JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 28, 2020

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Volume 28, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

January 2020

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 (six 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], St.Martin Univ.,Olympia,WA,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 $160 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2020 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

Martin Bohner Department of Mathematics and Statistics, Missouri S&T Rolla, MO 65409-0020, USA [email protected] web.mst.edu/~bohner Difference equations, differential equations, dynamic equations on time scale, applications in economics, finance, biology.

George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,U.S.A Tel.901-678-3144 e-mail: [email protected] Approximation Theory, Real Analysis, Wavelets, Neural Networks, Probability, Inequalities.

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

J. Marshall Ash Department of Mathematics De Paul University 2219 North Kenmore Ave. Chicago, IL 60614-3504 773-325-4216 e-mail: [email protected] Real and Harmonic Analysis

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Dumitru Baleanu Department of Mathematics and Computer Sciences, Cankaya University, Faculty of Art and Sciences, 06530 Balgat, Ankara, Turkey, [email protected]

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Dartmouth College 8000 Cummings Hall, Hanover, NH 03755-8000 603-646-3843 (X 3546 Secr.) e-mail:[email protected] Approximation Theory and Neural Networks

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H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany

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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

Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3R4 Canada tel.250-472-5313; office,250-4776960 home, fax 250-721-8962 [email protected] Real and Complex Analysis, Fractional Calculus and Appl., Integral Equations and Transforms, Higher Transcendental Functions and Appl.,q-Series and q-Polynomials, Analytic Number Th.

Svetlozar (Zari) Rachev, Professor of Finance, College of Business, and Director of Quantitative Finance Program, Department of Applied Mathematics & Statistics Stonybrook University 312 Harriman Hall, Stony Brook, NY 11794-3775 tel: +1-631-632-1998, [email protected] Alexander G. Ramm Mathematics Department Kansas State University Manhattan, KS 66506-2602 e-mail: [email protected] Inverse and Ill-posed Problems, Scattering Theory, Operator Theory, Theoretical Numerical Analysis, Wave Propagation, Signal Processing and Tomography

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Fractional: Differential EquationsOperators-Fourier Transforms, Special functions, Approximations, and Applications

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.1, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

GEHRING INEQUALITIES ON TIME SCALES MARTIN BOHNER AND SAMIR H. SAKER Abstract. In this paper, we first prove a new dynamic inequality based on an application of the time scales version of a Hardy-type inequality. Second, by employing the obtained inequality, we prove several Gehring-type inequalities on time scales. As an application of our Gehring-type inequalities, we present some interpolation and higher integrability theorems on time scales. The results as special cases, when the time scale is equal to the set of all real numbers, contain some known results, and when the time scale is equal to the set of all integers, the results are essentially new.

1. Introduction Let I be a fixed cube with sides parallel to the coordinate axes and let f and g be nonnegative measurable functions defined on I. The classical integral H¨older inequality Z  1 Z 1 Z p q p q |f (x)g(x)|dx ≤ |f (x)| dx |g(x)| dx , I

I

I

where 1/p + 1/q = 1, shows that there is a natural scale of inclusion for the Lp (I)-spaces, when the underlying space I has a finite measure |I|. In 1972, Muckenhoupt [14] proved the first simplest reverse integral (mean) inequality, which can be considered as a reverse inclusion, of the form Z 1 w(x)dx ≤ κ essinf x∈I w(x), (1.1) |I| I where w is a nonnegative measurable function defined on I. A function verifying (1.1) is called an A1 -weight Muckenhoupt function. In [14] (see also [13]), it is proved that any A1 -weight Muckenhoupt function belongs to Lr (I), for 1 ≤ r < s and s depending on κ and the dimension of the space. In 1973, Gehring [8] extended the result of Muckenhoupt for reverse mean inequalities. We say that w satisfies a Gehring condition (or a reverse H¨older inequality) if there exists p > 1 and a constant κ > 0 such that for every cube I with sides parallel to the coordinate axes, we have  Z 1/p Z 1 κ p w (x)dx ≤ w(x)dx. |I| I |I| I In this case we write w ∈ RHp . A well known result obtained by Gehring [8] states that if w ∈ RHp , then w satisfies a higher integrability condition, namely 1991 Mathematics Subject Classification. 26D07, 42B25, 42C10, 34N05. Key words and phrases. Gehring’s inequality, reversed H¨ older inequality, Hardy-type inequality, interpolation, higher integrability, time scales. 1

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MARTIN BOHNER AND SAMIR H. SAKER

for sufficiently small ε > 0, q = p + ε, we have for any cube I,



1 |I|

Z

q

1/q

 ≤

w (x)dx I

κ |I|

Z

p

w (x)dx

1/p .

I

In other words, Gehring’s result states that w ∈ RHp implies that there exists ε > 0 such that w ∈ RHp+ε . The proof of Gehring’s inequality is based on the use of the Calder´ on–Zygmund decomposition and the scale structure of Lp spaces. In [12], the author extended Gehring’s inequality by means of connecting it to the real method of interpolation by considering maximal operators, and via rearrangements reinterpreted the underlying estimates through the use of Kfunctionals. This technique allowed to quantify in a precise way, via reiteration, how Calder´on–Zygmund decompositions have to be reparameterized in order to characterize different Lp -spaces. Reverse integral inequalities (cf. [8, 9]) and its many variants and extensions are important in qualitative analysis of nonlinear PDEs, in the study of weighted norm inequalities for classical operators of harmonic analysis, as well as in functional analysis. These inequalities also appear in different fields of analysis such as quasiconformal mappings, weighted Sobolev embedding theorems, and regularity theory of variational problems (see [11]). In recent years, the study of dynamic inequalities on time scales has received a lot of attention. For details, we refer to the books [2, 3, 5, 6] and the recent paper [1] and the references cited therein. The general idea in studying dynamic inequalities on time scales is to prove a result for an inequality, where the domain of the unknown function is a so-called time scale T, which is an arbitrary nonempty closed subset of the real numbers R. This idea goes back to its founder Stefan Hilger [10]. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus, i.e., when T = R, T = N, and T = q N0 = {q t : t ∈ N0 } with q > 1. The study of dynamic inequalities on time scales helps avoid proving results twice – once for differential inequalities and once again for difference inequalities. Following this trend and to develop the study of dynamic inequalities on time scales, we aim in this paper to prove Gehring-type inequalities on time scales, which contain the classical integral inequalities of Gehring’s type and their discrete versions as special cases. We believe that the reverse dynamic inequalities on time scales will be, just like in the classical case, similarly important for the analysis of dynamic equations on time scales. The rest of the paper is organized as follows: In Section 2, we recall some definitions and notations related to time scales which will be used throughout the paper. Section 3 features some auxiliary results, in particular, a time scales version of Hardy’s inequality. In Section 4, we present the proofs of our Gehringtype inequalities on time scales and give some interpolation results as well as some higher integrability theorems for monotone nonincreasing functions on time scales, see Section 5. As special cases, we offer discrete versions of the Gehring inequalities. To the best of the authors’ knowledge, nothing is known regarding the discrete analogues of Gehring inequalities or even their extensions, and thus the presented discrete inequalities are essentially new.

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GEHRING INEQUALITIES ON TIME SCALES

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2. Time Scales Preliminaries We assume that the reader is familiar with time scales as presented in the monographs [5, 6]. For concepts concerning general measure and integration on time scales, see [6, Chapter 5] and [4, 7]. Here, we only state four facts that are essentially used in the proofs of our results. For a function f : T → R, where T is a time scale, we denote the delta derivative by f ∆ and the forward shift by f σ = f ◦σ, where σ is the time scales jump operator. The time scales product rule says that for two differentiable functions f and g, the product f g is differentiable with (f g)∆ = f ∆ g + f σ g ∆ .

(2.1)

On the other hand, the time scales integration by parts rule says that for two integrable functions f, g : T → R and a, b ∈ T, we have Z b Z b f (t)g ∆ (t)∆t. f ∆ (t)g(σ(t))∆t = f (b)g(b) − f (a)g(a) − (2.2) a

a

We also need the time scales chain rule which says that if f : R → R is continuously differentiable and g : T → R is delta differentiable, then f ◦ g : T → R is delta differentiable with Z 1 (2.3) (f ◦ g)∆ = g ∆ f 0 (hg σ + (1 − h)g)dh. 0

Finally, we need the time scales H¨ older inequality which says that for two nonnegative integrable functions f, g : T → R and a, b ∈ T and p, q > 1 with 1/p + 1/q = 1, we have Z b 1/p Z b 1/q Z b p q (2.4) f (t)g(t)∆t ≤ f (t)∆t f (t)∆t , a

a

a

and p, q are called the corresponding exponents. Throughout this paper, we assume that the functions in the statements of the theorems are nonnegative and rd-continuous functions, delta differentiable, locally delta integrable, and the integrals considered are assumed to exist (finite, i.e., convergent). 3. Auxiliary Results In this section, we give some auxiliary results that are used in the proofs of our main results. Definition 3.1. Throughout this paper, we suppose that T is a time scale with 0 ∈ T, and we let T > 0 with T ∈ T. For any function f : (0, T ] → R which is ∆-integrable, nonnegative, and nonincreasing, we define the average function Af : (0, T ] → R by Z 1 t f (s)∆s for all t ∈ (0, T ]. (3.1) Af (t) := t 0 Some simple facts about Af are given next. Lemma 3.2. If f : (0, T ] → R is ∆-integrable, nonnegative, and nonincreasing, then (3.2)

Af ≥ f.

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MARTIN BOHNER AND SAMIR H. SAKER

Proof. Due to 1 Af (t) = t

t

Z 0

1 f (s)∆s ≥ t

Z

t

f (t)∆s = f (t), 0

(3.2) follows immediately.



Lemma 3.3. If f : (0, T ] → R is ∆-integrable, nonnegative, and nonincreasing, then so is Af . Proof. In this proof, we write F = Af for brevity. We show that F inherits the nonincreasing nature of f . Let t1 < t2 . Then  Z t1 Z t2 Z 1 1 t1 f (s)∆s f (s)∆s − f (s)∆s + F (t1 ) − F (t2 ) = t1 0 t2 0 t1  Z  Z t2 t2 − t1 1 t1 1 = f (s)∆s − f (s)∆s t2 t1 0 t2 − t1 t1  Z  Z t2 t2 − t1 1 t1 1 ≥ f (t1 )∆s − f (t1 )∆s = 0, t2 t1 0 t2 − t1 t1 completing the proof.



Now we present a Hardy inequality (see also [3, Corollary 1.5.1]) which, for completeness, we prove in our special setting. Theorem 3.4. If q > 1 and f : (0, T ] → R is ∆-integrable, nonnegative, and nonincreasing, then q  q Af q . (3.3) A [(Af )σ ]q ≤ q−1 Proof. In this proof, we write F = Af for brevity. Using Lemma 3.3, the chain rule shows that Z 1 q ∆ (2.3) ∆ (F ) = qF (hF σ + (1 − h)F )q−1 dh 0 (3.4) Z 1 ≤ qF ∆ (hF σ + (1 − h)F σ )q−1 dh = qF ∆ (F σ )q−1 . 0

Moreover, since Z tF (t) =

t

f (s)∆s, 0

the product rule yields (3.5)

(2.1)

f (t) = F (σ(t)) + tF ∆ (t).

Now, putting u(t) = t and v(t) = F q (t), we use integration by parts to find Z t Z t q (F (σ(s))) ∆s = u∆ (s)v(σ(s))∆s 0 0 Z t (2.2) = u(t)v(t) − lim u(s)v(s) − u(s)v ∆ (s)∆s s→0+ 0 Z t = tF q (t) − sv ∆ (s)∆s 0

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GEHRING INEQUALITIES ON TIME SCALES

Z ≥

5

t

sv ∆ (s)∆s

− 0

(3.4)

≥

t

Z

sF ∆ (s)F q−1 (σ(s))∆s

−q 0

(3.5)

=

t

Z

[f (s) − F (σ(s))] F q−1 (σ(s))∆s Z t Z t q−1 (F (σ(s)))q ∆s f (s)F (σ(s))∆s + q −q

−q

0

=

0

0

so that, by using H¨older’s inequality with exponents q and q/(q − 1), Z t Z t q (q − 1) (F (σ(s))) ∆s ≤ q f (s)(F (σ(s)))q−1 ∆s 0

0

Z

(2.4)

≤

q

1/q Z

t q

(f (s)) ∆s

t q

(q−1)/q

(F (σ(s))) ∆s

,

0

0

resulting in (3.3).



In the main results of this paper, we assume that there exists a constant λ ≥ 1 such that (3.6)

σ(t) ≤ λt

for all

t ∈ T.

We now apply the time scales chain rule to obtain some estimates that will be used later. Lemma 3.5. Let x(t) = t. If 0 < γ < 1, then
∆ 1 − γ (3.7) x1−γ ≥ , σγ and if γ > 1 and (3.6) holds, then
∆ (1 − γ)λγ (3.8) x1−γ ≥ . σγ Proof. By the chain rule, we obtain Z 1
dh (2.3) 1−γ ∆ ∆ x (t) = (1 − γ)x (t) γ 0 (hx(σ(t)) + (1 − h)x(t)) Z 1 dh = (1 − γ) . (hσ(t) + (1 − h)t)γ 0 Thus, if 0 < γ < 1, then Z 1
dh 1−γ 1−γ ∆ x (t) ≥ (1 − γ) = , γ (σ(t))γ 0 (hσ(t) + (1 − h)σ(t)) which is (3.7), and if γ > 1 and (3.6) holds, then Z 1
∆ dh 1 − γ (3.6) (1 − γ)λγ x1−γ (t) ≥ (1 − γ) = ≥ , γ tγ (σ(t))γ 0 (ht + (1 − h)t) which is (3.8).



Lemma 3.6. If F is nonnegative and nondecreasing and γ > 1, then (3.9)

(F γ )∆ ≥ γF ∆ F γ−1 .

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Proof. Again we apply the chain rule to see that Z 1 (2.3) (hF σ + (1 − h)F )γ−1 dh (F γ )∆ = γF ∆ 0 Z 1 (hF + (1 − h)F )γ−1 dh ≥ γF ∆ =

∆

γF F

0 γ−1

,

which shows (3.9).

 4. Main Results

RT We say that f : (0, T ] → R belongs to Lp∆ (0, T ] provided 0 |f (t)|p ∆t < ∞. The first theorem will be used later in the proof of the Gehring inequality. Theorem 4.1. If f ∈ Lp∆ (0, T ] for p > 1 is nonnegative and nonincreasing, then, for any q ∈ (0, p), we have (4.1)

Af p ≤

q (p − q)λp/q A [(Af q )σ ]p/q . [Af q ]p/q + p p

Proof. From the Hardy inequality, see (3.3), we see that the second integral on the right-hand side of (4.1) is finite. Now, we consider this integral. Then, for 0 < q < p, we put Z t p γ = > 1 and F (t) = f q (s)∆s. q 0 Using the notation from Lemma 3.5, we have #p/q Z " Z σ(s) 1 (p − q)λp/q t q f (τ )∆τ ∆s pt σ(s) 0 0 #γ Z " Z σ(s) 1 (γ − 1)λγ t = f q (τ )∆τ ∆s γt σ(s) 0 0  Z  (γ − 1)λγ t F (σ(s)) γ = ∆s γt σ(s) 0 Z 1 t γ (1 − γ)λγ F (σ(s)) = − ∆s γt 0 (σ(s))γ Z (3.8)
∆ 1 t γ ≥ − F (σ(s)) x1−γ (s)∆s γt 0 Z F γ (s)x1−γ (s) F γ (t)x1−γ (t) 1 t γ ∆ (2.2) = lim − + (F ) (s)x1−γ (s)∆s γt γt γt 0 s→0+       Z 1 t 1−γ γ ∆ 1 F (s) γ 1 F (t) γ s (F ) (s)∆s + lim s − = γt 0 γt s→0+ s γ t   Z γ t ∆ γ−1 (3.9) 1 γF (s)F (s) 1 F (t) ≥ ∆s − γ−1 γt 0 s γ t Z t 1 1 = f q (s) [Af q (s)]γ−1 ∆s − [Af q (t)]γ t 0 γ

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≥ = =

7

Z 1 t q 1 f (s) [f q (s)]γ−1 ∆s − [Af q (t)]γ t 0 γ Z 1 1 t q [f (s)]γ ∆s − [Af q (t)]γ t 0 γ q Af p (t) − [Af q (t)]p/q p

from which (4.1) follows.



Now, we are ready to state and prove our first time scales version of Gehring’s mean inequality for monotone functions. Theorem 4.2 (Gehring Inequality I). Assume (3.6). If f ∈ Lq∆ (0, T ] for q > 1 is nonnegative and nonincreasing such that Af q ≤ κ [Af ]q

(4.2) then f ∈

Lp∆ (0, T ]

for some

κ > 0,

for any p > q satisfying

(4.3)

qκp/q

κ ˜ :=

p − (p − q)(λκ)p/q



p p−1

p > 0,

and in this case, Af p ≤ κ ˜ [Af ]p .

(4.4)

Proof. Assuming (4.2), we find  Z t p/q Z (4.1) q 1 1 t p q f (s)∆s ≤ f (s)∆s t 0 p t 0 #p/q Z " Z σ(s) (p − q)λp/q t 1 + f q (τ )∆τ ∆s pt σ(s) 0 0 " #p  Z t p Z Z σ(s) (4.2) q (p − q)λp/q t p/q 1 p/q 1 ≤ κ f (s)∆s + κ f (τ )∆τ ∆s p t 0 pt σ(s) 0 0  Z t p  p Z t (3.3) q (p − q)(λκ)p/q p p/q 1 ≤ κ f (s)∆s + f p (s)∆s p t 0 pt p−1 0 so that, due to (4.3), 1 t

Z 0

t

 Z t p 1 f (s)∆s ≤ κ ˜ f (s)∆s , t 0 p

from which (4.4) follows.



As a special case of Theorem 4.2 when T = R, we get the classical Gehring inequality (see Section 1) with λ = 1. In the case when T = N, we have the following result with λ = 2. Corollary 4.3 (Discrete Gehring Inequality I). Let q > 1 and {an }n∈N0 be a nonnegative and nonincreasing sequence such that !q n−1 n−1 1X q 1X ai ≤ κ ai . n n i=0

i=0

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MARTIN BOHNER AND SAMIR H. SAKER

Then, for p > q, we have n−1

1X p ai ≤ κ ˜ n i=0

n−1

1X ai n

!p ,

i=0

provided qκp/q

κ ˜ :=

p − (p − q)(2κ)p/q



p p−1

p > 0.

It is natural to ask what happens if in (4.4) we fix p > 1 and consider the improvement to this inequality that would result from lowering the exponent on the right-hand side. The following result gives an answer. Theorem 4.4. Suppose that the assumptions of Theorem 4.2 hold and define κ ˜ as in (4.3). Then, for all 0 < r < 1, we have (4.5)

Af p ≤ κ [Af r ]p/r ,

where

κ := κ ˜ 1/θ

with

θ :=

1− 1 r

−

1 p 1. p

Proof. Note first that θ ∈ (0, 1) and 1−θ θ + = 1. p r Then, by the H¨older inequality with exponents p/(1 − θ) and r/θ, we have  Z t 1/p (4.4) Z κ ˜ 1/p t 1 p ≤ f (s)∆s f (s)∆s t 0 t 0 Z κ ˜ 1/p t 1−θ f (s)f θ (s)∆s = t 0 Z t (1−θ)/p Z t θ/r (2.4) κ ˜ 1/p p r ≤ f (s)∆s f (s)∆s t 0 0  Z t (1−θ)/p  Z t θ/r 1 1/p 1 p r f (s)∆s f (s)∆s = κ ˜ t 0 t 0 so that, by dividing, we find  Z t θ/p  Z t θ/r 1 p 1/p 1 r f (s)∆s ≤κ ˜ f (s)∆s , t 0 t 0 i.e., (4.5).



By Theorem 4.4, under the assumptions of Theorem 4.2, if f ∈ Lr∆ (0, T ] for 0 < r < 1, then f ∈ Lp∆ (0, T ] for p > 1. But in the general case when p 6= r, Lp∆ (0, T ] neither includes nor is included in Lr∆ (0, T ]. The following theorem gives some results for Lp∆ (0, T ]-interpolation. Theorem 4.5. Suppose that 0 < p0 < p1 < ∞ and that 0 < θ < 1. (i) If p = (1 − θ)p0 + θp1 and f ∈ Lp∆0 (0, T ] ∩ Lp∆1 (0, T ], then f ∈ Lp∆ (0, T ] and Af p ≤ [Af p0 ]1−θ [Af p1 ]θ .

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GEHRING INEQUALITIES ON TIME SCALES

(ii) If p =

1 1−θ + pθ p0 1

9

and f ∈ Lp∆0 (0, T ] ∩ Lp∆1 (0, T ], then f ∈ Lp∆ (0, T ] and Af p ≤ [Af p0 ](1−θ)p/p0 [Af p1 ]θp/p1 .

Proof. For (i), we apply the H¨ older inequality with exponents 1/(1 − θ) and 1/θ to see that Z Z 1 t p 1 t (1−θ)p0 f (s)∆s = f (s)f θp1 (s)∆s t 0 t 0 1−θ  Z t θ  Z t (2.4) 1 1 f p0 (s)∆s f p1 (s)∆s , ≤ t 0 t 0 which shows (i). For (ii), we apply the H¨older inequality with exponents 1/(1−γ) and 1/γ, where θp (1 − θ)p γ := so that 1 − γ = , p1 p0 to see that Z Z 1 t (1−θ)p 1 t p f (s)∆s = f (s)f θp (s)∆s t 0 t 0 1−γ  Z t γ  Z t (2.4) 1 1 f (1−θ)p/(1−γ) (s)∆s f θp/γ (s)∆s ≤ t 0 t 0  Z t (1−θ)p/p0  Z t θp/p1 1 1 p0 p1 = f (s)∆s f (s)∆s , t 0 t 0 which shows (ii).



In the following, we give a new proof of Gehring’s mean inequality on time scales. The inequality will be proved by using a condition similar to the condition (1.1) due to Muckenhoupt. In fact, we do not assume that the reverse H¨older inequality holds. Theorem 4.6 (Gehring Inequality II). Assume (3.6). If f : (0, T ] → T is nonnegative and nonincreasing such that Af σ ≤ νf

(4.6) then f ∈ (4.7)

for some

ν > 1,

Lp∆ (0, T ]

for p ∈ [1, α/(α − 1)), where α = λν, and we have α A(f p )σ ≤ ν˜ [Af σ ]p , where ν˜ := > 0. α − p(α − 1)

Proof. For this proof, we put Z t F (t) := f σ (s)∆s,

l(t) = log(t),

L(t) = log(F (t)).

0

By the chain rule, we get 1 ∆ l (t) α

(2.3)

=

(3.6)

≤ ≤

1 λν

Z

1 λν

Z

1

0

dh hσ(t) + (1 − h)t

1

dh

hσ(t) + (1 − h) σ(t) λ 1 λ 1 · = λν σ(t) νσ(t) 0

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MARTIN BOHNER AND SAMIR H. SAKER (4.6)

≤ = ≤ (2.3)

=

f (σ(t)) F ∆ (t) = F (σ(t)) F (σ(t)) Z 1 dh F ∆ (t) 0 hF (σ(t)) + (1 − h)F (σ(t)) Z 1 dh F ∆ (t) 0 hF (σ(t)) + (1 − h)F (t) L∆ (t),

and hence, by integrating,  1/α   t 1 1 F (t) log = l(t) − l(σ(s)) ≤ L(t) − L(σ(s)) = log σ(s) α α F (σ(s)) so that 1 f (σ(s)) ≤ σ(s)

σ(s)

Z 0

F (σ(s) f (σ(τ ))∆τ = ≤ σ(s)



σ(s) t

1/α

F (t) , σ(s)

and by integrating again, putting γ := p(1 − 1/α) ∈ (0, 1), and using the notation from Lemma 3.5, we obtain Z Z F p (t) t ∆s 1 t p f (σ(s))∆s ≤ 1+p/α p(1−1/α) t 0 t 0 (σ(s)) Z t (3.7)
∆ F p (t) x1−γ (s)∆s ≤ 1+p/α (1 − γ)t 0   F (t) p t1−γ F p (t) 1 , = = 1−γ t (1 − γ)t1+p/α proving (4.7).



As a special case of Theorem 4.6 when T = N, we have the following result. Corollary 4.7 (Discrete Gehring Inequality II). Let {an }n∈N0 be a nonnegative and nonincreasing sequence. If there exists a constant ν > 1 such that n

1X ai ≤ νan , n i=1

then, for p ∈ [1, α/(α − 1)], where α = 2ν, we have " n #p n 1X p 1X α ai ≤ ν˜ ai , where ν˜ := . n n α − p(α − 1) i=1

i=1

5. Higher Integrability In the following, as an application of Gehring’s inequality (4.7), we prove a higher integrability theorem for monotone nonincreasing functions. First notice that for all nonnegative and nonincreasing functions f ∈ Lq∆ (0, T ] with q > 1, we always have Z Z Z 1 t q 1 t q−1 f q−1 (t) t q (5.1) Af (t) = f (s)∆s = f (s)f (s)∆s ≥ f (s)∆s. t 0 t 0 t 0

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11

Let us now consider the class of nonnegative and nonincreasing functions f ∈ Lq∆ (0, T ] that satisfy the reverse of (5.1), namely Af q ≤ ηf q−1 Af

(5.2)

for some

η > 1.

Theorem 5.1. Assume (3.6). If f ∈ Lq∆ (0, T ] for q > 1 is nonnegative and nonincreasing such that (5.2) holds, then f ∈ Lp∆ (0, T ] for p ∈ [q, q +c], c ∈ (q, η), and we have (5.3) A(f p )σ ≤ η˜ [Af q ]p/q ,

where

η˜ :=

ληq 1+p/q ληq − pq (ληq − 1)

with

ηq =

ηq . q−1

Proof. In this proof, we write F = Af q for brevity. By using the H¨older inequality with exponents q/(q − 1) and q, we obtain Z Z t Z σ(s) (5.2) η 1 1 t q−1 F (σ(s))∆s ≤ (f (σ(s))) · f (τ )∆τ ∆s t 0 t 0 σ(s) 0 #1/q !q Z t (q−1)/q "Z t Z σ(s) (2.4) η 1 ≤ (f (σ(s)))q ∆s f (τ )∆τ ∆s t 0 σ(s) 0 0 Z t (q−1)/q Z t 1/q (3.3) ηq q q ≤ (f (s)) ∆s (f (s)) ∆s (q − 1)t 0 0 Z ηq t q = f (s)∆s = ηq F (t), t 0 i.e., AF σ ≤ ηq F.

(5.4)

Since F is also nonnegative and nonincreasing (see Lemma 3.3), it satisfies the assumptions of Theorem 4.6, and thus r  Z t Z 1 t 1 (5.5) [F (σ(s))]r ∆s ≤ η˜q F (σ(s))∆s t 0 t 0 with αq η˜q = αq − r(αq − 1)

and

αq = ληq

for

  αq p r = ∈ 1, . q αq − 1

Noting that (5.6)

F (t) =

1 t

t

Z

f q (s)∆s ≥ f q (t),

0

we obtain 1 t

Z

t p

(f (σ(s))) ∆s

=

0 (5.6)

≤ (5.5)

≤ (5.4)

≤

21

1 t

Z

1 t

Z

t

(f q (σ(s)))r ∆s

0 t

(F (σ(s)))r ∆s  Z t r 1 σ η˜q F (s)∆s t 0 0

η˜q ηq r [F (t)]r = η˜ [F (t)]r

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MARTIN BOHNER AND SAMIR H. SAKER

=

p/q  Z t 1 q f (s)∆s η˜ , t 0

proving (5.3).



In Theorem 5.1, if T = R, then we have that σ(t) = t, αq = ηq , and we get the following result. Corollary 5.2. Let η > 1 and q > 1. Then every nonnegative nonincreasing function f satisfying Z t Z t q q−1 f (x)dx ≤ ηf (t) f (x)dx 0

0

belongs to Lp∆ (0, T ](0, T ] for p ∈ [q, q + c] and c ∈ (q, η), and we have  Z t p/q Z 1 t p 1 q f (x)dx ≤ η˜ f (x)dx , t 0 t 0 where p  η˜ :=

+1 q ηq q−1 ηq p ηq q−1 − q ( q−1 −

1)

.

In Theorem 5.1, if T = N, then we have that σ(t) = t + 1, and by choosing λ = 2, we get the following result. Corollary 5.3. Let η > 1 and q > 1. Suppose {an }n∈N0 is a nonnegative and nonincreasing sequence satisfying n−1 X

aqi ≤ ηaq−1 n

i=0

n−1 X

ai .

i=0

Then, for p ∈ [q, q + c], c ∈ (q, η), we have n−1

1X p ai ≤ η˜ n i=0

where

!p/q ,

i=0

 p +1 q ηq q−1 ηq ηq 2 q−1 − pq (2 q−1 − 2

η˜ :=

n−1

1X q ai n



1)

.

References [1] Ravi P. Agarwal, Martin Bohner, and Samir Saker. Dynamic Littlewood-type inequalities. Proc. Amer. Math. Soc., 143(2):667–677, 2015. [2] Ravi P. Agarwal, Donal O’Regan, and Samir Saker. Dynamic inequalities on time scales. Springer, Cham, 2014. [3] Ravi P. Agarwal, Donal O’Regan, and Samir H. Saker. Hardy type inequalities on time scales. Springer, Cham, 2016. [4] Ravi P. Agarwal, Victoria Otero-Espinar, Kanishka Perera, and Dolores R. Vivero. Basic properties of Sobolev’s spaces on time scales. Adv. Difference Equ., 14 pages, Art. ID 38121, 2006. [5] Martin Bohner and Allan Peterson. Dynamic equations on time scales. Birkh¨ auser Boston, Inc., Boston, MA, 2001. An introduction with applications. [6] Martin Bohner and Allan Peterson. Advances in dynamic equations on time scales. Birkh¨ auser Boston, Inc., Boston, MA, 2003.

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[7] Alberto Cabada and Dolores R. Vivero. Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral: application to the calculus of ∆-antiderivatives. Math. Comput. Modelling, 43(1-2):194–207, 2006. [8] Frederick W. Gehring. The Lp -integrability of the partial derivatives of a quasiconformal mapping. Acta Math., 130:265–277, 1973. [9] Frederick W. Gehring. The Lp -integrability of the partial derivatives of quasiconformal mapping. Bull. Amer. Math. Soc., 79:465–466, 1973. [10] Stefan Hilger. Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math., 18(1-2):18–56, 1990. [11] Tadeusz Iwaniec. On Lp -integrability in PDEs and quasiregular mappings for large exponents. Ann. Acad. Sci. Fenn. Ser. A I Math., 7(2):301–322, 1982. [12] Mario Milman. A note on Gehring’s lemma. Ann. Acad. Sci. Fenn. Math., 21(2):389–398, 1996. [13] Benjamin Muckenhoupt. Hardy’s inequality with weights. Studia Math., 44:31–38, 1972. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. [14] Benjamin Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972. Missouri University of Science and Technology, Department of Mathematics and Statistics, Rolla, Missouri 65409-0020, USA E-mail address: [email protected] Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt E-mail address: [email protected]

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Basin of attraction of the fixed point and period-two solutions of a certain anti-competitive map Emin Beˇso, Senada Kalabuˇsi´c and Esmir Pilav Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina [email protected]; [email protected]; [email protected] Naida Muji´c Faculty of Electrical Engineering, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina [email protected]; February 26, 2019 Abstract We investigate the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the difference equation xn+1 =

γx2n−1 Cx2n−1 + xn

where the parameters γ, C are positive numbers and the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that x−1 + x0 > 0. We determine the basin of attraction of fixed point and period-two solutions. The associated map is not defined at the (0, 0). However, we show that there exist period two solutions on the axis that are locally asymptotically stable and two continuous invariant curves passing through the point (0, 0), which are boundaries of the basins of attractions of these period two solutions, such that every solution starting on these two curves or in the region between these two curves is attracted to the point (0, 0).

Key Words: Basin of attraction; difference equation; global attractivity; global stable manifold; monotonicity; MSC(2010): Primary: 39A10,39A23, 39A30; Secondary: 37E05

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February 26, 2019

1

2

Introduction and Preliminaries

In this paper we consider the following quadratic rational difference equation of second order xn+1 =

γx2n−1 Cx2n−1 + xn

(1)

We assume that γ, C > 0 and initial conditions x−1 , x0 are positive real numbers, such that x0 + x−1 > 0. Notice that the map associated to this equation is not defined at the point (0, 0). The second iterate of the map associated to Equation (1) is competitive map. We call such map anti-competitive. See [7, 8]. Theory of competitive systems and maps in the plane have been extensively developed and main results are given in [2, 6, 11, 13, 14]. Equation (1) is a special case of the difference equation xn+1 =

βxn xn−1 + γx2n−1 + δxn , Bxn xn−1 + Cx2n−1 + Dxn

n = 0, 1, 2, . . .

(2)

where the parameters β, γ, δ, B, C, D are nonnegative numbers which satisfy B + C + D > 0 and the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that Bxn xn−1 + Cx2n−1 + Dxn > 0 for all n ≥ 0. Locally stability of the equilibrium points of (2) has been studied in [10]. In this paper we describe global behavior of solutions of Equation (1). Equation (1) is related to the difference equation xn+1 =

γxn−1 , n = 0, 1, . . . Bxn + Cxn−1

(3)

where the parameters γ, B and C are positive real numbers and the initial conditions x−1 , x0 are arbitrary nonnegative numbers such that x−1 + x0 > 0, see [1, 9]. As we will see in this paper Equation (1) has very different behaviour than Equation (3) showing that introduction of quadratic terms can significantly change behaviour of the equation. We prove that parametric space splits into four regions given by 0 < γ < 1, 1 < γ < 3, γ = 3 and γ > 3. By using results from [3, 13] we obtain global result in each of these four regions, different than global results for Equation (3). For example in Section 3 we show that there exist two increasing continues invariant curves passing through the point (0, 0) which are the boundaries of basins of attractions of the period-two solutions such that every solution that starts on these two curves or in the region between these two curves is attracted to the point (0, 0). We now present some basic notation about competitive map in the plane. Consider a first order system of difference equations of the form  xn+1 = f (xn , yn ) , n = 0, 1, 2, . . . , (x−1 , x0 ) ∈ I × I (4) yn+1 = g(xn , yn ) where f, g : I × I → I are continuous functions on an interval I ⊂ R, f (x, y) is non-decreasing in x and nonincreasing in y, and g(x, y) is non-increasing in x and non-decreasing in y. Such system is called competitive. One may associate a competitive map T to a competitive system (4) by setting T = (f, g) and considering T on B = I × I. A point x ∈ B is a fixed point of T if T (x) = x, and a minimal period-two point if T 2 (x) = x and T (x) 6= x. A period-two point is either a fixed point or a minimal period-two point. In a similar fashion one can define a minimal period p point. The orbit of x ∈ B is the sequence {T ` (x)}∞ `=0 . A minimal period-two orbit is an orbit {x` }∞ `=0 for which x0 6= x1 and x0 = x2 . The basin of attraction of a fixed point x is the set of all y such that T n (y) → x. A fixed point x is a global attractor on a set A if A is a subset of the basin of attraction of x. A fixed point x is a saddle point if T is differentiable at x, and the eigenvalues of the Jacobian matrix of T at x are such that one of them lies in the interior of the unit circle in R2 , while the other eigenvalue lies in the exterior of the unit circle. If T = (T1 , T2 ) is a map on R ⊂ R2 , define the sets RT (−, +) := {(x, y) ∈ R : T1 (x, y) ≤ x, T2 (x, y) ≥ y } and RT (+, −) := {(x, y) ∈ R : T1 (x, y) ≥ x, T2 (x, y) ≤ y }. If v = (u, v) ∈ R2 , we denote with Q` (v), ` ∈ {1, 2, 3, 4}, the four quadrants in R2 relative to v, i.e., Q1 (v) = {(x, y) ∈ R2 : x ≥ u, y ≥ v}, Q2 (v) = {(x, y) ∈ R2 : x ≤ u, y ≥ v}, and so on. Define the South-East partial order se on R2 by (x, y) se (s, t) if and only if x ≤ s and y ≥ t. Similarly, we define the North-East partial order ne on R2 by (x, y) ne (s, t) if and only if x ≤ s and y ≤ t. A stronger inequality

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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. A map T is competitive if T (x) se T (y) whenever x se y, and T is strongly competitive if x se y implies T (x) − T (y) ∈ {(u, v) : u > 0, v < 0}. If T is differentiable, a sufficient condition for T to be strongly competitive is that the Jacobian matrix of T at any x ∈ B has the sign configuration   + − . − + For additional definitions and results (e.g., repeller, hyperbolic fixed points, stability, asymptotic stability, stable and unstable manifolds) see [6, 14] for competitive maps, and [9, 11] for difference equations. This paper is structured as follows. In Section 2 we prove linearized stability results. Depending on parameter γ we determine the nature of equilibrium point and period-two solutions and then we prove convergence result for period-two solution. In Section 3 we describe completely global behaviour of Equation (1).

2

Linearized stability analysis and convergence result

In this section we prove linearized stability and convergence results for Equation(1). Theorem 1 If γ > 1 then Equation (1) has the unique equilibrium point x ¯ which is given by x ¯=

γ−1 C

and x ¯ is a) locally asymptotically stable if γ > 3. b) a non-hyperbolic point if γ = 3; c) a saddle point if 1 < γ < 3; Proof. The proof follows from the well known linearized stability theorem, see [10]. 2 Theorem 2 For the Equation (1) the following holds: (a) For all values of parameters Equation (1) has prime period-two solution n γo 0, C which is locally asymptotically stable. (b) If γ > 3 then Equation (1) has prime period-two solution ( ) p p γ − (γ − 3)(γ + 1) + 1 γ + (γ − 3)(γ + 1) + 1 , 2C 2C which is a saddle point. Proof. (a) It is easy to check that {0, Cγ } is period two solution for all values of parameters. This period two solution always exists.

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(b) Assume that γ > 3. If . . . a, b, a, a, b, . . . is a period two solution, then this solution satisfies the following system of algebraic equations γb2 Cb2 + a γa2 . Ca2 + b

b = a =

Straightforward calculations shows that under the condition γ > 3 the unique solution of this system is given by p p γ − (γ − 3)(γ + 1) + 1 γ + (γ − 3)(γ + 1) + 1 a= , b= . 2C 2C By using linearized stability theorem, it is easy to see that this period two solution is a saddle point, see [10]. Note that it is not possible to obtain period two solution {0, Cγ } by solving the previous system of algebraic equations. 2 Now, we show that every solution of Equation (1) converges to a period-two solution (not necessarily minimal). Let F (u, v) =

γv 2 . Cv 2 + u

It is easy to see that Fx0 = −

γv 2 (Cv 2

+ u)

2

and Fy0 =

2γuv 2

(Cv 2 + u)

Set un = xn−1 and vn = xn for n = 0, 1, . . .

(5)

We can rewrite Equation (1) in the equivalent form: un+1 vn+1

= vn =

(6)

γu2n Cu2n +

vn

for n = 0, 1, . . . . Let T be the map associated to Equation (1):  T (u, v) = (v, F (v, u)) = v,

γu2 Cu2 + v

 .

(7)

then (un+1 , vn+1 ) = T (un , vn )

(8)

It is easy to see that 2

T (u, v) = T (T (u, v)) = (T21 (u, v), T22 (u, v))

!
γv 2 Cu2 + v γu2 , Cu2 + v C 2 u2 v 2 + Cv 3 + γu2

from which it follows that (u2n+2 , v2n+2 ) = T 2 (u2n , v2n )

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which is equivalent to (x2n+1 , x2n+2 ) = T 2 (x2n−1 , x2n ). The Jacobian matrix of the map T has the form: JT (u, v) =

!

0

1

2uvγ (cu2 +v)2

γ − (cuu2 +v) 2

(10)

2

The determinant of (10) is given by detJT (u, v) = −

2γuv (cu2 + v)

The Jacobian matrix of the map T 2 has the form:  JT 2 (u, v) = 

2uvγ (Cu2 +v)2 2uv 3 γ 2 − (γu2 +Cv 2 (Cu2 +v))2

(11)

2



2

γ − (Cuu2 +v) 2 u2 v (2Cu2 +3v )γ 2 (γu2 +V v 2 (Cu2 +v))

(12)

 2

The determinant of (12) is given by detJT 2 (u, v) =

4γ 3 u3 v 2

(13)

2

(Cu2 + v) (Cv 2 (Cu2 + v) + γu2 )

The equilibrium curves of the map T 2 are given by   C1 := (x, y) ∈ [0, ∞)2 : T21 (x, y) = x = (x, y) ∈ [0, ∞)2 : y = γx − Cx2 and

( 

2

2



C2 := (x, y) ∈ [0, ∞) : T22 (x, y) = y =

√ y γ − Cy

)

(x, y) ∈ [0, ∞) : x = p Cy(Cy − γ) + γ

By direct inspection of Equation (13) we obtain the following result: Lemma 1 The map T 2 is competitive on [0, ∞)2 \ {(0, 0)} and strongly competitive on (0, ∞)2 . It is easy to see that the following holds. Lemma 2 For all x−1 , x0 ∈ [0, ∞), such that x−1 + x0 > 0, the following holds xn ≤

γ C

for n ≥ 1.

By using very powerful Theorem 1.5 from [4] and Lemma 2, we obtain the following convergence result. Theorem 3 Every solution of Equation (1) converges to a period-two solution or to zeros.

3

Global behavior

In this section we consider the following four parametric regions γ > 3, 1 < γ < 3, γ = 3 and 0 < γ < 1. We completely describe the global behaviour of Equation (1) in these regions. The following theorem details the case γ > 3. Theorem 4 Assume that γ > 3. Then system (8) has a unique equilibrium point E(¯ u, u ¯) which is locally asymptotically stable and there exist two prime period-two solutions: {P1 (¯ u1 , v¯1 ), P2 (¯ v1 , u ¯1 )} which is locally asymptotically stable and {P3 (¯ u2 , v¯2 ), P4 (¯ v2 , u ¯2 )} which is a saddle point, where u ¯1 = 0,

v¯1 =

γ γ−1 and u ¯= C C

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and

p p (γ − 3)(γ + 1) γ + 1 + (γ − 3)(γ + 1) u ¯2 = and v¯2 = 2C 2C Furthermore, global stable manifold of the periodic solution {P3 , P4 } is given by W s ({P3 , P4 }) = W s (P3 ) ∪ W s (P4 ) where W s (P3 ) and W s (P4 ) are continuous increasing curves, invariant under the map T 2 and T (W s (P3 )) = W s (P4 ), and divide the first quadrant into two connected components, namely γ+1−

W1− := {x ∈ R \ W s (P3 ) : ∃y ∈ W s (P3 ) with y se x} W1+ := {x ∈ R \ W s (P3 ) : ∃y ∈ W s (P3 ) with x se y} and W2− := {x ∈ R \ W s (P4 ) : ∃y ∈ W s (P4 ) with y se x} W2+ := {x ∈ R \ W s (P4 ) : ∃y ∈ W s (P4 ) with x se y} respectively. In addition, W s (P3 ) is passing through the point P3 and W s (P4 ) is passing through the point P4 and the following holds: i) If (u0 , v0 ) ∈ W s (P3 ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P3 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P4 . ii) If (u0 , v0 ) ∈ W s (P4 ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P4 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P3 . iii) If (u0 , v0 ) ∈ W1+ (the region above W s (P3 )) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P1 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P2 . iv) If (u0 , v0 ) ∈ W2− (the region below W s (P4 )) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P2 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P1 . v) If (u0 , v0 ) ∈ W1− ∩ W2+ (the region between W1− and W2+ ) then the sequence {(un , vn )} is attracted to E. See Figure 1. Proof. Theorem 1 implies that there exists a unique equilibrium point E(¯ x, x ¯) which is locally asymptotically stable. Theorem 2 implies that the periodic solution {P1 , P2 } is locally asymptotically stable and {P3 , P4 } is a saddle point. In view of (12) the map T 2 (u, v) = T (T (u, v)) is competitive on R = R2+ \ {(0, 0)} and strongly competitive on int(R). It is easy to see that at each point, the Jacobian matrix of T 2 has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively.

Figure 1: Visual illustration of Theorem 4.

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In view of Theorem 3 we have that all solutions converge to period-two solution. Hence, all conditions of Theorem 4 in [13] are satisfied, which yields the existence of the global stable manifolds W s (P3 ) and W s (P4 ) which are the graphs of the strictly decreasing functions of the first coordinate on an interval. By Theorem 4 in [13], we have that if (u0 , v0 ) ∈ W s (P3 ) then (u2n , v2n ) = T 2n (u0 , v0 ) → P3 as n → ∞ which implies that (u2n+1 , v2n+1 ) = T (T 2n (u0 , v0 )) → T (P3 ) = P4 as n → ∞ from which it follows the statement i). The proof of the statement ii) is similar to the proof of the statement i). Take (u0 , v0 ) ∈ W1+ ∩ R. By Theorem 4 in [13], we have that there exists n0 > 0 such that, T 2n (u0 , v0 ) ∈ int(Q2 (P3 ) ∩ R), n > n0 . In view of Theorem 1 in [11], since P3 is a saddle point, we obtain that for all (u0 , v0 ) ∈ int(Q2 (P3 ) ∩ R), there exists r0 > 0 such that (u0 , v0 ) se P3 − r0 v1 and T 2 (P3 − r0 v1 ) se P3 − r0 v1 . By monotonicity T 2n+2 (P3 − r0 v1 ) se T 2n (P3 − r0 v1 )  P3 . In view of Lemma 2 we have that
h γ 2 \ {(0, 0)}. T [0, ∞)2 \ {(0, 0)} ⊂ 0, C From this and the fact that P1  P3  E  P4  P2 we have that T 2n (P3 − r0 v1 ) → P1 as n → ∞. By monotonicity we have that P1 se T 2n (u0 , v0 ) se T 2n (P3 −r0 v1 )  P3 which implies that T 2n (u0 , v0 ) → P1 and T 2n+1 (u0 , v0 ) = T (T 2n (u0 , v0 )) → T (P1 ) = P2 as n → ∞ which proves the statement iii). Take (u0 , v0 ) ∈ W2− ∩ R. By Theorem 4 in [13], we have that there exists n1 > 0 such that, T 2n (u0 , v0 ) ∈ int(Q4 (P4 ) ∩ R), n > n1 . In view of Theorem 1 in [11], since P4 is a saddle point, we obtain that for all (u0 , v0 ) ∈ int(Q4 (P4 ) ∩ R), there exists r1 > 0 such that P4 + r1 v1 se (u0 , v0 ) and P4 + r1 v1 se T 2 (P4 + r1 v1 ). The rest of the proof of the statement iv) is similar to the proof of the statement iii) and we skip it here. Now, we show that each orbit starting in the region W1− ∩ W2+ converges to E. Take (u0 , v0 ) ∈ W1− ∩ W2+ . By Theorem 4 in [13],we have that there exists n2 > 0 such that, T 2n (u0 , v0 ) ∈ int(Q4 (P3 ) ∩ Q2 (P4 ) ∩ R) = [[P3 , P4 ]], for n > n2 . Since P3 and P4 are the saddle points and E is locally asymptotically stable, in view of Corollary 2 [12] we have that T 2n (u0 , v 0 ) → E and T 2n+1 (u0 , v 0 ) = T (T 2n (u0 , v 0 )) → T (E) = E as n → ∞ for all (u0 , v 0 ) ∈ [[P3 , E]] and that T 2n (u00 , v 00 ) → E and T 2n+1 (u00 , v 00 ) = T (T 2n (u00 , v 00 )) → T (E) = E as n → ∞ for all (u00 , v 00 ) ∈ [[E, P4 ]]. Then there exist the points (u00 , v00 ) ∈ [[P3 , E]] and (u000 , v000 ) ∈ [[E, P4 ]] such that (u00 , v00 ) se T 2n2 +2 (u0 , v0 ) se (u000 , v000 ). By monotonicity of the map T 2 we have that T 2n (u0 , v0 ) → E and T 2n+1 (u0 , v0 ) = T (T 2n (u0 , v0 )) → T (E) = E as n → ∞ for all (u0 , v0 ) ∈ W1− ∩ W2+ . This completes the proof of statement v) of the Theorem. 2 The following theorem considers the case 1 < γ < 3. Theorem 5 Assume that 1 < γ < 3. Then system (8) has a unique equilibrium point E(¯ u, u ¯) which is a saddle point and prime period-two solution {P1 (¯ u1 , v¯1 ), P2 (¯ v1 , u ¯1 )} which is locally asymptotically stable, where u ¯1 = 0,

v¯1 =

γ γ−1 and u ¯= . C C

Global stable manifold W s (E), which is continuous increasing curve, divides the first quadrant into two connected components W − (E) := {x ∈ R \ W s (E) : ∃y ∈ W s (E) with y se x} W + (E) := {x ∈ R \ W s (E) : ∃y ∈ W s (E) with x se y} such that R2+ = W − (E) ∪ W + (E) ∪ W s (E). In addition, W s (E) passing through the point E and the following holds: i) Every initial point (u0 , v0 ) in W s (E) is attracted to E. ii) If (u0 , v0 ) ∈ W + (E) (the region below W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P2 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P1 .

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iii) If (u0 , v0 ) ∈ W − (E) (the region above W s (E)) then the subsequence of even-indexed terms {(x2n , v2n )} is attracted to P1 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P2 . See Figure 1.

Figure 2: Visual illustration of Theorem 5 . Proof. Theorem 1 implies that there exists a unique equilibrium point E(¯ x, x ¯) which is a saddle point. Theorem 2 implies that the period-two solution {P1 , P2 } is locally asymptotically stable. Similar as in the proof of Theorem 4 all conditions of Theorem 4 in [13] are satisfied, which yields the existence of the global stable manifold W s (E) which is the graph of the strictly increasing function. Take (u0 , v0 ) ∈ W + ∩ R. By Theorem 4 in [13], we have that there exists n0 > 0 such that, T 2n (u0 , v0 ) ∈ int(Q2 (E) ∩ R), n > n0 . In view of Theorem 1 in [11], since E is a saddle point, we obtain that for all (u0 , v0 ) ∈ int(Q2 (E) ∩ R), there exists r0 > 0 such that (u0 , v0 ) se E − r0 v1 se E and T 2 (E − r0 v1 ) se E − r0 v1 . By monotonicity T 2n+2 (E − r0 v1 ) se T 2n (E − r0 v1 )  E. In view of Lemma 2 we have that T n (u, v) ∈ [0, γ/C)2 \ {(0, 0)}. From this and the fact that P1  E  P2 we have that T 2n (E − r0 v1 ) → P1 as n → ∞. By monotonicity, P1 se T 2n (u0 , v0 ) se T 2n (E −r0 v1 )  E which implies that T 2n (u0 , v0 ) → E and T 2n+1 (u0 , v0 ) = T (T 2n (u0 , v0 )) → T (E) = E as n → ∞ which proves the statement ii). The proof of the statement iii) is similar and we skip it here. 2 Now, we assume that γ = 3. The following theorem holds. Theorem 6 Assume that γ = 3. Then System (8) has a unique equilibrium point E(¯ u, u ¯) which is a non-hyperbolic and prime period-two solution {P1 (¯ u1 , v¯1 ), P2 (¯ v1 , u ¯1 )} which is locally asymptotically stable, where u ¯1 = 0,

v¯1 =

3 2 and u ¯= . C C

There exists a continuous increasing curve CE which is a subset of the basin of attraction of E and it divides the first quadrant into two connected invariant components W − (E) := {x ∈ R \ CE : ∃y ∈ CE with y se x} W + (E) := {x ∈ R \ CE : ∃y ∈ CE with x se y} such that the following holds: i) Every initial point (u0 , v0 ) in CE is attracted to E. ii) If (u0 , v0 ) ∈ W + (E) (the region above CE ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P1 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P2 .

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ii) If (u0 , v0 ) ∈ W − (E) (the region below CE ) then the subsequence of even-indexed terms {(x2n , v2n )} is attracted to P2 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P1 . See Figure 3.

Figure 3: Visual illustration of Theorem 6 . Proof. Theorem 1 implies that there exists a unique equilibrium point E(¯ x, x ¯) which is non-hyperbolic. Theorem 1(c) implies that the periodic solution {P1 , P2 } is locally asymptotically stable. Similar as in the proof of Theorem 4 all conditions of Theorem 4 in [13] are satisfied, which yields the existence a continuous increasing curve CE which is a subset of the basin of attraction of E and for every x ∈ W+ there exists n0 ∈ N such that T n (x) ∈ int Q2 (E) for n ≥ n0 and for every x ∈ W− there exists n1 ∈ N such that T n (x) ∈ int Q4 (E) for n ≥ n1 . 3 Set U (t) = t(3 − Ct). It is easy to see that (t, U (t)) se E if t ∈ [ 2C ,u ¯] and E se (t, U (t)) if t ∈ [¯ u, C3 ] and U (¯ u) = u ¯. In view of Lemma 2 we have that
h γ 2 \ {(0, 0)}. T [0, ∞)2 \ {(0, 0)} ⊂ 0, C One can show that T 2 (t, U (t)) − (t, U (t)) =

 0,

t(Ct − 3)(Ct − 2)3 Ct(Ct − 3)2 + 1



which implies that T 2 (t, U (t)) se (t, U (t)) if t < u ¯ and (t, U (t)) se T 2 (t, U (t)) if t > u ¯. By monotonicity 2n if t < u ¯ then we obtain that T (t, U (t)) → P1 as n → ∞ and if t > u ¯ then we have that T 2n (t, U (t)) → P2 as n → ∞. If (u0 , v 0 ) ∈ intQ2 (E) then there exists t1 such that P1 se (u0 , v 0 ) se (t1 , U (t1 )) se E. By monotonicity of the map T 2 we obtain that P1 se T 2n (u0 , v 0 ) se T 2n (t1 , U (t1 )) se E which implies that T 2n (u0 , v 0 ) → P1 and T 2n+1 (u0 , v 0 ) → T (P1 ) = P2 as n → ∞ which proves the statement ii). If (u00 , v 00 ) ∈ intQ4 (E) then there exists t1 such that E  (t2 , U (t2 )) se (u00 , v 00 ) se P2 . By monotonicity of the map T 2 we obtain that E se T 2n (t2 , U (t2 )) se T 2n (u00 , v 00 ) se P2 which implies that T 2n (u00 , v 00 ) → P2 and T 2n+1 (u00 , v 00 ) → T (P2 ) = P1 as n → ∞ which proves the statement iii), and completes the proof of the Theorem. 2 First we notice the following. Theorem 3 and Lemma 2 imply that T 2n (x0 , y0 ) is asymptotic to either P1 = (0, Cγ ) or P2 = ( Cγ , 0) or (0, 0), for all (x0 , y0 ) ∈ R \ {(0, 0)}. Let B(P1 ) be the basin of attraction of P1 and B(P2 ) be the basin of attraction of P2 with respect to the map T 2 . Let C + denote the boundary of B(P1 ) considered as a subset of int Q1 (0, 0) (the first quadrant relative to (0, 0)) and C − denote the boundary of B(P2 ) considered as a subset of int Q1 (0, 0). It is easy to see that (0, 0) ∈ C + and (0, 0) ∈ C − . Now, similarly to the proof of the of Claim 1 and Claim 2 in [5], one can prove that the following lemma holds.

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Lemma 3 Let C + and C − be the sets defined above. Then the sets C + and C − are invariant under the map T 2 and they are the graphs of continuous strictly increasing functions. Further, C + ∪ C − ⊂ B(0, 0). The following theorem details the existence two invariant strictly increasing curves passing through the point (0, 0), such that every solution that stars on these two curves or in the region between these two curves is attracted to the point (0, 0). Theorem 7 Assume that 0 < γ < 1. Then there exists prime period-two solution {P1 (¯ u1 , v¯1 ), P2 (¯ v1 , u ¯1 )} which is locally asymptotically stable, where γ u ¯1 = 0, v¯1 = C Furthermore, there exist sets C + and C − which are continuous increasing curves, invariant under the map T 2 and T (C + ) = C − , and divide the first quadrant into two connected components, namely W1− := {x ∈ R \ C + : ∃y ∈ C + with y se x}

and

W1+ := {x ∈ R \ C + : ∃y ∈ C + with x se y}

W2− := {x ∈ R \ C − : ∃y ∈ C − with y se x}

and

W2+ := {x ∈ R \ C − : ∃y ∈ C − with x se y}

and

respectively. In addition, C + and C − passing through the point (0, 0) and the following holds: i) If (u0 , v0 ) ∈ W1+ (the region above C + ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P1 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P2 . ii) If (u0 , v0 ) ∈ W2− (the region below C − ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P2 , and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P1 . iii) If (u0 , v0 ) ∈ (C + ∪ C + ) ∪ (W1− ∩ W2+ ) (the region between C + and C − ) then the sequence {(un , vn )} is attracted to (0, 0). Proof. The proof follows from Lemma 3, and it is similar to the proof of Theorem 4, so we skip it. 2 Based on a series of numerical simulations we pose the following hypothesis. Conjecture 1 Suppose that all assumptions of the Theorem 7 are satisfied, then the following holds: C + = C−.

References [1] A.M. Amleh, D.A.Georgiou, E.A.Grove, and G.Ladas, On the recursive sequence xn+1 = α + J.Math.Anal.Appl.1999, 233 , 790–515.

xn+1 xn .

[2] A. Brett and M. R. S. Kulenovi´c, Basins of attraction of equilibrium points of monotone difference equations. Sarajevo J. Math.2010, 5(18) , 211–233. [3] E. Camouzis and G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures; Chapman and Hall/CRC Press, London/Boca Raton, FL 2008. [4] E. Camouzis and G. Ladas, When does local asymptotic stability imply global attractivity in rational equations? 2006,12(8), 863–885. [5] V.Hadˇziabdi´c, M.R.S.Kulenovi´c and E.Pilav, Dymanics of a Two-dimensional Competitive System of Rational Difference Equations with Quadratic Terms. Adv.Difference Equ.2014, 2014:301,32 p. [6] M. Hirsch and H. Smith, Monotone dynamical systems. Handbook of differential equations: ordinary differential equations. Vol. II, 239–357, Elsevier B. V., Amsterdam, 2005.

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February 26, 2019

11

[7] S. Kalabuˇsi´c and M. R. S. Kulenovi´c, Dynamics of Certain Anti-competitive Systems of Rational Difference Equations in the Plane. J. Difference Equations Appl. 2011, 17, 1599–1615, [8] S. Kalabuˇsi´c, M. R. S. Kulenovi´c and E. Pilav, Global Dynamics of Anti-Competitive Systems in the Plane, J. Difference Equ. Appl.2013, 19 1849-1871. [9] M. R. S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman& Hall/CRC Press, Boca Raton, 2002. [10] M. R. S. Kulenovic, E. Pilav and E. Sili´c, Local Dynamics and Global Attractivity of a Certain Second Order Quadratic Fractional Difference Equation, Adv. Difference Equ. 2014, 32 pp. [11] M. R. S. Kulenovi´c and O. Merino, Competitive-Exclusion versus Competitive-Coexistence for Systems in the Plane. Discrete Contin. Dyn. Syst. Ser. B 2006, 6, 1141–1156. [12] M. R. S. Kulenovi´c and O. Merino, Global bifurcation for discrete competitive systems in the plane Discrete Contin. Dyn. Syst. Ser. B 12 (2009), 133–149. [13] M. R. S. Kulenovi´c and O. Merino, Invariant Manifolds for Competitive Discrete Systems in the Plane, Int. J. of Bifurcations and Chaos 2010, 20, 2471–2486. [14] H. L. Smith, Planar Competitive and Cooperative Difference Equations,J. Difference Equ. Appl. 1998, 3, 335–357.

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Fourier Approximation Schemes of Stochastic Pseudo-Hyperbolic Equations with Cubic Nonlinearity and Regular Noise Tengjin Zhaoa , Chenzhong Wanga , Quanyong Zhub , Zhiyue Zhanga∗ a School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS Nanjing Normal University, Nanjing 210023, China, b School of Technology, Lishui University, Lishui, 323000, China

November 13, 2018

Abstract The pseudo-hyperbolic equation with cubic nonlinearity and additive space-time noise is discussed. The space-time noise is assumed to be Gaussian in time and possesses a Fourier series expansion in space. First, we prove the existence and uniqueness of the approximate strong solutions of the equation and show that the truncated Fourier solution which can be approximated by the truncated finite-dimensional system, is an approximate solution. Second, a new transformation is used to convert pseudo-hyperbolic equation into a system of equations, which can construct an infinitesimal generator with good properties. After analyzing the related total energy evolution, we obtain that the energy growth will not blow-up in the limited time. Finally, we present a Fourier scheme of a procedure for its numerical approximation and give the stability and convergence analysis of the scheme.

keyword: thermal convection equation, Fourier coefficients, cubic-type nonlinearities; stochastic; energy

1

Introduction

Stochastic differential equations (SDEs) can model many natural phenomena with white noise and engineering applications, such as epidemiology, economics and so on [15, 1, 14, 3, 43, 23, 22]. SDEs hold for the important original work ∗ Corresponding

author. E-mail address: [email protected].

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of Ito [12] as well as books [7, 27]. The shorter accounts of stochastic dynamic systems on stability, filtering, and control [18, 13] are rather unsuited for the study. Stability of SDEs has been well studied by researchers [25, 16, 19]. Since the analytical solution is difficult to obtain, different numerical methods have been introduced such as [30, 33, 8, 29]. The common theoretical basis is the stochastic Ito-Taylor expansion in terms of multiple Wiener integrals [15]. The analysis of the linear SDEs is well investigated such as [20, 6, 28, 24]. In the recent past, the nonlinear SDEs are researched. In [21] a class of fully nonlinear SDEs is studied by using the stochastic characteristic method. In [11] a strong convergence result under less restrictive conditions is proved by using Euler-Maruyama method. In [10], the exponential stability of the multidimensional nonlinear SDEs with variable delays is investigated. Nonlinear filtering equations have developed based on a classification where the measure term is either deterministic or random [39]. Consider the semi-linear stochastic pseudo-hyperbolic equation with cubictype nonlinearities perturbed by additive space-time random noise W [40]:  ∂ 2 (u + ut )   dt + B(u + ut )dt + b·dW (t, x),  d(u + ut ) = σ 2  ∂2x (1.1) u(0, x) = u0 , ut (0, x) = ut0 , 0 < x < L,     u(t, 0) = u(t, L) = 0, ut (t, 0) = ut (t, L) = 0, 0 < t < T, where b ∈ R1 is an overall noise intensity parameter. B(u) = u(a1 − a2 kuk2L2 ) is cubic-type with real parameters a2 > 0 and a1 [38]. The space-time Q-regular noise W (t, x) is as follows: W (t, x) =

+∞ X n=1

r αn Wn (t)

+∞  nπx  X 2 sin αn Wn (t)en (x) = L L n=1

(1.2)

with independent and identically distributed Wiener process q Wn ∈ N
(0, t), P+∞ where trace(Q) = n=1 αn2 < +∞. We know that en (x) = L2 sin nπx ,n ≥ L 1 are the eigenfunctions of the Laplace operator which form an orthonormal 2 2 system in H = L2 (0, L) and satisfy in one-dimensional, ∆en (x) = − nLπ2 en . The main contribution of this paper is to discuss the Fourier solution u(t, x) and its numerical approximations by truncated Fourier series [41]. We construct an infinitesimal generator with good properties and convert into the equations which can be easily solved. The rest of the paper is organized as follows. In section 2, we verify the existence and uniqueness of solution and give a finite-dimensional system of the SDEs. In section 3, we estimate the truncated total energy. In section 4 we show numerical methods to find those Fourier coefficients. In the last section 5 numerical experiments are provided which support our results.

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2

Existence and Uniqueness of Approximate Strong Solutions and Fourier-Series Solutions

In general, it is difficult to solve nonlinear equations. However, taking the equations into system of equations can avoid lots of complex calculations in infinitesimal generators and energy estimation. Let v = u + ut , Eqs (1.1) becomes    v = u + ut ,   (2.1)
dv ∂2v dW (t, x)  = σ 2 2 + v a1 − a2 kvk2L2 + b· .  dt ∂x dt It can be rewritten as d dt

! u v

=

−1 0

1

!

σ2 ∂ 2 ∂x2

! u v

+

! 2

0

(a1 −a2

)+

v v L2 ! 0

! 0 dW . (2.2) b dt

From the definitions of strong solution and approximate strong solution[36], we obtain that conditions of the strong solutions of (2.1) exist and the uniqueness is that all operators are globally Lipschitz-continuous. Under conditions weaker than global Lipschitz-continuity, we can also achieve a result of the strong solutions. 2

Lemma 2.1. For all a2 ≥ 0, the mapping v ∈ H → 7 B(v) = v(a1 − a2 kvkL2 ) satisfies the angle condition on H. In other words, for all u, v ∈ H, we have F (u, v) :=< B(u) − B(v), u − v >H ≤ a1 ku − vk2H ,

(2.3)

specially < B(v), v >H ≤

  kvk2H a1 − a2 kvk2H ≤ a1 kvk2H . 2

Proof. Denoting f (u) := kuk2H u and g(u, v) :=< f (u) − f (v), u − v >H which is symmetric. Then we obtain that

2
kuk2H + kvk2H ku − vk2H , 2g(u, v) = kuk2H +kvk2H ku−vk2H + kuk2H −kvk2H g(u, v) ≥

kuk2H + kvk2H ku − vk2H . 2

Now using (2.3), the above inequality and the definition of B, we have kuk2H + kvk2H ku − vk2H + a1 ku − vk2H ≤ a1 ku − vk2H , 2 kvk2H < B(v), v >H ≤ −a2 kvk2H + a1 kvk2H ≤ a1 kvk2H , by setting u = (0, 0). 2

F (u, v) ≤ −a2

Then the proof is completed. 3

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From Lemma 2.1 and Theorem 3 in [36], the unique approximate strong and continuous solution of Eqs (2.2) exists. Theorem 2.2. Assumptions of definitions of strong and approximate strong solution [36] are satisfied with Ekv(0, ·)k2H L2 (a1 +1), we investigate the behavior of related energy functional, which is defined at time t ≥ 0 by E(t) =

σ2 a1 + 1 a2 kvx (t, ·)k2H − kv(t, ·)k2H + kv(t, ·)k4H . 2 2 4

(3.1)

This energy functional is indeed nonnegative and finite almost surely (a.s.) as one can see from the following theorem. For its proof, we take the functional in terms of its Fourier coefficients ck by  +∞  1 X σ 2 n2 π 2 a2 V (t) := V (cvk (t) : k ∈ N ) = −a −1 c2vn (t)+ 1 2 n=1 L2 4

+∞ X

!2 c2vn (t)

,

n=1

for t ≥ 0. It is easy to know that V ≥ 0 for all sequences (cvk (t))k and acts as a Lyapunov functional. Besides, E(t) = V (t) for all t ≥ 0. Theorem 3.1. Assume that e(0) = EV (cvk (0) : k ∈ N ) < ∞, σ 2 π 2 ≥ P∞ 2 L2 (a1 + 1) and trace(Q) = n=1 αn < ∞. Then, the total expected energy of the original system (2.1) is linearly bounded in time by " +∞ # X σ 2 n2 π 2 √ 2 2 2 2 e(t) = EV (cvk (t) : k ∈ N ) ≤ e(0)+2 [ −a1−1]cvn (t)+ a2 (b β ) t, L2 n=1 where β 2 =

P∞

n=1

αn2 + 2 max αn2 . n∈N

Proof. The truncated infinitesimal generator can be rewritten # " N N N N X X X ∂ b2 X 2 ∂ 2 σ 2 n2 π 2 ∂ 2 L= [cvn−cun ] + αn 2 + − + a1−a2 cvk cvn . ∂cun 2 n=1 ∂cvn n=1 L ∂cvn n=1 k=1

We express Eqs (3.1) in terms of its truncated Fourier coefficients cvk by N 1 X σ 2 n2 π 2 a2 Ve (t) : = Ve (cvk (t) : k ∈ N ) = [ −a1−1]c2vn (t) + 2 2 n=1 L 4

N X

!2 c2vn (t)

,

n=1

for t ≥ 0. Then, after calculating the infinitesimal generator, we estimate the

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energy of the system (3.1) as follow: LVe = LVe1 + LVe2 =

N X

" (cvn − cun ) cun −

n=1

n=1

+

N X

N

X σ 2 n2 π 2 − a1 + a2 c2vk 2 L

#2

! cvn

k=1

 N N  N N b2 X 2 X 2 1 X 2 X σ 2 n2 π 2 − a + αn α a cvn (t) + 2c2vn (t). 1 n 2 2 2 n=1 L 2 n=1 n=1 n=1

  +∞ N 2 X
3 5b2 X 2 σ 2 n2 π 2 1 0 2 2 2 cun + cun ≤ √ − a + a b β . αn 1 2 N 4 L2 12 3a2 n=1 n=1 P+∞ From the estimate in [38] and denoting β 2 = n=1 αn2 + 2 max αn2 , we obtain LVe1 ≤

n∈N

LVe2 ≤ b2

+∞ X

αn2

n=1




3 σ 2 n2 π 2 2 2 − a1 + a2 b2 βN 2 L



1 12a2

 12

5 . 6

Consequently, Dynkin formula says that  2 2 2    12 +∞ h i X
3 σ n π 1 5 2 2 2 eN (t) =E Ve (t) ≤ e(0)+ 2b2 αn2 − a , +2a b β 1 2 N 2 L 12a 6 2 n=1 for t ≥ 0. Since eN ≥ 0 is increasing in N and uniformly bounded in time t for any t ∈ [0, T ], we know that limN →+∞ en (t) = e(t), and 0 ≤ e(t) ≤e(0) + 2b

2

+∞ X n=1

αn2



   12
3 5 σ 2 n2 π 2 1 2 2 2 − a1 t + 2a2 b βN t, L2 12a2 6

as e(0) < ∞, σ 2 π 2 ≥ L2 (a1 + 1) and trace(Q) =

P∞

n=1

αn2 < ∞.

More precisely, for T < ∞, ∀ 0 ≤ t ≤ T, ∃ K1 , K2 ≥ 0 (Ekv(t, ·)k2H + K0 )eK1 T ≥ Ekv(t, ·)k2H ≥ Eku(t, ·)k2H . In fact, if σ 2 π 2 > L2 (a1 +1), we can know that the following mentioned estimates of second moments have linearly bounded ones (in time). For T < ∞, ∃ c ≥ 0, 0 ≤ t ≤ T , Eku(t, ·)k2H ≤ Ekv(t, ·)k2H ≤ Ekv(0, ·)k2H + ct.

4

Numerical Methods for Fourier Coefficients

The truncated Fourier series u e, ve in Eqs (2.5) satisfy the Eqs (2.1). Since the explicit solution is unknown, we take advantage of numerical approximations. Along partitions 0 = t0 < t1 < t2 < · · · < tnT = T of interval [0, T ] with the step sizes hn = tn+1 − tn , and 0 = x0 < x1 < x2 < · · · < xnL = L of interval [0, L] with the step sizes dn = xn+1 − xn . 6

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For each fixed xm , let us consider the forward Euler method for cvk ! N X −σ 2 k 2 π 2 2 + a1 − a2 cvk (n + 1) =hn cvk (n) cvl (n) + cvk (n) + bk 4Wnk , L2 l=1 (4.1) where 4Wnk = Wk (tn+1 ) − Wk (tn ) ∈ N (0, hn ). Other one is backward Euler method ! N X −σ 2 k 2 π 2 2 +a1 −a2 cvk (n+1) =hn cvk (n+1) cvl (n+1) + cvk (n) + bk 4Wnk . L2 l=1 (4.2) In our opinion, the best approach is linear-implicit Euler-type method ! N X −σ 2 k 2 π 2 2 + a1 − a2 cvl (n) + cvk (n) + bk 4Wnk . cvk (n+1) =hn cvk (n+1) L2 l=1 (4.3) PN After calculating the cvk , we can obtain ve(tn+1 , xm ) = n=1 cvn (t)en (x). Then uN can be calculated u e(tn+1 , xm+1 ) =dn (e v (tn+1 , xm ) − u e(tn+1 , xm )) + u e(tn+1 , xm ). We note that Eqs (4.1) has a disadvantage that is lacking of stability and monotonicity deficits. A slight disadvantage of Eqs (4.2) is that we have to solve locally implicit algebraic equations at each iteration step n, which results in a lot of calculation and time. An advantage of methods (4.2) and (4.3) is very well stability and moment dissipativity behavior, and they keep some monotonicity properties [34, 35]. Theorem 4.1. Consider the forward Euler method that cvk (n + 1) = 1 + hn



cvk (n) + bk 4Wnk , PN − a1 + a2 l=1 c2vl (n)

σ 2 k2 π 2 L2

(4.4)

where n ∈ N, bk = bαk and 4wnk ∈ N (0, hn ). If σ 2 π 2 ≥ L2 (a1 + 1), their second moments is linearly bounded in time,   E ku(tn , ·)k2H < +∞. i h 2 2 Proof. Suppose that 1 + hn σLπ2 − (a1 + 1) > 0. The Eqs (4.4) is finite due to the linear-implicit character of method (4.3). From Eqs (2.6), it follows  cuk (n + 1) − cuk (n)   = [cvk (t) − cuk (t)] ,   hn " # +∞ 2 2 2 X c (n + 1) − c (n) −σ k π Wk (tn + 1) − Wk (tn )  vk vk   = +a1 −a2 c2vl (t) cvk + bk .  hn L2 hn l=1

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It remains to consider the second moments. We estimate cvk by Eqs (4.4) cvk (n + 1) = 1 + hn

h

cvk (n) + bk 4wnk i, PN σ 2 k2 π 2 − a1 + a2 l=1 c2vl (n) L2



 2

 2 E [ck (n+1)] = E  h

[ck (n)] + 1 + hn



σ 2 k2 π 2 L2

b2k hn

− a1 + a2

 i2  . 2 l=1 cvl (n)

PN

Since denominator is less than one, we have 2

2

2

E [cvk (n + 1)] ≤E [cvk (n)] + (bk )2 hn ≤ E [cvk (0)] + (bk )2 tn+1 . From Eqs (2.5), we obtain N X

N N X   X   2 2 2 E kcvk (n)k ≤ E kcvk (0)k + b αk2 tn .

k=1

Since

P+∞

k=1

αk2 ,

k=1

P+∞

k=1

k=1

kcvk (0)k2 < ∞, we obtain that, as N → ∞ and h → 0

+∞ +∞ +∞ X   X   X   E kv(tn , ·)k2H = E kcvk (n)k2 ≤ E kcvk (0)k2 + b2 αk2 tn < ∞. k=1

k=1

k=1

Recall the definition in [38], let chk denote the numerical approximation of the k-th Fourier coefficients ck . The numerical approximation ch = (chk )k=1,2,···,N is said to be mean consistent with rate r0 iff there are a constant C0 = C0 (T ) and a positive continuous function or functional V such that ∀n=0,1, · · ·, nT − 1 : kE[c(n + 1)]−E[ch (n + 1)]kN ≤C0 V (c(n))hrn0 along any (nonrandom) partitions with sufficiently small step sizes hn ≤ δ ≤ 1, where k·k is the Euclidean vector norm in RN , provided that one has nonrandom data c(n) = ch (n). Lemma 4.2. The method (LIM) governed by Eqs (4.3) is mean consistent with rate r0 = 1.5. The similar results may be found in [38].

5

Numerical Experiments

Under the condition that σ 2 π 2 > L2 (a1 + 1), we present the results of systematic numerical simulations for solutions of the SDEs. The order is de
fined by order = lg kE[c(n + 1)] − E[ch (n + 1)]kN . The ratio is defined by 8

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ratio = 1 − the noise.



kEk kE+k ,

where E is the total energy of the system at t = 0 and  is

Case 5.1. We consider the simple initial data with  1   x, x < L, 2 u(0, x) = 1   L − x, x > L. 2 0.5

0.6

0.45 0.5 0.4 0.4

0.35 0.3

0.3

0.25 0.2

0.2 0.15

0.1

0.1 0 0.05 0

0

20

40

60

80

−0.1

100

0

20

40

60

80

100

Figure 1: The numerical results at the Figure 2: The numerical results at the times t = 2 with ratio ≈ 1% times t = 2 with ratio≈ 5% 0.5

1400

0.45 1200 0.4 1000

0.35 0.3

800

0.25 600

0.2 0.15

400

0.1 200 0.05 0

0

20

40

60

80

0

100

0

50

100

150

200

Figure 3: The numerical results at the Figure 4: The total energy with differtimes t = 2 with ratio > 10% ent random terms at t = 10 with ratio ≈ 1% 2000

4000

1800 3500 1600 1400

3000

1200 2500 1000 800

2000

600 1500 400 200

0

50

100

150

1000

200

0

50

100

150

200

Figure 5: The total energy with dif- Figure 6: The total energy with differferent random terms at t = 10 with ent random terms at t = 10 with ratio ratio≈ 5% > 10% The parameters T = 2, ∆t = 0.05, ∆x = 0.01, L = 1, a1 = 0.1, a2 = 1 and σ = 9 are chosen over the region [0, 1]. In Figure 1, 2 and 3, the lines of ”·” and ”◦” respectively denote the initial value u(0, x) and the terminal value u(2, x). Figure 1 shows that the wave dissipates at time t = 2. Figure 2, 3 show 9

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0.5

0.6

0.45 0.5 0.4 0.4

0.35 0.3

0.3

0.25 0.2

0.2 0.15

0.1

0.1 0 0.05 0

0

20

40

60

80

−0.1

100

0

20

40

60

80

100

Figure 7: The numerical results at the Figure 8: The numerical results at the times t = 10 with ratio ≈ 1% times t = 10 with ratio≈ 5% 0.5

1800

0.45 1600 0.4 1400

0.35 0.3

1200

0.25 1000

0.2 0.15

800

0.1 600 0.05 0

0

20

40

60

80

400

100

0

50

100

150

200

Figure 9: The numerical results at the Figure 10: The total energy with dtimes t = 10 with ratio > 10% ifferent random terms at t = 10 with ratio ≈ 1% 4

2200

10

2000

9

x 10

8

1800

7 1600 6 1400 5 1200

4

1000

800

3

0

50

100

150

2

200

0

50

100

150

200

Figure 11: The total energy with d- Figure 12: The total energy with different random terms at t = 10 with ifferent random terms at t = 10 with ratio≈ 5% ratio > 10%

Table 1: Order of convergence ∆x ∆t 0.01 0.05 0.01 0.1 0.05 0.05 0.05 0.1

in space and time for the Euclidean vector norm ratio order ratio order 1% 4.1677 5% 3.0281 1% 4.014 5% 2.6964 1% 4.5498 5% 3.4972 1% 4.9379 5% 2.6441

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the influence of noise enhancement on waveform. When the ratio > 10% , the waveform was destroyed. Figure 4, 5 and 6 present the total energy evolution and show that the total energy stablely declines. However the total energy is linearly bounded in time. These results are in good agreement with the Theorem 3.1. Similarly, Table 1 presents the numerical results of the linear-implicit Euler-type schemes, which are in good agreement with Lemma 4.2.

Case 5.2. In the second case, the initial data is same to the case 1. Using L = 1, a1 = 1, a2 = 1, b = 1 and σ = 9, we present the numerical solution at the terminal time T = 10.

Table 2: Order of convergence ∆x ∆t 0.01 0.05 0.01 0.1 0.05 0.05 0.05 0.1

in space and time for the Euclidean vector norm ratio order ratio order 1% 4.5093 5% 3.3822 1% 4.1886 5% 2.5853 1% 4.0957 5% 3.8375 1% 4.4245 5% 3.2689

Figure 7 shows that the wave dissipates at time t = 10. Figure 8, 9 show the influence of noise enhancement on waveform. When the ratio > 10% , the waveform was destroyed. Figure 11, 10 and 12 present the numerical results of the total energy evolution and show that the total energy stablely declines. With the increase of the noise, the downward trend is not significant but vibrate. These results are in good agreement with the Theorem 4.2. Similarly, Table 2 presents the numerical results of the linear-implicit Euler-type schemes, which are in good agreement with Lemma 4.2.

Acknowledgements The project is supported by the National Natural Science Foundation of China grants No.11471166 and No.11426134, Natural Science Foundation of Jiangsu Province grant No. BK20141443 and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

References [1] A. Benveniste and M. Metivier. Adaptive algorithms and stochastic approximations. Springer Berlin Heidelberg, 1990.

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An iterative algorithm for solving split feasibility problems and fixed point problems in p-uniformly convex and smooth Banach spaces P. Chuasuk1 , A. Farajzadeh2 and A. Kaewcharoen3∗ 2

Department of Mathematics, Faculty of Science, Razi University Kermanshah 67419, Iran, E-mail: [email protected]

1,3

Department of Mathematics, Faculty of Science, Naresuan University,

Phitsanulok 65000, Thailand, E-mails: [email protected], [email protected]

Abstract In this paper, we introduce an iterative process for approximation of a common fixed point for a finite family of multi-valued Bregman relatively nonexpansive mappings with a solution of the split feasibility problems in p-uniformly convex and uniformly smooth Banach spaces. We prove the strong convergence theorems of the proposed iterative process in p-uniformly convex and uniformly smooth Banach spaces and present the numerical results to verify the efficiency and implementation of our results.

Keywords: Bregman relatively nonexpansive mappings; strong convergence theorems; uniformly convex Banach spaces; uniformly smooth Banach spaces; split feasibility problems.

1

Introduction

Let E1 and E2 be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let C and Q be nonempty closed convex subsets of E1 and E2 respectively, A : E1 → E2 be a bounded linear operator and A∗ : E2∗ → E1∗ be the adjoint of A. The split feasibility problem (SFP) is to find a point x ∈ C such that Ax ∈ Q.

(1.1)

Note that the inverse image of the set Q under A is a convex set. Hence the problem 1.1 can be written in case that the intersection C ∩ A−1 (Q) is nonempty. We will denote the nonempty solution set of (1.1) by Ω = C ∩ A−1 (Q). Therefore Ω is a closed convex subset of E1 . In 1994, Censor and Elfving [8] introduced the SFP (1.1) in finite-dimensional Hilbert spaces for modelling inverse problems which arise from phase retrievals, medical image reconstruction. Various algorithms ∗ Corresponding

author.

1

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have been invented to solve the SFP (1.1) ( see [2, 6, 11, 25, 28, 29] and the references therein). In particular, Byrne [6] introduced a so-called CQ algorithm, taking an initial point x0 arbitrarily and construct the sequence {xn } by xn+1 = PC (xn − γA∗ (I − PQ )Axn ), n ≥ 1, 2 where 0 < γ < ∥A∥ 2 , and PC denotes the projection onto a set C. That is, PC (x) = arg miny∈C ∥x − y∥. Recently, Sch¨ opfer et al. [19] solved the SFP (1.1) in p-uniformly convex real Banach spaces which are also uniformly smooth using the following algorithm: for x1 ∈ E1 and n ≥ 1, set

xn+1 = ΠC JEq ∗ [JEp 1 xn − tn A∗ JEp 2 (Axn − PQ (Axn ))],

(1.2)

1

where ΠC denotes the the Bregman projection and J the duality mapping. Clearly the above algorithm covers the Byrne’ CQ algorithm [6]. They used algorithm (1.2) for obtaining the weak convergence result in a p-uniformly convex real Banach spaces which are uniformly smooth with the condition that the duality mapping of E is sequentially weak-to-weak-continuous. In 2014, Wang [26] studied the following multiple-sets split feasibility problem (MSSFP) (see [11]): find x ∈ E1 satisfying x∈

r ∩

Ci , Ax ∈

i=1

r+s ∩

Qj

(1.3)

j=r+1

where r, s are two given integers, Ci , i = 1, ..., r, is a closed convex subset of E1 , and Qj , j = r+1, ..., r+s, is a closed convex subset in E2 . Wang [26] modified the above algorithm (1.2) and proved the strong convergence theorem using an idea appeared in [13] and the following algorithm: for any initial guess x0 , define {xn } recursively by   yn = Tn xn    Dn = {u ∈ E : ∆p (yn , u) ≤ ∆p (xn , u)} (1.4)  En = {u ∈ E : ⟨xn − u, JEp (x0 ) − JEp (xn )⟩ ≥ 0}    xn+1 = ΠDn ∩En (x0 ), where Tn is defined, for each n ∈ N, by { ΠCi(n) (x), 1 ≤ i(n) ≤ r Tn x = JEq ∗ [JEp 1 (x) − tn A∗ JEp 2 (I − PQi(n) )Ax], r + 1 ≤ i(n) ≤ r + s,

(1.5)

1

i : N → I is the cyclic control mapping i(n) = n mod (r + s) + 1, and tn satisfies 0 < t ≤ tn ≤

(

1 q ) q−1 . q Cq ∥A∥

For better comparison of (1.5) with (1.2), we state a version of (1.2) for solving problem (1.3): xn+1 = ΠCi(n) JEq ∗ [JEp 1 (xn ) − tn A∗ JEp 2 (Axn − PQi(n) (Axn ))], 1

(1.6)

where i : N → I is the cyclic control mapping i(n) = n mod (r + s) + 1. In 1967, Bregman [3] has discovered an elegant and effective technique for the use of the Bregman distance function ∆p in the process of designing and analyzing feasibility and optimization algorithms.

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This opened a growing area of research in which Bregman’s technique is applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving variational inequality problems, equilibrium problems, fixed point problems for nonlinear mappings, and so on (see [16, 4, 17] , and the references therein). Recently, Shehu et al. [22] studied split feasibility problems and fixed point problems concerning left Bregman strongly nonexpansive mappings: find an element x ∈ E1 satisfying x ∈ C ∩ F (T ) such that Ax ∈ Q.

(1.7)

Shehu et al. [22] proposed the following algorithm: for a fixed u ∈ E1 , let {xn }∞ n=1 be iteratively generated by u1 ∈ E1 , { xn = ΠC JEq ∗ [JEp 1 (un ) − tn A∗ JEp 2 (Aun − PQ (Aun ))] 1 (1.8) un+1 = ΠC JEq ∗ (αn JEp 1 (u) + (1 − αn )JEp 1 (T xn )), n ≥ 1, 1

where {αn } is a sequence in (0, 1). Moreover Shehu et al. [22] proved the strong convergence of the sequence generated by (1.8) for solving problem (1.7) in p-uniformly convex real Banach spaces which are also uniformly smooth. In 2014, Pang et al. [9] showed that the class of Bregman relatively nonexpansive mappings embraces properly the class of Bregman strongly nonexpansive mappings. Very recently, Shahzad and Zegeye [21] introduced the class of multi-valued Bregman relatively nonexpansive mappings which includes the class of single-valued Bregman relatively nonexpansive mappings. Hence, the class of multi-valued Bregman relatively nonexpansive mappings is a more general class of mappings and gave a example of a multi-valued Bregman relatively nonexpansive mappings. Moreover, Shahzad and Zegeye [21] proved that if C is a nonempty closed convex subset of int(domf ) where f : E → R is a uniformly Frechet differentiable and totally convex on bounded subsets of E and T : C → CB(C) is a Bregman relatively nonexpansive mapping, then F (T ) is closed and convex. Our aim in this paper is to construct an iterative scheme for solving problem (1.7) which is also a fixed point of a multi-valued Bregman relatively nonexpansive mapping T in p-uniformly convex real Banach spaces which are also uniformly smooth and then prove the strong convergence theorems of the sequences generated by our scheme under some suitable assumptions.

2

Preliminaries

Let 1 < q ≤ 2 ≤ p with defined by

1 p

+

1 q

= 1. The modulus of smoothness of E is the function ρE (τ ) : [0, ∞) → [0, ∞)

ρE (τ ) = sup

{1 2

} (∥x + y∥ + ∥x − y∥) − 1 : ∥x∥ ≤ 1, ∥y∥ ≤ 1 .

E is called to be uniformly smooth if ρE (τ ) =0 τ and E is called to be q-uniformly smooth if there exists a Cq > 0 such that ρE (τ ) ≤ Cq τ q for any τ > 0. The modulus of convexity of E is the function δE : (0, 2] → [0, 1] defined by { } x+y δE (ϵ) := inf 1 − ∥ ∥ : ∥x∥ = ∥y∥ = 1; ϵ = ∥x − y∥ . 2 lim

τ →0

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E is called to be uniformly convex if δE (ε) > 0 for all ε ∈ (0, 2] and p-uniformly convex if there is a Cp > 0 so that δE (ε) ≥ Cp εp for any ε ∈ (0, 2]. The Lp space is 2-uniformly convex for 1 < p ≤ 2 and p-uniformly convex for p ≥ 2. Lemma 2.1. [27] Let x, y ∈ E. If E is q-uniformly smooth, then there exists a Cq > 0 such that ∥x − y|∥q ≤ ∥x∥q − q⟨y, JEq (x)⟩ + Cq ∥y∥q . It is known that if E is p-uniformly convex and uniformly smooth, then its dual E ∗ is q-uniformly smooth and uniformly convex. Moreover the duality mapping JEp is one-to-one, single-valued and JEp = (JEq ∗ )−1 where JEq ∗ is the duality mapping of E ∗ (see [10, 14]). ∗

Definition 2.2. The duality mapping JEp : E → 2E is defined by JEp (x) = {¯ x ∈ E ∗ : ⟨x, x ¯⟩ = ∥x∥p , ∥¯ x∥p = ∥x∥p−1 }. The duality mapping JEp is said to be weak-to-weak continuous if xn ⇀ x ⇒ ⟨JEp xn , y⟩ → ⟨JEp x, y⟩ holds for any y ∈ E. We observe that lp (p > 1) has such a property, but Lp (p > 2) does not have this property. Let f : E → (−∞, +∞] be a convex function and x ∈ int(dom)f . The function f is said to be Gˆ ateaux differentiable at x if f (x + ty) − f (x) exists for any y ∈ E. lim t t→0+ Definition 2.3. Let f : E → R be a Gˆ ateaux differentiable convex function. The Bregman distance with respect to f is defined as: ′

∆f (x, y) = f (y) − f (x) − ⟨f (x), y − x⟩, x, y ∈ E. It is worth noting that the duality mapping JEp is in fact the derivative of the function fp (x) = ( p1 )∥x∥p . Then the Bregman distance with respect to fp is given by 1 1 ∥x∥p − ⟨JEp x, y⟩ + ∥y∥p q p 1 = (∥y∥p − ∥x∥p ) + ⟨JEp x, x − y⟩ p 1 = (∥x∥p − ∥y∥p ) − ⟨JEp x − JEp y, x⟩. q

∆p (x, y) =

In general, the Bregman distance is not a metric due to the absence of symmetry, but it has some distancelike properties. The following are some of important properties of the Bregman distance which are needed in the sequel ∆p (x, y) = ∆p (x, z) + ∆p (z, y) + ⟨z − y, JEp x − JEp z⟩,

(2.1)

∆p (x, y) + ∆p (y, x) = ⟨x − y, JEp x − JEp y⟩.

(2.2)

and

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For the p-uniformly convex space, the metric and Bregman distance has the following relation (see [20]): τ ∥x − y∥p ≤ ∆p (x, y) ≤ ⟨x − y, JEp x − JEp y⟩,

(2.3)

where τ > 0 is some fixed number. Let C be a nonempty closed convex subset of E. The metric projection PC x = arg min ∥x − y∥, x ∈ E, y∈C

is the unique minimizer of the norm distance which can be characterized by a variational inequality: ⟨JEp (x − PC x), z − PC x⟩ ≤ 0, ∀z ∈ C.

(2.4)

Similar to the metric projection, the Bregman projection is defined as ΠC x = arg min ∆p (x, y), x ∈ E, y∈C

which is well-defined and the minimizer of it is unique (for more details see [19]). The Bregman projection can also be characterized by a variational inequality: ⟨JEp (x) − JEp (ΠC x), z − ΠC x⟩ ≤ 0, ∀z ∈ C,

(2.5)

∆p (ΠC x, z) ≤ ∆p (x, z) − ∆p (x, ΠC x), ∀z ∈ C.

(2.6)

from which one has Following [1] and [7], we use of the function Vp : E ∗ × E → [0, +∞) associated with fp which is defined by Vp (¯ x, x) =

1 1 ∥¯ x∥q − ⟨¯ x, x⟩ + ∥x∥p , ∀x ∈ E, x ¯ ∈ E∗. q p

x), x) for all x ∈ E ∗ and y ∈ E. x, x) = ∆p (JEq ∗ (¯ Then Vp is nonnegative and Vp (¯ Moreover, by the subdifferential inequality, ′

⟨f (¯ x), x − x ¯⟩ ≤ f (¯ x) − f (x).

(2.7)

′

With f (x) = 1q ∥x∥q , x ∈ E ∗ , then f (x) = JEq ∗ , we have ⟨JEq ∗ (x), y⟩ ≤

1 1 ∥x − y∥q − ∥x∥q , ∀x, y ∈ E ∗ . q q

(2.8)

Using (2.8), we have for all x ¯, y¯ ∈ E ∗ and x ∈ E that 1 1 ∥¯ x + y¯∥q − ⟨¯ x + y¯, x⟩ + ∥x∥p q p 1 1 q q ≥ ∥¯ x∥ + ⟨¯ y , JE ∗ (¯ x)⟩ − ⟨¯ x + y¯, x⟩ + ∥x∥p q p 1 1 = ∥¯ x∥q − ⟨¯ x, x⟩ + ∥x∥p + ⟨¯ y , JEq ∗ (¯ x)⟩ q p

Vp (¯ x + y¯, x) =

+ ⟨¯ y , JEq ∗ (¯ x)⟩ − ⟨¯ y , x⟩ 1 1 x∥q − ⟨¯ x, x⟩ + ∥x∥p + ⟨¯ y , JEq ∗ (¯ x) − x⟩ = ∥¯ q p = Vp (¯ x, x) + ⟨¯ y , JEq ∗ (¯ x) − x⟩.

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In other words, Vp (¯ x, x) + ⟨¯ y , JEq ∗ (¯ x) − x⟩ ≤ Vp (¯ x + y¯, x),

(2.9)

for all x ∈ E and x ¯, y¯ ∈ E ∗ (see, for example, [23],[24]). Let C be a nonempty closed convex subset of a smooth Banach space E and let T be a mapping from C into itself. A point p ∈ C is said to be an asymptotic fixed point [16] of T if there exists a sequence {xn }n∈N in C which converges weakly to p and limn→∞ ∥xn − T xn ∥ = 0. We denote the set of all asymptotic fixed points of T by Fˆ (T ). Definition 2.4. Let C be a nonempty convex subset of int(domf ). A mapping T : C → int(domf ) with F (T ) ̸= ∅ is called to be (i) Bregman quasi-nonexpansive if ∆p (T x, x ¯) ≤ ∆p (x, x ¯), ∀x ∈ C, x ¯ ∈ F (T ); (ii) Bregman relatively nonexpansive if F (T ) = Fˆ (T ), ∆p (T x, x ¯) ≤ ∆p (x, x ¯), ∀x ∈ C, x ¯ ∈ F (T ); (iii) left Bregman strongly nonexpansive with respect to a nonempty Fˆ (T ) if ∆p (T x, x ¯) ≤ ∆p (x, x ¯), ∀x ∈ C, x ¯ ∈ Fˆ (T ), and if whenever {xn } ⊂ C is bounded, x ¯ ∈ Fˆ (T ) and lim (∆p (xn , x ¯) − ∆p (T xn , x ¯)) = 0,

n→∞

it follows that lim ∆p (xn , T xn ) = 0.

n→∞

It is obvious that any left Bregman strongly nonexpansive mapping is a Bregman relatively nonexpansive mapping, but the converse is not true in general. Pang et al. [9] showed that there exists a Bregman relatively nonexpansive mapping which is not a Bregman strongly nonexpansive mapping. Let N (C) and CB(C) denote the families of nonempty subsets and nonempty closed bounded subsets of C, respectively. The Hausdorff metric on CB(C) is defined by

H(A, B) = max{sup dist(x, B), sup dist(y, A)}, x∈A

y∈B

for all A, B ∈ CB(C) where dist(x, B) = inf{∥x − y∥ : y ∈ B} is the distance from a point x to a subset B. Recall that a multi-valued mapping T : C → CB(C) is said to be (i) nonexpansive if H(T x, T y) ≤ ∥x − y∥, for all x, y ∈ C; (ii) quasi-nonexpansive if F (T ) ̸= ∅ and H(T x, T p) ≤ ∥x − p∥, for all x ∈ C and p ∈ F (T ).

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Let T : C → CB(C). A point p ∈ C is said to be a fixed point of T if p ∈ F (T ) where F (T ) = {p ∈ T : p ∈ T p}. A point p ∈ C is said to be an asymptotic fixed point [16] of T if there exists a sequence {xn }n∈N in C which converges weakly to p and limn→∞ dist(xn , T xn ) = 0. Definition 2.5. [21] Let T : C → CB(C) is said to be Bregman relatively nonexpansive if the following conditions are satisfied: (A1) F (T ) is nonempty; (A2) ∆p (z, x ¯) ≤ ∆p (x, x ¯) for z ∈ T x, x ∈ C and x ¯ ∈ F (T ); (A3) F (T ) = Fˆ (T ). The following is the example of a multi-valued Bregman relatively nonexpansive mapping appeared in [21]: Example 2.6. [21] Let I = [0, 1], X = Lp (I), 1 < p < ∞ and C = {f ∈ X : f (x) ≥ 0, ∀x ∈ I}. Let T : C → CB(C) be defined by { {h ∈ C : f (x) − 21 ≤ h(x) ≤ f (x) − 14 , ∀x ∈ I} if f (x) > 1, ∀x ∈ I (2.10) {0}, otherwise. Then T is defined by (2.10) is a multi-valued Bregman relatively nonexpansive mapping. We next state the following lemmas which will be used in the sequel. Lemma 2.7. [5] Let E be a Banach space and f : E → R a Gˆ ateaux differentiable function which is locally uniformly convex on E. Let {xn }n∈N and {yn }n∈N be bounded sequences in E. Then the following assertions are equivalent (i) limn→∞ Df (xn , yn ) = 0; (ii) limn→∞ ∥xn − yn ∥ = 0. Lemma 2.8. [12] Let E be a Banach space, let r > 0 be a constant, and let f : E → R be a uniformly convex function on bounded subsets of E. Then f

n (∑ k=0

n ) ∑ αk xk ≤ αk f (xk ) − αi αj ρr (∥xi − yi ∥), k=0

for all i, j ∈ {0, 1, 2, ..., n}, xk ∈ Br , αk ∈ (0, 1), and k = 0, 1, 2, ..., n with gauge of uniform convexity of f .

∑n k=0

αk = 1,where ρr is the

Lemma 2.9. [27] Let {an } be a sequence of nonnegative real numbers satisfying an+1 ≤ (1 − αn )an + αn σn + γn , n ≥ 1, where (i) {αn } ⊂ [0, 1], n → ∞.

∑

αn = ∞; (ii) lim sup σn ≤ 0; (iii) γn ≥ 0; (n ≥ 1),

55

∑

γn < 0. Then, an → 0 as

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3

Main results

In this section, we introduce an iterative process for approximation of a common fixed point for a finite family of multi-valued Bregman relatively nonexpansive mappings with a solution of the split feasibility problems in p-uniformly convex and uniformly smooth Banach spaces and prove the strong convergence theorems of the proposed iterative process in p-uniformly convex and uniformly smooth Banach spaces Theorem 3.1. Let E1 and E2 be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let C and Q be nonempty closed convex subsets of E1 and E2 , respectively, A : E1 → E2 be a bounded linear operator and A∗ : E2∗ → E1∗ be the adjoint of A. Suppose that SFP has a nonempty solution set Ω. Let {Ti }N i=1 be a finite family of multi-valued bregman relative nonexpansive mappings of C into CB(C) such that F = ∩N i=1 F (Ti ) ∩ Ω ̸= ∅. Let u1 ∈ E1 and the sequence {xn } be generated by { xn = Πc JEq ∗ [JEp 1 (un ) − tn A∗ JEp 2 (Aun − PQ (Aun ))] 1 (3.1) ∑N (0) (i) (i) (i) un+1 = Πc JEq ∗ (αn JEp 1 (xn ) + i=1 αn JEp 1 (zn )) , zn ∈ Ti xn , 1 ∑N (i) (i) where {αn } ⊂ [a.b] ⊂ (0, 1) for all i = 0, 1, ..., N such that i=0 αn = 1. Suppose the following conditions are satisfied: (i) Σ∞ n=1 αn = 0 for all i = 0, 1, ..., N. ( q 1 q−1 . (ii) 0 < t ≤ tn ≤ k < Cq ∥A∥ q) (i)

∗ Then the sequence {xn }∞ n=1 converges strongly to an element x ∈ F .

Proof. Let x∗ ∈ Ω. Suppose that wn = Aun − PQ (Aun ) and vn = JEq ∗ [JEp 1 (un ) − tn A∗ JEp 2 (Aun − PQ (Aun ))], 1 ∀n ≥ 1. Therefore xn = Πc vn , ∀n ≥ 1. It follows that ⟨JEp 2 (wn ), Aun − Ax∗ ⟩ = ∥Aun − PQ (Aun )∥p + ⟨JEp 2 (wn ), PQ (Aun ) − Ax∗ ⟩ ≥ ∥Aun − PQ (Aun )∥p = ∥wn ∥p .

(3.2)

By Lemma 2.1, we obtain that ∆p (xn , x∗ ) ≤ ∆p (vn , x∗ ) = ∆p (JEq ∗ [JEp 1 (un ) − tn A∗ JEp 2 (wn )], x∗ ) 1

1 1 = ∥JEp 1 (un ) − tn A∗ JEp 2 (wn )∥q − ⟨JEp 1 (un ), x∗ ⟩ + tn ⟨JEp 2 (wn ), Ax∗ ⟩ + ∥x∗ ∥p q p 1 Cq (tn ∥A∥)q p ≤ ∥JEp 1 (un )∥q − tn ⟨Aun , JEp 2 (wn )⟩ + ∥JE2 (wn )∥q q q 1 − ⟨JEp 1 (un ), x∗ ⟩ + tn ⟨JEp 2 (wn ), Ax∗ ⟩ + ∥x∗ ∥p p 1 1 = ∥un ∥p − ⟨JEp 1 (un ), x∗ ⟩ + ∥x∗ ∥p + tn ⟨Aun , JEp 2 (wn )⟩ q p Cq (tn ∥A∥)q p + ∥JE2 (wn )∥q q Cq (tn ∥A∥)q p = ∆p (un , x∗ ) + tn ⟨JEp 2 (wn ), Ax∗ − Aun ⟩ + ∥JE2 (wn )∥q q ( Cq (tn ∥A∥)q ) = ∆p (un , x∗ ) + tn − ∥wn ∥p . q

56

(3.3)

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Using the condition(ii), we have ∆p (xn , x∗ ) ≤ ∆p (un , x∗ ) ∀n ≥ 1. Now, using (3.1), we have ∆p (xn+1 , x∗ ) ≤ ∆p (un+1 , x∗ ) ≤ αn(0) ∆p (xn , x∗ ) +

N ∑

αn(i) ∆p (zn(i) , x∗ )

i=1

≤ αn(0) ∆p (xn , x∗ ) +

N ∑

αn(i) ∆p (xn , x∗ )

i=1 ∗

= ∆p (xn , x ).

(3.4)

This shows that {∆p (xn , x∗ )} is a bounded decreasing sequence. Hence the limn→∞ ∆p (xn , x∗ ) exists and ∑N (0) (i) (i) thus limn→∞ (∆p (xn , x∗ ) − ∆p (xn+1 , x∗ )) = 0. Let yn = JEq ∗ (αn JEp 1 (xn ) + i=1 αn JEp 1 (zn )), n ≥ 1. 1 Therefore ∆p (xn+1 , x∗ ) ≤ ∆p (un+1 , x∗ ) = Vp (αn(0) JEp 1 (xn ) +

N ∑

αn(i) JEp 1 (zn(i) ), x∗ )

i=1

≤ Vp (αn(0) JEp 1 (xn ) + +

αn(0) ⟨JEp 1 (xn )

−

N ∑

αn(i) JEp 1 (zn(i) ) − αn(0) (JEp 1 (xn ) − JEp 1 (x∗ )), x∗ )

i=1 p JE1 (x∗ ), yn

= Vp (αn(0) JEp 1 (x∗ ) +

N ∑

− x∗ ⟩

αn(i) JEp 1 (zn(i) ), x∗ ) + αn(i) ⟨JEp 1 (xn ) − JEp 1 (x∗ ), yn − x∗ ⟩

i=1

= αn(0) Vp (JEp 1 (x∗ ), x∗ ) + + =

αn(0) ⟨JEp 1 (xn ) N ∑

−

N ∑

αn(i) Vp (JEp 1 (zn(i) ), x∗ )

i=1 p ∗ JE1 (x ), yn

− x∗ ⟩

αn(i) ∆p (zn(i) , x∗ ) + αn(0) ⟨JEp 1 (xn ) − JEp 1 (x∗ ), yn − x∗ ⟩

i=1

≤

N ∑

αn(i) ∆p (xn , x∗ ) + αn(0) ⟨JEp 1 (xn ) − JEp 1 (x∗ ), yn − x∗ ⟩.

(3.5)

i=1

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By Lemma 2.8, we obtain that ∆p (xn+1 , x∗ ) ≤ ∆p (un+1 , x∗ ) = Vp (αn(0) JEp 1 (xn ) +

N ∑

αn(i) JEp 1 (zn(i) ), x∗ )

i=1

=

N ∑ 1 ∗ q ∥x ∥ − ⟨αn(0) JEp 1 (xn ) + αn(i) JEp 1 (zn(i) ), x∗ ⟩ q i=1

N ∑ 1 + ∥αn(0) JEp 1 (xn ) + αn(i) JEp 1 (zn(i) )∥p p i=1

=

N ∑ 1 ∗ q ∥x ∥ − αn(0) ⟨JEp 1 (xn ), x∗ ⟩ − αn(i) ⟨JEp 1 (zn(i) ), x∗ ⟩ q i=1

N ∑ 1 + ∥αn(0) JEp 1 (xn ) + αn(i) JEp 1 (zn(i) )∥p p i=1

≤

N ∑ 1 ∗ q αn(i) ⟨JEp 1 (zn(i) ), x∗ ⟩ ∥x ∥ − αn(0) ⟨JEp 1 (xn ), x∗ ⟩ − q i=1

N ∑ 1 1 + αn(0) ∥JEp 1 (xn )∥p + αn(i) ∥JEp 1 (zn(i) )∥p − αn(i) αn(j) ρr (∥JEp 1 (xn ) − JEp 1 (zn(i) )∥) p p i=1

= αn(0) Vp (JEp 1 (xn ), x∗ ) +

N ∑

αn(i) Vp (JEp 1 (zn(i) ), x∗ ) − αn(i) αn(j) ρr (∥JEp 1 (xn ) − JEp 1 (zn(i) )∥)

i=1

= αn(0) ∆p (xn , x∗ ) +

N ∑

αn(i) ∆p (zn(i) , x∗ ) − αn(i) αn(j) ρr (∥JEp 1 (xn ) − JEp 1 (zn(i) )∥)

i=1

≤ αn(0) ∆p (xn , x∗ ) +

N ∑

αn(i) ∆p (xn , x∗ ) − αn(i) αn(j) ρr (∥JEp 1 (xn ) − JEp 1 (zn(i) )∥)

i=1

= ∆p (xn , x∗ ) − αn(i) αn(j) ρr (∥JEp 1 (xn )p − JEp 1 (zn(i) )∥). Thus αn(i) αn(j) ρr (∥JEp 1 (xn )∥p − JEp 1 (zn(i) )∥) ≤ ∆p (xn , x∗ ) − ∆p (xn+1 , x∗ ).

(3.6)

Then, from (3.6), we have αn(i) αn(j) ρr (∥JEp 1 (xn ) − JEp 1 (zn(i) )∥) → 0, n → ∞. By the property of ρr , we have lim ∥JEp 1 (xn ) − JEp 1 (zn(i) )∥ = 0.

n→∞

Since JEq ∗ is norm-to-norm uniformly continuous on bounded subsets of E1∗ , we have 1

lim ∥xn − zn(i) ∥ = 0.

n→∞ (i)

Since d(xn , Ti xn ) ≤ ∥xn − zn ∥, we have lim d(xn , Ti xn ) = 0,

n→∞

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for each i = {1, 2, ..., N }. Since {xn } is bounded, there exists a subsequence {xnj } of {xn } that converges weakly to z. Since Ti is a multi-valued Bregman relative nonexpansive mapping, we obtain z ∈ F (Ti ), for each i ∈ {1, 2, ..., N } and hence z ∈ ∩N i=1 F (Ti ). We now show that z ∈ Ω. From (3.3), we obtain that ( C (t ∥A∥)q ) q n ∥Aun − PQ (Aun )∥p ≤ ∆p (un , x∗ ) − ∆p (xn , x∗ ). q

(3.7)

∆p (un+1 , x∗ ) ≤ ∆p (xn , x∗ ).

(3.8)

( C (t ∥A∥)q ) q n ∥Aun − PQ (Aun )∥p ≤ ∆p (xn−1 , x∗ ) − ∆p (xn , x∗ ). q

(3.9)

From (3.4), we have

Putting (3.7) into (3.8), we have

By condition (ii) and (3.9), we have ( Cq k q−1 ∥A∥q ) 0 0 is sufficiently large number. It then follows from (2.4) that ∥(I − PQ )Az∥p = ⟨JEp 2 (Az − PQ (Az)), Az − PQ (Az)⟩ = ⟨JEp 2 (Az − PQ (Az)), Az − Aunj ⟩ + ⟨JEp 2 (Az − PQ (Az)), Aunj − PQ (Aunj )⟩ + ⟨JEp 2 (Az − PQ (Az)), PQ (Aunj ) − PQ (Az)⟩ ≤ ⟨JEp 2 (Az − PQ (Az)), Az − Aunj ⟩ + ⟨JEp 2 (Az − PQ (Az)), Aunj − PQ (Aunj )⟩. Also, since Aunj ⇀ Az, we have that lim ∥(I − PQ )Az∥ = 0.

n→∞

Thus Az ∈ Q. This implies that z ∈ Ω and hence z ∈ F (T ) ∩ Ω.Furthermore, we have ∆p (xn , yn ) ≤

αn(0) ∆p (xn , xn )

+

N ∑

αn(i) ∆p (xn , zn(i) ).

(3.12)

i=1 (i)

Since ∥xn − zn ∥ → 0 as n → ∞ and {zni } is a bounded sequence. By Lemma 2.7, we obtain that (i) limn→∞ ∆p (xn , zn ) = 0. From (3.12), it follows that ∥xn − yn ∥ → 0, n → ∞.

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Let p ∈ F (T ) ∩ Ω. We next show that lim supn→∞ ⟨JEp 1 (xn ) − JEp 1 (p), yn − p⟩ ≤ 0. To show the inequality lim supn→∞ ⟨JEp 1 (xn ) − JEp 1 (p), yn − p⟩ ≤ 0, we choose a subsequence {xnj } of {xn } such that lim sup⟨JEp 1 (xn ) − JEp 1 (p), xn − p⟩ = lim ⟨JEp 1 (xn ) − JEp 1 (p), xnj − p⟩ = 0. n→∞

n→∞

Since ∥xn − yn ∥ → 0 as n → ∞ and (2.5), we obtain that lim sup⟨JEp 1 (xn ) − JEp 1 (p), yn − p⟩ ≤ lim sup⟨JEp 1 (xn ) − JEp 1 (p), xn − p⟩ = 0. n→∞

(3.13)

n→∞

Using (3.13), (3.5) and Lemma 2.9, we obtain that ∆p (xn , p) → 0, n → ∞. Hence, xn → p as n → ∞. Corollary 3.2. Let E1 and E2 be two Lp spaces with 2 ≤ p < ∞. Let C and Q be nonempty closed convex subsets of E1 and E2 , respectively, A : E1 → E2 be a bounded linear operator and A∗ : E2∗ → E1∗ be the adjoint of A. Suppose that SFP has a nonempty solution set Ω. Let {Ti }N i=1 be a finite family of multi-valued Bregman relative nonexpansive mappings of C into CB(C) such that F = ∩N i=1 F (Ti ) ∩ Ω ̸= ∅. Let u1 ∈ E1 and the sequence {xn } be generated by { xn = Πc JEq ∗ [JEp 1 (un ) − tn A∗ JEp 2 (Aun − PQ (Aun ))] 1 (3.14) ∑N (i) (i) (i) (0) un+1 = Πc JEq ∗ (αn JEp 1 (xn ) + i=1 αn JEp 1 (zn )) , zn ∈ Ti xn , 1

(i) {αn }

⊂ [a.b] ⊂ (0, 1) for all i = 0, 1, ..., N such that where are satisfied:

∑N i=0

(i)

αn = 1. Suppose the following conditions

(i) Σ∞ n=1 αn = 0 for all i = 0, 1, ..., N ( q 1 q−1 . (ii) 0 < t ≤ tn ≤ k < Cq ∥A∥ q) (i)

∗ Then the sequence {xn }∞ n=1 converges strongly to an element x ∈ F .

If we assume that each Ti , i = 1, 2, ..., N, in Theorem 3.1 is a Bregman relative nonexpansive singlevalued mapping, we obtain the following corollary: Corollary 3.3. Let E1 and E2 be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let C and Q be nonempty closed convex subsets of E1 and E2 , respectively, A : E1 → E2 be a bounded linear operator and A∗ : E2∗ → E1∗ be the adjoint of A. Suppose that SFP has a nonempty solution set Ω. Let {Ti }N i=1 be a finite family of single-valued Bregman relative nonexpansive mappings of C into C such that F = ∩N i=1 F (Ti ) ∩ Ω ̸= ∅. Let u1 ∈ E1 and the sequence {xn } be generated by { xn = Πc JEq ∗ [JEp 1 (un ) − tn A∗ JEp 2 (Aun − PQ (Aun ))] 1 (3.15) ∑N (0) (i) un+1 = Πc JEq ∗ (αn JEp 1 (xn ) + i=1 αn JEp 1 (Ti xn )), 1

(i)

where {αn } ⊂ [a.b] ⊂ (0, 1) for all i = 0, 1, ..., N such that are satisfied:

∑N i=0

(i)

αn = 1. Suppose the following conditions

(i) Σ∞ n=1 αn = 0 for all i = 0, 1, ..., N ( q 1 q−1 . (ii) 0 < t ≤ tn ≤ k < Cq ∥A∥ q) (i)

∗ Then the sequence {xn }∞ n=1 converges strongly to an element x ∈ F .

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4

Numerical Example

In this section, we present the numerical example supporting our main result. All codes are written in Matlab2013b. Example 4.1. Let E1 = L2 ([0, 1]) = E2 with the inner product given by ∫

1

⟨f, g⟩ =

f (t)(g(t)dt. 0

Suppose that C := {x ∈ L2 ([0, 1]) : ⟨x, a⟩ = b}, where a = 2t2 and b = 0. Therefore { b − ⟨a, x⟩ } PC (x) = max 0, a + x. ∥a∥22 Let Q := {x ∈ L2 ([0, 1]) : ⟨x, c⟩ ≥ d}, where c =

t 3

and d = −2. It follows that PQ (x) :=

d − ⟨c, x⟩ c + x. ∥c∥22

Define

x(t) . 2 Then A is a bounded linear operator with ∥A∥ = 2 and A∗ = A. Suppose that { {h ∈ C : f (x) − 34 ≤ h(x) ≤ f (x) − 13 , ∀x ∈ I} if f (x) > 1, ∀x ∈ I T1 (f ) {0}, otherwise, A : L2 ([0, 1] → L2 ([0, 1] by (Ax)(t) =

{

and T2 (f )

{g ∈ C : f (x) − {0}, otherwise.

1 2

≤ g(x) ≤ f (x) − 14 , ∀x ∈ I} if f (x) > 1, ∀x ∈ I

(4.1)

(4.2)

In [21], we obtain that T1 and T2 are multi-valued Bregman relative nonexpansive mappings. Consider the problem: find x ∈ F (T ) ∩ C such that Ax ∈ Q. (4.3) We see that the set of solutions of problem (4.3) is nonempty, since x = 0 is in the set of solutions. Let (1) (2) (1) (2) (0) 12n−1 3 1 1 , αn = 12n−1 αn = 12n 36n , and αn = 18n for all n ≥ 1. Put zn = xn − 4 and zn = xn − 2 . Using the iterative method (3.1), we obtain that { xn = Πc [un − tn A∗ (Aun − PQ (Aun ))] (4.4) 3 12n−1 1 1 un+1 = Πc ( 12n (xn ) + 12n−1 36n (xn − 4 ) + 18n (xn − 2 )), n ≥ 1. We make different choices of u1 and tn and take

∥xn+1 −xn ∥ ∥x2 −x1 ∥

62

< 10−6 as our stopping criterion.

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Case 1 tn = 0.001 and u1 = t. We have the numerical analysis tabulated in Table 1 and show in Figure 1.

Table 1 No.of iteration 2 3 4 5 6 7 8 9

Example 4.1: Case 1 ∥xn+1 − xn ∥2 ∥un+1 − un ∥2 0.45960659 0.45871914 0.03706339 0.03979071 0.00089775 0.00150921 0.00002339 0.00002339 0.00000070 0.00000210 0.00000043 0.00000141 0.00000030 0.00000100 0.00000023 0.00000075

Figure 1. Example 4.1: Case 1.

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Case 2 tn = 0.0002 and u1 = t2 . We have the numerical analysis tabulated in Table 2 and show in Figure 2.

Table 2 No.of iteration 2 3 4 5 6 7 8 9

Example 4.1: Case 2 ∥xn+1 − xn ∥2 ∥un+1 − un ∥2 0.30581518 0.30563219 0.02659138 0.02659287 0.0008344 0.00110931 0.00002409 0.00002388 0.00000053 0.00000064 0.00000009 0.00000028 0.00000006 0.00000020 0.00000005 0.00000015

4.jpg 4.bb Figure 2. Example 4.1: Case 2.

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[16] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. 17 313-318, Marcel Dekker, New York, NY, USA, 1996. [17] S. Reich, S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear Anal. 73 122–135 (2010) [18] V. Mart´in-M´ arquez, S. Reich, S. Sabach, Bregman strongly nonexpansive operators in reflexive Banach spaces. J. Math. Anal. Appl. 400 597-614 (2013) [19] F. Sch¨opfer, Iterative regularization method for the solution of the split feasibility problem in Banach spaces, PhD thesis, Saarbrucken (2007) [20] F. Schpfer, T. Schuster, A.K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Probl. 2008;24:055008. [21] N. Shahzad, H. Zegeye, Convergence theorem for common fixed points of a finite family of multivalued Bregman relatively nonexpansive mappings. Fixed Point Theory Appl. 2014, 2014:152 [22] Y. Shehu, O. S. Iyiola, C. D. Enyi, An iterative algorithm for solving split feasibility problems and fixed point problems in Banach spaces.Numer. Algor. DOI 10.1007/s11075-015-0069-4. [23] Y. Shehu, Strong convergence theorem for Multiple Sets Split Feasibility Problems in Banach Spaces, Under review: Numerical Functional Analysis and Optimization [24] Y. Shehu, A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces, Under review: Quaestiones Mathematicae [25] F. Wang and H.K. Xu, Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem, Journal of Inequalities and Applications, vol. 2010, Article ID102085, 13 pages, 2010. [26] F. Wang, A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces,” Numer. Funct. Anal. Optim. 35 99-110 (2014) [27] H.K. Xu, Inequalities in Banach spaces with applications,” Nonlinear Anal. 16(2), 1127-1138 (1991) [28] H.K. Xu, A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems. 22 (6) 2021-2034 (2006) [29] J. Zhao and Q. Yang, Several solution methods for the split feasibility problem, Inverse Problems. 21 (5) 1791- 1799 (2005)

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Expressions and Dynamical Behavior of Rational Recursive Sequences E. M. Elsayed1,2 and Marwa M. Alzubaidi1,3 King Abdulaziz University, Faculty of Science, Mathematics Department, P.O. Box 80203, Jeddah 21589,Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University,Mansoura35516, Egypt. 3 Mathematics Department , The University College of Duba, University of Tabuk, Tabuk, Saudi Arabia. E-mail: [email protected], [email protected]. 1

ABSTRACT In this paper, we study the qualitative behavior of the rational recursive sequences xn+1 =

xn−11 , ±1 ± xn−2 xn−5 xn−8 xn−11

n = 0, 1, 2, ...,

where the initial conditions are arbitrary real numbers. Also, we give the numerical examples of some cases of difference equations and obtained some related graphs and figures using by Matlab. Keywords: Difference Equation, Recursive sequence, Local stability, Periodicity. Mathematics Subject Classification: 39A10. –––––––––––––––––––

1. INTRODUCTION Difference equations and dynamic equations on time scales have an immense possibility for applications in engineering, physics, biology, economics, etc. Lately, considerable attentiveness has been devoted to the oscillation theory of the various classes of equations,see e.g. [1]-[42] and the references cited therein. In this study, we are interested with the behavior of the solution of difference equations xn+1 =

xn−11 , ±1 ± xn−2 xn−5 xn−8 xn−11

n = 0, 1, 2, ...,

(1)

where the initial conditions are arbitrary real numbers. For some outcome in this study for examples: Cinar [8-10] obtained the solutions of the difference equations xn+1 =

xn−1 , 1 + xn xn−1

xn+1 =

xn−1 , −1 + xn xn−1

xn+1 =

axn−1 . 1 + bxn xn−1

Cinar et al. [11] gave the form of the solution of the difference equation xn+1 =

xn−3 . −1 + xn xn−1 xn−2 xn−3

Elabbasy et al. [13] solved the following problem αxn−k

xn+1 =

k

.

β + γ Π xn−i i=0

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In [14] Elsayed studied the difference equation xn+1 =

xn−5 . −1 + xn−2 xn−5

Elsayed [21-22] obtained the solutions of the following difference equations xn+1 =

xn−7 , ±1 ± xn−1 xn−3 xn−5 xn−7

xn+1 =

xn−9 . ±1 ± xn−4 xn−9

Elsayed [23] investigated the Solution of difference equations xn+1 =

xn−3 . ±1 ± xn−1 xn−3

Elsayed and Iricanin [24] has got the solution of the difference equation xn+1 = max {An /xn , xn−1 } . Ibrahim [26] studied the third order rational difference equation xn+1 =

xn xn−2 . xn−1 (a + bxn xn−2 )

In [30] Kent et al studied the Behavior of solutions of the difference equation xn+1 = xn xn−2 − 1. Let I be some interval of real numbers and let F : I k+1 → I, be a continuously differentiable function. Then for every set of initial condition x−k , x−k+1 , ..., x0 ∈ I, the difference equation

xn+1 = F (xn , xn−1 , xn−2 , ..., xn−k ),

n = 0, 1, ...,

(2)

has a unique solution {xn }∞ n=−k .

Definition 1. A point x ∈ I is called an equilibrium point of Eq.(2) if x = F (x), that is, xn = x f or all

n ≥ −k.

is a solution of Eq.(2), or equivalently, x is a fixed point of F. Definition 2. (Periodicity) A sequence {xn }∞ n=−k is said to be periodic with period p if xn+p = xn for all n ≥ −k. Linearized Stability Analysis

Suppose that the function F is continuously differentiable in some open neighborhood of an equilibrium point x∗ . Let ∂F pi = (x, x, ..., x) f or i = 0, 1, ..., k, ∂ui denote the partial derivatives of F (u0 , u1 , ....uk ) evaluated at the equilibrium x of Eq.(2). Then the equation yn+1 = p0 yn + p1 yn−1 + ... + pk yn−k ,

68

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(3)

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is called the linearized equation associated of Eq.(2) about the equilibrium point x and the equation λk+1 − p0 λk − ... − pk−1 λ − pk = 0,

(4)

is called the characteristic equation of Eq.(3) about x. The following result known as the Linear Stability Theorem is very useful in determining the local stability character of the equilibrium point x of Eq.(2). Definition 3. The equilibrium point x is said to be hyperbolic if |F (x)| 6= 1. If |F (x)| = 1, x is non hyperbolic.

Theorem A. [31] Assume that p0 , p2 , ..., pk are real numbers such that |p0 | + |p1 | + ... + |pk | < 1,

or

k X i=1

|pi | < 1.

Then all roots of Eq.(4) lie inside the unit disk.

2. THE FIRST EQUATION XN+1 =

XN −11 1+XN −2 XN −5 XN −8 XN −11

In this part, we obtain the following special case of Eq.(1) in the form: xn+1 =

xn−11 , 1 + xn−2 xn−5 xn−8 xn−11

(5)

where the initial values are arbitrary real numbers. Theorem 2.1. Let {xn }∞ n=−11 be a solution of difference equation (5). Then for n = 0, 1, ... x12n−11

n−1 Y

= p

n−1 n−1 Y Y 1 + 4ipkf c 1 + 4imheb 1 + 4ildga , x12n−10 = m , x12n−9 = l , 1 + (4i + 1) pkf c 1 + (4i + 1) mheb 1 + (4i + 1) ldga i=0 i=0 i=0

n−1 Y

x12n−8

n−1 n−1 Y 1 + (4i + 1) mheb Y 1 + (4i + 1) ldga 1 + (4i + 1) pkf c , x12n−7 = h , x12n−6 = g , = k 1 + (4i + 2) pkf c 1 + (4i + 2) mheb 1 + (4i + 2) ldga i=0 i=0 i=0

x12n−5

= f

x12n−2

= c

n−1 Y

n−1 n−1 Y 1 + (4i + 2) mheb Y 1 + (4i + 2) ldga 1 + (4i + 2) pkf c , x12n−4 = e , x12n−3 = d , 1 + (4i + 3) pkf c 1 + (4i + 3) mheb 1 + (4i + 3) ldga i=0 i=0 i=0

n−1 Y

n−1 n−1 Y 1 + (4i + 3) mheb Y 1 + (4i + 3) ldga 1 + (4i + 3) pkf c , x12n−1 = b , x12n = a , 1 + (4i + 4) pkf c 1 + (4i + 4) mheb 1 + (4i + 4) ldga i=0 i=0 i=0

where x−11 = p, x−10 = m, x−9 = l, x−8 = k, x−7 = h, x−6 = g, x5 = f, x−4 = e, x−3 = d, x−2 = c, −1 Y αi = 1. x−1 = b, x0 = a and i=0

Proof. For n = 0, the result holds. Now, assume that n > 0 and that our assumption holds for n − 1. That is, n−2 Y

x12n−23

n−2 n−2 Y Y 1 + 4ipkf c 1 + 4imheb 1 + 4ildga , x12n−22 = m , x12n−21 = l , = p 1 + (4i + 1) pkf c 1 + (4i + 1) mheb 1 + (4i + 1) ldga i=0 i=0 i=0

x12n−20

= k

x12n−17

= f

n−2 Y

n−2 n−2 Y 1 + (4i + 1) mheb Y 1 + (4i + 1) ldga 1 + (4i + 1) pkf c , x12n−19 = h , x12n−18 = g , 1 + (4i + 2) pkf c 1 + (4i + 2) mheb 1 + (4i + 2) ldga i=0 i=0 i=0

n−2 Y

n−2 n−2 Y 1 + (4i + 2) mheb Y 1 + (4i + 2) ldga 1 + (4i + 2) pkf c , x12n−16 = e , x12n−15 = d , 1 + (4i + 3) pkf c 1 + (4i + 3) mheb 1 + (4i + 3) ldga i=0 i=0 i=0

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n−2 Y

x12n−14 = c

n−2 n−2 Y 1 + (4i + 3) mheb Y 1 + (4i + 3) ldga 1 + (4i + 3) pkf c , x12n−13 = b , x12n−12 = a . 1 + (4i + 4) pkf c 1 + (4i + 4) mheb 1 + (4i + 4) ldga i=0 i=0 i=0

Now, it follows from Eq. (5) that x12n−11

=

x12n−23 1 + x12n−14 x12n−17 x12n−20 x12n−23 n−2 Y

1+4ipkf c 1+(4i+1)pkf c

p

=

i=0

Y

n−2

1+c

i=0

p

Y

n−2

1+(4i+3)pkf c f 1+(4i+4)pkf c

i=0

n−2 Y

1+4ipkf c 1+(4i+1)pkf c

n−2 Y

1 + pkf c

n−2 Y

=p

i=0

n−2 Y

1 + 4ipkf c = p 1 + (4i + 1) pkf c i=0

µ

1+4ipkf c 1+(4i+1)pkf c

i=0

1+4ipkf c 1+(4i+1)pkf c

i=0

1+4ipkf c 1+(4i+4)pkf c

Y

n−2 1+(4i+1)pkf c p 1+(4i+2)pkf c

i=0

i=0

=

Y

n−2

1+(4i+2)pkf c k 1+(4i+3)pkf c

1 + (4n − 4) pkf c 1 + (4n − 3) pkf c

µ

1 pkf c 1+ 1+(4n−4)pkf c

¶

¶

Therefore, we have x12n−11 = p Similarly x12n−7

=

n−1 Y

1 + 4ipkf c . 1 + (4i + 1) pkf c i=0

x12n−19 1 + x12n−10 x12n−13 x12n−16 x12n−19 n−2 Y

1+(4i+1)mheb 1+(4i+2)mheb

h

=

i=0

Y

n−1

1+m

Y

n−2

1+4imheb 1+(4i+1)mheb b

i=0

i=0

n−2 Y

h

Y

n−2

1+(4i+3)mheb 1+(4i+4)mheb e

Y

n−2 1+(4i+2)mheb 1+(4i+3)mheb h

i=0

1+(4i+1)mheb 1+(4i+2)mheb

i=0

1+(4i+1)mheb 1+(4i+2)mheb

i=0

=

n−1 Y

1 + mheb

1+4imheb 1+(4i+1)mheb

i=0

n−2 Y i=0

n−2 Y

1 + (4i + 1) mheb = h 1 + (4i + 2) mheb i=0 n−2 Y

= h

1+(4i+1)mheb 1+(4i+4)mheb

1 + (4i + 1) mheb 1 + (4i + 2) mheb i=0

!

Ã

1+

µ

1 + (4n − 3) mheb 1 + (4n − 2) mheb

1 mheb 1+(4n−3)mheb

¶

.

Hence, we have n−1 Y

x12n−7 = h

1 + (4i + 1) mheb . 1 + (4i + 2) mheb i=0

Similarly, other relations can be obtained and thus, the proof has been proved.

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Theorem 2.2. Eq.(5) has unique equilibrium point which is the number zero and this equilibrium is not locally asymptotically stable. Also, x is non hyperbolic. Proof. For the equilibrium points of Eq.(5), we can write x=

x , 1 + x4

Then x + x5 = x, or x5 = 0. Then the unique equilibrium point of Eq.(5) is x = 0. Let f : (0, ∞)4 → (0, ∞) be a function defined by F (u, v, w, t) =

u . 1 + uvwt

Then it follows that, Fu (u, v, w, t) = Fw (u, v, w, t) =

1

2,

(1 + uvwt) −u2 vt

(1 + uvwt)2

Fv (u, v, w, t) =

, Ft (u, v, w, t) =

−u2 wt

2,

(1 + uvwt) −u2 vw

(1 + uvwt)2

,

we see that Fu (x, x, x, x) = 1, Fv (x, x, x, x) = 0, Fw (x, x, x, x) = 0, Ft (x, x, x, x) = 0. The proof follows by using Theorem A. By Definition 3, x is non hyperbolic. Theorem 2.3. Every positive solution of Eq.(5) is bounded and lim xn = 0. n→∞

Proof. It is following by Eq.(5) that xn+1 =

xn−11 ≤ xn−11 . 1 + xn−2 xn−5 xn−8 xn−11

Then xn+1 < xn−11 , Then the subsequences bounded from above by

∞ {x12n−11 }n=0

,

∞ {x12n−10 }n=0

f or all ,

n≥0

∞ {x12n−9 }n=0

∞

, ..., {x12n }n=0 are decreasing and so are

M = max {x−11 , x−10 , x−9 , x−8 , x−7 , x−6 , x−5 , x−4 , x−3 , x−2 , x−1 , x0 } .

3. THE SECOND EQUATION XN+1 =

XN −11 −1+XN −2 XN −5 XN −8 XN −11

In this part, we give the solution of the recursive equation in the form: xn+1 =

xn−11 , −1 + xn−2 xn−5 xn−8 xn−11

(6)

where the initial values are arbitrary real numbers with x−2 x−5 x−8 x−11 6= 1, x−1 x−4 x−7 x−10 6= 1, x0 x−3 x−6 x−9 6= 1.

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Theorem 3.1. Let {xn }∞ n=−11 be a solution of difference equation (6). Then for n = 0, 1, ... x12n−11 x12n−8 x12n−5 x12n−2

p m l n , x12n−10 = n , x12n−9 = n, (−1 + pkf c) (−1 + mheb) (−1 + ldga) = k (−1 + pkf c)n , x12n−7 = h (−1 + mheb)n , x12n−6 = g (−1 + ldga)n , f e d = n , x12n−4 = n , x12n−3 = n, (−1 + pkf c) (−1 + mheb) (−1 + ldga) = c (−1 + pkf c)n , x12n−1 = b (−1 + mheb)n , x12n = a (−1 + ldga)n , =

where x−11 = p, x−10 = m, x−9 = l, x−8 = k, x−7 = h, x−6 = g, x5 = f, x−4 = e, x−3 = d, x−2 = c, x−1 = b, x0 = a. Proof. For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. That is p

x12n−20

= k (−1 + pkf c) , x12n−19 = h (−1 + mheb) , x12n−18 = g (−1 + ldga) f e d = , n−1 , x12n−116 = n−1 , x12n−15 = (−1 + pkf c) (−1 + mheb) (−1 + ldga)n−1

x12n−14

(−1 + pkf c)

x12n−22 =

n−1 ,

(−1 + mheb)

n−1

= c (−1 + pkf c)

n−1

x12n−21 =

l

=

x12n−17

n−1 ,

m

x12n−23

(−1 + ldga)n−1

n−1

n−1

, x12n−13 = b (−1 + mheb)

, n−1

, x12n−12 = a (−1 + ldga)

n−1

,

.

Now, it follows from Eq.(6) that x12n−11

= = =

x12n−23 −1 + x12n−14 x12n−17 x12n−20 x12n−23 −1 + c (−1 +

p (−1+pkf c)n−1 f pkf c)n−1 (−1+pkf k (−1 c)n−1

p

n−1

(−1 + pkf c)

(−1 + pkf c)

Then x12n−11 =

+ pkf c)n−1

p (−1+pkf c)n−1

.

p n. (−1 + pkf c)

Similarly x12n−6

= =

= Therefore, we have

x12n−18 −1 + x12n−9 x12n−12 x12n−15 x12n−18

g (−1 + ldga)n−1

−1 +

l (−1+ldga)n a (−1

+ ldga)

n−1

d g (−1 (−1+ldga)n−1

n−1

+ ldga)

n−1

g (−1 + ldga)

−1 + ldga (−1 + ldga)−1

.

x12n−6 = g (−1 + ldga)n .

The same other relations can be proved and thus, the proof has been completed. √ Theorem 3.2. Eq.(6) has three equilibrium points which are 0, ± 4 2 and these equilibrium points are not locally asymptotically stable. Proof. The proof is the same as Theorem 2.2.

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Theorem 3.3. Eq.(6) has a periodic solutions of period twelve if f pkf c = mheb = ldga = 2 and will be take the form {p, m, l, k, h, g, f, e, d, c, b, a, p, m, l, k, h, g, f, e, d, c, b, a, ...} . Proof. Assume that there exists a prime twelve solutions p, m, l, k, h, g, f, e, d, c, b, a, p, m, l, k, h, g, f, e, d, c, b, a, ..., of Eq.(6) ,we have from Eq.(6) that m l p , n, m = n, l = (−1 + pkf c) (−1 + mheb) (−1 + ldga)n n n n k = k (−1 + pkf c) , h = h (−1 + mheb) , g = g (−1 + ldga) , e d f f = n, e = n, d = n, (−1 + pkf c) (−1 + mheb) (−1 + ldga) c = c (−1 + pkf c)n , b = b (−1 + mheb)n , a = a (−1 + ldga)n , p =

or

(−1 + pkf c)n = 1,

(−1 + mheb)n = 1,

(−1 + ldga)n = 1

Then pkf c = mheb = ldga = 2. Second let pkf c = mheb = ldga = 2. Then we have from Eq.(6) that x12n−11 x12n−7 x12n−3

= p, x12n−10 = m, x12n−9 = l, x12n−8 = k, = h, x12n−6 = g, x12n−5 = f, x12n−4 = e, = d, x12n−2 = c, x12n−1 = b, x12n = a.

Therefore we have a period twelve solutions and the proof is complete.

4. THE THIRD EQUATION XN+1 =

XN −11 1−XN −2 XN −5 XN −8 XN −11

In this section we examine the following equation xn+1 =

xn−11 , 1 − xn−2 xn−5 xn−8 xn−11

(7)

where the initial conditions are arbitrary positive real numbers. Theorem 4.1. Let {xn }∞ n=−11 be a solution of difference equation (7). Then for n = 0, 1, ... x12n−11

n−1 Y

= p

n−1 n−1 Y Y 1 − 4ipkf c 1 − 4imheb 1 − 4ildga , x12n−10 = m , x12n−9 = l , 1 − (4i + 1) pkf c 1 − (4i + 1) mheb 1 − (4i + 1) ldga i=0 i=0 i=0

n−1 Y

x12n−8

n−1 n−1 Y 1 − (4i + 1) mheb Y 1 − (4i + 1) ldga 1 − (4i + 1) pkf c , x12n−7 = h , x12n−6 = g , = k 1 − (4i + 2) pkf c 1 − (4i + 2) mheb 1 − (4i + 2) ldga i=0 i=0 i=0

x12n−5

= f

x12n−2

= c

n−1 Y

n−1 n−1 Y 1 − (4i + 2) mheb Y 1 − (4i + 2) ldga 1 − (4i + 2) pkf c , x12n−4 = e , x12n−3 = d , 1 − (4i + 3) pkf c 1 − (4i + 3) mheb 1 − (4i + 3) ldga i=0 i=0 i=0

n−1 Y

n−1 n−1 Y 1 − (4i + 3) mheb Y 1 − (4i + 3) ldga 1 − (4i + 3) pkf c , x12n−1 = b , x12n = a , 1 − (4i + 4) pkf c 1 − (4i + 4) mheb 1 − (4i + 4) ldga i=0 i=0 i=0

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where x−11 = p, x−10 = m, x−9 = l, x−8 = k, x−7 = h, x−6 = g , x5 = f, x−4 = e, x−3 = d, x−2 = c, x−1 = b, x0 = a and δpkf c 6= 1, δmheb 6= 1, δldga 6= 1 for δ = 1, 2, 3, .... Proof. The proof is similar as the proof of the Theorem 2.1.

Theorem 4.2. Eq.(7) has unique equilibrium point which is the number zero and this equilibrium is not locally asymptotically stable.

5. THE FOURTH EQUATION XN+1 = Here we obtain a form of the solutions of the equation xn+1 =

XN −11 −1−XN −2 XN −5 XN −8 XN −11

xn−11 −1 − xn−2 xn−5 xn−8 xn−11

(8)

where the initial values are arbitrary non zero real numbers with x−2 x−5 x−8 x−11 6= −1, x−1 x−4 x−7 x−10 6= −1, x0 x−3 x−6 x−9 6= −1. Theorem 5.1. Suppose {xn }∞ n=−11 be a solution of difference equation xn+1 = n = 0, 1, ... x12n−11 x12n−8 x12n−5 x12n−2

xn−11 −1−xn−2 xn−5 xn−8 xn−11 ,Then

for

p m l , x12n−10 = , x12n−9 = , (−1 − pkf c)n (−1 − mheb)n (−1 − ldga)n n n n = k (−1 − pkf c) , x12n−7 = h (−1 − mheb) , x12n−6 = g (−1 − ldga) , f e d = n , x12n−4 = n , x12n−3 = n, (−1 − pkf c) (−1 − mheb) (−1 − ldga) = c (−1 − pkf c)n , x12n−1 = b (−1 − mheb)n , x12n = a (−1 − ldga)n , =

where x−11 = p, x−10 = m, x−9 = l, x−8 = k, x−7 = h, x−6 = g , x5 = f, x−4 = e, x−3 = d, x−2 = c, x−1 = b, and x0 = a. √ Theorem 5.2 Eq.(8) has three equilibrium points which are 0, ± 4 −2 and these equilibrium points are not locally asymptotically stable. Proof. The proof as the proof of Theorem 3.3. Theorem 5.3. Eq.(8) has a periodic solutions of period twelve if f pkf c = mheb = ldga = −2 and will be take the form {p, m, l, k, h, g, f, e, d, c, b, a, p, m, l, k, h, g, f, e, d, c, b, a, ...} .

6. NUMERICAL EXAMPLES To verify the results of this paper, we consider some numerical examples as follows. Example 6.1 The graph of the difference equation (5) and the case when x−11 = 3.3, x−10 = 1.7, x−9 = 2.6, x−8 = 5, x−7 = 3, x−6 = 11, x5 = 6, x−4 = 2, x−3 = 7, x−2 = 9, x−1 = 4.6 and x0 = 1.6.shown in Figure 1. plot of x(n+1)=x(n-11)/(1+x(n-2)x(n-5)x(n-8)x(n-11)

12 10

x(n)

8 6 4 2 0

0

10

20

30

40

50

60

70

80

n

Figure 1.

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Example 6.2. In Figure 2, we show that for Eq.(5) that x−11 = 4.1, x−10 = 2, x−9 = 3.2, x−8 = 6, x−7 = −1, x−6 = 2.4, x5 = 1, x−4 = 4.2, x−3 = 7, x−2 = 11, x−1 = 4 and x0 = −2. plot of x(n+1)=x(n-11)/(1+x(n-2)x(n-5)x(n-8)x(n-11)

12 10

x(n)

8 6 4 2 0 -2

0

10

20

30

40

50

60

70

80

n

Figure 2. Example 6.3. The graph is shown of the solutions of Eq.(6) where x−11 = 3, x−10 = −2, x−9 = 9, x−8 = −5, x−7 = 8, x−6 = 2, x5 = 4, x−4 = 4, x−3 = −4, x−2 = −1/30, x−1 = −1/32 and x0 = −1/36.in Figure 3. plot of x(n+1)=x(n-11)/(-1+x(n-2)x(n-5)x(n-8)x(n-11)

10

x(n)

5

0

-5

0

10

20

30

40

50

60

70

80

n

Figure 3. Example 6.4. Figure 4 shows the behavior of difference equation.(6) when we choose x−11 = 5, x−10 = −2, x−9 = 6, x−8 = −1, x−7 = 4, x−6 = −11, x5 = 6, x−4 = 2, x−3 = 7, x−2 = −1/15, x−1 = −1/8 and x0 = −1/231. plot of x(n+1)=x(n-11)/(-1+x(n-2)x(n-5)x(n-8)x(n-11)

10 5

x(n)

0 -5 -10 -15

0

10

20

30

40

50

60

70

80

n

Figure 4.

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n−11 Example 6.5. The diagram of the difference equation defined by xn+1 = 1−xn−2 xxn−5 xn−8 xn−11 shows the period thirty six solutions since x−11 = 1, x−10 = 3.5, x−9 = −4, x−8 = 6, x−7 = −2, x−6 = 2.4, x5 = −1, x−4 = 1.2, x−3 = 8, x−2 = 10, x−1 = −3 and x0 = 4. in Figure 5

plot of x(n+1)=x(n-11)/(1-x(n-2)x(n-5)x(n-8)x(n-11)

10 8

x(n)

6 4 2 0 -2 -4

0

10

20

30

40

50

60

70

80

n

Figure 5. Example 6.6. See Figure 6, we suppose for Eq.(7), that x−11 = 4.3, x−10 = 8.1, x−9 = −3, x−8 = 2.7, x−7 = −1, x−6 = 2.4, x5 = 3, x−4 = 1.5, x−3 = 11, x−2 = −2, x−1 = 5 and x0 = −2. plot of x(n+1)=x(n-11)/(1-x(n-2)x(n-5)x(n-8)x(n-11)

15

x(n)

10

5

0

-5

0

10

20

30

40

50

60

70

80

n

Figure 6. Example 6.7.(see Figure 7) shows the period thirty six solutions of Eq.(8) since x−11 = 3, x−10 = 9, x−9 = −6, x−8 = 2, x−7 = 1, x−6 = 4, x5 = 5, x−4 = −4, x−3 = 3, x−2 = −1/15, x−1 = 1/18 and x0 = 1/36. plot of x(n+1)=x(n-11)/(-1-x(n-2)x(n-5)x(n-8)x(n-11)

10

x(n)

5

0

-5

-10

0

10

20

30

40

50

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70

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n

Figure 7.

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Example 6.8. (See Figure 8) , we suppose for difference equation (8), that x−11 = 11, x−10 = 3, x−9 = −5, x−8 = −2, x−7 = 4, x−6 = 2, x5 = 9, x−4 = −2, x−3 = 7, x−2 = 1/99, x−1 = 1/12 and x0 = 1/35. plot of x(n+1)=x(n-11)/(-1-x(n-2)x(n-5)x(n-8)x(n-11)

15 10

x(n)

5 0 -5 -10

0

10

20

30

40

50

60

70

80

n

Figure 8.

REFERENCES 1. M. A. E. Abdelrahman and O. Moaaz,ON The New Class of The Nonlinear Rational Difference Equations, Electronic Journal of Mathematical Analysis and Applications, 6 (1) (2018), 117-125. 2. R P Agarwal, Difference Equations and Inequalities. 1st edition, Marcel Dekker, New York, 1992, 2nd edition, 2000. 3. R. P. Agarwal,and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Advanced Studies in Contemporary Mathematics, 17(2) (2008), 181-201. 4. M. Aloqeili, Dynamics of a Rational Difference Equation, Appl. Math. Comput., 176 (2006), 768-774. 5. A. M. Amleh, J. Hoag and G. Ladas, A difference equation with eventually periodic solutions, Comput. Math. Appl., 36 (10-12), 1998, 401-404. 6. L. Berezansky, E. Braverman, and E. Liz. Sufficient conditions for the global stability of nonautonomous higher order difference equations, Journal of Difference Equations and Applications,vol. 11, no. 9, pp. 785— 798, 2005. 7. L. Berg, Oscillating solutions of rational difference equations, Rostocker Mathematisches Kolloquium, vol. 58, pp. 31—35, 2004. 8. C. Cinar, On the positive solutions of the difference equation xn+1 = xn−1 /1 + xn xn−1 , Appl. Math. Comput., 150 (2004), 21—24. 9. C. Cinar, On the difference equation xn+1 = xn−1 /−1+xn xn−1 , Appl. Math. Comput., 158 (2004), 813—816 10. C. Cinar, On the positive solutions of the difference equation xn+1 = axn−1 /1+bxn xn−1 Appl.Math. Comp., 156:587-590, 2004. 11. C. Cinar, R Karatas and I Yalcinkaya, On solutions of the difference equation xn+1 = xn−3 / − 1 + xn xn−1 xn−2 xn−3 Mathematica Bohemica, 132(3):257-261, 2007. 12. R. Devault, V. L. Kocic, and D. Stutson, Global behavior of solutions of the nonlinear difference equation xn+1 = pn + xn−1 /xn , J. Differ. Equ. Appl., 11 (8) (2005), 707-719. Tkn−k 13. E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation xn+1 = β+γ αx , Journal x i=0

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of Concrete and Applicable Mathematics, 5(2) (2007),101 113. 14. E. M. Elsayed, On The Difference Equation xn+1 = xn−5 / − 1 + xn−2 xn−5 , Inter. J. Contemp. Math. Sci., 3 (33) (2008), 1657-1664. 15. E. M. Elsayed, A Solution Form of a Class of Rational Difference Equations, International Journal of Nonlinear Science, 8 (4) (2009), 402-411. 16. E. M. Elsayed, On the Global attractivity and the solution of recursive sequence. Studia Scientiarum Mathematicarum Hungarica, 47 (3) (2010), 401-418.

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17. E. M. Elsayed, On the global attractivity and the periodic character of a recursive sequence, Opuscula Mathematica, 30 (4) (2010), 431—446. 18. E. M. Elsayed, On the Solutions of a Rational System of Difference Equations, Fasciculi Mathematici, 45 (2010), 25-36. 19. E. M. Elsayed, Solution and Behavior of a Rational Difference Equations, Acta Universitatis Apulensis, 23 (2010), 233-249. 20. E. M. Elsayed, Dynamics of Recursive Sequence of Order Two, Kyung-pook Mathematical Journal, 50 (2010), 483-497. 21. E. M. Elsayed, Behavior of a rational recursive sequences, Stud. Univ. Babe¸s-Bolyai Math.vol LVI (1) (2011), 27—42. 22. E. M. Elsayed, Solution of a Recursive Sequence of Order Ten, General Mathematics vol. 19, no. 1 (2011), 145-162. 23. E. M. Elsayed, On the Solution of Some Difference Equations, Eur. J. Pure Appl. Math., 4 (3) (2011), 287-303. 24. E. M. Elsayed, B. Iricanin and S. Stevic, On The Max-Type Equation, Ars Combinatoria, 95 (2010), 187—192 25. E. A. Grove , G. Ladas, Periodicities in Nonlinear Difference Equations. Chapman & Hall / CRC Press (2005). 26. T. Ibrahim, On the third order rational difference equation xn+1 = xn xn−2 /xn−1 (a + bxn xn−2 ) , Int. J. Contemp. Math. Sciences, 4 (27) (2009), 1321-1334. 27. R. Karatas: On solutions of the difference equation xn+1 = (−1)n xn−4 /1 + (−1)n xn xn−1 xn−2 xn−3 xn−4 , Selcuk J. Appl. Math., 8 (1) (2007), 51- 56. 28. R. Karatas, C. Cinar and D. Simsek, On Positive Solutions of The Difference Equation xn+1 = xn−5 /1 + xn−2 xn−5 ,Int. J. Contemp. Math. Sci., 1 (2006), 495-500. 2k+2 Y 29. R. Karatas and C. Cinar, On the solutions of the difference equation xn+1 = αxn−(2k+2) / − a + xn−i i=0

30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

,Int. J. Contemp. Math. Sciences. 2 (31) (2007), 1505-1509. C. Kent, W. Kosmala and S. Stevic, Long-Term Behavior of Solutions of the Difference Equation, Abstract and Applied Analysis vol 2010 (2010), Article ID 152378, 17 pages. V. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht,1993. K. Liu, P. Li, F. Han and W. Zhong, Global Dynamics of Nonlinear Difference Equation xn+1 = A + xn /xn−1 xn−2 , Journal of Computational Analysis and Applications,24 (6) (2018), 1125-1132. H. Ma, H. Feng, J. Wang, W. Ding, Boundedness and Asymptotic Behavior of Positive Solutions for Difference Equations of Exponential form, J. Nonlinear Sci. Appl., 8 (2015), 893-899. R. Mostafaei and N. Rastegar,On a recurrence relation, QScience Connect,10 (2014),1-11. W. T. Patula and H. D. Voulov, On a Max Type Recurrence Relation with Periodic Coefficients, Journal of Difference Equations and Applications 10 (3) (2004), 329-338. M. Saleh, and M. Aloqeili, On the rational difference equation yn+1 = A + yn−k /yn ,, Appl. Math. Comp., 171 (2) (2005), 862-869. H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Diff. Eq. Appl., 15(3) (2009), 215—224. S. Stevic, More on a rational recurrence relation, Appl. Math. E-Notes, 4 (2004), 80- 84. S. Stevic, A note on periodic character of a higher order difference equation, Rostock. Math. Kolloq., 61 (2006), 21—30. N. Touafek and E. M. Elsayed, On The Solution of some Difference Equations, Hokkaido Math. J., 44 (1) (2015), 29-45. I. Yalçınkaya, B. D. Iricanin and C. Cinar, On a max-type difference equation. Discrete Dynamics in Nature and Society, 2007 (2007), 10 pages, Article ID 47264, doi: 1155/2007/47264. Y. Yazlik, On the solutions and behavior of rational difference equations, J. Comp. Anal. Appl., 17 (2014), 584—594.

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Some fixed point theorems of non-self contractive mappings in complete metric spaces ∗

Dangdang Wang, Chuanxi Zhu , Zhaoqi Wu Department of Mathematics, Nanchang University, Nanchang, 330031, P. R. China Abstract In this paper, we establish some fixed point theorems for non-self mappings, which solve the problem 1 in [1], satisfying special contractive conditions in complete metric spaces. Keyword Fixed point; non-self mapping; contractive mapping; complete metric spaces

1

Introduction The aim of this paper is to answer an open problem of Rus [1]. We give a non-self mapping T satisfying

receptively four contractive conditions such that T has a unique fixed point. This is a solution for the open problem. An open problem in [1] as following: Let (X, d) be a metric space, Y a non-empty bounded and closed subset of X and T : Y → X a non-self operator. We suppose that there exists a sequence (xn )n∈N ∗ such that T n (xn ) is defined for all n ∈ N ∗ . In which additional conditions on T we have: (a) FT ̸= ∅? (b) FT = {x∗ }? where FT := {x ∈ X|x = T x}. In this paper, we give following marks. (1) MT (Y ) = sup{d(x, y)|x, y ∈ Y }; (2) ET (Y ) = sup{d(x, T x)|x ∈ Y }; (3) NT (y) = sup{d(x, T y)|x, y ∈ Y }. In (2), we can easy to obtain: i) if X ⊂ Y , then ET (X) ≤ ET (Y ); ii) ET (Y ) = ET (Y ). Lemma 1 [4] Let an , bn ∈ R+ , n ∈ N . We suppose that: (i)Σ∞ k=0 ak < ∞; (ii)bn → 0 as n → ∞. Then Σ∞ k=0 an−k bk → 0 as n → ∞. 1∗ Correspondence

author. Chuanxi Zhu. Email address: [email protected]. Tel:+8613970815298.

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2

Fixed point theorems In this section, we give some non-self contractions as follows. Let (X, d) be a metric space, Y be a non-empty bounded and closed subset of X. Suppose that T : Y → X

be a non-self mapping satisfied following condition: (W1) d(T x, T y) ≤ ad(x, y) + bd(x, T x) + cd(y, T y), for all x, y ∈ Y , where a, b, c ∈ R+ and a + b + c < 1; (W2) d(T x, T y) ≤ bd(x, T y) + cd(y, T x), for all x, y ∈ Y , where b, c ∈ R+ ; (W3) d(T x, T y) ≤ ad(x, y) + bd(x, T y) + cd(y, T x), for all x, y ∈ Y , where a, b, c ∈ R+ and a < 1; (W4) d(T x, T y) ≤ a1 d(x, y) + a2 d(x, T x) + a3 d(y, T y) + a4 d(x, T y), for all x, y ∈ Y , where a1 , a2 , a3 , a4 ∈ R+ and a1 + a2 + a3 < 1. Lemma 2 Let (X, d) be a metric space, Y be a bounded and non-empty closed subset of X. If T : Y → X satisfying (W1), then T is a non-self α-graphic contraction with α = a + c. Proof Let x ∈ Y such that T x ∈ Y , we get d(T 2 x, T x) ≤ ad(T x, x) + bd(T x, T 2 x) + cd(x, T x), so d(T 2 x, T x) ≤

a+c d(x, T x). 1−b

Theorem 1 Let (X, d) be a metric space, Y be a non-empty bounded and closed subset of X. T : Y → X be a non-self mapping satisfying (W1). We suppose that there exists a sequence (xn )n∈N ∗ such that T n (xn ) is defined for all n ∈ N ∗ . Then (i) T has a unique fixed point; (ii) T n−1 (xn ) → x∗ and T n (xn ) → x∗ as n → +∞; (iii) d(x, x∗ ) ≤

1+b 1−a d(x, T x),

∀ x ∈ Y , i.e. MT (Y ) ≤

1+b 1−a ET (Y

).

Proof (i)+(ii) Let Y1 := T (Y ), Y2 := T (Y1 ∩ Y ), · · · , Yn+1 := T (Yn ∩ Y ), n ∈ N ∗ . We remark that: (1) Yn+1 ⊂ Yn , ∀ n ∈ N ∗ ; (2) T n (xn ) ∈ Yn , ∀ n ∈ N ∗ , so Yn ̸= ∅. Since T satisfying (W1), we have that: M (Yn+1 ) = M (T (Yn ∩ Y )) = M (T (Yn ∩ Y )) ≤ aM (Yn ∩ Y ) + (b + c)ET (Yn ∩ Y ) ≤ · · ·

(2.1)

≤ an+1 M (Y ) + an (b + c)ET (Y ) + · · · + a(b + c)ET (Yn−1 ∩ Y ) + (b + c)ET (Yn ∩ Y ).

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On the other hand, from Lemma 2, we get ET (Yn ∩ Y ) = ET (T (Yn−1 ∩ Y ∩ Y ) = ET (T (Yn−1 ∩ Y ) ∩ Y ) a+c = sup{d(T x, T 2 x)|x ∈ Yn−1 ∩ Y, T x ∈ Y } ≤ ET (Yn−1 ∩ Y ) 1−b a+c n ) ET (Y ), n ∈ N ∗ . ≤ ··· ≤ ( 1−b n Because of a + b + c < 1, so ( a+c 1−b ) → 0, n → +∞, i.e. ET (Yn ∩ Y ) → 0, n → +∞.

Let an = an and bn = (b + c)ET (Yn ∩ Y ), by lemma 1, we have M (Yn+1 ) → 0 as n → +∞. From Cantor intersection lemma, we get Y∞ := ∩n∈N Yn ̸= ∅, M (Y∞ ) = 0 and T (Y∞ ∩ Y ) ⊂ Y∞ . From Y∞ ̸= ∅ and M (Y∞ ) = 0, we have that Y∞ = x∗ , i.e. Y∞ be a single point set. Otherwise, T n (xn ) ∈ Yn and T n−1 (xn ) ∈ Yn−1 ∩ Y , this implies that {T n (xn )}n∈N and {T n−1 (xn )}n∈N are fundamental sequences. Since Yn , n ∈ N are closed, so we get T n−1 (xn ) → x∗ and T n (xn ) → x∗ as n → +∞. Also because of T is continuous, then T n (xn ) → T (x∗ ). Therefore, T (x∗ ) = x∗ . (iii) Let x ∈ Y , by using (W1) we have d(x, x∗ ) ≤ d(x, T x) + d(T x, x∗ ) ≤ d(x, T x) + ad(x, x∗ ) + bd(x, T x) + cd(d(x, T x), T d(x, T x)), so d(x, x∗ ) ≤

1+b d(x, T x), ∀ x ∈ Y. 1−a

´ c − Reich − Rus operator. And then, Theorem 1 Remark 1 Let b = c in Theorem 1, then T is a non-self Ciri´ generalizes Theorem 5 in Rus [1]. At the same time, this theorem gives an answer to the Problem 1 of [1]. For (W4), we give a Lemma as following: Lemma 3 Let (X, d) be a metric space, Y be a non-empty bounded and closed subset of X. Define T : Y → X be a non-self mapping. Then NT (Yn ∩ Y ) → 0, as n → ∞, where Yn = T (Yn−1 ∩ Y ). Proof From the definitions of NT and Yn , we have sup{d(x, T y)|x, y ∈ Yn ∩ Y } = NT (Yn ∩ Y ) = NT (T (Yn−1 ∩ Y ) ∩ Y ) = NT (T (Yn−1 ∩ Y ) ∩ Y ) = sup{d(T x, T 2 y)|x, y ∈ Yn−1 ∩ Y }. Since Yn−1 ∩Y ⊂ Yn ∩Y , so d(T x, T 2 y) ≤ d(x, T y), for all x, y ∈ Yn−1 ∩Y . Hence, NT (Yn ∩Y ) ≤ NT (Yn−1 ∩Y ). 3 81

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By the density of real numbers we can get, there exists k ∈ R+ and k < 1, such that NT (Yn ∩ Y ) ≤ kNT (Yn−1 ∩ Y ). And then, NT (Yn ∩ Y ) ≤ kNT (Yn−1 ∩ Y ) ≤ · · · ≤ k n NT (Y ) → 0, as n → +∞.

Theorem 2 Let (X, d) be a metric space, Y be a non-empty bounded and closed subset of X. T : Y → X be a non-self mapping satisfying (W3). We suppose that there exists a sequence (xn )n∈N ∗ such that T n (xn ) is defined for all n ∈ N ∗ . Then (i) T has a unique fixed point; (ii) T n−1 (xn ) → x∗ and T n (xn ) → x∗ as n → +∞. Proof Let Y1 := T (Y ), Y2 := T (Y1 ∩ Y ), · · · , Yn+1 := T (Yn ∩ Y ), n ∈ N ∗ . We remark that: (1) Yn+1 ⊂ Yn , ∀ n ∈ N ∗ ; (2) T n (xn ) ∈ Yn , ∀ n ∈ N ∗ , so Yn ̸= ∅. Since T satisfying (W3), we have that: M (Yn+1 ) = M (T (Yn ∩ Y )) = M (T (Yn ∩ Y )) ≤ aM (Yn ∩ Y ) + (b + c)NT (Yn ∩ Y ) ≤ aM (Yn ) + (b + c)NT (Yn ∩ Y ) ≤ · · · ≤ an+1 M (Y ) + an (b + c)NT (Y ) + · · · + a(b + c)NT (Yn−1 ∩ Y ) + (b + c)NT (Yn ∩ Y ). Let an = an and bn = (b + c)NT (Yn ∩ Y ), by lemma 3 we have M (Yn+1 ) → 0 as n → +∞, and the proof is similar with the proof of Theorem 1. This is the complete proof. For (W3), when a = 0, it becomes condition (W2). Thence, we have the following Corollary: Corollary 1 Let (X, d) be a metric space, Y be a non-empty bounded and closed subset of X. Define T : Y → X be a non-self mapping satisfying (W3), the conclusions of Theorem 2 remain holds. Theorem 3 Let (X, d) be a metric space, Y be a non-empty bounded and closed subset of X. T : Y → X be a non-self mapping satisfying (W4). We suppose that there exists a sequence (xn )n∈N ∗ such that T n (xn ) is defined for all n ∈ N ∗ . Then (i) T has a unique fixed point; (ii) T n−1 (xn ) → x∗ and T n (xn ) → x∗ as n → +∞; (iii) d(x, x∗ ) ≤

1+b 1−a d(x, T x),

∀ x ∈ Y , i.e. MT (Y ) ≤

1+b 1−a ET (Y

).

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Proof (i)+(ii) Let Y1 := T (Y ), Y2 := T (Y1 ∩ Y ), · · · , Yn+1 := T (Yn ∩ Y ), n ∈ N ∗ . We remark that: (1) Yn+1 ⊂ Yn , ∀ n ∈ N ∗ ; (2) T n (xn ) ∈ Yn , ∀ n ∈ N ∗ , so Yn ̸= ∅. Since T satisfying (W1), we have that: M (Yn+1 ) = M (T (Yn ∩ Y )) = M (T (Yn ∩ Y )) ≤ a1 M (Yn ∩ Y ) + (a2 + a3 )ET (Yn ∩ Y ) + a4 NT (Yn ∩ Y ) ≤ ··· ≤ ≤ an+1 M (Y ) + [an1 (a2 + a3 )ET (Y ) + · · · + a1 (a2 + a3 )ET (Yn−1 ∩ Y ) + (a2 + a3 )ET (Yn ∩ Y )] 1 + [an1 · a4 NT (Y ) + · · · + a1 · a4 NT (Yn−1 ∩ Y ) + a4 NT (Yn ∩ Y )] = an+1 M (Y ) + ΦET + ΦNT , 1 where ΦET = an1 (a2 + a3 )ET (Y ) + · · · + a1 (a2 + a3 )ET (Yn−1 ∩ Y ) + (a2 + a3 )ET (Yn ∩ Y ), ΦNT = an1 · a4 NT (Y ) + · · · + a1 · a4 NT (Yn−1 ∩ Y ) + a4 NT (Yn ∩ Y ). For x ∈ Y , such that T x ∈ T (Y ), we have d(T 2 x, T x) ≤ a1 d(T x, x) + a2 d(T x, T 2 x) + a3 d(x, T x) + a4 d(T x, T x), so d(T 2 x, T x) ≤

a1 + a3 d(x, T x). 1 − a2

(2.2)

Thence ET (Yn ∩ Y ) = ET (T (Yn−1 ∩ Y ∩ Y ) = ET (T (Yn−1 ∩ Y ) ∩ Y ) a1 + a3 = sup{d(T x, T 2 x)|x ∈ Yn−1 ∩ Y, T x ∈ Y } ≤ ET (Yn−1 ∩ Y ) 1 − a2 a1 + a3 n ≤ ··· ≤ ( ) ET (Y ), n ∈ N ∗ . 1 − a2 1 +a3 n Because of a + b + c < 1, so ( a1−a ) → 0, n → +∞, i.e. ET (Yn ∩ Y ) → 0, n → +∞. 2

Let an = an1 and bn = (a2 + a3 )ET (Yn ∩ Y ), by lemma 1 we have ΦET → 0 as n → +∞. From Lemma 3, we know, NT (Yn ∩ Y ) → 0 as n → +∞. ′

′

Let an = an1 and bn = a4 NT (Yn ∩ Y ), by lemma 1 we obtain ΦNT → 0 as n → +∞. In summary, we get M (Yn+1 ) → 0 as n → +∞, and the proof is similar with the proof of Theorem 1. Although let a = 0 in (W4), it becomes (W2), because different proof process details of the transformation, so we give separately the proof of Theorem 1 and Theorem 3. Acknowledgements The author would like to thank the editors and the referees for their constructive comments and suggestions. The research was supported by the Natural Science Foundation of China (11771198,11361042,11071108, 11461045,11701259), the Natural Science Foundation of Jiangxi Province of China (20132BAB201001, 20142BAB211016) and the Scientific Program of the Provincial Education Department of Jiangxi (GJJ150008) and the Innovation Program of the Graduate student of Nanchang University(colonel-level project)(cx2016148). 5 83

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References [1] I. A., Rus, M. A., Serban, Some fixed point theorems for nonself generalized contraction. Miskolc Math. Notes, 17 (2017), 1021-1031. [2] S., Reich, A. J., Zasavski, A fixed point theorem for Matkowski contractions, Fixed Point Theory, 8 (2007), 303-307. [3] A. Amini-Harandi, M., Fakhar, M., Goli, Some fixed point theorems for non-self mappings of contractive type with applicatons to endpoint theory, J. Fixed Point Theory Appl., 2017 (2017), 10pages. [4] I. A., Rus, M. A., Serban, Some generalizations of a Cauchy lemma and applizations, Topicsnin Mathematicas, computer sciece and philosophy, 2008 (2008), 173-181. [5] A., Amini-Harandi, Endpoints of set-valued contractions in metric spaces, Nonlinear Anal., 72 (2010), 132-134. [6] A., Petrusel, I. A., Rus, M. A., Serban,Fixed points, fixed sets and iterated multifunction systems for nonself multivalued operators, Set-Valued Var. Anal., 23 (2015), 223-237. [7] B. E., Rhoades, A fixed point theorem for non-self-mappings, Math. Japon., 23 (1979), 457-459. [8] X. H., Mu, C. X., Zhu, Z. Q., Wu, New multipled common fixed point theorems in Menger PM-space, Fixed Point Theory Appl., 2015 (2015), 136 pages. [9] C. R., Ji, C. X., Zhu, Z. Q., Wu, Some new fixed point theorems in generalized probabilistic metric spaces, J. of Nonlinear Science and Appl., 9 (2016), 3735-3743. [10] Q., Tu, C. X., Zhu, Z. Q., Wu, Some new coupled fixed point theorems in partially ordered complete probabilistic metric spaces, J. of Nonlinear Science and Appl., 9 (2016), 1116-1128. [11] C. X. Zhu, Z. B., Xu, Inequalities and solution of an operator equation, Appl. Math. Lett., 21 (2008), 607-611. [12] C. X., Zhu, Research on some problems for nonlinear operators, Nonlinear Anal., 71 (2009), 4568-4571. [13] C. X.,Zhu, C. F., Chen, Calculations of random fixed point intex, J. Math. Anal. Appl., 339 (2008), 839-844.

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Double-framed soft sets in B-algebras Jung Mi Ko1 and Sun Shin Ahn2,∗ 1

Department of Mathematics, Gangneung-Wonju National University, Gangneung 25457, Korea 2

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

Abstract. The notion of a double-framed soft (normal) subalgebra in a B-algebra is introduced and related properties are investigated. We consider characterizations of a double-framed soft (normal) subalgebra and establish a new double-framed soft subalgebra from old one. Also, we show that the int-uni double-framed soft of two double framed soft subalgebras is a double framed soft subalgebra.

1. Introduction Molodtsov [11] 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 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. [10] described the application of soft set theory to a decision making problem. Jun [5] discussed the union soft sets with applications in BCK/BCI-algebras. Jun et al. [6] introduced the notion of double-framed soft sets, and applied it to BCK/BCI-algebras. They discussed double-frame soft algebras and investigated some related properties. We refer the reader to the papers [3, 4, 14] for further information regarding algebraic structures/properties of soft set theory. On the while, Y. B. Jun, E. H. Roh and H. S. Kim [7] introduced a new notion, called a BH-algebra. J. Neggers and H. S. Kim [12] introduced a new notion, called a B-algebra. C. B. Kim and H. S. Kim [9] introduced the notion of a BG-algebra which is a generalization of B-algebras. S. S. Ahn and H. D. Lee [1] classified the subalgebras by their family of level subalgebras in BG-algebras. In this paper, we introduce the notion of a double-framed soft (normal) subalgebra in a B-algebra and investigate some related properties. We consider characterizations of a double-framed soft (normal) subalgebra and establish a new double-framed soft subalgebra from old one. Also, we show that the int-uni double-framed soft of two double framed soft subalgebras is a double framed soft subalgebra. 2. Preliminaries A B-algebra [12] is a non-empty set X with a constant 0 and a binary operation “ ∗ ” satisfying axioms: 0

2010 Mathematics Subject Classification: 06F35; 03G25; 06D72. Keywords: γ-exclusive set, double-framed soft (normal) subalgebra, B-algebra. Correspondence: Tel: +82 2 2260 3410, Fax: +82 2 2266 3409 (S. S. Ahn). 0 E-mail: [email protected] (J. M. Ko); [email protected] (S. S. Ahn). 0 This study was supported by Gangneung-Wonju National University. 0

∗

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Jung Mi Ko and Sun Shin Ahn (B1) x ∗ x = 0, (B2) x ∗ 0 = x, (B) (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)) for any x, y, z in X. For brevity we call X a B-algebra. In X we can define a binary relation “ ≤ ” by x ≤ y if and only if x ∗ y = 0. An algebra (X; ∗, 0) of type (2, 0) is called a BH-algebra if it satisfies (B1), (B2) and (BH) x ∗ y = y ∗ x = 0 imply x = y for any x, y ∈ X. An algebra (X; ∗, 0) of type (2, 0) is called a BG-algebra if it satisfies (B1), (B2) and (BG) (x ∗ y) ∗ (0 ∗ y) = x for any x, y ∈ X. Proposition 2.1. [2, 12] Let (X; ∗, 0) be a B-algebra. Then (i) the left cancellation law holds in X, i.e., x ∗ y = x ∗ z implies y = z, (ii) if x ∗ y = 0, then x = y for any x, y ∈ X, (iii) if 0 ∗ x = 0 ∗ y, then x = y for any x, y ∈ X, (iv) 0 ∗ (0 ∗ x) = x, for all x ∈ X, (v) x ∗ (y ∗ z) = (x ∗ (0 ∗ z)) ∗ y for all x, y, z ∈ X. A non-empty subset S of a B-algebra X is called a subalgebra of X if x ∗ y ∈ S for any x, y ∈ S. A non-empty subset N of X is said to be normal if (x ∗ a) ∗ (y ∗ b) ∈ N for any x ∗ y, a ∗ b ∈ N . Then any normal subset N of a B-algebra X is a subalgebra of X, but the converse need not be true ([13]). A non-empty subset X of a B-algebra X is a called a normal subalgebra of X if it is both a subalgebra and normal. Molodtsov [11] 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 be 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. 3. Double-framed soft normal subalgebras In what follows let X denote a B-algebra unless otherwise specified. Definition 3.1. A double-framed pair ⟨(α, β); X⟩ is called a double-framed soft set over U, where α and β are mappings from X to P(U ). Definition 3.2. A double-framed soft set ⟨(α, β); X⟩ over U is called a double-framed soft subalgebra over U if it satisfies :

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Double-framed soft sets in B-algebras (3.1) (∀x, y ∈ X) (α(x ∗ y) ⊇ α(x) ∩ α(y), β(x ∗ y) ⊆ β(x) ∪ β(y)) . Example 3.3. Let X be the set of parameters where X := {0, 1, 2, 3} is a B-algebra with the following Cayley table:

∗ 0 1 2 3 Let ⟨(α, β); X⟩ be a double-framed soft set over U

0 1 0 2 1 0 2 3 3 1 defined

2 1 3 0 2 as

  τ3 α : X → P(U ), x 7→ τ  1 τ2 and

  γ3 β : X → P(U ), x 7→ γ  2 γ1

3 3 2 1 0 follows: if x = 0, if x = 3, if x = {1, 2},

if x = 0, if x = 3, if x = {1, 2}

where τ1 , τ2 , τ3 , γ1 , γ2 and γ3 are subsets of U with τ1 ⊊ τ2 ⊊ τ3 and γ1 ⊋ γ2 ⊋ γ3 It is easy to show that ⟨(α, β); X⟩ is a double-framed soft subalgebra over U. Lemma 3.4. Every double-framed soft subalgebra ⟨(α, β); X⟩ over U satisfies the following condition: (3.2) (∀x ∈ X) (α(x) ⊆ α(0), β(x) ⊇ β(0)) . □

Proof. Straightforward. Proposition 3.5. For a double-framed soft subalgebra ⟨(α, β); X⟩ over U, the following are equivalent: (i) (∀x ∈ X) (α(x) = α(0), β(x) = β(0)) . (ii) (∀x, y ∈ X) (α(y) ⊆ α(x ∗ y), β(y) ⊇ β(x ∗ y)) .

Proof. Assume that (ii) is valid. Taking y := 0 in (ii) and using (B2), we have α(0) ⊆ α(x ∗ 0) = α(x) and β(0) ⊇ β(x ∗ 0) = β(x). It follows from Lemma 3.4 that α(x) = α(0) and β(x) = β(0). Conversely, suppose that α(x) = α(0) and β(x) = β(0) for all x ∈ X. Using (3.1), we have α(y) = α(0) ∩ α(y) = α(x) ∩ α(y) ⊆ α(x ∗ y), β(y) = β(0) ∪ β(y) = β(x) ∪ β(y) ⊇ β(x ∗ y) for all x, y ∈ X. This completes the proof.

□

For two double-framed soft sets ⟨(α, β); X⟩ and ⟨(f, g); X⟩ over U, the double-framed soft int-uni set of ⟨(α, β); X⟩ ˜ f, β ∪ ˜ g); X⟩ where and ⟨(f, g); X⟩ is defined to be a double-framed soft set ⟨(α∩ ˜ f : X → P(U ), x 7→ α(x) ∩ f (x), α∩ ˜ g : X → P(U ), x 7→ β(x) ∪ g(x). β∪ ˜ f, β ∪ ˜ g); X⟩ . It is denoted by ⟨(α, β); X⟩ ⊓ ⟨(f, g); X⟩ = ⟨(α∩

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Jung Mi Ko and Sun Shin Ahn Theorem 3.6. The double-framed soft int-uni set of two double-framed soft subalgebras ⟨(α, β); X⟩ and ⟨(f, g); X⟩ over U is a double-framed soft subalgebra over U. Proof. For any x, y ∈ X, we have ˜ f )(x ∗ y) =α(x ∗ y) ∩ f (x ∗ y) ⊇ (α(x) ∩ α(y)) ∩ (f (x) ∩ f (y)) (α∩ ˜ f )(x) ∩ (α∩ ˜ f )(y) =(α(x) ∩ f (x)) ∩ (α(y) ∩ f (y)) = (α∩ and ˜ g)(x ∗ y) =β(x ∗ y) ∪ g(x ∗ y) ⊆ (β(x) ∪ β(y)) ∪ (g(x) ∪ g(y)) (β ∪ ˜ g)(x) ∪ (β ∪ ˜ g)(y). =(β(x) ∪ g(x)) ∪ (β(y) ∪ g(y)) = (β ∪ Therefore ⟨(α, β); X⟩ ⊓ ⟨(f, g); X⟩ is a double-framed soft subalgebra over U.

□

For two double-framed soft sets ⟨(α, β); X⟩ and ⟨(f, g); X⟩ over U, the double-framed soft uni-int set of ⟨(α, β); X⟩ ˜ f, β ∩ ˜ g); X⟩ where and ⟨(f, g); X⟩ is defined to be a double-framed soft set ⟨(α∪ ˜ f : X → P(U ), x 7→ α(x) ∪ f (x), α∪ ˜ g : X → P(U ), x 7→ β(x) ∩ g(x). β∩ ˜ f, β ∩ ˜ g); X⟩ . It is denoted by ⟨(α, β); X⟩ ⊔ ⟨(f, g); X⟩ = ⟨(α∪ The following example shows that the double-framed soft uni-int set of two double-framed soft subalgebras ⟨(α, β); X⟩ and ⟨(f, g); X⟩ over U may not be a double-framed soft subalgebra over U. Example 3.7. Let E = X be the set of parameters, and let U = Z be the initial universe set, where X = {0, 1, 2, 3, 4, 5} is a B-algebra [12] with the following table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 2 0 1 4 5 3

2 1 2 0 5 3 4

3 3 4 5 0 1 2

4 4 5 3 2 0 1

5 5 3 4 1 2 0

Let ⟨(α, β); X⟩ and ⟨(f, g); X⟩ be double-framed soft sets over U defined, respectively, as follows: { Z if x ∈ {0, 4}, α : X → P(U ), x 7→ 9Z if x ∈ {1, 2, 3, 5}, { β : X → P(U ), x 7→

{ f : X → P(U ), x 7→ and

{ g : X → P(U ), x 7→

if x ∈ {0, 4}, if x ∈ {1, 2, 3, 5},

7Z Z

Z 3Z

if x ∈ {0, 5}, if x ∈ {1, 2, 3, 4},

if x ∈ {0, 5}, if x ∈ {1, 2, 3, 4},

2Z Z

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Double-framed soft sets in B-algebras It is routine to verify that ⟨(α, β); X⟩ and ⟨(f, g); X⟩ are double-framed soft subalgebras over U. But ⟨(α, β); X⟩ ⊔ ˜ f, β ∩ ˜ g); X⟩ is not a double-framed soft subalgebra over U , since (α∪ ˜ f )(4 ∗ 5) = (α∪ ˜ f )(2) = ⟨(f, g); X⟩ = ⟨(α∪ ˜ f )(4) ∩ (α∪ ˜ f )(5) and/or (β ∩ ˜ g)(4 ∗ 5) = (β ∩ ˜ g)(2) = Z ∩ Z = Z ⊈ 7Z ∪ 2Z = α(2) ∪ f (2) = 9Z ∪ 3Z = 3Z ⊉ Z = (α∪ ˜ g)(4) ∪ (β ∩ ˜ g)(5). (β ∩ Let ⟨(α, β); A⟩ and ⟨(f, g); B⟩ be double-framed soft sets over a common universe U. Then ⟨(α, β); A⟩ is called ˜ ⟨(f, g); B⟩ , if a double-framed soft subset of ⟨(f, g); B⟩ , denoted by ⟨(α, β); A⟩ ⊆ (i) A ⊆ B, (ii) (∀e ∈ A)

(

α(e) and f (e) are identical approximations, β(e) and g(e) are identical approximations.

) .

Theorem 3.8. Let ⟨(α, β); A⟩ be a double-framed soft subset of a double-framed soft set ⟨(f, g); B⟩ . If ⟨(f, g); B⟩ is a double-framed soft subalgebra over U , then so is ⟨(α, β); A⟩. Proof. Let x, y ∈ A. Then x, y ∈ B, and so α(x) ∩ α(y) = f (x) ∩ f (y) ⊆ f (x ∗ y) = α(x ∗ y), β(x) ∪ β(y) = g(x) ∪ g(y) ⊇ g(x ∗ y) = β(x ∗ y). Hence ⟨(α, β); A⟩ is a double-framed soft subalgebra over U.

□

The converse of Theorem 3.8 is not true as seen in the following example. Example 3.9. Let (U = Z, X) where X = {0, 1, 2, 3} is a B-algebra as in Example 3.3. For a subalgebra {0, 3}, define a double-framed soft set ⟨(α, β); {0, 3}⟩ over U as follows: { Z α : {0, 3} → P(U ), x 7→ 2Z and

{ β : {0, 3} → P(U ), x 7→

27Z 9Z

if x = 0, if x = 3, if x = 0, if x = 3,

Then ⟨(α, β); {0, 3}⟩ is a double-framed soft subalgebra over U . Take B := X and define a double-framed soft set ⟨(f, g); B⟩ over U as follows:

and

 Z    72Z f : B → P(U ), x 7→  4Z   2Z  27Z    3Z g : B → P(U ), x 7→  Z   9Z

if if if if

if if if if

x = 0, x = 1, x = 2, x = 3,

x = 0, x = 1, x = 2, x = 3.

Then ⟨(f, g); B⟩ is not a double-framed soft subalgebra over U since f (0 ∗ 2) = f (1) = 72Z ⊉ f (0) ∩ f (2) = Z ∩ 4Z = 4Z and/or g(1 ∗ 3) = g(2) = Z ⊈ g(1) ∪ g(3) = 3Z ∪ 9Z = 3Z.

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Jung Mi Ko and Sun Shin Ahn For a double-framed soft set ⟨(α, β); X⟩ over U and two subsets γ and δ of U, the γ-inclusive set and the δ-exclusive set of ⟨(α, β); X⟩, denoted by iX (α; γ) and eX (β; δ), respectively, are defined as follows: iX (α; γ) := {x ∈ X | γ ⊆ α(x)} and eX (β; δ) := {x ∈ X | δ ⊇ β(x)} , respectively. The set DFX (α, β)(γ,δ) := {x ∈ X | γ ⊆ α(x), δ ⊇ β(x)} is called a double-framed including set of ⟨(α, β); X⟩ . It is clear that DFX (α, β)(γ,δ) = iX (α; γ) ∩ eX (β; δ). Theorem 3.10. For a double-framed soft set ⟨(α, β); X⟩ over U, the following are equivalent: (i) ⟨(α, β); X⟩ is a double-framed soft subalgebra over U. (ii) For every subsets γ and δ of U with γ ∈ Im(α) and δ ∈ Im(β), the γ-inclusive set and the δ-exclusive set of ⟨(α, β); X⟩ are subalgebras of X. Proof. Assume that ⟨(α, β); X⟩ is a double-framed soft subalgebra over U. Let x, y ∈ X be such that x, y ∈ iX (α; γ) and x, y ∈ eX (β; δ) for every subsets γ and δ of U with γ ∈ Im(α) and δ ∈ Im(β). It follows from (3.1) that α(x ∗ y) ⊇ α(x) ∩ α(y) ⊇ γ and β(x ∗ y) ⊆ β(x) ∪ β(y) ⊆ δ. Hence x ∗ y ∈ iX (α; γ) and x ∗ y ∈ eX (β; δ), and therefore iX (α; γ) and eX (β; δ) are subalgebras of X. Conversely, suppose that (ii) is valid. Let x, y ∈ X be such that α(x) = γx , α(y) = γy , β(x) = δx and β(y) = δy . Taking γ = γx ∩ γy and δ = δx ∪ δy imply that x, y ∈ iX (α; γ) and x, y ∈ eX (β; δ). Hence x ∗ y ∈ iX (α; γ) and x ∗ y ∈ eX (β; δ), which imply that α(x ∗ y) ⊇ γ = γx ∩ γy = α(x) ∩ α(y) and β(x ∗ y) ⊆ δ = δx ∪ δy = β(x) ∪ β(y). Therefore ⟨(α, β); X⟩ is a double-framed soft subalgebra over U.

□

Corollary 3.11. If ⟨(α, β); X⟩ is a double-framed soft algebra over U, then the double-framed including set of ⟨(α, β); X⟩ is a subalgebra X. For any double-framed soft set ⟨(α, β); X⟩ over U, let ⟨(α∗ , β ∗ ); X⟩ be a double-framed soft set over U defined by {

∗

α : X → P(U ), x 7→ β ∗ : X → P(U ), x 7→

{

α(x) η

if x ∈ iX (α; γ), otherwise,

β(x) ρ

if x ∈ eX (β; δ), otherwise,

where γ, δ, η and ρ are subsets of U with η ⊊ α(x) and ρ ⊋ β(x). Theorem 3.12. If ⟨(α, β); X⟩ is a double-framed soft subalgebra over U, then so is ⟨(α∗ , β ∗ ); X⟩ . Proof. Assume that ⟨(α, β); X⟩ is a double-framed soft subalgebra over U. Then iX (α; γ) and eX (β; δ) are subalgebras of X for every subsets γ and δ of U with γ ∈ Im(α) and δ ∈ Im(β), by Theorem 3.10. Let x, y ∈ X. If x, y ∈ iX (α; γ), then x ∗ y ∈ iX (α; γ). Thus α∗ (x ∗ y) = α(x ∗ y) ⊇ α(x) ∩ α(y) = α∗ (x) ∩ α∗ (y). If x ∈ / iX (α; γ) or y ∈ / iX (α; γ), then α∗ (x) = η or α∗ (y) = η. Hence α∗ (x ∗ y) ⊇ η = α∗ (x) ∩ α∗ (y).

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Double-framed soft sets in B-algebras Now, if x, y ∈ eX (β; δ), then x ∗ y ∈ eX (β; δ). Thus β ∗ (x ∗ y) = β(x ∗ y) ⊆ β(x) ∪ β(y) = β ∗ (x) ∪ β ∗ (y). If x ∈ / eX (β; δ) or y ∈ / eX (β; δ), then β ∗ (x) = ρ or β ∗ (y) = ρ. Hence β ∗ (x ∗ y) ⊆ ρ = β ∗ (x) ∪ β ∗ (y). Therefore ⟨(α∗ , β ∗ ); X⟩ is a double-framed soft subalgebra over U.

□

Let ⟨(α, β); X⟩ and ⟨(α, β); Y ⟩ be double-framed soft sets over U, where X, Y are B-algebras. The (α∧ , β∨ )product of ⟨(α, β); X⟩ and ⟨(α, β); Y ⟩ is defined to be a double-framed soft set ⟨(αX∧Y , βX∨Y ); X × Y ⟩ over U in which αX∧Y : X × Y → P(U ), (x, y) 7→ α(x) ∩ α(y), βX∨Y : X × Y → P(U ), (x, y) 7→ β(x) ∪ β(y). Theorem 3.13. For any B-algebras X and Y as sets of parameters, let ⟨(α, β); X⟩ and ⟨(α, β); Y ⟩ be doubleframed soft subalgebras over U. Then the (α∧ , β∨ )-product of ⟨(α, β); X⟩ and ⟨(α, β); Y ⟩ is also a double-framed soft subalgebra over U. Proof. Note that (X × Y, ⊛, (0, 0)) is a B-algebra. For any (x, y), (a, b) ∈ X × Y, we have αX∧Y ((x, y) ⊛ (a, b)) = αX∧Y (x ∗ a, y ∗ b) = α(x ∗ a) ∩ α(y ∗ b) ⊇ (α(x) ∩ α(a)) ∩ (α(y) ∩ α(b)) = (α(x) ∩ α(y)) ∩ (α(a) ∩ α(b)) = αX∧Y (x, y) ∩ αX∧Y (a, b) and βX∨Y ((x, y) ⊛ (a, b)) = βX∨Y (x ∗ a, y ∗ b) = β(x ∗ a) ∪ β(y ∗ b) ⊆ (β(x) ∪ β(a)) ∪ (β(y) ∪ β(b)) = (β(x) ∪ β(y)) ∪ (β(a) ∪ β(b)) = βX∨Y (x, y) ∪ βX∨Y (a, b) Hence ⟨(αX∧Y , βX∨Y ); E × F ⟩ is a double-framed soft subalgebra over U.

□

Definition 3.14. A double-framed soft set ⟨(α, β); X⟩ over U is said to be double-framed soft normal of a B-algebra X if it satisfies: (3.3) (∀x, y, a, b ∈ X)(α((x ∗ a) ∗ (y ∗ b)) ⊇ α(x ∗ y) ∩ α(a ∗ b), β((x ∗ a) ∗ (y ∗ b)) ⊆ β(x ∗ y) ∪ β(a ∗ b)). A double-framed soft ⟨(α, β); X⟩ over U is called a double-framed soft normal subalgebra of a B-algebra X if it satisfies (3.1) and (3.3).

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Jung Mi Ko and Sun Shin Ahn Example 3.15. Let (U = Z, X) where X = {0, 1, 2, 3} is a B-algebra as in Example 3.3. Let ⟨(α, β); X⟩ be a double-framed soft set over U defined as follows:

{

α : X → P(U ), x 7→ and

{ β : X → P(U ), x 7→

Z 2Z

if x ∈ {0, 3}, if x ∈ {1, 2},

if x ∈ {0, 3}, if x ∈ {1, 2},

3Z Z

It is easy to check that ⟨(α, β); X⟩ is a double-framed soft normal over U. Proposition 3.16. Every double-framed soft normal (f˜, X) of a B-algebra X is a double-framed soft subalgebra of X. Proof. Put y := 0, b := 0 and a := y in (3.3). Then α((x ∗ y) ∗ (0 ∗ 0)) ⊇ α(x ∗ 0) ∩ α(y ∗ 0) and β((x ∗ y) ∗ (0 ∗ 0)) ⊆ β(x ∗ 0) ∪ β(y ∗ 0) for any x, y ∈ X. Using (B2) and (B1), we have α(x ∗ y) ⊇ α(x) ∩ α(y) and β(x ∗ y) ⊆ β(x) ∪ β(y). Hence ⟨(α, β); X⟩ is a double-framed soft subalgebra over U .

□

The converse of Proposition 3.16 may not be true in general (Example 3.17). Example 3.17. Let E = X be the set of parameters, and let U = X be the initial universe set, where X = {0, 1, 2, 3, 4, 5} is a B-algebra as in Example 3.7. Let ⟨(α, β); X⟩ be double-framed soft set over U defined as follows:

and

  γ3 α : X → P(U ), x 7→ γ  2 γ1   τ1 β : X → P(U ), x 7→ τ  2 τ3

if x = 0, if x = 5, if x ∈ {1, 2, 3, 4},

if x = 0, if x = 5, if x ∈ {1, 2, 3, 4},

where γ1 , γ2 , γ3 , τ1 , τ2 and τ3 are subsets of U with γ1 ⊊ γ2 ⊊ γ3 and τ1 ⊊ τ2 ⊊ τ3 . It is routine to verify that ⟨(α, β); X⟩ is a double-framed soft subalgebra over U. But it is not double-framed soft normal over U since since α(1) = α((1 ∗ 3) ∗ (4 ∗ 2)) = γ1 ⊉ α(1 ∗ 4) ∩ α(3 ∗ 2) = α(5) ∩ α(5) = γ2 and/or β(1) = β((1 ∗ 3) ∗ (4 ∗ 2)) = τ3 ⊈ β(1 ∗ 4) ∪ β(3 ∗ 2) = β(5) ∪ β(5) = τ2 . Theorem 3.18. For a double-framed soft set ⟨(α, β); X⟩ over U, the following are equivalent: (i) ⟨(α, β); X⟩ is a double-framed soft normal subalgebra over U. (ii) For every subsets γ and δ of U with γ ∈ Im(α) and δ ∈ Im(β), the γ-inclusive set and the δ-exclusive set of ⟨(α, β); X⟩ are normal subalgebras of X. □

Proof. Similar to Theorem 3.10.

Proposition 3.19. Let a double-framed soft set ⟨(α, β); X⟩ over U of a B-algebra X be double-framed soft normal. Then α(x ∗ y) = α(y ∗ x) and β(x ∗ y) = β(y ∗ x) for any x, y ∈ X.

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Double-framed soft sets in B-algebras Proof. Let x, y ∈ X. By (B1) and (B2), we have α(x∗y) = α((x∗y)∗(x∗x)) ⊇ α(x∗x)∩α(y ∗x) = α(0)∩α(y ∗x) = α(y ∗ x). Interchanging x with y, we obtain α(y ∗ x) ⊇ α(x ∗ y). By (B1) and (B2), we have β(x ∗ y) = β((x ∗ y) ∗ (x ∗ x)) ⊆ β(x ∗ x) ∪ β(y ∗ x) = β(0) ∪ β(y ∗ x) = β(y ∗ x). Interchanging x with y, we obtain β(y ∗ x) ⊆ β(x ∗ y).

□

Theorem 3.20. Let ⟨(α, β); X⟩ be a double-framed soft normal subalgebra of a B-algebra X. Then the set X(α,β) := {x ∈ X|α(x) = α(0), β(x) = β(0)} is a normal subalgebra of X. Proof. It is sufficient to show that X(α,β) is normal. Let a, b, x, y ∈ X be such that x∗y ∈ X(α,β) and a∗b ∈ X(α,β) . Then α(x ∗ y) = α(0) = α(a ∗ b), β(x ∗ y) = β(0) = β(a ∗ b). Since ⟨(α, β); X⟩ is a double-framed soft normal subalgebra of X, we have α((x∗a)∗(y∗b)) ⊇ α(x∗y)∩α(a∗b) = α(0) and β((x∗a)∗(y∗b)) ⊆ β(x∗y)∪β(a∗b) = β(0). Using (3.2), we conclude that α((x∗a)∗(y ∗b)) = α(0) and β((x∗a)∗(y ∗b)) = β(0). Hence (x∗a)∗(y ∗b) ∈ X(α,β) . □

This completes the proof.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

S. S. Ahn and H. D. Lee, Fuzzy subalgebras of BG-algebras, Comm. Kore. Math. Soc. 19 (2004), 243-251. J. R. Cho and H. S. Kim, On B-algebras and quasi groups, Quasigroups and related systems 8(2001), 1-6. F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621-2628. Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408-1413. Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Bull. Korean Math. Soc. 50 (2013), 1937-1956. Y. B Jun and S. S. Ahn, Double-framed soft sets with applications in BCK/BCI-algebras, J. Appl. Math. (2012), 1-15. Y. B. Jun, E. H. Roh and H. S. Kim, On BH-algebras, Sci. Mathematica 1 (1998), 347-354. Y. B. Jun, E. H. Roh and H. S. Kim, On fuzzy B-algebras, Czech. Math. J. 52 (2002), 375-384. C. B. Kim and H. S. Kim, On BG-algebras, Demon. Math. 41 (2008), 497-505. 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. D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19-31. J. Neggers and H. S. Kim, On B-algebras, Mate. Vesnik 54(2002), 21-29. J. Neggers and H. S. Kim, A fundamental theorem of B-homomorphism for B-algebras, Intern. Math. J. 2(2002), 207-214. K. S. Yang and S. S. Ahn, Union soft q-ideals in BCI-algebras, Applied Mathematical Scineces 8(2014), 2859-2869.

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APPLICATIONS OF DOUBLE DIFFERENCE FRACTIONAL ORDER OPERATORS TO ORIGINATE SOME SPACES OF SEQUENCES ANU CHOUDHARY AND KULDIP RAJ

Abstract. In the present article, we introduce and study some sequence spaces by means of double difference fractional order operators, Orlicz function and four dimensional bounded regular matrix. We make an effort to study some topological and algebraic properties of these sequence spaces. Some inclusion relations between newly formed sequence spaces are also establish. Finally, we study several results under the suitable choice of order γ.

1. Introduction and Preliminaries Let ($k,l , νk,l ) be a double sequence of seminormed spaces such that $k−1,l−1 ⊆ $k,l for all non-negative integers k and l. A sequence space X is called solid or normal if and only if it contains all such sequences y = (yk,l ) corresponding to each of which there is a sequence x = (xk,l ) ∈ X such that |yk,l | ≤ |xk,l | for all non negative integers k and l. Let Q be a normal sequence space and Ω2 denotes the set of all double complex sequences. Define a linear space Ω2 ($k,l ) = {x = (xk,l ) ∈ Ω2 : xk,l ∈ $k,l for all non-negative integers k and l}. Let ν and ν 0 be seminorms on a linear space X. Then ν is said to be stronger than ν 0 if whenever (xk,l ) is a sequence such that ν(xk,l ) → 0, then also ν 0 (xk,l ) → 0. If each is stronger than the other, then ν and ν 0 are said to be equivalent. A double sequence has Pringsheim limit L (denoted by P − lim x = L) provided that given  > 0 there exist n ∈ N such that |xk,l − L| <  whenever k, l > n (see [11]). A double sequence x = (xk,l ) is bounded if there exists a positive number n such that |xk,l | < n for all k and l. Some initial works on double sequences is due to Bromwich [5]. Later on, the double sequences were studied in (see [12], [13]) and operators on sequence spaces were studied in (see [1], [9]). The fractional difference operator ∆(γ) for a positive proper fraction γ on single sequence is defined as ∞ X Γ(γ + 1) ∆(γ) (xk ) = (−1)m xk−m , m!Γ(γ − m + 1) m=0 where Γ(γ) denotes the Euler gamma function of a real number γ or generalized factorial function (see [2], [3]). For γ ∈ / {0, −1, −2, −3, · · · }, Γ(γ) can be expressed as an improper integral, Z ∞ Γ(γ) = e−s sγ−1 ds. 0

2010 Mathematics Subject Classification. 40A05, 40A30. Key words and phrases. Orlicz function, fractional double difference operator, double sequence, paranormed space. 1

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For x ∈ Ω2 and a positive proper fraction γ, the double difference operator of fractional order γ is defined as (1.1)

(γ) ∆2 (xk,l )

=

∞ X ∞ X

(−1)m+n

m=0 n=0

Γ(γ + 1)2 xk−m,l−n . m!n!Γ(γ − m + 1)Γ(γ − n + 1)

The above defined infinite series can be reduced to finite series if γ is a positive integer (see [4]). Throughout the text it is assumed that (xk,l ) = 0 for any negative integers k and l. An Orlicz function M is a function, which is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) −→ ∞ as x −→ ∞. An Orlicz function M is said to satisfy ∆2 -condition for all values of u, if there exists R > 0 such that M (2u) ≤ RM (u), u ≥ 0. The idea of Orlicz function was used by Lindenstrauss and Tzafriri [7] to define the following sequence space: ∞ o n  |x |  X k < ∞, for some ρ > 0 `M = x = (xk ) ∈ w : M ρ k=1

known as an Orlicz sequence space. The space lM is a Banach space with the norm,     ∞ X |xk | kxk = inf ρ > 0 : M ≤1 . ρ k=1

A sequence M = (Mk ) of Orlicz functions is called a Musielak-Orlicz function (see [8], [10]). Remark 1.1. (1) Let M = (Mk,l ) be a Musielak Orlicz function and q be a non-negative integer. Then for a real number d ∈ [0, ∞), we have (i)M(qx) ≤ qM(x) (ii) M(dx) ≤ (1 + bdc)M(x), where b.c denotes the greatest integer function. (2) For a complex number α, |α|pk,l ≤ max{1, |α|L } and |ak,l + bk,l |pk,l ≤ D(|ak,l |pk,l + |bk,l |pk,l ), where L = sup pk,l < ∞ and D = max(1, 2L−1 ). k,l

Let A = (aijkl ) be a four-dimensional infinite matrix of scalars. For all i, j ∈ N0 , where N0 = N ∪ {0}, the sum ∞,∞ X yi,j = aijkl xk,l k,l=0,0

is called the A-means of the double sequence (xk,l ). A double sequence (xk,l ) is said to be A-summable to the limit L if the A-means exist for all i, j in the sense of Pringsheim’s convergence p,q X P - lim aijkl xk,l = yi,j and P - lim yi,j = L. p,q→∞

i,j→∞

k,l=0,0

A four-dimensional matrix A is said to be bounded-regular (or RH-regular) if every bounded P -convergent sequence is A-summable to the same limit and the A-means are also bounded. Theorem 1.2. (Robison [14] and Hamilton [6]) The four dimensional matrix A is RHregular if and only if

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3

(RH1 ) P - lim aijkl = 0 for each k and l, i,j

(RH2 ) P - lim i,j

(RH3 ) P - lim i,j

(RH4 ) P - lim i,j

(RH5 )

∞,∞ X

∞,∞ X

|aijkl | = 1,

k,l=1,1 ∞ X

|aijkl | = 0 for each l,

k=1 ∞ X

|aijkl | = 0 for each k,

l=1

|aijkl | < ∞ for all i, j ∈ N.

k,l=1,1

A real valued function g defined on a linear space X is called a paranorm, if it satisfies the following conditions for all x, y ∈ X and for all scalars β (i) g(λ) = 0, where λ is the zero element of X (ii) g(−x) = g(x) (iii) g(x + y) ≤ g(x) + g(y) (iv) If (βn ) is a sequence of scalars with βn → 0 as n → 0 and xn is a sequence in X such that g(xn − x) → 0 as n → ∞ for some x ∈ X, then g(βn xn − βx) → 0 as n → ∞. Let M = (Mk,l ) be a Musielak Orlicz function, A = (aijkl ) be a nonnegative fourdimensional bounded-regular matrix, u = (uk,l ) be any double sequence of strictly positive (γ) real numbers, p = (pk,l ) be a bounded double sequence of positive real numbers, ∆2 denotes the double difference operator of fractional order γ. In this paper we define the following sequence space (γ) Q[∆2 , p, ν, u, A, M] =  pk,l    ∞,∞    (γ) X uk,l ∆2 xk,l 2 ∈ Q, for some ρ > 0 . x = (xk,l ) ∈ Ω ($k,l ) : aijkl Mk,l νk,l ρ i,j k,l=0,0

Remark 1.3. (1) Let M = (Mk,l ) be a Musielak Orlicz function and ρ = ρ1 + ρ2 . Then (γ) for x = (xk,l ) and y = (yk,l ) ∈ Q[∆2 , p, ν, u, A, M], we have  pk,l   ∞,∞ (γ) (γ) X uk,l ∆2 (xk,l ) + uk,l ∆2 (yk,l ) aijkl Mk,l νk,l ρ

k,l=0,0

 ∞,∞ X

   pk,l (γ) uk,l ∆2 xk,l ≤ D aijkl Mk,l νk,l ρ1 k,l=0,0    pk,l  ∞,∞ (γ) X uk,l ∆2 yk,l + aijkl Mk,l νk,l ρ2 k,l=0,0

for all non-negative integers i and j and for some ρ1 , ρ2 > 0. (2) Let M = (Mk,l ) be a Musielak Orlicz function and d ∈ C. Then for L = sup pk,l < ∞, k,l

we have

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4 ∞,∞ X k,l=0,0

ANU CHOUDHARY AND KULDIP RAJ

   pk,l (γ) uk,l ∆2 (dxk,l ) aijkl Mk,l νk,l ρ  ∞,∞    pk,l  (γ) X uk,l ∆2 (xk,l ) ≤ max{1, (1 + b|d|c)L } aijkl Mk,l νk,l ρ k,l=0,0

for all non-negative integers i and j and for some ρ > 0. 0 (3) Let M = (Mk,l ) and M0 = (Mk,l ) be two Musielak Orlicz functions. Then pk,l    ∞,∞ (γ) X uk,l ∆2 (xk,l ) 0 aijkl (Mk,l + Mk,l ) νk,l ρ

k,l=0,0

+

 ∞,∞ X

pk,l    (γ) uk,l ∆2 (xk,l ) aijkl Mk,l νk,l ρ k,l=0,0    pk,l  ∞,∞ (γ) X uk,l ∆2 (xk,l ) 0 aijkl Mk,l νk,l ρ

≤ D

k,l=0,0

for all non-negative integers i and j and for some ρ > 0. 0 (4) Let M = (Mk,l ) be a Musielak Orlicz function. Let ν = (νk,l ) and ν 0 = (νk,l ) be two sequences of seminorms. Then pk,l    ∞,∞ (γ) X uk,l ∆2 (xk,l ) 0 aijkl Mk,l (νk,l + νk,l ) ρ k,l=0,0

 ∞,∞ X

 pk,l   (γ) uk,l ∆2 (xk,l ) aijkl Mk,l νk,l ≤ D ρ k,l=0,0 pk,l     ∞,∞ (γ) X uk,l ∆2 (xk,l ) 0 + aijkl Mk,l νk,l ρ k,l=0,0

for all non-negative integers i and j and for some ρ > 0. (γ)

The main goal of this paper is to introduce the double difference operator ∆2 of frac(γ) tional order γ. In this study, being an application of double difference operator ∆2 , some new difference double sequence spaces of fractional order have been introduced and subsequently, their topological and algebraic properties have been discussed in detail. Infact, this study involves new results obtained under different suitable choice of γ. 2. Main Results Theorem 2.1. Let M = (Mk,l ) be a Musielak Orlicz function, ν = (νk,l ) be a sequence of seminorms and u = (uk,l ) be a double sequence of strictly positive real numbers. Then (γ) the sequence space Q[∆2 , p, ν, u, A, M] is a linear space over the complex field C. Proof. This is a routine matter, so we omit it.



Theorem 2.2. Let M = (Mk,l ) be a Musielak Orlicz function, ν = (νk,l ) be a sequence of seminorms and u = (uk,l ) be a double sequence of strictly positive real numbers. Then the

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5

(γ)

sequence space Q[∆2 , p, ν, u, A, M] is a paranormed space with paranorm g defined by  p  ∞,∞    pk,l  N1  (γ) X k,l uk,l ∆2 (xk,l ) g(x) = inf (ρ) L : aijkl Mk,l νk,l ≤ 1, for some ρ > 0 , ρ k,l=0,0

where N = max{1, L} and L = sup pk,l < ∞. k,l (γ)

Proof. (i) Clearly g(x) ≥ 0, for x = (xk,l ) ∈ Q[∆2 , p, ν, u, A, M]. Since Mk,l (0) = 0, we get g(0) = 0. (ii) g(−x) = g(x). (γ) (iii) Let x = (xk,l ), y = (yk,l ) ∈ Q[∆2 , p, ν, u, A, M], then there exist ρ1 > 0 and ρ2 > 0 such that pk,l  N1  ∞,∞    (γ) X uk,l ∆2 (xk,l ) ≤1 aijkl Mk,l νk,l ρ1 k,l=0,0

and  ∞,∞ X k,l=0,0

 pk,l  N1   (γ) uk,l ∆2 (yk,l ) aijkl Mk,l νk,l ≤ 1. ρ2

Now for ρ = ρ1 + ρ2 and by using Minkowski’s inequality, we have  ∞,∞  pk,l  N1   (γ) (γ) X uk,l ∆2 (xk,l ) + uk,l ∆2 (yk,l ) aijkl Mk,l νk,l ρ1 + ρ2 k,l=0,0

 ≤  +

ρ1 ρ1 + ρ2

 ∞,∞ X

ρ2 ρ1 + ρ2

 ∞,∞ X

k,l=0,0

k,l=0,0

pk,l  N1    (γ) uk,l ∆2 xk,l aijkl Mk,l νk,l ρ1  pk,l  N1   (γ) uk,l ∆2 yk,l aijkl Mk,l νk,l ρ2

≤ 1. Hence, g(x + y)  ∞,∞     pk,l  N1 (γ) (γ) X pk,l uk,l ∆2 (xk,l ) + uk,l ∆2 (yk,l ) L = inf (ρ1 +ρ2 ) : aijkl Mk,l νk,l ≤ ρ k,l=0,0

 1, for some ρ > 0

 ∞,∞    pk,l  N1   (γ) X pk,l uk,l ∆2 (xk,l ) L : aijkl Mk,l νk,l ≤ inf (ρ1 ) ≤ 1, for some ρ1 > 0 ρ1 k,l=0,0

+

  ∞,∞    pk,l  N1  (γ) X pk,l uk,l ∆2 (yk,l ) L inf (ρ2 ) : aijkl Mk,l νk,l ≤ 1, for some ρ2 > 0 ρ2 k,l=0,0

= g(x) + g(y). (iv) Finally, we show that scalar multiplication is continuous. In order to show this, let us consider a complex number σ. Then by definition we have

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g(σx) =

 p  ∞,∞    pk,l  N1  (γ) X k,l σuk,l ∆2 (xk,l ) L inf (ρ) : aijkl Mk,l νk,l ≤ 1, ρ > 0 , ρ

=

 ∞,∞    pk,l  N1   (γ) X pk,l uk,l ∆2 (xk,l ) L : aijkl Mk,l νk,l ≤ 1, t > 0 , inf (|σ|t) t

k,l=0,0

k,l=0,0

where t =

ρ |σ| .

Hence the proof.



0 Theorem 2.3. Let M = (Mk,l ) and M0 = (Mk,l ) be two Musielak Orlicz functions, u = (uk,l ) be a double sequence of strictly positive real numbers and A = (aijkl ) be a nonnegative four-dimensional bounded-regular matrix. Then (γ)

(γ)

(γ)

Q[∆2 , p, ν, u, A, M] ∩ Q[∆2 , p, ν, u, A, M0 ] ⊆ Q[∆2 , p, ν, u, A, M + M0 ]. (γ)

(γ)

Proof. Suppose x = (xk,l ) ∈ Q[∆2 , p, ν, u, A, M] ∩ Q[∆2 , p, ν, u, A, M0 ]. This implies that  ∞,∞  pk,l    (γ) X uk,l ∆2 xk,l aijkl Mk,l νk,l ρ i,j k,l=0,0

and  ∞,∞ X k,l=0,0

pk,l     (γ) uk,l ∆2 xk,l 0 aijkl Mk,l νk,l ρ i,j

both are in Q. Now by using part (3) of Remark 1.3, we have pk,l    ∞,∞ (γ) X uk,l ∆2 xk,l 0 aijkl (Mk,l + Mk,l ) νk,l ρ k,l=0,0

+

 ∞,∞ X

 pk,l   (γ) uk,l ∆2 xk,l aijkl Mk,l νk,l ρ k,l=0,0 pk,l     ∞,∞ (γ) X uk,l ∆2 xk,l 0 aijkl Mk,l νk,l . ρ

≤ D

k,l=0,0

Since Q is normal,

 ∞,∞ X k,l=0,0 (γ)

pk,l     (γ) uk,l ∆2 xk,l 0 aijkl (Mk,l + Mk,l ) νk,l ∈ Q. ρ i,j

Then x = (xk,l ) ∈ Q[∆2 , p, ν, u, A, M + M0 ]. Hence the proof.



Theorem 2.4. Suppose that M = (Mk,l ) be a Musielak Orlicz function, ν = (νk,l ) and 0 ν 0 = (νk,l ) be two double sequences of seminorms. Then (γ)

(γ)

(γ)

Q[∆2 , p, ν, u, A, M] ∩ Q[∆2 , p, ν 0 , u, A, M] ⊆ Q[∆2 , p, ν + ν 0 , u, A, M]. Proof. One can easily obtain the proof by using part (4) of Remark 1.3. So, we omit it.  0 Theorem 2.5. Let M = (Mk,l ) be a Musielak Orlicz function. If ν = (νk,l ) and ν 0 = (νk,l ) 0 be two double sequences of seminorms such that (νk,l ) is stronger than (νk,l ) for each k (γ)

(γ)

and l, then Q[∆2 , p, ν, u, A, M] ⊆ Q[∆2 , p, ν 0 , u, A, M].

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(γ)

Proof. Consider a double sequence x = (xk,l ) ∈ Q[∆2 , p, ν, u, A, M]. Then  ∞,∞    pk,l  (γ) X uk,l ∆2 xk,l aijkl Mk,l νk,l ∈ Q. ρ i,j k,l=0,0

0 Since each νk,l is stronger than corresponding νk,l , we have a natural number Nk,l corre0 sponding to each pair of non-negative integer k and l such that νk,l (w) ≤ Nk,l νk,l (w). Let 0 N = max{Nk,l }. Then νk,l (w) ≤ N νk,l (w) for all non-negative integers k and l. Thus,     (γ) (γ) uk,l ∆2 xk,l uk,l ∆2 xk,l 0 νk,l ≤ N νk,l . ρ ρ

From Remark 1.1, we have pk,l    ∞,∞ (γ) X uk,l ∆2 xk,l 0 aijkl Mk,l νk,l ρ k,l=0,0

∞,∞ X

L

≤ max{1, N }

k,l=0,0

Since Q is normal,

 ∞,∞ X k,l=0,0

   pk,l (γ) uk,l ∆2 xk,l aijkl Mk,l νk,l . ρ

 pk,l    (γ) uk,l ∆2 xk,l 0 aijkl Mk,l νk,l ∈ Q. ρ i,j (γ)

This implies x = (xk,l ) ∈ Q[∆2 , p, ν 0 , u, A, M].

 0

Corollary 2.6. Let M = (Mk,l ) be a Musielak Orlicz function. If ν = (νk,l ) and ν = 0 0 (νk,l ) be two double sequences of seminorms such that (νk,l ) is equivalent to (νk,l ) for each k and l. Then (γ) (γ) Q[∆2 , p, ν, u, A, M] = Q[∆2 , p, ν 0 , u, A, M]. 0 Theorem 2.7. Suppose M = (Mk,l ) and M0 = (Mk,l ) be two Musielak Orlicz function such that Mk,l (1) is finite for each k and l. Let A = (aijkl ) be a nonnegative fourdimensional bounded-regular matrix. Then (γ)

(γ)

Q[∆2 , p, ν, u, A, M0 ] ⊆ Q[∆2 , p, ν, u, A, M ◦ M0 ]. (γ)

Proof. Consider x = (xk,l ) ∈ Q[∆2 , p, ν, u, A, M0 ]. So, pk,l   ∞,∞    (γ) X uk,l ∆2 xk,l 0 ∈ Q. aijkl Mk,l νk,l ρ i,j k,l=0,0

Since each (Mk,l ) is continuous and Mk,l (0) = 0 for each k and l, we choose ς ∈ (0, 1) corresponding to an arbitrary  > 0 such that Mk,l (s) <  for 0 ≤ s ≤ ς. Let us take    (γ) uk,l ∆2 xk,l 0 sk,l = Mk,l νk,l ρ and (2.1)

∞,∞ X

     
pk,l X
pk,l X
pk,l aijkl Mk,l sk,l = aijkl Mk,l sk,l + aijkl Mk,l sk,l , 1

k,l=1,1

2

where the first summation is over sk,l ≤ ς and the second is taken over sk,l > ς. For sk,l ≤ ς, we have Mk,l (sk,l ) <  and hence   X
pk,l X aijkl Mk,l sk,l < aijkl []pk,l . 1

1

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8

ANU CHOUDHARY AND KULDIP RAJ

Now by using Part (2) of Remark 1.1, we have (2.2)

X

  ∞,∞ X X
pk,l aijkl Mk,l sk,l < max{1, L } aijkl ≤ max{1, L } aijkl .

1

1

 For sk,l > ς, we have sk,l
1, then the following statements holds: (i) Every positive solution {(xn , yn )} of (1) is bounded and persists.     kB (kA2 +B ) kA(kB 2 +A) (ii) The interval A, k2 AB−1 × B, k2 AB−1 is invariant set for (1). ∗ Department of Mathematics, University of Azad Jammu and [email protected] † Department of Mathematics, Faculty of Science, Jazan University, [email protected] ‡ Department of Mathematics, Faculty of Science, Jazan University, Jazan, § Department of Mathematics, Faculty of Science, Jazan University, Jazan,

104

Kashmir,

Muzaffarabad

13100,

Pakistan,

Jazan, Kingdom of Saudi Arabia, e-mail:

e-mail:

ab-

maliahmedi-

Kingdom of Saudi Arabia, e-mail: [email protected] Kingdom of Saudi Arabia, e-mail: [email protected]

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Proof. (i) If {(xn , yn )} be a positive solution of (1) then xn ≥ A, yn ≥ B, n = 0, 1, · · · .

(2)

From (1) and (2), one gets 1 1 yn , yn+1 ≤ B + xn . kA kB

(3)

xn+1 ≤ A +

B 1 A 1 + 2 xn−1 , yn+1 ≤ B + + 2 yn−1 . kA k AB kB k AB

(4)

ςn+1 = A +

B 1 A 1 + 2 ςn−1 , %n+1 = B + + 2 %n−1 . kA k AB kB k AB

(5)

xn+1 ≤ A + Moreover, from (3), one gets

Now consider

Therefore, solution {(ςn , %n )} of (5) is given by r ςn = c1 r %n = d 1

1 2 k AB 1 k 2 AB

!n

r

+ c2 !n + d2

!n


kB kA2 + B + , − k 2 AB − 1 !n r
kA kB 2 + A 1 − + , k 2 AB k 2 AB − 1 1 2 k AB

(6)

where c1 , c2 , d1 , d2 depend upon ς−1 , ς0 , %−1 , %0 . Assuming ABk 2 > 1, then (6) implies that {ςn } and {%n } are bounded. Now considering solution {(ςn , %n )} of (6) for which ς−1 = x−1 , ς0 = x0 , %−1 = y−1 , %0 = y0 ,     kB (kA2 +B ) kA(kB 2 +A) where x−1 , x0 ∈ A, k2 AB−1 and y−1 , y0 ∈ B, k2 AB−1 . From (4) and (7) one gets

(7)



kB kA2 + B kA kB 2 + A , yn ≤ . xn ≤ k 2 AB − 1 k 2 AB − 1

(8)

From (2) and (8), we get A ≤ xn ≤



kB kA2 + B kA kB 2 + A , B ≤ y ≤ , n = 0, 1, · · · . n k 2 AB − 1 k 2 AB − 1

Proof. (ii) Follows from induction.     kB (kA2 +B ) kA(kB 2 +A) Theorem 2. System (1) has a unique positive equilibrium point (¯ x, y¯) ∈ A, k2 AB−1 × B, k2 AB−1 if k

2



2k 2 B(kA2 + B) k 2 AB − 1



   kB(kA2 + B) 2kB(kA2 + B) −A −B − A < 1. k 2 AB − 1 k 2 AB − 1

Proof. Consider x=A+

y x , y=B+ . kx ky

(9)

(10)

From (10), y = kx(x − A), x = ky(y − B). Defining S(x) = ks(x)(s(x) − B) − x,

(11)

s(x) = kx(x − A),

(12)

where 105

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    kB (kA2 +B ) kB (kA2 +B ) and x ∈ A, k2 AB−1 . We claim that S(x) = 0 has a unique solution x ∈ A, k2 AB−1 . From (11) and (12) one gets S 0 (x) = 2ks(x)s0 (x) − kBs0 (x) − 1,

(13)

s0 (x) = 2kx − kA.

(14)

and   kB (kA2 +B ) be a solution of S(x) = 0 then from (11) and (12) one gets Now if x ¯ ∈ A, k2 AB−1 ks(¯ x)(s(¯ x) − B) = x ¯,

(15)

s(¯ x) = k¯ x(¯ x − A).

(16)

where In view of (14), (15) and (16), equation (13) becomes S 0 (x)

= k 2 (2kx(x − A) − B) (2x − A) − 1,      2 2kB(kA2 + B) 2k B(kA2 + B) kB(kA2 + B) 2 −A −B − A − 1. ≤ k k 2 AB − 1 k 2 AB − 1 k 2 AB − 1

(17)

0 Now assume that ¯ ∈   (9) hold then from (17) one gets S (x) < 0. Hence S(x) = 0 has a unique positive solution x 2 kB (kA +B ) A, k2 AB−1 .

Theorem 3. If 1 kB 2 + A 1 kA2 + B + < 1, + < 1, kA A(k 2 AB − 1) kB B(k 2 AB − 1)     kB (kA2 +B ) kA(kB 2 +A) then equilibrium (¯ x, y¯) ∈ A, k2 AB−1 × B, k2 AB−1 of the system (1) is locally asymptotically stable.

(18)

Proof. The linearized system of (1) about (¯ x, y¯) is Φn+1 = EΦn , where 

xn xn−1 .. .

      xn−k Φn =   yn   yn−1   .  .. yn−k





0 1 .. .

             , E =  01     ky¯   0     .   .. 0

− k2y¯x¯2 0 .. .

... ... .. .

− k2y¯x¯2 0 .. .

− k2y¯x¯2 0 .. .

1 k¯ x

0 0 0 .. .

... ... ... .. .

1 0 0 .. .

0

...

0

0 .. .

0 0 .. .

... ... .. .

0 0 .. .

0 0 .. .

0 0 0 .. .

0 0 1 .. .

0 − k2x¯y¯2 0 .. .

... ... ... .. .

0 − k2x¯y¯2 0 .. .

0 − k2x¯y¯2 0 .. .

0

0

0

...

1

0

       .      

Let us denote 2k + 2 eigenvalues of E as κ1 , κ2 , . . . , κ2k+2 and D = diag(m1 , m2 , . . . , m2k+2 ) be a diagonal matrix, where m1 = mk+2 = 1, mi = mk+1+i = 1 − i, i = 2, 3, · · · , k + 1, and  0 <  < min

1 k+1

 1−

1 kB 2 + A − kA A(K 2 AB − 1)

 ,

1 k+1

  1 kA2 + B 1− − < 1. kB B(K 2 AB − 1)

Since D is invertible and by computing DED−1 , one gets

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

− k2y¯x¯2 m1 m−1 2 0 .. .

0 m2 m−1 1 .. .

      0 1  mk+2 m−1 1  ky¯  0   ..  . DED−1 =

0

0 0 0 .. .

... ... ... ... .. .

0

...

−1 1 k¯ x m1 mk+2

0 .. . 0 0 mk+3 m−1 k+2 .. .

... ...

− k2y¯x¯2 m1 m−1 k 0 .. .

− k2y¯x¯2 m1 m−1 k+1 0 .. .

mk+1 m−1 k 0 0 .. .

0 0 0 .. .

0

0

0 0 .. .

... ... .. .

0 0 .. .

... ... ... .. .

0

...

− k2x¯y¯2 mk+2 m−1 k+3

0

0 0 .. . 0 − k2x¯y¯2 mk+2 m−1 2k+1 0 .. .

0 0 .. .

.

(19)

      0  −1 x ¯ − k2 y¯2 mk+2 m2k+2    0   ..  .

m2k+2 m−1 2k+1

0

From m1 > m2 > · · · > mk+1 > 0 and mk+2 > mk+3 > · · · > m2k+2 > 0, one has −1 −1 −1 −1 −1 m2 m−1 1 < 1, m3 m2 < 1, · · · , mk+1 mk < 1, mk+3 mk+2 < 1, mk+4 mk+3 < 1, · · · , m2k+2 m2k+1 < 1.

Also, y¯ y¯ 1 −1 m1 m−1 m1 m−1 2 + · · · + 2 2 m1 mk+1 + k+2 2 2 k x ¯ k x ¯ k¯ x

  1 y¯ 1 1 + 2 2 + ··· + , k¯ x k x ¯ 1 − 2 1 − (k + 1)   1 y¯ 1 < + 2 , k¯ x k¯ x 1 − (k + 1)   1 kB 2 + A 1 < + < 1. 2 kA A (k AB − 1) 1 − (k + 1) =

And 1 x ¯ x ¯ −1 −1 mk+2 m−1 1 + 2 2 mk+2 mk+3 + · · · + 2 2 mk+2 m2k+2 k y¯ k y¯ k y¯

  1 x ¯ 1 1 = + + ··· + , k y¯ k 2 y¯2 1 − 2 1 − (k + 1)   x ¯ 1 1 + , < k y¯ k y¯2 1 − (k + 1)   1 kA2 + B 1 < + < 1. 2 kB B (k AB − 1) 1 − (k + 1)

Since E has the same eigenvalues as DED−1 and hence −1 −1 −1 max{m2 m−1 1 , · · · , mk+1 mk , mk+3 mk+2 , · · · , m2k+2 m2k+1 ,   y¯ 1 1 1 1 + + ··· + , + k¯ x k2 x ¯2 1 − 2 1 − (k + 1) k y¯   x ¯ 1 1 + ··· + } < 1. (20) k 2 y¯2 1 − 2 1 − (k + 1)     kB (kA2 +B ) kA(kB 2 +A) Thus equation (20) implies that (¯ x, y¯) ∈ A, k2 AB−1 × B, k2 AB−1 of (1) is locally asymptotically stable.

max

1≤n≤2k+2

|κn | ≤ kDED−1 k∞

=

    kB (kA2 +B ) kA(kB 2 +A) Theorem 4. Equilibrium (¯ x, y¯) ∈ A, k2 AB−1 × B, k2 AB−1 of (1) is globally asymptotically stable. Proof. Let {(xn , yn )} be arbitrary solution of (1). Also let lim supxn = L1 , lim infxn = l1 , lim supyn = L2 , lim infyn = n→∞

n→∞

n→∞

n→∞

l2 where li , Li ∈ (0, ∞), i = 1, 2. Then from (1) one gets L1 ≤ A +

L2 l2 , l1 ≥ A + . kl1 kL1 107

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And L2 ≤ B +

L1 l1 , l2 ≥ B + . kl2 kL2

(22)

From (21), we have Ak(L1 − l1 ) ≤ L2 − l2 .

(23)

Bk(L2 − l2 ) ≤ L1 − l1 .

(24)

From (22), we get

From (23) and (24), we get (ABk 2 − 1)(L1 − l1 ) ≤ 0, which implies that l1 = L1 . Similarly it is easy to prove that l2 = L2 . Theorem 5. Assuming {(xn , yn )} x, y¯) as n → ∞, where (¯ x, y¯) ∈     be a positive solution of (1) such that (xn , yn ) → (¯ kB (kA2 +B ) kA(kB 2 +A) A, k2 AB−1 × B, k2 AB−1 . Then, the error vector ξn satisfying lim

n→∞

p kξn+1 k n kξn k = |κE|, lim = |κE|, n→∞ kξn k

where κE are the characteristic roots of E. Proof. If {(xn , yn )} be any solution of (1) such that (xn , yn ) → (¯ x, y¯) as n → ∞. To find error term one has k X y¯ y¯ 1 yn − = − ¯) + k (yn − y¯) , ! (xn−i − x k k k¯ x X X X i=1 xn−i xn−i k¯ x xn−i

xn+1 − x ¯ =

i=1

i=1

i=1

k X xn x ¯ 1 x ¯ − = (x − x ¯ ) − ! (yn−i − y¯) . n k k k k y¯ X X X i=1 yn−i yn−i k y¯ yn−i

yn+1 − y¯ =

i=1

i=1

i=1

Denote 1n = xn − x ¯ and 2n = yn − y¯, one has 1n+1 =

k X

Ani 1n−i + Bn 2n , 2n+1 = Cn 1n +

k X

i=1

i=1

where An1 = An2 = · · · = Ank = − k¯ x

y¯ k X

! , Bn = xn−i

i=1

Cn =

1 k X

Dni 2n−i ,

,

xn−i

i=1

, Dn1 = Dn2 = · · · = Dnk = −

k X i=1

yn−i

1 k X

i=1

k y¯

x ¯ k X

!. yn−i

i=1

Taking the limits, we obtain lim An1 = lim An2 = · · · = lim Ank = −

n→∞

lim Cn =

n→∞

n→∞

n→∞

y¯ k2 x ¯2

, lim Bn = n→∞

1 , k¯ x

1 x ¯ , lim Dn1 = lim Dn2 = · · · = lim Dnk = − 2 2 . n→∞ n→∞ k y¯ n→∞ k y¯ 108

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Hence we have system (1.10) of [8] where ξn+1 = Eξn , 1n  1n−1   ..  .  1   n−k where ξn =   2n  2  n−1   .  .. 2n−k to linearized system 

3

0 − k2y¯x¯2  1  0   ..  ..   .  .    0  0 , E =  1   0  ky¯   0  0    .  .. .  .  . 0 0 of (1) about (¯ x, y¯). 



(25)

... ... .. .

− k2y¯x¯2 0 .. .

− k2y¯x¯2 0 .. .

1 k¯ x

0 .. .

0 0 .. .

... ... .. .

0 0 .. .

0 0 .. .

... ... ... .. .

1 0 0 .. .

0 0 0 .. .

0 0 1 .. .

0 − k2x¯y¯2 0 .. .

... ... ... .. .

0 − k2x¯y¯2 0 .. .

0 − k2x¯y¯2 0 .. .

...

0

0

0

0

...

1

0

        . This is similar      

Conclusion

In the present work, dynamics of following higher-order anti-competitive system is studied: xn+1 = A +

yn k X

, yn+1 = B +

xn−i

i=1

xn k X

.

yn−i

i=1

Our investigations if ABk 2 > 1, then {(xn , yn )} of this system is bounded and persists and the region   reveal that 2 2 kB (kA +B ) kA(kB +A) 1 kB 2 +A 1 kA2 +B A, k2 AB−1 × B, k2 AB−1 is invariant set. It is proved that if kA + A(k 2 AB−1) < 1 and kB + B(k 2 AB−1) < 1     kB (kA2 +B ) kA(kB 2 +A) then equilibrium (¯ x, y¯) ∈ A, k2 AB−1 × B, k2 AB−1 of the system is locally asymptotically stable. Finally     kA(kB 2 +A) kB (kA2 +B ) × B, k2 AB−1 of (1) are also global dynamics and rate of convergence that converges to (¯ x, y¯) ∈ A, k2 AB−1 demonstrated. Acknowledgements A. Q. Khan research is supported by the Higher Education Commission(HEC) of Pakistan.

References [1] E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, Chapman and Hall/CRC Press, Boca Raton, (2004). [2] V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993). [3] A. S, Kurbanli, On the behavior of positive solutions of the system of rational difference equations xn+1 = xn−1 yn−1 1 yn xn−1 −1 , yn+1 = xn yn−1 −1 , zn+1 = yn zn , Advances in Difference Equations, 2011, 2011:40. [4] E. M. Elsayed, T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe Journal of Mathematics and Statistics, 44(6)(2015):1361-1390. [5] R. DeVault, G. Ladas, S. W. Schultz, On the recursive sequence xn+1 = Mathematical Society, 126(11)(1998):3257-3261.

A xn

+

1 xn−2 ,

Proceedings of the American

xn , Applied Mathematics Letters, 16(2003):173-178. [6] R. M. Abu-Saris, R. DeVault, Global stability of xn+1 = A + xn−k

[7] Q. Zhang, L. Yang, D. Liao, Global asymptotic behavior of positive solutions to the system of rational difference equations, Journal of Southwest University, 7(2012):12-15. [8] M. Pituk, More on Poincare’s and Perron’s theorems for difference equations, Journal of Difference Equations and Applications, 8(2002):201-216.

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Stability of a modified within-host HIV dynamics model with antibodies Ali Al-Qahtania , Shaban Alya , Ahmed Elaiwb and E. Kh. Elnaharyc a Department of Mathematics, Faculty of Science, King Khalid University, Saudi Arabia. b Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia. c Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt. Email: a m [email protected] (A. Elaiw) Abstract We investigate a modified HIV infection model with antibodies and latency. The model consider saturated HIV-CD4+ T cells and HIV-macrophages incidence rates. We show that the solutions of the proposed model are nonnegative and bounded. We established that the global stability of the three steady states of the model depend on threshold parameters R0 and R1 . Using Lyapunov function, we established the global stability of the steady states of the model. The theoretical results are confirmed by numerical simulations. The results show that antibodies can reduce the HIV infection.

1

Introduction

Constructing and analyzing of within-host human immunodeficiency virus (HIV) dynamics models have become one of the hot topics during the last decades [1]-[18]. These works can help researchers for better understanding the HIV dynamical behavior and providing new suggestions for clinical treatment. A vast of the mathematical models presented in the literature have focused on modeling the interaction between three main compartments, uninfected CD4+ T cells (s), infected cells (u) and free HIV particles (p). Other models have differentiated between latent and active infected cells [19]-[23], an HIV mathematical model has been presented by inroducing a new variable (w) for the latently infected cells as: s˙ = ρ − δs − λsp,

(1)

w˙ = λsp − (α + β) w,

(2)

u˙ = βw − au,

(3)

p˙ = ku − gp,

(4)

where, ρ is the creation rate of the uninfected CD4+ T cells, δ, α, a and g are the death rate constants of the four compartments s, w, u and p, respectively. The term βw represents the activation rate of the latently infected cells. The HIV-CD4+ T cell incidence rate is given by λsp. Parameter k represents the rate constant of free virus production. Sun et. al. [24] have modified the above model by considering the saturated infection rate

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λsp s+p

as: s˙ = ρ − δs − w˙ =

λsp , s+p

(5)

λsp − (α + β) w, s+p

(6)

u˙ = βw − au,

(7)

p˙ = ku − gp,

(8)

Model (5)-(8) consider one type of target cells (CD4+ T cells). Moreover, the model does not acount the presence of the antibodies which are important in reducing the HIV infection. To have more accurate HIV model we improve model (5)-(8) by taking into account the dynamics of HIV with two target cells, CD4+ T cells and macrophages and antibodies. The global stability of the model is proven by using Lyapunov method.

2

The modified HIV

We propose the following model: s˙ i = ρi − δi si − w˙ i =

λi si p , si + p

λi si p − (αi + βi ) wi , si + p

u˙ i = βi wi − ai ui , p˙ =

i = 1, 2,

(9)

i = 1, 2,

(10)

i = 1, 2,

2 X ki ui − gp − µpz,

(11) (12)

i=1

z˙ = rpz − ζz.

(13)

where, z (t) represents the populations of the antibody immune cells. The antibodies are proliferated and die at rates rpz and ζz, respectively. The HIV particles are killed by antibodies at rate µpz.

2.1

Preliminaries.

Lemma 1. The solutions of model (9)-(13) with the initial conditions si (0) , wi (0) , ui (0) , p (0) and z (0) are nonnegativite and bounded for t ≥ 0. Proof. We have s˙ i |si =0 = ρi > 0, p˙ |p=0 =

w˙ i |wi =0 =

λi si p ≥ 0 ∀ si ≥ 0, p ≥ 0, si + p

2 X ki ui ≥ 0 ∀ ui ≥ 0,

u˙ i |ui =0 = βi wi ≥ 0 ∀ wi ≥ 0, i = 1, 2

z˙ |z=0 = 0.

i=1

This shows the nonnegativity of the model’s solutions. Now we let Gi (t) = si (t) + wi (t) + ui (t), then G˙ i = ρi − δi si − αi wi − ai ui ≤ ρi − κi (si + wi + ui ) = ρi − κi Gi , where κi = min {δi , αi , ai } , i = 1, 2. Hence 0 ≤ Gi (t) ≤ Mi where, Mi = are all bonded. Let G3 (t) = p (t) + µr z (t), then G˙ 3 (t) =

ρi κi .

therefore si (t) , wi (t) and ui (t)

2 2 2  X X µζ µ  X ki ui − gp − ki Mi − κ3 G3 (t) , z≤ ki Mi − κ3 p + z = r r i=1 i=1 i=1

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where κ3 = min {g, ζ}. Hence p (t) ≤ M3 and z (t) ≤ M4 for t ≥ 0 where, M3 =

1 κ3

2 X ki Mi and M4 =

rM3 µ .

So

i=1

that, there is a bounded subset of D Γ = {(s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ H : 0 ≤ si + wi + ui ≤ Mi , 0 ≤ p ≤ M3 , 0 ≤ w ≤ M4 } . is positively invariant with respect to system (9)-(13). Lemma 2. For system (9)-(13) there exist two bifurcation parameters R0 and R1 with R0 > R1 such that (i) if R0 ≤ 1, then the system has only one steady state Π0 , (ii) if R1 ≤ 1 < R0 , then the system has only two steady states Π0 and Π1 , (iii) if R1 > 1, then the system has three steady states Π0 ,Π1 and Π2 . Proof. Let λi si p = 0, si + p

(14)

λi si p − (αi + βi ) wi = 0, si + p

(15)

βi wi − ai ui = 0,

(16)

2 X ki ui − gp − µpz = 0,

(17)

ρi − δi si −

i=1

rpz − ζz = 0.

(18)

Eq. (18) we obtain two possible solutions, z = 0 or p = ζr . First, we consider the case z = 0, then from Eqs. (15)-(16) we can get: λi si p λi βi si p wi = , ui = , (19) (αi + βi ) (si + p) ai (αi + βi ) (si + p) where s0i =

ρi δi .

From Eq. (17) we obtain ! ki λi βi si − 1 gp = 0. a g (αi + βi ) (si + p) i=1 i

2 X

Eq. (20) has two possible solutions p = 0 or

2 X

ki λi βi si ai g(αi +βi )(si +p)

(20)

= 1.

i=1

If p = 0, then substituting it in Eq. (19) leads to the uninfected steady state Π0 = (s01 , s02 , 0, 0, 0, 0, 0, 0). If p 6= 0, we have 2 X ki λi βi si − 1 = 0. (21) a g (α i + βi ) (si + p) i=1 i Eq. (14) implies that s± i where, s0i =

ρi δi , ϕi

=

λi δi

1 = 2



s0i


− ϕi p ±

 q 2 0 0 (ϕi p − si ) + 4si p ,

+ + + 1, i = 1, 2. Clearly if p > 0 then s− i < 0 and si > 0, then we choose si = si   q
1 2 si = s0i − ϕi p + (ϕi p − s0i ) + 4s0i p . 2

(22)

Substituting from Eqs. (14) and (19) into Eq. (17) we get 2 X

ki βi (ρi − δi si ) − gp = 0. d (α 3i i + βi ) i=1 112

(23)

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Since si is a function of p then from Eq. (23) we can define a function H1 (p) as: 2 X

ki β i (ρi − δi si (p)) − gp = 0. a (α i + βi ) i=1 i

H1 (p) =

(24)

We need to show that there exists a p > 0 such that H1 (p) = 0. It is clear that, if p = 0, then si = s0i and 2 X ki ρi βi ˆi = si (ˆ p) > 0 and H1 (0) = 0 and when p = pˆ = ai g(αi +βi ) > 0, we have s i=1 2 X H1 (ˆ p) = −

ki βi δi sˆi < 0. a g (αi + βi ) i=1 i

Since H1 (p) is continuous for all p ≥ 0, we obtain 0

H1 (0) = g

! ki λi βi −1 . a g (αi + βi ) i=1 i

2 X

0

Therefore, H1 (0) > 0, if 2 X

ki λi βi >1 a g (αi + βi ) i i=1

(25)

It means that if condition (25) is satisfied, then there exists p˜ ∈ (0, pˆ) such that H1 (˜ p) = 0. From Eqs. (19) and (22), we have s˜i , w ˜i , u ˜i , p˜ > 0. Thus, an infection steady state without antibodies Π1 = (˜ s1 , s˜2 , w ˜1 , w ˜2 , u ˜1 , u ˜2 , p˜, 0) 2 X ki λi βi exists when ai g(αi +βi ) > 1.Now we can define i=1

R0 =

2 2 X X R0i =

ki λi βi , a g (αi + βi ) i i=1

i=1

Now if z 6= 0, then from Eqs. Eqs. (14)-(16),     s 2 0 1 ζ ζ 4ζsi  ϕi − s0i s¯i =  s0i − ϕi + + , 2 r r r u ¯i =

Thus, z¯ > 0 when

2 X

λi βi s¯i p¯ , ai (αi + βi ) (¯ si + p¯) ki λi βi s¯i ai g(αi +βi )(¯ si +p) ¯

z¯ =

g µ

w ¯i =

2 X

λi s¯i p¯ , (αi + βi ) (¯ si + p¯) !

ki λi βi s¯i −1 , a g (α si + p¯) i + βi ) (¯ i=1 i

> 1. Let us define the parameter R1 as:

i=1

R1 =

2 X

ki λi βi s¯i , a g (α si + p¯) i + βi ) (¯ i=1 i

If R1 > 1, then z¯ = µg (R1 − 1) > 0 and exists an infection steady state with antibodies Π2 = (¯ s1 , s¯2 , w ¯1 , w ¯2 , u ¯1 , u ¯2 , p¯, z¯) if R1 > 1.

2.2

Global properties

We will use the following function throughout the paper, F : (0, ∞) −→ [0, ∞) as F (q) = q − 1 − ln q. Theorem 1. The steady state Π0 is globally asymptotically stable when R0 ≤ 1.

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Proof. Define W01 = where, γi =

ki βi ai (αi +βi ) .

We evaluate

dW01 dt

  2 X ζ (αi + βi ) ui + p + z, γi wi + β r i i=1 along the solutions of system (9)-(13) as:

  2 X dW01 µ (αi + βi ) = u˙ i + p˙ + z˙ γi w˙ i + dt β r i i=1   X 2 2 X λi si p (αi + βi ) µ = γi − (αi + βi ) wi + (βi wi − ai ui ) + ki ui − gp − µpz + (rpz − ζz) . si + p βi r i=1 i=1

(26)

Eq. (26) can be simplified as 2

2

X λi si p X µζ dW01 µζ = − gp − z≤ z γi γi λi p − gp − dt si + p r r i=1 i=1 ! 2 X ki λi βi µζ µζ =g −1 p− z = g (R0 − 1) p − z. a g (α + β ) r r i i i=1 i dW01 01 If R0 ≤ 1, then dW dt ≤ 0 holds in Γ. Moreover, dt = 0 when p = 0 and z = 0. Hence the largest compact invariant set in Γ is   dW01 Q1 = (s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Γ | =0 dt

= {(s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Γ | p = 0, z = 0} . LaSalle’s invariance principle yields limt→+∞ p (t) = 0 and limt→+∞ z (t) = 0. One can get limit equations: s˙ i = ρi − δi si ,

(27)

w˙ i = − (αi + βi ) wi ,

(28)

u˙ i = βi wi − ai ui .

(29)

Define a function W02 by W02 =

    2 X si (αi + βi ) γi s0i F + w + u i i . s0i βi i=1

Then 2

   s0 (αi + βi ) 1 − i s˙ i + w˙ i + u˙ i si βi    2 X s0i (αi + βi ) = γi 1 − (ρi − δi si ) − (αi + βi ) wi + (βi wi − ai ui ) si βi i=1
2 2 2 X X si − s0i = − γi δ i − k i ui . si i=1 i=1

X dW02 = γi dt i=1

dW02 0 02 Therefore, dW dt ≤ 0 holds in Q1 and dt = 0 if and only if si = si and ui = 0. There is the largest compact invariant set in Q1 :   dW02 Q2 = (s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Q1 | =0 dt  = (s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Q1 | si = s0i , wi ≥ 0, ui = 0 .

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In Q2 , from Eq. (29) we get βi wi − ai (0) = 0, and then wi = 0. So   dW02 =0 Q2 = (s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Q1 | dt  = (s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Q1 | si = s0i , wi = 0, ui = 0 = {Π0 } . Hence, if R0 ≤ 1, all solution trajectories in Γ approach the uninfected steady state Π0 . Theorem 2. The steady state Π1 is globally asymptotically stable when R1 ≤ 1 < R0 . Proof. We introduce         Zsi 2 X (α + β ) w ˜ (τ + p ˜ ) w (α + β ) u p µ i i i i i i i   W1 = γi si − s˜i − dτ + w ˜i F + u ˜i F + p˜F + z. λi τ p˜ w ˜i βi u ˜i p˜ r i=1 s˜i

Evaluating

dW1 dt

along the trajectories of system (9)-(13):

2

         (αi + βi ) w ˜i (si + p˜) w ˜i (αi + βi ) u ˜i µ p˜ 1− s˙ i + 1 − w˙ i + 1− u˙ i + 1 − p˙ + z˙ λi si p˜ wi βi ui p r         2 X λi si p w ˜i λi si p (αi + βi ) w ˜i (si + p˜) − (αi + βi ) wi = γi 1 − ρi − δi si − + 1− λi si p˜ si + p wi si + p i=1 !  X     2 (αi + βi ) µ u ˜i p˜ + ki ui − gp − µpz + (rpz − ζz) . (30) 1− (βi wi − ai ui ) + 1 − βi ui p r i=1

X dW1 = γi dt i=1

Simplify Eq. (30) as:    2 X dW1 λi si p λi si p w ˜i (αi + βi ) w ˜i (si + p˜) = γi ρi − δi si − ρi − δi si − − + (αi + βi ) w ˜i dt λ s p ˜ s + p s + p w i i i i i i=1    ζ ai (αi + βi ) ai (αi + βi ) p˜ u ˜i − gp + g p˜ + µ p˜ − z. u ˜i − ui − (αi + βi ) wi + ui βi βi p r

(31)

From the conditions of Π1 , we obtain ρi = δi s˜i + (αi + βi ) w ˜i , g p˜ =

2 X ki u ˜i , i=1

λi =

λi s˜i p˜ = (αi + βi ) w ˜i , s˜i + p˜

(αi + βi ) w ˜i (˜ si + p˜) , s˜i p˜

ai (αi + βi ) u ˜i = (αi + βi ) w ˜i , βi

(αi + βi ) w ˜i (si + p˜) s˜i (si + p˜) = , λi si p˜ si (˜ si + p˜)

then, we have      2 X dW1 si s˜i (si + p˜) si + p˜ p p (si + p˜) si + p = γi δi s˜i 1 − − + + (αi + βi ) w ˜i −1 − + + dt s˜i si (˜ si + p˜) s˜i + p˜ p˜ p˜ (si + p) si + p˜ i=1   s˜i (si + p˜) si w ˜i p (˜ si + p˜) wi u ˜i ui p˜ si + p + (αi + βi ) w ˜i 5 − − − − − + µ (˜ p − p¯) z. si (˜ si + p˜) s˜i wi p˜ (si + p) w ˜ i ui u ˜i p si + p˜ Eq. (32) becomes " 2 2 2 X dW1 δi p˜ (si − s˜i ) si (p − p˜) = γi − − (αi + βi ) w ˜i dt si (si + p) p˜ (si + p) (si + p˜) i=1   s˜i (si + p˜) si w ˜i p (˜ si + p˜) wi u ˜i ui p˜ si + p + (αi + βi ) w ˜i 5 − − − − − + µ (˜ p − p¯) z. si (˜ si + p˜) s˜i wi p˜ (si + p) w ˜i ui u ˜i p si + p˜

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(32)

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Using the rule v u n n uY 1X n ai ≥ t ai , n i=1 i=1 we obtain

s˜i (si + p˜) si w ˜i p (˜ si + p˜) wi u ˜i ui p˜ si + p + + + − 5 ≥ 0. + si (˜ si + p˜) s˜i wi p˜ (si + p) w ˜i ui u ˜i p si + p˜

Now we show that if R1 ≤ 1 then p˜ ≤

ζ r

= p¯. This can be shown if we prove that

sgn (¯ si − s˜i ) = sgn (˜ p − p¯) = sgn (R1 − 1) . Suppose that, sgn (¯ p − p˜) = sgn (¯ si − s˜i ). (ρi − δi s¯i ) − (ρi − δi s˜i ) =

  λi s¯i p¯ λi s˜i p˜ (¯ p − p˜) s2i (¯ si − s˜i ) p˜2 − = λi + . s¯i + p¯ s˜i + p˜ (¯ si + p¯) (¯ si + p˜) (¯ si + p˜) (˜ si + p˜)

This yields, sgn (˜ si − s¯i ) = sgn (¯ si − s˜i ), which leads to contradiction and then sgn (˜ p − p¯) = sgn (¯ si − s˜i ) . 2 X ki λi βi s˜i Using the condition for the steady state Π1 we have ai g(αi +βi )(˜ si +p) ˜ = 1, then i=1 2 X ki λi βi s˜i ki λi βi s¯i − a g (α + β ) (¯ s + p ¯ ) a g (α + βi ) (˜ si + p˜) i i i i i i i=1 i=1   2 X (¯ si − s˜i ) p˜ + (˜ p − p¯) s˜i ki λi βi . = a g (αi + βi ) (¯ si + p¯) (˜ si + p˜) i=1 i

R1 − 1 =

2 X

(33)

From (33) we get sgn (R1 − 1) = sgn (˜ p − p¯). So that, if R1 ≤ 1 then p˜ ≤ ζr = p¯. So that, if R1 ≤ 1 then dW1 dW1 ˜i , wi = w ˜i , ui = u ˜i , p = p˜, z = 0. Hence the largest compact dt ≤ 0 holds in Γ and dt = 0 when si = s invariant subset in Γ is   dW1 Q3 = (s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Γ | =0 dt = {(s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Γ | si = s˜i , wi = w ˜i , ui = u ˜i , p = p˜, z = 0} = {Π1 } . It follows that, if R1 ≤ 1 then Π1 is GAS in Γ by LIP. Theorem 3. The steady state Π2 is globally asymptotically stable when R1 > 1. Proof. Define         Zsi 2 z  X w (α + β ) u p µ (α + β ) w ¯ (τ + p ¯ ) i i i i  i i i dτ + w ¯i F + u ¯i F + p¯F + z¯F . W2 = γi si − s¯i − λi τ p¯ w ¯i βi u ¯i p¯ r z¯ i=1 s¯i

Then

dW2 dt

is given as:

2

         (αi + βi ) w ¯i (si + p¯) w ¯i (αi + βi ) u ¯i p¯ µ z¯  1− s˙ i + 1 − w˙ i + 1− u˙ i + 1 − p˙ + 1− z˙ λi si p¯ wi βi ui p r z       2 X (αi + βi ) w ¯i (si + p¯) λi si p w ¯i λi si p = γi 1 − ρi − δi si − + 1− − (αi + βi ) wi λi si p¯ si + p wi si + p i=1 !      X 2 (αi + βi ) u ¯i p¯ µ z¯  + 1− (βi wi − ai ui ) + 1 − ki ui − gp − µpz + 1− (rpz − ζz) . (34) βi ui p r z i=1

X dW2 = γi dt i=1

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Eq. (34) can be simplified as:    2 X dW2 λi si p λi si p w ¯i (αi + βi ) w ¯i (si + p¯) = ρi − δi si − − + (αi + βi ) w ¯i γi ρi − δi si − dt λ s p ¯ s + p s + p w i i i i i i=1  u ¯i ai (αi + βi ) ai (αi + βi ) p¯ µζ − (αi + βi ) wi + u ¯i − ui − gp + g p¯ − µp¯ z+ z¯. ui βi βi p r Using conditions of Π2 we get ρi = δi s¯i + (αi + βi ) w ¯i , λi =

(αi + βi ) w ¯i (¯ si + p¯) , s¯i p¯

ai (αi + βi ) u ¯i = (αi + βi ) w ¯i , βi g p¯ =

2 X ki u ¯i − µ¯ pz¯,

(αi + βi ) w ¯i (si + p¯) s¯i (si + p¯) = , λi si p¯ si (¯ si + p¯) 2

gp =

i=1

pX ki u ¯i − µp¯ z p¯ i=1

Then, we have      2 X dW2 s¯i (si + p¯) si + p¯ p p (si + p¯) si + p si = γi δi s¯i 1 − − + + + (αi + βi ) w ¯i −1 − + dt s¯i si (¯ si + p¯) s¯i + p¯ p¯ p¯ (si + p) si + p¯ i=1   ¯i si p (¯ si + p¯) wi u ¯i s¯i (si + p¯) w p¯ui si + p + (αi + βi ) w ¯i 5 − − − − − . si (¯ si + p¯) wi s¯i p¯ (si + p) w ¯ i ui p¯ ui si + p¯

(35)

Eq. (35) becomes " 2 2 2 X δi p¯ (si − s¯i ) si (p − p¯) dW2 = γi − − (αi + βi ) w ¯i dt si (¯ si + p¯) p¯ (si + p) (si + p¯) i=1   ¯i p (¯ si + p¯) wi u ¯i ui p¯ si + p s¯i (si + p¯) si w + (αi + βi ) w ¯i 5 − − − − − . si (¯ si + p¯) s¯i wi p¯ (si + p) w ¯i ui u ¯i p si + p¯ It follows that, Hence

dW2 dt

≤ 0 for all si , wi , ui , p, z > 0 and

dW2 dt

= 0 when si = s¯i , wi = w ¯i , ui = u ¯i , p = p¯, z = z¯.

  dW2 Q4 = (s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Γ | =0 dt = {(s1 , s2 , w1 , w2 , u1 , u2 , p, z) ∈ Γ | si = s¯i , wi = w ¯i , ui = u ¯i , p = p¯, z = z¯} = {Π2 } . It follows that, if R1 > 1 then Π2 is GAS in Γ by LIP.

3

Simulations

We support our results by numerical simulations using the values of the parameters given in Table 1. ¯1, λ ¯ 2 and r we have three cases to show its effect on the stability of the system. For the parameters λ We assume that ε1 = ε2 = 0 (there is no treatment). The initial condtions are considered to be: s1 (0) = 500, s2 (0) = 20, w1 (0) = 1, w2 (0) = 0.3, u1 (0) = 20, u2 (0) = 0.2, p (0) = 90, z (0) = 40. ¯ 1 = 0.002, λ ¯ 2 = 0.00001 and r = 0.0001. Then, R0 = 0.2469 < 1 and Case (I) R0 ≤ 1. We consider λ R1 = 0.1062 < 1. This means that Π0 is GAS. From Figures 1-8 we can see that the trajectory of the system converges the steady state Π0 (830, 24.6, 0, 0, 0, 0, 0, 0). ¯ 1 = 0.02, λ ¯ 2 = 0.0005 and r = 0.0001. In this case, R0 = 2.5694 and Case (II) R1 ≤ 1 < R0 . Choosing λ R1 = 0.7141 < 1 and Π1 exists with Π1 = (448.116, 17.9, 2.949, 0.436, 32.439, 0.218, 650.956, 0). According to Theorem 2, Π1 is GAS. Figures 1-8 show the validity of the theoretical results of Theorem 2.

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Table 1: The values of parameters of the models. Parameter ρ1 δ1 α1 a1 β1

Value 11.537 0.0139 0.57 0.1 1.1

Parameter ρ2 δ2 α2 a2 β2

Value 0.03198 0.001 0.5 0.02 0.01

Parameter k1 g ζ h λ2

Value 10 0.5 0.05 0.5 varied

Parameter k2 µ f λ1 r

Value 5 0.01 0.5 varied varied

¯ 1 = 0.02, λ ¯ 2 = 0.0005 and r = 0.002. Then we get R0 = 2.5694 > 1 and R1 = Case (III) R1 > 1. We take λ 2.3288 > 1. Figures 1-8 show that, the steady state Π2 (762.485, 19.254, 0.521, 0.348, 5.735, 0.174, 50, 66.438) is GAS which confirm the results of Theorem 3. 24

900

700

Uninfected macrophages

Uninfected CD4+ T cells

23 800

Case (I) Case (II) Case (III)

600

500

400

Case (I) Case (II) Case (III)

21 20 19 18

0

100

200

300

400 Time

500

600

700

800

Figure 1: The concentration of uninfected CD4+ T cells.

0

200

400

600 Time

800

1000

0.5 Latently infected macrophages

3.5 3 2.5 Case (I) Case (II) Case (III)

2 1.5 1

0.4

0.3

0.2

Case (I) Case (II) Case (III)

0.1

0.5 0

1200

Figure 2: The concentration of uninfected macrophages.

4 Latently infected CD4+ T cells

22

0 0

100

200

300

400 Time

500

600

700

800

Figure 3: The concentration of latently infected CD4+ T cells.

118

0

100

200

300

400 Time

500

600

700

800

Figure 4: The concentration of latently infected macrophages.

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0.25 Productively infected macrophages

Productively infected CD4+ T cells

40 35 30 25 Case (I) Case (II) Case (III)

20 15 10 5 0

0.15

0.1

100

200

300

400 Time

500

600

700

Case (I) Case (II) Case (III)

0.05

0 0

800

Figure 5: The concentration of productively infected CD4+ T cells.

0

100

200

300

400 Time

500

600

700

800

Figure 6: The concentration of productively infected macrophages.

1000

200 Case (I) Case (II) Case (III)

800

Case (I) Case (II) Case (III)

150

600

B cells

Free virus

0.2

100

400 50 200

0

0

100

200

300

400 Time

500

600

700

0

800

Figure 7: The concentration of HIV.

0

100

200

300

400 Time

500

600

700

800

Figure 8: The concentration of B cells.

References [1] D. Wodarz and M. A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24, (2002), 1178-1187. [2] L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331. [3] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996) 74-79. [4] M. A. Nowak, and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, 2000. [5] A. M. Elaiw, A. M. Althiabi, M. A. Alghamdi and N. Bellomo, Dynamical behavior of a general HIV-1 infection model with HAART and cellular reservoirs, Journal of Computational Analysis and Applications, 24(4) (2018), 728-743. [6] A. M. Elaiw, A. A. Raezah and A. S. Alofi, Stability of delay-distributed virus dynamics model with cell-tocell transmission and CTL immune response, Journal of Computational Analysis and Applications, 25(8) (2018), 1518-1531. [7] A. S. Perelson, and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. [8] P. K. Roy, A. N. Chatterjee, D. Greenhalgh, and Q. J.A. Khan, Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model, Nonlinear Anal. Real World Appl., 14 (2013), 1621-1633.

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[9] Huang, Y., S. L. Rosenkranz, H.Wu, Modeling HIV dynamic and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity, Mathematical Biosciences, 184(2), pp. 165-186, 2003. [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] 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. [12] A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications, 11 (2010), 2253-2263. [13] 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. [14] 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. [15] S. Liu, and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Mathematical Biosciences and Engineering, 7(3) (2010), 675-685. [16] A. M. Elaiw and N. H. AlShamrani, Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Mathematical Methods in the Applied Sciences, 40(3) (2017), 699-719. [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 2016; 39:4-31. [18] X. Wang, A. M. Elaiw, X. Song, Global properties of a delayed HIV infection model with CTL immune response, Applied Mathematics and Computation, 218 (2012), 9405-9414. [19] D.S. Callaway, and A.S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. [20] B. Buonomo, and C. Vargas-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. [21] H. Wang, R. Xu, Z. Wang, H. Chen, Global dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Analysis: Modelling and Control, 20 (1) (2012), 21-37. [22] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol. 66, (2004), 879-883. [23] 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, (2015), 161-190. [24] Q. Sun, L. Min and Y. Kuang,“ Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model,” IET Systems Biology, vol. 9, pp. 95 -103, 2015.

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FOURIER SERIES OF TWO VARIABLE HIGHER-ORDER FUBINI FUNCTIONS LEE CHAE JANG, GWAN-WOO JANG, DAE SAN KIM, AND TAEKYUN KIM

Abstract. In this paper, we consider the two variable higher-order Fubini functions and investigate their Fourier series expansions. In addition, we will express those functions in terms of Bernoulli functions and obtain as a consequence the corresponding polynomial identities for the two variable higher-order Fubini polynomials.

1. Introduction (r)

For each nonnegative integer r, the two variable Fubini polynomials Fm (x; y) of order r are defined by ∞ X ext tm (r) = F (x; y) , r m m! (1 − y(et − 1)) m=0

(see [4, 7]).

(1.1) (r)

However, in this paper y will be an arbitrary but fixed nonzero real number, and hence Fm (x; y) are polynomials in x, for each 0 6= y ∈ R. (1) In the case of r = 1, Fm (x; y) = Fm (x; y) are called two variable Fubini polynomials and they were (r) (r) introduced by Kilar and Simsek in [4]. For x = 0, Fm (y) = Fm (0; y) are called Fubini polynomials of (r) (r) (r) (r) order r, and Fm = Fm (1) = Fm (0; 1) Fubini numbers of order r. Further, Fm (x; 1) are called ordered (r) (r) (r) Bell polynomials of order r and they are denoted by Obm (x); Fm (1) = Fm (0; 1) are also called ordered (r) (r) (r) Bell numbers of order r and they are also denoted by Obm . Thus Obm (x) and Obm are respectively given by ∞ X ext tm (r) , = Ob (x) r m m! (2 − et ) m=0

(1.2)

∞ m X 1 (r) t = Ob , r m m! (2 − et ) m=0

(1.3)

(see [1, 3, 5]). (r) As we see from (1.1), Fm (x; y) are Appell polynomials and hence d (r) (r) F (x; y) = mFm−1 (x; y), dx m

(1.4)

(m ≥ 1).

Also, we have (r) (r) (r−1) yFm (x + 1; y) = (y + 1)Fm (x; y) − Fm (x; y),

(m ≥ 0).

(1.5)

2010 Mathematics Subject Classification. 11B83, 42A16. Key words and phrases. Fourier series, two variable higher-order Fubini function, two variable higher-order Fubini polynomial, Bernoulli function. 1

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Fourier series of two variable higher-order Fubini functions

Indeed, ∞  X

(r) (r) Fm (x + 1; y) − Fm (x; y)

m=0

=

 tm m!

ext (et − 1) r (1 − y(et − 1))

ext ext − r r−1 (1 − y(et − 1)) (1 − y(et − 1)) ∞  tm 1 X  (r) (r−1) Fm (x; y) − Fm (x; y) . = y m=0 m! 1 = y

(1.6)

!

The identity (1.5) follows from this. In turn, from (1.4) and (1.5), we obtain  1  (r) (r) (r) (r−1) Fm (1; y) − Fm (y) = Fm (y) − Fm (y) , y Z 1  1  (r) (r) (r) Fm+1 (1; y) − Fm+1 (y) Fm (x; y)dx = m+1 0   1 (r) (r−1) = Fm+1 (y) − Fm+1 (y) . (m + 1)y As is well-known, the Bernoulli polynomials Bm (x) are given by ∞ X t tm xt e = B (x) , m et − 1 m! m=0

(1.7)

(1.8)

(see [2]).

(1.9)

For any real number x, the fractional part of x is denoted by < x >= x − [x] ∈ [0, 1). We also need the following facts about Bernoulli functions Bm (< x >): (a) for m ≥ 2, ∞ X

Bm (< x >) = −m!

n=−∞,n6=0

e2πinx , (2πin)m

(1.10)

(b) for m = 1, −

∞ X n=−∞,n6=0

e2πinx = 2πin



B1 (< x >), for x ∈ R − Z, 0, for x ∈ Z.

(1.11) (r)

In this paper, we will consider the two variable higher-order Fubini functions Fm (< x >; y), for each 0 6= y ∈ R, and derive their Fourier series expansions. In addition, we will express those functions in terms of Bernoulli functions and obtain as a consequence the corresponding polynomial identities for the two variable higher-order Fubini polynomials. For some related to Fourier series, we refer the reader to [5, 6, 8].

2. Main results In this section, we assume that m ≥ 1, r ≥ 1, and 0 6= y ∈ R. For convenience, we set  1  (r) (r) (r) (r−1) ∆(r) Fm (y) − Fm (y) . m (y) = Fm (1; y) − Fm (y) = y

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3

We note here that (r) (r) Fm (1; y) = Fm (y) ⇔ ∆(r) m (y) = 0

(2.2)

(r) (r−1) ⇔ Fm (y) = Fm (y),

and Z

1 (r) Fm (x; y)dx =

0

1 (r) ∆ (y). m + 1 m+1

(2.3) (r)

Before we move on our discussion for Fourier series expansions of Fm (< x >; y), in passing we note the following:  ∞  X y 1 r+k−1 m k (r) F ( ) = k y , (2.4) (1 − y)r m 1 − y k k=0

from which, by letting y =

1 2,

we get (r) Ob(r) m = Fm (1) =

 ∞  1 X r + k − 1 km . k 2r 2k

(2.5)

k=0

Indeed, we may see (2.4) from   m ∞ X 1 y t (r) Fm = (1 − yet )−r r (1 − y) m=0 1 − y m!  ∞  X r + k − 1 k kt y e = k k=0  ∞  ∞ X r + k − 1 k X m tm = y k k m! m=0 k=0 !  ∞ ∞  X X r + k − 1 m k tm = k y . k m! m=0

(2.6)

k=0

(r) Fm (< x >; y) is a periodic function on R with (r) Fm (< x >; y) is continuous from those (r, m) with

period 1 and piecewise C ∞ . Further, in view of (2.2), (r−1) (r) (r) (y)), and ∆m (y) = 0 (or equivalently Fm (y) = Fm (r) is discontinuous with jump disconitinuities at integers for those (r, m) with ∆m (y) 6= 0 (or equivalently (r) (r−1) Fm (y) 6= Fm (y)). (r) The Fourier series of Fm (< x >; y) is ∞ X

Cn(m,r,y) e2πinx

(2.7)

n=−∞

where Cn(m) = Cn(m,r,y) =

Z

1 (r) Fm (< x >; y)e−2πinx dx

0

Z =

(2.8)

1 (r) Fm (x; y)e−2πinx dx.

0 (m)

Now, we would like to determine the Fourier coefficients Cn .

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Fourier series of two variable higher-order Fubini functions

Case 1 : n 6= 0. Cn(m) =

1

Z

(r) Fm (x; y)e−2πinx dx

0 1

i1 1 1 h (r) Fm (x; y)e−2πinx + 2πin 2πin 0

Z

m 1 (r) (F (r) (1; y) − Fm (y)) + =− 2πin m 2πin m (m−1) 1 Cn ∆(r) (y). = − 2πin 2πin m

Z

=−



0 1

 ∂ (r) Fm (x; y) e−2πinx dx ∂x

(2.9)

(r) Fm−1 (x; y)e−2πinx dx

0

Thus we have shown that 1 m (m−1) Cn ∆(r) (y), − 2πin 2πin m

Cn(m) =

(2.10)

from which by induction on m we get m

Cn(m) = −

1 X (m + 1)j (r) ∆ (y). m + 1 j=1 (2πin)j m−j+1

(2.11)

Case 2: n = 0. (m)

C0

Z =

1 (r) Fm (x; y)dx =

0 (r)

(r)

1 (r) ∆ (y). m + 1 m+1

(r)

(2.12)

(r)

Assume first that ∆m (y) = 0. Then Fm (1; y) = Fm (y). As Fm (< x >; y) is piecewise C ∞ and (r) (r) continuous, the Fourier series of Fm (< x >; y) converges uniformly to Fm (< x >; y), and (r) Fm (< x >; y)

 m X 1 (m + 1) j (r) − ∆ (y) e2πinx m + 1 j=1 (2πin)j m−j+1 n=−∞,n6=0    m  ∞ 2πinx X X m + 1 e 1 1 (r) (r)  ∆m−j+1 (y) −j! = ∆ (y) + m + 1 m+1 m + 1 j=1 j (2πin)j 1 (r) = ∆ (y) + m + 1 m+1

∞ X



(2.13)

n=−∞,n6=0

  m X m+1 (r) ∆m−j+1 (y)Bj (< x >) j j=0,j6=1  B1 (< x >), for x ∈ R − Z, + ∆(r) (y) × m 0, for x ∈ Z.

1 = m+1

We are ready to state our first result. Theorem 2.1. For positive integers r, l, and 0 6= y ∈ R, we let  1  (r) (r) (r) (r) (r−1) Fl (y) − Fl (y) . ∆l (y) = Fl (1; y) − Fl (y) = y

(2.14)

(r)

Assume that ∆m (y) = 0. Then we have the following.

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5

(r)

(a) Fm (< x >; y) has the Fourier series expansion (r) Fm (< x >; y)

1 (r) = ∆ (y) + m + 1 m+1

 m X 1 (m + 1) j (r) − ∆ (y) e2πinx , m + 1 j=1 (2πin)j m−j+1 

∞ X n=−∞,n6=0

(2.15)

for all x ∈ R, where the convergence is uniform. (b) (r) Fm (< x >; y) =

1 m+1

  m X m+1 (r) ∆m−j+1 (y)Bj (< x >), j

(2.16)

j=0,j6=1

for all x ∈ R. (r)

(r)

(r)

(r)

Assume next that ∆m (y) 6= 0. Then Fm (1; y) 6= Fm (y). Hence Fm (< x >; y) is piecewise C ∞ , and (r) discontinuous with jump discontinuities at integers. Thus the Fourier series of Fm (< x >; y) converges (r) pointwise to Fm (< x >; y), for x ∈ R − Z, and converges to 1 (r) 1 (r) (r) (F (y) + Fm (1; y)) = Fm (y) + ∆(r) (y), 2 m 2 m for x ∈ Z. We are now ready to state our second result.

