, x >∈ I × S n−1 . Theorem 3.3 ([23]). There exists a real Banach space X such that E n can be imbedding as a convex cone C with vertex θ into X. Furthermore the following conditions hold true: (1) the imbedding j is isometric, (2) addition in X induces addition in E n , (3) multiplication by nonnegative real number in X induces the corresponding operation in E n , (4) C − C is dense in X, (5) C is closed. A fuzzy-number-valued function F˜ : [a, b] → E n is said to satisfy the condition (H) on [a, b], if for any x1 < x2 ∈ [a, b] there exists u ∈ E n such that f (x2 ) = f (x1 ) + u. We call u is the H-difference of F˜ (x2 ) and F˜ (x1 ), denoted F˜ (x2 ) −H F˜ (x1 ) ([10]). For brevity, we always assume that the condition (H) is satisfied when dealing with the operation of subtraction of fuzzy numbers throughout this paper. Definition 3.4 ([24, 25]). A fuzzy-number-valued function F˜ : T → E n is said to be fuzzy Henstock integrable on T if there exists a fuzzy number A˜ ∈ E n such that for every ε > 0 there is a function δ(x) > 0 such that for any δ−fine division Π = {ξi , [xi−1 , xi ]} of T , we have X ˜ F˜ (ξi )(xi − xi−1 )) < ε. D(A, i

We write (F H)

R T

˜ F˜ (x)dx = A.

Definition 3.5 ([22]). A fuzzy-number-valued function F˜ : T → E n is said to be Pettis ( HenstockPettis) integrable on T if [F (t)]r is Pettis ( Henstock-Pettis) integrable on T for every r ∈ [0, 1], and there exists a fuzzy number A˜ ∈ E n such that for any x ∈ S n−1 we have Z σ(x, [A]r ) = (L) σ(x, [F (t)]r )dt T

(σ(x, [A]r ) = (H)

Z

σ(x, [F (t)]r )dt).

T

We write A˜ = (F P )

R T

F˜ (t)dt (A˜ = (F HP )

R T

F˜ (t)dt).

Remark 3.6. In particular, if F˜ is degenerated into F : T → Rn and A˜ is degenerated into A ∈ Rn , then σ(x, [A]r ) =< x, A > . Remark 3.7. When n = 1, if the fuzzy-number-valued function F˜ : T → E 1 is Kaleva integrable on T (refer to [25]), then F˜ is also Pettis integrable. Remark 3.8. When n = 1, if the fuzzy-number-valued function F˜ : T → E 1 is fuzzy Henstock integrable on T (refer to the Definition 3.2 of [24]), then F˜ is also fuzzy Henstock-Pettis integrable. A fuzzy-number-valued function F˜ : T → E n is said to be measurable on T iff the compact convex set-valued function F r : T → Pk (Rn ) is measurable for any r ∈ [0, 1]. Definition 3.9. Let F˜ : T → E n be a measurable fuzzy-number-valued function, F˜ is said to be fuzzy Aumann-Henstock-Petiss integrable on T if Z Z (F AHP ) [F (t)]r dt = {(HP ) f (t)dt|f ∈ sHP [F (t)]r } T

T

r determines a unique fuzzy number A˜ ∈ E n , where sH [F (t)] denotes the family of all fuzzy HenstockR r ˜ Petiss integrable selections of [F (t)] . We write (F AH) T F˜ (t)dt = A.

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Theorem 3.10. Let F˜ : T → E n be a fuzzy Aumann-Henstock-Pettis integrable function on T . If the set-valued function [F (t)]r is measurable and the measurable selections of [F (t)]r are Henstock-Pettis integrable for any r ∈ [0, 1], then for every x ∈ S n−1 we have Z Z
σ x, (AHP ) [F (t)]r dt = (H) σ(x, [F (t)]r )dt. T

T

Proof. Since the measurable selections of [F (t)]r are Henstock-Pettis integrable for any r ∈ [0, 1], we have Z Z
σ x, (AHP ) [F (t)]r dt = (H) σ(x, [F (t)]r )dt T

for any x ∈ S

n−1

T

, r ∈ [0, 1].

Furthermore, by Theorem 2.10, we can easily obtain the following Theorem 3.11. Theorem 3.11. Let F˜ : T → E n be a fuzzy-number-valued function. If all measurable selections of [F (t)]r are integrable and σ(x, [F (t)]r ) is Henstock integrable for any r ∈ [0, 1], then R Henstock-Pettis r (AHP ) T [F (t)] dt is a compact convex set. Theorem 3.12. Let F˜ : T → E n be a fuzzy-number-valued function on T . If F˜ is fuzzy HenstockPettis integrable on T , then each measurable selection of [F (t)]r is Henstock-Pettis integrable for any r ∈ [0, 1] and t ∈ T . Proof. Since [F (t)]r is Henstock-Pettis integrable on T, for any r ∈ [0, 1] and t ∈ T , by Lemma 3 of [5], the conclusion holds. ˜ B ˜ ∈ E n , then A˜ ⊂ B ˜ if and only if σ(x, [A]r ) ≤ σ(x, [B]r ) for any r ∈ [0, 1] Theorem 3.13. If A, n−1 and x ∈ S . ˜ then for any r ∈ [0, 1] and x ∈ S n−1 we have Proof. Necessity: If A˜ ⊂ B, σ(x, [A]r )

= sup{< x, a > |a ∈ [A]r } ≤ sup{< x, b > |b ∈ [B]r } = σ(x, [B]r ).

(3.1)

Sufficiency: If σ(x, [A]r ) ≤ σ(x, [B]r ) for any r ∈ [0, 1] and x ∈ S n−1 , then for every a ∈ A, by the Lemma 2.3 we have < x, a >≤ σ(x, A) ≤ σ(x, B), thus a ∈ [B]r , that is [A]r ⊂ [B]r . Hence, ˜ A˜ ⊂ B. R Theorem 3.14. Let F˜ : T → E n be a fuzzy-number-valued function on T. If the integration (F H) T F˜ (t)dt exists, then the following statements are equivalent: (1) F˜ is fuzzy Henstock-Pettis integrable on T ; (2) For every Henstock-Pettis integrable function f ∈ sHP ([F (t)]1 ), there exists a fuzzy-number˜ : T → E n such that F˜ (t) = G(t) ˜ + f˜(t) and G ˜ is fuzzy Pettis integrable on T ; valued function G (3) For every f, h ∈ sH ([F (t)]1 ), h − f is Pettis integrable; (4) F˜ (x) is fuzzy Aumann-Henstock-Pettis integrable on T and for any x ∈ S n−1 , we have Z Z r σ(x, (AHP ) [F (t)] dt) = (H) σ(x, [F (t)]r )dt (r ∈ [0, 1]). T

T

Proof. (1) ⇒ (2): For every f (t) ∈ sH ([F (t)]1 ), since [F (t)]1 is Henstock-Pettis integrable on T, by Theorem 3.10, wa can infer that f (t) is Henstock-Pettis integrable on T . Define [G(t)]r = [F (t)]r − f (t), [G]r : T → Pk (Rn ), then for any x ∈ S n−1 , t ∈ T we have σ(x, [G(t)]r ) ≥ 0, and σ(x, [F (t)]r ) = σ(x, [G(t)]r )+ < x, f (t) > . 7 1073

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By Theorem 3.12, [G(t)]r is Pettis integrable. In addition, we can prove that {[G(t)]r , r ∈ [0, 1]} determines a fuzzy number. In fact, {[G(t)]r } satisfies the following conditions: (i) [G(t)]r is a nonempty compact convex set; (ii) if 0 ≤ r1 ≤ r2 ≤ 1, then σ(x, [G(t)]r1 )

= σ(x, [F (t)]r1 )− < x, f (t) > ≥ σ(x, [F (t)]r2 )− < x, f (t) > = σ(x, [G(t)]r2 )

(3.2)

That is [G(t)]r1 ⊇ [G(t)]r2 ; (iii) for any {rm } converging increasingly to r ∈ (0, 1], since for any x ∈ S n−1 we have σ(x, [F (t)]rm ) ↓ σ(x, [F (t)]r ), so σ(x, [G(t)]rm )

= σ(x, [F (t)]rm )− < x, f (t) > ↓ σ(x, [F (t)]r )− < x, f (t) > = σ(x, [G(t)]r ).

(3.3)

(2) ⇒ (3): Let f ∈ sHP ([F (t)]1 ), [G(t)]r = [F (t)]r − f (t). If h ∈ sH ([F (t)]1 ), then g = h − f is a measurable selection of [G]r , and −σ(−x, [G(t)]r ) ≤< x, g(t) >≤ σ(x, [G(t)]r ). R For every E ∈ L, we denotes wE = (P ) E [G(t)]r dt ∈ Pk (Rn ), then Z −σ(−x, wE ) = −(L) σ(−x, [G(t)]r )dt E Z ≤ (L) < x, g(t) > dt ZE ≤ (L) σ(x, [G(t)]r )dt = σ(x, wE ).

(3.4)

E

On the other hand, wE is compact and R its support function σ(x, wE ) is Lipschitz continuous uniformly with respect to x, therefore x → (L) E < x, g(t) > dt is continuous uniformly, it follows that g = h−f is Pettis integrable. (3) ⇒ (2): For f ∈ sHP ([F (t)]1 ), define [G(t)]r = [F (t)]r − f (t), then by assumption, each measurable selection g of [G(t)]r is Pettis integrable, and by Theorem 3.12, [G(t)]r is also Pettis integrable on T . Furthermore, we can prove {[G(t)]r , r ∈ [0, 1]} determines a fuzzy number similar to ˜ is Pettis integrable on T . (1) ⇒ (2). It shows that G (2) ⇒ (4) For f ∈ sH ([F (t)]1 ), the set-valued function [G(t)]r = [F (t)]r − f (t) is Pettis integrable on T . By the Theorem 3.12, [G(t)]r is Aumann-Henstock-Pettis integrable on T , and Z Z r (P ) [G(t)] dt = {(P ) g(t)dt|g ∈ sP ([G(t)]r )}. T

Note that (P )

R T

T

[G(t)]r dt is a compact convex set, then Z Z Z (AHP ) [F (t)]r dt = (P ) [G(t)]r dt + (HP ) f (t)dt T

T

T

is also a compact convex set. 8 1074

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R r We can R prover that {(AHP ) T [F (t)] dt | r ∈ [0, 1]} determines a unique fuzzy number. In fact, {(AHP ) T [F (t)] R dt | r ∈ [0, 1]} satisfies the following conditions: (i) (AHP ) T [F (t)]r dt is a compact convex set; (ii) if 0 ≤ r1 ≤ r2 ≤ 1, then Z Z Z r1 r1 σ(x, (AHP ) [F (t)] dt) = σ(x, (P ) [G(t)] dt + (HP ) f (t)dt) T T ZT Z = σ(x, (P ) [G(t)]r1 dt) + σ(x, (HP ) f (t)dt) ZT ZT ≥ σ(x, (P ) [G(t)]r2 dt) + σ(x, (HP ) f (t)dt) T T Z = σ(x, (AHP ) [F (t)]r2 dt). (3.5) T

(iii) for any {rm } converging increasingly to r ∈ (0, 1], since for every x ∈ S n−1 we have σ(x, [G(t)]rm ) ↓ σ(x, [G(t)]r ). Consequently, Z

σ(x, (AHP ) [F (t)]rm dt) T Z Z = σ(x, (P ) [G(t)]rm dt) + σ(x, (HP ) f (t)dt) T T Z Z ↓ σ(x, (P ) [G(t)]r dt) + σ(x, (HP ) f (t)dt) T T Z = σ(x, (AHP ) [F (t)]r dt). T

Thus, {(AHP ) T [F (t)]r dt | r ∈ [0, 1]} determines a unique fuzzy number. That is, F˜ (x) is fuzzy Aumann-Henstock-Pettis integrable on T , and Z Z σ(x, (AHP ) [F (t)]r dt) = (H) σ(x, [F (t)]r )dt. R

T

T

(4) ⇒ (1): Since F˜ (t) is fuzzy Aumann-Henstock-Pettis integrable on T , so [F (t)]r is AumannHenstock-Pettis integrable on T for any r ∈ [0, 1]. By Theorem 3.12, [F (t)]r is Henstock-Pettis integrable on T . Similar to the proof we can prove that {[F (t)]r , r ∈ [0, 1]} determines R of (2) ⇒ (4), n ˜ a unique fuzzy number (F AHP ) T F (t)dt ∈ E , i.e., F˜ (t) is fuzzy Henstock-Pettis integrable on T. ˜ : T → E n , f ∈ sHP ([F (t)]1 , then the fuzzy Henstock-Pettis Corollary 3.15. If F˜ : T → E n , G ˜ ˜ and integration of F could be translated into the Henstock-Pettis integration of G, Z Z Z ˜ (F HP ) F˜ (t)dt = (F P ) G(t)dt + (HP ) f (t)dt. T

T

T

Theorem 3.16. Let F˜ : T → E be a measurable fuzzy-number-valued function, σ(x, [F (t)]r ) Henstock integrable on T . If F˜ is fuzzy Henstock-Pettis integrable on T , then [G(t)]r = [F (t)]r − f (t) is Pettis integrable on T for any measurable selection f of [F (t)]1 , and Z Z Z (H) σ(x, [F (t)]r )dt = (L) σ(x, [G(t)]r )dt + (H) < x, f (t) > dt. n

T

for every t ∈ T, x ∈ S

n−1

T

T

. 9 1075

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Proof. Suppose f is a measurable selection of [F (t)]1 , [G(t)]r = [F (t)]r − f (t). Since the support function σ(x, [G(t)]r ) is a Henstock integrable set-valued function and < x, f (t) > is Henstock integrable, we see that [F (t)]r = [G(t)]r + f (t). On the other hand, since [G(t)]r has at least one Bochner integrable selection (the null function), for every E ∈ L there is wE ∈ Rn with Z σ(x, wE ) = (L) σ(x, [G(t)]r )dt E

for any x ∈ S n−1 . Hence, we have for every x ∈ S n−1 Z Z σ(x, wT ) + (H) < x, f (t) > dt = (H) σ(x, [F (t)]r )dt 6= ±∞. T

T

It follows that σ(x, wT ) 6= ±∞ for all x ∈ S n−1 . By Banach-Steinhaus Theorem, wT ∈ Pk (Rn ). And we get that every wE is bounded, thus [G(t)]r is Pettis integrable on T .

4

The Koml´ os-type convergence theorem for the Fuzzy HenstockPettis integrals and a fuzzy integral inclusion

The Koml´ os’s classical theorem (see[11]) yields that from any L1 −bounded sequence of real functions one can extract a subsequence such that the arithmetic averages of all its subsequence converge point almost everywhere. In [20], the author extended these results by providing a Koml´os-type theorem for set-valued functions under Henstock-Pettis integrability assumptions. In this section, we extend the Koml´os theorem to the case of the fuzzy-number-valued Henstock-Pettis integrals. As an application, an existence theorem for a fuzzy integral inclusion involving the fuzzy Henstock-Pettis integral is obtained. Definition 4.1. A sequence (F˜n )n of fuzzy-number-valued functions is said to be Koml´os convergent (K−convergent for short) to a fuzzy-number-valued function F˜ if for every subsequence (F˜kn )n there exists a µ−null set N ⊂ T , such that for all t ∈ T \N , n

σ(x, [F (t)]r ) = lim σ(x, n

1X [Fk (t)]r ). n i=1 i

Theorem 4.2. Let F˜n : T → E n be a sequence of (F HP )−integrable functions. Suppose (i) there exists a real Henstock integrable function f¯, such that f¯(t) ≤ σ(x, [Fn (t)]r ), and

Z sup(H) n∈N

∀t ∈ T, ∀n ∈ N;

σ(x, [Fn (t)]r )dt < +∞;

T

(ii) there exists a function h : T × R → [0, +∞) such that, for every t ∈ T , h(t, ·) is convex and compact, and a countable measurable partition (Bm )m of T satisfying: R (a) supn (H) Bm |σ(x, [Fn (t)]r )|dt < +∞; R (b) supn (H) Bm h(t, [Fn (t)]r )dt. Then there exist function F˜ and a subsequence of (F˜n )n which K−converges to R a (F HP )−integrable r ˜ F . Moreover, Bm h(t, [F (t)] )dt exist for each m ∈ N. 10 1076

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Proof. Consider the convex Y ⊂ E n , let the function g(t, C) = σ(x, C) be continuous on Y . By ˜ ˜ (ii) R and defn r5.1, the fuzzy sequence (Fn )n K−converges to F which is (F HP )−integrable and h(t, [F (t)] )dt exist for each m ∈ N. Bm Pn for every n ∈ N. We are By (i), the function −f¯(t) + σ(x, n1 i=1 [Fki (t)]r ) is Henstock integrable Pn now able to apply Fatou’s Lemma to the sequence (−f¯(t) + σ(x, n1 i=1 [Fki (t)]r ))n , and have Z (−f¯(t) + σ(x, [F (t)]r )dt T

Z

n

1X [Fk (t)]r ))dt n n i=1 i T Z Z n 1X [Fki (t)]r )dt = (H) (−f¯(t)dt + lim inf(H) σ(x, n n T T i=1 Z Z σ(x, [Fn (t)]r )dt ≤ +∞. ≤ (H) (−f¯(t)dt + sup

≤ lim inf(H)

(−f¯(t) + σ(x,

n∈N

T

(4.1)

T

Consequently, −f¯(t)+σ(x, [F (t)]r is (H)-integrable and, since f¯(t) is (H)-integrable, the H-integrability of σ(x, [F (t)]r follows. Every measurable selection f˜ of F˜ is HP −integrable, so we have −σ(−x, [F (t)]r ) ≤ hx, f (t)i ≤ σ(x, F (t)]r ),

a.e.t ∈ T.

Rb For every [a, b] ⊂ T , there exist A, such that hx, Ai = (H) a hx, f (t)i. Thus every measurable selection of F˜ is Hestock-Pettis integrable. Finally, by implication (4) ⇒ (1) in Theorem 3.14, we have Z Z F˜ (t)dt. F˜n (t)dt = (F HP ) lim (F HP ) n→∞

T

T

Corollary 4.3. Let (F˜n )n be a sequence of (F HP )−integrable functions satisfying hypothesis (i) in Theorem 4.2 and for every n ∈ N there exist F˜n0 (t), such that F˜n (t) ⊂ F˜n0 a.e. Then there exist a F HP −integrable function F˜ and a subsequence of (F˜n )n which K−converges to F˜ . Proof. Let Bm = {t ∈ T |m − 1 ≤ D(F˜n0 (t), ˜0) < m, ∀m ∈ N} satisfy hypothesis (ii) in Theorem 4.2. Then, for every m ∈ N, we have Z Z sup(H) |σ(x, [Fn (t)]r )|dt ≤ (H) |σ(x, [Fn0 (t)]r )|dt n∈N Bm Bm Z ≤ (H) D(F˜n0 (t), ˜0) < +∞. Bm

By Theorem 4.2, the conclusion holds. Theorem 4.4. Let (F˜n )n be a sequence of (F HP )−integrable functions satisfying hypothesis (i) in Theorem 4.2 and (ii0 ) there exists a measurable countable partition (Bm )m of T such that, for each m ∈ N, Z sup(H) D(F˜n (t), ˜0)dt < +∞. n∈N

Bm

Then there exist aR (F HP )−integrable function F˜ and a subsequence of (F˜n )n which K−converges ˜ to F Moreover, (H) Bm D(F˜ (t), ˜ 0)dt < +∞ for every m ∈ N. 11 1077

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In the sequel, using above Koml´ os-type convergence theorem, we give an existence theorem of a fuzzy integral inclusion as the following: t

Z

F˜ (s, x(s))ds,

x(t) ∈ ξ + (F HP )

t∈T

0

Theorem 4.5. Let U an open subset of fuzzy number space (E n , D), and F HP -integrable function ˜ : T → E n satisfy F˜ : T × U → E n and Γ ˜ ˜ (1) F (t, x) ⊂ Γ(t), ∀t ∈ T, ∀x ∈ U ; (2) F˜ (t, x) is upper semi-continuous for t ∈ T ; (3) σ(x, [F (·, x)]r ) is measurable for every x ∈ U . Rt ˜ Then, for every fixed ξ ∈ U , there exist t0 ∈ T such that ξ + (F HP ) 0 0 Γ(s)ds ⊂ U and t

Z

F˜ (s, x(s))ds

x(t) ∈ ξ + (F HP ) 0

has a solution in C([0, t0 ], E n ). ˜ : T → E n such that F˜ (t) = Proof. By Theorem 3.14, for all f˜ ∈ sHP ([F (t)]1 ), there exists G ˜ ˜ ˜ ˜ G(t) + f (t), and G is fuzzy Pettis integrable on T , then f is measurable. Fixing ξ ∈ U , we consider the open subset U1 and U2 of E n such that ξ ∈ U1 and U1 + U2 ⊂ U . R (·) Rt Since (F HP ) 0 f˜(t)dt is continuous, there exist t1 ∈ T such that (F HP ) 0 f˜(t)dt ∈ U2 for every t ∈ [0, t1 ]. We define a fuzzy-number-valued function F˜0 : [0, t1 ] × U1 −→ E n as the following: F˜0 (t, x) = (−1) · f˜ + F˜ (t, x + (F HP )

Z

t

f˜(τ )dτ ),

0

which satisfies the following conditions: ˜ (1) F˜ 0 (t, x) ⊂ G(t), ∀t ∈ T, ∀x ∈ U ; (2) for evry t ∈ T , F˜ 0 (t, x) is upper semi-continuous; (3) σ(x, [F 0 (·, x)]r ) is measurable for every x ∈ U . Rt ˜ Then we obtain that there exist t0 ∈ [0, t1 ] such that ξ + (F P ) 0 0 G(s)ds ∈ U1 , the integral inclusion t

Z

F˜ 0 (s, y(s))ds

y(t) ∈ ξ + (F HP )

(4.2)

0

has a solution in C([0, t0 ], E n ) and the set of solution is compact in C([0, t0 ], E n ). Therefore, we have Z

t0

ξ + (F HP )

˜ Γ(s)ds = ξ + (F HP )

Z

0

t0

t0

Z

f˜(s)ds + (F P )

0

˜ G(s)ds ⊂U

0

and we find y(t) ∈ C([0, t0 ], E n ) such that Z y(t) ∈ ξ + (F P )

t

(−1) · f˜(s) + F˜ (s, y(s) + (F HP )

0

That is Z y(t) + (F HP ) Thus x(·) = y(·) + (F HP )

Z

s

f˜(τ )dτ )ds,

0

t

f˜(s)ds ∈ ξ + (F HP )

Z

t

F˜ (s, y(s) + (F HP )

Z

0

0

R (·)

f˜(τ )dτ is a solution of the integral inclusion.

0

s

f˜(τ )dτ )ds.

0

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5

Conclusions

In this paper, we study the Henstock-Pettis integral of compact convex set-valued functions and fuzzy-number-valued function and the K−convergence theorem of fuzzy Henstock-Pettis integrals. We emphasize that the outcomes of the second part in our paper are different from the results in L. Di Piazza’s paper [5]. In the future research, we shall deals with a new derivative and Hestock-Pettis-∆integral for fuzzy-number-valued functions on time scales. Also, we shall study and investigate fuzzy differential equations and fuzzy integral equations with ∆H −derivative and F HP − ∆−integral on time scales.

Acknowledgements The authors thanks to the National Nature Science Foundation of China (Grant No. 11161041, No. 61472056 and No. 61262022), the PhD Research Startup Foundation of Chongqing University of Posts and Telecommunications(No. A2014-90, A2016-13) and Chongqing Municipal Science and technology project (cstc2015jcyjA00015).

References [1] K. EI. Amri, C. Hess, On the Pettis integral of closed valued multifunctions. Set-Valued Anal. 8 (2000), 329-360. [2] B. Bongiorno, L. Di Piazza, K. Musial, A decomposition theorem for the fuzzy Henstock integral. Fuzzy Sets and Syst. 200 (2012) 36-47. [3] M. Cicho´ n, Convergence theorems for the Henstock-Kurzweil-Pettis integral. Aata. Math. Hungar. 92 (2001) 75-82. [4] P. Diamond, P. Kloeden, Metric Space of Fuzzy Fets: Theory and Applications, World Scientific, Singapore, 1994. [5] L. Dipiazza, K. Musial, Set-valued Kurzweil-Henstock-Pettis integral. Set-valued Anal. 13 (2005) 167-169. [6] Zengtai Gong, On the problem of characterizing derivatives for the fuzzy-valued functions(II). Fuzzy Sets and Syst. 145 (2004) 381-393. [7] Zengtai Gong, Yabin Shao, The controlled convergence theorems for the strong Henstock integrals of fuzzy-number-valued functions. Fuzzy Sets and Syst. 160 (2009) 1528-1546. [8] Zengtai Gong, Liangliang Wang, The Henstock-Stieltjes integral for fuzzy-number-valued functions. Info. Sci. 188 (2012) 276-297. [9] R. Henstock, Theory of Integration, Butterworths, London, 1963. [10] O. Kaleva, Fuzzy differential equations. Fuzzy Sets and Syst. 24 (1987) 301-317. [11] J. Koml´ os, A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 (1967), 217-229. [12] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7 (1957) 418-446. [13] H. Lebesgue, Integral, longueur, maire. Annali Math, Pura Applic. 7 (1902) 231-359. 13 1079

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[14] P. Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989. [15] Baolin Li, Haide Gou, On the existence of weak solutions of nonlinear integral equations in Banach spaces. Mediterr. J. Math. 13 (2016), 2633-2643. [16] Qiang Ma, Yabin Shao, Zizhong Chen, A Kind of Generalized Fuzzy Integro-Differential Equations of Mixed Type and Strong Fuzzy Henstock Integrals, J. Comput. Anal. and Appl. 23(1) (2017), 92-107. [17] Qiang Ma, Yabin Shao, Zizhong Chen, Existence of Generalized Solutions for Fuzzy Impulsive Retarded Differential Equations, J. Comput. Anal. and Appl. In press. [18] K. Musial, Adecomposition theorem for Banach space valued fuzzy Henstock. Fuzzy Sets and Syst. 259 (2015) 21-28. [19] M. Puri, D. Ralescu, Fuzzy random variables. Math. Anal. Appl. 114 (1986) 409-422. [20] B. Satco, A Koml´ os-type theorem for the set-valued Henstock-Kurzweil-Pettis integral and applications. Czechoslovak Math. J. 56 (2006) 1029-1047. [21] S. Schwabik, Guoju Ye, Topics in Banach Space Integration, World Scientific, Singapore, 2005. [22] Yabin Shao, Zengtai Gong, Zizhong Chen, On the existence of generalized weak solutions to discontinuous fuzzy differential equations, Submitted. [23] Congxin Wu, Ming Ma, On embedding problem of fuzzy number spaces: Part 2. Fuzzy Sets and Syst. 45 (1992) 189-202. [24] Congxin Wu, Zengtai Gong, On Henstock integral of fuzzy-number-valued functions(I). Fuzzy Sets and Syst. 120 (2001) 523-552. [25] Congxin Wu, Zengtai Gong, On Henstock integral of interval-valued functions and fuzzy-numbervalued functions. Fuzzy Sets and Syst. 115 (2000) 377-391. [26] X. Xue, Y. Fu, Caratheodory solution of fuzzy differential equations. Fuzzy Sets and Syst. 125 (2002) 239-243. [27] Bokan Zhang, Congxin Wu, On the representation of n-dimensional fuzzy numbers and their informational content. Fuzzy Sets and Syst. 128 (2002) 227-235. [28] H. Ziat, Convergence theorems for Pettis integrable multifunctions. Bull. Polish Acad. Sci. Math. 45 (1997) 123-137. [29] H. Ziat, On a characterization of Pettis integrable multifunctions. Bull. Polish Acad. Sci. Math. 48 (2000) 227-230.

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Some Inequalities for Riemann Diamond Integrals on Time Scales Xuexiao Youa,b , Dafang Zhaoa,c,∗, Wei Liuc , Guoju Yec a School

of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China. b College of Computer and Information, Hohai University, Nanjing, Jiangsu 210098, P. R. China. c College of Science, Hohai University, Nanjing, Jiangsu 210098, P. R. China.

Abstract In this paper, we investigate the Diamond integral on time scales. By using Darboux approach, we define the Riemann Diamond integral on time scales and prove the corresponding theorems. Our results extend and improve the corresponding results on inequality of [8]. Keywords: Diamond integral, generalized H¨older’s inequality, generalized Jensen’s inequality, time scales 2010 MSC: 26D15, 26E70

1. Introduction The theory of time scales was born in 1988 with the Ph.D. thesis of Stefan Hilger, done under the supervision of Bernd Aulbach [9]. The aim of this theory was to unify various definitions and results from the theories of discrete and continuous dynamical systems, and to extend such theories to more general classes of dynamical systems. It has been extensively studied on various aspects by several authors [1,4,5,6,7,10,14,16]. Two versions of the calculus on time scales, the delta and nabla calculus, are now standard in the theory of time scales [5,6]. In 2006, the Diamond-alpha ∗ Corresponding

author Email addresses: [email protected] (Xuexiao You), [email protected] (Dafang Zhao), [email protected] (Wei Liu), [email protected] (Guoju Ye)

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integral on time scales was introduced by Sheng, Fadag, Henderson, and Davis [16], as a linear combination of the delta and nabla integrals. The Diamondalpha integral reduces to the standard delta integral for α = 1 and to the standard nabla integral for α = 0. We refer the reader to [2,3,11,12,13,15,16] for a complete account of the recent Diamond-alpha integral on time scales. In 2015, the Diamond integral on time scales, as a refined version of the diamondalpha integral, was introduced by Artur M. C. Brito da Cruz et al., [8]. In this paper we define and study the Riemann Diamond integral on time scales. Basic properties of the theory are proved. The paper is organized as follows. Section 2 contains basic concepts of time scales theory. In Section 3, definition of the Riemann diamond integral will be introduced. We will investigate basic properties of the Riemann diamond integral. In Section 4, we will establish generalized H¨older’s inequality, CauchySchwarz’s inequality, Minkowski’s inequality and Jensen’s inequality on time scales.

2. Preliminaries Let T be a time scale, i.e. a nonempty closed subset of R. For a, b ∈ T we define the closed interval [a, b]T by [a, b]T = {t ∈ T : a ≤ t ≤ b}. The open and half-open intervals are defined in an similar way. For t ∈ T we define the forward jump operator σ : T → T by σ(t) = inf{s ∈ T : s > t} where inf ∅ = sup T, while the backward jump operator ρ : T → T is defined by ρ(t) = sup{s ∈ T : s < t} where sup ∅ = inf T. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. If σ(t) = t, we say that t is right-dense, while if ρ(t) = t, we say that t is left-dense. A point t ∈ T is dense if it is right and left dense; isolated if it is right and left scattered. The forward graininess function µ : T → [0, ∞) and the backward graininess function η : T → [0, ∞) are defined by µ(t) = σ(t) − t, η(t) = t − ρ(t) for all t ∈ T respectively. If sup T is finite and left-scattered, then we define Tk := T\ sup T, otherwise Tk := T; if inf T is finite and right-scattered,

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then Tk := T\ inf T, otherwise Tk := T. We set Tkk := Tk

T

Tk .

A function f : T → R is called regulated provided its right-sided limits exist (finite) at all right-dense point of T and its left-sided limits exist (finite) at all left-dense point of T. A function f : T → R is called rd-continuous provided it is continuous at all right-dense points in T and its left-sided limits exist (finite) at all left-dense points in T. Assume f : T → R is a function and let t ∈ Tk . Then we define f ∆ (t) to be the number (provided it exists) with the property that given any ε > 0, there exists a neighborhood U of t such that |f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s)| ≤ |σ(t) − s| for all s ∈ U . We call f ∆ (t) the delta derivative of f at t and we say that f is delta differentiable on Tk provided f ∆ (t) exists for all t ∈ Tk . let t ∈ Tk . We define f ∇ (t) to be the number with the property that given any ε > 0, there exists a neighborhood U of t such that |f (ρ(t)) − f (s) − f ∇ (t)(ρ(t) − s)| ≤ |ρ(t) − s| for all s ∈ U . We call f ∇ (t) the nabla derivative of f at t and we say that f is nabla differentiable on Tk provided f ∇ (t) exists for all t ∈ Tk . Let t, s ∈ T and define µt,s := σ(t) − s and ηt,s := ρ(t) − s. We define f

♦α

(t) to be the number with the property that given any ε > 0, there exists a

neighborhood U of t such that |α(f (σ(t)) − f (s))ηt,s + (1 − α)(f (ρ(t)) − f (s))µt,s − f ♦α (t)µt,s ηt,s | ≤ |µt,s ηt,s | for all s ∈ U . We call f ♦α (t) the diamond-α derivative of f at t and we say that f is diamond-α differentiable on Tkk provided f ♦α (t) exists for all t ∈ Tkk . The real function γ(t) := lim

s→t

σ(t) − s . σ(t) + 2t − 2s − ρ(t)

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3. The Riemann Diamond integral A partition of [a, b]T is any finite ordered subset P = {t0 , t1 , . . . , tn } ⊂ [a, b]T , where a = t0 < t1 < . . . < tn = b. Each partition P = {t0 , t1 , . . . , tn } of [a, b]T decomposes it into subintervals [ti−1 , ti )T , i = 1, 2, . . . , n, such that for i 6= j one has [ti−1 , ti )T ∩ [tj−1 , tj )T = ∅. By P([a, b]T ) we denote the set of all partitions of [a, b]T . Let Pn , Pm ∈ P([a, b]T ). If Pn ⊂ Pm we call Pn a refinement of Pm . If Pn , Pm are independently chosen, then the partition Pn ∪ Pm is a common refinement of Pn and Pm . Let f : [a, b]T → R be a real-valued bounded function on [a, b]T . We denote M = sup{γ(t)f (t) : t ∈ [a, b)T }, m = inf{γ(t)f (t) : t ∈ [a, b)T }, M = sup{(1 − γ(t))f (t) : t ∈ (a, b]T }, m = inf{(1 − γ(t))f (t) : t ∈ (a, b]T }, and for 1 ≤ i ≤ n, Mi = sup{γ(t)f (t) : t ∈ [ti−1 , ti )T }, mi = inf{γ(t)f (t) : t ∈ [ti−1 , ti )T }, Mi = sup{(1−γ(t))f (t) : t ∈ (ti−1 , ti ]T }, mi = inf{(1−γ(t))f (t) : t ∈ (ti−1 , ti ]T }, Let γ(t) ∈ [0, 1]. The upper Darboux ♦-sum of f with respect to the partition P , denoted by U (f, P ), is defined by U (f, P ) =

n X (Mi + Mi )(ti − ti−1 ), i=1

while the lower Darboux ♦-sum of f with respect to the partition P , denoted by L(f, P ), is defined by L(f, P ) =

n X (mi + mi )(ti − ti−1 ). i=1

Note that U (f, P ) ≤

n X (M + M )(ti − ti−1 ) = (M + M )(b − a) i=1

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and

n X (m + m)(ti − ti−1 ) = (m + m)(b − a).

L(f, P ) ≥

i=1

(m + m)(b − a) ≤ L(f, P ) ≤ U (f, P ) ≤ (M + M )(b − a).

Thus, we have Definition

Let I = [a, b]T , where a, b ∈ T. The upper Darboux

3.1

♦−integral of f from a to b is defined by b

Z

f (t)♦t = a

inf

P ∈P([a,b]T )

U (f, P );

The lower Darboux ♦−integral of f from a to b is defined by b

Z

f (t)♦t = Rb

sup

L(f, P ).

P ∈P([a,b]T )

a

Rb

f (t)♦t, then we say that f is Riemann ♦−integrable on Rb [a, b]T , and the common value of the integrals, denoted by a f (t)♦t, is called If

a

f (t)♦t =

a

the Riemann ♦− integral. Definition

Let I = [a, b]T , where a, b ∈ T. The upper Darboux

3.2

∆−integral of f from a to b is defined by b

Z

f (t)∆t = a

inf

P ∈P([a,b]T )

U (f, P )

where U (f, P ) denote the upper Darboux sum of f with respect to the partition P and U (f, P ) =

n X

Mi (ti − ti−1 ), Mi = sup{f (t) : t ∈ [ti−1 , ti )T }.

i=1

The lower Darboux ∆−integral of f from a to b is defined by Z

b

f (t)∇t = a

sup

L(f, P ).

P ∈P([a,b]T )

where L(f, P ) denote the lower Darboux sum of f with respect to the partition P and L(f, P ) =

n X

mi (ti − ti−1 ), mi = inf{f (t) : t ∈ [ti−1 , ti )T }.

i=1

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If

Rb

f (t)∆t =

Rb

f (t)∆t, then we say that f is ∆−integrable on [a, b]T , and the Rb common value of the integrals, denoted by a f (t)∆t, is called the Riemann ∆− a

a

integral. Similarly, we can give the definition of the Riemann ∇− integral. We can easily get the following two theorems. Theorem 3.1

If γf : [a, b]T → R is Riemann ∆−integrable and (1 − γ)f :

[a, b]T → R Riemann ∇−integrable on the interval [a, b]T , then f : [a, b]T → R is Riemann ♦−integral on [a, b]T and b

Z

Z f (t)♦t =

a

b

Z

b

(1 − γ(t))f (t)∇t.

γ(t)f (t)∆t + a

a

Theorem 3.2 Let f : [a, b]T → R is Riemann ♦−integrable on the interval [a, b]T . (1) If γ(t) ≡ 1, then f is Riemann ∆−integrable on [a, b]T . (2) If γ(t) ≡ 0, then f is Riemann ∇−integrable on [a, b]T . (3) If 0 < γ(t) < 1, then f is Riemann ∆−integrable and Riemann ∇−integrable on [a, b]T . The proofs of the following two Theorem are standard and similar to [6, Theorem 5.5 and Theorem 5.6]. Theorem 3.3

Let L(f, P ) = U (f, P ) for some P ∈ P([a, b]T ), then the

function f is Riemann ♦−integrable on the interval [a, b]T and Z

b

f (t)♦t = L(f, P ) = U (f, P ). a

Theorem

3.4

(Cauchy criterion) Let f : [a, b]T → R be a bounded

function on the interval [a, b]T . Then the function f is Riemann ♦−integrable on the interval [a, b]T if and only if for every  > 0 there exists a partition P ∈ P([a, b]T ) such that U (f, P ) − L(f, P ) < . The following Lemma can be found in [7]. Lemma 3.5 Let I = [a, b]T be a closed (bounded) interval in T. For every δ > 0 there is a partition Pδ = {t0 , t1 , . . . , tn } ∈ P([a, b]T ) such that for each i one has: ti − ti−1 ≤ δ or ti − ti−1 > δ ∧ ρ(ti ) = ti−1 .

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The next theorem gives another Cauchy criterion for integrability. Theorem 3.6 A bounded function f on [a, b]T is Riemann ♦−integrable if and only if for each  > 0 there exists δ > 0 such that Pδ ∈ P([a, b]T ) implies U (f, Pδ ) − L(f, Pδ ) < . Proof. If for each  > 0 there exists δ > 0 such that Pδ ∈ P([a, b]T ) implies U (f, Pδ ) − L(f, Pδ ) < , then we have that f is integrable on [a, b]T by Theorem 3.4. Conversely, suppose that f is Riemann ♦−integrable on [a, b]T . If γ(t) ≡ 1 or γ(t) ≡ 0, then f is Riemann ∆−integrable or ∇−integrable on [a, b]T . Therefore condition holds from [6, Theorem 5.9]. Now, let 0 < γ(t) < 1, f is Riemann ♦− integrable, then γf is Riemann ∆−integrable and (1 − γ)f is Riemann ∇−integrable. For each  > 0 there exists δ 0 > 0 and δ 00 > 0 such that Pδ0 ∈ P([a, b]T ), Pδ00 ∈ P([a, b]T ) we have U (γf, Pδ0 ) − L(γf, Pδ0 )
1; then

b

Z

|h(t)||f (t)g(t)|♦t ≤

Z

a

b

 p1  Z |h(t)||f (t)| ♦t

b

p

 q1 |h(t)||g(t)|q ♦t .

a

a

Proof For nonnegative real numbers α, β and p, q such that 1 p

1 q

α p

p > 1, we have the well-known Young’s inequality α β ≤

+

1 1 p+q β q.

= 1 with

Without loss of generality, we suppose that Z b  Z b  |h(t)||f (t)|p ♦t |h(t)||g(t)|q ♦t 6= 0. a

a

Let α(t) = R b a

|h(t)||f (t)|p |h(t)||f (t)|p ♦t

, β(t) = R b a

|h(t)||g(t)|q |h(t)||g(t)|q ♦t

.

Consequently we have that 1 1 Z b |h(t)| q |g(t)| |h(t)| p |f (t)| R  p1  R  q1 ♦t b b a p ♦t q ♦t |h(t)||f (t)| |h(t)||g(t)| a a Z b 1 1 = α p (t)β q (t)♦t a

Z

b

≤

 α(t) p

a

Z = a

=

b

β(t)  ♦t q

|h(t)||f (t)|p

1 p

+

Rb a

|h(t)||f (t)|p ♦t

+

1 |h(t)||g(t)|q  ♦t R q b |h(t)||g(t)|q ♦t a

1 1 + = 1, p q

which completes the proof. For the particular case p = q = 2 in Theorem 4.1, we obtain the Cauchy-Schwarz’s inequality. Theorem 4.2 (Generalized Cauchy-Schwarz’s Inequality) Let f, g, h be ♦−integrable on the interval [a, b]T , then s Z b Z b  Z |h(t)||f (t)g(t)|♦t ≤ |h(t)||f (t)|2 ♦t a

a

b

 |h(t)||g(t)|2 ♦t .

a

Theorem 4.3 (Generalized Minkowski’s inequality) Let f, g, h be ♦−integrable on the interval [a, b]T and p > 1, then Z b  p1 Z b  p1  Z p p |h(t)||f (t) + g(t)| ♦t ≤ |h(t)||f (t)| ♦t + a

a

b

 p1 |h(t)||g(t)|p ♦t .

a

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Proof It follows from Theorem 4.1 that Z b |h(t)||f (t) + g(t)|p ♦t a

Z

b

|h(t)||f (t) + g(t)|p−1 (|f (t) + g(t)|)♦t

= a

Z

b

|h(t)||f (t) + g(t)|p−1 (|f (t)| + |g(t)|)♦t

≤ a

Z

b

|h(t)||f (t) + g(t)|p−1 |f (t)|♦t +

=

|h(t)||f (t) + g(t)|p−1 |g(t)|♦t

a

a

≤

b

Z

nZ

b

 q o q1  Z |h(t)| |f (t) + g(t)|p−1 ♦t

a

=

|h(t)||f (t)|p ♦t

 p1

a b

nZ + nZ

b

 q o q1  Z |h(t)| |f (t) + g(t)|p−1 ♦t

a b

b

|h(t)||g(t)|p ♦t

 p1

a

o q1 n Z p |h(t)||f (t) + g(t)| ♦t

a

b p

|h(t)||f (t)| ♦t

 p1

+

Z

b

 p1 |h(t)||g(t)|p ♦t .

a

a

Dividing both sides by b

nZ

|h(t)||f (t) + g(t)|p ♦t

o q1

,

a

we arrive to Minkowskis inequality: Z b  p1  Z Z b  p1 p p ≤ |h(t)||f (t)| ♦t + |h(t)||f (t) + g(t)| ♦t a

Theorem

a

b

 p1 |h(t)||g(t)|p ♦t .

a

4.4 (Jensen’s inequality) Let a, b ∈ T and c, d ∈ R. If g :

[a, b]T → (c, d) is rd-continuous and f : (c, d) → R is continuous and convex, then f

 R b g(t)♦t  a

b−a

Rb ≤

a

f (g(t))♦t . b−a

Proof Let x0 ∈ (c, d). Then for each x ∈ (c, d), there exists β such that f (x) − f (x0 ) ≥ β(x − x0 ).

11

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Rb

Let x0 =

a

g(t)♦t , b−a

Thus, b

Z

f (g(t))♦t − (b − a)f a

 R b g(t)♦t  a

b−a

b

Z

f (g(t))♦t − (b − a)f (x0 )

= a b

Z

(f (g(t)) − f (x0 ))♦t

= a

Z

b

≥ β

(g(t) − x0 )♦t a

Z

b

g(t)♦t − (b − a)x0 = 0,

= β a

which completes our proof. Similarly, we have the following Generalized Jensen’s inequality. (Generalized Jensen’s inequality) Let a, b ∈ T and c, d ∈ R. Rb If g : [a, b]T → (c, d), h : [a, b]T → R is rd-continuous with a |h(t)|♦t > 0 and Theorem 4.5

f : (c, d) → R is continuous and convex, then  R b |h(t)|g(t)♦t  R b |h(t)|f (g(t))♦t f aR b . ≤ a Rb |h(t)|♦t |h(t)|♦t a a References [1] G. A. Anastassiou, Frontiers in Time Scales and Inequalities, Series on Concrete and Applicable Mathematics, Vol.17, World Scientific, 2015. [2] M. R. Sidi Ammi, R. A. C. Ferreira, D. F. M. Torres, Diamond−α Jensen’s inequality on time scales, J. Inequal. Appl., 2008, Art. ID 576876, 13 pp. [3] M. R. Sidi Ammi, D. F. M. Torres, H¨older’s and Hardy’s two dimensional diamond-alpha inequalities on time scales, An. Univ. Craiova Ser. Mat. Inform., 2010, 37(1):1–11. [4] M. Bohner, G. S. Guseinov, The convolution on time scales, Abstr. Appl. Anal., 2007, Art. ID 58373, 24 pp.

12

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[5] M. Bohner, A. Peterson, Dynamic equations on time scales. An introduction with applications, Birkhauser, Boston, MA, 2001. [6] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2004. [7] A. M. C. Brito da Cruz, N. Martins, D. F. M. Torres, Symmetric differentiation on time scales, Appl. Math. Lett., 2013, 26(2):264–269. [8] A. M. C. Brito da Cruz, N. Martins, D. F. M. Torres, The Diamond Integral on Time Scales, Bull. Malays. Math. Sci. Soc., 2015, 38(4):1453–1462. [9] S.

Hilger,

Eın

Maßkettenkalk¨ ul

mıt

Anwendung

auf

Zentrumsmannıgfaltıgkeıten, Ph. D. Thesis, Universt¨at W¨ urzburg, 1988. [10] S. Hilger, Analysis on measure chains-A unified approach to continuous and discrete calculus, Results Math. 1990, (18):18–56. [11] A. B. Malinowska, D. F. M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales, Dynam. Systems Appl., 2009, 18(3-4):469–481. [12] D. Mozyrska, D. F. M. Torres, Diamond−α polynomial series on time scales, Int. J. Math. Stat., 2009, 5:92–101. [13] D. Mozyrska, D. F. M. Torres, A Study of Diamond-alpha dynamic equations on regular time scales, Afr. Diaspora J. Math., 2009, 8(1):35–47. [14] A. Peterson, B. Thompson, Henstock-Kurzweil Delta and Nabla Integrals, J. Math. Anal. Appl. 2006, 323:162–178. [15] J. W. Rogers. Jr., Q. Sheng, Notes on the diamond−α dynamic derivative on time scales, J. Math. Anal. Appl. 2007, 326(1):228–241. [16] Q. Sheng, M. Fadag, J. Henderson, J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl., 2006, 7(3):395–413. 13

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CONTINUITY AND CONTINUOUS HOMOGENEOUS SELECTIONS OF SET-VALUED METRIC GENERALIZED INVERSE IN BANACH SPACES SHAOQIANG SHANG1∗ AND YUNAN CUI2 Abstract. In this paper, upper semicontinuity and continuity for the setvalued metric generalized inverses T ∂ in Banach spaces are investigated by metric projection operator. Moreover, criteria for the set-valued metric generalized inverses to have continuous homogeneous selections are given. Finally, the relation of continuity and continuous selection of the set-valued metric generalized inverse are given.

1. Introduction and preliminaries Let (X, k · k) be a real Banach space. Let S(X) and B(X) denote the unit sphere and the unit ball of X, respectively. By X ∗ we denote the dual space of X. Let T denote a linear bounded operator from subspace of X into Banach space Y . Let D(T ), R(T ) and N (T ) denote the domain, range and null space of T , respectively. Let L be a subspace of X. The set-valued mapping PL : X → L   PL (x) = z ∈ L : kx − zk = dist(x, L) := inf kx − yk y∈L

is said to be the metric projection operator from X onto L. A subspace L is said to be proximinal if PL (x) 6= ∅ for all x ∈ X. Continuity of metric projection operator is an important content in geometry of Banach spaces. Moreover, metric projection operator plays an important role in the optimization, computational mathematics, theory of equation and control theory. The concept of generalized inverses has been extensively studied in the last decades, which has its genetic in the context of the so-called ”ill-posed” linear problems. If N (T ) 6= {0} or R(T ) 6= Y , the operator equation T x = y is generally ill-posed, i.e., there exists y0 ∈ Y such that kT x − y0 k 6= 0 for any x ∈ D(T ). In order to solve the best approximation problems for ill-posed linear operator equations in Banach spaces, it is necessary to study the set-valued metric generalized inverses of linear operators between Banach spaces. In 1974, Nashed and Votruba [9] introduced the concept of the set-valued metric generalized inverse of a linear operator between Banach spaces and they raised the following research Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 46B20. Key words and phrases. metric projection operator, 2-Chebyshev subspace, continuous homogeneous selection, set-valued metric generalized inverse. 1

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suggestion ”The problem of obtaining selections with nice properties for the metric generalized inverse merits study”. Moreover, it is well known that set-valued metric generalized inverses is a set-valued mapping. Hence upper semicontinuity and continuity of set-valued metric generalized inverses merit study. In 2015, Shang and Cui [8] gave a criteria for upper semicontinuity of the set-valued metric generalized inverses in approximative compact spaces. In this paper, upper semicontinuity and continuity for the set-valued metric generalized inverses T ∂ in Banach spaces are investigated by metric projection operator. Moreover, criteria for the set-valued metric generalized inverses to have continuous homogeneous selection are given. Finally, the relation of continuity and continuous selections of the set-valued metric generalized inverse are given. First let us recall some definitions that will be used in the further part of the paper. Definition 1.1. (see [7]) A subspace L ⊂ X is said to be k-Chebyshev subspace if L is proximinal and for any x ∈ X, we have dim (span {x − PL (x)}) ≤ k. I. Singer defined the k-strictly convex spaces in [15]. He proved that if X is reflexive and k-strictly convex, then every closed subspace of X is k-Chebyshev subspace. Moreover, it is easy to see that if L is a Chebyshev subspace, then L is k-Chebyshev. Definition 1.2. (see [4]) Set-valued mapping F : X → Y is said to be upper semicontinuous at x0 , if for each norm open set W with F (x0 ) ⊂ W , there exists a norm neighborhood U of x0 such that F (x) ⊂ W for all x in U . F is called lower semicontinuous at x0 , if for any y ∈ F (x0 ) and any {xn }∞ n=1 in X with xn → x0 , there exists yn ∈ F (xn ) such that yn → y as n → ∞. F is called continuous at x0 , if F is upper semicontinuous and is lower semicontinuous at x0 . Definition 1.3. (see [14]) A closed subspace N of X is said to be a topologically complemented subspace of X, if there exists a closed subspace M of X such that M ⊕ N = X. Definition 1.4. A subspace L ⊂ X is said to be maximal subspace of X if there exists x∗ ∈ S(X ∗ ) such that L = {x ∈ X : x∗ (x) = 0}. Definition 1.5. (see [6]) A Banach space X is said to be nearly convex, if every closed convex set of S(X) is compact. Definition 1.6. (see [9]) A point x0 ∈ D(T ) is said to be the best approximative solution to the operator equation T x = y, if kT x0 − yk = inf {kT x − yk : x ∈ D(T )} and





kx0 k = min kvk : v ∈ D(T ), kT v − yk = inf kT x − yk . x∈D(T )

Definition 1.7. (see [9]) Let X, Y be Banach spaces, T be a linear bounded operator from subspace of X to Y and D(T ) be the domain of T . The set-valued mapping T ∂ : D(T ∂ ) → X defined by T ∂ (y) = {x0 ∈ D(T ) : x0 is a best approximative solution to T (x) = y}

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3

for any y ∈ D(T ∂ ), is said to be the (set-valued) metric generalized inverse of T, where D(T ∂ ) = {y ∈ Y : T (x) = y has a best approximative solution in X}. 2. Continuity of the set-valued metric generalized inverse in Banach spaces Theorem 2.1. Let X be a nearly convex space, Y be a Banach space, T be a linear bounded operator from subspace of X into Y , D(T ) be a closed subspace of X and R(T ) be a 2-Chebyshev maximal subspace of Y . Then (1) T ∂ is upper semicontinuous on Y if and only if PN (T ) is upper semicontinuous on D(T ); (2) T ∂ is continuous if and only if T T ∂ is lower semicontinuous and PN (T ) is continuous on D(T ). Proof. (1) ”⇒” We first will prove that the metric projector operator PR(T ) is continuous and PR(T ) (y) is a line segment. In fact, by Lemma 1 of [8], we know that if closed subspace H is a 2-Chebyshev subspace of X, then PH (x) is a line segment for any x ∈ X. Hence PR(T ) (y) is a line segment. Since R(T ) is an 2-Chebyshev maximal subspace of Y , there exists f ∈ S(X ∗ ) such that R(T ) = {y ∈ Y : f (y) = 0}. Let y ∈ Y . Pick z ∈ R(T ) and h ∈ S(Y ). Then there exists α ∈ R such that y−z = αh. It is easy to see that α = f (y)/f (h). Then y − z = (f (y)/f (h))h. Hence ky − zk = |f (y)| / |f (h)| ≥ |f (y)|. Then it is easy to see that z ∈ PR(T ) (y) if and only if h ∈ Af . Hence PR(T ) (y) = y − f (y)Af , where Af = {y ∈ S(Y ) : f (y) = 1}. Suppose that metric projection operator PR(T ) is not upper semicontinuous at y0 . Then there exist a sequence {yn }∞ n=1 ⊂ Y and an open set W ⊃ PR(T ) (y0 ) such that PR(T ) (yn ) 6⊂ W and yn → y0 as n → ∞. Hence there exists zn ∈ PR(T ) (yn ) such that zn ∈ / W . Then zn = yn − f (yn )hn , where hn ∈ Af . Since PR(T ) (y) is a line segment and PR(T ) (y) = y − f (y)Af , we obtain that Af is a line segment. ∞ Hence there exists a subsequence {hnk }∞ k=1 of {hn }n=1 such that hnk → h0 ∈ Af as k → ∞. Let z0 = y0 − f (y0 )h0 . Then z0 ∈ PR(T ) (y0 ) and lim znk = lim (ynk − f (ynk )hnk ) = y0 − f (y0 )h0 = z0 ,

k→∞

k→∞

a contradiction. This implies that PR(T ) is upper semicontinuous. Let yn → y0 as n → ∞. Pick z0 ∈ PR(T ) (y0 ). Then there exists h0 ∈ Af such that z0 = y0 − f (y0 )h0 . Hence zn = yn − f (yn )h0 ∈ PR(T ) (xn ) and lim zn = lim (yn − f (yn )h0 ) = y0 − f (y0 )h0 = z0

n→∞

n→∞

This implies that PR(T ) is lower semicontinuous at y0 . Hence we obtain that PR(T ) is continuous. Pick y0 ∈ Y . Suppose that T ∂ is not upper semicontinuous at y0 . Then there ∂ exist a sequence {yn }∞ n=1 ⊂ Y , yn → y0 ∈ Y and norm open set W with T (y0 ) ⊂ ∂ ∂ W such that T (yn ) 6⊂ W for all n ∈ N . Hence there exists xn ∈ T (yn ) ⊂ X such that xn ∈ / W . Since T is a bounded linear operator, we obtain that N (T ) is

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a closed subspace of D(T ). Let T : D(T )/N (T ) → R(T ),

T [x] = T x,

where [x] ∈ D(T )/N (T ) and x ∈ D(T ). Then it is easy to see that R(T ) = R(T ). Moreover, R(T ) = R(T ). In fact, suppose that R(T ) 6= R(T ). Then there exists y 0 ∈ R(T ) such that y 0 ∈ / R(T ). It is easy to see that {y ∈ R(T ) : ky 0 − yk = dist(y 0 , R(T ))} = ∅. This implies that R(T ) is not a 2-Chebyshev subspace of Y , a contradiction. Since R(T ) = R(T ), we obtain that R(T ) is a Banach space. Moreover, it is easy to see that T is a bounded linear operator and N (T ) = {0}. This implies that the bounded linear operator T is both injective and surjective. −1 Therefore, by the inverse operator theorem, we obtain that the operator T is a bounded linear operator. −1 Let PR(T ) (y0 ) = [y(1, 0), y(2, 0)]. Since T is a bounded linear operator and [y(1, 0), y(2, 0)] is a compact set, we obtain that the infimum n −1 o

inf T (z) : z ∈ [y(1, 0), y(2, 0)] is attainable on [y(1, 0), y(2, 0)]. Let



n o

−1

−1 A(0) = y ∈ [y(1, 0), y(2, 0)] : T (y) = inf{ T (z) : z ∈ [y(1, 0), y(2, 0)]} . It is easy to see that A(0) is a closed set. Moreover, if z1 ∈ A(0) and z2 ∈ A(0), then





−1

−1

−1

T (λz + (1 − λ)z ) ≤ λ T (z ) + (1 − λ) T (z )

1 2 1 2 o n −1

= inf T (z) : z ∈ [y(1, 0), y(2, 0)] , where λ ∈ [0, 1]. This implies that the set A(0) is a closed convex set. Hence there exist z(1, 0) ∈ [y(1, 0), y(2, 0)] and z(2, 0) ∈ [y(1, 0), y(2, 0)] such that A(0) = [z(1, 0), z(2, 0)]. Let PR(T ) (yn ) = [y(1, n), y(2, n)]. Then there exist z(1, n) ∈ [y(1, n), y(2, n)] and z(2, n) ∈ [y(1, n), y(2, n)] such that A(n)

n n −1 oo

−1

= y ∈ [y(1, n), y(2, n)] : T (y) = inf T (z) : z ∈ [y(1, n), y(2, n)] = [z(1, n), z(2, n)]. Since PR(T ) is continuous, we may assume without loss of generality that lim y(1, n) = z1 ∈ [y(1, 0), y(2, 0)] and

n→∞

lim y(2, n) = z2 ∈ [y(1, 0), y(2, 0)].

n→∞

(2.1) We claim that [z1 , z2 ] ∈ [z(1, 0), z(2, 0)]. Otherwise, we may assume without loss of generality that z1 ∈ / [z(1, 0), z(2, 0)]. Hence there exists r > 0 such that

n −1 o

−1



T (z1 ) > inf T (z) : z ∈ [y(1, 0), y(2, 0)] + 4r.

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5

−1

Since T is a bounded linear operator and y(1, n) → z1 as n → ∞, we may assume without loss of generality that

n −1 o

−1

(2.2)

T (z(1, n)) > inf T (z) : z ∈ [y(1, 0), y(2, 0)] + 2r for every n ∈ N . Since the metric projection operator PR(T ) is continuous, there −1 exists z(n) ∈ [y(1, n), y(2, n)] such that z(n) → z(1, 0) as n → ∞. Since T is a bounded linear operator and z(n) → z(1, 0) as n → ∞, we have



n −1 o

−1

−1

lim T (z(n)) = T (z(1, 0)) = inf T (z) : z ∈ [y(1, 0), y(2, 0)] . n→∞

Therefore, by formula (2.2), we may assume without loss of generality that



−1

−1

T (z(1, n)) > T (z(n)) + r for all n ∈ N , a contradiction. −1 −1 −1 Pick x(1, 0) ∈ T (z1 ), x(2, 0) ∈ T (z2 ), x(1, n) ∈ T (z(1, n)) and x(2, n) ∈ −1 −1 −1 T (z(2, n)). Then we have [x(1, n)] = T (z(1, n)) and [x(2, n)] = T (z(2, n)). −1 Since T is a bounded linear operator, by formula (2.1), we have

−1

−1 k[x(1, n) − x(1, 0)]k = k[x(1, n)] − [x(1, 0)]k ≤ T (z(1, n)) − T z1

−1 ≤ T kz(1, n) − z1 k → 0 as n → ∞. Hence we may assume without loss of generality that x(1, n) → x(1, 0) as n → ∞. Similarly, we may assume without loss of generality that x(2, n) → x(2, 0) as n → ∞. Moreover, by the definition of set-valued metric generalized inverse, there exists a sequence {λn }∞ n=1 ⊂ [0, 1] such that xn = λn x(1, n) + (1 − λn )x(2, n) − πN (T ) (λn x(1, n) + (1 − λn )x(2, n)) , where πN (T ) (λn x(1, n) + (1 − λn )x(2, n)) ∈ PN (T ) (λn x(1, n) + (1 − λn )x(2, n)) . We may assume without loss of generality that λn → λ as n → ∞. Then lim λn x(1, n) + (1 − λn )x(2, n) = λz1 + (1 − λ)z2 ∈ [z1 , z2 ].

n→∞

(2.3)

Since PN (T ) is upper semicontinuous, by formula (2.3), we obtain that for any ε > 0, there exists n0 ∈ N such that πN (T ) (λn x(1, n) + (1 − λn )x(2, n)) ∈

∪

B(x, ε)

x∈PN (T ) (λz1 +(1−λ)z2 )

whenever n > n0 . This implies that
dist {λn x(1, n) + (1 − λn )x(2, n)}∞ n=1 , PN (T ) (λz1 + (1 − λ)z2 ) = 0. Hence, for any k > 0, there exists hnk ∈ PN (T ) (λz1 + (1 − λ)z2 ) such that

πN (T ) (λn x(1, nk ) + (1 − λn )x(2, nk )) − hn < 1 . k k k k Moreover, there exists r > 0 such that

(2.4)

λz1 + (1 − λ)z2 − PN (T ) (λz1 + (1 − λ)z2 ) ⊂ rS(X).

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Since X is a nearly convex space, we obtain that λz1 + (1 − λ)z2 − PN (T ) (λz1 + (1 − λ)z2 ) is compact. Then the set PN (T ) (λz1 + (1 − λ)z2 ) is compact. Hence we may assume without loss of generality that hnk → h ∈ PN (T ) (λz1 + (1 − λ)z2 ) as n → ∞. Therefore, by formula (2.4), we have lim πN (T ) (λnk x(1, nk ) + (1 − λnk )x(2, nk )) = h ∈ PN (T ) (λz1 + (1 − λ)z2 ).

k→∞

This implies that xk

= λnk x(1, nk ) + (1 − λnk )x(2, nk ) − πN (T ) (λnk x(1, nk ) + (1 − λnk )x(2, nk )) → λz1 + (1 − λ)z2 − h ∈ λz1 + (1 − λ)z2 − PN (T ) (λz1 + (1 − λ)z2 ).

Moreover, by the definition of set-valued metric generalized inverse, we obtain that λz1 + (1 − λ)z2 − PN (T ) (λz1 + (1 − λ)z2 ) ⊂ T ∂ (y) ⊂ W . Since W is a norm open set, we have xk ∈ W for k large enough, a contradiction. ”⇐” Suppose that PN (T ) is not upper semicontinuous on D(T ). Then there exist {xn }∞ n=1 ⊂ D(T ), x0 ∈ D(T ) and a norm open set W such that xn → x0 , PN (T ) (x0 ) ⊂ W and PN (T ) (xn ) 6⊂ W . Hence there exists πN (T ) (xn ) ∈ PN (T ) (xn ) such that πN (T ) (xn ) ∈ / W . We claim that there exists δ > 0 such that B(z, 2δ) ⊂ W.

∪ z∈PN (T ) (x0 )

Otherwise, there exists zn ∈ PN (T ) (x0 ) such that B(zn , 1/n) 6⊂ W . Since PN (T ) (x0 ) is compact, we may assume that zn → z0 ∈ PN (T ) (x0 ) as n → ∞. Hence there exists η > 0 such that B(z0 , 4η) ⊂ W . Moreover, there exists n0 ∈ N such that 1/n0 < η and kzn0 − z0 k ≤ η. Hence, for any z ∈ B(zn0 , 1/n0 ), we have 1 kz − z0 k ≤ kz − zn0 k + kzn0 − z0 k ≤ + η < η + η < 4η. n0 This implies that z ∈ W . Then B(zn0 , 1/n0 ) ⊂ W , a contradiction. Let yn = T xn and y0 = T x0 . Then T ∂ (yn ) = xn − PN (T ) (xn ),

T ∂ (y0 ) = x0 − PN (T ) (x0 ) and

lim yn = y0 .

n→∞

Since PN (T ) (x0 ) ⊂ W , we obtain that T ∂ (y0 ) = x0 − PN (T ) (x0 ) ⊂ x0 − W . We claim that xn − πN (T ) (xn ) ∈ / x0 − B(z, δ) ∪ z∈PN (T ) (x0 )

whenever kxn − x0 k < δ. In fact, suppose that xn −πN (T ) (xn ) ∈ x0 −

∪

B(z,

z∈PN (T ) (x0 )

δ) whenever kxn − x0 k < δ. Then
πN (T ) (xn ) = xn − xn − πN (T ) (xn )   ∈ xn − x0 − B(z, δ) ∪ z∈PN (T ) (x0 )

=

∪ z∈PN (T ) (x0 )

⊂

B(z, δ) + (xn − x0 ) B(z, 2δ) ⊂ W,

∪ z∈PN (T ) (x0 )

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a contradiction. Since xn −πN (T ) (xn ) ∈ / x0 −

∪ z∈PN (T ) (x0 )

7

B(z, δ) whenever kxn − x0 k

< δ, we obtain that T ∂ is not upper semicontinuous at y0 , a contradiction. (2) ”⇒” Let y0 ∈ Y and yn → y0 as n → ∞. Then, by the previous proof, there exist z(1, n) ∈ PR(T ) (yn ) and z(2, n) ∈ PR(T ) (yn ), z(1, 0) ∈ PR(T ) (y0 ) and z(2, 0) ∈ PR(T ) (y0 ) such that

n −1 o

−1 [z(1, n), z(2, n)] = z : T (z) = inf{ T (y) : y ∈ PR(T ) (yn )} and

o n −1

−1

[z(1, 0), z(2, 0)] = z : T (z) = inf{ T (y) : y ∈ PR(T ) (y0 )} . Moreover, by the previous proof, we may assume without loss of generality that lim z(1, n) = z1 ∈ [z(1, 0), z(2, 0)] and lim z(2, n) = z2 ∈ [z(1, 0), z(2, 0)].

n→∞

n→∞

(2.5) From the previous proof, there exist [x(1, n), x(2, n)] ⊂ X and [x(1, 0), x(2, 0)] ⊂ X such that T {x : x ∈ [x(1, n), x(2, n)]} = [z(1, n), z(2, n)] and T {x : x ∈ [x(1, 0), x(2, 0)]} = [z(1, 0), z(2, 0)]. Moreover, by the definition of set-valued metric generalized inverse, we obtain that T T ∂ (y0 ) = [z(1, 0), z(2, 0)] and T T ∂ (yn ) = [z(1, n), z(2, n)]. Since the set-valued mapping T T ∂ is lower semicontinuous, by formula (2.5), we have lim z(1, n) = z1 = z(1, 0) and lim z(2, n) = z2 = z(2, 0). n→∞

n→∞

Therefore, by the previous proof, we obtain that lim x(1, n) = x(1, 0) and lim x(2, n) = x(2, 0),

n→∞

n→∞

(2.6)

Moreover, by the definition of set-valued metric generalized inverse, we obtain that for any x ∈ T ∂ (y0 ), there exist λ ∈ [0, 1] and h ∈ PN (T ) (λx(1, 0) + (1 − λ)x(2, 0)) such that x = λx(1, 0) + (1 − λ)x(2, 0) − h. Since PN (T ) is continuous, there exists hn ∈ PN (T ) (λx(1, n) + (1 − λ)x(2, n)) such that hn → h as n → ∞. Therefore, by formula (2.6), we obtain that lim λx(1, n) + (1 − λ)x(2, n) = λx(1, 0) + (1 − λ)x(2, 0).

n→∞

This implies that lim (λx(1, n) + (1 − λ)x(2, n) − hn ) = λx(1, 0) + (1 − λ)x(2, 0) − h = x. (2.7)

n→∞

Noticing that λx(1, n) + (1 − λ)x(2, n) − hn ∈ T ∂ (yn ) and formula (2.7), we obtain that T ∂ is lower semicontinuous at y0 . Therefore, by (1), we obtain that T ∂ is upper semicontinuous at y0 . Hence T ∂ is continuous at y0 . ”⇐” Let y0 ∈ Y and yn → y0 as n → ∞. Then, by the previous proof, there exist x0 ∈ X and {xn }∞ n=1 ⊂ X such that PR(T ) (y0 ) = T x0 , PR(T ) (yn ) = T xn and

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xn → x0 as n → ∞. Then T ∂ (y0 ) = x0 − PN (T ) (x0 ) and T ∂ (yn ) = xn − PN (T ) (xn ). Since T ∂ is continuous, we obtain that for any x0 − z ∈ x0 − PN (T ) (x0 ), there exists xn − zn ∈ xn − PN (T ) (xn ) such that xn − zn → x0 − z as n → ∞, where zn ∈ PN (T ) (xn ). Hence, for any z ∈ PN (T ) (x0 ), there exists zn ∈ PN (T ) (xn ) such that zn → z0 as n → ∞. This implies that PN (T ) is lower semicontinuous at y0 . Therefore, by (1), we obtain that PN (T ) is continuous at y0 . We next will prove that T T ∂ is lower semicontinuous. Let y0 ∈ Y and yn → y0 as n → ∞. Pick z ∈ T T ∂ (y0 ). Then there exists x0 ∈ D(T ) such that T x0 = y0 and x0 ∈ T ∂ (y0 ). Since T ∂ is continuous, we obtain that T ∂ is lower semicontinuous. Hence there exists xn ∈ T ∂ (yn ) such that xn → x0 as n → ∞. This implies that T xn → T x0 = y0 as n → ∞. Since xn ∈ T ∂ (yn ), we obtain that T xn ∈ T T ∂ (yn ). Hence T T ∂ is lower semicontinuous at y0 , which completes the proof.  3. Continuous selections of the set-valued metric generalized inverse in Banach spaces Theorem 3.1. Let X, Y be Banach spaces, T be a linear bounded operator from subspace of X into Y , D(T ) be a closed subspace of X, N (T ) be a topologically complemented subspace of D(T ) and R(T ) be a proximinal subspace of Y . Then the following statements are equivalent: (1) T ∂ has a continuous homogeneous selection on Y ; (2) PN (T ) has a continuous homogeneous selection on D(T ) and the set-valued mapping T T ∂ has a continuous homogeneous selection on Y . Proof. (2) ⇒ (1). Since N (T ) is a topologically complemented subspace of D(T ), there exists a closed subspace M (T ) of D(T ) such that M (T ) ⊕ N (T ) = D(T ). Moreover, by M (T ) ⊕ N (T ) = D(T ), we obtain that T −1 (y) ∩ M (T ) is singleton for any y ∈ R(T ). Define the mapping G : R(T ) → M (T ) such that G(y) = T −1 (y) ∩ M (T ),

y ∈ R(T ).

Since T be a linear bounded operator from D(T ) into R(T ), by M (T ) ⊕ N (T ) = D(T ), we obtain that G is a linear bounded operator from R(T ) into M (T ). Let T T σ be a continuous homogeneous selection of T T ∂ and πN (T ) be a continuous homogeneous selection of PN (T ) . Therefore, by the definition of set-valued metric generalized inverse, we obtain that the mapping G ◦ T T σ − πN (T ) ◦ G ◦ T T σ :

Y → D(T )

is a continuous homogeneous selection of T ∂ . (1) ⇒ (2). Since T ∂ has a continuous homogeneous selection on Y , by the definition of set-valued metric generalized inverse, we obtain that N (T ) is a proximinal subspace of D(T ). Let T σ be a continuous homogeneous selection of T ∂ . Pick x ∈ D(T ). Let πN (T ) (x) = x − T σ T (x). We next will prove that πN (T ) is a continuous homogeneous selection of PN (T ) . In fact, since
T πN (T ) (x) = T (x − T σ T (x)) = T x − T T σ T (x) = T x − T x = 0, (3.1)

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we obtain that πN (T ) (x) ∈ N (T ). Moreover, by the definition of set-valued metric generalized inverse, we obtain that kT σ T (x)k = dist (x, N (T )). Therefore, by x − πN (T ) (x) = x − x + T σ T (x) = T σ T (x) and formula (3.1), we obtain that πN (T ) (x) ∈ PN (T ) (x). Moreover, since T σ is a continuous homogeneous selection of T ∂ and πN (T ) (x) = x − T σ T (x), we obtain that πN (T ) is a homogeneous selection. This implies that PN (T ) has a continuous homogeneous selection on D(T ). Since T σ is a continuous homogeneous selection of T ∂ , we obtain that T T σ is a continuous homogeneous selection of T T ∂ , which completes the proof.  Definition 3.2. (see [5]) A nonempty subset C of X is said to be approximatively compact if for any {yn }∞ n=1 ⊂ C and any x ∈ X satisfying kx − yn k → inf y∈C kx − yk as n → ∞, there exists a subsequence of {yn }∞ n=1 converging to an element in C. X is called approximatively compact if every nonempty closed convex subset of X is approximatively compact. Definition 3.3. (see [5]) A Banach space X is be said to be strictly convex if for any x, y ∈ S(X) and kx + yk = 2 we have x = y. Theorem 3.4. Let X be approximatively compact and strictly convex, Y be a Banach spaces, T be a linear bounded operator from subspace of X into Y , D(T ) be a closed subspace of X, N (T ) be a topologically complemented subspace of D(T ) and R(T ) be a proximinal subspace of Y . Then the following statements are equivalent: (1) T ∂ has a continuous homogeneous selection on Y ; (2) T T ∂ has a continuous homogeneous selection on Y . Proof. Since X is approximatively compact, we obtain that PN (T ) is upper semicontinuous. Since X is a strictly convex space, we obtain that PN (T ) is single value mapping. This implies that PN (T ) is continuous. Therefore, by Theorem 3.1, it is easy to see that Theorem 3.4 is true, which completes the proof.  Theorem 3.5. Let X be approximatively compact and strictly convex, Y be a Banach space, T be a linear bounded operator from subspace of X into Y , D(T ) be a closed subspace of X and R(T ) be a proximinal subspace of Y . Then the following statements are equivalent: (1) T ∂ has a continuous selection on Y ; (2) T T ∂ has a continuous selection on Y . Proof. (2)⇒(1). Since X is approximatively compact, we obtain that PN (T ) is upper semicontinuous. Since X is a strictly convex space, we obtain that PN (T ) is a single value mapping. This implies that PN (T ) is a continuous and single value mapping. Let T T σ be a continuous selection of T T ∂ and f −1 T T σ be a selection of T −1 T T σ . This implies that the mapping f −1 T T σ − PN (T ) f −1 T T σ is a selection of T ∂ . We next will prove that if yn → y as n → ∞, then lim [f −1 T T σ (yn ) − PN (T ) f −1 T T σ (yn )] = f −1 T T σ (y) − PN (T ) f −1 T T σ (y). (3.2)

n→∞

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In fact, by the proof of Theorem 2.1, there exists a sequence {xn }∞ n=1 ⊂ D(T ) such that xn ∈ f −1 T T σ (yn ) − PN (T ) f −1 T T σ (yn ) + N (T ) and

lim xn = f −1 T T σ (y).

n→∞

Since the mapping PN (T ) and f −1 T T σ are single value mappings, we obtain that
T xn = T f −1 T T σ (yn ) − PN (T ) f −1 T T σ (yn ) . Hence there exists a sequence {zn }∞ n=1 ⊂ N (T ) such that xn − PN (T ) (xn ) − zn = f −1 T T σ (yn ) − PN (T ) f −1 T T σ (yn ). Moreover, by the definition of set-valued metric generalized inverse, we have



xn − PN (T ) (xn ) − zn = f −1 T T σ (yn ) − PN (T ) f −1 T T σ (yn ) = dist (xn , N (T )) . This implies that PN (T ) (xn )+zn ⊂ PN (T ) (xn ). Since the mapping PN (T ) is a single value mapping, we have zn = 0. Hence xn − PN (T ) (xn ) = f −1 T T σ (yn ) − PN (T ) f −1 T T σ (yn ). Since PN (T ) is a continuous single value mapping, by xn → f −1 T T σ (y) as n → ∞, we obtain that lim [xn − PN (T ) (xn )] = f −1 T T σ (y) − PN (T ) f −1 T T σ (y).

n→∞

Noticing that xn − PN (T ) (xn ) = f −1 T T σ (yn ) − PN (T ) f −1 T T σ (yn ), we obtain that (3.2) is true. Hence f −1 T T σ − PN (T ) f −1 T T σ is a continuous selection of T ∂ . (1)⇒(2). Let T σ be a continuous selection of T ∂ . Then T T σ is a continuous selection of T T ∂ , which completes the proof.  Theorem 3.6. Let X be approximatively compact and strictly convex, Y be a Banach spaces, T be a linear bounded operator from subspace of X into Y , D(T ) be a closed subspace of X, N (T ) be a topologically complemented subspace of D(T ) and R(T ) be a 2-Chebyshev maximal subspace of Y . Then the following statements are equivalent: (1) For any x ∈ T ∂ (y), there exists a selection T σ of T ∂ such that T σ (y) = x and T σ is continuous at y; (2) T T ∂ is lower semicontinuous at y. Proof. (1)⇒(2). From the proof of Theorem 2.1, there exist z(1) ∈ PR(T ) (y) and z(2) ∈ PR(T ) (y) such that T T ∂ (y) = [z(1, y), z(2, y)] ⊂ PR(T ) (y). Since N (T ) is a topologically complemented subspace of D(T ), there exists a closed subspace M (T ) of D(T ) such that M (T ) ⊕ N (T ) = D(T ). Pick x(1, y) ∈ −1 −1 T (z(1, y)) and x(2, y) ∈ T (z(2, y)). Then, by M (T ) ⊕ N (T ) = D(T ), there exist G(y(1)), G(y(2)) ∈ M (T ) and h(y(1)), h(y(2) ∈ N (T ) such that x(1, y) = G(y(1)) + h(y(1)) and x(2, y) = G(y(2)) + h(y(2)). Therefore, by x(1, y) ∈ T

−1

(z(1, y)) and x(2, y) ∈ T

−1

(z(2, y)), we obtain that

T (G(y(1))) = z(1, y) and T (G(y(2))) = z(2, y).

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Hence, for any z ∈ T T ∂ (y), there exists λ ∈ [0, 1] such that z = T (λG(y(1)) + (1 − λ)G(y(2))) . Pick x ∈ T σ (y). Then there exists a selection T σ of T ∂ such that T σ (y) = x and T σ is continuous at y. This implies that T x = T (λG(y(1)) + (1 − λ)G(y(2))) = z. Therefore, by the definition of set-valued metric generalized inverse, we have T σ (y) = λG(y(1)) + (1 − λ)G(y(2)) − PN (T ) (λG(y(1)) + (1 − λ)G(y(2))) . We next will prove that set-valued mapping T T ∂ is lower semicontinuous at y. Let yn → y as n → ∞. Then, by M (T ) ⊕ N (T ) = D(T ) and the previous proof, there exist G(yn (1)) ∈ M (T ) and G(yn (2)) ∈ M (T ) such that T (G(yn (1))) = z(1, yn ),

T (G(yn (2))) = z(2, yn )

and T T ∂ (yn ) = [z(1, yn ), z(2, yn )] ⊂ PR(T ) (yn ). Hence there exists a sequence {λn }∞ n=1 ⊂ [0, 1] such that T σ (yn ) = λn G(yn (1))+(1−λn )G(yn (2))−PN (T ) (λn G(yn (1)) + (1 − λn )G(yn (2))) . Since λn G(yn (1)) + (1 − λn )G(yn (2)) ∈ M (T ),

λG(y(1)) + (1 − λ)G(y(2)) ∈ M (T ),

PN (T ) (λn G(yn (1)) + (1 − λn )G(yn (2))) ∈ N (T ) and PN (T ) (λG(y(1)) + (1 − λ)G(y(2))) ∈ N (T ), by M (T ) ⊕ N (T ) = D(T ) and T σ (yn ) → T σ (y) as n → ∞, we obtain that lim λn G(yn (1)) + (1 − λn )G(yn (2)) = λG(y(1)) + (1 − λ)G(y(2)).

n→∞

This implies that lim T (λn G(yn (1)) + (1 − λn )G(yn (2))) = T (λG(y(1)) + (1 − λ)G(y(2))) = z.

n→∞

where T (λn G(yn (1)) + (1 − λn )G(yn (2))) ∈ T T σ (yn ). Hence the set-valued mapping T T ∂ is lower semicontinuous at y. (2)⇒(1). Let x ∈ T ∂ (y). Then, by the previous proof, there exists λ ∈ [0, 1] such that x = λG(y(1)) + (1 − λ)G(y(2)) − PN (T ) (λG(y(1)) + (1 − λ)G(y(2))) . Define the set-valued mapping Y → T T ∂ (Y ) such that F (z) = c, where kc − T xk =

inf

h∈T T ∂ (z)

kh − T xk and c ∈ T T ∂ (z).

It is easy to see that F (y) = T x. Let f be a selection of F . Moreover, let yn → y and f (yn ) = cn . Since T T ∂ is lower semicontinuous at y, we obtain that cn → T x as n → ∞. This implies that f is continuous at y. Since N (T ) is a topologically complemented subspace of D(T ), there exists a closed subspace M (T ) of D(T )

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such that M (T ) ⊕ N (T ) = D(T ). Define the mapping G : R(T ) → M (T ) such that G(y) = T −1 (y) ∩ M (T ), y ∈ R(T ). Then, by the previous proof, we obtain that G is a linear bounded operator from R(T ) into M (T ). Since M (T ) ⊕ N (T ) = D(T ), we have x = x1 − x2 , where x1 ∈ M (T ) and x2 ∈ N (T ). Since X is a strictly convex space, we obtain that PN (T ) is a single value mapping. Since x ∈ T ∂ (y), by the definition of set-valued metric generalized inverse, we have  kxk = kx1 − x2 k = inf kzk : z ∈ T −1 x = T −1 x1 = inf {kx1 − hk : h ∈ N (T )} . Therefore, by x1 ∈ M (T ) and x2 ∈ N (T ), we obtain that x2 = PN (T ) (x1 ). Since f is a selection of F and F (y) = T x, we have G ◦ f (y) = G(T x) = x1 Define the mapping T σ = G ◦ f − PN (T ) ◦ G ◦ f. Then T σ is a selection of T ∂ . Moreover, by G ◦ f (y) = G(T x) = x1 , we have T σ (y) = G ◦ f (y) − PN (T ) ◦ G ◦ f (y) = G(T x) − PN (T ) (G(T x)) = x1 − PN (T ) (x1 ) = x. Since X is approximatively compact, we obtain that PN (T ) is upper semicontinuous. Since PN (T ) is a single value mapping, we obtain that PN (T ) is continuous. Since f and G is continuous at y, we obtain that T σ = G ◦ f − PN (T ) ◦ G ◦ f is continuous at y, which completes the proof.  4. Relation of continuity and continuous selections of the set-valued metric generalized inverse in Banach spaces Theorem 4.1. Let X be approximatively compact and strictly convex, Y be a Banach space, T be a linear bounded operator from subspace of X into Y , D(T ) be a closed subspace of X, N (T ) be a topologically complemented subspace of D(T ) and R(T ) be a 2-Chebyshev maximal subspace of Y . Then the following statements are equivalent: (1) For any x ∈ ∪y∈Y T ∂ (y), there exists a selection T σ of T ∂ such that T σ (y) = x and T σ is continuous at y; (2) T T ∂ is lower semicontinuous; (3) T ∂ is continuous. Proof. Since X is approximatively compact, we obtain that PN (T ) is upper semicontinuous. Since X is a strictly convex space, we obtain that PN (T ) is single value mapping. This implies that PN (T ) is continuous. Therefore, by Theorem 2.1, we obtain that (2)⇔ (3) is true. Moreover, by Theorem 3.5, we obtain that (1)⇔ (2) is true, which completes the proof.  Acknowledgement. This research is supported by ”China Natural Science Fund under grant 11401084” and ”China Natural Science Fund under grant 11561053”.