(2.17)

Theorem 2.2. For positive integers r, l, and 0 6= y ∈ R, we let  1  (r) (r) (r) (r) (r−1) ∆l (y) = Fl (1; y) − Fl (y) = Fl (y) − Fl (y) . y

(2.18)

(r)

Assume that ∆m (y) 6= 0. Then we have the following. (a)   ∞ m X X 1 (m + 1)j (r) (r) − 1 ∆ (y) + ∆ (y) e2πinx m + 1 m+1 m + 1 j=1 (2πin)j m−j+1 n=−∞,n6=0 ( (r) for x ∈ R − Z, Fm (< x >; y), = (r) 1 (r) for x ∈ Z. Fm (y) + 2 ∆m (y),

(2.19)

(b)  m  1 X m+1 (r) (r) ∆m−j+1 (y)Bj (< x >) = Fm (< x >; y) m + 1 j=0 j

(2.20)

for all x ∈ R − Z; 1 m+1

  m X m+1 1 (r) (r) ∆m−j+1 (y)Bj (< x >) = Fm (y) + ∆(r) (y) j 2 m

(2.21)

j=0,j6=1

for all x ∈ Z. We remark that the case of y = 1 had been treated in the previous paper [?]. From Theorems 2.1 and 2.2, we have  m  1 X m+1 (r) (r) Fm (< x >; y) = ∆m−j+1 (y)Bj (< x >), (2.22) m + 1 j=0 j

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Fourier series of two variable higher-order Fubini functions

for all x ∈ R − Z and 0 6= y ∈ R. We immediately obtain the following polynomial identities from this observation. Corollary 2.3. We have the following polynomial identities for two variable higher-order Fubini polynomials  Pm m+1
 (r) (r) (r) 1 (a) Fm (x; y) = m+1 Fm−j+1 (1; y) − Fm−j+1 (0; y) Bj (x), j=0 j  Pm m+1
 (r) (r) (r−1) 1 (b) yFm (x; y) = m+1 Fm−j+1 (y) − Fm−j+1 (y) Bj (x). j=0 j For x = 0, we have the following identities for higher-order Fubini polynomials. Corollary 2.4. We have the following polynomial identities for higher-order Fubini polynomials  Pm m+1
 (r) (r) (r) 1 (a) Fm (y) = m+1 B F (1; y) − F (0; y) , j m−j+1 m−j+1 j=0 j  Pm m+1
 (r) (r) (r−1) 1 (b) yFm (y) = m+1 j=0 j Bj Fm−j+1 (y) − Fm−j+1 (y) . Finally, for y = 1, we get the following identities for higher-order ordered Bell polynomials. Corollary 2.5. We have the following polynomial identities for higher-order ordered Bell polynomials Pm m+1
 (r) (r) (r) 1 (a) Obm (x) = m+1 Ob m−j+1 (1) − Obm−j+1 Bj (x), j=0 j   Pm m+1
(r) (r) (r−1) 1 (b) Obm (x) = m+1 Ob − Ob m−j+1 m−j+1 Bj (x), j=0 j

References 1. R. L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics. A foundation for computer science,Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1989. xiv+625 pp. ISBN: 0-201-14236-8 1 2. D. S. Kim, T. Kim, H.-I. Kwon, Fourier series r-derangement and higher-order derangement functions, Adv. Stud. Contemp. Math., 28(1)(2018), 1–11. 3. G.-W.Jang, T. Kim, Some identities of oredered Bell numbers arising from differential equations, Adv. Stud. Contemp. Math., 27(3)(2017), 385–397. 1 4. D. S. Kim, T. Kim, H.-I. Kwon, A note on degenerate Fubini polynomials,Proc. Jangjeon Math. Soc., 20(4)(2017),521– 717. 1 5. T. Kim, D. S. Kim, D. V. Dolgy, J.-W. Park, Fourier series of sums of products of ordered Bell and poly-Bernoulli functions, J. Inequal. Appl., (2017) 2017:84, 17 pp. 1, 1 6. E. Kreyszig Advanced engineering mathematics. Tenth edition, John Wiley and Sons, Inc., New York, 2011. ISBN:9780-470-64613-7 1 7. M. Mursan, G. Toader, A generalization of Fubini’s numbers, Studia Univ. Babes-Bolyai Math., 31(1986), 60-65. 8. D. G. Zill, M. R. Cullen Advanced Engineering Mathematics. Third edition, Jones and Bartlett Publishers 40 Tall Pine Drive Sudbury, MA 01776, 2006. ISBN: 978-0-7637-3914-0 1 Graduate School of Education, Konkuk University, Seoul 143-701, 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, 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]

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WEIGHTED COMPOSITION OPERATORS FROM DIRICHLET TYPE SPACES TO SOME WEIGHTED-TYPE SPACES MANISHA DEVI, AJAY K. SHARMA AND KULDIP RAJ

Abstract. Let K : [0, ∞) → [0, ∞) be a right continuous increasing function and ν : D → (0, ∞) be any continuous function. In this paper by considering K and ν as weight functions, we characterize the boundedness and compactness of weighted composition operators from Dirichlet type spaces to some weighted-type spaces.

1. Introduction and Preliminaries Let D be the open unit disk and ∂D be its boundary in the complex plane C. Let H(D) denotes the class of all holomorphic functions on D, S(D) be the class of all holomorphic self-maps of D and H ∞ be the space of all bounded analytic functions on D. Let dA(z) = dxdy drdθ π = r π be the normalized area measure on D. A continuous function ν : D −→ (0, ∞) is called weight. For ν(z) = ν(|z|), z ∈ D, weight is radial and weight is a standard weight if lim ν(z) = 0. |z|→1−

For weight ν, the Bers-type space Aν is the collection of all f ∈ H(D) such that sup ν(z)|f (z)| < ∞ z∈D

and with the norm kf kAν = sup ν(z)|f (z)|, z∈D

it is a non-separable Banach space. The closure of the set of polynomials in Aν forms a separable Banach space. This set is denoted by Aν,0 and contains exactly of those f ∈ Aν such that lim ν(z)|f (z)| = 0.

|z|→1−

The Bloch-type space Bν on D with the weight ν is the space of all holomorphic functions f on D such that sup ν(z)|f 0 (z)| < ∞. z∈D

2010 Mathematics Subject Classification. 47B33, 30D55, 30H05, 30E05. Key words and phrases. Weighted composition operator, Dirichlet type space, Bloch-type space, Berstype spaces, boundedness, compactness.

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The little Bloch-type space Bν,0 is the closure of the set of polynomials in Bν and contains all those f ∈ Bν such that lim ν(z)|f 0 (z)| = 0

|z|→1

and with the norm kf kBν = |f (0)| + sup ν(z)|f 0 (z)| < ∞, z∈D

both Bν and Bν,0 form Banach spaces. For more information about these spaces one may refer [20] and references therein. Let ϕ ∈ S(D) and ψ be an analytic map on D. The operator Cϕ so called as the composition operator and is defined as Cϕ f = f ◦ ϕ, f ∈ H(D). The operator Mψ which is called as the multiplication operator is defined by Mψ f = ψ · f , f ∈ H(D). For f ∈ H(D), the weighted composition operator on H(D) is defined by (Wψ,ϕ f )(z) = ψ(z)f (ϕ(z)), where ψ ∈ H(D), ϕ ∈ S(D) and z ∈ D. It can be easily seen that for ψ ≡ 1, the operator reduced to Cϕ . If ϕ(z) = z, operator get reduced to Mψ . This operator is basically a linear transformation of H(D) defined by (Wψ,ϕ f )(z) = ψ(z)f (ϕ(z)) = (Mψ Cϕ f )(z), for f in H(D) and z in D. The basic problem is to give the function-theoretic characterization when between various function spaces ψ and ϕ induce bounded or compact weighted composition operator. Various holomorphic functions spaces on various domains have been studied for the the boundedness and compactness of weighted composition operators acting on them. Moreover, a number of papers have been studied on these operators acting on different spaces of holomorphic functions on various domains for more detail (see [1], [5], [7]-[11], [13], [15], [19]). Consider a function K : [0, ∞) → [0, ∞) which is right continuous and increasing. The Dirichlet type space DK consists of all functions f ∈ H(D) such that Z
2 2 kf kDK = |f (0)| + |f 0 (z)|2 K 1 − |z|2 dA(z) < ∞. D

For more about the Dirichlet type spaces we refer ([2], [3], [4], [12], [14], [16]). In this paper we consider function K as a weight function satisfying the following two conditions: Rt (a) K1 (t) = 0 K(s) ds s ≈ K(t), 0 < t < 1 ; (b) K2 (t) = t

R∞ t

K(s) ds s2 ≈ K(t), t > 0.

From condition (b), we get that K(2t) ≈ K(t) for 0 < t < 1. Also there exist C > 0 sufficiently small for which t−C K1 (t) is increasing and K2 (t)tC−1 is decreasing (see [4],

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WEIGHTED COMPOSITION OPERATORS FROM DIRICHLET TYPE SPACES

[17], [18]). This paper is entirely devoted to characterize the boundedness and compactness of operator Wψ,ϕ from Dirichlet type spaces to the Bers-type space and Bloch-type space. Throughout this paper, C will represents a constant which may differ from one occurrence to another. The notation A . B means that there exist C > 0 such that A ≤ CB. We write A ≈ B if A . B and B . A. The paper is organized in a systematic manner. Section 1 covers the introduction and literature part. Lemmas that are used to formulate our main theorems are kept in Section 2. Section 3 contains the boundedness and compactness of the oparator Wψ,ϕ : DK → Bν . Section 4 considers the boundedness and compactness of the oparator Wψ,ϕ : DK → Aν .

2. Auxiliary Results To arrive at the main results we use some lemmas, as given below Lemma 2.1. [4] Let K be a weight function. Then for any w ∈ D and ε > 0, we have (1 − |z|2 )ε/2

fz (w) = p

K(1 − |z|2 )(1 − w¯ z )1+ε/2

is in DK . Moreover, sup kfz kDK ≈ 1, z∈D

and fz converges to zero uniformly on compact subsets of D as |z| → 1− . The following two lemmas can be proved easily by following the Lemma 2.1 and [4]. Lemma 2.2. [4] Let K be a weight function. Then for every f ∈ DK we have kf kDK , |f (z)| ≤ C p K(1 − |z|2 )(1 − |z|2 )

z ∈ D.

Lemma 2.3. [4] Let K be a weight function and n be a positive integer. Then for every f ∈ DK we have |f (n) (z)| ≤ C p

kf kDK K(1 − |z|2 )(1 − |z|2 )n+1

,

z ∈ D.

The following criterion characterize the compactness. It was given for the first time in [6]. Since the proof is standard, so we omit it. Lemma 2.4. Let ν be the standard weight and the operator Wψ,ϕ : DK → Bν is bounded. Then Wψ,ϕ : DK → Bν is compact if and only if for any bounded sequence (fn )n∈N in DK which converges to zero uniformly on compact subsets of D, we have lim kWψ,ϕ fn kBν = 0.

n→∞

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MANISHA DEVI, AJAY K. SHARMA AND KULDIP RAJ

3. Boundedness and compactness of weighted composition operator from Dirichlet type space to Bloch-type space Theorem 3.1. Let ν and K be two weight functions, ψ ∈ H(D) and ϕ be a self analytic map on D. Then the operator Wψ,ϕ : DK → Bν is bounded if and only if the following conditions are satisfied: ν(z)|ψ 0 (z)|

(i) M1 = sup p

K(1 − |ϕ(z)|2 )(1 − |ϕ(z)|2 )

z∈D

ν(z)|ψ(z)ϕ0 (z)|

(ii) M2 = sup p z∈D

< ∞;

K(1 − |ϕ(z)|2 )2 (1 − |ϕ(z)|2 )2

< ∞.

Furthermore, if the operator Wψ,ϕ : DK → Bν is bounded, then M1 + M2 . kWψ,ϕ kDK →Bν . 1 + M1 + M2 . Proof. First suppose that condition (i) and (ii) hold. Using Lemma 2.2 we have, ν(z)|(Wψ,ϕ f )0 (z))| ≤ ν(z)|ψ 0 (z)||f (ϕ(z))| + ν(z)|ψ(z)ϕ0 (z)||f 0 (ϕ(z))| .

ν(z)|ψ 0 (z)| p

K(1 − |ϕ(z)|2 )(1 − |ϕ(z)|2 ) ν(z)|ψ(z)ϕ0 (z)|

!

+p kf kDK . K(1 − |ϕ(z)|2 )(1 − |ϕ(z)|2 )2

(3.1) Also,

|(Wψ,ϕ f )(0)| = |ψ(0)||f (ϕ(0))| |ψ(0)| kf kDK . .p K(1 − |ϕ(0)|2 )(1 − |ϕ(0)|2 )

(3.2)

From conditions (i), (ii) and equations (3.1) and (3.2), we get kWψ,ϕ f kBν = |ψ(0)||f (ϕ(0))| + sup ν(z)|(Wψ,ϕ f )0 (z)| z∈D

 .

|ψ(0)| p

K(1 − |ϕ(0)|2 )(1 − |ϕ(0)|2 )
. 1 + M1 + M2 kf kDK .

 + M1 + M2 kf kDK

Therefore, Wψ,ϕ : DK → Bν is bounded and (3.3)

kWψ,ϕ kDK →Bν . 1 + M1 + M2 .

Conversely, suppose that Wψ,ϕ : DK → Bν is bounded. Let z = ϕ(ζ), ζ ∈ D and (3.4)

gz (w) = τz (w)fz (w),

where fz (w) is defined in Lemma 2.1 and τz (w) is defined as (3.5)

τz (w) = 1 −

1 − |z|2 . 1 − z¯w

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WEIGHTED COMPOSITION OPERATORS FROM DIRICHLET TYPE SPACES

Then τz ∈ H ∞ as   1 − |z|2 sup |τz (w)| ≤ sup 1 + ≤ 3. 1 − |z||w| w∈D w∈D Therefore, gz ∈ DK and sup kgz kDK . 1. From equation (3.5) we have, w∈D

(3.6)

τz (z) = 0

and τz0 (w) =

−¯ z (1 − |z|2 ) . (1 − z¯w)2

Thus, τz0 (z) =

(3.7)

−¯ z . (1 − |z|2 )

Therefore, gz (z) = 0, using the value of fz (z) and from (3.7), we obtain gz0 (z) = τz0 (z)fz (z) + τz (z)fz0 (z) −¯ z =p . 2 K(1 − |z| )(1 − |z|2 )2 Using the above fact, we get kWψ,ϕ kDK →Bν & kWψ,ϕ gϕ(ζ) kBν 0 ≥ ν(ζ)|ψ 0 (ζ)gϕ(ζ) (ϕ(ζ)) + ψ(ζ)ϕ0 (ζ)gϕ(ζ) (ϕ(ζ))| 0 ≥ ν(ζ)|ψ(ζ)ϕ0 (ζ)gϕ(ζ) (ϕ(ζ))|

ν(ζ)|ψ(ζ)ϕ0 (ζ)||ϕ(ζ)|

≥p

K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )2

.

When δ ∈ (0, 1) is fixed, we have (3.8)

sup |ϕ(ζ)|>δ

ν(ζ)|ψ(ζ)ϕ0 (ζ)| p . kWψ,ϕ kDK →Bν . K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )2

Taking fz (w) ≡ 1 ∈ DK , implies that (3.9)

sup ν(w)|ψ 0 (w)| = kWψ,ϕ (1)kBν . kWψ,ϕ kDK →Bν . w∈D

Again taking f (w) = w ∈ DK , using the asymptotic estimate (3.9) and boundedness of ϕ, we get (3.10)

sup ν(w)|ψ(w)ϕ0 (w)| . kWψ,ϕ kDK →Bν . w∈D

Using (3.10) and the compactness of ϕ, we easily get sup |ϕ(ζ)|≤δ

ν(ζ)|ψ(ζ)ϕ0 (ζ)| p 

(3.11)

.

K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )2  1 p kWψ,ϕ kDK →Bν . K(1 − δ 2 )(1 − δ 2 )2

131

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MANISHA DEVI, AJAY K. SHARMA AND KULDIP RAJ

Further, from (3.8) and (3.11), we obtain (3.12)

ν(ζ)|ψ(ζ)ϕ0 (ζ)| sup p . kWψ,ϕ kDK →Bν . K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )2 ζ∈D

Again, for fz as defined in Lemma 2.1, we have kWψ,ϕ kDK →Bν & kWψ,ϕ fϕ(ζ) kBν 0 ≥ ν(ζ) ψ 0 (ζ)fϕ(ζ) (ϕ(ζ)) + ψ(ζ)ϕ0 (ζ)fϕ(ζ) (ϕ(ζ)) ν(ζ)|ψ 0 (ζ)|

≥p

K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )

(1 + /2)ν(ζ)|ψ(ζ)||ϕ0 (ζ)||ϕ(ζ)| + p . K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )2 By using the boundedness of ϕ, we get ν(ζ)|ψ 0 (ζ)| p K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 ) (3.13)

ν(ζ)|ψ(ζ)||ϕ0 (ζ)||ϕ(ζ)| ≤ kWψ,ϕ kDK →Bν + C p . K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )2

Taking the supremum over ζ ∈ D in (3.13) and using (3.12), we get (3.14)

sup p ζ∈D

ν(ζ)|ψ 0 (ζ)| K(1 − |ϕ(ζ)|2 )(1 − |ϕ(ζ)|2 )

. kWψ,ϕ kDK →Bν .

From (3.12) and (3.14), we obtain M1 + M2 . kWψ,ϕ kDK →Bν .

(3.15)

Hence, from (3.3) and (3.15), we get M1 + M2 . kWψ,ϕ kDK →Bν . 1 + M1 + M2 .  Theorem 3.2. Let ν be a standard weight, ψ ∈ H(D) and ϕ be a self analytic map on D. Let K be a weight function. Assume that Wψ,ϕ : DK → Bν is bounded. Then the operator Wψ,ϕ : DK → Bν is compact if and only if the following conditions are satisfied: (i)

ν(z)|ψ 0 (z)| p = 0; |ϕ(z)|→1 K(1 − |ϕ(z)|2 )(1 − |ϕ(z)|2 )

(ii)

ν(z)|ψ(z)ϕ0 (z)| p = 0. |ϕ(z)|→1 K(1 − |ϕ(z)|2 )(1 − |ϕ(z)|2 )2

lim

lim

Proof. First suppose that (i) and (ii) hold. Let (fn )n∈N be a bounded sequence of functions in DK that converges to zero uniformly on compact subset of D. To prove the compactness of Wψ,ϕ , we have to show that kWψ,ϕ fn kBν → 0 as n → ∞. Condition (i) and (ii) implies that for any ε > 0, there exists δ ∈ (0, 1) such that (3.16)

ν(z)|ψ 0 (z)| p

K(1 − |ϕ(z)|2 )(1 − |ϕ(z)|2 )

132

m 1 lim sup 4 @ m t!1 j=1 i=1 j (t)

i (s)

where

8 0 2 > > Zt m Zt m < Y X B 6 pi (s) exp @ Pk pj (t) 1 + m 4 Pk (t) = > > i=1 j=1 : (t) (s) j i

with

P0 (t) = m i

(1:26)

"

m Y

#1=m

p` (t)

`=1

m Y

#

ci ( i ) ;

i=1

(1:27)

31=m 9 > > = C 7 ; 1 (u) duA ds5 > > ; 1

;

is given by (1.14) and ci ( i ) by (1.15). Then all solutions of Eq.(1.1) oscillate. In 2018 Attia et al [3] established the following oscillation conditions. Assume that Z t X n 1 0 < := lim inf pk (s)ds ; t!1 e g(t) k=1

and

lim sup t!1

where Q(t) =

n X n X

k=1 i=1

or

pi (t)

Z

t

Rt

Q(v)dv + c( )e

g(t) Rt

t

pk (s)e i (t)

lim sup t!1

Z

Z

gk (t)

Pn

i=1

g(t)

Pn

i=1

pi (s)ds

pi (s)ds+( ( )

t

Rt

Q1 (v)dv + c( )e

g(t)

140

g(t)

Pn

i=1

)

> 1;

R gk (t) Pn k (s)

pi (s)ds

`=1

p` (u)du

ds;

2 (0; ( ));

> 1;

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G. M . M OREM EDI 1 , H. JAFARI 1 , AND I. P. STAVROULAKIS

6

where Q1 (t) =

n X n X

Z

pi (t)

k=1 i=1

and

Rt

t

pk (s)e i (t)

gk (t)

Z

q` = lim inf t!1

or

0

lim sup @ t!1

n Y

j=1

n Z Y

Rk (s) = e

gk (s)

Pn

i=1

pi (u)du

n X i=1

and

0


1 ; nn

2 (0; ( ));

1 : e

pi (s)ds i (t)

In this paper we further investigate the problem and derive oscillation conditions which essentially improve all the above mentioned conditions. 2. Main Results Our main results are the following two theorems Theorem 1. Assume that there exist non-decreasing functions i 2 C ([t0 ; 1) ; R+ ) such that (1.11) is satis…ed and for some k 2 N 2 0 1 31=m t i (t) Z Z m m Y 6Y 1 B C 7 lim sup (2:1) pi (s) exp @ Pk (u) duA ds5 > m; 4 m t!1 j=1 i=1 j (t)

where

2

6 Pk (t) = P (t) 41 +

Zt

i (t)

i (s)

0

B P (s) exp @

Zt

i (s)

0

B P (u) exp @

with P0 (t) = P (t) =

m X

Zu

i (u)

Pk

1

1

1

3

C C 7 ( ) d A duA ds5

(2:2)

pi (t) :

i=1

Then all solutions of Eq.(1:1) oscillate.

Proof. Suppose for the sake of contradiction that Eq.(1.1) has a non-oscillatory solution x (t) : Since x (t) is also a solution to (1.1), we con…ne ourselves only to the case that x (t) is an eventually positive solution of Eq.(1.1). Then there exists t1 > t0 such that x (t) > 0; x ( i (t)) > 0; x ( i (t)) > 0: Thus, from Eq.(1.1) it

141

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OSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIO NS W ITH NON-M ONOTONE ARGUM ENT7

follows that x0 (t) into account that

i

0 for all t t1 and therefore x (t) is non-increasing and taking (t) t; it follows x0 (t) +

m X

pi (t) x (t)

0; t

t1 :

(2:3)

i=1

Dividing the last inequality by x (t) and integrating from i (t) to t for su¢ ciently large t; we have 0 1 Zt X m B C (2:4) x ( i (t)) x (t) exp @ p` ( ) d A ; i = 1; 2; :::; m: i (t)

`=1

Dividing (1.1) by x (t) and integrating from i (s) to t; s t, we obtain 1 0 Zt X m x ( ` (u)) C B duA ; i = 1; 2; :::; m: x ( i (s)) = x (t) exp @ p` (u) x (u)

(2:5)

`=1 i (s)

Combining the last two relations, we obtain 1 1 0 0 Zu X Zt X m m C C B B p` ( ) d A duA : p` (u) exp @ x ( i (s)) x (t) exp @

(2:6)

`=1 i (u)

`=1 i (s)

Now, integrating (1.1) from i (t) to t and using (2.6) for su¢ ciently large have 0 0 2 Zu X Zt X Zt X m m m B B 6 p` ( ) d p` (u) exp @ p` (s) exp @ x ( i (t)) x (t) 41 + i (t)

`=1

i (s)

`=1

i (u)

`=1

t; we 1

1

3

C C 7 A duA ds5 :

(2:7) Multiplying the last inequality by pi (t) [cf.10,3,4] and taking the sum over i ( i = 1; 2; :::; m) ; we have x0 (t) + P1 (t) x (t)

where

2

6 P1 (t) = P (t) 41 +

Zt

i (t)

0

B P (s) exp @

Zt

i (s)

0; t

t1 ; 0

B P (u) exp @

Zu

i (u)

Pm

(2:8) 1

1

3

C C 7 P0 ( )d A duA ds5 :

Observe that (2.8) resembles with (2.3), where i=1 pi (t) [= P (t) = P0 (t)] is replaced by P1 (t) ; and following the same steps as from (2.3) to (2.8), for su¢ ciently large t we …nd x0 (t) + P2 (t) x (t) where

2

6 P2 (t) = P (t) 41 +

Zt

i (t)

0

B P (s) exp @

Zt

i (s)

0; 0

B P (u) exp @

142

(2:9) Zu

i (u)

1

1

3

C C 7 P1 ( )d A duA ds5 :

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G. M . M OREM EDI 1 , H. JAFARI 1 , AND I. P. STAVROULAKIS

8

;1

Repeating the above procedure, it follows by induction, that for su¢ ciently large t x0 (t) + Pk (t) x (t)

0;

(2:10)

where Pk (t) is given by 0 0 2 Zu Zt Zt B B 6 Pk P (u) exp @ P (s) exp @ Pk (t) = P (t) 41 + i (u)

i (s)

i (t)

Dividing (2.10) by x (t) and integrating from large t; we get 0 x(

i

(s))

1(

i

(s) to

i

(t) ; s

1

1

3

C C 7 )d A duA ds5 :

t; for su¢ ciently

1 Zi (t) C B Pk (u) duA : x ( i (t)) exp @

(2:11)

i (s)

On the other hand, integrating (1.1) from x(

j (t)) = x (t) +

j

(t) to t for su¢ ciently large t; we have

m Zt X i=1

pi (s) x (

i

(s)) ds:

(2:12)

j (t)

Combining (2.12) with (2.11) and using the arithmetic mean-geometric mean inequality, we obtain 1 131=m 0 2 0 "m #1=m m Zi (t) Zt Y Y C C7 B 6 B : Pk (u) duA dsA5 pi (s) exp @ x ( j (t)) m x ( i (t)) 4 @ i=1

i=1

i (s)

j (t)

Now, taking the product on both sides of the last inequality, we …nd 2 0 1 31=m 3 2 2 3 t i (t) Z Z m m m m Y Y 7 6Y B C 7 6Y 7: pi (s) exp @ Pk (u) duA ds5 x ( j (t)) mm 4 x ( j (t))5 6 4 5 4 j=1

j=1

j=1

i=1

j (t)

i (s)

Hence

lim sup t!1

m Y

j=1

2 6 4

m Y

i=1

which contradicts (2.1).

Zt

j (t)

0

1 31=m Zi (t) B C 7 pi (s) exp @ Pk (u) duA ds5

1 mm

i (s)

For the next theorem we need the following lemma (See [39,13,26,27,4]). Lemma 1. Let there exist non-decreasing functions i 2 C ([t0 ; 1) ; R+ ) such that condition (1.11) is ful…lled and equation (1.1) has an eventually positive solution x : [t0 ; +1) ! (0; +1). Then lim inf t!1

where

i

x (t) x ( i (t))

ci ( i ) ; i = 1; 2; : : : ; m;

and ci ( i ) are given by (1.14) and (1.15).

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Theorem 2. Assume that there exist non-decreasing functions i 2 C ([t0 ; 1) ; R+ ) such that (1.11) is satis…ed and for some k 2 N 1 131=m 0 2 0 " # t i (t) Z Z m m m Y 6Y B Y 1 C C7 B Pk (u) duA dsA5 > m 1 pi (s) exp @ lim sup ci ( i ) ; 4 @ m t!1 j=1 i=1 i=1 i (s)

j (t)

where Pk (u) is given by (2.2), of Eq.(1.1) oscillate.

i

(2:13) by (1.14) and ci ( i ) by (1.15). Then all solutions

As in the proof of Theorem 1, we assume, for the sake of contradiction, that Eq.(1.1) has a non-oscillatory solution x (t) and derive (2.11) and (2.12). Combining (2.12) with (2.11) and using the arithmetic mean-geometric mean inequality for su¢ ciently large t, we get 0 1 31=m 2 "m #1=m m Zt Zi (t) Y Y B C 7 6 pi (s) exp @ Pk (u) duA ds5 : x ( j (t)) x (t)+m x ( i (t)) 4 i=1

i=1

j (t)

i (s)

Taking the product on both sides of the last inequalities and using Lemma 1, as in proof of [4, Theorem 2], we …nd 0 1 31=m 2 Zi (t) m Zt m Y Y B C 7 6 pi (s) exp @ Pk (u) duA ds5 lim sup 4 t!1

j=1

i=1

j (t)

i (s)

2

1 6 61 mm 4 "

1 1 mm

lim inf Q t!1 m

xm (t)

i=1

m Y

#

x(

i

(t))

3 7 7 5

ci ( i )

i=1

which contradicts (2.13). Remark 1. It is clear that the left-hand sides of both conditions (2.1) and (2.13) are identically the same and also the right-hand side of (2.13) reduces to (2.1) when ci ( i ) = 0:Thus, it seems that Theorem 2 is exactly the same as Theorem 1, when ci ( i ) = 0: One may notice, however, that the condition (1.14) is required in Theorem 2 but not in Theorem 1. In the case of monotone arguments we have the following theorem. Theorem 3. Let i be non-decreasing functions and for some k 2 N 1 131=m 8 2 0 0 > t i (t) > Z Z m m < Y 6Y C C7 B B lim sup Pk (u) duA dsA5 > 4 @pi (s) exp @ > t!1 > j=1 i=1 : m1m 1 j (t) i (s)

144

1 mm

or m Q

ci ( i )

i=1

MOREMEDI et al 136-151

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G. M . M OREM EDI 1 , H. JAFARI 1 , AND I. P. STAVROULAKIS

10

where

2

6 Pk (t) = P (t) 41 +

0

Zt

i (t)

with

Zt

B P (s) exp @

i (s)

0

B P (u) exp @

P0 (t) = P (t) =

m X

Zu

Pk

1(

i (u)

;1

1

1

3

C C 7 )d A duA ds5 :

pj (t) ;

j=1

i

is given by (1.14), and ci ( i ) by (1.15). Then all solutions of (1.1) osillate. 3. Corollaries and Examples In the case m = 2; Eq.(1.1) reduces to the equation x0 (t) + p1 (t) x (

1

(t)) + p2 (t) x (

2

(t)) = 0:

(3:1)

From Theorems 1 and 2 the following corollary is immediate Corollary 1. Assume that (1.11) holds and for for k 2 N 0 1 31=2 8 2 > t i (t) > Z Z 2 2 < Y 6Y B C 7 > pi (s) exp @ Pk (u) duA ds5 lim sup 4 > t!1 > j=1 i=1 : j (t) i (s) where,

2

6 Pk (t) = P (t) 41 + and for i = 1; 2; Eq.(3.1) oscillate.

i

Zt

0

i (t)

B P (s) exp @

Zt

i (s)

0

B P (u) exp @

Zu

Pk

i (u)

1=2

1 4

1 4

1

2

6 Pk (t) = p(t) 41 +

; ci ( i )

i=1

1(

1

1

3

C C 7 )d A duA ds5 ;

P0 (t) = 2 (p1 (t) p2 (t)) ; is given by (1.14) and ci ( i ) by (1.15). Then all solutions of

Corollary 2. Assume that there exist a non-decreasing function (t) (t) t and for some k 2 N 0 1 8 Zt Z(t) 1 < B C or p (s) exp @ Pk (u) duA ds > lim sup : t!1 1 c( ) (t) (s) where

or 2 Q

Zt

(t)

0

B p (s) exp @ 0
0:

(3:4)

with the retarded argument 8 t 1; t 2 [3n; 3n + 1]; < 3t + (12n + 3) ; t 2 [3n + 1; 3n + 2]; (t) = : 5t (12n + 13) ; t 2 [3n + 2; 3n + 3]:

For this equation, as in [6,21,4], one may choose the funtion 8 t 2 [3n; 3n + 1]; < t 1; 3n; t 2 [3n + 1; 3n + 2:6]; (t) = : 5t (12n + 13) ; t 2 [3n + 2:6; 3n + 3]:

If we choose tn = 3n + 3; (cf. [6, Example 1] and [21, Example 4.2]), then for k = 1; the condition (2.1) of Theorem 1 (or the condition (3.2) of Corollary 2) reduces to 1 0 1 0 3n+2 3n+3 Z Z Z(t) Zt C B C B P1 (u) duA ds; p exp @ lim lim sup p exp @ P1 (u) duA ds n!1 3n+2

t!1

(s)

(t)

where

P1 (t)

2

6 = p 41 +

lim sup t!1

h where P1 = p 1 +

Zt

e

B p exp @

e6pe

Zt

(s)

0

B p exp @

3n+3 Z

5s (12n+13)

p

epe

p

e

5 0

(t)

6pep

0

3n+3 Z

3n+2

= p 1+ Therefore

B p exp @

(t)

2

6 p 41 +

0

Zt

p

1 Z(t) B C p exp @ P1 (u) duA ds

Zu

(u)

1

1

3

C C 7 pd A duA ds5 1

3

C 7 p exp(p)duA ds5

p e5P1 5P1

1 ;

(s)

pep

e

5

5s (12n+13)

e

p

p e5P1 5P1

i

:For p = 0:255, P1 1

0:484721; and so

1:082293 > 1:

Therefore all solutions of Eq.(3:4) oscillate. Observe, however, that when we consider the conditions stated in [6], [37] [21], [27], [7], [1] and[4] for the above equation (3.4), we obtain the following:

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12

Z

1. Observe that, for tn = 3n + 3; (Z ) Z 3n+3 (3n+3) p exp pd ds = (3n+3)

(s)

3n+3

p exp

(Z

;1

3n+2

pd

5s (12n+13))

3n+2

)

ds =

e5p 1 5

and condition (1.7) reduces to e5p 1 > 1: 5 But, for p = 0:255 e5p 1 0:51574 < 1 5 therefore the condition (1.7) is not satis…ed. 2. Similarly, in the condition (1.8), a = lim inf t!1

Zt

p (s) ds = lim

3n+3 Z

n!1 3n+2

(t)

and c (a) = c (p) =

1

p

and, as before, (1.8) reduces to

p 1 2

pds = p

2p

p2

:

p 1 p e5p 1 1 2p p2 >1 5 2 Taking p = 0:255 the left-hand side of (1.8) is equal to 0:51574 while the right-hand side is 0:95345: Therefore this condition is not satis…ed. 3: The condition (1.12) reduces to 1 1 0 0 Z Z(t) Zt C C B B lim sup p exp @ p exp @ pduA d A ds > 1;

(3:5)

t!+1

(s)

(t)

( )

and, as in [20,Example 4.2], the choice of tn = 3n + 3; leads to the inequality p

e5pe 1 > 1: p 5e Observe, however, that for p = 0:255,

(3:6)

p

e5pe 1 0:64849 < 1: 5ep Therefore the condition (3.6) is not satis…ed. 4. The condition (1.13), for k = 2; reduces to 0 1 Zt Z(t) B C lim sup p exp @ p 2 ( ) d A ds > 1

c( );

(3:7)

t!1

(t)

(s)

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OSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIO NS W ITH NON-M ONOTONE ARGUM ENT 13

where

( ) = 1; and for tn = 3n + 3; as before, it leads to p e5p 1 1 2p p2 1 p > 5 2 For p = 0:255; we have e5p 1 0:51574; 5 while the right-hand side 1 c (p) 0:95345: 2

Therefore the condition (3.7) is not satis…ed. 5. The condition (1.16) for r = 1 reduces to lim sup t!1

Zt

pa1 (h (t) ; (s)) ds > 1;

(3:8)

h(t)

where h (t) =

0 t 1 Z (t) and a1 (t; s) = exp @ pduA : s

That is, to the condition

lim sup t!1

Zt

(t)

0

1 Z(t) B C p exp @ pd A ds > 1;

(3:9)

(s)

and, as before, for tn = 3n + 3 and p = 0:255; we have e5p 1 0:51574 < 1: 5 Therefore the condition (3.8) is not satis…ed.

(3:10)

6. Similarly, condition (1.20) for r = 1 reduces to 0 1 Z(t) Zt 1 + ln B C lim sup p exp @ pd A ds > t!1

0

;

(3:11)

0

(t)

(s)

where 0 is the smaller root of the equation = ep : As before, for tn = 3n + 3 and p = 0:255; we have e5p 1 0:51574; 5 while 1 + ln 0 0:94664 0

Therefore the condition (3.11) is not satis…ed. 7. For k = 1; condition (1.26) reduces to

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G. M . M OREM EDI 1 , H. JAFARI 1 , AND I. P. STAVROULAKIS

lim sup t!1

Zt

(t)

1 Z(t) C B p exp @ P1 (u) duA ds > 1: 0

(3:12)

(s)

If we choose tn = 3n + 3, 8 8 0 1 9 0 3n+3 > > > Z Zt Zt = < < B C B p exp @ p exp @ pduA ds = p 1 + P1 (t) = p 1 + > > > ; : : = p 1+

e

3n+2

(s)

(t)

6p

e

3n+3 Z

5s (12n+13)

p

:

5

;1

9 > = C pduA ds > ; 1

and, as before, (3.12) reduces to p e5P1 5P1 For p = 0:255 we …nd P1

1 > 1:

0:424232 and so p e5P1 5P1

1

0:882491 < 1:

Therefore the condition (3,12) is not satis…ed We conclude, therefore, that for p = 0:255 no one of the conditions (1.7), (1.8), (1.12), (1.13) for k = 2; (1.16) and (1.20) for r = 1; and (1.26) is satis…ed. It should be also pointed out that not only for this value of p = 0:255 but for all values of p > 0:255 ; especially for all values of p 2 [0:255; 0:358]; (cf. [21, Example 4.2]), p e5P1 5P1

1 >1

and therefore all solutions of (3.4) oscillate. Observe, however, that for p = 0:358 e5p 1 5

0:99789 < 1;

also for p = 0:3 p

e5pe 1 5ep e5p 1 5 and for p = 0:263, P1

0:974101 < 1;

0:696337 < 0:912993

1 + ln

0

;

0

0:44944 and so p e5P1 5P1

1

0:99024 < 1:

Therefore for all values of p 2 [0:255; 0:358] the conditions of Corollary 2 are satis…ed and so all solutions to Eq.(3.4) oscillate, while no one of the above mentioned conditions is satis…ed for these values of p 2 [0:255; 0:358]:

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OSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIO NS W ITH NON-M ONOTONE ARGUM ENT 15

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;1

[25] Kopladatze, R. G. and Chanturija T. A., On the oscillatory and monotonic solutions of …rst order di¤erential equations with deviating arguments, Di¤erentsial’nye Uravneniya, 18 (1982), 1463-1465. [26] Kopladatze, R. G. and Kvinikadze, G., On the oscillation of solutions of …rst order delay di¤erential inequalities and equations, Georgian Math. J., 1 (1994), 675-685. [27] Kopladatze, R. G., Speci…c properties of solutions of …rst order di¤erential equations with several delay arguments, J. Contemp. Math. Anal. 50 (2015), 229-235. [28] Kwong, M. K., Oscillation of …rst-order delay equations, J. Math. Anal. Appl., 156 (1991), 274-286. [29] Ladas G., Sharp conditions for oscillations caused by delay, Applicable Anal., 9 (1979), 93-98. [30] Ladas, G., Lakshmikantham, V. and Papadakis, J. S., Oscillations of higher-order retarded di¤erential equations generated by retarded arguments, Delay and Functional Di¤erential Equations and Their Applications, Academic Press, New York, 1972, pp. 219-231. [31] Ladas, G. and Stavroulakis, I. P., Oscillations caused by several retarded and advanced arguments, J. Di¤erential Equations, 44 (1982), 134-152. [32] Ladde, G. S., Lakshmikantham, V. and Zhang, B. G., Oscillation Theory of Di¤ erential Equations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics, 110, Marcel Dekker, Inc., New York, 1987. [33] Li, B., Oscillations of …rst order delay di¤erential equations, Proc. Amer. Math. Soc., 124 (1996), 3729-3737. [34] Myshkis, A. D., Linear homogeneous di¤erential equations of …rst order with deviating arguments, Uspehi Matem. Nauk (N.S.), 5 (1950), 160-162 (in Russian). [35] G.M.Moremedi and I.P. Stavroulakis, A survey on the oscillation of solutions of di¤erential equations with a monotone or non-monotone argument, Springer Proc. Math. Stat. (to appear). [36] S…cas, Y.G. and Stavroulakis, I. P., Oscillation criteria for …rst-order delay equations, Bull. London Math. Soc., 35 (2003), 239-246. [37] Stavroulakis, I. P., Oscillation criteria for delay and di¤erence equations with non-monotone arguments, Appl. math. Comput. 226 (2014), 661-672. [38] Wang, Z. C., Stavroulakis, I. P. and Qian, X. Z., A survey on the oscillation of solutions of …rst order linear di¤erential equations with deviating arguments, Appl. Math. E-Notes, 2 (2002), 171-191. [39] Yu, J. S., Wang, Z. C., Zhang, B. G. and Qian, X. Z., Oscillations of di¤erential equations with deviating arguments, Panamer. Math. J., 2 (1992), 59-78. [40] Zhou, Y. and Yu, Y. H., On the oscillation of solutions of …rst order di¤erential equations with deviating arguments, Acta Math. Appl. Sinica (English Ser.), 15 1 Department of Mathematical Sciences, University of South Africa, 0003, South Africa E-mail address, M. Moremedi: [email protected]

E-mail address, H. jafari: [email protected] Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece E-mail address, I. P. Stavroulakis: [email protected]

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HyersUlam stability of second-order nonhomogeneous linear dierence equations with a constant stepsize Masakazu Onitsuka

Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan e-mail: [email protected] August 7, 2018

Abstract The present paper is concerned with HyersUlam stability of the second-order linear dierence equation ∆2h x(t) + α∆h x(t) + βx(t) = f (t) on hZ, where ∆h x(t) = (x(t + h) − x(t))/h and hZ = {hk| k ∈ Z} for the stepsize h > 0; α and β are real numbers; f (t) is a real-valued function on hZ. The purpose of this paper is to nd an explicit HUS constant for the second-order linear dierence equation whose characteristic equation has real roots. It is claried that an HUS constant changes by the inuence of the stepsize.

Keywords: 2010 Mathematics Subject Classication:

HyersUlam stability; HUS constant; second-order linear dierence equation; stepsize. Primary 39B82; Secondary 39A06; 65Q10.

1 Introduction HyersUlam stability is originated from in the eld of functional equations. In 1940, this problem was posed by Ulam [32, 33].

In the next year, it was solved by Hyers [9].

After that, there has been an

increasing interest in studying HyersUlam stability of functional equations, dierential equations and dierence equations (see [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 31, 34, 36]). In this paper, we will deal with HyersUlam stability of the second-order nonhomogeneous linear dierence equation

∆2h x(t) + α∆h x(t) + βx(t) = f (t) on

hZ,

where

∆h x(t) = for the stepsize

h > 0; α and β

x(t + h) − x(t) h

are real numbers;

and

(1.1)

hZ = {hk| k ∈ Z}

f (t) is a real-valued function on hZ.

If

1 − αh + βh2 = 0

holds, then we no longer have a second-order dierence equation. For this reason, we assume that

1 − αh + βh2 ̸= 0.

(1.2)

It is well-known that the global existence and uniqueness of solutions of (1.1) are guaranteed for the

We say that (1.1) has  HyersUlam stability on hZ if there exists a constant K > 0 with the following property: Let ε > 0 be a given arbitrary constant. If a function ϕ : hZ → R 2 satises ∆h ϕ(t) + α∆h ϕ(t) + βϕ(t) − f (t) ≤ ε for all t ∈ hZ, then there exists a solution x : hZ → R of (1.1) such that |ϕ(t) − x(t)| ≤ Kε for all t ∈ hZ. We call such K an  HUS constant for (1.1) on hZ. In addition, we call the minimum of HUS constants for (1.1) on hZ the  best HUS constant. Recently, initial-value problem.

the best HUS constant of various functional equations and linear operators has been discovered by Popa

1

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and Ra³a (see [28, 29, 30] and the references cited therein). When linear dierential equation

h → 0,

(1.1) becomes the second-order

x′′ + αx′ + βx = f (t),

(1.3)

that is, (1.1) is an approximation of the ordinary dierential equation (1.3). In 2010, Li and Shen [18] proved that (1.3) has HUS on a nite interval

I

if characteristic equation has two dierent positive roots.

In 2014, Xue [35] extended their results. Since the solution of the dierence equation with small stepsize is a good approximate solution of the dierential equation, studying HyersUlam stability of dierence equation (1.1) will contribute to computer science. In 2018, the author [22] dealt with HyersUlam stability of the rst-order nonhomogeneous linear dierence equation

∆h x(t) − ax(t) = f (t)

(1.4)

hZ, where a is a real number and f (t) is a real-valued function on hZ. We say that (1.4) has  Hyers Ulam stability on hZ if there exists a constant K > 0 with the following property: Let ε > 0 be a given arbitrary constant. If a function ϕ : hZ → R satises |∆h ϕ(t) − aϕ(t) − f (t)| ≤ ε for all t ∈ hZ, then there exists a solution x : hZ → R of (1.4) such that |ϕ(t) − x(t)| ≤ Kε for all t ∈ hZ. Noticing that if f (t) ≡ 0 with a = 0 or a = −2/h, then (1.4) does not have HyersUlam stability on hZ (see [21]); if a = −1/h, then we no longer have a rst-order dierence equation. For this reason, we assume that on

a ̸= 0, −

1 h

and

−

2 . h

In [22], the author proved that (1.4) has HyersUlam stability on (1.4) on

hZ

is

B(a, h) =

 1    |a| ,   

1 , |a + 2/h|

This constant is rewritten as

B(a, h) = Let

hZ,

and the best HUS constant for

1 0 0 or a < −2/h. Then (1.4) has HyersUlam stability with an HUS constant B(a, h) on hZ, where B(a, h) is the constant given by (1.5). Furthermore, if a function ϕ : hZ → R satises |∆h ϕ(t) − aϕ(t) − f (t)| ≤ ε for all t ∈ hZ, then { } t − t+h h lim ϕ(t)(ah + 1)− h − ∆−1 h f (t)(ah + 1)

t→∞

exists, and there exists a unique solution [ − t+h h x(t) = ∆−1 h f (t)(ah + 1) { }] t t − t+h h + lim ϕ(t)(ah + 1)− h − ∆−1 f (t)(ah + 1) (ah + 1) h h t→∞

of

(1.4)

such that |ϕ(t) − x(t)| ≤ B(a, h)ε for all t ∈ hZ.

Theorem B

.

(see [22, Corollary 2.6]) Suppose that −1/h < a < 0 or −2/h < a < −1/h. Then (1.4) has HyersUlam stability with an HUS constant B(a, h) on hZ, where B(a, h) is the constant given by (1.5). Furthermore, if a function ϕ : hZ → R satises |∆h ϕ(t) − aϕ(t) − f (t)| ≤ ε for all t ∈ hZ, then { } t − t+h h (t)(ah + 1) lim ϕ(t)(ah + 1)− h − ∆−1 f h

t→−∞

2

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exists, and there exists a unique solution [ − t+h h x(t) = ∆−1 h f (t)(ah + 1) { }] t t − t+h h + lim ϕ(t)(ah + 1)− h − ∆−1 (ah + 1) h f (t)(ah + 1) h t→−∞

of

(1.4)

such that |ϕ(t) − x(t)| ≤ B(a, h)ε for all t ∈ hZ.

Remark 1.1. We can conrm that the best HUS constant for (1.4) on hZ is greater than or equal to B(a, h) by the following example. Consider the rst-order nonhomogeneous linear dierence equation ∆h ϕ(t) − aϕ(t) − f (t) = ε(−1) on

hZ,

where

ε>0

and

m ∈ {1, 2}.

mt h

(1.6)

Let

− ϕ0 (t) = (ah + 1) h ∆−1 h f (t)(ah + 1) t

t+h h

,

mt

ϕm (t) = and

ϕ(t) = ϕ0 (t) + ϕm (t)

for all

t ∈ hZ.

ε(−1) h {(−1)m − 1} /h − a

Then

ϕ(t)

is a solution of (1.6). Now we will check this fact.

Since

f (t)(ah + 1)−

holds,

ϕ0 (t)

t+h h

t

is a solution of (1.4). From

∆h (−1)

mt h

=

} (−1)m − 1 m(t+h) mt mt 1{ (−1) h − (−1) h = (−1) h , h h

we have

(1.7)

mt

∆h ϕm (t) = That is,

= ∆h ϕ0 (t)(ah + 1)− h } t+h t 1{ = ϕ0 (t + h)(ah + 1)− h − ϕ0 (t)(ah + 1)− h h t+h ϕ0 (t + h) − (ah + 1)ϕ0 (t) = (ah + 1)− h h t+h = (∆h ϕ0 (t) − aϕ0 (t))(ah + 1)− h

ϕm (t)

mt ε {(−1)m − 1} (−1) h = ε(−1) h + aϕm (t). {(−1)m − 1} − ah

is a solution of (1.6) with

f (t) ≡ 0.

Using the above facts, we obtain

∆h ϕ(t) − aϕ(t) = ∆h (ϕ0 (t) + ϕm (t)) − a(ϕ0 (t) + ϕm (t)) = f (t) + ε(−1) This means that

ϕ(t)

mt h

.

is a solution of (1.6). Therefore,

|∆h ϕ(t) − aϕ(t) − f (t)| = ε holds for all

t ∈ hZ.

Since

ϕ0 (t)

is a solution of (1.4), and

(ah + 1)t/h

is a solution of (1.4) with

f (t) ≡ 0,

the general solution of (1.4) is written as t

x(t) = c(ah + 1) h + ϕ0 (t) for all

t ∈ hZ, where c is an arbitrary hZ. When c = 0, we have

constant. Noticing that

c=0

holds if and only if

|ϕ(t) − x(t)|

is

bounded on

|ϕ(t) − x(t)| = |ϕm (t)| = for all

t ∈ hZ

and

m ∈ {1, 2}.

ε |a + {1 − (−1)m } /h|

This means that the best HUS constant for (1.4) on

equal to

{ max

1 1 , |a| |a + 2/h|

hZ

is greater than or

} = B(a, h).

3

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Remark

1.2. Theorems A, B and Remark 1.1 imply that the best HUS constant for (1.4) on

hZ is B(a, h)

given by (1.5).

hZ. In addition, we will nd an x(t) of (1.1) such that |ϕ(t) − x(t)| is less than or equal to HUS constant multiplied by ε on hZ, where ϕ(t) is a function satisfying ∆2h ϕ(t) + α∆h ϕ(t) + βϕ(t) − f (t) ≤ ε on hZ. In the next section, we will present main theorems and their proofs, and give an HUS constant for (1.1) on hZ. In Section 3, we will classify HUS constants for (1.1) on hZ by coecients α and β . For illustration of the The purpose of this paper is to nd an HUS constant for (1.1) on

explicit solution

obtained results, we will take an example.

2 HUS constant for the second-order linear dierence equations We can easily see that the quadratic equation

λ2 + αλ + β = 0

(2.1)

is the characteristic equation for the second-order homogeneous linear dierence equation

∆2h x(t) + α∆h x(t) + βx(t) = 0

(2.2)

hZ, where α and β are real numbers with (1.2). In fact, we consider the funtion x(t) = (λh + 1)t/h on hZ, where λ is a root of (2.1). Notice that since (1.2), non of λ is equal to −1/h. On the other hand, t/h 2 2 t/h if λ ̸= −1/h then (1.2) holds. Clearly, ∆h x(t) = λ(λh + 1) and ∆h x(t) = λ (λh + 1) hold on hZ. Therefore, if (2.1) holds then x(t) is a solution of (2.2). Conversely, (2.1) is satised whenever x(t) is a t/h solution of (2.2) on hZ. Thus, (λh + 1) is a solution of (2.2) on hZ if and only if (2.1) holds. on

Throughout this paper, we dene

Λ1 = {λ ∈ R| λ > 0}, and

Λ3 =

Λ2 =

{ } 1 λ ∈ R − < λ < 0 , h

{ } 2 1 λ ∈ R − < λ < − , h h

Λ4 =

{ } 2 λ ∈ R λ < − . h

First, the following simple result is obtained by using Theorems A and B.

Theorem 2.1. Suppose that

∪ has real roots λ1 and λ2 with λi ∈ 4j=1 Λj for i ∈ {1, 2}. Then (1.1) has HyersUlam stability with an HUS constant B(λ1 , h)B(λ2 , h) on hZ, where B(·, h) is the constant given by (1.5). (2.1)

Proof.

Assume that a function

for all

t ∈ hZ.

Let

ϕ : hZ → R satises 2 ∆h ϕ(t) + α∆h ϕ(t) + βϕ(t) − f (t) ≤ ε

ψ(t) = ∆h ϕ(t) − λ1 ϕ(t)

for

t ∈ hZ.

From

λ1 + λ2 = −α, λ1 λ2 = β

and the above

assumption, we get the inequality

|∆h ψ(t) − λ2 ψ(t) − f (t)| = ∆2h ϕ(t) + α∆h ϕ(t) + βϕ(t) − f (t) ≤ ε for all

t ∈ hZ.

Using Theorems A and B, we can nd a solution

u : hZ → R

of

∆h u(t) − λ2 u(t) = f (t) such that

|ψ(t) − u(t)| ≤ B(λ2 , h)ε

for all

t ∈ hZ.

(2.4)

Namely, we have

|∆h ϕ(t) − λ1 ϕ(t) − u(t)| ≤ B(λ2 , h)ε for all

t ∈ hZ.

(2.3)

Using Theorems A and B again, there exists a solution

∆h v(t) − λ1 v(t) = u(t)

(2.5)

v : hZ → R

of (2.6)

4

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such that

|ϕ(t) − v(t)| ≤ B(λ1 , h)B(λ2 , h)ε

for all

t ∈ hZ.

Since

u(t)

is a solution of (2.4), we have

∆2h v(t) + α∆h v(t) + βv(t) = ∆2h v(t) − (λ1 + λ2 )∆h v(t) + λ1 λ2 v(t) = ∆h (∆h v(t) − λ1 v(t)) − λ2 (∆h v(t) − λ1 v(t)) = ∆h u(t) − λ2 u(t) = f (t) for all

t ∈ hZ.

Therefore we can conclude that

v(t)

is a solution of (1.1).

More explicitly, we can obtain the following result.

Theorem 2.2. Let ε > 0 be a given arbitrary constant, and let B(·, h) be the constant given by Dene

− F (t) = ∆−1 h f (t)(λ2 h + 1)

for t ∈ hZ. Suppose that ϕ : hZ → R satises

(2.1)

(1.5).

t+h h

has real roots λ1 and λ2 with λi ∈

∪4 j=1

Λj for i ∈ {1, 2}. If a function

2 ∆h ϕ(t) + α∆h ϕ(t) + βϕ(t) − f (t) ≤ ε

for all t ∈ hZ, then one of the following holds: (i) if λ1 , λ2 ∈ Λ1 ∪ Λ4 , then the limiting values { } t c1 = lim (∆h ϕ(t) − λ1 ϕ(t))(λ2 h + 1)− h − F (t) t→∞

and d1 = lim

t→∞

{ } t t − t+h h h ϕ(t)(λ1 h + 1)− h − ∆−1 h (F (t) + c1 ) (λ2 h + 1) (λ1 h + 1)

exist, and there exists a unique solution { } t+h t t h (λ h + 1)− h x(t) = ∆−1 (F (t) + c ) (λ h + 1) + d (λ1 h + 1) h 1 2 1 1 h of

(1.1)

such that |ϕ(t) − x(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ;

(ii) if λ1 ∈ Λ1 ∪ Λ4 and λ2 ∈ Λ2 ∪ Λ3 , then the limiting values } { t c2 = lim (∆h ϕ(t) − λ1 ϕ(t))(λ2 h + 1)− h − F (t) t→−∞

and d2 = lim

t→∞

} { t+h t t h (λ h + 1)− h (F (t) + c ) (λ ϕ(t)(λ1 h + 1)− h − ∆−1 h + 1) 2 2 1 h

exist, and there exists a unique solution { } t t − t+h h h x(t) = ∆−1 + d2 (λ1 h + 1) h h (F (t) + c2 ) (λ2 h + 1) (λ1 h + 1) of

(1.1)

such that |ϕ(t) − x(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ;

(iii) if λ1 , λ2 ∈ Λ2 ∪ Λ3 , then the limiting values c2 and { } t+h t t h (λ h + 1)− h d3 = lim ϕ(t)(λ1 h + 1)− h − ∆−1 (F (t) + c ) (λ h + 1) 2 2 1 h t→−∞

exist, and there exists a unique solution { } t t − t+h h h x(t) = ∆−1 + d3 (λ1 h + 1) h h (F (t) + c2 ) (λ2 h + 1) (λ1 h + 1) of

(1.1)

such that |ϕ(t) − x(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ. 5

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Proof.

on

hZ.

Assume that a function

ϕ : hZ → R satises 2 ∆h ϕ(t) + α∆h ϕ(t) + βϕ(t) − f (t) ≤ ε

ψ(t) = ∆h ϕ(t)−λ1 ϕ(t) for t ∈ hZ. Using the above assumption with λ1 +λ2 = −α, λ1 λ2 = β , t ∈ hZ. we prove case (i). Using λ2 ∈ Λ1 ∪ Λ4 and Theorem A, we see that } { t lim ψ(t)(λ2 h + 1)− h − F (t) t→∞ { } t = lim (∆h ϕ(t) − λ1 ϕ(t))(λ2 h + 1)− h − F (t) = c1

Let

we have (2.3) for First

t→∞

exists, and there exists a unique solution t

u(t) = (F (t) + c1 ) (λ2 h + 1) h of (2.4) such that

|ψ(t) − u(t)| ≤ B(λ2 , h)ε

for all

t ∈ hZ.

That is, (2.5) holds on

hZ.

Using

λ1 ∈ Λ1 ∪ Λ4

and Theorem A again, we conclude that the limiting value

lim

t→∞

} { t+h t t h (λ h + 1)− h = d1 ϕ(t)(λ1 h + 1)− h − ∆−1 (F (t) + c ) (λ h + 1) 1 2 1 h

exists, and there exists a unique solution

{ } t t − t+h h h v(t) = ∆−1 + d1 (λ1 h + 1) h h (F (t) + c1 ) (λ2 h + 1) (λ1 h + 1) |ϕ(t) − v(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ. Using the same argument as in the v(t) is a solution of (1.1). Noticing that v(t) is a unique solution of (1.1) such that |ϕ(t) − v(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ. Next we prove case (ii). Using λ2 ∈ Λ2 ∪ Λ3 and Theorem B, we see that the limiting value { } t lim ψ(t)(λ2 h + 1)− h − F (t) t→−∞ { } t = lim (∆h ϕ(t) − λ1 ϕ(t))(λ2 h + 1)− h − F (t) = c2

of (2.6) such that

proof of Theorem 2.1, we see that

t→−∞

exists, and there exists a unique solution t

u(t) = (F (t) + c2 ) (λ2 h + 1) h of (2.4) such that (2.5) holds for all

t ∈ hZ.

Using

λ1 ∈ Λ1 ∪ Λ4

and Theorem A, we can conclude that

the limiting value

lim

t→∞

{ } t+h t t h (λ h + 1)− h ϕ(t)(λ1 h + 1)− h − ∆−1 (F (t) + c ) (λ h + 1) = d2 2 2 1 h

exists, and there exists a unique solution

{ } t+h t t h (λ h + 1)− h v(t) = ∆−1 + d (F (t) + c ) (λ h + 1) 1 2 (λ1 h + 1) h 2 2 h of (2.6) such that

|ϕ(t) − v(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ. Repeating the same argument as in v(t) is a unique solution of (1.1) such that |ϕ(t) − v(t)| ≤ B(λ1 , h)B(λ2 , h)ε

the proof of Theorem 2.1, for all

t ∈ hZ.

We prove case (iii). As in the same argument of the preceding paragraph, using Theorem B, we see that

c2

λ2 ∈ Λ2 ∪ Λ3

and

exists, and there exists a unique solution t

u(t) = (F (t) + c2 ) (λ2 h + 1) h 6

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hZ. Using λ1 ∈ Λ2 ∪ Λ3 and Theorem B again, we can nd { } t+h t t h (λ h + 1)− h lim ϕ(t)(λ1 h + 1)− h − ∆−1 h + 1) = d3 (F (t) + c ) (λ 2 1 2 h

of (2.4) such that (2.5) holds on value

the limiting

t→−∞

and a unique solution

{ } t+h t t h (λ h + 1)− h v(t) = ∆−1 (F (t) + c ) (λ h + 1) + d (λ1 h + 1) h 2 2 1 3 h

|ϕ(t) − v(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ. By the same argument as in the proof v(t) is a unique solution of (1.1) such that |ϕ(t)−v(t)| ≤ B(λ1 , h)B(λ2 , h)ε for all t ∈ hZ.

of (2.6) such that of Theorem 2.1,

A natural question now arises. Is

B(λ1 , h)B(λ2 , h)

the best HUS constant for (1.1) on

hZ?

A partial

answer to this question is as follows.

Theorem 2.3. Suppose that best HUS constant for

(1.1)

∪ has real roots λ1 and λ2 with λi ∈ 4j=1 Λj for i ∈ {1, 2}. Then the on hZ is greater than or equal to { } 1 1 . max , |λ1 λ2 | |(λ1 + 2/h)(λ2 + 2/h)| (2.1)

Before to prove this theorem, we will give a lemma.

Lemma 2.1. Suppose that

(2.1)

has two roots λ1 and λ2 with λi ̸= −1/h for i ∈ {1, 2}. Dene − F (t) = ∆−1 h f (t)(λ2 h + 1)

and

} { t+h t t Y (t; λ1 , λ2 ) = ∆h−1 F (t)(λ2 h + 1) h (λ1 h + 1)− h (λ1 h + 1) h

for t ∈ hZ. Then Y (t; λ1 , λ2 ) is a solution of Proof.

t+h h

(2.7)

(1.1).

Since

F (t)(λ2 h + 1) h (λ1 h + 1)− t

t+h h

= ∆h Y (t; λ1 , λ2 )(λ1 h + 1)− h } t+h t 1{ = Y (t + h; λ1 , λ2 )(λ1 h + 1)− h − Y (t; λ1 , λ2 )(λ1 h + 1)− h h t+h 1 = {Y (t + h; λ1 , λ2 ) − (λ1 h + 1)Y (t; λ1 , λ2 )}(λ1 h + 1)− h h t+h = (∆h Y (t; λ1 , λ2 ) − λ1 Y (t; λ1 , λ2 ))(λ1 h + 1)− h t

holds, we have

t

∆h Y (t; λ1 , λ2 ) − λ1 Y (t; λ1 , λ2 ) = F (t)(λ2 h + 1) h for all

t ∈ hZ.

Using this equality, we obtain

∆2h Y (t; λ1 , λ2 ) − λ1 ∆h Y (t; λ1 , λ2 ) t

= ∆h F (t)(λ2 h + 1) h } t+h t 1{ = F (t + h)(λ2 h + 1) h − F (t)(λ2 h + 1) h h( ) t+h 1 1 = F (t + h) − F (t) (λ2 h + 1) h h λ2 h + 1 ) ( t+h λ2 = ∆h F (t) + F (t) (λ2 h + 1) h λ2 h + 1 t

= f (t) + λ2 F (t)(λ2 h + 1) h = f (t) + λ2 (∆h Y (t; λ1 , λ2 ) − λ1 Y (t; λ1 , λ2 )) for all

t ∈ hZ.

This means that

Y (t; λ1 , λ2 )

is a solution of (1.1).

7

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Proof of Theorem 2.3.

on

hZ,

φ(t) satisfying 2 ∆h φ(t) + α∆h φ(t) + βφ(t) − f (t) ≤ ε

We have only to show that for a given

we nd an explicit solution

x(t)

of (1.1) such that

{

1 1 , |λ1 λ2 | |(λ1 + 2/h)(λ2 + 2/h)|

|φ(t) − x(t)| = max for all

}

t ∈ hZ.

We now consider the second-order dierence equation

∆2h φ(t) + α∆h φ(t) + βφ(t) − f (t) = ε(−1) on

hZ,

where

ε>0

and

m ∈ {1, 2}.

mt h

(2.8)

Let mt

φm (t) = {

and

ε(−1) h }{ } m (−1)m − 1 (−1) − 1 − λ1 − λ2 h h

φ(t) = Y (t; λ1 , λ2 ) + φm (t) for all t ∈ hZ, where Y (t; λ1 , λ2 ) is the Y (t; λ1 , λ2 ) is a solution of (1.1) from Lemma 2.1. Now, we will

here that

(2.8). From (1.7), we have

{ ∆2h (−1)

mt h

=

(−1)m − 1 h

function given by (2.7). Note check that

φ(t)

is a solution of

}2 (−1)

mt h

.

Using this, we get

∆2h φm (t) + α∆h φm (t) + βφm (t) ] [{ }2 mt (−1)m − 1 ε(−1) h (−1)m − 1 { } { } +β = +α (−1)m − 1 (−1)m − 1 h h − λ1 − λ2 h h = ε(−1) for all

t ∈ hZ.

mt h

That is,

φm (t)

is a solution of (2.8) with

f (t) ≡ 0.

Using the above facts, we obtain

∆2h φ(t) + α∆h φ(t) + βφ(t) = ∆2h Y (t; λ1 , λ2 ) + α∆h Y (t; λ1 , λ2 ) + βY (t; λ1 , λ2 ) + ∆2h φm (t) + α∆h φm (t) + βφm (t) = f (t) + ε(−1) This means that

holds for all

φ(t)

t ∈ hZ.

mt h

is a solution of (2.8). Therefore,

2 ∆h φ(t) + α∆h φ(t) + βφ(t) − f (t) = ε

Let

x0 (t)

be the general solution of (1.1) with t

t

c1 (λ1 h + 1) h + c2 (λ2 h + 1) h where

c1

and

c2

.

are arbitrary constants. Since

or

f (t) ≡ 0.

That is,

t

x0 (t)

is written by

t

c1 (λ1 h + 1) h + c2 t(λ1 h + 1) h ,

Y (t; λ1 , λ2 )

is a solution of (1.1), the general solution of

(1.1) is written as

x(t) = x0 (t) + Y (t; λ1 , λ2 ) t ∈ hZ. c1 = c2 = 0, we for all

Noticing that

c1 = c2 = 0

holds if and only if

|φ(t) − x(t)|

is bounded on

hZ.

When

have

ε |φ(t) − x(t)| = |φm (t)| = m m 1 − (−1) λ1 + λ2 + 1 − (−1) h h 8

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for all

t ∈ hZ

and

m ∈ {1, 2}.

This means that the best HUS constant for (1.1) on

equal to

{ max

1 1 , |λ1 λ2 | |(λ1 + 2/h)(λ2 + 2/h)|

hZ

is greater than or

} .

Theorems 2.1 and 2.3 imply the following result.

Corollary 2.4. Suppose that

∪ ∪ has real roots λ1 and λ2 . If λ1 , λ2 ∈ Λ1 Λ2 or λ1 , λ2 ∈ Λ3 Λ4 , then (1.1) has HyersUlam stability with the best HUS constant B(λ1 , h)B(λ2 , h) on hZ, where B(·, h) is the constant given by (1.5).

Proof.

(2.1)

From Theorem 2.1, (1.1) has HyersUlam stability with an HUS constant

{

Since

max if

λ1 , λ2 ∈ Λ1

∪

Λ2 ,

max λ1 , λ2 ∈ Λ3

∪

Λ4 ,

=

B(λ1 , h)B(λ2 , h) on hZ.

1 |λ1 λ2 |

and

{

if

1 1 , |λ1 λ2 | |(λ1 + 2/h)(λ2 + 2/h)|

}

1 1 , |λ1 λ2 | |(λ1 + 2/h)(λ2 + 2/h)|

} =

1 |(λ1 + 2/h)(λ2 + 2/h)|

we conclude that

{ max

1 1 , |λ1 λ2 | |(λ1 + 2/h)(λ2 + 2/h)|

From Theorem 2.3 it follows that

B(λ1 , h)B(λ2 , h)

} = B(λ1 , h)B(λ2 , h).

is the best HUS constant.

From Corollary 2.4, we obtain the following.

Corollary 2.5. Suppose that

∪ has exactly one real root λ with λ ∈ 4j=1 Λj . Then (1.1) has Hyers Ulam stability with the best HUS constant B 2 (λ, h) on hZ, where B(·, h) is the constant given by (1.5). (2.1)

3 Classication of HUS constants by the coecients According to Theorem 2.1, we see that the following fact.

Remark

3.1. An HUS constant for (1.1) on

B(λ1 , h)B(λ2 , h) =

where

λ1

and

λ2

hZ

is rewritten as

 1     |λ λ2 | 1    1 |λ1 (λ2 + 2/h)|      1   |(λ1 + 2/h)(λ2 + 2/h)|

are real roots of (2.1) satisfying

if

λ1 , λ2 ∈ Λ1 ∪ Λ2 ,

if

λ1 ∈ Λ1 ∪ Λ2 , λ2 ∈ Λ3 ∪ Λ4 ,

if

λ1 , λ2 ∈ Λ3 ∪ Λ4 ,

λi ̸= 0, −1/h

and

−2/h

for

i ∈ {1, 2}.

Unfortunately, HUS constants in the right-hand side of the equation are implicit expressions. In this section, we will decide HUS constants more explicitly. To be specic, we will classify HUS constants for (1.1) on

hZ

by coecients

S=

α

and

β.

Let

S

be the set

{ } α2 1 1 2 4 (α, β) ∈ R2 β ≤ , β ̸= α − 2 , β ̸= α − 2 , β ̸= 0 . 4 h h h h

9

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Since

β = α/h − 1/h2

is the tangent line to the curve

β = α2 /4

at

(2/h, 1/h2 ), S

is divided into three

sets as follows (see Figure 1):

{ } α2 1 2 2 1 S1 = (α, β) ∈ R α − 2 < β ≤ , α < , β ̸= 0 , h h 4 h } { 2 1 1 4 S2 = (α, β) ∈ R2 β < α − 2 , β ̸= α − 2 , β ̸= 0 , h h h h } { 1 2 α 1 2 2 4 2 S3 = (α, β) ∈ R α − 2 < β ≤ , α > , β ̸= α − 2 . h h 4 h h h

and

PSfrag repla ements

β = 2α/h − 4/h2 is the tangent line to the curve β = α2 /4 at (4/h, 4/h2 ); S1 ∩ S2 , S2 ∩ S3 S3 ∩ S1 are empty sets; S = S1 ∪ S2 ∪ S3 holds. The above-mentioned sets are used without notice

Note that

in this paper.

S3

4

h2 1

h2

S1

2

1

1

h

h2

h

3

h

h

S2

4

h2 Figure 1: The sets



4

S1 , S2

and

S3

on the

(α, β)

plane.

The obtained result is as follows.

Corollary 3.1. If (α, β) ∈ S , then (

B

has HyersUlam stability with an HUS constant ) ( ) √ √ −α + α2 − 4β −α − α2 − 4β ,h B ,h 2 2 (1.1)

on hZ, where B(·, h) is the constant given by

(1.5).

(i) if (α, β) ∈ S1 , then the best HUS constant for (ii) if (α, β) ∈ S2 , then an HUS constant for

(1.1)

Furthermore, one of the following holds: (1.1)

on hZ is 1/|β|;

on hZ is

1 ( ) ; √ β + −α + α2 − 4β /h

(iii) if (α, β) ∈ S3 , then the best HUS constant for

(1.1)

on hZ is

1 . |β − 2α/h + 4/h2 |

Proof.

Suppose that

(α, β) ∈ S . µ1 =

Let

−α +

√ α2 − 4β 2

and

µ2 =

−α −

√

α2 − 4β . 2

10

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β ≤ α2 /4 holds. By β ̸= α/h − 1/h2 , (1.2) is satised, and 2 therefore, we have µ1 ̸= −1/h ̸= µ2 . From β ̸= 2α/h − 4/h we see that µ1 ̸= −2/h ̸= µ2 . In addition, ∪4 by β ̸= 0, non of µ1 and µ2 are equal to 0. Therefore, µ1 , µ2 ∈ j=1 Λj . Using Theorem 2.1, (1.1) has HyersUlam stability with an HUS constant B (µ1 , h) B (µ2 , h). Next, we will show that the assertions (i)(iii). We consider the case (α, β) ∈ S1 . From Then

µ1

µ2

and

are real roots of (2.1) since

α2 1 1 α− 2 (α − 2/h)2 = |α − 2/h|.

This means that

√ 2 √ − α2 − 4β < α − < α2 − 4β. h Using this inequality we obtain

µ2 < − That is,

µ1 ∈ Λ1 ∪ Λ2 , µ2 ∈ Λ3 ∪ Λ4 .

1 < µ1 . h

From Remark 3.1, an HUS constant for (1.1) on

hZ

is

1 1 ) . ( = √ |µ1 (µ2 + 2/h)| β + −α + α2 − 4β /h (α, β) ∈ S3 . Using the same argument in the proof of the case (α, β) ∈ S1 , α > 2/h, we obtain µ2 ≤ µ1 < −1/h. This and β ̸= 2α/h − 4/h2 imply that µ1 , Corollary 2.4 and Remark 3.1, the best HUS constant for (1.1) on hZ is

Finally, we consider the case we have (3.1). By using

µ2 ∈ Λ3 ∪ Λ4 .

From

1 1 1 = = . 2 |(µ1 + 2/h)(µ2 + 2/h)| |µ1 µ2 + 2(µ1 + µ2 )/h + 4/h | |β − 2α/h + 4/h2 | This completes the proof of Corollary 3.1. For illustration of the obtained result, we will present an example.

Example.

We consider the second-order linear dierence equation

∆2h x(t) + 3∆h x(t) + x(t) = f (t) on

hZ,

(3.2)

where (1.2) and

1 ̸=

6 4 − h h2 11

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hold. Since

(3, 1) ∈ S ,

Corollary 3.1 implies that (3.2) has HyersUlam stability. Moreover, xing the

h = 1/3 then (3, 1) ∈ S1 , and therefore, (√ the) best HUS (3, 1) ∈ S2 . So, we get an HUS constant 1/ 5 − 2 . If h = 3 HUS constant for (3.2) is 9/5.

stepsize gives an HUS constant. For example, if constant for (3.2) is one. If then

(3, 1) ∈ S3 ,

Remark

h=1

then

and thus, the best

3.2. Under the assumption that

(α, β) is included in the rst quadrant and S , if the stepsize h so that (α, β) ∈ S1 . On the other hand, if the stepsize is suciently large, then we can choose a h so that (α, β) ∈ S3 . From Corollary 3.1 and Example 3, we see that the best HUS constant for (1.1) on hZ is aected by the stepsize. In other words, it is concluded is suciently small, then we can choose a

that the best HUS constant changes by the inuence of the stepsize.

Acknowledgment The author was supported by JSPS KAKENHI Grant Number JP17K14226.

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Choquet-Iyengar type advanced inequalities George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we extend advanced known Iyengar type inequalities to Choquet integrals setting with respect to distorted Lebesgue measures and for monotone functions.

2010 Mathematics Subject Classi…cation: 26D10, 26D15, 28A25. Keywords and phrases: Choquet integral, distorted Lebesgue measure, analytic inequality, Iyengar inequality.

1

Background - I

In the year 1938, Iyengar [7] proved the following interesting inequality. Theorem 1 Let f be a di¤ erentiable function on [a; b] and jf 0 (x)j Z

b

f (x) dx

a

2

1 (b 2

a) (f (a) + f (b))

M1 (b a) 4

M1 . Then 2

(f (b) f (a)) : (1) 4M1

In 2001, X.-L. Cheng [3] proved that Theorem 2 Let f 2 C 2 ([a; b]) and jf 00 (x)j Z

a

b

f (x) dx

1 (b 2

a) (f (a) + f (b)) + M2 (b 24

where 1

= f 0 (a)

M2 . Then 1 (b 8

(b a) 16M2

3

a)

2

a) (f 0 (b)

2 1;

f 0 (a))

(2)

2 (f (b) f (a)) + f 0 (b) : (b a)

In 2006, [6], the authors proved: 1

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Theorem 3 Let f 2 C 2 ([a; b]) and jf 00 (x)j I=

Z

b

1 (b 2

f (x) dx

a

M . Set

a) (f (a) + f (b)) +

Then

1 (b 8

2

a) (f 0 (b)

f 0 (a)) :

(3)

;

(4)

3

M M (b a) + 24 3 " 3 M (b a) M b a 24 3 2 where a

=

1 2M

f0

b

=

1 2M

f 0 (b)

3 a

3 b

+

I

3

+

a

a+b 2

b

b

2

f 0 (a) + a+b 2

f0

3

a

+

b

a 4

b

a 4

#

;

(5)

:

(6)

In 1996, Agarwal and Dragomir [1] obtained a generalization of (1): Theorem 4 Let f : [a; b] ! R be a di¤ erentiable function such that for all x 2 [a; b] with M > m we have m f 0 (x) M . Then Z

b

f (x) dx

a

(f (b)

f (a)

m (b

1 (b 2

a) (f (a) + f (b))

a)) (M (b 2 (M m)

a)

f (b) + f (a))

:

(7)

In [9], Qi proved Theorem 5 Let f : [a; b] ! R be a twice di¤ erentiable function such that for all x 2 [a; b] with M > 0 we have jf 00 (x)j M . Then Z

b

a

f (x) dx

(f (a) + f (b)) (b 2

1 + Q2 (f 0 (b) 8

a) +

f 0 (a)) (b

2

a)

3

M (b a) 24

3Q2 ;

1

(8)

where 2

Q =

f 0 (a) + f 0 (b) M 2 (b

2

2

(f 0 (b)

a)

2

f (b) f (a) b a 2

f 0 (a))

:

(9)

Finally in 2005, Zheng Liu, [8], proved the following:

2

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Theorem 6 Let f : [a; b] ! R be a di¤ erentiable function such that f 0 is integrable on [a; b] and for all x 2 [a; b] with M > m we have m Then Z b

f 0 (x) x

f 0 (a) a

M and m

(f (a) + f (b)) (b 2

f (x) dx

a

1 + 3P 2 48

a) +

1 + P2 8

(M

3

(m + M ) (b

a)

f 0 (b) b

f 0 (x) x

(f 0 (b)

M:

(10)

f 0 (a)) (b

2

a)

3

m) (b 48

a)

1

3P 2 ;

(11)

2:

(12)

where P2 =

f 0 (a) + f 0 (b) m 2

M 2

(b

2

a)

f 0 (b)

2

f (b) f (a) b a

f 0 (a)

2

m+M 2

(b

a)

In [2] we extended (1) for Choquet integrals. Motivated by these results we extend here Theorems 2-6 to the Choquet integrals setting.

2

Background - II

In the next assume that (X; F) is a measurable space and (R+ ) R is the set of all (nonnegative) real numbers. We recall some concepts and some elementary results of capacity and the Choquet integral [4, 5]. De…nition 7 A set function : F ! R+ is called a non-additive measure (or capacity) if it satis…es (1) (?) = 0; (2) (A) (B) for any A B and A; B 2 F: The non-additive measure is called concave if (A [ B) + (A \ B)

(A) + (B) ;

(13)

for all A; B 2 F. In the literature the concave non-additive measure is known as submodular or 2-alternating non-additive measure. If the above inequality is reverse, is called convex. Similarly, convexity is called supermodularity or 2-monotonicity, too. First note that the Lebesgue measure for an interval [a; b] is de…ned by ([a; b]) = b a, and that given a distortion function m, which is increasing (or non-decreasing) and such that m (0) = 0, the measure (A) = m ( (A)) is a distorted Lebesgue measure. We denote a Lebesgue measure with distortion m

3

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by = m . It is known that m is concave (convex) if m is a concave (convex) function. The family of all the nonnegative, measurable function f : (X; F) ! (R+ ; B (R+ )) + + is denoted as L+ 1 , where B (R ) is the Borel -…eld of R . The concept of the integral with respect to a non-additive measure was introduced by Choquet [4]. De…nition 8 Let f 2 L+ 1 . The Choquet integral of f with respect to nonadditive measure on A 2 F is de…ned by Z Z 1 (C) f d := (fx : f (x) tg \ A) dt; (14) A

0

where the integral Ron the right-hand side is a RRiemann integral. R Instead of (C) X f d , we shall write (C) f d . If (C) f d < 1, we say that f is Choquet integrable and we write Z L1C ( ) = f : (C) f d < 1 : The next lemma summarizes the basic properties of Choquet integrals [5]. Lemma 9 RAssume that f; g 2 L1C ( ). (1) (C) 1A d = (A), A 2 F: R (2) 2 R+ , we have (C) fd = R (Positive homogeneity) For all (C) f d : R (3) R (Translation invariance) For all c 2 R, we have (C) A (f + c) d (C) A f d + c (A) : (4) (Monotonicity in the integrand) If f g, then we have Z Z (C) f d (C) gd :

R (Monotonicity in the set function) If , then we have (C) f d (5) (Subadditivity) If is concave, then Z Z Z (C) (f + g) d (C) f d + (C) gd :

=

R (C) f d :

(Superadditivity) If

is convex, then Z Z Z (C) (f + g) d (C) f d + (C) gd :

(6) (Comonotonic additivity) If f and g are comonotonic, then Z Z Z (C) (f + g) d = (C) f d + (C) gd ; where we say that f and g are comonotonic, if for any x; x0 2 X, then (f (x)

f (x0 )) (g (x)

g (x0 ))

0:

4

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We next mention the amazing result from [10], which permits us to compute the Choquet integral when the non-additive measure is a distorted Lebesgue measure. Theorem 10 Let f be a nonnegative and measurable function on R+ and = m be a distorted Lebesgue measure. Assume that m (x) and f (x) are both continuous and m (x) is di¤ erentiable. When f is an increasing (non-decreasing) function on R+ , the Choquet integral of f with respect to m on [0; t] is represented as Z Z t (C) fd m = m0 (t x) f (x) dx; (15) [0;t]

0

however, when f is a decreasing (non-increasing) function on R+ , the Choquet integral of f is Z Z t (C) fd m = m0 (x) f (x) dx: (16) [0;t]

0

Remark 11 We denote by (t; x) :=

m0 (t x) , when f is increasing (non-decreasing), m0 (x) , when f is decreasing (non-increasing).

(17)

So for f continuous and monotone we can combine (15) and (16) into (C)

Z

fd

m

[0;t]

3

=

Z

t

(t; x) f (x) dx:

(18)

0

Main Results

We present the following advanced Choquet-Iyengar type inequalities: The next is based on Theorem 2. Theorem 12 Here f : R+ ! R+ is a monotone twice continuously di¤ erentiable function on R+ , m is a distorted Lebesgue measure, where m is such that m (0) = 0, m is increasing and thrice continuously di¤ erentiable on R+ , t 2 R+ . Then 00 i) if f is increasing and (m0 (t ) f ) (x) M2 , 8 x 2 [0; t], M2 > 0, we have that Z t (C) f (x) d m (x) [m0 (t) f (0) + m0 (0) f (t)] + 2 [0;t] t2 [(m0 (0) f 0 (t) 8

m0 (t) f 0 (0)) + (m00 (t) f (0) M2 3 t 24

t 16M2

2 1

;

m00 (0) f (t))] (19)

5

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where 1

2 (m0 (0) f (t)

= (m0 (t) f 0 (0) + m0 (0) f 0 (t))

m0 (t) f (0)) t

(m00 (t) f (0) + m00 (0) f (t)) ;

(20)

00

0

ii) if f is decreasing and (m f ) (x) M3 , 8 x 2 [0; t], M3 > 0, we have that Z t (C) f (x) d m (x) (m0 (0) f (0) + m0 (t) f (t)) + 2 [0;t] t2 [(m00 (t) f (t) 8

m00 (0) f (0)) + (m0 (t) f 0 (t) M3 3 t 24

t 16M3

2 1

m0 (0) f 0 (0))]

;

(21)

where 1

2 [m0 (t) f (t)

= [m00 (t) f (t) + m00 (0) f (0)]

t

Z

(22)

00

M2 , 8 x 2 [0; t],

f (x) d

t2 (m0 (t 8 Z (C) f (x) d

0

(x)

[0;t]

t [(m0 (0) f 0 (t) 8

0

(m0 (t

) f ) (t)

m

) f ) (x)

t (m0 (t) f (0) + m0 (0) f (t)) + 2

(x)

m

[0;t]

2

+

[m0 (t) f 0 (t) + m0 (0) f 0 (0)] : Proof. i) If f is increasing and (m0 (t M2 > 0, we have that (C)

m0 (0) f (0)]

) f ) (0)

t [m0 (t) f (0) + m0 (0) f (t)] + 2

m0 (t) f 0 (0)) + (m00 (t) f (0) M2 3 t 24

=

t 16M2

2 1

m00 (0) f (t))]

(by (2) & (15))

;

(23)

where 1

= (m0 (t

0

) f ) (0)

2 (m0 (0) f (t)

m0 (t) f (0)) t

(m0 (t) f 0 (0) + m0 (0) f 0 (t))

2 (m0 (0) f (t)

t (m (t) f (0) + m (0) f (t)) : 00

00

+ (m0 (t

0

) f ) (t) =

m0 (t) f (0)) (24)

6

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00

ii) If f is decreasing and (m0 f ) (x) M3 , 8 x 2 [0; t], M3 > 0, we have that Z t (C) f (x) d m (x) (m0 (0) f (0) + m0 (t) f (t)) + 2 [0;t] t2 0 (m0 f ) (t) 8 (C)

Z

f (x) d

m

=

t (m0 (0) f (0) + m0 (t) f (t)) + 2

(x)

[0;t]

t2 [(m00 (t) f (t) 8

0

(m0 f ) (0)

m00 (0) f (0)) + (m0 (t) f 0 (t) t 16M3

M3 3 t 24

m0 (0) f 0 (0))] 2

1

(by (2) & (16))

;

(25)

where 1

= [m00 (t) f (t) + m00 (0) f (0)] + [m0 (t) f 0 (t) + m0 (0) f 0 (0)] 2 [(m0 f ) (t)

(m0 f ) (0)] t

:

(26)

The theorem is proved. The next result is based on Theorem 3. Theorem 13 Here f : R+ ! R+ is a monotone twice continuously di¤ erentiable function on R+ , m is a distorted Lebesgue measure, where m is such that m (0) = 0, m is increasing and thrice continuously di¤ erentiable on R+ , t 2 R+ . Then 00 i) if f is increasing and (m0 (t ) f ) (x) M1 , 8 x 2 [0; t], M1 > 0, we call: Z t I1 = (C) f (x) d m (x) [m0 (t) f (0) + m0 (0) f (t)] + 2 [0;t] t2 [(m0 (0) f 0 (t) 8 and (1) 0

=

m0 (t) f 0 (0)) + (m00 (t) f (0)

1 2M1

t 2

m00

+ (m00 (t) f (0) and (1) t

=

+ m00

f

t 2

t 2

+ m0

m00 (0) f (t))] ; f0

(27)

t 2

t m0 (t) f 0 (0))] + ; 4

(28)

1 [( m00 (0) f (t) + m0 (0) f 0 (t)) 2M1 t 2

f

t 2

m0

t 2

f0

t 2

t + ; 4

(29)

7

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and we obtain t3 M1 + 24 3 " M 1 t3 M1 t 24 3 2

(1) 0

M1

3

+

(1) t

3

t 2

(1) 0

+

3

I1 3 (1) t

#

;

(30)

00

ii) if f is decreasing and (m0 f ) (x) M2 , 8 x 2 [0; t], M2 > 0, we call: Z t I2 = (C) f (x) d m (x) (m0 (0) f (0) + m0 (t) f (t)) + 2 [0;t] t2 [(m00 (t) f (t) 8 and (2) 0

m00 (0) f (0)) + (m0 (t) f 0 (t)

1 2M2

=

t 2

m00

t 2

f

m0 (0) f 0 (0))]

t 2

+ m0

t 2

f0

t (m00 (0) f (0) + m0 (0) f 0 (0))] + ; 4 and (2) t

m00

=

(31)

(32)

1 [(m00 (t) f (t) + m0 (t) f 0 (t)) 2M2

t 2

f

t 2

t 2

+ m0

f0

t 2

t + ; 4

(33)

and we obtain: M 2 t3 M2 + 24 3 " M 2 t3 M2 t 24 3 2

(2) 0

3

(2) 0

+

(2) t

3

t 2

+

3

I2 3 (2) t

#

:

(34)

00

Proof. i) Here f is increasing and (m0 (t ) f ) (x) M1 , 8 x 2 [0; t], M1 > 0: We call Z t I1 = (C) f (x) d m (x) (m0 (t) f (0) + m0 (0) f (t)) + 2 [0;t]

(C)

Z

t2 (m0 (t 8 f (x) d

[0;t]

t2 [(m0 (0) f 0 (t) 8

0

(m0 (t

) f ) (t) m

(x)

0

) f ) (0) =

t [m0 (t) f (0) + m0 (0) f (t)] + 2

m0 (t) f 0 (0)) + (m00 (t) f (0)

m00 (0) f (t))] :

(35)

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We set (1) 0

=

1 2M1

(m0 (t

1 2M1

m00

) f)

0

t 2

t 2

f

t 2

(1) t

=

1 2M1

t 2

+ m0

0

(m0 (t

) f ) (0) +

t = 4

(37) t 2

0

(m0 (t

) f ) (t)

(36)

t 2

f0

t m0 (t) f 0 (0))] + ; 4

+ (m00 (t) f (0) and

0

(m0 (t

) f)

+

t = 4

(38)

1 [( m00 (0) f (t) + m0 (0) f 0 (t)) 2M1 t 2

+ m00

f

t 2

t 2

m0

f0

t 2

t + : 4

By Theorem 3 and (15) we get M1 t3 + 24 3 " M 1 t3 t M1 24 3 2 M1

(1) 0

3

+

(1) t

3

t 2

(1) 0

+

3

I1 3 (1) t

#

:

(39)

00

ii) Next f is decreasing and (m0 f ) (x) M2 , 8 x 2 [0; t], M2 > 0: We call Z t I2 = (C) f (x) d m (x) (m0 (0) f (0) + m0 (t) f (t)) + 2 [0;t] t2 0 0 (m0 f ) (t) (m0 f ) (0) = 8 Z t (C) f (x) d m (x) (m0 (0) f (0) + m0 (t) f (t)) + 2 [0;t] t2 [(m00 (t) f (t) 8 We set (2) 0

=

m00 (0) f (0)) + (m0 (t) f 0 (t)

1 2M2

m00

t 2

t 2

f

+ m0

t 2

(40)

m0 (0) f 0 (0))] : f0

t 2

(41)

t (m00 (0) f (0) + m0 (0) f 0 (0))] + ; 4 and (2) t

=

1 [(m00 (t) f (t) + m0 (t) f 0 (t)) 2M2

(42)

9

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m00

t 2

t 2

f

t 2

+ m0

t 2

f0

t + : 4

By Theorem 3 and (16) we get M 2 t3 M2 + 24 3 " M 2 t3 M2 t 24 3 2

(2) 0

3

3

+

(2) t

3

t 2

(2) 0

+

I2 3 (2) t

#

:

(43)

The theorem is proved. The next result is based on Theorem 4. Theorem 14 Here f : R+ ! R+ is a monotone di¤ erentiable function on R+ , m is a distorted Lebesgue measure, where m is such that m (0) = 0, m is increasing and twice di¤ erentiable on R+ , t 2 R+ . Then 0 i) if f is increasing, and m1 (m0 (t ) f ) (x) M1 , 8 x 2 [0; t], where M1 > m1 , we obtain: (C)

Z

f (x) d

m

(x)

[0;t]

(m0 (0) f (t)

m0 (t) f (0)

m1 t) (M1 t m0 (0) f (t) + m0 (t) f (0)) : 2 (M1 m1 )

ii) if f is decreasing, and m2 m2 , we obtain: (C)

Z

f (x) d

m

(x)

[0;t]

(m0 (t) f (t)

m0 (0) f (0)

t (m0 (t) f (0) + m0 (0) f (t)) 2

0

(m0 f ) (x)

M2 , 8 x 2 [0; t], where M2 >

t (m0 (0) f (0) + m0 (t) f (t)) 2

m2 t) (M2 t m0 (t) f (t) + m0 (0) f (0)) : 2 (M2 m2 ) 0

Proof. i) Here f is increasing and m1 (m0 (t ) f ) (x) where M1 > m1 : We get, by Theorem 4 and (15), that (C)

Z

f (x) d

m

(x)

[0;t]

(m0 (0) f (t)

m0 (t) f (0)

[0;t]

f (x) d

m

M1 , 8 x 2 [0; t],

m1 t) (M1 t m0 (0) f (t) + m0 (t) f (0)) : 2 (M1 m1 ) 0

(C)

(x)

(45)

t (m0 (t) f (0) + m0 (0) f (t)) 2

ii) Next f is decreasing and m2 (m0 f ) (x) M2 > m2 . We get, by Theorem 4 and (16), that Z

(44)

(46)

M2 , 8 x 2 [0; t], where

t (m0 (0) f (0) + m0 (t) f (t)) 2 10

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(m0 (t) f (t)

m0 (0) f (0)

m2 t) (M2 t m0 (t) f (t) + m0 (0) f (0)) : 2 (M2 m2 )

(47)

The theorem is proved. The next result is based on Theorem 5. Theorem 15 Here f : R+ ! R+ is a monotone twice di¤ erentiable function on R+ , m is a distorted Lebesgue measure, where m is such that m (0) = 0, m is increasing and thrice di¤ erentiable on R+ , t 2 R+ . Then 00 i) if f is increasing, and (m0 (t ) f ) (x) M1 , 8 x 2 [0; t], M1 > 0, we call: Q21 = h i2 0 0 ( m00 (t) f (0) + m0 (t) f 0 (0)) + ( m00 (0) f (t) + m0 (0) f 0 (t)) 2 m (0)f (t) t m (t)f (0) M12 t2

( m00 (0) f (t) + m0 (0) f 0 (t) + m00 (t) f (0)

2

m0 (t) f 0 (0))

and we obtain (C)

Z

f (x) d

m

t [m0 (t) f (0) + m0 (0) f (t)] + 2

(x)

[0;t]

1 + Q21 t2 8

!

( m00 (0) f (t) + m0 (0) f 0 (t) + m00 (t) f (0) M1 t 3 1 24

m0 (t) f 0 (0))

3Q21 ;

00

ii) if f is decreasing, and (m0 f ) (x)

(49)

M2 , 8 x 2 [0; t], M2 > 0, we call:

Q22 = h

(m00 (0) f (0) + m0 (0) f 0 (0) + m00 (t) f (t) + m0 (t) f 0 (t)) M22 t2

[m00 (t) f (t) + m0 (t) f 0 (t)

m00 (0) f (0)

2

m0 (t)f (t) m0 (0)f (0) t 2 0 0

m (0) f (0)]

i2

(50)

and we obtain (C)

Z

f (x) d

[0;t]

1 + Q22 t2 8

!

m

(x)

t [m0 (0) f (0) + m0 (t) f (t)] + 2

(m00 (t) f (t) + m0 (t) f 0 (t) M2 t 3 1 24

m00 (0) f (0)

3Q22 :

m0 (0) f 0 (0)) (51)

11

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00

Proof. i) If f is increasing, and (m0 (t M1 > 0, we set: 0

(m0 (t

0

) f ) (0) + (m0 (t

Q21 =

M12 t2

) f ) (t) 0

(m0 (t

) f ) (x)

2

) f ) (t)

(m0 (t

M1 , 8 x 2 [0; t], (m0 (t

)f )(t) 0

(m0 (t

) f ) (0)

( m00 (t) f (0) + m0 (t) f 0 (0)) + ( m00 (0) f (t) + m0 (0) f 0 (t)) M12 t2

2

)f )(0)

t

( m00 (0) f (t) + m0 (0) f 0 (t) + m00 (t) f (0)

=

2

m0 (0)f (t) m0 (t)f (0) t 2 0 0 m (t) f (0))

2

2

:

(52)

By Theorem 5 and (15) we derive (C)

Z

f (x) d

m

t [m0 (t) f (0) + m0 (0) f (t)] + 2

(x)

[0;t]

1 + Q21 t2 8

!

( m00 (0) f (t) + m0 (0) f 0 (t) + m00 (t) f (0) M1 t 3 1 24

3Q21 :

00

ii) If f is decreasing, and (m0 f ) (x) 0

0

h

M22 t2

(53)

M2 , 8 x 2 [0; t], M2 > 0, we set:

(m0 f ) (0) + (m0 f ) (t) Q22 =

2 0

(m0 f ) (t)

(m0 f )(t) (m0 f )(0)

[m00 (t) f (t) + m0 (t) f 0 (t)

2

t 0

(m0 f ) (0)

=

2

(m00 (0) f (0) + m0 (0) f 0 (0) + m00 (t) f (t) + m0 (t) f 0 (t)) M22 t2

m0 (t) f 0 (0))

m00 (0) f (0)

2

(54)

m0 (t)f (t) m0 (0)f (0) t 2 0 0

m (0) f (0)]

i2

:

By Theorem 5 and (16) we derive (C)

Z

f (x) d

[0;t]

1 + Q22 t2 8

!

m

(x)

t [m0 (0) f (0) + m0 (t) f (t)] + 2

(m00 (t) f (t) + m0 (t) f 0 (t) M2 t 3 1 24

m00 (0) f (0)

3Q22 :

m0 (0) f 0 (0)) (55)

The theorem is proved. Finally we apply Theorem 6 to obtain:

12

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Theorem 16 Here f : R+ ! R+ is a monotone and continuously di¤ erentiable function on R+ , m is a distorted Lebesgue measure, where m is such that m (0) = 0, m is increasing and twice continuously di¤ erentiable on R+ , t 2 R+ . We have i) If f is increasing, and 0

(m0 (t

m1

0

(m0 (t

) f ) (0)

(m0 (t x

) f ) (x)

) f ) (x) x

and

0

(m0 (t

m1

) f ) (t) t

0

M1 ;

(56)

M1 ;

(57)

8 x 2 [0; t], with m1 < M1 , we set: 0

(m0 (t P12 =

0

) f ) (0) + (m0 (t

M1 m1 2 2 t 2

) f ) (t)

2

0

(m0 (t

(m0 (t

2

)f )(0)

t

(m0 (t

) f ) (t)

(m0 (t

)f )(t)

0

(m1 +M1 ) t 2

) f ) (0)

2

:

(58) Then (C)

Z

f (x) d

m

((m0 (t

(x)

) f ) (0) + (m0 (t 2

[0;t]

1 + P12 8

0

(m0 (t

) f ) (t)

(M1

1 + 3P12 48

0

(m0 (t

) f ) (0) t2

m1 ) t3 1 48

) f ) (t))

t+

(m1 + M1 ) t3

3P12 :

(59)

ii) If f is decreasing, and m2

0

0

(m0 f ) (x)

and m2

(m0 f ) (0) x

0

0

(m0 f ) (t) t

(m0 f ) (x) x

M2 ;

(60)

M2 ;

(61)

8 x 2 [0; t], with m2 < M2 , we set: 0

0

(m0 f ) (0) + (m0 f ) (t) P22 =

M2 m2 2 2 t 2

Then (C)

Z

[0;t]

f (x) d

0

(m0 f ) (t)

m

(x)

2

(m0 f )(t) (m0 f )(0)

2

t 0

(m0 f ) (0)

(m2 +M2 ) t 2

2:

(62)

((m0 f ) (0) + (m0 f ) (t)) t+ 2 13

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1 + P22 8

0

(m0 f ) (t)

0

(m0 f ) (0) t2 (M2

m2 ) t3 1 48

1 + 3P22 48

(m2 + M2 ) t3

3P22 :

(63)

t Example 17 A well-known distortion function is m (t) = 1+t , t 2 R+ . We 1 0 have m (0) = 0, m (t) 0, m (t) = (1+t)2 > 0, that is m is strictly increasing. 3

We have that m00 (t) = 2 (1 + t) , m(3) (t) = 6 (1 + t) n+1 (n+1) get that m(n) (t) = ( 1) n! (1 + t) , 8 n 2 N:

4

, and in general we

References [1] R.P. Agarwal, S.S. Dragomir, An application of Hayashi’s inequality for di¤ erentiable functions, Computers Math. Applic., 6 (1996), 95-99. [2] G. Anastassiou, Choquet integral analytic inequalities, submitted, 2018. [3] Xiao-Liang Cheng, The Iyengar-type inequality, Applied Math. Letters 14 (2001), 975-978. [4] G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble), 5 (1953), 131-295. [5] D. Denneberg, Non-additive Measure and Integral, Kluwer Academic Publishers, Boston, 1994. [6] I. Franjic, J. Pecaric, I. Peric, Note on an Iyengar type inequality, Applied Math. Letters 19 (2006), 657-660. [7] K.S.K. Iyengar, Note on an inequality, Math. Student 6, (1938), 75-76. [8] Zheng Liu, Note on Iyengar’s inequality, Univ. Beograd Publ. Elektrotechn. Fak., Ser. Mat. 16 (2005), 29-35. [9] F. Qi, Further generalizations of inequalities for an integral, Univ. Beograd Publ. Elektrotechn. Fak., Ser. Mat. 8 (1997), 79-83. [10] M. Sugeno, A note on derivatives of functions with respect to fuzzy measures, Fuzzy Sets Syst., 222 (2013), 1-17.

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SOME RESULTS ABOUT I STATISTICALLY PRE-CAUCHY SEQUENCES WITH AN ORLICZ FUNCTION HAF I·ZE GÜMܸ S , ÖMER K I·S ¸ I·, AND EKREM SAVA¸ S Abstract. In this study, we de…ne the concept of I statistically convergence for di¤erence sequences and we use an Orlicz function to obtain more general results. We also show that an I statistically convergent sequence with an Orlicz function is I statistically pre-Cauchy .

1. Introduction In this part, we give a short literature data about I statistical convergence, statistical pre-Cauchy sequences and di¤erence sequence spaces. As is known, convergence is one of the basic notions of Mathematics and statistical convergence extends the notion. It is easy to see that any convergent sequence is statistically convergent but not conversely. Statistical convergence was given by Zygmund [35] in Warsaw in 1935 and then it was formally introduced by Fast [16] and Steinhaus [33], independently. Later it was reintroduced by Schoenberg [32] : Even now, this concept has very much applications in di¤erent areas such as number theory by Erdös and Tenenbaum [10] ; measure theory by Miller [26] and summability theory by Freedman and Sember [17] : Statistical convergence is also applied to approximation theory by Gadjiev and Orhan [18], Anastassiou and Duman [1] and Sakao¼ glu and Ünver [19]. If we want to brie‡y remember this concept by using the characteristic function, we should give the following de…nitions: De…nition 1.1. Let E be a subset of N; the set of all natural numbers. The natural density of E is de…ned by d(E) := lim n

n 1 P n j=1

E (j)

whenever the limit exists where (E) is characteristic function of E. De…nition 1.2. ([16]) A number sequence (xn ) is statistically convergent to x provided that for every " > 0; d fn 2 N : jxn

xj

"g = lim n

1 jfk n

n : jxk

xj

"gj = 0

2000 Mathematics Subject Classi…cation. 40G15, 40A35. Key words and phrases. I convergence, di¤erence sequences, statistical pre-cauchy sequences, Orlicz function. A small summary of this article was presented at the International Conference on Recent Advances in Pure and Applied Mathematics held in Ku¸sadas¬in 2017 and the …rst author supported by Necmettin Erbakan University Scienti…c Research Commission under research project number 172518001-186. 1

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HAF I·ZE GÜM Ü S ¸ , ÖM ER K I·S ¸ I·, AND EKREM SAVA S ¸

or equivalently there exists a subset K N with d(E) = 1 and n0 (") such that n > n0 (") and n 2 K imply that jxn xj < ": In this case we write st lim xn = x: Statistical convergent sequences are generally denoted by S: I convergence has emerged as a kind of generalization form of many types of convergence. This means that, if we choose di¤erent ideals we will have di¤erent convergences such as usual convergence and statistical convergence as we will see from the examples below. In 2000, Koystro et. al. [24] introduced this concept in a metric space and then many concepts studied for statistical convergence have moved to ideal convergence. Before de…ning I convergence, the de…nitions of ideal and …lter will be needed. De…nition 1.3. A non-empty family of sets I 2N is called an ideal if and only if i) ; 2 I, ii) for each A; B 2 I we have A [ B 2 I and iii) for each A 2 I and each B A we have B 2 I: An ideal is called non-trivial if N 2 = I and non-trivial ideal is called admissible if fng 2 I for each n 2 N .

De…nition 1.4. A non-empty family of sets F 2N is a …lter in N if and only if i) ; 2 = F; ii) for each A; B 2 F we have A \ B 2 F and iii) for each A 2 F and each B A we have B 2 F: If I is a non-trivial ideal in N (i.e., N 2 = I), then the family of sets is a …lter in N.

F (I) = fM

N : 9A 2 I : M = N n Ag

Remark 1.1. Generally we will use ideals in our proofs but if the notion is more familiar for …lters, we will use the notion of …lter. De…nition 1.5. ([24]) Let I 2N be a proper ideal on N: The real sequence x = (xn ) is said to be I convergent to x 2 R provided that for each " > 0; A(") = fk 2 N : jxn

xj

"g 2 I:

The set of all I convergent sequences usually denoted by cI . More investigations in this direction and more applications can be found in Kostyrko, Salát and Wilezy´nski’s paper. We just want to give some well known examples which we mentioned before. Example 1.1. If I = I f =fA gence.

N : A is …niteg then we have the usual conver-

Example 1.2. If I = I d =fA N : d(A ) = 0g then we have the statistical convergence where d is the asymptotic density of A: Following the statistical convergence and I convergence located an important role in this area, Das, Sava¸s and Ghosal [6] have introduced the concept of I statistical convergence as follows and they extend the important summability methods statistical convergence and I convergence using ideals. De…nition 1.6. ([6])A sequence x = (xn ) is said to be I statistically convergent to L for each " > 0 and > 0; 1 2 I: n 2 N : jfk n : jxk Lj "gj n

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SOM E RESULTS ABOUT

I STATISTICALLY PRE-CAUCHY SEQUENCES W ITH AN ORLICZ FUNCTION3

We will denote the set of all I statistically convergent sequences by SI : Before giving information about the de…nitions and works of pre-Cauchy sequences, lets remember the de…nition of an Orlicz function. Orlicz function is a function M : [0; 1) ! [0; 1) which is continuous, non decreasing and convex with M (0) = 0, M (x) > 0 for x > 0 and M (x) ! 1 as x ! 1. An Orlicz function M satis…es the 2 -condition if there exits a constant K > 0 such that M (2u) KM (u) for all u 0. We want to give a little note here that if convexity of Orlicz function M is replaced by M (x + y) = M (x) + M (y) then we get the modulus function which is familiar to us. Lindendstrauss and Tzafriri [25] used the idea of Orlicz function to de…ne the following sequence space. lM :=

x2w:

1 P

M

jxn j

n=1

< 1 for some

>0

which called an Orlicz sequence space. lM is a Banach space with the norm kxk := inf

>0:

1 P

M

jxn j

1 :

n=1

The notion of statistically pre-Cauchy for real sequences was introduced by Connor, Fridy and Kline [4] in 1994. They proved that statistically convergent sequences are statistically pre-Cauchy and any bounded statistical pre-Cauchy sequence with nowhere dense set of limit points is statistically convergent. Khan and Lohani [20] handled this concept in a di¤erent way with the Orlicz function. More works on statistically pre-Cauchy sequences are found in Dutta, E¸si and Tripathy [8] , Dutta and Tripathy [9] and Khan and Tabassum [21] . As an expected result, in 2012, Khan, Ebedullah and Ahmad [22] de…ned preCauchy sequences for I convergence and they introduced the concept of I preCauchy sequence. They established the criterion for arbitrary sequence to be I pre-Cauchy and they also gave another criterion for I convergence. De…nition 1.7. ([22]) Let x = (xn ) be a sequence and let M be an Orlicz function then x is I pre-Cauchy if and only if I for some

> 0:

lim n

1 P M n2 k;j n

jxk

xj j

=0

Yamanc¬and Gürdal [34] , Ojha and Srivastava [27] and Saha et. al. [28] have some studies about this new de…niton. De…nition 1.8. ([7]) A sequence x = (xn ) is said to be I statistically pre-Cauchy if for any " > 0 and > 0; n2N:

1 jf(j; k) : jxk n2

xj j

"g ; j; k

nj

2 I:

In another direction, in 1981, l1 ( ); c( ) and c0 ( ) di¤erence sequence spaces de…ned by K¬zmaz [23] where l1 ; c and c0 are bounded, convergent and null sequence spaces, respectively. In this study the sequence x = ( xn ) de…ned by ( xn ) = (xn xn+1 ) for all n 2 N and some relations between these spaces for example c0 ( ) c( ) l1 ( ) were obtained. In Et and Çolak’s paper [11] K¬zmaz’s

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4

m

results generalized for

c0 ( c( l1 ( where m 2 N and m P m xn = ( 1)v v=0

m m v

sequences such that, m

m

) = fx = (xn ) : ) = fx = (xn ) : m ) = fx = (xn ) :

x 2 c0 g x 2 cg m x 2 l1 g

m

x=(

m

m 1

xn ) = (

m

m 1

xn

xn+1 ) i.e.

xn+v . They proved that these spaces are Banach spaces with

the norm

k:k =

m P

i=1

jxi j + k

m

xk1 :

Following these de…nitions, Et [12] ; Et and Çolak [11] ; Et and Ba¸sar¬r [13] ; Ayd¬n and Ba¸sar [2] ; Bekta¸s et. al. [3] ; Et and E¸si [14] ; Sava¸s [31] and many others searched various properties of this concept. Et and Nuray [15] have introduced the m statistical convergence and the set of all m statistical convergent sequences was denoted by S( m ): Following this study, Gümü¸s and Nuray [19] have extended m statistical convergence to m ideal convergence. De…nition 1.9. ([19]) Let I 2N be a non-trivial ideal in N. The sequence x = (xn ) of real numbers is said to be I convergent to x 2 R if for each " > 0 the set fn 2 N : j xn The space of all

xj

"g 2 I.

I convergent sequences is denoted by cI ( ).

Before we get to the part where our main results are, we would like to give some expressions that have already been proved before about I convergence and I convergence, without moving away from our aim. At the same time it will be interesting to move these expressions to I statistical convergence. Proposition 1.1. Let I

2N be an ideal in N and ( xn ) be a real sequence. Then c( )

cI ( ):

Note that the inverse of this proposition is not generally true as can be seen from the following example. Example 1.3. For the di¤ erence sequence

x = ( xn ) =

1; 0;

n is square ; n is not square

x 2 cId ( ) but x 2 = c( ): De…nition 1.10. Let I be an ideal in N. If fn + 1 : n 2 Ng 2 I for any A 2 I, then I is said to be a translation invariant ideal. Corollary 1.1. If I is translation invariant and (xn ) 2 cI then (xn+1 ) 2 cI . Example 1.4. Id is a translation invariant ideal. Proposition 1.2. If I cI ( ).

2N is an admissible translation invariant ideal then cI

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SOM E RESULTS ABOUT

I STATISTICALLY PRE-CAUCHY SEQUENCES W ITH AN ORLICZ FUNCTION5

2. Main Results In this section, we de…ne I statistical convergent and Cauchy sequences and we give some inclusion theorems. De…nition 2.1. A sequence x = (xn ) is said to be L provided that n2N:

1 jfk n

In our paper, the set of all by SI ( ).

n : j xk

Lj

I statistically pre-

I statistically convergent to "gj

2 I.

I statistically convergent sequences will be denoted

Now, lets evaluate this new de…nition for the If ideal in the example mentioned above. Example 2.1. For the ideal I = If ; SIf ( ) = S( ). De…nition 2.2. A sequence x = (xn ) is said to be for any " > 0 and > 0; n2N:

1 jf(j; k) : j xk n2

Theorem 2.1. An Cauchy.

";

j; k

ngj

I statistically convergent sequence is

Proof. Let x = (xn ) be given. We know that A=

xj j

I statistically pre-Cauchy if, 2 I. I statistically pre-

I statistically convergent to L: Let " > 0 and

n2N:

1 n k n

n : j xk

Lj

"o 2

2 I.

Then for all n 2 Ac where c stands for the complement, 1 n "o "o 1 n < i.e. >1 k n : j xk Lj k n : j xk Lj < n 2 n 2 Writing Bn = k n : j xk Lj < 2" we observe that for j; k 2 Bn ; " " j xk xj j j xk Lj + j xj Lj < + = ": 2 2 Hence Bn

Bn

f(j; k) : j xk jBn j n

2

xj j < ";

1 jf(j; k) : j xk n2

j; k

i.e. Let

1

xj j < ";

j; k

xj j < ";

ngj

:

ng which implies j; k

Thus for all n 2 Ac ; 1 jf(j; k) : j xk n2

> 0 be

jBn j n

ngj :

2

> (1

)2

1 jf(j; k) : j xk xj j "; j; k ngj < 1 (1 )2 : n2 > 0 be given. Choosing > 0 so that 1 (1 )2 < 1 we see that 8n 2 Ac ; 1 jf(j; k) : j xk xj j "; j; k ngj < 1 n2

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6

and so 1 jf(j; k) : j xk n2 Since A 2 I, we have the proof. n2N:

xj j

";

j; k

ngj

A:

1

Theorem 2.2. Let x = (xn ) be a sequence and M be Orlicz function. Then x is I statistically pre-Cauchy if and only if 1 P j xk xj j I lim 2 M = 0 for some > 0: n n k;j n Proof. First suppose that I

limn

For each " > 0 and n 2 N we have P j xk xj j 1 M = n12 n2 k;j n

1 n2

+

1 n2

1 n2

n2N:

P

P

= 0 for some

> 0:

M

j xk

xj j

M

j xk

xj j

M

j xk

xj j

k;j n j xk xj j "

P

k;j n j xk xj j " 1 n2

jf(j; k) : j xk

jf(j; k) : j xk

1 n2

xj j

k;j n j xk xj j 0 be given. Let > 0 be such that M ( ) < 2" : Since Orlicz function is bounded, there exists an integer B such that M (x) < B2 for all x 0: Then for each n 2 N; P P j xk xj j j xk xj j 1 M = n12 M n2 k;j n

k;j n j xk xj j
0 be given. Then choosing "; > 0 such that 2" + for each n 2 Ac ; j xk xj j 1 P M < 1 2 n k;j n i.e.

(

1 P n2N: 2 M n k;j n

j xk

2 I.

xj j

1

)

B 2


0; n : j xk

1 n

1 n

j xk

n P

M

Lj

= 0 for some

j xk Lj

= 0 for some

> 0: We have,

k=1 n P

M

j xk Lj

+

k=1 j xk Lj 0:

n : j xk

"gj

Lj

1 n

n P

M

j xk Lj

k=1 j xk Lj "

"gj :

n2N:

1 n

n P

M

j xk Lj

M ("):

k=1

Due to the statement we accepted at the beginning of the theorem, right hand side belongs to the ideal. As we know from the second expression of ideal, left hand side is also in ideal and this proves the theorem. Since the second part of the theory is very similar to the second part of the previous theorem, we can easily prove. References [1] G. A. Anastassiou and O. Duman, Statistical Korovkin theory for multivariate stochastic processes, Stoch. Anal. Appl. 28 (2010), no. 4, 648-661. [2] C. Ayd¬n and F. Ba¸sar, Some new di¤erence sequence spaces, Appl. Math.Comput., 157(3) (2004), 677-693. [3] Ç.A. Bekta¸s, M. Et and R. Çolak, Generalized di¤erence sequence spaces and their dual spaces, J.Math.Anal.Appl. 292 (2004), 423-432.

186

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.1, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

8

HAF I·ZE GÜM Ü S ¸ , ÖM ER K I·S ¸ I·, AND EKREM SAVA S ¸

[4] J. Connor, J. A. Fridy and J. Kline, Statistically pre-Cauchy sequences, Analysis, 14 (1994), 311-317. [5] J. Connor, R-type summability methods, Cauchy criteria, P-sets and statistical convergence, Proc. Amer. Math. Soc. 115(2), (1992), 319-327. [6] P. Das, E. Sava¸s and S. K. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Letters 24 (2011), 1509-1514. [7] P. Das and E. Sava¸s, On I statistically pre-Cauchy sequences, Taiwanese Jour. of Math., 18 (1), (2014), 115-126. [8] A. J. Dutta, a. E¸si and B. C. Tripathy, Statistically convergence triple sequence spaces de…ned by Orlicz function, Jour. of Math. Anal. 4(2) (2013), 16-22. [9] A. J. Dutta and B. C. Tripathy, Statistically pre-Cauchy Fuzzy real-valued sequences de…ned by Orlicz function, Proyecciones Jour. of Math, 33(3) (2014), 235-243. [10] P.Erdös and G. Tenenbaum, Sur les densities de certaines suites d’entiers, Proc. London. Math. Soc.3(59), (1989), 417-438. [11] M. Et and R. Çolak, On some generalized di¤erence sequence spaces, Soochow Journal Of Mathematics, 21(4) (1995), 377-386 [12] M. Et, On some di¤erence sequence spaces, Do¼ga-Tr. J.of Mathematics 17 (1993), 18-24. [13] M. Et and M. Ba¸sar¬r, On some new generalized di¤erence sequence spaces, Periodica Math. Hungarica 35 (3) (1997), 169-175. [14] M. Et. and A. E¸si, On Köthe- Toeplitz duals of generalized di¤erence sequence spaces, Malaysian Math. Sci. Soc. 23 (2000), 25-32. [15] M. Et and F. Nuray, m Statistical convergence, Indian J.Pure Appl. Math. 32(6) (2001), 961-969. [16] H. Fast, Sur la Convergence Statistique, Coll. Math. 2 (1951), 241-244. [17] A.R. Freedman, J. Sember and M. Raphael, Some Cesàro-type summability spaces, Proc. London Math. Soc. (3) 37 no. 3 (1978), 508–520. [18] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), no. 1, 129-138. [19] H. Gümü¸s and F. Nuray, m Ideal Convergence, Selçuk J. Appl. Math.12(2) (2011), 101-110. [20] V. A. Khan and Q. M. D. Lohani, Statistically pre-Cauchy sequences and Orlicz functions, Southeast Asian Bulletin of Math., 31 (2007), 1107-1112. [21] V. A. Khan and S. Tabassum, Statistically pre-Cauchy double sequences, Southeast Asian Bulletin of Mathematics 36: (2012), 249–254. [22] V. A. Khan, K. Ebadullah and A. Ahmad, I pre-Cauchy sequences and Orlicz functions, Jour. of Mat. Anal. 3 (1) (2012), 21-26. [23] H. K¬zmaz, On certain sequence spaces, Canad. Math. Bull. 24(2) (1981), 169-176. ¼ [24] P. Kostyrko, T.Salát and W. Wilezy´n ski, I convergence, Real Anal. Exchange, 26, 2 (2000), 669-686. [25] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 101 (1971), 379-390. [26] H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. of the Amer. Math. Soc.Vol. 347, No.5, (1995), 1811-1819. [27] S. Ojha and P. D. Srivastava, I statistically pre-Cauchy sequences of fuzzy numbers, Annals of Fuzzy Math. and Informatics [28] S. Saha, B. Nath and S. Roy, I statistically pre-Cauchy triple sequences of fuzzy real numbers, Inter. Jour. of Management and Fuzzy Systems, 2 (2) (2016), 15-21. [29] I·. Sakao¼ g lu and M. Ünver, Statistical approximation for multivariable integrable functions, Miskolc Math. Notes 13 (2012), no.2, 485-491. [30] E. Sava¸s, On some new sequence spaces in 2-normed spaces using ideal convergence and Orlicz function, Jour. of Inequal. and Appl. (2010), 8 pages. [31] E. Sava¸s and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), 826-830. [32] I. J. Schoenberg, The integrability of certain functions and related summability methods, The American Mathematical Monthly 66 (1959), 361-375. [33] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloquium Athematicum 2 (1951), 73-74. [34] U. Yamanc¬and M. Gürdal, I statistically pre-Cauchy double sequences, Global Journal of Mathematical Analysis, 2 (4) (2014) 297-303.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.1, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

SOM E RESULTS ABOUT

I STATISTICALLY PRE-CAUCHY SEQUENCES W ITH AN ORLICZ FUNCTION9

[35] Zygmund, A., Trigonometric Series, Cam. Uni. Press, Cambridge, UK., (1979).

188

CUMUS ET AL 180-188

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 1, 2020

Gehring Inequalities On Time Scales, Martin Bohner and Samir H. Saker,………………11 Basin of Attraction of the Fixed Point and Period-Two Solutions of a Certain Anti-Competitive Map, Emin Bešo, Senada Kalabušić, and Esmir Pilav,……………………………………24 Fourier Approximation Schemes of Stochastic Pseudo-Hyperbolic Equations with Cubic Nonlinearity and Regular Noise, Tengjin Zhao, Chenzhong Wang, Quanyong Zhu, and Zhiyue Zhang,……………………………………………………………………………………..35 An Iterative Algorithm for Solving Split Feasibility Problems and Fixed Point Problems in p-Uniformly Convex and Smooth Banach Spaces, P. Chuasuk, A. Farajzadeh, and A. Kaewcharoen,……………………………………………………………………………..49 Expressions and Dynamical Behavior of Rational Recursive Sequences, E. M. Elsayed and Marwa M. Alzubaidi,……………………………………………………………………..67 Some Fixed Point Theorems of Non-Self Contractive Mappings in Complete Metric Spaces, Dangdang Wang, Chuanxi Zhu, and Zhaoqi Wu,…………………………………………79 Double-Framed Soft Sets in B-Algebras, Jung Mi Ko and Sun Shin Ahn,………………..85 Applications of Double Difference Fractional Order Operators to Originate Some Spaces of Sequences, Anu Choudhary and Kuldip Raj,………………………………………………94 On the Dynamics of Higher-Order Anti-Competitive System:𝑥𝑥𝑛𝑛+1 = 𝐴𝐴 + ∑𝑘𝑘 𝑥𝑥𝑛𝑛 𝑘𝑘 ∑𝑖𝑖=1 𝑦𝑦𝑛𝑛−𝑖𝑖

𝑦𝑦𝑛𝑛

𝑖𝑖=1 𝑥𝑥𝑛𝑛−𝑖𝑖

, 𝑦𝑦𝑛𝑛+1 = 𝐵𝐵 +

, A. Q. Khan, M. A. El-Moneam, E. S. Aly, and M. A. Aiyashi,…………………104

Stability of a Modified Within-Host HIV Dynamics Model with Antibodies, Ali Al-Qahtani, Shaban Aly, Ahmed Elaiw, and E. Kh. Elnahary,…………………………………………110 Fourier Series of Two Variable Higher-Order Fubini Functions, Lee Chae Jang, Gwan-Woo Jang, Dae San Kim, and Taekyun Kim,……………………………………………………121 Weighted Composition Operators from Dirichlet Type Spaces to Some Weighted-Type Spaces, Manisha Devi, Ajay K. Sharma, and Kuldip Raj,………………………………………….127 Oscillation Criteria for Differential Equations with Several Non-Monotone Deviating Arguments, G. M. Moremedi, H. Jafari, and I. P. Stavroulakis,………………………….136

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 1, 2020 (continued)

Hyers-Ulam Stability of Second-Order Nonhomogeneous Linear Difference Equations with a Constant Stepsize, Masakazu Onitsuka,……………………………………………………152 Choquet-Iyengar Type Advanced Inequalities, George A. Anastassiou,…………………..166 Some Results About Δ𝐼𝐼-Statistically Pre-Cauchy Sequences with an Orlicz Function, Hafize Gümüş, Ömer Kişi, and Ekrem Savaș,………………………………………………………180

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Contribution to a Book 3. M.K.Khan, Approximation properties of beta operators,in(title of book in italics) Progress in Approximation Theory (P.Nevai and A.Pinkus,eds.), Academic Press, New York,1991,pp.483-495.

11. All acknowledgements (including those for a grant and financial support) should occur in one paragraph that directly precedes the References section. 12. Footnotes should be avoided. When their use is absolutely necessary, footnotes should be numbered consecutively using Arabic numerals and should be typed at the bottom of the page to which they refer. Place a line above the footnote, so that it is set off from the text. Use the appropriate superscript numeral for citation in the text. 13. After each revision is made please again submit via email Latex and PDF files of the revised manuscript, including the final one. 14. Effective 1 Nov. 2009 for current journal page charges, contact the Editor in Chief. Upon acceptance of the paper an invoice will be sent to the contact author. The fee payment will be due one month from the invoice date. The article will proceed to publication only after the fee is paid. The charges are to be sent, by money order or certified check, in US dollars, payable to Eudoxus Press, LLC, to the address shown on the Eudoxus homepage. No galleys will be sent and the contact author will receive one (1) electronic copy of the journal issue in which the article appears.

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On invariance and solutions of some fifth-order rational recursive sequences M. Folly-Gbetoula

∗

and D. Nyirenda

†

Abstract We study the fifth-order difference equations of the form xn−4 xn−2 xn+1 = , n = 0, 1, . . . , xn−1 (an + bn xn−4 xn−2 ) where an and bn are real sequences, using the method of Lie group analysis. In particular, nontrivial vector fields associated with the group of point transformations are derived and exact solutions obtained. Closed form formulas for the solutions to the recursive sequences are given explicitly. This work is a generalization of a result by Elsayed [E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Computational Analysis and Applications, 15(1) (2013), 73–81].

Keywords: Difference equation; Symmetry; Reduction; Group invariant; Periodicity Mathematics Subjet Classification: 39A10; 39A13; 39A90

1

Introduction

Over a century ago, Sophus Lie [7] developed an algorithm based on the invariance of the ordinary differential equations under their symmetry group. Maeda [8, 9] observed that the Lie Symmetry approach can be applied to ordinary difference equations. Recently, Hydon [3] utilized a similar method to come up with some interest-provoking results. It is now a foregone conclusion that Lie’s method can be used to find symmetries and conservation laws of recursive sequences, even in the context of variational equations. In this paper, we obtain the vector fields of xn−4 xn−2 , (1) xn+1 = xn−1 (an + bn xn−4 xn−2 ) where an and bn are random real sequences, and then proceed to find the solutions in closed form. Our work extends the work by Elsayed [1], where the formulas of the solutions of the difference equations xn−4 xn−2 n = 0, 1, . . . , (2) xn+1 = xn−1 (± ± xn−4 xn−2 ) in which the initial conditions x−4 , x−3 , x−2 , x−1 , x0 are arbitrary non-zero real numbers, were obtained. For related work, see [2, 4, 10]. ∗

School of Mathematics, University of the Witwatersrand, Johannesburg, X3, Wits 2050, South Africa Email: [email protected] † School of Mathematics, University of the Witwatersrand, Johannesburg, X3, Wits 2050, South Africa Email: [email protected]

1

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1.1

Background on Lie analysis

In this section, we briefly discuss some key ideas on Lie group analysis of difference equations. For a broader comprehension of the concepts, refer to [3, 11]. The definitions and notation are taken from the same source [3, 11]. Let x? = X(x; ε) (3) be a one parameter Lie group of transformations. Definition 1.1 An infinitely differentiable function F is an invariant function of the Lie group of point transformation (3) if and only if, for any group transformations, F (x) = F (x? ).

(4)

Definition 1.2 The infinitesimal generator of the one-parameter Lie group of point transformation (3) is the operator n X ∂ , (5) X = X(x) = ξ(x) × ∆ = ξi (x) ∂xi i=1 where ∆ is the gradient operator. Theorem 1.1 F (x) is invariant under the Lie group of transformations (3) if and only if XF (x) = 0.

(6)

Consider the forward fifth-order recursive sequence un+5 = Φ(n, un , . . . , un+4 )

(7)

for some smooth function Φ. Suppose the one-parameter Lie group of point transformations is of the form n∗ =n,

u∗n+k = un+k + εS k ξ(n, un )+)(ε2 ),

k = 0, . . . , 5,

(8)

where ξ denotes the characteristic, ε (ε is small enough) is the group parameter and S : n 7→ n + 1 is the shift forward operator. The symmetry condition is given by u∗n+5 = Φ(n, u∗n , . . . , u∗n+4 ),

(9)

whenever (7) is true. The substitution of (8) in (9) yields the linearized symmetry condition: S 5 ξ(n, un ) − XΦ = 0

(10)

where X, the vector fields of (7), is given by ∂ ∂ ∂ X =ξ(n, un ) + ξ(n + 1, un+1 ) + · · · + ξ(n + 4, un+4 ) . (11) ∂un ∂un+1 ∂un+4 Despite the fact that (10) looks simple, its solution finding process is highly involving. In our work, we will use the canonical coordinate [5] Z dun Sn = (12) ξ(n, un ) to lower the order of the difference equation under investigation. 2

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2

Main results

Let un+5 = Φ =

un un+2 , un+3 (An + Bn un un+2 )

(13)

where An and Bn are random real sequences, be the forward recursive equation equivalent to (2). Substituting (13) in (10), we have that S 5ξ +

An un (Sξ) un un+2 (S 3 ξ) An un+2 ξ − − = 0. 2 2 un+3 (An + Bn un un+2 ) un+3 (An + Bn un un+2 ) un+3 (An + Bn un un+2 )2 (14)

We act the differential operator L=

Φun ∂ ∂ − ∂un Φun+3 ∂un+3

to eliminate the first term in (14). This leads to   An (An + Bn un un+2 ) (S 3 ξ)0 − (S 3 ξ) + Bn un (Sξ) − (An + Bn un un+2 )ξ 0 + ξ=0 un

(15)

after simplification. The differentiation of (15) with respect to un twice, keeping un+3 fixed, yields − (An + Bn un un+2 )ξ (3) +

An (2) 2An 0 2An ξ − 2 ξ + 3 ξ = 0. un un un

(16)

Split (16) by comparing powers of un+2 ; we have ( un+2 term : other terms :

u3n ξ (3) − un 2 ξ (2) + 2un ξ 0 − 2ξ = 0, ξ (3) = 0.

(17)

Equations in (17) further simplify to un 2 ξ (2) − 2un ξ 0 + 2ξ = 0.

(18)

It is clear that the solution of (16) is ξ (n, un ) = fn un + gn un 2

(19)

for some arbitrary functions fn and gn of n. Using characteristic’s expression as given in (19), we reduce equation (14) to the following difference equation Bn gn+3 un un+2 un+3 2 + Bn (fn+3 + fn+5 )un un+2 un+3 − An gn un un+3 + gn+5 un un+2 − An (fn + fn+2 + fn+3 + fn+5 )un+3 − An gn+1 un+2 un+3 + An gn+3 un+3 2 = 0.

(20)

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which then splits into a system (by comparing products of powers of shifts of un ) as follows: un+3 un un+2 un un+3 un un+2 un+3 un un+2 un+3 2 un+2 un+3 un+3 2

terms : fn + fn+2 + fn+3 + fn+5 = 0 terms : gn+5 = 0 terms : gn = 0 terms : fn+3 + fn+5 = 0 terms : gn+3 = 0 terms : gn+1 = 0 terms : gn+3 = 0.

(21a) (21b) (21c) (21d) (21e) (21f) (21g)

Thus, the ‘final constraint’ is given by: fn + fn+2 = 0, gn = 0.

(22a) (22b)

Solving (22) for f , we obtain two independent solutions given by exp(±nπ/2). Therefore, the characteristics are ξ1 =αn un , ξ2 = α ¯ n un ,

(23)

and so the prolonged infinitesimal generators admitted by (13) are X1 =αn un ∂un + αn+1 un+1 ∂un+1 + αn+2 un+2 ∂un+2 + αn+3 un+3 ∂un+3 + αn+4 un+4 ∂un+4 , (24a) X2 =¯ αn un ∂un + α ¯ n+1 un+1 ∂un+1 + α ¯ n+2 un+2 ∂un+2 + α ¯ n+3 un+3 ∂un+3 + α ¯ n+4 un+4 ∂un+4 . (24b) Observe that α = exp(iπ/2) and α ¯ is its complex conjugate. Using the generator X1 , we have the canonical coordinate Z 1 dun Sn = = n ln |un |. (25) n α un α Thanks to the form of (22), the invariant function V˜n is constructed as follows V˜n = Sn αn + Sn+2 αn+2

(26)

since X1 V˜n = αn + αn+2 = 0 and X2 V˜n = α ¯n + α ¯ n+2 = 0. For rational difference equations, it is convenience to use |Vn | = exp{−V˜n },

(27)

i.e., Vn = ±1/(un un+2 ) but we will be using the one with plus sign: Vn = 1/un un+2 . We then substitute (27) into equation (13) to get the third-order linear difference equation Vn+3 = An Vn + Bn .

(28)

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The iteration of equation (28) leads to ! n−1 ! n−1 n−1 Y X Y V3n+j =Vj A3k1 +j + B3l+j A3k2 +j , k1 =0

l=0

j = 0, 1, 2.

(29)

k2 =l+1

Invoking (25), (26) and (27), we have that |un | = exp (αn Sn ) n−1 n−1 1 X n k2 ˜ 1 X n k1 ˜ n n α α ¯ Vk 1 − α ¯ α Vk2 = exp α c1 + α ¯ c2 − 2 k =0 2 k =0 1

!

2

! n−1 X 1 αn α ¯ k1 ln |Vk1 | + α ¯ n αk2 ln |Vk2 | 2 k =0 1 =0 2 !

n−1 X

1 = exp αn c1 + α ¯ n c2 + 2k = exp Hn +

n−1 X

Re(γ(n, k1 )) ln |Vk1 | ,

(30)

k1 =0

in which Hn = αn c1 + α ¯ n c2 and γ(n, k) = αn α ¯k . It is worthwhile to mention that the function γ satisfies the following: γ(0, 1) = α ¯ , γ(1, 0) = α, γ(n, n) = 1, γ(n + 2, k) = −γ(n, k), γ(n, k + 2) = −γ(n, k), γ(4n, k) = γ(0, k), γ(n, 4k) = γ(n, 0).

(31)

From the expression of un given in (30) and from the above properties (31), note that ! 4n+j−1 X |u4n+j | = exp Hj + Re(γ(j, k1 )) ln |Vk1 | . (32) k1 =0

For j = 0, we have |u4n | = exp(H0 + ln |V0 | − ln |V2 | + . . . + ln |V4n−4 | − ln |V4n−2 |) n−1 Y V4s = exp(H0 ) V4s+2 . s=0

(33)

By setting n = 0 in (30), we get exp(H0 ) = u0 and so u4n = u0

n−1 Y s=0

V4s . V4s+2

(34)

We have omitted the absolute function because it can be shown, using (27), that there is no need for it. In a similar way, we have that u4n+j =uj

n−1 Y s=0

V4s+j , for any j = 0, 1, 2, 3. V4s+j+2

(35)

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This equation implies that u12n+j = uj = uj

3n−1 Y s=0 n−1 Y s=0

V4s+j V4s+j+2

V12s+j V12s+4+j V12s+j+8 V12s+j+2 V12s+j+6 V12s+j+10

which now holds for j = 0, 1, 2, . . . , 11. For j = 0, we have u12n = u0

n−1 Y s=0

V12s V12s+4 V12s+8 . V12s+2 V12s+6 V12s+10

(36)

Using (29) in (36), we have that

u12n = u0

V0

n−1 Y s=0

4s−1 Q

A3k1 +

k1 =0

V2

4s−1 Q

4s−1 P l=0

A3k1 +2 +

4s−1 P

k1 =0

4s−1 Q

B3l

4s Q

V1

A3k2

k1 =0

k2 =l+1 4s−1 Q

B3l+2

l=0

A3k1 +1 +

A3k2 +2

V0

k2 =l+1

4s+1 Q

A3k1 + k1 =0 

4s P

4s Q

B3l+1

l=0

A3k2 +1

k2 =l+1

4s+1 P

B3l

l=0

4s+1 Q

A3k2

k2 =l+1

 4s+1 Q V2 A3k1 +2 + B3l+2 A3k2 +2 k1 =0 l=0 k2 =l+1   × 4s+2 4s+2 4s+2 P Q Q V1 A3k1 +1 + B3l+1 A3k2 +1

= u0

n−1 Y s=0

4s−1 Q

4s+1 Q

4s+1 P

k1 =0

l=0

A3k1 + u0 u2

k1 =0 4s−1 Q k1 =0

4s−1 P l=0

A3k1 +2 + u2 u4

4s−1 P

4s−1 Q

B3l

k2 =l+1

4s Q

A3k2

k1 =0

k2 =l+1 4s−1 Q

A3k2 +2

4s+1 Q

l=0

B3l+1

l=0

A3k1 + u0 u2 l=0 k2 =l+1 k1 =0   4s+1 4s+1 4s+1 P Q Q A3k1 +2 + u2 u4 B3l+2 A3k2 +2 k1 =0 l=0 k2 =l+1  . × 4s+2 4s+2 4s+2 Q P Q A3k1 +1 + u1 u3 B3l+1 A3k2 +1 k1 =0

B3l+2

4s P

A3k1 +1 + u1 u3

4s+1 P l=0

4s Q

A3k2 +1

k2 =l+1

B3l

4s+1 Q

A3k2

k2 =l+1

k2 =l+1

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Hence x12n−4 is equal to 4s 4s 4s 4s−1 4s−1 4s−1 Q P Q Q P Q a + x x b a3k2 +1 a + x x b a 3k1 +1 −3 −1 3l+1 −4 −2 3l 3k2 n−1 Y k1 =0 3k1 k1 =0 l=0 k2 =l+1 l=0 k2 =l+1 x−4 4s−1 4s−1 4s−1 4s+1 4s+1 4s+1 Q P Q Q P Q s=0 a3k1 +2 + x−2 x0 b3l+2 a3k2 +2 a3k1 + x−4 x−2 b3l a3k2 k1 =0 l=0 k2 =l+1 k1 =0 l=0 k2 =l+1   4s+1 4s+1 4s+1 Q Q P b3l+2 a3k2 +2 a3k1 +2 + x−2 x0 k2 =l+1 k1 =0 l=0  . × 4s+2 4s+2 4s+2 Q Q P b3l+1 a3k2 +1 a3k1 +1 + x−3 x−1 k1 =0

k2 =l+1

l=0

For j = 1, we have 4s−1 4s−1 4s−1 4s 4s 4s Q P Q Q P Q V A + B A V A + B A n−1 Y 1 k1 =0 3k1 +1 l=0 3l+1 k2 =l+1 3k2 +1 2 k1 =0 3k1 +2 l=0 3l+2 k2 =l+1 3k2 +2

u12n+1 = u1

4s Q

V0

s=0

4s P

A3k1 +

k1 =0

V0

4s+2 Q

l=0 4s+2 P

A3k1 +

k1 =0

× V2

4s+2 Q

A3k1 +2 +

4s+2 P

k1 =0

A3k2

V1

4s+2 Q

4s+1 Q

A3k1 +1 +

4s+1 P

k1 =0

k2 =l+1

B3l

l=0

4s Q

B3l

B3l+1

l=0

4s+1 Q

A3k2 +1

k2 =l+1

A3k2

k2 =l+1 4s+2 Q

B3l+2

l=0

A3k2 +2

k2 =l+1

so that x12n−3 is equal to 4s−1 4s−1 4s−1 4s 4s 4s Q P Q Q P Q a + x x b a a + x x b A3k2 +2 −3 −1 3l+1 3k2 +1 3k1 +2 −2 0 3l+2 n−1 Y k1 =0 3k1 +1 l=0 k2 =l+1 k1 =0 l=0 k2 =l+1 x−3 4s 4s+1 4s+1 4s+1 4s 4s Q Q P Q P Q s=0 a3k1 + x−4 x−2 b3l A3k1 +1 + x−3 x−1 b3l+1 a3k2 +1 a3k2 k1 =0 4s+2 Q

×

l=0

a3k1 + x−4 x−2

k1 =0 4s+2 Q

4s+2 P

4s+2 Q

b3l

l=0

a3k1 +2 + x−2 x0

k1 =0

4s+2 P

k1 =0

k2 =l+1

b3l+2

k2 =l+1

a3k2

k2 =l+1

l=0

l=0

4s+2 Q

. a3k2 +2

k2 =l+1

For j = 2, we have

u12n+2 = u2

n−1 Y s=0

V2

4s−1 Q

A3k1 +2 +

4s−1 P

k1 =0

V1

4s Q

l=0 4s P

A3k1 +1 +

k1 =0

V1 ×

4s+2 Q

A3k1 +1 +

k1 =0

V0

4s+3 Q k1 =0

l=0

A3k2 +1 V2

k2 =l+1

B3l+2

4s+3 P

4s Q

B3l+1

l=0

A3k1 +

A3k2 +2

k2 =l+1

l=0 4s+2 P

4s−1 Q

B3l+2

4s+2 Q

V0

4s+1 Q

A3k1 +

k1 =0 4s+1 Q k1 =0

A3k1 +2 +

4s+1 P l=0

4s+1 P l=0

4s+1 Q

B3l

A3k2

k2 =l+1

B3l+2

4s+1 Q

A3k2 +2

k2 =l+1

A3k2 +1

k2 =l+1

B3l

4s+3 Q

A3k2

k2 =l+1

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so that

x12n−2 = x−2

4s−1 Q

n−1 Y

k1 =0

s=0

4s Q

4s−1 P

a3k1 +2 + x−2 x0

l=0

a3k1 +1 + x−3 x−1

k1 =0 4s+2 Q

×

4s+2 P

k1 =0

4s P

a3k1 + x−4 x−2

4s+2 Q 4s+3 Q

b3l

a3k1 + x−4 x−2

4s+1 Q

4s+1 P

b3l

l=0 4s+1 P

a3k1 +2 + x−2 x0

k1 =0

4s+1 Q

a3k2

k2 =l+1 4s+1 Q

b3l+2

l=0

k2 =l+1

a3k2 +1

k2 =l+1

4s+3 P

.

a3k2

k2 =l+1

l=0

k1 =0

a3k2 +1

k2 =l+1

b3l+2

4s+1 Q k1 =0

4s Q

b3l+1

l=0

4s+3 Q

a3k2 +2

k2 =l+1

l=0

a3k1 +1 + x−3 x−1

4s−1 Q

b3l+2

Following similar substitutions as above where ui = xi−4 and Vi = for x12n+j−4 with j = 3, 4, 5, . . . , 11;

1 , xi−4 xi−2

we deduce that

x12n−1 =

x−1

4s Q

n−1 Y s=0

a3k1 + x−4 x−2

k1 =0 4s Q

4s P

a3k1 +2 + x−2 x0

a3k1 +2 + x−2 x0

a3k1 +1 + x−3 x−1

b3l+1

l=0

b3l+1

a3k1 + x−4 x−2

4s+1 Q

a3k2 +1

k2 =l+1

4s+2 P

4s+2 Q

b3l

l=0

a3k2

k2 =l+1

a3k2 +2

k2 =l+1

4s+3 P

4s+1 P l=0

k1 =0

4s+2 Q

b3l+2

a3k1 +1 + x−3 x−1

4s+2 Q

a3k2 +2

k2 =l+1

l=0

k1 =0

4s+1 Q k1 =0

4s Q

b3l+2

4s+2 P

k1 =0

a3k2

k2 =l+1

l=0

4s+2 Q 4s+3 Q

4s Q

b3l

l=0

k1 =0

×

4s P

4s+3 Q

, a3k2 +1

k2 =l+1

x12n =

x0

n−1 Y

4s Q k1 =0

s=0

4s+1 Q

a3k1 + x−4 x−2

4s+3 Q 4s+3 Q k1 =0

a3k1 +2 + x−2 x0

4s+1 Q

b3l

l=0

k1 =0

b3l

l=0

l=0

a3k2

b3l+2

4s+3 Q

a3k1 +2 + x−2 x0

4s+1 P

k1 =0 4s+2 Q k1 =0

k2 =l+1

4s+3 P

4s+3 P

4s+1 Q

a3k2 +1

k2 =l+1

4s+1 P

a3k1 + x−4 x−2

4s Q

b3l+1

l=0

k1 =0

×

4s P

a3k1 +1 + x−3 x−1

a3k1 +1 + x−3 x−1

a3k2 +2

k2 =l+1

l=0 4s+2 P l=0

4s+1 Q

b3l+2 b3l+1

4s+2 Q

a3k2 +1

k2 =l+1

a3k2

k2 =l+1 4s+3 Q

, a3k2 +2

k2 =l+1

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x12n+1 = 4s 4s+2 4s 4s+2 4s 4s+2 Q Q P P Q Q a a3k2 b b a + x x a + x x 3k +2 3l+2 3l 3k +2 −2 0 3k −4 −2 n−1 2 1 1 Y k1 =0 k2 =l+1 k2 =l+1 l=0 l=0 k1 =0 x1 4s+1 4s+1 4s+1 4s+2 4s+2 4s+2 Q P Q Q P Q s=0 a3k1 +1 + x−3 x−1 b3l+1 a3k2 +1 a3k1 +2 + x−2 x0 b3l+2 a3k2 +2 k1 =0

l=0

4s+3 Q

×

a3k1 +1 + x−3 x−1

k1 =0

k2 =l+1

4s+3 P

b3l+1

l=0

4s+4 Q

a3k1 + x−4 x−2

4s+4 Q

b3l

l=0

l=0

k2 =l+1

a3k2 +1

k2 =l+1

4s+4 P

k1 =0

k1 =0

4s+3 Q

,

a3k2

k2 =l+1

x12n+2 =

x2

4s+1 Q

n−1 Y s=0

a3k1 + x−4 x−2

4s+1 P

k1 =0 4s+1 Q

4s+1 P

k1 =0

×

4s+1 Q

b3l+2

l=0

4s+3 Q

a3k1 +2 + x−2 x0

4s+4 Q

a3k1 +1 + x−3 x−1

4s+4 P

4s+3 Q

a3k2 +2

k1 =0

b3l+1

4s+2 P

b3l+1

l=0

a3k1 + x−4 x−2

4s+2 Q

a3k2 +1

k2 =l+1

4s+3 P l=0

b3l

4s+3 Q

a3k2

k2 =l+1

a3k2 +2

k2 =l+1

l=0

a3k1 +1 + x−3 x−1

k1 =0

4s+3 Q

b3l+2

l=0

k1 =0

4s+2 Q

k2 =l+1

4s+3 P

k1 =0

a3k2

k2 =l+1

l=0

a3k1 +2 + x−2 x0

4s+1 Q

b3l

4s+4 Q

, a3k2 +1

k2 =l+1

x12n+3 = 4s+1 4s+1 4s+1 4s+2 4s+2 4s+2 Q P Q Q P Q a + x x b a a3k2 +2 b a + x x 3k +1 −3 −1 3l+1 3k +1 3l+2 3k +2 −2 0 n−1 1 2 1 Y k1 =0 l=0 k2 =l+1 k2 =l+1 l=0 k1 =0 x3 4s+2 4s+3 4s+3 4s+3 4s+2 4s+2 Q Q P Q P Q s=0 a3k1 + x−4 x−2 a3k1 +1 + x−3 x−1 b3l+1 a3k2 +1 b3l a3k2 k1 =0 4s+4 Q

×

l=0

a3k1 + x−4 x−2

k1 =0 4s+4 Q k1 =0

a3k1 +2 + x−2 x0

4s+4 P

b3l

l=0 4s+4 P l=0

k1 =0

k2 =l+1

b3l+2

4s+4 Q

l=0

k2 =l+1

a3k2

k2 =l+1 4s+4 Q

, a3k2 +2

k2 =l+1

9

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x12n+4 = 4s+1 4s+3 4s+1 4s+3 4s+1 4s+3 Q Q P P Q Q a a3k2 b b a + x x a + x x 3k +2 3l+2 3l 3k +2 −2 0 3k −4 −2 n−1 2 1 1 Y k1 =0 k2 =l+1 k2 =l+1 l=0 l=0 k1 =0 x4 4s+2 4s+2 4s+2 4s+3 4s+3 4s+3 Q P Q Q P Q s=0 a3k1 +1 + x−3 x−1 b3l+1 a3k2 +1 a3k1 +2 + x−2 x0 b3l+2 a3k2 +2 k1 =0

l=0

4s+4 Q

×

a3k1 +1 + x−3 x−1

k1 =0

k2 =l+1

4s+4 P

b3l+1

l=0

4s+5 Q

a3k1 + x−4 x−2

4s+5 Q

b3l

l=0

l=0

k2 =l+1

a3k2 +1

k2 =l+1

4s+5 P

k1 =0

k1 =0

4s+4 Q

,

a3k2

k2 =l+1

x12n+5 =

x5

4s+2 Q

n−1 Y s=0

a3k1 + x−4 x−2

4s+2 P

k1 =0 4s+2 Q

4s+2 P

k1 =0

×

4s+2 Q

b3l+2

l=0

4s+4 Q

a3k1 +2 + x−2 x0

4s+5 Q

a3k1 +1 + x−3 x−1

4s+5 P

4s+4 Q

a3k2 +2

k1 =0

b3l+1

4s+3 P

b3l+1

l=0

a3k1 + x−4 x−2

4s+3 Q

a3k2 +1

k2 =l+1

4s+4 P l=0

b3l

4s+4 Q

a3k2

k2 =l+1

a3k2 +2

k2 =l+1

l=0

a3k1 +1 + x−3 x−1

k1 =0

4s+4 Q

b3l+2

l=0

k1 =0

4s+3 Q

k2 =l+1

4s+4 P

k1 =0

a3k2

k2 =l+1

l=0

a3k1 +2 + x−2 x0

4s+2 Q

b3l

4s+5 Q

, a3k2 +1

k2 =l+1

x12n+6 = 4s+2 4s+2 4s+2 4s+3 4s+3 4s+3 Q P Q Q P Q a + x x b a a3k2 +2 b a + x x 3k +1 −3 −1 3l+1 3k +1 3l+2 3k +2 −2 0 n−1 1 2 1 Y k1 =0 l=0 k2 =l+1 k2 =l+1 l=0 k1 =0 x6 4s+3 4s+4 4s+4 4s+4 4s+3 4s+3 Q Q P Q P Q s=0 a3k1 + x−4 x−2 a3k1 +1 + x−3 x−1 b3l+1 a3k2 +1 b3l a3k2 k1 =0 4s+5 Q

×

l=0

a3k1 + x−4 x−2

k1 =0 4s+5 Q k1 =0

a3k1 +2 + x−2 x0

4s+5 P

b3l

l=0 4s+5 P l=0

k1 =0

k2 =l+1

b3l+2

4s+5 Q

l=0

k2 =l+1

a3k2

k2 =l+1 4s+5 Q

, a3k2 +2

k2 =l+1

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x12n+7 = 4s+2 4s+4 4s+2 4s+4 4s+2 4s+4 Q Q P P Q Q a a3k2 b b a + x x a + x x 3k +2 3l+2 3l 3k +2 −2 0 3k −4 −2 n−1 2 1 1 Y k1 =0 k2 =l+1 k2 =l+1 l=0 l=0 k1 =0 x7 4s+3 4s+3 4s+3 4s+4 4s+4 4s+4 Q P Q Q P Q s=0 a3k1 +1 + x−3 x−1 b3l+1 a3k2 +1 a3k1 +2 + x−2 x0 b3l+2 a3k2 +2 k1 =0

l=0

4s+5 Q

×

a3k1 +1 + x−3 x−1

k1 =0

4s+5 P

k2 =l+1

b3l+1

l=0

4s+6 Q

a3k1 + x−4 x−2

k1 =0

k1 =0

4s+5 Q 4s+6 Q

b3l

l=0

k2 =l+1

a3k2 +1

k2 =l+1

4s+6 P

l=0

,

a3k2

k2 =l+1

where x1 , x2 , x3 , x4 , x5 , x6 and x7 are given as follows:

x1 = x4 =

x−4 x−2 x−3 x−1 x−1 x0 (a0 + b0 x−4 x−2 ) , x2 = , x3 = , x−1 (a0 + b0 x−4 x−2 ) x0 (a1 + b1 x−3 x−1 ) x−4 (a2 + b2 x−2 x0 )

x−4 x−2 x0 (a1 + b1 x−3 x−1 ) x−3 x−4 (a2 + b2 x−2 x0 ) , x5 = , x−3 x−1 (a0 a3 + (b0 a3 + b3 )x−4 x−2 ) x0 (a0 + b0 x−4 x−2 )(a1 a4 + (b1 a4 + b4 )x−3 x−1 ) x6 =

x−3 x−1 (a0 a3 + (b0 a3 + b3 )x−4 x−2 ) , x−4 (a1 + b1 x−3 x−1 )(a5 a2 + (b2 a5 + b5 )x−2 x0 )

and x7 =

x−2 x0 (a0 + b0 x−4 x−2 )(a1 a4 + (b1 a4 + b4 )x−3 x−1 ) . x−3 (a2 + b2 x−2 x0 )(a6 a3 a0 + (a6 a3 b0 + a6 b3 + b6 )x−4 x−2 )

We now turn our attention to special cases in the subsequent sections.

3

The case an and bn are 1-periodic

Let an = a and bn = b, where a, b ∈ R. We simply carry out a substitution and find the following solution:

x12n−4 = x−4

n−1 Y s=0

a4s + bx−4 x−2 a4s

+ bx−2 x0

4s−1 P

l=0 4s−1 P

al a4s+1 + bx−3 x−1

al

a4s+2

+ bx−4 x−2

x12n−3 = x−3

s=0

a4s + bx−3 x−1 a4s+1

4s−1 P

al a4s+2 + bx−2 x0

l=0 4s+1 P

l=0

n−1 Y

4s P

al

a4s+3

4s+1 P

+ bx−3 x−1

l=0

al

l=0 4s P

+ bx−4 x−2

al

a4s+1 + bx−2 x0 a4s+2

l=0

+ bx−3 x−1

4s P

al

l=0 4s+2 P

, al

l=0

al a4s+3 + bx−4 x−2

l=0 4s+1 P l=0

al

a4s+3

+ bx−2 x0

4s+2 P

l=0 4s+2 P

al ,

al

l=0

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x12n−2 = x−2

n−1 Y

a4s + bx−2 x0 a4s+1

s=0

4s−1 P

4s+1 P

al a4s+2 + bx−4 x−2

l=0 4s P

al

+ bx−3 x−1

a4s+2

+ bx−2 x0

l=0 4s+1 P

l=0

x12n−1 = x−1

n−1 Y s=0

a4s+1 + bx−4 x−2 a4s+1

+ bx−2 x0

4s P

al

a4s+3

x12n = x0

s=0

a4s+2

+ bx−4 x−2

al

a4s+3

4s+1 P

x12n+1 = x1

s=0

a4s+1 + bx−2 x0 a4s+2

+ bx−3 x−1

4s P

+ bx−3 x−1

al

a4s+3

+ bx−2 x0

x12n+2 = x2

s=0

a4s+2 + bx−2 x0

l=0 4s+1 P

x12n+3 = x3

s=0

a4s+3 + bx−4 x−2

x12n+4 = x4

a4s+2 + bx−2 x0

s=0 a4s+3

l=0 4s+2 P

4s+1 P

+ bx−3 x−1

al a4s+4 + bx−3 x−1

x12n+5 = x5

s=0

a4s+3 + bx−4 x−2

l=0 4s+2 P

a4s+3

+ bx−2 x0

a4s+5

+ bx−4 x−2

al a4s+4 + bx−2 x0

al a4s+5 + bx−3 x−1

al

a4s+4

+ bx−2 x0

al a4s+4 + bx−3 x−1

al

a4s+5

l=0

+ bx−4 x−2

4s+3 P

4s+3 P l=0 4s+4 P

al , al

al

l=0 4s+4 P

4s+4 P

al a4s+5 + x−4 x−2 al a4s+5 + bx−2 x0

, al

l=0 4s+4 P

l=0 4s+4 P

al , al

l=0

al a4s+5 + bx−3 x−1

al

4s+3 P

l=0

, al

l=0

l=0 4s+3 P

l=0 4s+3 P

al

4s+3 P

a4s+6

+ bx−4 x−2

l=0

4s+2 P

l=0 4s+2 P

+ bx−2 x0

l=0

al a4s+4 + bx−4 x−2

, al

l=0

4s+2 P

al a4s+3 + bx−2 x0

l=0

n−1 Y

a4s+4

l=0 4s+3 P

l=0

4s+1 P

al

l=0 4s+3 P

4s+3 P

al a4s+4 + bx−3 x−1

al

l=0 4s+3 P

al a4s+4 + bx−4 x−2

, al

l=0

4s+2 P

al a4s+3 + bx−3 x−1

l=0

n−1 Y

al

l=0 4s+2 P

l=0

4s+2 + bx−3 x−1 n−1 Ya

+ bx−3 x−1

l=0

4s+1 P

al

l=0

4s+2 P

al a4s+3 + bx−4 x−2

l=0 4s+1 P

a4s+2 + bx−4 x−2

a4s+4

l=0

l=0

n−1 Y

al

al a4s+4 + bx−4 x−2

l=0 4s+2 P

l=0 4s+3 P

4s+2 P

l=0

l=0

n−1 Y

+ bx−4 x−2

al a4s+3 + bx−2 x0

l=0 4s+2 P

+ bx−4 x−2

al a4s+2 + bx−2 x0

l=0 4s+1 P

a4s+4

4s+2 P

l=0

4s+1 P

al a4s+2 + bx−3 x−1

l=0 4s P

4s P

a4s+1 + bx−3 x−1

al

l=0

l=0

n−1 Y

al a4s+3 + bx−3 x−1

4s+4 P l=0 4s+5 P

al , al

l=0

al a4s+5 + bx−2 x0 al

a4s+6

4s+4 P

+ bx−3 x−1

al

l=0 4s+5 P

, al

l=0

12

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x12n+6 = x6

n−1 Y s=0

a4s+3 + bx−3 x−1 a4s+4

+ bx−4 x−2

4s+2 P l=0 4s+3 P

al a4s+4 + bx−2 x0 al

a4s+5

4s+3 P

+ bx−3 x−1

l=0

x12n+7 = x7

n−1 Y

a4s+3 + bx−2 x0

s=0 a4s+4

4s+2 P

+ bx−3 x−1

al a4s+6 + bx−4 x−2

l=0 4s+4 P

al

a4s+6

+ bx−2 x0

l=0 4s+5 P

l=0

al a4s+5 + bx−4 x−2

l=0 4s+3 P

al

a4s+5

+ bx−2 x0

l=0

al ,

al

l=0

4s+4 P

l=0 4s+4 P

4s+5 P

al a4s+6 + bx−3 x−1

al

a4s+7

+ bx−4 x−2

l=0

4s+5 P l=0 4s+6 P

al , al

l=0

where x1 , x2 , x3 , x4 , x5 , x6 , x7 are given by

x1 = x4 =

x−4 x−2 , x−1 (a + bx−4 x−2 )

x−3 x−1 , x0 (a + bx−3 x−1 )

x−4 x−2 x0 (a + bx−3 x−1 ) , x−3 x−1 (a2 + (ab + b)x−4 x−2 ) x6 =

and x7 =

3.1

x2 =

x5 =

x3 =

x−1 x0 (a + bx−4 x−2 ) , x−4 (a + bx−2 x0 )

x−4 x−3 (a + bx−2 x0 ) , x0 (a + bx−4 x−2 )(a2 + (ab + b)x−3 x−1 )

x−3 x−1 (a2 + (ab + b)x−4 x−2 ) x−4 (a + bx−3 x−1 )(a2 + (ab + b)x−2 x0 )

x−2 x0 (a + bx−4 x−2 )(a2 + (ab + b)x−3 x−1 ) . x−3 (a + bx−2 x0 )(a3 + (a2 b + ab + b)x−4 x−2 )

The case a = 1

The solution, which appears for b = ±1 in Theorems 1 and 6 of [1], is given by x12n−4 = x−4

n−1 Y s=0

x12n−3 = x−3

n−1 Y s=0

x12n−2 = x−2

n−1 Y s=0

x12n−1 = x−1

n−1 Y s=0

1 + 4sbx−4 x−2 1 + (4s + 1)bx−3 x−1 1 + (4s + 2)bx−2 x0 , 1 + 4sbx−2 x0 1 + (4s + 2)bx−4 x−2 1 + (4s + 3)bx−3 x−1

1 + (4s + 1)bx−2 x0 1 + (4s + 3)bx−4 x−2 1 + 4sbx−3 x−1 , 1 + (4s + 1)bx−4 x−2 1 + (4s + 2)bx−3 x−1 1 + (4s + 3)bx−2 x0

1 + 4sbx−2 x0 1 + (4s + 2)bx−4 x−2 1 + (4s + 3)bx−3 x−1 , 1 + (4s + 1)bx−3 x−1 1 + (4s + 2)bx−2 x0 1 + (4s + 4)bx−4 x−2

1 + (4s + 1)bx−4 x−2 1 + (4s + 2)bx−3 x−1 1 + (4s + 3)bx−2 x0 , 1 + (4s + 1)bx−2 x0 1 + (4s + 3)bx−4 x−2 1 + (4s + 4)bx−3 x−1 13

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x12n = x0

n−1 Y s=0

x12n+1 = x1

1 + (4s + 1)bx−3 x−1 1 + (4s + 2)bx−2 x0 1 + (4s + 4)bx−4 x−2 , 1 + (4s + 2)bx−4 x−2 1 + (4s + 3)bx−3 x−1 1 + (4s + 4)bx−2 x0

n−1 Y s=0

x12n+2 = x2

n−1 Y s=0

x12n+3 = x3

n−1 Y s=0

x12n+4 = x4

n−1 Y s=0

x12n+5 = x5

n−1 Y s=0

x12n+6 = x6

n−1 Y s=0

x12n+7 = x7

n−1 Y s=0

1 + (4s + 1)bx−2 x0 1 + (4s + 3)bx−4 x−2 1 + (4s + 4)bx−3 x−1 , 1 + (4s + 2)bx−3 x−1 1 + (4s + 3)bx−2 x0 1 + (4s + 5)bx−4 x−2

1 + (4s + 2)bx−4 x−2 1 + (4s + 3)bx−3 x−1 1 + (4s + 4)bx−2 x0 , 1 + (4s + 2)bx−2 x0 1 + (4s + 4)bx−4 x−2 1 + (4s + 5)bx−3 x−1

1 + (4s + 2)bx−3 x−1 1 + (4s + 3)bx−2 x0 1 + (4s + 5)bx−4 x−2 , 1 + (4s + 3)bx−4 x−2 1 + (4s + 4)bx−3 x−1 1 + (4s + 5)bx−2 x0

1 + (4s + 2)bx−2 x0 1 + (4s + 4)bx−4 x−2 1 + (4s + 5)bx−3 x−1 , 1 + (4s + 3)bx−3 x−1 1 + (4s + 4)bx−2 x0 1 + (4s + 6)bx−4 x−2

1 + (4s + 3)bx−4 x−2 1 + (4s + 4)bx−3 x−1 1 + (4s + 5)bx−2 x0 , 1 + (4s + 3)bx−2 x0 1 + (4s + 5)bx−4 x−2 1 + (4s + 6)bx−3 x−1

1 + (4s + 3)bx−3 x−1 1 + (4s + 4)bx−2 x0 1 + (4s + 6)bx−4 x−2 , 1 + (4s + 4)bx−4 x−2 1 + (4s + 5)bx−3 x−1 1 + (4s + 6)bx−2 x0

1 + (4s + 3)bx−2 x0 1 + (4s + 5)bx−4 x−2 1 + (4s + 6)bx−3 x−1 , 1 + (4s + 4)bx−3 x−1 1 + (4s + 5)bx−2 x0 1 + (4s + 7)bx−4 x−2

where x1 , x2 , x3 , x4 , x5 , x6 , x7 are given by

x1 =

x−4 x−2 , x−1 (1 + bx−4 x−2 )

x4 = x6 =

x2 =

x−3 x−1 , x0 (1 + bx−3 x−1 )

x−4 x−2 x0 (1 + bx−3 x−1 ) , x−3 x−1 (1 + 2bx−4 x−2 )

x−3 x−1 (1 + 2bx−4 x−2 ) x−4 (1 + bx−3 x−1 )(1 + 2bx−2 x0 )

x5 =

x3 =

x−1 x0 (1 + bx−4 x−2 ) , x−4 (1 + bx−2 x0 )

x−4 x−3 (1 + bx−2 x0 ) , x0 (1 + bx−4 x−2 )(1 + 2bx−3 x−1 )

and x7 =

x−2 x0 (1 + bx−4 x−2 )(1 + 2bx−3 x−1 ) . x−3 (1 + bx−2 x0 )(1 + 3bx−4 x−2 )

14

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3.2

The case a = −1

The solution, which appears for b = ± in Theorems 3 and 8 of [1], is given by x12n−4 = x−4 , x12n−3 = x−3 , x12n−2 = x−2 , x12n−1 = x−1 , x12n = x0 , x12n+1 =

x−4 x−2 x−3 x−1 x−1 x0 (−1 + bx−4 x−2 ) , x12n+2 = , x12n+3 = , x−1 (−1 + bx−4 x−2 ) x0 (−1 + bx−3 x−1 ) x−4 (−1 + bx−2 x0 ) x−4 x−2 x0 (−1 + bx−3 x−1 ) x−4 x−3 (−1 + bx−2 x0 ) , x12n+5 = , x−3 x−1 x0 (−1 + bx−4 x−2 ) x−2 x0 x−3 x−1 , x12n+7 = . x12n+6 = x−4 (−1 + bx−3 x−1 ) x−3 (−1 + bx−2 x0 )

x12n+4 =

4

The case an and bn are 3-periodic

The 3-periodicity of the sequences yields the following solution:

x12n−4 = x−4

n−1 Y s=0

a4s 0 + b0 x−4 x−2 a4s 2

+ b2 x−2 x0

4s−1 P

l=0 4s−1 P

al0 a4s+1 + b1 x−3 x−1 1 a4s+2 0

al2

+ b0 x−4 x−2

x12n−3 = x−3

s=0

a4s 1 + b1 x−3 x−1

4s−1 P

a4s+1 + b2 x−2 x0 2

al1

+ b0 x−4 x−2

al0

a4s+2 1

+ b1 x−3 x−1

l=0

x12n−2 = x−2

n−1 Y s=0

a4s 2 + b2 x−2 x0 a4s+1 1

4s−1 P

al2 a4s+2 + b0 x−4 x−2 0

l=0 4s P

+ b1 x−3 x−1

x12n−1 = x−1

s=0

a4s+1 + b0 x−4 x−2 0 a4s+1 2

+ b2 x−2 x0

al1

a4s+2 2

+ b2 x−2 x0

x12n = x0

s=0

a4s+1 + b1 x−3 x−1 1

4s P

al2

a4s+3 0

+ b0 x−4 x−2

a4s+2 0

+ b0 x−4 x−2

al2 a4s+3 + b0 x−4 x−2 0

l=0 4s+1 P

al1

a4s+3 2

al0 a4s+3 + b1 x−3 x−1 1

al2

al0

a4s+3 1

+ b1 x−3 x−1

l=0

l=0 4s+2 P

l=0

, al2

a4s+4 0

+ b0 x−4 x−2

4s+2 P l=0 4s+3 P

al1 , al0

l=0

4s+1 P

al1 a4s+3 + b2 x−2 x0 2 al0

a4s+4 1

al1

4s+2 P

+ b1 x−3 x−1

al2

l=0 4s+3 P

, al1

l=0

al2 a4s+4 + b0 x−4 x−2 0

l=0 4s+2 P

al0

l=0

4s+1 P

4s+1 P

4s+2 P

l=0 4s+2 P

+ b2 x−2 x0

l=0

al1 a4s+2 + b2 x−2 x0 2

l=0 4s+1 P

4s P

, al1

l=0

l=0 4s+1 P

al0 a4s+2 + b1 x−3 x−1 1

l=0 4s P

4s P

+ b1 x−3 x−1

l=0

l=0

n−1 Y

a4s+3 1

al2

l=0 4s+2 P

l=0

l=0

n−1 Y

al0

4s+1 P

l=0

l=0 4s P

a4s+1 0

al1 a4s+2 + b2 x−2 x0 2

l=0 4s+1 P

l=0

n−1 Y

4s P

a4s+4 2

+ b2 x−2 x0

4s+3 P

l=0 4s+3 P

al0 ,

al2

l=0

15

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x12n+1 = x1

n−1 Y s=0

4s P

a4s+1 + b2 x−2 x0 2 a4s+2 1

l=0 4s+1 P

+ b1 x−3 x−1

4s+2 P

al2 a4s+3 + b0 x−4 x−2 0 al1

a4s+3 2

+ b2 x−2 x0

l=0 4s+2 P

l=0

x12n+2 = x2

n−1 Y s=0

a4s+2 2

+ b2 x−2 x0

l=0 4s+1 P

al2

x12n+3 = x3

s=0

a4s+3 0

a4s+4 0

al0

a4s+4 1

4s+2 P

x12n+4 = x4

s=0 a4s+3 1

4s+1 P

+ b1 x−3 x−1

al1

a4s+4 2

+ b2 x−2 x0

l=0 4s+3 P

l=0

x12n+5 = x5

n−1 Y

s=0 a4s+3 2

+ b2 x−2 x0

l=0 4s+2 P

al2

a4s+5 0

x12n+6 = x6

s=0

a4s+4 0

al1 a4s+4 + b2 x−2 x0 2

l=0 4s+3 P

+ b0 x−4 x−2

al0

a4s+5 1

4s+3 P

+ b1 x−3 x−1

x12n+7 = x7

s=0

+ b2 x−2 x0 a4s+3 2

4s+2 P

+ b1 x−3 x−1 a4s+4 1

a4s+5 2

+ b2 x−2 x0

al0 a4s+5 + b1 x−3 x−1 1

al2

a4s+6 0

+ b0 x−4 x−2

l=0 4s+3 P

al1 a4s+5 + b2 x−2 x0 2

l=0

al1 a4s+5 + b2 x−2 x0 2 al0

al0 ,

al2

4s+4 P l=0 4s+5 P

al1 , al0

a4s+6 1

4s+4 P

+ b1 x−3 x−1

al2

l=0 4s+5 P

, al1

l=0

al2 a4s=6 + b0 x−4 x−2 0 al1

a4s+6 2

+ b0 x−2 x0

l=0 4s+4 P

4s+5 P

l=0 4s+5 P

al0 ,

al0

l=0

4s+4 P

l=0

4s+4 P

l=0

l=0

al2 a4s+5 + b0 x−4 x−2 0

, al1

l=0

l=0 4s+4 P

l=0

n−1 Y

al1

l=0 4s+4 P

+ b0 x−4 x−2

, al0

al2

l=0 4s+4 P

l=0 4s+4 P

l=0

4s+2 P

a4s+3 + b1 x−3 x−1 1

+ b1 x−3 x−1

al2 a4s+5 + b0 x−4 x−2 0

4s+3 P

al0 a4s+4 + b1 x−3 x−1 1

l=0

n−1 Y

a4s+5 1

l=0

4s+2 P

a4s+3 + b0 x−4 x−2 0

al1

l=0

4s+3 P

al2 a4s+4 + b0 x−4 x−2 0

l=0 4s+4 P

4s+3 P

l=0

l=0 4s+2 P

+ b1 x−3 x−1

al0

l=0 4s+3 P

l=0

4s+2 n−1 Y a2 + b2 x−2 x0

al1 a4s+4 + b2 x−2 x0 2

l=0 4s+3 P

+ b0 x−4 x−2

al1 a4s+3 + b2 x−2 x0 2

l=0 4s+2 P

+ b0 x−4 x−2

+ b0 x−4 x−2

l=0

4s+1 P

a4s+2 + b1 x−3 x−1 1

a4s+5 0

4s+3 P

l=0

4s+2 P

al0 a4s+3 + b1 x−3 x−1 1

l=0

n−1 Y

al2

l=0

4s+1 P

a4s+2 + b0 x−4 x−2 0

al0 a4s+4 + b1 x−3 x−1 1

al0 a4s+6 + b1 x−3 x−1 1

al2 a4s+7 + b0 x−4 x−2 0

4s+5 P l=0 4s+6 P

al1 al0

l=0

where x1 , x2 , x3 , x4 , x5 , x6 and x7 are given as follows:

16

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x1 = x4 =

x−4 x−2 , x−1 (a0 + b0 x−4 x−2 )

x2 =

x−3 x−1 , x0 (a1 + b1 x−3 x−1 )

x−4 x−2 x0 (a1 + b1 x−3 x−1 ) , x−3 x−1 (a20 + (b0 a0 + b0 )x−4 x−2 ) x6 =

x5 =

x3 =

x−1 x0 (a0 + b0 x−4 x−2 ) , x−4 (a2 + b2 x−2 x0 )

x−3 x−4 (a2 + b2 x−2 x0 ) , x0 (a0 + b0 x−4 x−2 )(a21 + (b1 a1 + b1 )x−3 x−1 )

x−3 x−1 (a20 + (b0 a0 + b0 )x−4 x−2 ) , x−4 (a1 + b1 x−3 x−1 )(a22 + (b2 a2 + b2 )x−2 x0 )

and x7 =

5

x−2 x0 (a0 + b0 x−4 x−2 )(a21 + (b1 a1 + b1 )x−3 x−1 ) . x−3 (a2 + b2 x−2 x0 )(a30 + (a20 b0 + a0 b0 + b0 )x−4 x−2 )

Conclusion

In this paper, we derived symmetry generators for the difference equations (2) and explicit formulas for the solutions of the equations were also obtained. Our solution generalised Theorems 1, 3, 6 and 8 of Elsayed [1].

References [1] E. M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Computational Analysis and Applications, 15(1) (2013), 73–81. [2] M. Folly-Gbetoula, Symmetry, reductions and exact solutions of the difference equation un+2 = (aun )/(1 + bun un+1 ), J. of Diff. Equations Appl., 23(6) (2017). [3] P. E. Hydon, Difference Equations by Differential Equation Methods, Cambridge University Press (2014). [4] T. F. Ibrahim and M. A. El-Moneam, Global stability of a higher-order difference equation, Iran J. Sci. Technol. Trans. Sci., 41(1) (2017), 51–58. [5] N. Joshi and P. Vassiliou, The existence of Lie Symmetries for First-Order Analytic Discrete Dynamical Systems, J. of Math. Anal. Appl., 195 (1995), 872-887 (1995). [6] D. Levi, L. Vinet and P. Winternitz, Lie group formalism for difference equations, J. Phys. A: Math. Gen. , 30 (1997), 633-649. [7] S. Lie, Classification und Integration von gewohnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten I , Math. Ann., 22 (1888), 213– 253.

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[8] S. Maeda, Canonical structure and symmetries for discrete systems, Math. Japonica, 25 (1980) 405–420. [9] S. Maeda, The similarity method for difference equations, IMA J. Appl. Math., 38 (1987), 129–134. [10] D. Nyirenda and M. Folly-Gbetoula, Invariance analysis and exact solutions of some sixth-order difference equations, J. Nonlinear Sci. Appl., 10 (2017), 6262-6273. [11] P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer, New York (1993). [12] G. R. W. Quispel and R. Sahadevan, Lie symmetries and the integration of difference equations, Physics Letters A, 184 (1993), 64-70.

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On some conditions for p-valency Mamoru Nunokawa1 , Janusz Sok´ ol2 1

University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan E-Mail:mamoru− [email protected] 2 University of Rzesz´ow, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzesz´ow, Poland E-Mail:[email protected] Abstract In this paper we consider analytic functions in the unit disc D such that |f (p) (z)| is bounded in D. We present several sufficient conditions for function to be p-valent starlike, convex or strongly starlike of a certain order. Key Words and Phrases. univalent functions; starlike; convex; close-to-convex 2010 Mathematics Subject Classification.Primary 30C45, Secondary 30C80 1. Introduction A function f analytic in a domain D ∈ C is called p-valent in D, if for every complex number w, the equation f (z) = w has at most p roots in D, so that there exists a complex number w0 such that the equation f (z) = w0 has exactly p roots in D. We denote by H the class of functions f (z) which are holomorphic in the open unit unit D = {z ∈ C : |z| < 1}. Denote by A(p), p ∈ N = {1, 2, . . .}, the class of functions f (z) ∈ H given by p

f (z) = z +

∞ X

an z n ,

(z ∈ D).

n=p+1

Let A = A(1). Let S denote the class of all functions in A which are univalent. Also let Sp∗ (α) and Cp (α) be the subclasses of A(p) consisting of all p-valent functions which are starlike and convex of order α, 0 ≤ α < 1, defined as   0   zf (z) ∗ Sp (α) = f (z) ∈ A(p) : Re > α, z ∈ D , pf (z)  Cp (α) = f (z) ∈ A(p) : zf 0 (z)/p ∈ Sp∗ (α) . Note that S1∗ (0) = S ∗ and C1 (0) = C, where S ∗ and C are usual classes of starlike and convex functions respectively. The well-known Noshiro-Warschawski theorem [1, 10], says that if f ∈ H satisfies  (1.1) Re eiα f 0 (z) > 0, (z ∈ D) for some real α, then f (z) is univalent in D. Ozaki [5], generalized the above theorem for f ∈ A(p): if  (1.2) Re eiα f (p) (z) > 0, (z ∈ D) for some real α, then f (z) is at most p-valent in D. Also in [3, 454] it was shown that if f ∈ A(p), p ≥ 2, and 3π (1.3) (z ∈ D), | arg{f (p) (z)}| < 4 then f is at most p-valent in D. The above results (1.1), (1.2) and (1.3) describe some consequences of a certain conditions on Re{f (p) (z)}, or | arg{f (p) (z)}|. It is the purpose of this paper is to consider analytic functions with bounded modulus of a certain order of derivative, like |f 00 (z)|, and to present some implications of this hypothesis. 219

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2. Main results A function f (z) ∈ H is said to subordinate a function g ∈ H in the unit disc E, written f ≺ g if and only if there exits an analytic function w ∈ H such that w(0) = 0, |w(z)| < 1 and f (z) = g[w(z)] for z ∈ E. Therefore f ≺ g in E implies f (E) ⊂ g(E). In particular if g is univalent in E then f ≺ g if and only if f (0) = g(0) and f (E) ⊂ g(E). The idea of subordination was used for defining many of the classes of functions studied in geometric function theory. In [9] Tuneski proved the following theorem. Theorem 2.1. If f (z) ∈ A, 0 < k ≤ 1 |f 00 (z)| ≤ k, then

(z ∈ D),

kz zf 0 (z) ≺1+ , f (z) 2−k

(z ∈ D).

In [6] it was proved a weaker result |f 00 (z)| ≤ 1,

(z ∈ D)

implies that f (z) is univalent in D . Applying Theorem 2.1, Tuneski in [9] obtained the following corollaries. Corollary 2.2. If f (z) ∈ A, 0 ≤ α < 1 and 2(1 − α) , 2−α  0  zf (z) Re > α, f (z)

|f 00 (z)| ≤ then

(z ∈ D), (z ∈ D).

The result is sharp. Corollary 2.3. If f (z) ∈ A, 0 < α ≤ 1 and 2 sin(απ/2) |f 00 (z)| ≤ , (z ∈ D), 1 + sin(απ/2) then  0  arg zf (z) < απ , (z ∈ D). f (z) 2 The result is sharp. In [9] Tuneski proved also the following result. Theorem 2.4. If f (z) ∈ A, 0 < k ≤ 1 |f 00 (z)| ≤ k,

(z ∈ D),

then f 0 (z) ≺ 1 + kz,

(z ∈ D).

Theorem 2.4 implies the following corollary. Corollary 2.5. If f (z) ∈ A, 0 ≤ α < 1 and 1−α , (z ∈ D), 2−α   zf 00 (z) Re 1 + 0 > α, (z ∈ D). f (z) |f 00 (z)| ≤

then

The result is sharp. 220

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We also need the following result. Theorem 2.6. [9] If f (z) ∈ A, 0 < λ ≤ 1 |f 0 (z) − 1| ≤ λ, then

where

(z ∈ D),

 0  arg zf (z) < απ , (z ∈ D), f (z) 2   p √ λ 2 −1 λ 1 − (λ2 /4) + 1 − λ2 . α = sin π 2

In [2] it was proved the following result. Theorem 2.7. [2] Let f (z) ∈ A(p). Suppose that there exists a positive integer j, 1 ≤ j ≤ p, such that  (j+1)  zf (z) > 0, (z ∈ D). j + Re (j) f (z) Then we have  (j)  zf (z) j − 1 + Re > 0, (z ∈ D). f (j−1) (z) 3. Main results Now we are going to make use of Theorem 2.1, Corollary 2.2 and of Theorem 2.7 to obtain the following theorem. Theorem 3.1. [2] Let f (z) ∈ A(p). Suppose that |f (p+1) (z)| < p!,

(z ∈ D).

Then f (z) is p-valently convex and p-valently starlike in D. Proof. If we put g(z) =

1 (p−1) f (z), p!

g(0) = g 0 (0) − 1 = 0,

(z ∈ D),

then it follows that

|f (p+1) (z)| < 1, (z ∈ D). p! From Theorem 2.1 and Corollary 2.2, we have  0   (p)  zg (z) zf (z) Re = Re > 0, (z ∈ D) g(z) f (p−1) (z) and so, we have  (p)  zf (z) p − 1 + Re > p − 1 ≥ 0, (z ∈ D). f (p−1) (z) From Theorem 2.7, it follows that  00   0  zf (z) zf (z) 1 + Re > 0 and Re > 0, (z ∈ D). 0 f (z) f (z) |g 00 (z)|
0 ⇒ ∀k ∈ {1, . . . , p} : Re > 0. Re f (p−1) (z) f (k−1) (z) Therefore, if we put 2f (p−2) (z) := G(z) = z 2 + · · · ∈ A(2), p! then zf (p−1) (z) zG0 (z) = (p−2) , G(z) f (z)

(z ∈ D)

and so (3.1) also implies that  0   (p−1)  zG (z) zf (z) Re = Re > 0, (p−2) G(z) f (z)

(z ∈ D).

This shows that G(z) or 2f (p−2) (z)/p! is 2-valently starlike in D. Theorem 3.3. If f (z) ∈ A(p), 0 < α ≤ 1, 1 ≤ p and 1 |f (p+1) (z)| ≤ , 2

(z ∈ D),

then  k + Re

zf (k+1) (z) f (k) (z)

 > 0,

(z ∈ D)

for all k, k ∈ {1, 2, . . . , p − 1}. 222

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Proof. If we put g(z) =

1 (p−1) f (z), p!

then it follows that

g(0) = g 0 (0) − 1 = 0,

zg 00 (z) zf (p+1) (z) = , g 0 (z) f (p) (z)

(z ∈ D),

(z ∈ D).

From Corollary 2.5, we have  1 + Re

zg 00 (z) g 0 (z)



 = 1 + Re

zf (p+1) (z) f (p) (z)

 (z ∈ D)

> 0,

and so, we have  p + Re

zf (p+1) (z) f (p) (z)

 > 0,

(z ∈ D).

> 0,

(z ∈ D)

Applying Theorem 2.7 gives finally  k + Re

zf (k+1) (z) f (k) (z)



for all k, k ∈ {1, 2, . . . , p − 1}. It completes the proof.  From Theorem 3.3, we have |f

(p+1)

1 (z)| ≤ , 2

(z ∈ D)

⇒

  zf 00 (z) Re 1 + 0 > 0, f (z)

(z ∈ D),

this suggests the following question. Open problem. What is the best value of α(p) such that   1 zf 00 (z) (p+1) |f (z)| ≤ , (z ∈ D) ⇒ Re 1 + 0 > α(p), 2 f (z)

(z ∈ D).

If p = 1, then the function f (z) = z + z 2 /4 shows that the best value of α(p) is 0. Theorem 3.4. If f (z) ∈ A(p), 0 < λ ≤ 1 and if |f (p) (z) − p!| < p!λ,

(3.3)

(z ∈ D),

then

 0  arg zf (z) < απ , (z ∈ D), (3.4) f (z) 2 where   p 2 λ√ −1 2 2 α = sin (3.5) λ 1 − (λ /4) + 1−λ . π 2 This means that f (z) is strongly starlike of order α in D. Proof. If we put g(z) =

1 (p−1) f (z), p!

g(0) = g 0 (0) − 1 = 0,

(z ∈ D),

then from (3.3), we have (p) f (z) |g (z) − 1| = − 1 < λ, p! 0

(z ∈ D).

From Theorem 2.6, we have  (p)  arg zf (z) < απ , (p−1) f (z) 2 223

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where α has the form (3.5). Let us put p(z) = Then it follows that

zf (p−1) (z) , f (p−2) (z)

(z ∈ D).

zp0 (z) zf (p) (z) zf (p−1) (z) = 1 + (p−1) − (p−2) p(z) f (z) f (z)

or 1+

zf (p) (z) zp0 (z) = p(z) + . f (p−1) (z) p(z)

From Theorem 2.6, we have   (p) απ zf (z) < arg , f (p−1) (z) 2

(z ∈ D),

this gives    (p)  (p) zf (z) zf (z) απ (3.6) arg 1 + f (p−1) (z) < arg f (p−1) (z) < 2 , If there exists a point z0 ∈ D, such that απ | arg {p(z)} | < , (|z| < |z0 |) 2 and απ | arg {p(z0 )} | = , 2 then from [4], we have z0 p0 (z0 ) = iαk, p(z0 ) where k is a real number such that   1 1 k≥ a+ 2 a when p(z0 ) = ia, while for p(z0 ) = −ia, such that   1 1 k≤− a+ , 2 a

(z ∈ D).

where p1/α (z0 ) = ±ia, a > 0. For the case p1/α (z0 ) = ia, we have 1+

z0 f 00 (z0 ) z0 p0 (z0 ) = p(z ) + 0 f 0 (z0 ) p(z0 )   z0 p0 (z0 ) = p(z0 ) 1 + 2 p (z0 )   1 α = (ia) 1 + iαk (ia)α   1 α iαπ/2 iπ(1−α)/2 = a e 1+e αk α . a

Thus, it is trivial that   z0 f (p) (z0 ) απ arg 1 + (p−1) ≥ f (z0 ) 2 since we have



iπ(1−α)/2

arg 1 + e

1 αk α a

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where

  1 1 a+ . k≥ 2 a This contradicts (3.6) and for the case p1/α (z0 ) = −ia, applying the same method as the above, we would have   z0 f (p) (z0 ) απ arg 1 + (p−1) ≤− f (z0 ) 2 which also contradicts (3.6). Applying the same method repeatedly once again, we can complete the proof of Theorem 3.4.  We now note that Pommerenke [7] and Sakaguchi [8] showed the following. Lemma 3.5. [7] If f and h are analytic in D, and h is convex and univalent in D, with  0  arg f (z) ≤ απ , (z ∈ D), h0 (z) 2 for some real α, 0 ≤ α ≤ 1, then   απ f (z ) − f (z ) 2 1 arg ≤ , h(z2 ) − h(z1 ) 2 for all z1 , z2 ∈ D. Putting z1 = 0, z2 = z in Lemma 3.5 gives  0  arg f (z) ≤ απ , z ∈ D (3.7) h0 (z) 2

(z ∈ D),

  απ f (z) arg ≤ , h(z) 2

⇒

(z ∈ D).

Therefore, applying Theorem 3.4 and (3.7) we can deduce the following corollary. Corollary 3.6. If f (z) ∈ A(p) is such that Z 0

is a convex function, and if (3.8) for some λ, 0 < λ ≤ 1, then (3.9)

z

f (t) dt t

|f (p) (z) − p!| < p!λ,

(z ∈ D),

) ( απ f (z) , < arg R z f (t) 2 dt 0

(z ∈ D),

t

where α is given in (3.5). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. Jap., 2(1)(1934-35) 129–135. M. Nunokawa, On the theory of multivalent functions, Tsukuba J. Math. 11(2)(1987) 273–286. 35–43. M. Nunokawa, A note on multivalent functions, Tsukuba J. Math. 13(2)(1989) 453–455. 35–43. M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A 69(7)(1993) 234–237. S. Ozaki, On the theory of multivalent functions , Sci. Rep. Tokyo Bunrika Daigaku Sect. A 2(1935) 167–188. S. Ozaki, I. Ono, T. Umezawa, On a General Second Order Derivative, Science Reports of the Tokyo Kyoiku Daigaku, 5( 124)(1956) 111–114. Ch. Pommerenke, On close to-convex functions, Trans. Amer. Math. Soc. 114(1)(1965) 176–186. K. Sakaguchi, On certain univalent mapping, J. Math. Soc. Japan, 11(1959) 72-75. N. Tuneski, On some simple sufficient conditions for univalence. Math. Bohemica, 126(1)(2001) 229–236. S. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38(1935) 310–340.

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Fractional Cauchy Euler Di¤erential Equation

By Al-Horani, M. Khalil, R. and Aldarawi, I. Department of Mathematics, The University of Jordan, Amman, Jordan e-mail: [email protected], [email protected], [email protected]

Abstract In this paper we give general solution of fractional linear di¤erential equations and fractional Cauchy Euler equation. Since there are many de…nitions for fractional derivatives, we use the conformable derivative to get exact solutions. Factorizing polynomials of the fractional di¤erential operators is the key method to get such solutions. Some speci…c examples on both types of equations are presented. ————————————————————————————— Key Words and Phrases:Conformable, Cauchy Euler, Conformable Linear Differential equations, Conformable Cauchy Euler Equation. AMS Classi…cation Number : 26A33

1. Introduction Many authors have solved many well known di¤erential equations like the Conformable Fractional Heat equation, Bessel equation, Legendre equation and many more. [1], [4], [5], [6], [7], [9] and [10]. The Cauchy Euler equation is a well known important type of ordinary di¤erential equation. In This paper we give the procedure and justi…cation of how to handle the Cauchy Euler equation, but the fractional one. However, there are many de…nitions available in the literature for fractional derivatives. The main ones are the Riemann Liouville de…nition and the Caputo de…nition, see [8] . 1

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(i) Riemann - Liouville De…nition. For

dn ) dtn

1

Da (f )(t) =

2 [n

(n

Zt

1; n); the

(t

derivative of f is

f (x) dx: x) n+1

a

(ii) Caputo De…nition. For

2 [n

Da (f )(t)=

1; n); the

1 (n

)

Zt

derivative of f is

f (n) (x) dx: (t x) n+1

a

Such de…nitions have many setbacks such as (i) The Riemann-Liouville derivative does not satisfy Da (1) = 0 (Da (1) = 0 for the Caputo derivative), if is not a natural number. (ii) All fractional derivatives do not satisfy the known formula of the derivative of the product of two functions: Da (f g) = f Da (g) + gDa (f ). (iii) All fractional derivatives do not satisfy the known formula of the derivative of the quotient of two functions: f Da (g) : g2 (iv) All fractional derivatives do not satisfy the chain rule: Da (f =g) =

Da (f

gDa (f )

g)(t) =f ( ) (g(t)) g ( ) (t):

(v) All fractional derivatives do not satisfy: D D f = D

+

f , in general.

(vi) All fractional derivatives, specially Caputo de…nition, assumes that the function f is di¤erentiable. We refer the reader to [3] for more results on Caputo and Riemann - Liouville De…nitions. Recently, the authors in [ 2 ], gave a new de…nition of fractional derivative which is a natural extension to the usual …rst derivative. So many papers since then were written, and many equations were solved using such de…nition. The de…nition goes as follows: Given a function f : [0; 1) ! R. Then for all t > 0; 2 (0; 1); let T (f )(t) = lim

f (t + "t1

)

f (t)

; " T is called the conformable fractional derivative of f of order Let f ( ) (t) stands for T (f )(t): "!0

If f is

:

di¤erentiable in some (0; b); b > 0, and lim f ( ) (t) exists; then t!0+

de…ne f ( ) (0) = lim+ f ( ) (t): t!0

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According to this de…nition, we have the following properties, [2 ], 1. T (1) = 0, 2. T (tp ) = ptp

for all p 2 R,

3. T (sin at) = at1

cos at;

at1

4. T (cos at) = 5. T (eat ) = at1

a 2 R,

sin at;

eat ;

a2R

a 2 R.

Further, many functions behave as in the usual derivative. Here are some formulas 1 T ( t )=1 1

1

)=e t ; 1 1 T (sin t ) = cos( t ); T (e

t

T (cos

1

t )=

1 sin( t ):

We will use the conformable fractional derivative for the Cauchy Euler equation. But …rst, we present the linear fractional case with constant coe¢ cients.

2. Conformable Linear Di¤erential equations Let us write y (n ) to denote the T T :::T (y); n-times.

derivative of y; n times. That is y (n

)

=

Theorem 1. Let y (n

)

+ an

1y

(n 1)

+ ::: + a1 y + a0 y = 0

(1)

Consider the equation r(n

)

+ an

If r1 =

1r

(n 1)

1 ; :::; rn

rk e

yk = e

+ ::: + a1 r + a0 = 0

=

n

are the real roots of ( ) then yh = c1 y1 + ::: + cn yn where

t

:

Proof. Let T n = T T :::T form (T (n

)

( )

+ an

1T

(n 1)

n-times. Then equation (1) can be written in the

+ ::: + a1 r + a0 I)y = 0

(2)

d dx

(where T = ): Now, if we let D = T n

(D + an

1D

n 1

+ ::: + a0 I)y = 0

The polynomial (Dn + an (D

1 )(D

then (2) becomes

2 ):::(D

n 1

+ ::: + a0 I)y = 0 , factorizes to

n )y

=0

1D

(3)

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Now, y will be a solution to (3) if y 2 ker(D k n , noting that (D k )81 commutes with (D j ) for all i and j . Thus y is the solution for (3) if (D

1 )y

However

= 0 or (D

(D Dy y

So Hence

2 )y

= 0 or :::or (D

= 0 implies y = 0 k y = 0 k

ke

t

if

is real.

k

1e

t

ne

Consequently, yh = c1 e +:::+cn e Now, replacing T by r we get 1 )y

=0

k )y

yk = e

(r

n )y

i)

= 0 or (r

2 )y

t

, if all the roots are real and distinct.

= 0 or :::or (r

n )y

=0

Thus the roots are r1 =

1

, r2 =

, :::, rn =

2

n

and the general solution is yh = c1 er1 e

t

+ ::: + cn ern e

t

There are two other cases for the roots to be considered: (1) (i)If one of the root is repeated, say 1 , 2-times. That is (T t

t

a factor of (3). Then y1 = e 1 e ; y2 = solutions for the di¤erential equation (3). Proof. We have to show t 2 t e 1e = 0 1)

(T

e

1e

2 1)

is

t

, are two independent

Indeed: (T

1)

(T

= (T

1) 1)

= (T

t

T (

1)

e

e

t

1e

e

1e

t

1e

=0 t

)

t

+

1(

t

1(

e

1e

t

e

1e

t

)=0

t

)

1(

t

e

1e

t

)=0

t

e 1e = 0 Similarly one can show that if t t t y1 = e 1 e ; y1 ; :::; ( )k 1 y1

= (T

1)

1

is repeated k-times then

are independent solutions. (ii)There is a root, say 1 = a + ib , a; b 2 R: Then t t t t and y2 = ea sin b y1 = ea cos b are two solutions of (3) associated with Indeed:

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Since Then y1 = e(a+ib)

1

t

= a + ib is a root, then y2 = e(a

and

ib) t

1

=a

ib is a root.

are solutions of (3)

But y1 = ea

t

y2 = ea

t

(cos b (cos b

t

+ i sin b

t

i sin b

t

)

t

)

Place y1 + y2 is a solution (the equation being homogenous) and y1 a solution too. So t t t t and ye2 = y1 y2 = 2iea sin b ye1 = y1 + y2 = 2ea cos b

y2 is

are solutions of the homogenous equation (3).

Consequentely t t t 1 t 1 and ye e2 = ye2 = ea sin b ye e1 = ye1 = ea cos b 2 2i are two independent solutions for the equation.

Example 1 T2 y + T y

2y = 0

(i)

Solution. Consider the associated equation r2 + r (r

2)(r + 1) = 0

Hence Thus

2=0

1

= 2;

2

y1 = c1 e2

= t

1 and y2 = c2 e

t

One can easily check that these are solutions of (i).See …gure (1) y 1 .5 e+4

1 .2 5 e+4

1 e+4

7500

5000

2500

0 0

1 .2 5

2 .5

3 .7 5

5 x

Fig.1 y = c1 e2

t

+ c2 e

230

t

;

= 0:5; c1 > 0

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3. Conformable Cauchy Euler Equation The standard form of the classical homogenous Cauchy Euler equation of order 2 is: x2 y 00 + a1 xy 0 + a0 y = 0 Now the conformable Cauchy Euler equation of order 2 can be written as x2 T 2 y + a1 x T y + a0 y = 0

(1)

Now we will give the procedure how to solve (1):

Procedure Put y = x Then

r

T 2 y = T (T y) = T (T x r ) = T ( rx

r

)

= r( r

r 2

)x

a1 T y = a1 (T x r ) = a1 rx

r

Thus x2 T 2 y = x2 (

2

(r 1)

a1 x T y = a1 rx a0 y = a0 x Hence x2 (

2

1))x r x

r(r

x

2

r

r

r(r

1))x

2

:x

r

+ a1 rx r x r x

x + a0 x

r

=0

So r

x

2

r(r

1) + a1 r + a0 = 0

Solve 2

r(r

1) + a1 r + a0 = 0

to get r = r1 ; r = r2 . Assume r1 ; r2 are reals . Then y1 = x yh = c1 x

r1

r1

; y2 = x

+ c2 x

r2

are two independent solutions of (1) and

r2

Remark. The case of conformable Cauchy Euler Equation of any order can be handled in the same way as the case of order 2: Example 2. Solve x2 y (2

)

+ x2 y (

)

y = 0; y(1) = 1; y ( ) (1) = 1 2

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Solution. Put y = x r and substitute in the equation to get 1 2 r(r 1) + r =0 2 1 Take = we get 2 1 1 1 r(r 1) + r =0 4 2 2 r(r 1) + 2r 2 = 0 r2 +r 2 = 0 (r+2)(r 1) = 0 r1 = 2; r2 = 1 y1 = x

2 2

1

; y2 = x 2

( =

1 ) 2

p 1 yh = c1 + c2 2 x x y(1) = c1 + c2 1

y 2 (x) = c1 ( 1)x 1

So y 2 (1) = yh =

1

1 2

c1 +

+ c2

1 2

c2 3 = 1 . Hence c2 = 2 =) c c1 = 2 2

4p 1 2 + x. 3x 3

1 3

See …gure (2).

0

y

1.25

2.5

x 5

3.75

0

-5

-10

-15

-20

-25

-30

Fig.2 yh =

1 4p 2 3x + 3 x;

=

1 2

4. Conclusion Conformable fractional derivative can be applied to solve linear di¤erential equation with variable coe¢ cients as an example Cauchy Euler equation.

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References [1] Anderson, Dr, Positive Green’s Functions for Boundary Value Problems with Conformable Derivatives.Mathematical Analysis, Approximation Theory and Their Applications 1(2016)63-74 [2] Hosseini, K., Bekir, A., Kaplan, M. and Güner,O. On a new technique for solving the nonlinear conformable time-fractional di¤erential equations.Optical and Quantum Electronics, 49(2017) [3] Kaplan, M. Applications of two reliable methods for solving a nonlinear conformable time-fractional equation.Optical and Quantum Electronics, 49(2017)312[4] Khalil, R. and Abu Hammad, M.Abel. S Formula And Wronskian For Conformable Fractional Di¤erential Equations. I.J. Di¤erential Equations and Applications, 13 (2014) 177-183. [5] Khalil,R. M. Al horani, M. Yousef, A. Sababheh,M. Anew de…nition of fractional derivative, Journal of Computational Apllied Mathematics, 264 (2014), 65-70. [6]Abu Hammad, M. and Khalil, R. Legendre fractional di¤erential equation and Legender fractional polynomials I.J of Applied Mathematical Research, 3 (3) (2014) 214-219. [7]Abu Hammad, M. and Khalil, R..Conformable fractional Heat di¤erential equation M Abu Hammad, R Khalil - Int. J. Pure Appl. Math, Volume 94 ( 2014,) 215-221 [8] Kilbas, A. Srivastava, H. and Trujillo,J. Theory and Applications of Fractional Di¤erential Equations, Elsevier, Amsterdam, 2006. [9] Kurt, A. Cenesiz, Y. and Tas¸bozan, O. Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform,C¸ ankaya University Journal of Science and Engineering ,V.13, (2016) 018–023 [10] Hosseini, K., Bekir, A., Kaplan, M. and Güner,O. On a new technique for solving the nonlinear conformable time-fractional di¤erential equations.Optical and Quantum Electronics, 49(2017)

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Applications of neutrosophic sets in B-algebras Sun Shin Ahn Department of Mathematics Education, Dongguk University, Seoul 04620, Korea Abstract. The notions of a neutrosophic subalgebra and a neutrosophic normal subalgebra of a B-algebra are introduced and characterizations of them are discussed. We show that the homomorphic preimage of a neutrosophic subalgebra of a B-algebra is a neutrosophic subalgebra, and the onto homomorphic image of a neutrosophic subalgebra of a B-algebra is a neutrosophic subalgebra.

1. Introduction Zadeh [12] introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. As a generalization of fuzzy sets, Atanassov [2] introduced the degree of nonmembership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood). Y. B. Jun, E. H. Roh and H. S. Kim [4] introduced a new notion, called a BH-algebra. J. Neggers and H. S. Kim [9] introduced a new notion, called a B-algebra. C. B. Kim and H. S. Kim [7] introduced the notion of a BG-algebra which is a generalization of B-algebras. S. S. Ahn and H. D. Lee [1] classified the subalgebras by their family of level subalgebras in BG-algebras. In this paper, we introduce the notions of a neutrosophic subalgebra and a neutrosophic normal subalgebra of a B-algebra and discuss characterizations of them. We show that the homomorphic preimage of a neutrosophic subalgebra of a B-algebra is a neutrosophic subalgebra, and the onto homomorphic image of neutrosophic image of a neutrosophic subalgebra of a B-algebra is a neutrosophic subalgebra.

2. Preliminaries A B-algebra ([9]) is a non-empty set X with a constant 0 and a binary operation “ ∗ ” satisfying axioms: (B1) x ∗ x = 0, (B2) x ∗ 0 = x, (B) (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)) for any x, y, z in X. For brevity we call X a B-algebra. In X we can define a binary relation “ ≤ ” by x ≤ y if and only if x ∗ y = 0. An algebra (X; ∗, 0) of type (2, 0) is called a BH-algebra if it satisfies (B1), (B2) and (BH) x ∗ y = y ∗ x = 0 imply x = y for any x, y ∈ X. 0

2010 Mathematics Subject Classification: 06F35; 03G25; 03B52. Keywords: B-algebra; (normal) subalgebra; neutrosophic subalgebra; neutrosophic normal subalgebra. 0 E-mail: [email protected]

0

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Sun Shin Ahn An algebra (X; ∗, 0) of type (2, 0) is called a BG-algebra if it satisfies (B1), (B2) and (BG) (x ∗ y) ∗ (0 ∗ y) = x for any x, y ∈ X. Proposition 2.1.([3, 9]) Let (X; ∗, 0) be a B-algebra. Then (i) the left cancellation law holds in X, i.e., x ∗ y = x ∗ z implies y = z, (ii) if x ∗ y = 0, then x = y for any x, y ∈ X, (iii) if 0 ∗ x = 0 ∗ y, then x = y for any x, y ∈ X, (iv) 0 ∗ (0 ∗ x) = x, for all x ∈ X, (v) x ∗ (y ∗ z) = (x ∗ (0 ∗ z)) ∗ y for all x, y, z ∈ X. Theorem 2.2.([7]) If (X; ∗, 0) is a B-algebra, then it is a BG-algebra. Proposition 2.3.([7]) Every BG-algebra is a BH-algebra. Let (X; ∗X , 0X ) and (Y ; ∗Y , 0Y ) be B-algebras. A mapping φ : X → Y is called a homomorphism if φ(x∗X y) = φ(x) ∗Y φ(y) for any x, y ∈ X. A non-empty subset S of X is called a subalgebra of X if x ∗ y ∈ S for any x, y ∈ X. A non-empty subset N of X is said to be normal if (x ∗ a) ∗ (y ∗ b) ∈ N for any x ∗ y, a ∗ b ∈ N . Then any normal subset N of a B-algebra X is a subalgebra of X, but the converse need not be true ([10]). A non-empty subset X of a B-algebra X is a called a normal subalgebra of X if it is both a subalgebra and a normal set. Definition 2.4. Let X be a space of points (objects) with generic elements in X denoted by x. A simple valued neutrosophic set A in X is characterized by a truth-membership function TA (x), an indeterminacy-membership function IA (x), and a falsity-membership function FA (x). Then a simple valued neutrosophic set A can be denoted by A := {⟨x, TA (x), IA (x), FA (x)⟩|x ∈ X}, where TA (x), IA (x), FA (x) ∈ [0, 1] for each point x in X. Therefore the sum of TA (x), IA (x), and FA (x) satisfies the condition 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3. For convenience, “simple valued neutrosophic set” is abbreviated to “neutrosophic set” later. Definition 2.5. Let A be a neutrosophic set in a B-algebra X and α, β, γ ∈ [0, 1] with 0 ≤ α + β + γ ≤ 3 and an (α, β, γ)-level set of X denoted by A(α,β,γ) is defined as A(α,β,γ) = {x ∈ X|TA (x) ≤ α, IA (x) ≥ β, FA (x) ≤ γ}. For any family {ai |i ∈ Λ}, we define ∨

and

{ai |i ∈ Λ} :=

∧

{ max{ai |i ∈ Λ} sup{ai |i ∈ Λ} {

{ai |i ∈ Λ} :=

if Λ is finite, otherwise

min{ai |i ∈ Λ}

if Λ is finite,

inf{ai |i ∈ Λ}

otherwise.

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Applications of neutrosophic sets in B-algebras 3. Neutrosophic subalgebras in B-algebras Definition 3.1. A neutrosophic set A in a B-algebra X is called a neutrosophic subalgebra of X if it satisfies: (NSS) TA (x ∗ y) ≤ max{TA (x), TA (y)}, IA (x ∗ y) ≥ min{IA (x), IA (y)}, and FA (x ∗ y) ≤ max{FA (x), FA (y)}, for any x, y ∈ X. Proposition 3.2. Every neutrosophic subalgebra of a B-algebra X satisfies the following conditions: (3.1) TA (0) ≤ TA (x), IA (0) ≥ IA (x), and FA (0) ≤ FA (x) for any x ∈ X. □

Proof. Straightforward. Example 3.3. Let X := {0, 1, 2, 3} be a B-algebra with the following table: ∗ 0 1 2 3 Define a neutrosophic set A in X as follows:

0 0 1 2 3

{

TA (x) =

FA (x) =

2 2 3 0 1

3 3 2 1 0

0.13, if x ∈ {0, 2} 0.84, otherwise,

{ IA (x) =

1 1 0 3 2

0.82, if x ∈ {0, 2}

0.15, otherwise, { 0.13, if x ∈ {0, 2} 0.84, otherwise.

It is easy to check that A is a neutrosophic subalgebra of X. Theorem 3.4. Let A be a neutrosophic set in a B-algebra X and let α, β, γ ∈ [0, 1] with 0 ≤ α + β + γ ≤ 3. Then A is a neutrosophic subalgebra of X if and only if all of (α, β, γ)-level set A(α,β,γ) are subalgebras of X when A(α,β,γ) ̸= ∅. Proof. Assume that A is a neutrosophic subalgebra of X. Let α, β, γ ∈ [0, 1] be such that 0 ≤ α + β + γ ≤ 3 and A(α,β,γ) ̸= ∅. Let x, y ∈ A(α,β,γ) . Then TA (x) ≤ α, TA (y) ≤ α, IA (x) ≥ β, IA (y) ≥ β and FA (x) ≤ γ, FA (y) ≤ γ. Using (NSS), we have TA (x ∗ y) ≤ max{TA (x), TA (y)} ≤ α, IA (x ∗ y) ≥ min{IA (x), IA (y)} ≥ β, and FA (x ∗ y) ≤ max{FA (x), FA (y)} ≤ γ. Hence x ∗ y ∈ A(α,β,γ) . Therefore A(α,β,γ) is a subalgebra of X. Conversely, all of (α, β, γ)-level set A(α,β,γ) are subalgebras of X when A(α,β,γ) ̸= ∅. Assume that there exist at , bt , ai , bi ∈ X and af , bf ∈ X such that TA (at ∗ bt ) > max{TA (at ), TA (bt )}, IA (ai ∗ bi ) < min{IA (ai ), IA (bi )} and FA (af ∗ bf ) > max{FA (af ), FA (bf )}. Then TA (at ∗ bt ) > α1 ≥ max{TA (at ), TA (bt )}, IA (ai ∗ bi ) < β1 ≤ min{IA (ai ), IA (bi )} and FA (af ∗ bf ) > γ1 ≥ max{FA (af ), FA (bf )} for some α1 , γ1 ∈ [0, 1) and β1 ∈ (0, 1]. Hence at , bt , ai , bi ∈ A(α1 ,β1 ,γ1 ) , and af , bf ∈ A(α1 ,β1 ,γ1 ) . But at ∗ bt , ai ∗ bi ∈ / A(α1 ,β1 ,γ1 ) , and af ∗ bf ∈ / A(α1 ,β1 ,γ1 ) , which is a contradiction. Hence TA (x ∗ y) ≤ max{TA (x), TA (y)}, IA (x ∗ y) ≥ min{IA (x), IA (y)}, and FA (x ∗ y) ≤ max{TA (x), TA (y)}, for any x, y ∈ X. Therefore A is a neutrosophic subalgebra of X.

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Sun Shin Ahn Since [0, 1] is a completely distributive lattice with respect to the usual ordering, we have the following theorem. Theorem 3.5. If {Ai |i ∈ N} is a family of neutrosopic subalgebras of a B-algebra X, then ({Ai |i ∈ N}, ⊆) forms a complete distributive lattice. Theorem 3.6. Let A be a neutrosophic subalgebra of a B-algebra X. If there exists a sequence {an } in X such that limn→∞ TA (an ) = 0, limn→∞ IA (an ) = 1, and limn→∞ FA (an ) = 0, then TA (0) = 0, IA (0) = 1, and FA (0) = 0. Proof. By Proposition 3.2, we have TA (0) ≤ TA (x), IA (0) ≥ IA (x), and FA (0) ≤ FA (x) for all x ∈ X. Hence we have TA (0) ≤ TA (an ), IA (0) ≥ IA (an ), and FA (0) ≤ FA (an ) for every positive integer n. Therefore 0 ≤ TA (0) ≤ limn→∞ TA (an ) = 0, 1 = limn→∞ IA (an ) ≤ IA (0) ≤ 1, and 0 ≤ FA (0) ≤ limn→∞ FA (an ) = 0. Thus we have □

TA (0) = 0, IA (0) = 1, and FA (0) = 0. Proposition 3.7. If every neutrosophic subalgebra A of a B-algebra X satisfies the condition (3.2) TA (x ∗ y) ≤ TA (y), IA (x ∗ y) ≥ IA (y), FA (x ∗ y) ≤ FA (y), for any x, y ∈ X, then TA , IA , and FA are constant functions.

Proof. It follows from (3.2) that TA (x) = TA (x ∗ 0) ≤ TA (0), IA (x) = IA (x ∗ 0) ≥ IA (0), and FA (x) = FA (x ∗ 0) ≤ FA (0) for any x ∈ X. By Proposition 3.2, we have TA (x) = TA (0), IA (x) = IA (0), and FA (x) = FA (0) for any x ∈ X. Hence TA , IA , and FA are constant functions.

□

Definition 3.8. A neutrosophic set A in a B-algebra X is said to be neutrosophic normal of X if it satisfies: (NSN) TA ((x ∗ a) ∗ (y ∗ b)) ≤ max{TA (x ∗ y), TA (a ∗ b)}, IA ((x ∗ a) ∗ (y ∗ b)) ≥ min{IA (x ∗ y), IA (a ∗ b)}, and FA ((x ∗ a) ∗ (y ∗ b)) ≤ max{FA (x ∗ y), FA (a ∗ b)}, for any x, y, a, b ∈ X. A neutrosophic set A in a B-algebra X is called a neutrosophic normal subalgebra of X if it satisfies (NSS) and (NSN). Example 3.9. Let X := {0, 1, 2, 3} be a B-algebra ([8]) with the following table: ∗ 0 1 2 3 Define a neutrosophic set A in X as follows:

0 0 1 2 3

{

TA (x) =

FA (x) =

2 1 3 0 2

3 3 2 1 0

0.12, if x ∈ {0, 3} 0.76, otherwise,

{ IA (x) =

1 2 0 3 1

0.73, if x ∈ {0, 3}

0.14, otherwise, { 0.12, if x ∈ {0, 3} 0.76, otherwise.

It is easy to check that A is a neutrosophic normal subalgebra of X.

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Applications of neutrosophic sets in B-algebras Proposition 3.10. Every neutrosophic normal of a B-algebra X is a neutrosophic subalgebra of X. Proof. Let A be neutrosophic normal of X. Put y := 0, b := 0 and a := y in (NSN). Then TA ((x ∗ y) ∗ (0 ∗ 0)) ≤ max{TA (x∗0), TA (y∗0)}, IA ((x∗y)∗(0∗0)) ≥ min{IA (x∗0), IA (y∗0)}, and FA ((x∗y)∗(0∗0)) ≤ max{FA (x∗0), FA (y∗ 0)}. Using (B2) and (B1), we have TA (x∗y) ≤ max{TA (x), TA (y)}, IA (x∗y) ≥ min{IA (x), IA (y)}, and FA (x∗y) ≤ max{FA (x), FA (y)}, for any x, y ∈ X. Hence A is a neutrosophic subalgebra of X.

□

The converse of Proposition 3.10 may not be true in general (see Example 3.11). Example 3.11. Let X = {0, 1, 2, 3, 4, 5} be a B-algebra ([10]) with the following table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 2 0 1 4 5 3

2 1 2 0 5 3 4

3 3 4 5 0 1 2

4 4 5 3 2 0 1

5 5 3 4 1 2 0

Define a neutrosophic set A in X as follows:

   0.12, if x = 0 TA (x) = 0.23, if x = 5   0.52 otherwise,    0.58, if x = 0 IA (x) = 0.13, if x = 5   0.11, otherwise,    0.12, if x = 0 FA (x) = 0.23, if x = 5   0.52 otherwise.

It is easy to check that A is a neutrosophic subalgebra of X. But it is not neutrosophic normal of X, since TA (1) = TA ((1 ∗ 3) ∗ (4 ∗ 2)) = 0.52 ≰ max{TA (1 ∗ 4), TA (3 ∗ 2)} = max{TA (5), TA (5)} = 0.23, and/or IA (1) = IA ((1∗3)∗(4∗2)) = 0.11 ≱ min{IA (1∗4), IA (3∗2)} = min{IA (5), IA (5)} = 0.13, and/or FA (1) = FA ((1∗3)∗(4∗2)) = 0.52 ≰ max{FA (1 ∗ 4), FA (3 ∗ 2)} = max{FA (5), FA (5)} = 0.23. Theorem 3.12. Let A be a neutrosophic set in a B-algebra X and let α, β, γ ∈ [0, 1] with 0 ≤ α+β +γ ≤ 3. Then A is a neutrosophic normal subalgebra of X if and only if all of (α, β, γ)-level set A(α,β,γ) are normal subalgebras of X when A(α,β,γ) ̸= ∅. □

Proof. Similar to Theorem 3.4.

Proposition 3.13. Let A be a neutrosophic normal subalgebra of a B-algebra X. Denote that XT := {x ∈ X|TA (x) = TA (0)}, XI := {x ∈ X|IA (x) = IA (0)}, and XF := {x ∈ X|FA (x) = FA (0)}. Then XT , XI , and XF are normal subalgebras of X.

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Sun Shin Ahn Proof. It is sufficient to show that XT , XI , and XF are normal. Let a, b, x, y ∈ X be such that x ∗ y, a ∗ b ∈ XT . Then TA (x∗y) = TA (0) = TA (a∗b). Since A is a neutrosophic normal subalgebra of X, we have TA ((x∗a)∗(y∗b)) ≤ max{TA (x∗y), TA (a∗b)} = TA (0). By Proposition 3.2, we get TA ((x∗a)∗(y∗b)) = TA (0). Hence (x∗a)∗(y∗b) ∈ XT . Therefore XT is a normal subalgebra of X. Similarly, XI , XF are normal subalgebras of X. This completes the □

proof.

Definition 3.14. Let A and B be neutrosophic sets of a set X. The union of A and B is defined to be a neutrosophic set ˜ B := {⟨x, TA∪B (x), IA∪B (x), FA∪B (x)⟩|x ∈ X}, A∪ where TA∪B (x) = min{TA (x), TB (x)}, IA∪B (x) = max{IA (x), IB (x)}, FA∪B (x) = min{FA (x), FB (x)}, for all x ∈ X. The intersection of A and B is defined to be a neutrosophic set ˜ B := {⟨x, TA∩B (x), IA∩B (x), FA∩B (x)⟩|x ∈ X}, A∩ where TA∩B (x) = max{TA (x), TB (x)}, IA∩B (x) = min{IA (x), IB (x)}, FA∩B (x) = max{FA (x), FB (x)}, for all x ∈ X. Theorem 3.15. The intersection of two neutrosophic subalgebras of a B-algebra X is a also a neutrosophic subalgebra of X. Proof. Let A and B be neutrosophic subalgebras of X. For any x, y ∈ X, we have TA∩B (x ∗ y) = max{TA (x ∗ y), TB (x ∗ y)} ≤ max{max{TA (x), TA (y)}, max{TB (x), TB (y)}} = max{max{TA (x), TB (x)}, max{TA (y), TB (y)}} = max{TA∩B (x), TA∩B (y)}, IA∩B (x ∗ y) = min{IA (x ∗ y), IB (x ∗ y)} ≥ min{min{IA (x), IA (y)}, min{IB (x), IB (y)}} = min{min{IA (x), IB (x)}, min{IA (y), IB (y)}} = min{IA∩B (x), IA∩B (y)}, and FA∩B (x ∗ y) = max{FA (x ∗ y), FB (x ∗ y)} ≤ max{max{FA (x), FA (y)}, max{FB (x), FB (y)}} = max{max{FA (x), FB (x)}, max{FA (y), FB (y)}} = max{FA∩B (x), FA∩B (y)}. ˜ B is a neutrosophic subalgebra of X. Hence A∩

□

˜ ı∈N Ai . Corollary 3.16. If {Ai |i ∈ N} is a family of neutrosophic subalgebras of a B-algebra X, then so is ∩ The union of any set of neutrosophic subalgebras of a B-algebra X need not be a neutrosophic subalgebra of X.

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Applications of neutrosophic sets in B-algebras Example 3.17. Let X = {0, 1, 2, 3, 4, 5} be a B-algebra as in Example 3.11. Define neutrosophic sets A and B of X as follows: TA (x) =

{ 0.11, if x ∈ {0, 4} {

IA (x) =

FA (x) =

TB (x) =

IB (x) = and

0.73

0.82, if x ∈ {0, 4}

0.12, otherwise, { 0.11, if x ∈ {0, 4} 0.73 otherwise, { 0.13, if x ∈ {0, 5} 0.74 otherwise, { 0.83, if x ∈ {0, 5} 0.13, otherwise, {

FB (x) =

otherwise,

0.13, if x ∈ {0, 5} 0.74

otherwise.

˜ B is not a neutrosophic subalgebra It is easy to check that A and B are neutrosophic subalgebras of X. But A∪ of X, since TA∪B (4 ∗ 5) =TA∪B (2) = min{TA (2), TB (2)} = 0.73 ≰ max{TA∪B (4), TA∪B (5)} = max{min{TA (4), TB (4)}, min{TA (5), TB (5)}} = 0.13, and/or IA∪B (4 ∗ 5) =IA∪B (2) = max{IA (2), IB (2)} = 0.13 ≱ min{IA∪B (4), IA∪B (5)} = min{max{IA (4), IB (4)}, max{IA (5), IB (5)}} = 0.82, and/or FA∪B (4 ∗ 5) =FA∪B (2) = min{FA (2), FB (2)} = 0.73 ≰ max{FA∪B (4), FA∪B (5)} = max{min{FA (4), FB (4)}, min{FA (5), FB (5)}} = 0.13. Let f : X → Y be a function of sets. If M = {⟨y, TM (y), IM (y), FM (y)⟩|y ∈ Y } is a neutrosophic set of a set Y , then the preimage of M under f is defined to be a neutrosophic set f −1 (M ) := {⟨x, f −1 (TM )(x), f −1 (IM )(x), f −1 (FM )(x)⟩|x ∈ X} of X, where f −1 (TM )(x) = TM (f (x)), f −1 (IM )(x) = IM (f (x)) and f −1 (FM )(x) = FM (f (x)) for all x ∈ X. Theorem 3.18. Let f : X → Y be a homomorphism of B-algebras. If M = {⟨y, TM (y), IM (y), FM (y)⟩|y ∈ Y } is a neutrosophic subalgebra of Y , then the preimage of M under f is a neutrosophic subalgebra of X.

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Sun Shin Ahn Proof. Let f −1 (M ) be the preimage of M under f . For any x, y ∈ X, we have f −1 (TM (x ∗ y)) =TM (f (x ∗ y)) = TM (f (x) ∗ f (y)) ≤ max{TM (f (x)), TM (f (y))} = max{f −1 (TM )(x), f −1 (TM )(y)}, f −1 (IM (x ∗ y)) =IM (f (x ∗ y)) = IM (f (x) ∗ f (y)) ≥ min{IM (f (x)), IM (f (y))} = min{f −1 (IM )(x), f −1 (IM )(y)}, and f −1 (FM (x ∗ y)) =FM (f (x ∗ y)) = FM (f (x) ∗ f (y)) ≤ max{FM (f (x)), FM (f (y))} = max{f −1 (FM )(x), f −1 (FM )(y)}. Hence f −1 (M ) is a neutrosophic subalgebra of X.

□

Let f : X → Y be an onto function of sets. If A is a neutrosophic set of X, then the image of A under f is defined to be a neutrosophic set f (A) := {⟨y, f (TA )(y), f (IA )(y), f (FA )(y)⟩|y ∈ Y } of Y , where f (TA )(y) =

∧ x∈f −1 (y)

TA (x), f (IA )(y) =

∨

x∈f −1 (y) IA (x),

and f (FA )(y) =

∧ x∈f −1 (y)

FA (x).

Theorem 3.19. For an onto homomorphism f : X → Y of B-algebras, let A be a neutrosophic set of X such that (3.3) (∀C ⊆ X)(∃x0 ∈ C)(TA (x0 ) =

∧

TA (z), IA (x0 ) =

z∈C

∨ z∈C

IA (z), FA (x0 ) =

∧ z∈C

FA (z)).

If A is a neutrosophic subalgebra of a B-algebra X, then the image of A under f is a neutrosophic subalgebra of Y. Proof. Let f (A) be the image of A under f . Let a, b ∈ Y . Then f −1 (a) ̸= ∅ and f −1 (b) ̸= ∅ in X. By (3.3), there exist xa ∈ f −1 (a) and xb ∈ f −1 (b) such that ∧ TA (xa ) = TA (z), IA (xa ) = z∈f −1 (a)

∧

TA (xb ) =

TA (w), IA (xb ) =

∧

= max{

TA (z),

FA (z),

z∈f −1 (a)

IA (w), FA (xb ) =

∧

FA (w).

w∈f −1 (b)

∨

∧

TA (w)} = max{f (TA )(a), f (TA )(b)},

w∈f −1 (b)

z∈f −1 (a)

IA (x) ≥ IA (xa ∗ xb ) ≥ min{IA (xa ), IA (xb )}

x∈f −1 (a∗b)

= min{

∧

TA (x) ≤ TA (xa ∗ xb ) ≤ max{TA (xa ), TA (xb )}

x∈f −1 (a∗b)

f (IA )(a ∗ b) =

∨ w∈f −1 (b)

∧

f (TA )(a ∗ b) =

IA (z), FA (xa ) =

z∈f −1 (a)

w∈f −1 (b)

Thus

∨

∨

z∈f −1 (a)

IA (z),

∨

IA (w)} = min{f (IA )(a), f (IA )(b)},

w∈f −1 (b)

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Applications of neutrosophic sets in B-algebras and f (FA )(a ∗ b) =

∧

FA (x) ≤ FA (xa ∗ xb ) ≤ max{FA (xa ), FA (xb )}

x∈f −1 (a∗b)

= max{

∧

FA (z),

z∈f −1 (a)

∧

FA (w)} = max{f (FA )(a), f (FA )(b)}.

w∈f −1 (b)

□

Hence f (A) is a neutrosophic subalgebra of Y . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

S. S. Ahn and H. D. Lee, Fuzzy subalgebras of BG-algebras, Comm. Kore. Math. Soc. 19 (2004), 243-251. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems 20 (1986), 87–96. J. R. Cho and H. S. Kim, On B-algebras and Related Systems, 8(2001), 1-6. Y. B. Jun, E. H. Roh and H. S. Kim, On BH-algebras, Sci. Mathematica 1 (1998), 347-354. Y. B. Jun, E. H. Roh and H. S. Kim, On fuzzy B-algebras, Czech. Math. J. 52 (2002), 375-384. M. Khan, S. Anis, F. Smarandache and Y. B. Jun, Neutrosophic N -structures and their applications in semigroups, Ann. Fuzzy Math. Inform., (to appear). C. B. Kim and H. S. Kim, On BG-algebras, Demon. Math. 41 (2008), 497-505. Y. H. Kim and S. J. Yeom, Qutient B-algebras via fuzzy normal B-algebras, Honam Math. J. 30 (2008), 21-32. J. Neggers and H. S. Kim, On B-algebras, Mate. Vesnik 54(2002), 21-29. J. Neggers and H. S. Kim, A fundamental theorem of B-homomorphism for B-algebras, Intern. Math. J. 2(2002), 207-214. F. Smarandache, Neutrosophy, Neutrosophic Probablity, Sets, and Logic, Amer. Res. Press, Rehoboth, USA, 1998. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338-353.

.

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Investigating Some Properties of a Fourth Order Di erence Equation M. B. Almatra…1 , E. M. Elsayed1;2 and Faris Alzahrani:1 1 Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mails: mmutra…@taibahu.edu.sa, [email protected], [email protected]. ABSTRACT The principal purpose of this paper is to present some qualitative behavior of the following fourth order di¤erence equation: bxn 1 ; n = 0; 1; :::; xn+1 = axn 1 cxn 1 dxn 3 where the parameters a; b; c and d are positive real numbers and the initial conditions x arbitrary non zero real numbers.

3;

x

2;

x

1

and x0 are

Keywords: stability, periodicity, global attractor, di¤erence equations. Mathematics Subject Classi…cation: 39A10. ——————————————————

1. INTRODUCTION This paper will provide a detailed study in terms of the local, global stability and obtain the form of the solutions of the following di¤erence equation xn+1 = axn where the initial conditions x constants:.

3;

x

2;

x

1

bxn 1

cxn

1

1

dxn

;

n = 0; 1; :::;

(1)

3

and x0 are arbitrary non zero real numbers and a; b; c; d are positive

A huge number of researchers has concentrated on studying and investigating nonlinear di¤erence equations in recent years. In particular, they have highlighted the boundedness, the global attractivity and the periodic behaviour of some certain types of di¤erence equations. For instance: Elsayed et al.19 studied the global attractor, local stability, periodic solutions and boundedness of the following recursive equation: xn+1 =

axn xn 2 : bxn + cxn 3

Cinar5 investigated the solution of the di¤erence equation xn+1 =

axn 1 1 + bxn xn

: 1

Ibrahim24 presented some relevant results of the di¤erence equation xn+1 =

xn

xn xn 2 (a + bxn xn 1

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Elsayed16 analyzed the global stability and examined the periodic solution of the following di¤erence equation:

xn+1 = axn

l

+

bxn cxn

l

dxn

l

: k

Elabbasy et al.8 investigated the global stability, periodicity character and gave the solution of special case of the di¤erence equation bxn : xn+1 = axn cxn dxn 1 Yang et al.36 examined the global and local stability of the equilibrium points of the following recursive equation: xn+1 =

axn 1 + bxn c + dxn 1 xn

2

:

2

Simsek et al.33 obtained the solution of the following di¤erence equation xn 3 1 + xn

xn+1 =

: 1

Abo-Zeid et al.1 gave a detailed study about the convergence and the periodicity of the solutions of the di¤erence equation Axn 1 xn+1 = : (B Cxn xn 2 ) Tolly et al.35 illustrated some properties of the solution of the following recursive equation: yn+1 =

byn yn

1

ayn 1 + cyn 1 yn

2

+d

:

Other relevant consequences of rational di¤erence equations can be obtained in refs.9 -.12 Now, some relevant results and de…nitions will be introduced here to be used in our discussion. Let I be some interval of real numbers and the function f has continuous partial derivatives on I k+1 where I =I I I (k + 1 times): Then, for initial conditions x k ; x k+1 ; :::; x0 2 I; it is easy to see that the di¤erence equation xn+1 = f (xn ; xn 1 ; :::; xn k ); n = 0; 1; :::; (2) k+1

has a unique solution fxn g1 n=

k:

A point x 2 I is called an equilibrium point of Eq.(2) if x = f (x; x; :::; x).

That is, xn = x for n

0; is a solution of Eq.(2), or equivalently, x is a …xed point of f .

Definition 1.1. (Stability) (i) The equilibrium point x of Eq.(2) is locally stable if for every x k ; x k+1 ; :::; x 1 ; x0 2 I with jx

k

xj + jx

k+1

xj + ::: + jx0

> 0; there exists

> 0 such that for all

xj < ;

we have jxn

xj
0; such that for all x k ; x k+1 ; :::; x 1 ; x0 2 I with jx

k

xj + jx

k+1

xj + ::: + jx0

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we have lim xn = x:

n!1

(iii) The equilibrium point x of Eq.(2) is global attractor if for all x

k ; x k+1 ,...,

x

1 ; x0

2 I; we have

lim xn = x:

n!1

(iv) The equilibrium point x of Eq.(2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Eq.(2). (v) The equilibrium point x of Eq.(2) is unstable if x is not locally stable. The linearized equation of Eq.(2) about the equilibrium x is the linear di¤erence equation yn+1 =

k X @f (x; x; :::; x)

@xn

i=0

yn i :

(3)

i

Now assume that the characteristic equation associated with Eq.(3) is p( ) = p0 where pi =

k

+ p1

k 1

+ ::: + pk

1

+ pk = 0;

(4)

@f (x; x; :::; x) : @xn i

Theorem A [12]: Assume that pi 2 R ; i = 1; 2; ::: and k 2 f0; 1; 2; :::g. Then k X i=1

jpi j < 1;

is a su¢ cient condition for the asymptotic stability of the di¤ erence equation yn+k + p1 yn+k

1

+ ::: + pk yn = 0; n = 0; 1; :::

Next, we introduce a fundamental theorem to prove the global attractor of the …xed points. Theorem B [26]: Let g : [a; b]k+1 ! [a; b], be a continuous function, where k is a positive integer, and where [a; b] is an interval of real numbers. Consider the di¤erence equation xn+1 = g(xn ; xn

1 ; :::; xn k );

n = 0; 1; ::: :

(5)

Suppose that g satis…es the following conditions. (1) For each integer i with 1 z1 ; z2 ; :::; zi 1 ; zi+1 ; :::; zk+1 :

i

k + 1; the function g(z1 ; z2 ; :::; zk+1 ) is weakly monotonic in zi for …xed

(2) If m; M is a solution of the system m = g(m1 ; m2 ; :::; mk+1 );

M = g(M1 ; M2 ; :::; Mk+1 );

then m = M , where for each i = 1; 2; :::; k + 1; we set mi =

m; M;

if g is non-decreasing in zi ; ; if g is non-increasing in zi ;

Mi =

M; m;

if g is non-decreasing in zi ; : if g is non-increasing in zi .

Then there exists exactly one equilibrium point x of Equation (5), and every solution of Equation (5) converges to x.

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2. LOCAL STABILITY OF THE EQUILIBRIUM POINT This section is devoted to give a detailed description about the local stability of the …xed point. The equilibrium point of Eq.(1) is given by the following equation: bx

x = ax

cx

dx

;

from which we have x=

b 1) (c

(a 2

where a 6= 1 and c 6= d: Suppose that f : (0; 1)

d)

;

! (0; 1) de…ned as following: bu

f (u; v) = au

cu

dv

:

(6)

Then, @f (u; v) =a @u

b(cu dv) bcu bdv =a+ ; (cu dv)2 (cu dv)2

@f (u; v) = @v

bu( d) = (cu dv)2

(cu

(7)

bdu : dv)2

(8)

Next, we calculate equations (7) and (8) at the equilibrium point as follows: bdx bd d(a 1) @f (x; x) =a+ =a+ =a+ = @u (cx dx)2 (c d)2 x (c d) @f (x; x) = @v

(cx

bdx = dx)2

(c

bd = d)2 x

d(a 1) = (c d)

p0 ;

p1 :

Now, the linearized di¤erence equation of Eq.(1) about the …xed point is given by yn+1 + p0 yn

1

+ p1 y n

3

= 0:

Theorem 1. Assume that jac

dj + d ja

1j < jc

dj :

Then the …xed point of Eq.(1) is locally asymptotically stable. Proof. By using Theorem A we notice that Eq.(1) is asymptotically stable if jp0 j + jp1 j < 1: Hence, we have a+

d(a 1) + (c d)

d(a 1) < 1; (c d)

which can be rearranged as follows:

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ja(c

d) + d(a

1)j + j d(a

1)j < j(c

d)j :

Therefore, jac

dj + d ja

1j < jc

dj :

This completes the proof.

3. GLOBAL STABILITY OF THE EQUILIBRIUM POINT The global attractivity character of the considered equation will be presented in this section. Theorem 2. The equilibrium point of Eq.(1) is a global attractor if a < 1: 2

Proof. Suppose that p and q are two real numbers and let f : [p; q] ! [p; q] be a function de…ned by Eq.(6). Then, equations (7) and (8) tell us that f (u; v) is increasing in u and decreasing in v. Now, we assume that (m; M ) is a solution of the following system: m = f (m; M ), and M = f (M; m): Substituting this into Eq.(6) gives

m = am M

bm ; dM bM : cM dm

cm

= aM

Then, cm2

dmM = acm2

admM

bm;

(9)

cM 2

dmM = acM 2

admM

bM:

(10)

Subtracting Eq.(9) from Eq.(10) yields c(m2

M 2 ) = ac(m2

M 2 ) + b (M

m) :

Hence, we obtain (m

M ) [c(1

a)(m + M ) + b] = 0:

Thus, when a < 1, then we have m = M: We conclude from Theorem B that the equilibrium point is a global attractor of Eq.(1).

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4. PERIODICITY OF THE SOLUTION This section will present a theorem which shows that Eq.(1) has no periodic solution. Theorem 3. Eq.(1) has no prime period two solutions. Proof. We will use contradiction to prove this theorem. Assume that Eq.(1) has a positive prime period two solutions given as follows: :::; p; q; p; q; :::: Then, bp

p = ap

cp

dp bq

q = aq

cq

dq

:

(11)

:

(12)

Equations (11) and (12) can be written as follows: p(a

1) =

q(a

1) =

b c

d b

c

d

;

;

which implies that p = q and this contradicts the fact that p 6= q:

5. SPECIAL CASE OF EQ.(1) In this section we will study the solution of the following special case: xn+1 = xn where the initial conditions x Theorem 4. Let for n = 0; 1; :::

1 fxn gn= 3

3;

x

2;

1

x

xn 1

xn

1

1

xn

;

n = 0; 1; 2; :::;

(13)

3

and x0 are nonzero real numbers with x

be the solution of Eq. (13) satisfying x

= r; x

3

2

= l; x

3 1

6= x

1

and x

6= x0 :

= k and x0 = h: Then

nk ; k r nh ; h l

x4n

3

= nk

(n

1)r

n(n

1)

x4n

2

= nh

(n

1)l

n(n

1)

x4n

=

(n + 1)k

nr

n2

nk

1

x4n

=

(n + 1)h

nl

n2

; k r nh : h l

Proof. For n = 0 the result holds. Now, we assume that n > 0 and our assumption satis…es for n x4n

7

= (n

1)k

(n

2)r

(n

1)(n

2)

(n 1)k ; k r

x4n

6

= (n

1)h

(n

2)l

(n

1)(n

2)

(n 1)h ; h l

248

2

1: That is

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x4n

5

= nk

(n

1)r

(n

1)2

(n 1)k ; k r

x4n

4

= nh

(n

1)l

(n

1)2

(n 1)h : h l

Next, it follows form Eq. (13) that x4n

= x4n

3

= nk

5

x4n

5

5

x4n

x4n

(n

nk

1)r

(n

= nk

(n

1)r

= nk

(n

1)r

= nk

(n

1)r

= nk

(n

1)r

(n 1)k k r nk (n 1)r

1)2

(n

1)r

; 7

(n 1)k k r

1)2

(n

((n

(n 1)k

1)2

(n 1)k k r

(n

2)r

(n

1)(n

2)

(n 1)k k r )

2) +

(n 1)h h l

;

(n 1)k (nk nr + r)(k n r + 1) ; k r (k r)(k n r + 1) 2nk k nr + r ; (n 1)2 k r n(nk rn + r) ; k r nk n(n 1) : k r 1)2

(n

Also, we obtain from Eq. (13)

x4n

2

= x4n = nh

nh

4

x4n

4

4

x4n

x4n

(n

(n

1)l

1)l

= nh

(n

1)l

= nh

(n

1)l

= nh

(n

1)l

= nh

(n

1)l

(n

(n

; 6

1)2

(n 1)h h l nh (n 1)l

1)2

(n 1)h h l

(n

(n

1)2

(n 1)h h l

1)h + (n

(n 1)h (nh nl + l)(h h l (h l)(h l 2nh h nl + l (n 1)2 ; h l n(nh nl + l) ; h l nh n(n 1) : h l (n

1)2

249

2)l + (n

1)(n

;

l n + 1) ; n + 1)

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Next, we will prove the third part of the theorem. Eq.(13) gives x4n

1

= x4n

3

= nk

x4n

3

3

x4n

x4n

(n

1)r

; 5

n(n

nk k r (n 1)r

1) nk

[nk

(n

1)r

n(n

nk k r]

1)

= nk

(n

1)r

n(n

1)

= nk

(n

1)r

n(n

1)

nk k

r nk

k

r

+

n(n

1)

[nk

(n

(k

+k

nk k r

1)r

(n

1)2

n r)(nk nr + r)(k (k r)(nk nr + r) n

r = (n + 1)k

nr

(n 1)k k r ]

r) n2

;

; nk k

r

:

Finally, we prove the last part of the theorem. Eq.(13) leads to x4n

= x4n

2

= nh

x4n

2

2

x4n

x4n

(n

1)l

; 4

n(n

1) nh

[nh

(n

1)l

n(n

nh h l (n 1)l nh h l]

1)

= nh

(n

1)l

n(n

1)

= nh

(n

1)l

n(n

1)

n(n

1)

[nh

(n

nh h l

1)l

(n

1)2

(n 1)h h l ]

;

nh (nh nl + l)(h l n)(h l) + ; h l (h l)(nh nl + 1) nh nh + h l n = (n + 1)h nl n2 : h l h l

Hence, the proof has done.

6. NUMERICAL SOLUTIONS This section shows some numerical examples that con…rm the results we obtained in this paper. Example 1. Let x 3 = 0:2; x stability is shown as follows:

2

= 5; x

1

= 1; x0 = 2; a = 0:5; b = 1; c = 6 and d = 1: Then, the local

plot of x(n+1)= ax(n-1)-(bx(n-1))/(cx(n-1)-dx(n-3)) 5 4 3 2

x(n)

1 0 -1 -2 -3 -4

0

10

20

30

40

50

n

Figure 1. This …gure shows the local stability of Eq.(1).

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Example 2. Assume that x 3 = 0:2; x global stability is illustrated as follows:

2

= 3; x

1

= 0:1; x0 = 2; a = 0:1; b = 1; c = 2 and d = 9: Then, the

plot of x(n+1)= ax(n-1)-(bx(n-1))/(cx(n-1)-dx(n-3)) 3 2.5 2 1.5

x(n)

1 0.5 0 -0.5 -1 -1.5

0

10

20

30

40

50

n

Figure 2. This …gure presents a global stability of Eq.(1).

Example 3. This example presents the solution of Eq.(1) when we suppose that x 1; x0 = 0:5; a = b = 1; c = 0:5 and d = 9: See Figure 3.

3

= 0:2; x

2

= 3; x

1

=

=

7; x

2

=

plot of x(n+1)= ax(n-1)-(bx(n-1))/(cx(n-1)-dx(n-3)) 7

6

5

x(n)

4

3

2

1

0

0

10

20

30

40

50

n

Figure 3. This …gure shows the solutions of Eq.(1) when x 3 = 0:2; x 2 = 3; x 1 = 1; x0 = 0:5; a = b = 1; c = 0:5 and d = 9: Example 4. This example illustrates the solution of Eq.(13) when we assume that x

251

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5; x

1

= 0:5; x0 = 8. See Figure 4.

plot of x(n+1)= ax(n-1)-(bx(n-1))/(cx(n-1)-dx(n-3)) 50

0

x(n)

-50

-100

-150

-200

-250

0

10

20

30

40

50

n

Figure 4.

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16. E. M. Elsayed, Dynamics of a Recursive Sequence of Higher Order, Communications on Applied Nonlinear Analysis, 16 (2) (2009), 37-50. 17. E. M. Elsayed, On The Global Attractivity and The Periodic Character of a Recursive Sequence, Opuscula Mathematica, 30 (4) (2010), 431-446. 18. E. M. Elsayed, A. Alghamdi, Dynamics and Global Stability of Higher Order Nonlinear Di¤erence Equation, Journal of Computational Analysis and Applications, 21 (3) (2016),493-503. 19. E. M. Elsayed, A. Alotaibi and H. A. Almaylabi, The Behavior and Closed Form of The Solutions of Some Di¤erence Equations, Journal of Computational and Theoretical Nanoscience, 13 (1-10) (2016). 20. E. M. Elsayed, F. Alzahrani, and H. S. Alayachi, Formulas and properties of some class of nonlinear di¤erence equations, Journal of Computational Analysis and Applications, 24 (8) (2018), 1517-1531. 21. E. M. Elsayed and M. M. El-Dessoky, Dynamics and Behavior of a Higher Order Rational Recursive Sequence, Advances in Di¤erence Equations, 2012 (69) (2012), 1-16. 22. E. M. Elsayed, M. Ghazel, and A. E. Matouk, Dynamical Analysis Of The Rational Di¤erence Equation xn+1 = Cxn 3 =(A + Bxn 1 xn 3 ), Journal of Computational Analysis and Applications, 23 (3) (2017), 496-507. 23. E. M. Elsayed, S. R. Mahmoud and A. T. Ali, Expression and Dynamics of The Solutions of Some Rational Recursive Sequences, Iranian Journal of Science & Technology, 38A3 (2014), 295-303. 24. T. Ibrahim, On The Third Order Rational Di¤erence Equation xn+1 = (xn xn 2 )=(xn 1 (a + bxn xn 2 )); Int. J. Contemp. Math. Sciences, 4 (27) (2009), 1321-1334. 25. R. Karatas, Global Behavior of a Higher Order Di¤erence Equation, International Journal of Contemporary Mathematical Sciences, 12 (3) (2017),133-138. 26. A. Khaliq, F. Alzahrani and E. M. Elsayed, Global Attractivity of a Rational Di¤erence Equation of Order Ten, Journal of Nonlinear Science and Applications, 9 (2016), 4465-4477. 27. A. Khaliq, and E. Elsayed, The Dynamics and Solution of Some Di¤erence Equations, Journal of Nonlinear Sciences and Applications, 9 (3) (2016), 1052-1063. 28. Y. Kostrov, On a Second-Order Rational Di¤erence Equation with a Quadratic Term, International Journal of Di¤erence Equations, 11 (2) (2016),179-202. 29. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. 30. K. Liu, P. Li, F. Han, and W. Zhong, Global Dynamics of Nonlinear Di¤erence Equation xn+1 = A + xn =xn 1 xn 2 , Journal of Computational Analysis and Applications, 24 (6) (2018), 1125-1132. 31. M. A. Obaid, E. M. Elsayed and M. M. El-Dessoky, Global Attractivity and Periodic Character of Di¤erence Equation of Order Four, Disc. Dyn. Nat. Soc., Volume 2012 (2012), Article ID 746738, 20 pages. 32. M. Saleh and M. Aloqeili, On The Di¤erence Equation yn+1 = A + ynyn k , Appl. Math. Comput., 176 (1), (2006), 359 363. xn 3 33. D. Simsek, C. Cinar and I. Yalcinkaya, On The Recursive Sequence xn+1 = 1+x , Int. J. Contemp. Math. n 1 Sci., 1 (10) (2006), 475-480. 34. Y. Su and W. Li, Global Asymptotic Stability of a Second-Order Nonlinear Di¤erence Equation, Applied Mathematics and Computation, 168 (2005), 981-989. 35. D. Tolly, Y. Yazlik and N. Taskara, Behavior of Positive Solutions of a Di¤erence Equation, 35 (3-4) (2017), 217-230. 36. X. Yang, W. Su, B. Chen, G. M. Megson and D. J. Evans, On The Recursive Sequence xn+1 = (axn 1 + bxn 2 )=(c + dxn 1 xn 2 ); Appl. Math. Comp., 162 (2005), 1485-1497. 37. X. Yan and W. Li , Global Attractivity In The Recursive Sequence xn+1 = xn =( xn 1 ), Appl. Math. Comp., 138 (2-3) (2003), 415-423. 38. E. Zayed and M. El-Moneam, On The Rational Recursive Sequence xn+1 = Axn + Bxn k + ( xn + xn k )=(Cxn + Dxn k ); Acta Applicandae Mathematicae, 111 (3) (2010), 287–301.

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A necessary condition for eventually equilibrium or periodic to a system of difference equations Wirot Tikjhaa,b,⋆ and Kunrasatree Piasu a a Faculty of Science and Technology,Pibulsongkram Rajabhat University, Phitsanulok,Thailand 65000. b Centre of Excellence in Mathematics, PERDO, CHE, Thailand. ⋆ Corresponding author. E-mail: [email protected] March 1, 2018 Abstract In this paper we consider the behavior of a special case of piecewise linear systems of difference equations with initial condition in first quadrant. We found a necessary condition that the solutions become equilibrium point or periodic with prime period 4 without using stability theorems. We constructed inductive statement to represent the behavior of the system and we apply useful lemmas in the proof of main theorem. Key words: Difference equation, Periodic solution, Stability, Equilibrium point, Piecewise linear system of difference equation. 2010 Mathematics Subject Classification: 39A10 and 65Q10.

1

Introduction

To investigate stability of system of difference equations requires theorems that involve Jacobian matrix. So the functions of the system must are differentiable. Unfortunately, piecewise linear systems of difference equations are the system with absolute value. So we can not apply the stability theorem to the piecewise linear systems. In 1978 Lozi [1] hypothesized a simplified version of H´enon’s transformation by using system of difference equation with absolute value and Lozi’s Piecewise Linear Model admits a strange attractor with a specific parameter and initial condition. Then, Devaney [2, 3] investigated Gingerbreadman map and he was shown Gingerbreadman map, a map with absolute value, being chaotic in certain regions. Moreover, Ladas’s open problem was mentioned an in article [4] as the system of difference equations: xn+1 = |xn | + ayn + b, yn+1 = xn + c|yn | + d, n = 0, 1, . . . where the initial condition (x0 , y0 ) ∈ R2 and the parameters a, b, c, and d ∈ {−1, 0, 1}. He suggests to investigate boundedness character of solutions, the 1

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global stability, and periodic nature of the solutions.There are several authors studied this open problem such as Grove et. al [4] found that every solution of a specific system is eventually periodic with period 3, Tikjha et. al [5, 6] found that the character of system is eventually periodic with some period and equilibrium point respectively. As mentioned above, we can not apply the stability theorems to this open problem. The common idea of proofs of the above systems of piecewise linear articles is to separate initial condition into few regions and find some characters of solution to the system of each region and then establishing lemmas and finally summarizing the behaviors of each system to be a theorem. Our ultimate goals is to know complete global character of system: xn+1 = |xn | − yn − b, yn+1 = xn − |yn | + 1, n = 0, 1, . . . (1) where the initial condition (x0 , y0 ) ∈ R2 and the parameters b is any positive number. In this article we will focus to a special case of System(1) when b = 3 with initial condition are some points in the first quadrant.

2

Preliminaries

The following definitions [7] are used in this article. A system of difference equations of the first order is a system of the form xn+1 = f (xn , yn ), yn+1 = g(xn , yn ), n = 0, 1, ...

(2)

where f and g are continuous functions which map R2 into R. A solution of the System(2) is a sequence {(xn , yn )}∞ n=0 which satisfies the system for all n ≥ 0. If we prescribe an initial condition (x0 , y0 ) ∈ R2 then x1 = f (x0 , y0 ), y1 = g(x0 , y0 ) x2 = f (x1 , y1 ), y2 = g(x1 , y1 ) .. . and so the solution {(xn , yn )}∞ n=0 of the System(2) exists for all n ≥ 0 and is uniquely determined by the initial condition (x0 , y0 ). A solution of the System(2) which is constant for all n ≥ 0 is called an equilibrium solution. If (xn , yn ) = (¯ x, y¯) for all n ≥ 0 is an equilibrium solution of the System(2), then (¯ x, y¯) is called an equilibrium point, or simply an equilibrium of the System(2). A solution {(xn , yn )}∞ n=0 of a system of difference equations is called eventually periodic with prime period p or eventually prime period p solution if

2

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there exists an integer N > 0 and p is the smallest positive integer such that {(xn , yn )}∞ n=0 is periodic with period p; that is, (xn+p , yn+p ) = (xn , yn ) for all n ≥ N.

(3)

The p consecutive point of the solution is called a p-cycle of System(2). We denote   x 0 , y0  x 1 , y1     x 2 , y2  x 3 , y3 as 4-cycle which consists of (x0 , y0 ), (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) in xy plain.

3

Main Results

In this section we will investigate behaviors of the following system: xn+1 = |xn | − yn − 3, yn+1 = xn − |yn | + 1, n = 0, 1, . . . .

(4)

From System(4) and by simple calculations,     −5, −1 1, −3  3, −5    and P4.2 =  1, −1 P4.1 =   5, −1 −1, 1 3, 5 −3, −1 are two 4-cycles of System(4) and equilibrium point is (−1, −1). For convenience in the later part of the proof, we let S := {(x, y)|x + 21 ≤ y ≤ x + 1}, an := 22n+3 −1 22n+2 +1 22n+2 −1 2n+4 − 1, Bn+2 := {(x, y)|x + 22n+3 , un := 22n+2 , ln := 22n+2 , δn = 2 2n+4 2n+3 2 −1 2 −1 22n+3 ≤ y < x + 22n+4 }. The proof of main theorem requires the following two lemmas. Lemma 1. Let {(xn , yn )}∞ n=1 be a solution of System(4) If there is positive integer N such that xN = −yN − 2 < 0 and yN < 0 then (xN +1 , yN +1 ) is equilibrium point (−1, −1). Proof. The proof is obvious. Lemma 2. Let {(xn , yn )}∞ n=1 be a solution of System(4) If there is positive integer N such that xN = yN − 2 ≥ 0 then {(xn , yn )}∞ n=N +6 are in P4.1 . Proof. With condition xN = yN − 2 ≥ 0 by simple calculation, we have (xN +1 , yN +1 ) = (−5, −1) ∈ P4.1 . The following theorem provides a necessary condition of equilibrium point or prime period 4 to System(4) with initial condition in first quadrant.

3

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Theorem 1. Let {(xn , yn )}∞ n=0 be a solution of System(4) and x0 , y0 ≥ 0. If (x0 , y0 ) ∈ S − Bn+2

(5)

for all integer n ≥ −1, then {(xn , yn )}∞ n=0 is eventually equilibrium point or the prime period 4 solution(P4.1 or P4.2 ). Proof. Let (x0 , y0 ) ∈ S − Bn+2 for all integer n ≥ −1. Then x0 + 21 ≤ y0 and y0 ≤ x0 + 1, so we have x1 = x0 − y0 − 3 < 0 and y1 = x0 − y0 + 1 ≥ 0, x2 = −2x0 + 2y0 − 1 ≥ 0 and y2 = −3, x3 = −2x0 + 2y0 − 1 ≥ 0 and y3 = −2x0 + 2y0 − 3. If y0 ≥ x0 + 32 then y3 ≥ 0 and so (x4 , y4 ) = (−1, 3) and (x6 , y6 ) = (−5, −1) ∈ P4.1 . Suppose that y0 < x0 + 23 then y3 < 0 and so (x4 , y4 ) = (−1, −4x0 +4y0 −3) If y4 = −4x0 + 4y0 − 3 < 0 then we have (x5 , y5 ) ∈ B1 . This contradicts Condition(5). Suppose that y4 ≥ 0, so x5 = 4x0 − 4y0 + 1 < 0 and y5 = 4x0 − 4y0 + 3 ≤ 0, x6 = −8x0 + 8y0 − 7 and y6 = 8x0 − 8y0 + 5 < 0. If x6 < 0, that is x6 = −y6 − 2 < 0, then applying Lemma(1), we have (x7 , y7 ) = (−1, −1). Suppose that x6 ≥ 0, that is x0 + 87 ≤ y0 < x0 + 32 , then (x7 , y7 ) = (−16x0 + 16y0 − 15, −1). If x7 < 0, then x0 + 78 ≤ y0 < x0 + 15 16 , and so (x8 , y8 ) ∈ B2 . This contradicts 15 Condition(5). Suppose that x7 ≥ 0. That is x0 + 16 ≤ y0 < x0 + 23 , so x8 = −16x0 + 16y0 − 17 and y8 = −16x0 + 16y0 − 15 ≥ 0. 17 ≤ y0 < x0 + 32 . Applying Lemma(2), (x9 , y9 ) ∈ P4.1 . If x8 ≥ 0 that is x0 + 16 15 17 Suppose that x8 < 0 that is x0 + 16 ≤ y0 < x0 + 16 . We have x9 = 32x0 −32y0 + 29 < 0 and y9 = −1, x10 = −32x0 + 32y0 − 31 and y10 = 32x0 − 32y0 + 29 < 0. 31 If x10 < 0 that is x0 + 15 16 ≤ y0 < x0 + 32 . We have (x11 , y11 ) = (−1, −1). 31 Suppose that x10 ≥ 0 that is x0 + 32 ≤ y0 < x0 + 17 16 . We have a closed form of inductive statement on n ≥ 1 and let P (n) be the following statement: For (x0 , y0 ) ∈ Rn = {(x, y)|x + an ≤ y < x + un }, then x4n+6 ≥ 0 and so x4n+7 = −22n+4 x0 + 22n+4 y0 − δn y4n+7 = −1. If (x0 , y0 ) ∈ Bn+2 = {(x, y)|x + an ≤ y < x + ln+1 }, then x4n+7 < 0. If (x0 , y0 ) ∈ Rn − Bn+2 = {(x, y)|x + ln+1 ≤ y < x + un }, then x4n+7 ≥ 0 and so x4n+8 = −22n+4 x0 + 22n+4 y0 − δn − 2 y4n+8 = −22n+4 x0 + 22n+4 y0 − δn ≥ 0. If (x0 , y0 ) ∈ Rn∗ = {(x, y)|x + un+1 ≤ y < x + un }, then x4n+8 ≥ 0 and so x4n+9 = −5 y4n+9 = −1. If (x0 , y0 ) ∈ (Rn − Bn+2 ) − Rn∗ = {(x, y)|x + ln+1 ≤ y < x + un+1 }, then x4n+8 < 0 and so x4n+9 = 22n+5 x0 − 22n+5 y0 + 2δn − 1 < 0 y4n+9 = −1 x4n+10 = −22n+5 x0 + 22n+5 y0 − 2δn − 1 y4n+10 = 22n+5 x0 − 22n+5 y0 + 2δn − 1 < 0.

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˜ n = {(x, y)|x + ln+1 ≤ y < x + an+1 }, then x4n+10 < 0 and so If (x0 , y0 ) ∈ R x4n+11 = −1 y4n+11 = −1. If (x0 , y0 ) ∈ Rn+1 = {(x, y)|x + an+1 ≤ y < x + un+1 }, then x4n+10 ≥ 0. 31 We shall first show that P (1) is true. For (x0 , y0 ) ∈ R1 = {(x, y)|x + 32 ≤y< 17 x + 16 } and δ1 = 63, we have x10 = −32x0 + 32y0 − 31 ≥ 0 and so x4(1)+7 = x11 = −22(1)+4 x0 + 22(1)+4 y0 − δ1 y4(1)+7 = y11 = −1. 31 If (x0 , y0 ) ∈ B3 = {(x, y)|x + 32 ≤ y < x + 63 64 }, then x11 = −64x0 + 64y0 − 63 < 0. 63 If (x0 , y0 ) ∈ R1 − B3 = {(x, y)|x + 64 ≤ y < x + 17 16 }, then x11 = −64x0 + 64y0 − 63 ≥ 0 and so x4(1)+8 = x12 = −64x0 + 64y0 − 65 = −22(1)+4 x0 + 22(1)+4 y0 − δ1 − 2 y4(1)+8 = y12 = −64x0 + 64y0 − 63 = −22(1)+4 x0 + 22(1)+4 y0 − δ1 ≥ 0. 65 If (x0 , y0 ) ∈ R1∗ = {(x, y)|x + 64 ≤ y < x + 17 16 }, then x12 = −64x0 + 64y0 − 65 ≥ 0 and so x4(1)+9 = x13 = −5 y4(1)+9 = y13 = −1. 63 If (x0 , y0 ) ∈ (R1 − B3 ) − R1∗ = {(x, y)|x + 64 ≤ y < x + 65 64 }, then x12 = −64x0 + 64y0 − 65 < 0 and so x4(1)+9 = x13 = 128x0 − 128y0 + 125 = 22(1)+5 x0 − 22(1)+5 y0 + 2δ1 − 1 < 0 y4(1)+9 = y13 = −1 x4(1)+10 = x14 = −128x0 + 128y0 − 127 = −22(1)+5 x0 + 22(1)+5 y0 − 2δ1 − 1 y4(1)+10 = y14 = 128x0 − 128y0 + 125 = 22(1)+5 x0 − 22(1)+5 y0 + 2δ1 − 1 < 0. ˜ 1 = {(x, y)|x + 63 ≤ y < x + 127 }, then x14 = −128x0 + If (x0 , y0 ) ∈ R 64 128 128y0 − 127 < 0 and so x4(1)+11 = x15 = −1 y4(1)+11 = y15 = −1. 65 If (x0 , y0 ) ∈ R2 = {(x, y)|x + 127 128 ≤ y < x + 64 }, then x4(1)+10 = −128x0 + 128y0 − 127 ≥ 0. Therefore P (1) is true, as required. Suppose P (k) is true for a positive integer k. If (x0 , y0 ) ∈ Rk+1 = {(x, y)|x+ 2k+4 22k+5 −1 ≤ y < x + 2 22k+4+1 }, then 22k+5 x4k+10 = −22k+5 x0 +22k+5 y0 −2δk −1 ≥ 0 and y4k+10 = 22k+5 x0 −22k+5 y0 + 2δk − 1 < 0 and so x4(k+1)+7 = x4k+11 = −22k+6 x0 + 22k+6 y0 − (4δk + 3) = −22(k+1)+4 x0 + 22(k+1)+4 y0 − δk+1 y4(k+1)+7 = y4k+11{ = −1. } If (x0 , y0 ) ∈ Bk+3 = (x, y)|x +

22k+5 −1 22k+5

≤y 0, give (rather general and typical examples of H-functions, not reducible to Gfunctions):     (α1 , A1 ), . . . , (α` , A` ) : z =` Ψm (αj , Aj )1,` ; (βj , Bj )1,m ; z ` Ψm (β1 , B1 ), . . . , (βm , Bm ) ∞ X Γ(α1 + nA1 ) . . . Γ(α` + nA` ) z n = , Γ(β1 + nB1 ) . . . Γ(βm + nBm ) n! n=0

2010 Mathematics Subject Classification: Primary 30C45, 30C50; Secondary 30C80 Key words and phrases: Coefficient estimates, Faber polynomial expansion, Wright hypergeometric functions, Subordinate * Corresponding author 1

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where 1 +

m P

Bn −

n=1

` P

An ≥ 0 (`, m ∈ N = {1, 2, . . .}) and for suitably bounded

n=1

values of |z|. Now the linear operator is introduced comprising of the generalized hypergeometric function from Srivastava [19] (see [7]) and Wright [24]. Let `, m ∈ N and suppose that the parameters α1 , A1 , . . . , α` , A` and β1 , B1 , . . . , βm , Bm are also positive real numbers. Then, corresponding to a function   ` Φm (αj , Aj )1,` ; (βj , Bj )1,m ; z defined by     ` Φm (αj , Aj )1,` ; (βj , Bj )1,m ; z = Ωz ` Ψm (αj , Aj )1,` ; (βj , Bj )1,m ; z ,  ` −1  m  Q Q where Ω = Γ(αj ) Γ(βj ) , we consider a linear operator j=1

j=1

  W (αj , Aj )1,` ; (βj , Bj )1,m : A −→ A defined by the following Hadamard product     W (αj , Aj )1,` ; (βj , Bj )1,m f (z) := z ` Φm (αj , Aj )1,` ; (βj , Bj )1,m ; z ∗ f (z). We observe that, for f (z) of the form (1.1), we have ∞ X  W (αj , Aj )1,` ; (βj , Bj )1,m f (z) := z + ϕn an z n ,



n=2

where

Ω Γ(α1 + A1 (n − 1)) . . . Γ(α` + A` (n − 1)) . (n − 1)! Γ(β1 + B1 (n − 1)) . . . Γ(βm + Bm (n − 1)) If, for convenience, we write   ` Wm f (z) = W (α1 , A1 ), . . . , (α` , A` ); (β1 , B1 ), . . . , (βm , Bm ) f (z). ϕn =

The Koebe one-quarter theorem [6] ensures that the image of U under every univalent function f ∈ S contains a disk of radius 1/4. Therefore, every function f ∈ S has an inverse f −1 , which is defined by  

1 −1 −1 f f (z) = z (z ∈ U) and f f (w) = w |w| < r0 (f ) ; r0 (f ) = , 4 where (1.2) g(w) = f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 − (5a32 − 5a2 a3 + a4 )w4 + · · · =: w +

∞ X

bn wn .

n=2

A function f ∈ S is said to be bi-univalent in U if both f and f −1 are univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1.1). Determination of the bounds for the coefficients an is an important problem in geometric function theory as they give information about the geometric properties of these functions. For a brief history and interesting examples in the class Σ, see [13]. Recently, many researchers introduced and investigated subclasses of bi-univalent functions and obtained bounds for the initial coefficients, see,

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for example, [4, 15, 20–22, 25, 27]. But the coefficient problem for each of the Taylor-Maclaurin coefficients |an | (n ∈ N\{1, 2, 3}, is still an open problem. A function f (z) is said to be quasi-subordinate to φ(z) in the open unit disk U if there exist analytic functions ψ(z) and w(z), with w(0) = 0 such that |ψ(z)| ≤ 1, |w(z)| < 1 and f (z) = ψ(z)φ(w(z)). Denote this quasi-subordination by f (z) ≺q˜ φ(z). For ψ(z) = 1, the quasi-subordination reduces to the subordination (see [17, 18]). Throughout this paper, we let φ(z) is analytic function in the unit disk U with φ(0) = 1 such that φ(z) = 1 + C1 z + C2 z 2 + C3 z 3 + · · ·

(C1 > 0)

and assume that the function ψ(z) is analytic in the unit disk U and |ψ(z)| ≤ 1 such that ψ(z) = D0 + D1 z + D2 z 2 + D3 z 3 + · · · . l,m l,m Recently, Cho et al., [5] introduced subclasses GΣ (γ, λ, φ) and BΣ (γ, λ, φ) of Σ and only obtained estimates on the coefficients |a2 | and |a3 | for functions in these subclasses.

Definition 1.1. [5] A function f ∈ Σ given by (1.1) is said to be in the class l,m (γ, λ, φ) if the following conditions are satisfied: GΣ ! l f (z))0 1 z(Wm − 1 ≺q˜ (φ(z) − 1) l f (z) + λz(W l f (z))0 γ (1 − λ)Wm m and 1 γ

l g(w))0 w(Wm −1 l g(w) + λw(W l g(w))0 (1 − λ)Wm m

! ≺q˜ (φ(w) − 1),

where γ ∈ C\{0}, 0 ≤ λ < 1, z, w ∈ U and the function g is given by (2.1). Definition 1.2. [5] A function f ∈ Σ given by (1.1) is said to be in the class l,m (γ, λ, φ) if the following conditions are satisfied: BΣ ! l f (z))0 1 z 1−λ (Wm − 1 ≺q˜ (φ(z) − 1) l f (z)]1−λ γ [Wm and 1 γ

l g(w))0 w1−λ (Wm −1 l g(w)]1−λ [Wm

! ≺q˜ (φ(w) − 1),

where γ ∈ C\{0}, λ ≥ 0, z, w ∈ U and the function g is given by (2.1). Lemma 1.3. [6] Let u(z) be analytic P in then unit disk U with u(0) = 0 and |u(z)| < 1 and suppose that u(z) = ∞ n=1 pn z . Then |pn | 5 1 (n ∈ N). Lemma 1.4. [9] Let the function w in the Schwarz function is given by w(z) = ∞ P wn z n , where z ∈ U. Then for every complex number s, n=1

|w2 + sw12 | ≤ 1 + (|s| − 1)|w12 |.

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Faber [8] introduced the Faber polynomials, which play an important role in various areas of mathematical sciences, especially in geometric function theory. By using the Faber polynomial expansion of functions f ∈ S of the form (1.1), the coefficients of its inverse map g = f −1 may be expressed, (see for details [1] and [2]), ∞ X 1 −n −1 g(w) = f (w) = w + (1.3) K (a2 , a3 , · · · , an )wn , n n−1 where bn = −n Kn−1 =

n=2 1 −n n Kn−1 (a2 , a3 , . . . , an ), and

(−n)! (−n)! an−1 + an−3 a3 (−2n + 1)!(n − 1)! 2 (2(−n + 1))!(n − 3)! 2 (−n)! (−n)! + an−4 a4 + an−5 2 (−2n + 3)!(n − 4)! (2(−n + 2))!(n − 5)! 2 (−n)! ×[a5 + (−n + 2)a23 ] + an−6 [a6 + (−2n + 5)a3 a4 ] (−2n + 5)!(n − 6)! 2 X n−j + a2 Vj j≥7

such that Vj with 7 ≤ j ≤ n is a homogeneous polynomial in the variables −n a2 , a3 , · · · , an , (see for details [2]). In particular, the first three terms of Kn−1 are 1 −3 1 −4 1 −2 K1 = −a2 , K2 = 2a22 − a3 , K = −(5a32 − 5a2 a3 + a4 ). 2 3 4 3 In general, for any p ∈ Z = {0, ±1, ±2, · · · }, an expansion of Knp is (see for details [1, 23] or [2, page 349]) Knp = pan+1 +

p! p! p(p − 1) 2 Dn + D3 + · · · + Dn , 2 (p − 3)!3! n (p − n)!n! n

where (1.4)

Dnm (a2 , a3 , · · · , an ) =

∞ X m!(a2 )µ1 · · · (an )µn µ1 ! · · · µ n !

n=1

and the sum is taken over all nonnegative integers µ1 , ..., µn satisfying  µ1 + µ2 + · · · + µn = m, µ1 + 2µ2 + · · · + nµn = n. We noth that it is clear that Dnn (a2 , a3 , · · · , an ) = an2 . Lemma 1.5. [2, Equation (1.6) and (1.7)] Let f (z) = z + a2 z 2 + a3 z 3 + · · · ∈ S, and k ∈ Z then we have the following expansion   ∞ X zf 0 (z) f (z) k n+k−1 =1− Fn−1 (a2 , a3 , . . . , an )z n−1 f (z) z n=1  ∞  X n−1 k =1+ 1+ Kn−1 (a2 , a3 , . . . , an )z n−1 , k n=2

n+k−1 where the first Faber polynomials Fn−1 (a2 , a3 , . . . , an ) are given by

F1k+1 (a2 ) = (1 + λ)a2 ,

F2k+2 (a2 , a3 ) =

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Several researchers have solved coefficient estimates problem for various subclasses of bi-univalent functions by using Faber polynomial expansions, see for example [10, 11, 20, 26]. In the present paper, by using the Faber polynomial expansions we obtain estimates of coefficients |an | where n ≥ 3, of functions in l,m l,m the subclasses GΣ (γ, λ, φ) and BΣ (γ, λ, φ) of Σ with various special cases. 2. Main results l,m First, we can write that the Faber polynomial expansion for f ∈ GΣ (γ, λ, φ) given by (1.1) is in the form of

(2.1) l z(Wm f (z))0 −1 l f (z) + λz(W l f (z))0 (1 − λ)Wm m

1 γ

! =

∞ 1X Fn−1 (ϕ2 a2 , ϕ3 a3 , · · · , ϕn an )z n−1 , γ n=2

where F1 (ϕ2 a2 ) = (1 − λ)ϕ2 a2 , F2 (ϕ2 a2 , ϕ3 a3 ) = (λ2 − 1)(ϕ2 a2 )2 + 2(1 − λ)ϕ3 a3 .

In general, Fn−1 (ϕ2 a2 , ϕ3 a3 , · · · , ϕn an ) = (1 − λ)(n − 1)ϕn an +

n−2 X

  Kl−1 (1 + λ)ϕ2 a2 , (1 + 2λ)ϕ3 a3 , · · · , (1 + lλ)ϕl+1 al+1 (1 − λ)(n − l − 1)ϕn−l an−l .

l=1

Then to simplify, we define: F (z) ≺q˜ (φ(z) − 1)

(2.2)

and G(w) ≺q˜ (φ(w) − 1),

where 1 F (z) = γ

!

l z(Wm f (z))0 l f (z)+λz(W l f (z))0 (1−λ)Wm m

1 F (z) = γ

−1

l z 1−λ (Wm f (z))0 l f (z)]1−λ [Wm

! −1

1 and G(w) = γ

l w(Wm g(w))0 l g(w)+λw(W l g(w))0 (1−λ)Wm m

1 and G(w) = γ

l w1−λ (Wm g(w))0 l g(w)]1−λ [Wm

! −1 ,

! −1 .

In addition, by definition of quasi-subordinate there exist functions ψ P P∞analytic n and v(z) = n , so that and u, v : U → U, where u(z) = ∞ p z q z n n n=1 n=1 (2.3)

F (z) = ψ(z)[φ(u(z)) − 1]

and G(w) = ψ(w)[φ(v(w)) − 1],

where by equation (1.4) we have (2.4) ψ(z)[φ(u(z)) − 1] = [C1 p1 z + (C1 p2 + C2 p21 )z 2 + · · · ][D0 + D1 z + D2 z 2 + · · · ] X X ∞ X n ∞ = Ck Dnk (p1 , p2 , · · · , pn )z n Dn z n n=1 k=1

n=0

and (2.5)

ϕ(w)h(v(w)) =

X ∞ X n

Ck Dnk (q1 , q2 , · · ·

n=1 k=1

, qn )w

n

X ∞

Dn w n .

n=0

Now, we obtain the following coefficient estimates for these subclasses.

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l,m Theorem 2.1. Let the function f ∈ GΣ (γ, λ, φ) be given by (1.1) and D0 6= 0. If ak = 0 for 2 ≤ k ≤ n − 1, then
|γ| C1 + |Dn−1 | , n ≥ 3. |an | ≤ (1 − λ)(n − 1)ϕn l,m Theorem 2.2. Let the function f ∈ BΣ (γ, λ, φ) be given by (1.1) and D0 6= 0. If ak = 0 for 2 ≤ k ≤ n − 1, then
|γ| C1 + |Dn−1 |
, |an | ≤ n ≥ 3. λ + (n − 1) ϕn l,m Theorem 2.3. Let the function f ∈ GΣ (γ, λ, φ) be given by (1.1) and C1 ≥ |C2 |. Then

√ |γD0 | C1 C1 |a2 | ≤ q . |γD0 |C12 (λ2 − 1)ϕ22 + 2(1 − λ)ϕ3 + (C1 − |C2 |)(1 − λ)2 ϕ22 l,m (γ, λ, φ) be given by (1.1) and C1 ≥ Theorem 2.4. Let the function f ∈ BΣ |C2 |. Then

√ |γD0 |C1 2C1 |a2 | ≤ q . |γD0 |C12 (λ − 1)(λ + 2)ϕ22 + 2(λ + 2)ϕ3 + 2(C1 − |C2 |)(1 + λ)2 ϕ22

Remark 2.5. (1) If we take ψ(z) = 1 in Theorem 2.1, then we obtain estimates of coefficients |an | (n ≥ 3) for subclass defined by Murugusundaramoorthy in [14, Theorem 2.2]. (2) If we take ψ(z) = 1 in Theorem 2.3, then we obtain an improvement of the estimates obtained for |a2 | by Murugusundaramoorthy in [14, Theorem 2.2]. (3) By setting λ = 0, γ = 1 and ` = 2, m = 1 with A1 = A2 = B1 = α1 = α2 = β1 = 1, in Theorem 2.3, we get ϕn = 1 and then we obtain an improvement of the estimates obtained for |a2 | by Algahtani in [3, Theorem 2.5]. (4) By setting λ = γ = 1 and ` = 2, m = 1 with A1 = A2 = B1 = α1 = α2 = β1 = 1, in Theorem 2.4, we get ϕn = 1 and then we obtain an improvement of the estimates obtained for |a2 | by Algahtani in [3, Theorem 2.2]. (5) By setting λ = 0, γ = 1 and ` = 2, m = 1 with α1 = 2 and A1 = A2 = B1 = α2 = β1 = 1, (W12 f (z) = zf 0 (z)) in Theorem 2.3, then we obtain an improvement of the estimates obtained for |a2 | by Algahtani in [3, Theorem 2.8]. (6) By setting ψ(z) = 1, λ = 0, γ = 1 and ` = 2, m = 1 with α1 = 2 and A1 = A2 = B1 = α2 = β1 = 1, in Theorem 2.3, then we obtain an improvement of the estimates obtained for |a2 | by Algahtani in [3, Theorem 2.9]. (7) By setting ψ(z) = 1, λ = 1, γ = 1 and ` = 2, m = 1 with α1 = a, α2 = b, β1 = c, in Theorem 2.4, then we obtain an improvement of the estimates obtained for |a2 | by Omar et al., in [16, Theorem 1]. (8) By setting ψ(z) = 1, λ = 0, γ = 1 and ` = 2, m = 1 with A1 = A2 = B1 = α1 = α2 = β1 = 1, in Theorem 2.3, we get ϕn = 1 and then we

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obtain an improvement of the estimates obtained for |a2 | by Ali et al., in [4, Corollary 2.1]. (9) By setting ψ(z) = 1, λ = γ = 1 and ` = 2, m = 1 with A1 = A2 = B1 = α1 = α2 = β1 = 1, in Theorem 2.4, we get ϕn = 1 and then we obtain an improvement of the estimates obtained for |a2 | by Ali et al., in [4, Theorem 2.1]. (10) Theorem 2.3 and Theorem 2.4 are improvements of the results obtained by Cho et al. [5], respectively. 3. Proof of Theorems Proof of Theorem 2.1. For this work, let  l f (z))0 1 z(Wm F (z) = − 1 l f (z) + λz(W l f (z))0 γ (1 − λ)Wm m and G(w) =

 l g(w))0 w(Wm 1 − 1 . l g(w) + λw(W l g(w))0 γ (1 − λ)Wm m

l,m (γ, λ, φ), we have the expansion (2.1) and for the For the function f ∈ GΣ inverse map g = f −1 , considering (1.2), we get that

(3.1)

∞ 1X G(w) = Fn−1 (ϕ2 b2 , ϕ3 b3 , · · · , ϕn bn )wn−1 . γ n=2

Comparing the coefficients of (2.1) and (2.4), we conclude (3.2) " n−2   X 1 (1 − λ)(n − 1)ϕn an + Kl−1 (1 + λ)ϕ2 a2 , (1 + 2λ)ϕ3 a3 , · · · , (1 + lλ)ϕl+1 al+1 γ l=1 # n−1 t XX × (1 − λ)(n − l − 1)ϕn−l an−l = Dn−1 + Ck Dtk (p1 , p2 , · · · , pt )Dn−(t+1) . t=1 k=1

Similarly, from (3.1) and (2.5), we have (3.3) " n−2   X 1 (1 − λ)(n − 1)ϕn bn + Kl−1 (1 + λ)ϕ2 b2 , (1 + 2λ)ϕ3 b3 , · · · , (1 + lλ)ϕl+1 bl+1 γ l=1 # n−1 t XX × (1 − λ)(n − l − 1)ϕn−l bn−l = Dn−1 + Ck Dtk (q1 , q2 , · · · , qt )Dn−(t+1) . t=1 k=1

For ak = 0 where 2 ≤ k ≤ n − 1 and D0 6= 0, we have p2 = p3 = · · · = pn−2 = 0 and q2 = q3 = · · · = qn−2 = 0. So from (3.2) and also from equation (1.3) and (3.3) we get, respectively, (3.4)

1 (1 − λ)(n − 1)ϕn an = C1 pn−1 + Dn−1 γ

and 1 1 (3.5) (1 − λ)(n − 1)ϕn bn = − (1 + λ)(n − 1)ϕn an = C1 qn−1 + Dn−1 . γ γ

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By solving either of the equations (3.4) and (3.5) for an and using Lemma 1.3, we obtain
|γ| C1 + |Dn−1 | |γ||C1 pn−1 + Dn−1 | |an | = ≤ (1 − λ)(n − 1)ϕn (1 − λ)(n − 1)ϕn and this completes the proof.



Proof of Theorem 2.2. Let F (z) =

 l f (z))0 1  z 1−λ (Wm − 1 l f (z)]1−λ γ [Wm

and G(w) =

 l g(w))0 1  w1−λ (Wm − 1 . l g(w)]1−λ γ [Wm

l,m For the function f ∈ BΣ (γ, λ, φ), by Lemma1.5 we have

(3.6) F (z) =

 ∞  1X n−1 λ 1+ Kn−1 (ϕ2 a2 , ϕ3 a3 , . . . , ϕn an )z n−1 . γ λ n=2

For its inverse map g = f −1 , regarding the equality (1.2) we have (3.7)

 ∞  n−1 1X λ 1+ Kn−1 (ϕ2 b2 , ϕ3 b3 , . . . , ϕn bn )wn−1 . G(w) = γ λ n=2

Comparing the coefficients of (3.6), and (2.4), we conclude that (3.8)   t n−1 XX 1 n−1 λ Ck Dtk (p1 , p2 , · · · , pt )Dn−(t+1) . 1+ Kn−1 (ϕ2 a2 , ϕ3 a3 , . . . , ϕn an ) = Dn−1 + γ λ t=1 k=1

Similarly, from (3.7) and (2.5), we have (3.9)   n−1 t XX 1 n−1 λ 1+ Kn−1 (ϕ2 b2 , ϕ3 b3 , . . . , ϕn bn ) = Dn−1 + Ck Dtk (q1 , q2 , · · · , qt )Dn−(t+1) . γ λ t=1 k=1

Since ak = 0 where 2 ≤ k ≤ n − 1, and D0 6= 0 from (3.8) and (3.9) we get, respectively,
1 λ + (n − 1) ϕn an = C1 pn−1 + Dn−1 γ and −


1 λ + (n − 1) ϕn an = C1 qn−1 + Dn−1 . γ

By solving either of the above equations for an and using Lemma 1.3, we conclude the desired results and this completes the proof. 

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Proof of Theorem 2.3. For n = 2 and n = 3 in (3.2) and (3.3), respectively, we obtain (1 − λ)ϕ2 a2 = γD0 C1 p1 ,

(3.10)

(λ2 − 1)ϕ22 a22 + 2(1 − λ)ϕ3 a3 = γD0 [C1 p2 + C2 p21 ] + γD1 C1 p1 ,

(3.11)

−(1 − λ)ϕ2 a2 = γD0 C1 q1 ,

(3.12) 2

(3.13) (λ −

1)ϕ22 a22

+ 2(1 − λ)ϕ3 (2a22 − a3 ) = γD0 [C1 q2 + C2 q12 ] + γD1 C1 q1 .

From (3.10) and (3.12), we get p1 = −q1 .

(3.14)

Adding (3.11) and (3.13) and using (3.14) we obtain  2   C2 2  C2 2 p 1 + q2 + q . 2(λ − 1)ϕ22 + 4(1 − λ)ϕ3 a22 = γD0 C1 p2 + C1 C1 1 By using Lemma 1.4 we have 2   2(λ − 1)ϕ22 + 4(1 − λ)ϕ3 |a2 |2 ≤ |γD0 |C1 |p2 + |C2 | p21 | + |q2 + C2 q12 | C1 C1  |C2 | − C1
2  ≤ |γD0 |C1 2 + 2 |p1 | C1    (|C2 | − C1 )(1 − λ)2 ϕ22 |a22 | = |γD0 |C1 2 + 2 . |γD0 |2 C13 After simplification we have   2 2 2 2 2 |γD0 |C1 2(λ − 1)ϕ2 + 4(1 − λ)ϕ3 + 2(C1 − |C2 |)(1 − λ) ϕ2 |a2 |2 ≤ 2|γD0 |2 C13 , which implies that |a2 |2 ≤



|γD0 |C12 (λ2

−

1)ϕ22

|γD0 |2 C13 + 2(1 − λ)ϕ3 + (C1 − |C2 |)(1 − λ)2 ϕ22

and this completes the proof.



Proof of Theorem 2.4. For n = 2 and n = 3 in (3.8) and (3.9), respectively, we obtain (1 + λ)ϕ2 a2 = γD0 C1 p1 , (λ − 1)(λ + 2) 2 2 ϕ2 a2 + (λ + 2)ϕ3 a3 = γD0 [C1 p2 + C2 p21 ] + γD1 C1 p1 , 2 −(1 + λ)ϕ2 a2 = γD0 C1 q1 ,   (λ − 1)(λ + 2) 2 ϕ2 + 2(λ + 2)ϕ3 a22 − (λ + 2)ϕ3 a3 = γD0 [C1 q2 + C2 q12 ] + γD1 C1 q1 . 2 With similar method to Theorem 2.3 we get the desired results and this completes the proof.  Acknowledgements This research was supported 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).

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E. ANALOUEI ADEGANI, N. E. CHO, A. MOTAMEDNEZHAD AND M. JAFARI

References [1] H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math. 130 (2006), 179-222. [2] H. Airault and J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math. 126 (2002), 343-367. [3] O. Algahtani, Estimates of initial coefficients for certain subclasses of biunivalent functions involving quasi-subordination, J. Nonlinear Sci. Appl. 10 (2017), 1004-1011. [4] R. M. Ali, S. K. Lee, V. Ravichandran and S. Subramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25 (2012), 344-351. [5] N. E. Cho, G. Murugusundaramoorthy and K. Vijaya, Bi-univalent functions of complex order based on quasi-subordinate conditions involving Wright hypergeometric functions, J. Comput. Anal. Appl. 24 (2018), 58-70. [6] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, 1983. [7] J. Dziok and R. K. Raina, Families of analytic functions associated with the Wright generalized hypergeometric function, Demonstratio Math. 37 (2004), 533-542. ¨ [8] G. Faber, Uber polynomische Entwickelungen, Math. Ann. 57 (1903) 389408. [9] F. R. Keogh and E. P. Merkes, A coeffcient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8-12. [10] S. G. Hamidi and J. M. Jahangiri, Faber polynomial coefficients of bisubordinate functions, C. R. Math. Acad. Sci. Paris. 354 (2016), 365-370. [11] J. M. Jahangiri, S. G. Hamidi and S. A. Halim, Coefficients of bi-univalent functions with positive real part derivatives, Bull. Malays. Math. Sci. Soc. 37 (2014), 633-640. [12] S. Kanas, Seong-A. Kim and S. Sivasubramanian, Verification of Brannan and Clunie’s conjecture for certain subclasses of bi-univalent function, Ann. Polonici Mathematici. 113 (3) (2015), 295-304. [13] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63-68. [14] G. Murugusundaramoorthy, Subclasses of bi-univalent functions of complex order based on subordination conditions involving Wright hypergeometric, J. Math. Fund. Sci. 47 (2015), 60-75. [15] G. Murugusundaramoorthy, T. Janani and N. E. Cho, Bi-univalent functions of complex order based on subordinate conditions involving HurwitzLerch zeta function, East Asian Math. J. 32 (2016), 47-59. [16] R. Omar, S. A. Halim and A. Janteng, Subclasses of bi-univalent functions associated with Hohlov operator, WASET Int. J. Math. Comput. Sci. 11 (2017), 428-431. [17] M. S. Robertson, Quasi-subordinate functions, In: Mathematical Essays Dedicated to A. J. MacIntyre, Ohio University Press, Athens, OH (1970), 311-330. [18] M. S. Robertson, Quasi-subordination and coefficient conjecture, Bull. Amer. Math. Soc. 76 (1970), 1-9.

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BI-UNIVALENT FUNCTIONS

11

[19] H. M. Srivastava, Some Fox’s-Wright generalized hypergeometric functions and associated families of convolution operators, Appl. Anal. Discrete Math. 1 (2007), 56-71. [20] H. M. Srivastava, S. S. Eker and R. M. Ali, Coeffcient bounds for a certain class of analytic and bi-univalent functions, Filomat. 29 (2015), 1839-1845. [21] H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma-Minda type, RACSAM. (2017). https://doi.org/10.1007/s13398-017-0416-5 [22] H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and biunivalent functions, Appl. Math. Lett. 23 (2010), 1188-1192. [23] P. G. Todorov, On the Faber polynomials of the univalent functions of class Σ, J. Math. Anal. Appl. 162 (1991), 268-276. [24] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London. Math. Soc. 46 (1946), 389-408. [25] Q-H. Xu, H-G. Xiao and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput. 218 (2012), 11461-11465. [26] A. Zireh, E. Analouei Adegani and S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions defined by subordination, Bull. Belg. Math. Soc. Simon Stevin. 23 (2016), 487-504. [27] A. Zireh and E. Analouei Adegani, Coefficient estimates for a subclass of analytic and bi-univalent functions, Bull. Iranian Math. Soc. 42 (2016), 881-889. 1

Faculty of Mathematical Sciences, University of Shahrood, P.O.Box 31636155, Shahrood, Iran E-mail address: [email protected], [email protected] 2 Department of Applied Mathematics Pukyong National University Busan 608737, Korea E-mail address: [email protected] 3 Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran E-mail address: [email protected]

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Conformable Fractional Approximation by Choquet integrals George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we present the conformable fractional quantitative approximation of positive sublinear operators to the unit operator. These are given a precise Choquet integral interpretation. Initially we start with the study of the conformable fractional rate of the convergence of the well-known Bernstein-Kantorovich-Choquet and Bernstein-Durrweyer-Choquet polynomial Choquet-integral operators. Then we study in the fractional sense the very general comonotonic positive sublinear operators based on the representation theorem of Schmeidler (1986) [11]. We continue with the conformable fractional approximation by the very general direct Choquetintegral form positive sublinear operators. The case of convexity is also studied throughly and the estimates become much simpler. All approximations are given via inequalities involving the modulus of continuity of the approximated function’s higher order conformable fractional derivative.

2010 AMS Mathematics Subject Classi…cation: 26A33, 41A17, 41A25, 41A35, 41A36. Keywords and Phrases: Jackson type inequality, Choquet integral, Conformable Fractional derivative, comonotonicity of functions and operators, BernsteinKantorovich-Choquet and Bernstein-Durrmeyer-Choquet operators, convexity.

1

Introduction

G. Choquet (1953) ([4]), introduced the capacities and his integral. Initially these were applied to statistical mechanics and potential theory, and they gave rise to the study of non-additive measure theory. Slowly but steady these ideas of Choquet started to attract economists especially after the very important 1

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work of Shapley (1953) ([13]) in the study of cooperative games. Capacities and Choquet integrals became main stream to Decision theorists since 1989 when D. Schmeidler ([12]) was the …rst to use them in an axiomatic model of choice with non-additive beliefs. The expected utility results are strengthned by the use of Choquet capacities instead of probability measures. In now days Choquet integral has wide applications, among others, to decision making under risk and uncertainty, in …nance, in economics, in portofolio problems and in insurance. Our motivation also comes from the foundations of Bayesian decision theory and subjective probability. Because of the paramount importance of Choquet integral, we decided to research the related positive sublinear operators approximation, part of it is exhibited in this work in the conformable fractional sense.

2

Background - I

Next we present brie‡y about the Choquet integral, see also [8]. We make De…nition 1 Consider 6= ? and let C be a -algebra of subsets in . (i) (see, e.g., [14], p. 63) The set function : C ! [0; +1] is called a monotone set function (or capacity) if (?) = 0 and (A) (B) for all A; B 2 C, with A B. Also, is called submodular if (A [ B) + (A \ B)

(A) + (B) , for all A; B 2 C:

is called bounded if ( ) < +1 and normalized if ( ) = 1: (ii) (see, e.g., [14], p. 233, or [4]) If is a monotone set function on C and if f : ! R is C-measurable (that is, for any Borel subset B R it follows f 1 (B) 2 C), then for any A 2 C, the Choquet integral is de…ned by (C)

Z

A

fd =

Z

0

+1

(F (f ) \ A) d +

Z

0 1

[ (F (f ) \ A)

(A)] d ;

where we used the notation F (f ) = f! 2 : f (!) g. Notice that if f 0 R0 on A, then in the above formula we get 1 = 0: The integrals on the right-hand side are the usual Riemann integral. R The function f will be called Choquet integrable on A if (C) A f d 2 R. Next we list some well known properties of the Choquet integral.

Remark 2 If : C ! [0; +1] is a monotone set function, then the following properties hold:

2

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R R (i) For all a 0 we have (C) A af d = a (C) A f d (if f 0 then see, e.g., [14], Theorem 11.2, (5), p. 228 and if f is arbitrary sign, then see, e.g., [5], p. 64, Proposition 5.1, (ii)). (ii) For all c 2 R and f of arbitrary sign, we have (see, e.g., [14], pp. R R 232-233, or [5], p. 65) (C) A (f + c) d = (C) A f d + c (A) : If is submodular too, then for all f; g of arbitrary sign and lower bounded, we have (see, e.g., [5], p. 75, Theorem 6.3) Z Z Z (C) (f + g) d (C) f d + (C) gd : A

A

R

A

R

(iii) If f g on A then (C) A f d (C) A gd (see, e.g., [14], p. 228, Theorem 11.2, (3) if f; g 0 and p. 232 if f; g are of arbitrary sign). R R (iv) Let f 0. If A B then (C) A f d (C) B f d . In addition, if is …nitely subadditive, then Z Z Z (C) fd (C) f d + (C) fd : A[B

A

R

B

(v) It is immediate that (C) A 1 d (t) = (A) : (vi) The formula (A) = (M (A)), where : [0; 1] ! [0; 1] is an increasing and concave function, with (0) = 0, (1) = 1 and M is a probability measure (or only …nitely additive) on a -algebra on (that is, M (?) = 0, M ( ) = 1 and M is countably additive), gives simple examples of normalized, monotone and submodular set functions (see, e.g., [5], pp. 16-17, Example 2.1). Such of set functions are also called distorsions of countably normalized, additive measures (or distorted measures). For a simple example, we can take (t) = p 2t (t) = t: 1+t ; If the above function is increasing, concave and satis…es only (0) = 0, then for any bounded Borel measure m, (A) = (m (A)) gives a simple example of bounded, monotone and submodular set function. (vii) If is a countably additive bounded measure, then the Choquet integral R (C) A f d reduces to the usual Lebesgue type integral (see, e.g., [5], p. 62, or [14], p. 226). R (viii) If f 0, then (C) fd 0. A p (ix) Let = M , where M is the Lebesgue measure on [0; +1), then is a monotone and submodular set function, furthermore is strictly positive, see [7]. (x) If = RN , N 2 N, we call strictly positive if (A) > 0, for any open subset A RN : We need some possibility theory: De…nition 3 ([6]) For the all subsets of :

6= ?, the power set P ( ) denotes the family of

3

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(i) A function : ! [0; 1] with the property sup f (s) : s 2 g = 1, is called possibility distribution on : (ii) P : P ( ) ! [0; 1] is called possibility measure, if it satis…es P (?) = 0, P ( ) = 1, and P ([i2I Ai ) = supfP (Ai ) : i 2 Ig for all Ai , and any I, an at most countable family of indices. Note that if A; B , A B, then the last property implies P (A) P (B) and that P (A [ B) P (A) + P (B). Any possibility distribution on , induces the possibility measure P : P ( ) ! [0; 1] ; P (A) = supf (s) : s 2 Ag, A . Also, if f : ! R+ , then the possibilistic integral of f on A with respect to P is de…ned by R (P os) A f dP = supff (t) (t) : t 2 Ag (see [6], chapter 1). Note that any possiblity measure is normalized, monotone and submodular. From (A [ B) = maxf (A) ; (B)g we get monotonicity, and from (A \ B) minf (A) ; (B)g we derive the submodularity.

3

Background - II

We make De…nition 4 ([2]) Let f : [a; b] [0; 1) ! R and 2 (0; 1]. We say that f is an -fractional continuous function, i¤ 8 " > 0 9 > 0 : for any x; y 2 [a; b] such that jx y j we get that jf (x) f (y)j ". We mention Theorem 5 ([2]) Over [a; b] [0; 1), 2 [0; 1], an -fractional continuous function is a uniformly continuous function and vice versa, a uniformly continuous function is an -fractional continuous function. We need De…nition 6 ([2]) Let [a; b] modulus of continuity: ! 1 (f; ) :=

[0; 1), sup x;y2[a;b]: jx y j

2 [0; 1]. We de…ne the

jf (x)

f (y)j ,

> 0:

-fractional

(1)

Properties ([2]): 1) ! 1 (f; 0) = 0: 2) ! 1 (f; ) ! 0 as # 0, i¤ f is in the set of all -fractional continuous functions, denoted as f 2 C ([a; b] ; R) (= C ([a; b] ; R)). 3) ! 1 is 0 and non-decreasing on R+ :

4

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4) ! 1 is subadditive: ! 1 (f; t1 + t2 )

! 1 (f; t1 ) + ! 1 (f; t2 ) :

(2)

! 1 (f; t1 ) + ::: + ! 1 (f; tn ) ;

(3)

5) ! 1 is continuous on R+ : 6) Clearly it holds ! 1 (f; t1 + ::: + tn ) for t = t1 = ::: = tn , we obtain ! 1 (f; nt) = n! 1 (f; t) : 7) Let

0,

(4)

2 = N, we get ! 1 (f; t)

( + 1) ! 1 (f; t) :

(5)

We notice that ! 1 (f; ) is …nite when f is uniformly continuous on [a; b] [0; 1). We need De…nition 7 ([9], [10]) Let f : [0; 1) ! R. The conformable derivative for 2 (0; 1] is given by D f (t) := lim

f t + "t1

f (t) "

"!0

;

-fractional

(6)

D f (0) = lim D f (t) : t!0+

If f is di¤ erentiable, then D f (t) = t1

f 0 (t) ;

(7)

where f 0 is the usual derivative. We de…ne Dn f = Dn 1 (D f ), D0 f = f: If f : [0; 1) ! R is -di¤erentiable at t0 > 0, at t0 , see [10]. We need

2 (0; 1], then f is continuous

De…nition 8 ([2]) Here C+ ([a; b]) := ff : [a; b] [0; 1) ! R+ , continuous functionsg: Let LN : C+ ([a; b]) ! C+ ([a; b]), operators, 8 N 2 N, such that (i) LN ( f ) = LN (f ) ; 8 0; 8f 2 C+ ([a; b]) ; (8) (ii) if f; g 2 C+ ([a; b]) : f

g; then

LN (f )

LN (g) , 8N 2 N;

(9)

(iii) LN (f + g)

LN (f ) + LN (g) ; 8 f; g 2 C+ ([a; b]) :

(10)

We call fLN gN 2N positive sublinear operators. 5

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We need Theorem 9 ([2]) Let 2 (0; 1], [a; b] [0; 1). Suppose f is -conformable fractional di¤ erentiable on [a; b]. D f is continuous on [a; b]. Let an x 2 [a; b] such that D f (x) = 0, and LN from C+ ([a; b]) into itself, positive sublinear op+1 2( +1) erators. Assume that LN (1) = 1 and LN j xj (x) ; LN ( x) (x) > 0, 8 N 2 N. Then !1 jLN (f ) (x) LN j

xj

2( +1)

D f; LN (

x)

2( +1)

(x)

f (x)j

+1

+1

(x)

+

1 LN ( 2

2( +1)

x)

2( +1)

(x)

; 8 N 2 N: (11)

We make Remark 10 ([2]) By [2], we get that LN j

xj

+1

(x)

2( +1)

LN (

As N ! +1, by (11) and (12), and LN ( LN (f ) (x) ! f (x) :

x)

2( +1)

x)

(x)

1 2

:

(12)

(x) ! 0, we obtain that

We need Theorem 11 ([2]) Let 2 (0; 1], n 2 N. Suppose f is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and Dn f is continuous on [a; b]. k For a …xed x 2 [a; b] we have D f (x) = 0, k = 1; :::; n. Let positive sublinear operators fLN gN 2N from C+ ([a; b]) into itself, such that LN (1) = 1, and LN j

n( +1)

xj

(x) ; LN j !1

jLN (f ) (x) LN j

f (x)j

n( +1)

xj

(x)

(n+1)( +1)

xj

Dn f; LN j

(x) > 0, 8 N 2 N. Then (n+1)( +1)

xj

(x)

(n+1)( +1)

n n! +1

1 + LN j (n + 1)

(13) (n+1)( +1)

xj

(x)

n (n+1)( +1)

;

8 N 2 N: We make

6

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Remark 12 ([2]) By [2], we get that LN j

n( +1)

xj

(x)

(n+1)( +1)

LN j

xj

(n+1)( +1)

As N ! +1, by (13), (14), and LN j LN (f ) (x) ! f (x) :

xj

(x)

n n+1

:

(14)

(x) ! 0, we derive that

We also need De…nition 13 Let f 2 C ([a; b]). We de…ne the usual …rst modulus of continuity of f as: ! 1 (f; ) := sup jf (x) f (y)j , > 0: (15) x;y2[a;b]: jx yj

We need Theorem 14 ([3]) Let 2 (0; 1] and n 2 N. Suppose f 2 C+ ([a; b]) is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and x 2 (a; b), and Dn f is continuous on [a; b]. Let 0 < h min (x a; b x) and assume n jD f j is convex over [a; b]. Furthermore assume that Dk f (x) = 0, k = 1; :::; n. Let fLN gN 2N from C+ ([a; b]) into itself, positive sublinear operators such that: LN (1) = 1, 8 N 2 N: Then jLN (f ) (x)

f (x)j

! 1 (Dn f; h) b1 (n + 1)! n+1 h

LN j

(n+1)

xj

(x) ; 8 N 2 N: (16)

We have Theorem 15 ([3]) All as in Theorem 14. Additionally assume that (n+1)( +1) LN j xj (x) > 0; 8 N 2 N. Then jLN (f ) (x)

f (x)j

! 1 (Dn f; h) b1 (n + 1)! n+1 h

LN j

(n+1)( +1)

xj

(x)

+1

;

(17) 8 N 2 N: An application of Theorem 15 follows: Theorem 16 ([3])Let fLN gN 2N from C+ ([a; b]) into itself, positive sublinear (n+1)( +1)

operators: LN (1) = 1, 8 N 2 N: Also x 2 (a; b) and LN j xj (x) > 0; 8 N 2 N: Here 2 (0; 1] and n 2 N. Suppose f 2 C+ ([a; b]) is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and Dn f is 7

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+1

(n+1)( +1)

continuous on [a; b]. Assume here that 0 < LN j xj (x) n min (x a; b x) ; 8 N 2 N : N N 2 N, and assume jD f j is convex over [a; b]. Furthermore assume that Dk f (x) = 0, k = 1; :::; n. Then b1 jLN (f ) (x)

f (x)j

! 1 Dn f; LN j

(n+1)( +1)

xj

(n + 1)!

+1

(x)

; (18)

n+1

8 N 2 N : N N 2 N: (n+1)( +1) If LN j xj (x) ! 0, then LN (f ) (x) ! f (x), as N ! +1: An application of Theorem 14 follows: Theorem 17 ([3]) Let fLN gN 2N from C+ ([a; b]) into itself, positive sublinear (n+1)

operators: LN (1) = 1, 8 N 2 N: Also LN j xj (x) > 0; 8 N 2 N: Here 2 (0; 1]; n 2 N and x 2 (a; b); [a; b] [0; 1): Suppose f 2 C+ ([a; b]) is n times conformable -fractional di¤ erentiable on [a; b], and Dn f is continuous (n+1) on [a; b]. Let 0 < LN j xj (x) min (x a; b x) ; 8 N N ; N; n N 2 N, and assume jD f j is convex over [a; b]. Furthermore assume that Dk f (x) = 0, k = 1; :::; n. Then b1 jLN (f ) (x)

f (x)j

! 1 Dn f; LN j (n + 1)!

(n+1)

xj

(x) ;

n+1

(19)

8 N N , where N; N 2 N: (n+1) If LN j xj (x) ! 0, then LN (f ) (x) ! f (x), as N ! +1:

4

Background - III

We mention De…nition 18 ([7]) Let I = [0; 1], BI the -algebra of all Borel measurable subsets of I, ( N;x )N 2N, x2I will be the collection of the family N;x = f N;k;x gN k=0 ; of monotone, submodular and strictly positive set functions N;k;x on BI : Let f : [0; 1] ! R+ be a BI -measurable function which is bounded, and call N N k pN;k (x) = xk (1 x) , for any x 2 [0; 1]. k The Bernstein-Kantorovich-Choquet operators are de…ned by the formula

KN;

(f ) (x) = N;x

N X

k=0

If

N;k;x

(C) pN;k (x)

R

(k+1) (N +1) k (N +1)

N;k;x

h

f (t) d

(k+1) k (N +1) ; (N +1)

= , for all N; x; k, we will denote KN;

(t) i ; 8 x 2 [0; 1] : (20)

N;k;x

N;x

(f ) := KN; (f ) :

8

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Theorem 19 ([7]) Suppose that N;k;x = := M is the Lebesgue measure on [0; 1] : Then jKN; (f ) (x)

f (x)j

p

M , for all N; k and x, where

! p x (1 x) 1 p f; + ; N N

2! 1

(21)

8 N 2 N, x 2 [0; 1], f 2 C+ ([0; 1]), above ! 1 is over [0; 1] : Remark 20 By [7] we have that KN; (j

p x (1 x) 1 p + ; 8 N 2 N: N N

xj) (x) m 1

Let m > 1, notice that j

xj m

j

xj

=j

(22)

1, therefore m 1

xj j

xj

j

xj ;

hence m

KN; (j

xj ) (x)

KN; (j

xj) (x) ;

that is KN; (j

p

m

xj ) (x)

x (1 x) 1 p + ; 8 x 2 [0; 1] ; N 2 N; m N N

1:

(23)

Notice that KN; (1) = 1, 8 N 2 N. Clearly KN; operators are positive sublinear operators from C+ ([0; 1]) into itself. We mention De…nition 21 ([8]) Here we consider measures of possibility. Denoting pN;k (x) = N N k xk (1 x) , let us de…ned k

N;k

pN;k (t)

(t) := kk N

N

(N

N

k)

k

=

N k

tk (1 kk N

N

N

t)

k N

(N

k)

k

; k = 0; :::; N:

(24) By convention we assume that 0 = 1, so that the cases k = 0, and k = N make k sense. By considering the root N of p0N;k (x), it is clear that 0

maxfpN;k (t) : t 2 [0; 1]g = k k N which implies that each

N;k

N

(N

N

k)

k

N k

;

is a possibility distribution on [0; 1] : 9

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Denoting by P N;k the possibility measure induced by N;k and n;x := N := fP N;k gN k=0 (that is N is independent of x), we de…ne the nonlinear BernsteinDurrmeyer-Choquet polynomial operators with respect to the set functions in N given by the formula DN;

N

N X

(f ) (x) :=

pN;k (x)

R1

k=0

8 x 2 [0; 1], N 2 N, f 2 C+ ([0; 1]) :

N

k

f (t) tk (1 t) dP N;k (t) ; R1 N k k (C) 0 t (1 t) dP N;k (t)

(C)

0

(25)

Remark 22 Above P N;k is bounded, monotone, submodular and strictly positive, N 2 N, k = 0; 1; :::; N . Notice that DN; N (1) = 1, 8 N 2 N. Clearly DN; N operators are positive sublinear operators mapping C+ ([0; 1]) into itself. We mention Theorem 23 ([8]) For every f 2 C+ ([0; 1]), x 2 [0; 1] and N 2 N f1g, we have ! p p p p 1+ 2 x (1 x) + 2 x 1 p + ; (26) jDN; N (f ) (x) f (x)j 2! 1 f; N N where ! 1 is on [0; 1] : Remark 24 By [8] we have that p p p p x (1 x) + 2 x 1+ 2 1 p + ; 8N 2N DN; N (j xj) (x) N N m 1

Let m > 1, notice that j

xj m

j

xj

f1g: (27)

1, therefore

=j

m 1

xj j

xj

j

xj ;

hence DN;

N

(j

m

xj ) (x)

DN;

N

(j

xj) (x) ;

that is DN; 8N 2N

N

(j

m

xj ) (x)

f1g; 8 x 2 [0; 1], m

1+

p

2

p

x (1 x) + p N

p p 2 x

+

1 ; N

(28)

1:

We make

10

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Remark 25 When x 2 [0; 1], then the max (x (1 x)) = 41 , at x = 12 . Therefore it holds p x (1 x) 1 1 1 p p + ; + (29) N N N 2 N 8 x 2 [0; 1] ; 8 N 2 N. Similarly, it holds p p p p 1+ 2 x (1 x) + 2 x 1 p + N N 8 x 2 [0; 1] ; 8 N 2 N

p 1+3 2 1 p + ; N 2 N

(30)

f1g.

Corollary 26 (to Theorem 19) It holds kKN; (f )

f k1

1 1 2! 1 f; p + N 2 N

;

(31)

8 N 2 N, f 2 C+ ([0; 1]) : Corollary 27 (to Theorem 23) It holds kDN; 8N 2N

5

N

(f )

f k1

2! 1

! p 1+3 2 1 f; p + ; N 2 N

(32)

f1g, f 2 C+ ([0; 1]) :

Main Results

Here …rst we apply some of the main theorems mentioned inpsection 3 to the Bernstein-Kantorovich-Choquet operators KN; , where := M , with M the Lebesgue measure on [0; 1]. More precisely here it is

KN; (f ) (x) =

N X

(C) pN;k (x)

k=0

R

h

(k+1) (N +1) k (N +1)

f (t) d (t) i ; (k+1) k ; (N +1) (N +1)

(33)

8 x 2 [0; 1] ; 8 N 2 N, f 2 C+ ([0; 1]) : It follows applications to Bernstein-Durremeyer-Choquet operators DN; see (25). In particular we need (a variation of Theorem 11):

N

,

Theorem 28 ([2]) Let 2 (0; 1] and n 2 N : n 1. That is n1 1: Suppose f is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and Dn f is continuous on [a; b]. For a …xed x 2 [a; b] we have Dk f (x) = 0, 11

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k = 1; :::; n. Let positive sublinear operators fLN gN 2N from C+ ([a; b]) into itself, such that LN (1) = 1, 8 N 2 N; and > 0. Then jLN (f ) (x) n

LN (j

xj

) (x) +

! 1 (Dn f; ) n n!

f (x)j

1 LN j (n + 1)

(n+1)

xj

(x) ;

(34)

8 N 2 N: We present Theorem 29 Let 2 (0; 1] and n 2 N : n 1. Suppose f is n times conformable -fractional di¤ erentiable on [0; 1], and Dn f is continuous on [0; 1]. For a …xed x 2 [0; 1] we have Dk f (x) = 0, k = 1; :::; n. Then 1 ! q n+1 x(1 x) 1 n +N ! 1 D f; N jKN; (f ) (x) f (x)j 2 r ! x (1 x) 1 1 4 + + N N (n + 1)

!1

Dn f;

1 p 2 N

+

1 N

1 n+1

n n!

"

n n!

r

1 1 p + N 2 N

n 3 ! n+1 x (1 x) 1 5 + N N

1 + (n + 1)

n n+1

1 1 p + N 2 N

#

;

(35) 8 N 2 N: Notice that lim KN; (f ) (x) = f (x) : N !1

Proof. By (34) we have jKN; (f ) (x)

f (x)j

! 1 (Dn f; ) n n!

1 n KN; (j xj ) (x) + KN; j (n + 1) " r ! ! 1 (Dn f; ) x (1 x) 1 1 + + n n! N N (n + 1) (choose q

x(1 x) N

q

:= +

1 N

x(1 x) N

+

1 N

1 n+1

> 0, then

n+1

n n+1

(n+1)

xj

(23)

(x)

r

x (1 x) 1 + N N

=

q

x(1 x) N

+

1 N,

!# and

(36) n

=

) !1 =

n

D f;

q

x(1 x) N

+

1 N

1 n+1

!

n n!

12

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2 r ! x (1 x) 1 1 4 + + N N (n + 1) !1

1 p 2 N

Dn f;

1 n+1

1 N

+

"

n n!

r

x (1 x) 1 + N N

1 1 p + N 2 N

1 + (n + 1)

n 3 ! n+1 (29) 5 n n+1

1 1 p + N 2 N

#

;

(37) proving the claim. We continue with Theorem 30 All as in Theorem 29. Then !1 j(DN;

N

(f )) (x)

(1+

n

D f;

p

p

2)

p x(1 x)+ 2x p N

f (x)j

" 1 (n + 1)

"

1 n+1

!

n n!

p

1+

+

1 N

2

p

1+

p x (1 p N

p 2 x (1 p N

Dn f;

!1

x) +

p

x) +

p 1+3 p 2 2 N

+

2x

p

1 N

1 + N

!

n 3 # n+1 2x 1 5 + N 1 n+1

(38)

n n!

2

+

3

n ! " # n+1 p p 1 + 3 2 1 1 1 + 3 2 1 5; 4 p p + + + N (n + 1) N 2 N 2 N

8 N 2 N f1g. Notice that lim DN; N !+1

N

(f ) (x) = f (x) :

Proof. By (34) we have jDN; DN;

N

(f ) (x)

f (x)j

! 1 (Dn f; ) n n!

1 (n+1) DN; N j xj (x) (n + 1) " ! p p p 1+ 2 x (1 x) + 2x ! 1 (Dn f; ) 1 p + + n n! N N !# p p p 1+ 2 x (1 x) + 2x 1 1 p + (n + 1) N N N

(j

n

xj

(28)

) (x) +

(39)

13

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(choose n+1

=

(1+

:= (1+

p

p

p

2)

p x(1 x)+ 2x p N

p

2)

p x(1 x)+ 2x p N

!1

+

1 N,

D f;

n

and

(1+

n

1 n+1

1 N

+

p

=

> 0, then p p (1+ 2) px(1 N

p

p x(1 x)+ 2x p N

2)

=

+

1 n+1

1 N

n n!

"

p p x (1 1+ 2 p N " p p x (1 1+ 2 1 p (n + 1) N !1

Dn f;

2

p 4 1 +p3 2 + 1 N 2 N

!

x) +

p

x) +

p

p 1+3 p 2 2 N

+

p x)+ 2x

1 N

2x

2x

1 + N 1 + N

1 n+1

!

+

1 N

n n+1

)

!

+

n 3 # n+1 (30) 5

(40)

n n!

n 3 " # n+1 p 1 1+3 2 1 5; p + + (n + 1) N 2 N

8 N 2 N f1g, proving the claim. Next we apply Theorem 14. We give

1 Theorem 31 Let 2 (0; 1] and n 2 N such that (n + 1) 1, that is n+1 1. Suppose f 2 C+ ([0; 1]) is n times conformable -fractional di¤ erentiable on [0; 1], and x 2 (0; 1), and Dn f is continuous on [0; 1]. Let N 2 N such that p1 + 1 min (x; 1 x) and assume jDn f j is convex over [0; 1]. Furthermore N 2 N assume that Dk f (x) = 0, k = 1; :::; n. Then

j(KN; (f )) (x)

f (x)j

! 1 Dn f; 2p1N + (n + 1)!

n+1

1 N

;

(41)

8 N N , N 2 N: It holds lim KN; (f ) (x) = f (x) : N !+1

Proof. By (16) we get jKN; (f ) (x)

! 1 (Dn f; h) (n+1) KN; j xj (n + 1)! n+1 h ! r ! 1 (Dn f; h) x (1 x) 1 (29) + (n + 1)! n+1 h N N f (x)j

(23)

(x)

14

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! 1 (Dn f; h) (n + 1)! n+1 h (setting h :=

1 p 2 N

+

1 N

1 1 p + N 2 N

=

(42)

> 0) ! 1 Dn f; 2p1N +

1 N

n+1

(n + 1)!

;

proving the claim. We continue with Theorem 32 Let x 2 (0; 1) and N 2 N The rest as in Theorem 31. Then

f1g :

(1+3 p

N

(f )) (x)

f (x)j

2)

2 N

! 1 Dn f; j(DN;

p

(1+3 p

p

+ N1 2)

2 N

(n + 1)!

+

n+1

min (x; 1

x).

1 N

;

(43)

8 N N , N 2 N f1g. It holds lim DN; N (f ) (x) = f (x) : N !+1

Proof. We use Theorem 14: By (16) we get jDN;

N

(28) ! 1 (Dn f; h) (n+1) DN; N j xj (x) n+1 (n + 1)! h ! p p p 1+ 2 x (1 x) + 2x 1 (30) ! 1 (Dn f; h) p + (n + 1)! n+1 h N N ! p 1+3 2 ! 1 (Dn f; h) 1 p + (44) (n + 1)! n+1 h N 2 N

(f ) (x)

(setting h :=

(1+3 p

p

2 N

f (x)j

2)

+

1 N

> 0) ! 1 Dn f;

=

(1+3 p

p

2 N

(n + 1)!

2)

+

1 N

n+1

;

proving the claim. We need De…nition 33 Let be a set, and let f; g : ! R be bounded functions. We say that f and g are comonotonic, if for every !; ! 0 2 , (f (!)

f (! 0 )) (g (!)

g (! 0 ))

0:

(45)

15

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We also need the famous Schmeidler’s Representation Theorem (Schmeidler 1986) Theorem 34 ([11]) Denote with L1 (A) the vector space of A-measurable bounded real valued functions on , where A 2 is a -algebra. Given a real functional : L1 (A) ! R, assume that for f; g 2 L1 (A): (i) (cf ) = c (f ), 8 c > 0; (ii) f g, implies (f ) (g), and (iii) (f + g) = (f ) + (g), for any comonotonic f; g: Then (A) := (1A ), 8 A 2 A, de…nes a …nite monotone set function on A, and is the Choquet integral with respect to , i.e. Z (f ) = (C) f (t) d (t) ; 8 f 2 L1 (A) : (46)

Above 1A denotes the characteristic function on A.

Next we give nice interpretations of Theorems 9, 11, 16, 17 involving Choquet integrals and based on Theorem 34. We make Remark 35 Consider here [a; b] R+ , B = B ([a; b]) is the Borel -algebra on [a; b], and L1 (B) is the vector space of B-measurable bounded real valued functions on [a; b]. Let (LN )N 2N be a sequence of positive sublinear operators from L1 (B) into C+ ([a; b]), and x 2 [a; b]. That is here LN ful…lls the positive homogenuity, monotonicity and subadditivity properties, see (8)-(10). Assume LN (1) = 1, 8 N 2 N. Clearly here L1 (B) C+ ([a; b]), where [a; b] [0; 1): In particular we treat LN jC+ ([a;b]) , just denoted for simplicity by LN , 8 N 2 N. It is clear that LN ( ) (x) : L1 (B) ! R is a functional, 8 N 2 N. It has the properties: (i) LN (cf ) (x) = cLN (f ) (x) ; 8 c > 0; 8 f 2 L1 (B) ; (47) (ii) f

g, implies LN (f ) (x)

LN (g) (x) ; where f; g 2 L1 (B) ;

(48)

and (iii) LN (f + g) (x)

LN (f ) (x) + LN (g) (x) ; 8 f; g 2 L1 (B) :

(49)

For comonotonic f; g 2 L1 (B), we further assume that LN (f + g) (x) = LN (f ) (x) + LN (g) (x) :

(50)

In that case LN is called comonotonic. 16

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By Theorem 34 we get that: N;x

(A) := LN (1A ) (x) , 8 A 2 B, 8 N 2 N;

de…nes a …nite monotone set function on B, and Z b LN (f ) (x) = (C) f (t) d

N;x

(51)

(t) ;

(52)

a

8 f 2 L1 (B), 8 N 2 N. In particular (52) is valid for any f 2 C+ ([a; b]). Furthermore malized, that is N;x ([a; b]) = 1, 8 N 2 N. We give

N;x

is nor-

Theorem 36 Let 2 (0; 1], [a; b] [0; 1). Suppose f is R+ valued and is -conformable fractional di¤ erentiable on [a; b], with D f being continuous on [a; b]. Let x 2 [a; b] such that D f (x) = 0, and (LN )N 2N be a sequence of positive sublinear comonotonic operators from L1 (B) into C+ ([a; b]). We assume Rb Rb +1 2( +1) that LN (1) = 1, and (C) a jt xj d N;x (t) > 0, (C) a (t x) d N;x (t) > 0, 8 N 2 N. Then !1 jLN (f ) (x) 2

4 (C)

Z

a

f (x)j

b

jt

D f; (C)

xj

+1

d

N;x

!

(t)

+1

1 + 2

Rb a

2( +1)

(t

(C)

x)

Z

b

(t

d

N;x

2( +1)

x)

(t)

d

2( +1)

N;x

a

! 2(

+1)

3

5;

(t)

(53)

8 N 2 N. Rb As (C) a (t

2( +1)

x)

d

N;x

(t) ! 0, N ! 1, we get that

f (x) :

lim LN (f ) (x) =

N !+1

Proof. By Theorems 9, 34. Theorem 37 Let 2 (0; 1], n 2 N. Suppose f is R+ valued and is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and Dn f is continuous on [a; b]. For a …xed x 2 [a; b] we have Dk f (x) = 0, k = 1; :::; n: Let positive sublinear comonotonic operators fLN gN 2N from L1 (B) into C+ ([a; b]), such Rb Rb n( +1) (n+1)( +1) that (C) a jt xj d N;x (t), (C) a jt xj d N;x (t) > 0, 8 N 2 N. Then !1 jLN (f ) (x)

f (x)j

Dn f; (C)

Rb a

jt

(n+1)( +1)

xj

n n!

d

N;x

(t)

(n+1)( +1)

(54)

17

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2

4 (C) 1 (n + 1)

(C)

Z

n( +1)

jt

a

Z

xj

b

a

8 N 2 N. Rb (n+1)( As (C) a jt xj lim LN (f ) (x) = f (x) :

b

(n+1)( +1)

jt

+1)

d

xj

d

N;x

N;x

d

!

+1

(t)

N;x

+ n ! (n+1)(

+1)

3

5;

(t)

(t) ! 0, when N ! 1, we get that

N !+1

Proof. By Theorems 11, 34. We continue with Theorem 38 Let fLN gN 2N from L1 (B) into C+ ([a; b]) positive sublinear comonotonic operators, such that LN (1) = 1, 8 N 2 N. Additionally assume Rb (n+1)( +1) that (C) a jt xj d N;x (t) > 0, 8 N 2 N; x 2 (a; b). Here 2 (0; 1], and n 2 N. Suppose f 2 C+ ([a; b]) is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and Dn f is continuous on [a; b]. Assume here Rb +1 (n+1)( +1) d N;x (t) 0 < (C) a jt xj min (x a; b x), 8 N 2 N : n N N 2 N, and assume jD f j is convex over [a; b]. Furthermore assume that Dk f (x) = 0, k = 1; :::; n. Then b1 jLN (f ) (x)

! 1 Dn f; (C)

f (x)j

Rb a

jt

(n+1)( +1)

xj

(n + 1)!

n+1

d

N;x

(t)

+1

; (55)

8 N N ; N; N 2 N. Rb (n+1)( If (C) a jt xj 1.

+1)

d

N;x

(t) ! 0, then LN (f ) (x) ! f (x) as N !

Proof. By Theorems 16, 34. Theorem 39 Let fLN gN 2N from L1 (B) into C+ ([a; b]) positive sublinear comonotonic operators, such that LN (1) = 1, 8 N 2 N. Additionally assume Rb (n+1) that (C) a jt xj d N;x (t) > 0, 8 N 2 N; x 2 (a; b). Here 2 (0; 1], and n 2 N. Suppose f 2 C+ ([a; b]) is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and Dn f is continuous on [a; b]. Assume here 0 < Rb (n+1) (C) a jt xj d N;x (t) min (x a; b x), 8 N 2 N : N N 2 N,

and assume jDn f j is convex over [a; b]. Furthermore assume that Dk f (x) = 0, k = 1; :::; n. Then Rb (n+1) b1 ! 1 Dn f; (C) a jt xj d N;x (t) jLN (f ) (x) f (x)j ; (56) (n + 1)! n+1 18

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8 N N ; where N; N 2 N. Rb (n+1) If (C) a jt xj d N;x (t) ! 0, then LN (f ) (x) ! f (x) as N ! 1. Proof. By Theorems 17, 34. We make

Remark 40 Consider again [a; b] R+ , B = B ([a; b]) the Borel -algebra on [a; b]. For each N 2 N and each x 2 [a; b] consider the monotone set functions N;x ; B ! R+ . We assume that all N;x are normalized, that is N;x ([a; b]) = 1, and submodular. Here we consider the operators TN : C+ ([a; b]) ! C+ ([a; b]) given by the formula Z b TN (f ) (x) = (C) f (t) d N;x (t) ; (57) a

8 N 2 N; 8 x 2 [a; b] : Infact here N;x are chosen so that TN (C+ ([a; b])) C+ ([a; b]) : We notice here that hold: (i) TN ( f ) (x) = TN (f ) (x) ; 8 0;

(58)

(ii) f

g, implies TN (f ) (x)

TN (g) (x) ;

(59)

TN (f ) (x) + TN (g) (x) ;

(60)

and (iii) TN (f + g) (x)

8 N 2 N; 8 x 2 [a; b] ; 8 f; g 2 C+ ([a; b]) : Clearly TN are positive sublinear operators, compare to (8)-(10). We also have that TN (1) = 1, 8 N 2 N: We give Theorem 41 Let 2 (0; 1], [a; b] [0; 1). Suppose f is -conformable fractional di¤ erentiable on [a; b]. D f is continuous on [a; b]. Let an x 2 [a; b] such Rb Rb +1 2( +1) that D f (x) = 0. Assume (C) a jt xj d N;x (t), (C) a (t x) d N;x (t) > 0, 8 N 2 N. Then !1 jTN (f ) (x) 2

4 (C)

Z

a

f (x)j

b

jt

D f; (C)

xj

+1

d

N;x

!

(t)

+1

1 + 2

Rb a

2( +1)

(t

(C)

x)

Z

a

b

(t

d

N;x

2( +1)

x)

(t)

d

2( +1)

N;x

! 2(

(t)

+1)

3

5;

(61) 19

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8 N 2 N. Rb As N ! 1, and (C) a (t f (x) :

2( +1)

x)

d

(t) ! 0, we obtain TN (f ) (x) !

N;x

Proof. By Theorem 9. Theorem 42 Let 2 (0; 1], n 2 N. Suppose f is n times conformable fractional di¤ erentiable on [a; b] [0; 1) and takes values on R+ : Dn f is continuous on [a; b]. For a …xed x 2 [a; b] we have Dk f (x) = 0, k = 1; :::; n: AsRb Rb n( +1) (n+1)( +1) sume that (C) a jt xj d N;x (t), (C) a jt xj d N;x (t) > 0, 8 N 2 N. Then Dn f; (C)

!1 jTN (f ) (x)

f (x)j 2

4 (C) 1 (n + 1)

(C)

Z

b

Z

jt

8 N 2 N. Rb As N ! 1, and (C) a jt f (x) :

(n+1)( +1)

jt

xj

b

a

a

n( +1)

jt

a

Rb

xj

N;x

(t)

(n+1)( +1)

n n!

d

(n+1)( +1)

xj

(n+1)( +1)

xj

d

d

N;x

d

!

+1

(t)

N;x

N;x

+ n ! (n+1)(

+1)

3

5;

(t)

(62)

(t) ! 0, we get TN (f ) (x) !

Proof. By Theorem 11. We continue with Rb (n+1)( +1) Theorem 43 Assume (C) a jt xj d N;x (t) > 0, 8 N 2 N; x 2 (a; b). Here 2 (0; 1], and n 2 N. Suppose f 2 C+ ([a; b]) is n times conformable -fractional di¤ erentiable on [a; b] [0; 1), and Dn f is continRb +1 (n+1)( +1) uous on [a; b]. Assume here that 0 < (C) a jt xj d N;x (t) min (x a; b x), 8 N 2 N : N N 2 N, and assume jDn f j is convex over k [a; b]. Furthermore assume that D f (x) = 0, k = 1; :::; n. Then b1 jTN (f ) (x)

! 1 Dn f; (C)

f (x)j

Rb a

jt

(n+1)( +1)

xj

(n + 1)!

n+1

d

N;x

(t)

+1

; (63)

8 N 2 N : N N 2 N. Rb (n+1)( If (C) a jt xj

+1)

d

N;x

(t) ! 0, then TN (f ) (x) ! f (x) as N ! 1.

Proof. By Theorem 16. 20

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Rb (n+1) Theorem 44 Assume (C) a jt xj d N;x (t) > 0, 8 N 2 N. Here 2 (0; 1], n 2 N and x 2 (a; b); [a; b] [0; 1): Suppose f 2 C+ ([a; b]) is n times conformable -fractional di¤ erentiable on [a; b], and Dn f is continuous on [a; b]. Rb (n+1) Let 0 < (C) a jt xj d N;x (t) min (x a; b x), 8 N N ; N; N 2 N, and assume jDn f j is convex over [a; b]. Furthermore assume that Dk f (x) = 0, k = 1; :::; n. Then b1 jTN (f ) (x)

! 1 Dn f; (C)

f (x)j

Rb a

jt

(n + 1)!

(n+1)

xj

n+1

d

N;x

(t) ;

(64)

8 N N ; where N; N 2 N. Rb (n+1) If (C) a jt xj d N;x (t) ! 0, then TN (f ) (x) ! f (x) as N ! 1. Proof. By Theorem 17.

References [1] G. Anastassiou, Moments in probability and approximation theory, Pitman Research Notes in Mathematics Series, Longman Group UK, New York, NY, 1993. [2] G. Anastassiou, Conformable Fractional Approximation by Max-Product Operators, Studia Mathematica Babes Bolyai, 63 (1) (2018), 3-22. [3] G. Anastassiou, Conformable Fractional Approximations by Max-Product Operators using Convexity, Arabian Journal of Mathematics, accepted for publication, 2018. [4] G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble), 5 (1954), 131-295. [5] D. Denneberg, Non-additive Measure and Integral, Kluwer, Dordrecht, 1994. [6] D. Dubois and H. Prade, Possibility Theory, Plenum Press, New York, 1988. [7] S. Gal, Uniform and Pointwise Quantitative Approximation by Kantorovich-Choquet type integral Operators with respect to monotone and submodular set functions, Mediterranean Journal of Mathematics, 14 (2017), no. 5, Art. 205, 12 pp. [8] S. Gal and S. Trifa, Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators, Carpathian J. Math., 33 (2017) (1), 49-58. 21

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[9] M. Abu Hammad and R. Khalil, Abel’s formula and Wronskian for conformable fractional di¤ erential equations, International J. Di¤erential Equations Appl., 13, No. 3 (2014), 177-183. [10] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new de…nition of fractional derivative, J. Computational Appl. Math., 264 (2014), 65-70. [11] D. Schmeidler, Integral representation without additivity, Proceedings of the American Mathematical Society, 97 (1986), 255-261. [12] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica, 57 (1989), 571-587. [13] Lloyd S. Shapley, A Value for n-person Games, in H.W. Kuhn, A.W. Tucker, Contributions to the Theory of Games, Annals of Mathematical Studies 28, Princeton University Press, (1953), 307-317. [14] Z. Wang, G.J. Klir, Generalized Measure Theory, Springer, New York, 2009.

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The Minkowski Inequality and the Brunn-Minkowski Inequality for Dual Orlicz Mixed Affine Quermassintegrals Tongyi Ma (College of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, P.R.China)

Abstract In this paper, the Orlicz version of the classical dual Cauchy-Kubota formula is given and the concept of dual affine quermassintegrals is extended to dual Orlicz mixed affine quermassintegrals in the framework of Orlicz Brunn-Minkowski theory. Some inequalities for dual Orlicz mixed affine quermassintegrals are obtained, such as dual Orlicz-Minkowski inequality and dual Orlicz-BrunnMinkowski inequality. Keywords: Orlicz Brunn-Minkowski theory, integral geometry, dual affine quermassintegral.

1

Introduction

We work in Euclidean space Rn , and use voli (·) to denote the i-dimensional volume. The unit sphere in Rn is written by S n−1 . In the projection of convex body K, quermassintegrals are important geometric invariants and have different definitions in many areas of mathematics. In the theory of mixed volumes quermassintegrals are usually called simple mixed volumes. The reader should refer to [24] and fn−i , of a star body K. Suppose [26] for details. Lutwak [21] introduced the dual quermassintegrals, W f f W0 = voln (K) and Wn = ωn . If 0 < i < n, then ∫ ωn f Wn−i (K) = voli (K ∩ ξ)dµi (ξ), (1.1) ωi G(n,i) where the Grassmann manifold G(n, i) is endowed with the probability Haar measure µi , voli (K ∩ ξ) is the i-dimensional volume of slice of K by an i-dimensional subspace ξ ⊂ Rn and ωi = π i/2 /Γ(1 + i/2) denotes the i-dimensional volume of the unit ball in Ri . The quermassintegrals are connected with the projections of convex bodies, while the dual quermassintegrals are closely related to the cross sections of star bodies, which is proved in [11] that they are the only rotation invariant continuous star valuations with the corresponding homogeneity. Zhang [28] showed that the dual quermassintegrals have the same kind of kinematic formulas as the quermassintegrals. Affine quermassintegrals [16] is an important geometric invariants in the projection of convex body. e n−i (K), of a star body K containing the Lutwak [15] introduced the dual affine quermassintegrals, Φ AMS Subject Classification: 52A39, 52A40, 52A22. This work was Supported by the National Natural Science Foundation of China(Grant No. 11561020) and was partly supported by the National Natural Science Foundation of China (Grant No. 11371224). E-mail: [email protected], matongyi− [email protected] (Tongyi Ma).

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e 0 (K) = voln (K) and Φ e n (K) = ωn . If 0 < i < n, then origin in its interior. Suppose Φ e n−i (K) = ωn Φ ωi

(∫

) n1 voli (K ∩ ξ)n dµi (ξ)

.

(1.2)

G(n,i)

Grinberg [6] showed that both the affine quermassintegrals and the dual affine quermassintegrals are invariant under volume-preserving affine transformations. However, the dual affine quermassintegrals of star bodies received more considerable attention, see [6, 2, 16, 26, 27]. The aim of this paper is to study them further. Some opened articles [9, 13, 17, 18, 23, 25], Gardner’ work [3] and the classical Brunn-Minkowski theory of convex bodies (see, e.g., [4, 26]) were generalized to the Orlicz space, which is called the Orlicz Brunn-Minkowski theory and further extend the Lp -Brunn-Minkowski theory (see, e.g., [19, 20, 12]). We considers a non-zero convex function ϕ : [0, ∞) → [0, ∞) in this paper. It is strictly increasing with ϕ(0) = 0. Suppose that C is the class of convex and strictly increasing functions ϕ : [0, ∞) → [0, ∞), where lim ϕ(t) = +∞, and ϕ(0) = 0. Note that Son denotes the set of star bodies in Rn containing the t→∞ origin in their interiors. The dual Orlicz mixed volume, Ve−ϕ (K, L), of K, L ∈ Son is defined by e −ϕ ε ⋄ L) − voln (K) −ϕ′r (1) voln (K + Ve−ϕ (K, L) = lim , n ε ε→0+

(1.3)

e −ϕ ε ⋄ L denotes the Orlicz where ϕ′r (1) is the right derivative of a real-valued function ϕ at 1 and K + radial harmonic combination of K and L. It follows from (1.3) that the dual Orlicz mixed volume Ve−ϕ has the following integral representation: ( ) ∫ 1 ρK e V−ϕ (K, L) = ϕ ρK (u)n dS(u), (1.4) n S n−1 ρL In [5, 10, 22, 31, 20], the dual mixed volume is extended to the dual Lp -mixed volume. If ϕ(t) = tp , 1 ≤ p < ∞, then ( )p ∫ 1 ρK (u) e V−p (K, L) = ρnK (u)dS(u). (1.5) n S n−1 ρL (u) Recently, Zhao [30] introduced the notion of dual Orlicz mixed quermassintegrals for 0 ≤ i ≤ n and established its integral representation. If K, L ∈ Son and ϕ ∈ C, then ′ f f e f−ϕ,i (K, L) = −ϕr (1) lim Wi (K +−ϕ ε ⋄ L) − Wi (K) , and W n − i ε→0+ ε

f−ϕ,i (K, L) = 1 W n

∫

) ρK ϕ ρK (u)n−i dS(u), i = 0, 1, · · · , n. ρL S n−1

(1.6)

(

(1.7)

In this paper, we first established the Orlicz version of the classical dual Cauchy-Kubota formula (1.1) ∫ ωn (i) f W−ϕ,n−i (K, L) = Ve (K ∩ ξ, L ∩ ξ)dµi (ξ), (1.8) ωi G(n,i) −ϕ (i) where Ve−ϕ (K ∩ ξ, L ∩ ξ) is the dual Orlicz mixed volume of the (i)-dimensional star bodies K ∩ ξ and L ∩ ξ in the subspace ξ ∈ G(n, i). For i = 1, 2, · · · , n, we further consider the following formula.

e ϕ,n−i (K, L) = Φ =

]1/n ωn [ e (i) E(V−ϕ (K ∩ ξ, L ∩ ξ)n ) ωi [∫ ] n1 ωn (i) Ve−ϕ (K ∩ ξ, L ∩ ξ)n dµi (ξ) , ωi G(n,i)

(1.9)

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e ϕ,n−i (K, L) is known as the dual Orlicz mixed affine quermassintegrals. and Φ Let ϕ(t) = tp with p ≥ 1. Then e p,n−i (K, L) = ωn Φ ωi

] n1 (i) n e V−p (K ∩ ξ, L ∩ ξ) dµi (ξ) ,

[ ∫

(1.10)

G(n,i)

where Ve−p (K ∩ ξ, L ∩ ξ) denotes the dual Lp -mixed volume of K ∩ ξ and L ∩ ξ in the subspace ξ ∈ G(n, i). e ϕ,n−i (K, K)/ϕ(1) = Φ e n−i (K) is just the classical dual affine quermassinteTaking L = K in (1.9), Φ grals of K. On the basis of the above concepts, one aim of this paper is to establish the following dual OrliczMinkowski inequality for dual Orlicz mixed affine quermassintegrals. (i)

Theorem 1.1. Suppose K, L ∈ Son , n ≥ 3 and ϕ ∈ C. Then for 2 ≤ i ≤ n, (( e )1 ) Φn−i (K) i e ϕ,n−i (K, L) ≥ Φ e n−i (K)ϕ . Φ e n−i (L) Φ

(1.11)

If K and L are convex bodies containing the origin in their interiors, then equality holds in the inequality (1.11) if and only if K and L are dilations. As an application of Theorem 1.1, we prove a uniqueness theorem of convex bodies. The other aim of this paper is to prove Orlicz radial sum versions of the dual Brunn-Minkowski inequality for dual Orlicz mixed affine quermassintegrals. Theorem 1.2. Suppose K, L ∈ Son and ϕ ∈ C. Then for 2 ≤ i ≤ n, (( e )1 ) )1 ) (( e e −ϕ L) i e −ϕ L) i Φn−i (K + Φn−i (K + ϕ +ϕ ≤ ϕ(1). e n−i (K) e n−i (L) Φ Φ

(1.12)

If K and L are convex bodies containing the origin in their interiors, then equality holds in the inequality (1.12) if and only if K and L are dilations. In order to prove Theorems 1.1 and 1.2, we use the integral-geometric technique, motivated by Furstenberg and Tzkoni [1], Grinberg [7], Ma [22], Gardner and Hug, et al. [5] and Zhu et al. [31].

2

Preliminaries

Let Kn denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space Rn . We write Kon for the set of convex bodies containing the origin in their interiors. The support function of K ∈ Kon , hK = h(K, ·) : Rn \{o} → [0, ∞), is defined by h(K, x) = max{⟨x, y⟩ : y ∈ K}, where x ∈ Rn \{o}. For K ∈ Kon , its polar body, K ∗ ∈ Kon , is defined by K ∗ = {x ∈ Rn : ⟨x, y⟩ ≤ 1, for any y ∈ K}. It is easily known that (K ∗ )∗ = K for K ∈ Kon , and for c > 0 we have (cK)∗ = c−1 K ∗ . If K is a compact set in Rn , then the radial function ρK of K is defined by ρK (x) = max{λ ≥ 0 : λx ∈ K} for x ∈ Rn \{o}. If ρK is continuous then we call K a star body (about the origin). Two star bodies K and L are dilates (of one another) if ρK (u)/ρL (u) is independent of u ∈ S n−1 . It is easy to see that for K, L ∈ Son , K ⊆ L if and only if ρK ≤ ρL and for c > 0 and x ∈ Rn \{o}, ρ(cK, x) = cρ(K, x). More generally, for T ∈ GL(n) the radial function of the image T K = {T y : y ∈ K} of K is given by (see [26]) ρ(T K, x) = ρ(K, T −1 x), for x ∈ Rn \{o}, (2.1) where GL(n) denotes the linear transformation group on Rn , and T −1 is the inverse of T .

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e −ϕ β ⋄ L of For K, L ∈ Son , α, β ≥ 0 (not both zero) and ϕ ∈ C, the Orlicz radial combination α ⋄ K + K and L is defined by (see [22]) { ( ) ( ) } 1 1 e −ϕ β ⋄ L, u)−1 = inf λ > 0 : αϕ ρ(α ⋄ K + + βϕ ≤ ϕ(1) , u ∈ S n−1 . (2.2) λρK (u) λρL (u) e −ϕ β ⋄ L, u) is defined by Note that for all u ∈ S n−1 , ρ(α ⋄ K + (

e −ϕ β ⋄ L, u) ρ(α ⋄ K + αϕ ρK (u)

)

(

e −ϕ β ⋄ L, u) ρ(α ⋄ K + + βϕ ρL (u)

) = ϕ(1).

e −ϕ β ⋄L is the Lp -radial harmonic combination α⋄K + e −p β ⋄L, If ϕ(t) = tp with 1 ≤ p < ∞, then α⋄K + and correspondingly Ve−ϕ (K, L) is the dual Lp -mixed volume Ve−p (K, L). See [20] for more details. Lemma 2.1. Let K, L ∈ Son and α, β ≥ 0. If ϕ ∈ C, then for T ∈ GL(n), e −ϕ β ⋄ L) = α ⋄ T K + e −ϕ β ⋄ T L. T (α ⋄ K + Proof. From (2.2) and (2.1), we have for u ∈ S n−1 , { ( ) ( ) } 1 1 e −ϕ β ⋄ T L, u)−1 = inf λ > 0 : αϕ ρ(α ⋄ T K + + βϕ ≤ ϕ(1) λρT K (u) λρT L (u) { ( ) ( ) } 1 1 = inf λ > 0 : αϕ + βϕ ≤ ϕ(1) λρK (T −1 u) λρL (T −1 u) e −ϕ β ⋄ L, T −1 u)−1 = ρ(α ⋄ K + e −ϕ β ⋄ L), u)−1 . = ρ(T (α ⋄ K + Thus

e −ϕ β ⋄ L) = α ⋄ T K + e −ϕ β ⋄ T L. T (α ⋄ K + 

Lemma 2.2. Let K, L ∈ Son , ϕ ∈ C. Then for each ξ ∈ G(n, i), i = 1, · · · , n − 1 and ε > 0, e −ϕ ε ⋄ L) ∩ ξ = (K ∩ ξ)+ e −ϕ ε ⋄ (L ∩ ξ). (K + Proof. Fixed ξ ∈ G(n, i), and let S i−1 = S n−1 ∩ ξ. For any u ∈ S i−1 and Q ∈ Son , we get ρQ (u) = e −ϕ ε ⋄ L to u ∈ S i−1 , it follows that ρQ∩ξ (u). Applying the definition of K + (

e −ϕ ε ⋄ L) ∩ ξ, u) ρ((K + ϕ ρK∩ξ (u)

)

( ) e −ϕ ε ⋄ L) ∩ ξ, u) ρ((K + + εϕ = ϕ(1). ρL∩ξ (u)

e −ϕ ε ⋄ (L ∩ ξ) defined in ξ, we have On the other hand, from (K ∩ ξ)+ ) ( ) ( e −ϕ ε ⋄ (L ∩ ξ), u) e −ϕ ε ⋄ (L ∩ ξ), u) ρ((K ∩ ξ)+ ρ((K ∩ ξ)+ + εϕ = ϕ(1). ϕ ρK∩ξ (u) ρL∩ξ (u) e −ϕ ε ⋄ L) ∩ ξ and (K ∩ ξ)+ e −ϕ ε ⋄ (L ∩ ξ) is a same star body in ξ. Thus, (K +



Lemma 2.3. (see [22]) Suppose K, L ∈ Son and ϕ ∈ C. Then (( )1 ) voln (K) n Ve−ϕ (K, L) ≥ voln (K)ϕ , voln (L)

(2.3)

with equality if and only if K and L are dilates of each other. 4 297

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Taking ϕ(t) = tp with p ≥ 1. The above dual Orlicz-Minkowski inequality is Lutwak’s Lp -dual Minkowski inequality (see [20]): p n+p Ve−p (K, L) ≥ voln (K) n voln (L)− n ,

(2.4)

with equality holds if and only if K and L are dilations. Lemma 2.4. (see [8]) Suppose that µ is a probability measure on a space X and f : X → I ⊂ R is a µ-integrable function, where I is a possibly infinite interval. Jensen’s inequality states that if ϕ : I → R is a convex function, then

(∫ ) ( ) ϕ f (x) dµ(x) ≥ ϕ f (x)dµ(x) .

∫ X

(2.5)

X

If ϕ is strictly convex, the equality holds in every inequality if and only if f (x) is constant for µ-almost all x ∈ X.

3

The generalized dual Cauchy-Kubota formula

In this section, we prove the probabilistic essence of dual Orlicz mixed quermassintegrals. We first see the dual Cauchy-Kubota formula. For K ∈ Son , ∫ fn−i (K) = ωn W voli (K ∩ ξ)dµi (ξ), i = 1, · · · , n − 1. (3.1) ωi G(n,i) Theorem 3.1. Suppose K, L ∈ Son and ϕ ∈ C. Then for each i = 1, · · · , n − 1, ∫ (i) f−ϕ,n−i (K, L) = ωn W Ve (K ∩ ξ, L ∩ ξ)dµi (ξ). ωi G(n,i) −ϕ Proof. By (1.6), (3.1) and Lemma 2.2, we have ′ f f e f−ϕ,n−i (K, L) = −ϕr (1) lim Wn−i (K +−ϕ ε ⋄ L) − Wn−i (K) W + i ε ε→0 ∫ e −ϕ ε ⋄ L) ∩ ξ) − voli (K ∩ ξ) −ϕ′r (1) ωn voli ((K + = · lim+ dµi (u) i ωi G(n,i) ε→0 ε ∫ e −ϕ ε ⋄ L ∩ ξ)) − voli (K ∩ ξ) −ϕ′r (1) ωn voli ((K ∩ ξ + = · lim+ dµi (u). i ωi G(n,i) ε→0 ε

From (1.3), we have f−ϕ,n−i (K, L) = ωn W ωi

∫ G(n,i)

(i) Ve−ϕ (K ∩ ξ, L ∩ ξ)dµi (ξ).

f−ϕ,i (K, L) is the expectation of the random variable Up to a constant, the quantity W (i) Ve−ϕ (K ∩ ·, L ∩ ·) : G(n, i) → (0, ∞),



(i) ξ 7→ Ve−ϕ (K ∩ ξ, L ∩ ξ),

which is defined on the probability space (G(n, i), B, µi ) (where B is the Borel sigma-algebra on G(n, i)). Taking ϕ(t) = tp with p > 0 in Theorem 3.1, we have the formula ∫ (i) f−p,n−i (K, L) = ωn W Ve (K ∩ ξ, L ∩ ξ)dµi (ξ). ωi G(n,i) −p

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For K ∈ Son , we extend the dual Cauchy-Kubota formula to 1 ≤ q ≤ i < n, ∫ ωn f f (n−q) (K ∩ ξ)dµn−q (ξ), Wi (K) = W ωn−q G(n,n−q) i−q

(3.2)

f (n−q) denotes the (i-q)th dual harmonic quermassintegral in the subspace ξ. where W i−q It follows from (3.2) and (1.6) that we have the following theorem. Theorem 3.2. Suppose K, L ∈ Son and ϕ ∈ Φ1 or ϕ ∈ Φ2 . Then for 1 ≤ q ≤ i < n, ∫ f−ϕ,i (K, L) = ωn f (n−q) (K ∩ ξ, L ∩ ξ)dµn−q (ξ), W W ωn−q G(n,n−q) −ϕ,i−q f (n−q) (K ∩ ξ, L ∩ ξ) denotes the dual Orlicz harmonic mixed quermassintegral of the (n-q)where W −ϕ,i−q dimensional star bodies K ∩ ξ and L ∩ ξ in the subspace ξ.

4

Inequalities of dual Orlicz mixed affine quermassintegrals

e ϕ,1 (K, L), · · · , Φ e ϕ,n (K, L) are SL(n)-invariant. Here, In this section, we first show that the quantities Φ (i) (i) E(Ve−ϕ (K ∩ ·, L ∩ ·)n ) is the expectation of Ve−ϕ (K ∩ ·, L ∩ ·)n . Theorem 4.1. Suppose K, L ∈ Son and ϕ ∈ C. Then for T ∈ SL(n), there holds e ϕ,i (T K, T L) = Φ e ϕ,i (K, L), i = 1, 2, · · · , n. Φ Proof. Suppose ξ ∈ G(n, n − i). For S n−i−1 = S n−1 ∩ ξ, if T ∈ SL(n) = {T ∈ GL(n) : det T = 1}, then for u ∈ S n−i−1 and Q ∈ Son , we get ρT Q (u) = ρT Q∩ξ (u). For x ∈ Rn \{o}, let ⟨x⟩ = x/||x||. From (1.4) and (2.1), we obtain ( ) ∫ 1 ρT K∩ξ (u) (n−i) Ve−ϕ (T K ∩ ξ, T L ∩ ξ) = ϕ ρn−i T K∩ξ (u)dSn−i−1 (u) n − i S n−1 ∩ξ ρT L∩ξ (u) ( ) ∫ 1 ρT K (u) = ϕ ρn−i T K (u)dSn−i−1 (u) n − i S n−1 ∩ξ ρT L (u) ( ) ∫ 1 ρK (⟨T −1 u⟩) −1 = ϕ ρn−i u⟩)dSn−i−1 (⟨T −1 u⟩) K (⟨T n − i S n−1 ∩ξ ρL (⟨T −1 u⟩) ( ) ∫ 1 ρK∩ξ (v) ϕ ρn−i = K∩ξ (v)dSn−i−1 (v) n − i S n−1 ∩ξ ρL∩ξ (v) (n−i) = Ve−ϕ (K ∩ ξ, L ∩ ξ),

where Sn−i−1 denotes n − i − 1-dimensional spherical Lebesgue measure. Thus, from (1.9), it follows that (∫ ) n1 [ ]n ω n (n−i) e ϕ,i (T K, T L) = Φ Ve−ϕ (T K ∩ ξ, T L ∩ ξ) dµn−i (ξ) ωn−i G(n,n−i) (∫ ) n1 [ ]n ωn (n−i) e = V−ϕ (K ∩ ξ, L ∩ ξ) dµn−i (ξ) ωn−i G(n,n−i) e ϕ,i (K, L). = Φ  6 299

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To prove Theorem 1.1 and Theorem 1.2, the next three lemmas are needed. Lemma 4.2. (see [14]) Suppose K ∈ Kon and ξ ∈ G(n, i), then K ∗ ∩ ξ = (K|ξ)∗ . Lemma 4.3. (see [12]) Suppose K1 , K2 ∈ Kon and 2 ≤ k ≤ n − 1. If K1 |ξ and K1 |ξ are dilations for each ξ ∈ G(n, k), then K1 and K2 are dilations. Lemma 4.4. Suppose K1 , K2 ∈ Kon and 2 ≤ k ≤ n − 1. If K1 ∩ ξ and K1 ∩ ξ are dilations for each ξ ∈ G(n, k), then K1 and K2 are dilations. Proof. If both K1 ∩ ξ and K2 ∩ ξ are dilations for each ξ ∈ G(n, k), then K1 ∩ ξ = a(K2 ∩ ξ) for a > 0. If follows from Lemma 4.2 that (K1∗ |ξ)∗ = a(K2∗ |ξ)∗ = (a−1 K2∗ |ξ)∗ . Thus, K1∗ |ξ = a−1 K2∗ |ξ. From  Lemma 4.3, we know K1∗ = ab K2∗ . Therefore, K1 = cK2 for some c > 0. The normalized dual affine quermassintegrals measure of K are defined by ( )n ωn n ∗ e [voli (K ∩ ·)] dµi , (4.1) dΦi (K, ·) = e n−i (K) ωi Φ e ∗ (K, ·) is a probability measure on where dµi is the normalized Haar measure on G(n, i). Obviously, Φ i G(n, i). e ϕ,0 = Ve−ϕ (K, L), Φ e 0 (K) = voln (K), and Φ e 0 (L) = voln (L). It Proof of Theorem 1.1. Note that Φ follows directly from Lemma 2.3 that the case when i = n. Now, we consider the case when 2 ≤ i ≤ n − 1. By (1.9), (2.3), (4.1), (2.5) and H¨older’s inequality, it follows that

=

≥

= ≥ = ≥

=

e ϕ,n−i (K, L) Φ e n−i (K) Φ ] n1 [∫ ( )n ωn (i) Ve (K ∩ ξ, L ∩ ξ) dµi (ξ) e n−i (K) G(n,i) −ϕ ωi Φ [∫ (( ] n1 )1 ) ωn voli (K ∩ ξ) i n n (voli (K ∩ ξ)) ϕ dµi (ξ) e n−i (K) G(n,i) voli (L ∩ ξ) ωi Φ [∫ (( ] n1 ) 1i ) vol (L ∩ ξ) i e ∗i (K, ξ) ϕn dΦ voli (K ∩ ξ) G(n,i) [∫ ] ( )1 voli (K ∩ ξ) i e ∗ ϕ dΦi (K, ξ) voli (L ∩ ξ) G(n,i) [( )n ∫ ] 1 ωn n( ni+1 n − ϕ (voli (K ∩ ξ)) ni ) (voli (L ∩ ξ)) ( ni ) dµi (ξ) e n−i (K) ωi Φ G(n,i) ( ) e n−i (K) ni+1 e n−i (L)− 1i i Φ Φ ϕ e n−i (K)n Φ ( )1  e n−i (K) i Φ . ϕ e n−i (L) Φ

If K and L are dilations, then the equality holds in (1.11) is obvious. Conversely, let K, L ∈ Kon . Together the equality conditions of the dual Brunn-Minkowski inequality (2.3), Jensen’s inequality (2.5) with H¨older’s inequality, we know equality holds in inequality (1.11) if and only if K ∩ ξ and L ∩ ξ are dilations for each ξ ∈ G(n, n − i). Therefore, Lemma 4.4 can reduce that K and L are dilations.  Let ϕ(t) = tp with p ≥ 1. An immediate consequence of Theorem 1.1 is: 7 300

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Corolloary 4.5. Suppose K, L ∈ Son . Then for p ≥ 1 and 2 ≤ i ≤ n, e n−i (L)− pi . e p,n−i (K, L) ≥ Φ e n−i (K)1+ pi Φ Φ

(4.2)

If K, L ∈ Kon , then equality holds in the inequality (4.2) if and only if K and L are dilations. A direct consequence of the dual Orlicz-Minkowski inequality is the following uniqueness. Corolloary 4.6. Suppose ϕ ∈ C with ϕ(1) = 1, and U ⊂ Kon (n ≥ 3) such that K, L ∈ U. If for 2 ≤ i ≤ n, there holds e ϕ,n−i (M, K) = Φ e ϕ,n−i (M, L), for all M ∈ U, Φ

(4.3)

e ϕ,n−i (K, M ) e ϕ,n−i (L, M ) Φ Φ = , for all M ∈ U, e n−i (K) e n−i (L) Φ Φ

(4.4)

or

then K = L. Proof. Suppose (4.3) holds. If we take K for M , then by (1.9), (1.2), and ϕ(1) = 1, we have e n−i (K) = ϕ(1)Φ e n−i (K) = Φ e ϕ,n−i (K, K) = Φ e ϕ,n−i (K, L). Φ (

Thus

e n−i (K) Φ 1 = ϕ(1) ≥ ϕ  e n−i (L) Φ

) 1i  ,

with equality if and only if K and L are dilates of each other. Since ϕ is strictly increasing on (0, ∞), we e n−i (K) ≤ Φ e n−i (L), with equality if and only if K and L are dilates of each other. have Φ e n−i (K) ≥ Φ e n−i (L). Therefore, Φ e n−i (K) = Φ e n−i (L), this obtains If let L for M we similarly get Φ voli (K ∩ ξ) = voli (L ∩ ξ), and from the equality conditions of the dual Orlicz-Minkowski inequality we obtain that K and L are dilates of each other. Since K ∩ ξ and L ∩ ξ have the same volume, this implies K = L. Further, suppose that (4.4) holds. Similarly, we get 1 = ϕ(1) = Therefore,

e ϕ,n−i (K, K) e ϕ,n−i (L, K) Φ Φ = . e n−i (K) e n−i (L) Φ Φ (

e n−i (L) Φ 1 = ϕ(1) ≤ ϕ  e Φn−i (K)

) 1i  ,

with equality if and only if K and L are dilates of each other. Since ϕ is strictly increasing on (0, ∞), we e n−i (L) ≥ Φ e n−i (K), with equality if and only if K and L are dilates of each other. have Φ e n−i (L) ≤ Φ e n−i (K). Therefore, Φ e n−i (L) = Φ e n−i (K) can obtain that K Taking L for M , obviously Φ and L are dilates of each other. Since K ∩ ξ and L ∩ ξ have the same volume, this gets K = L.  e Proof of Theorem 1.2. For the convenience, define Kϕ = K +−ϕ L. From Lemma 2.2, we have for ˜ −ϕ L) ∩ ξ = (K ∩ ξ)+ ˜ −ϕ (L ∩ ξ). Note that Kϕ ∩ ξ ∈ Son implies that for ξ ∈ G(n, n − i), Kϕ ∩ ξ = (K + n−i−1 u∈S , ) ( ) ( ρKϕ ∩ξ (u) ρKϕ ∩ξ (u) +ϕ = ϕ(1). (4.5) ϕ ρK∩ξ (u) ρL∩ξ (u)

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[ ] e n−i (Kϕ ) . By (1.2), (4.5), (1.4), (2.3), (4.1) and (2.5) we obtain Suppose λϕ = ωn / ωi Φ ϕ(1) [∫ = λϕ

] n1 n

(ϕ(1)voli (Kϕ ∩ ξ)) dµi (ξ)

G(n,i)

[∫

(

= λϕ G(n,i)

(i) (i) Ve−ϕ (Kϕ ∩ ξ, K ∩ ξ) + Ve−ϕ (Kϕ ∩ ξ, L ∩ ξ)

] n1

)n

dµi (ξ)

{∫

≥

= ≥ =

≥

[ (( (( } n1 ) 1i ) ) 1i )]n vol (K ∩ ξ) vol (K ∩ ξ) i ϕ i ϕ λϕ voli (Kϕ ∩ ξ)n ϕ +ϕ dµi (ξ) voli (K ∩ ξ) voli (L ∩ ξ) G(n,i) (( } n1 {∫ [ (( )1 ) ) 1 )]n voli (Kϕ ∩ ξ) i voli (Kϕ ∩ ξ) i e ∗i (Kϕ , ξ) +ϕ dΦ ϕ voli (K ∩ ξ) voli (L ∩ ξ) G(n,i) (( [ (( )1 ) ) 1 )] ∫ voli (Kϕ ∩ ξ) i voli (Kϕ ∩ ξ) i e ∗ (Kϕ , ξ) +ϕ dΦ ϕ i voli (K ∩ ξ) voli (L ∩ ξ) G(n,i) (( (( )1 ) ) 1i ) ∫ ∫ voli (Kϕ ∩ ξ) i vol (K ∩ ξ) i ϕ ∗ e (Kϕ , ξ) + e ∗ (Kϕ , ξ) ϕ dΦ dΦ ϕ i i vol (K ∩ ξ) vol (L ∩ ξ) i i G(n,i) G(n,i)     ) 1i ) 1i ∫ ( ∫ ( voli (Kϕ ∩ ξ) voli (Kϕ ∩ ξ)   e ∗ (Kϕ , ξ) e ∗ (Kϕ , ξ) ϕ dΦ dΦ  + ϕ . i i voli (K ∩ ξ) voli (L ∩ ξ) G(n,i)

(4.6)

G(n,i)

From H¨older inequality and (1.2), we get   ) 1i ∫ ( voli (Kϕ ∩ ξ)  e ∗i (Kϕ , ξ) dΦ ϕ  voli (K ∩ ξ) G(n,i)

[( = ϕ [( ≥ ϕ

]

)n ∫

ωn e n−i (Kϕ ) ωi Φ ωn

n( ni+1 ni )

(voli (Kϕ ∩ ξ))

1 n(− ni )

(voli (K ∩ ξ))

dµi (ξ)

G(n,i)

)n (∫

) ni+1 ni n

(voli (Kϕ ∩ ξ)) dµi (ξ)

e n−i (Kϕ ) ωi Φ

(4.7)

G(n,i)

(∫

1 ] )− ni

n

×

(voli (K ∩ ξ)) dµi (ξ) G(n,i)

(

e n−i (Kϕ ) Φ = ϕ e n−i (K) Φ Similarly,

  ϕ

) 1i  .

∫

G(n,i)

(

voli (Kϕ ∩ ξ) voli (L ∩ ξ)

) 1i



(

e n−i (Kϕ ) Φ e ∗i (Kϕ , ξ) dΦ  ≥ ϕ e n−i (L) Φ

) 1i  .

(4.8)

Together (4.6), (4.7) with (4.8), this yields ( ( )1  )1  e n−i (Kϕ ) i e n−i (Kϕ ) i Φ Φ  + ϕ  ≤ ϕ(1). ϕ e n−i (K) e n−i (L) Φ Φ 9 302

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Finally, we give the equality conditions. Suppose that K and L are dilations. with equality in (1.12) is obvious. Conversely, Let K, L ∈ Kon . From the equality conditions of the dual Orlicz-Minkowski inequality of star bodies, Jensen’s inequality (2.5) and H¨older’s inequality, we obtain that equality holds in inequality (1.12) if and only if K ∩ ξ and L ∩ ξ are dilations for each ξ ∈ G(n, n − i). Therefore, Lemma 4.4 can get that K and L are dilations.  If ϕ(t) = tp with p ≥ 1, then we get: Corolloary 4.7. Let K, L ∈ Son . If p ≥ 1 and 2 ≤ i ≤ n, then p e n−i (K + e n−i (K)− pi + Φ e n−i (L)− pi . e −p L)− i ≥ Φ Φ

(4.9)

If K, L ∈ Kon , then equality holds in the inequality (4.9) if and only if K and L are dilations. An immediate consequence of the inequality (4.9) is: Corolloary 4.8. Let K, L ∈ Son . If p ≥ 1 and 2 ≤ i ≤ n, then ) ( )p ( p e n−i (K + e n−i (K)Φ e n−i (L) 2i ≤ 1 Φ e n−i (K) pi + Φ e n−i (L) pi . e −p L) i ≤ Φ 2Φ 2

(4.10)

If K, L ∈ Kon , with equality in (4.10) if and only if K = L. Proof. By (4.9) and the arithmetic-geometric-harmonic mean inequality, we have e n−i (K + e −p L) i 2Φ

p

≤ ≤ ≤

1 p e n−i (K) i Φ

2 +

(

1 p e n−i (L) i Φ

)p e n−i (K)Φ e n−i (L) 2i Φ ) p 1 (e e n−i (L) pi . Φn−i (K) i + Φ 2 

We see easily that equality holds in the inequality (4.10) if and only if K = L. e ϕ,i (K, L) and W f−ϕ,i (K, L). The next result is a relationship between Φ Theorem 4.9. Suppose K, L ∈ Son and i = 1, 2, · · · , n − 1. Then e ϕ,i (K, L) ≥ W f−ϕ,i (K, L), Φ

(4.11)

(n−i) with equality if and only if Ve−ϕ (K ∩ ξ, L ∩ ξ) is constant for all ξ ∈ G(n, n − i).

Proof. Notice that Ve−ϕ (K ∩·, L∩·) is positive on G(n, n−i) and that µn−i is a probability measure on G(n, n − i). Hence, it follows from Jensen’s inequality (2.5) that (∫ ) n1 ∫ (n−i) (n−i) Ve−ϕ (K ∩ ξ, L ∩ ξ)n dµn−i (ξ) ≥ Ve−ϕ (K ∩ ξ, L ∩ ξ)dµn−i (ξ), (n−i)

G(n,n−i)

G(n,n−i)

(n−i) Ve−ϕ (K

with equality if and only if ∩ ξ, L ∩ ξ) is constant for all ξ ∈ G(n, n − i). This inequality and e ϕ,i (K, L) and W f−ϕ,i (K, L) can easily yield the desired inequality. the definitions of Φ  Conflict of Interests The author declare that they have no competing interests. Authors’ Contribution All authors contributed equally to the paper and read and approved its final version. Acknowledgment This work is supported by the National Natural Science Foundations of China (Grant No.11561020) and is partly supported by the National Natural Science Foundations of China (Grant No.11371224). 10 303

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References [1] H. Furstenberg, I. Tzkoni, Spherical harmonics and integral geometry. Israel J Math, 1971, 10: 327-338. [2] R. J. Gardner, Geometric Tomography, Encyclopedia of Mathematics and Its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. [3] R. J. Gardner, D. Hug, W. Weil, The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities. J Differential Geom, 2014, 97: 427-476. [4] R. J. Gardner, Geometric Tomography. Cambridge: Cambridge University Press, 2006. [5] R. J. Gardner, D. Hug, W. Weil, et al., The dual Orlicz-Brunn-Minkowski theory. Journal of Mathematical Analysis and Applications, 2015, 430(2): 810-829. [6] E. L. Grinberg, Isoperimetric inequalities and identities for k-dimensional cross-sections of a convex bodies. London Mathematical Society, 1990, 22: 478-484. [7] E. L. Grinberg, Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies. Math. Ann., 1991, 291: 75-86. [8] G. H. Hardy, J. E. Littlewood, G. P´ olya. Inequalities, Cambridge Univ. Press, London, 1934. [9] C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz Minkowski problem. Adv. Math., 2010, 224: 2485-2510. [10] H. L. Jin, S. F. Yuan and G. S. Leng, On the dual Orlicz mixed volumes, Chin. Ann. Math., 2015, 36B(6): 1019-1026. [11] D. Klain, Star valuations and dual mixed volumes. Adv. Math., 1996, 121(1): 80-101. [12] D. Y. Li, D. Zou, G. Xiong, Orlicz mixed affine quermassintegrals. Science China Mathematics, 2015, 58(8): 1715-1722. [13] M. Ludwig, General affine surface areas. Adv. Math., 2010, 224: 2346-2360. [14] E. Lutwak, Inequalities for Hadwiger’s harmonic quermassintegrals. Math. Ann., 1988, 280: 165-175. [15] E. Lutwak, Dual mixed volumes. Pacific Journal of Mathematics, 1975, 58(2): 531-538. [16] E. Lutwak, A general isepiphanic inequality. Proceedings of the AmericanMathematical Society, 1984, 90(3): 415-421. [17] E. Lutwak, D. Yang, G. Zhang, Orlicz projection bodies. Adv. Math., 2010, 223: 220-242. [18] E. Lutwak, D. Yang, G. Zhang, Orlicz centroid bodies. J. Differential Geom., 2010, 84: 365-387. [19] E. Lutwak, The Brunn-Minkowski-Firey theory, I: Mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38: 131-150. [20] E. Lutwak, The Brunn-Minkowski-Firey theory, II: Affine and geominimal surface areas. Adv. Math., 1996, [21] [22] [23] [24] [25]

118: 244-294. E. Lutwak, Intersection bodies and dual mixed volumes. Adv. Math., 1988, 71: 232-261. T. Y. Ma, The minimal dual Orlicz surface area. Tawanese Journal of Mathematics, 2016, 20(2): 287-309. D. L. Ren, Topics in Integral Geometry. Singapore: World Scientific, 1994. L. A. Santal´ o, Integral geometry and geometric probability. Reading, MA: Addison-Wesley, 1976. L. A. Santal´ o, Integral Geometry and Geometric Probability. Cambridge: Cambridge University Press, 2004.

[26] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge University Press, Second Expanded Edition, 2014. [27] J. Yuan, G. S. Leng, Inequalities for dual affine quermassintegrals. Journal of Inequalities and Applications, 2006, 2006(1): 1-7. [28] G. Y. Zhang, Dual kinematic formulas. Tran. Amer. Math., 1999, 351(3): 985-995. [29] C. J. Zhao, Reverse Lp -dual Minkowski’s inequality. Differential Geometry and its Applications, 2015, 40: 243-251. [30] C. J. Zhao, Orlicz dual mixed volumes, Results. Math., 2015, 68: 93-104. [31] B. Zhu, J. Zhou, W. Xu, Dual Orlicz-Brunn-Minkowski theory. Adv. Math., 2014, 264: 700-725.

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Existence and convergence for fixed points of a strict pseudo-contraction in CAT(0) spaces Narongrit Puturonga , Kasamsuk Ungchittrakoola,b,∗ 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 The purposes of this paper are to introduce and study some existence and convergence theorems for fixed points of a strict pseudo-contraction in the framework of complete CAT(0) spaces. By using available properties in the spaces together with some appropriate conditions of the mapping and under certain assumptions, we can create some suitable sets to be used to construct an iterative projection algorithm to guarantee the existence fixed points for a strict pseudo-contraction. The method allows us to obtain a strong convergence iteration for finding some fixed points of a strict pseudo-contraction in the framework of complete CAT(0) spaces. Keywords: Strict pseudo-contraction; Iterative projection technique; CAT(0) space

1. Introduction Let (X, d) be a metric space, and x, y ∈ X with l = d(x, y). A geodesic path from x to y is an isometry γ : [0, l] → X such that γ(0) = x and γ(l) = y. The image of a geodesic path is called a geodesic segment. When it is unique this geodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A geodesic triangle 4(x1 , x2 , x3 ) in a geodesic space X consists of three points x1 , x2 , x3 of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle ¯ 1 , x2 , x3 ) = 4(¯ 4(x1 , x2 , x3 ) is the triangle 4(x x1 , x ¯2 , x ¯3 ) in the Euclidean space E2 such that d(xi , xj ) = dE2 (¯ xi , x ¯j ) for all i, j = 1, 2, 3. A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. ¯ be a comparison triangle for 4. Then 4 is CAT(0) : Let 4 be a geodesic triangle in X and let 4 ¯ said to satisfy the CAT(0) inequality if for all x, y ∈ 4 and all comparison points x ¯, y¯ ∈ 4, d(x, y) ≤ dE2 (¯ x, y¯). If x, y1 , y2 are points in a CAT(0) space and if y0 is the midpoint of the segment [y1 , y2 ], then the CAT(0) inequality implies that d2 (x, y0 ) ≤

1 2 1 1 d (x, y1 ) + d2 (x, y2 ) − d2 (y1 , y2 ). 2 2 4

(1.1)

∗ Corresponding

author. Tel.:+66 55963250; fax:+66 55963201. Email addresses: [email protected] (Kasamsuk Ungchittrakool), [email protected] (Narongrit Puturong)

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This is the (CN) - inequality of Bruhat and Tits [5]. In fact ([3] p.163), a geodesic space is a CAT(0) space if and only if it satisfies the (CN) - inequality. The study of CAT(0) spaces, Kirk [15, 16] first studied the fixed point theory in CAT(0) spaces. Since then, many authors have developed the fixed point theory for single-valued and set-valued mappings in the setting of CAT(0) spaces. Dhompongsa et al. [7] proved that a nonexpansive mapping from a nonempty bounded closed convex subset of a CAT(0) space to the family of nonempty compact subsets of the CAT(0) space has a fixed point under suitable conditions. In 2008, Berg and Nikolaev [2] introduced the concept of quasilinearization. In 2010, Kakavandi and Amini [13] introduced the concept of dual space for CAT(0) spaces. In 2012, Dehghan and Rooin [6] presented a characterization of metric projection in CAT(0) spaces. In 2014, Lu et al. [19] establish generalized CAT(0) versions of the Fan-Browder fixed point theorem. In the same year, Ungchittrakool [22] has discovered some significant inequalities for a strict pseudo-contraction in the framework of Hilbert spaces that has resulted in creating the important sets and the iterative shrinking projection technique to ensure the existence for fixed points of a strict pseudo-contraction in the terminology of Browder and Petryshyn [4]. Inspired and motivated by the significance of the problems mentioned above, we will pay attention to investigate and establish the existence theorem for fixed points of the mapping called strict pseudo-contraction mappings and some related mappings in complete CAT(0) spaces by employing suitable structure of certain sets based on the shrinking projection technique. 2. Preliminaries Recall that a metric space (X, d) is said to be a geodesic space if every two points of X are joining by a geodesic and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. We write (1 − t)x ⊕ ty for the unique point z in the geodesic segment joining from x to y such that d(z, x) = td(x, y) and d(z, y) = (1 − t)d(x, y). We also denote by [x, y] the geodesic segment joining from x to y, that is [x, y] = {(1 − t)x ⊕ ty : t ∈ [0, 1]}. A subset C of a CAT(0) space is convex if [x, y] ⊆ C for all x, y ∈ C. In 1976, Lim in [18] introduced the concept of 4-convergence, and Kirk and Panyanak [17] has obtained some results in CAT(0) spaces which is every similar for weak convergence in Banach space setting. Next , we present the concept of 4-convergence and collect some basic properties. Let {xn } be a bounded sequence in a CAT(0) space X. For x ∈ X, we set r(x, {xn }) = lim sup d(x, xn ). n→∞

The asymptotic radius r({xn }) of {xn } is given by r({xn }) = inf{r(x, {xn }) : x ∈ X}, the asymptotic radius rC ({xn }) of {xn } with respect to C ⊂ X is given by rC ({xn }) = inf{r(x, {xn }) : x ∈ C}, the asymptotic center A({xn }) of {xn } is the set A({xn }) = {x ∈ X : r(x, {xn }) = r({xn })} and the asymptotic center AC ({xn }) of {xn } with respect to C ⊂ X is the set AC ({xn }) = {x ∈ C : r(x, {xn }) = rC ({xn })}. It is known from Proposition 7 of [8] that in a CAT(0) space, A({xn }) consists of exactly one point. A subset of a CAT(0) space equipped with the induced metric, is a CAT(0) space if and only if it is convex ([3], p.167).

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Definition 2.1 ([17], Definition 3.1). A sequence {xn } in a CAT(0) space X is said to be 4converge to x ∈ X if x is the unique asymptotic center of {un } for every subsequence {un } of {xn }. In this case, we write 4 − lim xn = x and x is called the 4-limit of {xn }. n→∞

Lemma 2.2 ([17], Opial’s property). Let X be a complete CAT(0) space and a sequence {xn } in X such that {xn } 4-converge to x and given y ∈ X with y 6= x. Then we have lim sup d(xn , x) < lim sup d(xn , y). n→∞

n→∞

It is known from [17] that, the uniqueness of asymtotic center implies that CAT(0) space X satisfies Opial’s property. Let X be a complete CAT(0) space. Bijan Ahmadi Kakavandi [12] introduced the properties of 4-convergence, i.e., every closed convex subset of X is 4-closed in the sense that it contains all 4-limit point of every 4-convergent sequence. Lemma 2.3 ([20], Lemma 3.5). Every bounded closed convex set in a complete CAT(0) space always has a 4-convergent subsequence. Lemma 2.4 ([9], Proposition 2.1). If C is a closed convex subset of a complete CAT(0) space and if {xn } is a bounded sequence in C, then the asymptotic center of {xn } is in C. Recall that a subset K of a metric space X is said to be 4- compact if every sequence in K has a 4- convergent subsequence. Lemma 2.5 ([17], Proposition 3.6). Every bounded closed convex set in a complete CAT(0) space is 4- compact. Let C be a closed convex subset of a CAT(0) space X and {xn } be a bounded sequence in C. We use the following notation {xn } * w ⇐⇒ Φ(w) = inf x∈C Φ(x) where Φ(x) = lim sup d(xn , x). n→∞

Also, we have {xn } * w ⇐⇒ AC ({xn }) = {w}.

Lemma 2.6 ([20], Proposition 3.12). Let {xn } be a bounded sequence in a CAT(0) space X and let C be a closed convex subset of X which contain {xn }. Then 4 − lim xn = x implies that n→∞

{xn } * x.

Berg and Nikolaev [2] have introduced the concept of quasilinearization as follows. Let us − → formally denote a pair (a, b) ∈ X × X by ab and call it a vector. Then quasilinearization is the map h·, ·i : (X × X) × (X × X) → R defined by D→ − − →E 1 (2.1) ab, cd = {d2 (a, d) + d2 (b, c) − d2 (a, c) − d2 (b, d)} for all a, b, c, d ∈ X. 2 D− D− D → − →E D→ − − →E D− → − →E → − →E − E D− → → − E D− → − →E → → It is easily seen that ab, cd = cd, ab , ab, cd = − ba, cd and − ax, cd + xb, cd = ab, cd for all a, b, c, d, x ∈ X. D− → → −E We say that X satisfies the Cauchy - Schwarz inequality if ab, cd ≤ d(a, b)d(c, d) for all a, b, c, d ∈ X. It is known ([2], Corollary 3) that a geodesically connected metric space is CAT(0) space if and only if it satisfies the Cauchy- Schwarz inequality. Definition 2.7 ([3], Proposition 2.4). Let (X, d) be a metric space and C ⊆ X. The distance function d(x, C) : X → C is defined by d(x, C) = inf d(x, c) for any x ∈ X. c∈C

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Lemma 2.8 ([3], Proposition 2.4). Let C be a closed convex subset of a complete CAT(0) space X and x ∈ X. Then there exists a unique point p ∈ C such that d(x, C) = d(x, p) = inf d(x, y). y∈C

Definition 2.9 ([3], Proposition 2.4). Let C be a closed convex subset of a complete CAT(0) space X and PC : X → C is defined by PC x = p such that p satisfies Lemma 2.8. PC is said to be the metric projection from X onto C. Dehghan and Rooin [6] presented monotone and a characterization of metric projection in CAT(0) spaces as follows: T of C where C is a subset of CAT(0) space (X, d) is said to be monotone if D A−self-mapping −−→E − → xy, T xT y ≥ 0 for all x, y ∈ C. Aslo, it is nonexpansive if d(T x, T y) ≤ d(x, y) for all x, y ∈ C. Lemma 2.10 ([6], Lemma 2.1). Let X be a CAT(0) space, x, y ∈ X, λ ∈ [0, 1] and z = λx⊕(1−λ)y. → − → ≤ λ h− → − → for all w ∈ X. Then h− zy, zwi xy, zwi Lemma 2.11 ([6], Theorem 2.2). Let C be a nonempty convex subset of a CAT(0) space X, x ∈ X → − → ≥ 0 for all y ∈ C. and u ∈ C. Then u = PC x if and only if h− xu, uyi Lemma 2.12 ([6], Proposition 2.4). Let C be a nonempty closed convex subset of a complete CAT(0) space X. Then PC : X → C is monotone and nonexpansive. Lemma 2.13 ([23], Lemma 2.10). Let X be a CAT(0) space. For any u, v ∈ X and t ∈ [0, 1], let ut = tu ⊕ (1 − t)v. Then, for all x, y ∈ X, −→ − → −→ − → −→ (1) h− u→ t x, ut yi ≤ t hux, ut yi + (1 − t) hvx, ut yi; → ≤ t h− → − → + (1 − t) h− → − → and h− → ≤ t h− → − → + (1 − t) h− → − → (2) h− u→ x, − uyi ux, uyi vx, uyi u→ x, − vyi ux, vyi vx, vyi. t

t

Lemma 2.14 ([3], Proposition 2.2). Let X be a CAT(0) space, p, q, r, s ∈ X and λ ∈ [0, 1]. Then d[λp ⊕ (1 − λ)q, λr ⊕ (1 − λ)s] ≤ λd(p, r) + (1 − λ)d(q, s). Lemma 2.15 ([10], Lemma 2.5). Let X be a CAT(0) space, x, y, z ∈ X and λ ∈ [0, 1]. Then d2 (λx ⊕ (1 − λ)y, z) ≤ λd2 (x, z) + (1 − λ)d2 (y, z) − λ(1 − λ)d2 (x, y). Definition 2.16 ([1], Definition 3.2.2). Let X be a complete CAT(0) space and let f be a function of X into (−∞, ∞]. Then f is said to be weakly lower semicontinuous on X if and only if for any x0 ∈ X, {xn } * x0 implies that f (x0 ) ≤ lim inf f (xn ). n→∞

Lemma 2.17 ([1], Corollary 3.2.4). Let C be a nonempty closed convex subset of a complete CAT(0) space X. The distance function d(x, C) as well as its square d2 (x, C) are weakly lower semicontinuous. We first introduce the definition of k-strict pseudo-contraction in CAT(0) spaces. Definition 2.18. Let (X, d) be a CAT(0) space and C be a nonempty subset of X. The mapping T : C → C is said to be a k-strict pseudo-contraction in the terminology of Browder and Petryshyn [4] if for all x, y ∈ C there exists k ∈ (−∞, 1) that D−such −→ −−→E 2 2 2 d (T x, T y) ≤ d (x, y) + k{d (x, T x) − 2 xT x, yT y + d2 (y, T y)}. Lemma 2.19. Let C be a nonempty closed convex subset of a CAT(0) space X,and T : C → C be a k-strict pseudo-contraction, then T satisfies the Lipschitz condition with Lipschitz constant 1+k , 1} for all x, y ∈ C. That is L = max{ 1−k 1+k , 1}d(x, y) for all x, y ∈ C. d(T x, T y) ≤ max{ 1−k

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Proof. Let C be a nonempty closed convex subset of a CAT(0) space X. For T : C → C be a k-strict pseudo-contraction, we have D−−→ −−→E d2 (T x, T y) ≤ d2 (x, y) + k{d2 (x, T x) − 2 xT x, yT y + d2 (y, T y)} = d2 (x, y) + k{d2 (x, T x) − d2 (x, T y) − d2 (y, T x) + d2 (x, y) + d2 (T x, T y) + d2 (y, T y)} (2.2) By simple calculation from (2.2) we have that (1 − k)d2 (T x, T y) ≤ (1 + k)d2 (x, y) + k{d2 (x, T x) − d2 (x, T y) − d2 (y, T x) + d2 (y, T y)} D −−→E → − = (1 + k)d2 (x, y) + 2k − xy, T yT x . (2.3) Since X satisfies the Cauchy-Schwarz inequality, it follows from (2.3), we get that (1 − k)d2 (T x, T y) − 2kd(x, y)d(T x, T y) − (1 + k)d2 (x, y) ≤ 0.

(2.4)

Next, we will divide the proof into two cases. Case 1. k ≤ 0. Notice that k ≤ 0 ⇔ 2k ≤ 0 ⇔ 1 + k ≤ 1 − k ⇔ (2.4), we have

1+k 1−k

1+k ≤ 1 ⇔ max{ 1−k , 1} = 1. Since k ≤ 0, from

(1 − k)d2 (T x, T y) + 2kd(x, y)d(T x, T y) − (1 + k)d2 (x, y) ≤ (1 − k)d2 (T x, T y) − 2kd(x, y)d(T x, T y) − (1 + k)d2 (x, y) ≤ 0. Thus (1 − k)d2 (T x, T y) + 2kd(x, y)d(T x, T y) − (1 + k)d2 (x, y) ≤ 0. Solving this quadratic inequality, we obtain 1+k }d(x, y) for all x, y ∈ C. It implies that d(T x, T y) ≤ d(x, y) or d(T x, T y) ≤ { 1−k 1+k d(T x, T y) ≤ d(x, y) = max{ 1−k , 1}d(x, y) for all x, y ∈ C. Case 2. 0 ≤ k < 1. 1+k 1+k ≥ 1 ⇔ max{ 1−k , 1} = We have 1 − k > 0 and then k ≥ 0 ⇔ 2k ≥ 0 ⇔ 1 + k ≥ 1 − k ⇔ 1−k 2 2 Similarly case 1, we have (1 − k)d (T x, T y) − 2kd(x, y)d(T x, T y) − (1 + k)d (x, y) ≤ 0. 1+k 1+k }d(x, y) = max{ 1−k , 1}d(x, y) for all x, y ∈ C. It implies that d(T x, T y) ≤ { 1−k Therefore, the desired result.

1+k 1−k .

In this paper, we denote that Fix(T ) is the set of fixed point of T such that Fix(T ) = {x ∈ C : T x = x}. Lemma 2.20 ([11], Theorem 2.3). Let C be a closed convex subset of a CAT(0) space X and T : C → C be a k - strict pseudo-contraction mapping. If Fix(T ) 6= ∅, then Fix(T ) is closed and convex so that the projection PF ix(T ) is well defined. Lemma 2.21 ([19], Lemma 2.2). Let (E, d) be a complete metric space. Then E is a geodesic space if and only if for every x, y ∈ E, there exists z ∈ E such that d(x, z) = d(z, y) = 21 d(x, y). Lemma 2.22 ([3], p.163). A geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality. Let (E, d) be a CAT(0) space and D ⊆ E. Niculescu and Roventa [21] introduced the notion of a convex hull of D as follows : ∞ [ co(D) = Dn where D0 = D and for n ≥ 1, the set Dn consists of all points in E which lie n=0

on geodesics which start and end in Dn−1 .

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co(D) ¯ denote the closure of the convex hull. It is easy to see that in a CAT(0) space, the closure of the convex hull will be convex and hence it is the smallest closed convex set containing D ([1], p.31). Definition 2.23 ([19], Definition 2.2). Let D be a nonempty subset of a CAT(0) [ space (E, d). A set-valued mapping G : D → 2E is called to be a KKM mapping if co(F ) ⊂ G(x) for every x∈F

F ∈ hDi where hDi denotes the class of all nonempty finite subsets of D. Lemma 2.24 ([14], Lemma 1.8). Suppose X is a complete CAT(0) space and K is a nonempty subset of X. Let G : K → 2K be a mapping such that for each x ∈ K, G(x) be 4-closed. Suppose that m [ (1) each x1 , ..., xm ∈ K, co({x1 , ..., xm }) ⊂ G(xi ), i=1

(2) there exists x0 ∈ K such that G(x0 ) is 4- compact. \ Then G(x) 6= ∅. x∈K

Lemma 2.25. Let (E, d) be a complete CAT(0) space, K be a nonempty 4-compact subset of E, and F, G : E → 2E be two set-valued mappings such that (1) for every y ∈ E, F (y) ⊆ G(y) and G(y) is convex; (2) for every x ∈ E, F −1 (x) is open in E; (3) for every y ∈ K, F (y) 6= ∅; −1 (4) there exists a point x0 ∈ E such that co(E\G ¯ (x0 )) ⊆ K. Then, there exists yb ∈ E such that yb ∈ G(b y ). Proof. Suppose the contrary. Then, for every y ∈ E, y 6∈ G(y). Now let us define two set-valued e Fe : E → 2E by mappings G, −1 e G(x) = co(E\G ¯ (x)) and Fe (x) = co(E\F −1 (x)) for all x ∈ E.

By using (1) and F (y) ⊆ G(y) for every y ∈ E, we have F −1 (x) = {y ∈ E : x ∈ F (y)} ⊆ {y ∈ E : x ∈ G(y)} = G−1 (x). Then, E\G−1 (x) ⊆ E\F −1 (x) for every x ∈ E. It implies that co(E\G−1 (x)) ⊆ co(E\F −1 (x)). −1 By using (2), we have co(E\F −1 (x)) is closed in E. Since co(E\G ¯ (x)) is the smallest closed −1 e set containing co(E\G−1 (x)). Then co(E\G ¯ (x)) ⊆ co(E\F −1 (x)). Therefore G(x) ⊆ Fe(x) for every x ∈ E. [ e is a KKM mapping. That is, for every A ∈ hEi , co(A) ⊆ e G(x). OtherWe next show that G wise, there exist A ∈ hEi and a point y ∈ co(A) such that y 6∈ (E\G−1 (x)) ⊆ E, we have co(E\G−1 (x)) =

∞ [

[

e G(x) =

x∈A

[

x∈A −1 (co(E\G ¯ (x))). For

x∈A

(E\G−1 (x))n where (E\G−1 (x))0 = E\G−1 (x)

n=0

and for n ≥ 1, (E\G−1 (x))n consists of all points in E which lie on geodesics which start and end in (E\G−1 (x))n−1 . ∞ [ [ [ [ −1 e Let us consider, y 6∈ G(x) = (co(E\G ¯ (x))) = clE { (E\G−1 (x))n }. Since [ x∈A

clE (E\G−1 (x)) ⊆

[

x∈A

clE {

∞ [

x∈A

(E\G−1 (x))n }. It implies that

n=0

x∈A

y 6∈

[

n=0

x∈A

clE (E\G−1 (x)) = E\

x∈A

\

intE G−1 (x).

x∈A

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It follows that y ∈

\

G−1 (x). Therefore A ⊆ G(y). Since G(y) is convex by (1), and co(A) is the

x∈A

smallest convex set containing A. We get that y ∈ co(A) ⊆ G(y), which is a contradiction. Hence e is a KKM mapping. By the definition of G, e G(x) e G is 4-closed in E for every x ∈ E. By using −1 e e 0 ) is (4), there exists a point x0 ∈ E such that G(x0 ) = co(E\G ¯\ (x0 )) ⊆ K, it implies that G(x e e 0 ) ⊆ K. Therefore, we have 4-compact. Then, by Lemma 2.24, we get that ∅ 6= G(x) ⊆ G(x x∈E

∅ 6= K ∩ (

\

e G(x)) ⊆K ∩(

x∈E

Taking y0 ∈ K ∩ (

\

\

Fe(x)).

x∈E

Fe(x)), we have y0 ∈ K and x 6∈ F (y0 ) for every x ∈ E. Hence, we have

x∈E

F (y0 ) = ∅ which contradicts (3). Therefore, there exists yb ∈ E such that yb ∈ G(b y ). This completes our proof. Remark 2.26. If F = G, then (4) of Lemma 2.25 can be replaced by the following equivalent condition: (4)∗ there exists a point x0 ∈ E such that co(E\F −1 (x0 )) ⊆ K. 3. Main Results In this section, motivated by Ungchittrakool [22]. We discuss the existence and convergence for fixed point of a strict pseudo-contraction in the terminology of Browder and Petryshyn in the framework of complete CAT(0) spaces. Lemma 3.1. Let C be a bounded closed convex subset of a complete CAT(0) space (X, d). Then (C, d) is a complete CAT(0) space. Proof. Let C be a bounded closed convex subset of complete CAT(0) space (X, d). Notice that, a subset of a CAT(0) space equipped with the induced metric, is a CAT(0) space if and only if it is convex. This implies that (C, d) is a CAT(0) space. Since C is closed subset of complete metric space (X, d), then (C, d) is complete metric space. Therefore, we have (C, d) is a complete CAT(0) space. Lemma 3.2. Let C be a bounded closed convex subset of a complete CAT(0) space X. Let T be a k- strict pseudo-contraction the terminology of Browder and Petryshyn. Then, there exists an D in−− −→E − − → element x0 ∈ C such that xx0 , xT x0 ≥ 0 for all x ∈ C. Proof. Let C be a bounded closed convex subset a complete CAT(0) space (X, d). We claim D of − −−→E − − → that there exists an element x0 ∈ C such that xx0 , xT x0 ≥ 0 for all x ∈ C. For any y ∈ C, we D −→E → − assume that the set {x ∈ C : − xy, xT y < 0} is nonempty. We also define two set-valued mappings D −→E → − F, G : C → 2C by F (y) = G(y) = {x ∈ C : − xy, xT y < 0}. We first show that G(y) is convex and F −1 (x) is an open set. Step1. To show that G(y) is convex. Let x1 , x2 ∈ G(y) and ut = tx1 ⊕ (1 − t)x2 such that t ∈ [0, 1]. So, we have x1 , x2 ∈ C, that ut ∈ C.

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D −−−→E Let us consider − u→ t y, ut T y , by Lemma 2.13, we get that D

−−−→E − u→ t y, ut T y D D E D−−−→ E D−−−→ E −−−→E −→ −−−→ −→ −→ ≤t − x→ 1 y, ut T y + (1 − t) x2 y, ut T y = t ut T y, x1 y + (1 − t) ut T y, x2 y D−−−→ E D−−−→ E D−−−→ E D−−−→ E −→ −→ −→ ≤ t{t x1 T y, − x→ 1 y + (1 − t) x2 T y, x1 y } + (1 − t){t x1 T y, x2 y + (1 − t) x2 T y, x2 y } D−−−→ E D−−−→ E D −−→ E D−−−→ E −→ −→ 2 − = t2 x1 T y, − x→ x2 T y, − x→ 1 y + t(1 − t) x2 T y, x1 y + (1 − t) 2 y + t(1 − t) x1 T y, x2 y D−−−→ E D−−−→ E −−→ −→ −→ = t2 x1 T y, − x→ 1 y + t(1 − t){hx2 x1 , x1 yi + x1 T y, x1 y } D−−−→ E D−−−→ E −−→ −→ −→ + (1 − t)2 x2 T y, − x→ 2 y + t(1 − t){hx1 x2 , x2 yi + x2 T y, x2 y } D−−−→ E D−−−→ E D −−→ E D−−−→ E −→ −→ 2 − = t2 x1 T y, − x→ x2 T y, − x→ 1 y + t(1 − t) x1 T y, x1 y + (1 − t) 2 y + t(1 − t) x2 T y, x2 y → −−→ −−→ −→ −−→ −→ + t(1 − t){h− x− 2 x1 , x1 x2 i + hx2 x1 , x2 yi} + t(1 − t) hx1 x2 , x2 yi D−−−→ E D−−−→ E −→ −−→ −−→ = t x1 T y, − x→ 1 y + (1 − t) x2 T y, x2 y − t(1 − t) hx1 x2 , x1 x2 i → −→ −−→ −→ − t(1 − t) h− x− 1 x2 , x2 yi + t(1 − t) hx1 x2 , x2 yi D−−−→ E D−−−→ E − → ≤ t x1 T y, − x→ y + (1 − t) x T y, x y 1 2 2 < 0.

Therefore ut ∈ G(y), that is G(y) is convex. Step 2. To show that F −1 is an open set. D D (x)−− −→E →E → − − → −1 xy, xT y ≥ 0} is a For F (x) = {y ∈ C : xy, xT y < 0}, we show that C\F −1 (x) = {y ∈ C : − D −−→E −→, − closed set. Let {y } ⊆ C\F −1 (x) such that y → y . Then − xy xT y ≥ 0. We will show that n

n

0

n

n

y0 ∈ C\F −1 (x). By Lemma 2.19, T is a Lipschitzian map. Inparticular, T is continuous. It follows that D −−→E D−→ −−−→E D−−→ −−−→E D−→ −−−→E D−→ −−−−−→E D−−→ −−−→E −→, − 0≤ − xy n xT yn = xy0 , xT yn + y0 yn , xT yn = xy0 , xT y0 + xy0 , T y0 T yn + y0 yn , xT yn D −−→E →, − xy ≤ − 0 xT y0 + d(x, y0 )d(T y0 , T yn ) + d(y0 , yn )d(x, T yn ) D −−→E 1+k →, − , 1}d(x, y0 )d(y0 , yn ) + d(y0 , yn )d(x, T yn ), ≤ − xy 0 xT y0 + max{ 1−k D −−→E →, − for all n ∈ N. Taking the limit in both sides, we get that − xy 0 xT y0 ≥ 0. That is y0 ∈ C\F −1 (x). Hence C\F −1 (x) is a closed set in C, therefore F −1 (x) is an open set. D −−−→E − − → We next show that there exists an element x0 ∈ C such that xx0 , xT x0 ≥ 0 for all x ∈ C. By assumption, we have F (y) 6= ∅ for every y ∈ C, and by Lemma 2.5, we have C is 4- compact. Notice that there exists a point z ∈ C such that C\F −1 (z) ⊆ C. Also, co(C\F −1 (z)) is the smallest convex set containing C\F −1 (z). Then, we get that there exists a point z ∈ C such that co(C\F −1 (z)) ⊆ C where C is a nonempty 4-compact subset of C. Also, by Lemma 3.1, we have (C, d) is a complete CAT(0) space. D By Lemma 2.25 and Remark 2.26, we have x0 ∈ C such that −−−→E →, − x0 ∈ G(x0 ). This implies that 0 = − x− x x T x < 0. This is a contradiction. We obtain that 0 0 0 0 D E D −−→ −−→E −→, − − −→, − {x ∈ C : − xx xT x < 0} = ∅. Therefore xx xT x0 ≥ 0 for all x ∈ C. 0 0 0 Lemma 3.3. Let C be a bounded closed convex subset of a complete CAT(0) space X. Let T be a k- strict pseudo-contraction the terminology of Browder and Petryshyn. Then, there exists an D in−− −−→E − − → element x0 ∈ C such that xx0 , x0 T x0 ≥ 0 for all x ∈ C.

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Proof. Let C be a bounded closed convex subset of complete CAT(0) space (X, d). By Lemma 3.1, we have (C, d) is a complete CAT(0) space. By Lemma 3.2, we have D −−→E −→, − x0 ∈ C such that − xx (3.1) 0 xT x0 ≥ 0 for all x ∈ C. Also, for any u, z ∈ C and 0 < t < 1 and since C is convex, we have yt = (1 − t)u ⊕ tz ∈ C. Then, for x0 ∈ C we have yt = (1 − t)x0 ⊕ tz ∈ C. (3.2) D−−−−−−−−−−−−−→ −−−→E By using (3.1) and (3.2), we have 0 ≤ {(1 − t)x0 ⊕ tz}x0 , yt T x0 . By Lemma 2.10, we have D−−−−−−−−−−−−−→ −−−→E D D −−→E −−→E →, − →, − 0 ≤ {(1 − t)x ⊕ tz}x , y T x ≤ t − zx y T x . Since t > 0, it follows that 0 ≤ − zx y Tx . 0

0

t

0

0

t

0

0

t

0

By Lemma 2.19, T is a Lipschitzian map. Inparticular, T is continuous. Also yt → x0 as t → 0. It follows that D D E D E −−→E →, − −→ −−→ −→ −−−−→ −→ −−−−→ 0≤ − zx 0 yt T x0 = hzx0 , yt x0 i + zx0 , x0 T x0 ≤ d(z, x0 )d(yt , x0 ) + zx0 , x0 T x0 , D −−−→E →, − for 0 < t < 1. Taking the limit in both sides, we get that − zx 0 x0 T x0 ≥ 0 as t → 0. D −−−→E −→, − Therefore, − xx x T x ≥ 0 for all x ∈ C. 0

0

0

Lemma 3.4. Let (X, d) be a complete CAT(0) space and T be a k-strict pseudo-contraction in the terminology of Browder and Petryshyn with domain D(T ) and range R(T ). Then for all x, y ∈ D(T ) the following inequalities hold and are equivalent : D D D−−→ −−→E −−→E −→E − − → xT → − 2k 2 xy, xy, x − 1−k yT y − 2 T xy, yT y ; (1) d2 (x, T x) + d2 (y, T y) ≤ 1−k D D D−−→ −−→E −→E −−→E − − → xT → − 2 2 (2) d2 (x, T x) + d2 (y, T y) ≤ 1−k x − 1−k yT y + 2 xT x, yT y ; xy, xy, D−−→ −−→E D−−→ −−→E D−−→ −−→E 2 2 1+k ) xT x, yT y ; xT y, xT x − 1−k T xy, yT y − 2( 1−k (3) d2 (x, T x) + d2 (y, T y) ≤ 1−k (4) d2 (x, d2 (y, T y) D−−T→x)−+ −→E D−−→ −−→E ≤ xT y, xT x − T xy, yT y +

1+k 2 2 {d (x, T x)

D−−→ −−→E − 2 xT x, yT y + d2 (y, T y)}.

Proof. We first show that (2) holds.

D−−→ −−→E D−−→ −−→E d2 (x, T x) + d2 (y, T y) = d2 (x, T x) − 2 xT x, yT y + d2 (y, T y) + 2 xT x, yT y = d2 (x, T x) − d2 (x, T y) − d2 (T x, y) + d2 (x, y) + d2 (T x, T y) D−−→ −−→E + d2 (y, T y) + 2 xT x, yT y ≤ d2 (x, T x) − d2 (x, T y) − d2 (T x, y) + d2 (x, y) + d2 (x, y) D−−→ −−→E D−−→ −−→E + kd2 (x, T x) − 2k xT x, yT y + kd2 (y, T y) + d2 (y, T y) + 2 xT x, yT y .

By simple calculation from the inequality above we get that (1 − k){d2 (x, T x) + d2 (y, T y)} ≤ d2 (x, T x) + d2 (y, T y) + d2 (y, x) + d2 (x, y) − d2 (y, T x) D−−→ −−→E − d2 (x, T y) + 2(1 − k) xT x, yT y . Dividing throughout with (1 − k) we have that 1 {d2 (x, T x) + d2 (y, T y) + d2 (y, x) + d2 (x, y) 1−k D−−→ −−→E − d2 (y, T x) − d2 (x, T y)} + 2 xT x, yT y D−−→ −−→E −→E −→E 2 D− 2 D− → − → − xy, xy, = xT x − yT y + 2 xT x, yT y . 1−k 1−k

d2 (x, T x) + d2 (y, T y) ≤

313

(3.3)

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Then, (2) is true. Next, we observe that −

D−−→ −−→E D D−−→ −−→E −→E −→E −→E 2 D− 2 D− → − → − → − xy, xy, yT y + 2 xT x, yT y = − yT y + 2 − xy, yT y + 2 yT x, yT y 1−k 1−k D D−−→ −−→E −−→E 2 → yT } − = {2 − y − 2 T xy, yT y xy, 1−k D−−→ −−→E −→E −2k D− → − xy, = yT y − 2 T xy, yT y . (3.4) 1−k

Substituting (3.4) in (3.3), we get (1) holds; D that is E D that D−−→ −−→E −−→E −→ − − → xT → − 2 2k xy, xy, d2 (x, T x) + d2 (y, T y) ≤ 1−k x − 1−k yT y − 2 T xy, yT y , and hence (1) and (2) are equivalent. We next show that (3) is true. Let us consider D−−→ −−→E D−−→ −−→E −→E 2 D− 2 2 D−−→ −−→E 2 D−−→ −−→E → − { xT y, xT x + T yy, xT x } = xy, xT y, xT x − xT x, yT y . xT x = 1−k 1−k 1−k 1−k (3.5)

−

D−−→ −−→E D−−→ −−→E −−→E 2 D− 2 2 D−−→ −−→E 2 D−−→ −−→E → yT { T xy, yT y + xT x, yT y } = − xy, T xy, yT y − xT x, yT y . y =− 1−k 1−k 1−k 1−k (3.6)

Combining (3.5) and (3.6), we have −→E −→E 2 D− 2 D−−→ −−→E 2 D− 2 D−−→ −−→E 4 D−−→ −−→E → − → − xT x − yT y = xy, xy, xT y, xT x − T xy, yT y − xT x, yT y . 1−k 1−k 1−k 1−k 1−k We get that D−−→ −−→E −→E −−→E 2 D− 2 D− → − → yT xT x − y + 2 xT x, yT y xy, xy, 1−k 1−k D−−→ −−→E 2 D−−→ −−→E 2 D−−→ −−→E 4 D−−→ −−→E xT y, xT x − T xy, yT y − xT x, yT y + 2 xT x, yT y = 1−k 1−k 1−k 2 D−−→ −−→E 2 D−−→ −−→E 1 + k D−−→ −−→E ) xT x, yT y . = xT y, xT x − T xy, yT y − 2( 1−k 1−k 1−k This shows that (3) is true. We get that (2) and (3) are equivalent. Next, we will show that (3) and (4) are equivalent. We will show that (3) implies (4). Since 1−k 2 > 0, for (3) is true, we get that D−−→ −−→E D−−→ −−→E D−−→ −−→E 1−k ){d2 (x, T x) + d2 (y, T y)} ≤ xT y, xT x − T xy, yT y − (1 + k) xT x, yT y ( 2 D−−→ −−→E D−−→ −−→E D−−→ −−→E 1+k )]{d2 (x, T x) + d2 (y, T y)} ≤ xT y, xT x − T xy, yT y − (1 + k) xT x, yT y [1 − ( 2 D−−→ −−→E D−−→ −−→E D−−→ −−→E d2 (x, T x) + d2 (y, T y) ≤ xT y, xT x − T xy, yT y − (1 + k) xT x, yT y 1+k ){d2 (x, T x) + d2 (y, T y)} +( 2 D−−→ −−→E D−−→ −−→E = xT y, xT x − T xy, yT y +(

D−−→ −−→E 1+k ){d2 (x, T x) + d2 (y, T y) − 2 xT x, yT y }. 2

Then, we get that (4) holds. By using a similar method, we get that (4) implis (3). That is, we get that (3) and (4) are equivalent. This completes our proof.

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Lemma 3.5. Let (X, d) be a complete CAT(0) space and T be a k-strict pseudo-contraction in the terminology and Petryshyn domain D(T ) and range R(T ). If, there exists u ∈ D(T ) D of Browder D−−→ −−with −−→E →E − → such that xu, uT u ≥ 0 and T xu, uT u ≥ 0 for some x ∈ D(T ), the following inequalities hold :

d2 (x, T x) ≤

 D  −−→E  − → xT 2  xu, x ,  1−k      D   −−→E → xT   2 −  x xu,  1−k  D−−→ −−→E    2 xT u, xT x

(

,

if k ∈D[0, 1); −−→ −−→E or if xT x, uT u ≤ 0;

if

D−−→ −−→E xT x, uT u ≥ 0 and k ∈ [0, 1);

if

D−−→ −−→E xT x, uT u ≥ 0 and k ∈ [−1, 0);

1−k

   D−−→ −−→E    2   1−k xT u, xT x ,        D−−→ −−→E  xT u, xT x ,

if k ∈ (−∞, −1].

Proof. If k ∈ [0, 1), then k < 1 ⇔ 0 < 1 − kD and note that 0 ≤ D2k, so we have −→E −−→ −−→E − → − 2k 2k ≥ 0 ⇔ ≤ 0. 3.4(1), xu, uT u ≥ 0 and T xu, uT u ≥ 0, we get that − By Lemma 1−k 1−k D−−→ −−→E −→E −−→E 2 D− 2k D− → − → uT d2 (x, T x) ≤ d2 (x, T x) + d2 (u, T u) ≤ xT x − u − 2 T xu, uT u xu, xu, 1−k 1−k −−→E 2 D− → xu, xT x . ≤ (3.7) 1−k D−−→ −−→E D −→E → − If xT x, uT u ≤ 0, then by Lemma 3.4(2) and − xu, uT u ≥ 0, we get that D−−→ −−→E −→E −−→E 2 D− 2 D− → − → uT xu, xu, d2 (x, T x) ≤ d2 (x, T x) + d2 (u, T u) ≤ xT x − u + 2 xT x, uT u 1−k 1−k E → − − 2 D− → xT x . ≤ xu, 1−k Before we prove the next case, ( let us consider the following ⇔ 0 ≤ 1 + k < 2. k ∈ [−1, 1) ⇔ −1 ≤ k < 1 ⇔ 1 ≥ −k > −1 ⇔ 2 ≥ 1 − k > 0 ⇔ 1+k Therefore, we have 2( 1−k ) ≥ 1 + k ≥ 0 and then −2(

1 1−k

1+k ) ≤ 0 whenever k ∈ [−1, 1). 1−k

≥ 21 .

(3.8)

D−−→ −−→E D−−→ −−→E If xT x, uT u ≥ 0 and k ∈ [0, 1), then it follows from (3.8), Lemma 3.4(3) and T xu, uT u ≥ 0, we get that 2 D−−→ −−→E 2 D−−→ −−→E 1 + k D−−→ −−→E ) xT x, uT u xT u, xT x − T xu, uT u − 2( d2 (x, T x) ≤ d2 (x, T x) + d2 (u, T u) ≤ 1−k 1−k 1−k D E −−→ −−→ 2 ≤ xT u, xT x . 1−k  D −−→E → xT  2 − D−−→ −−→E xu, x 1−k D−−→ −−→E From (3.7), we can conclude in this case that d2 (x, T x) ≤ If xT x, uT u ≥  2 xT u, xT x . D−−→1−k −−→E 0 and k ∈ [−1, 0), then by (3.8), Lemma 3.4(3) and T xu, uT u ≥ 0, we get that d2 (x, T x) ≤

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D−−→ −−→E D−−→ −−→E xT u, xT x . Finally, if k ∈ (−∞, −1], then by using lemma 3.4(4) and T xu, uT u ≥ 0, we get that D−−→ −−→E D−−→ −−→E 1 + k d2 (xT x, uT u) d2 (x, T x) ≤ d2 (x, T x) + d2 (u, T u) ≤ xT u, xT x − T xu, uT u + 2 D−−→ −−→E ≤ xT u, xT x . 2 1−k

This completes our proof. Every iteration process generated by the shrinking projection method for a k-strict pseudocontraction T in the terminology of Browder and Petryshn is well defined even if T is fixed point free. Lemma 3.6. Let (X, d) be a complete CAT(0) space and C be nonempty closed and convex subset of X. Let T : C → C be a k-strict pseudo-contraction in the terminology of Browder and Petryshyn, that is for all x, y ∈ X there exists an element k ∈ (−∞, 1) such that D−−→ −−→E d2 (T x, T y) ≤ d2 (x, y) + k[d2 (x, T x) − 2 xT x, yT y + d2 (y, T y)]. Let x0 ∈ X, C1 = C and {xn } be a sequence in C generated by ( xn =P D Eo nCn (x0 ), (3.9) → −−−−→ , 2 , 1} − x− Cn+1 = z ∈ Cn : d2 (xn , T xn ) ≤ max { 1−k n z, xn T xn for all n ∈ N. Then, Cn is nonempty closed convex subsets of X and consequently, {xn } is well defined for every n ∈ N. Proof. Clearly, C1 is nonempty. Suppose that Cm is nonempty for some m ∈ N . We wish to show that Cm+1 is nonempty. Since Cm ⊂ Cm−1 ⊂ ... ⊂ C1 , we have that C1 , C2 , ..., Cm are nonempty. Next, we will show that C1 , C2 , ..., Cm are closed and convex. It is sufficient to show that Cm is closed and convex. It is not hard to show that for any {zk } ⊆ Cm such that zk → z0 , we have z0 ∈ Cm . We get that Cm is closed. We next show that Cm is convex. Notice that a subset of a CAT(0) space, equipped with the induced metric, is a CAT(0) space if and only if it is convex. Thus, we will show that (Cm , d) is a complete CAT(0) space. Let each x, y ∈ Cm , we have x, y ∈ C. By Lemma 3.1, we have (C, d) is a complete CAT(0) space and thus, it is a geodesic space, hence x, y are joined by a geodesic. Since x, y are arbitrary, thus we have x, y ∈ Cm are joined by a geodesic. Hence Cm is a geodesic space. Since Cm is closed subset of complete metric space (C, d), then (Cm , d) is a complete metric space. It follows from Lemma 2.21 that, for every y, z ∈ Cm , there exists p ∈ Cm such that d(y, p) = d(p, z) = 21 d(y, z). Now, we claim that Cm satisfies the (CN) inequality. In fact, let x, y, z ∈ Cm and p ∈ Cm with d(y, p) = d(p, z) = 21 d(y, z). Let α and β be two numbers satisfy α + β ≥ 1. Then α2 + β 2 ≥ 21 (α + β)2 ≥ 21 with equality if and only if α = β= 21 . d(y, p) 2 d(p, z) 2 1 By this fact and by the triangle inequality, we get that ( ) +( ) ≥ . That is, d(y, z) d(y, z) 2 1 2 2 2 2 d (y, z) ≤ d (y, p) + d (p, z). It follows from the above inequality that, setting x = p, we get that 2 2 d (x, y) + d (x, z) ≥ 2d2 (x, p) + 21 d2 (y, z), this implies that Cm satisfies the (CN) inequality. By Lemma 2.22, we know that (Cm , d) is a CAT(0) space. By above, we have (Cm , d) is a complete metric space. Then (Cm , d) is a complete CAT(0) space. This implies that Cm is convex subset of X. Thus, we have Cm is closed and convex. Finally, put r = max{d(x0 , xi ), d(x0 , T xi ) : i = 1, 2, ..., m} and Br = {z ∈ X : d(x0 , z) ≤ r}. Obviously C ∩ Br is a nonempty bounded closedDconvex subset −→E → − of X. It follows from Lemma 3.3 that there exists an element u ∈ C ∩ Br such that − yu, uT u ≥ 0 for all y ∈ C ∩ Br . In particular, we have D D−−−→ −−→E −−→E − x→ T xi u, uT u ≥ 0 (3.10) i u, uT u ≥ 0 and

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for every i = 1, 2, ..., m. 2 2 , 1} = 1−k . Case I. max { 1−k 2 2 2 ⇔ 1 ≤ 1−k ⇔ 1 − k ≤ 2 ⇔ −1 ≤ k ⇔ k ∈ [−1, 1), it follows from Notice that max { 1−k , 1} = 1−k (3.10) and Lemma 3.5 that  (  if k ∈D[0, 1);  2 D−→ −−−→E   xi u, xi T xi , −−−→ −−→E  1−k  or if xi T xi , uT u ≤ 0;     D  −−−→E  →  2 − D−−−→ −−→E x u, xi T xi i 1−k D E d2 (xi , T xi ) ≤ , if xi T xi , uT u ≥ 0 and k ∈ [0, 1); −−→ −−−→  2 − xi T u, xi T xi  1−k        D −−→ −−−→E D−−−→ −−→E    2 − if xi T xi , uT u ≥ 0 and k ∈ [−1, 0) 1−k xi T u, xi T xi , for every i = 1, 2, ..., m. This shows that u ∨ T u ∈ Cm+1 . 2 , 1} = 1. Case q. max { 1−k 2 2 , 1} = 1−k ≤ 1 ⇔ 2 ≤ 1 − k ⇔ k ≤ −1 ⇔ k ∈ (−∞, −1], it follows from (3.9) Notice that max { 1−k D−−−→ −−−→E 2 and Lemma 3.5 that d (xi , T xi ) ≤ xi T u, xi T xi , if k ∈ (−∞, −1] for every i = 1, 2, ..., m. This shows that T u ∈ Cm+1 . By Case I and Case q, we can conclude that u ∨ T u ∈ Cm+1 . Hence Cm+1 is nonempty. By induction on n, therefore the desired result. Theorem 3.7. Let all the assumptions be the same as in Lemma 3.6. Then, the following are equivalent : ∞ \ (1) Cn is nonempty; n=1

(2) {xn } is bounded; (3) F ix(T ) is nonempty. Proof. [(1) ⇒ (2)] Let u ∈

∞ \

Cn . By Lemma 2.12, it follows from the nonexpansiveness of PCn

n=1

that d(xn , u) = d(PCn x0 , PCn u) ≤ d(x0 , u). This shows that xn is bounded. [(2) ⇒ (3)] Suppose that xn is bounded, we first claim that 0 ≤ d2 (xn+1 , xn ) ≤ d2 (xn+1 , x0 ) − −−−−−→ 2 d (xn , x0 ). Since xn = PCn x0 , by Lemma 2.11, we have h− x−0− x→ n , xn xn+1 i ≥ 0 for all xn+1 ∈ − − − → − − − − → − − − → − − − → − − − → − − − − − → Cn . So, we have hx0 xn , x0 xn+1 i − hx0 xn , x0 xn i = hx0 xn , xn xn+1 i ≥ 0 and hence d2 (x0 , xn ) = −−−→ −−−→ −−−−→ h− x−0− x→ n , x0 xn i ≤ hx0 xn , x0 xn+1 i. By Lemma 2.14, 2.15 and using (2.1), we have 1 1 1 1 d2 (xn , xn+1 ) ≤ 2d2 (xn , xn+1 ) = 4d2 ( x0 ⊕ xn , x0 ⊕ xn+1 ) 2 2 2 2 ≤ d2 (xn , x0 ) + d2 (xn+1 , x0 ) + d2 (x0 , x0 ) + d2 (xn , xn+1 ) − d2 (x0 , xn+1 ) − d2 (x0 , xn ) →, − = d2 (x , x ) + d2 (x , x ) + 2 h− x−−−x x−− x→i n

0

n+1

0

n+1 0

0 n

−−→ −−−→ = d2 (xn , x0 ) + d2 (xn+1 , x0 ) − 2 h− x− 0 xn+1 , x0 xn i ≤ d2 (xn , x0 ) + d2 (xn+1 , x0 ) − 2d2 (x0 , xn ).

(3.11)

This shows that {d(xn , x0 )} is nondecreasing and with is the bounded of {xn }, we have lim d(xn , x0 ) n→∞ −→ = 0. Since x exists. From (3.11), we get that d2 (xn+1 , x n →E∞. Thus − x−n− x−n+1 n+1 ∈ Dn ) → 0 as −−−−→ − − − → − − 2 2 2 , 1} x x , 1}d(x , x , x T x ≤ max{ C , we have d (x , T x ) ≤ max{ )d(x , T x ). n+1

n

n

1−k

n n+1

n

n

1−k

n

n+1

n

n

Thus d(xn , T xn ) → 0 as n → ∞. Since {xn } is bounded and by Lemma 2.3, we have 4 − lim xnj = w. Since d(xnj , T xnj ) → 0 as j → ∞, then we get that j→∞

Φ(x) = lim sup d(xnj , x) = lim sup d(T xnj , x) for all x ∈ C. j→∞

(3.12)

j→∞

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By taking x = T w in (3.12), we have Φ(T w)2 = lim sup d2 (T xnj , T w) j→∞

D−−−−−→ −−−→E ≤ lim sup{d2 (xnj , w) + k[d2 (xnj , T xnj ) − 2 xnj T xnj , wT w + d2 (w, T w)]} j→∞

D−−−−−→ −−−→E = lim sup d2 (xnj , w) + k lim sup[d2 (xnj , T xnj ) + 2 T xnj xnj , wT w + d2 (w, T w)] j→∞

j→∞

2

≤ lim sup d (xnj , w) + k lim sup d2 (xnj , T xnj ) j→∞

j→∞

+ 2k lim sup[d(T xnj , xnj )d(w, T w)] + k lim sup d2 (w, T w) j→∞

2

j→∞

2

= Φ(w) + k d (w, T w).

(3.13)

Since −∞ < k < 1, we can choose a real number λ ∈ [0, 1] be such that max{0, k} < λ < 1. By Lemma 2.15, we have ¡ ¢ ¡ ¢ ¡ ¢ d2 xnj , λw ⊕ (1 − λ)T w ≤ λd2 xnj , w + (1 − λ)d2 xnj , T w − λ(1 − λ)d2 (w, T w) Taking the superior limit on both sides of the above inequality, we get that 2

2

2

Φ(λw ⊕ (1 − λ)T w) ≤ λΦ(w) + (1 − λ)Φ(T w) − λ(1 − λ)d2 (w, T w) Since 4 − lim xnj = w. By using (3.13) and Lemma 2.2, we have j→∞

2

2

2

2

Φ(w) ≤ Φ(λw ⊕ (1 − λ)T w) ≤ λΦ(w) + (1 − λ)Φ(T w) − λ(1 − λ)d2 (w, T w) ³ ´ 2 2 ≤ λΦ(w) + (1 − λ) Φ(w) + kd2 (w, T w) − λ(1 − λ)d2 (w, T w) 2

2

= λΦ(w) + (1 − λ)Φ(w) + (1 − λ)kd2 (w, T w) − λ(1 − λ)d2 (w, T w) 2

= Φ(w) + (1 − λ)(k − λ)d2 (w, T w) . This implies that (1−λ)(λ−k)d2 (w, T w) ≤ 0. Since max{0, k} < λ < 1, we have (1−λ)(λ−k) > 0. This implies that T w = w, that is w ∈ F ix(T ) 6= ∅. [(3) ⇒ (1)]DSuppose that F ix(T ) 6= ∅. We claim that F ix(T ) ⊂ Cn for all n ∈ N . If w ∈ F ix(T ), − → −−−→E then we have ab, wT w = 0 for all a, b ∈ X. Taking u = w in the proof of Lemma 3.6, it is not hard to observe that all inequalities are satisfied. This implies that w ∈ Cn for all n ∈ N . Therefore ∞ \ F ix(T ) ⊂ Cn 6= ∅. n=1

Theorem 3.8. Let all the assumptions be the same as in Theorem 3.7 Then, if

∞ \

Cn 6= ∅ (⇔ {xn }

n=1

is bounded ⇔ F ix(T ) 6= ∅),then the sequence {xn } generated by (3.9) converges strongly to some points of C and its strong limit point is a member of F ix(T ), that is lim xn = PF ix(T ) x0 ∈ F ix(T ). n→∞

Proof. If

∞ \

Cn 6= ∅, then Theorem 3.7 ensures that {xn } is bounded sequence in C. By Lemma

n=1

2.3, we have {xnj } ⊆ {xn } such that 4 − lim xnj = u. By the proof of Theorem 3.7[(2) ⇒ (3)], we j→∞

have u ∈ F ix(T ). By Lemma 2.20, we have PF ix(T ) is well defined. Also, PF ix(T ) x0 ∈ F ix(T ) ⊂ Cn , we observe that d(xn , x0 ) = d(PCn x0 , x0 ) ≤ d(PF ix(T ) x0 , x0 )

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(3.14)

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for all n ∈ N . Since {d(xn , x0 )} is nondecreasing, we get that lim d(xn , x0 ) exists. Since 4 − n→∞

lim xnj = u, by using Lemma 2.6, we have {xnj } * u. By Definition 2.16 and Lemma 2.17, we

j→∞

get that d(u, {x0 }) ≤ lim inf j→∞ d(xnj , {x0 }). By Lemma 2.8, there exists an x0 ∈ {x0 } such that d(u, x0 ) = d(u, {x0 }) ≤ lim inf d(xnj , x0 )

(3.15)

j→∞

By using (3.14) and (3.15), we get that d(u, x0 ) ≤ lim inf d(xnj , x0 ) = lim d(xn , x0 ) ≤ d(PF ix(T ) x0 , x0 ). j→∞

n→∞

(3.16)

Taking into account u ∈ F ix(T ), from (3.16), we have d(u, x0 ) ≤ d(PF ix(T ) x0 , x0 ) ≤ d(u, x0 ). This implies that d(u, x0 ) = d(PF ix(T ) x0 , x0 ). By Lemma 2.8, we obtain that u = PF ix(T ) x0 . Therefore {xn } * PF ix(T ) x0 and d(xn , x0 ) → d(PF ix(T ) x0 , x0 ). Consequently, from (3.11), we get that d2 (xn , PF ix(T ) x0 ) ≤ d2 (PF ix(T ) x0 , x0 ) − d2 (xn , x0 ) → 0 as n → ∞. This completes our proof. Remark 3.9. The results in this section extend and improve the corresponding Theorem 3.4 and 3.5 in [22] in the case of an iterative projection technique in a Hilbert space. 4. Conclusion In the present paper, we study some existence and convergence theorems for fixed points of a strict pseudo-contraction by using an iterative projection technique with some suitable conditions. We obtain the sufficient conditions for the existence and convergence theorem for the fixed points of strict pseudo-contraction mappings in complete CAT(0) spaces. Acknowledgements The authors would like to thank, Naresuan University, Phitsanulok 65000, Thailand. References [1] M. Bacak, Convex Analysis and Optimization in Hadamard spaces, Walter de Gruyter GmbH, Berlin (2014). [2] I. D. Berg, I. G. Nikolaev, Quasilinearization and curvature of Alexandrov spaces, Geom Dedicata, 133, 195-218 (2008). [3] M. Bridson, A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer, Berlin (1999). [4] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20, 197-228 (1967). [5] F. Bruhat, J. Tits, Groupes re ductifs sur un corps local. I. Donees radicielles valuees, Inst., Hautes Etudes Sci. Publ. Math., 41, 5-251 (1972). [6] H. Dehghan, J. Rooin, A characterization of metric projection in CAT(0) spaces, International Conference on Functional Equation, Geometric Functions and Applications(ICFGA 2012), Payame Noor University, Tabriz, Iran, 10-12 May 2012. [7] S. Dhompongsa, A. Kaewkhao, B. Panyanak, Lim’s theorems for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl., 312, 478-487 (2005).

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[8] S. Dhompongsa, W. A. Kirk, B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal., 65, 762-772 (2006). [9] S. Dhompongsa, W. A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal., 8, 35-45 (2007). [10] S. Dhompongsa, B. Panyanak, On 4-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56, 2572-2579 (2008). [11] A. Gharajelo, H. Dehghan, Convergence theorems for strict pseudo-contractions in CAT(0) metric spaces, Filomat 31(7), 1967-1971 (2017). doi 10.2298/FIL1707967G. [12] B. A. Kakavandi, Weak topologies in complete CAT(0) metric spaces, Proc. Amer. Math. Soc. 141, 1029 - 1039 (2012). [13] B. A. Kakavandi, M. Amini, Duality and subdifferential for convex functions on complete CAT(0) metric spaces, Nonlinear Anal., 73, 3450-3455 (2010). [14] H. Khatibzadeh, V. Mohebbi, Monotone and pseudo-monotone equilibrium problems in Hadamard spaces, https://arxiv.org/abs/1611.01829. [15] W. A. Kirk, Geodesic geometry and fixed point theory, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc. Abierta, vol. 64, pp. 195-225. Univ. Sevilla Secr. Publ.,Seville (2003). [16] W. A. Kirk, Geodesic geometry and fixed point theory II, in International Conference on Fixed Point Theory and Application, pp.113-142. Yokohama Publ., Yokohama (2004). [17] W. A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68, 3689-3696 (2008). [18] T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Soc., 60, 179-182 (1976). [19] H. Lu, D. Lan, Q. Hu, G. Yuan, Fixed point theorems in CAT(0) spaces with applications, J. Inequa. Appl., 320, 1-26 (2014). [20] B. Nanjaras, B. Panyanak, Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl., 2010:doi:10.1155/2010/268780. [21] C. P. Niculescu, I. Roventa, Fan’s inequality in geodesic spaces, Appl. Math. Lett. 22, 1529-1533 (2009). [22] K. Ungchittrakool, Existence and convergence of fixed points for a strict pseudo-contraction via an iterative shrinking projection technique, J. Nonlinear Convex Anal., 15, 693-710 (2014). [23] R. Wangkeeree, P. Preechasilp, Viscosity approximation method, for nonexpansive mappings in CAT(0) spaces, J. Inequal. Appl., 93, 1-15 (2013).

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ON THE CEU-DEGREE OF SIMILARITY IN INTERNATIONAL TRADE BY USING THE CHOQUET INTEGRAL EXPECTED UTILITY LEE-CHAE JANG, JACOB WOOD

Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea E-mail : [email protected] Department of Business James Cook University Singapore, 149 Sims Road, Singapore 387380 E-mail : [email protected]

Abstract. Recently, we considered the Choquet integrals with respected to a fuzzy measure and the Choquet-expected utility(CEU)which was represented by preference functionals. We note that the CEU provides a useful tool to calculate the subjective capacity of trade values between Korea and some countries in Wood-Jang [4,10]. In this paper, by using the Choquet-expected utility in Wood-Jang [4] and the degree of similarity in Biswas [1], we define the CEU-degree of the similarity related with the CEU of trade values between Korea and some countries. In particular, we investigate some applications of the CEU-degree of similarity related with the CEU of trade values.

1. Introduction By using fuzzy sets and Choquet integrals in [1,2,4,5,6,10], many researchers have studied the concept of Choquet intgeral expected utility and its related areas(see[3,4,8.9.11,12]). Recently, Wood-Jang [6,7] studied some applications of the Choquet integral as imprecise market premium functionals with respect to an imprecise set function which was an interval-valued measure of risk and the Choquet integral with respect to a fuzzy measure of a utility function. In 1995, Biswas [1] investigated a student’s evaluation on the space of fuzzy sets which include data information for the students respective classes. In this paper, by using the degree of similarity in Biswas [1], we define the CEU-degree of the similarity which is related to the CEU for the trade values that exist between Korea and some of its important trading partners (such as Korea-USA, Korea-New Zealand, KoreaIndia, and Korea-Turkey). In particular, we investigate the evaluation of the CEU-degree of similarity which is related with the CEU of trade values CEU (u(a)) of a utility u from an act a on S for specified HS product codes for animal product exports between Korea and selected trading partners for years 2010-2013. We note that we include the dates used in our previous studies [4,10]. In particular, we investigate the following applications: 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. Choquet integral, Choquet expected utility, fuzzy neasure, the degree of similarity. 1

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(1) we calculate CEU-degree of contribution from an economic value perspective, for animal exports with HS product code i = 1, 2, 3, 4, 5 between Korea and selected trading partners for years 2010-2013 and (2) we compare these values with the USA and other trading partners in terms of CEUdegrees (13), (14), and (15) of the similarity which is related to the relationships and characterizations involved in the value of international trade between Korea and each of the four countries analyzed in this study(see[14]).

2. Preliminaries and definitions Let S be a finite set of states of nature and F (S) be the set of all fuzzy sets A = {(s, mA (s)) | s ∈ S, mA −→ [0, 1] is a function}. Recall that mA is called a membership function of A.

Definition 2.1. ([4-7,9,10,11,13]) (1) A real-valued function µ on S the subsets of is called a fuzzy measure if it satisfies (i) µ(∅) = 0, µ(S) = 1, (ii) A ⊂ B ⇒ µ(A) ≤ µ(B).

(1)

(2) The Choquet integrals with respect to a fuzzy measure µ of A ∈ F (S) is defined by Z Z 1 (C) fA dµ = µ({s ∈ S|fA (s) ≥ α})dα, (2) 0

Definition 2.2. ([4-7,9,10,11,13]) (1) Let A ∈ F (S). The Choquet integrals with respect to a fuzzy measure µ of a fuzzy set A = (S, fA ) is defined by where the integral on the right-hand side is an ordinary one. (2) Let S = {s1 , s2 , · · · , sn } be a finite set. The discrete Choquet integral with respect to a fuzzy measure µ is defined by Z n h i X (C) mA dµ = fA (s(i) ) µ(E (i) ) − µ(E (i+1) ) , (3) i=1

where E

(i)

= {s ∈ S|mA (s) ≥ mA (si )} for i = 1, 2, · · · , n. By convention, let E n+1 = ∅.

By using the Choquet integral, we consider the Choquet expected utility(CEU) of a utility u from an act a as follows. Definition 2.3. ([4]) Let u : X −→ [0, 1] be a utility and a be an act from S to X. The Choquet expected utility(CEU) with respect to a fuzzy measure µ of utility u from act a is defined by Z CEU (u(a)) = (C) u(a(s))dµ(s). (4)

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We note that if mA (s) = u(a(s)) and A = (S, mA ), then A ∈ F (S), that is, A is a fuzzy set. From Definition 2.1(3) and Definition 2.2 with a finite set S, we get CEU (u(a)) as follows: n h i X CEU (u(a)) = u(a(s(i) )) µ(E (i) ) − µ(E (i+1) ) . (5) i=1

where E

(i)

= {s ∈ S|u(a(s)) ≥ u(a(s(i) )} for all i = 1, 2, · · · , n.

3. CEU-fuzzy marks and CEU-degree of similarity In this section, we consider the CEU of a utility on a set of trade values (in USD) that represent the trading relationship that Korea shares with selected trading partners(i.e. KoreaUSA, Korea-New Zealand, Korea-India, and Korea-Turkey). We also examine these respective trading relationships by incorporating a clearly defined set of Harmonized System (HS) product code product categories (i.e. HS Codes i = 1, 2, 3, 4, 5) for each individual year that is under review (i.e. 2010, 2011, 2012, 2013). We note that the product code definitions have been provided by the UN Comtrade’s online database and the relevant categories are defined as follows(see[14]): 1. Live animals; animal products. 2. Meat and edible meat offal. 3. Fish and crustaceans, mollusks and other aquatic invertebrates. 4. Dairy produce; birds’ eggs; natural honey; edible products of animal origin, not elsewhere specified or included. 5. Products of animal origin, not elsewhere specified or included. Firstly, we denote that HSPC=HS Product Code, s=Year, a(s)=Trade Value, u(a(s))=the utility of a(s), CEU (u, a)=the Choquet Expected Utility of u from a. By using the trade values in tables A1 A4, we can calculate the Choquet integral of an utility on the set of trade values (in USD) that represent Korea’s trading relationship with a particular country for years 2010, 2012, 2012, 2013. Let s1 = 2010, s2 = 2011, s3 = 2012, s4 = 2013. If we define a fuzzy measure µ on S as follows(see[4]): µ(E (4) ) = µ1 ({s(4) }) = 0.1, µ(E (3) ) = µ1 ({s(3) , s(4) }) = 0.3, µ(E (2) ) = µ1 ({s(2) , s(3) , s(4) }) = 0.6, µ(E (1) ) = µ1 ({s(4) , s(3) , s(2) , s(1) }) = 1, (6) p a and if a(s) is the trade value of s and u(a) = 100141401 , then we obtain the following CEU (u(a)) as follows: CEU (u(a))

=

4 X

  u(a(s(i) )) µ(E (i) ) − (µ(E (i+1) )

i=1

=

0.4u(a(s(1) )) + 0.3u(a(s(2) )) + 0.2u(a(s(3) )) + 0.1u(a(s(4) )).

(7)

By using (5), we calculate the four tables A1 ∼ A4 as follows(see [4]): By using four tables, we get the four X-fuzzy sets X : {1, 2, 3, 4, 5} → [0, 1] by X = {(i, mX (i))|i = 1, 2, 3, 4, 5} (i.e., USA-fuzzy set U , NZ-fuzzy set N , IN-fuzzy set I, TR-fuzzy set T ) defined by U = {(1, 0.05664), (2, 0.04483), (3, 0.93879), (4, 0.20821), (5, 0.04858)}

(8)

N = {(1, 0.00533), (2, 0.00000), (3, 0.78873), (4, 0.15976), (5, 0.01557)}

(9)

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I = {(1, 0.00154), (2, 0.00000), (3, 0.04570), (4, 0.00000), (5, 0.00000)}

(10)

T = {(1, 0.00264), (2, 0.00887), (3, 0.00368), (4, 0.00470), (5, 0.00000)}

(11)

Definition 3.1. ([1]) The degree of similarity between X-fuzzy set and Y -fuzzy set is defined by S(X, Y ) =

ˆ · Yˆ X ˆ · X, ˆ Yˆ · Yˆ } max{X

(12)

where ˆ =< mX (1), mX (2), mX (3), mX (4), mX (5) >, X

(13)

Yˆ =< mY (1), mY (2), mY (3), mY (4), mY (5) > are vectors and ˆ · Yˆ X

= mX (1) · mY (1) + mX (2) · mY (2) + mX (3) · mY (3) + mX (4) · mY (4) + mX (5) · mY (5). (14) By using Definition 3.1, we define the degree of similarity between X-fuzzy set and Y -fuzzy set is called the CEU-degree of similarity as follows. Definition 3.2. If X and Y are elements of {U, N, I, T }, then the degree of similarity between X-fuzzy set and Y -fuzzy set is called the CEU-degree of similarity. From Definition 3.1 and Definition 3.2, we get the CEU-degree of similarity between Xfuzzy set and Y -fuzzy set where X and Y are elements of {U, N, I, T }. Example 3.1. (1) From Definition 3.1 and Definition 3.2, we get the CEU-degree of similarity between U -fuzzy set and N -fuzzy set as follows: S(U, N ) =

ˆ ·N ˆ U 0.7747737841 = = 0.83174152. ˆ ˆ ˆ ˆ max{0.932255903, 0.6478891043} max{U · U , N · N }

(15)

(2) From (8) and (10), we get the CEU-degree of similarity between U -fuzzy set and I-fuzzy set as follows: ˆ · Iˆ U 0.0429899286 (16) S(U, I) = = = 0.0461138712. ˆ ˆ ˆ ˆ max{0.932255903, 0.0020908616} max{U · U , I · I} (3) From (8) and (11), we get the CEU-degree of similarity between U -fuzzy set and T -fuzzy set as follows: ˆ · Tˆ U 0.004979328 = S(U, T ) = = 0.0053411601. (17) ˆ ·U ˆ , Tˆ · Tˆ} max{0.932255903, 0.0001219413} max{U By using four information with those of the CEU-degrees of similarity (13), (14), and (15), we understand the exact difference of similarity between the USA and each of the other three trading partners. By using the CEU-degrees of similarity between USA and another country, we are able to provide a useful plan to find a more effective method of improving the value of international trade between Korea and each of the four countries analyzed in this study. We provide information that may well be of interest to international business practitioners that want a clearer understanding of the relationship and characterizations related to the value of international trade between Korea and each of the four countries measured.

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Table A1: The CEU for animal product exports between Korea and the USA for years 2010-2013 HSPC 1

2

3

4

5

s s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4

a(s)(USD) 286892 = a(s(1) ) 330299 = a(s(2) ) 358496 = a(s(3) ) 364918 = a(s(4) ) 997539 = a(s(4) ) 376805 = a(s(3) ) 30005 = a(s(1) ) 272884 = a(s(2) ) 74866073 = a(s(1) ) 95654573 = a(s(2) ) 100141401 = a(s(4) ) 99871717 = a(s(3) ) 3722326 = a(s(1) ) 4323214 = a(s(2) ) 5016833 = a(s(4) ) 4910771 = a(s(3) ) 235669 = a(s(2) ) 359747 = a(s(3) ) 101795 = a(s(1) ) 863858 = a(s(4) )

u(a(s)) CEU(i,U SA) (u(a)) 0.05352 0.05743 0.05664 0.05983 0.06037 0.09981 0.06034 0.04483 0.01731 0.05220 0.86464 0.97734 0.93879 1.00000 0.99865 0.19280 0.20778 0.20821 0.22382 0.22145 0.04851 0.05994 0.04858 0.05994 0.09088

Remark 3.1. As demonstrated in (13) (14) and (15) this study compares the similarities that exist between Korea and its respective trading partners. As such, our study details the following information: Korea − USA : Korea − NZ : Korea − India : Korea − Turkey = 1 : 0.832 : 0.046 : 0.005 (18) (2) Given a situation whereby Korea spends 10 million USD as a means of developing a strong trading relationship between itself and its US trading partner, we are able to also ascertain the level of support that is needed to develop effective trading ties with other countries, for example: NewZealand : 8, 320, 000USD India : 460, 000USD

(19)

Turkey50, 000USD.

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Table A2: The CEU for animal product exports between Korea and New Zealand for years 2010-2013 HSPC

s s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4

1

2

3

4

5

a(s)(USD) 6650 = a(s(4) ) 4497 = a(s(3) ) 1589 = a(s(1) ) 2779 = a(s(2) ) 0 = a(s(1) ) 0 = a(s(2) ) 0 = a(s(3) ) 0 = a(s(4) ) 70759196 = a(s(2) ) 91263506 = a(s(4) ) 70763937 = a(s(3) ) 46632301 = a(s(1) ) 165773 = a(s(3) ) 113751 = a(s(1) ) 148756 = a(s(2) ) 277350 = a(s(4) ) 0 = a(s(1) ) 0 = a(s(2) ) 218022 = a(s(3) ) 393025 = a(s(4) )

u(a(s)) CEU(i,N Z) (u(a)) 0.00815 0.00670 0.00533 0.00398 0.00527 0.00000 0.00000 0.00000 0.00000 0.00000 0.84059 0.95464 0.78873 0.84062 0.68240 0.04069 0.03370 0.15976 0.03854 0.05263 0.00000 0.00000 0.01557 0.04666 0.00265

Table A3: the CEU for Animal product expert between Korea and India for years 2010-2013 HSPC 1

2

3

4

5

s s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4

a(s)(USD) u(a(s)) CEU (u(a)) (3) 1050 = a(s ) 0.00324 1300 = a(s(4) ) 0.00360 0.00264 450 = a(s(1) ) 0.00212 700 = a(s(2) ) 0.00264 35432 = a(s(3) ) 0.01881 50639 = a(s(4) ) 0.02249 0.00887 2656 = a(s(1) ) 0.00515 8230 = a(s(2) ) 0.00907 8695 = a(s(4) ) 0.009318 5247 = a(s(3) ) 0.00724 0.00368 0 = a(s(1) ) 0.00000 1865 = a(s(2) ) 0.00432 0 = a(s(1) ) 0.00000 21614 = a(s(3) ) 0.01469 0.00470 30938 = a(s(4) ) 0.01758 0 = a(s(2) ) 0.00000 (1) 0 = a(s ) 0.00000 0 = a(s(2) ) 0.00000 0.00000 0 = a(s(3) ) 0.00000 0 = a(s(4) ) 0.00000

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Table A4: The CEU for animal product exports between Korea and Turkey for years 2010-2013 HSPC 1

2

3

4

5

s s1 s2 s3 S4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4 s1 s2 s3 s4

a(s)(USD) 0 = a(s(1) ) 6900 = a(s(4) ) 150 = a(s(2) ) 300 = a(s(3) ) 0 = a(s(1) ) 0 = a(s(2) ) 0 = a(s(3) ) 0 = a(s(4) ) 0 = a(s(1) ) 672952 = a(s(3) ) 2532837 = a(s(4) ) 199874 = a(s(2) ) 0 = a(s(1) ) 0 = a(s(2) ) 0 = a(s(3) ) 0 = a(s(4) ) 0 = a(s(1) ) 0 = a(s(2) ) 0 = a(s(3) ) 0 = a(s(4) )

u(a(s)) CEU (u(a)) 0.00000 0.00830 0.00154 0.00122 0.00173 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.08198 0.04570 0.15904 0.04468 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

4. Conclusions The Choquet expected utility(see Definition 2.3) is a useful tool which can be used to calculate the evaluation of the contribution of animal exports between Korea and selected trading partners. By using the Choquet expected utility, we obtained Tables A1 ∼ A4 in [10]. From these Tables A1 ∼ A4, we gave four X- fuzzy sets (8),(9),(10),(11) which are representations of the evaluation of contribution to animal exports for HP product codes i = 1, 2, 3, 4, 5 between Korea and selected trading partners for years 2010-2013. By using these X-fuzzy sets, we obtained three CEU-degrees (13), (14), and (15) of similarity. From three CEU-degrees (13), (14), and (15) of similarity, we can clearly understand the difference of similarity that exists between the USA and each of three countries measured in the study. By using CEU-degrees of similarity between the USA and a respective trading partner, we are able to provide a more effective method of improving the value of international trade between Korea and its trading partners. We also provide valuable information that can be used to compare the USA and another countries as was the case with the three CEU-degrees (13), (14), and (15) of similarity that is related with the relationship and characterizations of the international trade values that exist between Korea and its respective trading partner.

References [1] R. Biswas, An apllication of fuzzy sets in students’ evaluation, Fuzzy Sets Syst. 74 (1995) 187-194. [2] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131-295.

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[3] I. Gilboa, D. Schmeidler, Maximin expected utility with non-unique prior, J. Math. Econ. 18 (1989) 141-153. [4] L.C. Jang, J. Wood, The application of the Choquet integral expected utility in international trade, Advan. Stud. Contem. Math., 27(2) (2017) 159-173. [5] L.C. Jang, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences 183 (2012) 151-158. [6] L.C. Jang, A note on the interval-valued fuzzy integral by means of an interval-representable pseudomultiplication and their convergence properties, Fuzzy Sets Syst. 137 (2003) 11-26. [7] L. Mangelsdorff, M. Weber, Testing Choquet expected utility J. Econ. Behavior and Organization 25 (1994) 437-457. [8] D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica 57(3) (1989) 571-587. [9] J. Wood, L.C. Jang, A note on Choquet integrals and imprecise market premium functionals, Proc. Jangjeon Math.Soc., 18(4) (2015) 601-608. [10] J. Wood, L.C. Jang, A study on the Choquet integral with respect to a capacity and its applications, Global J. Pure Applied Math., 12(2) (2016) 1593-1599. [11] L. Xuechang, Entropy, distance maesure and similarity measure of fuzzy sets and their relations, Fuzzy Sets Syst., 52 (1992) 201-227. [12] W. Zeng, H. Li, Relationship btween similarity maesure and entropy of interval-valued fuzzy sets , Fuzzy Sets Syst., 57 (2006) 1477-1484. [13] D. Zhang, Subjectiv ambiguity, expected utility and Choquet expected utility, Econ. Theory 20 (2002) 159-181. [14] WTO (2016). WTO Regional Trade Database.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.2, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

The general solution of a mixed cubic-quartic functional equation and the Ulam stability of matrix fuzzy normed spaces Yali Ding1,∗, Tian-Zhou Xu1 , John Michael Rassias2 (1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China) (2. Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece)

Abstract In this paper, we consider the following new type cubic-quartic (CQ) functional equation λ2 + λ λ2 − λ [f (x + y) + f (x − y)] + [f (−x − y) + f (y − x)] 2 2 + (λ4 + λ3 − λ2 − λ)f (x) + (λ4 − λ3 − λ2 + λ)f (−x) + (1 − λ2 )[f (y) + f (−y)],

f (λx + y) + f (λx − y) =

where λ ≥ 2 is a fixed integer. We investigate the general solution of the functional equation, and then, using the fixed point method, we prove some stability results for this functional equation in matrix fuzzy normed spaces. Keywords Ulam stability; Cubic-quartic mapping; Cubic-quartic functional equation; Matrix fuzzy normed spaces. Mathematics Subject Classification(2010) 39B82; 39B52; 46H25.

1

Introduction

Throughout this paper, N stands for the set of all positive integers, R and C stand for the sets of reals and complex numbers, respectively. N0 := N ∪ {0}, R+ := [0, ∞), and Nm0 denotes the set of all positive integers greater than or equal to a given m0 ∈ N. The study of stability problems for functional equations is related to a question of Ulam [14] concerning the stability of group homomorphisms. Subsequently, the partial result of Ulam’s problem was proved by Hyers [8], The solution of Hyers was generalized by Rassias [13] for approximate linear mappings by allowing the Cauchy p

p

difference ||f (x + y) − f (x) − f (y)|| to be controlled by ϵ (∥x∥ + ∥y∥ ). In 1994, a further generalization was obtained by G˘avrut¸a [10], who replaced ϵ(∥x∥p + ∥y∥p ) by a general control function φ(x, y). This new idea is known as the Hyers-Ulam-Rassias stability of functional equations. Park [7] considered the following cubic-quartic functional equation f (2x + y) + f (2x − y) = 3f (x + y) + 3f (x − y) + f (−x − y) + f (y − x) + 18f (x) + 6f (−x) − 3f (y) − 3f (−y), (1.1) and investigated the orthogonally stability of (1.1). Very recently, Song [5] proved Ulam stability of this equation (1.1) in matrix intuitionistic fuzzy normed spaces. For more interesting discussions and generalizations of the original problem of Ulam have been investigated, see for instance [1, 2, 9, 11, 12, 15] and the references therein. ∗ Corresponding

author. E-mail addresses: [email protected] (Y. Ding), [email protected] (T.Z. Xu), [email protected] (J.M. Rassias).

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In the present paper, we introduce a new mixed type cubic and quartic functional equation: λ2 + λ λ2 − λ [f (x + y) + f (x − y)] + [f (−x − y) + f (y − x)] 2 2 + (λ4 + λ3 − λ2 − λ)f (x) + (λ4 − λ3 − λ2 + λ)f (−x) + (1 − λ2 )[f (y) + f (−y)],

f (λx + y) + f (λx − y) =

(1.2)

where λ ≥ 2 is a fixed integer. One can see that the functional equation (1.1) is a special case of (1.2) when we take the integer λ = 2. Every solution of the functional equation (1.2) is said to be a cubic-quartic mapping. The aim of this paper is to discuss the general solution and then establish the Ulam stability of (1.2). More precisely, we discuss the Ulam stability of (1.2) in matrix fuzzy normed spaces by applying the fixed point method.

2

Preliminaries

In this section, we recall some basic facts concerning fuzzy normed spaces, matrix fuzzy normed spaces and some useful results. Definition 2.1 ( [4]) Let X be a real vector space. A function N : X × R → [0, 1] is said to be a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R: t (1) N (x, t) = 0 for t ≤ 0; (2) x = 0 if and only if N (x, t) = 1 for all t > 0; (3) N (cx, t) = N (x, |c| ) if c ̸= 0; (4)

N (x + y, s + t) ≥ min {N (x, s), N (y, t)}; (5) N (x, ·) is a non-decreasing function on R and limt→∞ N (x, t) = 1; (6) N (x, ·) is continuous on R for x ̸= 0. In this case (X, N ) is called a fuzzy normed vector space. Definition 2.2 ( [4]) Let (X, N ) be a fuzzy normed space. A sequence xn in X is said to be convergent if there exists x ∈ X such that limn→∞ N (xn −x, t) = 1(t > 0). A sequence xn in X is called Cauchy if for each ϵ > 0 and t > 0, there exists n0 ∈ N such that N (xm − xn , t) > 1 − ϵ (m, n ≥ n0 ). If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space. We will use the following notations: Mm,n (X) is the set of all m × n matrices in X; When m = n, the matrix Mm,n (X) will be written as Mn (X); ej ∈ M1,n (R) denote the row vector whose jth component is 1 and the other components are zero; Eij ∈ Mn (R) is that (i, j)-component is 1 and the other components are zero; Eij ⊗ x ∈ Mn (X) is that (i, j)-component is x and the other components are zero. Let (X, ∥·∥) be a normed space. Note that (X, {∥·∥n }) is a matrix normed space if and only if (Mn (X), ∥·∥n ) is a normed space for each positive integer n and ∥AxB∥k ≤ ∥A∥ ∥B∥ ∥x∥n holds for A ∈ Mk,n (R), x = [xij ] ∈ Mn (X) and B ∈ Mn,k (R), and that (X, {∥·∥n }) is a matrix Banach space if and only if X is a Banach space and (X, {∥·∥n }) is a matrix normed space. ( ) x 0 For x ∈ Mn (X), y ∈ Mk (X), x ⊕ y := , we introduce the concept of matrix fuzzy normed spaces. 0 y Let X, Y be vector space. For a given mapping h : X → Y and a given positive integer n, define hn : Mn (X) → Mn (Y ) by hn ([xij ]) := [h(xij )] for all [xij ] ∈ Mn (X). Definition 2.3 ( [6, 15]) Let (X, N ) be a fuzzy normed space. (1) (X, {Nn }) is called a matrix fuzzy normed space if for each positive integer n, (Mn (X), Nn ) is a fuzzy t ) for all t > 0, A ∈ Mk,n (R), x = [xij ] ∈ Mn (X) and B ∈ Mn,k (R) normed space and Nk (AxB, t) ≥ Nn (x, ∥A∥·∥B∥ with ||A|| · ||B|| ̸= 0. (2) (X, {Nn }) is called a matrix fuzzy Banach space if (X, N ) is a fuzzy Banach space and (X, {Nn }) is a matrix fuzzy normed space.

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Lemma 2.1 ( [6]) Let (X, {Nn }) be a matrix fuzzy normed space. Then (1) Nn (Ekl ⊗ x, t) = N (x, t) for all t > 0, x ∈ X, ∑n (2) For all [xij ] ∈ Mn (X) and t = i,j=1 tij , N (xkl , t) ≥ Nn ([xij ], t) ≥ min {N (xij , tij ) : i, j = 1, 2, . . . , n} , { } t N (xkl , t) ≥ Nn ([xij ], t) ≥ min N (xij , 2 ) : i, j = 1, 2, . . . , n . n (3) limn→∞ xn = x if and only if limn→∞ xijn = xij for xn = [xijn ], x = [xij ] ∈ Mk (X). Theorem 2.1 ( [3]) Let (E, d) be a complete generalized metric space and J : E → E be a strictly contractive mapping, that is d(Jx, Jy) ≤ Ld(x, y), ∀x, y ∈ E for some 0 < L < 1. Then, for each given element x ∈ E, either d(J n x, J n+1 x) = +∞, ∀n ≥ 0 or d(J n x, J n+1 x) < ∞, ∀n ≥ n0 , for some natural number n0 . Moreover, if the second alternative holds, then (1) The sequence {J n x} is convergent to a fixed point y ∗ of J; (2) y ∗ is the unique fixed point of J in the set E ′ = {y ∈ E|d(J n0 x, y) < +∞} and d(y, y ∗ ) ≤ for all y ∈ E ′ .

3

1 1−L d(y, Jy)

General solution of the functional equation (1.2)

In this section, we investigate the general solution of the mixed cubic-quartic functional equation (1.2). Throughout this section, let X be a vector space over Q, Y be a vector space, and λ ∈ N2 . Some basic facts on n-additive symmetric mappings can be found in [12]. Lemma 3.1 If an odd mapping f : X → Y satisfies (1.2), then f is of the form f (x) = A3 (x) for all x ∈ X, where A3 (x) is the diagonal of the 3-additive symmetric map A3 : X 3 → Y . Proof.

Using the oddness of f , we have f (0) = 0 and f (−x) = −f (x) for all x ∈ X. (1.2) with y = 0 yields f (λx) = λ3 f (x).

(3.1)

f (λx + y) + f (λx − y) = λ[f (x + y) + f (x − y)] + 2λ(λ2 − 1)f (x),

(3.2)

Applying (3.1) to (1.2), we obtain

From (3.2), by Theorems 3.4 and 3.5 in [12], f is a generalized polynomial function of degree at most 3: f (x) = A3 (x) + A2 (x) + A1 (x) + A0 (x),

(3.3)

where A0 (x) = A0 is an arbitrary element of Y , and Ai is the diagonal of the i-additive symmetric map Ai : X i → Y for i = 1, 2, 3. By f (0) = 0 and f (−x) = −f (x) for all x ∈ X, we get A0 (x) = A0 = 0 and A2 (x) = 0 for all x ∈ X. By f (λx) = λ3 f (x) and Ai (rx) = ri Ai (x) whenever x ∈ X and r ∈ Q, we obtain A1 (x) = 0 for all x ∈ X. Therefore, f (x) = A3 (x) for all x ∈ X.  Lemma 3.2 If an even mapping f : X → Y satisfies (1.2), then f is of the form f (x) = A4 (x) for all x ∈ X, where A4 (x) is the diagonal of the 4-additive symmetric map A4 : X 4 → Y .

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Proof.

In view of the evenness of f , we have f (−x) = f (x) for all x ∈ X. Let y = 0 in (1.2), we obtain f (λx) = λ4 f (x).

(3.4)

The rest of the proof is similar to the proof of Lemma 3.1.  Theorem 3.1 A mapping f : X → Y satisfies (1.2) for all x, y ∈ X if and only if f is the form f (x) = A4 (x) + A3 (x),

(3.5)

where Ai is the diagonal of the i-additive symmetric map Ai : X i → Y for i = 3, 4. Proof. Assume that f satisfies the functional equation (1.2), we decompose f into the odd part and the even part by putting f (x) − f (−x) f (x) + f (−x) fo (x) = , fe (x) = , (3.6) 2 2 then, f (x) = fo (x) + fe (x) for all x ∈ X. It is easy to show that the mapping fo and fe satisfy (1.2). Therefore our assertion follows immediately from Lemmas 3.1 and 3.2. Conversely, assume that f (x) = A4 (x) + A3 (x) for all x ∈ X, where Ai (x) is the diagonal of the i-additive symmetric map Ai : X i → Y for i = 3, 4. Using A4 (x + y) + A4 (x − y) = 2A4 (x) + 2A4 (y) + 12A2,2 (x, y), A3 (x + y) + A3 (x − y) = 2A3 (x) + 6A1,2 (x, y),

(3.7)

Ai (rx) = ri Ai (x), i ∈ {3, 4}, r ∈ Q, Ai,j (rx, sy) = ri sj Ai,j (x, y), i ∈ {1, 2}, r, s ∈ Q, by a simple computation, one can see that f satisfies (1.2), which complete the proof of Theorem 3.1.

4



Stability of the functional equation (1.2) Throughout this section, let (X, {Nn }) be a matrix fuzzy normed space, (Y, {Nn }) be a matrix fuzzy Banach

space, λ ∈ N2 and n ∈ N. Using the fixed point method, we prove the Ulam stability of the CQ-functional equation (1.2) in matrix fuzzy normed spaces. Now before taking up the main subject, for a given mapping f : X → Y , we define the difference operator Df : X 2 → Y , and Dfn : Mn (X 2 ) → Mn (Y ). λ2 + λ λ2 − λ [f (a + b) + f (a − b)] − [f (−a − b) + f (b − a)] 2 2 4 3 2 4 3 2 − (λ + λ − λ − λ)f (a) − (λ − λ − λ + λ)f (−a) − (1 − λ2 )[f (b) + f (−b)],

(Df )(a, b) :=f (λa + b) + f (λa − b) −

(Dfn )([xij ], [yij ]) :=fn (λ[xij ] + [yij ]) + fn (λ[xij ] − [yij ]) −

λ2 + λ [fn ([xij ] + [yij ]) + fn ([xij ] − [yij ])] 2

λ2 − λ [fn (−[xij ] − [yij ]) + fn ([yij ] − [xij ])] − (λ4 + λ3 − λ2 − λ)fn ([xij ]) 2 − (λ4 − λ3 − λ2 + λ)fn (−[xij ]) − (1 − λ2 )[fn ([yij ]) + fn (−[yij ])]

−

for all a, b ∈ X, x = [xij ], y = [yij ] ∈ Mn (X). Theorem 4.1 Let φ1 : X 2 → [0, ∞) be a function such that for some real number α with 0 < α < 1, a b α φ1 ( , ) ≤ 4 φ1 (a, b), λ λ λ 332

a, b ∈ X.

(4.1)

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Suppose that f : X → Y is an even function with f (0) = 0 and such that Nn (Dfn ([xij ], [yij ]), t) ≥

t+

t , i,j=1 φ1 (xij , yij )

∑n

t > 0, x = [xij ], y = [yij ] ∈ Mn (X).

(4.2)

Then there exists a unique quartic mapping Q : X → Y such that Nn (fn ([xij ]) − Qn ([xij ]), t) ≥ Proof.

2λ4 (1

2λ4 (1 − α)t ∑n , − α)t + αn2 i,j=1 φ1 (xij , 0)

t > 0, x = [xij ] ∈ Mn (X).

When n = 1, (4.2) is equivalent to N (Df (a, b), t) ≥

t t + φ1 (a, b)

,

t > 0, a, b ∈ X.

(4.3)

Putting b = 0 in (4.3), we obtain that N (2f (λa) − 2λ4 f (a), t) ≥ Hence

t , t + φ1 (a, 0)

a t t N (f (a) − λ4 f ( ), ) ≥ , λ 2 t + φ1 ( λa , 0)

t > 0, a ∈ X.

(4.4)

t > 0, a ∈ X.

(4.5)

Using (4.1) we get a N (f (a) − λ4 f ( ), t) ≥ λ t+

t

, α 2λ4 φ1 (a, 0)

t > 0, a ∈ X.

(4.6)

Consider the set E1 = {g : X → Y, g(0) = 0}, and introduce the generalized metric d1 : { } t d1 (g, h) := inf ϵ ∈ R+ : N (g(a) − h(a), ϵt) ≥ , t > 0, a ∈ X , t + φ1 (a, 0) where, as usual, inf ∅ = +∞. It is easy to prove that (E1 , d1 ) is a complete generalized metric space. Now, let us consider the linear mapping J1 : E1 → E1 such that a J1 g(a) = λ4 g( ), λ

g ∈ E1 , a ∈ X.

It is easy to see that J1 is a strictly contractive self-mapping of E1 with the Lipschitz constant L = α. Indeed, given g, h ∈ E1 , let ϵ ∈ (0, ∞) be an arbitrary constant with d1 (g, h) = ϵ. From the definition of d1 , it follows that t N (g(a) − h(a), ϵt) ≥ , t > 0, a ∈ X. t + φ1 (a, 0) Hence a a a αϵt a N (J1 g(a) − J1 h(a), αϵt) = N (λ4 g( ) − λ4 h( ), αϵt) = N (g( ) − h( ), 4 ) λ λ λ λ λ α t λ4 t ≥ α ≥ , t > 0, a ∈ X. a t + φ ( , 0) t + φ 1 λ 1 (a, 0) λ4 So, d1 (g, h) = ϵ implies that d(J1 g, J1 h) ≤ αϵ. This means that d1 (J1 g, J1 h) ≤ αd1 (g, h) for all g, h ∈ E1 , thus J1 is a strictly contractive self-mapping, and the Lipschitz constant L = α. It follows from (4.6) that N (f (a) − J1 f (a), t) ≥ thus we have that d1 (f, J1 f ) ≤

α 2λ4

t

t+

, α 2λ4 φ1 (a, 0)

t > 0, a ∈ X,

< +∞.

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According to Theorem 2.1, we deduce the existence of a fixed point of J1 , that is, the existence of a mapping Q : X → Y such that Q(a) = J1 Q(a) = λ4 Q( λa ), i.e., Q( λa ) = d1 (J1l f, Q) → 0(l → +∞), which implies lim N (J1l f (a) − Q(a), t) = 1,

1 1−L d1 (J1 f, f )

implies the inequality d1 (f, Q) ≤

N (f (a) − Q(a), t) ≥ Replacing a and b by

a λl

and

b λl

for each a ∈ X. Moreover, we have

t > 0, a ∈ X.

l→+∞

Also, d1 (f, Q) ≤

1 λ4 Q(a)

α 2λ4 (1−α) ,

2λ4 (1 − α)t , 2λ4 (1 − α)t + αφ1 (a, 0)

(4.7)

which means that

t > 0, a ∈ X.

(4.8)

in (4.3), respectively, we have

) ( ) ( a b t a b 4l N λ Df ( l , l ), t = N Df ( l , l ), 4l ≥ λ λ λ λ λ

t λ4l t λ4l

+ φ1 ( λal , λbl )

,

t > 0, a, b ∈ X.

(4.9)

It follows from (4.1) that αl a b , ) ≤ φ1 (a, b), a, b ∈ X, λl λl λ4l ( ) a b t 4l N λ Df ( l , l ), t ≥ , t > 0, a, b ∈ X. λ λ t + αl φ1 (a, b) φ1 (

thus

Letting l → +∞ in (4.10), we obtain ( ) a b N λ4l Df ( l , l ), t → 1, λ λ

(4.10)

t > 0, a, b ∈ X,

(4.11)

which means N (DQ(a, b), t) = 1,

t > 0, a, b ∈ X.

(4.12)

Thus, DQ(a, b) = 0 for all a, b ∈ X. By the definition of Q, it is clear that Q(−a) = Q(a) for all a ∈ X. Then by Lemma 3.1, the mapping Q is quartic. Assume that there exists another quartic function F : X → Y which satisfies (4.8). Then it is clear that F ( λa ) = λ14 F (a), and while a = 0, we have F (a) = 0, thus J1 F (a) = λ4 F ( λa ) = F (a) for all a ∈ X, i.e., F is a fixed point of J1 . By (4.8) we get N (f (a) − F (a), t) ≥

2λ4 (1 − α)t , 2λ4 (1 − α)t + αφ1 (a, 0)

t > 0, a ∈ X.

′

α Hence, d1 (f, F ) ≤ 2λ4 (1−α) . So, F ∈ E1 = {g ∈ E1 , d1 (f, g) < ∞}. By Theorem 2.1, Q is the unique fixed point in E1 , which means that Q = F .

By Lemma 2.1 and (4.8), we have {

} t N (fn ([xij ]) − Qn ([xij ]), t) ≥ min N (f (xij ) − Q(xij ), 2 ) : i, j = 1, 2, ..., n n } { 2λ4 (1 − α)t : i, j = 1, 2, ..., n ≥ min 2λ4 (1 − α)t + αn2 φ1 (xij , 0) 2λ4 (1 − α)t ∑n ≥ 4 2λ (1 − α)t + αn2 i,j=1 φ1 (xij , 0) for all x = [xij ] ∈ Mn (X), t > 0. This completes the proof.



Theorem 4.2 Let φ2 : X 2 → [0, ∞) be a function such that for some real number α with 0 < α < λ, a b α φ2 ( , ) ≤ 4 φ2 (a, b), λ λ λ 334

a, b ∈ X.

(4.13) Yali Ding et al 329-336

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Suppose that f : X → Y is an odd function such that Nn (Dfn ([xij ], [yij ]), t) ≥

t+

t , i,j=1 φ2 (xij , yij )

∑n

t > 0, x = [xij ], y = [yij ] ∈ Mn (X).

(4.14)

Then there exists a unique cubic mapping C : X → Y such that Nn (fn ([xij ]) − Cn ([xij ]), t) ≥ Proof.

2λ3 (λ − α)t ∑n , 2λ3 (λ − α)t + αn2 i,j=1 φ2 (xij , 0)

t > 0, x = [xij ] ∈ Mn (X).

The proof is similar to the proof of Theorem 4.1. 

Theorem 4.3 Let φ : X 2 → [0, ∞) be a function such that for some real number α with 0 < α < 1, a b α φ( , ) ≤ 4 φ(a, b), λ λ λ

a, b ∈ X.

(4.15)

Suppose that f : X → Y is a function such that f (0) = 0, and for all x = [xij ], y = [yij ] ∈ Mn (X), satisfying Nn (Dfn ([xij ], [yij ]), t) ≥

t+

∑n

t

i,j=1

φ(xij , yij )

,

t > 0.

(4.16)

Then there exist a unique cubic mapping C : X → Y and a unique quartic mapping Q : X → Y such that Nn (fn ([xij ]) − Qn ([xij ]) − Cn ([xij ]), t) ≥

λ4 (1 − α)t ∑n , λ4 (1 − α)t + αn2 i,j=1 ψ(xij , 0)

where ψ(a, b) := φ(a, b) + φ(−a, −b) for all a, b ∈ X. Proof.

Let fe (a) = 21 (f (a) + f (−a)), it is easy to see that fe (0) = 0, fe (−a) = fe (a). 1 1 N (Dfe (a, b), t) = N ( Df (a, b) + Df (−a, −b), t) = N (Df (a, b) + Df (−a, −b), 2t) 2 2 t ≥ min{N (Df (a, b), t), N (Df (−a, −b), t)} ≥ . t + ψ(a, b)

t . From (4.15), it follows that ψ( λa , λb ) ≤ Let fo (a) = 12 (f (a) − f (−a)), we can get N (Dfo (a, b), t) ≥ t+ψ(a,b) α λ4 ψ(a, b). It is easy to check that all conditions of Theorems 4.1 and 4.2 hold, by the proofs of Theorems 4.1

and 4.2, we know that there exist a quartic mapping Q : X → Y and a cubic mapping C : X → Y such that N (fe (a) − Q(a), t) ≥ and N (fo (a) − C(a), t) ≥

2λ4 (1 − α)t , − α)t + αψ(a, 0)

t > 0, a ∈ X,

2λ3 (λ − α)t , 2λ3 (λ − α)t + αψ(a, 0)

t > 0, a ∈ X.

2λ4 (1

Therefore N (f (a) − C(a) − Q(a), t) = N (fe (a) − Q(a) + fo (a) − C(a), t) t t ≥ min{N (fe (a) − Q(a), ), N (fo (a) − C(a), )} 2 2 λ4 (1 − α)t λ3 (λ − α)t ≥ min{ 4 , 3 } λ (1 − α)t + αψ(a, 0) λ (λ − α)t + αψ(a, 0) λ4 (1 − α)t = 4 , t > 0, a ∈ X. λ (1 − α)t + αψ(a, 0)

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(4.17)

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Using a proof method similar to Theorem 3.10 in [11], we can prove the uniqueness of C and Q. By Lemma 2.1 and (4.17), we have

{ } t N (fn ([xij ]) − Qn ([xij ] − Cn ([xij ]), t) ≥ min N (f (xij ) − Q(xij ) − C(xij ), 2 ) : i, j = 1, 2, ..., n n { } λ4 (1 − α)t ≥ min : i, j = 1, 2, ..., n λ4 (1 − α)t + αn2 ψ(xij , 0) λ4 (1 − α)t ∑n ≥ 4 λ (1 − α)t + αn2 i,j=1 ψ(xij , 0)

for all x = [xij ] ∈ Mn (X), t > 0. This completes the proof.



References [1] A. Bahyrycz, J. Brzd¸ek, E. Jablo´ nska, R. Malejki, Ulam’s stability of a generalization of the Fr´echet functional equation, J. Math. Anal. Appl., 442(2016), 537-553. [2] A. Bahyrycz, K. Ciepli´ nski, On an equation characterizing multi-Jensen-quadratic mappings and its HyersUlam stability via a fixed point method, J. Fixed Point Theory Appl., 18(2016), 737-751. [3] J.B. Diaz, 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. [4] A.K. Mirmostafaee, M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159(2008), 720-729. [5] A.M. Song, The Ulam stability of matrix intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Syst., 32(2017), 629-641. [6] C. Park, D. Shin, J.R. Lee, Fuzzy stability of functional inequalities in matrix fuzzy normed spaces, J. Ineq. Appl., 2013(2013), Artical ID 547, 13 pages. [7] C. Park, Orthogonal stability of a cubic-quartic functional equation, J. Nonlinear Sci. Appl., 5(2012), 28-36. [8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27(1941), 222-224. [9] M.E. Gordji, M.B. Savadkouhi, Stability of a mixed type cubic-quartic functional equation in nonArchimedean spaces, Applied Mathematics Letters, 23(10)(2010), 1198-1202. [10] P. G˘avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431-436. [11] T.Z. Xu, J.M. Rassias, W.X. Xu, Intuitionistic fuzzy stability of a genereal mixed additive-cubic equation, J. Math. Phys., 51(2010), 063519, 21 pages. [12] T.Z. Xu, J.M. Rassias, W.X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malays. Math. Sci. Soc., 35(3)(2012), 633-649. [13] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72(1978), 297-300. [14] S.M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960. [15] Z.H. Wang, P.K. Sahoo, Stability of ACQ-functional equation in various matrix normed spaces, J. Nonlinear Sci. Appl., 8(2015), 64-85.

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A High-Accuracy Collocation Method for Solving Mixed Boundary Value Problems on Nonsmooth Boundaries∗ Xin Luo

†

Chuan-Long Wang‡

Abstract By potential theory, the mixed Dirichlet-Neumann boundary value problem for the Laplacian is converted into the boundary integral equations (BIEs) with logarithmic singularity. Then the resulting system of the integral equations is solved by the Sidi-Israeli quadrature method (SIQM) with a Sigmoidal transformation. The convergence of numerical solutions by SIQM is proved based on Anselone’s collective compact theory. Furthermore, a convergence estimate of the solution error is presented, which possesses high accuracy order O (h3max ), where hmax is the mesh size. Finally, The efficiency of the method is illustrated by examples. Keyword : Boundary value problem, collective compact theory, singularity, integral equations

1

Introduction

Consider the following mixed Dirichlet-Neumann boundary value problem for the Laplacian  ∆u = 0, in Ω,    u|ΓDi = fi , i = 1, 2, · · · , p, (1.1)  ∂u   |Γ = gj , j = 1, 2, · · · , q, ∂n Nj where Ω is a simply connected region with the piecewise-smooth boundary Γ = ΓD ∪ ΓN , and ΓD = ∪pi=1 ΓDi and ΓN = ∪qj=1 ΓNj . Here, fi and gj are given on ΓDi and ∗

This work is supported by Project (NO. KYTZ201505) Supported by the Scientific Research Foundation of CUIT † College of Applied Mathematics, Chengdu University of Information Technology, Chengdu 610225, P.R. China, corresponding author: [email protected] ‡ Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, P.R. China

1

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ΓNj respectively, and ∂u/∂n denotes the derivative of u with respect to the outward normal vector n. By the potential theory [17], the solution of Eq. (1.1) can be represented as a single-layer potential of the form Z 1 u(P ) = − ln |P − Q|z(Q)dSQ , P ∈ Ω, (1.2) π Γ where z is an unknown function called the ”the single layer” density. From the jump condition for the normal derivative of the single layer potential at the boundary, we then have the following boundary integral equations (BIEs)  p Z  1X   − ln |P − Q|zDj (Q)dSQ    π j=1 ΓDj     q Z  X  1   ln |P − Q|zNj (Q)dSQ = fi , P ∈ ΓDi , i = 1, 2, · · · , p, −   π Γ N j=1 j (1.3) p Z X  ∂ ln |P − Q| 1    zDj (Q)dSQ zNi (P ) −   π j=1 ΓDj ∂nP      q Z   1X ∂ ln |P − Q|   zNj (Q)dSQ = gi , P ∈ ΓNi , i = 1, 2, · · · , q,  −π ∂nP j=1 ΓNj where zDj := z|ΓDj and zNj := z|ΓNj are sought on ΓDi and ΓNj , respectively. Once zDj and zNj are solved from the Eq. (1.3), the solution u(P ) can be computed by p Z q Z 1X 1X u(P ) = − ln |P − Q|zDj (Q)dSQ − ln |P − Q|zNj (Q)dSQ , P ∈ Ω. π j=1 ΓDj π j=1 ΓNj

(1.4) Even for the boundary data fi and gi are smooth, the solutions zDj and zNj may not be smooth. We denote by Pi , i = 0, 1 of the two interface points of the boundary Γ and by βi with 0 < βi < 2π, i = 0, 1 the interior angle of Γ at Pi . In fact, from [1, 2] it follows that around Pi we have u(P ) = c(Θ)rπ/(2βi ) + smoother terms,

P ∈ Ω,

(1.5)

where (r, Θ) are the polar coordinates centered at Pi . Then , using (1.2) to define a ¯ the single z is the difference between the potential not only in Ω but also in R2 \Ω, normal derivatives of u on Γ from inside and outside Γ. Therefore, near Pi , i = 0, 1, we get z(P ) = crmin{π/(2βi ), π/(4π−2βi )}−1 + smoother terms, P ∈ Ω. (1.6) Hence, zDj and zNj have this behavior near the corners Pi . To smooth these irregularities, in the next section we will introduce a smoothing parameterization ψγ (t), 2

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which improves the behavior of the unknown function z by incorporating the Jacobian of the transformation. In fact, the new unknown function will be z(ψγ (t))|ψγ0 (t)|, whose smoothness degree at the corner depends upon a smoothing parameter: the larger its value, the smoother the transformed density. There exist numerical methods for approximately solving mixed value problems on polygonal domains by means of boundary integral equations (see [18, 19]). They are based on the collocation method, and in general no error estimates are available [20]. After that, the proof of asymptotic error estimates for the finite element Galerkin approximation of the boundary integral equations for a mixed Dirichlet-Neumann boundary value problem for the Laplacian in a plane polygonal domain is given in [21]. This was a generalization of [22], where the case of a domain with a smooth boundary was treated. In [6], the trigonometric collocation method which uses a mesh grading transformation and a cosine approximating space is proposed for solving the mixed boundary value problems on domains with curved polygonal boundaries, the complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in which each arc of the polygon has associated with it a periodic Sobolev space. Inspired by the technique developed in [6], A collocation method using Chebyshev polynomial expansions as approximants and the zeros of Chebyshev polynomials as collocation nodes is applied to solved (1.3) [2]. From [5], we know that the Sidi transformation [3] is the important one of ”integral” sigmoidal transformations, which can yield fast convergence of the collocation solution by smoothing the singularities of the exact solution. Hence, we apply Sidi-Israeli quadrature method [4] and trapezoidal rule with Sidi transformation [3, 16] to calculate the integrals with weakly singular kernels and continuous kernels in (1.3) respectively. This paper is organized as follows: in Section 2, the convergence analysis is carried out based on the theory of collectively compact operators [7, 8, 9, 10] for closed curved polygons. in Section 3, a convergence estimate of the solution error is given. Numerical examples are provided to verify the theoretical results in Section 4, and conclusions are made in Section 5.

2 2.1

Collocation method for the boundary integral equations Discretization for integral operators

In [4], high-accuracy numerical quadrature methods based on the appropriate EulerMaclaurin expansions of trapezoidal rule approximations are proposed for the singular and weakly singular Fredholm integral equations. These integral equations are used in the solution of planar elliptic boundary value problems such as those that arise in free surface flows, elasticity, potential theory, conformal mapping, etc. Let the functions G(x, t) = log |x − t|g(x) + g˜(x) are periodic with period T = b − a , and that they are 2m times differentiable on R\{t+kT }∞ k=−∞ . Then the Sidi-Israeli quadrature

3

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formula [4] for integrals with kernel G(x, t) can be described by ¾ ½X n h Qn [G(x, t)] = h G(xj , t)+˜ g (t)+log( )g(t) , h = (b−a)/n, xj = a+jh, (2.1) 2π j=1 xj 6=t

and Z

b

G(x, t)dx − Qn [G(x, t)] = 2 a

m−1 X µ=1

0

ζ (−2µ) (2µ) 2µ+1 g h + O (h2m ), as h → 0, (2µ)!

where ζ(z) is a Riemann function. Define the following boundary integral operators on ΓDj and ΓNj Z 1 ln |x − y|zj (y)dsy , x ∈ ΓDi , i, j = 1, · · · , p, Uij zj (x) = − π ΓDj Z ∂ ln |x − y| 1 Mij zj (x) = − zj (y)dsy , x ∈ ΓNi , i, j = 1, · · · , q, π ΓNj ∂nx Z 1 ln |x − y|zj (y)dsy , x ∈ ΓDi , i = 1, · · · , p, j = 1, · · · , q, Vij zj (x) = − π ΓNj Z ∂ ln |x − y| 1 zj (y)dsy , x ∈ ΓNi , i = 1, · · · , q, j = 1, · · · , p. Wij zj (x) = − π ΓDj ∂nx Assume that ΓDj or ΓNj can be described by the parameter mapping: xj (t) = 0 0 0 (xj1 (t), xj2 (t)) : [0, 1] → ΓDj (or ΓNj ) with |xj (t)| = [|xj1 (t)|2 + |xj2 (t)|2 ]1/2 > 0. In order to degrade the singularities at corners, we apply the Sidi transformation [3, 16] to the parameter mapping, which is defined by Rt (sinπτ )γ dτ 0 : [0, 1] → [0, 1], γ ≥ 1. ψγ (t) = R 1 (2.2) (sinπτ )γ dτ 0 Define the following ”smoothing parameterization” ( (1) αi (t) = xi (ψγ (t)) ∈ ΓDi α(t) = (2) αi (t) = xi (ψγ (t)) ∈ ΓNi

t ∈ [−1, 1], t ∈ [−1, 1],

(2.3)

Thus, we can rewrite equations (1.3) as a p × q matrix integral equation system  p Z q Z 1 X X 1   (2) (1)  v(t, s)¯ zj (s)ds = fi (t), t ∈ [0, 1], i = 1, 2, · · · , p, u(t, s)¯ zj (s)ds +    j=1 0 j=1 0 q Z 1 p Z 1 X X   (2) (1)   m(t, s)¯ zj (s)ds = gi (t), i = 1, 2, · · · , q, w(t, s)¯ zj (s)ds + zi (t) +   0 0 j=1

j=1

(2.4) 4

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where

1 (1) (1) u(t, s) = − ln |αi (t) − αj (s)|, π 1 (1) (2) v(t, s) = − ln |αi (t) − αj (s)|, π (2)

0

(2)

(1)

(2)

0

(2.5) (2.6) (2)

(1)

1 αi2 (t)[αi1 (t) − αj1 (s)] − αi1 (t)[αi2 (t) − αj2 (s)] w(t, s) = − , (2) (1) (2) (1) π [αi1 (t) − αj1 (s)]2 + [αi2 (t) − αj2 (s)]2  0 0 (2) (2) (2) (2) (2) (2)  1 αi2 (t)[αi1 (t)−αj1 (s)]−αi1 (t)[αi2 (t)−αj2 (s)]   , as t 6= s, − (2) (2) (2) (2)  [αi1 (t)−αj1 (s)]2 +[αi2 (t)−αj2 (s)]2  π m(t, s) = 0 00 0 00   (2) (2) (2) (2)  αi2 (t)αi1 (t)−αi1 (t)αi2 (t) 1   − 2π , as t = s. 0 0 (2) (2)

(2.7)

(2.8)

(αi1 (t))2 +(αi2 (t))2

and (1)

z¯j (s) = zDj (xj (ψγ (s)))|x0j (ψγ (s))|ψγ0 (s), (2) z¯j (s) = zNj (xj (ψγ (s)))|x0j (ψγ (s))|ψγ0 (s), (k) (k) (k) (2) (1) fi (t) = f (αi (t)), gi (t) = g(αi (t)), αi (t) = (αi1 (t), αi2 (t)), k = 1, 2. Lemma 2.1. Although zDj (s) and zNj (s) have singularities at endpoints s = 0 (2) (1) and s = 1, z¯j (s) and z¯j (s) have no singularities by Sidi transformation at s = 0 and s = 1. Proof. Let dj = min{π/(2βj ), π/(4π − 2βj )} − 1 in (1.6), then we have −1/2 ≤ dj < 0. Suppose that zDN (s) = sdj ϕj (s) near s = 0, where zDN = zDj or zNj , and the function ϕj (s) is differentiable enough on [0, 1] with ϕj (0) 6= 0. Using Taylor’s formula, we can obtain zDN (s) =

(i) l X ϕj (0) i=0

i!

si+dj + O(sl+dj +1 ), as s → 0+ .

(2.9)

From [3], we have ψγ (s) ∼

∞ X

²i s

γ+2i+1

i=0

,

ψγ0 (s)

∼

∞ X

δi sγ+2i , as s → 0+ , ²0 , δ0 > 0.

(2.10)

i=0 (k)

By substituting (2.9) and (2.10) into the expression of z¯j (s), k = 1, 2, we have (k)

z¯j (s) = c1 ϕj (0)s(γ+1)dj +γ (1 + O(s2 )) as s → 0+ ,

(2.11)

where c1 is a constant. Also assume that zDN (s) = (1−s)dj ϕj (s) near s = 1. Similarly, we have (k)

z¯j (s) = c2 ϕj (0)(1 − s)(γ+1)dj +γ (1 + O((1 − s)2 )) as s → 1− ,

(2.12)

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where c2 is a constant independent of s. By (2.11), (2.12) and dj ≥ − 12 , we can obtain (γ + 1)(dj + 1) − 1 ≥ 0 for γ ≥ 1. The proof is completed. ¤ Now we can rewrite the Eqs. (2.4) as follows · ¸ · (1) ¸ · ¸ U V z f = , (2.13) (2) W I +M g z where

f = (f1 , f2 , · · · , fp ), g = (g1 , g2 , · · · , gq ), (1) (1) (1) (2) (2) (2) z (1) = (¯ z1 , z¯2 , · · · , z¯p )T , z (2) = (¯ z1 , z¯2 , · · · , z¯q )T , p,q q,p q,q U = [Uij ]p,p i,j=1 , V = [Vij ]i,j=1 , W = [Wij ]i,j=1 , M = [Mij ]i,j=1 .

Let U = A + B, where A = diag(A11 , A22 , · · · , App ) and B = [Bij ]p,p i,j=1 , where Z 1 (1) (1) Aii z¯i (t) = a(t, s)¯ zi (s)ds, 0

with the kernel

1 aii (t, s) = − ln |2e−1/2 sin(π(t − s))|, π

and

Z (1) Bij z¯j (t)

with the kernel bij (t, s) =

1

=

(1)

b(t, s)¯ zj (s)ds,

0

 (1) (1) αi (t)−αj (s)   − π1 ln | 2e−1/2 |, as i = j, sin(π(t−s))  

(1) − π1 ln |αi (t)

−

(1) αj (s)|,

(2.14)

as i 6= j.

In the subsequent analysis we will focus on the singularity of the kernels bij (t, s). Obviously, if ΓDi ∩ ΓDj = ∅, bij (t, s) are continuous in [0, 1]2 , and if ΓDi ∩ ΓDj 6= ∅, bij (t, s) have singularities at the points (t, s) = (0, 1) and (t, s) = (1, 0). For convenience of analysis, we only discuss the case in which (t, s) = (1, 0). Defining the following function ˜bij (t, s) = bij (t, s) sinγ (πt),

γ ≥ 1,

ΓDi ∩ ΓDj 6= ∅.

(2.15)

∂ k ˜bij (t,s) (k = 1, 2) Lemma 2.2. Let ˜bij (t, s) be defined by (2.15), then ˜bij (t, s) and ∂t k are smooth on [0, 1]2 . Proof. By the continuity of ˜bii (t, s) in (2.14) and the boundness of sinγ (πt), we can immediately complete the proof for the case i = j. Hence, we only consider the case in which j − i = 1. Let ΓDi−1 ∩ ΓDi = Pi = (0, 0) and θi ∈ (0, 2π) be the corresponding interior angle. Then we have (1)

(1)

ln |αi (t) − αi−1 (s)| =

1 (1) (1) (1) (1) ln[(|αi (t)| − |αi−1 (s)|)2 + 4|αi (t)||αi−1 (s)|sin2 (θi /2)] 2 (2.16) 6

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which shows the kernel bi−1,i (t, s) has a logarithmic singularity at (t, s) = (1, 0). (1) (1) Suppose that a0 (t) = |αi−1 (t)| and a1 (s) = |αi (s)|, we have a0 (0) = a1 (0) = 0. If θi ∈ (0, π) ∪ (π, 2π), then bi−1,i (1, 0) = 0. If (t, s) 6= (1, 0), then we can obtain ˜bi−1,i (t, s) = − 1 sinγ (πt) ln[a2 (t) + a2 (s) − 2a0 (t)a1 (s) cos θi−1 ] 0 1 2π 1 = − sinγ (πt) ln[a20 (t) + a21 (s)] 2π 1 − sinγ (πt) ln[1 − 2a0 (t)a1 (s) cos θi−1 /(a20 (t) + a21 (s))] 2π = $1 (t, s) + $2 (t, s).

(2.17)

Since |2a0 (t)a1 (s) cos θi−1 /(a20 (t) + a21 (s))| ≤ | cos θi−1 | < 1, the function $2 (t, s) and its second derivative are bounded. Noting that ψγ(k) (0) = ψγ(k) (1) = 0, we have

(k)

(k)

a¯i (0) = a¯i (1) = 0,

k = 0, · · · , γ,

¯i = i − 1 or i, k = 1, · · · , γ.

Let (t, s) ∈ [ε/2, ε] × [1 − ε, 1 − ε/2] for all ε > 0, we have |$1 (t, s)| = O(εγ | ln ε|), so $1 (t, s) is also bounded. In addition, we have (1)0

0

2a0 (s)|αi−1 (ψγ (s))|ψγ (s) ∂ 1 | $1 (t, s)| ≤ | sinγ (πt) | ∂t 2π a20 (t) + a21 (s) = O(εγ )O(ε2γ )/O(ε2γ ) = O(εγ ) and

This shows then

∂2 | 2 $1 (t, s)| = O(εγ−1 ). ∂t ∂ k ˜bij (t,s) ∂tk

(k = 0, 1, 2) are also continuous in [0, 1]2 . At last, if θi−1 = π,

˜bi−1,i (t, s) = − 1 sinγ (πt) ln(a0 (t) + a1 (s)), (2.18) π we can use the same method mentioned above to prove ˜bi−1,i (t, s) and its second derivative are bounded. The proof of Lemma 2.2 is completed. ¤ Let hj = 1/nj (nj ∈ N ) and tj = sj = (j − 1/2)hj (j = 1, ..., nj ) be the mesh sizes and nodes respectively. By the trapezoidal or the midpoint rule [11] we construct the h Nystr¨om’s approximation operator Bijj of the integral operator Bij , defined by h (1) (Bijj z¯j )(t)

= hj

nj X

(1)

bij (t, sj )¯ zj (sj ), t ∈ [0, 1], i = 1, ..., p,

(2.19)

j=1

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which has the error bounds [3, 11] h

(1)

(1)

(Bij z¯j )(t) − (Bijj z¯j )(t) = O (h2` j ), for ΓDi ∩ ΓDj = ∅, ` ∈ N, and

h

(1)

(1)

(Bij z¯j )(t) − (Bijj z¯j )(t) = O (hωj ), for ΓDi ∩ ΓDj ∈ {Pj },

(2.20) (2.21)

where (see [3])   min{(γ + 1)(dj + 1), γ + 1}, γ odd, ω=



(2.22) min{(γ + 1)(dj + 1), 2(γ + 1)}, γ even.

For the logarithmically singular operators Aii , by the Sidi-Israeli quadrature formula, we can also construct the approximate operator Ahiii ,       n j X  ¯ ¯ h i (1) hi (1) −1/2 (Aii z¯j )(t) = − ln ¯2e sin π(t − sj )¯z¯j (sj )  π    (2.23)  j=1  sj 6=t

¢ (1) o hi n ¡ ln 2πe−1/2 hi /(2π) z¯j (t) − π

(i = 1, ..., ni ),

which has the error bounds [4] (1) (Ahiii z¯j )(t)

−

(1) (Aii z¯j )(t)

2`−1 2 X ζ 0 (−2µ) (1) (2µ) 2µ+1 =− [¯ z ] hi + O(h2` i ), π µ=1 (2µ)! j

t ∈ {ti },

where ζ 0 (t) is the derivative of the Riemann zeta function. By the trapezoidal or the midpoint rule, we can also construct the Nystr¨om’s h h h approximation operators Vij j , Wijj and Mijj for the continuous operators Vij , Wij and Mij , that is, nj X hj χij (t, sj )¯ zj (sj ), t ∈ [0, 1], (2.24) (Ξij z¯j )(t) = hj j=1 ω which have the error bounds O(h2` j ) or O(hj ). Here, Ξij = Vij , Wij or Mij , χij (t, s) = vij (t, s), wij (t, s) or mij (t, s). Now we write the discrete equations for (2.13) are ¸ · (1)h ¸ · h ¸ · h f z U Vh , (2.25) = h h h (2)h gh W I +M z

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where h

n

p U h = Ah + B h , Ah = diag(Ah111 , ..., Appp ), Ahiii = [a(ti , sj )]i,j=1 , h h h n ,n p p,q j j p p j B h = [Bij ]i,j=1 , Bij = [bij (ti , sj )]i,j=1 , V h = [Vij ]i,j=1 , h h h np ,nq nq ,np Vij j = [vij (ti , sj )]i,j=1 , W h = [Wijj ]q,p Wijj = [wij (ti , sj )]i,j=1 , i,j=1 , hj p,q hj nq ,nq h M = [Mij ]i,j=1 , Mij = [mij (ti , sj )]i,j=1 , (1)h (1)h (1)h (1)h z (1)h = (z1 1 (t1 ), ..., z1 1 (tn1 ), ..., zp p (t1 ), ..., zp p (tnp ))T , (2)h (2)h (2)h (2)h z (2)h = (z1 1 (t1 ), ..., z1 1 (tn1 ), ..., zq q (t1 ), ..., zq q (tnq ))T , h h f h = (f1h1 (t1 ), ..., f1h1 (tn1 ), ..., fp p (t1 ), ..., fp p (tnp ))T , h h g h = (g1h1 (t1 ), ..., g1h1 (tn1 ), ..., gq q (t1 ), ..., gq q (tnq ))T .

Let

·

Uh Vh W h Ih + M h

¸

· =

Ah 0 0 Ih

¸

· +

Bh V h W h Mh

¸ ,

then (2.25) is equivalent to µ· h ¸ · ¸¶ · (1)h ¸ · ¸ E1 0 (Ah )−1 B h (Ah )−1 V h z (Ah )−1 f h + = . Wh Mh gh 0 E2h z (2)h

2.2

(2.26)

(2.27)

The collectively compact convergence

For the convenience of the analysis of the existence and convergence of numerical solutions, we first introduce the subspaces and some special operators to be used. Define the subspace C0 [0, 1] = {v(t) ∈ C[0, 1] : v(t)(sinγ (πt))−1 ∈ C[0, 1]} of the space C[0, 1] with the norm ||v||∗ = max0≤t≤1 |v(t)(sinγ (πt)))−1 |. Let S hj = span{ej (t), j = 1, ..., nj } ⊂ C0 [0, 1] be a piecewise linear function subspace with the basis nodes nj {ti }i=1 , where ej (t) are the basis functions satisfying ej (ti ) = δji . Also define a prolonnj P h h vj ej (t), ∀v = (v1 , ..., vnj ) ∈ 0, and x21 + x22 < 1,

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−3

3.5

−4

x 10

1.2

3

x 10

1

2.5

Errors

Errors

0.8 2

0.6

1.5 0.4 1 0.2

0.5

0 −0.4

−0.3

−0.2

−0.1 0 0.1 x coordinate of the points

0.2

0.3

0 −0.4

0.4

−0.3

−0.2

−0.1 0 0.1 x coordinate of the points

0.2

0.3

0.4

Figure 2: Left: Errors of u by 2 × 25 boundary nodes; Right: Errors of u by 2 × 26 boundary nodes. subject to the boundary conditions ΓD1 : u = 1,

on x21 + x22 = 1, for x1 > 0, x2 > 0; ΓD2 : u = 0, x2 = 0; ∂u = 0, x1 = 0. ΓN1 : ∂n

2 The analytical solution of this problem is u = π2 arctan( 1−x2x2 −x 2 ). 1 2 Let each boundary be divided into 2k (k = 3, · · · , 8) segments. The errors and error ratio of the interior points P1 = (0.1, 0.1), P2 = (0.8, 0.1) and P3 = (0.1, 0.7) using n (= 3 × 2k , k = 3, · · · , 8) nodes by transformation ψ6 (t) are listed in Table 2. In addition, the numerical solution u of the interior points along the curve segment L : x1 = 0.7cos( π2 t), x2 = 0.7sin( π2 t) are computed, where t = 0.05 : 0.01 : 0.95. The plots of computed errors are shown in Figure 3 (b) to Figure 4. From the numerical results of Table 1 and Table 2 we can see that EOC ≈ 3.

Table 2: The Errors of u . 3 × 23 1.167-03 EOC(P1 ) − u errn (P2 ) 7.409-03 EOC(P2 ) − u errn (P3 ) 1.997-02 EOC(P3 ) − n

errnu (P1 )

3 × 24 1.223-04 3.255 9.133-04 3.020 1.694-03 3.559

3 × 25 1.341-05 3.188 1.691-05 5.755 3.282-05 5.689

3 × 26 1.675-06 3.001 2.421-06 2.805 1.062-06 4.950

3 × 27 2.093-07 3.000 3.025-07 3.000 1.313-07 3.016

3 × 28 2.617-08 3.000 3.781-08 3.000 1.641-08 3.000

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−6

1

6

0.9

5.5

0.8

5

0.7

4.5 u=1 ∂u ∂n

0.5

4 Errors

0.6 =0

3.5

0.4

3

0.3

2.5

0.2

2 u=0

0.1 0

x 10

0

0.2

0.4

1.5 0.6

0.8

1

1

0

0.1

0.2

0.3 0.4 0.5 x coordinate of the points

0.6

0.7

Figure 3: Left: The contour Γ for Example 2; Right: Errors of u by 3 × 26 boundary nodes.

−7

5

−8

x 10

6.5

x 10

6

4.5

5.5 4 5 4.5 Errors

Errors

3.5 3

4 3.5

2.5

3 2 2.5 1.5 1

2 0

0.1

0.2

0.3 0.4 0.5 x coordinate of the points

0.6

0.7

0

0.1

0.2

0.3 0.4 0.5 x coordinate of the points

0.6

0.7

Figure 4: Left: Errors of u by 3 × 27 boundary nodes; Right: Errors of u by 3 × 28 boundary nodes.

5

Conclusions

In this paper, the convergence and error of SIQM for the boundary integral equations of the mixed Dirichlet-Neumann boundary value problem for the Laplacian are studied on nonsmooth boundaries. Especially, in order to provide a good accuracy in the solution near the singular points, the Sidi transformation is used for the boundary integral equations of problems (1.1). The numerical results show that the presented algorithm has a high accuracy of O (h3max ), which coincides with our theoretical analysis.

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References [1] M. Costabel and E.P. Stephan, On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp. 50 (1987), pp. 461–478. [2] L.Scuderi, A chebyshev polynomial collocation BIEM for mixed boundary value problems on nonsmooth boundaries, J. Integral Equations Appl. 14 (2002), pp. 179–221. [3] A. Sidi, A new variable transformation for numerical integration, Int Ser Numer Math. 112 (1993), pp. 359–373. [4] A. Sidi and M. Israeli, Quadrature methods for periodic singular and weakly singular Fredholm integral equation J. Sci. Comput. 3 (1988), pp. 201–231. [5] David Elliott, Sigmoidal Transformations and the Trapezoidal Rule, J. Austral. Math. Soc. B 40 (1998) 77-137. [6] J. Elschner, Y. Jeon, I.H. Sloan and E.P. Stephan, The collocation method for mixed boundary value problem on domains with curved polygonal boundaries, Numer. Math. 76 (1997), 335–381. [7] P.M. Anselone, Singularity subtraction in numerical solution of integral equations, J. Austral Math. Soc. 22 (1981), pp. 408–418. [8] P.M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs, NJ. 1971. [9] P.M. Anselone and T.W. Palmer, Collectively compact sets of linear operators, Pacific J. Math. 25 (1968), pp. 417–422. [10] P.M. Anselone and M.L. Treuden, Regular operator approximation theory, Pacific J. Math. 120 (1985), pp. 257–268. [11] P. Davis, Methods of Numerical Integration Second edition, Academic Press, New York, 1984. [12] J. Huang and T. L¨ u, The mechanical quadrature methods and their extrapolation for solving BIE of Steklov eigenvalue problems, J. Comput. Math., 22 (2004), pp. 719-726. [13] F. Chatelin, Spectral approximation of linear operator, Academic Press, New York, 1983. [14] J. Huang and Z. Wang, Extrapolation algorithms for solving mixed boundary integral equations of the Helmholtz equation by mechanical quadrature methods, SIAM J. Sci. Comput. 31 (2009), pp. 4115–4129.

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[15] J. Huang, G. Zeng, X.-M. He and Z.-C. Li,Splitting extrapolation algorithm for first kind boundary integral equations with singularities by mechanical quadrature methods, Adv. Comput. Math., 36 (2012), pp. 79-97. [16] A. Sidi, Exitension of a class of periodizing variable transformations for numerical integration, Math. Comp. 75 (2005), pp. 327–343. [17] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, UK, 1997. [18] M. A. Jaswon, G. T. Symm, Integral equation methods in potential theory and elastostatics, Acaademic Press. Now York-San Francisco-London 1977. [19] N. Papamicheal, G. T. Symm, Numerical techniques for two-dimensional Laplacian problems, Comp. Math. Appl. Mech. Eng. 6 (1975). 175-194. [20] S. Pr¨ossdorf, G. Schmidt, Notwendige und hinreichende Bedingungen f¨ ur die Konvergenzdes Kollokationsverfahrens bei singul¨aren Integralgleichungen, Math. Nachr. 89 (1979), 203-215. [21] M. Costabel and E. Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximations, Math. Models and Meth. Mech. 15 (1985), 175-251. [22] W. L. Wendland, E. Stephan, G. C. Hsiao, On the integal equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. in the Appl. Sci. 1 (1979), 265-321.

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Adaptive Modified Function Projective Synchronization of Chaotic Dynamical System with Different Order M. M. El-Dessoky1,2 , Ebraheem Alzahrani1 and N. A. Almohammadi1 1

Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E-mail: [email protected]; [email protected]; [email protected]

Abstract: This work present the adaptive modified function projective synchronization of two systems with different order, which is a further extension of many existing synchronization schemes, such as function projection synchronization, modified projective synchronization and so on. Based on Lyapunov direct method of stability, an adaptive control is proposed to realize the modified function projective synchronization. Finally, numerical results are provided to illustrate the effectiveness of the obtained result.

1

Introduction

In the last few years, control and synchronization of chaos have generate much interest according to its application in secure communications [2]. Synchronization of chaotic systems means that two or more systems adjust each other to a common dynamical behavior. Up to now, many different kind of synchronization were studied such as: complete and anti synchronization, generalized synchronization, projective synchronization [9]-[39]. Recently, projective synchronization has a lot of attention because it obtain faster communication. Modifief projective synchronization is one of the important projective synchronization methods. It means that the drive and response systems could be synchronized up to constant scaling matrix [28]-[31] .Later, a new projective synchronization method called function projective synchronization where the responses of the synchronized dynamical states synchronize up to a scaling function [32]-[37]. More recently, researcher introduces a new type of synchronization phenomenon, modified function projective synchronization ,where the drive and response systems could be synchronized up to a desired scaling function matrix [38]-[39]. In recent years, most of researches for the synchronization assumed that the drive and response are identical or different systems with the same order. But in the real systems, especially in biology and social systems the synchronization is applied even though the oscillators haven’t the same order. Hence, studying the synchronization of two systems with different order plays significant role in application. 1

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The rest of this paper is as the following: The Liu chaotic and hyperchaotic dynamical systems are introduced in Section 2. Section 3 gives the definition of MFPS. In Section 4, an adaptive modified projective synchronization of Liu chaotic and hyperchaotic systems is proposed based on Lyapunov direct method of stability. Section 5 gives the numerical result and the conclusion is obtained in the last Section.

2

The Liu (chaotic and hyperchaotic) systems

The Liu hyperchaotic system is defined by:  x˙ = a(y − x),     y˙ = bx + kxz + ew,  z˙ = −cz − hx2 + mw,    w˙ = −dy,

(1)

where x, y, z and w are the state vectors, and a, b, c, d, e, k, h and m are constant parameters.It can be generate a chaotic attractor for the parameters a = 10, b = 40, c = 2.5, d = 2.5, e = 1, k = 1, h = 4, and m = 1 in Figure 1 and the chaotic motions of Liu system are illustrated in Figure 2.

Figure 1: Liu hyperchaotic system at a=10, b=40, c=2.5, d=2.5, e=1, m=1, k=1 and h=4 The Liu chaotic system is given by:   x˙ = a(y − x), y˙ = bx − kxz,   z˙ = −cz + hx2 ,

(2)

where x, y, and z are the state vectors, and the parameters a, b, c, h and k are positive real constants. A chaotic attractor for the parameters a = 10, b = 40, c = 2.5, k = 1 and h = 4 is shown in Figure 3, and the system states responses in time domain are shown in Figure 4.

3

The modified function projective synchronization scheme

We define the drive and the response systems as follows: x˙ = χ(x), y˙ = Ψ(y) + U (t, x, y), 2

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Figure 2: The behavior of the trajectories of the Liu hyper chaotic system

Figure 3: Phase portrait of Liu chaotic system at a=10, b=40, c=2.5, k=1 and h=4.

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Figure 4: The behavior of the trajectories of the Liu chaotic system. where x, y are the state variables, χ, Ψ : Rn → Rn are continuous nonlinear functions and U (t, x, y) is a control function. Let the error state be e = y − Λ(t)x where Λ(t) = diag{β1 (t), β2 (t), . . . , βn (t)} is n-order diagonal matrix where βi = ηi1 x + ηi2 , (i = 1, 2, . . . , n), η ∈ R. Definition 1. (MFPS) We say that the drive system and the response system are modified function projective synchronization (MFPS), if there is a scaling function Λ(t), such that lim kek = 0.

t→+∞

4

Modified function projective synchronization between Liu chaotic and hyperchaotic systems

Following the scheme of Zheng in [39], we apply this scheme to achieve the MFPS between Liu chaotic and hyperchaotic systems with different order. The Liu hyperchaotic system is defined below as a drive (or master) system:  x˙ 1     y˙ 1  z ˙ 1    w˙ 1

= a(y1 − x1 ), = bx1 + kx1 z1 + ew1 , = −cz1 − hx21 + mw1 , = −dy1 ,

(3)

where x1 , y1 , z1 and w1 are the state vectors. Moreover, the Liu system as the response (or slave) system is given by::   x˙ 2 = a(y2 − x2 ) + u1 , y˙ 2 = bx2 − kx2 z2 + u2 , (4)   2 z˙2 = −cz2 + hx2 + u3 , 4

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where x2 , y2 and z2 are the state vectors, and ui , (i = 1, 2, 3) are the controller to be determined later. Since the order of the drive system is greater than the response system, we must increase the order of the response system by structure a state vector. Based on the method in [39], we structure a state variable w2 = 12 x22 , then the response system become:  x˙ 2     y˙ 2  z˙2    w˙ 2

= a(y2 − x2 ) + u1 , = bx2 − kx2 z2 + u2 ,

(5)

= −cz2 + hx22 + u3 , = a(y2 − x2 )x2 + u4 .

Let the error state vector be expressed by:  e1 = x2 − (η11 x1 + η12 )x1 ,     e2 = y2 − (η21 y1 + η22 )y1 ,  e3 = z2 − (η31 z1 + η32 )z1 ,    e4 = w2 − (η41 w1 + η42 )w1 ,

(6)

Moreover, the error dynamical system can be described by:  e˙ 1 = ay2 − ax2 − 2η11 ax1 y1 + 2η11 ax21 − η21 ay1 + η12 ax1 + u1 ,    e˙ = bx − kx z − 2η bx y − 2η kx y z − 2η ey w − η bx − η kx z − η ew + u , 2 2 2 2 21 1 1 21 1 1 1 21 1 1 22 1 22 1 1 22 1 2 2 2 2 2  e ˙ = −cz + hx + 2η cz + 2η hz x − 2η mz w + η cz + η hx − η mw + u , 3 2 31 1 31 1 1 31 1 1 32 1 32 32 1 3  2 1   e˙ 4 = ay2 x2 − ax22 + 2η41 dy1 w1 + η42 y1 + u4 .

(7)

Now, the aim is to design the control function ui (t), (i = 1, 2, 3, 4) to achieve the MFPS. Consider the following Lyapunov function: V =

1 2 (e + e22 + e23 + e24 ), 2 1

which is a positive definite function, then the time derivative of the Lyapunov function is given as follows: V˙ = e1 e˙1 + e2 e˙2 + e3 e˙3 + e4 e˙4 . Moreover, V˙ = e1 (ay2 − ax2 − 2η11 ax1 y1 + 2η11 ax21 − η12 ay1 + η12 ax1 + u1 ), + e2 (bx2 − kx2 z2 − 2η21 bx1 y1 − 2η21 kx1 y1 z1 − 2η21 ey1 w1 − η22 bx1 − η22 kx1 z1 − η22 ew1 + u2 ), + e3 (−cz2 + hx22 + 2η31 cz12 + 2η31 hz1 x21 − 2η31 mz1 w1 + η32 cz1 + η32 hx21 − η32 mw1 + u3 ),

(8)

+ e4 (ay2 x2 − ax22 + 2η41 dy1 w1 + η42 y1 + u4 ).

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Thus, we choose the controller as the following:  u1 = −ay2 + 2η11 ax1 y1 − η11 ax21 + η12 ay1 ,       u2 = −bx2 + kx2 z2 + 2η21 bx1 y1 + 2η21 kx1 y1 z1 + 2η21 ey1 w1 + η22 bx1 + η22 kx1 z1 + η22 ew1 − by2 + η21 by12 + η22 by1 ,    u3 = −hx22 − η31 cz12 − 2η31 hz1 x21 + 2η31 mz1 w1 − η32 hx21 + η32 mw1 ,    u4 = −ay2 x2 + ax22 − 2η41 dy1 w1 − η42 y1 − dw2 + dη41 w12 + dη42 w1 ,

(9)

by this choice, the time derivative of Lyapunov function is: V˙ = e1 (−ax2 + η11 ax21 + η12 ax1 ) + e2 (−by2 + η21 by12 + η22 by1 ) + e3 (−cz2 + η31 cz12 + η32 cz1 ) + e4 (−dw2 + η41 dw12 + η42 dw1 ), = −(ae21 + be22 + ce23 + de24 ),

(10)

= −eT P e, where P = diag[a, b, c, d]. Obviously, the origin of the error dynamical system is asymptotically stable since V˙ is negative definite. Thus, the drive and the response systems are achieving the MFPS.

5

Numerical results

In this section, we show a numerical simulation to verify the influence of the synchronization controller (9). We assume that the initial states of the drive and the response systems are [x1 (0), y1 (0), z1 (0), w1 (0)]T = [2.4, 2.2, 0.8, 0]T and [x2 (0), y2 (0), z2 (0), w2 (0)]T = [0.2, 0.1, 3, 6]T . These numerical simulation are presented in Figure 5. Firstly, when the scaling functions are given by: β1 = 3x1 + 4, β2 = 1.5y1 + 2, β3 = 2z1 + 4 and β4 = w1 + 7, we get adaptive modified function projective synchronization (MFPS) in Figure 5 (a) . Furthermore, Figure 5 (b) shows the generalized function projective synchronization (GFPS) when the scaling functions are given by β1 = 3x1 , β2 = 1.5y1 , β3 = 2z1 , and β4 = w1 . Also, we get the modified projective synchronization (MPS) according to the constants β1 = 4, β2 = 2, β3 = 1, and β4 = 7 shown in Figure 5 (c) . The complete synchronization error of the drive and response systems are displayed in Figure 5 (d) when the scaling function is simplified to βi = +1, (i = 1, 2, 3, 4) with ηi1 = 0, ηi2 = +1, (i = 1, 2, 3, 4). Finally, if we choose the scaling function βi = −1, (i = 1, 2, 3, 4) in which ηi1 = 0, and ηi2 = −1, (i = 1, 2, 3, 4) we gained the anti-phase synchronization between the two systems in Figure 5 (e). From these results, they clearly show that the synchronization errors e = [e1 , e2 , e3 , e4 ]T are converge to zero as time goes to infinity.

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(a)

(b)

(c)

(d)

(e)

Figure 5: The errors between Liu (chaotic and hyperchaotic) systems for (a) MFPS (b) GFPS (c) MPS (d) Complete synchronization (e) Anti-phase synchronization.

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6

Conclusion

In this paper, we have introduced a modified function projective synchronization between two chaotic systems with different dimensional. The Liu chaotic system (third order) and Liu hyperchaotic system (fourth order) are chosen to illustrate the proposed technique. The results show that we can apply the MFPS between the two systems if we increased the order. By using adaptive control method, some conditions are derived for the stability of the error proved according to Lyapunov direct method of stability. Finally, the graphical presentation of the numerical results with error states tending to zero as time becomes large, clearly exhibit that the applied adaptive control method is effective and convenient to achieve global synchronization among non identical chaotic systems with different order.

Acknowledgement 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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[11] N. Rulkov, M. Sushchik, L. Tsimring and H. Abarbanel, Generalized synchronization of chaos in directionally coupled chaotic systems, Physical Review Letters, Vol.51 (2), (1995), 980- 994. [12] S. Yang and C. Duan, Generalized synchronization in chaotic systems, Chaos, Solitons and Fractals, Vol.9 (10), (1998), 1703-1707. [13] X. Yang, A framework for synchronization theory Chaos, Solitons and Fractals, Vol.11 (9), (2000), 1365-1368. [14] E. Bai and K. Lonngren, Sequential synchronization of two Lorenz system using active control, Chaos, Solitons and Fractals, Vol.11 (7), (2000), 1041-1044. [15] E. M. Elabbasy, H. N. Agiza and M. M. El-Dessoky, Global chaos synchronization for four-scroll attractor by nonlinear control, Scientific Research and Essay, Vol.1 (3), (2006), 65-71. [16] E. M. Elabbasy and M. M. El-Dessoky, Adaptive coupled synchronization of coupled chaotic dynamical systems, Applied Sciences Research, Vol.2 (2), (2007), 88-102. [17] E. M. Elabbasy and M. M. El-Dessoky, Synchronization of Van Der Pol oscillator and chen chaotic dynamical system, Chaos,Solitons and Fractals, Vol.36 (5), (2008), 1425-1435. [18] J. Huang, Adaptive synchronization between different hyperchaotic systems with fully uncertain parameters Physics Letters A, Vol.372 (27-28), (2008), 4799-4804. [19] G. Li, Generalized synchronization of chaos based on suitable separation, Chaos, Solitons and Fractals, Vol.39 (5), (2009), 2056-2062. [20] A. Loria, Master-slave synchronization of fourth order Lu chaotic oscillators via linear output feadback IEEE Transactions on circuits and systems, Vol.57 (3), (2010), 213-217. [21] M. M. El-Dessoky and M. T. Yassen, Adaptive feedback control for chaos control and synchronization for new chaotic dynamical system, Mathematical Problems in Engineering, Vol. 2012, (2012), Article ID 347210, 12 pages, doi:10.1155/2012/347210 [22] C. H. Yang and C. L. Wu, Nonlinear dynamic analysis and synchronization of four-dimensional Lorenz-Stenflo system and its circuit experimental implementation, Abstract and Applied Analysis, Vol. 2014, (2014), Article ID 213694, 17 pages. [23] K. Vishal and S. Agrawal, On the dynamics, existence of chaos, control and synchronization of a novel complex chaotic system, Chinese Journal of Physics, Vol.55 (2), (2017), 519-532. [24] J. Petereit and A. Pikovsky, Chaos synchronization by nonlinear coupling, Communications in Nonlinear Science and Numerical Simulation, Vol.44 (C), (2017), 344-351. [25] K. Ojo, S. Ogunjo and A. Olagundoye, Projective synchronization via active control of identical chaotic oscillators with parametric and external excitation, International Journal of Nonlinear Science, Vol.24 (2), (2017), 76-83. [26] M. M. El-Dessoky, Synchronization and anti-synchronization of a hyperchaotic Chen system, Chaos, Solitons and Fractals, Vol.39 (4), (2009), 1790-1797.

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Dual log-Minkowski inequality for star bodies Tongyi Ma (School of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, P.R.China)

Abstract We validate a modified dual log-Minkowski inequality and prove some variants of the dual logMinkowski inequality for star bodies in Rn containing the origin in their interior. In addition, we point out that the equivalence between the dual log-Minkowski inequality and the dual log-BrunnMinkowski inequality. Keywords: Dual cone-volume measure, L0 -Minkowski problem, dual log-Brunn-Minkowski inequality, dual log-Minkowski inequality.

1

Introduction

The classical Brunn-Minkowski theory of convex bodies was placed in a larger theory by Lutwak’s Lp -Minkowski problem [13, 14]. Therefore, many classical results for convex bodies became a part of the extended Lp -Brunn-Minkowski-Firey theory, while many other results of the extended theory bring new and original insight in convex geometric analysis. One such strikingly new behavior is due to the log-Brunn-Minkowski inequality [2]. That is, let K, L be convex bodies that contain the origin in their interiors and 0 ≤ λ ≤ 1, the log-Minkowski combination which is defined by (1 − λ) · K +0 λ · L = ∩u∈Sn−1 {x ∈ Rn : x · u ≤ hK (u)1−λ hL (u)λ },

(1.1)

where, x · u denotes the standard inner product of x and u in Rn , Sn−1 denotes the unit sphere in Rn and hK denotes the support function of convex body. B¨or¨oczky, Lutwak, Yang and Zhang [2] conjectured that for origin-symmetric convex bodies K and L in Rn with 0 ≤ λ ≤ 1, voln ((1 − λ) · K +0 λ · L) ≥ voln (K)1−λ voln (L)λ ,

(1.2)

where voln (·) denotes the n-dimensional volume of body in Rn . They call (1.2) as the log-BrunnMinkowski inequality. Note that while the inequality (1.2) is not true for general convex bodies, it implies the classical Brunn-Minkowski inequality for origin-symmetric convex bodies. In [2], B¨or¨oczky, et al. proved the inequality (1.2) when n = 2 and showed that (1.2) is equivalent to the logarithmic Minkowski inequality (log-Minkowski inequality) for all n, that is ( ( ) ) ∫ hK (u) 1 voln (K) ln d¯ vL (u) ≥ ln , (1.3) hL (u) n voln (L) Sn−1 AMS Subject Classification: 52A30, 52A40. This work was Supported by the National Natural Science Foundation of China(Grant No.11561020) and was partly supported by the National Natural Science Foundation of China (Grant No.11371224). Email: [email protected], matongyi− [email protected] (Tongyi Ma).

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where dvL (u) = n1 hL (u)dSL (u) is the cone-volume measure of L, d¯ vL (u) = voln1(L) dvL (u) and SL is surface area measure of L on Sn−1 . In [23], Stancu proved some variants of the log-Minkowski inequality for general convex bodies without the symmetry assumption. The dual Lp -Brunn-Minkowski theory for star bodies developed by Lutwak [15, 16] and received considerable attention, see [1, 4, 6, 10, 11, 17, 20, 21, 22, 25]. Recently, Gardner, et al. [7] established dual log-Minkowski inequality as follows. If K and L be star bodies in Rn containing the origin in their interior, then ( ) ( ) ∫ ρK (u) 1 voln (K) ln de v K (u) ≤ ln , (1.4) ρL (u) n voln (L) Sn−1 v K is the dual cone-volume probability measure with equality if and only if K and L are dilatates, where de of K (see definition (2.11)). In the present paper, we prove a modified dual log-Minkowski inequality and obtain the double dual log-Minkowski inequality through the Gibbs’ inequality. Secondly, we prove an analogue of the dual log-Minkowski inequality. In addition, we point out the equivalence between the dual log-Minkowski inequality and the dual log-Brunn-Minkowski inequality. Our first result is the following dual log-Minkowski inequality: Theorem 1.1. Let K and L be star bodies in Rn containing the origin in their interior. Then (

∫ ln Sn−1

(e ) ) ( ) V−1 (K, L) ρK (u) 1 voln (K) de v −1 (K, L; u) ≥ ln ≥ ln , ρL (u) voln (K) n voln (L)

(1.5)

with equality if and only if K and L are dilates, where de v−1 (K, L; ·) is the dual mixed volume measure ∫ 1 e V−1 (K, L) = n−1 de v−1 (K, L; u) and de v −1 (K, L; u) = de v−1 (K, L; u). e−1 (K,L) V

S

Secondly, we obtain the following double log-Minkowski inequality. Theorem 1.2. Let K and L be star bodies in Rn containing the origin in their interior. Then (

∫ ln Sn−1

(e ( ) ) ∫ ) V−1 (K, L) ρK (u) ρK (u) v K (u) ≤ ln ln v −1 (K, L; u), de ≤ de ρL (u) voln (K) ρL (u) Sn−1

(1.6)

with equality in inequality if and only if K and L are dilates. Further, we prove an analogue of the dual log-Minkowski inequality. In what follows, we will denote ∫ ( ) ρK (u) de vK (u) ρK Sn−1 ρL (u) ∫ , = ρL average de vK (u) Sn−1 (

ρK ρL

)

(

max

= max n−1 u∈S

ρK (u) ρL (u)

)

( and

ρK ρL

)

( min

= min n−1 u∈S

) ρK (u) . ρL (u)

Theorem 1.3. Let K and L be star bodies in Rn containing the origin in their interior with L ⊆ K. Then (

∫ ln Sn−1

(

)

ρK ρL

)

1 ρK (u) average v K (u) ≥ ( ) · ln de ρK ρL (u) n ρL

(

) voln (K) , voln (L)

(1.7)

max

with equality if and only if K = λL, where 0 < λ ≤ 1.

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In general, if K, L ∈ Son , then

( ) ρK ) ( ) ρL 1 ρK (u) voln (K) average v K (u) ≥ ( ) · ln ln de ρK ρL (u) n voln (L) Sn−1 ρL max ( ) ] ρK [( ) ] [ ρL ρK average , + ln · 1− ( ) ρK ρL min

∫

(

ρL

(1.8)

max

with equality if and only if K is homothetic to L. Finally, we point out the equivalence between the dual log-Minkowski inequality (1.4) and the dual log-Brunn-Minkowski inequality (2.8). We give a different proof with Wang and Liu [24].

2

Notation and preliminaries The support function hK : Rn → R, of a compact, convex set K ⊂ Rn is defined, for x ∈ Rn , by hK (x) = max(x · y : y ∈ K),

(2.1)

and uniquely determines the convex set. Let Kon be the set of convex bodies in Rn containing the origin in their interior. If L is a compact star-shaped (about the origin) in Rn , its radial function, ρL = ρ(L, ·) : Rn \{o} → [0, +∞), is defined by ρK (x) = max{λ ≥ 0 : λx ∈ L}, x ∈ Rn \{o}.

(2.2)

If ρL is positive and continuous, then L will be called a star body (about the origin). Let Son denotes the set of star bodies in Rn containing the origin in their interior. Two star bodies K and L are said to be dilates (of one another) if ρK (u)/ρL (u) is independent of u ∈ Sn−1 . Obviously, for a pair K, L ∈ Son , we have ρK ≤ ρL , if and only if, K ⊆ L.

(2.3)

b pµ ⋄ If K, L ∈ Son and λ, µ ≥ 0 (not both zero), then, for p ≥ 1, the harmonic Lp -combination, λ ⋄ K + L ∈ Son is defined by (see [14]) b p µ ⋄ L, ·)−p = λρ(K, ·)−p + µρ(L, ·)−p . ρ(λ ⋄ K +

(2.4)

For p ≥ 1 and K, L ∈ Son , the dual mixed volume, Ve−p (K, L), is defined b p ε ⋄ L) − voln (K) n voln (K + − Ve−p (K, L) = lim . ε→0 p ε The following integral representation for the dual mixed volume Ve−p is obtained (see [14]): If p ≥ 1 and K, L ∈ Son , then ∫ 1 e V−p (K, L) = ρ(K, u)n+p ρ(L, u)−p dS(u), n Sn−1 where dS is the spherical Lebesgue measure on Sn−1 . This integral representation, together the H¨older inequality with the polar coordinate formula, immediately gives the dual Lp -Minkowski inequality: If p ≥ 1 and K, L ∈ Son , then Ve−p (K, L)n ≥ voln (K)n+p voln (L)−p ,

(2.5)

with equality if and only if K and L are dilates. 3 366

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Using the dual Lp -Minkowski inequality, we can obtain the following dual Lp -Brunn-Minkowski inequality (see [14]). Suppose K, L ∈ Son , λ, µ > 0 and p ≥ 1, then b p µ ⋄ L)−p/n ≥ λvoln (K)−p/n + µvoln (L)−p/n , voln (λ ⋄ K +

(2.6)

with equality if and only if K and L are dilates. Note that definition (2.4) makes sense for all p > 0. The case p = 0 is the limiting case given by ˆ 0 λ ⋄ L, ·) = ρ(K, ·)1−λ ρ(L, ·)λ , 0 ≤ λ ≤ 1, ρ((1 − λ) ⋄ K +

(2.7)

it is called the radial log-Minkowski-combination. Similarly, the inequality (2.6) makes sense for all p > 0. The case p = 0 is the limiting case given by an dual log-Brunn-Minkowski inequality. Namely, if K, L ∈ Son , then for all λ ∈ [0, 1], ˆ 0 λ ⋄ L) ≤ voln (K)1−λ voln (L)λ , voln ((1 − λ) ⋄ K +

(2.8)

with equality if and only if K and L are dilates. If K ∈ Son , then de vK (u) =

1 n ρ (u)dS(u) n K

(2.9)

is the dual cone-volume measure of K and de v−1 (K, L; u) =

1 n+1 ρ (u)ρ−1 L (u)dS(u) n K

(2.10)

is the dual mixed volume measure with (n + 1) copies of K and (−1) copies of L. Note that we usually ∫ write Ve−1 (K, L) = Sn−1 de v−1 (K, L; u). The dual cone-volume measure of a star body K in Rn with voln (K) is the Borel probability measure veK in Sn−1 defined by ρnK (u) dS(u). nvoln (K)

de vK =

(2.11)

And the normalized dual mixed cone measure of a star bodies K, L in Rn with Ve−1 (K, L) is the Borel probability measure ve−1 (K, L; ·) on Sn−1 defined by de v −1 (K, L; u) =

3

1 Ve−1 (K, L)

de v−1 (K, L; u).

(2.12)

Proofs of dual log-Minkowski type results In this section, we will prove the theorems mentioned in Section 1. Proof of Theorem 1.1.

Consider the function GK,L (p) : [1, ∞] → R defined by

GK,L (p) =

(

∫

1 Ve−1 (K, L)

Sn−1

ρK (u) ρL (u)

p ) n+p

de vK (u).

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Through using L’Hˆopital’s rule, we obtain (

(∫ lim ln(GK,L (p))

n+p

p→∞

Sn−1

= lim ln

ρK (u) ρL (u)

p ) n+p

de vK (u)

)n+p

Ve−1 (K, L)

p→∞

n ( )− n+p ) (u) ρK (u)n+1 ρL (u)−1 ρρK dS(u) n+p L (u) ∫ = lim ln p→∞ ρ (u)n+1 ρL (u)−1 dS(u) Sn−1 K ( )−n (∫ ) ρK (u) n+1 −1 ρ (u) ρ (u) ln dS(u) K L n−1 ρL (u) S ∫ = ln exp ρ (u)n+1 ρL (u)−1 dS(u) Sn−1 K )−n ( ∫ ρK (u) n+1 −1 dS(u) ρ (u) ρ (u) ln K L n−1 ρL (u) S ∫ = n+1 −1 ρ (u) ρL (u) dS(u) Sn−1 K ( ) ∫ ρK (u) ρK (u) n ln de vK (u). =− ρL (u) Ve−1 (K, L) Sn−1 ρL (u)

(∫

Sn−1

Thus, we have [

( ) ] ρK (u) ρK (u) exp − ln de vK (u) ρL (u) Ve−1 (K, L) Sn−1 ρL (u) ]p+n ( ) p [ ∫ ρK (u) p+n 1 de vK (u) , = lim p→∞ V e−1 (K, L) Sn−1 ρL (u) n

∫

(3.1)

and it follows from H¨older’s inequality that ) p ) p+n (∫ )− np p ρK (u) p+n de vK (u) de vK (u) ρL (u) Sn−1 Sn−1 ∫ ρK (u) ≤ de vK (u) = Ve−1 (K, L). Sn−1 ρL (u) (∫

Note that

∫ Sn−1

(

(3.2)

de vK (u) = voln (K), (2.10) and (2.12), together (3.1) with (3.2), we have (

∫ ln Sn−1

) (e ) ρK (u) V−1 (K, L) de v −1 (K, L; u) ≥ ln . ρL (u) voln (K)

According to the condition of equality in H¨older’s inequality, we easily see that with equality in the above inequality if and only if K and L are dilates. Using dual Minkowski’s inequality (2.5), we have the second inequality in the theorem, this is, ( ) ) ( ) (e ∫ ρK (u) V−1 (K, L) 1 voln (K) ln . (3.3) de v −1 (K, L; u) ≥ ln ≥ ln ρL (u) voln (K) n voln (L) Sn−1 From the condition of equality in dual Minkowski’s inequality, we know that with equality if and only if K and L are dilates. Which completes the proof of the theorem.  Remark 3.1. Our first inequality in (1.5) can be written as ( ) (e ) ∫ ρK (u) ρK (u) V−1 (K, L) Ve−1 (K, L) ln ln de v K (u) ≥ . ρL (u) voln (K) voln (K) Sn−1 ρL (u)

(3.4)

Use dual Minkowski’s inequality in (3.4), we have ( ) ( )1/n ( ) ∫ ρK (u) ρK (u) 1 voln (K) voln (K) ln de v K (u) ≥ ln . ρL (u) n voln (L) voln (L) Sn−1 ρL (u)

(3.5)

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Proof of Theorem 1.2. We consider Gibbs’ inequality from information theory (see [3], (8.57), p. 252-253): If p and q are probability density functions on a measure space (X, ν), then ∫ ∫ p ln pdν ≥ p ln qdν, (3.6) with equality if and only if p = q almost everywhere (a.e.). By taking pdν =

1 ρL (u) 1 · de v−1 (K, L; u) and qdν = de v−1 (K, L; u) e ρK (u) voln (K) V−1 (K, L)

(and later reversing the two measures above so that the first is qdν and the second is pdν), we obtain the double inequality as follows. (

∫ ln Sn−1

(e ( ) ) ∫ ) ρK (u) V−1 (K, L) ρK (u) v K (u) ≤ ln ln v −1 (K, L; u). de ≤ de ρL (u) voln (K) ρL (u) Sn−1

(3.7)

According to the condition of equality in Gibbs’ inequality (3.6), we obtain that with equality in inequality (3.7) if and only if )n ( ρL (u) voln (K) 1 n 1 voln (K) ρnK (u) = ⇐⇒ ρL (u) = ρK (u) n n Ve−1 (K, L) Ve−1 (K, L) almost everywhere (a.e.) on Sn−1 . Integrating both sides of the last equation over Sn−1 with the sphere Lebesgue measure dS(u), we get ( )n voln (L) voln (K) = . voln (K) Ve−1 (K, L) From the condition of equality in the dual Lp -Minkowski inequality (2.5) (p = 1), we see that with equality in inequality (3.7) if and only if K and L are dilates.  Remark3.2. The proof of Theorem 1.2 can be seen that we provide a new proof for the dual Minkowski inequality itself. In fact, it is consistent with the idea of splitting mentioned by Gardner, Hug and Weil in [8] and [9]. A natural idea is to give a proof of the dual log-Minkowski inequality similar to the proof of the Theorem 1.1. However, as such, we obtain again the left-hand side inequality of (1.6) due to the following lemma: Lemma 3.3. Let K, L ∈ Son , then (∫ exp

) ρK (u) de v K (u) ρL (u) Sn−1 ( ( ) 1 )p+n ∫ 1 ρK (u) p+n = lim de vK (u) . p→∞ voln (K) Sn−1 ρL (u) ln

(3.8)

The proof follows the same idea used in deriving (3.1). From H¨older’s inequality, we have (∫

(

) 1 )p+n ( ∫ )1−(p+n) ρK (u) p+n de vK (u) de vK (u) ρL (u) Sn−1 Sn−1 ∫ ρK (u) ≤ de vK (u) = Ve−1 (K, L). Sn−1 ρL (u)

(3.9)

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Lemma 3.3, together with (3.9), implies that ) ) ( (e ∫ ρK (u) V−1 (K, L) de . ln v K (u) ≤ ln ρL (u) voln (K) Sn−1 It will be convenient to invoke the logarithmic mean L(x, y) of two positive numbers x, y, which is given by { x−y for x ̸= y lnx−ln y , L(x, y) = (3.10) x, for x = y. To prove Theorem 1.3, the following Hadamard type inequality for positive log-convex functions will be used [12]. Lemma 3.4. Let f be a positive, integrable, log-convex function on [a, b]. Then ∫ b 1 f (t)dt ≤ L (f (a), f (b)) . b−a a

(3.11)

Suppose f has two derivative. The equality holds in the inequality (3.11) if and only if f (t) = c almost everywhere (a.e.) or

f ′ (t) f (t)

= c almost everywhere (a.e.), where c is the constant.

The condition of the equality holds in the inequality (3.11) is that we supplements. Indeed, since f is ′ (t) log-convex function on [a, b], and then f (t) and ff (t) are monotonically increasing at the same time. So, we have ∫b ′ f (x)dx f (b) − f (a) L(f (a), f (b)) = = ∫af (b) 1 ln f (b) − ln f (b) dx f (a) x ∫b ′ ∫ b 1 x=f(t) a f (x)dx ≥ = ∫ b f ′ (t) f (t)dx b−a a dt a f (t) ∫b f (t)dt . (3.12) = a∫ b 1dt a Thus, the inequality (3.11) is transformed into ∫ b ∫ b ∫ f ′ (x)dx 1dx ≥ a

a

∫

b

f (t)dt a

a

b

f ′ (t) dt. f (t)

(3.13)

′

(t) Note that f (t) and ff (t) are monotonically increasing at the same time, According to the condition of equality in Chebyshev’s inequality, we see with equality in inequality (3.11) if and only if f (t) = c or f ′ (t) ct  f (t) = c. Namely, f (t) = c or f (t) = e .

Consider the case L ⊆ K and the function )q ( ) ( ∫ ρK (u) ρK (u) ln de vK (u), q ∈ R. F (q) : q 7→ ρL (u) ρL (u) Sn−1 ) ( (u) Apparently, F (q) is non-negative. If u 7→ ln ρρK is zero on Sn−1 , then F (q) is identically zero. Now, L (u) Proof of Theorem 1.3.

we assume that this is not the case, which also implies F (1) ≥ F (0) > 0. If F (1) = F (0), the conclusion is trivial (as using (3.7), K must be equal to L), and then, we assume F (1) > F (0). d2 A simple verification shows that F (q) is a log-convex function, this is because dq 2 ln F (q) ≥ 0. By employing Hadamard type inequality (3.11) for positive log-convex functions [12], we have that ( )q ( ) ] ∫ 1 [∫ F (1) − F (0) ρK (u) ρK (u) ( )≥ ln de vK (u) dq. (3.14) ρL (u) ρL (u) ln F (1)/F (0) 0 Sn−1 7 370

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Using Fubini-Tonelli’s theorem, the following inequality [

F (1) − F (0) ( ) F (0) ≥ F (1) · exp − ∫ ρK (u) − 1 de vK (u) Sn−1 ρL (u)

is true. Note that

] (3.15)

( ) ( ) ρK (u) ρK (u) ln · · − 1 de vK (u) ρL (u) ρL (u) F (1) − F (0) ( ) ( ) = ∫ ∫ ρK (u) ρK (u) − 1 de vK (u) − 1 de vK (u) Sn−1 ρL (u) Sn−1 ρL (u) ( ) ρK ≤ ln , ρL max ∫

Sn−1

then combining (3.15) and (3.16), we have ( ) ∫ ρK (u) ln de v K (u) ρL (u) Sn−1 [ ( ) ] e ( ) ∫ ρK V−1 (K, L) ρK (u) ≥ exp − ln v −1 (K, L; u), · ln de ρL max voln (K) Sn−1 ρL (u)

(3.16)

(3.17)

it follows from (3.3) that (

∫ ln Sn−1

(

)

ρK ρL

)

ρK average 1 de ln v K (u) ≥ ( ) ρ ρL n K ρL

(

) voln (K) . voln (L)

max

Now we discuss the conditions of equality in inequality (1.7), and the discussion is split into two cases. Assuming that F (q) is identically zero, then ρK (u) = ρL (u) for all u’s with respect to Sn−1 if and only if K = L. Case 1. According to the conditions of equality in Hadamard type inequality (3.11) and inequality (3.3), we see with equality in inequality (1.7) if and only if { F (q) = c for q ≥ 1, (3.18) K = λL for λ > 0. From the definition of function F (q), (3.18) is equivalent to { ρK (u) n−1 , ρL (u) = 1 for any u ∈ S K = λL for λ > 0.

(3.19)

Namely, K = L. Case 2. According to the conditions of equality in Hadamard type inequality (3.11) and inequality (3.3), we see with equality in inequality (1.7) if and only if { ′ F (q) F (q) = c for q ≥ 1, (3.20) K = λL for λ > 0. Using mean value theorem for multiple integral [5, 19], there is a u0 ∈ Sn−1 , such that (3.18) is equivalent to { ρ (u ) 0 K n−1 , ρL (u0 ) = cq for a u0 ∈ S (3.21) K = λL for λ > 0. Since L ⊆ K, K = λL with 0 < λ ≤ 1. As mentioned above, we see with equality in inequality (1.7) if and only if K = λL with 0 < λ ≤ 1. 8 371

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Assume now that K and L are arbitrary star bodies. If L is not included in K, there exists a ˜ := λL ⊆ K. By using (1.7) for L ˜ and K. Thus, λ, 0 < λ < 1, such that L ( ) ρK ( ) ( ) ∫ ρL 1 voln (K) ρK (u) average de · ln (3.22) ln v K (u) − ln λ ≥ ( ) ρK ρL (u) n λn voln (L) Sn−1 ρL

or (

∫ ln Sn−1

( ) ρK ) ( ) ρL ρK (u) 1 voln (K) average de · ln v K (u) ≥ ( ) ρK ρL (u) n voln (L) ρL max ( ) ( ) ρK + ln λ ·

( Taking λ = minu∈Sn−1

max

ρL

1− (

) ρK (u) ρL (u)

ρK ρL

average ) .

(3.23)

max

will suffice, we now obtain the second inequality.

The claim that the homothety of K and L is the only case of equality follows from the first part.  ( ) ( ) ( ) ρK ρK , and Remark3.5. Note that, if L ⊆ K then (1.8) implies (1.7). Also, ρρK ρL ρL L average max min ( ) ρK (u) depend only on the values of the ratio ρL (u) on Sn−1 . We conclude this paper by pointing out that the equivalence between inequalities (1.4) and (2.8). We give a different proof with Wang and Liu [24]. For any K ∈ Son , define the real numbers RK and rK by RK = max ρK (u), n−1

rK = min ρK (u). n−1

u∈S

(3.24)

u∈S

Note that the definition of Son is such that 0 < rK ≤ RK < ∞, for all K ∈ Son . Theorem 3.6. For K, L ∈ Son , the dual log-Brunn-Minkowski inequality (2.8) and the dual logMinkowski inequality (1.4) are equivalent. Proof. Suppose that K and L are fixed star bodies in Son . For 0 ≤ λ ≤ 1, let ˆ 0 λ ⋄ L, Qλ = (1 − λ) ⋄ K + λ i.e., the radial function of star body Qλ is qλ := ρQλ = ρ1−λ K ρL . Since q0 and q1 are the radial functions of star bodies, we have Q0 = K and Q1 = L. λ Suppose that we have the dual log-Minkowski inequality (1.4) for K and L. Now ρQλ = ρ1−λ K ρL a.e. n−1 with respect to S , and thus ∫ ρK (u)1−λ ρL (u)λ 1 ρQλ (u)n ln dS(u) 0= nvoln (Qλ ) Sn−1 ρQλ (u) ∫ 1 ρK (u) ρQ (u)n ln = (1 − λ) dS(u) nvoln (Qλ ) Sn−1 λ ρQλ (u) ∫ 1 ρL (u) +λ ρQ (u)n ln dS(u) nvoln (Qλ ) Sn−1 λ ρQλ (u) ∫ ∫ ρQ (u) ρQ (u) ln λ = −(1 − λ) de v Qλ (u) − λ ln λ de v Qλ (u) ρ (u) ρL (u) n−1 n−1 K S S 1 voln (Qλ ) 1 voln (Qλ ) ≤ −(1 − λ) ln − λ ln n voln (K) n voln (L) 1−λ λ 1 voln (K) voln (L) = ln . (3.25) n voln (Qλ )

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This gives the dual log-Brunn-Minkowski inequality (2.8). Suppose now that we have the dual log-Brunn-Minkowski inequality (2.8) for K and L. Namely, ( )λ voln (L) ˆ voln ((1 − λ) ⋄ K +0 λ ⋄ L) ≤ voln (K) . voln (K)

(3.26)

Using the polar coordinates formula of volume, the radial log-Minkowski-combination (2.7) and the Borel probability measure (2.11), it follows from (3.26) that (

∫ Sn−1

ρL (u) ρK (u)

(

)nλ

)nλ

( v K (u) ≤ de

voln (L) voln (K)

)λ .