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References 1. S. Shang and Y. Cui, 2-strict convexity and continuity of the set-valued metric generalized inverse in Banach spaces, Abstr. Appl. Anal 2014 (2014) no. 1, 1–8. 2. L. Chen and L. Cheng, Analytic characterizations of of Mazur intersection property via convex functions, J. Funct. Anal. 262 (2012) no. 1, 4731–4745. 3. N.W. Jefimow and S.B. Stechkin, Approximative compactness and Chebyshev sets, Sov Math, 2 (1961) no. 1, 1226–1228. 4. J. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990. 5. S. Chen, H. Hudzik, W. Kowalewski, Y. Wang, M. Wisla, Approximative compactness and continuity of metric projector in Banach spaces and applications, Sci Sin Math, 50(2007) no. 2, 75–84. 6. Sekowski, Stachura, Noncompact smoothness and noncompact convexity, Atti. Semin. Mat. Fis. Univ. Modena 36 (1988)no.1, 329–333. 7. I. Singer, On the set of best approximation of an element in a normed linear space, Rev. Roumaine Math. Pures Appl. 5(1960), no.1, 383–402. 8. S. Shang and Y. Cui, Approximative compactness and continuity of the set-valued metric generalized inverse in Banach spaces, J. Math. Anal. Appl. 422 (2015) no. 2, 1361–1375. 9. M.Z. Nashed , G.F. Votruba, A unified apporoach to generalized inverses of linnear operator: II Extremal and proximinal properties, Bull Amer Math Soc, 80 (1974) no. 1, 831–835. 10. H. Hudzik, Y.W. Wang and W. Zheng, Criteria for the Metric Generalized Inverse and its Selections in Banach Spaces, Set-Valued Anal. 16 (2008) no. 1, 51–65. 11. Y.W. Wang and J. Liu, Metric generalized inverse of linear manifold and extremal solution of linear inclusion in Banach spaces, J. Math. Anal. Appl. 302 (2005) no. 2, 360–371. 12. G. Chen and Y. Xue, Perturbation analysis for the operator equation T x = b in Banach spaces, J. Math. Anal. Appl. 212 (1997) no. 1, 107–125. 13. Y.W. Wang and H. Zhang, Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces, Linear Algebra Appl. 426 (2007) no. 1, 1–11. 14. Y.W. Wang, Generalized Inverse of Operator in Banach Spaces and Applications, Science Press, Beijing, 2005. 1

Department of Mathematics, Northeast Forestry University, Harbin 150040, P. R. China E-mail address: [email protected] 2

Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, P. R. China E-mail address: [email protected]

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Locally and globally small Riemann sums and Henstock-Stieltjes integral of fuzzy-number-valued functions

b

Muawya Elsheikh Hamida ∗, Luoshan Xu a , Zengtai Gong b a School of Mathematical Science, Yangzhou University, Yangzhou 225002, China College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R.China

Abstract: In this paper, we introduce and study locally and globally small Riemann sums with respect to α on [a, b] for fuzzy-number-valued functions and obtain some of it, s characterizations. Also, we shall prove two main theorems: (i) If a fuzzy-number-valued functions f˜(x) is Henstock-Stieltjes integrable on [a, b] then it has (LSRS) and the converse is always true. (ii) If a fuzzy-number-valued functions f˜(x) is Henstock-Stieltjes integrable on [a, b] then it has (GSRS) and the converse is always true. Finally, by Egorov, s Theorem, we obtain the dominated convergence theorem for globally small Riemann sums (GSRS) with respect to α on [a, b] for fuzzy-number-valued functions. Keywords : Fuzzy numbers; fuzzy integrals; Henstock-Stieltjes integral; locally small Riemann sums (LSRS) with respect to α on [a, b]; globally small Riemann sums (GSRS) with respect to α on [a, b].

1

Introduction

Since the concept of fuzzy sets was firstly introduced by Zadeh in 1965 [21], it has been studied extensively from many different aspects of the theory and applications, such as fuzzy topology, fuzzy analysis, fuzzy decision making and fuzzy logic, information science and so on. It, s well known that the concept of the Stieltjes integral for fuzzy-number-valued functions was originally introduced by Nanda [12] in 1989. Nonetheless, as Wu et al. [17] pointed out that the existence of supremum and infimum for a finite set of fuzzy numbers wasn, t easy at first thought. That is, Nanda, s concept of fuzzy Riemann-Stieltjes (F RS) integral in [12] was incorrect. In 1998, Wu [18] introduced the notion of (F RS) integral by means of the representation theorem of fuzzy-number-valued functions, whose membership function could be obtained by solving a nonlinear programming problem, but it, s difficult to calculate and extend to the higher-dimensional space. In 2006, Ren et al. proposed the notion of two types of (F RS) integral for fuzzy-number-valued functions [13, 14] and showed that a continuous fuzzy-numbervalued function was (F RS) integrable with respect to a real-valued increasing function. Gong et al. [2] defined and discussed the (HS) integral for fuzzy-number-valued functions and proved two convergence theorems for sequences of the (F HS) integrable functions in 2012. The locally and globally small Riemann sums have been introduced by many authors from different points of views. In 1986, Schurle characterized the Lebesgue integral in (LSRS) (locally small Riemann sums) property [15]. The (LSRS) property has been used to characterized the Perron (P ) integral on [a, b] [16]. By considering the equivalency between the (P ) integral and the Henstock-Kurzweil (HK) integral, the (LSRS) property has been used to characterized the (HK) integral on [a, b] [10]. The (LSRS) property brought a research to have global characterization on the Riemann sums of an (HK) integrable function on [a, b]. This research has been done by considering the following fact: Every (HK) integrable function on [a, b] is measurable, however, there is no guarantee the boundedness of the function. A measurable function f is (HK) integrable on [a, b] depends on it behaves on the set of x in which |f (x)| is large, i.e. |f (x)| ≥ N ∗ Corresponding author. Tel.: +8613218977118. E-mail address: [email protected], [email protected] (M.E. Hamid), [email protected] (L.S. Xu) and [email protected] (Z.T. Gong).

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J. COMPUTATIONAL AND APPLICATIONS, VOL. 25, NO.6, 2018,sums COPYRIGHT 2018 EUDOXUS PRESS, LLC M.E. Hamid, L.S. Xu,ANALYSIS Z.T. Gong: Locally and globally small Riemann and Henstock-Stieltjes integral...

for some N . This fact has been characterized in (GSRS) (globally small Riemann sums) property [10]. The (GSRS) property involves one characteristic of the primitive of an (HK) integrable function. That is the primitive of the (HK) integral on [a, b] is ACG∗ (generalized strongly absolutely continuous) on [a, b]. This is not a simple concept. In 2015, Indrati [8] introduced a countably Lipschitz condition of a function which is simpler than the ACG∗ , and proved that the (HK) integrable function or it, s primitive could be characterized in countably Lipschitz condition. Also, by considering the characterization of the (HK) integral in the (GSRS) property, it showed that the relationship between (GSRS) property and countably Lipschitz condition of an (HK) integrable function on [a, b]. In 2018, Hamid et al. [6] investigated locally and globally small Riemann sums for fuzzy-number-valued functions and proved two main theorems: (1) A fuzzy-number-valued functions f˜(x) is Henstock integrable on [a, b] if and only if f˜(x) has (LSRS). (2) A fuzzy-number-valued functions f˜(x) is Henstock integrable on [a, b] if and only if f˜(x) has (GSRS). In this paper, we introduce and study the locally and globally small Riemann sums with respect to α on [a, b] for fuzzy-number-valued functions. We show that a fuzzy-number-valued functions is Henstock-Stieltjes integrable with respect to α on [a, b] iff it has (LSRS) with respect to α on [a, b]. Also it is shown that a fuzzy-numbervalued functions is Henstock-Stieltjes integrable on [a, b] iff it has (GSRS) with respect to α on [a, b]. Finally, by Egorov, s Theorem, we get the dominated convergence theorem for globally small Riemann sums (GSRS) with respect to α on [a, b] for fuzzy-number-valued functions. The rest of this paper is organized as follows, in Section 2 we shall review the relevant concepts and properties of fuzzy sets and the definition of Henstock-Stieltjes integrals for fuzzy-number-valued functions. Section 3 is devoted to discussing the locally small Riemann sums (LSRS) with respect to α on [a, b] for fuzzy-number-valued functions. In Section 4 we shall investigate the globally small Riemann sums (GSRS) with respect to α on [a, b] for fuzzy-number-valued functions.

2

Preliminaries

Definition 2.1 [7, 10] Let δ : [a, b] → R+ be a positive real-valued function. P = {[xi−1 , xi ]; ξi } is said to be a δ-fine division, if the following conditions are satisfied: (1) a = x0 < x1 < x2 < ... < xn = b; (2) ξi ∈ [xi−1 , xi ] ⊂ (ξi − δ(ξi ), ξi + δ(ξi ))(i = 1, 2, · · · , n). For brevity, we write P = {[u, v]; ξ}, where [u, v] denotes a typical interval in P and ξ is the associated point of [u, v]. Definition 2.2 [4] Let α : [a, b] → R be an increasing function. A real function f : [a, b] → R is Henstock-Stieltjes (HS) integrable to A ∈ R with respect to α on [a, b] if for every ε > 0, there is a function δ(x) > 0 , such that for any δ-fine division P = {[ui , vi ]; ξi }n i=1 we have n X f (ξi )[α(vi ) − α(ui )] − A < ε.

(2.1)

i=1

We write (HS)

Rb

f (x)dα = A , and f ∈ HSα [a, b].

a

For the results about fuzzy number space E 1 . we recall that E 1 = {u : R → [0, 1] : u satisfies (1)-(4) below}: (1) u is normal, i.e., there exists a x0 ∈ R such that u(x0 ) = 1; (2) u is a convex fuzzy set, i.e., u(rx + (1 − r)y) > min(u(x), u(y)), x, y ∈ R, r ∈ [0, 1]; (3) u is upper semi-continuous; (4) cl{x ∈ R : u(x) > 0} is compact, where clA denotes the closure of A. For 0 < r 6 1, denote [u]r = {x : u(x) > r}. Then from (1)-(4), it follows that the r−level set [u]r is a close interval for all r ∈ [0, 1] (refer to [1, 3, 5, 9, 11, 19, 20]). We write ur = [u]r = [ur− , ur+ ] or [u− (r), u+ (r)]. For u, v ∈ E 1 , the addition and scalar multiplication are defined by the equations: r r [u + v]r = [u]r + [v]r , i.e., ur− + v− = [u + v]r− and ur+ + v+ = [u + v]r+ ;

1108

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J. COMPUTATIONAL AND APPLICATIONS, VOL. 25, NO.6, 2018,sums COPYRIGHT 2018 EUDOXUS PRESS, LLC M.E. Hamid, L.S. Xu,ANALYSIS Z.T. Gong: Locally and globally small Riemann and Henstock-Stieltjes integral...

[ku]r = k[u]r , i.e., [ku]r− = min{kur− , kur+ } and [ku]r+ = max{kur− , kur+ }, respectively. r r Define D(u, v) = sup d([u]r , [v]r ) = sup max{|ur− − v− |, |ur+ − v+ |}, where d is Hausdorff metric. Furtherr∈[0,1]

r∈[0,1]

more, we write + k˜ ukE 1 = D(˜ u, ˜ 0) = sup max{|u− λ |, |uλ |}. λ∈[0,1]

Notice that k · kE 1 = D(·, ˜ 0) doesn’t stands for the norm of E 1 . 1 r r For u, v ∈ E , u 6 v means ur− 6 v− , ur+ 6 v+ (see [1, 3, 5, 9, 11, 19, 20]). Using the results of [1, 3, 5, 9, 11, 19, 20], we recall that: (1) (E 1 , D) is a complete metric space, (2) D(u + w, v + w) = D(u, v), (3) D(u + v, w + e) 6 D(u, w) + D(v, e), (4) D(ku, kv) = |k|D(u, v), k ∈ R, (5) D(u + v, ˜ 0) 6 D(u, ˜ 0) + D(v, ˜ 0), (6) D(u + v, w) 6 D(u, w) + D(v, ˜ 0), where ˜ 0 = χ{0} and u, v, w, e ∈ E 1 . Definition 2.3 [2] Let α : [a, b] → R be an increasing function. A fuzzy-number-valued function f˜(x) is said to e ∈ E1 be fuzzy Henstock-Stieltjes (F HS) integrable with respect to α on [a, b] if there exists a fuzzy number H such that for every ε > 0, there is a function δ(x) > 0 such that for any δ-fine division P = {[ui , vi ]; ξi }n i=1 , we have n X
e < ε. D f˜(ξi )[α(vi ) − α(ui )], H (2.2) i=1

We write (F HS)

Rb

e , and f˜ ∈ F HSα [a, b]. f˜(x)dα = H

a

Definition 2.4 [6] A fuzzy-number-valued function f˜ : [a, b] → E 1 is said to be have locally small Riemann sums or (LSRS) if for every ε > 0 there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have

X

f˜(ξ)(v − u)

E1

< ε,

(2.3)

whenever P = {[u, v]; ξ} is a δ-fine division of an interval [r, s] ⊂ (t − δ(t), t + δ(t)), t ∈ [r, s] and Σ sums over P .

3

Locally small Riemann sums and Henstock-Stieltjes integral of fuzzy-number-valued functions

In this section, we shall define locally small Riemann sums with respect to α on [a, b] for fuzzy-numbervalued functions. Furthermore, we prove that a fuzzy-number-valued functions is Henstock-Stieltjes integrable with respect to α on [a, b] if and only if it has (LSRS) with respect to α on [a, b]. We begin with the following definition. Definition 3.1 Let α : [a, b] → R be an increasing function. A fuzzy-number-valued function f˜ : [a, b] → E 1 is said to be have locally small Riemann sums (or LSRS) with respect to α on [a, b], if for every ε > 0 there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have

X

˜(ξ)[α(v) − α(u)] < ε,

f (3.1)

E1

whenever P = {[u, v]; ξ} is a δ-fine division of an interval [r, s] ⊂ (t − δ(t), t + δ(t)), t ∈ [r, s] and Σ sums over P . If there exists a z ∈ E 1 such that x = y + z, then we call z the H− difference of x and y, denoted by x − y. According to the additivity of F HS, we have the following Lemma. Lemma 3.1 [2] Let f˜ ∈ F HSα [a, b] and F˜ be the primitive of f˜(x) then F˜ satisfies the H− difference. 1109

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J. COMPUTATIONAL AND APPLICATIONS, VOL. 25, NO.6, 2018,sums COPYRIGHT 2018 EUDOXUS PRESS, LLC M.E. Hamid, L.S. Xu,ANALYSIS Z.T. Gong: Locally and globally small Riemann and Henstock-Stieltjes integral...

Lemma 3.2 (Henstock Lemma). Let α : [a, b] → R be an increasing function. If a fuzzy-number-valued function f˜ : [a, b] → E 1 is Henstock-Stieltjes integrable with respect to α on [a, b] with primitive F˜ , i.e., for every ε > 0 there is a positive function δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have  X X f˜(ξ)[α(v) − α(u)], F˜ (u, v) < ε. (3.2) D Then for any sum of parts

P

from

P

, we have

1

D

X

f˜(ξ)[α(v) − α(u)],

1

X

 F˜ (u, v) < ε.

(3.3)

1

The proof is similar to the Theorem 3.7 [10]. Theorem 3.1 Let α : [a, b] → R be an increasing function. If f˜(x) is Henstock-Stieltjes integrable with respect to α on [a, b] then f˜(x) it has LSRS with respect to α on [a, b]. Proof Let F˜ be the primitive of f˜(x). Given ε > 0 there is a δ(ξ) > 0 such that for any δ-fine division P = {[u, v]; ξ} of [a, b], we have X  X f˜(ξ)[α(v) − α(u)], D F˜ (u, v) < ε. (3.4) Where F˜ (u, v) = F˜ (v) − F˜ (u). By the continuity of F˜ at ξ,
D F˜ (u), F˜ (v) < ε

whenever

[u, v] ⊂ (ξ − δ(ξ), ξ + δ(ξ)).

Therefore for t ∈ [a, b] and any δ-fine division P = {[u, v]; ξ} of [r, s] ⊂ (t − δ(t), t + δ(t)), we have

X    X

X ˜(ξ)[α(v) − α(u)], ˜(ξ)[α(v) − α(u)] ˜ (u, v) + D F˜ (r), F˜ (s)

≤ D f F f

1

E


0 there is a positive function δ(ξ) > 0 such that whenever P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } are two δ-fine divisions, we have X  X ˜ ˜ D f (ξ1 )[α(v1 ) − α(u1 )], f (ξ2 )[α(v2 ) − α(u2 )] < ε. (3.5) (P1 )

(P2 )

Theorem 3.2 Let α : [a, b] → R be an increasing function. If a fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS with respect to α on [a, b] then f˜(x) is Henstock-Stieltjes integrable with respect to α on any closed sub-interval C ⊂ (a, b). (Where C = [r, s]). Proof A fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS with respect to α on [a, b] means that for every ε > 0 there is a δ(ξ) > 0 such that for every t ∈ [a, b], we have



X ˜(ξ)[α(v) − α(u)] < ε,

f (3.6)

E1

whenever P = {[u, v]; ξ} is a δ-fine division of an interval C ⊂ (t − δ(t), t + δ(t)), t ∈ C and Σ sums over P . (i) If there t ∈ [a, b] with C ⊂ (t − δ(t), t + δ(t)) we have the following discussion: (1) If t ∈ C then for every ε > 0 there is a two δ-fine divisions P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } on C, such that X  X ˜ ˜ D f (ξ1 )[α(v1 ) − α(u1 )], f (ξ2 )[α(v2 ) − α(u2 )] < ε. (3.7) (P1 )

(P2 )

According to the Cauchy criterion, then f˜(x) is Henstock-Stieltjes integrable on C. 1110

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J. COMPUTATIONAL AND APPLICATIONS, VOL. 25, NO.6, 2018,sums COPYRIGHT 2018 EUDOXUS PRESS, LLC M.E. Hamid, L.S. Xu,ANALYSIS Z.T. Gong: Locally and globally small Riemann and Henstock-Stieltjes integral...

(2) If t ∈ / C then there is a closed interval E ⊂ (t − δ(t), t + δ(t)), with the result that t ∈ E and C ⊂ E (where E = [g, h] ). As a result, for every ε > 0 there is a two δ-fine divisions P1 = {[u1 , v1 ]; ξ1 }, P2 = {[u2 , v2 ]; ξ2 } on E, such that X  X D f˜(ξ1 )[α(v1 ) − α(u1 )], f˜(ξ2 )[α(v2 ) − α(u2 )] < ε. (3.8) (P1 )

(P2 )

According to the Cauchy criterion, then f˜(x) is Henstock-Stieltjes integrable on E. Because C ⊂ E and f˜(x) is Henstock-Stieltjes integrable on E then f˜(x) is Henstock-Stieltjes integrable on C. (ii) If C * (t − δ(t), t + δ(t)) then there is a positive function δ on [a, b] which resulted in the presence that P = {(Ci , ti ) : i = 1, 2, · · · , k} is a δ-fine division of the interval C. It follows that f˜(x) is Henstock-Stieltjes integrable on Ci for i = 1, 2, · · · , k. Then f˜(x) is Henstock-Stieltjes integrable on C. This completes the proof.  Corollary 3.1 Let α : [a, b] → R be an increasing function. If a fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS with respect to α on [a, b] then f˜(x) is Henstock-Stieltjes integrable with respect to α on C for any simple set C ⊂ (a, b). Notice that a simple set C means that there exists finite closed sub-interval Ci which belongs to (a, b) such k S that C = Ci . i=1

Theorem 3.3 Let α : [a, b] → R be an increasing function. If a fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS with respect to α on [a, b] then f˜(x) is Henstock-Stieltjes integrable with respect to α on [a, b]. Proof A fuzzy-number-valued function f˜ : [a, b] → E 1 has LSRS with respect to α on [a, b], then for every ε > 0 there is δ ∗ (ξ) > 0 such that for every t ∈ [a, b], we have

X

˜(ξ)[α(v) − α(u)] < ε,

f (3.9)

E1

∗

whenever P = {[u, v]; ξ} is a δ -fine division of an interval C ⊂ (t − δ(t), t + δ(t)), t ∈ C and Σ sums over P . According to the Corollary 3.1, f˜(x) is Henstock-Stieltjes integrable on C for any simple set C ⊂ (a, b). T S Rows set {Ei }, Ei Ej = φ, ∀i 6= j with property (a, b) = Ei , Ei is a closed interval. Thus for a bove ε > 0, there is a positive numbers n0 with property [ µ{[a, b] − Ei } < ε, (3.10) i≤n0

where µ is Lebesgue measure. For any i, there is a positive function δi such that for any δi -fine division on Ei , we have X  Z D f˜(ξ)[α(v) − α(u)], (HS) f˜(x)dα < ε.

(3.11)

Ei

Define a positive function δ by the formula:  ∗ 1   min{δ (ξ), 2 d(ξ, ∂[a, b])} δ(ξ) = ∗   min{δ (ξ), δi (ξ)},

if ξ ∈

S

Ei ,

i>n0

if ξ ∈

S

Ei .

i≤n0

T For each C = {C} = {C1 , C2 , · · · , Ck } with Cj = Ei Q (where Q = [u, v]), for one i ≤ n0 and one Q with {[u, v]; ξ} is a δ-fine division and ξ ∈ (a, b), we have (i) If Cj = Ei for i ≤ n0 . Because f˜(x) is Henstock-Stieltjes integrable on Ei and f˜(x) is Henstock-Stieltjes k S integrable on Cj consequently f˜(x) is Henstock-Stieltjes integrable on Cj . Selected a positive function δ∗ with j=1

δ∗ (ξ) = min{δj (ξ) : j = 1, 2, · · · , k}, then for each δ∗ -fine division P = {[u, v]; ξ} on

k S

Cj , we have

j=1

 Z D (HS)

k S

f˜(x)dα,

X

f˜(ξ)[α(v) − α(u)]

 < ε.

(3.12)

Cj

j=1

1111

Elsheikh Hamid et al 1107-1115

J. COMPUTATIONAL AND APPLICATIONS, VOL. 25, NO.6, 2018,sums COPYRIGHT 2018 EUDOXUS PRESS, LLC M.E. Hamid, L.S. Xu,ANALYSIS Z.T. Gong: Locally and globally small Riemann and Henstock-Stieltjes integral...

Thus obtained:

X

Z

˜(x)dα

C (HS) f

C

≤

 Z D (HS)

k S

E1

f˜(x)dα,

X

Cj

 X k X

˜(ξ)[α(v) − α(u)]

f˜(ξ)[α(v) − α(u)] + f

j=1

j=1


N there is a δn (ξ) > 0 and for every δn -fine division P = {[u, v]; ξ} of [a, b], we have

X

˜(ξ)[α(v) − α(u)] < ε,

f (4.1)

1

E

kf˜(ξ)kE 1 >n

where the

P

is taken over P and for which kf˜(ξ)kE 1 > n.

Theorem 4.1 Let α : [a, b] → R be an increasing function and let f˜(x) be Henstock-Stieltjes integrable to F˜ (a, b) with respect to α on F HSα [a, b] and F˜n (a, b) the integral of f˜n (x) on F HSα [a, b], where f˜n (x) = f˜(x)

when f˜(x) E 1 6 n and ˜ 0 otherwise. If F˜n (a, b) → F˜ (a, b) as n → ∞ then f˜(x) has GSRS with respect to α on [a, b]. Proof Given ε > 0 there is a δn (ξ) > 0 such that for every δn -fine division P = {[u, v]; ξ} of [a, b], we have D

X


f˜n (ξ)[α(v) − α(u)], F˜n (a, b) < ε, 1112

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J. COMPUTATIONAL AND APPLICATIONS, VOL. 25, NO.6, 2018,sums COPYRIGHT 2018 EUDOXUS PRESS, LLC M.E. Hamid, L.S. Xu,ANALYSIS Z.T. Gong: Locally and globally small Riemann and Henstock-Stieltjes integral...

where (F HS)

Rb a

f˜n (x)dα = F˜n (a, b). D

X


f˜(ξ)[α(v) − α(u)], F˜ (a, b) < ε,

(4.3)


D F˜n (a, b), F˜ (a, b) < ε.

(4.4)

Rb

where (F HS) a f˜(x)dα = F˜ (a, b). Choose N so that whenever n > N Therefore for n > N and δn -fine division P = {[u, v]; ξ} of [a, b], we have

X  X X

˜(ξ)[α(v) − α(u)] = D ˜n (ξ)[α(v) − α(u)], ˜(ξ)[α(v) − α(u)]

f f f

1 E

kf˜(ξ)kE 1 >n

X

6

D


N and a suitably chosen δ-fine division P = {[u, v]; ξ}, we have   ˜ ˜ D Fn (a, b), Fm (a, b)     X X f˜(ξ)[α(v) − α(u)], F˜m (a, b) 6 D F˜n (a, b), f˜(ξ)[α(v) − α(u)] + D kf˜(ξ)kE 1 6m

kf˜(ξ)kE 1 6n

+ <



X kf˜(ξ)kE 1 >n

f˜(ξ)[α(v) − α(u)]

E1

+

X kf˜(ξ)kE 1 >m

f˜(ξ)[α(v) − α(u)]

E1

4ε.

That is, F˜n (a, b) converge to a fuzzy number, say F˜ (a, b), as n → ∞. Again, for suitably chosen N and δ(ξ) and for every δ-fine division P = {[u, v]; ξ}, we have X    D f˜(ξ)[α(v) − α(u)], F˜ (a, b) 6 D F˜ (a, b), F˜N (a, b)

  X X

˜(ξ)[α(v) − α(u)] + D F˜N (a, b), f˜(ξ)[α(v) − α(u)] + f

kf˜(ξ)kE 1 6N


N

E1

3ε.

That is, f˜(x) is fuzzy Henstock-Stieljes integrable on [a, b]. This completes the proof.



Theorem 4.3 Let α : [a, b] → R be an increasing function and let f˜n (x) ∈ F HSα [a, b], n = 1, 2, 3 · · · and satisfy: (1) lim f˜n (x) = f˜(x) almost everywhere in [a, b]; n→∞ (2) there exists a Lebesgue-Stieljes integrable (Henstock-Stieljes integrable) function h(x) on [a, b] such that
D f˜n (x), f˜m (x) < h(x). (4.5) Then, f˜n (x) has GSRS with respect to α on [a, b] uniformly for any n. Naturally, f˜ is (F HS) integrable with respect to α on [a, b]. Furthermore, Z b Z b lim (F HS) f˜n (x)dα = (F HS) f˜(x)dα. (4.6) n→∞

a

a

1113

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Rx Proof Let ε > 0. Since H(x) = (LS) a h(t)dα is absolutely continuous with respect to stieljes measurable P on [a, b], there exists a positive number η > 0 such that |H(bi ) − H(ai )| < ε whenever {[ai , bi ]} is a finite
P collection of non-overlapping intervals in [a, b] that satisfy α(bi ) − α(ai ) < η. Since lim f˜n (x) = f˜(x) almost n→∞ everywhere in [a, b], and D(f˜n , f˜)

= sup max{|(fn (x))r− − (f (x))r− |, |(fn (x))r+ − (f (x))r+ |} r∈[0,1] r

r

r

r

= sup max{|(fn (x))−k − (f (x))−k |, |(fn (x))+k − (f (x))+k |} rk ∈[0,1]

is a sequence of Lebesgue-Stieljes measurable functions, where rk ∈ [0, 1] is the set of rational numbers, by Egorov’s Theorem, there exists an open set G with LS(G) < η such that lim f˜n (x) = f˜(x) uniformly for n→∞ x ∈ [a, b] \ G. Then, there is an natural number N , such that for any n, m > N , and for any x ∈ [a, b] \ G, we have D(f˜n (x), f˜m (x)) < ε. Since h(x) is Henstock-Stieljes integrable on [a, b], there is a δh (ξ) > 0 such that for any δh -fine division P = {[u, v]; ξ} of [a, b], we have X Z b < ε. h(t)dα (4.7) h(ξ)[α(v) − α(u)] − (LS) a

Define δ(ξ) =

 δh (ξ),

if ξ ∈ [a, b] \ G,

δ(ξ), satisfying (ξi − δ(ξi ), ξi + δ(ξi )) ⊂ G,

if ξ ∈ [a, b].

Then, it follows that for a δ-fine division P0 = {[xi−1 , xi ]; ξi } of [a, b], X  X D f˜n (ξi )[α(xi ) − α(xi−1 )], f˜m (ξi )[α(xi ) − α(xi−1 )]  X  X ˜ ˜ ≤ D fn (ξi )[α(xi ) − α(xi−1 )], fm (ξi )[α(xi ) − α(xi−1 )] ξi ∈[a,b]\G

+

D

X

f˜n (ξi )[α(xi ) − α(xi−1 )],

ξi ∈G

≤

X

ξi ∈[a,b]\G

X

 f˜m (ξi )[α(xi ) − α(xi−1 )]

ξi ∈G

X

D f˜n (ξi ), f˜m (ξi ) [α(xi ) − α(xi−1 )] + D f˜n (ξi ), f˜m (ξi ) [α(xi ) − α(xi−1 )] ξi ∈G

ξi ∈[a,b]\G


0 such that for any D(Fn [a, b], A) N1 δN1 -fine division P = {[u, v]; ξ} of [a, b], for any n > NN1 , we have X  D f˜n (ξ)[α(v) − α(u)], F˜n [a, b]   X  ≤ D F˜n [a, b], F˜N1 [a, b] + D f˜N1 (ξ)[α(v) − α(u)], F˜N1 [a, b] X  X + D f˜n (ξ)[α(v) − α(u)], f˜N1 (ξ)[α(v) − α(u)]
0, z ∈ U . denoted by K = f ∈ An : Re zff 0 (z) We need the following. Definition 1.1 ([6]) Let D ⊂ C, z0 ∈ D be a fixed point and let the functions f, g ∈ H (D). The function f is said to be fuzzy subordinate to g and write f ≺F g or f (z) ≺F g (z), if are satisfied the conditions f (z0 ) = g (z0 ) and Ff (D) f (z) ≤ Fg(D) g (z), z ∈ D. Definition 1.2 ([7, Definition 2.2]) Let ψ : C3 × U → C and h univalent in U , with ψ (a, 0; 0) = h (0) = a. If p is analytic in U , with p (0) = a and satisfies the fuzzy differential subordination Fψ(C3 ×U ) ψ(p(z), zp0 (z) , z 2 p00 (z); z) ≤ Fh(U ) h(z),

z ∈ U,

(1.1)

then p is called a fuzzy solution of the fuzzy differential subordination. The univalent function q is called a fuzzy dominant of the fuzzy solutions of the fuzzy differential subordination, if Fp(U ) p(z) ≤ Fq(U ) q(z), z ∈ U , for all p satisfying (1.1). A fuzzy dominant qe that satisfies Fqe(U ) q˜(z) ≤ Fq(U ) q(z), z ∈ U , for all fuzzy dominants q of (1.1) is said to be the fuzzy best dominant of (1.1). Rz 1 Lemma 1.1 ([5, Corollary 2.6g.2, p. 66]) Let h ∈ An and L [f ] (z) = G (z) = 1 1 0 h (t) t n −1 dt, z ∈ U. If n nz   00 (z) 1 Re zh h0 (z) + 1 > − 2 , z ∈ U, then L (f ) = G ∈ K. Lemma 1.2 ([8]) Let h be a convex function with h(0) = a, and let γ ∈ C∗ be a complex number with Re γ ≥ 0. If p ∈ H[a, n] with p (0) = a, ψ : C2 × U → C, ψ (p (z) , zp0 (z) ; z) = p (z) + γ1 zp0 (z) an analytic function in U and   1 0 Fψ(C2 ×U ) p(z) + zp (z) ≤ Fh(U ) h(z), z ∈ U, (1.2) γ Rz then Fp(U ) p(z) ≤ Fg(U ) g(z) ≤ Fh(U ) h(z), z ∈ U, where g(z) = nzγγ/n 0 h(t)tγ/n−1 dt, z ∈ U. The function q is convex and is the fuzzy best dominant. 1116

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Lemma 1.3 ([8]) Let g be a convex function in U and let h(z) = g(z)+nαzg 0 (z), z ∈ U, where α > 0 and n is a positive integer. If p(z) = g(0)+pn z n +pn+1 z n+1 +. . . , z ∈ U, is holomorphic in U and Fp(U ) (p(z) + αzp0 (z)) ≤ Fh(U ) h(z), z ∈ U, then Fp(U ) p(z) ≤ Fg(U ) g(z), z ∈ U, and this result is sharp. We need the following diferential operators. Definition 1.3 (Ruscheweyh [9]) For f ∈ An , m, n ∈ N, the operator Rm is defined by Rm : An → An , R0 f (z) = f (z) , R1 f (z) = zf 0 (z) , ... 0 (m + 1) R f (z) = z (Rm f (z)) + mRm f (z) , z ∈ U. m+1

Remark 1.1 If f ∈ An , f (z) = z +

P∞

j=n+1

aj z j , then Rm f (z) = z +

P∞

(n+j−1)! j j=n+1 n!(j−1)! aj z ,

z ∈ U.

Definition 1.4 ([1]) For f ∈ An , m, n ∈ N, the operator I (m, λ, l) f (z) is defined by the following  λ, l ≥ 0,  m P∞ λ(j−1)+l+1 infinite series I (m, λ, l) f (z) = z + j=n+1 aj z j . l+1 0

Remark 1.2 We have I (0, λ, l) f (z) = f (z), (l + 1) I (m + 1, λ, l) f (z) = (l + 1 − λ) I (m, λ, l) f (z)+λz (I (m, λ, l) f (z)) , z ∈ U. α α Definition 1.5 ([4]) Let α, λ, l ≥ 0, m, n ∈ N. Denote by RIm,λ,l the operator given by RIm,λ,l : An → An , α m RIm,λ,l f (z) = (1 − α)R f (z) + αI (m, λ, l) f (z), z ∈ U.

P∞ Remark 1.3 If f ∈ An , f (z) = z + j=n+1 aj z j , then n  m o P∞ (n+j−1)! α + (1 − α) aj z j , z ∈ U. RIm,λ,l f (z) = z + j=n+1 α 1+λ(j−1)+l l+1 n!(j−1)!

2

Main results α Using the operator RIm,λ,l we define the class RIFδ (α, m, λ, l) and we study fuzzy subordinations.

Definition 2.1 The class RIFδ (α, m, λ, l) contains all the functions f ∈ An which satisfy the inequality
0 α F(RI α f )0 (U ) RIm,λ,l f (z) > δ, z ∈ U, m,λ,l

(2.1)

where δ ∈ (0, 1], α, λ, l ≥ 0 and m, n ∈ N. Theorem 2.1 RIFδ (α, m, λ, l) is a convex set. P∞ Proof. Let f1 , f2 ∈ RIFδ (α, m, λ, l), fk (z) = z + j=n+1 ajk z j , k = 1, 2, z ∈ U. We show that the function h (z) = η1 f1 (z) + η2 f2 (z) is in the class RIFδ (α, m, λ, l) , where η1 and η2 are nonnegative such that η1 + η2 = 1. 0 Differentiating, we obtain h0 (z) = (µ1 f1 + µ2 f2 ) (z) = µ1 f10 (z) + µ2 f20 (z), z ∈ U , and  0  0  0  0 α α α α RIm,λ,l h (z) = RIm,λ,l (µ1 f1 + µ2 f2 ) (z) = µ1 RIm,λ,l f1 (z) + µ2 RIm,λ,l f2 (z) , so we have also  0  0 α α F(RI α h)0 (U ) RIm,λ,l h (z) = F(RI α (µ1 f1 +µ2 f2 ))0 (U ) RIm,λ,l (µ1 f1 + µ2 f2 ) (z) = m,λ,l m,λ,l   0  0  α α f2 (z) = F(RI α (µ1 f1 +µ2 f2 ))0 (U ) µ1 RIm,λ,l f1 (z) + µ2 RIm,λ,l m,λ,l

F F

0

0

α α f1 (z)) )+F f2 (z)) ) 0 0 (µ1 (RIm,λ,l (µ2 (RIm,λ,l α α f2 ) (U ) f1 ) (U ) (µ2 RIm,λ,l (µ1 RIm,λ,l

α (RIm,λ,l

2

0

α f1 (z)) +F (RIm,λ,l f1 ) (U ) (RI α 0

m,λ,l

2

.

 0 α f1 (z) ≤ 1 and (α, m, λ, l), we have δ < F(RI α f1 )0 (U ) RIm,λ,l m,λ,l  0 α 0 RI f (z) ≤ 1, z ∈ U . m,λ,l 2 f2 ) (U )

Since f1 , f2 ∈ δ < F(RI α m,λ,l

=

0

α f2 (z)) (RIm,λ,l f2 ) (U ) 0

RIFδ

0

0

α α f1 (z)) +F f2 (z)) 0 0 (RIm,λ,l (RIm,λ,l α α f2 ) (U ) f1 ) (U ) (RIm,λ,l (RIm,λ,l Therefore δ < ≤ 1 and we obtain that 2  0 α δ δ < F(RI α h)0 (U ) RIm,λ,l h (z) ≤ 1, which means that h ∈ RIF (α, m, λ, l) and RIFδ (α, m, λ, l) is a convex m,λ,l set.

F

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1 Theorem 2.2 Consider g a convex function in U and h (z) = g (z) + c+2 zg 0 (z) , with z ∈ U, c > 0. When R z f ∈ RIFδ (α, m, λ, l) and G (z) = zc+2 tc f (t) dt, z ∈ U, then c+1 0
0 α f (z) ≤ Fh(U ) h (z) , z ∈ U, (2.2) F(RI α f )0 (U ) RIm,λ,l m,λ,l  0 α G (z) ≤ Fg(U ) g (z), z ∈ U, and this result is sharp. implies F(RI α G)0 (U ) RIm,λ,l m,λ,l Rz Proof. Differentiating with respect to z the following relation z c+1 G (z) = (c + 2) 0 tc f (t) dt. We obtain  0 α α α (c + 1) G (z) + zG0 (z) = (c + 2) f (z) and (c + 1) RIm,λ,l G (z) + z RIm,λ,l G (z) = (c + 2) RIm,λ,l f (z) , z ∈ U.  0  00  0 1 α α α Differentiating again we get RIm,λ,l G (z) + c+2 z RIm,λ,l G (z) = RIm,λ,l f (z) , z ∈ U, and the fuzzy differential subordination (2.2) becomes    
0
00 1 1 α α 0 α FRIm,λ,l G(U ) RIm,λ,l G (z) + z RIm,λ,l G (z) zg (z) . ≤ Fg(U ) g (z) + (2.3) c+2 c+2

Consider


0 α (2.4) p (z) = RIm,λ,l G (z) , z ∈ U,     1 1 zp0 (z) ≤ Fg(U ) g (z) + c+2 zg 0 (z) , evidently p ∈ H [1, n] and from relation (2.3) we get Fp(U ) p (z) + c+2 z ∈ U.  0 α G (z) ≤ Fg(U ) g (z), z ∈ U, and g is the fuzzy best By using Lemma 1.3 we obtain F(RI α G)0 (U ) RIm,λ,l m,λ,l dominant. Rz c Theorem 2.3 Let h (z) = 1+(2β−1)z , β ∈ [0, 1) and c > 0. If λ, l ≥ 0, m, n ∈ N and Ic (f ) (z) = zc+2 t f (t) dt, c+1 1+z 0 z ∈ U, then h i ∗ Ic RIFβ (α, m, λ, l) ⊂ RIFβ (α, m, λ, l) , (2.5) where β ∗ = 2β − 1 +

(c+2)(2−2β) n

R1 0

t

c+2 −1 n

t+1

dt.

  1 zp0 (z) ≤ Proof. A similar proof with Theorem 2.2 for the convex function h get us Fp(U ) p (z) + c+2 fh(U ) h (z) , with p (z) defined in (2.4).  0 α Applying Lemma 1.2 we obtain F(RI α G)0 (U ) RIm,λ,l G (z) ≤ Fg(U ) g (z) ≤ Fh(U ) h (z) , where g (z) = m,λ,l c+2 R z c+2 −1 1+(2β−1)t (c+2)(2−2β) R z t n −1 c+2 n dt = 2β − 1 + t c+2 c+2 1+t t+1 dt. Since g is convex and g (U ) is symmetric with 0 0 nz

nz

n

respect to the real axis, we have

n

0

FI(m,λ,l)G(U ) (I (m, λ, l) G (z)) ≥ min Fg(U ) g (z) = Fg(U ) g (1) |z|=1

and β ∗ = g (1) = 2β − 1 +

(c+2)(2−2β) n

R1 0

t

c+2 −1 n

t+1

(2.6)

dt. The inclusion (2.5) follows from (2.6).

Theorem 2.4 Consider g a convex function with g(0) = 1 and h(z) = g(z) + zg 0 (z), z ∈ U. For λ, l ≥ 0, m, n ∈ N, f ∈ An and the fuzzy differential subordination
0 α F(RI α f )0 (U ) RIm,λ,l f (z) ≤ Fh(U ) h (z) , z ∈ U, (2.7) m,λ,l α holds, we obtain FRIm,λ,l f (U )

α RIm,λ,l f (z) z

≤ Fg(U ) g(z), z ∈ U, and this result is sharp.

α RIm,λ,l f (z) , z

α z ∈ U , we have p ∈ H[1, n] and differentiating the relation RIm,λ,l f (z) = 0 α zp(z), z ∈ U, we obtain RIm,λ,l f (z) = p(z) + zp0 (z), z ∈ U.

Proof. For p(z) =



The fuzzy differential subordination (2.7) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z) = Fg(U ) (g(z) + zg 0 (z)) , RI α

f (z)

z ∈ U, and applying Lemma 1.3, we deduce F(RI α f )0 (U ) m,λ,l ≤ Fg(U ) g(z), z ∈ U. z m,λ,l   00 (z) Theorem 2.5 Let h be a holomorphic function with Re 1 + zh > − 12 , z ∈ U, and h(0) = 1. When the 0 h (z) fuzzy differential subordination
0 α F(RI α f )0 (U ) RIm,λ,l f (z) ≤ Fh(U ) h (z) , z ∈ U, (2.8) m,λ,l RI α

m,λ,l α holds, where λ, l ≥ 0, m, n ∈ N, f ∈ An , then FRIm,λ,l f (U ) z R 1 z q(z) = 1 1 0 h(t)t n −1 dt is convex and it is the fuzzy best dominant.

f (z)

≤ Fq(U ) q(z), z ∈ U. The function

nz n

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Proof. Since Re



1+

zh00 (z) h0 (z)



> − 12 , z ∈ U, from Lemma 1.1, we deduce that q (z) =

1 1

nz n

Rz 0

1

h(t)t n −1 dt is

a convex function and verifies the differential equation associated to the fuzzy differential subordination (2.8) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant.  0 α RIm,λ,l f (z) α , p ∈ H[1, n] and differentiating it, we obtain RIm,λ,l f (z) = p(z) + zp0 (z), Keeping p(z) = z z ∈ U and (2.8) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U. α By using Lemma 1.3, we deduce FRIm,λ,l f (U )

α RIm,λ,l f (z) z

≤ Fq(U ) q(z), z ∈ U.

−2 4 00 with h (0) = 1, h0 (z) = (1+z) . Example 2.1 Let h (z) = 1−z 2 and h (z) = (1+z)3  00     1+z  2 zh (z) 1−ρ 1−ρ cos θ−iρ sin θ Since Re h0 (z) + 1 = Re 1−z = Re 1+ρ cos θ+iρ sin θ = 1+2ρ cos θ+ρ2 > 0 > − 12 , the function h is 1+z convex in U . 2 3 f (z) = 31 R1 f (z) + Let f (z) = z − z 2 , z ∈ U . For n = 1, m = 1, l = 1, λ = 2, α = 32 , we obtain RI1,2,1 2  2 0 RI 3 f (z) 1 2 2 0 0 2 3 = 1 − 4z and 1,2,1 = 1 − 2z. Function 3 I (1, 2, 1) f (z) = 3 zf (z) + 3 zf (z) = z − 2z . Then RI1,2,1 f (z) z R z 2 ln(1+z) q (z) = z1 0 1−t . 1+t dt = −1 + z     2 ln(1+z) , z ∈ U, imply F (1 − 2z) ≤ F −1 + , From Theorem 2.5 we have FU (1 − 4z) ≤ FU 1−z U U 1+z z z ∈ U.

Theorem 2.6 Let g be a convex function with g (0) = 1 and h (z) = g (z) + zg 0 (z), z ∈ U . When the fuzzy differential subordination !0 α f (z) zRIm+1,λ,l α ≤ Fh(U ) h (z) , z ∈ U (2.9) FRIm,λ,l f (U ) α RIm,λ,l f (z) α holds, for λ, l ≥ 0, m, n ∈ N, f ∈ An , then FRIm,λ,l f (U ) sharp.

Proof. Diferentiating p(z) =   α zRIm+1,λ,l f (z) 0 p (z) + z · p0 (z) = . RI α f (z)

α RIm+1,λ,l f (z) α RIm,λ,l f (z)

α RIm+1,λ,l f (z) α RIm,λ,l f (z)

≤ Fg(U ) g (z), z ∈ U, and this result is 0

we get p0 (z) =

α f (z)) (RIm+1,λ,l α RIm,λ,l f (z)

− p (z) ·

α f (z)) (RIm+1,λ,l α RIm,λ,l f (z)

0

and

m,λ,l

In this conditions, relation (2.9) becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z) = Fg(U ) (g(z) + zg 0 (z)) , z ∈ U, α and aplying Lemma 1.3, we obtain FRIm,λ,l f (U )

α RIm+1,λ,l f (z) α RIm,λ,l f (z)

≤ Fg(U ) g(z), z ∈ U.

Theorem 2.7 For a convex function g, with g(0) = 1, consider h(z) = g(z) + zδ g 0 (z), z ∈ U. The fuzzy differential subordination !  α δ−1
RIm,λ,l f (z) 0 α α FRIm,λ,l RIm,λ,l f (z) ≤ Fh(U ) h (z) , z ∈ U, (2.10) f (U ) z is satisfied, when α, λ, l, δ ≥ 0, m, n ∈ N, f ∈ An , and implies the following fuzzy sharp differential subordination  α  RIm,λ,l f (z) δ α FRIm,λ,l ≤ Fg(U ) g (z), z ∈ U. f (U ) z Proof. Differentiating p(z) = 1 0 δ zp (z),



α RIm,λ,l f (z) z

δ

, z ∈ U, we get

z ∈ U. Evidently p ∈ H[1, n], and (2.10) becomes Fp(U ) p(z)

z ∈ U.

α By using Lemma 1.3, we have FRIm,λ,l f (U )



α RIm,λ,l f (z) z

δ

  0 α RIm,λ,l f (z) δ−1 α RI f (z) m,λ,l z
+ 1δ zp0 (z) ≤ Fh(U ) h(z) = Fg(U ) 

= p(z) +
g(z) + zδ g 0 (z) ,

≤ Fg(U ) g(z), z ∈ U.

  00 (z) Theorem 2.8 Consider an holomorphic function h which satisfies the inequality Re 1 + zh > − 12 , z ∈ U, h0 (z) and h(0) = 1. When the fuzzy differential subordination !  α δ−1
0 RIm,λ,l f (z) α α FRIm,λ,l RIm,λ,l f (z) ≤ Fh(U ) h (z) , z ∈ U, (2.11) f (U ) z α holds, with α, λ, l, δ ≥ 0, m, n ∈ N, f ∈ An , then FRIm,λ,l f (U ) Rz δ δ −1 h(t)t n dt. δ 0



α RIm,λ,l f (z) z

δ

≤ Fq(U ) q (z), z ∈ U, and q(z) =

nz n

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Proof. Since Re



1+

zh00 (z) h0 (z)



> − 12 , z ∈ U, from Lemma 1.1, we deduce that q (z) =

Rz

δ δ

nz n

0

δ

h(t)t n −1 dt is

a convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.11) q (z) + 1δ zq 0 (z) = h (z), therefore it is the fuzzy best dominant.  α   α   0 RIm,λ,l f (z) δ RIm,λ,l f (z) δ−1 α Consider p(z) = , z ∈ U, then p ∈ H[1, n]. Differentiating, we obtain RI f (z) = m,λ,l z z
1 1 0 0 p(z)+ δ zp (z), z ∈ U, and the fuzzy differential subordination (2.11) becomes Fp(U ) p(z) + δ zp (z) ≤ Fh(U ) h(z),  α  Rz δ RIm,λ,l f (z) δ α z ∈ U. Aplying Lemma 1.2, we get FRIm,λ,l ≤ Fq(U ) q(z) and q(z) = δ δ 0 h(t)t n −1 dt, z ∈ U, f (U ) z nz n is the best dominant. Theorem 2.9 For a convex function g, with g (0) = 1, define the function h (z) = g (z) + zδ g 0 (z), z ∈ U . If α, λ, l, δ ≥ 0, m, n ∈ N, f ∈ An and the fuzzy differential subordination   0  0  α α α RI f (z) RI f (z) 2 RIm,λ,l f (z) m,λ,l m+1,λ,l z   α FRIm,λ,l −2   2  f (U )  α α δ RI f (z) RI f (z) α m,λ,l m+1,λ,l RIm+1,λ,l f (z)  +z  α holds, then FRIm,λ,l f (U ) z

δ+1  δ

2

(2.12)

≤ Fg(U ) g (z), z ∈ U, and this result is sharp.

α RIm,λ,l f (z) α f (z)) (RIm+1,λ,l

2

, we get 

α RIm,λ,l f (z) z2 2 δ (RI α m+1,λ,l f (z))

0

α f (z)) (RIm,λ,l

0

−2

α f (z)) (RIm+1,λ,l



. α f (z)) (RIm+1,λ,l The fuzzy differential subordination (2.12) becomes

z 0 Fp(U ) p(z) + zδ p0 (z) ≤ Fh(U ) h(z)F g(U ) g(z) + δ g (z) , z ∈ U, and by using Lemma 1.3, we derive  α FRIm,λ,l f (U ) z

2

α RIm,λ,l f (z)

α f (z)) (RIm+1,λ,l

2

+

 2  ≤ Fh(U ) h (z) , z ∈ U, (z)



α f (z)) (RIm+1,λ,l

α RIm,λ,l f (z)

(z)

α RIm+1,λ,l f

α RIm,λ,l f (z)

Proof. Differentiating p(z) = z p (z) + zδ p0 (z) = z δ+1 δ

α RIm,λ,l f

α RIm,λ,l f (z)

α RIm+1,λ,l f (z)

≤ Fg(U ) g(z), z ∈ U.

  00 (z) Theorem 2.10 Let h be a holomorphic function with properties Re 1 + zh > − 12 , z ∈ U, and h(0) = 1. h0 (z) The fuzzy differential subordination   0  0  α α α f (z) f (z) RI RI 2 RIm,λ,l f (z) m+1,λ,l m,λ,l  z  α FRIm,λ,l −2   2  f (U )  α α δ RIm,λ,l f (z) RIm+1,λ,l f (z) α RIm+1,λ,l f (z)  α RIm,λ,l f (z)

δ+1  (2.13)  2  ≤ Fh(U ) h (z) , z ∈ U, δ α RIm+1,λ,l f (z)   α RIm,λ,l f (z) α implies the following fuzzy differential subordination FRIm,λ,l z ≤ Fq(U ) q (z), z ∈ U, where 2 f (U ) α f (z)) (RIm+1,λ,l R δ z α, λ, l, δ ≥ 0, m, n ∈ N, f ∈ An , and q(z) = δ δ 0 h(t)t n −1 dt. +z

nz n

Proof. Since Re



1+

zh00 (z) h0 (z)



> − 12 , z ∈ U, from Lemma 1.1, we deduce that q (z) =

δ

Rz

δ nz n

0

δ

h(t)t n −1 dt is

a convex function and verifies the differential equation associated to the fuzzy differential subordination (2.13) q (z) + 1δ zq 0 (z) = h (z), therefore it is the fuzzy best dominant. α RIm,λ,l f (z)

, z ∈ U, p ∈ H[1, n]. Since   0 0 α α α f (z)) f (z)) RIm,λ,l f (z) (RIm,λ,l (RIm+1,λ,l z 0 δ+1 z2 p (z) + δ p (z) = z δ + δ , z ∈ U, from α α 2 α RIm,λ,l f (z) − 2 RIm+1,λ,l f (z) f (z)) ( ) (RIm+1,λ,l   α
RIm,λ,l f (z) z 0 α (2.13) we have Fp(U ) p(z) + δ p (z) ≤ Fh(U ) h(z), z ∈ U, and from Lemma 1.2, we obtain FRIm,λ,l ≤ 2 f (U ) z α f (z)) (RIm+1,λ,l R δ z Fq(U ) q(z), z ∈ U, and the best dominant is q (z) = δ δ 0 h(t)t n −1 dt. Considering p(z) = z

α f (z)) (RIm+1,λ,l

2

α RIm,λ,l f (z) 2 α RIm+1,λ,l f (z)

nz n

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Theorem 2.11 For a convex function g with g(0) = 0 define h(z) = g(z) + zδ g 0 (z), z ∈ U. If α, λ, l, δ ≥ 0, m, n ∈ N, f ∈ An and the fuzzy differential subordination   0 α RI f (z) m,λ,l  2δ + 2 α FRIm,λ,l + f (U ) z α δ RIm,λ,l f (z) 00   0 2  α α RI f (z) RI f (z) m,λ,l m,λ,l z     −    ≤ Fh(U ) h (z) , z ∈ U, α α δ RIm,λ,l f (z) RIm,λ,l f (z) 

3

 holds, then the following result is sharp F

α f (U ) RIm,λ,l

0

α

f (z)) (RI z 2 RIm,λ,l α m,λ,l f (z)

(2.14)

 ≤ Fg(U ) g (z), z ∈ U.

0

α f (z)) (RIm,λ,l α f (z) , we deduce that p ∈ H[0, 1] and differentiating it, we obtain RIm,λ,l "  2 # 0 0 00 α α α 3 RI f (z) RI f (z) RI f (z) ( ) ( ) ( ) m,λ,l m,λ,l m,λ,l z p (z) + zδ p0 (z) = z 2 δ+2 , z ∈ U. α α α f (z) + δ f (z) − f (z) δ RIm,λ,l RIm,λ,l RIm,λ,l

The fuzzy differential subordination becomesFp(U ) p(z) + 1δ zp0 (z) ≤ Fh(U ) h(z) = Fg(U ) g(z) + zδ g 0 (z) 0 α 2 (RIm,λ,l f (z)) α and by using Lemma 1.3, we deduce FRIm,λ,l ≤ Fg(U ) g(z), z ∈ U, and this result is sharp. α f (U ) z f (z) RI

Proof. Considering p(z) = z 2

m,λ,l

  00 (z) Theorem 2.12 Consider h a holomorphic function with Re 1 + zh > − 12 , z ∈ U, and h(0) = 0. If the 0 h (z) fuzzy differential subordination   0 α RI f (z) m,λ,l δ + 2  2 α FRIm,λ,l + f (U ) z α δ RIm,λ,l f (z) 00   0 2  α α RI f (z) f (z) RI m,λ,l m,λ,l z     −    ≤ Fh(U ) h (z) , z ∈ U, α α δ RIm,λ,l f (z) RIm,λ,l f (z) 

3

 α holds, for α, λ, l, δ ≥ 0, m, n ∈ N, f ∈ An , then FRIm,λ,l f (U ) R δ z q(z) = δ δ 0 h(t)t n −1 dt.

(RI α f (z)) z 2 RIm,λ,l α m,λ,l f (z)

0

(2.15)

 ≤ Fq(U ) q (z), z ∈ U, where

nz n

Proof. Since Re



1+

zh00 (z) h0 (z)



> − 12 , z ∈ U, from Lemma 1.1, we deduce that q (z) =

δ

Rz

δ nz n

0

δ

h(t)t n −1 dt is

a convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.15) q (z) + 1δ zq 0 (z) = h (z), therefore it is the fuzzy best dominant. 0 (RI α f (z)) Considering p(z) = z 2 RIm,λ,l , z ∈ U, p ∈ H[0, n]. Differentiating it, we obtain p (z) + zδ p0 (z) = α m,λ,l f (z) "  2 # 0 00 0 α α α
3 RI f (z) RI f (z) RI f (z) ( ) ( ) ( ) m,λ,l m,λ,l m,λ,l z δ+2 , z ∈ U, and (2.15) becomes Fp(U ) p(z) + 1δ zp0 (z) ≤ z 2 δ RI α f (z) + δ f (z) − f (z) RI α RI α m,λ,l

m,λ,l

m,λ,l

Fh(U ) h(z), z ∈ U.   0 α 2 (RIm,λ,l f (z)) α ≤ Fq(U ) q(z), z ∈ U, and the best dominant is Applying Lemma 1.2, we get FRIm,λ,l z α f (U ) RIm,λ,l f (z) R δ z q(z) = δ δ 0 h(t)t n −1 dt. nz n

Theorem 2.13 Let h(z) = g(z) + zg 0 (z), z ∈ U , where g is a convex function such that g(0) = 1. When the fuzzy differential subordination    00 α α  RIm,λ,l f (z) · RIm,λ,l f (z)    α FRIm,λ,l 1 − z∈U (2.16)   ≤ Fh(U ) h (z) ,  f (U ) 0 2   α RIm,λ,l f (z)  α holds, for α, λ, l ≥ 0, m, n ∈ N, f ∈ An , then the followinf result is sharp FRIm,λ,l f (U )



α RIm,λ,l f (z) 0

α z (RIm,λ,l f (z))

Fg(U ) g(z), z ∈ U. 1121

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.6, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Proof. Let p(z) =

00

α RIm,λ,l f (z) 0

α f (z)) z (RIm,λ,l

. We deduce that p ∈ H[1, n] and differentiating it, we get 1−

α α RIm,λ,l f (z)·(RIm,λ,l f (z)) 0 2

α f (z)) ] [(RIm,λ,l

p (z) + zp0 (z) , z ∈ U. The fuzzy differential subordination becomes Fp(U ) (p(z) + zp0 (z))≤ Fh(U ) h(z) =Fg(U ) (g(z) + zg 0 (z)) and α RIm,λ,l f (z) α we apply Lemma 1.3 to deduce the following sharp result RIm,λ,l ≤ Fg(U ) g(z), z ∈ U. 0 f (U ) α z (RIm,λ,l f (z))   00 (z) Theorem 2.14 Let h be a holomorphic function with h(0) = 1 and Re 1 + zh > − 12 , z ∈ U. The fuzzy h0 (z) differential subordination    00 α α  RIm,λ,l f (z) · RIm,λ,l f (z)    α 1 − z ∈ U, (2.17) FRIm,λ,l   ≤ Fh(U ) h(z),  f (U ) 0 2   α RIm,λ,l f (z)  α induce FRIm,λ,l f (U )



α RIm,λ,l f (z) α f (z)) z (RIm,λ,l

≤ Fq(U ) q(z), z ∈ U, where q(z) =

0

Rz

1 1 nz n

0

1

h(t)t n −1 dt, for α, λ, l ≥ 0,

m, n ∈ N, f ∈ An . Proof. Since Re



zh00 (z) h0 (z)

1+



> − 12 , z ∈ U, from Lemma 1.1, we deduce that

1 1 nz n

Rz 0

1

h(t)t n −1 dt is a

convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.17) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant. 00 α α α RIm,λ,l f (z)·(RIm,λ,l f (z)) RIm,λ,l f (z) = Consider p(z) = 0 , z ∈ U, p ∈ H[1, n]. Differentiating it, we obtain 1− 0 2 α α z (RIm,λ,l f (z)) [(RIm,λ,l f (z)) ] p (z)+zp0 (z) , z ∈ U, and the fuzzy differential subordination (2.17) is written Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U.   α RIm,λ,l f (z) α ≤ Fq(U ) q(z), z ∈ U, and the best dominant is By using Lemma 1.2, we get FRIm,λ,l 0 f (U ) α z (RIm,λ,l f (z)) R 1 z q(z) = 1 1 0 h(t)t n −1 dt. nz n

Example 2.2 Let h (z) =

1−z 1+z

a convex function in U with h (0) = 1 and Re



zh00 (z) h0 (z)

 + 1 > − 12 .

2

3 Let f (z) = z − z 2 , z ∈ U . For n = 1, m = 1, l = 1, λ = 2, α = 32 , we obtain RI1,2,1 f (z) = z − 2z 2 , z ∈ U . 2  2 0  2 00 2 RI 3 f (z) 1−2z 3 3 0 = z−2z Then RI1,2,1 f (z) = 1 − 4z and RI1,2,1 f (z) = −4,  1,2,1 2 z(1−4z) = 1−4z , 3 f (z) z RI1,2,1

2 3 f (z)· RI1,2,1



1−



2 3 RI1,2,1 f (z)

00

0  2

2 3 RI1,2,1 f (z)

=1−

Using Theorem 2.14 we obtain FU z ∈ U.

(z−2z2 )·(−4)



(1−4z)2 8z 2 −4z+1 (1−4z)2



=

8z 2 −4z+1 . (1−4z)2

≤ FU



1−z 1+z



We have q (z) = , z ∈ U, induce FU

1 z

Rz



1−t dt 0 1+t

1−2z 1−4z



= −1 +

2 ln(1+z) . z

 ≤ FU −1 +

2 ln(1+z) z



,

Theorem 2.15 Consider h(z) = g(z) + zg 0 (z), z ∈ U, where g is a convex function with g(0) = 1. When the fuzzy differential subordination h 
0 i2
00 α α α α FRIm,λ,l RI f (z) + RI f (z) · RI f (z) ≤ Fh(U ) h(z), z∈U (2.18) f (U ) m,λ,l m,λ,l m,λ,l  α holds, for α, λ, l ≥ 0, m, n ∈ N, f ∈ An , then the following result is sharp FRIm,λ,l f (U )

α α RIm,λ,l f (z)·(RIm,λ,l f (z)) z

0

 ≤

Fg(U ) g(z), z ∈ U. 0

Proof. Let p(z) =

α α RIm,λ,l f (z)·(RIm,λ,l f (z)) z 

Differentiating it, we obtain



. We deduce that p ∈ H[1, n]. 0 2  00 α α α RIm,λ,l f (z) + RIm,λ,l f (z) · RIm,λ,l f (z) = p (z) + zp0 (z) , z ∈ U , and

the fuzzy differential subordination becomes Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U) h(z) = Fg(U ) (g(z) + zg0 (z)) . 0 α α RIm,λ,l f (z)·(RIm,λ,l f (z)) α By using Lemma 1.3, we obtain the following result sharp FRIm,λ,l ≤ Fg(U ) g(z), f (U ) z z ∈ U.

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  00 (z) Theorem 2.16 For a holomorphic function h which satisfies the inequality Re 1 + zh > − 12 , z ∈ U, and h0 (z) h(0) = 1, the fuzzy differential subordination h 
0 i2
00 α α α α FRIm,λ,l RI f (z) + RI f (z) · RI f (z) ≤ Fh(U ) h(z), z ∈ U, (2.19) f (U ) m,λ,l m,λ,l m,λ,l  α induce FRIm,λ,l f (U )

α α RIm,λ,l f (z)·(RIm,λ,l f (z)) z

0

 ≤ Fq(U ) q(z), z ∈ U, where q(z) =

1 1 nz n

Rz 0

1

h(t)t n −1 dt, and α, λ, l ≥

0, m, n ∈ N, f ∈ An Proof. Since Re



1+

zh00 (z) h0 (z)



> − 12 , z ∈ U, from Lemma 1.1, we deduce that q (z) =

1 1 nz n

Rz 0

1

h(t)t n −1 dt is

a convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.19) q (z) + zq 0 (z) = h (z), therefore it is the fuzzy best dominant. 0 α RI α f (z)·(RIm,λ,l f (z)) , z ∈ U, evidently p ∈ H[1, n]. Let p(z) = m,λ,l z   00 0 2 α α α + RIm,λ,l f (z) · RIm,λ,l f (z) , z ∈ U, and (2.19) means We have p (z) + zp0 (z) = RIm,λ,l f (z) Fp(U ) (p(z) + zp0 (z)) ≤ Fh(U ) h(z), z ∈ U.  α Applying Lemma 1.2, we get FRIm,λ,l f (U ) R 1 z dominant is q(z) = 1 1 0 h(t)t n −1 dt.

α α RIm,λ,l f (z)·(RIm,λ,l f (z)) z

0

 ≤ Fq(U ) q(z), z ∈ U, and the best

nz n

Example 2.3 Let h (z) =

1−z 1+z

a convex function in U with h (0) = 1 and Re



zh00 (z) h0 (z)

 + 1 > − 12 . 2

3 f (z) = z − 2z 2 , Consider f (z) = z − z 2 , z ∈ U . For n = 1, m = 1, l = 1, λ = 2, α = 32 , we obtain RI1,2,1 z ∈ U.  0 2 2 3 3  2 0  2 00 f (z)· RI1,2,1 f (z) RI1,2,1 (z−2z2 )(1−4z) 3 3 = = 8z 2 − 6z + 1, Then RI1,2,1 f (z) = 1 − 4z, RI1,2,1 f (z) = −4, z z  0 2  2 00 2 2
2 3 3 3 RI1,2,1 f (z) + RI1,2,1 f (z) · RI1,2,1 f (z) = (1 − 4z) + z − 2z 2 · (−4) = 24z 2 − 12z + 1. Rz 2 ln(1+z) . We have q (z) = z1 0 1−t 1+t dt = −1 + z  

2 2 From Theorem 2.16 we obtain FU 24z − 12z + 1 ≤ FU 1−z ≤ 1+z , z ∈ U, induce FU 8z − 6z + 1   FU −1 + 2 ln(1+z) , z ∈ U. z

z Theorem 2.17 Consider h(z) = g(z) + 1−δ g 0 (z), z ∈ U , where g is a convex function with g(0) = 1. If the fuzzy differential subordination   0  0   !δ α α α RI f (z) RI f (z) RI f (z) m+1,λ,l m,λ,l z    m+1,λ,l α −δ FRIm,λ,l   ≤ Fh(U ) h(z), f (U )  α α α RIm,λ,l f (z) 1−δ RIm+1,λ,l f (z) RIm,λ,l f (z)

(2.20) z ∈ U, holds,for α, λ, l ≥ 0, δ ∈ (0, 1), m,  n ∈ N, f ∈ An , then the following result is sharp  δ α RIm+1,λ,l f (z) α FRIm,λ,l · RI α z f (z) ≤ Fg(U ) g(z), z ∈ U. f (U ) z m,λ,l

 δ RI α f (z) z · . We deduce that p ∈ H[1, n] and differentiating the function Proof. Let p(z) = m+1,λ,l α z RIm,λ,l f (z)    δ α 0 0 α α f (z)) RIm+1,λ,l f (z) (RIm+1,λ,l f (z)) (RIm,λ,l z 1 p, we obtain RI α f (z) = p (z) + 1−δ zp0 (z) , z ∈ U. α α 1−δ RIm+1,λ,l f (z) − δ RIm,λ,l f (z) m,λ,l     1 z The fuzzy differential subordination means Fp(U ) p(z) + 1−δ zp0 (z) ≤ Fh(U ) h(z) = Fg(U ) g(z) + 1−δ g 0 (z)   δ  α RIm+1,λ,l f (z) z α and by using Lemma 1.3, we obtain the following sharp result FRIm,λ,l f (U ) · RI α f (z) ≤ z m,λ,l

Fg(U ) g(z), z ∈ U.   00 (z) Theorem 2.18 Let h be an holomorphic function with h(0) = 1 and Re 1 + zh > − 12 , z ∈ U.The fuzzy h0 (z) differential subordination    0   0 !δ α α α RI f (z) RI f (z) RIm+1,λ,l f (z)  m+1,λ,l m,λ,l z   α FRIm,λ,l −δ   ≤ Fh(U ) h(z), f (U )  α α α RIm,λ,l f (z) 1−δ RIm+1,λ,l f (z) RIm,λ,l f (z) (2.21) 1123

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 α z ∈ U, induce FRIm,λ,l f (U )

α RIm+1,λ,l f (z) z

·



z

δ 

α f (z) RIm,λ,l

≤ Fq(U ) q(z), z ∈ U, where q(z) =

for α, λ, l ≥ 0, δ ∈ (0, 1) , m, n ∈ N, f ∈ An .   00 (z) > − 12 , z ∈ U, from Lemma 1.1, we deduce that q (z) = Proof. Since Re 1 + zh 0 h (z)

1−δ

Rz

1−δ nz n

0

1−δ

Rz

1−δ nz n

0

h(t)t

h(t)t

1−δ n −1

1−δ n −1

dtt

is a convex function and verifies the differential equation asscociated to the fuzzy differential subordination (2.21) 1 zq 0 (z) = h (z), therefore it is the fuzzy best dominant. q (z) + 1−δ δ  RI α f (z) z , z ∈ U, p ∈ H[1, n]. · Consider p(z) = m+1,λ,l α z RIm,λ,l f (z)   δ α  0 0 α α f (z)) f (z)) RIm+1,λ,l f (z) (RIm,λ,l (RIm+1,λ,l 1 − δ Differentiating it, we get RI α z f (z) = p (z) + 1−δ zp0 (z) , α α f (z) f (z) 1−δ RIm+1,λ,l RIm,λ,l m,λ,l   1 z ∈ U, and (2.21) becomes Fp(U ) p(z) + 1−δ zp0 (z) ≤ Fh(U ) h(z), z ∈ U.  δ   α RIm+1,λ,l f (z) z α ≤ Fq(U ) q (z), z ∈ U, and the best From Lemma 1.2, we get FRIm,λ,l · α f (U ) f (z) z RIm,λ,l R 1−δ z dominant is q(z) = 1−δ h(t)t n −1 dt. 1−δ 0 nz

n

References [1] A. Alb Lupa¸s, A new comprehensive class of analytic functions defined by multiplier transformation, Mathematical and Computer Modelling 54, 2011, 2355–2362. [2] A. Alb Lupa¸s, Gh. Oros, On special fuzzy differential subordinations using S˘ al˘ agean and Ruscheweyh operators, Applied Mathematics and Computation, Volume 261, 2015, 119-127. [3] A. Alb Lupa¸s, A Note on Special Fuzzy Differential Subordinations Using Generalized Salagean Operator and Ruscheweyh Derivative, Journal of Computational Analysis and Applications, Vol. 15, No. 8, 2013, 1476-1483. [4] A. Alb Lupas, Some differential subordinations using Ruscheweyh derivative and a multiplier transformation, Computational Analysis: Contributions from AMAT 2015, Proceedings Volume, Springer - New York, Chapter 8, 97-117. [5] S.S. Miller, P.T. Mocanu, Differential Subordinations. Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 225, Marcel Dekker Inc., New York, Basel, 2000. [6] G.I. Oros, Gh. Oros, The notion of subordination in fuzzy sets theory, General Mathematics, vol. 19, No. 4 (2011), 97-103. [7] G.I. Oros, Gh. Oros, Fuzzy differential subordinations, Acta Universitatis Apulensis, No. 30/2012, pp. 55-64. [8] G.I. Oros, Gh. Oros, Dominant and best dominant for fuzzy differential subordinations, Stud. Univ. BabesBolyai Math. 57(2012), No. 2, 239-248. [9] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115.

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About some differential sandwich theorems involving a multiplier transformation and Ruscheweyh derivative Alb Lupa¸s Alina Department of Mathematics and Computer Science University of Oradea 1 Universitatii street, 410087 Oradea, Romania [email protected] Abstract m,n IRλ,l n

In this work we study a new operator defined as the Hadamard product of the multiplier transform,n m,n mation I (m, λ, l) and Ruscheweyh derivative R , given by IRλ,l : A → A, IRλ,l f (z) = (I (m, λ, l) ∗ Rn ) f (z) n+1 and An = {f ∈ H (U ) : f (z) = z + an+1 z + ..., z ∈ U } is the class of normalized analytic functions with A1 = A. The purpose of this paper is to derive certain subordination and superordination results involving m,n the operator IRλ.l and we establish differential sandwich-type theorems.

Keywords: analytic functions, differential operator, differential subordination, differential superordination. 2010 Mathematical Subject Classification: 30C45.

1

Introduction

Let H (U ) be the class of analytic function in the open unit disc of the complex plane U = {z ∈ C : |z| < 1}. Let H (a, n) be the subclass of H (U ) consisting of functions of the form f (z) = a + an z n + an+1 z n+1 + . . . . Let An = {f ∈ H(U ) : f (z) = z + an+1 z n+1 + . . . , z ∈ U } and A = A1 . Let the functions f and g be analytic in U . We say that the function f is subordinate to g, written f ≺ g, if there exists a Schwarz function w, analytic in U , with w(0) = 0 and |w(z)| < 1, for all z ∈ U, such that f (z) = g(w(z)), for all z ∈ U . In particular, if the function g is univalent in U , the above subordination is equivalent to f (0) = g(0) and f (U ) ⊂ g(U ). Let ψ : C3 × U → C and h be an univalent function in U . If p is analytic in U and satisfies the second order differential subordination ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z), for z ∈ U, (1.1) then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (1.1). A dominant qe that satisfies qe ≺ q for all dominants q of (1.1) is said to be the best dominant of (1.1). The best dominant is unique up to a rotation of U .
Let ψ : C2 × U → C and h analytic in U . If p and ψ p (z) , zp0 (z) , z 2 p00 (z) ; z are univalent and if p satisfies the second order differential superordination h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z),

z ∈ U,

(1.2)

then p is a solution of the differential superordination (1.2) (if f is subordinate to F , then F is called to be superordinate to f ). An analytic function q is called a subordinant if q ≺ p for all p satisfying (1.2). An univalent subordinant qe that satisfies q ≺ qe for all subordinants q of (1.2) is said to be the best subordinant. Miller and Mocanu [6] obtained conditions h, q and ψ for which the following implication holds h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z) ⇒ q (z) ≺P p (z) . P∞ ∞ For two functions f (z) = z + j=2 aj z j and g(z) = z + j=2 bj z j analytic in the open unit disc U , the Hadamard product P∞ (or convolution) of f (z) and g (z), written as (f ∗ g) (z) is defined by f (z) ∗ g (z) = (f ∗ g) (z) = z + j=2 aj bj z j . Definition 1.1 [5] For f ∈ A, m ∈ N∪ {0}, λ, l ≥ 0, the multiplier transformation I (m, λ, l) f (z) is defined by m P∞ aj z j . the following infinite series I (m, λ, l) f (z) := z + j=2 1+λ(j−1)+l 1+l 0

Remark 1.1 We have (l + 1) I (m + 1, λ, l) f (z) = (l + 1 − λ) I (m, λ, l) f (z) + λz (I (m, λ, l) f (z)) ,

1125

z ∈ U.

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Remark 1.2 For l = 0, λ ≥ 0, the operator Dλm = I (m, λ, 0) was introduced and studied by Al-Oboudi , which reduced to the S˘ al˘ agean differential operator S m = I (m, 1, 0) for λ = 1. Definition 1.2 (Ruscheweyh [8]) For f ∈ A and n ∈ N, the Ruscheweyh derivative Rn is defined by Rn : A → A, R0 f (z) = f (z) , R1 f (z) = zf 0 (z) , ... 0 (n + 1) Rn+1 f (z) = z (Rn f (z)) + nRn f (z) , z ∈ U. Remark 1.3 If f ∈ A, f (z) = z +

P∞

j=2

aj z j , then Rn f (z) = z +

P∞

(n+j−1)! j j=2 n!(j−1)! aj z

for z ∈ U .

m,n Definition 1.3 ([2]) Let λ, l ≥ 0 and n, m ∈ N. Denote by IRλ,l : A → A the operator given by the m,n Hadamard product of the multiplier transformation I (m, λ, l) and the Ruscheweyh derivative Rn , IRλ,l f (z) = n (I (m, λ, l) ∗ R ) f (z) , for any z ∈ U and each nonnegative integers m, n. m P∞ P∞  (n+j−1)! 2 j m,n Remark 1.4 If f ∈ A and f (z) = z + j=2 aj z j , then IRλ,l f (z) = z + j=2 1+λ(j−1)+l l+1 n!(j−1)! aj z , z ∈ U.

Using simple computation one obtains the next result. Proposition 1.1 [1]For m, n ∈ N and λ, l ≥ 0 we have  0 m,n+1 m,n m,n (n + 1) IRλ,l f (z) − nIRλ,l f (z) = z IRλ,l f (z) .

(1.3)

The purpose of this paper is to derive the several subordination and superordination results involving a differential operator. Furthermore, we studied the results of Selvaraj and Karthikeyan [10], Shanmugam, Ramachandran, Darus and Sivasubramanian [11] and Srivastava and Lashin [12]. In order to prove our subordination and superordination results, we make use of the following known results. Definition 1.4 [7] Denote by Q the set of all functions f that are analytic and injective on U \E (f ), where E (f ) = {ζ ∈ ∂U : lim f (z) = ∞}, and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂U \E (f ). z→ζ

Lemma 1.1 [7] Let the function q be univalent in the unit disc U and θ and φ be analytic in a domain D containing q (U ) with φ (w) 6= 0 when w ∈ q (U ). Q (z) = zq 0 (z) φ (q (z)) and h (z) = θ (q (z)) + Q (z).  Set  zh0 (z) Suppose that Q is starlike univalent in U and Re Q(z) > 0 for z ∈ U . If p is analytic with p (0) = q (0), p (U ) ⊆ D and θ (p (z)) + zp0 (z) φ (p (z)) ≺ θ (q (z)) + zq 0 (z) φ (q (z)) , then p (z) ≺ q (z) and q is the best dominant. Lemma 1.2 [4] Let the function q be convex univalent  in the open unit disc U and ν and φ be analytic in a 0 (q(z)) domain D containing q (U ). Suppose that Re νφ(q(z)) > 0 for z ∈ U and ψ (z) = zq 0 (z) φ (q (z)) is starlike univalent in U . If p (z) ∈ H [q (0) , 1] ∩ Q, with p (U ) ⊆ D and ν (p (z)) + zp0 (z) φ (p (z)) is univalent in U and ν (q (z)) + 0 zq (z) φ (q (z)) ≺ ν (p (z)) + zp0 (z) φ (p (z)) , then q (z) ≺ p (z) and q is the best subordinant.

2

Main results We begin with the following

Theorem 2.1 Let

m,n z (IRλ,l f (z)) m,n f (z) IRλ,l

0

∈ H (U ) and let the function q (z) be analytic and univalent in U such that

q (z) 6= 0, for all z ∈ U . Suppose that  Re

zq 0 (z) q(z)

is starlike univalent in U . Let

ξ 2µ 2 q 00 (z) q 0 (z) q (z) + q (z) + 1 + z −z β β q (z) q (z)

 > 0,

(2.1)

for α, ξ, β, µ ∈ C, β 6= 0, z ∈ U and m,n+1 IRλ,l f (z)
m,n ψλ,l (α, ξ, µ, β; z) := α − ξn + µn2 + (n + 1) (ξ − 2nµ − β) m,n IRλ,l f (z)

1126

(2.2)

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+µ (n + 1)

2

m,n+1 IRλ,l f (z)

!2

m,n IRλ,l f (z)

+ β (n + 1)

m,n+2 m,n+1 (n + 2) IRλ,l f (z) − (n + 1) IRλ,l f (z) m,n+1 m,n (n + 1) IRλ,l f (z) − nIRλ,l f (z)

.

If q satisfies the following subordination 2

m,n ψλ,l (α, β, µ; z) ≺ α + ξq (z) + µ (q (z)) + β

for α, ξ, β, µ ∈ C, β 6= 0, then

 0 m,n z IRλ,l f (z) m,n IRλ,l f (z)

zq 0 (z) , q (z)

(2.3)

≺ q (z) ,

(2.4)

and q is the best dominant. 0

Proof. Let the function p be defined by p (z) := 0

(n + 1)

m,n+1 f (z)) (IRλ,l m,n IRλ,l f (z)

− (n + 1)

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

·

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

m,n f (z)) (IRλ,l m,n IRλ,l f (z)

, z ∈ U , z 6= 0, f ∈ A. We have p0 (z) =

0

.

By using the identity (1.3), we obtain m,n+1 m,n+2 m,n+1 IRλ,l f (z) (n + 2) IRλ,l f (z) − (n + 1) IRλ,l f (z) zp0 (z) − (n + 1) = (n + 1) . m,n m,n+1 m,n p (z) IRλ,l f (z) (n + 1) IRλ,l f (z) − nIRλ,l f (z)

(2.5)

β , it can be easily verified that θ is analytic in C, φ is By setting θ (w) := α + ξw + µw2 and φ (w) := w analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. 0 0 2 (z) (z) Also, by letting Q (z) = zq 0 (z) φ (q (z)) = β zqq(z) and h (z) = θ (q (z))+Q (z) = α+ξq (z)+µ (q (z)) +β zqq(z) , we find that Q (z) is starlike univalent in U .  0 2 0 00 0 (z) (z) (z) (z) ξ 2µ 2 and zh + βz qq(z) − βz qq(z) We have h0 (z) = ξq 0 (z) + 2µq (z) q 0 (z) + β qq(z) Q(z) = β q (z) + β q (z) + 1 + 00

0

(z) (z) z qq(z) − z qq(z) .

 0   (z) We deduce that Re zh = Re βξ q (z) + Q(z) By using (2.5), we obtain

2µ 2 β q

 00 0 (z) (z) (z) + 1 + z qq(z) − z qq(z) > 0.


0 IRm,n+1 f (z) 2 (z) α + ξp (z) + µ (p (z)) + β zpp(z) = α − ξn + µn2 + (n + 1) (ξ − 2nµ − β) IRλ,lm,n f (z) + λ,l  m,n+1 2 m,n+2 m,n+1 IRλ,l f (z) (n+2)IRλ,l f (z)−(n+1)IRλ,l f (z) 2 µ (n + 1) + β (n + 1) . IRm,n f (z) (n+1)IRm,n+1 f (z)−nIRm,n f (z) λ,l

λ,l

2

λ,l

0 (z) β zpp(z)

2

0

(z) ≺ α + ξq (z) + µ (q (z)) + β zqq(z) . 0 m,n z (IRλ,l f (z)) By an application of Lemma 1.1, we have p (z) ≺ q (z), z ∈ U, i.e. IRm,n f (z) ≺ q (z), z ∈ U and q is the λ,l best dominant.

By using (2.3), we have α + ξp (z) + µ (p (z)) +

m,n 1+Az Corollary 2.2 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) holds. If f ∈ A and ψλ,l (α, β, µ; z) ≺ α + ξ 1+Bz +  2 β(A−B)z m,n 1+Az + (1+Az)(1+Bz) , for α, β, µ, ξ ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψλ,l is defined in (2.2), then µ 1+Bz 0

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

≺

1+Az 1+Bz ,

Proof. For q (z) =

and

1+Az 1+Bz

1+Az 1+Bz ,

is the best dominant.

−1 ≤ B < A ≤ 1 in Theorem 2.1 we get the corollary.

m,n Corollary 2.3 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) holds. If f ∈ A and ψλ,l (α, β, µ; z) ≺ α +  γ  2γ m,n 2βγz 1+z 1+z + µ 1−z + 1−z is defined in (2.2), then ξ 1−z 2 , for α, β, µ, ξ ∈ C, 0 < γ ≤ 1, β 6= 0, where ψλ,l     0 m,n γ γ z (IRλ,l f (z)) 1+z 1+z ≺ 1−z , and 1−z is the best dominant. IRm,n f (z) λ,l

Proof. Corollary follows by using Theorem 2.1 for q (z) =



1+z 1−z

γ

, 0 < γ ≤ 1. 0

(z) Theorem 2.4 Let q be analytic and univalent in U such that q (z) 6= 0 and zqq(z) be starlike univalent in U . Assume that   2µ 2 ξ Re q (z) q 0 (z) + q (z) q 0 (z) > 0, for ξ, β, µ ∈ C, β 6= 0. (2.6) β β

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If f ∈ A,

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

m,n m,n ∈ H [q (0) , 1] ∩ Q and ψλ,l (α, β, µ; z) is univalent in U , where ψλ,l (α, β, µ; z) is as

defined in (2.2), then 2

α + ξq (z) + µ (q (z)) + implies q (z) ≺

βzq 0 (z) m,n ≺ ψλ,l (α, β, µ; z) q (z)

 0 m,n z IRλ,l f (z) m,n IRλ,l f (z)

(2.7)

z ∈ U,

,

(2.8)

and q is the best subordinant. 0

m,n z (IRλ,l f (z)) , m,n IRλ,l f (z) β := w it can be

Proof. Let the function p be defined by p (z) :=

z ∈ U , z 6= 0, f ∈ A.

By setting ν (w) := α + ξw + µw2 and φ (w) easily verified that ν is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}.    0  0 0 (q(z)) (q(z)) 2 0 Since νφ(q(z)) = Re βξ q (z) q 0 (z) + 2µ = q (z)q(z)[ξ+2µq(z)] , it follows that Re νφ(q(z)) β β q (z) q (z) > 0, for α, β, µ ∈ C, µ 6= 0. 0 2 2 βzp0 (z) (z) By using (2.5) and (2.7) we obtain α + ξq (z) + µ (q (z)) + βzq q(z) ≺ α + ξp (z) + µ (p (z)) + p(z) . 0 m,n z (IRλ,l f (z)) Using Lemma 1.2, we have q (z) ≺ p (z) = IRm,n , z ∈ U, and q is the best subordinant. f (z) λ,l

0

m,n z (IRλ,l f (z)) Corollary 2.5 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.6) holds. If f ∈ A, IRm,n ∈ H [q (0) , 1] ∩ Q and λ,l f (z) 2  β(A−B)z m,n 1+Az 1+Az + (1+Az)(1+Bz) α + ξ 1+Bz + µ 1+Bz ≺ ψλ,l (α, β, µ; z) , for α, β, ξ, µ ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where

m,n ψλ,l is defined in (2.2), then

Proof. For q (z) =

1+Az 1+Bz ,

1+Az 1+Bz

≺

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

0

, and

1+Az 1+Bz

is the best subordinant.

−1 ≤ B < A ≤ 1 in Theorem 2.4 we get the corollary. 0

m,n z (IRλ,l f (z)) Corollary 2.6 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.6) holds. If f ∈ A, IRm,n ∈ H [q (0) , 1] ∩ Q and λ,l f (z)   γ 2γ m,n m,n 2βγz 1+z 1+z + µ 1−z + 1−z α + ξ 1−z is defined 2 ≺ ψλ,l (α, β, µ; z) , for α, β, µ, ξ ∈ C, β 6= 0, 0 < γ ≤ 1, where ψλ,l  γ   0 m,n γ z (IRλ,l f (z)) 1+z 1+z in (2.2), then 1−z ≺ IRm,n is the best subordinant. , and 1−z f (z) λ,l



γ

1+z Proof. For q (z) = 1−z , 0 < γ ≤ 1 in Theorem 2.4 we get the corollary. Combining Theorem 2.1 and Theorem 2.4, we state the following sandwich theorem.

Theorem 2.7 Let q1 and q2 be analytic and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all z ∈ U , zq 0 (z) zq 0 (z) with q11(z) and q22(z) being starlike univalent. Suppose that q1 satisfies (2.1) and q2 satisfies (2.6). If f ∈ A, 0 m,n z (IRλ,l f (z)) m,n ∈ H [q (0) , 1] ∩ Q and ψλ,l (α, β, µ; z) is as defined in (2.2) univalent in U , then m,n IR f (z) λ,l

2

α + ξq1 (z) + µ (q1 (z)) +

βzq10 (z) βzq20 (z) 2 m,n ≺ ψλ,l (α, β, µ; z) ≺ α + ξq2 (z) + µ (q2 (z)) + , q1 (z) q2 (z) 0

for α, β, µ, ξ ∈ C, β 6= 0, implies q1 (z) ≺ subordinant and the best dominant. For q1 (z) =

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

≺ q2 (z), and q1 and q2 are respectively the best

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the following corollary. 0

m,n z (IRλ,l f (z)) Corollary 2.8 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) and (2.6) hold. If f ∈ A, IRm,n ∈ H [q (0) , 1]∩ λ,l f (z)  2  2 β(A1 −B1 )z β(A2 −B2 )z m,n 1+A1 z 1+A1 z 1+A2 z 1+A2 z Q and α+ξ 1+B +µ 1+B + (1+A ≺ ψλ,l (α, β, µ; z) ≺ α+ξ 1+B +µ 1+B + (1+A , 1z 1z 1 z)(1+B1 z) 2z 2z 2 z)(1+B2 z)

m,n 1+A1 z for α, β, µ, ξ ∈ C, β 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψλ,l is defined in (2.2), then 1+B ≺ 1z 0 m,n z (IRλ,l f (z)) 1+A2 z 1+A1 z 1+A2 z ≺ 1+B , hence 1+B and 1+B are the best subordinant and the best dominant, respectively. IRm,n f (z) 2z 1z 2z λ,l

For q1 (z) =



1+z 1−z

γ1

, q2 (z) =



1+z 1−z

γ2

, where 0 < γ1 < γ2 ≤ 1, we have the following corollary. 1128

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m,n z (IRλ,l f (z)) Corollary 2.9 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.1) and (2.6) hold. If f ∈ A, IRm,n ∈ H [q (0) , 1]∩ f (z) λ,l  γ1 2γ1  γ2 2γ2   m,n 1+z 1+z 1+z 1+z 1z 2z Q and α + ξ 1−z + µ 1−z + 2βγ + µ 1−z + 2βγ 1−z 2 ≺ ψλ,l (α, β, µ; z) ≺ α + ξ 1−z 1−z 2 , for   γ1 γ2 0 m,n z (IRλ,l f (z)) m,n 1+z 1+z α, β, µ, ξ ∈ C, β 6= 0, 0 < γ1 < γ2 ≤ 1, where ψλ,l is defined in (2.2), then 1−z ≺ ≺ IRm,n , 1−z λ,l f (z)  γ1  γ2 1+z 1+z hence 1−z and 1−z are the best subordinant and the best dominant, respectively.

We have also Theorem 2.10 Let

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

0

∈ H (U ) , f ∈ A, z ∈ U , m, n ∈ N, λ, l ≥ 0 and let the function q (z) be convex

and univalent in U such that q (0) = 1, z ∈ U . Assume that   α+β q 00 (z) Re +z 0 > 0, β q (z)

(2.9)

for α, β ∈ C, β 6= 0, z ∈ U, and m,n ψλ,l (α, β; z) := β (n + 1) (n + 2)

β (n + 1)

m,n+1 IRλ,l f (z)

2

m,n+2 IRλ,l f (z) m,n IRλ,l f (z)

!2 + (α − β) (n + 1)

m,n IRλ,l f (z)

−

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

(2.10)

− αn.

If q satisfies the following subordination m,n ψλ,l (α, β; z) ≺ αq (z) + βzq 0 (z) ,

for α, β ∈ C, β 6= 0, z ∈ U, then

 0 m,n z IRλ,l f (z) m,n IRλ,l f (z)

≺ q (z) ,

(2.11)

z ∈ U,

(2.12)

and q is the best dominant. 0

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

Proof. Let the function p be defined by p (z) :=

, z ∈ U , z 6= 0, f ∈ A. The function p is

analytic in U and p (0) = 1 We have p0 (z) = (n + 1)

m,n+1 f (z)) (IRλ,l

0

m,n IRλ,l f (z)

− (n + 1)

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

·

m,n f (z)) (IRλ,l m,n IRλ,l f (z)

0

.

By using the identity (1.3), we obtain 0

zp (z) = (n + 1) (n + 2)

m,n+2 IRλ,l f (z) m,n IRλ,l f (z)

− (n + 1)

m,n+1 IRλ,l f (z)

2

m,n IRλ,l f (z)

!2 − (n + 1)

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

.

(2.13)

By setting θ (w) := αw and φ (w) := β, it can be easily verified that θ is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. Also, by letting Q (z) = zq 0 (z) φ (q (z)) = βzq 0 (z) , we find that Q (z) is starlike univalent in U. 0 Let h (z) = θ(q (z))+ Q (z)= αq (z) + βzq  (z). 00 zh0 (z) q (z) We have Re Q(z) = Re α+β > 0. β + z q 0 (z) By using (2.13), we obtain αp (z) + βzp0 (z) = β (n + 1) (n + 2)  m,n+1 2 IRλ,l f (z) IRm,n+1 f (z) 2 β (n + 1) + (α − β) (n + 1) IRλ,lm,n f (z) − αn. IRm,n f (z) λ,l

m,n+2 IRλ,l f (z) − m,n f (z) IRλ,l

λ,l

By using (2.11), we have αp (z) + βzp0 (z) ≺ αq (z) + βzq 0 (z) . 0 m,n z (IRλ,l f (z)) From Lemma 1.1, we have p (z) ≺ q (z), z ∈ U, i.e. IRm,n ≺ q (z), z ∈ U, and q is the best dominant. f (z) λ,l

1+Az 1+Bz , z ∈ U, −1 ≤ B < 1+Az α 1+Bz + β(A−B)z , for α, β (1+Bz)2

Corollary 2.11 Let q (z) = f ∈ A and

m,n ψλ,l

(α, β; z) ≺ 0 m,n z (IRλ,l f (z)) (2.10), then IRm,n ≺ f (z) λ,l

1+Az 1+Bz ,

and

1+Az 1+Bz

A ≤ 1, m, n ∈ N, λ, l ≥ 0. Assume that (2.9) holds. If m,n ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψλ,l is defined in

is the best dominant. 1129

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Proof. For q (z) =

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, in Theorem 2.10 we get the corollary.

 γ m,n 1+z Corollary 2.12 Let q (z) = 1−z , m, n ∈ N, λ, l ≥ 0. Assume that (2.9) holds. If f ∈ A and ψλ,l (α, β; z) ≺  γ  γ 0 m,n z (IRλ,l f (z)) m,n 2βγz 1+z 1+z α 1−z + 1−z , for α, β ∈ C, 0 < γ ≤ 1, β 6= 0, where ψλ,l is defined in (2.10), then IRm,n ≺ 2 1−z f (z) λ,l  γ  γ 1+z 1+z , and 1−z is the best dominant. 1−z Proof. Corollary follows by using Theorem 2.10 for q (z) =



1+z 1−z

γ

, 0 < γ ≤ 1.

Theorem 2.13 Let q be convex and univalent in U such that q (0) = 1. Assume that   α 0 Re q (z) > 0, for α, β ∈ C, β 6= 0. β If f ∈ A,

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

(2.14)

0

m,n m,n ∈ H [q (0) , 1] ∩ Q and ψλ,l (α, β; z) is univalent in U , where ψλ,l (α, β; z) is as defined

in (2.10), then m,n αq (z) + βzq 0 (z) ≺ ψλ,l (α, β; z)

implies q (z) ≺

 0 m,n z IRλ,l f (z) m,n f (z) IRλ,l

,

(2.15)

δ ∈ C, δ 6= 0, z ∈ U,

(2.16)

and q is the best subordinant. 0

Proof. Let the function p be defined by p (z) :=

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

, z ∈ U , z 6= 0, f ∈ A. The function p is

analytic in U and p (0) = 1. By setting ν (w) := αw and φ (w) := β it can be easily verified that ν is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}.  0    0 ν (q(z)) (q(z)) 0 0 =α = Re α Since νφ(q(z)) β q (z), it follows that Re φ(q(z)) β q (z) > 0, for α, β ∈ C, β 6= 0. Now, by using (2.15) we obtain αq (z) + βzq 0 (z) ≺ αp (z) + βzp0 (z) , z ∈ U. From Lemma 1.2, we have 0 m,n z (IRλ,l f (z)) q (z) ≺ p (z) = IRm,n , z ∈ U, and q is the best subordinant. f (z) λ,l

1+Az Corollary 2.14 Let q (z) = 1+Bz , −1 ≤ B < A ≤ 1, z ∈ U, m, n ∈ N, λ, l ≥ 0. Assume that (2.14) holds. 0 m,n z (IRλ,l f (z)) m,n 1+Az If f ∈ A, IRm,n f (z) ∈ H [q (0) , 1] ∩ Q, and α 1+Bz + β(A−B)z ≺ ψλ,l (α, β; z) , for α, β ∈ C, β 6= 0, (1+Bz)2 λ,l

0

m,n −1 ≤ B < A ≤ 1, where ψλ,l is defined in (2.10), then

Proof. For q (z) =

1+Az 1+Bz ,

1+Az 1+Bz

≺

m,n z (IRλ,l f (z)) m,n IRλ,l f (z)

, and

1+Az 1+Bz

is the best subordinant.

−1 ≤ B < A ≤ 1, in Theorem 2.13 we get the corollary.

γ  0 m,n z (IRλ,l f (z)) 1+z , m, n ∈ N, λ, l ≥ 0. Assume that (2.14) holds. If f ∈ A, IRm,n Corollary 2.15 Let q (z) = 1−z ∈ λ,l f (z) γ  γ  m,n m,n 2βγz 1+z 1+z + 1−z ≺ ψλ,l (α, β; z) , for α, β ∈ C, 0 < γ ≤ 1, β 6= 0, where ψλ,l is H [q (0) , 1] ∩ Q and α 1−z 2 1−z  γ   0 m,n γ z (IRλ,l f (z)) 1+z 1+z defined in (2.10), then 1−z ≺ IRm,n , and 1−z is the best subordinant. f (z) λ,l

 γ 1+z Proof. Corollary follows by using Theorem 2.13 for q (z) = 1−z , 0 < γ ≤ 1. Combining Theorem 2.10 and Theorem 2.13, we state the following sandwich theorem. Theorem 2.16 Let q1 and q2 be convex and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all z ∈ U . 0 m,n f (z)) z (IRλ,l m,n Suppose that q1 satisfies (2.9) and q2 satisfies (2.14). If f ∈ A, IRm,n ∈ H [q (0) , 1]∩Q , and ψλ,l (α, β; z) f (z) λ,l

m,n is as defined in (2.10) univalent in U , then αq1 (z) + βzq10 (z) ≺ ψλ,l (α, β; z) ≺ αq2 (z) + βzq20 (z) , for α, β ∈ C, 0 m,n z (IRλ,l f (z)) β 6= 0, implies q1 (z) ≺ IRm,n ≺ q2 (z), z ∈ U, and q1 and q2 are respectively the best subordinant and the λ,l f (z) best dominant.

For q1 (z) =

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the following corollary. 1130

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1+A1 z Corollary 2.17 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.9) and (2.14) hold for q1 (z) = 1+B and q2 (z) = 1z 0 m,n z IR f (z) ( ) m,n 1+A2 z 1+A1 z λ,l 1 −B1 )z ≺ ψλ,l (α, β; z) ≺ ∈ H [q (0) , 1] ∩ Q and α 1+B + β(A 1+B2 z , respectively. If f ∈ A, IRm,n f (z) (1+B z)2 1z 1

λ,l

β(A2 −B2 )z , z ∈ U, for (1+B2 z)2 0 m,n z (IRλ,l f (z)) 1+A1 z ≺ then 1+B m,n z IR f (z) 1 λ,l

1+A2 z α 1+B + 2z

m,n α, β ∈ C, β 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψλ,l is defined in

1+A1 z 1+A2 z 1+A2 z , z ∈ U, hence 1+B and 1+B are the best subordinant and the best (2.2), ≺ 1+B 2z 1z 2z dominant, respectively. γ1 γ2   1+z 1+z For q1 (z) = 1−z , q2 (z) = 1−z , where 0 < γ1 < γ2 ≤ 1, we have the following corollary.

 γ1 1+z Corollary 2.18 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.9) and (2.14) hold for q1 (z) = 1−z and q2 (z) = γ2 γ1  γ1   0 m,n z IR f (z) ( λ,l ) m,n 1+z 1+z 1+z 1z , respectively. If f ∈ A, IRm,n + 2βγ ≺ ψλ,l (α, β; z) ∈ H [q (0) , 1] ∩ Q and α 1−z 1−z 1−z 2 1−z λ,l f (z)  γ2 γ2  m,n 1+z 1+z 2z ≺ α 1−z + 2βγ , z ∈ U, for α, β ∈ C, β 6= 0, 0 < γ1 < γ2 ≤ 1, where ψλ,l is defined in (2.2), 1−z 2 1−z  γ1       0 m,n γ γ γ 2 1 2 z (IRλ,l f (z)) 1+z 1+z 1+z 1+z then 1−z ≺ 1−z ≺ IRm,n , z ∈ U, hence 1−z and 1−z are the best subordinant and the λ,l f (z) best dominant, respectively.

References [1] A. Alb Lupas, Some Differential Sandwich Theorems using a multiplier transformation and Ruscheweyh derivative, Electronic Journal of Mathematics and Its Application, Vol. 1, No. 2 (2015), 76-86. [2] A. Alb Lupas, About some differential sandwich theorems using a multiplier transformation and Ruscheweyh derivative, Journal of Computational Analysis and Applications, Vol. 21, No.7, 2016, 1218-1224. [3] F.M. Al-Oboudi, On univalent functions defined by a generalized S˘ al˘ agean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [4] T. Bulboac˘a, Classes of first order differential superordinations, Demonstratio Math., Vol. 35, No. 2, 287292. [5] A. C˘ata¸s, On certain class of p-valent functions defined by new multiplier transformations, Proceedings Book of the International Symposium on Geometric Function Theory and Applications, August 20-24, 2007, TC Istanbul Kultur University, Turkey, 241-250. [6] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 10, 815-826, October, 2003. [7] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker Inc., New York, 2000. [8] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [9] G. St. S˘al˘agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [10] C. Selvaraj, K.T. Karthikeyan, Differential Subordination and Superordination for Analytic Functions Defined Using a Family of Generalized Differential Operators, An. St. Univ. Ovidius Constanta, Vol. 17 (1) 2009, 201-210. [11] T.N. Shanmugan, C. Ramachandran, M. Darus, S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions involving a linear operator, Acta Math. Univ. Comenianae, 16 (2007), no. 2, 287-294. [12] H.M. Srivastava, A.Y. Lashin, Some applications of the Briot-Bouquet differential subordination, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), no. 2, Article 41, 7 pp. (electronic).

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Global stability of a quadratic anti-competitive system of rational difference equations in the plane with Allee effects. V. Hadˇziabdi´c† M. R. S. Kulenovi´c‡1

2

and E. Pilav§3

† Division of Mathematics Faculty of Mechanical Engineering, University of Sarajevo, Bosnia and Herzegovina ‡

Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA §

Department of Mathematics University of Sarajevo, Sarajevo, Bosnia and Herzegovina

Abstract. We investigate global dynamics of the following systems of difference equations  2 yn   xn+1 = a+x 2 n , n = 0, 1, 2, . . . 2  xn  yn+1 = b+y 2 n

where the parameters a, b are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers. We find all possible dynamical scenario for this system. We show that this system has substantially different behavior than the corresponding linear fractional system. Keywords. Competitive map, global stable manifold, monotonicity, period-two solution.

AMS 2010 Mathematics Subject Classification: Primary: 39A10, 39A30 Secondary: 37E99, 37D10

1

Introduction and Preliminaries

We investigate global dynamics of the following systems of difference equations  2 yn   xn+1 = a+x 2 n , n = 0, 1, . . . 2  x  yn+1 = n b+y 2

(1)

n

where the parameters a, b are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers. System (1) is related to an anti-competitive system considered in [21] γ1 yn β2 xn , yn+1 = , n = 0, 1, ..., (2) A1 + xn A2 + yn where the parameters A1 , γ1 , A2 and β2 are positive numbers and the initial conditions (x0 , y0 ) are arbitrary nonnegative numbers. In the classification of all linear fractional systems in [3], System (2) was mentioned as system (16, 16). The main result on the global behavior of System (2) is summarized in the following theorem, see [21]. xn+1 =

Theorem 1

(a) If β2 γ1 − A1 A2 < 0, then E0 (0, 0) is a unique equilibrium and it is globally asymptotically stable.

(b) If β2 γ1 − A1 A2 > 0, then there exist two equilibrium points, namely a repeller E0 and an interior saddle E+ . There exists a set C ⊂ R = [0, ∞) × [0, ∞) which is invariant subset of the basin of attraction of E+ . The set C is a graph of a strictly increasing continuous function of the first variable on an interval, and E0 ∈ C and separates R into two connected and invariant components, namely W−

:

= {x ∈ R\C : ∃y ∈ C with x se y} ,

W+

:

= {x ∈ R\C : ∃y ∈ C with y se x} .

1

Corresponding author, e-mail: [email protected] Partially supported by Maitland P. Simmons Foundation 3 Partially supported by FMON grant 05-39-3087-18/16 2

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which satisfy: i) If (x0 , y0 ) ∈ W+ , then lim (x2n , y2n ) = (∞, 0) and

n→∞

lim (x2n+1 , y2n+1 ) = (0, ∞) .

n→∞

ii) If (x0 , y0 ) ∈ W− , lim (x2n , y2n ) = (0, ∞) and

n→∞

lim (x2n+1 , y2n+1 ) = (∞, 0) .

n→∞

(c) If β2 γ1 − A1 A2 = 0, then, i. E0 (0, 0) is the unique equilibrium, and every point of the positive semiaxes is a period-two point. Orbits of β2 x), for some x > 0. period-two solutions consist of the points (x, 0) and (0, A 2 ii. All minimal period-two solutions and the equilibrium are stable but not asymptotically stable. iii. There exists a family of strictly increasing curves C0 , Cx and C x for x > 0, that emanate from E0 , Ex := (x, 0) β2 x) respectively, such that the curves are pairwise disjoint, the union of all the curves equals and E x := (0, A 2 2 R+ , and solutions with initial point in C0 converge to E0 , solutions with initial point in Cx have even-indexed terms converging to Ex and odd-indexed terms converging to E x , and, solutions with initial point in C x have even-indexed terms converging to E x and odd-indexed terms converging to Ex . As we will show in this paper System (1) has very different behavior than System (2), showing that introduction of quadratic terms can significantly change behavior of the system. As we will show in Section 4 there are three dynamic scenarios for System (1), each different than one of the three scenarios for System (2). For example System (1) always possesses the unique period-two solution which substantially effects the global behavior. Second System (1) exhibits the Allee’s effect which is nonexistent in System (2). Third major difference between two systems lies in the techniques of the proof used in two results. While the results about the global stable and unstable manifolds in [18, 19, 20] were sufficient for the proofs of global dynamics of System (2), these results are not effective in the case of System (1) as the eigenvectors which correspond to the period-two solution of System (1) are parallel to the coordinate axes. Thus we used new techniques based on the properties of the basins of attraction of the period-two solution or the points at infinity (0, ∞) and (∞, 0). Furthermore, we used the real algebraic geometry to prove some basic facts about the local stabilty of the equilibrium points and the period-two solutions. Our results show that the introduction of quadratic terms in the linear fractional systems of difference equations change substantially their behavior, see [2, 10] for similar results. In particular, introduction of quadratic terms creates the Allee’s effect and introduces the periodic solutions. The rest of this section contains some known results about competitive systems. Section 2 gives some basic facts about the global behavior of System (1). Section 3 presents local stability analysis of the equilibrium solutions and the period-two solution. Finally, Section 4 gives complete global dynamics of System (1). A first order system of difference equations  xn+1 = f (xn , yn ) , n = 0, 1, ..., (x0 , y0 ) ∈ R, (3) yn+1 = g (xn , yn ) where R ⊂ R2 , (f, g) : R → R, f, g are continuous functions is competitive if f (x, y) is non-decreasing in x and non-increasing in y, and g (x, y) is non-increasing in x and non-decreasing in y. System (3) where the functions f and g have monotonic character opposite of the monotonic character in competitive system will be called anti-competitive. In other words (3) is anti-competitive if f (x, y) is non-increasing in x and nondecreasing in y, and g (x, y) is non-decreasing in x and non-increasing in y. Consider a partial ordering  on R2 . Two points v, w ∈ R2 are said to be related if v  w or w  v. Also, a strict inequality between points may be defined as v ≺ w if v  w and v 6= w. A stronger inequality may be defined as v = (v1 , v2 )  w = (w1 , w2 ) if v  w with v1 6= w1 and v2 6= w2 . For u, v in R2 , the order interval Ju, vK is the set of all x ∈ R2 such that u  x  v. The interior of a set A is deoned as intA. A map T on a nonempty set S ⊂ R2 is a continuous function T : S → S. The map T is monotone if v  w implies T (v)  T (w) for all v, w ∈ S, and it is strongly monotone on S if v ≺ w implies that T (v)  T (w) for all v, w ∈ S. The map is strictly monotone on S if v ≺ w implies that T (v) ≺ T (w) for all v, w ∈ S. Clearly, being related is invariant under iteration of a strongly monotone map. Throughout this paper we shall use the North-East ordering (NE) for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x1 , y1 ) ne (x2 , y2 ) if x1 ≤ x2 and y1 ≤ y2 and the South-East (SE) ordering defined as (x1 , y1 ) se (x2 , y2 ) if x1 ≤ x2 and y1 ≥ y2 . A map T on a nonempty set S ⊂ R2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive. A map T on a nonempty set S ⊂ R2 which second iterate T 2 is monotone with respect to the North-East ordering is called anti-cooperative and a map which second iterate T 2 is monotone with respect to the South-East ordering is called anti-competitive. A map T that corresponds to System (3) is defined as T = (f, g). An equilibrium x of anti-competitive system (3) is said to

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be nonhyperbolic of stable (resp. unstable) type if one of the eigenvalues of the Jacobian matrix evaluated at x is by absolute value 1 and the second one is by absolute value less (resp. bigger) than 1. Next, we give three results for competitive maps in the plane. There is an extensive literature on competitive systems in the plane, see [1, 2, 4, 5, 6, 8, 9, 18, 19, 20, 23] for different examples of planar competitive systems and their applications. The following definition is from [24]. Definition 1 Let S be a nonempty subset of R2 . A competitive map T : S → S is said to satisfy condition (O+) if for every x, y in S, T (x) ne T (y) implies x ne y, and T is said to satisfy condition (O−) if for every x, y in S, T (x) ne T (y) implies y ne x. The following theorem was proved by de Mottoni-Schiaffino [7] for the Poincar´e map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [23]. Theorem 2 Let S be a nonempty subset of R2 . If T is a competitive map for which (O+) holds then for all x ∈ S, {T n (x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T . If instead (O−) holds, then for all x ∈ S, {T 2n } is eventually componentwise monotone. If the orbit of x has compact closure in S, then its omega limit set is either a period-two orbit or a fixed point. The following result is from [24], with the domain of the map specialized to be the cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O−). Theorem 3 Let R ⊂ R2 be the cartesian product of two intervals in R. Let T : R → R be a C 1 competitive map. If T is injective and detJT (x) > 0 for all x ∈ R then T satisfies (O+). If T is injective and detJT (x) < 0 for all x ∈ R then T satisfies (O−). The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess, see [11, 19], and is helpful for determining the basins of attraction of the equilibrium points. Corollary 1 If the nonnegative cone of  is a generalized quadrant in Rn , and if T has no fixed points in Ju1 , u2 K other than u1 and u2 , then the interior of Ju1 , u2 K is either a subset of the basin of attraction of u1 or a subset of the basin of attraction of u2 .

2

Some Basic Facts

Let

y2 x2 , T2 (x, y) = . 2 a+x b + y2 The map T (x, y) associated to system (1) is given by   x2 y2 , , (x, y) ∈ [0, ∞) × [0, ∞) T (x, y) = (T1 (x, y), T2 (x, y)) = a + x2 b + y 2 T1 (x, y) =

and the Jacobian matrix of the map T at the point (x, y) is given by:   2 2y − (x2xy 2 +a)2 x2 +a . JT (x, y) =  2 y 2x − (y2x 2 +b)2 y 2 +b

(4)

(5)

Determinant of the Jacobian matrix (5) is given by det JT (x, y) = −

4xy(bx2 + a(y 2 + b)) , (x2 + a)2 (y 2 + b)2

(6)

and the trace of the Jacobian matrix (5) is given by  trJT (x, y) = −

2y 2 x 2yx2 + 2 2 2 (y + b) (x + a)2

 .

(7)

The map T 2 is given by T 2 (x, y) = (F (x, y), G(x, y)), where x4

F (x, y) = (b + y 2 )2



,

y4 2 a+x2

(

)

G(x, y) = (a + x2 )2

+a

y4 

x4 (b+y2 )2

. +b

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The Jacobian matrix of T 2 is given as  JT 2 (x, y) =

A C

B D

 ,

(9)

where  (2x2 +a)y4 + a (x2 +a)3 A=  2 , 2 y4 2 (y + b) +a (x2 +a)2 4x3

4xy C=−

4



2

y +b


2 

2





2 
2  4 y 2y + by 2 + a x2 + a ,
2 (y 2 + b)3 y 4 + a (x2 + a)2

4x4 x2 + a B=−

2

2

2x + a x + b y + b
2 (x2 + a) x4 + b (y 2 + b)2

,

3



 (2y2 +b)x4 + b (y2 +b)3 D= 2 .  2 x4 2 +b (x + a) (y2 +b)2 4y 3


2 

(10)

(11)

Determinant of the Jacobian matrix (9) is det JT 2 (x, y) =

A (a +

x2 ) (b

+

y2 )

a (a +

x2 )2

+ y4


2

b (b + y 2 )2 + x4


2 ,

(12)

where 
2 3
A = 16x3 y 3 a a + x2 b + x4   
2
2

+by 4 a a + x2 + b2 + 2ab2 y 2 a + x2 + 2b2 y 6 + by 8 a b + y 2 + bx2 . The following lemma summarizes some basic facts about System (1). Lemma 1 Let (xn , yn ) := T n (x0 , y0 ) be any solution of System (1). Then (i) Assume that x0 = 0 and y0 > 0. Then the following holds: (i.1) x2n = 0 and y2n−1 = 0 for all n ∈ N. √ √ 3 3 (i.2) If 0 < y0 < a2 b, then 0 < x1 < ab2 and y2n+2 ≤ y2n for all n ∈ N. √ √ 3 3 (i.3) If y0 > a2 b, then x1 > ab2 and y2n ≤ y2n+2 for all n ∈ N. (ii) Assume that x0 > 0 and y0 = 0. Then the following holds (ii.1) x2n−1 = 0 and y2n = 0 for all n ∈ N. √ √ 3 3 (ii.2) If 0 < x0 < ab2 , then y1 < a2 b and x2n+2 ≤ x2n for all n ∈ N. √ √ 3 3 (ii.3) If x0 > ab2 , then y1 > a2 b and x2n ≤ x2n+2 for all n ∈ N. (iii) For all n > 0 we have xn yn < 1. Proof. We prove statement (i.1). Take x0 = 0 and y0 > 0. The statement (i) follows from     a2 by0 − y04 y4 (0, y0 ) − T 2 (0, y0 ) = (0, y0 ) − 0, 20 = 0, , a b a2 b √ 3 √ √ y2 y 2 − a4 b 2 3 3 x1 − ab2 = 0 − ab2 = 0 a a and monotonicity of T 2 . Similarly, we prove (ii.1). Proofs of (i.2), (ii.2), (i.3) and (ii.3) are immediate. Take x0 , y0 ∈ [0, ∞) × [0, ∞). Let (xn , yn ) := T n (x0 , y0 ). The proof of the statement (iii) follows from the fact xn+1 yn+1 =

yn2 x2n x2n yn2 = < 1. a + x2n b + yn2 a + x2n b + yn2 2

Lemma 2 The map T is injective.

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Proof. We have to prove that T (x1 , y1 ) = T (x2 , y2 ) ⇒ x1 = x2

and

y1 = y2 .

Let x1 , y1 , x2 , y2 > 0. Then  T (x1 , y1 ) − T (x2 , y2 ) =

x22 y12 + ay12 − x21 y22 − ay22 y22 x21 + bx21 − bx22 − x22 y12 , (x21 + a) (x22 + a) (y12 + b) (y22 + b)

 .

From T (x1 , y1 ) − T (x2 , y2 ) = 0 we have x22 y12 + ay12 − x21 y22 − ay22 = 0

(13)

y22 x21 + bx21 − bx22 − x22 y12 = 0.

(14)

and Equation (13) implies that x22 y12 + ay12 − ay22 . y22

x21 =

(15)

Substituting x21 from (15) into equation (14) we have
(y1 − y2 ) (y1 + y2 ) bx22 + ay22 + ab = 0, y22 2

from which it follows that y1 = y2 . Substituting in (15) we have x1 = x2 . This proves Lemma.

Theorem 4 Every bounded solution of System (1) converges to a period-two solution. Proof. In view of Lemma 2 the map T associated to System (1) is injective which implies that the map T 2 is also injective. Relation (12) implies that determinant of the Jacobian matrix (9) is positive for all x ∈ (0, ∞) × (0, ∞). By using Theorem 3 we have that the condition (O+) is satisfied for the map T 2 (T 2 is competitive). Theorem 2 implies that ∞ ∞ ∞ ∞ odd and even subsequences {x2n }∞ n=0 , {x2n+1 }n=−1 , {y2n }n=0 , {y2n+1 }n=−1 of any solution {(xn , yn )}n=0 are eventually monotonic, from which the proof follows. 2

3

The local stability of the equilibrium solutions and the period-two solution

The equilibrium points (¯ x, y¯) of System (1) satisfy equations y¯2 =x ¯, a+x ¯2

x ¯2 = y¯. b + y¯2

(16)

By eliminating x ¯ from (16) we get
y¯9 + 3b¯ y 7 + 2a¯ y 6 + 3b2 y¯5 + (4ab − 1)¯ y 4 + b3 + a2 y¯3 + 2ab2 y¯2 + a2 b¯ y = 0.

(17)

Similarly, we can eliminate variable y¯ from system (16) to obtain
3 x ¯9 + 3a¯ x7 + 2b¯ x6 + 3a2 x ¯5 + (4ab − 1)¯ x4 + a3 + b2 x ¯ + 2a2 b¯ x2 + ab2 x ¯ = 0.

(18)

In view of Descartes’ Rule of Signs we obtain that Eq. (18) has zero equilibrium always and either zero, one or two positive equilibrium points if 4ab − 1 < 0. By using (16) all its real roots are positive numbers. These equilibrium points will be denoted E0 (0, 0), E(¯ x, y¯), ESW (¯ x, y¯) and EN E (¯ x, y¯). Lemma 3 Let ∆1 = 186624b2 a15 + 55296b5 a13 + 2657664b4 a12 + 4096b8 a11 + 1619712b3 a11 + 754688b7 a10 − 12500b2 a10 + 10767632b6 a9 + 55296b10 a8 − 11550400b5 a8 + 2657664b9 a7 + 1980000b4 a7 + 1619712b8 a6 − 84375b3 a6 + 186624b12 a5 − 12500b7 a5

(19)

and ∆2 = 4ab − 1. Then the following holds:

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a) If ∆2 ≥ 0, then equation (18) has one real root and four pairs of conjugate imaginary roots. Consequently, System (1) has one equilibrium point E0 (0, 0); b) If ∆2 < 0, and ∆1 < 0, then equation (18) has three distinct real roots and three pairs of conjugate imaginary roots. Consequently, System (1) has three equilibrium points E0 (0, 0); ESW (¯ x, y¯) and EN E (¯ x, y¯); c) If ∆2 < 0, and ∆1 > 0, then equation (18) has four pairs of conjugate imaginary roots and one real root. Consequently, System (1) has one equilibrium point E0 (0, 0); d) If ∆2 < 0 and ∆1 = 0 then equation (18) has three pairs of conjugate imaginary roots and one real root of multiplicity two and one root of multiplicity one. Consequently, System (1) has two equilibrium points E0 (0, 0) and E(¯ x, y¯). Proof. Let ∆ be discriminant of
f˜(x) = x9 + 3ax7 + 2bx6 + 3a2 x5 + (4ab − 1)x4 + a3 + b2 x3 + 2a2 bx2 + ab2 x. Then ∆ = a2 b∆1 . The rest of the proof follows from the fact that equation (18) has at most three real roots and Theorem 5.1 from [13]. 2

Period-two solution {(Φ, Ψ), T (Φ, Ψ)} satisfies the system F (Φ, Ψ) = Φ, G(Φ, Ψ) = Ψ, which is equivalent to   3 2
2 Φ Φ − b+Ψ

    Ψ4 Φ4 3 2
2 +a = 0, Ψ Ψ − a + Φ +b = 0. (20) (a + Φ2 )2 (b + Ψ2 )2 √ √ 3 2 3 2 For Φ = 0 we have  Ψ = a√b, and for Ψ = 0 we have Φ = 0 or Φ = ab . Hence, we have two minimal  Ψ√= 0 or 3 2 3 period-two points P1 0, a b) and P2 ab2 , 0 . Lemma 4 The period-two solution {P1 , P2 } is a saddle point with corresponding eigenvectors which are coordinate axes.  Proof. The proof follows from the fact that JT 2 (P1 ) =

0 0

0 4



2

.

Lemma 5 Let CF := {(x, y) : F (x, y) = x} and CG := {(x, y) : G(x, y) = y} be the period-two curves, that is the √ 3 curves which intersection is a period-two solution. Then for all y > a2 b there exists exactly one xG (y) > 0 such that √ 3 G(xG (y), y) = y and for all x > ab2 there exists exactly one yF (x) > 0 such that F (x, yF (x)) = x. Furthermore, xG (y) and yF (x) are continuous functions and x0G (y) > 0, yF0 (x) > 0. Proof. Since F (x, y) = x and G(x, y) = y if and only if

2 3

2 −xy 4 a3 + 2a2 x2 + ax4 + b2 + x a + x2 x − ab2 − 2abxy 2 a + x2 − 2bxy 6 − xy 8 = 0,  
2
2 2

2 − b + y2 a b − y 3 − 2abx2 b + y 2 − 2ax6 − x8 = 0, −x4 a2 + b b + y 2 √ 3 respectively, in view of Descartes’ Rule of √Signs we have that for all y > a2 b there exists exactly one xG (y) > 0 3 such that G(xG (y), y) = y and for all x > ab2 there exists exactly one yF (x) > 0 such that F (x, yF (x)) = x. Taking derivatives of F (x, y) = x with respect to x we get yF0 (x) = From F (x, y) = x we have that b + y 2


2

x3

= (

Fx0 (x, y)

y4 2 a+x2

)

=

1 − Fx0 (x, y) . Fy0 (x, y)

, which implies +a



3 4y 4 a + 2x2 + 4a a + x2 y 4 (a + x2 ) + a (a + x2 )3

> 1.

Since Fy0 (x, y) < 0 we get x0F (y) > 0. Taking derivatives of G(x, y) = y with respect to y we get x0G (y) =

1 − G0y (x, y) . G0x (x, y)

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From G(x, y) = y we have that b + y 2


2

x3

=

y4 2 a+x2

(

)

G0y (x, y) =

, which implies +a

(b +

y2 )

4x4 y 2
+ 4 > 4. b (b + y 2 )2 + x4 2

Since G0x (x, y) < 0 we get x0G (y) > 0.

Theorem 5 If T has a period-two solution {(Φ, Ψ), T (Φ, Ψ)}, then it is unstable. If µ1 and µ2 , (0 < µ1 < µ2 ) are the eigenvalues of JT 2 (Φ, Ψ) then µ1 > 0 and µ2 > 1. All period-two-solutions are ordered with respect to the North-East ordering. Proof. Since Fx0 (Φ, Ψ)

=

and G0y (Φ, Ψ) =



3 4Ψ4 a + 2Φ2 + 4a a + Φ2 Ψ4 (a + Φ2 ) + a (a + Φ2 )3

>1

4Φ4 Ψ2
+ 4 > 4, (b + Ψ2 ) b (b + Ψ2 )2 + Φ4

we obtain trJT 2 (Φ, Ψ) = µ1 + µ2 > 5. The rest of the proof follows from the fact that detJT 2 (Φ, Ψ) = µ1 µ2 > 0 and Lemma 5.

2

Theorem 6 If map T has a minimal period-two point {(Φ1 , Ψ1 ), T (Φ1 , Ψ1 )}, which is non-hyperbolic, then Dis(p) = 0, where Dis(p) is the discriminant of polynomial p(x) := p16 x16 + p15 x15 + · · · + p1 x + p0 , where the coefficients pi , i = 0, ..., 16 are given in appendix A. If {(Φ1 , Ψ1 ), T (Φ1 , Ψ1 )} and {(Φ2 , Ψ2 ), T (Φ2 , Ψ2 )} are two period-two points such that T has no other period-two points in [[(Φ1 , Ψ1 ), (Φ2 , Ψ2 )]] = {(x, y) : (Φ1 , Ψ1 ) ne (x, y) ne (Φ2 , Ψ2 )}, Dis(f˜) 6= 0 and Dis(p) 6= 0, then one of them is a saddle point and the other one is repeller. ˜ y) = 0}, where Proof. Period-two solution curves CF = {(x, y) ∈ R : F˜ (x, y) = 0} and CG = {(x, y) ∈ R : G(x, F˜ (x, y) = − a3 b2 x − 2a3 bxy 2 − a3 xy 4 − 2a2 b2 x3 − 4a2 bx3 y 2 + a2 x4 − 2a2 x3 y 4 − ab2 x5 − 2abx5 y 2 + 2ax6 − ax5 y 4 − b2 xy 4 − 2bxy 6 + x8 − xy 8 , ˜ y) = − a2 b3 y − 2a2 b2 y 3 − a2 by 5 − a2 x4 y − 2ab3 x2 y − 4ab2 x2 y 3 G(x, − 2abx2 y 5 − 2ax6 y − b3 x4 y − 2b2 x4 y 3 + b2 y 4 − bx4 y 5 + 2by 6 − x8 − y + y 8 , are algebraic curves. By using software Mathematica one can see that the resultant of the polynomials F˜ (x, y) and ˜ y) in variable y is given by G(x, ˜ =x20 a + x2 R(F˜ , G)


16

ab2 − x3





a3 x2 + 2a2 bx + 3a2 x4 + ab2 + 4abx3 + 3ax6 + b2 x2 + 2bx5 + x8 − x3 p(y) =x19 (a + x2 )16 (ab2 − x3 )f˜(x)p(x). 2

The rest of the proof is the same as the proof of Theorem 15 in [10] so we skip it. It is easy to see that the following holds: Lemma 6 The equilibrium point E0 is locally asymptotically stable.

Let C1 := {(x, y) : T1 (x, y) = x} and C2 := {(x, y) : T2 (x, y) = y} be the equilibrium curves, that is the curves which intersection is an equilibrium solution. Then for all x ≥ 0 there exist exactly one y1 (x) > 0 such that T1 (x, y1 (x)) = x and exactly one y2 (x) > 0 such that T2 (x, y2 (x)) = y. Furthermore, it can be seen that y1 (x) and y2 (x) are continuous increasing functions.

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Theorem 7 Assume that ∆1 < 0 and ∆2 < 0. Then there exist two positive equilibrium solutions ESW and EN E and the following holds true: (i) The equilibrium solution ESW is repeller. (ii) The equilibrium solution EN E is a saddle point. Proof. The existence of the equilibrium solution follows from Lemma 3. (i) Since the map T is anti-competitive then by results in [15], the eigenvalues λ1 and λ2 of the Jacobian matrix associated to the map T at ESW (¯ x, y¯) ∈ int(R2+ ) are real and distinct, and the following holds |λ2 | < −λ1 . By using (6), we see that detJT (¯ x, y¯) = λ1 λ2 < 0, which implies λ2 > 0. Let   e f . (21) JT (¯ x, y¯) = g h Taking derivatives of T1 (x, y) = x and T2 (x, y) = y with respect to x in the neighborhood of x ¯, we have y10 (¯ x) − y20 (¯ x) =

p(1) (λ1 − 1)(λ2 − 1) 1−e g − = = , f 1−h f (1 − h) f (1 − h)

(22)

where p(λ) = λ2 − (e + f )λ + (eh − f g) is the characteristic equation of (21). One can see that y10 (¯ x) − y20 (¯ x) < 0. Since λ1 < 0, and f > 0, h < 0, we obtain λ2 > 1. In view of |λ2 | < −λ1 we have λ1 < −1 from which the proof follows. (ii) Now, we consider the equilibrium point EN E (¯ x, y¯). Same as in the previous case we have that y10 (¯ x) − y20 (¯ x) > 0 2 2 2 which implies that 0 < λ2 < 1. By Theorem 5 we have λ1 + λ2 > 5 from which it follows λ1 > 4, i.e. λ1 < −2. This completes the proof. 2

Theorem 8 Assume that ∆2 < 0 and ∆1 = 0. Then there exist one positive equilibrium point E(¯ x, y¯) which is a non-hyperbolic equilibrium point of unstable type. If λ1 and λ2 are the eigenvalues of the Jacobian matrix associated to the map T at E(¯ x, y¯) ∈ int(R2+ ) then λ1 < −1 and λ2 = 1. Proof. In view of Lemmas 6 and 7 from [1], the curves CF and CG intersect tangentially at E(¯ x, y¯) (i.e. y10 (¯ x)−y20 (¯ x) = 0) ˜ ˜ if and only if x ¯ is zero of f (x) of multiplicity greater then one. By Lemma 3, x ¯ is a root of f (x) of multiplicity two. In view of (λ1 − 1)(λ2 − 1) y10 (¯ x) − y20 (¯ x) = = 0, (23) f (1 − h) 2

we obtain λ2 = 1. Since |λ2 | < −λ1 we have λ1 < −1.

4

The global behavior

Let R = [0, ∞)2 , CF = {(x, y) ∈ R : F (x, y) = x}, CG = {(x, y) ∈ R : G(x, y) = y} and RT 2 (−, −)

=

{ (x, y) ∈ R : F (x, y) < x, G(x, y) < y },

RT 2 (+, −)

=

{ (x, y) ∈ R : F (x, y) > x, G(x, y) < y },

RT 2 (+, +)

=

{ (x, y) ∈ R : F (x, y) > x, G(x, y) > y },

RT 2 (−, +)

=

{ (x, y) ∈ R : F (x, y) < x, , G(x, y) > y }.

By Lemma 5, CF na CG are the graphs of continuous strictly increasing functions yF and yG , i.e. CF = { (x, yF (x)) : √ 3 ab2 } and CG = { (x, yG (x)) : x ≥ 0 }. In view of Lemma 4 [15] we have that T (RT 2 (+, −)) ⊆ RT 2 (−, +) and T (RT 2 (−, +)) ⊆ RT 2 (+, −) and T 2 (RT 2 (−, +)) ⊆ RT 2 (−, +) and T 2 (RT 2 (+, −)) ⊆ RT 2 (+, −). Since T 2 is competitive map, by using (iii) of Lemma 1, we obtain T 2 (x0 , y0 ) → (0, ∞) if (x0 , y0 ) ∈ RT 2 (−, +) and T 2 (˜ x0 , y˜0 ) → (∞, 0) if (˜ x0 , y˜0 ) ∈ RT 2 (+, −).

x≥

Lemma 7 int[[P1 , P2 ]] ⊂ B(E0 ).

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Proof. If P ∈ int[[P1 , P2 ]] then there exist x0 and y0 such that P1 ≺se (0, y0 ) ≺se P ≺se (x0 , 0) ≺se P2 . Since T 2 is competitive map we have P1 ≺se T 2n (0, y0 ) ≺se T 2n (P ) ≺se T 2n (x0 , 0) ≺se P2 . From Lemma 1 we obtain T 2n (0, y0 ) → E0 and T 2n (x0 , 0) → E0 as n → ∞ from which the proof follows.

4.1

2

The case ∆2 ≥ 0 or (∆2 < 0 and ∆1 > 0)

In this case, by Lemma 3, there exists one equilibrium point, E0 which is locally asymptotically stable and the minimal period two solution {P1 , P2 } which is a saddle point. In this case we have that yF (x) < yG (x) for x ≥ 0 and RT 2 (−, −)

=

{ (x, y) ∈ R : yF (x) < y < yG (x) },

RT 2 (+, −)

=

{ (x, y) ∈ R : yF (x) > y } ⊆ B(∞, 0),

RT 2 (−, +)

=

{ (x, y) ∈ R : y > yG (x) } ⊆ B(0, ∞).

Let B(0, ∞) denote the basin of attraction of (0, ∞) and B(∞, 0) denote the basin of attraction of (∞, 0) with respect to the map T 2 . In view of Theorem 4 and iii) of Lemma 1 it is clear that {T n (x0 , y0 )} is either asymptotic to {(0, ∞), (∞, 0)} or converges to a period-two solution, for all (x0 , y0 ) ∈ R = [0, ∞)2 . Let S1 denote the boundary of B(0, ∞) and let S2 denote the boundary of B(∞, 0). It is easy to see that P1 ∈ S1 , P2 ∈ S2 , and S1 , S2 ⊆ RT 2 (−, −). Similarly as in [10] it follows that T 2 (S1 ) ⊆ S1 , T 2 (S2 ) ⊆ S2 and T (S1 ) = S2 , T (S2 ) = S1 . Further, S1 and S2 are the graphs of continuous strictly increasing functions. Since S1 , S2 ⊆ RT 2 (−, −), we have, by the uniqueness of the global stable manifold of the map T 2 , that W s (P1 ) = S1 and W s (P2 ) = S2 . Theorem 9 Assume that ∆2 ≥ 0. Then System (1) possesses one equilibrium point E0 (0, 0) and one minimal period-two solution {P1 , P2 }. Equilibrium E0 is locally asymptotically stable and {P1 , P2 } is a saddle point. Global stable manifold W s ({P1 , P2 }), which is a union of two continuous increasing curves S1 and S2 , divides the first quadrant such that the following holds: i) Every initial point (x0 , y0 ) ∈ R such that (˜ x0 , y˜0 ) se (x0 , y0 ) se (¯ x0 , y¯0 ) for some (˜ x0 , y˜0 ) ∈ S1 and (¯ x0 , y¯0 ) ∈ S2 is attracted to E0 . ii) If (x0 , y0 ) ∈ R such that (x0 , y0 ) se (˜ x0 , y˜0 ) for some (˜ x0 , y˜0 ) ∈ S1 then the subsequence of even-indexed terms {(x2n , y2n )} is asymptotic to (0, ∞), and the subsequence of odd-indexed terms {(x2n+1 , y2n+1 )} is asymptotic to (∞, 0). iii) If (x0 , y0 ) ∈ R such that (¯ x0 , y¯0 ) se (x0 , y0 ) for some (¯ x0 , y¯0 ) ∈ S2 then the subsequence of even-indexed terms {(x2n , y2n )} is asymptotic to (∞, 0), and the subsequence of odd-indexed terms {(x2n+1 , y2n+1 )} is asymptotic to (0, ∞). See Figure 1 (a) for visual illustration. Proof. Since S1 is invariant under T 2 and subset of RT 2 (−, −) we have that if (x0 , y0 ) ∈ S1 then T 2n+2 (x0 , y0 ) ne T 2n (x0 , y0 ). This implies that subsequences {x2n } and {y2n } are decreasing and since they are bounded sequences, they are convergent. It must be that T 2n (x0 , y0 ) → P1 as n → ∞. By the uniqueness of the global stable manifold of T 2 we obtain W s (P1 ) = S1 . Similarly we get W s (P2 ) = S2 from which it follows that W s ({P1 , P2 }) = S1 ∪ S2 . Take (x0 , y0 ) ∈ R, (˜ x0 , y˜0 ) ∈ S1 and (¯ x0 , y¯0 ) ∈ S2 such that (˜ x0 , y˜0 ) se (x0 , y0 ) se (¯ x0 , y¯0 ). By monotonicity of T 2 we 2n 2n 2n 2n 2n have T (˜ x0 , y˜0 ) se T (x0 , y0 ) se T (¯ x0 , y¯0 ). Since T (˜ x0 , y˜0 ) → P1 and T (¯ x0 , y¯0 ) → P2 as n → ∞ and by the uniqueness of the global stable manifold we obtain that T 2n (x0 , y0 ) eventually enters int[[P1 , P2 ]]se . So√it is enough to √ 3 ¯0 > 0 and 3 a2 b > y¯0 > 0 prove that [[P1 , P2 ]]se ⊆ B(E0 ). Indeed, for (x0 , y0 ) ∈ int[[P1 , P2 ]]se there exist ab2 > x 2n 2n ¯0 , 0). By Lemma 1 we have T (0, y¯0 ) → E0 and T (x ¯0 , 0) → E0 as n → ∞. By such that (0, y¯0 ) se (x0 , y0 ) se (x monotonicity of T 2 we get T 2n (x0 , y0 ) → E0 from which the proof follows. By construction of the sets S1 and S2 the statements ii) and iii) are valid. 2

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4.2

The case ∆2 < 0 and ∆1 < 0

In this case, by Lemma 3, there exist three equilibrium points E0 , ESW and EN E . By Lemma 6 E0 is locally asymptotically stable and by Theorem 7 ESW is repeller and EN E is a saddle point. If Q2 (ESW ) = {(x, y) : 0 ≤ x ≤ x ¯SW and y ≥ x ¯SW } and Q4 (ESW ) = {(x, y) : x ≥ x ¯SW and 0 ≤ y ≤ x ¯SW }, then one can see that Q2 (ESW ) ⊆ RT 2 (−, +) and Q4 (ESW ) ⊆ RT 2 (+, −). Let B(0, ∞) denote the basin of attraction of (0, ∞) and B(∞, 0) denote the basin of attraction of (∞, 0) with respect to the map T 2 . In view of Theorem 4 and iii) of Lemma 1 it is easy to see that {T n (x0 , y0 )} is either asymptotic to (0, ∞) or (∞, 0) or converges to a period-two solution, for all (x0 , y0 ) ∈ R = [0, ∞)2 . Let S1 denote the boundary of B(0, ∞) considered as a subset of Q3 (ESW ) and S2 denote the boundary of B(∞, 0) considered as a subset of Q3 (ESW ). It follows that P1 , ESW ∈ S1 and P2 , ESW ∈ S2 . It is easy to see S1 , S2 ⊆ RT 2 (−, −), from which, similarly as in [10], it follows that T 2 (S1 ) ⊆ S1 , T 2 (S2 ) ⊆ S2 and T (S1 ) = S2 , T (S2 ) = S1 . Further, S1 and S2 are the graphs of continuous strictly increasing functions. Since S1 , S2 ⊆ RT 2 (−, −), we have, by the uniqueness of the global stable manifold of T 2 , that W s (P1 ) = S1 and W s (P2 ) = S2 . Lemma 8 B(E0 ) = {P ∈ [0, ∞)2 : Pe ≺se P ≺se P for Pe ∈ S1 and P ∈ S2 }. Proof. Assume that Pe ≺se P ≺se P for Pe ∈ S1 and P ∈ S2 . By monotonicity of T we get T 2n (Pe) ≺se T 2n (P ) ≺se T 2n (P ). Since T 2n (Pe) → P1 and T 2n (P ) → P2 as n → ∞. By the uniqueness of the global stable manifold we have that T 2n eventually enters int[[P1 , P2 ]]. The rest of the proof follows from Lemma 7. 2

Lemma 9 Assume that Dis(P ) 6= 0. Then System (1) does not have minimal period-two solution. Proof. For contradiction, assume that P is a minimal period-two solution of System (1). It is clear from previous discussions that P ∈ (Q1 (ESW ) ∩ Q3 (EN E )) ∪ Q1 (EN E ). Furthermore, assume that P ∈ Q1 (ESW ) ∩ Q3 (EN E ) and T has no other minimal period-two solutions in [[ESW , P ]]ne . Since ESW is a repeller by Lemma 6 we obtain that P is a saddle point. The map T 2 satisfy all conditions of Theorem 5 [20], which yields the existence of the global stable manifolds W s ({P, T (P )}), which is the union of two curves W s (P ) and W s (T (P )). Since W s (T (P )) = T (W s (P )) we have that these two curves have a common endpoint ESW and there exists minimal period-two solution {P˜ , T (P˜ )} such that P ne P˜ ne EN E and the curve W s (P ) has the second endpoint at P˜ while the curve W s (T (P )) has the second endpoint at T (P˜ ). Furthermore, the minimal period-two solution {P˜ , T (P˜ )} is a repeller. Since all positive period-two solutions are ordered with respect to the North-East ordering it must be W s (T (P )) ne W s (P ), i.e. W s (T (P )) ⊂ Q3 (ESW ) which is in contradiction to W s (T (P )) ⊂ Q1 (ESW ). Similarly, we have contradiction if P ∈ Q1 (EN E ). Hence, T has no minimal period-two solutions. 2

Theorem 10 Assume that ∆2 < 0 and ∆1 < 0. Then System (1) has three equilibrium solutions E0 ≺ne ESW ≺ne EN E , where E0 is locally asymptotically stable, ESW is a repeller and EN E is a saddle point and the minimal period-two solution {P1 , P2 } which is a saddle point. In this case there exist three invariant continuous curves W s (EN E ), W s (P1 ), W s (P2 ), which have end point at ESW and they are graphs of increasing functions. Every solution {(xn , yn )} which starts below W s (EN E ) ∪ W s (P1 ) in South-East ordering is asymptotic to (0, ∞) and every solution {(xn , yn )} which starts above W s (EN E ) ∪ W s (P2 ) in South-East ordering is asymptotic to (∞, 0). Every solution {(xn , yn )} which starts below W s (P2 ) and above W s (P1 ) in South-East ordering converges to E0 . The first quadrant of the initial conditions Q1 = {(x0 , y0 ) : x0 ≥ 0, y0 ≥ 0} is the union of six disjoint basins of attraction, i.e. Q1 = B(0, ∞) ∪ B(∞, 0) ∪ B(E0 ) ∪ B({P1 , P2 }) ∪ B(EN E ) ∪ B(ESW ), where B(ESW ) = {ESW },

B(EN E ) = W s (EN E ),

B({P1 , P2 }) = W s (P1 ) ∪ W s (P2 ),

B(0, ∞) ={(x, y)|(x, y) se (˜ x0 , y˜0 )

for some

(˜ x0 , y˜0 ) ∈ W s (EN E ) ∪ W s (P1 )},

B(∞, 0) ={(x, y)|(˜ x1 , y˜1 ) se (x, y)

for some

(˜ x1 , y˜1 ) ∈ W s (EN E ) ∪ W s (P2 )},

B(E0 ) ={(x, y)|(˜ x1 , y˜1 ) se (x, y) se (˜ x2 , y˜2 )

for some

(˜ x1 , y˜1 ) ∈ W s (P1 ), (˜ x2 , y˜2 ) ∈ W s (P2 )}.

Proof. The proof which follows from previous discussions and Theorem 5 [20] will be ommited. See Figure 1 (c) for visual illustration. 2

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Figure 1: Visual illustration of (a) Theorem 9 (b) Theorem 11 and (c) Theorem 10.

4.3

The case ∆2 < 0 and ∆1 = 0

Let S1 and S2 be defined as in the previous case where ESW = EN E = E. In this case, by Lemma 3, there exist two equilibrium points E0 and E. By Lemma 6 we have that E0 is locally asymptotically stable and E is a non-hyperbolic equilibrium point of unstable type. Let Se1 denote the boundary of B(0, ∞) considered as a subset of Q1 (E) and Se2 denote the boundary of B(∞, 0) considered as a subset of Q1 (E). It is clear that E ∈ Se1 ∪ Se2 . Furthermore, one can see Se1 , Se2 ⊆ RT 2 (−, −). Similarly as in [10], we have that T 2 (S¯1 ) ⊆ Se1 , T 2 (Se2 ) ⊆ Se2 and T (Se1 ) = Se2 , T (Se2 ) = Se1 . Further, Se1 and Se2 are the graphs of continuous strictly increasing functions. Theorem 11 Assume that ∆2 < 0 and ∆1 = 0. Then System (1) possesses two equilibrium solutions E0 and E and one minimal period-two solution {P1 , P2 }. Equilibrium E0 is locally asymptotically stable, E is non-hyperbolic of unstable type and {P1 , P2 } is a saddle point. Global stable manifold W s ({P1 , P2 }) is the union of two continuous increasing curves S1 and S2 and the following holds i) Every initial point (x0 , y0 ) ∈ R such that (˜ x0 , y˜0 ) se (x0 , y0 ) se (¯ x0 , y¯0 ) for some (˜ x0 , y˜0 ) ∈ S1 and (¯ x0 , y¯0 ) ∈ S2 is attracted to E0 . ii) Every initial point (x0 , y0 ) ∈ R such that (˜ x0 , y˜0 ) se (x0 , y0 ) se (¯ x0 , y¯0 ) for some (˜ x0 , y˜0 ) ∈ Se1 and (¯ x0 , y¯0 ) ∈ Se2 is attracted to E. iii) If (x0 , y0 ) ∈ R such that (x0 , y0 ) se (˜ x0 , y˜0 ) for some (˜ x0 , y˜0 ) ∈ S1 ∪Se1 then the subsequence of even-indexed terms {(x2n , y2n )} is asymptotic to (0, ∞), and the subsequence of odd-indexed terms {(x2n+1 , y2n+1 )} is asymptotic to (∞, 0). iv) If (x0 , y0 ) ∈ R such that (¯ x0 , y¯0 ) se (x0 , y0 ) for some (¯ x0 , y¯0 ) ∈ S2 ∪Se2 then the subsequence of even-indexed terms {(x2n , y2n )} is asymptotic to (∞, 0), and the subsequence of odd-indexed terms {(x2n+1 , y2n+1 )} is asymptotic to (0, ∞). See Figure 1 (b) for visual illustration. Proof. The proof of the statement i) is the same as the proof of the statement i) of Theorem 9. Since Se1 is invariant under T 2 and subset of RT 2 (−, −) we have that if (x0 , y0 ) ∈ Se1 then T 2n+2 (x0 , y0 ) ne T 2n (x0 , y0 ). This implies that subsequences {x2n } and {y2n } are decreasing and since they are bounded sequences, they are convergent. It must be that T 2n (x0 , y0 ) → E as n → ∞. Since T is continuous map and E is an equilibrium point we obtain T 2n+1 (x0 , y0 ) → E as n → ∞ and Se2 = T (Se1 ). Similarly we obtain that if (x0 , y0 ) ∈ Se2 then T 2n (x0 , y0 ) ∈ Se2 , T 2n (x0 , y0 ) → E as n → ∞. Further, T 2n+1 (x0 , y0 ) ∈ Se1 , T 2n+1 (x0 , y0 ) → E as n → ∞. Take (x0 , y0 ) ∈ R and (˜ x0 , y˜0 ) ∈ Se1 and (¯ x0 , y¯0 ) ∈ Se2 such that (˜ x0 , y˜0 ) se (x0 , y0 ) se (¯ x0 , y¯0 ). By monotonicity of T 2 we have T 2n (˜ x0 , y˜0 ) se T 2n (x0 , y0 ) se T 2n (¯ x0 , y¯0 ). Since T 2n (˜ x0 , y˜0 ) → E and T 2n (¯ x0 , y¯0 ) → E as n → ∞ we obtain that T 2n (x0 , y0 ) → E, which implies the statement ii). The statements iii) and iv) follow by construction of the sets Se1 and Se2 . 2

Remark 1 The major results of this paper, Theorems 9 - 11, are actually the general results for general anti-competitive system (3). In fact, any anti-competitive system (3) with same configuration and local stability of the equilibrium and

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period-two solutions will have the same global dynamics. So System (1) is actually an example of a global dynamics described in Theorems 9 - 11. Remark 2 In [22] we consider system    xn+1

=

x2 n 2 a+yn

  yn+1

=

2 yn b+x2 n

,

n = 0, 1, . . .

(24)

where the parameters a, b are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers, and obtain global dynamics similar to one described in Theorems 9 - 11, with the major difference that P1 and P2 are saddle point equilibrium solutions. Since the eigenvectors of the linearized system at P1 and P2 are parallel to the coordinate axes one can not apply at this time the results from [19, 20] to prove the existence of stable manifolds at these two points. However, existence of stable manifolds at these two points can be proved as in Theorems 9 - 11, where these two manifolds are obtained as Julia sets of the points (∞, 0) and (0, ∞). Thus all results in [22] are correct with this adjustment of the proof.

A

Values of coefficients pi for i = 0, ..., 16. p16 =a4 b4 − 4a3 b3 + 6a2 b2 − 4ab + 1 p15 =4a3 b5 − 12a2 b4 + 12ab3 − 4b2 p14 =6a2 b6 − 12ab5 + 7a5 b4 + 6b4 − 20a4 b3 + 18a3 b2 − 4a2 b − a p13 =4ab7 − 4b6 + 26a4 b5 − 52a3 b4 + 24a2 b3 + 4ab2 − 2b p12 =b8 + 38a3 b6 − 40a2 b5 + 21a6 b4 − 10ab4 − 40a5 b3 + 12b3 + 10a4 b2 + 8a3 b + a7 + a2 p11 =28a2 b7 − 4ab6 + 72a5 b5 − 16b5 − 87a4 b4 − 30a3 b3 + 26a2 b2 + 4a6 b + 10ab + 7a5 + 1 p10 =5a8 + 35b4 a7 − 40b3 a6 − 12b2 a5 + 101b6 a4 + 32ba4 − 44b5 a3 + 13a3 − 100b4 a2 + 11b8 a + 16b3 a + 4b7 + 11b2 p9 =2b9 + 76a3 b7 + 10a2 b6 + 110a6 b5 − 58ab5 − 68a5 b4 − 6b4 − 102a4 b3 + 34a3 b2 + 18a7 b + 28a2 b + 21a6 + 3a p8 =10a9 + 35b4 a8 − 20b3 a7 + 2b2 a6 + 145b6 a5 + 58ba5 − 12b5 a4 + 15a4 − 163b4 a3 + 33b8 a2 − 20b3 a2 + 8b7 a + 13b2 a + 8b6 + 4b p7 =8ab9 + b8 + 104a4 b7 + 16a3 b6 + 100a7 b5 − 66a2 b5 − 22a6 b4 − 33ab4 − 66a5 b3 + 29a4 b2 + 32a8 b + 22a3 b + 22a7 + 3a2 p6 =10a10 + 21b4 a9 − 4b3 a8 + 35b2 a7 + 120b6 a6 + 48ba6 + 8b5 a5 + 6a5 − 90b4 a4 + 43b8 a3 − 42b3 a3 + 6b7 a2 − b2 a2 + 16b6 a + 4ba + b10 + 2b5 + 1 p5 =28ba9 + 54b5 a8 + 9a8 + 10b3 a6 + 76b7 a5 + 16b2 a5 + 6b6 a4 + 10ba4 − 22b5 a3 + a3 + 10b9 a2 − 37b4 a2 + 4b8 a − 10b3 a + b2 p4 =5a11 + 7b4 a10 + 33b2 a8 + 56b6 a7 + 14ba7 + 4b5 a6 + a6 − 5b4 a5 + 26b8 a4 − 18b3 a4 + 6b7 a3 − b2 a3 + 14b6 a2 + 2ba2 + b10 a − 2b5 a + 2b9 + b4 p3 =12ba10 + 16b5 a9 + a9 + b4 a8 + 20b3 a7 + 28b7 a6 + 3b2 a6 + 4ba5 + 2b5 a4 + 4b9 a3 − 9b4 a3 + 8b8 a2 − 6b3 a2 − 4b7 a − 2b2 a + b6 p2 =a12 + b4 a11 + 11b2 a9 + 13b6 a8 + 6b4 a6 + 6b8 a5 + 2b3 a5 + 6b7 a4 + 3b2 a4 + 3b6 a3 + 2b5 a2 + 2b9 a − b4 a + b8 p1 =2ba11 + 2b5 a10 + 4b3 a8 + 4b7 a7 + 4b4 a4 + 4b8 a3 − 2b3 a3 − 2b7 a2 p0 =ab8 + 2a5 b7 + a9 b6 + a2 b4 + 2a6 b3 + a10 b2

References [1] S. Basu and O. Merino, On the Behavior of Solutions of a System of Difference Equations, Comm. Appl. Nonlinear Anal. 16 (2009), no. 1, 89–101.

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[2] A. Brett and M. R. S. Kulenovi´c, Two Species Competitive Model with the Allee Effect, Adv. Difference Equ., Volume 2014 (2014):307, 28 p. [3] E. Camouzis, M. R. S. Kulenovi´c, G. Ladas, and O. Merino, Rational Systems in the Plane, J. Difference Equ. Appl. 15 (2009), 303-323. [4] D. Clark and M. R. S. Kulenovi´c, On a Coupled System of Rational Difference Equations, Comput. Math. Appl. 43(2002), 849-867. [5] D. Clark, M. R. S. Kulenovi´c, and J.F. Selgrade, Global Asymptotic Behavior of a Two Dimensional Difference Equation Modelling Competition, Nonlinear Anal., TMA 52(2003), 1765-1776. [6] J. M. Cushing, S. Levarge, N. Chitnis and S. M. Henson, Some discrete competition models and the competitive exclusion principle , J. Difference Equ. Appl. 10(2004), 1139-1152. [7] P. deMottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol. 11 (1981), 319–335. [8] J. E. Franke and A.-A. Yakubu, Mutual exclusion verses coexistence for discrete competitive systems, J. Math. Biol. 30 (1991), 161-168. [9] J. E. Franke and A.-A. Yakubu, Geometry of exclusion principles in discrete systems, J. Math. Anal. Appl. 168 (1992), 385-400. [10] V. Hadˇziabdi´c, M. R. S. Kulenovi´c and E. Pilav, Dynamics of a two-dimensional competitive system of rational difference equations with quadratic terms, Adv. Difference Equ. (2014), 2013:301, 32 p. [11] P. Hess, Periodic-parabolic boundary value problems and positivity. Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; 1991. viii+139 pp. [12] M. Hirsch and H. Smith, Monotone dynamical systems. Handbook of differential equations: ordinary differential equations. Vol. II, 239-357, Elsevier B. V., Amsterdam, 2005. [13] S. Janson, Resultant and discriminant of polynomials, http://www2.math.uu.se/ svante/ papers/, 2010. [14] S. Kalabuˇsi´c and M. R. S. Kulenovi´c, Dynamics of Certain Anti-competitive Systems of Rational Difference Equations in the Plane, J. Difference Equ. Appl., 17(2011), 1599–1615. [15] S. Kalabuˇsi´c, M. R. S. Kulenovi´c and E. Pilav, Global Dynamics of Anti-Competitive Systems in the Plane, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20(2013), 477-505 [16] M. R. S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, London, 2001. [17] M. R. S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman& Hall/CRC Press, Boca Raton, 2002. [18] M. R. S. Kulenovi´c and O. Merino, Competitive-Exclusion versus Competitive-Coexistence for Systems in the Plane, Discrete Contin. Dyn. Syst. Ser. B 6(2006), 1141-1156. [19] M. R. S. Kulenovi´c and O. Merino, Global Bifurcation for Competitive Systems in the Plane, Discrete Contin. Dyn. Syst. B 12(2009), 133-149. [20] M. R. S. Kulenovi´c and O. Merino, Invariant Manifolds for Competitive Discrete Systems in the Plane, Int. J. of Bifurcations and Chaos, 20(2010), 2471-2486. [21] M. R. S. Kulenovi´c and M. Nurkanovi´c, Basins of Attraction of an Anti-competitive System of Difference Equations in the Plane, Comm. Appl. Nonlinear Anal., 19(2012), 41–53. [22] M. R. S. Kulenovi´c and M. Pilling, Global Dynamics of a Certain Two-dimensional Competitive System of Rational Difference Equations with Quadratic Terms, J. Comput. Anal. Appl., 19(2015), 156–166. [23] H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations 64 (1986), 165-194. [24] H. L. Smith, Planar Competitive and Cooperative Difference Equations,J. Difference Equ. Appl. 3(1998), 335-357.

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Solutions to Periodic Sylvester Matrix Equations Based on Matrices Splitting ∗ Lingling Lv, †Chaofei Han, ‡Lei Zhang,

§

Abstract New iterative algorithms are introduced to solve periodic Sylvester matrix equations in this paper. The iterative algorithms are based on the principle of matrix splitting and gradient iteration method. Detailed iterative steps for solving equations are presented and their convergence property are strictly verified. A numerical test is employed to prove the correctness and effectiveness of the iterative algorithms. Keywords: Periodic Sylvester matrix equations; iterative algorithm; matrix splitting.

1

Introduction

Analysis and design of time-varying systems are more charllenging than that of time-invariant dynamic systems since their coefficients are changing according to time. Take stability and stabilization for example, the stability concepts and criterion for (linear) time-varying systems are very difficult to characterize as they generally have no direct relationship with their coefficients (see [18] and [19] for detailed introductions). The periodic linear system as a special case of linear time-varying systems is thus important since it helps to understand that methods built for time-invariant systems can be generalized to time-varying setting. On the other, periodic linear systems also have important applications in engineering since they can be frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes. For example, Caswell analyzed in [1] the periodic models that must trace the effects of parameter changes and they applied the method to periodic system for periodic environments, and Verstraete introduced in [13] a picture to analyse the density matrix renormalization group (DMRG) numerical method from a quantum information prespective, which leads to a variational formulation of DMRG that allows for dramatic improvements in the case of problems with periodic boundary conditions. Therefore, in recent years, periodic linear systems have attracted significant attention in the literature. Periodic Sylvester matrix equations play a major role in the analysis and design of discrete-time periodic linear systems. A general form of the periodic Sylvester matrix equation is as follows At Xt + Xt+1 Bt = Ct ,

(1)

At Xt+1 + Xt Bt = Ct ,

(2)

and where the coefficient matrices At , Bt , Ct ∈ Rn×n , t = 0, 1, · · · , are given matrices and Xt ∈ Rn×n are unknown matrices. These matrices are periodic with period T , i.e., At+T = At , Bt+T = Bt , Ct+T = Ct and Xt+T = Xt . In [8], Korotyaev showns that it is related with the periodic matrix-valued Jacobi operators. We have shown recently that the aboveperiodic Sylvester matrix equation are helpful in the design of periodic ∗ This work is supported by the Programs of National Natural Science Foundation of China (Nos. 11501200, U1604148, 61402149), Innovative Talents of Higher Learning Institutions of Henan (No. 17HASTIT023), China Postdoctoral Science Foundation (No. 2016M592285). † Institute of Electric power, North China University of Water Resources and Electric Power, Zhengzhou 450045, P. R. China. Email: lingling [email protected] (Lingling Lv). ‡ Institute of electric power, North China University of Water Resources and Electric Power, Zhengzhou 450045, P. R. China. Email: [email protected] (Chaofei Han). § Computer and Information Engineering College, Henan University, Kaifeng 475004, P. R. China. Email: [email protected] (Lei Zhang). Corresponding author.

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Luenberger observers [10] and output regulator [11]. For more applications of this class of periodic Sylvester matrix equations, see [10], [11] and the refereces therein. The (generalized periodic) Sylvester matrix equations were first put forward by Sylvester and applied in mathematical control theory. As the development of science and technology, they become more and more important. Now many scholars and experts have analyzed the existence and uniqueness of solutions to the Sylvester matrix equation. In [12], Sreedhar proposed an elegant and simple method for computing the periodic solution of Sylvester matrix equations. In [3], Chen used the matrix sign function to solve periodic Sylvester equations. In [16], Zhang offered a finite iterative algorithm for solving the complex generalized coupled Sylvester matrix equations. In [2], the author pays attention to solving the Lyapunov matrix equations and Sylvester matrix equations in control theory by numerical methods. [4] constructs an iterative algorithm to solve the generalized coupled Sylvester matrix equations over reflexive matrices. In [9], a comprehensive theory of the matrix linear equation AX + XB = C is presented. In [6], Gu applied Jacobi iteration of solving linear equations to solve Sylvester matrix equations. In [5], M Dehghan propose two iterative algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of the Sylvester matrix equation AX + XB = C. Furthermore, it should be pointed that gradient iterative algorithm is attracting more and more researchers. Many experts apply it to solve the Sylvester matrix equations and a lot of cases show that it is a better way to solve matrix equations. In [17], zhang present a gradient iterative algorithm for solving coupled matrix equations based on the hierarchical identification principle. In [14], Li study solutions of general matrix equations by using the iterative method and present gradient iterative algorithms by applying the hierarchical identification principle. In [7], Hoskins discussed an iterative method for solving the matrix equation XA + AY = F and compared it with existing techniques. In [15], an iterative algorithm is construct to solve the general coupled matrix equations over reflexive matrix solution. Of course, researchers have given numerical examples to demonstrate the correctness of the proposed algorithm. However, to the best of our knlowedge, the iterative algorithms for the periodic Sylvester matrix equations have not been fully researched in the literature. Therefore, in this paper, we dedicate to give iterative algorithms for solving equations (1) and (2). The iterative algorithms are based on the principle of matrix splitting and gradient iteration method. Detailed iterative steps for solving equations are presented and their convergence property are strictly verified. A numerical test is employed to prove the correctness and effectiveness of the iterative algorithms. The rest of this paper is arranged in the following ways. In section 2, new iterative algorithms are proposed to solve the Sylvester matrix equations and the convergences are validated. In section 3, a numerical example is provided to verify the correctness of the iterative algorithm. And in section 4, we draw some conclusions.

2 2.1

Main results Iterative algorithm for equation (1) ′

′′

Firstly, for t = 0, 1, · · · , T − 1, define Wt and Wt as ′

Wt = Ct − Xt+1 Bt

(3)

′′

Wt = Ct − At Xt Rewrite matrices At and Bt as

(4)

′

At = αt In×n + Tt ,

(5)

′′

Bt = βt In×n + Tt ,

(6) ′

′′

where αt and βt are arbitrary constant numbers, In×n is the unit matrix and Tt , Tt are the remaining matrices of At , Bt . Based on the above division, it is easy to obtain that ′

′

Wt = (αt In×n + Tt )Xt

(7)

2

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′′

Wt = Xt+1 (βt In×n + Tt )

(8)

( ) ′ ′ ′ ′ Xt (k) = Xt (k − 1) + θt αt Ct − Xt+1 (k − 1)Bt − At Xt (k − 1)

(9)

( ) ′′ ′′ ′′ ′′ Xt (k) = Xt (k − 1) + θt βt Ct − Xt+1 (k − 1)Bt − At Xt (k − 1)

(10)

Construct the following iteration:

Where k is iterative step and bigger than 1. Further, let ′

′′

Xt (k) + Xt (k) 2 1 = Xt (k − 1) + θt (αt + βt ) (Ct − Xt+1 (k − 1)Bt − At Xt (k − 1)) 2

Xt (k) =

(11)

In addition, denote Rt (k) = ∥Xt (k) − Xt (k − 1)∥

(12)

Algorithm 1 (An iterative algorithm for equation (1)) ′

′′

1. Set error upper limit ε, freely select initial matrices Xt (0) and Xt (0), calculate ′

Xt (0) =

′′

Xt (0) + Xt (0) 2

2. Choose parameters of αt , βt and θt for t = 0, 1, · · · , T − 1, calculate λt =

T −1 ∑ t=0

T −1 T −1 ∑ ∑ 1 1 ∥(It − θt (αt + βt )At )∥ + ∥ θt (αt + βt )∥ ∥Bt ∥) 2 2 t=0 t=0

(13)

k := 0; 3. If λt < 1 for t = 0, 1, · · · , T − 1, go to next step; else, return to step 2. 4. Set k=k + 1, according to (9),(10),(11), compute Xt (k); Further more, compute Rt (k) by (12). 5. If Rt (k) ≤ ε, stop; else, go to step 4. The convergence of the iterative algorithm will be proved by the following theorem. Theorem 1 If equation (1) has solutions Xt∗ and λt shown in (13) is less than 1, the iterative sequence of Xt (k) generated by Algorithm 1 converges to the true solution Xt∗ , which means, for any initial Xt (0), there is lim Xt (k) = Xt∗ k→∞

¯ t (k)=Xt (k) − Xt∗ , where Xt∗ act as the real matrix, Xt (k) is the iterative Proof. Define error matrix X solution to k by the algorithm, then ¯ t′ (k) = Xt ′ (k) − Xt∗ X (14) ¯ t′′ (k) = Xt ′′ (k) − Xt∗ X

(15)

′ ¯ t+1 ¯ t′ (k − 1)) ¯ t′ (k) = X ¯ t′ (k − 1) + 1 θt (αt + βt )(−X (k − 1)Bt − At X X 2

(16)

We can easily get

3

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.6, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC ′′ ¯ t′′ (k) = X ¯ t′′ (k − 1) + 1 θt (αt + βt )(−X ¯ t+1 ¯ t′′ (k − 1)) X (k − 1)Bt − At X 2

(17)

Then, we get ¯ t′ (k) + X ¯ t′′ (k) X ¯ Xt (k) = 2 ¯ t+1 (k − 1)Bt − At X ¯ t (k − 1)) ¯ t (k − 1) + 1 θt (αt + βt )(−X =X 2 ¯ t (k − 1) − 1 θt αt X ¯ t+1 (k − 1)Bt − 1 θt βt X ¯ t+1 (k − 1)Bt =X 2 2 1 ¯ t (k − 1)) − 1 θt βt At X ¯ t (k − 1)) − θt αt At X 2 2 1 1 ¯ t (k − 1) − 1 θt αt X ¯ t+1 (k − 1)Bt = (It − θt αt At − θt βt At )X 2 2 2 1 ¯ t+1 (k − 1)Bt − θt βt X 2 1 ¯ t (k − 1) − 1 θt (αt + βt )X ¯ t+1 (k − 1)Bt = (It − θt (αt + βt )At )X 2 2 Let ¯ t (k)∥ = ∥(It − 1 θt (αt + βt )At )X ¯ t (k − 1) − 1 θt (αt + βt )X ¯ t+1 (k − 1)Bt ∥ ∥X 2 2 1 ¯ t (k − 1)∥ + ∥ 1 θt (αt + βt )X ¯ t+1 (k − 1)Bt ∥ ≤ ∥(It − θt (αt + βt )At )X 2 2 1 ¯ t (k − 1)∥ + ∥ 1 θt (αt + βt )∥∥X ¯ t+1 (k − 1)∥∥Bt ∥ ≤ ∥(It − θt (αt + βt )At )∥∥X 2 2 So we can obtain T −1 ∑

¯ t (k)∥ ≤ ∥X

t=0

T −1 ∑ t=0

1 ¯ t (k − 1)∥ + ∥ 1 θt (αt ∥(It − θt (αt + βt )At )∥∥X 2 2

¯ t+1 (k − 1)∥∥Bt ∥ + βt )∥∥X ≤

T −1 ∑ t=0

T −1 T −1 ∑ ∑ 1 1 ¯ t (k − 1)∥ + ∥X ∥ θt (αt ∥(It − θt (αt + βt )At )∥ 2 2 t=0 t=0 T −1 ∑

+ βt )∥

¯ t+1 (k − 1)∥ ∥X

T −1 ∑

t=0 T −1 ∑

≤(

t=0

+ βt )∥

∥Bt ∥

t=0

T −1 ∑ 1 1 ∥(It − θt (αt + βt )At )∥ + ∥ θt (αt 2 2 t=0

T −1 ∑ t=0

∥Bt ∥)

T −1 ∑

¯ t (k − 1)∥ ∥X

t=0

According to assumption λt < 1, where λt are shown in (14), we can obtain that T −1 ∑

¯ t (k)∥ ≤ λt ∥X

t=0

T −1 ∑

¯ t (k − 1)∥ ≤ λ2t ∥X

T −1 ∑

t=0

¯ t (k − 2)∥ ≤ . . . ≤ λkt ∥X

t=0

T −1 ∑

¯ t (0)∥ ∥X

(18)

t=0

By controlling parameters of αt , βt , θt to make λt < 1. When k is towards infinity, ¯ t (k) = 0 lim X

k→∞

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So

lim Xt (k) = Xt∗

k→∞

2.2

Iterative algorithm for equation (2)

On periodic Sylvester matrix equation (2),we can also build an convergent algorithm which is similar to Algorithm 1. Firstly construct the following iteration: ( ) ′ ′ ′ ′ Xt (k) = Xt (k − 1) + θt αt Ct − Xt (k − 1)Bt − At Xt+1 (k − 1)

(19)

( ) ′′ ′′ ′′ ′′ Xt (k) = Xt (k − 1) + θt βt Ct − Xt (k − 1)Bt − At Xt+1 (k − 1)

(20)

Let ′

′′

Xt (k) + Xt (k) 2 1 = Xt (k − 1) + θt (αt + βt ) (Ct − Xt (k − 1)Bt − At Xt+1 (k − 1)) 2

Xt (k) =

(21)

Algorithm 2 (An iterative algorithm for equation (2)) ′

′′

1. Set error upper limit ε, arbitrary select initial matrices Xt (0) and Xt (0), calculate Xt (0) as ′

Xt (0) =

′′

Xt (0) + Xt (0) 2

2. Choose parameters of αt , βt and θt for t = 0, 1, · · · , T − 1, calculate λt according to (13), and set k := 0; 3. If λt < 1 for t = 0, 1, · · · , T − 1, go to next step; else, return to step 2. 4. Set k=k + 1, according to (19),(20),(21), compute Xt (k). 5. Compute Rt (k) according to (12). If Rt (k) ≤ ε, stop; else, go to step 4. We can make use of the following theorem to prove the convergence of the iterative algorithm. Theorem 2 If equation (2) has solutions Xt∗ and λt shown in (13) is less than 1, the iterative sequence of Xt (k) generated by Algorithm 2 converges to the true solutions Xt∗ , which means, for any initial Xt (0), there is lim Xt (k) = Xt∗ k→∞

Proof. According to Algorithm 2, we can acquire the following results ′ ¯ t′ (k − 1)Bt − At X ¯ t+1 ¯ t′ (k) = X ¯ t′ (k − 1) + 1 θt (αt + βt )(−X (k − 1)) X 2 ′′ ¯ t′′ (k) = X ¯ t′′ (k − 1) + 1 θt (αt + βt )(−X ¯ t′′ (k − 1)Bt − At X ¯ t+1 X (k − 1)) 2 ¯ t (k) = X ¯ t (k − 1) + 1 θt (αt + βt )(−X ¯ t (k − 1)Bt − At X ¯ t+1 (k − 1)) X 2

(22) (23) (24)

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Let ¯ t (k)∥ = ∥(It − 1 θt (αt + βt )At )X ¯ t+1 (k − 1) − 1 θt (αt + βt )X ¯ t (k − 1)Bt ∥ ∥X 2 2 1 ¯ t+1 (k − 1)∥ + ∥ 1 θt (αt + βt )X ¯ t (k − 1)Bt ∥ ≤ ∥(It − θt (αt + βt )At )X 2 2 1 ¯ t+1 (k − 1)∥ + ∥ 1 θt (αt + βt )∥∥X ¯ t (k − 1)∥∥Bt ∥ ≤ ∥(It − θt (αt + βt )At )∥∥X 2 2 Further, let T −1 ∑

¯ t (k)∥ ≤ ∥X

t=0

T −1 ∑ t=0

1 ¯ t+1 (k − 1)∥ + ∥ 1 θt (αt ∥(It − θt (αt + βt )At )∥∥X 2 2

¯ t (k − 1)∥∥Bt ∥ + βt )∥∥X ≤

T −1 ∑ t=0

T −1 T −1 ∑ ∑ 1 1 ¯ t+1 (k − 1)∥ + ∥(It − θt (αt + βt )At )∥ ∥X ∥ θt (αt 2 2 t=0 t=0

+ βt )∥

T −1 ∑

¯ t (k − 1)∥ ∥X

t=0

∥Bt ∥

t=0

T −1 ∑ 1 1 ∥ θt (αt ∥(It − θt (αt + βt )At )∥ + 2 2 t=0

T −1 ∑

≤(

t=0

+ βt )∥

T −1 ∑

T −1 ∑

∥Bt ∥)

t=0

T −1 ∑

¯ t (k − 1)∥ ∥X

t=0

According to assumption λt < 1, where λt are shown in (13), we can obtain that T −1 ∑

¯ t (k)∥ ≤ λt ∥X

T −1 ∑

t=0

¯ t (k − 1)∥ ≤ λ2t ∥X

T −1 ∑

t=0

¯ t (k − 2)∥ ≤ . . . ≤ λkt ∥X

t=0

T −1 ∑

¯ t (0)∥ ∥X

(25)

t=0

When k is towards infinity and λ < 1, we can obtain ¯ t (k) = 0 lim X k→∞

So

lim Xt (k) = Xt∗

k→∞

3

A numerical example

In this section, we will give an example to illustrate the correctness and effectiveness of the iterative algorithm. Example 1 In this example, we consider the following periodic Sylvester matrix equation with T = 3: At Xt + Xt+1 Bt = Ct For given matrices

[ A0 = [ B0 = [ C0 =

0.5 0.3 12.2 0.6

]

[

]

[

] 5.2 2.8 , A1 = , A2 = −3.1 5.3 ] [ ] [ ] −0.2 1.1 −0.4 2.1 −1.6 , B1 = , B2 = 1.0 0.3 1.0 0.7 2.5 ] [ ] [ ] 10.6 25.6 21.4 37.4 30.2 , C1 = , C2 = 1.6 24.4 7.4 1.2 15.1

2.1 0.8 −1.0 1.3

3.2 1.3 0.9 3.1

6

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Set the corresponding parameters as follows: θ0 = 0.22, α0 = 1, β0 = 1 θ1 = 0.44, α1 = 1, β1 = 0 θ2 = 0.44, α2 = 0, β2 = 1 ε = 0.0000001 By applying the iterative algorithm given in Algorithm 1 with X0 (0) = X1 (0) = X2 (0) = 10−6 1(2), we can compute the sequences X0 (k) , X1 (k) and X2 (k) and finally obtain the convergent solution as [ ] 2.2792084 2.1443643 ∗ X0 = −0.0053164232 2.8578095 [ ] 3.8959794 3.0173837 ∗ X1 = 0.91457065 4.3687527 [ ] 3.7974423 2.1832934 ∗ X2 = 1.8076325 3.3274102 In order to demonstrate the convergent effectiveness,we define the relative iteration error as v u ∑T −1 2 u ∥Xt (k) − Xt∗ ∥ δ(k) = t t=0 . ∑T −1 ∗ 2 t=0 ∥Xt ∥ The varying trajectory of relative iteration error with the time is shown in 1. It is cleared that δ(k) decreases quickly and converges to zero as k increases.