(3.27)

Therefore ∫ Sn−1

ρL (u) ρK (u)

de v K (u) − 1

λ

( ≤

voln (L) voln (K)

)λ

−1

λ

.

Taking the limit on both sides of the last inequality as λ → 0, we get (

∫ lim

Sn−1

)nλ

ρL (u) ρK (u)

v K (u) − 1 de

λ

λ→0

( ≤ lim

voln (L) voln (K)

)λ

−1

λ

λ→0

.

We are easy to prove the function f (x) = a x−1 is uniformly continuous on (0, ∞) for 0 < a ≤ 1, and x the Bernoulli’s inequality leads to the function f (x) = a x−1 is uniform boundness for a > 1. From the definition (3.24), we have x

(

ρL ρK

)nλ

−1

λ

( ≤

RL rK

)nλ

−1

λ

.

Using Lebesgue dominated convergence theorem we know that the order of the integral and the limit can be changed. Therefore, we can obtain (

∫ lim

ρL (u) ρK (u)

ax −1 x

−1

λ

Sn−1 λ→0

Since limx→0

)nλ

( de v K (u) ≤ lim

λ→0

voln (L) voln (K)

)λ

λ

−1

.

= ln a, then (

∫ ln Sn−1

ρL (u) ρK (u)

)n

( de v K (u) ≤ ln

) voln (L) . voln (K)

(3.28)

This is the dual log-Minkowski inequality (1.4), which completes the proof. Acknowledgment The authors are deeply indebted to Professor Li Yin, Professor Xiaoming Zhang, Doctor Xuezhong Wang, Doctor Denghui Wu and Doctor Yibin Feng for many invaluable suggestions in preparing the manuscript. The authors also would like to thank the referee for the many suggested improvements.

References [1] A. Bernig, The isoperimetrix in the dual Brunn-Minkowski theory, Adv. Math. 254 (2014) 1-14. [2] K. J. B¨ or¨ oczky, E. Lutwak, D. Yang, and G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012) 1974-1997. [3] T. M. Cover, J. A. Thomas, Elements of Information Theory, second edition, Wiley-Interscience, Hoboken, NJ, 2006.

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[4] P. Dulio, R.J. Gardner and C. Peri, Characterizing the dual mixed volume via additive functionals, arXiv preprint arXiv: 1312.4072, 2013. [5] J. Fan, B. Yang, The mean value theorem for multiple integral, Mathematics in Practice and Theory 37 (12) (2007) 197-200 (in Chinese). [6] R. J. Gardner, D. Hug, W. Weil, The Orlicz-Brunn-Minkowski theory: a general framework, addi-tions, and inequalities, J. Differential Geom. 97 (2014) 427-476. [7] R. J. Gardner, D. Hug, W. Weil, et al., The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl. 430(2)(2015) 810-829. [8] R. J. Gardner, The dual Brunn-Minkowski theory for bounded borel sets: Dual affine quermassintegrals and inequalities, Adv. Math. 216 (2007) 358-386. [9] R. J. Gardner and S. Vassallo, Inequalities for dual isoperimetric deficits, Mathematika. 45 (1998) 269-285. [10] R. J. Gardner and S. Vassallo, Stability of inequalities in the dual Brunn-Minkowski theory, J. Math. Anal. Appl. 231 (1999) 568-587. [11] R. J. Gardner and S. Vassallo, The Brunn-Minkowski inequality, Minkowskis first inequality, and their duals’, J. Math. Anal. Appl. 245 (2000) 502-512. [12] P. M. Gill, C. E. M. Pearce, J. Pe˘cari´c Hadamard’s inequality for r-convex functions, J. Math. Anal. Appl. 215 (1997) 461-470. [13] E. Lutwak, The Brunn-Minkowski-Firey theory. I: mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993) 131-150. [14] E. Lutwak, The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas, Adv. Math. 118 (1996) 244-294. [15] E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975) 531-538. [16] E. Lutwak, Intersection bodies and dual mixed volume, Adv. Math. 71 (1988) 232-261. [17] E. Lutwak, Centered bodies and dual mixed volumes, Proc. London. Math. Soc. 60 (1990) 365-391. [18] E. Lutwak, D. Yang, G. Zhang, Orlicz centroid bodies, J. Differential Geom. 84 (2010), 365-387. [19] H. Liu, The inverse proposition about mid-value theorem of multiple integral, Journal of Huanggang teachers college (Natural Science Edition). 17(1)(1997) 39-42 (in Chinese). [20] T. Ma, The minimal dual Orlicz surface area, Taiwanese Journal of Mathematics 20(2)(2016) 287-309. [21] T. Ma, W. Wang, Dual Orlicz geominimal surface area, Journal of Inequalities and Applications 2016(1)(2016) 1-13. [22] E. Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006) 566-598. [23] A. Stancu, The logarithmic Minkowski inequality fornon-symmetric convex bodies, Adv. Appl. Math. 73(2016) 43-58. [24] W. Wang, L. Liu, The dual Log-Brunn-Minkowski inequalities, Taiwanese Jouranl of Mathematics 20(4)(2016), 909-919. [25] G. Zhang, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345 (1994) 777-801.

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Subalgebra and ideal-type hyper values in BCK/BCI-algebras Young Bae Jun1 and Sun Shin Ahn2,∗ 1

Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea 2

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

Abstract. The notions of subalgebra-type hyper value and ideal-type hyper value are introduced, and related properties are investigated. The relation between subalgebra-type hyper value and ideal-type hyper value is considered. Conditions for a pair (α, β) in [0, 1] × [0, 1] to be subalgebra-type hyper value and ideal-type hyper value are discussed. For a hyperfuzzy structure, conditions for its level sets to be S-energetic, I-energetic, right vanished and right stable are founded. 1. Introduction Jun et al. [3] introduced the notion of energetic (resp. right vanish, right stable) subsets in BCK/BCI-algebras, and investigated several related properties. Ghosh et al. [1] introduced the concept of hyperfuzzy sets which is a generalization of fuzzy sets and interval-valued fuzzy sets. Jun et al. [4] and Song et al. [6] applied hyper structure to BCK/BCI-algebras, and discussed hyperfuzzy subalgebras and hyperfuzzy ideals in BCK/BCI-algebras. In this article, we introduce the concepts of subalgebra-type hyper value and ideal-type hyper value, and investigate several properties. We discuss the relation between subalgebra-type hyper value and ideal-type hyper value. We provide an example to show that any subalgebra-type hyper value is not an ideal-type hyper value. We consider conditions for a pair (α, β) in [0, 1] × [0, 1] to be subalgebra-type hyper value and ideal-type hyper value. Given a hyperfuzzy structure, we find conditions for its level sets to be S-energetic, I-energetic, right vanished and right stable. 2. Preliminaries By a BCI-algebra we mean a system X := (X, ∗, 0) in which the following axioms hold: (I) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (II) (x ∗ (x ∗ y)) ∗ y = 0, (III) x ∗ x = 0, (IV) x ∗ y = y ∗ x = 0 ⇒ x = y for all x, y, z ∈ X. If a BCI-algebra X satisfies 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra. We can define a partial ordering ≤ by (∀x, y ∈ X) (x ≤ y ⇐⇒ x ∗ y = 0). 0

2010 Mathematics Subject Classification: 08F35; 03G25; 03B52. Keywords: hyperfuzzy subalgebra; hyperfuzzy ideal; subalgebra-type hyper value; ideal-type hyper value; energetic subset; right vanished subset; right stable subset. ∗ Correspondence: Tel: +82 2 2260 3410, Fax: +82 2 2266 3409 (S. S. Ahn). 0 E-mail: [email protected] (Y. B. Jun); [email protected] (S. S. Ahn). 0

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Y. B. Jun and S. S. Ahn In a BCK/BCI-algebra X, the following hold: (∀x ∈ X) (x ∗ 0 = x),

(2.1)

(∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y).

(2.2)

A non-empty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. A subset I of a BCK/BCI-algebra X is called an ideal of X if 0 ∈ I,

(2.3)

(∀x ∈ X)(∀y ∈ I) (x ∗ y ∈ I ⇒ x ∈ I) .

(2.4)

We refer the reader to the books [2] and [5] for further information regarding BCK/BCI-algebras. By a fuzzy structure over a nonempty set X we mean an ordered pair (X, ρ) of X and a fuzzy set ρ on X. ˜ Let X be a nonempty set. A mapping µ ˜ : X → P([0, 1]) is called a hyperfuzzy set over X (see [1]), where ˜ P([0, 1]) is the family of all nonempty subsets of [0, 1]. An ordered pair (X, µ ˜) is called a hyper structure over X. Given a hyper structure (X, µ ˜) over a nonempty set X, we consider two fuzzy structures (X, µ ˜inf ) and (X, µ ˜sup ) over X in which µ ˜inf : X → [0, 1], x 7→ inf{˜ µ(x)}, µ ˜sup : X → [0, 1], x 7→ sup{˜ µ(x)}. Given a nonempty set X, let BK (X) and BI (X) denote the collection of all BCK-algebras and all BCI-algebras, respectively. Also B(X) := BK (X) ∪ BI (X). In what follows, let (X, ∗, 0) ∈ B(X) unless otherwise specified. Definition 2.1 ([4]). For any (X, ∗, 0) ∈ B(X), a fuzzy structure (X, µ) over (X, ∗, 0) is called a • fuzzy subalgebra of (X, ∗, 0) with type 1 (briefly, 1-fuzzy subalgebra of (X, ∗, 0)) if (∀x, y ∈ X) (µ(x ∗ y) ≥ min{µ(x), µ(y)}) ,

(2.5)

• fuzzy subalgebra of (X, ∗, 0) with type 2 (briefly, 2-fuzzy subalgebra of (X, ∗, 0)) if (∀x, y ∈ X) (µ(x ∗ y) ≤ min{µ(x), µ(y)}) ,

(2.6)

• fuzzy subalgebra of (X, ∗, 0) with type 3 (briefly, 3-fuzzy subalgebra of (X, ∗, 0)) if (∀x, y ∈ X) (µ(x ∗ y) ≥ max{µ(x), µ(y)}) ,

(2.7)

• fuzzy subalgebra of (X, ∗, 0) with type 4 (briefly, 4-fuzzy subalgebra of (X, ∗, 0)) if (∀x, y ∈ X) (µ(x ∗ y) ≤ max{µ(x), µ(y)}) .

(2.8)

It is clear that every 3-fuzzy subalgebra is a 1-fuzzy subalgebra and every 2-fuzzy subalgebra is a 4-fuzzy subalgebra. Definition 2.2 ([4]). For any (X, ∗, 0) ∈ B(X) and i, j ∈ {1, 2, 3, 4}, a hyper structure (X, µ ˜) over (X, ∗, 0) is called an (i, j)-hyperfuzzy subalgebra of (X, ∗, 0) if (X, µ ˜inf ) is an i-fuzzy subalgebra of (X, ∗, 0) and (X, µ ˜sup ) is a j-fuzzy subalgebra of (X, ∗, 0).

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Subalgebra and ideal-type hyper values in BCK/BCI-algebras Given a hyper structure (X, µ ˜) over X and α, β ∈ [0, 1], we consider the following sets (see [6]): U (˜ µinf ; α) := {x ∈ X | µ ˜inf (x) ≥ α}, L(˜ µinf ; α) := {x ∈ X | µ ˜inf (x) ≤ α}, U (˜ µsup ; β) := {x ∈ X | µ ˜sup (x) ≥ β}, L(˜ µsup ; β) := {x ∈ X | µ ˜sup (x) ≤ β}. Definition 2.3 ([6]). A fuzzy structure (X, µ) over (X, ∗, 0) is called a • fuzzy ideal of (X, ∗, 0) with type 1 (briefly, 1-fuzzy ideal of (X, ∗, 0)) if (∀x ∈ X) (µ(0) ≥ µ(x)) ,

(2.9)

(∀x, y ∈ X) (µ(x) ≥ min{µ(x ∗ y), µ(y)}) ,

(2.10)

• fuzzy ideal of (X, ∗, 0) with type 2 (briefly, 2-fuzzy ideal of (X, ∗, 0)) if (∀x ∈ X) (µ(0) ≤ µ(x)) ,

(2.11)

(∀x, y ∈ X) (µ(x) ≤ min{µ(x ∗ y), µ(y)}) ,

(2.12)

• fuzzy ideal of (X, ∗, 0) with type 3 (briefly, 3-fuzzy ideal of (X, ∗, 0)) if it satisfies (2.9) and (∀x, y ∈ X) (µ(x) ≥ max{µ(x ∗ y), µ(y)}) ,

(2.13)

• fuzzy ideal of (X, ∗, 0) with type 4 (briefly, 4-fuzzy ideal of (X, ∗, 0)) if it satisfies (2.11) and (∀x, y ∈ X) (µ(x) ≤ max{µ(x ∗ y), µ(y)}) .

(2.14)

It is clear that every 3-fuzzy ideal is a 1-fuzzy ideal and every 2-fuzzy ideal is a 4-fuzzy ideal. Definition 2.4 ([6]). For any i, j ∈ {1, 2, 3, 4}, a hyper structure (X, µ ˜) over (X, ∗, 0) is called an (i, j)-hyperfuzzy ideal of (X, ∗, 0) if (X, µ ˜inf ) is an i-fuzzy ideal of (X, ∗, 0) and (X, µ ˜sup ) is a j-fuzzy ideal of (X, ∗, 0).

3. Subalgebra and ideal-type hyper values Definition 3.1 ([3]). A nonempty subset A of (X, ∗, 0) is said to be S-energetic if it satisfies: (∀a, b ∈ X) (a ∗ b ∈ A ⇒ {a, b} ∩ A ̸= ∅) . Let A be a proper subset of X containing 0. Then there exists a ∈ X \ A, and so a ∗ a = 0 ∈ A but {a} and A are disjoint. Thus every proper subset A of X containing 0 cannot be S-energetic. Theorem 3.2. Given a hyper structure (X, µ ˜) over (X, ∗, 0), if it is a (4, 1)-hyperfuzzy subalgebra of (X, ∗, 0), then its nonempty level subsets U (˜ µinf ; α) and L(˜ µsup ; β) are S-energetic subsets of (X, ∗, 0) for all (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1].

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Y. B. Jun and S. S. Ahn Proof. Assume that U (˜ µinf ; α) and L(˜ µsup ; β) are nonempty for every (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1]. If x ∗ y ∈ U (˜ µinf ; α) and a ∗ b ∈ L(˜ µsup ; β) for all x, y, a, b ∈ X, then α≤µ ˜inf (x ∗ y) ≤ max{˜ µinf (x), µ ˜inf (y)} and β≥µ ˜sup (a ∗ b) ≥ min{˜ µsup (a), µ ˜sup (b)}. It follows that µ ˜inf (x) ≥ α or µ ˜inf (y) ≥ α, that is, x ∈ U (˜ µinf ; α) or y ∈ U (˜ µinf ; α) and µ ˜sup (a) ≤ β or µ ˜sup (b) ≤ β, that is, a ∈ L(˜ µsup ; β) or b ∈ L(˜ µsup ; β). Hence {x, y} ∩ U (˜ µinf ; α) ̸= ∅ and {a, b} ∩ L(˜ µsup ; β) ̸= ∅. Therefore U (˜ µinf ; α) and L(˜ µsup ; β) are S-energetic subsets of (X, ∗, 0) for all (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1].

□

Corollary 3.3. Given a hyper structure (X, µ ˜) over (X, ∗, 0), if it is a (2, 1)-hyperfuzzy (resp., (2, 3)-hyperfuzzy and (4, 3)-hyperfuzzy ) subalgebra of (X, ∗, 0), then its nonempty level subsets U (˜ µinf ; α) and L(˜ µsup ; β) are Senergetic subsets of (X, ∗, 0) for all (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1]. □

Proof. Straightforward. Definition 3.4 ([3]). A nonempty subset A of (X, ∗, 0) is said to be I-energetic if it satisfies: (∀x, y ∈ X) (y ∈ A ⇒ {x, y ∗ x} ∩ A ̸= ∅) .

Theorem 3.5. Given a hyper structure (X, µ ˜) over (X, ∗, 0), if it is a (4, 1)-hyperfuzzy ideal of (X, ∗, 0), then its nonempty level subsets U (˜ µinf ; α) and L(˜ µsup ; β) are I-energetic subsets of (X, ∗, 0) for all (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1]. Proof. Let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that U (˜ µinf ; α) and L(˜ µsup ; β) are nonempty. Let x, y, a, b ∈ X be such that y ∈ U (˜ µinf ; α) and b ∈ L(˜ µsup ; β). Then α≤µ ˜inf (y) ≤ max{˜ µinf (y ∗ x), µ ˜inf (x)} and β≥µ ˜sup (b) ≥ min{˜ µsup (b ∗ a), µ ˜sup (a)}. Hence µ ˜inf (y ∗ x) ≥ α or µ ˜inf (x) ≥ α, i.e., y ∗ x ∈ U (˜ µinf ; α) or x ∈ U (˜ µinf ; α) and µ ˜sup (b ∗ a) ≤ β or µ ˜sup (a) ≤ β, i.e., b ∗ a ∈ L(˜ µsup ; β) or a ∈ L(˜ µsup ; β). It follows that {x, y ∗ x} ∩ U (˜ µinf ; α) ̸= ∅ and {a, b ∗ a} ∩ L(˜ µsup ; β) ̸= ∅. Therefore U (˜ µinf ; α) and L(˜ µsup ; β) are I-energetic subsets of (X, ∗, 0) for all (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1].

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Subalgebra and ideal-type hyper values in BCK/BCI-algebras Corollary 3.6. Given a hyper structure (X, µ ˜) over (X, ∗, 0), if it is a (2, 1)-hyperfuzzy (resp., (2, 3)-hyperfuzzy and (4, 3)-hyperfuzzy ) subalgebra of (X, ∗, 0), then its nonempty level subsets U (˜ µinf ; α) and L(˜ µsup ; β) are Ienergetic subsets of (X, ∗, 0) for all (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1]. □

Proof. Straightforward.

Definition 3.7. Given a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. Then (α, β) is called a subalgebra-type hyper value for (X, µ ˜) if the following assertion is valid.

(

(∀x, y ∈ X)

µ ˜inf (x ∗ y) ≤ α ⇒ min{˜ µinf (x), µ ˜inf (y)} ≤ α,

)

µ ˜sup (x ∗ y) ≥ β ⇒ max{˜ µsup (x), µ ˜sup (y)} ≥ β

.

(3.1)

Example 3.8. Let X = {0, 1, 2, 3, 4} be a set with the binary operation ∗ which is given in Table 1.

Table 1. Cayley table for the binary operation “∗” ∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 1 3 4

2 0 0 0 3 4

3 0 1 2 0 4

4 0 0 0 3 0

˜) be a hyper structure over (X, ∗, 0) in which µ ˜ is given as Then (X, ∗, 0) is a BCK-algebra (see [5]). Let (X, µ follows:

  [0.5, 0.53)      (0.3, 0.58] ˜ µ ˜ : X → P([0, 1]), x 7→ [0.3, 0.44) ∪ [0.45, 0.58)    (0.4, 0.5] ∪ [0.60, 0.68]    [0.2, 0.63]

if if if if if

x = 0, x = 1, x = 2, x = 3, x = 4.

It is routine to verify that every pair (α, β) ∈ [0.2, 0.5] × [0.53, 0.68) is a subalgebra-type hyper value for (X, µ ˜). Theorem 3.9. For a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. If (X, µ ˜) is a (1, 4)-hyperfuzzy subalgebra of (X, ∗, 0), then (α, β) is a subalgebra-type hyper value for (X, µ ˜). Proof. Let x, y, a, b ∈ X be such that µ ˜inf (x ∗ y) ≤ α and µ ˜sup (a ∗ b) ≥ β. Since (X, µ ˜) is a (1, 4)-hyperfuzzy subalgebra of (X, ∗, 0), we have α≥µ ˜inf (x ∗ y) ≥ min{˜ µinf (x), µ ˜inf (y)} and β≤µ ˜sup (a ∗ b) ≤ max{˜ µsup (a), µ ˜sup (b)}. □

Hence (α, β) is a subalgebra-type hyper value for (X, µ ˜).

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Y. B. Jun and S. S. Ahn Corollary 3.10. For a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. If (X, µ ˜) is a (1, 2)-hyperfuzzy (resp., (3, 2)-hyperfuzzy and (3, 4)hyperfuzzy ) subalgebra of (X, ∗, 0), then (α, β) is a subalgebra-type hyper value for (X, µ ˜). □

Proof. Straightforward.

Theorem 3.11. Let (X, µ ˜) be a hyper structure over (X, ∗, 0). If (α, β) is a subalgebra-type hyper value for (X, µ ˜), then L(˜ µinf ; α) and U (˜ µsup ; β) are S-energetic subsets of (X, ∗, 0). Proof. Let x, y, a, b ∈ X be such that x∗y ∈ L(˜ µinf ; α) and a∗b ∈ U (˜ µsup ; β). Then µ ˜inf (x∗y) ≤ α and µ ˜sup (a∗b) ≥ β. Since (α, β) is a subalgebra-type hyper value for (X, µ ˜), it follows from (3.1) that min{˜ µinf (x), µ ˜inf (y)} ≤ α and max{˜ µsup (a), µ ˜sup (b)} ≥ β. Hence µ ˜inf (x) ≤ α or µ ˜inf (y) ≤ α and µ ˜sup (a) ≥ β or µ ˜sup (b) ≥ β, that is, x ∈ L(˜ µinf ; α) or y ∈ L(˜ µinf ; α) and a ∈ U (˜ µsup ; β) or b ∈ U (˜ µsup ; β). Thus {x, y} ∩ L(˜ µinf ; α) ̸= ∅ and {a, b} ∩ U (˜ µsup ; β) ̸= ∅, and therefore L(˜ µinf ; α) and U (˜ µsup ; β) are S-energetic subsets of (X, ∗, 0).

□

Combining Theorems 3.9 and 3.11, we have the following corollary. Corollary 3.12. For a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. If (X, µ ˜) is a (1, 4)-hyperfuzzy subalgebra of (X, ∗, 0), then L(˜ µinf ; α) and U (˜ µsup ; β) are S-energetic subsets of (X, ∗, 0). Definition 3.13. Given a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. Then (α, β) is called an ideal-type hyper value for (X, µ ˜) if the following assertion is valid.

( (∀x, y ∈ X)

µ ˜inf (y) ≤ α ⇒ min{˜ µinf (y ∗ x), µ ˜inf (x)} ≤ α,

)

µ ˜sup (y) ≥ β ⇒ max{˜ µsup (y ∗ x), µ ˜sup (x)} ≥ β

.

(3.2)

Example 3.14. In Example 3.8, the pair (α, β) is an ideal-type hyper value for (X, µ ˜). Example 3.15. Let X = {0, 1, 2, a, b} be a set with the binary operation ∗ which is given in Table 2.

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Subalgebra and ideal-type hyper values in BCK/BCI-algebras

Table 2. Cayley table for the binary operation “∗” ∗ 0 1 2 a b

0 0 1 2 a b

1 0 0 2 a a

2 0 1 0 a b

a a b a 0 1

b a a a 0 0

˜) be a hyper structure over (X, ∗, 0) in which µ ˜ is given as Then (X, ∗, 0) is a BCI-algebra (see [5]). Let (X, µ follows:

  [0.54, 0.72)      (0.58, 0.64] ˜ µ ˜ : X → P([0, 1]), x 7→ [0.56, 0.72)    (0.60, 0.68]    [0.60, 0.64)

if if if if if

x = 0, x = 1, x = 2, x = a, x = b.

If we take (α, β) ∈ (0.54, 0.60] × [0.64, 0.72), then (α, β) is an ideal-type hyper value for (X, µ ˜). We consider a relation between subalgebra-type hyper value and ideal-type hyper value. Theorem 3.16. Let (X, µ ˜) be a hyper structure over (X, µ ˜) ∈ BK (X) such that (∀x ∈ X) (˜ µinf (0) ≥ µ ˜inf (x), µ ˜sup (0) ≤ µ ˜sup (x)) .

(3.3)

Then every ideal-type hyper value for (X, µ ˜) is a subalgebra-type hyper value for (X, µ ˜). Proof. Let (α, β) be an ideal-type hyper value for (X, µ ˜). Assume that µ ˜inf (x ∗ y) ≤ α and µ ˜sup (a ∗ b) ≥ β for x, y, a, b ∈ X. Using (3.2), (2.2) and (3.3), we have α ≥ min{˜ µinf ((x ∗ y) ∗ x), µ ˜inf (x)} = min{˜ µinf ((x ∗ x) ∗ y), µ ˜inf (x)} = min{˜ µinf (0 ∗ y), µ ˜inf (x)} = min{˜ µinf (0), µ ˜inf (x)} = µ ˜inf (x) and β ≤ max{˜ µsup ((a ∗ b) ∗ a), µ ˜sup (a)} = max{˜ µsup ((a ∗ a) ∗ b), µ ˜sup (a)} = max{˜ µsup (0 ∗ b), µ ˜sup (a)} = max{˜ µsup (0), µ ˜sup (a)} = µ ˜sup (a). It follows that min{˜ µinf (x), µ ˜inf (y)} ≤ µ ˜inf (x) ≤ α and max{˜ µsup (a), µ ˜sup (b)} ≥ µ ˜sup (a) ≥ β. □

Therefore (α, β) is a subalgebra-type hyper value for (X, µ ˜).

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Y. B. Jun and S. S. Ahn The converse of Theorem 3.16 is not true in general as seen in the following example. Example 3.17. Let X = {0, 1, a, b, c} be a set with the binary operation ∗ which is given in Table 3.

Table 3. Cayley table for the binary operation “∗” ∗ 0 1 a b c

0 0 1 a b c

1 0 0 a a a

a a a 0 1 1

b a a 0 0 1

c a a 0 0 0

˜) be a hyper structure over (X, ∗, 0) in which µ ˜ is given as Then (X, ∗, 0) is a BCI-algebra (see [5]). Let (X, µ follows:

  [0.51, 0.55)      (0.48, 0.63] ˜ µ ˜ : X → P([0, 1]), x 7→ [0.45, 0.58)    (0.41, 0.5] ∪ [0.60, 0.63]    [0.35, 0.65]

if if if if if

x = 0, x = 1, x = a, x = b, x = c.

If we take (α, β) ∈ (0.41, 0.45) × (0.63, 0.65], then (α, β) is a subalgebra-type hyper value for (X, µ ˜), but it is not an ideal-type hyper value for (X, µ ˜) since µ ˜inf (b) = 0.41 ≤ α and min{˜ µinf (b ∗ a), µ ˜inf (a)} = 0.45 ≰ α and/or µ ˜sup (c) = 0.65 ≥ β and max{˜ µsup (c ∗ a), µ ˜sup (a)} = 0.63 ≱ β. We provide conditions for a pair (α, β) to be an ideal-type hyper value. Theorem 3.18. Given a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. If (X, µ ˜) is a (1, 4)-hyperfuzzy ideal of (X, ∗, 0), then (α, β) is an ideal-type hyper value for (X, µ ˜). Proof. Let x, y, a, b ∈ X be such that µ ˜inf (y) ≤ α and µ ˜sup (b) ≥ β. Since (X, µ ˜) is a (1, 4)-hyperfuzzy ideal of (X, ∗, 0), it follows that α≥µ ˜inf (y) ≥ min{˜ µinf (y ∗ x), µ ˜inf (x)} and β≤µ ˜sup (b) ≤ max{˜ µsup (b ∗ a), µ ˜sup (a)}. □

Therefore (α, β) is an ideal-type hyper value for (X, µ ˜).

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Subalgebra and ideal-type hyper values in BCK/BCI-algebras Corollary 3.19. Given a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. If (X, µ ˜) is a (1, 2)-hyperfuzzy (resp., (3, 2)-hyperfuzzy and (3, 4)hyperfuzzy ) ideal of (X, ∗, 0), then (α, β) is an ideal-type hyper value for (X, µ ˜). □

Proof. Straightforward.

Theorem 3.20. Let (X, µ ˜) be a hyper structure over (X, ∗, 0). If (α, β) is an ideal-type hyper value for (X, µ ˜), then L(˜ µinf ; α) and U (˜ µsup ; β) are I-energetic subsets of (X, ∗, 0). Proof. Let x, y, a, b ∈ X be such that y ∈ L(˜ µinf ; α) and b ∈ U (˜ µsup ; β). Then µ ˜inf (y) ≤ α and µ ˜sup (b) ≥ β. Since (α, β) is an ideal-type hyper value for (X, µ ˜), it follows from (3.2) that min{˜ µinf (y ∗ x), µ ˜inf (x)} ≤ α and max{˜ µsup (b ∗ a), µ ˜sup (a)} ≥ β. Hence µ ˜inf (y ∗ x) ≤ α or µ ˜inf (x) ≤ α and µ ˜sup (b ∗ a) ≥ β or µ ˜sup (a) ≥ β, that is, y ∗ x ∈ L(˜ µinf ; α) or x ∈ L(˜ µinf ; α) and b ∗ a ∈ U (˜ µsup ; β) or a ∈ U (˜ µsup ; β). Thus {y ∗ x, x} ∩ L(˜ µinf ; α) ̸= ∅ and {b ∗ a, a} ∩ U (˜ µsup ; β) ̸= ∅, and therefore L(˜ µinf ; α) and U (˜ µsup ; β) are I-energetic subsets of (X, ∗, 0).

□

Combining Theorems 3.18 and 3.20, we have the following corollary. Corollary 3.21. Given a hyper structure (X, µ ˜) over (X, ∗, 0), let (α, β) ∈ Λα × Λβ ⊆ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. If (X, µ ˜) is a (1, 4)-hyperfuzzy ideal of (X, ∗, 0), then L(˜ µinf ; α) and U (˜ µsup ; β) are I-energetic subsets of (X, ∗, 0). Definition 3.22 ([3]). A nonempty subset A of (X, ∗, 0) is said to be right vanished if it satisfies: (∀x, y ∈ X) (x ∗ y ∈ A ⇒ x ∈ A) .

(3.4)

A is said to be right stable if A ∗ X := {a ∗ x | a ∈ A, x ∈ X} ⊆ A. Lemma 3.23 ([6]). If (X, µ ˜) is a (4, 1)-hyperfuzzy ideal of (X, ∗, 0), then (∀x, y ∈ X) (x ≤ y ⇒ µ ˜inf (x) ≤ µ ˜inf (y), µ ˜sup (x) ≥ µ ˜sup (y)) .

(3.5)

Theorem 3.24. Given a hyper structure (X, µ ˜) over (X, ∗, 0) ∈ BK (X) and (α, β) ∈ [0, 1] × [0, 1], if (X, µ ˜) is a (4, 1)-hyperfuzzy ideal of (X, ∗, 0), then L(˜ µinf ; α) and U (˜ µsup ; β) are right stable subsets of (X, ∗, 0) whenever they are nonempty.

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Y. B. Jun and S. S. Ahn Proof. Let (α, β) ∈ [0, 1] × [0, 1] be such that L(˜ µinf ; α) and U (˜ µsup ; β) are nonempty. Let x, a, b ∈ X be such that a ∈ L(˜ µinf ; α) and b ∈ U (˜ µsup ; β). Then µ ˜inf (a) ≤ α and µ ˜sup (b) ≥ β. Since a ∗ x ≤ a and b ∗ x ≤ b, it follows from Lemma 3.23 that µ ˜inf (a ∗ x) ≤ µ ˜inf (a) ≤ α and µ ˜sup (b ∗ x) ≥ µ ˜sup (b) ≥ β, that is, a ∗ x ∈ L(˜ µinf ; α) and b ∗ x ∈ U (˜ µsup ; β). Hence L(˜ µinf ; α) ∗ X ⊆ L(˜ µinf ; α) and U (˜ µsup ; β) ∗ X ⊆ U (˜ µsup ; β). Therefore L(˜ µinf ; α) and U (˜ µsup ; β) are right stable subsets of (X, ∗, 0).

□

Corollary 3.25. Given a hyper structure (X, µ ˜) over (X, ∗, 0) ∈ BK (X) and (α, β) ∈ [0, 1] × [0, 1], if (X, µ ˜) is a (2, 1)-hyperfuzzy (resp., (2, 3)-hyperfuzzy and (4, 3)-hyperfuzzy ) ideal of (X, ∗, 0), then L(˜ µinf ; α) and U (˜ µsup ; β) are right stable subsets of (X, ∗, 0) whenever they are nonempty. □

Proof. Straightforward.

Theorem 3.26. Given a hyper structure (X, µ ˜) over (X, ∗, 0) ∈ BK (X) and (α, β) ∈ [0, 1] × [0, 1], if (X, µ ˜) is a (4, 1)-hyperfuzzy ideal of (X, ∗, 0), then U (˜ µinf ; α) and L(˜ µsup ; β) are right vanished subsets of (X, ∗, 0) whenever they are nonempty. Proof. Let (α, β) ∈ [0, 1]×[0, 1] be such that U (˜ µinf ; α) and L(˜ µsup ; β) are nonempty. Assume that x∗y ∈ U (˜ µinf ; α) and a ∗ b ∈ L(˜ µsup ; β) for any x, y, a, b ∈ X. Using Lemma 3.23 implies that α≤µ ˜inf (x ∗ y) ≤ µ ˜inf (x), that is, x ∈ U (˜ µinf ; α) and β≥µ ˜sup (a ∗ b) ≥ µ ˜sup (a), that is, a ∈ L(˜ µsup ; β). Hence U (˜ µinf ; α) and L(˜ µsup ; β) are right vanished subsets of (X, ∗, 0).

□

Corollary 3.27. Given a hyper structure (X, µ ˜) over (X, ∗, 0) ∈ BK (X) and (α, β) ∈ [0, 1] × [0, 1], if (X, µ ˜) is a (2, 1)-hyperfuzzy (resp., (2, 3)-hyperfuzzy and (4, 3)-hyperfuzzy ) ideal of (X, ∗, 0), then U (˜ µinf ; α) and L(˜ µsup ; β) are right vanished subsets of (X, ∗, 0) whenever they are nonempty. □

Proof. Straightforward. References

[1] J. Ghosh and T. K. Samanta, Hyperfuzzy sets and hyperfuzzy group, Int. J. Advanced Sci Tech. 41 (2012), 27–37. [2] Y. S. Huang, BCI-algebra, Science Press, Beijing, 2006. [3] Y. B. Jun, S. S. Ahn and E. H. Roh, Energetic subsets and permeable values with applications in BCK/BCIalgebras, Appl. Math. Sci. 7 (2013), no. 89, 4425–4438. [4] Y. B. Jun, K. Hur and K. J. Lee, Hyperfuzzy subalgebras of BCK/BCI-algebras, Ann. Fuzzy Math. Inform. 15 (2018), no. 1, 17–28. [5] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa Co., Seoul, 1994. [6] S. Z. Song, S. J. Kim and Y. B. Jun, Hyperfuzzy ideals in BCK/BCI-algebras, Mathematics 2017, 5, 81.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 2, 2020

On Invariance and Solutions of Some Fifth-Order Rational Recursive Sequences, M. FollyGbetoula and D. Nyirenda,…………………………………………………………………201 On Some Conditions for p-Valency, Mamoru Nunokawa and Janusz Sokól,……………..219 Fractional Cauchy Euler Differential Equation, M. Al-Horani, R. Khalil, and I. Aldarawi,226 Applications of Neutrosophic Sets in B-Algebras, Sun Shin Ahn,………………………..234 Investigating Some Properties of a Fourth Order Difference Equation, M. B. Almatrafi, E. M. Elsayed, and Faris Alzahrani,………………………………………………………………243 A Necessary Condition for Eventually Equilibrium or Periodic to a System of Difference Equations, Wirot Tikjha and Kunrasatree Piasu,…………………………………………..254 Bi-Univalent Functions Associated with Wright Hypergeometric Functions, E. Analouei Adegani, N. E. Cho, A. Motamednezhad, and M. Jafari,………………………………….261 Conformable Fractional Approximation by Choquet Integrals, George A. Anastassiou,…272 The Minkowski Inequality and the Brunn-Minkowski Inequality for Dual Orlicz Mixed Affine Quermassintegrals, Tongyi Ma,……………………………………………………………294 Existence and Convergence for Fixed Points of a Strict Pseudo-Contraction in CAT(0) Spaces, Narongrit Puturong and Kasamsuk Ungchittrakool,……………………………………….305 On the CEU-Degree of Similarity in International Trade by Using the Choquet Integral Expected Utility, Lee-Chae Jang and Jacob Wood,…………………………………………………..321 The General Solution of a Mixed Cubic-Quartic Functional Equation and the Ulam Stability of Matrix Fuzzy Normed Spaces, Yali Ding, Tian-Zhou Xu, and John Michael Rassias,……329 A High-Accuracy Collocation Method for Solving Mixed Boundary Value Problems on Nonsmooth Boundaries, Xin Luo and Chuan-Long Wang,………………………………..337 Adaptive Modified Function Projective Synchronization of Chaotic Dynamical System with Different Order, M. M. El-Dessoky, Ebraheem Alzahrani, and N. A. Almohammadi,……354 Dual log-Minkowski Inequality for Star Bodies, Tongyi Ma,………………………….…..364

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 2, 2020 (continued) Subalgebra and Ideal-Type Hyper Values in BCK/BCI-Algebras, Young Bae Jun and Sun Shin Ahn,…………………………………………………………………………………………375

Volume 28, Number 3 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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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 (six 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], St.Martin Univ.,Olympia,WA,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.

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.

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Control problems for semilinear impulsive differential control systems Ah-ran Park1 and Jin-Mun Jeong2,∗ 1,2

Department of Applied Mathematics, Pukyong National University Busan 48513, Republic of Korea

Abstract In this paper, we establish the approximate controllability for the semilinear impulsive differential equation in relation to the the corresponding linear control system based on the regularity for the equation under natural assumptions such as the local Lipschitz continuity of nonlinear term. Keywords: approximate controllability, semilinear equation, ,impulsive differential equation, local lipschitz continuity, controller operator, reachable set AMS Classification Primary 35B37; Secondary 93C20

1

Introduction

In this paper, we are concerned with the approximate controllability for the semilinear impulsive control system in Hilbert spaces:  0  x (t) + Ax(t) = f (t, x(t)) + (Bu)(t), t ∈ (0, T ], t = tk ,     k = 1, 2, · · · , m, (1.1) − −  ∆x(tk ) = x(t+ k = 1, 2, · · · , m,  k ) − x(tk ) = Ik (x(tk )),    x(0) = x . 0 Let H be identified with its dual space we may write V ⊂ H ⊂ V ∗ densely and the corresponding injections be continuous. Here, A is the operator associated with a sesquilinear form a(·, ·) defined on V × V satisfying G˚ arding’s inequality: (Au, v) = a(u, v),

u, v ∈ V

Email: 1 [email protected], 2,∗ [email protected]( Corresponding author) This work was supported by a Research Grant of Pukyong National University(2019Year).

1

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2 where V is a Hilbert space such that V ⊂ H ⊂ V ∗ . Then −A generates an analytic semigroup in both H and V ∗ (see [1, Theorem 3.6.1]) and so the equation (1.1) may be considered as an equation in H as well as in V ∗ . The nonlinear operator f from [0, T ] × V to H is assumed to be locally Lipschitz continuous with respect to the second variable. Let U be a Banach space of control variables and the controller operator B be a bounded linear operator from the Banach space L2 (0, T ; U ) to L2 (0, T ; H). The impulsive condition − − ∆x(tk ) = x(t+ k ) − x(tk ) = Ik (x(tk )),

k = 1, 2, · · · , m,

is a combination of traditional evolution systems. Let x(t; f, u) be a solution of the equation (1.1) associated with a nonlinear term f and a control u. We will show the approximate controllability for the equation (1.1), namely that the reachable set RT (f ) = {x(T ; f, u) : u ∈ L2 (0, T ; U )} is a dense subset of H. This kind of equations arise naturally in biology, in physics, control engineering problem, etc. In the first part of this paper we establish the wellposedness and regularity property for the following equation:  0  x (t) + Ax(t) = f (t, x(t)) + k(t), t ∈ (0, T ], t = tk ,     k = 1, 2, · · · , m, (1.2) + − −  ∆x(t ) = x(t ) − x(t ) = I (x(t )), k = 1, 2, · · · , m,  k k k k k    x(0) = x . 0 The regularity for the semilinear heat equations has been developed as seen in Barbu [2] and [3, 4, 5, 6]. In this paper, based on the regularity for (1.2), we intend to establish the approximate controllability for (1.1). Approximate controllability for semilinear control systems can be founded in [7-15]. Similar considerations of linear and semilinear systems have been dealt with in many references, linear problems in the book [15] and Nakagiri [14], semilinear cases with the uniform bounded nonlinear term in [16], and with the uniform Lipschtz continuous nonlinear term in [3, 17, 18, 19]. However, there are few papers treating the systems with local Lipschipz continuity, we can just find a recent article Wang [20]. Among these literatures, in [17, 20], they assumed that the semigroup S(t) generated by A is compact in order to guarantee the compactness of the solution mapping, and investigated the approximate controllability for the equation (1.1). In this paper, in order to show that the main result of Naito [17] is extended to the nonlinear differential equation, we assume that the embedding D(A) ⊂ V is compact instead of the compact property of semigroup used in [17, 21]. Then by virtue of the result in Aubin [22], we can take advantage of the fact that the solution mapping u ∈ L2 (0, T ; U ) 7→ x(T ; f, u) is compact. Under natural assumptions such as the local Lipschtiz continuity of nonlinear term, we obtain the approximate controllability for the equation (1.1) when the corresponding linear system is approximately controllable. The paper is organized as follows. In section 2, the results of general linear evolution equations besides notations and assumptions are stated. In section 3, we investigate the

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3 approximate controllability for the problem (1.1). The approach used here is similar to that developed in [1, 3] on the general semilnear evolution equations, which is an important role to extend the theory of practical nonlinear partial differential equations.

2

Regularity for semilinear impulsive systems

The norm on V , H and V ∗ will be denoted by || · ||, | · | and || · ||∗ , respectively. We assume that V has a stronger topology than H and, for brevity, we may regard that ||u||∗ ≤ |u| ≤ ||u||,

∀u ∈ V.

(2.1)

Let a(·, ·) be a bounded sesquilinear form defined in V × V and satisfying G˚ arding’s inequality Re a(u, u) ≥ ω1 ||u||2 − ω2 |u|2 , (2.2) where ω1 > 0 and ω2 is a real number. Let A be the operator associated with this sesquilinear form: (Au, v) = a(u, v), u, v ∈ V. Then −A is a bounded linear operator from V to V ∗ by the Lax-Milgram Theorem. The realization of A in H which is the restriction of A to D(A) = {u ∈ V : Au ∈ H} is also denoted by A. Then we consider the following sequence D(A) ⊂ V ⊂ H ⊂ V ∗ ⊂ D(A)∗ ,

(2.3)

where each space is dense in the next one which continuous injection. It is also well known that A generates an analytic semigroup S(t) in both H and V ∗ . For the sake of simplicity, we assume that ω2 = 0 and hence the closed half plane {λ : Re λ ≥ 0} is contained in the resolvent set of A. If X is a Banach space, L2 (0, T ; X) is the collection of all strongly measurable square integrable functions from (0, T ) into X and W 1,2 (0, T ; X) is the set of all absolutely continuous functions on [0, T ] such that their derivative belongs to L2 (0, T ; X). C([0, T ]; X) will denote the set of all continuously functions from [0, T ] into X with the supremum norm. Let the solution spaces W(T ) and W1 (T ) of strong solutions be defined by W(T ) = L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H), W1 (T ) = L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ). Here, we note that by using interpolation theory, we have W(T ) ⊂ C([0, T ]; V ),

W1 (T ) ⊂ C([0, T ]; H).

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4 Thus, there exists a constant M0 > 0 such that ||x||C([0,T ];V ) ≤ M0 ||x||W(T ) ,

||x||C([0,T ];H) ≤ M0 ||x||W1 (T ) .

(2.4)

The semigroup generated by −A is denoted by S(t) and there exists a constant M such that |S(t)| ≤ M, ||s(t)||∗ ≤ M. Let f be a nonlinear mapping from V into H. We need to impose the following conditions on nonlinear term f . Assumption (F). There exists a function L : R+ → R such that L(r1 ) ≤ L(r2 ) for r1 ≤ r2 and |f (t, x)| ≤ L(r), |f (t, x) − f (t, y)| ≤ L(r)||x − y|| hold for any t ∈ [0, T ], ||x|| ≤ r and ||y|| ≤ r. Assumption (I). The functions Ik : V → H are continuous and there exist positive constants L(Ik ) and β ∈ (1/3, 1] such that |Aβ Ik (x)| ≤ L(Ik )||x||,

|Aβ Ik (x) − Ik (y)| ≤ L(Ik )||x − y||,

k = 1, 2, · · · , m

for each x, y ∈ V , and ||x(t− k )|| ≤ K,

k = 1, 2, · · · , m.

From now on, we establish the following results on the local solvability of the following equation;  0  x (t) + Ax(t) = f (t, x(t)) + k(t), t ∈ (0, T ], t 6= tk ,     k = 1, 2, · · · , m, (2.5) − −  ∆x(tk ) = x(t+  k ) − x(tk ) = Ik (x(tk )), k = 1, 2, · · · , m,    x(0) = x . 0 Let us rewrite (F x)(t) = f (t, x(t)) for each x ∈ L2 (0, T ; V ). Then there is a constant, denoted again by L(r), such that √ ||F x||L2 (0,T ;H) ≤ L(r) T , ||F x1 − F x2 ||L2 (0,T ;H) ≤ L(r)||x1 − x2 ||L2 (0,T ;V ) hold for x1 , x2 ∈ Br (T ) = {x ∈ L2 (0, T ; V ) : ||x||L2 (0,T ;V ) ≤ r}. Here, we note that by using interpolation theory, we have that for any t > 0, L2 (0, t; V ) ∩ W 1,2 (0, t; V ∗ ) ⊂ C([0, t]; H). Thus, for any t > 0, there exists a constant c > 0 such that ||x||C([0,t];H) ≤ c||x||L2 (0,t;V )∩W 1,2 (0,t;V ∗ ) .

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(2.6)

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5 Let 0 = t0 < t1 < · · · < tk < · · · < tm = T. Then by Assumption (I) and (2.5), it is immediately seen that x ∈ W 1,2 (ti , ti+1 ; V ∗ ),

i = 0, · · · , m − 1.

Thus by virtue of Assumption (I) and (2.6), we may consider that there exists a constant C3 > 0 such that max {|x(t)| : x is a solution of (2.5)} ≤ C3 ||x||L2 (0,T :V ) .

(2.6)

0≤t≤T

With the notations (2.2), (2.3), we have (V, V ∗ )1/2,2 = H,

(D(A), H)1/2,2 = V,

where (V, V ∗ )1/2,2 denotes the real interpolation space between V and V ∗ (Section 1.3.3 of [23]). From now on, we establish the following results on the solvability of the equation (2.5). Theorem 2.1. 1) Let Assumption (F) be satisfied. Assume that x0 ∈ H, k ∈ L2 (0, T ; V ∗ ). Then, there exists a time T0 ∈ (0, T ) such that the equation (2.5) admits a solution x ∈ W1 (T0 ) ⊂ C([0, T0 ]; H).

(2.7)

2) Under Assumption (F) for the nonlinear mapping f , there exists a unique solution x of (2.5) such that x ∈ W1 (T ) ≡ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C([0, T ]; H),

T > 0.

for any x0 ∈ H, k ∈ L2 (0, T ; V ∗ ). Moreover, there exists a constant C1 such that ||x||W1 (T ) ≤ C1 (1 + |x0 | + ||k||L2 (0,T ;V ∗ ) ),

(2.8)

where C1 is a constant depending on T . 3) Let Assumptions (F) and (I) be satisfied and (x0 , k) ∈ H × L2 (0, T ; V ). Then the solution x of the equation (2.5) belongs to x ∈ W1 ≡ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) and the mapping H × L2 (0, T ; V ∗ ) 3 (x0 , k) 7→ x ∈ W1 (T )

(2.9)

is continuous. Rt Corollary 2.1. Suppose that k ∈ L2 (0, T ; H) and x(t) = 0 S(t − s)k(s)ds for 0 ≤ t ≤ T . Then there exists a constant C2 such that √ ||x||L2 (0,T ;V ) ≤ C2 T ||k||L2 (0,T ;H) . (2.10)

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6 Proof. From Theorem 2.3 of [24], it follows that there exists a C > 0 such that ||x||L2 (0,T ;D(A)) ≤ C||k||L2 (0,T ;H) .

(2.11)

Moreover, we have ||x||2L2 (0,T ;H)

T

Z ≤M

Z t

0

0

t

T2 |k(s)| dsdt ≤ M 2 2

Z

T

|k(s)|2 ds.

(2.12)

0

Since (D(A), H)1/2,2 = V, there exists a constant C0 > 0 such that 1/2

||u|| ≤ C0 ||u||D(A) |u|1/2 .

(2.13)

√ Thus, by (2.11), (2.12) and (2.13), if C2 = C0 CT (M/2)1/4 , then the inequality (2.10) holds.

3

Approximate Controllability

Let U be a Banach space of control variables. Here B is a linear bounded operator from L2 (0, T ; U ) to L2 (0, T ; H), which is called a controller. Consider the following nonlinear impulsive control systems.  0   x (t) + Ax(t) = f (t, x(t)) + (Bu)(t), t ∈ (0, T ], (3.1) x(0) = x0 .   + − − ∆x(tk ) = x(tk ) − x(tk ) = Ik (x(tk )), k = 1, 2, · · · , m. Let x(T ; f, u) be a state value of the system (3.1) at time T corresponding to the nonlinear term f and the control u. Let S(·) be the analytic semigroup generated by −A. Then the solution x(t; f, u) can be written as Z t X x(t; f, u) = S(t)x0 + S(t − s){f (s, x(s, f, u)) + (Bu)(s)}ds + S(t − s)Ik (x(t− k )), 0

0 0, p ∈ L2 (0, T ; H) there exists a u ∈ L2 (0, T ; U ) such that (

ˆ − SBu| ˆ |Sp ≤ε ||Bu||L2 (0,t;H) ≤ q1 ||p||L2 (0,t;H) ,

0≤t≤T

where q is a constant independent of p. Assumption (F1) The nonlinear operator f is a nonlinear mapping of [0, T ] × H into H satisfying the following. There exists a constant L1 = L1 (r) > 0 such that |f (t, x) − f (t, y)| ≤ L1 ||x − y||,

t ∈ [0, T ],

hold for ||x|| ≤ r and ||y|| ≤ r. Assumption (H) We assume the following inequality condition: √ max{q, 1}{1 − M2 }−1 C2 L1 T < 1. where C2 is the constant in (2.10), X √ L(Ik ). M2 = C2 T L1 + (3β)−1/2 2(3β − 1)−1 C1−β C3 T 3β/2 0≤tk ≤T

Lemma 3.1. Let u1 and u2 be in L2 (0, T ; U ). Then under Assumption(B) and Assumption(F 1), one has that, for 0 ≤ t ≤ T , √ ||x(t : f, u1 ) − x(t : f, u2 )||L2 (0,T ;V ) ≤ {1 − M2 }−1 C2 t||Bu1 − Bu2 ||L2 (0,T ;H) .

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8 Proof. Let x1 (t) = x(t : f, u1 ) and x2 (t) = x(t : f, u2 ). Then for 0 ≤ t ≤ T ,we have Z t S(t − s){f (s, x1 (s)) − f (s, x2 (s))}ds x1 (t) − x2 (t) = 0 Z t + S(t − s){Bu1 − Bu2 }ds 0 X − + S(t − s){Ik (x1 (t− (3.4) k )) − Ik (x2 (tk ))}. 0≤tk ≤T

By Assumption(F 1) and (2.10), we obtain Z t √ S(t − s){f (s, x1 (s)) − f (s, x2 (s))}ds||L2 (0,t;V ) ≤ C2 tL1 ||x1 − x2 ||L2 (0,t;V ) . || 0

Moreover, by Lemma 2.5 of (2.11) and Theorem 3.1, we have Z t √ || S(t − s){Bu1 − Bu2 }ds||L2 (0,t;V ) ≤ C2 T ||Bu1 − Bu2 ||L2 (0,t;H) 0

and ||

X

− S(t − s){Ik (x1 (t− k )) − Ik (x2 (tk ))}||L2 (0,t;V )

0≤tk ≤t

≤ (3β)−1/2 2(3β − 1)−1 C1−β C3 t3β/2

X

− L(Ik )||x1 (t− k ) − x2 (tk )||L2 (0,t;V ) .

0≤tk ≤t

Thus, from (3.4) it follows that ||x(t; f, u1 ) − x(t; f, u2 )||L2 (0,T ;V ) √ √ ≤ C2 T ||Bu1 − Bu2 ||L2 (0,T ;H) + C2 T L1 ||x1 − x2 ||L2 (0,T ;V ) X − + (3β)−1/2 2(3β − 1)−1 C1−β C3 t3β/2 L(Ik )||x1 (t− k ) − x2 (tk )||L2 (0,T ;V ) . 0≤tk ≤t

Theorem 3.1. Under Assumptions (B),(F1), and (H) the system(4.1) is approximately controllable on [0, T ]. Proof. The reachable set for the system(4.1) is given by RT = {x(T ; f, u) : u ∈ L2 (0, T ; U )}. We will show that D(A) ⊂ RT (f ), i.e., for given ε > 0 and ξT ∈ D(A), there exists u ∈ L2 (0, T ; U ) such that |ξT − x(T ; f, u)| < ε, (3.5)

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9 where Z x(T ; , f, u) = S(T )x0 + 0

T

S(T − s){f (s, x(s, f, u)) + (Bu)(s)}ds X + S(T − s)Ik (x(t− k )).

(3.6)

0 |. k+1 2 k=0

4

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Let P± : E → E ± be the orthogonal projection of E onto E ± . We denote by τ the topology on E generated by the norm ( ) ∞ ∑ 1 | < P− u, ek > | . ∥u∥τ = max ∥P+ u∥, k+1 2 k=1 τ

Remark 2.1. Note that if un → − u, then P+ un → P+ u and P− un ⇀ P− u. τ

Definition 2.1. Let J ∈ C 1 (E), we say J is τ −upper semicontinuous if un − → u implies J(u) ≥ lim J(un ). n→∞

Definition 2.2. Let J ∈ C 1 (E), we say J ′ is weakly sequentially continuous, if un ⇀ u implies J ′ (un ) → J ′ (un ), as n → ∞. The purpose of this paper is to use the generalized saddle point theorem to solve some strongly indefinite problems with asymptotically linear nonlinearity. The following lemma is the generalized saddle point theorem taken from [9] and will play an important role in the proofs of our main results. Lemma 2.1. Assume that J ∈ C 1 (E, R) is τ −upper semicontinuous and J ′ is weakly sequentially continuous. If d = sup J < ∞,

b := inf+ J > sup J, E

M

∂M

then for some c ∈ [b, d], there is a sequence {un } ⊂ E such that J(un ) → c and J ′ (un ) → 0 as n → ∞.

(2.7)

Such a sequence is called a Palais-Smale sequence on the level c, or a (P S)c sequence.

3

Proofs of main results

Lemma 3.1. Assume that (V ) and (f1 ) − (f3 ) are satisfied. Then J is τ −upper semicontinuous, and J ′ is weakly sequentially continuous. τ

Proof. Let u(k) − → u and c = lim J(u(k) ). Then there is a subsequence, still denoted by k→∞

{u(k) } such that J(u(k) ) → c. By Remark 2.1 we have u(k)+ → u+

and u(k)− ⇀ u− ,

as k → ∞.

(3.1)

(k)

Passing to a subsequence if necessary, we have un → un for all n ∈ Z, as k → ∞, (k) hence, Fn (un ) → Fn (un ). Since Fn (u(k) ) ≥ 0, using the Fatou lemma we have ∑ ∑ (k) I(u) = lim Fn (u(k) Fn (u(k) (3.2) n ) ≤ lim n ) = lim I(u ). n∈Z

k→∞

k→∞

n∈Z

k→∞

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Combining (3.1) and (3.2), we have ∥u− ∥2 ∥u+ ∥2 − + I(u) 2( 2 ) ∥u(k)− ∥2 ∥u(k)+ ∥2 (k) ≤ lim − + I(u ) 2 2 k→∞ ( ) = lim −J(u(k) ) = −c.

−J(u) =

k→∞

So J(u) ≥ c and J is τ −upper semicontinuous. Finally, we show that J ′ is weakly sequentially continuous. Let u(k) ⇀ u in E, we have (k) that un → un for all n ∈ Z, as k → ∞. and there exists M > 0 such that ∥u(k) ∥ ≤ M and ∥u∥ ≤ M . By (f3 ), there exists constant C0 such that |fn (u)| ≤ C0 |u| for |u| ≤ M . ∑ 2 For any v ∈ E fix 0 < N ∈ N such that |n|>N |vn |2 < 16Cε2 M 2 . Therefore, we have 0

′

′

|I (u )v − I (u)v| ≤ | (k)

N ∑

(fn (u(k) n ) − fn (un ))vn | + |

≤ |

(fn (u(k) n ) − fn (un ))vn |

|n|>N

n=−N N ∑

∑

(k)  (fn (u(k) n ) − fn (un ))vn | + C0 (∥u ∥ + ∥u∥)

∑

 12 |vn |2 

|n|>N

n=−N

≤ |



N ∑

ε (fn (u(k) n ) − fn (un ))vn | + . 2 n=−N

(k)

Note that fn (un ) → fn (un ), as k → ∞, then there exists k0 such that for k ≥ k0 , |

N ∑

ε (fn (u(k) n ) − fn (un ))vn | < . 2 n=−N

So |I ′ (u(k) )v−I ′ (u)v| < ε, for all k ≥ k0 . By the definition of J ′ , then J ′ is weakly sequentially continuous.  Proof of Theorem 2.1. By (f2 ) and (f3 ), for any ε > 0, there exists Cε > 0 such that |fn (u)| ≤ Cε |u|,

|Fn (u)| ≤ Cε |u|2 .

For u ∈ E + , we have J(u) =

∑ 1 ∥u∥2 − Fn (un ) 2 n∈Z

1 ∥u∥2 − Cε ∥u∥2 2 1 = ( − Cε )∥u∥2 . 2

≥

6

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So inf E + J > −∞. For u ∈ E − , since F (u) ≥ 0, we have ∑ 1 J(u) = − ∥u∥2 − Fn (un ) 2 n∈Z 1 ≤ − ∥u∥2 . 2 For R large enough, we have inf+ J > sup J, E

sup J < ∞, M

∂M

where M = {u ∈ E − : ∥u∥ ≤ R}. By Lemma 2.1, for some c ∈ R, there is a sequence {u(k) } such that J(u(k) ) → c and J ′ (u(k) ) → 0 as k → ∞. Let u e(k) = u(k)+ − u(k)− , then ∥e u(k) ∥ = ∥u(k) ∥ and ∥u(k) ∥ = ∥e u(k) ∥ ≥ (J ′ (u(k) ), u e(k) ) ∑ = ∥u(k)+ ∥2 + ∥u(k)− ∥2 − fn (u(k) u(k) n )e n ≥ ∥u ∥ − (k) 2

∑

n∈Z (k)+ Cε |u(k) | n |(|un

+ |u(k)− |) n

n∈Z

≥ ∥u ∥ − Cε ∥u(k) ∥∥u(k)+ ∥ − Cε ∥u(k) ∥∥u(k)− ∥ = ∥u(k) ∥2 − Cε ∥u(k) ∥2 . (k) 2

It implies {u(k) } is bounded. Next we may extract a subsequence, still denoted by {u(k) }, such that u(k) ⇀ u and (k) un → un for all n ∈ Z. Moreover, we have (J ′ (u), v) = lim (J ′ (u(k) ), v) = 0, ∀v ∈ E, k→∞

so J ′ (u) = 0 and u is a homoclinic solution of (1.1).  Acknowledgments This work is Supported by National Natural Science Foundation of China(11526183), the Natural Science Foundation of Shanxi Province (2015021015) and Foundation of Yuncheng University(YQ-2017003, YQ-2014011).

References [1] A. Davydov, The theory of contraction of proteins under their excitation, J. Theor. Biol. 38 (1973) 559-569. 7

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[2] S. Flach, A. Gorbach, Discrete breakers-Advances in theory and applications, Phys. Rep. 467 (2008) 1-116. [3] S. Flash, C. Willis, Discrete breathers, Phys. Rep. 295 (1998) 181-264. [4] A. V. Gorbach, M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schr¨odinger model, Eur. Phys. J. D 29 (2004) 77-93. [5] D. Henning, G. P. Tsironis, Wave transmission in nonlinear lattics, Phys. Rep. 307 (1999) 333-432. [6] P. G. Kevrekidis, K. ϕ. Rasmussen, A. R. Bishop, The discrete nonlinear Schr¨odinger equation: a survey of recent results, Int. J. Mod. Phys. B 15 (2001) 2883-2900. [7] Y. S. Kivshar, G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego, 2003. [8] G. Li, A. Szulkin, An asymptotically periodic Schr¨odinger equation with indefinite linear part, Commun. Contemp. Math. 4 (2002) 763-776. [9] S. Liu, Z. Shen, Generalized saddle point theorem and asymptotically linear problems with periodic potential, Nonlinear Anal. 86 (2013):52-57. [10] A. Mai, G. Sun, Ground state solutions for second order nonlinear p-Laplacian difference equations with periodic coefficients, J. Comput. Anal. Appl.,2017, 22(7): 1288-1297. [11] A. Mai, Z. Zhou, Discrete solitons for periodic discrete nonlinear Schr¨odinger equations, Appl. Math. Comput. 222(2013): 34-41. [12] A. Pankov, N. Zakharchenko, On some discrete variational problems, Acta. Appl. Math. 65 (2000) 295-303. [13] W. Su, J. Schieffer, A. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42 (1979) 1698-1701. [14] G. Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. [15] A. A. Sukhorukov, Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett. 28 (2003) 2345-2347. [16] H. Shi, Gap solitons in periodic discrete Schr¨odinger equations with nonlinearity, Acta Appl. Math. 109 (2010) 1065-1075. [17] H. Shi, H. Zhang, Existence of gap solitons in periodic discrete nonlinear Schr¨odinger equations, J. Math. Anal. Appl. 361 (2010) 411-419. [18] M. Weintein, Excitation thresholds for nonlinear localized modes on lattices, Nonlinearity 12 (1999) 673-691. [19] M. Willem, Minimax Theorems, Birkh¨auser, Boston, 1996. 8

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APPROXIMATION OF ALMOST CAUCHY’S POINTS BY CAUCHY’S POINTS GWANG HUI KIM AND HWAN-YONG SHIN

Abstract. In this paper, we investigate Hyers–Ulam stability of Cauchy’s mean value points which is a extended and generalized version of I. R. Peter and D. Popa’s theorem [10] and then, as applications, we obtain Hyers-Ulam stability results of Lagrange’s mean value points which refine the result of P. Gˇ avrutˇ a, J. Huang and Y. Li [5].

1. Introduction The concept of Hyers–Ulam stability was raised by S. M. Ulam [11] in 1940. We are given a group G and a metric group G0 with metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if f : G → G0 satisfies d(f (xy), f (x)f (y)) < δ for all x, y ∈ G, then a homomorphism h : G → G0 exists with d(f (x), h(x)) < ε for all x ∈ G? Ulam’s question was partially solved by D. H. Hyers [6] in the case of approximately additive functions and when the groups in the question are Banach spaces. Due to the question of Ulam and the answer of Hyers, the stability of functional equations is called after their names. For more information of Hyers–Ulam stability, we can refer to [1, 2]. A similar problem of Ulam’s question can be formulated for the mean value points : “Assume that a function f satisfies a mean value theorem with a point η. If ξ is a point near to η, does there exists a function g near to f satisfying the same mean value theorem with the point ξ?” [10]. It seems that the first result to the previous question was given by D. H. Hyers and S. M. Ulam [7] in the case of differential expressions. Theorem 1.1. (D. H. Hyers, S. M. Ulam, 1954, [7]) Let f : R → R be n-times differentiable in a neighborhood N of a point η. Suppose that f (n) (η) = 0 and f (n) (x) changes sign at η. Then, for all ε > 0, there exists a δ > 0 such that for every function g : R → R which is n-times differentiable in N and satisfies |f (x) − g(x)| < δ for all x ∈ N , there exists a point ξ ∈ N such that g (n) (ξ) = 0 and |ξ − η| < ε. 2010 Mathematics Subject Classification. 39B52, 39B82, 54C65. Key words and phrases. mean value theorem, Cauchy’s mean value points, Lagrange’s mean value points, Hyers–Ulam stability. 1

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In 2003, M. Das, T. Riedel and P. K. Sahoo [3] proved the stability problem for Flett’s mean value points by using Theorem 1.1. Subsequently, some authors applied the idean from [3] to prove the Hyers–Ulam stability of various mean value points [5, 8, 9, 10]. Especially, P. Gˇavrutˇ a, S.-M. Jung and Y. Li [5] proved the following stability result of Lagrange’s mean value points which is a point η of a differentiable function f : [a, b] → R satisfying f (b)−f (a) = f 0 (η). b−a Theorem 1.2. (P. Gˇ avrutˇ a, S.-M. Jung, Y. Li, 2010, [5]) Let a, b, η be real numbers satisfying a < η < b. Assume that f : R → R is a twice continuously differentiable function and η is the unique Lagrange’s mean value point of f in an open interval (a, b) and moreover that f 00 (η) 6= 0. Suppose g : R → R is a differentiable function. Then, for a given ε > 0, there exists a δ > 0 such that if |f (x) − g(x)| < δ for all x ∈ [a, b], then there is a Lagrange’s mean value point ξ ∈ (a, b) of g with |ξ − η| < ε. Hereafter, Theorem 1.2 was generalized by I. R. Peter and D. Popa [10] by proving the stability of Cauchy’s mean value points which is a point η of two differentiable functions f, g : [a, b] → R satisfying (f (b) − f (a))g 0 (η) − (g(b) − g(a))f 0 (η) = 0. Let I be an open interval which contains the interval (a, b). Theorem 1.3. (I. R. Peter, D. Popa, 2013, [10]) Assume that f, g : I → R are continuously differentiable functions, η is the unique Cauchy’s mean value point of the pair (f, g) in I and f, g are twice continuously differentiable in a neighborhood of η, satisfying f 00 (η)(g(b) − g(a)) − g 00 (η)(f (b) − f (a)) 6= 0. Then, for every ε > 0 there exists δ > 0 such that, if f1 , g1 : (a, b) → R are continuously differentiable functions with the property that |f (x) − f1 (x)| < δ and |g(x) − g1 (x)| < δ for all x ∈ [a, b] there exists a Cauchy mean value point ξ ∈ (a, b) of (f1 , g1 ) with |η − ξ| < ε. In this paper, we prove Hyers–Ulam stability of Cauchy’s mean value points which is a extended and generalized version of Theorem 1.3 and then, as applications, we obtain the stability results of Lagrange’s mean value points which refine Theorem 1.2. 2. Hyers–Ulam Stability of Cauchy’s mean value points We now present a main theorem, which is a Hyers–Ulam stability of Cauchy’s mean value points for real-valued differentiable functions on [a, b].

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Theorem 2.1. Let f, g, f1 , g1 : [a, b] → R be countinuously differentiable functions and η be a Cauchy’s mean value point of the pair (f, g) in the interval (a, b) and N ⊆ (a, b) be a neighborhood of η. Suppose the following control function (f (b) − f (a))g 0 (x) − (g(b) − g(a))f 0 (x) changes sign at η. Then, for a given ε > 0, there exists a δ > 0 such that if |f (x) − f1 (x)| < δ and |g(x) − g1 (x)| < δ for all x ∈ N ∪ {a, b}, then there exists a point ξ ∈ N such that ξ is a Cauchy’s mean value point of (f1 , g1 ) with |ξ − η| < ε. Proof. Let ε > 0 be given and N ⊆ (a, b) be any neighborhood of η. Consider the auxiliary function Gf,g (x) : [a, b] → R corresponding to (f, g) defined by Gf,g (x) = (f (b) − f (a))g(x) − (g(b) − g(a))f (x) for all x ∈ [a, b]. Evidently Gf,g (x) is continuous on [a, b] and differentiable on [a, b]. Further, we have G0f,g (x) = (f (b) − f (a))g 0 (x) − (g(b) − g(a))f 0 (x),

x ∈ [a, b].

Since η is the Cauchy’s mean value point of (f, g), we get G0f,g (η) = 0. Thus it follows from the assumption that there exists a neighborhood (η − r, η + r) ⊆ N of η such that G0f,g (x) changes sign at η in (η − r, η + r) ⊆ N for some r > 0 with η − r > a. Then it follows from Theorem 1.1 that there exists a δ > 0 such that for any differentiable function H on [a, b] with |H(x) − Gf,g (x)| < δ for x in (η − r, η + r), there exists a point ζ ∈ (η − r, η + r) satisfying H 0 (ζ) = 0 and |ζ − η| < ε. For a continuous function f : [a, b] → R define Mf := max{|f (x)| : x ∈ [a, b]} and analogously Mg . Define Gf1 ,g1 (x) : [a, b] → R be the corresponding auxiliary function defined as Gf1 ,g1 (x) = (f1 (b) − f1 (a))g1 (x) − (g1 (b) − g1 (a))f1 (x) for all x ∈ [a, b]. For some fixed λ > 0, let δ := min

n

o δ ,λ . 4Mf + 4Mg + 4λ

and let f1 , g1 : [a, b] → R be any differentiable functions satisfying |f (x) − f1 (x)| < δ and |g(x)−g1 (x)| < δ for all x ∈ N ∪{a, b}. Then one can easy to see that Gf1 ,g1 (x) is differentiable

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in N . And it follows that |f1 (b) − f1 (a)| ≤ |f1 (b) − f (b)| + |f (b) − f (a)| + |f (a) − f1 (a)| ≤ 2λ + 2Mf . By the same reason we obtain that |g1 (b) − g1 (a)| ≤ 2λ + 2Mg . These yield that |Gf,g (x) − Gf1 ,g1 (x)| = |(f (b) − f (a))g(x) − (g(b) − g(a))f (x) −(f1 (b) − f1 (a))g1 (x) + (g1 (b) − g1 (a))f1 (x)| = |(f (b) − f (a))g(x) − (f1 (b) − f1 (a))g(x) +(f1 (b) − f1 (a))g(x) − (f1 (b) − f1 (a))g1 (x) +(g1 (b) − g1 (a))f1 (x) − (g1 (b) − g1 (a))f (x) +(g1 (b) − g1 (a))f (x) − (g(b) − g(a))f (x)| ≤ (|f (b) − f1 (b)| + |f (a) − f1 (a)|)|g(x)| +|f1 (b) − f1 (a)| · |g(x) − g1 (x)| +|g1 (b) − g1 (a)| · |f1 (x) − f (x)| +(|g1 (b) − g(b)| + |g1 (a) − g(a)|)|f (x)| ≤ (2Mg + |f1 (b) − f1 (a)| + |g1 (b) − g1 (a)| + 2Mf )δ ≤ (4Mf + 4Mg + 4λ)δ ≤ δ for all x ∈ (η − r, η + r) ⊆ N . Hence, there exists a point ξ ∈ (η − r, η + r) such that G0f1 ,g1 (ξ) = 0 and |ξ − η| < ε. We note that G0f1 ,g1 (ξ) = 0 implies (f1 (b) − f1 (a))g10 (ξ) − (g1 (b) − g1 (a))f10 (ξ) = 0. Hence, the point ξ is a Cauchy’s mean value point of (f1 , g1 ) and the proof is complete.



The following corollary is a refined result of Theorem 1.3. Corollary 2.2. Let f, g, f1 , g1 : [a, b] → R be countinuously differentiable functions and η be a Cauchy’s mean value point of the pair (f, g) in the interval (a, b) and N ⊆ (a, b) be a neighborhood of η. Suppose either η is unique Cauchy’s mean value point of (f, g) or f , g have second derivative at η such that (2.1)

[f (b) − f (a)]g 00 (η) 6= [g(b) − g(a)]f 00 (η).

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Then, for a given ε > 0, there exists a δ > 0 such that if |f (x)−f1 (x)| < δ and |g(x)−g1 (x)| < δ for all x ∈ N ∪ {a, b}, then there exists a point ξ ∈ N such that ξ is a Cauchy’s mean value point of (f1 , g1 ) with |ξ − η| < ε. Proof. Let Gf,g : [a, b] → R be defined as Gf,g (x) = (f (b) − f (a))(g(x) − g(a)) − (g(b) − g(a))(f (x) − f (a)) for all x ∈ [a, b]. Suppose η is a unique Cauchy’s mean value point of (f, g). Then we obtain that Gf,g (a) = Gf,g (b) and η ∈ (a, b) is a unique point such that G0f,g (η) = 0. These yield that G0f,g (x) changes sign at η. If f and g have second derivative and satisfy (2.1), we have G00f,g (η) 6= 0. Thus associating this fact and G0f,g (η) = 0, we get G0f,g (x) changes sign at η. Rewriting the fact Gf,g changes sign at η, we obtain (f (b) − f (a))g 0 (x) − (g(b) − g(a))f 0 (x) changes sign at η. By applying Theorem 2.1, we get the desired result.



If we take f1 , g1 : [a, b] → R by f1 := h and g1 := g in Theorem 2.1 and Corollary 2.2, then we get the following two corollaries. Corollary 2.3. Let f, g, h : [a, b] → R be differentiable and η be a Cauchy’s mean value point of the pair (f, g) in the interval (a, b) and N ⊆ (a, b) be a neighborhood of η. Suppose the following control function (f (b) − f (a))g 0 (x) − (g(b) − g(a))f 0 (x) changes sign at η. Then, for a given ε > 0, there exists a δ > 0 such that if |f (x) − h(x)| < δ for all x ∈ N ∪ {a, b}, then there exists a point ξ ∈ N such that ξ is a Cauchy’s mean value point of (g, h) with |ξ − η| < ε. Corollary 2.4. Let f, g, h : [a, b] → R be countinuously differentiable functions and η be a Cauchy’s mean value point of the pair (f, g) in the interval (a, b) and N ⊆ (a, b) be a neighborhood of η. Suppose either η is a unique Cauchy’s mean value point of (f, g) or f , g have second derivative at η such that (f (b) − f (a))g 00 (η) 6= (g(b) − g(a))f 00 (η). Then, for a given ε > 0, there exists a δ > 0 such that if |f (x)−h(x)| < δ for all x ∈ N ∪{a, b}, then there exists a point ξ ∈ N such that ξ is a Cauchy’s mean value point of (g, h) with |ξ − η| < ε. The following theorem is another type of Hyers-Ulam stability for Cauchy’s mean value points.

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Theorem 2.5. Let a, b, ξ be real numbers satisfying a < ξ < b. Assume that f, g : [a, b] → R are countinuously differentiable functions such that g 0 (x),

f 0 (x)g 00 (x) − f 00 (x)g 0 (x) 6= 0 g 0 (x)2

for all x ∈ [a, b]. If f 0 (ξ) f (b) − f (a)) − ≤ε 0 g (ξ) g(b) − g(a)

(2.2)

for some ε > 0, then there exists a Cauchy’s mean value point η of (f, g) on (a, b) satisfying ε 0 00 . |ξ − η| ≤ 00 (x)g 0 (x) minx∈[a,b] f (x)g (x)−f g 0 (x)2 Proof. Due to Cauchy’s mean value theorem, there exists a point η ∈ (a, b) such that f (b) − f (a) f 0 (η) = . g 0 (η) g(b) − g(a) Hence it follows from (2.2) that f 0 (ξ) f 0 (η) − 0 0 ≤ ε. g (ξ) g (η) If ξ = η then the proof is clear. Otherwise, we assume that a < η < ξ < b. Since f and g have second derivative on [a, b], by Lagrange’s mean value theorem, there exists a point ξ0 ∈ (η, ξ) such that  f 0 (ξ )g 00 (ξ ) − f 00 (ξ )g 0 (ξ )  f 0 (η) f 0 (ξ) 0 0 0 0 − 0 . (ξ − η) = 0 g 0 (ξ0 )2 g (η) g (ξ) Since f 0 , f 00 , g 0 , g 00 are continuous on [a, b], we obtain 0

0

f (η) f (ξ) g 0 (η) − g 0 (ξ) |ξ − η| = f 0 (ξ )g00 (ξ )−f 00 (ξ )g0 (ξ ) ≤ 0 0 0 0 g 0 (ξ0 )2

ε 0 00 f (x)g (x)−f 00 (x)g0 (x) , minx∈[a,b] g 0 (x)2

which complete the proof.



3. Applications to Lagrange’s mean value points In this section, we obtain stability results of Lagrange’s mean value points for the differentiable functions on [a, b]. Corollary 3.1. Let f, g : [a, b] → R be countinuously differentiable functions and η be a Lagrange’s mean value point of f in (a, b) and N ⊆ (a, b) be a neighborhood of η. Suppose the following control function f (b) − f (a) − (b − a)f 0 (x)

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changes sign at η. Then, for a given ε > 0, there exists a δ > 0 such that if |f (x) − g(x)| < δ for all x ∈ N ∪ {a, b} there exists a point ξ ∈ N such that ξ is a Lagrange’s mean value point of g with |ξ − η| < ε. Proof. Consider the auxiliary function Gf (x) : [a, b] → R corresponding to f defined by Gf (x) = (f (b) − f (a))x − f (x)(b − a) for all x ∈ [a, b]. Then the proof goes through the same way as that of Theorem 2.1.



Example 3.2. Let f : [−2π, 2π] → R be defined by  cos x − 1, if x ≤ 0, f (x) = 1 − cos x, if x > 0. It is obvious to see that there exist three Lagrange’s mean value points −π, 0, π of f . Let Ni be a neighborhood of (−1)i π for each i = 1, 2. We can easily check that f (2π)−f (−2π)−(2π− (−2π))f 0 (x) = −4πf 0 (x) changes sign at ±π. Therefore, by Corollary 3.1, for each ε > 0, there exists a δ > 0 such that for every differentiable function g satisfying |f (x)−g(x)| < δ for all x ∈ Ni ∪ {±2π} then there exists a point ξi ∈ Ni such that ξi is a Lagrange’s mean value point of g and |ξi − (−1)i π| < ε. However, f (2π) − f (−2π) − (2π − (−2π))f 0 (x) = −4πf 0 (x) does not change sign at 0, and so we cannot apply Corollary 3.1 for the function f at the Lagrange’s mean value point 0. π π Let N := (− , ) and δ > 0 be given. And let g : [−2π, 2π] → R be defined by 4 4 δ g(x) := f (x) + (x3 − 4π 2 x) 1024 for all x ∈ [−2π, 2π]. Then δ δ |x3 − 4π 2 x| < ((2π)3 + 4π 2 (2π)) < δ 1024 1024 for all x ∈ N ∪ {±2π}. But, for all x ∈ N , the following inequlity holds |f (x) − g(x)| =

g(2π) − g(−2π) − g 0 (x) > 0. 4π Therefore, we can conclude that there is no Lagrange’s mean value point of g in N . The following refined result of Theorem 1.2 is obtained as a corollary of Corollary 3.1. Corollary 3.3. Let f, g : [a, b] → R be countinuously differentiable functions and η be a Lagrange’s mean value point of f in (a, b) and N ⊆ (a, b) be a neighborhood of η. Suppose either η is a unique Lagrange’s mean value point of f or f has second derivative at η with f 00 (η) 6= 0. Then, for a given ε > 0, there exists a δ > 0 such that if |f (x) − g(x)| < δ for all

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x ∈ N ∪ {a, b}, then there exists a point ξ ∈ N such that ξ is a Lagrange’s mean value point of g with |ξ − η| < ε. References [1] J. Brzdek, W. Fechner, M. Moslehian, J. Sikorska, Recent developments of the conditional stability of the homomorphism equation, Banach J. Math. Anal., 9 (2015) no. 3, 278-326. [2] K. Ciepli´ nski, Applications of fixed point theorems to the Hyers-Ulam stability of functional equations–a survey, Ann. Funct. Anal., 3 (2012), no. 1, 151-164. [3] M. Das, T. Riedel, P. K. Sahoo, Hyers-Ulam stability of Flett’s points, Appl. Math. Lett., 16(3) (2013), 269-271. [4] P. Gˇ avrutˇ a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. [5] P. Gˇ avrutˇ a, S.-M. Jung, Y. Li, Hyers–Ulam stability of mean value points, Ann. Funct. Anal., 295 (2010), no. 2, 68-74. [6] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci, U.S.A. 27, (1941), 222-224. [7] D. H. Hyers, S. M. Ulam, On the stability of differential expressions, Math. Mag., 28, (1954), 59-64. [8] H.-M. Kim, H.-Y. Shin, Approximation of almost Sahoo–Riedel’s points by Sahoo–Riedel’s points, Aequat. Mathe., 90, (2016), 809-815. [9] W. Lee, S. Xu, F. Ye, Hyers-Ulam stability of Sahoo-Riedel’s point, Appl. Math. Lett., 22, (2009), 16491652. [10] I. R. Peter and D. Popa, Stability of points in mean value theorems, Publ. Math. Debrecen, 83/3 (2013), 375-384. [11] S. M. Ulam, Problems in Modern Mathematics, Chapter 6 Wiley Interscience, New York, (1964). Gwang Hui Kim, Department of Mathematics, Kangnam Universaty,Yongin, Gyeonggi 16979, Republic of Korea E-mail address: [email protected] Hwan-Yong Shin, Department of Mathematics, Chungnam National University,99 Daehangno, Yuseong-gu, Daejeon 34134, Republic of Korea E-mail address: [email protected]

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Weak Galerkin Finite Element Method for Convection-Diffusion-Reaction Problems F. Z. Gaoa,1 , A. K. Hashimb,2 , S. C. Mohammedb,3 a

School of Math., Shandong Univ., Jinan, China of Math., College of Education for Pure Science Univ. of Basrah, Basrah, Iraq

b Dept.

Abstract In this paper, a weak Galerkin (WG) finite element method is proposed for solving the convection-diffusion-reaction problems. The main idea of WG finite element methods is the use of weak functions and their corresponding discrete weak derivatives in standard weak form of the model problem. We show that the continuous time WG finite element method preserves the energy conservation law as well the optimal order error estimate in L2 norm. Numerical experiment is conducted to confirm the theoretical results.

Keywords: WG finite element method, convection-diffusion-reaction equation, energy conservation law, error estimate.

1

Introduction

The convection-diffusion-reaction processes appear in many areas of science and technology. For example, fluid dynamics, heat and mass transfer hydrology and so on. In this paper, we consider the following convection-diffusion-reaction equation: ut − ∇ · (λ∇u) + b · ∇u + cu = f, u(x, 0) = 0, u(x, t)|Γ = g,

(x, t) ∈ Ω × (0, T ],

(1.1)

x ∈ Ω,

(1.2)

t ∈ (0, T ],

(1.3)

where Ω is a bounded region in R2 , with a Lipschitz continuous boundary Γ = ∂Ω, ut = ∂u ∂t , and ∇u denote the gradient of function u = u(x, t). Further λ > 0 is a diffusion coefficient, b is a convection coeffient and f, g are given functions. The standard weak form of equations (1.1) − (1.3) seeks u ∈ L2 (0, T ; H 1 (Ω)) such that u = g on ∂Ω × (0, T ) and (ut , v) + (λ∇u, ∇v) − (bu, ∇v) + (cu, v) = (f, v),

∀v ∈ H01 (Ω).

(1.4)

The WG finite element method refers to a general finite element technique for partial differential equation where the differential operators (e.g., gradient, divergence, curl, Laplacian) are approximated by weak forms. The method, first introduced by Wang and Ye [1] for solving a second order elliptic problems, is a newly developed finite element method. Since 1

E-mail address: [email protected] E-mail addresses: [email protected] 3 Corresponding author. E-mail address: [email protected] 2

1

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then, some WG finite element methods have been developed to solve other problems, such as parabolic equation [2, 3, 4], Stokes equations [5, 6], Helmholtz equation [7], Biharmonic equation [8, 9] and Navier-Stokes equations [10, 11], etc. In general, WG finite element formulations for partial differential equation can be derived naturally by replacing usual derivatives by variational forms. The implementations of all these possible extension are based on the computation of these weak operators. The rest of this paper is organized as follows. In section 2, we shall introduce some preliminaries and notations for Sobolev spaces. We define the weak gradient and discrete weak gradient operator and the weak finite element spaces and present semi-discrete WG finite element method for problem (1.1) − (1.3) in section 3 and section 4, respectively. In section 5, we prove the energy conservation law of the continuous time WG approximation, and in section 6 we present optimal order error estimate in L2 norm for the WG finite element approximations. Finally, we present a numerical example to verify theory.

2

Preliminaries and notations

We use standard definitions for the Sobolev spaces H m (Ω) and their associated inner products (·, ·)m,Ω , norms k · km,Ω , and seminorms | · |m,Ω for m ≥ 0 [12, 13]. For any integers m ≥ 0 the seminorm | · |m,Ω is given by X Z |v|m,Ω = ( |∂ α v|2 dΩ)1/2 , |α|=m Ω

with the usual notation α = (α1 , α2 ),

|α| = α1 + α2 ,

∂ α = ∂xα11 ∂xα22 .

The Sobolev norm k · kn,Ω , is given by kvkn,Ω

n X =( |v|2j,Ω )1/2 . j=0

The space H(div; Ω) is defined as the set of vector-valued functions on Ω which, together with their divergence, are square integrable; i.e, H(div; Ω) = {v : v ∈ [L2 (Ω)]2 , ∇ · v ∈ L2 (Ω)}. The norm in H(div; Ω) is defined by kvkH(div;Ω) = (kvk2 + k∇ · vk2 )1/2 .

3

A weak Gradient operator and its approximation

In this section we introduce a weak gradient operator defined on a space of generalized functions. Let K be any polygonal domain with interior K 0 and boundary ∂K. A weak function on the region K refers to vector-valued function v = {v0 , vb } such that v0 ∈ L2 (K) and vb ∈ H 1/2 (∂K). The first component v0 can be understood as the value of v in interior 2

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of K, and the second component vb is the value of v on the boundary of ∂K. Denote by W (K) the space of weak function associated with K ; i.e., W (K) := {v = {v0 , vb } : v0 ∈ L2 (K), vb ∈ H 1/2 (∂K)}.

(3.1)

Definition 3.1. For any v ∈ W (K), the weak gradient of v is defined as a linear functional ∇d v in the dual space of H(div, K) whose action on each q ∈ H(div, K) is given by Z Z Z ∇d v · qdK = − v0 ∇ · qdK + vb q · nds, (3.2) K

K

∂K

where n is the outward normal direction to ∂K. Next, we introduce a discrete weak gradient operator by defining ∇d in a polynomial subspace of H(div, K). To this end , for any non-negative integer r ≥ 0 denote by Pr (K) the set of polynomials on K with degree no more than r. Let V (K, r) ⊂ [Pr (K)]2 be a subspace of the space of vector-valued polynomials of degree r. A discrete weak gradient operator, denoted by ∇d,r , is defined so that ∇d,r v ∈ V (K, r) is the unique solution of the following equation Z Z Z ∇d,r v · qdK = − v0 ∇ · qdK + vb q · nds, ∀q ∈ V (K, r). (3.3) K

4

K

∂K

A weak Galerkin finite element scheme

Let Th be triangular partition of the domain Ω with mesh size h. Assume that the partition Th is shape regular so that the routine inverse inequality holds true (see[13]). In the general spirit of Galerkin procedure, we shall design a WG method for (1.4) by following two basic principles: first replacing H 1 (Ω) by a space of discrete weak functions defined on the finite element partition Th and the boundary of triangular elements; second replacing the classical gradient operator by a discrete weak gradient operator ∇d,r for weak functions on each triangle T . For each T ∈ Th . Denote by Pj (T 0 ) the set of polynomials with degree no more than j and Pℓ (∂T ) the set of polynomial on ∂T with degree no more than ℓ. A discrete weak function v = {v0 , vb } on T refers to a weak function v = {v0 , vb } such that v0 ∈ Pj (T 0 ) and vb ∈ Pℓ (∂T ) with j ≥ 0 and ℓ ≥ 0. Denote this space by W (T, j, ℓ), i.e., W (T, j, ℓ) = {v = {v0 , vb } : v0 ∈ Pj (T 0 ), vb ∈ Pℓ (∂T )}.

(4.1)

The corresponding finite element space would be defined by patching W (T, j, ℓ) over all the triangles T ∈ Th . In other words, the weak finite element space is given by Sh (j, ℓ) = {v = {v0 , vb } : {v0 , vb }|T ∈ W (T, j, ℓ), ∀T ∈ Th }.

(4.2)

Denote by Sh0 (j, ℓ) the subspace of Sh (j, ℓ) with vanishing boundary values on ∂Ω, i.e., Sh0 (j, ℓ) = {v = {v0 , vb } ∈ Sh (j, ℓ), vb |∂T ∩∂Ω = 0, ∀T ∈ Th }.

(4.3)

To investigate the approximation properties of the discrete weak space Sh (j, ℓ), we define three projections in this paper. The first two are local projections defined on each triangle T : 3

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one is Qh u = {Q0 u, Qb u}, the L2 projection of H 1 (T ) onto Pj (T 0 ) × Pj+1 (∂T ) and another is Rh , the L2 projection of [L2 (T )]2 onto V (T, r). The third projection Πh is assumed to exist and satisfy the following property: for q ∈ H(div, Ω) with mildly added regularity, Πh q ∈ H(div, Ω) such that Πh q ∈ V (T, r) on each T ∈ Th , and (∇ · q, v0 )T = (∇ · Πh q, v0 )T , ∀v0 ∈ Pj (T ).

(4.4)

It is easy to see the following two useful identities: ∇d,r (Qh u) = Rh (∇u), ∀u ∈ H 1 (T ),

(4.5)

and for any q ∈ H(div, Ω) X X (−∇ · q, v0 )T = (Πh q, ∇d,r v)T , ∀v = {v0 , vb } ∈ Sh0 (j, ℓ). T ∈Th

(4.6)

T ∈Th

Now for any u, v ∈ Sh (j, ℓ), we introduce the following bilinear form a(u, v) = (λ∇d,r u, ∇d,r v) − (bu0 , ∇d,r v) + (cu0 , v0 ), where (λ∇d,r u, ∇d,r v) = (bu0 , ∇d,r v) =

Z

ZΩ Z

(4.7)

λ∇d,r u · ∇d,r vdΩ, bu0 · ∇d,r vdΩ,

Ω

(cu0 , v0 ) =

cu0 v0 dΩ.

Ω

We pose the continuous time WG finite element method based on (3.3) and (1.4) which is to find uh (t) = {u0 (·, t), ub (·, t)}, belonging to Sh (j, ℓ)) for t > 0, satisfying ub = Qb g on ∂Ω, and the following equation ((uh )t , v0 ) + a(uh , v) = (f, v0 ),

∀v = {v0 , vb } ∈ Sh0 (j, ℓ),

(4.8)

where a(uh , v) = (λ∇d,r u, ∇d,r v) − (buh , ∇d,r v) + (cuh , v0 ), where, Qb g is an approximation of the boundary value in the polynomial space Pℓ (∂T ∩ ∂Ω). For simplicity, Qb g shall be taken as the standard L2 projection for each boundary segment.

5

Energy conservation property of WG

In this section, we investigate the energy conservation property of the semi-discrete WG finite element approximation uh . The solution u of the problem (1.1) − (1.3) has the following energy preserving property on each K ∈ Th [2]. Z

t+∆t Z

t−∆t

K

ut dxdt +

Z

t+∆t Z

t−∆t

q · ndsdt = ∂K

Z

t+∆t Z

t−∆t

f dxdt,

(5.1)

K

where q = −λ∇u + bu is the flow rate of heat energy. 4

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We claim that the semi-discrete WG for (1.1) − (1.3) preserves the energy conservation property in (5.1). Choosing in (4.8) the test function v = {v0 , vb = 0} so that v0 = 1 on K and v0 = 0 elsewhere. We then obtain by integration over the time period [t − ∆t, t + ∆t] Z

t+∆t Z

t−∆t

ut dxdt +

K

Z

t+∆t

a(uh , v)dt =

t−∆t

Z

t+∆t Z

t−∆t

f dxdt,

(5.2)

K

where Z

a(uh , v) =

Z

λ∇d,r uh · ∇d,r vdx − K

bu0 · ∇d,r vdx +

K

Z

cu0 dx. K

Using the definition of operators Rh and ∇d,r in (4.4), we obtain Z Z λ∇d,r uh · ∇d,r vdx = Rh (λ∇d,r uh ) · ∇d,r vdx K KZ = − ∇ · Rh (λ∇d,r uh )dx ZK = − Rh (λ∇d,r uh ) · nds,

(5.3)

∂K

and

Z

Z

bu0 · ∇d,r vdx =

Rh (bu0 ) · ∇d,r vdx

KZ

K

= − = −

ZK

∇ · Rh (bu0 )dx Rh (bu0 ) · nds.

(5.4)

∂K

Now substituting (5.3) and (5.4) into (5.2) yields Z

t+∆t Z

t−∆t

ut dxdt +

K

Z

t+∆t Z

t−∆t

Rh (−λ∇d,r uh + bu0 ) · nds =

∂K

Z

t+∆t Z

t−∆t

f dxdt,

K

which provides a numerical flux.

qh · n = Rh (−λ∇d,r uh + bu0 ) · n. The numerical flux qh · n can be verified to be continuous across the edge of each element K through a selection of the test function v = {v0 , vb } so that v0 ≡ 0 and vb are arbitrary.

6

Error analysis

In this section, we derive optimal order error estimate for the semi-discrete scheme (4.8) in L2 norm. Let us begin with proving the elliptic property of WG finite element method for equation (1.1).

5

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Lemma 6.1. Let Sh (j, ℓ) be the weak finite element space defined in (4.2) and a(uh , v) be the bilinear form given in (4.8). There exists positive constant α satisfying a(vh , vh ) ≥ α(k∇d,r vh k2 + kv0 k2 ), for all vh ∈ Sh (j, ℓ). Proof. Taking u = v in equation (4.8) we have a(vh , vh ) = (λ∇d,r v, ∇d,r v) − (bv0 , ∇d,r v) + (cv0 , v0 ).

(6.1)

Let A = kbkL∞ (Ω) and B = kckL∞ (Ω) be the L∞ -norm of the coefficients b and c, respectively and using Cauchy- Schwarz inequality we have. |(bv0 , ∇d,r v)| ≤ kbkL∞ (Ω) k∇d,r vkkv0 k, ≤ Ak∇d,r vkkv0 k

(6.2)

and |(cv0 , v0 )| ≤ kckL∞ (Ω) kv0 k2 ≤ Bkv0 k2 .

(6.3)

Substituting (6.2) and (6.3) into (6.1) we obtain a(vh , vh ) ≥ |λ|k∇d,r vk2 + Ak∇d,r vkkv0 k − Bkv0 k2 , by using Young-inequality, we have ǫA2 1 )k∇d,r vk2 + ( − B)kv0 k2 2ǫ 2 ≥ α1 k∇d,r vk2 + α2 kv0 k2

a(vh , vh ) ≥ (|λ| +

≥ α(k∇d,r vk2 + kv0 k2 ), where α = min{α1 , α2 }, which completes the proof. Lemma 6.2. ([2]) For u ∈ H 1+κ (Ω) with κ > 0, we have kΠh (λ∇u) − λRh (∇u)k ≤ Chκ kuk1+κ .

(6.4)

Lemma 6.3. ([14]) For u ∈ H 1+κ (Ω) with κ > 0, we have ku − Πh uk ≤ Chκ kuk1+κ .

(6.5)

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6.1

Continuous time WG finite element method

Our aim is to prove the following estimate in L2 norm for the semi-discrete approximation. Theorem 6.1. Let u ∈ H 1+κ (Ω) with κ > 0 and uh be the solutions of (1.1) − (1.3) and (4.8) respectively. Denote by e = uh − Qh u the difference between WG approximation and the L2 projection of the exact solution u = u(x, t). Then there exists a constant C such that Z t Z t kek2 + αkek2 ds ≤ ke(·, 0)k2 + Ch2κ kuk21+κ ds (6.6) 0

0

Proof. Let v = {v0 , vb } ∈ Sh0 (j, ℓ) be the testing function. By testing (1.1) − (1.3) against v0 , together with (4.6) we arrive at X X (f, v0 ) = (ut , v0 ) + (−∇ · (λ∇u), v0 )T + (∇ · (bu), v0 ) + (cu, v0 ) T ∈Th

T ∈Th

= (ut , v0 ) + (Πh (λ∇u), ∇d,r v) − (Πh (bu), ∇d,r v) + (cu, v0 ).

(6.7)

Adding and subtracting the term a(Qh u, v) ≡ (λ∇d,r (Qh u), ∇d,r v) − (b(Q0 ), ∇d,r v) + (c(Q0 u), v0 ), on the right hand side of the equation (6.7) and using (Qh ut , v0 ) = (ut , v0 ) we obtain (f, v0 ) = (Qh ut , v0 ) + (Πh (λ∇u) − λ∇d,r (Qh u), ∇d,r v) − (Πh (bu) − b(Q0 u), ∇d,r v) + (cu − c(Q0 u), v0 ) + (λ∇d,r (Qh u), ∇d,r v) − (b(Q0 ), ∇d,r ) + (c(Q0 u), v0 ), by using Rh (∇u) = ∇d,r (Qh u) for u ∈ H 1 and (4.8) we obtain ((uh )t , v0 ) + a(uh , v) = (Qh ut , v0 ) + (Πh (λ∇u) − λRh (∇u), ∇d,r v) − (Πh (bu) − b(Q0 u), ∇d,r v) + (cu − c(Q0 u), v0 ) + a(Qh u, v), which can be rewritten as ((uh − Qh )t , v0 ) + a(uh − Qh u, v) = (Πh (λ∇u) − λRh (∇u), ∇d,r v) −(Πh (bu) − b(Q0 u), ∇d,r v) + (cu − c(Q0 u), v0 ).

(6.8)

Equation (6.8) shall be called the error equation for the WG finite element method (4.8). Substituting v in (6.8) by e = {uh − Qh u} = {e0 , eb } = {u0 − Q0 u, ub − Qb u}, we have (et , e) + a(e, e) = (Πh (λ∇u) − λRh (∇u), ∇d,r e) − (Πh (bu) − b(Q0 u), ∇d,r e) + (cu − c(Q0 u), e). Hence 3 X 1 d 2 2 2 kek + βk∇d,r ek + αkek = I (i) , 2 dt

(6.9)

i=1

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where I (1) = (Πh (λ∇u) − λRh (∇u), ∇d,r e) I (2) = (Πh (bu) − bu0 , ∇d,r e) I (3) = (cu − c(Q0 u), e). To estimate I (1) , by Cauchy-Schwarz inequality and Young inequality, we have |I (1) | ≤

β 1 kΠh (λ∇u) − λRh (∇u)k2 + k∇d,r ek2 2β 2

by lemma (6.2), we have |I (1) | ≤ Ch2κ kuk21+κ +

β k∇d,r ek2 . 2

(6.10)

To estimate I (2) , by Cauchy-Schwarz inequality and Young inequality, we have |I (2) | ≤

1 β kΠh (bu) − bu0 k2 + k∇d,r ek2 , 2β 2

by lemma (6.3), we have |I (2) | ≤ Ch2κ kuk21+κ +

β k∇d,r ek2 . 2

(6.11)

To estimate I (3) , again by Cauchy-Schwarz inequality, Young inequality and lemma(6.3), we have α 1 kcu − c(Q0 u)k2 + kek2 2α 2 α 2κ 2 2 ≤ Ch kuk1+κ + kek . 2

|I (3) | ≤

(6.12)

Substituting (6.10), (6.11), and (6.12), into (6.9) we get 1 d α kek2 + βk∇d,r ek2 + αkek2 ≤ Ch2κ kuk21+κ + βk∇d,r ek2 + kek2 . 2 dt 2 It follows that d kek2 + αkek2 ≤ Ch2κ kuk21+κ . dt Thus, integrating with respect to t, we obtain Z t Z t kek2 + αkek2 ds ≤ ke(·, 0)k2 + Ch2κ kuk21+κ ds, 0

0

which completes the proof.

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6.2

Optimal order of error estimation in L2

To get an optimal order of error estimate in L2 , the idea, similar to Wheeler’s projection as in [14, 15], is used where an elliptic projection Eh onto the discrete weak space Sh (j, ℓ) is defined as the following: Find Eh u ∈ Sh (j, ℓ) such that Eh u is the L2 projection of the trace of u on the boundary ∂Ω and (λ∇d,r Eh u, ∇d,r w) + (b · ∇d,r Eh u, w) = (−∇ · (λ∇u), w) + (−bu, ∇w) ∀w ∈ Sh0 (j, ℓ).

(6.13)

In view of the weak formulation of the convection-diffusion-reaction problem. −∇ · (λ∇u) + b · ∇u = F, in Ω,

(6.14)

u = g, on ∂Ω,

(6.15)

this defined may be expressed by using that Eh u is the WG finite element approximation of the solution of the corresponding convection-diffusion problem with exact solution u. Lemma 6.4. (see[1]) Assume that problem (6.14)− (6.15) has the H 1+s (Ω) regularity (s ∈ (0, 1]). Let u ∈ H 1+κ (Ω) be the exact solution of (6.14)−(6.15), and Eh u be a WG approximation of u defined in (6.13). Let Qh u = {Q0 u, Qb u} be the L2 projection of u in the corresponding finite element space. Then there exists a constant C such that kQ0 u − Eh uk ≤ C(hκ+1 kF − Q0 F k + hκ+s kukκ+1 ) and k∇d,r (Qh u − Eh uk ≤ Chκ kukκ+1 . Theorem 6.2. Under the assumption of Theorem (6.1) and the assumption that the corresponding convection-diffusion problem has the H 1+s regularity (s ∈ (0, 1]), there exists a constant C such that Z t kuh (t) − Qh u(t)k ≤ kuh (0) − Qh u(0)k + Chκ+s (kψkκ+1 + kut kκ+1 ds) 0 Z t s+1 + Ch ( (kft − Q0 ft k + kutt − Q0 utt k)ds) 0

s+1

+ Ch

(kf (0) − Q0 f (0)k + kut (0) − Q0 ut (0)k)

(6.16)

Proof. The error in the problem (1.1) − (1.3) is written as a sum of two terms, uh (t) − Qh u(t) = θ(t) + ρ(t),

(6.17)

where θ = uh − Eh u,

ρ = Eh u − Qh u.

The error bound for ρ easily by lemma (6.4) as the following [2] kρk ≤ C(hs+1 (kf − Q0 f k + kut − Q0 ut k) Z t κ+s + h (kψkκ+1 + kut kκ+1 ds)).

(6.18)

0

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Now, to estimate θ, we note that by our definitions (θt , w) + a(θ, w) = ((uh )t , w) + a(uh , w) − (Eh ut , w) − a(Eh uh , w) = (f, w) − (Eh ut , w) − a(Eh uh , w) = (f, w) + (∇ · (λ∇u), w) + (b · ∇u, w) − (cu, w) − (Eh ut , w) = (ut , w) − (Eh ut , w) = (Qh ut , w) − (Eh ut , w) = −(ρt , w), which is ∀w ∈ Sh0 (j, ℓ), t > 0,

(θt , w) + a(θ, w) = −(ρt , w),

(6.19)

where we have used the fact that the operator Eh commutes with time differentiation. Since θ ∈ Sh0 (j, ℓ), we may choose w = θ in (6.19) and obtain (θt , θ) + a(θ, θ) = −(ρt , θ),

t > 0,

(6.20)

by using lemma (6.1) we have a(θ, θ) ≥ α(k∇d,r θk2 + kθ0 k2 ) > 0. Therefore 1d d kθk2 = kθk kθk ≤ kρt kkθk, 2 dt dt and integrating with respect to t, we obtain kθ(t)k ≤ kθ(0)k +

Z

t

kρt kds.

(6.21)

0

using lemma (6.3), we have |θ(0)k = kuh (0) − Eh u(0)k ≤ kuh (0) − Qh u(0)k + kEh u(0) − Qh u(0)k ≤ kuh (0) − Qh u(0)k + C(hs+1 (kf (0) − Q0 f (0)k + kut (0) − Q0 ut (0)k) + hκ+s kψkκ+1 ),

(6.22)

and since kρt k = kEh ut − Qh ut k ≤ C(hs+1 (kft − Q0 ft k + kutt − Q0 utt k) + hκ+s kut kκ+1 ).

(6.23)

Substituting (6.18) and (6.21) into (6.17), we have an optimal order of error estimate in L2 which completes the proof.

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7

Numerical result

In this section, we present some numerical results to illustrate the theoretical analysis in the previous section. We consider the following convection-diffusion-reaction problem. ut − ∇ · (D∇u) + b · ∇u + cu = f,

in Ω × J,

(7.1)

with homogeneous Dirichlet boundary condition and initial condition. The data for problem (7.1) taken as follows: let D = 100, Ω be a unit square, i.e., Ω = [0, 1] × [0, 1], time interval be J = (0, T ) = (0, 1), the absorption coefficient is c = 1 and the velocity vector has been taken as b = (cos( π3 ), sin( π3 )), we can get the initial and boundary conditions and source term f (x, t) according to the corresponding analysis solution of example. First, we partition the square domain Ω = (0, 1) × (0, 1) in to N × N sub-square uniformly. Then we divide each square element into two triangles by the diagonal line with a negative slopeso that we complete the construction of the triangular mesh let h = 1/N (N = 4, 8, 16, 32, 64) be mesh size for triangular meshes. In the example, the analytical solution is chosen as u = sin(πx)sin(πy)exp(−t). Numerical error results and convergence rate are listed in Table 7.1 and convergence rate in Figure 1. Table 7.1:numerical result h 1/4 1/8 1/16 1/32 1/64

L2 -error 3.7148e-00 9.4454e-01 2.3719e-01 5.9383e-02 1.4875e-02

L2 -order 1.97 1.99 2.00 2.00

1

10

||u−uh||0 2

0(h ) 0

10

−1

10

−2

10

−3

10

−4

10

1

10

2

3

10

10

4

10

Figure 1: Convergence rate for κ = 1 and s = 1.

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Acknowledgements The first author’s research is supported in part by National Natural Science Foundation of China (NSFC) no. 11871038, China Postdoctoral Science Foundation no.2014M560547, Fundamental Research Funds of Shandong University no.2017JC005.

References [1] J. Wank, X. Ye. A weak Galerkin finite element method for the second order elliptic problem. J. Comput. Appl. Math, 241(2013) 103-115. [2] Q.H. Li, J.P. Wang. Weak Galerkin finite element methods for parabolic equations. Numer. Meth. PDEs, 29 (2013) 2004-2024. [3] F. Gao, L. Mu. On L2 error estimate for weak Galerkin finite element method for parabolic problems. J. Comput. Math, 32 (2014) 195-204. [4] F. Gao, X. Wang. A modified weak Galerkin finite element method for a class of parabolic problems. J. Comp. Appl. Math, 271 (2014) 1-19. [5] L. Mu, J.P. Wang, X. Ye. A modified weak Galerkin finite element method for the stokes equations. J. Comp. Appl. Math, 275 (2015) 79-90. [6] J. Wang, X. Ye. A weak Galerkin finite element method for the stokes equations. Adv. Comput. Math, 42 (2016) 155-174. [7] L. Mu, J. Wang, X. Ye, S. Zhao. A numerical study on the weak Galerkin method for the Helmholtz equation. Commun. Comput. Phys, 15 (2014) 1461-1479. [8] L. Mu, J.P. Wang, X. Ye. Weak Galerkin finite element method for the biharmonic equation on polytopal meshes, Numer. Meth. PDEs. 30 (2014) 1003-1029. [9] R. Zhang, Q.L. Li. A weak Galerkin finite element scheme for the Biharmonic equations by using polynomials of reduced order. J. Sci. Comput, 64 (2015) 559-585. [10] J. Zhang, K. Zhang, J. Li, X. Wang. A weak Galerkin finite element method for the Navier-Stokes equation. Commun. Comput. Phys, 10 (2017) 1-14. [11] X. Lin, J. Li, Z. Chen. A weak Galerkin finite element method for the Navier-Stokes equation, J. Comp. Appl. Math, 333 (2018) 442-457. [12] R.A. Adams. Sobolev spaces. Academic Press, New York, 1975. [13] P.G.Ciarlet. The finite element method for elliptic problems. North-Holland, 1978. [14] V. Thom´ee. Galerkin finite element method for parabolic problems. (Springer Series in Computational Mathematics), Springer-Verlag, Berlin-Heidelberg, New York, Inc., Secancns, NJ, 1984. [15] M.F. Wheeler. A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM Journal on Numerical Analysis, 10 (1973) 723-759.

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The Generalized Moment Problem on White Noise Spaces A. S. Okb El Bab1 and Hossam A. Ghany2 1 Department of Mathematics, Faculty of Science, Azhar University, Naser City(11884),Cairo Egypt. 2 Department of Mathematics, Helwan University, Sawah Street(11282), Cairo Egypt. [email protected] Abstract Our purpose in this paper, is to derive the main properties of the generalized moment functions defined on some types of white noise spaces. A new version of Wick product on some spaces of generalized functions is introduced. Applying the direct connection between the theory of construction for hypercomplex systems and white noise analysis, we setup a framework to construct a lot of spaces of generalized functions connected with different examples of hypercomplex systems.

2010 Mathematics Subject Classification: 43A62, 60H40, 30G35. Keywords: White Noise; Wick product; moment function; generalized functions.

1

Introduction

In this paper, the main properties of the generalized moment functions defined on some types of white noise spaces are derived. A new version of Wick product with respect to nonGaussian measures, the associated Hermite transform and the characterization theorem for the constructed spaces of generalized functions are introduced. Let Q denotes a locally compact basis on the space Rn . The linear space of bounded continuous complex-valued functions Cb (Q) is complete normed space with respect to the norm kf k∞ = sup |f (x)|, x∈Q

where f define on Q. We will denote by Cb∞ (Q) the space of infinitely differential bounded functions on Q, and by S(Q) the linear subspace of Cb∞ (Q) formed by the set of functions on Q such that xα Dβ f (x) is bounded on Q, where α, β ∈ Zn+ . The space of continuous 1 437

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linear functional on S(Q) is called tempered distribution space and is denoted by S 0 (Q). There exist many works aims to investigate white noise spaces. Some of these works devoted to deal with the construction of spaces of test, generalized functions and operators acting in these spaces using the Wiener-Itˆo-Segal isomorphism and various riggings of the Fock space [2, 9]. Distribution play a crucial role in the study of PDEs and quantum field theory [5,11], where quantum field are defined as operator valued distributions. The contemporary theory of generalized functions of infinitely many variables originates from the works of Berezanskyi and Samoilenko [3] and Hida [9]. In [3], the spaces of test and generalized functions were constructed as infinite tensor products of one-dimensional spaces. In [9], the classical approach to the construction of the theory of generalized functions was, in fact, used, but all functions under consideration were functions of a point of the infinitedimensional space on which the Gaussian measure was defined; this measure played the same role as the Lebesgue measure in the classical theory of generalized functions. This paper is organized as follows: In section 2, we give the main properties of the generalized moment functions defined on the space of rabidly decreasing functions on Q. In section 3, a new way for constructing spaces of generalized functions is given. In section 4, we derive the main relations between the construction of hypercomplex system and the Theory of white noise analysis.

2

The moment problem on S(Q)

The elements of S(Q) are called rabidly decreasing functions and for each α, β ∈ Zn+ , S(Q) is equipped with the family of seminorms kf kα,β = sup |xα Dβ f (x)| x∈Q

In this section, we devoted to give a full description of the integral Z λ(x)dμ(λ), μ ∈ M+ (Q), φ(x) = Q

where λ : Q → C belongs to the linear space of bounded continuous complex-valued functions Cb (Q) and the measure μ belongs to the space of positive Radon measures M+ (Q). Let s = (sα )α∈Zn+ (s0 > 0) be an n-sequence of real numbers. We set Ls (xα ) = sα ,

α ∈ Zn+ .

The n-sequence s = (sα )α∈Zn+ is called quasi-positive definite if Ls is quasi-positive definite (i.e., Ls (f fˉ) ≥ 0 for all f ∈ S(Q)). The n-sequence s is called a generalized moment R sequence if there exists a Radon measure μ on Q such that xα ∈ L1 (μ) and sα = Q xα dμ(x) for all α ∈ Zn+ . When such measure exists, then it is called a representing measure of the sequence s. Let F = f1 , ..., fm be a finite family in S(Q), and QF = {q ∈ Q; fj (q) ≥ 0,

i = 1, ..., m}.

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Clearly, we have mj = supx∈Q fj (x) < ∞, setting fˆj (x) = m−1 j fj (x),

x ∈ Q if

mj > 0,

and fˆj = fj ,

mj = 0,

j = 1, ..., m. We define Fˆ = {0, 1, fˆ1 , ..., fˆm }, and we will denote by ΔF the set of all products of the form f1 ...fi (1 − g1 )...(1 − gi ) for functions f1 , ..., fi , g1 ..., gi ∈ Fˆ and integers i, j ≥ 1. Theorem 2.1. Let Π(F ) denote the convex set of all linear mappings L : S(Q) → R such that L(1) = 1 and L(f ) ≥ 0 for all f ∈ ΔF . Then we have 0 ≤ L(f ) ≤ 1 for all L ∈ Π(F) and f ∈ ΔF . Proof. Let f1 ...fk ∈ ΔF , where either fj ∈ Fˆ or 1 − fj ∈ Fˆ for all j = 1, ..., k. We have f1 ...fn = (1 − f1 ) + f1 (1 − f2 ) + ... + f1 ...fk−1 (1 − fk ) This implies L(1 − f ) ≥ 0, whence L(f ) ≤ 1. Remark. Let Γ+ (Q) be the positive cone generated by ΔF . From the previous proof we notice that if f ∈ ΔF , then 1 − f ∈ Γ+ (Q). Moreover, we notice that if f, g ∈ ΔF , then (1 − f )g ∈ Γ+ (Q). In particular, if L : S(Q) → R is positive on ΔF , then L((1 − f )g) ≥ 0 for all f, g ∈ ΔF . Finally, we notice that L(1) = 0 implies L = 0. Lemma 2.2. Let L be an extreme point of the convex set Π(F). Then L is multiplicative on S(Q). Proof. Suppose f ∈ ΔF be fixed. Sufficiently, we need to prove that L(f g) = L(f )L(g) f or

all

g ∈ ΔF

Let d = L(f ). We have the following possibilities: 1. If 0 < d < 1, we consider the linear functionals L1 (h) = d−1 L(f h) and L2 (h) = (1 − d)−1 L((1 − f )h), h ∈ S(Q). Clearly, L1 , L2 ∈ Π(F ). since L = dL1 + (1 − d)L2 and L is an extreme point of Π(F ), this implies L = L1 , whence L(f g) = L(f )L(g). 2. If d = 0, then the functional L0 (h) = L(f h) is positive on ΔF and L0 (1) = 0, applying the above remark implies L0 = 0, whence L(f g) = 0 = L(f )L(g). 3. If d = 1, we use the above discussion to the functional L1 (g) = L((1 − f )g), and obtain L(f g) = L(g) = L(f )L(g). 3 439

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Theorem 2.3. For every linear functional L ∈ Π(F ) there exists a uniquely probability measure μ on Q such that Z L(f ) = f dμ Q

for all f ∈ S(Q). Proof. Let L0 ∈ Π(F ) be an extreme point. Then L0 is multiplicative on S(Q), by the above lemma. Thus, for the sequence γ = (γ1 , ..., γn ) ∈ Rn defined by L0 (tj ) = γj , we have L0 (f ) = f (γ) for all f ∈ S(Q). But we have 0 ≤ L0 (f ) ≤ 1, f ∈ ΔF , by Theorem 2.1, we obtain that |L0 (f )| = |f (γ)| ≤ ||f ||Q = supt∈Q |f (t)|,

f ∈ S(Q).

P P If f ∈ Π(F ) is of the form L = j∈I cj Lj , where cj ≥ 0, j∈I cj = 1, Lj an extreme point of Π(F), then X X cj |Lj (f )| ≤ cj ||f ||Q = ||f ||Q , f ∈ S(Q). |L(f )| ≤ j∈I

j∈I

Let γ = (γα )α∈Zn+ (γ0 > 0 be a generalized moment sequence. Then the linear form L = γ0−1 is an element of Π(F ), and by using the result obtained from the above Theorem we have: Corollary 2.4. Let Q is compact and F = f0 = 1, f1 , ..., fm be a finite family which generates the space S(Q). An n-sequence of real numbers s = (sα )α∈Zn+ (s0 > 0) is a generalized moment sequence if and only if the linear form Ls is nonnegative on the set ΔF .

3

The spaces of generalized functions

This section is devoted to give the main relations between the construction of hypercomplex system and the Theory of white noise analysis. We will consider the following rigging of a Hilbert space H0 with positive and negative spaces H+ and H− : H− ⊇ H0 ⊇ H+ .

(3.1)

Let I+ 0 : H− −→ H+ be the canonical isometry transferring the negative space H− onto the positive space H+ . A biorthogonal basis (pn , qn )∞ n=0 in the space H0 can be understood as − ∞ ∞ sequences (pn )n=0 ⊂ H+ and (qn = I0 pn )n=0 ⊂ H− , where the first sequence is an orthogonal basis in the positive space H+ and the second is an orthogonal basis in the negative space H− . Hence, these systems of sequences pn and qn are biorthogonal: (pn , qn )H0 = δn,m hn , hn = kpn k2H+ = kqn k2H− , n, m ∈ Z+ , 4 440

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(3.2)

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for all ϕ ∈ H+ , ϕ=

∞ X

ϕ n pn , ϕ n =

∞ X

(ϕ, qn )H0 h−1 n ,

n=0

n=0

|ϕn |2 hn = kϕk2H+ < ∞,

(3.3)

|ξn |2 hn = kξk2H− < ∞,

(3.4)

for all ξ ∈ H− , ξ=

∞ X

ξn qn , ξn =

∞ X

(ξ, pn )H0 h−1 n ,

n=0

n=0

(ξ, ϕ)H0 =

∞ X

ξ n ϕn h n .

(3.5)

n=0

Let (pn )∞ n=0 be an arbitrary total sequence of vectors pn of a Hilbert space H0 . It is easy to prove that such sequence (hn )∞ n=0 of positive numbers hn exists for which the set of test functions ( ) ∞ ∞ X X ϕn pn | ϕn ∈ C : kϕk2H+ = |ϕn |2 hn < ∞ , (3.6) H+ = ϕ = n=0

n=0

with the corresponding scalar product is the positive space with respect to H0 . Note that, it is necessary to assume in addition the fulfilment of the following necessary and suffi(i) ∞ (i) cient condition on (pn )∞ ∈ H+ with finite n=0 : an arbitrary sequence (ϕ )i=0 of vectors ϕ (i) sequences of coordinates ϕn which is fundamental in H+ and converges to 0 in H0 must converge to 0 in H+ . This condition will always be fulfilled in our case. Similarly, for the negative space H− , by replacing pn by qn , we have the set of generalized functions as follows ( ) ∞ ∞ X X ξn qn | ξn ∈ C : kξk2H− = |ξn |2 hn < ∞ . (3.7) H− = ξ = n=0

n=0

As pointed out from [1-3], there exists a quasinuclear rigging such that, the zero space H0 is a hypercomplex system L2 (Q, dm(p))(p ∈ Q) and we assume that χ I+ χ : H+ −→ H1 ,

χ I− χ : H− −→ H−1 .

such that + hI− χ ξ, Iχ ϕiL2 (Q,dm(p)) = hξ, ϕiH0 ,

ξ ∈ H− , ϕ ∈ H+ .

+ So, we have a biunitary map {I− χ , Iχ }. This mapping transfers the rigging of the space H0 to a rigging of the hypercomplex space L2 (Q, dm(p)):

H− ⊇

⊇ H+ .

H0

  − y Iχ

  + y Iχ

χ H−1 ⊇ L2 (Q, dm(p)) ⊇ H1χ ,

(3.8)

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Hence, we consider the space H1χ is a positive space of the form ( ) ∞ ∞ X X ϕn χn | : kϕk2H χ = |ϕn |2 (n!)2 K n < ∞ , H1χ = ϕ = 1

n=0

n=0

(3.9)

where pn = χn , hn = (n!)2 K n , n ∈ Z+ , (K > 1 is a fixed sufficiently large number), and consists of continuous functions on Q. Similarly, for the space H1χ , we have ( ) ∞ ∞ X X χ = ξ= ξn χn | : kξk2H χ = |ξn |2 (n!)2 K n < ∞ , H−1 −1

n=0

n=0

(3.10)

χ − χ The system (χn , qnχ )∞ n=0 , where qn = I1 χn ∈ H−1 , is a biorthogonal basis of the space L2 (Q, dm(p)). It is essential to introduce the rigging of the hypercomplex space L2 (Q, dm(p)) by means of projective and inductive limits of Hilbert spaces which are constructed by rules of type (3.6), (3.8) and (3.9). For every q ∈ N, we define the Hilbert space of type (3.6): ( ) ∞ ∞ X X Hqχ = ϕ = ϕn χn ∈ H0 : kϕk2Hqχ = |ϕn |2 (n!)2 K qn < ∞ . n=0

n=0

(3.11)

Then, we have the rigging: χ ⊇ L2 (Q, dm(p)) ⊇ Hqχ ⊇ Ψχ , (Ψχ )0 ⊇ H−q

Ψχ = pr lim Hqχ = q∈N

χ H−q

with the action

=

(

ξ=

∞ X n=0

\

χ (Ψχ )0 = ind lim H−q =

Hqχ ,

q∈N

q∈N

χ = ξn qnχ : kξk2H−q

(ξ, ϕ)L2 (Q,dm(p)) =

∞ X

∞ X n=0

[

q∈N

|ξn |2 (n!)2 K −qn < ∞

ξn ϕn (n!)2 K qn ,

n=0

(3.12) χ H−q ,

)

,

(3.13)

χ ϕ ∈ Hqχ , ξ ∈ H−q .

To illustrate the above result, we give the following example Example 3.1. In the classical case when H0 := L2 (R, dx) with respect to the Lebesgue measure dx and ordinary convolution. Then, the generalized character χ(x, λ) = eλx (λ ∈ C) and χn (x) = xn (x ∈ R, n ∈ Z+ ). Therefore, the space (3.11) consists of entire functions ϕ(x) and ϕn (x) are the Taylor coefficients of ϕ(x). Formula (3.2) gives their representation χ , ϕ ∈ H1χ ). as the Fourier coefficients using the scalar product (ξ, ϕ)H0 , (ξ ∈ H−1 Remark. Obviously, such a generalization gives the possibility of constructing a lot of spaces of generalized functions connected with different examples of hypercomplex systems. 6 442

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4

The generalized Wick product

In this section, we devoted to introduce a new version of Wick product with respect to non-Gaussian measures, the associated Hermite transform and the characterization theorem for the constructed spaces of generalized functions. Wick is the first one introduced the product between two functions in white noise space, so this product carry his name [13]. He was used as a tool to renormalize certain infinite quantities in quantum field theory. Later on, the Wick product was considered, in a stochastic ordinary and partial differential equations (see, e.g., [6,8,10]). Under the assumption that kχk2H0 ≤ C n for some C > 0, we define a new χ Wick product, called χ-Wick product on the space H−q . Then, we give the definition of the χ . χ-Hermite transform and apply it to establish a characterization theorem for the space H−q P∞ P∞ χ χ χ Definition 4.1. Let ξ = ξ q , η = m m m=0 n=0 ηn qn ∈ H−q with ξm , ηn ∈ C. The χ-Wick product of ξ, η, denoted by ξ χ η, is defined by the formula ξ χ η =

∞ X

χ ξm ηn qm+n .

(4.1)

m,n=0

χ , Hqχ are closed under χ-Wick product. It is important to show that the spaces H−q χ Lemma 4.2. If ξ, η ∈ H−q and ϕ, ψ ∈ Hqχ , we have

χ (i ) ξ χ η ∈ H−q ,

(ii ) ϕ χ ψ ∈ Hqχ . ∞ ∞ P P χ χ ξm qm ,η= ηn qnχ ∈ H−q , then for some q1 ∈ N we have Proof. If ξ = m=0

n=0

∞ X

m=0

We note that

ξ χ η = where ζ l =

∞ P

2

|ξm | K

∞ X

−q1 m

< ∞ and

χ ξm ηn qm+n

∞ X

=

m,n=0

∞ X n=0

|ηn |2 K −q1 n < ∞.

∞ X

l=0

m+n=l

ξm ηn

!

qlχ

=

(4.2)

∞ X

ζ l qlχ ,

(4.3)

l=0

ξm ηn . With q = q1 + p we have

m+n=l ∞ X l=0

|ζ l |2 K −ql

2 ∞ X ∞ X = ξm ηn K −q1 l K −pl l=0 m+n=l ! ! ∞ ∞ ∞ X X X ≤ |ξm |2 K −q1 m |ηn |2 K −q1 n K −pl l=0

≤

∞ X l=0

m+n=l

K −pl

!

< ∞,

m+n=l

∞ X

m=0

|ξm |2 K −q1 m

!

∞ X n=0

|ηn |2 K −q1 n

!

(4.4)

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which proves (i ). The proof of (ii ) is similar.

The following important algebraic properties of the χ-Wick product follow directly from Definition 4.1. χ , we get Lemma 4.3. For each ξ, η, ζ ∈ H−q

(i ) ξ χ η = η χ ξ (Commutative law),

(ii ) ξ χ (η χ ζ) = (ξ χ η) χ ζ (Associative law), (ii ) ξ χ (η + ζ) = ξ χ η + ξ χ ζ (Distributive law).

χ and Remark. According to Lemmas 4.2. and 4.3., we can conclude that the spaces H−q χ Hq form topological algebras with respect to the χ-Wick product.

From the above arguement, the χ-Wick product satisfies all the ordinary algebraic rules for multiplication. But, there are some problems when limit operations are involved. To treat these situations it is convenient to apply a transformation, called the χ-Hermite transform, which converts χ-Wick products into ordinary (complex) products and convergence χ in H−q into bounded, pointwise convergence in a certain neighborhood of 0 in C. P∞ χ χ Definition 4.4. Let ξ = n=0 ξn qn ∈ H−q with ξn ∈ C. Then, the χ-Hermite transform of ξ, denoted by Hχ ξ, is defined by ∞ X Hχ ξ(z) = (4.5) ξn z n ∈ C (when convergent). n=0

In the following, we define for 0 < M, q < ∞ the neighborhoods of zero in C which denoted it by Oq,M (0): ( ) ∞ X n 2 qn 2 |z | K < M . Oq,M (0) = z ∈ C : (4.6) n=0

It is easy to see that

Note that, if ξ = estimate

P∞

q ≤ p, N ≤ M ⇒ Oq,N (0) ⊆ Oq,M (0).

χ n=0 ξn qn

∞ X n=0

∈

χ H−q ,z

|ξn ||z n | = ≤

∞ X n=0

∈ Oq,M (0) for some 0 < M, q < ∞, we have the qn

|ξn ||z n |K − 2 K

∞ X n=0

< M

|ξn |2 K −qn

∞ X n=0

< ∞.

(4.7)

! 12

|ξn |2 K −qn

qn 2

∞ X n=0

! 12

|z n |2 K qn

! 12

(4.8) 8

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The conclusion above can be stated as follows: χ Proposition 4.5. If ξ ∈ H−q , then Hχ ξ converges for all z ∈ Oq (M ) for all q, M < ∞. χ , then Proposition 4.6. If ξ, η ∈ H−q

Hχ (ξ χ η)(z) = Hχ ξ(z).Hχ η(z).

(4.9)

for all z such that Hχ ξ and Hχ η exist. Proof. The proof is an immediate consequence of Definitions 4.1. and 4.4. P χ χ Let ξ = ∞ n=0 ξn qn ∈ H−q , with ξn ∈ R. Then, the number ξ0 = Hχ ξ(0) ∈ R is called the generalized expectation of ξ and is denoted by E(ξ). Suppose that V 3 z 7→ f (z) ∈ C is an analytic function, where V is a neighborhood of E(ξ). Assume that the Taylor series of f around E(ξ) has coefficients in R. Then, the χ-Wick version f χ of f is defined by
χ χ 3 ξ 7→ f χ (ξ) = H−1 f ◦ Hχ (ξ) ∈ H−q . (4.10) H−q χ Example 4.7. If the function f : C → C is entire, then f χ is defined for all ξ ∈ H−q . For example, the χ-Wick exponential is defined by ∞ X 1 χ n ξ . exp (ξ) = j! j=0 χ

5

(4.11)

Concluding Remarks

The space of continuous linear functional on S(Q) are called tempered distributions, 0 0 and is denoted by S (Q). Let L ∈ S (Q) and α ∈ Zd+ . The weak derivative Dα L (or the derivative of the sense of distributions) is given by (Dα L)(f ) = (−1)|α| L(Dα f )

(5.1)

for f ∈ (Q). This corresponds to Dα L{g} = L{Dα g}. Note that distribution always has a weak derivative. A function f is completely monotonic if for each α ∈ Zn+ , (−1)|α| Dα f (x) ≥ 0 on Rn+ ; see [4, 7, 12] for many properties of completely monotonic functions. Bernstien’s R theorem asserts that f is completely monotonic if and only if f (x) = Rn e−x.t dμ(t) where μ is a positive measure supported on a subset of Rn+ . If assume that Q = Rn . So, x = (x1 , ..., xn ) ∈ Rn . Let xα be denote the product xα1 1 ... xαnn , Zn+ denote the set of nP tuples (α1 , ..., αn ) where each αi is a non-negative integer, |α| = ni=1 αi and Dα denote |α| the partial differential operator ∂xα1∂... ∂xαn . Then, we obtain the special case S(Q) = S(Rn ) n 1 is the space of rabidly decreasing function on Rn (so-called Schwartz space) and its dual 0 0 S (Q) = S (Rn ) is the space of tempered distribution on Rn . 9 445

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References [1] Yu. M. Berezansky and A. A. Kalyuzhnyi, Harmonic Analysis in Hypercomplex Systems, Naukova Dumka,Kyiv, Ukrania, 1992. (in Russian; English transl.: Kluwer: Dordrecht-Boston-London, 1996). [2] Yu. M. Berezansky and Yu. G. Kondratiev, Spectral methods in infinite dimensional analysis, vol. 1, 2, Kluwer, Dordrecht 1995. [3] Yu. M. Berezanskyi and Yu. S. Samoilenko, Nuclear spaces of functions of infinitely many variables, Ukr. Mat. Zh., 25, No. 6, 723-737 (1973). [4] C. Berg, J. P. R. Christensen, P. Ressel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions, Springer-Verlag: Berlin, Heidelberg, New York, 1984. [5] I. M. Gelfand and G. E. Shilov, Generalized Function, Academic Press,Inc., 1964. [6] H. A. Ghany, Exact solutions for stochastic generalized Hirota-Satsuma coupled KdV equations, Chinese Journal of Physics, vol. 49, no. 4, (2011) 926-940. [7] H. A. Ghany, Harmonic analysis in the product of commutative hypercomplex systems. Journal of Computational Analysis and Applications, 25 (5), (2018) 875-888. [8] H. A. Ghany and M. A. Qurashi, Travelling Solitary Wave Solutions for Stochastic Kadomtsev-Petviashvili Equation. Journal of Computational Analysis and Applications, 21 (1), (2015) 121-131. [9] T. Hida, Analysis of Brownian Functionals, Carleton Math. Lect. Notes, No. 13 (1975). [10] H. Holden, B. Øsendal, J. Ubøe and T. Zhang, Stochastic partial differential equations, Springer Science+Business Media, LLC, (2010). [11] J. Lighthill, Introduction to Fourier Analysis and Generalized Function,cambridge university press,1958. [12] A. S. Okb El Bab and H. A. Ghany, Harmonic analysis on hypergroups systems, AIP Conf. Proc., 312 (2010) 1309. [13] G. C. Wick, The evaluation of the collinear matrix, Phys. Rev 80 (1950), 268-272.

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QUADRATIC TYPE FUNCTIONAL INCLUSIONS ON SQUARE-SYMMETRIC GROUPOIDS AND HYERS-ULAM STABILITY GWANG HUI KIM AND HWAN-YONG SHIN

Abstract. We consider that a set-valued map F : X → P0 (Y ) satisfying the functional inclusion F (x ∗ y)♦F (x ∗ y −1 ) ⊆ σ♦ (F (x)♦F (y))(or σ♦ (F (x)♦F (y)) ⊆ σ♦ (F (x ∗ y)♦F (x ∗ y −1 ))) admits a unique selection f : X → Y satisfying the functional equation f (x ∗ y)  f (x ∗ y −1 ) = σ (f (x)  f (y)) in appropriate conditions, where (X, ∗), (Y, ) are square-symmetric groupoids and ♦ is the extension of  to the collection P0 (Y ) of all nonempty subsets of Y.

1. Introduction Let (X, ∗), (Y, ) be groupoids with binary operations. If the binary oepration ∗ satisfies the following inequality (x ∗ y) ∗ (x ∗ y) = (x ∗ x) ∗ (y ∗ y),

x, y ∈ X

then the operation ∗ is called square-symmetric. Note that the square symmetric ∗ implies that σ∗ (x) := x∗x is an endomorphism. A binary operation ∗ such that σ∗ is an automorphism of (X, ∗) is called divisible and the corresponding groupoid is said to be a divisible groupoid. The triple (Y, , d) is called a metric groupoid if (Y, ) is a groupoid, (Y, d) is a metric space and  is a continuous operation with respect to the topology of (Y, d). For a nonempty set Y we denote by P0 (Y ) the collection of all nonempty subsets of Y. The diameter of a set A ∈ P0 (Y ) is defined by δ(A) := sup{d(x, y)|x, y ∈ A}. The Lipschitz modulus of a function f : X → Y is the smallest real extended number L with the property d(f (x), f (y)) ≤ Ld(x, y),

x, y ∈ Y.

The Lipschitz modulus of a function f is denoted by Lipf . A selection of a set-valued mapping F : X → P0 is a single-valued map f : X → Y with the property f (x) ∈ F (x) for all x ∈ X. 2010 Mathematics Subject Classification. 39B72, 54C60. Key words and phrases. Hyers–Ulam stability; square-symmetric groupoid; functional inclusion. 1

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In a linear normed space (Y, k · k) we define the following families of sets c(Y ) := {A : A ∈ P0 (Y ), A is convex set} ccl(Y ) := {A : A ∈ P0 (Y ), A is closed and convex set} cc(Y ) := {A : A ∈ P0 (Y ), A is compact and convex set}. The theory of stability of functional equations had been formulated by Ulam [14]. In 1941, Hyers [3] had answered affirmatively the question of Ulam for Banach spaces and it represents the starting point of the Hyers–Ulam stability of functional equations. Let us recall the Hyers’ result. Theorem 1.1. [3] Let X be a linear normed space, Y a Banach space and ε > 0. If a function f : X → Y satisfies the following inequality (1.1)

kf (x + y) − f (x) − f (y)k ≤ ε,

x, y ∈ X

then there exists a unique additive function g : X → Y such that (1.2)

kf (x) − g(x)k ≤ ε,

x ∈ X.

Smajdor [13] and Gajda and Ger [2] observed an interesting connection between the stability of the Cauchy functional equation and set-valued functions satisfying F (x + y) ⊆ F (x) + F (y). If f : X → Y satisfies (1.1), then the set-valued mapping F : X → P0 defined by F (x) = f (x) + B(0, ε),

x ∈ X,

where B(0, ε) is the closed ball in Y centered at 0 and radius ε > 0, implies that F (x + y) ⊆ F (x) + F (y) for x, y ∈ X, and the function g from relation (1.2) is an additive selection of F . Naturally Gajda and Ger [2] considered under what conditions a set-valued mapping with F (x + y) ⊆ F (x) + F (y) admits an additive selection and they obtained the following theorem. Theorem 1.2. [2] Let (S, +) be a commutative semigroup with zero element, X a Banach space over R and F : S → ccl(X) a set-valued mapping with convex and closed values such that F (x + y) ⊆ F (x) + F (y) for x, y ∈ S and supx∈S δ(F (x)) < ∞. Then F admits a unique additive selection. For the last two decades, many mathematicians have developed Theorem 1.2 [6, 9, 10, 11] and investigated various properties of functional inclusion and its connectedness of Hyers– Ulam stability of functional equations [4, 5, 7, 8, 12].

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STABILITY OF SET-VALUED FUNCTIONAL EQUATIONS ON SYMMTRIC GROUPOIDS

3

The aim of this paper is to study some properties for set-valued mappings satisfying the following quadratic type functional inclusions σ♦ (F (x)♦F (y)) ⊆ F (x ∗ y)♦F (x ∗ y −1 ) F (x ∗ y)♦F (x ∗ y −1 ) ⊆ σ♦ (F (x)♦F (y)) and obtain Hyers-Ulam stability of functional equation. 2. Main results Let N0 = N ∪ {0}. Throughout this section, suppose that the operation ♦ satisfies the following condition : for all ε > 0 there exists η > 0 such that if δ(A), δ(B) < η, A, B ∈ P0 (Y ), then δ(A♦B) < ε and we assume that X and Y have unique identity idX and idY respectively. If the operation  satisfies that (x1  y1 )  (x2  y2 ) = (x1  x2 )  (y1  y2 ) for all x1 , x2 , y1 , y2 ∈ Y , then we say  is bisymmetric operation. Lemma 2.1. [9] If (Y, ) is a groupoid with a bisymmetric operation, then σ is increasing endomorphism of (P0 (Y ), ♦, ⊆). Now, we present the main theorem of this paper. Theorem 2.2. Let (X, ∗) be a square-symmetric divisible groupoid, (Y, , d) a complete metric bisymmetric divisible groupoid and F : X → P0 (Y ) with F (idX ) = {idY } a set-valued mapping such that (2.1)

σ♦ (F (x)♦F (y)) ⊆ F (x ∗ y)♦F (x ∗ y −1 )

for all x, y ∈ X. Assume that (2.2)

lim δ(F ◦ σ∗−m (x))Lip(σ2m ) = 0,

m→∞ σ♦2n ◦ F

and

◦ σ∗−n (x) ∈ cl(Y )

for all x ∈ X and n ∈ N0 . Then there exists a unique selection f : X → Y of F such that (2.3)

σ (f (x)  f (y)) = f (x ∗ y)  f (x ∗ y −1 )

for all x, y ∈ X.

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G.H. KIM AND H.-Y. SHIN

Proof. First we prove that there exists a selection of F satisfying (2.3). Consider the set valued mapping Fn : X → P0 (Y ) corresponding to F defined by (2.4)

F0 := F,

Fn := σ♦2n ◦ F ◦ σ∗−n .

for each n ∈ N. Letting x, y by σ∗−n−1 (x) in (2.14) respectively, we get σ♦2 ◦ F (σ∗−n−1 (x)) ⊆ F (σ∗−n (x))

(2.5)

for all x ∈ X. By composing σ2n to the both sides of (2.5) and using Lemma 2.1, we obtain σ♦2n+2 ◦ F ◦ σ∗−n−1 (x) ⊆ σ♦2n ◦ F ◦ σ∗−n (x) for all x ∈ X and n ∈ N0 . This means that {Fn (x)}∞ n=0 is a decreasing sequence of closed subsets of the Banach space Y . Let s, t ∈ Fm (x) for some fixed m ∈ N. Denoting σ−2m (s) = u, σ−2m (t) = v, we have d(s, t) = d(σ2m (u), σ2m (v)) ≤ Lip(σ2m ) · d(u, v) ≤ Lip(σ2m )δ(F ◦ σ∗−m (x)) and this implies that (2.6)

δ(Fm (x)) ≤ Lip(σ2m ) · δ(F ◦ σ∗−m (x))

for all x ∈ X. Taking the limit m → ∞ of (2.6), we find that lim δ(Fm (x))

m→∞

for all x ∈ X. It is follows from the Cantor intersection theorem in the complete metric spaces that ∞ \ (2.7) Fn (x) n=0

is singleton f (x). Since the function f : X → Y satisfies f (x) ∈ F0 (x) = F (x) for all x ∈ X, f is a selection of F . Putting x, y for σ∗−n (x) and σ∗−n (y), respectively in (2.14) and applying σ♦2n to the both sides of (2.14), we arrive at σ♦ (Fn (x)♦Fn (y)) ⊆ Fn (x ∗ y)♦Fn (x ∗ y −1 ) T for all x, y ∈ X and n ∈ N0 . Since {f (x)} = ∞ n=0 Fn (x), x ∈ X, we have σ (f (x)  f (y)) ∈ σ♦ (Fn (x)♦Fn (y)), for all x, y ∈ X, n ∈ N0 . Therefore, in view of (2.8), we get

(2.8)

(2.9)

d(σ (f (x)  f (y)), f (x ∗ y)  f (x ∗ y −1 )) ≤ δ(Fn (x ∗ y)  Fn (x ∗ y −1 ))

for all x, y ∈ X and n ∈ N0 . Taking the limit n → ∞ of (2.9), it is reduced to the equation (2.10)

σ (f (x)  f (y)) = f (x ∗ y)  f (x ∗ y −1 ), for all x, y ∈ X.

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To show the uniqueness of f , assume that g : X → Y is a selection of F such that σ (g(x)  g(y)) = g(x ∗ y)  g(x ∗ y −1 ), for all x, y ∈ X.

(2.11)

From (2.10) and (2.11), it follows that f (x) = σ2n ◦ f ◦ σ∗−n (x), g(x) = σ2n ◦ g ◦ σ∗−n (x) for all x ∈ X, n ∈ N. Hence, for x ∈ X and n ∈ N, we see that d(f (x), g(x)) = d(σ2n ◦ f ◦ σ∗−n (x), σ2n ◦ g ◦ σ∗−n (x)) = Liq(σ2n )d(f ◦ σ∗−n (x), g ◦ σ∗−n (x)) ≤ Liq(σ2n )δ(F ◦ σ◦−n (x)). Taking n → ∞, we arrive at the desired conclusion.



Next, we are going to establish another theorem about the inclusion (2.14). Theorem 2.3. Let (X, ∗) be a square-symmetric divisible groupoid, (Y, , d) a metric bisymmetric divisible groupoid and A a divisible subgroupoid of (P0 (Y ), ♦). Suppose that F : X → A with F (idX ) = {idY } is a set-valued mapping subject to the condition (2.14) and satisfying lim δ(F ◦ σ∗n (x))Lip(σ−2n ) = 0,

(2.12)

n→∞

x ∈ X.

Then F is single-valued mapping and (2.13)

σ♦ (F (x)♦F (y)) = F (x ∗ y)♦F (x ∗ y −1 ),

for all x, y ∈ X.

Proof. Consider the function Gn : X → A corresponding to F defined by G0 := F,

Gn := σ♦−2n ◦ F ◦ σ∗n

for each n ∈ N. Replacing x, y by σ∗n (x) in (2.12) respectively, and then composing on both sides by σ♦−2n−2 , we have σ♦−2n ◦ F ◦ σ∗n (x) ⊆ σ♦−2n−2 ◦ F ◦ σ∗n+1 (x) for all x, y ∈ X. This means that {Gn (x)}∞ n=0 is an increasing sequence of (A, ♦). By the similar argument in the proof of Theorem 2.2, we see that lim δ(Gn (x)) ≤ lim δ(F ◦ σ∗n (x))Lip(σ−2n ) = 0,

n→∞

n→∞

for all x ∈ X.

It implies that δ(Gn (x)) = 0 for every n ∈ N0 and Gn (x) is single-valued for all n ∈ N0 . Therefore, in view of (2.14), G0 = F satisfies (2.13) and the proof is completed. 

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Corollary 2.4. Let (X, ∗) be a square-symmetric divisible groupoid, (Z, k · k) a Banace space over R, p, q ∈ R, p + q 6= 0, p + q 6= 1, and F : X → c(Z) with F (idX ) = {0Z } a set-valued mapping such that (2.14)

p(p + q)F (x) + q(p + q)F (y)) ⊆ pF (x ∗ y) + qF (x ∗ y −1 )

for all x, y ∈ X. Assume that there exists M > 0 such that δ(F (x)) ≤ M, F◦

σ∗−n (x)

and

∈ cl(Y )

for all x, y ∈ X and n ∈ Z. Then there exists a unique selection f : X → Z of F such that (2.15)

p(p + q)f (x) + q(p + q)f (y) = pf (x ∗ y) + qf (x ∗ y −1 ),

x, y ∈ X.

Proof. Consider the operation  : Z × Z → Z is defined by x  y = px + qy,

x, y ∈ Z,

where p, q ∈ R are given real numbers. Then the triple (Z, , k · k) is a metric groupoid with a bisymmetric operation. For all U, V ∈ P0 (Z), the operation ♦ is naturally defined by U ♦V = pU + qV and we have σ♦ (U ) = (p + q)U and in general, σ♦n (U ) = (p + q)n U , for all n ∈ N. And we get Lip(σ2n ) = |p + q|2n ,

n ∈ Z.

If |p + q| < 1, we have 2n −n σ♦2n ◦ F ◦−n ∗ (x) = (p + q) F ◦ σ∗ (x) ∈ cl(Z)

and δ(F ◦ σ∗−n ) ≤ M |p + q|2n ,

x ∈ X, n ∈ N0 ,

thus, by Theorem 2.2, there exists a unique selection of F satisfying (2.15). If |p + q| > 1, we obtain δ(F ◦ σ∗n )Lip(σ−2n ) ≤

M , |p + q|2n

x ∈ X, n ∈ N0 .

By using Theorem 2.3, F is single-valued mapping satisfying (2.15). We arrive at the desired conclusion. 

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Corollary 2.5. Let (X, ∗) be a square-symmetric divisible groupoid, (Z, k · k) a Banach space over R, p, q, ε > 0, p + q < 1, and z ∈ Z. Assume that f : X → Z is a function satisfying kpf (x ∗ y) + qf (x ∗ y −1 ) − p(p + q)f (x) − q(p + q)f (y) − zk ≤ ε,

x, y ∈ X.

Then there exists a unique function g : X → Z satisfying (2.16)

pg(x ∗ y) + qg(x ∗ y −1 ) = p(p + q)g(x) + q(p + q)g(y) + z,

x, y ∈ X

and (2.17)

kf (x) − g(x)k ≤

ε , (1 − p − q)(p + q)

x ∈ X.

Proof. Consider the auxiliary set-valued mapping Gf : X → ccl(Z) corresponding to f defined by 1 Gf (x) = f (x) + (B(0, ε) − z), if x ∈ X − {idX } (1 − p − q)(p + q) and Gf (idX ) = {0Z }. Then we obtain p(p + q) (B(0, ε) − z) (1 − p − q)(p + q) q(p + q) +q(p + q)f (y) + (B(0, ε) − z) (1 − p − q)(p + q)

p(p + q)Gf (x) + q(p + q)Gf (y) = p(p + q)f (x) +

⊆ pf (x ∗ y) + qf (x ∗ y −1 ) + (B(0, ε) − z) (p + q)2 (B(0, ε) − z) (1 − p − q)(p + q) p (B(0, ε) − z) = pf (x ∗ y) + (1 − p − q)(p + q) q +qf (x ∗ y −1 ) + (B(0, ε) − z) (1 − p − q)(p + q) +

= pGf (x ∗ y) + qGf (x ∗ y −1 ) for all x, y ∈ X. By the definition of δ(Gf (x)), we have δ(Gf (x)) ≤

2ε (1 − p − q)(p + q)

for all x ∈ X. Since all conditions of Corollary 2.4 are equipped, Gf has a unique selection h : X → Z such that ph(x ∗ y) + qh(x ∗ y −1 ) = p(p + q)h(x) + q(p + q)h(y),

x, y ∈ X.

Defining the function g : X → Z as g(x) = h(x) +

z (1 − p − q)(p + q)

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for all x ∈ X, we see that the function g satisfies (2.16) and (2.17).



Next, we will introduce some theorems and corollaries which are obtained by the similar proofs of Theorem 2.2, 2.3 and Corollary 2.4, 2.5. Theorem 2.6. Let (X, ∗) be a square-symmetric divisible groupoid, (Y, , d) a complete metric bisymmetric divisible groupoid and F : X → P0 (Y ) with F (idX ) = {idY } a set-valued mapping such that F (x ∗ y)♦F (x ∗ y −1 ) ⊆ σ♦ (F (x)♦F (y))

(2.18)

for all x, y ∈ X. Assume that lim δ(F ◦ σ∗m (x))Lip(σ♦−2m ) = 0,

m→∞ σ♦2n ◦ F

and

◦ σ∗−n (x) ∈ cl(Y )

for all x ∈ X and n ∈ N0 . Then there exists a unique selection f : X → Y of F such that σ (f (x)  f (y)) = f (x ∗ y)  f (x ∗ y −1 ) for all x, y ∈ X. Theorem 2.7. Let (X, ∗) be a square-symmetric divisible groupoid, (Y, , d) a metric bisymmetric divisible groupoid and A a divisible subgroupoid of (P0 (Y ), ♦). Suppose that F : X → A with F (idX ) = {idY } is a set-valued mapping subject to the condition (2.18) and satisfying lim δ(F ◦ σ∗2n (x))Lip(σ♦2n ) = 0,

n→∞

x ∈ X.

Then F is single valued and σ♦ (F (x)♦F (y)) = F (x ∗ y)♦F (x ∗ y −1 ),

for all x, y ∈ X.

Corollary 2.8. Let (X, ∗) be a square-symmetric divisible groupoid, (Z, k · k) a Banace space over R, p, q ∈ R, p + q 6= 0, p + q 6= 1, and F : X → c(Z) with F (idX ) = {0Z } a set-valued mapping subject to the condition (2.18). Assume that there exists M > 0 such that δ(F (x)) ≤ M,

and

F ◦ σ∗−n (x) ∈ cl(Z) for all x, y ∈ X and n ∈ N0 . Then there exists a unique selection f : X → Z of F such that p(p + q)f (x) + q(p + q)f (y) = pf (x ∗ y) + qf (x ∗ y −1 ),

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Corollary 2.9. Let (X, ∗) be a square-symmetric divisible groupoid, (Z, k · k) a Banach space over R, p, q, ε > 0, p + q > 1, and z ∈ Z. Assume that f : X → Z is a function satisfying kpf (x ∗ y) + qf (x ∗ y −1 ) − p(p + q)f (x) − q(p + q)f (y) − zk ≤ ε,

x, y ∈ X.

Then there exists a unique function g : X → Z satisfying pg(x ∗ y) + qg(x ∗ y −1 ) = p(p + q)g(x) + q(p + q)g(y) + z,

x, y ∈ X

and kf (x) − g(x)k ≤

ε , (p + q − 1)(p + q)

x ∈ X.

Remark. If (X, +, ·) is a vector space and ∗ is defined by x ∗ y = x + y and z = 0, p = q = 1, y −1 = −y in Corollary 2.9, then it is a same result of Czerwik [1]. Acknowledgment The second author of this work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2015R1D1A1A01058083). References [1] S. Czerwik, On the stability of the quadratic mapping in normed space, Bull. Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59-64. [2] Z. Gajda, R. Ger. Subadditive multifunctions and Hyers–Ulam stability, Numer. Math. 80, (1987), 281291. [3] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, (1941), 222–224. [4] G.H. Kim, On the stability of homogeneous functional equations with degree t and n-variables, Mathematical Inequalities and Applications, 6, (2003), 675-688. [5] G.H. Kim, On the stability of functional equations on a square-symmetric groupoid, Nonlinear Analysis: Theory, Methods and Applications, 63, (2005), 2559-2568. [6] K. Nikodem, D. Popa, On selections of general linear inclusions, Publ. Math. Debrecen 75, (2009), 239-249. [7] M. Piszczek, On selections of set-valued inclusions in a single variable with applications to several variables, Results. Math., 64, (2013), 1-12. [8] M. Piszczek, The properties of functional inclusions and Hyers–Ulam stability, Aequat. Math., 85, (2013), 111-118. [9] D. Popa, Functional inclusions on square-symmetric grupoids and Hyers–Ulam stability, Math. Inequal. Appl., 3, (2004), 419–428. [10] D. Popa, A property of a functional inclusion connected with Hyers–Ulam stability, J. Math. Inequal., 4, (2009), 591-598. [11] J. Brzdek, D. Popa, B. Xu, Selections of set-valued maps satisfying a linear inclusion in a single variable, Nonlinear Analysis, 74, (2011), 324-330.

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[12] J. Sikorska, Set-valued Orthogonal Additivity,Set-Valued Var. Anal. 23, (2015), 547-557. [13] W. Smajdor, Superadditive set-valued functions, Glas. Math. 21, (1986), 343-348. [14] S. M. Ulam, Problems in Modern Mathematics, Chapter 6, Wiley Interscience, New York, (1964). Gwang Hui Kim, Department of Mathematics, Kangnam Universaty,Yongin, Gyeonggi 16979, Republic of Korea E-mail address: [email protected] Hwan-Yong Shin, Department of Mathematics, Chungnam National University,99 Daehangno, Yuseong-gu, Daejeon 34134, Republic of Korea E-mail address: [email protected]

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Explicit identities involving r-Bell polynomials Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper, we study differential equations arising from the generating functions of the r-Bell polynomials. We give explicit identities for the r-Bell polynomials. Key words : Differential equations, Bell polynomials, r-Bell polynomials. 2000 Mathematics Subject Classification : 05A19, 11B68, 11S40, 11S80, 11B83, 34A30, 65L99.

1

Introduction

The moments of the Poisson distribution are well-known to be connected to the combinatorics of the Bell and Stirling numbers(see [1, 4, 5]). As is well known, the Bell numbers Bn are given by the generating function ∞ ∑ t tn e(e −1) = (1.1) Bn . n! n=0 The Bell polynomials Bn (λ) are given by the generating function eλ(e

t

−1)

=

∞ ∑ n=0

Bn (λ)

tn . n!

(1.2)

The Bell polynomials Bn (λ) satisfy the relation Bn (λ) = Eλ [Z n ], n ∈ N, where Z is a Poisson random variable with parameter λ > 0. The r-Bell polynomials Gn (x, r) are defined by the exponential generating function: ∞ ∑

Gn (x, r)

n=0

t tn = ert+x(e −1) , (see [4]), n!

(1.3)

where, r may be real or complex numbers. Note that Bn (x) = Gn (x, 0). The first few examples of r-Bell polynomials Gn (x, r) are G0 (x, r) = 1, G1 (x, r) = r + x, G2 (x, r) = r2 + x + 2rx + x2 , G3 (x, r) = r3 + x + 3rx + 3r2 x + 3x2 + 3rx2 + x3 , G4 (x, r) = r4 + x + 4rx + 6r2 x + 4r3 x + 7x2 + 12rx2 + 6r2 x2 + 6x3 + 4rx3 + x4 , G5 (x, r) = r5 + x + 5rx + 10r2 x + 10r3 x + 5r4 x + 15x2 + 35rx2 + 30r2 x2 + 10r3 x2 + 25x3 + 30rx3 + 10r2 x3 + 10x4 + 5rx4 + x5 .

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From (1.2) and (1.3), we see that ∞ ∑

Gn (x, r)

n=0

t tn = e(e −1)x ert n! (∞ )( ∞ ) m ∑ ∑ tk mt = Bk (x) r k! m! m=0 k=0 ( ) ( ) ∞ n ∑ ∑ n tn . = Bk (x)rn−k n! k n=0

(1.4)

k=0

Comparing the coefficients on both sides of (1.4), we obtain n ( ) ∑ n Gn (x, r) = Bk (x)rn−k (n ≥ 0). k k=0

Similarly we also have Gn (x + y, r) =

n ( ) ∑ n k=0

k

Gk (x, r)Bn−k (y).

Recently, many mathematicians have have studied the differential equations arising from the generating function of special polynomials(see [2, 3, 6, 7, 8, 9]). In this paper, we study differential equations arising from the generating function of r-Bell polynomials. We give explicit identities for the r-Bell polynomials.

2

Explicit identities involving r-Bell polynomials

Differential equations arising from the generating functions of special polynomials are studied by many authors in order to give explicit identities for special polynomials(see [7, 8, 13]). In this section, we study differential equations arising from the generating functions of r-Bell polynomials. Let ∞ ∑ t tn F = F (t, x, r) = (2.1) Gn (x, r) = ert+(e −1)x , x, r ∈ C. n! n=0 Then, by (2.1), we have F (1) =

d d ( rt+(et −1)x ) F (t, x, r) = e dt dt t = ert+(e −1)x (r + xet ) = rert+(e

t

−1)x

+ xe(r+1)t+(e

(2.2) t

−1)x

= rF (t, x, r) + xF (t, x, r + 1), d (1) F = rF (1) (t, x, r) + xF (1) (t, x, r + 1) dt = r2 F (t, x, r) + x(2r + 1)F (t, x, r + 1) + x2 F (t, x, r + 2),

F (2) =

and

(2.3)

d (2) F dt = r2 F (1) (t, x, r) + x(2r + 1)F (1) (t, x, r + 1) + x2 F (1) (t, x, r + 2) ( ) = r3 F (t, x, r) + x r2 + (2r + 1)(r + 1) F (t, x, r + 1)

F (3) =

+ x2 (3r + 3)F (t, x, r + 2) + x3 F (t, x, r + 3).

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Continuing this process, we can guess that ( )N d F (N ) = F (t, x, r) dt =

N ∑

(2.4) ai (N, x, r)F (t, x, r + i), (N = 0, 1, 2, . . .).

i=0

Taking the derivative with respect to t in (2.4), we get ∑ dF (N ) = ai (N, x, r)F (1) (t, x, r + i) dt i=0 N

F (N +1) =

=

N ∑

ai (N, x, r) {(r + i)F (t, x, r + i) + xF (t, x, r + i + 1)}

i=0

=

N ∑

ai (N, x, r)(r + i)F (t, x, r + i)

i=0

+x

N ∑

(2.5) ai (N, x, r)F (t, x, r + (i + 1))

i=0

=

N ∑ (r + i)ai (N, x, r)F (t, x, r + i) i=0

+x

N +1 ∑

ai−1 (N, x, r)F (t, x, r + i).

i=1

On the other hand, by replacing N by N + 1 in (2.4), we get F (N +1) =

N +1 ∑

ai (N + 1, x, r)F (t, x, r + i).

(2.6)

i=0

Comparing the coefficients on both sides of (2.5) and (2.6), we obtain a0 (N + 1, x, r) = ra0 (N, x, r),

aN +1 (N + 1, x, r) = xaN (N, x, r),

(2.7)

ai (N + 1, x, r) = (r + i)ai−1 (N, x, r) + xai−1 (N, x, r), (1 ≤ i ≤ N ).

(2.8)

and

In addition, by (2.4), we get F (t, x, r) = F (0) (t, x, r) = a0 (0, x, r)F (t, x, r).

(2.9)

a0 (0, x, r) = 1.

(2.10)

By (2.9), we get

It is not difficult to show that rF (t, x, r) + xF (t, x, r + 1) = F (1) (t, x, r) =

1 ∑

(2.11) ai (1, x, r)F (t, x, r + 1)

i=0

= a0 (1, x, r)F (t, x, r) + a1 (1, x, r)F (t, x, r + 1).

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Thus, by (2.11), we also get a0 (1, x, r) = r,

a1 (1, x, r) = x.

(2.12)

From (2.7), we note that a0 (N + 1, x, r) = ra0 (N, x, r) = · · · = rN a0 (1, x, r) = rN +1 ,

(2.13)

and aN +1 (N + 1, x, r) = xaN (N, x, r) = · · · = xN a1 (1, x, r) = xN +1 .

(2.14)

For i = 1, 2, 3 in (2.8), we have a1 (N + 1, x, r) = x

N ∑

(r + 1)k a0 (N − k, x, r),

(2.15)

k=0

a2 (N + 1, x, r) = x

N −1 ∑

(r + 2)k a1 (N − k, x, r),

(2.16)

k=0

and a3 (N + 1, x, r) = x

N −2 ∑

(r + 3)k a2 (N − k, x, r).

(2.17)

k=0

Continuing this process, we can deduce that, for 1 ≤ i ≤ N, ai (N + 1, x, r) = x

N∑ −i+1

(r + i)k ai−1 (N − k, x, r).

(2.18)

k=0

Here, we note that the matrix ai (j, x, r)0≤i,j≤N +1 is given by   1 r r2 r3 · · · rN +1   · ··· ·  0 x ·   0 0 x 2 · · · · ·      ·  0 0 0 x3 · · · . . .. .. ..  .. . .  . . . . . .  0

0

0

0

···

xN +1

Now, we give explicit expressions for ai (N + 1, x, r). By (2.15), (2.16), and (2.17), we get a1 (N + 1, x, r) = x

N ∑

(r + 1) a0 (N − k1 , x, r) = k1

k1 =0

a2 (N + 1, x, r) = x

N −1 ∑

N ∑

(r + 1)k1 rN −k1 ,

k1 =0

(r + 2)k2 a1 (N − k2 , x, r)

k2 =0

= x2

N −1 N −1−k ∑ ∑ 2 k2 =0

(r + 1)k1 (r + 2)k2 rN −k2 −k1 −1 ,

k1 =0

and a3 (N + 1, x, r) =x

N −2 ∑

(r + 3)k3 a2 (N − k3 , x, r)

k3 =0

= x3

N −2 N −2−k ∑ ∑ 3 N −2−k ∑3 −k2 k3 =0

k2 =0

(r + 3)k3 (r + 2)k2 (r + 1)k1 rN −k3 −k2 −k1 −2 .

k1 =0

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Continuing this process, we have ai (N + 1, x, r) = xi

N∑ −i+1 N −i+1−k ∑ i ki =0

···

N −i+1−k ∑i −···−k2

ki−1 =0

k1 =0

( i ∏

) (r + l)kl

∑i

rN −i+1−

l=1

kl

(2.19)

.

l=1

Therefore, by (2.19), we obtain the following theorem. Theorem 2.1 For N = 0, 1, 2, . . . , the differential equation F (N ) =

N ∑

ai (N, x, r)eit F (t, x, r)

i=0

has a solution F = F (t, x, r) = ert+(e

t

−1)x

,

where a0 (N, x, r) = rN , aN (N, x, r) = xN , ai (N, x, r) = xi

N −i N −i−k ∑ ∑ i

···

ki =0 ki−1 =0

N −i−k i −···−k2 ∑

(

k1 =0

i ∏ (r + l)kl

) rN −i−

∑i l=1

kl

,

l=1

(1 ≤ i ≤ N ). From (2.1), we note that F (N ) =

∞ ∑ ( d )N tk F (t, x, r) = Gk+N (x, r) . dt k!

(2.20)

k=0

From Theorem 1 and (2.20), we can derive the following equation: ) (N ∞ ∑ ∑ tk Gk+N (x, r) = F (N ) = ai (N, x, r)eit F (t, x, r) k! i=0 k=0 )( ∞ ) (∞ N ∑ ∑ ∑ tl tm l = i Gm (x, r) ai (N, x, r) l! m! m=0 i=0 l=0 ( ) N ∞ ∑ k ( ) ∑ ∑ k k−m tk = ai (N, x, r) i Gm (x, r) m k! i=0 k=0 m=0 ) ( ( ) ∞ N ∑ k ∑ ∑ k k−m tk = i ai (N, x, r)Gm (x, r) . m k! i=0 m=0

(2.21)

k=0

By comparing the coefficients on both sides of (2.21), we obtain the following theorem. Theorem 2.2 For k, N = 0, 1, 2, . . . , we have N ∑ k ( ) ∑ k k−m Gk+N (x, r) = i ai (N, x, r)Gm (x, r), m i=0 m=0

(2.22)

where where a0 (N, x, r) = rN , i

ai (N, x, r) = x

aN (N, x, r) = xN ,

N −i N −i−k ∑ ∑ i ki =0 ki−1 =0

···

N −i−k i −···−k2 ∑ k1 =0

(

i ∏ (r + l)kl

) rN −i−

∑i l=1

kl

,

l=1

(1 ≤ i ≤ N ).

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Let us take k = 0 in (2.22). Then, we have the following corollary. Corollary 2.3 For N = 0, 1, 2, . . . , we have GN (x, r) =

N ∑

ai (N, x, r).

i=0

∑N it For N = 0, 1, 2, . . . , the functional equation F (N ) = i=0 ai (N, x, r)e F (t, x, r) has a solution t F = F (t, x, r) = ert+(e −1)x . Here is a plot of the surface for this solution. In Figure 1(left), we

Figure 1: The surface for the solution F (t, x, r) choose −3 ≤ x ≤ 1, −5 ≤ t ≤ 5, and r = −2. In Figure 1(right), we choose −3 ≤ x ≤ 3, −5 ≤ t ≤ 5, and r = 2. Acknowledgement: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092). REFERENCES 1. Roberto B. Corcino and Cristina B. Corcino, On generalized Bell polynomials, Discrete Dynamics in Nature and Society, 2011 (2011), Article ID 623456, 21 pages. 2. T. Kim, D.S. Kim, C.S. Ryoo, H.I. Kwon, Differential equations associated with Mahler and Sheffer-Mahler polynomials, submitted for publication. 3. T. Kim, D.S. Kim, Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations, J. Nonlinear Sci. Appl., 9(2016), 2086-2098. 4. I. Mez˝o, The r-Bell numbers, J. Integer Seq., 13 (2010), Article 10.9.8. 5. N. Privault, Genrealized Bell polynomials and the combinatorics of Poisson central moments, The Electronic Journal of Combinatorics, 18(2011), #54 6. C.S. Ryoo, Differential equations associated with tangent numbers, J. Appl. Math. & Informatics, 34 (2016), 487-494. 7. C. S. Ryoo, Differential equations associated with generalized Bell polynomials and their zeros, Open Mathematics, 14 (2016), 807-815. 8. C. S. Ryoo, A numerical investigation on the structure of the zeros of the degenerate Eulertangent mixed-type polynomials, J. Nonlinear Sci. Appl., 10 (2017), 4474-4484

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.3, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

A CLASS INVOLVING DERIVATIVES OF RATIO OF THE ANALYTIC FUNCTIONS JI HYANG PARK, VIRENDRA KUMAR, AND NAK EUN CHO Abstract. The class of functions defined using linear combination of the derivatives of ratio of the normalized analytic function with the identity function is considered in this manuscript. Further, the sharp bounds on the Hankel determinants and estimates on the higher order Schwarzian derivatives for the first three consecutive derivatives are investigated.

1. Introduction Let A be the family of functions f in the open unit disk D and satisfying the normalization conditions f (0) = 0 = f 0 (0) − 1. Let the collection S ⊂ A contains univalent functions in D. An analytic function f is subordinate to another analytic function g if there is an analytic function w with |w(z)| ≤ |z| and w(0) = 0 such that f (z) = g(w(z)) and we write f ≺ g. If g is univalent, then f ≺ g if and only if f (0) = g(0) and f (D) ⊆ g(D). The classes S ∗ and K of starlike and convex functions, respectively, are defined by Re (zf 0 (z)/f (z)) > 0 and Re (1 + zf 00 (z)/f 0 (z)) > 0. There are several sufficient conditions for functions to be univalent. Among them the simplest one is to verify Re f 0 (z) > 0 in z ∈ D. However, there are several other sufficient conditions for univalency were investigated in the recent years. Obradoviˇc [17] proved that if f ∈ A satisfy |f 00 (z)| < 1/2, then f is convex in D. Later, this condition was generalized by Frasin [7]. For 0 < γ ≤ 1, Tunseki [24] investigated the conditions on the expressions f 0 (z) − (1 − γ)f (z)/z and zf 00 (z) − γf 0 (z) for the sufficient conditions of starlikeness and convexity. Frasin [8] obtained some sufficient conditions on f 000 (z) for starlikeness and convexity. In particular, he proved that when the function f ∈ A with f 00 (0) = 0 satisfies |f 000 (z)| < 1, then f is starlike in D and if |f 000 (z)| < 1/2, then f is convex in D, see [8, Cororllary 2, Cororllary 3, p. 65]. Motivated by this, in 2010, Uyanik et al. [25] introduced and investigated a new subclass of A defined using the linear combination of the derivatives of ratio of the normalized analytic function with the identity function. For β1 , β2 ∈ C, λ > 0 and f ∈ A he defined V(β1 , β2 , λ) as follows:  0  00 β1 z f (z) + β2 z 2 f (z) ≤ λ. z z 2010 Mathematics Subject Classification. 30C45, 30C50, 30C80. Key words and phrases. Univalent function, Coefficient bound, Hankel determinant. 1

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He obtained sufficient condition for normalized analytic functions to be in the class V(β1 , β2 , λ). He proved that the nth coefficient of functions in this class is bounded by λ/((n − 1)(|β1 | + (n − 2)|β2 |)). It is well-known |an | ≤ 2 (n = 2, 3, · · · ). P∞ that the function in∗ the class P∞S satisfy 2 Moreover, if n=2 n|an | ≤ 1, then f ∈ S and if n=2 n |an | ≤ 1, then f ∈ K. There is another important quantity related to coefficients, called the Hankel determinant, which enable us to determine the necessary condition on coefficient functional for functions belonging to a given class of functions. For numbers n, q, the Hankel P∞given natural n determinant Hq,n (f ) of a function f (z) = n=0 an z , a1 = 1 is defined by means of the following determinant an an+1 · · · an+q−1 an+1 an+2 · · · an+q . Hq,n (f ) := .. .. .. .. . . . . a an+2(q−1) n+q−1 an+q · · · It is easy to see that the functional H2,1 (f ) = a3 − a22 is the well-known Fekete-Szeg¨o functional. However, the second Hankel determinant is given by H2,2 (f ) := a2 a4 − a23 . Further, the third Hankel determinant is H3, 1 (f ) := a3 (a2 a4 − a23 ) − a4 (a4 − a2 a3 ) + a5 (a3 − a22 ). The Hankel determinant Hq,n (f ) for the class S was investigated by Pommerenke [19] and Hayman [10]. For more details, see [4,5,11,13,19,21] and the references cited therein. The Schwarzian derivative of a locally univalent function f , defined by  00 0  2 f (z) 1 f 00 (z) S(f )(z) := − . f 0 (z) 2 f 0 (z) The Schwarzian derivative is an important quantity in Univalent Function Theory. Further properties were investigated by Nehari [16]. He obtained the necessary and sufficient conditions for f ∈ S. The higher order Schwarzian derivative [9, 23]), is defined by σ3 (f ) = S(f ) and for any integer n ≥ 4, it is given by σn+1 (f ) = (σn (f ))0 − (n − 1)σn (f )

f 00 . f0

Droff and Szynal [6] studied the higher order Schwarzian derivative for convex functions. Now σn (f )(0) =: Sn and S3 = σ3 (f )(0) = 6(a3 − a22 ), S4 = σ4 (f )(0) = 24(a4 − 3a2 a3 + 2a32 ) and S5 = σ5 (f )(0) = 24(5a5 − 20a2 a4 − 9a23 + 48a3 a22 − 24a42 ). The sharp bound on |Si | (i = 2, 3, 4), for f ∈ K, investigated by Droff and Szynal. The generalization of their work, recently, carried out in [3] by Cho et al. We shall investigate, the estimates on the Hankel determinants and the higher order Schwarzian derivatives by associating the functions of the class under consideration with the Carath´eodory functions. Now we recall those results which shall be needed for investigation of our results. Let P denote the class of Carath´eodory [1, 2] functions of the form ∞ X p(z) = 1 + pn z n (z ∈ D). (1.1) n=1

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P∞ n Let B be the class of analytic functions w(z) = (z ∈ D) and satisfying n=1 cn z the condition |w(z)| < 1 for z ∈ D. The function w ∈ B and p ∈ P are related as p(z) = (1 + w(z))/(1 − w(z)). Consider a functional Ψ(w) = |c3 + αc1 c2 + βc31 | for w ∈ B and α, β ∈ R. Lemma 1.1. [20, Lemma 2, p. 128] If w ∈ B, then for any real numbers α and β the following sharp estimate Ψ(w) ≤ Φ(α, β) holds, where   1, if (α, β) ∈ Ω1 ∪ Ω2 ,   if (α, β) ∈ Ω3 ∪ Ω4 ∪ Ω5 Φ(α, β) = |β|,  1/2   |α|+1  2 (|α| + 1) , if (α, β) ∈ Ω6 ∪ Ω7 . 3 3(|α|+β+1) Here the sets Ωi ’s are defined by Ω1 :=  {(α, β) ∈ R2 : |α| ≤ 1/2, −1 ≤ β ≤ 1} , 4 Ω2 := (α, β) ∈ R2 : 12 ≤ |α| ≤ 2, 27 (|α| + 1)3 − (|α| + 1) ≤ β ≤ 1 , Ω3 :=  {(α, β) ∈ R2 : |α| ≤ 2, β ≥ 1} , 1 (α2 + 8) , and Ω4 := (α, β) ∈ R2 : 2 ≤ |α| ≤ 4, β ≥ 12 Ω5 := (α, β) ∈ R2 : |α| ≥ 4, β ≥ 23 (|α| − 1) . 4 3 (|α| Ω6 := n(α, β) ∈ R2 : 21 ≤ |α| ≤ 2, − 32 (|α| + 1) ≤ β ≤ 27 o + 1) − (|α| + 1) , Ω7 := (α, β) ∈ R2 : |α| ≥ 2, − 23 (|α| + 1) ≤ β ≤

2|α|(|α|+1) α2 +2|α|+4

.

Lemma 1.2. [14,15, Libera and Zlotkiewicz] If p ∈ P has the form given by (1.1) with p1 ≥ 0, then 2p2 = p21 + x(4 − p21 ) (1.2) and 4p3 = p31 + 2p1 (4 − p21 )x − p1 (4 − p21 )x2 + 2(4 − p21 )(1 − |x|2 )y

(1.3)

for some x and y such that |x| ≤ 1 and |y| ≤ 1. ˆ γˆ and a Lemma 1.3. [22, Ravichandran and Verma] Let α ˆ , β, ˆ satisfy the inequalities 0 (3 2−2)β3β . 9 3β (β +β )(β +2β ) 1 +6β2 1

1

2

1

2

Proof. Since f ∈ V(β1 , β2 , λ), it follows that there exists a Schwarz function w(z) = c1 z + c2 z 2 + c3 z 3 + · · · ∈ B such that  0 00  f (z) f (z) 2 β1 z + β2 z = λw(z)). (2.1) z z In the view of interconnection w(z) = (p(z) − 1)/(p(z) + 1) ∈ B if and only if p ∈ P between the Schwarz function w and the Carath´eodory function p(z) = 1 + p1 z + p2 z 2 + p3 z 3 + · · · ∈ P, from (2.1), we get a2 =

λ(2p2 − p1 2 ) λp1 , a3 = , 2β1 8(β1 + β2 )

(2.2)

and a4 =

λ(4p3 − 4p1 p2 + p1 3 ) λ(8p4 − 8p1 p3 − 4p2 2 + 6p1 2 p2 − p1 4 ) , a5 = . 24(β1 + 2β2 ) 64(β1 + 3β2 )

(2.3)

(1) From (2.2), Using the result [see [12]], for any complex number µ, p2 − µp21 ≤ 2 max{1; |2µ − 1|},

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we have   λ β1 2 + 2µλ(β1 + β2 ) 2 |a3 − µa2 | = p2 − p1 4(β1 + β2 ) 2β1 2   2λ(β1 + β2 ) λ max 1, |µ| . = 2(β1 + β2 ) β1 2 2

(2.4)

The equality holds in case of the function f defined by (2.1) with choice of the function w(z) = z. (2) Using (2.2) and (2.3), we have a2 a4 − a3 2 =

 2 λ2 (β1 + 2β1 β2 + 4β2 2 )p1 4 2 192β1 (β1 + β2 ) (β1 + 2β2 ) −4(β1 2 + 2β1 β2 + 4β2 2 )p1 2 p2 − 12β1 (β1 + 2β2 )p2 2  +16(β1 + β2 )2 p1 p3 .

(2.5)

Putting equivalent expressions for p2 and p3 in terms of p1 from (1.2) and (1.3) in (2.5), we have a2 a4 − a3 2 =

 λ2 {−3β1 (β1 + 2β2 )(4 − p1 2 ) + 4(β1 + β2 )2 p1 2 } 2 192β1 (β1 + β2 ) (β1 + 2β2 )  ×(4 − p1 2 )x2 + 8p1 (4 − p1 2 )(β1 + β2 )2 (1 − |x|2 )y . (2.6)

Because p ∈ P, and the class P is invariant under rotation, without loss of any generality, we can set p1 = |p1 | =: s ∈ [0, 2]. Further, since |x| ≤ 1 and |y| ≤ 1 for some x, y ∈ C, using this facts and the triangle inequality in (2.6) we can write   3β1 (β1 + 2β2 )(4 − s2 ) + 4(β1 + β2 )2 s2 2 + s(1 − |x|2 ) , (2.7) |a2 a4 − a3 | ≤ T − x 8(β1 + β2 )2 2

where λ2 (4 − s2 ) T := . 24β1 (β1 + 2β2 ) We note that for s = p1 = 0, and s = p1 = 2 from (2.7), we have |a2 a4 − a3 2 | ≤ λ2 /4(β1 + β2 )2 and |a2 a4 − a3 2 | = 0, respectively. Now we assume that s ∈ (0, 2). Then, form (2.7), we obtain |a2 a4 − a3 2 | ≤

λ2 s(4 − s2 )F (a, b, c), 24β1 (β1 + 2β2 )

(2.8)

where F (a, b, c) := |a + bx + cx2 | + 1 − |x|2 , with 3β1 (β1 + 2β2 )(4 − s2 ) + 4(β1 + β2 )2 s2 . 8(β1 + β2 )2 s Here it is easily seen that ac = 0. Here we have two cases now: a := 0,

b := 0 and c := −

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√ (i) When 0 < β1 < β2 ( 3 − 1) and s∗ ≤ s < 2, we obtain |b| ≥ 2(1 − |c|). Therefore, by Lemma 1.4, we have λ2 (β1 + β2 )2 s(4 − s2 ) F (a, b, c) 24β1 (β1 + β2 )2 (β1 + 2β2 )   λ2 (β1 + β2 )2 s(4 − s2 ) 3β1 (β1 + 2β2 )(4 − s2 ) + 4(β1 + β2 )2 s2 = 24β1 (β1 + β2 )2 (β1 + 2β2 ) 8(β1 + β2 )2 s λ2 g(s), = 192β1 (β1 + β2 )2 (β1 + 2β2 )

|a2 a4 − a3 2 | ≤

where g : [s∗ , 2) → R is defined by g(s) := 3β1 (β1 + 2β2 )(4 − s2 ) + 4(β1 + β2 )2 (4 − s2 )s2 . Cleanly, g has maximum at p 2 −β1 2 − 2β1 β2 + β2 2 s = s1 := p 2 , β1 + 2β1 β2 + 4β2 2 we have λ2 g(s1 ) |a2 a4 − a3 | ≤ 192β1 (β1 + β2 )2 (β1 + 2β2 ) λ2 (β1 + β2 )2 . = 3β1 (β1 + 2β2 )(β1 2 + 2β1 β2 + 4β2 2 ) √ √ (ii) When 0 < β1 ≤ β2 ( 3 − 1) and 0 < s < s∗ , β1 > β2 ( 3 − 1) > 0 and 0 < s < 2, we obtain |b| < 2(1 − |c|). Therefore, by Lemma 1.4, we have 2

λ2 (β1 + β2 )2 s(4 − s2 ) F (a, b, c) 24β1 (β1 + β2 )2 (β1 + 2β2 ) λ2 (4 − s2 )s = 24β1 (β1 + 2β2 ) λ2 = h(s), 24β1 (β1 + 2β2 )

|a2 a4 − a3 2 | ≤

where the function h : (0, 2) → R is defined by h(s) := (4 − s2 )s. √ Further computation reveals that h has its maximum at s = s2 := 2/ 3, and thus we have √ 2 λ2 2 3λ 2 |a2 a4 − a3 | ≤ h(s2 ) = . 24β1 (β1 + 2β2 ) 27β1 (β1 + 2β2 ) Therefore, from (i) and (ii), we conclude that |a2 a4 − a3 2 | ≤

λ2 (β1 + β2 )2 . 3β1 (β1 + 2β2 )(β1 2 + 2β1 β2 + 4β2 2 )

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The equality holds in case of the function defined in (2.1) with w(z) =

z(u0 − 2z 2 ) , 2 − u0 z

p p where u0 = 2 −β1 2 − 2β1 β2 + β2 2 / β1 2 + 2β1 β2 + 4β2 2 . (3) To find the estimates on the functional |a2 a3 −a4 |, we shall express the coefficients (ai ) in terms of Schwarz’s coefficients (ci ). From (2.1), we have a2 =

c1 λ c2 λ c3 λ c4 λ , a3 = , a4 = , a5 = . β1 2(β1 + β2 ) 3(β1 + 2β2 ) 4(β1 + 3β2 )

(2.9)

Using (2.10), we get λ 2 c1 c2 λc3 |a2 a3 − a4 | = − + 2β1 (β1 + β2 ) 3(β1 + 2β2 ) 3λ(β1 + 2β2 ) λ − = c1 c2 + c3 3(β1 + 2β2 ) 2β1 (β1 + β2 ) λ Φ(µ, ν), = 3(β1 + 2β2 ) where Φ(µ, ν) := |c3 + µc1 c2 + νc1 3 | with µ := −

3λ(β1 + 2β2 ) , and ν := 0. 2β1 (β1 + β2 )

Assume that Ωi ’s are as defined in lemma 1.1 with µ and ν as given above. We now complete the proof in the following cases. (i) Suppose that 0 < λ ≤ β1 (β1 + β2 )/3(β1 + 2β2 ), then we see that −1/2 ≤ µ ≤ 1/2 and −1 ≤ ν ≤ 1. So, we conclude that (µ, √ ν) ∈ Ω1 . (ii) Let β1 (β1 + β2 )/3(β1 + 2β2 ) ≤ λ ≤ β1 (3 3 − 2)(β1 + β2 )/3(β1 + 2β2 ). Then we can easily verify that −2 ≤ µ ≤ −1/2 and (4/27)(|µ| + 1)3 − (|µ| + 1) ≤ ν ≤ 1 holds and √we get (µ, ν) ∈ Ω2 . (iii) Let β1 (3 3 − 2)(β1 + β2 )/3(β1 + 2β2 ) ≤ λ ≤ 4β1 (β1 + β2 )/3(β1 + 2β2 ). Now we see that −2 ≤ µ ≤ −1/2 and −2(|µ| + 1)/3 ≤ λ ≤ (4/27)(|µ| + 1)3 − (|µ| + 1) hold for all such positive values of λ and so (µ, ν) ∈ Ω6 . (iv) Let λ ≥ 4β1 (β1 + β2 )/3(β1 + 2β2 ). Then we see that the conditions µ ≤ −2 and −(2/3)(|µ| + 1) ≤ ν ≤ 2|µ|(|µ| + 1)/(µ2 + |µ| + 4) hold. Therefore, (µ, ν) ∈ Ω7 . Now by using Lemma 1.1, the cases (i) and (ii), we conclude that if √ 0 < λ ≤ β1 (3 3 − 2)(β1 + β2 )/3(β1 + 2β2 ), then Φ(µ, ν) ≤ 1. Further, the (iii) and (iv) hold, then √ Φ(µ, ν) ≤ (2β1 (β1 + β2 ) + 3λ(β1 + 2β2 ))/3 3β1 (β1 + β2 ) √ for λ ≥ β1 (3 3 − 2)(β1 + β2 )/3(β1 + 2β2 ). The result is sharp in case of the function f defined by (2.1) with choice of the Schwarz function w(z) = z 3 and w(z) = z(t1 +

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z)/(1 + t1 z), respectively, where  t1 =

|µ| − 1 3(|µ| − 1 − ν)

 12 .

This ends the proof. Remark 2.2. In the case, when β1 and β2 are real numbers, from the result [25, Corollary 2.1, p.383], we conclude that λ |an | ≤ . (n − 1)(|β1 | + (n − 2)|β2 |) Using the above results, we deduce the following estimates on the third Hankel determinant: Corollary 2.3. Let 0 < β1 < 1, 0 < β2 < 1 and f ∈ V(β1 , β2 ; λ). Then the following holds:   τ 1 λ2 , 0 < λ ≤ λ1 ; τ2 λ, λ1 < λ ≤ λ2 ; |H3, 1 (f )| ≤  τ, λ ≥ λ2 , 3 where 24λ(β1 + β2 )2 (β1 + 2β2 )(β1 + 3β2 ) + β1 (β1 2 + 2β1 β2 + 4β2 2 ) ×(5β1 + 6β2 )(5β1 + 14β2 )

τ1 :=

2(αg3 + βh3 )B12

,

6λ2 (β1 + β2 )(β1 + 2β2 )(β1 + 3β2 ) + 4λβ1 (β1 + 3β2 )(β1 2 + 2β1 β2 + 4β2 2 ) +9β1 (β1 + 2β2 )2 (β1 2 + 2β1 β2 + 4β2 2 )

τ2 :=

β1 (β1 + 2β2 )2 (β1 + 3β2 )(β1 2 + 2β1 β2 + 4β2 2 )

and τ3 :=

µ2 (1296µ4 + 3456µ3 + 2304µ2 + 1740µ + 1015) . 5184(12µ + 7)

Theorem 2.4. Let f ∈ V(β1 , β2 ; λ), then the following sharp inequalities hold: (1) If 0 < β1 < 1 and 0 < β2 < 1, then ( |S3 | ≤

3λ , β1 +β2 6λ2 , β1 2

0

β1 2 ; 2(β1 +β2 )

β1 2 . 2(β1 +β2 )

(2) (a) If either of the set of conditions 0 < λ ≤ λ∗ or λ1 ∗ ≤ λ ≤ λ∗∗ and {9(β1 + 2β2 )λ + 2β1 (β1 + β2 )}[{9(β1 + 2β2 )λ + 2β1 (β1 + β2 )}2 − 27β1 2 (β1 + β2 )2 ] ≤ 324(β1 + 2β2 )λ2 (β1 + β2 )3 holds, then |S4 | ≤

24λ . 3(β1 + 2β2 )

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A CLASS INVOLVING DERIVATIVES OF RATIO OF THE ANALYTIC FUNCTIONS ∗

∗

(b) If either of the set of conditions λ2 ≤ λ ≤ 4λ1 and λ ≥ 4λ1 ∗ holds, then 48λ4 |S4 | ≤ . β1 3

9 4

q

β1 (β1 +2β) 6

9

≤ β1 + β2 or

(3) If 0 < β1 < 1, 0 < β2 < 1 and 0 < λ < 3β1 (β1 + 2β2 )/8(β1 + 3β3 ), then |S5 | ≤

720λ . β1 + 3β2

Proof. Let f ∈ V(β1 , β2 ; λ). Then, to find the estimates on the higher order Schwarzian derivatives, we shall express the coefficients (ai ) in terms of Schwarz’s coefficients (ci ). From (2.1), we have a2 =

c1 λ c3 λ c4 λ c2 λ , a4 = , a5 = . , a3 = β1 2(β1 + β2 ) 3(β1 + 2β2 ) 4(β1 + 3β2 )

(2.10)

Using first part of Theorem 2.1, we have |S3 | = 6|a3 − a2 2 )|   3λ 2(β1 + β2 )λ ≤ max 1, . β1 + β2 β1 2 The function for the equality holds by (2.1) with the choice w(z) = z. Now we consider the estimate on |S4 |. From (2.10), we obtain S4 = 24(a4 − 3a2 a3 + 2a3 2 )   24λ 6(β1 + 2β2 )λ2 3 9(β1 + 2β2 )λ = c1 − c1 c2 + c3 3(β1 + 2β2 ) 2β1 (β1 + β2 ) β1 3 24λ Υ(µ, ν) = 3(β1 + 2β2 ) where Υ(µ, ν) := c3 + µc1 c2 + νc1 3 with µ := −

9(β1 + 2β2 )λ 6(β1 + 2β2 )λ2 , and ν := . 2β1 (β1 + β2 ) β1 3

Assume that Ωi ’s are as defined in Lemma 1.1 with µ and ν as given above. We now complete the proof with the following cases. (i) Suppose that 0 < λ ≤ λ1 ∗ . In this case, we see that −1/2 ≤ µ ≤ 1/2 holds. Moreover, −1 ≤ ν ≤ 1 holds if and only if 0 < λ ≤ λ2 ∗ , where λ1 ∗ := β1 (β1 +β2 )/9(β1 + p 2β2 ) and λ2 ∗ := β1 β1 /6(β1 + 2β2 ). Thus, for all 0 < λ ≤ min{λ1 ∗ , λ2 ∗ }, we conclude that (µ, ν) ∈ Ω1 . (ii) Next suppose that λ1 ∗ < λ ≤ 4λ1 ∗ , then we see that the condition −2 ≤ µ ≤ −1/2 holds. Further, (4/27)(µ + 1)3 − (µ + 1) ≤ ν ≤ 1 holds if and only if 0 < λ ≤ λ2 ∗

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and {9(β1 + 2β2 )λ + 2β1 (β1 + β2 )}[{9(β1 + 2β2 )λ + 2β1 (β1 + β2 )}2 − 27β1 2 (β1 + β2 )2 ] ≤ 324(β1 + 2β2 )λ2 (β1 + β2 )3 . (2.11) So, if λ1 ∗ ≤ λ ≤ λ∗∗ := min{4λ1 ∗ , λ2 ∗ } and (2.11) hold, then (µ, ν) ∈ Ω2 . p √ (iii) Let λ2 ∗ ≤ λ ≤ λ3 ∗ := 4(β1 + β2 )/9(β1 + 2β2 ) and β1 + β2 ≥ 9 β1 (β1 + 2β2 )/4 6. Then, we can easily verify that |µ| ≤ 2 and ν ≥ 1. Therefore, (µ, ν) ∈ Ω5 . (iv) Let 4λ1 ∗ ≤ λ ≤ 8λ1 ∗ . Now we see that −4 ≤ µ ≤ −2 and ν ≥ (µ2 + 8)/12 hold for all such positive values of λ and hence (µ, ν) ∈ Ω6 . (v) Let λ ≥ 8λ1 ∗ . Then we see that µ ≤ −4 and ν ≥ 2(|µ| − 1), so (µ, ν) ∈ Ω7 . Now by using Lemma 1.1 and the cases (i) and (ii), we conclude that if 0 < λ ≤ min{λ1 ∗ , λ2 ∗ } or λ1 ∗ ≤ λ ≤ λ∗∗ and (2.11) hold, then Υ(µ, ν) ≤ 1. Further, from the cases (iii) − (v) p and Lemma 1.1, √we conclude that Υ(µ, ν) ≤ ν, for λ2 ∗ ≤ λ ≤ λ3 ∗ and β1 + β2 ≥ 9 β1 (β1 + 2β2 )/4 6 or λ ≥ 4λ1 ∗ . The result is sharp in case of the function f defined by (2.1) with choice of the Schwarz function w(z) = z 3 and w(z) = z, respectively. This completes the proof. Now we find the estimate on |S5 |. Using 2.2 and 2.3, we get S5 = 24(5a5 − 20a2 a4 − 9a23 + 48a22 a3 − 24a42 ) −720λ ˆ 2 p2 − p4 )] [ˆ γ p4 + a ˆp22 + 2ˆ αp1 p3 − (3/2)βp = 1 β1 + 3β2 1 −720λ ˆ Ψ(ˆ γ, a ˆ, α ˆ , β), (2.12) = β1 + 3β2 ˆ := γˆ p4 + a ˆ 2 p2 − p4 with the parameters γˆ , a where Ψ(ˆ γ, a ˆ, α ˆ , β) ˆp22 + 2ˆ αp1 p3 − (3/2)βp ˆ, α ˆ 1 1 ˆ and β are given by   15 27λ 10λ 36λ2 36λ3 β1 + 3β2 + + + γˆ := + , 15 8(β1 + 3β2 ) 8(β1 + β2 )2 β1 (β1 + 2β2 ) β1 2 (β1 + β2 ) β1 4   8λ 3 β1 + 3β2 + , a ˆ := 6 β1 (β1 + 2β2 ) β1 + 3β2   β1 + 3β2 5 9λ α ˆ := + 5 2(β1 + 3β2 ) 2(β1 + β2 )2 and   2(β1 + 3β2 ) 45 27λ 20λ 72λ2 ˆ β := . + + + 45 4(β1 + 3β2 ) 2(β1 + β2 )2 β1 (β1 + 2β2 ) β1 2 (β1 + β2 ) We assume that 0 < β1 < 1 and 0 < β2 < 1 and 0 < λ < 3β1 (β1 + 2β2 )/8(β1 + 3β3 ). Under these conditions, it is a simple matter to verify that 0 < α ˆ < 1 and 0 < a ˆ < 1. Moreover, with these restrictions all conditions of Lemma 1.3 are fulfilled and thus, we ˆ ≤ 2. Thus, the result follows from (2.12). get |Ψ(ˆ γ, a ˆ, α ˆ , β)|

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11

Acknowledgement The third author was supported 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).

References ¨ [1] C. Carath´eodory, Uber den variabilit¨atsbereich der fourier’schen konstanten von positiven harmonischen funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193–217. ¨ [2] C. Carath´eodory, Uber den variabilit¨atsbereich der coeffizienten von potenzreihen, die gegebene werte nicht annehmen, Math. Ann. 64 (1907), no. 1, 95–115. [3] N. E. Cho, V. Kumar and V. Ravichandran, Sharp bounds on the higher order Schwarzian derivatives for Janowski classes, submitted. [4] N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko and Y. J. Sim The bounds of some determinants for starlike functions of order alpha, Bull. Malays. Math. Sci. Soc. (2017), 13 pp. [5] N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko and Y. J. Sim, Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha, J. Math. Ineq. 11, 2 (2017), 429–439. [6] M. Dorff and J. Szynal, Higher order Schwarzian derivatives for convex univalent functions, Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 15 (2009), 7–11. [7] B. A. Frasin, New sufficient conditions for univalence, Gen. Math. 17 (2009), no. 3, 91–98. [8] B. A. Frasin, New sufficient conditions for analytic and univalent functions, Acta Univ. Apulensis Math. Inform. No. 17 (2009), 61–67. [9] R. Harmelin, Aharonov invariants and univalent functions, Israel J. Math. 43 (1982), no. 3, 244– 254. [10] W. K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc. 18 (1968), no. 3, 77–94. [11] A. Janteng, S. A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. (Ruse) 1 (2007), no. 13-16, 619–625. [12] F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12. [13] S. K. Lee, V. Ravichandran and S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. Art. ID. 281 (2013), 17 pp. [14] R. J. Libera and E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), no. 2, 225–230. [15] R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivatives in P, Proc. Amer. Math. Soc. 87 (1983), no. 2, 251–257. [16] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551. [17] M. Obradoviˇc, Simple suients conditions for starlikeness, Matematicki Vesnik. 49 (1997), 241–244. [18] R. Ohno and T. Sugawa, Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions, Kyoto J. Math. (advance publication, 9 June 2017), doi: 10.1215/21562261-2017-0015. [19] C. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41 (1966), 111–122. [20] D. V. Prokhorov and J. Szynal, Inverse coefficients for (α, β)-convex functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 35 (1981), 125–143. [21] C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108–112.

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[22] V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353, 6 (2015), 505–510. [23] E. Schippers, Distortion theorems for higher order Schwarzian derivatives of univalent functions, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3241–3249. [24] N. Tuneski, Some simple sufficient conditions for starlikeness and convexity, Appl. Math. Lett. 22 (2009), no. 5, 693–697. [25] N. Uyanik, S. Owa, E. Kadioˇ glu, Some properties of functions associated with close-to-convex and starlike of order α, Appl. Math. Comput. 216 (2010), no. 2, 381–387. (Ji Hyang Park) Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea E-mail address: [email protected] (Virendra Kumar) Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea E-mail address: [email protected] (Nak Eun Cho) Corresponding Author, Department of Applied Mathematics, Pukyong National University, Busan 48513, Korea E-mail address: [email protected]

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EXPLICIT FORMULAE OF CAUCHY POLYNOMIALS WITH A q PARAMETER IN TERMS OF r-WHITNEY NUMBERS F. A. SHIHA

Abstract. The Cauchy polynomials with a q parameter were recently defined, and several arithmetical properties were studied. In this paper, we establish explicit formulae for computing the Cauchy polynomials with a q parameter in terms of r-Whitney numbers of the first kind. We also obtain several properties and combinatorial identities.

AMS (2010) Subject Classification: 05A15, 05A19, 11B73, 11B75. Key Words. Cauchy numbers and polynomials, r-Whitney numbers, Stirling numbers. 1. Introduction The Cauchy polynomials of the first kind cn (z) are defined by Z 1 (1.1) cn (z) = (x − z)n dx, 0

and the Cauchy polynomials of the second kind cˆn (z) are defined by Z 1 (1.2) cˆn (z) = (−x + z)n dx, 0

Qn−1

where (y)n = i=0 (y − i) is the falling factorial with (y)0 = 1. The exponential generating function of these polynomials are ∞ X tn t (1.3) cn (z) = . z n! (1 + t) ln(1 + t) n=0 (1.4)

∞ X

cˆn (z)

n=0

t(1 + t)z tn = . n! (1 + t) ln(1 + t)

(see [7, 4]). When z = 0, cn (0) = cn and cˆn (0) = cˆn are the Cauchy numbers of the first and second kind (see [2, 9, 12, 8]). Recently [5] obtained a representation of the integer values of Cauchy polynomials in terms of r-Stirling numbers of the first kind sr (n, k) [3]. For all integers n, r ≥ 0, n X 1 , (1.5) cn (r) = sr (n + r, k + r) k+1 k=0

(1.6)

cˆn (−r) =

n X

(−1)k sr (n + r, k + r)

k=0

1 . k+1

1

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Given variables y and m and a positive integer k, define the generalized rising and falling factorials of order k with increment m by [y|m]k =

k−1 Y

(y + jm),

[y|m]0 = 1

(y − jm),

(y|m)0 = 1.

j=0

(y|m)k =

k−1 Y j=0

Komatsu [6] introduced the Poly-Cauchy polynomials and numbers with a q parameter, and the Cauchy polynomials and numbers with a q parameter as special cases. Let q be a real number with q 6= 0, Komatsu [6] defined the Cauchy polynomials with a q parameter of the first kind cqn (z) by Z 1 q (1.7) cn (z) = (x − z|q)n dx 0

and the Cauchy polynomials with a q parameter of the second kind cˆqn (z) by Z 1 (1.8) cˆqn (z) = (−x + z|q)n dx. 0

The exponential generating functions are (1.9)

∞ X

cqn (z)

n=0

(1.10)

∞ X

k ∞  −z X ln(1 + qt) 1 1 tn = (1 + qt) q , n! q k! k + 1 k=0

cˆqn (z)

n=0

k ∞  z X tn ln(1 + qt) 1 1 − = (1 + qt) q . n! q k! k + 1 k=0

= cˆqn are the Cauchy numbers with q parameter and = If z = 0, then of the first and second kind, respectively. If q = 1, then c1n (z) = cn (z) and cˆ1n (z) = cˆn (z). The r-Whitney numbers of the first and second kind were introduced by Mez¨o [10]. For non-negative integers n and k with 0 ≤ k ≤ n, let w(n, k) = wq,r (n, k) denote the r-Whitney numbers of the first kind, which are defined by n X (1.11) q n (x)n = w(n, k) (qx + r)k . cqn (0)

cqn

cˆqn (0)

k=0

Let W (n, k) = Wq,r (n, k) denote the r-Whitney numbers of the second kind, which are defined by n X (1.12) (qx + r)n = q k W (n, k) (x)k . k=0

Usually r is taken to be a non-negative integer and q a positive integer, but both may also be regarded as real numbers [11]. The exponential generating function of w(n, k) is given by [10]  k X −r tn ln(1 + qt) 1 , (1.13) w(n, k) = (1 + qt) q n! q k! n≥k

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2. Basic results Replace x by are given by (2.1)

x−r q

in (1.11), then the r-Whitney numbers of the first kind w(n, k)

(x − r|q)n =

n−1 Y

n X

j=0

k=0

(x − r − jq) =

w(n, k) xk ,

q 6= 0,

Using (1.7), we get the following theorem. Theorem 1. The Cauchy polynomials with q parameter of the first kind cqn (r), q 6= 0 can be written explicitly as n X 1 . (2.2) cqn (r) = w(n, k) k+1 k=0

The first few polynomials are cq0 (r) = 1, cq1 (r) = −r + 12 , cq2 (r) = r2 + (q − 1)r − 12 q + 31 , cq3 (r) = −r3 − 32 (2q − 1)r2 + (−2q 2 + 3q − 1)r + q 2 − q + 14 , 1 2 3 cq4 (r) = r4 +(6q−2)r3 +(11q 2 −9q+2)r2 +(6q 3 −11q 2 +6q−1)r−3q 3 + 11 3 q − 2 q+ 5 . Remark 1. If r = 0, then cqn (0) = cqn are the Cauchy numbers with q parameter of the first kind [6] Z 1 n X 1 cqn = q n−k s(n, k) (x|q)n dx = , k + 1 0 k=0

where s(n, k) are the Stirling numbers of the first kind. If q = 1, we have c1n (r) = cn (r) and w1,r (n, k) are reduced to sr (n + r, k + r), and hence we obtain the explicit formula (1.5). From (1.13), we can easily derive the exponential generating function of cqn (r) as follows: n ∞ ∞ X X X tn 1 tn cqn (r) = w(n, k) n! n=0 k + 1 n! n=0 k=0

∞ X ∞ X

tn 1 n! k + 1 k=0 n=k  k ∞ −r X ln(1 + qt) 1 1 q = (1 + qt) q k! k + 1 k=0   ∞ k+1 −r X ln(1 + qt) 1 q = (1 + qt) q q (k + 1)! ln(1 + qt) k=0 −r k ∞  q(1 + qt) q X ln(1 + qt) 1 = ln(1 + qt) q k! =

w(n, k)

k=1

−r q

=

q(1 + qt) ln(1 + qt)



 1 (1 + qt) q − 1 .

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When r = 0, we get the exponential generating function of cqn ∞   X 1 tn q cqn (1 + qt) q − 1 = n! ln(1 + qt) n=0 According to (2.1), (2.3)

(−x − r|q)n =

n−1 Y

n X

j=0

k=0

(−x − r − jq) =

w(n, k) (−1)k xk ,

q 6= 0.

Using (1.7), we get the following theorem. Theorem 2. The Cauchy polynomials with q parameter of the second kind cˆqn (r), q 6= 0 can be written explicitly as cˆqn (−r) =

(2.4)

n X

(−1)k w(n, k)

k=0

1 . k+1

The first few polynomials are cˆq0 (r) = 1, cˆq1 (r) = r − 12 , cˆq2 (r) = r2 − (q + 1)r + 12 q + 31 , cˆq3 (r) = r3 − 32 (2q + 1)r2 + (2q 2 + 3q + 1)r − q 2 − q − 14 , 1 2 3 cˆq4 (r) = r4 −(6q+2)r3 +(11q 2 +9q+2)r2 −(6q 3 +11q 2 +6q+1)r+3q 3 + 11 3 q + 2 q+ 5 . Remark 2. If r = 0, then cˆqn (0) = cˆqn are the Cauchy numbers with q parameter of the second kind [6] Z 1 n X (−1)k cˆqn = q n−k s(n, k) (−x|q)n dx = , k+1 0 k=0

Similarly, we can obtain the exponential generating function of cˆqn (r): k ∞ ∞  X r X tn 1 ln(1 + qt) 1 q q cˆn (r) = (1 + qt) − n! q k! k + 1 n=0 k=0 (2.5) r   −1 q(1 + qt) q = 1 − (1 + qt) q . ln(1 + qt) And ∞ X

(2.6)

n=0

Replace x by are given by (2.7)

x−r q

xn =

cˆqn

  −1 q tn = 1 − (1 + qt) q . n! ln(1 + qt)

in (1.12), then the r-Whitney numbers of the second kind W (n, k)

n X

W (n, k)(x − r|q)k =

k=0

n X k=0

W (n, k)

k−1 Y

(x − r − jq),

q 6= 0.

j=0

Thus, the relation between cqn (r), cˆqn (r) and W (n, k) can be obtained as follows: Z 1X Z 1 n n X 1 q (2.8) W (n, k) ck (r) = W (n, k)(x − r|q)k dx = xn dx = n + 1 0 0 k=0

k=0

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CAUCHY POLYNOMIALS WITH q PARAMETER IN TERMS OF r-WHITNEY NUMBERS 5

(2.9) n X

W (n, k) cˆqk (−r) =

Z

n 1X

Z

0 k=0

k=0

1

W (n, k)(−x − r|q)k dx =

(−1)n xn dx =

0

(−1)n n+1

Cheon et al. [1] gave the following representation of w(n, k) in terms of s(n, k) n   X n w(n, k) = (−1)n−i q i−k [r|q]n−i s(i, k). i i=k

Hence, Corollary 1. The Cauchy polynomials cqn (r) can be computed by using s(n, k) as follows: n X n   X n 1 cqn (r) = (−1)n−i q i−k [r|q]n−i s(i, k) k+1 i k=0 i=k (2.10) n X i   X n 1 (−1)n−i q i−k [r|q]n−i s(i, k) = . i k+1 i=0 k=0

When q = 1, we obtain the identity n   X n cn (r) = (−1)n−i [r|1]n−i ci . i i=0

(2.11)

The r-Whitney numbers wq,r (n, k) satisfy the following identity [1].   n X n−j n (2.12) wq,r+s (n, k) = (−1) [r|q]n−j wq,s (j, k), j j=k

hence, we obtain the following theorem. Theorem 3. For n ≥ 0, we have cqn (r

(2.13)

  n X n−j n + s) = (−1) [r|q]n−j cqj (s). j j=0

Proof. cqn (r

n X

1 k+1 k=0   n X n X 1 n−j n = (−1) [r|q]n−j wq,s (j, k) j k+1

+ s) =

wq,r+s (n, k)

k=0 j=k

j n X X

  n 1 [r|q]n−j wq,s (j, k) j k+1 j=0 k=0   n X n = (−1)n−j [r|q]n−j cqj (s). j j=0 =

(−1)n−j



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Remark 3. For s = 0, we get (2.14)

cqn (r)

  n X n−j n = (−1) [r|q]n−j cqj . j j=0

For q = 1, we get (2.15)

cn (r + s) =

  n X n (−1)n−j [r|1]n−j cj (s). j j=0

Acknowledgement 1. The author thank Prof. Istv´ an Mez¨ o for reading carefully the paper and giving helpful suggestions. References [1] G. S. Cheon and J. H. Jung, r-Whitney numbers of Dowling lattices, Discrete Math. 312, 2337–2348 (2012). [2] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [3] A. Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984). [4] K. Kamano and T. Komatsu, Poly-Cauchy polynomials, Mosc. J. Comb. Number Theory 3, 61–87 (2013). [5] T. Komatsu and I. Mez¨ o, Several explicit formulae of Cauchy polynomials in terms of rStirling numbers, Acta Math. Hungar. 148 2, 522–529 (2016). [6] T. Komatsu, Poly-Cauchy numbers with a q parameter, Ramanujan J. 31, 353–371, (2013). [7] T. Komatsu, Poly-Cauchy numbers, Kyushu J. Math. 67, 143–153 (2013). [8] T. Komatsu, Sums of products of Cauchy numbers, including Poly-Cauchy numbers, J. Discrete Math. 2013, Article ID 373927, 10 pages (2013). [9] D. Merlini, R. Sprugnoli and M. C. Verri, The Cauchy numbers, Discrete Math. 306, 19061920 (2006). [10] I. Mez¨ o, A new formula for the Bernoulli polynomials, Results Math. 58, 329–335 (2010). [11] M. Shattuck, Identities for Generalized Whitney and Stirling Numbers, J. Integer Seq. 20, Article 17.10.4. (2017) [12] F-Z. Zhao, Sums of products of Cauchy numbers, Discrete Math. 309, 3830–3842 (2009). Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, EGYPT. E-mail address: [email protected], [email protected]

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Global dynamics of Chikungunya virus with two routes of infection A. M. Elaiwa , S. E. Almalkia,b and A. Hobinya a

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.

b

Jeddah College of Technology, Technical and Vocational Training Corporation

Emails: a m [email protected] (A. Elaiw), [email protected] (S. E. Almalki)

Abstract In this paper, we address the stability analysis of within-host Chikungunya virus (CHIKV) infection models with antibodies. We incorporate two modes of infections, attaching a CHIKV to a host monocyte, and contacting an infected monocyte with an uninfected monocyte. The global stability analysis of the equilibria are established using Lyapunov method. The existence and global stability of the steady states are determined by the basic reproduction number R0 . We have proven that the CHIKV-free equilibrium E0 is globally asymptotically stable when R0 ≤ 1, and the infected equilibrium E1 is globally asymptotically stable when R0 > 1. The theoretical results are confirmed by numerical simulations.

1

Introduction

During last decades, many researchers have developed and analyzed several mathematical models human pathogens (see e.g. [1]-[16]). Chikungunya virus (CHIKV) is an alphavirus causes chikungunya fever. CHIKV is a mosquito-transmitted and is transmitted by the Aedes albopictus and Aedes agypti mosquito. Most of authors develop the mathematical models to describe the disease transmission mosquito and human populations. Recently, Wang and Liu [16] have proved a mathematical model for the within-host CHIKV dynamics as: s˙ = β − δs − ηsy,

(1)

y˙ = ηsy − y,

(2)

p˙ = πy − cp − rxp,

(3)

x˙ = λ + ρxp − mx,

(4)

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Here, s, y, p and x are the concentrations of uninfected monocytes, infected monocytes, CHIKV pathogen and antibodies, respectively. β and δ represent the birth rate and death rate constants of the uninfected monocytes, respectively. The monocytes become infected at rate ηsy, where η is the infection rate constant. Constants , c and m represent, respectively, the death rate of the infected monocytes, CHIKV and antibodies. Constant π is the generation rate of the CHIKV from actively infected monocytes. Antibodies attack the CHIKV at rate rxp. Once antigen is encountered, the antibodies expand at a constant rate λ and proliferate at rate ρxp. In a very recent work, Elaiw et al. [17], [18] have studied the global stability analysis of a class of CHIKV dynamics models. The models presented in [16]-[18] assume that the uninfected monocyte becomes infected by contacting with CHIKV(CHIKV-to-monocyte transmission). Kristin and Mork [19] reported that the CHIKV can also spread by infected-to-monocyte transmission. Viral danamics models with both cellular and viral infections have been studied in several works [20]-[24]. However, the dynamics of CHIKV with two routes of infection did not studied before. Our aim is to propose and analyse a CHIKV dynamics model with two routes of infection. We calculate the basic reproduction number R0 , and construct Lyapunov functions to prove the global stability of the equilbria.

2

CHIKV dynamics model

We investigate the following CHIKV dynamics model with CHIKV-to-monocyte and infected-tomonocyte with two routes of infection: s˙ = β − δs − η1 sp − η2 sy,

(5)

y˙ = η1 sp + η2 sy − y,

(6)

p˙ = πy − cp − rxp,

(7)

x˙ = λ + ρxp − mx.

(8)

Here, the uninfected monocytes become infected at rate (η1 y + η2 p)s, where η1 and η2 are the CHIKVmonocyte and infected-monocyte incidence constants, respectively.

2.1

Nonnegativity and boundedness

Lemma 1 There exist M1 , M2 , M3 > 0, such that the following compact set is positively invariant for system (5)-(8) Γ1 = {(s, y, p, x) ∈ R4≥0 : 0 ≤ s, y ≤ M1 , 0 ≤ p ≤ M2 , 0 ≤ x ≤ M3 }

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Proof. We have s˙ |s=0 = β > 0, p˙ |p=0 = πy ≥ 0,

y˙ |y=0 = η1 sp ≥ 0, for all y ≥ 0,

for all s, p ≥ 0,

x˙ |x=0 = λ > 0.

Thus R4≥0 positively invariant with respect to system (5)-(8). Let us define F1 (t) = s(t) + y(t), r F2 (t) = p(t) + x(t). ρ Then from Eqs. (5)-(8) we get F˙1 (t) = β − δs(t) − y(t) ≤ β − σ1 (s(t) + y(t)) = β − σ1 F1 (t), where, σ1 = min{δ, }. Hence F1 (t) ≤ M1 , if F1 (0) ≤ M1 , where M1 =

β σ1 .

It follows that 0 ≤

s(t), y(t) ≤ M1 if 0 ≤ s(0) + y(0) ≤ M1 . Moreover, we have r mr F˙2 (t) = πy(t) − cp(t) + λ − x(t) ρ ρ   r r ≤ πM1 + λ − σ2 p(t) + x(t) ρ ρ r = πM1 + λ − σ2 F2 (t), ρ where, σ2 = min{c, m}. Hence F2 (t) ≤ M2 , if F2 (0) ≤ M2 , where M2 =

πM1 + ρr λ . σ2

Since p(t) and x(t)

are all nonnegative, then 0 ≤ p(t) ≤ M2 and x(t) ≤ M3 if 0 ≤ p(0) + ρr x(0) ≤ M2 , where M3 =

2.2

ρM2 r .

Equilbria

We define the basic reproduction number R0 =

(η1 πm + η2 cm + η2 rλ)β . δ(cm + rλ)

Lemma 2 (i) if R0 ≤ 1, then there exists only one equilbrium E0 ∈ Γ1 (ii) if R0 > 1, then there exist ◦

◦

two equilbria E0 ∈ Γ1 and E1 ∈ Γ1 , where Γ1 is the interior of Γ1 . Proof. Any equilbrium satisfying β − δs − η1 sp − η2 sy = 0,

(9)

η1 sp + η2 sy − y = 0,

(10)

πy − cp − rxp = 0,

(11)

λ + ρxp − mx = 0.

(12)

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By solving Eqs. (9)-(12) we we get two equilbria a CHIKV-free equilbrium E0 = (s0 , 0, 0, x0 ), where s0 =

β δ

and x0 =

λ m.

Moreover we have C1 p3 + C2 p2 + C3 p + C4 = 0,

where C1 = −cπη1 ρ2 − c2 η2 ρ2 , C2 = 2cmπη1 ρ + 2c2 mη2 ρ + πrη1 λρ + 2crη2 λρ − cπδρ2 + π 2 βη1 ρ2 + cπβη2 ρ2 C3 = −cm2 πη1 − c2 m2 η2 − mπrη1 λ − 2cmrη2 λ − r2 η2 λ2 + 2cmπδρ − 2mπ 2 βη1 ρ − 2cmπβη2 ρ + πrδλρ − πrβη2 λρ, C4 = −cm2 πδ + m2 π 2 βη1 + cm2 πβη2 − mπrδλ + mπrβη2 λ. Let define a function X(p) as: X(p) = C1 p3 + C2 p2 + C3 p + C4 = 0. Then we obtain X(0) = C4 ,   mr2 η2 λ2 m =− < 0. X ρ ρ The constant C4 can be written as   (η1 πm + η2 cm + η2 rλ)β −1 C4 = mπδ(cm + rλ) δ(cm + rλ) Then C4 > 0 if the following condition is satisfied (η1 πm + η2 cm + η2 rλ)β > 1, δ(cm + rλ)

(13)

then there exists p1 ∈ (0, m ρ ) such that X(p1 ) = 0. Therefore, if condition (13) is satisfied, then c(m − ρp1 ) + rλ > 0, η1 π(m − ρp1 ) + η2 c(m − ρp1 ) + η2 rλ p1 (c(m − ρp1 ) + rλ) λ y1 = > 0, x1 = > 0. π(m − ρp1 ) m − ρp1 s1 =

Then an infected equilbrium E1 = (s1 , y1 , p1 , x1 ) exists when R0 > 1. ◦

Now we show that E0 ∈ Γ1 and E1 ∈ Γ1 . Clearly, E0 ∈ Γ1 . From the equilbrium conditions of E1 we have β = δs1 + η1 s1 p1 + η2 s1 y1 ⇒ δs1 + y1 = β ⇒ 0 < s1
0 we have be easily shown that

dU0 dt

dU0 dt

≤ 0. Let W0 = {(s, y, p, x) :

dU0 dt

(14) = 0}. It can

= 0 at E0 . Appling LaSalle’s invariance principle, we get E0 is globally

asymptotically stable when R0 ≤ 1. ◦

Theorem 2 If R0 > 1, then E1 is globally asymptotically stable in Γ1 . Proof. Define         s y η1 s1 p1 p r η1 s1 p1 x U1 (s, y, p, x) = s1 G + y1 G + p1 G + x1 G . s1 y1 πy1 p1 ρ πy1 x1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.3, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC 1 Calculating dU dt along the trajectories of (5)-(8) we obtain       dU1 y1 s1 β − δs − η1 sp − η2 sy + 1 − η1 sp + η2 sy − y = 1− dt s y       p1 rη1 s1 p1 x1 η1 s1 p1 1− πy − cp − rxp + 1− λ + ρxp − mx + πy1 p ρπy1 x    s1 y1 p1 y η1 s1 p1 = 1− β − δs + η1 s1 p + η2 s1 y − η1 sp − η2 sy1 − y + y1 + y − η1 s1 p1 s y y1 py1    η1 s1 p1 η1 s1 p1 η1 s1 p1 rη1 s1 p1 rη1 s1 p1 x1 − cp + cp1 + rxp1 − x1 p + 1− λ − mx . πy1 πy1 πy1 πy1 ρπy1 x

Applying the equilbrium conditions for E1 β = η1 s1 p1 + η2 s1 y1 + δs1 , y1 = η1 s1 p1 + η2 s1 y1 , cp1 = πy1 − rx1 p1 , λ = mx1 − ρx1 p1 . we get    dU1 (s − s1 )2 s1 η1 s1 p1 + η2 s1 y1 = −δ + 1− dt s s spy1 s p1 y − η1 s1 p1 − η2 s1 y1 + η1 s1 p1 + η2 s1 y1 − η1 s1 p1 + η1 s1 p1 s1 p1 y s1 py1 η1 s1 p1 η1 s1 p1 η1 s1 p1 x1 rη1 s1 p1 m (x − x1 )2 rx1 p1 + rxp1 + rx1 p1 − . −2 πy1 πy1 πy1 x ρπy1 x

(15)

Eq. (15) can be simplified as:    dU1 (s − s1 )2 s1 s s1 spy1 = −δ + η2 s1 y1 2 − − + η1 s1 p1 3 − − − dt s s s1 s s1 p1 y   x x1 rη1 s1 p1 m (x − x1 )2 η1 s1 p1 rx1 p1 2 − − − − πy1 x1 x ρπy1 x   2 2 (s − s1 ) η2 y1 (s − s1 ) s1 spy1 p1 y = −δ − + η1 s1 p1 3 − − − s s s s1 p1 y py1 η1 s1 p1 (x − x1 )2 rη1 s1 p1 m (x − x1 )2 + rp1 − πy1 x ρπy1 x  2 (s − s1 ) η1 s1 p1 rλ (x − x1 )2 s1 = −(δ + η2 y1 ) − + η1 s1 p1 3 − − s πy1 ρx1 x s

p1 y py1



 spy1 p1 y − . s1 p1 y py1

We use the following arithmetic mean-geometric mean inequality rule. If ai ≥ 0, i = 1, 2, ..., n, then v u n n uY 1X n ai ≥ t ai , (16) n i=1

i=1

where equality holding if and only if a1 = a2 = ... = an . It follows that   1 s1 spy1 p1 y + + ≥ 1. 3 s s1 p1 y py1 Therefore,

dU1 dt

≤ 0 for all s, y, p, x > 0 and

dU1 dt

= 0 if and only if s = s1 , y = y1 , p = p1 and x = x1 . It

follows that the global stability of E1 is induced from LaSalle’s invariance principle. 6 486

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Table 1: The value of the parameters of model (5)-(8).

4

Parameter

Value

Parameter

Value

β

2

δ

0.1

η1 , η2

varied



0.5

π

4

c

0.1

r

0.5

λ

1.4

m

1

ρ

0.2

Numerical Simulations

We will use the values of the parameters given in Table 1. Moreover, we similate the system with three different initial values as: IV1: s(0) = 14.0, y(0) = 1.0, p(0) = 1.0, and x(0) = 1.0, IV2: s(0) = 8.0, y(0) = 2.0, p(0) = 3.0, and x(0) = 4.0, IV3: s(0) = 4.0, y(0) = 3.5, p(0) = 6.0, and x(0) = 7.0. Then we consider two sets of the values of η1 and η2 as follows: Set (I): We choose η1 = η2 = 0.001. The value of R0 is computed as R0 = 0.2400 < 1. Figure 1 shows that, the concentrations of the uninfected monocytes and B cells return to their values s0 = and x0 =

λ m

β δ

= 20

= 1.4, respectively. On the other hand, the concentrations of infected monocytes and

CHIKV particles are declining and reaching zero for the initial values IV1-IV3. This shows that, E0 is GAS which agrees with the result of Theorem 1. Set (II): We take η1 = η2 = 0.05. Then, we calculate R0 = 12.0 > 1. We comput the equilbria as E0 (20.0, 0, 0, 1.4) and E1 = (4.45, 3.10, 3.87, 6.22). Figure 1 shows that when R0 > 1, the states of the system tend to E1 for all the three initial values IV1-IV3. This confirms that the validity of Theorem 2.

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20

8 Set (I)

18

7 6

14

Infected monocytes

Uninfected monocytes

16

12 10 8

5 4 Set (II) 3 2

6

Set (II) 1

4

Set (I) 2 0

10

20

30

40

50

60

0 0

70

10

20

30

Time

40

50

60

70

Time

(a) Uninfected monocytes.

(b) Infected monocytes. 12

10 9

10

7

8

6

Antibodies

Free CHIKV particles

8

5 Set (II) 4

Set (II) 6

4 3 Set (I)

2

2

1 Set (I) 0 0

10

20

30

40

50

60

0 0

70

Time

10

20

30

40

50

60

70

Time

(c) Free CHIKV particles.

(d) Antibodies.

Figure 1: The simulation of trajectories of system (5)-(8).

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. Li and S. Fu, Global stability of a virus dynamics model with intracellular delay and CTL immune response, Mathematical Methods in the Applied Sciences, 38 (2015), 420-430. [3] A. M. Elaiw, N. H. AlShamrani and K. Hattaf, Dynamical behaviors of a general humoral immunity viral infection model with distributed invasion and production, International Journal of Biomathematics, 10(3), (2017) Article ID 1750035. [4] A. M. Elaiw,, A. A. Raezah, and K. Hattaf, Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response, International Journal of Biomathematics, Vol. 10, No. 5 (2017), 1750070.

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[5] A. M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with BeddingtonDeAngelis functional response, Mathematical Methods in the Applied Sciences, 36 (2013), 383394. [6] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications, 11 (2010), 2253-2263. [7] 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. [8] 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. [9] K. Wang, A. Fan, and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Analysis: Real World Applications, 11 (2010), 3131-3138. [10] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J, Layden, and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, Science, 282 (1998), 103-107. [11] L. Wang, M. Y. Li, and D. Kirschner, Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Mathematical Biosciences, 179 (2002) 207-217. [12] 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. [13] 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, (2015), 161-190. [14] A. M. Elaiw and N. H. AlShamrani, Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Mathematical Methods in the Applied Sciences, 40(3) (2017), 699-719. [15] M. Li, and H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T-cells, Applied Mathematical Modelling, 35(7) (2011) 3587-3595 [16] Y. Wang, X. Liu, Stability and Hopf bifurcation of a within-host chikungunya virus infection model with two delays, Mathematics and Computers in Simulation, 138 (2017), 31-48.

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[17] A. M. Elaiw, T. O. Alade and S. M. Alsulami, Stability of a within-host Chikungunya virus dynamics model with latency, Journal of Computational Analysis and Applications, 26(5) 2019, 777-790. [18] A. M. Elaiw, T. O. Alade and S. M. Alsulami, Analysis of latent CHIKV dynamics model with time delays, Journal of Computational Analysis and Applications, 27(1) 2019, 19-36. [19] Kristin M. Long and Mark T. Heise, Protective and Pathogenic Responses to Chikungunya Virus Infection, Curr Trop Med Rep. 2(1) (2015), 13-21. [20] J. Wang, J. Lang, X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Analysis: Real World Applications, 34 (2017), 75-96. [21] F. Li and J. Wang, Analysis of an HIV infection model with logistic target cell growth and cellto-cell transmission, Chaos, Solitons and Fractals, 81 (2015), 136-145. [22] 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. [23] 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. [24] Y. Yang, 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.

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Weighted norm inequalities of θ-type Calder__n-Zygmund operators and commutators on λ-central Morrey space and Shuangping Tao∗

Yanqi Yang

( College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070,Gansu, China, Email:[email protected] )

Abstract: In this paper, the weighted boundedness for θ-type Calder´on-Zygmund operators Tθ is established on the λ-central Morrey space. Futhermore, the weighted norm inequalities for commutators of [b, Tθ ] generated by Tθ and BMO functions on the weighted λ-central Morrey space is also given. Keywords: θ-type Calder´on-Zygmund operator; weighted λ-central Morrey space; commutator 2010 MR Subject Classification: 42B20, 42B25, 42B35.

1

Introduction and notation The theory of Calder´ on-Zygmund operators has played very important roles in modern harmonic

analysis with lots of extensive applications in the others fields of mathematics, which has been extensively studied (see [7-10, 16-17], for instance). In 1985, Yabuta introduced certain θ-type Calder´onZygmund operators to facilitate his study of certain classes of pseudodifferential operators (see [36]). Following the terminology of Yabuta, we recall the so-called θ-type Calder_n-Zygmund operators. Let θ be a non-negative and non-decreasing function on R+ = (0, ∞) satisfying Z 0

1

θ(t) dt < ∞. t

(1.1)

A measurable function K(·, ·) on Rn × Rn is said to be a θ-type Calder_n-Zygmund kernel if it satisfies |K(x, y)| ≤ C|x − y|−n

(1.2)

and  |x − x0 | |K(x, y) − K(x , y)| + |K(y, x) − K(y, x )| ≤ Cθ |x − y|−n , as |x − y| ≥ 2|x − x0 |. (1.3) |x − y| 0



0

Definition 1.1[36] Let Tθ be a linear operator from S(Rn ) into its dual S 0 (Rn ), where S(Rn ) denotes the Schwartz class. One can say that Tθ is a θ-type Calder_n-Zygmund operator if it satisfies the following conditions: (1) Tθ can be extended to be a bounded linear operator on L2 (Rn ) ; (2) there is a θ-type Calder_n-Zygmund kernel K(x, y) such that Z Tθ f (x) := K(x, y)f (y)dy, as f ∈ Cc∞ (Rn ) and x ∈ / suppf. Rn

*Corresponding author and Email:[email protected](by S. Tao); [email protected] (by Y. Yang)

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(1.4)

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It is easy to see that the classical Calder_n-Zygmund operator with standard kernel is a special case of θ-type operator Tθ as θ(t) = tδ with 0 < δ ≤ 1. Given a locally integrable function b, the commutator generated by Tθ and b is defined by Z [b(x) − b(y)]K(x, y)f (y)dy.

[b, Tθ ]f (x) = b(x)Tθ f (x) − Tθ (b · f )(x) =

(1.5)

Rn

Such type of operators is extensively applied in PDE with non-smooth area. Many authors concentrates on the boundedness of this operators on various function spaces, we refer the reader to see [19-20, 27, 29-30, 33-35] for its developments and applications. In [27], Quek-Yang established the boundedness of Tθ on spaces such as weighted Lebesgue spaces and weak Lebesgue spaces, weighted Hardy spaces and weak Hardy spaces. Ri-Zhang obtained the bounedness of Tθ on Hardy spaces with non-doubling measures and non-homogeneous metric measure spaces in [29-30]. Wang proved the boundedness of Tθ and [b, Tθ ] on the generalized weighted Morrey spaces in [33]. Inspired by the results mentioned previously, a natural and interesting problem is to consider whether the θ-type Calder´on-Zygmund operators Tθ and their commutators [b, Tθ ] are bounded on λ-central Morrey space or not. The purpose of this paper is to give an surely answer. On the other hand, the well-known Morrey spaces which introduced originally by Morrey [23] in relation to the study of partial differential equations, were widely investigated during last decades, including the study of classical operators of harmonic analysis in various generalizations of these spaces. Morrey-type spaces appeared to be quite useful in the study of the local behavior of the solutions of partial differential equations, a priori estimates and other topics. They are also widely used in applications to regularity properties of solutions to PDE including the study of Navier-Stokes equations (see [32] and references therein). The ideas of Morrey (see [23]) were further developed by Campanato in 1964 (see [11]). In 1975, Adam proved the boundedness of Riesz potential on the classical Morrey space in [1]. Later, in 1987, the boundedness of singular integrals and HardyLittlewood maximal functions on Morrey spaces was obtained By Chiarenza and Frasca in [13]. A more systematic study of these (and even more general) spaces, we refer the readers to see [2-3, 6, 26, 28, 31]. 0

In [5], Beurling introduced a pair of dual Banach spaces, Aq and B q with 1/q + 1/q 0 = 1. After that, Feichtinger found the folling way to describe B q as kf kB q = sup(2−kn/q kf χk kLq ) < ∞,

(1.6)

k≥0

where χ0 is the characteristic function of the unit ball defined by {x ∈ Rn : |x| ≤ 1} and χk is the characteristic function of the annulus, that is {x ∈ Rn : 2k−1 < |x| ≤ 2k } with k ∈ Z+ . By duality, the beurling algebra Aq can be written as kf kAq =

∞ X

0

2kn/q kf χk kLq < ∞.

(1.7)

k=0

Later, a new Hardy space HAq related to the Beurling algebra Aq was introduced by Chen and Lau (see [12]). Denotes B(0, R) be a cube centered at the origin with the side-length R > 0. Let 2 492

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fB(0,R) =

1 |B(0,R)|

R

B(0,R) f (x)dx

be the integral average of f on B. Then using duality, the dual space

q

of HA can be described by CBMOq with the following norm, kf kCBMOq =

1 |B(0, R)|

X R≥1

!1/q

Z

q

|f (x) − fB(0,R) | dx

< ∞.

(1.8)

B(0,R)

˙ q and they Later, Lu and Yang (see [21-22]) introduced the homogeneous new Hardy type space HA ˙ q can be written by proved that the dual space of HA q = kf kCBMO ˙

1 |B(0, R)|

X R≥0

!1/q

Z

q

|f (x) − fB(0,R) | dx

< ∞.

(1.9)

B(0,R)

q ˙ Obviously, the space of CBMO is the homogeneous central bounded mean oscillation space depending

on q and it can be regarded as an extention of the classical BMO since the famous John-Nirenberg inequality no longer hold in such space. Alverez, Lakey and Guzm´an-Partida introduced the λ-central bounded mean oscillation space and the λ-central Morrey space in 2000 (see [4]), respectively. Definition 1.2[4] Let λ < 1/n and 1 < q < ∞. Then we say that a function f ∈ Lqloc (Rn ) belongs q,λ ˙ to the λ-central bounded mean oscillation space CBMO (Rn ) if kf kCBMO q,λ = ˙

X R>0

1 |B(0, R)|1+λq

!1/q

Z

q

|f (x) − fB(0,R) | dx

< ∞.

(1.10)

B(0,R)

Definition 1.3[4] Let λ ∈ R and 1 < q < ∞. Then the λ-central Morrey space B˙ q,λ (Rn ) is defined of all functions f ∈ Lqloc (Rn ) by the following norm kf kB˙ q,λ =

X R>0

1 |B(0, R)|1+λq

!1/q

Z

|f (x)|q dx

< ∞.

(1.11)

B(0,R)

It is very important to study the weighted norm inequalities for some integral operators on classical Lp spaces, one may see [14, 24-25] et al. for more details. In 2009, Komori-Furuya and Shirai (see [18]) defined the weighted Morrey space and showed the boundedness of some classical integral operators and their commutators on the weighted Morrey spaces. In this paper, we will prove the weighted boundedness of θ-type Calder´ on-Zygmund operator Tθ on the weighted λ-central Morrey space. Before giving the main results, we introduce the following definitions. Definition 1.4[37] Let λ ∈ R and 1 < q < ∞. Then the weighted λ-central Morrey space n B˙ q,λ ω1 ,ω2 (R ) is defined by   X n f ∈ B˙ q,λ = ˙ q,λ ω1 ,ω2 (R ) : kf kB ω1 ,ω2 

R>0

1 ω1 (B(0, R))1+λq

Z B(0,R)

 

!1/q |f (x)|q ω2 (x)dx

0

where the definition of fB,ω is fB,ω =

1 ω(B)

R B

!1/q

Z

q

|f (x) − fB,ω | ω(x)dx

< ∞,

(1.13)

B(0,R)

f (x)ω(x)dx.

Definition 1.6[25] We say a non-negative function ω(x) belongs to the Muckenhoupt class Ap with 1 < p < ∞ if there exist a constant C > 1 such that 

1 |Q|



Z ω(x)dx Q

1 |Q|

Z

1−p0

ω(x)

p−1 < ∞,

dx

Q

where 1/p + 1/p0 = 1 and [ω]Ap denotes the infimum of C. Moreover, we define A∞ =

S

1 0 independent of f , such that, for any f ∈ B˙ ωp2 ,λ2 , k[b, Tθ ]f kB˙ p,λ ≤ CkbkCBMO p1 ,λ1 kf k ˙ p2 ,λ2 . ˙ B ω

ω

ω

Let us give some necessary notations. Throughout the paper C will denote a positive constant whose value may change at each appearance. In the following, unless otherwise stated, for any real number p > 1, we denote p0 by 1/p + 1/p0 = 1. Moreover, we say that a weight ω satisfies the doubling condition if there exists a constant D, such that for any cube Q ∈ Rn , we have ω(2Q) ≤ Dω(Q). For simplicity, we denote ω ∈ ∆2 if ω satisfies the doubling condition. 2

Preliminary Lemmas Lemmas 2.1[15] If ω ∈ Ap for some 1 ≤ p < ∞, then ω ∈ ∆2 . More precisely, for all α > 1, we

have ω(αQ) ≤ αnp [ω]Ap ω(Q). Lemmas 2.2[27] Let 1 < p < ∞ and ω ∈ Ap . Then, the θ-type Calder_n-Zygmund operator Tθ is bounded on Lpω . Lemmas 2.3[18] If ω ∈ ∆2 , then there exists a constant D > 1 such that for any cube B, ω(2B) ≥ Dω(B). 4 494

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Lemmas 2.4[37] If ω ∈ Ap for some 1 ≤ p < ∞, then for any k ∈ Z+ , s < 0 and any cube B ∈ Rn , ω(2k B)s ≤ D1ks ω(B)s , where D1 is a positive constant which belongs to the interval (1, 2). 3

Proof of Theorems

Proof of Theorem 1.1. For a fixed cube B = B(0, R), we may decompose f = f1 + f2 with f1 = f χ2B . Then we obain 1 ω(B)1+λp

Z

Z 1 |T (f )(x)| ω(x)dx ≤ |T (f1 )(x)|p ω(x)dx ω(B)1+λp B B Z 1 |T (f2 )(x)|p ω(x)dx =: C(I1 + I2 ). ω(B)1+λp B p

From Lemma 2.1 and Lemma 2.2, we have Z 1 |T (f1 )(x)|p ω(x)dx ω(B)1+λp B Z 1 ≤ |f (x)|p ω(x)dx ω(B)1+λp 2B ω(2B)1+λp ≤ Ckf kpB˙ p,ω . ω ω(B)1+λp

I1 =

As 1 + λp ≥ 0, by using Lemma 2.1, then there exists a constant C > 0 independent of f such that I1 ≤ Ckf kpB˙ p,ω .

(3.1)

ω

On the other hand, by using Lemma 2.4, we can also get (3.1) with an similar argument in the case of 1 + λp < 0. Next let’s estimate I2 . Noting that x ∈ B and y ∈ (2B)c , then there exists a constant C > 0 such that |y| < C|x − y|. Thus, we have Z Z |Tθ (f2 )| ≤ |K(x, y)|f (y)|dy ≤ C Rn

1/|y|n |f (y)|dy

|y|>2r

Furthermore, by using Definition 1.6 and the H¨older’s inequality, we can get Z

n

1/|y| |f (y)|dy = |y|>2r

≤

∞ X j=1

∞ Z X j=1

1 |2j B|

Z

p

1/|y|n |f (y)|dy

2j r2r |x0 − y| !p Z |f (x)| dy |b(x) − bB,ω |p ≤C n |x − y| 0 |y|>2r !p Z |f (x)| +C dy|b(y) − bB,ω |dy . n |y|>2r |x0 − y| Thus, we can decompose II as 1 ω(B)1+λp

Z

1 ≤ ω(B)1+λp

Z

1 + ω(B)1+λp

Z

II =

|[b, Tθ ]f2 (x)|p ω(x)dx

B

|y|>2r

|y|>2r

|f (x)| dy|b(y) − bB,ω |dy |x0 − y|n

B

Z

B

!p

|f (x)| dy |x0 − y|n

Z

|b(x) − bB,ω |p ω(x)dx !p ω(x)dx

= II1 + II2 . For II1 , by the same estimate as in the proof of Theorem 1.1, we can obtain that Z ∞ X 1/|y|n |f (y)|dy ≤ Ckf kB˙ λ2 ,q2 ω(2j+1 B)λ2 , ω2

|y|>2r

j=1

which implies II1 ≤ kf kp˙ λ2 ,q2 Bω2

≤

kf kp˙ λ2 ,q2 Bω 2

Z ∞ X ω(2j+1 B)λ2 p j=1 ∞ X j=1

ω(B)1+λp

Bω2

=

Z

p1

|b(x) − bB,ω | ω(x)dx

ω

≤ Ckf kp˙ λ2 ,q2 kbkp

p ,λ1

1 ˙ CBMO ω

Z ×

ω(x) B

˙ CBMO ω

2

p/p1

B

p/p1 +λ1 p+1−p/p1 p1 ,λ1 ω(B)

Ckf kp˙ λ2 ,q2 kbkp ˙ p1 ,λ1 Bω CBMO Bω2

B

ω(2j+1 B)λ2 p ω(B)1+λp

≤ Ckf kp˙ λ2 ,q2 kbkp

|b(x) − bB,ω |p ω(x)dx p1 −p p

1−p/p1 p1 dx p1 − p

∞ X ω(2j+1 B)λ2 p j=1

ω(B)1+λp

∞ X ω(2j+1 B)λ2 p j=1

ω(B)λ2 p

,

where in the last inequality we use the fact λ2 < 0 and Lemma 2.4. 7 497

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Next, we turn to estimate II2 . Noticing that 1/p = 1/p1 + 1/p2 , by using the H¨ older’s inequality, then we have Z ∞ Z X |f (y)| |f (y)| |b(y) − bB,ω |dy = |b(y) − bB,ω |dy n n j j+1 r |y| |y|>2r |y| j=1 2 r a > 0. Let t = log b/a > 0, then from (1.1)-(1.3) one has √ √ sinh(t) ab H(a, b) = , L(a, b) = ab , (3.1) cosh(t) t √ √ T Q(a, b) = abI0 (t), A(a, b) = ab cosh(t), (3.2) log T Q(a, b) − log H(a, b) (3.3) log A(a, b) − log H(a, b) log I0 (t) + log cosh(t) 1 1 = f (t) + , = 2 log cosh(t) 2 2 I0 (t) cosh(t) − 1 T Q(a, b) − H(a, b) = = 2g(t), (3.4) A(a, b) − H(a, b) cosh2 (t) − 1 T Q(a, b)L(a, b) − H(a, b)A(a, b) (3.5) A2 (a, b) − H(a, b)A(a, b) sinh(t)I0 (t) − t = 2h(t), = t[cosh2 (t) − 1] T Q2 (a, b) − H(a, b)A(a, b) (3.6) L(a, b)A(a, b) − H(a, b)A(a, b) t[I02 (t) − 1] = = 2λ(t), sinh(t) cosh(t) − t where, f (t), g(t), h(t) and λ(t) are given by Lemma 2.8, Lemma 2.9, Lemma 2.10 and Lemma 2.11, respectively.

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8

Therefore, Theorem 3.1 follows easily from (3.3)-(3.6), and Lemma 2.8, Lemma 2.9, Lemma 2.10 and Lemma 2.11.  From Theorem 3.1, (3.1) and (3.2), we get Corollary 3.2 immediately. Corollary 3.2. The double inequalities cosh1/2 (t) < I0 (t)
0.

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Symmetric identities for Dirichlet-type multiple twisted (h, q)-l-function and higher-order generalized twisted (h, q)-Euler polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper we investigate some interesting symmetric identities for multiple twisted (h, q)-l-function and higher-order generalized twisted (h, q)-Euler polynomials in complex field. Key words : Symmetric properties, power sums, Euler numbers and polynomials, multiple twisted (h, q)-l-function, higher-order generalized twisted (h, q)-Euler numbers and polynomials. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Euler numbers and polynomials possess many interesting properties and arising in many areas of mathematics, mathematical physics and statistical physics. Many mathematicians have studied in the area of the q- extension of Euler numbers and polynomials(see [1-10]). Y. He studied several identities of symmetry for Carlitz’s q-Bernoulli numbers and polynomials in complex field(see [2]). D. Kim et al.[3] derived some identities of symmetry for (h, q)-extension of higher-order Euler numbers and polynomials. D. V. Dolgy et al.[1] derived some identities of symmetry for higher-order generalized q-Euler polynomials. In this paper, we present a systemic study of the generalized twisted (h, q)-Euler numbers and polynomials of higher-order by using the multiple twisted(h, q)-l-function. Throughout this paper, the notations N, Z, R, and C denote the sets of positive integers, integers, real numbers, and complex numbers, respectively, and Z+ := N ∪ {0}. We assume that q ∈ C with |q| < 1. Throughout this paper we use the notation: [x]q =

1 − qx (cf. [1, 2, 3, 5]) . 1−q

Note that limq→1 [x] = x. Let χ be a Dirichlet character with conductor d ∈ N with d ≡ 1 (mod 2) and ε be the pN -th root of unity(see [8, 9, 10]). T. Kim introduced the multiple q-Euler zeta function which interpolates higher-order q-Euler polynomials at negative integers as follows(see [4, 5]): ζq,r (s, x) =

[2]rq

∞ ∑ m1 ,··· ,mr

∑r

∑r

(−1) j=1 mj q j=1 mj , [m1 + · · · + mr + x]sq =0

(1)

where s ∈ C and x ∈ R, with x ̸= 0, −1, −2, . . .. Recently, D. V. Dolgy et al.[1] considered some symmetric identities for higher-order generalized q-Euler polynomials. The generalized Euler polynomials of order r ∈ N attached to χ are also defined by the generating function: )r ( d−1 ∞ ∑ ∑ χ(l)(−1)l e(x+l)t tm (r) = Em,χ (x) . (2) 2 dt e +1 m! m=0 l=0

(r)

(r)

(r)

When x = 0, En,χ = En,χ (0) are called the generalized Euler numbers En,χ attached to χ.

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For h ∈ Z, α, k ∈ N, and n ∈ Z+ , we introduced the higher order twisted q-Euler polynomials with weight α as follows(see [7]): n ( ) ∑ [2]kq n q αlx . (−1)l α n αl+h (1 − q ) (1 + εq ) · · · (1 + εq αl+h−k+1 ) l

(α) en,q,ε E (h, k|x) =

l=0

(α) (α) en,q,ε en,q,ε In the special case, x = 0, E (h, k|0) = E (h, k) are called the higher-order twisted q-Euler numbers with weight α.

We consider the higher order generalized q-Euler polynomials of order r attached to χ twisted by ramified roots of unity as follows(see [8]): ( r ) ∞ ∞ ∑r ∑r ∑ ∏ ∑ tn (r) mj r j=0 (−ζ) En,χ,ζ,q (x) = 2 χ(mi ) e[x+ j=1 mj ]q t . n! m ,...,m =0 n=0 i=1 1

r

(r)

(r)

In the special case x = 0, the sequence En,χ,ζ,q (0) = En,χ,ζ,q are called the n-th generalized q-Euler numbers of order r attached to χ twisted by ramified roots of unity. (h,k) As is well known, the higher-order generalized twisted (h, q)-Euler polynomials En,χ,q,ε (x) attached to χ are defined by the following generating function to be ∞ ∑

(h,k) Feχ,q,ε (t, x) = [2]kq

(−1)m1 +···+mk q

m1 ,··· ,mk =0



×

k ∏

∑k

j=1 (h−j+1)mj

εm1 +···+mk

 χ(mj ) e[m1 +···+mk +x]q t

(3)

j=1 ∞ ∑

=

(h,k) En,χ,q,ε (x)

n=0

tn , n!

(h,k)

(h,k)

where h ∈ Z and k ∈ N. When x = 0, En,χ,q,ε = En,χ,q,ε (0) are called the higher-order generalized (h,k)

(h,k)

twisted (h, q)-Euler numbers En,χ,q,ε attached to χ. Observe that if q → 1, ε → 1, then En,χ,q,ε → (k) (h,k) (k) En,χ and En,χ,q,ε (x) → En,χ (x). By using (3) and Cauchy product, we have (h,k) (x) = En,χ,q,ε

n ( ) ∑ n

l

l=0

(h,k)

(h,k) + [x]q )n , = (q x Eχ,q,ε q lx El,χ,q,ε [x]n−l q

(4)

(h,k)

(h,k)

with the usual convention about replacing (Eχ,q,ε )n by En,χ,q,ε . By using complex integral and (3), we can also obtain the Dirichlet-type multiple twisted (h, q)-l-function as follows: (h,k) lχ,q,ε (s, x) =

=

1 Γ(s) [2]kq

∫

∞

(h,k) Feχ,q,ε (−t, x)ts−1 dt 0 (∏ ) ∑k ∑k ∑k k ∞ j=1 (h−j+1)mj ε j=1 mj (−1) j=1 mj ∑ j=1 χ(mj ) q

[m1 + · · · + mk + x]sq

m1 ,··· ,mk =0

(5) ,

where s ∈ C and x ∈ R, with x ̸= 0, −1, −2, . . .. By using Cauchy residue theorem, the value of Dirichlet-type multiple twisted (h, q)-l-function at negative integers is given explicitly by the following theorem: Theorem 1. Let k ∈ N and n ∈ Z+ . We obtain (h,k) (h,k) lχ,q,ε (−n, x) = En,χ,q,ε (x).

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The purpose of this paper is to obtain some interesting identities of the power sums and (h,k) the higher-order generalized twisted (h, q)-Euler polynomials En,χ,q,ε (x) attached to χ using the symmetric properties for Dirichlet-type multiple twisted (h, q)-l-function. In this paper, if we take χ0 = 1, ε = 1, then [3] is the special case of this paper. If we take ε = 1 in all equations of this article, then [1] are the special case of our results. 2. Symmetry identities for Dirichlet-type multiple twisted (h, q)-l-function In this section, by using the similar method of [1, 2, 3], expect for obvious modifications, we investigate some symmetric identities for higher-order generalized twisted (h, q)-Euler polynomials (h,k) En,χ,q,ε (x) attached to χ using the symmetric properties for Dirichlet-type multiple twisted (h, q)l-function. We assume that χ is a Dirichlet character with conductor d ∈ N with d ≡ 1 (mod 2) and ε be the pN -th root of unity. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z, k ∈ N and n ∈ Z+ , we obtain certain symmetry identities for Dirichlet-type multiple twisted (h, q)-l-function. Observe that [xy]q = [x]qy [y]q for any x, y ∈ C. In (5), we derive next result by substitute w2 w2 x + (j1 + · · · + jk ) for x in and replace q and ε by q w1 and εw1 , respectively. w1 w2 1 (h,k) lχ,qw1 ,εw1 (s, w2 x + (j1 + · · · + jk )) k w1 [2]qw1 ) (∏ ∑ ∑k ∑k k w1 k ∞ j=1 (h−j+1)mj εw1 j=1 mj χ(m ) q (−1) j=1 mj ∑ j j=1 [ ]s = w1 (m1 + · · · + mk ) + w1 w2 x + w2 (j1 + · · · + jk ) m1 ,··· ,mk =0 w1 q w1 ) (∏ ∑k ∑k ∑k k m w (h−j+1)m w 1 j ∞ j=1 (−1) j=1 j ε 1 j=1 mj ∑ j=1 χ(mj ) q = [w1 (m1 + · · · + mk ) + w1 w2 x + w2 (j1 + · · · + jk )]sq m1 ,··· ,mk =0 [w1 ]sq (∏ ) ∑k ∑k ∑k k dw ∞ 2 −1 (−1) j=1 mj χ(m ) q w1 j=1 (h−j+1)mj εw1 j=1 mj ∑ ∑ j j=1 = [w1 ]sq [w1 (m1 + · · · + mk ) + w1 w2 x + w2 (j1 + · · · + jk )]sq m1 ,··· ,mk =0 i1 ,··· ,ik =0   dw ∞ k 2 −1 ∑k ∑k ∑ ∑ ∏ = [w1 ]sq (−1) j=1 mj (−1) j=1 ij  χ(ij )

(6)

m1 ,··· ,mk =0 i1 ,··· ,ik =0 j=1 ∑k ∑k ∑k ∑ dw1 w2 j=1 (h−j+1)mj w1 j=1 (h−j+1)ij dw1 w2 j=1 mj w1 k j=1 ij

×q q ε ε ( )−1 × [w1 w2 (x + dm1 + · · · + dmk ) + w1 (i1 + · · · + ik ) + w2 (j1 + · · · + jk )]sq Thus, from (6), we can derive the following equation. [w2 ]sq [2]kqw1

dw 1 −1 ∑

l=1 jl

j1 ,··· ,jk =0

×

k ∏

) χ(jl ) q w2

∑k

l=1 (h−l+1)jl

εw2

l=1

jl

w2 (j1 + · · · + jk )) w1 ( k )( k ) dw dw 2 −1 1 −1 ∑ ∑ ∏ ∏ ∑k (jl +il +ml ) l=1 (−1) χ(jl ) χ(il )

∞ ∑

+

m1 ,··· ,mk =0 i1 ,··· ,ik =0 j1 ,··· ,jk =0 ∑k ∑k ∑ dw1 w2 k l=1 (h−l+1)jl l=1 (h−l+1)il w2 l=1 (h−l+1)ml w1 ∑k

l=1

(7)

l=1

q

q

∑k

∑k

l=1

(h,k) lχ,qw1 ,εw1 (s, w2 x

= [w1 ]sq [w2 ]sq ×q

(−1)

(

∑k

∑k

× εdw1 w2 l=1 ml εw1 l=1 il εw2 l=1 jl ( )−1 × [w1 w2 (x + dm1 + · · · + dmk ) + w1 (i1 + · · · + ik ) + w2 (j1 + · · · + jk )]sq

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By using the same method as (7), we have

[w1 ]sq [2]kqw2

dw 2 −1 ∑

(−1)

l=1 jl

j1 ,··· ,jk =0

×

k ∏

) χ(jl ) q w1

∑k

l=1 (h−l+1)jl

εw1

∑k l=1

jl

l=1

w1 (j1 + · · · + jk )) w2 ( k )( k ) dw dw 2 −1 1 −1 ∑ ∑ ∏ ∏ ∑k (j +i +m ) χ(jl ) χ(il ) (−1) l=1 l l l

(h,k) lχ,qw2 ,εw2 (s, w1 x ∞ ∑

= [w1 ]sq [w2 ]sq ×q

(

∑k

+

m1 ,··· ,mk =0 j1 ,··· ,jk =0 i1 ,··· ,ik =0 ∑k ∑ ∑k dw1 w2 l=1 (h−l+1)ml w2 k l=1 (h−l+1)il w1 l=1 (h−l+1)jl

q

∑k

l=1

(8)

l=1

q

∑k

∑k

×ε ε ε ( )−1 × [w1 w2 (x + dm1 + · · · + dmk ) + w1 (j1 + · · · + jk ) + w2 (i1 + · · · + ik )]sq dw1 w2

l=1 ml w2

l=1 il w1

l=1 jl

Therefore, by (7) and (8), we have the following theorem. Theorem 2. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z , we obtain ( k ) dw 1 −1 ∑ ∏ ∑k ∑k ∑k s k j l [w2 ]q [2]qw2 (−1) l=1 χ(jl ) q w2 l=1 (h−l+1)jl εw2 l=1 jl j1 ,··· ,jk =0

l=1

) ( w2 (h,k) (j1 + · · · + jk ) × lχ,qw1 ,εw1 s, w2 x + w1 ( k ) dw −1 2 ∑ ∏ ∑k ∑k ∑k s k j l [w1 ]q [2]qw1 (−1) l=1 χ(jl ) q w1 l=1 (h−l+1)jl εw1 l=1 jl j1 ,··· ,jk =0 (h,k)

× lχ,qw2 ,εw2

(9)

l=1

( ) w1 s, w1 x + (j1 + · · · + jk ) w2

By (9) and Theorem 1, we obtain the following theorem. Theorem 3. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z, k ∈ N and n ∈ Z+ , we obtain ( k ) dw 1 −1 ∑ ∏ ∑k ∑k ∑k s k j l [w2 ]q [2]qw2 χ(jl ) q w2 l=1 (h−l+1)jl εw2 l=1 jl (−1) l=1 j1 ,··· ,jk =0

l=1

( ) w2 (h,k) × En,χ,qw1 ,εw1 w2 x + (j1 + · · · + jk ) w1 ( k ) dw −1 2 ∑ ∏ ∑k ∑k ∑k s k j l = [w1 ]q [2]qw1 (−1) l=1 χ(jl ) q w1 l=1 (h−l+1)jl εw1 l=1 jl j1 ,··· ,jk =0 (h,k)

× En,χ,qw2 ,εw2

(10)

l=1

) ( w1 w1 x + (j1 + · · · + jk ) . w2

From (4), we note that (h,k) (h,k) En,χ,q,ε (x + y) = (q x+y En,χ,q,ε + [x + y]q )n = (h,k)

n ( ) ∑ n xi (h,k) q Ei,χ,q,ε (y)[x]n−i . q i i=0

(11)

(h,k)

with the usual convention about replacing (Eχ,q,ε )n by En,χ,q,ε . By (11), we have

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dw 1 −1 ∑

(−1)

∑k l=1

jl

( k ∏

j1 ,··· ,jk =0

=

dw 1 −1 ∑ n ( ) ∑ n

i

i=0

=

(−1)

(

∑k l=1

jl

w2 i(j1 +···+jk )

∑k

l=1 (h−l+1)jl

) χ(jl ) q w2

(−1)

(

∑k

l=1 jl

k ∏

l=1 (h−l+1)jl

[ ) χ(jl ) q w2

ε

w2

∑k

(h,k) Ei,χ,qw1 ,εw1 (w2 x)

j1 ,··· ,jk =0

×

k ∏

w2

l=1

q

dw 1 −1 ∑ n ( ∑

χ(jl ) q

l=1

j1 ,··· ,jk =0

×

)

(

∑k

jl

l=1

εw2

(h,k) En,χ,qw1 ,εw1

∑k l=1

) w2 w2 x + (j1 + · · · + jk ) w1

jl

]n−i w2 (j1 + · · · + jk ) w1 q w1

∑k

l=1 (h−l+1)jl

εw2

∑k l=1

jl

l=1

]i [ n w2 (n−i) ∑kl=1 jl (h,k) w2 (j1 + · · · + jk ) q En−i,χ,qw1 ,εw1 (w2 x) w1 i q w1

i=0

)

(12) Hence we have the following theorem. Theorem 4. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z, k ∈ N and n ∈ Z+ , we obtain dw 1 −1 ∑

(−1)

∑k l=1

jl

( k ∏

j1 ,··· ,jk =0 n ( ∑

) χ(jl ) q

w2

∑k

l=1 (h−l+1)jl

ε

w2

(

∑k l=1

jl

(h,k) En,χ,qw1 ,εw1

l=1

) w2 w2 x + (j1 + · · · + jk ) w1

) n (h,k) [w2 ]iq [w1 ]−i q En−i,χ,q w1 ,εw1 (w2 x) i i=0 ( k ) dw 1 −1 ∑ ∏ ∑k ∑k ∑k j l × (−1) l=1 χ(jl ) q w2 l=1 (n+h−l−i+1)jl εw2 l=1 jl [j1 · · · + jk ]iqw2 . =

j1 ,··· ,jk =0

l=1

For each integer n ≥ 0, let (h,k) Sn,i,χ,q,ε (w)

w−1 ∑

=

(−1)

(

∑k l=1

jl

j1 ,··· ,jk =0

k ∏

) χ(jl ) q

∑k

l=1 (n+h−l−i+1)jl

ε

∑k l=1

jl

[j1 · · · + jk ]iq .

l=1

(h,k)

The above sum Sn,i,χ,q,ε (w) is called the alternating generalized (h, q)-power sums. By Theorem 4, we have ( k ) dw 1 −1 ∑ ∏ ∑k ∑k ∑k k n j l [2]qw2 [w1 ]q (−1) l=1 χ(jl ) q w2 l=1 (h−l+1)jl εw2 l=1 jl j1 ,··· ,jk =0

l=1

(h,k)

× En,χ,qw1 ,εw1 =

[2]kqw2

n ( ) ∑ n i=0

i

) ( w2 (j1 + · · · + jk ) w2 x + w1 (h,k)

(13)

(h,k)

[w2 ]iq [w1 ]n−i En−i,χ,qw1 ,εw1 (w2 x)Sn,i,χ,qw2 ,εw2 (dw1 ) q

By using the same method as in (13), we obtain dw 2 −1 ∑

[2]kqw1 [w2 ]nq

(−1)

j1 ,··· ,jk =0 (h,k)

× En,χ,qw2 ,εw2 =

[2]kqw1

n ( ) ∑ n i=0

i

∑k l=1

( jl

k ∏

) χ(jl ) q w1

∑k

l=1 (h−l+1)jl

εw1

∑k l=1

jl

l=1

) ( w1 (j1 + · · · + jk ) w1 x + w2 (h,k)

(14)

(h,k)

[w1 ]iq [w2 ]n−i En−i,χ,qw2 ,εw2 (w1 x)Sn,i,χ,qw1 ,εw1 (dw2 ) q

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Therefore, by (13), (14), and Theorem 3, we have the following theorem. Theorem 5. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z, k ∈ N and n ∈ Z+ , we obtain n ( ) ∑ n (h,k) (h,k) k [2]qw2 [w2 ]iq [w1 ]n−i En−i,χ,qw1 ,εw1 (w2 x)Sn,i,χ,qw2 ,εw2 (dw1 ) q i i=0 n ( ) ∑ n (h,k) (h,k) = [2]kqw1 [w1 ]iq [w2 ]n−i En−i,χ,qw2 ,εw2 (w1 x)Sn,i,χ,qw1 ,εw1 (dw2 ). q i i=0 By Theorem 5, we obtain the interesting symmetric identity for the higher-order generalized (h,k) twisted (h, q)-Euler numbers En,χ,q,ε in complex field. Corollary 6. Let w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2). For h ∈ Z, k ∈ N and n ∈ Z+ , we obtain n ( ) ∑ n (h,k) (h,k) k [2]qw2 [w2 ]iq [w1 ]n−i Sn,i,χ,qw2 ,εw2 (dw1 )En−i,χ,qw1 ,εw1 q i i=0 n ( ) ∑ n (h,k) (h,k) = [2]kqw1 [w1 ]iq [w2 ]n−i Sn,i,χ,qw1 ,εw1 (dw2 )En−i,χ,qw2 ,εw2 . q i i=0

REFERENCES 1. D. V. Dolgy, D.S. Kim, T.G. Kim, J.J. Seo, Identities of Symmetry for Higher-Order Generalized q-Euler Polynomials, Abstract and Applied Analysis, 2014(2014), Article ID 286239, 6 pages. 2. Yuan He,

Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials,

Adv.

Difference Equ., 246(2013), 10 pages. 3. D. Kim, T. Kim, J.-J. Seo, Identities of symmetric for (h, q)-extension of higher-order Euler polynomials, Applied Mathemtical Sciences 8 (2014), 3799-3808. 4. T. Kim,

New approach to q-Euler polynomials of higher order,

Russ. J. Math. Phys.

17(2010), 218-225. 5. T. Kim, Barnes type multiple q-zeta function and q-Euler polynomials, J. phys. A : Math. Theor. 43(2010) 255201(11pp). 6. H. Y. Lee, N. S. Jung, J. Y. Kang, C. S. Ryoo, Some identities on the higher-order-twisted q-Euler numbers and polynomials with weight α, Adv. Difference Equ., 2012:21(2012), 10pp. 7. E.-J. Moon, S.-H. Rim, J.-H. Jin, S.-J. Lee, On the symmetric properties of higher-order twisted q-Euler numbers and polynomials, Adv. Difference Equ., 2010, Art ID 765259, 8pp. 8. C. S. Ryoo, On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity, Proc. Jangjeon Math. Soc. 13(2010), 255-263. 9. C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, Adv. Stud. Contemp. Math., 21(2011), 47-54. 10. Y. Simsek, q-analogue of twisted l-series and q-twisted Euler numbers, Journal of Number Theory, 110(2005), 267-278.

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An efficient m-step Levenberg-Marquardt method for systems of nonlinear equations∗ Liang Chen† School of Mathematical Sciences, Huaibei Normal University, Huaibei 235000, P.R. China Yanfang Ma School of Computer Science and Technology, Huaibei Normal University, Huaibei, Anhui 235000, PR China

Abstract In this paper, we propose an efficient m-step Levenberg-Marquardt method for systems of nonlinear equations. At every iteration, the efficient m-step LM method computes not only the classical LM step, but also m−1 approximate LM steps with frozen (JkT Jk +λk I)−1 JkT . Also, we employ m−1 line searches for m−1 approximate LM steps for better numerical performance. Under the local error bound condition which is weaker than nonsingularity, the efficient m-step LM method has been proved to have (m + 1)th convergence order. The global convergence has also been given by trust region technique. Numerical results show that the efficient m-step LM method is efficient and could save many calculations of the Jacobian especially for large scale problems. Keywords: Unconstrained optimization; Systems of nonlinear equations; Levenberg-Marquardt method; Trust region MSC2010: 65K05; 90C30

1

Introduction

It’s a well-known problem in science and engineering that is to find the solutions of systems of nonlinear equations F (x) = 0, (1) where F : D ⊂ Rn → Rn is continuously differentiable function. Due to the nonlinearity of F (x), (1) may have no solutions. Throughout the paper, we let that the solution set of (1) is nonempty and denote it by X ∗ , and in all cases k · k refers to the 2-norm. There are many numerical methods to approximate the solutions of (1) because the exact solutions is difficult to find. A classical numerical method is Newton method which computes the trial step −1 dN k = −Jk Fk

at every iteration, where Fk = F (xk ) and Jk = F 0 (xk ) is the Jacobian. And the Newton method has quadratic rate of convergence under the condition that J(x) is Lipschitz continuous and nonsingular at the solution of (1). However, the Newton method will be failed when Jk is singular or near singular. To overcome these disadvantages, a large number of researchers have presented many modifications of Newton ∗ The work is supported by the Anhui Provincial Natural Science Foundation (1508085MA14, 1708085MF159), the Natural Science Foundation of the Anhui Higher Education Institutions (KJ2017A375) and the Major Teaching Reform Project of Anhui Higher Education Revitalization Plan (2014ZDJY058). † Corresponding author. Email: [email protected], [email protected]. Tel: +86 157 5613 7533

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method [1]. One of them is the Levenberg-Marquardt method (LM) [2, 3], which is a famous numerical method with computing the linear equation
JkT Jk + λk I d = −JkT Fk (2) to obtain the LM trail step dk = − JkT Jk + λk I


−1

JkT Fk

(3)

at every iteration, where λk > 0 is the LM parameter. It is well-known that the LM method has quadratic convergence as the Newton method if the Jacobian matrix is nonsingular and Lipschitz continuous at the solution. A large number of researchers have focused on this system and many efficient solution techniques are available [4–7]. As we all known, the cost of Jacobian computations is expensive when F (x) is complicated or n is quite large. Recently, to save Jacobian calculations and achieve a fast convergence rate, Fan [8] presented a modified Levenberg-Marquardt method (MLM) with cubic convergence. At every iteration, the MLM method solves not only the linear equations (2) to obtain the LM step (3), but also the linear equations
JkT Jk + λk I d = −JkT Fk,1 to obtain the approximate LM step dk,1 = − JkT Jk + λk I


−1

JkT Fk,1

(4)

δ

with Fk,1 = F (xk,1 ), xk,1 = xk + dk , λk = µk kFk k , µk > 0 and δ ∈ [1, 2], and the trial step is LM sM = dk + dk,1 . k

Fan use JkT Jk + λk I


−1

JkT in stead of 

T

δ

J (xk,1 ) J (xk,1 ) + µk+1 kF (xk,1 )k I

−1

T

J (xk,1 )

(5)

in (4), which does not involve the calculation of J (xk,1 ). Since Jk has been used in (3), the cost of Jacobian calculations will be saved. Similarly, to save more Jacobian calculations, based on the MLM method, Yang [9] presented a high-order Levenberg-Marquardt method (HLM) with biquadratic convergence by solving another linear equations
JkT Jk + λk I d = −JkT Fk,2 (6) to obtain another approximate LM step dk,2 = − JkT Jk + λk I


−1

JkT Fk,2

(7)


−1 T δ with Fk,2 = F (xk,2 ), xk,2 = xk,1 +dk,1 , λk = µk kFk k , µk > 0 and δ ∈ [1, 2]. Yang still use JkT Jk + λk I Jk  −1 T δ T in stead of (5) in (4), J (xk,2 ) J (xk,2 ) + µk+2 kF (xk,2 )k I J (xk,2 ) in (7) respectively, which does not need to compute J (xk,1 ) and J (xk,2 ). The trial step of the HLM method is sHLM = dk + dk,1 + dk,2 . k Furthermore, to save more Jacobian calculations and achieve a faster convergence rate, Fan [10] presented a Shamanskii-like Levenberg-Marquardt (SLM) method with (m + 1)th convergence by solving m − 1 linear equations
JkT Jk + λk I d = −JkT Fk,i with i = 1, · · · , m − 1 (8) to obtain m − 1 approximate LM steps dk,i = − JkT Jk + λk I

544


−1

JkT Fk,i

(9)

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δ

where Fk,i = F (xk,i ), xk,i = xk,i−1 + dk,i−1 with xk,0 = xk , dk,0 = dk , λk = µk kFk k , µk > 0 and δ ∈ [1, 2].  −1
−1 T T δ T Jk in stead of J (xk,i ) J (xk,i ) + µk+i kF (xk,i )k I J (xk,i ) in (9), which Fan still use JkT Jk + λk I does not need to compute J (xk,i ) (i = 1, 2, · · · , m − 1). The trial step of the SLM method is sSLM = dk,0 + dk,1 + · · · + dk,m−1 = k

m−1 X

dk,i .