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0

5

10

15

20

25

30

Figure 1: The changed trend of relative error

4

Conclusions

In this paper, we introduce a new iterative algorithm to solve a kind of periodic Sylvester matrix equation. The iterative algorithm is proven to converge the exact solutions in finite iteration steps without round-off errors. Finally, we give a numerical example to check the convergence and performance of the iterative algorithm.

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References [1] Caswell H, Shyu E. Sensitivity analysis of periodic matrix population models.[J]. Theoretical Population Biology, 2012, 82(4):329. [2] Chen T, Francis B A. Optimal Sampled-Data Control Systems[M]. Springer-Verlag New York, Inc. 1995. [3] Chen X. Solving the (generalized) periodic Sylvester equation with the matrix sign function[J]. Mathematica Numerica Sinica, 2012, 34(2):153-162. [4] Dehghan M, Hajarian M. An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation[J]. Applied Mathematics & Computation, 2008, 202(2):571-588. [5] Dehghan M, Hajarian M. Two algorithms for finding the Hermitian reflexive and skew-Hermitian solutions of Sylvester matrix equations[J]. Applied Mathematics Letters, 2011, 24(4):444-449. [6] Gu C, Jiang X. Improved gradient iteration algorithm for solving Sylvester matrix equations[J]. Communication on Applied Mathematics and Computation, 2014, 28(4): 432-439. [7] Hoskins W D, Meek D S, Walton D J. The numerical solution of the matrix equation XA + AY = F [J]. Bit Numerical Mathematics, 1977, 17(2):184-190. [8] Korotyaev E, Kutsenko A. Lyapunov functions for periodic matrix-valued Jacobi operators[J]. Mathematics, 2007. [9] Kucera V. The matrix equation AX + XB = C[J]. Siam Journal on Applied Mathematics, 1974, 26(1):15-25. [10] Lv L, Zhang L. On the periodic Sylvester equations and their applications in periodic Luenberger observers design[J]. Journal of the Franklin Institute, 2016, 353(5): 1005-1018. [11] Lv L L, Zhang L. Parametric solutions to the discrete periodic regul ator equations[J]. Journal of the Franklin Institute, 2016, 353(5): 1089-1101. [12] Sreedhar J, Van Dooren P. Periodic Schur forms and some matrix equations[C]// 1994. [13] Verstraete F, Porras D, Cirac J I. Density matrix renormalization group and periodic boundary conditions: a quantum information perspective.[J]. Physical Review Letters, 2004, 93(22):227205. [14] Xie L,Ding J,Ding F.Gradient based iterative solutions for general linear matrix equations. Computers Mathe-matics with Applications. 2009 [15] Yin F, Guo K, Huang G X. An iterative algorithm for the generalized reflexive solutions of the general coupled matrix equations[J]. Journal of Inequalities and Applications, 2013, 2013(1):952974-952974. [16] Zhang H. A finite iterative algorithm for solving the complex generalized coupled Sylvester matrix equations by using the linear operators[J]. Journal of the Franklin Institute, 2016, http://dx.doi.org/10.1016/j.jfranklin.2016.12.011. [17] Zhang L, Science F O. A Gradient Iterative Algorithm for Solving the Coupled Sylvester Matrix Equations[J]. Value Engineering, 2014. [18] Zhou B. On asymptotic stability of linear time-varying systems[J]. Automatica, 2016, 68: 266-276. [19] Zhou B, Egorov A V. Razumikhin and Krasovskii stability theorems for time-varying time-delay systems[J]. Automatica, 2016, 71: 281-291.

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FOURIER SERIES OF SUMS OF PRODUCTS OF EULER AND GENOCCHI FUNCTIONS AND THEIR APPLICATIONS TAEKYUN KIM1 , DAE SAN KIM2 , DMITRY V. DOLGY3 , AND JIN-WOO PARK4,∗

Abstract. We study three types of sums of products of Euler and Genocchi functions and derive Fourier series expansions for them. Further, we will be able to express each of those functions in terms of Bernoulli functions.

1. Introduction The Genocchi polynomials Gm (x) are given by the generating function ∞ X tm 2t xt e = Gm (x) , (see [1-5, 7, 11]). t e +1 m! m=0

The first few Genocchi polynomials are as follows: G0 (x) = 0, G1 (x) = 1, G2 (x) = 2x − 1, G3 (x) = 3x2 − 3x, G4 (x) = 4x3 − 6x2 + 1, G5 (x) = 5x4 − 10x3 + 5x, G6 (x) = 6x5 − 15x4 + 15x2 − 3, G7 (x) = 7x6 − 21x5 + 35x3 − 21x. The Euler polynomials Em (x) are defined by the generating function ∞ X 2 tm xt e = Em (x) , (see [2-4, 6-8, 10]). t e +1 m! m=0

When x = 0, Em (0) = Em are called the Euler numbers. From the relation Gm (x) = mEm−1 (x) (m ≥ 1), we have deg Gm (x) = m − 1 (m ≥ 1), Gm = mEm−1 (m ≥ 1), G0 = 0, G1 = 1, G2m+1 = 0 (m ≥ 1), and G2m 6= 0 (m ≥ 1). Moreover, d Gm (x) = mGm−1 (x), (m ≥ 1), dx Gm (x + 1) + Gm (x) = 2mxm−1 , (m ≥ 0). From these, we have Gm (1) + Gm (0) = 2δm,1 , (m ≥ 0), 2010 Mathematics Subject Classification. 11B68, 11B83, 42A16. Key words and phrases. Fourier series, Euler polynomial, Euler function, Genocchi polynomial, Genocchi function. ∗ Corresponding author. 1

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Fourier series of sums of products of Euler and Genocchi functions

Z

1

1 (Gm+1 (1) − Gm+1 (0)) m+1 2 = (−Gm+1 (0) + δm,0 ) m + 1 0, for m even,  = 2 − Gm+1 , for m odd. m+1

Gm (x)dx = 0

For any real number x, let < x >= x − bxc ∈ [0, 1) denote the fractional part of x. P∞ m t Let Bm (x) denote the Bernoulli polynomials given by et −1 etx = m=0 Bm (x) tm! . Then we recall the following about Bernoulli functions Bm (< x >): (a) for m ≥ 2, Bm (< x >) = −m!

∞ X

e2πinx , (2πin)m n=−∞ n6=0

(b) for m = 1, ∞ X e2πinx − = 2πin n=−∞

(

B1 (< x >),

for x ∈ / Z,

0,

for x ∈ Z.

n6=0

Fourier series expansion of higher-order Bernoulli functions were treated in the recent paper [9]. Here we will study three types of sums of products of Euler and Genocchi functions and derive Fourier series expansions for them. Further, we will be able to express each of those functions in terms of Bernoulli functions. 2. Sums of products of Euler and Genocchi functions of the first type Let αm (x) =

m−1 X

Ek (x)Gm−k (x), (m ≥ 2).

k=0

Note that deg αm (x) = m − 1. Then we will consider the function αm (< x >) =

m−1 X

Ek (< x >)Gm−k (< x >), (m ≥ 2)

k=0

defined on (−∞, ∞), which is periodic with period 1. The Fourier series of αm (< x >) is ∞ X

2πinx A(m) , n e

n=−∞

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where A(m) = n

Z

1

αm (< x >)e−2πinx dx

0

Z =

1

αm (x)e−2πinx dx.

0

To proceed further, we need the following α0 (x) =

m−1 X

(kEk−1 (x)Gm−k (x) + (m − k)Ek (x)Gm−k−1 (x))

k=0

=

m−1 X

kEk−1 (x)Gm−k (x) +

k=1

=

m−2 X

(m − k)Ek (x)Gm−k−1 (x)

k=0

m−2 X

m−2 X

k=0

k=0

(k + 1)Ek (x)Gm−1−k (x) +

=(m + 1)

m−2 X

(m − k)Ek (x)Gm−1−k (x)

Ek (x)Gm−1−k (x)

k=0

=(m + 1)αm−1 (x). 0 So, αm (x) = (m + 1)αm−1 (x). From this, 0  αm+1 (x) =αm (x), m+2 Z 1 1 (αm+1 (1) − αm+1 (0)) , αm (x)dx = m +2 0

and αm (1) − αm (0) =

m−1 X

(Ek (1)Gm−k (1) − Ek Gm−k )

k=0

=

m−1 X

((−Ek + 2δ0,k ) (−Gm−k + 2δm−1,k ) − Ek Gm−k )

k=0

=

m−1 X

(−2Ek δm−1,k − 2δ0,k Gm−k + 4δk,0 δm−1,k )

k=0

= − 2Em−1 − 2Gm + 4δm−1,0 = − 2 (Em−1 + Gm ) = − 2(m + 1)Em−1 . Recall that E2n = 0 (n ≥ 1), E2n−1 6= 0 (n ≥ 1), and E0 = 1. So αm (0) = α(1) ⇔ m = 2n + 1 (n ≥ 1).

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Fourier series of sums of products of Euler and Genocchi functions

Also,

Z

1

1 (−2(m + 2)Em ) m+2 = − 2Em .

αm (x)dx = 0

(m)

Now, we are going to determine the Fourier coefficients An . Case 1: n 6= 0.

A(m) = n

Z

1

αm (x)e−2πinx dx

0

=−

1 1 1  αm (x)e−2πinx 0 + 2πin 2πin

Z

1 0 αm (x)e−2πinx dx

0

Z 1 m+1 1 =− (αm (1) − αm (0)) + αm−1 (x)e−2πinx dx 2πin 2πin 0 m + 1 (m−1) 2(m + 1) A + Em−1 = 2πin n 2πin  m+1 m (m−2) 2m 2(m + 1) = A + Em−2 + Em−1 2πin 2πin n 2πin 2πin 2

(m + 1)2 (m−2) X 2(m + 1)k A + Em−k = (2πin)2 n (2πin)k k=1

=··· =

m−1 (m + 1)m−1 (1) X 2(m + 1)k A + Em−k (2πin)m−1 n (2πin)k k=1

=

m−1 X k=1

(1)

where An =

R1 0

2(m + 1)k Em−k , (2πin)k

e−2πinx dx = 0.

Case 2: n = 0. (m)

A0

Z

1

αm (x)dx = −2Em .

= 0

αm (< x >), (m ≥ 2) is piecewise C ∞ . Moreover, αm (< x >) is continuous for all odd integers ≥ 3 and is discontinuous with jump discontinuities at integers for all even integers ≥ 2. Assume the first that m is an odd integer ≥ 3 . Then αm (0) = αm (1). αm (< x >) is piecewise C ∞ , and continuous. So the Fourier series of αm (< x >) converges

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uniformly to αm (< x >), and ∞ X

m−1 X

n=−∞ n6=0

k=1

αm (< x >) = − 2Em +

m−1 X

= − 2Em − 2

k=1 m−1 X

= − 2Em − 2

k=2

×

2(m + 1)k Em−k (2πin)k

! e2πinx

   ∞ 2πinx X m+1 e   Em−k −k! k k (2πin) n=−∞ n6=0

 m+1 Em−k Bk (< x >) − 2(m + 1)Em−1 k

( B1 (< x >),

for x ∈ / Z, for x ∈ Z.

0,

Now, we can state our first theorem. Theorem 2.1. Let m be an odd integer ≥ 3. Then we have the following. Pm−1 (i) k=0 Ek (< x >)Gm−k (< x >) has the Fourier series expansion m−1 X

Ek (< x >)Gm−k (< x >)

k=0

= − 2Em +

∞ X

m−1 X

n=−∞ n6=0

k=1

2(m + 1)k Em−k (2πin)k

! e2πinx ,

for all x ∈ (−∞, ∞). Here the convergence is uniform. (ii) m−1 X

Ek (< x >)Gm−k (< x >)

k=0

= − 2Em − 2

m−1 X k=2

 m+1 Em−k Bk (< x >). k

Here Bk (< x >) is the Bernoulli function. Assume next that m is an even integer ≥ 2. Then αm (0) 6= αm (1). Hence αm (< x >) is piecewise C ∞ , and discontinuous with jump discontinuities at integers. The Fourier series of αm (< x >) converges pointwise to αm (< x >), for x ∈ / Z, and converges to 1 (αm (0) + αm (1)) 2 =αm (0) − (m + 1)Em−1 =

m−1 X

Ek Gm−k − (m + 1)Em−1 ,

k=0

for x ∈ Z. Hence we have the following theorem. Theorem 2.2. Let m be an even integer ≥ 2. Then we have the following.

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Fourier series of sums of products of Euler and Genocchi functions

(i) − 2Em +

=

∞ X

m−1 X

n=−∞ n6=0

k=1

2(m + 1)k Em−k (2πin)k

! e2πinx

 Pm−1 for x ∈ / Z, k=0 Ek (< x >)Gm−k (< x >), Pm−1 E G − (m + 1)E , for x ∈ Z. k m−k m−1 k=0

Here the convergence is pointwise. (ii) − 2Em − 2

m−1 X k=1

=

m−1 X

 m+1 Em−k Bk (< x >) k

Ek (< x >)Gm−k (< x >), for x ∈ / Z;

k=0

− 2Em − 2

m−1 X k=2

=

m−1 X

 m+1 Em−k Bk (< x >) k

Ek Gm−k − (m + 1)Em−1 , for x ∈ Z.

k=0

Here Bk (< x >) is the Bernoulli function. 3. Sums of products of Euler and Genocchi functions of the second type Let βm (x) =

m−1 X k=0

1 Ek (x)Gm−k (x), (m ≥ 2). k!(m − k)!

Then we will consider the function βm (< x >) =

m−1 X k=0

1 Ek (< x >)Gm−k (< x >), (m ≥ 2) k!(m − k)!

defined on (−∞, ∞), which is periodic with period 1. The Fourier series of βm (< x >) is ∞ X

Bn(m) e2πinx ,

n=−∞

where Bn(m)

Z

1

βm (< x >)e−2πinx dx

= 0

Z =

1

βm (x)e−2πinx dx.

0

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To proceed further, we need to observe the following.  m−1 X k m−k 0 βm (x) = Ek−1 (x)Gm−k (x) + Ek (x)Gm−k−1 (x) k!(m − k)! k!(m − k)! k=0

=

m−1 X k=1

=

m−2 X k=0

=2

m−2 X 1 1 Ek−1 (x)Gm−k (x) + Ek (x)Gm−k−1 (x) (k − 1)!(m − k)! k!(m − k − 1)! k=0

1 Ek (x)Gm−1−k (x) + k!(m − 1 − k)!

m−2 X k=0

m−2 X k=0

1 Ek (x)Gm−1−k (x) k!(m − 1 − k)!

1 Ek (x)Gm−1−k (x) k!(m − 1 − k)!

=2βm−1 (x). 0 So βm (x) = 2βm−1 (x), and from this we obtain  0 βm+1 (x) = βm (x). 2

Thus

1

Z

βm (x)dx = 0

1 (βm+1 (1) − βm+1 (0)) , 2

and βm (1) − βm (0) =

m−1 X k=0

=

m−1 X k=0

=

m−1 X k=0

1 (Ek (1)Gm−k (1) − Ek Gm−k ) k!(m − k)! 1 ((−Ek + 2δk,0 )(−Gm−k + 2δm−1,k ) − Ek Gm−k ) k!(m − k)! 1 (−2Ek δm−1,k − 2δk,0 Gm−k + 4δk,0 δm−1,k ) k!(m − k)!

2Em−1 2Gm 4δm−1,0 − + (m − 1)! m! m! 2Em−1 2mEm−1 =− − (m − 1)! m! 4 =− Em−1 . (m − 1)! =−

So, βm (0) = βm (1) ⇐⇒ Em−1 = 0 ⇐⇒ m = 2n + 1 (n ≥ 1). Also, Z 0

1

1 βm (x)dx = (βm+1 (1) − βm+1 (0)) 2 2 =− Em . m!

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Fourier series of sums of products of Euler and Genocchi functions (m)

Now, we are ready to determine the Fourier coefficients Bn . Case 1 : n 6= 0. Z 1 βm (x)e−2πinx dx Bn(m) = 0

=−

1 1  1 βm (x)e−2πinx 0 + 2πin 2πin

Z

1 0 βm (x)e−2πinx dx

0

Z 1 1 2 βm−1 (x)e−2πinx dx (βm (1) − βm (0)) + 2πin 2πin 0 2 4 = Bn(m−1) + Em−1 2πin 2πin(m − 1)!   2 2 4 4 (m−2) = B + Em−2 + Em−1 2πin 2πin n 2πin(m − 2)! 2πin(m − 1)!  2 2 X 2 2k+1 (m−2) Bn = + Em−k 2πin (2πin)k (m − k)!

=−

k=1

=··· m−1  m−1 X 2k+1 2 Em−k Bn(1) + = 2πin (2πin)k (m − k)! k=1

=

m−1 X k=1

k+1

2 Em−k , (2πin)k (m − k)!

R1 = 0 e−2πinx dx = 0. where Case 2 : n = 0. Z (m) B0 = (1) Bn

1

2 Em . m! 0 βm (< x >), (m ≥ 2) is piecewise C ∞ . Moreover, βm (< x >) is continuous for all odd integers ≥ 3 and discontinuous with jump discontinuities at integers for all even integers ≥ 2. Assume first that m is an odd integer ≥ 3. Then βm (0) = βm (1). βm (< x >) is piecewise C ∞ , and continuous. So the Fourier series of βm (< x >) converges uniformly to βm (< x >), and ! ∞ m−1 X X 2 2k βm (< x >) = − Em + 2 E e2πinx k (m − k)! m−k m! (2πin) n=−∞ βm (x)dx = −

k=1

n6=0

=−

2 2 Em − m! m!

m−1 X k=1

    ∞ 2πinx X e m   2k Em−k −k! k k (2πin) n=−∞ n6=0

m−1 X

  2 2 k m =− Em − 2 Em−k Bk (< x >) m! m! k k=2  4 B1 (< x >), for x ∈ / Z, − Em−1 × 0, for x ∈ Z. (m − 1)! We can now state our first theorem.

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Theorem 3.1. Let m be an odd integer ≥ 3. Then we have the following. (i) m−1 X k=0

1 Ek (< x >)Gm−k (< x >) k!(m − k)!

has the Fourier series expansion m−1 X k=0

1 Ek (< x >)Gm−k (< x >) k!(m − k)!

∞ X 2 =− Em + 2 m! n=−∞ n6=0

m−1 X k=1

2k Em−k (2πin)k (m − k)!

! e2πinx

for all x ∈ (−∞, ∞). Here the convergence is uniform. (ii) m−1 X

1 Ek (< x >)Gm−k (< x >) k!(m − k)! k=0   m−1 2 2 X k m =− Em − 2 Em−k Bk (< x >) m! m! k k=2

Here Bk (< x >) is the Bernoulli function. Assume next that m is an even integer ≥ 2. Then βm (0) 6= βm (1). Hence βm (< x >) is piecewise C ∞ , and discontinuous with jump discontinuities at integers. The Fourier series of βm (< x >) converges pointwise to βm (< x >), for x ∈ / Z, and converges to 1 2 (βm (0) + βm (1)) = βm (0) − Em−1 2 (m − 1)! =

m−1 X k=0

2 1 Ek Gm−k − Em−1 , k!(m − k)! (m − 1)!

for x ∈ Z. Now, we can state our second theorem. Theorem 3.2. Let m be an even integer ≥ 2. Then we have the following. (i) ∞ X 2 − Em + 2 m! n=−∞ n6=0

( =

m−1 X k=1

2k Em−k k (2πin) (m − k)!

! e2πinx

Pm−1

1 Ek (x)Gm−k (x), for x ∈ / Z, Pm−1 k=0 1 k!(m−k)! 2 E G − E , for x ∈ Z. k=0 k!(m−k)! k m−k (m−1)! m−1

Here the convergence is pointwise.

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Fourier series of sums of products of Euler and Genocchi functions

(ii)   m−1 2 2 X k m − Em − 2 Em−k Bk (< x >) m! m! k k=1

=

m−1 X k=0

−

1 Ek (< x >)Gm−k (< x >), for x ∈ / Z; k!(m − k)!

  m−1 2 2 X k m Em − 2 Em−k Bk (< x >) m! m! k k=2

=

m−1 X k=0

1 2 Ek Gm−k − Em−1 , for x ∈ Z. k!(m − k)! (m − 1)!

Here Bk (< x >) is the Bernoulli function.

4. Sums of products of Euler and Genocchi functions of the third type Pm−1 1 Ek (x)Gm−k (x), (m ≥ 3). Then we will consider the Let γm (x) = k=1 k(m−k) function

γm (< x >) =

m−1 X k=1

1 Ek (< x >)Gm−k (< x >) k(m − k)

defined on (−∞, ∞), which is periodic with period 1. The Fourier series of γm (< x >) is ∞ X

Cn(m) e2πinx ,

n=−∞

where

Cn(m)

Z

1

γm (< x >)e−2πinx dx

= 0

Z =

1

γm (x)e−2πinx dx.

0

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To proceed further, we need to note the following. 0 γm (x) =

m−1 X k=1

=

m−1 X k=1

=

m−2 X k=0

=

=

1 {kEk−1 (x)Gm−k (x) + (m − k)Ek (x)Gm−k−1 (x)} k(m − k)

m−2 X 1 1 Ek−1 (x)Gm−k (x) + Ek (x)Gm−k−1 (x) m−k k k=1

m−2 X 1 1 Ek (x)Gm−1−k (x) + Ek (x)Gm−1−k (x) m−1−k k k=1

1 Gm−1 (x) + m−1

m−2 X k=1

m−2 X 1 1 Ek (x)Gm−1−k (x) + Ek (x)Gm−1−k (x) m−1−k k k=1

1 Gm−1 (x) + (m − 1) m−1

m−2 X k=1

1 Ek (x)Gm−1−k (x) k(m − 1 − k)

1 = Gm−1 (x) + (m − 1)γm−1 (x). m−1 0 (x) = So γm

1 m−1 Gm−1 (x)



1 m

+ (m − 1)γm−1 (x), and from this, we have

 γm+1 (x) −

0 1 Gm+1 (x) = γm (x). m(m + 1)

Since Gm+1 (1) + Gm+1 (0) = 2δm,0 ,  1 Z 1 1 1 γm (x)dx = γm+1 (x) − Gm+1 (x) m m(m + 1) 0 0   1 1 = γm+1 (1) − γm+1 (0) − (Gm+1 (1) − Gm+1 (0)) m m(m + 1)   1 2 = Gm+1 , γm+1 (1) − γm+1 (0) + m m(m + 1) and γm (1) − γm (0) =

m−1 X k=1

=

m−1 X k=1

=

m−1 X k=1

=−

1 (Ek (1)Gm−k (1) − Ek Gm−k ) k(m − k)

1 ((−Ek + 2δk,0 ) (−Gm−k + 2δm−1,k ) − Ek Gm−k ) k(m − k) 1 (−2Ek δm−1,k ) k(m − k)

2Em−1 . m−1

So γm (0) = γm (1) ⇐⇒ Em−1 = 0 ⇐⇒ m = 2n + 1, (n ≥ 1).

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Fourier series of sums of products of Euler and Genocchi functions

Also, Z 0

1

  1 2Em 2 γm (x)dx = − + Gm+1 m m m(m + 1)   1 2Em 2Em = − + m m m =0. (m)

Now, we are going to determine the Fourier coefficients Cn . Case 1 : n 6= 0. Z 1 (m) Cn = γm (x)e−2πinx dx 0

1 1 1  γm (x)e−2πinx 0 + =− 2πin 2πin

Z

1

0 γm (x)e−2πinx dx  Z  1 1 1 (γm (1) − γm (0)) + Gm−1 (x) + (m − 1)γm−1 (x) e−2πinx dx =− 2πin 2πin 0 m−1 Z 1 1 1 (γm (1) − γm (0)) + Gm−1 (x)e−2πinx dx =− 2πin 2πin(m − 1) 0 Z m−1 1 γm−1 (x)e−2πinx dx + 2πin 0 2Em−1 2 m − 1 (m−1) = + Φm + C , 2πin(m − 1) 2πin(m − 1) 2πin n Pm−2 k−1 Gm−k where Φm = k=1 (m−1) , and, for l ≥ 2, (2πin)k ( P Z 1 l−1 Gl−k+1 , for n 6= 0, 2 k=1 (l)k−1 −2πinx (2πin)k Gl (x)e dx = 2Gl+1 − l+1 , for n = 0. 0 0 1

Continuing our argument, we have m − 1 (m−1) 2Em−1 2 Cn(m) = Cn + + Φm 2πin 2πin(m − 1) 2πin(m − 1)   m − 1 m − 2 (m−2) 2Em−2 2 = Cn + + Φm−1 2πin 2πin 2πin(m − 2) 2πin(m − 2) 2Em−1 2 + Φm + 2πin(m − 1) 2πin(m − 1) =

2 2 X (m − 1)2 (m−2) X 2(m − 1)j−1 2(m − 1)j−1 C + E + Φ m−j n j (m − j) j (m − j) m−j+1 (2πin)2 (2πin) (2πin) j=1 j=1

=··· =

m−2 m−2 X 2(m − 1)j−1 (m − 1)! (2) X 2(m − 1)j−1 C + E + Φm−j+1 m−j n m−2 j (2πin) (2πin) (m − j) (2πin)j (m − j) j=1 j=1

=−

m−2 m−2 X 2(m − 1)j−1 X 2(m − 1)j−1 (m − 1)! + E + Φm−j+1 , m−j m−1 j (2πin) (2πin) (m − j) (2πin)j (m − j) j=1 j=1

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13

R1 (2) 1 where Cn = 0 (x − 12 )e−2πinx dx = − 2πin . Here we note that m−2 X j=1

=

m−2 X j=1

=

=

=

m−j−1 X (m − j)k−1 Gm−j−k+1 2(m − 1)j−1 (2πin)j (m − j) (2πin)k k=1

m−2 X m−j−1 X j=1

=

2(m − 1)j−1 Φm−j+1 (2πin)j (m − j)

2 m 2 m 2 m

k=1

2(m − 1)j+k−2 Gm−j−k+1 (2πin)j+k (m − j)

m−2 X m−1 X j=1 m−1 X s=2 m−1 X s=1

(m)s−1 G s (m − j) m−s+1 (2πin) s=j+1 s−1 X (m)s−1 1 G m−s+1 (2πin)s m −j j=1

(m)s Gm−s+1 (Hm−1 − Hm−s ). (2πin)s m − s + 1

Thus Cn(m) = −

m−2 (m)s (m − 1)! 2 X + Em−s (2πin)m−1 m s=1 (2πin)s (m − s)

+

m−1 2 X (m)s Gm−s+1 (Hm−1 − Hm−s ) m s=1 (2πin)s m − s + 1

=−

m−2 (m − 1)! (m)s 2 X + Em−s m−1 (2πin) m s=1 (2πin)s (m − s)

m−2 2 X (m)s Gm−s+1 (m − 1)! (Hm−1 − Hm−s ) − (Hm−1 − 1) m s=1 (2πin)s m − s + 1 (2πin)m−1   m−2 (m − 1)! 2 X (m)s Em−s Gm−s+1 =− H + (H − H ) + . m−1 m−1 m−s (2πin)m−1 m s=1 (2πin)s m − s + 1 m−s

+

Case 2 : n = 0. (m) C0

Z =

1

γm (x)dx = 0. 0

γm (< x >), (m ≥ 2) is piecewise C ∞ . Moreover, γm (< x >) is continuous for all odd integers ≥ 3, and discontinuous with jump discontinuities at integers for all even integers ≥ 2. Assume first that m is an odd integer ≥ 3. Then γm (0) = γm (1). γm (< x >) is piecewise C ∞ , and continuous. So the Fourier series of γm (< x >) converges

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uniformly to γm (< x >), and ∞  X (m − 1)! γm (< x >) = H − m−1 m−1 (2πin) n=−∞ n6=0

 ) m−2 2 X (m)s Gm−s+1 Em−s + (Hm−1 − Hm−s ) + e2πinx m s=1 (2πin)s m − s + 1 m−s   ∞ X e2πinx   =Hm−1 −(m − 1)!  (2πin)m−1 n=−∞ n6=0

−

2 m

m−2 X s=1

    ∞ X e2πinx  Em−s  m Gm−s+1 (Hm−1 − Hm−s ) +  −s! s m−s+1 m−s (2πin)s n=−∞ n6=0

=Hm−1 Bm−1 (< x >) −

2 m

m−2 X s=2

2Em−1 × Bs (< x >) − × m−1



  m Gm−s+1 Em−s (Hm−1 − Hm−s ) + m−s+1 m−s s

B1 (< x >), for x ∈ / Z, 0, for x ∈ Z.

Now, we can state our first theorem. Theorem 4.1. Let m be an odd integer ≥ 3. Then we have the following. (i) m−1 X k=1

1 Ek (< x >)Gm−k (< x >) k(m − k)

has the Fourier series expansion m−1 X

1 Ek (< x >)Gm−k (< x >) k(m − k) k=1 ∞  X (m − 1)! = − H m−1 m−1 (2πin) n=−∞ n6=0

 ) m−2 Gm−s+1 Em−s 2 X (m)s (Hm−1 − Hm−s ) + e2πinx , + m s=1 (2πin)s m − s + 1 m−s for all x ∈ (−∞, ∞). Here the convergence is uniform. (ii) m−1 X

1 Ek (< x >)Gm−k (< x >) k(m − k) k=1  m−2    2 X m Gm−s+1 Em−s =Hm−1 Bm−1 (< x >) − (Hm−1 − Hm−s ) + Bs (< x >). m s=2 s m−s+1 m−s Here Bs (< x >) is the Bernoulli function.

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15

Assume next that m is an even integer ≥ 4. Then γm (0) 6= γm (1). Hence γm (< x >) is piecewise C ∞ and discontinuous with jump discontinuities at integers. The Fourier series of γm (< x >) converges pointwise to γm (< x >), for x ∈ / Z, and converges to 1 Em−1 (γm (0) + γm (1)) = γm (0) − 2 m−1 m−2 X 1 = Ek Gm−k , k(m − k) k=1

for x ∈ Z. Now, we can state our second theorem. Theorem 4.2. Let m be an even integer ≥ 4. Then we have the following. (i) ∞  X (m − 1)! H − m−1 m−1 (2πin) n=−∞ n6=0

 ) m−2 2 X (m)s Em−s Gm−s+1 + (Hm−1 − Hm−s ) + e2πinx m s=1 (2πin)s m − s + 1 m−s ( Pm−1 1 / Z, k=1 k(m−k) Ek (< x >)Gm−k (< x >), for x ∈ Pm−2 = 1 E G , for x ∈ Z. k=1 k(m−k) k m−k Here the convergence is pointwise. (ii) −

=

 m−1    Gm−s+1 2 X m Em−s (Hm−1 − Hm−s ) + Bs (< x >) m s=1 s m−s+1 m−s

m−1 X k=1

−

=

1 Ek (< x >)Gm−k (< x >), for x ∈ / Z; k(m − k)

 m−1    2 X m Em−s Gm−s+1 (Hm−1 − Hm−s ) + Bs (< x >) m s=2 s m−s+1 m−s

m−2 X k=1

1 Ek Gm−k , for x ∈ Z. k(m − k) References

[1] T. Agoh, Convolution identities for Bernoulli and Genocchi polynomials, Electron. J. Combin., 21 (2014), no. 1, Paper 1.65, 14 pp. [2] A. Bayad, Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Math. Comp., 80 (2011), no. 276, 2219-2221. [3] A. Bayad and T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20 (2010), no. 2, 247-253. [4] Y. He and T. Kim, General convolution identities for Apostol-Bernoulli Euler and Genocchi polynomials, J. Nonlinear Sci. Appl., 9 (2016), no. 6, 4780-4797. [5] D. S. Kim and T. Kim, Some identities involving Genocchi polynomials and numbers, Ars Combin., 121 (2015), 403-412. [6] T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal. , 2008 (2008), 2008, Art. ID 581582, 11 pp.

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Fourier series of sums of products of Euler and Genocchi functions

[7] T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys. 15 (2008), no. 4, 481486. [8] T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. 17 (2008), no. 2, 131-136. [9] T. Kim, D. S. Kim, S.-H. Rim and D.-V. Dolgy, Fourier series of higher-order Bernoulli functions and their applications, J. Inequal. Appl. 2017 (2017), 2017:8. [10] Q. M. Luo, Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order, Adv. Stud. Contemp. Math. (Kyungsang), 10 (2005), no. 1, 63-70. [11] H. Ma¨iga, Identities and congruences for Genocchi numbers, Advances in non-Archimedean analysis, Contemp. Math., 551, Amer. Math., Providence, RI, (2011), pp. 207-220. 1

Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected] 2

Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea. E-mail address: [email protected] 1

Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea. E-mail address: [email protected] 2 Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbukdo, 712-714, Republic of Korea. E-mail address: [email protected]

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MAJORIZATION PROPERTIES FOR CERTAIN FAMILIES OF ANALYTIC FUNCTIONS IN THE UNIT DISK ADEL A. ATTIYA, M. F. YASSEN, AND MAHER I. ABDELHAFIZ Abstract. The main object of this paper is to introduce the majorization properties for certain families of analytic functions associated with generalized Srivastava-Attiya operator in the unit disk. Also, some applications of our results are discussed which give a number of new results.

1. Introduction Let A(p) denote the class of functions f (z) of the from f (z) = z p +

(1.1)

1 X

ak+p z k+p ;

k=1

which are analytic in the open unit disk U = fz 2 C : jzj < 1g: Also, let A = A(1). De…nition 1.1. Let f and F be analytic functions in U; f is said to be majorized by F in U (see [15], [18]), written f F , z 2U, if there exists a function ', analytic in U such that (1.2)

j'(z)j

1 and

f (z) = '(z)F (z)

(z 2 U):

Noting that the concept of majorization is closely related to the concept of quasi-subordination between analytic functions (see [18]). De…nition 1.2. Let f and F be analytic functions. The function f is said to be subordinate to F , written f F; if there exists a function w analytic in U with w(0) = 0 and jw(z)j < 1; and such that f (z) = g(w(z)); in particular, if F is univalent, then f F if and only if f (0) = F (0) and f (U) F (U) : 2010 Mathematics Subject Classi…cation. 30C45. Key words and phrases. Analytic functions, Hurwitz-Lerch Zeta function, Majorization , Subordination relation, Srivastava-Attiya operator . 1

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2

ADEL A. ATTIYA, M. F. YASSEN, AND MAHER I. ABDELHAFIZ

A general Hurwitz-Lerch Zeta function (z; s; b) de…ned by (cf., e.g., [20, P. 121 et seq.])

(1.3)

(z; s; b) =

1 X k=0

(b 2 C n Z0 ; Z0 = Z [ f0g = f0; U; Re(s) > 1 when jzj = 1 ):

zk ; (k + b)s 1;

2; ::: g; s 2 C when z 2

Many authors studied and invistagated various properties of (z; s; b), see e.g. [2], [6], [5], [7], [8], [14], [10], [11], [19], [21], [22] and [17]. Now, let us de…ne, the operator Js;b (f ) which has been introduced by Srivastava and Attiya [19] (1.4)

Js;b (f ) (z) = Gs;b (z) f (z) z 2 U; f 2 A; b 2 CnZ0 ; s 2 C

where

Gs;b (z) = (1 + b)s

(1.5) and

(z; s; b)

b

s

denotes the Hadamard product .

Moreover, Attiya and Hakami [2] de…ned the function Gs;b;t by Gs;b;t = 1 + (t + b)s z (z; s; 1 + t + b)

(1.6)

we denote by

z 2 U; b 2 CnZ0 ; s 2 C; t 2 R ; t Js;b (f ) : A(p) ! A(p);

t Attiya and Hakami [2] de…ned the operator Js;b (f )(z) by:

(1.7)

t Js;b (f )(z) = z p Gs;b;t f (z)

z 2 U; f 2 A(p); b 2 CnZ0 ; s 2 C; t 2 R ;

where denotes the Hadamard product Noting that

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MAJORIZATION PROPERTIES FOR CERTAIN FAMILIES...

(1.8)

t (f )(z) = z p + Js;b

we note that

1 X k=1

t+b k+t+b

3

s

ak+p z k+p (z 2 U)

1 Js;b (f ) = Js;bp (f ) ;

(1.9)

where Js;bp (f ) introduced by Liu [13] . t (f ) generalizes many operators e.g., Srivastava and The operator Js;b Attiya operator [19], Liu operator [13], Alexander operator [1], Libera operator [12], Bernardi operator [4] and Jung-Kim-Srivastava integral operator [9]. Now, we begin by the following lemma due to Attiya and Hakami [2]. Lemma 1.1. Let f (z) 2 A(p); then (1.10)

t z Js+1;b f (z)

0

t = (t + b)Js;b f (z)

(t + b

t p)Js+1;b f (z);

z 2 U; b 2 CnZ0 ; s 2 C; t 2 R

De…nition 1.3. A function f (z) 2 A(p) is said to be in the class n;t Ss;b;p (A; B; ) if it satis…es 0 1 (n+1) t z J (f )(z) s+1;b 1 + Az 1 (1.11) 1+ @ ; p + nA (n) t 1 + Bz Js+1;b (f )(z)

where n 2 N0 = f0; 1; :::g; 1 C; t 2 R and b 2 C n Z0 :

B

1)

ADEL A. ATTIYA et al 1169-1177

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4

ADEL A. ATTIYA, M. F. YASSEN, AND MAHER I. ABDELHAFIZ n;t (6) S n;t ;1;p (A; B; ) =I ;p (A; B; ) (

real ;

> 0)

2. Main Results To introduce our results, we need the following lemma which can be proved by using Lemma 1.1 and induction . Lemma 2.1. Let f (z) 2 A(p); then

(2.1)

t z Js+1;b

(n+1)

t (f )(z) = (t + b) Js;b

(n)

(f )(z)

(t + n + b

t p) Js+1;b

(n)

(f )(z)

n 2 N0 ; z 2 U; b 2 CnZ0 ; s 2 C; t 2 R

We begin by proving the following main result. n Theorem 2.1. Let the function g(z) 2 Ss;b (A; B; ), if (n)

(2.2)

Js+1;b (f )(z)

then (2.3)

t Js+1;b

(n)

(n)

Js+1;b (g)(z); t Js+1;b

(f )(z)

(n)

(z 2 U);

(g)(z) (jzj

r0 );

where f (z) 2 A(p) and r0 = r0 ( ; b; A; B) is the positive root and the smallest of the equation (2.4) r3 j (A B)+(t+b)Bj [jt + bj+2jBj]r2 [j (A B)+(t+b)Bj+2]r+jt + bj = 0; ( 1

B 0) and b = 1 in Theorem 2.1, we get the

Corollary 2.6. Let the function g(z) 2 I n (A; B; ), if (2.31) then (2.32)

I t ;p

(n)

j I t ;p

(n)

(n)

(f )(z)

I t ;p

(f )(z)j

j I t ;p

(g)(z);

(n)

(z 2 U;

(g)(z)j (jzj

> 0);

r0 );

where f (z) 2 A(p) and r0 = r0 ( ; A; B) is the smallest positive root of the equation r3 j (A B)+(t+1)Bj [j1 + tj+2jBj]r2 [j (A B)+(1+t)Bj+2]r+j1 + tj = 0; ( 1

B 0 for all 1 6 i 6 n, we have (

n ∑ i=1

)( wi ai

n ∑ wi i=1

ai

)

(a + b)2 6 4ab

(

n ∑

)2 wi

.

(1.1)

i=1

This inequality can be regarded as a reverse version of weighted arithmeticharmonic mean inequality. Applications of this inequality arise in convergence analysis for numerical methods and statistics. Various generalizations, variations, refinements and equivalences of this inequality in several settings have been investigated. Let us focus on an integral version of (1.1): ∗ Corresponding

author. Email: [email protected]

1

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Theorem 1.1 (see e.g. [4]). Let J be a real interval equipped with a probability measure µ. For any continuous function f : J → R such that Range(f ) ⊆ [a, b] for some a, b > 0, it holds that (∫ )2 ∫ (a + b)2 f 2 dµ 6 f dµ . (1.2) 4ab J J Over the years, Kantorovich type inequalities were obtained in the contexts of matrices and operators, see e.g. [5, 7, 12, 14] and references therein. A matrix analogue of the inequality (1.1) involving Hadamard product (entrywise product, denoted by ⊙) is given as follows. Theorem 1.2 ([13], Theorem 2.2). For each i = 1, 2, . . . , n, let Ai and Wi be positive definite matrices of the same size such that 0 < aI 6 Ai 6 bI. Then ( n ) n n n ∑ ∑ ∑ 1 1 1 1 a2 + b2 ∑ −1 2 2 2 2 Wi Ai Wi ⊙ Wi A i Wi 6 Wi ⊙ Wi . (1.3) 2ab i=1 i=1 i=1 i=1 Note that the constant bound (a2 + b2 )/(2ab) of the matrix case (1.3) is slightly different to that of scalar case (a2 + b2 )/(4ab) in (1.1) and (1.2). The inequality (1.3) can be viewed as a reverse of the Fiedler’s inequality A ◦ A−1 > I which holds for any positive definite matrix A (see [6]). Kantorovich type inequality in which the operator product is replaced by an operator mean was considered in [15, 17]. In this paper, we establish certain integral inequalities of Kantorovich type for continuous fields of positive operators on a Hilbert space. The inequalities (1.1) and (1.2) are generalized in many ways in terms of Bochner integrals of operator-valued functions defined on a locally compact Hausdorff space equipped with a finite Radon measure. Instead of the Hadamard product in Theorem 1.2, we consider the (Hilbert) tensor product and Kubo-Ando operator mean. Our results include discrete inequalities as special cases. This paper consists of four sections. Section 2 provides fundamental facts about continuous fields of operators and its integrability. Section 3 deals with Kantorovich type integral inequalities involving tensor products of continuous fields of operators. In Section 4, we recall Kubo-Ando theory of operator means and then derive Kantorovich type inequalities involving operator means.

2

Continuous field of operators and Bochner integrability

Throughout, let H be a complex Hilbert space. Denote by B(H) and B(H)+ the C∗ -algebra of all bounded linear operators on H and its positive cone, respectively. Let A and A+ be a unital C∗ -subalgebra of B(H) and its positive 2

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cone, respectively. Capital letters always denote operators on a Hilbert space. In particular, I denotes the identity operator, where the underlying space is clear from the context. The spectrum of an operator A is expressed as Sp(A). As usual, the operator norm of an operator A is denoted by ∥A∥. For selfadjoint elements A, B ∈ A, the expression A 6 B indicates that B − A is a positive element, while A > 0 means that A is positive and invertible. Let us denote the supremum norm of a real-valued function f defined on a set E by ∥f ∥∞,E . The symbol ∥·∥1 denotes the L1 -norm on a given set, which is clear from the context. The next lemma asserts the continuity of the map A 7→ f (A). Here, f (A) is the continuous functional calculus of f on Sp(A). Lemma 2.1. Let ∆ be a nonempty compact subset of C and let f : ∆ → C be a continuous function. Let A be the subset of A consisting of all operators whose spectra are contained in ∆. Then the map sending A ∈ A to f (A) ∈ A is continuous. Proof. Let ϵ > 0. Weierstrass’ approximation theorem guarantees the existence of a polynomial p such that ϵ . 3

∥f − p∥∞,∆
0.

Theorem 3.2 can be extended in the following way: Theorem 3.6. Let (At )t∈Ω be a filed in C(Ω; A+ , [a, b]). Let (Wt )t∈Ω be a field in C(Ω; A+ ) such that the function t 7→ ∥Wt ∥ is integrable on Ω. Let f be a continuous real-valued function defined on [a, b] ∪ [1/b, 1/a] such that (i) f (x)f (1/x) 6 1 for all x ∈ [a, b], (ii) f ([a, b]) ⊆ [a, b] or f ([a, b]) ⊆ [1/b, 1/a]. Then (∫ )⊗2 ∫ ∫ 1 1 1 1 a2 + b2 2 Wt2 f (At )Wt2 dµ(t) ⊗s Wt2 f (A−1 )W dµ(t) 6 W dµ(t) . t t t 2ab Ω Ω Ω (3.5) 1

1

2 Proof. Since Sp(At−1 ) ⊆ [1/b, 1/a] for each t, the function t 7→ Wt2 f (A−1 t )Wt is Bochner integrable by Proposition 2.3. The assumption also implies that

−1 f (A−1 t ) 6 f (At )

for each t ∈ Ω. The inequality (3.5) now follows from Theorem 3.2. Note that the constant (a2 + b2 )/(2ab) is not affected. Theorem 3.6 is reduced to Theorem 3.2 by setting f (x) = x or f (x) = 1/x. Corollary 3.7. Let 0 < a < b. Consider three continuous functions ϕ : Ω → [a, b], g : [a, b] → (0, ∞) and f : [a, b] ∪ [1/b, 1/a] → R. Suppose that (i) f (x)f (1/x) 6 1 for all x ∈ [a, b], (ii) f ([a, b]) ⊆ [a, b] or f ([a, b]) ⊆ [1/b, 1/a]. Then we have the bound ∥(f g) ◦ ϕ∥1 6

∥g ◦ ϕ∥21 a2 + b2 . 2ab ∥(f ◦ ϕ1 )(g ◦ ϕ)∥1

Proof. It is a special case of Theorem 3.6 when A = C.

4

Kantorovich type integral inequalities involving operator means

In this section, we establish integral analogues of Kantorovich inequality involving operator means. First of all, we recall some fundamental facts about operator means [11]; see also [9, Ch. 5]. 8

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Definition 4.1. A binary operation σ : B(H)+ × B(H)+ → B(H)+ is called a connection if the following conditions hold for all A, B, C, D ∈ B(H)+ : (i) (joint) monotonicity: A 6 C, B 6 D =⇒ A σ B 6 C σ D (ii) transformer inequality: C(A σ B)C 6 (CAC) σ (CBC) (iii) (joint) continuity from above: for any sequences (An ), (Bn ) in B(H)+ , if An ↓ A and Bn ↓ B, then An σ Bn ↓ A σ B. Here, Xn ↓ X indicates that (Xn ) is a decreasing sequence converging strongly to X. It follows that every connection σ satisfies the following properties: X(A σ B)X = (XAX) σ (XBX), (A + B) σ (C + D) > (A σ C) + (B σ D)

(4.1) (4.2)

for all A, B, C, D > 0 and X > 0. A mean is a connection σ with idempotent property A σ A = A for all A > 0. Recall also that a continuous function f : [0, ∞) → R is said to be operator monotone if the condition 0 6 A 6 B implies f (A) 6 f (B). Such f is said to be super-multiplicative if f (xy) > f (x)f (y) for all x, y > 0. Proposition 4.2 ([11]). There is a one-to-one correspondence between operator connections and operator monotone functions from [0, ∞) to itself such that f (A) = I σ A,

A ∈ B(H)+ .

(4.3)

Moreover, σ is an operator mean if and only if f (1) = 1. Such f in this proposition is called the representing function of σ. Every operator connection σ admits an integral representation (see e.g. [3]) ∫

1

AσB =

A !t B dν(t),

A, B ∈ B(H)+

0

for some finite Radon measure ν on the interval [0, 1]. Here, !t denotes the t-weighted harmonic mean. Hence if A, B ∈ A+ , then A σ B ∈ A+ since the integral is a limit of finite sums. Lemma 4.3 ([2]). For any operator connection σ and A, B ∈ B(H)+ , we have ∥A σ B∥ 6 ∥A∥ σ ∥B∥. Here, σ on the right hand side is the induced connection on [0, ∞) defined by (a σ b)I = aI σ bI for any a, b ∈ [0, ∞). Lemma 4.4. Let σ be an operator connection with associated super-multiplicative operator-monotone function. Then for all positive operators A, B, C, D, we have (A σ C) ⊗s (B σ D) 6 (A ⊗s B) σ (C ⊗s D).

(4.4)

9

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Proof. By a continuity argument using the monotonicity and the continuity from 1 1 above of a connection, we may assume that A, B > 0. Putting X = A− 2 CA− 2 − 12 − 21 and Y = B DB , we have from properties (4.1) and (4.3) that

1

1

(A σ C) ⊗ (B σ D) = (A ⊗ B) 2 [(I σ X) ⊗ (I σ Y )](A ⊗ B) 2 1

1

= (A ⊗ B) 2 [f (X) ⊗ f (Y )](A ⊗ B) 2 1

1

6 (A ⊗ B) 2 [f (X ⊗ Y )](A ⊗ B) 2 1

1

= (A ⊗ B) 2 [I σ (X ⊗ Y )](A ⊗ B) 2 = (A ⊗ B) σ (C ⊗ D).

Now, using property (4.2) yields

(A σ C) ⊗ (B σ D) + (B σ D) ⊗ (A σ C) 6 (A ⊗ B) σ (C ⊗ D) + (B ⊗ A) σ (D ⊗ C) 6 [(A ⊗ B) + (B ⊗ A)] σ [(C ⊗ D) + (D ⊗ C)] .

The following result can be regarded as a Kantorovich type integral inequality concerning an operator mean.

Theorem 4.5. Let (At )t∈Ω be a filed in C(Ω; A+ , [a, b]). Let (Wt )t∈Ω be a field in C(Ω; A+ ) such that the function t 7→ ∥Wt ∥ is integrable on Ω. Let σ be a mean associated with a super-multiplicative representing function. Then ∫

1

∫

1

Wt2 (At σ Bt )Wt2 dµ(t) ⊗s Ω

6

Ω 2

1

1

−1 2 Wt2 (A−1 t σ Bt )Wt dµ(t)

a + b2 2ab

(∫

)⊗2 Wt dµ(t)

(4.5) .

Ω

Proof. The upper semicontinuity of σ, and the continuity of the maps t 7→ At and t 7→ Bt together imply the measurability of the map t 7→ At σBt . Note that ∥At σBt ∥ 6 b by the monotonicity and the idempotency of σ, and the 1

1

norm estimation in Lemma 4.3. It follows that the map t 7→ Wt2 (At σ Bt )Wt2 is 1

1

−1 2 Bochner integrable by Lemma 2.2. Similarly, the map t 7→ Wt2 (A−1 t σ Bt )Wt

10

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is Bochner integrable. Now, we have ∫ ∫ 1 1 1 1 −1 2 Wt2 (A−1 Wt2 (At σBt )Wt2 dµ(t) ⊗s t σ Bt )Wt dµ(t) Ω Ω ∫ ( ∫ ( ) ) 1 1 1 1 1 1 1 1 −1 2 2 2 2 2 2 2 6 Wt At Wt σ Wt Bt Wt dµ(t) ⊗s Wt2 A−1 dµ(t) t Wt σ Wt Bt Wt Ω

Ω

(since σ satisfies the transformer inequality) [∫ ] ∫ 1 1 1 1 2 2 2 2 6 Wt At Wt dµ(t) σ Wt Bt Wt dµ(t) Ω Ω ] [∫ ∫ 1 1 1 1 −1 −1 2 2 2 2 Wt Bt Wt dµ(t) ⊗s Wt At Wt dµ(t) σ Ω Ω [∫ ] ∫ 1 1 1 1 2 dµ(t) 6 Wt 2 At Wt 2 dµ(t) ⊗s Wt 2 A−1 W t t Ω Ω [∫ ] ∫ 1 1 1 1 σ Wt 2 Bt Wt 2 dµ(t) ⊗s Wt 2 Bt−1 Wt 2 dµ(t) Ω

a2 + b2 6 2ab a2 + b2 2ab

=

(∫

)⊗2 Wt dµ(t)

(∫

Ω

)⊗2 Wt dµ(t)

Ω

a2 + b2 σ 2ab

(∫

(by (4.2))

(by Lemma 4.4)

)⊗2 Wt dµ(t)

(by Theorem 3.2)

Ω

.

Ω

Theorem 4.5 is reduced to Theorem 3.2 by putting At = Bt for all t ∈ Ω. Corollary 4.6. Let (At )t∈Ω and (Bt )t∈Ω be two fields in C(Ω; A+ , [a, b]). Let w : Ω → [0, ∞) be an integrable continuous function. Let σ be an operator mean associated with a super-multiplicative representing function. Then ∫

∫ w(t)(At σ Bt ) dµ(t) ⊗s

Ω

−1 w(t)(A−1 t σ Bt ) dµ(t) 6

Ω

a2 + b2 ∥w∥21 I. (4.6) 2ab

Proof. From Theorem 4.5, put Wt = w(t)I for all t ∈ Ω. Theorem 4.7. Let 0 < a 6 1 6 b. Let (At )t∈Ω be a field in C(Ω; A+ , [a, b]). Let (Wt )t∈Ω be a field in C(Ω; A+ ) such that the function t 7→ ∥Wt ∥ is integrable. For any super-multiplicative operator-monotone function f : [0, ∞) → [0, ∞) such that f (1) = 1, we have ∫

1 2

1 2

∫

1 2

Wt f (At )Wt dµ(t) ⊗s Ω

1

2 Wt f (A−1 t )Wt

Ω

a2 + b2 dµ(t) 6 2ab

(∫

)⊗2 Wt dµ(t) . Ω

(4.7) Proof. Proposition 4.2 guarantees the existence of an operator mean σ such that f (A) = I σ A for all A > 0. The inequality (4.7) now follows from Theorem 4.5 by considering I σAt instead of At σ Bt .

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Corollary 4.8. Let 0 < a 6 1 6 b and α ∈ [−1, 1]. Let (At )t∈Ω be a field in C(Ω, ; A+ , [a, b]), and let (Bt )t∈Ω be a field in C(Ω, ; A+ ) such that At Bt = Bt At for each t ∈ Ω. Then ∫

∫

A−α t Bt dµ(t) 6

Aα t Bt dµ(t) ⊗s Ω

Ω

a2 + b2 2ab

(∫

)⊗2 Bt dµ(t) .

(4.8)

Ω

Proof. It suffices to assume that α ∈ [0, 1]. The famous L¨owner-Heinz states that the function f (x) = xα is operator monotone (see e.g. [9, Ch.4]). Note that f is also super-multiplicative and f (1) = 1. The inequality 4.8 now follows by replacing Wt by Bt in Theorem 4.7. The case A = C in Corollary 4.8 reads as follows. Corollary 4.9. Let 0 < a 6 1 6 b and α ∈ [−1, 1]. Let ϕ : Ω → [a, b] and g : Ω → (0, ∞) be continuous functions. We have ∥gϕα ∥1 6

a2 + b2 ∥g∥21 . 2ab ∥gϕ−α ∥1

The next result is a generalization of Theorem 1.2 in the context of operators in which the constant bound is given by (a2 + b2 )/(2ab). Corollary 4.10. Let (At )t∈Ω be a field in C(Ω; A+ , [a, b]). If µ(Ω) = 1, then ∫ A2t dµ(t) ⊗s I 6 Ω

a2 + b2 2ab

(∫

)⊗2 At dµ(t)

.

(4.9)

Ω

Proof. From Corollary 4.8, put α = 1 and At = Bt for all t ∈ Ω. Remark 4.11. Discrete versions for all results in this paper can be obtained by putting Ω to be a finite space endowed with the counting measure. For example, a discrete version of Theorem 4.5 is as follows: For each i = 1, 2, . . . , n, let Ai and Bi be operators in A+ whose spectra are contained in [a, b], and let Wi ∈ A+ . Let σ be an operator mean associated with a super-multiplicative operator-monotone function. Then n ∑ i=1

1 2

1 2

Wi (Ai σ Bi )Wi ⊗s

n ∑

1 2

Wi (A−1 i

1

σ Bi−1 )Wi2

i=1

a2 + b2 6 2ab

(

n ∑

)⊗2 Wi

.

i=1

Acknowledgements The author would like to thank Thailand Research Fund for financial supports. 12

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References [1] C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis, SpringerVerlag, New York, 2006. [2] Y. M. Arlinskii, Theory of operator means, Ukrainian Math. J., 42(6), 723-730 (1990). [3] P. Chansangiam, W. Lewkeeratiyutkul, Operator connections and Borel measures on the unit interval, ScienceAsia, 41, 273-279 (2015). [4] S. S. Dragomir, A survey on Cauchy-Bunyakowsky-Schwarz type discrete inequalities, J. Inequal. Pure and Appl. Math., 4(3), Art. 63 (2003). [5] S. S. Dragomir, Operator Inequalities of the Jensen, Cˇebyˇsev and Gr¨ uss Type, Springer, New York, 2012. [6] M. Fiedler, Uber eine ungleichung fur positiv definite matrizen, Math. Nachrichten, 23, 197-199 (1961) [7] M. Fujii, H. Zuo, N. Cheng, Generalization on Kantorovich inequality, J. Math. Inequal., 7, 517-522 (2013). [8] F. Hansen, J. Peˇcari´c, I. Peri´c, Jensen’s operator inequality and its converses, Math Scand., 100, 61-73 (2007). [9] F. Hiai, D. Petz, Introduction to Matrix Analysis and Applications, Springer, New Delhi, 2014. [10] L. V. Kantorovich, Functional analysis and applied mathematics, Uspehi Mat. Nauk (N.S.), 3, 89-185 (1948). (in Russian) [11] F. Kubo, T. Ando, Means of positive linear operators Math. Ann., 246, 205-224 (1980). [12] S. Liu, H. Neudecker, Several matrix Kantorovich-type inequalities, J. Math. Anal. Appl., 197, 23-26 (1996). [13] J. S. Mathru, J. S. Aujla, Hadamard product versions of the Chebyshev and Kantorovich inequalities, J. Ineq. Pure Applied Math, 10, Article 51 (2009). [14] M. S. Moslehian, Recent developments of the operator Kantorovich inequality, Expositiones Mathematicae, 30, 376-388 (2012). [15] R. Nakamoto, M. Nakamura, Operator mean and Kantorovich inequality, Math. Japon., 44, 495-498 (1996). [16] G. K. Pedersen, Analysis Now, Springer-Verlag, New York, 1989. [17] T. Yamazaki, An extension of Kantorovich inequality to n-operators via the geometric mean by Ando–Li–Mathias, Linear Algebra Appl., 416, 688695 (2006).

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Characteristic fuzzy sets and conditional fuzzy subalgebras G. Muhiuddina,∗ and Shuaa Aldhafeerib a b

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Department of Mathematics, College of basic education, Public authority for applied education and training, Kuwait

Abstract.

The notion of characteristic fuzzy sets is introduced.

Using this notion,

conditions for a subset of BCK/BCI-algebra to be a subalgebra are discussed. The notion of conditional fuzzy subalgebras is introduced, and several properties are investigated. Given a subalgebra of BCK/BCI-algebras, conditions for the characteristic fuzzy set to be a conditional fuzzy subalgebra of several types are provided.

1. Introduction The notions of “membership” and “quasicoincidence” of fuzzy points and fuzzy sets were introduced by Pu and Liu in [13]. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, played a vital role to generate some different types of fuzzy subgroups, called (α, β)-fuzzy subgroups, introduced by Bhakat and Das [1]. In particular, (∈, ∈ ∨ q )fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. Recently, these notions are applied to several algebraic structures, for example, near rings (see [2]), hypernear-rings (see [3]), hemirings (see [4]), lattices (see [9]), pseudo-BL algebras (see [15]), and BL-algebras (see [16]) etc. In BCK/BCI-algebras, many research articles have been published on (α, β)-fuzzy subalgebras (see [6], [7], [8], [11], [12] and [14]) which is an important and useful generalization of the well-known concepts, called fuzzy subalgebras. In this paper, we define characteristic fuzzy sets, as a generalization of crisp characteristic function, and conditional fuzzy subalgebra. Using this notion, we discuss conditions * Corresponding author. 2010 Mathematics Subject Classification: 06F35, 03G25, 06D72. Keywords: BCK/BCI-algebras, characteristic fuzzy set, conditional fuzzy subalgebra. E-mail: [email protected] (G. Muhiuddin), [email protected] (Shuaa Aldhafeeri)

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2

for a subset of BCK/BCI-algebra to be a subalgebra. Given a subalgebra of BCK/BCIalgebras, we provide conditions for the characteristic fuzzy set to be a conditional (∈, q)fuzzy subalgebra, a conditional (q, ∈)-fuzzy subalgebra, a conditional (q, q)-fuzzy subalgebra, a conditional (∈, ∈ ∧ q )-fuzzy subalgebra, and a conditional (q, ∈ ∧ q )-fuzzy subalgebra. 2. Preliminaries By a BCI-algebra we mean an algebra (X, ∗, 0) of type (2, 0) satisfying the axioms: (a1) (a2) (a3) (a4)

((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (x ∗ (x ∗ y)) ∗ y = 0, x ∗ x = 0, x ∗ y = y ∗ x = 0 ⇒ x = y,

for all x, y, z ∈ X. We can define a partial ordering ≤ by x ≤ y if and only if x ∗ y = 0. If a BCI-algebra X satisfies the axiom (a5) 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra. A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. We refer the reader to the books [5] and [10] for further information regarding BCK/BCI-algebras. A fuzzy set µ in a set X of the form   t ∈ (0, 1] if y = x, µ(y) :=  0 if y 6= x, is said to be a fuzzy point with support x and value t and is denoted by xt . For a fuzzy point xt and a fuzzy set µ in a set X, Pu and Liu [13] introduced the symbol xt αµ, where α ∈ {∈, q , ∈ ∨ q , ∈ ∧ q }. To say that xt ∈ µ (resp. xt q µ), we mean µ(x) ≥ t (resp. µ(x) + t > 1), and in this case, xt is said to belong to (resp. be quasi-coincident with) a fuzzy set µ. To say that xt ∈ ∨ q µ (resp. xt ∈ ∧ q µ), we mean xt ∈ µ or xt q µ (resp. xt ∈ µ and xt q µ). To say that xt α µ, we mean xt αµ does not hold, where α ∈ {∈, q, ∈ ∨ q , ∈ ∧ q }. A fuzzy set µ in a BCK/BCI-algebra X is called a fuzzy subalgebra of X if it satisfies: (2.1)

µ(x ∗ y) ≥ min{µ(x), µ(y)}

for all x, y ∈ X.

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Characteristic fuzzy sets and conditional fuzzy subalgebras

3

A fuzzy set µ in X is said to be an (α, β)-fuzzy subalgebra of X, where α, β ∈ {∈, q , ∈ ∨ q , ∈ ∧ q } and α 6= ∈ ∧ q , if it satisfies the following condition: xt1 α µ, yt2 α µ ⇒ (x ∗ y)min{t1 ,t2 } β µ.

(2.2)

for all x, y ∈ X and t1 , t2 ∈ (0, 1]. Lemma 2.1 ([7]). A fuzzy set µ in X is an (∈, ∈ ∨ q )-fuzzy subalgebra of X if and only if it satisfies: (2.3)

(∀x, y ∈ X) (µ(x ∗ y) ≥ min{µ(x), µ(y), 0.5}) .

3. Characteristic fuzzy sets In what follows, let X denote a BCK/BCI-algebra and ε, δ ∈ [0, 1] with ε > δ unless otherwise specified. (ε,δ)

For a non-empty subset S of X, define a characteristic fuzzy set µS   ε if x ∈ S, (ε,δ) µS (x) :=  δ otherwise. (ε,δ)

In particular, the characteristic fuzzy set µS acteristic function χS of S in X.

in X as follows:

in X with ε = 1 and δ = 0 is the char-

Theorem 3.1. For any non-empty subset S of X, the following are equivalent: (1) S is a subalgebra of X. (ε,δ) (2) The characteristic fuzzy set µS is a fuzzy subalgebra of X. Proof. Assume that S is a subalgebra of X and let x, y ∈ X. If x, y ∈ S, then x ∗ y ∈ S and so n o (ε,δ) (ε,δ) (ε,δ) µS (x ∗ y) = ε = min µS (x), µS (y) . (ε,δ)

If x ∈ / S or y ∈ / S, then µS

(ε,δ)

(x) = δ or µS

(y) = δ. Hence n o (ε,δ) (ε,δ) (ε,δ) µS (x ∗ y) ≥ δ = min µS (x), µS (y) .

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4 (ε,δ)

Therefore µS

is a fuzzy subalgebra of X. (ε,δ)

(ε,δ)

Conversely, suppose that (2) is valid. Let x, y ∈ S.oThen µS (x) = ε and µS (y) = ε. n (ε,δ) (ε,δ) (ε,δ) It follows that µS (x ∗ y) ≥ min µS (x), µS (y) = ε. Thus x ∗ y ∈ S, and therefore S is a subalgebra of X.  (ε,δ)

Theorem 3.2. If S is a subalgebra of X, then the characteristic fuzzy set µS (∈, ∈ ∨ q )-fuzzy subalgebra of X.

is an

Proof. Assume that S is a subalgebra of X. For any x, y ∈ X, if x, y ∈ S, then x ∗ y ∈ S and so n o (ε,δ) (ε,δ) (ε,δ) µS (x ∗ y) = ε ≥ min µS (x), µS (y), 0.5 . (ε,δ)

If x ∈ / S or y ∈ / S, then µS

(ε,δ)

(x) = δ or µS

(y) = δ. Hence n o (ε,δ) (ε,δ) (ε,δ) µS (x ∗ y) ≥ δ ≥ min µS (x), µS (y), 0.5 . (ε,δ)

It follows from Lemma 2.1 that µS

is an (∈, ∈ ∨ q )-fuzzy subalgebra of X.



The converse of Theorem 3.2 is not true in general as seen in the following example.

Example 3.3. Let X = {0, a, b, c, d} be a BCK-algebra with the following Cayley table: ∗ 0 a b c d 0 0 0 0 0 0 a a 0 0 0 0 b b b 0 0 0 c c c b 0 0 d d d c b 0 (ε,δ)

For a subset S = {0, c, d} of X, consider a characteristic fuzzy set µS in X with ε = 0.7 (ε,δ) and δ = 0.5. Then µS is an (∈, ∈ ∨ q )-fuzzy subalgebra of X, but S is not a subalgebra of X since d ∗ c = b ∈ / S. (ε,δ)

Theorem 3.4. Assume that ε ≤ 0.5. If the characteristic fuzzy set µS fuzzy subalgebra of X then S is a subalgebra of X.

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Characteristic fuzzy sets and conditional fuzzy subalgebras (ε,δ)

5

(ε,δ)

Proof. Let x, y ∈ S. Then µS

(x) = ε = µS (y). Using Lemma 2.1, we have n o (ε,δ) (ε,δ) (ε,δ) µS (x ∗ y) ≥ µS (x), µS (y), 0.5 = {ε, 0.5} = ε,

and so x ∗ y ∈ S. Therefore S is a subalgebra of X.



Corollary 3.5. A non-empty subset S of X is a subalgebra of X if and only if the characteristic function χS of S is an (∈, ∈ ∨ q )-fuzzy subalgebra of X. Proof. Clearly, we can find the necessity by taking ε = 1 and δ = 0 in Theorem3.2. Conversely, suppose that the characteristic function χS of S is an (∈, ∈ ∨ q )-fuzzy subalgebra of X. Let x, y ∈ S. Then χS (x) = 1 = χS (y), which implies from (2.3) that χS (x ∗ y) ≥ min{χS (x), χS (y), 0.5} = min{1, 0.5} = 0.5. Hence x ∗ y ∈ S, and therefore S is a subalgebra of X. (ε,δ)

Theorem 3.6. For a subset S of X, let µS or ε + δ ≤ 1, then S is a subalgebra of X.



is an (∈, q)-fuzzy subalgebra of X. If δ ≤ 0.5 (ε,δ)

(ε,δ)

Proof. Let x, y ∈ S and assume that δ ≤ 0.5. Then µS (x) = ε > δ and µS (y) = ε > δ, (ε,δ) (ε,δ) (ε,δ) that is, xδ ∈ µS and yδ ∈ µS . Hence (x ∗ y)δ = (x ∗ y)min{δ,δ} q µS , which implies (ε,δ) (ε,δ) that µS (x ∗ y) + δ > 1. Since δ ≤ 0, 5, it follows that µS (x ∗ y) > 1 − δ ≥ δ. Thus (ε,δ) µS (x ∗ y) = ε and x ∗ y ∈ S. Therefore S is a subalgebra of X. (ε,δ)

(ε,δ)

(ε,δ)

Now, suppose that ε + δ ≤ 1. Then µS (x) = ε = µS (y), and so xε ∈ µS and (ε,δ) (ε,δ) (ε,δ) yε ∈ µS . Hence (x ∗ y)ε = (x ∗ y)min{ε,ε} q µS , which implies that µS (x ∗ y) + ε > 1. (ε,δ) (ε,δ) Therefore µS (x∗y) > 1−ε ≥ δ, and thus µS (x∗y) = ε, that is, x∗y ∈ S. Consequently, S is a subalgebra of X.  (ε,δ)

Theorem 3.7. Let ε > 0.5. If the characteristic fuzzy set µS of X, then S is a subalgebra of X. (ε,δ)

Proof. Let x, y ∈ S. Then µS (ε,δ)

µS (ε,δ)

(ε,δ)

(x) = ε = µS

(y), which implies that (ε,δ)

(x) + ε = ε + ε > 1 and µS (ε,δ)

is a (q, ∈)-fuzzy subalgebra

(y) + ε = ε + ε > 1,

(ε,δ)

that is, xε q µS and yε q µS . Since µS is a (q, ∈)-fuzzy subalgebra of X, it follows (ε,δ) (ε,δ) that (x ∗ y)ε = (x ∗ y)min{ε,ε} ∈ µS and so that µS (x ∗ y) = ε, that is, x ∗ y ∈ S. Therefore S is a subalgebra of X.  (ε,δ)

Theorem 3.8. Assume that ε > 0.5 and ε + δ ≤ 1. If the characteristic fuzzy set µS a (q, q)-fuzzy subalgebra of X, then S is a subalgebra of X.

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(ε,δ)

Proof. Let x, y ∈ S. Then µS (ε,δ)

µS (ε,δ)

(ε,δ)

(x) = ε = µS

(y), which implies that (ε,δ)

(x) + ε = ε + ε > 1 and µS (ε,δ)

(y) + ε = ε + ε > 1,

(ε,δ)

that is, xε q µS and yε q µS . Since µS is a (q, q)-fuzzy subalgebra of X, it follows that (ε,δ) (ε,δ) (ε,δ) (x∗y)ε = (x∗y)min{ε,ε} q µS . Hence µS (x∗y) > 1−ε ≥ δ, and therefore µS (x∗y) = ε. This proves that x ∗ y ∈ S, and S is a subalgebra of X.  (ε,δ)

Theorem 3.9. Assume that ε + δ ≤ 1. If the characteristic fuzzy set µS ∈ ∧ q )-fuzzy subalgebra of X, then S is a subalgebra of X.

is an (∈,

(ε,δ)

Proof. Assume that ε + δ ≤ 1 and the characteristic fuzzy set µS is an (∈, ∈ ∧ q )(ε,δ) (ε,δ) (ε,δ) fuzzy subalgebra of X. Let x, y ∈ S. Then µS (x) = ε = µS (y), and so xε ∈ µS and (ε,δ) (ε,δ) (ε,δ) yε ∈ µS . Hence (x∗y)ε = (x∗y)min{ε,ε} ∈ ∧ q µS , that is, (x∗y)ε = (x∗y)min{ε,ε} ∈ µS (ε,δ) (ε,δ) (ε,δ) and (x ∗ y)ε = (x ∗ y)min{ε,ε} q µS . Hence µS (x ∗ y) ≥ ε and µS (x ∗ y) + ε > 1. If (ε,δ) (ε,δ) (ε,δ) µS (x ∗ y) ≥ ε, then µS (x ∗ y) = ε and thus x ∗ y ∈ S. If µS (x ∗ y) + ε > 1, then (ε,δ) (ε,δ) µS (x ∗ y) > 1 − ε ≥ δ and so µS (x ∗ y) = ε, which shows that x ∗ y ∈ S. Therefore S is a subalgebra of X.  (ε,δ)

Theorem 3.10. Assume that ε > 0.5 and ε + δ ≤ 1. If the characteristic fuzzy set µS is a (q, ∈ ∧ q )-fuzzy subalgebra or a (q, ∈ ∨ q )-fuzzy subalgebra of X, then S is a subalgebra of X. (ε,δ)

Proof. Let x, y ∈ S. Then µS (ε,δ)

µS (ε,δ)

that is, xε q µS

(ε,δ)

(x) = ε = µS

(y), which implies that (ε,δ)

(x) + ε = ε + ε > 1 and µS (ε,δ)

and yε q µS

(ε,δ)

. If µS

(y) + ε = ε + ε > 1,

is a (q, ∈ ∧ q )-fuzzy subalgebra of X, then (ε,δ)

(x ∗ y)ε = (x ∗ y)min{ε,ε} ∈ ∧ q µS (ε,δ)

(ε,δ)

,

(ε,δ)

that is, µS (x ∗ y) ≥ ε and µS (x ∗ y) + ε > 1. If µS (x ∗ y) ≥ ε, then x ∗ y ∈ S. If (ε,δ) (ε,δ) (ε,δ) µS (x ∗ y) + ε > 1, then µS (x ∗ y) > 1 − ε ≥ δ and so µS (x ∗ y) = ε. Thus x ∗ y ∈ S, and therefore S is a subalgebra of X. (ε,δ)

(ε,δ)

If µS is a (q, ∈ ∨ q )-fuzzy subalgebra of X, then (x ∗ y)ε = (x ∗ y)min{ε,ε} ∈ ∨ q µS , (ε,δ) (ε,δ) (ε,δ) (ε,δ) and so that (x ∗ y)ε ∈ µS or (x ∗ y)ε q µS . If (x ∗ y)ε ∈ µS , then µS (x ∗ y) = ε (ε,δ) (ε,δ) and so x ∗ y ∈ S. If (x ∗ y)ε q µS , then µS (x ∗ y) + ε > 1. Since ε + δ ≤ 1, it follows (ε,δ) (ε,δ) that µS (x ∗ y) > 1 − ε ≥ δ and so that µS (x ∗ y) = ε. Thus x ∗ y ∈ S. Therefore S is a subalgebra of X. 

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Lemma 3.11. We have the following relations among the types of (∈, ∈ ∨ q ), (∈ ∨ q , ∈), (∈ ∨ q , q), (∈ ∨ q , ∈ ∧ q ), and (∈ ∨ q , ∈ ∨ q ): (∈, ∈ ∨ q ) KS

(∈ ∨ q , ∈ ∨ q )

(3.1)

3;

(∈ ∨ q , ∈) ks

KS

ck

(∈ ∨ q , ∈ ∧ q )

+3 (∈ ∨ q , q)

Combining Lemma 3.11 and Theorem 3.4, we have the following corollary. (ε,δ)

Corollary 3.12. Assume that ε ≤ 0.5. If the characteristic fuzzy set µS is any one of an (α, β)-fuzzy subalgebra of X with (α, β) ∈ {(∈ ∨ q , ∈), (∈ ∨ q , ∈ ∧ q ), (∈ ∨ q , ∈ ∨ q )}, then S is a subalgebra of X. (ε,δ)

Theorem 3.13. Assume that ε + δ ≤ 1. If the characteristic fuzzy set µS q)-fuzzy subalgebra of X, then S is a subalgebra of X.

is a (∈ ∨ q ,

Proof. If S is not a subalgebra of X, then there exists a, b ∈ S such that a ∗ b ∈ / S. Thus (ε,δ) (ε,δ) (ε,δ) (ε,δ) (ε,δ) µS (a) = ε = µS (b) and µS (a ∗ b) = δ. Hence aε ∈ µS and bε ∈ µS , which (ε,δ) (ε,δ) (ε,δ) imply that aε ∈ ∨ q µS and bε ∈ ∨ q µS . Since µS (a ∗ b) + ε = δ + ε ≤ 1, we have (ε,δ) (a ∗ b)ε q µS . This is a contradiction, and so S is a subalgebra of X.  4. Conditional (α, β)-fuzzy subalgebras We begin with a definition. Definition 4.1. Let R := {ρ ∈ (0, 1] | ρ has relations to ε and/or δ} . A characteristic (ε,δ) fuzzy set µS in X is called an R-conditional (α, β)-fuzzy subalgebra of X, where α, β ∈ {∈, q , ∈ ∨ q , ∈ ∧ q } and α 6= ∈ ∧ q , if it satisfies the following condition:   (ε,δ) (ε,δ) (ε,δ) (∀x, y ∈ X) (∀ρ1 , ρ2 ∈ R) xρ1 α µS , yρ2 α µS (4.1) . ⇒ (x ∗ y)min{ρ1 ,ρ2 } β µS Example 4.2. (1) Let X = {0, 1, 2, 3, 4} be a set with the following Cayley table: ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 1 2 2 2 0 2 0 3 3 1 3 0 3 4 4 4 2 4 0

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Then X is a BCK-algebra (see [10]). If we take R = {ρ ∈ (0, 1] | 0.3 < ρ ≤ 0.7}, (ε,δ) then µS with S := {0, 2, 4} is an R-conditional (∈, ∈ ∧ q )-fuzzy subalgebra of X where δ = 0.2 and ε = 0.7. (2) For a fixed element a of a BCI-algebra X, let S := {x ∈ X | a ∗ (a ∗ x) = x}. (ε,δ)

For δ = 0.3 and ε = 0.6, if we consider R1 = {ρ ∈ (0, 1] | ρ > 0.4} then µS is an R1 -conditional (∈, q)-fuzzy subalgebra of X. If we take R2 = {ρ ∈ (0, 1] | ρ ≤ 0.6} then (ε,δ) µS is an R2 -conditional (q, ∈)-fuzzy subalgebra of X. (3) Let X = {0, 1, 2, a, b} be a set with the following Cayley table: ∗ 0 1 2 a b 0 0 0 0 a a 1 1 0 1 a a 2 2 2 0 a a a a a a 0 0 b b a b 1 0 Then X is a BCI-algebra (see [5, 10]). Consider R = {ρ ∈ (0, 1] | 0.3 < ρ ≤ 0.9}, Then (ε,δ) µS with S := {0, 1, 2} is an R-conditional (q, q)-fuzzy subalgebra of X where δ = 0.1 and ε = 0.7. (4) Let X be a BCI-algebra and let S := {x ∗ a | x ∈ X} for a fixed element a ∈ X. (ε,δ) Consider R = {ρ ∈ (0, 1] | 0.3 < ρ ≤ 0.7}. Then µS is an R-conditional (q, ∈ ∧ q )-fuzzy subalgebra of X with δ = 0.1 and ε = 0.7. Theorem 4.3. Let R := {ρ ∈ (0, 1] | ρ > δ and ε + ρ > 1} . If S is a subalgebra of X, (ε,δ) then µS is an R-conditional (∈, q)-fuzzy subalgebra of X. (ε,δ)

(ε,δ)

(ε,δ)

Proof. Let x, y ∈ X and ρ1 , ρ2 ∈ R be such that xρ1 α µS and yρ2 α µS . Then µS (x) ≥ (ε,δ) ρ1 > δ and µS (y) ≥ ρ2 > δ, which imply that x, y ∈ S. Thus x ∗ y ∈ S, and so (ε,δ) µS (x ∗ y) = ε. Hence (ε,δ)

µS

(x ∗ y) + min{ρ1 , ρ2 } = ε + min{ρ1 , ρ2 } > 1, (ε,δ)

that is, (x ∗ y)min{ρ1 ,ρ2 } q µS of X.

(ε,δ)

. Therefore µS

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If we take ε = 1 and δ = 0 in Theorems 3.6 and 4.3, then we have the following corollary. Corollary 4.4. A non-empty subset S of X is a subalgebra of X if and only if the characteristic function χS of S is an (∈, q)-fuzzy subalgebra of X. Theorem 4.5. Let R := {ρ ∈ (0, 1] | ε ≥ ρ and δ ≤ 1 − ρ} . If S is a subalgebra of X, (ε,δ) then µS is an R-conditional (q, ∈)-fuzzy subalgebra of X. (ε,δ)

(ε,δ)

(ε,δ)

Proof. Let x, y ∈ X and ρ1 , ρ2 ∈ R be such that xρ1 q µS and yρ2 q µS . Then µS (x)+ (ε,δ) (ε,δ) (ε,δ) ρ1 > 1 and µS (y) + ρ2 > 1, which imply that µS (x) > 1 − ρ1 ≥ δ and µS (u) > (ε,δ) (ε,δ) 1 − ρ2 ≥ δ. Hence µS (x) = ε = µS (y), and so x, y ∈ S. Since S is a subalgebra of X, (ε,δ) (ε,δ) we have x ∗ y ∈ S. Thus µS (x ∗ y) = ε ≥ min{ρ1 , ρ2 }, and hence (x ∗ y)min{ρ1 ,ρ2 } ∈ µS . (ε,δ) Therefore µS is an R-conditional (q, ∈)-fuzzy subalgebra of X.  If we take ε = 1 and δ = 0 in Theorems 3.7 and 4.5, then we have the following corollary. Corollary 4.6. A non-empty subset S of X is a subalgebra of X if and only if the characteristic function χS of S is a (q, ∈)-fuzzy subalgebra of X. Theorem 4.7. Let R := {ρ ∈ (0, 1] | δ ≤ 1 − ρ < ε} . If S is a subalgebra of X, then the (ε,δ) characteristic fuzzy set µS is an R-conditional (q, q)-fuzzy subalgebra of X. (ε,δ)

(ε,δ)

(ε,δ)

Proof. Let x, y ∈ X and ρ1 , ρ2 ∈ R be such that xρ1 q µS and yρ2 q µS . Then µS (x)+ (ε,δ) (ε,δ) (ε,δ) ρ1 > 1 and µS (y) + ρ2 > 1, which imply that µS (x) > 1 − ρ1 ≥ δ and µS (y) > (ε,δ) (ε,δ) 1 − ρ2 ≥ δ. It follows that µS (x) = ε = µS (y) and so that x, y ∈ S. Since S is a (ε,δ) subalgebra of X, we have x ∗ y ∈ S and so µS (x ∗ y) = ε. Thus (ε,δ)

µS

(x ∗ y) + min{ρ1 , ρ2 } = ε + min{ρ1 , ρ2 } > 1, (ε,δ)

that is, (x ∗ y)min{ρ1 ,ρ2 } q µS

(ε,δ)

. This shows that µS

is a (q, q)-fuzzy subalgebra of X. 