(10)

i=0

If we consider the MLM method as two-step Levenberg-Marquardt method and the HLM method as threestep Levenberg-Marquardt method respectively, then, the Shamanskii-like Levenberg-Marquardt method can
−1 T be considered as m-step Levenberg-Marquardt method. Also, it is easy to see that JkT Jk + λk I Jk is computed in all of the classical LM step (3) and the approximate LM step (4), (7), (9) respectively. So, we
−1 T can consider JkT Jk + λk I Jk is frozen in the two-step LM method, three-step LM method and m-step LM method. To accelerate the MLM method and for better numerical performance, Fan [11] proposed an accelerated version of the MLM (AMLM) method by employing a line search for the approximate LM step dk,1 and computed the trial step by LM sAM = dk,0 + αk,1 dk,1 , (11) k where αk,1 ∈ [1, α ˆ 1 ] is step size with α ˆ 1 > 1 is a positive constant. For the same purpose, based on the AMLM method, Chen [12] compute the linear equation (6) with xk,2 = xk,1 + αk,1 dk,1 to obtain an approximate LM step d¯k,2 . By employing another line search for the approximate LM step d¯k,2 , Chen presented a new modified Levenberg-Marquardt (NMLM) method. The trial step of the NMLM method is M LM sN = dk,0 + αk,1 dk,1 + αk,2 d¯k,2 , k

(12)

where αk,2 ∈ [1, α ˆ 2 ] is step size with α ˆ 2 > 1 is a positive constant. Now, motivated by (10), (11) and (12), we will employ m − 1 line searches for approximate LM step dk,i by solving linear equation (8) with xk,i = xk,i−1 + αk,i−1 dk,i−1 and present an efficient m-step LevenbergMarquardt method with trial step as sk = dk,0 + αk,1 dk,1 + · · · + αk,m−1 dk,m−1 ,

(13)

where αk,i ∈ [1, α ˆ ] are step size with α ˆ > 1 (i = 1, · · · , m − 1) is a positive constants. It is quite clear that the above new LM method will reduce to the classical Levenberg-Marquardt method while m = 1, the AMLM method while m = 2 and the NMLM method while m = 3 respectively. We will organize the rest of this paper as follow: In Section 2, we first give the new modified LevenbergMarquardt method which is called efficient m-step Levenberg-Marquardt algorithm. In Section 3, we derive the global convergence of the new algorithm by using trust region technique. Then we derive the convergence order of the algorithm under the local error bound condition in Section 4. Finally, some numerical results of the new algorithm are given in Section 5.

2

The efficient m-step Levenberg-Marquardt algorithm

In this section, we first present the efficient m-step Levenberg-Marquardt algorithm by using trust region technique, then prove the global convergence.

2.1

The motivation

We take 2

Φ(x) = kF (x)k

(14)

as the merit function for (1). It is easy to see that dk,i (i = 0, · · · , m − 1) is not only the minimizer of the convex minimization problem 2

2

min kFk,i + Jk dk + λk kdk , ϕk,i (d) ,

d∈Rn

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but also a solution of the trust region problem 2

min kFk,i + Jk dk ,

d∈Rn

where ∆k,i

(16)

s.t. kdk 6 ∆k,i ,


−1 T

Jk Fk,i . From the result given by Powell in [13], we have = kdk,i k = − JkT Jk + λk I (

T

)

J Fk,i

T

k

kFk,i k − kFk,i + Jk dk,i k > Jk Fk,i min kdk,i k ,

J T Jk . k 2

2

(17)

Moreover, similar to Fan proposed in [11], if dk,i is a descent direction of the merit function Φ (x) at xk,i , then more reduction of Φ (x) at xk,i could be expected. So we may perform many line searches at xk,i along dk,i by solving the problem 2 min kF (xk,i + αdk,i )k . α>0

By Taylor extension, replace J (xk,i ) with Jk for save Jacobian calculations, the above problem could be approximated by 2 min kF (xk,i ) + αJk dk,i k . α>0

The above problem is equivalent to 2

2

max kFk,i k − kFk,i + αJk dk,i k , φ (α) ,

(18)

α>0

where
φ (α) = −dTk,i JkT Jk dk,i α2 + 2dTk,i JkT Jk + λk I dk,i α is a quadratic function of α, and attains its maximum at
λk dTk,i dk,i dTk,i JkT Jk + λk I dk,i =1+ T T , α ˜ k,i = T T dk,i Jk Jk dk,i dk,i Jk Jk dk,i provided that Jk dk,i 6= 0. We bound α ˜ k,i ∈ [1, α ˆ ] with α ˆ > 1 is a positive constant because of α ˜ k,i may be very large if Jk dk,i is close to 0. The problem (18) now is equivalent to 2

2

max kFk,i k − kFk,i + αJk dk,i k , φ (α) .

(19)

α∈[1,α] ˆ

And we have 2

2

2

2

kFk,i k − kFk,i + αk,i Jk dk,i k > kFk,i k − kFk,i + Jk dk,i k

2.2

(20)

The algorithm

Now, we define the actual reduction of Φ (x) at the kth iteration as 2

2

Aredk = kFk k − kF (xk + dk,0 + αk,1 dk,1 + · · · + αk,m−1 dk,m−1 )k .

(21)

where dk,i are computed by (9). Note that the predicted reduction cannot be defined as usual definition 2 2 kFk k − kFk + Jk (dk,0 + αk,1 dk,1 + · · · + αk,m−1 dk,m−1 )k , because it cannot be proven to be nonnegative, which is required for the global convergence in the trust region method. Hence, we define the new modified predicted reduction as m−1  X 2 2 Predk = kFk,i k − kFk,i + αk,i Jk dk,i k , (22) i=0

with αk,0 = 1.

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Lemma 2.1. Let the predicted reduction is defined by (22), then (

T

)

J Fk,0

T

k

Predk > Jk Fk,0 min kdk,0 k ,

J T Jk , k

(23)

where m > 1. Proof. From (17) and (20), we have Pred =

m−1 X

2

2

kFk,i k − kFk,i + αk,i Jk dk,i k



i=0

>

m−1 X

2

2

kFk,i k − kFk,i + Jk dk,i k



i=0

(

T

)!

J Fk,i

T

k

Jk Fk,i min kdk,i k ,

>

J T Jk k i=0 (

T

)

J Fk,0

T

k

> Jk Fk,0 min kdk,0 k ,

J T Jk . k m−1 X

Then (23) holds. The proof is completed. Now, we present the efficient m-step Levenberg-Marquardt algorithm. Algorithm 2.2 (The efficient m-step Levenberg-Marquardt algorithm). Input: Given x0 ∈ Rn , µ1 > µ > 0, 0 < p0 6 p1 6 p2 < 1, 1 6 δ 6 2, ε > 0, α ˆ > 1 and m > 1. Step 1. Set xk,0 = xk , dk,0 = dk and k := 0.

Step 2. Compute Fk = Fk,0 = F (xk,0 ), Jk = J (xk,0 ). If JkT Fk < ε, then stop. compute
δ JkT Jk + λk I d = −JkT Fk,i with λk = µk kFk k ,

Otherwise (24)

where xk,i = xk,i−1 + αk,i−1 dk,i−1 to obtain dk,i , i = 0, 1, · · · , m − 1. Set sk =

m−1 X

αk,i dk,i ,

(25)

i=0

where αk,0 = 1, αk,i (i = 1, · · · , m − 1) is the step size obtained by solving (19). Step 3. Compute rk = Aredk /Predk . Set  xk+1 =

xk + sk , xk ,

if rk > p0 , otherwise.

(26)

Step 4. Update µk+1 as µk+1

  4µk , µk ,  =  max µ4k , µ ,

if rk < p1 , if rk ∈ [p1 , p2 ] , if rk > p2 .

(27)

Step 5. Set k = k + 1, and go to Step 2. Remark 2.3. (a) Notice that, µk should be no less than a positive constant µ to prevent the steps from being too large when the sequence {xk } is near the solution. (b) Fan set δ ∈ (0, 2] in [11], but here, we still set δ ∈ [1, 2] as usual in [8–10, 12] for stable and preferable.

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3

The global convergence

To study the global convergence of Algorithm 2.2, we need the following assumptions. Assumption 3.1. Let F (x) is continuously differentiable, and both F (x) and its Jacobian J(x) are Lipschitz continuous, i.e., there exist positive constant L1 and L2 such that kJ (y) − J(x)k 6 L1 ky − xk ,

∀x, y ∈ Rn

(28)

kF (y) − F (x)k 6 L2 ky − xk ,

∀x, y ∈ Rn .

(29)

and By the Lipschitzness of the Jacobian proposed by (28), we have 2

kF (y) − F (x) − J(x) (y − x)k 6 L1 ky − xk ,

∀x, y ∈ Rn .

(30)

Theorem 3.2. Under the conditions of Assumption 3.1, Algorithm 2.2 will terminates in finite iterations or satisfies

(31) lim JkT Fk = 0. k→∞

Proof. By contradiction, suppose there exist a positive τ and infinite many k such that

T

Jk Fk > τ.

(32)

Let T1 , T2 be the sets of the indices as follow:

 T1 = k | JkT Fk > τ , n o

τ T2 = k | JkT Fk > and xk+1 6= xk . 2 It is easy to see that T1 is infinite. In the following, we will derive the contradictions whether T2 is finite or infinite. Case 1: T2 is finite. Then the set

 T3 = k | JkT Fk > τ and xk+1 6= xk is also k˜ be the largest index of T3 . Then it is easy to see that xk+1 = xk holds for all n finite. Let o k ∈ k > k˜ | k ∈ T1 . Define the indices set n

T4 = k > k˜ | JkT Fk > τ

o and xk+1 = xk .

T

If k ∈ T4 , we can deduce that Jk+1 Fk+1 > τ and xk+2 = xk+1 . Hence, we have xk+1 ∈ T4 . By induction,

˜ which means rk < p0 . Now, we obtain we know that JkT Fk > τ and xk+1 = xk hold for all k > k, λk → +∞ and µk → +∞

(33)

and, due to (24), (25) and (27),


−1 T

dk,0 = − JkT Jk + λk I Jk Fk,0 → 0.

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Moreover, it follows from (29) and (30) that


−1 T

Jk Fk,i kdk,i k = − JkT Jk + λk I

i−1

X



−1 −1

T

T T T Jk Jk αk,j dk,j 6 Jk Jk + λk I Jk Fk,0 + Jk Jk + λk I

j=0

2

i−1


−1 T

X

α d + L1 JkT Jk + λk I Jk k,j k,j

j=0

2

X

i−1

L1 L2 X

i−1



αk,j dk,j + 6 kdk,0 k + αk,j dk,j

λk j=0

j=0

 2 i−1 i−1 X X L1 L2  6 kdk,0 k + αk,j kdk,j k + αk,j kdk,j k λk j=0 j=0 with i = 1, · · · , m − 1. Hence, by induction, we obtain kdk,i k 6 O (kdk,0 k) .

(34)

Note that 2

2

kFk,i+1 k − kFk,i + αk,i Jk dk,i k = (kFk,i+1 k + kFk,i + αk,i Jk dk,i k) (kFk,i+1 k − kFk,i + αk,i Jk dk,i k) 

2 

2



i−1



i i X X X









αk,j dk,j αk,j dk,j + L1 6 2 Fk,0 + Jk αk,j dk,j + L1



j=0



j=0 j=0  

2

2

X

X

i i−1



 

× L1 αk,j dk,j + L1 αk,j dk,j



j=0

j=0

(35)

with i = 0, 1, · · · , m − 1. It’s clear that while i = 0 and αk,0 = 1,   2 2 2 4 2 kFk,1 k − kFk,0 + Jk dk,0 k 6 2L1 kFk,0 + Jk dk,0 k kdk,0 k + L21 kdk,0 k = O kdk,0 k . It follows from (21), (22), (29), (35) and Lemma 2.1 that Aredk − Predk |rk − 1| = Predk   P 2 2 m−1 kF k,i+1 k − kFk,i + αk,i Jk dk,i k i=0   6 P m−1 2 2 kFk,i k − kFk,i + αk,i Jk dk,i k i=0   2 O kdk,0 k  → 0,  6 T

J T Fk,0 min kdk,0 k , kJkTFk,0 k k kJk Jk k which implies that rk → 1. In view of the updating rule of µk , we know that there exists a positive constant µ ¯ > µ such that µk < µ ¯ holds for all sufficiently large k, which is a contradiction to (33) .

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Case 2: T2 is infinite. It follows from (23) and (29) that   X X 2 2 2 2 2 kFk k − kFk+1 k kFk k − kFk+1 k > kF1 k > k∈T2

k

(

T

)

J Fk,0

T

k

p0 Jk Fk,0 min kdk,0 k , p0 Predk > >

J T Jk k k∈T2 k∈T2   X p0 τ τ > min kdk,0 k , 2 . 2 2L2 X

X

(36)

k∈T2

which implies kdk,0 k → 0,

k ∈ T2 .

(37)

µk → +∞,

k ∈ T2 .

(38)

Then by the definition of dk,0 , we have

Moreover, it follows from (28), (29), (34) and (36) that X

T

JkT Fk − Jk+1 Fk+1 k∈T2

6

X




T






JkT Fk − JkT Fk+1 − Jk+1 Fk+1 − JkT Fk+1 k∈T2

6

X L2 JkT ksk k − L1 kFk+1 k ksk k k∈T2

6 L1 L2 c˙

X

kdk,0 k < +∞,

k∈T2

with some constants c˙ > 0, which together with (32) implies there exists a sufficiently large kˆ such that

T

Jk Fk > τ

and

X

T

τ JkT Fk − Jk+1 Fk+1 < . 2

k∈T2

Hence we can derive that JkT Fk >

τ 2

ˆ Combining (37) with (38), we have for all k > k.

kdk,0 k → 0

and

µk → +∞.

(39)

In the same way as proved in Case 1, we can also obtain that rk → 1. Hence, there exists a positive constant µ ¯ such that µk < µ ¯ holds for all sufficiently large k, which is contradicted to (39). The proof is completed.

4

The local convergence

In this section, we assume that xk → x∗ ∈ X ∗ and the sequence {xk } lies on some neighbourhood of x∗ , i.e., there exist a positive constant b1 < 1 such that x ∈ N (x∗ , b1 ). We give some assumptions which the local convergence theory required. Assumption 4.1. (a) F (x) is continuously differentiable, and Jacobian J(x) is Lipschitz continuous on N (x∗ , b1 ), i.e., there exist a positive constant L1 such that kJ (y) − J(x)k 6 L1 ky − xk ,

∀x, y ∈ N (x∗ , b1 ) = {x | kx − x∗ k 6 b1 } .

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(b) kF (x)k provides a local error bound on some neighborhood of x∗ ∈ X ∗ , i.e., there exist a positive constant c > 0 such that kF (x)k > c dist (x, X ∗ ) , ∀x ∈ N (x∗ , b1 ) . (41) Since the condition of nonsingularity of J(x) is too strong, the Assumption 4.1 (b) provides a weak local error bound condition, which implies that the converse is not necessarily true [4]. By (40), we have 2

kF (y) − F (x) − J(x) (y − x)k 6 L1 ky − xk ,

∀x, y ∈ N (x∗ , b1 ) ,

(42)

and ∀x, y ∈ N (x∗ , b1 ) ,

kF (y) − F (x)k 6 L2 ky − xk ,

(43)

where L2 is a positive constant. There exists a positive constant ω > 0 if F (x) provides a local error bound which proposed by Behling and Iusem in [14], then rank (J (˜ x)) = rank (J (x∗ )) , ∀˜ x ∈ N (x∗ , ω) ∩ X ∗ . Let b ∈ (0, 1) and
b1 = min {ω, b}. Without loss of generality, we further assume that xk,i , i = 0, 1, · · · , m − 1 lie in N x∗ , b21 . In the following, we denote x ¯k ∈ X ∗ such that k¯ xk − xk k = dist (xk , X ∗ ) = inf ∗ ky − xk k . y∈X

Hence, we have k¯ xk − x∗ k 6 k¯ xk − xk k 6 k + kxk − x∗ k 6 2 kxk − x∗ k 6 b1 , which implies that x ¯k ∈ N (x∗ , b1 ). Lemma 4.2. Let Assumption 4.1 hold, then

 
−1 T

T

−δ Jk 6 O k¯ xk − xk k 2 .

Jk Jk + λk I

(44)

Proof. Suppose rank (J (¯ xk )) = r for all x ¯k ∈ N (x∗ , b1 ) ∩ X ∗ and the SVD of J (¯ xk ) is   ¯T  ¯ k,1
Σ Vk,1 ¯k Σ ¯ k V¯ T = U ¯k,1 , U ¯k,2 ¯k,1 Σ ¯ k,1 V¯ T , J (¯ xk ) = U =U T k k,1 0 V¯k,2 ¯ k,1 = diag (¯ where Σ σk,1 , σ ¯k,2 , · · · , σ ¯k,r ) with σ ¯k,1 > σ ¯k,2 > · · · > σ ¯k,r > 0. The   Σk,1  Σk,2 Jk =Uk Σk VkT = (Uk,1 , Uk,2 , Uk,3 )  0

corresponding SVD of Jk is T  Vk,1 T  Vk,2 T Vk,3

T T =Uk,1 Σk,1 Vk,1 + Uk,2 Σk,2 Vk,2 ,

where Σk,1 = diag (σk,1 , σk,2 , · · · , σk,r ) with σk,1 > σk,2 > · · · > σk,r > 0, and Σk,2 = diag(σk,r+1 , σk,r+2 , · · · , σk,r+q ) with σk,r+1 > σk,r+2 > · · · > σk,r+q > 0. We will neglect the subscript k if the context is clear in the following, and write Jk as Jk = U1 Σ1 V1T + U2 Σ2 V2T . (45) By the theory of matrix perturbation [15] and the Lipschitzness of Jk , we have


¯ 1 , Σ2 , 0 6 Jk − J¯k 6 L1 k¯

diag Σ1 − Σ xk − xk k ,

(46)

which yields

¯ 1 6 L1 k¯

Σ1 − Σ xk − xk k Hence

−1

λ Σ2 = k

kΣ2 k µk kFk k

δ

6

kΣ2 k 6 L1 k¯ xk − xk k .

and

L1 k¯ xk − xk k

1−δ

δ

mcδ k¯ xk − xk k

551

(47)

= L1 m−1 c−δ k¯ xk − xk k

(48)

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Since for any positive σi (i = 1, 2, · · · , r), we have σi σi 1 √ = √ , 6 σi2 + λk 2σi λk 2 λk which implies


−1 δ 1 1 1

2 −δ 6 m− 2 c− 2 k¯ Σ1 6 q xk − xk k 2 .

Σ1 + λk I 2 δ 2 µk kFk k Combining (48) and (49) with δ ∈ [1, 2], we have


−1

Σ21 + λk I Σ1


−1 T
−1 

T Jk = (V1 , V2 , V3 ) 

Jk Jk + λk I Σ22 + λk I Σ2

 
−1

Σ1 Σ21 + λk I




  −1 2 6 

 Σ2 + λk I Σ2

0


−1



6 Σ21 + λk I Σ1 + λ−1 k Σ2

(49)



U1T

  T  U2



U3T

0 

1 δ 1 −δ 1−δ 6 m− 2 c− 2 k¯ xk − xk k 2 + L1 m−1 c−δ k¯ xk − xk k 2  −δ 6O k¯ xk − xk k 2 .

The proof is completed.

4.1

Properties of the trial step

Firstly, we investigate the properties of dk,i , and hence sk . Lemma 4.3. Under the condition of Assumption 4.1, for sufficiently large k, we have kdk,i k 6 ci dist (xk , X ∗ ) ,

i = 0, 1, · · · , m − 1,

where ci are some positive constants. Proof. The proof of dk,0 can be found in Lemma 1 of [11], thus kdk,0 k 6 c0 dist (xk , X ∗ ) .

(50)

Now we prove i > 1. From (24), (42), (44) and (50), we obtain


−1 T

kdk,i k = − JkT Jk + λk I Jk Fk,i

i−1

X



−1 T −1 T

T

6 JkT Jk + λk I Jk Fk,0 + J J + λ I J J α d k k,i k,i k k

k k

j=0

2

i−1

X


−1 T

T

+ L1 Jk Jk + λk I Jk αk,i dk,i

j=0

 2 i−1 i−1   X X −δ 6 kdk,0 k + αk,i kdk,i k + L1  αk,i kdk,i k O k¯ xk − xk k 2 j=0

j=0 ∗

6 ci dist (xk , X ) , with i = 1, · · · , m − 1, for some positive constant ci . The proof is completed.

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Lemma 4.3 indicates that the trail step

m−1

m−1

X

X

ksk k = αk,i dk,i 6 αk,i kdk,i k 6 c¨ dist (xk , X ∗ ) ,

i=0

i=0

for some positive constants c¨.

4.2

Boundedness of the LM parameter

Lemma 4.4. Under the conditions of Assumption 4.1, there exists a positive µ ¯ > µ such that µk 6 µ ¯ holds for all sufficiently large k. Proof. Following the result given in [10, Lemma 2], we have the following inequalities for all sufficiently large k, 2

2

kFk,i k − kFk,i + Jk dk,i k > c¯i kFk,i k min {kdk,i k , k¯ xk,i − xk,i k} , where c¯i are some positive constants, i = 0, 1, · · · , m − 1. In fact, if k¯ xk,i − xk,i k 6 kdk,i k, by (41), (42) and the fact that dk,i is the solution of (16), we have kF (xk,i )k − kFk,i + Jk dk,i k > kFk,i k − kFk,i + Jk (¯ xk,i − xk,i )k > kFk,i k − kFk,i + Jk,i (¯ xk,i − xk,i )k − kJk − Jk,i k k¯ xk,i − xk,i k 2

> c k¯ xk,i − xk,i k − L1 k¯ xk,i − xk,i k − L1 k¯ xk,i − xk,i k

i−1 X

αk,j kdk,j k

j=0

> c¯i k¯ xk,i − xk,i k ,

(51)

for some c¯i > 0 when k is sufficiently large. In the other case when k¯ xk,i − xk,i k > kdk,i k, we have



kdk,i k

F + J (¯ x − x ) kFk,i k − kFk,i + Jk dk,i k > kFk,i k − k,i k k,i k,i

k¯ xk,i − xk,i k kdk,i k > (kFk,i k − kFk,i + Jk (¯ xk,i − xk,i )k) k¯ xk,i − xk,i k kdk,i k > c¯i k¯ xk,i − xk,i k k¯ xk,i − xk,i k > c¯i kdk,i k .

(52)

Combining (51) with (52), we obtain 2

2

kFk,i k − kFk,i + Jk dk,i k = (kFk,i k + kFk,i + Jk dk,i k) (kFk,i k − kFk,i + Jk dk,i k) > c¯i kFk,i k min {kdk,i k , k¯ xk,i − xk,i k} . Together with (20), we have 2

2

2

2

kFk,i k − kFk,i + αk,i Jk dk,i k > kFk,i k − kFk,i + Jk dk,i k > c¯i kFk,i k min {kdk,i k , k¯ xk,i − xk,i k} .

(53)

Hence, it follows from (22) and Lemma 4.3, we have Predk > O (k¯ xk − xk k kdk,0 k) . Since dk,0 is a minimizer of (15), we have the following results from (43) and Lemma 4.3 that kFk,0 +Jk (αk,0 dk,0 + · · · + αk,i dk,i )k 6 kFk,0 + αk,0 Jk dk,0 k + kJk k (αk,1 kdk,1 k + · · · + αk,i kdk,i k) 6 c˜i k¯ xk − xk k ,

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with i = 1, · · · , m − 1 for some positive constants c˜i > 0. Also, follows from (35), we have   2 2 2 kFk,i+1 k − kF (xk,i ) + αk,i Jk dk,i k 6 O k¯ xk − xk k kdk,0 k which implies that   2 xk − xk k kdk,0 k Aredk − Predk O k¯ 6 |rk − 1| = →0 Predk O (k¯ xk − xk k kdk,0 k) holds for sufficiently large k. Hence rk → 1. Therefore there exists a positive µ ¯ > µ such that µk 6 µ ¯ holds for all sufficiently large k. The proof is completed.

4.3

Convergence order of m-step Levenberg-Marquardt algorithm

We now prove the convergence order of m-step LM algorithm based on the results obtained in the above two subsections. By the SVD of Jk proposed in (45), we have dk,i = −V1 Σ21 + λk I


−1

Σ1 U1T Fk,i − V2 Σ22 + λk I


−1

Σ2 U2T Fk,i ,

(54)

F (xk,i ) + Jk dk,i
−1
−1 =Fk,i − U1 Σ1 Σ21 + λk I Σ1 U1T Fk,i − U2 Σ2 Σ22 + λk I Σ2 U2T Fk,i

−1 T −1 T =λk U1 Σ21 + λk I U1 Fk,i + λk U2 Σ22 + λk I U2 Fk,i + U3 U3T Fk,i ,

(55)

with i = 0, · · · , m − 1. Lemma 4.5. Under the condition of Assumption 4.1, if xk,i ∈ N (x∗ , b1 /2), then we have  

i+1 xk − xk k ; (a) U1 U1T Fk,i 6 O k¯  

i+2 xk − xk k ; (b) U2 U2T Fk,i 6 O k¯  

i+2 (c) U3 U3T Fk,i 6 O k¯ xk − xk k ; with i = 0, · · · , m − 1. Proof. We will prove this lemma by an induction process. For i = 1, 2, the results have been shown by Fan and Chen respectively (see [11, 12]), and we have     2 3 kdk,1 k 6 O k¯ xk − xk k , kFk,1 + Jk dk,1 k 6 O k¯ xk − xk k ,     3 4 kdk,2 k 6 O k¯ xk − xk k , kFk,2 + Jk dk,2 k 6 O k¯ xk − xk k . Assuming the truth for some i − 1, we obtain the induction hypothesis:     i i+1 kdk,i−1 k 6 O k¯ xk − xk k , kF (xk,i−1 ) + Jk dk,i−1 k 6 O k¯ xk − xk k .

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Turning now to the case for i. It follows from above induction hypothesis that kFk,i k = kF (xk,i−1 + αk,i−1 dk,i−1 )k 2

2 6 kFk,i−1 + αk,i−1 Jk,i−1 dk,i−1 k + L1 αk,i−1 kdk,i−1 k 2

2 6 kFk,i−1 + Jk,i−1 dk,i−1 k + L1 αk,i−1 kdk,i−1 k

2

2 6 kFk,i−1 + Jk dk,i−1 k + kJk,i−1 − Jk k kdk,i−1 k + L1 αk,i−1 kdk,i−1 k



X

i−1 2 2

α d 6 kFk,i−1 + Jk dk,i−1 k + L1 k,j k,j kdk,i−1 k + L1 αk,i−1 kdk,i−1 k

j=0     i+1 i 6 O k¯ xk − xk k + L1 k¯ xk − xk k O k¯ xk − xk k   2i 2 + L1 αk,i−1 O k¯ xk − xk k   i+1 6 O k¯ xk − xk k .

So, we have  

i+1

U1 U1T Fk,i 6 kFk,i k 6 O k¯ xk − xk k . Moreover, the local error bound condition implies that   i+1 k¯ xk,i − xk,i k 6 c−1 kFk,i k 6 O k¯ xk − xk k .

(56)

Let q¯k = −Jk+ Fk,i . Then q¯k is the least squares solution of kmin Fk,i + Jk qk. It follows from (40), (42), (56) and Lemma 4.3 that

U3 U3T Fk,i = kFk,i + Jk q¯k k 6 kFk,i + Jk (¯ xk,i − xk,i )k 6 kFk,i + Jk,i (¯ xk,i − xk,i )k + k(Jk,i − Jk ) (¯ xk,i − xk,i )k



i−1

X 2

k¯ α d 6L1 k¯ xk,i − xk,i k + L1 k,j k,j

xk,i − xk,i k

j=0     2i+2 i+2 6O k¯ xk − xk k + O k¯ xk − xk k   i+2 =O k¯ xk − xk k . (57)



Let J˜k = U1 Σ1 V1T and q˜k = −J˜k+ Fk,i . Since q˜k is the least squares solution of min Fk,i + J˜k q , deducing from (40), (42),(47), (56) and Lemma 4.3 that




U2 U2T + U3 U3T Fk,i





= Fk,i + J˜k q˜k 6 Fk,i + J˜k (¯ xk,i − xk,i )





6 kFk,i + Jk,i (¯ xk,i − xk,i )k + J˜k − Jk,i (¯ xk,i − xk,i )


2 6L1 k¯ xk,i − xk,i k + Jk − Jk,i − U2 Σ2 V2T (¯ xk,i − xk,i )

2 6L1 k¯ xk,i − xk,i k + k(Jk − Jk,i ) (¯ xk,i − xk,i )k + U2 Σ2 V2T (¯ xk,i − xk,i )

i−1

X

2

6L1 k¯ xk,i − xk,i k + L1 αk,j dk,j xk,i − xk,i k + L1 k¯ xk − xk k k¯ xk,i − xk,i k

k¯

j=0

      2i+2 i+2 i+2 6O k¯ xk − xk k + O k¯ xk − xk k + O k¯ xk − xk k   i+2 6O k¯ xk − xk k . (58)

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Due to the orthogonality of U2 and U3 , combining (57) and (58), we know that  

i+2

U2 U2T Fk,i 6 O k¯ xk − xk k . The proof is completed. Now, we are ready to give the estimations of dk,m−1 and kF (xk,m−1 ) + Jk dk,m−1 k. Lemma 4.6. Under the condition of Assumption 4.1, for sufficiently large k, we have m

(a) kdk,m−1 k 6 O (k¯ xk − xk k );   m+1 (b) kF (xk,m−1 ) + Jk dk,m−1 k 6 O k¯ xk − xk k . Proof. By (46), we have kΣ1 k

−1

1 1 , = 6 σr σ ¯r − L1 k¯ xk − xk k

which implies 2 . σ ¯r When δ ∈ [1, 2], it then follows from Lemma 4.4, Lemma 4.5, (48), (54) and (55) that


−1
−1

kdk,m−1 k = −V1 Σ21 + λk I Σ1 U1T F (xk,m−1 ) − V2 Σ22 + λk I Σ2 U2T F (xk,m−1 )



T

−1

U2 F (xk,m−1 ) 6 kΣ1 k U1T F (xk,m−1 ) + λ−1 k Σ2   m m+2−δ 6 O (k¯ xk − xk k ) + O k¯ xk − xk k −1

kΣ1 k

6

m

= O (k¯ xk − xk k ) , and kF (xk,m−1 ) + Jk dk,m−1 k


−1 T
−1 T

= λk U1 Σ21 + λk I U1 F (xk,m−1 ) + λk U2 Σ22 + λk I U2 F (xk,m−1 ) + U3 U3T F (xk,m−1 )

−1





6 λk Σ21 U1T F (xk,m−1 ) + U2T F (xk,m−1 ) + U3T F (xk,m−1 )       m+δ m+1 m+1 6 O k¯ xk − xk k + O k¯ xk − xk k + O k¯ xk − xk k   m+1 6 O k¯ xk − xk k . The proof is completed. Based on the results above, we obtain the convergence rate of Algorithm 2.2. Theorem 4.7. Under the conditions of Assumptions 4.1, the convergence rate of Algorithm 2.2 is (m + 1)th. Proof. It follows from Lemma 4.3 and Lemma 4.6 that c k¯ xk+1 − xk+1 k 6 kF (xk+1 )k = kF (xk + sk )k = kF (xk,m−1 + αk,m−1 dk,m−1 )k 2

2 6 kF (xk,m−1 ) + αk,m−1 J (xk,m−1 ) dk,m−1 k + L1 αk,m−1 kdk,m−1 k 2 6 kF (xk,m−1 ) + J (xk,m−1 ) dk,m−1 k + L1 αk,m−1 kdk,m−1 k

2 2

2 6 kF (xk,m−1 ) + Jk dk,m−1 k + k(J (xk,m−1 ) − Jk ) dk,m−1 k + L1 αk,m−1 kdk,m−1 k

m−2

X

2 2 6 kF (xk,m−1 ) + Jk dk,m−1 k + L1 αk,j dk,j

kdk,m−1 k + L1 αk,m−1 kdk,m−1 k

j=0

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6 kF (xk,m−1 ) + Jk dk,m−1 k + L1

m−2 X

2

2 αk,j kdk,j k kdk,m−1 k + L1 αk,m−1 kdk,m−1 k

j=0



m+1





m+1

6 O k¯ xk − xk k + O k¯ xk − xk k   m+1 6 O k¯ xk − xk k , with m > 1. Hence we have



  2m + O k¯ xk − xk k

  m+1 k¯ xk+1 − xk+1 k 6 O k¯ xk − xk k ,

(59)

which means that {xk } generated by m-step LM method converges to the solution set X ∗ with (m + 1)th order. The proof is completed. Since k¯ xk − xk k 6 k¯ xk+1 − xk+1 k + ksk k , we obtain from (59) that k¯ xk − xk k 6 2 ksk k holds for sufficiently large k. By Lemma 4.3, we finally have   m+1 ksk+1 k 6 O ksk k , which indicates that {xk } converges to some solution of (1) with Q-order m + 1. This result is stronger than the convergence to the solution set.

5

Numerical results

We will compute some singular problems, which come from [16] with the same forms as in [17], to test Algorithm 2.2, and compare it with the general LM algorithm (LM), the SLM method which has presented in [10] with m = 4. We compute these test problems with different initial points and different size,
−1 T Fˆ (x) = F (x) − J (x∗ ) A AT A A (x − x∗ ) , where F (x) is the standard nonsingular test function, x∗ is its root, and A ∈ Rn×k has full column rank with 1 6 k 6 n. Obviously, Fˆ (x∗ ) = 0 and 
−1 T  Jˆ (x∗ ) = J (x∗ ) I − A AT A A has rank n − k. A disadvantage of these problems is that Fˆ (x) may have roots that are not roots of F (x). We chose the rank of Jˆ (x∗ ) to be n − 1 and n − 2, respectively, by using A ∈ Rn×1 , and A∈R

n×2

,

T

A =



AT = (1, 1, · · · , 1) 1 1 1 −1

1 1 ··· 1 −1 · · ·

1 ±1

 .

Set p0 = 0.0001, ˜ = 10−8 , µ1 = 1, δ = 1 for all the tests. The stopping criteria for 1 = 0.25, p2 = 0.75, m

p T −5

the Algorithm is Jk Fk < 10 or the iteration number exceeds 100 (n + 1). The points x0 , 10x0 , 100x0 in the third column of the tables are the starting points, where x0 was suggested by Mor´e et. al in [16]. “NF” and “NJ” represent the number of function calculations and Jacobian calculations, respectively. If the method failed to find the solution in 100 (n + 1) iterations, we denoted it by the sign “-”, and if the iterations had underflows or overflows, we denoted it by “OF”. We also denote ”TIME” represents the running time of the problem. All codes are written in MATLAB R2012 programming environment on a personal PC with Inter(R) Core(TM) i5-4590 CPU, 3.30GHz, 4GB RAM, using Windows 7 operation system.

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Table 1: Results on the first singular test set with rank(F 0 (x∗ ))= n − 1 Problem 8 9

n 3000 3000

10

3000

11

3000

13

3000

14

3000

x0 1 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

Algorith LM NF/NJ/F/TIME 9/9/1.7993e-05/16.0927 1/1/6.9349e-06/0.39072 3/3/4.9157e-03/4.6299 5/5/2.9136e-02/8.8327 7/7/1.8841e-05/113.7354 9/9/1.1683e-05/146.8708 10/10/9.479e-09/163.7237 20/10/2.2123e-04/34.464 38/26/1.9482e-03/68.8421 37/22/3.0277e-03/65.9771 9/9/1.4397e-04/16.3563 14/14/1.4123e-04/26.2765 17/17/2.5192e-04/32.2734 12/12/3.6595e-05/22.8178 18/18/4.3039e-05/35.2549 24/24/2.5066e-05/47.6055

Algorithm SLM with m = 4 NF/NJ/F/TIME 17/5/6.0835e-06/27.9715 1/1/6.9349e-06/0.38856 9/3/1.8731e-03/14.9589 13/4/2.8697e-02/22.3504 13/4/1.5023e-05/90.6983 17/5/1.2381e-05/115.8011 21/6/1.4677e-10/142.4809 77/6/2.3676e-04/128.9774 161/17/1.3971e-03/269.0105 145/16/2.45e-04/236.8702 17/5/8.5893e-05/27.4254 25/7/2.604e-04/41.0561 33/9/8.9702e-05/54.5243 25/7/4.4361e-06/41.4525 37/10/4.9713e-06/62.3313 49/13/2.9067e-06/82.7621

Algorithm 2.2 with m = 4 NF/NJ/F/TIME 17/5/1.6373e-05/45.5972 1/1/6.9349e-06/0.38503 9/3/1.468e-03/16.2261 13/4/2.3369e-02/24.1051 13/4/1.7169e-05/96.5679 21/6/7.057e-06/155.4833 21/6/1.0401e-13/159.7523 161/15/1.9047e-04/297.2913 165/17/2.8922e-03/340.8817 129/12/4.9012e-04/272.922 17/5/1.8655e-04/44.4526 29/8/5.5618e-05/79.057 37/10/2.8248e-05/102.0578 25/7/1.3946e-05/65.4345 37/10/2.4413e-05/98.7529 49/13/2.1752e-05/132.2612

Table 2: Results on the first singular test set with rank(F 0 (x∗ ))= n − 2 Problem 8 9

n 3000 3000

10

3000

11

3000

13

3000

14

3000

x0 1 1 10 100 1 10 100 1 10 100 1 10 100 1 10 100

Algorith LM NF/NJ/F/TIME 9/9/1.7993e-05/15.758 1/1/6.9349e-06/0.38195 3/3/4.9157e-03/4.5209 5/5/2.9136e-02/8.6748 7/7/1.8841e-05/113.166 9/9/1.1683e-05/145.9313 17/12/5.585e-06/210.7712 20/10/2.2124e-04/33.8705 37/24/2.3572e-03/65.5213 46/26/3.0342e-03/80.6403 9/9/1.4397e-04/16.2237 14/14/1.4123e-04/26.2398 17/17/2.5192e-04/32.1537 12/12/3.6595e-05/22.7153 18/18/4.3039e-05/34.929 24/24/2.5066e-05/47.1261

Algorithm SLM with m = 4 NF/NJ/F/TIME 17/5/6.0835e-06/27.7172 1/1/6.9349e-06/0.38875 9/3/1.8731e-03/14.8385 13/4/2.8697e-02/21.854 13/4/1.5023e-05/89.9889 17/5/1.2381e-05/115.1672 21/6/5.259e-06/141.4457 77/6/2.3676e-04/125.7451 149/16/1.9236e-03/245.8551 137/15/2.5525e-04/227.3232 17/5/8.5893e-05/27.5697 25/7/2.604e-04/41.4618 33/9/8.9702e-05/55.034 25/7/4.4361e-06/41.5594 37/10/4.9713e-06/62.0526 49/13/2.9067e-06/82.8124

Algorithm 2.2 with m = 4 NF/NJ/F/TIME 17/5/1.6373e-05/44.3831 1/1/6.9349e-06/0.38878 9/3/1.468e-03/16.9752 13/4/2.3369e-02/25.3346 13/4/1.7169e-05/96.959 21/6/7.057e-06/155.429 21/6/5.2587e-06/156.9107 149/14/1.9077e-04/288.4942 165/17/2.8922e-03/353.1612 129/13/4.9734e-04/280.1359 17/5/1.8655e-004/43.5593 29/8/5.5618e-05/77.8504 37/10/2.8248e-05/100.1965 25/7/1.3946e-05/67.8233 37/10/2.4413e-05/102.6231 49/13/2.1752e-05/135.4006

From table 1 and table 2, we can see that, though Algorithm 2.2 take more running time than the SLM method to compute step size αk,i , Algorithm 2.2 still almost always outperforms or at least performs as well as the SLM method whether on the first singular test set or on the second test set, which indicate that the line search really makes the method more efficient and contributes a lot to the numerical performance. That would be great helpful for the real application of the method and especially useful for the large scale problems.

6

Conclusions

In this work, to save more Jacobian calculations, we presented the efficient m-step LM method for systems of nonlinear equations. At every iteration, we compute m − 1 approximate LM steps with frozen
−1 T JkT Jk + λk I Jk and employ m − 1 line search for better numerical performance. The efficient m-step LM method have been proved to have (m + 1)th convergence order under the local error bound condition

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which is weaker than nonsingularity. Numerical results show that the efficient m-step LM method saved more Jacobian calculations although the calculations of function are more.

References [1] W. Sun, Y. Yuan, Optimization theory and methods: Nonlinear programming, springer, 2006. [2] K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quarterly of Applied Mathematics 2 (1944) 164–168. [3] D. W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, Journal of the Society for Industrial and Applied Mathematics 11 (1963) 431–441. [4] N. Yamashita, M. Fukushima, On the Rate of Convergence of the Levenberg-Marquardt Method, Springer Vienna, Vienna, 2001, pp. 239–249. [5] A. Fischer, P. K. Shukla, M. Wang, On the inexactness level of robust Levenberg-Marquardt methods, Optimization 59 (2) (2010) 273–287, 65H10 (90C30). [6] K. Amini, F. Rostami, Three-steps modified levenberg-marquardt method with a new line search for systems of nonlinear equations, Journal of Computational and Applied Mathematics 300 (2016) 30–42. [7] P. Kazemi, R. J. Renka, A Levenberg-Marquardt method based on sobolev gradients, Nonlinear Analysis: Theory, Methods & Applications 75 (16) (2012) 6170–6179. [8] J. Fan, The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence, Mathematics of Computation 81 (2012) 447–466. [9] X. Yang, A higher-order Levenberg-Marquardt method for nonlinear equations, Applied Mathematics and Computation 219 (22) (2013) 10682–10694, 65H10. [10] J. Fan, A shamanskii-like Levenberg-Marquardt method for nonlinear equations, Computational Optimization and Applications 56 (1) (2013) 63–80. [11] J. Fan, Accelerating the modified Levenberg-Marquardt method for nonlinear equations, Mathematics of Computation 83 (2014) 1173–1187. [12] L. Chen, A modified levenberg–marquardt method with line search for nonlinear equations, Computational Optimization and Applications (2016) 1–27. [13] M. Powell, Convergence properties of a class of minimization algorithms, in: O. Mangasarian, , R. Meyer, , S. Robinson (Eds.), Nonlinear Programming 2, Academic Press, 1975, pp. 1 – 27. [14] R. Behling, A. Iusem, The effect of calmness on the solution set of systems of nonlinear equations, Mathematical Programming 137 (1-2) (2013) 155–165. [15] G. W. Stewart, J. Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing, Academic Press Inc., Boston, MA, 1990. [16] J. J. Mor´e, B. S. Garbow, K. H. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software 7 (1981) 17–41. [17] R. B. Schnabel, P. D. Frank, Tensor methods for nonlinear equations, SIAM Journal on Numerical Analysis 21 (5) (1984) 815–843.

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OPTIMAL BOUNDS FOR A TOADER TYPE MEAN USING ARITHMETIC AND GEOMETRIC MEANS∗ WEI-MAO QIAN1,2 , WEN ZHANG3 , AND YU-MING CHU4,∗∗

Abstract. In the aritcle, we prove that the double inequalities αA(a, b)+(1−α)G(a, b) < T [A(a, b), G(a, b)] < βA(a, b) + (1 − β)G(a, b) and G[λa + (1 − λ)b, λb + (1 − λ)a] < T [A(a, b), G(a, b)] < G[µa + (1 − µ)b, µb + (1 −pµ)a] hold for all a, b > 0 with a 6= b √ if and only if α ≤ 1/2, β ≥ 2/π, λ ≤ (1 − 1 − 4/π 2 )/2 and µ ≥ 1/2 − 2/4 if α, β ∈ R and λ, µ ∈ (0, 1/2), and find new bounds for the complete elliptic integral √ R E(r) = 0π/2 (1 − r2 sin2 θ)1/2 dθ (0 < r < 1) of the second kind, where G(a, b) = ab, R π/2 p a2 cos2 θ + b2 sin2 θdθ/π and A(a, b) = (a + b)/2 are respectively the T (a, b) = 2 0 geometric, Toader and arithmetic means of a and b.

1. Introduction Let r ∈ (0, 1) and a, b > 0. Then the complete elliptic integrals K(r) and E(r) [1-24] of the first and second kind, Toader mean T (a, b) [25-34], geometric mean G(a, b) [35-41] and arithmetic mean A(a, b) [42-50] are respectively given by Zπ/2 K(r) = (1 − r2 sin2 θ)−1/2 dθ,

Zπ/2 E(r) = (1 − r2 sin2 θ)1/2 dθ,

0

0

T (a, b) =

2 π

π/2

Z

p

a2 cos2 θ + b2 sin2 θdθ,

(1.1)

0

G(a, b) =

√ ab,

A(a, b) =

a+b . 2

(1.2)

It is well known that K(0+ ) = E(0+ ) = π/2,

K(1− ) = +∞,

E(1− ) = 1,

K(r) is strictly increasing and E(r) is strictly decreasing on (0, 1), K(r) and E(r) satisfy the derivatives formulas [51, Appendix E, p. 474-475] dK(r) E(r) − (1 − r2 )K(r) = , dr r(1 − r2 )

dE(r) E(r) − K(r) = , dr r

and T (a, b) can be rewritten as

T (a, b) =

          

q 1−


b 2 a

a,  q 2b E 1− π


a 2 b

2a π E

 ,

a > b, a = b,

(1.3)

 ,

a < b.

2010 Mathematics Subject Classification. Primary: 26E60; Secondary: 33E05. Key words and phrases. Toader mean, arithmetic mean, geometric mean. ∗ The research was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485), the Science and Technology Research Program of Zhejiang Educational Committee (Grant No. Y201635325) and the Natural Science Foundation of Huzhou City (Grant No. 2018YZ07). ∗∗ Corresponding author: Yu-Ming Chu, Email: [email protected]. 1

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Optimal Bounds for A Toader Type Mean Using Arithmetic and Geometric Means

Recently, the bounds for the Toader mean T (a, b) have attracted the attention of several researchers. Barnard et. al. [52], and Alzer and Qiu [53] proved that the double inequality Mp1 (a, b) < T (a, b) < Mp2 (a, b) holds for all a, b > 0 with a 6= b if and only if p1 ≤ 3/2 and p2 ≥ log √ 2/(log π − log 2) = 1.5349 · · · , where Mp (a, b) = [(ap + bp )/2]1/p (p 6= 0) and M0 (a, b) = ab is the pth power mean. Very recently, Song et. al. [54] proved that the double inequality Mq1 (a, b) < T [A(a, b), Q(a, b)] < Mq2 (a, b)

√ 2/2)] = holds for all a, b > 0 with a 6= b if and only if q ≤ 2 log 2/[2 log π − log 2 − 2 log E( 1 p 1.3930 · · · and q2 ≥ 3/2, where Q(a, b) = (a2 + b2 )/2 is the quadratic mean of a and b. Let a, b > 0 with a 6= b. Then from (1.1) and (1.2) together with G(a, b) < A(a, b) we clearly see that the function λ → R(λ) = G[λa + (1 − λ)b, λb + (1 − λ)a] is continuous and strictly increasing on [0, 1/2], and   1 R(0) = G(a, b) < T [A(a, b), G(a, b)] < A(a, b) = R . 2 It is the aim of this article to find the best possible parameters α, β ∈ R and λ, µ ∈ (0, 1/2) such that the double inequalities αA(a, b) + (1 − α)G(a, b) < T [A(a, b), G(a, b)] < βA(a, b) + (1 − β)G(a, b), G[λa + (1 − λ)b, λb + (1 − λ)a] < T [A(a, b), G(a, b)] < G[µa + (1 − µ)b, µb + (1 − µ)a] hold for all a, b > 0 with a 6= b holds for all a, b > 0 with a 6= b. 2. Lemmas Lemma 2.1. (See [51, Theorem 3.21(1)]) The function r 7→ [E(r) − (1 − r2 )K(r)]/r2 is strictly increasing from (0, 1) onto (π/4, 1). Lemma 2.2. (See [51, Exercise 3.43(11)]) The function r 7→ [K(r) − E(r)]/r2 is strictly increasing from (0, 1) onto (π/4, +∞). Lemma 2.3. (See [51, Theorem 3.21(7)]) The function r 7→ (1 − r2 )λ K(r) is strictly decreasing from (0, 1) onto (0, π/2) if λ ≥ 1/4. Lemma 2.4. (See [51, Theorem 1.25]) Let a, b ∈ R with a < b, f, g : [a, b] → R be continuous on [a, b] and differentiable on (a, b), and g 0 (x) 6= 0 on (a, b). If f 0 (x)/g 0 (x) is increasing (decreasing) on (a, b), then so are the functions f (x) − f (a) , g(x) − g(a)

f (x) − f (b) . g(x) − g(b)

If f 0 (x)/g 0 (x) is strictly monotone, then the monotonicity in the conclusion is also strict. √ Lemma 2.5. The function r 7→ 1 − r2 [E(r) − K(r)]/r2 is strictly increasing from (0, 1) onto (−π/4, 0). Proof. Let

√ 1 − r2 [E(r) − K(r)] f (r) = , r2 g(r) = [K(r) − E(r)] − [E(r) − (1 − r2 )K(r)]. Then it follows from (2.1), (2.2), L’Hˆopital rule, and Lemmas 2.1 and 2.3 that √ [ 1 − r2 (E(r) − K(r))]0 K(r) − 2E(r) π − + √ f (1 ) = 0, f (0 ) = lim+ = lim+ =− , 2 2r 4 r→0 r→0 2 1−r 1 f 0 (r) = √ g(r), r3 1 − r2 g(0+ ) = 0,

561

(2.1) (2.2)

(2.3) (2.4) (2.5)

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g 0 (r) =

r3 E(r) − (1 − r2 )K(r) >0 1 − r2 r2

(2.6)

for r ∈ (0, 1). Therefore, Lemma 2.5 follows easily from (2.3)-(2.6).



Lemma 2.6. The function r 7→ E(r)[K(r) − E(r)]/r2 is strictly increasing from (0, 1) onto (π 2 /8, +∞). Proof. Let p E(r)[K(r) − E(r)] , h (r) = E(r) − 1 − r2 K(r). 1 r2 Then from Lemma 2.2 and (2.7) we clearly see that h(r) =

h(0+ ) =

π2 , 8

h(1− ) = +∞,

h1 (0+ ) = 0,

(2.7)

(2.8)

√ E(r) + 1 − r2 K(r) h (r) = h1 (r), r3 (1 − r2 ) √ r(1 − 1 − r2 ) K(r) − E(r) 0 √ >0 h1 (r) = r2 1 − r2 0

(2.9) (2.10)

for r ∈ (0, 1). Therefore, Lemma 2.6 follows easily from (2.8)-(2.10).



3. Main Results Theorem 3.1. The double inequality αA(a, b) + (1 − α)G(a, b) < T [A(a, b), G(a, b)] < βA(a, b) + (1 − β)G(a, b) holds for all a, b > 0 with a 6= b if and only if α ≤ 1/2 and β ≥ 2/π = 0.6366 · · · . Proof. Since A(a, b), T (a, b) and G(a, b) are symmetric and homogeneous of degree 1, without loss of generality, we assume that a > b > 0 and r = (a − b)/(a + b) ∈ (0, 1). Then (1.2) and (1.3) lead to p 2 T [A(a, b), G(a, b)] = A(a, b)E(r), G(a, b) = A(a, b) 1 − r2 , π √ 2 1 − r2 T [A(a, b), G(a, b)] − G(a, b) π E(r) − √ = . (3.1) A(a, b) − G(a, b) 1 − 1 − r2 Let p p 2 (3.2) F1 (r) = E(r) − 1 − r2 , F2 (r) = 1 − 1 − r2 , π √ 2 E(r) − 1 − r2 F1 (r) √ F (r) = = π . (3.3) F2 (r) 1 − 1 − r2 Then Lemma 2.5, (3.2) and (3.3) lead to F1 (0+ ) = F2 (0+ ) = 0, √ 2 1 − r2 [E(r) − K(r)] F10 (r) = + 1, F20 (r) π r2

(3.4) (3.5)

F10 (r) 1 2 = , F (1− ) = . (3.6) 0 F2 (r) 2 π It follows from Lemmas 2.4 and 2.5 together with (3.3)-(3.5) that F (r) is strictly increasing on (0, 1). Therefore, Theorem 3.1 follows from (3.1), (3.3) and (3.6) together with the monotonicity of F (r).  F (0+ ) = lim

r→0+

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Optimal Bounds for A Toader Type Mean Using Arithmetic and Geometric Means

Theorem 3.2. Let λ, µ ∈ (0, 1/2). Then the double inequality G[λa + (1 − λ)b, λb + (1 − λa)] < T [A(a, b), G(a, b)] < G[µa + (1 − µ)b, µb + (1 − µa)] p 2 holds for all √ a, b > 0 with a 6= b if and only if λ ≤ (1 − 1 − 4/π )/2 = 0.1144 · · · and µ ≥ 1/2 − 2/4 = 0.1464 · · · . Proof. We assume that a > b > 0, r = (a − b)/(a + b) ∈ (0, 1) and p ∈ (0, 1/2). Then (1.2) and (1.3) lead to G[pa + (1 − p)b, pb + (1 − pa)] − T [A(a, b), G(a, b)]   p 2 1 − (1 − 2p)2 r2 − E(r) = A(a, b) π A(a, b) H(r), =p 1 − (1 − 2p)2 r2 + π2 E(r) where H(r) = 1 − (1 − 2p)2 r2 −

(3.7)

4 2 E (r), π2

H(0+ ) = 0,

(3.8)

4 H(1− ) = 4p(1 − p) − 2 , π H 0 (r) = 2rH1 (r),

(3.9) (3.10)

where

4 E(r)[K(r) − E(r)] − (1 − 2p)2 . π2 r2 It follows from Lemma 2.6 and (3.11) that 1 H1 (0+ ) = − (1 − 2p)2 , 2 H1 (1− ) = +∞. We divide the proof into four √ cases. Case 1 p = µ0 = 1/2 − 2/4. Then (3.12) becomes H1 (r) =

(3.11)

(3.12) (3.13)

H1 (0+ ) = 0.

(3.14)

From Lemma 2.6, (3.11) and (3.14) we clearly see that H1 (r) > 0

(3.15)

for all r ∈ (0, 1). Therefore, T [A(a, b), G(a, b)] < G[µ0 a + (1 − µ0 )b, µ0 b + (1 − µ0 )a] follows from (3.7), (3.8), p (3.10) and (3.15). Case 2 p = λ0 = (1 − 1 − 4/π 2 )/2. Then (3.9) and (3.12) lead to H(1− ) = 0,

(3.16)

2

π −8 < 0. (3.17) 2π 2 From Lemma 2.6, (3.10), (3.11), (3.13) and (3.17) we know that there exists r0 ∈ (0, 1) such that H(r) is strictly decreasing on (0, r0 ) and strictly increasing on (r0 , 1). Therefore, H1 (0+ ) = −

T [A(a, b), G(a, b)] > G[λ0 a + (1 − λ0 )b, λ0 b + (1 − λ0 )a] follows from (3.7), (3.8) and (3.16) √ together with the piecewise monotonicity of H(r). Case 3 0 < p = µ∗ < 1/2 − 2/4. Then (3.12) leads to H1 (0+ ) < 0.

(3.18)

Equations (3.7), (3.8) and (3.10) together with inequality (3.18) imply that there exists small enough δ0 ∈ (0, 1) such that T [A(a, b), G(a, b)] > G[µ∗ a + (1 − µ∗ )b, µ∗ b + (1 − µ∗ )a] for all a > b > 0 with (a − b)/(a + b) ∈ (0, δ0 ).

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Case 4 (1 −

p 1 − 4/π 2 )/2 < p = λ∗ < 1/2. Then (3.9) leads to H(1− ) > 0.

(3.19)

Equation (3.7) and inequality (3.19) imply that there exists small enough δ1 ∈ (0, 1) such that T [A(a, b), G(a, b)] < G[λ∗ a + (1 − λ∗ )b, λ∗ b + (1 − λ∗ )a] for all a > b > 0 with (a − b)/(a + b) ∈ (1 − δ1 , 1).  From Theorems 3.1 and 3.2 we get Corollary 3.3 immediately. Corollary 3.3. The double inequality s (   )  π p 4 π 1+ 1 + 1 − r2 , − 1 r2 < E(r) max 4 2 π2 ( ) √ π p
2π 2 < min 1 + −1 1 − r2 , 2−r 2 4 holds for all r ∈ (0, 1).

References [1] W.-M. Qian and Y.-M. Chu, Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters, J. Inequal. Appl., 2017, 2017, Article 274, 10 pages. [2] Y.-M. Chu, Y.-F. Qiu and M.-K. Wang, H¨ older mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct., 2012, 23(7), 521–527. [3] Y.-M. Chu, M.-K. Wang, Y.-P. Jiang and S.-L. Qiu, Concavity of the complete elliptic integrals of the second kind with respect to H¨ older means, J. Math. Anal. Appl., 2012, 395(2), 637–642. [4] Y.-M. Chu, M.-K. Wang and Y.-F. Qiu, On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function, Abstr. Appl. Anal., 2011, 2011, Article ID 697547, 7 pages. [5] Y.-M. Chu, M.-K. Wang, S.-L. Qiu and Y.-P. Jiang, Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl., 2012, 63(7), 1177–1184. [6] Y.-M. Chu and T.-H. Zhao, Convexity and concavity of the complete elliptic integrals with respect to Lehmer mean, J. Inequal. Appl., 2015, 2015, Article 396, 6 pages. [7] T.-R. Huang, S.-Y. Tan, X.-Y. Ma and Y.-M. Chu, Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl., 2018, 2018, Article 239, 11 pages. [8] M.-K. Wang and Y.-M. Chu, Asymptotical bounds for complete elliptic integrals of the second kind, J. Math. Anal. Appl., 2013, 402(1), 119–126. [9] M.-K. Wang and Y.-M. Chu, Refinements of transformation inequalities for zero-balanced hypergeometric function, Acta Math. Sci., 2017, 37B(3), 607–622. [10] M.-K. Wang and Y.-M. Chu, Landen inequalities for a class of hypergeometric functions with applications, Math. Inequal. Appl., 2018, 21(2), 521–537. [11] M.-K. Wang, Y.-M. Chu and Y.-P. Jiang, Ramanujan’s cubic transformation inequalities for zerobalanced hypergeometric functions, Rocky Mountain J. Math., 2016, 46(2), 679–691. [12] M.-K. Wang, Y.-M. Chu and S.-L. Qiu, Some monotonicity properties of generalized elliptic integrals with applications, Math. Inequal. Appl., 2013, 16(3), 671–677. [13] M.-K. Wang, Y.-M. Chu, S.-L. Qiu and Y.-P. Jiang, Convexity of the complete elliptic integrals of the first kind with respect to H¨ older means, J. Math. Anal. Appl., 2012, 388(2), 1141–1146. [14] M.-K. Wang, Y.-M. Chu and Y.-Q. Song, Asymptotical formulas for Gaussian and generalized hypergeometric functions, Appl. Math. Comput., 2016, 276, 44–60. [15] M.-K. Wang, Y.-M. Li and Y.-M. Chu, Inequalities and infinite product formula for Ramanujan generalized modular equation function, Ramanujan J., 2018, 46(1), 189–200. [16] M.-K. Wang, S.-L. Qiu and Y.-M. Chu, Infinite series formula for H¨ ubner upper bound function with applications to Hersch-Pfluger distortion function, Math. Inequal. Appl., 2018, 21(3), 629–648. [17] M.-K. Wang, S.-L. Qiu, Y.-M. Chu and Y.-P. Jiang, Generalized Hersch-Pfluger distortion function and complete elliptic integrals, J. Math. Anal. Appl., 2012, 385(1), 221–229. [18] Zh.-H. Yang and Y.-M. Chu, A monotonicity property involving the generalized elliptic integral of the first kind, Math. Inequal. Appl., 2017, 20(3), 729–735. [19] Zh.-H. Yang, Y.-M. Chu and M.-K. Wang, Monotonicity criterion for the quotient of power series with applications, J. Math. Anal. Appl., 2015, 428(1), 587–604. [20] Zh.-H. Yang, Y.-M. Chu and W. Zhang, Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean, J. Inequal. Appl., 2016, 2016, Article 176, 10 pages. [21] Zh.-H. Yang, Y.-M. Chu and W. Zhang, Accurate approximations for the complete elliptic integral of the second kind, J. Math. Anal. Appl., 2016, 438(2), 875–888.

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[22] Zh.-H. Yang, W.-M. Qian and Y.-M. Chu, Monotonicity properties and bounds involving the complete elliptic integrals of the first kind, Math. Inequal. Appl., 2018, 21(4), 1185–1199. [23] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl., 2018, 462(2), 1714–1726. [24] T.-H. Zhao, M.-K. Wang, W. Zhang and Y.-M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018, 2018, Article 251, 15 pages. [25] Gh. Toader, Some mean values related to the arithmetic-geometric mean, J. Math. Anal. Appl., 1998, 218(2), 358–368. [26] Y.-M. Chu, M.-K. Wang, S.-L. Qiu and Y.-F. Qiu, Sharp generalized Seiffert mean bounds for Toader mean, Abstr. Appl. Anal., 2011, 2011, Article ID 605259, 8 pages. [27] Y.-M. Chu and M.-K. Wang, Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal., 2012, 2012, Article ID 830585, 11 pages. [28] W.-M. Qian, Z.-H. Zhang and Y.-M. Chu, Sharp bounds for Toader-Qi mean in terms of harmonic and geometric means, J. Math. Inequal., 2017, 11(1), 121–127. [29] Y.-M. Chu, M.-K. Wang and S.-L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci., 2012, 122(1), 41–51. [30] Y.-M. Chu and M.-K. Wang, Optimal Lehmer mean bouns for the Toader mean, Results Math., 2012, 61(3-4), 223–229. [31] J.-F. Li, W.-M. Qian and Y.-M. Chu, Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means, J. Inequal. Appl., 2015, Article 277, 9 pages. [32] Y.-Q. Song, W.-D. Jiang, Y.-M. Chu and D.-D. Yan, Optimal bounds for Toader mean in terms of arithmetic and contraharmonic means, J. Math. Inequal., 2013, 7(4), 751–757. [33] W.-H. Li and M.-M. Zheng, Some inequalities for bounding Toader mean, J. Funct. Spaces Appl., 2013, Article ID 394194, 5 pages. [34] Y.-M. Chu, M.-K. Wang and X.-Y. Ma, Sharp bounds for Toader mean in terms of contraharmonic mean with applications, J. Math. Inequal., 2013, 7(2), 161–166. [35] W.-M. Gong, Y.-Q. Song, Y.-M. Chu and M.-K. Wang, A sharp double inequality between Seiffert, arithmetic, and geoemtric means, Abstr. Appl. Anal., 2012, 2012, Article ID 684834, 7 pages. [36] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means, Math. Inequal. Appl., 2012, 15(2), 415–422. [37] M.-K. Wang, Z.-K. Wang and Y.-M. Chu, An optimal double inequality between geometric and identric means, Appl. Math. Lett., 2012, 25(3), 471–475. [38] Y.-M. Chu, C. Zong and G.-D. Wang, Optimal convex combination bounds of Seiffert and geometric means for arithmetic mean, J. Math. Inequal., 2011, 5(2), 429–434. [39] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, An optimal double inequality between Seiffert and geometric means, J. Appl. Math., 2011, 2011, Article ID 261237, 6 pages. [40] Y.-M. Chu and M.-K. Wang, Optimal inequalities between harmonic, geometric, logarithmic, and arithmetic-geometric means, J. Appl. Math., 2011, 2011, Article ID 618929, 9 pages [41] B.-Y. Long and Y.-M. Chu, Optimal inequalities for generalized logarithmic, arithmetic, and geometric means, J. Inequal. Appl., 2010, 2010, Article ID 806825, 10 pages. [42] A. Iqbal, M. Adil Khan, S. Ullah, Y.-M. Chu and A. Kashuri, Hermite-Hadamard type inequalities pertaining conformable fractional integrals and their applications, AIP Advances, 2018, 8, Article ID 075101, 18 pages, DOI: 10.1063/1.5031954. [43] Y.-M. Chu, M. Adil Khan, T. Ali and S. S. Dragomir, Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017, 2017, Article 93, 12 pages. [44] M. Adil Khan, S. Begum, Y. Khurshid and Y.-M. Chu, Ostrowski type inequalities involving conformable fractional integrals, J. Inequal. Appl., 2018, 2018, Article 70, 14 pages. [45] M. Adil Khan, Y. Khurshid, T.-S. Du and Y.-M. Chu, Generalized of Hermite-Hadamard type inequalities via conformable fractional integrals, J. Funct. Spaces, 2018, 2018, Article ID 5357463, 12 pages. [46] M. Adil Khan, Y.-M. Chu, T. U. Khan and J. Khan, Some new inequalities of Hermite-Hadamard type for s-convex functions with applications, Open Math., 2017, 15, 1414–1430. [47] M. Adil Khan, Z. M. Al-sahwi and Y.-M. Chu, New estimations for Shannon and Zipf-Mandelbrot entropies, Entropy, 2018, 20, Article 608, 10 pages, DOI: 10.3390/e2008608. [48] M. Adil Khan, Y.-M. Chu, A. Kashuri and R. Liko, Hermite-Hadamard type fractional integral inequalities for MT(r;g,m,ϕ) -preinvex functions, J. Comput. Anal. Appl., 2019, 26(8), 1487–1503. [49] Y.-M. Chu, Y.-F. Qiu, M.-K. Wang and G.-D. Wang, The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert mean, J. Inequal. Appl., 2010, 2010, Article ID 436457, 7 pages. [50] X.-M. Zhang and Y.-M. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain J. Math., 2010, 40(3), 1061–1068. [51] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Song, New York, 1997. [52] R. W. Barnard, K. Pearce and K. C. Richards, An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal., 2000, 31(3),693–699.

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[53] H. Alzer and S.-L. Qiu, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math., 2004, 172(2), 289–312. [54] Y.-Q. Song, T.-H. Zhao, Y.-M. Chu and X.-H. Zhang, Optimal evaluation of a Toader-type mean by power mean, J. Inequal. Appl., 2015, Article 408, 12 pages. Wei-Mao Qian, 1 College of Science, Hunan City University, Yiyang 413000, Hunan, China; 2 School of Continuing Education, Huzhou Vocational and Technological College, Huzhou 313000, Zhejiang, China E-mail address: [email protected] Wen Zhang, 3 Friedman Brain Institute, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA E-mail address: [email protected] Yu-Ming Chu (Corresponding author), 4 Department of Mathematics, Huzhou University, Huzhou 313000, China E-mail address: [email protected]

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Addition Theorem For Exton’s q–Exponential Functions Mahmoud Jafari Shah Belaghi Bah¸ce¸sehir University, Istanbul, Turkey [email protected] Nuri Kuruo˘ glu ˙ Istanbul Geli¸sim University, Istanbul, Turkey [email protected] Abstract. In this paper, we study about the q-exponential function which was introduced by Exton. We propose the addition theorem for this q–exponential function and also Continued fraction representation for this q–exponential function is given. Keywords. Exton’s q-Exponential Function, Symmetric q–derivative, Symmetric q-Binomial. Mathematics Subject Classification. 11B65, 33D05.

1

Introduction

The qe–derivative (or symmetric q–derivative) of a function f (x) is defined [3] as −1 e q f (x) = f (qx) − f (q x) D −1 (q − q )x

where q = 6 ±1. This qe–derivative is invariant under inversion of basis. For any number α, the qe–derivative of powers of x are given by e q xα = [α]qe xα−1 D where [α]qe =

q α − q −α q − q −1

and it is called symmetric q–number. In the case, if α is a positive integer we have [α]qe =

q α − q −α = q 1−α (1 + q 2 + q 4 + · · · + q 2α−2 ). q − q −1

Relation between q–number and symmetric q–number is [α]qe =

q α − q −α = q 1−α [α]q2 q − q −1

(1)

α

−1 where [α]q = qq−1 is called q–number. With easy calculation, one can see [5] that

[α] e1 = [α]qe.

(2)

q

[−α]qe = −[α]qe. β

[α + β]qe = q [α]qe + q

(3) −α

[β]qe.

(4)

Furthermore, the qe–analogue of factorial, denoted by [n]qe !, is defined [1] as ( 1 if n = 0, [n]qe ! = [n]qe × [n − 1]qe × · · · × [1]qe if n = 1, 2, . . .

(5)

and by using (1), we may also write the qe–factorial as follows n

[n]qe ! = [n]q2 ! q −( 2 )

(6)

where [n]q ! = [n]q × [n − 1]q × · · · × [1]q for n = 1, 2, . . . . The qe–analogue of (a − x)n , denoted by (a − x)nqe , is defined [3] as  1
Q (a − x)nqe = n−1  a − xq 1−n+2i

n = 0, n = 1, 2, . . . .

(7)

i=0

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The qe–analogue in (7) is invariant under inversion of basis and one can see that (a − x)nqe = (−1)n (x − a)nqe .

(8)

The qe–derivative of (x − a)nqe is founded [3] as e q (x − a)n = [n]qe (x − a)n−1 . D qe qe

(9)

The qe–Taylor series expansion of (a + x)nqe about x = 0 is n   X n n (a + x)qe = an−k xk k qe

(10)

k=0


[n]qe! where nk qe = [k]qe! [n−k] are called symmetric q–binomial coefficients. Formula (10) is called Gauss’s q e! qe–binomial formula (see [3], p. 100). The object of study in this paper is the q–exponential function which was introduced by Exton (see [6] or [4], p. 128) as ∞ X 1 n 21 (n2 ) , x∈C (11) x q E(q, x) = [n]q ! n=0 n

−1 where [n]q = qq−1 . This q–exponential function is invariant under inversion of basis and unfortunately, there is no known addition theorem for it. Our goal is to give the addition theorem for this q–exponential function and also represent it as a continued fractions.

2

Some Identities

Definition 1. For any number α, we define (a + x)α qe = where (a + x)∞ q := limn→∞

Qn

j=0 (a

(a + q 1−α x)∞ q2

(12)

(a + q 1+α x)∞ q2

+ q j x).

Theorem 1. For any numbers α and β, α β (a + x)α+β = (a + q −β x)α qe (a + q x)qe . qe

Proof. The result will be obtained directly by using the definition of (a + x)α q˜ , which is given in (12). Corollary 1. For any number α, (a + x)−α qe =

1 (a + x)α qe

Proof. The result will be obtained by using (12). Proposition 1. For 1 ≤ j ≤ n − 1, the qe–Pascal rule is       n n−j n − 1 −j n − 1 =q +q j qe j − 1 qe j qe
Proof. Let us expand the symmetric q–binomial coefficient nj , then we have qe   [n]qe! n = j qe [j]qe! [n − j]qe! =

[n − 1]qe! [n]qe [j]qe! [n − j]qe!

[n − 1]qe! (q n−j [j]qe + q −j [n − j]qe) [j]qe! [n − j]qe!     n−j n − 1 −j n − 1 =q +q j − 1 qe j qe =

which completes the proof. We used (4) in the third line.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.3, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Lemma 1. For any number x and positive integer r,     −x x+r−1 = (−1)r r qe r qe Proof. To prove the lemma we make a use of (5) and (3), then we may write   [−x]qe! −x = r qe [r]qe! [−x − r]qe! [x]qe [x + 1]qe . . . [x + r − 1]qe [r]qe!   [x + r − 1]qe! x+r−1 = (−1)r = (−1)r [x − 1]qe! [r]qe! r qe

= (−1)r

which completes the proof. The following theorem is a symmetric version of Heine’s q–binomial formula. Theorem 2. For any number x and positive integer n, the following equation holds  ∞  X 1 n+j−1 = xj . (1 − x)nqe j q e j=0 Proof. To prove the Theorem we make a use of Corollary 1 and Lemma 1, then we may write   ∞  ∞  X X −n n+j−1 1 −n j (−x) = xj = (1 − x) = qe j j (1 − x)nqe qe qe j=0 j=0 which completes the proof. In the next theorem, qe–analogue of Vandermonde’s identity is given. Theorem 3. For any m, n, r ∈ N0 

m+n r

 =q

mr

qe

 r    X m n q −(m+n)k k qe r − k qe

k=0

Proof. We make a use of Theorem 1 to write that m n (1 + x)m+n = (1 + q −n x)m qe (1 + q x)qe . qe

Using the qe–binomial formula in (10) for both sides of the above formula, and then we obtain m+n X r=0

m+n r



xr =

qe

m   X m

(q −n x)r

n   X n

(q m x)r r r qe qe r=0 r=0   r    m+n X X m n mr −(m+n)k = q q xr . k r − k q e q e r=0 k=0

The proof is complete by comparing coefficients of xr . The following corollary is the special case of Vandermonde’s identity. Corollary 2. For any positive integer n, n  2 X n k=0

k

q

n(n−2k)

qe

 =

 2n n qe

(13)

Proof. Take m = r = n in Theorem 3 and make a use of the identity corollary.

569

n k qe




=

n n−k qe




to prove the

Mahmoud Jafari Shah Belaghi et al 567-572

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.3, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Corollary 3. For any positive integer n,   n  2 X n 2n [2k]qfn = [n]qfn . k qe n qe

(14)

k=0

Proof. Let us change the base q to q −1 in Corollary 2 to obtain   n  2 X n 2n q −n(n−2k) = k qe n qe

(15)

k=0

because of the identity

n 1 k e q




=

n k qe.




Now by comparing equations (13) and (15) we can write

n  2 X n k=0

k

and also


q n(n−2k) − q −n(n−2k) = 0

n  2 X n

k

k=0

since [α]qe =

q α − q −α q − q −1 .

(16)

qe

qe

[n − 2k]qfn = 0

(17)

Then we make a use of equations (3) and (4) to rewrite the equation (17) as n  2  X n k=0 n  X k=0 n  X k=0

k

qe

2 n  k n k

qe

 2 q 2nk [n]qfn + q n [−2k]qfn = 0,  2 q 2nk [n]qfn − q n [2k]qfn = 0,

2 qe

[2k]qfn = [n]qfn

n  2 X n k=0

k

2

q −n

+2nk

.

qe

The proof will be complete if we apply the identity in (15) to the right side of the last equation.

3

qe–Exponential Functions

In this section, we study about the q–exponential functions (11) which was introduced by Exton. Let us consider E(q 2 , x), then we have E(q 2 , x) =

∞ X

n 1 xn q ( 2 ) , [n]q2 ! n=0

x ∈ C.

(18)

Now we make a use of (6) to rewrite the above formula as follows E(q 2 , x) =

∞ X

1 n x , [n]qe! n=0

x ∈ C.

(19)

We use a different notation for the Exton’s q–exponential function as exqe

2

:= E(q , x) =

∞ X

1 n x , [n] qe! n=0

x ∈ C.

(20)

One can see that this qe–exponential function (20) is invariant under inversion of basis and its qe–derivative is equal to itself, that means exe1 = exqe

(21)

e q ex = ex D qe qe

(22)

q

The next theorem is about the product of two qe–exponential functions. 570

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.3, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Theorem 4. For any x and y, the following equation holds (x+y)qe

exqe eyqe = eqe

(23)

where (x + y)nqe is defined in (10). Proof. We use (20) to expand both exqe and eyqe, therefore we obtain exqe eyqe =

∞ X

∞ 1 n X 1 n x y [n]qe! [n]qe! n=0 n=0 ∞ X n X

1 xk y n−k [k] ! [n − k] ! q e q e n=0 k=0 ∞ n   X 1 X n xk y n−k = [n] ! k q e qe n=0 =

k=0

=

∞ X

1 (x + y)nqe [n] ! q e n=0 (x+y)qe

= eqe and the proof is complete.

4

Continued Fractions

A continued fraction is an expression of the form b1

a0 +

,

b2

a1 + a2 +

b3 a3 + · · ·

where a0 , a1 , a2 , . . . and b1 , b2 , b3 , . . . are two sequences of real or complex numbers. We use the following symbol for the above continued fraction  ∞  bn a0 + . (24) an n=1

K

The following theorem is the convergent theorem of continued fractions (See [7], p. 126). 1 ∞ Theorem 5. If an > 0 for n > 1 then the continued fraction Kn=1 an converges if and only if the P∞ series n=1 an diverges.

4.1

Continued Fraction Representation of qe–Exponential Functions

The q–exponential functions exq and Eqx can be written as infinite product form as follows exq =

1 , (1 − (1 − q)x))∞ q

Eqx = (1 + (1 − q)x))∞ q .

In this section, we want to show that the qe–exponential function also can be written as infinite product form. Let us consider the qe–derivative of a function f (x) which is defined as −1 e q f (x) = f (qx) − f (q x) . D −1 (q − q )x

(25)

Take f (x) = eqx qe , therefore we can write (25) as follows 2

q

eqx qe

=

eqqe x − exqe

(q − q −1 )x

571

,

(26)

Mahmoud Jafari Shah Belaghi et al 567-572

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.3, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

e q eqx = q eqx . Now by easy manipulation, we may write (26) as because D qe qe exqe

eqx qe Let us define g(x) :=

ex q e eqx q e

2

eqqe x

−

eqx qe

= (1 − q 2 )x.

(27)

and then we may write (27) as g(x) = (1 − q 2 )x +

1 . g(qx)

(28)

Iterating the formula in (28) infinity many times to obtain 1

g(x) = (1 − q 2 )x +

.

1

(1 − q 2 )qx + (1 − q 2 )q 2 x +

(29)

1 (1 −

q 2 )q 3 x

+ ···

Now by using continued fractions symbol which is defined in (24), we may rewrite (29) as follows  ∞  1 g(x) = (1 − q 2 )x + (1 − q 2 )q n x n=1 or  ∞  1 1 = . g(x) n=0 (1 − q 2 )q n x

K

K

(30)

(31)

By using Theorem 5, one can see that the continued fraction in the right hand side of equation (31) is converge, if x < 0 and q > 1. ex q e Substitute x with q −1 x in the equation (31) and then replace g(x) = eqx to obtain q e  ∞  −1 1 x eqe = eqqe x . (32) 2 )q n−1 x (1 − q n=0

K

Iterating this formula k times to obtain exqe =

k Y

exqe −k

k→∞

K



j=1 n=0

In the case, if k → ∞, we have

because if q > 1, then we have lim eqqe

∞

=

∞ Y

 −k 1 eqqe x . 2 n−j (1 − q )q x

∞

K



j=1 n=0 x

1 2 (1 − q )q n−j x

(33)

 (34)

= 1.

References [1] Ernst, T., A Comprehensive Treatment of Q-calculus, Springer, 2012. [2] Gasper, G., Rahman, M., Basic hypergeometric series, Vol. 96 Cambridge university press, 2004. [3] Kac, V., Cheung, P., Quantum Calculus, Springer, 2002. [4] Exton, H., q-Hypergeometric Functions and ApplicationsEllis, Horwood, Chichester, 1983. [5] McAnally, D. S., q-exponential and q-gamma functions. I. q-exponential functions. Journal of Mathematical Physics, Vol. 36, no. 1, pp. 546–573, 1995. [6] Exton, H. Basic circular functions. In Indagationes Mathematicae (Proceedings), Vol. 84, no. 2, pp. 165–171, North-Holland. 1981. [7] Khrushchev, S.V.: Orthogonal Polynomials and Continued Fractions from Euler’s Point of View, Encyclopedia of Mathematics and Its Applications, Vol. 122. Cambridge. Cambridge University Press 2008.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 3, 2020

Control Problems for Semilinear Impulsive Differential Control Systems, Ah-ran Park and Jin-Mun Jeong,…………………………………………………………………………………397 Homoclinic Solutions for a Class of Difference Equations with Asymptotically Linear Nonlinearity, Ali Mai and Guowei Sun,………………………………………………………409

Approximation of Almost Cauchy's Points by Cauchy's Points, Gwang Hui Kim and Hwan-Yong Shin,……………………………………………………………………………………………417 Weak Galerkin Finite Element Method for Convection-Diffusion-Reaction Problems, F. Z. Gao, A. K. Hashim, and S. C. Mohammed,…………………………………………………………425 The Generalized Moment Problem on White Noise Spaces, A. S. Okb El Bab and Hossam A. Ghany,………………………………………………………………………………………….437 Quadratic Type Functional Inclusions on Square-Symmetric Groupoids and Hyers-Ulam Stability, Gwang Hui Kim and Hwan-Yong Shin,…………………………………………….447 Explicit Identities Involving r-Bell Polynomials, Cheon Seoung Ryoo,………………………457 A Class Involving Derivatives of Ratio of the Analytic Functions, Ji Hyang Park, Virendra Kumar, and Nak Eun Cho,……………………………………………………………………..463 Explicit Formulae of Cauchy Polynomials with a q Parameter in Terms of r-Whitney Numbers, F. A. Shiha,……………………………………………………………………………………..475 Global Dynamics of Chikungunya Virus with Two Routes of Infection, A. M. Elaiw, S. E. Almalki, and A. Hobiny,……………………………………………………………………….481 Weighted Norm Inequalities of 𝜃𝜃-Type Calderón–Zygmund Operators and Commutators on 𝜆𝜆 Central Morrey Space, Yanqi Yang and Shuangping Tao,…………………………………….491 Stability of Latent CHIKV Infection Model with CHIKV-to-Monocyte and Infected-toMonocyte Transmissions, A. M. Elaiw, S. E. Almalki, and A. Hobiny,……………………….502 Optimal Bounds for Toader Mean in Terms of Geometric and Contraharmonic Means, Wei-Mao Qian, Wen Zhang, and Yu-Ming Chu,………………………………………………………… 514

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 3, 2020 (continued) Optimal Bounds for Toader-Qi Mean with Applications, Wen-Mao Qian, Wen Zhang, and YuMing Chu,……………………………………………………………………………………526 Symmetric Identities for Dirichlet-Type Multiple Twisted (h, q)-l-Function and Higher-Order Generalized Twisted (h, q)-Euler polynomials, C. S. Ryoo,………………………………… 537 An Efficient m-Step Levenberg-Marquardt Method for Systems of Nonlinear Equations, Liang Chen and Yanfang Ma,……………………………………………………………………….543 Optimal Bounds for a Toader Type Mean Using Arithmetic and Geometric Means, Wei-Mao Qian, Wen Zhang, and Yu-Ming Chu,…………………………………………………….….560 Addition Theorem For Exton's q-Exponential Functions, Mahmoud Jafari Shah Belaghi and Nuri Kuruoglu,………………………………………………………………………………………567

Volume 28, Number 4 ISSN:1521-1398 PRINT,1572-9206 ONLINE

August 2020

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg Lotharstr.65, D-47048 Duisburg, Germany e-mail:[email protected] Fourier Analysis, Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

Xiao-Jun Yang State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China Local Fractional Calculus and Applications, Fractional Calculus and Applications, General Fractional Calculus and Applications, Variable-order Calculus and Applications, Viscoelasticity and Computational methods for Mathematical [email protected]

Jessada Tariboon Department of Mathematics, King Mongkut's University of Technology N. Bangkok 1518 Pracharat 1 Rd., Wongsawang, Bangsue, Bangkok, Thailand 10800 [email protected], Time scales, Differential/Difference Equations, Fractional Differential Equations

Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310 USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.4, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

HERMITE-HADAMARD TYPE INEQUALITIES INVOLVING CONFORMABLE FRACTIONAL INTEGRALS∗ YOUSAF KHURSHID1,2 , MUHAMMAD ADIL KHAN2 , AND YU-MING CHU3,∗∗

Abstract. In the article, we establish an identity and several new HermiteHadamard type inequalities for conformable fractional integrals. As applications, we provide some inequalities for certain bivariate means and present the error estimations for the trapezoidal formula. The given results are the generalization of previously results.

1. Introduction A real-valued function f : I → R is said to be convex if the inequality (1.1)

f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y)

holds for all x, y ∈ I and λ ∈ [0, 1]. If inequality (1.1) holds in the reverse direction, then we say that f is a concave function on the interval I. The word “convexity” is one of the most important, natural and fundamental notations in mathematics. Convex functions were presented by Johan Jensen over 100 years ago. Over the past few years, many generalizations and extensions have been made for convexity. These extensions and generalizations in the theory of inequalities have made valuable contributions in many areas of mathematics. Some new generalized concepts in this point of view are quasi-convex [1], strongly convex [2], approximately convex [3], logarithmically convex [4], midconvex functions [5], pseudo-convex [6], ϕ-convex [7], λ-convex [8], h-convex [9], delta-convex [10], Schur convex [11-17] and and others [18-24]. Let I ⊆ R be an interval and h : I ⊆ R → R be a convex function. Then the well-known Hermite-Hadamard inequality [25-33] for convex functions states that the double inequality   Z a2 a1 + a2 1 h(a1 ) + h(a2 ) (1.2) h ≤ h(x)dx ≤ 2 a2 − a1 a1 2 holds for all a1 , a2 ∈ I with a1 6= a2 . If the function h is concave on I, then both the inequalities in (1.2) hold in the reverse direction. It gives an estimate from both sides of the mean value of a convex function and also ensure the integrability of convex function. It is also a matter of great interest and one has to note that some 2010 Mathematics Subject Classification. Primary: 26D15, 26A33; Secondary: 26A51, 26A42. Key words and phrases. Hermite-Hadamard inequality, convex function, conformable fractional integral, mean, midpoint formula. ∗ The research was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485) and the Science and Technology Research Program of Zhejiang Educational Committee (Grant no. Y201635325). ∗∗ Corresponding author: Yu-Ming Chu, Email: [email protected]. 1

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of the classical inequalities for means can be obtained from Hadamard’s inequality under the utility of peculiar convex functions h. These inequalities for convex functions play a crucial role in analysis and as well as in other areas of pure and applied mathematics. In the last 60 years, many efforts have gone on generalizations extensions and variants of Hermite-Hadamard’s inequality (see [34-36]). Recently, the authors in [37] defined the conformable fractional derivative as follows: for a function h : [0, ∞) → R the (conformable) fractional derivative of order 0 < α ≤ 1 of h at s > 0 was defined by h(s + s1−α ) − f (s) , →0 

Dα (h)(s) = lim

if the conformable fractional derivative of h of order α exists, then we say that h is α-differentiable. The fractional derivative at 0 is defined as hα (0) = lims→0+ hα (s). Now we recall some results for the conformable fractional derivative. Theorem 1.1. Let α ∈ (0, 1] and h1 , h2 be α-differentiable at a point s > 0. Then dα n (s ) = nsn−α dα s for all n ∈ R; dα (c) = 0 dα s for all constant c ∈ R; dα dα dα (a1 h1 (s) + a2 h2 (s)) = a1 (h1 (s)) + a2 (h2 (s)) dα s dα s dα s for all constants a1 , a2 ∈ R; dα dα dα (h1 (s)h2 (s)) = h1 (s) (h2 (s)) + h2 (s) (h1 (s)); dα s dα s dα s dα dα s



h1 (s) h2 (s)

 =

h2 (s) ddααs (h1 (s)) − h1 (s) ddααs (h2 (s)) (h2 (s))2

;

dα dα (h1 (h2 )(s)) = h01 (h2 (s)) (h2 (s)) dα s dα s if h1 differentiable at h2 (s). If in addition h1 is differentiable, then one has d dα (h1 (s)) = s1−α (h1 (s)). dα s ds Definition 1.2. (Conformable fractional integral) Let α ∈ (0, 1] and 0 ≤ a1 < a2 . Then the function h1 : [a1 , a2 ] → R is said to be α-fractional integrable on [a1 , a2 ] if the integral Z a2 Z a2 h1 (x)dα x := h1 (x)xα−1 dx a1

a1

exists and is finite. All α-fractional integrable functions on [a1 , a2 ] is indicated by L1α ([a1 , a2 ]).

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Remark 1. Let α ∈ (0, 1]. Then Iαa1 (h1 )(s) = I1a1 (sα−1 h1 ) =

Z

s

a1

h1 (x) dx, x1−α

where the integral is the usual Riemann improper integral. Anderson [38] established the conformable integral version of Hermite-Hadamard inequality as follows: Theorem 1.3. If α ∈ (0, 1] and h : [a1 , a2 ] → R is an α-fractional differentiable function such that Dα h is increasing, then one has Z a2 h(a1 ) + h(a2 ) α h(x)dα x ≤ . α − a aα 2 1 a1 2 Moreover, if the function h is decreasing on [a1 , a2 ], then we have   Z a2 α a1 + a2 ≤ α h h(x)dα x. 2 a2 − aα 1 a1 In particular, if α = 1, then this reduces to the classical Hermite-Hadamard inequality. Due to the great importance of Hermite-Hadamard inequality, in recent years many mathematician have shown their interest for generalizations, extensions and variants for this inequality. In the article, we deal with the conformable integral version of Hermite-Hadamard inequality investigated by Anderson [38]. We shall establish an identity for the left side of the inequality and discuss their particular case. By applying Jensen’s inequality, power mean inequality and the convexity of the functions xα−1 and −xα (x > 0, α ∈ (0, 1]) in the identity, we obtain inequalities for conformable integral version of Hermite-Hadamard inequality. By using particular classes of convex functions we find new inequalities for special bivariate means. We also apply the results for error estimations of the mid point formula, for some related results (see [39-42]). 2. Main Results We begin this section with the following Lemma 2.1, which is needed for the establishment of our main results. Lemma 2.1. Let α ∈ (0, 1], a1 , a2 ∈ R+ with a1 < a2 and h : [a1 , a2 ] → R be an α-fractional differentiable function. Then the identity   Z a2 a1 + a2 α (2.1) h − α h(x)dα x 2 a2 − aα 1 a1 " Z  2α−1  α−1  1 2−s (a2 − a1 ) sa2 sa2 α 2−s = a1 + − a1 a1 + α 2(aα 2 2 2 2 0 2 − a1 )   2α−1 Z 1  2−s sa2 1−α 1−s 1+s ×Dα (h) a1 + s dα s + a1 + a2 2 2 2 2 0 #  α−1    1+s 1−s 1+s α 1−s 1−α −a2 a1 + a2 × Dα (h) a1 + a2 s dα s . 2 2 2 2 holds if Dα (h) ∈ L1α ([a1 , a2 ]).

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Proof. Integrating by parts, we have 2α−1  α−1    Z 1  2−s sa2 sa2 2−s sa2 α 2−s −a1 a1 + a1 + ×Dα (h) a1 + ds 2 2 2 2 2 2 0  2α−1 α−1  1−s 1+s 1+s 1−s − aα a1 + a2 a + a 1 2 2 2 2 2 2 0   1−s 1+s ×Dα (h) a1 + a2 ds 2 2 α    Z 1  2−s sa2 2−s sa2 0 = a1 + − aα h a + ds 1 1 2 2 2 2 0 α    Z 1  1−s 1+s 1−s 1+s α 0 + a1 + a2 − a2 h a1 + a2 ds 2 2 2 2 0
 α  2−s sa2 1 sa2 2−s α h 2 a1 + 2 a1 + = − a1 a2 −a1 2 2 0 2
α−1   Z 1  sa2 2−s sa2 a2 − a1 h 2−s 2 a1 + 2 − α a1 + ds a2 −a1 2 2 2 0 2
1  α  1−s 1+s 1−s 1+s α h 2 a1 + 2 a2 + a1 + a2 − a2 a2 −a1 2 2 0 2
 α−1  Z 1  1+s 1−s 1+s 1−s a2 − a1 h 2 a1 + 2 a2 − a1 + a2 ds α a2 −a1 2 2 2 0 2 " # α    Z a1 +a 2 2 a1 + a2 2 a1 + a2 α = − a1 h −α h(s)dα s a2 − a1 2 2 a Z

1



+

2 + a2 − a1

"

aα 2 −



a1 + a2 2

α 2(aα 2 − a1 ) h = a2 − a1



# α    Z a2 a1 + a2 h −α h(s)dα s a1 +a2 2 2

a1 + a2 2



2α − a2 − a1

Z

a2

h(x)dα x, a1

where we have used the change of variable x = (1 − s)a1 + sa2 and then multiplying a2 −a1 both sides by 2(a  α −aα ) to get the desired result in (2.1). 2

1

Remark 2. By putting α = 1 in (2.1), we get the identity   Z a2 a1 + a2 1 h − h(x)dx 2 a2 − a1 a1 a2 − a1 = 4

" Z1

0

sh



   # Z1 sa2 2−s 1−s 1+s 0 + a1 ds − (1 − s)h a1 + a2 ds . 2 2 2 2

0

0

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Theorem 2.2. Let α ∈ (0, 1], a1 , a2 ∈ R+ with a1 < a2 and h : [a1 , a2 ] → R be an α-differentiable function on (a1 , a2 ). Then the inequality   Z a2 a1 + a2 α h(x)dα x (2.2) − α α h 2 a2 − a1 a1 " a2 − a1 |h0 (a1 )| |h0 (a2 )| α α α ≤ [13a − 19a ] + [19aα 2 1 2 − 21a1 ] α 2(aα 96 96 2 − a1 )  0   # 0 11|h0 (a1 )| + 5|h0 (a2 )| α−1 α−1 α α−1 2|h (a1 )| + |h (a2 )| −a1 a2 + (a1 a2 + a1 a2 ) 12 96 holds if Dα (h) ∈ L1α ([a1 , a2 ]) and |h0 | is convex on [a1 , a2 ]. Proof. Let ϕ1 = xα−1 and ϕ2 = −xα (x > 0, α ∈ (0, 1]). Then we clearly see that the functions ϕ1 and ϕ2 are convex. Now using Lemma 2.1 and the convexity of ϕ1 , ϕ2 and |h0 |, we have   Z a2 α h a 1 + a 2 − h(x)dα x α 2 aα − a 2 1 a1 " Z  α    1 2−s sa2 2−s sa2 a2 − a1 α 0 ds a1 + − a1 h a1 + ≤ α 2(aα 2 2 2 2 0 2 − a1 )  α    # Z 1 0 1−s 1−s 1+s 1 + s α h + a2 − a1 + a2 a1 + a2 ds 2 2 2 2 0 " Z      α+1−1 1 a2 − a1 sa2 2−s sa2 2−s α 0 = a + a + ds − a h 1 1 1 α 2(aα 2 2 2 2 0 2 − a1 )  α    # Z 1 1 − s 1 + s 1 − s 1 + s h0 + aα a1 + a2 a1 + a2 ds 2 − 2 2 2 2 0 " Z   α−1     1 a2 − a1 s s 2−s sa2 2−s 2−s α 0 ds ≤ a1 + a2 a1 + a2 −a1 h a1 + α 2(aα 2 2 2 2 2 2 0 2 − a1 )     # Z 1 0 1−s 1−s α 1+s α 1+s α + a2 − a1 + a2 h a1 + a2 ds 2 2 2 2 0 " Z       1 a2 − a1 sa2 sa2 2 − s α−1 s α−1 2−s 2−s α 0 ≤ a + a a + a + −a h ds 1 1 1 α 2(aα 2 1 2 2 2 2 2 2 0 2 − a1 )    #  Z 1 1 + s 1 + s 1 − s 1 − s h0 aα aα a1 + a2 ds + aα 1 + 2 2 − 2 2 2 2 0 " Z    " 1 2 − s α−1 s α−1 2−s sa2 2−s 0 b−a α a1 + a2 a1 + − a1 |h (a1 )| ≤ α α 2(a2 − a1 ) 0 2 2 2 2 2 #     # Z 1 s 0 1−s α 1+s α 1−s 0 1+s 0 α + |h (a2 )| ds+ a2 − a + a |h (a1 )| + |h (a2 )| ds . 2 2 1 2 2 2 2 0 Evaluating all the above integrals, we have the following " Z    " 1 2 − s α−1 s α−1 2−s sa2 2−s 0 a2 − a1 α a1 + a2 a1 + − a1 |h (a1 )| α α 2(a2 − a1 ) 0 2 2 2 2 2

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#     Z 1 1−s α 1+s α s 0 1−s 0 1+s 0 α a2 − + |h (a2 )| ds+ a + a |h (a1 )| + |h (a2 ))| ds 2 2 1 2 2 2 2 0 " a2 − a1 15 α 0 11 α−1 7 11 0 = a2 |h0 (a1 )| − aα a1 aα−1 |h0 (a1 )| 1 |h (a1 )| + 2 α ) 32 a1 |h (a1 )| + 96 a1 2(aα − a 12 96 2 1 5 α 0 1 5 α−1 11 1 α−1 0 0 a2 |h (a1 )| − aα a1 a2 |h0 (a2 )| − aα |h0 (a2 )| |h (a1 )| + aα 1 a2 1 |h (a2 )| + 96 6 96 96 6 1 5 1 1 1 1 + a1 aα−1 |h0 (a2 )| − aα aα−1 |h0 (a2 )| + aα |h0 (a1 )| − aα |h0 (a1 )| |h0 (a2 )| + aα 2 96 32 2 12 1 2 4 2 12 1 # 1 α 0 3 α 0 1 α 0 7 α 0 − a2 |h (a1 )| + a2 |h (a2 )| − a1 |h (a2 )| − a2 |h (a2 )| 6 4 6 12 " a2 − a1 |h0 (a1 )| |h0 (a2 )| α α α [13a − 19a ] + [19aα = 2 1 2 − 21a1 ] α 2(aα 96 96 2 − a1 )  0   # 0 0 0 2|h (a )| + |h (a )| 11|h (a )| + 5|h (a )| 1 2 1 2 α−1 −aα + (a1 aα−1 + aα−1 a2 ) . 1 a2 2 1 12 96

+

 Remark 3. By putting α = 1 in (2.2), we obtain the inequality which is proved by Kirmaci in [43]   Z a2 (a2 − a1 )(|h0 (a1 )| + |h0 (a2 )|) 1 h a 1 + a 2 − ≤ h(x)dx . 2 a2 − a1 8 a1

Theorem 2.3. Let q > 1, α ∈ (0, 1], a1 , a2 ∈ R+ with a1 < a2 and h : [a1 , a2 ] → R be an α-differentiable function on (a1 , a2 ). Then the inequality   Z a2 a1 + a2 α (2.3) − α h(x)dα x α h 2 a2 − a1 a1 " 1 (a2 − a1 ) 1− 1 (A1 (α)) q {A2 (α)|h0 (a1 )|q + A3 (α)|h0 (a2 )|q } q ≤ α α 2(a2 − a1 ) # 1− q1

+ (B1 (α))

1

{B2 (α)|h0 (a1 )|q + B3 (α)|h0 (a2 )|q } q

holds if Dα (h) ∈ L1α ([a1 , a2 ]) and |h0 |q is convex on [a1 , a2 ], where   (a1 + a2 )α+1 − (2a1 )α+1 − aα A1 (α) = 1, 2α (α + 1)(a2 − a1 )   (2a2 )α+1 − (a1 + a2 )α+1 α B1 (α) = a2 − , 2α (α + 1)(a2 − a1 )   (a1 + a2 )α+1 (a2 − a1 )(α + 2) + (a1 + a2 ) A2 (α) = α+1 2 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2)   α+1 2a1 (a2 − a1 )(α + 2) + a1 3aα − − 1, (a2 − a1 )(α + 1) (α + 2)(a2 − a1 ) 4   α+1 (a1 + a2 ) (a2 − a1 )(α + 2) + (a1 + a2 ) B2 (α) = α+1 2 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2)

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aα+2 2 2 ) (α +

+

2aα+2 1 2 ) (α + 1)(α

−

aα 2 , (a2 − a1 1)(α + 2) 4   (a1 + a2 )α+1 (a2 − a1 )(α + 2) − (a1 + a2 ) A3 (α) = α+1 2 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2) −

aα 1 , (a2 − a1 + 2) 4   3aα (a1 + a2 )α+1 (a2 − a1 )(α + 2) + (a1 + a2 ) B3 (α) = 2 + α+1 4 2 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2)   α+1 (α + 2)(a2 − a1 ) + a2 2a2 . − (a2 − a1 )(α + 1) (α + 2)(a2 − a1 ) −

Proof. It follows from Lemma 2.1 that   Z a2 α h a 1 + a 2 − h(x)dα x α α 2 a2 − a1 a1 " 2α−1 α−1     (a − a ) Z 1  2 − s sa2 sa2 sa2 2−s 2 1 α 2−s a + a + a + = −a D (h) ds 1 1 1 α 1 α 2(aα 2 2 2 2 2 2 0 2 − a1 ) 2α−1  α−1 #   Z 1  1+s 1 + s 1 + s 1 − s 1 − s 1−s a1 + a2 a1 + a2 a1 + a2 ds + −aα Dα (h) 2 2 2 2 2 2 2 0 " Z      α 1 0 2−s (a2 − a1 ) 2−s sa2 sa2 ds ≤ a + − aα a1 + 1 1 h α α 2(a2 − a1 ) 0 2 2 2 2  α    # Z 1 0 1−s 1−s 1+s 1 + s α h + a2 − a1 + a2 a1 + a2 ds . 2 2 2 2 0 From the power-mean inequality and the convexity |h0 |q we get α    Z 1  2−s sa2 2−s sa2 α 0 ds a1 + − a1 h a1 + 2 2 2 2 0 Z 1  α  1− q1 2−s sa2 α ≤ a1 + − a1 ds 2 2 0  Z 1  α    q  q1 0 2−s sa2 sa2 2−s × a+ h a + ds , − aα 1 1 2 2 2 2 0  α    Z1  0 1−s 1−s 1+s 1+s α a2 − a1 + a2 h a1 + a2 ds 2 2 2 2 0

 1 1− q1  α  Z  1 − s 1 + s ≤ aα a1 + a2 ds 2 − 2 2 0

 q1  α    q Z1  1 − s 1 + s 1 + s 1 − s h0 aα a1 + a2 a1 + a2 ds , × 2 − 2 2 2 2 0 α    q Z 1  2−s sa2 2−s sa2 α 0 a1 + − a1 h a1 + ds 2 2 2 2 0 

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α   2−s sa2 2−s 0 s 0 q q − aα a1 + |h (a )| + |h (a )| ds 1 2 1 2 2 2 2 0 α  Z 1  2−s sa2 2−s − aα = |h0 (a1 )|q a1 + ds 1 2 2 2 0  α Z 1  2−s sa2 α s 0 q − a1 a1 + ds +|h (a2 )| 2 2 2 0   (a1 + a2 )α+1 (a2 − a1 )(α + 2) + (a1 + a2 ) = |h0 (a1 )|q α+1 2 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2) !   2aα+1 (a2 − a1 )(α + 2) + a1 3aα 1 1 − − (a2 − a1 )(α + 1) (α + 2)(a2 − a1 ) 4   (a2 − a1 )(α + 2) − (a1 + a2 ) (a1 + a2 )α+1 0 q +|h (a2 )| 2α+1 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2) ! 2aα+2 aα 1 1 − − , (a2 − a1 )2 (α + 1)(α + 2) 4 Z

1



≤

Z1 

aα 2

 −

1−s 1+s a1 + a2 2 2

α    q 0 1−s 1+s h a1 + a2 ds 2 2

0

  α   Z1  1−s 1+s 1−s 0 1+s 0 q q aα − a + a |h (a )| + |h (a )| ds ≤ 1 2 1 2 2 2 2 2 2 0 0

q

Z1 

= |h (a1 )|

aα 2

 −

1+s 1−s a1 + a2 2 2

α 

1−s ds 2

0 0

q

Z1 

+|h (a2 )|

aα 2

 −

1+s 1−s a1 + a2 2 2

α 

1+s ds 2

0

  (a1 + a2 )α+1 (a2 − a1 )(α + 2) + (a1 + a2 ) 2α+1 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2) ! aα+2 aα 2 2 − + (a2 − a1 )2 (α + 1)(α + 2) 4   3aα (a1 + a2 )α+1 (a2 − a1 )(α + 2) + (a1 + a2 ) 2 + α+1 4 2 (α + 1)(a2 − a1 ) (a2 − a1 )(α + 2)  ! 2aα+1 (α + 2)(a2 − a1 ) + a2 2 − , (a2 − a1 )(α + 1) (α + 2)(a2 − a1 )

= |h0 (a1 )|q

+|h0 (a2 )|q

where we have used the facts that α    Z 1  2−s sa2 (a1 + a2 )α+1 − (2a1 )α+1 α a1 + − a1 ds = − aα 1, 2 2 2α (α + 1)(a2 − a1 ) 0

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Z1 

aα 2

 −

1+s 1−s a1 + a2 2 2

α    (2a2 )α+1 − (a1 + a2 )α+1 α ds = a2 − . 2α (α + 1)(a2 − a1 )

0

Hence, we get the desired inequality (2.3).



Remark 4. Let α = 1. Then inequality (2.3) leads to   Z a2 1 h a 1 + a 2 − h(x)dx 2 a2 − a1 a1  1− q1 " 1 1 a2 − a1 ≤ {A2 (1)|h0 (a1 )|q + A3 (1)|h0 (a2 )|q } q 2 4 # 1

+ {B2 (1)|h0 (a1 )|q + B3 (1)|h0 (a2 )|q } q , where A2 (1) =

(a1 + a2 )2 (4a2 − 2a1 ) − 8a21 (3a2 − 2a1 ) − 18a1 (a2 − a1 )2 , 24(a2 − a1 )2

(a1 + a2 )2 (4a2 − 2a1 ) − 4a32 − 6a2 (a2 − a1 )2 , 24(a2 − a1 )2 (a1 + a2 )2 (2a2 − 4a1 ) − 8a31 − 6a1 (a2 − a1 )2 A3 (1) = , 24(a2 − a1 )2 (a1 + a2 )2 (4a2 − 2a1 ) − 8a22 (4a2 − 3a1 ) + 18a2 (a2 − a1 )2 . B3 (1) = 24(a2 − a1 )2 B2 (1) =

Theorem 2.4. Let q > 1, α ∈ (0, 1], a1 , a2 ∈ R+ with a1 < a2 and h : [a1 , a2 ] → R be an α-differentiable function on (a1 , a2 ). Then the inequality   Z a2 α h a 1 + a 2 − (2.4) h(x)d x α α α 2 a2 − a1 a1 "   #  C (α) (a2 − a1 ) C (α) 2 1 ≤ + B1 (α)h0 A1 (α)h0 α 2(aα A1 (α) B1 (α) 2 − a1 ) holds if Dα (h) ∈ L1α ([a1 , a2 ]) and |h0 |q is concave on [a1 , a2 ], where   (a1 + a2 )α+1 − (2a1 )α+1 A1 (α) = − aα 1, 2α (α + 1)(a2 − a1 )   (2a2 )α+1 − (a1 + a2 )α+1 B1 (α) = aα − , 2 2α (α + 1)(a2 − a1 )   (α + 2) − 1 C1 (α) = (a1 + a2 )α+2 α+1 2 (α + 1)(a2 − a1 )   α+2 2a1 (a2 − a1 )(α + 2) + (a1 + a2 ) aα − + 1 (3a1 + a2 ), 2 (α + 2)(a2 − a1 ) (α + 1) 4   α+2 (a1 + a2 ) (a2 − a1 )(α + 2) + (a1 + a2 ) C2 (α) = α+1 2 (a2 − a1 )(α + 1) (α + 2)(a2 − a1 )   aα 2 α+1 (a1 + a2 ) + 2(α + 2)(a2 − a1 ) −a2 + (a1 + 3a2 ). (α + 2)(a2 − a1 )2 (α + 1) 4

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10

Proof. It follows from [44] and the concavity of |h0 |q that |h0 | is also concave. Making use of Lemma 2.1 and Jensen’s integral inequality we get   Z a2 α h a 1 + a 2 − h(x)d x α α α 2 a2 − a1 a1 " Z  α    a −a 1 2−s sa2 2−s sa2 2 1 α 0 a1 + − a1 h a1 + ds = α 2(aα 2 2 2 2 0 2 − a1 )  α    Z 1 1−s 1+s 1−s 1+s α 0 a2 − a1 + a2 h a1 + a2 ds + 2 2 2 2 0 " Z      α 1 a2 − a1 2−s 2−s sa2 sa2 α 0 ≤ − a1 h ds a1 + a1 + α 2(aα 2 2 2 2 0 2 − a1 )  α    # Z 1 0 1−s 1−s 1+s 1 + s α h + a2 − a1 + a2 a1 + a2 ds , 2 2 2 2 0 α    Z 1  0 2−s 2−s sa2 sa2 ds a1 + − aα h a + 1 1 2 2 2 2 0 Z 1  α  2−s sa2 ≤ a1 + − aα 1 2 2 0 α   R 
 1 sa2 sa2 2−s 2−s α a + − a a + ds  1 1 1 2 2 2 2  0 0  α   h   R1 sa2 2−s α − a1 ds 2 a1 + 2 0   C1 (α) = A1 (α)h0 , A1 (α) α     Z 1 0 1−s 1+s 1+s 1−s a + a h a + a aα − 1 2 1 2 ds 2 2 2 2 2 0  α  Z 1 1−s 1+s α ≤ a2 − a1 + a2 2 2 0     R α
 1 1+s 1−s 1+s 1−s α a + a a + a ds  a − 1 2 1 2 2 2 2 2 2  0      h0  α   R1 α 1+s a2 − 1−s ds 2 a1 + 2 a2 0   C2 (α) = B1 (α)h0 , B1 (α) where we used the facts that α    Z 1  2−s sa2 (a1 + a2 )α+1 − (2a1 )α+1 α − aα a1 + − a1 ds = A1 (α) = 1, 2 2 2α (α + 1)(a2 − a1 ) 0 Z1  0

aα 2

 −

α    1−s 1+s (2a2 )α+1 − (a1 + a2 )α+1 α a1 + a2 ds = B1 (α) = a2 − , 2 2 2α (α + 1)(a2 − a1 ) α   Z 1  2−s sa2 2−s sa2 α a1 + − a1 a1 + ds 2 2 2 2 0

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  (α + 2) − 1 = C1 (α) = (a1 + a2 )α+2 α+1 2 (α + 1)(a2 − a1 )   α+2 2a1 (a2 − a1 )(α + 2) + (a1 + a2 ) aα − + 1 (3a1 + a2 ) 2 (α + 2)(a2 − a1 ) (α + 1) 4 and

α      1+s 1−s 1+s 1−s α a1 + a2 a1 + a2 ds a2 − 2 2 2 2 0   (a1 + a2 )α+2 (a2 − a1 )(α + 2) + (a1 + a2 ) = C2 (α) = α+1 2 (a2 − a1 )(α + 1) (α + 2)(a2 − a1 )   aα (a1 + a2 ) + 2(α + 2)(a2 − a1 ) + 2 (a1 + 3a2 ). −aα+1 2 2 (α + 2)(a2 − a1 ) (α + 1) 4 Z

1

 Remark 5. Let α = 1. Then inequality (2.4) becomes   Z a2 1 h a1 + a2 − h(x)dx 2 a2 − a1 a1 "   (a2 − a1 ) 0 (a1 + a2 )3 (a2 − a1 ) − 2a31 (4a2 − 2a1 ) + 3a1 (3a1 + a2 )(a2 − a1 )2 ≤ h 8 3(a2 − a1 )  # (a1 + a2 )3 (a2 − a1 )(2a2 − a1 ) − 2a22 (7a2 − 5a1 ) + 3a2 (a1 + 3a2 )(a2 − a1 )2 0 +h . 3(a2 − a1 ) Theorem 2.5. Let q > 1, α ∈ (0, 1], a1 , a2 ∈ R+ with a1 < a2 and h : [a1 , a2 ] → R be an α-differentiable function. Then the inequality   Z a2 a1 + a2 α − α (2.5) h(x)dα x h 2 a2 − aα 1 a1 "    # (a2 − a1 ) F1 (α) F2 (α) 0 0 ≤ D1 (α)h + E1 (α)h α 2(aα D1 (α) E1 (α) 2 − a1 ) holds if Dα (h) ∈ L1α ([a1 , a2 ]) and |h0 |q is concave on [a1 , a2 ], where α−1 −5aα a2 + 2aα−1 a1 + aα 1 + 2a1 2 2 , 12 aα − aα 1 E1 (α) = 2 , 4 α−1 2 α+1 α−1 2 −27a1 − 2aα + 5a1 aα a2 + 3aα+1 2 + 5a1 1 a2 + 11a1 a2 2 F1 (α) = , 96 α+1 a1 aα + 2aα+1 − 2aα 2 − a1 1 a2 2 . F2 (α) = 12 Proof. It follows from [44] and the concavity of |h0 |q that |h0 | is also concave. Making use of Lemma 2.1 and Jensen’s integral inequality one has   Z a2 α h a 1 + a 2 − h(x)d x α α α 2 a2 − a1 a1 " Z  α    1 a2 − a1 2−s sa2 2−s sa2 α 0 ≤ a1 + − a1 h a1 + ds α 2(aα 2 2 2 2 0 2 − a1 )

D1 (α) =

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YOUSAF KHURSHID1,2 , MUHAMMAD ADIL KHAN2 , AND YU-MING CHU3,∗∗

  α    # 0 1−s 1−s 1+s 1+s α a2 − h + a1 + a2 a1 + a2 ds 2 2 2 2 0 " Z      1 2 − s α saα a2 − a1 2−s sa2 2 α 0 ≤ h ds a + − a a + 1 1 α 2(aα 2 1 2 2 2 0 2 − a1 )   #   Z 1 0 1−s 1−s α 1+s α 1+s α + a2 − a1 + a2 h a1 + a2 ds 2 2 2 2 0 " Z       1 a2 − a1 2 − s α−1 s α−1 2−s sa2 sa2 2−s α 0 ≤ h ds a a a + −a a + + 1 1 1 α 2(aα 2 1 2 2 2 2 2 2 0 2 − a1 )     # Z 1 1 − s 1 + s 1 − s 1 + s h0 + aα aα aα a1 + a2 ds , 2 − 1 + 2 2 2 2 2 0      Z 1  2−s sa2 2−s sa2 2 − s α−1 s α−1 α 0 ds a + a2 a1 + − a1 h a1 + 2 1 2 2 2 2 2 0 Z 1     2−s sa2 2 − s α−1 s α−1 ≤ a1 + a2 a1 + − aα 1 2 2 2 2 0     R 
 1 sa2 sa2 2−s α−1 s α−1 2−s 2−s α a + a a + − a a + ds  1 1 1 1 2 2 2 2 2 2 2  0      h0    R1 sa2 s α−1 2−s 2−s α−1 α a + a a + − a ds 1 1 2 1 2 2 2 2 0   F1 (α) = D1 (α)h0 , D1 (α)     Z 1 0 1−s 1−s α 1+s α 1+s α h a2 − a1 + a2 a1 + a2 ds 2 2 2 2 0   Z 1 1 − s 1 + s ≤ aα + aα aα 2 − 2 1 2 2 0    R 
 1 1+s α 1+s α 1−s α 1−s α α a + a a + a a − 2 1 2 1 2 ds  2 2 2 2  0     h0    R1 α 1−s α 1+s α a − a + a ds 2 1 2 2 2 0   F2 (α) 0 , = E1 (α)f E1 (α) where we have used the facts that    Z 1  2 − s α−1 s α−1 2−s sa2 a1 + a2 a1 + − aα 1 ds 2 2 2 2 0 Z

1

α−1 −5aα a2 + 2aα−1 a1 + aα 1 + 2a1 2 2 , 12   Z1  1−s α 1+s α aα − aα 1 α a2 − a1 + a2 ds = E1 (α) = 2 , 2 2 4 0     Z 1  2 − s α−1 s α−1 2−s sa2 2−s sa2 α a + a2 a1 + − a1 a1 + ds = F1 (α) 2 1 2 2 2 2 2 0

= D1 (α) =

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13 α−1 2 2 α−1 −27aα+1 − 2aα + 5a1 aα a2 + 3aα+1 1 a2 + 11a1 a2 2 + 5a1 1 2 96     Z 1  1−s α 1+s α 1−s 1+s aα − a + a a + a ds 1 2 2 2 1 2 2 2 2 0

=

and

= F2 (α) =

α+1 a1 aα + 2aα+1 − 2aα 2 − a1 1 a2 2 . 12

 Remark 6. Let α = 1. Then inequality (2.5) leads to   Za2 1 a1 + a2 h(x)dx − (2.6) h 2 a2 − a1 a1

"

≤

a2 − a1 0 h 8



   # a2 + 2a1 0 a1 + 2a2 + h . 3 3

Note that inequality (2.6) is an improvement of the inequality obtained by Pearce and Peˇcari´c in [44] due to |h0 | is concave on [a1 , a2 ] and "     # a2 − a1 0 a2 + 2a1 0 a1 + 2a2 h + h 8 3 3 "     #   a2 − a1 1 0 a2 + 2a1 1 0 a1 + 2a2 a2 − a1 0 a1 + a2 = h + h ≤ h . 4 2 3 2 3 4 2

3. Applications to Special Bivariate Means Let a, b > 0 with a 6= b. Then the arithmetic mean A(a, b) [45-50], logarithmic mean L(a, b) [51-55] and (α, r)-th generalized logarithmic mean L(α,r) (a, b) [56-59] are defined by  1/r a+b b−a α(br+α − ar+α ) A(a, b) = , L(a, b) = , L(α,r) (a, b) = , 2 log b − log a (r + α)(bα − aα ) respectively. Recently, the bivariate means have been the subject of intensive research [60-74] and many remarkable inequalities for the bivariate means and related special functions can be found in the literature [75-97]. Making use of Theorems 2.2 and 2.3 together with the convexity of the functions xr and 1/x (x > 0) we get some new inequalities for the arithmetic, logarithmic and generalized means immediately. Theorem 3.1. Let a1 , a2 ∈ R+ with a1 < a2 . Then the inequality r (3.1) A (a1 , a2 ) − Lr(α,r) (a1 , a2 ) " |a2 |r−1 r(a2 − a1 ) |a1 |r−1 α α α [13a − 19a ] + [19aα ≤ 2 1 2 − 21a1 ] α 2(aα 96 96 2 − a1 )   2|a1 |r−1 + |a2 |r−1 α α−1 −a1 a2 12

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+(a1 aα−1 2

+

aα−1 a2 ) 1



11|a1 |r−1 + 5|a2 |r−1 192

#

holds for all r > 1 and α ∈ (0, 1]. Remark 7. Let α = 1. Then inequality (3.1) leads to |Ar (a1 , a2 ) − Lrr (a1 , a2 )| ≤

r(a2 − a1 ) A(|a1 |r−1 , |a2 |r−1 ), 4

which was proved by Kirmaci in [43]. Theorem 3.2. Let a1 , a2 ∈ R+ with a1 < a2 and r > 1. Then the inequality r A (a1 , a2 ) − Lr(α,r) (a1 , a2 )) " n o q1 r(a2 − a1 ) 1− 1 ≤ (A1 (α)) q A2 (α)|a1 |(r−1)q + A3 (α)|a2 |(r−1)q α α 2(a2 − a1 ) # n o1 1− q1 (r−1)q (r−1)q q + (B1 (α)) B2 (α)|a1 | + B3 (α)|a2 | holds for all q > 1 and α ∈ (0, 1]. Theorem 3.3. Let a1 , a2 ∈ R+ with a1 < a2 . The the inequality r (3.2) A (a1 , a2 ) − Lr(α,r) (a1 , a2 ) " (a2 − a1 ) |a1 |−2 |a2 |−2 α ≤ [13aα [19bα − 21aα 2 − 19a1 ] + 1] α α 2(a2 − a1 ) 96 96 α−1 −aα 1 a2



2|a1 |−2 + |a2 |−2 12

 +

(a1 aα−1 2

+

aα−1 a2 ) 1



11|a1 |−2 + 5|a2 |−2 192

#

holds for all α ∈ (0, 1]. Remark 8. Let α = 1. Then inequality (3.2) reduces to |Ar (a1 , a2 ) − Lrr (a1 , a2 )| ≤

(a2 − a1 ) A(|a1 |−2 , |a2 |−2 ), 4

which was proved by Kirmaci in [43]. Theorem 3.4. Let a1 , a2 ∈ R+ with a1 < a2 . Then the inequality r A (a1 , a2 ) − Lr(α,r) (a1 , a2 )) " 1 (a2 − a1 ) 1− 1  (A1 (α)) q A2 (α)|a1 |−2q + A3 (α)|a2 |−2q q ≤ 2(bα − aα ) 1 # 1 1− q1  + (B1 (α)) B2 (α)|a1 |−2q + B3 (α)|a2 |−2q q holds for all q > 1 and α ∈ (0, 1].

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4. Applications to Mid-Point Formula Let P be the partition of the points a1 = y0 < y1 < ... < yn−1 < yn = a2 of the interval [a1 , a2 ] and consider the quadrature formula Z b h(x)dα x = Tα (h, P ) + Eα (h, P ), a

where Tα (h, P ) =

n−1 X i=0

 h

yi + yi+1 2




α yi+1 − yiα , α

is the midpoint version and Eα (h, P ) denotes the associated approximation error. In this section, we shall present some new estimates for the midpoint formula. Theorem 4.1. Let α ∈ (0, 1], a1 , a2 ∈ R+ with a1 < a2 and h : [a1 , a2 ] → R be an α-differentiable function. Then the inequality " n−1 X (yi+1 − yi ) |h0 (yi )|   |h0 (yi+1 )|   α α 13yi+1 − 19yiα + 19yi+1 − 21yiα |Eα (h, P )| ≤ 2α 96 96 i=0  0   # 0 11|h0 (yi )| + 5|h0 (yi+1 )| α−1 α−1 α α−1 2|h (yi )| + |h (yi+1 )| −yi yi+1 + (yi yi+1 + yi yi+1 ) 12 12 holds if Dα (h) ∈ L1α ([a1 , a2 ]) and |h0 | is convex on [a1 , a2 ]. Proof. Applying Theorem 2.2 on the subinterval [yi , yi+1 ] (i = 0, 1, ..., n − 1) of the partition P , we have   α Z yi+1 α h yi + yi+1 (yi+1 − yi ) − h(x)d x α 2 α yi "  |h0 (yi+1 )|   (yi+1 − yi ) |h0 (yi )|  α α ≤ 13yi+1 − 19yiα + 19yi+1 − 21yiα 2α 96 96   #  0 0 11|h0 (yi )| + 5|h0 (yi+1 )| α−1 α−1 α α−1 2|h (yi )| + |h (yi+1 )| +(yi yi+1 +yi yi+1 ) , −yi yi+1 12 12 Z a2 h(x)dα x − Tα (h, P ) a1

n−1 (Z
)   α X yi+1 yi+1 − yiα yi + yi+1 = h(x)dα x − h 2 α yi i=0 (Z )
  α n−1 yi+1 X yi+1 − yiα yi + yi+1 ≤ h(x)dα x − h yi 2 α i=0 " n−1 X (yi+1 − yi ) |h0 (yi )|   |h0 (yi+1 )|   α α ≤ 13yi+1 − 19yiα + 19yi+1 − 21yiα 2α 96 96 i=0  0   # 0 11|h0 (yi )| + 5|h0 (yi+1 )| α−1 α−1 α α−1 2|h (yi )| + |h (yi+1 )| −yi yi+1 +(yi yi+1 +yi yi+1 ) . 12 12 

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Theorem 4.2. Let q > 1, α ∈ (0, 1], a1 , a2 ∈ R+ with a1 < a2 and h : [a1 , a2 ] → R be an α-differentiable function. Then the inequality " n−1 X (yi+1 − yi ) 1 1− 1 (A1 (α)) q {A2 (α)|h0 (yi )|q + A3 (α)|h0 (yi+1 )|q } q |Eα (h, P )| ≤ 2α i=0 # 1− q1

+ (B1 (α))

1

{B2 (α)|h0 (yi )|q + B3 (α)|h0 (yi+1 )|q } q

holds if Dα (h) ∈ L1α ([a1 , a2 ]) and |h0 |q is convex on [a1 , a2 ]. Proof. The proof is analogous to that of Theorem 4.1 only by using Theorem 2.3.  References [1] M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex, Tamkang J. Math., 2010, 41(4), 353–359. [2] S. S. Dragomir and K. Nikodem, Jensen’s and Hermite-Hadamard’s type inequalities for lower and strongly convex functions on normed spaces, Bull. Iranian Math. Soc., 2018, 44(5), 1337–1349. [3] J. Mak´ o and Z. P´ ales, Approximate Hermite-Hadamard type inequalities for approximately convex functions, Math. Inequal. Appl., 2013, 16(2), 507–526. [4] J. Park, Some Hermite-Hadamard-like type inequalities for logarithmically convex functions, Int. J. Math. Anal., 2013, 7(45-48), 2217–2233. [5] P. Spurek and Ja. Tabor, Numerical verification of condition for approximately midconvex functions, Aequationes Math., 2012, 83(3), 223–237. [6] P. Mereau and J.-G. Paquet, Second order conditions for pseudo-convex functions, SIAM J. Anal. Math., 1974, 27, 131–137. [7] S. S. Dragomir, Inequalities of Hermite-Hadamard type for ϕ-convex functions, Konuralp J. Math., 2016, 4(1), 54–67. [8] M. Adamek, A characterization of λ-convex functions, JIPAM. J. Inequal. Pure Appl. Math., 2004, 5(3), Article 71, 5 pages. [9] M. Bombardelli and S. Varoˇsanec, Properties of h-convex functions related to the HermiteHadamard-Fej´ er inequalities, Comput. Math. Appl., 2009, 58(9), 1869–1877. [10] T. Rajba, On strong delta-convexity and Hermite-Hadamard type inequalities for deltaconvex functions of higher order, Math. Inequal. Appl., 2015, 18(1), 267–293. [11] Y.-M. Chu and T.-C. Sun, The Schur harmonic convexity for a class of symmetric functions, Acta Math. Sci., 2010, 30B(5), 1501–1506. [12] Y.-M. Chu, W.-F. Xia and T.-H. Zhao, Schur convexity for a class of symmetric functions, Sci. China Math., 2010, 53(2), 465–474. [13] W.-F. Xia and Y.-M. Chu, On Schur convexity of some symmetric functions, J. Inequal. Appl., 2010, 2010, Article ID 543250, 12 pages. [14] W.-F. Xia and Y.-M. Chu, Necessary and sufficient conditions for the Schur harmonic convexity or concavity of the extended mean values, Rev. Un. Mat. Argentina, 2011, 52(1), 121–132. [15] W.-F. Xia and Y.-M. Chu, The Schur convexity of Gini mean values in the sense of harmonic mean, Acta Math. Sci., 2011, 31B(3), 1103–1112. [16] Y.-M. Chu, W.-F. Xia and X.-H. Zhang, The schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications, J. Multivariate Anal., 2012, 105, 412–421. [17] Y.-M. Chu, G.-D. Wang and X.-H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function, Math. Nachr., 2011, 284(5-6), 653–663. [18] W.-D. Jiang, M.-K. Wang, Y.-M. Chu, Y.-P. Jiang and F. Qi, Convexity of the generalized sine function and the generalized hyperbolic sine function, J. Approx. Theory, 2013, 174, 1–9.

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[43] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 2004, 147(1), 137–146. [44] C. E. M. Pearce and J. E. Peˇ cari´ c, Inequalities for diffrentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 2000, 13(2), 51–55. [45] H.-Z. Xu, Y.-M. Chu and W.-M. Qian, Sharp bounds for the S´ andor-Yang means in terms of arithmetic and contra-harmonic means, J. Inequal. Appl., 2018, 2018, Article 127, 13 pages. [46] W.-F. Xia, W. Janous and Y.-M. Chu, The optimal convex combination bounds of arithmetic and harmonic means in terms of power mean, J. Math. Inequal., 2012, 6(2), 241–248. [47] W.-M. Gong, Y.-Q. Song, M.-K. Wang and Y.-M. Chu, A sharp double inequality between Seiffert, arithmetic, and geometric means, Abstr. Appl. Anal., 2012, 2012, Article ID 684834, 7 pages. [48] Y.-M. Chu and M.-K. Wang, Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal., 2012, 2012, Article ID 830585, 11 pages. [49] Y.-M. Chu, Y.-F. Qiu, M.-K. Wang and G.-D. Wang, The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert’s mean, J. Inequal. Appl., 2010, 2010, Article ID 436457, 7 pages. [50] W.-F. Xia, Y.-M. Chu and G.-D. Wang, The optimal upper and lower power mean bounds for a convex combiantion of the arithmetic and logarithmic means, Abstr. Appl. Anal., 2010, 2010, Article ID 604804, 9 pages. [51] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means, Math. Inequal. Appl., 2012, 15(2), 415–422. [52] Y.-M. Chu and M.-K. Wang, Optimal inequalities betwen harmonic, geometric, logarithmic, and arithmetic-geometric means, J. Appl. Math., 2011, 2011, Article ID 618929, 9 pages. [53] Y.-M. Chu, S.-S. Wang and C. Zong, Optimal lower power mean bound for the convex combination of harmonic and logarithmic means, Abstr. Appl. Anal., 2011, 2011, Article ID 520648, 9 pages. [54] Y.-F. Qiu, M.-K. Wang, Y.-M. Chu and G.-D. Wang, Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, J. Math. Inequal., 2011, 5(3), 301–306. [55] Y.-M. Chu and W.-F. Xia, Two double inequalities between power mean and logarithmic mean, Comput. Math. Appl., 2010, 60(1), 83–89. [56] Y.-M. Chu and B.-Y. Long, Best possible inequalities between generalized logarithmic mean and classical means, Abstr. Appl. Anal., 2010, 2010, Article ID 303286, 13 pages. [57] W.-M. Qian and Y.-M. Chu, Best possible bounds for Yang mean using generalized logarithmic mean, Math. Probl. Eng., 2016, 2016, Article ID 8901258, 7 pages. [58] Y.-M. Li, B.-Y. Long and Y.-M. Chu, Sharp bounds for the Neuman-S´ andor mean in terms of generalized logarithmic mean, J. Math. Inequal., 2012, 6(4), 567–577. [59] Y.-M. Chu, M.-K. Wang and G.-D. Wang, The optimal generalized logarithmic mean bounds for Seiffert’s mean, Acta Math. Sci., 2012, 32B(4), 1619–1626. [60] Y.-M. Chu, Y.-M. Li, W.-F. Xia and X.-H. Zhang, Best possible inequalities for the harmonic mean of error function, J. Inequal. Appl., 2014, 2014, Article 525, 9 pages. [61] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, A best-possible double inequality between Seiffert and harmonic means, J. Inequal. Appl., 2011, 2011, Article 94, 7 pages. [62] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, A sharp double inequality between harmonic and identric means, Abstr. Appl. Anal., 2011, 2011, Article ID 657935, 7 pages. [63] Y.-M. Chu, M.-K. Wang and S.-L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci., 2012, 122(1), 41–51. [64] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the arithmeticgeometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl., 2018, 462(2), 1714–1726. [65] W.-M. Qian and Y.-M. Chu, Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters, J. Inequal. Appl., 2017, 2017, Article 274, 10 pages. [66] Y.-M. Chu and M.-K. Wang, Optimal Lehmer mean bounds for the Toader mean, Results Math., 2012, 61(3-4), 223–229. [67] M.-K. Wang, Y.-M. Chu, Y.-F. Qiu and S.-L. Qiu, An optimal power mean inequality for the complete elliptic integrals, Appl. Math. Lett., 2011, 24(6), 887–890. [68] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, An optimal double inequality between Seiffert and geoemtric means, J. Appl. Math., 2011, 2011, Article ID 261237, 6 pages.

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[69] Y.-M. Chu, Y.-F. Qiu and M.-K. Wang, Sharp power mean bounds for the combination of Seiffert and geoemtric means, Abstr. Appl. Anal., 2010, 2010, Article ID 108920, 12 pages. [70] B.-Y. Long and Y.-M. Chu, Optimal power mean bounds for the weighted geometric mean of classical means, J. Inequal. Appl., 2010, 2010, Article ID 905697, 6 pages. [71] Y.-M. Chu, B.-Y. Long and B.-Y. Liu, Bounds of the Neuman-S´ andor mean using power and identric means, Abstr. Appl. Anal., 2013, 2013, Article ID 832591, 6 pages. [72] M.-K. Wang, Y.-M. Chu and Y.-F. Qiu, Some comparison inequalities for generalized Muirhead and identric means, J. Inequal. Appl., 2010, 2010, Article ID 295620, 10 pages. [73] M.-K. Wang, Z.-K. Wang and Y.-M. Chu, An double inequality between geometric and identric means, Appl. Math. Lett., 2012, 25(3), 471–475. [74] W.-M. Qian, X.-H. Zhang and Y.-M. Chu, Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means, J. Math. Inequal., 2017, 11(1), 121–127. [75] Zh.-H. Yang, W.-M. Qian and Y.-M. Chu, Monotonicity properties and bounds involving the complete elliptic integrals of the first kind, Math. Inequal. Appl., 2018, 21(4), 1185–1199. [76] T.-H. Zhao, M.-K. Wang, W. Zhang and Y.-M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018, 2018, Article 251, 15 pages. [77] T.-R. Huang, S.-Y. Tan, X.-Y. Ma and Y.-M. Chu, Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl., 2018, 2018, Article 239, 11 pages. [78] M.-K. Wang, S.-L. Qiu and Y.-M. Chu, Infinite series formula for H¨ ubner upper bound function with applications to Hersch-Pfluger distortion function, Math. Inequal. Appl., 2018, 21(3), 629–648. [79] M.-K. Wang and Y.-M. Chu, Landen inequalities for a class of hypergeometric functions with applications, Math. Inequal. Appl., 2018, 21(2), 521–537. [80] Zh.-H. Yang and Y.-M. Chu, A monotonicity property involving the generalized elliptic integral of the first kind, Math. Inequal. Appl., 2017, 20(3), 729–735. [81] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, Monotonicity rule for the quotient of two function and its applications, J. Inequal. Appl., 2017, 2017, Article 106, 13 pages. [82] M.-K. Wang, Y.-M. Chu and Y.-Q. Song, Asymptotical formulas for Gaussian and generalized hypergeometric functions, Appl. Math. Comput., 2016, 276, 44–60. [83] M.-K. Wang, Y.-M. Chu and Y.-P. Jiang, Ramanujan’s cubic transformaation inequalities for zero-balanced hypergeometric functions, Rocky Mountain J. Math., 2016, 46(2), 679–691. [84] Y.-M. Chu, M.-K. Wang and Y.-F. Qiu, On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function, Abstr. Appl. Anal., 2011, 2011, Article ID 697547, 7 pages. [85] M.-K. Wang and Y.-M. Chu, Asymptotical bounds for complete elliptic integrals of the second kind, J. Math. Anal. Appl., 2013, 402(1), 119–126. [86] Y.-M. Chu, Y.-F. Qiu and M.-K. Wang, H¨ older mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct., 2012, 23(7), 521–527. [87] Y.-M. Chu, M.-K. Wang, S.-L. Qiu and Y.-P. Jiang, Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl., 2012, 63(7), 1177–1184. [88] M.-K. Wang, S.-L. Qiu, Y.-M. Chu and Y.-P. Jiang, Generalized Hersch-Pfluger distortion function and complete elliptic integrals, J. Math. Anal. Appl., 2012, 385(1), 221–229. [89] M.-K. Wang, Y.-M. Li and Y.-M. Chu, Inequalities and infinite product formula for Ramanujan generalized modular equation function, Ramanujan J., 2018, 46(1), 189–200. [90] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On rational bounds for the gamma function, J. Inequal. Appl., 2017, 2017, Article 210, 17 pages. [91] T.-H. Zhao and Y.-M. Chu, A class of logarithmically completely monotonic functions associated with gamma function, J. Inequal. Appl., 2010, 2010, Article ID 392431, 11 pages. [92] T.-H. Zhao, Y.-M. Chu and H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011, 2011, Article ID 896483, 13 pages. [93] T.-R. Huang, B.-W. Han, X.-Y. Ma and Y.-M. Chu, Optimal bounds for the generalized Euler-Mascheronic constant, J. Inequal. Appl., 2018, 2018, Article 118, 9 pages. [94] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the error function, Math. Inequal. Appl., 2018, 21(2), 469–479. [95] Y.-Q. Song, P.-G. Zhou and Y.-M. Chu, Inequalities for the Gaussian hypergeometric function, Sci. China Math., 2014, 57(11), 2369–2380.

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[96] Zh.-H. Yang, W. Zhang and Y.-M. Chu, Sharp Gautschi inequality for parameter 0 < p < 1 with applications, Math. Inequal. Appl., 2017, 20(4), 1107–1120. [97] Zh.-H. Yang, Y.-M. Chu and W. Zhang, Accurate approximations for the complete elliptic of the second kind, J. Math. Anal. Appl., 2016, 438(2), 875–888. Yousaf Khurshid, 1 College of Science, Hunan City University, Yiyang 413000, Hunan, China; 2 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan E-mail address: [email protected] Muhammad Adil Khan, 2 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan E-mail address: [email protected] Yu-Ming Chu (Corresponding author), versity, Huzhou 313000, Zhejiang, China E-mail address: [email protected]

3 Department

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Neutrosophic BCC-ideals in BCC-algebras Sun Shin Ahn Department of Mathematics Education, Dongguk University, Seoul 04620, Korea Abstract. The notions of a neutrosophic subalgebra and a neutrosohic ideal of a BCC-algebra are introduced and consider characterizations of a neutrosophic subalgebra and a neutrosophic ideal. We define the notion of a neutrosophic BCC-ideal of a BCC-algebra, and investigated some properties of it.

1. Introduction Y. Kormori [8] introduced a notion of a BCC-algebras, and W. A. Dudek [4] redefined the notion of BCCalgebras by using a dual from of the ordinary definition of Y. Kormori. In [6], J. Hao introduced the notion of ideals in a BCC-algebra and studied some related properties. W. A. Dudek and X. Zhang [5] introdued a BCC-ideals in a BCC-algebra and described connections between such BCC-ideals and congruences. S. S. Ahn and S. H. Kwon [2] defined a topological BCC-algebra and investigated some properties of it. Zadeh [10] introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. As a generalization of fuzzy sets, Atanassov [3] introduced the degree of nonmembership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood). Jun et. al [7] introduced the notions of a neutrosophic N -subalgebras and a (closed) neutrosophic N -ideal in a BCK/BCI-algebras and investigated some related properties. subalgebras In this paper, we introduce the notions of a neutrosophic subalgebra and a neutrosohic ideal of a BCC-algebra and consider characterizations of a neutrosophic subalgebra and a neutrosophic ideal. We define the notion of a neutrosophic BCC-ideal of a BCC-algebra, and investigate some properties of it. 2. Preliminaries By a BCC-algebra [4] we mean an algebra (X, ∗, 0) of type (2,0) satisfying the following conditions: for all x, y, z ∈ X, (a1) ((x ∗ y) ∗ (z ∗ y)) ∗ (x ∗ z) = 0, (a2) 0 ∗ x = 0, (a3) x ∗ 0 = x, (a4) x ∗ y = 0 and y ∗ x = 0 imply x = y. For brevity, we also call X a BCC-algebra. In X, we can define a partial order “≤” by putting x ≤ y if and only if x ∗ y = 0. Then ≤ is a partial order on X. 0

2010 Mathematics Subject Classification: 06F35; 03G25; 03B52. Keywords: BCC-algebra; ; (BCC-)ideal; neutrosophic subalgebra; neutrosophic (BCC-)ideal. 0 E-mail: [email protected]

0

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Sun Shin Ahn A BCC-algebra X has the following properties: for any x, y ∈ X, (b1) x ∗ x = 0, (b2) (x ∗ y) ∗ x = 0, (b3) x ≤ y ⇒ x ∗ z ≤ y ∗ z and z ∗ y ≤ z ∗ x. Any BCK-algebra is a BCC-algebra, but there are BCC-algebras which are not BCK-algebra [4]. Note that a BCC-algebra is a BCK-algebra if and only if it satisfies: (b4) (x ∗ y) ∗ z = (x ∗ z) ∗ y, for all x, y, z ∈ X. Let (X, ∗, 0X ) and (Y, ∗, 0Y ) be BCC-algebras. A mapping φ : X → Y is called a homomorphism if φ(x ∗X y) = φ(x) ∗Y φ(y) for all x, y ∈ X. A non-empty subset S of a BCC-algebra X is called a subalgebra of X if x ∗ y ∈ S whenever x, y ∈ S. A non-empty subset I of a BCI-algebra X is called an ideal [6] of X if it satisfies: (c1) 0 ∈ I, (c2) x ∗ y, y ∈ I ⇒ x ∈ I for all x, y ∈ X. I is called an BCC-ideal [5] of X if it satisfies (c1) and (c3) (x ∗ y) ∗ z, y ∈ I ⇒ x ∗ z ∈ I, for all x, y, z ∈ X. Theorem 2.1. [6] In a BCC-algebra, an ideal is a subalgebra. Theorem 2.2. [5] In a BCC-algebra, a BCC-ideal is an ideal. Corollary 2.3. [5] Any BCC-ideal of a BCC-algebra is a subalgebra. Definition 2.4. Let X be a space of points (objects) with generic elements in X denoted by x. A simple valued neutrosophic set A in X is characterized by a truth-membership function TA (x), an indeterminacy-membership function IA (x), and a falsity-membership function FA (x). Then a simple valued neutrosophic set A can be denoted by A := {⟨x, TA (x), IA (x), FA (x)⟩|x ∈ X}, where TA (x), IA (x), FA (x) ∈ [0, 1] for each point x in X. Therefore the sum of TA (x), IA (x), and FA (x) satisfies the condition 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3. For convenience, “simple valued neutrosophic set” is abbreviated to “neutrosophic set” later. Definition 2.5. Let A be a neutrosophic set in a B-algebra X and α, β, γ ∈ [0, 1] with 0 ≤ α + β + γ ≤ 3 and an (α, β, γ)-level set of X denoted by A(α,β,γ) is defined as A(α,β,γ) = {x ∈ X|TA (x) ≤ α, IA (x) ≥ β, FA (x) ≤ γ}. For any family {ai |i ∈ Λ}, we define ∨

{ai |i ∈ Λ} :=

{ max{ai |i ∈ Λ} sup{ai |i ∈ Λ}

606

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Neutrosophic BCC-ideals in BCC-algebras and

∧

{ {ai |i ∈ Λ} :=

min{ai |i ∈ Λ}

if Λ is finite,

inf{ai |i ∈ Λ}

otherwise.

3. Neutrosophic BCC-ideals In what follows, let X be a BCC-algebra unless otherwise specified. Definition 3.1. A neutrosophic set A in a BCC-algebra X is called a neutrosophic subalgebra of X if it satisfies: (NSS) TA (x ∗ y) ≤ max{TA (x), TA (y)}, IA (x ∗ y) ≥ min{IA (x), IA (y)}, and FA (x ∗ y) ≤ max{FA (x), FA (y)}, for any x, y ∈ X. Proposition 3.2. Every neutrosophic subalgebra of a BCC-algebra X satisfies the following conditions: (3.1) TA (0) ≤ TA (x), IA (0) ≥ IA (x), and FA (0) ≤ FA (x) for any x ∈ X. □

Proof. Straightforward. Example 3.3. Let X := {0, 1, 2, 3} be a BCC-algebra [6] with the following table: ∗ 0 1 2 3

0 0 1 2 3

Define a neutrosophic set A in X as follows:

1 0 0 1 3 {

TA : X → [0, 1], x 7→ { IA : X → [0, 1], x 7→ and

{ FA : X → [0, 1], x 7→

2 0 0 0 3

3 0 1 1 0

0.12 if x ∈ {0, 1, 2} 0.83 if x = 3, 0.81 if x ∈ {0, 1, 2} 0.14 if x = 3, 0.12 0.83

if x ∈ {0, 1, 2} if x = 3.

It is easy to check that A is a neutrosophic subalgebra of X. Theorem 3.4. Let A be a neutrosophic set in a BCC-algebra X and let α, β, γ ∈ [0, 1] with 0 ≤ α + β + γ ≤ 3. Then A is a neutrosophic subalgebra of X if and only if all of (α, β, γ)-level set A(α,β,γ) are subalgebras of X when A(α,β,γ) ̸= ∅. Proof. Assume that A is a neutrosophic subalgebra of X. Let α, β, γ ∈ [0, 1] be such that 0 ≤ α + β + γ ≤ 3 and A(α,β,γ) ̸= ∅. Let x, y ∈ A(α,β,γ) . Then TA (x) ≤ α, TA (y) ≤ α, IA (x) ≥ β, IA (y) ≥ β and FA (x) ≤ γ, FA (y) ≤ γ. Using (NSS), we have TA (x ∗ y) ≤ max{TA (x), TA (y)} ≤ α, IA (x ∗ y) ≥ min{IA (x), IA (y)} ≥ β, and FA (x ∗ y) ≤ max{FA (x), FA (y)} ≤ γ. Hence x ∗ y ∈ A(α,β,γ) . Therefore A(α,β,γ) is a subalgebra of X. Conversely, all of (α, β, γ)-level set A(α,β,γ) are subalgebras of X when A(α,β,γ) ̸= ∅. Assume that there exist at , bt , ai , bi ∈ X and af , bf ∈ X such that TA (at ∗ bt ) > max{TA (at ), TA (bt )}, IA (ai ∗ bi ) < min{IA (ai ), IA (bi )}

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Sun Shin Ahn and FA (af ∗ bf ) > max{FA (af ), FA (bf )}. Then TA (at ∗ bt ) > α1 ≥ max{TA (at ), TA (bt )}, IA (ai ∗ bi ) < β1 ≤ min{IA (ai ), IA (bi )} and FA (af ∗ bf ) > γ1 ≥ max{FA (af ), FA (bf )} for some α1 , γ1 ∈ [0, 1) and β1 ∈ (0, 1]. Hence at , bt , ai , bi ∈ A(α1 ,β1 ,γ1 ) , and af , bf ∈ A(α1 ,β1 ,γ1 ) . But at ∗ bt , ai ∗ bi ∈ / A(α1 ,β1 ,γ1 ) , and af ∗ bf ∈ / A(α1 ,β1 ,γ1 ) , which is a contradiction. Hence TA (x ∗ y) ≤ max{TA (x), TA (y)}, IA (x ∗ y) ≥ min{IA (x), IA (y)}, and FA (x ∗ y) ≤ max{TA (x), TA (y)}, for any x, y ∈ X. Therefore A is a neutrosophic subalgebra of X.

□

Since [0, 1] is a completely distributive lattice with respect to the usual ordering, we have the following theorem. Theorem 3.5. If {Ai |i ∈ N} is a family of neutrosopic subalgebras of a BCC-algebra X, then ({Ai |i ∈ N}, ⊆) forms a complete distributive lattice. Theorem 3.6. Let A be a neutrosophic subalgebra of a BCC-algebra X. If there exists a sequence {an } in X such that limn→∞ TA (an ) = 0, limn→∞ IA (an ) = 1, and limn→∞ FA (an ) = 0, then TA (0) = 0, IA (0) = 1, and FA (0) = 0. Proof. By Proposition 3.2, we have TA (0) ≤ TA (x), IA (0) ≥ IA (x), and FA (0) ≤ FA (x) for all x ∈ X. Hence we have TA (0) ≤ TA (an ), IA (0) ≥ IA (an ), and FA (0) ≤ FA (an ) for every positive integer n. Therefore 0 ≤ TA (0) ≤ limn→∞ TA (an ) = 0, 1 = limn→∞ IA (an ) ≤ IA (0) ≤ 1, and 0 ≤ FA (0) ≤ limn→∞ FA (an ) = 0. Thus we have □

TA (0) = 0, IA (0) = 1, and FA (0) = 0. Proposition 3.7. If every neutrosophic subalgebra A of a BCC-algebra X satisfies the condition (3.2) TA (x ∗ y) ≤ TA (y), IA (x ∗ y) ≥ IA (y), FA (x ∗ y) ≤ FA (y), for any x, y ∈ X, then TA , IA , and FA are constant functions.

Proof. It follows from (3.2) that TA (x) = TA (x ∗ 0) ≤ TA (0), IA (x) = IA (x ∗ 0) ≥ IA (0), and FA (x) = FA (x ∗ 0) ≤ FA (0) for any x ∈ X. By Proposition 3.2, we have TA (x) = TA (0), IA (x) = IA (0), and FA (x) = FA (0) for any x ∈ X. Hence TA , IA , and FA are constant functions.

□

Theorem 3.8. Every subalgebra of a BCC-algebra X can be represented as an (α, β, γ)-level set of a neutrosophic subalgebra A of X. Proof. Let S be a subalgebra of a BCC-algebra X and let A be a neutrosophic subalgebra of X. Define a neutrosophic set A in X as follows:

{

TA : X → [0, 1], x 7→ { IA : X → [0, 1], x 7→ { FA : X → [0, 1], x 7→

α1 α2

if x ∈ S otherwise,

β1 β2

if x ∈ S otherwise,

γ1 γ2

if x ∈ S otherwise,

where α1 , α2 , γ1 , γ2 ∈ [0, 1) and β1 , β2 ∈ (0, 1] with α1 < α2 , β1 > β2 , γ1 < γ2 , and 0 ≤ α1 + β1 + γ1 ≤ 3, 0 ≤ α2 + β2 + γ2 ≤ 3. Obviously, S = A(α1 ,β1 ,γ1 ) . We now prove that A is a neutrosophic subalgebra of X. Let x, y ∈ X. If x, y ∈ S, then x ∗ y ∈ S because S is a subalgebra of X. Hence TA (x) = TA (y) = TA (x ∗ y) = α1 ,

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Neutrosophic BCC-ideals in BCC-algebras IA (x) = IA (y) = IA (x ∗ y) = β1 , FA (x) = FA (y) = FA (x ∗ y) = γ1 and so TA (x ∗ y) ≤ max{TA (x), TA (y)}, IA (x ∗ y) ≥ min{IA (x), IA (y)}, FA (x ∗ y) ≤ max{FA (x), FA (y)}. If x ∈ S and y ∈ / S, then TA (x) = α1 , TA (y) = α2 , IA (x) = β1 , IA (y) = β2 , FA (x) = γ1 , FA (y) = γ2 and so TA (x ∗ y) ≤ max{TA (x), TA (y)} = α2 , IA (x ∗ y) ≥ min{IA (x), IA (y)} = β2 , FA (x ∗ y) ≤ max{FA (x), FA (y)} = γ2 . Obviously, if x ∈ / A and y ∈ / A, then TA (x ∗ y) ≤ max{TA (x), TA (y)} = α2 , IA (x ∗ y) ≥ min{IA (x), IA (y)} = β2 , FA (x ∗ y) ≤ max{FA (x), FA (y)} = γ2 . Therefore □

A is a neutrosophic subalgebra of X. Definition 3.9. A neutrosophic set A in a BCC-algebra X is said to be neutrosophic ideal of X if it satisfies: (NSI1) TA (0) ≤ TA (x), IA (0) ≥ IA (x), and FA (0) ≤ FA (x) for any x ∈ X;

(NSI2) TA (x) ≤ max{TA (x ∗ y), TA (y)}, IA (x) ≥ min{IA (x ∗ y), IA (y)}, and FA (x) ≤ max{FA (x ∗ y), FA (y)}, for any x, y ∈ X. Proposition 3.10. Every neutrosophic ideal of a BCC-algebra X is a neutrosophic subalgebra of X. Proof. Let A be a neutrosophic ideal of X. Put x := x ∗ y and y := x in (NSI2). Then we have TA (x ∗ y) ≤ max{TA ((x ∗ y) ∗ x), TA (x)}, IA (x ∗ y) ≥ min{IA ((x ∗ y) ∗ x), IA (x)}, and FA (x ∗ y) ≤ max{FA ((x ∗ y) ∗ x), FA (x)}. It follows from (b2) and (NSI1) that TA (x ∗ y) ≤ max{TA ((x ∗ y) ∗ x), TA (x)} = max{TA (0), TA (x)} ≤ max{TA (x), TA (y)}, IA (x ∗ y) ≥ min{IA ((x ∗ y) ∗ x), IA (x)} = max{IA (0), IA (x)} ≥ max{IA (x), IA (y)}, and FA (x ∗ y) ≤ max{FA ((x ∗ y) ∗ x), FA (x)} = max{FA (0), FA (x)} ≤ max{FA (x), FA (y)}. Thus A is a neutrosophic □

subalgebra of X.

Theorem 3.11. Let A be a neutrosophic set in a BCC-algebra X and let α, β, γ ∈ [0, 1] with 0 ≤ α + β + γ ≤ 3. Then A is a neutrosophic ideal of X if and only if all of (α, β, γ)-level set A(α,β,γ) are ideals of X when A(α,β,γ) ̸= ∅. Proof. Assume that A is a neutrosophic ideal of X. Let α, β, γ ∈ [0, 1] be such that 0 ≤ α + β + γ ≤ 3 and A(α,β,γ) ̸= ∅. Let x, y ∈ X be such that x∗y, y ∈ A(α,β,γ) . Then TA (x∗y) ≤ α, TA (y) ≤ α, IA (x∗y) ≥ β, IA (y) ≥ β, and FA (x ∗ y) ≤ γ, FA (y) ≤ γ. By Definition 3.9, we have TA (0) ≤ TA (x) ≤ max{TA (x ∗ y), TA (y)} ≤ α, IA (0) ≥ IA (x) ≥ min{IA (x ∗ y)), IA (y)} ≥ β, and FA (0) ≤ FA (x) ≤ max{FA (x ∗ y), TA (y)} ≤ γ. Hence 0, x ∈ A(α,β,γ) . Therefore A(α,β,γ) is an ideal of X. Conversely, suppose that there exist a, b, c ∈ X such that TA (0) > TA (a), IA (0) < IA (b), and FA (0) > FA (c). Then there exist at , ct ∈ [0, 1) and bt ∈ (0, 1] such that TA (0) > at ≥ TA (a), IA (0) < bt ≤ IA (b) and FA (0) > ct ≥ FA (c). Hence 0 ∈ / A(at ,bt ,ct ) , which is a contradiction. Therefore TA (0) ≤ TA (x), IA (0) ≥ IA (x) and FA (0) ≤ FA (x) for all x ∈ X. Assume that there exist at , bt , ai , bi , af , bf ∈ X such that TA (at ) > max{TA (at ∗ bt ), TA (bt )}, IA (ai ) < min{IA (ai ∗ bi ), IA (bi )}, and FA (af ) > max{TA (af ∗ bf ), TA (bf )}. Then there exist st , sf ∈ [0, 1) and si ∈ (0, 1] such that TA (at ) > st ≥ max{TA (at ∗ bt ), TA (bt )}, IA (ai ) < si ≤ min{IA (ai ∗ bi ), IA (bi )}, and FA (af ) > sf ≥ max{TA (af ∗bf ), TA (bf )}. Hence at ∗bt , bt , ai ∗bi , af ∗bf ∈ A(st ,si ,sf ) , and bt , bi , bf ∈ A(st ,si ,sf ) . But a t , ai ∈ / A(st ,si ,sf ) and af ∈ / A(st ,si ,sf ) . This is a contradiction. Therefore TA (x) ≤ max{TA (x ∗ y), TA (y)}, IA (x) ≥ min{IA (x ∗ y)), IA (y)} and FA (x) ≤ max{FA (x ∗ y), FA (y)}, for any x, y ∈ X. Therefore A is a neutrosophic ideal □

of X Proposition 3.12. Every neutrosophic ideal A of a BCC-algebra X satisfies the following properties:

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Sun Shin Ahn (i) (∀x, y ∈ X)(x ≤ y ⇒ TA (x) ≤ TA (y), IA (x) ≥ IA (y), FA (x) ≤ FA (y)), (ii) (∀x, y, z ∈ X)(x∗y ≤ z ⇒ TA (x) ≤ max{TA (y), TA (z)}, IA (x) ≥ min{IA (y), IA (z)}, FA (x) ≤ max{FA (y), FA (z)}). Proof. (i) Let x, y ∈ X be such that x ≤ y. Then x ∗ y = 0. Using (NSI2) and (NSI1), we have TA (x) ≤ max{TA (x ∗ y), TA (y)} = max{TA (0), TA (y)} = TA (y), IA (y) ≥ min{IA (x ∗ y), IA (y)} = min{IA (0), IA (y)} = IA (y), and FA (x) ≤ max{FA (x ∗ y), FA (y)} = max{FA (0), FA (y)} = FA (y). (ii) Let x, y, z ∈ X be such that x ∗ y ≤ z. By (NSI2) and (NSI1). we get TA (x ∗ y) ≤ max{TA ((x ∗ y) ∗ z), TA (z)} = max{TA (0), TA (z)} = TA (z), IA (x∗y) ≥ min{IA ((x∗y)∗z), IA (z)} = min{IA (0), IA (z)} = IA (z), and FA (x ∗ y) ≤ max{FA ((x ∗ y) ∗ z), FA (z)} = max{FA (0), FA (z)} = FA (z). Hence TA (x) ≤ max{TA (x ∗ y), TA (y)} ≤ max{TA (y), TA (z)}, IA (x) ≥ min{IA (x ∗ y), IA (y)} ≥ min{IA (y), IA (z)}, and FA (x) ≤ max{FA (x ∗ y), FA (y)} ≤ □

max{FA (y), FA (z)}. The following corollary is easily proved by induction.

Corollary 3.13. Every neutrosophic ideal A of a BCC-algebra X satisfies the following property: ∨n ∧n ∨n (3.3) (· · · (x ∗ a1 ) ∗ · · · ) ∗ an = 0 ⇒ TA (x) ≤ k=1 TA (ak ), IA (x) ≥ k=1 IA (ak ), FA (x) ≤ k=1 FA (ak ), for all x, a1 , · · · , an ∈ X. Definition 3.14. Let A and B be neutrosophic sets of a set X. The union of A and B is defined to be a neutrosophic set ˜ B := {⟨x, TA∪B (x), IA∪B (x), FA∪B (x)⟩|x ∈ X}, A∪ where TA∪B (x) = min{TA (x), TB (x)}, IA∪B (x) = max{IA (x), IB (x)}, FA∪B (x) = min{FA (x), FB (x)}, for all x ∈ X. The intersection of A and B is defined to be a neutrosophic set ˜ B := {⟨x, TA∩B (x), IA∩B (x), FA∩B (x)⟩|x ∈ X}, A∩ where TA∩B (x) = max{TA (x), TB (x)}, IA∩B (x) = min{IA (x), IB (x)}, FA∩B (x) = max{FA (x), FB (x)}, for all x ∈ X. Theorem 3.15. The intersection of two neutrosophic ideals of a BCC-algebra X is a also a neutrosophic ideal of X. Proof. Let A and B be neutrosophic ideals of X. For any x ∈ X, we have TA∩B (0) = max{TA (0), TB (0)} ≤ max{TA (x), TB (x)} = TA∩B (x), IA∩B (0) = min{TA (0), TB (0)} ≥ min{IA (x), IB (x)} = IA∩B (x), and FA∩B (0) = max{FA (0), FB (0)} ≤ max{FA (x), FB (x)} = FA∩B (x). Let x, y ∈ X. Then we have TA∩B (x) = max{TA (x), TB (x)} ≤ max{max{TA (x ∗ y), TA (y)}, max{TB (x ∗ y), TB (y)}} = max{max{TA (x ∗ y), TB (x ∗ y)}, max{TA (y), TB (y)}} = max{TA∩B (x ∗ y), TA∩B (y)},

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Neutrosophic BCC-ideals in BCC-algebras IA∩B (x) = min{IA (x), IB (x)} ≥ min{min{IA (x ∗ y), IA (y)}, min{IB (x ∗ y), IB (y)}} = min{min{IA (x ∗ y), IB (x ∗ y)}, min{IA (y), IB (y)}} = min{IA∩B (x ∗ y), IA∩B (y)}, and FA∩B (x) = max{FA (x), FB (x)} ≤ max{max{FA (x ∗ y), FA (y)}, max{FB (x ∗ y), FB (y)}} = max{max{FA (x ∗ y), FB (x ∗ y)}, max{FA (y), FB (y)}} = max{FA∩B (x ∗ y), FA∩B (y)}. ˜ B is a neutrosophic ideal of X. Hence A∩

□

˜ ı∈N Ai . Corollary 3.16. If {Ai |i ∈ N} is a family of neutrosophic ideals of a BCC-algebra X, then so is ∩ The union of any set of neutrosophic ideals of a BCC-algebra X need not be a neutrosophic ideal of X. Example 3.17. Let X = {0, 1, 2, 3, 4} be a BCC-algebra [5] with the following table: ∗ 0 1 2 3 4

0 0 1 2 3 4

Define neutrosophic sets A and B of X as follows:

1 0 0 2 3 3

2 0 1 0 1 4

{

TA : X → [0, 1], x 7→ { IA : X → [0, 1], x 7→ { FA : X → [0, 1], x 7→ { TB : X → [0, 1], x 7→ { IB : X → [0, 1], x 7→ and

{ FB : X → [0, 1], x 7→

3 0 0 0 0 3

4 0 0 0 0 0

0.12, if x ∈ {0, 1} 0.74 otherwise, 0.63, if x ∈ {0, 1} 0.11 otherwise, 0.12, if x ∈ {0, 1} 0.74 otherwise, 0.13, if x ∈ {0, 2} 0.63 otherwise, 0.75, if x ∈ {0, 2} 0.14 otherwise, 0.13, if x ∈ {0, 2} 0.63 otherwise.

It is easy to check that A and B are neutrosophic ideals of X.

˜ B is not a neutrosophic ideal of But A∪

X, since TA∪B (3) = min{TA (3), TB (3)} = 0.63 ≰ max{TA∪B (3 ∗ 2), TA∪B (2)} = max{TA∪B (1), TA∪B (2)} = max{min{TA (1), TB (1)}, min{TA (2), TB (2)}} = max{0.12, 0.13} = 0.13.

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Sun Shin Ahn Definition 3.18. A neutrosophic set A in a BCC-algebra X is said to be a neutrosophic BCC-ideal of X if it satisfies (NSI1) and (NSI3) TA (x ∗ z) ≤ max{TA ((x ∗ y) ∗ z), TA (y)}, IA (x ∗ z) ≥ min{IA ((x ∗ y) ∗ z), IA (y)}, and FA (x ∗ z) ≤ max{FA ((x ∗ y) ∗ z), FA (y)}, for any x, y, z ∈ X. Lemma 3.19. Every neutrosophic BCC-ideal of a BCC-algebra X is a neutrosophic ideal of X. Proof. Let A be a neutrosophic BCC-ideal of a BCC-algebra X. Put z := 0 in (NSI3). By (a3), we have TA (x ∗ 0) = TA (x) ≤ max{TA ((x ∗ y) ∗ 0), TA (y)} = max{TA (x ∗ y), TA (y)}, IA (x ∗ 0) = IA (x) ≥ min{IA ((x ∗ y) ∗ 0), IA (y)} = min{IA (x∗y), IA (y)}, and FA (x∗0) = FA (x) ≤ max{FA ((x∗y)∗0), FA (y)} = max{FA (x∗y), FA (y)}, for any x, y ∈ X. Hence A is a neutrosophic ideal of X.

□

Corollary 3.20. Every neutrosophic BCC-ideal of a BCC-algebra X is a neutrosophic subalgebra of X. The converse of Proposition 3.10 and Lemma 3.19 need not be true in general (see Example 3.21). Example 3.21. Let X = {0, 1, 2, 3, 4} be a BCC-algebra as in Example 3.17. Define a neutrosophic set A of X as follows: {

0.13 0.83

TA : X → [0, 1], x 7→ {

if x ∈ {0, 1, 2, 3} if x = 4,

0.82 if x ∈ {0, 1, 2, 3} 0.11 if x = 4,

IA : X → [0, 1], x 7→ and {

0.13 if x ∈ {0, 1, 2, 3} 0.83 if x = 4,

FA : X → [0, 1], x 7→

It is easy to check that A is a neutrosophic subalgebra of X, but not a neutrosophic ideal of X, since TA (4) = 0.83 ≰ max{TA (4 ∗ 3), TA (3)} = max{TA (3), TA (3)} = 0.13. Consider a neutrosophic set B of X which is given by { 0.14 if x ∈ {0, 1}, TB : X → [0, 1], x 7→ 0.84 if x ∈ {2, 3, 4} {

0.85 if x ∈ {0, 1} 0.12 if x ∈ {2, 3, 4},

IB : X → [0, 1], x 7→ and { FB : X → [0, 1], x 7→

0.14 if x ∈ {0, 1} 0.84 if x ∈ {2, 3, 4}.

It is easy to show that B is a neutrosophic ideal of X, but not a neutrosophic BCC-ideal of X, since TB (4 ∗ 3) = TB (3) = 0.84 ≰ max{TB ((4 ∗ 1) ∗ 3), TB (1)} = max{TB (0), TB (1)} = 0.14.

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Neutrosophic BCC-ideals in BCC-algebras Example 3.22. Let X = {0, 1, 2, 3, 4, 5} be a BCC-algebra [5] with the following table: ∗ 0 1 2 3 4 5

0 0 1 2 3 4 5

Define a neutrosophic set A of X as follows:

1 0 0 2 2 4 5 {

TA : X → [0, 1], x 7→ { IA : X → [0, 1], x 7→ and

{ FA : X → [0, 1], x 7→

2 0 0 0 1 4 5

3 0 0 0 0 4 5

4 0 0 1 1 0 5

5 0 1 1 1 1 0

0.43 if x ∈ {0, 1, 2, 3, 4} 0.55 if x = 5, 0.54 if x ∈ {0, 1, 2, 3 4} 0.42 if x = 5, 0.43 if x ∈ {0, 1, 2, 3, 4} 0.55 if x = 5.

It is easy to check that A is a neutrosophic BCC-ideal of X. Theorem 3.23. Let A be a neutrosophic set in a BCC-algebra X and let α, β, γ ∈ [0, 1] with 0 ≤ α + β + γ ≤ 3. Then A is a neutrosophic BCC-ideal of X if and only if all of (α, β, γ)-level set A(α,β,γ) are BCC-ideals of X when A(α,β,γ) ̸= ∅. □

Proof. Similar to Theorem 3.11.

Proposition 3.24. Let A be a neutrosophic BCC-ideal of a BCC-algebra X. Then XT := {x ∈ X|TA (x) = TA (0)}, XT := {x ∈ X|IA (x) = IA (0)}, and XF := {x ∈ X|FA (x) = FA (0)} are BCC-ideals of X. Proof. Clearly, 0 ∈ XT . Let (x ∗ y) ∗ z, y ∈ XT . Then TA ((x ∗ y) ∗ z) = TA (0) and TA (y) = TA (0). It follows from (NSI3) that TA (x ∗ z) ≤ max{TA ((x ∗ y) ∗ z), TA (y)} = TA (0). By (NSI1), we get TA (x ∗ z) = TA (0). Hence x ∗ z ∈ XT . Therefore XT is a BCC-ideal of X. By a similar way, XI and XF are BCC-ideals of X.

□

References [1] S. S. Ahn, Applications of soft sets to BCC-ideals in BCC-algebras, J. Comput. Anal. Appl. (to appear). [2] S. S. Ahn and S. H. Kwon, Toplogical properties in BCC-algerbras, Commun. Korean Math. Soc. 23(2) (2008), 169-178. [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems 20 (1986), 87–96. [4] W. A. Dudek, On constructions of BCC-algebras, Selected Papers on BCK- and BCI-algebras 1 (1992), 93-96. [5] W. A. Dudek and X. Zhang, On ideals and congruences in BCC-algeras, Czecho Math. J. 48 (1998), 21-29. [6] J. Hao, Ideal Theory of BCC-algebras, Sci. Math. Japo. 3 (1998), 373-381. [7] Y. B. Jun, F. Smarandache and H. Bordbar, Neutrosophic N -structures applied to BCK/BCI-algebras, Information, (to submit). [8] Y. Kormori, The class of BCC-algebras is not a varity, Math. Japo. 29 (1984), 391-394.

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Sun Shin Ahn [9] F. Smarandache, Neutrosophy, Neutrosophic Probablity, Sets, and Logic, Amer. Res. Press, Rehoboth, USA, 1998. [10] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-353.

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Global Dynamics and Bifurcations of Two Second Order Difference Equations in Mathematical Biology M. R. S. Kulenovi´c12 and Sarah Van Beaver Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA Abstract. We investigate the global behavior of two difference equations with exponential nonlinearities xn+1 = be−cxn + pxn−1 ,

n = 0, 1, . . .

where the parameters b, c are positive real numbers and p ∈ (0, 1) and xn+1 = a + bxn−1 e−xn ,

n = 0, 1, . . .

where the parameters a, b are positive numbers. The the initial conditions x−1 , x0 are arbitrary nonnegative numbers. The two equations are well known mathematical models in biology which behavior was studied by other authors and resulted in partial global dynamics behavior. In this paper, we complete the results of other authors and give the global dynamics of both equations. In order to obtain our results we will prove several results on global attractivity and boundedness and unboundedness for general second order difference equations xn+1 = f (xn , xn−1 ),

n = 0, 1, . . .

which are of interest on their own. Keywords. attractivity, difference equation, invariant sets, period doubling, periodic solutions, stable set .

AMS 2010 Mathematics Subject Classification: 39A20, 39A28, 39A30, 92D25

1

Introduction and Preliminaries

We investigate the global behavior of the system of difference equations xn+1 = be−cxn + pyn ,

yn+1 = xn ,

n = 0, 1, . . .

(1)

where the parameters b and c are positive real numbers, p ∈ (0, 1), and the initial conditions x−1 , x0 are arbitrary nonnegative numbers. This system can be rewritten in the form of the second order difference equation xn+1 = be−cxn + pxn−1 ,

n = 0, 1, . . .

(2)

In [5], the authors originally studied this model to describe the synchrony of ovulation cycles of the Glaucous-winged Gulls. The model assumed that there is an infinite breeding season as well as the number of gulls available to breed is infinite. The value of c is a positive number representing the colony density. The parameter b is the number of birds per day ready to begin ovulating. The parameter p is the probability that a bird will begin to ovulate and 1 − e−cxn is the probability of delaying ovulation. In making the model, the authors assumed that the delay only occurs for birds entering the system, not birds switching between different segments of the cycle. Note the authors state that the bifurcation of two-cycle solutions is the same as ovulation synchrony with the value of c increasing. In [5], they used the local bifurcation theory to come to the conclusion that there exists a unique equilibrium such that for sufficiently small values of c, the equilibrium branch is locally asymptotically stable. Additionally, for large enough values of c, there exists a two-cycle branch that will be locally asymptotically stable. In this paper we will improve these results by making them global. Using the results of Camouzis and Ladas, see [2] and [6], we are able to find the global dynamics of (1), which was not completed in [5]. We will show that Equation (1) exhibits global period doubling bifurcation described by Theorem 5.1 in [11], which shows that global dynamics of Equation (1) changes from global asymptotic stability of the unique equilibrium solution to the global asymptotic stability of the minimal period-two solution within its basin of attraction, as the parameter passes through the critical value. By using a similar method, we investigate the dynamics of xn+1 = a + bxn−1 e−xn ,

n = 0, 1, . . .

(3)

where the parameters a, b are positive real numbers and the the initial conditions x−1 , x0 are arbitrary nonnegative numbers. As it was mentioned in [8], Equation (3) could be considered as a mathematical model in biology where a represent the 1 2

Corresponding author, e-mail: [email protected] Partially supported by Maitland P. Simmons Foundation

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constant immigration and b represent the population growth rate. In this paper we find a simpler equivalent condition to √ √ a+ a2 +4a −a+ a2 +4a 2 √ e < b in [8] for the existence of a minimal period-two solution. We split the results into the two cases of 2 a+

a +4a

b ≥ ea and b < ea . While using a similar method as in [9] to establish the existence of a period-two solution when b < ea , we are able to find the global dynamics of Equation (3). By using new results for general second order difference equation we will prove the existence of unbounded solutions for the case when b ≥ ea . Similar as for Equation (1) we will show that Equation (3) exhibits global period doubling bifurcation described by Theorem 5.1 in [11]. In addition, we give the precise description of the basins of attractions of all attractors of both Equations (1) and (3). The rest of the paper is organized as follows. In the rest of this section we introduce some known results about monotone systems in the plane needed for the proofs of the main results as well as some new results about the existence of unbounded solutions. Section 2 gives the global dynamics of Equation (1) and Section 3 gives the global dynamics of Equation (3). The next result, which is combination of two theorems from [2] and [6], is important for the global dynamics of general second order difference equation. Theorem 1 Let I be a set of real numbers and f : I × I → I be a function which is either non-increasing in the first variable and non-decreasing in the second variable or non-decreasing in both variables. Then, for every solution {xn }∞ n=−1 of the equation xn+1 = f (xn , xn−1 ) , x−1 , x0 ∈ I, n = 0, 1, . . . (4) ∞ the subsequences {x2n }∞ n=0 and {x2n−1 }n=0 of even and odd terms of the solution are eventually monotonic.

We now give some basic notions about monotone maps in the plane. Consider a partial ordering  on R2 where x, y ∈ R2 are said to be related if x  y or y  x. Also, a strict inequality between points may be defined as x ≺ y if x  y and x 6= y. A stronger inequality may be defined as x = (x1 , x2 )  y = (y1 , y2 ) if x  y with x1 6= y1 and x2 6= y2 . A map T on a nonempty set R ⊂ R2 is a continuous function T : R → R. The map T is monotone if x  y implies T (x)  T (y) for all x, y ∈ R, and it is strongly monotone on R if x ≺ y implies that T (x)  T (y) for all x, y ∈ R. The map is strictly monotone on R if x ≺ y implies that T (x) ≺ T (y) for all x, y ∈ R. 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 R ⊂ 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. If T is differentiable map on a nonempty set R, a sufficient condition for T to be strongly monotone with respect to the SE ordering is that the Jacobian matrix at all points x has the sign configuration # " + − , (5) sign (JT (x)) = − + provided that R is open and convex. For x ∈ R2 , define Q` (x) for ` = 1, . . . , 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction. Basin of attraction of a fixed point (¯ x, y¯) of a map T , denoted as B((¯ x, y¯)), is defined as the set of all initial points (x0 , y0 ) for which the sequence of iterates T n ((x0 , y0 )) converges to (¯ x, y¯). Similarly, we define a basin of attraction of a periodic point of period p. The next five results, from [12, 11], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by H. L. Smith in [14, 13]. Theorem 2 Let T be a competitive map on a rectangular region R ⊂ R2 . Let x ∈ R be a fixed point of T such that ∆ := R ∩ int (Q1 (x) ∪ Q3 (x)) is nonempty (i.e., x is not the NW or SE vertex of R), and T is strongly competitive on ∆. Suppose that the following statements are true. a. The map T has a C 1 extension to a neighborhood of x. b. The Jacobian JT (x) of T at x has real eigenvalues λ, µ such that 0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis. Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T . We shall see in Theorem 4 that the situation where the endpoints of C are boundary points of R is of interest. The following result gives a sufficient condition for this case.

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Theorem 3 For the curve C of Theorem 2 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied. i. The map T has no fixed points nor periodic points of minimal period-two in ∆. ii. The map T has no fixed points in ∆, det JT (x) > 0, and T (x) = x has no solutions x ∈ ∆. iii. The map T has no points of minimal period-two in ∆, det JT (x) < 0, and T (x) = x has no solutions x ∈ ∆. For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 2 reduces just to |λ| < 1. This follows from a change of variables [14] that allows the Perron-Frobenius Theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis. The next result is useful for determining basins of attraction of fixed points of competitive maps. Theorem 4 (A) Assume the hypotheses of Theorem 2, and let C be the curve whose existence is guaranteed by Theorem 2. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely W− := {x ∈ R \ C : ∃y ∈ C with x se y}

and

W+ := {x ∈ R \ C : ∃y ∈ C with y se x} ,

(6)

such that the following statements are true. (i) W− is invariant, and dist(T n (x), Q2 (x)) → 0 as n → ∞ for every x ∈ W− . (ii) W+ is invariant, and dist(T n (x), Q4 (x)) → 0 as n → ∞ for every x ∈ W+ . (B) If, in addition to the hypotheses of part (A), x is an interior point of R and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q1 (x) ∪ Q3 (x) except for x, and the following statements are true. (iii) For every x ∈ W− there exists n0 ∈ N such that T n (x) ∈ int Q2 (x) for n ≥ n0 . (iv) For every x ∈ W+ there exists n0 ∈ N such that T n (x) ∈ int Q4 (x) for n ≥ n0 . If T is a map on a set R and if x is a fixed point of T , the stable set W s (x) of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set   x ∈ R : there exists {xn }0n=−∞ ⊂ R s.t. T (xn ) = xn+1 , x0 = x, and lim xn = x n→−∞

s

When T is non-invertible, the set W (x) may not be connected and made up of infinitely many curves, or W u (x) may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on R, the sets W s (x) and W u (x) are the stable and unstable manifolds of x. Theorem 5 In addition to the hypotheses of part (B) of Theorem 4, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis. If the curve C of Theorem 2 has endpoints in ∂R, then C is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in R that is tangential to E µ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W u (x) in R are fixed points of T . Remark 1 We say that f (u, v) is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative D1 f negative and first partial derivative D2 f positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of Equation (4) follows from the fact that if f is strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to Equation (4) is a strictly competitive map on I × I, see [11]. Set xn−1 = un and xn = vn in Equation (4) to obtain the equivalent system un+1 = vn , vn+1 = f (vn , un )

n = 0, 1, . . . .

Let T (u, v) = (v, f (v, u)). The second iterate T 2 is given by T 2 (u, v) = (f (v, u), f (f (v, u), v)) and it is strictly competitive on I × I, see [12]. Remark 2 The characteristic equation of Equation (4) at an equilibrium point (¯ x, x ¯): λ2 − D1 f (¯ x, x ¯)λ − D2 f (¯ x, x ¯) = 0,

(7)

has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ, whenever f is strictly decraesing in first and increasing in second variable. Thus the applicability of Theorems 2-5 depends on the existence or nonexistence of minimal period-two solutions.

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We now present theorems relating to the existence of unbounded solutions of Equation (4). The original result was obtained in [4]. Here we give an improved version of Theorem 2.1 in [4] taking out the extraneous conditions of requiring a continuity of f and the existence of an equilibrium solution. Additionally, we have extended the results in [4] to obtain a theorem in which the function f is nondecreasing in both arguments. Theorem 6 Assume that the function f : I × I → I is nonincreasing in the the first variable and nondecreasing in the second variable, where I ⊂ R is an interval. Assume there exists numbers L, U ∈ I such that L < U which satisfy f (U, L) ≤ L

(8)

and f (L, U ) ≥ U,

(9)

where at least one inequality is strict. If x−1 ≤ L and x0 ≥ U, then the corresponding solution x2n−1 ≤ L

x2n ≥ U,

and

{xn }∞ n=−1

satisfies

n = 0, 1, . . .

If, in addition, f is continuous and Equation (4) has no minimal period-two solution then, lim x2n = ∞

lim x2n−1 = −∞.

and/or

n→∞

n→∞

Similarily, if x−1 ≥ U and x0 ≤ L, then the corresponding solution {xn }∞ n=−1 satisfies x2n−1 ≥ U

x2n ≤ L,

and

n = 0, 1, . . .

If, in addition, f is continuous and Equation (4) has no minimal period-two solution then, lim x2n−1 = ∞

and/or

n→∞

lim x2n = −∞.

n→∞

Proof. Assume that x−1 ≤ L and x0 ≥ U. Then by using the monotonicity of f (nonincreasing in the first variable and nondecreasing in the second variable) and conditions (8) and (9) we obtain x1 = f (x0 , x−1 ) ≤ f (U, L) ≤ L and x2 = f (x1 , x0 ) ≥ f (L, U ) ≥ U. By using induction it follows that x2n−1 ≤ L and x2n ≥ U for all n = 0, 1, . . . where at least one inequality is strict. In view of ∞ Theorem 1 both sequences {x2n }∞ n=0 and {x2n−1 }n=0 are eventually monotonic. Assume that f is a continuous function and there is no minimal period-two solution. We will consider a few cases based on the properties of the interval I. First suppose there exist a ∈ R such that I = [a, ∞) and a < L. Then {x2n−1 }∞ n=0 will be convergent as the subsequence is bounded in [a, L]. If {x2n }∞ n=0 converges, this would create a contradiction as there would exist a minimal period-two solution. Therefore, lim x2n = ∞.

n→∞

Next suppose that for some b ∈ R, both I = (−∞, b] and U < b. Here {x2n }∞ n=0 will be convergent as the subsequence is bounded in the interval of [U, b]. So {x2n−1 }∞ n=0 cannot converge as there is no minimal period-two solution resulting in lim x2n−1 = −∞.

n→∞

If I = (−∞, ∞), then similar to the two cases above, at most one subsequence can converge as there is no minimal period-two solution. So either lim x2n = ∞ or lim x2n−1 = −∞. n→∞

n→∞

with the option of both occurring. Finally, we will prove that I cannot be I = [a, b] where a, b ∈ R. Suppose that I = [a, b] such that a < L < U < b and a, b ∈ R. Since xn ∈ [a, b] for all n, both subsequences would be convergent. As limn→∞ x2n−1 = p < limn→∞ x2n = q for some p, q ∈ R, there exists a period-two solution, which is a contradiction. The case when x−1 ≥ U and x0 ≤ L will follow similarly to the proof used here. 2 Many examples of the use of Theorem 6 are provided in [4]. Theorem 7 Assume that f : I × I → I is a function which is nondecreasing in both variables, where I ⊂ R is an interval. Assume there exists numbers L, U ∈ I such that L < U where f (L, L) ≤ L

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and f (U, U ) ≥ U

(11)

are satisfied, where at least one inequality is strict. If x−1 , x0 ≤ L, then the corresponding solution satisfies xn ≤ L, n = 0, 1, . . .

{xn }∞ n=−1

of Equation (4)

If, in addition, f is continuous and Equation (4) has no minimal period-two solution, then either xn converges to an equilibrium point or lim x2n−1 = −∞ and/or lim x2n = −∞. n→∞

n→∞

If x−1 , x0 ≥ U, then the corresponding solution {xn }∞ n=−1 satisfies xn ≥ U,

n = 0, 1, . . .

If, in addition, f is continuous and Equation (4) has no period-two solution, then either xn converges to an equilibrium point or lim x2n−1 = ∞ and/or lim x2n = ∞. n→∞

n→∞

Proof. Assume that x−1 , x0 ≤ L. Then by using the monotonicity of f (both variables are nondecreasing) and conditions (10) and (11) we obtain x1 = f (x0 , x−1 ) ≤ f (L, L) ≤ L

and

x2 = f (x1 , x0 ) ≤ f (L, L) ≤ L.

By using induction it follows that x2n−1 , x2n ≤ L for all n = 0, 1, . . . with at least one inequality being strict. In view of ∞ Theorem 1 both sequences {x2n }∞ n=0 and {x2n−1 }n=0 are eventually monotonic. We can assume that f is continuous and that there is no minimal period-two solution. We can choose the value of L such that at most one equilibrium is included in the region. Note the subsequences may converge to the equilibrium point if present. We will break this proof into cases for different intervals I assuming that the subsequences do not converge to an equilibrium point. First suppose that either I = [a, ∞) or I = [a, b] for some a, b ∈ R such that a < L < U < b. As both subsequences are less than L, then xn ∈ [a, L] for every n. As a consequence, both subsequences will be convergent. Thus, limn→∞ x2n−1 = p and limn→∞ x2n = q. If p = q, we get a contradiction as the subsequences do not converge to an equilibrium point. Otherwise, p 6= q, so (p, q) is a period-two solution, which is a contradiction as well. Thus, for I = [a, ∞) or I = [a, b], there must be an equilibrium point present. Next suppose that either I = (−∞, a] or I = (−∞, ∞). Now xn ∈ (−∞, L] for all n. At least one subsequence must be decreasing as the subsequences do not converge to an equilibrium point. Furthermore since there is no period-two solution, the subsequences cannot be bounded below resulting in either lim x2n = −∞

n→∞

or

lim x2n−1 = −∞.

n→∞

with the possibility of both options occurring. Now assume that x−1 , x0 ≥ U. Then by using the monotonicity of f and conditions (10) and (11) we obtain x1 = f (x0 , x−1 ) ≥ f (U, U ) ≥ U and x2 = f (x1 , x0 ) ≥ f (U, U ) ≥ U. By using induction it follows that x2n−1 , x2n ≥ U for all n = 0, 1, . . . with at least one inequality being strict. In view of ∞ Theorem 1 both sequences {x2n }∞ n=0 and {x2n−1 }n=0 are eventually monotonic. Assume that f is continuous and that there is no minimal period-two solution. We can choose the value of U such that at most one equilibrium is included in the region. Note the subsequences may converge to the equilibrium point if present. We will break this proof into cases for different intervals I assuming that the subsequences do not converge to an equilibrium point. First suppose that either I = (−∞, b] or I = [a, b] for some a, b ∈ R such that a < L < U < b. As both subsequences are greater than U , then xn ∈ [U, b] for every n. As a consequence, both subsequences will be convergent. Thus, limn→∞ x2n−1 = p and limn→∞ x2n = q. If p = q, we get a contradiction as the subsequences do not converge to an equilibrium point. Otherwise, p 6= q, so (p, q) is a period-two solution, which is a contradiction as well. Thus, for I = (−∞, b] or I = [a, b], there must be an equilibrium point present. Next suppose that either I = [a, ∞] or I = (−∞, ∞). Now xn ∈ [U, ∞) for all n. At least one subsequence must be increasing as the subsequences do not converge to an equilibrium point. Furthermore since there is no period-two solution, the subsequences cannot be bounded above resulting in either lim x2n = ∞

n→∞

or

lim x2n−1 = ∞.

n→∞

2

with the option of both occurring. Now we give few examples which illustrate all possible scenarios of Theorem 7.

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Example 1 Consider the difference equation xn+1 = x2n + x2n−1 ,

n = 1, 2, . . .

where x−1 , x0 ∈ R+ , and xn ≥ 0 for n = 1, 2, . . .. Here f (u, v) = u2 + v 2 is increasing in both variables. The equilibrium points are x0 = 0 and x+ = 1/2. The linearized difference equation is zn+1 = 2xzn + 2xzn−1 and the characteristic equation is λ2 = 2xλ + 2x. The zero equilibrium x0 is locally asymptotically stable. For the equilibrium point x+ , λ2 = λ + 1, so that √ √ √ λ1,2 = 1±2 5 . As 1+2 5 > 1 and 1−2 5 ∈ (−1, 0), then x+ is a saddle point. There is no minimal period-two solution as φ = ψ 2 + φ2

and

ψ = φ2 + ψ 2

implies φ = ψ. Now we want to find a L < U that satisfies the conditions (10) and (11). Condition (10) f (L, L) ≤ L implies 2L2 ≤ L, which simplifies to L ≤ 1/2. As well, f (U, U ) ≥ U if 2U 2 ≥ U , which simplifies to U ≥ 1/2. We can choose at least one of these inequalities to be strict. From Theorem 7, we can conclude that every solution with x1 , x0 ≤ L converges to 0, while every solution with x−1 , x0 ≥ U is eventually increasing and tends toward ∞. As L < 1/2 < U are arbitrary this conclusion holds for every case where x−1 , x0 ≤ L or x−1 , x0 ≥ U . These results do not give conclusions when x−1 ≤ L and x0 ≥ U or x−1 ≥ U and x0 ≤ L. In this case one may use theory of monotone maps as in [3]. Example 2 Consider the difference equation xn+1 = x2n + x2n−1 + a,

n = 1, 2, . . .

where a > 1/8, xn ≥ 0, and x−1 , x0 ∈ R. Here f (u, v) = u2 + v 2 + a is increasing in both variables. There is no equilibrium points as the discriminant of the equilibrium equation 1 − 8a < 0 and no minimal period-two solution exists as φ = ψ 2 + φ2 + a

and

ψ = φ2 + ψ 2 + a

implies φ = ψ. We can find U that satisfies the conditions (10) and (11) of Theorem 7. As f (U, U ) ≥ U simplifies to 2U 2 + a ≥ U , which always holds, every solution will be eventually increasing and tends to ∞. Example 3 Consider the difference equation xn+1 = x5n + x5n−1 , 5

n = 1, 2, . . .

5

where x−1 ,√x0 ∈ R. The function f (u, v) = u + v is increasing in both variables. The equilibrium points are x0 = 0 and x± = ±1/ 4 2. The characteristic equation at the equilibrium solution x ¯ is λ2 = 5x4 λ + 5x4 . For the equilibrium point x0 , 2 λ = 0 so√that λ1,2 = 0 and x0 is locally asymptotically stable. For the equilibrium point x± , λ2 = 5/2λ + 5/2, so that √ √ 5± 65 5+ 65 5− 65 λ1,2 = . As > 1 and ∈ (−1, 0), then the equilibrium points x± are saddle points. There is no minimal 4 4 4 period-two solution as φ = ψ 5 + φ5

and

ψ = φ5 + ψ 5

implies φ = ψ. Now we want to find L < U that satisfies the conditions of Theorem 7. Clearly f (L, L) ≤ L if 2L5 ≤ L, which simplifies √ √ √ 4 4 to L ≤ 1/ 2 if L > 0 and to L ≤ −1/ 2 if L < 0. As well, f (U, U ) ≥ U if 2U 5 ≥ U , which simplifies to U ≥ 1/ 4 2. We can choose at least one of these inequalities to be strict. From Theorem 7, we can conclude that every solution with x1 , x 0 ≤ √ L, L > 0 converges to 0, while every solution with x−1 , x0 ≥ U is eventually increasing and tends toward ∞. As L < 1/ 4 2 < U are arbitrary we conclude that  when x ¯− < x−1 , x0 < x ¯+ ,  0 ∞ when x−1 , x0 > x ¯+ , lim xn = n→∞  −∞ when x−1 , x0 < x ¯− . Theorem 7 does not apply when x−1 ≤ L and x0 ≥ U or x−1 ≥ U and x0 ≤ L. In this cases one can use the results from [3]. Example 4 Consider the difference equation xn+1 =

bx2n−1 ax2n , + 2 1 + xn 1 + x2n−1 2

n = 1, 2, . . .

2

au bv where a, b > 0 and x−1 , x0 ∈ R. The function f (u, v) = 1+u 2 + 1+v 2 is increasing in both variables. One equilibrium point is x0 = 0. The non-zero equilibrium point satisfies the quadratic equation 1 + x2 − (a + b)x = 0 which has real solutions if (a + b)2 − 4 ≥ 0. If a + b < 2, then there only exist x0 , if a + b = 2, then there exists x0 and x, and if a + b > 2, then there exist

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2ax 2bx three equilibrium points x0 < x− < x+ . The characteristic equation at the equilibrium solution x ¯ is λ2 = (1+x 2 )2 λ + (1+x2 )2 . 2 For the equilibrium point x0 , λ = 0 so that λ1,2 = 0 and thus, x0 is locally asymptotically stable. The conditions for local stability of the equilibrium points x ¯± are quite involved and can be found p in [1]. In particular x− will either be a saddle point, repeller, or non-hyperbolic depending on whether 2a(a + b) + (a − b) (a + b)2 − 4 is greater than, less than, or equal to 0, and the equilibrium point x+ is either locally asymptotically stable or non-hyperbolic when it exists. 2 ≤ L, Now we want to find a L < U that satisfies the conditions (10) and (11) of Theorem 7. First f (L, L) ≤ L if (a+b)L 1+L2 2 which simplifies to 0 ≤ 1 + L − (a + b)L. This will occur when L < L− or L > L+ where we can set L− = x− and L+ = x+ . 2 ≥ U , which simplifies to 0 ≥ 1 + U 2 − (a + b)U. This occurs when U− < U < U+ where we can As well, f (U, U ) ≥ U if (a+b)U 1+U 2 set U− = x− and U+ = x+ . For both L and U to exist, we need L < L− to satisfy L < U. From Theorem 7, we can conclude that every solution with x1 , x0 ≤ L converges to 0, while every solution with x−1 , x0 ≥ U converges to x+ . Note that in the region where L and U exist, no minimal period-two solutions exists. All the period-two solutions are located in the region which is the union of the second and the fourth quadrant with respect to x− .

2

Global Dynamics of Equation (1)

In this section we present the global dynamics of Equation (1).

2.1

Local stability results

We begin by observing that the function f (u, v) = be−cu + pv is decreasing in the first variable and increasing in the second ∞ ∞ variable and so by Theorem 1, for every solution {xn }∞ n=−1 of Equation (1) the subsequences {x2n }n=0 and {x2n−1 }n=0 are eventually monotonic. ∂f b b where 0 < x < 1−p . Note that ∂u (x, x) = −cbe−cx = Equation (1) has a unique positive equilibrium point xecx = 1−p ∂f −c(1 − p)x and ∂v (x, x) = p. The characteristic equation of Equation (1) is λ2 + (1 − p)cxλ − p = 0. Applying local stability test [10] we obtain b Lemma 1 Equation (1) has a unique positive equilibrium solution xecx = 1−p . 1 i) If x < c , then the equilibrium point x is locally asymptotically stable. ii) If x > 1c , then the equilibrium point x is a saddle point. iii) If x = 1c , then the equilibrium point x is non-hyperbolic of the stable type (with eigenvalues λ1 = −1 and λ2 = p).

Proof. i) Equilibrium point x is locally asymptotically stable if |(1 − p)cx| < 1 − p < 2. As p ∈ (0, 1) then 1 − p < 2 holds. As (1 − p)cx > 0, then x is stable if (1 − p)cx < 1 − p ⇔ cx < 1 ⇔ x
|1 − p| , then the equilibrium point x is a saddle point. As (1 − p)cx is positive, we obtain (1 − p)cx > 1 − p ⇔ cx > 1 ⇔ x >

1 . c

So the equilibrium point x is a saddle point if x > 1c . iii) The equilibrium point x is non-hyperbolic if |(1 − p)cx| = |1 − p| . We see that cx = 1 ⇔ x = 1c . The characteristic equation at the equilibrium becomes λ2 + (1 − p)λ − p = 0, 2

with eigenvalues λ1 = −1 and λ2 = p.

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2.2

Periodic solutions

In this section we present results about existence and uniqueness of the minimal period-two solution of Equation (1). Theorem 8 If x > 1c , then Equation (1) has a unique minimal period-two solution: φ, ψ, φ, ψ, . . . (φ 6= ψ, φ > 0 and ψ > 0) .

Proof. Let {φ, ψ} be a minimal period-two solution of Equation (1), where φ and ψ are distinct positive real numbers. Then we have φ = be−cψ + pφ, ψ = be−cφ + pψ, (12) where φ 6= ψ. This implies ψ= Let F (φ) = be

−cbe−cφ 1−p

be−cφ , 1−p

φ = be

+ (p − 1)φ. The equilibrium point x = F (x) = be

−cbe−cx 1−p

−cbe−cφ 1−p

b e−cx 1−p

+ pφ.

will be a zero of F as

+ (p − 1)x = be−cx + (p − 1)x = 0.

−cb

Note that F (0) = be 1−p > 0 since b > 0. Additionally, as φ approaches ∞, then F (φ) approaches −∞. Notice graphically, the the function F begins above the x-axis and ends approaching −∞. As the function F crosses the x-axis at least once at x, then F must cross the x-axis at least three times when F 0 (x) > 0. This will result in the existence of a minimal period-two solution. We want to prove that F 0 (x) > 0 holds true for some values of parameters. Observe that the derivative of F is F 0 (φ) =

−cφ b2 c2 −cφ −cbe e e 1−p + (p − 1) 1−p

so that when x is substituted F 0 (x) = xbc2 e−cx + (p − 1). Then F 0 (x) > 0 when x > F 0 (x) = xbc2 e−cx + (p − 1) > 0 ⇔ c2 x >

1 c

as

1 − p cx 1 1 e ⇔ c2 x > ⇔ x > . b x c

Thus when x > 1c , there will be a minimal period-two solution. Next we want to prove that the period-two solution is unique. Rewritting (12) we obtain φecψ =

b = ψecφ ⇔ φe−cφ = ψe−cψ . 1−p

Let g(x) = xe−cx . As g 0 (x) = e−cx (1 − cx), then the global maximum of g is attatined at x = 1c . For each y value there will 1 . This will happen when be two corresponding x values when g(x) < g( 1c ) = ce xe−cx
0. ce

Let G(x) = ecx − ecx and notice that G(0) = 1. The derivative of G will be G0 (x) = c(ecx − e). Notice G0 (x) ≤ 0 when ecx ≤ e such that x ≤ 1c , and G0 (x) > 0 when x > 1c . Thus, G(x) > 0 on [0, 1c ) ∪ ( 1c , ∞) where G( 1c ) = 0 is a global minimum. Thus when the period-two solution exists, it is unique. 2

2.3

Global stability results

In view of Theorem 1 every bounded solution of Equation (1) converges to either an equilibrium solution or a minimal period-two solution. Lemma 2 The solutions of Equation (1) are bounded. Proof. Equation (1) implies xn+1 = be−cxn + pxn−1 ≤ b + pxn−1 ,

n = 0, 1, . . . .

Consider the difference equation of un+1 = b + pun−1 , n = 0, 1, . . . . √ √ b + C1 ( p)n + C2 (− p)n . As n → ∞, then un → The solution of Equation (13) is un = 1−p b result, see [7] xn ≤ un ≤ 1−p +  = U for n = 0, 1, ... and some + > 0 when x0 ≤ u0 .

622

(13) b . 1−p

In view of difference inequality 2

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Theorem 9 (i) If x ¯ > 1c , then the equilibrium solution x ¯ is a saddle point and the minimal period-two solution {φ, ψ}, φ < ψ is globally asymptotically stable within the basin of attraction B(φ, ψ) = [0, ∞)2 \ W s (¯ x, x ¯), where W s (¯ x, x ¯) is the global stable manifold of (¯ x, x ¯). (ii) If x ¯ ≤ 1c , then the equilibrium solution x ¯ is globally asymptotically stable. Proof. Using Theorem 1 every bounded solution of Equation (1) converges to an equilibrium solution or period-two solution. By Lemma 2, every solution of Equation (1) is bounded so that all solutions converge to either an equilibrium solution or to the unique period-two solution {φ, ψ}, φ < ψ. When x > 1c , then x is a saddle point, by Lemma 1 part (ii), and has the global stable W s (x, x) and global unstable manifolds W u (x, x), where W s (x, x) is the graph of a non-decreasing function and W u (x, x) is the graph of a non-increasing function, which has endpoints at (φ, ψ) and (ψ, φ). Every initial point (x−1 , x0 ) which starts south east of W s (x, x) is attracted to (ψ, φ), while every initial point (x−1 , x0 ) which starts north west of W s (x, x) is attracted to (φ, ψ), see Theorems 2, 4. In this case in view of Theorem 1 global attractivity of period-two solution implies its local stability since the convergence is monotonic. When x ≤ 1c , the equilibrium solution is locally and so globally asymptotically stable by Lemma 1 part (i) and part (iii) . 2 Remark 3 For instance, case i) of Theorem 9 holds when b = 1, p = .5, c = 2, case ii) holds when b = 1, p = .5, c = 1 and when b = 1, p = (e − 1)/e, c = 1.

3

Global Dynamics of Equation (3)

In this section we present global dynamics of Equation (3).

3.1

Local stability results

First, notice that the function f (u, v) = a + bve−u is decreasing in the first variable and increasing in the second variable. By ∞ ∞ Theorem 1, for all solutions {xn }∞ n=−1 of Equation (3) the subsequences {x2n }n=0 and {x2n−1 }n=0 are eventually monotonic. ∂f −x a Equation (3) has a unique positive equilibrium point x = 1−be and −x where a < x. Note that ∂u (x, x) = −bxe ∂f −x x) = be . The characteristic equation of Equation (3) is (x, ∂v λ2 + bxe−x λ − be−x = 0. a Lemma 3 Equation (3) has a unique positive equilibrium solution x = 1−be −x . √ a+ a2 +4a , then the equilibrium solution x is locally asymptotically stable. i) If x < √2 a+ a2 +4a ii) If x > , then the equilibrium solution x is a saddle point. 2 √ a+ a2 +4a iii) If x = , then the equilibrium solution x is non-hyperbolic of stable type (with eigenvalues λ1 = −1 and 2 λ2 = be−x ).

Proof. i) The equilibrium point x is locally asymptotically stable if −x −x bxe < 1 − be < 2. As be−x > 0, then 1 − be−x < 2 holds true. So rearranging the other inequality we obtain bxe−x < 1 − be−x ⇔ be−x (x + 1) < 1 ⇔ x + 1 < Therefore, the equilibrium x is locally asymptotically stable if x < ex =

ex b

1 x ex e ⇔x< − 1. b b

− 1. As x = a + bxe−x we have

bx . x−a

(14)

Then we can equivalently write the condition to be locally asymptotically stable as x
1 − be−x ⇔ be−x (x + 1) > 1 ⇔ x + 1 >

x>

ex −1⇔x> b

bx x−a

b

ex b

− 1. By using (14), the inequality can then equivalently be written as √ x a + a2 + 4a −1⇔x> − 1 ⇔ x2 − xa − a > 0 ⇔ x > . x−a 2

So the equilibrium point x is a saddle point if x >

iii) The equilibrium point x is non-hyperbolic point if −x −x bxe = 1 − be . We see that 1 x ex e ⇔x= − 1. b b

bxe−x = 1 − be−x ⇔ be−x (x + 1) = 1 ⇔ x + 1 = In view of (14) this can be rewritten as ex −1⇔x= x= b

bx x−a

b

x a+ −1⇔x= − 1 ⇔ x2 − xa − a = 0 ⇔ x = x−a

√

a2 + 4a . 2

The characteristic equation at the equilibrium point will become λ2 + (1 − be−x )λ − be−x = 0, 2

with eigenvalues λ1 = −1 and λ2 = be−x ∈ (0, 1).

3.2

Periodic solutions

In this section we present results about existence and uniqueness of minimal period-two solutions of Equation (3). √ a+ a2 +4a Theorem 10 Assume that b < ea . If x > , then Equation (3) has minimal period-two solution: 2 φ, ψ, φ, ψ, . . .

(φ 6= ψ and φ > 0, ψ > 0) .

Proof. We want to find for which values of x there exists a minimal period-two solution (φ, ψ) where φ and ψ are distinct positive real numbers. A period-two solution satisfies φ = a + bφe−ψ ,

ψ = a + bψe−φ ,

(15)

where φ and ψ are distinct real numbers. Rewritting ψ and then substituting into φ we obtain ψ= Let F (φ) = a + φ(be

−

a 1−be−φ

a , 1 − be−φ

φ = a + bφe

− 1). The equilibrium point x = F (x) = a + x(be

−

a 1−be−x

ex (x−a) b

−

a 1−be−φ

.

(16)

will be a zero of F as

− 1) = a + x(be−x − 1) = 0.

Now F (a) = a + a(be

−

a 1−be−a

− 1) = abe

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is positive as a and b are positive constants. As φ approaches ∞, then F approaches −∞ assuming that b < ea . When F 0 (x) > 0 then F will cross the x−axis at least three times resulting in a minimal period-two solution. Thus, we want to prove when F 0 (x) > 0 holds. Taking the derivative of F we have F 0 (φ) = (be so that F 0 (x) =

−a x

+

x3 b2 e−2x . a

−

a 1−be−φ

− 1) +

a φab2 e−φ − 1−be −φ e −φ 2 (1 − be )

Then F 0 (x) > 0 hold true when

x3 b2 e−2x −a axb + > 0 ⇔ x4 b2 e−2x > a2 ⇔ x2 be−x > a ⇔ x2 b > ⇔ x(x − a) > ax2 − xa − a > 0. x a x−a √ a+ a2 +4a Thus, when x > , there will be a minimal period-two solution. 2 √ a+ a2 +4a , then When x > 2 √ 2a a a + a2 + 4a √ ⇔ > 1 − be−x > 1 − be−x 2 a + a2 + 4a √ √ √ −a + a2 + 4a x −a + a2 + 4a a+ a2 2 +4a √ √ ⇔ e 0 ⇔

1 1 e−1 m+1 > be−(m+1) ⇔ > ⇔ > e−1 . m+2 m+2 m+1 m+2

This proves that g 0 (m + 1) > 0. As the derivative changes from negative to positive around the critical point, it will be a local minima. Note that g(a) > 0 and as x approaches ∞, g(x) approaches ∞. Since m is the only critical point, each y value will have two x values with the exception at m. This results in the fact that there can only be one period-two solution. 2 Proposition 1 If b ≥ ea , there are no minimal period-two solutions. Proof. Assume that {φ, ψ} is a period-two solution. Then {φ, ψ} satisfies (15) and so it satisfies (16) as well. a x − Let F (φ) = a + φ(be 1−be−φ − 1). The equilibrium point x = e (x−a) will be a zero of F as b F (x) = a + x(be

−

a 1−be−x

− 1) = a + x(be−x − 1) = 0.

We see that F (a) = a + a(be

−

a 1−be−a

− 1) = abe

−

a 1−be−a

which is a positive value as a and b are positive constants. As φ approaches ∞, then F approaches ∞ as b ≥ ea . As the function begins above the x-axis at a and approaches ∞, F will cross the x-axis an even number of times. Since F (x) = 0 is one of the points that lie on the x-axis and the only equilibrium point, there cannot be a minimal period-two solution. 2

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3.3

Global stability results

By Theorem 1 every bounded solution of Equation (1) converges to either an equilibrium solution or a minimal period-two solution. Lemma 4 The solutions of Equation (3) are bounded if b < ea . Proof. By Equation (3), xn+1 = a + bxn−1 e−xn ≤ a + bxn−1 ,

n = 0, 1, ....

Consider the difference equation of un+1 = a + bun−1 ,

n = 0, 1, .... √ √ Suppose that b < e . The solution of Equation (17) is un = + C1 ( b)n + C2 (− b)n . As n → ∞, then un → a of difference inequality result, see [7] xn ≤ un ≤ 1−b +  = U for n = 0, 1, ... when x0 ≤ u0 , where  > 0. a

a 1−b

(17) a . 1−b

In view 2

Theorem 11 Consider Equation (3). √ a+ a2 +4a (i) If b < ea and x > , then there exists a period-two solution that is locally asymptotically stable and the 2 equilibrium point, x, that is is a saddle point. The unique period-two solution attracts all solutions which start off the global stable manifold of W s (E(x, x)). √ a+ a2 +4a (ii) If b < ea and x < , then the equilibrium solution, x, is globally asymptotically stable. 2 √ 2 a+ a +4a , then the equilibrium solution, x, is non-hyperbolic of the stable type and is global attractor. (iii) If b < ea and x = 2 Proof. (i) Using Theorem 1 every bounded solution of Equation (3) converges to an equilibrium solution or period-two solution. By Lemma 4, when b < ea every solution of Equation (3) is bounded √ such that all solutions will converge to either an a+ a2 +4a a equilibrium solution or period-two solution. If b < e and x > , then x will be a saddle point by Lemma 3 2 part (ii), and there will be a minimal period-two solution by Theorem 10. In view of Theorems 2, 4 there exist the global stable manifold W s (x, x) and global unstable manifold W u (x, x), where W s (x, x) is the graph of a non-decreasing function and W u (x, x) is the graph of a non-increasing function, which has endpoints at (φ, ψ) and (ψ, φ). Every initial point (x−1 , x0 ) which starts south east of W s (x, x) is attracted to (ψ, φ), while every initial point (x−1 , x0 ) which starts north west of W s (x, x) is attracted to (φ, ψ). √ a+ a2 +4a (ii) When b < ea and x < , then x is locally asymptotically stable by Lemma 3 part (i). Since [a, U ]2 is invariant 2 box and (x, x) is the only fixed point then, by Theorem 2.1 in [11] is global attractor and so globally asymptotically stable. √ a+ a2 +4a (iii) Moreover, when b < ea and x = , x will be non-hyperbolic of the stable type by Lemma 3 part (iii). Since 2 [a, U ]2 is invariant box and (x, x) is the only fixed point then, by Theorem 2.1 in [11] is global attractor and so globally asymptotically stable. 2 Theorem 12 If b ≥ ea , then Equation (3) has unbounded solutions. Proof. We will use Theorem 6 to prove this theorem. The conditions of (8) and (9) of Theorem 6 become f (U, L) = a + bLe−U ≤ L

and

f (L, U ) = a + bU e−L ≥ U.

and

a ≥ U (1 − be−L ).

These inequalities can be reduced to a ≤ L(1 − be−U )

U L x Any value of L and U such that 1−be −U ≤ 1−be−L will satisfy the theorem. Let G(x) = 1−be−x . There is a vertical asymptote −x at 1 − be = 0 that is at x = ln(b). In interval (ln(b), ∞) we can find L and U that satisfies these inequalities. As b ≥ ea then ln(b) ≥ a so that (ln(b), ∞) is part of the domain of difference equation (3). An example of where this holds is when L = a + . Using the fact that b ≥ ea and  is small, then b ≥ ea+ . By condition (9) the inequality holds true as

a + bU e−(a+) ≥ U ⇔ ea+ ≤

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We will use condition (8) and b ≥ ea to find the criteria for U based on our L. Thus, a + b(a + )e−U ≤ (a + ) ⇔ eU ≥

a +  b(a + ) ea (a + ) ⇔ eU ≥ ⇔ U ≥ a + ln .   


Let U be such that U > a + ln a+ . It holds that U ≥ L. Overall, as f is continuous and there is no minimal period-two  solution by Proposition 1, using Theorem (6) some solutions will approach ∞. 2 Remark 4 For √ instance,√case i) of Theorem 11 holds when a = 1, b = 2, case ii) holds when a = 4, b = 2 and case iii) holds e1+ 3 , and the conditions of Theorem 12 holds when a = .5, b = 2. when a = 2, b = √3−1 3+1 In conclusion, Equations (1) and (3) exhibit the global period doubling bifurcation described by Theorem 5.1 in [11]. Checking the conditions of Theorem 5.1 in [11] is exactly the content of Lemmas 1-3 and Theorems 10-12.

References [1] A. Bilgin, M. R. S. Kulenovi´c, and E. Pilav, Basins of Attraction of Period-two Solutions of Monotone Difference Equations, Adv. Difference Equ. (2016) 25 pp. [2] A.M. Amleh, E. Camouzis, and G. Ladas, On the Dynamics of a Rational Difference Equation, Part 1, Int. J. Difference Equ.Vol. 3 (1) (2008), 1-35. [3] A. Brett and M. R. S. Kulenovi´c, Basins of attraction of equlilibrium points of monotone difference equations, Sarajevo J. Math., Vol. 5 (18) (2009), 211-233. [4] D. Burgi´c, S. Kalabu˘si´c, and M. R. S. Kulenovi´c, Nonhyperbolic Dynamics for Competitive Systems in the Plane and Global Period-doubling Bifurcations, Adv. Dyn. Syst. Appl. 3 (2008), 229-249. [5] D. Burton and S. Henson, A note on the onset of synchrony in avian ovulation cycles, J. Difference Equ. Appl., Vol. 20 (4) (2014), 664–668. [6] E. Camouzis and G. Ladas, When does local asymptotic stability imply global attractivity in rational equations? J. Difference Equ. Appl., 12(2006), 863–885. [7] S. Elaydi, An introduction to difference equations. Third edition. Undergraduate Texts in Mathematics. Springer, New York, 2005. [8] H. El-Metwally, E.A. Grove, G. Ladas, R. Levins, and M. Radin, On the Difference Equation xn+1 = α + βxn−1 e−xn , Nonlinear Analysis TMA Vol. 47 (2001), 4623-4634. [9] N. Fotiades and G. Papaschinopoulos, Existence uniqueness and attractivity of prime period two solution for a difference equation of exponential form, Appl. Math. Comp., 218 (2012), 11648-11653 [10] 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. [11] M. R. S. Kulenovi´c and O. Merino, Global bifurcations for competitive system in the plane, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), 133-149. [12] M. R. S. Kulenovi´c and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Int. J. Bifur. Chaos 20 (2010),2471-2486. [13] H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations 64 (1986), 163-194. [14] H. L. Smith, Planar competitive and cooperative difference equations. J. Difference Equ. Appl. 3 (1998), 335–357.

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Bounds for the Real Parts and Arguments of Normalized Analytic Functions Defined by the Srivastava-Attiya Operator Young Jae Sim1 , Oh Sang Kwon1 , Nak Eun Cho2,∗ and H. M. Srivastava3,4 1 Department

of Mathematics, Kyungsung University, Busan 48434, Republic of Korea E-Mail: [email protected] [email protected]

2 Department

of Applied Mathematics, Pukyong National University, Busan 48513, Republic of Korea E-Mail: [email protected] ∗ Corresponding Author

3 Department

of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada 4 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China E-Mail: [email protected] Abstract In this paper, we derive some bounds for the real parts and arguments of the functionals given by 0 (f )(z) zJs,b

Js,b (f )(z)

,

Js,b (f )(z) z

and

Js,b (f )(z) Js+1,b (f )(z)

(z ∈ D),

where Js,b is the widely-investigated Srivastava-Attiya operator defined on the class of normalized analytic functions f in the open unit disk D := {z : z ∈ C

and |z| < 1}

with suitable real parameters s and b. These results reduce upon specialization to some well-known inclusion relationships for several classes of functions with given geometric properties. We also make a comparison between one of the results obtained here and an already known result for some specific cases. 2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C55. Key Words and Phrases. Analytic functions; Univalent functions; Starlike functions; Convex functions; Srivastava-Attiya operator; Strongly starlike functions; Strongly convex functions; Principle of differential subordination; Inclusion relationships. 1. Introduction and Preliminaries Let A denote the class of functions f normalized by f (z) = z +

∞ X

an z n ,

(1.1)

n=2

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2

Young Jae Sim, Oh Sang Kwon, Nak Eun Cho and H. M. Srivastava

which are analytic in the open unit disk D := {z : z ∈ C

and |z| < 1}.

The general Hurwitz-Lerch Zeta function Φ(z, s, b) is defined by Φ(z, s, b) =

∞ X n=0

zn (b + n)s

Z− 0;


b∈C\ s ∈ C when |z| < 1; R{s} > 1 when |z| = 1 . It is known that the function Φ(z, s, b) reduces to such more familiar functions of Analytic Number Theory as the Riemann and the Hurwitz Zeta functions, Lerch’s Zeta function, the Polylogarithmic function and the Lipschitz-Lerch Zeta function (see, for details, [12]). Srivastava and Attiya [11] introduced the linear operator Js,b : A → A defined by (b ∈ C \ Z− 0 ; s ∈ C),

Js,b (f )(z) = Gs,b (z) ∗ f (z)

where the symbol ∗ denotes the Hadamard product (or convolution) of analytic functions and the function Gs,b is defined by Gs,b (z) = (b + 1)s [Φ(z, s, b) − b−s ]. For a f ∈ A of the form given by (1.1), we get  ∞  X b+1 s Js,b (f )(z) = z + an z n b+n

(z ∈ D).

(1.2)

n=2

Srivastava and Attiya [11] showed that (see also the recent work by Srivastava et al. [13]) J0,b (f )(z) = f (z), Z z f (t) J1,0 (f )(z) = dt =: A(f )(z), t 0 Z z 1+γ tγ−1 f (t)dt =: Jγ (f )(z) J1,γ (f )(z) = zγ 0 and Jσ,1 (f )(z) = z +

∞  X n=2

2 n+1

σ

an z n =: I σ (f )(z)

(γ > −1)

(σ > 0),

where A, Jγ and I σ are the familiar Alexander [1], Bernardi [2] and Jung-Kim-Srivastava [4] integral operators, respectively. From the equation (1.2), we can obtain the following recurrence relation: 0 zJs+1,b (f )(z) = (1 + b)Js,b (f )(z) − bJs+1,b (f )(z).

(1.3)

For α ∈ [0, 1) and β ∈ (0, 1], let Ωα,β denote a subset of C defined by n π o Ωα,β = w : w ∈ C and |arg (w − α)| < β . 2 ∗ We denote by S (α, β) and C(α, β) the classes of functions f ∈ A satisfying the following conditions: zf 0 (z) zf 00 (z) ∈ Ωα,β and 1 + 0 ∈ Ωα,β (∀ z ∈ D), f (z) f (z)

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Bounds of Real Parts and Arguments of Normalized Analytic Functions

3

respectively. The function f in the classes S ∗ (α, β) and C(α, β) is called starlike of order β and type α in D and strongly convex of order β and type α in D, respectively. We note that S ∗ (α, 1) ≡ S ∗ (α)

and C(α, 1) ≡ C(α),

which are the well-known classes of starlike functions of order α in D and convex functions of order α in D. Wilken and Feng [15] showed that f ∈ C(α, 1) implies that f ∈ S ∗ (β, 1), where    1 1 − 2α   α 6=    22−2α [1 − 22α−1 ] 2 (1.4) β := β(α) =     1 1   α= .  2 log 2 2 Nunokawa et al. [8] investigated relations between γ ∈ (0, 1) and δ ∈ (0, 1) so that S ∗ (α, γ) implies that C(β, δ), where β is given by (1.4). We will discuss this relation in Section 4. The relation given above can be represented by using the operator Js,b as follows: 0 (f )(z) zJs,b

Js,b (f )(z)

∈ Ωα,γ =⇒

0 (f )(z) zJs+1,b

Js+1,b (f )(z)

∈ Ωβ,δ

(z ∈ D),

(1.5)

for s = −1 and b = 0. In the present paper, we will consider the implication given in (1.5) for suitable values of s and b in R. We also consider other similar problems associated with (1.5), which are related to the forms given by Js,b (f )(z) Js,b (f )(z) and . z Js+1,b (f )(z) We say that f is subordinate to F in D, written as f ≺ F or as f (z) ≺ F (z) in D, if and
only if f (z) = F ω(z) for some Schwarz function ω(z) with ω(0) = 0 and |ω(z)| < 1 for z ∈ D. It is well known that, if F is univalent in D, then f ≺ F is equivalent to f (0) = F (0) and f (D) ⊂ F (D) (see, for details, [10, p. 36]). Let ψ : C2 → C and let h be univalent in D. If p is analytic in D and satisfies the following differential subordination:
ψ p(z), zp0 (z) ≺ h(z) (z ∈ D), then p is called a solution of the differential subordination. A univalent function q is called a dominant of the solutions of the differential subordination (or, simply, a dominant) if p ≺ q in D for all solutions p. A function qe is called best dominant if qe ≺ q in D for all dominants q. We recall the following lemmas which are required in our present investigation. Lemma 1. (see Hallenbeck and Ruscheweyh [3]; see also [6, p. 71]) Let h be convex in D with h(0) = a, γ 6= 0 and R{γ} = 0. If p is analytic in D with the form given by p(z) = a + cn z n + cn+1 z n+1 · · · and p(z) +

zp0 (z) ≺ h(z) γ

630

(n ∈ N := {1, 2, 3, · · · }) (z ∈ D),

(1.6)

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then p(z) ≺ q(z) ≺ h(z)

(z ∈ D),

where

Z z γ h(t)t(γ/n)−1 dt. nz γ/n 0 The function q is convex and is the best dominant of (1.6). q(z) =

Lemma 2. (see Miller and Mocanu [5]) If −1 5 B < A 5 1, β > 0 and the complex number γ satisfies the inequality: (1 − A)β R{γ} = − , 1−B then the following differential equation: zq 0 (z) 1 + Az q(z) + = (z ∈ D) βq(z) + γ 1 + Bz has a univalent solution in D given by   γ z β+γ (1 + Bz)β(A−B)/B   R − (B = 6 0)  z   β 0 tβ+γ−1 (1 + Bt)(A−B)β/B dt β q(z) =   z β+γ exp(βAz) γ    (B = 0).  β R z tβ+γ−1 exp(βAt)dt − β 0 If the function p(z) given by p(z) = 1 + c1 z + c2 z 2 + · · · is analytic in D and satisfies the following subordination condition: zp0 (z) 1 + Az p(z) + ≺ (z ∈ D), βp(z) + γ 1 + Bz then 1 + Az p(z) ≺ q(z) ≺ (z ∈ D), 1 + Bz and q is the best dominant of (1.7).

(1.7)

The generalized hypergeometric function q Fs is defined by q Fs (z) = q Fs (α1 , · · · , αq ; β1 , · · · , βs ; z) =

∞ X (α1 )n · · · (αq )n z n (β1 )n · · · (βs )n n!

(z ∈ D),

(1.8)

n=0

Z− 0,

Z− 0

where αj ∈ C (j = 1, · · · , q), βj ∈ C \ := {0, −1, −2, · · ·} (j = 1, · · · , s), q 5 s + 1, q, s ∈ N0 , and (α)n is the Pochhammer symbol defined by (α)0 = 1

and

(α)n =

Γ(α + n) = α(α + 1) · · · (α + n − 1) Γ(α)

(n ∈ N),

Γ(z) being the Gamma function of the argument z. We recall following well-known identities for the Gaussian hypergeometric function 2 F1 , that is, the special case of (1.8) when q − 1 = s = 1: Lemma 3. (see [14, pp. 285 and 293]) For real or complex numbers a, b and c (c ∈ / Z− 0 ), the following identities hold true:

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Bounds of Real Parts and Arguments of Normalized Analytic Functions

Z

1

(i)

tb−1 (1 − t)c−b−1 (1 − zt)−a dt =

0

5

Γ(b)Γ(c − b) 2 F1 (a, b; c; z) Γ(c)

when R{c} > R{b} > 0; (ii) 2 F1 (a, b; c; z) = 2 F1 (b, a; c; z) ;  z (iii) 2 F1 (a, b; c; z) = (1 − z)−a 2 F1 a, c − b; c; z−1 . The following lemmas will also be required in our present investigation. Lemma 4. (see Wilken and Feng [15]) Let ν be a positive measure on [0, 1] and let h be a complex-valued function defined on D × [0, 1] such that h(·, t) is analytic in D for each t ∈ [0, 1] and that h(z, ·) is ν-integrable on [0, 1] for all z ∈ D. In addition, suppose that R{h(z, t)} > 0, h(−r, t) is real and   1 1 (|z| 5 r < 1; t ∈ [0, 1]). = R h(z, t) h(−r, t) If the function H is defined by Z

1

H(z) =

h(z, t)dν(t), 0

then  R

1 H(z)

 =

1 H(−r)

(|z| 5 r < 1).

Lemma 5. (see Nunokawa [7]) Let the function P be analytic in D, P (0) = 1, P (z) 6= 0 in D and suppose that there exists a point z0 ∈ D such that
arg P (z) < π δ (|z| < |z0 |) 2 and
arg P (z0 ) = π δ 2

(δ > 0).

Then z0 P 0 (z0 ) = ikδ, P (z0 )

(1.9)

where 1 k= 2



1 k5− 2



1 a+ a

 when


π arg P (z0 ) = δ 2

when


π arg P (z0 ) = − δ, 2

and 1 a+ a



and where 1

P (z0 ) δ = ±ia with a > 0.

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2. Bounds for the Real Parts In this section, we investigate the bounds for the real parts of normalized analytic functions defined by the Srivastava-Attiya operator Js,b . Theorem 1. Let f ∈ A and ( R

0 (f )(z) zJs,b

)

Js,b (f )(z)

(z ∈ D),

>α

where s ∈ R, 0 5 α < 1 and b = −α. Then ) (    0 zJs+1,b (f )(z) 1 −1 R > −b + (b + 1) 2 F1 1, 2 − 2α; b + 2; Js+1,b (f )(z) 2

(2.1)

(z ∈ D).

(2.2)

This result is sharp. Proof. Let us define a function p : D → C by p(z) =

0 zJs+1,b (f )(z)

Js+1,b (f )(z)

.

Then p is analytic in D with p(0) = 1. Thus, from the recurrence relation (1.3), we have 0 (f )(z) zJs,b

Js,b (f )(z)

= p(z) +

zp0 (z) . p(z) + b

(2.3)

From (2.1), the above relation shows that p(z) +

1 + (1 − 2α)z zp0 (z) ≺ . p(z) + b 1−z

(2.4)

Also, from Lemma 2 with A = 1 − 2α, B = −1, β = 1 and γ = b, we find that 1 p(z) ≺ −b (z ∈ D), Q(z)

(2.5)

where Q is defined by Z Q(z) =

1 b



t 0

1 − zt 1−z

−2(1−α) dt.

By applying Lemma 3, we have   Γ(b + 1) z Q(z) = . 2 F1 2 − 2α, 1; b + 2; Γ(b + 2) z−1 Moreover, the function Q is represented as follows: Z 1 Q(z) = g(t, z)dµ(t), 0

where g(t, z) =

1−z 1 − (1 − t)z

and dµ(t) =

Γ(b + 1) t1−2α (1 − t)b+2α−1 dt Γ(2 − 2α)Γ(b + 2α)

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Bounds of Real Parts and Arguments of Normalized Analytic Functions

7

with t ∈ [0, 1] and z ∈ D. We note that dµ(t) is a positive measure on [0, 1]. We can easily verify that the assertions hold true: (i) (ii) (iii) (iv)

g(·, t) is analytic in D for each t ∈ [0, 1]; g(z, ·) is integrable with respect to µ on [0, 1]; R {g(z, t)} > 0 for all z ∈ D and t ∈ [0, 1]; g(−r, t) is real for all r and for t ∈ [0, 1].

Indeed, we have  R

1 g(z, t)





zt =R 1+ 1−z

 =1−

tr 1 = , 1+r g(−r, t)

for |z| 5 r < 1 and t ∈ [0, 1]. Therefore, by applying Lemma 4, we obtain     −1 1 Γ(b + 2) r R = (|z| 5 r < 1). 2 F1 2 − 2α, 1; b + 2; Q(z) Γ(b + 1) 1+r

(2.6)

Letting r → 1− in (2.6) we conclude that the inequality (2.2) holds true from the relation (2.5). The sharpness of this result follows from the fact that the function Q is the best dominant of (2.4).  Theorem 2. Let f ∈ A and suppose that   Js,b (f )(z) >α R z

(z ∈ D),

where s ∈ R, 0 5 α < 1 and b > −1. Then     Js+1,b (f )(z) (1 − α)(b + 1) 1 >1− R 2 F1 1, 1; b + 3; z b+2 2

(2.7)

(z ∈ D).

(2.8)

This result is sharp. Proof. Let us define a function p : D → C by p(z) =

Js+1,b (f )(z) z

(z ∈ D).

Then we have Js,b (f )(z) 1 = p(z) + zp0 (z). z b+1 From (2.7), we see that p(z) +

zp0 (z) ≺ h(z) b+1

(z ∈ D),

where 1 + (1 − 2α)z (z ∈ D). 1−z Thus, by applying Lemma 1 with γ = b + 1 and h given above, we have p(z) ≺ q(z) in D, where q is a convex function in D defined by Z b + 1 z 1 + (1 − 2α)t b q(z) = b+1 t dt, 1−t z 0 h(z) =

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Young Jae Sim, Oh Sang Kwon, Nak Eun Cho and H. M. Srivastava

which, in view of Lemma 3, yields   z 2(1 − α)(b + 1)z . q(z) = 1 + 2 F1 1, 1; b + 3; (b + 2)(1 − z) z−1 Since the function q is convex with real coefficients, by the subordination relation: p(z) ≺ q(z)

(z ∈ D),

we obtain the inequality (2.8) by letting z → −1+. The sharpness of this result follows from the fact that the function q is the best dominant of the differential subordination given by zp0 (z) p(z) + ≺ h(z) (z ∈ D). b+1  We recall the following special case due to Prajapat and Bulboac˘a [9, Corollary 2.10]. Theorem 3. Let f ∈ A and suppose that   Js,b (f )(z) R >α Js+1,b (f )(z)

(z ∈ D),

where s ∈ R, 0 5 α < 1 and b = −α. Then      Js+1,b (f )(z) 1 −1 R > 2 F1 1, 2 − 2α; b + 2; Js+2,b (f )(z) 2

(z ∈ D).

(2.9)

This result is sharp. 3. Bounds for the Arguments For given α ∈ [0, 1), let the parameters β1 , β2 and β3 be real numbers defined by    1 −1 (b = −α), β1 = β1 (α, b) := −b + (b + 1) 2 F1 1, 2 − 2α; b + 2; 2   (1 − α)(b + 1) 1 β2 = β2 (α, b) := 1 − (b > −1) 2 F1 1, 1; b + 3; b+2 2 and    1 −1 β3 = β3 (α, b) := 2 F1 1, 2 − 2α; b + 2; (b = −α). 2

(3.1) (3.2)

(3.3)

We note that βj < 1 (j = 1, 2, 3). We also note that βj = α

(j = 1, 2, 3).

These inequalities are immediate consequences of Lemma 1 or 2 with 1 + Az 1 + (1 − 2α)z h(z) = = (A = 1 − 2α; B = −1) 1 + Bz 1−z such that R{h(z)} > α (z ∈ D). In this section, we investigate the bounds for the arguments of normalized analytic functions defined by the Srivastava-Attiya operator Js,b . In order to get our results, we need the following propositions.

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Bounds of Real Parts and Arguments of Normalized Analytic Functions

9

Proposition 1. Let w1 , w2 , w3 ∈ C satisfy the following conditions: (i) R{w1 } > 0 and I{w1 } < 0; (ii) 0 < arg(w3 ) 5 arg(w2 ) < π2 ; (iii) |w3 | 5 |w2 |. Then the inequality: arg(w1 + w3 ) 5 arg(w1 + w2 )

(3.4)

holds true. Proof. First of all, we consider a case for which arg(w3 ) = arg(w2 ). In this case, we let w1 = x + iy,

w2 = R2 eiθ

and w3 = R3 eiθ ,

where x > 0, y < 0 and R2 = R3 . Then the inequality (3.4) is equivalent to y + R2 sin θ y + R3 sin θ = . x + R2 cos θ x + R3 cos θ Furthermore, since x > 0 and θ ∈ (0, π/2), the above inequality is equivalent to (R2 − R3 )(x sin θ − y cos θ) = 0. Therefore, it follows from x > 0 and y < 0 that the above inequality holds true. To complete the proof of Proposition 1, let Ω ⊂ C be defined by n o Ω = Reiψ ∈ C : 0 < R 5 R2 and 0 < ψ 5 arg(w2 ) . Letting w3 ∈ Ω, we suppose that `1 be a straight line through the points −w1 and w2 and `2 be a straight line through the points −w1 and w3 . From Condition (ii) of Proposition 1, we can take the unique intersection point denoted by w e3 ∈ Ω of `1 and `2 . For this point, we have


arg w3 − (−w1 ) = arg w e3 − (−w1 ) = arg w2 − (−w1 ) , which completes the proof of Proposition 1.



The demonstration of Proposition 2 below is fairly straightforward. Proposition 2. Let w1 and w2 be in C \ {0}. Then arg(w1 + w2 ) = min {arg(w1 ), arg(w2 )} . Theorem 4. Let β ∈ R be the parameter β1 given by (3.1). Suppose also that f ∈ A and ! 0 (f )(z) π zJs,b − α (z ∈ D), (3.5) arg < γ 2 Js,b (f )(z) where s ∈ R, b = −β, 0 5 α < 1 and 0 < γ < 1. Then ! 0 π zJs+1,b (f )(z) arg − β < δ 2 Js+1,b (f )(z)

(z ∈ D; 0 < δ < 1),

where 0 < δ < 1 and (

2 γ = min δ, arctan π

δ(1 − β)(x1+δ + xδ−1 0 0 ) δ 2(β − α)[(1 − β)x0 + β + b]

636

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Young Jae Sim, Oh Sang Kwon, Nak Eun Cho and H. M. Srivastava

and x0 ∈ (0, 1) is the root of the following equation: (1 − β)(x2 − 1)xδ = (β + b)[1 − δ − (1 + δ)xδ ].

(3.6)

Proof. Let us define the functions p and P : D → C by p(z) =

0 zJs+1,b (f )(z)

Js+1,b (f )(z)

and P (z) =

p(z) − β . 1−β

Then the functions p and P are analytic in D with p(0) = P (0) = 1. From the recurrence relation (1.3), we have 0 (f )(z) zJs,b zp0 (z) = p(z) + . (3.7) Js,b (f )(z) p(z) + b We now assume that there exists a point z0 ∈ R such that

arg P (z) = arg p(z) − β < π δ 2 for |z| < |z0 | and

arg P (z0 ) = arg p(z0 ) − β = π δ. 2 Consider the case when

π arg P (z0 ) = arg p(z0 ) − β = δ. 2 Then, by Lemma 5, we have z0 P 0 (z0 ) z0 p0 (z0 ) = = iδk, P (z0 ) p(z0 ) − β where   1 1 k= a+ 2 a with a > 0. Also, from (3.7) and (3.8), we have ! 0 (f )(z ) z0 Js,b 0 −α arg Js,b (f )(z0 )   z0 p0 (z0 ) = arg p(z0 ) + −α p(z0 ) + b  
p(z0 ) − α z0 p0 (z0 ) 1 = arg p(z0 ) − β + arg + · p(z0 ) − β p(z0 ) − β p(z0 ) + b   π iδk(1 − β) −δ = δ + arg 1 − β + (β − α)(ia) + . 2 (1 − β)(ia)δ + β + b

(3.8)

(3.9)

(3.10)

Let us define w1 , w2 and w3 by w1 = 1 − β + (β − α)(ia)−δ , w2 = and w3 =

iδk(1 − β) (1 − β)(ia)δ + β + b iδk(1 − β) iπ

[(1 − β)aδ + β + b]e 2 δ

637

.

Young Jae Sim ET AL 628-645

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Bounds of Real Parts and Arguments of Normalized Analytic Functions

We note that R {w1 } = 1 − β +

11

π  β−α cos δ >0 2 (β − α)aδ

and

π  β−α sin δ < 0. 2 (β − α)aδ Also, from the inequality β + b = 0, we can easily verify that the inequality arg(w2 ) = arg(w3 ) holds true. Furthermore, the inequality |w2 | = |w3 | is true, since I {w1 } = −

|w2 |2 = =

δ 2 k 2 (1 − β)2 (1 − β)2 a2δ + 2(1 − β)(β + b)aδ cos

π 2


δ + (β + b)2

δ 2 k 2 (1 − β)2 [(1 − β)aδ + β + b]2

= |w3 |2 . Therefore, by applying Proposition 1 with (3.10), we have ! 0 (f )(z ) z0 Js,b 0 −α arg Js,b (f )(z0 ) ! π iδk(1 − β) = δ + arg 1 − β + (β − α)(ia)−δ + iπ 2 [(1 − β)aδ + β + b]e 2 δ    iπ iδk π β−α − iπ δ δ + = δ + arg e 2 e2 + 2 (1 − β)aδ (1 − β)aδ + β + b   iπ iδ(a + a−1 ) β−α δ = arg e 2 + + . (1 − β)aδ 2[(1 − β)aδ + β + b]

(3.11)

Let us now put w4 =

β−α δ(a + a−1 ) . + i (1 − β)aδ 2[(1 − β)aδ + β + b]

Then  arg (w4 ) = arctan

 (1 − β)δ g(a) , 2(β − α)

where g : (0, ∞) → R is a function defined by g(x) =

x + x−1 . 1 − β + (β + b)x−δ

Differentiating the function g with respect to x, we have xδ+2 [1 − β + (β + b)x−δ ]2 g 0 (x) = h(x), where the function h : (0, ∞) → R is defined by h(x) = (x2 − 1)[(1 − β)xδ + β + b] + δ(β + b)(x2 + 1). Since the function h is continuous on (0, ∞) with h(0) = −(β + b)(1 − δ) < 0

and h(1) = 2δ(β + b) > 0,

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there exists an x0 ∈ (0, 1) such that h(x0 ) = 0, which is equivalent to the equation given by (3.6). Differentiating the function h twice with respect to x, we find that h00 (x) = (2 + δ)(1 − β)(1 + δ)xδ + δ(1 − δ)(1 − β)xδ−2 + 2(1 + δ)(β + b) > 0 for all x ∈ (0, 1). Since h(x) > 0 for x ∈ (1, ∞), it follows from the convexity of h(x) on (0, 1) that the function g 0 (x) vanishes only at x0 ∈ (0, 1). Furthermore, we can easily verify that g(x0 ) is the minimum value of g(x) on (0, ∞). Therefore, we have  arg (w4 ) = arctan = arctan

 (1 − β)δ g(x0 ) 2(β − α) δ(1 − β)(x1+δ + xδ−1 0 0 )   δ 2(β − α) (1 − β)x0 + β + b

! .

(3.12)

Finally, from (3.11), (3.12) and Proposition 2, we have

arg

0 (f )(z ) z0 Js,b 0

!

−α Js,b (f )(z0 )  iπ  = arg e 2 δ + w4 o nπ δ, arg(w4 ) = min !) (2 δ−1 ) + x π δ(1 − β)(x1+δ 0 0 = min δ, arctan 2 2(β − α)[(1 − β)xδ0 + β + b] π = γ, 2

which leads to a contradiction to the hypothesis (3.5). For the case when

π arg P (z0 ) = arg p(z0 ) − β = − δ, 2 Lemma 5 yields z0 P 0 (z0 ) z0 p0 (z0 ) = = iδk, P (z0 ) p(z0 ) − β

(3.13)

where 1 k5− 2

  1 a+ a

639

(3.14)

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Bounds of Real Parts and Arguments of Normalized Analytic Functions

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with a > 0. We also have 0 (f )(z ) z0 Js,b 0

arg

Js,b (f )(z0 )

! −α




p(z0 ) − α z0 p0 (z0 ) 1 + · p(z0 ) − β p(z0 ) − β p(z0 ) + b



= arg p(z0 ) − β + arg   π (1 − β)(−ia)δ + β − α iδk = − δ + arg + 2 (1 − β)(−ia)δ (1 − β)(−ia)δ + β + b " !# π iδ e k(1 − β) −δ =− δ + arg 1 − β + (β − α)(ia) + , 2 (1 − β)(ia)δ + β + b where

a + a−1 e . k := −k > 2 Therefore, from the proof of the first case, we have ! 0 (f )(z ) z0 Js,b 0 π arg − α 5 − γ, Js,b (f )(z0 ) 2 which also leads to a contradiction to the hypothesis (3.5). This completes the proof of Theorem 4.  Theorem 5. Let β ∈ R be the parameter β2 given by (3.2). Let f ∈ A and suppose that   arg Js,b (f )(z) − α < π γ (z ∈ D), (3.15) 2 z where s ∈ R, b > −1, 0 5 α < 1 and 0 < γ < 1. Then   arg Js+1,b (f )(z) − β < π δ 2 z where 2 γ = δ + arctan π

(

(z ∈ D; 0 < δ < 1),

)
−2(b + 1)(β − α) sin π2 δ + δ(1 − β)(xδ+1 + xδ−1 0 ) 0 
 , 2(b + 1) (1 − β)xδ0 + (β − α) cos π2 δ

and x0 ∈ (0, 1) is the unique zero of the function h defined by h(x) = Cxδ (x2 − 1) + AC(δ + 1)x2 + δBx + AC(δ − 1) with

π  β−α cos δ , 1−β 2

π  β−α sin δ 1−β 2

(3.16)

δ . 2(b + 1)

(3.17)

Proof. Let us define the functions p and P : D → C by Js+1,b (f )(z) p(z) − β p(z) = and P (z) = . z 1−β Then the functions p and P are analytic in D with p(0) = P (0) = 1. We also have

(3.18)

A=

B=

and

Js,b (f )(z) 1 = p(z) + zp0 (z). z b+1

640

C=

(3.19)

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We now assume that there exists a point z0 ∈ R such that

arg P (z) = arg p(z) − β < π δ, 2 for |z| < |z0 | and

arg P (z0 ) = arg p(z0 ) − β = π δ. 2 We consider the case when

π arg P (z0 ) = arg p(z0 ) − β = δ. 2 Then, by Lemma 5, we have the relations given by (3.8) and (3.9) with a > 0. From (3.18) and (3.19), we have   Js,b (f )(z0 ) arg −α z0 !
z0 p0 (z0 ) p(z0 ) − α
+ = arg p(z0 ) − β + arg p(z0 ) − β (b + 1) p(z0 ) − β   β−α π iδk = δ + arg 1 + + 2 b+1 (1 − β)(ia)δ  
β−α π δk sin δ + δ π 2 b+1 (1−β)a = δ + arctan −
 2 1 + β−α δ cos π δ (1−β)a

=

2

π δ + arctan g(a) , 2


(3.20)

where

−B + C(xδ+1 + xδ−1 ) , xδ + A and A, B and C are positive constants given by (3.17). Differentiating the function g(x) with respect to x, we have a2−δ (aδ + A)2 g 0 (x) = h(x), where h is given by (3.16). Simple calculations show that g(x) =

h(0) = −AC(1 − δ) < 0

and h(x) = h(1) = δ(2AC + B) > 0

(x = 1)

and h00 (x) = C[(δ + 2)(δ + 1)xδ + δ(1 − δ)xδ−2 ] + 2AC(δ + 1) > 0 Similar methods as in the proof of Theorem 4 would yield (0 < x < ∞),

g(x) = g(x0 )

(0 < x < 1). (3.21)

where x0 is the unique zero of h(x) on (0, ∞). Therefore, by (3.20) and (3.21), we obtain   Js,b (f )(z0 ) arg −α z0 ! δ−1 π −2(b + 1)(β − α) sin(δπ/2) + δ(1 − β)(xδ+1 + x ) 0 0 = δ + arctan 2 2(b + 1)[(1 − β)xδ0 + (β − α) cos(δπ/2)] π = γ, 2

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which provides a contradiction to the hypothesis (3.15). For the case when

π arg P (z0 ) = arg p(z0 ) − β = − δ, 2 we have the relations given by (3.13) and (3.14) with a > 0. Therefore, we have   Js,b (f )(z0 ) −α arg z0   
β−α π δe k sin δ + δ π 2 b+1 (1−β)a = −  δ + arctan −
 β−α π 2 1 + (1−β)aδ cos 2 δ π
 5− δ + arctan g(a) , 2 where a + a−1 e . k := −k > 2 Therefore, from (3.22) and (3.21), we have   Js,b (f )(z0 ) π arg − α 5 − γ, z0 2

15

(3.22)

which also provides a contradiction to the hypothesis (3.15). This evidently completes the proof of Theorem 5.  Next, for given suitable real of the parameters s and b and for f ∈ A, we define a function p : D → C by Js+1,b (f )(z) p(z) = . Js+2,b (f )(z) Then, by using the recurrence relation (1.3), we obtain Js,b (f )(z) zp0 (z) = p(z) + Js+1,b (f )(z) (b + 1)p(z)

(z ∈ D).

(3.23)

By applying the same methods as in the proof of Theorem 4 to the differential equation (3.23) instead of (3.7), we can establish the following argument property associated with the Srivastava-Attiya operator. Theorem 6. Let β ∈ R be the parameter β3 given by (3.3). Also let f ∈ A and   π J (f )(z) s,b arg − α < γ (z ∈ D), Js+1,b (f )(z) 2 where s ∈ R, 0 5 α < 1, b = −α and 0 < γ < 1. Then   arg Js+1,b (f )(z) − β < π δ 2 Js+2,b (f )(z)

(z ∈ D),

where 0 < δ < 1 and (

2 γ = min δ, arctan π

δ(1 − β)(x1+δ + xδ−1 0 0 ) 2(β − α)(b + 1)[(1 − β)xδ0 + β]

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Young Jae Sim, Oh Sang Kwon, Nak Eun Cho and H. M. Srivastava

and x0 is the root in the interval (0, 1) of the following equation: [(1 − β)xδ + β](1 − x2 ) = βδ(x2 + 1).

(3.24)

4. Numerical and Computational Analysis Let s = −1 and b = 0. Since the following equalities: Js,b (f )(z) = zf 0 (z)

and Js+1,b (f )(z) = f (z)

hold true for f ∈ A, it follows from Theorem 4 that f ∈ C(α, γ1 ) implies that f ∈ S ∗ (β, δ), where !) ( δ(1 − β)(x1+δ + xδ−1 2 0 0 ) (4.1) γ1 = min δ, arctan π 2(β − α)[(1 − β)xδ0 + β] and x0 ∈ (0, 1) is the root of the following equation: (1 − β)(x2 − 1)xδ = β[1 − δ − (1 + δ)xδ ]). On the other hand, Nunokawa et al. [8] showed that f ∈ C(α, γ2 ) implies that f ∈ S ∗ (β, δ), where ! δ−1 ) + x 2 δ(1 − β)(x1+δ 0 0 γ2 = arctan , (4.2) π (1 − β)xδ0 + β and x0 ∈ (0, 1) is the root of the following equation: (1 − β)(x2 − 1)xδ = β(1 − δ − (1 + δ)x2 ).

(4.3)

As it does not seem to be so easy to compare the values γ1 and γ2 for the whole ranges of the parameters α ∈ (0, 1) and δ ∈ (0, 1), we will compare them here in several particular cases of α and δ. Thus, if we fix α = 21 , then we have β=

1 . 2 log 2

With the aid of Mathematica, we can thus obtain Table 1 (see below) which gives the approximate values of γ1 ∈ (0, 1) and γ2 ∈ (0, 1) defined by (4.1) and (4.2), respectively, when δ is given by δ=

j 10

(j = 1, 2, · · · , 9).

As we see from Table 1, we can verify that the results in this paper would significantly improve the results in the earlier work [8] for the special cases considered above. Finally, we give another table (Table 2 below) which gives the approximate values of γ defined in Theorem 5 and Theorem 6, respectively, when δ is given by δ=

j 10

(j = 1, 2, · · · , 9).

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δ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

γ1 0.44897 0.43647 0.41317 0.38021 0.33781 0.28596 0.22487 0.15544 0.07952

17

γ2 0.27427 0.28576 0.28270 0.26598 0.23485 0.18916 0.13310 0.07889 0.03626

Table 1. The Approximate Values of γ1 and γ2

δ Theorem 5 (γ) Theorem 6 (γ) 0.9 0.75302 0.28582 0.75151 0.27787 0.8 0.7 0.67933 0.26303 0.59106 0.24205 0.6 0.5 0.49662 0.21506 0.4 0.39926 0.18205 0.30036 0.14316 0.3 0.2 0.20061 0.09896 0.1 0.10041 0.05062 Table 2. The Approximate Values of γ in Theorem 5 and Theorem 6

Acknowledgements The first-named author was supported by a research grant of the National Research Foundation of the Republic of Korea (NRF) funded by the Government of the Republic of Korea (MSIP; Ministry of Science, ICT & Future Planning) (Grant No. NRF-2017R1C1B5076778). The third-named author was supported by the Basic Science Research Program through the National Research Foundation of the Republic of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2016R1D1A1A09916450). References [1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math. 17 (1915), 12–22. [2] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429–446. [3] D. J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc. 52 (1975), 191–195. [4] I. B. Jung, Y. C. Kim and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operator, J. Math. Anal. Appl. 176 (1993), 138–147. [5] S. S. Miller and P. T. Mocanu, Univalent solutions of Briot-Bouquet differential subordinations, J. Differ. Equ. 58 (1985). 297–309.

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[6] S. S. Miller and P. T. Mocanu, Differential Subordination: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, No. 225, Marcel Dekker Incorporated, New York and Basel, 2000. [7] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan. Acad. Ser. A Math. Sci. 69 (1993), 234–237. [8] M. Nunokawa, T. Hayami and S. Owa, On the order of strongly starlikeness and order of starlikeness of a certain convex functions, Surikaisekikenkyusho Kokyuroku 1772 (2011), 85–90. [9] J. K. Prajapat and T. Bulboac˘ a, Differential subordination for subclasses of analytic functions involving Srivastava-Attiya operator, Thai J. Math. 15 (2017), 655–672. [10] Ch. Pommerenke, Univalent Functions, Vandenhoeck und Ruprecht, G¨ ottingen, 1975. [11] H. M. Srivastava and A. A. Attiya, An integral operator associated with the Hurwitz-Lerch zeta function and differential subordination, Integral Transforms Spec. Funct. 18 (2007), 207–216. [12] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001. [13] H. M. Srivastava, A. Prajapati and P. Gochhayat, Third-order differential subordination and differential superordination results for analytic functions involving the Srivastava-Attiya operator, Appl. Math. Inform. Sci. 12 (2018), 469–481. [14] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, Fourth Edition, Cambridge University Press, Cambridge, London and New York, 1927. [15] D. R. Wilken and J. Feng, A remark on convex and starlike functions, J. London Math. Soc. (Ser. 2) 21 (1980), 287–290.

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SHARP BOUNDS FOR THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS∗ XIAO-HUI ZHANG1,2 , YU-MING CHU3,∗∗ , AND WEN ZHANG4

Abstract. In the article, we prove that α = 3, β = log 4/(π/2 − log 4) = 7.51371 · · · , γ = 1/4 and δ = 1 + log 2 − π/2 = 0.122351 · · · are the best possible constants such that the double inequalities β+1 4 α+1 4 log 0 < K(r) < log 0 , β + r2 r α + r2 r     4 4 1 1 log 0 − γ r02 < E(r) < 1 + log 0 − δ r02 1+ 2 r 2 r √ R π/2 0 2 √ dθ hold for all r ∈ (0, 1), where r = 1 − r , and K(r) = 0 and 1−r 2 sin2 θ R π/2 p 2 2 E(r) = 0 1 − r sin θdθ are the complete elliptic integrals of the first and second kinds.

1. Introduction The complete elliptic integrals K(r) and E(r) [1-5] of the first and the second kinds are respectively defined by ( R π/2 K(r) = 0 √ dθ2 2 , 1−r sin θ

K(0) =

π 2,

K(1) = ∞

and 

E(r) = E(0) =

R π/2 p 1 − r2 sin2 θdθ, 0 π E(1) = 1. 2,

It is well known that the function r → K(r) is strictly increasing from (0, 1) onto (π/2, ∞) and the function r → E(r) is strictly decreasing from (0, 1) onto (1, π/2). The complete elliptic integrals K(r) and E(r) are the particular cases of the Gaussian hypergeometric function [6-15] F (a, b; c; x) =

∞ X (a)n (b)n xn (c)n n! n=0

(−1 < x < 1),

where (a)0 = 1 for a 6= 0, (a)n = a(a + 1)(a R+ 2) · · · (a + n − 1) = Γ(a + n)/Γ(a) ∞ is the shifted factorial function and Γ(x) = 0 tx−1 e−t dt (x > 0) is the gamma 2010 Mathematics Subject Classification. Primary: 33E15; Secondary: 33C05. Key words and phrases. complete elliptic integrals, monotonicity, bound. ∗ The research was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11771400, 11301127, 11701176, 11626101, 11601485), the Science and Technology Research Program of Zhejiang Educational Committee (Grant no. Y201635325) and the Natural Science Foundation of Zhejiang Sci-Tech University (Grant No. 16062023Y). ∗∗ Corresponding author: Yu-Ming Chu, Email: [email protected]. 1

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function [16-21]. Indeed,
  ∞ 1 2 π 1 1 π X 2 n 2n 2 r , K(r) = F , ; 1; r = 2 2 2 2 n=0 (n!)2
1
  ∞ 1 π 1 1 π X − 2 n 2 n 2n 2 E(r) = F − , ; 1; r = r . 2 2 2 2 n=0 (n!)2 The complete elliptic integrals play a very important role in the study of geometric function theory and they have numerous applications in various problems of physics and engineering. In particular, Many remarkable inequalities and elementary approximations for the complete elliptic integrals can be found in the literature [22-34]. In the sequel, we will use the symbols K and E for K(r) and E(r), respectively. √ Throughout this paper we let r0 = 1 − r2 for 0 < r < 1. Then we use the symbols K0 and E 0 for K(r0 ) and E(r0 ), respectively. Carlson and Gustafson [35] proved that the double inequality 1
0 for r ∈ (0, 1) and F is strictly increasing on (0, 1). It is easy to see that the limiting value F (0) = −a, and by [53, 112.01] F (1− ) = 0.  Theorem 2.2. Let β = log 4/(π/2 − log 4) = 7.51371 · · · . Then there exists s0 ∈ (0, 1) such that the function G(r) = (β + r2 )K − (β + 1) log(4/r0 ) is strictly increasing on (0, s0 ) and strictly decreasing on (s0 , 1) with the limiting values G(0+ ) = 0 = G(1− ). In particular, the inequality (2.2)

(β + 1) log(4/r0 ) 0 we clearly see that there exists r2 ∈ (0, 1) such that l is negative on (0, r2 ) and positive on (r2 , 1). We conclude that h is strictly decreasing on (0, r2 ) and strictly increasing on (r2 , 1). This together with the values h(0+ ) = 2π − 2 + (π/2 − 1)β = 1.05827 · · · , h(0.8) = −0.760875 · · · and h(1− ) = ∞ implies that there exists 0 < r3 < r4 < 1 such that h is positive on (0, r3 ) ∪ (r4 , 1) and negative on (r3 , r4 ). Hence g is strictly increasing on (0, r3 ) and (r4 , 1), and strictly decreasing on (r3 , r4 ). Since g(0+ ) = 0 = g(1− ), we conclude that there exists s0 ∈ (0, 1) such that g is positive on (0, s0 ) and negative on (s0 , 1). Therefore, the function G is strictly increasing on (0, s0 ) and strictly decreasing on (s0 , 1). It is easy to see that G(0+ ) = 0 and lim G(r) = lim (a + 1)(K − log(4/r0 )) − r02 K = 0. r→1−

r→1−

 Proof of Theorem 1.1. Inequality (1.2) follows from inequality (2.2) and the right-hand side inequality of (2.1) immediately. Lemma 2.3. The function 5 1 u(r) = (1 + r2 )E − r02 K − r2 + r4 2 2 is negative on (0, 1). Proof. Let the functions f and g be defined by f (r) = 3E + 2r2 − 5, 3(E − K) + 4. r2 Then Applying the derivative formulas we get d u(r) = rf (r), dr g(r) =

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5

d f (r) = rg(r). dr Since the function r 7→ (E−K)/r2 is strictly decreasing from (0, 1) onto (−∞, −π/4) (see [52, 3.43(11)]), the function g is strictly decreasing from (0, 1) onto (−∞, (16 − 3π)/4). Then from (16 − 3π)/4) > 0 we know that there exists r0 ∈ (0, 1) such that rg(r) is positive on (0, r0 ) and negative on (r0 , 1). Hence f is strictly increasing on (0, r0 ) and strictly decreasing on (r0 , 1). It is easy to see that f (0+ ) = 3π/2 − 5 < 0 and f (1− ) = 0. We conclude that there exists r1 ∈ (0, 1) such that rf (r) is negative on (0, r1 ) and positive on (r1 , 1). Therefore, the function u is strictly decreasing on (0, r1 ) and strictly increasing on (r1 , 1). Then from the facts that u(0+ ) = 0 = u(1− ) we get u(x) < 0 for all r ∈ (0, 1).  Theorem 2.4. The function 4 E −1 1 − log 0 r02 2 r is strictly decreasing from (0, 1) onto (−1/4, −δ) with δ = 1 + log 2 − π/2 = 0.122351 · · · . H(r) =

Proof. Differentiation yields 5 1 d rr04 H(r) = (1 + r2 )E − r02 K − r2 + r4 = u(r) < 0 dr 2 2 by Lemma 2.3. Hence, the function H is strictly decreasing on (0, 1). We clearly see that H(0+ ) = π/2 − 1 − log 2 = −δ. Let

1 4 h1 (r) = E − 1 − r02 log 0 , h2 (r) = r02 . 2 r Then h1 (1− ) = 0 = h2 (1− ), and by l’Hospital’s rule one has   h0 (r) 1 r02 K 4 1 1 E + K − log H(1− ) = lim 01 = lim − 2 + =− , r2 r0 2r 4 4 r→1− h2 (r) r→1− 2 where the last equality follows from the facts (see [52, 3.21(7) and (3.25)] or [53, 112.01] that   4 02 lim r K = 0, lim K − log 0 = 0. r r→1− r→1−  Proof of Theorem 1.2. Inequality (1.3) follows easily from Theorem 2.4 immediately. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U. S. Government Printing Office, Washington, 1964. [2] M.-K. Wang, Y.-M. Chu, S.-L. Qiu and Y.-P. Jiang, Convexity of the complete elliptic integrals of the first kind with respect to H¨ older means, J. Math. Anal. Appl., 2012, 388(2), 1141–1146. [3] Y.-M. Chu, M.-K. Wang, Y.-P. Jiang and S.-L. Qiu, Concavity of the complete elliptic integrals of the second kind with respect to H¨ older means, J. Math. Anal. Appl., 2012, 395(2), 637–642.

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[4] Y.-M. Chu, M.-K. Wang, S.-L. Qiu and Y.-P. Jiang, Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl., 2012, 63(7), 1177–1184. [5] M.-K. Wang, S.-L. Qiu, Y.-M. Chu and Y.-P. Jiang, Generalized Hersch-Pfluger distortion function and complete elliptic integrals, J. Math. Anal. Appl., 2012, 385(1), 221–229. [6] Zh.-H. Yang, Y.-M. Chu and M.-K. Wang, Monotonicity criterion for the quotient of power series with applications, J. Math. Anal. Appl., 2015, 428(1), 587–604. [7] M.-K. Wang, Y.-M. Chu and Y.-P. Jiang, Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions, Rocky Mountain J. Math., 2016, 46(2), 679–691. [8] M.-K. Wang, Y.-M. Chu and Y.-Q. Song, Asymptotical formulas for Gaussian and generalized hypergeometric functions, Appl. Math. Comput., 2016, 276, 44–60. [9] M.-K. Wang and Y.-M. Chu, Refinenemts of transformation inequalities for zero-balanced hypergeometric functions, Acta Math. Sci., 2017, 37B(3), 607–622. [10] M.-K. Wang, Y.-M. Li and Y.-M. Chu, Inequalities and infinite product formula for Ramanujan generalized modular equation function, Ramanujan J., 2018, 46(1), 189–200. [11] M.-K. Wang and Y.-M. Chu, Landen inequalities for a class of hypergeometric functions with applications, Math. Inequal. Appl., 2018, 21(2), 521–537. [12] M.-K. Wang, S.-L. Qiu and Y.-M. Chu, Infinite series formula for H¨ ubner upper bound function with applications to Hersch-Pfluger distortion function, Math. Inequal. Appl., 2018, 21(3), 629–648. [13] T.-H. Zhao, M.-K. Wang, W. Zhang and Y.-M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018, 2018, Article 251, 15 pages. [14] Y.-F. Yang, Special values of hypergeometric functions and periods of CM elliptic curves, Trans. Amer. Math. Soc., 2018, 370(9), 6433–6467. [15] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the arithmeticgeometric mean and complete elliptic integral of the first kind, J. Math. Anal. Appl., 2018, 462(2), 1714–1726 [16] T.-H. Zhao and Y.-M. Chu, A class of logarithmically completely monotonic functions associated with a gamma function, J. Inequal. Appl., 2010, 2010, Article ID 392431, 11 pages. [17] T.-H. Zhao, Y.-M. Chu and H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011, 2011, Article ID 896483, 13 pages. [18] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On rational bounds for the gamma function, J. Inequal. Appl., 2017, 2017, Article 210, 17 pages. [19] Zh.-H. Yang, W. Zhang and Y.-M. Chu, Sharp Gautschi inequality for parameter 0 < p < 1 with applications, Math. Inequal. Appl., 2017, 20(4), 1107–1120. [20] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the error function, Math. Inequal. Appl., 2018, 21(2), 469–479. [21] T.-R. Huang, B.-W. Han, X.-Y. Ma and Y.-M. Chu, Optimal bounds for the generalized Euler-Mascheronic constant, J. Inequal. Appl., 2018, 2018, Article 118, 9 pages. [22] Y.-M. Chu, M.-K. Wang and Y.-F. Qiu, On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function, Abstr. Appl. Anal., 2011, 2011, Article ID 697547, 7 pages. [23] M.-K. Wang, Y.-M. Chu, Y.-F. Qiu and S.-L. Qiu, An optimal power mean inequality for the complete elliptic integrals, Appl. Math. Lett., 2011, 24(6), 887–890. [24] Y.-M. Chu, M.-K. Wang, Y.-P. Jiang and S.-L. Qiu, Monotonicity, convexity, and inequalities invoving the Agard distortion function, Abstr. Appl. Anal., 2011, 2011, Article ID 671765, 8 pages. [25] Y.-M. Chu and M.-K. Wang, Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal., 2012, 2012, Article ID 830585, 11 pages. [26] Y.-M. Chu, M.-K. Wang and S.-L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci., 2012, 122(1), 41–51. [27] Y.-M. Chu and M.-K. Wang, Optimal Lehmer mean bounds for the Toader mean, Results Math., 2012, 61(3-4), 223–229. [28] Y.-M. Chu, Y.-F. Qiu and M.-K. Wang, H¨ older mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct., 2012, 23(7), 521–527. [29] M.-K. Wang and Y.-M. Chu, Asymptotical bounds for complete elliptic integrals of the second kind, J. Math. Anal. Appl., 2013, 402(1), 119–126.

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[30] Y.-M. Chu and T.-H. Zhao, Concavity of the error functions with respect to H¨ older means, Math. Inequal. Appl., 2016, 19(2), 589–595. [31] W.-M. Qian and Y.-M. Chu, Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters, J. Inequal. Appl., 2017, 2017, Article 274, 10 pages. [32] Zh.-H. Yang, W.-M. Qian and Y.-M. Chu, Monotonicity properties and bounds involving the complete elliptic integrals of the first kind, Math. Inequal. Appl., 2018, 21(4), 1185–1199. [33] T.-R. Huang, S.-Y. Tan, X.-Y. Ma and Y.-M. Chu, Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl., 2018, 2018, Article 239, 11 pages. [34] Zh.-H. Yang, Y.-M. Chu and W. Zhang, Accurate approximations for the complete elliptic of the second kind, J. Math. Anal. Appl., 2016, 438(2), 875–888. [35] B. C. Carlson and J. L. Gustafson, Asymptotic expansion of the first elliptic integral, SIAM J. Math. Anal., 1985, 16(5), 1072–1092. [36] R. K¨ uhnau, Eine Methode, die Positivit¨ at einer Funktion zu pr¨ ufen, Z. Angew. Math. Mech., 1994, 74(2), 140–143. [37] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (Ed.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010. [38] B.-Y. Long and Y.-M. Chu, Optimal power mean bounds for the weighted geometric mean of calssical means, J. Inequal. Appl., 2010, 2010, Article ID 905679, 6 pages. [39] W.-F. Xia, Y.-M. Chu and G.-D. Wang, The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means, Abstr. Appl. Anal., 2010, 2010, Article 604804, 9 pages. [40] Y.-M. Chu and W.-F. Xia, Two optimal double inequalities between power mean and logarithmic mean, Comput. Math. Appl., 2010, 60(1), 83–89. [41] Y.-M. Chu, Y.-F. Qiu and M.-K. Wang, Sharp power mean bounds for the combination of Seiffert and geometric means, Abstr. Appl. Anal., 2010, 2010, Article ID 108920, 12 pages. [42] Y.-M. Chu, M.-K. Wang and Y.-F. Qiu, An optimal double inequality between power-type Heron and Seiffert means, J. Inequal. Appl., 2010, Article ID 146945, 11 pages. [43] Y.-M. Chu, S.-S. Wang and C. Zong, Optimal lower power mean bound for the convex combination of harmonic and logarithmic means, Abstr. Appl. Anal., 2011, 2011, Article ID 520648, 9 pages. [44] G.-D. Wang, X.-H. Zhang and Y.-M. Chu, A power mean inequality for the Gr¨ otzsch ring function, Math. Inequal. Appl., 2011, 14(4), 833–837. [45] Y.-M. Li, B.-Y. Long, Y.-M. Chu and W.-M. Gong, Optimal inequalities for power mean, J. Appl. Math., 2012, 2012, Article ID 182905, 8 pages. [46] Y.-M. Li, B.-Y. Long and Y.-M. Chu, Sharp bounds by the power mean for the generalized Heronian mean, J. Inequal. Appl., 2012, 2012, Article 129, 9 pages. [47] W.-F. Xia, W. Janous and Y.-M. Chu, The optimal convex combination bounds of arithmetic and harmonic means in terms of power mean, J. Math. Inequal., 2012, 6(2), 241–248. [48] Y.-M. Li, B.-Y. Long and Y.-M. Chu, A best possible double inequality for power mean, J. Appl. Math., 2012, 2012, Article ID 379785, 12 pages. [49] Y.-M. Chu and B.-Y. Long, Bounds of the Neuman-S´ andor mean using power and identric means, Abstr. Appl. Anal., 2013, 2013, Article ID 832591, 6 pages. [50] G.-D. Wang, X.-H. Zhang and Y.-M. Chu, A power mean inequality involving the complete elliptic integrals, Rocky Mountain J. Math., 2014, 44(5), 1661–1667. [51] Y.-M. Chu, Zh.-H. Yang and L.-M. Wu, Sharp power mean bounds for S´ andor mean, Abstr. Appl. Anal., 2015, 2015, Article ID 172867, 5 pages. [52] G. D. Anderson, M. K. Vamanamurthy and M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997. [53] P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and Scientists, Springer-Verlag, New York, 1971.

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8

XIAO-HUI ZHANG1,2 , YU-MING CHU3,∗∗ , AND WEN ZHANG4

Xiao-Hui Zhang, 1 College of Science, Hunan City University, Yiyang 413000, Hunan, China; 2 Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, Zhejiang, China E-mail address: [email protected] Yu-Ming Chu (Corresponding author), versity, Huzhou 313000, Zhejiang, China E-mail address: [email protected]

3 Department

of Mathematics, Huzhou Uni-

Wen Zhang, 4 Friedman Brain Institute, Icahn School of Medicine at Mount Sinai, New York, NY 10029, USA E-mail address: [email protected]

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Symmetric identities for the second kind q-Bernoulli polynomials C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In [9], we studied the second kind q-Bernoulli numbers and polynomials. By using these numbers and polynomials, we investigated the zeros of the second kind q-Bernoulli polynomials. In this paper, by applying the symmetry of the fermionic p-adic q-integral on Zp , we give recurrence identities the second kind q-Bernoulli polynomials and the sums of powers of consecutive q-odd integers. Key words : Symmetric properties, the sums of powers of consecutive q-odd integers, the second kind Bernoulli numbers and polynomials, the second kind q-Bernoulli numbers and polynomials. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Bernoulli numbers, Bernoulli polynomials, q-Bernoulli numbers, q-Bernoulli polynomials, the second kind Bernoulli number, the second kind Bernoulli polynomials, Euler numbers, Euler polynomials, Genocchi numbers, Genocchi polynomials, tangent numbers, tangent polynomials, and Bell polynomials were studied by many authors (see for details [1-11]). Bernoulli numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. In [8], by using the second kind Euler numbers Ej and polynomials Ej (x), we investigated the alternating sums of powers of consecutive odd integers. Let k be a positive integer. Then we obtain Tj (k − 1) =

k−1 ∑

(−1)n (2n + 1)j =

n=0

(−1)k+1 Ej (2k) + Ej . 2

In [9], we introduced the second kind q-Bernoulli numbers Bn,q and polynomials Bn,q (x). By using computer, we observed an interesting phenomenon of ‘scattering’ of the zeros of the second kind q-Bernoulli polynomials Bn,q (x) in complex plane. Also we carried out computer experiments for doing demonstrate a remarkably regular structure of the complex roots of the second kind q-Bernoulli polynomials Bn,q (x). The outline of this paper is as follows. We introduce the second kind qBernoulli numbers Bn,q and polynomials Bn,q (x). In Section 2, we obtain the sums of powers of consecutive q-odd integers. Finally, we give recurrence identities the second kind q-Bernoulli polynomials and the sums of powers of consecutive q-odd integers. Throughout this paper, we always make use of the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, R denotes the set of real numbers, C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally 1 assume that |q| < 1. If q ∈ Cp , we normally assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function},

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the p-adic q-integral was defined by [1, 2, 3, 4, 6] ∫

p −1 1 ∑ Iq (g) = g(x)dµq (x) = lim g(x)q x . N →∞ [pN ] Zp x=0 N

The bosonic integral was considered from a physical point of view to the bosonic limit q → 1, as follows: ∫ pN −1 1 ∑ I1 (g) = lim Iq (g) = g(x)dµ1 (x) = lim N g(x) (see [1, 2, 3, 5]). (1.1) q→1 N →∞ p Zp x=0 By (1.1), we easily see that I1 (g1 ) = I1 (g) + g ′ (0), cf. [1, 2, 3, 4, 6, 7],

(1.2)

dg(x) where g1 (x) = g(x + 1) and g ′ (0) = . dx x=0 First, we introduce the second kind Bernoulli numbers Bn and polynomials Bn (x). The second kind Bernoulli numbers Bn and polynomials Bn (x) are defined by means of the following generating functions (see [7]):

∞ ∑ 2tet tn F (t) = 2t = Bn , e − 1 n=0 n!

and

∞ 2tet xt ∑ tn F (x, t) = 2t e = Bn,q (x) , e −1 n! n=0

respectively. The second kind q-Bernoulli polynomials, Bn,q (x) are defined by means of the generating function: Fq (x, t) =

∞ (log q + 2t)et xt ∑ tn e = . B (x) n,q qe2t − 1 n! n=0

(1.3)

The second kind q-Bernoulli numbers Bn,q are defined by means of the generating function: Fq (t) =

∞ ∑ (log q + 2t)et tn = Bn,q . 2t qe − 1 n! n=0

(1.4)

In (1.2), if we take g(x) = q x e(2x+1)t , then we have ∫ (log q + 2t)et q x e(2x+1)t dµ1 (x) = . q h e2t − 1 Zp

(1.5)

for |t| ≤ p− p−1 . In (1.2), if we take g(x) = e2nxt , then we also have ∫ 2nt e2nxt dµ1 (x) = 2nt . e −1 Zp 1

(1.6)

It will be more convenient to write (1.2) as the equivalent bosonic integral form ∫ ∫ g(x + 1)dµ1 (x) = g(x)dµ1 (x) + g ′ (0), (see [1,2,3,4,6]). Zp

(1.7)

Zp

For n ∈ N, we also derive the following bosonic integral form by (1.7), ∫

∫ Zp

g(x + n)dµ1 (x) =

Zp

g(x)dµ1 (x) +

n−1 ∑ k=0

655

g ′ (k), where g ′ (k) =

dg(x) . dx x=k

(1.8)

RYOO 654-659

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.4, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

In [9], we introduced the second kind q-Bernoulli numbers Bn,q and polynomials Bn,q (x) and investigate their properties. The following elementary properties of the second kind q-Bernoulli numbers Bn,q and polynomials Bn,q (x) are readily derived form (1.1), (1.2), (1.3) and (1.4). We, therefore, choose to omit details involved. Theorem 1(Witt formula). For q ∈ Cp with |1 − q|p < p− p−1 , we have ∫ q x (2x + 1)n dµ1 (x) = Bn,q , 1

Zp

∫ Zp

q y (x + 2y + 1)n dµ1 (y) = Bn,q (x).