If we take ε = 1 and δ = 0 in Theorems 3.8 and 4.7, then we have the following corollary. Corollary 4.8. A non-empty subset S of X is a subalgebra of X if and only if the characteristic function χS of S is a (q, q)-fuzzy subalgebra of X. Since the (q, ∈ ∨ q )-fuzzy subalgebra is induced by a (q, ∈)-fuzzy subalgebra or a (q, q)fuzzy subalgebra, we have the following corollary by using Theorems 4.5 and 4.7. Corollary 4.9. Let R be any one of {ρ ∈ (0, 1] | δ ≤ 1 − ρ < ε} and {ρ ∈ (0, 1] | ε ≥ ρ and δ ≤ 1 − ρ} . (ε,δ)

If S is a subalgebra of X, then the characteristic fuzzy set µS (q, ∈ ∨ q )-fuzzy subalgebra of X.

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Theorem 4.10. Let R := {ρ ∈ (0, 1] | δ < ρ ≤ ε and 1 − ρ < ε} . If S is a subalgebra of (ε,δ) X, then the characteristic fuzzy set µS is an R-conditional (∈, ∈ ∧ q )-fuzzy subalgebra of X. (ε,δ)

(ε,δ)

Proof. Let x, y ∈ X and ρ1 , ρ2 ∈ R be such that xρ1 ∈ µS and xρ2 ∈ µS . Then (ε,δ) (ε,δ) (ε,δ) (ε,δ) µS (x) ≥ ρ1 > δ and µS (y) ≥ ρ2 > δ, which imply that µS (x) = ε = µS (y). Hence (ε,δ) x, y ∈ S. Since S is a subalgebra of X, we have x ∗ y ∈ S. Hence µS (x ∗ y) = ε ≥ (ε,δ) min{ρ1 , ρ2 }, i.e., (x ∗ y)min{ρ1 ,ρ2 } ∈ µS . Now, (ε,δ)

µS

(x ∗ y) + min{ρ1 , ρ2 } = ε + min{ρ1 , ρ2 } > 1

(ε,δ)

(ε,δ)

and so (x ∗ y)min{ρ1 ,ρ2 } q µS . Therefore (x ∗ y)min{ρ1 ,ρ2 } ∈ ∧ q µS is an (∈, ∈ ∧ q )-fuzzy subalgebra of X.

(ε,δ)

, and consequently µS



If we take ε = 1 and δ = 0 in Theorems 3.9 and 4.10, then we have the following corollary. Corollary 4.11. A non-empty subset S of X is a subalgebra of X if and only if the characteristic function χS of S is an (∈, ∈ ∧ q )-fuzzy subalgebra of X. Theorem 4.12. Let R := {ρ ∈ (0, 1] | ε ≥ ρ and ε + ρ > 1 ≥ δ + ρ} . If S is a subalgebra (ε,δ) of X, then the characteristic fuzzy set µS is an R-conditional (q, ∈ ∧ q )-fuzzy subalgebra of X. (ε,δ)

(ε,δ)

(ε,δ)

Proof. Let x, y ∈ X and ρ1 , ρ2 ∈ R be such that xρ1 q µS and yρ2 q µS . Then µS (x)+ (ε,δ) (ε,δ) (ε,δ) ρ1 > 1 and µS (y) + ρ2 > 1, which imply that µS (x) > 1 − ρ1 ≥ δ and µS (y) > (ε,δ) (ε,δ) 1 − ρ2 ≥ δ. Hence µS (x) = ε = µS (y), and so x, y ∈ S. Since S is a subalgebra of X, we have x ∗ y ∈ S and thus (ε,δ)

µS (ε,δ)

(x ∗ y) = ε ≥ min{ρ1 , ρ2 }, (ε,δ)

that is, (x ∗ y)min{ρ1 ,ρ2 } ∈ µS . Now, µS (x ∗ y) + min{ρ1 , ρ2 } = ε + min{ρ1 , ρ2 } > 1, and (ε,δ) (ε,δ) (ε,δ) so (x ∗ y)min{ρ1 ,ρ2 } q µS . Hence (x ∗ y)min{ρ1 ,ρ2 } ∈ ∧ q µS , and µS is a (q, ∈ ∧ q )-fuzzy subalgebra of X.  If we take ε = 1 and δ = 0 in Theorems 3.10 and 4.12, then we have the following corollary. Corollary 4.13. A non-empty subset S of X is a subalgebra of X if and only if the characteristic function χS of S is an (q, ∈ ∧ q )-fuzzy subalgebra of X. Before ending our research, we pose an open question. (ε,δ)

Question. Given a subalgebra S of X, when will the characteristic fuzzy set µS be a

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(1) (2) (3) (4)

conditional conditional conditional conditional

11

(∈ ∨ q , ∈)-fuzzy subalgebra of X? (∈ ∨ q , q)-fuzzy subalgebra of X? (∈ ∨ q , ∈ ∨ q )-fuzzy subalgebra of X? (∈ ∨ q , ∈ ∧ q )-fuzzy subalgebra of X? 5. Conclusion

we have introduced the notions of characteristic fuzzy sets, as a generalization of crisp characteristic function, and conditional fuzzy subalgebra. Using this notion, we have discussed conditions for a subset of BCK/BCI-algebra to be a subalgebra. Given a subalgebra of BCK/BCI-algebras, we have provided conditions for the characteristic fuzzy set to be a conditional (∈, q)-fuzzy subalgebra, a conditional (q, ∈)-fuzzy subalgebra, a conditional (q, q)-fuzzy subalgebra, a conditional (∈, ∈ ∧ q )-fuzzy subalgebra, and a conditional (q, ∈ ∧ q )-fuzzy subalgebra. On the basis of these results, we will apply the notions of characteristic fuzzy sets and conditional fuzzy substructures to ideal and filter theory in several algebraic structures, for example, BCK/BCI-algebras, M V -algebras, BL-algebras, M T L-algebras, residuated lattices, R0 -algebras, lattice implication algebras, EQ-algebras etc. 6. Acknowledgements The authors would like to express their sincere thanks to the anonymous referee(s) for a careful checking of the details and for helpful comments.

References [1] S. K. Bhakat and P. Das, (∈, ∈ ∨ q)-fuzzy subgroup, Fuzzy Sets and Systems 80 (1996), 359–368. [2] B. Davvaz, (∈, ∈ ∨ q )-fuzzy subnear-rings and ideals, Soft Computing 10(3) (2006) 206–211. [3] B. Davvaz and P. Corsini, Generalized fuzzy hyperideals of hypernear-rings and many valued implications, J. Intell. Fuzzy Systems 17 (2006) 241–251. [4] W. A. Dudek, M. Shabir and M. Irfan Ali, (α, β)-fuzzy ideals of hemirings, Comput. Math. Appl. 58 (2009) 310–321. [5] Y. S. Huang, BCI-algebra, Science Press, Beijing, 2006. [6] Y. B. Jun, On (α, β)-fuzzy ideals of BCK/BCI-algebras, Sci. Math. Jpn. 60(3) (2004), 613–617. [7] Y. B. Jun, On (α, β)-fuzzy subalgebras of BCK/BCI-algebras, Bull. Korean Math. Soc. 42(4) (2005), 703–711.

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[8] Y. B. Jun, Fuzzy subalgebras of type (α, β) in BCK/BCI-algebras, Kyungpook Math. J. 47 (2007), 403–410. [9] O. Kazanci and B. Davvaz, More on fuzzy lattices, Comput. Math. Appl. 64 (2012) 2917–2925. [10] J. Meng and Y. B. Jun, BCK-algebra, Kyungmoon Sa Co. Seoul, 1994. [11] G. Muhiuddin and Abdullah M. Al-roqi, Classifications of (alpha, beta)-fuzzy ideals in BCK/BCIalgebras, Journal of Mathematical Analysis, 7 (6) (2016) 75–82. [12] G. Muhiuddin and Abdullah M. Al-roqi, Subalgebras of BCK/BCI-algebras based on (alpha, beta)type fuzzy sets, J. Comput. Anal. Appl., 18 (6) (2015) 1057–1064. [13] P. M. Pu and Y. M. Liu, Fuzzy topology I, Neighborhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980), 571–599. [14] J. Zhan, Y. B. Jun and B. Davvaz, On (∈, ∈ ∨ q )-fuzzy ideals of BCI-algebras, Iran. J. Fuzzy Syst. 6(1) (2009), 81–94. [15] J. Zhan, Y. B. Jun and W. A. Dudek, On (∈, ∈ ∨ q )-fuzzy filters of pseudo-BL algerbas, Bull. Malays. Math. Sci. Soc. 33(1) (2010) 57–67. [16] J. Zhan and Y. Xu, Some types of generalized fuzzy filters of BL-algebras, Comput. Math. Appl. 56 (2008) 1604–1616.

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Some common fixed point theorems for two pairs of self maps in dislocated metric spaces Peiguang Wang College of Electronic and Information Engineering, Hebei University, Baoding, Hebei 071002, P. R. China E-mail: [email protected] Akbar Zada Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan E-mail: [email protected], [email protected] Rahim Shah Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan E-mail: safeer [email protected], [email protected] Tongxing Li 1 School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P. R. China LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P. R. China E-mail: [email protected] Abstract. The purpose of this paper is to establish some common fixed point theorems for two pairs of self mappings in dislocated metric spaces which generalize, extend, and improve related results in the literature. Some applications of new results are provided. Keywords: Dislocated metric space; fixed point; weakly compatible maps; common fixed point.

Mathematics Subject Classification 2010: 47H10, 54H25. 1

Corresponding author. E-mail address: [email protected], Tel.: +86 13869959692, Fax: +86 539 8766320.

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Introduction and preliminaries

In 2000, Hitzler and Seda [4] presented the concept of dislocated metric space and generalized the well-known Banach contraction mapping principle in complete dislocated metric spaces. In recent years, the study of dislocated metric spaces has always attracted interest of researchers; see, for instance, [1–3, 5–8, 10–16] and the references cited therein. One of the main reasons for this lies in the fact that dislocated metric spaces play very important roles not only in topology but also in other branches of science involving mathematics especially in logic programming and electronics engineering. In the present paper, we prove some common fixed point theorems in the setting of dislocated metric spaces for two pairs of weakly compatible self mappings which generalize, extend, and improve related results reported in the literature. We need the following auxiliary definitions and results. Definition 1.1 [4] Let X be a nonempty set and let d : X × X → [0, ∞) be a function satisfying the following conditions: (c1 ) d(ξ, η) = d(η, ξ) for all ξ, η ∈ X ; (c2 ) d(ξ, η) = d(η, ξ) = 0 implies that ξ = η; (c3 ) d(ξ, η) ≤ d(ξ, ζ) + d(ζ, η) for all ξ, η, ζ ∈ X . Then d is called dislocated metric (or d-metric) on X . The nonempty set X together with d-metric, i.e., (X , d), is called a dislocated metric space. Definition 1.2 [4] A sequence {ξn } in a d-metric space (X , d) is called Cauchy sequence if for given ϵ > 0, there exists an n0 ∈ N such that d(ξm , ξn ) < ϵ for all m, n ≥ n0 . Definition 1.3 [4] A sequence {ξn } in a d-metric space (X , d) converges with respect to d (or in d) if there exists a ξ ∈ X such that lim d(ξn , ξ) = 0.

n→∞

In this case, ξ is called a limit of sequence {ξn } and we write ξn → ξ as n → ∞. Definition 1.4 [4] A d-metric space (X , d) is called complete if every Cauchy sequence in it is convergent with respect to d.

2

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Definition 1.5 [4] Let (X , d) be a d-metric space. A mapping T : X → X is called contraction if there exists a λ ∈ [0, 1) such that d(T ξ, T η) ≤ λd(ξ, η) for all ξ, η ∈ X . Lemma 1.6 [11] Let (X , d) be a d-metric space. If g : X → X is a contraction function, then {g n (ξ0 )} is a Cauchy sequence for each ξ0 ∈ X . Lemma 1.7 [4] Limits in a d-metric space are unique. Definition 1.8 [9] Let A and S be mappings from a metric space (X , d) into itself. Then, A and S are said to be weakly compatible if they commute at their coincident points; that is, Aξ = Sξ for some ξ ∈ X yields ASξ = SAξ. Theorem 1.9 [4] Let (X , d) be a complete dislocated metric space and let T : X → X be a contraction mapping. Then, T has a unique fixed point. Remark 1.10 [3] It is easy to verify that in a d-metric space, the following statements hold. (i) A subsequence of a Cauchy sequence in d-metric space is a Cauchy sequence. (ii) A Cauchy sequence in d-metric space with a convergent subsequence is also convergent. (iii) Limits of a convergent sequence are unique. (iv) A d-metric d is continuous, i.e., ξn → ξ and ηn → η imply that d(ξn , ηn ) → d(ξ, η) as n → ∞.

2

Main results

In this section, we prove some fixed point theorems in d-metric spaces. Theorem 2.1 Let (X , d) be a complete dislocated metric space. Assume that A, B, S, T : X → X are continuous self mappings satisfying the conditions: (i) T (X ) ⊂ A(X ) and S(X ) ⊂ B(X ); (ii) the pairs (S, A) and (T, B) are weakly compatible; (iii) d(Sξ, T η) ≤ a1 [d(Aξ, T η) + d(Bη, Sξ)] + a2 [d(Bη, T η) + d(Aξ, Sξ)] + a3 d(Aξ, Bη) + a4 [d(Aξ, Sξ) + d(Aξ, T η)] + a5 [d(Aξ, Bη) + d(Bη, T η)] for all ξ, η ∈ X , where a1 , a2 , a3 , a4 , a5 ≥ 0 and 0 ≤ 4a1 + 2a2 + a3 + 3a4 + 2a5 < 1. Then A, B, S, and T have a unique common fixed point. 3

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Proof. Define two sequences {ξn } and {ηn } by η2n := Bξ2n+1 = Sξ2n and η2n+1 := Aξ2n+2 = T ξ2n+1 for n = 0, 1, 2, . . . . If η2n = η2n+1 for some n, then Bξ2n+1 = T ξ2n+1 . Therefore, ξ2n+1 is a coincident point of B and T . Also, if η2n+1 = η2n+2 for some n, then Aξ2n+2 = Sξ2n+2 . Hence, ξ2n+2 is a coincidence point of A and S. Suppose now that η2n ̸= η2n+1 for all n. Then, we conclude that d(η2n , η2n+1 ) = d(Sξ2n , T ξ2n+1 ) ≤ a1 [d(Aξ2n , T ξ2n+1 ) + d(Bξ2n+1 , Sξ2n )] +a2 [d(Bξ2n+1 , T ξ2n+1 ) + d(Aξ2n , Sξ2n )] +a3 d(Aξ2n , Bξ2n+1 ) + a4 [d(Aξ2n , Sξ2n ) + d(Aξ2n , T ξ2n+1 )] +a5 [d(Aξ2n , Bξ2n+1 ) + d(Bξ2n+1 , T ξ2n+1 )] ≤ a1 [d(η2n−1 , η2n+1 ) + d(η2n , η2n )] +a2 [d(η2n , η2n+1 ) + d(η2n−1 , η2n )] +a3 d(η2n−1 , η2n ) + a4 [d(η2n−1 , η2n ) + d(η2n−1 , η2n+1 )] +a5 [d(η2n−1 , η2n ) + d(η2n , η2n+1 )] ≤ a1 [d(η2n−1 , η2n ) + d(η2n , η2n+1 ) +d(η2n−1 , η2n ) + d(η2n , η2n+1 )] +a2 [d(η2n , η2n+1 ) + d(η2n−1 , η2n )] + a3 d(η2n−1 , η2n ) +a4 [d(η2n−1 , η2n ) + d(η2n−1 , η2n ) + d(η2n , η2n+1 )] +a5 [d(η2n−1 , η2n ) + d(η2n , η2n+1 )] = (2a1 + a2 + a3 + 2a4 + a5 )d(η2n−1 , η2n ) +(2a1 + a2 + a4 + a5 )d(η2n , η2n+1 ). Therefore, we get d(η2n , η2n+1 ) ≤ Let h=

2a1 + a2 + a3 + 2a4 + a5 d(η2n−1 , η2n ). 1 − (2a1 + a2 + a4 + a5 ) 2a1 + a2 + a3 + 2a4 + a5 < 1. 1 − (2a1 + a2 + a4 + a5 )

Then d(ηn , ηn+1 ) ≤ hd(ηn−1 , ηn ). 4

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Similarly, we have d(ηn−1 , ηn ) ≤ hd(ηn−2 , ηn−1 ). Continuing this process, we obtain d(ηn , ηn+1 ) ≤ hn d(η0 , η1 ). Now, for any m, n satisfying m > n, using triangle inequality, we get d(ηn , ηm ) ≤ d(ηn , ηn+1 ) + d(ηn+1 , ηn+2 ) + · · · + d(ηm−1 , ηm ) ≤ hn d(η0 , η1 ) + hn+1 d(η0 , η1 ) + · · · + hm−1 d(η0 , η1 ) ≤ (hn + hn+1 + hn+2 + · · · )d(η0 , η1 ) hn d(η0 , η1 ). = 1−h Since h ∈ [0, 1), hn → 0 as n → ∞, which shows that {ηn } is a Cauchy sequence in the complete dislocated metric space (X , d). Hence, there exists a point p ∈ X such that limn→∞ ηn = p and lim Sξ2n = lim Bξ2n+1 = lim T ξ2n+1 = lim Aξ2n+2 = p.

n→∞

n→∞

n→∞

n→∞

Since T (X ) ⊂ A(X ), there exists a point v ∈ X such that p = Av. Therefore, d(Sv, p) = d(Sv, T ξ2n+1 ) ≤ a1 [d(Av, T ξ2n+1 ) + d(Bξ2n+1 , Sv)] +a2 [d(Bξ2n+1 , T ξ2n+1 ) + d(Av, Sv)] +a3 d(Av, Sv) + a4 [d(Av, Sv) + d(Au, T ξ2n+1 )] +a5 [d(Av, Bξ2n+1 ) + d(Bξ2n+1 , T ξ2n+1 )]. Taking n → ∞, we get d(Sv, p) ≤ a1 [d(p, p) + d(p, Sv)] + a2 [d(p, p) + d(p, Sv)] +a3 d(p, Sv) + a4 [d(p, Sv) + d(p, p)] + a5 [d(p, p) + d(p, p)] ≤ (2a1 + 2a2 + 2a4 + 4a5 )d(p, Sv) + (a1 + a2 + a3 + a4 )d(p, Sv) = (3a1 + 3a2 + a3 + 3a4 + 4a5 )d(p, Sv), which is a contradiction, and so Sv = Av = p. Again, since S(X ) ⊂ B(X ), there exists a point u ∈ X such that p = Bu. We claim now that p = T u. If 5

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p ̸= T u, then d(p, T u) = d(Sv, T u) ≤ a1 [d(Av, T u) + d(Bu, Sv)] + a2 [d(Bu, T u) + d(Av, Sv)] +a3 d(Av, Bu) + a4 [d(Av, Sv) + d(Av, T u)] +a5 [d(Av, Bu) + d(Bu, T u)] = a1 [d(p, T u) + d(p, p)] + a2 [d(p, T u) + d(p, p)] + a3 d(p, p) +a4 [d(p, p) + d(p, T u)] + a5 [d(p, p) + d(p, T u)] ≤ (3a1 + 3a2 + 2a3 + 3a4 + 3a5 )d(p, T u), which is a contradiction, and thus p = T u. Hence, we have Sv = Av = T u = Bu = p. Since (S, A) are weakly compatible, SAv = ASv implies that Sp = Ap. Next, we show that p is the fixed point of S. If Sp ̸= p, then d(Sp, p) = d(Sp, T u) ≤ a1 [d(Ap, T u) + d(Bu, Sp)] + a2 [d(Bu, T u) + d(Ap, Sp)] +a3 d(Ap, Bu) + a4 [d(Ap, Sp) + d(Ap, T u)] +a5 [d(Ap, Bu) + d(Bu, T u)] = a1 [d(Sp, p) + d(p, Sp)] + a2 [d(p, p) + d(Sp, Sp)] + a3 d(Sp, p) +a4 [d(Sp, Sp) + d(Sp, p)] + a5 [d(Sp, p) + d(p, p)] ≤ (2a1 + 4a2 + a3 + 3a4 + 3a5 )d(Sp, p), which is a contradiction, and so Sp = p. This yields Ap = Sp = p. Again, (T, B) are weakly compatible, and hence T Bu = BT u implies that T p = Bp. Now, we show that p is the fixed point of T . If T p ̸= p, then d(p, T p) = d(Sp, T p) ≤ a1 [d(Ap, T p) + d(Bp, Sp)] + a2 [d(Bp, T p) + d(Ap, Sp)] +a3 d(Ap, Sp) + a4 [d(Ap, Sp) + d(Ap, T p)] +a5 [d(Ap, Bp) + d(Bp, T p)] = a1 [d(p, T p) + d(T p, p)] + a2 [d(T p, T p) + d(p, p)] +a3 d(p, T p) + a4 [d(p, T p) + d(p, T p)] +a5 [d(p, T p) + d(T p, T p)] ≤ (2a1 + 4a2 + a3 + 2a4 + 3a5 )d(p, T p),

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which is a contradiction, and hence p = T p. Therefore, we have Ap = Bp = Sp = T p = p, which shows that p is the common fixed point of the self mappings A, B, S, and T . Uniqueness. Suppose that v ̸= u are two common fixed points of the mappings A, B, S, and T . Then, we have d(v, u) = d(Sv, T u) ≤ a1 [d(Av, T u) + d(Bu, Sv)] + a2 [d(Bu, T u) + d(Av, Sv)] +a3 d(Av, Bu) + a4 [d(Av, Sv) + d(Av, T u)] +a5 [d(Av, Bu) + d(Bu, T u)] = a1 [d(v, u) + d(u, v)] + a2 [d(u, u) + d(v, v)] + a3 d(v, u) +a4 [d(v, v) + d(v, u)] + a5 [d(v, u) + d(u, u)] ≤ (2a1 + 4a2 + a3 + 3a4 + 3a5 )d(v, u), which is a contradiction, and therefore v = u. The proof is complete.  Letting A = B = I (an identity mapping), we can derive the following result from Theorem 2.1. Corollary 2.2 Let (X , d) be a complete dislocated metric space. If S, T : X → X are continuous self mappings satisfying d(Sξ, T η) ≤ a1 [d(ξ, T η) + d(η, Sξ)] + a2 [d(η, T η) + d(ξ, Sξ)] + a3 d(ξ, η) +a4 [d(ξ, Sξ) + d(ξ, T η)] + a5 [d(ξ, η) + d(η, T η)] for all ξ, η ∈ X , where a1 , a2 , a3 , a4 , a5 ≥ 0 and 0 ≤ 4a1 +2a2 +a3 +3a4 +2a5 < 1, then S and T have a unique common fixed point. If a4 = a5 = 0 and S = T , then Corollary 2.2 reduces to the following result obtained by Isufati [6]. Corollary 2.3 Let (X , d) be a complete dislocated metric space. If T : X → X is a continuous self mapping satisfying d(T ξ, T η) ≤ a1 [d(ξ, T η) + d(η, T ξ)] + a2 [d(η, T η) + d(ξ, T ξ)] + a3 d(ξ, η) for all ξ, η ∈ X , where a1 , a2 , a3 ≥ 0 and 0 ≤ 4a1 + 2a2 + a3 < 1, then T has a unique fixed point. 7

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Letting a4 = a5 = 0 in Theorem 2.1, we get the following result reported by Panthi and Jha [11]. Corollary 2.4 Let (X , d) be a complete dislocated metric space. If A, B, S, T : X → X are continuous self mappings satisfying the conditions: (i) T (X ) ⊂ A(X ) and S(X ) ⊂ B(X ); (ii) the pairs (S, A) and (T, B) are weakly compatible; (iii) d(Sξ, T η) ≤ a1 [d(Aξ, T η) + d(Bη, Sξ)] + a2 [d(Bη, T η) + d(Aξ, Sξ)] + a3 d(Aξ, Bη) for all ξ, η ∈ X , where a1 , a2 , a3 ≥ 0 and 0 ≤ 4a1 + 2a2 + a3 < 1, then A, B, S, and T have a unique common fixed point. Remark 2.5 Our results improve those obtained by Aage and Salunke [1, 2], Jha and Panthi [7], Jha et al. [8], Rao and Rangaswamy [12], and Shrivastava et al. [14].

3

Further results without any continuity requirement

In this section, we prove some fixed point theorems without any continuity requirement in d-metric spaces. Theorem 3.1 Let (X , d) be a complete dislocated metric space. Suppose that A, B, S, T : X → X are self mappings satisfying the conditions: (i) T (X ) ⊂ A(X ) and S(X ) ⊂ B(X ); (ii) the pairs (S, A) and (T, B) are weakly compatible; (iii) d(Sξ, T η) ≤ a1 [d(Aξ, T η) + d(Bη, Sξ)] + a2 [d(Bη, T η) + d(Aξ, Sξ)] + a3 d(Aξ, Bη) + a4 [d(Aξ, Sξ) + d(Aξ, T η)] + a5 [d(Aξ, Bη) + d(Bη, T η)] for all ξ, η ∈ X , where a1 , a2 , a3 , a4 , a5 ≥ 0 and 0 ≤ 4a1 + 2a2 + a3 + 3a4 + 2a5 < 1, then A, B, S, and T have a unique common fixed point. Proof. Let ξ0 ∈ X be arbitrary. Choose ξ1 ∈ X such that Bξ1 = Sξ0 . Again, choose ξ2 ∈ X such that Aξ2 = T ξ1 . Continuing this process, choose ξn ∈ X such that Sξ2n = Bξ2n+1 and T ξ2n+1 = Aξ2n+2 for n = 0, 1, 2, . . .. To simplify, we consider the sequence {ηn } which is defined by η2n := Sξ2n and η2n+1 := T ξ2n+1 for n = 0, 1, 2, . . .. Next, we claim that {ηn } is a Cauchy

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sequence. Indeed, for n ≥ 1, we have d(η2n , η2n+1 ) = d(Sξ2n , T ξ2n+1 ) ≤ a1 [d(Aξ2n , T ξ2n+1 ) + d(Bξ2n+1 , Sξ2n )] +a2 [d(Aξ2n , Sξ2n ) + d(Bξ2n+1 , T ξ2n+1 )] +a3 d(Aξ2n , Bξ2n+1 ) + a4 [d(Aξ2n , Sξ2n ) + d(Aξ2n , T ξ2n+1 )] +a5 [d(Aξ2n , Bξ2n+1 ) + d(Bξ2n+1 , T ξ2n+1 )] ≤ a1 [d(η2n−1 , η2n+1 ) + d(η2n , η2n )] +a2 [d(η2n−1 , η2n ) + d(η2n , η2n+1 )] +a3 d(η2n−1 , η2n ) + a4 [d(η2n−1 , η2n ) + d(η2n−1 , η2n+1 )] +a5 [d(η2n−1 , η2n ) + d(η2n , η2n+1 )] ≤ a1 [d(η2n−1 , η2n ) + d(η2n , η2n+1 ) +d(η2n , η2n+1 ) + d(η2n+1 , η2n )] +a2 [d(η2n−1 , η2n ) + d(η2n , η2n+1 )] + a3 d(η2n−1 , η2n ) +a4 [d(η2n−1 , η2n ) + d(η2n−1 , η2n ) + d(η2n , η2n+1 )] +a5 [d(η2n−1 , η2n ) + d(η2n , η2n+1 )] = (a1 + a2 + a3 + 2a4 + a5 )d(η2n−1 , η2n ) +(3a1 + a2 + a4 + a5 )d(η2n , η2n+1 ). Hence, we conclude that d(η2n , η2n+1 ) ≤ hd(η2n−1 , η2n ), where

a1 + a2 + a3 + 2a4 + a5 ∈ [0, 1). 1 − (3a1 + a2 + a4 + a5 ) This implies that {ηn } is a Cauchy sequence in X . Then, by Remark 1.10, {Sξ2n }, {Bξ2n+1 }, {T ξ2n+1 }, and {Aξ2n+2 } are also Cauchy sequences. Assume that Sξ is a complete subspace of X , the sequence {Sξ2n } converges to some Sa such that a ∈ X . So, {ηn }, {Bξ2n+1 }, {T ξ2n+1 }, and {Aξ2n+2 } also converge to Sa. Since SX ⊂ BX , there exists a v ∈ X such that Sa = Bv. We show that Bv = T v. In fact, we have h=

d(Sξ2n , T v) ≤ a1 [d(Aξ2n , T v) + d(Bv, Sξ2n )] +a2 [d(Aξ2n , Sξ2n ) + d(Bv, T v)] +a3 d(Aξ2n , Bv) + a4 [d(Aξ2n , Sξ2n ) + d(Aξ2n , T v)] +a5 [d(Aξ2n , Bv) + d(Bv, T v)]. 9

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Letting n → ∞, we get d(Bv, T v) ≤ a1 [d(Bv, T v) + d(Bv, Bv)] + a2 [d(Bv, Bv) + d(Bv, T v)] +a3 d(Bv, Bv) + a4 [d(Bv, Bv) + d(Bv, T v)] +a5 [d(Bv, Bv) + d(Bv, T v)] ≤ (a1 + a2 + a3 + a4 + a5 )d(Bv, Bv) +(a1 + a2 + a4 + a5 )d(Bv, T v) ≤ (3a1 + 3a2 + 2a3 + 3a4 + 3a5 )d(Bv, T v). Therefore, d(Bv, T v) = 0, which implies that T v = Bv. Since T X ⊂ AX , there exists a u ∈ X such that T v = Au. We show that Su = Au. Indeed, we have d(Su, Au) = d(Su, T v) ≤ a1 [d(Au, T v) + d(Bv, Su)] + a2 [d(Au, Su) + d(Bv, T v)] +a3 d(Au, Bv) + a4 [d(Au, Su) + d(Au, T v)] +a5 [d(Au, Bv) + d(Bv, T v)] ≤ a1 [d(Au, Au) + d(Au, Su)] + a2 [d(Au, Su) + d(Au, Au)] +a3 d(Au, Au) + a4 [d(Au, Su) + d(Au, Au)] +a5 [d(Au, Au) + d(Au, Au)] ≤ a1 [d(Au, Su) + d(Su, Au) + d(Au, Su)] +a2 [d(Au, Su) + d(Au, Su) + d(Su, Au)] +a3 [d(Au, Su) + d(Su, Au)] +a4 [d(Au, Su) + d(Au, Su) + d(Su, Au)] +a5 [d(Au, Su) + d(Su, Au) + d(Au, Su) + d(Su, Au)] = (3a1 + 3a2 + 2a3 + 3a4 + 4a5 )d(Au, Su). Hence, d(Su, Au) = 0, which yields Au = Su, and so Bv = T v = Au = Su. By virtue of the fact that (S, A) are weakly compatible, we deduce that ASu = SAu, which yields AAu = ASu = SAu = SSu. The weak compatibility of B and T implies that BT v = T Bv, from which it follows that BBv = BT v = T Bv = T T v. Let us show that Bv is a fixed point of T .

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In fact, we have d(Bv, T Bv) = d(Su, T Bv) ≤ a1 [d(Au, T Bv) + d(BBv, Su)] +a2 [d(Au, Su) + d(BBv, T Bv)] +a3 d(Au, BBv) + a4 [d(Au, Su) + d(Au, T Bv)] +a5 [d(Au, BBv) + d(BBv, T Bv)] ≤ a1 [d(Bv, T Bv) + d(T Bv, Bv)] +a2 [d(Bv, Bv) + d(T Bv, T Bv)] +a3 d(Bv, T Bv) + a4 [d(Bv, Bv) + d(Bv, T Bv)] +a5 [d(Bv, T Bv) + d(T Bv, T Bv)] ≤ 2a1 d(Bv, T Bv) + a2 [d(Bv, T Bv) + d(T Bv, Bv) +d(T Bv, Bv) + d(Bv, T Bv)] + a3 d(Bv, T Bv) +a4 [d(Bv, T Bv) + d(T Bv, Bv) + d(Bv, T Bv)] +a5 [d(Bv, T Bv) + d(T Bv, Bv) + d(Bv, T Bv)] = (2a1 + 4a2 + a3 + 3a4 + 3a5 )d(Bv, T Bv), which yields d(Bv, T Bv) = 0, and so T Bv = Bv. Therefore, Bv is a fixed point of T . It follows that BBv = T Bv = Bv, which implies that Bv is also a fixed point of B. On the other hand, we get d(SBv, Bv) = d(SBv, T Bv) ≤ a1 [d(ABv, T Bv) + d(BBv, SBv)] + a2 [d(ABv, SBv) +d(BBv, T Bv)] + a3 d(ABv, BBv) + a4 [d(ABv, SBv) +d(ABv, T Bv)] + a5 [d(ABv, BBv) + d(BBv, T Bv)] ≤ a1 [d(Bv, Bv) + d(Bv, SBv)] + a2 [d(SBv, Bv) +d(Bv, SBv)] + a3 d(Bv, Bv) + a4 [d(Bv, Bv) +d(Bv, Bv)] + a5 [d(Bv, Bv) + d(Bv, Bv)] ≤ a1 [d(Bv, SBv) + d(SBv, Bv) + d(Bv, SBv)] +a2 [d(SBv, Bv) + d(Bv, SBv)] + a3 [d(Bv, SBv) +d(SBv, Bv)] + a4 [d(Bv, SBv) + d(SBv, Bv) +d(Bv, SBv) + d(SBv, Bv)] + a5 [d(Bv, SBv) +d(SBv, Bv) + d(Bv, SBv) + d(SBv, Bv)] = (3a1 + 2a2 + 2a3 + 4a4 + 4a5 )d(SBv, Bv), 11

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which implies that d(Bv, SBv) = 0, and hence SBv = Bv. Therefore, Bv is a fixed point of S. It follows that ABv = SBv = Bv, which shows that Bv is also a fixed point of A. Then Bv is a common fixed point of A, B, S, and T. Uniqueness. Let w, u ∈ X be two fixed points such that Aw = Bw = Sw = T w and Au = Bu = Su = T u. If d(w, u) ̸= 0, then d(w, u) = d(Sw, T u) ≤ a1 [d(Aw, T u) + d(Bu, Sw)] + a2 [d(Bu, T u) + d(Aw, Sw)] +a3 d(Aw, Bu) + a4 [d(Aw, Sw) + d(Aw, T u)] +a5 [d(Aw, Bu) + d(Bu, T u)] = a1 [d(w, u) + d(u, w)] + a2 [d(u, u) + d(w, w)] + a3 d(w, u) +a4 [d(w, w) + d(w, u)] + a5 [d(w, u) + d(u, u)] ≤ (2a1 + 4a2 + a3 + 3a4 + 3a5 )d(w, u), which is a contradiction. Hence, d(v, u) = 0, which implies that v = u. The proof is complete.  Remark 3.2 One can derive from Theorem 3.1 a number of fixed point theorems for self mappings A, B, S, and T . For example, we have the following result by letting a4 = a5 = 0. Corollary 3.3 Let (X , d) be a complete dislocated metric space. If A, B, S, T : X → X are self mappings satisfying the conditions: (i) T (X ) ⊂ A(X ) and S(X ) ⊂ B(X ); (ii) the pairs (S, A) and (T, B) are weakly compatible; (iii) d(Sξ, T η) ≤ a1 [d(Aξ, T η) + d(Bη, Sξ)] + a2 [d(Bη, T η) + d(Aξ, Sξ)] + a3 d(Aξ, Bη) for all ξ, η ∈ X , where a1 , a2 , a3 ≥ 0 and 0 ≤ 4a1 + 2a2 + a3 < 1, then A, B, S, and T have a unique common fixed point. Remark 3.4 Our results generalize, extend, and improve those obtained by Bennani et al. [3].

4

Examples

The following examples illustrate theoretical results obtained in the previous sections. 12

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Example 4.1 Assume that X = [0, 1], d is a usual metric, and define the mappings A, B, S, and T by Aξ = ξ,

Bξ = ξ,

Sξ = 0,

and T ξ =

1 ξ. 12

Let

1 1 1 1 1 , a2 = , a3 = , a4 = , and a5 = . 20 24 28 30 34 Then A, B, S, and T satisfy all assumptions of Theorem 2.1. As a matter of fact, 0 ∈ X is the unique common fixed point of the mappings A, B, S, and T . a1 =

Example 4.2 Let X = [0, 1], d(ξ, η) = |ξ| + |η|, and define the mappings A, B, S, and T by Aξ = ξ,

Bξ = ξ,

Sξ = 0,

ξ and T ξ = . 6

Set

1 1 1 1 1 , a2 = , a3 = , a4 = , and a5 = . 25 28 32 36 40 Then A, B, S, and T satisfy all assumptions of Theorem 3.1. In fact, 0 ∈ X is the unique common fixed point of the mappings A, B, S, and T . a1 =

5

Acknowledgements

The authors express their sincere gratitude to the editors for useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by NNSF of P. R. China (Grant Nos. 11271106 and 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P. R. China.

References [1] C. T. Aage, J. N. Salunke, Some results of fixed point theorem in dislocated quasi-metric spaces, Bull. Marathwada Math. Soc., 9 (2008) 1–5. 13

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[2] C. T. Aage, J. N. Salunke, The results on fixed points in dislocated and dislocated quasi-metric space, Appl. Math. Sci., 2 (2008) 2941–2948. [3] S. Bennani, H. Bourijal, S. Mhanna, D. E. Moutawakil, Some common fixed point theorems in dislocated metric spaces, J. Nonlinear Sci. Appl., 8 (2015) 86–92. [4] P. Hitzler, A. K. Seda, Dislocated topologies, J. Electr. Eng., 51 (2000) 3–7. [5] P. Hitzler, Generalized metrics and topology in logic programming semantics, Ph.D. Thesis, National University of Ireland (University College, Cork), 2001. [6] A. Isufati, Fixed point theorems in dislocated quasi-metric space, Appl. Math. Sci., 4 (2010) 217–223. [7] K. Jha, D. Panthi, A common fixed point theorem in dislocated metric space, Appl. Math. Sci., 6 (2012) 4497–4503. [8] K. Jha, K. P. R. Rao, D. Panthi, A common fixed point theorem for four mappings in dislocated quasi-metric space, Int. J. Math. Sci. Eng. Appl., 6 (2012) 417–424. [9] G. Jungck, B. E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29 (1998) 227–238. [10] S. Shukla, S. Balasubramanian, M. Pavlovi´c, A generalized Banach fixed point theorem, Bull. Malays. Math. Sci. Soc., 39 (2016) 1529– 1539. [11] D. Panthi, K. Jha, A common fixed point of weakly compatible mappings in dislocated metric space, Kathmandu Univ. J. Sci. Eng. Tech., 8 (2012) 25–30. [12] K. P. R. Rao, P. Rangaswamy, Common fixed point theorem for four mappings in dislocated quasi-metric space, Nepali Math. Sci. Rep., 30 (2010) 70–75. [13] I. R. Sarma, P. S. Kumari, On dislocated metric spaces, Int. J. Math. Arch., 3 (2012) 72–77. 14

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[14] R. Shrivastava, Z. K. Ansari, M. Sharma, Some results on fixed points in dislocated and dislocated quasi-metric spaces, J. Adv. Stud. Topol., 3 (2012) 25–31. [15] R. K. Vats, Weakly compatible maps in metric spaces, J. Indian Math. Soc., 69 (2002) 139–143. [16] F. M. Zeyada, G. H. Hassan, M. A. Ahmed, A generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spaces, Arab. J. Sci. Eng., 31 (2006) 111–114.

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN NON-ARCHIMEDEAN BANACH SPACES SUNGSIK YUN Abstract. In this paper, we solve the quadratic ρ-functional inequalities kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

   

x+y

, + f (x − y) − 2f (x) − 2f (y) ρ 4f ≤

2 where ρ is a fixed non-Archimedean number with |ρ| < |2|, and





4f x + y + f (x − y) − 2f (x) − 2f (y)

2

(0.1)

(0.2)

≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k, where ρ is a fixed non-Archimedean number with |ρ| < 1. Furthermore, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in non-Archimedean Banach spaces.

1. Introduction and preliminaries A valuation is a function | · | from a field K into [0, ∞) such that 0 is the unique element having the 0 valuation, |rs| = |r| · |s| and the triangle inequality holds, i.e., |r + s| ≤ |r| + |s|, ∀r, s ∈ K. A field K is called a valued field if K carries a valuation. The usual absolute values of R and C are examples of valuations. Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by |r + s| ≤ max{|r|, |s|}, ∀r, s ∈ K, then the function | · | is called a non-Archimedean valuation, and the field is called a nonArchimedean field. Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. A trivial example of a non-Archimedean valuation is the function | · | taking everything except for 0 into 1 and |0| = 0. Throughout this paper, we assume that the base field is a non-Archimedean field, hence call it simply a field. Definition 1.1. ([8]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function k · k : X → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: (i) kxk = 0 if and only if x = 0; (ii) krxk = |r|kxk (r ∈ K, x ∈ X); 2010 Mathematics Subject Classification. Primary 46S10, 39B62, 39B52, 47S10, 12J25. Key words and phrases. Hyers-Ulam stability; non-Archimedean normed space; quadratic ρ-functional inequality.

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S. YUN

(iii) the strong triangle inequality kx + yk ≤ max{kxk, kyk},

∀x, y ∈ X

holds. Then (X, k · k) is called a non-Archimedean normed space. Definition 1.2. (i) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called Cauchy if for a given ε > 0 there is a positive integer N such that kxn − xm k ≤ ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that kxn − xk ≤ ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. The stability problem of functional equations originated from a question of Ulam [18] concerning the stability of group homomorphisms. The functional equation f (x+y) = f (x)+f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [7] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [11] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation f x+y = 12 f (x) + 12 f (y) is called the Jensen equation. 2 The functional equation f (x+y)+f (x−y) = 2f (x)+2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [17] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [5] noticed that the theorem of Skof is still true if therelevant domain E1 is replaced by an  x+y x−y Abelian group. The functional equation 2f 2 + 2 2 = f (x) + f (y) is called a Jensen type quadratic equation. The stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 3, 4, 9, 10, 12, 13, 14, 15, 16, 19, 20]). In Section 2, we solve the quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in non-Archimedean Banach spaces. In Section 3, we solve the quadratic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in non-Archimedean Banach spaces. Throughout this paper, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| 6= 1. 2. Quadratic ρ-functional inequality (0.1) in non-Archimedean normed spaces Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |2|. In this section, we solve the quadratic ρ-functional inequality (0.1) in non-Archimedean normed spaces.

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Lemma 2.1. If a mapping f : G → Y satisfies kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

   

x+y ≤

ρ 4f + f (x − y) − 2f (x) − 2f (y)

2

(2.1)

for all x, y ∈ G, then f : G → Y is quadratic. Proof. Assume that f : G → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get k2f (0)k ≤ |ρ|kf (0)k. So f (0) = 0. Letting y = x in (2.1), we get kf (2x) − 4f (x)k ≤ 0 and so f (2x) = 4f (x) for all x ∈ G. Thus x 1 = f (x) 2 4

 

f

(2.2)

for all x ∈ G. It follows from (2.1) and (2.2) that kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

   

x+y + f (x − y) − 2f (x) − 2f (y)

≤

ρ 4f 2 = |ρ|kf (x + y) + f (x − y) − 2f (x) − 2f (y)k and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ G.



Now, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (2.1) in non-Archimedean Banach spaces. Theorem 2.2. Let r < 2 and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying

x+y + f (x − y) − 2f (x) − 2f (y)

2 r + θ(kxk + kykr ) (2.3)



kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤

ρ 4f







for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

2 θkxkr r |2|

(2.4)

for all x ∈ X. Proof. Letting x = y = 0 in (2.3), we get kf (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = x in (2.3), we get kf (2x) − 4f (x)k ≤ 2θkxkr

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(2.5)

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for all x ∈ X. So f (x) − 4f x2 ≤ |2|2r θkxkr for all x ∈ X. Hence

   

l x x

m

4 f (2.6) − 4 f

m 2l 2

         

x x x x ≤ max

4l f l − 4l+1 f l+1

, · · · ,

4m−1 f m−1 − 4m f m

2 2  2 2  

 

    



x x x x = max |4|l

f l − 4f l+1

, · · · , |4|m−1

f m−1 − 4f m

2 2 2 2 ( )

≤ max

|4|l |4|m−1 , · · · , |2|rl |2|r(m−1)

2 2 θ θkxkr = (r−2)l r kxkr r |2| |2| |2|

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.6) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := n→∞ lim 4n f ( n ) 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.4). It follows from (2.3) that kQ(x + y) + Q(x − y) − 2Q(x) − 2Q(y)k

       

x−y x y

x+y + f − 2f − 2f = n→∞ lim |4|n

f 2n 2n 2n 2n

       

x+y x−y x y

|4|n θ n

(kxkr + kykr ) + f − 2f − 2f + lim ≤ n→∞ lim |4| |ρ| 4f 2n+1 2n 2n 2n n→∞ |2|nr

 

x+y

= |ρ| 4Q + Q(x − y) − 2Q(x) − 2Q(y)

2 for all x, y ∈ X. So

   

x+y

kQ(x + y) + Q(x − y) − 2Q(x) − 2Q(y)k ≤ ρ 4Q + Q(x − y) − 2Q(x) − 2Q(y)

2 for all x, y ∈ X. By Lemma 2.1, the mapping h : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (2.4). Then we have

   

q x x

q

kQ(x) − T (x)k = 4 Q q − 4 T 2 2q          

x x x x

2 q ≤ max

4q Q q − 4q f q

,

4q T − 4 f θkxkr ,

≤ q q (r−2)q+r 2 2 2 2 |2| which tends to zero as q → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mapping satisfying (2.4).  Theorem 2.3. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.3). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

2θ kxkr |4|

for all x ∈ X.

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Proof. It follows from (2.5) that



1 2θ

f (x) − f (2x) ≤ kxkr

|4|

4

for all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.



3. Quadratic ρ-functional inequality (0.2) Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < 1. In this section, we solve the quadratic ρ-functional inequality (0.2) in non-Archimedean normed spaces. Lemma 3.1. If a mapping f : G → Y satisfies





x+y

4f + f (x − y) − 2f (x) − 2f (y)

2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k

(3.1)

for all x, y ∈ G, then f : G → Y is quadratic. Proof. Assume that f : G → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get kf(0)k ≤ |ρ|k2f (0)k. So f (0) = 0.

x Letting y = 0 in (3.1), we get 4f 2 − f (x) ≤ 0 and so x = f (x) 2

 

4f

(3.2)

for all x ∈ G. It follows from (3.1) and (3.2) that kf (x + y) + f (x − y) − 2f (x) − 2f (y)k





x+y

= 4f + f (x − y) − 2f (x) − 2f (y)

2 ≤ |ρ|kf (x + y) + f (x − y) − 2f (x) − 2f (y)k and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ G.



Now, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (3.1) in non-Archimedean Banach spaces. Theorem 3.2. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping satisfying

 

x+y

≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k

4f + f (x − y) − 2f (x) − 2f (y)

2 + θ(kxkr + kykr ) (3.3) for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤ θkxkr

(3.4)

for all x ∈ X.

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Proof. Letting x = y = 0 in (3.3), we get k2f (0)k ≤ |ρ|kf (0)k. So f (0) = 0. Letting y = 0 in (3.3), we get

 

x

4f

≤ θkxkr (3.5) − f (x)

2 for all x ∈ X. So

   

l x x

m

4 f (3.6) − 4 f

m 2l 2

         

x x x x ≤ max

4l f l − 4l+1 f l+1

, · · · ,

4m−1 f m−1 − 4m f m

2 2  2 2  

 

    



x x x x = max |4|l

f l − 4f l+1

, · · · , |4|m−1

f m−1 − 4f m

2 2 2 2 ( ) l m−1 |4| θ |4| ≤ max , · · · , r(m−1) θkxkr = (r−2)l kxkr rl |2| |2| |2| for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.6) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.6), we get (3.4). The rest of the proof is similar to the proof of Theorem 2.2.  Theorem 3.3. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (3.3). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

|2|r θ kxkr |4|

(3.7)

for all x ∈ X. Proof. It follows from (3.5) that



|2|r θ 1

f (x) − f (2x) ≤ kxkr

|4|

4

for all x ∈ X. Hence



1 1 m

f (2l x) − f (2 x) (3.8)

l m 4 4

     

1 

1

1  1 ≤ max

l f 2l x − l+1 f 2l+1 x

, · · · ,

m−1 f 2m−1 x − m f (2m x)

4 4 4 4 (





)    

1

1 1

1 l l+1 m−1 m

x − f (2 x) = max

f 2 x − f 2 x , · · · ,

f 2 l m−1 |4| 4 |4| 4 |2|r(m−1) |2|r θ |2|rl r r , · · · , kxkr |2| θkxk = ≤ max |4|l+1 |4|(m−1)+1 |2|(2−r)l+2 (

)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.8) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES

{ 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 Q(x) := lim n f (2n x) n→∞ 4 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.8), we get (3.7). The rest of the proof is similar to the proofs of Theorems 2.2 and 3.2.  Acknowledgments This research was supported by Hanshin University Research Grant. References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] L. C˘adariu, L. G˘ avruta and P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [4] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [6] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [8] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.–TMA 69 (2008), 3405–3408. [9] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [10] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [11] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [12] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [13] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [14] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [15] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [16] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [17] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [18] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [19] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [20] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122. Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Republic of Korea E-mail address: [email protected]

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A Fast Inversion Free Iterative Algorithm for Solving X + A∗X −1A = I Duanmei Zhou∗ College of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi, P.R. China Guoliang Chen Department of Mathematics, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Dongchuan RD 500, Shanghai 200241, P.R. China Fengqiu Zhang Daqing Secondary Construction Vocational and Technical School, Daqing, 163000, P.R. China Guoxing Wu Department of Mathematics, Northeast Forestry University, Harbin 150040, P.R. China

Abstract We introduce a new inversion free variant of the basic fixed point iteration method for obtaining a maximal positive definite solution of the nonlinear matrix equation X + A∗ X −1 A = I with A normal. It has fewer operations and matrix-matrix multiplications than the existing algorithms. We derive convergence conditions for the iteration and some numerical results to illustrate the behavior of the new algorithm. AMS classification: 15A24; 65F10; 65F35 Key words: Nonlinear matrix equation; Hermitian positive definite solution; Fixed point iteration method; Convergence rate.

1

Introduction

Consider the nonlinear matrix equation X + A∗ X −1 A = I,

(1.1)

where A is an n × n complex normal matrix and I the identity matrix. Here A∗ stands for the conjugate transpose of A. Nonlinear matrix equation (1.1) has many applications. It often arises in control theory, dynamic programming, ladder networks, stochastic filtering, statistics, and etc.; see [1, 3–11, 14–23] and the references therein. It is well known that X is a solution of (1.1) if and only if it solves X = I − A∗ (I − A∗ X −1 A)−1 A. Assuming that A is invertible, we can write the above equation as X = F ∗ (R + X −1 )−1 F + I, where F = A−∗ A and R = −A−∗ A−1 . This is a special case of the discrete algebraic Riccati equation X − F ∗ (R + X −1 )−1 F − I = 0, ∗ E-mail

address: [email protected].

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where I = I ∗ and R = R∗ is invertible. For more details about the discrete algebraic Riccati equation, we refer to [2, 13]. In [17], Zhan proposed the following inversion free iteration ( Xn+1 = I − A∗ Yn A, M1 : Yn+1 = Yn (2I − Xn Yn ), starting from X0 = Y0 = I. In [11], Guo and Lancaster proposed the following inversion free iteration ( Yn+1 = Yn (2I − Xn Yn ), M2 : Xn+1 = I − A∗ Yn+1 A, starting from X0 = Y0 = I. When A is a nonsingular matrix, Monsalve and Raydan proposed in [16] the following inversion free iteration ( X0 = AA∗ , M3 : Xn+1 = 2Xn − Xn A−∗ (I − Xn )A−1 Xn , n = 0, 1, . . . to solve the minimal solution. The maximal solution of (1.1) can be obtained through X+ = I − Y− , where Y− is the minimal solution of the dual equation Y + AY −1 A∗ = I. M3 generates a Hermitian sequence of Xn . The implementation of iteration M3 involves three matrix-matrix multiplications per iteration and the inverse operation of A at the beginning only. In [8], El-Sayed and Al-Dbiban proposed an algorithm that avoids the matrix inversion for every iteration, called an inversion free variant of the basic fixed point iteration. ( Yn+1 = (I − Xn )Yn + I, M4 : Xn+1 = I − A∗ Yn+1 A, starting from X0 = Y0 = I. It is important to notice that M1 and M2 generate a Hermitian sequence and require four matrix-matrix multiplications per iteration, while M4 requires three matrix-matrix multiplications per iteration but does not generate a Hermitian sequence. If A is a normal matrix, we will prove that M4 generates a Hermitian sequence. And we propose the following algorithm. ( Y0 = I, M5 : Yn+1 = (AYn )∗ (AYn ) + I, n = 0, 1, . . . . The algorithm indicated by M5 is an inverse-free iterative method. Notice that it only requires to compute X = I − A∗ Y A at the end of the process, and only needs two matrix-matrix multiplications per iteration. Therefore it is clearly inexpensive. By an inductive argument, it is also worth noticing that in M5, Yn is a Hermitian matrix for all n. The following notations will be used throughout the paper. Let Cn×n be the set of n×n complex matrices. The notation B ≥ 0(B > 0) means that B is a Hermitian positive semi-definite (definite) matrix. Moreover, B ≥ C(B > C) is used as a different notation for B − C ≥ 0(B − C > 0). This induces a partial ordering on the Hermitian matrices. The symbols ρ(A) and kAk denote the spectral radius and the spectral norm of a square matrix A, respectively.

2

Conditions for the Existence of Solutions

The following lemmas are needed for our purpose. Lemma 2.1. For Algorithm M4, if A is normal, then AXn = Xn A, AYn = Yn A, A∗ Xn = Xn A∗ , A∗ Yn = Yn A∗ for n = 0, 1, . . ..

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Proof. Since Y1 = Y0 = X0 = I, AY0 = Y0 A,

AY1 = Y1 A,

AX0 = X0 A.

Because X1 = I − A∗ Y1 A = I − A∗ A, Y2 = (I − X1 )Y1 + I = A∗ A + I, and AA∗ = A∗ A, AY2 = A(A∗ A + I) = (A∗ A + I)A = Y2 A. That is, AYn = Yn A is true for n=0, 1, 2. So, assume that AYn = Yn A is true for n = k. Now we prove that AYn = Yn A when n = k + 1. In fact AYk+1 = A((I − Xk )Yk + I) = AA∗ Yk AYk + A = Yk+1 A. This completes the induction for n = k + 1. Therefore, AYn = Yn A for n = 0, 1, 2, . . .. We also have AXn+1 = A(I − A∗ Yn+1 A) = A − A∗ Yn+1 AA + A = Xn+1 A for n = 1, 2, . . .. The proof of A∗ Xn = Xn A∗ and A∗ Yn = Yn A∗ are similar to that of AXn = Xn A and AYn = Yn A, respectively. Lemma 2.2. If A is normal, then Algorithm M4 generates a Hermitian sequence. Proof. Since Y1 = Y0 = X0 = I, Y1 , Y0 , and X0 are Hermitian. Assume that Yn is Hermitian for n = k. Then ∗ Yk+1 = ((I − Xk )Yk + I)∗ = (A∗ Yk AYk + I)∗ = Yk A∗ Yk A + I. If A is a normal matrix, then according to Lemma 2.1, ∗ Yk+1 = A∗ Yk AYk + I = Yk+1 .

This completes the induction for n = k + 1. Since Xn = I − A∗ Yn A, Xn∗ = I − A∗ Yn∗ A = I − A∗ Yn A = Xn for n = 1, 2, . . .. This completes the proof. Lemma 2.3. For Algorithm M4, if A is normal, then {Xn , Yn , Xn+1 , Yn+1 } is a commuting family, n = 0, 1, 2, . . .. Proof. Since Y1 = Y0 = X0 = I, X1 = I −A∗ A, it is easy to check that {X0 , Y0 , X1 , Y1 } is a commuting family. Now we prove that {Xn , Yn , Xn+1 , Yn+1 } is a commuting family for n = 1, 2, . . .. Since Xn = I − A∗ Yn A, Yn+1 = (I − Xn )Yn + I = A∗ Yn AYn + I. According to Lemma 2.1, Yn+1 Yn = (A∗ Yn AYn + I)Yn = A∗ Yn AYn Yn + Yn = Yn A∗ Yn AYn + Yn = Yn Yn+1 ,

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

and Yn+1 Xn+1 = Yn+1 (I − A∗ Yn+1 A) = Yn+1 − Yn+1 A∗ Yn+1 A = Yn+1 − A∗ Yn+1 AYn+1 = Xn+1 Yn+1 . This implies that Yn Xn = Xn Yn for n = 1, 2, . . .. From (2.1) and Lemma 2.1, Yn+1 Xn = Yn+1 (I − A∗ Yn A) = Yn+1 − Yn+1 A∗ Yn A = Yn+1 − A∗ Yn AYn+1 = Xn Yn+1 ,

(2.2)

and Xn+1 Yn = (I − A∗ Yn+1 A)Yn = Yn − A∗ Yn+1 AYn = Yn − Yn A∗ Yn+1 A = Yn Xn+1 . It follows from (2.2) and Lemma 2.1 that Xn+1 Xn = (I − A∗ Yn+1 A)Xn = Xn − A∗ Yn+1 AXn = Xn − Xn A∗ Yn+1 A = Xn Xn+1 . This completes the proof. Lemma 2.4. If 0 < M ≤ N , 0 < P ≤ Q, and {M, N, P, Q} is a commuting family, then M P ≤ N Q. Proof. Since M , N , P , Q are positive definite matrices, and {M, N, P, Q} is a commuting family. By Theorem 2.5.5 in [12], there is a unitary U such that U ∗ M U = Λ, U ∗ N U = Ω, U ∗ P U = Σ, U ∗ M U = Γ, where Λ, Ω, Σ, and Γ are diagonal. Since M , N , P , Q are positive definite matrices, Λ, Ω, Σ, and Γ are positive diagonal matrices. Because {M, N, P, Q} is a commuting family, 1

1

1

1

1

1

1

1

M P = U ΛΣU = U Λ 2 ΣΛ 2 U = P 2 M P 2 ≤ P 2 N P 2 = N 2 P N 2 . Since 0 < P ≤ Q,

1

1

1

1

N 2 P N 2 ≤ N 2 QN 2 = N Q.

(2.3) (2.4)

Combining (2.3) and (2.4), we have M P ≤ N Q. This completes the proof. Now, we prove that the sequence {Xn } in Algorithm M4 is monotone decreasing and converges to the −1 maximal solution X+ , and the sequence {Yn } in Algorithm M4 is monotone increasing and converges to X+ . Theorem 2.5. Let A be normal. If the nonlinear matrix equation (1.1) has a positive definite solution, and the two sequences {Xn } and {Yn } are determined by Algorithm M4, then {Xn } is monotone decreasing and −1 converges to the maximal solution X+ , and {Yn } is monotone increasing and converges to X+ .

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Proof. We will prove that I = X0 ≥ X1 ≥ · · · ≥ Xn ≥ X+ and −1 I = Y0 ≤ Y1 ≤ · · · ≤ Yn ≤ X+ .

Since X+ is a solution of (1.1), i.e., −1 X0 = I ≥ X+ and I ≤ X+ . Also

−1 X+ = I − A∗ X+ A,

X1 = I − A∗ A ≤ I = X0

and −1 X1 = I − A∗ A ≥ I − A∗ X+ A = X+ ,

i.e., I = X0 ≥ X1 ≥ X+ . −1 −1 For the sequence {Yn } we have Y0 = Y1 = I, and since I ≤ X+ , then Y0 = Y1 ≤ X+ . From Lemmas 2.3 and 2.4, on the one hand, Y2 = (I − X1 )Y1 + I = A∗ A + I ≥ I = Y1 = Y0 ; on the other hand −1 −1 Y2 = (I − X1 )Y1 + I ≤ (I − X+ )X+ + I = X+ , −1 i.e., Y0 = Y1 ≤ Y2 ≤ X+ . Assume that the above inequalities are true for n = k, i.e.,

I = X0 ≥ X1 ≥ · · · ≥ Xk ≥ X+

(2.5)

−1 I = Y0 ≤ Y1 ≤ · · · ≤ Yk ≤ X+ .

(2.6)

and Now we prove inequalities for n = k + 1. From (2.5) and (2.6), the sequences {Xn } and {Yn } are Hermitian positive definite for i = 0, 1, . . . , k. According to Algorithm M4, Lemmas 2.3 and 2.4, (2.5), and (2.6), we get Yk+1 = (I − Xk )Yk + I ≥ (I − Xk−1 )Yk−1 + I = Yk , −1 −1 Yk+1 = (I − Xk )Yk + I ≤ (I − X+ )X+ + I = X+ , −1 . Concerning the sequence {Xn }, we have i.e., Yk ≤ Yk+1 ≤ X+

Xk − Xk+1 = A∗ (Yk+1 − Yk )A since Yk+1 ≥ Yk . Hence Xk ≥ Xk+1 . Therefore, −1 Xk+1 = I − A∗ Yk+1 A ≥ I − A∗ X+ A = X+ ,

i.e., Xk ≥ Xk+1 ≤ X+ . This completes the induction for n = k + 1. Thus, I = X0 ≥ X1 ≥ · · · ≥ Xn ≥ X+ and −1 I = Y0 ≤ Y1 ≤ · · · ≤ Yn ≤ X+

are true for all n. Therefore, These are convergent sequences, i.e., lim Xn and lim Yn exist. Taking limit n→∞

n→∞

−1 in Algorithm M4 leads to Y = X −1 and X = I − A∗ X −1 A. Moreover, as each Xn ≥ X+ and Yn ≤ X+ , −1 then X = X+ and Y = X+ , respectively. This completes the proof.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

According to Theorem 2.5, we know that the sequence {Yn } determined by Algorithm M4 is monotone −1 increasing and converges to X+ . From Lemmas 2.1 and 2.3, if A is normal, then {A, Yn , Yn+1 } is a commuting family, n = 0, 1, 2, . . .. So we can amend Algorithm M4 and obtain the following algorithm. ( Y0 = I, M5 : Yn+1 = (AYn )∗ (AYn ) + I, n = 0, 1, . . . . In this algorithm, M5 generates a Hermitian sequence, and requires two matrix-matrix multiplications per iteration. Notice that it only requires to compute X = I −A∗ Y A at the end of the process. It is an inverse-free iterative method. If A is normal, from [9, Theorem 11], the nonlinear matrix equation (1.1) has a solution if and only if ρ(A) ≤ 21 . Therefore, the nonlinear matrix equation (1.1) has a solution if and only if kAk ≤ 21 . Lemma 2.6. Let A be normal. Assume that nonlinear matrix equation (1.1) has a positive definite solution and the sequence {Yn } is determined by Algorithm M5. Then {Yn } satisfies kAYn k < 1 for every n = 0, 1, . . .. Proof. Since A is normal and the nonlinear matrix equation (1.1) has a positive definite solution, kAk ≤ 12 . Because Y0 = I, kAY0 k = kAk ≤ 21 < 1. For Y1 we have Y1 = A∗ A + I, thus kAY1 k = kA(A∗ A + I)k ≤ kAk3 + kAk < 1. That is, the inequality holds for n = 0, 1. So, assume that the inequality satisfies n = k, i.e., kAYk k < 1. Now we prove the inequality when n = k + 1. kAYk+1 k = kA ((AYk )∗ (AYk ) + I) k = kA(AYk )∗ (AYk ) + Ak ≤ kAkkAYk k2 + kAk < kAk + kAk ≤ 1. This completes the induction for n = k + 1 and the lemma. Lemma 2.7. Let A be normal. The maximal solution X+ of the nonlinear matrix equation (1.1) commutes with A. Proof. If A is normal, by [18], we have X+ =

i 1h 1/2 I + (I − 4A∗ A) . 2

So, AX+ = X+ A. This completes the proof. Theorem 2.8. Let A be normal. If the nonlinear matrix equation (1.1) has a positive definite solution, then the sequence {Yn } determined by Algorithm M5 satisfies −1 −1 −1 kYn+1 − X+ k ≤ kAX+ kkYn − X+ k

for all n large enough. Proof. Since the nonlinear matrix equation (1.1) has a positive definite solution, X+ = is the maximal solution. Then −1 X+ + A∗ X+ A = I.

1 2

h i 1/2 I + (I − 4A∗ A)

−1 Multiplying by X+ on the right, we obtain −1 −1 −1 X+ = (X+ A)∗ AX+ + I.

By Lemma 2.7, −1 −1 ∗ −1 X+ = (AX+ ) (AX+ ) + I.

By Algorithm M5, Yn+1 = (AYn )∗ (AYn ) + I.

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(2.7)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

−1 Subtracting X+ from both sides of (2.7) we have that −1 −1 Yn+1 − X+ = (AYn )∗ (AYn ) + I − X+ −1 ∗ −1 = (AYn )∗ (AYn ) + I − (AX+ ) (AX+ )+I




−1 ∗ −1 = (AYn )∗ (AYn ) − (AX+ ) (AX+ )

= =

−1 ∗ −1 ∗ (AYn ) (AYn ) − (AX+ ) (AYn ) + (AX+ ) (AYn ) −1 ∗ ∗ −1 ∗ −1 (Yn − X+ ) A (AYn ) + (AX+ ) A(Yn − X+ ). ∗

(2.8) −

−1 ∗ −1 (AX+ ) (AX+ )

−1 Taking norms in (2.8) and recalling that lim Yn = X+ and kAk ≤ 21 , we have that n→∞

kYn+1 −

−1 X+ k

−1 ∗ ∗ −1 ∗ −1 = k(Yn − X+ ) A (AYn ) + (AX+ ) A(Yn − X+ )k
−1 −1 ≤ kYn − X+ kkAk kAYn k + kAX+ k −1 −1 ≤ kAX+ kkYn − X+ k.

This completes the proof.

3

Numerical Experiments

To illustrate the performance of our method described in the previous section, in this section several interesting examples are given, which were carried out using MATLAB on a PC computer. We report the number of required iterations (denoted as IT), the norm of the residual (denoted as Res), the computing time in seconds (denoted as CPU), and the number of matrix-matrix (denoted as MM) multiplications required when the process is stopped. Assume A is normal and (1.1) has a solution. Then i 1h 1/2 X+ = I + (I − 4A∗ A) (3.1) 2 is the maximal solution, and if A is nonsingular, i 1h 1/2 X− = I − (I − 4A∗ A) . 2

(3.2)

is the minimal solution [18, Theorem 4.1]. This allows us to test the local convergence behavior of the five methods by taking the computed X+ by using (3.1) as the accurate maximal solution and X− by using (3.2) as the accurate minimal solution. In our implementation, −1 −1 kYn − X+ k ≤ kX+ k × 10−9 is used as the termination criterion for M1, M2, M4, and M5, and kXn − X− k ≤ kX− k × 10−9 is used as the termination criterion for M3. We compare our iteration M5, with the inverse-free methods M1, M2, M3 and M4 for solving (1.1). In all cases we describe also the initial guess for which convergence is guaranteed. Since M3 converges to the minimal solution, we use M3 to find the minimal solution Y− of the dual equation Y + AY −1 A∗ = I. We can obtain the maximal solution of (1.1) through X+ = I − Y− . The value of ”Res” in our tables reports kF (Xn )kF = kXn + A∗ Xn−1 A − IkF for M1, M2, M3, M4 and M5 when the process is stopped. Experiment 3.1. In this test, the matrix A is from [8] using Example 3.1   0.2 −0.1 −0.5 0.1  1  −0.1 0.6 −0.5 0.7 A= 32 −0.5 −0.5 0.1 0.8 0.1 0.7 0.8 0.5

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Table 1: Performance of M1, M2, M3, M4 and M5 to solve (1) for Experiment 3.1. Scheme IT kF (Xn )kF MM M1 6 8.7390e-12 24 M2 4 2.9322e-14 16 M3 7 1.7493e-11 21 M4 4 1.1655e-13 12 M5 3 1.1655e-13 6 Since A is normal and kAk = 0.0412 ≤ 0.5, from [9, Theorem 11], the nonlinear matrix equation (1.1) has a solution. In Table 1, we can see that M1 requires more matrix-matrix multiplications than the other methods to achieve convergence. M3 requires more iterations than the other four methods to satisfy the stopping criterion. M5 carries out fewer iterations and matrixCmatrix multiplications than all the other methods. For this experiment, we could say that M5 is the best option. Experiment 3.2. In this test, let 

4 −1 −1 4 −1 1   .. .. A=  . . 14   −1

    ..  ∈ Cm×m . .  4 −1 −1 4

Table 2: Iterations and the number of matrix-matrix products for Experiment 3.2. M1 M2 M3 M4 M5 m kAk IT MM IT MM IT MM IT MM IT MM 4 0.4013 29 116 16 64 15 45 22 66 21 42 8 0.4200 33 132 18 72 17 51 26 78 25 50 16 0.4261 34 136 19 76 18 54 27 81 26 52 32 0.4279 35 140 19 76 18 54 28 84 27 54 64 0.4284 35 140 19 76 18 54 28 84 27 54 128 0.4285 35 140 19 76 18 54 28 84 27 54 256 0.4286 35 140 19 76 18 54 28 84 27 54 512 0.4286 35 140 19 76 18 54 28 84 27 54 1024 0.4286 35 140 19 76 28 84 27 54

m 4 8 16 32 64 128 256 512 1024

Table 3: CPU time(s) for Experiment 3.2. M1 M2 M3 M4 0.002191 0.000315 0.000634 0.000986 0.002479 0.000821 0.000937 0.001263 0.005854 0.001868 0.003523 0.002451 0.012124 0.008553 0.006721 0.006701 0.019125 0.014864 0.017950 0.022326 0.075103 0.046237 0.075727 0.065126 0.522263 0.300470 0.479174 0.383668 3.557083 2.176928 20.980815 2.392791 46.442456 24.082342 27.282594

M5 0.000246 0.000568 0.000956 0.004385 0.013876 0.049925 0.290019 1.886120 21.436297

According to Table 2, we know that kAk ≤ 0.5. Since A is normal and A is nonsingular, the nonlinear matrix equation X +A∗ X −1 A = I has a solution. In this test, the matrix sequence of Xn in M3 is badly scaled

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m 4 8 16 32 64 128 256 512 1024

Table 4: Errors M1 M2 1.9737e-10 5.8016e-11 1.8449e-10 6.3914e-11 3.2512e-10 5.5185e-11 2.5631e-10 9.4206e-11 3.8878e-10 1.4330e-10 5.6750e-10 2.0941e-10 8.1479e-10 3.0082e-10 1.1608e-9 4.2868e-10 1.647e-19 6.0853e-10

for Experiment 3.2. M3 M4 2.2961e-10 1.3549e-10 2.1623e-10 9.5667e-11 1.8543e-10 1.4121e-10 3.0952e-10 1.1751e-10 4.6222e-10 1.8003e-10 6.7030e-10 2.6386e-10 9.5949e-10 3.7953e-10 1.3654e-09 5.4118e-10 7.6847e-10

M5 1.3549e-10 9.5667e-11 1.4121e-10 1.1751e-10 1.8003e-10 2.6386e-10 3.7953e-10 5.4118e-10 7.6847e-10

when m = 1024. In Table 2, we can see that M1 requires more iterations and matrix-matrix multiplications than the other methods to satisfy the stopping criterion. We can also observe that M5 needs more iterations than M2 and M3 to reach convergence, but it carries out fewer matrix-matrix multiplications than M1, M2, M4. Table 2 shows that M5 and M3 require the same number of matrix-matrix multiplications, but M5 considerably outperforms M3 in CPU time from Table 3. From Table 3 we observe that M5 outperforms the other methods in CPU time. From our numerical results, we can see that M1 is the most expensive iteration out of the five methods. For this experiment, we could say that M5 is the best option.

Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 11501126, No. 11471122), the Youth Natural Science Foundation of Jiangxi Province (No. 20151BAB211011), and the Foundation for Supporting Local Colleges and Universities of China - Applied Mathematics Innovative Team Building.

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On the Ulam-Hyers Stability of Some Differential Equations involving Hadamard Fractional DerivativesI Xiulan Yua,∗, Zhuoyan Gaoa , Mengmeng Lib a College

of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030031, China b Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China

Abstract In this paper, we show the Ulam-Hyers stability and Ulam-Hyers-Rassias stability criterion for nonlinear Hadamard fractional relaxation differential equations on compact and unbounded time intervals. More explicit Ulam-Hyers stability and Ulam-Hyers-Rassias results are presented by virtue of estimation of Mittag-Leffler functions. Keywords: Hadamard fractional derivative, Relaxation differential equations, Ulam-Hyers stability, Ulam-Hyers-Rassias stability, Mittag-Leffler functions.

1. Introduction The widely application of fractional differential equations arise in various areas of physics and engineering (see [1, 2, 3, 4]). During the past decades, fractional differential equations has been more and more recognized as an alternative model to the classical differential equations. There are many interesting advance on the theory analysis for Caputo type and Riemann-Liouville type fractional differential equations as well as Hadamard type fractional differential equations (see, for example, [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). The well-known Ulam stability problem of functional equations originated are posed in 1940. Numerous monographs and special issues have appeared devoted to the theory of Ulam stability for functional equations and differential equations (see for example [16, 17, 18, 19, 20, 21, 22, 23]). Recently, Li and Wang [24] explore some fundamental properties of continuity, integrable estimation, asymptotic property on Mittag-Leffler functions for a Hadamard fractional differential equation with constant coefficient and present existence results for such equation by using fixed point theorems. However, to our knowledge, Ulam’s stability results for nonlinear Hadamard fractional differential equation with constant coefficient have not been investigated extensively. Especially, there are few research on the Ulam’s stability for this kind of equation on noncompact interval. I This

work was partially supported the Youth Science Foundation of Shanxi University of Finance and Economic. author. Email addresses: [email protected] (Xiulan Yu), [email protected] (Zhuoyan Gao), [email protected] (Mengmeng Li) ∗ Corresponding

Preprint submitted to Journal of Computational Analysis and Applications

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In this paper, we investigate Ulam’s type stability of Hadamard fractional differential equations with constant coefficient λ ∈ R \ {0} of the type: α H D1+ y(x)

where

α H D1+

= λy(x) + f (x, y(x)), 0 < α < 1, x ∈ J = (1, e] or (e, ∞),

(1)

denotes the left-sided Hadamard fractional derivative of order α with the low limit 1

(see Definition 2.2), and nonlinear term f : J × R → R is a given function satisfying some certain conditions. Let ε > 0 and φ : J → R+ be a continuous function. Set J¯ := [1, e] or [e, ∞). Consider equation (1) and the following inequalities: |H D1α+ z(x) − λz(x) − f (x, z(x))| ≤ ε, 0 < α < 1, x ∈ J,

(2)

|H D1α+ z(x) − λz(x) − f (x, z(x))| ≤ εφ(x), 0 < α < 1, x ∈ J.

(3)

and

Definition 1.1. Equation (1) is Ulam-Hyers stable if there exists a constant c > 0 such that for each ¯ R) of inequality (2) there exists a solution y ∈ Cγ,ln (J, ¯ R) ε > 0 and for each solution z ∈ Cγ,ln (J, of equation (1) with |z(x) − y(x)| ≤ cε, x ∈ J. ¯ R) is a solution of inequality (2) if and only if there exists Remark 1.2. A function z ∈ Cγ,ln (J, ¯ R) such that (i) |h(x)| ≤ ε, x ∈ J, (ii) a function h ∈ Cγ,ln (J,

α H D1+ z(x)

= λz(x) + f (x, z(x)) +

h(x), x ∈ J. Definition 1.3. Equation (1) is Ulam-Hyers-Rassias stable stable if there exists a constant c > 0 ¯ R) of inequality (3) there exists a solution such that for each ε > 0 and for each solution z ∈ Cγ,ln (J, ¯ R) of equation (1) with y ∈ Cγ,ln (J, |z(x) − y(x)| ≤ cεφ(x), x ∈ J. ¯ R) is a solution of inequality (3) if and only if there exists a Remark 1.4. A function z ∈ Cγ,ln (J, ˜ ∈ Cγ,ln (J, ˜ ¯ R) such that (i) |h(x)| function h ≤ εφ(x), x ∈ J, (ii)

α H D1+ z(x)

= λz(x) + f (x, z(x)) +

˜ h(x), x ∈ J. The rest of this paper is organized as follows. In Section 2, some notations and preparation results are given. In Section 3, some useful remarks on bounded and unbounded time intervals are presented. Section 4 is devoted to to give Ulam-Hyers stability and Ulam-Hyers-Rassias stability criteria of the equation (1) on bounded and unbounded time intervals respectively. Finally, the reason on the equation (1) is not necessary Ulam-Hyers-Rassias stable is analysed.

2

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2. Preliminaries Let Y be a Banach space endowed with the norm ∥ · ∥Y . Set J := (1, e] or (e, ∞). Denote C(J, Y ) be the Banach space of all continuous functions from J into Y with the norm ∥y∥C = supx∈J ∥y(x)∥Y . ¯ Y ) := {y(x) : y : J → Y is continuous such that For 0 < µ < 1, we denote the set Cµ,ln (J, ( x )µ ¯ Y )}. Following [1, Theorem 3.29], Cµ,ln (J, ¯ Y ) is a Banach space with the norm ln a y(x) ∈ C(J, ∥y∥Cµ,ln = ∥(ln x)µ y(x)∥C = sup ∥(ln x)µ y(x)∥Y . x∈J¯

The following definitions and lemmas will be used in this paper. Definition 2.1. (see [1, p.110, (2.7.1)]) The left-sided Hadamard fractional integral of order α ∈ R+ of function y(t) are defined by (H J α a+ y)(t) =

1 Γ(α)

)α−1 ∫ t( t ds ln y(s) , (0 < a < t ≤ b), s s a

where Γ(·) is the Gamma function. Definition 2.2. (see [1, p.111, (2.7.7)]) The left-sided Hadamard fractional derivative of order α ∈ [n − 1, n), n ∈ Z+ of function y(t) are defined by (H Dα a+ y)(t) =

1 Γ(n − α)

( t

d dt

)n ∫ t ( )n−α+1 t ds ln y(s) , (0 < a < t ≤ b). s s a

Lemma 2.3. (see [4, Theorem 2.3]) Let α, β ∈ (0, 1] and β < 1 + α be arbitrary. Then the following statements hold: (i) For all z > 0, we have Eα,β (z) :=

∞ ∑ k=0

where S(v, z) =

1 zk 1 1−β = z α exp(z α ) + Γ(αk + β) α

∫

∞

S(v, z)dv, 0

1 v sin(π(1 − β)) − z sin(π(1 − β + α)) 1 1−β v α exp(−v α ) . πα v 2 − 2vz cos(πα) + z 2

(ii) For all z < 0, we have

∫

∞

Eα,β (z) =

S(v, z)dv, 0

where S(v, z) =

1 v sin(π(1 − β)) − z sin(π(1 − β + α)) 1 1−β v α exp(−v α ) . πα v 2 − 2vz cos(πα) + z 2

We note that Eα (z) = Eα,1 (z). By virtue of Lemma 2.3, Li and Wang [24] derived the following useful results for two-parameter Mittag-Leffler function. Lemma 2.4. (see [24, Theorem 2.11]) Let λ > 0 be arbitrary, α, β ∈ (0, 1] and β < 1 + α. Denote { } α sin(βπ)Γ(2α − β + 1) α| sin(π(β − α))|Γ(α − β + 1) ω(α, β, λ) = max , . λ2 απ sin2 (πα) λαπ sin2 (πα) 3

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For all x ∈ (1, ∞), we have ω(α, α, λ) 1 (ln x)α−1 Eα,α (λ(ln x)α ) − 1 λ 1−α α α exp(λ ln x) ≤ . α (ln x)α+1 In particular, for all x ∈ (1, ∞), Eα (λ(ln x)α ) − 1 exp(λ α1 ln x) ≤ ω(α, 1, λ) . α (ln x)α Further, we give the following integral estimation. Lemma 2.5. Let λ > 0 be arbitrary, α ∈ (0, 1], we have (i) For all x ∈ (1, e], we have ∫ x ( ) dt 1−α 1 α−1 α α α α J1 := (ln x − ln t) Eα,α (λ(ln x − ln t) ) − λ Eα (λ(ln x) ) exp(−λ ln t) t 1 Eα,α (λ) Eα (λ) ≤ + . α λ (ii) For all x ∈ (e, ∞), we have ∫ x ( ) dt 1 J2 := (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α ) − (ln x)α−1 Eα,α (λ(ln x)α ) exp(−λ α ln t) t 1 ( ) 1 1 2 Eα,α (λ) exp(λ α ) + + ω(α, α, λ) + := M (α, λ). ≤ α αλ α λ α1 Proof. (i) For all x ∈ (1, e], we obtain ∫ x ∫ x 1−α 1 dt α−1 dt α α J1 ≤ Eα,α (λ) Eα (λ) (ln x − ln t) exp(−λ ln t) +λ t t 1 1 Eα,α (λ) Eα (λ) + , ≤ α λ where we use the decreasing property of Eα,α (z) for z > 0. (ii) For all x > e, by using Lemma 2.4, we have ∫ x e( ) dt 1 α−1 α α−1 α α I1 := (ln x − ln t) Eα,α (λ(ln x − ln t) ) − (ln x) Eα,α (λ(ln x) ) exp(−λ ln t) 1 t ∫ x ( ) e 1 1 1−α dt α−1 α (ln x − ln t) Eα,α (λ(ln x − ln t) ) − λ α exp(λ α (ln x − ln t)) ≤ α t 1 ∫ x ) e 1 1−α 1 1 dt λ α exp(λ α (ln x − ln t)) − (ln x)α−1 Eα,α (λ(ln x)α ) exp(−λ α ln t) + t 1 α ∫ xe ∫ xe 1 ω(α, α, λ) dt ω(α, α, λ) dt ≤ + exp(−λ α ln t) α+1 t α+1 (ln x − ln t) (ln x) t 1 1 ∫ xe ∫ xe 1 ω(α, α, λ) dt dt ≤ + ω(α, α, λ) exp(−λ α ln t) α+1 t t 1 (ln x − ln t) 1 ω(α, α, λ) ω(α, α, λ) ≤ + 1 α λα ) ( 1 1 . = ω(α, α, λ) + α λ α1 4

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According to Lemma 2.4 again, we have (ln x)α−1 Eα,α (λ(ln x)α ) ≤

1 1 1−α ω(α, α, λ) + λ α exp(λ α ln x). α+1 (ln x) α

Thus, using the decreasing property of Eα,α (z) for z > 0 again, one has ∫ x( ) dt 1 α−1 α α−1 α I2 := (ln x − ln t) Eα,α (λ(ln x − ln t) ) − (ln x) Eα,α (λ(ln x) ) exp(−λ α ln t) x t e ∫ ∫ x x 1 dt α−1 α dt α−1 α (ln x − ln t) Eα,α (λ(ln x − ln t) ) + (ln x) Eα,α (λ(ln x) ) exp(−λ α ln t) ≤ x t xe t e ∫ x dt ≤ Eα,α (λ) (ln x − ln t)α−1 x t e ∫ x ( ) 1−α 1 1 ω(α, α, λ) λ α dt α ln t) + α (ln x − ln t)) + exp(−λ exp(λ α+1 x (ln x) α t e ∫ x ∫ x 1 1 1 1−α dt dt Eα,α (λ) ω(α, α, λ) + exp(−λ α ln t) + λ α exp(λ α (ln x − ln t)) ≤ x α (ln x)α+1 xe t α t e 1

≤ ≤

1 1 Eα,α (λ) ω(α, α, λ) exp(−λ α ) 1 1−α − + λ α λ− α exp(λ α ) + 1 1 α α α α λ λ 1 Eα,α (λ) ω(α, α, λ) exp(λ α ) + + . 1 α αλ λα

From above, we obtain 1

Eα,α (λ) exp(λ α ) J2 ≤ I1 + I2 ≤ + + ω(α, α, λ) α αλ

(

1 2 + 1 α λα

) .

The proof is completed To end this section, we recall the following inequality which will be used in the sequel. Lemma 2.6. (see [25, Lemma 23]) If λ, v, w > 0, then for any t > a, a > 0, we have ( )1−v ∫ t ( )v−1 ( t t s )λ−1 ( s )−w ds ln ln ln ≤ Cw−λ , a s a a s a where C is a positive constant independent of the time variable t.

3. Some useful lemmas and remarks Now we plan to give the following integral estimation. Lemma 3.1. Let λ > 0, z ∈ Cγ,ln ([1, ∞), R) be a solution of inequality (2). Then z is a solution of the following inequality: ∫ x dt α−1 α z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) E (λ(ln x − ln t) )f (t, z(t)) α,α t 1 1 ∫ x ω(α, α, λ)ε dt εxλ α ≤ + , α+1 t αλ 1 (ln x − ln t) + where c0 = H J 1−α 1+ z(1 ).

5

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Proof. According to the Remark 1.2, we have α H D1+ z(x)

= λz(x) + f (x, z(x)) + h(x), x ∈ (1, ∞).

By [1, p.234,(4.1.89)-(4.1.95)] or [26, p.182, (7.2.60)-(7.2.64)], we obtain ∫ x dt α−1 α z(x) = (ln x) Eα,α (λ(ln x) )c0 + (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α )f (t, z(t)) t 1 ∫ x dt + (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α )h(t) , x ∈ (1, ∞). t 1 For all 1 < x < ∞, using Lemma 2.4, one has ∫ x dt α−1 α z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) Eα,α (λ(ln x − ln t) )f (t, z(t)) t 1 ∫ x dt = (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α )h(t) t ∫1 x ( ) 1 dt 1 1−α ≤ (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α ) − λ α exp(λ α (ln x − ln t)) h(t) α t 1 ∫ x 1 dt 1 1−α + λ α exp(λ α (ln x − ln t))h(t) α t 1 ∫ x ∫ x 1 dt 1 1−α ω(α, α, λ)ε dt +ε λ α exp(λ α (ln x − ln t)) ≤ α+1 t (ln x − ln t) α t 1 ∫1 x 1 ω(α, α, λ)ε dt ε ε ≤ + exp(λ α ln x) − 1 α+1 t αλ λα 1 (ln x − ln t) 1 ∫ x ω(α, α, λ)ε dt εxλ α ≤ + . α+1 t αλ 1 (ln x − ln t) The proof is completed. Remark 3.2. Note that for some fixed point x0 and x0 > δ > 1, we have ∫ x0 ∫ δ 1 dt 1 dt 1 = lim = lim [(ln x0 − ln δ)−α − (ln x0 )−α ] = ∞, α+1 t α+1 t δ→x δ→x (ln x − ln t) (ln x − ln t) α 0 0 0 0 1 1 which yields that it is not possible to obtain some explicit estimation in this case. Next, we divide our time interval (1, ∞) into two subintervals (1, e] and (e, ∞). Remark 3.3. Let λ > 0, z ∈ Cγ,ln ([1, e], R) be a solution of inequality (2). Then z is a solution of the following inequality: ∫ x dt α−1 α z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) E (λ(ln x − ln t) )f (t, z(t)) α,α t 1 ∫ x ( ) 1−α 1 dt ≤ (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α ) − λ α Eα (λ(ln x)α ) exp(−λ α ln t) h(t) t 1 ∫ x 1−α 1 dt + λ α Eα (λ(ln x)α ) exp(−λ α ln t)h(t) t ( 1 ) Eα,α (λ) Eα (λ) εEα (λ) ≤ ε + + α λ λ ( ) Eα,α (λ) 2Eα (λ) ≤ ε + , x ∈ (1, e], α λ where we use Lemma 2.5(i), Remark 1.2 and Eα,α (z) is an increasing function for z > 0. 6

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Remark 3.4. Let λ > 0, z ∈ Cγ,ln ([e, ∞), R) be a solution of inequality (2). Then z is a solution of the following inequality: ∫ x dt α−1 α z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) E (λ(ln x − ln t) )f (t, z(t)) α,α t 1 ∫ x ( ) 1 dt ≤ (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α ) − (ln x)α−1 Eα,α (λ(ln x)α ) exp(−λ α ln t) h(t) t 1 ∫ x 1 dt + (ln x)α−1 Eα,α (λ(ln x)α ) exp(−λ α ln t)h(t) t 1 ∫ x ( ) 1 1 1 1−α dt ω(α, α, λ) α ln t) + α α (ln x − ln t)) exp(−λ λ exp(λ ≤ εM (α, λ) + ε α+1 (ln x) α t 1 ≤ εM (α, λ) + (

εω(α, α, λ)

1

εx α

+ 1 1 λα λα ) 1 ω(α, α, λ) x α + , = ε M (α, λ) + 1 1 λα λα

where we use Lemma 2.4, Lemma 2.5(ii) and Remark 1.2. Remark 3.5. Let λ < 0, z ∈ Cγ,ln ([1, ∞), R) be a solution of inequality (2). Then z is a solution of the following inequality: ∫ x dt α−1 α z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) E (λ(ln x − ln t) )f (t, z(t)) α,α t 1 ε(ln x)α ≤ , x ∈ (1, ∞), Γ(α + 1) where we use the fact Eα,α (z) ≤

1 Γ(α)

for z < 0.

Remark 3.6. Let λ < 0, z ∈ Cγ,ln ([1, ∞), R) be a solution of inequality (3). Then z is a solution of the following inequality: ∫ x dt α−1 α z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) Eα,α (λ(ln x − ln t) )f (t, z(t)) t 1 ∫ x ˜ dt = (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α )h(t) t 1 ∫ x ε dt ≤ (ln x − ln t)α−1 φ(t) , x ∈ (1, ∞), Γ(α) 1 t where we use the fact Eα,α (z) ≤

1 Γ(α)

for z < 0 again.

4. Main results 4.1. Ulam-Hyers stability results We introduce the following assumptions: (A1 ) f : (1, e] × R → R be a function such that f (·, y(·)) ∈ Cγ ln [1, e], 1 − α ≤ γ < 1. (A2 ) There exists L > 0 such that |f (x, y) − f (x, z)| ≤ L|y − z| for each x ∈ J and all y, z ∈ R. (A3 ) ω = 1 − LEα,α (λ)B[1 − γ, α] > 0. 7

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Theorem 4.1. Assume that (A1 ), (A2 ), and (A3 ) are satisfied. Then equation (1) with λ > 0 is Ulam-Hyers stable on J = (1, e]. Proof. Let z ∈ Cγ,ln ([1, e], R) be a solution of inequality (2). By (A1 ), (A2 ), and (A3 ), one can apply Banach fixed point theorem to derive   Dα+ y(x) = λy(x) + f (x, y(x)), 0 < α < 1, x ∈ J, H 1  J 1−α z(1+ ) = c , H

0

1+

has the unique solution ∫

x

y(x) = (ln x)α−1 Eα,α (λ(ln x)α )c0 +

(ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α )f (t, y(t)) 1

dt . t

By using Lemma 2.5(i) and Remark 3.3, we have

= ≤ + ≤ ≤

|(z(x) − y(x))(ln x)γ | ( ) ∫ x dt α−1 α (ln x)γ z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) Eα,α (λ(ln x − ln t) )f (t, y(t)) t 1 ) ( ∫ x dt α−1 α (ln x)γ z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) Eα,α (λ(ln x − ln t) )f (t, z(t)) t 1 ∫ x dt (ln x)γ (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α )(f (t, z(t)) − f (t, y(t))) t 1 ) ( ∫ x dt Eα,α (λ) 2Eα (λ) + + LEα,α (λ) (ln x)γ (ln x − ln t)α−1 (ln t)−γ ∥z − y∥Cγ,ln ε(ln x)γ α λ t 1 ( ) Eα,α (λ) 2Eα (λ) ε(ln x)γ + + LEα,α (λ)B[1 − γ, α]∥z − y∥Cγ,ln , α λ

which yields that ∥z − y∥Cγ,ln

ε ≤ ω

(

Eα,α (λ) 2Eα (λ) + α λ

Thus, 1 |z(x) − y(x)| ≤ cε, c = ω

(

) (ln x)γ .

Eα,α (λ) 2Eα (λ) + α λ

) > 0.

The proof is completed. Remark 4.2. Let λ < 0. Assume that (A1 ) and (A2 ) are satisfied. One can use the above similar methods via Remark 3.5 to check that the equation (1) is Ulam-Hyers-Rassias stable on J = (1, ∞) provided by ρ = 1 −

LB[1−γ,α] Γ(α)

> 0. That is,

|z(x) − y(x)| ≤ cεφ(x), x ∈ (1, ∞), c = Example 4.3. Let α = 23 , λ = 2 3

1 2

ε > 0, φ(x) = (ln x)α . ρΓ(α + 1)

and γ = 12 . Consider the fractional order differential equation

H D1+ y(x)

=

1 1 y(x) + sin2 y(x), x ∈ (1, e], l > 0, 2 l

(4)

and the inequality 2 1 1 |H D13+ z(x) − z(x) − sin2 z(x)| ≤ ε, x ∈ (1, e]. 2 l

(5)

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Let z ∈ Cγ,ln ([1, e], R) be a solution of inequality (5). Then there exists a function h(x) = ε ln x ∈ Cγ,ln ([1, e], R) such that |h(x)| ≤ ε, x ∈ (1, e], and

2 3

H D1+ z(x)

= 21 z(x) + 1l sin2 z(x) + h(x), x ∈ (1, e].

Define f (x, y(x)) = 1l sin2 (x), x ∈ (1, e] and L = 2l . Obviously, (A1 ) and (A2 ) hold. Moreover, ( ) [ ] we choose l = 6E 32 , 23 12 B 12 , 23 , then ] ( ) [ 1 2 2 2 1 B , = > 0, ω = 1 − E 23 , 23 l 2 2 3 3 which implies that (A3 ) holds. According to Theorem 4.2, we have ( ( ) ( )) 3ε 1 1 |z(x) − y(x)| ≤ 1.5E 23 , 23 + 4E 23 . 2 2 2 ( ( ) ( )) Thus, equation (4) is Ulam-Hyers stable on (1, e] with c = 32 1.5E 23 , 23 12 + 4E 23 12 . 4.2. Ulam-Hyers-Rassias stability result Next, we introduce the following assumptions: (B1 ) Let λ < 0 and γ = 1 − α. (B2 ) f : J × R → R is jointly continuous and there exists L(·) ∈ C([1, ∞), R+ ) such that |f (x, y) − f (x, z)| ≤ L(x)|y − z| for each x ∈ J and all y, z ∈ R, where L(·) satisfying ∫ x dt e x)α−1 , C e > 0. (ln x)γ (ln x − ln t)α−1 (ln t)−γ L(t) ≤ C(ln t 1

(6)

(B3 ) There exists a φ(·) ∈ C([1, ∞), R+ ) such that ∫ x dt b b > 0. C (ln x − ln t)α−1 φ(t) ≤ Cφ(t), t 1 (B4 ) ω ′ = 1 −

e C Γ(α)

(7)

> 0.

Theorem 4.4. Let λ < 0. Assume that (B1 ), (B2 ), (B3 ) and (B4 ) are satisfied. Then equation (1) is Ulam-Hyers-Rassias stable on J = (1, ∞). Proof. Note that the fact Eα,α (z) ≤

≤

≤

≤ ≤

1 Γ(α)

for z < 0. By Remark 3.6, one can obtain

|(z(x) − y(x))(ln x)γ | ( ) ∫ x dt α−1 α (ln x)γ z(x) − (ln x)α−1 Eα,α (λ(ln x)α )c0 − (ln x − ln t) E (λ(ln x − ln t) )f (t, z(t)) α,α t 1 ∫ x dt + (ln x)γ (ln x − ln t)α−1 Eα,α (λ(ln x − ln t)α )(f (t, z(t)) − f (t, y(t))) t 1 ∫ 1 x dt (ln x)γ (ln x − ln t)α−1 εφ(t) Γ(α) 1 t ∫ x 1 dt + (ln x)γ (ln x − ln t)α−1 L(t)|y − z| Γ(α) 1 t ∫ x ε(ln x)γ b 1 dt (ln x)γ (ln x − ln t)α−1 (ln t)−γ L(t) ∥y − z∥Cγ,ln Cφ(x) + Γ(α) Γ(α) 1 t e ε(ln x)γ b C Cφ(x) + (ln x)γ+α−1 ∥y − z∥Cγ,ln . Γ(α) Γ(α) 9

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This yields that

(

e C 1− Γ(α)

) ∥y − z∥Cγ,ln ≤

This implies that |y(x) − z(x)| ≤

b Cε ω ′ Γ(α)

ε(ln x)γ b Cφ(x). Γ(α)

φ(x), x ∈ J.

The proof is completed. Example 4.5. Let α = 21 , λ = − 12 and γ = 21 . Consider the fractional order differential equation 1 2

H D1+ y(x)

1 1 = − y(x) + 3 sin2 y(x), x ∈ (1, ∞), l > 0, 2 lx

(8)

and the inequality 1 1 1 |H D12+ z(x) − z(x) − 3 sin2 z(x)| ≤ εφ(x), x ∈ (1, ∞). 2 lx

Define f (x, y(x)) =

1 lx3

sin2 (x) and L(x) =

1 lx3 ,

(9)

x ∈ (1, ∞). Let z ∈ Cγ,ln ([1, ∞), R) be a solution

of inequality (9). There exists a function h(x) = xε (ln x)α−1−γ ∈ Cγ,ln ([1, ∞), R) such that |h(x)| ≤ ε(ln x)α−1−γ := φ(x), x ∈ (1, ∞). Moreover, (5) via Lemma 2.6 reduces to ∫ x 1 1 1 2 dt 1 2C √ (ln x) 2 (ln x − ln t)− 2 (ln t)− 2 3 ≤ 3(ln x)− 2 , x ∈ (1, ∞), lt t l 1 and (9) reduces to ∫ x dt ˆ ˆ (ln x)γ (ln x − ln t)α−1 ε(ln x)α−1−γ ≤ Cε(ln x)α−1−γ = Cε(ln x)−1 , Cˆ > 0, x ∈ (1, ∞). t 1 From above, (B1 ), (B2 ) and (B3 ) hold. Now we choose l =

Γ(0.5) √ 4 3C

√ 3

and ω ′ = 1 − 2Cl

=

1 2

> 0, which

implies that (B4 ) holds. According to Theorem 4.4, we have |z(x) − y(x)| ≤

ˆ 2Cε (ln x)−1 . Γ(0.5)

Thus, equation (8) is Ulam-Hyers-Rassias stable on (1, ∞) with c =

ˆ 2C Γ(0.5)

and φ(x) = ε(ln x)−1 , x ∈

(1, ∞). 4.3. Final remarks Let λ > 0. Assume that (B1 ) and (B2 ) are satisfied. It seems that we can not use the above approach to discuss Ulam-Hyers-Rassias stability of the equation (1) with λ > 0 on J = (e, ∞). In fact, by Remark 3.4, one has ( |(z(x) − y(x))(ln x) | γ

≤

εM (α, λ) +

εω(α, α, λ) 1

λα

1

+

εx α 1

λα

) (ln x)γ

∫ x dt γ α−1 α (ln x) (ln x − ln t) + Eα,α (λ(ln x − ln t) )L(t)|y(t) − z(t)| t ( 1 ) 1 εω(α, α, λ) εx α ≤ εM (α, λ) + + 1 (ln x)γ + Υ(x)∥y − z∥Cγ,ln , (10) 1 λα λα 10

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where

∫ x dt Υ(x) = (ln x)γ Eα,α (λ(ln x)α ) (ln x − ln t)α−1 L(t)(ln t)−γ . t 1

Set L(t) = t−w , w > 0. According to Lemma 2.6, we have ∫ x γ α α−1 −γ −w dt Υ(x) = (ln x) Eα,α (λ(ln x) ) (ln x − ln t) (ln t) t t ∫1 x (1−γ)−1 −w dt γ α α−1 (ln t) t = (ln x) Eα,α (λ(ln x) ) (ln x − ln t) t 1 α Eα,α (λ(ln x) ) (C > 0) ≤ Cw1−γ (ln x)1−α−γ ≤

λ

1−α α

where we use the fact

Cw1−γ λ α1 +γ x , α 1 1 1−α α exp(λ α ln x) αλ lim x→∞ (ln x)α−1 Eα,α (λ(ln x)α )

Dividing (ln x)γ by (10) we obtain ( |(z(x) − y(x))| ≤

εM (α, λ) +

εω(α, α, λ) λ

1 α

= 1.

1

+

εx α λ

1 α

) +

Υ(x) ∥y − z∥Cγ,ln . (ln x)γ

(11)

Obviously, Υ(x) λ lim = x→∞ (ln x)γ Thus, the term

Υ(x) (ln x)γ ∥y

1−α α

1

cw1−γ xλ α +γ lim = ∞. x→∞ (ln x)γ α

− z∥Cγ,ln in (11) does not vanish. Therefore, the equation (1) with λ > 0 is

not necessary Ulam-Hyers-Rassias stable on J = (e, ∞).

References [1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., 2006. [2] D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional dynamics and control, Springer, 2012. [3] K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, 2010. [4] R. Gorenflo, J. Loutchko, Y. Luchko, Computation of the Mittag-Leffler function Eα,β (z) and its derivative, Fract. Calc. Appl. Anal., 5(2002), 491-518. Correction: Fract. Calc. Appl. Anal., 6(2003), 111-112. [5] I. Podlubny, Fractional differential equations, Academic Press, 1999. [6] R. Hilfer, Application of fractional calculus in physics, World Scientific Publishing Company, Singapore, 2000. [7] V. E. Tarasov, Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media, Springer, HEP, 2011. [8] Y. Zhou, J. Wang, L. Zhang, Basic theory of fractional differential equations, World Scientific, Singapore, 2016. [9] B. Ahmad, S. K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Frac. Calc. Appl. Anal., 17(2014), 348-360.

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ckan, Y. Zhou, A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal., [10] J. Wang, M. Feˇ 19(2016), 806-831 [11] J. Wang, A. G. Ibrahim, M. Feˇ ckan, Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on banach spaces, Appl. Math. Comput., 257(2015), 103-118. [12] J. Wang, Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 39(2015), 85-90. [13] J. Wang, X. Li, M. Feˇ ckan, Y. Zhou, Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal., 92(2013), 2241-2253. [14] B. Ahmad, S. K. Ntouyas, Initial value problems of fractional order Hadamard-type functional differential equations, Electronic J. Differential Equations, 77(2015), 1-9. ˇ Existence and stability of fractional differential equations with Hadamard deriva[15] J. Wang, Y. Zhou, M. Medved, tive, Topol. Meth. Nonlinear Anal., 41(2013), 113-133. [16] L. C˘ adariu, Stabilitatea Ulam-Hyers-Bourgin pentru ecuatii functionale, Ed. Univ. Vest Timi¸soara, Timi¸sara, 2007. [17] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of functional equations in several variables, Birkh¨ auser, 1998. [18] S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Palm Harbor, 2001. [19] J. Wang, Y. Zhou, M. Feˇ ckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comp. Math. Appl., 64(2012) 3389-3405. [20] J. Wang, M. Feˇ ckan, A general class of impulsive evolution equations, Topol. Meth. Nonlinear Anal., 46(2015), 915-934. [21] J. Wang, M. Feˇ ckan, Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395(2012), 258-264. [22] D. Popa, I. Ra¸sa, Hyers-Ulam stability of some differential equations and differential operators, Handbook of Functional Equations, Springer, 2014, 301-322. [23] J. Brzd¸ek, L. Cˇ adariu, K. Ciepli´ nski, Ulam’s type stability and fixed points methods, J. Function Spaces, 2014(2014), Art. ID 829419, 16 pages. [24] M. Li, J. Wang, Analysis of nonlinear Hadamard fractional differential equations via properties of Mittag-Leffler functions, J. Appl. Math. Comput., 51(2016), 487-508. [25] M. D. Kassim, N. E. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abst. Appl. Anal., 2013(2013), Art. ID 605029, 12 pages. [26] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer, 2014.

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LINEARLY STABLE PERIODIC SOLUTIONS FOR LAGRANGIAN EQUATION FENG WANG1,2

AND

SHENGJUN LI3,4

Abstract. In this paper we study the existence and uniqueness of linearly stable periodic solutions for the Lagrangian equation. The proof is based on the eigenvalue theory combined with degree theory. Compared with those results in the literature, our conditions are weaker.

1. Introduction This paper is devoted to the study of the existence and uniqueness of linear stablity of periodic solutions for the following nonlinear scalar Lagrangian equation (1.1)

x ¨ + g(t, x) = 0,

where g(t, x) : R × R → R is a T -periodic function in t and is semilinear in the following sense: There exist T -periodic functions φ, Φ ∈ L1 (0, T ) such that φ(t) ≤ gx (t, x) ≤ Φ(t), uniformly in t ∈ [0, T ]. We say that a T -periodic solution ψ(t) of (1.1) is linearly stable if the linearized equation (1.2)

y¨ + (gx (t, ψ(t))y = 0

is stable. But it is not sufficient to guarantee that ψ(t) is Lyapunov stable as (1.1) is a conservative system, Lyapunov stability of ψ(t) cannot be determined by linearized equation (1.2) and involves higher order approximations of (1.1). Based on this idea, a practical method, now known as the third order approximation, has been developed by Ortega based on the Birkhoff normal forms and the Moser’s twist theorem [17]. After there has been considerable progress on this topic. We refer the reader to [2, 3, 4, 5]. However, an “almost” necessary condition for ψ(t) to be stable is that it is linearly stable. In this direction, it is worth mentioning the example found by Chu [3]. That is, the equilibrium x(t) = 0 of the motion of 2010 Mathematics Subject Classification. Primary 34C25. Key words and phrases. Lagrangian equation; linear stability; periodic solutions; existence; uniqueness. This work was sponsored by Qing Lan Project of Jiangsu Province, and was supported by the National Natural Science Foundation of China (Grant No. 11501055 and No. 11401166), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 15KJB110001), Jiangsu Planned Projects for Postdoctoral Research Funds. Shengjun Li is supported by the National Natural Science Foundation of China (Grant No.11461016), the Scientific Research Foundation of Hainan University (kyqd1544), Hainan Natural Science Foundation(Grant No.20167246). 1

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a pendulum with variable length and relativistic effects  0 x0 √ + l(t) sin x = 0, l(t) > 0, l ∈ C(R/T Z), 1 − x02 is stable if its linearized equation x ¨ + l(t)x = 0 is stable. In this paper it is shown how a topological invariant, the index of an oscillation, can be used to obtain linear stability results. Actually, it will be proved that the index characterizes linearly stable in certain case (see section 2). Our study is mainly motivated by [7], where a result is the following. Theorem 1.1. If there exists a > 0 and Φ ∈ L1 (0, T ) such that (1.3)

a < gx (t, x) ≤ Φ(t),

for all x and a.e. t ∈ [0, T ], and kΦkp < K(2p∗ ) for some p ∈ [1, +∞], then (1.1) has a unique T -periodic solution which is linearly stable. It must be noticed that the strict positiveness assumption of gx (t, x) is crucial for this result since this implied the monotonicity of the nonlinearity and then a method of lower and upper coupled with the monotone iterative technique was used to get existence and linear stability. Unfortunately, in some interesting problems we find that the constant gx (t, x) changes sign and Theorem 1.1 cannot be applied. It should be pointed out that Ortega has presented the index characterized asymptotic stability in dissipative case. See the references [15, 16] and the surveys [18, 19]. In this paper, we try to obtain similar results for the conservative case following the ideas in [15, 16, 18, 19], see Theorem 2.3 below. The purpose of the present paper is to extend Theorem 1.1 which hold when gx (t, x) changes sign, we prove the following theorems. Theorem 1.2. Suppose that g(t, x) ∈ C 1 (R × R) satisfies the following semilinearity condition: there exist T -periodic functions φ, Φ ∈ L1 (0, T ) such that (1.4)

φ(t) ≤ gx (t, x) ≤ Φ(t),

uniformly in t. Furthermore, assume (1.5)

φ(t) > 0 and λ1 (Φ) > 0,

here φ(t) denotes the average of φ(t) over a period and λ1 is antiperiodic eigenvalue. Then (1.1) has a unique T -periodic solution which is linearly stable. Let us recall a lower bound for λ1 (Φ) from [20]. Let us define the positive part of a function Φ as Φ+ = max{Φ, 0}. If the Lp norm kΦ+ kp satisfies kΦ+ kp ≤ K(2p∗ ), p∗ = p/p − 1, then (see (13) in [20]) λ1 (Φ) ≥

 π 2  T

1−

kΦ+ kp  . K(2p∗ )

Here K(q) is the best Sobolev constant in the following inequality: Ckxk2q ≤ kxk ˙ 22 for all x ∈ H01 (0, T ),

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where H01 (0, T ) is a Sobolev space of all the T -periodic absolutely continuous funcRT tions x such that 0 x˙ 2 (t)dt < ∞ with the norm Z T Z T  21 2 kxk1,T = x˙ (t)dt + x2 (t)dt . 0

0

Explicitly, K(q) =

  

2π qT 1+2/q



2 2+q

1−2/q  4 T

Γ( q1 ) Γ( 21 + q1 )

2

, if 1 ≤ q < ∞, if q = ∞.

,

See Talenti [13]. Thus we have the following Corollary 1.3. Assume all conditions of Theorem 1.2 hold except (1.5) and assume φ(t) > 0 and kΦ+ kp < K(2p∗ ), 1 ≤ p ≤ +∞. Then the same conclusion holds. In particular, when p = +∞, we arrive at the usual criterion π2 kΦ+ k∞ < K(2) = 2 . T 2. Linear stability and index 2.1. Hill’s equation and eigenvalue theory. To each function a ∈ L1 (R/T Z), we associate a linear equation (2.1)

x ¨ + a(t)x = 0,

which is called Hill’s equation and there are many studies about it. The book by Magnus and Winkler [11] is a classical reference. Now we recall some standard notions in the theory of Hill’s equations. Denote by Ψ(t) = φ1 (t) + iφ2 (t) the complex-valued solution of (2.1) with the initial data: Ψ(0) = 1 and Ψ0 (0) = i, where φ1 and φ2 are respectively the real and imaginary parts of Ψ. Let   φ1 (t) φ2 (t) M (t) = φ˙ 1 (t) φ˙ 2 (t) be employed for the fundamental matrix solution of X˙ = A(t)X, X(0) = I2 , where the column vector function X(t) = (x(t), x0 (t))T , I2 is the 2 × 2 identity matrix and A(t) is the matrix function   0 1 A(t) = . −a(t) 0 Liouville’s theorem implies that the matrix solution M (t) always satisfies det M (t) = 1. This property motivates our interest in the matrix associated with (2.1) is  φ1 (T ) M (T ) = φ˙ 1 (T )

symplectic group. The monodromy φ2 (T ) φ˙ 2 (T )

 .

Then M is symplectic, i.e., det M = 1. The eigenvalues ρi , i = 1, 2, of M are called the Floquet multipliers of (2.1). They satisfy ρ1 · ρ2 = 1. We can classify (2.1)

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into three types, according to the Floquet multipliers, as either hyperbolic when |ρ1,2 | 6= 1, or elliptic when |ρ1,2 | = 1 but ρ1,2 6= ±1, or parabolic when ρ1,2 = ±1, respectively. Next we introduce some notations on eigenvalues. Consider the eigenvalue problems (2.2)

x00 + (λ + a(t))x = 0

subject to the periodic boundary condition (2.3)

x(0) − x(T ) = x0 (0) − x0 (T ) = 0,

or to the anti-periodic boundary condition (2.4)

x(0) + x(T ) = x0 (0) + x0 (T ) = 0.

We use D D λD 1 (a) < λ2 (a) < · · · < λn (a) · · ·

to denote all eigenvalues of (2.2) with the Dirichlet boundary condition (D) : (2.5)

x(0) = x(T ) = 0.

The following are standard results for eigenvalue theory. See, e.g. Reference [11]. A partial generalization of these results to the one-dimensional p-Laplacian with periodic potentials is given in Reference [14]. Theorem 2.1. There exist two sequences {λn (a) : n ∈ N} and {λn (a) : n ∈ Z+ } of the reals such that (P1 ) they have the following order: −∞ < λ0 (a) < λ1 (a) ≤ λ1 (a) < · · · < λn (a) ≤ λn (a) < · · · and λn (a) → +∞, λn (a) → +∞ as n → ∞. (P2 ) λ is an eigenvalue of (2.2)-(2.3) if and only if λ = λn (a) or λn (a) for some even integer n; λ is an eigenvalue of (2.2)-(2.4) if and only if λ = λn (a) or λn (a) for some odd integer n. (P3 ) (Continuity) λD are continuous functions of q with n (a), λn (a), and λn (a) RT respect to the L1 -metric on q’s: d(a1 , a2 ) = 0 |a1 (t) − a2 (t)|dt. (P4 ) the eigenvalues λn (a) and λn (a) can be recovered from the Dirichlet eigenvalues in the following way: for any n ∈ N, D λn (a) = min{λD n (at0 ) : t0 ∈ R}, λn (a) = max{λn (at0 ) : t0 ∈ R},

here at0 (t) denotes the translation of a(t) : at0 (t) ≡ a(t + t0 ). (P5 ) (Comparison) the comparison results hold for all of these eigenvalues. If a1 ≥ a2 then (2.6)

D λn (a1 ) ≤ λn (a2 ), λn (a1 ) ≤ λn (a1 ), λD n (q1 ) ≤ λn (q2 ),

for any n ∈ N. If a1 (t) ≥ a2 (t) for all t, and a1 (t) > a2 (t) for t in a subset of positive measure, then all of the inequalities in (2.6) are strict. (P6 ) (Nodal structure) The eigenfunction of λ0 (a) do not vanish everywhere. For n ∈ N, the eigenfunctions of λn (a) or λn (a) have exactly n − 1 zeros in the intervals of the form (t0 , t0 + T ).

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2.2. Definition of the index via the Poincar´ e map. In this subsection we assume uniqueness for the initial value problem associated to (1.1). Given ξ = (ξ1 , ξ2 ) ∈ R2 , let x(t; ξ) be the solution of (1.1) satisfying x(0) = ξ1 ,

x0 (0) = ξ2 .

The Poincar´ e map is defined as the mapping PT : DT ⊂ R2 → R2 ,

PT (ξ) = (x(T ; ξ), x(T ˙ ; ξ)),

2

where DT = {ξ ∈ R : x(t; ξ) is defined in [0, T ]}. The standard theory of the Cauchy problem says that DT is open in R2 and PT is a homeomorphism between DT and PT (DT ). In addition, the fixed points of PT correspond to the initial conditions of the T -periodic solutions and the search of T -periodic solutions is reduced to the study of the equation in R2 , ξ = PT (ξ). Let x be a T -periodic solution of 1.1 and ξ0 = (x(0), x0 (0)). The solution x is said to be isolated (periodic T ) if ξ0 is an isolated fixed point of PT . In such case the index of x is defined in terms of the following formula indT (x) = i[PT , ξ0 ], where i refers to the definition of the local fixed point index in the plane employed in [1]. For more information on the index of periodic solutions see [8] and [12]. 2.3. Connection between linear stability and index. In this subsection we proof a few lemmas that are crucial for the proofs of the main results. First we need the following. Definition 2.2. Given a T -periodic solution x(t) of (1.1). It will be said that x(t) is non-degenerate of periodic T if the variational equation is (2.7)

y¨ + gx (t, x)y = 0

has no periodic solutions different from zero of periodic T . Theorem 2.3. Assume that x is a nondegenerate T -periodic solution of (1.1) such that the inequality below holds (2.8)

λ1 (gx (t, x)) ≥ 0,

for t ∈ R. Then x is linearly stable if and only if indT (x)=1. Remark 2.4. Notice that a nondegenerate solution is always isolated and it is assumed that x is a nondegenerate in order to employ linearization techniques. We do not know if Theorem 2.3 is still valid when nondegenerate is replaced by degenerate. Because the computation of the index in the degenerate case ρ1 = ρ2 = 1 is more delicate the previous technique does not work and the index of x depends not only on (2.7) but also on the nonlinear terms of the Taylor expansion of g. Some methods about computation in the degenerate case can be found in [?] or [9] for more details. The crucial step in proof of Theorem 1.2 is the following observation on the Hill equation (2.1).

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Lemma 2.5. Assume that λ1 (a) > 0.

(2.9)

Then problem (2.1) does not admit any negative Floquet multipliers. In particular, (2.1) does not admit any nontrivial subharmonic periodic solution of order 2. Proof. Suppose that there is a nontrivial solution of (2.1) with a negative Floquet multiplier, i.e. x(t + T ) = ρx(t), t ∈ R for some ρ < 0. Hence, there exists t0 ∈ [0, T ] with x(t0 ) = x(t0 + T ) = 0. Thus the corresponding x is a nontrivial solution of the following Dirichlet boundary value problem  x ¨(t) + a(t)x = 0, x(t0 ) = x(T + t0 ) = 0. That is, x is an eigenfunction associated with eigenvalue λD k (a) = 0 for some k ≥ 1 of  x ¨(t) + (λ + a(t))x = 0, x(t0 ) = x(T + t0 ) = 0, D and hence λ1 (a) ≤ λD 1 (a) ≤ λk (a) = 0, contradicting (2.9). Proof of Theorem 2.3. When the inequality in (2.8) is not strict it is elementary to show that x is linearly stable and indT (x) = 1. Therefore it will be assumed that the inequality in (2.8) is strict, at least on a set of positive measure. Denote by ρ1 , ρ2 (|ρ1 | ≥ |ρ2 |) the Floquet multipliers of (2.1). By Lemma 2.5 the multipliers are either conjugate complex or real and positive. In the elliptic case,

ρ1 = ρ2 if and only if indT (x) = sign{det(I2 − M (T ))} = sign{|1 − ρ1 |2 } = 1. In the hyperbolic case, 0 < ρ1 < 1 < ρ2 if and only if indT (x) = sign{det(I2 − M (T ))} = sign{(1 − ρ1 )(1 − ρ2 )} = −1. The parabolic case is excluded because x is nondegenerate 1 cannot be a Floquet multipler. The conclusion now follows from the well-known principle of stability for Hill equation (see Theorem 7.2 in [6]) that periodic system (2.1) is stable in the sense of Lyapunov if and only if (2.1) is elliptic, or is parabolic (ρ1 = ρ2 = ±1) with further property that all solutions of (2.1) satisfy x(t + T ) = x(t), the T -periodic solutions in case ρ1 = ρ2 = 1, or x(t + T ) = −x(t), the T -anti-periodic solutions in case ρ1 = ρ2 = −1. 3. Proof of Theorem 1.2 The proof of existence is based on the following two lemmas. Lemma 3.1. Assume that (3.1)

a(t) > 0 and λ1 (a) > 0.

Then Hill equation (2.1) has only the trivial T -periodic solution.

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Proof. Suppose on the contrary that (2.1) admits a nontrivial T -periodic solution x(t). We claim that x(t) vanishes at some t0 ∈ [0, T ]. If not, then x(t) 6= 0 for all t ˙ ) x(0) ˙ in R. By the periodic boundary conditions, we have x(T ˙ ) = x(0) ˙ and x(T x(T ) = x(0) . Dividing (2.1) by x(t) and integrating by part gives that Z T Z T x(t) ˙ 2 + a(t)dt = 0, 2 0 0 x(t) which contradicts the hypothesis a(t) > 0. So x(t) has a zero in [0,T]. We may assume that x(0) = 0 so that x(0) = x(T ) = 0. Thus the corresponding x is a nontrivial solution of the Dirichlet boundary value problem (2.1)-(2.5) That is, x is an eigenfunction associated with eigenvalue λD k (a) = 0 for some k ≥ 1 of (2.2)-(2.5), D and hence λ1 (a) ≤ λD 1 (a) ≤ λk (a) = 0, contradicting (3.1). Lemma 3.2. Under the conditions of Lemma 3.1, the Hill equation (2.1) is stable. Proof. The proof will be completed using Theorem 2.3. To this end, we compute the local index and consider following parametric equation Lλ = x ¨ + [λa(t) + (1 − λ)a0 ]x = 0,

λ ∈ [0, 1],

2

where 0 < a0 < (π/T ) . Let aλ = λa(t) + (1 − λ)a0 . Since aλ = λa(t) + (1 − λ)a0 > 0, and λ1 (aλ ) ≥ λ λ1 (a) + (1 − λ)λ1 (a0 ) > 0, it follows from Lemma 3.1 that Lλ x = 0 does not admit a nontrivial T -periodic solution. Let B be the -ball of 0, then Lλ x = 0 has no T -periodic solution on ∂B for λ ∈ [0, 1]. By the homotopy invariance properties of the topological degree, we have that ind(L1 , 0) = deg(L1 , B , 0) = deg(L λ , B , 0) 0 1 = 1. = deg(L0 , B , 0) = sgn −a0 0 The conclusion follows from Theorem 2.3. Proof of Theorem 1.2 In order to show that the conditions are sufficient, we divide the proof into two steps. Step 1: Uniqueness. Suppose that x1 (t) and x2 (t) are two T -periodic solutions of (1.1). Then (3.2)

[x1 (t) − x2 (t)]00 + [g(t, x1 (t)) − g(t, x2 (t))] = 0.

Setting x e(t) = x1 (t) − x2 (t), we obtain, from (3.2), that (3.3)

x e00 (t) + β(t)e x(t) = 0,

2) where β(t) = g(t,xx11)−g(t,x . It follows from Lemma 3.1 that x e(t) ≡ 0, which implies −x2 that x1 (t) ≡ x2 (t) for all t ∈ R. Step 2: Existence and linearly stable. Without loss of generality, we may assume that g(t, 0) = 0, for otherwise we can reduce both sides of Eq. (1.1) by g(t, 0). A natural choice for the parametrized equation in applying homotopy invariance property is to take H defined by

(3.4)

Hλ (x) = x ¨(t) + λg(t, x) + (1 − λ)Φ(t)x = 0,

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FENG WANG

AND

SHENGJUN LI

in which Φ(t) is as in Theorem 1.2. We claim that there is R > 0 such that equation (3.4) has no solution on ∂BR in L∞ [0, T ] for all λ ∈ [0, 1]. Suppose the assertion is not true. Let xn = xn (t) be a sequence of T -periodic solutions such that kxn k → ∞ and λn ∈ [0, 1] be the corresponding sequence. Let zn (t) = xkxnn(t)k . First, dividing (3.4) by kxn k, then multiplying by ϕ(t) ∈ CT2 and finally integrating by parts we have that Z Tn h i o zn ϕ¨ + λn g(t, xn ) + (1 − λn )Φ(t)xn · ϕ/kxn k dt = 0. 0

The conditions of Theorem 1.2 imply that {[λn g(t, xn ) + (1 − λn )Φ(t)xn ]/kxn k} is bounded and hence is pre-compact in weak star topology in L1 [0, T ]. Thus there is a subsequence such that g(t, xn )/xn * α(t) and λn → λ. Taking the limit as n → ∞, one obtains that zn → z, Z T (z ϕ¨ + zω(t)ϕ)dt = 0, 0

where ω(t) = λα(t) + (1 − λ)Φ(t) satisfying the conditions of Lemma 3.1. It follows from Lemma 3.1 that z(t) ≡ 0, which contradicts kz(t)k = 1. This shows the boundedness of the periodic solutions of (3.4). By the standard argument one can verify that the C 1 -norm is bounded independently of λ. Next, by applying the homotopy invariance property, we have that deg(H1 , BR , 0) = deg(H0 , BR , 0). From the same reasonings of Lemma 3.2 one proves that deg(H0 , BR , 0) = 1. This completes the existence of T -periodic solution x, and linear stability can be obtained by Lemma 3.2. 4. Final remark Theorem 2.3 can be applied to other kinds of problems. In particular, to the piecewise linear equation x ¨ + µ(t)x+ − ν(t)x− = 0,

(4.1)

where x+ = max{x, 0}, x− = max{−x, 0}, µ, ν ∈ L1 (0, T ). The equation (4.1) is very popular since a series of works of Lazer and McKenna [10] as a simple mathematical model for vertical oscillations of a long-span suspension bridge. The following result can be proved in a way similar to Theorem 1.2 Equation (4.1) has a unique T -periodic solution which is linearly stable if µ(t) > 0, ν(t) > 0 and k max{µ(t), ν(t)}kp < K(2p∗ ), 1 ≤ p ≤ +∞. t∈R

References 1. S. Chow, J.K. Hale, Methods of Bifurcation Theory, Spring-Verlag, 1982. 2. J. Chu, M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst. 21 (2008) 1071-1094. 3. J. Chu, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations 247 (2009) 530-542. 4. J. Chu, M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl. 355 (2009) 830-838.

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5. J. Chu, J. Lei, M. Zhang, Lyapunov stability for conservative systems with lower degree of freedom, Discrete Contin. Dyn. Syst. 16 (2011) 423-443. 6. J.K. Hale, Ordinary Differential Equations, Second ed., Krieger, New York, 1980. 7. P.J. Torres, M. Zhang, A monotone iterative scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr. 251 (2003), 101-107. 8. M.A. Krasnosel’skii, The operator of translation along the trayectories of differential equations, Trans. Math. Monograph 19. American Mathematical Society, Providence, 1968. 9. M.A. Krasnosel’skii, A.I. Perov, A.I. Povolotskiy, and P.P. Zabreiko, Plane Vector Fields, Academic Press, 1966. 10. A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connection with nonlinear analysis, SIAM Rev. 32 (1990) 537-578. 11. W. Magnus, S. Winkler, Hill’s Equations, Wiley, New York, 1966. 12. F. Nakajima, Index theorems and bifurcations in Duffing’s equations, In ”Recent Topics in Nonlinear PDE II.” edited by Masuda and Mimura, 133-162. North Holland, 1985. 13. G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353-372. 14. M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. London Math. Soc. 64 (2001) 125-143. 15. R. Ortega, Stability and index of periodic solutions of an equation of Duffing equation, Boll. Un. Mat. Ital. 3-B (1989) 533-546. 16. R. Ortega, Topological degree and stability of periodic solutions for certain differential equations, J. London Math. Soc. 42 (1990) 505-516. 17. R. Ortega, The twist coefficient of periodic solutions of a time-dependent Newton’s equation, J. Dynam. Differential Equations, 4 (1992) 651-665. 18. R. Ortega, A criterion for asymptotic stability based on topological degree, in Proceedings of the First World Congress of Nonlinear Analysis, Tampa, 1992. 19. R. Ortega, Some applications of the topological degree to stability theory, in Topological Methods in Differential Equations and Inclusions (Montreal, PQ, 1994), A. Granas and M. Frigon (Eds.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Dordrecht: Kluwer, 472 (1995) 377-409. 20. M. Zhang, W. Li, A Lyapunov-type stability criterion using Lα norms, Proc. Amer. Math. Soc. 130 (2002) 3325-3333. 1 School of Mathematics and Physics, Changzhou University, Changzhou, Jiangsu, 213164, China 2

Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, China

3 College of Information Sciences and Technology, Hainan University, Haikou, 570228, China 4

School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, China E-mail address: [email protected] (F. Wang)

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On a system of three max-type nonlinear difference equations Chang-you Wang1, 3, 4, Yu-qian Zhou 2, Shuang Pan 1, 3, Rui Li 3

*

1. Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065 P.R. China 2. School of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, P.R. China 3. College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065 P. R. China 4. College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065 P. R. China Abstract: The primary focus of this paper is to investigate the boundedness and asymptotic behavior of the following symmetric system of max-type difference equations

xn +1 = max{c,

ynp znp xnp }, = max{ , }, = max{ , }, n ∈ ` 0 , y c z c n +1 n +1 znq−1 xnq−1 ynq−1

where the parameters c, p, q ∈ (0, ∞) and the initial conditions x−1 , x0 , y−1 , y0 , z−1 , z0 are arbitrary positive real numbers. Our main results considerably improve results appearing in the literature (see, Stević, (2014) [29]). Keywords: max-type system, difference equations, boundedness, global attractivity.

1. Introduction In last few decades there has been a great interest in studying nonlinear difference equations and systems for developing some new techniques which can be used in investigating the models describing real life situations in biology, control theory, economics, etc. (see, e.g., [1-15] and the references therein). Recently, the so-called max-type difference equation has attracted more and more attention. However, the maxima operator is not a smooth function in n-dimensional real vector space so that the techniques which use *

Corresponding author at: College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065 P. R. China. Email addresses: [email protected] (Rui LI). 1

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derivatives could be of almost no use, so the study of max-type systems of difference equations become more difficult. Some studies of these difference equations have been presented in [16-25]. Paper [26] is one of the first such papers on max-type difference equations. It studies positive solutions of the difference equation xn +1 = max{a,

xnp }, n ∈ ` 0 , xnp−1

(1.1)

where initial values x−1 , x0 , and parameters a and p are positive numbers. In [27], Stevo Stević studied the boundedness character of positive solutions to the following max-type difference equation,

xn = max{ A,

xnp−1 }, n ∈ ` 0 , xnr − k

(1.2)

where k ∈ N \ {1} , the parameters A and r are positive and p is a nonnegative real number. As an extension of (1.2), Stevo Stević studied the boundedness character and global attractivity of positive solutions of the following symmetric system of max-type difference equation

xn +1 = max{c,

ynp }, xnp−1

yn +1 = max{c,

xnp }, n ∈ ` 0 , ynp−1

(1.3)

where c, p ∈ (0, +∞) (see [28]). Above results motivated Stevo Stević to continuously investigate the behavior of positive solutions of the following max-type system of differences

xn +1 = max{c,

ynp znp xnp y c z c }, = max{ , }, = max{ , }, n ∈ ` 0 , n +1 n +1 znp−1 xnp−1 ynp−1

(1.4)

where the parameters c and p are positive real numbers. It is proved that system (1.4) is permanent when p ∈ (0, 4) and so forth (see [29]). Motivated by works [26-29], the primary focus of this paper is to investigate the boundedness character and global attractivity of the following max-type difference equations

ynp znp xnp xn +1 = max{c, q }, yn +1 = max{c, q }, zn +1 = max{c, q }, n ∈ ` 0 , zn −1 xn −1 yn −1

(1.5)

2

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where c, p, q ∈ (0, +∞) . It is obvious that the paper can be considered as a continuation of studying special cases of the next systems of difference equations

xn +1 = max{ An ,

ynp− m znp− m xnp− m y A z A }, = max{ , }, = max{ , }, n ∈ ` 0 , n +1 n n +1 n znq− k xnq− k ynq− k

where m, k ∈ N , p, q ∈ (0, +∞) , and

( An )n∈N

is a sequence of positive numbers. For more 0

related papers in this research area, see, for example, [30-33] and the references therein. The rest of the paper is organized as follows. In Section 2, we will focus our attention on the buondedness character of solutions of system (1.5) by developing new iterative method and inequality technique. In Section 3, we will investigate the asymptotic behavior of solutions of system (1.5). Then we show an example and carry out numerical simulations in Section 4, from which it can be seen that all simulations agree with the theoretical results. We finally conclude our paper in Section 5.

2. Boundedness character of solutions This section is devoted to analyzing the boundedness of the positive solutions to the maxtype difference systems (1.5). Theorem 2.1. Assume that f (λ ) = λ 2 − pλ + q and (a) there is λ1 > 1 such that f (λ1 ) = 0 , or (b) there is λ1 = λ2 = 1 such that f (λ1 ) = f (λ2 ) = 0 . Then the system (1.5) has positive unbounded solutions. Proof. Obviously, from (1.5), we can easily see that

xn +1 ≥

ynp , znq−1

yn +1 ≥

znp xnp z , ≥ . n +1 xnq−1 ynq−1

(2.1)

By taking logarithm in (2.1), for any n ∈ ` 0 , we obtain

ln xn +1 ≥ p ln yn − q ln zn −1 , ln yn +1 ≥ p ln zn − q ln xn −1 , ln zn +1 ≥ p ln xn − q ln yn −1 . (2.2) Moreover, it follows that ln xn +1 yn +1 zn +1 ≥ p ln xn yn zn − q ln xn −1 yn −1 zn −1 .

(2.3)

Let vn = ln xn yn zn , where n ≥ −1 , then inequality (2.3) becomes vn +1 ≥ pvn − qvn −1 , n ∈ ` 0 .

(2.4)

By hypothesis (a), we have that f (λ1 ) = 0 and λ1 > 1 . Let 3

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f1 (λ ) =

f (λ ) =λ+a, λ − λ1

(2.5)

then it follows that f (λ ) = (λ + a )(λ − λ1 ) .

(2.6)

Thus, we can obtain p = λ1 − a and q = − aλ1 . Set un = vn + avn -1 , n ∈ ` 0 .

(2.7)

Then inequation (2.4) can be written in the following form vn +1 − pvn + qvn −1 = vn +1 − (λ1 − a)vn − aλ1vn −1 = vn +1 + avn − λ1 (vn + avn −1 )

(2.8)

= un +1 − λ1un ≥ 0. That is un +1 ≥ λ1un .

(2.9)

v0 ≥| a || v−1 | .

(2.10)

Let v−1 , v0 be chosen such that

This, along with (2.9), yields to

un +1 ≥ λ1n u0 , and u0 > 0 .

(2.11)

Letting n → ∞ in (2.11), from assumption (a) λ1 > 1 and u0 > 0 , it follows that un = vn + avn -1 → +∞ as n → +∞ .

Hence

{vn }n≥−1

(2.12)

is unbounded. As vn = ln xn yn zn , it follows that xn yn zn → ∞ as n → ∞ ,

(2.13)

which along with xn2 + yn2 + zn2 ≥ 3 3 xn yn zn implies

xn2 + yn2 + zn2 → +∞ , from which it follows that the sequence

(2.14)

{( xn , yn , zn )}n≥−1 is unbounded. 4

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By hypothesis (b), we have p = 2, q = 1 . Then from (2.1) we get xn +1 y ≥ n , yn zn −1

yn +1 z ≥ n , zn xn −1

zn +1 x ≥ n . xn yn −1

(2.15)

Moreover, one has xn +1 yn +1 yn +1 xn yn zn x yz ≥ ≥ " ≥ 0 0 0 , n ∈ N0 . xn yn zn xn −1 yn −1 zn −1 x−1 y−1 z−1

(2.16)

and consequently xn yn zn ≥ (

y0 x0 z0 n ) x0 y0 z0 , n ∈ N 0 . x−1 y−1 z−1

(2.17)

If we choose the initial conditions x−1 , y−1 , z−1 , x0 , y0 , z0 such that x0 y0 z0 > x−1 y−1 z−1 > 0 then

we obtain (2.13) and consequently (2.14), which implies that the sequence {(xn , yn , zn )}n≥−1 is

unbounded, and then the proof of Theorem 2.1 is completed. Next, we study the different cases concerning with the boundedness of positive solutions to the systems (1.5). Theorem 2.2. If c > 0 , p > 0 and p 2 < 4q , then the solutions to system (1.5) are bounded. Proof. Assume that ( xn , yn , zn )n≥−1 is a positive solution to systems (1.5). Then the following

estimate obviously holds min{xn , yn , zn } ≥ c, n ∈ ` 0 .

(2.18)

Due to the symmetry among {xn } , { yn } and {zn } , as long as we prove the boundedness of {xn } , other sequences { yn } and {zn } can be proved as well.

From systems (1.5), it follows that yp c p z p −q xn +1 = max{c, qn } = max{c, q , n −pq1 }, n ∈ ` 0 . zn −1 zn −1 xn − 2 2

(2.19)

Case1. When p 2 ≤ q , we get

xn +1 ≤ max{c,

1 c

q− p

,

1 c

pq − p 2 + q

}.

(2.20)

Thus, the sequence {xn }n ≥−1 is bounded. Case2. When p 2 > q , let sequence

{al }l ≥0

be defined as follows 5

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al +1 = q /( p − al ), a0 = 0, l ∈ N 0 .

(2.21)

From (1.5) and (2.21), we have yp c p z p −q xn +1 = max{c, qn } = max{c, q , n −pq1 } zn −1 zn −1 xn − 2 2

= max{c, ( = max{c, (

c z

q/ p n −1

)p,(

c

)p,( q/ p

z

( p −q / p ) p n −1 q /( p − q / p ) n−2

)

x

}

c

)( p −q / p ) p , ( q /( p − q / p )

zn −1

xn − 2

c

c

xnp−−2q /( p − q / p ) ( p − q / p ) p ) } ynq−3

= "" = max{c, ( = max{c, ( = max{c, ( = max{c, (

z

q/ p n −1

c znq−/1p c znq−/1p c znq−/1p

,( ,( ,( ,(

q /( p − q / p ) n−2

x

," , (

xnp−−(3a3kk−−11) y

q n −3 k

(2.22) ) p − a3 k −2 ， ")( p − q / p ) ) p }

c

ynp−(3pk− a3 k −1 ) −q p − a3k-2 c , ) ， ")( p − q / p ) ) p } q q yn − 3 k zn − (3k +1)

c

ynp−−3ak3 k p −a3k-1 ) ， ")( p − q / p ) ) p } znq−(3k +1)

c

znp−−(3a3kk++11)

," , ( q /( p − q / p )

xn − 2

," , ( q /( p − q / p )

xn − 2

xnq−/(2p − q / p )

," , (

p − a3 k −1

xnq−(3k + 2)

) p − a3k ， ")( p − q / p ) ) p }.

From the monotonicity of g ( x) = q /( p − x) on the interval (0, p) along with the fact 0 = a0 < a1 = q / p , it follows that the sequence

{al }

is increasing as far as al ≤ p for

* every l ∈ Ν 0 . Hence, we have liml →+∞ al = x , x* ∈ (0, p ] and x* is the solution of the

following equation

f ( x) = x( p − x) − q = 0 .

(2.23)

However the equation (2.23) has no real roots existing in (0, p] when p 2 < 4q , which is contradiction. Hence there is l0 ∈ ` such that al0 −1 < p and al0 ≥ p . If l0 = 3k , then by using (2.18) in (2.22) it follows that

xn +1 = max{c, (

c znq−/ 1p

,(

c

," , ( q /( p − q / p )

xn − 2

ynp−−3ak3 k p − a3k-1 ) ， ")( p − q / p ) ) p } znq−(3k +1)

(2.24)

c c 1 ")( p − a1 ) ) p } ≤ max{c, ( a1 , ( a2 ," , ( q − p + a3 k ) p − a3k-1 ， c c c for n > 3k , from which the boundedness of

{ xn }n≥−1

follows in this case.

If l0 = 3k + 1 , then it follows that 6

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xn +1 = max{c, (

c znq−/ 1p

,(

c xnq−/(2p − q / p )

," , (

znp−−(3a3kk++11) xnq− (3k + 2)

) p − a3k ， ")( p − q / p ) ) p }

(2.25)

c c 1 ≤ max{c, ( a1 , ( a2 ," , ( q − p + a3 k +1 ) p − a3k ， ")( p − a1 ) ) p } c c c

for n ≥ 3k + 1 , from which the boundedness of

{ xn }n≥−1

follows in this case.

If l0 = 3k + 2 , then it follows that xn +1 = max{c, (

c z

q/ p n −1

,(

c q /( p − q / p ) n−2

x

," , (

xnp−−(3a3kk++22) y

q n − 3( k +1)

) p − a3k+1 ， ")( p − q / p ) ) p }

(2.26)

c c 1 ≤ max{c, ( a1 , ( a2 ," , ( q − p + a3 k +2 ) p − a3k+1 ， ")( p − a1 ) ) p } c c c

for n ≥ 3k + 1 , from which the boundedness of

{ xn }n≥−1 follows in this case.

Combined the case 1 p2 ≤ q and the case 2 q < p2 < 4q , we can obtain that the sequence {xn}n≥−1 is bounded when p2 0 , q > 0 and p = 1 . Then the solutions to systems (1.5)

are bounded. Proof. Assume that {( xn , yn , zn )} is any positive solution to systems (1.5) in particular p = 1 .

We can easily know that xn ≥ c, yn ≥ c, zn ≥ c . Therefore, we have

xn+1 ≤ max{c,

yn }, cq

yn+1 ≤ max{c,

zn x }, zn +1 ≤ max{c, qn }, n ∈ N 0 . q c c

(2.27)

From the above (2.27), it follows that xn +1 ≤ max{c,

yn c z c c x } ≤ max{c, q , n2−q1 } ≤ max{c, q , 2 q , n3−q2 } . q c c c c c c

(2.28)

vn +1 = max{c,

c c vn − 2 , , }, n = 1, 2," and v1 = x1 , v0 = x0 , v−1 = x−1 . c q c 2 q c 3q

(2.29)

Set

Assume that {vn } is the solution to (2.29). Then vn is greater than xn for any n > 2 . Case 1. c > 1 .

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3q+1 3 q +1 3 q +1 (a). If v−1 ≤ c , v0 ≤ c and v1 ≤ c , from (2.29) we can obtain that c q > 1 and

vn − 2 < c , so v2 = c, v5 = c, v8 = c," , which implies that v3n −1 = c . Moreover, v4 = c , v7 = c , c 3q v10 = c," , which implies that v3n +1 = c and similarly v3n = c . Hence, the boundedness of

{vn }n≥−1

follows in this case.

3q +1 3q+1 3q+1 (b). If v−1 > c , v0 > c and v1 > c , from (2.29) we can obtain that c < z5 < z2 < z−1 .

Through iteration, we can get that {v3n −1}

is monotonically decreasing. Additionally,

vn ≥ c for any n = 3k −1, k ∈N , we can obtain that {v3n −1} is bounded. Similarly, {v3n } and {v3n +1} are bounded as well. Hence, the boundedness of

{vn }n≥−1

follows in this case.

3q+1 3 q +1 3 q +1 (c). If v−1 ≤ c , v0 ≤ c and v1 > c , from above proof we can obtain that v3n−1 = c ,

v3n = c and {v3n +1} is monotonically decreasing. Additionally vn ≥ c , we can obtain the

boundedness of

{vn }n≥−1

follows in this case.

3q+1 3 q +1 (d). If v−1 ≤ c , v0 > c

3 q +1 and v1 ≤ c , from above proof we can obtain that

v3n −1 = c , v3n +1 = c and {v3n } is monotonically decreasing. Additionally vn ≥ c , we can

obtain the boundedness of

{vn }n≥−1

follows in this case.

3q+1 3 q +1 3 q +1 (e). If v−1 > c , v0 ≤ c and v1 ≤ c , from above proof we can obtain that v3n = c ,

v3n +1 = c and {v3n −1} is monotonically decreasing. Additionally vn ≥ c , we can obtain the

boundedness of

{vn }n≥−1

follows in this case.

3q+1 3 q +1 3 q +1 (f). If v−1 ≤ c , v0 > c and v1 > c , from above proof we can obtain that v3n −1 = c ,

{v3n } and {v3n +1} are monotonically decreasing. Additionally vn ≥ c , we can obtain the

boundedness of

{vn }n≥−1

follows in this case.

3q+1 3 q +1 3 q +1 (g). If v−1 > c , v0 ≤ c and v1 > c , from above proof we can obtain that v3n = c ,

{v3n −1} and {v3n +1} are monotonically decreasing. Additionally vn ≥ c , we can obtain the

boundedness of

{vn }n≥−1

follows in this case.

3q+1 3 q +1 (h). If v−1 > c , v0 > c

3 q +1 and v1 ≤ c , from above proof we can obtain that

v3n +1 = c , {v3n −1} and {v3n } are monotonically decreasing. Additionally vn ≥ c , we can

obtain the boundedness of

{vn }n≥−1

follows in this case.

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Due to the bounededness of {vn}n≥−1 and xn ≤ vn , we can obtain the boundedness of {xn } . Similarly { yn } and {zn } are bounded as well. Hence, every positive solution to systems (1.5) is bounded. Case 2. 0 < c ≤ 1 . (a)

If q ≥ 1 , in fact xn ≥ c , from (1.5) it follows that

xn +1 = max{c,

yn z c } = max{c, q , q n −1q } q zn −1 zn −1 zn −1 xn − 2

(2.30)

1 1 1 = max{c, q , q −1 q } ≤ max{c, q −1 , 2 q −1 } zn −1 zn −1 xn − 2 c c c

for n ∈ N , which means that {xn } is bounded. (b). If 0 < q < 1 , let sequence

{al }l ≥0

be defined as follows

al +1 = al − bl , bl +1 = qal , a1 = 1 − q, b1 = q, l ∈ ` .

(2.31)

Thus, from (1.5) we have xn +1 = max{c,

c znq−1

,

z1n−−1q c zna1−1 c } = max{ , , } xnq− 2 znq−1 xnb1− 2

c1− q xn(1−−2q ) − q c c a1 xna1−−2b1 = max{c, q , q , q (1− q ) } = max{c, q , b1 , qa1 } zn −1 xn − 2 yn −3 zn −1 xn − 2 yn −3 c

= max{c,

c znq−1

,

c1− q c (1− q ) − q yn[(1−−3 q ) − q ]− q (1− q ) c c a1 c a1 −b1 yna−2 −3 b2 c }= max{ , , , , } , , xnq− 2 ynq−(13− q ) znq−[(14 − q )− q ] znq−1 xnb1− 2 ynqa−13 znqa−24

= "" −b

xn −3 k(3−1k −1) xn −3 k(3−2k −1)3 k −2 c a1 c a1 −b1 c a2 −b2 c c a1 c a2 c a3 c " , } = max{c, q , b1 , b2 , b3 " , }= max{ , , , , ynb3−k3−1k zn −1 xn − 2 yn −3 zn − 4 ynb3−k3−1k znq−1 xnb1− 2 ynb2−3 znb3− 4 a

c

= max{c, = max{c,

c znq−1 c znq−1

a

,

yna−3 k3−k1 −b3 k −1 yna−3 k3k c a1 c a1 −b1 c a2 −b2 c c a1 c a2 c a3 c , , " , }= max{ , , , , " , } xnb1− 2 ynb2−3 znb3− 4 znb3−k(3k +1) znq−1 xnb1− 2 ynb2−3 znb3− 4 znb3−k(3k +1)

,

zna3−k(3−kb3+k1) zna3−k3+k1 +1 c a1 c a1 −b1 c a2 −b2 c c a1 c a2 c a3 c , , " , }= max{ , , , , " , } xnb1− 2 ynb2−3 znb3− 4 xnb3−k(3+1 k + 2) znq−1 xnb1− 2 ynb2−3 znb3− 4 xnb3−k(3+1 k + 2)

(2.32) for every k ∈ N . From (2.31), we can deduce al +1 − al + qal −1 = 0, l ∈ N .

(2.33)

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It is easy to see that the general solution of difference equation (2.33) is al = c1λ1l + c2 λ2l ,

c1 , c2 ∈ R ,

(2.34)

where λ1,2 = (1 ± 1 − 4q ) / 2 . The fact λ1,2 < 1 along with (2.34) implies that the sequence lim l →+∞ al = 0 . From this and (2.31) we get lim l →+∞ bl = 0 .

Now note that from (2.32) it follows that xn +1 ≤ max{c,

c znq−1

a

,

xn −3 k(3−1k −1) c a1 c a2 c a3 , , " , }, xnb1− 2 ynb2−3 znb3− 4 ynb3−k3−1k

(2.35)

,

yna−3 k3k c a1 c a2 c a3 , , " , }, xnb1− 2 ynb2−3 znb3− 4 znb3−k(3k +1)

(2.36)

or

x n +1 ≤ max{c,

c znq−1

or zna3−k(3+1k +1) c a1 c a2 c a3 xn +1 ≤ max{c, q , b1 , b2 , b3 " , b3 k +1 } . zn −1 xn − 2 yn −3 zn − 4 xn − (3k + 2) c

The convergence of of

{ xn }n≥−1 .

(2.37)

{al }l ≥−1 and {bl }l ≥−1 along with (2.35)-(2.37) implies the boundedness

Since systems (1.5) is symmetric, the boundedness of

boundedness of

{ yn }n≥−1

and

{ xn }n≥−1 implies

the

{ zn }n≥−1 , as claimed.

Theorem 2.4. Assume that c > 0 , p ∈(0,1) , then the solutions to systems (1.5) are bounded. Proof. Assume that ( xn , yn , zn )n≥−1 is a positive solution to systems (1.5). Then the following

estimate obviously holds min{xn , yn , zn } ≥ c, n ∈ ` 0 .

(2.38)

Hence 2

yp cp zp xn +1 ≤ max{c, nq } ≤ max{c, q , pqn −+1q } c c c 2

3

xp cp cp ≤ max{c, q , pq + q , p2 qn+−pq2 + q }. c c c

(2.39)

Let {vn } be the solutions of the following difference equation (2.40) and v−1 = x−1 , v0 = x0 , v1 = x1 .

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vn +1 = max{c,

1

3

vnp−2

1

, , 2 }, n = 1, 2,3," . 2 c q − p c pq − p + q c p q + pq + q

(2.40)

Since p ∈ (0,1) , the following function f ( x) = max{c,

1

1

xp

3

(2.41)

, , 2 } 2 c q − p c pq − p + q c p q + pq + q

is a concave function for sufficiently large x . Thus, it follows that there is a fixed point x* , * such that f ( x ) < x for x > x* . It is easy to see that if v2 ∈ (0, x ] , the sequence {vn }n ≥ 2 is * bounded above by x* and if v2 > x , it is non-increasing and bounded below by x* .

Hence the sequence {vn } is bounded and consequently the sequence {xn } is bounded too. In the same way, we can prove that the sequence { yn } and {zn } are bounded as well. Hence, every solution to systems (1.5) is bounded as claimed.

3. Asymptotic behavior of solutions This section is devoted to analyzing the asymptotic behavior of solutions to system (1.5) for c > 1 and c ∈ (0,1] . Theorem 3.1. Assuming 0 < p ≤ 1 , when c ∈ ( 0 ,1] then {( xn , y n , z n )} converges to ( x* , y* , z* ) = (1,1,1) , while c > 1 , {( xn , yn , zn )} converges to ( x* , y* , z * ) = (c, c, c) .

Proof.

Case 1. c ∈ (0,1] .

Due to the positivity of the solution to {( xn , yn , zn )}

with initial data x0 , x−1 > 0 ,

y0 , y−1 > 0 and z0 , z−1 > 0 , the systems (1.5) can be transformed to the following systems (3.1) s t r by the change of xn = e n , yn = e n , zn = e n , where sn ≥ 0, tn ≥ 0, rn ≥ 0, n > k and ln c < 0 .

sn+1 = max{ln c, ptn − qrn−1} ≤ max{ln c, ptn } ≤ ptn , tn +1 = max{ln c, prn − qsn−1} ≤ max{ln c, prn } ≤ prn ,

(3.1)

rn+1 = max{ln c, psn − qtn−1} ≤ max{ln c, psn } ≤ psn . Obviously, from systems (3.1), inequality (3.2) follows sn +3 ≤ p 3s n , tn +3 ≤ p 3tn , rn +3 ≤ p 3 rn .

(3.2)

For p ≤ 1 , s n + 3 < s n can be obtained from inequality (3.2). The sequences {s3n }, { s 3 n +1 } and {s3n + 2 } are monotone decreasing. In addition, as sn > 0 , we can get that lim n→∞ sn = 0 s and lim n→∞ xn = lim n→∞ e n = 1 . Then {xn } converges to 1. { yn } and {zn } can be proved

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in the same way. Therefore, when c ∈ (0,1] , if p ≤ 1 , then {(xn , yn , zn )} converges to ( x* , y* , z * ) = (1,1,1) .

Case 2. c>1.

Correspondingly, systems (1.5) can be transformed to systems (3.3) by the change of xn = c sn , yn = ctn , zn = c rn , where sn ≥ 1, tn ≥ 1, rn ≥ 1 due to xn ≥ c, yn ≥ c, zn ≥ c . sn+1 = max{1, ptn − qrn−1}, tn+1 = max{1, prn − qsn−1},

(3.3)

rn+1 = max{1, psn − qtn−1}. Furthermore, systems (3.3) can be written as systems (3.4). sn+1 − 1 = max{0, ptn − qrn−1 − 1}, tn+1 − 1 = max{0, prn − qsn−1 − 1},

(3.4)

rn+1 − 1 = max{0, psn − qtn−1 − 1}. Similar proof has been fully given in case c ∈ (0,1] and here the rest proof is omitted. Then the result is much alike with the one in above case: lim sn = 1 , lim tn = 1 , lim rn = 1 , n →∞ n →∞ n →∞

(3.5)

and lim xn = lim c sn = c , n →∞

n →∞

lim yn = lim ctn = c , lim zn = lim c rn = c . n →∞

n →∞

n →∞

n →∞

(3.6)

Therefore, assume 0 < p ≤ 1 and c > 1, then ( xn , y n , z n ) converges to (x*, y*, z*) = (c, c, c) . Hence Theorem 3.1 is proved completely.

4. Simulation experiment In this section, some numerical simulations are given to support our theoretical analysis. As examples, we consider the following difference equations

xn +1 = max{0.5,

xn +1 = max{1,

yn0.5 zn0.5 zn0.5 }, = max{0.5, }, = max{0.5, }, n ∈ ` 0 , y y n +1 n +1 zn2−1 xn2−1 xn2−1

yn }, zn2−1

yn +1 = max{1,

zn x }, zn +1 = max{1, 2n }, n ∈ ` 0 , 2 xn −1 yn −1

(4.1)

(4.2)

and

xn +1 = max{1.5,

yn0.5 zn0.5 xn0.5 }, = max{1.5, }, = max{1.5, }, n ∈ ` 0 . y z n +1 n +1 zn2−1 xn2−1 yn2−1

(4.3)

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By employing Matlab R2013b, we solve the numerical solutions of the above equations, which are shown respectively in the following Figures. More precisely, the initial conditions of (4.1) are that x−1 = 1.5 , x0 = 1.2 , y−1 = 0.5 , y0 = 0.8 , z−1 = 1 , and z0 = 1.5 . It is easy to show that the equations (4.1) satisfy the conditions of Theorem 2.2. Fig.4.1 shows that the solutions of the equations (4.1) are bounded. The initial conditions of (4.2) are that x−1 = 1.5 , x0 = 1.2 , y−1 = 0.5 , y0 = 0.8 , z−1 = 2.5 , and z0 = 3 . It is easy to show that the equations (4.2) satisfy the conditions of Theorem 2.3 and Theorem 3.1. Figure 4.2 shows the solutions to equations (4.2) are bounded and globally attractive. The initial conditions of equations (4.3) are that x−1 = 1.5 , x0 = 1.2 , y−1 = 0.5 , y0 = 0.8 , z−1 = 1 , and z0 = 1.5 . It is easy to show that the equations (4.3) satisfy the conditions of Theorem 2.4 and Theorem 3.1. Figure 4.3 shows the solutions to the equations (4.3) are bounded and globally attractive.

12 x(n) y(n) z(n)

c=0.5,p=0.5,q=2 10

x(n)/y(n)/z(n)

8

6

4

2

0 -20

0

20

40 n

60

80

100

Figure 4.1. the solutions to equation (4.1)

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5

x(n) y(n) z(n)

4.5

c=1,p=1,q=2 4

x (n )/y (n )/z (n )

3.5

3

2.5

2

1.5

1

0.5 -20

0

20

40 n

60

80

100

Figure4.2. the solutions to equation (4.2)

4.5

x(n) y(n) z(n)

4

c=1.5,p=0.5,q=2

x(n)/y(n)/z(n)

3.5

3

2.5

2

1.5

1

0.5 -20

0

20

40 n

60

80

100

Figure 4.3. the solutions to equation (4.3)

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5. Conclusion It is obvious that the system of three max-type difference equations (1.5) is the extension of the models in [26-29]. In this paper, we have dealt with the problem of boundedness character and global attractivity for a class of max-type difference system. And we have obtained some sufficient conditions which ensure the boundedness character and global attractivity of the max-type system. Especially, the sufficient conditions that we obtained are very simple, which provide flexibility for the application and analysis of max-type difference system. These results generalize and improve some previous works. In addition, we present the use of a new iteration method for symmetric systems of max-type difference equations. This technique is a powerful tool for solving various difference equations and it can be applied to other nonlinear differential equations in mathematical physics. Computations are performed using the software package Matlab R2013b. Finally, some numerical examples are given to show the validity of the obtained theoretic results

Acknowledgement This work is supported Science Fund for Distinguished Young Scholars (cstc2014jc yjjq40004) of China, the National Nature Science Fund (Project nos. 11372366, 11301043 and 61503053) of China, the Natural Science Foundation Project of CQ CSTC (Grant nos. cstc2015jcyjBX0135 and cstc2015jjA20016) of China, the Postdoctoral Science Foundation (Grant no. 2016m602663) of China, and the excellent talents project of colleges and universities in Chongqing of China.

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The convexity of n-dimensional fuzzy mappings and the saddle point conditions of the fuzzy optimization problems Hong-Xia Lia,∗ , Yu-Juan Baia , Zeng-Tai Gongb a College of Mathematics and Statistics, LongDong University, QingYang, Gansu 745000, China b College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070 China

Abstract The purpose of this work is to consider the optimization problem of n-dimensional fuzzy number valued functions. Firstly, the differentiability and convexity of n-dimensional fuzzy number valued function are discussed by means of the support function and a new order relation, which is built in the aid of the support function and the order of vector. Secondly, the fuzzy Lagrange function of fuzzy nonlinear programming is presented and weak duality theorems are obtained. At last, the saddle point of fuzzy lagrangian function is defined, the sufficient and necessity conditions of saddle point are given. Keywords Fuzzy numbers; fuzzy programming; saddle point; duality theorem 1. Introduction Since the concept and operations of fuzzy set were introduced by Zadeh, many studies have focused on the theoretical aspects and applications of fuzzy sets, one of the main stream is the fuzzy optimization in operation research. In 1970, Bellman and Zadeh[1] inspired the development of fuzzy optimization by providing the aggregation operators, which combined the fuzzy goals and fuzzy decision space. After this motivation and inspiration, there came out a lot of articles dealing with the fuzzy optimization problems and the insightful survey can be seen in [3, 7, 11]. The duality of fuzzy linear programming was first studied by Rodder and Zimmermann[9] who considered the economic interpretation of the dual variables. Zhong and Shi[20] presented a parametric approach for duality in fuzzy multi criteria and multi constrainted linear programming which extended fuzzy linear programming approaches. Wu[14]formulate the fuzzy primal and dual linear programming problems with fuzzy coefficients by using fuzzy scalar product, prove the weak and strong duality theorems. Wu[15] discuss the saddle-point optimality conditions in fuzzy optimization problems by introducing the fuzzy scalar product. In Wu[16], under a general setting partial ordering, the duality theorems and saddle point optimality of fuzzy nonlinear programming problems are derived. Zhang[21]discuss the saddle-points and minimax theorems under fuzzy environment, obtain the KT conditions for fuzzy programming and consider the /perturbed0convex fuzzy programming. Gong[5] propose the fuzzy Lagrangian function of a fuzzy optimization problem by considering a total ordering on the set of fuzzy numbers, and the saddle point of fuzzy Lagrangian function with its optimality condition were dicussed.Howere,the fuzzy number in these research above is on the real line,which is one dimensional.There are few studies on n-dimensional fuzzy numbers, maybe the ranking of n-dimensional fuzzy numbers has been a bottleneck for researchers. The differentiability of fuzzy mappings from an open subset of a normed space into the n-dimension fuzzy number space E n was developed by Puri and Ralescu[8], which generalized and extended the concept of Hukuhara differentiability for set mappings. Wang and Wu[12] proposed the directional derivative,differential and sub-differential of fuzzy mappings from Rn into E using Hukuhara difference. Hai[6] characterize the generalized difference of n-dimensional fuzzy number valued functions by means of support functions and give the order relation s by the aid of support function. Yan[18] give the order relation on E considering the left and right endpoints and weights, which is a total order. Based on this, we give the order relation of n- dimensional fuzzy numbers by means of the support function and the order of vector, this order is partial and practical. †

This work is supported by National Natural Science Foundation (11461062) and the Scientific Research Project for Higher Education of Gansu Province (2015A-144). ∗ Corresponding Author: Hong-Xia Li. Tel.: +8613830467235. E-mail addresses: [email protected] 1480

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

The purpose of this work is to consider the optimization problem of n-dimensional fuzzy number valued function. First, we present the terminology used in the present paper, and give the order relation of n-dimensional fuzzy-number-valued function. In section 3, the differentiability and convexity are introduced and its relations is studied. For nonlinear fuzzy programming problem, the weak duality theorem the saddle point of fuzzy lagrangian function is is presented, further, the sufficient and necessity condition of saddle point are obtained. 2. Definitions and preliminaries In this section,basic definitions and operations for fuzzy numbers are presented. Definition 2.1[17] X = Rn , n ≥ 1 is the real n-dimensional Euclidean space, A fuzzy number is a mapping u e : Rn → [0, 1] with the following properties: (1) u e is a normal fuzzy set, i.e.there exists x0 ∈ Rn such that u e(x0 ) = 1, (2) u e is a convex fuzzy set, i.e. u e(λx + (1 − λ)y) ≥ min{e u(x), u e(y)} for any x, y ∈ Rn and λ ∈ [0, 1]. (3) u e is upper semi-continuous. (4) [e u]0 = cl(suppe u) = {x ∈ Rn : u e(x) > 0} is compact. n we will denote E the set of fuzzy numbers.It is clear that any u e ∈ E n , r ∈ [0, 1], [e u]r = {x ∈ Rn : u e(x) ≥ r} denoted as r−level cut is a compact convex set. Further, we give the representation theorem of these compact convex sets. Theorem 2.2[17] Let u e ∈ E n , then r (1)[e u] is a nonempty compact convex subset of Rn for any r ∈ [0, 1], (2)[e u]r1 ⊆ [e u]r2 , for 0 ≤ r2 ≤ r1 ≤ 1, T u]rk = [e u]r , (3)If rk > 0 and rk is a nondecreasing sequence converging to r ∈ (0, 1], then ∞ k=1 [e r n Conversely, if {[A] ⊆ R : r ∈ [0, 1]} satisfies the conditions (1)-(3),then there exists a unique u e ∈ En S r r 0 r 0 such that [e u] = [A] for each r ∈ (0, 1] and [e u] = cl( r∈(0,1] [e u] ) ⊆ [A] . Let u e, ve ∈ E n and k ∈ R, the addition u e + ve and scalar multiplication ke u is defined as: for any x ∈ Rn , (e u + ve)(x) = sup min{e u(s), ve(t)}, s+t=x

x (ke u)(x) = u e( ), k 6= 0, 0e u=e 0, k where e 0(x) = 1 when x = 0, e 0(x) = 0 when x 6= 0. It is easy to get that the addition u e + ve and scalar multiplication ke u have the level cut: [e u + ve]r = [e u]r + [e v ]r = {x + y : x ∈ [e u]r , y ∈ [e u]r }, [ke u]r = k[e u]r = {kx : x ∈ [e u]r }. The Hausdorff distance D : E n × E n → [0, +∞) is defined by D(e u, ve) = sup d([e u]r , ve]r ), r∈[0,1]

where d is Harsdorff metric given by d(A, B) = inf{ε : N (A, ε) ⊃ B, N (B, ε) ⊃ A}, and N (A, ε) = {x ∈ Rn : d(x, A) = inf y∈A d(x, y) ≤ ε}is the ε− neighborhood of A. Then (E n , D) is a complete metric space, and satisfies D(e u + w, e ve + w) e = D(e u, ve), D(ke u, ke v ) = |k|D(e u, ve) for any u e, ve, w e ∈ E n and k ∈ R. [17] n Definition 2.3 Let u e ∈ E , the support function of u e is defined by u e∗ (r, p) = sup ha, pi, (r, p) ∈ I × S n−1 , a∈[e u]r

where I = [0, 1], S n−1 {x ∈ Rn : kxk = 1} be the unit sphere of Rn and h·, ·ibe the inner product in P= n n R , that is hx, yi = i=1 xi yi , where x P = (x1 , x2 , · · · , xn ) ∈ Rn , y = (y1 , y2 , · · · , yn ) ∈ Rn . Also, assume that u e = (e u1 , u e2 , · · · , u en ), then hx, u ei = ni=1 xi u ei , u ei ∈ E n . [6,17] n Theorem 2.4 Let u e ∈ E , then support functione u∗ satisfy: (1)e u∗ (r, p + q) ≤ u e∗ (r, p) + u e∗ (r, q) for p, q ∈ S n−1 . ∗ ∗ (2)e u (r, kp) = ke u (r, p), k ≥ 0. 1481

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Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

(3)e u∗ is uniformly bounded on I × S n−1 , and |e u∗ (r, p)| ≤ supa∈eu]0 kak, ∗ (4)e u (r, p) is nonincreasing and left continuous on r ∈ [0, 1], right continuous at r = 0 for each fixed n−1 p∈S . (5)e u∗ (r, ·) is uniformly Lipschitz continuous for r ∈ [0, 1], that is |e u∗ (r, p) − ve∗ (r, q)| ≤ ( sup kak)kx − yk, a∈e u]0

(6) u∗ (r, p) − ve∗ (r, p)|, d([e u]r , [e v ]r ) = sup |e p∈S n−1

for any r ∈ [0, 1], u e, u e ∈ En. (7)(−e u)∗ (r, p) = u e∗ (r, −p). Theorem 2.5 Let u e, ve ∈ E n , then ∗ (1)(se u + te v ) (r, p) = se u∗ (r, p) + te v ∗ (r, p), s, t ≥ 0, ∗ (2)D(e u, ve) = supr∈[0,1] ke u (r, p) − ve∗ (r, p)k = supr∈[0,1] supp∈S n−1 |e u∗ (r, p) − ve∗ (r, p)|. ∗ ∗ ∗ Proof (1)We prove that (e u + ve) (r, p) = u e (r, p) + ve (r, p) firstly. From the definition of support function, (e u + ve)∗ (r, p) = =

sup ha, pi = a∈[e u+e v ]r

sup b∈[e u]r ,c∈[e v ]r

sup

ha, pi =

a∈[e u]r +[e v ]r

sup

hb + c, pi

b∈[e u]r ,c∈[e v ]r

(hb, pi + hc, pi) = u e∗ (r, p) + ve∗ (r, p).

in addition,for any k ≥ 0, a (ke u)∗ (r, p) = sup ha, pi = sup ha, pi = sup kh , pi = ke u∗ (r, p), a k r r r a∈k[e u] a∈[ke u] ∈[e u] k

therefore,we get (1). (2) D(e u, ve) = sup d([e u]r , [e u]r ) = sup r∈[0,1]

sup |e u∗ (r, p) − ve∗ (r, p)| = sup ke u∗ (r, p) − ve∗ (r, p)k.

r∈[0,1] p∈S n−1

r∈[0,1]

we denote by Kn and KCn the spaces of (nonempty) compact convex sets of Rn respectively.The generalized Hukuhara difference of two set A, B ∈ KCn (gH-difference for short)is defined in[4,6,10] as follows: ( (a)A = B + C, A gH B = C ⇔ or(b)B=A+(-1)C where A + B = {x + y : x ∈ A, y ∈ B}, kA = {kx : x ∈ A}, k ∈ R. Stefanini[10] extent the generalized Hukuhara difference to the fuzzy case. For any u e, ve ∈ E n , the generalized Hukuhara difference(gHdifference for short)is the fuzzy number w, e if it exist,then ( (a)e u = ve + w, e u e gH ve = w e⇔ or(b)e v=u e + (−1)w. e From the theorem 2.4,it is easy to have the follows: Theorem 2.6 Let u e, ve ∈ E n , u e gH ve = w, e Then w e∗ (r, p) = u e∗ (r, p) − ve∗ (r, p), r ∈ [0, 1], p ∈ S n−1 Proof Since u e gH ve = w, e then either (a)e u = ve + w, e or (b)e v=u e + (−1)w. e For (a), from 2.5(1),we have u e∗ (r, p) = ve∗ (r, p) + w e∗ (r, p). For (b), ve∗ (r, p) = u e∗ (r, p) + (−1)w e∗ (r, p) = u e∗ (r, p) + w e∗ (r, −p), then w e∗ (r, p) = −w e∗ (r, −p) = u e∗ (r, p) − ve∗ (r, p) 1482

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

for any r ∈ [0, 1], p ∈ S n−1 . Definition 2.7[18] Let u e, ve ∈ E, u e  ve denoted as Z Z 1 r(e u− (r) + u e+ (r))dr ≤

0

0

[e u− (r), u e+ (r)], [e v ]r

[e u]r

1

r(e v − (r) + ve+ (r))dr,

[e v − (r), ve+ (r)], r

where = = ∈ [0, 1]. For any x = (x1 , x2 , · · · , xn ), y = (y1 , y2 , · · · , yn ) ∈ Rn , we define: x ≤ y if and only if xi ≤ yi for any i(i = 1, 2, · · · , n), and x < y means x ≤ y and there exist m, such that xm < ym (m = 1, 2, · · · , n). Definition 2.8 For any u e, ve ∈ E n , we say that u e  ve if (τ (e u1 ), τ (e u2 ), · · · , τ (e un ) ≤ (τ (e v1 ), τ (e v2 ), · · · , τ (e vn ), R1 + n−1 , e− = (y , y , · · · , y ) ∈ S n−1 where τ (e ui ) = 0 r(e u∗ (r, e+ e∗ (r, e− 1 2 n i )−u i ))dr, ei = (x1 , x2 , · · · , xn ) ∈ S i andxj = 1, yj = −1 when j = i, xj = yj = 0 when j 6= i(i, j = 1, 2, · · · , n). we say that u e ≺ ve if u e  ve and there exist i(i = 1, 2, · · · , n), such that τ (e ui ) < τ (e vi ). Particularly, Def 2.8 is just as Def 2.7 when n = 1, it means the Def 2.8 is the extension of the Def 2.7. u e  ve also denoted as ve  u e. min(e u, ve) = w e if and only if τ (w ei ) = min(τ (e ui ), τ (e vi ))(i = 1, 2, · · · , n). In the follows, we denote (τ (e u1 ), τ (e u2 ), · · · , τ (e un )) = H[e u], 3. Differentiability and convexity Wang and Wu[12] present the directional derivative of the fuzzy mapping F : Rn → E, that is characterized by the directional derivative of the real function. Below we give the differentiability of F : Rn → E n , and transformed it into the differentiability of functional in Banach space, then defined gradient and convexity and studied its relations. Definition 3.1 Let F : M (⊂ Rn ) → E n be a fuzzy-number-valued function, x0 = (x01 , x02 , · · · , x0n ) ∈ intM. If there exist u e1 , u e2 , · · · , u en ∈ E n , such that P uj ) D(F (x), F (x0 ) + nj=1 (xj − x0j )e = 0, lim 0 d(x, x ) x→x0 then we call F is differentiable at x0 , and denote5F (x0 ) = (e u1 , u e2 , · · · , u en ) the gradient of F at x0 , where x = (x1 , x2 , · · · , xn ). Theorem 3.2 Let F : M (⊂ Rn ) → E n be a fuzzy-number-valued function, x0 = (x01 , x02 , · · · , x0n ) ∈ intM. F is differentiable at x0 if and only if F (x)∗ (r, p) = F (x0 )∗ (r, p) +

n X

(xj − x0j )e u∗j (r, p) + o(d(x, x0 ))

(3.1)

j=1

for any r ∈ [0, 1] and p ∈ S n−1 . Proof F is differentiable at x0 , if and only if lim

D(F (x), F (x0 ) +

x→x0

Pn

j=1 (xj d(x, x0 )

− x0j )e uj )

= 0,

if and only if lim

supr∈[0,1] supp∈S n−1 |F (x)∗ (r, p) − F (x0 )∗ (r, p) −

Pn

j=1 (xj

− x0j )e u∗j (r, p)|

d(x, x0 )

x→x0

if and only if lim

|F (x)∗ (r, p) − F (x0 )∗ (r, p) −

x→x0

Pn

d(x, x0 )

j=1 (xj

− x0j )e u∗j (r, p)|

= 0,

=0

for any r ∈ [0, 1] and p ∈ S n−1 , if and only if F (x)∗ (r, p) = F (x0 )∗ (r, p) +

n X

(xj − x0j )e u∗j (r, p) + o(d(x, x0 ))

j=1

1483

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Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

for any r ∈ [0, 1] and p ∈ S n−1 . Theorem 3.3 Let F : M (⊂ Rn ) → E n be a fuzzy-number-valued function, F is differentiable at x, if x is a local minimum solution, then ∇F (x) = (e 0, e 0, · · · , e 0). Proof Since x is a local minimum solution, then there exist a δ > 0, such that F (x)  F (x) for any S x ∈ (x, δ) ∩ M. that is 1

Z τ (F (x)i ) =

r[F (x)

∗

0

(r, e+ i )

− F (x)

∗

(r, e− i )]dr

Z ≥ 0

1

− ∗ r[F (x)∗ (r, e+ i ) − F (x) (r, ei )]dr = τ (F (x)i )

for 1 ≤ i ≤ n. From the theorem 3.2 and arbitrariness of x, Z 1 r[e u∗j (r, e+ τ ((e uj )i ) = e∗j (r, e− i )−u i )]dr = 0(i, j = 1, 2, · · · , n), 0

0, e 0, · · · , e 0). thus u ej = e 0, and ∇F (x) = (e Definition 3.4 Let F : M (⊂ Rn ) → E n be a fuzzy-number-valued function,M be a convex set. Then F (x) is said to be a convex fuzzy-number-valued function on M if for any x, y ∈ M, λ ∈ [0, 1], such that λx + (1 − λ)y ∈ M, we have F (λx + (1 − λ)y)  λF (x) + (1 − λ)F (y).