Theorem 2. For any positive integer n, we have Bn,q (x) =

n ( ) ∑ n

k

k=0

Bk,q xn−k .

Theorem 3(Distribution Relation). For any positive integer m, we obtain Bn,q (x) = m

n−1

m−1 ∑

( i

q Bn,qm

i=0

2i + x + 1 − m m

) for n ≥ 0.

2. Symmetry identities for the second kind q-Bernoulli polynomials In this section, we assume that q ∈ Cp . In [2], Kim investigated interesting properties of symmetry p-adic invariant integral on Zp for Bernoulli polynomials and Bernoulli polynomials. By using same method of [3], expect for obvious modifications, we obtain recurrence identities the second kind q-Bernoulli polynomials. By (1.7), we obtain (∫ ) ∫ 1 x n (2x+2n+1)t x (2x+1)t q q e dµ1 (x) − q e dµ1 (x) h log q + 2t Zp Zp ∫ (2.1) n Zp q x e(2x+1)t dµ1 (x) = ∫ . q nx e2ntx dµ1 (x) Zp By (1.8), we obtain 1 h log q + 2t

(∫ q q e Zp

=

)

∫ x n (2x+2n+1)t

∞ ∑

(n−1 ∑

k=0

i=0

dµ1 (x) − )

i

q (2i + 1)

k

x (2x+1)t

q e Zp

dµ1 (x) (2.2)

tk . k!

For each integer k ≥ 0, let Ok,q (n) = 1k + q3k + q 2 5k + q 3 7k + · · · + q n (2n + 1)k . The above sum Ok,q (n) is called the sums of powers of consecutive q-odd integers. From the above and (2.2), we obtain (∫ ) ∫ ∞ ∑ 1 tk tk x n (2x+2n+1)t x (2x+1)t q q e dµ1 (x) − q e dµ1 (x) = Ok,q (n − 1) . log q + 2t k! k! Zp Zp

(2.3)

k=0

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Thus, we have ( ∫ ∫ ∞ ∑ n x k q q (2x + 2n + 1) dµ1 (x) − Zp

k=0

Zp

) x

k

q (2x + 1) dµ1 (x)

∞

∑ tk tk = (log q + 2t)Ok,q (n − 1) . k! k! k=0

k

t in the above equation, we arrive at the following theorem: k! Theorem 4. Let k be a positive integer. Then we obtain

By comparing coefficients

q n Bn,q (2n) − Bn,q = log qOk,q (n − 1) + 2kOk−1,q (n − 1).

(2.4)

Remark 5. For the sums of powers of consecutive odd integers, we have lim (log qOk,q (n − 1) + 2kOk−1,q (n − 1)) = 2k

q→1

n−1 ∑

(2i + 1)k−1 = Bk (2n) − Bk for k ∈ N.

i=0

By using (2.1) and (2.3), we arrive at the following theorem: Theorem 6. Let n be positive integer. Then we have ∫ ∞ ∑ n Zp q x e(2x+1)t dµ1 (x) tm ∫ = . (O (n − 1)) m,q m! q nx e2ntx dµ1 (x) Zp m=0

(2.5)

Let w1 and w2 be positive integers. By using (1.5) and (1.6), we have ∫ ∫ q (w1 x1 +w2 x2 ) e(w1 (2x1 +1)+w2 (2x2 +1)+w1 w2 x)t dµ1 (x1 )dµ1 (x2 ) Zp Zp ∫ q w1 w2 x e2w1 w2 xt dµ1 (x) Zp

(2.6)

(log q + 2t)ew1 t ew2 t ew1 w2 xt (q w1 w2 e2w1 w2 t − 1) = . (q w1 e2w1 t − 1)(q w2 e2w2 t − 1) By using (2.4) and (2.6), after elementary calculations, we obtain ( )( ∫ ) ∫ w1 Zp q w2 x2 e(2x2 +1)(w2 t) dµ1 (x2 ) 1 w1 x1 (w1 (2x1 +1)+w1 w2 x)t ∫ a= q e dµ1 (x1 ) w1 Zp q w1 w2 x e2w1 w2 tx dµ1 (x) Zp ( ) ( ) ∞ ∞ ∑ tm 1 ∑ tm m m = Om,qw2 (w1 − 1)w2 . Bm,qw1 (w2 x)w1 w1 m=0 m! m! m=0 By using Cauchy product in the above, we have   ∞ m ( ) ∑ ∑ m tm j−1 m−j  a= Bj,qw1 (w2 x)w1 Om−j,qw2 (w1 − 1)w2  . j m! m=0 j=0

(2.7)

(2.8)

Again, by using the symmetry in (2.7), we have ( )( ∫ ) ∫ w2 Zp q w1 x1 e(2x1 +1)(w1 t) dµ1 (x1 ) 1 w2 x2 (w2 (2x2 +1)+w1 w2 x)t ∫ a= q e dµ1 (x2 ) w2 Zp q w1 w2 x e2w1 w2 tx dµ1 (x) Zp ( ) ( ) ∞ ∞ ∑ tm 1 ∑ tm m m = . Bm,qw2 (w1 x)w2 Om,qw1 (w2 − 1)w1 w2 m=0 m! m! m=0 Thus we have



 m ( ) ∑ m tm  a= . Bj,qw2 (w1 x)w2j−1 Om−j,qw1 (w2 − 1)w1m−j  m! j m=0 j=0 ∞ ∑

657

(2.9)

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tm on the both sides of (2.8) and (2.9), we arrive at the following theorem: m! Theorem 7. Let w1 and w2 be positive integers. Then we obtain

By comparing coefficients

m ( ) ∑ m Bj,qw1 (w2 x)w1j−1 Om−j,qw2 (w1 − 1)w2m−j j j=0 m ( ) ∑ m = Bj,qw2 (w1 x)w2j−1 Om−j,qw1 (w2 − 1)w1m−j , j j=0

where Bk,q (x) and Om,q (k) denote the second kind q-Bernoulli polynomials and the sums of powers of consecutive q-odd integers, respectively. By using Theorem 2, we have the following corollary: Corollary 8. Let w1 and w2 be positive integers. Then we have j ( )( ) m ∑ ∑ m j w1m−k w2j−1 xj−k Bk,qw2 Om−j,qw1 (w2 − 1) j k j=0 k=0

j ( m ∑ ∑

=

j=0 k=0

)( ) m j wj−1 w2m−k xj−k Bk,qw1 Om−j,qw2 (w1 − 1). j k 1

By using (2.6), we have ( )( ∫ ) ∫ w1 Zp q w2 x2 e(2x2 +1)(w2 t) dµ1 (x2 ) 1 w1 w2 xt w1 x1 (2x1 +1)w1 t ∫ a= e q e dµ1 (x1 ) w1 q w1 w2 x e2w1 w2 tx dµ1 (x) Zp Zp  ( ) w −1 ∫ 1 ∑ 1 w1 w2 xt = e q w1 x1 e(2x1 +1)w1 t dµ1 (x1 )  q w2 j e(2j+1)(w2 t)  w1 Zp j=0 =

w∑ 1 −1

q w2 j

q w1 x1 e

=

 

n=0

2x1 +1+w2 x+(2j+1)

w2 w1

)

(2.10)

(w1 t)

dµ1 (x1 )

Zp

j=0 ∞ ∑

(

∫

w∑ 1 −1

( q w2 j Bn,qw1

j=0

w2 w2 x + (2j + 1) w1

)

 w1n−1 

tn . n!

Again, by using the symmetry property in (2.10), we also have ( )( ∫ ) ∫ w2 Zp q w1 x1 e(2x1 +1)(w1 t) dµ1 (x1 ) 1 w1 w2 xt w2 x2 (2x2 +1)w2 t ∫ e q e dµ1 (x2 ) a= w2 q w1 w2 x e2w1 w2 tx dµ1 (x) Zp Zp   ( ) w −1 ∫ 2 ∑ 1 w1 w2 xt = q w2 x2 e(2x2 +1)w2 t dµ1 (x2 )  e q w1 j e(2j+1)(w1 t)  w2 Zp j=0 =

w∑ 2 −1

q w1 j

=

n=0

 

q w2 x2 e

2x2 +1+w1 x+(2j+1)

w1 w2

)

w∑ 2 −1

( q w1 j Bn,qw2

w1 x + (2j + 1)

j=0

By comparing coefficients

w1 w2

(2.11)

(w2 t)

dµ1 (x2 )

Zp

j=0 ∞ ∑

(

∫

)

 w2n−1 

tn . n!

tn on the both sides of (2.10) and (2.11), we have the following theorem. n!

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Theorem 9. Let w1 and w2 be positive integers. Then we obtain ( ) w∑ 1 −1 w2 w1n−1 q w2 j Bn,qw1 w2 x + (2j + 1) w 1 j=0 =

w∑ 2 −1 j=0

q

w1 j

Bn,qw2

(2.12)

( ) w1 w1 x + (2j + 1) w2n−1 . w2

Substituting w1 = 1 into (2.12), we arrive at the following corollary. Corollary 10. Let w2 be positive integer. Then we obtain ( ) w∑ 2 −1 x − w2 + (2j + 1) n−1 j . Bn,q (x) = w2 q Bn,qw2 w2 j=0 This last result(Corollary 10) is shown to yield the known Distribution Relation of the second kind q-Bernoulli polynomials(Theorem 3). Acknowledgment This work was supported by 2019 Hannam University Research Fund. REFERENCES 1. Kim, T.(2002). q-Volkenborn integration, Russ. J. Math. Phys., v.9, pp. 288-299. 2. Kim, T.(2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 3. Kim, T.(2008) Symmetry p-adic invariant integral on Zp for Bernoulli and Euler polynomials, Journal of Difference Equations and Applications, v.12, pp. 1267-1277. 4. Kim, T., Jang, L. C., Pak, H. K.(2001). A note on q-Euler and Genocchi numbers, Proc. Japan Acad., v.77 A, pp. 139-141. 5. Liu, G.(2006). Congruences for higher-order Euler numbers, Proc. Japan Acad., v.82 A, pp. 30-33. 6. Ryoo, C.S., Kim, T., Jang, L. C.(2007). Some relationships between the analogs of Euler numbers and polynomials, Journal of Inequalities and Applications, v.2007, ID 86052, pp. 122. 7. Ryoo, C.S.(2011). Distribution of the roots of the second kind Bernoulli polynomials, Journal of Computational Analysis and Applications, v.13, pp. 971-976. 8. Ryoo, C.S.(2011). On the alternating sums of powers of consecutive odd integers, Journal of Computational Analysis and Applications, v.13, pp. 1019-1024. 9. Ryoo, C.S.(2013). Calculating zeros of q-extension of the second kind Bernoulli polynomials, Journal of Computational Analysis and Applications, v.15, pp. 248-254. 10. Ryoo, C.S.(2016). Differential equations associated with generalized Bell polynomials and their zeros, Open Mathematics, v.14, pp. 807-815. 11. Ryoo, C.S.(2016) Differential equations associated with tangent numbers, J. Appl. Math. & Informatics, v.34, pp. 487-494.

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On some finite difference methods on the Shishkin mesh for the singularly perturbed problem

∗

Quan Zheng†, Ying Liu, Jie He College of Sciences, North China University of Technology, Beijing 100144, China

Abstract: This paper studies the convergence behavior of three finite difference schemes on the Shishkin mesh to solve the singularly perturbed two-point boundary value problem. Three new error estimates are proved for the hybrid finite difference scheme that combines the midpoint upwind scheme on the coarse part with the central difference scheme on the fine part, the midpoint upwind scheme and the simple upwind scheme, respectively. Finally, numerical experiments illustrate that these error estimates are sharp and the convergence is uniform with respect to the perturbation parameter. Keywords: Singularly perturbed boundary value problem; Finite difference scheme; Piecewise equidistant mesh; Error estimate; Uniform convergence

1

Introduction Consider the singularly perturbed two-point boundary value problem:  Lu(x) := −εu00 (x) + b(x)u0 (x) + c(x)u(x) = f (x), x ∈ (0, 1), u(0) = A, u(1) = B,

(1)

where 0 < ε  1 is a small perturbation parameter, A and B are given constants, and the functions b(x), c(x) and f (x) are sufficiently smooth satisfying 0 < β < b(x) < β ∗ and 0 ≤ c(x) < γ ∗ , where β, β ∗ and γ ∗ are constants. Under these conditions, the singularly perturbed problem (1) has a unique solution that possesses a boundary layer at x = 1 (see [1–4]). Among various numerical methods to solve singularly perturbed problems, finite difference schemes on layer-adapted meshes for the singularly perturbed two-point boundary value problem have been widely studied in the literature, see [1–10]. The simple upwind scheme was proved to have the error estimate O(N −1 ) on the coarse part and O(N −1 ln N ) on the fine part on the Shishkin mesh and the error estimate O(N −1 ) on the whole interval on the BakhvalovShishkin mesh, see, e.g., [5, 6]. The central difference scheme on the Shishkin mesh was proved ∗ †

This paper is supported by Natural Science Foundation of China (No. 11471019). E-mail: [email protected] (Q. Zheng).

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to have the convergence O(N −2 ln2 N ) on the whole nodes by discrete Green’s functions, although the computed solution had small oscillations on the coarse part, see [3, 7]. In order to avoid oscillation, the midpoint upwind scheme on the Shishkin mesh was constructed and the convergence O(max{N −2 , N −5+4i/N ln N }) were proved for i = 1, · · · , N in [8]. The midpoint upwind scheme on the Bakhvalov-Shishkin mesh was shown to be of order O(N −2 ) on the coarse part and O(N −1 ) on the fine part in [9]. To improve the convergence behaviour of boundary layer, the hybrid finite difference scheme was proposed and the convergence O(N −2 ) on the coarse part and O(N −2 ln2 N ) on the fine part were proved for (1.1) with c(x) ≡ 0 in [8] and with geeneral c(x) ≥ 0 in [4]. In this paper, we construct the hybrid finite difference scheme on the Shishkin mesh to slove (1) with c(x) ≥ 0, not only give the suitable conditions especially for the c(x) to guarantee an associated M -matrix and the discrete maximum principle, but also obtain a better error estimate. Furthermore, new error estimates for the midpoint upwind scheme and the simple upwind scheme are also obtained. Finally, the convergence behaviours according to these new error estimates for these schemes are confirmed by numerical experiments. Note: Throughout the paper, the nontrivial case ε ≤ CN −1 is considered, C denotes a generic positive constant that is independent of both perturbation parameter ε and mesh parameter N , and C can take different values at each occurrence, even in the same argument.

2

Error estimates on the Shishkin mesh

Lemma 1 (see [1–3] The solution u(x) of (1) can be decomposed as u(x) = S(x) + E(x) on [0, 1], where the smooth part S satisfies LS(x) = f (x) and | S (i) (x) |≤ C, 0 ≤ i ≤ q, while the layer part E satisfies LE(x) = 0 and |

E (i) (x)

|≤

Cε−i exp

  β(1 − x) , 0 ≤ i ≤ q, − ε

where the maximal order q depends on the smoothness of theodata.  n 1 4ε Let N be a positive even integer and τ = min 2 , β ln N . Since the singularly perturbed problem is considered, we generally take ε ≤ CN −1 and τ =

4ε β

ln N . Choose 1 − τ be the

transition point. Divide [0, 1 − τ ] and [1 − τ, 1] uniformly into N/2 subintervals, respectively. Then the Shishkin mesh is:  N 2(1 − τ )   i, 0 ≤ i ≤ , N   2 xi = i N  1 − 2τ 1 − , ≤ i ≤ N. N 2

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Lemma 2 Denote hi = xi − xi−1 for (2), then N −1 ≤ hi < 2N −1 and hN/2+i = i = 1, 2, · · · , N/2.

8ε −1 N ln N for β



We construct the following hybrid finite difference scheme:  − N N −εD+ D− uN + b i−1/2 D ui + ci−1/2 ui−1/2 = fi−1/2 , 0 < i ≤ N/2, i LN uN := i −εD+ D− uN + b D0 uN + c uN = f , N/2 < i < N, i

i

i i

i

i

(3)

N uN 0 = A, uN = B, N N uN uN i − ui−1 i+1 − ui − N , D ui = , D+ D− uN where define as a discrete operator, = i = hi+1 hi
N N − N
uN uN 2 D+ uN i−1 + ui i+1 − ui−1 i − D ui , D0 uN , uN = , b = b x , bi = b (xi ), i−1/2 i−1/2 i = i−1/2 hi+1 + hi
hi+1 + hi 2 fi−1/2 = f xi−1/2 and so on.

LN

D+ uN i

The scheme (3) is slightly different from the scheme (2.86) in [4] in the discretization of cu at xi−1/2 . The scheme (3) gives the following expression: LN uN i

 bi−1/2 ci−1/2 ci−1/2 bi−1/2 2ε 2ε 2ε N N N − 2 )ui − ( hi (hi +hi+1 ) + hi − 2 )ui−1 , hi+1 (hi +hi+1 ) ui+1 + ( hi hi+1 + hi + = bi bi −( 2ε 2ε 2ε N N N hi+1 (hi +hi+1 ) − hi +hi+1 )ui+1 + ( hi hi+1 + ci )ui − ( hi (hi +hi+1 ) + hi +hi+1 )ui−1 .

γ∗ N 4β ∗ and > , then the operator LN β ln N β defined by (3) on (2) satisfies the discrete comparison principle, i.e., let {vi } and {wi } are mesh

Lemma 3 (Discrete comparison principle) If N >

functions, if v0 ≤ w0 , vN ≤ wN and LN vi ≤ LN wi for i = 1, 2, · · · , N − 1, then vi ≤ wi for all i. Proof. Under the conditions of Lemma 3, the coefficient matrix associated with LN by the above expression is clearly an (N − 1) × (N − 1) strictly diagonally dominant matrix, and has positive diagonal entries and non-positive off diagonal entries. So it is an irreducible M -matrix. Hence, the operator satisfies the discrete comparison principle.



So, the scheme (3) on (2) has a unique solution and the function wi is defined as a barrier function for vi by Lemma 3.  i  Y βhj Lemma 4 Set Z0 = 1, define the mesh function Zi = 1+ for i = 1, 2, · · · , N . Then 2ε j=1

the operator

LN

of (3) satisfies LN Zi ≥

C Zi for i = 1, 2, · · · , N − 1. max{ε, hi }

Proof. Clearly D + Zi = Hence −εD+ D− Zi = −

β β Zi and D− Zi = Zi . 2ε 2ε + βhi


β 2 hi 2ε D + Zi − D − Zi = − Zi , hi+1 + hi (hi+1 + hi ) (2ε + βhi )

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and

hi+1 D+ Zi + hi D− Zi D Zi = = hi+1 + hi 0



β β 2 hi+1 hi + 2ε + βhi 2ε (hi+1 + hi ) (2ε + βhi )

 Zi .

Thus, from (3), by using c(x) ≥ 0 and b(x) > β > 0, we have β 2 hi β Zi Zi + bi−1/2 · (hi+1 + hi ) (2ε + βhi ) 2ε + βhi   βhi β bi−1/2 − Zi , i = 1, 2, · · · , N/2, = 2ε + βhi hi+1 + hi   β 2 hi β β 2 hi+1 hi N L Zi ≥ − + Zi Zi + bi (hi+1 + hi ) (2ε + βhi ) 2ε + βhi 2ε (hi+1 + hi ) (2ε + βhi )   β βhi ≥ bi − Zi , i = N/2 + 1, · · · , N − 1, 2ε + βhi hi+1 + hi

LN Z i ≥ −

and obtain the result.  Lemma 5 For the Shishkin mesh (2), there exists a constant C such that  N  Y βhj −1 1+ ≤ CN −4(1−i/N ) for N/2 ≤ i < N. 2ε

j=i+1

Proof. By Lemma 4.1(b) in [1] and noting hj = h for j = N/2 + 1, · · · , N , we have   N  N  Y Y −1 βhj −1 βh −1 = 1+ ≤ e−β(1−xi )/(2ε+βh) = e−4(1−i/N ) ln N/(1+4N ln N ) . 1+ 2ε 2ε

j=i+1

j=i+1

Then as the proof of Lemma 3.2 in [8], the result is proved.



Lemma 6 Assuming that u(x) be sufficiently smooth function defined on [0, 1], for the truncation error of (3) on the Shishkin mesh "Z to solve (1), thereZexists a constant C such# that xi xi+1 N L (ui ) − (Lu)(xi−1/2 ) ≤ C (|u000 (t)| + |u00 (t)|)dt , i = 1, · · · , N/2, ε|u000 (t)|dt + hi xi−1 xi−1 Z xi+1   N L (ui ) − (Lu) (xi ) ≤ Chi ε u(4) (t) + u000 (t) dt, i = N/2 + 1, · · · , N − 1. xi−1

Proof. The results follow by noting that c(x)u(x) contributes
ci−1/2 (u(xi−1 ) + u(xi )) /2 − u xi−1/2 ≤ Chi

Z

xi

00 u (t) dt

xi−1

for i = 1, 2, · · · , N/2 to the truncation error in the Lemma 2.4 in [8] and zero for i = N/2 + 1, · · · , N − 1 to the truncation error in Theorem 3.2 in [8], respectively.  Similarly, the numerical solution can also be split into the smooth part and the layer part N N N N N N N by uN i = Si + Ei , where Si satisfies L Si = fi−1/2 , i = 1, 2, · · · , N/2, L Si = fi , i = N = S , and E N satisfies LN E N = 0, i = 1, 2, · · · , N − 1, N/2 + 1, · · · , N − 1, S0N = S0 and SN N i i N = E , therefore E0N = E0 and EN N

N N ui − uN i ≤ Si − Si + Ei − Ei .

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γ∗ N 4β ∗ and > , then for the smooth part of the solutions of (1) and (3) β ln N β on the Shishkin mesh, there exists a constant C such that

Lemma 7 If N >

Si − SiN ≤ CN −2 for all i. Proof. By Lemma 1 and Lemma 6, we have   LN (Si ) − (LS)(x N ) ≤ C(hi + hi+1 )(ε + hi ), i = 1, · · · , N/2, i−1/2 N L (Si − Si ) =  LN (S ) − (LS) (x ) ≤ Ch (h + h )(ε + 1), i = N/2 + 1, · · · , N − 1. i i i i i+1
Set wi = C0 N −1 ε + N −1 xi for all i, where constant C0 is chosen sufficiently large. Then  −1 (ε + N −1 ) + c −1 (ε + N −1 ), i = 1, · · · , N/2, b i−1/2 C0 N i−1/2 (wi−1 + wi ) /2 ≥ CN LN wi = b C N −1 (ε + N −1 ) + c w ≥ CN −1 (ε + N −1 ), i = N/2 + 1, · · · , N − 1. i

0

i

i


Therefore, LN wi ≥ LN Si − SiN for i = 1, · · · , N − 1. Clearly, w0 = 0 = S0 − S0N and
N . By Lemma 3, w is a barrier function for S − S N wN = C0 N −1 ε + N −1 ≥ 0 = SN − SN i i i and then the proof is completed.  N 4β ∗ γ∗ and > , then for the layer part of the solutions of (1) and (3) Lemma 8 If N > β ln N β on the Shishkin mesh, there exists a constant C such that Ei − EiN ≤ CN −2 for i = 0, 1, · · · , N/2. Proof. For i = 0, 1, · · · , N/2, from Lemma 1, we have β(1 − xN/2 ) β(1 − xi ) β(1 − xi ) − − ε 2ε 2ε |Ei | ≤ Ce ≤ Ce ≤ Ce = CN −2 . −

(5)

Recall the function Zi in Lemma 4. Now et ≥ 1 + t for all t ≥ 0. So,  N  N Y Y βhj −1 Zi = 1+ ≥ e−βhj /(2ε) = e−β(1−xi )/(2ε) . ZN 2ε j=i+1

(6)

j=i+1

Zi for all i, where constant C0 is chosen sufficiently large. From Lemma 4, we ZN N have L Yi = C0 /ZN · LN Zi ≥ 0 = LN EiN for i = 1, · · · , N − 1. By (6) and Lemma 1, N β β . Thus, by Y0 = C0 Z0 /ZN ≥ C0 e− 2ε ≥ C0 e− ε ≥ |E (0)| = E0N and YN = C0 ≥ |E (1)| = EN

Let Yi = C0

Lemma 3, we have  N  Y N βhj −1 Ei ≤ Yi = C 1+ for all i. 2ε

(7)

j=i+1

By Lemma 5, we have N Ei ≤ C

N Y j=N/2+1

  βhj −1 1+ ≤ CN −2 for i = 0, 1, · · · , N/2. 2ε

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Consequently, combining this inequality with (5), the proof is completed.  γ∗ N 4β ∗ Lemma 9 If N > and > , then for the layer part of the solutions of (1) and (3) β ln N β on the Shishkin mesh, there exists a constant C such that n o Ei − EiN ≤ C max N −2 , N −6+4i/N ln2 N for i = N/2 + 1, · · · , N. Proof. By Lemmas 6, 1 and 2, we have N
L Ei − EiN = LN (Ei ) − (LE) (xi ) Z xi+1   ≤ Chi ε E (4) (t) + E 000 (t) dt xi−1   Z xi+1 β(1 − t) −3 dt ≤ Chi ε exp − ε xi−1 ε βhi −β(1−xi )/ε = Cε−3 hi · sinh ·e β ε ≤ Cε−3 h2i e−β(1−xi )/(2ε) ≤ Cε−1 N −2 ln2 N

  N  N  Y Y βhj −1 βhj −1 , since e−β(1−xi )/(2ε) ≤ 1+ 1+ 2ε 2ε j=i+1

j=i+1 −1

= Cε

N

−2

2

ln N · Zi /ZN .

 Set φi = C0 N −2 + N −2 ln2 N · Zi /ZN for i = N/2, · · · , N , where constant C0 is chosen
sufficiently large. By Lemma 4, we have LN φi ≥ CN −2 ln2 N/ZN · LN Zi ≥ LN Ei − EiN N by Lemma 8 and for i = N/2 + 1, · · · , N − 1. Clearly, φN/2 ≥ C0 N −2 ≥ EN/2 − EN/2 N N φN ≥ 0 = EN − EN . Thus, φi is a barrier function for Ei − Ei by Lemma 3. And by Lemma 5, the result follows. 

N 4β ∗ γ∗ and > , the hybrid finite difference scheme (3) β ln N β on the Shishkin mesh (2) for (1) satisfies: n o −2 −6+4i/N 2 ui − uN ≤ C max N , N ln N for i = 1, · · · , N. (8) i

Theorem 1 Assuming that N >

Furthermore,   CN −2 , 0 ≤ i ≤ ph N, ui − uN i ≤  CN −2 ln2 N, p N < i ≤ N,

(9)

h

1 ≈ 0.8161. 2e Proof. From (4) and Lemmas 7, 8 and 9, we have the error estimate (8).

where ph = 1 −

Furthermore, since N −6+4i/N ln2 N = N −2 N −4+4i/N ln2 N , we consider the function f (x) = x−4+4ph ln2 x, x > 1.

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From f 0 (x) = x−4+4ph −1 ln x [(−4 + 4ph ) ln x + 2] = 0, we have x = e1/(2−2ph ) . So, max{f (x)} = x>1

then N −4+4ph ln2 N ≤ Therefore, (9) is proved.



Theorem 2 Assuming that N >

1 , 4(1 − ph )2 e2 1 = 1. 4(1 − ph )2 e2

γ∗ , the midpoint upwind scheme on the Shishkin mesh (2) for β

(1) satisfies:   CN −2 , 0 ≤ i ≤ pm N, ui − uN i ≤  CN −1 ln N, p N < i ≤ N,

(10)

m

1 3 − ≈ 0.6580. 4 4e Proof. Under the hypothesis of Theorem 2, the matrix associated with the midpoint upwind

where pm =

scheme is an M -matrix.  In [8], it is shown that Ei − EiN ≤ C max N −2 , N −5+4i/N ln N for i = N/2, · · · , N . Combining this with (3.1) and (3.2) in [8] yields the result: n o −2 −5+4i/N ui − uN ≤ C max N , N ln N for all i. i Further, as the proof of Theorem 1, Theorem 2 follows.

(11)



Theorem 3 The simple upwind scheme on the Shishkin mesh (2) for (1) satisfies:  CN −1 , 0 ≤ i ≤ ps N, ui − uN i ≤ CN −1 ln N, p N < i ≤ N, s

(12)

1 ≈ 0.8161. 2e Proof. The matrix associated with the simple upwind scheme is an M -matrix. From Lemma 2.95 in [3], we know LN (Ei − EiN ) ≤ Cε−1 N −1 e−β(1−xi )/ε .  Set a new φi = C0 N −1 + N −1 ln N · Zi /ZN for i = N/2, · · · , N , where constant C0 is chosen sufficiently large. It is easy to verify that Ei − E N ≤ φi for i = N/2, · · · , N , by the

where ps = 1 −

i

discrete comparison principle. −1 ln N for i = N/2 + 1, · · · , N . As the proof of and hi = 4ε βN  βh −1 ≤ CN −2(1−i/N ) for N/2 ≤ i < N . Thus Ei − EiN ≤ Lemma 5, we have 1 + 2εj j=i+1  C max N −1 , N −3+2i/N ln N for i = N/2, · · · , N . Combining this inequality with Lemma 2.86

Note that τ =

2ε β ln N N  Q

and Corollary 2.95 in [3], we have n o −1 ui − uN , N −3+2i/N ln N for all i. i ≤ C max N

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Consequently, as the proof of Theorem 1, the proof is completed.



Remark. In this paper, for the hybrid scheme and the midpoint upwind scheme, the condition N >

γ∗ β ,

not in [4] and [8], is added in Theorems 1 and 2 for c(x) ≥ 0 in (1). Moreover, the 1 2,

constants ph , pm and ps are much larger than

and the factor C of new error estimates are

uniform to ε and N .

3

Numerical results

Example 1 (see [10]). Consider the singularly perturbed problem  −εy 00 + 1 y 0 + 1 y = f (x), 0 < x < 1, x+1 x+2  − 1ε y(0) = 1 + 2 , y(1) = e + 2, 1

1

where f (x) is chosen such that y(x) = ex + 2− ε (x + 1)1+ ε is the exact solution. The numerical results of Example 1 by the hybrid scheme (3) on (2) are shown in Table 1, max0 pm N 1.476 0.3578 0.582 0.2550 0.629 0.1735 0.635 0.1091 0.635 0.0655 0.635 0.0381 0.634 0.0216 0.632 0.0120

order —— 0.489 0.556 0.670 0.736 0.782 0.819 0.848

const 2.065 2.355 2.670 2.877 3.023 3.127 3.191 3.225

667

i ≤ pm N 0.0058 5.6849e-4 1.5350e-4 3.8792e-5 9.7056e-6 2.4265e-6 6.0664e-7 1.5166e-7

order —— 3.351 1.889 1.984 1.999 2.000 2.000 2.000

ε = 10−10 const i > pm N 1.476 0.3578 0.582 0.2550 0.629 0.1735 0.636 0.1091 0.636 0.0655 0.636 0.0381 0.636 0.0216 0.636 0.0120

order —— 0.489 0.556 0.670 0.736 0.782 0.819 0.848

const 2.065 2.355 2.670 2.878 3.023 3.127 3.191 3.225

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Table 2 shows the uniform convergence of second-order on [0, x[pm N ] ] and almost first-order on (x[pm N ] , 1], and agrees with Theorem 2. Table 3. The numerical results of the simple upwind scheme on the Shishkin mesh i ≤ ps N 0.2228 0.1123 0.0516 0.0226 0.0097 0.0043 0.0019 8.5786e-4

N 16 32 64 128 256 512 1024 2048

ε = 10−6 const i > ps N 3.565 0.2409 3.594 0.1540 3.303 0.0968 2.888 0.0599 2.484 0.0356 2.187 0.0205 1.942 0.0116 1.757 0.0064

order —— 0.988 1.122 1.191 1.220 1.174 1.178 1.147

order —— 0.646 0.670 0.693 0.751 0.796 0.822 0.858

i ≤ ps N 0.2228 0.1123 0.0516 0.0226 0.0097 0.0043 0.0019 8.5793e-4

const 1.390 1.422 1.490 1.581 1.644 1.686 1.709 1.719

ε = 10−10 const i > ps N 3.565 0.2409 3.594 0.1540 3.303 0.0968 2.888 0.0599 2.484 0.0356 2.187 0.0205 1.942 0.0116 1.757 0.0064

order —— 0.988 1.122 1.191 1.220 1.174 1.178 1.147

order —— 0.646 0.670 0.693 0.751 0.796 0.822 0.858

const 1.390 1.422 1.490 1.581 1.644 1.686 1.709 1.719

Table 3 shows the uniform convergence of first-order on [0, x[ps N ] ] and almost first-order on (x[ps N ] , 1], and verifies Theorem 3. The log2-log2 graphs of errors to illustrate the convergence orders for the hybrid scheme on [0, 1 − τ ], (1 − τ, x[ph N ] ] and (x[ph N ] , 1] on the Shishkin mesh are shown in Fig. 1 (a), those for the midpoint upwind scheme and the simple upwind scheme are in Figs. 1 (b) and (c). 0

0

-5

-5

-10

-10

-2 -3 -4

-15

log2(Error)

log2(Error)

log2(Error)

-5

-15

-6 -7 -8 -9

[0,1-τ] (1-τ,x [p

-20

(x[p

-20

]

h

[0,1-τ] (1-τ,x [p

N]

,1]

h

(x[p

N]

-25

m

N]

]

m

,1]

-25 4

5

6

7

8

9

10

11

-10

[0,1-τ] (1-τ,x [p

-11

(x[p

N]

]

s

N]

,1]

s

N]

-12 4

5

log2(N)

6

7

8

9

10

11

log2(N)

4

5

6

7

8

9

10

11

log2(N)

Fig. 1 The log2-log2 graphs of errors on the Shishkin mesh for: (a) the hybrid scheme, (b) the midpoint upwind scheme, (c) the simple upwind scheme.

4

Conclusions In this paper, the hybrid finite difference scheme is constructed, which is slightly different

from the schemes in [4] and [8]. The new estimates on the Shishkin mesh, which are O(N −2 ) for 1 1 ≤ i ≤ ph N and O(N −2 ln2 N ) for ph N < i < N with ph = 1− 2e for the hybrid finite difference

scheme, O(N −2 ) for 1 ≤ i ≤ pm N and O(N −1 ln N ) for pm N < i < N with pm = the midpoint upwind scheme, and with ps = 1 −

1 2e

O(N −1 )

for 1 ≤ i ≤ ps N and

O(N −1 ln N )

3 4

−

1 4e

for

for ps N < i < N

for the simple upwind scheme, are better than those in [4–6, 8]. The numerical

example strongly support our results.

References [1] R. B. Kellogg, A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comput. 32 (1978) 1025-1039.

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[2] J. J. H. Miller, E. O’Riordan, G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996. [3] H.-G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2008. [4] T. Linß. Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Lecture Notes in Mathematics, Springer-Verlag, 2010. [5] J.J.H. Miller, E. O’Riordan, G.I. Shishkin. On piecewise-uniform meshes for upwind-and centraldifference operators for solving singularly perturbed problems, IMA J. Numer. Anal. 15 (1995) 89-99. [6] H.-G. Roos, T. Linß, Sufficient conditions for uniform convergence on layer-adapted grids, Computing. 63 (1999) 27-45. [7] V. B. Andreev and N. V. Kopteva, Investigation of difference schemes with an approximation of the first derivative by a central difference relation, Comput. Math. Math. Phys., 36 (1996) 1065-1078. [8] M. Stynes, H.-G. Roos, The midpoint upwind scheme, Appl. Numer. Math. 23 (1997) 361-374. [9] Q. Zheng, X.-L. Feng, X.-Z. Li, ε-uniform convergence of the midpoint upwind scheme on the Bakhvalov-Shishkin mesh for singularly perturbed problems, J. Comput. Anal. Appl. 17 (2014) 40-47. [10] M. K. Kadalbajoo, K. C. Patidar, ε-Uniformly convergent fitted mesh finite difference methods for general singular perturbation problems, Appl. Math. Comput. 179 (2006) 248-266.

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¨ PROBLEM RELATED TO SIMPLE LOGISTIC FEKETE SZEGO ACTIVATION FUNCTION C. RAMACHANDRAN AND D. KAVITHA Abstract. In this present paper, we introduce the new subclass of analytic univalent functions associated with quasi-subordination in the field of sigmoid functions. We obtained the coefficient bounds and Fekete-Szego inequality belongs to the defined class. Also, we extracted the new subclasses from the dened class of analytic functions. Mathematics Subject Classification: Primary:30C45; Secondary:30C50,33E99 Keywords: Univalent functions, Sigmoid function, Subordination, Quasi-subordination, Fekete-Szeg¨ o Inequality.

1. Introduction and preliminaries Sigmoid function playing an important role in the branch of special functions which is the part of logistic activation function developed in eighteenth century. The theory of special functions has been developed by C. F.Gauss, C. G. J. Jacobi, F. Klein and many others in nineteenth century. However, in the twentieth century , from the perspective of fundamental science sigmoid functions are of special interest in abstract areas such as approximation theory, functional analysis, topology, differential equations and probability theory and so on. A typical applications of the sigmoid function includes neural networks, image processing, artificial networks, biomathematics, chemistry, geoscience, probability theory, economics etc., We can find the similar kind of functions called gompertz function and ogee function which are used in modelling systems to saturate at more values of time period. The evaluation process of sigmoid function in many ways especially by truncated series expansion method was seen in [4, 10]. Recently Ramachandran et al. [13] discssed the problem of Hankel determinant for the subclass of analytic and univalent functions. The sigmoid function is of the form 1 (1.1) h(z) = 1 + e−z is differentiable and has the following properties:

1

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• Bound output real numbers between 0 and 1, it leads to the probability theory. • It maps a very large input domain to a small range of outputs. • It never loses information because it is an injective function. • It increases monotonically. The above properties permit us to use sigmoid function in the univalent function theory. In computational networks, this sigmoid function leads the output as digital numbers 1 for ON and 0 for OFF. Kannan et al.[6] brought out contrast enhancement using modified sigmoid function provides the highest measure of contrast and can be effectively used for further analysis of sports color images. Let A be the class of functions f (z) which are analytic in the open disk U = {z : z ∈ C : |z| < 1} is of the form: f (z) = z +

∞ X

an z n

(z ∈ U).

(1.2)

n=2

and normalized by f (0) = f 0 (0) − 1 = 0 and let S be a class of all functions in A consisting of univalent functions in U . If f (z) and g(z) be analytic in U, we say that the function f (z) is subordinate to g(z) in U, and write f (z) ≺ g(z), z ∈ U if there exits a Schwarz function ω(z), which is analytic in U with ω(0) = 0 and |ω(z)| < 1 (z ∈ U) such that f (z) = g(ω(z)), z ∈ U. In particular, if the function g is univalent in U, then We have that f ≺ g or f (z) ≺ g(z), z ∈ U if and only if f (0) = g(0) and f (U) ⊆ g(U) defined by[11]. In the year 1970, Robertson [15] introduced the concept of quasi-subordination. For, two analytic functions f (z) and g(z), the function f (z) is quasi-subordinate to g(z) in the open unit disc U, written by f (z) ≺q g(z). If there exist an analytic function ϕ and ω, with |ϕ(z)| ≤ 1, ω(0) = 0 and |ω(z)| < 1 such that f (z) = ϕ(z)g(ω(z)), (z ∈ U).

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Observe that if ϕ(z) ≡ 1, then f (z) = g(ω(z)), so that f (z) ≺ g(z) in U. Furthermore, if ω(z) = z, then f (z) = ϕ(z)g(z), said to be that f (z) is majorized by g(z) and symbolically written as f (z)  g(z) in U. Hence it is obvious that the quasi-subordination is a generalization of subordination as well as majorization [2, 8, 16]. Haji Mohd and Darus [5] introduced the concepts of q-starlike and q-convex functions as follows: Definition 1. Let the class Sq∗ (ϕ) consists of functions f ∈ A satisfies the quasisubordination  0  zf (z) − 1 ≺q ϕ(z) − 1; (z ∈ U). (1.3) f (z) Example 1. A function f ∈ U → C defined by   0 zf (z) − 1 = z(ϕ(z) − 1) ≺q ϕ(z) − 1; f (z)

(z ∈ U).

belongs to the class Sq∗ (ϕ). Definition 2. Let the class Cq (ϕ) consists of functions f ∈ A satisfies the quasisubordination  00  zf (z) ≺q ϕ(z) − 1; (z ∈ U). (1.4) f 0 (z) Example 2. A function f ∈ U → C defined by  00  zf (z) = z(ϕ(z) − 1) ≺q ϕ(z) − 1; f 0 (z)

(z ∈ U).

belongs to the class Cq (ϕ). To prove our main results, we need the following lemmas: Lemma 1. [7] Let ω be the analytic function in D, with ω(0) = 0, |ω(z)| < 1 and ω(z) = ω1 z + ω2 z 2 + . . . , then |ω2 − νω12 | ≤ max[1; |ν|], where ν ∈ C. The result is sharp for the functions ω(z) = z 2 or ω(z) = z. Lemma 2. [3] Let ω be the analytic function in D, with ω(0) = 0, |ω(z)| < 1 and ω(z) = ω1 z + ω2 z 2 + . . . , then  1, n=1 |ωn | ≤ 2 1 − |ω1 | , n ≥ 2. The result is sharp for the functions ω(z) = z 2 or ω(z) = z.

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Lemma 3. [11] Let ϕ be an analytic function with positive real part in D, with |ϕ(z)| < 1 and let ϕ(z) = c0 + c1 z + c2 z 2 + . . . Then |c0 | ≤ 1 and |cn | ≤ 1 − |c0 |2 ≤ 1, for n > 0. Lemma 4. [4] Let h be the sigmoid function defined in (1.1) and !m ∞ ∞ X (−1)m X (−1)n n Φ(z) = 2h(z) = 1 + z , m 2 n! m=1 n=1

(1.5)

then Φ(z) ∈ P, |z| < 1 where Φ(z) is a modified sigmoid function. Lemma 5. [4] Let ∞ X (−1)m Φn,m (z) = 1 + 2m m=1

∞ X (−1)n n=1

n!

!m zn

then |Φn,m (z)| < 2. Lemma 6. [4] If Φ(z) ∈ P and it is starlike, then f is a normalized univalent function of the form (1.2). Taking m = 1, Joseph et al [4] remarked the following: Remark 1. Let Φ(z) = 1 +

∞ X

cn z n

n=1 n+1

where cn = [4].

(−1) 2n!

then |cn | ≤ 2, n = 1, 2, 3 . . . this result is sharp for each n see

Motivated by the earlier works of Ramachandran et al. [14], we define the class of function involving quasi-subordination in terms of sigmoid functions. Definition 3. A function f ∈ A is in the class Mqα,λ,β (Φn,m ) if  0 α   β 00 0 zf (z) zf (z) zf (z) +λ 1+ 0 − 1 ≺q Φ(z) − 1 (1 − λ) f (z) f (z) f (z)

(1.6)

here 0 < β ≤ 1, 0 ≤ α ≤ 1, 0 ≤ λ ≤ 1. With various choices of the parameters, the class Mqα,λ,β (Φn,m ) reduces to the following new classes, (1) Mq0,0,1 (Φn,m ) ≡ Sq∗ (Φn,m ), (2) Mq0,1,1 (Φn,m ) ≡ Cq (Φn,m ), (3) Mq0,λ,1 (Φn,m ) ≡ Mqλ (Φn,m ). In this present paper, we determine the coefficient estimates including a FeketeSzeg¨o inequality of functions belonging to the above defined class and the class

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involving majorization. This result can assist us to represent various geometric interpretation as well as behaviours of the functions in complex domain. Let f (z) be of the form (1.2), ϕ(z) = c0 + c1 z + c2 z 2 + . . . . and ω(z) = ω1 z + ω2 z 2 + . . . , throughout this paper unless otherwise mentioned. ¨ Inequality 2. Fekete-Szego In this section we obtain the first two coefficient estimates and the Fekete-Szeg¨o Inequality for the class Mqα,λ,β (Φn,m ). Theorem 1. If f (z) ∈ Mqα,λ,β (Φn,m ) , then |a2 | ≤

1 , 2[α + β(1 + λ)]

  α(α − 3) + β(β − 1)(1 + λ)2 + 2αβ(1 + λ) − 2β(1 + 3λ) 1 , |a3 | ≤ max 1, 4[α + β(1 + 2λ)] 4[α + β(1 + λ)]2 and for any complex number µ, we have   µ[α + β(1 + 2λ)] 1 , |a3 − µa22 | ≤ max 1, Λ + 4[α + β(1 + 2λ)] [α + β(1 + λ)]2 where Λ=

α(α − 3) + β(β − 1)(1 + λ)2 + 2αβ(1 + λ) − 2β(1 + 3λ) . 4[α + β(1 + λ)]2

(2.1)

Proof. Since f ∈ A belongs to the class Mqα,λ,β (Φn,m ), then from (1.6) we have 

0

zf (z) f (z)

α 

 β 00 0 zf (z) zf (z) +λ 1+ 0 − 1 = ϕ(z)(Φ(z) − 1), z ∈ U. (2.2) (1 − λ) f (z) f (z)

The modified sigmoid function Φ(z) can be expressed as 1 1 1 5 1 779 7 Φ(z) = 1 + z − z 3 + z − z6 + z + ... 2 24 240 64 20160 since ϕ(z) as defined earlier, now we obtain     ω1 ω2 2 ω3 ω13 2 3 ϕ(z)(Φ(ω(z)) − 1) = (c0 + c1 z + c2 z + . . . .) z+ z + − z + ... 2 2 2 24 c ω c 0 ω1 c 1 ω1  2 0 2 = z+ + z + ... 2 2 2 (2.3)

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C. RAMACHANDRAN AND D. KAVITHA 0

00

f (z) zf (z) by replacing the equivalent expressions of f (z), and 0 in (2.2) and the f (z) f (z) simple calculation yields the following,  0 α   β 0 00 zf (z) zf (z) zf (z) (1 − λ) +λ 1+ 0 − 1 = [α + β(1 + λ)]a2 z f (z) f (z) f (z)     α(α − 3) β(β − 1) 2 + 2[α + β(1 + 2λ)]a3 + + (1 + λ) + αβ(1 + λ) − β(1 + 3λ) a22 z 2 + . . . 2 2 (2.4) Equating right hand side part of (2.3) and (2.4), we get, c0 ω1 , a2 = 2[α + β(1 + λ)]

(2.5)

and 1 {c1 ω1 + c0 [ω2 4[α + β(1 + 2λ)]    α(α − 3) + β(β − 1)(1 + λ)2 + 2αβ(1 + λ) − 2β(1 + 3λ) 2 − ω1 c 0 . 4[α + β(1 + λ)]2

a3 =

(2.6)

Using the hypothesis of Lemma 3 and the well-known inequality of Lemma 2, for n > 0 |cn | ≤ 1 − |c0 |2 ≤ 1. and |ω1 | ≤ 1 we have, 1 , 2[α + β(1 + λ)] and for any µ ∈ C, we obtain from (2.5) and (2.6)      1 µ[α + β(1 + 2λ)] 2 2 a3 − µa2 = c1 ω 1 + c0 ω 2 − Λ + ω1 c 0 . 4[α + β(1 + 2λ)] [α + β(1 + λ)]2 |a2 | ≤

Since ϕ(z) is analytic and bounded in U, using [11], for some y, |y| ≤ 1 : |c0 | ≤ 1

and

c1 = (1 − c20 )y.

Now, replacing the value of c1 as defined above, we get      1 µ[α + β(1 + 2λ)] 2 2 a3 −µa2 = yω1 + c0 ω2 − Λ + ω1 + yω1 c20 . 4[α + β(1 + 2λ)] [α + β(1 + λ)]2 (2.7)

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If c0 = 0 then |a3 − µa22 | ≤

1 |ω1 ||y| = . 4[α + β(1 + 2λ)] 4[α + β(1 + 2λ)]

If c0 6= 0 then |a3 −

µa22 |

  µ[α + β(1 + 2λ)] 1 max 1, Λ + ≤ 4[α + β(1 + 2λ)] [α + β(1 + λ)]2

and the result is sharp.

(2.8) 

Further setting µ = 0 in (2.8) we get the bound on |a3 |. This completes the proof of the Theorem 1. Corollary 1. Let α = 0, Sq∗ (Φn,m ) then we have,

λ = 0 and β = 1 the class Mqα,λ,β (Φn,m ) reduced to

c 0 ω1 , 2   2µ − 1 1 2 . |a3 − µa2 | ≤ max 1, 4 2 a2 =

and

Corollary 2. Let α = 0, Cq (Φn,m ) then we have,

λ = 1 and β = 1 the class Mqα,λ,β (Φn,m ) reduced to

c 0 ω1 , 4   3µ − 2 1 2 . |a3 − µa2 | ≤ max 1, 12 4 a2 =

and

Corollary 3. Let α = 0 and β = 1 the class Mqα,λ,β (Φn,m ) reduced to Mqλ (Φn,m ) then we have, c 0 ω1 a2 = , 2(1 + λ) and   2µ(1 + 2λ) − (1 + 3λ) 1 2 . |a3 − µa2 | ≤ max 1, 4(1 + 2λ) 2(1 + λ)2 Theorem 2. If f ∈ A, such that the function  β  0 α  0 00 zf (z) zf (z) zf (z) (1 − λ) +λ 1+ 0 − 1  Φ(z) − 1, f (z) f (z) f (z)

z∈U

then, |a2 | ≤

1 , 2|α + β(1 + λ)|

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and for any complex number µ   1 µ[α + β(1 + 2λ)] . |a3 − µa22 | ≤ max 1, Λ + 4[α + β(1 + 2λ)] [α + β(1 + λ)]2 Proof. Taking ω(z) = z in the proof of Theorem 1, we get the desired result.



3. Conclusion Finding the estimates for various subclasses of analytic functions with normalization is the most important role of geometric function theory. These estimates characterise the behaviours of functions in complex domain. This characterisation provides a tool using the sigmoid function in wide range of fields like image processing, digital communications, neural sciences etc., References [1] R. M. Ali, V. Ravichandran & N. Seenivasagan, Coefficient bounds for p-valent functions, Appl. Math. Comput. vol. 187, 2007, no. 1, pp. 35–46. [2] O. Altintas & S. Owa, Majorizations and Quasi-subordinations for certain analytic functions, Proceedings of the Japan AcademyA, vol.68, 1992, no.7, pp. 181–185. [3] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Spinger, New York, 1983. [4] O. A. Fadipe-Joseph, A. T. Oladipo & U. A. Ezeafulukwe, Modified sigmoid function in univalent function theory, International Journal of Mathematical Sciences and Engineering Application, vol.7, 2013, No. 7 ,pp. 313–317. [5] M. Haji Mohd & M. Darus, Fekete-Szego problems for quasi-subordination classes, Abstr. Appl. Anal., 2012, Art. ID 192956, pp. 14 [6] P. Kannan, S. Deepa, & R. Ramakrishnan, Contrast Enhancement of Sports Images Using Two Comparative Approaches, Amer. Jour. Int. Sys., vol.2, 2012, no. 6, pp. 141–147. [7] F. R. Keogh & E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., vol.20, 1969, pp. 8–12. [8] S. Y. Lee, Quasi-subordinate functions and coefficient conjectures, Journal of Korean Mathematical Socity, vol.12, no. 1, Art. ID 19750, pp. 43-50. [9] W. C. Ma & D. Minda, A Unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, & S. Zhang(Eds.), Int. Press, 1994, pp. 157-169. [10] G. Murugusundaramoorthy, T. Janani, Sigmoid function in the space of univalent λ-pseudo starlike functions, IJPAM, Vol.101, 2015, no.1, pp. 33–41. [11] Z. Nehari, Conformal mappings, McGraw-Hill, New York, 1952. [12] C. Ramachandran & R. Ambrose Prabhu, Applications of geometric function theory related to mechanical systems, Rev. Tec. Ing. Univ. Zulia, Vol.39,2016, no.1, pp. 177–184. [13] C. Ramachandran & K. Dhanalakshmi, The Fekete Szeg¨ o poblem for a subclass of analytic functions related to sigmoid function,IJPAM, vol.113, 2017, no.3, pp. 389–398. [14] C. Ramachandran, S. Sivasubramanyan, H.M. Srivastava & A. Swaminathan, Coefficient inequalities for certain subclasses of ananlytic functions and their applications involving the Owa-srivastsva operator of fractional calculus, Math.Ineq. and Appl.

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[15] M. S. Robertson, Quasi-subordination and Coefficient conjectures, Bull. Amer. S-Bolyai Math. Soc. vol.76, 1970, pp. 1–9. [16] F. Y. Ren, S. Owa & S. Fukui, Some inequalities on quasi-subordinate functions, Bulletin of Australian Mathematical Socity, vol. 43, 1991, no.2, pp. 317–324. [17] S. Owa, Properties of certain integral operators, Southeast Asian Bulletin of Mathematics, Vol. 24, 2000, no.3, pp. 411-419. Department of Mathematics, University College of Engineering Villupuram,Kakuppam, Villupuram 605103, Anna University, Tamilnadu, India E-mail address: [email protected] Department of Mathematics, IFET College of Engineering, Villupuram 605108, Tamilnadu, India E-mail address: [email protected]

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On the second kind twisted q-Euler numbers and polynomials of higher order Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea (k)

Abstract : In this paper, we introduce the second kind twisted q-Euler polynomials En,ω,q (x) of order k. We also get interesting properties related to the second kind twisted q-Euler numbers and polynomials. Finally, we construct twisted q-zeta function of order which interpolates the second kind twisted q-Euler numbers of higher order at negative integer. Key words : Euler numbers, Euler polynomials, the second kind Euler numbers and polynomials, q-zeta function, twisted q-Euler numbers and polynomials, twisted q-Euler numbers and polynomials of higher order, twisted q-zeta function. 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Recently, mathematicians have studied Euler numbers, Euler polynomials, the second kind Euler numbers and the second kind Euler polynomials (see [1-9]). These numbers and polynomials possess many interesting properties and arising in many areas of mathematics, applied mathematics, and (k) physics. In this paper, we introduce the second kind twisted q-Euler numbers En,ω,q and polynomials (k) En,ω,q (x) of higher order. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Qp denotes the field of rational numbers, N denotes the set of natural numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . For g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}, the fermionic p-adic invariant integral on Zp of the function g ∈ U D(Zp ) is defined by ∫ I−1 (g) = From (1.1), we note that

Zp

g(x)dµ−1 (x) = lim

N p∑ −1

N →∞

∫

g(x)(−1)x , see [1, 3].

(1.1)

x=0

∫ Zp

g(x + 1)dµ−1 (x) +

Zp

g(x)dµ−1 (x) = 2g(0).

(1.2)

(k)

First, we introduce the second kind q-Euler numbers En,q of higher order k. The second kind q-Euler (k) numbers En,q of higher order k are defined by the generating function: ( )k ∑ ∞ n 2et (k) t = E , (| log q + 2t| < π). (1.3) n,q qe2t + 1 n! n=0 Let Tp = ∪N ≥1 CpN = limN →∞ CpN , where CpN = {ω|ω p

N

= 1} is the cyclic group of order pN .

For ω ∈ Tp , we denote by ϕω : Zp → Cp the locally constant function x 7−→ ω x . We introduce the second kind twisted q-Euler polynomials En,ω,q (x) as follows: ∞ ∑ tn 2et xt e = E (x) . n,w,q ωqe2t + 1 n! n=0

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(1.4)

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In [5], we obtain the second kind twisted q-Euler numbers En,ω,q polynomials En,ω,q (x) and investigate their properties. Theorem 1. For positive integers n, ω ∈ Tp , we have ∫ ϕω (x)q x (2x + 1)n dµ−1 (x) = En,ω,q , ∫ Zp

Zp

ϕω (y)q y (x + 2y + 1)n dµ−1 (y) = En,ω,q (x).

2. The second kind twisted q-Euler polynomials of higher order The main purpose of this section is to study a systemic properties of the second kind twisted q-Euler numbers and polynomials of higher order. In this section, we assume that q ∈ Cp . We (k)

(k)

construct the second kind twisted q-Euler numbers En,ω,q and polynomials En,ω,q (x) of higher order k. We use the notation m m m ∑ ∑ ∑ ··· = . k1 =0

kn =0

k1 ···kn =0

The binomial formulae are known as ( ) n ( ) ∑ n n n(n − 1) . . . (n − i + 1) (1 − a)n = (−a)i , where = , i i i! i=0 and

) ) n ( n ( ∑ ∑ 1 −n n+i−1 i −n i = (1 − a) (−a) = a (1 − a)n i i i=0 i=0

Now, using multiple of p-adic q-integral, we introduce the second kind twisted q-Euler polynomials (k) En,w,q (x) of higher order : For k ∈ N, we define ∞ ∑

(k) (x) En,ω,q

tn n!

∫n=0 ∫ ··· ω x1 +···+xk q x1 +x2 +···+xk e(x+2x1 +2x2 +···+2xk +k)t dµ−1 (x1 ) · · · dµ−1 (xk ). |

Zp k

(2.1)

Zp

{z } times

By using Taylor series of e(x+2x1 +2x2 +···+2xk +k)t in the above equation, we obtain (∫ ) ∫ ∞ ∑ tn x1 +···+xk x1 +···+xk n ··· ω q (x + 2x1 + · · · + 2xk + k) dµ−1 (x1 ) · · · dµ−1 (xk ) n! Zp Zp n=0 =

∞ ∑

(k) En,w,q (x)

n=0

tn . n!

tn on the above equation, we arrive at the following theorem. n! Theorem 2. For positive integers n and k, we have ∫ ∫ (k) En,ω,q (x) = ··· ω x1 +···+xk q x1 +···+xk (x + 2x1 + · · · + 2xk + k)n dµ−1 (x1 ) · · · dµ−1 (xk ). (2.2)

By comparing coefficients

Zp

Zp

(k)

By (2.1), the second kind twisted q-Euler polynomials of higher order, En,ω,q (x) are defined by means of the following generating function ( )k ∞ ∑ 2et tn (k) xt (k) Fω,q (x, t) = e = E (x) . (2.3) n,ω,q ωqe2t + 1 n! n=0

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(k)

Again, by using (2,1), the second kind twisted q-Euler numbers of higher order, En,ω,q are defined by the following generating function (

2et ωqe2t + 1

)k =

∞ ∑

(k) En,ω,q

n=0

tn , n!

|t + log q|
0, we can derive the following Eq. (3.1) form the Mellin transformation (k)

of Fω,q (x, t). 1 Γ(s)

∫

∞

∞ ∑

(k) ts−1 Fω,q (x, −t)dt = 2k

0

a1 ,··· ,ak

(−1)a1 +···+ak ω a1 +···+ak q a1 +···+ak (2a1 + · · · + 2ak + k + x)s =0

(3.1)

For s, x ∈ C with R(x) > 0, we define the multiple twisted q-Euler zeta function as follows: Definition 8. For s, x ∈ C with R(x) > 0, we define (k) (s, x) = 2k ζω,q

∞ ∑ a1 ,··· ,ak

(−1)a1 +···+ak ω a1 +···+ak q a1 +···+ak . (2a1 + · · · + 2ak + k + x)s =0

(3.2)

For s = −l in (3.2) and using (2.6), we arrive at the following theorem. Theorem 9. For positive integer l, we have (k)

(k) (−l, x) = El,ω,q (x). ζω,q

By (2.4), we have ∞ ∑

(k) En,ω,q

n=0

tn = n!

(

2et ωqe2t + 1

)k = 2k

) ∞ ( ∑ m+k−1 (−1)m ω m q m e(2m+k)t . m m=0

By using Taylor series of e(2m+k)t in the above, we have ( ) ) ∞ ∞ ∞ ( n ∑ ∑ ∑ t m + k − 1 tn (k) En,ω,q = 2k (−1)m ω m q m (2m + k)n . n! n=0 n! m n=0 m=0 By comparing coefficients

tn n!

(k) En,ω,q

in the above equation, we have ) ∞ ( ∑ m+k−1 =2 (−1)m ω m q m (2m + k)n . m m=0 k

(3.3)

By using (3.3), we define twisted q-Euler zeta function as follows: Definition 10. For s ∈ C, we define (k) ζω,q (s)

) ∞ ( ∑ m + k − 1 (−1)m ω m q m =2 . m (2m + k)s m=0 k

(k)

(3.4)

(k)

The function ζω,q (s) interpolates the number En,ω,q at negative integers. Substituting s = −n with n ∈ Z+ into (3.4), and using (3.3), we obtain the following theorem:

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Theorem 11. Let n ∈ Z+ , We have (k) (k) ζω,q (−n) = En,ω,q .

Further, by (3.2) and (3.4), we have ∞ ∑ a1 ,··· ,ak

) ∞ ( ∑ (−1)a1 +···+ak ω a1 +···+ak q a1 +···+ak m + k − 1 (−1)m ω m q m = . (2a1 + · · · + 2ak + k)s m (2m + k)s =0 m=0

Acknowledgment This paper has been supported by the 2019 Hannam University Research Fund. REFERENCES 1. Kim, T.(2008). Euler numbers and polynomials associated with zeta function, Abstract and Applied Analysis, Art. ID 581582. 2. Liu, G.(2006). Congruences for higher-order Euler numbers, Proc. Japan Acad. , v. 82 A, pp. 30-33. 3. Ryoo, C.S., Kim, T., Jang, L.C.(2007). Some relationships between the analogs of Euler numbers and polynomials, Journal of Inequalities and Applications, v.2007, ID 86052, pp. 122. 4. Ryoo, C.S.(2010). Calculating zeros of the second kind Euler polynomials, Journal of Computational Analysis and Applications, v.12, pp. 828-833. 5. Ryoo, C.S.(2013). Exploring the zeros of the second kind twisted q-Euler polynomials, Far East Journal of Mathematical Sciences, v.77, pp. 195-203. 6. Ryoo, C.S.(2014). Note on the second kind Barnes’ type multiple q-Euler polynomials, Journal of Computational Analysis and Applications, v.16, pp. 246-250. 7. Ryoo, C.S.(2015). On the second kind Barnes-type multiple twisted zeta function and twisted Euler polynomials, Journal of Computational Analysis and Applications, v.18, pp. 423-429. 8. Ryoo, C.S.(2016). Differential equations associated with generalized Bell polynomials and their zeros, Open Mathematics, v.14, pp. 807-815. 9. Ryoo, C.S.(2016) Differential equations associated with tangent numbers, J. Appl. Math. & Informatics, v.34, pp. 487-494.

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HERMITE-HADAMARD INEQUALITY AND GREEN’S FUNCTION WITH APPLICATIONS∗ YING-QING SONG1 , YU-MING CHU2,∗∗ , MUHAMMAD ADIL KHAN3 , AND ARSHAD IQBAL3

Abstract. In the article, we derive the Hermite-Hadamard inequality by using Green’s function, establish some Hermite-Hadamrd type inequalities for the class of monotonic as well as convex functions, and give applications for means, mid-point and trapezoid formulae.

1. Introduction Convexity plays an important role in different fields of pure and applied sciences such as statistics, optimization theory, economics and finance etc. The fundamental justification for the significance of convexity is its meaningful relationship with the theory of inequalities. Many useful inequalities have been obtained by using convexity. Among those inequalities, the most extensively and intensively attractive inequality in the last decades is the well known Hermite-Hadamard inequality [1-9], which can be stated as follows: the double inequality   Z α2 α1 + α2 1 ψ(α1 ) + ψ(α2 ) (1.1) ψ ≤ ψ(x)dx ≤ 2 α2 − α1 α1 2 holds if the function ψ : [α1 , α2 ] → R is a convex function. If ψ is a concave function then (1.1) holds in the reverse direction. The Hermite-Hadamard inequality gives an upper as well as lower estimations for the integral mean of any convex function defined on closed and bounded interval which involves the the endpoints and midpoint of the domain of the function. Also inequality (1.1) provides the necessary and sufficient condition for the function to be convex. There are several applications of the Hermite-Hadamard inequality in the geometry of Banach spaces [10] and nonlinear analysis [11]. Some peculiar convex functions can be used in (1.1) to obtain classical inequalities for means. For some comprehensive surveys on various generalizations and developments of inequality (1.1) we recommend [12]. Due to the great importance of the convexity and the Hermite-Hadamard inequlity, in the recent years many generalizations, refinements and extensions can be found in the literature [13-37] In the article, we give a new proof for the Hermite-Hadamard inequality by using Green’s function, obtain some refinements of the Hermite-Hadamard inequality 2010 Mathematics Subject Classification. Primary: 26D15; Secondary: 26A51, 26E60. Key words and phrases. Hermite-Hadamard inequality, Green’s function, convexity. ∗ The research was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485) and the Science and Technology Research Program of Zhejiang Educational Committee (Grant No. Y201635325). ∗∗ Corresponding author: Yu-Ming Chu, Email: [email protected]. 1

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for monotonic functions as well as convex functions. At the end, we give some applications for means, mid-point and trapezoid formulae. 2. Main Results In order to obtain our main results we need to establish a lemma, which we present in this section. Lemma 2.1. Let G be the Green’s function defined on [α1 , α2 ] × [α1 , α2 ] by ( α1 − µ , α1 ≤ µ ≤ λ; (2.1) G(λ, µ) = α1 − λ, λ ≤ µ ≤ α2 . Then any ψ ∈ C 2 ([α1 , α2 ]) can be expressed as (2.2)

ψ(x) = ψ(α1 ) + (x − α1 )ψ 0 (α2 ) +

Z

α2

G(x, µ)ψ 00 (µ)dµ.

α1

Proof. By using the techniques of integration by parts in easily obtain (2.2).

Rb a

G(t, µ)ψ 00 (µ)dµ, we can 

The following Theorem 2.2 give a new proof for the Hermite-Hadamard inequality. Theorem 2.2. Let ψ ∈ C 2 ([α1 , α2 ]). Then the double inequality   Z α2 1 ψ(α1 ) + ψ(α2 ) α1 + α2 ≤ ψ(x)dx ≤ (2.3) ψ 2 α2 − α1 α1 2 holds if ψ is convex on [α1 , α2 ]. Proof. Let x = (α1 + α2 )/2. Then (2.2) leads to      Z α2  α1 + α2 α1 + α2 α1 + α2 = ψ(α1 ) + − α1 ψ 0 (α2 ) + , µ ψ 00 (µ)dµ, ψ G 2 2 2 α1      Z α2  α1 + α2 α2 − α1 α1 + α2 0 (2.4) ψ , µ ψ 00 (µ)dµ. = ψ(α1 )+ ψ (α2 )+ G 2 2 2 α1 Taking integral of (2.2) with respect to x and dividing by α2 − α1 , we get  2  Z α2 1 1 α2 − α12 ψ(x)dx = ψ(α1 ) + − α1 (α2 − α1 ) ψ 0 (α2 ) α2 − α1 α1 α2 − α1 2 Z α2 Z α2 1 + G(x, µ)ψ 00 (µ)dµdx, α2 − α1 α1 α1   Z α2 1 α2 − α1 ψ(x)dx = ψ(α1 ) + ψ 0 (α2 ) α2 − α1 α1 2 Z α2 Z α2 1 (2.5) + G(x, µ)ψ 00 (µ)dµdx. α2 − α1 α1 α1 Subtracting (2.5) from (2.4) we obtain   Z α2 α1 + α2 1 ψ − ψ(x)dx 2 α2 − α1 α1

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 Z α2 Z α2 α1 + α2 1 G(x, µ)ψ 00 (µ)dµdx , µ ψ 00 (µ)dµ − 2 α2 − α1 α1 α1 α1   Z α2   Z α2 α1 + α2 1 G = G(x, µ)dx ψ 00 (µ)dµ. ,µ − 2 α2 − α1 α1 α1

Z = (2.6)

α2



G

Note that Z

α2

G(x, µ)dx =

(2.7) α1

 G If α1 ≤ µ ≤

µ2 α2 − 1 + α1 α2 − α2 µ, 2 2

 ( α1 − µ , α1 + α2 , µ = α1 −α2 2 , 2

2 α1 ≤ µ ≤ α1 +α ; 2 α1 +α2 ≤ µ ≤ α 2. 2

α1 +α2 , 2

then   Z α2 α1 + α2 1 G G(x, µ)dx ,µ − 2 α2 − α1 α1   2 1 α2 µ = α1 − µ − − 1 + α1 α2 − α2 µ α2 − α1 2 2 2

(2.8) If

α1 +α2 2

=

− (µ − α1 ) ≤ 0. 2(α2 − α1 )

≤ µ ≤ α2 , then   Z α2 α1 + α2 1 G ,µ − G(x, µ)dx 2 α2 − α1 α1   2 1 α12 α1 − α2 µ − − + α1 α2 − α2 µ = 2 α2 − α1 2 2 2

(2.9)

=

− (α2 − µ) ≤ 0. 2(α2 − α1 )

From the convexity of ψ we know that ψ 00 (µ) ≥ 0. Therefore, the first inequality of (2.3) follows easily from (2.6), (2.8) and (2.9). Next, we prove second inequality of (2.3). Let x = α2 . Then (2.2) gives Z α2 ψ(α2 ) = ψ(α1 ) + (α2 − α1 )ψ 0 (α2 ) + G(α2 , µ)ψ 00 (µ)dµ, α1

(2.10)

ψ(α1 ) + ψ(α2 ) 1 1 = ψ(α1 ) + (α2 − α1 )ψ 0 (α2 ) + 2 2 2

Z

α2

G(α2 , µ)ψ 00 (µ)dµ.

α1

It follows from (2.5) and (2.10) that

(2.11)

Z α2 1 ψ(α1 ) + ψ(α2 ) − ψ(x)dx 2 α2 − α1 α1  Z α2  Z α2 1 1 = G(α2 , µ) − G(x, µ)dx ψ 00 (µ)dµ. 2 α2 − α1 α1 α1

From (2.1) one has (2.12)

G(α2 , µ) = α1 − µ

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4 YING-QING SONG1 , YU-MING CHU2,∗∗ , MUHAMMAD ADIL KHAN3 , AND ARSHAD IQBAL3

if α1 ≤ µ ≤ α2 . It follows from (2.7) and (2.12) that Z α2 1 1 G(x, µ)dx G(α2 , µ) − 2 α2 − α1 α1  2  α1 − µ 1 µ α12 = − − + α1 α2 − α2 µ 2 α2 − α1 2 2
1 = (α1 − µ)(α2 − α1 ) − µ2 + α12 − 2α1 α2 + 2α2 µ 2(α2 − α1 ) (2.13)

=

1 ((µ − α1 )(α2 − µ)) ≥ 0. 2(α2 − α1 )

Therefore, Z α2 ψ(α1 ) + ψ(α2 ) 1 − ψ(x)dx ≥ 0 2 α2 − α1 α1 follows from (2.11) and (2.13) together with ψ 00 (µ) ≥ 0.



Next, we give some refinements of the Hermite-Hadamard inequality for the class of monotonic and convex functions. Theorem 2.3. Let ψ ∈ C 2 ([α1 , α2 ]). Then the following statements are true: (1) If |ψ 00 | is increasing, then      Z α2 (α − α )2  α1 + α2 1 α1 + α2 2 1 − |ψ 00 | + |ψ 00 (α2 )| ; ψ(x)dx ≤ ψ 2 α2 − α1 α1 48 2 (2) If |ψ 00 | is decreasing, then      Z α2 (α − α )2  α1 + α2 1 α1 + α2 2 1 − ψ(x)dx ≤ |ψ 00 (α1 ) | + |ψ 00 | ; ψ 2 α2 − α1 α1 48 2 (3) If |ψ 00 | is convex, then   Z α2 α1 + α2 1 − ψ(x)dx ψ 2 α2 − α1 α1    n o n  α + α  o (α2 − α1 )2 00 α1 + α2 00 00 00 1 2 ≤ max ψ , |ψ (α1 )| + max ψ , |ψ (α2 )| . 48 2 2 Proof. (1) By using (2.6) we have   Z α2 α1 + α2 1 ψ − ψ(x)dx 2 α2 − α1 α1    Z α1 +α Z α2 2  2 1 α1 + α2 = ,µ − G(x, µ)dx ψ 00 (µ)dµ G 2 α2 − α1 α1 α1   Z α2   Z α2 α1 + α2 1 + G ,µ − G(x, µ)dx ψ 00 (µ)dµ α1 +α2 2 α2 − α1 α1 2 "Z α1 +α2 # Z α2 2 −1 2 00 2 00 (µ − α1 ) ψ (µ)dµ + (α2 − µ) ψ (µ)dµ . = α1 +α2 2(α2 − α1 ) α1 2

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Taking absolute and using triangular inequality we obtain   Z α2 α1 + α2 1 ψ(x)dx − ψ 2 α2 − α1 α1 1 ≤ 2(α2 − α1 )

"Z

α1 +α2 2

2

(µ − α1 ) |ψ 00 (µ)|dµ +

α1

Z

α2 α1 +α2 2

# 2

(α2 − µ) |ψ 00 (µ)|dµ

# "   Z α1 +α2 Z α2 2 00 α1 + α2 1 2 2 ψ (α2 − µ) dµ (µ − α1 ) dµ + |ψ 00 (α2 )| ≤ α1 +α2 2(α2 − α1 ) 2 α1 2     1 1 α1 + α2 1 00 3 00 3 |ψ | × (α2 − α1 ) + |ψ (α2 )| × (α2 − α1 ) = 2(α2 − α1 ) 2 24 24 (2.14)

=

    α1 + α2 (α2 − α1 )2 |ψ 00 | + |ψ 00 (α2 )| . 48 2

Similarly we can prove part (2). For part (3), using (2.14) and the fact that every convex function ψ defined on the interval [α1 , α2 ] is bounded above by max{ψ(α1 ), ψ(α2 )}, we obtain   Z α2 α1 + α2 1 − ψ(x)dx ψ 2 α2 − α1 α1 ≤

 Z α1 +α2     2 α1 + α2 1 00 (µ − α1 )2 dµ , |ψ (α )| max ψ 00 1 2(α2 − α1 ) 2 α1    Z α2   00 α1 + α2 00 2 + max ψ (α2 − µ) dµ , |ψ (α2 )| α1 +α2 2 2

         00 α1 + α2 00 α1 + α2 (α2 − α1 )2 00 00 = max ψ , |ψ (α1 )| + max ψ , |ψ (α2 )| . 48 2 2  Theorem 2.4. Let ψ ∈ C 2 ([α1 , α2 ]). Then the following statements are true: (1) If |ψ 00 | is increasing, then Z α2 ψ(α1 ) + ψ(α2 ) |ψ 00 (α2 )|(α2 − α1 )2 1 − ψ(x)dx ≤ ; 2 α2 − α1 α1 12 (2) If |ψ 00 | is decreasing, then Z α2 ψ(α1 ) + ψ(α2 ) |ψ 00 (α1 )|(α2 − α1 )2 1 ≤ − ψ(x)dx ; 2 α2 − α1 α1 12 (3) If |ψ 00 | is a convex function, then Z α2 ψ(α1 ) + ψ(α2 ) | max{|ψ 00 (α1 )|, |ψ 00 (α2 )|}|(α2 − α1 )2 1 ≤ − ψ(x)dx . 2 α2 − α1 α1 12

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Proof. It follows from (2.11) that Z α2 Z α2 1 1 ψ(α1 ) + ψ(α2 ) ψ(x)dx = (µ − α1 )(α2 − µ)ψ 00 (µ)dµ. − 2 α2 − α1 α1 2(α2 − α1 ) α1 Taking absolute and using triangular inequality one has Z α2 ψ(α ) + ψ(α ) 1 1 2 ψ(x)dx − 2 α2 − α1 α1 (2.15)

≤

1 2(α2 − α1 )

Z

α2

((µ − α1 )(α2 − µ)) |ψ 00 (µ)|dµ.

α1

Since (µ − α1 )(α2 − µ) ≥ 0 and |ψ 00 | is increasing, therefore Z α2 ψ(α1 ) + ψ(α2 ) 1 − ψ(x)dx 2 α2 − α1 α1 Z α2 |ψ 00 (α2 )| ≤ (µ − α1 )(α2 − µ)dµ 2(α2 − α1 ) α1 |ψ 00 (α2 )|(α2 − α1 )2 . 12 Similarly we can prove part (2) For part (3), using (2.15) and the fact that every convex function f defined on the interval [α1 , α2 ] is bounded above by max{f (α1 ), f (α2 )}, we have Z α2 ψ(α ) + ψ(α ) 1 1 2 ψ(x)dx − 2 α2 − α1 α1 =

max{|ψ 00 (α1 )|, |ψ 00 (α2 )|} ≤ 2(α2 − α1 )

Z

α2

((µ − α1 )(α2 − µ)) dµ. α1



3. Applications to Means A bivariate function M : (0, ∞) × (0, ∞) 7→ (0, ∞) is said to be a mean if min{a, b} ≤ M (a, b) ≤ max{a, b}, M (a, b) = M (b, a) and M (λa, λb) = λM (a, b) for all a, b, λ ∈ (0, ∞). Let a, b > 0 with a 6= b. Then the arithmetic mean A(a, b) [38-43], logarithmic mean L(a, b) [44-48] and (α, r)-th generalized logarithmic mean L(α,r) (a, b) [49-52] are defined by  1/r b−a α(br+α − ar+α ) a+b A(a, b) = , L(a, b) = , L(α,r) (a, b) = , 2 log b − log a (r + α)(bα − aα ) respectively. Recently, the bivariate means have been the subject of intensive research [53-67] and many remarkable inequalities for the bivariate means and related special functions can be found in the literature [68-90]. In this section we present several new inequalities the arithmetic, logarithmic and generalized logarithmic means by using our results.

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Theorem 3.1. Let 0 < α1 < α2 . Then the following statements are true: (1) if r ≥ 2, then " # r(r − 1)(α − α )2  α + α r−2 r 2 1 1 2 r−2 r (3.1) A (α1 , α2 ) − Lr (α1 , α2 ) ≤ + α2 ; 48 2 (2) if r < 2 and r 6= 0, −1, then " # r(r − 1)(α − α )2  α + α r−2 r 2 1 1 2 r−2 r (3.2) A (α1 , α2 ) − Lr (α1 , α2 ) ≤ + α1 . 48 2 Proof. Let ψ(x) = xr (x > 0) and r ≥ 2. Then we clearly see that |ψ 00 | is increasing and inequality (3.1) follows easily from Theorem 2.3(1). Similarly, we can prove inequality (3.2).  Theorem 3.2. Let 0 < α1 < α2 . Then the following statements are true: (1) if r ≥ 2, then |A(α1r , α2r ) − Lrr (α1 , α2 )| ≤

r(r − 1)(α2 − α1 )2 α2r−2 ; 48

(2) if r < 2 and r 6= 0, −1, then |A(α1r , α2r ) − Lrr (α1 , α2 )| ≤

r(r − 1)(α2 − α1 )2 α1r−2 . 48

Proof. By using Theorem 2.4 and the same arguments as given in the proof of Theorem 3.1, we can obtain the desired results.  Theorem 3.3. The inequalities "

# 1 3 + α3 , (α1 + α2 ) 1 −1 A (α1 , α2 ) − L−1 (α1 , α2 )      (α2 − α1 )2 8 1 8 1 ≤ max , + max , 24 (α1 + α2 )3 α23 (α1 + α2 )3 α13 (α − α )2 −1 2 1 A (α1 , α2 ) − L−1 (α1 , α2 ) ≤ 24

8

hold for all α1 , α2 ∈ R+ with α1 < α2 . Proof. Let x > 0 and ψ(x) = 1/x. Then we clearly see that |ψ 00 | is decreasing and convex and Theorem 3.3 follows easily from Theorem 2.3(2) and (3).  Theorem 3.4. The inequality (α − α )2
2 1 A α1−1 , α2−1 − L−1 (α1 , α2 ) ≥ 6α13 holds for all α1 , α2 ∈ R+ with α1 < α2 . Proof. Similar proof as in Theorem 3.3 but use Theorem 2.4 instead of Theorem 2.3 

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8 YING-QING SONG1 , YU-MING CHU2,∗∗ , MUHAMMAD ADIL KHAN3 , AND ARSHAD IQBAL3

4. Applications to Trapezoidal and Mid-Point Formulae In this section we provide some new error estimations for the trapezoidal and mid-point formulae. Let d be a division a = x0 < x1 < · · · < xn−1 < xn = b of the interval [a, b] and consider the quadrature formula Z b ψ(x)dx = T(ψ, d) + E(ψ, d), a

where T(ψ, d) =

n−1 X i=0

ψ(xi ) + ψ(xi+1 ) (xi+1 − xi ) 2

for the trapezoidal version and T(ψ, d) =

n−1 X

ψ

i=0

x + x  i i+1 (xi+1 − xi ) 2

for the midpoint version and E(ψ, d) denotes the associated approximation error. Theorem 4.1. Let d be a division a = x0 < x1 < · · · < xn−1 < xn = b of the interval [a, b], ψ ∈ C 2 ([a, b]) and E(ψ, d) be the trapezoidal error. Then one has (4.1)

|E(ψ, d)| ≤

n−1 X i=0

3

|ψ 00 (xi+1 )| (xi+1 − xi ) 12

00

if |ψ | is an increasing function; |E(ψ, d)| ≤

n−1 X i=0

3

max{|ψ 00 (xi )|, |ψ 00 (xi+1 )|} (xi+1 − xi ) 12

00

if |ψ | is a decreasing function; |E(ψ, d)| ≤

n−1 X i=0

3

|ψ 00 (xi+1 )| (xi+1 − xi ) 12

00

if |ψ | is a convex function. Proof. Applying Theorem 2.4 on each subinterval [xi , xi+1 ] (i = 0, 1, 2, · · · , n − 1) of the division d, we have Z xi+1 ψ(xi ) + ψ(xi+1 ) |ψ 00 (xi+1 )|(xi+1 − xi )2 1 ≤ − ψ(x)dx . 2 xi+1 − xi 12 xi

Multiplying both sides by xi+1 − xi and taking summation we obtain Z  n−1   b n−1 X  |ψ 00 (xi+1 )| X |ψ 00 (xi+1 )| 3 3 ψ(x)dx − T(ψ, d) ≤ (xi+1 − xi ) ≤ (x − x ) i+1 i , a 12 12 i=0

i=0

which is equivalent to (4.1). Similarly we can prove other parts.



Theorem 4.2. Let d be a division a = x0 < x1 < · · · < xn−1 < xn = b of the interval [a, b], ψ ∈ C 2 ([a, b]) and E(ψ, d) be the mid-point error. Then one has    n−1  1 X 3 00 xi+1 + xi 00 |E(ψ, d)| ≤ (xi+1 − xi ) ψ + ψ (xi+1 ) 48 i=0 2

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if |ψ 00 | is an increasing function; |E(ψ, d)| ≤

   n−1 xi+1 + xi  00 1 X 3 ψ (x ) + (xi+1 − xi ) ψ 00 i 48 i=0 2

if |ψ 00 | is a decreasing function;  n−1 x  1 X i+1 + xi 00 3 (xi+1 − xi ) max{ ψ 00 , ψ (xi ) } 48 i=0 2  x  i+1 + xi 00 + max{ ψ 00 , ψ (xi+1 ) } 2 if |ψ 00 | is convex function. |E(ψ, d)| ≤

Proof. The proof is analogous to the proof of Theorem 4.1.



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[68] Zh.-H. Yang, W.-M. Qian and Y.-M. Chu, Monotonicity properties and bounds involving the complete elliptic integrals of the first kind, Math. Inequal. Appl., 2018, 21(4), 1185–1199. [69] T.-H. Zhao, M.-K. Wang, W. Zhang and Y.-M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018, 2018, Article 251, 15 pages. [70] T.-R. Huang, S.-Y. Tan, X.-Y. Ma and Y.-M. Chu, Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl., 2018, 2018, Article 239, 11 pages. [71] M.-K. Wang, S.-L. Qiu and Y.-M. Chu, Infinite series formula for H¨ ubner upper bound function with applications to Hersch-Pfluger distortion function, Math. Inequal. Appl., 2018, 21(3), 629–648. [72] M.-K. Wang and Y.-M. Chu, Landen inequalities for a class of hypergeometric functions with applications, Math. Inequal. Appl., 2018, 21(2), 521–537. [73] Zh.-H. Yang and Y.-M. Chu, A monotonicity property involving the generalized elliptic integral of the first kind, Math. Inequal. Appl., 2017, 20(3), 729–735. [74] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, Monotonicity rule for the quotient of two function and its applications, J. Inequal. Appl., 2017, 2017, Article 106, 13 pages. [75] M.-K. Wang, Y.-M. Chu and Y.-Q. Song, Asymptotical formulas for Gaussian and generalized hypergeometric functions, Appl. Math. Comput., 2016, 276, 44–60. [76] M.-K. Wang, Y.-M. Chu and Y.-P. Jiang, Ramanujan’s cubic transformaation inequalities for zero-balanced hypergeometric functions, Rocky Mountain J. Math., 2016, 46(2), 679–691. [77] Y.-M. Chu, M.-K. Wang and Y.-F. Qiu, On Alzer and Qiu’s conjecture for complete elliptic integral and inverse hyperbolic tangent function, Abstr. Appl. Anal., 2011, 2011, Article ID 697547, 7 pages. [78] M.-K. Wang and Y.-M. Chu, Asymptotical bounds for complete elliptic integrals of the second kind, J. Math. Anal. Appl., 2013, 402(1), 119–126. [79] Y.-M. Chu, Y.-F. Qiu and M.-K. Wang, H¨ older mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct., 2012, 23(7), 521–527. [80] Y.-M. Chu, M.-K. Wang, S.-L. Qiu and Y.-P. Jiang, Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl., 2012, 63(7), 1177–1184. [81] M.-K. Wang, S.-L. Qiu, Y.-M. Chu and Y.-P. Jiang, Generalized Hersch-Pfluger distortion function and complete elliptic integrals, J. Math. Anal. Appl., 2012, 385(1), 221–229. [82] M.-K. Wang, Y.-M. Li and Y.-M. Chu, Inequalities and infinite product formula for Ramanujan generalized modular equation function, Ramanujan J., 2018, 46(1), 189–200. [83] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On rational bounds for the gamma function, J. Inequal. Appl., 2017, 2017, Article 210, 17 pages. [84] T.-H. Zhao and Y.-M. Chu, A class of logarithmically completely monotonic functions associated with gamma function, J. Inequal. Appl., 2010, 2010, Article ID 392431, 11 pages. [85] T.-H. Zhao, Y.-M. Chu and H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011, 2011, Article ID 896483, 13 pages. [86] T.-R. Huang, B.-W. Han, X.-Y. Ma and Y.-M. Chu, Optimal bounds for the generalized Euler-Mascheronic constant, J. Inequal. Appl., 2018, 2018, Article 118, 9 pages. [87] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the error function, Math. Inequal. Appl., 2018, 21(2), 469–479. [88] Y.-Q. Song, P.-G. Zhou and Y.-M. Chu, Inequalities for the Gaussian hypergeometric function, Sci. China Math., 2014, 57(11), 2369–2380. [89] Zh.-H. Yang, W. Zhang and Y.-M. Chu, Sharp Gautschi inequality for parameter 0 < p < 1 with applications, Math. Inequal. Appl., 2017, 20(4), 1107–1120. [90] Zh.-H. Yang, Y.-M. Chu and W. Zhang, Accurate approximations for the complete elliptic of the second kind, J. Math. Anal. Appl., 2016, 438(2), 875–888.

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Ying-Qing Song, College of Science, Hunan City University, Yiyang 413000, China E-mail address: [email protected] Yu-Ming Chu (Corresponding author), Department of Mathematics, Huzhou University, Huzhou 313000, China E-mail address: [email protected] Muhammad Adil Khan, Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan E-mail address: [email protected] Arshad Iqbal, Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan E-mail address: [email protected]

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On the Applications of the Girard-Waring Identities Tian-Xiao He1 and Peter J.-S. Shiue2∗ 1

Department of Mathematics Illinois Wesleyan University Bloomington, IL 61702-2900, USA 2 Department of Mathematical Sciences University of Nevada, Las Vegas Las Vegas, Nevada, 89154-4020, USA

Abstract We present here some applications of Girard-Waring identities. Many various identities for things like elementary mathematics and other advanced mathematics come from those identities. AMS Subject Classification: 05A15, 05A05, 15B36, 15A06, 05A19, 11B83. Key Words and Phrases: Girard-Waring identities, Recursive sequences, Fibonacci numbers and polynomials, Lucas numbers, Chebysheve polynomials of the first kind, Chebysheve polynomials of the second kind.

1

Introduction

In this paper, we are concerned with the applications of the following GirardWaring identities:   X n−k n n k n x +y = (−1) (x + y)n−2k (xy)k (1) n−k k 0≤k≤[n/2]

and xn+1 − y n+1 = x−y

X

(−1)

k

0≤k≤[n/2]



 n−k (x + y)n−2k (xy)k . k

(2)

∗ This work was completed while on sabbatical leave from University of Nevada, Las Vegas, and the author would like to thank UNLV for its support.

1

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Albert Girard published these identities in Amsterdam in 1629 and Edward Waring published similar material in Cambridge in 1762-1782. These may be derived from
the earlier work of Sir Isaac Newton. It worth noting that n n−k k (−1) n−k k is an integer because       n n−k n−k n−k−1 = + n−k k k k−1     n−k n−k−1 =2 − . k k The proofs of formulas (1) and (2) can be seen in Comtet [3] (P. 198) and the survey paper by Gould [6]. Recently, Shapiro and one the authors [8] gave a different proof of (2) by using Riordan arrays. There are some alternative forms of formula (1). As an example, we give the following one. If x + y + z = 0, then (1) gives   X n n−k xn + y n = (−1)k (−z)n−2k (xy)k n−k k 0≤k≤[n/2]   X n − k n−2k n n n−k n = (−1) z + (−1) z (xy)k , k n−k 1≤k≤[n/2]

which implies X

xn + y n − (−1)n z n =

(−1)n−k

1≤k≤[n/2]

  n − k n−2k n z (xy)k . n−k k

Thus, when n is even, we have formula n

n

X

n

x +y −z =

n−k

  n − k n−2k n z (xy)k , n−k k

(3)

n−k

  n − k n−2k n z (xy)k , n−k k

(4)

(−1)

1≤k≤[n/2]

while for odd n we have n

n

n

x +y +z =

X

(−1)

1≤k≤[n/2]

where x + y + z = 0. Particularly, if n = 3, then x3 + y 3 + z 3 = 3xyz,

(5)

which will be shown in Corollary ?? and applied in the following examples. The formulas (3) and (4) can be considered as analogies of the results for the case of xy + yz + zx = 0 shown in Ma [11]. Draim and Bicknell [4] use sums

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and products of two roots of a quadratic equation to derive a class of GirardWaring identities. Using Girard-Waring formulas to derive combinatorial identities is also an attractive topic. For instance, Filipponi [5] uses GirardWaring formula (1) to derive some unusual binomial Fibonacci identities. Furthermore, some well-known identities can be re-derived by√using Girard2 Waring √ formulas. As an example, we substitute x = u + u − 4, y = 2 u − u − 4, and z = −x − y into (1) and obtain the following identity shown on page 57 of Riordan [13]: m X

(−1)k

k=0

  h i p p n − k n−2k n u = 2−n (u + u2 − 4)n + (u − u2 − 4)n n−k k (6)

for n = 1, 2, . . ., where m = [n/2]. In particular, if u = 2, above identity (6) reduces to   m X n − k n−2k k n (−1) 2 =2 n−k k k=0

for n = 1, 2, . . .. It worth mentioning that Vasil’ev and Zelevinskii [18] denoted the function shown on the right-hand side of (6) by Qn and obtained (see (4’) on Page 57 of [18])   (2k − 1)π Qn (x) = Π1≤k≤n x − 2 cos , 2n which implies √ (2k − 1)π 2 = m Π1≤k≤m cos 4m 2 for m ≥ 1 (see (d) on Page 58 of [18]). In the next section, we present some applications of Girard-Waring identities to the trigonometric identities. In section 3, some applications of Girard-Waring identities to the linear recurrence relations of order 2 will be given.

2

Girard-Waring identities and trigonometric identities

Girard-Waring identities can be applied to construct many interesting trigonometric identities related to the roots of some quadratic equations.

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Our idea of the first application of Girard-Waring identities can be presented as follows: In formulas (1) and (2), there are two terms x + y, and xy. If we consider x and y the two roots r1 and r2 of a given quadratic equation, ax2 + bx + c = 0, then we have the sums of, and differences of, n-th powers of the roots of the quadratic equation. Therefore we have r1 + r2 = − ab =: p and r1 r2 = ac =: q. Thus formula (1) and (2) give:   X n − k n−2k k k n n n (−1) p q (7) r1 + r2 = n−k k 0≤k≤[n/2]

and r1n+1 − r2n+1 = r1 − r2

X

(−1)k

0≤k≤[n/2]

  n − k n−2k k p q . k

(8)

We first consider a simple quadratic equation x2 + c = 0. Then two roots, r1 and r2 , of the equation satisfy r1 + r2 = 0

and r1 r2 = c.

From (7) we have the identity r1n

+

r2n

=

  n n−k (−1) (r1 + r2 )n−2k (r1 r2 )k , n−k k

X

k

0≤k≤[n/2]

which implies r12` + r22` = 2(−c)`

(9)

and r12`+1 + r22`+1 = 0. For instance, if c = −3, then r1 = 2 cos(π/6) and r2 = 2 cos(5π/6). From (9) we obtain cos

2`

π  6

+ cos

2`



5π 6



 ` 3 =2 . 4

π 2 5π = 1.5 and cos4 6 +cos 6



π 4 5π = When ` = 1 and 2, when cos2 6 +cos 6 9/8, respectively. √ b2 −4ac Consider a quadratic equation ax2 + bx + c = 0, we have x = −b± 2a . 2 If b − 4ac < 0, then x = A ± Bi = ρ(cos θ ± i sin θ), where θ = tan−1 B A.

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Then two roots r1 and r2 are: and r2 = ρ(cos θ − i sin θ),

r1 = ρ(cos θ + i sin θ)

which implies p = r1 + r2 = 2ρ cos θ and q = r1 r2 = ρ. Thus, equation (7) gives   X n n−k (−1)k r1n + r2n = ρn (2 cos θ)n−2k , n−k k 0≤k≤[n/2]

which implies 2 cos nθ =

X

(−1)k

0≤k≤[n/2]

  n n−k (2 cos θ)n−2k . n−k k

Note that     n−k n n−k−1 n = , n−k k k k−1

k ≥ 1.

Thus n 1n cos nθ = (2 cos θ)n − (2 cos θ)n−2 2 1     n n−4 n n−3 n−4 n−6 (2 cos θ) − (2 cos θ) + ··· . + 2 1 3 2 Similarly, from (8) we have X

sin(n + 1)θ = sin θ

  n−k (−1) (2 cos θ)n−2k . k k

0≤k≤[n/2]

Example 2.1 On Page 50 of Comtet [3], it can be seen that sin(n + 1)θ = Un (cos θ), sin θ where Un (x) are the Chebyshev polynomials of the second kind. Thus,   X n−k Un (cos θ) = (−1)k (2 cos θ)n−2k . k 0≤k≤[n/2]

On Page 88 of Comtet [3], we also find that 2 cos θ 1 0 0 1 2 cos θ 1 0 sin(n + 1)θ 0 1 2 cos θ 1 Un (cos θ) = = sin θ 0 0 1 2 cos θ ... ... ... ...

702

... ... ... ... ...

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Hence, the determinant of the tridiangonal matrix on the rightmost side of the above equation is equal to 2 cos θ 1 0 0 . . .   1 2 cos θ 1 0 . . . X k n−k = 0 1 2 cos θ 1 . . . (2 cos θ)n−2k . (−1) k 0 0 1 2 cos θ . . . 0≤k≤[n/2] ... ... ... ... ... Recall that the Chebyshev polynomials of the first kind Tn (x) are defined by Tn (x) = cos (n cos−1 x). Thus, Tn (x) = cos (n cos−1 x)     1 n n n−3 = (2x)n − (2x)n−2 + (2x)n−4 − · · · 2 1 2 1 From Page 88 of [3], cos θ 1 0 0 ... 1 2 cos θ 1 0 ... 1 2 cos θ 1 ... Tn (cos θ) = cos nθ = 0 0 0 1 2 cos θ . . . ... ... ... ... ... Thus, x 1 0 0 ...

1 0 0 2x 1 0 1 2x 1 0 1 2x ... ... ...

... ... ... ... ...

.

    1 n n n−3 n n−2 n−4 = + (2x) − ··· . 2 (2x) − 1 (2x) 2 1

From [19] (see Page 696),    (2k − 1)π n−1 n . Tn (x) = 2 Πk=1 x − cos 2n Since Tn (cos θ) = cos nθ, the above formula implies that    (2k − 1)π cos nθ = 2n−1 Πnk=1 cos θ − cos . 2n

(10)

Remark 2.1 It is well known (see, for example, [19]) that    kπ n n Un (x) = 2 Πk=1 x − cos . n+1

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Thus,    kπ sin(n + 1)θ n n . = Un (cos θ) = 2 Πk=1 cos θ − cos sin θ n+1

(11)

Substituting different values of θ into (11), we may obtain a type of trigonometric identities. For instance, let θ = π/2. Then (11) yields   π kπ n n sin(n + 1) = −2 Πk=1 cos . 2 n+1 If n = 2m, m = 0, 1, , 2 . . ., because of sin(2m + 1)π/2 = (−1)m , the last equation implies     kπ kπ m 2m 2m 2 m m m (−1) = 2 Πk=1 cos = 4 (−1) Πk=1 cos , 2m + 1 2m + 1 where in the last step we use the fact       kπ (2m − k + 1)π kπ = − cos π − = − cos cos 2m + 1 2m + 1 2m + 1 for k = m + 1, m + 2, . . . , 2m. Thus we obtain the identity   kπ 1 2 Πm cos = m. k=1 2m + 1 4 Other identities can be obtained by substituting θ = π/6, π/4, π/3, etc. Recall also that cosh x =

ex + e−x 2

and

sinh x =

ex − e−x . 2

Let r1 = ex and r2 = e−x . Then (7) gives   1 X n−k k n cosh(nx) = (−1) (2 cosh x)n−2k 2 n−k k 0≤k≤[n/2]

and sinh(nx) = sinh x

X 0≤k≤[n/2]

  n n−k (−1) (2 cosh x)n−2k . n−k k k

Other applications of Girard-Waring identities to the product expansions of trigonometric functions similar to the results shown in [2] will be presented in the author’s further work.

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8

Girard-Waring Identities and linear recurrence relations of order 2

Girard-Waring identities can be applied to construct the expressions of the linear recursive sequences of order 2.

Note that xn + y n = (x + y) xn−1 + y n−1 − xy xn−2 + y n−2 (n ≥ 2). Let wn (x, y) = xn + y n . Then wn (x, y) = (x + y)wn−1 (x, y) − xywn−2 (x, y), n ≥ 2,

(12)

with the initial conditions w0 (x, y) = 2 and w1 (x, y) = x + y. The characteristic equation of the above recurrence relation is t2 − (x + y)t + xy = 0. Thus t = x, y Proposition 3.1. Let an (x, y) = p(x, y)an−1 (x, y)+q(x, y)an−2 (x, y), n ≥ 2, with given a0 (x, y) and a1 (x, y). Then an (x, y) a1 (x, y) − β(x, y)a0 (x, y) n a1 (xy) − α(x, y)a0 (x, y) n = α (x, y) − β (x, y), α(x, y) − β(x, y) α(x, y) − β(x, y0 where α(x, y) 6= β(x, y) are the roots of the characteristic equation t2 − p(x, y)t − q(x, y) = 0. By using this proposition 3.1, the solution of (12) is wn (x, y) = xn + y n . Example 3.1 The generalized Lucas polynomials (Lucas 1891, see Swamy [16]) Vn (x, y) are defined by Vn (x, y) = xVn−1 (x, y) + yVn−2 (x, y),

V0 (x, y) = 2, V1 (x, y) = x.

The characteristic equation is t2 − xt − y = 0. Thus p x ± x2 + 4y t= . 2 By Proposition 3.1 and the Girard-Waring identity (1)   X n n − k n−2k k n n Vn (x, y) = α (x, y) + β (x, y) = x y . n−k k 0≤k≤[n/2]

Example 3.2 Dickson polynomials of the first kind of degree n (Dickson 1897, see Lidl, Mullen, and Turnwald [10]) are defined by Dn (x, a) = xDn−1 (x, a) − aDn−2 (x, a),

705

D0 (x, a) = 2, D1 (x, a) = x.

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Thus from Proposition 3.1 and the Girard-Waring identity (1), Vn (x, −a) = Dn (x, a)   X n n−k = (−a)k xn−2k . n−k k 0≤k≤[n/2]

Example 3.3 For the Lucas polynomials (see Bicknell [1]) {Ln (x) = Vn (x, 1)}, i.e., let y = 1 in {Vn (x, y)}, we have   X n n − k n−2k Ln (x) = x . n−k k 0≤k≤[n/2]

Note that Dn (x, −1) = Ln (x). For the Lucas numbers Ln = Ln (1), we have   X n−k n Ln = . n−k k 0≤k≤[n/2]

Example 3.4 The Chebysheve polynomials of the first kind (Chehysher 1821-1894, see Rivlin [14] and Zwillinger [19]) are defined by Tn (x) = 2xTn−1 (x) − Tn−2 (x)

n ≥ 2,

with the initial conditions T0 (x) = 1 and T1 (x) = x. Thus, from Proposition 3.1 and the Girard-Waring identity (1), we have   1 X n−k k n Tn (x) = (2x)n−2k . (−1) 2 n−k k 0≤k≤[n/2]

Note that from (2), we have xn+1 − y n+1 xn − y n xn−1 − y n−1 = (x + y) − xy , x−y x−y x−y

n ≥ 2.

Let Wn+1 (x, y) = (xn+1 − y n+1 )/(x − y). Then Wn (x, y) = (x + y)Wn−1 (x, y) − xyWn−2 (x, y),

n ≥ 2,

(13)

with the initial conditions W0 (x, y) = 0 and W1 (x, y) = 1. Thus   X xn − y n k n−1−k Wn (x, y) = = (−1) (x + y)n−1−k (xy)k . x−y k 0≤k≤[(n−1)/2]

Remark 3.1 From the expression of Wn (x, y) and noting the initial condition W0 (x, y) = 0, we know {Wn (x, y)} is a linear divisibility sequence.

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More precisely, from authors’ recent work [9], if x and y be distinct real (or complex) numbers, then sequence (Wn (x, y)) is a second order linear homogenous recursive sequence with W0 = 0 and W1 = 1 and a linear divisibility sequence of order 2. For instance, when r 6= 1 and a1 = 1, the geometric sequence {sn = a1 (1 − rn )/(1 − r) = (1 − rn )/(1 − r)}n≥1 is a linear divisibility sequence because sn = Wn (1, r). Example 3.5 The Generalized Fibonacci polynomials Fn (x, y) ( see Swammy [17]) are defined by Fn (x, y) = xFn−1 (x, y) + yFn−2 (x, y) (n ≥ 2) with the initial conditions F0 (x, y) = 0 and F1 (x, y) = 1. From Proposition 3.1 and the Girard-Waring identity (2), we have   X n − 1 − k n−1−k k Fn (x, y) = x y . k 0≤k≤[(n−1)/2]

Thus, for the Fibonacci polynomials   X n − 1 − k n−1−k Fn (x) = Fn (x, 1) = x . k 0≤k≤[(n−1)/2]

For Fibonacci sequence {Fn } Fn = Fn (1) =

X 0≤k≤[(n−1)/2]

  n−1−k . k

For the Pell sequence {Pn } Pn = Fn (2) =

X 0≤k≤[(n−1)/2]



 n − 1 − k n−1−k 2 . k

References [1] M. Bicknell, A primer for the Fibonacci numbers VII, Fibonacci Quart. 8 (1970) pp. 407–420. [2] G. Brouwer, A generalization of the angle doubling formulas for trigonometric functions, Math. Magazine, 90 (2017), No.1, 12–18. [3] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. [4] N. A. Draim and M. Bicknell, Sums of n-th powers of roots of a given quadratic equation, Fibonacci Quart. 4 (1966), no. 3, 170–178. [5] P. Filipponi, Some binomial Fibonacci identities. Fibonacci Quart. 33 (1995), no. 3, 251–257.

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[6] H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart. 37 (1999), no. 2, 135–140. [7] R. P. Grimaldi, Fibonacci and Catalan numbers, An introduction, John Wiley & Sons, Inc., Hoboken, NJ, 2012. [8] T.-X. He and L. W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra Appl. 507 (2016), 77–95. [9] T.-X. He and P. J.-S. Shiue, An approach to the construction of linear divisibility sequences of high orders, J. Integer Sequences, 20 (2017), Artical 17. 9. 3. [10] R. Lidl, G. L. Mullen, and G. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 65. Longman Scientific & Technical and John Wiley & Sons, Inc., 1993. [11] X. Ma, A generalization of the Kummer identity and its application to Fibonacci-Lucas sequences, Fibonacci Quart. 36 (1998), no. 4, 339– 347. [12] R. W. D. Nickalls, Vi´ete, Descartes and the cubic equation, Math. Gazette, 90 (2006), July, 203–208. [13] J. Riordan, Combinatorial identities, John Wiley & Sons, Inc. New York-London-Sydney, 1968. [14] T. J. Rivlin, Chebyshev Polynomials, New York: Wiley, 1990. [15] M. Saul and T. Andreescu, Symmelltry in algebra, part lll, Quantum, 8 (1998), no. 6, July/August, 41–42. [16] M. N. S. Swamy, On a class of generalized polynomials, Fibonacci Quart. 35 (1997), no. 4, 329–334. [17] M. N. S. Swamy, Generalized Fibonacci and Lucas polynomials and their associated diagonal polynomials, Fibonacci Quart. 37 (1999), no. 3, 213–222. [18] N. Vasil’ev and A. Zelevinskii, Chebyshev polynomials and recurrence relations [Kvant 1982, no. 1, 12–19], Kvant selecta: algebra and analysis, II, 51–61, Math. World, 15, Amer. Math. Soc., Providence, RI, 1999. [19] D. Zwillinger (Ed.), CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, 1995.

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Local Fractional Taylor Formula George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we derive an appropiate local fractional Taylor formula. We provide a complete description of the formula.

2010 AMS Mathematics Subject Classification: 26A33. Keywords and Phrases: Local fractional derivative, Riemann-Liouville fractional derivative, Fractional Taylor formula.

1

Introduction

In [3], [4] was first introduced the local fractional derivative and presented an incomplete local fractional Taylor formula, all done by the use of RiemannLiouville fractional derivative. Similar work was done in [1], but again with some gaps. The author is greatly motivated by the pioneering work of [1]-[4] and presents a local fractional Taylor formula in a complete suitable form and without any gaps.

2

Main Results

We mention Definition 1 ([5], pp. 68, 89) Let x, x0 ∈ [a, b], f ∈ C ([a, b]). The RiemannLioville fractional derivative of a function f of order q (0 < q < 1) is defined as   q Dx+ f (x0 ) , x0 > x, q 0 Dx f (x ) = = q Dx− f (x0 ) , x0 < x ( R x0 0 −q d 1 (x − t) f (t) dt, x0 > x, dx0 x Rx (1) d Γ (1 − q) − dx0 x0 (t − x0 )−q f (t) dt, x0 < x. 1

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We need Definition 2 ([3]) The local fractional derivative of order q (0 < q < 1) of a function f ∈ C ([a, b]) is defined as Dq f (x) = lim Dxq (f (x0 ) − f (x)) . 0

(2)

x →x

More generally we define Definition 3 (see also [1]) Let N ∈ Z+ , 0 < q < 1, the local fractional derivative of order (N + q) of a function f ∈ C N ([a, b]) is defined by ! N X f (n) (x) 0 n N +q q 0 D f (x) = lim Dx f (x ) − (x − x) . (3) x0 →x n! n=0 If N = 0, then Definition 3 collapses to Definition 2. We need Definition 4 (related to Definition 3) Let f ∈ C N ([a, b]), N ∈ Z+ . Set ! N X f (n) (x) 0 n 0 q 0 (x − x) . F (x, x − x; q, N ) := Dx f (x ) − n! n=0

(4)

Let x0 − x := t, then x0 = x + t, and F (x, t; q, N ) =

Dxq

N X f (n) (x) n t f (x + t) − n! n=0

! .

(5)

We make Remark 5 Here x0 , x ∈ [a, b], and a ≤ x + t ≤ b, equivalently a − x ≤ t ≤ b − x. From a ≤ x ≤ b, we get a − x ≤ 0 ≤ b − x. We assume here that F (x, ·; q, N ) ∈ C 1 ([a − x, b − x]). Clearly, then it holds DN +q f (x) = F (x, 0; q, N ) ,

(6)

and DN +q f (x) exists in R. We make Remark 6 We observe that: I) Let x0 > x (x0 − x > 0) then f (x0 ) −

N X f (n) (x) 0 n ([2]) (x − x) = n! n=0

2

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" Dx−q

N X f (n) (x) 0 n f (x ) − (x − x) n! n=0

!#

0

Dxq

Dx−q (F (x, x0 − x; q, N )) = 1 Γ (q)

1 Γ (q)

x0 −x

Z

x0

Z

(x0 − z)

q−1

F (x, z − x; q, N ) dz =

x

F (x, t; q, N )

−q+1 dt

(x0 − x − t)

0

=

=

(integration by parts)  x0 −x Z 1 q−1 0 F (x, t; q, N ) (x − x − t) dt + Γ (q) 0 1 Γ (q)

x0 −x

Z 0

(7)

q

dF (x, t; q, N ) (x0 − x − t) dt. dt q

Thus, f (x0 ) − 1 Γ (q + 1)

N X DN +q f (x) 0 f (n) (x) 0 n q (x − x) = (x − x) + n! Γ (q + 1) n=0 x0 −x

Z 0

dF (x, t; q, N ) 0 q (x − x − t) dt, for x0 > x, dt

(8)

N ∈ Z+ . II) Let x0 < x (x0 − x < 0): We have similarly, f (x0 ) −

N X f (n) (x) 0 n ([2]) (x − x) = n! n=0

"

!# N X f (n) (x) 0 n f (x ) − (x − x) = n! n=0 Z x 1 q−1 Dx−q (F (x, x0 − x; q, N )) = (z − x0 ) F (x, z − x; q, N ) dz = Γ (q) x0 Z 0 1 q−1 (x − x0 + t) F (x, t; q, N ) dt = Γ (q) x0 −x Dx−q

0

Dxq

(9)

(integration by parts)  0 Z 1 0 q−1 F (x, t; q, N ) (t + x − x ) dt − Γ (q) x0 −x 1 Γ (q)

Z

0

x0 −x

q

dF (x, t; q, N ) (t + x − x0 ) dt = dt q

(10)

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 Z x0 −x q q dF (x, t; q, N ) (t + x − x0 ) (x − x0 ) 1 1 F (x, 0; q, N ) + dt = Γ (q) q Γ (q) 0 dt q Z x0 −x dF (x, t; q, N ) 1 1 q q DN +q f (x) (x − x0 ) + (t − x0 + x) dt. Γ (q + 1) Γ (q + 1) 0 dt (11) Conclusion: We have proved that (N ∈ Z+ ) I) N X f (n) (x) 0 DN +q f (x) 0 n q f (x0 ) = (x − x) + (x − x) + n! Γ (q + 1) n=0 1 Γ (q + 1)

x0 −x

Z 0

dF (x, t; q, N ) 0 q (x − x − t) dt, when x0 > x, dt

(12)

and II) f (x0 ) = 1 Γ (q + 1)

N q X f (n) (x) 0 DN +q f (x) (x − x0 ) n (x − x) + + n! Γ (q + 1) n=0

Z

x0 −x

0

dF (x, t; q, N ) q (t − x0 + x) dt, when x0 < x. dt

(13)

We have derived Theorem 7 Let f ∈ C N ([a, b]), N ∈ Z+ . Here x, x0 ∈ [a, b], and F (x, ·; q, N ) ∈ C 1 ([a − x, b − x]). Then N X f (n) (x) 0 DN +q f (x) 0 n q f (x ) = (x − x) + |x − x| + n! Γ (q + 1) n=0 0

1 Γ (q + 1)

x0 −x

Z 0

(14)

dF (x, t; q, N ) q |(x0 − x) − t| dt. dt

In particular we get Corollary 8 (to Theorem 7, N = 0) Let f ∈ C ([a, b]); x, x0 ∈ [a, b], and F (x, ·; q, 0) ∈ C 1 ([a − x, b − x]). Then f (x0 ) = f (x) + 1 Γ (q + 1)

Z 0

x0 −x

Dq f (x) 0 q |x − x| + Γ (q + 1)

(15)

dF (x, t; q, 0) q |(x0 − x) − t| dt. dt

4

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References [1] F.B. Adda, J.Cresson, About Non-differential Functions, J. Math. Anal. & Appl., 263 (2001), 721-737. [2] F.B. Adda, J. Cresso, Fractional differential equations and the schr¨ odinger equation, Applied Math. & Computation, 161 (2005), 323-345. [3] K.M. Kolwankar, Local fractional calculus: a Review, arXiv: 1307:0739v1 [nlin.CD] 2 Jul. 2013. [4] K.M. Kolwankar and A.D. Gangal, Local fractional calculus: a calculus for fractal space-time, Fractals: theory and applications in engineering, 171-181, London, New York, Springer, 1999. [5] I. Podlubny, Fractional Differential equations, Academic Press, San Diego, New York, 1999.

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ON VORONOVSKAJA TYPE ESTIMATES OF BERNSTEIN-STANCU OPERATORS RONGRONG XIA AND DANSHENG YU∗

Abstract. In the present paper, we obtain the Voronovaskaja-type results of approximation by a type of Bernstein-Stancu operators with shifted knots.

1. Introduction and The Main Results For any f (x) ∈ C[0,1] , the corresponding Bernstein operators Bn (f, x) are defined as follows:   n X k Bn (f ; x) = f pn,k (x), n k=0
n−k n k where pn,k (x) = k x (1 − x) , k = 0, 1, · · · , n. The approximation properties of Bernstein operators for continuous functions or functions of smoothness have been investigated extensively. Among them, many authors have studied the Voronovaskaja-type asymptotical estimates (see [5]-[7], [13] ). Stancu ([11]) generalized the Bernstein operators to the following so called BernsteinStancu operators:   n X k+α pn,k (x). (1.1) Bn,α,β (f ; x) = f n+β k=0

It was showed that Bn,α,β (f ; x) converges to continuous function f (x) uniformly in [0, 1] for α,β satisfying 0 ≤ α ≤ β. Recently, Gadjiev and Ghorbanalizadeh ([4]) further generalized Bernstein-Stancu operators by using shifted knots as follows:   n   k + α1 n + β2 n X Sn,α,β (f ; x) = f qn,k (x), (1.2) n n + β1 k=0

k  n−k n α2 n+α2 α2 n+α2 where x ∈ n+β , − x , k = 0, 1, · · · , n, and , q (x) = x − n,k n+β2 k n+β2 n+β2 2 αk , βk , k = 1, 2 are positive real numbers satisfying 0 ≤ α1 ≤ β1 , 0 ≤ α2 ≤ β2 . They estimated the approximation rate of approxiamtion by Sn,α,β (f, x) for continuous functions in An . In fact, they established the following: h

i




2010 Mathematics Subject Classification. 41A25, 41A35. Key words and phrases. Bernstein operators with shifted knots, Voronovaskaja-type results. * Corresponding author. 1

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RONGRONG XIA AND DANSHENG YU∗

2

Theorem 1.1. ([4]) Let f be continuous function on [0, 1] . Then the following inequalities hold:  r !  2 n+α  4(β2 −β1 )2 n+β 2 +n  3 2  , if (β2 − β1 ) ≥ (α2 − α1 ) ,  2 ω f, (n+β1 )2 |Sn,α,β (f, x) − f (x)| ≤  r    4(α2 −α1 )2 +n 3   , if (β2 − β1 ) ≤ (α2 − α1 ) . 2 ω f, (n+β )2 1

i h n+α2 α2 , In Theorem 1.1, the approximation properties of Sn,α,β (f, x) in An := n+β n+β2 are 2 considered. As we know, Sn,α,β is positive and linear in the set An . Although, Sn,α,β is still definable on [0, 1]\An , but it is not positive in this case. Then, a natural problem is whether Sn,α,β (f, x) can be used to approximate the continuous functions on the whole interval [0, 1]. Wang, Yu and Zhou ([14]) give a positive answer by establishing the following: Theorem 1.2. Let f be a continuous function on [0, 1] , λ ∈ [0, 1] be a fixed positive number. Then there exists a positive constant C only depending on λ, α1 , α2 , β1 and β2 such that   δ 1−λ (x) |Sn,α,β (f, x) − f (x)| ≤ Cωϕλ f, n √ , (1.3) n p where ϕ(x) = x(1 − x), δn (x) := ϕ(x) + √1n , and     λ λ f x + hϕ (x) − f x − hϕ (x) . ωϕλ (f, t) := sup sup 2 2 0 1: (n) Theorem 1.3
Let f : R → R be n (> 1) times differentiable function such that f (x) ≤ M ∀x ∈ (a, b) . Then, for every x ∈ [a, b]  Zb n−1 1 X  1 M (x − a)n+1 + (b − x)n+1 f (x) + Fk − f (y)dy ≤ · , n n(n + 1)! b−a b−a k=1

a

where Fk is defined by

Fk ≡ Fk (f ; n; x; a; b) ≡

n − k f (k−1) (a)(x − a)k − f (k−1) (b)(x − b)k · . k! b−a

For n = 2, Theorem 1.3 gives  1 f (x) + 2 ≤

2

 Zb (x − a)f (a) + (b − x)f (b) 1 − f (y)dy b−a b−a a "
2 # 2 x − a+b M (b − a) 1 2 + . 4 12 (b − a)2

Preliminaries

Associated with differentiable mappings, there has been extensive research in the literature on related results. Over the past few decades, many studies on obtaining sharp bounds of Ostrowski’s tpye inequalities have been conducted. Most of the calculations within these sharp bounds depend mainly on the magnitudes of Lebesgue norms of derivatives of given functions. In [5]–[8], Dragomir and Wang obtained the following bounds on the deviation of an absolutely continuous mapping f , defined over the interval [a, b], from its integral mean  h

i ′  b−a 2 a+b 2 kf k∞  + x − f ′ ∈ L∞ [a, b];  2 2 b−a ,    Zx   1   f ′ ∈ Lp [a, b], f′ 1 f (x) − f (t)dt q+1 q+1 1/q k kp ≤ (b − x) , (x − a) +  1 1 b−a q+1 b−a   p + q = 1, p > 1; a    ′    b−a + x − a+b  kf k1 , f ′ ∈ L1 [a, b]. 2 2 b−a In [9], Masjed-Jamei and Dragomir provided the following analogues of the Ostrowski’s inequality for a differentiable function f whose first derivative f ′ is bounded, bounded from below, and bounded from above in terms of two functions α, β ∈ C[a, b] as follows: ◦

Theorem 2.1 Let f : I → R, where I is an interval, be a function differentiable in the interior I ◦

of I, and let [a, b] ⊂ I. For any α, β ∈ C[a, b] and x ∈ [a, b], we have the following three cases: 2

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1◦ If α(x) ≤ f ′ (x) ≤ β(x), then  x  Z Zb Zb 1  1 (t − a) α(t)dt + (t − b) β(t)dt ≤ f (x) − f (t)dt b−a b−a a x a  x  Z Zb 1  ≤ (t − a) β(t)dt + (t − b) α(t)dt ; b−a a

◦

(2.1)

x

′

2 If α(x) ≤ f (x), then 1 b−a

"Zx

(t − a)α(t)dt +

a

Zb

(t − b)α(t)dt

x

− max{x − a, b − x} f (b) − f (a) −

Zb

α(t)dt

!#

α(t)dt

!#

a

1 ≤ f (x) − b−a 1 ≤ b−a

"Zx

Zb

f (t)dt

a

(t − a)α(t)dt +

a

Zb

(t − b)α(t)dt

x

+ max{x − a, b − x} f (b) − f (a) −

Zb

,

(2.2)

,

(2.3)

a ◦

′

3 If f (x) ≤ β(x), then 1 b−a

"Zx

(t − a)β(t)dt +

a

Zb

(t − b)β(t)dt

x

− max{x − a, b − x}

Zb

!#

β(t)dt − f (b) + f (a)

a

≤ f (x) − 1 ≤ b−a

1 b−a

"Zx

Zb

f (t)dt

a

(t − a)β(t)dt +

a

Zb

(t − b)β(t)dt

x

+ max{x − a, b − x}

Zb

!#

β(t)dt − f (b) + f (a)

a

3

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The listed inequalities in Theorem 2.1 are significant as they improve all previous results in which the Lebesgue norms of f ′ come into play when handling the bounds calculations. In this case, the required computations in bounds are just in terms of pre-assigned functions. For other related general results, the reader may be refer to [3], [15], [16], [18], [1], and [2]. In this paper, motivated by [9], new integral inequalities of Ostrowski type are obtained. Namely, under certain conditions on f ′ , we give the lower and upper bounds for the difference h 1 E(f ; h) = [f (a) + f (b)] + (1 − h) f (x) − 2 b−a

Zb

f (t)dt,

(2.4)

a

b−a where h ∈ [0, 1] and x ∈ [a+h b−a 2 , b−h 2 ]. Our results provides range of estimates including those given by [9] and [5]–[8]. Utilizing general Peano kernel, we recapture the three inequalities (2.1)– (2.3) obtained by [9]. Some special cases of our result and applications to numerical quadrature rules are also given.

3

Main Results

In order to formulate our main results, we need a kernel K(t; · ) : [a, b] → R defined by (
t − a + h b−a , t ∈ [a, x], 2 K(t; x) =
b−a t − b − h 2 , t ∈ (x, b],

(3.1)

  b−a for all h ∈ [0, 1] and x ∈ a + h b−a 2 , b − h 2 . Also, for two functions α, β ∈ C[a, b], such that α(t) ≤ β(t) for each t ∈ [a, b], we define the functions A(t; · ) : [a, b] → R and B(t; · ) : [a, b] → R by o    1 n A(t; x) = 1 − sgn K(t; x) β(t) + 1 + sgn K(t; x) α(t) (3.2) 2 and o    1 n B(t; x) = 1 − sgn K(t; x) α(t) + 1 + sgn K(t; x) β(t) , (3.3) 2 respectively. We note that  
−1, t ∈ a, a + h b−a ,   2      1, t ∈ a + h b−a , x , 2 sgn K(t; x) = (3.4)
b−a  −1, t ∈ x, b − h ,   2     1, t ∈ b − h b−a 2 ,b ,

b−a and equal to zero at t = a + h b−a 2 and t = b − h 2 . Obviously, (2.4) provides range of estimates including those introduced by [9] and [5]–[8]. For

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instance, when h = 0, h = 1/2, and h = 1, (2.4) can be, respectively, reduced to E(f ; 0) = f (x) −

1 b−a

Zb

f (t)dt

(x ∈ [a, b]),

(3.5)

a

1 1 E(f ; 1/2) = [f (a) + f (b) + 2f (x)] − 4 b−a 1 1 E(f ; 1) = [f (a) + f (b)] − 2 b−a

Zb

Zb

   3a + b a + 3b x∈ , , 4 4

f (t)dt

a

f (t)dt.

(3.6)

(3.7)

a

Theorem 3.1 Let f : I → R, where I is an interval, be a function differentiable in the interior ◦

◦

I of I, and let [a, b] ⊂ I. Also, let E(f ; h), K(t; x), A(t; x), and B(t; x) be given by (2.4), (3.1), (3.2), and (3.3), respectively. b−a For any α, β ∈ C[a, b], h ∈ [0, 1], and x ∈ [a + h b−a 2 , b − h 2 ], we have the following three cases: 1◦ If α(x) ≤ f ′ (x) ≤ β(x), then Zb

1 b−a

a

1 K(t; x)A(t; x)dt ≤ E(f ; h) ≤ b−a

Zb

K(t; x)B(t; x)dt;

(3.8)

a

2◦ If α(x) ≤ f ′ (x), then 1 b−a

(Zb

K(t; x)α(t)dt − L(x, h) f (b) − f (a) −

a

1 ≤ b−a where

(Zb

Zb

α(t)dt

!)

≤ E(f ; h)

a

K(t; x)α(t)dt + L(x, h) f (b) − f (a) −

a

Zb

α(t)dt

!)

,

(3.9)

a

  b−a b−a b−a L(x, h) = max K(t; x) = max x − a − h ,b − x − h ,h ; 2 2 2 t∈[a,b]

(3.10)

3◦ If f ′ (x) ≤ β(x), then 1 b−a

(Zb

K(t; x)β(t)dt − L(x, h)

a

1 ≤ b−a

Z

!)

b

β(t)dt − f (b) + f (a)

a

(Zb

K(t; x)β(t)dt + L(x, h)

Z

a

a

b

≤ E(f ; h) !)

β(t)dt − f (b) + f (a)

,

where L(x, h) is defined by (3.10). 5

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(3.11)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.4, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Proof. By considering the kernel K(t; x) in (3.1), we have Zb a

  Zb Zb
α(t) + β(t) 1 K(t; x) f ′ (t) − dt = K(t; x)f ′ (t)dt − K(t; x) α(t) + β(t) dt 2 2 a

a

= (b − a)E(f ; h) −

1 2

Zb a


K(t; x) α(t) + β(t) dt,

(3.12)

because of Zb

    Zb  Zx  b−a b−a ′ f (x)dt + t− b−h f ′ (x)dt K(t; x)f (t)dt = t− a+h 2 2 ′

a

x

a

=

Zb a

    b−a b−a tf ′ (t)dt − a + h (f (x) − f (a)) − b − h (f (b) − f (x)) 2 2

= (b − a)



 Zb h [f (a) + f (b)] + (1 − h) f (x) − f (t)dt 2 a

= (b − a)E(f ; h). Now, for the first inequality (3.8), the given assumption α(x) ≤ f ′ (x) ≤ β(x) yields ′ f (t) − α (t) + β (t) ≤ β (t) − α (t) . 2 2

(3.13)

Therefore, from (3.12) and (3.13), we get b Z   Zb
(b − a)E(f ; h) − 1 K(t; x) α(t) + β(t) dt = K(t; x) f ′ (t) − α(t) + β(t) dt 2 2 a

a

=

1 2

Zb a


K(t; x) β(t) − α(t) dt,

i.e., 1 − 2

Zb a


K(t; x) β(t) − α(t) dt + 1 2 1 ≤ 2

Zb a

Zb a


K(t; x) α(t) + β(t) dt ≤ (b − a)E(f ; h)


K(t; x) β(t) − α(t) dt + 1 2

Zb a


K(t; x) α(t) + β(t) dt.

Since K(t; x) = K(t; x) sgn K(t; x), and A(t; x) and B(t; x) are defined by (3.2) and (3.3), respectively, the previous inequalities reduce to (3.8). 6

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For the second case, when α(x) ≤ f ′ (x), we have Zb

′

K(t; x) (f (t) − α(t)) dt

=

a

Zb

K(t; x)f (t) −

Zb

(b − a)E(f ; h) −

Zb

′

a

=

K(t; x)α(t)dt

a

K(t; x)α(t)dt.

a

Hence,

Zb (b − a)E(f ; h) − K(t; x)α(t)dt

b Z K(x, t) (f ′ (t) − α(t)) dt a Z x
K(t; x) f ′ (t) − α(t) dt a  Z b
max |K(t; x)| f ′ (t) − α(t) dt t∈[a,b] a ! Z

≤

a

≤ ≤

b

=

L(x, h) f (b) − f (a) −

α(t)dt ,

a

where L(x, h) is defined by (3.10). Then, (3.14) gives (3.9). Finally, for the third case, when f ′ (x) ≤ β(x), we have Zb

′

K(t; x) (f (t) − β(t)) dt

=

a

Zb

K(t; x)f (t) −

Zb

(b − a)E(f ; h) −

Zb

′

a

=

K(t; x)β(t)dt

a

K(t; x)β(t)dt,

a

from which, as before, we obtain Zb (b − a)E(f ; h) − K(t; x)β(t)dt

b Z K(x, t) (f ′ (t) − β(t)) dt a Z x
K(t; x) β(t) − f ′ (t) dt

≤

a

≤

a



≤

max |K(t; x)|

t∈[a,b]

Z L(x, h)

=

a

i.e., (3.11). The proof of this theorem is completed.

Z

a

b

b


β(t) − f ′ (t) dt

 β(t)dt − f (b) + f (a) ,



7

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(3.14)

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Remark 3.1 According to (3.10) and max{u, v} = 12 (u + v + |u − v|), we can see that a + b 1 b−a (1 − h) + x − if h ≤ . L(x, h) = 2 2 2

This expression holds also for h > 21 , but only when |x| > 2h − 1. However, for |x| ≤ 2h − 1, the

= = = = = =

-

-

Figure 1: The function x 7→ L(x, h) for h = 0, 0.25, 0.5, 0.65, 0.8, and 1. function x 7→ L(x, h) is a constant, i.e., L(x, h) =

b−a h. 2

This function on [a, b] = [−1, 1] for different value of h is presented in Figure 1. Now, we consider cases with constant functions α on [a, b]. According to (3.2), (3.3), and (3.4), we get  (β0 , α0 ),     
 (α0 , β0 ), A(t; x), B(t; x) =  (β0 , α0 ),      (α0 , β0 ),

and β, i.e., when α(x) = α0 and β(x) = β0 
, t ∈ a, a + h b−a 2  t ∈ a + h b−a 2 ,x ,
t ∈ x, b − h b−a , 2  t ∈ b − h b−a 2 ,b ,

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so that the corresponding bounds in (3.8) become B

(1)

1 = b−a

Zb

K(t; x)A(t; x)dt

a

 2  1 = − b β0 − a2 α0 − 2(bβ0 − aα0 )x + (β0 − α0 )x2 2(b − a)  1 1 + aα0 + bβ0 − (α0 + β0 )x h − (b − a)(β0 − α0 )h2 2 4

(3.15)

and B

(1)

1 = b−a

Zb

K(t; x)B(t; x)dt

a

 2  1 = a β0 − b2 α0 + 2(bα0 − aβ0 )x + (β0 − α0 )x2 2(b − a)  1 1 + bα0 + aβ0 − (α0 + β0 )x h + (b − a)(β0 − α0 )h2 . 2 4 Also, 1 b−a

Z

a

b

1 K(t; x)dt = b−a

(Z

x a



(3.16)

    ) Z b b−a b−a t− a+h dt + t− b−h dt 2 2 x

1 = (1 − h)(2x − a − b), 2 so that we can find the corresponding lower and upper bounds in the inequalities (3.9) and   α0 f (b) − f (a) (2) B = (1 − h)(2x − a − b) − L(x, h) − α0 , 2 b−a   α0 f (b) − f (a) (2) B = (1 − h)(2x − a − b) + L(x, h) − α0 , 2 b−a   β0 f (b) − f (a) (3) B = (1 − h)(2x − a − b) − L(x, h) β0 − , 2 b−a   β0 f (b) − f (a) (3) B = (1 − h)(2x − a − b) + L(x, h) β0 − , 2 b−a

(3.11): (3.17) (3.18) (3.19) (3.20)

where L(x, h) is defined by (3.10). Thus, for constant functions α and β on [a, b], we get the following result: Corollary 3.1 Under the assumptions of Theorem 3.1 with α(x) = α0 and β(x) = β0 , we have: 1◦ If α0 ≤ f ′ (x) ≤ β0 , then B (1) ≤ E(f ; h) ≤ B 2◦ If α0 ≤ f ′ (x), then B (2) ≤ E(f ; h) ≤ B ◦

′

(3)

3 If f (x) ≤ β0 , then B ≤ E(f ; h) ≤ B where the bounds are given in (3.15)–(3.19).

(2)

(3)

(1)

;

;

,

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4

Some Applications in Numerical Integration

Inequalities of Ostrowski’s type have attracted considerable interest over the years. Many authors have worked on this subject and proved many extensions and generalizations, including applications in numerical integration (cf. [4]). These inequalities can be considered as error estimates of certain elementary quadrature rules in some classes of functions. Beside the bounds of (3.5)–(3.7), in this section we consider also ones for h = 1/3, 1/4, 2/3, and 3/4, i.e., 1 1 E(f ; 1/3) = [f (a) + f (b) + 4f (x)] − 6 b−a

Zb

1 1 E(f ; 1/4) = [f (a) + f (b) + 6f (x)] − 8 b−a

Zb

f (t)dt

   5a + b a + 5b x∈ , , 6 6

f (t)dt

   7a + b a + 7b x∈ , , 8 8

(4.1)

a

a

1 1 E(f ; 2/3) = [f (a) + f (b) + f (x)] − 3 b−a

Zb

f (t)dt

   2a + b a + 2b x∈ , , 3 3

a

1 1 E(f ; 3/4) = [3f (a) + 3f (b) + 2f (x)] − 8 b−a

Zb

f (t)dt



x∈

a



5a + 3b 3a + 5b , 8 8



,

respectively. For x = (a + b)/2, E(f ; 1/3), given before by (4.1), represents the error in the well-known Simpson formula (cf. [13, pp. 343–350]). In order to get the corresponding estimates of (2.4), i.e.,  h 1 E(f ; h) = f (a) + f (b) + (1 − h)f (x) − 2 b−a

Zb

f (t)dt

   b−a b−a x∈ a+h ,b − h , 2 2

a

for different values of h, we use here Corollary 3.1 (Case 1◦ ). Case h = 0. Here, the value of x can be arbitrary in [a, b]. Then, B (1) and B B (1) = − and B

(1)

=

(1)

reduce to

 2  1 b β0 − a2 α0 − 2(bβ0 − aα0 )x + (β0 − α0 )x2 2(b − a)

 2  1 a β0 − b2 α0 + 2(bα0 − aβ0 )x + (β0 − α0 )x2 , 2(b − a)

so that, under the condition α0 ≤ f ′ (x) ≤ β0 , for each x ∈ [a, b], we have B

(1)

1 ≤ f (x) − b−a

Zb

f (t)dt ≤ B

(1)

.

a

10

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(4.2)

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For the symmetric bounds of the first derivative f ′ (|f ′ (x)| ≤ β0 ), i.e., if α0 = −β0 , (4.2) reduces to "
2 # Zb x − a+b 1 1 2 f (x) − f (t)dt ≤ + (b − a)β0 , b−a 4 (b − a)2 a

which is, in fact, the original Ostrowski inequality (1.1). Otherwise, (4.2) for x = (a + b)/2 gives the error estimate for the midpoint rule,   Zb 1 a + b 1 f ≤ (b − a)(β0 − α0 ), − f (t)dt 8 2 b−a a

while for x = b it gives the error estimate for the so-called endpoint rule 1 1 (b − a)α0 ≤ f (b) − 2 b−a

Zb

f (t)dt ≤

1 (b − a)β0 . 2

a

Case h = 1. Here x must be (b − a)/2! Taking h = 1 in (3.15) and (3.15), for the trapezoidal rule (3.7), we obtain the same bound as for the midpoint rule, Zb 1 1  f (a) + f (b) − 1 f (t)dt ≤ (b − a)(β0 − α0 ). 2 b−a 8 a

Case 0 < h < 1. Now we take x = (a + b)/2 in (3.15) and (3.15). Since, in that case,
1 (1) −B (1) = B = (b − a) 1 − 2h + 2h2 (β0 − α0 ), 8 we get   Zb  h b − a1  f (a) + f (b) + (1 − h)f a + b − 1 f (t)dt ≤ − h + h2 (β0 − α0 ), 2 2 b−a 4 2 a

provided that α0 ≤ f ′ (x) ≤ β0 for x ∈ [a, b]. For h = 1/2, 1/3, 1/4, 2/3, and 3/4, the inequality (4.3) reduces to     Zb 1 f (a) + 2f a + b + f (b) − 1 ≤ b − a (β0 − α0 ), f (t)dt 4 2 b − a 16 a     Zb 1 a + b 1 5(b − a) f (a) + 4f + f (b) − f (t)dt ≤ (β0 − α0 ), 6 2 b − a 72 a     Zb 1 5(b − a) a + b 1 f (a) + 6f + f (b) − f (t)dt ≤ (β0 − α0 ), 8 2 b−a 64 a     Zb 1 a + b 1 13(b − a) + f (b) − f (t)dt ≤ (β0 − α0 ), 3 f (a) + f 2 b−a 144 a

11

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(4.3)

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and

    Zb 1 5(b − a) 1 a + b 3f (a) + 2f + 3f (b) − f (t)dt ≤ (β0 − α0 ), 8 2 b−a 64 a

respectively.

5

Conclusion

Inspired and motivated by the work of Masjed-Jamei and Dragomir [9], new integral inequalities of Ostrowski type are obtained with bounds are just in terms of pre-assigned functions. Our results provides a generalization of error bounds that is independent of Lebesgue norms including those given by [9] and [5]–[8]. We utilize general Peano kernel to recapture the three inequalities (3.8), (3.9), and (3.11), obtained in [9]. Some special cases and applications to numerical quadrature rules are also proposed.

References [1] W.G. Alshanti and A. Qayyum, A note on new Ostrowski type inequalities using a generalized kernel, Bull. Math. Anal. Appl. 9(1) (2017), 74–91. [2] W.G. Alshanti, A. Qayyum and M.A. Majid, Ostrowski type inequalities by using general quadratic kernel, J. Ineq. Spec. Funct. 8(4) (2017), 111–135. [3] P. Cerone, A new Ostrowski type inequality involving integral means over end intervals, Tamkang J. Math. 33 (2002), 109–118. [4] S. S. Dragomir and Th. M. Rassias (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002. [5] S. S. Dragomir and S. Wang, A new inequality Ostrowski’s type in L1 norm and applications to some special means and some numerical quadrature rules. Tamkang J. Math. 28 (1997), 239–244. [6] S. S. Dragomir and S. Wang, An inequality Ostrowski-Gr¨ uss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl. 33 (1997), 15–22. [7] S. S. Dragomir and S. Wang, A new inequality Ostrowski’s type in Lp norm, Indian J. Math. 40 (1998), 299–304. [8] S. S. Dragomir and S. Wang, Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and to some numerical quadrature rules, Appl. Math. Lett. 11 (1998), 105–109. [9] M. Masjed-Jamei and S. S. Dragomir, An analogue of the Ostrowski inequality and applications, Filomat 28 (2014), 373–381.

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[10] G. V. Milovanovi´c, On some integral inequalities. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Math. Fiz. No. 498–541 (1975), 119–124. [11] G. V. Milovanovi´c and J. E. Peˇcari´c, On generalization of the inequality of A. Ostrowski and some related applications. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Math. Fiz. No. 544–576 (1976), 155–158. [12] G. V. Milovanovi´c, On some functional inequalities. Univ. Beograd. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Math. Fiz. No. 599 (1977), 1–59. [13] G. V. Milovanovi´c, Numerical Analysis, Part I, University of Niˇs, Niˇs, 1979 (Serbian). ¨ [14] A. Ostrowski, Uber die Absolutabweichung einer differentienbaren Funktion von ihrem Integralimittelwert. Comment. Math. Helv. 10 (1938), 226–227. [15] N. Ujevi´c, New bounds for the first inequality of Ostrowski-Gr¨ uss type and applications, Comput. Math. Appl. 46 (2003), 421–427. [16] N. Ujevi´c, A generalization of Ostrowski’s inequality and applications in numerical integration, Appl. Math. Lett. 17 (2004), 133–137. [17] A. Qayyum, M. Shoaib and I. Faye, Some new generalized results on Ostrowski type integral inequalities with application, J. Comput. Anal. Appl. 19 (2015), 693–712. [18] L. Zeng, More on Ostrowski type inequalities for some s-convex functions in the second sense, Demonstratio Mathematica. 49(4) (2016), 398–412. (W. G. Alshanti) Department of General Studies, Jubail University College, Saudi Arabia E-mail address: [email protected] (G. V. Milovanovi´c) Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia & Faculty of Sciences and Mathematics, University of Niˇ s, 18000 Niˇ s, Serbia E-mail address: [email protected]

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ITERATES OF CHENEY-SHARMA TYPE OPERATORS ON A TRIANGLE WITH CURVED SIDE ˘ TEODORA CATINAS ¸ ∗ , DIANA OTROCOL∗∗

Abstract. We consider some Cheney–Sharma type operators as well as their product and Boolean sum for a function defined on a triangle with one curved side. Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of these operators. Keywords: Triangle with curved side, Cheney-Sharma operators, contraction principle, weakly Picard operators. MSC 2010 Subject Classification: 41A36, 41A25, 39B12, 47H10.

1. Cheney-Sharma type operators We recall some results regarding Cheney-Sharma type operators on a triangle with one curved side, introduced in [6]. Similar operators were introduced and studied in [3], [4], [5] and [9]. We consider the standard triangle T˜h with vertices V1 = (0, h), V2 = (h, 0) and V3 = (0, 0), with two straight sides Γ1 , Γ2 , along the coordinate axes, and with the third side Γ3 (opposite to the vertex V3 ) defined by the one-to-one functions f and g, where g is the inverse of the function f, i.e., y = f (x) and x = g(y), with f (0) = g(0) = h, for h > 0. Also, we have f (x) ≤ h and g(y) ≤ h, for x, y ∈ [0, h] . Let F be a real-valued function defined on Teh and (0, y), (g(y), y), respectively, (x, 0), (x, f (x)) be the points in which the parallel lines to the coordinate axes, passing through the point (x, y) ∈ Teh , intersect the sides Γi , i = 1, 2, 3. (See Figure 1.) In [6], we have obtained the following extensions of Cheney-Sharma operator of second kind, to the case of functions defined on Teh :   m P (1.1) , y , (Qxm F )(x, y) = qm,i (x, y)F i g(y) m (Qyn F )(x, y)

=

i=0 n P

  qn,j (x, y)F x, j f (x) , n

j=0

with qm,i (x, y) = qn,j (x, y) =

m i−1 1 x x x x (1 − g(y) )[1 − g(y) + (m − i)β]m−i−1 , i (1+mβ)m−1 g(y) ( g(y) + iβ)
y y y y n j−1 1 (1 − f (x) )[1 − f (x) + (n − j)b]n−j−1 , j (1+nb)n−1 f (x) ( f (x) + jb)




∗ Babe¸ s-Bolyai University, Faculty of Mathematics and Computer Science, M. Kog˘ alniceanu St. 1, RO-400084 Cluj Napoca, Romania, E-mail: [email protected]. ∗∗ T. Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj Napoca, Romania; Technical University of Cluj Napoca, Memorandumului St. 28, RO-400114, Cluj Napoca, Romania, E-mail: [email protected].

1

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2

V

2

(x,f(x))

Γ3

Γ1 (x,y)

(0,y)

V

Γ2

(x,0)

3

(g(y),y)

V1

Figure 1. Triangle T˜h . where ∆xm =

n

o n o f (x) y i g(y) 0, m and ∆ = j i = n m n j = 0, n

are uniform partitions of the intervals [0, g(y)] and [0, f (x)] and m, n ∈ N, β, b ∈ R+ . Remark 1.1. As the Cheney-Sharma operator of second kind interpolates a given function at the endpoints of the interval, we may use the operators Qxm and Qyn as interpolation operators on Teh . Theorem 1.2. [6] If F is a real-valued function defined on Teh then the following properties hold: (i) Qxm F = F on Γ1 ∪ Γ3 ; (ii) Qyn F = F on Γ2 ∪ Γ3 ; (iii) (Qxm eij ) (x, y) = xi y j , i = 0, 1; j ∈ N; (iv) (Qyn eij ) (x, y) = xi y j , i ∈ N; j = 0, 1, where eij (x, y) = xi y j , i, j ∈ N. 1 2 Let Pmn = Qxm Qyn , respectively, Pnm = Qyn Qxm be the products of the operators y x Qm and Qn . We have n m X g(y)     X
f (i m ) g(y) 1 , y F i , j , (1.2) Pmn qm,i (x, y) qn,j i g(y) F (x, y) = m m n i=0 j=0

respectively, 2 Pnm F




(x, y) =

m X n X

   g(j f (x) )  qm,i x, j f (x) qn,j (x, y) F i mn , j f (x) . n n

i=0 j=0

Theorem 1.3. If F is a real-valued function defined on Teh then 1 (i) (Pmn F )(Vi ) = F (Vi ), i = 1, ..., 3; 1 (Pmn F )(Γ3 ) = F (Γ3 ), 2 (ii) (Pnm F )(Vi ) = F (Vi ), i = 1, ..., 3; 2 (Pnm F )(Γ3 ) = F (Γ3 ). We consider the Boolean sums of the operators Qxm and Qyn , (1.3)

1 Smn := Qxm ⊕ Qyn = Qxm + Qyn − Qxm Qyn , 2 Snm := Qyn ⊕ Qxm = Qyn + Qxm − Qyn Qxm .

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Theorem 1.4. If F is a real-valued function defined on Teh , then 1 Smn , F ∂ Teh = F ∂ Te h 2 Smn F ∂ Teh = F . ∂ Teh

2. Weakly Picard operators We recall some results regarding weakly Picard operators that will be used in the sequel (see, e.g., [21]). Let (X, d) be a metric space and A : X → X an operator. We denote by FA := {x ∈ X | A(x) = x}-the fixed points set of A; I(A) := {Y ⊂ X | A(Y ) ⊂ Y, Y 6= ∅}-the family of the nonempty invariant subsets of A; 0

A := 1X , A1 := A, ..., An+1 := A ◦ An , n ∈ N. Definition 2.1. The operator A : X → X is a Picard operator if there exists x∗ ∈ X such that: (i) FA = {x∗ }; (ii) the sequence (An (x0 ))n∈N converges to x∗ for all x0 ∈ X. Definition 2.2. The operator A is a weakly Picard operator if the sequence (An (x))n∈N converges, for all x ∈ X, and the limit (which may depend on x) is a fixed point of A. Definition 2.3. If A is a weakly Picard operator then we consider the operator A∞ , A∞ : X → X, defined by A∞ (x) := lim An (x). n→∞

Theorem 2.4. An operator S A is a weakly Picard operator if and only if there exists a partition of X, X = Xλ , such that λ∈Λ

(a) Xλ ∈ I(A), ∀λ ∈ Λ; (b) A|Xλ : Xλ → Xλ is a Picard operator, ∀λ ∈ Λ. 3. Iterates of Cheney-Sharma type operators We study the convergence of the iterates of the Cheney-Sharma type operators (1.1) and of their product and Boolean sum operators, using the weakly Picard operators technique and the contraction principle. The same approach for some other linear and positive operators lead to similar results in [1], [2], [7], [8], [22][24]. The limit behavior for the iterates of some classes of positive linear operators were also studied, for example, in [10]-[20]. In the papers [10]-[12] were introduced new methods for the study of the asymptotic behavior of the iterates of positive linear operators. These techniques enlarge the class of operators for which the limit of the iterates can be calculated. Let F be a real-valued function defined on Teh , h ∈ R+ . First we study the convergence of the iterates of the Cheney–Sharma type operators given in (1.1).

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Theorem 3.1. The operators Qxm and Qyn are weakly Picard operators and F (g(y), y) − F (0, y) x + F (0, y) , g(y) F (x, f (x)) − F (x, 0) (Qy,∞ y + F (x, 0) . n F ) (x, y) = f (x)

(Qx,∞ m F ) (x, y) =

(3.1) (3.2)

Proof. Taking into account the interpolation properties of Qxm and Qyn (from Theorem 1.2), let us consider the following sets: (3.3) (1)

X ϕ|

= {F ∈ C(Teh ) | F (0, y) = ϕ|Γ1 , F (g(y), y) = ϕ|Γ3 }, for y ∈ [0, h],

(2)

= {F ∈ C(Teh ) | F (x, 0) = ψ|Γ2 , F (x, f (x)) = ψ|Γ3 }, for x ∈ [0, h],

Γ1 , ϕ|Γ3

X ψ|

Γ2 , ψ|Γ3

and for ϕ, ψ ∈ C(Teh ) we denote by ϕ|Γ3 − ϕ|Γ1 x + ϕ|Γ1 , Γ1 g(y) 3 ψ| − ψ|Γ2 (2) F ψ| , ψ| (x, y) = Γ3 y + ψ|Γ2 . Γ2 Γ3 f (x) (1)

F ϕ|

, ϕ|Γ

(x, y) =

We have the following properties: (1)

(2)

(i) X ϕ|

Γ1 , ϕ|Γ3

and X ψ|

(2)

(1)

is an invariant subset of Qxm and X ψ|

(ii) X ϕ|

Γ2 , ψ|Γ3

Γ1 , ϕ|Γ3

subset of (iii) C(Teh ) =

for ϕ, ψ ∈ C(Teh ) and n, m ∈ N ; (1) ∪ ∪ X ϕ| , ϕ| and C(Teh ) = Γ1

of C(Teh ); (1) (1) (iv) F ϕ| , ϕ| ∈ X ϕ| F

is an invariant

∗

Qyn ,

ϕ∈C(Teh )

Γ1 Qx m

are closed subsets of C(Teh );

Γ2 , ψ|Γ3

Γ3

X ψ|

Γ2 , ψ|Γ3

(2)

(2)

Γ1 , ϕ|Γ3

Γ3

(2)

ψ∈C(Teh )

∈ X ψ|

∩ FQxm and F ψ|

Γ2 , ψ|Γ3

Γ2 , ψ|Γ3

are partitions

∩ FQyn , where

and FQyn denote the fixed points sets of Qxm and Qyn .

The statements (i) and (iii) are obvious. (ii) By linearity of Cheney-Sharma operators and Theorem 1.2, it follows that (1) (1) (2) (2) ∀F ϕ| , ϕ| ∈ X ϕ| , ϕ| and ∀F ψ| , ψ| ∈ X ψ| , ψ| we have Γ1

Γ3

Γ1

Γ3

Γ2

(1) Γ1 , ϕ|Γ3

(2) Γ2 , ψ|Γ3

Γ1 , ϕ|Γ3

(2)

and X ψ|

Γ2 , ψ|Γ3

Γ3

(x, y) = F ϕ|

Γ1 , ϕ|Γ3

(2)

Qyn F ψ| (1)

Γ2

(1)

Qxm F ϕ|

So, X ϕ|

Γ3

(x, y) = F ψ|

Γ2 , ψ|Γ3

(x, y), (x, y).

are invariant subsets of Qxm and, respectively, of

Qyn , for ϕ, ψ ∈ C(Teh ) and n, m ∈ N∗ ; (iv) We prove that Qxm |X (1)

ϕ|Γ , ϕ|Γ 1 3

(1)

: X ϕ|

Γ1 , ϕ|Γ3

(1)

→ X ϕ|

Γ1 , ϕ|Γ3

and Qyn |X (2)

ψ|Γ , ψ|Γ 2 3

(2)

: X ψ|

(2)

Γ2 , ψ|Γ3

→ X ψ|

Γ2 , ψ|Γ3

are contractions for ϕ, ψ ∈ C(Teh ) and n, m ∈ N∗ .

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Let F, G ∈ X ϕ|

Γ1 , ϕ|Γ3

. From (1.1) and (3.3) we get

|Qxm (F )(x, y) − Qxm (G)(x, y)| = = |Qxm (F − G)(x, y)| ≤ ≤ |qm,0 (x; y) [F (0, 0) − G(0, 0)]| m X h  i   ig(y) jf (x) + qm,i (x; y) F m , y − G x, n i=1 m X h    i jf (x) = qm,i (x; y) F ig(y) , y − G x, m n i=1

≤

=

m X

qm,i (x; y) kF − Gk∞

i=1 "m X

# qm,i (x; y) − qm,0 (x; y) kF − Gk∞

i=0

  h im−1  x x 1 − 1 − g(y) 1 − g(y)(1+mβ) kF − Gk∞   m−1  1 ≤ 1 − 1 − 1+mβ kF − Gk∞ , =

where k·k∞ denotes the Chebyshev norm. Hence, (3.4)

kQxm (F )(x, y) − Qxm (G)(x, y)k∞ ≤   m−1  (1) 1 ≤ 1 − 1 − 1+mβ kF − Gk∞ , ∀F, G ∈ X ϕ|

Γ1 , ϕ|Γ3

i.e., Qxm |X (1)

ϕ|Γ , ϕ|Γ 1 3

is a contraction for ϕ ∈ C(Teh ).

Analogously, we prove that Qyn |X (2)

ψ|Γ , ψ|Γ 2 3

On the other hand, (2)

X ψ|

Γ2 , ψ|Γ3

,

ϕ|Γ − ϕ|Γ 3 1 g(y)

is a contraction for ψ ∈ C(Teh ). (1)

(·) + ϕ|Γ1 ∈ X ϕ|

Γ1 , ϕ|Γ3

and

ψ|Γ − ψ|Γ 3 2 f (x)

(·) + ψ|Γ2 ∈

are fixed points of Qxm and Qyn , i.e., ϕ|Γ − ϕ|Γ 3 1 g(y)

(·) + ϕ|Γ1 ,

ψ|Γ − ψ|Γ 3 2 f (x)

(·) + ψ|Γ2 .

(x, y) :=

ϕ|Γ − ϕ|Γ 3 1 g(y)

Γ3 − ϕ|Γ1

 ϕ|

 (·) + ϕ| Γ1 = g(y)  ψ| − ψ|  Γ3 Γ2 Qyn (·) + ψ| = Γ f (x) 2

Qxm

(1)

From the contraction principle, F ϕ|

Γ1 , ϕ|Γ3

unique fixed point of

Qxm

in

(1) X ϕ| , ϕ| Γ Γ 1

and

3

Qxm |X (1)

x + ϕ|Γ1 is the

is a Picard operator,

ϕ|Γ , ϕ|Γ 1 3

with (Qx,∞ m F ) (x, y) =

F (g(y),y)−F (0,y) x g(y)

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6 (2)

and, similarly, F ψ|

Γ2 , ψ|Γ3

Qyn

in

(2) X ψ| , ψ| Γ Γ 2

and

(x, y) :=

ψ|Γ − ψ|Γ 3 2 f (x)

Qyn |X (2)

y + ψ|Γ2 is the unique fixed point of

is a Picard operator, with

ψ|Γ , ψ|Γ 2 3

3

(Qy,∞ n F ) (x, y) =

F (x,f (x))−F (x,0) y f (x)

+ F (x, 0) .

Consequently, taking into account (ii), by Theorem 2.4, it follows that the operators Qxm and Qyn are weakly Picard operators.  Further we study the convergence of the product and Boolean sum operators given in (1.2) and (1.3). 1 Theorem 3.2. The operator Pmn is a weakly Picard operator and
F (g(y), y) 1,∞ (3.5) Pmn F (x, y) = x. g(y)

Proof. Let Xα = {F ∈ C(Teh ) | F (g(y), y) = α}, and denote by Fα (x, y) :=

α∈R

α x. g(y)

We remark that: (i) Xα is a closed subset of C(Teh ); 1 (ii) Xα is an invariant subset of Pmn , for α ∈ R and n, m ∈ N∗ ; (iii) C(Teh ) = ∪Xα is a partition of C(Teh ); α

1 1 , where FP 1 . (iv) Fα ∈ Xα ∩ FPmn denote the fixed points sets of Pmn mn The statements (i) and (iii) are obvious. (ii) Similarly with the proof of Theorem 3.1, by linearity of Cheney-Sharma 1 , for operators and Theorem 1.3, it follows that Xα is an invariant subset of Pmn ∗ α ∈ R and n, m ∈ N ; (iv) We prove that 1 Pmn : Xα → Xα Xα

is a contraction for α ∈ R and n, m ∈ N∗ . Let F, G ∈ Xα . From [2, Lemma 8] and (3.4), it follows that 1 1 1 Pmn (F )(x, y) − Pmn (G)(x, y) = Pmn (F − G)(x, y)   m−1  n−1  mβ nb ≤ 1 − 1+mβ kF − Gk∞ , 1+nb 1 so, Pmn is a contraction for α ∈ R. Xα 1 From the contraction principle we have that Fα is the unique fixed point of Pmn 1 in Xα and Pmn X is a Picard operator, so (3.5) holds. Consequently, taking into α 1 account (ii), by Theorem 2.4, it follows that the operators Pmn is a weakly Picard operator.  2 Remark 3.3. Similar results can be obtained for the operator Pmn . 1 Theorem 3.4. The operator Smn is a weakly Picard operator and
−F (0,y) (x,0) 1,∞ Smn F (x, y) = g(y) x + F (x,f (x))−F y + F (x, 0) + F (0, y) . f (x)

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Proof. The proof follows the same steps as the proof of Theorem 3.2, using the following inequality kSmn (F )(x, y) − Smn (G)(x, y)k∞   m−1  n−1  m−1  n−1  mβ mβ nb nb ≤ 1 − 1+mβ + 1+nb − 1+mβ kF − Gk∞ , 1+nb 1 for proving that Smn is a contraction.



2 Remark 3.5. We have a similar result for the operator Snm .

References [1] Agratini, O., Rus, I.A., Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Caroline 44 (2003), 555-563. [2] Agratini, O., Rus, I.A., Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Anal. Forum 8(2) (2003), 159-168. [3] Blaga, P., C˘ atina¸s, T., Coman, G., Bernstein-type operators on triangle with one curved side, Mediterr. J. Math. 9 (2012), No. 4, 833-845. [4] Blaga, P., C˘ atina¸s, T., Coman, G., Bernstein-type operators on a square with one and two curved sides, Studia Univ. Babe¸s–Bolyai Math. 55 (2010), No. 3, 51-67. [5] Blaga, P., C˘ atina¸s, T., Coman, G., Bernstein-type operators on triangle with all curved sides, Appl. Math. Comput. 218 (2011), 3072-3082. [6] C˘ atina¸s, T., Extension of some Cheney-Sharma type operators to a triangle with one curved side, 2017, submitted. [7] C˘ atina¸s, T., Otrocol, D., Iterates of Bernstein type operators on a square with one curved side via contraction principle, Fixed Point Theory 14 (2013), No. 1, 97-106. [8] C˘ atina¸s, T., Otrocol, D., Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15 (2013), No. 7, 1240-1246. [9] Coman, G., C˘ atina¸s, T., Interpolation operators on a triangle with one curved side, BIT Numerical Mathematics 50 (2010), No. 2, 243-267. [10] Gavrea, I., Ivan, M., The iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010), 366-368. [11] Gavrea, I., Ivan, M., The iterates of positive linear operators preserving the constants, Appl. Math. Lett. 24 (2011), No. 12, 2068-2071. [12] Gavrea, I., Ivan, M., On the iterates of positive linear operators, J. Approx. Theory 163 (2011), No. 9, 1076-1079. [13] Gonska, H., Kacso, D., Pit¸ul, P., The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal. 1 (2006), 403-423. [14] Gonska, H., Pit¸ul, P., Ra¸sa, I., Over-iterates of Bernstein-Stancu operators, Calcolo 44 (2007), 117-125. [15] Gonska, H., Ra¸sa, I., The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar. 111 (2006), No. 1-2, 119-130. [16] Gonska, H., Ra¸sa, I., On infinite products of positive linear operators reproducing linear functions, Positivity 17 (2013), No. 1, 67-79. [17] Gw´ o´ zd´ z-Lukawska, G., Jachymski, J., IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J. Math. Anal. Appl. 356(2) (2009), 453-463. [18] Karlin, S. , Ziegler, Z., Iteration of positive approximation operators, J. Approx. Theory 3 (1970), 310-339. [19] Kelisky, R.P., Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511-520. [20] Ra¸sa, I., C0 -Semigroups and iterates of positive linear operators: asymptotic behaviour, Rend. Circ. Mat. Palermo, Ser. II, Suppl. 82 (2010), 123-142. [21] Rus, I.A., Picard operators and applications, Sci. Math. Jpn. 58 (2003), 191-219. [22] Rus, I.A., Iterates of Stancu operators, via contraction principle, Studia Univ. Babe¸s–Bolyai Math. 47 (2002), No. 4, 101-104. [23] Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004), 259-261.

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[24] Rus, I.A., Fixed point and interpolation point set of a positive linear operator on C(D), Studia Univ. Babe¸s–Bolyai Math. 55 (2010), No. 4, 243-248.

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Contemporary Concepts of Neutrosophic Fuzzy Soft BCK -submodules R. S. Alghamdi1 and Noura Omair Alshehri2 1

Department of Mathematics, Faculty of Science, King Abdulaziz University,

Jeddah, Saudi Arabia; [email protected], [email protected] 2

Department of Mathematics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia; [email protected]

Abstract In this paper, we introduce the concept of neutrosophic fuzzy soft translations and neutrosophic fuzzy soft extensions of neutrosophic fuzzy soft BCK-submodules and discusse the relation between them. Also, we define the notion of neutrosophic fuzzy soft multiplications of neutrosophic fuzzy soft BCK-submodules. Finally, we investigate some resultes.

Keywords: BCK -algebras, BCK -modules, soft sets, fuzzy soft sets, neutrosophic sets, neutrosophic soft sets, neutrosophic fuzzy soft BCK-submodules, neutrosophic fuzzy soft translations, neutrosophic fuzzy soft multiplications and neutrosophic fuzzy soft extensions.

1

Introduction

Fuzzy set theory which was developed by Zadeh [23] is an appropriate theory for dealing with vagueness. It is consedered as the one of theories can be handled with uncertainties. Combining fuzzy set models with other mathematical models has attracted the attention of many researchers. Intervalvalued fuzzy sets [24], hesitant fuzzy sets [21] , intuitionistic fuzzy sets [3, 4], Intutionistic Fuzzy BCK-submodules [5] and (,  ∨ q)-fuzzy BCK-submodules [2] are some of the researches that have dealt this subject. Neutrosophic algebraic structure is a very recent study. It was applied in many fields in order to solve problems related to uncertainties and indeterminacy where they happens to be one of the major factors in almost all real-world problems. Neutrosophic set is a generalizations of the fuzzy set especially of intuitionistic fuzzy set. The intuitionistic fuzzy set has the degree of non-membership

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as introduced by K. Atanassov [3]. Smarandache in 1998 [19] has introduced the degree of indeterminacy as an independent component and defined the neutrosophic set on three components: truth, indeterminacy and falsity. The concept of BCK-algebra was first initiated by Imai and Iseki [8]. In 1994, the notion of BCKmodules was introduced by H. Abujable, M. Aslam and A. Thaheem as an action of BCK-algebras on abelian group [1]. BCK-modules theory then was developed by Z. perveen, M. Aslam and A. Thaheem [18]. Bakhshi [6] presented the concept of fuzzy BCK-submodules and investigated their properties. Recently, H. Bashir and Z. Zahid applied the theory of soft sets on BCK-modules in [12]. Translations, multiplications and extensions are very interested mathematical tools. They are types of operations that researchers like to apply with fuzzy set theory. In this paper, we introduce the concept of neutrosophic fuzzy soft translations and neutrosophic fuzzy soft extensions of neutrosophic fuzzy soft BCK-submodules and discusse the relation between them. Also, we define the notion of neutrosophic fuzzy soft multiplications of neutrosophic fuzzy soft BCK-submodules. Finally, we investigate some resultes.

2

Preliminaries

In this section, some preliminaries from the soft set theory, neutrosophic soft sets, BCK-algebras and BCK-modules are induced. Definition 2.1.[17] Let U be an initial universe and E be a set of parameters. Let P (U ) denote the power set of U and let A be a nonempty subset of E. A pair FA = (F, A) is called a soft set over U , where A ⊆ E and F : A → P (U ) is a set-valued mapping, called the approximate function of the soft set (F, A). It is easy to represent a soft set (F, A) by a set of ordered pairs as follows: (F, A) = {(x, F (x)) : x ∈ A} Definition 2.2.[20] A neutrosophic set A on the universe of discourse U is defined as A = {(x, TA (x) , IA (x) , FA (x)) , x ∈ U } where TA : X → ]− 0, 1+ [ is a truth membership function, IA : U → ]− 0, 1+ [ is an indeterminate membership function, and FA : X → ]− 0, 1+ [ is a false membership function and − 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3+ . From philosophical point of view, the neutrosophic set takes the value from real standard or nonstandard subsets of ]− 0, 1+ [. But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of ]− 0, 1+ [. Hence we consider the neutrosophic set which takes the value from the subset of [0, 1]. Definition 2.3.[13] Let U be an initial universe set and E be a set of parameters. Consider A ⊂ E. Let P (U ) denotes the set of all neutrosophic sets of U . The collection (F, A) is termed to be the neutrosophic soft set (NSS) over U , where F is a mapping given by F : A → P (U ) . 2 746

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Definition 2.4.[8, 9] An algebra (X, ∗, 0) of type (2, 0) is called BCK-algebra if it satisfying the following axioms: (BCK -1) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, (BCK -2) (x ∗ (x ∗ y)) ∗ y = 0, (BCK -3) x ∗ x = 0, (BCK -4) 0 ∗ x = 0, (BCK -5) x ∗ y = 0 and y ∗ x = 0 imply x = y, for all x, y, z ∈ X. A partial ordering “≤ ” is defined on X by x ≤ y ⇔ x ∗ y = 0. A BCK -algebra X is said to be bounded if there is an element 1 ∈ X such that x ≤ 1, for all x ∈ X, commutative if it satisfies the identity x ∧ y = y ∧ x, where x ∧ y = y ∗ (y ∗ x), for all x, y ∈ X and implicative if x ∗ (y ∗ x) = x, for all x, y ∈ X. Definition 2.5.[1] Let X be a BCK -algebra. Then by a left X-module (abbreviated X-module), we mean an abelian group M with an operation X × M → M with (x, m) 7−→ xm satisfies the following axioms for all x, y ∈ X and m, n ∈ M : (i) (x ∧ y)m = x(ym), (ii) x(m + n) = xm + xn, (iii) 0m = 0. If X is bounded and M satisfies 1m = m, for all m ∈ M, then M is said to be unitary. A mapping µ : X → [0, 1] is called a fuzzy set in a BCK -algebra X. For any fuzzy set µ in X and any t ∈ [0, 1], we define set U (µ; t) = µt = {x ∈ X|µ(x) ≥ t}, which is called upper t-level cut of µ. Definition 2.6.[6] A fuzzy subset µ of M is said to be a fuzzy BCK -submodule if for all m, m1 , m2 ∈ M and x ∈ X, the following axioms hold: (FBCKM1) µ(m1 + m2 ) ≥ min{µ(m1 ), µ(m2 )}, (FBCKM2) µ(−m) = µ(m), (FBCKM3) µ(xm) ≥ µ(m). Definition 2.7.[6] Let M , N be modules over a BCK-algebra X. A mapping f : M → N is called BCK-module homomorphism if (1) f (m1 + m2 ) = f (m1 ) + f (m2 ) , (2) f (xm) = xf (m) for all m, m1 , m2 ∈ M and x ∈ X. A BCK-module homomorphism is said to be monomorphism (epimorphism) if it is one to one (onto). If it is both one to one and onto, then we say that it is an isomorphism. Definition 2.8.[12] Let (F, A) and (G, B) be two soft modules over M and N respectively, f : M → N , g : A → B be two functions. Then we say that (f, g) is a soft BCK-homomorphism if the following conditions are satisfied: (1) f is a homomorphism from M onto N , 3 747

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(2) g is a mapping from A onto B, and (3) f (F (x)) = G(g(x)) for all x ∈ A.

3

Neutrosophic fuzzy soft BCK-submodules

Definition 3.1. A neutrosophic fuzzy soft set (F, A) over a BCK-module M is said to be a neutrosophic fuzzy soft BCK-submodule over M if for all m, m1 , m2 ∈ M , x ∈ X and ε ∈ A the following axioms hold : (NFSS1) TF (ε) (m1 + m2 ) ≥ min{TF (ε) (m1 ), TF (ε) (m2 )}, IF (ε) (m1 + m2 ) ≥ min{IF (ε) (m1 ), IF (ε) (m2 )}, FF (ε) (m1 + m2 ) ≤ max{FF (ε) (m1 ), FF (ε) (m2 )}, (NFSS2) TF (ε) (−m) = TF (ε) (m), IF (ε) (−m) = IF (ε) (m), FF (ε) (−m) = FF (ε) (m), (NFSS3) TF (ε) (xm) ≥ TF (ε) (m), IF (ε) (xm) ≥ IF (ε) (m), FF (ε) (xm) ≤ FF (ε) (m). Example 3.2. Let X = {0, a, b, c, d} be a set along with a binary operation ∗ defined in Table 1, then (X, ∗, 0) forms a commutative BCK-algebra which is not bounded (see [16]). Let M = {0, a, b, c} be a subset of X along with an operation + defined by Table 2. Then (M, +) forms a commutative group. Table 3 explains the action of X on M under the operation xm = x ∧ m for all x ∈ X and m ∈ M . Consequently, M forms an X-module (see [11]).

∗

0

a

b

c

d

0

0

0

0

0

0

+

0

a

b

a

a

0

a

0

a

0

0

a

b

b

b

0

0

b

a

a

c

c

b

a

0

d

b

d

d

d

d

d

0

c

Table 1

∧

0

a

b

c

c

0

0

0

0

0

b

c

a

0

a

0

a

0

c

b

b

0

0

b

b

b

c

0

a

c

0

a

b

c

c

b

a

0

d

0

0

0

0

Table 2

Table 3

Let A = {0, a}. Define a neutrosophic fuzzy soft set (F, A) over M as shown in Table 4

Consequently, a routine exercise of calculations show that (F, A) forms a neutrosophic fuzzy soft BCK-submodule over M. 4 748

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(F, A)

0

a

b

c

TF (0)

0.9

0.7

0.8

0.7

IF (0)

0.8

0.5

0.6

0.5

FF (0)

0.1

0.1

0.1

0.1

TF (a)

0.5

0.2

0.3

0.2

IF (a)

0.3

0.1

0.3

0.1

FF (a)

0.1

0.5

0.4

0.5

Table 4

For the sake of simplicity, we shall use the symbols N F S(M ) and N F SS(M ) for the set of all neutrosophic fuzzy soft sets over M and the set of all neutrosophic fuzzy soft BCK-submodules over M , respectively. Theorem 3.3. A neutrosophic fuzzy soft set (F, A) ∈ N F SS(M ) if and only if (i) TF (ε) (xm) ≥ TF (ε) (m),

IF (ε) (xm) ≥ IF (ε) (m),

FF (ε) (xm) ≤ FF (ε) (m),

(ii) TF (ε) (m1 − m2 ) ≥ min{TF (ε) (m1 ), TF (ε) (m2 )}, IF (ε) (m1 − m2 ) ≥ min{IF (ε) (m1 ), IF (ε) (m2 )}, FF (ε) (m1 − m2 ) ≤ max{FF (ε) (m1 ), FF (ε) (m2 )}. for all m, m1 , m2 ∈ M , x ∈ X and ε ∈ A. Proof. Let (F, A) be a neutrosophic fuzzy soft BCK-submodule over M then by the definition(3.1) condition (i) is hold. (ii) TF (ε) (m1 − m2 ) = TF (ε) (m1 + (−m2 )) ≥ min{TF (ε) (m1 ), TF (ε) (−m2 )} = min{TF (ε) (m1 ), TF (ε) (m2 )}, IF (ε) (m1 − m2 ) = IF (ε) (m1 + (−m2 )) ≥ min{IF (ε) (m1 ), IF (ε) (−m2 )} = min{IF (ε) (m1 ), IF (ε) (m2 )}, FF (ε) (m1 − m2 ) = FF (ε) (m1 + (−m2 )) ≤ max{FF (ε) (m1 ), FF (ε) (−m2 )} = max{FF (ε) (m1 ), FF (ε) (m2 )}. Conversely suppose (F, A) satisfies the conditions (i),(ii). Then we have by (i) TF (ε) (−m) = TF (ε) ((−1)m) ≥ TF (ε) (m), and TF (ε) (m) = TF (ε) ((−1)(−1)m) ≥ TF (ε) (−m). Thus, TF (ε) (m) = TF (ε) (−m). Similarly for IF (ε) (−m) = IF (ε) (m) and FF (ε) (−m) = FF (ε) (m). TF (ε) (m1 + m2 ) = TF (ε) (m1 − (−m2 )) ≥ min{TF (ε) (m1 ), TF (ε) (−m2 )} = min{TF (ε) (m1 ), TF (ε) (m2 )}, IF (ε) (m1 + m2 ) = IF (ε) (m1 − (−m2 )) ≥ min{IF (ε) (m1 ), IF (ε) (−m2 )} = min{IF (ε) (m1 ), IF (ε) (m2 )}, FF (ε) (m1 + m2 ) = FF (ε) (m1 − (−m2 )) ≤ max{FF (ε) (m1 ), FF (ε) (−m2 ){= max{FF (ε) (m1 ), FF (ε) (m2 )}. 5 749

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Hence (F, A) is a neutrosophic fuzzy soft BCK-submodule over M. Theorem 3.4. A neutrosophic fuzzy soft set (F, A) ∈ N F SS(M ) if and only if for all m, m1 , m2 ∈ M , x, y ∈ X and ε ∈ A the following statements hold: (i) TF (ε) (0) ≥ TF (ε) (m),

IF (ε) (0) ≥ IF (ε) (m),

FF (ε) (0) ≤ FF (ε) (m),

(ii) TF (ε) (xm1 − ym2 ) ≥ min{TF (ε) (m1 ), TF (ε) (m2 )}, IF (ε) (xm1 − ym2 ) ≥ min{IF (ε) (m1 ), IF (ε) (m2 )}, FF (ε) (xm1 − ym2 ) ≤ max{FF (ε) (m1 ), FF (ε) (m2 )}. Proof. Let (F, A) ∈ N F SS(M ) then by theorem (3.3) and since 0m = 0 for all m ∈ M,we have

(i) TF (ε) (0) = TF (ε) (0m) ≥ TF (ε) (m), IF (ε) (0) = IF (ε) (0m) ≥ IF (ε) (m), and FF (ε) (0) = FF (ε) (0m) ≤ FF (ε) (m). (ii) TF (ε) (xm1 − ym2 ) ≥ min{TF (ε) (xm1 ), TF (ε) (ym2 )} ≥ min{TF (ε) (m1 ), TF (ε) (m2 )}. Similarly for IF (ε) (xm1 − ym2 ) ≥ min{IF (ε) (m1 ), IF (ε) (m2 )}, and FF (ε) (xm1 − ym2 ) ≤ max{FF (ε) (m1 ), FF (ε) (m2 )}. Conversely suppose (F, A) satisfies (i),(ii), then we have TF (ε) (0) ≥ TF (ε) (m), IF (ε) (0) ≥ IF (ε) (m)and FF (ε) (0) ≤ FF (ε) (m). Then TF (ε) (xm) = TF (ε) (x(m − 0)) ≥ min{TF (ε) (m), TF (ε) (0)} = TF (ε) (m). Similarly for IF (ε) (xm) ≥ IF (ε) (m) and FF (ε) (xm) ≤ FF (ε) (m). Also, TF (ε) (m1 − m2 ) = TF (ε) (1m1 − 1m2 ) ≥ min{TF (ε) (m1 ), TF (ε) (m2 )}. Similarly for IF (ε) (m1 − m2 ) ≥ min{IF (ε) (m1 ), IF (ε) (m2 )} and FF (ε) (m1 − m2 ) ≤ max{FF (ε) (m1 ), FF (ε) (m2 )}. Hence (F, A) is a neutrosophic fuzzy soft BCK-submodule over M.

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Definition 3.5. Let (F, A) be a neutrosophic fuzzy soft set over a BCK-module M and α ∈ [0, ⊥]  such that ⊥= 1 − sup FF (ε) (m) : m ∈ M, ε ∈ A .Then T˜α [(F, A)] = (G, ATα ) is called a neutrosophic fuzzy soft α-translation of (F, A) if it satisfies:  
T
T
T G (ε) = TF (ε) α (m) , IF (ε) α (m) , FF (ε) α (m) , for all ε ∈ A, m ∈ M where: TF (ε)


T

IF (ε)


T

FF (ε)


T

α α α

(m) = TF (ε) (m) + α, (m) = IF (ε) (m) , (m) = FF (ε) (m) − α.

Theorem 3.6. A neutrosophic fuzzy soft set (F, A) is said to be a neutrosophic fuzzy soft BCKsubmodule over M if and only if the α-translation neutrosophic fuzzy soft set T˜α [(F, A)] is a neutrosophic fuzzy soft BCK-submodule over M for all α ∈ [0, ⊥]. Proof. Let (F, A) be a neutrosophic fuzzy soft BCK-submodule over M and α ∈ [0, ⊥] , then by Theorem (3.3) TF (ε)


T

FF (ε)


T

(xm) = TF (ε) (xm) + α ≥ TF (ε) (m) + α = TF (ε)

α


T

(xm) = FF (ε) (xm) − α ≤ FF (ε) (m) − α = FF (ε)

α

α


T α

(m) , (m) ,

for all m ∈ M, x ∈ X. Also, for all m1 , m2 ∈ M we have TF (ε)


T α

(m1 − m2 ) = TF (ε) (m1 − m2 ) + α  ≥ min TF (ε) (m1 ) , TF (ε) (m2 ) + α  = min TF (ε) (m1 ) + α, TF (ε) (m2 ) + α n o
T
T = min TF (ε) α (m1 ) , TF (ε) α (m2 ) ,

and FF (ε)


T α

(m1 − m2 ) = FF (ε) (m1 − m2 ) − α  ≤ max FF (ε) (m1 ) , FF (ε) (m2 ) − α  = max FF (ε) (m1 ) − α, FF (ε) (m2 ) − α o n
T
T = max FF (ε) α (m1 ) , FF (ε) α (m2 ) .

Hence T˜α [(F, A)] is a neutrosophic fuzzy soft BCK-submodule over M. Conversely, assume that T˜α [(F, A)] is a neutrosophic fuzzy soft BCK-submodule over M for some α ∈ [0, ⊥] . Then for all m ∈ M, x ∈ X TF (ε) (xm) + α = TF (ε)


T

(xm) ≥ TF (ε) α


T α

(m) = TF (ε) (m) + α

=⇒ TF (ε) (xm) ≥ TF (ε) (m) . 7 751

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Also, FF (ε) (xm) − α = FF (ε)


T α

(xm) ≤ FF (ε)


T α

(m) = FF (ε) (m) − α

=⇒ FF (ε) (xm) ≤ FF (ε) (m) . Now let m1 , m2 ∈ M , then
T TF (ε) (m1 − m2 ) + α = TF (ε) α (m1 − m2 ) o n
T
T ≥ min TF (ε) α (m1 ) , TF (ε) α (m2 )  = min TF (ε) (m1 ) + α, TF (ε) (m2 ) + α  = min TF (ε) (m1 ) , TF (ε) (m2 ) + α  =⇒ TF (ε) (m1 − m2 ) ≥ min TF (ε) (m1 ) , TF (ε) (m2 ) , and
T FF (ε) (m1 − m2 ) − α = FF (ε) α (m1 − m2 ) n o
T
T ≤ max FF (ε) α (m1 ) , FF (ε) α (m2 )  = max FF (ε) (m1 ) − α, FF (ε) (m2 ) − α  = max FF (ε) (m1 ) , FF (ε) (m2 ) − α  =⇒ FF (ε) (m1 − m2 ) ≤ max FF (ε) (m1 ) , FF (ε) (m2 ) . Hence by Theorem (3.3), (F, A) is a neutrosophic fuzzy soft BCK-submodule over M. Definition 3.7. Let (F, A) and (G, B) be two neutrosophic fuzzy soft sets over a BCK-module M. If A ⊂ B and TF (ε) (m) ≤ TG(ε) (m), IF (ε) (m) ≤ IG(ε) (m), FF (ε) (m) ≥ FG(ε) (m), ∀ε ∈ A and m ∈ M . Then we say that (G, B) is a neutrosophic fuzzy soft extinsion of (F, A). Definition 3.8. Let (F, A) and (G, B) be two neutrosophic fuzzy soft sets over a BCK-module M. Then (G, B) is a neutrosophic fuzzy soft s-extinsion of (F, A) if the following assertions hold: (i) (G, B) is a neutrosophic fuzzy soft extinsion of (F, A). (ii) If (F, A) is a neutrosophic fuzzy soft BCK-submodule over M, then so (G, B) . Theorem 3.9. Let (F, A) be a neutrosophic fuzzy soft BCK-submodule over M and α ∈ [0, ⊥] . Then the neutrosophic fuzzy soft α-translation T˜α [(F, A)] is a neutrosophic fuzzy soft s-extinsion of (F, A).
T Proof. Since T˜α [(F, A)] is an α-translation, we know that TF (ε) α (m) ≥ TF (ε) (m),
T
T IF (ε) α (m) = IF (ε) (m) and FF (ε) α (m) ≤ FF (ε) (m) for all m ∈ M, ε ∈ A. Hence T˜α [(F, A)] is a neutrosophic fuzzy soft extinsion of (F, A) . According to Theorem (3.6), T˜α [(F, A)] is a neutrosophic fuzzy soft s-extinsion of (F, A). 8 752

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The converse of Theorem (3.9) is not true in general as seen in the following example: Example 3.10. Let X = {0, a, b, c} along with a binary operation ∗ defined in Table 5, then (X, ∗, 0) forms a bounded implicative BCK-algebra (see [16]). Let M = {0, a} be a subset of X with a binary operation + defined by x + y = (x ∗ y) ∨ (y ∗ x). Then M is a commutative group as shown in table 6. Define scalar multiplication (X, M ) → M by xm = x ∧ m for all x ∈ X and m ∈ M that is given in Table 7. Consequently, M forms an X-module (see [11]).

∗

0

a

b

c

∧

0

a

0

0

0

0

0

0

0

0

a

a

0

a

0

+

0

a

a

0

a

b

b

b

0

0

0

0

a

b

0

0

c

c

b

a

0

a

a

0

c

0

a

Table 5

Table 6

Table 7

Let A = M. Define a neutrosophic fuzzy soft set (F, A) over M as shown in Table 8.

(F, A)

0

a

TF (0)

0.9

0.5

IF (0)

0.8

0.6

FF (0)

0.1

0.3

TF (a)

0.3

0.3

IF (a)

0.2

0.2

FF (a)

0.3

0.5

Table 8

Then (F, A) is a neutrosophic fuzzy soft BCK-submodule over M. Let (G, B) be a neutrosophic fuzzy soft set over M given by Table 9.

Then (G, B) is also a neutrosophic fuzzy soft BCK-submodule over M. Since TF (ε) (m) ≥ TG(ε) (m) , IF (ε) (m) ≥ IG(ε) (m) and FF (ε) (m) ≤ FG(ε) (m) for all m ∈ M and ε ∈ A ⊂ B, hence (F, A) is a neutrosophic fuzzy soft s-extension of (G, B), but since IF (0) (0) = 0.8 6= IG(0) (0) = 0.7 then (F, A) is not a neutrosophic fuzzy soft α-translation of (G, B) for all α ∈ [0, ⊥] . 9 753

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(G, B)

0

a

TG(0)

0.5

0.3

IG(0)

0.7

0.6

FG(0)

0.1

0.4

TG(a)

0.2

0.2

IG(a)

0.1

0.1

FG(a)

0.4

0.5

Table 9

Definition 3.11. Let (F, A) be a neutrosophic fuzzy soft set over a BCK-module M and υ ∈ [0, 1] . ˜ υ [(F, A)] = (G, mυ (A)) is defined A neutrosophic fuzzy soft υ-multiplication of (F, A) denoted by M as:



G (ε) = mυ TF (ε) (m) , mυ IF (ε) (m) , mυ FF (ε) (m) , where
mυ TF (ε) (m) = TF (ε) (m) .υ,
mυ IF (ε) (m) = IF (ε) (m) ,
mυ FF (ε) (m) = FF (ε) (m) .υ, for all ε ∈ A and m ∈ M . Theorem.3.12.

If (F, A) ∈ N F SS(M ), then the neutrosophic fuzzy soft υ-multiplication

˜ υ [(F, A)] ∈ N F SS(M ) for all υ ∈ [0, 1] . M Proof.

Assume that (F, A) is a neutrosophic fuzzy soft BCK-submodule over M and let

m, m1 , m2 ∈ M , x ∈ X and ε ∈ A. Then

mυ TF (ε) (xm) = TF (ε) (xm) .υ ≥ TF (ε) (m) .υ = mυ TF (ε) (m) ,

mυ IF (ε) (xm) = IF (ε) (xm) ≥ IF (ε) (m) = mυ IF (ε) (m) ,

mυ FF (ε) (xm) = FF (ε) (xm) .υ ≤ FF (ε) (m) .υ = mυ FF (ε) (m) . Moreover,
mυ TF (ε) (m1 − m2 ) = TF (ε) (m1 − m2 ) .υ  ≥ min TF (ε) (m1 ) , TF (ε) (m2 ) .υ  = min TF (ε) (m1 ) .υ, TF (ε) (m2 ) .υ 

= min mυ TF (ε) (m1 ) , mυ TF (ε) (m2 ) , 10 754

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mυ IF (ε) (m1 − m2 ) = IF (ε) (m1 − m2 )  ≥ min IF (ε) (m1 ) , IF (ε) (m2 ) 

= min mυ IF (ε) (m1 ) , mυ IF (ε) (m2 ) ,
mυ FF (ε) (m1 − m2 ) = FF (ε) (m1 − m2 ) .υ  ≤ max FF (ε) (m1 ) , FF (ε) (m2 ) .υ  = max FF (ε) (m1 ) .υ, FF (ε) (m2 ) .υ 

= max mυ FF (ε) (m1 ) , mυ FF (ε) (m2 ) . ˜ υ [(F, A)] is a neutrosophic fuzzy soft BCK-submodule over M. Therefore by Theorem (3.3), M The converse of Theorem (3.12) is not true in general as seen in the following example: Example 3.13. Consider a BCK-algebra X = {0, a, b, c} and X-module M = {0, a} that are defined in Example 3.10. Table 10 defines a neutrosophic fuzzy soft set (F, A) over M

(F, A)

0

a

TF (0)

0.3

0.4

IF (0)

0.7

0.5

FF (0)

0.1

0.5

TF (a)

0.1

0.1

IF (a)

0.1

0.1

FF (a)

0.5

0.6

Table 10

If we take υ = 0, then the υ-multiplication is a neutrosophic fuzzy soft BCK-submodule over M since

m0 TF (ε) (xm) = 0 = m0 TF (ε) (m) ,

m0 IF (ε) (xm) ≥ m0 IF (ε) (m) ,

m0 FF (ε) (xm) = 0 = m0 FF (ε) (m) , and


m0 TF (ε) (m1 − m2 ) = 0 = min m0 TF (ε) (m1 ) , m0 TF (ε) (m2 ) ,


m0 IF (ε) (m1 − m2 ) ≥ min m0 IF (ε) (m1 ) , m0 IF (ε) (m2 ) ,


m0 FF (ε) (m1 − m2 ) = 0 = min m0 FF (ε) (m1 ) , m0 FF (ε) (m2 ) , 11 755

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for all m, m1 , m2 ∈ M and x ∈ X. But if we take m1 = 0, m2 = a and ε = 0 then  TF (0) (0 + a) = TF (0) (a) = 0.4  min TF (0) (0) , TF (0) (a) = 0.3. Hence (F, A) is not a neutrosophic fuzzy soft BCK-submodule over M. Theorem.3.14. A neutrosophic fuzzy soft set (F, A) is said to be a neutrosophic fuzzy soft ˜ υ [(F, A)] is a BCK-submodule over M if and only if the υ-multiplication neutrosophic fuzzy set M neutrosophic fuzzy soft BCK-submodule over M for all υ ∈ (0, 1] . Proof. Let (F, A) be a neutrosophic fuzzy soft BCK-submodule over M then by Theorem (3.12) ˜ υ [(F, A)] is a neutrosophic fuzzy soft BCK-submodule over M for all υ ∈ (0, 1] . M ˜ υ [(F, A)] is a neutrosophic fuzzy soft BCK-submodule over M Now let υ ∈ (0, 1] be such that M and let m, m1 , m2 ∈ M , x ∈ X and ε ∈ A. Then

TF (ε) (xm) .υ = mυ TF (ε) (xm) ≥ mυ TF (ε) (m) = TF (ε) (m) .υ,

IF (ε) (xm) = mυ IF (ε) (xm) ≥ mυ IF (ε) (m) = IF (ε) (m) ,

FF (ε) (xm) .υ = mυ FF (ε) (xm) ≤ mυ FF (ε) (m) = FF (ε) (m) .υ, and since υ 6= 0, then TF (ε) (xm) ≥ TF (ε) (m) and FF (ε) (xm) ≤ FF (ε) (m) . Now
TF (ε) (m1 − m2 ) .υ = mυ TF (ε) (m1 − m2 ) 

≥ min mυ TF (ε) (m1 ) , mυ TF (ε) (m2 )  = min TF (ε) (m1 ) .υ, TF (ε) (m2 ) .υ  = min TF (ε) (m1 ) , TF (ε) (m2 ) .υ, which means that  TF (ε) (m1 − m2 ) ≥ min TF (ε) (m1 ) , TF (ε) (m2 ) . Similarly,  FF (ε) (m1 − m2 ) ≤ max FF (ε) (m1 ) , FF (ε) (m2 ) . Hence (F, A) is a neutrosophic fuzzy soft BCK-submodule over M.

4

Ismorphism

Theorem

Of

Neutrosophic

Fuzzy

Soft

BCK-

submodules Definition 4.1. Let M and N be two BCK-modules over a BCK-algebra X. Let f : M −→ N be a BCK-submodule homomorphism and let (F, A) , (G, B) be two neutrosophic fuzzy soft BCKsubmodule over M and N respectively. Then the image of (F, A) is a neutrosophic fuzzy soft set over N defined as follows for all x ∈ M, y ∈ N and ε ∈ A.
f (F (ε)) (x) = Tf (F ) (y), If (F ) (y), Ff (F ) (y) = (f (TF ) (y) , f (IF ) (y) , f (FF ) (y)) , 12 756

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where 

sup TF (x) 0  sup IF (x) f (IF ) (y) = 0  inf FF (x) f (FF ) (y) = 0 f (TF ) (y) =

if x ∈ f −1 (y) , otherwise if x ∈ f −1 (y) , otherwise if x ∈ f −1 (y) , otherwise

and the preimage of (G, B) is a neutrosophic fuzzy soft set over M defined as
f −1 (G (δ)) (y) = Tf −1 (G) (x), If −1 (G) (x), Ff −1 (G) (x) = (TG (f (x)) , IG (f (x)) , FG (f (x))) , where δ ∈ B. Theorem 4.2. Let (X, ∗, 0) be a BCK-algebra, M and N are modules of X. A mapping f : M −→ N is a BCK-submodule homomorphism and (F, A) ∈ N F SS(N ), then the inverse image
f −1 (F ) , A ∈ N F SS(M ). Proof. Since (F, A) is a neutrosophic fuzzy soft BCK-submodule over N. Let m ∈ M, ε ∈ A then by Theorem (3.4) Tf −1 (F ) (0) = TF (ε) (f (0)) = TF (ε) (0) ≥ TF (ε) (f (m)) = Tf −1 (F ) (m), If −1 (F ) (0) = IF (ε) (f (0)) = IF (ε) (0) ≥ IF (ε) (f (m)) = If −1 (F ) (m), Ff −1 (F ) (0) = FF (ε) (f (0)) = FF (ε) (0) ≤ FF (ε) (f (m)) = Ff −1 (F ) (m). Now let m1 , m2 ∈ M, x, y ∈ X, and ε ∈ A, then Tf −1 (F ) (xm1 − ym2 ) = TF (ε) (f (xm − ym2 )) = TF (ε) (xf (m1 ) − yf (m2 ))  ≥ min TF (ε) (f (m1 )), TF (ε) (f (m2 ))  = min Tf −1 (F ) (m1 ), Tf −1 (F ) (m2 ) . Similarly for  If −1 (F ) (xm1 − ym2 ) ≥ min If −1 (F ) (m1 ), If −1 (F ) (m2 ) , and  Ff −1 (F ) (xm1 − ym2 ) ≤ max Ff −1 (F ) (m1 ), Ff −1 (F ) (m2 ) .
Hence f −1 (F ) , A is a neutrosophic fuzzy soft BCK-submodule over M. Theorem.4.3. Let (X, ∗, 0) be a BCK-algebra, M and N are modules of X. A mapping f : M −→ N is a BCK-submodule epimorphism. If (F, A) is a neutrosophic fuzzy soft set over N such
that f −1 (F ) , A ∈ N F SS(M ), then (F, A) ∈ N F SS(N ).

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Proof. Assume that f −1 (F ) , A is a neutrosophic fuzzy soft BCK-submodule over M. Let n ∈ N then there exist m ∈ M such that f (m) = n. Then for all ε ∈ A TF (ε) (n) = TF (ε) (f (m)) = Tf −1 (F ) (m) ≤ Tf −1 (F ) (0) = TF (ε) (f (0)) = TF (ε) (0) , IF (ε) (n) = IF (ε) (f (m)) = If −1 (F ) (m) ≤ If −1 (F ) (0) = IF (ε) (f (0)) = IF (ε) (0) , FF (ε) (n) = FF (ε) (f (m)) = Ff −1 (F ) (m) ≥ Ff −1 (F ) (0) = FF (ε) (f (0)) = FF (ε) (0) . Let m, m ` ∈ M, n, n ` ∈ N such that f (m) = n and f (m) ` =n ` and x, y ∈ X then TF (ε) (xn − y` n) = TF (ε) (xf (m) − yf (m)) ` = TF (ε) (f (xm − y m)) ` = Tf −1 (F ) (xm − y m) `  ≥ min Tf −1 (F ) (m), Tf −1 (F ) (m) `  = min TF (ε) (f (m)) , TF (ε) (f (m)) `  = min TF (ε) (n) , TF (ε) (` n) . Similarly for  IF (ε) (xn − y` n) ≥ min IF (ε) (n) , IF (ε) (` n) , and  FF (ε) (xn − y` n) ≤ max FF (ε) (n) , FF (ε) (` n) . Hence according to Theorem (3.4), (F, A) is a neutrosophic fuzzy soft BCK-submodule over N. Theorem.4.4. Let (X, ∗, 0) be a BCK-algebra, M and N are modules of X. A mapping f : M −→ N is a BCK-submodule epimorphism and let (F, A) be a neutrosophic fuzzy soft BCK-submodule over M. Then the homomorphic image (f (F ) , A) is a neutrosophic fuzzy soft BCK-submodule over N. Proof. Assume that (F, A) is a neutrosophic fuzzy soft BCK-submodule over M. Let n ∈ N then there exist m ∈ M such that f (m) = n. Then Tf (F ) (n) = f (TF ) (n) = sup TF (m) ≤ sup T (0) = f (TF ) (0) = Tf (F ) (0), If (F ) (n) = f (IF ) (n) = sup IF (m) ≤ sup I(0) = f (IF ) (0) = If (F ) (0), Ff (F ) (n) = f (FF ) (n) = inf FF (m) ≥ inf F (0) = f (FF ) (0) = Ff (F ) (0).

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Let m1 , m2 ∈ M, n1 , n2 ∈ N such that f (m1 ) = n1 and f (m2 ) = n2 and x, y ∈ X then Tf (F ) (xn1 − yn2 ) = f (TF ) (xn1 − yn2 ) = sup TF (xm1 − ym2 ) ≥ sup{min {TF (m1 ), TF (m2 )}} = min {sup TF (m1 ), sup TF (m2 )} = min {f (TF ) (n1 ) , f (TF ) (n2 )} = min{Tf (F ) (n1 ) , Tf (F ) (n2 )}. Similarly for If (F ) (xn1 − yn2 ) ≥ min{If (F ) (n1 ) , If (F ) (n2 )}, and Ff (F ) (xn1 − yn2 ) ≤ max{Ff (F ) (n1 ) , Ff (F ) (n2 )}. Hence by Theorem (3.4), (f (F ), A) is a neutrosophic fuzzy soft BCK-submodule over N. Corollary 4.5. Let f : M −→ N be a homomorphism of BCK-submodules and (F, A) is a neutrosophic fuzzy soft set over N. If (F, A) is a neutrosophic fuzzy soft BCK-submodule, then so is
f −1 (F ) , ATα for any α-translation T˜α [(F, A)] of (F, A) with α ∈ [0, ⊥]. Proof. Directly by Theorem(3.6) and Theorem(4.2). Joining Theorems (3.6), (4.3) and (4.4) we have the following corollaries: Corollary 4.6. Let f : M −→ N be an epimorphism of BCK-submodules and (F, A) is a neutrosophic fuzzy soft set over N. If the inverse image of a neutrosophic fuzzy soft α-translation of (F, A) is a neutrosophic fuzzy soft BCK-submodule for some α ∈ [0, ⊥] , then so is (F, A) . Corollary 4.7. Let f : M −→ N be an epimorphism of BCK-submodules and (F, A) is a neutrosophic fuzzy soft BCK-submodule over M, then the homomorphic image of a neutrosophic fuzzy soft α-translation of (F, A) is a neutrosophic fuzzy soft BCK-submodule over N for any α ∈ [0, ⊥] . Using Theorems (3,14), (4.2), (4.3) and (4.4), we deduce the following results: Corollary 4.8. Let f : M −→ N be a homomorphism of BCK-submodules and (F, A) is a neutrosophic fuzzy soft BCK-submodule over N, then the inverse image of a neutrosophic fuzzy soft υ-multiplication of (F, A) is a neutrosophic fuzzy soft BCK-submodule over M for any υ-multiplication of (F, A) with υ ∈ [0, 1] . Corollary 4.9. Let f : M −→ N be an epimorphism of BCK-submodules. If the inverse image of a neutrosophic fuzzy soft υ-multiplication of (F, A) is a neutrosophic fuzzy soft BCK-submodule over M for some υ ∈ (0, 1] , then (F, A) is a neutrosophic fuzzy soft BCK-submodule over N. Corollary 4.10. Let f : M −→ N be an epimorphism of BCK-submodules and (F, A) is a neutrosophic fuzzy soft BCK-submodule over M, then the homomorphic image of a neutrosophic 15 759

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fuzzy soft υ-multiplication of (F, A) is a neutrosophic fuzzy soft BCK-submodule over N for any υ ∈ (0, 1] .

5

Conclusion

Translations, multiplications and extensions are very interested mathematical tools. They are types of operations that researchers like to apply with fuzzy set theory. In this paper, the concept of neutrosophic fuzzy soft translations and neutrosophic fuzzy soft extensions of neutrosophic fuzzy soft BCK-submodules were introduced and the relation between them were discussed. Also, the notion of neutrosophic fuzzy soft multiplications of neutrosophic fuzzy soft BCK-submodules was defined. Finally, some results were investigated.

6

Compliance with Ethical Standards

Conflict of Interest: The authors declare that there is no conflict of interests regarding the publication of this paper. Ethical Approval: This artical does not contain any studies with human participants or animals performed by any of the authors.

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[8] Imai and K. Iseki, On axiom systems of propositional calculi, XIV, Proc. Japan Academy, 42 (1996), 19-22. [9] K. Iseki, An algebraic related with a propositional calculus, Proc. Japan Academy, 42 (1996), 26-29. [10] W. B. V. Kandasamy, Florentin Smarandache, Some neutrosophic algebraic structures and neutrosophic N -algebraic structures, Hexis, Phoenix, Arizona, 2006. [11] A. Kashif and M. Aslam, “Homology theory of BCK-modules”, Southeast Asian Bulletin ofMathematics, vol. 38,no. 1, pp. 61–72, 2014. [12] A. Kashif, H. Bashir and Z. Zahid, On soft BCK- modules, Punjab University, Journal of Mathematics (ISSN 1016-2526), Vol. 50(1)(2018) pp. 67-78. [13] P. K. Maji, Neutrosophic soft set, Annals of Fuzzy Mathematics and Informatics, Volume 5, No. 1, (January 2013), pp. 157-168. [14] P.K. Maji, R. Biswas and R. Roy, Soft set theory, Comput. Math. Appl. 45(2003) 555-562. [15] 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. [16] J. Meng, Jie and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co., Seoul, 1994. [17] D. Molodtsov, Soft set theory first results, Comput. Math. Appl. 37(1999) 19-31. [18] Z. Perveen, M. Aslam and A. B. Thaheem, On BCK- modules, Southeast Asian Bulletin of Mathematics, 30 (2006), 317-329.Bulletin of Mathematics, 30 (2006), 317-329. [19] F. Smarandache, Neutrosophy, Neutrosophic Probability, Set, and Logic, Amer. Res. Press, Rehoboth, USA, 105 p., 1998. [20] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy sets. Int. J. Pure Appl. Math. 24, 287–297 (2005). [21] V. Torra, Y.Narukawa,”On hesitant fuzzy sets and decision”. The 18th IEEE international Conference on Fuzzy Systems, Jeju Island, Korea, (2009), pp.1378-1382. [22] Y. Yin and J. Zhan, The characterizations of hemirings in terms of fuzzy soft h-ideals, Neural Computing and Applications 21(2012) S43-S57. [23] L. A. Zadeh, Fuzzy sets, Inform. and Conntrol 8 (1965) 338-353.

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[24] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning”. Information Sciences 8(3), (1975), pp.199-249.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 4, 2020

Hermite-Hadamard Type Inequalities Involving Conformable Fractional Integrals, Yousaf Khurshid, Muhammad Adil Khan, and Yu-Ming Chu,………………………………………585 Neutrosophic BCC-Ideals in BCC-Algebras, Sun Shin Ahn,……………………………...…605 Global Dynamics and Bifurcations of Two Second Order Difference Equations in Mathematical Biology, M. R. S. Kulenović and Sarah Van Beaver,…………………………………….….615 Bounds for the Real Parts and Arguments of Normalized Analytic Functions Defined by the Srivastava-Attiya Operator, Young Jae Sim, Oh Sang Kwon, Nak Eun Cho, and H. M. Srivastava,……………………………………………………………………………………628 Sharp Bounds for the Complete Elliptic Integrals of the First and Second Kinds, Xiao-Hui Zhang, Yu-Ming Chu, and Wen Zhang,……………………………………………………..646 Symmetric Identities for the Second Kind q-Bernoulli Polynomials, C. S. Ryoo,…………..654 On Some Finite Difference Methods on the Shishkin Mesh for the Singularly Perturbed Problem, Quan Zheng, Ying Liu, and Jie He,…………………………………………………………..660 Fekete Szegő Problem Related to Simple Logistic Activation Function, C. Ramachandran and D. Kavitha,………………………………………………………………………………………670 On the Second Kind Twisted q-Euler Numbers and Polynomials of Higher Order, Cheon Seoung Ryoo,…………………………………………………………………………………………679 Hermite-Hadamard Inequality and Green's Function with Applications, Ying-Qing Song, YuMing Chu, Muhammad Adil Khan, and Arshad Iqbal,………………………………………685 On the Applications of the Girard-Waring Identities, Tian-Xiao He and Peter J.-S. Shiue,…698 Local Fractional Taylor Formula, George A. Anastassiou,…………………………………..709 On Voronovskaja Type Estimates of Bernstein-Stancu Operators, Rongrong Xia and Dansheng Yu,…………………………………………………………………………………………….714 Double-sided Inequalities of Ostrowski’s Type and Some Applications, Waseem Ghazi Alshanti and Gradimir V. Milovanović,…………………………………………………………………724

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 4, 2020 (continued) Iterates of Cheney-Sharma Type Operators on a Triangle with Curved Side, Teodora Cătinaș and Diana Otrocol,………………………………………………………………………………737 Contemporary Concepts of Neutrosophic Fuzzy Soft BCK-submodules, R. S. Alghamdi and Noura Omair Alshehri,………………………………………………………………………745

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.5, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

ON HARMONIC MULTIVALENT FUNCTIONS DEFINED BY A NEW DERIVATIVE OPERATOR ˘ ADRIANA CATAS ¸ 1∗ , ROXANA S ¸ ENDRUT ¸ IU2

Abstract. In the present paper, we define and investigate a new class of multivalent harmonic functions in the open unit disc U = {z ∈ C : |z| < 1}, under certain conditions involving a new generalized differential operator. Coefficient inequalities, distortion bounds and a covering result are also obtained.

Keywords: differential operator, harmonic function, coefficient bounds. 2000 Mathematical Subject Classification: 30C45. 1. Introduction A continuous complex-valued function f = u + iv defined in a simply connected complex domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simple connected domain we can write f = h + g¯, where h and g are analytic in D. A necessary and sufficient condition for f to be univalent and sense preserving in D is that |h0 (z)| > |g 0 (z)|, z ∈ D. (See [4] for more details.) Denote by SH (p, n), (p, n ∈ N = {1, 2, . . .}) the class of functions f = h + g¯ that are harmonic multivalent and sense-preserving in the unit disc U for which f (0) = fz (0) − 1 = 0. Then for f = h + g¯ ∈ SH (p, n) we may express the analytic functions h and g as (1.1)

h(z) = z p +

∞ X

ak z k ,

∞ X

g(z) =

k=p+n

bk z k ,

|bp+n−1 | < 1.

k=p+n−1

Let S˜H (p, n, m), (p, n ∈ N, m ∈ N0 ∪{0}) denote the family of functions fm = h+¯ gm that are harmonic in D with the normalization ∞ ∞ X X (1.2) h(z) = z p − |ak |z k , gm (z) = (−1)m |bk |z k , |bp+n−1 | < 1. k=p+n

k=p+n−1

2. Coefficient bounds for the new classes ALH (p, m, δ, α, λ, l) and f H (p, m, δ, α, λ, l) AL We propose for the beginning a new generalized differential operator as follows. Definition 2.1. Let H(U ) denote the class of analytic functions in the open unit disc U = {z ∈ C : |z| < 1} and let A(p) be the subclass of the functions belonging 1

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A. C˘ ata¸s, R. S ¸ endrut¸iu

P∞ to H(U ) of the form h(z) = z p + k=p+n ak z k . For m ∈ N0 , λ ≥ 0, δ ∈ N0 , l ≥ 0 we m define the generalized differential operator Iλ,δ (p, l) on A(p) (2.1)

∞ X

m Iλ,δ (p, l)h(z) = (p + l)m z p +

[p + λ(k − p) + l]m C(δ, k)ak z k ,

k=p+n

 (2.2)

C(δ, k) =

k+δ−1 δ

 =

Γ(k + δ) . Γ(k)Γ(δ + 1)

Remark 2.2. When λ = 1, p = 1, l = 0, δ = 0 we get S˘al˘agean differential operator [10]; p = 1, m = 0 gives Ruscheweyh operator [9]; p = 1, l = 0, δ = 0 implies AlOboudi differential operator of order m (see [1]); λ = 1, p = 1, l = 0 operator (2.1) reduces to Al-Shaqsi and Darus differential operator [2] and when p = 1, l = 0 we reobtain the operator introduced by Darus and Ibrahim in [5]. Definition 2.3. Let f ∈ SH (p, n), p ∈ N. Using the operator (2.1) for f = h + g¯ given by (1.1) we define the differential operator of f as (2.3) (2.4)

m m m (p, l)g(z) Iλ,δ (p, l)f (z) = Iλ,δ (p, l)h(z) + (−1)m Iλ,δ ∞ X

m Iλ,δ (p, l)h(z) = (p + l)m z p +

[p + λ(k − p) + l]m C(δ, k)ak z k

k=p+n

(2.5)

m Iλ,δ (p, l)g(z)

=

∞ X

[p + λ(k − p) + l]m C(δ, k)bk z k .

k=p+n−1

Remark 2.4. When λ = 1, l = 0, δ = 0 the operator (2.3) reduces to the operator introduced earlier in [7] by Jahangiri et al. Definition 2.5. A function f ∈ SH (p, n) belongs to the class ALH (p, m, δ, α, λ, l) if ( m+1 ) Iλ,δ (p, l)f (z) 1 (2.6) Re ≥ α, 0 ≤ α < 1, m (p, l)f (z) p+l Iλ,δ m where Iλ,δ f is defined by (2.3), for m ∈ N0 . Finally, we define the subclass

(2.7)

f H (p, m, δ, α, λ, l) ≡ ALH (p, m, δ, α, λ, l) ∩ S˜H (p, n, m). AL

Remark 2.6. The class ALH (p, m, δ, α, λ, l) includes a variety of well-known subclasses of SH (p, n). For example, letting n = 1 we get ALH (1, 1, 0, α, 1, 0) ≡ HK(α) in [6], for n = 1, ALH (1, m − 1, 0, α, 1, 0) ≡ SH (t, u, α) in [11], ALH (p, n + p, 0, α, 1, 0) ≡ SHp (n, α) in [8] and n = 1, ALH (1, m, δ, α, 1, 0) ≡ MH (m, δ, α) in [3]. Theorem 2.7. Let f = h + g¯ be given by (1.1). If (2.8)

∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak |+ (p + l)m+1 (1 − α)

k=p+n

776

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.5, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

On harmonic multivalent functions defined by a new derivative operator ∞ X

+

k=p+n−1

3

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk | ≤ 1, (p + l)m+1 (1 − α)

with λn ≥ α(p + l), where dp,k (m, λ, l) = [p + λ(k − p) + l]m

(2.9)

then f ∈ ALH (p, m, δ, α, λ, l). 1 Proof. Using the fact that p+l Re w ≥ α if and only if |(p + l) − (p + l)α + w| ≥ |(p + l) + (p + l)α − w|, it is sufficient to show that m+1 m |(p + l)(1 − α)Iλ,δ (p, l)f (z) + Iλ,δ (p, l)f (z)|−

(2.10)

m+1 m −|(p + l)(1 + α)Iλ,δ (p, l)f (z) − Iλ,δ (p, l)f (z)| ≥ 0. m+1 m Substituting Iλ,δ (p, l)f (z) and Iλ,δ (p, l)f (z) in (2.10) yields by (2.8) m+1 m |(p + l)(1 − α)Iλ,δ (p, l)f (z) + Iλ,δ (p, l)f (z)|− m+1 m −|(p + l)(1 + α)Iλ,δ (p, l)f (z) − Iλ,δ (p, l)f (z)| >  ∞  X [(p + l)(1 − α) + λ(k − p)]d (m, λ, l)C(δ, k) p,k |ak |− > 2(p + l)m+1 (1 − α) 1 −  (p + l)m+1 (1 − α) k=p+n  ∞  X [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) − |bk | . m+1  (p + l) (1 − α) k=p+n−1

The last expression is nonnegative by (2.8) and therefore the proof is complete.



Remark 2.8. The harmonic function ∞ X (p + l)m+1 (1 − α) (2.11) f (z) = z p + xk z k + [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) k=p+n

∞ X

(p + l)m+1 (1 − α) yk z k , [(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) k=p+n−1 P∞ P∞ where k=p+n |xk | + k=p+n−1 |yk | = 1, 0 ≤ α < 1, m ∈ N0 , λn ≥ α(p + l), λ ≥ 0 and dp,k (m, λ, l) is given in (2.9), show that the coefficient bound expressed by (2.8) is sharp. +

f H (p, m, δ, α, λ, l) Theorem 2.9. Let fm = h + g¯m be given by (1.2). Then fm ∈ AL if and only if ∞ X [(p + l)(1 − α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |ak |+ (2.12) (p + l)m+1 (1 − α) k=p+n

+

∞ X k=p+n−1

[(p + l)(1 + α) + λ(k − p)]dp,k (m, λ, l)C(δ, k) |bk | ≤ 1, (p + l)m+1 (1 − α)

where λn ≥ α(p + l), 0 ≤ α < 1, m ∈ N0 , λ ≥ 0 and dp,k (m, λ, l) is given in (2.9).

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A. C˘ ata¸s, R. S ¸ endrut¸iu

f H (p, m, δ, α, λ, l) ⊂ ALH (p, m, δ, α, λ, l), we only need to prove the Proof. Since AL f H (p, m, δ, α, λ, l). ”only if” part of the theorem. For this part we consider that fm ∈ AL Then ( m+1 ) Iλ,δ (p, l)f (z) Re = m (p, l)f (z) − α(p + l) Iλ,δ

Re

 ∞ X  m+1 p  (p + l) (1 − α)z − [(p + l)(1 − α) + λ(k − p)]ξ(m, λ, l; δ, k)ak z k    k=p+n

∞ ∞ X X   k 2m  mzp − (p + l) ξ(m, λ, l; δ, k)a z + (−1) ξ(m, λ, l; δ, k)bk z k  k  k=p+n

2m

(−1) − (p +

l)m z p

∞ X

k=p+n−1

[(p + l)(1 + α) + λ(k − p)]ξ(m, λ, l; δ, k)bk

k=p+n−1 ∞ X

−

k

ξ(m, λ, l; δ, k)ak z + (−1)

k=p+n

2m

∞ X

     

zk

ξ(m, λ, l; δ, k)bk

≥ 0,

 

 zk  

k=p+n−1

where ξ(m, λ, l; δ, k) = dp,k (m, λ, l)C(δ, k). The above required condition must hold for all values of z in U . Upon choosing the values of z on the positive real axis where 0 ≤ |z| = r < 1, we must have (p + l)m+1 (1 − α) −

∞ X

[(p + l)(1 − α) + λ(k − p)]ξ(m, λ, l; δ, k)|ak |rk−p

k=p+n

(2.13) (p + l)m −

∞ X

ξ(m, λ, l; δ, k)|ak |rk−p +

k=p+n ∞ X

(p +

∞ X

− ξ(m, λ, l; δ, k)|bk |rk−p

k=p+n−1

[(p + l)(1 + α) + λ(k − p)]ξ(m, λ, l; δ, k)|bk |rk−p

k=p+n−1 ∞ X l)m − k=p+n

ξ(m, λ, l; δ, k)|ak |r

k−p

+

∞ X

≥ 0. ξ(m, λ, l; δ, k)|bk |r

k−p

k=p+n−1

If the condition (2.12) does not hold, then the numerator in (2.13) is negative for r sufficiently close to 1. Hence there exists a z0 = ro in (0, 1) for which the quotient in f H (p, m, δ, α, λ, l) (2.13) is negative. This contradicts the required condition for f ∈ AL and so the proof is complete.  3. Distortion bounds The following theorem gives the distortion bounds for f H (p, m, δ, α, λ, l) which yields a covering result for this class. AL

778

functions

in

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.5, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

On harmonic multivalent functions defined by a new derivative operator

5

f H (p, m, δ, α, λ, l), with 0 ≤ α < 1, λn ≥ α(p + l), m ∈ N0 , Theorem 3.1. Let f ∈ AL λ ≥ 0. Then for |z| = r < 1 one obtains (p + l)m+1 (1 − α) · [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p)   [(p + l)(1 + α) + λ(n − 1)]dp,n+p−1 (m, λ, l)C(δ, n + p − 1) · 1− |b | rn+p p+n−1 (p + l)m+1 (1 − α) and

(3.1) |f (z)| ≤ (1+|bp+n−1 |rn−1 )rp +

(p + l)m+1 (1 − α) · [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p)   [(p + l)(1 + α) + λ(n − 1)]dp,n+p−1 (m, λ, l)C(δ, n + p − 1) |b | · 1− rn+p . p+n−1 (p + l)m+1 (1 − α) |f (z)| ≥ (1 − |bp+n−1 |rn−1 )rp −

Proof. We only prove the left-hand inequality. The proof for the right-hand inequality is similar and will be omitted. f H (p, m, δ, α, λ, l). Taking the absolute value of f we have Let f ∈ AL ∞ ∞ ∞ X X X |f (z)| = |z p − ak z k + (−1)m bk z¯k | ≥ (1 − |bp+n−1 |rn−1 )rp − (|ak | + |bk |)rp+n k=p+n

k=p+n−1

= (1 − |bp+n−1 |rn−1 )rp −

k=p+n

(p + l)m+1 (1 − α) · [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p)

∞ X [(p + l)(1 − α) + λn]dp,p+n (m, λ, l)C(δ, p + n) · (|ak | + |bk |)rp+n ≥ (p + l)m+1 (1 − α) k=p+n

(p + l)m+1 (1 − α) · [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p)   [(p + l)(1 + α) + λ(n − 1)]dp,n+p−1 (m, λ, l)C(δ, n + p − 1) · 1− |b | rn+p . p+n−1 (p + l)m+1 (1 − α) (1 − |bp+n−1 |rn−1 )rp −

The bounds given in Theorem 3.1 for the functions f of the form (1.2) also hold for the functions of the form (1.1) if the coefficient condition (2.8) is satisfied. The f H (p, m, δ, α, λ, l) is sharp and the equality occurs for upper bound given for f ∈ AL the function (p + l)m+1 (1 − α) · [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p)   [(p + l)(1 + α) + λ(n − 1)]dp,n+p−1 (m, λ, l)C(δ, n + p − 1) · 1− |bp+n−1 | rn+p , (p + l)m+1 (1 − α) f (z) = z + |bp+n−1 |¯ zp +

where |bp+n−1 | ≤

(p+l)m+1 (1−α) [(p+l)(1+α)+λ(n−1)]dp,n+p−1 (m,λ,l)C(δ,n+p−1)



The following covering result follows from the left-hand inequality in Theorem 3.1.

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6

A. C˘ ata¸s, R. S ¸ endrut¸iu

f H (p, m, δ, α, λ, l), then Corollary 3.2. If the function f ∈ AL  [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p) − (p + l)m+1 (1 − α) − w: |w|< [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p)  α [(p + l)(1 − α) + λn]dp,n+p (m, λ, l)C(δ, n + p) − Ep,δ (m, λ, l) − · |bp+n−1 | ⊂f (U ) [p(1 − α) + λn + l]dp,n+p (m, λ, l)C(δ, n + p) α (m, λ, l) = [(p + l)(1 + α) + λ(n − 1)]dp,n+p−1 (m, λ, l)C(δ, n + p − 1). where Ep,δ References [1] F. M. Al-Oboudi, On univalent functions defined by a generalized S˘ al˘ agean operator, Inter. J. of Math. and Mathematical Sci., 27(2004), 1429-1436. [2] K. Al-Shaqsi, M. Darus, An operator defined by convolution involving polylogarithms functions, Journal of Math. and Statistics, 4(1)(2008), 46-50. [3] K. Al-Shaqsi, M. Darus, On Harmonic Functions Defined by Derivative Operator, Journal of Inequalities and Applications, vol. 2008, Article ID 263413, doi: 10.1155/2008/263413. [4] J. Clunie and T. Sheil-Small, Harmonic Univalent Functions, Ann. Acad. Sci. Fenn, Ser. A I. Math. 9(1984), 3-25. [5] M. Darus, R. W. Ibrahim, On new classes of univalent harmonic functions defined by generalized differential operator, Acta Universitatis Apulensis, 18(2009), 61-69. [6] J. M. Jahangiri, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie-Sklowdowska Sect. A, 52(1998), 57-66. [7] J. M. Jahangiri, G. Murugusundaramoorthy and K. Vijaya S˘ al˘ agean type harmonic univalent functions South. J. Pure Appl. Math., 2(2002), 77-82. [8] Om P. Ahuja and J. M. Jahangiri, Multivalent harmonic starlike functions with missing coefficients, Math. Sci. Res. J., 7(9)(2003), 347-352. [9] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109115. [10] Gr. S ¸ t. S˘ al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, Heidelberg and New York, 1013(1983), 362-372. [11] Sibel Yalcin, A new class of S˘ al˘ agean-type harmonic univalent functions Appl. Math. Letters, 18(2005), 191-198. 1

Department of Mathematics and Computer Sciences, University of Oradea, ˘t Str. Universita ¸ ii, No.1, 410087 Oradea, Romania ∗ Corresponding author: [email protected] 2

Faculty of Environmental Protection, University of Oradea, Str. B-dul Gen. Magheru, No.26, 410048 Oradea, Romania E-mail address: [email protected]

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GOOD AND SPECIAL WEAKLY PICARD OPERATORS PROPERTIES FOR A CLASS OF DISCRETE LINEAR OPERATORS ˘ LOREDANA-FLORENTINA IAMBOR, ADRIANA CATAS ¸

Abstract. Based on the results of the weakly Picard operators theory, in this paper we study the good and special convergence of the iterates of a general class of positive linear operators of discrete type introduced by O.Agratini and I.A. Rus ([1]).

2010 AMS Mathematics Subject Classification: 47H10, 41A36. Keywords and phrases: linear positive operators, weakly Picard operators, good and special Picard operators. 1. Introduction and Preliminaries The study of the convergence of the sequence of successive approximations is realized in metric spaces. That is, for (X, d) metric space and A : X → X an operator, for any x ∈ X can be considered the sequence: (1)

(Am (x))m∈N , x ∈ X

where A0 = 1X and Am = Am−1 ◦ A for m ∈ N∗ . Investigating the properties of sequence (1), L. d’Apuzzo introduced in 1976 (see [3]) the good and special convergence, giving necessary and sufficient conditions for this kind of convergence (see [2]). In paper [3], she considers the good and special convergence of type M, as a particular case, in

which the sequence (d (Am (x) , A∞ (x)))m∈N , (respectively, d Am (x) , Am−1 (x) m∈N ) is strictly decreasing for any x. I.A. Rus introduced, in paper (see [8]), the good and special weakly Picard operators . In what follow, let (X, d) be a metric space and A : X → X an operator. In this paper we will use the following notations: P (X) := { Y ⊂ X| Y 6= ∅} ; FA := { x ∈ X| A (x) = x} - the fixed point set of A; I (A) := { Y ∈ P (X)| A (Y ) ⊂ Y } - the family of the nonempty invariant subsets of A. Definition 1. (I.A. Rus - [6], [7], [8]) Let (X, d) be a metric space. 1) An operator A : X → X is weakly Picard operator (briefly WPO) if the sequence of successive approximations (Am (x0 ))m∈N converges for all x0 ∈ X and the limit (which may depend on x0 ) is a fixed point of A. 2) If the operator A : X → X is WPO and FA = {x∗ }, then by definition the operator A is Picard operator (briefly PO). 3) If the operator A : X → X is WPO, then can be considered the operator A∞ defined by A∞ : X → X, A∞ (x) := lim Am (x). m→∞ 1

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˘ LOREDANA-FLORENTINA IAMBOR, ADRIANA CATAS ¸

2

The basic result in the WPO’s theory is the following: Theorem 1. (Characterization theorem - [6], [7], [8])SAn operator A : X → X is WPO if and only if there exits a partition of X, X = Xλ , such that: λ∈Λ

(a) Xλ ∈ I (A) , ∀λ ∈ Λ; (b) A|Xλ : Xλ → Xλ is PO, ∀λ ∈ Λ. Definition 2. Let (X, d) be a metric space and A : X → X a WPO. ∞
P 1) A : X → X is good WPO, if the series d Am−1 (x) , Am (x) converges, for m=1

all x ∈ X (see [8]). In the case that the sequence d Am−1 (x) , Am (x) m∈N∗ is strictly decreasing for all x ∈ X, the operator A is good WPO of type M (see [3]). ∞ P 2) A : X → X is special WPO, if the series d (Am (x) , A∞ (x)) converges, m=1

for all x ∈ X(see [8]). When the sequence (d (Am (x) , A∞ (x)))m∈N∗ is strictly decreasing for all x ∈ X, A is special WPO of type M (see [3]). In 2015, S. Mure¸san and L.F. Iambor obtained the following result regarding to good and special weakly Picard operators. Theorem 2. ([5]) Let (X, d) be a metric space and A : X → X a WPO. If A is special WPO then A is good WPO. In the paper [4], A.Bica and L.F. Galea(Iambor) introduced the notions of uniform good and special weakly Picard operators like this: Definition 3. (A.Bica, L.F. Galea - [4]) Let (X, d) be a metric space and F ⊂ { A| A : X → X} a family of operators on X. We say that F is a family of uniform special (good) WPO’s if for any A ∈ F , A is special (good) WPO and there exist the functionals ϕ : X → R+ and ψ, ψ 0 : F → R+ such that ϕ is continuous and ∞ P d (Am (x) , A∞ (x)) 6 ψ (A) · ϕ (x) , ∀ x ∈ X, ∀ A ∈ F m=1

(respectively,

∞ P


d Am (x) , Am−1 (x) 6 ψ 0 (A) · ϕ (x) , ∀ x ∈ X, ∀ A ∈ F ).

m=1

In what follow, we present the general class of linear positive operators of discrete type and some properties of these operators investigated by O. Agratini and I.A. Rus in [1]. At first they construct an approximation process of discrete type acting on the space C ([a, b]) endowed with the Chebyshev norm k.k. For each integer n > 1 they consider the following: (i) A net on [a, b] named ∆n is fixed (a = xn,0 < xn,1 < ... < xn,n = b). (ii) A system (ψn,k )k=0,n is given, where every ψn,k belongs to C ([a, b]). They assume that it is a blending system with a certain connection with ∆n , more precisely the following conditions hold: n n
P P ψn,k > 0, k = 0, n , ψn,k = e0 , xn,k ψn,k = e1 k=0

k=0

Definition 4. (O.Agratini, I.A. Rus - [1]) The operators Ln : C ([a, b]) → C ([a, b]) defined by n P ψn,k (x) f (xn,k ) Ln (f ) (x) = k=0

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GOOD AND SPECIAL WEAKLY PICARD OPERATORS PROPERTIES FOR A CLASS OF DISCRETE LINEAR OPERATORS 3

are called the operators of discrete type. The operators of discrete type Ln , have the following properties: 1) Ln , n ∈ N are positive linear operators; 2) Ln (e0 ) = e0 and Ln (e1 ) = e1 . Theorem 3. (O.Agratini, I.A. Rus - [1]) Let Ln , n ∈ N, such that ψn,0 (a) = ψn,n (b) = 1. Let us denote un := min [Φn,0 (x) + Φn,n (x)]. x∈[a,b]

If the un > 0 the iterates sequence (Lm n )m>1 verifies lim (Lm n f ) (x) = f (a) +

m→∞

f (a)−f (b) b−a

(x − a) , f ∈ C ([a, b])

uniformly on [a, b]. Theorem 4. (O.Agratini, I.A. Rus - [1]) Let Ln , n ∈ N, such that ψn,0 (a) = ψn,n (b) = 1. Then the operator Ln is weakly Picard operator for every n ∈ N and not

∗ L∞ n (f ) = c1 (f ) e1 + c2 (f ) = f (x) , f ∈ C ([a, b])

where c1 (f ) =

f (b)−f (a) b−a

and c2 (f ) =

bf (a)−af (b) . b−a

The convergence exists on the space (C [a, b] , k.k∞ ). In the application of Characterization theorem of weakly Picard operator, it was considerate the partition of C ([a, b]): S C ([a, b]) := Xα,β α,β∈R

where Xα,β = {f ∈ C ([a, b]) : f (a) = α, f (b) = β} , α, β ∈ R. Proposition 5. (O. Agratini, I.A. Rus - [1]) The operators of discrete type satisfied the following contraction property relative to above partition: (2)

kLn (f ) − Ln (g)k∞ 6 (1 − un ) kf − gk∞ , ∀ f, g ∈ Xα,β , α, β ∈ R

where un = min [Φn,0 (x) − Φn,n (x)], un > 0. x∈[a,b]

2. Main results In this section, we will investigate some properties of the iterates of discrete type of operators in sense of good and special convergence. Theorem 6. The operators of discrete type Ln , n ∈ N are special WPO and good WPO of type M on C ([a, b]). From Theorem 3, we have that Ln , n ∈ N is weakly Picard operator. Let f ∈ C ([a, b]). Then f ∈ Xf (a),f (b) and according to (1) we infer that Ln is contraction on Xf (a),f (b) . So, the operator Ln , n ∈ N is special WPO of type M on Xf (a),f (b) . Finally, we get that Ln , n ∈ N is special WPO of type M on C ([a, b]). From Theorem 2, any special WPO is good WPO. Then we have that Ln , n ∈ N is good WPO of type M on C ([a, b]). Theorem 7. The family of the operators of discrete type {Ln : n ∈ N∗ } is family of uniform special and good WPO’s on C [a, b]. Proof. obtain the estimation: 1 Using the inequality (1), we 1 Ln (f ) (x) − L∞ (f ) (x) = L (f ) (x) − L1n (L∞ n n n (f )) (x) 6 ∞ 6 (1 − un ) |f (x) − Ln (f ) (x)| = (1 − un ) |f (x) − c1 (f ) e1 − c2 (f )| 6

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4

6 (1 − un ) · C, ∀ x ∈ [a, b], where C = diam (Im f ) + 2 max {|f (a)| , |f (b)|}, with diam (Im f ) = max {|f (x) − f (y)| : x, y ∈ [a, b]}. The constant C was obtained using the following technique: • If x = a then: f (b)−f (a) bf (a)−af (b) (x) − · a − |f (x) − L∞ (f ) (x)| 6 = f n b−a b−a = f (x) − f (a)(b−a) = |f (x) − f (a)| 6 diam (Im f ) b−a • If x = b then: f (b)−f (a) (b) |f (x) − L∞ · b − bf (a)−af = n (f ) (x)| 6 f (x) − b−a b−a = f (x) − f (b)(b−a) = |f (x) − f (b)| 6 diam (Im f ) b−a • If x ∈ [a, b] then: f (b)−f (a) (b) |f (x) − L∞ · x − bf (a)−af = n (f ) (x)| 6 f (x) − b−a b−a b−x x−a = f (x) − b−a · f (b) + b−a · f (a) 6 b−x 6 |f (x) − f (b)| + |f (b)| · 1 − x−a + b−a b−a · |f (a)| 6 6 diam (Im f ) + 2 max {|f (a)| , |f (b)|} . By induction, for m ∈ N∗ , we have:
∞ 1 m−1 (f ) (x) − L1n (L∞ |Lm n (f )) (x) 6 n (f ) (x) − Ln (f ) (x)| = Ln Ln m 6 (1 − un ) · C, ∀ x ∈ [a, b] . ∞ P ∞ Then, |Lm n (f ) (x) − Ln (f ) (x)| 6 m=1h i 2 m 6 lim C (1 − un ) + (1 − un ) + ... + (1 − un ) = m→∞ h i 1−(1−un )m 1−un = lim C (1 − un ) · 6 C · un (2) un m→∞ On the other hand, we have: n P 1 Ln (f ) (x) − L0n (f ) (x) = ψn,k (x)f (xn,k ) − f (x) = k=0 n n P P ψn,k (x) = C 0 e0 , ∀ x 6 [a, b], where = ψn,k (x) [f (xn,k ) − f (x)] 6 C 0 k=0

k=0

C 0 = diam (Im f ) = max {|f (x) − f (y)| : x, y ∈ [a, b]} . By m ∈ N, we minduction, for have:

Ln (f ) (x) − Lm−1 = L1n Lm−1 (f ) (x) (f ) (x) − L1n Lm−2 (f ) (x) 6 n n n m−1 6 (1 − un ) · C 0 e0 , ∀ x ∈ [a, b] ∞ P m−1 Lm Then, (f ) (x) 6 n (f ) (x) − Ln m=1 h i 2 m−1 6 6 lim C 0 e0 1 + (1 − un ) + (1 − un ) + ... + (1 − un ) m→∞

1 6 C 0 e0 1−u , ∀ f ∈ C [a, b] (3). n Now, the property of uniform and special WPO follows from the estimations (2) and (3). For instance, for the property of uniform special WPO we have:

ϕ : C [a, b] → R+ , ϕ (f ) = diam (Im f ) + 2 max {|f (a)| , |f (b)|} and ψ : {Ln : n ∈ N∗ } → R+ , ψ (Ln ) =

1−un un ,

∀ n ∈ N∗

and for the property of uniform good WPO we have:

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GOOD AND SPECIAL WEAKLY PICARD OPERATORS PROPERTIES FOR A CLASS OF DISCRETE LINEAR OPERATORS 5

ϕ0 : C [a, b] → R+ , ϕ0 (f ) = diam (Im f ) and ψ 0 : {Ln : n ∈ N∗ } → R+ , ψ 0 (Ln ) =

1 1−un e0 ,

∀ n ∈ N∗ .

It is easy to prove that ϕ, ϕ0 : C [a, b] → R+ , ϕ (f ) = diam (Im f )+2 max {|f (a)| , |f (b)|} and ϕ0 (f ) = diam (Imf ) are seminorms on C [a, b] and ϕ (f − g) 6 2 kf − gkC + 2 kf kC , ϕ0 (f − g) 6 2 kf − gkC since |ϕ (f ) − ϕ (g)| 6 ϕ (f − g) , ∀ f, g ∈ C [a, b] and |ϕ0 (f ) − ϕ0 (g)| 6 ϕ0 (f − g) , ∀ f, g ∈ C [a, b] we infer the ϕ, ϕ0 are continuous.  References [1] O. Agratini, I.A. Rus Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Carolinae, 44(3), 2003, 555-563. [2] L. D’Apuzzo, On the convergence of the method of succesive approximation in metric spaces, Ann. Istit. Univ. Navale Napoli, 45/46(1976/1977)(in Italian). [3] L. D’Apuzzo, On the notion of good and special convergence of the method of succesive approximations, Ann. Istit. Univ. Navale Napoli, 45/46(1976/1977), 123-138, (in Italian). [4] A. Bica, L.F. Galea On Picard iterative properties of the Bernstein operators and an application to fuzzy numbers, Comm. in Math. Analysis, 5(1), 2008, 8-19. [5] S. Mure¸san, L.F. Iambor, On good and special weakly Picard operators, Analele Universitatii Oradea, Fasc. Matematica, Tom XXII, Issue No.1, pp.5-10, 2015. [6] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, 2001, (in Romanian). [7] I. A. Rus, Weakly Picard Operators and Applications, Seminar on Fixed Point Theory ClujNapoca, Volume2, 41-58, 2001. [8] I. A. Rus, Picard operators and applications, Sci. Math.Japon., 58 (1), pp.191-219, 2003. Department of Mathematics and Computer Science, University of Oradea, Str. Uni˘t versita ¸ ii No.1, 410087, Oradea, Romania, E-mail address: [email protected], [email protected]

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General Iyengar type Inequalities George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we present general Iyengar type inequalities with respect to Lp norms, with 1 p 1. The method is based on the generalized Taylor’s formula.

2010 Mathematics Subject Classi…cation: 26D10, 26D15. Key Words and Phrases: Iyengar inequality, Taylor formula.

1

Introduction

We are motivated by the following famous Iyengar inequality (1938), [2]. Theorem 1 Let f be a di¤ erentiable function on [a; b] and jf 0 (x)j Z

b

f (x) dx

a

2

1 (b 2

M (b a) 4

a) (f (a) + f (b))

2

(f (b)

M . Then 2

f (a)) : (1) 4M

Main Results

We present the following Iyengar type inequalities: Theorem 2 Let n 2 N, f 2 AC n ([a; b]) (i.e. f (n 1) 2 AC ([a; b]), absolutely continuous functions). We assume that f (n) 2 L1 ([a; b]). Then i) Z

b

f (x) dx

a

n X1 k=0

h 1 f (k) (a) (t (k + 1)! f (n)

L1 ([a;b])

(n + 1)! 8 t 2 [a; b] ;

h

(t

k+1

a)

n+1

a)

+ ( 1)k f (k) (b) (b

+ (b

n+1

t)

i

;

k+1

t)

i (2)

1

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ii) at t = Z

a+b 2 ,

the right hand side of (2) is minimized, and we get: n X1

b

f (x) dx

a

k=0

k+1

1 (b a) (k + 1)! 2k+1 f (n)

(b

L1 ([a;b])

(n + 1)!

h

i f (k) (a) + ( 1)k f (k) (b) n+1

a) 2n

;

(3)

iii) if f (k) (a) = f (k) (b) = 0, for all k = 0; 1; :::; n Z

f (n)

b

f (x) dx

L1 ([a;b])

n+1

(b

a) 2n

(n + 1)!

a

1, we obtain ;

(4)

which is a sharp inequality, iv) more generally, for j = 0; 1; 2; :::; N 2 N, it holds Z

b

f (x) dx

a

n X1 k=0

b

1 (k + 1)!

f (n)

a

k+1

N b

L1 ([a;b])

(n + 1)!

a

h

n+1

N

v) if f (k) (a) = f (k) (b) = 0, k = 1; :::; n Z

b

b

f (x) dx

N

a

f (n)

a

b

L1 ([a;b])

(n + 1)!

a

h

j n+1 + (N

N

h

b

b

f (x) dx

2

a

f (n)

a

L1 ([a;b])

(n + 1)!

i

;

i

;

j)

i f (k) (b) (5)

1; from (5) we get:

[jf (a) + (N n+1

n+1

j)

j) f (b)]

j n+1 + (N

n+1

j)

for j = 0; 1; 2; :::; N 2 N; vi) when N = 2 and j = 1, (6) turns to Z

k+1

j k+1 f (k) (a) + ( 1)k (N

(6)

(f (a) + f (b))

(b

n+1

a) 2n

;

(7)

vii) when n = 1 (without any boundary conditions), we get from (7) that Z

a

b

f (x) dx

b

a 2

(f (a) + f (b))

kf 0 k1;[a;b]

2

(b

a) 4

;

(8)

a similar to Iyengar inequality (1). 2

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Proof. Here n 2 N and f (n 1) is absolutely continuous on [a; b]. We assumed that f (n) := f (n) < +1: 1;[a;b]

L1 ([a;b])

By [1], we have the following generalized Taylor’s formulae: n X1

f (x)

k=0

f (k) (a) (x k!

1

k

a) =

(n

1)!

Z

x

n 1

(x

t)

f (n) (t) dt

(9)

a

and f (x)

n X1 k=0

f (k) (b) (x k!

1

k

b) =

(n

1)!

Z

x

n 1

(x

t)

f (n) (t) dt;

(10)

b

8 x 2 [a; b] : Then we get n X1

f (x)

k=0

f (k) (a) (x k!

f (n)

k

a)

1;[a;b]

n

(x

n!

a) ;

(11)

8 x 2 [a; b] ; and n X1

f (x)

k=0

1 (n

1)!

Z

f (k) (b) (x k!

b

(t

n 1

x)

f

b)

(n)

k

1

=

(n

(n

x

1;[a;b]

n! that is f (x)

n X1 k=0

f (k) (b) (x k!

8 x 2 [a; b] : We call

x

b)

(b

Z

1)!

b

(t

n 1

x)

f (n) (t) dt

n

x) ;

1;[a;b]

n!

f (n) (t) dt =

x

1;[a;b]

n!

f (n) :=

t)

b

f (n)

k

n 1

(x

1

(t) dt

f (n)

1)!

Z

(b

n

x) ;

:

(12)

(13)

So we have (x

n

a)

f (x)

n X1 k=0

f (k) (a) (x k!

k

a)

(x

n

a)

(14)

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and (b

n

x)

n X1

f (x)

k=0

f (k) (b) (x k!

b)

k

n

(b

x) ;

(15)

8 x 2 [a; b] : Therefore it holds n X1 k=0

f (k) (a) (x k!

k

a)

n

(x

a)

n X1

f (x)

k=0

f (k) (a) (x k!

k

a) + (x

n

a)

(16)

and n X1 k=0

f (k) (b) (x k!

b)

k

n

(b

x)

n X1

f (x)

k=0

f (k) (b) (x k!

k

b) + (b

n

x) ; (17)

8 x 2 [a; b] : Let any t 2 [a; b], then n X1 k=0

f (k) (a) (t (k + 1)! n X1 k=0

and

n X1 k=0

k+1

a)

(n + 1)

f (k) (a) (t (k + 1)!

f (k) (b) (t (k + 1)! n X1 k=0

b)

k+1

a)

+

Adding (18) and (19), we obtain: (n 1 h X 1 f (k) (a) (t (k + 1)!

k+1

t

f (x) dx

+

k+1

a)

n+1

(t

(n + 1)

(n + 1) b)

a)

a

k+1

f (k) (b) (t (k + 1)!

Z

n+1

(t

a)

Z

n+1

(b

t)

;

(18)

b

f (x) dx

(n + 1)

f

(k)

n+1

(b

t)

(b) (t

b)

:

k+1

k=0

(n + 1) (n 1 X k=0

h

(t

h

(t

n+1

+ (b

h 1 f (k) (a) (t (k + 1)! (n + 1)

8 t 2 [a; b] :

n+1

a)

t)

k+1

a)

n+1

a)

(19)

t

f

+ (b

Z

i

i

b

f (x) dx

a

(k)

k+1

(b) (t n+1

t)

)

i

b) ;

i

)

+ (20)

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Consequently we derive: Z

n X1

b

f (x) dx

a

k=0

h 1 f (k) (a) (t (k + 1)! (n + 1)

8 t 2 [a; b] : Let us consider g (t) := (t

h

k+1

a)

n+1

(t

a)

n+1

+ (b

a)

n

n+1

+ (b

t)

n+1

t)

Hence

n

g 0 (t) = (n + 1) [(t

+ ( 1)k f (k) (b) (b

a)

i

k+1

t)

;

i (21)

; 8 t 2 [a; b] :

(22)

n

(b

t) ] = 0,

n

giving (t a) = (b t) and t a = b t, that is t = a+b 2 the only critical number here. n+1 n+1 = (b 2a)n , which is the We have g (a) = g (b) = (b a) , and g a+b 2 minimum of g over [a; b]. Consequently the right hand side of (21) is minimized when t = a+b 2 , with kf (n) k1;[a;b] (b a)n+1 value . Assuming f (k) (a) = f (k) (b) = 0, for k = 0; 1; :::; n (n+1)! 2n 1, then we obtain that Z

f (n)

b

f (x) dx

1;[a;b]

n+1

(b

a) 2n

(n + 1)!

a

;

(23)

which is a sharp inequality. When t = a+b 2 , then (21) becomes Z

n X1

b

f (x) dx

a

k=0

f (n)

1;[a;b]

i f (k) (a) + ( 1)k f (k) (b)

k+1

h

(b

a) 2n

1 (b a) (k + 1)! 2k+1

(n + 1)!

n+1

:

Next let N 2 N, j = 0; 1; 2; :::; N and tj = a + j a + bNa ; :::; tN = b: Hence it holds tj

b

a=j

a N

, (b

tj ) = (N

j)

b

a

b

a N

(24) b a N

, that is t0 = a, t1 =

; j = 0; 1; 2; :::; N:

(25)

We notice that (tj

n+1

a)

+ (b

n+1

tj )

=

N

n+1

h

j n+1 + (N

n+1

j)

i

;

(26)

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j = 0; 1; 2; :::; N; and (k = 0; 1; :::; n 1) h k+1 f (k) (a) (tj a) + ( 1)k f (k) (b) (b "

f

(k)

(a) j

b

k+1

a

j = 0; 1; 2; :::; N: By (21) we get b

f (x) dx

a

k (k)

+ ( 1) f

k+1

N

Z

k+1

a N

b

n X1 k=0

h

b

k+1

a N b

1;[a;b]

(n + 1)!

h

N k+1

b

f (x) dx

a

b

1;[a;b]

(n + 1)!

j)

N

b

f (x) dx

b

h

f (n)

b

1;[a;b]

(n + 1)!

a 2

a 2

a

;

k+1

i

;

i

j) f (b)]

j n+1 + (N

n+1

j)

for j = 0; 1; 2; :::; N: When N = 2 and j = 1, then (29) becomes Z

(27)

1, then (28) becomes

n+1

a

i

=

(28)

n+1

j n+1 + (N

[jf (a) + (N

N

a

j)

#

h

n+1

a

k+1

a

j)

N

b

f (n)

j)

=

f (k) (a) j k+1 + ( 1)k f (k) (b) (N

j = 0; 1; 2; :::; N: If f (k) (a) = f (k) (b) = 0, k = 1; :::; n Z

(b) (N

i

b

k+1

f (k) (a) j k+1 + ( 1)k f (k) (b) (N

1 (k + 1)!

f (n)

k+1

tj )

2=

;

(29)

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

n+1

i

1;[a;b]

(n + 1)!

(b

n+1

a) 2n

:

(30)

And, if n = 1 (without any boundary conditions), we get from (30) that Z

a

b

f (x) dx

b

a 2

(f (a) + f (b))

kf 0 k1;[a;b]

2

(b

a) 4

;

(31)

which a similar inequality to Iyengar inequality (1). We give

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Theorem 3 Let f 2 AC n ([a; b]), n 2 N. Then i) Z

b

f (x) dx

a

n X1 k=0

h 1 f (k) (a) (t (k + 1)! f (n)

L1 ([a;b])

Z

a+b 2 ,

b

f (x) dx

a

+ ( 1)k f (k) (b) (b

n

[(t

n! 8 t 2 [a; b] ; ii) at t =

k+1

a)

k+1

t)

i

(32)

n

a) + (b

t) ] ;

the right hand side of (32) is minimized, and we get: n X1 k=0

h

k+1

(b a) 1 (k + 1)! 2k+1 f (n)

L1 ([a;b])

n!

i f (k) (a) + ( 1)k f (k) (b) n

(b a) ; 2n 1

iii) if f (k) (a) = f (k) (b) = 0, for all k = 0; 1; :::; n Z

f (n)

b

f (x) dx

L1 ([a;b])

n!

a

(33) 1, we obtain n

(b a) ; 2n 1

(34)

which is a sharp inequality, iv) more generally, for j = 0; 1; 2; :::; N 2 N, it holds Z

a

b

f (x) dx

n X1 k=0

1 (k + 1)! f (n)

b

a

k+1

N b

L1 ([a;b])

n!

h

j k+1 f (k) (a) + ( 1)k (N

a

b

f (x) dx

[j n + (N

N

b

a

f (n)

L1 ([a;b])

a

b

n!

a N

n

j) ] ;

(35)

1; from (35) we get:

[jf (a) + (N

N

i f (k) (b)

n

v) if f (k) (a) = f (k) (b) = 0, k = 1; :::; n Z

k+1

j)

j) f (b)]

n

[j n + (N

n

j) ] ;

(36)

for j = 0; 1; 2; :::; N 2 N; vi) when N = 2 and j = 1, (36) turns to Z

a

b

f (x) dx

(b

a) 2

(f (a) + f (b))

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f (n)

n

(b a) ; (37) n! 2n 1 vii) when n = 1 (without any boundary conditions), we get from (37) that Z

b

b

f (x) dx

a

kf 0 kL1 ([a;b]) (b

(f (a) + f (b))

2

a

L1 ([a;b])

a) :

(38)

Proof. Here n 2 N and f (n 1) is absolutely continuous on [a; b]. Hence f (n) exists almost everywhere and f (n) 2 L1 ([a; b]). By (9) we get n X1

f (x)

k=0

1 (n

1)!

Z

f (k) (a) (x k!

k

a)

1

=

(n

1)!

Z

x

n 1

(x

t)

a n 1

x

n 1

(x

t)

(x a) (n 1)!

f (n) (t) dt

a

(x a) (n 1)!

f (n)

L1 ([a;b])

Z

b

f (n) (t) dt

(39)

a

n 1

=

f (n) (t) dt

:

That is n X1

f (x)

f (k) (a) (x k!

k=0

f (n)

k

a)

L1 ([a;b])

(n

1)!

(x

n 1

a)

;

(40)

8 x 2 [a; b] : By (10) we get f (x)

n X1 k=0

Z

1 (n

1)!

f (k) (b) (x k!

b

(t

n 1

x)

f

b)

(n)

k

=

1 (n

n 1

Z

a

b

x

(t) dt

(n

n 1

(x

t)

f (n) (t) dt =

b

Z

1

x

(b x) (n 1)!

1)!

Z

1)!

b

(t

n 1

x)

f (n) (t) dt

x

(41)

n 1

(b x) f (n) (t) dt = (n 1)!

f (n)

L1 ([a;b])

:

That is f (x)

n X1 k=0

f (k) (b) (x k!

8 x 2 [a; b] : Set

b)

f (n)

k

(n

f (n) :=

L1 ([a;b])

(n

L1 ([a;b])

1)!

1)!

(b

n 1

x)

;

(42)

:

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Hence f (x)

n X1

f (k) (a) (x k!

a)

n X1

f (k) (b) (x k!

b)

k=0

and f (x)

k=0

n 1

;

(43)

n 1

;

(44)

k

(x

a)

k

(b

x)

8 x 2 [a; b] : As in the proof of Theorem 2 we get: Z

b

f (x) dx

a

n X1 k=0

h 1 f (k) (a) (t (k + 1)! [(t

n

k+1

a)

n

k+1

+ ( 1)k f (k) (b) (b

t)

i

n

a) + (b

t) ] ;

(45)

8 t 2 [a; b] : The rest of the proof is similar to the proof of Theorem 2. We continue with Theorem 4 Let f 2 AC n ([a; b]), n 2 N; p; q > 1 : Lq ([a; b]). Then i) Z

b

f (x) dx

a

n X1 k=0

h 1 f (k) (a) (t (k + 1)! f (n)

(n

1)! n +

8 t 2 [a; b] ; ii) at t = Z

a+b 2 ,

1 p

k+1

a)

Lq ([a;b])

(p (n

1) + 1)

1 p

h

f (x) dx

a

n X1 k=0

k+1

(b a) 1 (k + 1)! 2k+1 f (n)

(n

1 q

= 1, and f (n) 2

k+1

+ ( 1)k f (k) (b) (b

(t

1 n+ p

a)

1)! n +

1 p

h

+ (b

t) i 1

n+ p

t)

i

(46)

;

i f (k) (a) + ( 1)k f (k) (b) (b

Lq ([a;b]) 1

(p (n

1) + 1) p

iii) if f (k) (a) = f (k) (b) = 0, for all k = 0; 1; :::; n

a

+

the right hand side of (46) is minimized, and we get:

b

Z

1 p

f (n)

b

f (x) dx (n

1)! n +

1 p

1 n+ p

a) 2n

(47)

1, we obtain (b

Lq ([a;b])

(p (n

;

1 q

1

1) + 1) p

1 n+ p

a) 2n

1 q

;

(48)

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which is a sharp inequality, iv) more generally, for j = 0; 1; 2; :::; N 2 N, it holds Z

n X1

b

f (x) dx

a

k=0

f (n) (n

1 (k + 1)!

(p (n

h

k+1

a N

1) + 1)

1 p

a

b

b

f (x) dx

a

f (n) (n

1)! n +

b

Lq ([a;b])

1 p

1

(p (n

j)

i f (k) (b)

i

;

(49)

i

;

(50)

1; from (49) we get:

a

h

1 n+ p

N

1) + 1) p

j)

1 n+ p

1

j n+ p + (N

[jf (a) + (N

N

a

h

1 n+ p

N

v) if f (k) (a) = f (k) (b) = 0, k = 1; :::; n Z

k+1

j k+1 f (k) (a) + ( 1)k (N

b

Lq ([a;b])

1 p

1)! n +

b

j) f (b)]

1 n+ p

1

j n+ p + (N

j)

for j = 0; 1; 2; :::; N 2 N; vi) when N = 2 and j = 1, (50) turns to Z

b

(b

f (x) dx

a

f (n) (n

1)! n +

a) 2

(f (a) + f (b))

1 p

1 n+ p

(b

Lq ([a;b]) 1

(p (n

a) 2n

1) + 1) p

;

1 q

(51)

vii) when n = 1 (without any boundary conditions), we get from (51) that Z

b

b

f (x) dx

a 2

a

(f (a) + f (b))

1+

n X1 k=0

f (k) (a) (x k! 1

1 (n

1)!

k

a) Z

=

1 (n

x

(x (n 1)! a Z x p(n (x t)

1)! n 1

t) 1)

1 p

dt

a

Z

a) 1

1 p

x

(x

:

(52)

2p

Proof. Here f (n) 2 Lq ([a; b]), where p; q > 1, such that we get f (x)

1 1+ p

kf 0 kLq ([a;b]) (b

1 p

n 1

t)

+

1 q

= 1. By (9)

f (n) (t) dt

a

f (n) (t) dt Z

x

q

f (n) (t) dt

1 q

a

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p(n

(x (n

a)

1)+1 p

1)! (p (n

1) + 1)

f (n)

1 p

Lq ([a;b])

:

(53)

That is f (x)

n X1 k=0

f (k) (a) (x k!

f (n)

k

Lq ([a;b])

a)

(n

1)! (p (n

(x

1 p

1) + 1)

n

a)

1 q

;

(54)

8 x 2 [a; b] : By (10) we get n X1

f (x)

k=0

f (k) (b) (x k!

b)

1 (n 1 (n

1)!

Z

1)!

Z

k

=

(n

b

x)

p(n 1)

x)

dt

x p(n

(b (n

b

n 1

(t

x)

f (n) (t) dt

x

f (n) (t) dt

x

b

(t

1)! n 1

(t

Z

1

x)

1)! (p (n

! p1

Z

! q1

q

f (n) (t) dt

x

1)+1 p

1) + 1)

b

f (n)

1 p

Lq ([a;b])

:

(55)

That is f (x)

n X1 k=0

f (k) (b) (x k!

b)

f (n)

k

(n

8 x 2 [a; b] : Set

1)! (p (n

f (n) := (n

Lq ([a;b])

Lq ([a;b]) 1

1)! (p (n

and

1) + 1)

1 p

(b

;

n

x)

1 q

;

(56)

(57)

1) + 1) p

m := n

1 > 0: q

n X1

f (k) (a) (x k!

a)

n X1

f (k) (b) (x k!

b)

(58)

So, we can write f (x)

k=0

and f (x)

k=0

8 x 2 [a; b] :

k

(x

a) ;

m

(59)

k

(b

x) ;

m

(60)

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As in the proof of Theorem 2 we obtain: Z

b

f (x) dx

a

n X1 k=0

h 1 f (k) (a) (t (k + 1)! (m + 1) f (n)

(n

1)! (p (n

h

(t

k+1

a)

m+1

a)

Lq ([a;b])

1) + 1)

1 p

n+

1 p

+ ( 1)k f (k) (b) (b m+1

+ (b

t)

h

a)

(t

1 n+ p

i

k+1

t)

i

=

+ (b

(61) 1 n+ p

t)

i

;

(62)

8 t 2 [a; b] : The rest of the proof is similar to the proof of Theorem 2.

References [1] G.A. Anastassiou, S.S. Dragomir, On some estimates of the remainder in Taylor’s formula, J. Math. Anal. Appl., 263 (2001), 246-263. [2] K.S.K. Iyengar, Note on an inequality, Math. Student, 6 (1938), 75-76.

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A NOTE ON THE APPROXIMATE SOLUTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY G-BROWNIAN MOTION F. Faizullah1 , R. Ullah2, Jihen Majdoubi3, I. Tlili4, I. Khan5, Ghaus ur Rahman6 1Department of Mathematics, Swansea University, Singleton Park SA2 8PP UK 2Department of Mathematics, Women University, Swabi, Pakistan.

3Department of computer science college of science and humanities at Alghat Majmaah University, P.O. Box 66 Majmaah 11952, Kingdom of Saudi Arabia 4Department of Mechanical and Industrial Engineering, College of Engineering, Majmaah University, P.O. Box 66 Majmaah 11952, Saudi Arabia 5Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 6Department of Mathematics & Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan

Abstract. By using the Caratheodory approximation method, the current article presents the analysis of exact and approximate solutions for stochastic differential equations (SDEs) in the framework of G-Brownian motion. In view of the non-linear growth and non-Lipschitz conditions, the boundedness of the Caratheodory approximate solutions Y q (t), q ≥ 1 in the space 2 ([t0 , T ]; Rn ) has been determined. Estimate for the difference between the exact solution Y (t) MG and the Caratheodory approximate solutions Y q (t) has been derived.

Keywords: G-Brownian motion, non-linear growth and non-Lipschitz conditions, Caratheodory approximation procedure, bounded solutions, stochastic differential equations MSC: 60H20, 60H10, 60H35, 62L20. 1. Introduction Stochastic differential equations (SDEs) are employed by several and diverse scientific disciplines such as chemistry, statistical physics, biology and engineering. In finance and economics, they are utilized to find out the risk measures and stochastic volatility problems. SDEs describe heavy traffic behavior of communication networks and control systems [16]. Mathematics use the concept of SDEs to incorporate random fluctuations in the model when one investigates the evolution of the number of cells in an organism infected by a virus. The weather and climate can be modeled by these equations. The clarification of fluid through porous structures and water catchment can be modeled by SDEs [17]. They are used to describe the motion of wildlife [4]. SDEs play an important role to study the animal’s swarm, such as schooling of fish, flocking of birds or herding of mammals, to find resource of food in noisy and obstacle environment [30]. In physics, SDEs are used to study and model the effect of random variations on distinct physical processes. A large literature is available on the applications of SDEs in numerous fields of engineering such as computer engineering [16, 22], mechanical engineering [26, 28, 29], random vibrations [3, 24], stability theory [25] and wave processes [27]. In general, one can not find the explicit solutions for non-linear SDEs, so we have to present and study the analysis for the solutions of these equations. Moreover, the developments of computational techniques are very important for solving several demanding problems, for instance to find the optimal construction of a design and to determine input data from fundamental principles. Therefore it is valuable to know computational accuracy, which leads us to convergence results and estimates for the difference between exact and approximate solutions. 1

Corresponding author email (I. Khan): [email protected]

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The aim of the current article is to investigate estimates for the difference between exact and approximate solutions for SDEs driven by G-Brownian motion with Caratheodory approximation procedure. In view of growth and Lipschitz conditions, the existence-uniqueness results for G-SDEs was studied by Peng [20, 21] and Gao [15]. Later, Bai and Lin [1] established the existence theory for G-SDEs with integral Lipschitz coefficients. Subject to some discontinuous coefficients, the said theory was generalized by Faizullah [11]. Let 0 ≤ t0 ≤ t ≤ T < ∞. Consider the following SDE in the framework of G-Brownian motion (1.1)

dY (t) =κ(t, Y (t))dt + λ(t, Y (t))dhW, W i(t) + µ(t, Y (t))dW (t),

with initial value Y (t0 ) ∈ Rn . The given coefficients g(., x), h(., x) and w(., x) belong to space MG2 ([t0 , T ]; Rn ), for all x ∈ Rn . SDE (1.1) in the integral form is expressed as the following Z t Z t Y (t) = Y (t0 ) + κ(s, Y (s))ds + λ(s, Y (s))dhW, W i(s) t0 t0 (1.2) Z t + µ(s, Y (s))dW (s), t0

on t ∈ [t0 , T ]. Its solution is a process Y ∈ MG2 ([t0 , T ]; Rn ) and satisfying SDE (1.2). The rest of the current paper contains three sections. Building on the previous notions of G-expectation, section 2 presents the fundamental definitions and results of G-Browinian motion, sub-expectation, Grownwall’s inequality, Doobs martingale inequality, G-Itˆo’s integral and H¨ older’s inequality etc. Section 3 reveals the Caratheodory approximate solutions procedure for SDEs driven by G-Brownian motion. This section give an important result, which shows that the Caratheodory approximate solutions are bounded. Section 4 derives estimates for the difference between approximate and exact solutions to SDEs driven by G-Brownian motion. 2. Preliminaries We present some basic results and notions required for the subsequent sections of the current article. We don’t give detailed literature on basic notions of G-expectation, so readers are suggested to consult the more depth oriented papers [9, 13, 18, 20, 21]. Let Ω be a given basic non-empty set. Assume H be a space of linear real functions defined on Ω so that (i) 1 ∈ H (ii) for every n ≥ 1, Y1 , Y2 , ..., Yn ∈ H and ϕ ∈ Cb.Lip (Rn ) it satisfies ϕ(Y1 , Y2 , ..., Yn ) ∈ H i.e., subject to Lipschitz bounded functions, H is stable. Then (Ω, H, E) is a sub-expectation space, where E is a sub-expectation defined as follows. Definition 2.1. A functional E : H → R satisfying the below four features is known as a subexpectation. Let X, Y ∈ H, then (1) (2) (3) (4)

Monotonicity: E(X) ≤ E(Y ) if X ≤ Y . Constant preservation: E(M1 ) = M1 , for all M1 ∈ R. Positive homogeneity: E(N1 Y ) = N1 E(Y ), for all N1 ∈ R+ . Sub-additivity: E(X) + E(Y ) ≥ E(X + Y ). 2

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Moreover, let Ω be the space of all Rn -valued continuous paths (wt )t≥0 starting from zero. In addition, assume that subject to the below distance, Ω is a metric space ∞ X 1 ( max |w1 − wt2 | ∧ 1). ρ(w , w ) = 2i t∈[0,k] t 1

2

i=1

Fix T ≥ 0 and set L0ip (ΩT ) = {φ(Wt1 , Wt2 , ..., Wtm ) : m ≥ 1, t1 , t2 , ..., tm ∈ [0, T ], φ ∈ Cb.Lip (Rm×n ))}, 0 where W is the canonical process, L0ip (Ωt ) ⊆ L0ip (ΩT ) for t ≤ T and L0ip (Ω) = ∪∞ n=1 Lip (Ωn ). 1

The completion of L0ip (Ω) under the Banach norm E[|.|p ] p , p ≥ 1 is denoted by LpG (Ω), where LpG (Ωt ) ⊆ LpG (ΩT ) ⊆ LpG (Ω) for 0 ≤ t ≤ T < ∞. Generated by the canonical process {W (t)}t≥0 , the filtration is represented as Ft = σ{Ws , 0 ≤ s ≤ t}, F = {Ft }t≥0 . Suppose πT = {t0 , t1 , ..., tN }, 0 ≤ t0 ≤ t1 ≤ ... ≤ tN ≤ ∞ be a division of [0, T ]. For p ≥ 1, MGp,0 (0, T ) denotes a set of the processes given by (2.1)

N −1 X

ηt (w) =

ξi (w)I[ti ,ti+1 ] (t),

i=0

where ξi ∈ LpG (Ωti ), i = 0, 1, ..., N − 1. Furthermore, the completion of MGp,0 (0, T ) with the below given norm is indicated by MGp (0, T ), p ≥ 1 Z T kηk = { E[|ηs |p ]ds}1/p . 0

Definition 2.2. An n-dimensional stochastic process {W (t)}t≥0 is called a G-Brownian motion if (1) W (0) = 0. (2) For any t, m ≥ 0, Wt+m −Wt is G-normally distributed and independent from Wt1 , Wt2 , ........Wtn , for n ∈ N and 0 ≤ t1 ≤ t2 ≤, ..., ≤ tn ≤ t, Definition 2.3. Let ηt ∈ MG2,0 (0, T ) having the form (2.1). Then the G-quadratic variation process {hW it }t≥0 and G-Itˆo’s integral I(η) are respectively defined by Z t 2 hW it = Wt − 2 Ws dW (s), Z I(η) = 0

0

T

ηs dW (s) =

N −1 X

ξi (Wti+1 − Wti ).

i=0

The following two lemmas can be found in the book [19]. They are known as H¨ older’s and Gronwall’s inequalities respectively, . Lemma 2.4. Assume m, n > 1 such that Z

1 m

µZ

b

αβ ≤ a

+ b

1 n

= 1 and β ∈ L2 then αβ ∈ L1 and

m

¶ m1 µZ

b

n

|β|

|α|

¶ n1 .

a

a 3

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Lemma 2.5. Let α(t) ≥ 0 and β(t) be continuous real functions defined on [a, b]. If for all t ∈ [a, b], Z b β(t) ≤ K + α(s)β(s)ds, a

where K ≥ 0, then

Rt

β(t) ≤ Ke

a

α(s)ds

,

for all t ∈ [a, b]. The following lemma, known as Doob’s martingale inequality, is borrowed from [15]. Lemma 2.6. Assume [c, d] be a bounded interval of R+ . Consider an Rn valued G-martingale {X(t) : t ≥ 0}. Then we have E( sup |Y (t)|p ) ≤ ( c≤t≤d

p p ) E(|Y (d)|P ), p−1

where p > 1 and Y (t) ∈ LpG (Ω, Rd ). In particular, if p = 2 then E(supc≤t≤d |Y (t)|2 ) ≤ 4E(|Y (d)|2 ). 3. Caratheodory approximate solutions We now present the Caratheodory approximation procedure for equation (1.2). Let q ≥ 1 be any positive integer. For t ∈ [t0 − 1, t0 ], we set Y q (t) = Y0 and for t ∈ [t0 , T ], Z t Z t 1 1 q q λ(s, Y q (s − ))dhW, W i(s) Y (t) = Y0 + κ(s, Y (s − ))ds + q q t0 t0 (3.1) Z t 1 + µ(s, Y q (s − ))dW (s). q t0 The approximate solutions Z q (.) can be determined step-by-step on the intervals [t0 , t0 + 1q ], (t0 + 2 1 1 q , t0 + q ] and son on with the following procedure. For t ∈ [t0 , t0 + q ], we have Z Y q (t) = Y0 + Z

Z

t

t0

κ(s, Y0 )ds +

t

t0

λ(s, Y0 )dhW, W i(s)

t

+ t0

µ(s, Y0 )dW (s),

and for t ∈ (t0 + 1q , t0 + 2q ], Z t Z t 1 1 1 q κ(s, Y (s − ))ds + λ(s, Y q (s − ))dhW, W i(s) Y (t) = Y (t0 + ) + q q q t0 + 1q t0 + 1q Z t 1 µ(s, Y q (s − ))dW (s), + 1 q t0 + q

q

q

etc. All through this article, we assume two conditions, described as follows. Let M be a positive constant. For any t ∈ [t0 , T ] and κ(t, 0), λ(t, 0), µ(t, 0) ∈ L2 , (3.2)

|κ(t, 0)|2 + |λ(t, 0)|2 + |µ(t, 0)|2 ≤ M, 4

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which is weakened linear growth condition. Let t ∈ [t0 , T ]. For every u, v ∈ Rn , there exists a concave non-decreasing function Ψ(.) : R+ → R+ with Ψ(0) = 0 and for s > 0, Ψ(s) > 0 such that (3.3)

|κ(t, u) − κ(t, v)|2 + |λ(t, u) − λ(t, v)|2 + |µ(t, u) − µ(t, v)|2 ≤ Ψ(|u − v|2 ),

R ds = ∞ and for all s ≥ 0, C, D > 0, Ψ(s) ≤ C + Ds. Assumption (3.3) is a nonwhere 0+ Ψ(s) uniform Lipschitz condition. Subject to conditions (3.2) and (3.3), we assume that problem (1.1) has a unique solution Y (t) ∈ MG2 ([t0 , T ]; Rn ) [1]. Lemma 3.1. Let assumptions (3.2) and (3.2) are satisfied. For every q ≥ 1 and any T > 0, sup E(|Y q (t)|2 ) ≤ N1 ,

(3.4)

t0 ≤t≤T

where N1 = H1 eH2 (T −t0 ) , H1 = 4E|Y0 |2 + 8T (T + 2)(2M + C), H2 = 8(T + 2)D and M, C, D are arbitrary positive constants. P P Proof. In view of the inequality | 4i=1 ci |2 ≤ 7 4i=1 |ci |2 , from (3.1) we derive Z t Z t 1 1 2 q 2 2 q λ(s, Y q (s − ))dhW, W i(s)|2 |Y (t)| ≤ 4|Y0 | + 4| κ(s, Y (s − ))ds| + 4| q q t0 t0 Z t 1 µ(s, Y q (s − ))dW (s)|2 . + 4| q t0 Apply G-subexpectation on both sides. Then by virtue of the Doob’s martingale, Holder’s and BDG [5] inequalities we have Z t Z t 1 1 2 q 2 2 q E|λ(s, Y q (s − ))|2 ds E( sup |Y (s)| ) ≤ 4E(|Y0 | ) + 4T E|κ(s, Y (s − ))| ds + 4T q q t0 ≤s≤t t0 t0 Z t 1 + 16 E|µ(s, Y q (s − ))|2 ds q t0 Z t 1 ≤ 4E(|Y0 |2 ) + 8T E[|κ(s, Y q (s − )) − κ(s, 0)|2 + |κ(s, 0)|2 ]ds q t0 Z t 1 + 8T E[|λ(s, Y q (s − )) − λ(s, 0)|2 + |λ(s, 0)|2 ]ds q t0 Z t 1 + 32 E[|µ(s, Y q (s − )) − µ(s, 0)|2 + |µ(s, 0)|2 ]ds. q t0 Using (3.2) and (3.3), we derive Z E( sup |Y q (s)|2 ) ≤ 4E(|Y0 |2 ) + 16T 2 M + 32T M + 8(T + 2) t0 ≤s≤t

t

t0 t

Z

1 E[Ψ(|Y q (s − )|2 )]ds q

1 [C + DE|Y q (s − )|2 ]ds q t0 Z t ≤ 4E(|Y0 |2 ) + 16T 2 M + 32T M + 8T (T + 2)C + 8(T + 2)D E[ sup |Y q (r)|2 ]ds ≤ 4E(|Y0 |2 ) + 16T 2 M + 32T M + 8(T + 2)

t0

t0 ≤r≤s

5

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By virtue of the Grownwall’s inequality, we derive E( sup |Y q (s)|2 ) ≤ H1 eH2 (t−t0 ) , t0 ≤s≤t

where H1 = 4E|Y0 |2 + 8T (T + 2)(2M + C) and H2 = 8(T + 2)D. Consequently, supposing t = T , we obtain E( sup |Y q (s)|2 ) ≤ H1 eH2 (T −t0 ) = N1 . t0 ≤s≤T

The proof stands completed.