(3.2)

we call F (x) is a strictly convex fuzzy-number-valued function on M, if for any x, y ∈ M, x 6= y, λ ∈ [0, 1],such that λx + (1 − λ)y ∈ M, we have F (λx + (1 − λ)y) ≺ λF (x) + (1 − λ)F (y). Theorem 3.5 Let M be an open convex set, F : M → E n , and F is differentiable, then F is convex if and only if F (x)  F (y) + h∇F (y), x − yi (3.3) for any x, y ∈ M. Proof Assume that F is convex, then for any x, y ∈ M, λ ∈ (0, 1), we have F (λx + (1 − λ)y)  λF (x) + (1 − λ)F (y), that is H[F (λx + (1 − λ)y)] ≤ λH[F (x)] + (1 − λ)H[F (y)]. thus τ (F (λx + (1 − λ)y)i ) ≤ λτ (F (x)i ) + (1 − λ)τ (F (y)i ) for any 1 ≤ i ≤ n, that is Z 1

− ∗ r[F (λx + (1 − λ)y)∗ (r, e+ i ) − F (λx + (1 − λ)y) (r, ei )]dr

0 1

Z ≤λ

∗

r[F (x) 0

(r, e+ i )

∗

− F (x)

(r, e− i )]dr

Z + (1 − λ) 0

1

− ∗ r[F (y)∗ (r, e+ i ) − F (y) (r, ei )]dr.

Since F is differentiable, then F (λx + (1 − λ)y)∗ (r, p) − F (y)∗ (r, p) =

n X

λ(xj − yj )e vj∗ (r, p) + λokx − yk,

j=1

for any r ∈ [0, 1] and p ∈ S n−1 , where ∇F (y) = (e v1 , ve2 , · · · , ven ), therefore, Z

1

∗

r[F (λx + (1 − λ)y) 0

(r, e+ i )

∗

− F (λx + (1 − λ)y)

1484

(r, e− i )]dr

Z − 0

1

− ∗ r[F (y)∗ (r, e+ i ) − F (y) (r, ei )]dr

Hong-Xia Li et al 1480-1489

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO.8, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

Z

1

≤ λ(

r[F (x)

∗

0

thus Z

1

0

Z ≤ λ(

− F (x)

∗

(r, e− i )]dr

Z

r[F (x)

1

− 0

− ∗ r[F (y)∗ (r, e+ i ) − F (y) (r, ei )]dr),

n n X X + ∗ r[ λ(xj − yj )e vj (r, ei ) − λ(xj − yj )e vj∗ (r, e− i )]dr j=1

1

0

(r, e+ i )

∗

j=1

(r, e+ i )

that is τ(

− F (x)

n X

∗

(r, e− i )]dr

Z − 0

1

− ∗ r[F (y)∗ (r, e+ i ) − F (y) (r, ei )]dr),

(xj − yj )(e vj )i ) ≤ τ (F (x)i ) − τ (F (y)i ),

j=1

We have

n X

(xj − yj )e vj  F (x) − F (y),

j=1

therefore F (x)  F (y) + h∇F (y), x − yi. Conversely, assume that for any x(1) , x(2) ∈ M, we have F (x(2) )  F (x(1) ) + h∇F (x(1) ), x(2) − x(1) i. Let y be a point between the x(1) and x(2) , then y = λx(1) + (1 − λ)x(2) for some λ ∈ (0, 1), and y ∈ M since M is a convex set. Based on the assumption, we have F (x(1) )  F (y) + h∇F (y), x(1) − yi, F (x(2) )  F (y) + h∇F (y), x(2) − yi, that is H[F (x(1) )] ≥ H[F (y)] + H[h∇F (y), x(1) − yi],

(3.4)

H[F (x(2) )] ≥ H[F (y)] + H[h∇F (y), x(2) − yi].

(3.5)

τ (F (x(1) )i ) ≥ τ (F (y)i ) + τ (h∇F (y), x(1) − yi)i ),

(3.6)

τ (F (x(2) )i ) ≥ τ (F (y)i ) + τ (h∇F (y), x(2) − yi)i )

(3.7)

From (3.4)and (3.5),

for 1 ≤ i < n. Multiple(3.6),(3.7)by λ, (1 − λ) respectively, and then add the result, we have λτ (F (x(1) )i ) + (1 − λ)τ (F (x(2) )i ) ≥ τ (F (y)i ), that is F (λx(1) + (1 − λ)x(2) )  λF (x(1) ) + (1 − λ)F (x(2) ), thus F is a convex fuzzy-number-valued function. 4. The duality and the saddle point Duality plays an important role in the development of optimization theory and algorithm. In this section, the duality theory of fuzzy nonlinear programming is introduced, and the weak duality theorems are obtained. At the same time, the Lagrange function of fuzzy nonlinear programming and saddle point are defined, and then discusses the relation between the saddle point of Lagrange function and the optimal solution of prime problem and dual problem and given saddle point optimality conditions.

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Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

Let X ⊂ Rn be an open set, F (x), Gi (x)(i = 1, 2, · · · , m) be fuzzy-valued functions on X, now we consider the following primal fuzzy optimization: ( min F (x), (F P ) (4.1) Gj (x)  e 0(j = 1, 2, · · · , m), where S = {x ∈ X|Gj (x)  e 0(j = 1, 2, · · · , m)} is the set of feasible solutions for problem (F P ), and denote x ∈ S the feasible solution for problem (F P ). We define the fuzzy-valued Lagrangian function for the primal problem as follow: L(x, u) = F (x) +

m X

uj Gj (x)

j=1

for all x ∈ S and all (u1 , u2 , · · · , um ) ∈ Rm , uj ≥ 0(j = 1, 2, · · · , m). Now we define the dual fuzzy optimization problem as follow: ( max L(u) (F D) uj ≥ 0(j = 1, 2, · · · , m).

(4.2)

where L(u) = minx L(x, u), u = (u1 , u2 , · · · , um ) ∈ Rm . Theorem 4.1 (Weak Duality Theorem) Let x ∈ X(⊂ Rn ), u ∈ Y (⊂ Rm ) be the feasible solution of problems (FP) and (FD) respectively. then F (x)  L(u). Proof From the definition of L(u), we have L(u) = min L(x, u) = min(F (x) + x

x

m X

uj Gj (x))  F (x) +

j=1

that is τ (L(u)i ) ≤ τ (F (x)i ) +

m X

uj Gj (x).

(4.3)

j=1

m X

uj τ (Gj (x)i ),

j=1

for 1 ≤ i < n. Since x and u is the feasible solution of problems (FP) and (FD) respectively, that is uj ≥ 0 and Gj (x)  e 0(j = 1, 2, · · · , m), thus τ (Gj (x)i ) ≤ 0(i = 1, 2, · · · , n), then we have τ (L(u)i ) ≤ τ (F (x)i )for 1 ≤ i < n, then L(u)  F (x). From above it follows easily: Proposition 4.2 For the problems (FP) and (FD), we have min{F (x)|Gj (x)  e 0, x ∈ X, j = 1, 2, · · · , m}  max{L(u)|u ≥ 0}. Proposition 4.3 Assume that F (x)  L(u), where x ∈ {x|Gj (x)  e 0, x ∈ X, j = 1, 2, · · · , m}, u ≥ 0, then x and u are the optimal solutions of problems (FP) and (FD) respectively. Definition 4.4 Let x ∈ X(⊂ Rn ), u ∈ Y (⊂ Rm ), then (x, u) is called a saddle point of the fuzzy-valued Lagrangian function L : X × Y → E n if and only if L(x, u)  L(x, u)  L(x, u)

(4.4)

holds for every (x, u) ∈ X × Y. Theorem 4.5 Let (x, u) be a saddle point of the fuzzy-valued Lagrangian function L(x, u), then x and u are the optimal solutions of problems (FP) and (FD) respectively. 1486

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Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

Proof Assume that (x, u) be a saddle point, we are going to prove x ∈ S firstly. It follows easily from the definition of saddle point, L(x, u)  L(x, u) holds for all u ∈ Rm , that is F (x) +

m X

uj Gj (x)  F (x) +

j=1

so τ (F (x)i ) +

m X

m X

uj Gj (x),

j=1

uj τ (Gj (x)i ) ≤ τ (F (x)i ) +

j=1

m X

uj τ (Gj (x)i ),

j=1 m X

(uj − uj )τ (Gj (x)i ) ≤ 0.

(4.5)

j=1

holds for any 1 ≤ i < n, Let uk = uk + 1 and uj = uj , j 6= k, from (5.5)we have τ (Gk (x)i ) ≤ 0(k = 1, 2, · · · , m).It says that x is a feasible solution of (FP). Now we’ll prove x and u are the optimal solutions of problems (FP) and (FD) respectively. Let uj (j = 1, 2, · · · , m) in (4.5) be taken as 0, then Pm (−u j )τ (Gj (x)i ) ≤ 0. since uj ≥ 0, τ (Gj (x)i ) ≤ 0, then we have j=1 m X

(−uj )τ (Gj (x)i ) = 0.

(4.6)

j=1

P From the right inequality of (4.4), τ (F (x)i ) ≤ τ (F (x)i ) + m j=1 (uj τ (Gj (x)i ) holds for all x ∈ S. So F (x)  L(u). That is x and u are the optimal solutions of problems (FP) and (FD) from the proposition 4.3. Lemma 4.6[16] Let X be a nonempty convex set in a real vector space Rn , F : X → R, G = (G1 , G2 , · · · , Gn ), Gi : Rn → R(i = 1, 2, · · · , n) be convex functions. We consider the following conditions. Condition a: F (x) < 0 and G(x) ≤ 0 for some x ∈ X; Condition b: u0 F (x) + hu, G(x)i ≥ 0 for all x ∈ X, (u0 , u) ≥ 0 and (u0 , u) 6= 0. If x does not satisfy Condition a, then Condition b has a solution (u0 , u) when x substitute x. Theorem 4.7 Let X ⊂ Rn be a nonempty convex sets, F : X → E n , Gj : X → E n (j = 1, 2, · · · , m) be convex fuzzy-valued functions, x be an optimal solution of problem (F P ), assume that there exist x , such that Gj (x)  e 0, then there exists u ≥ 0,such that L(x, u)  F (x) holds for every x ∈ X. Proof x be an optimal solution of (F P ), then F (x)  F (x) for any x ∈ X, that is Z τ (F (x)i ) =

1

r[F (x) 0

∗

(r, e+ i )

∗

− F (x)

(r, e− i )]dr

1

Z ≥ 0

− ∗ r[F (x)∗ (r, e+ i ) − F (x) (r, ei )]dr = τ (F (x)i )

for 1 ≤ i < n, Since F : X → E n , Gj : X → E n (j = 1, 2, · · · , m) be convex fuzzy-valued functions, then τ (F (x)i ), τ (Gj (x)i )(j = 1, 2, · · · , m, i = 1, 2, · · · , n) are convex real-valued functions. Therefore we consider the following systems: Z 1 Z 1 − − ∗ ∗ r[F (x)∗ (r, e+ ) − F (x) (r, e )]dr − r[F (x)∗ (r, e+ i i i ) − F (x) (r, ei )]dr < 0, 0

0

1

Z 0

− ∗ r[G∗j (r, e+ i ) − Gj (r, ei )]dr ≤ 0(j = 1, 2, · · · , m),

the system has no solution on X, then from lemma 4.6, there exists (u0 , u) ≥ 0 and (u0 , u) 6= 0 such that u0 (τ (F (x)i ) − τ (F (x)i )) +

m X

uj τ (Gj (x)i ) ≥ 0

j=1

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Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

P for every x ∈ X. assume that u0 = 0, then m j=1 uj τ (Gj (x)i ) ≥ 0 holds for every x ∈ X, since there exists x, such that τ (Gj (x)i ) ≤ 0 for 1 ≤ i ≤ n, then uj = 0(j = 1, 2, · · · , m), it contracts (u0 , u) 6= 0, thus u0 > 0. dividing the inequality by u0 , we have τ (F (x)i ) − τ (F (x)i ) +

m X

u0j τ (Gj (x)i ) ≥ 0,

j=1

where u0j =

uj u0 (j

= 1, 2, · · · , m). then F (x) +

m X

u0j Gj (x)  F (x).

j=1

denote u by u0 , and then there exists u ≥ 0, such that L(x, u)  F (x). Theorem 4.8 Let X ⊂ Rn be a nonempty convex set, F : X → E n , Gj : X → E n (j = 1, 2, · · · , m) be convex fuzzy-valued functions, x be a optimal solution of problem (F P ), assume there exists x, such that Gj (x)  e 0, then there exists u ≥ 0, such that (x, u) be a saddle point of the fuzzy-valued Lagrangian function L(x, u). Proof Let x be an optimal solution of problem (F P ), from the theorem 4.7, there exists u ≥ 0 such that L(x, u)  F (x) holds for every x ∈ X. then L(x, u) = F (x) +

m X

(4.7)

uj Gj (x)  F (x),

j=1

and uj ≥ 0, Gj (x)  e 0(j = 1, 2, · · · , m), so m X

uj Gj (x) = e 0.

(4.8)

j=1

That is L(x, u) = F (x) +

m X

uj Gj (x) = F (x),

j=1

From (4.7), we have L(x, u)  L(x, u).

(4.9)

From the definition of L(x, u), we have L(x, u) = F (x) +

m X

uj Gj (x),

j=1

and Gj (x)  e 0, uj ≥ 0(j = 1, 2, · · · , m), then L(x, u)  F (x) = L(x, u).

(4.10)

the (4.9),(4.10) indicate that (x, u) be a saddle point of the fuzzy-valued Lagrangian function L(x, u). 5. Conclusion In this article, we introduced the convexity and differentiability of n-dimensional fuzzy-number-valued function by means of a new order relation, which Pave a way for n-dimensional fuzzy optimization problem, so the saddle point optimal condition can be implemented.The n-dimensional fuzzy number valued function has been embedded into a complete Banach space, which is expressed by its support function, but the order of its support function is improper for fuzzy number, and establish this new order relationship more grasp the location information of fuzzy numbers. In the future work, we will discuss the application of n-dimensional fuzzy optimization in practice. 1488

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Hong-Xia Li and Zeng-Tai Gong: The convexity of n-dimensional fuzzy mappings and the saddle point ...

References [1] R.E. Bellman, L.A. Zadeh, Decision making in a fuzzy environment, Management Science 17 (1970) 141-164. [2] B. Bede, L. Stefanini, Generalized differentiablity of fuzzy-valued functions, Fuzzy Sets and Systems 230 (2013) 119-141. [3] M. Delgado, J. Kacporzyk, J.L. Verdegay, M.A. Vila, Fuzzy optimization: Recent Advances, New Advances, New York: Physica-Verlag, 1994. [4] Z.T. Gong, H.Yang, Ill-posed fuzzy initial-boundary value problems based on generalized differentiability and regularization, Fuzzy Sets Syst. 295 (2016) 99-113. [5] Z.T. Gong, H.X. Li, Saddle point optimality conditions in fuzzy optimization problems, Advances in Intelligent and Soft Computing 54 (2009) 7-14. [6] S.X. Hai, Z.T. Gong, H.X. Li, Generalized differentiability for n-dimensional fuzzy-number-valued functions and fuzzy optimization, Information Sciences 374 (2016) 151-163. [7] Y.J. Lai, C.L. Hwang, Fuzzy multiple objective decision making: methods and applications, Lecture Notes in Economics and Mathematical Systems, New York: Springer-Verlag, 1992. [8] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552-558. [9] W. Rodder, H.J. Zimmermann, Duality in fuzzy linear programming, In Extremal Methods and System Analysis, (Edited by A.V.Fiacco and K.O.Kortanek) pp.415-429, Berlin, 1980. [10] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst. 161 (2010) 1564-1584. [11] R. Slowe´ nski, J. Teghem, Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty, Boston: Kluwer Academic Publishers, 1990. [12] G.X. Wang, C.X. Wu, Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming, Fuzzy Sets Syst. 138 (2003) 559-591. [13] C.X. Wu, M. Ma, J.X. Fang, Structure theory of fuzzy analysis, Guiyang: Guizhou Scientific Publication, 1994.(in Chinese). [14] H.C. Wu, Duality theorems in fuzzy linear programming problems with fuzzy coefficients, Fuzzy Optimization and Decision Making 2 (2003) 61-73. [15] H.C. Wu, Saddle point optimality conditions in fuzzy optimization problems, Fuzzy Sets and Systems 14 (2003) 131-141. [16] H.C. Wu, Duality theorems in fuzzy mathematical programming problems based on the concept of necessity, Fuzzy Sets and Systems 139 (2003) 363-377. [17] C.X. Wu, M. Ma, The Basic of fuzzy analysis, BeiJing: National Defence Industrial Press, 1991. [18] H. Yan, J.P. Xu, A class of convex fuzzy mappings, Fuzzy Sets and Systems 129 (2002) 47-56. [19] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 331-353. [20] Y. Zhong, Y. Shi, Duality in fuzzy multi-criteria and multi-constraint level linear programming: A parametric approach, Fuzzy Sets and Systems 132 (2002) 335-346. [21] C. Zhang, Duality theory in fuzzy mathematical programming problems with fuzzy coefficients, Computers and Mathematics with Application 49 (2005) 1709-1730.

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Complex harmonic poles in the evolution of macromolecules depolymerization E.F. Doungmo Goufo* and S. Mugisha

∗*

Department of Mathematical Sciences, University of South Africa, Florida Campus, South Africa.

Abstract The full comprehension and handling of the phenomenon of shattering, sometime happening during the process of polymer chain degradation [29, 32], remains unsolved when using the traditional evolution equations describing the degradation. This traditional model has been proved to be very hard to handle as it involves evolution of two intertwined quantities. moreover, the explicit form of its solution is, in general, impossible to obtain. In this article, we explore the possibility of generalizing evolution equation modeling the polymer chain degradation and analyze the model with β derivative. We consider the general case where the breakup rate depends on the size of the chain breaking up. In the process, the alternative version of Sumudu integral transform is used to provide an explicit form of the general solution representing the evolution of polymer sizes distribution. In particular, we show that this evolution exhibits existence of complex periodic properties due to the presence of cosine and sine functions governing the solutions. Numerical simulations are performed for some particular cases and proves that such a system describing the polymer chain degradation contains complex and simple harmonic poles whose effects are given by these functions or a combination of them. This result may be crucial in the ongoing research to better handle and explain the phenomenon of shattering. Keywords: β- derivative; depolymerization; replicated fractional poles; simple and complex harmonic motion; shattering

1

Introduction, motivation and Justification

Depolymerization is the process where polymers or biopolymers are converted into monomers or mixtures of monomers. Polymers range from familiar synthetic plastics such as polystyrene (also called styrofoam) to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Historically, products arising from the linkage of repeating units by covalent chemical bonds have been the primary focus of polymer science; emerging important areas of the science now focus on non-covalent links. Polyisoprene of latex rubber and the polystyrene of styrofoam are examples of polymeric natural/biological and synthetic polymers, respectively. In biological contexts, essentially all biological macromolecules, i.e. proteins (polyamides), nucleic ∗

Email: [email protected] [email protected]

∗∗

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acids (polynucleotides), and polysaccharides are purely polymeric, composed in large part of polymeric components, for instance, isoprenylated/lipid-modified glycoproteins, where small lipidic molecule and oligosaccharide modifications occur on the polyamide backbone of the protein. Today, it is widely known that the Newtonian concept of derivative can no longer satisfy all the complexity of the natural occurrences. A couple of complex phenomena and features happening in some areas of sciences or engineering are still (partially) unexplained by the traditional existing methods and remain open problems. Usually in mathematical modeling of a natural phenomenon that changes, the evolution is described by a family of time-parameter operators, that map an initial given state of the system to all subsequent states that takes the system during the evolution. A widely devotion has been predominantly offered to way of looking at that evolution in which time’s change is described as transitions from one state to another. Hence, this is how the theory of semigroups was developed [16, 25], providing the mathematicians with very interesting tools to investigate and analyze resulting mathematical models. However, most of the phenomena scientists try to analyze and describe mathematically are complex and very hard to handle. Some of them like depolymerization, the rock fractures and fragmentation processes are difficult to analyze [11, 33] and often involve evolution of two intertwined quantities: the number of particles and the distribution of mass among the particles in the ensemble [15, 20, 28]. Then, though linear, they display non-linear features such as phase transition (called “shattering”) causing the appearance of a “dust” of “zero-size” particles with nonzero mass. The phenomena of “shattering” remain (partially) unexplained by traditional models. Another example is the groundwater flowing within a leaky aquifer. Recall that an aquifer is an underground layer of water-bearing permeable rock or unconsolidated materials (gravel, sand, or silt) from which groundwater can be extracted using a water well. Then, how do we explain accurately the observed movement of water within the leaky aquifer? As an attempt to answer this question, Hantush [17, 18] proposed an equation with the same name and his model has since been used by many hydro-geologists around the world. However, it is necessary to note that the model does not take into account all the non-usual details surrounding the movement of water through a leaky geological formation. Indeed, due to the deformation of some aquifers, the Hantush equation is not able to account for the effect of the changes in the mathematical formulation. Hence, all those non-usual features are beyond the usual models’ resolutions and need other techniques and methods of modeling with more parameters involved. Furthermore, time’s evolution and changes occurring in some systems do not happen on the same manner after a fixed or constant interval of time and do not follow the same routine as one would expect. For instance, a huge variation can occur in a fraction of second causing a major change that may affect the whole system’s state forever. Indeed, it has turned out recently that many phenomena in different fields, including sciences, en-

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gineering and technology can be described very successfully by the models using fractional order differential equations [4, 6, 9, 10, 13, 14, 19, 22, 27]. Hence, differential equations with fractional derivative have become a useful tool for describing nonlinear phenomena that are involved in many branches of chemistry, engineering, biology, ecology and numerous domains of applied sciences. Many mathematical models, including those in acoustic dissipation, mathematical epidemiology, continuous time random walk, biomedical engineering, fractional signal and image processing, control theory, Levy statistics, fractional phase-locked loops, fractional Brownian, porous media, fractional filters motion and nonlocal phenomena have proved to provide a better description of the phenomenon under investigation than models with the conventional integer-order derivative [6, 22, 26]. One of the attempts to enhance mathematical models was to introduce the concept of derivative with fractional order. There exist in the literature number of definitions of fractional derivatives, including Riemann–Liouville and Caputo derivatives respectively defined as  n Z x 1 d α Dx (f (x)) = (x − t)n−α−1 f (t) dt, (1) Γ (n − α) dx 0 n − 1 < α ≤ n and Dxα (f (x)) =

1 Γ (n − α)

Z

0

x

(x − t)

n−α−1



d dt

n

f (t) dt,

(2)

n − 1 < α ≤ n. A new fractional derivative with no singular kernel was recently proposed by Caputo et al. in [7]. However, Caputo fractional derivative [8], for instance, is the one mostly used for modelling real world problems in the field [4, 6, 13–15, 20, 28]. However, this derivative exhibits some limitations like not obeying the traditional chain rule; which chain rule represents one of the key elements of the match asymptotic method [20, 28]. Recall that the match asymptotic method has never been used to solve any kind of fractional differential equations because of the nature and properties of fractional derivatives. Hence, the conformable fractional derivative was proposed [2, 21]. This fractional derivative is theoretically very easier to handle and obeys the chain rule. But it also exhibits a huge failure that is expressed by the fact that the fractional derivative of any differentiable function at the point zero is zero. This does not make any sense in a physical point of view and then, a modified new version, the β–derivative was proposed in order to skirt the noticed weakness. The main aim of this new derivative was, first of all, to extend the well-known match asymptotic method to the scope of the fractional differential equation and later to describe the boundary layers problems within the folder of fractional calculus. The β–derivative was defined as [1, 15, 20]:    1−β 1 −g(t) g t+ε(t+ Γ(β) )    lim for all t ≥ 0, 0 < β ≤ 1 ε ε→0 A β D g(t) = (3) t 0    g(t) for all t ≥ 0, β = 0,

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where g is a function such that g : [0, ∞) → R and Γ the gamma-function Z ∞ Γ(ζ) = tζ−1 e−t dt. 0

If the above limit of exists then g is said to be β−differentiable. β d Note that for β = 1, we have A 0 Dt g(t) = dt g(t). Moreover, unlike other derivatives with fractional parameters, the β–derivative of a function can be locally defined at a certain point, the same way like the first order derivative. For a general order, let us say mβ, the mβ–derivative of g is defined as   β A (m−1)β A mβ A D g(t) = D D g(t) for all t ≥ 0, m ∈ N, 0 < β ≤ 1 (4) t t 0 t 0 0

Notice that the mβ–derivative of a given function provides information about the previous n − 1–derivatives of the same function. For instance we have   β A β A 2β A 0 Dt g(t) =0 Dt 0 Dt g(t)   1−β " −β 1−β #  (5) 1 1 1 0 00 (1 − β) t + g + t+ g . = t+ Γ (β) Γ (β) Γ (β)

This gives the β–derivative a unique property of memory, that is not provided by any other derivative. It is also easy to verify that for β = 1, we recover the second derivative of g. For more properties and details on this new derivative, the readers can consult the reference [1, 15, 20, 28]. 1.1

The kinetic equation

The evolution of the sizes distribution occurring during polymer chain degradation is well known [12, 15, 32] to be described by the following integrodifferential equation ∂ g(x, t) = −g(x, t) ∂t

Z

0

x

H(y, x − y)dy + 2

Z∞

g(y, t)H(x, y − x)dy,

x, t > 0.

(6)

x

Expressing the solution of equation (6) in its explicit form is very hard since fragmentation (or polymer chain degradation) processes, as explained in the previous section, are difficult to analyse as they involve evolution of two intertwined quantities: the distribution of mass among the particles in the ensemble and the number of particles in it. That is why, though linear, they display non-linear features such as “shattering” phenomena which they cannot fully explain [11, 15, 33]. Then, in order to have a broader idea about the evolution of polymer chain degradation and maybe trying to understand the phenomenon of shattering as described here above, we explore the possibility of extending the analysis by considering the β–derivative defined in the previous section. This yields the following integrodifferential equation:

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A β 0 Dt g(x, t) = −g(x, t)

Z

0

x

Hβ (y, x − y)dy + 2

Z∞

g(y, t)Hβ (x, y − x)dy,

x, t > 0.

(7)

x

subject to the initial condition g(x, 0) = g0 (x),

x>0

(8)

where g(x, t) represents the density of x-groups (i.e. groups of size x) at time t and Hβ (x, y) gives the average fragmentation rate, that is, the average number at which clusters of size x + y undergo splitting to form an x-group and a y-group.

Some useful properties in the β−differentiation

2

Recall that there is a growing problem about the choice of the type of fractional derivative to use among the large number of its existing versions. We already mentioned the incapacity of most of them to explicitly provide the variation of the functions. Moreover, many models using fractional derivatives are not easy to handle analytically. The β–derivative allows us to palliate some insufficiencies of other fractional derivatives and then, we were able to successfully extend the well-known match asymptotic method [20, 28] to the scope of the fractional differential equation and also describe the boundary layers problems within the scope of fractional calculus. Next we recall some properties of the β–derivative all proved in [15, 20, 28]. Theorem 2.1. Assuming that, a given function, say g : [a, ∞) → R is β−differentiable at a given point, say t0 ≥ a, β ∈ (0, 1], then g is also continuous at t0 . Theorem 2.2. Assuming that f is β−differentiable on an open interval (a, b) then 1. If

A β 0 D t f (t) < 0

for all t ∈ (a, b) then f is decreasing on (a, b);

2. If

A β 0 D t f (t) > 0

for all t ∈ (a, b) then f is increasing on (a, b);

3. If

A β 0 D t f (t) = 0

for all t ∈ (a, b) then f is constant on (a, b).

Theorem 2.3. Assuming that, g 6= 0 and f are two β−differentiable functions with β ∈ (0, 1] then the following relations are satisfied 1.

A β 0 D t (af

2.

A β 0 D t (c) = 0 A β 0 D t (f

β

β

A (t) + bg(t)) = aA 0 D t (f (t)) + b0 D t (g (t)) for all real numbers a and b;

for any given constant c; β

β

A (t) g(t)) = g (t) A 0 D t (f (t)) + f (t) 0 D t (g (t)) ;   β A β β f (t) g(t)A 0 D t (f (t))−f (t)0 D t (g(t)) 4. A D = . 2 0 t g(t) g (t)

3.

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Theorem 2.4. Let f : [a, ∞) → R be a function such that f is differentiable and also β−differentiable. Let g be a function defined in the range of f and also differentiable, then we have the following rule A β 0 D t (gof (t)) =

 t+

1 Γ (β)

1−β

f 0 (t)g 0 (f (t))

(9)

Definition 2.1. Let f : [a, ∞) → R be a given function, then we propose that the β− integral of f is β−1 Z t 1 A β f (ξ)dξ (10) ξ+ a I t (f (t)) = Γ (β) a The above operator is the inverse operator of the proposed fractional derivative. We shall present to underpin this statement by the following theorem. h i A β A β Theorem 2.5. 0 Dt 0 I t f (t) = f (t) for all t ≥ 0 with f a given continuous and differentiable function. Proof. [1, Theorem 7] Theorem 2.6.

A β a It

h i Dtβ f (t) = f (t) − f (a)

(11)

for all t ≥ a with f a given continuous and differentiable function. Proof. [1, Theorem 8]

3

Solutions to the model

Note that these above models (6) and (7) are well applicable in many branches of natural sciences, including physics, chemistry, engineering, biology, ecology, just to name a few, and in numerous domains of applied sciences, such as the rock fractures and break of droplets. Various types of fragmentation equations have been comprehensively analyzed in numerous works (see, e.g., [12, 30, 33]). In the domain of polymer science, the fragmentation dynamics has also been of considerable interest, since degradation of bonds or depolymerisation results in fragmentation, see [5, 23, 32]. In [23], the authors used statistical arguments to find and analyze the size distribution of the model. The authors in [5] analysed the model in combination with the inverse process, that is, the coagulation process, and provided a similar result for the size distribution. However, the classical fragmentation model (6) has been proved to be unable to fully describe some bizarre phenomena observed in such a degradation process, like for instance shattering as described above and also in [11, 23, 32, 33]. Recall that shattering is a phenomenon seen as an explosive or dishonest Markov process, see e.g. [3, 24] and has been associated with an infinite cascade of breakup events creating a ‘dust’ of particles of zero size which, however, carry non-zero mass. Hence, to have explicit solutions to the model, we consider

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the case where the breakup rate depends on the size of the chain breaking and takes the form Hβ (x, y) = (x + y)ν , ν ∈ R (12)

Substituting in equation (7) yields

Dtβ (g(x, t)) = −xν+1 g(x, t) + 2

Z∞

y ν g(y, t)dy,

0≤β ≤1

(13)

x

Taking the the modified Sumudu transform Sβ (see the Appendix below) of both sides of equation (13) yields Z∞   β ν+1 β Sβ Dt g(x, t), r = −x Gs (x, r) + 2 y ν Gβs (y, r)dy, x

where Gβs (x, r) represents the the modified Sumudu transform Sβ (g(x, t), r) of g(x, t). Using the relation (23) of Appendix, we obtain Z∞ r −2 (Gβs (x, r) − g0 (x)) = −xν+1 Gβs (x, r) + 2 y ν Gβs (y, r)dy, x

rearranged to have
2

1 + xν+1 r Gβs (x, r) − 2r 2

Z∞

y ν Gβs (y, r)dy = g0 (x).

(14)

x

Next, it is important to mention that considering the differential equation (13), it is implicitly required that the function ξ −→ g(ξ, t) is integrable, in the sense of Lebesgue, on any interval [, ∞) for  > 0 and almost every ξ > 0. Obviously, the same assertion applies to the functions ξ −→ g0 (ξ) and ξ −→ Gβs (ξ, r), 0 ≤ β ≤ 1. This allows us to put Z∞ 2 y ν Gβs (y, r)dy (15) Z(x, r) = −2r x

knowing that the integrand will be integrable over any interval [, ∞) and the integral will be absolutely continuous at each x > 0. The substitution of Z(x, r) into (14) yields the partial differential equation   1 + xν+1 r 2 ∂x Z(x, r) + Z(x, r) = g0 (x). (16) 1 + r 2 xν Choosing the constant in the general solution so as to have solutions converging to zero at ∞, we obtain its solution given as Z∞ ν ξ g0 (ξ) σr,ν (ξ) Z(x, r) = 2r 2e−σr,ν (x) e dξ 1 + r 2 ξ ν+1 x

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where σr,ν (x) =

Zx 0


2 2r 2 ξ ν 2 ν+1 ν+1 dξ = ln 1 + r x . 1 + r 2 ξ ν+1

(17)

Thus, substituting Z(x, r) into (15) yields the solution of (14) given as Gβs (x, r) =

−1 xν



2r 2 xν e−σr,ν (x) 2 ν+1 1+r x

 Zx

g0 (x) ξ ν g0 (ξ) σr,ν (ξ) e dξ + 2 ν+1 1+r ξ 1 + r 2 xν+1

∞

g0 (x) = − 1 + r 2 xν+1

2r 2 (1 + r 2 xν+1 )

2 ν+1 +1

Zx

(18) ξ ν 1 + r2ξ

∞


2 −1 ν+1 ν+1

g0 (ξ)dξ

Applying the inverse of the modified Sumudu transform, which coincides with the inverse Sumudu transform, we are finally lead to the solution of the model (13), given by g(x, t) = Sβ−1 (Gβs (x, r), t)   2 −1   Zx 2 2 ν+1 ν+1 r (1 + r ξ ) 1 , t − 2 ξ ν g0 (ξ)Sβ−1  , t dξ = g0 (x)Sβ−1 2 2 ν+1 ν+1 +1 1+r x (1 + r 2 xν+1 ) ∞   2 −1 Zx 2 2 ν+1 ν+1 √ r (1 + r ξ ) = g0 (x) cos(t xν+1 ) − 2 ξ ν g0 (ξ)Sβ−1  , t dξ 2 ν+1 +1 (1 + r 2 xν+1 ) ∞

(19)

Remark 3.1. The expression g(x, t) in (19) is well-defined only if the integral Zx

∞

converges.



ξ ν g0 (ξ)Sβ−1 

2

2 ν+1

r (1 + r ξ

(1 + r 2 xν+1 )

)

2 ν+1 −1

2 ν+1 +1



, t dξ

We are now capable of taking some specific values of ν to see the exact expression of the solution. • For ν = 1, expression (19) becomes g(x, t) = g0 (x)Sβ−1



 Zx 1 , t − 2 ξg0 (ξ)Sβ−1 1 + r 2 x2

= g0 (x) cos xt −

t sin xt x

Zx

∞

r2 (1 + r 2 x2 )

2

!

, t dξ (20)

ξg0 (ξ)dξ

∞

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• For ν = −3, expression (19) becomes g(x, t) = g0 (x)Sβ−1



 Zx  
1 −1 2 2 −2 −2 , t − 2 ξg (ξ)S r 1 + r ξ , t dξ 0 β 1 + r 2 x−2

t = g0 (x) cos − 2 x

Zx

∞

ξg0 (ξ)

ξt sin ξt 2

dξ

(21)

∞

t = g0 (x) cos − x

Zx

t tξ 2 g0 (ξ) sin dξ ξ

∞

3

g1(x,t) when g0(x) = 1/x

40 30 20

0

1

g (x,t)

10

−10 −20 −30 −40 15 3

10

2.5 2 5

1.5 1

t

Fig. 1.

4

0

0.5

x

g(x, t) when ν = 1 and g0 (x) = 1/x3

Concluding remarks

We have explored the possibility of using new and alternative methods to generalize evolution equation modeling the polymer chain degradation. In the process, a modified version of the Sumudu transform is exploited to perform analysis of the system endowed the β−derivative and where the breakup rate depends on the size of the chain breaking up. Explicit forms of the solutions in some particular cases showed that the dynamics of this evolution exhibits complex periodic properties due to the presence of cosine and sine functions, as shown in Figs. 1 to 6, plotted for a positive value (ν = 1) and a negative value (ν = −3) of ν. Figs. 1 to 3 represent the solution for ν = 1 with initial condition g0 (x) = 1/x3 : Fig.1 is the 2−D surface plot while Fig. 2 and 3 are respectively its cross

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10

g1(t)

0

−10

−20 x = 0.5 x=1 x = 1.5 x=2 x = 2.5 x=3

−30

−40

0

2

4

6

8

10

12

14

t

Fig. 2.

g(x, t) as a function of t when ν = 1 and g0 (x) = 1/x3 , for a few values of x

g1(x) when g0(x) = 1/x3 40

30

20

g1(x)

10

0

−10

−20

−30

−40 0.5

t=0 t=π t = 2π t = 3π t = 4π

1

1.5

2

2.5

3

x

Fig. 3.

g(x, t) as a function of x when ν = 1 and g0 (x) = 1/x3 , for a few values of t : 0, π, 2π, 3π, 4π

section and longitudinal section drawn for some specific values of the size x and time t. A similar reasoning applies to Figs. 4 to 6, but this time with ν = −3. This infers existence of complex and simple harmonic poles in the dynamics of polymer chain degradation whose effects are characterized by these functions or a combination of them. This work improved the preceding one with the inclusion of a more general expression of the breakup rate derivative and β−derivative. This work might be a breakthrough that may lead to

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0

30 25 20

g (x,t)

15

2

10 5 0 −5 −10 15 3

10

2.5 2 5

1.5 1 0

t

Fig. 4.

0.5

x

g(x, t) when ν = −3 and g0 (x) = 1/x3

g2(t) when g0(x) = 1/x3 30

25

20

g2(t)

15

10

5 x = 0.5 x=1 x = 1.5 x=2 x = 2.5 x=3

0

−5

−10

0

2

4

6

8

10

12

14

t

Fig. 5.

g(x, t) as a function of t when ν = −3 and g0 (x) = 1/x3 , for a few values of x

a better understanding of bizarre phenomena happening in some dynamics such as the phenomenon of shattering.

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25 t=0 t=π t = 2π t = 3π t = 4π

g2(x)

20

15

10

5

0 0.5

1

1.5

2

2.5

3

x

Fig. 6.

g(x, t) as a function of x when ν = −3 and g0 (x) = 1/x3 , for a few values of t : 0, π, 2π, 3π, 4π

Appendix: The new Sumudu integral transform Definition: Let g be a function defined in (0, ∞), then, we define the modified Sumudu transform of g as

Sβ (g(t), u) =

Z

0

∞

1 t+ Γ(β)

β−dβe

1 −t e u g(t)dt, u

(22)

where dβe is the smallest integer greater or equal to β. Since β ∈ (0, 1] in this article then, β − dβe = β − 1.

An important property of the modified Sumudu transform: If S(g(t), u) is the well known Sumudu transform of g defined in [31] as S(g(t), u) =

Z

0

∞

  1 t exp − g(t)dt, u u

then, we have the following relation: X 1 1 S(g(t), u) − g (k) (0) n n−k u u n−1

β n−1 Sβ ( A (t), u) = 0 Dt g

(23)

k=0

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Proof. By definition we have β−1 Z ∞ 1 A β n−1 t+ Sβ (0 Dt g (t), u) = Γ(β) 0     1−β  1 n−1 n−1   β−1  t + ε t + g − g (t) Γ(β)  1 t   t+ 1  dt lim exp −  ε→0 u u  Γ(β) ε =

Z

0

∞

1 t+ Γ(β)

β−1

  1 t exp − u u

(24)

!  1−β 1 g n−1 (t + η) − g n−1 (t) t+ lim dt η→0 Γ(β) η

 1−β 1 where we have put η = ε t + Γ(β) −→ 0 as ε −→ 0. Hence, making use of the well known property of Sumudu transform S(g(t), u) [31], we obtain X 1 1 S(g(t), u) − g (k) (0), un un−k n−1

β n−1 Sβ ( A (t), u) = S(g n(t), u) = 0 Dt g

k=0

which concludes the proof.

Acknowledgments: This work was partially supported by the grant No: 105932 from the National Research Foundation (NRF) and a grant from the Simons Foundation. References [1] A. Atangana and E.F. Doungmo Goufo, Extension of Match Asymptotic Method to Fractional Boundary Layers Problems, Mathematical Problems in Engineering, Volume 2014, Article ID 107535, (2014). http://dx.doi.org/10.1155/2014/107535 [2] M. Abu Hammad and R. Khalil, Conformable fractional Heat differential equation, International Journal of Pure and Applied Mathematics, vol. 94, no. 2, pp. 215–221, 2014. [3] W.J. Anderson, Continuous-Time Markov Chains. An Applications-Oriented Approach, Springer Verlag, New York, 1991. [4] E.G. Bazhlekova, Subordination principle for fractional evolution equations, Fractional Calculus & Applied Analysis, vol. 3, Number 3, pp. 213 – 230, 2000. [5] R. Blatz, J.N. Tobobsky, Statistical investigation of fragments of polymer molecules, J. Phys. Chem., V.49. P.77., 1945. [6] D. Brockmann, L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming L´evy flights with fluctuations, Phys. Review Lett. 98, No 17, Article ID 178301, 2007; DOI:10.1103/PhysRevLett.98.178301. [7] M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1:2, pp. 1–13, 2015. [8] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Ast. Soc. 13, No 5, 529539, 1967; Reprinted in: Fract. Calc. Appl. Anal. 11, No 1, pp. 3-14, 2008.

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¨ [9] E. Demirci, A. Unal and N. Ozalp , A fractional order seir model with density dependent death rate, Hacettepe Journal of Mathematics and Statistics, 40, No2, 287–295, 2011 [10] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 2010. [11] E. F. Doungmo Goufo, Non-local and Non-autonomous Fragmentation-Coagulation Processes in Moving Media, PhD thesis, North-West University, South Africa, 2014. [12] E.F. Doungmo Goufo, S.C. Oukouomi Noutchie, Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates. Comptes Rendus Mathematique, C.R Acad. Sci Paris, Ser, I, (2013); http://dx.doi.org/10.1016/j.crma.2013.09.023. [13] Doungmo Goufo EF, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos, Vol. 26, No 8, 2016 http://dx.doi.org/10.1063/1.4958921 [14] E. F. Doungmo Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fractional Calculus and Applied Analysis, 2015 (in press) [15] E.F. Doungmo Goufo, Evolution equations with a parameter and application to transport-convection differential equations, Turkish Journal Of Mathematics, 2016, DOI: 10.3906/mat-1603-107, Available online: 27.06.2016 at http://journals.tubitak.gov.tr/math/accepted.htm [16] K-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194), Springer, 2000 [17] M. S. Hantush, Analysis of data from pumping tests in leaky aquifers, Transactions, American Geophysical Union, vol. 37,no. 6, pp. 702-714, 1956. [18] M. S. Hantush, C.E. Jacob, Non-steady radial flow in an infinite leaky aquifer, Transactions, American Geophysical Union, vol. 36, no. 1, pp. 95-100, 1955. [19] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific, Singapore, 1999. [20] J. Kestin, L.N. Persen, The transfer of heat across a turbulent boundary layer at very high prandtl numbers. Int. J. Heat Mass Transfer 5, 1962, pp.355–371. [21] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014. [22] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Sci. B.V., Amsterdam, 2006. [23] H. Mark, R Simha, Degradation of long chain molecules, Trans Faraday 35, 611–618, 1940 [24] J.R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998. [25] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, 1983. [26] S. Pooseh, H.S Rodrigues, Torres D.F.M., Fractional derivatives in dengue epidemics. In: Simos, T.E., Psihoyios, G., Tsitouras, C., Anastassi, Z. (eds.) Numerical Analysis and Applied Mathematics, ICNAAM, American Institute of Physics, Melville, 2011, pp. 739–742. [27] Pr¨ uss J., Evolutionary Integral Equations and Applications, Birkh¨auser, Basel–Boston–Berlin, 1993. [28] H. Schlichting, Boundary-Layer Theory (7 ed.) New York (USA): McGraw-Hill, 1979. [29] G. T., Tsao, Structures of Cellulosic Materials and their Hydrolysis by Enzymes, Perspectives in Biotechnology and Applied Microbiology, 205–212, 1986 [30] W. Wagner, Explosion phenomena in stochastic coagulation-fragmentation models, Ann. Appl. Probab. 15 (3) 2081–2112, 2005. [31] G. K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology 24, 35–43, 1993. [32] R.M. Ziff, E.D. McGrady, The kinetics of cluster fragmentation and depolymerization, J. Phys. A 18 3027–3037, 1985. [33] R.M. Ziff, E.D. McGrady, Shattering transition in fragmentation, Phys. Rev. Lett. 58 (9) 892–895, 1987.

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A …xed point convergence theorem with applications in left multivariate fractional calculus George A. Anastassiou1 and Ioannis K. Argyros2 1 Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] 2 Department of Mathematics Sciences Cameron University Lawton, Ok 73505, USA [email protected] Abstract A …xed point theorem is given under general conditions on the operators involved in a Banach space setting. The results …nd applications in left multivariate fractional calculus.

2010 AMS Subject Classi…cation: 65G99, 65H10, 26A33, 47J25, 47J05. Key Words and Phrases: Fixed point, Banach space, semi-local convergence, left multivariate fractional calculus.

1

Introductions

Numerous problems can be formulated as an equation like R (x) = 0;

(1.1)

where R is a continuous operator de…ned on a subset of a Banach space B1 with values in a Banach space B2 using Mathematical Modelling [1], [7], [11], [12], [16], [18]. The solutions denoted by x can be found in explicit form only in special cases. That is why most solution methods for these equations are usually iterative. Let L (B1 ; B2 ) denote the space of bounded linear operators from B1 into B2 . Let also A ( ) : ! L (B1 ; B1 ) be a continuous operator. Set F = LR;

(1.2)

1

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where L 2 L (B2 ; B1 ). We shall approximate x using a sequence fxn g generated by the …xed point scheme: xn+1 := xn + zn , A (xn ) zn + F (xn ) = 0 , zn = Q (zn ) := (I A (xn )) zn F (xn ) ;

(1.3)

where x0 2 . The sequence fxn g de…ned by xn+1 = Q (xn ) = Q(n+1) (x0 )

(1.4)

exists. In case of convergence we write: Q1 (x0 ) := lim (Qn (x0 )) = lim xn : n!1

(1.5)

n!1

Many methods in the literature can be considered special cases of method (1.3). We can choose A to be: A (x) = F 0 (x) (Newton’s method), A (x) = F 0 (x0 ) (Modi…ed Newton’s method), A (x) = [x; g (x) ; F ] ; g : ! B1 (Ste¤ensen’s method). Many other choices for A can be found in [1-20] and the references there in. Therefore, it is important to study the convergence of method (1.3) under generalized conditions. In particular, we present the semi-local convergence of method (1.3) using only continuity assumptions on operator F and for a so general operator A as to allow applications to left multivariate fractional calculus and other areas. The rest of the paper is organized as follows: Section 2 contains the semilocal convergence of method (1.3). In the concluding Section 3, we suggest some applications to left multivariate fractional calculus.

2

Convergence

Let B (w; ), B (w; ) stand, respectively for the open and closed balls in B1 with center w 2 B1 and of radius > 0. We present the semi-local convergence of method (1.3) in this section. Theorem 2.1 Let F : B1 ! B2 , A ( ) : ! L (B1 ; B1 ) and x0 2 be as de…ned in the Introduction. Suppose: there exist 0 2 (0; 1), 1 2 (0; 1), 0 such that for each x; y 2 :=

0

+

1

kF (x0 )k kI kF (y)

F (x)

< 1;

(2.1)

;

(2.2)

A (x)k

0;

A (x) (y

x)k

(2.3) 1

ky

xk

(2.4)

2

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and B (x0 ; )

;

(2.5)

where =

:

1

Then, sequence fxn g generated for x0 2

(2.6)

by

xn+1 = xn + Q1 (0) ; Qn (z) := (I

A (xn )) z

F (xn )

(2.7)

is well de…ned in B (x0 ; ), remains in B (x0 ; ) for each n = 0; 1; 2; ::: and converges to x which is the only solution of equation F (x) = 0 in B (x0 ; ). Moreover, an apriori error estimate is given by the sequence f n g de…ned by 0

:= ,

n

= Tn1 (0) , Tn (t) =

0

+

(2.8)

1 n 1

for each n = 1; 2; ::: and satisfying lim

n!1

n

= 0:

(2.9)

Furthermore, an aposteriori error estimate is given by the sequence f by 1 n := Hn (0) , Hn (t) = t + 1 pn 1 ; qn := kxn

x0 k

n

;

ng

der…ned (2.10) (2.11)

where pn

1

:= kxn

xn

1k

for each n = 1; 2; :::

(2.12)

Proof. We shall show using mathematical induction the following assertion is true: (An ) xn 2 X and n 0 are well de…ned and such that n

+ pn

1

n 1:

(2.13)

By the de…nition of , (2.3)-(2.6) we have that there exists r [7, pp. 3]) such that 0 + kF (x0 )k = r

(Lemma 1.4

and k 0r

k 0

! 0 as k ! 1:

That is (Lemma 1.5 [7, pp. 4]) x1 is well de…ned and p0 We need the estimate: T1 ( 0

r) = 0r

+

1

0

(

r) +

= G0 ( )

1 0

r=

r.

= r:

3

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That is (Lemma 1.4 [7, pp. 3]) 1

exists and satis…es

1

+ p0

r+r =

=

0:

Hence (I0 ) is true. Suppose that for each k = 1; 2; :::; n; assertion (Ik ) is true. We must show: xk+1 exists and …nd a bound r for pk . Indeed, we have in turn that 0 k + 1 1 k k 1 k = 0 k + 1 k 1 = Tk ( That is there exists r r=

k:

1 k

such that

k

0r

k)

+

1

k 1

k

and (

0

+

m+1

=

i 1)

r!0

(2.14)

as i ! 1: The induction hypothesis gives that k X1

qk

k X1

pm

m=0

m

k

;

m=0

so xk 2 B (x0 ; ) and x1 satis…es kI A (x1 )k 0 (by (2.3)). Using the induction hypothesis, (1.3) and (2.4), we get kF (xk )k = kF (xk )

F (xk

1 pk 1

1) 1

A (xk k 1

1 ) (xk

xk

1 )k

(2.15)

k

leading together with (2.14) to: 0r

+ kF (xk )k

which implies xk+1 exists and pk r that k+1 Tk+1 ( k r) = Tk ( so

k+1

k. k)

r; It follows from the de…nition of r=

k

r+r =

k

r;

exists and satis…es k+1

+ pk

k

so the induction for (In ) is completed. Let j k. Then, we obtain in turn that kxj+k

xk k

j X

pi

i=k

j X

j

j+1

=

k

j+k

k:

(2.16)

i=k

We also have using induction that k+1

= Tk+1

k+1

Tk+1 (

k)

k

:::

k+1

:

(2.17)

4

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Hence, by (2.1) and (2.17) lim

= 0, so fxk g is a complete sequence in a

k

k!1

Banach space X and as such it converges to some x . By letting j ! 1 in (2.16), we conclude that x 2 B (xk ; k ). Moreover, by letting k ! 1 in (2.15) and using the continuity of F we get that F (x ) = 0. Notice that Hk (

k)

Tk (

k)

k;

so the apriori bound exists. That is k is smaller in general than k . Clearly, the conditions of the theorem are satis…ed for xk replacing x0 (by (2.16)). Hence, by (2.8) x 2 B (xn ; n ), which completes the proof for the aposteriori bound. Remark 2.2 (a) It follows from the proof of Theorem 2.1 that the conclusions hold, if A ( ) is replaced by a more general continuous operator A : ! B1 . (b) In the next section some applications are suggested for special choices of the "A" operators with 0 := 0 and 1 := 1 :

3

Applications to left multivariate fractional calculus

Our presented earlier semi-local convergence results, see Theorem 2.1, apply in the next two multivariate fractional settings given that the following inequalities are ful…lled: k1 A (x)k1 (3.1) 0 2 (0; 1) ; and (F (y) where

0;

1

! F (x)) i

x)

1

ky

xk ;

(3.2)

2 (0; 1), furthermore =

for all x; y 2

A (x) (y

k Q

i=1

0

+

1

2 (0; 1) ;

(3.3)

[ai ; bi ], where ai < ai < bi < bi , i = 1; :::; k.

! ! Above i is the unit vector in Rk , k 2 N, i = 1, and k k is a norm in Rk . The speci…c functions A (x), F (x) will be described next. I) Consider the left multidimensional Riemann-Liouville fractional integral of order = ( 1 ; :::; k ) ( i > 0, i = 1; :::; k): Ia+ f (x) =

1 k Q

i=1

( i)

Z

x1

a1

:::

Z

xk

ak

k Y

(xi

ti )

i

1

f (t1 ; :::; tk ) dt1 :::dtk ;

i=1

(3.4) 5

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where

is the gamma function, f 2 L1

x = (x1 ; :::; xk ) 2

k Q

k Q

[ai ; bi ] , a = (a1 ; :::; ak ), and

i=1

[ai ; bi ] :

i=1

By [6], we get that Ia+ f is a continuous function on

k Q

[ai ; bi ]. Further-

i=1

more by [6] we get that Ia+ is a bounded linear operator, which is a positive operator, plus that Ia+ f (a) = 0: k Q In particular, Ia+ f is continuous on [ai ; bi ] : i=1

Thus there exist x1 ; x2 2

over all x 2

k Q

i=1

k Q

i=1

[ai ; bi ] such that

Ia+ f (x1 ) = min Ia+ f (x) ; Ia+ f (x2 ) = max Ia+ f (x) ;

(3.5)

Ia+ f (x1 ) > 0:

(3.6)

[ai ; bi ] :

We assume that Hence Ia+ f

1;

Here, we de…ne

k Q

[ai ;bi ]

= Ia+ f (x2 ) > 0:

Jf (x) = mf (x) , 0 < m < for any x 2

k Q

i=1

(3.7)

i=1

1 ; 2

(3.8)

[ai ; bi ] :

Therefore the equation Jf (x) = 0, x 2

k Y

[ai ; bi ] ;

(3.9)

i=1

has the same solutions as the equation F (x) :=

Jf (x) = 0; 2 Ia+ f (x2 )

x2

k Y

[ai ; bi ] :

(3.10)

i=1

Notice that Ia+

f 2 Ia+ f (x2 )

!

(x) =

Ia+ f (x) 2 Ia+ f (x2 )

k Y 1 < 1; x 2 [ai ; bi ] : 2 i=1

(3.11)

6

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Call

k Y Ia+ f (x) , 8x2 [ai ; bi ] : 2 Ia+ f (x2 ) i=1

A (x) := We notice that 0
> ;

D a G (x1 ) ; 2D a G (x2 ) (3.50)

12

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equivalently,

2 D a G (x2 ) +

k m Y (bi ai ) i i (mi i + 1) i=1

!

k P

mi

@ G mk 1 :::@x @xm 1 k i=1

< D a G (x1 ) ; (3.51) 1

which is possible for small and small (bi ai ), all i = 1; :::; k: That is 2 (0; 1), ful…lling (3.3). So our numerical method converges and solves (3.32).

References [1] S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newtontype method, J. Math. Anal. Applic. 366, 1, (2010), 164-174. [2] G. Anastassiou, Fractional Di¤ erentiation Inequalities, Springer, New York, 2009. [3] G. Anastassiou, Fractional Representation Formulae and Right Fractional Inequalities, Mathematical and Computer Modelling, 54, (10-12), (2011), 3098-3115. [4] G. Anastassiou, Intelligent Mathematics: Springer, Heidelberg, 2011.

Computational Analysis,

[5] G. Anastassiou, Advanced Inequalities, World Scienti…c Publ. Corp., Singapore, New York, 2011. [6] G. Anastassiou, On left multidimensional Riemann-Liouville fractional integral, J. of Computational Analysis and Applications, accepted, 2015. [7] G. Anastassiou, I.K. Argyros, Studies in Computational Intelligence, 624, Intelligent Numerical Methods: Applications to Fractional Calculus, Springer, Heidelberg, 2016. [8] I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298, (2004), 374-397. [9] I.K. Argyros, Convergence and Applications of Newton-type iterations, Springer-Verlag Publ., New York, 2008. [10] I.K. Argyros, On a class of Newton-like methods for solving nonlinear equations, J. Comput. Appl. Math. 228, (2009), 115-122. [11] I.K. Argyros, A semilocal convergence analysis for directional Newton methods, Math. Comp., AMS 80, (2011), 327-343. 13

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[12] I.K. Argyros, Y.J. Cho, S. Hilout, Numerical methods for equations and its applications, CRC Press/Taylor and Fracncis Group, New York, 2012. [13] I.K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton’s method, J. Complexity 28, (2012), 364-387. [14] J.A. Ezquérro, J.M. Gutiérrez, M.A. Hernández, N. Romero, M.J. Rubio, The Newton method: From Newton to Kantorovich (Spanish), Gac. R. Soc. Mat. Esp. 13, (2010), 53-76. [15] J.A. Ezquérro, M.A. Hernández, Newton-type methods of high order and domains of semilocal and global convergence, Appl. Math. Comput. 214, 1, (2009), 142-154. [16] L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [17] A.A. Magréñán, Di¤ erent anomalies in a Jarratt family of iterative root …nding methods, Appl. Math. Comput. 233, (2014), 29-38. [18] A.A. Magréñán, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 215-224. [19] F.A. Potra, V. Ptak, Nondiscrete induction and iterative processes, Pitman Publ., London, 1984. [20] P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26, (2010), 3-42.

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Stability of delay-distributed virus dynamics model with cell-to-cell transmission and CTL immune response A. M. Elaiw, A. A. Raezah and A. S. Alofi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Emails: a m [email protected] (A. Elaiw) Abstract In this paper, we study the stability analysis of a virus dynamics model with CTL immune response and with both cell-to-cell and virus-to-cell transmissions. The model contains three types of distributed time delays. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number (R0 ) and the CTL immune response activation number (R1 ). By using suitable Lyapunov functionals, we show that if R0 ≤ 1, then the infection-free steady state E0 is globally asymptotically stable; if R1 ≤ 1 < R0 , then the CTL-inactivated infection steady state E1 is globally asymptotically stable; if R1 > 1, then the CTL-activated infection steady state E2 is globally asymptotically stable. Numerical simulations are conducted to support the theoretical results. Keywords: Virus dynamics; CTL immune response; Global stability; time delay; cell-to-cell transmission.

1

Introduction

During the past decades, several mathematical models have been proposed to describe the dynamical behavior of many human viruses such as HIV, HBV, HCV and HTLV-I (see e.g. [1]-[27]). Studying the global stability of the model’s equilibria has become one of the most important features which help us to better understanding of the virus dynamics. Thus, several researchers have devoted extensive efforts to study the global stability of virus dynamics models (see e.g. [2]-[13]). All the above mentioned works focus on cell-free viral spread in a compartment such as the bloodstream. Recently, some viral infection models have been proposed to model both virus-to-cell and cell-to-cell transmissions (see [28]-[29]). The viral infection model with cell-to-cell transmission and distributed time delay has been proposed in [29] as: T˙ (t) = λ − dT (t) − β1 T (t)V (t) − β2 T (t)T ∗ (t), Z ∞ T˙ ∗ (t) = [β1 T (t − s)V (t − s)ds + β2 T (t − s)T ∗ (t − s)] f (s)e−µ1 s ds − µ1 T ∗ (t),

(1) (2)

0

V˙ (t) = bT ∗ (t − s)ds − cV (t),

(3)

where, T (t), T ∗ (t) andV (t) are the concentrations of the uninfected cells which are susceptible to infection, infected cells that produces viruses, and free virus particles at time t, respectively; β1 is the virus-to-cell infection rate constant; β2 is the cell-to-cell infection rate constant; µ1 and c are death rate constants of the infected cells and viruses, respectively; b is the average number of viruses that bud out from an infected cell. e−µ1 s is the survival rate of infected cells during the time delay s, where s is assumed to be distributed according to a probability distribution f (s).

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It is observed that, all the viral infection models with cell-to-cell transmission did not consider the effect of immune response. The immune response is universal and necessary to eliminate or control the disease after viral infection. The Cytotoxic T Lymphocyte (CTL) cells are responsible to attack and kill the infected cells. Several viral infection models have been introduced in the literature to model the CTL immune response to several diseases [23]-[27]. However, in [23]-[27], only virus-to-cell transmission has been considered. Therefore, our aim in this paper is to propose and analyze a delay-distributed virus dynamics model with virus-to-cell and cell-to-cell transmissions and takes into account the CTL immune response.

2

The model

In this section, we propose a virus dynamics model with cell-to-cell transmission and CTL immune response. T˙ (t) = λ − dT (t) − β1 T (t)V (t) − β2 T (t)T ∗ (t), Z ∞ ∗ ˙ f1 (s)e−µ1 s [β1 T (t − s)V (t − s) + β2 T (t − s)T ∗ (t − s)] ds − µ1 T ∗ (t) − pT ∗ (t)Z(t), T (t) = 0 Z ∞ ˙ V (t) = b e−µ2 s f2 (s)T ∗ (t − s)ds − cV (t), 0 ∗

˙ Z(t) = kT (t)Z(t) − qZ(t),

(4) (5) (6) (7)

where, Z(t) is the concentration of CTL immune cells at time t. The infected cells are killed by the CTL immune response with rate pT ∗ (t)Z(t), where p is constant. The CTLs are proliferated at a rate kT ∗ (t)Z(t) and die at a rate qZ(t). All the other variables and parameters of the model have the same meanings as given in (1)-(3). Let us assume that the probability distribution function fi (s) satisfy fi (s) > 0, i = 1, 2 and Z∞

Z∞ fi (s)ds = 1,

0

fi (u)e`u du < ∞, i = 1, 2,

0

where ` > 0. Denote

Z ηi =

∞

fi (s)e−µi s ds,

i = 1, 2.

0

Thus 0 < ηi ≤ 1. Define the Banach space of fading memory type Cα = {φ ∈ C((−∞, 0], R) : φ(θ)eαθ is uniformly continuous for θ ∈ (−∞, 0] and kφk < ∞} where α is a positive constant and kφk = sup |φ(θ)| eαθ . Let θ≤0

Cα+ = {φ ∈ Cα : φ(θ) ≥ 0 f or θ ∈ (−∞, 0]}. The initial conditions for system (4)-(7) are given as: T (θ) = ϕ1 (θ), T ∗ (θ) = ϕ2 (θ), V (θ) = ϕ3 (θ), Z(θ) = ϕ4 (θ), for θ ∈ (−∞, 0], ϕi ∈ Cα+ , i = 1, ..., 4.

(8)

By the fundamental theory of functional differential equations [33], system (4)-(7) with initial conditions (8) has a unique solution.

2.1

Non-negativity and boundedness of solutions

We show the non-negativity and boundedness of the solutions of model (4)-(7).

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Lemma 1. The solutions (T (t), T ∗ (t), V (t), Z(t)) of model (4)-(7) with initial conditions (8) are non-negative and ultimately bounded. Proof : First we prove T (t) > 0 for all t ≥ 0. Assume the contrary and let t1 > 0 such that T (t1 ) = 0. Then from Eq. (4), we have T˙ (t1 ) = λ > 0. Therefore T (t) < 0 for t ∈ (t1 − ε, t1 ) and ε > 0 is sufficiently small. This contradicts with the fact of T (t) > 0 for t ∈ [0, t1 ). It follows that T (t) > 0 for t ≥ 0. From Eqs. (5)-(7), we have Rt

T ∗ (t) = ϕ2 (0)e− 0 (µ1 +pZ(ζ))dζ Z t R Z ∞ − ηt (µ1 +pZ(ζ))dζ + e f1 (s)e−µ1 s [β1 T (η − s)V (η − s) + β2 T (η − s)T ∗ (η − s)] dsdη, 0 0 Z ∞ Z t f2 (s)e−µ2 s T ∗ (ζ − s)dsdζ, e−c(t−ζ) V (t) = ϕ3 (0)e−ct + b −

Z(t) = ϕ4 (0)e

Rt 0

0 (q−kT ∗ (ζ))dζ

0

,

which yield that T ∗ (t) ≥ 0, V (t) ≥ 0 and Z(t) ≥ 0for all t ≥ 0. Next we show the boundedness of the solutions. From Eq. (7) we have lim sup T (t) ≤ t→∞

Z F (t) =

∞

f1 (s)e−µ1 s T (t − s)ds + T ∗ (t) +

0

λ . Let d

p Z(t). k

Then F˙ (t) =

Z

∞

f1 (s)e−µ1 s [λ − dT (t − s) − β1 T (t − s)V (t − s) − β2 T (t − s)T ∗ (t − s)] ds− Z ∞ pq −µ1 s + β1 T (t − s)V (t − s)f1 (s)e ds + β2 T (t − s)T ∗ (t − s)f1 (s)e−µ1 s ds − µ1 T ∗ (t) − Z(t) k 0 0 Z ∞ pq = λη1 − d f1 (s)e−µ1 s T (t − s)ds − µ1 T ∗ (t) − Z(t) k Z 0∞  p ≤λ−σ f1 (s)e−µ1 s T (t − s)ds + T ∗ (t) + Z(t) = λ − σF (t), k 0 Z 0∞

R∞ λ . Since 0 f1 (s)e−µ1 s T (t − s)ds > 0 , T ∗ ≥ 0 and Z ≥ 0, σ λ ∗ then lim supt→∞ T (t) ≤ L1 and lim supt→∞ Z(t) ≤ L2 , where L1 = and L2 = kp L1 . From Eq. (6) we have σ Z ∞ V˙ = b e−µ2 s f2 (s)T ∗ (t − s)ds − cV (t) ≤ bη2 L1 − cV (t) ≤ bL1 − cV (t). where, σ = min{d, µ1 , q}. Hence, lim supt→∞ F (t) ≤

0

Thus lim sup V (t) ≤ L3 , where L3 = t→∞

2.2

bL1 . Therefore, T (t), T ∗ (t), V (t) and Z(t) are ultimately bounded.  c

Steady States

Lemma 1. (i) If R0 ≤ 1, then there exists only positive steady state E0 , (i) if R1 ≤ 1 < R0 , then there exist only two positive steady states E0 and E1 , (ii) if R1 > 1, then there exist three positive steady states E0 , E1 and E2 . The proof. Let the R.H.S of system (4)-(7) be equal zero 0 = λ − dT − β1 T V − β2 T T ∗ , ∗

(9) ∗

∗

0 = η1 (β1 T V + β2 T T ) − µ1 T − pT Z,

(10)

0 = η2 bT ∗ − cV,

(11)

∗

(12)

0 = kT Z − qZ.

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Solving Eqs.

(9)-(12) we find that the system has three steady states, infection-free steady state E0 = λ (T0 , 0, 0, 0, 0), where T0 = , CTL-inactivated infection steady state E1 (T1 , T1∗ , V1 , 0) and CTL-activated infecd tion steady state E2 (T2 , T2∗ , V2 , Z2 ), where dc (R0 − 1), (bβ1 η2 + cβ2 ) kλc q bη2 ∗ T2 = , T2∗ = , V2 = T , kdc + bqβ1 η2 + qβ2 c k c 2 T1 =

T0 , R0

and R0 =

T1∗ =

T0 η1 (bβ1 η2 + β2 c) , µ1 c

R1 =

V1 = Z2 =

bη2 T1∗ , c

µ1 (R1 − 1) , p

kdc R0 , q(bβ1 η2 + β2 c) + kdc

where R0 represents the basic infection reproduction number which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process and R1 represents the immune response activation number which expresses the CTL load during the lifespan of a CTL cell. Clearly R0 > R1 . 

2.3

Global stability analysis

In this section, we study the global stability of all the steady states of system (4)-(7) employing the method of Lyapunov function. We will use the follwing function g(x) = x − 1 − ln x and the notation (T, T ∗ , V, Z) = (T (t), T ∗ (t), V (t), Z(t)). Theorem 1. If R0 ≤ 1, then E0 is GAS. Proof. Define a Lyapunov functional L as follows:   T p 1 β1 T0 ∗ L(T, T , V, Z) = T0 g V + Z + T∗ + T0 η1 c η1 k Z ∞ Z s 1 + f1 (s)e−µ1 s [β1 T (t − θ)V (t − θ) + β2 T (t − θ)T ∗ (t − θ)]dθds η1 0 0 Z Z s bβ1 T0 ∞ f2 (s)e−µ2 s T ∗ (t − θ)dθds. + c 0 0 Calculating the derivative of L along the solutions of the system (4)-(7), we obtain   dL T0 = 1− (λ − dT − β1 T V − β2 T T ∗ ) dt T Z ∞  1 + f1 (s)e−µ1 s [β1 T (t − s)V (t − s) + β2 T (t − s)T ∗ (t − s)] ds − µ1 T ∗ − pT ∗ Z η1 0  Z ∞  p β1 T0 b e−µ2 s f2 (s)T ∗ (t − s)ds − cV + [kT ∗ Z − qZ] + c η1 k 0 Z ∞ 1 + f1 (s)e−µ1 s [β1 T V + β2 T T ∗ − β1 T (t − s)V (t − s) − β2 T (t − s)T ∗ (t − s)] ds η1 0 Z bβ1 T0 ∞ f2 (s)e−µ2 s [T ∗ − T ∗ (t − s)] ds + c 0     T0 bβ1 T0 η2 µ1 pq = 1− (λ − dT ) + β2 T0 + − T∗ − Z T c η1 η1 k (T − T0 )2 µ1 pq = −d + (R0 − 1)T ∗ − Z. T η1 η1 k

(13)

dL ≤ 0 for all T, T ∗ , Z > 0. Thus the solutions of system (4)-(7) limit to M , the largest dt  dL invariant subset of (T, T ∗ , V, Z) : dL dt = 0 . Clearly, it follows from Eq. (13) that dt = 0 if and only if T = T0 , If R0 ≤ 1, then

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T ∗ = 0 and Z = 0. Noting that M is invariant, for each element of M we have T ∗ = 0 and Z = 0, then T˙ ∗ = 0. From Eq. (5) we drive that Z ∞ f1 (s)e−µ1 s β1 T0 V (t − s)ds. 0 = T˙ ∗ = 0

It follows that V = 0. Hence = 0 if and only if T = T0 , T ∗ = 0, V = 0 and Z = 0. LaSalle’s invariance principle implies that E0 is GAS when R0 ≤ 1.  Theorem 2. If R1 ≤ 1 < R0 , then E1 is GAS. Proof. Define the following Lyapunov functional      ∗ T β1 T1 V1 V 1 T p U (T, T ∗ , V, Z) = T1 g + V g + T1∗ g + Z 1 T1 η1 T1∗ bη2 T1∗ V1 η1 k  Z Z s  T (t − θ)V (t − θ) β1 T1 V1 ∞ f1 (s)e−µ1 s g dθds + η1 T1 V1 0 0  Z s  Z T (t − θ)T ∗ (t − θ) β2 T1 T1∗ ∞ g f1 (s)e−µ1 s dθds + η1 T1 T1∗ 0 0  Z s  ∗ Z T (t − θ) β1 T1 V1 ∞ g f2 (s)e−µ2 s dθds. + η2 T1∗ 0 0 dL dt

The time derivative of U along the trajectories of (4)-(7) is given by   dU T1 = 1− (λ − dT − β1 T V − β2 T T ∗ ) dt T   Z ∞  T∗ 1 1 − 1∗ + f1 (s)e−µ1 s [β1 T (t − s)V (t − s) + β2 T (t − s)T ∗ (t − s)] ds − µ1 T ∗ − pT ∗ Z η1 T 0    Z ∞  V1 p β1 T1 V1 −µ2 s ∗ (kT ∗ Z − qZ) 1 − b f (s)e T (t − s)ds − cV + + 2 ∗ bη2 T1 V η1 k 0    Z β1 T1 V1 ∞ TV T (t − s)V (t − s) T (t − s)V (t − s) + ds f1 (s)e−µ1 s − + ln η1 T1 V1 T1 V1 TV 0    Z β2 T1 T1∗ ∞ TT∗ T (t − s)T ∗ (t − s) T (t − s)T ∗ (t − s) + f1 (s)e−µ1 s − + ln ds η1 T1 T1∗ T1 T1∗ TT∗ 0  ∗  ∗  Z β1 T1 V1 ∞ T ∗ (t − s) T (t − s) T + − + ln f2 (s)e−µ2 s ds. (14) η2 T1∗ T1∗ T∗ 0 Collecting terms of Eq. (14) and applying the steady state condtions for E1 : λ − dT1 = β1 T1 V1 + β2 T1 T1∗ =

cµ1 µ1 ∗ T1 = V1 , η1 bη1 η2

we get   dU d T1 2 = − (T − T1 ) + (β1 T1 V1 + β2 T1 T1∗ ) 1 − dt T T Z ∞ Z ∗ β1 T1 V1 T (t − s)V (t − s)T1 β2 T1 T1∗ ∞ T (t − s)T ∗ (t − s) ds − ds − f1 (s)e−µ1 s f1 (s)e−µ1 s ∗ η1 T1 V1 T η1 T1 T ∗ 0 0   Z Z β1 T1 V1 ∞ V1 T ∗ (t − s) β1 T1 V1 ∞ T (t − s)V (t − s) −µ1 s − f2 (s)e−µ2 s ds + f (s)e ln ds 1 η2 V T1∗ η1 TV 0 0   Z β2 T1 T1∗ ∞ T (t − s)T ∗ (t − s) + f1 (s)e−µ1 s ln ds η1 TT∗ 0  ∗  Z β1 T1 V1 ∞ T (t − s) p  ∗ q + f2 (s)e−µ2 s ln ds + T1 − Z + 2β1 T1 V1 + β2 T1 T1∗ . η2 T∗ η1 k 0

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Consider the following equalities         Ti T (t − s)V (t − s) T (t − s)V (t − s)Ti∗ Vi T ∗ + ln , = ln + ln ln TV Ti Vi T ∗ T V Ti∗       T (t − s)T ∗ (t − s) Ti T (t − s)T ∗ (t − s) = ln + ln ln , TT∗ Ti T ∗ T  ∗      T (t − s) Vi T ∗ (t − s) V Ti∗ ln = ln + ln , i = 1, 2. T∗ V Ti∗ Vi T ∗

(15)

Using Eq. (15) with i = 1 we get    dU d T1 T1 2 = − (T − T1 ) − (β1 T1 V1 + β2 T1 T1∗ ) − 1 − ln dt T T T    Z ∞ ∗ T (t − s)V (t − s)T1 T (t − s)V (t − s)T1∗ β1 T1 V1 f1 (s)e−µ1 s − 1 − ln ds − η1 T1 V1 T ∗ T1 V1 T ∗ 0    Z β2 T1 T1∗ ∞ T (t − s)T ∗ (t − s) T (t − s)T ∗ (t − s) − f1 (s)e−µ1 s − 1 − ln ds η1 T1 T ∗ T1 T ∗ 0    Z β1 T1 V1 ∞ V1 T ∗ (t − s) V1 T ∗ (t − s) p  ∗ q − f2 (s)e−µ2 s Z − 1 − ln ds + T1 − η2 V T1∗ V T1∗ η1 k 0     Z d T (t − s)V (t − s)T1∗ T1 β1 T1 V1 ∞ 2 = − (T − T1 ) − (β1 T1 V1 + β2 T1 T1∗ ) g f1 (s)e−µ1 s g ds − T T η1 T1 V1 T ∗ 0     Z Z β2 T1 T1∗ ∞ T (t − s)T ∗ (t − s) β1 T1 V1 ∞ V1 T ∗ (t − s) −µ2 s − f1 (s)e−µ1 s g ds − f (s)e g ds 2 η1 T1 T ∗ η2 V T1∗ 0 0 p β1 bqη2 + β2 qc + dck (R1 − 1) Z. + η1 (β1 bqη2 + β2 qc)k Hence, if R1 ≤ 1, then we obtain that dU ≤ 0 and then solutions of system (4)-(7) limit to M , the largest dt  dU ∗ invariant subset of (T, T , V, Z) : dt = 0 . It can be seen that, dU dt = 0 if and only if T (t − s)V (t − s)T1∗ T (t − s)T ∗ (t − s) V1 T ∗ (t − s) T1 = = = = 1. T T1 V1 T ∗ T1 T ∗ V T1∗ LaSalle’s invariance principle implies the global stability of E1 .  Theorem 3. If R1 > 1, then E2 is GAS. Define the following Lyapunov functional    ∗     T T Z 1 ∗ β1 T2 V2 V p ∗ W (T, T , V, Z) = T2 g Z2 g + T2 g + V2 g + ∗ ∗ T2 η1 T2 bη2 T2 V2 η1 k Z2  Z Z s  β1 T2 V2 ∞ T (t − θ)V (t − θ) + f1 (s)e−µ1 s g dθds η1 T2 V2 0 0   Z Z s β2 T2 T2∗ ∞ T (t − θ)T ∗ (t − θ) + f1 (s)e−µ1 s dθds g η1 T2 T2∗ 0 0  Z ∞ Z s  ∗ β1 T2 V2 T (t − θ) + f2 (s)e−µ2 s g dθds. η2 T2∗ 0 0

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The time derivative of W along the trajectories of (4)-(7) is given by   dW T2 = 1− (λ − dT − β1 T V − β2 T T ∗ ) dt T   Z ∞  1 T∗ f1 (s)e−µ1 s (β1 T (t − s)V (t − s) + β2 T (t − s)T ∗ (t − s)) ds − µ1 T ∗ − pT ∗ Z + 1 − 2∗ η1 T 0    Z ∞    β1 T2 V2 V2 p Z2 −µ2 s ∗ + 1 − b f (s)e T (t − s)ds − cV + 1 − (kT ∗ Z − qZ) 2 bη2 T2∗ V η k Z 1 0    Z TV T (t − s)V (t − s) T (t − s)V (t − s) β1 T2 V2 ∞ −µ1 s f1 (s)e − + ln ds + η1 T2 V2 T2 V2 TV 0    Z TT∗ T (t − s)T ∗ (t − s) T (t − s)T ∗ (t − s) β2 T2 T2∗ ∞ f1 (s)e−µ1 s − + ln ds + η1 T2 T2∗ T2 T2∗ TT∗ 0  ∗  ∗  Z β1 T2 V2 ∞ T T ∗ (t − s) T (t − s) + f2 (s)e−µ2 s − + ln ds. η2 T2∗ T2∗ T∗ 0 Using the following steady state conditions for E2 λ − dT2 = β1 T2 V2 + β2 T2 T2∗ =

p ∗ µ1 ∗ T2 Z2 + T , η1 η1 2

T2∗ =

q , k

V2 =

bqη2 , ck

we get   dW d T2 2 = − (T − T2 ) + (β1 T2 V2 + β2 T2 T2∗ ) 1 − dt T T Z Z ∗ ∗ β2 T2 T2∗ ∞ T (t − s)V (t − s)T β1 T2 V2 ∞ 2 −µ1 s T (t − s)T (t − s) ds − ds f1 (s)e−µ1 s f (s)e − 1 η1 T2 V2 T ∗ η1 T2 T ∗ 0 0   Z Z β1 T2 V2 ∞ β1 T2 V2 ∞ T (t − s)V (t − s) V2 T ∗ (t − s) − f2 (s)e−µ2 s ds + f1 (s)e−µ1 s ln ds η2 V T2 η1 TV 0 0   Z β2 T2 T2∗ ∞ T (t − s)T ∗ (t − s) + f1 (s)e−µ1 s ln ds η1 TT∗ 0   ∗ Z ∞ T (t − s) β1 T2 V2 ds + 2β1 T2 V2 + β2 T2 T2∗ . f2 (s)e−µ2 s ln + η2 T∗ 0 Using Eq. (15) with i = 2 we get     Z dW d T2 β1 T2 V2 ∞ T (t − s)V (t − s)T2∗ 2 ∗ −µ1 s = − (T − T2 ) − (β1 T2 V2 + β2 T2 T2 ) g − ds f1 (s)e g dt T T η1 T2 V2 T ∗ 0     Z Z β2 T2 T2∗ ∞ T (t − s)T ∗ (t − s) V2 T ∗ (t − s) β1 T2 V2 ∞ −µ2 s − f1 (s)e−µ1 s g ds − f (s)e g ds. 2 η1 T2 T ∗ η2 V T2∗ 0 0 Noting that T, T ∗ , V, Z > 0, we have that dW ≤ 0. The solutions of model (4)-(7) converge to M , the largest  dt dW ∗ invariant subset of (T, T , V, Z) : dt = 0 . We have dW dt = 0 if and only if T = T2 and g = 0 i.e., T (t − s)V (t − s)T2∗ T (t − s)T ∗ (t − s) V2 T ∗ (t − s) T2 = = 1. = = T T2 V2 T ∗ T2 T ∗ V T2∗

(16)

If T = T2 , then from Eq. (16) we get T ∗ = T2∗ and V = V2 . The set M is invariant and for any element belongs to M satisfies T ∗ = T2∗ and T˙ ∗ = 0 = η1 (β1 T2 V2 + β2 T2 T2∗ ) − µ1 T2∗ − pT2∗ Z, ∗ ∗ which gives Z = Z2 . Therefore, dW dt = 0 if and only if T = T2 , T = T2 , V = V2 and Z = Z2 . The global asymptotic stability of E2 follows from LaSalle’s invariance principle. 

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3

Numerical simulations

In this section, we perform numerical simulations for the model (4)-(7) with particular distribution functions f1 (s) and f2 (s) as: f1 (s) = δ(s − s1 ), f2 (s) = δ(s − s2 ), where δ(.) is the dirac delta function, s1 and s2 are positive constants. Then, we can see that, Z∞

∞

Z

δ(s − si )e−µi s ds = e−µi si , i = 1, 2,

fi (s)ds = 1, ηi = 0

0

Z

∞

δ(s − s1 )e−µs φ(t − s)ds = e−µs1 φ(t − s1 ),

0

for any function φ. With such choice, model (4)-(7) leads to: T˙ (t) = λ − dT (t) − β1 T (t)V (t) − β2 T (t)T ∗ (t), T˙ ∗ (t) = [β1 T (t − s1 )V (t − s1 ) + β2 T (t − s1 )T ∗ (t − s1 )] e−µ1 s1 − µ1 T ∗ (t) − pT ∗ (t)Z(t), V˙ (t) = be−µ2 s2 T ∗ (t − s2 ) − cV (t), ˙ Z(t) = kT ∗ (t)Z(t) − qZ(t).