¤

In a similar way as lemma 3.1, we can prove the following result. Lemma 3.2. Subject to the growth condition (3.2), for any T > 0, sup E(|Y (t)|2 ) ≤ N1 ,

(3.5)

t0 ≤t≤T

where N1 is a positive constant.

4. Estimates for the difference between exact and Caratheodory approximate solutions We first give an important result. Then in view of weakened growth and non-uniform Lipschitz conditions, we derive an estimate for the difference between the approximate and exact solutions to problem (1.1). Lemma 4.1. Let 0 ≤ r < t ≤ T . Suppose that the assumptions of lemma 3.1 are satisfied. For all q≥1 E[|Z q (t) − Z q (u)|2 ] ≤ G1 (t − u),

(4.1)

where G1 = 12(T + 2)(M + C + DN1 ), M , C, D and N1 are positive constants. Proof. In view of the fundamental inequality | t ≤ T , from (3.1) we derive Z |Y q (t) − Y q (u)|2 ≤ 3| Z

t

u t

+ 3| u

P3

2 i=1 ci |

≤7

1 κ(s, Y q (s − ))ds|2 + 3| q

Z

P3

t

u

2 i=1 |ci | ,

for any q ≥ 1 and 0 ≤ r
β1 or α2 < β2 .   1   1  β2 −α2 k+1 β1 −α1 k+1 (ii) A , is unstable. γ2 γ1 Proof. (i.1) The linearized system of    0 xn      xn−1   1    .  ..   ..  .       x   0  n−k  Xn =   , J(0,0) =   0  yn         0  yn−1    .   .  .   ..  .   yn−k J( 0, 0) about (0, 0) is

(1) about (0, 0) becomes: Xn+1 = J(0,0) Xn where  0 . . . 0 αβ11 0 0 . . . 0 0  0 ... 0 0 0 0 ... 0 0   .. . . .. .. .. . . .. .. ..  . . . . . . . .  .   0 ... 1 0 0 0 ... 0 0   . The characteristic equation of 0 . . . 0 0 0 0 . . . 0 αβ22    0 ... 0 0 1 0 ... 0 0   .. . . .. .. .. . . .. .. ..  . . . . . . . .  . 

0 0 ... 0

λ

2k+2

 −

0

0

α1 α2 + β1 β2



0

... 1

λk+1 +

0

α1 α2 = 0. β1 β2

(2)

If α1 < β1 and α2 < β2 then all roots of (2) lie inside unit disk. So O(0, 0) of system (1) is locally asymptotically stable. (i.2) It is easy to show that if α1 > β1 or α2 > β2 then O(0, 0) is unstable.   1   1  β2 −α2 k+1 β1 −α1 k+1 (ii). The linearized system of (1) about A , becomes: Xn+1 = JA Xn where γ2 γ1

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JA = E(2k+2)×(2k+2)



0

0

...

0

1

γ1 x ¯y¯k α1

γ1 x ¯y¯k α1

...

γ1 x ¯y¯k α1

γ1 x ¯y¯k α1

        =       

1 .. .

0 .. .

... .. .

0 .. .

0 .. .

0 .. .

0 .. .

... .. .

0 .. .

0 .. .

0

0

...

1

0

0

0

...

0

0

γ2 y¯x ¯k α2

γ2 y¯x ¯k α2

...

γ2 y¯x ¯k α2

γ2 y¯x ¯k α2

0

0

...

0

1

0 .. .

0 .. .

... .. .

0 .. .

0 .. .

1 .. .

0 .. .

... .. .

0 .. .

0 .. .

         .       

0 0 ... 0 0 0 0 ... 1 0 Let λ1 , λ2 , . . . , λ2k+2 denote the 2k + 2 eigenvalues of matrix E. Let D = diag(d1 , d2 , . . . , d2k+2 ) be a diagonal matrix, where d1 = dk+2 = 1, di = dk+1+i = 1 − i, i = 2, 3, · · · , k + 1 for 0 <  < 1. Clearly, D is invertible. In computing DED−1 , we obtain that 

DED−1 =

0   d2 d−1 1   ..  .    0   γ2 y¯x¯k dk+2 d−1 1  α2   0   ..  .  0

0 0 .. . 0

... ... ... ...

γ2 y¯x ¯k dk+2 d−1 2 α2

... ... .. .

0 .. . 0

0 0 .. . dk+1 d−1 k γ2 y¯x ¯k dk+2 d−1 k α2

γ2 y¯x ¯k dk+2 d−1 k+1 α2

0 .. . 0

0 .. . 0

...

γ1 x ¯y¯k d1 d−1 k+2 α1

γ1 x ¯y¯k d1 d−1 k+3 α1

0 .. . 0 0 dk+3 d−1 k+2 .. . 0

0 ..

.

0 0 0 .. . 0

d1 d−1 k+1 0 .. . 0

... ... .. . ... ... ... .. . ...

γ1 x ¯y¯k d1 d−1 2k+1 α1

0 .. . 0 0 0 .. . d2k+2 d−1 2k+1

.

γ1 x ¯y¯k d1 d−1 2k+2 α1 

0 .. . 0 dk+2 d−1 2k+2 0 .. . 0

(3)

               

−1 From d1 > d2 > · · · > dk+1 > 0 and dk+2 > dk+3 > · · · > d2k+2 > 0 it implies that d2 d−1 1 < 1, d3 d2 < −1 −1 −1 1, · · · , dk+1 d−1 k < 1 and dk+3 dk+2 < 1, dk+4 dk+3 < 1, · · · , d2k+2 d2k+1 < 1. Furthermore,

γ1 x ¯y¯k d1 d−1 γ1 x ¯y¯k d1 d−1 γ1 x ¯y¯k d1 d−1 γ1 x ¯y¯k d1 d−1 k+2 k+3 2k+1 2k+2 + + ··· + + = α1 α1 α1 α1   1 γ1 x ¯y¯k 1 1 1 + 1+ + ··· + + > 1. 1 − (k + 1) α1 1 − 2 1 − k 1 − (k + 1)

d1 d−1 k+1 +

Also γ2 y¯x ¯k dk+2 d−1 γ2 y¯x ¯k dk+2 d−1 γ2 y¯x ¯k dk+2 d−1 γ2 y¯x ¯k dk+2 d−1 k+1 1 2 k + + ··· + + + dk+2 d−1 2k+2 = α2 α2 α2 α2   γ2 y¯x ¯k 1 1 1 1 1+ + ··· + + + > 1. α2 1 − 2 1 − k 1 − (k + 1) 1 − (k + 1)

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It is well-known fact that E has the same eigenvalues as DED−1 . Hence, we obtain 1 −1 −1 −1 |λm | ≤ kDED−1 k∞ = max{d2 d−1 1 , · · · , dk+1 dk , dk+3 dk+2 , · · · , d2k+2 d2k+1 , 1≤m≤2k+2 1 − (k + 1)   k 1 1 γ1 x ¯y¯ 1 1 1+ , + + ··· + + α1 1 − 2 1 − k 1 − (k + 1) 1 − (k + 1)   k γ2 y¯x ¯ 1 1 1 + 1+ } > 1. + ··· + + α2 1 − 2 1 − k 1 − (k + 1)   1   1  β1 −α1 k+1 β2 −α2 k+1 , of system (1) is unstable. This implies that A γ2 γ1 max

3

Periodicity nature and existence of unbounded solutions

In this section, we will study the periodicity nature and existence of unbounded solutions of system (1). Let us denote a1 = γ1 y−k y1−k · · · y0 , a2 = γ2 x−k x1−k · · · x0 to study the periodicity nature of positive solution of system (1). Theorem 1. If a1 = β1 − α1 and a2 = β2 − α2 , then system (1) has prime period-(k+1) solutions. Proof. From system (1) and a1 = β1 − α1 , a2 = β2 − α2 , we have x1 =

α2 y−k α1 x−k α1 x−k α2 y−k = x , y = = y−k . = = 1 −k k k β1 − a1 β 2 − a2 Y Y β 1 − γ1 β 2 − γ2 y−i x−i i=0

i=0

α1 x1−k α1 x1−k α1 x1−k α1 x1−k = = = = x1−k , k β − γ y y y · · · y β − γ y y · · · y y β Y 1 1 1 0 −1 1 1 0 −1 1 − a1 1−k 1−k −k β1 − γ1 y1−i

x2 =

i=0

y2 =

α2 y1−k α2 y1−k α2 y1−k α2 y1−k = = = y1−k . = k β2 − γ2 x1 x0 x−1 · · · x1−k β2 − γ2 x0 x−1 · · · x1−k x−k β2 − a2 Y β 2 − γ2 x1−i i=0

By induction, one has xk+2 =

α1 x1 α1 x1 α1 x1 α1 x1 = = = = x1 , k β1 − γ1 yk+1 yk yk−1 · · · y1 β1 − γ1 y0 y−1 · · · y1−k y−k β1 − a1 Y β 1 − γ1 yk+1−i i=0

yk+2 =

α 2 y1 α 2 y1 α 2 y1 α2 y1 = = = = y1 . k β2 − γ2 xk+1 xk xk−1 · · · x1 β2 − γ2 x0 x−1 · · · x1−k x−k β 2 − a2 Y β2 − γ2 xk+1−i i=0

Theorem 2. Assume that β1 < α1 , β2 < α2 . Then, every positive solution {(xn , yn )} of system (1) tends to ∞ as n → ∞.

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(c)

(b)

(a)

Figure 1: Plots for system (5) Proof. From system (1), it follows that xn+1 =

α2 yn−k α1 xn−k α2 yn−k α1 xn−k > xn−k , yn+1 = > yn−k . ≥ ≥ k k β1 β2 Y Y β 1 − γ1 yn−i β 2 − γ2 xn−i i=0

(4)

i=0

From first equation of (4), we have x(k+1)n+1 > x(k+1)n−k , and x(k+1)n+(k+2) > x(k+1)n+1 . Hence, the subsequences {x(k+1)n+1 }, · · · , {x(k+1)n+(k+1) } are increasing, i.e., the sequence {xn } is increasing. So, xn → ∞ as n → ∞. Similarly, from second equation of (4) one gets: y(k+1)n+1 > y(k+1)n−k and y(k+1)n+(k+2) > y(k+1)n+1 . Hence, the subsequences {y(k+1)n+1 }, · · · , {y(k+1)n+(k+1) } are increasing, i.e., the sequence {yn } is increasing. So, yn → ∞ as n → ∞.

4

Numerical simulations

In this section we will present numerical simulations to verify theoretical results. Example 1. If α1 = 50, β1 = 63, γ1 = 4, α2 = 90, β2 = 122, γ2 = 2 then system (1) with x−5 = 3.9, x−4 = 1.5, x−3 = 12.4, x−2 = 11.9, x−1 = 1.6, x0 = 2.9, y−5 = 2.6, y−4 = 3.8, y−3 = 5.8, y−2 = 3.5, y−1 = 3.1, y0 = 0.9 can be written as: xn+1 =

90yn−5 50xn−5 , yn+1 = . 63 − 4yn yn−1 yn−2 yn−3 yn−4 yn−5 122 − 2xn xn−1 xn−2 xn−3 xn−4 xn−5

(5)

Moreover, in Fig. 1 the plot of xn is shown in Fig. 1a, the plot of yn is shown in Fig. 1b and global attractor of system (5) is shown in Fig. 1c. Example 2. If α1 = 15.5, β1 = 17, γ1 = 27, α2 = 11.2, β2 = 12, γ2 = 23, then system (1) with x−8 = 1.9, x−7 = 1.7, x−6 = 2.5, x−5 = 0.9, x−4 = 1.5, x−3 = 10.4, x−2 = 6.9, x−1 = 0.6, x0 = 2.9, y−8 = 2.8, y−7 = 1.6, y−6 = 1.8, y−5 = 2.6, y−4 = 2.8, y−3 = 2.8, y−2 = 3.5, y−1 = 2.1, y0 = 1.6 can be written as 15.5xn−8 , 17 − 27yn yn−1 yn−2 yn−3 yn−4 yn−5 yn−6 yn−7 yn−8 11.2yn−8 = . 12 − 23xn xn−1 xn−2 xn−3 xn−4 xn−5 xn−6 xn−7 xn−8

xn+1 = yn+1

(6)

Moreover, in Fig. 2 the plot of xn is shown in Fig. 2a, the plot of yn is shown in Fig. 2b and global attractor of system (6) is shown in Fig. 2c. 812

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(b)

(a)

(c)

Figure 2: Plots for system (6)

5

Conclusion and future work

This work is related to the qualitative behavior of a system of higher-order rational difference equations. We have proved that under some restrictions  to parameters, system (1) has a boundary equilibrium O(0, 0)  1   1   k+1 β2 −α2 k+1 1 , β1γ−α in the closed first quadrant R2+ . and the unique positive equilibrium point A γ2 1 We have analyzed the local stability about equilibria, periodicity nature of positive solutions and existence of unbounded solutions of system (1). Finally, theoretical results are verified numerically. Besides the local properties, the global stability of under consideration system (1), which is our further aim to study. Acknowledgements A. Q. Khan research is supported by the Higher Education Commission(HEC) of Pakistan.

References [1] I. Bajo, E. Liz, Global behaviour of a second-order nonlinear difference equation, Journal of Difference Equations and Applications, 17(10)(2011):1471-1486. [2] M. Aloqeili, Dynamics of a rational difference equation, Applied Mathematics and Computation, 176(2)(2006):768-774. [3] E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, Chapman and Hall/CRC Press, Boca Raton, (2004). [4] H. Sedaghat, Nonlinear difference equations:theory with applications to social science models, Kluwer Academic Publishers, Dordrecht, (2003). [5] V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher-order with applications, Kluwer Academic Publishers, Dordrecht, (1993). [6] E. Camouzis, G. Ladas, Dynamics of third-order rational difference equations:with open problems and conjectures, Chapman and Hall/HRC, Boca Raton, (2007). [7] V. L. Kocic, G. Ladas, Global attractivity in a second order nonlinear difference equations, Journal of Mathematical Analysis and Applications, 180(1993):144-150.

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ON APPROXIMATING THE GENERALIZED EULER-MASCHERONI CONSTANT∗ TI-REN HUANG1 , BO-WEN HAN2 , XIAO-YAN MA2 , AND YU-MING CHU3,∗∗

Abstract. In the article, we provide several sharp bounds for the the generalized Euler-Mascheroni constant, which are the generalizations of the previously results on the Euler-Mascheroni constant.

1. Introduction It is well known that the sequence 1 1 1 (1.1) γn = 1 + + + · · · + − log n 2 3 n is convergent towards the Euler-Mascheroni constant (1.2)

γ = 0.57721566490115328 · · · .

The Euler-Mascheroni constant has been involved in a variety of mathematical formulas and results [1-6], many special functions are closely related to the EulerMascheroni constant [7-63]. Recently, the bounds for γn − γ have attracted the attention of many researchers. Alzer [64] proved that the double inequality 1 1 ≤ γn − γ ≤ 2n + 1 2n holds for n ≥ 1. In [65], T´ oth proved that the two-sided inequality 1 1 (1.3) < γn − γ ≤ 2n + 25 2n + 13 takes place for n ≥ 1. Chen [66] proved that α = (2γ − 1)/(1 − γ) and β = 1/3 are the best possible constants such that the double inequality 1 1 (1.4) ≤ γn − γ < 2n + α 2n + β holds for n ≥ 1. 2010 Mathematics Subject Classification. Primary: 11Y60; Secondary: 40A05, 33B15. Key words and phrases. Euler-Mascheroni constant, gamma function, psi function, asymptotic expansion. ∗ The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 11301127, 11701176, 11626101, 11401531, 11601485), the Science and Technology Research Program of Zhejiang Educational Committee (Grant no. Y201635325), the Natural Science Foundation of Zhejiang Province (Grant No. LQ17A010010) and the Science Foundation of Zhejiang Sci-Tech University (Grant No. 14062093-Y). ∗∗ Corresponding author: Yu-Ming Chu, Email: [email protected]. 1

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2

In [67], Qiu and Vuorinen proved that the double inequality 1 1 λ µ < γn − γ ≤ − − 2n n2 2n n2 holds for n ≥ 1 if and only if λ ≥ 1/12 and µ ≤ γ − 1/2. Let a > 0. Then the generalized Euler-Mascheroni constant γ(a) is defined by   1 1 1 a+n−1 (1.6) γ(a) = lim + + ··· + − log , n→∞ a a+1 a+n−1 a

(1.5)

which was introduced by Knopp [68]. We clearly see that γ(1) = γ. Recently, the generalized Euler-Mascheroni constant γ(a) has been the subject of intensive research [69-71]. In [70], Sˆınt˘ am˘ arian introduced the sequences (1.7)

xn =

1 1 a+n 1 + + ··· + − log , a a+1 a+n−1 a

1 1 a+n−1 1 + + ··· + − log , a a+1 a+n−1 a and proved that the double inequalities 1 1 (1.9) ≤ γ(a) − xn ≤ , 2(n + a) 2(n + a − 1) (1.8)

(1.10)

yn =

1 1 ≤ yn − γ(a) ≤ 2(n + a) 2(n + a − 1)

hold for n ≥ 1. In [71], Berinde and Mortici established Theorems 1.1 and 1.2 as follows. Theorem 1.1. The double inequalities 1 1 (1.11) < γ(a) − xn < , 2(n + a) − 41 2(n + a) − 31 (1.12)

1 2(n + a) −

4 3

< yn − γ(a)
0 and n ≥ 2. Theorem 1.2. (a) The inequality (1.13)

1 2(n + a) −

1 3

+

1 18n

≤ γ(a) − xn

holds for a ≥ 13/30 and any integer n ≥ 1. (b) The inequality (1.14)

1 2(n + a) −

5 3

+

1 18n

≤ yn − γ(a)

holds for a ≥ 17/30 and n ≥ 1. The main purpose of this article is to generalize inequalities (1.4) and (1.5) to the generalized Euler-Mascheroni constant γ(a). Our main results are the following Theorems 1.3 and 1.4.

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Theorem 1.3. Let a > 0, n ≥ 1. Then one has (1) the double inequality 1 1 ≤ γ(a) − xn < 2(n + a) − α1 2(n + a) − β1

(1.15)

holds with the best possible constants (1.16)

α1 = 2(1 + a) −

1 , ψ(1 + a) − log(1 + a)

β1 =

1 ; 3

(2) the two-sided inequality 1 1 ≤ yn − γ(a) < 2(n + a) − α2 2(n + a) − β2

(1.17)

is valid with the best possible constants α2 = 2(1 − d),

(1.18)

β2 =

5 , 3

where d = max{f˜2 (a), f˜2 (1 + a), f˜2 (2 + a)},

f˜2 (x) =

1 − x. 2(ψ(x + 1) − log(x))

Theorem 1.4. Let a > 0, n ≥ 1. Then the double inequalities (1.19)

(1.20)

1 α3 1 β3 + ≤ γ(a) − xn < + , 2(n + a) (n + a)2 2(n + a) (n + a)2 1 α4 1 β4 + < yn − γ(a) ≤ + 2(n + a − 1) (n + a − 1)2 2(n + a − 1) (n + a − 1)2

hold with the best possible constants (1.21)

α3 = (1 + a)2 [log(1 + a) − ψ(1 + a)] − α4 = −

(1.22)

1 , 12

1+a , 2

β3 =

1 , 12

a β4 = a2 [ψ(a) − log(a)] + . 2

2. Lemmas In order to prove our main results, we need the following formulas and lemmas. For x > 0, the classical gamma function Γ(x) and psi function ψ(x) [72-84] are defined as Z ∞ Γ0 (x) , Γ(x) = tx−1 e−t dt, ψ(x) = Γ(x) 0 respectively. The psi function ψ(x) has the following recurrence and asymptotic formulas [85] (2.1) ψ(n + x) =

(2.2)

1 1 1 1 1 + + ··· + + + + ψ(x), (n − 1) + x (n − 2) + x 2+x 1+x x

ψ(x) ∼ log(x) −

1 1 1 1 − + − + ··· 2x 12x2 120x4 252x6

816

(x → ∞)

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4

According to (2.1) and the definitions of xn and yn given in (1.7) and (1.8), we clearly see that xn and yn can be rewritten as n+a (2.3) xn = ψ(n + a) − ψ(a) − log , a yn = ψ(n + a) − ψ(a) − log

(2.4)

n+a−1 . a

It follows from (1.6) and (2.2) that (2.5)

γ(a) = lim yn n→∞

= lim (ψ(n + a) − log(n + a − 1) + log(a) − ψ(a)) = log(a) − ψ(a). n→∞

Therefore, (2.6)

γ(a) − xn = log(n + a) − ψ(n + a),

(2.7)

yn − γ(a) = ψ(n + a) − log(n + a − 1).

Lemma 2.1. The function (2.8)

f1 (x) =

1 − 2x log(x) − ψ(x)

is strictly decreasing from (1, ∞) onto (−1/3, 1/γ − 2). The function 1 − 2x (2.9) f2 (x) = ψ(x + 1) − log(x) is strictly decreasing from [2, ∞) onto (1/3, f2 (2)]. Proof. Differentiating f1 (x) gives 2

(log(x) − ψ(x)) f10 (x) = ψ 0 (x) −

1 − 2(log(x) − ψ(x))2 . x

It follows from the inequalities 1 1 1 1 1 ψ 0 (x) − < 2 + 3 − + , x 2x 6x 30x5 42x7 1 1 1 log(x) − ψ(x) > + − 2x 12x2 120x4 given in [86] that (2.10)

2

(log(x) − ψ(x)) f10 (x)
− x 2x 12x2 120x4 252x6 for x > 0 given in [86] that 2

2 (ψ(x + 1) − ln(x)) f20 (x) < −

F2 (x) , 3175200x12

where F2 (x) = 3217636 + 17887632(x − 2) + 39443124(x − 2)2 +47009928(x − 2)3 + 33797841(x − 2)4 + 15180480(x − 2)5 +4189500(x − 2)6 + 652680(x − 2)7 + 44100(x − 2)8 > 0 for x ≥ 2. Therefore, f2 (x) is a strictly decreasing function on [2, ∞). The limit limx→∞ f2 (x) = 1/6 follows from the asymptotic formula (2.2).  Remark 1. Qi et. al. [87] proved that the function f2 (x) defined by (2.9) is strictly decreasing on (12/5, ∞). The following Lemma 2.2 can be found in [88, 89]. Lemma 2.2. The function x 2 is strictly decreasing from (0, ∞) onto (−1/12, 0) and completely monotonic on (0, ∞). (2.12)

f3 (x) = x2 (ψ(x) − log(x)) +

3. Proof of Theorems 1.3 and 1.4 Proof of Theorem 1.3. From (2.6) we clearly see that inequality (1.15) can be rewritten as 1 −β < − 2(n + a) < −α. log(n + a) − ψ(n + a) It follows from Lemma 2.1 that the sequence 1 f1 (n + a) = − 2(n + a) log(n + a) − ψ(n + a) is strictly decreasing, which leads to the conclusion that 1 1 − = lim f1 (n) < f1 (n) ≤ f1 (1) = − 2(1 + a). 3 n→∞ log(1 + a) − ψ(1 + a) Therefore, 1 1 , β1 = α1 = 2(1 + a) − ψ(1 + a) − log(1 + a) 3 are the best possible constants such that inequality (1.15) holds. From (2.7) we clearly see that inequality (1.17) is equivalent to β 1 α − (n + a − 1) ≤ 1 − . 1− < 2 2(ψ(n + a) − log(n + a − 1)) 2

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It follows from Lemma 2.1 that the sequence f˜2 (n + a − 1) =

1 − (n + a − 1) 2(ψ(n + a) − log(n + a − 1))

is strictly decreasing for n ≥ 2, which leads to the conclusion that n o 1 = lim f˜2 (n) < f˜2 (n) ≤ max f˜2 (a), f˜2 (1 + a), f˜2 (2 + a) = d. 6 n→∞ Therefore, α2 = 2(1 − d),

(3.1)

β2 =

5 3

are the best possible constants such that inequality (1.17) holds. Proof of Theorem 1.4. From (2.6) and (2.7) we know that inequalities (1.19) and (1.20) can be rewritten as α3 ≤ (n + a)2 (log(n + a) − ψ(n + a)) −

(n + a) < β3 , 2

α4 < (n + a − 1)2 (ψ(n + a − 1) − log(n + a − 1)) +

(n + a − 1) ≤ β4 , 2

respectively. It follows from Lemma 2.2 that the sequence (n + a − 1) f˜3 (n + a − 1) = (n + a − 1)2 (ψ(n + a − 1) − log(n + a − 1)) + 2 is strictly decreasing for n ∈ N. Note that lim f3 (n) = −

n→∞

1 . 12

Therefore, α3 = (1 + a)2 [log(1 + a) − ψ(1 + a)] −

1+a , 2

β3 =

1 , 12

a 1 , β4 = (a)2 [ψ(a) − log(a)] + 12 2 are the best possible constants such that inequalities (1.19) and (1.20) hold. α4 = −

Remark 2. (1) Let a = 1. Then Theorem 1.3(2) leads to inequality (1.4) with the best possible constants α = (2γ − 1)/(1 − γ) and β = 1/3. (2) Let a = 1. Then inequality (1.20) becomes inequality (1.5) with the best possible constants α = 1/12 and β = γ − 1/2. (3) From Theorem 1.3 we know that both the upper bounds 1/[2(n + a) − 1/3] for γ(a) − xn and 1/[2(n + a) − 5/3] for yn − γ(a) given in (1.11) and (1.12) are sharp for any a > 0.

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[51] Y.-M. Chu and B.-Y. Long, Best possible inequalities between generalized logarithmic mean and classical means, Abstr. Appl. Anal., 2010, 2010, Article ID 303286, 13 pages. [52] W.-M. Qian and Y.-M. Chu, Best possible bounds for Yang mean using generalized logarithmic mean, Math. Probl. Eng., 2016, 2016, Article ID 8901258, 7 pages. [53] Y.-M. Li, B.-Y. Long and Y.-M. Chu, Sharp bounds for the Neuman-S´ andor mean in terms of generalized logarithmic mean, J. Math. Inequal., 2012, 6(4), 567–577. [54] Y.-M. Chu, M.-K. Wang and G.-D. Wang, The optimal generalized logarithmic mean bounds for Seiffert’s mean, Acta Math. Sci., 2012, 32B(4), 1619–1626. [55] Y.-M. Chu, Y.-M. Li, W.-F. Xia and X.-H. Zhang, Best possible inequalities for the harmonic mean of error function, J. Inequal. Appl., 2014, 2014, Article 525, 9 pages. [56] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, A best-possible double inequality between Seiffert and harmonic means, J. Inequal. Appl., 2011, 2011, Article 94, 7 pages. [57] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, A sharp double inequality between harmonic and identric means, Abstr. Appl. Anal., 2011, 2011, Article ID 657935, 7 pages. [58] Y.-M. Chu, Y.-F. Qiu and M.-K. Wang, Sharp power mean bounds for the combination of Seiffert and geoemtric means, Abstr. Appl. Anal., 2010, 2010, Article ID 108920, 12 pages. [59] B.-Y. Long and Y.-M. Chu, Optimal power mean bounds for the weighted geometric mean of classical means, J. Inequal. Appl., 2010, 2010, Article ID 905697, 6 pages. [60] M.-K. Wang, Z.-K. Wang and Y.-M. Chu, An double inequality between geometric and identric means, Appl. Math. Lett., 2012, 25(3), 471–475. [61] Y.-M. Chu, B.-Y. Long and B.-Y. Liu, Bounds of the Neuman-S´ andor mean using power and identric means, Abstr. Appl. Anal., 2013, 2013, Article ID 832591, 6 pages. [62] M.-K. Wang, Y.-M. Chu and Y.-F. Qiu, Some comparison inequalities for generalized Muirhead and identric means, J. Inequal. Appl., 2010, 2010, Article ID 295620, 10 pages. [63] W.-M. Qian, X.-H. Zhang and Y.-M. Chu, Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means, J. Math. Inequal., 2017, 11(1), 121–127. [64] H. Alzer, Inequalities for the gamma and polygamma functions, Abh. Math. Sem. Univ. Hamburg, 1998, 68, 363–372. [65] L. T´ oth, Problem E3432, Amer. Math. Monthly, 1991, 98(3), 264–264. [66] C.-P. Chen, Inequalities for the Euler-Mascheroni constant, Appl. Math. Lett., 2010, 23(2), 161–164. [67] S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions, with applications, Math. Comput., 2005, 74(250), 723–742. [68] K. Knopp, Theory and Applications of Infinite Series, Dover Publications, New York, 1990. [69] C. Mortici, Improved convergence towards generalized Euler-Mascheroni constant, Appl. Math. Comput., 2010, 215(9), 3443-3448. [70] A. Sˆınt˘ am˘ arian, A generalization of Euler’s constant, Numer. Algorithms, 2007, 46(2), 141– 151. [71] V. Berinde and C. Mortici, New sharp estimates of the generalized Euler-Mascheroni constant, Math. Inequal. Appl., 2013, 16(1), 279–288. [72] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On rational bounds for the gamma function, J. Inequal. Appl., 2017, 2017, Article 210, 17 pages. [73] T.-H. Zhao and Y.-M. Chu, A class of logarithmically completely monotonic functions associated with gamma function, J. Inequal. Appl., 2010, 2010, Article ID 392431, 11 pages. [74] T.-H. Zhao, Y.-M. Chu and H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011, 2011, Article ID 896483, 13 pages. [75] M. Adil Khan, A. Iqbal, M. Suleman and Y.-M. Chu, Hermite-Hadamard typ inequalities for fractional integrals via Green’s function, J. Inequal. Appl., 2018, 2018, Article 161, 15 pages. [76] M. Adil Khan, Y.-M. Chu, A. Kashuri, R. Liko and G. Ali, Conformable fractional integrals version of Hermite-Hadamard inequalities and their applications, J. Funct. Spaces, 2018, 2018, Article ID 6928130, 9 pages. [77] M. Adil Khan, Y. Khurshid, T.-S. Du and Y.-M. Chu, Generalization of Hermite-Hadamard type inequalities via conformable fractional integrals, J. Funct. Spaces, 2018, 2018, Article ID 5357463, 12 pages. [78] A. Iqbal, M. Adil Khan, S. Ullah, Y.-M. Chu and A. Kashuri, Hermite-Hadamard type inequalities pertaining conformable fractional integrals and their applications, AIP Advances, 2018, 8, Article ID 075101, 18 pages, DOI: 10.1063/1.5031954.

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TI-REN HUANG1 , BO-WEN HAN2 , XIAO-YAN MA2 , AND YU-MING CHU3,∗∗

[79] M. Adil Khan, Z. M. Al-sahwi and Y.-M. Chu, New estimations for Shannon and ZipfMandelbrot entropies, Entropy, 2018, 20, Article 608, 10 pages, DOI: 10.3390/e2008608. [80] M. Adil Khan, Y.-M. Chu, A. Kashuri and R. Liko, Hermite-Hadamard type fractional integral inequalities for MT(r;g,m,ϕ) -preinvex functions, J. Comput. Anal. Appl., 2019, 26(8), 1487–1503. [81] M. Adil Khan, Y.-M. Chu, T. U. Khan and J. Khan, Some new inequalities of HermiteHadamard type for s-convex functions with applications, Open Math., 2017, 15, 1414–1430. [82] Y.-M. Chu, M. Adil khan, T. Ali and S. S. Dragomir, Inequalities for α-fractional differentiable functions, J. Inequal. Appl, 2017, 2017, Article 93, 12 pages. [83] Y.-Q. Song, M. Adil Khan, S. Zaheer Ullah and Y.-M. Chu, Integral inequalities involving strongly convex functions, J. Funct. Spaces, 2018, 2018, Article ID 6595921, 8 pages. [84] M. Adil Khan, S. Begum, Y. Khurshid and Y.-M. Chu, Ostrowski type inequalities involving conformable fractional integrals, J. Inequal. Appl., 2018, 2018, Article 70, 14 pages. [85] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U. S. Government Printing Office, Washington, 1964. [86] C.-P. Chen and F. Qi, The best lower and upper bounds of harmonic sequence, RGMIA Res. Rep. Coll., 2003, 6(2), Article 14, 5 pages. [87] F. Qi, R.-Q. Cui, C.-P. Chen and B.-N. Guo, Some completely monotonic functions involving polygamma functions and an applications, J. Math. Anal. Appl., 2005, 310(1), 303–308. [88] S.-L. Qiu and M. Vuorinen, Some properties of the gamma and psi functions, with applications, Math. Comput., 2005, 74(250), 723–742. [89] C.-P. Chen, F. Qi and H. M. Srivastava, Some properties of functions related to the gamma and psi functions, Integral Transforms Spec. Funct., 2010, 21(1-2), 153–166. Ti-Ren Huang, College of Science, Hunan City University, Yiyang 413000, China E-mail address: [email protected] Bo-Wen Han, Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China E-mail address: [email protected] Xiao-Yan Ma, Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China E-mail address: [email protected] Yu-Ming Chu (Corresponding author), Department of Mathematics, Huzhou University, Huzhou 313000, China E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.5, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

General study on Volterra integral equations of the second kind in space with weight function M. E. Nasr 1 2

1,2

and M. F. Jabbar1

Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

Department of Mathematics, Collage of Science and Arts -Al Qurayyat, Jouf University, Kingdom of Saudi Arabia [email protected] ,

and [email protected]

Abstract This paper is devoted to present a new and simple algorithm to prove that the function ϕn (x) is a good approximation to the solution ϕ(x) for Volterra integral equations (VIEs) of the second kind in the space L2p(x) [0, 2π] with weight function p(x). This approximation is discussed in details with help of the Valle´e-Poussin’s and F`ejer’s, operators. Special attention is given to study the convergence analysis and estimation of an upper bound for the error of the approximated solution. Key-Words: Volterra integral equations; Valle´e-Poussin’s and F`ejer’s operators; Convergence analysis; 1. Introduction In this paper, we present the approximate solution for Volterra integral equations (VIEs) of the second kind in the space L2p(x) [0, 2π] with weight function p(x) ≥ 1 where p(x) is a summable function on [0, 2π] Z ϕ(x) = f (x) + λ

x

0 ≤ x, y ≤ 2π,

k(x, y)ϕ(y)dy,

(1)

0

where the functions f (x), k(x, y) belong to L2p(x) [0, 2π] and are 2π-periodic functions,

1 λ

is a regular

value of the kernel k(x, y) and the kernel k(x, y) satisfies the following conditions Rx 1 1. { 0 p(y)|k(x, y)|2 dy} 2 = χ(x) ∈ L2p(x) [0, 2π]; 2. |λ|kk(x, y)kL2p < 1, where Z kk(x, y)kL2p = kk(x, y)kL2

p(x)

[0,2π]

2π

Z

= 0

x

 21 p(x)p(y)|k(x, y)|2 dydx .

0

The simplicity of finding a solution for Fredholm integral equations (FIEs) of the second kind with degenerate kernel naturally leads one to think of replacing the given equation (1) by FIE with degenerate kernel, see [1, 2, 8, 9]. The solution of the new equation is taken as an approximate solution of the original equation. The study employs Dzyadyk’s method which is based on the linear polynomial operator ([3]-[5]). Eq.(1) can be written in the new form Z ϕ(x) = f (x) + λ

x

˜ y)ϕ(y)dy, k(x,

(2)

0

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where ( ˜ y) = e(x, y)k(x, y), k(x,

e(x, y) =

1,

for y ≤ x,

0,

for y > x.

(3)

˜ y) in (2) satisfies the following conditions (A∗ ) From (3), it is found that the kernel k(x, R 2π ˜ y)|2 dy} 21 = ρ(x) ∈ L2 [0, 2π]; 1. { 0 p(y)|k(x, p(x) ˜ y)kL2 < 1, 2. |λ|kk(x, p where ˜ y)kL2 [0,2π] = kk(x, p

Z

2π

0

2π

Z

 21 ˜ y)| dydx . p(x)p(y)|k(x, 2

0

Now, instead of Eqs.(1) and (2), let us solve the following equations Z 2π ˜ y); x]ϕn (y)dy, ϕn (x) = Un (f ; x) + λ Un [k(.,

0 ≤ x, y ≤ 2π,

(4)

0

˜ y); x] will mean that the operator Un acts on k(x, ˜ y) as a function of x and at the The notation Un [k(., same time, the variable y plays the role of the parameter. ˜ y); x] are both trigonometric polynomials of order n Now, since the functions Un (f ; x) and Un [k(., with respect to x, the solution ϕn (x) of the Eq.(4) will also be trigonometric polynomial of order n in x. It is well known that the problem of determination of the solution of Fredholm integral equation of the second kind with degenerate kernel is reduced to the solution of corresponding system of algebraic equations [11]. In this study, it will be proved that the function ϕn (x) is a good approximation to the solution ϕ(x) of Eq.(1) on the space L2p(x) [0, 2π]. This approximation is discussed in details for Valle´e-Poussin’s and F`ejer’s operators. 2. Preliminaries Starting from the known linear polynomial operators Un (g; x) which are good approximation to the function g(x) in the space L2p(x) , and have the form: Un (g; x) =

1 π

2π

Z

g(t)Un (x − t)dt = 0

where

1 π

Z

2π

g(x − t)Un (t)dt,

(5)

0

n

Un (x) =

1 X (n) + λk cos(kx), 2

(6)

k=1

(n) λk

are constants which define the method of approximation.

Theorem 1. [6] ˜ y)kL2 < 1, and f (x) belongs to L2 , then the For k(x, y) belongs to L2p [0, 2π], such that |λ|kk(x, p(x) p integral equation Z ϕ(x) = f (x) + λ

2π

k(x, y)ϕ(y)dy, 0

has an unique solution ϕ(x) in L2p(x) [0, 2π]. 825

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Now, with the help of the following theorem we will find the condition by which the equation (4) has an unique solution. Theorem 2. [6] If A and B are two bounded linear operators in Banach space E, while A has an inverse and kBkE kA−1 kE < 1, then the operator (A + B) has also an inverse and k(A + B)−1 kE ≤ kA−1 kE (1 − kBkE kA−1 kE )−1 . To find this condition, we write both of Eqs.(2) and (4) in the operator form ˜ = f, (I − λK)ϕ where ˜ = Kϕ

Z

π

˜ y)ϕ(y)dy, k(x,

˜ (I − λUn (K))ϕ n = fn ,

˜ n= Un (K)ϕ

Z

π

˜ y); x]ϕn (y)dy. Un [k(.,

−π

−π

˜ = A, It is obvious that I − λK

˜ − Un (K)) ˜ = B, are two bounded linear operators in the space λ(K

L2p(x) . ˜ has an inverse for each λ such that It is well-known that the operator I − λK

1 λ

˜ is a regular value of K

[6]. So Eq.(2) has an unique solution and we can write ϕ = (I + λR)f = f + λRf, ˜ −1 = (I + λR) and R is the resolvent of the operator K. ˜ From theorem 2 if |λ|k(I − where (I − λK) ˜ −1 kE kK ˜ − Un (K)k ˜ E < 1, then (I − λUn (K)) ˜ has also an inverse, thereby Eq.(4) has an unique λK) solution and can be written in the form ϕn = (I + λRn )fn = fn + λRn fn , ˜ −1 = I + λRn and Rn is the resolvent of the operator Un (K). ˜ where (I − λUn (K)) Now, we return to the functional representation of resolvents R(x, y; λ); Rn (x, y; λ) and equations (2) and (4). Knowing the resolvent R(x, y; λ), we at once obtain the solution of the original equation (2) with an arbitrary right hand side f (x) in the following form 2π

Z ϕ(x) = f (x) + λ

R(x, y; λ)f (y)dy. 0

Also, the solution of Eq.(4) can be represented through the resolvent as follows 2π

Z ϕn (x) = fn (x) + λ

Rn (x, y; λ)fn (y)dy. 0

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Theorem 3. For any kernel k(x, y) ∈ L2p [0, 2π], if the linear polynomial operator Un of order n is defined in L2p(x) and if the function f (x) ∈ L2p(x) , then Z

b



Z

b

Un [k(., y); x]f (y)dy.

k(., y)f (y)dy; x =

Un

a

a

The proof of this theorem is very similar to the proof of a theorem in [4]. 3. Auxiliary definitions and theorems Definition 1. The averaged-modulus of continuity of the kernel k(x, y) ∈ L2p [0, 2π] is defined as follows 1 wL2p (k; t) = wL2p (t) = sup 2π |s|≤t

2π

Z

 21

x−s

Z

2

p(x)p(y)[k(x − s, y) − k(x, y)] dxdy 0

.

(7)

0

Lemma 1. The function wL2p (t) has the following properties: 1. wL2p (t) → 0 for t → 0; 2. wL2p (t) is positive and monotonic increasing; 3. wL2p (t1 +2 ) ≤ wL2p (t1 ) + wL2p (t2 ); 4. wL2p (t) is continuous; 5. for any positive real number η, the following inequality holds wL2p (ηt) ≤ (1 + η)wL2p (t). ˜ y) = e(x, y)k(x, y) Also, by the averaged-modulus of continuity with respect to x and y of a function k(x, defined in [0, 2π], we mean the following function ΩL2p (t) 1 ΩL2p (k; t) = ΩL2p (t) = sup 2π |s|≤t

Z

2π

2π

Z

2

p(x)p(y)[k(x, y)[e(x − s, y) − e(x, y)]] dxdy 0

 12 .

(8)

0

It is evident that the function ΩL2p (t) satisfies the above properties of the modulus of continuity (1-5). Definition 2. The value of the following norm ˜ = δ(k; ˜ Un ) = kUn (k(., ˜ y); x) − k(x, ˜ y)kL2 = δn (k) p

Z 0

2π

Z

2π

˜ y); x) − k(x, ˜ y)]2 dxdy p(x)p(y)[Un (k(.,

 12 ,

0

(9) will play an important role in estimating the error arising from replacement of Eq.(1) by Eq.(4). ˜ Un ). The following theorem provides an estimate of δ(k, 827

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Theorem 4. ˜ y) ∈ L2 [0, 2π], and for any linear polynomial operator Un (g; x), we always have For any kernel k(x, p the following inequality  Z π 1 1 ˜ δn (k) ≤ 2 wL2p ( ) + ΩL2p ( ) [n|t| + 1]|Un (t)|dt. n n −π

(10)

Proof. Using Minkowski inequality and equalities (5) and (7), we obtain ˜ = kUn (k(., ˜ y); x) − k(x, ˜ y)kL2 = δn (k) p

Z π

1 ˜ − t, y) − k(x, ˜ y)]Un (t)dt = [k(x

π −π

1 = π

"Z

1 ≤ π

Z

1 ≤ π

Z

1 ≤ π

Z

π

L2p

Z

π

Z p(x)p(y)

−π

−π

−π

π

2π

Z

Z

2π

0

π

0 2π

Z

Z

2π

0

π

0 2π

Z

Z

2π

0 2π

+ 0

Z

0

 12 # p(x)p(y)e(x − t, y)[k(x − t, y) − k(x, y)]2 dydx dt

2π

Z 0

π

2π

Z

Z

2π

|Un (t)| −π

0

"Z

2π Z

+ 0

Z

 12 p(x)p(y)k(x, y)[e(x − t, y) − e(x, y)] dydx 2

|Un (t)| −π

 12 p(x)p(y)[e(x − t, y)k(x − t, y) − e(x, y)k(x, y)] dydx dt 2

|Un (t)| −π

 21 ˜ − t, y) − k(x, ˜ y)] dydx dt p(x)p(y)[k(x 2

|Un (t)| −π

"Z

1 ≤ π

#1 2 2 ˜ − t, y) − k(x, ˜ y)) dydx Un (t)(k(x

π

2π

 12 p(x)p(y)k(x, y)[e(x − t, y) − e(x, y)] dydx 2

0

 12 # p(x)p(y)e(x − t, y)[k(x − t, y) − k(x, y)] dydx dt 2

0

π

≤2

|Un |[wL2p (t) + ΩL2p (t)]dt ≤ −π Z π 1 1 ≤ 2 wL2p ( ) + ΩL2p ( ) [n|t| + 1]|Un (t)|dt. n n −π Definition 3. ˜ y) as follows We define the error of approximation of k(x, ∗ ˜ L2 = kk(x, ˜ y) − T ∗ (x, y)kL2 En,m (k) n,m p p Z π Z π 1 2 2 ˜ = inf p(x)p(y)[k(x, y) − Tn,m (x, y)] dxdy , Tn,m (x,y)

−π

−π

∗ ˜ L2 = kk(x, ˜ y) − T ∗ (x, y)kL2 En,∞ (k) n,∞ p p Z π Z π 1 2 2 ˜ = inf p(x)p(y)[k(x, y) − Tn,∞ (x, y)] dxdy , Tn,∞ (x,y)

−π

−π

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∗ ˜ L2 = kk(x, ˜ y) − T ∗ (x, y)kL2 E∞,m (k) ∞,m p p Z π Z π 1 2 2 ˜ = inf p(x)p(y)[k(x, y) − T∞,m (x, y)] dxdy , T∞,m (x,y)

−π

−π

∗ (x, y) denotes the trigonometric polynomial in x of order n and in y of order m of best where Tn,m ˜ y) in the metric L2 [0, 2π], T ∗ (x, y) denotes the trigonometric polynomial in x approximation of k(x, p

n,∞

˜ y) in the metric L2 [0, 2π], T ∗ (x, y) denotes the trigonometric of order n of best approximation of k(x, p ∞,m ˜ polynomial in y of order m of best approximation of k(x, y) in the metric L2 [0, 2π]. The estimates of p

∗ (k) ˜ L2 , E ∗ (k) ˜ L2 and E ∗ (k) ˜ L2 tend to zero as n → ∞, m → ∞ are how rapidly the quantities En,m n,∞ ∞,m p p p

given in [10], where ∗ ˜ L2 → 0, En,m (k) p ∗ ˜ L2 ≥ E ∗ (k) ˜ L2 , En,m (k) n,∞ p p

n, m → ∞, ∗ ˜ L2 ≥ E ∗ (k) ˜ L2 En,m (k) ∞,m p p

then ∗ ˜ L2 → 0, En,∞ (k) p

as

n → ∞,

(11)

∗ ˜ L2 → 0, E∞,m (k) p

as

m → ∞.

(12)

Now, we will mention the bounds of the norm (9) for various linear polynomial operators Un as the following cases: Case 1: Valle` e-Poussin’s method [5]: Un = Vn , we have 1 π

√ 1 2 3 |Vn (t)|dt ≤ + , 3 π −π

Z

π

(13)

from Eq.(10) and definition 3, we get ∗ ˜ L2 En,∞ (k) p

  1 1 ≤ 12π wL2p ( ) + ΩL2p ( ) . n n

(14)

By using the inequality (13) and considering that the method of Valle`e-Poussin’s Vn leaves trigonometric polynomial of order n invariant, then ˜ Vn ) = kk(x, ˜ y) − Vn (k(., ˜ y); x)kL2 δn (k; p ˜ y) − T ∗ (x, y) − Vn [k(., ˜ y) − T ∗ (., y); x]kL2 = kk(x, n,∞ n,∞ p Z π Z πZ π h i1 ∗ ˜ − t, y) − T ∗ (x − t, y)]2 dydx 2 dt (15) ˜ L2 + 1 ≤ En,∞ (k) |V (t)| p(x)p(y)[ k(x n n,∞ p π −π −π −π   Z π 1 ∗ ˜ L2 ' 2.436E ∗ (k) ˜ L2 , ≤ 1+ |Vn (t)|dt En,∞ (k) n,∞ p p π −π and from (14) we get ˜ Vn ) ≤ 29.232π[wL2 ( 1 ) + ΩL2 ( 1 )]. δn (k; p p n n 829

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Case 2: F´ ejer’s method [5]: Un = Fn , we have

1 π Z

Z

π

|Fn (t)|dt = 1,

π

(1 + n|t|)|Fn (t)|dt < 6(1 + ln n), We let n0 =

√

(17)

−π

∀n ≥ 3.

(18)

−π

n ∗ 2 , ai (y), bi (y), ai (y)

and b∗i (y) denote the corresponding coefficients of Fourier series in ˜ y) and Vn0 [k(., ˜ y); x]. Then, the variable x of the functions k(x, ˜ y); x) − Fn [Vn0 (k(., ˜ y); x)]kL2 kVn0 (k(., p

2n0

X i

∗ ∗ [ai (y)cos ix + bi (y)sin ix] =

2

n i=1 Lp

" 0   # 12 " 0 # 21 2n 2n 2

X

X i

[a∗i (y)cos ix + b∗i (y)sin ix]2 ≤

n i=1

i=1

2 Lp

1 1 " 2n0   # 2 " 2n0 #2

X i 2

X ∗2

∗2 ≤ [ai (y) + bi (y)]

n i=1

i=1

2 Lp

" 2n0 # 12 X 1 ˜ y)kL2 ≤ √1 (2n0 ) 32 kk(x, ˜ y)kL2 i2 kk(x, ≤√ p p πn πn i=1

≤√

1 1

π n4

˜ y)kL2 . kk(x, p

Thereby ˜ Fn ) = kk(x, ˜ y) − Fn (k(., ˜ y); x)kL2 δ(k; p ˜ y) − Vn0 (k(., ˜ y); x) + Vn0 (k(., ˜ y); x) − Fn (Vn0 (k(., ˜ y); x)) + Fn (Vn0 − k); ˜ x)kL2 = kk(x, p ˜ y) − Vn0 (k(., ˜ y); x)kL2 + kFn (Vn0 − k); ˜ x)kL2 + kVn0 (k(., ˜ y); x) − Fn (Vn0 (k(., ˜ y); x))kL2 ≤ kk(x, p p p   Z π 1 1 ˜ L2 + √ ˜ ≤ 1+ |Fn (t)|dt (2.5)En∗0 ,∞ (k) , 1 kk(x, y)kL2 p p π −π 4 πn (19) from Eqs.(17) and (19), we get ˜ Fn ) ≤ 5E ∗0 (k) ˜ L2 + √ 1 kk(x, ˜ y)kL2 . δ(k; 1 n ,∞ p p π n4

(20)

Also, from Eqs.(18) and (10), we have ˜ Fn ) ≤ 12(1 + ln n)[wL2 ( 1 ) + ΩL2 ( 1 )]. δ(k; (21) p p n n ˜ → 0 as n → ∞ for Valle´e-Poussin’s and F´ejer’s Now from (16), (20) and (21) it is clear that δn (k) ˜ y) ∈ L2 [0, 2π], wL2 ( 1 ) = o(1/ln n) and ΩL2 ( 1 ) = o(1/ln n). methods for every periodic function k(x, p

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7

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p

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Definition 4. The following quantities will play an important role in estimating the error of our approximation

Z 2π

˜ ˜ (22) ξ(k; Un ; ϕ) = ξn = k(x, y)[ϕ(y) − Un (ϕ; y)]dy

, 0

L2p

γm = γm (Un ; ϕ) =

m X

(n)

|1 − λi |Ei−1 (ϕ)L2p ,

(23)

i=1

where En (ϕ)L2p = inf kϕ(x) − Tn (x)kL2p , Tn

Tn (x) is a trigonometric polynomial of order n in x, m ≤ n. Theorem 5. ˜ y) ∈ L2 [0, 2π] and for linear polynomial operator Un (g; x) the following inequality For any kernel k(x, p holds

Z

˜ ˜ ξn (k) = ξn (k; Un ; ϕ) =

2π

0

˜ k(x, y)[ϕ(y) − Un (ϕ; y)]dy

L2p

r ≤

∗ ˜ L2 kϕ(y) E∞,m (k) p

− Un (ϕ; y)kL2p +

2 γm (Un ; ϕ) π

Z

2π

0

 12 h i ˜ y)kL2 + E ∗ (k) ˜ L2 , p(x)dx kk(x, ∞,m p p (24)

for any positive integer m ≤ n. Proof. For any function ϕ(x) ∈ L2p with Fourier coefficients ci and di in view of Bunyakovskii inequality and p(x) ≥ 1, we obtain Z 1 π [ϕ(t) − T (t)]cos(i(x − t))dt |ci cos ix + di sin ix| = inf i−1 Ti−1 (t) π −π Z π  1 Z π  12 2 1 [cos(i(x − t))]2 2 ≤ inf p(t)[ϕ(t) − Ti−1 (t)dt] dt . π Ti−1 (t) −π p(t) −π r Z π 1 2 2 2 ≤ inf p(t)|ϕ(t) − Ti−1 (t)| dt π Ti−1 (t) −π r 2 ∗ ≤ E (ϕ) 2 , π i−1 Lp therefore r kci cos ix + di sin ixkL2p ≤ Letting T∞,m (x, y) =

m X

2 ∗ E (ϕ) 2 π i−1 Lp

Z

1

π

2

p(x)dx

.

−π

ai (x)cos iy + bi (x)sin iy,

i=0

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∗ ˜ L2 E∞,m (k) p

m

X

˜

= inf k(x, y) − ai (x)cos iy + bi (x)sin iy

ai ,bi i=0

,

L2p

and taking into consideration (23) and using Bunyakovskii inequality, we obtain

Z 2π

˜ ˜

ξn = ξn (k; Un ; ϕ) = k(x, y)[ϕ(y) − Un (ϕ; y)]dy

0

"Z

2π

=

p(x) 0

≤

2π

Z

hZ

Z + 0 "Z

˜ y)[ϕ(y) − Un (ϕ; y)]dy k(x,

2

2

dx

0 2π

2π

hZ

˜ y) − T∞,m (x, y)||ϕ(y) − Un (ϕ; y)|dy |k(x, i 2 i 12 ˜ ˜ (k(x, y) + T∞,m (x, y) − k(x, y))(ϕ(y) − Un (ϕ; y))dy dx p(x)

0 2π

inf

T∞,m (x,y)

2π

≤

0

2π

Z p(x)

inf T∞,m (x,y)

0

hZ

L2p

#1

2π

˜ y) − T∞,m (x, y)||ϕ(y) − Un (ϕ; y)|dy |k(x,

#1

2

2

dx

0

hZ

2π

˜ y) + T∞,m (x, y) − k(x, ˜ y)). (k(x, ! m i2 i 1 X 2 (n) (1 − λi )(ci cos iy + di sin iy) dy dx

+

0

p(x)

inf

T∞,m (x,y)

0

i=1 ∗ ˜ L2 kϕ(y) − Un (ϕ; y)kL2 ≤ E∞,m (k) p p r Z 2π  21 h i 2 ˜ y)kL2 + E ∗ (k) ˜ L2 . + kk(x, γm (Un ; ϕ) p(x)dx ∞,m p p π 0

4. The approximate solution and its error bounds The following theorem shows that for sufficiently good linear methods Un (g; x), the difference between the polynomials ϕn (x) and the original solution ϕ(x) is sufficiently small. Theorem 6. ˜ y) in Eq.(2) satisfies the assumptions (A∗ ), all functions appearing in (2) are If the kernel k(x, ˜ Un ) < 1 and if Eq.(1) 2π−periodic in x and y, then any linear polynomial operator Un (g; x), if |λ|Rδ(k; is replaced by Eq.(4), the following inequality holds

in which

˜ kϕ(x) − ϕn (x)kL2p ≤ (1 + αn (k))kϕ(x) − Un (ϕ; x)kL2p ,

(25)

# ˜ Un ; ϕ) ξ( k; ˜ Un )], ˜ = |λ|R δ(k; ˜ Un ) + /[1 − |λ|Rδ(k; αn (k) kϕ(x) − Un (ϕ; x)kL2p

(26)

"

˜ Un ) and ξ(k; ˜ Un ; ϕ) are defined in (9) and (22), respectively, and R = 1 + |λ|kR(x, y)kL2 , where δ(k; p ˜ y). where R(x, y) denotes the resolvent of the kernel k(x, 832

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Proof. Using theorem 3, and Eq.(2), we represent the solution ϕn (x) of Eq.(4) in the form Z 2π  ˜ ϕn (x) = Un (f ; x) + λUn k(., y)ϕn (y)dy; x 0 Z 2π  Z 2π ˜ y)[ϕn (y) − ϕ(y)]dy + ˜ y)ϕn (y)dy; x = Un (f ; x) + λUn k(., k(., 0 0   Z 2π Z 2π ˜ y); x][ϕn (y) − ϕ(y)]dy + Un f (.) + λ ˜ y)ϕ(y)dy; x =λ Un [k(., k(., 0 0 Z 2π ˜ y); x][ϕn (y) − ϕ(y)]dy + Un (ϕ; x), =λ Un [k(.,

(27)

0

it follows that Z ϕn (x) − Un (ϕ; x) = λ

2π

˜ y)[ϕn (y) − Un (ϕ; y)]dy + gn (x), k(x,

(28)

0

where Z gn (x) =λ

2π

˜ y); x) − k(x, ˜ y)][ϕn (y) − ϕ(y)]dy + λ [Un (k(.,

Z

π

˜ y)[Un (ϕ; y) − ϕ(y)]dy. k(x,

−π

0

Thus, by Eqs.(9), (10) and (22) we get the estimate

Z 2π

˜ ˜

kgn (x)kL2p ≤ |λ| [Un (k(., y); x) − k(x, y)][ϕn (y) − ϕ(y)]dy

0

L2p

2π

˜ k(x, y)[Un (ϕ; y) − ϕ(y)]dy

2 0 Lp h i ˜ Un ) kϕn (x) − Un (ϕ; x)kL2 + kUn (ϕ; x) − ϕ(x)kL2 + |λ|ξ(k; ˜ Un ; ϕ). ≤ |λ|δ(k; p p

Z

+ |λ|

(29)

˜ y)kL2 < 1, Eq.(28) has an unique solution given by In view of |λ|kk(x, p Z 2π ϕn (x) − Un (ϕ; x) = gn (x) + λ R(x, y)gn (y)dy. 0

Therefore h i kϕn (x)−Un (ϕ; x)kL2p ≤ kgn (x)kL2p 1 + |λ|kR(x, y)kL2p | = Rkgn (x)kL2p h i ˜ Un )[kϕ(x) − Un (ϕ; x)kL2 + kϕn (x) − Un (ϕ; x)kL2 ] + ξ(k; ˜ Un ; ϕ) . ≤ R|λ| δ(k; p p ˜ Un ) < 1, we obtain Taking into consideration |λ|Rδ(k; kϕn (x) − Un (ϕ; x)kL2p ≤

˜ Un )kUn (ϕ; x) − ϕ(x)kL2 + ξ(k; ˜ Un ; ϕ)] |λ|R[δ(k; p ˜ Un ) 1 − |λ|Rδ(k;

Therefore kϕ(x) − ϕn (x)kL2p ≤ kϕ(x) − Un (ϕ; x)kL2p + kϕn (x) − Un (ϕ; x)kL2p ≤ kϕ(x) − Un (ϕ; x)kL2p +

˜ Un )kUn (ϕ; x) − ϕ(x)kL2 + ξ(k; ˜ Un ; ϕ)] |λ|R[δ(k; p ˜ Un ) 1 − |λ|Rδ(k;

˜ ≤ (1 + αn (k))kϕ(x) − Un (ϕ; x)kL2p , where αn is given by (26). Thus, the inequality (25) is proved. 833

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5. The results It is well-known that in [7], one cannot achieve an error less than the corresponding to the best ˜ means that, the rate of approximation. The error estimate in (25) with rate of convergence αn (k), convergence of ϕn (x) to ϕ(x) is comparable with the rate of convergence of the best approximate, which means that the error estimate (25) is optimal. Applying theorem 6, and also the corresponding results from section 3, we obtain the following results: In the case of the application of Valle´ e-Poussin’s method: From [10] and (25) we obtain kϕ(x) − ϕn (x)kL2p

√ 4 2 3 ∗ ∗ ˜ ˜ )En (ϕ)L2p ≤ (1 + αn (k))(2.5)E ≤ (1 + αn (k))( + n (ϕ)L2p , 3 π

where by (15) we have ˜ ≤|λ|R αn (k)

∗ (k) ˜ L2 + E ∗ (k) ˜ L2 2.5En,∞ ∞,m p p , ∗ ˜ 1 − λR(2.5)E (k)L2 n,∞

p

˜ → 0 as n → ∞ for all ϕ(x) ∈ L2 , k(x, ˜ y) ∈ L2 [0, 2π]. then αn (k) p(x) p(x) In the case of the application of F´ ejer’s method: ˜ in the relation (25) will not tend to zero for any solution ϕ(x), but will tend to zero The quantity αn (k) only under the condition that ”the solution ϕ(x) belongs to some subclasses of integrable functions”. Restricting ourselves to the Holder classes W (r) H β (L2p ) where r is a non-negative integer and 0 < β ≤ 1, we obtain the following case: ˜ → 0 as n → ∞ considering (20), (21) and [10], it is sufficient that the following In order that αn (k) conditioned be satisfied ϕ(x) ∈ W (0) H β (L2p ),

i.e. r = 0,

0 < β ≤ 1,

1 w( )L2p = o(1/ln n), n

1 Ω( )L2p = o(1/ln n). n

6. Conclusion and remarks In this article, we presented the approximate solutions of the Volterra integral equations of the second kind in the space L2p(x) [0, 2π] with weight function p(x) with the help of the Valle´e-Poussin’s and F`ejer’s operators. In the same time, we proved that the function ϕn (x) is a good approximation to the exact solution ϕ(x) for the Volterra integral equations. From the obtained approximate solutions using ADM, we can conclude that the proposed approach is easy to implement and computationally very attractive. A good agreement between the theoretical study with the obtained approximate solutions have been obtained.

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References [1] M. A. Abdou, M. E. Nasr and M. A. Abdel-Aty, Study of the Normality and Continuity for the Mixed Integral Equations with Phase-Lag Term, Inter. J. of Math. Analysis, 11 (2017), 787–799. https://doi.org/10.12988/ijma.2017.7798 [2] M. A. Abdou, M. E. Nasr and M. A. Abdel-Aty, A study of normality and continuity for mixed integral equations, J. of Fixed Point Theory Appl., 20(1) (2018). https://doi.org/10.1007/s11784018-0490-0 [3] V. K. Dzyadyk, V. T. Gavrilyuk and O. I. Stepanets, On the best approximations of Holder class functions by Rogozinski polynomials, Dokl. Akad. Nauk Ukr. SSR. Ser. A, 3, 1969. [4] V. K. Dzyadyk, On the approximations of linear methods to the approximations by polynomials of functions which are solution of Fredholm integral equation of the second kind I and II, Urain. Matem. Zh. 22, 1970. [5] V. K. Dzyadyk, Approximation Methods for Solutions of Differential and Integral Equations, The Netherlands, VSP, 1995. [6] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover Pubns, 1999. [7] P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan (India), 1960 (Translated from the Russian). [8] M. E. Nasr and M. F. Jabbar, An Approximate Solution for Volterra Integral Equations of the Second Kind in Space with Weight Function, Inter. J. of Math. Analysis, 11 (2017), 849–861. [9] M. E. Nasr and M. A. Abdel-Aty, Analytical discussion for the mixed integral equations, J. of Fixed Point Theory Appl., 20(3) (2018). https://doi.org/10.1007/s11784-018-0589-3 [10] A. F. Timan, Theory of Approximations of Functions of a Real Variable, Dover, New York, 1994. [11] F. G. Tricomi, Integral Equations, Dover, New York, 1985.

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A Modified SSDP Method for Nonlinear Semidefinie Programming∗ Jianling Li†

Chunting Lu

Hui Zhang

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, China

Abstract In this paper, we investigate nonlinear semidefinite programming and propose a modified sequential semidefinite programming (SSDP for short) algorithm without a penalty function or a filter. At each iteration, the search direction is yielded by solving a linear semidefinite programming subproblem and a quadratic semidefinite programming subproblem. The nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. Under some appropriate conditions, the global convergence of the proposed algorithm is shown. Some preliminary numerical results are reported. Key words nonlinear semidefinite programming; sequential semidefinite programming; nonmonotone line search; global convergence

1

Introduction Consider the following nonlinear semidefinite programming (NLSDP): min f (x) s.t. G(x)  0,

(1.1)

where f : Rn → R is assumed to be a smooth and real value function, G : Rn → Sm is a smooth and matrix value function. Sm represents the set of all real symmetric matrices. The symbol A  B means that A − B is a negative semidefinite matrix. Nonlinear semidefinite programming has many real-world applications, such as engineering design, optimal structure design, optimal robust control and robust feedback control design (see [1]-[4]). In recent years, the investigation of NLSDP has attracted much attention. The main solution methods for NLSDP are augmented Lagrange method [5]-[10], interior point method [11]-[15], SSDP method [16]-[21]. In this paper, our focus is on SSDP method. Correa and Ramirez in [16] proposed an SSDP algorithm. At each iteration, the search direction is generated by solving a traditional quadratic semidefinite programming (QSDP for short) subproblem. A subdifferentiable penalty function is used as a merit function to design line search. Under some conditions, the algorithm is globally convergent. However, it is not easy for the choice of an appropriate penalty parameter. Gomez in [17] proposed a filter-type SSDP algorithm for nonlinear semidefinite programming problem. For each iteration point, by solving a trust-region type QSDP subproblem ∗

Project supported by the National Natural Science Foundation (No. 11561005), the National Science Foundation of Guangxi (No. 2016GXNSFAA380248 † Corresponding author. E-mail: [email protected] 836

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to get search direction. When objective function value or the constraint violation function is improved, the trial point is accepted by filter. Chen in [21] proposed a trust region SSDP method without a penalty function or a filter. The search direction is obtained by solving trust region QSDP subproblem. Whether the trial point is accepted or not depends on the decline of the objective function or constraint violation function. In all above SSDP algorithms, the traditional QSDP subproblem, which generated the search direction, may be incompatible. Motivated by the idea of modified SQP methods for nonlinear programming, in this paper, we proposed a modified SSDP algorithm for NLSDP (1.1). At each iteration, the search direction is yielded by solving a linear semidefinite programming (LSDP for short) subproblem and a modified QSDP subproblem. Nonmonotone line search technique is used to determine step size. The paper is organized as follows. In the next section, the algorithm is described in detail. The global convergence is shown in Section 3. Some preliminary numerical results are reported in Section 4 and some concluding remarks are given in the final section.

2

Description of Algorithm

In this section, we first restate some concepts and notations about nonlinear semidefinite programming, and then describe the proposed algorithm. Let G(x) : Rn → Sm be a matrix value function, we use the notation   ∂G(x) ∂G(x) T DG(x) = ,···, (2.1) ∂x1 ∂xn for its differential operator evaluated at x. For any d = (d1 , · · · , n) ∈ Rn , DG(x)d is defined by DG(x)d =

n X

di

i=1

∂G(x) . ∂xi

(2.2)

DG(x)∗

The adjoint operator of the linear operator DG(x) satisfies  T ∂G(x) ∂G(x) ∂G(x) , Y >, < , Y >, · · · , < ,Y > , ∀ Y ∈ Sm . DG(x)∗ Y = < ∂x1 ∂x2 ∂xn

(2.3)

where < A, B > means the inner product of the matrix A and B. Definition 2.1 [16] Let x e ∈ Rn be a feasible point of NLSDP (1.1), if there exists Ye ∈ Sm satisfying the following KKT conditions ∇x L(˜ x, Y˜ ) = ∇f (˜ x) + DG(˜ x)∗ Y˜ = 0, Y˜  0,

< G(˜ x), Y˜ >= 0,

(2.4) (2.5)

where L : Rn × Sm → R is the Lagrangian function of NLSDP (1.1), that is, L(x, λ, Y ) = f (x)+ < Y, G(x) >, then x e is called a KKT point of NLSDP (1.1), the matrix Ye is called a Lagrangian multiplier associated with x e. 837

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Let xk ∈ Rn be the current iterate point. In order to generate search directions, we borrow the ideas in [22] and construct the following linear semidefinite programming (LSDP (xk ) for short): min z s.t. G(xk ) + DG(xk )d  zIm , z ≥ 0,

(2.6)

where Im is the m order identity. Obviously, the feasible set of LSDP(xk )(2.6) is not empty, so T there exists an optimal solution of (2.6). Let (dˆk , zk )T be an optimal solution of (2.6), then we construct a quadratic semidefinite programming (QSDP (xk , Hk ) for short) as follows: min ∇f (xk )T d + 12 dT Hk d

d∈Rn

s.t.

G(xk ) + DG(xk )d  zk Im .

(2.7)

If Hk is a symmetric positive definite matrix, then the solution of QSDP(xk , Hk ) (2.7) is unique. To measure the degree of feasibility at the iterate point, we define the degree of constraint violation as follows: h(x) = (λ1 (G(x)))+ ,

(2.8)

where λ1 (·) is the largest eigenvalue of a matrix, (α)+ = max{0, α}. Obviously, h(x) = 0 is equivalent with that x is a feasible point of NLSDP (1.1). Let dk be the solution of QSDP(xk , Hk ) (2.7). Similar to the idea of filter method, we hope that the search direction dk can improve the feasibility of the iterate point or the value of the objective function. In other words, if dk satisfies 1 ∇f (xk )T dk ≤ − (dk )T Hk dk , 2

(2.9)

and t satisfies f (xk + tdk ) ≤

max {f (xk−j )} − tα(dk )T Hk dk ,

(2.10)

0≤j≤m(k)

h(xk + tdk ) ≤ β

max {h(xk−j )},

(2.11)

0≤j≤m(k)

where α ∈ (0, 12 ), m(0) = 0, m(k) = min{m(k − 1) + 1, M }, M is a positive integer, then the corresponding trial step xk + tdk is accepted. If dk does not satisfy (2.9), that is, ∇f (xk )T dk > − 12 (dk )T Hk dk ,

(2.12)

then let t = 1. If the following inequality h(xk + dk ) ≤ β

max {h(xk−j )} 0≤j≤m(k)

(2.13)

hold, then the corresponding trial step xk + dk is accepted. Based on the above strategy, we now present the new algorithm in detail. 838

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Algorithm A S0. Given x0 ∈ Rn , H0 = Im , α ∈ (0, 12 ), σ ∈ (0, 1), β ∈ ( 12 , 1), m(0) = 0, apositiveinteger M . Let k := 0. T S1. Solve LSDP(xk ) (2.6) to get a solution (dˆk , zk )T . If dˆk = 0 and zk 6= 0, stop. S2. Solve QSDP (xk , Hk ) (2.7) to get the solution dk . If dk = 0, stop. S3. If dk satisfies (2.9), then let tk be the first number in the sequence of {1, σ, σ 2 , · · ·} satisfying the following inequality f (xk + tdk ) ≤

max {f (xk−j )} − tα(dk )T Hk dk ,

0≤j≤m(k)

(2.14)

and go to S4; otherwise, let tk = 1 and go to S4. S4. Let xk+1 = xk + tk dk . If the following inequality h(xk+1 ) ≤ β

max {h(xk−j )},

(2.15)

0≤j≤m(k)

holds, then set m(k + 1) = min{m(k) + 1, M }. Update Hk such that Hk+1 is a positive definite matrix. Let k = k + 1 and go to S1; otherwise, go into the restoration phase to obtain a new point xk+1 . Let k = k + 1 and go to S1. Remark. In the restoration phase, our aim is to decrease the value of h(x). The restoration algorithm is similar to the one given by Long et al. [23].

3

Global Convergence

In this section, we first show that Algorithm A is well-defined, and then show the global convergence. To this end, the following assumptions are necessary. A1

The iterate {xk } remains in a closed, bounded subset X .

A2 The objective function f (x) and the constraint function G(x) are twice continuously differentiable in Rn . A3

There exist two constants 0 < a ≤ b such that akdk2 ≤ dT Hk d ≤ bkdk2 for any d ∈ Rn .

In what follows, we analyze the feasibility of Algorithm A. To this end, it is necessary to extend the definition of infeasible stationary point for nonlinear programming [24] to nonlinear semidefinite programming. Definition 3.1

Let x e ∈ Rn be an infeasible point of N LSDP (1.1), if

min max{λ1 (G(e x) + DG(e x)d), 0} = max{λ1 (G(e x)), 0} = h(e x),

d∈Rn

(3.1)

then x e is called an infeasible sationary of N LSDP (1.1). 839

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Lemma 3.1 Supposed that the assumptions A1-A3 hold, if Algorithm A terminates at xk , then xk is either an infeasible stationary point or a KKT point of N LSDP (1.1). Proof. The proof is divided into two cases. Case A. If Algorithm A terminates in S1, then dˆk = 0 and zk 6= 0. We know from LSDP (xk ) (2.6) that zk = h(xk ), so h(xk ) 6= 0, which implies xk is an infeasible point of N LSDP (1.1). In the following, we prove that xk is an infeasible stationary point of N LSDP (1.1), namely, xk satisfies: min max{λ1 (G(xk ) + DG(xk )d), 0} = max{λ1 (G(xk )), 0} = h(xk ).

d∈Rn

By contradiction, suppose that the conclusion is not true. So there exists dk,0 ∈ Rn such that zb := max{λ1 (G(xk ) + DG(xk )dk,0 ), 0} < h(xk ).

(3.2)

T

Clearly, (dk,0 , zˆ)T is a feasible solution of LSDP (xk ) (2.6). Note that zk is a solution of LSDP (xk ) (2.6), so we obtain zk ≤ zb < h(xk ),

(3.3)

this contradicts zk = h(xk ). Therefore, xk is an infeasible stationary point of N LSDP (1.1). Case B. If Algorithm A terminates in S2, then the solution dk of QSD(xk , Hk ) (2.7) is zero, i.e., dk = 0. Further, dk = 0 satisfies KKT condition of QSDP (xk , Hk ) (2.7), that is to say, there exists Yk ∈ Sm , such that ∇f (xk ) + DG(xk )∗ Yk = 0,

(3.4)

k

Yk  0,

G(x )  zk Im ,

(3.5)

< G(xk ) − zk Im , Yk >= 0.

(3.6)

In what follows, we prove that zk = 0. By contradiction, supposed that zk 6= 0, obviously, (0T , zk )T is a solution of LSDP (xk ) (2.6) from (3.5). Therefore, xk is an infeasible point of N LSDP (1.1). Since Algorithm A does not stop in S1, zk < h(xk ). On the other hand, it follows from (3.5) that λ1 (G(xk )) ≤ zk . In view of zk > 0, we obtain h(xk ) = max{λ1 (G(xk )), 0} ≤ zk . This contradict zk < h(xk ). Therefore, zk = 0. Substituting zk = 0 into (3.5), and conbining with (3.4) and (3.6), we know that xk is a KKT point of N LSDP (1.1).  Lemma 3.2 If dk satisfies the inequality (2.9), then the line search (2.14) is performed. Proof. It is sufficient to show that there exsits t ∈ (0, 1) such that (2.14) hold. 840

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In view of ∇f (xk )T dk ≤ − 12 (dk )T Hk dk , so in combination with the positive definiteness of Hk , we know that there exists dk = 6 0 such that ∇f (xk )T dk < 0. For convinence, denote f (xl(k) ) =

max {f (xk−j )}.

(3.7)

0≤j≤m(k)

By contradiction, if the conclusion is not true, then for all t ∈ (0, 1), we have f (xk + tdk ) − f (xl(k) ) > −tα(dk )T Hk dk ≥ 2tα∇f (xk )T dk . From (3.7), it is obvious that

(3.8)

f (xl(k) ) ≥ f (xk ), so combining with (3.8), we have

f (xk + tdk ) − f (xk ) ≥ f (xk + tdk ) − f (xl(k) ) > 2tα∇f (xk )T dk ,

(3.9)

equivalently, [f (xk +tdk )−f (xk )] t

> 2α∇f (xk )T dk .

(3.10)

Let t → 0+ , taking the limit for the both sides, it follows that ∇f (xk )T dk ≥ 2α∇f (xk )T dk . This implies α ∈ [ 12 , ∝) due to ∇f (xk )T dk < 0,. This contradicts α ∈ (0, 12 ). Hence, the desired result holds.  Lemma 3.3 Supposed that the assumptions A1-A3 hold, then there exists t¯ > 0 such that tk ≥ t¯ for k sufficiently large,. Proof. According to Algorithm A, without loss of generality, suppose that the search direction dk satisfies (2.10), that is, 1 ∇f (xk )T dk ≤ − (dk )T Hk dk . 2 By Taylor expansion, (3.7) and the assumptions A1-A3, we have f (xk + tk dk ) − f (xl(k) ) + tk α(dk )T Hk dk = f (xk ) + tk ∇f (xk )T dk + 12 t2k (dk )T ∇2 f (y k )dk − f (xl(k) ) + tk α(dk )T Hk dk ≤ f (xk ) + tk ∇f (xk )T dk + 21 t2k (dk )T ∇2 f (y k )dk − f (xk ) + tk α(dk )T Hk dk = tk ∇f (xk )T dk + 12 t2k (dk )T ∇2 f (y k )dk + tk α(dk )T Hk dk ≤ − 21 tk (dk )T Hk dk + 12 t2k (dk )T ∇2 f (y k )dk + tk α(dk )T Hk dk ≤ −atk ( 21 − α)kdk k2 + 21 t2k M kdk k2 , where y k is between xk and xk + tk dk , M is a positive integer such that k∇2 f (x)k ≤ M . Let t¯ = a(1−2α) > 0, so (2.10) holds for tk ≥ t¯ and α ∈ (0, 21 ). M

(3.11)



Lemma 3.4 Supposed that the assumptions A1-A3 hold, {xk } is an infinite sequence generated by Algorithm A, then lim h(xk ) = 0. k→∞

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Proof. Since m(k + 1) ≤ m(k) + 1, we have h(xl(k+1) ) =

{h(xk+1−j )} ≤

max

0≤j≤m(k+1)

max

{h(xk+1−j )} = max{h(xk+1 ), h(xl(k) )} = h(xl(k) ),

0≤j≤m(k)+1

this implies that the sequence {h(xl(k) )} is not increasing for k. Combining with h(xl(k) ) ≥ 0, we conclude that {h(xl(k) )} is convergent. By Algorithm A, we have h(xk+1 ) ≤ β

max {h(xk−j )} = βh(xl(k) ).

(3.12)

0≤j≤m(k)

Replace k by l(k) − 1. we obtain h(xl(k) ) ≤ βh(xl(l(k)−1) ), which together with β ∈ ( 12 , 1) gives that

(3.13)

lim h(xl(k) ) = 0. Further, by (3.12), we can conclude

k→∞

lim h(xk ) = 0.



k→∞

Theorem 3.1 Supposed that the assumptions A1-A3 hold, {xk } is an infinite sequence generated by Algorithm A, dk is the solution of QSDP (xk , Hk ) (2.7). If the multiplier corresponding e ⊆ {1, 2, · · ·} such that lim dk = 0. to dk is uniform bounded, then there exists K e k∈K

Proof. By the assumption A1, we know that {xk } is bounded, so there exists an infinite index set K ⊆ {1, 2, · · ·}, such that {xk }K is convergent. Let lim xk = x∗ . k∈K

We consider the following two cases: Case 1. The index set K0 = {k ∈ K | ∇f (xk )T dk ≤ − 21 (dk )T Hk dk } is infinite. By (2.14), we obtain f (xk+1 ) = f (xk + tk dk ) ≤ f (xl(k) ) − tk α(dk )T Hk dk ≤ f (xl(k) ), ∀ k ∈ K0 .

(3.14)

Since m(k + 1) ≤ m(k) + 1, we obtain f (xl(k+1) ) ≤

max

{f (xk+1−j )} = max{f (xk+1 ), f (xl(k) )} = f (xl(k) ).

(3.15)

0≤j≤m(k)+1

This implies that the sequence {f (xl(k) )} is not increasing. Combining with the boundedness of {f (xl(k) )}, it follows that {f (xl(k) )}K0 is convergent. For {l(k) − 1, k ∈ K0 }, we obtain f (xl(k) ) ≤ f (xl(l(k)−1) ) − tl(k)−1 α(dl(k)−1 )T Hl(k)−1 dl(k)−1 .

(3.16)

Since {f (xl(k) )} is convergent, we have lim tl(k)−1 α(dl(k)−1 )T Hl(k)−1 dl(k)−1 = 0, K0

By Lemma 3.3, we know that there exists t¯ > 0 such that tl(k)−1 ≥ t¯ > 0, so by the assumption A3, we obtain lim dl(k)−1 = 0. (3.17) K 0

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The uniform continuity of f (x) implies that lim f (xl(k)−1 ) = lim f (xl(k) ). K0

(3.18)

K0

Let ˆl(k) = l(k + M + 2), it is not difficult to prove by induction that for any given j ≥ 1 , ˆ

lim kdl(k)−j k = 0,

(3.19)

K0

ˆ

lim f (xl(k)−j ) = lim f (xl(k) ). K0

ˆ

For any k ∈ K0 , we obtain xk+1 = xl(k) −

ˆ l(k)−k−1 P j=1

1 and (3.19), we get lim K0

kxk+1

−

(3.20)

K0

ˆ xl(k) k

ˆ tˆl(k)−j dl(k)−j . Note that ˆl(k) − k − 1 ≤ M +

= 0. So it follows from the convergence of {f (xl(k) )} and

the uniform continuity of f (x) that lim f (xk+1 ) = lim f (xl(k) ). K0

K0

So let k (∈ K0 ) → ∞, taking the limit in (3.14), we have lim tk α(dk )T Hk dk = 0.

(3.21)

K0

e = K0 and the conclusion Similar to the proof of (3.17), we obtain lim dk = 0. Hence, let K K0

follows. Case 2. The index set K0 = {k ∈ K | ∇f (xk )T dk ≤ − 21 (dk )T Hk dk } is finite, which implies that K1 = {k ∈ K | ∇f (xk )T dk > − 12 (dk )T Hk dk } is infinite. By contradiction, supposed that the conclusion is not true, then lim dk = 6 0. So there exist K1

kdk k

K2 ⊆ K1 and a constant ε > 0, such that > ε for k ∈ K2 . Since dk is the solution of QSDP (xk , Hk ) (2.7), by KKT condition of QSD(xk , Hk ) (2.7) , it follows that there exists a positive semidefinite matrix Yk such that ∇f (xk ) + Hk dk + DG(xk )∗ Yk = 0, k

k

(3.22)

k

Tr((G(x ) + DG(x )d − zk Im )Yk ) = 0,

(3.23)

f > 0 such that kYk kF ≤ M f. According to the assumption of Theorem 3.1, there exists M k By Lemma 3.4, we know lim h(x ) = 0, hence there exists k0 > 0, such that k→∞

h(xk ) ≤ combining with

1 aε2 , for k (∈ K2 ) > k0 , f 2M m

(3.24)

kdk k > ε and the assumption A3, we obtain h(xk ) ≤

1 (dk )T Hk dk . ˜ 2M m

(3.25)

It follows from (2.2) that k

k

Tr(DG(x )d Yk ) =

n X

∂G(xk ) Tr(( dki )Yk ) ∂xi i=1

n X

n

X ∂G(xk ) ∂G(xk ) = Tr( Yk )dki = < , Yk > dki . ∂xi ∂xi i=1 i=1 (3.26) 843

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It follows from (3.23) that Tr((DG(xk )dk )Yk ) = Tr((G(xk ) − zk Im )Yk ),

(3.27)

so (3.26) and (3.27) give rise to n X i=1


dki = Tr((G(xk ) − zk Im )Yk ). ∂xi

(3.28)

By (3.22) and (3.28), we have ∇f (xk )T dk = −(dk )T Hk dk − (DG(xk )∗ Yk )T dk n X ∂G(xk ) = −(dk )T Hk dk − , Yk > dki < ∂xi i=1

k T

k

= −(d ) Hk d + Tr((G(xk ) − zk Im )Yk ).

(3.29)

By N eumann Inequality, we obtain k

Tr((G(x ) − zk Im )Yk ) ≤

m X

λi (G(xk ) − zk Im )λi (Yk )

i=1

≤

m X

λi (G(xk ) − zk Im )kYk kF

i=1

≤

m X

f λi (G(xk ) − zk Im )M

i=1

≤

m X

f, λi (G(xk ))M

(3.30)

i=1

the last inequality above is due to zk ≥ 0. According to the definition (2.8) of h(xk ) and (3.30), we obtain fmh(xk ) ≤ 1 (dk )T Hk dk . Tr((G(xk ) − zk Im )Yk ) ≤ M 2

(3.31)

Substituting (3.31) into (3.29), it follows that 1 ∇f (xk )T dk ≤ − (dk )T Hk dk , 2 which contradicts the definition of K1 . Hence, the conclusion is true.



Theorem 3.2 Supposed that {xk } is an infinite sequence generated by Algorithm A, and the assumptions in Theorem 3.1 hold, then any accumulation point of {xk } is a KKT point of N LSDP (1.1). Proof. Supposed that x∗ is an accumulation point of {xk }, then there exists K ⊆ {1, 2, · · ·} , such that lim xk = x∗ . In view of the assumption A3, without loss of generality, we suppose k∈K

that lim Hk = H∗ . k∈K

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By Lemma 3.4, we have lim h(xk ) = h(x∗ ) = 0, which means that x∗ is a feasible point of k∈K

N LSDP (1.1). e ⊆ {1, 2, · · ·} such that lim dk = d∗ = 0. By the proof of By Theorem 3.1, there exists K e K

e ⊆ K. Theorem 3.1, we know that K According to KKT conditions of QSDP (2.7), we obtain ∇f (xk ) + Hk dk + DG(xk )∗ Yk = 0, Yk  0,

Tr((G(xk ) + DG(xk )dk − zk Im )Yk ) = 0.

e → ∞, taking the limit, we obtain Let k(∈ K) ∇f (x∗ ) + DG(x∗ )∗ Y∗ = 0, Y∗  0,

< G(x∗ ), Y∗ >= 0.

This implies that x∗ is a KKT point of N LSDP (1.1).

4



Numerical experiments

In this section, preliminary numerical experiments of Algorithm A is implemented. Algorithm A was coded by Matlab (2014a) and run on the computer with Windows 7 (64 bite), Intel(R) Core(TM) i7-4790 CPU @ 3.60GHz 3.60GHz, RAM: 4.00GB. In the numerical experiments, the parameters are chosen as follows: α = 0.25, β = 0.85, σ = 0.5, M = 3. And the termination criteria of Algorithm A is: k dk k≤ 10−4 . The test problem is chosen from [11]. Problem 1.

Nearest Correlation Matrix (NCM) Problem: min s.t

f (X) = 21 kX − Ck2F X  I, Xii = 1, i = 1, 2, ..., m,

(4.1)

where C ∈ Sm is a given matrix, X ∈ Sm ,  is a scalar. In the implementation,  = 10−3 , C is generated randomly, which diagonal elements are 1. We test ten times for every fixed dimensionality. We compare Algorithm A with the ones in [11] (denoted by Algo. YYH) and [14] ( denoted by Algo. YYY ). The numerical results are listed in Table 1. The meaning of the notations in Table 1 are described as follows: n: m: A − Iter :

the dimensionality of independent variable; the dimensionality ofG(x); the average number of evaluation of iterations. Table 1.

Numerical results of NCM 845

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5

n

m

x0

10

5

(0.5, ..., 0.5)T

45

10

(0.5, ..., 0.5)T

105

15

(0.5, ..., 0.5)T

190

20

(0.5, ..., 0.5)T

Algorithm Algorithm A Algo. YYY Algo. YYH Algorithm A Algo. YYY Algo. YYH Algorithm A Algo. YYY Algo. YYH Algorithm A Algo. YYY Algo. YYH

A-Iter 15 8 9 15 8 10 17 10 11 17 11 12

Concluding remarks

In this paper, we have presented a new SSDP algorithm for nonlinear semidefinite programming. Two subproblems, which are constructed skillfully, are solved to generate the search directions. The nonmonotone line search ensures that the objective function or constraint violation function is sufficiently reduced. The global convergence of the proposed algorithm is shown under some mild conditions. The preliminary numerical results show that the proposed algorithm is effective.

References [1] Konno, H., Kawadai, N. and Wu, D.: Estimation of failure probability using semi-definite logitmodel. Computational Management Science, 1: 59-73 (2003) [2] Apkarian, P., Noll, D and Tuan, D.: Fixed-order HI control design via partially argmented Lagrangian method. International Journal of Control, 11: 1137-1148 (2003) [3] Kanno, Y. and Takewaki, I.: Sequential semidefinite program for maximum robustness design of structures under load uncertainty. Journal of Optimization Theroy and Applications, 130: 265-287 (2006) [4] Freund, R. W., Jarre, F. and Vogelbusch, C. H.: Nonlinear semidefinite programming: sensitivity, convergence and an application in passive reduce-order modeling. Mathematical Programming, 109: 581-611 (2007) [5] Sun, J., Zhang, L. W. and Wu, Y.: Properties of the augmented Lagrangian in nonlinear semidefinite optimization. Journal of Optimization Theory and Applications, 129: 437-456 (2006) [6] Noll, D.: Local convergence of an augmented Lagrangian method for matrix inequality constrained programming. Optimization Methods and Software, 22: 777õ802 (2007) [7] Sun, D. F., Sun, J. and Zhang, L. W.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Mathwmatical Programming, 114: 349-391 (2008) [8] Luo, H. Z., Wu, H. X. and Chen, G. T.: On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming. Journal of Global Optimization, 54: 599-618 (2012) 846

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[9] Wu, H. X., Luo, H. Z. and Ding, X. D.: Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming[J]. Computational Optimization and Applications, 56: 531-558 (2013) [10] Wu, H. X., Luo, H. Z. and Yang, J. F.: Nonlinear separation approach for the augmented Lagrangianin nonlinear semidenite programming. Journal of Global Optimization, 59(4): 695-727 (2014) [11] Yamashita, H., Yabe, H. and Harada, K.: A primalõdual interior point method for nonlinear semidefinite programming. Mathematical Programming, 135: 89-121 (2012) [12] Yamashita, H. and Yabe, H.: Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming. Mathematical Programming, 132: 1-30 ( 2012) [13] Aroztegui, M., Herskovits, J. and Roche, J. R.: A feasible direction interior point algorithm for nonlinear semidefinite programming. Structral and Multidisciplinary Optimization, 50(6): 1019-1035 (2014) [14] Yamakawa, Y., Yamashita, N. and Yabe, H.: A differentiable merit function for the shifted perturbed Karush-Kuhn-Tucker conditions of the nonlinear semidefinite programming. Pacific Journal of Optimization, 11: 557-579 ( (2015)) [15] Li, J. L., Yang, Z. P. and Jian, J. B.: A globally convergent QP-free algorithm for nonlinear semidefinite programming. Journal of Inequalities and Applications, 145: 1-21 (2017), [16] Correa, R. and Ramirez, H.: A global algorithm for nonlinear semidefinite programming. SIAM Journal on Optimization, 15: 303-318 (2004) [17] Gomez, W. and Ramirez, H.: A filter algorithm for nonlinear semidefinite programming. Computation and Applied Mathematics, 29: 297-32 (2010) [18] Zhu, Z. B. and Zhu, H. L.: A filter method for nonlinear semidefinite programming with global convergence. Acta Mathematica Sinica (English Series), 30: 1810-1826 (2014) [19] Li, C. J. and Sun, W. Y.: On filter-successive linearization methods for nonlinear semidefinite programming. Science in China (Series A), Mathematics, 52: 2341-2361 (2009) [20] Zhao, Q.: Global convergence of new method for semidefinite programming. South East Asian Journal of Mathematics and Mathematical Sciences, 3: 189-198 (2013) [21] Chen, Z. W. and Miao, S. C.: A penalty-free method with trust region for nonlinear semidefinite programming. Asia-Pacific Journal of Operational Reasearch, 32(1): 1-24 (2015) [22] Zhang, J. L. and Zhang, X. S.: A robust SQP method for optimization with inequality constraints. Journal of Computational Mathematics, 21(2): 247-256 (2003) [23] Long, J., Ma, C. F. and Nie, P. Y.: A new filter method for solving nonlinear complementarity problems. Application Mathematics and Computation, 185: 705-718 (2007) [24] Liu, X. W. and Yuan, Y. X.: A robust algorithm for optimization with general equality and inequality constraints. SIAM Journal on Optimization, 22(2): 517-534 (2000)

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Approximation by Sublinear and Max-product Operators using Convexity George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we consider quantitatively using convexity the approximation of a function by general positive sublinear operators with applications to Maxproduct operators. These are of Bernstein type, of Favard-Szász-Mirakjan type, of Baskakov type, of Meyer-Köning and Zeller type, of sampling type, of Lagrange interpolation type and of Hermite-Fejér interpolation type. Our results are both: under the presence of smoothness and without any smoothness assumption on the function to be approximated which ful…lls a convexity property.

2010 AMS Mathematics Subject Classi…cation: 41A17, 41A25, 41A36. Keywords and Phrases: positive sublinear operators, Max-product operators, modulus of continuity, convexity.

1

Background

We make Remark 1 Let f 2 C ([a; b]), x0 2 (a; b), 0 < h jf (t) f (x0 )j is convex in t 2 [a; b]. By Lemma 8.1.1, p. 243 of [1] we have that jf (t)

f (x0 )j

! 1 (f; h) jt h

min (x0

x0 j ; 8 t 2 [a; b] ;

a; b

x0 ), and

(1)

where ! 1 (f; h) :=

sup jf (x)

f (y)j ;

(2)

x;y2[a;b] jx yj h

the …rst modulus of continuity of f . 1

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We also make Remark 2 Let f 2 C n ([a; b]), n 2 N, x0 2 (a; b), 0 < h min (x0 and f (n) (t) f (n) (x0 ) is convex in t 2 [a; b]. We have that f (t) =

n X f (k) (x0 )

k!

k=0

(t

a; b

k

x0 ),

x0 ) + It ;

(3)

f (n) (x0 ) dtn ::: dt1 :

(4)

where It =

Z

Z

t

t1

:::

x0

x0

Z

tn

1

f (n) (tn )

x0

Assuming f (k) (x0 ) = 0, k = 1; :::; n, we get f (t)

f (x0 ) = It :

(5)

By Lemma 8.1.1, p. 243 of [1] we have f (n) (t)

f (n) (x0 )

! 1 f (n) ; h jt h

x0 j ; 8 t 2 [a; b] :

(6)

Furthermore it holds ! 1 f (n) ; h jt x0 jn+1 ; 8 t 2 [a; b] : h (n + 1)!

jIt j

(7)

Hence we derive that (5)

jf (t)

f (x0 )j

! 1 f (n) ; h jt x0 jn+1 ; 8 t 2 [a; b] : h (n + 1)!

(8)

We have proved the following results: Theorem 3 Let f 2 C ([a; b]), x 2 (a; b), 0 < h jf ( ) f (x)j is convex over [a; b]. Then jf ( )

f (x)j

! 1 (f; h) j h

min (x

a; b

xj ; over [a; b] :

x), and

(9)

Theorem 4 Let f 2 C n ([a; b]), n 2 N, x 2 (a; b), 0 < h min (x a; b x), and f (n) ( ) f (n) (x) is convex over [a; b]. Assume that f (k) (x) = 0, k = 1; :::; n: Then jf ( )

f (x)j

! 1 f (n) ; h j xjn+1 ; over [a; b] : h (n + 1)!

(10)

We give 2

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De…nition 5 Call C+ ([a; b]) := ff : [a; b] ! R+ and continuousg : Let LN from C+ ([a; b]) into C+ ([a; b]) be a sequence of operators satisfying the following properties (see also [6], p. 17): (i) (positive homogeneous) LN ( f ) = LN (f ) ; 8 (ii) (Monotonicity) if f; g 2 C+ ([a; b]) satisfy f

0; 8 f 2 C+ ([a; b]) ;

(11)

g; then

LN (f )

LN (g) , 8 N 2 N;

(12)

(iii) (Subadditivity) LN (f + g)

LN (f ) + LN (g) ; 8 f; g 2 C+ ([a; b]) :

(13)

We call LN positive sublinear operators. We make Remark 6 As in [6], p. 17, we get that for f; g 2 C+ ([a; b]) jLN (f ) (x)

LN (g) (x)j

LN (jf

gj) (x) ; 8 x 2 [a; b] :

(14)

From now on we assume that LN (1) = 1, 8 N 2 N. Hence it holds jLN (f ) (x)

f (x)j

LN (jf ( )

f (x)j) (x) ; 8 x 2 [a; b] ; 8 N 2 N,

(15)

see also [6], p. 17. We obtain the following results: Theorem 7 Let f 2 C+ ([a; b]), x 2 (a; b), 0 < h min (x a; b x), and jf ( ) f (x)j is a convex function over [a; b]. Let fLN gN 2N positive sublinear operators from C+ ([a; b]) into itself, such that LN (1) = 1, 8 N 2 N. Then jLN (f ) (x)

f (x)j

! 1 (f; h) LN (j h

xj) (x) ; 8 N 2 N:

(16)

Proof. By (9) and (15). Theorem 8 Let f 2 C n ([a; b] ; R+ ), n 2 N, x 2 (a; b), 0 < h min (x a; b x), and f (n) ( ) f (n) (x) is convex over [a; b]. Assume that f (k) (x) = 0, k = 1; :::; n: Let fLN gN 2N positive sublinear operators from C+ ([a; b]) into itself, such that LN (1) = 1, 8 N 2 N. Then jLN (f ) (x)

f (x)j

! 1 f (n) ; h LN j h (n + 1)!

n+1

xj

(x) ; 8 N 2 N:

(17)

3

850

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Proof. By (10) and (15). We continue with Theorem 9 Let f 2 C+ ([a; b]), x 2 (a; b), 0 < LN (j xj) (x) min (x a; b x), 8 N 2 N, and jf ( ) f (x)j is a convex function over [a; b]. Here LN are positive sublinear operators from C+ ([a; b]) into itself, such that LN (1) = 1, 8 N 2 N. Then jLN (f ) (x) If LN (j

f (x)j

! 1 (f; LN (j

xj) (x)) ; 8 N 2 N:

(18)

xj) (x) ! 0, then LN (f ) (x) ! f (x), as N ! +1:

Proof. By (16). Theorem 10 Let f 2 C n ([a; b] ; R+ ), n 2 N, x 2 (a; b), 0 < LN j (n)

n+1

xj

(x)

(n)

min (x a; b x), 8 N 2 N, and f ( ) f (x) is convex over [a; b]. Assume that f (k) (x) = 0, k = 1; :::; n: Here fLN gN 2N are positive sublinear operators from C+ ([a; b]) into itself, such that LN (1) = 1, 8 N 2 N. Then jLN (f ) (x) If LN j

n+1

xj

f (x)j

! 1 f (n) ; LN j

n+1

xj

(x) ; 8 N 2 N:

(n + 1)!

(19)

(x) ! 0, then LN (f ) (x) ! f (x), as N ! +1:

Proof. By (17). Next we combine both Theorems 7, 8: Theorem 11 Let f 2 C n ([a; b] ; R+ ), n 2 Z+ , x 2 (a; b), 0 < h min (x a; b x), and f (n) ( ) f (n) (x) is convex over [a; b]. Assume that f (k) (x) = 0, k = 1; :::; n: Let fLN gN 2N positive sublinear operators from C+ ([a; b]) into itself, such that LN (1) = 1, 8 N 2 N. Then jLN (f ) (x)

f (x)j

! 1 f (n) ; h LN j h (n + 1)!

n+1

xj

(x) ; 8 N 2 N; n 2 Z+ : (20)

The initial conditions f (k) (x) = 0, k = 1; :::; n; are void when n = 0: In this article we study under convexity quantitatively the approximation properties of Max-product operators to the unit. These are special cases of positive sublinear operators. We present also results regarding the convergence to the unit of general positive sublinear operators under convexity. Special emphasis is given to our study about approximation under the presence of smoothness. Our work is inspired from [6]. Under our convexity conditions the derived convergence inequalities are elegant and compact with very small constants. 4

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2

Main Results

Here we apply Theorem 11 to Max-product operators. We make Remark 12 We start with the Max-product Bernstein operators ([6], p. 10) WN k pN;k (x) f N (M ) BN (f ) (x) = k=0 ; 8 N 2 N, (21) WN k=0 pN;k (x)

W N N 1 xk (1 x) , x 2 [0; 1] ; stands for maximum, and f 2 k C+ ([0; 1]) = ff : [0; 1] ! R+ is continuousg ; where R+ := [0; 1): (M ) Clearly BN is a positive sublinear operators from C+ ([0; 1]) into itself with (M ) BN (1) = 1: By [6], p. 31, we have pN;k (x) =

(M )

BN

(j

xj) (x)

p

6 , 8 x 2 [0; 1] , 8 N 2 N. N +1

(22)

6 , 8 x 2 [0; 1] , m; N 2 N. N +1

(23)

And by [2] we get: (M )

BN

m

(j

xj ) (x)

p

Denote by n C+ ([0; 1]) = ff : [0; 1] ! R+ , n-times continuously di¤erentiableg ; n 2 Z+ :

We present n Theorem 13 Let f 2 C+ ([0; 1]), n 2 Z+ , x 2 (0; 1) and N 2 N : 0 < p 1 min (x; 1 x), and f (n) ( ) f (n) (x) is convex over [0; 1]. Assume N +1

that f (k) (x) = 0, k = 1; :::; n: Then (M ) BN

It holds

(f ) (x) (M )

lim BN

N !+1

6! 1 f (n) ; pN1+1

f (x)

(n + 1)!

; 8N 2N:N

N :

(24)

(f ) (x) = f (x) :

Proof. By (20) we get (M )

BN

(f ) (x)

f (x)

! 1 f (n) ; h (M ) B j h (n + 1)! N

n+1

xj

(23)

(x)

! 1 f (n) ; h 6 p = h (n + 1)! N +1 5

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(setting h :=

p 1 ) N +1

6! 1 f (n) ; pN1+1 (n + 1)!

;

(25)

proving the claim. We make Remark 14 Here we focus on the truncated Favard-Szász-Mirakjan operators (M ) TN

(f ) (x) =

WN

k k=0 sN;k (x) f N WN k=0 sN;k (x)

k

; x 2 [0; 1] ; N 2 N, f 2 C+ ([0; 1]) ; (26)

sN;k (x) = (Nk!x) , see also [6], p. 11. By [6], p. 178-179 we have (M )

TN

(j

xj) (x)

3 p ; 8 x 2 [0; 1] ; 8 N 2 N: N

(27)

3 p ; 8 x 2 [0; 1] ; 8 m; N 2 N: N

(28)

And by [2] we get (M )

TN

(M )

The operators TN = 1, 8 N 2 N.

m

(j

xj ) (x)

(M )

are positive sublinear from C+ ([0; 1]) into itself with TN

(1)

We give n Theorem 15 Let f 2 C+ ([0; 1]), n 2 Z+ , x 2 (0; 1) and N 2 N : 0 < 1 p 1 x), and f (n) ( ) f (n) (x) is convex over [0; 1]. Assume min (x; N

that f (k) (x) = 0, k = 1; :::; n: Then (M ) TN

It holds

(f ) (x) (M )

lim TN

N !+1

f (x)

3! 1 f (n) ; p1N (n + 1)!

; 8N 2N:N

N :

(29)

(f ) (x) = f (x) :

Proof. By (20) we get (M )

TN

(f ) (x)

f (x)

! 1 f (n) ; h (M ) T j h (n + 1)! N

n+1

xj

(28)

(x)

! 1 f (n) ; h 3 p = h (n + 1)! N

6

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(setting h :=

p1 ) N

3! 1 f (n) ; p1N

;

(n + 1)!

(30)

proving the claim. We make Remark 16 Next we study the truncated Max-product Baskakov operators (see [6], p. 11) (M ) UN

WN

k k=0 bN;k (x) f N WN k=0 bN;k (x)

(f ) (x) =

where

bN;k (x) =

; x 2 [0; 1] ; f 2 C+ ([0; 1]) ; N 2 N, (31)

N +k k

xk

1

N +k

(1 + x)

:

From [6], pp. 217-218, we get (x 2 [0; 1]) (M )

UN

(j

xj) (x)

p

12 , N N +1

2, N 2 N:

(32)

And as in [2], we obtain (m 2 N) (M )

UN

m

(j

p

xj ) (x)

12 , N N +1

(M )

(M )

Also it holds UN (1) (x) = 1, and UN C+ ([0; 1]) into itself, 8 N 2 N.

2, N 2 N; 8 x 2 [0; 1] :

(33)

are positive sublinear operators from

We give n Theorem 17 Let f 2 C+ ([0; 1]), n 2 Z+ , x 2 (0; 1) and N 2 N f1g : 0 < 1 p min (x; 1 x), and f (n) ( ) f (n) (x) is convex over [0; 1]. Assume N +1

that f (k) (x) = 0, k = 1; :::; n: Then (M ) UN

It holds

(f ) (x) (M )

lim UN

N !+1

f (x)

12! 1 f (n) ; pN1+1 (n + 1)!

; 8N 2N:N

N :

(34)

(f ) (x) = f (x) :

Proof. By (20) we get (M )

UN

(f ) (x)

f (x)

! 1 f (n) ; h (M ) U j h (n + 1)! N

n+1

xj

(33)

(x)

7

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! 1 f (n) ; h 12 p = h (n + 1)! N +1 (setting h :=

p 1 ) N +1

12! 1 f (n) ; pN1+1 (n + 1)!

;

(35)

proving the claim. We make Remark 18 Here we study Max-product Meyer-Köning and Zeller operators (see [6], p. 11) de…ned by (M ) ZN

(f ) (x) =

W1

k k=0 sN;k (x) f N +k W1 k=0 sN;k (x)

; 8 N 2 N, f 2 C+ ([0; 1]) ;

N +k xk , x 2 [0; 1]. k By [6], p. 253, we get that p p 8 1+ 5 x (1 x) (M ) p ZN (j xj) (x) , 8 x 2 [0; 1] , N 3 N

(36)

sN;k (x) =

4:

(37)

And by [2], we derive that (M ) ZN

8 x 2 [0; 1], N The ceiling

m

(j

xj ) (x)

8 1+ 3

p

5

p

x (1 x) p , N

(38)

4; 8 m 2 N.

p 8(1+ 5) 3

= 9, and using a basic calculus technique (see [4]) we

p

get that g (x) := (1 x) x has an absolute maximum over (0; 1] : g p 2 That is (1 x) x 3p , 8 x 2 [0; 1] : 3 Consequently it holds (M )

ZN

(j

m

xj ) (x)

6 p p ; 3 N

1 3

=

2 p . 3 3

(39)

8 x 2 [0; 1], 8 N 2 N : N 4; 8 m 2 N. (M ) (M ) Also it holds ZN (1) = 1, and ZN are positive sublinear operators from C+ ([0; 1]) into itself, 8 N 2 N. We give

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n Theorem 19 Let f 2 C+ ([0; 1]), n 2 Z+ , x 2 (0; 1) and N 2 N : N 4 1 with 0 < pN min (x; 1 x), and f (n) ( ) f (n) (x) is convex over [0; 1].

Assume that f (k) (x) = 0, k = 1; :::; n: Then (M )

ZN

(f ) (x)

p

f (x)

6 3 (n + 1)!

1 ! 1 f (n) ; p N

; 8N 2N:N

N : (40)

It holds

(M )

lim ZN

N !+1

(f ) (x) = f (x) :

Proof. By (20) we get (M )

ZN

(f ) (x)

! 1 f (n) ; h (M ) Z j h (n + 1)! N

f (x)

n+1

xj

(39)

(x)

! 1 f (n) ; h 6 p p = h (n + 1)! 3 N (setting h :=

p1 ) N

p

6 3 (n + 1)!

1 ! 1 f (n) ; p N

;

(41)

proving the claim. We make Remark 20 Here we mention the Max-product truncated sampling operators (see [6], p. 13) de…ned by WN

; x 2 [0; ] ;

(42)

f : [0; ] ! R+ , continuous, and WN

; x 2 [0; ] ;

(43)

(M ) WN

(M ) KN

(f ) (x) :=

(f ) (x) :=

sin(N x k ) f kN k=0 Nx k WN sin(N x k ) k=0 Nx k sin2 (N x k ) f kN (N x k )2 WN sin2 (N x k ) k=0 (N x k )2

k=0

f : [0; ] ! R+ , continuous. By convention we take sin(0) = 1; which implies for every x = kN , k 2 0 sin(N x k ) f0; 1; :::; N g that we have N x k = 1: We de…ne the Max-product truncated combined sampling operators (see also [5]) WN k N;k (x) f N (M ) MN (f ) (x) := k=0 ; x 2 [0; ] ; (44) WN k=0 N;k (x)

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f 2 C+ ([0; ]) ; where (M ) MN

(f ) (x) :=

8 < W (M ) (f ) (x) , if N (M ) : KN (f ) (x) , if

N;k

(x) :=

N;k

(x) :=

sin(N x k ) ; Nx k sin(N x k ) Nx k

2

:

(45)

By [6], p. 346 and p. 352 we get (M )

MN

(j

xj) (x)

2N

,

(46)

and by [3] (m 2 N) we have (M )

MN

m

m

(j

xj ) (x)

(M )

2N (M )

Also it holds MN (1) = 1, and MN C+ ([0; ]) into itself, 8 N 2 N.

, 8 x 2 [0; ] , 8 N 2 N:

(47)

are positive sublinear operators from

We give Theorem 21 Let f 2 C n ([0; ] ; R+ ), n 2 Z+ , x 2 (0; ) and N 2 N : 0 < 1 min (x; x), and f (n) ( ) f (n) (x) is convex over [0; ]. Assume that N (k) f (x) = 0, k = 1; :::; n: Then (M )

MN

n+1

(f ) (x)

f (x)

! 1 f (n) ;

2 (n + 1)!

1 N

;

(48)

8 N 2 N : N N ; n 2 Z+ : (M ) It holds lim MN (f ) (x) = f (x) : N !+1

Proof. By (20) we have: (M )

MN

(f ) (x)

f (x)

! 1 f (n) ; h (M ) MN j h (n + 1)! ! 1 f (n) ; h h (n + 1)!

(setting h :=

1 N)

n+1

2 (n + 1)!

n+1

xj

(47)

(x)

n+1

2N

=

! 1 f (n) ;

1 N

;

(49)

proving the claim. We make

10

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k)+1) Remark 22 The Chebyshev knots of …rst kind xN;k := cos (2(N 2 2(N +1) ( 1; 1), k 2 f0; 1; :::; N g; 1 < xN;0 < xN;1 < ::: < xN;N < 1, are the roots of the …rst kind Chebyshev polynomial TN +1 (x) := cos ((N + 1) arccos x), x 2 [ 1; 1] : De…ne (x 2 [ 1; 1])

hN;k (x) := (1

x xN;k )

TN +1 (x) (N + 1) (x xN;k )

2

;

(50)

the fundamental interpolation polynomials. The Max-product interpolation Hermite-Fejér operators on Chebyshev knots of the …rst kind (see p. 12 of [6]) are de…ned by WN hN;k (x) f (xN;k ) (M ) ; 8 N 2 N, (51) H2N +1 (f ) (x) = k=0WN k=0 hN;k (x)

for f 2 C+ ([ 1; 1]), 8 x 2 [ 1; 1] : By [6], p. 287, we have (M )

H2N +1 (j

xj) (x)

2 ; 8 x 2 [ 1; 1] , 8 N 2 N: N +1

(52)

2m ; 8 x 2 [ 1; 1] , 8 m; N 2 N: N +1

(53)

And by [3], we get that (M )

H2N +1 (j

m

xj ) (x)

(M )

(M )

Notice H2N +1 (1) = 1, and H2N +1 maps C+ ([ 1; 1]) into itself, and it is WN a positive sublinear operator. Furthermore it holds k=0 hN;k (x) > 0, 8 x 2 [ 1; 1]. We also have hN;k (xN;k ) = 1, and hN;k (xN;j ) = 0, if k 6= j, and (M ) H2N +1 (f ) (xN;j ) = f (xN;j ), for all j 2 f0; 1; :::; N g, see [6], p. 282. We give Theorem 23 Let f 2 C n ([ 1; 1] ; R+ ), n 2 Z+ , x 2 ( 1; 1) and let N 2 N : 0 < N 1+1 min (x + 1; 1 x), and f (n) ( ) f (n) (x) is convex over [ 1; 1]. Assume that f (k) (x) = 0, k = 1; :::; n: Then (M )

H2N +1 (f ) (x)

f (x)

2n+1 (n + 1)!

! 1 f (n) ;

1 N +1

;

(54)

8 N N , N 2 N; n 2 Z+ : (M ) It holds lim H2N +1 (f ) (x) = f (x) : N !+1

Proof. By (20) we get (M )

H2N +1 (f ) (x)

f (x)

! 1 f (n) ; h (M ) H j h (n + 1)! 2N +1

n+1

xj

(53)

(x)

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! 1 f (n) ; h h (n + 1)! (setting h :=

1 N +1 )

2n+1 (n + 1)!

2n+1 N +1

! 1 f (n) ;

=

1 N +1

;

(55)

proving the claim. We make Remark 24 Let f 2 C+ ([ 1; 1]). Let the Chebyshev knots of second kind k xN;k = cos N 2 [ 1; 1], k = 1; :::; N; N 2 N f1g, which are the roots N 1 of ! N (x) = sin (N 1) t sin t, x = cos t 2 [ 1; 1]. Notice that xN;1 = 1 and xN;N = 1: De…ne k 1 ( 1) ! N (x) lN;k (x) := ; (56) (1 + k;1 + k;N ) (N 1) (x xN;k ) QN N 2, k = 1; :::; N , and ! N (x) = k=1 (x xN;k ) and i;j denotes the Kronecher’s symbol, that is i;j = 1, if i = j, and i;j = 0, if i 6= j. The Max-product Lagrange interpolation operators on Chebyshev knots of second kind, plus the endpoints 1, are de…ned by ([6], p. 12) WN lN;k (x) f (xN;k ) (M ) LN (f ) (x) = k=1WN ; x 2 [ 1; 1] : (57) k=1 lN;k (x) By [6], pp. 297-298 and [3], we get that (M )

LN

(j

m

xj ) (x)

2m+1 2 ; 3 (N 1)

(58)

8 x 2 ( 1; 1) and 8 m 2 N; 8 N 2 N, N 4: (M ) We see that LN (f ) (x) 0 is well de…ned and continuous for any x 2 PN [ 1; 1]. Following [6], p. 289, because 1, 8 x 2 [ 1; 1], for k=1 lN;k (x) = WN any x there exists k 2 f1; :::; N g : lN;k (x) > 0, hence k=1 lN;k (x) > 0. We have that lN;k (xN;k ) = 1, and lN;k (xN;j ) = 0, if k 6= j. Furthermore it holds (M ) (M ) LN (f ) (xN;j ) = f (xN;j ), all j 2 f1; :::; N g ; and LN (1) = 1. (M ) By [6], pp. 289-290, LN are positive sublinear operators. Finally we present Theorem 25 Let f 2 C n ([ 1; 1] ; R+ ), n 2 Z+ , x 2 ( 1; 1) and let N 2 N : N 4, with 0 < N 1 1 min (x + 1; 1 x), and f (n) ( ) f (n) (x) is convex over [ 1; 1]. Assume that f (k) (x) = 0, k = 1; :::; n: Then (M )

LN

(f ) (x)

f (x)

2n+2 2 3 (n + 1)!

! 1 f (n) ;

1 N

1

;

(59)

12

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8N 2N:N N 4; n 2 Z+ : (M ) It holds lim LN (f ) (x) = f (x) : N !+1

Proof. Using (20) we get: (M )

LN

(f ) (x)

f (x)

! 1 f (n) ; h (M ) L j h (n + 1)! N

! 1 f (n) ; h h (n + 1)! (setting h :=

1 N

2n+2 2 3 (N 1)

n+1

xj

(58)

(x)

=

1)

2n+2 2 3 (n + 1)!

! 1 f (n) ;

1 N

1

;

(60)

proving the claim.

References [1] G. Anastassiou, Moments in probability and approximation theory, Pitman Research Notes in Mathematics Series, Longman Group UK, New York, NY, 1993. [2] G. Anastassiou, Approximation by Sublinear Operators, Acta Mathematica Universitatis Comenianae, 87(2018)(2), 237-250. [3] G. Anastassiou, Approximation by Max-Product Operators, Fasc. Math. 60(2018), 5-28. [4] G. Anastassiou, Approximation of Fuzzy numbers by Max-product operators, Transylvanian Journal of Mathematics and Mechanics, 9(2017)(2),117-123. [5] G. Anastassiou, Approximations by Multivariate Sublinear and Max-product Operators under Convexity, Demonstratio Mathematica, 51(2018), 85-105. [6] B. Bede, L. Coroianu, S. Gal, Approximation by Max-Product type Operators, Springer, Heidelberg, New York, 2016.

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Symmetric identities for Carlitz’s generalized twisted q-Bernoulli numbers and polynomials associated with p-adic invariant integral on Zp Cheon Seoung Ryoo Department of Mathematics, Hannam University, Daejeon 34430, Korea

Abstract : In this paper, we study the symmetry for the Carlitz’s generalized twisted q-Bernoulli polynomials βn,χ,q,ζ (x). We obtain some interesting identities of the power sums and the Carlitz’s generalized twisted q-Bernoulli polynomials βn,χ,q,ζ (x) using the symmetric properties for the p-adic invariant integral on Zp . Key words : Symmetric properties, power sums, Bernoulli numbers and polynomials, Carlitz’s generalized twisted q-Bernoulli numbers and polynomials, p-adic invariant integral on Zp . 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Bernoulli polynomials, q-Bernoulli polynomials, the second kind Bernoulli polynomials, Euler polynomials, tangent polynomials, and Bell polynomials were studied by many authors( see [1, 3, 4, 5, 6, 7, 8, 9, 10]). Recently, Y. He obtained several identities of symmetry for Carlitz’s qBernoulli numbers and polynomials in complex field(see [1]). D. Kim et al.[3] studied some identities of symmetry for generalized Carlitz’s q-Bernoulli numbers and polynomials by using the p-adic integrals on Zp in p-adic field. The purpose of this paper is to obtain some interesting identities of the power sums and Carlitz’s generalized twisted q-Bernoulli polynomials βn,χ,q,ζ (x) using the symmetric properties for the p-adic invariant integral on Zp . Let p be a fixed prime number. Throughout this paper we use the notation: [x]q =

1 − qx , 1−q

[x]−q =

1 − (−q)x (cf. [1-4]) . 1+q

Hence, limq→1 [x] = x for any x with |x|p ≤ 1 in the present p-adic case. Let g ∈ U D(Zp ) = {g|g : Zp → Cp is uniformly differentiable function}. For g ∈ U D(Zp ) the p-adic invariant integral on Zp is defined by Kim as follows: ∫

p −1 1 ∑ I1 (g) = g(x)dµ1 (x) = lim N g(x), (cf. [2, 3, 4]) . N →∞ p Zp x=0 N

(1.1)

Let a fixed positive integer d with (p, d) = 1, set X = Xd = lim(Z/dpN Z), ←− N ∪ ∗ X = a + dpZp ,

X1 = Zp ,

0 0, Next D∗a anchored at a ∈ R, see [10], p. 50.

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The author in [7], pp. 82-83, proved the following left Caputo fractional Landau inequality: Let 0 < ν ≤ 1, f ∈ AC 2 ([0, b]) (i.e. f 0 ∈ AC ([0, b]), ν+1 absolutely continuous functions), ∀ b > 0. Suppose kf k∞,R+ < +∞, D∗0 f ∈ L∞ (R+ ), and

ν+1

ν+1

D∗a f (3) ≤ D∗0 f ∞,R+ , ∀ a ≥ 0. ∞,[a,+∞) Then ν   ν+1 ν 1   ν+1   ν+1

2 − 1

Dν+1 f (Γ (ν + 2)) ν+1 kf k∞,R+ , kf k∞,R+ ≤ (ν + 1) ∗0 ∞,R+ ν (4) that is kf 0 k∞,R+ is finite. The last inequality is another inspiration. The author’s monographs [2], [3], [4], [5], [6], [8], motivate and support largely this work too. See also [1]. Under the point of view of local fractional differentiation the author examines the broad area of analytic inequalities and produces a variety of well-known inequalities in a local fractional setting over a negative domain to all possible directions.

0

2

Background

We mention Definition 1 ([11]) Let x, x0 ∈ [a, b], f ∈ C ([a, b]). The Riemann-Liouville (R-L) fractional derivative of a function f of order q (0 < q < 1) is defined as  q  Dx+ f (x0 ) , x0 > x, Dxq f (x0 ) = = q Dx− f (x0 ) , x0 < x ( R x0 0 −q d 1 (x − t) f (t) dt, x0 > x, dx0 x R (5) x −q d Γ (1 − q) − dx (t − x0 ) f (t) dt, x0 < x, 0 x0 the left and right R-L fractional derivatives, respectively. We need Definition 2 ([11], [12]) The local fractional derivative of order q (0 < q < 1) of a function f ∈ C ([a, b]) is defined as Dq f (x) = lim Dxq (f (x0 ) − f (x)) . 0 x →x

(6)

More generally we define

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Definition 3 ([9]) Let N ∈ Z+ , 0 < q < 1, the local fractional derivative of order (N + q) of a function f ∈ C N ([a, b]) is defined by ! N X f (n) (x) 0 n N +q q 0 D f (x) = lim Dx f (x ) − (x − x) . (7) x0 →x n! n=0 If N = 0, then Definition 3 collapses to Definition 2. We need Definition 4 (related to Definition 3) Let f ∈ C N ([a, b]), N ∈ Z+ . Set ! N X f (n) (x) 0 n 0 q 0 F (x, x − x; q, N ) := Dx f (x ) − (x − x) . n! n=0

(8)

Let x0 − x := t, then x0 = x + t, and F (x, t; q, N ) =

Dxq

N X f (n) (x) n t f (x + t) − n! n=0

! .

(9)

We make Remark 5 Here x0 , x ∈ [a, b], and a ≤ x + t ≤ b, equivalently a − x ≤ t ≤ b − x. From a ≤ x ≤ b, we get a − x ≤ 0 ≤ b − x. We assume here that F (x, ·; q, N ) ∈ C 1 ([a − x, b − x]). Clearly, then it holds DN +q f (x) = F (x, 0; q, N ) ,

(10)

and DN +q f (x) exists in R. We would need: Theorem 6 ([9]) Let f ∈ C N ([a, b]), N ∈ Z+ . F (x, ·; q, N ) ∈ C 1 ([a − x, b − x]). Then

Here x, x0 ∈ [a, b], and

N X DN +q f (x) 0 f (n) (x) 0 n q f (x ) = (x − x) + |x − x| + n! Γ (q + 1) n=0 0

1 Γ (q + 1)

x0 −x

Z 0

(11)

dF (x, t; q, N ) q |(x0 − x) − t| dt. dt

Corollary 7 (to Theorem 6, N = 0) Let f ∈ C ([a, b]), x, x0 ∈ [a, b], and F (x, ·; q, 0) ∈ C 1 ([a − x, b − x]). Then f (x0 ) = f (x) + 1 Γ (q + 1)

Z 0

x0 −x

Dq f (x) 0 q |x − x| + Γ (q + 1)

(12)

dF (x, t; q, 0) q |(x0 − x) − t| dt. dt 3

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We make Remark 8 Let f ∈ C N ([a, b]), N ∈ Z+ . Here x, x0 ∈ [a, b] : x0 < x, and F (x, ·; q, N ) ∈ C 1 ([a − x, b − x]), 0 < q < 1. By Theorem 6 we get N X DN +q f (x) f (n) (x) 0 n q (x − x) + (x − x0 ) − f (x ) = n! Γ (q + 1) n=0 0

1 Γ (q + 1)

Z

0

dF (x, t; q, N ) q (t − x0 + x) dt. dt

x0 −x

(13)

Clearly then we get: Let f ∈ C N ([a, 0]), a < 0, N ∈ Z+ , F (0, ·; q, N ) ∈ C 1 ([a, 0]), 0 < q < 1. Then, for any x ∈ [a, 0], we derive f (x) =

N X f (n) (0) n DN +q f (0) q x + (−x) − n! Γ (q + 1) n=0

1 Γ (q + 1)

0

Z

x

dF (0, t; q, N ) q (t − x) dt. dt

(14)

In this article we will use a lot (14). Remark 9 Let f ∈ C N ([a, 0]), N ∈ Z+ , a < 0, x ∈ [a, 0]; F (0, ·; q, N ) ∈ C 1 ([a, 0]), 0 < q < 1. Then, by (14), we have f (x) =

−

N X f (n) (0) n DN +q f (0) q x + (−x) n! Γ (q + 1) n=0

1 Γ (q + 1)

Z

0

x

(15)

dF (0, t; q, N ) q (t − x) dt. dt

Assume that f (n) (0) = 0, n = 0, 1, ..., N, and DN +q f (0) = 0 (= F (0, 0; q, N ) = D0q f (0)). Then Z 0 1 dF (0, t; q, N ) q −f (x) = (t − x) dt, (16) Γ (q + 1) x dt ∀ x ∈ [a, 0] . Here it is F (0, t; q, N ) = D0q (f (t)) ∈ C 1 ([a, 0]) , where D0q is the right Riemann-Liouville fractional derivative. Let a ≤ x ≤ w ≤ 0, then Z 0 1 dF (0, t; q, N ) q −f (w) = (t − w) dt. Γ (q + 1) w dt

(17)

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Consider p1 , q1 > 1 :

1 p1

+

1 q1

= 1. Then

1 |f (w)| = Γ (q + 1)

0

Z

w

dF (0, t; q, N ) (t − w)q dt ≤ dt

 q11 Z 0  p11 dF (0, t; q, N ) q1 qp 1 dt (t − w) dt = dt w w qp1 +1 Z q  q1 0 1 dF (0, t; q, N ) 1 1 (−w) p1 dt = 1 Γ (q + 1) (qp1 + 1) p1 dt w

1 Γ (q + 1)

Z

0

qp1 +1

1 1 (−w) p1 q1 , 1 (z (w)) Γ (q + 1) (qp1 + 1) p1

where

0

Z z (w) :=

w

all a ≤ x ≤ w ≤ 0, and z (0) = 0. From Z 0

we get

dF (0, t; q, N ) q1 dt, dt

w

−z (w) =

(18)

(19)

dF (0, t; q, N ) q1 dt, dt

dF (0, w; q, N ) q1 , −z (w) = (−z (w)) = dw

(20)

dF (0, w; q, N ) 1 = (−z 0 (w)) q1 . dw

(21)

dF (0, w; q, N ) ≤ |f (w)| dw

(22)

0

0

and

Therefore we obtain

1 Γ (q + 1) (qp1 + 1) Hence it holds

1 p1

(−w)

qp1 +1 p1

1

1

(z (w)) q1 (−z 0 (w)) q1 .

0

dF (0, w; q, N ) dw ≤ (23) |f (w)| dw x Z 0 qp1 +1 1 1 (−w) p1 (z (w) (−z 0 (w))) q1 dw ≤ 1 p1 x Γ (q + 1) (qp1 + 1) Z 0  p11 Z 0  q11 1 qp1 +1 0 (−w) dw z (w) (−z (w)) dw = 1 x x Γ (q + 1) (qp1 + 1) p1 ! p1  1 qp +2 1 (−x) 1 z 2 (w) 0 q1 1 − | = (24) 1 qp1 + 2 2 x Γ (q + 1) (qp1 + 1) p1 Z

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2

1 1

(−x)

1

qp1 +2 p1

(z (x)) q1 1

.

2 q1

Γ (q + 1) (qp1 + 1) p1 (qp1 + 2) p1 We have proved that 0

Z

dF (0, w; q, N ) dw ≤ |f (w)| dw

x q+ p2

(−x)

 q21 dF (0, w; q, N ) q1 dw . dw

0

Z

1 1

1

2 q1 Γ (q + 1) [(qp1 + 1) (qp1 + 2)] p1

x

(25)

We have established the following negative domain Lp -Opial type local right fractional inequality: Theorem 10 Let p1 , q1 > 1 : p11 + q11 = 1; f ∈ C N ([a, 0]), N ∈ Z+ , a < 0, x ∈ [a, 0]; F (0, ·; q, N ) ∈ C 1 ([a, 0]), 0 < q < 1. Assume that f (n) (0) = 0, n = 0, 1, ..., N, and DN +q f (0) = 0 (= F (0, 0; q, N ) = D0q f (0)). [Here it is F (0, t; q, N ) = D0q (f (t)) ∈ C 1 ([a, 0]), where D0q is the right Riemann-Liouville fractional derivative]. Then Z 0 dF (0, t; q, N ) dt ≤ |f (t)| dt x q+ p2

(−x)

0

Z

1 1

1

2 q1 Γ (q + 1) [(qp1 + 1) (qp1 + 2)] p1

x

 q21 dF (0, t; q, N ) q1 dt , dt

(26)

⇔ it holds

q dD (f (t)) dt ≤ |f (t)| 0 dt x Z 0 q  q21 q+ 2 dD0 (f (t)) q1 (−x) p1 , 1 1 dt dt x 2 q1 Γ (q + 1) [(qp1 + 1) (qp1 + 2)] p1 Z

0

(27)

∀ x ∈ [a, 0] . The case p1 = q1 = 2 follows: Corollary 11 All as in Theorem 10, with p1 = q1 = 2. Then Z 0 dF (0, t; q, N ) dt ≤ |f (t)| dt x q+1

(−x) p 2Γ (q + 1) (q + 1) (2q + 1)

Z

0

x

(28)

! dF (0, t; q, N ) 2 dt , dt

⇔

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it holds

q dD (f (t)) dt ≤ |f (t)| 0 dt x 2 ! Z 0 q q+1 (−x) dD0 (f (t)) p dt , dt 2Γ (q + 1) (q + 1) (2q + 1) x Z

0

(29)

∀ x ∈ [a, 0] . We make Remark 12 Let f1 , f2 according to the assumptions of Theorem 10. Then Z 0 dF1 (0, t1 ; q, N ) 1 q −f1 (x1 ) = (t1 − x1 ) dt1 , (30) Γ (q + 1) x1 dt1 ∀ x1 ∈ [a1 , 0] , a1 < 0; 1 −f2 (x2 ) = Γ (q + 1)

Z

0

x2

dF2 (0, t2 ; q, N ) q (t2 − x2 ) dt2 , dt2

(31)

∀ x2 ∈ [a2 , 0] , a2 < 0. Here it is Fi (0, ti ; q, N ) = D0q (fi (ti )) ∈ C 1 ([ai , 0]) , i = 1, 2; where D0q is the right Riemann-Liouville fractional derivative. Consider p1 , q1 > 1 : p11 + q11 = 1. Hence Z 0 dFi (0, ti ; q, N ) 1 (ti − xi )q dti , |fi (xi )| ≤ Γ (q + 1) xi dti

(32)

i = 1, 2; ∀ xi ∈ [ai , 0] . We get by H¨ older’s inequality: |f1 (x1 )| ≤ 1 Γ (q + 1)

Z

0

(t1 − x1 )

qp1

 p11 Z

0

dt1 x1

x1

 q11 dF1 (0, t1 ; q, N ) q1 dt1 ≤ dt1

(33)

qp1 +1

1 (−x1 ) p1 dF1 (0, t1 ; q, N )

,

Γ (q + 1) (qp1 + 1) p11 dt1 q1 ,[a1 ,0]

∀ x1 ∈ [a1 , 0] . Similarly, we obtain qq1 +1

1 (−x2 ) q1 dF2 (0, t2 ; q, N )

|f2 (x2 )| ≤ ,

Γ (q + 1) (qq1 + 1) q11 dt2 p1 ,[a2 ,0]

(34)

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∀ x2 ∈ [a2 , 0] . Therefore we have 1

|f1 (x1 )| |f2 (x2 )| ≤

(−x1 )

qp1 +1 p1

(−x2 )

qq1 +1 q1

(35) 1 1 (Γ (q + 1)) (qp1 + 1) p1 (qq1 + 1) q1



dF2 (0, t2 ; q, N )

dF1 (0, t1 ; q, N )



≤



dt1 dt2 q1 ,[a1 ,0] p1 ,[a2 ,0] 2

1

1

(using Young’s inequality for a, b ≥ 0, a p1 b q1 ≤ pa1 + qb1 ) # " qq +1 qp +1 1 (−x2 ) 1 (−x) 1 + 1 1 2 p1 q1 (Γ (q + 1)) (qp1 + 1) p1 (qq1 + 1) q1



dF1 (0, t1 ; q, N )

dF2 (0, t2 ; q, N )



,



dt1 dt2 q1 ,[a1 ,0] p1 ,[a2 ,0]

(36)

∀ xi ∈ [ai , 0] , i = 1, 2. So far we have established |f1 (x1 )| |f2 (x2 )| h

(−x)qp1 +1 p1

+

(−x2 )qq1 +1 q1

i≤

1 2

1

1

(37)

(Γ (q + 1)) (qp1 + 1) p1 (qq1 + 1) q1



dF1 (0, t1 ; q, N )



dt1

q1 ,[a1 ,0]



dF2 (0, t2 ; q, N )



dt2

,

p1 ,[a2 ,0]

∀ xi ∈ [ai , 0] , i = 1, 2. The denominator of left hand side of (37) can be zero only when x1 = 0 and x2 = 0. By integrating (37) over [a1 , 0] × [a2 , 0] we get Z 0Z 0 |f (x1 )| |f2 (x2 )| dx1 dx2 a1 a2 h 1 qp i≤ (38) 1 1 2 (−x) 1 +1 (−x2 )qq1 +1 a1 a2 (Γ (q + 1)) (qp1 + 1) p1 (qq1 + 1) q1 + p q 1

1



dF1 (0, t1 ; q, N )



dt1

q1 ,[a1 ,0]



dF2 (0, t2 ; q, N )



dt2

.

p1 ,[a2 ,0]

We have proved the following negative domains local right fractional HilbertPachpatte inequality: Theorem 13 Let p1 , q1 > 1 :

1 1 p1 + q 1

= 1; i = 1, 2 for fi ∈ C N ([ai , 0]), N ∈ Z+ , (n)

ai < 0; Fi (0, ·; q, N ) ∈ C 1 ([ai , 0]), 0 < q < 1. Assume that fi (0) = 0, n = 0, 1, ..., N, and DN +q fi (0) = 0 , i = 1, 2 (i.e. Fi (0, 0; q, N ) = D0q fi (0) = 0). [Here it is Fi (0, ti ; q, N ) = D0q (fi (ti )) ∈ C 1 ([ai , 0]), where D0q is the right Riemann-Liouville fractional derivative]. Then Z 0Z 0 |f (x1 )| |f2 (x2 )| dx1 dx2 a1 a2 h 1 qp i≤ (39) 1 1 2 (−x) 1 +1 (−x2 )qq1 +1 a1 a2 (Γ (q + 1)) (qp1 + 1) p1 (qq1 + 1) q1 + p q 1

1

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dF1 (0, t1 ; q, N )

dF2 (0, t2 ; q, N )



,



dt1 dt2 q1 ,[a1 ,0] p1 ,[a2 ,0] ⇔ it holds Z 0Z 0 |f (x1 )| |f2 (x2 )| dx1 dx2 a1 a2 h 1 qp i≤ 1 1 2 (−x) 1 +1 (−x2 )qq1 +1 a1 a2 (Γ (q + 1)) (qp1 + 1) p1 (qq1 + 1) q1 + p1 q1

q

dD0 (f1 (t1 ))



dt1

q1 ,[a1 ,0]

q

dD0 (f2 (t2 ))



dt2

(40)

.

p1 ,[a2 ,0]

We make Remark 14 Let f ∈ C N ([a, 0]), a < 0, N ∈ Z+ , F (0, ·; q, N ) ∈ C 1 ([a, 0]), 0 < q < 1. Then for any x ∈ [a, 0], we have f (x) =

N X f (n) (0) n DN +q f (0) q x + (−x) n! Γ (q + 1) n=0

1 − Γ (q + 1)

Z

0

x

(41)

dF (0, t; q, N ) q (t − x) dt. dt

(n)

Assume that f (0) = 0, n = 0, 1, ..., N. Here DN +q f (0) = F (0, 0; q, N ) = D0q f (0), where D0q is the right Riemann-Liouville fractional derivative. So far we have f (x) =

DN +q f (0) q (−x) + R (x) , Γ (q + 1)

where R (x) := −

1 Γ (q + 1)

0

Z

dF (0, t; q, N ) q (t − x) dt. dt

x

We also assume that D0q f ∈ C 1 ([a, 0]). We can rewrite  Z 0 d q 1 q D f (t) (t − x) dt. R (x) = − Γ (q + 1) x dt 0

(42)

(43)

(44)

We notice that 1 |R (x)| ≤ Γ (q + 1)

Z

0

x

d q D f (t) (t − x)q dt ≤ dt 0

q+1

d q

1 (−x)

D f (t) .

Γ (q + 1) dt 0 ∞,[a,0] q + 1

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That is

q+1

d q (−x)

, |R (x)| ≤ D0 f (t)

Γ (q + 2) dt ∞,[a,0]

(45)

∀ x ∈ [a, 0] . Hence, it holds Z 0 Z 0 Z DN +q f (0) 0 q f (x) dx = R (x) dx = (−x) dx + Γ (q + 1) a a a q+1

DN +q f (0) (−a) Γ (q + 1) q + 1

Z

0

+

R (x) dx = a

Therefore, we get Z 0 f (x) dx − a

DN +q f (0) q+1 (−a) + Γ (q + 2)

DN +q f (0) q+1 (−a) = Γ (q + 2)

Z

Z

(46)

0

R (x) dx. a

0

R (x) dx.

(47)

a

Consequently, we derive Z 0 Z 0 (D0q f ) (0) q+1 f (x) dx − |R (x)| dx ≤ (−a) ≤ Γ (q + 2) a a

d q

d q

Z 0

D f (t)

D f (t) q+2 dt 0 dt 0 ∞,[a,0] ∞,[a,0] (−a) q+1 (−x) dx = Γ (q + 2) Γ (q + 2) q+2 a

d q

q+2

D f (t) (−a) dt 0 ∞,[a,0] = . Γ (q + 3)

(48)

(49)

We have proved the following negative domain local right fractional comparison of means results: Theorem 15 Let f ∈ C N ([a, 0]), a < 0, N ∈ Z+ , D0q f ∈ C 1 ([a, 0]), 0 < q < 1. Assume f (n) (0) = 0, n = 0, 1, ..., N. Then

q+2 Z 0 q d q dt D0 f (t) ∞,[a,0] (−a) (D f ) (0) q+1 0 ≤ f (x) dx − (−a) , (50) Γ (q + 2) Γ (q + 3) a ⇔ Z 1 (−a)

0

a

q+1

d Dq f (t) (−a) 0 (D0q f ) (0) dt ∞,[a,0] q f (x) dx − (−a) ≤ . Γ (q + 2) Γ (q + 3)

(51)

We make Remark 16 All as in Theorem 10. Then Z 0 1 dF (0, t; q, N ) q −f (x) = (t − x) dt, Γ (q + 1) x dt

(52)

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∀ x ∈ [a, 0] . Let p1 , q1 > 1 :

1 p1

+

1 q1

= 1. Thus

1 Γ (q + 1)

0

Z

1 |f (x)| ≤ Γ (q + 1)

x

dF (0, t; q, N ) (t − x)q dt ≤ dt

 q11 Z 0  p11 dF (0, t; q, N ) q1 qp1 dt (t − x) dt ≤ dt x x qp1 +1

dF (0, t; q, N ) (−x) p1 1

1 .

Γ (q + 1) dt q1 ,[a,0] (qp1 + 1) p1 Z

0

That is

qp1 +1 p1

(−x)

|f (x)| ≤

1

Γ (q + 1) (qp1 + 1) p1



dF (0, t; q, N )

,

dt q1 ,[a,0]

(53)

(54)

∀ x ∈ [a, 0] . Therefore q1

|f (x)|

q1 (q+1)−1

(−x)

≤

q1

q

(Γ (q + 1)) 1 (qp1 + 1) p1



dF (0, t; q, N ) q1

.

dt q1 ,[a,0]

(55)

Consequently, it holds Z

0

q

|f (x)| 1 dx ≤ a



dF (0, t; q, N ) q1

h iq1 . 1

dt q1 ,[a,0] q1 (q + 1) Γ (q + 1) (qp1 + 1) p1 q1 (q+1)

(−a)

(56)

That is (q+1)

kf kq1 ,[a,0] ≤

(−a)

1

1

Γ (q + 1) (qp1 + 1) p1 (q1 (q + 1)) q1



dF (0, t; q, N )

. (57)

dt q1 ,[a,0]

We have proved the following negative domain local right fractional Poincare inequality: Theorem 17 All as in Theorem 10. Then (q+1)

kf kq1 ,[a,0] ≤

(−a)

Γ (q + 1) (qp1 + 1)

1 p1

(q1 (q + 1))

1 q1



q 0

(D0 (f ))

.

(58)

q1 ,[a,0]

We make

11

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Remark 18 All as in Theorem 10, plus r > 0. By (54) we have (−x)

|f (x)| ≤

qp1 +1 p1 1

Γ (q + 1) (qp1 + 1) p1



dF (0, t; q, N )

,

dt q1 ,[a,0]

(59)

∀ x ∈ [a, 0] . Hence it holds r |f (x)| ≤ h

  r q+ p1

(−x)

1

1

Γ (q + 1) (qp1 + 1) p1

r

0 ir (D0q (f ))

.

(60)

q1 ,[a,0]

Consequently, we get 0

Z a

  r q+ p1 +1

(−a)

1

r ir h  |f (x)| dx ≤ h 1 Γ (q + 1) (qp1 + 1) p1 r q+

1 p1



+1

r

0 i (D0q (f ))

.

q1 ,[a,0]

(61) We have proved the following negative domain local ritgh fractional Sobolev type inequality: Theorem 19 All as in Theorem 10, plus r > 0. Then q+ p1 + r1

kf kr,[a,0] ≤

(−a)

1

h  1 Γ (q + 1) (qp1 + 1) p1 r q +

1 p1



+1



q 0 (f ))

(D

0 i q 1 r

.

(62)

1 ,[a,0]

References [1] F.B. Adda, J. Cresson, Fractional differentiation equations and the Schr¨ odinger equation, Applied Math. & Computation, 161 (2005), 323-345. [2] G.A. Anastassiou, Quantitative Approximations, CRC press, Boca Raton, London, New York, 2001. [3] G.A. Anastassiou, Fractional differentiation inequalities, Springer, Heidelberg, New York, 2009. [4] G.A. Anastassiou, Probabilistic Inequalities, World Scientific, Singapore, New York, 2010. [5] G.A. Anastassiou, Advanced Inequalities, World Scientific, Singapore, New York, 2010. [6] G.A. Anastassiou, Intelligent Mathematics: Springer, Heidelberg, New York, 2011.

Computational Analysis,

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[7] G.A. Anastassiou, Advances on Fractional Inequalities, Springer, Heidelberg, New York, 2011. [8] G.A. Anastassiou, Intelligent Comparisons: Analytic Inequalities, Springer, Heidelberg, New York, 2016. [9] G.A. Anastassiou, Local Fractional Taylor Formula, J. of Computational Analysis and Applications, accepted, 2018. [10] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Heidelberg, New York, 2010. [11] K.M. Kolwankar, Local fractional calculus: a Review, arXiv: 1307:0739v1 [nlin.CD] 2 Jul. 2013. [12] K.M. Kolwankar and A.D. Gangal, Local fractional calculus: a calculus for fractal space-time, Fractals: theory and applications in engineering, 171181, London, New York, Springer, 1999. [13] Z. Opial, Sur une in´egalit´e, Ann. Polon. Math. 8 (1960), 29-32. ¨ [14] A. Ostrowski, Uber die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227.

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Approximate controllability for semilinear integro-differential control equations in Hilbert spaces Yong Han Kang1 , Jin-Mun Jeong2,∗ and Ah-ran Park3 1

Institute of Liberal Education, Catholic University of Daegu Gyeongsan, 712-702, South Korea E-mail:[email protected]

2,3

Department of Applied Mathematics, Pukyong National University Busan 48513, South Korea E-mail: ∗ [email protected](Corresponding author), [email protected]

Abstract This paper deals with the approximate controllability for a class of semilinear integro-differential functional control equations, which is provided under general sufficient conditions on the system operator, controller and nonlinear terms. Our used tool is applying results similar to Fredholm alternative for nonlinear operators under restrictive assumptions. Finally, a simple example to which our main result can be applied is given. Keywords: approximate controllability, semilinear control equations, integro-differential control equations, controller, Fredholm alternative. AMS Classification: Primary 93B05, 35F25

1

Introduction

In this paper, we deal with the approximate controllability for semilinear integro-differential functional control equations in the form ( Rt d 0 < t ≤ T, dt x(t) = Ax(t) + 0 k(t − s)g(s, x(s), u(s))ds + Bu(t), (1.1) x(0) = x0 Email: [email protected](A. park), *[email protected](J. jeong), [email protected](E. Y. Ju) This work was supported by a Research Grant of Pukyong National University(2021 Year).

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in a Hilbert space H, where k belongs to L2 (0, T )(T > 0) and g is a nonlinear mapping as detailed in Section 2. The principal operator A generates an analytic semigroup (S(t))t≥0 and B is a bounded linear operator from another Hilbert space U to H. The controllability problem is a question of whether is possible to steer a dynamic system from an initial state to an arbitrary final state using the set of admissible controls. Naito [13] was the first to deal with the range condition argument of controller in order to obtain the approximate controllability of a semilinear control system. In [3, 9, 17, 18], they have studied continuously about controllability of semilinear systems dominated by linear parts(in case g ≡ 0) by assuming that S(t) is compact operator for each t > 0 as matters connected with [13]. Another approach used to obtain sufficient conditions for approximate solvability of nonlinear equations is a fixed point theorem combined with technique of operator transformations by configuring the resolvent as seen in [2] The controllability for various nonlinear equations has been studied by many authors, for example, see [5, 6, 12] for local controllability of neutral functional differential systems with unbounded delay, [10, 14] for neutral evolution integrodifferential systems with state dependent delay. Sukavanam and Tomar [15] studied the approximate controllability for the general retarded initial value problem by assuming that the Lipschitz constant of the nonlinear term is less then 1, and Wang [17] for general retarded semilinear equations assuming the growth condition of the nonlinear term and the compactness of the semigroup. In this paper, authors want to use a different method than the previous one. Our used tool is the theorems similar to the Fredholm alternative for nonlinear operators under restrictive assumption, which is on the solution of nonlinear operator equations λT (x) − F (x) = y in dependence on the real number λ, where T and F are nonlinear operators defined a Banach space X with values in a Banach space Y . In order to obtain the approximate controllability for a class of semilinear integro-differential functional control equations, it is necessary to suppose that T acts as the identity operator while F related to the nonlinear term of (1.1) is completely continuous In Section 2, we introduce regularity properties for (1.1). Since we apply the Fredholm theory in the proof of the main theorem, we assume some compactness of the embedding between intermediate spaces. Then by virtue of Aubin [1], we can show that the solution mapping of a control space to the terminal state space is completely continuous. Based on Section 2, it is shown the sufficient conditions on the controller and nonlinear terms for approximate controllability for (1.1) by using the Fredholm theory. Finally, a simple example to which our main result can be applied is given.

2

Semilinear functional equations

Let V and H be complex Hilbert spaces forming a Gelfand triple V ,→ H ≡ H ∗ ,→ V ∗ 2

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by identifying the antidual of H with H. Therefore, for the brevity, we may regard that ||u||∗ ≤ |u| ≤ ||u|| for all u ∈ V , where the notations | · |, || · || and || · ||∗ denote the norms of H, V and V ∗ , respectively as usual. Let a(u, v) be a bounded sesquilinear form defined in V × V satisfying G˚ arding’s inequality Re a(u, u) ≥ c0 ||u||2 − c1 |u|2 ,

c0 > 0,

c1 ≥ 0.

Let A be the operator associated with this sesquilinear form: (Au, v) = −a(u, v),

u, v ∈ V.

Then A is a bounded linear operator from V to V ∗ . The realization of A in H which is the restriction of A to D(A) = {u ∈ V : Au ∈ H} is also denoted by A. For the sake of simplicity we assume that c1 = 0 and hence the closed half plane {λ : Re λ ≥ 0} is contained in the resolvent set of A. It is known that A generates an analytic semigroup S(t) in both H and V ∗ . As seen in Lemma 3.6.2 of [16], there exists a constant M > 0 such that |S(t)x| ≤ M |x| and ||S(t)x||∗ ≤ M ||x||∗ ,

(2.1)

The following initial value problem for the abstract linear parabolic equation ( dx(t) 0 < t ≤ T, dt = Ax(t) + k(t), x(0) = x0 .

(2.2)

By virtue of Theorem 3.3 of [4](or Theorem 3.1 of [9]), we have the following result on the corresponding linear equation (2.2). Proposition 2.1. Suppose that the assumptions for the principal operator A stated above are satisfied. Then the following properties hold: 1) For x0 ∈ V and k ∈ L2 (0, T ; H), T > 0, there exists a unique solution x of (2.2) belonging to L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H) ⊂ C([0, T ]; V ) and satisfying ||x||L2 (0,T ;D(A))∩W 1,2 (0,T ;H) ≤ C1 (||x0 || + ||k||L2 (0,T ;H) ),

(2.3)

where C1 is a constant depending on T . 2) Let x0 ∈ H and k ∈ L2 (0, T ; V ∗ ), T > 0. Then, there exists a unique solution x of (2.2) belonging to L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C([0, T ]; H) and satisfying ||x||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C1 (|x0 | + ||k||L2 (0,T ;V ∗ ) ),

(2.4)

where C1 is a constant depending on T . 3

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By virtue of Proposition 2.1, we have the following lemma. Rt Lemma 2.1. Suppose that k ∈ L2 (0, T ; H) and x(t) = 0 S(t − s)k(s)ds for 0 ≤ t ≤ T . Then there exists a constant C2 such that

and

||x||L2 (0,T ;H) ≤ C2 T ||k||L2 (0,T ;H) ,

(2.5)

√ ||x||L2 (0,T ;V ) ≤ C2 T ||k||L2 (0,T ;H) .

(2.6)

Consider the following initial value problem for the abstract semilinear parabolic equation ( Rt d dt x(t) = Ax(t) + 0 k(t − s)g(s, x(s), u(s))ds + Bu(t), (2.7) x(0) = x0 . Let U be a Hilbert space and the controller operator B be a bounded linear operator from U to H. Let g : R+ × V × U → H be a nonlinear mapping satisfying the following: Assumption (F). (i)

For any x ∈ V , u ∈ U the mapping g(·, x, u) is strongly measurable;

(ii)

There exist positive constants L0 , L1 , L2 such that (a)

u 7→ g(t, x, u) is an odd mapping (g(·, x, −u) = −g(·, x, u));

(b)

for all t ∈ R+ , x, x ˆ ∈ V , and u, u ˆ ∈ U, |g(t, x, u) − g(t, x ˆ, u ˆ)| ≤ L1 ||x − x ˆ|| + L2 ||u − u ˆ||U , |g(t, 0, 0)| ≤ L0 .

For x ∈ L2 (0, T ; V ), we set Z

t

k(t − s)g(s, x(s), u(s))ds

f (t, x, u) = 0

where k belongs to L2 (0, T ). Lemma 2.2. Let Assumption (F) be satisfied. Assume that x ∈ L2 (0, T ; V ) for any T > 0. Then f (·, x, u) ∈ L2 (0, T ; H) and √ ||f (·, x, u))||L2 (0,T ;H) ≤ L0 ||k||L2 (0,T ) T / 2 √ + ||k||L2 (0,T ) T (L1 ||x||L2 (0,T ;V ) + L2 ||u||L2 (0,T ;U ) ). (2.8) Moreover if x, x ˆ ∈ L2 (0, T ; V ), then ||f (·, x, u) − f (·, x ˆ, u ˆ)||L2 (0,T ;H) √ ˆ||L2 (0,T ;V ) + L2 ||u − u ˆ||L2 (0,T ;U ) ). ≤ ||k||L2 (0,T ) T (L1 ||x − x

(2.9)

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The proof is easily from Assumption (F), and using the H¨older inequality. By virtue of Theorem 2.1 of [8], we have the following result on (2.7). Proposition 2.2. Let Assumption (F) be satisfied. Then there exists a unique solution x of (2.7) such that x ∈ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ∗ ) ⊂ C([0, T ]; H) for any x0 ∈ H. Moreover, there exists a constant C3 such that ||x||L2 (0,T ;V )∩W 1,2 (0,T ;V ∗ ) ≤ C3 (|x0 | + ||u||L2 (0,T ;U ) ).

(2.10)

Corollary 2.1. Assume that the embedding D(A) ⊂ V is completely continuous. Let Assumption (F) be satisfied, and xu be the solution of equation (2.7) associated with u ∈ L2 (0, T ; U ). Then the mapping u 7→ xu is completely continuous from L2 (0, T ; U ) to L2 (0, T ; V ). Proof. If u is bounded in L2 (0, T ; U ), then so is xu in L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H) by (2.8). Since D(A) is compactly embedded in V by assumption, the embedding L2 (0, T ; D(A)) ∩ W 1,2 (0, T ; H) ⊂ L2 (0, T ; V ) is completely continuous in view of Theorem 2 of [1], the mapping u 7→ xu is completely continuous from L2 (0, T ; U ) to L2 (0, T ; V ).

3

Approximate controllability

Throughout this section, we assume that D(A) is compactly embedded in V . Let x(T ; f, u) be a state value of the system (2.7) at time T corresponding to the nonlinear term f and the control u. We define the reachable sets for the system (2.7) as follows: RT (f ) = {x(T ; f, u) : u ∈ L2 (0, T ; U )}, RT (0) = {x(T ; 0, u) : u ∈ L2 (0, T ; U )}. Definition 3.1. The system (2.7) is said to be approximately controllable in the time interval [0, T ] if for every desired final state x1 ∈ H and  > 0 there exists a control function u ∈ L2 (0, T ; U ) such that the solution x(T ; f, u) of (2.7) satisfies |x(T ; f, u) − x1 | < , that is, if RT (f ) = H where RT (f ) is the closure of RT (f ) in H, then the system (2.9) is called approximately controllable at time T . Let us introduce the theory of the degree for completely continuous perturbations of the identity operator, which is the infinite dimensional version of Borsuk’s theorem. Let 0 ∈ D be a bounded open set in a Banach space X, D its closure and ∂D its boundary. The number d[I − T ; D, 0] is the degree of the mapping I − T with respect to the set D and the point 0 (see Fuˇcik et al. [7] or Lloid [11]). 5

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Theorem 3.1. (Borsuk’s theorem) Let D be a bounded open symmetric set in a Banach space X, 0 ∈ D. Suppose that T : D → X be odd completely continuous operator satisfying T (x) 6= x for x ∈ ∂D. Then d[I − T ; D, 0] is odd integer. That is, there exists at least one point x0 ∈ D such that (I − T )(x0 ) = 0. Definition 3.2. Let T be a mapping defined by on a Banach space X with value in a real Banach space Y . The mapping T is said to be a (K, L, α)-homeomorphism of X onto Y if (i) (ii)

T is a homeomorphism of X onto Y ; there exist real numbers K > 0, L > 0, and α > 0 such that L||x||αX ≤ ||T (x)||Y ≤ K||x||αX ,

∀x ∈ X.

Lemma 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and F : X → Y a continuous operator satisfying ||F (x)||Y = N ∈ R+ . α ||x||X →∞ ||x||X lim sup

N N Then if |λ| ∈ / [K , L ] ∪ {0} then

lim

||x||X →∞

||λT (x) − F (x)||Y = ∞.

Proof. Suppose that there exist a constant M > 0 and a sequence {xn } ⊂ X such that ||λT (xn ) − F (xn )||Y ≤ M

(3.1)

as xn → ∞. From (3.1) it follows that λT (xn ) F (xn ) − → 0. α ||xn ||X ||xn ||αX Hence, we have lim sup n→∞

|λ|||T (xn )||Y = N, ||xn ||αX

N N , L ]. and so, |λ|K ≥ N ≥ |λ|L. It is a contradiction with |λ| ∈ / [K

Proposition 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and F : X → Y an odd completely continuous operator. Suppose that for λ 6= 0, lim

||x||X →∞

||λT (x) − F (x)||Y = ∞.

(3.2)

Then λT − F maps X onto Y . 6

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Proof. We follow the proof Theorem 1.1 in Chapter II of Fuˇcik et al. [7]. Suppose that there exists y ∈ Y such that λT (x) = y. Then from (3.2) it follows that F T −1 : Y → Y is an odd completely continuous operator and y ||y − F T −1 ( )||Y = ∞. λ ||y||Y →∞ lim

Let y0 ∈ Y . There exists r > 0 such that y ||y − F T −1 ( )||Y > ||y0 ||Y ≥ 0 λ for each y ∈ Y satisfying ||y||Y = r. Let Yr = {y ∈ Y : ||y||Y < r} be a open ball. Then by view of Theorem 3.1, we have d[y − F T −1 ( λy ); Yr , 0] is an odd number. For each y ∈ Y satisfying ||y||Y = r and t ∈ [0, 1], there is y y ||y − F T −1 ( ) − ty0 ||Y ≥ ||y − F T −1 ( )||Y − ||y0 ||Y > 0 λ λ and hence, by the homotopic property of degree, we have y y d[y − F T −1 ( ); Yr , y0 ] = d[y − F T −1 ( ); Yr , 0] 6= 0. λ λ Hence, by the existence theory of the Leray-Schauder degree, there exists a y1 ∈ Yr such that y1 y1 − F T −1 ( ) = y0 . λ We can choose x0 ∈ X satisfying λT (x0 ) = y1 , and so, λT (x0 ) − F (x0 ) = y0 . Thus, it implies that λT − F is a mapping of X onto Y . Combining Lemma 3.1. and Proposition 3,1, we have the following results. Corollary 3.1. Let T be an odd (K, L, α)-homeomorphism of X onto Y and F : X → Y an odd completely continuous operator satisfying ||F (x)||Y = N ∈ R+ . α ||x|| ||x||X →∞ X lim sup

N N Then if |λ| ∈ / [K , L ] ∪ {0} then λT − F maps X onto Y . Therefore, if N = 0, then for all λ 6= 0 the operator λT − F maps X onto Y .

First we consider the approximate controllability of the system (2.7) in case where the controller B is the identity operator on H under Assumption (F) on the nonlinear operator f in Section 2. Hence, noting that H = U , we consider the linear system given by ( d dt y(t) = Ay(t) + u(t), (3.3) y(0) = x0 , 7

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and the following semilinear control system ( d dt x(t) = Ax(t) + f (t, x(t), v(t)) + v(t), x(0) = x0 .

(3.4)

Theorem 3.2. Assume that lim sup ||u||→∞

||f (·, xu , u)||L2 (0,T ;H) < 1. ||u||L2 (0,T ;H)

(3.5)

Under the Assumption (F) we have RT (0) ⊂ RT (f ). Therefore, if the linear system (3.3) with f = 0 is approximately controllable, then so is the semilinear system (3.4). Proof. Let x(t) be solution of (3.4) corresponding to a control u. First, we show that there exist a v ∈ L2 (0, T ; H) such that ( v(t) = u(t) − f (t, x(t), v(t)), 0 < t ≤ T, v(0) = u(0). Let us define an operator F : L2 (0, T ; H) → L2 (0, T ; H) as F v = −f (·, xv , v). Then by Corollary 2.1, F is a compact mapping from L2 (0, T ; H) to itself, and we have lim ||λI(v) − F (v)||L2 (0,T ;H) = ∞,

||v||→∞

where the identity operator I on L2 (0, T ; H) is an odd (1, 1, 1)-homeomorphism. Thus, from (3.5) and Corollary 3.1, if λ ≥ 1 then λI − F maps L2 (0, T ; H) onto itself. Hence, we have showed that there exists a v ∈ L2 (0, T ; H) such that v(t) = u(t) − f (t, y(t), v(t)). Let y and x be solutions of (3.3) and (3.4) corresponding to controls u and v, respectively. Then, equation (3.4) is rewritten as d x(t) = Ax(t) + f (t, x(t), v(t)) + v(t), 0 < t ≤ T dt = Ax(t) + f (t, x(t), v(t)) + u(t) − f (t, y(t), v(t)) = Ax(t) + u(t) 8

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with x(0) = x0 , which means Z

t

S(t − s){f (s, x(s), v(s)) + v(s)}ds

x(t) =S(t)x0 + 0

Z

t

S(t − s)u(s)ds = y(t),

=S(t)x0 + 0

where y be solution of (3.3) corresponding to a control u. Therefore, we have proved that RT (0) ⊂ RT (f ). Corollary 3.2. Let us assume that √
||k||L2 (0,T ) T L1 C3 + L2 < 1, where C3 is the constant in Proposition 2.2. Under the Assumption (F), we have RT (0) ⊂ RT (f ) in case where B ≡ I. Proof. By Lemma 2.2 and Proposition 2.2, we have ||F u||L2 (0,T ;H) = ||f (·, xu , u)||L2 (0,T ;H) √ √ ≤ L0 ||k||L2 (0,T ) T / 2 + ||k||L2 (0,T ) T (L1 ||x||L2 (0,T ;V ) + L2 ||u||L2 (0,T ;U ) ) √  √
≤ L0 ||k||L2 (0,T ) T / 2 + ||k||L2 (0,T ) T L1 C3 |x0 | + ||u||L2 (0,T ;U ) + L2 ||u||L2 (0,T ;U ) . Hence, we have lim sup ||u||→∞

√
||F (u)||L2 (0,T ;H) ≤ ||k||L2 (0,T ) T L1 C3 + L2 . ||u||L2 (0,T ;U )

Thus, from Theorem 3.2, it follows that if λ ≥ 1 then λI − F maps L2 (0, T ; H) onto itself, and so, by the same argument as in the proof of theorem it holds that RT (0) ⊂ RT (f ). From now on, we consider the initial value problem for the semilinear parabolic equation (2.7). Let U be some Hilbert space and the controller operator B be a bounded linear operator from U to H. Assumption (B) There exists a constant β > 0 such that R(f ) ⊂ R(B) and ||Bu|| ≥ β||u||, Consider the linear system given by ( d dt y(t) y(0)

∀u ∈ L2 (0, T ; U ).

= Ay(t) + Bu(t), = x0 .

(3.6)

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Theorem 3.3. Under the Assumptions (3.5), (B) and (F), we have RT (0) ⊂ RT (f ). Therefore, if the linear system (3.6) with f = 0 is approximately controllable, then so is the semilinear system (2.7). Proof. Let y be a solution of the linear system (3.6) with f = 0 corresponding to a control u, and let x be a solutions of the semilinear system (3.4) corresponding to a control v. Set v(t) = u(t) − B −1 f (t, x(t), v(t)). Then, system (2.9) is rewritten as d x(t) = Ax(t) + f (t, x(t), v(t)) + Bv(t), 0 < t ≤ T dt = Ax(t) + f (t, x(t), v(t)) + Bu(t) − f (t, x(t), v(t)) with x(0) = x0 . Hence, we have Z

t

S(t − s){f (s, x(s), v(s)) + v(s))}ds

x(t) =S(t)x0 + 0

Z

t

S(t − s)u(s))ds = y(t).

=S(t)x0 + 0

Thus, we obtain that RT (0) ⊂ RT (f ).

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Convergence theorems and approximating endpoints for multivalued Suzuki mappings in hyperbolic spaces Preeyalak Chuadchawna1 , Ali Farajzadeh2 and Anchalee Kaewcharoen3∗ 2

3

Department of Mathematics, Razi University, Kermanshah, 67149, Iran E-mail: [email protected] 1,3 Department of Mathematics, Faculty of Science, Naresuan University Phitsanulok 65000, Thailand Center of Excellence in Nonlinear Analysis and Optimization, Faculty of Science Naresuan University Phitsanulok 65000, Thailand E-mails: [email protected]; [email protected]

November 13, 2018 Abstract

The objective of this paper is to determine a modified SP-iteration process for multi-valued mappings and to establish the convergence theorems for sequences generated by modified SP-iteration processes involving multi-valued Suzuki mappings converging to endpoints in uniformly convex hyperbolic spaces. The numerical example for supporting our main result is also presented. Keywords: modified SP-iteration; ∆-convergence theorem; strong convergence theorem; endpoint; hyperbolic space. MSC: Primary 47H10; Seconday 54H25.

1

Introduction

The distance from u in a metric space (X, d) to a nonempty subset E of X is defined by dist(u, E) := inf{d(u, v) : v ∈ E}. It is denoted by K(E) the family of nonempty compact subsets of E. The Hausdorff distance on K(E) is defined by H(U, V ) := max{sup dist(u, V ), sup dist(v, U )} for all U, V ∈ K(E). u∈U ∗ Corresponding

v∈V

author.

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For an element x in E, if x ∈ T (x), then x is said to be a fixed point of T . Moreover, if {x} = T (x), then x is said to be an endpoint of T . It denote by F ix(T ) the set of all fixed points of T and by End(T ) the set of all endpoints of T . We can see that for every a multi-valued mapping T , End(T ) ⊂ F ix(T ) and whenever t is a single-valued mapping, End(T ) = F ix(T ). The notion of endpoints for multi-valued mappings is significant notion which put between the notion of fixed points for single-valued mappings and the notion of fixed points for multi-valued mappings. Aubin and Siegel [3] were first studied the existence of endpoints for special kind of contractive mappings on complete metric spaces. The endpoint results for several types of contractive mappings have been quickly developed and many of papers have showed (see, e.g.,[9],[18],[20],[21]). On the other hand, Panyanak [15] presented the existence of endpoints for multi-valued nonexpansive mappings in uniformly convex Banach spaces. Next, Kudtha and Panyanak [13] proved the existence of endpoints for Suzuki mappings in uniformly convex hyperbolic spaces. Recently, Panyanak [16] established the convergence theorems to an endpoint for modified Ishikawa iteration of multi-vaued nonexpansive mappings in uniformly convex Banach spaces. Motivated and inspired by above mention, we prove the convergence results to an endpoint for modified SP-iteration of multi-valued Suzuki mappings in uniformly convex hyperbolic spaces. The numerical example for supporting our main result is also presented.

2

Preliminaries

For this paper, we work in the setting of a hyperbolic space which is defined by Kohlenbach [12]. Definition 2.1 A hyperbolic space [12] is a metric space (X, d) together with a mapping W : X 2 × [0, 1] → X satisfying the following statements: (W1) d(u, W (x, y, α)) ≤ (1 − α)d(u, x) + αd(u, y); (W2) d(W (x, y, α), W (x, y, β)) = |α − β|d(x, y); (W3) W (x, y, α) = W (y, x, (1 − α)); (W4) d(W (x, z, α), W (y, w, α)) ≤ (1 − α)d(x, y) + αd(z, w), for all x, y, u, z, w ∈ X and α, β ∈ [0, 1]. If x, y ∈ X and α ∈ [0, 1], then we use the notion (1−α)x⊕αy for W (x, y, α). A hyperbolic space (X, d, W ) is said to be uniformly convex [14] if for any r > 0 and ε ∈ (0, 2] there exists a δ ∈ (0, 1] such that for all u, x, y ∈ X, we have 1 d(W (x, y, ), u) ≤ (1 − δ)r, 2 provided d(x, u) ≤ r, d(y, u) ≤ r and d(x, y) ≥ εr. A mapping η : (0, ∞) × (0, 2] → (0, 1] which provides such δ = η(r, ε) for given r > 0 and ε ∈ (0, 2] is well known as a modulus of uniformly convexity of 2

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X. We call η monotone if it decreases with r (for a fixed ε), i.e., for any given ε > 0 and for any r2 ≥ r1 > 0, we have η(r2 , ε) ≤ η(r1 , ε). A nonempty subset E of a hyperbolic space X is convex if W (x, y, α) ∈ E for any x, y ∈ E and α ∈ [0, 1]. Obviously, uniformly convex Banach spaces are uniformly convex hyperbolic spaces, CAT(0) spaces are also uniformly convex hyperbolic spaces, [14]. Definition 2.2 [7] A multi-valued mapping T : E → CB(E) is called to be a Suzuki mapping if 1 dist(x, T (x)) ≤ d(x, y) implies H(T (x), T (y)) ≤ d(x, y) 2

(1)

for all x, y ∈ X. Definition 2.3 [1] A multi-valued mapping T : E → CB(E) is said to satisfy condition (Eµ ) provided that dist(x, T (y)) ≤ µdist(x, T (x)) + d(x, y), for all x, y ∈ E. We say that T satisfies condition (E) whenever T satisfies condition (Eµ ) for some µ ≥ 1. Lemma 2.4 [6] If E is a nonempty closed convex subset of X and T : E → CB(E) is a multi-valued Suzuki mapping, then T satisfies the condition (E3 ). We need the following definition of convergence in hyperbolic spaces [5] which is called ∆-convergence. Let {xn } be a bounded sequence in a hyperbolic space X. Define a function r(·, {xn }) : X → [0, ∞) by r(x, {xn }) = lim sup d(x, xn ), for all x ∈ X. n→∞

The asymptotic radius of a bounded sequence {xn } with respect to a nonempty subset K of X is defined and denoted by rK ({xn }) = inf{r(x, {xn }) : x ∈ K}. The asymptotic center of a bounded sequence {xn } with respect to a nonempty subset K of X is defined and denoted by ACK ({xn }) = {x ∈ X : r(x, {xn }) ≤ r(y, {xn }), for all y ∈ K}. Recall that a sequence {xn } in X is said to ∆-converge to x ∈ X if x is the unique asymptotic center of {un } for every subsequence {un } of {xn }. In this case, we write ∆-limn→∞ xn = x and call x the ∆-lim of {xn }. The sequence {xn } is called to be regular relative to E if r(E, {xn }) = r(E, {xnj }) for every subsequence {xnj } of {xn }. It is known that every bounded sequence in a Banach space has a regular subsequence (see [8]). The proof is metric in nature and carries over to the present setting without change. 3

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Lemma 2.5 [4] Let (X, d, W ) be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η and E is a closed convex subset of X if {xn } is a bounded sequence in E, then the asymptotic center of {xn } is in E. Lemma 2.6 [10] Let (X, d, W ) be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity η. Let x ∈ X and {αn } be a sequence in [a, b] for some a, b ∈ (0, 1). If {xn } and {yn } are sequences in X such that lim supn→∞ d(xn , x) ≤ c, lim supn→∞ d(yn , x) ≤ c, limn→∞ d(W (xn , yn , αn ), x) = c for some c ≥ 0, then lim d(xn , yn ) = 0. n→∞

Lemma 2.7 [11] Every bounded sequence in a complete CAT(0)(and hence hyperbolic) space has a ∆-convergent subsequence. Lemma 2.8 [7] If {xn } is a bounded sequence in complete uniformly convex hyperbolic space (X, d, W ) with A({xn }) = {p}, {un } is a subsequence of {xn } with A({un }) = {u} and the sequence {d(xn , u)} converges, then p = u. Definition 2.9 [8] Let E be a nonempty subset of a metric space (X, d) and x ∈ X. The radius of E relative to x is defined by rx (E) := sup{d(x, y) : y ∈ E}. The diameter of E is defined by diam(E) := sup{d(x, y) : x, y ∈ E}. Definition 2.10 [2] Let T : E → CB(E) be a multi-valued mapping. A sequence {xn } in E is called an approximate fixed point sequence (resp. an approximate endpoint sequence) for T if limn→∞ dist(xn , T (xn )) = 0 (resp. limn→∞ rxn (T (xn )) = 0). A mapping T is said to have the approximate fixed point property (resp. the approximate endpoint property) if it has an approximate fixed point sequence (resp. an approximate endpoint sequence) in E. Lemma 2.11 [15] Let E be a nonempty subset of X, {xn } be a sequence in E and T : E → K(E) be a multi-valued mapping. Then rxn (T (xn )) → 0 if and only if dist(xn , T (xn )) → 0 and diam(T (xn )) → 0. Lemma 2.12 [13] Let E be a nonempty bounded closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and T : E → K(E) be a multi-valued Suzuki mapping. Then T has an endpoint if and only if T has the approximate endpoint property. Next, we also need the following definitions that will be used in the next section. A sequence {xn } in E is said to be Fejér monotone with respect to E if d(xn+1 , q) ≤ d(xn , q) for all q ∈ E and n ∈ N. 4

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Definition 2.13 [13] Let E be a nonempty subset of a hyperbolic space X. A mapping T : E → K(E) is said to satisfy condition (J) if there exists a nondecreasing function h : [0, ∞) → [0, ∞) with h(0) = 0, h(r) > 0 for r ∈ (0, ∞) such that rx (T (x)) ≥ h(dist(x, End(T ))) for all x ∈ E. The mapping T is called semicompact if for any sequence {xn } in E such that lim rxn (T (xn )) = 0,

n→∞

there exists a subsequence {xnj } of {xn } and q ∈ E such that limn→∞ xnj = q.

3

Main results

For this part, we start by introducing the notion of the modified SP-iteration process for multi-valued mappings. Notice that it is an improvement of the one so called the SP-iteration process given in Phuengrattana and Suantai [17]. They [17] also showed that SP-iteration process is a generalized version and the sequence generated by the SP-iteration process converges faster than Ishikawa for the class of nondecreasing and continuous functions. Let X be a hyperbolic space and E be a nonempty convex subset of X, {αn }, {βn }, {γn } be sequences in [0, 1] and T : E → K(E) be a multi-valued mapping. The sequence generated by the modified SP-iteration is defined by z1 ∈ E,   yn = W (un , zn , γn ) (2) wn = W (vn , yn , βn )   zn+1 = W (xn , wn , αn ), where un ∈ T (zn ) such that d(zn , un ) = rzn (T (zn )), vn ∈ T (yn ) such that d(vn , yn ) = ryn (T (yn )) and xn ∈ T (wn ) such that d(xn , wn ) = rwn (T (wn )). We need the following important Lemmas that will be used in the sequel. Lemma 3.1 Let E be a nonempty bounded closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and T : E → K(E) be a multi-valued Suzuki mapping. If {zn } is a sequence in E, then the following holds: ∆

zn −→ z, dist(zn , T (zn )) → 0 and diam(T (zn )) → 0 imply z ∈ End(T ). Proof. From Lemma 2.5, we obtain that z ∈ E. For each n ∈ N, we can choose wn ∈ T (zn ) such that d(zn , wn ) = dist(zn , T (zn )). By passing throught a subsequence, we may assume that {zn } is regular relative to E. Let A(E, {zn }) = {z} and r = r(E, {zn }). By similar way in the proof of Lemma 2.12, we obtain that z ∈ End(T ). 5

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Lemma 3.2 Let E be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and T : E → K(E) be a multi-valued Suzuki mapping with End(T ) ̸= ∅. Let {zn } be a sequence generated by the modified SP-iteration process (2). Then limn→∞ d(zn , q) exists for each q ∈ End(T ). Proof. Let T be a multi-valued Suzuki mapping and q ∈ End(T ). Therefore, 1 dist(q, T (q)) = 0 ≤ d(q, yn ), 2

(3)

1 dist(q, T (q)) = 0 ≤ d(q, wn ), 2

(4)

and

1 dist(q, T (q)) = 0 ≤ d(q, zn ), 2 for all n ∈ N. This implies that

(5)

H(T (q), T (yn )) ≤ d(q, yn ),

(6)

H(T (q), T (wn )) ≤ d(q, wn ),

(7)

H(T (q), T (zn )) ≤ d(q, zn ).

(8)

and Using (2) and (8), we obtain that d(yn , q)

= ≤ = ≤ ≤ ≤

d(W (un , zn , γn ), q) (1 − γn )d(un , q) + γn d(zn , q) (1 − γn )dist(un , T (q)) + γn d(zn , q) (1 − γn )H(T (zn ), T (q)) + γn d(zn , q) (1 − γn )d(zn , q) + γn d(zn , q) d(zn , q).

(9)

d(W (vn , yn , βn ), q) (1 − βn )d(vn , q) + βn d(yn , q) (1 − βn )dist(vn , T (q)) + βn d(yn , q) (1 − βn )H(T (yn ), T (q)) + βn d(yn , q) (1 − βn )d(yn , q) + βn d(yn , q) d(yn , q) ≤ d(zn , q).

(10)

Next, using (2), (6) and (9) d(wn , q) = ≤ = ≤ ≤ ≤

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Again, using (2), (7) and (10) d(zn+1 , q) = ≤ = ≤ ≤ ≤ ≤

d(W (xn , wn , αn ), q) (1 − αn )d(xn , q) + αn d(wn , q) (1 − αn )dist(xn , T (q)) + αn d(wn , q) (1 − αn )H(T (wn ), T (q)) + αn d(wn , q) (1 − αn )d(wn , q) + αn d(wn , q) d(wn , q) d(zn , q).

(11)

This shows that sequence {d(zn , q)} is decreasing and bounded below. Thus limn→∞ d(zn , q) exists for each q ∈ End(T ). Next, we prove ∆-convergence theorem for a multi-valued mapping in hyperbolic spaces. Theorem 3.3 Let E be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and T : E → K(E) be a multi-valued Suzuki mapping with End(T ) ̸= ∅. Let {zn } be a sequence generated by the modified SP-iteration process (2). Then {zn } ∆-converges to an endpoint of T . Proof. First we will prove that rzn (T (zn )) → 0. Let q ∈ End(T ). Since T is a multi-valued Suzuki mapping and 1 dist(q, T (q)) = 0 ≤ d(q, zn ) 2 for all n ∈ N, then

H(T (q), T (zn )) ≤ d(q, zn ).

From Lemma 3.2, we know that for each q ∈ End(T ), limn→∞ d(zn , q) exists. Let limn→∞ d(zn , q) = t ≥ 0. If t = 0, then d(zn , un ) ≤ = ≤ ≤

d(zn , q) + d(q, un ) d(zn , q) + dist(T (q), un ) d(zn , q) + H(T (q), T (zn )) d(zn , q) + d(zn , q).

Taking n → ∞ on above inequality, we have lim rzn (T (zn )) = lim d(zn , un ) = 0.

n→∞

n→∞

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If t > 0, then d(yn , q)

= ≤ = ≤ ≤ ≤

d(W (un , zn , γn ), q) (1 − γn )d(un , q) + γn d(zn , q) (1 − γn )dist(un , T (q)) + γn d(zn , q) (1 − γn )H(T (zn ), T (q)) + γn d(zn , q) (1 − γn )d(zn , q) + γn d(zn , q) d(zn , q).

Letting limsup as n → ∞ on the both sides of above inequality, we have lim sup d(yn , q) ≤ lim sup d(zn , q) ≤ t. n→∞

(12)

n→∞

From (11), we have d(zn+1 , q) ≤ d(wn , q). Then we obtain that t ≤ lim inf d(zn+1 , q) ≤ lim inf d(wn , q). n→∞

n→∞

(13)

From the proof in (10), we have d(wn , q) ≤ d(yn , q). Taking liminf as n → ∞ on above inequality and using (13), t ≤ lim inf d(yn , q). n→∞

(14)

Combine (12) and (14), we obtain that lim d(W (un , zn , γn ), q) = lim d(yn , q) = t.

n→∞

n→∞

(15)

Since d(un , q) = dist(un , T (q)) ≤ H(T (zn ), T (q)) ≤ d(zn , q), this implies that lim sup d(un , q) ≤ t.

(16)

n→∞

By (15), (16), limn→∞ d(zn , q) = t together with Lemma 2.6, we have lim d(un , zn ) = 0.

n→∞

(17)

From the condition of the modified SP-iteration, so lim rzn (T (zn )) = lim d(un , zn ) = 0.

n→∞

n→∞

(18)

Hence by the both cases we can conclude that rzn (T (zn )) → 0. It follows from Lemma 2.11, we have dist(zn , T (zn )) → 0 and diam(T (zn )) → 0. 8

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To show that {zn } ∆-converges to an endpoint of T . Now we prove that Wω (zn ) := ∪{sn }⊂{zn } AC(E, {sn }) ⊂ End(T ) and Wω (zn ) consists of exactly one point. Let s ∈ Wω (zn ). Therefore there exists a subsequence {sn } of {zn } such that AC(E, {sn }) = {s}. From Lemma 2.5 and Lemma 2.7, there exists a subsequence {tn } of {sn } such that ∆-limn→∞ tn = t ∈ E. Since dist(tn , T (tn )) → 0 and diam(T (tn )) → 0 and it follows from Lemma 3.1, we have t ∈ End(T ) and limn→∞ d(zn , t) exists by Lemma 3.2. Thus by Lemma 2.8 we have s = t ∈ End(T ). This shows that Wω (zn ) ⊂ End(T ). Next, we prove that Wω (zn ) consists of exactly one point. Let {sn } be a subsequence of {zn } such that AC(E, {sn }) = {s} and AC(E, {zn }) = {z}. Since s ∈ Wω (zn ) ⊂ End(T ) and from Lemma 3.2, we know that {d(zn , s)} exists. By Lemma 2.8, z = s. Therefore the proof is completed. Next, we present the following key lemma for proving the strong convergence theorem. Lemma 3.4 Let E be a nonempty closed subset of a complete hyperbolic space X and {wn } be a Fejér monotone sequence with respect to E. Then {wn } converges strongly to an element of E if and only if limn→∞ dist(wn , E) = 0. Proof. Assume that {wn } converges strongly to q ∈ E. Thus limn→∞ d(wn , q) = 0. Because 0 ≤ dist(wn , E) ≤ d(wn , q), therefore limn→∞ dist(wn , E) = 0. Conversely, suppose that limn→∞ dist(wn , E) = 0. Since {wn } is a Fejér monotone sequence with respect to E, we have d(wn+1 , q) ≤ d(wn , q) for all q ∈ E. Thus infq∈E d(wn+1 , q) ≤ infq∈E d(wn , q), which means that dist(wn+1 , E) ≤ dist(wn , E). Therefore limn→∞ dist(wn , E) exists. By hypothesis, we obtain that limn→∞ dist(wn , E) = 0. Next, we show that {wn } is a Cauchy sequence in E. Let r > 0. Since limn→∞ dist(wn , E) = 0, there exists n0 ∈ N such that r dist(wn , E) < for all n ≥ n0 . 2 Inparticular, inf{d(wn0 , q) : q ∈ E} < 2r . Therefore there exists q0 ∈ E such that d(wn0 , q0 ) < 2r . For any n, m ≥ n0 , we have d(wn+m , wn ) ≤ d(wn+m , q0 ) + d(q0 , wn ) ≤ d(wn0 , q0 ) + d(q0 , wn0 ) r r ≤ + = r. 2 2 This means that a sequence {wn } is a Cauchy sequence in E. Since E is a closed subset of a complete hyperbolic space X, we have E is also complete. Then {wn } must be convergent to a point in E.

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Theorem 3.5 Let E be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and T : E → K(E) be a multi-valued Suzuki mapping with End(T ) ̸= ∅. Let {zn } be a sequence generated by the modified SP-iteration process (2). If T satisfies condition (J), then {zn } converges strongly to an endpoint of T . Proof. First, we will show that End(T ) is closed. Let {zn } ⊆ End(T ) such that zn → z ∈ E. We will prove that z ∈ End(T ). Since T is a multi-valued Suzuki mapping, therefore T satisfies condition (E3 ). Then dist(zn , T z) ≤ 3dist(zn , T (zn )) + d(zn , z) → 0 as n → ∞. This implies that z ∈ T (z). Next, we show that {z} = T (z). Take any point w ∈ T (z). Since T is a multi-valued Suzuki mapping, 1 dist(zn , T (zn )) = 0 ≤ d(zn , z) implies that H(T (zn ), T (z)) ≤ d(zn , z). 2 Since zn ∈ End(T ), we have d(w, z) ≤ = ≤ ≤

d(w, zn ) + d(zn , z) dist(w, T (zn )) + d(zn , z) H(T (z), T (zn )) + d(zn , z) d(zn , z) + d(zn , z) → 0 as n → ∞.

Hence w = z. Because w ∈ T (z) is arbitrary, then T (z) = {z}, so z ∈ End(T ). Thus End(T ) is closed. Next, as in the proof of Theorem 3.3, we have rzn (T (zn )) → 0 and it follows from T satisfies condition (J), h(dist(zn , End(T ))) ≤ rzn (T (zn )) → 0. This implies that limn→∞ h(dist(zn , End(T ))) = 0. Since h : [0, ∞) → [0, ∞) is nondecreasing with h(0) = 0, h(r) > 0 for r ∈ (0, ∞), we obtain that limn→∞ dist(zn , End(T )) = 0. As in the proof of Lemma 3.2 implies that {zn } is Fejér monotone with respect to End(T ). By applying Lemma 3.4, we obtain the desired result. Theorem 3.6 Let E be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity and T : E → K(E) be a multi-valued Suzuki mapping with End(T ) ̸= ∅. Let {zn } be a sequence generated by the modified SP-iteration process (2). If T is semicompact, then {zn } converges strongly to an endpoint of T . Proof. As in the proof of Theorem 3.3, rzn (T (zn )) → 0 and T is semicompact, we may assume a subsequence znk → z for some z ∈ E. Again, as in the proof of Theorem 3.3, we obtain that rznk (T (znk )) → 0. By Lemma 2.11, we also get

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dist(znk , T (znk )) → 0 as k → ∞. Since T is a multi-valued Suzuki mapping, therefore T satisfies condition(E3 ). Because of dist(z, T (z)) ≤ d(z, znk ) + dist(znk , T (z)) ≤ d(z, znk ) + 3dist(znk , T (znk )) + d(znk , z) → 0 as k → ∞, we obtain that z ∈ T (z). Next, we show that {z} = T (z). Notice that 21 dist(z, T (z)) = 0 ≤ d(znk , z) for all k ∈ N. Since T is a multivalued Suzuki mapping, we have H(T (znk , T (z))) ≤ d(znk , z). We now let u ∈ T (z) and choose wnk ∈ T (znk ) so that d(u, wnk ) = dist(u, T (znk )). For all k ∈ N, we obtain that d(z, u)

≤ ≤ ≤ ≤

d(z, znk ) + d(znk , wnk ) + d(wnk , u) d(z, znk ) + rznk (T (znk )) + dist(u, T (znk )) d(z, znk ) + rznk (T (znk )) + H(T (z), T (znk )) d(z, znk ) + rznk (T (znk )) + d(z, znk ).

Taking limit as k → ∞, we get that z = u for all u ∈ T (z) and so {z} = T (z). Hence z ∈ End(T ). By Lemma 3.2, limn→∞ d(zn , q) exists for each q ∈ End(T ), it follows that zn → z as n → ∞. This completes the proof.

4

Numerical example

In this section, we give an example shows that there exists a mapping which is a multi-valued Suzuki mapping but is not a nonexpansive mapping. Furthermore, we illustrate that a sequence generated by the modified SP-iteration process (2) converges to an endpoint of the multi-valued Suzuki mapping. Example 4.1 Let X = R with metric defined by d(x, y) = |x−y| and E = [0, 3]. Define W : X 2 × [0, 1] → X by W (x, y, α) := αx + (1 − α)y for all x, y ∈ X and α ∈ [0, 1]. Then (X, d, W ) is a complete uniformly hyperbolic space with a monotone modulus of uniform convexity and E is a nonempty compact convex subset of X. Let T : E → K(E) defined by { {0}, z ̸= 3; Tz = {1}, z = 3. By [19] showed that the mapping T is a Suzuki mapping. But T is not a nonexpansive mapping if we take x = 2.9 and y = 3. Moreover, End(T ) = 1 . Therefore {0}. For initial point z0 = 0.1 and αn = βn = γn = √ 3n + 7 {αn }, {βn }, {γn } ⊆ [0, 1]. Set stop parameter to |zn − 0| ≤ 10−12 , where 0 is an endpoint of T . By using MATLAB, we compute the sequence generated by the modified SP-iteration process (2) converging to 0 as in Table 1 and Figure 1.

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iterate z0 z1 z2 z3 z4 : z20 z21 z22

the modified SP-iteration process 0.1 0.024068308483 0.006367770608 0.001780515413 0.000516698713 : 0.000000000008 0.000000000003 0.000000000000

Table 1: Sequences generated by SP-iteration process

Figure 1 Convergence of iterative sequences generated by SP-iteration process

Acknowledgement The first and the third authors would like to express their deep thanks to Naresuan University for supporting this research.

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References [1] A. Abkar, M.Eslamian, Common fixed point results in CAT(0) spaces, Nonlinear Anal., 74, 1835-1840(2011). [2] A. Amini-Harandi, Endpoints of set-valued contractions in metric spaces, Nonlinear Anal., 72, 132-134(2010). [3] J.P. Aubin, J. Siegel, Fixed points and stationary points of dissipative multi-valued maps, Proc. Amer. Math. Soc., 78, 391-398(1980). [4] S. Dhompongsa, W. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, Nonlinear Convex Anal., 8,3545(2007). [5] S. Dhompongsa, B. Panyanak, On ∆-convergence theorem in CAT(0) spaces, Comput. Math. Appl., 56(10), 2572-2579(2008). [6] R. Espinola, P. Lorenzo, A. Nicolae, Fixed points selections and common fixed points for nonexpansive-type mappings, Math. Anal. Appl., 382,503-515(2011). [7] J. García-Falset, E. Llorens-Fuster, T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, Math. Anal. Appl., 375, 185-195(2011). [8] K. Goebel, W.A. Kirk, Topics in metric fixed point theory, Cambridge Univ.Press, Cambridge (1990). [9] M.S. Kahn, K.R. Rao, Y.J. Cho, Common stationary points for setvalued mappings, Internat. Math. Math. Sci., 16, 733-736(1993). [10] A.R. Khan, H. Fukhar-ud-din, M.A.A. Kuan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl., 2012, 54(2012). [11] W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear anal., 68, 3689-3696(2008). [12] U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Am. Math. Soc., 357, 89-128(2005). [13] A. Kudtha, B. Panyanak, Common endpoints for Suzuki mappings in uniformly convex hyperbolic spaces, Thai.J. Math, 159-168, A.M.M.(2017). [14] L. Leustean, A quadratic rate of asymptotic regularity for CAT(0) spaces, Math. Anal. Appl., 235, 386-399(2007). [15] B. Panyanak, Endpoints of multivalued nonexpansive mappings in geodesic spaces, Fixed Point Theory Appl., 2015, 147(2015). [16] B. Panyanak, Approximating endpoints of multi-valued nonexpansive mappings in Banach spaces, Fixed Point Theory Appl.(2018). [17] W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, Comput. Appl. Math., 235,3006-3014(2011). 13

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[18] S.L. Singh, S.N. Mishra, Coincidence points, hybrid fixed and stationary points of orbitally weakly dissipative maps, Math. Japan., 39,451459(1994). [19] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, Math. Anal. Appl., 340, 10881095(2008). [20] Y. Yamamoto, A path following algorithm for stationary point problems, Oper. Res. Soc. Japan, 30,181-199(1987). [21] Y. Yamamoto, Fixed point algorithms for stationary point problems, Mathematical Programming(Tokyo, 1988), Math. Appl. (Japanese Ser.), SCIPRESS, Tokyo 6, 283-307(1989).

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 5, 2020

On Harmonic Multivalent Functions Defined by a New Derivative Operator, Adriana Cătaș and Roxana Șendruțiu,……………………………………………………………………………775 Good and Special Weakly Picard Operators Properties for a Class of Discrete Linear Operators, Loredana-Florentina Iambor and Adriana Cătaș,…………………………………………….781 General Iyengar type Inequalities, George A. Anastassiou,………………………………….786 A Note on the Approximate Solutions for Stochastic Differential Equations Driven by G-Brownian Motion, F. Faizullah, R. Ullah, Jihen Majdoubi, I. Tlili, I. Khan, and Ghaus Ur Rahman,……………………………………………………………………………………….798 Behavior of a System of Higher-Order Difference Equations, M. A. El-Moneam, A. Q. Khan, E. S. Aly, and M. A. Aiyashi,…………………………………………………………………….808 On Approximating the Generalized Euler-Mascheroni Constant, Ti-Ren Huang, Bo-Wen Han, Xiao-Yan Ma, and Yu-Ming Chu,…………………………………………………………….814 General Study on Volterra Integral Equations of the Second Kind in Space with Weight Function, M. E. Nasr and M. F. Jabbar,………………………………………………………824 A Modified SSDP Method for Nonlinear Semidefinite Programming, Jianling Li, Chunting Lu, and Hui Zhang,………………………………………………………………………………..836 Approximation by Sublinear and Max-product Operators using Convexity, George A. Anastassiou,……………………………………………………………………………………848 Symmetric Identities for Carlitz's Generalized Twisted q-Bernoulli Numbers and Polynomials Associated with p-Adic Invariant Integral on ℤ𝑝𝑝 , Cheon Seoung Ryoo,……………………..861

An Efficient Optimal Algorithm for High Frequency in Wavelet Based Image Reconstruction, Jingjing Liu and Guoxi Ni,……………………………………………………………….…..865 Negative Domain Local fractional Inequalities, George A. Anastassiou,……………………879

Approximate Controllability for Semilinear Integro-Differential Control Equations in Hilbert Spaces, Yong Han Kang, Jin-Mun Jeong, and Ah-ran Park,…………………………………892

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 5, 2020 (continued) Convergence Theorems and Approximating Endpoints for Multivalued Suzuki Mappings in Hyperbolic Spaces, Preeyalak Chuadchawna, Ali Farajzadeh, and Anchalee Kaewcharoen,903

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

SHARP INEQUALITIES BETWEEN TOADER AND NEUMAN MEANS∗ WEI-MAO QIAN1 , ZAI-YIN HE2 , AND YU-MING CHU3,∗∗

Abstract. In the article, we prove that the double inequalities α1 Q(a, b) + (1 − α1 )NGA (a, b) < T (a, b) < β1 Q(a, b) + (1 − β1 )NGA (a, b), α2 Q(a, b) + (1 − α2 )NQA (a, b) < T (a, b) < β2 Q(a, b) + (1 − β2 )NQA (a, b), α3 C(a, b) + (1 − α3 )NGA (a, b) < T (a, b) < β3 C(a, b) + (1 − β3 )NGA (a, b), α4 C(a, b) + (1 − α4 )NQA (a, b) < T (a, b) < β4 C(a, b) + (1 − β4 )NQA (a, b) √ 2 2− hold for all a, b > 0 with a 6= b if and only if√α1 ≤ 5/8, β √1 ≥ (16 − π )/[(4 √ π)π] = 0, 7758 · · · , α2 ≤ 1/4, β2 ≥ 1 − 2( 2π − 4)/[( 2 − log(1 + 2))π] = 2 )/[(8 − π)π] = 0.4016 · · · , 0.4708 · · · , α3 ≤ 5/14 = 0.3571 · · · , β√ 3 ≥ (16 − π √ α4 ≤ 1/10 and β4 ≥ 1 − 4(π − 2)/[(4 − 2 − log(1 + 2))π] = 0.1472 · · · , where Q(a, b), C(a, b) and T (a, b) are respectively the quadratic, contra-harmonic and Toader means, and NGA (a, b) and NQA (a, b) are the Neuman means.

1. Introduction Let p ∈ R, r ∈ (0, 1) and a, b > 0 with a 6= b. Then the complete elliptic integrals K(r) and E(r) [1-32] of the first and second kinds, geometric mean G(a, b), arithmetic mean A(a, b), quadratic mean Q(a, b), contra-harmonic mean C(a, b), second contra-harmonic e b), Toader mean T (a, b) [33-36], pth power mean mean C(a, b), centroidal mean C(a, Mp (a, b) [37-43], and Schwab-Borchardt mean SB(a, b) [44-48] of a and b are given by Z π/2 Z π/2 q
−1/2 K(r) = 1 − r2 sin2 t 1 − r2 sin2 (t)dt, dt, E(r) = 0

0

√ G(a, b) = ab, C(a, b) =

a+b , A(a, b) = 2

r Q(a, b) =

a2 + b 2 , 2

2 2 a3 + b 3 e b) = 2(a + ab + b ) , , C(a, 2 2 a +b 3(a + b) Z q 2 π/2 a2 cos2 (t) + b2 sin2 (t)dt, T (a, b) = π 0  p  2aE 1 − (b/a)2 /π, a > b, p  = 2 2bE a < b, 1 − (a/b) /π,

a2 + b 2 , a+b

C(a, b) =

(1.1)

2010 Mathematics Subject Classification. Primary: 26E60; Secondary: 33E05. Key words and phrases. Toader mean, Neuman mean, geometric mean, arithmetic mean, quadratic mean, contra-harmonic mean. ∗ The research was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485), the Science and Technology Research Program of Zhejiang Educational Committee (Grant No. Y201635325) and the Natural Science Foundation of Huzhou City (Grant No. 2018YZ07). ∗∗ Corresponding author: Yu-Ming Chu, Email: [email protected]. 1

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   ap +bp 1/p , 2 Mp (a, b) = √  ab,

p 6= 0, p=0

and

 √ b2 −a2  , a < b, √ (a/b) SB(a, b) = arccos a2 −b2  , a > b, cosh−1 (a/b) √ −1 respectively, where cosh (x) = log(x+ x2 − 1) is the inverse hyperbolic cosine functions. Recently, the bivariate means have attracted the attention of many researchers [49-82]. Neuman [83] introduced the Neuman mean   1 b2 N (a, b) = a+ , 2 SB(a, b) provided the explicit formulae for NAG (a, b)(a, b), NGA (a, b), NAQ (a, b) and NQA (a, b) as follows   −1 (v) 1 2 tanh , NAG (a, b) =: N [A(a, b), G(a, b)] = A(a, b) 1 + (1 − v ) 2 v p  arcsin(v) 1 , (1.2) NGA (a, b) =: N [G(a, b), A(a, b)] = A(a, b) 1 − v2 + 2 v   arctan(v) 1 NAQ (a, b) =: N [A(a, b), Q(a, b)] = A(a, b) 1 + (1 + v 2 ) , 2 v p  sinh−1 (v) 1 , (1.3) NQA (a, b) =: N [Q(a, b), A(a, b)] = A(a, b) 1 + v2 + 2 v where v = (a − b)/(a + b), tanh−1 (x) = log[(1 + x)/(1 − x)]/2 and sinh−1 (x) = log(x + √ x2 + 1) are the inverse hyperbolic tangent and sine functions, respectively. It is well known that the power mean Mp (a, b) is continuous and strictly increasing with respect to p ∈ R for fixed a, b > 0 with a 6= b and the inequalities e b) G(a, b) = M0 (a, b) < A(a, b) = M1 (a, b) < C(a, (1.4) < Q(a, b) = M2 (a, b) < C(a, b) < C(a, b) hold for all a, b > 0 with a 6= b. Barnard, Pearce and Richards [84], and Alzer and Qiu [85] proved that the double inequality M3/2 (a, b) < T (a, b) < Mlog 2/ log(π/2) (a, b) holds all a, b > 0 with a 6= b. In [86], the authors stated that the double inequality αQ(a, b) + (1 − α)A(a, b) < T (a, b) < βQ(a, b) + (1 − β)A(a, b)

√

(1.5)

is valid for all a, b > 0 with a 6= b if and only if α ≤ 1/2 and β ≥ (4 − π)/[( 2 − 1)π] = 0.6596 · · · . Neuman [83] presented the inequalities G(a, b) < NAG (a, b) < NGA (a, b) < A(a, b)

(1.6)

< NQA (a, b) < NAQ (a, b) < Q(a, b), α1 A(a, b) + (1 − α1 )G(a, b) < NGA (a, b) < β1 A(a, b) + (1 − β1 )G(a, b), α2 Q(a, b) + (1 − α2 )A(a, b) < NAQ (a, b) < β2 Q(a, b) + (1 − β2 )A(a, b), α3 A(a, b) + (1 − α3 )G(a, b) < NAG (a, b) < β3 A(a, b) + (1 − β3 )G(a, b), α4 Q(a, b) + (1 − α4 )A(a, b) < NQA (a, b) < β4 Q(a, b) + (1 − β4 )A(a, b) √ for all a, b > 0 with a 6= b if α1 ≤ 2/3, β1 ≥ π/4, α2 ≤ 2/3, √ β2 ≥√(π − 2)/[4(√ 2 − 1)] = 0.6890 · · · , α3 ≤ 1/3, β3 ≥ 1/2, α4 ≤ 1/3, β4 ≥ (log(1 + 2) + 2 − 2)/[2( 2 − 1)] = 0.3568 · · · .

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Li, Qian and Chu [87] proved that the double inequalities αNAQ (a, b) + (1 − α)A(a, b) < T (a, b) < βNAQ (a, b) + (1 − β)A(a, b), Q[λa + (1 − λ)b, λb + (1 − λ)a] < T (a, b) < Q[µa + (1 − µ)b, µb + (1 − µ)a] hold for all a, b > 0 with p β ≥ 4(4 − π)/[π(π − 2)] = √ a 6= b if and only if α ≤ 3/4 and 0.9753 · · · , λ ≤ 1/2 + 2/4 = 0.8535 · · · and µ ≥ 1/2 + 16/π 2 − 1/2 = 0.8940 · · · if λ, µ ∈ (1/2, 1). Qian, Song, Zhang and Chu [88] proved that the two-sided inequalities λ1 C(a, b) + (1 − λ1 )A(a, b) < T (a, b) < µ1 C(a, b) + (1 − µ1 )A(a, b) C[λ2 a + (1 − λ2 )b, λ2 b + (1 − λ2 )a] < T (a, b) < C[µ2 a + (1 − µ2 )b, µ2 b + (1 − µ2 )a] are valid for all 1 ≤ 1/8, µ1 ≥ 4/π − 1 = 0.2732 · · · , √ a, b > 0 with a 6= b if and only if λp λ2 ≤ 1/2 + 2/8 = 0.6767 · · · and µ2 ≥ 1/2 + (4 − π)/(3π − 4)/2 = 0.6988 · · · if λ2 , µ2 ∈ (1/2, 1). In [89], Song, Qian and Chu found that the inequalities e b) < NQA (a, b) < β1 A(a, b) + (1 − β1 )C(a, e b), α1 A(a, b) + (1 − α1 )C(a,

(1.7)

e 1−α2 (a, b) < NQA (a, b) < Aβ2 (a, b)C e 1−β2 (a, b), Aα2 (a, b)C e 3 a + (1 − α3 )b, α3 b + (1 − α3 )a] < NQA (a, b) < C[β e 3 a + (1 − β3 )b, β3 b + (1 − β3 )a] C[α √ √ take place √ if α1 ≥ 4 − 3[ 2 + log(1 + 2)]/2 = 0.5566 · · · , β1 ≤ 1/2, √ α2 ≥ √ if and only 2]/(2 log 2−log 3) = 0.5208 · · · , β 1−[log( 2+log(1+ 2))−log 2 ≤ 1/2, β3 ≥ 1/2+ 2/4 = q √ √ 0.8535 · · · and α3 ≤ 1/2 + 6[ 2 + log(1 + 2)] − 12/4 = 0.8329 · · · if α3 , β3 ∈ (1/2, 1). From (1.4)-(1.7) we clearly see that the inequalities NGA (a, b) < NQA (a, b)
0 with a 6= b. Motivated by inequality (1.8), in the article we deal with the optimality of the parameters α1 , α2 , α3 , α4 , β1 , β2 , β3 and β4 such that the double inequalities
0 with a 6= b. 2. Lemmas In order to prove our main results, we need several formulas and lemmas which we present in this section. The following formulas for K(r) and E(r) can be found in the literature [90]: E(r) − (1 − r2 )K(r) dE(r) E(r) − K(r) dK(r) = , = , dr r(1 − r2 ) dr r  √  rE(r) 2E(r) − (1 − r2 )K(r) d [K(r) − E(r)] 2 r = , E = , dr 1 − r2 1+r 1+r π K(0+ ) = E(0+ ) = , K(1− ) = ∞, E(1− ) = 1. 2

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Lemma 2.1. (See [90, Theorem 1.25]) Let −∞ < a < b < ∞, f, g : [a, b] → R be continuous on [a, b] and differentiable on (a, b), and g 0 (x) 6= 0 on (a, b). If f 0 (x)/g 0 (x) is increasing (decreasing) on (a, b), then so are the functions f (x) − f (a) , g(x) − g(a)

f (x) − f (b) . g(x) − g(b)

If f 0 (x)/g 0 (x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 2.2. The following  statements are true:  (1) The function r 7→ E(r) − (1 − r2 )K(r) /r2 is strictly increasing from (0, 1) onto (π/4, 1); (2) The function r 7→ K(r) is strictly increasing from (0, 1) onto (π/2, ∞); 2 (3) The function r 7→ [K(r) − E(r)]  /r is strictly2increasing  √ from (0, 1) onto (π/4, +∞); (4) The function r 7→ φ(r) = 3E(r) − 2(1 − r )K(r) / 1 + r2 is strictly increasing √ from (0, 1) onto (π/2, 3 2/2). Proof. Parts (1)-(3) can be found in [8, Theorem 3.21(1), (2) and Exercise 3.43(11)]. For part (4), it is not difficult to verify that √ π 3 2 φ(0+ ) = , φ(1+ ) = , (2.1) 2 2 E(r) − 2r2 E(r) − K(r) + 3r2 K(r) φ0 (r) = r(1 + r2 )3/2     2 E(r) − (1 − r )K(r) K(r) − E(r) r 2r3 = + . (2.2) r2 r2 (1 + r2 )3/2 (1 + r2 )3/2 It follows from (2.2) together with Lemma 2.2(1) and (3) that φ0 (r) > 0 for r ∈ (0, 1). Therefore, part (4) follows from (2.1) and (2.3). Lemma 2.3. The function 2r2 + 1 −

(2.3) 

√

  1 + r2 3E(r) − 2(1 − r2 )K(r) ϕ(r) = r2 √ is strictly decreasing from (0, 1) onto (3 − 6 2/π, 3/4). √   Proof. Let ϕ1 (r) = 2r2 + 1 − 2 1 + r2 3E(r) − 2(1 − r2 )K(r) /π, ϕ2 (r) = r2 . Then simple computations lead to ϕ1 (r) ϕ1 (0+ ) = ϕ2 (0+ ) = 0, ϕ(r) = , (2.4) ϕ2 (r) √ 6 2 , (2.5) ϕ(1− ) = 3 − π    0 2 2 ϕ1 (r) E(r) − (1 − r )K(r) 1 3E(r) − 2(1 − r )K(r) p √ =2− + 1 + r2 + K(r) . 2 ϕ02 (r) π r2 1+r (2.6) √ It is not difficult to verify that the function r 7→ 1 + r2 is strictly increasing on (0, 1). Then it follows from Lemma 2.2(1), (2) and (4) together with (2.6) that ϕ01 (r)/ϕ02 (r) is strictly decreasing on (0, 1) and 2 π

ϕ(0+ ) = lim

r→0+

ϕ01 (r) 3 = . ϕ02 (r) 4

(2.7)

Therefore, Lemma 2.3 follows from Lemma 2.1, (2.4), (2.5) and (2.7) together with the monotonicity of ϕ01 (r)/ϕ02 (r). 

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Lemma 2.4. The function 3r2 + 1 −

 3E(r) − 2(1 − r2 )K(r) r2 is strictly decreasing from (0, 1) onto (4 − 6/π, 9/4).   Proof. Let ψ1 (r) = 3r2 + 1 − 2 3E(r) − 2(1 − r2 )K(r) /π, ψ2 (r) = r2 . Then simple computations lead to ψ(r) =

2 π



ψ1 (0+ ) = ψ2 (0+ ) = 0,

ψ(r) =

ψ1 (r) , ψ2 (r)

(2.8)

6 ψ(1− ) = 4 − , (2.9) π   ψ10 (r) 1 E(r) − (1 − r2 )K(r) =3− + K(r) . (2.10) 0 ψ2 (r) π r2 From Lemma 2.2(1), (2) and (2.10) we know that ψ10 (r)/ψ20 (r) is strictly decreasing on (0, 1) and ψ 0 (r) 9 ψ(0+ ) = lim 10 = . (2.11) 4 r→0+ ψ2 (r) Therefore, Lemma 2.4 follows from Lemma 2.1, (2.8), (2.9) and (2.11) together with the monotonicity of ψ10 (r)/ψ20 (r). 

3. Main Results Theorem 3.1. The double inequality α1 Q(a, b) + (1 − α1 )NGA (a, b) < T (a, b) < β1 Q(a, b) + (1 − β1 )NGA (a, b) √ holds for all a, b > 0 with a 6= b if and only if α1 ≤ 5/8 and β1 ≥ (16 − π 2 )/[π(4 2 − π)] = 0.7758 · · · . Proof. Since Q(a, b), NGA (a, b) and T (a, b) are symmetric and homogenous of degree one. Without loss of generality, we assume that a > b. Let r = (a − b)/(a + b) ∈ (0, 1). Then from (1.1) and (1.2) one has   2 T (a, b) = A(a, b) 2E(r) − (1 − r2 )K(r) , (3.1) π p  arcsin(r) 1 NGA (a, b) = A(a, b) 1 − r2 + . (3.2) 2 r √ It follows from (3.1) and (3.2) together with Q(a, b) = A(a, b) 1 + r2 that h√ i   2 2E(r) − (1 − r2 )K(r) − 12 1 − r2 + arcsin(r) π r T (a, b) − NGA (a, b) h√ i = √ Q(a, b) − NGA (a, b) 1 + r2 − 1 1 − r2 + arcsin(r) 2

r

√   2r 1 + r2 − π4 r 2E(r) − (1 − r2 )K(r) √ √ =1− := 1 − F (r). (3.3) 2r 1 + r2 − r 1 − r2 − arcsin(r) √ √ √   Let f1 (r) = 2r 1 + r2 −4r 2E(r) − (1 − r2 )K(r) /π and g1 (r) = 2r 1 + r2 −r 1 − r2 − arcsin(r). Then simple computations lead to f1 (0+ ) = g1 (0+ ) = 0,

F (r) =

f1 (r) , g1 (r)

(3.4)

√   2r2 + 1 − π2 1 + r2 3E(r) − 2(1 − r2 )K(r) f10 (r) √ = g10 (r) 2r2 − 1 − r4 + 1

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ϕ(r) √ , (3.5) (2r2 − 1 − r4 + 1)/r2 where ϕ(r) is defined as in Lemma 2.3. √ It is easy to verify that the function r 7→ (2r2 − 1 − r4 + 1)/r2 is positive and strictly increasing on (0, 1), then (3.5) and Lemma 2.3 lead to the conclusion that f10 (r)/g10 (r) is strictly decreasing on (0,1). Hence from Lemma 2.1 and (3.4) we know that F (r) is strictly decreasing on (0,1). Moreover, √   2r 1 + r2 − π4 r 2E(r) − (1 − r2 )K(r) 3 √ √ lim (3.6) = , 2 2 + 8 r→0 2r 1 + r − r 1 − r − arcsin(r) √   √ 2r 1 + r2 − π4 r 2E(r) − (1 − r2 )K(r) 4( 2π − 4) √ √ √ lim = . (3.7) r→1− 2r 1 + r2 − r 1 − r2 − arcsin(r) π(4 2 − π) Therefore, Theorem 3.1 follows from (3.3), (3.6) and (3.7) together with the monotonicity of F (r).  =

Theorem 3.2. The double inequality α2 Q(a, b) + (1 − α2 )NQA (a, b) < T (a, b) < β2 Q(a, b) + (1 − β2 )NQA (a, b)

√ holds √ 0 with  √for all a, b >  a 6= b if and only if α2 ≤ 1/4 and β2 ≥ 1 − 2( 2π − 4)/ ( 2 − log(1 + 2))π = 0.4708 · · · . Proof. Since Q(a, b), NQA (a, b) and T (a, b) are symmetric and homogenous of degree one. Without loss of generality, we assume that a > b. Let r = (a − b)/(a + b) ∈ (0, 1). Then from (1.4) we have p  sinh−1 (r) 1 . (3.8) NQA (a, b) = A(a, b) 1 + r2 + 2 r √ It follows from (3.1) and (3.8) together with Q(a, b) = A(a, b) 1 + r2 that h√ i   −1 2 2E(r) − (1 − r2 )K(r) − 12 1 + r2 + sinh r (r) π T (a, b) − NQA (a, b) h√ i = √ −1 Q(a, b) − NQA (a, b) 1 + r2 − 12 1 + r2 + sinh r (r) √   2r 1 + r2 − π4 r 2E(r) − (1 − r2 )K(r) √ := 1 − G(r). (3.9) =1− r 1 + r2 − sinh−1 (r) √ √   Let f1 (r) = 2r 1 + r2 −4r 2E(r) − (1 − r2 )K(r) /π and g2 (r) = r 1 + r2 −arcsinh(r). Then simple computations lead to f1 (r) f1 (0+ ) = g2 (0+ ) = 0, G(r) = , (3.10) g2 (r) √   2r2 + 1 − π2 1 + r2 3E(r) − 2(1 − r2 )K(r) f10 (r) = = ϕ(r), (3.11) g20 (r) r2 where ϕ(r) is defined as in Lemma 2.3. It follows from Lemma 2.3 and (3.11) that f10 (r)/g20 (r) is strictly decreasing on (0,1). Then Lemma 2.1 and (3.10) lead to the conclusion that G(r) is strictly decreasing on (0,1). Moreover, √   2r 1 + r2 − π4 r 2E(r) − (1 − r2 )K(r) 3 √ lim = , (3.12) 4 r→0+ r 1 + r2 − sinh−1 (r) √   √ 2r 1 + r2 − π4 r 2E(r) − (1 − r2 )K(r) 2( 2π − 4) √ √  . lim = (3.13) √ r→1− r 1 + r2 − sinh−1 (r) 2 − log(1 + 2) π Therefore, Theorem 3.2 follows from (3.9), (3.12) and (3.13) together with the monotonicity of G(r). 

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Theorem 3.3. The double inequality α3 C(a, b) + (1 − α3 )NGA (a, b) < T (a, b) < β3 C(a, b) + (1 − β3 )NGA (a, b), holds for all a, b > 0 with a 6= b if and only if α3 ≤ 5/14 and β3 ≥ (16 − π 2 )/[π(8 − π)] = 0.4016 · · · . Proof. Without loss of generality, we assume that a > b. Let r = (a − b)/(a + b) ∈ (0, 1). Then it follows from (3.1), (3.2) and C(a, b) = A(a, b)(1 + r2 ) that h√ i   2 1 − r2 + arcsin(r) 2E(r) − (1 − r2 )K(r) − 12 π r T (a, b) − NGA (a, b) h√ i = C(a, b) − NGA (a, b) 1 + r2 − 12 1 − r2 + arcsin(r) r   2r(1 + r2 ) − π4 r 2E(r) − (1 − r2 )K(r) √ := 1 − H(r). (3.14) =1− 2r(1 + r2 ) − r 1 − r2 − arcsin(r) √   Let f2 (r) = 2r(1 + r2 ) − 4r 2E(r) − (1 − r2 )K(r) /π and g3 (r) = 2r(1 + r2 ) − r 1 − r2 − arcsin(r). Then simple computations lead to f2 (0+ ) = g3 (0+ ) = 0,

H(r) =

f2 (r) , g3 (r)

(3.15)

  3r2 + 1 − π2 3E(r) − 2(1 − r2 )K(r) f20 (r) √ = g30 (r) 3r2 − 1 − r2 + 1 ψ(r) √ , (3.16) (3r2 − 1 − r2 + 1)/r2 where ψ(r) is defined as in Lemma 2.4. √ It is easy to verify that the function r 7→ (3r2 − 1 − r2 + 1)/r2 is positive and strictly increasing on (0, 1). Then from Lemma 2.4 and (3.16) we know that f20 (r)/g30 (r) is strictly decreasing on (0,1). Hence Lemma 2.1 and (3.15) lead to the conclusion that H(r) is strictly decreasing on (0,1). Moreover,   2r(1 + r2 ) − π4 r 2E(r) − (1 − r2 )K(r) 9 √ lim = , (3.17) 14 r→0+ 2r(1 + r2 ) − r 1 − r2 − arcsin(r)   2r(1 + r2 ) − π4 r 2E(r) − (1 − r2 )K(r) 8(π − 2) √ = . (3.18) lim π(8 − π) r→1− 2r(1 + r2 ) − r 1 − r2 − arcsin(r) =

Therefore, Theorem 3.3 follows from (3.14), (3.17) and (3.18) together with the monotonicity of H(r). 

Theorem 3.4. The double inequality α4 C(a, b) + (1 − α4 )NQA (a, b) < T (a, b) < β4 C(a, b) + (1 − β4 )NQA (a, b) √ √   holds for all a, b > 0 with a 6= b if and only if α4 ≤ 1/10 and β4 ≥ 1−4(π−2)/ (4 − 2 − log(1 + 2))π = 0.1472. Proof. Without loss of generality, we assume that a > b. Let r = (a − b)/(a + b) ∈ (0, 1). Then it follows from (3.1), (3.8) and C(a, b) = A(a, b)(1 + r2 ) that h√ i   −1 2 2E(r) − (1 − r2 )K(r) − 12 1 + r2 + sinh r (r) π T (a, b) − NQA (a, b) h√ i = −1 C(a, b) − NQA (a, b) 1 + r2 − 1 1 + r2 + sinh (r) 2

2

4 r π

r

2



 2r(1 + r ) − 2E(r) − (1 − r )K(r) √ =1− := 1 − J(r). 2r(1 + r2 ) − r 1 + r2 − sinh−1 (r)

935

(3.19)

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√   Let f2 (r) = 2r(1+r2 )−4r 2E(r) − (1 − r2 )K(r) /π and g4 (r) = 2r(1+r2 )−r 1 + r2 − sinh−1 (r). Then simple computations lead to f2 (0+ ) = g4 (0+ ) = 0,

J(r) =

f2 (r) , g4 (r)

(3.20)

  3r2 + 1 − π2 3E(r) − 2(1 − r2 )K(r) f20 (r) √ = g40 (r) 3r2 − 1 + r2 + 1 ψ(r) √ , (3r2 − 1 + r2 + 1)/r2 where ψ(r) is defined as in Lemma 2.4. √ It is easy to verify that the function r 7→ (3r2 − 1 + r2 + 1)/r2 is positive and increasing on (0,1). Then from Lemma 2.4 and (3.21) we know that f20 (r)/g40 (r) is decreasing on (0,1). Hence Lemma 2.1 and (3.20) lead to the conclusion that strictly decreasing on (0,1). Moreover, =

2r(1 + r2 ) −

 2E(r) − (1 − r2 )K(r) 9 √ lim , = 10 r→0+ 2r(1 + r2 ) − r 1 + r2 − sinh−1 (r)   2r(1 + r2 ) − π4 r 2E(r) − (1 − r2 )K(r) 4(π − 2) √ √ √  . lim =  r→1− 2r(1 + r2 ) − r 1 + r2 − sinh−1 (r) 4 − 2 − log(1 + 2) π 4 r π

(3.21)

strictly strictly J(r) is



(3.22) (3.23)

Therefore, Theorem 3.4 follows from (3.19), (3.22) and (3.23) together with the monotonicity of J(r).  √ √ Let r0 = log(1 + 2), r∗ = r2 /(1 + 1 − r2 )2 . Then (1.1) and Theorems 3.1-3.4 lead to Corollary 3.5 immediately. Corollary 3.5. The double inequalities   p p √ p arcsin(r∗ ) π 2 2 ∗2 10 2 2 − r + 3(1 + 1 − r ) < E(r) 1−r + 64 r∗ √ √ p  p 2(16 − π 2 ) p arcsin(r∗ ) 2π − 4 √ √ < 2 − r2 + (1 + 1 − r2 ) 1 − r∗2 + , r∗ 4(4 2 − π) 2(4 2 − π) p   p √ p sinh−1 (r∗ ) π 2 2 2 − r2 + 3(1 + 1 − r2 ) 1 + r∗2 + < E(r) 32 r∗ √ √ √ p  p 2(8 − π( 2 + r0 )) p sinh−1 (r∗ ) 2π − 4 √ 2 − r2 + √ 1 + r∗2 + , < (1 + 1 − r2 ) r∗ 4( 2 − r0 ) 4( 2 − r0 )  p  p 20(2 − r2 ) arcsin(r∗ ) π √ + 9(1 + 1 − r2 ) 1 − r∗2 + < E(r) 112 1 + 1 − r2 r∗  p p arcsin(r∗ ) 16 − π 2 2 − r2 π−2 √ < + (1 + 1 − r2 ) 1 − r∗2 + , 2(8 − π) 1 + 1 − r2 8−π r∗  p  p 4(2 − r2 ) sinh−1 (r∗ ) π √ + 9(1 + 1 − r2 ) 1 + r∗2 + < E(r) 80 1 + 1 − r2 r∗ √ p  p 8 − π( 2 + r0 ) sinh−1 (r∗ ) 2 − r2 π−2 √ √ √ (1+ 1 − r2 ) < + 1 + r∗2 + . r∗ 2(4 − 2 − r0 ) 1 + 1 − r2 2(4 − 2 − r0 ) hold for all r ∈ (0, 1).

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9

4. Results and discussion In the article, we present the best possible parameters α1 , α2 , α3 , α4 , β1 , β2 , β3 and β4 such that the double inequalities α1 Q(a, b) + (1 − α1 )NGA (a, b) < T (a, b) < β1 Q(a, b) + (1 − β1 )NGA (a, b), α2 Q(a, b) + (1 − α2 )NQA (a, b) < T (a, b) < β2 Q(a, b) + (1 − β2 )NQA (a, b), α3 C(a, b) + (1 − α3 )NGA (a, b) < T (a, b) < β3 C(a, b) + (1 − β3 )NGA (a, b), α4 C(a, b) + (1 − α4 )NQA (a, b) < T (a, b) < β4 C(a, b) + (1 − β4 )NQA (a, b) hold for all a, b > 0 with a 6= b. Our results are the improvements and refinements of the previously results. 5. Conclusion We present several sharp bounds for the Toader mean in terms of the Neuman mean, quadratic mean and contraharmonic mean, and give new bounds for the complete elliptic integral of the second kind E(r). Our approach may have further applications in the theory of bivariate means and special functions.

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[43] W.-F. Xia, W. Janous and Y.-M. Chu, The optimal convex combination bounds of arithmetic and harmonic mean in terms of power mean, J. Math. Inequal., 2012, 6(2), 241–248. [44] E. Neuman and J. S´ andor, On the Schwab-Borchardt mean, Math. Pannon., 2003, 14(2), 253–266. [45] Z.-Y. He, Y.-M. Chu and M.-K. Wang, Optimal bounds for Neuman means in terms of harmonic and contraharmonic means, J. Appl. Math., 2013, 2013, Article ID 807623, 4 pages. [46] Y.-Y. Yang, W.-M. Qian, and Y.-M. Chu, Refinements of bounds for Neuman means with applications, J. Nonlinear Sci. Appl., 2016, 9(4), 1529–1540. [47] Y.-M. Chu and B.-Y. Long, Sharp inequalities between means, Math. Inequal. Appl., 2011, 14(3), 647–655. [48] Y.-M. Chu, M.-K. Wang and W.-M. Gong, Two sharp double inequalities for Seiffert mean, J. Inequal. Appl., 2011, 2011, Article 44, 7 pages. [49] B.-Y. Long and Y.-M. Chu, Optimal inequalities for generalized logarithmic, arithmetic, and geometric means, J. Inequal. Appl., 2010, 2010, Article ID 806825, 10 pages. [50] M.-K. Wang, Y.-M. Chu and Y.-F. Qiu, Some comparison inequalities for generalized Muirhead and identric means, J. Inequal. Appl., 2010, 2010, Article ID 295620, 10 pages. [51] Y.-M. Chu and B.-Y. Long, Best possible inequalities between generalized logarithmic mean and classical means, Abstr. Appl. Anal., 2010, 2010, Article ID 303286, 13 pages. [52] X.-M. Zhang and Y.-M. Chu, Convexity of the integral arithmetic mean of a convex function, Rocky Mountain J. Math., 2010, 40(3), 1061–1068. [53] Y.-M. Chu, Y.-F. Qiu, M.-K. Wang and G.-D. Wang, The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert’s mean, J. Inequal. Appl., 2010, 2010, Article ID 436457, 7 pages. [54] M.-K. Wang, Y.-F. Qiu and Y.-M. Chu, Sharp bounds for Seiffert means in terms of Lehmer means, J. Math. Inequal., 2010, 4(4), 581–586. [55] W.-F. Xia, Y.-M. Chu and G.-D. Wang, Necessary and sufficient conditions for the Schur harmonic convexity or concavity of the extended mean values, Rev. Un. Mat. Argentina, 2011, 52(1), 121–132. [56] W.-F. Xia and Y.-M. Chu, The Schur convexity of Gini mean values in the sense of harmonic mean, Acta Math. Sci., 2011, 31B(3), 1103–1112. [57] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, A sharp double inequality between harmonic and identric means, Abstr. Appl. Anal., 2011, 2011, Article ID 657935, 7 pages. [58] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On approximating the error function, Math. Inequal. Appl., 2018, 21(2), 469–479. [59] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, An optimal double inequality between Seiffert and geometric means, J. Appl. Math., 2011, 2011, Article ID 261237, 6 pages. [60] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, A best possible double inequality between Seiffert and harmonic means, J. Inequal. Appl., 2011, 2011, Article 94, 7 pages. [61] Y.-F. Qiu, M.-K. Wang, Y.-M. Chu and G.-D. Wang, Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, J. Math. Inequal., 2011, 5(3), 301–306. [62] Y.-M. Chu, C. Zong and G.-D. Wang, Optimal convex combiantion bounds of Seiffert and geometric means for the arithmetic mean, J. Math. Inequal., 2011, 5(3), 429–434. [63] M.-K. Wang, Z.-K. Wang and Y.-M. Chu, An optimal double inequality between geometric and identric means, Appl. Math. Lett., 2012, 25(3), 471–475. [64] Y.-M. Chu and S.-W. Hou, Sharp bounds for Seiffert mean in terms of contraharmonic mean, Abstr. Appl. Anal., 2012, 2012, Article ID 425175, 6 pages. [65] S.-L. Qiu, Y.-F. Qiu, M.-K. Wang and Y.-M. Chu, H¨ older mean inequalities for the generalized Gr¨ otzsch ring and Hersch-Pfluger distortion functions, Math. Inequal. Appl., 2012, 15(1), 237–245. [66] Y.-M. Chu, M.-K. Wang and G.-D. Wang, The optimal generalized logarithmic mean bounds for Seiffert’s mean, Acta Math. Sci., 2012, 32B(4), 1619–1626. [67] Y.-M. Chu, M.-K. Wang and Z.-K. Wang, Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means, Math. Inequal. Appl., 2012, 15(2), 415–422. [68] W.-M. Gong, Y.-Q. Song, M.-K. Wang and Y.-M. Chu, A sharp double inequality betwee Seiffert, arithmetic, and geometric means, Abstr. Appl. Anal., 2012, 2012, Article ID 648834, 7 pages.

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[69] H.-N. Hu, G.-Y. Tu and Y.-M. Chu, Optimal bounds for Seiffert mean in terms of oneparameter means, J. Appl. Math., 2012, 2012, Article ID 917120, 7 pages. [70] Y.-M. Li, B.-Y. Long and Y.-M. Chu, Sharp bounds for the Neuman-S´ andor mean in terms of generalized logarithmic mean, J. Math. Inequal., 2012, 6(4), 567–577. [71] Y.-M. Chu and B.-Y. Long, Bounds of the Neuman-S´ andor mean using power and identric means, Abstr. Appl. Anal., 2013, 2013, Article ID 832591, 6 pages. [72] Y.-M. Chu, S.-W. Hou and W.-F. Xia, Optimal convex combination bounds of centroidal and harmonic means for logarithmic and inentric means, Bull. Iranian Math. Soc., 2013, 39(2), 259–269. [73] Y.-M. Chu, M.-K. Wang and X.-Y. Ma, Sharp bounds for Toader mean in terms of contraharmonic mean with applications, J. Math. Inequal., 2013, 7(2), 161–166. [74] W.-F. Xia and Y.-M. Chu, Optimal inequaliites between Neuman-S´ andor, centroidal and harmonic means, J. Math. Inequal., 2013, 7(4), 593–600. [75] Y.-M. chu, Y.-M. Li, W.-F. Xia and X.-H. Zhang, Best possible inequalities for the harmonic mean of error function, J. Inequal. Appl., 2014, 2014, Article 525, 9 pages. [76] W.-M. Qian and Y.-M. Chu, Best possible bounds for Yang mean using generalized logarithmic mean, Math. Probl. Eng., 2016, 2016, Article ID 8901258, 7 pages. [77] W.-M. Qian, X.-H. Zhang and Y.-M. Chu, Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means, J. Math. Inequal., 2017, 11(1), 121–127. [78] H.-Z. Xu, Y.-M. Chu and W.-M. Qian, Sharp bounds for the S´ andor-Yang means in terms of arithmetic and contra-harmonic means, J. Inequal. Appl., 2018, 2018, Article 127, 13 pages. [79] Zh.-H. Yang, W. Zhang and Y.-M. Chu, Sharp Gautschi inequality for parameter 0 < p < 1 with applications, Math. Inequal. Appl., 2017, 20(4), 1107–1120. [80] Zh.-H. Yang, Y.-M. Chu and W. Zhang, Accurate approximations for the complete elliptic of the second kind, J. Math. Anal. Appl., 2016, 438(2), 875–888. [81] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, Monotonicity rule for the quotient of two function and its applications, J. Inequal. Appl., 2017, 2017, Article 106, 13 pages. [82] Zh.-H. Yang, W.-M. Qian, Y.-M. Chu and W. Zhang, On rational bounds for the gamma function, J. Inequal. Appl., 2017, 2017, Article 210, 17 pages. [83] E. Neuman, On a new bivariate mean, Aequat. Math., 2014, 88(3), 277–289. [84] R. W. Barnard, K. Pearce and K. C. Richards, An inequality involving the generalized hypergeometric function and the arc length of an ellipse, SIAM J. Math. Anal., 2000, 31(3), 693–699. [85] H. Alzer and S.-L. Qiu, Monotonicity theorems and inequalities for the complete elliptic integrals, J. Comput. Appl. Math., 2004, 172(2), 289–312. [86] Y.-M. Chu, M.-K. Wang and S.-L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci., 2012, 122(1), 41–51. [87] J.-F. Li, W.-M. Qian and Y.-M. Chu, Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means, J. Inequal. Appl., 2015, 2015, Article ID 277, 9 pages. [88] W.-M. Qian, Y.-Q. Song, X.-H. Zhang and Y.-M. Chu, Sharp bounds for Toader mean in terms of arithmetic and second contraharmonic means, J. Funct. Spaces, 2015, 2015, Article ID 452823, 5 pages. [89] Y.-Q. Song, W.-M. Qian and Y.-M. Chu, Optimal bounds for Neuman mean using arithmetic and centroidal means, J. Funct. Spaces, 2016, 2016, Article ID 5131907, 7 pages. [90] G. D. Anderson, M. K. Vamanamurthy and M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997. Wei-Mao Qian, College of Science, Hunan City University, Yiyang 413000, Hunan, China. E-mail address: [email protected] Zai-Yin He, College of Mathematics and Econometrics, Hunan University, Changsha 410082, Hunan, China E-mail address: [email protected] Yu-Ming Chu (Corresponding author), Department of Mathematics, Huzhou University, Huzhou 313000, Zhejiang, China E-mail address: [email protected]

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ON STRONGLY STARLIKENESS OF STRONGLY CONVEX FUNCTIONS ADEL A. ATTIYA, NAK EUN CHO, AND M. F. YASSEN Abstract. In this paper we introduce an argument property which gives an interesting relation between the classes of strongly convex and strongly starlike functions of order α and type β in the open unit disk. Also, the sufficient condition of starlikeness under certain restrictions is obtained.

1. Introduction Let A denote the class of functions f (z) of the form

(1.1)

f (z) = z +

∞ X

ak z k ,

k=1

which are analytic in the open unit disc U = {z ∈ C : |z| < 1}. The function f (z) is called strongly starlike of order β and type α and strongly convex of order β and type α, respectively if it satisfies (1.2)

  0 π zf (z) arg − α < β f (z) 2

and (1.3)

  00 π zf (z) arg 1 + < β, − α 2 f 0 (z)

where α ∈ [0, 1) and β ∈ (0, 1]. We denote by S ∗ (α, β) and C(α, β) the classes of functions satisfy the conditions (1.2) and (1.3) respectively. We note that both S ∗ (α, 1) = S ∗ (α) and C(α, 1) = C(α), are the well known classes of starlike functions of order α and convex functions of order α. MacGregor [2] Wilken and Feng [5] obtained the following result: 2010 Mathematics Subject Classification. 30C45. Key words and phrases. Analytic functions; Strongly convex functions; Strongly starlike functions. 1

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f (z) ∈ C(α) ⇒ f (z) ∈ S ∗ (β)

(0 ≤ α < 1),

where

(1.4)

β := β(α) =

  

1−2α 22−2α (1−22α−1) )

, α 6=

1 2

α = 21 .

1 , 2 log 2

Also, Nunokawa et al.[4] investigated a certain relation between S ∗ (α, β) and C(α, β). In the present paper, we obtain a relationship between strongly convex and strongly starlike functions by using the result given by Nunokawa [3]. In our investigation, we need the following lemma: Lemma 1.1. [3] Let P (z) be analytic in U, P (0) = 1, P (z) 6= 0 in U and suppose that there exists a point z0 ∈ U such that π |arg(P (z0 ))| = δ, 2 where 0 < δ. Then we have z0 P 0 (z0 ) = ikδ, P (z0 ) where 1 k≥ 2

  1 π a+ when arg(P (z0 )) = δ a 2

and 1 k≤− 2



1 a+ a



π when arg(P (z0 )) = − δ, 2

where (P (z0 ))1/δ = ±ia and a > 0. 2. Main Result Theorem 2.1. Let f (z) be analytic function defined by (1.1) and also, let (2.5)

f (z) ∈ C(α, γ)

(z ∈ U),

where 0 ≤ α < 1 and 0 < γ < 1. Then (2.6)

f (z) ∈ S(β, δ)

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(z ∈ U),

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3

where (2.7)

δ(1 − β)aδ−1 (a20 + 1) 0

2 β + (1 − β)aδ0 (β − α) + (1 − β)aδ0

2 γ= arctan π

! ,

β is defined by (1.4), 0 < δ < 1 and a0 is the positive root of the equation:

(2.8) (β − α)β (1 + δ) x2 − (1 − δ) + xδ (1 − β) (2β − α) x2 − 1
+ x2δ (1 − β)2 (1 − δ) x2 − (1 + δ) = 0, which satisfies (2.9)

aδ0 ≥

β 1−β

s

π  β − α π  2 csc δ + δ − csc 2 β 2

!!1/δ

Proof. Let p(z) =

z f 0 (z) , p(0) = 1 f (z)

and

p(z) 6= β

(z ∈ U).

Then we have 1+

z f 00 (z) zp0 (z) = p(z) + . f 0 (z) p(z)

If there exists z0 ∈ U such that π δ 2

|arg (P (z))| = |arg (p(z) − β)| < for |z| < |z0 | and |arg (P (z0 ))| = |arg (p(z0 ) − β)| =

π δ, 2

where P (z) =

p(z) − β . 1−β

Since P (0) = 1 and by using Lemma 1.1, we have z0 P 0 (z0 ) z0 p0 (z0 ) = = iδk. P (z0 ) p(z0 ) − β The first case, if arg (P (z0 )) = arg (p(z0 ) − β) =

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then we have   z f 00 (z) arg 1 + 0 −α f (z)    z0 p0 (z0 )/p(z0 ) β−α = arg (p(z0 ) − β) 1 + + p(z0 ) − β p(z0 ) − β   π iδk β−α = δ + arg 1 + + 2 β + (1 − β)(ia)δ (1 − β)(ia)δ   iδk β−α π + = δ + arg 1 + π π 2 β + (1 − β)aδ ei 2 δ (1 − β)aδ ei 2 δ   iδk (β − α) i π2 δ = arg e + −i π δ + βe 2 + (1 − β)aδ (1 − β)aδ 
 δk(1−β)aδ +δkβ cos( π2 δ ) π + sin 2 δ  2  (β+(1−β)aδ )  . ≥ arctan 
βδk sin( π2 δ )  β−α π + cos 2 δ − 2 (1−β)aδ (β+(1−β)aδ ) Since the function h(k) defined by 


 + sin 2   (β+(1−β)aδ )   h(k) = arctan  π
βδk sin( 2 δ )  β−α π + cos 2 δ − 2 (1−β)aδ (β+(1−β)aδ ) δk(1−β)aδ +δkβ cos( π2 δ )

π δ 2

is an increasing function of k (k ≥ 1), we have  zf 00 (z) arg 1 + 0 −α f (z) 
 (δ (1−β)aδ +δβ cos( π2 δ))(a+1/a) π + sin 2 δ  2  2(β+(1−β)aδ )   ≥ arctan 
βδ sin( π2 δ)(a+1/a)  . β−α π + cos 2 δ − 2 (1−β)aδ 2(β+(1−β)aδ ) 

Also, the function f (θ) defined by 

δ(1−β)aδ (a+1/a) 2

+

δβ(a+1/a) 2

cos θ + sin θ

2(β+(1−β)aδ )  2(β+(1−β)aδ ) f (θ) = arctan  β−α + cos θ − βδ(a+1/a)δ 2 sin θ (1−β)aδ 2(β+(1−β)a )

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STRONGLY STARLIKENESS OF STRONGLY CONVEX FUNCTIONS

5

is an increasing and continuous function of θ (0 < θ < π2 ) when aδ satisfies (2.9). Therefore, we have   zf 00 (z) (2.10) −α arg 1 + 0 f (z)   δ(1 − β) (a + 1/a) aδ ≥ arctan . 2 (β + (1 − β)aδ ) ((β − α) + (1 − β)aδ ) On the other hand, since the function g(x) defined by
δ(1 − β) x + x1 xδ (2.11) g(x) = 2 (β + (1 − β)xδ ) ((β − α) + (1 − β)xδ )

(x > 0),

takes its minimum value when x is defined by (2.8), we see that this contradicts the hypothesis of Theorem 2.1. The second case, if π arg (P (z0 )) = arg (p(z0 ) − β) = − δ, 2 then we have   z f 00 (z) arg 1 + 0 −α f (z)    z0 p0 (z0 )/p(z0 ) β−α = arg (p(z0 ) − β) 1 + + p(z0 ) − β p(z0 ) − β   iδk β−α π + = − δ + arg 1 + π π 2 β + (1 − β)aδ e−i 2 δ (1 − β)aδ e−i 2 δ   i δk (β − α) −i π2 δ = arg e + + π β ei 2 δ + (1 − β)aδ (1 − β)aδ 
 δk(1−β)aδ +δkβ cos( π2 δ ) π − sin 2 δ  2  (β+(1−β)aδ )  . = arctan 
βδk sin( π2 δ )  β−α π + cos 2 δ + 2 (1−β)aδ (β+(1−β)aδ ) Since the function h(k) defined by h(k) = arctan



! δk(1 − β)aδ + δkβ cos π2 δ − sin π2 δ

β−α + cos π2 δ + βδk sin π2 δ (1−β)aδ

is a decreasing function of k (k ≤ −1), we have

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ADEL A. ATTIYA, NAK EUN CHO, AND M. F. YASSEN

  zf 00 (z) arg 1 + 0 −α f (z) 
 (δ(1−β)aδ +δβ cos( π2 δ))(a+1/a) π − sin 2 δ  2 − 2(β+(1−β)aδ ) .  ≤ arctan  π
 βδ sin δ (a+1/a) ( ) β−α π 2 + cos δ − 2 δ 2 (1−β)a 2(β+(1−β)aδ ) Also, the function f (θ) defined by  δ

δ(1−β)a (a+1/a) 2

+

δβ(a+1/a) 2

cos θ + sin θ

2(β+(1−β)aδ )  2(β+(1−β)aδ ) f (θ) = − arctan  β−α + cos θ − βδ(a+1/a)δ 2 sin θ (1−β)aδ 2(β+(1−β)a )

  

is a decreasing and continuous function of θ (0 < θ < π2 ), when aδ satisfies (2.9). Therefore, we have   z f 00 (z) −α arg 1 + 0 f (z) !
δ(1 − β) a + a1 aδ ≤ − arctan . 2 (β + (1 − β)aδ ) ((β − α) + (1 − β)aδ ) Also, by using the function g(x) defind by (2.11) which contradicts hypothesis of Theorem 2.1. Therefore, it completes the proof of the theorem.  Putting f (z) instead of zf 0 (z) in Theorem 2.1, we have the following corollary Corollary 2.1. Let f (z) be analytic function defined by (1.1) and also, let (2.12)

f (z) ∈ S ∗ (α, γ)

(z ∈ U),

where 0 ≤ α < 1 and 0 < γ < 1. Then   π f (z) arg < δ (z ∈ U), (2.13) − β 2 A(z) Rz where A(z) = 0 (f (t)/t)dt is Alexander operator defined by Alexander [1], ! 2 2 δ(1 − β)aδ−1 (a + 1) 0
0
, (2.14) γ = arctan π 2 β + (1 − β)aδ0 (β − α) + (1 − β)aδ0

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STRONGLY STARLIKENESS OF STRONGLY CONVEX FUNCTIONS

7

β is defined by (1.4), 0 < δ < 1 and a0 is the positive root of the equation:

(2.15) (β − α)β (1 + δ) x2 − (1 − δ) + xδ (1 − β) (2β − α) x2 − 1
+ x2δ (1 − β)2 (1 − δ) x2 − (1 + δ) = 0. which satisfies (2.16) aδ0 ≥

β 1−β

s

π  β − α π  2 csc δ + − csc δ 2 β 2

!!1/δ .

Corollary 2.2. Let f (z) be analytic function defined by (1.1) and also, let f (z) ∈ C(α, γ)

(2.17)

(z ∈ U),

where 0 ≤ α < 1 and 0 < γ < 1. Then f (z) ∈ S(β, δ)

(2.18)

(z ∈ U),

where (2.19) 

 p δ β(β − α) 2   , γ= arctan   p p π β + β(β − α) (β − α) + β(β − α) and β is defined by (1.4). Proof. Let f (z) ∈ C(α, γ). Since the inequality (2.10) is satisfied when aδ satisfies (2.9), we have δ(1 − β) (a + 1/a) aδ 2 (β + (1 − β)aδ ) ((β − α) + (1 − β)aδ ) δ(1 − β)aδ . ≥ (β + (1 − β)aδ ) ((β − α) + (1 − β)aδ ) Then the function k(x) defined by δ(1 − β)x k(x) = (x > 0) (β + (1 − β)x) ((β − α) + (1 − β)x) √ β(β−α) takes its minimum value when x = 1−β . On the other hand , we have p β(β − α) ≥ 1−β

β 1−β

s

!! π  β − α π  csc2 δ + − csc δ . 2 β 2

Hence we have f (z) ∈ S(β, δ).



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ADEL A. ATTIYA, NAK EUN CHO, AND M. F. YASSEN

Acknowledgement. The first and the second authors of this research were supported by the Deanship of Scientific Research, Hail University, Hail, under research no. 0150323 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). References [1] J.W. Alexander, Functions which map the interior of the unit circle upon simple region, Ann. Math. 17(1915), 12-22. [2] T.H. MacGregor, A subordination for convex functions of order α J. London Math. Soc. (2)9(1974/75), 530–536. [3] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci. 69(1993), 234-237. [4] M. Nunokawa, S. Owa and H. Shiraishi, On a remark of strongly convex functions of order β and convex of order α, RIMS Kˆokyˆ uroku, Vol. 2010, No. 1717, 100-106. [5] D.R. Wilken and J.A. Feng, A remark on convex and starlike functions. J. London Math. Soc. 21(1980), no. 2, 287-290.

Department of Mathematics, Faculty of Science, University of Hail, Hail, Saudi Arabia, and, Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura, 35516, Egypt E-mail address: [email protected] Department of Applied Mathematics, Pukyong National University, Busan, Korea E-mail address: [email protected] Department of Mathematics, Faculty of Science, Damietta University, New Damietta, 34517, Egypt Current address: Department of Mathematics, Faculty of Sciences and Humanities Aflaj, Prince Sattam bin Abdulaziz University, Kingdom of Saudi Arabia E-mail address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Invariance analysis of a four-dimensional system of fourth-order difference equations with variable coefficients

Mensah Folly-Gbetoula∗ School of Mathematics, University of the Witwatersrand, 2050, Johannesburg, South Africa. Abstract A class of a four-dimensional system of difference equations is considered. A Lie symmetry analysis is performed and symmetries are derived. We use the differential invariant approach to obtain exact solutions. The link between the similarity variables and these symmetries is clearly given. Furthermore, we show the existence of periodic solutions for some specific coefficients. This work considerably extends some findings by El-Dessoky and Hobiny [M. M. El-Dessoky and A. Hobiny, J. Computational Analysis and Applications, 26:8 (2019), 1428–1439].

Keywords: System of difference equation; invariance analysis; group invariant solutions; periodicity MSC: 39A11, 39A05

1

Introduction

The group theoretical approach for finding exact solutions to differential equations is now well reported [2, 14] and its application to difference equations has sparked interest recently [6–8, 10–13]. This approach, commonly known as Lie symmetry analysis, permits one to lower the order of the difference equations via a convenient choice of canonical coordinates obtained using a group of transformations admitted by the equation. Its application to higher dimensional system of difference equations is somewhat new and the calculation one deals with when finding symmetries in the latter can become cumbersome. Hydon in [10] extends the idea of Maeda [16] by developing a systematic algorithm permitting one to obtain the Lie algebra of a difference equation. Several authors have studied difference equations from different approaches and some interesting results can be found in [3–5, 17] In this paper, inspired by the work in [1] where the authors study the behavior and existence of solutions of xn−3 yn−3 , yn+1 = ±1 ± xn−3 yn−2 zn−1 tn ±1 ± xn yn−3 zn−2 tn−1 zn−3 tn−3 = , tn+1 = , ±1 ± xn−1 yn zn−3 tn−2 ±1 ± xn−2 yn−1 zn tn−3

xn+1 = zn+1

(1)

we utilize Hydon’s idea in a slightly modified manner to investigate the solutions to xn−3 yn−3 , yn+1 = an + bn xn−3 yn−2 zn−1 tn cn + dn xn yn−3 zn−2 tn−1 zn−3 tn−3 = , tn+1 = , en + fn xn−1 yn zn−3 tn−2 gn + hn xn−2 yn−1 zn tn−3

xn+1 = zn+1

(2)

where (an )n∈N0 , (bn )n∈N0 , (cn )n∈N0 , (dn )n∈N0 , (en )n∈N0 , (fn )n∈N0 , (gn )n∈N0 and (hn )n∈N0 are non-zero sequences of real numbers. The solutions of (2) are derived after a series of steps. Firstly, we obtain the Lie algebra of (2). We make use of point symmetries and additional assumptions on the characteristics to allow us derive analytic expressions for the symmetry generators. Secondly, we lower the order via the invariants and finally, find the solutions. We have showed that results in [1] are special cases of our findings. ∗ [email protected]

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1.1

Preliminaries

In this section, we commence with some background necessary for understanding symmetry analysis. Note that throughout this paper, we utilize definitions and notation in [10, 14]. The notion of symmetry is strongly related to the notion of group transformations. Basically, it is a group of transformations that map a solution of a given equation onto another solution. Suppose G is a group of transformations acting on a manifold M. Certain subsets H of this group, called H-invariant, transform solutions onto themselves. Often times, for system of difference equations, the difference invariants of H are the new variables of the much simpler difference equations equivalent to the original system of equations. Let S i be the forward shift operator that maps n to n + i. We shall assume that a system of fourth order ordinary difference equations is of the form S p (uk ) =Ωk (n, [uk ]),

k = 1, 2, 3, 4,

(3)

where [ui ] denotes the dependent variable ui and its shifts. The invertible mapping (n, uk ) 7→ (n, u ˜k = k k 2 u + εQk (n, [u ]) + O(ε )), k = 1, 2, 3, 4, is a symmetry group of transformations if and only if it satisfies the following linearized symmetry condition S p (Qk ) − X (Ωk ) = 0,

k = 1, 2, 3, 4,

(4)

where X is the (p − 1)st prolongation of the symmetry generator X=

4 X

Qk

k=1

∂ , ∂uk

(5)

i. e., X = X [p−1] =

p−1 X 4 X

S j (Qk )

j=0 k=1

∂ . ∂Sj (uk )

(6)

We shall refer to Qk = Qk (n, un ) as characteristics and for simplicity we shall consider point transformations only, that is, Qk = Qk (n, uk ). Definition 1.1 [14] Let G be a connected group of transformations acting on a manifold M . A smooth real-valued function ζ : M → R is an invariant function for G if and only if X(ζ) = 0

for all

x ∈ M,

Without any lucky guess, the reduction of order can readily be done via the canonical coordinates [9] Z duk k s = , k = 1, 2, 3, 4. Qk (n, uk )

(7)

Eventually, the constraining restrictions on the constants in the characteristics, Qk , k = 1, 2, 3, 4, hint on a perfect choice of invariants.

2

Main results

To start, we consider the corresponding forward system xn yn xn+4 = Ω1 = , yn+4 = Ω2 = An + Bn xn yn+1 zn+2 tn+3 Cn + Dn xn+3 yn zn+1 tn+2 zn tn zn+4 = Ω3 = , tn+4 = Ω4 = , En + Fn xn+2 yn+3 zn tn+1 Gn + Hn xn+1 yn+2 zn+3 tn

(8)

where (An )n∈N0 , (Bn )n∈N0 , (Cn )n∈N0 , (Dn )n∈N0 , (En )n∈N0 , (Fn )n∈N0 , (Gn )n∈N0 and (Hn )n∈N0 are nonzero sequences of real numbers, equivalent to (2). 2

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2.1

Symmetries

To construct the characteristics of the system of fourth order difference equations (8), we must impose linearized symmetry criterion (4). This amounts to Bn x2n (tn+3 zn+2 (S 1 Q2 ) + tn+3 yn+1 (S 2 Q3 ) + yn+1 zn+2 (S 3 Q4 )) − An Q1 = 0, (An + Bn xn yn+1 zn+2 tn+3 )2 Dn yn2 (tn+2 xn+3 (S 1 Q3 ) + xn+3 zn+1 (S 2 Q4 ) + tn+2 zn+1 (S 3 Q1 )) − Cn Q2 S 4 Q2 + = 0, (Cn + Dn xn+3 yn zn+1 tn+2 )2 Fn zn2 (tn+1 xn+2 (S 3 Q2 ) + xn+2 yn+3 (S 1 Q4 ) + tn+1 yn+3 (S 2 Q1 )) − En Q3 S 4 Q3 + = 0, (En + Fn xn+2 yn+3 zn tn+1 )2 Hn t2n (xn+1 zn+3 (S 2 Q2 ) + xn+1 yn+2 (S 3 Q3 ) + yn+2 zn+3 (S 1 Q1 )) − Gn Q4 = 0. S 4 Q4 + (Gn + Hn xn+1 yn+2 zn+3 tn )2 S 4 Q1 +

(9a) (9b) (9c) (9d)

We act the operators ∂/∂xn −[(∂Ω1 /∂xn )/(∂Ω1 /∂yn+1 )]∂/∂yn+1 , ∂/∂yn −[(∂Ω2 /∂yn )(∂Ω2 /∂zn+1 )]∂/∂zn+1 , ∂/∂zn −[(∂Ω3 /∂zn )(∂Ω3 /∂yn+3 )]∂/∂yn+3 and ∂/∂tn −[(∂Ω4 /∂tn )(∂Ω4 /∂yn+2 )]∂/∂yn+2 on equations in (9), respectively, to get 0

0

(S 1 Q2 ) − Q1 + (1/zn+2 )(S 2 Q3 ) + (1/tn+3 )(S 3 Q4 ) + (2/xn )Q1 = 0

(10a)

20

− Q + (S 1 Q3 )0 + (2/yn )Q2 + (1/tn+2 )(S 2 Q4 ) + (1/xn+3 )(S 3 Q1 ) = 0 3

2 0

30

2

2 0

0

3

1

1

(10c)

1

1

(10d)

(S Q ) − Q + (2/zn )Q + (1/tn+1 )(S Q4 ) + (1/xn+2 )(S Q ) = 0 3

(10b)

2

(S Q ) − Q4 + (1/zn+3 )(S Q3 ) + (2/tn )Q4 + (1/xn+1 )(S Q ) = 0

after simplification. Note that 0 denotes the derivative with respect to the continuous variable. Next, we differentiate equations in (10) with respect to xn , yn , zn and tn , respectively. The latter leads to the differential equations 00

0

00

0

00

0

− Q1 + (2/xn )Q1 − (2/x2n )Q1 = 0, −Q2 + (2/yn )Q2 − (2/yn2 )Q2 = 0, 00

0

− Q3 + (2/zn )Q3 − (2/zn2 )Q3 = 0, −Q4 + (2/tn )Q4 − (2/t2n )Q4 = 0

(11)

whose solutions are given by Q1 (n, xn ) = α1 (n)xn 2 + β1 (n)xn , Q3 (n, zn ) = α3 (n)zn 2 + β3 (n)zn ,

Q2 (n, yn ) = α2 (n)yn 2 + β2 (n)yn ,

(12)

Q4 (n, tn ) = α4 (n)tn 2 + β4 (n)tn ,

for some functions αi and βi , respectively. We replace (12) and their shits in (9). Due to the fact that the αi ’s and βi ’s depend on the independent variable only, we equate all products of shifts of dependent variables xn , yn , zn and tn in the resulting equations to zero; this yields the ‘final constraints’ below β1 (n) + β2 (n + 1) + β3 (n + 2) + β4 (n + 3) = 0, α1 (n) = α2 (n) = α3 (n) = α4 (n) = 0,

(13)

with β1 (n) = β1 (n + 4), β2 (n) = β2 (n + 4), β3 (n) = β3 (n + 4), β4 (n) = β4 (n + 4). The reader can easily verify that the functions satisfying the above constraints are of the forms: αj (n) = 0, j = 1, 2, 3, 4; β1 (n) = c1 + c2 (−i)n + c3 (i)n + c4 (−1)n ; β2 (n) = c5 + c6 (−i)n + c7 (i)n + c8 (−1)n ; β3 (n) = c9 + c10 (−i)n + c11 (i)n + c12 (−1)n ; β4 (n) = (ic2 + c6 − ic10 )(−i)n + (c7 − ic3 + ic11 )(i)n + (c4 − c8 + c12 )(−1)n − c1 − c5 − c9 , (14)

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where the ci ’s, i = 1, . . . , 12, are arbitrary constants. Consequently, thanks to (5), (12) and (14), we obtain twelve symmetry generators: X1 = xn ∂xn − tn ∂tn , X2 = (−i)n (xn ∂xn + itn ∂tn ), X3 = in (xn ∂xn − itn ∂tn ), X4 = (−1)n (xn ∂xn + tn ∂tn ), X5 = yn ∂yn − tn ∂tn , X6 = (−i)n (yn ∂yn + tn ∂tn ), X7 = in (yn ∂yn + tn ∂tn ), X8 = (−1)n (yn ∂yn − tn ∂tn ), X9 = zn ∂zn − tn ∂tn , X10 = (−i)n (zn ∂zn − itn ∂tn ), X11 = in (zn ∂zn + itn ∂tn ), X12 = (−1)n (zn ∂zn + tn ∂tn ). (15) Note that for simplicity, we adopt the notation ∂x = ∂/∂x.

2.2

Reduction of order via symmetries and formulas for solutions

Using any linear combinations of the symmetries in (15) that involves all four independent variables xn , yn , zn and tn , say X = X1 + X2 + X3 = xn ∂xn + yn ∂yn + zn ∂zn − 3tn ∂tn , we derive the corresponding canonical coordinates Z Z Z Z dyn dzn dtn dxn , s2 (n) = , s3 (n) = , s4 (n) = . (16) s1 (n) = xn yn zn −3tn Inspired by the form of the equations in the final constraints (13), we construct the invariants: ˜ n =β1 (n)s1 (n) + β2 (n + 1)s2 (n + 1) + β3 (n + 2)s3 (n + 2) + β4 (n + 3)s4 (n + 3) = ln |xn yn+1 zn+2 tn+3 | X Y˜n =β1 (n + 3)s1 (n + 3) + β2 (n)s2 (n) + β3 (n + 1)s3 (n + 1) + β4 (n + 2)s4 (n + 2) = ln |xn+3 yn zn+1 tn+2 | Z˜n =β1 (n + 2)s1 (n + 2) + β2 (n + 3)s2 (n + 3) + β3 (n)s3 (n) + β4 (n + 1)s4 (n + 1) = ln |xn+2 yn+3 zn tn+1 | T˜n =β1 (n + 1)s1 (n + 1) + β2 (n + 2)s2 (n + 2) + β3 (n + 3)s3 (n + 3) + β4 (n)s4 (n) = ln |xn+1 yn+2 zn+3 tn |, obtained by replacing βi (n + j) by si (n + j)βi (n + j) in the left hand sides of equations in (13). ˜ n , Y˜n , Z˜n and T˜n are invariant functions. For Using Definition 1.1, the reader can easily confirm that X simplicity, we introduce the variables ˜ n ), Yn = exp(−Y˜n ), Zn = exp(−Z˜n ), Tn = exp(−T˜n ). Xn = exp(−X

(17)

Xn+1 =Hn + Gn Tn , Yn+1 = Bn + An Xn , Zn+1 = Dn + Cn Yn , Tn+1 = Fn + En Zn

(18a)

Thus

and so xn+4 =

Xn Yn Zn Tn xn , yn+4 = yn , zn+4 = zn , tn+4 = tn . Yn+1 Zn+1 Tn+1 Xn+1

(18b)

Straightforward iterations ( using equation (18a)) yield Xn+4 = Λxn + (Θxn )Xn , Yn+4 = ∆yn + (Θyn )Yn , Zn+4 = ∆zn + (Θzn )Zn , Tn+4 = ∆tn + (Θtn )Tn that is U4n+j = Uj

n−1 Y

! Θu4k1 +j

+

k1 =0

n−1 X

Λu4l+j

l=0

n−1 Y

! Θu4k2 +j

,

(19a)

k2 =l+1

for j = 0, 1, 2, 3 and (U, u) ∈ {(X, x), (Y, y), (Z, z), (T, t)}, where Λxn = Hn+3 + Gn+3 Fn+2 + Gn+3 En+2 Dn+1 + Gn+3 En+2 Cn+1 Bn , Θxn = Gn+3 En+2 Cn+1 An ; Λyn = Bn+3 + An+3 Hn+2 + An+3 Gn+2 Fn+1 + An+3 Gn+2 En+1 Dn , Θyn = An+3 Gn+2 En+1 Cn ; Λzn = Dn+3 + Cn+3 Bn+2 + Cn+3 An+2 Hn+1 + Cn+3 An+2 Gn+1 Fn , Θzn = Cn+3 An+2 Gn+1 En ; Λtn = Fn+3 + En+3 Dn+2 + En+3 Cn+2 Bn+1 + En+3 Cn+2 An+1 Hn , Θtn = En+3 Cn+2 An+1 Gn ;

(19b)

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Also, straightforward iterations (using equation (18b)) yield x4n+j = xj

n−1 Y k=0

n−1 n−1 n−1 Y Y4k+j Y Z4k+j Y T4k+j X4k+j , y4n+j = yj , z4n+j = zj , t4n+j = tj , Y4k+1+j Z4k+1+j T4k+1+j X4k+1+j k=0

k=0

(19c)

k=0

j = 0, 1, 2, 3. Combining equations in (19), we obtain the following solutions {xn } of the system of equations (8): ! !   s−1 s−1 s−1 Q Q x P x x Xj Θ4k1 +j + Λ4l+j Θ4k2 +j   n−1 Y  k1 =0 l=0 k2 =l+1  , j = 0, 1, 2,  ! ! x4n+j =xj   s−1 s−1 s−1 P Q Q y  y y s=0  Θ4k2 +j+1 Λ4l+j+1 Yj+1 Θ4k1 +j+1 + l=0

k1 =0

 x4n+3

 n−1 Y

  =x3  s=0 

X3

s−1 Q k1 =0

Y0

! Θx4k1 +3 s Q

k1 =0

+

s−1 P l=0

! Θy4k1

+

s P l=0

k2 =l+1

Λx4l+3 Λy4l

!

s−1 Q k2 =l+1

Θx4k2 +3

  ,  

!

s Q k2 =l+1

Θy4k2

(20)

where Θun and Λun , u ∈ {x, y, z, t} are defined in (19b); and X0 = 1/(x0 y1 z2 t3 ), X1 = H0 +G0 /(t0 x1 y2 z3 ), X2 = F0 G1 +H1 +(E0 G1 )/(t1 x2 y3 z0 ) X3 = D0 E1 G2 +F1 G2 +H2 +(C0 E1 G2 )/(t2 x3 y0 z1 ), Y0 = 1/(t2 x3 y0 z1 ), Y1 = B0 + A0 /(t3 x0 y1 z2 ), Y2 = A1 H0 + B1 + (A1 G0 )/(t0 x1 y2 z3 ), Y3 = A2 F0 G1 + B1 + (A2 E0 G1 )/(t1 x2 y3 z0 ). Recall that we forward shifted equation (2) thrice to obtain (8) whose solutions xn is giving in (20). Now, we go backward thrice and replace the capital letters in the right hand sides of equations in (19b) with lower cases letters to get the solutions xn corresponding to (8). In other words, solutions {xn } of the system of equations (2) is giving by ! s−1  s−1 s−1 Q x P Q x θ4i + x−3 y−2 z−1 t0 λx4l θ4i n−1 Y i=0 l=0 i=l+1 ! x4n−3 =x−3 s−1  s−1 s−1 Q P Q y y s=0 (a0 + b0 x−3 y−2 z−1 t0 ) θ4i+1 + x−3 y−2 z−1 t0 λy4l+1 θ4i+1 i=0

x4n−2 =x−2

n−1 Y s=0

(g0 + h0 t−3 x−2 y−1 z0 )

l=0

s−1 Q i=0

x θ4i+1

((a1 h0 + b1 )t−3 x−2 y−1 z0 + a1 g0 )

x4n−1 =x−1

s=0

+ t−3 x−2 y−1 z0 y θ4i+2

((f0 g1 + h1 )t−2 x−1 y0 z−3 + e0 g1 )

 + t−3 x−2 y−1 z0

x4n =x0

s=0

i=0

x θ4i+2

((a0 f0 g1 + a2 h1 + b2 )t−2 x−1 y0 z−3 + a2 e0 g1 )

((d0 e1 g2 + f1 g2 + h2 )t−1 x0 y−3 z−2 + c0 e1 g2 ) s Q i=0

y θ4i

s−1 P

!

s−1 Q

x θ4i+1

i=l+1 s−1 Q

λy4l+2

 + t−2 x−1 y0 z−3

s−1 Q

y θ4i+3

s−1 Q x λx4l+2 θ4i+2 i=l+1

x θ4i+3

 + t−1 x0 y−3 z−2

 + t−2 x−1 y0 z−3

s−1 P l=0

 + t−1 x0 y−3 z−2

s−1 P l=0

s P l=0

λy4l

s Q

! y θ4i+2

i=l+1

s−1 P l=0

s−1 Q

i=0



λx4l+1

l=0

s−1 Q

i=0

n−1 Y

s−1 P l=0

s−1 Q i=0

n−1 Y

i=l+1



!

s−1 Q y λy4l+3 θ4i+3 i=l+1

λx4l+3

s−1 Q

! x θ4i+3

i=l+1

! y θ4i

i=l+1

5

953

!

Folly-Gbetoula 949-961

.

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Similar computations yield s−1 Q y4n−3 =y−3

n−1 Y s=0

i=0

y θ4i

 + t−1 x0 y−3 z−2

(c0 + d0 t−1 x0 y−3 z−2 )

y4n−2 =y−2

s=0

(a0 + b0 t0 x−3 y−2 z−1 )

+ t−1 x0 y−3 z−2

y4n−1 =y−1

s=0

s−1 Q i=0

y θ4i+1

y4n =y0

+ t0 x−3 y−2 z−1

l=0

((b0 c1 + d1 )t0 x−3 y−2 z−1 + a0 c1 )

s−1 Q

z θ4i+2

+ t0 x−3 y−2 z−1

s−1 Q i=0

((a2 f0 g1 + a2 h1 + b2 )t−2 x−1 y0 z−3 + a2 e0 g1 )

y θ4i+2

i=0

s−1 Q z4n−3 =z−3

n−1 Y s=0

i=0

z θ4i

z θ4i+3

z4n−2 =z−2

s=0

n−1 Y s=0

s−1 P

(e0 + f0 t−2 x−1 y0 z−3 )

(c0 + d0 t−1 x0 y−3 z−2 )

((b0 c1 + d1 )t0 x−3 y−2 z−1 + a0 c1 )

z4n =z0

((b0 c1 e2 + d1 e2 + f2 )t0 x−3 y−2 z−1 + a0 c1 e2 )

((a1 c2 h0 + b1 c2 + d2 )t−3 x−2 y−1 z0 + a1 c2 g0 )

s=0

s Q

i=0

s−1 Q t4n−3 =t−3

n−1 Y s=0

i=0

t θ4i

(g0 + h0 t−3 x−2 y−1 z0 )

t θ4i

s−1 Q

t θ4i+3

t θ4i+1

i=l+1

λz4l+1 s−1 P

s−1 P

!

s−1 Q

z θ4i+1

i=l+1

λt4l+2 s−1 P

λt4l

+ t0 x−3 y−2 z−1 + t−3 x−2 y−1 z0

i=l+1

s−1 P

s−1 P l=0

s P

s−1 Q

λt4l

t θ4i+2

s−1 Q z λz4l+2 θ4i+2 i=l+1





!

s−1 Q

l=0

l=0

l=0 x θ4i+1

s−1 P

!

s−1 Q

+ t0 x−3 y−2 z−1

z θ4i+3

+ t−3 x−2 y−1 z0

+ t−3 x−2 y−1 z0

i=0

λt4l+1

l=0





s−1 Q

s−1 P



s−1 Q

i=0



z θ4i

l=0

i=0

!

i=l+1

+ t−1 x0 y−3 z−2 z θ4i+2

s−1 Q y λy4l+3 θ4i+3 i=l+1

i=l+1



s−1 Q

s−1 P

λt4l+3

!

s−1 Q

! t θ4i+3

i=l+1

s−1 Q z λz4l+3 θ4i+3 i=l+1

!