(17)

(18) (19)

e−µ1 s1 λ(bβ1 e−µ2 s2 + β2 c) kdc , R1 = R0 . cµ1 d q(bβe−µ2 s2 + β2 c) + kdc Now we perform some numerical simulations for model (17)-(19) with parameters values given in Table 1.

The parameters R0 and R1 become R0 =

Table 1: The values of the parameters of model (17 )-(19).

3.1

Parameter

Value

Parameter

Value

λ

10

c

3

d

0.01

q

0.1

p

0.1

β2

0.0001

b

10

µ1

0.9

s1

Varied

µ2

0.1

s2

Varied

β1 , k

Varied

Effect of the parameters β1 and k on the stability of steady states

To show the global stablity of the steady states we consider three different initial conditions: IC1: ϕ1 (θ) = 600, ϕ2 (θ) = 1, ϕ3 (θ) = 1, ϕ4 (θ) = 10, IC2: ϕ1 (θ) = 200, ϕ2 (θ) = 0.5, ϕ3 (θ) = 3, ϕ4 (θ) = 5, IC3: ϕ1 (θ) = 700, ϕ2 (θ) = 5, ϕ3 (θ) = 9, ϕ4 (θ) = 12, where, θ ∈ [− max{s1 , s2 }, 0]. In this case we choose s1 = 0.5, s2 = 0.9 and study the following subcases: (i): R0 < 1. We choose, β1 = 0.0001 and k = 0.008, then we compute R0 = 0.295489 and R1 = 0.194228. From Lemma 2 we have that the system has one steady state E0 . From Figures 1-4 we can see that, the concentration of uninfected cells is increasing and tends its normal value λ/d = 1000, while the concentrations of infected cells, free viruses and CTls are decaying and approaching zero. It means that, E0 is GAS and the virus will be removed. This result support the result of Theorem 1.

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(ii): R1 ≤ 1 < R0 . We choose β1 = 0.001 and k = 0.008, and then, R0 = 2.317257 and R1 = 0.455395. Lemma 2 state that the system has two steady states E0 and E1 . Figures 5-8 show that the numerical results are consistent with Theorem 2. We can see that, the solution of the system converges to the steady E1 (431.54, 4.03, 12.77, 0) for the initial conditions IC1-IC3. (iii): R1 > 1: In this case, we choose β1 = 0.001 and k = 0.03 and then R1 = 1.108600 > 1. According to Lemma 2, the system has three steady states E0 , E1 and E2 . From Figures 9-12 we can see that, the solutions of the system approach the steady state E2 (478.41, 3.33, 10.57, 0.98) for large t and for the initial conditions IC1-IC3. This support the result of Theorems 3.

3.2

Effect of the time delays on the stability of steady states

In this case, we consider the initial condition IC2. We take the values β1 = 0.001 and k = 0.03. Without loss of generality we let S = s1 = s2 . In Table 2, we present the values of R0 , R1 and the steady states of system (17 )-(19) with different values of S. From Table 2 we can see that, the values of R0 and R1 are decreased as S is increased. Using the values of the parameters given in Table 1, we obtain that the following: (i) if 0 ≤ S < 0.8447, then E2 exists and it is GAS, (ii) if 0.8447 ≤ S < 0.8868, then E1 exists and it is GAS, (iii) if S ≥ 0.8868, then E0 is GAS. Figures 13-16 show that the numerical results are also compatible with the results of Theorems 1-3. From a biological point of view, the intracellular delay plays a similar role as an antiviral treatment in eliminating the virus. We observe that, sufficiently large delay suppresses viral replication and clears the virus. This gives us some suggestions on new drugs to prolong the increase the intracellular delay period. Table 2: The values of steady states, R0 and R1 for model (17)-(19) with different values of the delay parameter S. Delay parameter

Steady states

R0

R1

S = 0.0

E2 (578.98, 1.11, 1.23, 27.89)

6.54

3.79

S = 0.2

E2 (626.03, 1.11, 1.01, 17.56)

4.40

2.76

S = 0.4

E2 (670.65, 1.11, 0.83, 9.87)

2.96

1.99

S = 0.7

E2 (731.69, 1.11, 0.61, 1.99)

1.64

1.2

S = 0.80

E2 (751.30, 1.11, 0.55, 0)

1.33

1

S = 0.9

E1 (904.23, 0.39, 0.18, 0)

1.11

0.85

S = 0.95

E0 (1000, 0, 0, 0)

1

0.78

S=1

E0 (1000, 0, 0, 0)

0.91

0.71

S = 1.5

E0 (1000, 0, 0, 0)

0.34

0.29

S=2

E0 (1000, 0, 0, 0)

0.13

0.12

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6

1000 IC1 IC2 IC3

900 800 700

4 Infected cells

Uninfected cells

IC1 IC2 IC3

5

600 500 400

3

2 300 200

1

100 0 0

200

400

600 Time

800

1000

0 0

1200

Figure 1: The evolution of uninfected cells with initial IC1-IC3 in case of R0 ≤ 1.

1

2

3

4

5 Time

6

7

8

9

10

Figure 2: The evolution of infected cells with initial IC1-IC3 in case of R0 ≤ 1.

35 IC1 IC2 IC3

15 IC1 IC2 IC3

30

25 10 CTLs

Free Virus

20

15 5

10

5

0 0

1

2

3

4

5 Time

6

7

8

9

0 0

10

10

20

30

40

50

60

70

80

90

Time

Figure 3: The evolution of free viruses with initial IC1-IC3 in case of R0 ≤ 1.

Figure 4: The evolution of CTLs with initial IC1IC3 in case of R0 ≤ 1.

800

30 IC1 IC2 IC3

700

IC1 IC2 IC3

25

600 Infected cells

Uninfected cells

20 500 400 300

15

10 200 5 100 0 0

100

200

300

400 Time

500

600

700

0 0

800

Figure 5: The evolution of uninfected cells with initial IC1-IC3 in case of R1 ≤ 1 < R0 .

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100

200

300

400 Time

500

600

700

800

Figure 6: The evolution of infected cells with initial IC1-IC3 in case of R1 ≤ 1 < R0 .

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90 IC1 IC2 IC3

80

30 IC1 IC2 IC3

25

70 60

CTLs

Free Virus

20 50 40 30

15

10

20 5 10 0 0

100

200

300

400 Time

500

600

700

0 0

800

Figure 7: The evolution of free viruses with initial IC1-IC3 in case of R1 ≤ 1 < R0 .

10

20

30

40

50 Time

60

70

80

90

100

Figure 8: The evolution of CTLs with initial IC1IC3 in case of R1 ≤ 1 < R0 .

25 IC1 IC2 IC3

800 IC1 IC2 IC3

700

20

Infected cells

Uninfected cells

600 500 400

15

10

300 200 5 100 0 0

200

400

600 Time

800

1000

0 0

1200

Figure 9: The evolution of uninfected cells with initial IC1-IC3 in case of R1 > 1.

200

400

600 Time

800

1000

1200

Figure 10: The evolution of infected cells with initial IC1-IC3 in case of R1 > 1.

15

80

IC1 IC2 IC3

IC1 IC2 IC3

70 60

10 CTLs

Free Virus

50 40 30 5 20 10 0 0

200

400

600 Time

800

1000

0 0

1200

Figure 11: The evolution of free viruses with initial IC1-IC3 in case of R1 > 1.

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100

200

300

400

500 Time

600

700

800

900

1000

Figure 12: The evolution of CTLs with initial IC1IC3 in case of R1 > 1.

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35

1000 S=0 S=0.4 S=0.8 S=1.0 S=1.4

900 800

25 Infected Cells

700 Uninfected cells

S=0.0 S=0.4 S=0.8 S=1.0 S=1.4

30

600 500 400 300

20 15 10

200 5 100 0 0

200

400

600 Time

800

1000

0 0

1200

Figure 13: The evolution of uninfected cells with different delay parameter S.

200

400

600 Time

800

1000

Figure 14: The evolution of infected cells with different delay parameter S.

100

16 S=0.0 S=0.4 S=0.8 S=1.0

90 80

S=0.0 S=0.4 S=0.8 S=1.0 S=1.4

14 12

70 10

60 CTLs

Free Virus

1200

50 40

8 6

30 4 20 2

10 0 0

200

400

600 Time

800

1000

0 0

1200

Figure 15: The evolution of free viruses with different delay parameter S.

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800

1000

1200

Figure 16: The evolution of CTLs with different delay parameter S.

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4

Acknowledgment

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

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Dynamical behavior of HIV-1 infection with saturated virus-target and infected-target incidences and delays A. M. Elaiw, A. A. Raezah and A. S. Alofi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Emails: a m [email protected] (A. Elaiw) Abstract This paper study the dynamical behavior of HIV-1 infection model with saturated virus-target and infected-target incidences. The model is incorporated by two types of intracellular distributed time delays. The model generalizes all the existing HIV-1 infection models with cell-to-cell transmission presented in the literature by considering saturated incidence rate. The nonnegativity and boundedness of the solutions of the model as well as global stability of the steady states are studied. The global stability are established using Lyapunov method. Using MATLAB we conduct some numerical simulations to confirm our results. The effect of the saturated incidence of the HIV-1 dynamics is shown. Keywords: HIV-1 dynamics; Global stability; time delay; cell-to-cell transfer.

1

Introduction

It is known that human immunodeficiency virus type 1 (HIV-1) infects the CD4+ T cells which play the central role in the immune system of the human body. Mathematical models that describe the dynamics of HIV-1 are helpful in understanding the virus dynamics and improving diagnosis and treatment strategies. The basic HIV-1 infection model has been given in [1] as: T˙ = ρ − dT − βT V T˙ ∗ = βT V − µT ∗ , V˙ = bT ∗ − cV,

(1) (2) (3)

where, T , T ∗ and V are the concentrations of the uninfected CD4+ T cells, infected cells, and free HIV-1 particles, respectively. The CD4+ T cells are replenished at rate ρ, die at rate dT and become infected at rate βT V , where β is the virus-target incidence rate constant. The infected cells are die at rate µ. The HIV-1 particles are produced from infected cells at rate bT ∗ and cleared at rate cV . Parameters ρ, d, β, µ, b and c are all positive. In model (1)-(3), the infection rate is given by bilinear incidence βT V . In case when the concentration of the viruses is high, this bilinear incidence may not describe the HIV-1 dynamics accurately. Therefore, the model has been modified to incorporate the saturated incidence rate [2]:   V T˙ = ρ − dT − βT (4) 1 + αV   V T˙ ∗ = βT − µT ∗ , (5) 1 + αV V˙ = bT ∗ − cV, (6)

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where, α is the saturation constant. Moreover, several works have been done to modify the basic model (1)-(3) by considering different effects such as: CTL immune response [3]-[5], humoral immune response [6]-[8], nonlinear incidence rate [9]-[11], intracellular time delay [10], [12], [13], [15], antiviral treatments [15]-[17], latently infected cells [18]-[19] and two types of target cells [20]-[22]. All the these works assume that the uninfected CD4+ T cells becomes infected due to HIV-1 contacts. Recently, it has been reported that the uninfected CD4+ T cells can also become infected due to direct contact with infected cells (see [23]-[26]). However, in [23]-[26], the rates of virus-target and infected-target infection are based on the mass action principle. The aim of this paper is to study the dynamical behavior of HIV-1 infection model with saturated virustarget and infected-target incidences. Both discrete and distributed time delays are incorporated. We study the global stability analysis of the model using Lyapunov method.

2

HIV-1 model with discrete delays

We formulate an HIV-1 infection model with saturated virus-target and infected-target incidences and two types of discrete time delays as: β1 T (t)V (t) β2 T (t)T ∗ (t) − , T˙ (t) = ρ − dT (t) − 1 + α1 V (t) 1 + α2 T ∗ (t)   β1 T (t − τ1 )V (t − τ1 ) β2 T (t − τ1 )T ∗ (t − τ1 ) T˙ ∗ (t) = e−δ1 τ1 + − µT ∗ (t), 1 + α1 V (t − τ1 ) 1 + α2 T ∗ (t − τ1 ) V˙ (t) = be−δ2 τ2 T ∗ (t − τ2 ) − cV (t).

(7) (8) (9)

Parameter τ1 represents for the time between the virus or the infected cell contacts with an uninfected CD4+ T cell, until it becomes infected but not yet producer cell. The parameter τ2 represents the time needed for new HIV-1 to be mature. The factor e−δ1 τ1 is the loss of CD4+ T cells during the interval [t − τ1 , t] while, e−δ2 τ2 represents the loss of infected cells during the interval [t − τ2 , t], where δ1 and δ2 are positive constants. The initial conditions for system (7)-(9) are given as: T (η) = ϕ1 (η), T ∗ (η) = ϕ2 (η), V (η) = ϕ3 (η), ϕj (η) ≥ 0, η ∈ [−τ, 0],

j = 1, 2, 3

(10)

where τ = max{τ1 , τ2 } and (ϕ1 (η), ϕ2 (η), ϕ3 (η)) ∈ C([−τ : 0), R3+ ), where C is the Banach space of continuous functions mapping the interval [−τ, 0) into R3+ . System (7)-(9) with initial conditions (10) has a unique solution [27].

2.1

Basic properties

The non-negativity and boundedness of the solutions of system (7)-(9) is established in the following lemma: Lemma 1. All solutions (T (t), T ∗ (t), V (t)) of model (7)-(9) with initial conditions (10) are non-negative and ultimately bounded. Proof : From Eq. (7), we have T˙ |T =0 = ρ > 0, therefore T (t) > 0 for t ∈ (0, $1 ) where (0, $1 ) is the maximal interval of existence of solution of system (7)-(9) with (10). Moreover, from Eqs. (8)-(9), we have   Z t β2 T (η − τ1 )T ∗ (η − τ1 ) ∗ −µt −δ1 τ1 −µ(t−η) β1 T (η − τ1 )V (η − τ1 ) dη ≥ 0, T (t) = e ϕ2 (0) + e e + 1 + α1 V (η − τ1 ) 1 + α2 T ∗ (η − τ1 ) 0 Z t V (t) = e−ct ϕ3 (0) + be−δ2 τ2 e−c(t−η) T ∗ (η − τ2 )dη ≥ 0, 0

for t ∈ [0, τ ]. By recursive argument we obtain T ∗ (t), V (t) ≥ 0 for all t ≥ 0.

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From Eq. (7) we know lim sup T (t) ≤ t→∞

ρ . Let F1 (t) = e−δ1 τ1 T (t − τ1 ) + T ∗ (t). Then d

  β1 T (t − τ1 )V (t − τ1 ) β2 T (t − τ1 )T ∗ (t − τ1 ) −δ1 τ1 ˙ ρ − dT (t − τ1 ) − F1 (t) = e − 1 + α1 V (t − τ1 ) 1 + α2 T ∗ (t − τ1 )   ∗ β2 T (t − τ1 )T (t − τ1 ) −δ1 τ1 β1 T (t − τ1 )V (t − τ1 ) +e + − µT ∗ (t) 1 + α1 V (t − τ1 ) 1 + α2 T ∗ (t − τ1 ) = ρe−δ1 τ1 − de−δ1 τ1 T (t − τ1 ) − µT ∗ (t)
≤ ρ − σ e−δ1 τ1 T (t − τ1 ) + T ∗ (t) = ρ − σF1 (t), where, σ = min{d, µ}. Hence, lim supt→∞ F1 (t) ≤

ρ ρ and then lim supt→∞ T ∗ (t) ≤ . From Eq. (9) we have σ σ

ρ ρ V˙ (t) = be−δ2 τ2 T ∗ (t − τ2 ) − cV (t) ≤ be−δ2 τ2 − cV (t) < b − cV (t). σ σ bρ . Therefore, T (t), T ∗ (t) and V (t) are all ultimately bounded. cσ t→∞ Now we prove the existence of the steady state of the model (7)-(9). Lemma 2. (i) If R0 ≤ 1, then there exists only positive steady state S0 , (ii) If 1 < R0 , then there exist two positive steady states S0 and S1 . The proof. Let the R.H.S of system (7)-(9) equal to zero

Thus lim sup V (t) ≤

β1 T V β2 T T ∗ − , 0 = ρ − dT − 1 + α1 V 1 + α2 T ∗   β2 T T ∗ β1 T V + 0 = e−δ1 τ1 − µT ∗ , 1 + α1 V 1 + α2 T ∗

(11) (12)

0 = e−δ2 τ2 bT ∗ − cV,

(13)

Solving Eqs. (11)-(13) we find that the system has two steady states, disease-free steady state S0 = (T0 , 0, 0), ρ where T0 = and endemic steady state S1 (T1 , T1∗ , V1 ), where d
√ µc (1 + α1 V1 ) be−δ2 τ2 + α2 cV1 −B + B 2 − 4AC ∗ T1 = −(δ τ +δ τ ) , T1 = , 2A be 1 1 2 2 [β1 (be−δ2 τ2 + α2 cV1 ) + β2 c (1 + α1 V1 )] be−δ2 τ2 T1∗ V1 = , c where A = µbe−δ2 τ2 (dα1 α2 + β1 α2 + β2 α1 ) , B = β2 (µc − ρα1 be−(δ1 τ1 +δ2 τ2 ) ) + β1 be−δ2 τ2 (µ − ρα2 e−δ1 τ1 ) + dµ(cα2 + α1 be−δ2 τ2 ),

(14)

C = dµc (1 − R0 ) . and


T0 e−δ1 τ1 bβ1 e−δ2 τ2 + β2 c , R0 = µc

where R0 represents the basic infection reproduction number. 

2.2

Global properties

In the following we established the global stability of the two steady states by of system (7)-(9) by constructing suitable Lyapunov functionals. Through the paper we will use the following function g(x) = x − 1 − ln x and the notation (T, T ∗ , V ) = (T (t), T ∗ (t), V (t)).

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Theorem 1. If R0 ≤ 1, then S0 is globally asymptotically stable. Proof. Define a Lyapunov functional    Z τ1  β1 T0 β1 T (t − η)V (t − η) β2 T (t − η)T ∗ (t − η) T 1 ∗ ∗ L1 (T, T , V ) = T0 g + −δ1 τ1 T + V + + dη T0 e c 1 + α1 V (t − η) 1 + α2 T ∗ (t − η) 0 Z τ2 bβ1 T0 −δ2 τ2 + e T ∗ (t − η)dη. c 0 We evaluate

dL1 along the solutions of the system (7)-(9), dt    dL1 T0 β1 T V β2 T T ∗ ρ − dT − = 1− − dt T 1 + α1 V 1 + α2 T ∗     1 β1 T (t − τ1 )V (t − τ1 ) β2 T (t − τ1 )T ∗ (t − τ1 ) ∗ + −δ1 τ1 e−δ1 τ1 + − µT e 1 + α1 V (t − τ1 ) 1 + α2 T ∗ (t − τ1 )  β2 T T ∗ β1 T0  −δ2 τ2 ∗ β1 T V be T (t − τ2 ) − cV + + + c 1 + α1 V 1 + α2 T ∗ ∗ β1 T (t − τ1 )V (t − τ1 ) β2 T (t − τ )T (t − τ1 ) bβ1 T0 −δ2 τ2 ∗ − [T − T ∗ (t − τ2 )] − + e 1 + α1 V (t − τ1 ) 1 + α2 T ∗ (t − τ1 ) c   T0 V2 T ∗2 = 1− (ρ − dT ) − α1 β1 T0 − α2 β2 T0 T 1 + α1 V 1 + α2 T ∗   −(δ1 τ1 +δ2 τ2 ) −δ1 τ1 µ T0 bβ1 e T0 β2 e + −δ1 τ1 + − 1 T∗ e µc µ (T − T0 )2 V2 T ∗2 µ = −d − α1 β1 T0 − α2 β2 T0 + −δ1 τ1 (R0 − 1)T ∗ . ∗ T 1 + α1 V 1 + α2 T e

(15)

dL1 dL1 ≤ 0 for all T, T ∗ , V > 0 and = 0 if and only if T = T0 , T ∗ = 0 and V = 0. If R0 ≤ 1, then dt dt  1 Let D0 = (T, T ∗ , V ) : dL dt = 0 . It is easy to show that S0 is the largest invariant subset of D0 . LaSalle’s invariance principle implies that S0 is globally asymptotically stable when R0 ≤ 1. Theorem 2. If 1 < R0 , then S1 is globally asymptotically stable. Proof. Define    ∗   T T β1 T1 V1 V 1 ∗ ∗ U (T, T , V ) = T1 g + −δ2 τ2 ∗ V1 g + −δ1 τ1 T1 g T1 e T1∗ be T1 (1 + α1 V1 ) V1  Z τ1  β1 T1 V1 T (t − η)V (t − η) (1 + α1 V1 ) + dη g 1 + α1 V1 0 T1 V1 (1 + α1 V (t − η))    Z τ2  ∗ Z τ1 T (t − η)T ∗ (t − η) (1 + α2 T1∗ ) T (t − η) β1 T1 V1 β2 T1 T1∗ + g g dη. dη + 1 + α2 T1∗ 0 T1 T1∗ (1 + α2 T ∗ (t − η)) 1 + α1 V1 0 T1∗

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Evaluating

dU1 dt

along the trajectories of (7)-(9) as:    dU1 T1 β1 T V β2 T T ∗ = 1− ρ − dT − − dt T 1 + α1 V 1 + α2 T ∗       ∗ 1 T1 β2 T (t − τ1 )T ∗ (t − τ1 ) −δ1 τ1 β1 T (t − τ1 )V (t − τ1 ) ∗ + −δ1 τ1 1 − ∗ e + − µT e T 1 + α1 V (t − τ1 ) 1 + α2 T ∗ (t − τ1 )  
V1 β1 T1 V1 1− be−δ2 τ2 T ∗ (t − τ2 ) − cV + −δ2 τ2 ∗ T1 (1 + α1 V1 ) be V   T V (1 + α1 V1 ) T (t − τ1 )V (t − τ1 ) (1 + α1 V1 ) β1 T1 V1 − + 1 + α1 V1 T1 V1 (1 + α1 V ) T1 V1 (1 + α1 V (t − τ1 ))   β1 T1 V1 T (t − τ1 )V (t − τ1 ) (1 + α1 V ) + ln 1 + α1 V1 T V (1 + α1 V (t − τ1 ))   ∗ ∗ T T (1 + α2 T1∗ ) β2 T1 T1 T (t − τ1 )T ∗ (t − τ1 ) (1 + α2 T1∗ ) + − 1 + α2 T1∗ T1 T1∗ (1 + α2 T ∗ ) T1 T1∗ (1 + α2 T ∗ (t − τ1 ))   β2 T1 T1∗ T (t − τ1 )T ∗ (t − τ1 ) (1 + α2 T ∗ ) + ln 1 + α2 T1∗ T T ∗ (1 + α2 T ∗ (t − τ1 ))  ∗   ∗ T ∗ (t − τ2 ) T (t − τ2 ) T β1 T1 V1 − + ln . + 1 + α1 V1 T1∗ T1∗ T∗

(16)

Collecting terms of Eq. (16) and applying the steady state conditions for S1 : ρ − dT1 =

β1 T1 V1 µ β2 T1 T1∗ cµ = −δ1 τ1 T1∗ = −(δ τ +δ τ ) V1 , + 1 + α1 V1 1 + α2 T1∗ e be 1 1 2 2

we get    dU1 d T1 β1 T1 V1 β2 T1 T1∗ β2 T1 T ∗ β1 T1 V 2 = − (T − T1 ) + 1 − + + + dt T T 1 + α1 V1 1 + α2 T1∗ 1 + α1 V 1 + α2 T ∗ ∗ ∗ ∗ ∗ β2 T1 T β1 T1 V1 β1 T1 T (t − τ1 )V (t − τ1 ) β2 T1 T (t − τ1 )T (t − τ1 ) − − + − T ∗ (1 + α1 V (t − τ1 )) T ∗ (1 + α2 T ∗ (t − τ1 )) 1 + α2 T1∗ 1 + α1 V1 β2 T1 T1∗ β1 T1 V1 V1 T ∗ (t − τ2 ) β1 T1 V β1 T1 V1 + − − + ∗ 1 + α2 T1 1 + α1 V1 V T1∗ 1 + α1 V1 1 + α1 V1     β2 T1 T1∗ β1 T1 V1 T (t − τ1 )V (t − τ1 ) (1 + α1 V ) T (t − τ1 )T ∗ (t − τ1 ) (1 + α2 T ∗ ) + + ln ln 1 + α1 V1 T V (1 + α1 V (t − τ1 )) 1 + α2 T1∗ T T ∗ (1 + α2 T ∗ (t − τ1 ))  ∗  β1 T1 V1 T (t − τ2 ) + ln . 1 + α1 V1 T∗ Consider the following equalities:       T (t − τ1 )V (t − τ1 ) (1 + α1 V ) T (t − τ1 )V (t − τ1 ) (1 + α1 V1 ) T1∗ T1 ln = ln + ln T V (1 + α1 V (t − τ1 )) T1 V1 (1 + α1 V (t − τ1 )) T ∗ T     ∗ 1 + α1 V V1 T + ln , + ln 1 + α1 V1 V T1∗       T (t − τ1 )T ∗ (t − τ1 ) (1 + α2 T ∗ ) T (t − τ1 )T ∗ (t − τ1 ) (1 + α2 T1∗ ) T1 ln = ln + ln T T ∗ (1 + α2 T ∗ (t − τ1 )) T1 T ∗ (1 + α2 T ∗ (t − τ1 )) T   1 + α2 T ∗ , + ln 1 + α2 T1∗  ∗      T (t − τ2 ) V1 T ∗ (t − τ2 ) V T1∗ + ln ln = ln . T∗ V T1∗ V1 T ∗

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(17)

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Using Eqs. (17) we get   dU1 d β1 T1 V1 (1 + α1 V1 ) V V 1 + α1 V 2 = − (T − T1 ) + − −1+ dt T 1 + α1 V1 (1 + α1 V ) V1 V1 1 + α1 V1      T∗ 1 + α2 T ∗ β1 T1 V1 T1 β2 T1 T1∗ (1 + α2 T1∗ ) T ∗ T1 − − 1 + − + − 1 − ln ∗ ∗ ∗ ∗ ∗ 1 + α2 T1 (1 + α2 T ) T1 T1 1 + α2 T1 1 + α1 V1 T T      V1 T ∗ (t − τ2 ) 1 + α1 V β1 T1 V1 V1 T ∗ (t − τ2 ) 1 + α1 V − 1 − ln + − − 1 − ln 1 + α1 V1 V T1∗ V T1∗ 1 + α1 V1 1 + α1 V1       ∗ T1 1 + α2 T ∗ T1 β2 T1 T1∗ 1 + α2 T ∗ β2 T1 T1 − 1 − ln − 1 − ln − − 1 + α2 T1∗ T T 1 + α2 T1∗ 1 + α2 T1∗ 1 + α2 T1∗    ∗ β1 T1 V1 T (t − τ1 )V (t − τ1 ) (1 + α1 V1 ) T1 T (t − τ1 )V (t − τ1 ) (1 + α1 V1 ) T1∗ − − 1 − ln 1 + α1 V1 T1 V1 (1 + α1 V (t − τ1 )) T ∗ T1 V1 (1 + α1 V (t − τ1 )) T ∗    ∗ ∗ ∗ β2 T1 T1 T (t − τ1 )T (t − τ1 ) (1 + α2 T1 ) T (t − τ1 )T ∗ (t − τ1 ) (1 + α2 T1∗ ) − − 1 − ln 1 + α2 T1∗ T1 T ∗ (1 + α2 T ∗ (t − τ1 )) T1 T ∗ (1 + α2 T ∗ (t − τ1 )) Then " " # # 2 2 α2 (T ∗ − T1∗ ) d β1 T1 V1 α1 (V − V1 ) β2 T1 T1∗ dU1 2 = − (T − T1 ) − − dt T 1 + α1 V1 (1 + α1 V ) (1 + α1 V1 ) V1 1 + α2 T1∗ (1 + α2 T ∗ ) (1 + α2 T1∗ ) T1∗          β1 T1 V1 1 + α1 V T1 T (t − τ1 )V (t − τ1 ) (1 + α1 V1 ) T1∗ V1 T ∗ (t − τ2 ) − +g g +g +g 1 + α1 V1 T T1 V1 (1 + α1 V (t − τ1 )) T ∗ 1 + α1 V1 V T1∗        ∗ ∗ ∗ ∗ β2 T1 T1 T (t − τ1 )T (t − τ1 ) (1 + α2 T1 ) 1 + α2 T T1 − +g +g g . (18) ∗ ∗ ∗ 1 + α2 T1 T T1 T (1 + α2 T (t − τ1 )) 1 + α2 T1∗ dU1 1 Since R0 > 1, then T, T ∗ , V > 0. From Eq. (18) we have dU dt ≤ 0 and dt = 0 ocurs at S1 . Let D1 =  1 (T, T ∗ , V ) : dU dt = 0 . It is clear that S1 is the largest invariant subset of D1 . Using LaSalle’s invariance principle we obtain that S1 is globally asymptotically stable when R0 > 1. 

3

HIV-1 model with distributed delays

In this section, we formulate an HIV-1 infection model with saturated virus-target and infected-target incidences and two types of distributed time delays: β1 T (t)V (t) β2 T (t)T ∗ (t) − , T˙ (t) = ρ − dT (t) − 1 + α1 V (t) 1 + α2 T ∗ (t)   Z ∞ β1 T (t − s)V (t − s) β2 T (t − s)T ∗ (t − s) T˙ ∗ (t) = f1 (s)e−δ1 s + ds − µT ∗ (t), 1 + α1 V (t − s) 1 + α2 T ∗ (t − s) 0 Z ∞ V˙ (t) = b f2 (s)e−δ2 s T ∗ (t − s)ds − cV (t).

(19) (20) (21)

0

Let us assume that the probability distribution functions fi (s) satisfy fi (s) > 0, i = 1, 2 and Z∞

Z∞ fi (s)ds = 1,

0

where ` > 0. Denote ηi = memory type

R∞ 0

fi (u)e`u du < ∞, i = 1, 2,

0

fi (s)e−δi s ds,

i = 1, 2, thus, 0 < ηi ≤ 1. Define the Banach space of fading

Cγ = {φ ∈ C((−∞, 0], R) : eαη φ(η) is uniformly continuous for η ∈ (−∞, 0] and kφk < ∞} where γ is a positive constant and kφk = sup |φ(η)| eγη . Let η≤0

Cγ+ = {φ ∈ Cγ : φ(η) ≥ 0 f or η ∈ (−∞, 0]}.

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The initial conditions for system (19)-(21) are given as: T (η) = ϕ1 (η), T ∗ (η) = ϕ2 (η), V (η) = ϕ3 (η), for η ∈ (−∞, 0], ϕi ∈ Cγ+ , i = 1, 2, 3.

(22)

System (7)-(9) with initial conditions (22) has a unique solution [27].

3.1

Basic properties

The non-negativity and boundedness of the solutions of model (19)-(21) will be established in the next lemma. Lemma 3. The solutions (T (t), T ∗ (t), V (t)) of model (19)-(21) with initial conditions (22) are non-negative and ultimately bounded. Proof : Similar to the proof of Lemma 1, one can show T (t) > 0 for all T (t) > 0 for t ∈ (0, $2 ), where (0, $2 ) is the maximal interval of existence of solution of system (19)-(21) with (22). From Eqs. (20)-(21), we have   Z ∞ Z t β1 T (η − s)V (η − s) β2 T (η − s)T ∗ (η − s) f1 (s)e−δ1 s + dsdη ≥ 0, T ∗ (t) = e−µt ϕ2 (0) + e−µ(t−η) 1 + α1 V (η − s) 1 + α2 T ∗ (η − s) 0 0 Z ∞ Z t −c(t−η) −ct f2 (s)e−δ2 s T ∗ (η − s)dsdη ≥ 0. e V (t) = e ϕ3 (0) + b 0

0

From Eq. (19) we have lim sup T (t) ≤ t→∞

R∞ ρ . Let F (t) = 0 f1 (s)e−δ1 s T (t − s)ds + T ∗ (t). Then d

∞

  β1 T (t − s)V (t − s) β2 T (t − s)T ∗ (t − s) − ds f1 (s)e−δ1 s ρ − dT (t − s) − 1 + α1 V (t − s) 1 + α2 T ∗ (t − s) 0   Z ∞ β1 T (t − s)V (t − s) β2 T (t − s)T ∗ (t − s) + f1 (s)e−δ1 s + ds − µT ∗ (t) ∗ (t − s) 1 + α V (t − s) 1 + α T 1 2 0 Z ∞ = ρη1 − d f1 (s)e−δ1 s T (t − s)ds − µT ∗ (t) Z 0∞  ≤ρ−σ f1 (s)e−δ1 s T (t − s)ds + T ∗ (t) = ρ − σF2 (t),

F˙2 (t) =

Z

0

R∞ ρ where, σ = min{d, µ}. Hence, lim supt→∞ F2 (t) ≤ . Since 0 f1 (s)e−δ1 s T (t − s)ds > 0 and T ∗ ≥ 0 , then σ ρ lim supt→∞ T ∗ (t) ≤ . From Eq. (21) we have σ Z ∞ ρ ρ ˙ V (t) = b f2 (s)e−δ2 s T ∗ (t − s)ds − cV (t) ≤ bη2 − cV (t) ≤ b − cV (t). σ σ 0 bρ . Therefore, T (t), T ∗ (t) and V (t) are ultimately bounded. cσ t→∞ The existence of the steady state of the model (19)-(21) will be shown in the next lemma. Lemma 4. (i) If R0 ≤ 1, then there exists only positive steady state S0 , (ii) if 1 < R0 , then there exist only two positive steady states S0 and S1 . The proof. Let the R.H.S of system (19)-(21) be equal zero

Thus lim sup V (t) ≤

β1 T V β2 T T ∗ 0 = ρ − dT − − , 1 + α1 V 1 + α2 T ∗   β1 T V β2 T T ∗ 0 = η1 + − µT ∗ , 1 + α1 V 1 + α2 T ∗

(23) (24)

0 = η2 bT ∗ − cV,

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Solving Eqs. (23)-(25) we find that the system has two steady states, disease-free steady state S0 = (T0 , 0, 0, 0), ρ where T0 = , and endemic steady state S1 (T1 , T1∗ , V1 ), where d √ µc (1 + α1 V1 ) (bη2 + α2 cV1 ) −B + B 2 − 4AC bη2 T1∗ T1 = , T1∗ = , V1 = , bη1 η2 [β1 (bη2 + α2 cV1 ) + β2 c (1 + α1 V1 )] 2A c where A = µbη2 (dα1 α2 + β1 α2 + β2 α1 ) , B = β2 (µc − ρα1 bη1 η2 ) + β1 bη2 (µ − ρα2 η1 ) + dµ(cα2 + α1 bη2 ),

(26)

C = dµc (1 − R0 ) , and R0 =

T0 η1 (bβ1 η2 + β2 c) , µc

where R0 represents the basic infection reproduction number. 

3.2

Global properties

In this section, we study the global stability of all the steady states of system (19)-(21) employing the method of Lyapunov function. Theorem 3. If R0 ≤ 1, then S0 is globally asymptotically stable. Proof. Define   T 1 β1 T0 L2 (T, T ∗ , V ) = T0 g V + T∗ + T0 η1 c  Z ∞ Z s 1 β1 T (t − η)V (t − η) β2 T (t − η)T ∗ (t − η) + + dηds f1 (s)e−δ1 s η1 0 1 + α1 V (t − η) 1 + α2 T ∗ (t − η) 0 Z Z s bβ1 T0 ∞ + f2 (s)e−δ2 s T ∗ (t − η)dηds. c 0 0 dL2 along the solutions of the system (19)-(21), we obtain dt    T0 β2 T T ∗ dL2 β1 T V = 1− − ρ − dT − dt T 1 + α1 V 1 + α2 T ∗   Z ∞  1 β2 T (t − s)T ∗ (t − s) ∗ −δ1 s β1 T (t − s)V (t − s) + + ds − µT f1 (s)e η1 0 1 + α1 V (t − s) 1 + α2 T ∗ (t − s)  Z ∞  β1 T0 + b f2 (s)e−δ2 s T ∗ (t − s)ds − cV c 0   Z ∞ 1 β1 T V β2 T T ∗ β1 T (t − s)V (t − s) β2 T (t − s)T ∗ (t − s) −δ1 s + − − ds + f1 (s)e η1 0 1 + α1 V 1 + α2 T ∗ 1 + α1 V (t − s) 1 + α2 T ∗ (t − s) Z ∞ bβ1 T0 + f2 (s)e−δ2 s [T ∗ − T ∗ (t − s)] ds c 0     T0 V2 T ∗2 µ T0 bβ1 η1 η2 T0 β2 η1 = 1− (ρ − dT ) − α1 β1 T0 − α2 β2 T0 + + − 1 T∗ T 1 + α1 V 1 + α2 T ∗ η1 µc µ (T − T0 )2 V2 T ∗2 µ = −d − α1 β1 T0 − α2 β2 T0 + (R0 − 1)T ∗ . (27) T 1 + α1 V 1 + α2 T ∗ η1

Calculating

dL2 If R0 ≤ 1, then ≤ 0 for all T, T ∗ , V > 0. Similar to the proof of Theorem 1 one can easily show that S0 is dt globally asymptotically stable when R0 ≤ 1. Theorem 4. If 1 < R0 , then S1 is globally asymptotically stable.

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Proof. We construct the following Lyapunov functional      ∗ β1 T1 V1 V T 1 T + V g U2 (T, T ∗ , V, Z) = T1 g + T1∗ g 1 T1 η1 T1∗ bη2 T1∗ (1 + α1 V1 ) V1  Z ∞ Z s  T (t − η)V (t − η) (1 + α1 V1 ) β1 T1 V1 f1 (s)e−δ1 s g + dηds η1 (1 + α1 V1 ) 0 T1 V1 (1 + α1 V (t − η)) 0  Z s  Z ∞ T (t − η)T ∗ (t − η) (1 + α2 T1∗ ) β2 T1 T1∗ −δ1 s g f (s)e + dηds 1 η1 (1 + α2 T1∗ ) 0 T1 T1∗ (1 + α2 T ∗ (t − η)) 0  Z ∞ Z s  ∗ T (t − η) β1 T1 V1 f2 (s)e−δ2 s g + dηds. η2 (1 + α1 V1 ) 0 T1∗ 0 dU2 dt

along the trajectories of (19)-(21) is given by    dU2 T1 β1 T V β2 T T ∗ = 1− ρ − dT − − dt T 1 + α1 V 1 + α2 T ∗  Z     ∞ ∗ 1 T1 β2 T (t − s)T ∗ (t − s) −δ1 s β1 T (t − s)V (t − s) ∗ f1 (s)e + 1− ∗ + ds − µT η1 T 1 + α1 V (t − s) 1 + α2 T ∗ (t − s) 0    Z ∞ V1 β1 T1 V1 f2 (s)e−δ2 s T ∗ (t − s)ds − cV 1− b + bη2 T1∗ (1 + α1 V1 ) V 0   Z ∞ β1 T1 V1 T (t − s)V (t − s) (1 + α1 V1 ) T V (1 + α1 V1 ) −δ1 s + f1 (s)e − ds η1 (1 + α1 V1 ) 0 T1 V1 (1 + α1 V ) T1 V1 (1 + α1 V (t − s))   Z ∞ β1 T1 V1 T (t − s)V (t − s) (1 + α1 V ) + f1 (s)e−δ1 s ln ds η1 (1 + α1 V1 ) 0 T V (1 + α1 V (t − s))   Z ∞ β2 T1 T1∗ T (t − s)T ∗ (t − s) (1 + α2 T1∗ ) T T ∗ (1 + α2 T1∗ ) −δ1 s ds + − f1 (s)e η1 (1 + α2 T1∗ ) 0 T1 T1∗ (1 + α2 T ∗ ) T1 T1∗ (1 + α2 T ∗ (t − s))   Z ∞ T (t − s)T ∗ (t − s) (1 + α2 T ∗ ) β2 T1 T1∗ −δ1 s f (s)e ln ds + 1 η1 (1 + α2 T1∗ ) 0 T T ∗ (1 + α2 T ∗ (t − s))  ∗   Z ∞ β1 T1 V1 T T ∗ (t − s) T ∗ (t − s) + f2 (s)e−δ2 s − + ln ds. η2 (1 + α1 V1 ) 0 T1∗ T1∗ T∗

We evaluate

Collecting terms of Eq. (28) and applying the steady state conditions for S1 : ρ − dT1 =

β1 T1 V1 µ β2 T1 T1∗ cµ = T1∗ = + V1 , 1 + α1 V1 1 + α2 T1∗ η1 bη1 η2

we get    d T1 β1 T1 V β1 T1 V1 β2 T1 T1∗ dU2 2 = − (T − T1 ) + 1 − + + ∗ dt T T 1 + α1 V1 1 + α2 T1 1 + α1 V Z ∞ β2 T1 T ∗ T1∗ β T (t − s)V (t − s) 1 + − f1 (s)e−δ1 s ds 1 + α2 T ∗ η1 T ∗ 0 1 + α1 V (t − s) Z ∞ T∗ β2 T1 T ∗ β1 T1 V1 β2 T (t − s)T ∗ (t − s) − 1∗ ds − + f1 (s)e−δ1 s η1 T 0 1 + α2 T ∗ (t − s) 1 + α2 T1∗ 1 + α1 V1 Z ∞ β1 T1 V1 V1 T ∗ (t − s) β1 T1 V β2 T1 T1∗ − f2 (s)e−δ2 s ds − + ∗ 1 + α2 T1 η2 (1 + α1 V1 ) 0 V T1∗ 1 + α1 V1   Z ∞ β1 T1 V1 β1 T1 V1 T (t − s)V (t − s) (1 + α1 V ) −δ1 s + + f1 (s)e ln ds 1 + α1 V1 η1 (1 + α1 V1 ) 0 T V (1 + α1 V (t − s))   Z ∞ β2 T1 T1∗ T (t − s)T ∗ (t − s) (1 + α2 T ∗ ) + f1 (s)e−δ1 s ln ds ∗ η1 (1 + α2 T1 ) 0 T T ∗ (1 + α2 T ∗ (t − s))  ∗  Z ∞ β1 T1 V1 T (t − s) + f2 (s)e−δ2 s ln ds. η2 (1 + α1 V1 ) 0 T∗

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Using Eq. (17) we get   d β1 T1 V1 (1 + α1 V1 ) V V 1 + α1 V dU2 2 = − (T − T1 ) + − −1+ dt T 1 + α1 V1 (1 + α1 V ) V1 V1 1 + α1 V1   ∗ ∗ ∗ ∗ ∗ β2 T1 T1 (1 + α2 T1 ) T T 1 + α2 T + − ∗ −1+ 1 + α2 T1∗ (1 + α2 T ∗ ) T1∗ T1 1 + α2 T1∗       β1 T1 V1 T1 T1 β2 T1 T1∗ T1 T1 − − 1 − ln − − 1 − ln 1 + α1 V1 T T 1 + α2 T1∗ T T   Z ∞ T (t − s)V (t − s) (1 + α1 V1 ) T1∗ β1 T1 V1 f1 (s)e−δ1 s − 1 ds − η1 (1 + α1 V1 ) 0 T1 V1 (1 + α1 V (t − s)) T ∗   Z ∞ β1 T1 V1 T (t − s)V (t − s) (1 + α1 V1 ) T1∗ + f1 (s)e−δ1 s ln ds η1 (1 + α1 V1 ) 0 T1 V1 (1 + α1 V (t − s)) T ∗   Z ∞ ∗ ∗ β2 T1 T1∗ −δ1 s T (t − s)T (t − s) (1 + α2 T1 ) f (s)e − − 1 ds 1 η1 (1 + α2 T1∗ ) 0 T1 T ∗ (1 + α2 T ∗ (t − s))   Z ∞ β2 T1 T1∗ T (t − s)T ∗ (t − s) (1 + α2 T1∗ ) −δ1 s + f (s)e ln ds 1 η1 (1 + α2 T1∗ ) 0 T1 T ∗ (1 + α2 T ∗ (t − s))    Z ∞ V1 T ∗ (t − s) V1 T ∗ (t − s) β1 T1 V1 f2 (s)e−δ2 s − 1 − ln ds − η2 (1 + α1 V1 ) 0 V T1∗ V T1∗       β1 T1 V1 1 + α1 V 1 + α1 V β2 T1 T1∗ 1 + α2 T ∗ 1 + α2 T ∗ − − 1 − ln − − 1 − ln 1 + α1 V1 1 + α1 V1 1 + α1 V1 1 + α2 T1∗ 1 + α2 T1∗ 1 + α2 T1∗ " # " # 2 2 d β2 T1 T1∗ β1 T1 V1 α1 (V − V1 ) α2 (T ∗ − T1∗ ) 2 = − (T − T1 ) − − T 1 + α1 V1 (1 + α1 V ) (1 + α1 V1 ) V1 1 + α2 T1∗ (1 + α2 T ∗ ) (1 + α2 T1∗ ) T1∗        Z ∞ T (t − s)V (t − s) (1 + α1 V1 ) T1∗ 1 + α1 V T1 β1 T1 V1 −δ1 s +g +g f1 (s)e g ds − η1 (1 + α1 V1 ) 0 T T1 V1 (1 + α1 V (t − s)) T ∗ 1 + α1 V1        Z ∞ T (t − s)T ∗ (t − s) (1 + α2 T1∗ ) 1 + α2 T ∗ β2 T1 T1∗ T1 − +g + g ds f1 (s)e−δ1 s g ∗ η1 (1 + α2 T1 ) 0 T T1 T ∗ (1 + α2 T ∗ (t − s)) 1 + α2 T1∗   Z ∞ V1 T ∗ (t − s) β1 T1 V1 f2 (s)e−δ2 s g ds. − η2 (1 + α1 V1 ) 0 V T1∗ Similar to the proof of Theorem 2, one can easily show that S1 is globally asymptotically stable. 

4

Numerical simulations

In order to illustrate our theoretical results, we will perform numerical simulations for system (7)-(9). We use the data given in Table 1. Table 1: The data of system (7)-(9). Parameter

Value

λ

10 cells mm−3 day−1 −1

Parameter

Parameter

τ1

Varied

τ2

Varied

d

0.01 day

β1

Varied

δ1

0.9 day−1

β2

0.0001 cells−1 mm3 day−1

δ2

0.1 day−1

α1

Varied

b

10 virus cells−1 day−1

α2

Varied

c

3 day−1

µ

0.9 day−1

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4.1

Effect of the parameter β1 on the stability of steady states

To show the global stability of the steady states we consider three different initial conditions: IC1: ϕ1 (η) = 600, ϕ2 (η) = 1, ϕ3 (η) = 1, IC2: ϕ1 (η) = 200, ϕ2 (η) = 0.5, ϕ3 (η) = 3, IC3: ϕ1 (η) = 700, ϕ2 (η) = 5, ϕ3 (η) = 9, where, η ∈ [− max{τ1 , τ2 }, 0]. In this case we choose τ1 = 0.5 day, τ2 = 0.9 day, α1 = 0.009 virus −1 mm3 , α2 = 0.005 cells −1 mm3 and study the following subcases for the initial conditions IC1-IC3.: (i) R0 ≤ 1. We choose, β1 = 0.0001 virus −1 mm3 day−1 , then we compute R0 = 0.2867 < 1. From Lemma 2 we have that the system has one steady state S0 . From Figures 1-3 we can see that, the concentration of uninfected CD4+ T cells is increasing and tends its normal value ρ/d = 1000, while the concentrations of infected cells and free HIV-1 are decaying and approaching zero for all the three initial conditions IC1-IC3. It means that, S0 is globally asymptotically stable and the virus will be removed. This result support the result of Theorem 1. (ii) R0 > 1. We take β1 = 0.001 virus −1 mm3 day−1 , and then, R0 = 2.2292 > 1. Lemma 2 state that the system has two positive steady states S0 and S1 . It is clear from Figures 4-6 that, both the numerical results and the theoretical results given in Theorem 2 are consistent. It is seen that, the solutions of the system converges to the steady S1 (491.6543, 3.6015, 10.9717), for all the three initial conditions IC1-IC3.

4.2

Effect of the saturation infection on the HIV-1 dynamics

In this case, we consider the initial condition IC2. We take the values τ1 = 0.5 day, τ2 = 0.9 day andβ1 = 0.001 virus−1 mm3 day−1 . Figures 7-9 show the effect of saturation infection. We observe that, as α1 and α2 are increased, both the virus-target and infected-target infection rates are decreased, and then the concentration of the CD4+ T cells are increased, while the concentrations of the infected cells and free HIV-1 particles are decreased.

4.3

Effect of the time delays on the stability of steady states

In this case, we consider the initial condition IC2. We take the values β1 = 0.001 virus −1 mm3 day−1 , α1 = 0.009 virus−1 mm3 and α2 = 0.005 cells −1 mm3 . Let us consider the case τ = τ1 = τ2 . The values of R0 and the steady states of system (7)-(9) with different values of τ are presented in Table 2. Table 2: The values of steady states, R0 for model (7)-(9) with different values of the delay parameter τ . Delay parameter

Steady states

R0

τ = 0.0

E1 = (319.8688, 7.5570, 25.1900)

3.8148

τ = 0.2

E1 = (373.1935, 5.8172, 19.0069)

3.1251

τ = 0.6

E1 = (517.7076, 3.1228, 9.8032)

2.0973

τ = 0.9

E1 = (670.8935, 1.6267, 4.9557)

1.5552

τ = 1.

E1 = (733.0161, 1.2061, 3.6377)

1.4077

τ =1.3431

E0 = (1000, 0, 0, 0)

1

τ = 1.5

E0 = (1000, 0, 0, 0)

0.8552

τ =2

E0 = (1000, 0, 0, 0)

0.5196

τ = 2.5

E0 = (1000, 0, 0, 0)

0.3157

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From Table 2 we can see that, R0 is decreased as τ is increased. Using the data given in Table 1, we get: (i) if 0 ≤ τ < 1.343070098, then S1 exists and it is globally asymptotically stable, (ii) if τ ≥ 1.343070098, then S0 is globally asymptotically stable. Figures 10-12 show that the numerical results are also compatible with the results of Theorems 1 and 2. It can be seen that when the time delay is increased, the system can be stabilized around the disease-free steady state S0 . This means that the delay plays a similar job as the antiviral treatment in clearing the HIV-1 from the plasma. 5

1000 IC1 IC2 IC3

900

IC1 IC2 IC3

4.5 4

800 Uninfected Cells

3.5 Infected Cells

700 600 500

3 2.5 2 1.5

400 1 300 200 0

0.5 200

400

600 Time

800

1000

0 0

1200

2

4

6

8

10

Time

Figure 1: The evolution of uninfected CD4+ T cells with initial IC1-IC3 in case of R0 ≤ 1.

Figure 2: The evolution of infected cells with initial IC1-IC3 in case of R0 ≤ 1.

15

700 IC1 IC2 IC3

IC1 IC2 IC3

600 500

Free Virus

Uninfected Cells

10

5

400 300 200 100

0 0

2

4

6

8

0 0

10

Time

Figure 3: The evolution of free HIV-1 with initial IC1-IC3 in case of R0 ≤ 1.

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100

200

300 Time

400

500

600

Figure 4: The evolution of uninfected CD4+ T cells with initial IC1-IC3 in case of R0 > 1.

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35 IC1 IC2 IC3

10

IC1 IC2 IC3

30 25 Free Virus

Infected Cells

8

6

20 15

4 10 2

0 0

5

100

200

300 Time

400

500

0 0

600

Figure 5: The evolution of infected cells with initial IC1-IC3 in case of R0 > 1.

200

300 Time

400

500

600

Figure 6: The evolution of free HIV-1 with initial IC1-IC3 in case of R0 > 1.

20

1000 α1=0.,α2=0. 900

α1=0.,α2=0.

18

α1=0.009,α2=0.005

α1=0.009,α2=0.005

800

α1=0.03,α2=0.8

16

α1=0.03,α2=0.8

α1=0.1,α2=1.

14

α1=0.1,α2=1.

12

α1=1.9,α2=1.8

α1=1.9,α2=1.8

700

Infected Cells

Uninfected Cells

100

600 500

10 8 6

400 4 300 200 0

2 200

400

600 Time

800

1000

0 0

1200

Figure 7: The evolution of uninfected CD4+ T cells with different saturation parameters α1 ,α2 .

100

200

300 Time

400

500

Figure 8: The evolution of infected cells with different saturation parameters α1 ,α2 .

1000

70

τ=0. τ=0.4 τ=0.8 τ=1.0 τ=1.4

α =0.,α =0. 1

2

900

α =0.009,α =0.005

60

1

50

2

α1=0.03,α2=0.8

800

α =0.1,α =1.

700

1

2

α1=1.9,α2=1.8

Uninfected Cells

Free Virus

600

40 30

600 500 400 300

20

200

10 100

0 0

100

200

300 Time

400

500

0 0

600

Figure 9: The evolution of free HIV-1 with different saturation parameters α1 ,α2 .

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200

400

600 Time

800

1000

1200

Figure 10: The evolution of uninfected CD4+ T cells with different delay parameter τ .

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40 τ=0. τ=0.4 τ=0.8 τ=1.0 τ=1.4

10

τ=0. τ=0.4 τ=0.8 τ=1. τ=1.4

35 30 25 Free Virus

Infected Cells

8

6

20 15

4 10 2 5 0 0

100

200

300

400 Time

500

600

700

0 0

800

Figure 11: The evolution of infected cells with different delay parameter τ .

5

100

200

300 Time

400

500

600

Figure 12: The evolution of free HIV-1 with different delay parameter τ .

Acknowledgment

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References [1] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996) 74-79. [2] X. Song, A.U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. [3] X. Shi, X. Zhou, and X. Son, Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Analysis: Real World Applications, 11 (2010), 1795-1809. [4] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses, SIAM Journal of Applied Mathematics, 73(3) (2013), 1280-1302. [5] A. M. Elaiw and N. H. AlShamrani, Mathematical analysis of a cell mediated immunity in a virus dynamics model with nonlinear infection rate and removal, Journal of Computational Analysis and Applications, 21(3) (2016), 578-586. [6] T. Wang, Z. Hu and F. Liao, Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, J. Math. Anal. Appl., 411 (2014) 63-74. [7] A. M. Elaiw and N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Analysis: Real World Applications, 26, pp. 161-190, 2015. [8] M. A. Obaid, Dynamical behaviors of a nonlinear virus infection model with latently infected cells and immune response, Journal of Computational Analysis and Applications, 21(1) (2016), 182-193. [9] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81.

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[10] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70(7) (2010), 2693-2708. [11] S. Xu, Global stability of the virus dynamics model with Crowley-Martin functional response, Electronic Journal of Qualitative Theory of Differential Equations, 2012, (2012), 1-9. [12] N.M. Dixit and A.S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay, J. Theoret. Biol., 226 (2004), 95-109. [13] M. Y. Li and H. Shu, Impact of intracellular delays and target cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (7) (2010), 2434-2448. [14] D. Huang, X. Zhang, Y. Guo, and H. Wang, Analysis of an HIV infection model with treatments and delayed immune response, Applied Mathematical Modelling, 40(4) (2016), 3081-3089. [15] C. Monica and M. Pitchaimani, Analysis of stability and Hopf bifurcation for HIV-1 dynamics with PI and three intracellular delays, Nonlinear Analysis: Real World Applications, 27 (2016), 55-69. [16] M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonlinear Analysis: Real World Applications, 17 (2014), 147-160. [17] A. M. Elaiw, and N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Mathematical Methods in the Applied Sciences, 39 (2016), 4-31. [18] B. Buonomo, and C. Varglobally asymptotically stable-De-Le, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, Journal of Mathematical Analysis and Applications, 385 (2012), 709-720. [19] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol. 66, (2004), 879-883. [20] A. M. Elaiw, I. A. Hassanien, and S. A. Azoz, Global stability of HIV infection models with intracellular delays, Journal of the Korean Mathematical Society, 49(4) (2012), 779-794. [21] A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences, 36 (2013), 383-394. [22] A. M. Elaiw, and N. A. Almuallem, Global properties of delayed-HIV dynamics models with differential drug efficacy in co-circulating target cells, Applied Mathematics and Computation, 265 (2015), 1067-1089. [23] F. Li and J. Wang, Analysis of an HIV infection model with logistic target cell growth and cell-to-cell transmission, Chaos, Solitons and Fractals, 81(2015), 136-145. [24] X. Lai and X. Zou , Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, Journal of Mathematical Analysis and Applications, 426 (2015), 563–584. [25] X. Lai, X. Zou, Modelling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM Journal of Applied Mathematics, 74 (2014), 898–917. [26] Y. Yanga, L. Zou and S. Ruanc, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Mathematical Biosciences, 270 (2015), 183-191. [27] J. K. Hale, and S. M. V. Lunel, Introduction to functional differential equations, Springer-Verlag, New York, (1993).

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Equations on Banach space valued functions of abstract g-fractional calculus George A. Anastassiou1 and Ioannis K. Argyros2 1 Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] 2 Department of Mathematical Sciences Cameron University Lawton, OK 73505, USA [email protected] Abstract The aim of this paper is utilize proper iterative methods for solving equations on Banach spaces. The di¤erentiability of the operator involved is not assumed neither the convexity of its domain. Applications of the semi-local convergence are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.

2010 AMS Subject Classi…cation Codes: 65G99, 65H10, 26A33, 46B25, 47J25, 47J05. Key Words and Phrases: iterative method, Banach space, semi-local convergence, Fractional Calculus, Bochner-type integral.

1

Introduction

Let B1 ; B2 stand for Banach space and let stand for an open subset of B1 . Let also U (z; ) := fu 2 B1 : ku zk < g and let U (z; ) stand for the closure of U (z; ). Many problems in Computational Sciences, Engineering, Mathematical Chemistry, Mathematical Physics, Mathematical Economics and other disciplines can written as F (x) = 0 (1.1) using Mathematical Modeling [1]-[17], where F : ! B2 is a continuous operator. The solution x of equation (1.1) is sought in closed form, but this is 1

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attainable only in special cases. That explains why most solution methods for such equations are usually iterative. There is a plethora of iterative methods for solving equation (1.1), more the [2, 6, 7, 9 - 13, 15, 16]. Newton’s method [6, 7, 11, 15, 16]: F 0 (xn )

xn+1 = xn

1

F (xn ) :

(1.2)

Secant method: xn+1 = xn

[xn

1 ; xn ; F ]

1

F (xn ) ;

where [ ; ; F ] denotes a divided di¤erence of order one on Newton-like method: En 1 F (xn ) ;

xn+1 = xn

(1.3) [7, 15, 16].

(1.4)

where En = E (F ) (xn ) and E : ! L (B1 ; B2 ) the space of bounded linear operators from B1 into B2 . Other methods can be found in [7], [11], [15], [16] and the references therein. In the present study we consider the new method de…ned for each n = 0; 1; 2; ::: by xn+1 = G (xn ) G (xn+1 ) = G (xn )

An 1 F (xn ) ;

(1.5)

where x0 2 is an initial point, G : B3 ! (B3 a Banach space), An = A (F ) (xn+1 ; xn ) = A (xn+1 ; xn ) and A : ! L (B1 ; B2 ). Method (1.5) generates a sequence which we shall show converges to x under some Lipschitztype conditions (to be precised in Section 2). Although method (1.5) (and Section 2) is of independent interest, it is nevertheless designed especially to be used in g-Abstract Fractional Calculus (to be precised in Section 3). As far as we know such iterative methods have not yet appeared in connection to solve equations in Abstract Fractional Calculus. In this paper we present the semi-local convergence of method (1.5) in Section 2. Some applications to Abstract g-Fractional Calculus are suggested in Section 3 on a certain Banach space valued functions, where all the integrals are of Bochner-type [8], [14].

2

Semi-local Convergence analysis

We present the semi-local convergence analysis of method (1.5) using conditions (M ): (m1 ) F : B1 ! B2 is continuous, G : B3 ! is continuous and A (x; y) 2 L (B1 ; B2 ) for each (x; y) 2 :

2

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(m2 ) There exist > 0 and each (x; y) 2 0 0 and

B1 such that A (x; y)

0

1

A (x; y)

1

1

2 L (B2 ; B1 ) for

:

Set

\ 0. 1 = (m3 ) There exists a continuous and nondecreasing function [0; +1) such that for each x; y 2 1 kF (x) (kx

F (y)

yk ; kx

A (x; y) (G (x) x0 k ; ky

G (y))k

x0 k) kG (x)

G (y)k :

(m4 ) There exists a continuous and nondecreasing function [0; +1) such that for each x 2 1 kG (x) (m5 ) For x0 2

0

G (x0 )k

0

(kx

and x1 = G (x0 ) 2 A (x1 ; x0 )

1

0

x0 k) kx there exists

F (x0 )

: [0; +1)3 !

0

: [0; +1) !

x0 k : 0 such that

:

(m6 ) There exists s > 0 such that ( ; s; s) < 1; 0

(s) < 1

and kG (x0 )

x0 k

s

1

q0

;

where q0 = ( ; s; s) : (m7 ) U (x0 ; s) . Next, we present the semi-local convergence analysis for method (1.5) using the conditions (M ) and the preceding notation. Theorem 2.1 Assume that the conditions (M ) hold. Then, sequence fxn g generated by method (1.5) starting at x0 2 is well de…ned in U (x0 ; s), remains in U (x0 ; s) for each n = 0; 1; 2; ::: and converges to a solution x 2 U (x0 ; s) of equation F (x) = 0. The limit point x is the unique solution of equation F (x) = 0 in U (x0 ; s) : Proof. By the de…nition of s and (m5 ), we have x1 2 U (x0 ; s). The proof is based on mathematical induction on k. Suppose that kxk xk 1 k q0k 1 and kxk x0 k s: 3

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We get by (1.5), (m2 )

(m5 ) in turn that G (xk )k = Ak 1 F (xk ) =

kG (xk+1 ) Ak 1 (F (xk ) Ak 1

1

(kxk

F (xk

kF (xk ) xk

( ; s; s) kG (xk )

1)

F (xk

1)

1 k ; kxk 1

G (xk

1 )k

Ak

1

Ak

(G (xk ) 1

(G (xk )

x0 k ; kyk

= q0 kG (xk )

G (xk G (xk

x0 k) kG (xk ) G (xk

1 )))

1 )k

1 ))k

G (xk q0k kx1

1 )k

x0 k

q0k (2.1)

and by (m6 ) kxk+1

x0 k = kG (xk ) 0

x0 k

(kxk

kG (xk )

x0 k) kxk 0

G (x0 )k + kG (x0 )

x0 k + kG (x0 )

(s) s + kG (x0 )

x0 k

x0 k

x0 k

s:

The induction is completed. Moreover, we have by (2.1) that for m = 0; 1; 2; ::: kxk+m

1 1

xk k

q0m k q : q0 0

It follows from the preceding inequation that sequence fG (xk )g is complete in a Banach space B1 and as such it converges to some x 2 U (x0 ; s) (since U (x0 ; s) is a closed ball). By letting k ! +1 in (2.1) we get F (x ) = 0. We also get by (1.5) that G (x ) = x . To show the uniqueness part, let x 2 U (x0 ; s) be a solution of equation F (x) = 0 and G (x ) = x : By using (1.5), we obtain in turn that G (xk ) + Ak 1 F (xk )

G (xk+1 )k = x

kx

Ak 1 kF (x ) 1 0

(kx

F (xk )

xk k ; kxk+1 q0 kG (x )

Ak (G (x )

x0 k ; kxk

G (xk ))k

x0 k) kG (x )

q0k+1 kx

G (xk )k

Ak 1 F (x )

G (xk )k

x0 k ;

so lim xk = x . We have shown that lim xk = x , so x = x : k!+1

k!+1

Remark 2.2 (1) Condition (m2 ) can become part of condition (m3 ) by considering 0 (m3 ) There exists a continuous and nondecreasing function ' : [0; +1)3 ! [0; +1) such that for each x; y 2 1 A (x; y)

1

[F (x)

F (y)

A (x; y) (G (x)

G (y))]

4

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' (kx

yk ; kx

x0 k ; ky

x0 k) kG (x)

G (y)k :

Notice that ' (u1 ; u2 ; u3 )

(u1 ; u2 ; u3 )

for each u1 0, u2 0 and u3 0. Similarly, a function '1 can replace 1 for the uniqueness of the solution part. These replacements are of Mysovskii-type [6], [11], [15] and in‡uence the weaking of the convergence criterion in (m6 ), error bounds and the precision of s. (2) Suppose that there exist > 0, 1 > 0 and L 2 L (B1 ; B2 ) with L 1 2 L (B2 ; B1 ) such that 1 L 1 kA (x; y)

Lk

1

and 2

1

:=

1

< 1:

Then, it follows from the Banach lemma on invertible operators [11], and L

1

kA (x; y)

1

Lk

1

=

2

1; 2 = N, with integral part [ ] = n 2 N: Let g : [a; b] ! R be a strictly increasing function, such that g 2 C 1 ([a; b]), g 1 2 C n ([g (a) ; g (b)]), and let f 2 C n ([a; b] ; X). It clear then we obtain that f g 1 2 C n ([g (a) ; g (b)] ; X). Let := [ ]= n (0 < < 1). (I) See [5]. Let h 2 C ([g (a) ; g (b)] ; X), we de…ne the X-valued left RiemannLiouville fractional integral as Z z 1 1 (J z0 h) (z) := (z t) h (t) dt; (3.1) ( ) z0 for g (a) z0 z g (b), where is the gamma function: We de…ne the subspace Cg(x) ([g (a) ; g (b)] ; X) of C n ([g (a) ; g (b)] ; X), where x 2 [a; b] : n o g(x) Cg(x) ([g (a) ; g (b)] ; X) := h 2 C n ([g (a) ; g (b)] ; X) : J1 h(n) 2 C 1 ([g (x) ; g (b)] ; X) : (3.2) So let h 2 Cg(x) ([g (a) ; g (b)] ; X); we de…ne the X-valued left g-generalized fractional derivative of h of order , of Canavati type, over [g (x) ; g (b)] as g(x) (n)

Dg(x) h := J1

h

0

:

Clearly, for h 2 Cg(x) ([g (a) ; g (b)] ; X), there exists Z z 1 d Dg(x) h (z) = (z t) (1 ) dz g(x) for all g (x) z g (b) : In particular, when f Dg(x) f

g

1

(z) =

g

(3.3)

h(n) (t) dt;

(3.4)

1

2 Cg(x) ([g (a) ; g (b)] ; X) we have that Z z 1 d (n) (z t) f g 1 (t) dt; (3.5) (1 ) dz g(x)

for all z : g (x) z g (b) : (n) n 0 We have that Dg(x) f g 1 = f g 1 and Dg(x) f g 1 = f g 1: 1 From [5] we have for f g 2 Cg(x) ([g (a) ; g (b)] ; X), where x 2 [a; b], (X-valued left fractional Taylor’s formula) that f (y)

f (x) =

n X1 k=1

f

g

1 (k)

k!

(g (x))

(g (y)

k

g (x)) +

(3.6)

6

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1 ( )

Z

g(y)

(g (y)

1

t)

Dg(x) f

g(x)

1

g

(t) dt; for all y 2 [a; b] : y

x:

Alternatively, for f g 1 2 Cg(y) ([g (a) ; g (b)] ; X), where y 2 [a; b], we can write (again X-valued left fractional Taylor’s formula) that: f (x)

f (y) =

n X1

f

g

1 (k)

k!

k=1

1 ( )

Z

g(x)

(g (x)

1

t)

Dg(y) f

g(y)

(g (y))

1

g

(g (x)

k

g (y)) +

(3.7)

(t) dt; for all x 2 [a; b] : x

y:

Here we consider f 2 C n ([a; b] ; X), such that f g 1 2 Cg(x) ([g (a) ; g (b)] ; X), for every x 2 [a; b]; which is the same as f g 1 2 Cg(y) ([g (a) ; g (b)] ; X), for every y 2 [a; b] (i.e. exchange roles of x and y); we write that as f g 1 2 Cg+ ([g (a) ; g (b)] ; X). We have that Z z 1 d (n) Dg(y) f g 1 (z) = (z t) f g 1 (t) dt; (3.8) (1 ) dz g(y) for all z : g (y) z g (b) : So here we work with f 2 C n ([a; b] ; X), such that f g 1 2 Cg+ ([g (a) ; g (b)] ; X) : We de…ne the X-valued left linear fractional operator 8P (k) (f g 1 ) (g(x)) k 1 > > nk=11 (g (y) g (x)) + > k! > > (g(y) g(x)) 1 > 1 > Dg(x) f g (g (y)) ; y > x; > ( +1) > > > > < (k) (3.9) (A1 (f )) (x; y) := Pn 1 (f g 1 ) (g(y)) k 1 (g (x) g (y)) + > k=1 k! > > 1 > > Dg(y) f g 1 (g (x)) (g(x) ( g(y)) ; x > y; > +1) > > > > > > : (n) f (x) , x = y: We may assume that (see [12], p. 3) k(A1 (f )) (x; x) f (n) g

1

(A1 (f )) (y; y)k = f (n) (x)

(g (x))

f (n) g

1

(g (y))

f (n) (y) =

jg (x)

g (y)j ;

(3.10)

where > 0; for any x; y 2 [a; b] : We make the following estimations: (i) case of y > x : We have that kf (y)

f (x)

(A1 (f )) (x; y) (g (y)

g (x))k =

7

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Z

1 ( )

g(y)

(g (y)

1

t)

Dg(x) f

g(x)

Dg(x) f

1

g

(g (y))

g

1

(t) dt

(g (y) g (x)) ( + 1)

(by [1], p. 426, Theorem 11.43) =

Z

1 ( )

g(y)

(g (y)

1

t)

Dg(x) f

g(x)

1

g

(t)

Dg(x) f

1

g

(g (y)) dt (3.11)

(by [8]) 1 ( )

Z

g(y)

(g (y)

1

t)

Dg(x) f

g(x)

1

g

(t)

Dg(x) f

(g (y))

1

1

g

(g (y)) dt

(we assume here that Dg(x) f

g

1

(t)

Dg(x) f

1

g

for every t; g (y) ; g (x) 2 [g (a) ; g (b)] such that g (y) 1

( ) 1

( )

Z

Z

g(y)

(g (y)

1

t)

(g (y)

t

jt

g (y)j ;

g (x) ;

1

(3.12)

> 0)

t) dt =

(3.13)

g(x)

g(y)

(g (y)

(g (y) g (x)) ( ) ( + 1)

+1

1

t) dt =

g(x)

:

(3.14)

We have proved that kf (y)

f (x)

(A1 (f )) (x; y) (g (y)

g (x))k

(g (y) g (x)) ( ) ( + 1) 1

+1

; (3.15)

for all x; y 2 [a; b] : y > x: (ii) Case of x > y : We observe that kf (y)

f (x)

(A1 (f )) (x; y) (g (y)

g (x))k =

kf (x) f (y) (A1 (f )) (x; y) (g (x) g (y))k = Z g(x) 1 1 (g (x) t) Dg(y) f g 1 (t) dt ( ) g(y) Dg(y) f 1 ( )

Z

g(x)

g(y)

(g (x)

t)

1

g

1

(g (x))

Dg(y) f

g

(g (x) g (y)) ( + 1) 1

(t)

=

Dg(y) f

(3.16) g

1

(g (x)) dt

8

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1 ( )

Z

g(x)

(g (x)

1

t)

g(y)

Dg(y) f

1

g

(t)

Dg(y) f

1

g

(g (x)) dt (3.17)

(we assume that Dg(y) f

g

1

(t)

Dg(y) f

1

g

(g (x))

2

jt

for all t; g (x) ; g (y) 2 [g (a) ; g (b)] such that g (x) t g (y) ; Z g(x) 2 1 (g (x) t) (g (x) t) dt = ( ) g(y) 2

( )

Z

g(x)

(g (x)

t) dt =

g(y)

(g (x) g (y)) ( ) ( + 1) 2

g (x)j ; 2

(3.18)

> 0) (3.19)

+1

:

We have proved that kf (y)

f (x)

(A1 (f )) (x; y) (g (y)

g (x))k

(g (x) g (y)) ( ) ( + 1)

+1

2

; (3.20)

for any x; y 2 [a; b] : x > y: Conclusion 3.1 Set kf (y)

f (x)

:= max (

1;

2 ).

Then

(A1 (f )) (x; y) (g (y)

g (x))k

jg (y) g (x)j ( ) ( + 1)

+1

; (3.21)

8 x; y 2 [a; b] (the case of x = y is trivially true). We may choose that ( ) < 1: Also we notice here that + 1 > 2: (II) See [5] again. Let h 2 C ([g (a) ; g (b)] ; X), we de…ne the X-valued right Riemann-Liouville fractional integral as Z z0 1 1 (t z) h (t) dt; (3.22) Jz0 h (z) := ( ) z for g (a) z z0 g (b) : We de…ne the subspace Cg(x) ([g (a) ; g (b)] ; X) of C n ([g (a) ; g (b)] ; X), where x 2 [a; b] : n o 1 Cg(x) ([g (a) ; g (b)] ; X) := h 2 C n ([g (a) ; g (b)] ; X) : Jg(x) h(n) 2 C 1 ([g (a) ; g (x)] ; X) : (3.23) So let h 2 Cg(x) ([g (a) ; g (b)] ; X); we de…ne the X-valued right g-generalized fractional derivative of h of order , of Canavati type, over [g (a) ; g (x)] as n 1

Dg(x) h := ( 1)

1 Jg(x) h(n)

0

:

(3.24)

9

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Clearly, for h 2 Cg(x) ([g (a) ; g (b)] ; X), there exists n 1

( 1) h (z) = (1

Dg(x)

d ) dz

Z

g(x)

(t

h(n) (t) dt;

z)

(3.25)

z

for all g (a) z g (x) g (b) : In particular, when f g 1 2 Cg(x) ([g (a) ; g (b)] ; X) we have that n 1

Dg(x)

f

g

for all g (a) z We get that

( 1) (z) = (1

1

g (x) n Dg(x)

d ) dz

Z

g(x)

(t

z)

f

1 (n)

g

(t) dt;

z

(3.26)

g (b) :

f

n

1

g

(z) = ( 1)

f

g

1 (n)

(z) ;

(3.27)

and 0 Dg(x)

f

g

1

(z) = f

g

1

(z) ;

(3.28)

for all z 2 [g (a) ; g (x)], see [5]. From [5] we have, for f g 1 2 Cg(x) ([g (a) ; g (b)] ; X), where x 2 [a; b], 1 (X-valued right fractional Taylor’s formula) that: f (y)

f (x) =

n X1

f

g

Z

g(x)

(t

g (y))

1

g(y)

(g (x))

k!

k=1

1 ( )

1 (k)

Dg(x)

f

g

1

(g (y)

k

g (x)) +

(t) dt; all a

y

x:

(3.29)

Alternatively, for f g 1 2 Cg(y) ([g (a) ; g (b)] ; X), where y 2 [a; b], 1 (again X-valued right fractional Taylor’s formula) that: f (x)

f (y) =

n X1

f

g

k=1

1 ( )

Z

g(y)

g(x)

(t

g (x))

1

Dg(y)

1 (k)

(g (y))

k! f

g

1

(g (x)

k

g (y)) +

(t) dt; all a

x

y:

(3.30)

Here we consider f 2 C n ([a; b] ; X), such that f g 1 2 Cg(x) ([g (a) ; g (b)] ; X), for every x 2 [a; b]; which is the same as f g 1 2 Cg(y) ([g (a) ; g (b)] ; X), for every y 2 [a; b] ; (i.e. exchange roles of x and y) we write that as f g 1 2 Cg ([g (a) ; g (b)] ; X). We have that Z g(y) n 1 ( 1) d (n) Dg(y) f g 1 (z) = (t z) f g 1 (t) dt; (1 ) dz z (3.31) 10

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for all g (a) z g (y) g (b) : So here we work with f 2 C n ([a; b] ; X), such that f g We de…ne the X-valued right linear fractional operator 8P 1 (k) k n 1 (f g ) (g(x)) > > (g (y) g (x)) > k=1 k! > > > 1 > (g (y)) (g(x) ( g(y)) > Dg(x) f g +1) > > > > < (k) (A2 (f )) (x; y) := Pn 1 (f g 1 ) (g(y)) k (g (x) g (y)) > k=1 k! > > > > Dg(y) f g 1 (g (x)) (g(y) ( g(x)) > +1) > > > > > > : (n) f (x) , x = y:

1

2 Cg ([g (a) ; g (b)] ; X) :

1 1

; x > y; (3.32)

1 1

; y > x;

We may assume that ([12], p. 3) k(A2 (f )) (x; x)

(A2 (f )) (y; y)k = f (n) (x)

f (n) (y)

jg (x)

g (y)j ; (3.33)

where > 0; for any x; y 2 [a; b] : We make the following estimations: (i) case of x > y : We have that kf (x)

f (y)

(A2 (f )) (x; y) (g (x)

g (y))k =

kf (y)

f (x)

(A2 (f )) (x; y) (g (y)

g (x))k =

(3.34)

kf (y) f (x) + (A2 (f )) (x; y) (g (x) g (y))k = Z g(x) 1 1 (t g (y)) Dg(x) f g 1 (t) dt ( ) g(y) Dg(x)

f

g

1

(g (y))

(g (x) g (y)) ( + 1)

(3.35)

(by [1], p. 426, Theorem 11.43) =

1 ( )

Z

g(x)

(t

1

g (y))

g(y)

Dg(x)

f

1

g

(t)

Dg(x)

f

1

g

(g (y)) dt

(by [8]) 1 ( )

Z

g(x)

(t

g (y))

g(y)

1

f

Dg(x)

1

g

(t)

Dg(x)

f

g

1

(g (y)) dt (3.36)

(we assume here that Dg(x)

f

g

1

(t)

Dg(x)

f

g

1

(g (y))

1

jt

g (y)j ; (3.37)

11

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for every t; g (y) ; g (x) 2 [g (a) ; g (b)] such that g (x) 1

( ) 1

( )

Z

Z

g(x)

(t

1

g (y))

(t

t

g (y) ;

1

> 0)

g (y)) dt =

g(y)

g(x)

(t

+1

(g (x) g (y)) ( ) ( + 1) 1

g (y)) dt =

g(y)

:

(3.38)

We have proved that kf (x)

f (y)

(A2 (f )) (x; y) (g (x)

(g (x) g (y)) ( ) ( + 1) 1

g (y))k

+1

; (3.39)

8 x; y 2 [a; b] : x > y: (ii) Case of x < y : We have that kf (x)

f (y)

(A2 (f )) (x; y) (g (x)

g (y))k =

kf (x) f (y) + (A2 (f )) (x; y) (g (y) g (x))k = Z g(y) 1 1 (t g (x)) Dg(y) f g 1 (t) dt ( ) g(x) Dg(y) 1 ( ) 1 ( )

Z

f

g(y)

(t

1

g (x))

g(y)

(t

1

g (x))

(g (x))

Dg(y)

g(x)

Z

1

g

Dg(y)

g(x)

f

f

(g (y) g (x)) ( + 1) 1

g 1

g

(t)

=

Dg(y)

(t)

(3.40)

f

Dg(y)

f

1

g

g

1

(g (x)) dt

(g (x)) dt (3.41)

(we assume that Dg(y)

f

g

1

(t)

Dg(y)

f

g

for any t; g (x) ; g (y) 2 [g (a) ; g (b)] : g (y) 2

( )

Z

(g (x))

t

g(y)

(t

1

g (x))

1

g (x) ; (t

2

2

jt

g (x)j ; (3.42)

> 0)

g (x)) dt =

g(x)

2

( )

Z

g(y)

(t

g (x)) dt =

(3.43)

g(x)

(g (y) g (x)) ( ) ( + 1) 2

+1

:

(3.44)

12

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We have proved that kf (x)

f (y)

(A2 (f )) (x; y) (g (x)

g (y))k

(g (y) g (x)) ( ) ( + 1)

+1

2

; (3.45)

8 x; y 2 [a; b] : x < y: Conclusion 3.2 Set kf (x)

f (y)

:= max ( 1 ;

2 ).

Then

(A2 (f )) (x; y) (g (x)

g (y))k

jg (x) g (y)j ( ) ( + 1)

+1

; (3.46)

8 x; y 2 [a; b] ((3.46) is trivially true when x = y). One may choose ( ) < 1: Here again + 1 > 2: Conclusion 3.3 Based on (3.10) and (3.21) of (I), and based on (3.33) and (3.46) of (II), using our numerical results presented earlier, we can solve numerically f (x) = 0: Some examples for g follow: g (x) = ex , x 2 [a; b] g (x) = sin x; g (x) = tan x; where x 2 2 + "; 2

R;

" ; with " > 0 small.

References [1] C.D. Aliprantis and K.C. Border, In…nite Dimensional Analysis, Springer, New York, 2006. [2] S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newtontype method, J. Math. Anal. Applic. 366, 1, (2010), 164-174. [3] G.A. Anastassiou, Strong Right Fractional Calculus for Banach space valued functions, Revista Proyecciones, Vol. 36, No. 1 (2017), 149-186. [4] G.A. Anastassiou, A strong Fractional Calculus Theory for Banach space valued functions, Nonlinear Functional Analysis and Applications (Korea), accepted for publication, 2017. [5] G.A. Anastassiou, Principles of general fractional analysis for Banach space valued functions, submitted for publication (2017).

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[6] I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298, (2004), 374-397. [7] I.K. Argyros, A. Magrenan, Iterative methods and their dynamics with applications, CRC Press, New York, 2017. [8] Bochner integral. Encyclopedia of Mathematics. URL: http://www.ency clopedia ofmath.org/index.php?title=Bochner_integral& oldid=38659. [9] M. Edelstein, On …xed and periodic points under contractive mappings, J. London Math. Soc. 37, (1962), 74-79. [10] J.A. Ezquerro, J.M. Gutierrez, M.A. Hernandez, N. Romero, M.J. Rubio, The Newton method: From Newton to Kantorovich (Spanish), Gac. R. Soc. Mat. Esp. 13, (2010), 53-76. [11] L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, New York, 1982. [12] G.E. Ladas and V. Lakshmikantham, Di¤ erential equations in abstract spaces, Academic Press, New York, London, 1972. [13] A. Magrenan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 215-224. [14] J. Mikusinski, The Bochner integral, Academic Press, New York, 1978. [15] F.A. Potra, V. Ptak, Nondiscrete induction and iterative processes, Pitman Publ., London, 1984. [16] P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26, (2010), 3-42. [17] G.E. Shilov, Elementary Functional Analysis, Dover Publications, Inc., New York, 1996.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 25, NO. 8, 2018

Kantorovich Type Integral Inequalities for Tensor Products of Continuous Fields of Positive Operators, Pattrawut Chansangiam,………………………………………………………1385 Characteristic fuzzy sets and conditional fuzzy subalgebras, G. Muhiuddin and Shuaa Aldhafeeri,…………………………………………………………………………………1398 Some common fixed point theorems for two pairs of self maps in dislocated metric spaces, Peiguang Wang, Akbar Zada, Rahim Shah, and Tongxing Li,……………………………1410 Quadratic 𝜌-functional inequalities in non-Archimedean Banach spaces, Sungsik Yun,…1425

A Fast Inversion Free Iterative Algorithm for Solving, 𝑋 + 𝐴∗ 𝑋 −1 𝐴= I, Duanmei Zhou,..1432

On the Ulam-Hyers Stability of Some Differential Equations involving Hadamard Fractional Derivatives, Xiulan Yu, Zhuoyan Gao, and Mengmeng Li,………………………………1442 Linearly stable periodic solutions for Lagrangian equation, Feng Wang and Shengjun Li,1454 On a system of three max-type nonlinear difference equations, Chang-you Wang, Yu-qian Zhou, Shuang Pan, and Rui Li,…………………………………………………………………1463 The convexity of n-dimensional fuzzy mappings and the saddle point conditions of the fuzzy optimization problems, Hong-Xia Li, Yu-Juan Bai, and Zeng-Tai Gong,………………1480 Complex harmonic poles in the evolution of macromolecules depolymerization, E.F. Doungmo Goufo and S. Mugisha,……………………………………………………………………1490 A fixed point convergence theorem with applications in left multivariate fractional calculus, George A. Anastassiou and Ioannis K. Argyros,…………………………………………1504 Stability of delay-distributed virus dynamics model with cell-to-cell transmission and CTL immune response, A. M. Elaiw, A. A. Raezah, and A. S. Alofi,………………………..1518 Dynamical behavior of HIV-1 infection with saturated virus-target and infected-target incidences and delays, A. M. Elaiw, A. A. Raezah and A. S. Alofi,…………………………………1532 Equations on Banach space valued functions of abstract g-fractional calculus, George A. Anastassiou and Ioannis K. Argyros,………………………………………………………1547