!

s Q

t θ4i

i=l+1

! t θ4i

i=l+1

 + t−3 x−2 y−1 z0

s−1 P l=0

λx4l+1

s−1 Q

! x θ4i+1

i=l+1

6

954

!

!

s Q

l=0

((d0 e1 + f1 )t−1 x0 y−3 z−2 + c0 e1 )

s−1 Q z λz4l+3 θ4i+3 i=l+1

z θ4i

+ t−1 x0 y−3 z−2 t θ4i+2

s−1 P

!

!

s−1 Q

i=0

n−1 Y

+ t−3 x−2 y−1 z0

λz4l



s−1 Q

s−1 Q y θ4i+2 λy4l+2 i=l+1



l=0

i=0

s−1 P

l=0

+ t−2 x−1 y0 z−3 z θ4i+1

z θ4i+2

i=l+1

+ t−2 x−1 y0 z−3

s P



s−1 Q

!

l=0

y θ4i+3

λz4l

l=0 t θ4i+1

s−1 Q

λz4l+2



l=0

i=0

z4n−1 =z−1

s−1 Q

+ t−2 x−1 y0 z−3

+ t−2 x−1 y0 z−3

s−1 Q

s−1 P

+ t−3 x−2 y−1 z0





i=0

n−1 Y

z θ4i

! y θ4i+1

i=l+1



s−1 Q

i=0

s=0

s−1 Q

l=0

((a1 c2 h0 + b1 c2 + d2 )t−3 x−2 y−1 z0 + a1 c2 g0 )

s Q

λy4l+1



z θ4i+1

i=l+1

l=0

((a1 h0 + b1 )t−3 x−2 y−1 z0 + a1 g0 )



s−1 P

!

s−1 Q

λz4l+1



i=0

n−1 Y

s−1 P l=0

i=0

n−1 Y

y θ4i

i=l+1



z θ4i+1

!

s−1 Q

λy4l

l=0

s−1 Q i=0

n−1 Y

s−1 P

Folly-Gbetoula 949-961

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

t4n−2 =t−2

(e0 + f0 t−2 x−1 y0 z−3 )

n−1 Y s=0

s−1 Q i=0

t θ4i+1

((f0 g1 + h1 )t−2 x−1 y0 z−3 + e0 g1 )



t4n−1 =t−1

s=0

+ t−2 x−1 y0 z−3

((d0 e1 + f1 )t−1 x0 y−3 z−2 + c0 e1 )

x θ4i+2

 + t−2 x−1 y0 z−3

t4n =t0

i=0

t θ4i+2

((b0 c1 e2 + d1 e2 + f2 )t0 x−3 y−2 z−1 + a0 c1 e2 )

s−1 Q i=0

s=0

s Q

i=0

x θ4i

i=l+1

s−1 P

+ t−1 x0 y−3 z−2

+ t0 x−3 y−2 z−1

+ t−1 x0 y−3 z−2 + t0 x−3 y−2 z−1

s−1 P

s−1 P

l=0

λx4l

s−1 Q

! t θ4i+2

i=l+1 s−1 Q

λx4l+3

! x θ4i+3

i=l+1

s−1 Q t λt4l+3 θ4i+3 i=l+1

l=0 s P

x θ4i+2

i=l+1

l=0



!

s−1 Q

λt4l+2



x θ4i+3

t θ4i+3



t θ4i+1

λx4l+2



s−1 Q

!

s−1 Q

l=0

((d0 e1 g2 + f1 g2 + h2 )t−1 x0 y−3 z−2 + c0 e1 g2 )



s−1 P l=0

s−1 Q

i=0

n−1 Y

λt4l+1

l=0

s−1 Q i=0

n−1 Y

s−1 P

!

!

s Q

x θ4i

i=l+1

(21a) Note that λxn = hn+3 + gn+3 fn+2 + gn+3 en+2 dn+1 + gn+3 en+2 cn+1 bn , θnx = gn+3 en+2 cn+1 an ; λyn = bn+3 + an+3 hn+2 + an+3 gn+2 fn+1 + an+3 gn+2 en+1 dn , θny = an+3 gn+2 en+1 cn ; λzn = dn+3 + cn+3 bn+2 + cn+3 an+2 hn+1 + cn+3 an+2 gn+1 fn , θnz = cn+3 an+2 gn+1 en ; λtn = fn+3 + en+3 dn+2 + en+3 cn+2 bn+1 + en+3 cn+2 an+1 hn , θnt = en+3 cn+2 an+1 gn .

2.3

(21b)

Case where an , bn , cn , dn , en , fn , gn and hn are periodic of period four

Suppose {an } = {a0 , a1 , a2 , a3 , a0 , . . . }, {bn } = {b0 , b1 , b2 , b3 , b0 , . . . }, {cn } = {c0 , c1 , c2 , c3 , c0 , . . . }, {dn } = {d0 , d1 , d2 , d3 , d0 , . . . }, {en } = {e0 , e1 , e2 , e3 , e0 , . . . }, {fn } = {f0 , f1 , f2 , f3 , f0 , . . . } and {gn } = {g0 , g1 , g2 , g3 , g0 , . . . }. Equations in (21) simplify to s

x4n−3 =x−3

n−1 Y s=0

(θ0x ) + x−3 y−2 z−1 t0 (λx0 )

s−1 P

(θ0x )l

l=0 s

(a0 + b0 x−3 y−2 z−1 t0 ) (θ1y ) + x−3 y−2 z−1 t0 (λy1 )

s−1 P

l

(θ1y )

l=0

x4n−2 =x−2

n−1 Y s=0

(g0 + h0 t−3 x−2 y−1 z0 )(θ1x )s + t−3 x−2 y−1 z0 (λx1 )

s−1 P

(θ1x )l

l=0 s

((a1 h0 + b1 )t−3 x−2 y−1 z0 + a1 g0 ) [θ2y ] + t−3 x−2 y−1 z0 (λy2 )

s−1 P

(θ2y )l

l=0 s

x4n−1 =x−1

n−1 Y s=0

((f0 g1 + h1 )t−2 x−1 y0 z−3 + e0 g1 ) (θ2x ) + t−2 x−1 y0 z−3 λx2

s−1 P

l

(θ2x )

l=0 s

((a0 f0 g1 + a2 h1 + b2 )t−2 x−1 y0 z−3 + a2 e0 g1 ) (θ3y ) + t−2 x−1 y0 z−3 (λy3 )

s−1 P

l

(θ3y )

l=0 s

x4n =x0

n−1 Y s=0

((d0 e1 g2 + f1 g2 + h2 )t−1 x0 y−3 z−2 + c0 e1 g2 ) (θ3x ) + t−1 x0 y−3 z−2 (λx3 )

s−1 P

l

(θ3x )

l=0

(θ0y )

s+1

+ t−1 x0 y−3 z−2 (λy0 )

s P

(θ0y )

l

l=0

7

955

Folly-Gbetoula 949-961

.

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

s

y4n−3 =y−3

(θ0y ) + t−1 x0 y−3 z−2 (λy0 )

n−1 Y s=0

s−1 P

l

(θ0y )

l=0 s−1 P

s

(c0 + d0 t−1 x0 y−3 z−2 ) (θ1z ) + t−1 x0 y−3 z−2 (λz1 )

(θ1z )

l

l=0 s

y4n−2 =y−2

(a0 + b0 t0 x−3 y−2 z−1 ) (θ1y ) + t0 x−3 y−2 z−1 (λy1 )

n−1 Y s=0

s−1 P

l

(θ1y )

l=0

((b0 c1 + d1 )t0 x−3 y−2 z−1 +

s a0 c1 ) (θ2z )

+ t0 x−3 y−2 z−1 (λz2 )

s−1 P

l

(θ2z )

l=0

y4n−1 =y−1

((a1 h0 + b1 )t−3 x−2 y−1 z0 +

n−1 Y s=0

s a1 g0 ) (θ2y )

+ t−3 x−2 y−1 z0 (λy2 )

s−1 P

(θ2y )

l

l=0

((a1 c2 h0 + b1 c2 + d2 )t−3 x−2 y−1 z0 +

s a1 c2 g0 ) (θ3z )

+ t−3 x−2 y−1 z0 (λz3 )

s−1 P

(θ3z )

l

l=0

y4n =y0

n−1 Y

((a2 f0 g1 + a2 h1 + b2 )t−2 x−1 y0 z−3 +

s a2 e0 g1 ) (θ3y )

+ t−2 x−1 y0 z−3 (λy3 )

s−1 P

l

(θ3y )

l=0 s+1 (θ0z )

s=0

+

t−2 x−1 y0 z−3 (λz0 )

s P

l (θ0z )

l=0

z4n−3 =z−3

s (θ0z )

n−1 Y s=0

+ t−2 x−1 y0 z−3 (λz0 )

s−1 P

l (θ0z )

l=0

(e0 +

s f0 t−2 x−1 y0 z−3 ) (θ1t )

+ t−2 x−1 y0 z−3 (λt1 )

s−1 P

l

(θ1t )

l=0

z4n−2 =z−2

n−1 Y s=0

(c0 +

s d0 t−1 x0 y−3 z−2 ) (θ1z )

+ t−1 x0 y−3 z−2 (λz1 )

s−1 P

l

(θ1z )

l=0

((d0 e1 + f1 )t−1 x0 y−3 z−2 +

s c0 e1 ) (θ2t )

+ t−1 x0 y−3 z−2 (λt2 )

s−1 P

l

(θ2t )

l=0 s

z4n−1 =z−1

n−1 Y s=0

((b0 c1 + d1 )t0 x−3 y−2 z−1 + a0 c1 ) (θ2z ) + t0 x−3 y−2 z−1 (λz2 )

s−1 P

(θ2z )

l

l=0 s

((b0 c1 e2 + d1 e2 + f2 )t0 x−3 y−2 z−1 + a0 c1 e2 ) (θ3t ) + t0 x−3 y−2 z−1 (λt3 )

s−1 P

l

(θ3t )

l=0 s

z4n =z0

n−1 Y

((a1 c2 h0 + b1 c2 + d2 )t−3 x−2 y−1 z0 + a1 c2 g0 ) (θ3z ) + t−3 x−2 y−1 z0 (λz3 )

s−1 P

(θ3z )

l

l=0 s+1

(θ0t )

s=0

+ t−3 x−2 y−1 z0 (λt0 )

s P

(θ0t )

l

l=0

t4n−3 =t−3

n−1 Y s=0

s (θ0t )

+ t−3 x−2 y−1 z0 (λt0 )

s−1 P

l (θ0t )

l=0 s

(g0 + h0 t−3 x−2 y−1 z0 ) (θ1x ) + t−3 x−2 y−1 z0 (λx1 )

s−1 P

l

(θ1x )

l=0 s

t4n−2 =t−2

n−1 Y s=0

(e0 + f0 t−2 x−1 y0 z−3 ) (θ1t ) + t−2 x−1 y0 z−3 (λt1 )

s−1 P

l

(θ1t )

l=0 s

((f0 g1 + h1 )t−2 x−1 y0 z−3 + e0 g1 ) (θ2x ) + t−2 x−1 y0 z−3 (λx2 )

s−1 P

l

(θ2x )

l=0 s

t4n−1 =t−1

n−1 Y s=0

((d0 e1 + f1 )t−1 x0 y−3 z−2 + c0 e1 ) (θ2t ) + t−1 x0 y−3 z−2 (λt2 )

s−1 P

l

(θ2t )

l=0 s

((d0 e1 g2 + f1 g2 + h2 )t−1 x0 y−3 z−2 + c0 e1 g2 ) (θ3x ) + t−1 x0 y−3 z−2 (λx3 )

s−1 P

(θ3x )

l

l=0

8

956

Folly-Gbetoula 949-961

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

s

t4n =t0

n−1 Y

((b0 c1 e2 + d1 e2 + f2 )t0 x−3 y−2 z−1 + a0 c1 e2 ) (θ3t ) + t0 x−3 y−2 z−1 (λt3 )

s−1 P

(θ3t )

l

l=0 s+1

(θ0x )

s=0

+ t0 x−3 y−2 z−1 (λx0 )

s P

,

(22)

l

(θ0x )

l=0

where

θ0u ,

λu0 ,

2.4

Case where an , bn , cn , dn , en , fn , gn and hn are constant

u = x, y, z, t are defined in (21b).

Suppose that an = a, bn = b, cn = c, dn = d, en = e, fn = f and gn = g. Equations in (22) simplify to   s−1 P s l (aceg) + x y z t (h + gf + ged + gecb) (aceg) n−1 −3 −2 −1 0 Y  l=0   x4n−3 =x−3   s−1 P s l s=0 (a + bx (aceg) −3 y−2 z−1 t0 ) (aceg) + x−3 y−2 z−1 t0 (b + ah + agf + aged) l=0

x4n−2 =x−2

n−1 Y s=0

(g + htx−2 y−1 z0 )(aceg)s + t−3 x−2 y−1 z0 (h + gf + ged + gecb)

s−1 P

(aceg)l

l=0 s

((ah + b)t−3 x−2 y−1 z0 + ag) (aceg) + t−3 x−2 y−1 z0 (b + ah + agf + aged)

s−1 P

(aceg)l

l=0 s

x4n−1 =x−1

n−1 Y s=0

((f g + h)t−2 x−1 y0 z−3 + eg) (aceg) + t−2 x−1 y0 z−3 (h + gf + ged + gecb)

s−1 P

l

(aceg)

l=0 s−1 P

s

((af g + a2 h + b)t−2 x−1 y0 z−3 + aeg) (aceg) + t−2 x−1 y0 z−3 (b + ah + agf + aged)

l

(aceg)

l=0 s

x4n =x0

n−1 Y

((deg + f g + h)t−1 x0 y−3 z−2 + ceg) (aceg) + t−1 x0 y−3 z−2 (h + gf + ged + gecb)

s−1 P

l

(aceg)

l=0

(aceg)

s=0

s+1

+ t−1 x0 y−3 z−2 (b + ah + agf + aged)

s P

(aceg)

l

l=0 s

y4n−3 =y−3

n−1 Y s=0

(aceg) + t−1 x0 y−3 z−2 (b + ah + agf + aged)

s−1 P

(aceg)

l

l=0 s−1 P

s

(c + dt−1 x0 y−3 z−2 ) (aceg) + t−1 x0 y−3 z−2 (d + cb + cah + cagf )

l

(aceg)

l=0 s

y4n−2 =y−2

n−1 Y s=0

(a + bt0 x−3 y−2 z−1 ) (aceg) + t0 x−3 y−2 z−1 (b + ah + agf + aged)

s−1 P

(aceg)

l

l=0 s−1 P

s

((bc + d)t0 x−3 y−2 z−1 + ac) (aceg) + t0 x−3 y−2 z−1 (d + cb + cah + cagf )

l

(aceg)

l=0 s

y4n−1 =y−1

n−1 Y s=0

((ah + b)t−3 x−2 y−1 z0 + ag) (aceg) + t−3 x−2 y−1 z0 (b + ah + agf + aged)

s−1 P

(aceg)

l

l=0 s

((ach + bc + d)t−3 x−2 y−1 z0 + acg) (aceg) + t−3 x−2 y−1 z0 (d + cb + cah + cagf )

s−1 P

(aceg)

l

l=0 s

y4n =y0

n−1 Y

((af g + ah + b)t−2 x−1 y0 z−3 + aeg) (aceg) + t−2 x−1 y0 z−3 (b + ah + agf + aged)

s=0

s−1 P

l

(aceg)

l=0 s+1

(aceg)

+ t−2 x−1 y0 z−3 (d + cb + cah + cagf )

s P

l

(aceg)

l=0 s

z4n−3 =z−3

n−1 Y s=0

(aceg) + t−2 x−1 y0 z−3 (d + cb + cah + cagf )

s−1 P

l

(aceg)

l=0 s

(e + f t−2 x−1 y0 z−3 ) (aceg) + t−2 x−1 y0 z−3 (f + ed + ecb + ecah)

s−1 P

l

(aceg)

l=0

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s

z4n−2 =z−2

(c + dt−1 x0 y−3 z−2 ) (aceg) + t−1 x0 y−3 z−2 (d + cb + cah + cagf )

n−1 Y s=0

s−1 P

l

(aceg)

l=0 s

((de + f )t−1 x0 y−3 z−2 + ce) (aceg) + t−1 x0 y−3 z−2 (f + ed + ecb + ecah)

s−1 P

(aceg)

l

l=0 s

z4n−1 =z−1

((bc + d)t0 x−3 y−2 z−1 + ac) (aceg) + t0 x−3 y−2 z−1 (d + cb + cah + cagf )

n−1 Y s=0

s−1 P

l

(aceg)

l=0 s

((bce + de + f )t0 x−3 y−2 z−1 + ace) (aceg) + t0 x−3 y−2 z−1 (f + ed + ecb + ecah)

s−1 P

l

(aceg)

l=0 s

z4n =z0

n−1 Y

((ach + bc + d)t−3 x−2 y−1 z0 + acg) (aceg) + t−3 x−2 y−1 z0 (d + cb + cah + cagf )

s−1 P

l

(aceg)

l=0 s+1

(aceg)

s=0

+ t−3 x−2 y−1 z0 (f + ed + ecb + ecah)

s P

(aceg)

l

l=0 s

t4n−3 =t−3

(aceg) + t−3 x−2 y−1 z0 (f + ed + ecb + ecah)

n−1 Y s=0

s−1 P

(aceg)

l

l=0 s

(g + ht−3 x−2 y−1 z0 ) (aceg) + t−3 x−2 y−1 z0 (h + gf + ged + gecb)

s−1 P

l

(aceg)

l=0 s

t4n−2 =t−2

(e + f t−2 x−1 y0 z−3 ) (aceg) + t−2 x−1 y0 z−3 (f + ed + ecb + ecah)

n−1 Y s=0

s−1 P

l

(aceg)

l=0 s

((f g + h)t−2 x−1 y0 z−3 + eg) (aceg) + t−2 x−1 y0 z−3 (h + gf + ged + gecb)

s−1 P

(aceg)

l

l=0 s

t4n−1 =t−1

((de + f )t−1 x0 y−3 z−2 + ce) (aceg) + t−1 x0 y−3 z−2 (f + ed + ecb + ecah)

n−1 Y s=0

s−1 P

l

(aceg)

l=0 s

((deg + f g + h)t−1 x0 y−3 z−2 + ceg) (aceg) + t−1 x0 y−3 z−2 (h + gf + ged + gecb)

s−1 P

l

(aceg)

l=0 s

t4n =t0

n−1 Y

((bce + de + f )t0 x−3 y−2 z−1 + ace) (aceg) + t0 x−3 y−2 z−1 (f + ed + ecb + ecah)

s−1 P

s+1

(aceg)

s=0

+ t0 x−3 y−2 z−1 (h + gf + ged + gecb)

s P

(aceg)

l

(aceg)

l=0

.

l

l=0

(23) 2.4.1

Case where a = 1, b = 1, c = 1, d = 1, e = 1, f = 1, g = 1 and h = 1 x

Here, θ = θy = θz = θt = 1 and λx = λy = λz = λt = 4. Thus, equations in (23) simplify to  n−1 Y  1 + (4s + 1)t−3 x−2 y−1 z0  1 + 4sx−3 y−2 z−1 t0 , x4n−2 = x−2 , 1 + (4s + 1)x−3 y−2 z−1 t0 1 + (4s + 2)t−3 x−2 y−1 z0 s=0 s=0 n−1 n−1 Y  1 + (4s + 3)t−1 x0 y−3 z−2  Y  1 + (4s + 2)t−2 x−1 y0 z−3  =x−1 , x4n = x0 , 1 + (4s + 3)t−2 x−1 y0 z−3 1 + (4s + 4)t−1 x0 y−3 z−2 s=0 s=0 n−1 n−1 Y  1 + 4st−1 x0 y−3 z−2  Y  1 + +(4s + 1)t0 x−3 y−2 z−1  =y−3 , y4n−2 = y−2 , 1 + (4s + 1)t−1 x0 y−3 z−2 1 + (4s + 2)t0 x−3 y−2 z−1 s=0 s=0 n−1 n−1 Y  1 + (4s + 2)t−3 x−2 y−1 z0  Y  1 + (4s + 3)t−2 x−1 y0 z−3  =y−1 , y4n = y0 , 1 + (4s + 3)t−3 x−2 y−1 z0 1 + (4s + 4)t−2 x−1 y0 z−3 s=0 s=0

x4n−3 =x−3 x4n−1 y4n−3 y4n−1

n−1 Y

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z4n−3 z4n−1 t4n−3 t4n−1

2.5

n−1 Y

 n−1 Y  1 + (4s + 1)t−1 x0 y−3 z−2  1 + 4st−2 x−1 y0 z−3 =z−3 , z4n−2 = z−2 , 1 + (4s + 1)t−2 x−1 y0 z−3 1 + (4s + 2)t−1 x0 y−3 z−2 s=0 s=0 n−1 n−1 Y  1 + (4s + 2)t0 x−3 y−2 z−1  Y  1 + (4s + 3)t−3 x−2 y−1 z0  =z−1 , z4n = z0 , 1 + (4s + 3)t0 x−3 y−2 z−1 1 + (4s + 4)t−3 x−2 y−1 z0 s=0 s=0 n−1 n−1 Y  1 + (4s + 1)t−2 x−1 y0 z−3  Y  1 + 4st−3 x−2 y−1 z0  , t4n−2 = t−2 , =t−3 1 + (4s + 1)t−3 x−2 y−1 z0 1 + (4s + 2)t−2 x−1 y0 z−3 s=0 s=0 n−1 n−1 Y  1 + (4s + 3)t0 x−3 y−2 z−1  Y  1 + (4s + 2)t−1 x0 y−3 z−2  =t−1 , t4n = t0 . 1 + (4s + 3)t−1 x0 y−3 z−2 1 + (4s + 4)t0 x−3 y−2 z−1 s=0 s=0

(24)

Case where a = c = h = −1 and b = d = e = f = g = 1

Here, θx = θy = θz = θt = 1 and λx = λy = λz = λt = 0. Thus, equations in (23) simplify to Theorem 2.2 in [1].

2.6

Case where a = c = e = g = −1 and b = d = f = h = 1

Here, θx = θy = θz = θt = 1 and λx = λy = λz = λt = 0. Thus, equations in (23) simplify to Theorem 2.3 in [1].

2.7

Case where a = b = c = d = e = f = g = 1 and h = −1

Here, θx = θy = θz = θt = 1 and λx = λy = λz = λt = 0. Thus, equations in (23) simplify to Theorem 3.1 in [1].

3

Existence of four periodic solutions

If x−3 y−2 z−1 t0 = x−2 y−1 z0 t−3 = x−1 y0 z−3 t−2 = x0 y−3 z−2 t−1 =

1−c 1−e 1−g 1−a = = = , b d f h

then θx = θy = θz = θt = geca and λx = λy = λz = λt =

b (1 − geca). 1−a

Thus, equations in (23) simplify to x4n−3 = x−3 , x4n−2 = x−2 , x4n−1 = x−1 , x4n = x0 , y4n−3 = y−3 , y4n−2 = y−2 , y4n−1 = y−1 , y4n = y0 , z4n−3 = z−3 , z4n−2 = z−2 , z4n−1 = z−1 , z4n = z0 , t4n−3 = t−3 , t4n−2 = t−2 , t4n−1 = t−1 , t4n = t0 and therefore all solutions of (8) are periodic with period four. Below are the figures of some numerical examples that illustrate two cases of systems where solutions are periodic with period four.

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Figure 1: Periodic solutions of (8) when a = 2, b = −1, c = 3, d = −2, e = 4, f = −3, g = 5, h = −4 with initial conditions x0 = 0.5, x1 = 0.75, x2 = −3/2, x3 = 0.4, y0 = 0.5, y1 = 2, y2 = 0.5, y3 = −2/3, z0 = 1/5, z1 = 5, z2 = 0.25, z3 = 1/3, t0 = 8, t1 = 5, t2 = 1, t3 = 4.

Figure 2: Periodic solutions of (8) when a = 0.5, b = 0.5, c = 0.75, d = 0.25, e = 6, f = −5, g = −1, h = 2 with initial conditions x0 = −0.5, x1 = −1/7, x2 = −1/4, x3 = 1.25, y0 = −0.125, y1 = 2, y2 = −1/5, y3 = 10, z0 = −0.8, z1 = 5, z2 = −1/3, z3 = 3.5, t0 = 10, t1 = 0.5, t2 = −1.28, t3 = 3.

References [1] M. M. El-Dessoky and A. Hobiny, On the existence and behavior of the solutions for some difference equations systems, J. Computational Analysis and Applications 26:8 (2019), 1428–1439. [2] G. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York (2002). [3] M. M. El-Dessoky, Solution for Rational Systems of Difference Equations of Order Three, Mathematics 4:53 (2016). [4] E. M. Elsayed, Solutions of rational difference systems of order two, Mathematical and Computer Modelling, 55 (2012), 378–384.

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[5] E. M. Elsayed and T.F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe Journal of Mathematics and Statistics, 44:6 (2015), 1361–1390. [6] M. Folly-Gbetoula, Symmetry, reductions and exact solutions of the difference equation un+2 = aun /(1+ bun un+1 ), J.Diff. Eq. and Appl., 23:6 (2017), 1017–1024. [7] M. Folly-Gbetoula and A.H. Kara, Symmetries, conservation laws, and ’integrability’ of difference equations, Advances in Difference Equations, 2014:224 (2014). [8] M. Folly-Gbetoula and D. Nyirenda, On some sixth-order rational recursive sequences, Journal of computational analysis and applications, 27:6 (2019), 1057–1069. [9] N. Joshi and P. Vassiliou, The existence of Lie Symmetries for First-Order Analytic Discrete Dynamical Systems, Journal of Mathematical Analysis and Applications, 195 (1995), 872–887. [10] P. E. Hydon, Difference Equations by Differential Equation Methods, Cambridge University Press, (2014). [11] P. E. Hydon, Symmetries and first integrals of ordinary difference equations, Proc. Roy. Soc. Lond. A, 456 (2000), 2835–2855. [12] N. Mnguni and M. Folly-Gbetoula, Invariance analysis of a third-order difference equation with variable coefficients, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 25 (2018), 63–73. [13] D. Nyirenda and M. Folly-Gbetoula, Invariance analysis and exact solutions of some sixth-order difference equations, J. Nonlinear Sci. Appl. 10 (2017), 6262–6273. [14] P. J. Olver, Applications of Lie Groups to Differential Equations, Second Edition, Springer, New York, (1993). [15] G. R. W. Quispel and R. Sahadevan, Lie symmetries and the integration of difference equations, Physics Letters A, 184 (1993), 64-70. [16] S. Maeda, The similarity method for difference equations, IMA J. Appl. Math. 38 (1987), 129–134. [17] S. Stevic, On a system of difference equation, Applied Mathematics and Computation, 218 (2011), 3372–3378.

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Dynamics of an anti-competitive system of difference equations J. Ma∗ A. Q. Khan†

Abstract In this paper, we study the dynamical properties of an anti-competitive system of second-order rational difference equations. The proposed work is considerably extended and improve some exiting results in the literature.

Keywords and phrases: difference equations; boundedness and persistence; asymptotic behavior 2010 AMS: 39A10, 40A05

1

Introduction

Axn−1 In [1], Hamza et al. have investigated the global behavior of the difference equation: xn+1 = B+Cx 2 , n = 0, 1, · · · , where n A, B, C and initial conditions x0 , x−1 are positive real numbers. Motivated by the above studies, our aim in this paper is to investigate the dynamical properties of the following anti-competitive system of second-order rational difference equations: α + βyn−1 α1 + β1 xn−1 xn+1 = , yn+1 = , n = 0, 1, · · · , (1) γ + δx2n γ1 + δ1 yn2

where α, β, γ, δ, α1 , β1 , γ1 , δ1 and the initial conditions x0 , x−1 , y0 , y−1 are positive real numbers. The rest of the paper is dedicated to investigate the boundedness and persistence, existence of unbounded solutions, existence and uniqueness of positive equilibrium point, local and global stability about the unique positive equilibrium point of the system (1).

2

Main results

2.1

Boundedness and persistence

Theorem 1. If ββ1 < γγ1 then every solution {(xn , yn )/xn , yn > 0} of the system (1) is bounded and persists. Proof. If {(xn , yn )/xn , yn > 0} is a solution of the system (1) then α β α1 β1 + yn−1 , yn+1 ≤ + xn−1 , n = 0, 1, · · · . γ γ γ1 γ1

(2)

xn+1 ≤

ββ1 α1 αβ1 ββ1 α α1 β + + xn−3 , yn+1 ≤ + + yn−3 , n = 0, 1, · · · . γ γγ1 γγ1 γ1 γγ1 γγ1

(3)

Φn+1 =

α α1 β ββ1 α1 αβ1 ββ1 + + Φn−3 , ξn+1 = + + ξn−3 , n = 0, 1, · · · . γ γγ1 γγ1 γ1 γγ1 γγ1

(4)

xn+1 ≤ From (2), one get

Consider

The solution {(Φn , ξn )} of (4) is s s s !n !n !n 4 ββ1 4 ββ1 4 ββ1 Φn = r1 + r2 − + r3 ι + r4 γγ1 γγ1 γγ1 s s s !n !n !n 4 ββ1 4 ββ1 4 ββ1 ξn = s1 + s2 − + s3 ι + s4 γγ1 γγ1 γγ1 ∗ College

s −ι

4

s −ι

4

ββ1 γγ1 ββ1 γγ1

!n + !n

αγ1 + βα1 , γγ1 − ββ1 (5)

α1 γ + αβ1 + , γγ1 − ββ1

of Science, University of Shanghai for Science and Technology, Shanghai, 200093, P. R. China, e-mail: [email protected] of Mathematics, University of Azad Jammu & Kashmir, Muzaffarabad 13100, Pakistan, e-mail: [email protected]

† Department

962

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where r1 , r2 , r3 , r4 , s1 , s2 , s3 , s4 depend upon the initial values Φ−3 , Φ−2 , Φ−1 , Φ0 , ξ−3 , ξ−2 , ξ−1 , ξ0 . Assuming that ββ1 < γγ1 then (5) implies that Φn and ξn are bounded. Now consider the solution {(Φn , ξn )} of (5) such that Φ−3 = x−3 , Φ−2 = x−2 , Φ−1 = x−1 , Φ0 = x0 , ξ−3 = y−3 , ξ−2 = y−2 , ξ−1 = y−1 , ξ0 = y0 .

(6)

From (3), (5) and (6) one get αγ1 + βα1 α1 γ + αβ1 +  = U1 + , yn ≤ +  = U2 + , γγ1 − ββ1 γγ1 − ββ1

xn ≤

(7)

where for large n,  is a sufficiently small number. In addition from (1) and (7), we get α(γγ1 − ββ1 )2 α ≥ = L1 . γ + δx2n γ(γγ1 − ββ1 )2 + δ(αγ1 + βα1 )2

(8)

α1 α1 (γγ1 − ββ1 )2 ≥ = L2 . γ1 + δ1 yn2 γ1 (γγ1 − ββ1 )2 + δ1 (α1 γ + β1 α)2

(9)

xn ≥

yn ≥

Finally, from (7), (8) and (9) one get L1 ≤ xn ≤ U1 , L2 ≤ yn ≤ U2 , n = 0, 1, · · · .

2.2

(10)

Existence of unbounded solution

Theorem 2. For solution {(xn , yn )/xn , yn > 0} of the system (1), the following statements hold: (i) If ββ1 > (γ + δU12 )(γ1 + δ1 U22 ) then xn → ∞ as n → ∞. (ii) If ββ1 > (γ + δU12 )(γ1 + δ1 U22 ) then yn → ∞ as n → ∞. Proof. (i) If {(xn , yn )/xn , yn > 0} is a solution of the system (1) then α + βyn−1 α β α + βyn−1 ≥ = + yn−1 . 2 2 2 γ + δxn γ + δU1 γ + δU1 γ + δU12

(11)

α1 + β1 xn−1 α1 + β1 xn−1 α1 β1 ≥ = + xn−1 . 2 2 2 γ1 + δ1 yn γ1 + δ1 U2 γ1 + δ1 U2 γ1 + δ1 U22

(12)

α1 β1 + xn−3 . γ1 + δ1 U22 γ1 + δ1 U22

(13)

xn+1 ≥

α βα1 ββ1 + + xn−3 . 2 2 2 2 γ + δU1 (γ + δU1 )(γ1 + δ1 U2 ) (γ + δU1 )(γ1 + δ1 U22 )

(14)

τn+1 =

α βα1 ββ1 + + τn−3 . γ + δU12 (γ + δU12 )(γ1 + δ1 U22 ) (γ + δU12 )(γ1 + δ1 U22 )

(15)

xn+1 =

yn+1 = From (12)

yn−1 ≥ Using (13) in (11), one get

Consider

The solution of (15) is s τn

= c1

c4

s !n !n ββ ββ 1 1 4 + c2 − 4 + c3 (γ + δU12 )(γ1 + δ1 U22 ) (γ + δU12 )(γ1 + δ1 U22 ) s !n ββ α(γ1 + δ1 U22 ) + βα1 1 −ι 4 + , 2 2 (γ + δU1 )(γ1 + δ1 U2 ) (γ + δU12 )(γ1 + δ1 U22 ) − ββ1 963

s ι4

ββ1 (γ + δU12 )(γ1 + δ1 U22 )

!n

MA-KHAN 962-967

+

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

where c1 , c2 , c3 , c4 depends on τ−3 , τ−2 , τ−1 , τ0 . Now if ββ1 > (γ + δU12 )(γ1 + δ1 U22 ) then {τn } is divergent. Hence by comparison xn → ∞ as n → ∞. (ii) Similarly from (11), we have α β + yn−3 . γ + δU12 γ + δU12

(16)

yn+1 ≥

α1 β1 α ββ1 + + yn−3 . 2 2 2 2 γ1 + δ1 U2 (γ + δU1 )(γ1 + δ1 U2 ) (γ + δU1 )(γ1 + δ1 U22 )

(17)

µn+1 =

ββ1 β1 α α1 + µn−3 . + 2 2 2 2 γ1 + δ1 U2 (γ + δU1 )(γ1 + δ1 U2 ) (γ + δU1 )(γ1 + δ1 U22 )

(18)

xn−1 ≥ Using (16) in (12), we get

Consider

The solution of (18) is given by s s !n !n ββ ββ 1 1 µn = c5 4 + c6 − 4 + c7 (γ + δU12 )(γ1 + δ1 U22 ) (γ + δU12 )(γ1 + δ1 U22 ) s !n ββ1 α1 (γ + δU12 ) + β1 α 4 c8 −ι , + 2 2 (γ + δU1 )(γ1 + δ1 U2 ) (γ + δU12 )(γ1 + δ1 U22 ) − ββ1

s ι4

ββ1 (γ + δU12 )(γ1 + δ1 U22 )

!n +

where c5 , c6 , c7 , c8 depends on µ−3 , µ−2 , µ−1 , µ0 . If ββ1 > (γ + δU12 )(γ1 + δ1 U22 ) then {µn } is divergent. Hence by comparison yn → ∞ as n → ∞.

2.3

Existence and uniqueness of positive equilibrium point

Theorem 3. If  α1 + β1 L1


γ1 + δ 1

and γ + 3δU1 2




(γ + δL21 )L1 − α β

2 !

(γ + δU12 )L1 − α β

2 !

(γ + δL21 )L1 − α , β

(19)

(γ + δU12 )U1 − α , β

(20)



2  γ + δU1 2 U1 − α

γ1 β 2 + 3δ1 β 3 β1

< 1,

(21)

then the system (1) has a unique positive equilibrium point Ω = (¯ x, y¯) ∈ [L1 , U1 ] × [L2 , U2 ]. Proof. Consider x=

α1 + β1 x α + βy , y= . 2 γ + δx γ1 + δ1 y 2

(22)

From (22), we have y=

(γ1 + δ1 y 2 )y − α1 (γ + δx2 )x − α , x= . β β1

Taking F (x) =

(γ1 + δ1 (h(x))2 )h(x) − α1 − x, β1

(23)

(γ + δx2 )x − α , β

(24)

where h(x) = and x ∈ [L1 , U1 ]. Now

F (L1 )

=

(γ1 + δ1 (h(L1 ))2 )h(L1 ) − α1 − L1 = β1

  2  (γ+δL21 )L1 −α (γ+δL21 )L1 −α γ1 + δ 1 − α1 β β

964

β1

− L1 . MA-KHAN 962-967

(25)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Assume that (19) hold then (25) implies that F (L1 ) > 0. Also,  2   (γ+δU12 )U1 −α (γ+δU12 )U1 −α − α1 γ + δ 1 1 2 β β (γ1 + δ1 (h(U1 )) )h(U1 ) − α1 F (U1 ) = − U1 = − U1 . β1 β1

(26)

Assuming (20) hold then from (26) one get F (U1 ) < 0. Hence, F (x) has at least one positive solution in x ∈ [L1 , U1 ]. Furthermore, F 0 (x) = h0 (x)

γ1 + 3δ1 (h(x))2 − 1, β1

(27)

where h0 (x) =

γ + 3δx2 . β

(28)

Let x ¯ be a solution of equation F (x) = 0, then from (23), (24) and (28) one get x ¯=

(γ1 + δ1 (h(¯ x))2 )h(¯ x) − α1 , β1

h(¯ x) =

(29)

(γ + δ x ¯2 )¯ x−α , β

(30)

γ + 3δ x ¯2 . β

(31)

h0 (¯ x) =

In view of (30) and (31), equation (27) takes the following form


2  γ + 3δ x ¯2 γ1 β 2 + 3δ1 γ + δ x ¯2 x ¯−α F 0 (¯ x) = − 1, β 3 β1 


2  γ + 3δU1 2 γ1 β 2 + 3δ1 γ + δU1 2 U1 − α − 1. ≤ β 3 β1

(32)

Assume that (21) hold then from (32) one get F 0 (¯ x) < 0.

2.4

Local stability

Theorem 4. For equilibrium Ω of the system (1), the following statements hold: (i) Ω of the system (1) is locally asymptotically stable if     2δ1 U22 1 ββ1 2δU12 2 1 + + 2δ U + < 1. 1 2 γ + δL21 γ1 + δ1 L22 γ1 + δ1 L22 γ + δL21

(33)

(ii) Ω of the system (1) is unstable if 2δL21 γ + δU12

 1+

2δ1 L22 γ1 + δ1 U22

 +

1 γ1 + δ1 U22



2δ1 L22 +

ββ1 γ + δU12

 > 1.

(34)

Proof. If (¯ x, y¯) is an equilibrium point of the system (1) then x ¯=

α1 + β1 x ¯ α + β y¯ , y¯ = . 2 γ + δx ¯ γ1 + δ1 y¯2

(35)

Consider the following transformation in order to construct the corresponding linearized form of the system (1): (xn+1 , xn , yn+1 , yn ) 7→ (f, f1 , g, g1 ), 965

(36) MA-KHAN 962-967

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where f=

α + βyn−1 α1 + β1 xn−1 , f1 = xn , g = , g1 = yn . 2 γ + δxn γ1 + δ1 yn2

(37)

The Jacobian matrix J|(¯x,¯y) about (¯ x, y¯) under the transformation (36) is given by   a 0 0 b  1 0 0 0   J|(¯x,¯y) =   0 a1 b1 0  , 0 0 1 0 where a=−

(38)

2δ x ¯2 β β1 2δ1 y¯2 , b = , a = , b = − . 1 1 γ + δx ¯2 γ + δx ¯2 γ1 + δ1 y¯2 γ1 + δ1 y¯2

(39)

The characteristic equation of J|(¯x,¯y) about (¯ x, y¯) is given by λ4 − (a + b1 )λ3 + ab1 λ2 − a1 b = 0.

(40)

Now, |a| + |b1 | + |ab1 | + |a1 b| = ≤ =

2δ x ¯2 2δ1 y¯2 4δδ1 x ¯2 y¯2 ββ1 + + + , 2 2 2 2 2 γ + δx ¯ γ1 + δ1 y¯ (γ + δ x ¯ )(γ1 + δ1 y¯ ) (γ + δ x ¯ )(γ1 + δ1 y¯2 ) 2δU12 2δ1 U22 4δδ1 U12 U22 ββ1 + + + , 2 2 2 2 2 γ + δL1 γ1 + δ1 L2 (γ + δL1 )(γ1 + δ1 L2 ) (γ + δL1 )(γ1 + δ1 L22 )     2δU12 2δ1 U22 1 ββ1 2 1 + + 2δ U + . 1 2 γ + δL21 γ1 + δ1 L22 γ1 + δ1 L22 γ + δL21

(41)

Assuming that (33) hold then from (41) one gets |a| + |b1 | + |ab1 | + |a1 b| < 1. Hence from Remark 1.3.1 of [2], Ω of (1) is locally asymptotically stable. Proof (ii). Using same manipulations as for the proof of (i) and assume that (34) hold then |a| + |b1 | + |ab1 | + |a1 b| = ≥ =

2δ1 y¯2 4δδ1 x ¯2 y¯2 ββ1 2δ x ¯2 + + + , 2 2 2 2 2 γ + δx ¯ γ1 + δ1 y¯ (γ + δ x ¯ )(γ1 + δ1 y¯ ) (γ + δ x ¯ )(γ1 + δ1 y¯2 ) 2δL21 2δ1 L22 4δδ1 L21 L22 ββ1 + + + , 2 2 2 2 2 γ + δU1 γ1 + δ 1 U 2 (γ + δU1 )(γ1 + δ1 U2 ) (γ + δU1 )(γ1 + δ1 U22 )     2δL21 2δ1 L22 1 ββ1 2 1 + + 2δ L + > 1. 1 2 γ + δU12 γ1 + δ1 U22 γ1 + δ1 U22 γ + δU12

(42)

Hence Ω of system (1) is unstable.

2.5

Global character

Now we will study the global dynamics of (1) about Ω by utilizing Theorem 1.16 of [3]. Theorem 5. Ω of the system (1) is a global attractor. Proof. If f (x, y) =

α+βy γ+δx2

and g(x, y) =

α1 +β1 x

then it is easy to examine that f (x, y) is non-increasing (resp. i h i α1 (γγ1 −ββ1 )2 α1 γ+αβ1 × γ1 (γγ1 −ββ , non-decreasing) in x (resp. y) ∀ (x, y) ∈ 2 2 γγ1 −ββ1 . 1 ) +δ1 (α1 γ+β1 α) h i 2 α(γγ1 −ββ1 ) αγ1 +βα1 , Also g(x, y) is non-decreasing (resp. non-increasing) in x (resp. y) ∀ (x, y) ∈ γ(γγ1 −ββ 2 2 γγ1 −ββ1 × 1 ) +δ(αγ1 +βα1 ) h i α1 (γγ1 −ββ1 )2 α1 γ+αβ1 γ1 (γγ1 −ββ1 )2 +δ1 (α1 γ+β1 α)2 , γγ1 −ββ1 . Let (m1 , M1 , m2 , M2 ) be a solution of the system hγ1 +δ1 y2

α(γγ1 −ββ1 )2 αγ1 +βα1 γ(γγ1 −ββ1 )2 +δ(αγ1 +βα1 )2 , γγ1 −ββ1

α + βm2 α + βM2 , M1 = . γ + δM12 γ + δm21

(43)

α1 + β1 M1 α1 + β1 m1 , M2 = . γ1 + δ1 M22 γ1 + δ1 m22

(44)

m1 = and m2 =

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From (43) and (44), we get

Setting

m1 (α + βm2 )(γ + δm21 ) = . M1 (γ + δM12 )(α + βM2 )

(45)

(α1 + β1 m1 )(γ1 + δ1 m22 ) m2 . = M2 (γ1 + δ1 M22 )(α1 + β1 M1 )

(46)

m1 m2 = a1 ≤ 1, = a2 ≤ 1. M1 M2

(47)

In view of (47), equations (45) and (46) then implies that βγ(a1 − a2 )M2 β1 γ1 (a2 − a1 )M1

= αδ(a1 − 1)a1 M12 + βδ(a1 a2 − 1)a1 M12 M2 − αγ(a1 − 1), = α1 δ1 (a2 − 1)a2 M22 + β1 δ1 (a1 a2 − 1)a2 M1 M22 − α1 γ1 (a2 − 1).

(48)

So right-hand sides of (48) are less then or equal to zero, and thus a1 − a2 ≤ 0, a2 − a1 ≤ 0. This implies that a1 ≤ a2 ≤ a1 , which hold if and only if a1 = a2 . In view of (48) it follows that a1 = a2 = 1 and thus m1 = M1 , m2 = M2 . Hence, by Theorem 1.16 of [3], Ω of the system (1) is a global attractor.

3

Conclusion

This work is related to the dynamical properties of an anti-competitive system of rational difference equations. We proved that if ββ1 < γγ1 then every solution {(xn, yn )/xn , yn > 0} of the system (1) is bounded andpersists. We   2  2 (γ+δL21 )L1 −α (γ+δL21 )L1 −α (γ+δU12 )L1 −α (γ+δU12 )U1 −α proved that if α1 +β1 L1 < γ1 + δ1 , α +β U > γ + δ 1 1 1 1 1 β β β β (γ+3δU1 2 )

  2 γ1 β 2 +3δ1 ((γ+δU1 2 )U1 −α)

and < 1 then system (1) has a unique positive equilibrium point Ω = (¯ x, y¯) ∈ β 3 β1 [L1 , U1 ] × [L2 , U2 ]. Furthermore method of Linearization is used to study the local stability about the unique positive  2δU 2 2δ1 U22 equilibrium point Ω. Linear stability analysis shows that Ω is locally asymptotically stable if γ+δL1 2 1 + γ1 +δ + 2 1 L2       1 2δL21 2δ1 L22 ββ1 ββ1 1 1 2 2 2δ1 U2 + γ+δL2 < 1 and unstable if γ+δU 2 1 + γ1 +δ1 U 2 + γ1 +δ1 U 2 2δ1 L2 + γ+δU 2 > 1. Finally global γ1 +δ1 L22 1 1 2 2 1 dynamics about Ω is also investigated. Acknowledgements J. Ma’s research is supported by the National Natural Science Foundations of China [grant number 11501364] while A. Q. Khan’s research is supported by the Higher Education Commission (HEC) of Pakistan.

References [1] A. E. Hamza, R. Khalaf-Allah, Dynamics of second-order rational difference equation, BAMS, 23(1)(2008):206-214. [2] V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993). [3] E. A. Grove, G. Ladas, Periodicities in monlinear difference equations, Chapman and Hall/CRC Press, Boca Raton, (2004).

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AN ITERATIVE SCHEME FOR SOLVING SPLIT SYSTEM OF MINIMIZATION PROBLEMS ANTENEH GETACHEW GEBRIE AND RABIAN WANGKEEREE Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Abstract. In this paper, we propose iterative algorithm for solving split system of minimization problems. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some example to numerically verify the efficiency and implementation of our method. Keywords: Minimization problem, strong convergence, Moreau-Yosida approximate, Hilbert space. AMS Subject Classification: 49J53, 49J52, 47J05, 90C25, 65K10.

1. Introduction Let H1 and H2 be real Hilbert spaces and let A : H1 → H2 be a bounded linear operator. Given nonempty closed convex subsets Ci (i = 1, . . . , N ) and Qi (i = 1, . . . , M ) of H1 and H2 , respectively. The multiple-set split feasibility problem (MSSFP) which was introduced by Censor et al. [10] is formulated as finding a point M N ∩ ∩ Qj . (1.1) Ci such that A¯ x∈ x ¯∈ i=1

j=1

In particular, if N = M = 1, then the MSSFP (1.1) is reduced to find a point x ¯ ∈ C such that A¯ x ∈ Q.

(1.2)

where C and Q are nonempty closed convex subsets of H1 and H2 , respectively. The problem (1.2) is known as the split feasibility problem (SFP) which was first introduced by Censor and Elfving [9] for modeling inverse problems in finite-dimensional Hilbert spaces. Many authors studied the SFP, see for example in [5, 9, 13, 14, 17, 24], and MSSFP, see for example in [10, 15, 19, 34, 35], provided the solution exists. The SFP and MSSFP arises in many fields in the real world, such as image reconstruction, modeling inverse problems, radiation therapy treatment planning and signal processing, and medical care; for details see [6, 7, 8] and the references therein. Throughout this paper, unless otherwise stated, we assume that H1 and H2 are real Hilbert spaces, A : H1 → H2 is nonzero bounded linear operator, I denotes the identity operator on a Hilbert space and R denotes set of real numbers. Let us consider the following problem: find x ∈ H1 with the property that min {f (x) + gλ (Ax)},

(1.3)

x∈H1

where f : H1 → R ∪ {+∞}, g : H2 → R ∪ {+∞} are two proper, convex, lower-semicontinuous functions and gλ is Moreau-Yosida approximate [26] of the function g of parameter λ given by gλ (y) = minu∈H2 {g(u) + 1 2 2λ ∥y − u∥ }. In [21], Moudafi and Thakur introduced a weakly convergent algorithm solving the (1.3) in case arg min f ∩ A−1 (arg min g) ̸= ∅. Note that if we take f = δC [defined as δC (x) = 0 if x ∈ C and +∞ otherwise], the indicator function of nonempty, closed and convex subset C of H1 and g = δQ , the indicator ∗ Corresponding

author: R. Wangkeeree. Email address: [email protected] (A.G Gebrie) and [email protected] (R. Wangkeeree). 1

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A.G GEBRIE AND R. WANGKEEREE

function of nonempty, closed and convex subset Q of H2 , then problem problem (1.3) is reduced to the following minimization problem: { } 1 min ∥(I − PQ )(Ax)∥2 (1.4) x∈C 2λ which, when C ∩ A−1 (Q) ̸= ∅, is equivalent to the split feasibility problem (SEP). It should also be noticed that (1.3) is equivalent to the problem of finding a point x ¯ ∈ H1 with the property x ¯ ∈ arg min f such that A¯ x ∈ arg min g.

(1.5)

Moudafi and Thakur [21] used the idea of Lopez et al. [17] to introduce a new way of selecting the step sizes given by √ ∥A∗ (I − proxλg )Ax∥2 + ∥(I − proxλµf )x∥2

θλµ (x) = with hλ (x) =

1 2 ∥(I

− proxλg )Ax∥2 and lλµ (x) =

1 2 ∥(I

− proxλµf )x∥2 where proxλf (x) = arg min {f (u) + u∈H1

1 2λ ∥u

− y∥2 } stands for the proximal mapping of f . They proposed the following split proximal algorithm, which generates, from an initial point x1 ∈ H1 assume that xn has been constructed and θλ (xn ) ̸= 0, then compute xn+1 via the rule ( ) xn+1 = proxλµn f xn − µn A∗ (I − proxλg )Axn (1.6) where stepsize µn = ρn

hλ (xn )+lλµn (xn ) 2 θλµ (xn ) n

with 0 < ρn < 4 and if θλµn (xn ) = 0, then xn+1 = xn is a solution

of (1.5) and the iterative process stops; otherwise, we set n := n + 1 and go to (1.6). Based on Moudafi and Thakur [21] many iterative algorithms are proposed for solving split minimization problem (1.5), see eg, Shehu and Iyiola in [28, 29, 30, 31], Shehu and Ogbuisi in [27], Shehu et al. in [32], Abbas et al. in [1]. Very recently, Shehu and Iyiola [29] proposed algorithm for solving (1.5) as follows:  u, x1 ∈ H1 ,    z = (1 − α )x + α u, n n n n ) (1.7) h(zn )+l(zn ) ( y = z − ρ (I − proxλf )zn + A∗ (I − proxλg )Azn ,  2 (z ) n n n θ  n  xn+1 = (1 − βn )zn + βn yn , where l(x) = 12 ∥(I −proxλf )x∥2 , h(x) = 21 ∥(I −proxλg )Ax∥2 and θ(x) = ∥(I −proxλf )x+A∗ (I −proxλg )Ax∥. It was shown that the sequence {xn } generated by iterative algorithm (1.7) converges strongly to the solution of problem (1.5) under the following conditions: ∞ ∑ (a) : 0 < αn < 1, lim αn = 0 and αn = ∞. n→∞

n=1

(b) : 0 < β ≤ βn ≤ δ < 1, (c) : 0 < ρn < 4, lim inf ρn (4 − ρn ) > 0. n→∞

To prove the strong convergence of iterative algorithm (1.7) the authors used simpler alternative proof without recourse to ‘two cases method’ of proof studied by other authors [1, 27, 30, 31, 32] and is also different from the approaches used in the proofs of [21, 28]. Motivated and inspired by results in [10, 21, 29], in this paper, we introduce and study the following split system of minimization problem (SSMP): finding a point x ¯ ∈ H1 with the property x ¯∈

N ∩

(arg min fi ) such that A¯ x∈

i=1

M ∩

(1.8)

(arg min gj )

j=1

where fi : H1 → R ∪ {+∞} and gj : H2 → R ∪ {+∞} are proper, lower semicontinuous convex functions, arg min fi = {¯ x ∈ H1 : fi (¯ x) ≤ fi (x), ∀x ∈ H1 }, arg min gj = {¯ y ∈ H2 : gj (¯ y ) ≤ gj (y), ∀y ∈ H2 } and i ∈ Φ = {1, . . . , N }, j ∈ Ψ = {1, . . . , M }. The solution set Γ of problem (1.8) is denoted by N M { } ∩ ∩ Γ= x ¯ ∈ H1 : x ¯∈ (arg min fi ) and A¯ x∈ (arg min gj ) . i=1

j=1

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3

Minimizers of any proper, lower semicontinuous function are exactly fixed points of its proximal mappings and proximal mappings are nonexpansive mapping (whose set of fixed points is closed and convex), we have that the set of minimizers of any proper, lower semicontinuous function is closed and convex. Therefore, since A bounded linear operator the solution set Γ of problem (1.8) is closed convex set. We assume Γ is nonempty. We propose an iterative scheme using extended form of selecting step sizes used to solve (1.5) to the context of solving split system of minimization problem (1.8). The iterative scheme is developed by computation of proximal of fi at zn and gj at Azn in a parallel setting under simple assumptions on step sizes. Moreover, the technique of the proof takes some steps of [29, 33] so that it takes few steps to complete the proof. Note that if fi = f for all i ∈ Φ and gj = g for all j ∈ Ψ, then problem (1.8) reduces to the problem of split minimization problem (1.5) considered in [1, 21, 27, 28, 29, 30, 31, 32]. This paper is organized in the following way. In Section 2, we collect some basic and useful lemmas for further study. In Section 3, we propose and analyze the convergence result of our algorithm. In Section 4, we give a numerical example to discuss performance of the proposed algorithm. 2. Preliminary In order to prove our main results, we recall some basic definitions and lemmas, which will be needed in the sequel. The symbols ” ⇀ ” and ” → ” denote weak and strong convergence, respectively. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. The metric projection on C is a mapping PC : H → C defined by PC (x) = arg min{∥y − x∥ : y ∈ C}, x ∈ H. Lemma 2.1. Let C be a closed convex subset of H. Given x ∈ H and a point z ∈ C, then z = PC (x) if and only if ⟨x − z, y − z⟩ ≤ 0, ∀y ∈ C. Let T : H → H. Then, (I): T is L-Lipschitz if there exists L > 0 such that ∥T x − T y∥ ≤ L∥x − y∥, ∀x, y ∈ H. If L ∈ (0, 1), then we call T a contraction. If L = 1, then T is called a nonexpansive mapping. (II): T is firmly nonexpansive if ∥T x − T y∥2 ≤ ∥x − y∥2 − ||(I − T )x − (I − T )y∥2 , ∀x, y ∈ H, which is equivalent to ∥T x − T y∥2 ≤ ⟨T x − T y, x − y⟩, ∀x, y ∈ H. If T is firmly nonexpansive, I − T is also firmly nonexpansive. (III): strongly monotone if there exists a constant α > 0 such that ⟨T x − T y, x − y⟩ ≥ α∥x − y∥2 for all x, y ∈ H. (IV): inverse strongly monotone if there exists a constant α > 0 such that ⟨T x − T y, x − y⟩ ≥ α∥T x − T y∥2 for all x, y ∈ H. Note that the proximal mapping of f is nonexpansive and firmly nonexpansive mapping. The minimizers of any proper, lower semicontinuous function are exactly fixed points of its proximal mappings. Many properties of proximal operator can be found in [12] and the references therein. Lemma 2.2. Let H be a real Hilbert space. Then, ∥x + y∥2 = ∥x∥2 + ∥y∥2 + 2⟨x, y⟩, ∀x, y ∈ H

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The following facts will be used several times in the paper. Lemma 2.3. [2] Let H be a real Hilbert space. Then, ∥(1 − α)x + αy∥2 = (1 − α)∥x∥2 + α∥y∥2 − α(1 − α)∥x − y∥2 , ∀α ∈ R, ∀x, y ∈ H. Let H be a real Hilbert space, {x1 , x2 , . . . , xd } ⊂ H and {λ1 , λ2 , . . . , λd } ⊂ [0, 1] with

d ∑

λi = 1. Then,

i=1

from [2, 37] one can see that

d d

∑

2 ∑

λ x ≤ λi ∥xi ∥2 ,

i i i=1

i=1

i.e., convexity of ∥.∥ . 2

Lemma 2.4. [18] Let {an } be the sequence of nonnegative numbers such that an+1 ≤ (1 − αn )an + αn δn , where {δn } is a sequence of real numbers bounded from above and 0 ≤ αn ≤ 1 and

∞ ∑

αn = ∞. Then it

n=1

holds that lim sup αn ≤ lim sup δn . n→∞

n→∞

3. Main result First we introduce the following settings which is an extension of settings introduced by Moudafi and Thakur [21]. Let λ > 0. For x ∈ H1 , (i): for each i ∈ Φ, define 1 li (x) = ∥(I − proxλfi )x∥2 and ∇li (x) = (I − proxλfi )x, 2 (ii): l(x) and ∇l(x) are defined as l(x) = lix (x) and so ∇l(x) = ∇lix (x) where ix is in Φ such that ix ∈ arg max{∥(I − proxλfi )x∥ : i ∈ Φ}, (iii): for each j ∈ Ψ, define 1 hj (x) = ∥(I − proxλgj )Ax∥2 and ∇hj (x) = A∗ (I − proxλgj )Ax, 2 (iv): for each j ∈ Ψ, define θj (x) = max{∥∇hj (x)∥, ∥∇l(x)∥}. It is easy to see that, for x ∈ H1 ∥∇li (x)∥ ≤ ∥∇lix (x)∥ = ∥∇l(x)∥, ∀i ∈ Φ and li (x) =

1 ∥∇li (x)∥2 , ∀i ∈ Φ. 2

In this section, we propose algorithm for solving SSMP (1.8) and we analyse the convergence of the iteration sequence generated by the algorithm by assuming that the solution set Γ is nonempty. In order to design the algorithm, we consider the parameter sequences satisfying the following conditions. Condition 1 (C1) : 0 < αn < 1, lim αn = 0 and n→∞

(C2) : 0 < β ≤ βn ≤ δ < 1,

∞ ∑

αn = ∞.

n=1

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(C3) : 0 < ξ ≤ ξnj ≤ 1 such that (C4) : 0 < δ ≤ δni ≤ 1 such that

M ∑ j=1 N ∑

5

ξnj = 1 for each n ≥ 1. δni = 1 for each n ≥ 1.

i=1

(C5) : 0 < ρn < 2δ, lim inf ρn (2δ − ρn ) > 0. n→∞

Throughout this paper, unless otherwise stated, Condition 1 refers to conditions (C1)-(C5) above. Using the definitions of ∇li , li , l, ∇l, hj , ∇hj and θj given in (i)-(iv), we are now in a position to introduce our algorithm. Algorithm 1 Initialization: Choose u, x1 ∈ H1 . Let {αn }, {βn }, {ρn }, {δni } and {ξnj } be real sequences satisfying Condition 1. Step 1: Evaluate zn = (1 − αn )xn + αn u. Step 2: For each j ∈ Ψ compute θj (zn ), hj (zn ) and l(zn ). Let Ψn = {j ∈ Ψ : θj (zn ) ̸= 0}. If Ψn = ∅, then zn is a solution of (1.8) and the iterative process stops, otherwise, go to Step 3. Step 3: For each j ∈ Ψ evaluate µjn = ρn ηnj where { 0, if j ∈ / Ψn j ηn = hj (zn )+l(zn ) , if j ∈ Ψn . θ 2 (zn ) j

Step 4: Evaluate wn = zn −

(∑

ξnj µjn

)∑

j∈Ψ

and t n = zn −

δni ∇li (zn )

i∈Φ

∑

ξnj µjn ∇hj (zn ).

j∈Ψ

Step 5: Evaluate yn = Step 6: Evaluate xn+1 = (1 − βn )zn + βn yn . Step 7: Set n := n + 1 and go to Step 1.

w n + tn . 2

Lemma 3.1. If Ψn = ∅, then zn is the solution of (1.8). Proof. Suppose Ψn = ∅ at some iteration n. Then, from Ψn = {j ∈ Ψ : θj (zn ) ̸= 0} = ∅, we have max{∥∇hj (zn )∥, ∥∇l(zn )∥} = 0, ∀j ∈ Ψ ⇔ ∥∇hj (zn )∥ = 0 = ∥∇l(zn )∥, ∀j ∈ Ψ, ⇔ ∥∇hj (zn )∥ = 0 = ∥∇li (zn )∥, ∀i ∈ Φ, ∀j ∈ Ψ, ⇔ A∗ (I − proxλgj )Azn = 0 = (I − proxλfi )zn , ∀i ∈ Φ, ∀j ∈ Ψ, and this implies that zn ∈ Γ.



√

Remark 3.2. Note that we can also use θj (x) = ∥∇hj (x)∥2 + ∥∇l(x)∥2 instead of θj (x) = max{∥∇hj (x)∥, ∥∇l(x)∥} and the proof for convergence will be the same. It is clear to see that √ max{∥∇hj (x)∥, ∥∇l(x)∥} ≤ ∥∇hj (x)∥2 + ∥∇l(x)∥2 . If Algorithm 1 does not stop, then we have the following strong convergence theorem for approximation of solution of problem (1.8).

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Theorem 3.3. The sequence {xn } generated by Algorithm 1 converges strongly to x ¯ ∈ Γ where x ¯ = PΓ u. Proof. Let x ¯ = PΓ u. Since proxλfi and proxλgj are firmly nonexpansive, I − proxλfi and I − proxλgj are also firmly nonexpansive, and since x ¯ verifies (1.8) (since minimizers of any function are exactly fixed-points of its proximal mapping), we have for all z ∈ H1 ⟨∇li (z), z − x ¯⟩ = ⟨(I − proxλfi )z, z − x ¯⟩ ≥ ∥(I − proxλfi )z∥2 = 2li (z)

(3.1)

⟨∇hj (z), z − x ¯⟩ = ⟨A∗ (I − proxλgj )Az, z − x ¯⟩ = ⟨(I − proxλgj )Az, Az − A¯ x⟩ ≥ ∥(I − proxλgj )Az∥2 = 2hj (z), ∀j ∈ Ψ.

(3.2)

and

Note that, for all z ∈ H1 , ∥∇l(z)∥ ≤ θj (z), ∥∇hj (z)∥ ≤ θj (z), ∀j ∈ Ψ, ∑

δni ∥∇li (z)∥2 ≤ ∥∇l(z)∥2 and

i∈Φ

∑

δni li (z) ≥ ζl(z).

i∈Φ

Using convexity of ∥.∥2 together with (3.1), we have )∑ ¯ ∥2 δni ∇li (zn ) − x ξnj µjn i∈Φ j∈Ψ

2

( ∑ j j ) ∑ i δn ∇li (zn ) = ∥zn − x ¯∥2 + ξn µn i∈Φ (j∈Ψ ∑ j j) ∑ i ¯⟩ δn ∇li (zn ), zn − x − 2⟨ ξn µn i∈Φ ( ∑j∈Ψ )2 ∑

2 ≤ ∥zn − x ¯∥2 + δni ∇li (zn ) ξnj µjn i∈Φ (j∈Ψ ∑ j j) ∑ i ¯⟩ −2 δn ⟨∇li (zn ), zn − x ξn µ n j∈Ψ )i∈Φ ( ∑

∑ 2 ≤ ∥zn − x ¯∥2 + δni ∇li (zn ) ξnj (µjn )2 i∈Φ j∈Ψ )∑ ( ∑ ¯⟩ −2 ξnj µjn δni ⟨∇li (zn ), zn − x j∈Ψ i∈Φ ( ∑ )∑

2 ≤ ∥zn − x ¯∥2 + ξnj (µjn )2 δni ∇li (zn ) i∈Φ (j∈Ψ ∑ j j) ∑ i δn li (zn ). −4 ξn µ n

∥wn − x ¯∥2 = ∥zn −

( ∑

j∈Ψ

(3.3)

i∈Φ

Similarly, using convexity of ∥.∥2 together with (3.2), we have ∑ j j ∥tn − x ¯∥2 = ∥zn − ξn µn ∇hj (zn ) − x ¯ ∥2 j∈Ψ ∑ j j ∑ j j = ∥zn − x ¯ ∥2 + ∥ ξn µn ∇hj (zn )∥2 − 2⟨ ξn µn ∇hj (zn ), zn − x ¯⟩ j∈Ψ j∈Ψ ∑ ∑ ≤ ∥zn − x ¯∥2 + ξnj (µjn )2 ∥∇hj (zn )∥2 − 2 ξnj µjn ⟨∇hj (zn ), zn − x ¯⟩ j∈Ψ j∈Ψ ∑ ∑ ≤ ∥zn − x ¯∥2 + ξnj (µjn )2 ∥∇hj (zn )∥2 − 4 ξnj µjn hj (zn ). j∈Ψ

(3.4)

j∈Ψ

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AN ITERATIVE SCHEME FOR SOLVING SPLIT SYSTEM OF MINIMIZATION PROBLEMS

Now,

( ∑

( ∑ )∑

2 δni ∇li (zn ) − 4 ξnj µjn δni li (zn ) j∈Ψ i∈Φ j∈Ψ i∈Φ ( ∑ ) ( ∑ ) i ≤ ξnj (µjn )2 ∥∇l(zn )∥2 − 4 ξnj µjn δnzn l(zn ) ( j∈Ψ (j∈Ψ ∑ j j 2) ∑ j j) ≤ ξn (µn ) ∥∇l(zn )∥2 − 4ζ ξn µn l(zn ) j∈Ψ ( j∈Ψ ) ( ) ∑ j ∑ j = ξn (ρn ηnj )2 ∥∇l(zn )∥2 − 4ζ ξn ρn ηnj l(zn ) j∈Ψ j∈Ψ ∑ j hj (zn )+l(zn ) ∑ j ( hj (zn )+l(zn ) )2 ∥∇l(zn )∥2 − 4ζ ξn ρn θ2 (zn ) l(zn ) = ξn ρn θ2 (zn ) ≤ = =

and

ξnj (µjn )2

7

)∑

(3.5)

j j j∈Ψn j∈Ψn 2 ∑ ∑ h (z )+l(z (h (z )+l(z )) ) ξnj j θn2 (zn ) n l(zn ) ρ2n ξnj j θn4 (zn ) n θj2 (zn ) − 4ζρn j j j∈Ψn j∈Ψn ∑ j (hj (zn )+l(zn ))2 ∑ j (hj (zn )+l(zn ))2 l(zn ) ρ2n ξn − 4ζρ ξ n n hj (zn )+l(zn ) θj2 (zn ) θj2 (zn ) j∈Ψn j∈Ψ n ) ∑ j( (hj (zn )+l(zn ))2 n) , ρn ξn ρn − hj (z4ζl(z 2 θj (zn ) n )+l(zn ) j∈Ψn

∑ j j ξn µn hj (zn ) ξnj (µjn )2 ∥∇hj (zn )∥2 − 4 j∈Ψ ∑ j∈Ψ ∑ j = ξnj ρn ηnj hj (zn ) ξn (ρn ηnj )2 ∥∇hj (zn )∥2 − 4 j∈Ψ j∈Ψ ∑ j ( hj (zn )+l(zn ) )2 ∑ j hj (zn )+(zn ) = ξn ρn θ2 (zn ) ∥∇hj (zn )∥2 − 4 ξn ρn θ2 (zn ) hj (zn ) ∑

≤ = ≤ =

j j j∈Ψn j∈Ψn ∑ j hj (zn )+l(zn ) ∑ j (hj (zn )+l(zn ))2 2 2 θj (zn ) − 4ρn ξn θ2 (zn ) hj (zn ) ρn ξn θj4 (zn ) j j∈Ψn j∈Ψn 2 2 ∑ ∑ hj (zn ) 2 j (hj (zn )+l(zn )) j (hj (zn )+l(zn )) ρn ξn − 4ρn ξn hj (zn )+l(zn ) θj2 (xn ) θj2 (zn ) j∈Ψn j∈Ψn ∑ j (hj (zn )+l(zn ))2 ∑ j (hj (zn )+l(zn ))2 hj (zn ) ξn − 4ζρ ξn ρ2n 2 n hj (zn )+l(zn ) θj (zn ) θj2 (zn ) j∈Ψn j∈Ψn ( ) ∑ j 4ζhj (zn ) (hj (zn )+l(zn ))2 ρn ξn ρn − hj (zn )+l(zn ) . θj2 (zn ) j∈Ψn

(3.6)

From convexity of ∥.∥2 and (3.3)-(3.6), we have ∥yn − x ¯∥2 = ∥ 12 (wn + tn ) − ¯∥2 ≤ 21 ∥wn − x ¯) ∥2 + 12 ∥tn − x ¯∥2 (x (hj (zn )+l(zn ))2 4ζl(zn ) ρn ∑ 2 j ≤ ∥zn − x ¯∥ + 2 ξn ρn − hj (zn )+l(zn ) θj2 (zn ) j∈Ψn ( ) ∑ (hj (zn )+l(zn ))2 4ζhj (zn ) + ρ2n ξnj ρn − hj (zn )+l(z θ 2 (zn ) n) j∈Ψn

= ∥zn − x ¯∥2 + ρn (ρn − 2ζ)

∑

j∈Ψn

(3.7)

j

ξnj

(hj (zn )+l(zn ))2 . θj2 (zn )

From (3.7) and (C5), we have ∥yn − x ¯∥ ≤ ∥zn − x ¯∥.

(3.8)

Using (3.8) and the definition of xn+1 , we get ∥xn+1 −¯ x∥2 = ∥(1 − βn )zn + βn yn − x ¯∥2 = ∥(1 − βn )(zn − x ¯) + βn (yn − x ¯)∥2 2 = (1 − βn )∥zn − x ¯∥ + βn ∥yn − x ¯∥2 − βn (1 − βn )∥zn − yn ∥2 2 ≤ ∥zn − x ¯∥ − βn (1 − βn )∥zn − yn ∥2 .

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(3.9)

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From (3.9) and the definition of zn , we get ∥xn+1 − x ¯∥ ≤ ∥zn − x ¯∥= (1 − αn )∥xn − x ¯∥ + αn ∥u − x ¯∥ ≤ max{∥xn − x ¯∥, ∥u − x ¯∥} .. .

(3.10)

≤ max{∥xn − x ¯∥, ∥u − x ¯∥} which shows that {xn } is bounded. Consequently, {yn }, {Ayn } and {zn } are all bounded. Now, ( ) 1 1 βn (xn+1 − zn ) = βn (1 − βn )zn + βn yn − zn = yn − zn and 1 2 ∥xn+1 βn

∥yn − zn ∥2 =

− zn ∥ 2 =

αn βn

(

∥xn+1 −zn ∥2 αn βn

) .

(3.11) (3.12)

Using (3.9) and (3.11), we have ∥xn+1 − x ¯∥2 ≤ ∥zn − x ¯∥2 −

1−βn βn ∥xn+1

− zn ∥2 .

(3.13)

From the definition of zn , we have ∥zn − x ¯∥2 = ∥(1 − αn )xn + αn u − x ¯ ∥2 2 2 2 ¯∥ + αn ∥u − x ¯, u − x ¯⟩ = (1 − αn ) ∥xn − x ¯∥2 + 2αn (1 − αn )⟨xn − x 2 2 = (1 − αn )∥xn − x ¯∥ + αn ∥u − x ¯∥2 + 2αn (1 − αn )⟨xn − x ¯, u − x ¯⟩

(3.14)

Thus, (3.13) and (3.14) gives ¯∥2 ∥xn+1 − x ¯∥2 ≤ (1 − αn )∥xn − x ¯∥2 + αn2 ∥u − x 2 n +2αn (1 − αn )⟨xn − x ¯, u − x ¯⟩ − 1−β βn ∥xn+1 − zn ∥ .

(3.15)

∥xn+1 − x ¯∥2 ≤ (1 − αn )∥xn − x ¯∥2 − αn Γn

(3.16)

That is, where

1 − βn ∥xn+1 − zn ∥2 . αn βn We know that {xn } is bounded and so it is bounded below. Hence, Γn is bounded below. Furthermore, using Lemma 2.4 and (C1), we have Γn = −αn ∥u − x ¯∥2 + 2(1 − αn )⟨¯ x − xn , u − x ¯⟩ +

lim sup ∥xn − x ¯∥ ≤ lim sup(−Γn ) = − lim inf Γn . n→∞

(3.17)

n→∞

n→∞

Therefore, lim inf Γn is a finite real number and by (C1), we have n→∞ ( ) 2 n x − xn , u − x ¯⟩ + 1−β lim inf Γn = lim inf 2⟨¯ αn βn ∥xn+1 − zn ∥ . n→∞

n→∞

Since {xn } is bounded, there exists a subsequence {xnk } of {xn } such that xnk ⇀ p in H1 and ( ) 1−β lim inf Γn = lim inf 2⟨¯ x − xnk , u − x ¯⟩ + αn βnnk ∥xnk +1 − znk ∥2 . n→∞

k→∞

k

Since {xn } is bounded and lim inf Γn is finite, we have that n→∞

n (C2), we have 1−β αn βn ≥ and (C2), we have

1−δ αn βn

> 0 and so we have that 0
0. n→∞

then the sequence {xn } generated by iterative algorithm  u, x1 ∈ H1 ,     z  n = (1  { − αn )xn + αn u,   ρn 0, if θ(zn ) = 0 µn = , h(zn )+l(zn ) ρ , if θ(zn ) ̸= 0.  2 n θ (zn )   ( )   yn = zn − 21 µn ∇l(zn ) + ∇h(zn ) ,    xn+1 = (1 − βn )zn + βn yn ,

(3.25)

converges strongly to x ¯ ∈ Ω1 where x ¯ = PΩ1 u. Proof. Setting fi = f for all i ∈ Φ and gj = g for all j ∈ Ψ in Theorem 3.3, we obtain the desired result.



Remark 3.5. Iterative algorithm (3.25) seems to share a similar structure with the proposed algorithm in [29]. However, the selection of the step-sizes and their restriction slightly different. The feasibility problem (convex feasibility problem), equilibrium problem and inclusion problem can be converted to the fixed point problem of firmly nonexpansive mapping. We can apply our algorithm to solve split system of feasibility problems (MSSFPs), split system of equilibrium problems and split system of inclusion problems. 1. Multiple-set split feasibility problem (1.1) by replacing proxλfi by projection mapping PCi and proxλgj by projection mapping PQj in the Algorithm 1, for all i′ ∈ Φ′ , i ∈ Φ = {1, 2, . . . , N } and j ∈ Ψ = {1, 2, . . . , M }.

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11

2. Split system of equilibrium problem: Let fi : H1 × H1 → R and gj : H2 × H2 → R be bifunctions where i ∈ Φ = {1, . . . , N }, j ∈ Ψ = {1, . . . , M }. Split system of equilibrium problem of a problem of find x ¯ ∈ H1 such that { fi (¯ x, x) ≥ 0, ∀x ∈ H1 , ∀i ∈ Φ, (3.26) gj (A¯ x, u) ≥ 0, ∀u ∈ H2 , , ∀j ∈ Ψ. Our iterative algorithm solves (3.26) by replacing proximal mappings by the resolvent operators associated to monotone equilibrium bifunctions, see [11, 3, 22]. 3. Split null point problem: Let Ti : H1 → 2H1 , Uj : H2 → 2H2 be maximal monotone mappings for all i ∈ Φ = {1, . . . , N } and j ∈ Ψ = {1, . . . , M }. The split system of inclusion problem is to find x ¯ ∈ H1 such that { 0 ∈ Ti (¯ x), ∀i ∈ Φ, (3.27) 0 ∈ Uj (A¯ x), ∀j ∈ Ψ. Our iterative algorithm solves (3.27) by replacing proximal mappings by the resolvent operators associated to the maximal monotone operators, see, [4, 25, 16, 20, 23, 36]. Our algorithm works for several split type problems and avoids the computational cost of finding operator norm. 4. Numerical results Now in this section we will consider SSMP (1.8) involving quadratic optimization problems. The algorithm has been coded in Matlab R2017a running on MacBook 1.1 GHz Intel Core m3 8 GB 1867 MHz LPDDR3. Let H1 = Rp and H2 = Rq . Consider 1 fi (x) = xT Bi x + xT Di , i ∈ Φ = {1, . . . , N }, 2 q ∑ g1 (u) = ∥u∥q and g2 (u) = h(uk ) k=1

where for each i ∈ Φ, Bi is invertible symmetric positive semidefinite p × p matrix and each Di are vectors in Rp , u = (u1 , u2 , . . . , uq ) ∈ Rq , ∥.∥q is the Euclidean norm in Rq and h(uk ) = max{|uk | − 1, 0} for k = 1, 2, . . . , q. Now for λ = 1, the proximal operators are given by proxλfi (x) = (I + Bi )−1 (x − Di ), i ∈ Φ, { ( ) 1 1 − ∥u∥ u, ∥u∥q ≥ 1 q proxλg1 (u) = 0, otherwise

(4.1)

and proxλg2 (u) = (proxλh (u1 ), proxλh (u2 ), . . . , proxλh (uq ))  if |uk | < 1  uk , sign(uk ), if 1 ≤ |uk | ≤ 2 proxλh (uk ) =  sign(uk − 1), if |uk | > 2. The proximal operator (4.1) is called the block soft thresholding obtained in de-noising model. We set Di = 0 (zero vector in Rp ) for all i ∈ Φ. Let N = 3, p = q, A is identity p × p matrix and B1 , B2 and B3 are randomly generated invertible symmetric positive semidefinite p × p matrices. Hence, with this setting, it is clear to see that Γ = {0}. In all the experiments we took δni = 6i and ξnj = 3j for i ∈ Φ = {1, 2, 3}, 1 j ∈ Ψ = {1, 2}, ρn = 10 as 0 < ρn < 2ζ for ζ = 61 . Table 1, 2 and 3 describe the average execution time in second (CPU-t(s)) and the number of iterations (Iter(n)) of our algorithm for this example. The stopping n+1 −xn ∥ criteria in the tables 1, 2 and 3 is defined as ∥x∥x ≤ TOL. 2 −x1 ∥ where

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Table 1. For p = q = 4, αn =

√1 , n+1

βn = 0.9, u = (1, 1, 1, 1), x1 = 10u. xn

Iter(n)

TOL CPU-t(s) x1n 10 5.4841 3.9817 3.2317 2.7825 2.4828 2.2691 .. .

1 2 3 4 5 6 7 .. . 23 24

10−3

0.0352

Table 2. For p = q = 100, αn = and x1 in R100 .

x2n x3n 10 10 5.4854 5.4842 3.9834 3.9820 3.2341 3.2314 2.7839 2.7820 2.4844 2.4827 2.2702 2.2690 .. .. . . 1.3796 1.3800 1.3796 1.3634 1.3638 1.3634

1 n+1 ,

∥xn+1 − xn ∥ x4n 10 9.0304 5.4852 3.0046 3.9826 1.5000 3.2326 0.8991 2.7831 0.5988 2.4838 0.4277 2.2699 0.3204 .. ... . 1.3799 0.0323 1.3637 0.0297

βn = 0.5 and randomly generated starting points u

Iter(n) TOL CPU-t(s) 1 2 3 4 5 .. .

∥xn+1 − xn ∥ 6383.1845 1519.9088 554.2387 247.0358 124.2771 .. .

15 16

1.5845 0.5736

Table 3. For p = q = 200, αn = u and x1 in R200 .

10−4

0.2923

1 10(n+1 ),

βn = 0.1 and randomly generated starting points

Iter(n) TOL CPU-t(s) 1 2 3 4 5 6 10−2 0.0093

∥xn+1 − xn ∥ 14554.8769 3475.8500 1270.6095 567.6027 286.1360 156.6170

From the tables 1-3 we can see that our proposed algorithm is efficient and easy to implement. References 1. M. Abbas, M. AlShahrani, Q. Ansari, O.S. Iyiola and Y. Shehu, Iterative methods for solving proximal split minimization problems, Numer. Algorithms pp. 1–23 (2018) 2. H.H. Bauschke, P.L. Combettes, L. Patrick and others, Convex analysis and monotone operator theory in Hilbert spaces, vol. 408. Springer (2011) 3. E. Blum, From optimization and variational inequalities to equilibrium problems, Math. Student 63, 123–145 (1994) 4. H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, vol. 5. Elsevier (1973)

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Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse probl. 21(6), 2071 (2005) 11. P.L. Combettes, S.A. Hirstoaga, and others, Equilibrium programming in hilbert spaces, J. Nonlinear Convex Anal. 6(1), 117–136 (2005) 12. P.L. Combettes and J.C. Pesquet, Proximal splitting methods in signal processing, In: Fixed-point algorithms for inverse problems in science and engineering, pp. 185–212. Springer (2011) 13. Y. Dang and Y. Gao, The strong convergence of a three-step algorithm for the split feasibility problem, Optim. Lett. 7(6), 1325–1339 (2013) 14. Y. Dang, Y. Gao and L. Li, Inertial projection algorithms for convex feasibility problem, J. Syst. Eng. Electron 23(5), 734–740 (2012) 15. A. Latif, J. Vahidi, and M. Eslamian, Strong convergence for generalized multiple-set split feasibility problem, Filomat 30(2), 459–467 (2016) 16. B. Lemaire, Which fixed point does the iteration method select? 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Springer (1999) 23. S. Plubtieng and W. Sriprad, A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in hilbert spaces, Fixed Point Theory Appl. 2009(1), 567147 (2009) 24. B. Qu, and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Probl. 21(5), 1655 (2005) 25. R. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149(1), 75–88 (1970) 26. R.T. Rockafellar and R.J.B. Wets, Variational analysis, vol. 317. Springer Science & Business Media (2009) 27. Y. Shehu, G. Cai and O.S. Iyiola, Iterative approximation of solutions for proximal split feasibility problems, Fixed Point Theory Appl. 2015(1), 123 (2015) 28. Y. Shehu and O.S. Iyiola, Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl. 19(4), 2483– 2510 (2017) 29. Y. Shehu and O.S. Iyiola, Strong convergence result for proximal split feasibility problem in Hilbert spaces, Optimization 66(12), 2275–2290 (2017) 30. Y. Shehu and O.S. Iyiola, Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems, Optimization 67(4), 475–492 (2018) 31. Y. Shehu and O.S. Iyiola, Nonlinear iteration method for proximal split feasibility problems, Math. Meth. Appl. Sci. (2018) 32. Y. Shehu and F.U.Ogbuisi, Convergence analysis for proximal split feasibility problems and fixed point problems, J. Appl. Math. Comp. 48(1-2), 221–239 (2015) 33. Y. Wang, F. Wang and H.K. Xu, Error sensitivity for strongly convergent modifications of the proximal point algorithm, J. Opt. Theory Appl. 168(3), 901–916 (2016) 34. M. Wen, J. Peng and Y. Tang, A cyclic and simultaneous iterative method for solving the multiple-sets split feasibility problem, J. Opt. Theory Appl. 166(3), 844–860 (2015) 35. H.K. Xu, A variable krasnosel’skii–mann algorithm and the multiple-set split feasibility problem, Inverse probl. 22(6), 2021 (2006) 36. Y. Yao, Y.J. Cho and Y.C. Liou, Iterative algorithms for variational inclusions, mixed equilibrium and fixed point problems with application to optimization problems, Cent. Eur. J. Math. 9(3), 640–656 (2011) 37. H. Zegeye and N. Shahzad, Convergence of mann’s type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl. 62(11), 4007–4014 (2011)

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Complex Korovkin Theory George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Let K be a compact convex subspace of C and C (K; C) the space of continuous functions from K into C. We consider bounded linear functionals from C (K; C) into C and bounded linear operators from C (K; C) into itself. We assume that these are bounded by companion real positive linear entities, respectively. We study quantitatively the rate of convergence of the approximation of these linearities to the corresponding unit elements. Our results are inequalities of Korovkin type involving the complex modulus of continuity and basic test functions.

2010 Mathematics Subject Classi…cation : 41A17, 41A25, 41A36. Keywords and phrases: positive linear functional, positive linear operator, complex functions, Korovkin theory, complex modulus of continuity.

1

Introduction

The study of the convergence of positive linear operators became more intensive and attractive when P. Korovkin (1953) proved his famous theorem (see [7], p. 14). Korovkin’s First Theorem. Let [a; b] be a compact interval in R and (Ln )n2N be a sequence of positive linear operators Ln mapping C ([a; b]) into itself. Assume that (Ln f ) converges uniformly to f for the three test functions f = 1; x; x2 . Then (Ln f ) converges uniformly to f on [a; b] for all functions of f 2 C ([a; b]). So a lot of authors since then have worked on the theoretical aspects of the above convergence. But R. A. Mamedov (1959) (see [8]) was the …rst to put Korovkin’s theorem in a quantitative scheme. Mamedov’s Theorem. Let fLn gn2N be a sequence of positive linear operators in the space C ([a; b]), for which Ln 1 = 1, Ln (t; x) = x + n (x), Ln t2 ; x = x2 + n (x). Then it holds kLn (f; x)

f (x)k1

3! 1 f;

p dn ;

1

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where ! 1 is the …rst modulus of continuity and dn = k n (x) 2x n (x)k1 : An improvement of the last result was the following. Shisha and Mond’s Theorem. (1968, see [10]). Let [a; b] R be a compact interval. Let fLn gn2N be a sequence of positive linear operators acting on C ([a; b]). For n = 1; 2; :::; suppose Ln (1) is bounded. Let f 2 C ([a; b]). Then for n = 1; 2; :::; it holds kLn f

f k1

kf k1 kLn 1

1k1 + kLn (1) + 1k1 ! 1 (f;

where n

:=

2

Ln (t

x)

(x)

1 2

1

n) ;

:

Shisha-Mond inequality generated and inspired a lot of research done by many authors worldwide on the rate of convergence of a sequence of positive linear operators to the unit operator, always producing similar inequalities however in many di¤erent directions, e.g., see the important work of H. Censka of 1983 in [6], etc. The author (see [1]) in his 1993 research monograph, produces in many directions best upper bounds for j(Ln f ) (x0 ) f (x0 )j, x0 2 Q Rn , n 1, compact and convex, which lead for the …rst time to sharp/attained inequalities of Shisha-Mond type. The method of proving is probabilistic from the theory of moments. His pointwise approach is closely related to the study of the weak convergence with rates of a sequence of …nite positive measures to the unit measure at a speci…c point. The author in [3], pp. 383-412 continued this work in an abstract setting: Let X be a normed vector space, Y be a Banach lattice; M X is a compact and convex subset. Consider the space of continuous functions from M into Y , denoted by C (M; Y ); also consider the space of bounded functions B (M; Y ). He studied the rate of the uniform convergence of lattice homomorphisms T : C (M; Y ) ! C (M; Y ) or T : C (M; Y ) ! B (M; Y ) to the unit operator I. See also [2]. Also the author in [4], pp. 175-188 continued the last abstract work for bounded linear operators that are bounded by companion real positive linear operators. Here the invoved functions are from [a; b] R into (X; k k) a Banach space. All the above have inspired and motivated the work of this article. Our results are of Shisha-Mond type, i.e., of Korovkin type. Namely here let K be a convex and compact subset of C and l be a linear functional from C (K; C) into C, and let e l be a positive linear functional from C (K; R) into R, such that jl (f )j e l (jf j), 8 f 2 C (K; C). Clearly then l is a bounded linear functional. Initially we create a quantitative Korovkin type theory over the last described setting, then we transfer these results to related bounded linear operators with similar properties.

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2

Background

We need Theorem 1 Let K (C; j j) and f a function from K into C. Consider the …rst complex modulus of continuity ! 1 (f; ) :=

sup jf (x)

f (y)j ,

> 0:

(1)

x;y2K jx yj
0, where f 2 U C (K; C) (uniformly continuous functions). (2)’ If K is open convex or compact convex, then ! 1 (f; ) is continuous on R+ in , for f 2 U C (K; C) : (3)’ If K is convex, then ! 1 (f; t1 + t2 )

! 1 (f; t1 ) + ! 1 (f; t2 ) , t1 ; t2 > 0;

(2)

that is the subadditivity property is true. Also it holds ! 1 (f; n )

n! 1 (f; )

(3)

and ! 1 (f;

)

d e ! 1 (f; )

( + 1) ! 1 (f; ) ;

(4)

where n 2 N, > 0, > 0, d e is the ceiling of the number. (4)’Clearly in general ! 1 (f; ) 0 and is increasing in > 0 and ! 1 (f; 0) = 0: (5)’If K is open or compact, then ! 1 (f; ) ! 0 as # 0, i¤ f 2 U C (K; C) : (6)’ It holds ! 1 (f + g; ) ! 1 (f; ) + ! 1 (g; ) ; (5) for

> 0, any f; g : K ! C, K

C is arbitrary.

Proof. (1)’ Here K is open convex. Let here f 2 U C (K; C), i¤ 8 " > 0, 9 > 0 : jx yj < implies jf (x) f (y)j < ". Let "0 > 0 then 9 0 > 0 : jx yj f (y)j < "0 , hence ! 1 (f; 0 ) "0 < 1: 0 with jf (x) Let > 0 arbitrary and x; y 2 K : jx yj . Choose n 2 N : n 0 > , and set xi = x + ni (y x), 0 i n. Notice that all xi 2 K. Then jf (x) jf (x)

f (x1 )j + jf (x1 )

n X1

f (y)j =

f (x2 )j + jf (x2 ) n! 1 (f;

since jxi

xi+1 j =

1 n

jx

(f (xi )

f (xi+1 ))

i=0

yj

1 n


0. Notice that jf (x)

f (y)j

jf (x)

f (z)j + jf (z)

f (y)j

! 1 (f; t1 ) + ! 1 (f; t2 ) :

Hence ! 1 (f; t1 + t2 )

! 1 (f; t1 ) + ! 1 (f; t2 ) ;

proving (3)’. Then by the obvious property (4)’we get 0

! 1 (f; t1 + t2 )

! 1 (f; t1 )

! 1 (f; t2 ) ;

and j! 1 (f; t1 + t2 )

! 1 (f; t1 )j

! 1 (f; t2 ) :

Let f 2 U C (K; C), then lim ! 1 (f; t2 ) = 0, by property (5)’. Hence ! 1 (f; ) t2 #0

is continuous on R+ : (5)’()) Let ! 1 (f; ) ! 0 as # 0. Then 8 " > 0; 9 > 0 with ! 1 (f; ) ". I.e. 8 x; y 2 K : jx yj we get jf (x) f (y)j ": That is f 2 U C (K; C). (() Let f 2 U C (K; C). Then 8 " > 0; 9 > 0 : whenever jx yj , x; y 2 K, it implies jf (x) f (y)j ". I.e. 8 " > 0; 9 > 0 : ! 1 (f; ) ". That is ! 1 (f; ) ! 0 as # 0: (6)’Notice that j(f (x) + g (x))

(f (y) + g (y))j

jf (x)

f (y)j + jg (x)

g (y)j :

That is property (6)’now is clear. We need Theorem 2 ([1], p. 208) Let (V1 ; k k) ; (V2 ; k k) be real normed vector spaces and Q V1 which is star- shaped relative to the …xed point x0 . Consider f : Q ! V2 with the properties: f (x0 ) = 0, and ks

tk

h implies kf (s)

Then, there exists a maximal such function (t) :=

kt

x0 k h

f (t)k

w; w; h > 0:

(6)

, namely w

! i;

! where i is any unit vector in V2 . That is kf (t)k k (t)k , all t 2 Q:

(7)

(8)

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Corollary 3 Let K Then jf (x)

(C; j j) be a compact convex subset, and f 2 C (K; C). f (x0 )j

! 1 (f; )

jx

x0 j

;

> 0;

(9)

8 x; x0 2 K: We make Remark 4 Let K (C; j j) be a compact subset and g 2 C (K; R). A linear functional I from C (K; R) into R is positive, i¤ I (g1 ) I (g2 ), whenever g1 g2 , where g1 ; g2 2 C (K; R) : Let us assume that I is a positive linear functional. Then by Riesz representation theorem, [9], p. 304, there exists a unique Borel measure on K such that Z I (g) = g (t) d (t) ; (10) K

8 g 2 C (K; R) : We make

Remark 5 Here initially we follow [5]. Suppose is a smooth path parametrized by z (t), t 2 [a; b] and f is a complex function which is continuous on . Put z (a) = u and z (b) = w with u; w 2 C. We de…ne the integral of f on u;w = as Z Z Z b f (z) dz := f (z (t)) z 0 (t) dt: (11) f (z) dz = a

u;w

By triangle inequality we have Z b Z f (z (t)) z 0 (t) dt f (z) dz = a

Z

a

b

jf (z (t))j jz 0 (t)j dt :=

Z

jf (z)j jdzj :

R (12) Inequalities (12) provide a typical example on linear functionals: clearly f (z) dz R induces a linear functional from C ( ; C) into C, and jf (z)j jdzj involves a positive linear functional from C ( ; R) into R. Thus, be given K a convex and compact subset of C and l be a linear functional from C (K; C) into C, it is not strange to assume that there exists a positive linear functional e l from C (K; R) into R, such that jl (f )j

e l (jf j) ; 8 f 2 C (K; C) :

(13)

Furthermore, we may assume that e l (1 ( )) = 1, where 1 (t) = 1, 8 t 2 K; l (c ( )) = c; 8c 2 C where c (t) = c, 8 t 2 K We call e l the companion functional to l. Here C is a vector space over the …eld of reals. The functional l is linear over R and the functional e l is linear over R. Next we study approximation properties of ln ; e ln pairs, n 2 N: 5

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

3

Main Results - I

First about linear functionals: We present the following quantitative approximation result of Korovkin type. Theorem 6 Here K is a convex and compact subset of C and ln is a sequence of linear functionals from C (K; C) into C, n 2 N. There is a sequence of companion positive linear functionals e ln from C (K; R) into R, such that jln (f )j

e ln (jf j) , 8 f 2 C (K; C) ; 8 n 2 N:

(14)

f (x0 )j

2! 1 f; e ln (j

(15)

Additionally, we assume that e ln (1 ( )) = 1 and ln (c ( )) = c; 8c 2 C 8 n 2 N: Then jln (f )

8 f 2 C (K; C) :

x0 j) ; 8 n 2 N; 8 x0 2 K,

Proof. We notice that jln (f )

f (x0 )j = jln (f ) (14)

jln (f ( )

f (x0 ) ( ))j

j e ln ! 1 (f; )

x0 j

ln (f (x0 ) ( ))j =

e ln (jf ( )

(by

! 1 (f; ) e ln 1 ( ) +

1 ln (j ! 1 (f; ) e ln (1( )) + e

by choosing

1 ! 1 (f; ) 1 + e ln (j

j

x0 j

=

x0 j) =

x0 j) = 2! 1 f; e ln (j

:= e ln (j

>0; (9))

f (x0 ) ( )j)

x0 j) ;

(16)

x0 j) ;

if e ln (j x0 j) > 0, that is proving (15). Next, we consider the case of e ln (j x0 j) = 0. By Riesz representation theorem, see (10) there exists a probability measure such that Z e ln (g) = g (t) d (t) ; 8 g 2 C (K; R) : (17) K

That is, here it holds

Z

K

jt

x0 j d (t) = 0;

which implies jt x0 j = 0, a.e, hence t x0 = 0, a.e, and t = x0 , a.e. Consequently (ft 2 K : t 6= x0 g) = 0. Hence = x0 , the Dirac measure with support only fx0 g : Therefore in that case e ln (g) = g (x0 ), 8 g 2 C (K; R). Thus, it holds e ! 1 f; ln (j x0 j) = ! 1 (f; 0) = 0, and e ln (jf ( ) f (x0 ) ( )j) = jf (x0 ) f (x0 )j = 0, giving jln (f ) f (x0 )j = 0. That is (15) is again true. 6

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Remark 7 We have that e ln (j

x0 j) =

Z

K

jt

x0 j d (t)

(by Schwarz’s inequality) Z

1 2

1d (t)

K

K

We give

e ln (1)

Z

1 2

Z

K

jt

2

jt

x0 j d (t) 1 2

2

x0 j d (t)

= e ln j

1 2

=

2

x0 j

1 2

:

(18)

Corollary 8 All as in Theorem 6. Then jln (f )

f (x0 )j

2! 1 f; e ln j

2

x0 j

1 2

; 8 n 2 N; 8 x0 2 K.

(19)

ln (j x0 j) ! 0, Conclusion 9 All as in Theorem 6. By (15) and/or (19), as e 2 e or ln j x0 j ! 0, as n ! +1, we obtain that ln (f ) ! f (x0 ) with rates, 8 x0 2 K. Next comes a more general quantitative approximation result of Korovkin type. Theorem 10 Here K is a convex and compact subset of C and ln is a sequence of linear functionals from C (K; C) into C, n 2 N. There is a sequence of companion positive linear functionals e ln from C (K; R) into R, such that e ln (jf j) , 8 f 2 C (K; C) ; 8 n 2 N:

(20)

ln (cg) = ce ln (g) ; 8 g 2 C (K; R) ; 8 c 2 C:

(21)

jln (f )j

Additionally, we assume that

Then, for any f 2 C (K; C), we have jln (f )

f (x0 )j

jf (x0 )j e ln (1 ( ))

1 + e ln (1 ( )) + 1 ! 1 f; e ln (j

x0 j) ; (22)

8 x0 2 K, 8 n 2 N: (Notice if e ln (1 ( )) = 1, then (22) collapses to (15). So Theorem 10 generalizes Theorem 6). By (22), as e ln (1 ( )) ! 1 and e ln (j x0 j) ! 0, then ln (f ) ! f (x0 ), as e n ! +1, with rates, and as here ln (1 ( )) is bounded. 7

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Proof. We observe that jln (f )

f (x0 )j = jln (f ) jln (f )

ln (f (x0 ) ( )) + ln (f (x0 ) ( ))

ln (f (x0 ) ( ))j + f (x0 ) e ln (1 ( ))

jln (f ( )

jf (x0 )j e ln (1 ( ))

ln (1 ( )) jf (x0 )j e

jf (x0 )j e ln (1 ( ))

jf (x0 )j e ln (1 ( ))

by choosing

f (x0 ) =

f (x0 ) ( ))j + jf (x0 )j e ln (1 ( ))

jf (x0 )j e ln (1 ( ))

1 +e ln (jf ( )

ln ! 1 (f; ) 1 +e

1

(23)

f (x0 ) ( )j) j

ln (! 1 (f; )) 1 ( ) + 1 +e

x0 j j

x0 j

1 1 + ! 1 (f; ) e ln (1 ( )) + e ln (j

1 + e ln (1 ( )) + 1 ! 1 f; e ln (j

if e ln (j x0 j) > 0: Next we consider the case of

:= e ln (j

e ln (j

f (x0 )j

=

x0 j) = x0 j) ;

x0 j) ;

(24)

x0 j) = 0:

(25)

By Riesz representation theorem there exists a positive …nite measure that Z e ln (g) = g (t) d (t) , 8 g 2 C (K; R) :

such (26)

K

That is

Z

K

jt

x0 j d (t) = 0;

(27)

which implies jt x0 j = 0, a.e., hence t x0 = 0, a.e, and t = x0 , a.e. on K. Consequently (ft 2 K : t 6= x0 g) = 0. That is = x0 M (where 0 < M := (K) = e ln (1 ( ))). Hence, in that case e ln (g) = g (x0 ) M . Consequently it holds ! 1 f; e ln (j x0 j) = 0, and the right hand side of (22) equals jf (x0 )j jM 1j. Also, it is e ln (jf ( ) f (x0 ) ( )j) = jf (x0 ) f (x0 )j M = 0. Hence from the …rst part of this proof we get jln (f ) ln (f (x0 ) ( ))j = 0, and ln (f ) = ln (f (x0 ) ( )) = f (x0 ) e ln (1 ( )) = M f (x0 ) : Consequently the left hand side of (22) becomes jln (f )

f (x0 )j = jM f (x0 )

f (x0 )j = jf (x0 )j jM

1j :

So that (22) becomes an equality, and both sides equal jf (x0 )j jM 1j in the extreme case of e ln (j x0 j) = 0: Thus inequality (22) is proved completely in all cases. We make 8

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Remark 11 By Schwartz’s inequality we get e ln (j

We give

e ln j

x0 j)

1 2

2

x0 j

1 2

e ln (1 ( ))

:

(28)

Corollary 12 All as in Theorem 10. Then jln (f )

jf (x0 )j e ln (1 ( ))

f (x0 )j

e ln (1 ( )) + 1 ! 1 f; e ln (1 ( ))

1 2

e ln j

8 x0 2 K, 8 n 2 N:

1 + 1 2

2

x0 j

;

(29)

Next we give another version of our Korovkin type result. Theorem 13 Here all are as in Theorem 10. Then, for any f 2 C (K; C), we have jln (f )

jf (x0 )j e ln (1 ( ))

f (x0 )j

1+ e ln (1 ( )) + 1 ! 1 f; e ln j

2

x0 j

1 2

;

(30) 8 x0 2 K, 8 n 2 N: 2 By (30), as e ln (1 ( )) ! 1 and e ln j x0 j ! 0, then ln (f ) ! f (x0 ), as n ! +1, with rates, and as here e ln (1 ( )) is bounded. Proof. Let t; x0 2 K and jf (t) 1+

f (x0 )j jt

> 0. If jt

! 1 (f; jt

x0 j

x0 j > , then

x0 j) = ! 1 f; jt

! 1 (f; )

jt

jt

x0 j

The estimate jf (t)

f (x0 )j

1+

2

1+

x0 j

2

2

2

!

!

x0 j

1

(31)

! 1 (f; ) :

! 1 (f; )

also holds trivially when jt x0 j . So (32) is true always, 8 t 2 K, for any x0 2 K: We can rewrite ! 2 j x0 j jf ( ) f (x0 )j 1+ ! 1 (f; ) : 2

(32)

(33)

As in the proof of Theorem 10 we have jln (f )

f (x0 )j

:::

jf (x0 )j e ln (1 ( ))

1 +

9

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e ln

! 1 (f; ) 1 ( ) +

2

j

x0 j 2

!!

=

1 ln (1 ( )) + 2 e ln j 1 + ! 1 (f; ) e

jf (x0 )j e ln (1 ( ))

jf (x0 )j e ln (1 ( ))

1 + ! 1 f; e ln j

by choosing

:= e ln j

2

e ln j

1 2

2

x0 j

if e ln j x0 j > 0. Next we consider the case of

2

x0 j

1 2

2

x0 j

2

x0 j

=

(34)

e ln (1 ( )) + 1 ;

;

(35)

= 0:

(36)

By Riesz representation theorem there exists a positive …nite measure that Z e ln (g) = g (t) d (t) , 8 g 2 C (K; R) :

such (37)

K

That is

Z

K

2

jt

2

x0 j d (t) = 0;

which implies jt x0 j = 0, a.e., hence t x0 = 0, a.e, and t = x0 , a.e. on K. Consequently (ft 2 K : t 6= x0 g) = 0. That is = x0 M (where 0 < e e M := (K) = ln (1 ( ))). Hence, in that case ln (g) = g (x0 ) M . Consequently it holds ! 1 f; e ln j

2

x0 j

1 2

= 0, and the right hand side of (30) equals

jf (x0 )j jM 1j. Also, it is e ln (jf ( ) f (x0 ) ( )j) = jf (x0 ) f (x0 )j M = 0. Hence from the …rst part of this proof we get: jln (f ) ln (f (x0 ) ( ))j = 0, and ln (f ) = ln (f (x0 ) ( )) = f (x0 ) e ln (1 ( )) = M f (x0 ) : Consequently the left hand side of (30) becomes jln (f )

f (x0 )j = jf (x0 )j jM

1j :

So that (30) is true again. The proof of the theorem is now complete. Corollary 14 Here all are as in Theorem 10. Then jln (f )

f (x0 )j

min ! 1 f; e ln (1 ( ))

8 x0 2 K, 8 n 2 N:

1 2

jf (x0 )j e ln (1 ( )) e ln j

2

x0 j

1 2

1 + e ln (1 ( )) + 1

; ! 1 f; e ln j

2

x0 j

1 2

; (38)

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Proof. By (29) and (30). So (29) is better that (30) only if e ln (1 ( )) < 1: We need

Theorem 15 Let K C convex, x0 2 K 0 (interior of K) and f : K ! R such that jf (t) f (x0 )j is convex in t 2 K. Furthermore let > 0 so that the closed disk D (x0 ; ) K. Then jf (t)

! 1 (f; )

f (x0 )j

jt

x0 j ; 8 t 2 K:

(39)

Proof. Let g (t) := jf (t) f (x0 )j, t 2 K, which is convex in t 2 K and g (x0 ) = 0. Then by Lemma 8.1.1, p. 243 of [1], we obtain g (t)

! 1 (g; )

jt

x0 j ; 8 t 2 K:

(40)

We notice the following jf (t1 )

f (x0 )j = jf (t1 )

f (t2 ) + f (t2 )

f (x0 )j

jf (t1 )

f (t2 )j + jf (t2 )

f (x0 )j ;

f (x0 )j

jf (t2 )

f (x0 )j

jf (t1 )

f (t2 )j :

(41)

f (x0 )j

jf (t1 )

f (x0 )j

jf (t1 )

f (t2 )j :

(42)

hence jf (t1 ) Similarly, it holds jf (t2 )

Therefore for any t1 ; t2 2 K : jt1 j jf (t1 )

f (x0 )j

jf (t2 )

t2 j

we get

f (x0 )j j

jf (t1 )

f (t2 )j

! 1 (f; ) :

(43)

That is ! 1 (g; )

! 1 (f; ) :

(44)

The last and (40) imply jf (t)

f (x0 )j

! 1 (f; )

jt

x0 j ; 8 t 2 K;

(45)

proving (39). We continue with a convex Korovkin type result: Theorem 16 All as in Theorem 10. Let x0 2 K 0 and assume that jf (t) f (x0 )j is convex in t 2 K. Let > 0,such that the closed disk D (x0 ; ) K. Then jln (f )

f (x0 )j

jf (x0 )j e ln (1 ( ))

1 + ! 1 f; e ln (j

x0 j) ; 8 n 2 N: (46)

11

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Proof. As in the proof Theorem 10 we have jln (f )

f (x0 )j

jf (x0 )j e ln (1 ( ))

:::

jf (x0 )j e ln (1 ( )) by choosing

1 +

jf (x0 )j e ln (1 ( ))

! 1 (f; ) e ln (j

f (x0 ) ( )j)

(47)

x0 j) =

1 + ! 1 f; e ln (j

:= e ln (j

(39)

1 +e ln (jf ( )

x0 j) ;

x0 j) > 0;

if the last is positive. The case of e ln (j x0 j) = 0 is treated similarly as in the proof of Theorem 10. The theorem is proved.

Theorem 17 All as in Theorem 16. Inequality (46) is sharp, in fact it is ! ! attained by f (t) = j jt x0 j, where j is a unit vector of (C; j j); t; x0 2 K: Proof. Indeed, f here ful…lls the assumptions of the theorem. We further notice that f (x0 ) = 0, and jf (t) f (x0 )j = jt x0 j is convex in t 2 K. The left hand side of (46) is jln (f )

f (x0 )j = jln (f )j = ln !e j ln (j

The right hand side of (46) is !1 f ; e ln (j

x0 j) = ! 1

sup

jt1

x0 j) = e ln (j

t1 ;t2 2K t2 j e ln (j x0 j)

! j j

! j j

x0 j ; e ln (j

x0 j

jjt1

x0 j

jt1

t2 j = e ln (j

sup

jt1

jt1

t1 ;t2 2K t2 j e ln (j x0 j)

sup

(21)

=

x0 j) :

! j jt1

t1 ;t2 2K t2 j e ln (j x0 j)

x0 j

! j jt2 jt2

(48)

x0 j) = x0 j =

x0 jj

(49)

x0 j) :

Hence we have found that

!1 f ; e ln (j

x0 j)

Clearly (46) is attained. The theorem is proved.

e ln (j

x0 j) :

(50)

12

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4

Main Results - II

Next we give results on linear operators: Let K be a compact convex subset of C. Consider L : C (K; C) ! C (K; C) e : C (K; R) ! C (K; R) a positve linear operator (i.e. for a linear operator and L e (f1 ) L e (f2 )) both over the …eld of R: f1 :f2 2 C (K; R) with f1 f2 we get L We assume that jL (f )j

e (jf j) , 8 f 2 C (K; C) ; L

e (jf j) (z), 8 z 2 K). (i.e. jL (f ) (z)j L e We call L the companion operator of L: Let x0 2 K. Clearly, then L ( ) (x0 ) is a linear functional from C (K; C) into e ( ) (x0 ) is a positive linear functional from C (K; R) into R. Notice C, and L e (jf j) (z) 2 R, 8 f 2 C (K; C) (thus jf j 2 C (K; R)). Here L (f ) (z) 2 C and L e (jf j) 2 C (K; R), 8 f 2 C (K; C) : L (f ) 2 C (K; C), and L Notice that C (K; C) = U C (K; C), also C (K; R) = U C (K; R) (uniformly continuous functions). e (j x0 jr ) (x0 ), r > 0, is a continuous function By [3], p. 388, we have that L in x0 2 K: After this preparation we transfer the main results from section 3 to linear operators. We have the following approximation results with rates of Korovkin type. Theorem 18 Here K is a convex and compact subset of C and Ln is a sequence of linear operators from C (K; C) into itself, n 2 N. There is a sequence of e n from C (K; R) into itself, such that companion positive linear operators L e n (jf j) , 8 f 2 C (K; C) ; 8 n 2 N L

(51)

e n (g) ; 8 g 2 C (K; R) ; 8 c 2 C Ln (cg) = cL

(52)

jLn (f )j

e n (jf j) (x0 ), 8 x0 2 K). (i.e. jLn (f ) (x0 )j L Additionally, we assume that

e n (g) (x0 ) ; 8 x0 2 K). (i.e. (Ln (cg)) (x0 ) = c L Then, for any f 2 C (K; C), we have j(Ln (f )) (x0 )

f (x0 )j

e n (1 ( )) (x0 ) jf (x0 )j L

e n (1 ( )) (x0 ) + 1 ! 1 f; L e n (j L

x0 j) (x0 ) ;

1 + (53)

8 x0 2 K, 8 n 2 N:

Proof. By Theorem 10. 13

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Corollary 19 All as in Theorem 18. Then kLn (f )

e n (1 ( )) kf k1;K L

f k1;K

e n (1 ( )) + 1 L

1;K

f k1;K

8 n 2 N: u e n (1 ( )) ! e n (j As L 1, L

1;K

e n (j ! 1 f; L

x0 j) (x0 )

e n (j 2! 1 f; L

x0 j) (x0 )

8 n 2 N: e n (1 ( )) = 1, 8 n 2 N, then If L kLn (f )

1

x0 j) (x0 )

+

1;K

1;K

;

(54)

;

(55)

u

1;K

u

! 0, then (by (54)) Ln (f ) ! f ,

e n (1 ( )) is bounded, and all the as n ! +1, where u means uniformly. Notice L suprema in (54) are …nite. We continue with

Theorem 20 Here all as in Theorem 18. Then, for any f 2 C (K; C), we have j(Ln (f )) (x0 )

e n (1 ( )) (x0 ) jf (x0 )j L

f (x0 )j

e n (1 ( )) (x0 ) + 1 ! 1 f; L en j L

2

x0 j

1 + 1 2

(x0 )

;

(56)

8 x0 2 K, 8 n 2 N:

Proof. By Theorem 13. Corollary 21 All as in Theorem 18. Then, for any f 2 C (K; C), we have kLn (f )

f k1;K

e n (1 ( )) + 1 L

1;K

8 n 2 N: e n (1 ( )) = 1, then If L kLn (f )

f k1;K

8 n 2 N: u en j e n (1 ( )) ! As L 1, L

f , as n ! +1.

e n (1 ( )) kf k1;K L

en j ! 1 f; L

x0 j

en j 2! 1 f; L

x0 j

2

x0 j

(x0 )

2

2

u

1;K

1

1;K

(x0 )

(x0 )

+

1 2

1;K

1 2

1;K

;

(57)

;

(58)

u

! 0, then (by (57)) Ln (f ) !

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We continue with a convex Korovkin type result: Theorem 22 All as in Theorem 18. Let a …xed x0 2 K 0 and assume that jf (t) f (x0 )j is convex in t 2 K. Let > 0,such that the closed disk D (x0 ; ) K. Then j(Ln (f )) (x0 )

f (x0 )j

e n (j +! 1 f; L

e n (1 ( )) (x ) jf (x0 )j L 0

x0 j) (x0 ) ; 8 n 2 N:

1 (59)

e n (1 ( )) (x ) ! 1, and L e n (j x j) (x ) ! 0, we get that (Ln (f )) (x ) ! As L 0 0 0 0 f (x0 ), as n ! +1; a pointwise convergence. Proof. By Theorem 16. Note: Theorem 22 goes throw if (51), (52) are valid only for the particular

x0 : We …nish with Proposition 23 All as in Theorem 22. Inequality (59) is sharp, in fact it is ! ! attained by f (t) = j jt x0 j, where j is a unit vector of C; x0 ; t 2 K: Proof. By Theorem 17. Note: Let K be a convex compact subset of a real normed vector space (V; k k1 ) and (X; k k2 ) is a Banach space. We can consider bounded linear functionals and bounded operators on C (K; X). This paper’s methodology can be applied to this more general setting and produce a similar Korovkin theory in full strength.

References [1] G.A. Anastassiou, Moments in Probability and Approximation Theory, Pitman Research Notes in Math., Vol. 287, Longman Sci. & Tech., Harlow, U.K., 1993. [2] G.A. Anastassiou, Lattice homomorphism- Korovkin type inequalities for vector valued functions, Hokkaido Math. J., 26 (1997), 337-364. [3] G.A. Anastassiou, Quantitative Approximations, Chapman &Hall / CRC, Boca Raton, New York, 2001. [4] G.A. Anastassiou, Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg, New York, 2018. [5] S.S. Dragomir, An integral representation of the remainder in Taylor’s expansion formula for analytic function on general domains, RGMIA Res. Rep. Call. 22 (2019), Art. 2, 14 pp., rgmia.org/v22.php.

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[6] H. Gonska, On approximation of continuously di¤ erentiable functions by positive linear operators, Bull. Austral. Math. Soc., 27 (1983), 73-81. [7] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Corp., Delhi, India, 1960. [8] R.G. Mamedov, On the order of the approximation of functions by linear positive operators, Dokl. Akad. Nauk USSR, 128 (1959), 674-676. [9] H.L. Royden, Real Analysis, 2nd edition, Macmillan, New York, 1968. [10] O. Shisha and B. Mond, The degree of convergence of sequences of linear positive operators, Nat. Acad. of Sci., 60 (1968), 1196-1200.

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN NON-ARCHIMEDEAN BANACH SPACES INHO HWANG Abstract. In this paper, we solve the additive ρ-functional inequalities

( ( ) ) x+y

(0.1) ∥f (x + y) + f (x − y) − 2f (x)∥ ≤ ρ 2f + f (x − y) − 2f (x) , 2 where ρ is a fixed non-Archimedean number with |ρ| < 1, and

(

) x+y

(0.2) + f (x − y) − 2f (x) ≤ ∥ρ(f (x + y) + f (x − y) − 2f (x))∥,

2f 2 where ρ is a fixed non-Archimedean number with |ρ| < |2|. Furthermore, we prove the Hyers-Ulam stability of the additive 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. ([6]) Let X be a vector space over a field K with a non-Archimedean valuation | · |. A function ∥ · ∥ : X → [0, ∞) is said to be a non-Archimedean norm if it satisfies the following conditions: (i) ∥x∥ = 0 if and only if x = 0; (ii) ∥rx∥ = |r|∥x∥ (r ∈ K, x ∈ X); (iii) the strong triangle inequality ∥x + y∥ ≤ max{∥x∥, ∥y∥},

∀x, y ∈ X

2010 Mathematics Subject Classification. Primary 46S10, 39B62, 39B52, 47S10, 12J25. Key words and phrases. Hyers-Ulam stability; non-Archimedean normed space; additive ρ-functional inequality.

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holds. Then (X, ∥ · ∥) 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 ∥xn − xm ∥ ≤ 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 ∥xn − x∥ ≤ 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 [13] 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 [5] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [10] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [4] 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) is called the Jensen type additive functional equation. 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 [12] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [3] 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 [2, 7, 8, ?, 11]). In this paper, we solve the additive ρ-functional inequalities (0.1) and (0.2) and prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (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| ̸= 1. 2. Additive ρ-functional inequality (0.1) in non-Archimedean normed spaces Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < 1. In this section, we solve the additive ρ-functional inequality (0.1) in non-Archimedean normed spaces. Lemma 2.1. If a mapping f : X → Y satisfies f (0) = 0 and

( ( ) )

x+y

∥f (x + y) + f (x − y) − 2f (x)∥ ≤ ρ 2f + f (x − y) − 2f (x)

2

(2.1)

for all x, y ∈ X, then f : X → Y is additive.

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Proof. Assume that f : X → Y satisfies (2.1). Letting y = x in (2.1), we get ∥f (2x) − 2f (x)∥ ≤ 0 and so f (2x) = 2f (x) for all x ∈ X. Thus ( ) x 1 = f (x) f (2.2) 2 2 for all x ∈ X. It follows from (2.1) and (2.2) that

( ( ) )

ρ 2f x + y + f (x − y) − 2f (x)

2

∥f (x + y) + f (x − y) − 2f (x)∥ ≤

= |ρ|∥f (x + y) + f (x − y) − 2f (x)∥ and so f (x + y) + f (x − y) = 2f (x) for all x, y ∈ X. It is easy to show that f is additive.



We prove the Hyers-Ulam stability of the additive ρ-functional inequality (2.1) in nonArchimedean Banach spaces. Theorem 2.2. Let r < 1 and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying f (0) = 0 and

(

(

)

)

x+y + f (x − y) − 2f (x)

2 + θ(∥x∥r + ∥y∥r )

∥f (x + y) + f (x − y) − 2f (x)∥ ≤

ρ 2f

(2.3)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that 2θ ∥x∥r |2|r

(2.4)

∥f (2x) − 2f (x)∥ ≤ 2θ∥x∥r

(2.5)

∥f (x) − A(x)∥ ≤ for all x ∈ X. Proof. Letting y = x in (2.3), we get

( ) for all x ∈ X. So f (x) − 2f x2 ≤ |2|2r θ∥x∥r for all x ∈ X. Hence

( ) ( )

l x

2 f x − 2m f

(2.6)

2l 2m

{ ( ) ( ) ( ) ( ) }

l

m−1 x x x x l+1

≤ max f − 2m f

2 f 2l − 2 f 2l+1 , · · · , 2 m−1 2 2m

( )

( { ( ) ) ( ) }

x x x x l m−1

= max |2| f − 2f , · · · , |2|

f 2m−1 − 2f 2m 2l 2l+1 { }

≤ max

|2|m−1 |2|l , · · · , |2|rl+r |2|r(m−1)+r

2θ∥x∥r =

2θ

|2|(r−1)l+r

∥x∥r

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.6) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.6), we get (2.4).

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It follows from (2.3) that

( ) ( ) ( )

x−y x x+y

+f − 2f ∥A(x + y) + A(x − y) − 2A(x)∥ = lim |2| f n n n→∞ 2 2 2n

( ) ( ) ( ) n

x−y x x+y n

+ lim |2| θ (∥x∥r + ∥y∥r ) 2f ≤ lim |2| |ρ| + f − 2f

n→∞ 2n+1 2n 2n n→∞ |2|nr

( )

x+y = |ρ| + A(x − y) − 2A(x)

2A

2 n

for all x, y ∈ X. So

( ) ) (

x+y

+ A(x − y) − 2A(x) ∥A(x + y) + A(x − y) − 2A(x)∥ ≤ ρ 2A

2

for all x, y ∈ X. By Lemma 2.1, the mapping A : X → Y is additive . Now, let T : X → Y be another additive mapping satisfying (2.4). Then we have

(

)

(

)

x x

− 2q T q 2 2q ( ) { ( ) ( ) ( ) }

q

x 2θ x q

, 2q T x − 2q f x ≤ A 2 ≤ max f − 2 ∥x∥r ,

2q 2q 2q 2q |2|(r−1)q+r

q ∥A(x) − T (x)∥ =

2 A

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping A : X → Y is a unique additive mapping satisfying (2.4).  Theorem 2.3. Let r > 1 and θ be nonnegative real numbers and let f : X → Y be a mapping satisfying f (0) = 0 and (2.3). Then there exists a unique additive mapping A : X → Y such that 2θ ∥f (x) − A(x)∥ ≤ ∥x∥r |2| for all x ∈ X. Proof. It follows from (2.5) that



f (x) − 1 f (2x) ≤ 2 θ∥x∥r

2 |2|

for all x ∈ X. Hence

1 ( l )

f 2 x − 1 f (2m x)

2l

m 2

} { ( ) ( ) ( )

1

1

1 1 l l+1 m−1 m

x − m f (2 x) ≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 2 2 2 2

( )

(

} { ( ) ) 1

1 1 1 l l+1 m−1 m

f 2 x − f 2 x , · · · , m−1 f 2 x − f (2 x) = max

l |2| 2 |2| 2 {

≤ max

|2|lr |2|r(m−1) , · · · , |2|l+1 |2|(m−1)+1

}

2θ∥x∥r =

2θ |2|(1−r)l+1

for all nonnegative integers m and l with m > l and all x ∈ X. The rest of the proof is similar to the proof of Theorem 2.2.

1000

∥x∥r



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3. Additive ρ-functional inequality (0.2) Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |2|. In this section, we solve the additive ρ-functional inequality (0.2) in non-Archimedean normed spaces. Lemma 3.1. If a mapping f : X → Y satisfies

( )

2f x + y + f (x − y) − 2f (x) ≤ ∥ρ(f (x + y) + f (x − y) − 2f (x))∥

2

(3.1)

for all x, y ∈ X, then f : X → Y is additive. Proof. Assume that f : X → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get ∥f (0)∥ ≤ 0. So f (0) = 0. ( ) Letting y = 0 in (3.1), we get 2f x2 − f (x) ≤ 0 and so ( )

2f

x 2

= f (x)

(3.2)

for all x ∈ X. It follows from (3.1) and (3.2) that

(

)



x+y + f (x − y) − 2f (x)

2 ≤ |ρ|∥f (x + y) + f (x − y) − 2f (x)∥

∥f (x + y) + f (x − y) − 2f (x)∥ =

2f

and so f (x + y) + f (x − y) = 2f (x) for all x, y ∈ X. It is easy to show that f is additive.



We prove the Hyers-Ulam stability of the additive ρ-functional inequality (3.1) in nonArchimedean Banach spaces. Theorem 3.2. Let r < 1 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that

(

)

2f x + y + f (x − y) − 2f (x) ≤ ∥ρ(f (x + y) + f (x − y) − 2f (x))∥

2

+ θ(∥x∥r + ∥y∥r )

(3.3)

for all x, y ∈ X. Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤ θ∥x∥r

(3.4)

for all x ∈ X. Proof. Letting x = y = 0 in (3.3), we get f (0) = 0. Letting y = 0 in (3.3), we get

( )

2f x − f (x) ≤ θ∥x∥r

2

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for all x ∈ X. So

( ) ( )

l x

2 f x − 2m f (3.6)

2l 2m

( ) { ( ) ( ) ( ) }

l

m−1 x x x x l+1

≤ max f − 2m f

2 f 2l − 2 f 2l+1 , · · · , 2 m−1 2 2m

( )

( { ( ) ) ( ) }

x x x x l m−1

= max |2| f , · · · , |2| − 2f

f 2m−1 − 2f 2m 2l 2l+1 { } |2|l |2|m−1 θ ≤ max , · · · , r(m−1) θ∥x∥r = (r−1)l ∥x∥r 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 {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n 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 > 1 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (3.3). Then there exists a unique additive mapping A : X → Y such that ∥f (x) − A(x)∥ ≤

|2|r θ ∥x∥r |2|

(3.7)

for all x ∈ X. Proof. It follows from (3.5) that

r

f (x) − 1 f (2x) ≤ |2| θ ∥x∥r

2 |2|

for all x ∈ X. Hence

1

f (2l x) − 1 f (2m x) (3.8)

2l

m 2

{ } ( ) ( )

1 ( l )

1 l+1

, · · · , 1 f 2m−1 x − 1 f (2m x) f 2 x − f 2 x ≤ max

2m−1

2l 2l+1 2m

( )

(

} { ( ) )

1 1

f 2l x − 1 f 2l+1 x , · · · ,

f 2m−1 x − 1 f (2m x) = max



l m−1 |2| 2 |2| 2 {

≤ max

|2|rl |2|r(m−1) , · · · , |2|l+1 |2|(m−1)+1

}

|2|r θ∥x∥r =

|2|r θ |2|(1−r)l+1

∥x∥r

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.8) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by A(x) := lim

1

n→∞ n

f (2n x)

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. 

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Competing interests The author declares that he has no competing interests. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [3] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [4] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [5] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [6] M.S. Moslehian and Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.–TMA 69 (2008), 3405–3408. [7] C. Park, C ∗ -ternary biderivations and C ∗ -ternary bihomomorphisms, Math. 6 (2018), Art. 30. [8] C. Park, Bi-additive s-functional inequalities and quasi-∗-multipliers on Banach algebras, Math. 6 (2018), Art. 31. [9] 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. [10] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [11] 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. [12] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [13] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Inho Hwang Department of Mathematics, Incheon National University, Incheon 22012, Korea Email address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Square root and 3rd root functional equations in C ∗ -algebras Choonkil Park ,Sun Young Jang∗ and, Jieun Ahn

Abstract. In this paper, we introduce a square root functional equation and a 3rd root functional equation. We prove the Hyers-Ulam stability of the square root functional equation and of the 3rd root functional equation in C ∗ -algebras.

1. Introduction and preliminaries The stability problem of functional equations was originated from a question of Ulam [7] concerning the stability of group homomorphisms. Hyers [5] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [6] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [3] by replacing the unbounded Cauchy difference by a general control function in the spirit of the Rassias’ approach. Definition 1.1. [2] Let A be a C ∗ -algebra and x ∈ A a self-adjoint element, i.e., x∗ = x. Then x is said to be positive if it is of the form yy ∗ for some y ∈ A. The set of positive elements of A is denoted by A+ . Note that A+ is a closed convex cone (see [2]). It is well-known that for a positive element x and a positive integer n there exists a unique positive 1 element y ∈ A+ such that x = y n . We denote y by x n (see [4]). In this paper, we introduce a square root functional equation   1 1 1 1 1 1 S x + y + x 4 y 2 x 4 + y 4 x 2 y 4 = S(x) + S(y) (1.1) and a 3rd root functional equation   1 1 1 1 1 1 T x + y + 3x 3 y 3 x 3 + 3y 3 x 3 y 3 = T (x) + T (y)

(1.2)

for all x, y ∈ A+ . Each solution of the square root functional equation is called a square root mapping and each solution of the 3rd root functional equation is called a 3rd root mapping. √ √ 1 1 Note that the functions S(x) = x = x 2 and T (x) = 3 x = x 3 in the set of non-negative real numbers are solutions of the functional equations (1.1) and (1.2), respectively. In this paper, we prove the Hyers-Ulam stability of the functional equations (1.1) and (1.2) in C ∗ -algebras. Throughout this paper, let A+ and B + be the sets of positive elements in C ∗ -algebras A and B, respectively. 0

2010 Mathematics Subject Classification: 46L05, 39B52. Keywords: Hyers-Ulam stability, C ∗ -algebra, square root functional equation, 3rd root functional equation. ∗ Corresponding author: Sun Young Jang (email: [email protected]). 0

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2. Stability of the square root functional equation In this section, we investigate the square root functional equation in C ∗ -algebras. Lemma 2.1. Let S : A+ → B + be a square root mapping satisfying (1.1). Then S satisfies S(4n x) = 2n S(x) +

for all x ∈ A

(2.1)

and all n ∈ Z.

Proof. Putting x = y = 0 in (1.1), we obtain S(0) = 0. Letting y = 0 in (1.1), we obtain S(40 x) = S(x) = 20 S(x) for all x ∈ A+ . First of all, we use the induction on n to prove the equality (2.1) for all positive integers n. Replacing y by x in (1.1), we get S(4x) = 2S(x)

(2.2)

+

for all x ∈ A . So the equality (2.1) holds for n = 1. Assume that S(4k x) = 2k S(x)

(2.3)

holds for a positive integer k. Replacing x by 4x in (2.3) and using (2.2), we obtain S(4k+1 x) = S(4k · 4x) = 2k S(4x) = 2k+1 S(x) for all x ∈ A+ . So the equality (2.1) holds for n = k + 1. Thus S(4n x) = 2n S(x)

(2.4)

+

for all x ∈ A and all positive integers n. Next, replacing x by 4−n x in (2.4), we obtain S(x) = S(4n · 4−n x) = 2n S(4−n x) for all x ∈ A+ and all positive integers n. So S(4n x) = 2n S(x) for all x ∈ A+ and all negative integers n. Therefore, S(4n x) = 2n S(x) for all x ∈ A+ and all n ∈ Z.



We prove the Hyers-Ulam stability of the square root functional equation in C ∗ -algebras. Theorem 2.2. Let f : A+ → B + be a mapping for which there exists a function ϕ : A+ ×A+ → [0, ∞) such that ∞ x y X ϕ(x, e y) := 2j ϕ j , j < ∞, (2.5) 4 4 j=1



 1 1 1 1 1 1

(2.6)

f x + y + x 4 y 2 x 4 + y 4 x 2 y 4 − f (x) − f (y) ≤ ϕ(x, y) for all x, y ∈ A+ . Then there exists a unique square root mapping S : A+ → A+ satisfying (1.1) and 1 e y) (2.7) kf (x) − S(x)k ≤ ϕ(x, 2 for all x ∈ A+ .

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Proof. Letting y = x in (2.6), we get kf (4x) − 2f (x)k ≤ ϕ(x, x) for all x ∈ A+ . It follows from (2.8) that

 x  x x

,

f (x) − 2f

≤ϕ 4 4 4 for all x ∈ A+ . Hence m

x  x  1 X j x x

l

2 ϕ j, j

2 f l − 2m f m ≤ 4 4 2 4 4

(2.8)

(2.9)

j=l+1

for all nonnegative all x ∈ A+ . It follows from (2.5) and (2.9) that  k integers
m and l with m > l and
x + the sequence 2 f 4k is Cauchy for all x ∈ A . Since B + is complete, the sequence 2k f 4xk converges. So one can define the mapping S : A+ → B + by x S(x) := lim 2k f k k→∞ 4 for all x ∈ A+ . By (2.8) and (2.9),



 1 1 1 1 1 1

S x + y + x 4 y 2 x 4 + y 4 x 2 y 4 − S(x) − S(y)

! 1 1 1 1 1 1

x  y  x + y + x4 y2 x4 + y4 x2 y4

k = lim 2 f −f k −f k

k→∞ 4k 4 4 x y ≤ lim 2k ϕ k , k = 0 k→∞ 4 4 + for all x, y ∈ A . So   1 1 1 1 1 1 S x + y + x 4 y 2 x 4 + y 4 x 2 y 4 − S(x) − S(y) = 0. Hence the mapping S : A+ → B + is a square root mapping. Moreover, letting l = 0 and passing the limit m → ∞ in (2.9), we get (2.7). So there exists a square root mapping S : A+ → B + satisfying (1.1) and (2.7). Now, let S 0 : A+ → B + be another square root mapping satisfying (1.1) and (2.7). Then we have

x  x 

kS(x) − S 0 (x)k = 2q S q − S 0 q 4 4

x

x  x   x 



≤ 2q S q − f q + 2q S 0 q − f q 4 4 4 4 2 · 2q  x x  ≤ ϕ e q, q , 2 4 4 + which tends to zero as q → ∞ for all x ∈ A . So we can conclude that S(x) = S 0 (x) for all x ∈ A+ . This proves the uniqueness of S.  Corollary 2.3. Let p > 21 and θ1 , θ2 be non-negative real numbers, and let f : A+ → B + be a mapping such that



 p p 1 1 1 1 1 1

(2.10)

f x + y + x 4 y 2 x 4 + y 4 x 2 y 4 − f (x) − f (y) ≤ θ1 (kxkp + kykp ) + θ2 · kxk 2 · kyk 2 for all x, y ∈ A+ . Then there exists a unique square root mapping S : A+ → B + satisfying (1.1) and kf (x) − S(x)k ≤

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for all x ∈ A+ . p

p

Proof. Define ϕ(x, y) = θ1 (kxkp + kykp ) + θ2 · kxk 2 · kyk 2 , and apply Theorem 2.2. Then we get the desired result.  Theorem 2.4. Let f : A+ → B + be a mapping for which there exists a function ϕ : A+ ×A+ → [0, ∞) satisfying (2.6) such that ϕ(x, e y) :=

∞ X

2−j ϕ(4j x, 4j y) < ∞

j=0 +

for all x, y ∈ A . Then there exists a unique square root mapping S : A+ → B + satisfying (1.1) and kf (x) − S(x)k ≤

1 ϕ(x, e x) 2

for all x ∈ A+ . Proof. It follows from (2.8) that



f (x) − 1 f (4x) ≤ 1 ϕ(x, x)

2 2 for all x ∈ A+ . The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 2.5. Let 0 < p < 21 and θ1 , θ2 be non-negative real numbers, and let f : A+ → B + be a mapping satisfying (2.10). Then there exists a unique square root mapping S : A+ → B + satisfying (1.1) and 2θ1 + θ2 kf (x) − S(x)k ≤ ||x||p 2 − 4p for all x ∈ A+ . p

p

Proof. Define ϕ(x, y) = θ1 (kxkp + kykp ) + θ2 · kxk 2 · kyk 2 , and apply Theorem 2.4. Then we get the desired result.  3. Stability of the 3rd root functional equation In this section, we investigate the 3rd root functional equation in C ∗ -algebras. Lemma 3.1. Let T : A+ → B + be a 3rd root mapping satisfying (1.2). Then T satisfies T (8n x) = 2n T (x) for all x ∈ A+ and all n ∈ Z. Proof. The proof is similar to the proof of Lemma 2.1.



We prove the Hyers-Ulam stability of the 3rd root functional equation in C ∗ -algebras. Theorem 3.2. Let f : A+ → B + be a mapping for which there exists a function ϕ : A+ ×A+ → [0, ∞) such that ∞ x y X ϕ(x, e y) := 2j ϕ j , j < ∞, 8 8 j=1



 1 1 1 1 1 1

(3.1)

f x + y + 3x 3 y 3 x 3 + 3y 3 x 3 y 3 − f (x) − f (y) ≤ ϕ(x, y)

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for all x, y ∈ A+ . Then there exists a unique 3rd root mapping T : A+ → A+ satisfying (1.2) and kf (x) − T (x)k ≤

1 ϕ(x, e y) 2

for all x ∈ A+ . Proof. Letting y = x in (3.1), we get kf (8x) − 2f (x)k ≤ ϕ(x, x)

(3.2)

+

for all x ∈ A . The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 3.3. Let p > 13 and θ1 , θ2 be non-negative real numbers, and let f : A+ → B + be a mapping such that



 p p 1 1 1 1 1 1

(3.3)

f x + y + 3x 3 y 3 x 3 + 3y 3 x 3 y 3 − f (x) − f (y) ≤ θ1 (kxkp + kykp ) + θ2 · kxk 2 · kyk 2 for all x, y ∈ A+ . Then there exists a unique 3rd root mapping T : A+ → B + satisfying (1.2) and kf (x) − T (x)k ≤

2θ1 + θ2 ||x||p 8p − 2

for all x ∈ A+ . p

p

Proof. Define ϕ(x, y) = θ1 (kxkp + kykp ) + θ2 · kxk 2 · kyk 2 , and apply Theorem 3.2. Then we get the desired result.  Theorem 3.4. Let f : A+ → B + be a mapping for which there exists a function ϕ : A+ ×A+ → [0, ∞) satisfying (3.1) such that ϕ(x, e y) :=

∞ X

2−j ϕ(8j x, 8j y) < ∞

j=0 +

for all x, y ∈ A . Then there exists a unique 3rd root mapping T : A+ → B + satisfying (1.2) and kf (x) − T (x)k ≤

1 ϕ(x, e x) 2

for all x ∈ A+ . Proof. It follows from (3.2) that



f (x) − 1 f (8x) ≤ 1 ϕ(x, x)

2 2 for all x ∈ A+ . The rest of the proof is similar to the proof of Theorem 2.2.



Corollary 3.5. Let 0 < p < 31 and θ1 , θ2 be non-negative real numbers, and let f : A+ → B + be a mapping satisfying (3.3). Then there exists a unique 3rd root mapping T : A+ → B + satisfying (1.2) and 2θ1 + θ2 kf (x) − T (x)k ≤ ||x||p 2 − 8p for all x ∈ A+ . p

p

Proof. Define ϕ(x, y) = θ1 (kxkp + kykp ) + θ2 · kxk 2 · kyk 2 , and apply Theorem 3.4. Then we get the desired result. 

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4. Square root and 3rd root functional equations in C ∗ -algebras We have defined a square root functional equation   1 1 1 1 1 1 S x + y + x 4 y 2 x 4 + y 4 x 2 y 4 = S(x) + S(y) and a 3rd root functional equation   1 1 1 1 1 1 T x + y + 3x 3 y 3 x 3 + 3y 3 x 3 y 3 = T (x) + T (y) for all x, y ∈ A+ . Each solution of the square root functional equation is called a square root mapping and each solution of the 3rd root functional equation is called a 3rd root mapping. It was shown that each square root mapping S : A+ → B + satisfies S(4n x) = 2n S(x) for all x ∈ A+ and all n ∈ Z and that each 3rd root mapping T : A+ → B + satisfies T (8n x) = 2n T (x) for all x ∈ A+ and all n ∈ Z. Moreover, we prove that there exists a square root mapping near a given approximate square root mapping and that there exists a 3rd root mapping near a given approximate 3rd root mapping by using the Hyer-Ulam-Rassias approach. Acknowledgments The authors would like to thank the useful comments of referees. J.S. Jang is supoorted by NRF 201807042748. Competing interests The authors declare that they have no competing interests.

References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] J. Dixmier, C ∗ -Algebras, North-Holland Publ. Com., Amsterdam, New York and Oxford, 1977. [3] P. G˘avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [4] K.R. Goodearl, Notes on Real and Complex C ∗ -Algebras, Shiva Math. Series IV , Shiva Publ. Limited, Cheshire, England, 1982. [5] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [6] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300. [7] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea Email address: [email protected] Sun Young Jang Department of Mathematics, University of Ulsan, Ulsan 44610, Korea Email address: [email protected]

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Jieun Ahn Department of Mathematics, University of Ulsan, Ulsan 44610, Republic of Korea Email address: [email protected]

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Approximation by Multivariate Sublinear and Max-product Operators, Revisited George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we study quantitatively the approximation of multivariate function by general multivariate positive sublinear operators with applications to multivariate Max-product operators. These are of Bernstein type, of Favard-Szász-Mirakjan type, of Baskakov type, of sampling type, of Lagrange interpolation type and of Hermite-Fejér interpolation type. Our results are both: under the presence of smoothness and without any smoothness assumption on the function to be approximated.

2010 AMS Mathematics Subject Classi…cation: 41A17, 41A25, 41A36, 41A63. Keywords and Phrases: multivariate positive sublinear operators, multivariate Max-product operators, multivariate modulus of continuity.

1

Background

Let Q be a compact and convex subset of Rk , k 2 N f1g and let x0 := (x01 ; :::; x0k ) 2 Q be …xed. Let f 2 C n (Q) and suppose that each nth partial derivative f = @@x f , where := ( 1 ; :::; k ), i 2 Z+ , i = 1; :::; k, and j j := Pk i=1 i = n, has relative to Q and the l1 -norm k k, a modulus of continuity ! 1 (f ; h) w, where h and w are …xed positive numbers. Here ! 1 (f ; h) :=

sup x;y2Q kx ykl h

jf (x)

f (y)j :

(1)

1

1

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The jth derivative of gz (t) = f (x0 + t (z x0 )), (z = (z1 ; :::; zk ) 2 Q) is given by 2 !j 3 k X @ gz(j) (t) = 4 (zi x0i ) f 5 (x01 + t (z1 x01 ) ; :::; x0k + t (zk x0k )) : @x i i=1

(2)

Consequently it holds n (j) X gz (0)

f (z1 ; :::; zk ) = gz (1) =

j!

j=0

+ Rn (z; 0) ;

(3)

gz(n) (0) dtn ::: dt1 :

(4)

where Rn (z; 0) :=

Z

Z

1

0

Z

t1

:::

0

tn

1

gz(n) (tn )

0

We apply Lemma 7.1.1, [1], pp. 208-209, to (f (x0 + t (z function of z, when ! 1 (f ; h) w: jf (x0 + t (z

x0 ))

f (x0 )j

t kz

w

h

x0 )) x0 k

f (x0 )) as a

;

(5)

all t 0, where d e is the ceiling function. For kz x0 k 6= 0, it follows from (2)

Z

0

1

Z

t1

:::

Z

tn

1

0

0

=

X

j j=n

since kz

0 @

jRn (z; 0)j X

j j=n

n! 1 !:::

x0 k =

n! 1 !:::

k!

Pk

i=1

n

k!

jz1

1

x01 j

::: jzk

x0k j

k

w

tn kz

h

1 x0 k A dtn :::dt1 (6)

Qk

x0i j i=1 jzi n kz x0 k

jzi

i

w

n

(kz

x0 k) = w (kz

x0 k) ;

x0i j. Above we denote (for h > 0 …xed):

(x) :=

Z

jxj

0

n 1

t (jxj t) dt; (x 2 R), h (n 1)!

(7)

equivalently n

(x) =

Z

0

jxj

Z

0

x1

:::

Z

xn

0

1

lx m n

h

dxn :::dx1 ;

(8)

see [1], p. 210-211.

2

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Therefore we have jRn (z; 0)j

w

n

(kz

x0 k) ; for all z 2 Q:

(9)

Also we have gz (0) = f (x0 ) : One obtains ([1], p. 210) 0 1 1 @X (jxj n (x) = n! j=0

n

jh)+ A ;

which is a polynomial spline function. Furthermore we get ([1], pp. 210-211)

n+1

n

(x)

(x) :=

n

1

n

(10)

n 1

jxj jxj h jxj + + (n + 1)!h 2n! 8 (n 1)!

!

;

(11)

with equality only at x = 0. Moreover, n is convex on R and strictly increasing on R+ , n 1. In case of Q := fx 2 R : kxk 1g, where k k is the l1 -norm in Rk we have 0 hence

n

(kz

kz

x0 k

x0 k)

n

kzk + kx0 k

1 + kx0 k , 8 z 2 Q;

(1 + kx0 k), and by convexity of (kz x0 k) kz x0 k

n

we get

(1 + kx0 k) ; (1 + kx0 k)

n

n

(12)

8 z 2 Q : kz x0 k 6= 0; and hence n

(kz

x0 k)

kz

n (1 + kx0 k) ; 8 z 2 Q: (1 + kx0 k)

x0 k

(13)

Let Q be a compact and convex subset of Rk , k 2 N f1g, x0 2 Q …xed, f 2 C n (Q). Then for j = 1; :::; n, we have ! k ! X Y j! (j) i (zi x0i ) gz (0) = f (x0 ) : (14) Qk i=1 i ! i=1 :=( 1 ;:::; k ); i 2Z+ ; i=1;:::;k, j j:=

If f (x0 ) = 0, for all

Pk

i=1

i =j

(j)

: j j = 1; :::; n; then gz (0) = 0, j = 1; :::; n, and by (3): f (z)

f (x0 ) = Rn (z; 0) ;

(15)

that is jf (z)

f (x0 )j

w

n

(kz

x0 k) , 8 z 2 Q;

(16)

3

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where x0 2 Q is …xed. Using (11) we derive n+1

kf (z)

f (x0 )k

kz x0 k (n + 1)!h

w

+

n

kz

n 1

x0 k kz x0 k +h 2n! 8 (n 1)!

!

; 8 z 2 Q: (17)

We have proved the following fundamental result: Theorem 1 Let (Q; k k), where k k is the l1 -norm, be a compact and convex subset of Rk , k 2 N f1g and let x0 2 Q be …xed. Let f 2 C n (Q), n 2 N, h > 0. We assume that f (x0 ) = 0, for all : j j = 1; :::; n: Then kf (z) n+1

kz x0 k (n + 1)!h

+

f (x0 )k kz

max ! 1 (f ; h) :j =nj

n

n 1

x0 k kz x0 k +h 2n! 8 (n 1)!

!

; 8 z 2 Q:

(18)

In conclusion we have Theorem 2 Let (Q; k k), where k k is the l1 -norm, be a compact and convex subset of Rk , k 2 N f1g and let x 2 Q (x = (x1 ; :::; xk )) be …xed. Let f 2 C n (Q), n 2 N, h > 0. We assume that f (x) = 0, for all : j j = 1; :::; n: Then kf (t)

f (x)k

max ! 1 (f ; h)

! n 1 n+1 n kt xk kt xk kt xk + +h (n + 1)!h 2n! 8 (n 1)! 0 Pk Pk n+1 kn kn 1 xi j i=1 jti i=1 jti + max ! 1 (f ; h) @ (n + 1)!h 2n! :j j=n hk n 2 + 8 (n 1)!

k X i=1

(19)

:j j=n

n 1

jti

xi j

!!

; 8 t 2 Q;

n

xi j

(20)

where t = (t1 ; :::; tk ) : Proof. By Theorem 1 and a convexity argument. We need De…nition 3 Let Q be a compact and convex subset of Rk , k 2 N we denote C+ (Q) = ff : Q ! R+ and continuousg :

f1g. Here

4

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Let LN : C+ (Q) ! C+ (Q), N 2 N, be a sequence of operators satisfying the following properties: (i) (positive homogeneous) LN ( f ) = LN (f ) , 8 (ii) (monotonicity) if f; g 2 C+ (Q) satisfy f

0, f 2 C+ (Q) ;

(21)

g, then

LN (f )

LN (g) , 8 N 2 N,

(22)

and (iii) (subadditivity) LN (f + g)

LN (f ) + LN (g) , 8 f; g 2 C+ (Q) :

(23)

We call LN positive sublinear operators. Remark 4 (to De…nition 3) Let f; g 2 C+ (Q). We see that f = f g + g jf gj + g. Then LN (f ) LN (jf gj) + LN (g), and LN (f ) LN (g) LN (jf gj). Similarly g = g f + f jg f j + f , hence LN (g) LN (jf gj) + LN (f ), and LN (g) LN (f ) LN (jf gj). Consequently it holds jLN (f ) (x)

LN (g) (x)j

LN (jf

gj) (x) , 8 x 2 Q:

(24)

In this article we treat LN : LN (1) = 1: We observe that (24)

jLN (f ) (x)

f (x)j = jLN (f ) (x)

LN (jf ( )

LN (f (x)) (x)j

f (x)j) (x) , 8 x 2 Q:

(25)

We give Theorem 5 Let Q be a compact and convex subset of Rk , k 2 N f1g and let x 2 Q be …xed. Let f 2 C n (Q; R+ ), n 2 N, h > 0. We assume that f (x) = 0, for all : j j = 1; :::; n: Let fLN gN 2N positive sublinear operators mapping C+ (Q) into itself, such that LN (1) = 1. Then jLN (f ) (x) kn (n + 1)!h

k X i=1

LN jti

hk n 2 + 8 (n 1)!

f (x)j

xi j

k X i=1

n+1

max ! 1 (f ; h) :j j=n

!

kn 1 + 2n!

(x)

n 1

LN jti

xi j

k X i=1

!!

(x)

LN (jti

n

!

xi j ) (x)

; 8 N 2 N:

(26)

5

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Proof. By Theorem 2, see De…nition 3, and by (25). We need The Maximum Multiplicative Principle 6 Here _ stands for maximum. Let i > 0, i = 1; :::; n; j > 0, j = 1; :::; m. Then _ni=1 _m j=1

= (_ni=1

i j

i)

_m j=1

j

:

(27)

Proof. Obvious. We make Remark 7 In [4], p. 10, the authors introduced the basic Max-product Bernstein operators WN k pN;k (x) f N (M ) BN (f ) (x) = k=0 ; N 2 N, (28) WN k=0 pN;k (x) where pN;k (x) =

N k

xk (1

N

k

x)

uous. In [4], p. 31, they proved that (M )

BN

(j

6 , 8 x 2 [0; 1] , 8 N 2 N. N +1

(29)

6 , 8 x 2 [0; 1] , 8 m; N 2 N. N +1

(30)

p

xj) (x)

; x 2 [0; 1] ; and f : [0; 1] ! R+ is contin-

And in [2] was proved that (M )

BN

(j

m

xj ) (x)

p

We will also use Corollary 8 (to Theorem 5, case of n = 1) Let Q be a compact and convex subset of Rk , k 2 N f1g and let x 2 Q. Let f 2 C 1 (Q; R+ ), h > 0. We assume that @f@x(x) = 0, for i = 1; :::; k: Let fLN gN 2N be positive sublinear operators from i C+ (Q) into C+ (Q) : LN (1) = 1, 8 N 2 N. Then jLN (f ) (x) "

k 2h

8 N 2 N.

k X i=1

LN (ti

2

xi )

f (x)j !

(x)

max ! 1

i=1;:::;k

1 + 2

k X i=1

@f ;h @xi

LN (jti

!

xi j) (x)

# h + ; 8

(31)

In this article we study quantitatively the approximation properties of multivariate Max-product operators to the unit. These are special cases of positive sublinear operators. We give also general results regarding the convergence to the unit of positive sublinear operators. Special emphasis is given in our study about approximation under di¤erentiability. Our work is motivated by [4]. 6

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2

Main Results k

From now on Q = [0; 1] , k 2 N We mention

f1g, except otherwise speci…ed.

! k De…nition 9 Let f 2 C+ [0; 1] , and N = (N1 ; :::; Nk ) 2 Nk . We de…ne the multivariate Max-product Bernstein operators as follows: (M )

B! (f ) (x) := N

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 pN1 ;i1 (x1 ) pN2 ;i2 (x2 ) :::pNk ;ik (xk ) f

ik i1 N1 ; :::; Nk

;

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 pN1 ;i1 (x1 ) pN2 ;i2 (x2 ) :::pNk ;ik (xk )

(32)

k

8 x = (x1 ; :::; xk ) 2 [0; 1] . Call Nmin := minfN1 ; :::; Nk g: (M ) k The operators B! (f ) (x) are positive sublinear and they map C+ [0; 1] N (M )

into itself, and B! (1) = 1: N See also [4], p. 123 the bivariate case. We also have (M )

B! (f ) (x) := N

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 pN1 ;i1 (x1 ) pN2 ;i2 (x2 ) :::pNk ;ik (xk ) f Qk N =1 _i =0 pN ;i (x )

ik i1 N1 ; :::; Nk

;

(33)

k

8 x 2 [0; 1] , by the maximum multiplicative principle, see (27). We make Remark 10 The coordinate Max-product Bernstein operators are de…ned as follows ( = 1; :::; k): (M )

BN

(g) (x ) :=

_N i =0 pN

;i

(x ) g

_N i =0 pN

;i

i N

(x )

;

(34)

8 N 2 N, and 8 x 2 [0; 1], 8 g 2 C+ ([0; 1]) := fg : [0; 1] ! R+ continuousg: Here we have pN

;i

(x ) =

N i

xi (1

N

x )

i

; for all

= 1; :::; k; x 2 [0; 1] : (35)

k

k

In case of f 2 C+ [0; 1] is such that f (x) := g (x ), 8 x 2 [0; 1] ; where x = (x1 ; :::; x ; :::; xk ) and g 2 C+ ([0; 1]), we get that (M )

(M )

B! (f ) (x) = BN N

(g) (x ) ;

(36)

7

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

by the maximum multiplicative principle (27) and simpli…cation of (33). Clearly it holds that (M )

k

B! (f ) (x) = f (x) , 8 x = (x1 ; :::; xk ) 2 [0; 1] : x 2 f0; 1g;

= 1; :::; k: (37)

N

We present k

k

Theorem 11 Let x 2 [0; 1] , k 2 N f1g, be …xed, and let f 2 C n [0; 1] ; R+ , n 2 N f1g. We assume that f (x) = 0, for all : j j = 1; :::; n: Then 1 !!! n+1 1 (M ) B! (f ) (x) f (x) 6 max ! 1 f ; p (38) N :j j=n Nmin + 1 "

k n+1 (n + 1)!

p

n n+1

1 Nmin + 1

kn + 2n!

p

1 Nmin + 1

kn 1 + 8 (n 1)!

p

n+2 n+1

1 Nmin + 1

#

;

! 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim B! (f ) (x) = f (x) : N

N !(1;:::;1)

Proof. By (26) we get: (36)

(M )

B! (f ) (x) "

kn (n + 1)!h

k X

(M ) BNi

i=1

+

p

hk 8 (n

6

1)!

Pk

i=1

(M )

(M )

(M ) BNi

(jti

i=1

n 1

xi j

!#

!

n

xi j ) (xi )

(39) (30)

(xi )

k n+1 kn hk n 1 + + =: ( ) : (n + 1)!h 2n! 8 (n 1)! (30)

n

xi j ) (xi )

1 Nmin +1

k X

kn 1 + 2n!

jti

i=1

BNi (jti p

(xi )

BNi

:j j=n

Next we choose h := p

k X

max ! 1 (f ; h)

Nmin + 1

Above notice

xi j

n 2

:j j=n

!

n+1

jti

max ! 1 (f ; h)

f (x)

N

1 n+1

Pk

i=1

p 6 Ni +1

, then hn =

p

p 6k , Nmin +1

1 Nmin +1

n n+1

etc.

and hn+1 =

1

: We have

Nmin +1

( )=6

max

:j j=n

!1

f ;

p

1 Nmin + 1

1 n+1

!!!

(40)

8

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"

k n+1 (n + 1)!

p

n n+1

1 Nmin + 1

kn + 2n!

p

kn 1 + 8 (n 1)!

1 Nmin + 1

p

n+2 n+1

1 Nmin + 1

#

;

proving the claim. We also give k

k

Proposition 12 Let x 2 [0; 1] , k 2 N f1g, be …xed and let f 2 C 1 [0; 1] ; R+ . @f (x) @xi

We assume that

= 0, for i = 1; :::; k. Then

@f 1 ;p 4 i=1;:::;k @xi Nmin + 1 # " 3k 1 3k 2 p p +p + ; 4 Nmin + 1 Nmin + 1 8 4 Nmin + 1

(M )

B! (f ) (x) N

f (x)

max ! 1

(41)

! 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: (M ) Also it holds ! lim B! (f ) (x) = f (x) : N

N !(1;:::;1)

Proof. By (31) we get: (36)

(M )

B! (f ) (x)

f (x)

N

"

k 2h

k X

(M ) BNi

(ti

!

2

xi )

(xi )

i=1

(next we choose h :=

p

1 2

1 Nmin +1

(30)

max ! 1

i=1;:::;k

"

3k

2

p

Nmin + 1

+ 3k

k X

1 + 2

(M ) BNi

@f ; @xi p

!

(jti

xi j) (xi )

i=1

, then h2 =

1 2

1

@f ;h @xi

max ! 1

i=1;:::;k

p

p

1 2

Nmin + 1

Nmin + 1

#

(42)

1 ) Nmin +1

1

1

h + 8

1 + 8

!!

p

(43) 1

Nmin + 1

1 2

#

;

proving the claim. We need Theorem 13 Let Q with k k the l1 -norm, be a compact and convex subset of Rk , k 2 N f1g, and f 2 C+ (Q); h > 0. We denote ! 1 (f; h) := sup jf (x) f (y)j, x;y2Q: kx yk h

the modulus of continuity of f . Let fLN gN 2N be positive sublinear operators from C+ (Q) into itself such that LN (1) = 1, 8 N 2 N. Then jLN (f ) (x)

f (x)j

! 1 (f; h) 1 +

1 LN (kt h

xk) (x)

9

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

k X

1 ! 1 (f; h) 1 + h

!!

LN (jti

xi j) (x)

i=1

;

(44)

8 N 2 N, 8 x 2 Q, where x := (x1 ; :::; xk ) ; t = (t1 ; :::; tk ) 2 Q: Proof. We have that ([1], pp. 208-209) jf (t)

f (x)j

kt

! 1 (f; h)

xk

! 1 (f; h) 1 +

h

kt

xk h

;

(45)

8 t; x 2 Q: By (25) we get: jLN (f ) (x)

f (x)j

! 1 (f; h) 1 +

LN (jf (t)

1 LN (kt h

f (x)j) (x)

(46)

xk) (x) , 8 N 2 N;

proving the claim. We give k

Theorem 14 Let f 2 C+ [0; 1] (M )

B! (f ) (x)

,k2N

(6k + 1) ! 1 f; p

f (x)

N

f1g. Then 1

;

Nmin + 1

(47)

! k 8 x 2 [0; 1] , 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: That is (M )

B! (f )

f

N

It holds that

1

(6k + 1) ! 1 f; p

1 Nmin + 1

:

(48)

(M )

lim

! N !(1;:::;1)

B! (f ) (x) = f (x), uniformly. N

Proof. We get that (use of (44)) (M ) B! N

(36)

(f ) (x)

f (x)

1 ! 1 (f; h) 1 + h

(29)

! 1 (f; h) 1 + (setting h :=

p

1 h

p

k X

(M ) BNi

i=1

(jti

!!

xi j) (xi )

6k Nmin + 1

(49)

1 ) Nmin +1

= ! 1 f; p

1 Nmin + 1

! k (6k + 1) ; 8 x 2 [0; 1] ; 8 N 2 Nk ;

proving the claim. We continue with 10

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

De…nition 15 ([4], p. 123) We de…ne the bivariate Max-product Bernstein type operators:

(M ) AN

(f ) (x; y) :=

_N i=0 8 (x; y) 2

N i

N i _N i=0 _j=0

:= f(x; y) : x

N

i

xi y j (1

j N i

i _N j=0

0, y

N

i j

N

x

y)

i j

x

N

j i N; N

f

; xi y j

(1

y)

i j

(50) 1g ; 8 N 2 N, and 8 f 2 C+ ( ) :

0, x + y

(M )

Remark 16 By [4], p. 137, Theorem 2.7.5 there, AN is a positive sublinear (M ) operator mapping C+ ( ) into itself and AN (1) = 1, furthermore it holds (M )

AN

(M )

(f )

AN

(M )

AN

(g)

(jf

gj) , 8 f; g 2 C+ ( ) , 8 N 2 N:

(M )

(M )

By [4], p. 125 we get that AN (f ) (1; 0) = f (1; 0), AN (M ) and AN (f ) (0; 0) = f (0; 0) : By [4], p. 139, we have that ((x; y) 2 ): (M )

(j

xj) (x; y) = BN

(M )

(j

yj) (x; y) = BN

AN

(51)

(f ) (0; 1) = f (0; 1),

(M )

(j

xj) (x) ;

(52)

(M )

(j

yj) (y) :

(53)

and AN

Working exactly the same way as (52), (53) are proved we also derive (m 2 N, (x; y) 2 ): (M ) (M ) m m AN (j xj ) (x; y) = BN (j xj ) (x) ; (54) and (M )

AN

(j

(M )

m

yj ) (x; y) = BN

(j

m

yj ) (y) :

(55)

We present Theorem 17 Let x := (x1 ; x2 ) 2 be …xed, and f 2 C n ( ; R+ ), n 2 N We assume that f (x) = 0, for all : j j = 1; :::; n. Then 1 !! n+1 1 (M ) AN (f ) (x1 ; x2 ) f (x1 ; x2 ) 6 max ! 1 f ; p :j j=n N +1 "

2n+1 (n + 1)!

1 p N +1

n n+1

2n 1 + n!

1 p N +1

2n 4 + (n 1)!

1 p N +1

f1g.

(56)

n+2 n+1

#

;

8 N 2 N: (M ) It holds lim AN (f ) (x1 ; x2 ) = f (x1 ; x2 ) : N !1

11

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Anastassiou 1011-1046

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Proof. By (26) we get (here x := (x1 ; x2 ) 2 (M )

AN "

2 X

2n (n + 1)!h

(f ) (x1 ; x2 )

(M ) AN

i=1

+

h2 (n 1)!

:j j=n

2 X

(M ) BN

6

p 1 N +1

( )=6 "

2n+1 (n + 1)!

1 p N +1

n n+1

(x)

:j j=n

2 X

(x)

(M ) BN

h2n 5 + (n 1)!

2 X

!

n

(jti

xi j ) (x)

(57)

(by (54), (55))

=

n+1

jti

i=1

(M ) AN

i=1

!#

n 1

2n (n + 1)!h !

1 n+1

2 X

2n 2 + n!

xi j

max ! 1 (f ; h) :j j=n p N +1

Next we choose h := We have

!

max ! 1 (f ; h)

jti

xi j ) (xi )

i=1

(30)

"

n

(jti

(M )

AN

i=1

max ! 1 (f ; h) 2n 2 n!

xi j

2 X

n 5

f (x1 ; x2 )

n+1

jti

):

xi j

(M ) BN

jti

i=1

!

(xi ) + n 1

xi j

!#

(xi )

(58)

2n 1 h2n 4 2n+1 + + =: ( ) : (n + 1)!h n! (n 1)! , then hn =

1 p N +1

max ! 1

f ;

2n 1 + n!

1 p N +1

:j j=n

p 1 N +1

n n+1

1 n+1

and hn+1 =

p 1 : N +1

!!

2n 4 + (n 1)!

(59) 1 p N +1

n+2 n+1

#

;

proving the claim. We also give Theorem 18 Let x := (x1 ; x2 ) 2 be …xed, and f 2 C 1 ( ; R+ ). We assume @f that @x (x) = 0, for i = 1; 2. Then i (M )

AN

(f ) (x1 ; x2 ) p 4

f (x1 ; x2 )

max ! 1

i=1;2

12 6 1 +p + N +1 N +1 8

p 4

@f 1 ;p @xi 4 N + 1

1 N +1

(60)

;

8 N 2 N: (M ) It holds lim AN (f ) (x1 ; x2 ) = f (x1 ; x2 ) : N !1

12

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Proof. By (31) we get (here x := (x1 ; x2 ) 2 (M )

AN "

1 h

2 X

(f ) (x1 ; x2 )

(M ) AN

i=1

(by (54), (55))

=

f (x1 ; x2 )

i=1;2

@f ;h @xi

! # 2 X 1 h (M ) 2 A (jti xi j) (x) + (61) (ti xi ) (x) + 2 i=1 N 8 " ! 2 @f 1 X (M ) 2 max ! 1 ;h B (ti xi ) (xi ) + i=1;2 @xi h i=1 N ! # 2 h 1 X (M ) B (jti xi j) (xi ) + 2 i=1 N 8 1 2

p 1 N +1

, then h2 =

(30)

max ! 1

i=1;2

12

max ! 1

!

(next we choose h :=

"

):

1 p N +1

1 2

@f ; @xi

1 2

1 p N +1

6 p N +1

+

p 1 ) N +1

!!

(62) 1 2

1 p N +1

1 + 8

#

;

proving the claim. We further obtain Theorem 19 Let f 2 C+ ( ). Then (M )

AN 8 (x1 ; x2 ) 2 That is

(f ) (x1 ; x2 )

f (x1 ; x2 )

13! 1 f; p

1 N +1

;

(63)

, 8 N 2 N. (M )

AN

(f )

f

13! 1 f; p

1;

1 N +1

;

(64)

8 N 2 N. (M ) It holds that lim AN (f ) = f; uniformly, 8 f 2 C+ ( ) : N !1

Proof. Using (44) (x := (x1 ; x2 ) 2 (M )

AN

1 ! 1 (f; h) 1 + h

) we get:

(f ) (x1 ; x2 ) 2 X

(M ) AN

(jti

i=1

f (x1 ; x2 ) !!

xi j) (x)

(by (52), (53))

=

13

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1 ! 1 (f; h) 1 + h

2 X

(M ) BN

i=1

! 1 (f; h) 1 + (setting h :=

(jti

!!

(29)

xi j) (xi )

2 6 p h N +1

(65)

p 1 ) N +1

= 13! 1 f; p

1 N +1

; 8 (x1 ; x2 ) 2

; 8 N 2 N;

proving the claim. We make Remark 20 The Max-product truncated Favard-Szász-Mirakjan operators (M ) TN

(f ) (x) =

WN

k k=0 sN;k (x) f N WN k=0 sN;k (x)

k

; x 2 [0; 1] ; N 2 N, f 2 C+ ([0; 1]) ; (66)

sN;k (x) = (Nk!x) , see also [4], p. 11. By [4], p. 178-179, we get that (M )

TN

(j

3 p ; 8 x 2 [0; 1] ; 8 N 2 N: N

(67)

3 p ; 8 x 2 [0; 1] ; 8 N; m 2 N: N

(68)

xj) (x)

And from [2] we have (M )

TN

(j

m

xj ) (x)

We make k

De…nition 21 Let f 2 C+ [0; 1]

, k 2 N

! f1g; and N = (N1 ; :::; Nk ) 2

Nk . We de…ne the multivariate Max-product truncated Favard-Szász-Mirakjan operators as follows: (M ) T! (f ) (x) := N

1 _N i1 =0

2 _N i2 =0

:::

k _N ik =0

sN1 ;i1 (x1 ) sN2 ;i2 (x2 ) :::sNk ;ik (xk ) f

ik i1 N1 ; :::; Nk

;

(69)

8 x = (x1 ; :::; xk ) 2 [0; 1] . Call Nmin := minfN1 ; :::; Nk g: (M ) k The operators T! (f ) (x) are positive sublinear mapping C+ [0; 1]

into

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 sN1 ;i1 (x1 ) sN2 ;i2 (x2 ) :::sNk ;ik (xk ) k

N

(M ) T! N

itself, and (1) = 1: We also have (M )

T! N

(f ) (x) := 14

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 sN1 ;i1 (x1 ) sN2 ;i2 (x2 ) :::sNk ;ik (xk ) f Qk N =1 _i =0 sN ;i (x )

ik i1 N1 ; :::; Nk

;

(70)

k

8 x 2 [0; 1] , by the maximum multiplicative principle, see (27). We make Remark 22 The coordinate Max-product truncated Favard-Szász-Mirakjan operators are de…ned as follows ( = 1; :::; k): (M ) TN

(g) (x ) :=

_N i =0 sN

;i

(x ) g

_N i =0 sN

;i

i N

;

(x )

(71)

8 N 2 N, and 8 x 2 [0; 1], 8 g 2 C+ ([0; 1]) : Here we have i

sN

;i

(x ) =

(N x ) ; i !

= 1; :::; k; x 2 [0; 1] :

k

(72) k

In case of f 2 C+ [0; 1] such that f (x) := g (x ), 8 x 2 [0; 1] ; where x = (x1 ; :::; x ; :::; xk ) and g 2 C+ ([0; 1]), we get that (M )

T! N

(M )

(f ) (x) = TN

(g) (x ) ;

(73)

by the maximum multiplicative principle (27) and simpli…cation of (70). We present k

k

Theorem 23 Let x 2 [0; 1] , k 2 N f1g, be …xed, and let f 2 C n [0; 1] ; R+ , n 2 N f1g. We assume that f (x) = 0, for all : j j = 1; :::; n: Then 1 !!! n+1 1 (M ) T! (f ) (x) f (x) 3 max ! 1 f ; p N :j j=n Nmin "

k n+1 (n + 1)!

1 p Nmin

n n+1

kn + 2n!

1 p Nmin

kn 1 + 8 (n 1)!

1 p Nmin

n+2 n+1

#

; (74)

! 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim T! (f ) (x) = f (x) : N !(1;:::;1)

N

Proof. By (26) we get: (M )

T! N

(73)

(f ) (x)

f (x)

max ! 1 (f ; h) :j j=n

15

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

"

kn (n + 1)!h

k X

(M ) TNi

jti

i=1

xi j

k X

hk n 2 + 8 (n 1)! p

3 Nmin

(M )

(M ) TNi

n 1

!#

!

n

(jti

xi j ) (xi )

i=1

xi j

(68)

n

TNi (jti

xi j ) (xi )

1 n+1

p 1 Nmin

Next we choose h :=

k X

kn 1 + 2n!

(75) (68)

(xi )

k n+1 kn hk n 1 + + =: ( ) : (n + 1)!h 2n! 8 (n 1)!

(M )

i=1

jti

i=1

:j j=n

Pk

(xi )

TNi

max ! 1 (f ; h)

Above notice that

!

n+1

Pk

i=1

p3 Ni

p 1 Nmin

, then hn =

p 3k , Nmin

n n+1

etc.

and hn+1 =

p 1 : Nmin

We have ( )=3

"

k n+1 (n + 1)!

max

n n+1

1 p Nmin

!1

:j j=n

kn + 2n!

f ;

1 p Nmin

1 p Nmin

1 n+1

kn 1 + 8 (n 1)!

!!! 1 p Nmin

n+2 n+1

#

; (76)

proving the claim. We also give k

k

Proposition 24 Let x 2 [0; 1] , k 2 N f1g, be …xed and let f 2 C 1 [0; 1] ; R+ . @f (x) @xi

We assume that

(M )

T! N

= 0, for i = 1; :::; k. Then

(f ) (x)

3k 2 2

p 4

f (x)

1 Nmin

+

@f 1 ;p 4 @xi Nmin

max ! 1

i=1;:::;k

3k 2

p

1 Nmin

+

1 8

p 4

1 Nmin

;

(77)

! 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: (M ) Also it holds ! lim T! (f ) (x) = f (x) : N !(1;:::;1)

N

Proof. By (31) we get: (M )

T! N

"

k 2h

k X i=1

(M ) TNi

(73)

(f ) (x)

f (x)

2

!

(ti

xi )

(xi )

max ! 1

i=1;:::;k

1 + 2

k X i=1

(M ) TNi

@f ;h @xi (jti

!

xi j) (xi )

h + 8

#

(78)

16

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

1 2

p 1 Nmin

(next we choose h :=

, then h2 =

(68)

max ! 1

i=1;:::;k

3k 2 2

p 4

1 Nmin

3k 2

+

p 1 ) Nmin

@f 1 ;p 4 @xi Nmin p

1 Nmin

+

1 8

p 4

1 Nmin

;

(79)

proving the claim. It follows k

Theorem 25 Let f 2 C+ [0; 1] (M )

T! N

(f ) (x)

,k2N

f (x)

f1g. Then

(3k + 1) ! 1 f; p

1 Nmin

! k 8 x 2 [0; 1] , 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: That is 1 (M ) T! (f ) f (3k + 1) ! 1 f; p N 1 Nmin (M ) lim T! ! N N !(1;:::;1)

It holds that

;

(80)

:

(81)

(f ) = f , uniformly.

Proof. We get that (use of (44)) (M ) T! N

(73)

(f ) (x)

f (x)

1 ! 1 (f; h) 1 + h

(67)

! 1 (f; h) 1 + (setting h :=

1 h

p

k X i=1

(M ) TNi

(jti

!!

xi j) (x)

3k Nmin

(82)

p 1 ) Nmin

= ! 1 f; p

1 Nmin

! k (3k + 1) ; 8 x 2 [0; 1] ; 8 N 2 Nk ;

proving the claim. We make Remark 26 We mention the truncated Max-product Baskakov operator (see [4], p. 11) (M ) UN

(f ) (x) =

WN

k k=0 bN;k (x) f N WN k=0 bN;k (x)

; x 2 [0; 1] ; f 2 C+ ([0; 1]) ; 8 N 2 N, (83)

17

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where N +k k

bN;k (x) =

xk

1

N +k

(1 + x)

:

(84)

From [4], pp. 217-218, we get (x 2 [0; 1]) (M )

UN

(j

p

xj) (x)

12 , N N +1

2, N 2 N:

(85)

And as in [2], we obtain (m 2 N) (M )

UN

m

xj ) (x)

(j

p

12 , N N +1 k

De…nition 27 Let f 2 C+ [0; 1]

2, N 2 N; 8 x 2 [0; 1] :

(86)

! f1g; and N = (N1 ; :::; Nk ) 2

, k 2 N

Nk . We de…ne the multivariate Max-product truncated Baskakov operators as follows: (M ) U! (f ) (x) := N

1 _N i1 =0

2 _N i2 =0

:::

k _N ik =0

bN1 ;i1 (x1 ) bN2 ;i2 (x2 ) :::bNk ;ik (xk ) f

ik i1 N1 ; :::; Nk

;

(87)

8 x = (x1 ; :::; xk ) 2 [0; 1] . Call Nmin := minfN1 ; :::; Nk g: (M ) k The operators U! (f ) (x) are positive sublinear mapping C+ [0; 1]

into

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 bN1 ;i1 (x1 ) bN2 ;i2 (x2 ) :::bNk ;ik (xk ) k

N

(M )

itself, and U! (1) = 1: N We also have (M )

U! N

(f ) (x) :=

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 bN1 ;i1 (x1 ) bN2 ;i2 (x2 ) :::bNk ;ik (xk ) f Qk N =1 _i =0 bN ;i (x )

ik i1 N1 ; :::; Nk

;

(88)

k

8 x 2 [0; 1] , by the maximum multiplicative principle, see (27). We make Remark 28 The coordinate Max-product truncated Baskakov operators are de…ned as follows ( = 1; :::; k): (M )

UN

(g) (x ) :=

_N i =0 bN

;i

(x ) g

_N i =0 bN

;i

i N

(x )

;

(89)

8 N 2 N, and 8 x 2 [0; 1], 8 g 2 C+ ([0; 1]) : Here we have bN

;i

(x ) =

N +i i

1

xi N +i

(1 + x )

;

= 1; :::; k; x 2 [0; 1] :

18

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

k

k

In case of f 2 C+ [0; 1] such that f (x) := g (x ), 8 x 2 [0; 1] ; where x = (x1 ; :::; x ; :::; xk ) and g 2 C+ ([0; 1]), we get that (M )

U! N

(M )

(f ) (x) = UN

(g) (x ) ;

(90)

by the maximum multiplicative principle (27) and simpli…cation of (89). We present k

k

Theorem 29 Let x 2 [0; 1] , k 2 N f1g, be …xed, and let f 2 C n [0; 1] ; R+ , n 2 N f1g. We assume that f (x) = 0, for all : j j = 1; :::; n: Then 1 !!! n+1 1 (M ) U! (f ) (x) f (x) 12 max ! 1 f ; p N :j j=n Nmin + 1 "

k n+1 (n + 1)!

p

n n+1

1 Nmin + 1

kn + 2n!

p

kn 1 + 8 (n 1)!

1 Nmin + 1

p

! k 8 N 2 (N f1g) , where Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim U! (f ) (x) = f (x) :

n+2 n+1

1

Nmin + 1 (91)

#

;

N

N !(1;:::;1)

Proof. By (26) we get: (M )

U! N

"

kn (n + 1)!h

k X

(M ) UNi

i=1

+

p

hk 8 (n

12 Nmin + 1

Above notice that

(90)

(f ) (x)

xi j

n 2

1)!

k X

(M )

1 Nmin +1

kn 1 + 2n! n 1

jti

i=1

UNi (jti p

(xi )

(M )

:j j=n i=1

:j j=n

UNi

max ! 1 (f ; h) Pk

max ! 1 (f ; h) !

n+1

jti

Next we choose h := p

f (x)

xi j

k X

(M ) UNi

(jti

xi j ) (xi )

i=1

!#

!

n

(92) (86)

(xi )

kn hk n 1 k n+1 + + =: ( ) : (n + 1)!h 2n! 8 (n 1)! (86)

n

xi j ) (xi )

1 n+1

Pk

i=1

, then hn =

p

p 12 Ni +1

1 Nmin +1

p 12k , Nmin +1

n n+1

etc.

and hn+1 =

1 : Nmin +1

We have ( ) = 12

max

:j j=n

!1

f ;

p

1 Nmin + 1

1 n+1

!!!

19

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"

k n+1 (n + 1)!

p

n n+1

1 Nmin + 1

kn + 2n!

p

kn 1 + 8 (n 1)!

1 Nmin + 1

p

n+2 n+1

1

Nmin + 1 (93)

#

;

proving the claim. We also give k

k

Proposition 30 Let x 2 [0; 1] , k 2 N f1g, be …xed and let f 2 C 1 [0; 1] ; R+ . @f (x) @xi

We assume that

@f 1 ;p 4 i=1;:::;k @xi Nmin + 1 " # 6k 2 6k 1 p p +p + ; 4 Nmin + 1 Nmin + 1 8 4 Nmin + 1

(M )

U! N

= 0, for i = 1; :::; k. Then

(f ) (x)

f (x)

(94)

max ! 1

! k 8 N 2 (N f1g) , where Nmin := minfN1 ; :::; Nk g: (M ) Also it holds ! lim U! (f ) (x) = f (x) : N !(1;:::;1)

N

Proof. By (31) we get: (90)

(M )

U! N

"

k 2h

k X

(M ) UNi

(f ) (x)

f (x)

2

!

(ti

xi )

(xi )

i=1

(next we choose h :=

p

1 2

1 Nmin +1

max ! 1

i=1;:::;k

"

i=1;:::;k

1 + 2

k X

(M ) UNi

!

(jti

xi j) (xi )

i=1

, then h2 =

(85)

@f ;h @xi

max ! 1

p

h + 8

#

(95)

1 ) Nmin +1

@f 1 ;p @xi 4 Nmin + 1

6k 2 6k 1 p p +p + 4 4 Nmin + 1 Nmin + 1 8 Nmin + 1

#

;

(96)

proving the claim. It follows k

Theorem 31 Let f 2 C+ [0; 1] (M )

U! N

(f ) (x)

! k 8 x 2 [0; 1] , 8 N 2 (N

f (x)

,k2N

f1g. Then

(12k + 1) ! 1 f; p

1 Nmin + 1

;

(97)

k

f1g) , where Nmin := minfN1 ; :::; Nk g: 20

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That is (M )

U! N

It holds that

(f )

f

1

(M )

lim

! N !(1;:::;1)

(12k + 1) ! 1 f; p

U! N

1

:

Nmin + 1

(98)

(f ) = f , uniformly.

Proof. We get that (use of (44)) (M ) U! N

(90)

(f ) (x)

f (x) (85)

! 1 (f; h) 1 + (setting h :=

p

k X

1 ! 1 (f; h) 1 + h 1 h

p

(M ) UNi

i=1

(jti

!!

xi j) (xi )

12k Nmin + 1

(99)

1 ) Nmin +1

= ! 1 f; p

1 Nmin + 1

! k (12k + 1) ; 8 x 2 [0; 1] ; 8 N 2 (N

k

f1g) ;

proving the claim. We make Remark 32 Here we mention the Max-product truncated sampling operators (see [4], p. 13) de…ned by WN sin(N x k ) k f N (M ) k=0 Nx k WN (f ) (x) := ; x 2 [0; ] ; (100) WN sin(N x k ) k=0

Nx k

f : [0; ] ! R+ , continuous, and WN (M ) KN

(f ) (x) :=

sin2 (N x k ) f kN (N x k )2 WN sin2 (N x k ) k=0 (N x k )2

k=0

; x 2 [0; ] ;

(101)

f : [0; ] ! R+ , continuous. By convention we talk sin(0) = 1; which implies for every x = 0 x k ) f0; 1; :::; N g that we have sin(N = 1: Nx k We de…ne the Max-product truncated combined sampling operators WN k N;k (x) f N (M ) MN (f ) (x) := k=0 ; x 2 [0; ] ; WN k=0 N;k (x) f 2 C+ ([0; ]) ; where (M )

MN

(f ) (x) :=

8 < W (M ) (f ) (x) , if N (M ) : KN (f ) (x) , if

N;k

(x) :=

N;k

(x) :=

sin(N x k ) ; Nx k sin(N x k ) Nx k

2

:

k N

, k 2

(102)

(103)

21

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Anastassiou 1011-1046

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

By [4], p. 346 and p. 352 we get (M )

MN

(j

xj) (x)

2N

,

(104)

and by [3] (m 2 N) we have (M )

MN

m

m

(j

xj ) (x)

2N

, 8 x 2 [0; ] , 8 N 2 N:

(105)

We give ! k De…nition 33 Let f 2 C+ [0; ] , k 2 N f1g; and N = (N1 ; :::; Nk ) 2 Nk . We de…ne the multivariate Max-product truncated combined sampling operators as follows: (M ) M! (f ) (x) := N

1 _N i1 =0

2 _N i2 =0

:::

k _N ik =0

N1 ;i1

(x1 )

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0

N2 ;i2 N1 ;i1

(x2 ) :::

(x1 )

Nk ;ik

N2 ;i2

(xk ) f

(x2 ) :::

i1 N1

Nk ;ik

; iN2 2 ; :::; iNk k

(xk )

;

(106) k 8 x = (x1 ; :::; xk ) 2 [0; ] . Call Nmin := minfN1 ; :::; Nk g: (M ) k The operators M! (f ) (x) are positive sublinear mapping C+ [0; ] into N

(M )

itself, and M! (1) = 1: N We also have (M )

M! N

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0

k

N1 ;i1

(x1 )

Qk

=1

(f ) (x) :=

N2 ;i2

(x2 ) :::

_N i =0

N ;i

Nk ;ik

(xk ) f

i1 N1

; iN2 2 ; :::; iNk k

;

(x ) (107)

8 x 2 [0; ] , by the maximum multiplicative principle, see (27). We make Remark 34 The coordinate Max-product truncated combined sampling operators are de…ned as follows ( = 1; :::; k): (M ) MN

(g) (x ) :=

_N i =0

N ;i

_N i =0

(x ) g N ;i

i N

(x )

;

8 N 2 N, and 8 x 2 [0; ], 8 g 2 C+ ([0; ]) : Here we have ( = 1; :::; k; x 2 [0; ]) 8 9 (M ) (M ) < sin(N x i ) ; if MN = = W ; N N x i 2 (x ) = : N ;i (M ) (M ) : sin(N x i ) ; if MN = KN : ; N x i

(108)

(109)

22

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k

k

In case of f 2 C+ [0; ] such that f (x) := g (x ), 8 x 2 [0; ] ; where x = (x1 ; :::; x ; :::; xk ) and g 2 C+ ([0; ]), we get that (M )

M! N

(M )

(f ) (x) = MN

(g) (x ) ;

(110)

by the maximum multiplicative principle (27) and simpli…cation of (107). We present k

k

Theorem 35 Let x 2 [0; ] , k 2 N f1g, be …xed, and let f 2 C n [0; ] ; R+ , n 2 N f1g. We assume that f (x) = 0, for all : j j = 1; :::; n: Then !! n 1 1 (k ) (M ) M! (f ) (x) f (x) max ! 1 f ; (111) 1 N 2 :j j=n (Nmin ) n+1 " # 2 (k ) k 1 1 + ; + n n+2 (n + 1)! (Nmin ) n+1 2n!Nmin 8 (n 1)! (Nmin ) n+1 ! 8 N = (N1 ; :::; Nk ) 2 Nk , where Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim M! (f ) (x) = f (x) : N

N !(1;:::;1)

Proof. By (26) we get: (M )

M! N

"

kn (n + 1)!h

k X

(M ) MNi

hk 8 (n

f (x)

xi j k X

n 2

1)!

:j j=n

Pk

i=1

Next we choose h := We have

(xi )

(M )

jti

MNi

i=1

max ! 1 (f ; h)

Above notice that

max ! 1 (f ; h) !

n+1

jti

i=1

+ 1 2Nmin

(110)

(f ) (x)

(M )

n 1

xi j

1 n+1

n

(105)

xi j ) (xi )

"

(M ) MNi

!#

!

n

(jti

xi j ) (xi )

i=1

(112) (105)

(xi )

1 Nmin

, then hn =

max ! 1 :j j=n

2

Pk

n

i=1 2Ni

n 1

(k ) ( )= 2

k X

kn 1 + 2n!

k n+1 n+1 kn n hk n 1 n 1 + + =: ( ) : (n + 1)!h 2n! 8 (n 1)!

MNi (jti 1 Nmin

:j j=n

n n+1

1

f ;

1

(Nmin ) n+1

(k ) 1 k + + n (n + 1)! (Nmin ) n+1 2n!Nmin 8 (n proving the claim. We also give

k n 2Nmin ,

etc.

and hn+1 =

!!

1 Nmin :

(113)

1 n+2

1)! (Nmin ) n+1

#

;

23

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k

Proposition 36 Let x 2 [0; ] , k 2 N f1g, be …xed and let f 2 C 1 ([0; ] ; R+ ). We assume that @f@x(x) = 0, for i = 1; :::; k. Then i (M )

M! N

(f ) (x) "

f (x)

@f 1 ;p @xi Nmin #

max ! 1

i=1;:::;k

2

k 1 (k ) p + + p 4Nmin 4 Nmin 8 Nmin

;

(114)

! 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: (M ) Also it holds ! lim M! (f ) (x) = f (x) : N

N !(1;:::;1)

Proof. By (31) we get: (M )

M! N

"

k 2h

k X

(M ) MNi

(110)

(f ) (x)

f (x)

2

(ti

xi )

max ! 1

i=1;:::;k

!

(xi )

i=1

1 2

1

(next we choose h :=

Nmin

, then h2 =

(105)

max ! 1

i=1;:::;k

"

1 + 2

k X

(M ) MNi

@f ;h @xi (jti

i=1

!

xi j) (xi )

# h + 8 (115)

1 Nmin )

@f 1 ;p @xi Nmin

2

k (k ) 1 p + + p 4Nmin 4 Nmin 8 Nmin

#

;

(116)

proving the claim. It follows k

Theorem 37 Let f 2 C+ [0; ] (M )

M! N

(f ) (x)

,k2N

f1g. Then

k 1 + 1 ! 1 f; 2 Nmin

f (x)

! k 8 x 2 [0; ] , 8 N 2 Nk , where Nmin := minfN1 ; :::; Nk g: That is k 1 (M ) M! (f ) f + 1 ! 1 f; N 2 Nmin 1 It holds

lim

! N !(1;:::;1)

(M )

M! N

;

:

(117)

(118)

(f ) = f , uniformly.

24

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Proof. We get that (use of (44)) (M ) M! N

(110)

(f ) (x)

f (x)

1 ! 1 (f; h) 1 + h

(104)

! 1 (f; h) 1 + (setting h :=

1 h

k X

(M ) MNi

i=1

(jti

!!

xi j) (xi )

k 2Nmin

(119)

1 Nmin )

= ! 1 f;

! k k + 1 ; 8 x 2 [0; ] ; 8 N 2 Nk ; 2

1 Nmin

proving the claim. We make Remark 38 Let f 2 C+ ([ 1; 1]). Let the Chebyshev knots of second kind k xN;k = cos N 2 [ 1; 1], k = 1; :::; N; N 2 N f1g, which are the roots N 1 of ! N (x) = sin (N 1) t sin t, x = cos t 2 [ 1; 1]. Notice that xN;1 = 1 and xN;N = 1: De…ne k 1 ( 1) ! N (x) lN;k (x) := ; (120) (1 + k;1 + k;N ) (N 1) (x xN;k ) QN N 2, k = 1; :::; N , and ! N (x) = k=1 (x xN;k ) and i;j denotes the Kronecher’s symbol, that is i;j = 1, if i = j, and i;j = 0, if i 6= j. The Max-product Lagrange interpolation operators on Chebyshev knots of second kind, plus the endpoints 1, are de…ned by ([4], p. 12) WN lN;k (x) f (xN;k ) (M ) LN (f ) (x) = k=1WN ; x 2 [ 1; 1] : (121) k=1 lN;k (x) By [4], pp. 297-298 and [3], we get that (M )

LN

(j

m

xj ) (x)

2m+1 2 ; 3 (N 1)

(122)

8 x 2 ( 1; 1) and 8 m 2 N; 8 N 2 N, N 4: (M ) We see that LN (f ) (x) 0 is well de…ned and continuous for any x 2 PN [ 1; 1]. Following [4], p. 289, because 1, 8 x 2 [ 1; 1], for k=1 lN;k (x) = WN any x there exists k 2 f1; :::; N g : lN;k (x) > 0, hence k=1 lN;k (x) > 0. We have that lN;k (xN;k ) = 1, and lN;k (xN;j ) = 0, if k 6= j. Furthermore it holds (M ) (M ) LN (f ) (xN;j ) = f (xN;j ), all j 2 f1; :::; N g ; and LN (1) = 1. (M ) By [4], pp. 289-290, LN are positive sublinear operators. We give 25

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k

De…nition 39 Let f 2 C+ [ 1; 1] k

, k 2N

! f1g; and N = (N1 ; :::; Nk ) 2

(N f1g) . We de…ne the multivariate Max-product Lagrange interpolation operators on Chebyshev knots of second kind, plus the endpoints 1, as follows: (M )

L! (f ) (x) := N

1 _N i1 =1

2 _N i2 =1

k ::: _N ik =1 lN1 ;i1 (x1 ) lN2 ;i2 (x2 ) :::lNk ;ik (xk ) f (xN1 ;i1 ; xN2 ;i2 ; :::; xNk ;ik ) ; Nk N2 1 _N i1 =1 _i2 =1 ::: _ik =1 lN1 ;i1 (x1 ) lN2 ;i2 (x2 ) :::lNk ;ik (xk )

(123) k 8 x = (x1 ; :::; xk ) 2 [ 1; 1] . Call Nmin := minfN1 ; :::; Nk g: (M ) k The operators L! (f ) (x) are positive sublinear mapping C+ [ 1; 1] into N

(M )

itself, and L! (1) = 1: N We also have (M )

L! (f ) (x) := N

1 _N i1 =1

2 _N i2 =1

:::

k _N ik =1 lN1 ;i1

(x1 ) lN2 ;i2 (x2 ) :::lNk ;ik (xk ) f (xN1 ;i1 ; xN2 ;i2 ; :::; xNk ;ik ) ; Qk N =1 _i =1 lN ;i (x ) (124) k 8 x = (x1 ; :::; x ; :::; xk ) 2 [ 1; 1] , by the maximum multiplicative principle, see (M ) (27). Notice that L! (f ) (xN1 ;i1 ; :::; xNk ;ik ) = f (xN1 ;i1 ; :::; xNk ;ik ). The last is N also true if xN1 ;i1 ; :::; xNk ;ik 2 f 1; 1g: We make Remark 40 The coordinate Max-product Lagrange interpolation operators on Chebyshev knots of second kind, plus the endpoints 1; are de…ned as follows ( = 1; :::; k): (M )

LN

(g) (x ) :=

_N i =1 lN

;i (x ) g (xN ;i N _i =1 lN ;i (x )

)

;

(125)

8 N 2 N, N 2; and 8 x 2 [ 1; 1], 8 g 2 C+ ([ 1; 1]) : Here we have ( = 1; :::; k; x 2 [ 1; 1]) i

lN

;i

(x ) =

1

( 1)

(1 +

i ;1

N

2, i = 1; :::; N and ! N

cos sin (N

N N

i 1

! N (x ) ; (126) ) (N 1) (x xN ;i ) QN (x ) = i =1 (x xN ;i ) ; where xN ;i = +

i ;N

2 [ 1; 1], i = 1; :::; N (N 1) t sin t , x = cos t . Notice that xN

2) are roots of ! N (x ) = 1, xN ;N = 1: ;1 =

k

k

In case of f 2 C+ [ 1; 1] such that f (x) := g (x ), 8 x 2 [ 1; 1] ; where x = (x1 ; :::; x ; :::; xk ) and g 2 C+ ([ 1; 1]), we get that (M )

(M )

L! (f ) (x) = LN N

(g) (x ) ;

(127)

26

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by the maximum multiplicative principle (27) and simpli…cation of (124). We present k

k

Theorem 41 Let x 2 ( 1; 1) , k 2 N f1g, be …xed, and let f 2 C n [ 1; 1] ; R+ , n 2 N f1g. We assume that f (x) = 0, for all : j j = 1; :::; n: Then n 1

(M )

L! (f ) (x) N

"

max ! 1 f ;

3

8k 2 1) n+1

:j j=n

2k

+

n

(n + 1)! (Nmin

2

(2k)

f (x)

n! (Nmin

1)

p

n+1

1 Nmin

(128)

1

1

+ 4 (n

n+2

1)! (Nmin

1) n+1

#

;

! 8 N = (N1 ; :::; Nk ) 2 Nk ; Ni 4, i = 1; :::; k; and Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim L! (f ) (x) = f (x) : N !(1;:::;1)

N

Proof. By (26) we get: (127)

(M )

L! (f ) (x)

f (x)

(M ) LNi

n+1

N

"

k X

kn (n + 1)!h

i=1

+

hk 8 (n

jti

n 2

1)!

2

3 (Nmin

1)

(M )

(M )

i=1

LNi (jti 1

Next we choose h :=

jti

i=1

:j j=n

Pk

(xi )

LNi

max ! 1 (f ; h)

Above we notice that etc.

:j j=n

!

xi j

k X

max ! 1 (f ; h)

Nmin 1

k X

kn 1 + 2n! n 1

xi j

(M ) LNi

(jti

(129) (122)

(xi )

k n+1 2n+2 k n 2n+1 hk n + + (n + 1)!h 2n! 8 (n (122)

n

xi j ) (xi ) 1 n+1

xi j ) (xi )

i=1

!#

!

n

, then hn =

Pk

1 n

2 =: ( ) : 1)!

2n+1 2 i=1 3(Ni 1) 1

Nmin 1

n n+1

2n+1 2 k 3(Nmin 1) ,

and hn+1 =

1 Nmin 1 :

We have

2

( )= "

3

max ! 1 f ; :j j=n

p

n+1

1 Nmin

1

k n+1 2n+2 1 k n 1 2n 1 1 k n 2n + + n (n + 1)! (Nmin 1) n+1 n! (Nmin 1) 4 (n 1)! (Nmin 1) n+2 n+1

(130) #

;

proving the claim. We also give

27

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k

k

Proposition 42 Let x 2 ( 1; 1) , k 2 N f1g, be …xed, and let f 2 C 1 [ 1; 1] ; R+ . @f (x) @xi

We assume that

= 0, for i = 1; :::; k. Then

(M )

L! (f ) (x)

f (x)

N

"

max ! 1

i=1;:::;k

@f 1 ;p @xi Nmin

(2=3)k 2 1 (4=3)(k )2 p + + p Nmin 1 (Nmin 1) 8 Nmin

1

(131)

1 #

;

! 8 N = (N1 ; :::; Nk ) 2 Nk ; Ni 4, i = 1; :::; k; and Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim L! (f ) (x) = f (x) : N

N !(1;:::;1)

Proof. By (31) we get: (127)

(M )

L! (f ) (x)

f (x)

N

"

k 2h

k X

(M ) LNi

(ti

!

2

xi )

(xi )

i=1

(next we choose h :=

1 2

1 Nmin 1

k X

1 + 2

(M ) LNi

(jti

i=1

, then h2 =

(122)

!

xi j) (xi )

h + 8

#

(132)

1 Nmin 1 )

@f 1 ;p @xi Nmin

max ! 1

i=1;:::;k

"

@f ;h @xi

max ! 1

i=1;:::;k

1

(4=3)(k )2 (2=3)k 2 1 p + + p Nmin 1 (Nmin 1) 8 Nmin

1

#

;

(133)

proving the claim. It follows k

k

Theorem 43 Let any x 2 [ 1; 1] , k 2 N f1g; and let f 2 C+ [ 1; 1] . Then 1 4 2k (M ) ! 1 f; ; (134) L! (f ) (x) f (x) 1+ N 3 (Nmin 1) and (M )

L! (f ) N

f

1+

1

2

4

3

k

! 1 f;

1 (Nmin

1)

;

(135)

! 8 N = (N1 ; :::; Nk ) 2 Nk ; Ni 4, i = 1; :::; k; and Nmin := minfN1 ; :::; Nk g: (M ) k We have that ! lim L! (f ) (x) = f (x), 8 x := (x1 ; :::; xk ) 2 [ 1; 1] ; uniformly.

N !(1;:::;1)

N

28

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Proof. We get that (use of (44)) (M ) L! N

(122)

(127)

(f ) (x)

f (x) k X

1 ! 1 (f; h) 1 + h

(setting h :=

i=1

1 ! 1 (f; h) 1 + h

22 3 (Ni

!!

2

1)

k X

(M ) LNi

!!

(jti

xi j) (x)

i=1

! 1 (f; h) 1 +

1 h

4 2k 3 (Nmin 1) (137)

1 Nmin 1 )

= ! 1 f;

1 (Nmin

1+

1)

4

2

3

k

k

; 8 x 2 ( 1; 1) ;

proving the claim. We make k)+1) Remark 44 The Chebyshev knots of …rst kind xN;k := cos (2(N 2 2(N +1) ( 1; 1), k 2 f0; 1; :::; N g; 1 < xN;0 < xN;1 < ::: < xN;N < 1, are the roots of the …rst kind Chebyshev polynomial TN +1 (x) := cos ((N + 1) arccos x), x 2 [ 1; 1] : De…ne (x 2 [ 1; 1])

hN;k (x) := (1

x xN;k )

TN +1 (x) (N + 1) (x xN;k )

2

;

(138)

the fundamental interpolation polynomials. The Max-product interpolation Hermite-Fejér operators on Chebyshev knots of the …rst kind (seep. 12 of [4]) are de…ned by WN hN;k (x) f (xN;k ) (M ) H2N +1 (f ) (x) = k=0WN ; 8 N 2 N, (139) k=0 hN;k (x) for f 2 C+ ([ 1; 1]), 8 x 2 [ 1; 1] : By [4], p. 287, we have (M )

H2N +1 (j

xj) (x)

2 ; 8 x 2 [ 1; 1] , 8 N 2 N: N +1

(140)

2m ; 8 x 2 [ 1; 1] , 8 m; N 2 N: N +1

(141)

And by [3], we get that (M )

H2N +1 (j (M )

m

xj ) (x)

(M )

Notice H2N +1 (1) = 1, and H2N +1 maps C+ ([ 1; 1]) into itself, and it is WN a positive sublinear operator. Furthermore it holds k=0 hN;k (x) > 0, 8 x 2 [ 1; 1]. We also have hN;k (xN;k ) = 1, and hN;k (xN;j ) = 0, if k 6= j, and (M ) H2N +1 (f ) (xN;j ) = f (xN;j ), for all j 2 f0; 1; :::; N g, see [4], p. 282. 29

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We need ! k De…nition 45 Let f 2 C+ [ 1; 1] , k 2 N f1g; and N = (N1 ; :::; Nk ) 2 Nk . We de…ne the multivariate Max-product interpolation Hermite-Fejér operators on Chebyshev knots of the …rst kind, as follows: H

(M ) ! 2 N +1

(f ) (x) :=

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 hN1 ;i1 (x1 ) hN2 ;i2 (x2 ) :::hNk ;ik (xk ) f (xN1 ;i1 ; xN2 ;i2 ; :::; xNk ;ik ) Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 hN1 ;i1 (x1 ) hN2 ;i2 (x2 ) :::hNk ;ik (xk )

;

(142)

k

8 x = (x1 ; :::; xk ) 2 [ 1; 1] . Call Nmin := minfN1 ; :::; Nk g: (M ) k The operators H ! (f ) (x) are positive sublinear mapping C+ [ 1; 1] 2 N +1 (M )

into itself, and H ! (1) = 1: 2 N +1 We also have H

(M ) ! 2 N +1

(f ) (x) :=

Nk N2 1 _N i1 =0 _i2 =0 ::: _ik =0 hN1 ;i1 (x1 ) hN2 ;i2 (x2 ) :::hNk ;ik (xk ) f (xN1 ;i1 ; xN2 ;i2 ; :::; xNk ;ik ) ; Qk N =1 _i =0 hN ;i (x ) (143) k 8 x = (x1 ; :::; x ; :::; xk ) 2 [ 1; 1] , by the maximum multiplicative principle, see (M ) (27). Notice that H ! (f ) (xN1 ;i1 ; :::; xNk ;ik ) = f (xN1 ;i1 ; :::; xNk ;ik ). 2 N +1

We make Remark 46 The coordinate Max-product interpolation Hermite-Fejér operators on Chebyshev knots of the …rst kind, are de…ned as follows ( = 1; :::; k): (M ) +1

H2N

(g) (x ) :=

_N i =0 hN

;i (x ) g (xN ;i N _i =0 hN ;i (x )

)

8 N 2 N, and 8 x 2 [ 1; 1], 8 g 2 C+ ([ 1; 1]) : Here we have ( = 1; :::; k; x 2 [ 1; 1]) hN

;i

(x ) = (1

x

xN

;i

)

TN +1 (x ) (N + 1) (x xN

;

(144)

2 ;i

)

;

(145)

i )+1) where the Chebyshev knots xN ;i = cos (2(N 2 ( 1; 1), i 2 f0; 1; :::; N g, 2(N +1) 1 < xN ;0 < xN ;1 < ::: < xN ;N < 1 are the roots of the …rst kind Chebyshev polynomial TN +1 (x ) = cos ((N + 1) arccos x ), x 2 [ 1; 1] : k

In case of f 2 C+ [ 1; 1] g 2 C+ ([ 1; 1]), we get that H

(M ) ! 2 N +1

k

such that f (x) := g (x ), 8 x 2 [ 1; 1] and (M ) +1

(f ) (x) = H2N

(g) (x ) ;

(146)

by the maximum multiplicative principle (27) and simpli…cation of (143). 30

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We present k

k

Theorem 47 Let x 2 [ 1; 1] , k 2 N f1g, be …xed, and let f 2 C n [ 1; 1] ; R+ , n 2 N f1g. We assume that f (x) = 0, for all : j j = 1; :::; n: Then H "

(M ) ! 2 N +1

(f ) (x)

2n

f (x)

8k 2

+

n

(n + 1)! (Nmin + 1) n+1

2 n 1

k

p

max ! 1 f ;

n+1

:j j=n

2k n! (Nmin + 1)

1 Nmin + 1

1

+ 4 (n

n+2

1)! (Nmin + 1) n+1

#(147) ;

! 8 N = (N1 ; :::; Nk ) 2 Nk , and Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim H ! (f ) (x) = f (x) : 2 N +1

N !(1;:::;1)

Proof. By (26) we get: H "

kn (n + 1)!h

k X i=1

(M ) ! 2 N +1

(146)

(f ) (x)

(M )

n+1

H2Ni +1 jti

xi j

k X

hk n 2 + 8 (n 1)!

f (x)

i=1

(xi )

(M )

1 n+1

1

:j j=n

+

kn 1 (M ) H2Ni +1 (jti 2n! n 1

xi j

!#

n

xi j ) (xi ) (148)

(141)

(xi )

k n+1 2n+1 k n 2n hk n 1 2n 1 + + =: ( ) : (n + 1)!h 2n! 8 (n 1)!

:j j=n

Next we choose h :=

!

H2Ni +1 jti

max ! 1 (f ; h)

Nmin + 1

max ! 1 (f ; h)

Nmin +1

, then hn =

n n+1

1 Nmin +1

and hn+1 =

1 Nmin +1 :

We have ( )= "

max ! 1 f ; :j j=n

n+1

(2k)

n

(n + 1)! (Nmin + 1) n+1

p

n+1

1 Nmin + 1

2n 1 k n + + n! (Nmin + 1) 4 (n

2n

(149) 2 n 1

k

n+2

1)! (Nmin + 1) n+1

#

;

proving the claim. We also give k

k

Proposition 48 Let x 2 [ 1; 1] , k 2 N f1g, be …xed, and let f 2 C 1 [ 1; 1] ; R+ . We assume that H

(M ) ! 2 N +1

@f (x) @xi

(f ) (x)

= 0, for i = 1; :::; k. Then f (x)

max ! 1

i=1;:::;k

@f 1 ;p @xi Nmin + 1

(150)

31

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"

k 2k 2 1 p + + p Nmin + 1 (Nmin + 1) 8 Nmin + 1

#

;

! 8 N = (N1 ; :::; Nk ) 2 Nk ; Nmin := minfN1 ; :::; Nk g: (M ) We have that ! lim H ! (f ) (x) = f (x) : 2 N +1

N !(1;:::;1)

Proof. By (31) we get H "

k 2h

k X

(M ) ! 2 N +1

(M ) H2Ni +1

(146)

(f ) (x)

f (x) !

2

(ti

xi )

(xi )

i=1

(next we choose h :=

p

1 , Nmin +1

k X

1 + 2

then h2 =

(141)

(M ) H2Ni +1

!

(jti

xi j) (xi )

i=1

# h + 8 (151)

1 Nmin +1 )

@f 1 ;p @xi Nmin + 1

max ! 1

i=1;:::;k

"

@f ;h @xi

max ! 1

i=1;:::;k

2k 2 k 1 p + + p Nmin + 1 (Nmin + 1) 8 Nmin + 1

#

;

(152)

proving the claim. It follows k

Theorem 49 Let f 2 C+ [ 1; 1] H

(M ) ! 2 N +1

(f ) (x)

,k2N

f (x)

f1g. Then

(2k + 1) ! 1 f;

1

;

Nmin + 1

(153)

! k 8 x 2 [ 1; 1] , and 8 N = (N1 ; :::; Nk ) 2 Nk , where Nmin := minfN1 ; :::; Nk g: That is H

(M ) ! 2 N +1

(f )

f

1

(2k + 1) ! 1 f;

1 Nmin + 1

;

(154)

We get that lim

! N !(1;:::;1)

H

(M ) ! 2 N +1

(f ) = f;

(155)

uniformly.

32

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Proof. We get that (use of (44)) (M ) H! 2 N +1

(146)

(f ) (x)

f (x) (140)

! 1 (f; h) 1 + (setting h :=

k X

1 ! 1 (f; h) 1 + h k h

(M ) H2Ni +1

(jti

i=1

!!

xi j) (xi )

2 (Nmin + 1)

(156)

1 Nmin +1 )

= ! 1 f;

1 Nmin + 1

k

(1 + 2k ) ; 8 x 2 [ 1; 1] ;

proving the claim. We make Remark 50 Let

(M ) ! N

denote any of the Max-product multivariate operators (M )

(M )

(M )

(M )

(M )

(M )

studied in this article: B! ; TN ; U! ; T! ; M! ; L! N N N N N observe that an important contraction property holds: (M ) ! N

(f )

and H

(M ) . ! 2 N +1

kf k1 ,

1

We

(157)

and (M ) ! N

(M ) ! N

(f )

(M ) ! N

1

i.e.

(f ) 1

and in general holds n

(f )

kf k1 ,

(158)

kf k1 ,

(159)

n 1

(M ) ! N

1

1

2

(M ) ! N

(M ) ! N

(f )

(f )

::: 1

kf k1 , 8 n 2 N.

(160)

We need the following Holder’s type inequality: Theorem 51 Let Q, with the l1 -norm k k, be a compact and convex subset of Rk , k 2 N f1g and L : C+ (Q) ! C+ (Q), be a positive sublinear operator and p f; g 2 C+ (Q), furthermore let p; q > 1 : p1 + 1q = 1. Assume that L ((f ( )) ) (s ) ; q L ((g ( )) ) (s ) > 0 for some s 2 Q. Then L (f ( ) g ( )) (s ) Proof. Let a; b

p

1

q

1

(L ((f ( )) ) (s )) p (L ((g ( )) ) (s )) q :

0, p; q > 1 :

1 p

ab

+

1 q

(161)

= 1. The Young’s inequality says

bq ap + : p q

(162)

33

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Then

f (s)

g (s)

p

(L ((f ( )) ) (s ))

1 p

1

q

(L ((g ( )) ) (s )) q

p

q

(f (s)) (g (s)) + ; 8 s 2 Q: p q p (L ((f ( )) ) (s )) q (L ((g ( )) ) (s )) Hence it holds

L (f ( ) g ( )) (s ) 1

p

(164)

1

q

(163)

(L ((f ( )) ) (s )) p (L ((g ( )) ) (s )) q p

q

(L ((f ( )) )) (s ) (L ((g ( )) )) (s ) 1 1 + = + = 1; for s 2 Q; p q p q p (L ((f ( )) ) (s )) q (L ((g ( )) ) (s )) proving the claim. By (161), under the assumption LN k we obtain n LN (k xk ) (x) LN k

n+1

(x) > 0, and LN (1) = 1,

xk

n+1

xk

(x)

n n+1

;

(165)

in case of n = 1 we derive LN (k

xk) (x)

r

LN k

2

xk

(x) :

(166)

We give Theorem 52 Let Q with k k the l1 -norm, be a compact and convex subset of Rk , k 2 N f1g, and f 2 C+ (Q). Let fLN gN 2N be positive sublinear operators from C+ (Q) into itself, such that LN (1) = 1, 8 N 2 N. We assume further that LN (kt xk) (x) > 0, 8 N 2 N. Then jLN (f ) (x)

f (x)j

2! 1 (f; LN (kt

xk) (x)) ;

(167)

8 N 2 N, x = (x1 ; :::; xk ) 2 Q; t = (t1 ; :::; tk ) 2 Q, where ! 1 (f; h) :=

sup x;y2Q: kx yk h

If LN (kt

jf (x)

f (y)j :

(168)

xk) (x) ! 0, then LN (f ) (x) ! f (x), as N ! +1:

Proof. By Theorem 13. We need Theorem 53 Let (Q; k k) ; where k k is the l1 -norm, be a compact and convex subset of Rk , k 2 N f1g, and let x 2 Q (x = (x1 ; :::; xk )) be …xed. Let f 2 C n (Q), n 2 N, h > 0. We assume that f (x) = 0, for all : j j = 1; :::; n:

34

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Let fLN gN 2N be positive sublinear operators from C+ (Q) into C+ (Q), such that LN (1) = 1, 8 N 2 N. Then jLN (f ) (x) 2 4

LN k

n+1

xk

(x) +

(n + 1)!h

f (x)j

LN (k

max ! 1 (f ; h) :j j=n

3

n

xk ) (x) h + LN k 2n! 8 (n 1)!

n 1

(x)5 ;

xk

(169)

8 N 2 N. Proof. By (19) and (25). It follows n+1

Theorem 54 All as in Theorem 53. Additionally assume that LN k > 0, 8 N 2 N. Then jLN (f ) (x) max ! 1 f ; :j j=n

1 2n!

f (x)j

1 LN k (n + 1)

n+1

xk

(x)

n 4 (n + 1)

3+

1 n+1

(x)

xk

n+1

LN k

xk

(x)

n n+1

;

(170)

8 N 2 N, x = (x1 ; :::; xk ) 2 Q, ! 1 as in (168) for f : n+1 If LN k xk (x) ! 0, then LN (f ) (x) ! f (x), as N ! +1: Proof. By Theorem 51 notice also that n 1

LN k

xk

We choose h :=

(x)

n+1

LN k

1 LN k (n + 1)

xk

n+1

xk

(x)

(x)

1 n+1

n 1 n+1

:

> 0:

(171)

(172)

That is n+1

(h (n + 1))

= LN k

n+1

xk

(x) :

(173)

We apply (169) to have (see also (165) and (171)). jLN (f ) (x) 2

n+1

6 LN k xk 4 (n + 1)!h

f (x)j

(x) +

max ! 1 (f ; h) :j j=n

LN k

n+1

xk

2n!

(x)

n n+1

+

(174)

35

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h 8 (n

1)!

LN

n+1

k

xk

(x)

n 1 n+1

= 1 n+1

1 n+1 LN k xk max ! 1 f ; (x) (n + 1) :j j=n " n n n+1 hn (n + 1) hn (n + 1) hn (n + 1) + + (n + 1)! 2n! 8 (n 1)!

1

max ! 1 f ; :j j=n

LN k

1 LN k (n + 1) n+1

xk

(x)

n n+1

=

1 n+1

1 n+1 LN k xk (x) (n + 1) :j j=n # " n n 1 n+1 (n + 1) (n + 1) 1 (n + 1) LN k + + (n + 1)! 2n! 8 (n 1)! (n + 1)n max ! 1 f ;

n 3 + 2n! 8 (n + 1)!

#

n+1

xk n+1

xk

(x)

(x)

n n+1

=

1 n+1

;

(175)

proving the claim. Final application for n = 1 follows: Corollary 55 Let (Q; k k) ; where k k is the l1 -norm, be a compact and convex subset of Rk , k 2 N f1g, and let x 2 Q (x = (x1 ; :::; xk )) be …xed. Let @f (x) = 0, i = 1; :::; k: Let fLN gN 2N be positive f 2 C 1 (Q). We assume that @x i sublinear operators from C+ (Q) into C+ (Q), such that LN (1) = 1, 8 N 2 N. 2 Assume that LN k xk (x) > 0, 8 N 2 N. Then jLN (f ) (x)

f (x)j

25 16

i=1;:::;k

LN k 8 N 2 N. If LN k

xk

2

@f 1 ; LN k @xi 2

max ! 1 xk

2

(x)

1 2

2

xk

(x)

;

1 2

(176)

(x) ! 0, then LN (f ) (x) ! f (x), as N ! +1:

References [1] G. Anastassiou, Moments in probability and approximation theory, Pitman Research Notes in Mathematics Series, Longman Group UK, New York, NY, 1993. [2] G. Anastassiou, Approximation by Sublinear Operators, submitted, 2017. [3] G. Anastassiou, Approximation by Max-Product Operators, submitted, 2017. [4] B. Bede, L. Coroianu, S. Gal, Approximation by Max-Product type Operators, Springer, Heidelberg, New York, 2016.

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NEW DYNAMIC INEQUALITIES ON TIME SCALES BY USING THE SNEAK-OUT PRINCIPLE S. H. SAKER1 , M. M. OSMAN1 AND I. ABOHELA2 Abstract. In this paper, we extend and improve some dynamic inequalities by using the sneak-out principle with di¤erent exponents on time scales. The main results can be used to formulate the corresponding discrete inequalities of Bennett and G-Erdmann type. 2010 Mathematics Subject Classi…cation:34A40, 34N05, 26D10, 26D15, 39A13. Key words and phrases. Hardy’s inequality, sneak-out principle, dynamic inequlities, time scales.

1. Introduction In 1967 Littlewood [9] formulated some problems concerning elementary inequalities for in…nite series in connection with some work on general theory of orthogonal series. One of the simplest (non-trivial) examples is the following inequality ! 1 1 n X X X 3 2 K a4n A2n , (1.1) an ak Ak n=1

n=1 P An = nk=1 ak .

k=1

One of such problems where an is a non-negative sequence and that has been proposed by Littlewood is to seek to know whether a constant K exists such that the inequality (1.1) holds. In other words, is it possible to get the term Ak out from the inner sum in (1.1) and if this happened what is the smallest value of K which preserves on the direction of the inequality? Bennett [4] proved this for the special case when the sequence an is decreasing, and he showed that the inequality (1.1) holds with K = 2: His proof based on the fact that an nAn (noting that an is decreasing) and the application of Cauchy’s inequality and the classical discrete Hardy’s inequality. The generalization of the Littlewood inequality (1.1) which has not been considered before is given by (1.2)

1 P

n=1

ap(p n

1)+1

Apk

2

n P

k=1

apk Ak

K

1 P

n=1

[apn An ]p ;

p > 1;

where K is a positive constant. Motivated by the work of Littlewood [9] Bennett and G-Erdmann [5] considered the inequality !p !p 1 1 1 1 X X X X (1.3) an Ak g k K( ; p) an Anp gk ; n=1

n=1

k=n

k=n

1

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S. H. SAKER 1 , M. M. OSMAN 1 AND I. ABOHELA 2

2

and determined the value of K for di¤erent values of p and : In particular, Bennett and G-Erdmann [5, Theorem 8] proved that if 1 and p 1; then ! !p p 1 1 1 1 X X X X p p an An (1.4) an Ak gk (1 + p) gk ; n=1

n=1

k=n

k=n

Pn

where gn is a non-negative sequence and An = k=1 ak ; for any n 2 N. In [5, Theorem 9] the authors proved that if p 1 and 0 1; then ! !p p 1 1 1 1 X X X X p p a n An (1.5) an Ak gk (1 + p) gk : n=1

n=1

k=n

Also in [5, Theorem 10] they proved that if p !p 1 1 X X 1+ p (1.6) an Ak g k 1+p+ p n=1

k=n

k=n

1 and 1 pX

n=1

1=p
1. We assume that the reader has a good background in time scale calculus. For dynamic inequalities on time scales, we refer the reader to the books [2, 3] and the papers [1, 7, 10, 11, 12, 13]. For instance, we recall some related results. Saker, O’Regan and Agarwal [13] proved a new inequality of Hardy type of the form Z 1 pZ 1 p ( (t) a) (p 1) p (A (t))p t g (t) t; p; > 1; (1.7) ( (t) a) 1 (t a)( 1)p a a Rt where A(t) := a g(s) s; for t 2 [a; 1)T and employed it in the proof of the extension of (1.2) on time scales. In particular they proved that if p; > 1 and g is a nonnegative rd-continuous and decreasing function, then (1.8) ! Z 1 Z (t) Z 1 (a(t))p(p 1)+1 p p p a (s)A (s) s t [ap (t)A (t)]p t; 2 p (p 1) (A (t)) a a a Rt where A(t) = a a(s) s; for t 2 [a; 1)T : Bohner and Saker in [7] employed the Minkowski inequality [6, Theorem 6.16] on time scales (1.9) 1 1 Z b Z b Z b 1=p p p p p p jh(t)j ju(t) + (t)j t jh(t)j ju(t)j t + jh(t)j j (t)j t ; a

a

a

where a; b 2 T; u; 2 Crd ([a; b]T , R); p > 1 and established the time scale versions of the inequalities (1.4), (1.5) and (1.6). In more precisely, they proved

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SOME NEW DYNAMIC INEQUALITIES ON TIME SCALES

that if a(t); g(t) are nonnegative rd-continuous functions on [t0 ; 1)T , then for 1 and p 1 Z 1 Z 1 Z 1 p g(s) s a(t) (A (t)) p t; (1.10) a(t) p (t) t (1 + p)p t

t0

t0

where (t) =

Z

1

(A (s)) g(s) s and A(t) =

(1.11)

1; p Z

1

a(t)

t

a(s) s;

t0

t

and if 0

Z

1; then p

(t) t

t0

(1 + p)

p

Z

1

a(t) (A (t))

p

Z

1

p

t:

g(s) s

t

t0

Also in [7] they proved that if 1=p < 0 and p 1; then (1.12) Z 1 Z 1 pZ 1 1+ p p p a(t) (t) t a(t) (A (t)) g(s) s 1+p+ p t0 t0 t

p

t:

Our aim in this paper is to apply the sneak-out principle which is given in the inequalities (1.10) and (1.11) to prove some new inequalities with di¤erent exponents for the given values of . Also we prove a new dynamic inequality which as special case improves the inequality (1.12).

2. Main Results Before we prove our main results, we brie‡y introduce some basic de…nitions and results concerning the delta calculus on time scales that will be used in the sequel; for more details we refer the reader to the book [6]. A time scale T is an arbitrary nonempty closed subset of the real numbers R. We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R: The forward jump operator and the backward jump operator are de…ned by (t) := inffs 2 T : s > tg. A function f : T ! R is said to be right–dense continuous (rd–continuous) provided f is continuous at right–dense points and at left–dense points in T; left hand limits exist and are …nite. The set of all such rd–continuous functions is denoted by Crd (T). The graininess function for a time scale T is de…ned by (t) := (t) t, and for any function f : T ! R the notation f (t) denotes f ( (t)): We de…ne the time scale interval [a; b]T by [a; b]T := [a; b] \ T: Recall the following product and quotient rules for the derivative of the product f g and the quotient f =g (where gg 6= 0, here g =g ) of two di¤erentiable function f and g (2.1)

(f g) = f g + f g

= f g + f g ; and

1049

f g

=

f g fg : gg

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S. H. SAKER 1 , M. M. OSMAN 1 AND I. ABOHELA 2

4

The chain rule formula on time scales [6] is given by (here x : T ! (0; 1) is assumed to be di¤erentiable) (2.2)

(x (t)) =

Z1

[hx + (1

1

h)x]

dhx (t);

2 R:

0

In this paper we will use the (delta) integral which we can de…ne as follows. If Rt G (t) = g(t), then the Cauchy (delta) integral of g is de…ned by a g(s) s := G(t) G(a): The integration by parts formula on time scales reads Z b Z b b u (t) (t) t: u(t) (t) t = [u(t) (t)]a (2.3) a

a

Hölder’s inequality [6, Theorem 6.13] states that any two rd-continuous functions u; : T ! R satisfy Z

(2.4)

a

Z

b

ju(t) (t)j t

b

ju(t)j

a

Z

1 q

q

t

a

b

1 p

p

j (t)j

t

;

where p > 1; p1 + 1q = 1 and a; b 2 T: Throughout this paper, we will assume that the functions in the statements of the theorems are nonnegative and rd-continuous functions and the integrals considered are assumed to exist. The following dynamic inequality of Copson’s type on time scales [3], will be used later to prove the main results. Theorem is rd-continuous function and de…ne R t 2.1. Assume that a : T ! R + A(t) = t0 a(s) s; t 2 T: Let ' : T ! R and de…ne (2.5)

(t) :=

Z

1

a(s)'(s) s;

t 2 T:

t

If k > 1 and 0 (2.6)

Z

1

t0

c < 1; then

a(t) (A (t))c

(t)

k

t

k

k 1

c

Z

1

a(t) (A (t))k

c

'k (t)

t:

t0

Our main results are given in the following. For simplicity, we de…ne Z 1 Z 1 (2.7) (t) := g(s) s; and (t) := (A (s)) g(s) s; t 2 T: t

t

Theorem 2.2. Let t0 2 T; (r q)=(p q) > 1: Then (2.8)

Z

1

a(t)

p

(t) t

1, p

K1 ( ; p; q; r)

t0

1 and q; r > 1 such that r > q and Z

1

((A (t))

2r q

(t))

p 2r q

t

;

t0

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SOME NEW DYNAMIC INEQUALITIES ON TIME SCALES

where K1 ( ; p; q; r) : =

"

(1 + r)r(p

q)

#

1 r q

(1 + q)q(p r) Z 1 2r q a r q (t) t

p q 2r q

t0

Proof. We …rst observe that Z 1 Z 1 p p ar a(t) (t) t =

q q

1

a

2r q 2(r q)

2(r p) 2r q

(t)

t

:

t0

r(p q) r q

(t)

Z

(t)

r p q

ar

q(r p) r q

(t)

(t)

t:

t0

t0

Applying Hölder’s inequality (2.4) with indices (r q)= (p q) and (r q)=(r p); we obtain r p p q Z 1 Z 1 Z 1 r q r q q p r a(t) (t) t : a(t) (t) t a(t) (t) t t0

t0

t0

By using (1.10) to the two integrals on the right-hand side with p = r and also with p = q, we get that p q Z 1 Z 1 Z 1 r r q r(p q) r p a(t) (t) t (1 + r) r q a(t) (A (t)) g(s) s t t0

t0

(1 + q)

q(r p) r q

t

Z

1

a(t) (A (t))

q

t0

Z

1

r p r q

q

g(s) s

t

:

t

Applying Hölder’s inequality (2.4) with indices (2r q)=r and (2r the integral Z 1 a(t) (A (t)) r ( (t))r t;

q)=(r

q) to

t0

also applying it again on the integral Z 1 a(t) (A (t))

q

( (t))q

t;

t0

with indices (2r q)=q and (2r q)=2(r q) and combining the result, we get that p Z 1 Z 1 2r q 2r q p a(t) ( (t)) t K1 ( ; p; q; r) ((A (t)) (t)) t ; t0

t0

which is the desired inequality (2.8). The proof is complete.

Proceeding as in the proof of Theorem 2.2 and using inequality (1.11) instead of (1.10), we can obtain the following result. Theorem 2.3. Let t0 2 T; 0 1, p 1 and q; r > 1 such that r > q and (r q)=(p q) > 1: Then p Z 1 Z 1 2r q 2r q p (2.9) a(t) (t) t K2 (p; q; r) ((A (t)) (t)) t ; t0

t0

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S. H. SAKER 1 , M. M. OSMAN 1 AND I. ABOHELA 2

6

where K2 (p; q; r) : =

"

(1 + r)r(p

q)

#

1 r q

(1 + q)q(p r) Z 1 2r q a r q (t)

Z

p q 2r q

t

t0

1

2r q 2(r q)

a

2(r p) 2r q

(t)

t

:

t0

The next result follows from Theorem 2.2 by choosing r = p and q = p Corollary 2.1. Let p 1 and 1: Then Z 1 Z 1 (2.10) a(t) p (t) t K1 ( ; p) ((A (t)) t0

(t))

p+1

1.

p p+1

;

t

t0

where

Z

p

K1 ( ; p) = (1 + p)

1

a

p+1

1 p+1

(t)

t

:

t0

Remark 2.1. In Theorem 2.2 when T = R; we have that Z 1 Z t Z (t) = A (s)g(s)ds; A(t) = a(s)ds and (t) = t

t0

1

g(s)ds;

t 2 R;

t

and then from (2.8) we obtain the following new integral inequality Z 1 Z 1 (2.11) a(t) p (t)dt K1 ( ; p; q; r) A (2r q) (t) ( (t))2r q dt t0

p 2r q

;

t0

where K1 ( ; p; q; r) : =

"

(1 + r)r(p

q)

#

1 r q

(1 + q)q(p r) Z 1 2r q a r q (t) dt

p q 2r q

t0

Z

1

a

2r q 2(r q)

2(r p) 2r q

(t) dt

:

t0

Remark 2.2. In Theorem 2.2 when T = N and n0 = 1; we have that (n) =

1 X

A (k)g(k); A(n) =

k=n

n X k=1

a(k); n 2 N;

and then from (2.8), we get the following discrete inequality of Bennett and GErdmann [5] type 0 !2r q 1 2rp q 1 1 1 X X X A (2.12) a(n) p (n) K1 ( ; p; q; r) @ A (2r q) (n) g(k) ; n=1

n=1

1052

k=n

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SOME NEW DYNAMIC INEQUALITIES ON TIME SCALES

where K1 ( ; p; q; r) : =

"

(1 + r)r(p

q)

(1 + q)q(p

r)

1 X

a

2r q r q

#

1 r q

!p

q 2r q

(n)

n=1

1 X

a

2r q 2(r q)

! 2(r

p) 2r q

(n)

n=1

:

Remark 2.3. Setting r = p and q = p 1 in (2.12) yields the following inequality p 0 !p+1 1 p+1 1 1 1 X X X A ; (2.13) a(n) p (n) K1 ( ; p) @ A (p+1) (n) g(k) n=1

n=1

k=n

where

K1 ( ; p) = (1 + p)p

1 X

ap+1 (n)

n=1

!

1 p+1

:

An improvement of the dynamic inequality (1.12) is obtained in the following Theorem. Theorem 2.4. Let t0 2 T; 1=p < 0; p and (r q)=(p q) > 1: Then Z 1 (2.14) a(t) (A (t)) p ( (t))p t t0

K3 ( ; p; q; r) Z

Z

1

a(t) (A (t))

1 and q; r > 1 such that r > q

(p r)

r

( (t))

p q r q

t

t0

1

a(t) (A (t))

(p q)

r p r q

q

( (t))

t

;

t0

where

K3 ( ; p; q; r) :=

1+r+ p 1+ p

r(p q) r q

1+q+ p 1+ p

q(r p) r q

:

Proof. In this proof for brivity, we set b(t) := (A (t)) g(t): Then the left hand side of (2.14) can be written in the form (2.15) Z 1 Z 1 Z 1 p b(s) p p p a(t) (A (t)) (t) t = a(t) (A (t)) t: s (A (s)) t0 t0 t R1 Integrating the term t (A (s)) b(s) s by parts, with u (s) = b(s) and (s) = (A (s)) ; we have Z 1 Z 1 1 (A (s)) b(s) s = u(s) (A(s)) jt u(s) (A(s)) s; t

t

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8

R1 where u(t) = t b(s) s = Z 1 (A (s)) b(s) s = t

(t); and so (note that A(t) A (t) and > 0) Z 1 (s) (A(s)) s (t) (A(t)) + t Z 1 (s) (A(s)) s: (t) (A (t)) + t

Using the following inequality (see [7, Lemma 2.2]) (2.16)

(f (t))

with f = A and

=

1

;

if 0

1; f

> 0;

; we observe that a(s) (A (s))

(A(s)) This gives us Z (2.17)

1

f (t) (f (t))

(A (s))

+1 ;

b(s) s

(note that 0

(t) (A (t))

+

t

1).

Z

1

t

a(s) (s) s: (A (s)) +1

Substitute (2.17) into (2.15) and using the Minkowski inequality [8, Theorem 2.1] Z b (2.18) jh(t)j ju(t) + (t)jp t a

2 4

Z

b

jh(t)j ju(t)jr

a

2 4

Z

a

b

Z

1 r

t

+

b

jh(t)j j (t)jr

a

Z

1 q

jh(t)j ju(t)jq

t

+

b

a

jh(t)j j (t)jq

1 r

t

a(t) (A (t))

(p r)

1

r

( (t))r

t

+

t0

q) r q

5

1 q

t

for r > q such that r; q > 1 and (r q)=(p q) > 1; we obtain Z 1 Z 1 p b(s) p a(t) (A (t)) s t (A (s)) t0 t Z 1 Z 1 a(s) (s) p a(t) (A (t)) (t) (A (t)) + s (A (s)) +1 t t0 " Z 1 Z 1

3 r(p

3 q(r

p) r q

5

:

p

a(t) (A (t))

t

p

(t)

1 r

r

t

t0

" Z

1

a(t) (A (t))

(p q)

q

( (t))

t0

1 q

t

+

Z

1

t0

where (t) :=

Z

1

t

a(t) (A (t))

p

(t)

q

# r(p r 1 q

t

q) q

# q(rr

a(s) (s) s: (A (s)) +1

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p) q

;

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9

SOME NEW DYNAMIC INEQUALITIES ON TIME SCALES

Applying Theorem 2.1 with 0 < c = p < 1; and '(t) = have Z 1 r (2.19) t a(t) (A (t)) p (t) t0

r

r 1+ p

= and

1

1

a(t) (A (t))r+

t0 rZ 1

r 1+ p Z

(2.20)

Z

q

q 1+ p

Z

1

Z

1+r+ p 1+ p

t;

q

t (p q)

( (t))q

t:

1

p

b(s) (A (s))

a(t) (A (t))

s

t

(p r)

1 r

( (t))r

t

t0

Z

1+q+ p 1+ p

1+q+ p 1+ p

(t)

a(t) (A (t))

t

1+r+ p 1+ p

( (t))r

t0

t0

=

p

a(t) (A (t))

From (2.19) and (2.20), we get that Z 1 Z 1 p a(t) (A (t))

"

t

+1

t0

t0

"

(p r)

a(t) (A (t))

; we

r

(t) (A (t))

p

+1

(t)= (A (t))

1

a(t) (A (t))

(p q)

# r(p r 1 q

r

( (t))

t

t0

r(p q) r q

Z

1

a(t) (A (t))

(p r)

q) q

# q(rr

p) q

p q r q

r

( (t))

t

t0

q(r p) r q

Z

1

a(t) (A (t))

(p q)

q

( (t))

r p r q

t

;

t0

which is the desired inequality (2.14). The proof is complete. Remark 2.4. As a special case of (2.14) when r = p; we get the inequality (1.12) which has been proved by Bohner and Saker. Remark 2.5. In Theorem 2.4 if T = N and r = p; then inequality (2.14) reduces to the discrete inequality (1.6) due to Bennett and G-Erdmann. References [1] R. P. Agarwal, M. Bohner and S. H. Saker, Dynamic Littlewood-type inequalities, Proc.Amer. Math. Soc. 143 (2015), 667–677. [2] R. P. Agarwal, D. O’Regan and S. H. Saker, Dynamic Inequalities on Time Scales, Springer Heidlelberg New York Drodrechet London, (2014). [3] R. P. Agarwal, D. O’Regan and S. H. Saker, Hardy Type Inequalities on Time Scales, Springer International Publishing, Cham Heidlelberg New York Drodrechet London, (2016).

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S. H. SAKER 1 , M. M. OSMAN 1 AND I. ABOHELA 2

10

[4] G. Bennett, Some elementary inequalities, Quart. J. Math. 2 (1987), 401–425. [5] G. Bennet and K.-G. Grosse-Erdmann, On series of positive terms, Houston Journal of Mathematics 31 (2005), 541–586. [6] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, (2001). [7] M. Bohner and S. H. Saker, Sneak-out principle on time scales, J. Math. Ineq. 10 (2) (2016), 393–403. [8] G. S. Chen, Some improvements of Minkowsk’s integral inequality on time scales, J. Ineq. Appl. 2013, 2013:318, 1–6. [9] J. E. Littlewood, Some new inequalities and unsolved problems, Inequalities (Editor O. Shisha), Academic Press, New York, 151–162 (1967). [10] S. H. Saker, R. R. Mahmoud, M. M. Osman and R. P. Agarwal, Some new generalized forms of Hardy’s type inequality on time scales, Mathematical Inequalities & Applications 20 (2017), 459–481. [11] S. H. Saker, M. M. Osman, D. O’Regan and R. P. Agarwal, Inequalities of Hardy-type and generalizations on time scales, Analysis: International mathematical journal of analysis and its applications 38 (1) (2018), 47–62. [12] S. H. Saker, M. M. Osman, D. O’Regan and R. P. Agarwal, Levinson type inequalities and their extensions via convexity on time scales, RACSAM 113 (1) (2019), 299–314. [13] S. H. Saker, D. O’Regan and R. P. Agarwal, Littlewood and Bennett inequalities on time scales, Mediterranean Journal of Mathematics 8 (2014), 1–15. 1

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, e-mails:[email protected], [email protected], 2 College of Engineering, Applied Science University, Kingdom of Bahrain, e-mail: [email protected]

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ADDITIVE-QUADRATIC FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES AND ITS STABILITY CHANG IL KIM AND GILJUN HAN∗

Abstract. In this paper, we investigate the functional inequality N (f (2x + y) + f (2x − y) − 6f (x) − 2f (−x) − f (y) − f (−y), t) ≥ N (f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y), kt) for some fixed real number k and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces.

1. Introduction In 1940, Ulam proposed the following stability problem (cf. [28]): “Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exist a constant c > 0 such that if a mapping f : G1 −→ G2 satisfies d(f (xy), f (x)f (y)) < c for all x, y ∈ G1 , then there exists an unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” In the next year, Hyers [13] gave a partial solution of Ulam, s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki ([1]) for additive mappings and by Rassias [22] for linear mappings to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problem of functional equations have been extensively investigated by a number of mathematicians (see [3], [4], [5], [10], and [18]). In 2008, for the first time, Mirmostafaee and Moslehian [15], [16] used the definition of a fuzzy norm in [2] to obtain a fuzzy version of the stability for the Cauchy functional equation (1.1)

f (x + y) = f (x) + f (y)

and the quadratic functional equation (1.2)

f (x + y) + f (x − y) = 2f (x) + 2f (y).

In [11], Gl´ anyi showed that if a mapping f : X −→ Y satisfies the following functional inequality (1.3)

k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k,

then f satisfies the Jordan-Von Neumann functional equation 2f (x) + 2f (y) − f (xy −1 ) = f (xy). 2010 Mathematics Subject Classification. 39B62, 39B72, 54A40, 47H10. Key words and phrases. Hyers-Ulam stability, additive-quadratic functional equation, fuzzy normed space, fixed point theorem. * Corresponding author. 1

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Gl´anyi [12] and Fechner [9] proved the Hyers-Ulam stability of (1.3). Park, Cho, and Han [21] proved the Hyers-Ulam stability of the following functional inequality: (1.4)

kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k.

Further, Park [20] proved the generalized Hyers-Ulam stability of the Cauchy additive functional inequality (1.4) in fuzzy Banach spaces using the fixed point method if f is an odd mapping. In this paper, we investigate the following functional inequality N (f (2x + y) + f (2x − y) − 6f (x) − 2f (−x) − f (y) − f (−y), t) (1.5)

≥ N (f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y), kt)

for some fixed nonzero real number k and prove the generalized Hyers-Ulam stability for (1.5) in fuzzy Banach spaces by fixed point methods. 2. preliminaries In this paper, we use the definition of fuzzy normed spaces given in [2], [16], and [17]. Definition 2.1. Let X be a real vector space. A function N : X × R −→ [0, 1] is called a fuzzy norm on X if for any x, y ∈ X and any 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 nondecreasing function of R and limt→∞ N (x, t) = 1; (N6) for any x 6= 0, N (x, ·) is continuous on R. In this case, the pair (X, N ) is called a fuzzy normed space. Let (X, N ) be a fuzzy normed space and {xn } a sequence in X. Then (i) {xn } is said to be Cauchy in (X, N ) if for any ε > 0, there exists an m ∈ N such that N (xn+p − xn , t) > 1 − ε for all n ≥ m, all positive integer p, and all t > 0 and (ii) {xn } is said to be convergent in (X, N ) 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 } in X and one denotes it by N − limn→∞ xn = x. Sequences of fuzzy numbers using the fuzzy metric or the fuzzy norm was studied by Das [6], [7], Tripathy et al. [23], Tripathy and Borgohain [24], [25], Tripathy and Dutta [26], Tripathy and Debnath [27] and others. Example 2.2. For example, it is well known that for any normed space (X, ||·||) and any nonnegative real number ε, the mapping NX : X × R −→ [0, 1], defined by  0, if t ≤ 0 NX (x, t) =  t , if t > 0 , t+ε||x||

is a fuzzy norm on X([16], [17], and [18]).

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It is well known that every convergent sequence in a fuzzy normed space is Cauchy. A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and a complete fuzzy normed space is called a fuzzy Banach space. In 1996, Isac and Rassias [14] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. Theorem 2.3. [8] Let (X, d) be a complete generalized metric space and let J : X −→ X be a strictly contractive mapping with some Lipschitz constant L with 0 < L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integer n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {J n 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 ∗ ) ≤ d(y, Jy) for all y ∈ Y . 1−L Throughout this paper, we assume that X is a linear space, (Y, N ) is a fuzzy Banach space, and (Z, N 0 ) is a fuzzy normed space. 3. Solutions of (1.5) In this section, we investigate the solution of (1.5) in fuzzy spaces. For any mapping f : X −→ Y , let Af (x, y) = f (2x + y) + f (2x − y) − 6f (x) − 2f (−x) − f (y) − f (−y), Bf (x, y) = f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y), Cf (x, y) = f (x + y) − f (x) − f (y), Df (x, y) = f (x − y) − f (x) + f (y), and

f (x) − f (−x) f (x) + f (−x) , fe (x) = . 2 2 Then fo is an odd mapping and fe is an even mapping. By (N5), we can easily prove the following lemma. fo (x) =

Lemma 3.1. Let αi : [0, ∞) −→ [0, ∞)(i = 1, 2, · · ·, n) be mappings and r a real number with r > 1 and y, z, z1 , z2 , ·, ·, ·, zn ∈ Y . Then we have the following : (1) If N (y, t) ≥ min{N (z, rk t), N (z1 , α1 (t)), N (z2 , α2 (t)), · · ·, N (zn , αn (t))} for all t > 0 and all k ∈ N, then N (y, t) ≥ min{N (z1 , α1 (t)), N (z2 , α2 (t)), · · ·, N (zn , αn (t))} for all t > 0. (2) If N (y, t) ≥ min{N (y, rt), N (z1 , α1 (t)), N (z2 , α2 (t)), · · ·, N (zn , αn (t))} for all t > 0 and αi (i = 1, 2, · · ·, n) is non-decreasing, then N (y, t) ≥ min{N (z1 , α1 (t)), N (z2 , α2 (t)), · · ·, N (zn , αn (t))} for all t > 0. (3) If N (y, t) ≥ N (y, rt) for all t > 0, then y = 0.

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We establish the following theorem using Lemma 3.1 : Theorem 3.2. Let f : X −→ Y be an odd mapping. Suppose that a and b are real numbers with a > 4 and b > 2. Then f is an additive mapping if and only if f satisfies the following inequality (3.1)

N (Af (x, y), t) ≥ min{N (Bf (x, y), at), N (Bf (y, 2x), bt)}

for all x, y ∈ X and all t > 0. Proof. Suppose that f is a solution of (3.1). Letting x = 0 and y = 0 in (3.1), we get f (0) = 0. Letting y = 0 in (3.1), by (N2), we get (3.2)

f (2x) = 2f (x)

for all x ∈ X. Letting y = 2y in (3.1), by (3.2), we have (3.3)

N (Bf (x, y), t) ≥ min{N (Bf (x, 2y), 2at), N (Bf (y, x), bt)}

for all x, y ∈ X and all t > 0. Putting x = 2x + y and y = x in (3.3), we get N (f (3x + y) + f (x + y) − 2f (2x + y), t) ≥ min{N (f (4x + y) + f (y) − 2f (2x + y), 2at), N (f (3x + y) − f (x + y) − 2f (x), bt)} n  (3.4) b  ≥ min N (f (4x + y) + f (y) − 2f (2x + y), 2at), N f (2x + y) − f (x + y) − f (x), t , 4  b o N f (3x + y) + f (x + y) − 2f (2x + y), t 2 for all x, y ∈ X and all t > 0. Since b > 2, by (3.4) and Lemma 3.1, we have N (f (3x + y) + f (x + y) − 2f (2x + y), t) n  b o ≥ min N (f (4x + y) + f (y) − 2f (2x + y), 2at), N f (2x + y) − f (x + y) − f (x), t 4 for all x, y ∈ X and all t > 0. Letting x = x + y and y = x in (3.3), by (3.5), we get (3.5)

N (f (2x + y) + f (y) − 2f (x + y), t) ≥ min{N (f (3x + y) − f (x − y) − 2f (x + y), 2at), N (f (2x + y) − f (y) − 2f (x), bt)} n  a  a   ≥ min N f (3x + y) + f (x + y) − 2f (2x + y), t , N f (2x + y) + f (y) − 2f (x + y), t , 2 2  o a  N f (x + y) − f (x − y) − 2f (y), t , N (f (2x + y) − f (y) − 2f (x), bt) (3.6) 2 n    ab  ≥ min N f (4x + y) + f (y) − 2f (2x + y), a2 t , N f (2x + y) − f (x + y) − f (x), t , 8  a   a  N f (2x + y) + f (y) − 2f (x + y), t , N f (x + y) − f (x − y) − 2f (y), t , 2 2 o N (f (2x + y) − f (y) − 2f (x), bt) for all x, y ∈ X and all t > 0. Since a > 4, by (3.6) and Lemma 3.1, we have n   N (f (2x + y) + f (y) − 2f (x + y), t) ≥ min N f (4x + y) + f (y) − 2f (2x + y), a2 t ,  ab   a  (3.7) N f (2x + y) − f (x + y) − f (x), t , N f (x + y) − f (x − y) − 2f (y), t , 8 2 o N (f (2x + y) − f (y) − 2f (x), bt)

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for all x, y ∈ X and all t > 0. Letting y = 2y in (3.7), by (3.2), we have

(3.8)

N (f (x + y) + f (y) − f (x + 2y), t) n  ab  ≥ min N (f (2x + y) + f (y) − 2f (x + y), a2 t), N 2f (x + y) − f (x + 2y) − f (x), t , 4 o N (f (x + 2y) − f (x − 2y) − 4f (y), at), N (Cf (x, y), bt) n  ab  ≥ min N (f (2x + y) + f (y) − 2f (x + y), a2 t), N f (y) + f (x + y) − f (x + 2y), t , 8 n ab o o N (f (x + 2y) − f (x − 2y) − 4f (y), at), N (Cf (x, y), min ,b t 8 n  a  ≥ min N (f (2x + y) + f (y) − 2f (x + y), a2 t), N f (y) + f (x + y) − f (x + 2y), t , 4  n a o o a   a   N f (x − 2y) − f (x − y) + f (y), t , N Df (x, y), t , N Cf (x, y), min ,b t 4 4 4

for all x, y ∈ X and all t > 0, because b > 2. Since a > 4, by (3.8), we have

(3.9)

n N (f (x + y) + f (y) − f (x + 2y), t) ≥ min N (f (2x + y) + f (y) − 2f (x + y), a2 t),  n a o o a   a   N f (x − 2y) − f (x − y) + f (y), t , N Df (x, y), t , N Cf (x, y), min ,b t 4 4 4

for all x, y ∈ X and all t > 0. Interchanging x and y in (3.9), we have

(3.10)

n N (f (2x + y) − f (x + y) − f (x), t) ≥ min N (f (x + 2y) + f (x) − 2f (x + y), a2 t), n a o o  a   a   ,b t N f (2x − y) − f (x − y) − f (x), t , N Df (x, y), t , N Cf (x, y), min 4 4 4 n  a2   a  ≥ min N f (x + 2y) − f (x + y) − f (y), t , N f (2x − y) − f (x − y) − f (x), t , 2 4 n a o o  a   ,b t N Df (x, y), t , N Cf (x, y), min 4 4 n  a4   a3  ≥ min N f (2x + y) − f (x + y) − f (x), t , N f (x − 2y) − f (x − y) + f (y), t , 2 8 n a o o  a   a   ,b t N f (2x − y) − f (x − y) − f (x), t , N Df (x, y), t , N Cf (x, y), min 4 4 4

for all x, y ∈ X and all t > 0. Hence by Lemma 3.1 and (3.10), we have

(3.11)

n  a  N (f (2x + y) − f (x + y) − f (x), t) ≥ min N f (2x − y) − f (x − y) − f (x), t , 4  n a o o a3   a   N f (x − 2y) − f (x − y) + f (y), t , N Df (x, y), t , N Cf (x, y), min ,b t 8 4 4

for all x, y ∈ X and all t > 0, because a > 4. By (3.11), we have

(3.12)

n  a2  N (f (2x + y) − f (x + y) − f (x), t) ≥ min N f (2x + y) − f (x + y) − f (x), 4 t , 2  a4   a3  N f (x + 2y) − f (x + y) − f (y), 5 t , N f (x − 2y) − f (x − y) + f (y), t , 2 8  na o   n a o o N Df (x, y), min , b t , N Cf (x, y), min ,b t 4 4

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for all x, y ∈ X and all t > 0. Thus by Lemma 3.1 and (3.12), we have N (f (2x + y) − f (x + y) − f (x), t) n  a4   a3  ≥ min N f (x + 2y) − f (x + y) − f (y), 5 t , N f (x − 2y) − f (x − y) + f (y), 3 t , (3.13) 2 2  na o   n a o o N Df (x, y), min , b t , N Cf (x, y), min ,b t 4 4 for all x, y ∈ X and all t > 0. Interchanging x and y in (3.13), we have

(3.14)

for all

(3.15)

for all (3.16) for all (3.17) for all

(3.18)

for all

for all

N (f (x + 2y) − f (x + y) − f (y), t) n  a3  a4   ≥ min N f (2x + y) − f (x + y) − f (x), 5 t , N f (2x − y) − f (x − y) − f (x), 3 t , 2 2  na o   n a o o N Df (x, y), min , b t , N Cf (x, y), min ,b t 4 4 n  a3  a8   ≥ min N f (x + 2y) − f (x + y) − f (y), 10 t , N f (x − 2y) − f (x − y) + f (y), 3 t , 2 2  na o   n a o o N Df (x, y), min , b t , N Cf (x, y), min ,b t 4 4 x, y ∈ X and all t > 0. By Lemma 3.1 and (3.14), we get n  a3  N (f (x + 2y) − f (x + y) − f (y), t) ≥ min N f (x − 2y) − f (x − y) + f (y), 3 t , 2 n a o o  na o   , b t , N Cf (x, y), min ,b t N Df (x, y), min 4 4 na o  n  a6   ,b t , ≥ min N f (x + 2y) − f (x + y) − f (y), 6 t , N Df (x, y), min 2 4  n a o o N Cf (x, y), min ,b t 4 x, y ∈ X and all t > 0. By Lemma 3.1 and (3.15), we get n  na o  N (f (x + 2y) − f (x + y) − f (y), t) ≥ min N Df (x, y), min ,b t , 4  n a o o N Cf (x, y), min ,b t 4 x, y ∈ X and all t > 0. Interchanging x and y in (3.16), we have n  na o  N (f (2x + y) − f (x + y) − f (x), t) ≥ min N Df (x, y), min ,b t , 4  n a o o N Cf (x, y), min ,b t 4 x, y ∈ X and all t > 0. Letting y = y − x in (3.17), we get n  na o  N (Cf (x, y), t) ≥ min N f (2x − y) − f (x) − f (x − y), min ,b t , 4  n a o o N Df (x, y), min ,b t 4 n  na o   n a o o ≥ min N Df (x, y), min , b t , N Cf (x, y), min ,b t 4 4 x, y ∈ X and all t > 0. Since min{ a4 , b} > 1, by Lemma 3.1 and (3.18), we have  na o   h n a oi2  N (Cf (x, y), t) ≥ N Df (x, y), min , b t ≥ N Cf (x, y), min ,b t 4 4 x, y ∈ X and all t > 0 and hence by Lemma 3.1, f is an additive mapping.

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The converse is trivial.



Theorem 3.3. Let f : X −→ Y be an even mapping. Suppose that k is a real number with k > 1. Then f is a solution of the following functional equation (3.19)

N (Af (x, y), t) ≥ N (Bf (x, y), kt)

for all x, y ∈ X if and only if f is a quadratic mapping. Proof. Suppose that f is a solution of (3.19). Letting x = 0 and y = 0 in (1.5), we have   N (f (0), t) ≥ N f (0), 4kt for all t > 0 and sicne 4k > 1, by Lemma 3.1, we get f (0) = 0. Letting y = 0 in (3.19), by (N2), we get (3.20)

f (2x) = 4f (x)

for all x ∈ X. Now, letting x = 2x in (3.19), by (3.20), we have N (f (4x + y) + f (4x − y) − 32f (x) − 2f (y), t) ≥ N (Af (x, y), kt) (3.21)

≥ N (Bf (x, y), k 2 t)

for all x, y ∈ X. Letting y = 2y in (3.21), by (3.19), we have (3.22)

N (Af (x, y), t) ≥ N (Bf (2y, x), 4k 2 t) = N (Af (y, x), 4k 2 t) ≥ N (Bf (x, y), 4k 3 t)

for all x, y ∈ X. Letting x = 2x in (3.22), by (3.19), we have N (f (4x + y) + f (4x − y) − 32f (x) − 2f (y), t) ≥ N (Bf (x, y), 4k 4 t) for all x, y ∈ X. Hence by induction, we get N (f (4x + y) + f (4x − y) − 32f (x) − 2f (y), t) ≥ N (Bf (x, y), 4n k n+3 t) for all x, y ∈ X and n ∈ N. Since k > 1, by Lemma 3.1 and (N5), we have f (4x + y) + f (4x − y) − 32f (x) − 2f (y) = 0 for all x, y ∈ X. Hence f is a quadratic mapping.



4. The generalized Hyers-Ulam stability for (1.5) Now, we will prove the generalized Hyers-Ulam stability for (1.5) in fuzzy normed spaces. Theorem 4.1. Assume that φ : X 3 −→ [0, ∞) is a function such that (4.1)

N 0 (φ(2x, 2y), t) ≥ N 0 (4Lφ(x, y), t)

for all x, y ∈ X, t > 0 and some real number L with 0 < L < 21 . Let f : X −→ Y be a mapping such that f (0) = 0 and (4.2)

N (Af (x, y), t) ≥ min{N (Bf (x, y), kt), N 0 (φ(x, y), t)}

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for all x, y ∈ X, t > 0 and some real number k with k > 32. Then there exists an unique additivequadratic mapping F : X −→ Y such that   1 t ≥ min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)} (4.3) N f (x) − F (x), 2(1 − 2L) for all x ∈ X and all t > 0. Proof. By (4.2), we get

(4.4)

n  k  k   N (Afo (x, y), t) ≥ min N Bfo (x, y), t , N Bfe (x, y), t , 2 2 o 0 0 N (φ(x, y), t), N (φ(−x, −y), t)

for all x, y ∈ X, t > 0 and (4.5)

n  k  k   N (Afe (x, y), t) ≥ min N Bfo (x, y), t , N Bfe (x, y), t , 2 2 o 0 0 N (φ(x, y), t), N (φ(−x, −y), t)

for all x, y ∈ X and all t > 0. Letting y = 0 in (4.4) and (4.5), by (N2), we have (4.6)

N (2fo (2x) − 4fo (x), t) ≥ min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)}

and (4.7)

N (2fe (2x) − 8fe (x), t) ≥ min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)}

for all y ∈ X and all t > 0. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞) | N (g(x) − h(x), ct) ≥ φo (x, t), ∀x ∈ X, ∀t > 0}, where φo (x, t) = min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)}. Then (S, d) is a complete metric space([19]). Define a mapping Jo : S −→ S by Jo g(x) = 21 g(2x) for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (4.1), we have   N (Jo g(x) − Jo h(x), 2cLt) = N g(2x) − h(2x), 4cLt ≥ φo (2x, 4Lt) ≥ φo (x, t) for all x ∈ X and all t > 0. Hence d(Jo g, Jo h) ≤ 2Ld(g, h) for any g, h ∈ S and by (4.6), we have d(Jo fo , fo ) ≤ 14 < ∞. By Theorem 2.3, there exists a mapping P : X −→ Y which is a fixed point of Jo such that   1 (4.8) N fo (x) − P (x), t ≥ φo (x, t) 4(1 − 2L) for all x ∈ X and all t > 0. Moreover, d(Jon fo , A) → 0 as n → ∞. That is, fo (2n x) n→∞ 2n

P (x) = N − lim

for all x ∈ X. Now, define a mapping Je : S −→ S by Je g(x) = 14 g(2x) for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (4.1), we have   N (Je g(x) − Je h(x), cLt) = N g(2x) − h(2x), 4cLt ≥ φo (2x, 4Lt) ≥ φo (x, t)

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for all x ∈ X and t > 0. Hence d(Je g, Je h) ≤ Ld(g, h) for any g, h ∈ S and by (4.7), we have d(Je fe , fe ) ≤ 18 < ∞. By Theorem 2.3, there exists a mapping Q : X −→ Y which is a fixed point of Je such that  N fe (x) − Q(x),

(4.9)

 1 t ≥ φo (x, t) 8(1 − L)

for all x ∈ X and all t > 0. Moreover, d(Jen fe , A) → 0 as n → ∞. That is, fe (2n x) n→∞ 22n

Q(x) = N − lim

(4.10)

for all x ∈ X. Replacing x, and y by 2n x and 2n y in (4.5), respectively, by (4.1), we have

(4.11)

 1  n 1  N 2n Afe (2n x, 2n y), t ≥ min N n Bfo (2n x, 2n y), 2n−1 kt , 2 2   1 1 o k  0 1  n n N 2n Bfe (2 x, 2 y), t , N φ(x, y), n t , N 0 φ(−x, −y), n t 2 2 L L

for all x, y ∈ X, t > 0, and all n ∈ N. By (N4) and (4.11), we have

(4.12)

N (AQ (x, y), t) n  1 t  1 t o ≥ min N AQ (x, y) − 2n Afe (2n x, 2n y), , N 2n Afe (2n x, 2n y), 2 2 2 2 n   1 t  1 n n n n ≥ min N AQ (x, y) − 2n Afe (2 x, 2 y), , N n Bfo (2 x, 2 y), 2n−2 kt , 2 2 2      1 k 1 1 o N 2n Bfe (2n x, 2n y), t , N 0 φ(x, y), n t , N 0 φ(−x, −y), n t 2 4 2L 2L n   1 t  1 n n n ≥ min N AQ (x, y) − 2n Afe (2 x, 2 y), , N n Bfo (2 x, 2n y), 2n−2 kt , 2 2 2  1 k   k  n n N 2n Bfe (2 x, 2 y) − BQ (x, y), t , N BQ (x, y), t , 2 8 8  1 o 1  0 0 N φ(x, y), n t , N φ(−x, −y), n t 2L 2L

for all x, y ∈ X, t > 0, and all n ∈ N. By (N4), we have 1  n n n B (2 x, 2 y), 2 t f o 2n n 1   o ≥ min N n Bfo (2n x, 2n y) − BP (x, y), 2n−1 t , N BP (x, y), 2n−1 t 2 N

(4.13)

for all x, y ∈ X, t > 0, and all n ∈ N. Letting n → ∞ in (4.13), by (N5), we have (4.14)

lim N

n→∞

 1  Bfo (2n x, 2n y), t = 1 2n 2

for all x, y ∈ X, t > 0, and all n ∈ N. Letting n → ∞ in (4.12), by (4.10) and (4.14), we have (4.15)

 k  N (AQ (x, y), t) ≥ N BQ (x, y), t 8

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for all x, y ∈ X and all t > 0. Since fe is even, by (4.10), Q is even and hence by (4.15) and Theorem 3.3, Q is a quadratic mapping. By (4.5) and (4.7), we have n  t  t o N (Bfe (x, 2y), t) ≥ min N Afe (y, x), , N 8fe (y) − 2fe (2y), 2 2 n  k   k  0 t (4.16) , ≥ min N Bfo (y, x), t , N Bfe (y, x), t , N φ(y, x), 4 4 2      o t t t N 0 φ(−y, −x), , N 0 φ(y, 0), , N 0 φ(−y, 0), 2 2 2 for all x, y ∈ X and t > 0. By (4.7) and (4.16), we have N (Bfe (x, y), t) = N (4Bfe (x, y), 4t)

(4.17)

≥ min{N (Bfe (2x, 2y), 2t), N (4Bfe (x, y) − Bfe (2x, 2y), 2t)} o n  k   k  ≥ min N Bfo (y, 2x), t , N Bfe (y, 2x), t , Φ1 (x, y, t) 2 2 o n  k2   k2  k   ≥ min N Bfo (y, 2x), t , N Bfo (x, y), t , N Bfe (x, y), t , Φ2 (x, y, t) 2 8 8

for all x, y ∈ X and all t > 0, where n Φ1 (x, y, t) = min N 0 (φ(y, 2x), t), N 0 (φ(−y, −2x), t), N 0 (φ(x + y, 0), t), N 0 (φ(−x − y, 0), t), N 0 (φ(x − y, 0), t), N 0 (φ(−x + y, 0), t),  t  0 t  0 t o t  0 , N φ(−x, 0), , N φ(y, 0), , N φ(−y, 0), N 0 φ(x, 0), 2 2 2 2 and n   k  k o Φ2 (x, y, t) = min Φ1 (x, y, t), N 0 φ(x, y), t , N 0 φ(−x, −y), t , 4 4 because k > 32. By Lemma 3.1 and (4.17), we have o n  k2  k   (4.18) N (Bfe (x, y), t) ≥ min N Bfo (y, 2x), t , N Bfo (x, y), t , Φ2 (x, y, t) 2 8 for all x, y ∈ X and all t > 0 and hence by (4.4) and (4.18), we have n  k   k2  N (Afo (x, y), t) ≥ min N Bfo (x, y), t , N Bfo (y, 2x), t , 2 4 (4.19)  o k  0 Φ1 x, y, t , N (φ(x, y), t), N 0 (φ(−x, −y), t) 2 for all x, y ∈ X, t > 0 and replacing x and y by 2n x and 2n y in (4.19), respectively, by (4.1), we have   N Afo (2n x, 2n y), 2n t n ≥ min N (Bfo (2n x, 2n y), 2n−1 kt), N (Bfo (2n y, 2n+1 x), 2n−2 k 2 t),    k 1  0 1 o Φ1 x, y, t , N 0 φ(x, y), t , N φ(−x, −y), t n n 2(2L) (2L) (2L)n for all x, y ∈ X, all t > 0 and all n ∈ N. Similar to Q, we have n  k 2 o k   (4.20) N (AP (x, y), t) ≥ min N BP (x, y), t , N BP (y, 2x), t 8 16

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for all x, y ∈ X and all t > 0. Cleraly, P is an odd mapping and since k > 32, by Theorem 3.2, P is an additive mapping. Let F = P + Q. Then F : X −→ Y is an additive-quadratic mapping. By (4.8) and (4.9), we have (4.3). Now, we show the uniqueness of F . Let H be another additive-quadratic mapping with (4.3). Since F and H are addiitve-quadratic mappings, we have 1 + 2n 1 − 2n 1 + 2n 1 − 2n F (2n x) + 2n+1 F (−2n x), H(x) = 2n+1 H(2n x) + 2n+1 H(−2n x), 2n+1 2 2 2 2 for all x ∈ X and all positive integer n. Hence by (4.3), (N3) and (N4), we have F (x) =

N (F (x) − H(x), t) n  22n o 22n   n n t , N F (−2 x) − H(−2 x), ≥ min N F (2n x) − H(2n x), t 1 + 2n 2n − 1 n  22n−1   n 22n−1  n t , N f (2 t , ≥ min N F (2n x) − f (2n x), x) − H(2 x), 1 + 2n 1 + 2n  22n−1   22n−1 o N F (−2n x) − f (−2n x), n t , N f (−2n x) − H(−2n x), n t 2 −1 2 −1 n  22n (1 − 2L)   n 22n (1 − 2L) o t , φo 2 x, t ≥ min φo 2n x, 1 + 2n 2n − 1 n    o 1 − 2L 1 − 2L   t ≥ min φo x, t , φ x, o (L)n + (2L)n (2L)n 1 − 1n 2

for all x ∈ X, t > 0, and all n ∈ N. Since 0 < L < have F (x) = H(x) for all x ∈ X.

1 2,

letting n → ∞ in the above inequality, we 

By Theorem 4.1, we can show that the following corollaries: Corollary 4.2. Let ε and p be real numbers with ε ≥ 0 and 0 < p < 21 . Let f : X −→ Y be a mapping such that n o t (4.21) N (Af (x, y), t) ≥ min N (Bf (x, y), kt), t + ε(kxk2p + kyk2p + kxkp kykp ) for all x, y ∈ X, all t > 0 and some real number k with k > 32. Then there exists an unique additive-quadratic mapping F : X −→ Y such that N (f (x) − F (x), t) ≥

(2 − 22p )t (2 − 22p )t + εkxk2p

for all x ∈ X and all t > 0. Corollary 4.3. Assume that φ : X 3 −→ [0, ∞) is a function with (4.1). Let f : X −→ Y be a mapping such that f (0) = 0 and (4.22)

N (rAf (x, y) + Bf (x, y), t) ≥ min{N (Bf (x, y), t), N 0 (φ(x, y), t)}

for all x, y ∈ X, all t > 0 and some real numbers r with |r| > 64. Then there exists an unique additive-quadratic mapping F : X −→ Y such that   1 N f (x) − F (x), t ≥ min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)} 2(1 − 2L) for all x ∈ X and all t > 0.

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Proof. By (N5) and (4.22), we have n  |r|   |r| o N (Af (x, y), t) ≥ min N rAf (x, y) + Bf (x, y), t , N Bf (x, y), t 2 2 n  |r| o |r|  0  ≥ min N Bf (x, y), t , N φ(x, y), t 2 2  o n   |r| ≥ min N Bf (x, y), t , N 0 φ(x, y), t 2 for all x, y ∈ X and all t > 0. Hence we have the results.



Corollary 4.4. Let ε and p be real numbers with ε ≥ 0 and 0 < p < 12 . Let f : X −→ Y be a mapping such that n o t (4.23) N (rAf (x, y) + Bf (x, y), t) ≥ min N (Bf (x, y), t), 2p 2p p p t + ε(kxk + kyk + kxk kyk ) for all x, y ∈ X, all t > 0 and some real number r with |r| > 64. Then there exists an unique additive-quadratic mapping F : X −→ Y such that N (f (x) − F (x), t) ≥

(2 − 22p )t (2 − 22p )t + εkxk2p

for all x ∈ X and all t > 0. Related with Theorem 4.1, we can also have the following theorem. The proof is similar to that of Theorem 4.1. Theorem 4.5. Assume that φ : X 3 −→ [0, ∞) is a function such that   x y  L (4.24) N0 φ , φ(x, y), t , t ≥ N0 2 2 2 for all x, y ∈ X, t > 0 and some real number L with 0 < L < 21 . Let f : X −→ Y be a mapping such that f (0) = 0 and (4.2). Then there exists an unique additive-quadratic mapping F : X −→ Y such that   L t ≥ min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)} N f (x) − F (x), 2(1 − L) for all x ∈ X and t > 0. Proof. Let φo (x, t) = min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)}. Letting x = (4.24), we have  x L  (4.25) N 2fo (x) − 4fo , t ≥ φo (x, t) 2 2

x 2

in (4.6) and (4.7), by

and (4.26)

 x L  N 2fe (x) − 8fe , t ≥ φo (x, t) 2 2

for all y ∈ X and t > 0. Consider the set S = {g | g : X −→ Y } and the generalized metric d on S defined by d(g, h) = inf{c ∈ [0, ∞) | N (g(x) − h(x), ct) ≥ φo (x, t), ∀x ∈ X, ∀t > 0}.

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  Then (S, d) is a complete metric space([19]). Define a mapping Jo : S −→ S by Jo g(x) = 2g x2 for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (4.1), we have    x x L  x L  N Jo g(x) − Jo h(x), cLt = N g −h , c t ≥ φo , t ≥ φo (x, t) 2 2 2 2 2 for all x ∈ X and t > 0. Hence d(Jo g, Jo h) ≤ Ld(g, h) for any g, h ∈ S. By (4.25), we have d(Jo fo , fo ) ≤ L4 < ∞ and by Theorem 2.3, there exists a mapping P : X −→ Y which is a fixed point of Jo such that   L N fo (x) − P (x), t ≥ φo (x, t) 4(1 − L) for all x ∈ X, all t > 0 and d(Jon fo , A) → 0 as n → ∞.  Now, define a mapping Je : S −→ S by Je g(x) = 4g x2 for all x ∈ X and all g ∈ S. Let g, h ∈ S and d(g, h) ≤ c for some c ∈ [0, ∞). Then by (4.1), we have  x x L  x L  N (Je g(x) − Je h(x), 2cLt) = N g −h , c t ≥ φo , t ≥ φo (x, t) 2 2 2 2 2 for all x ∈ X and t > 0. Hence d(Je g, Je h) ≤ 2Ld(g, h) and by (4.26), we have d(Je fe , fe ) ≤ L4 < ∞. By Theorem 2.3, there exists a mapping Q : X −→ Y which is a fixed point of Je such that   L N fe (x) − Q(x), t ≥ φo (x, t) 4(1 − 2L) for all x ∈ X, all t > 0 and d(Jen fe , A) → 0 as n → ∞. The rest of the proof is similar to Theorem 4.1.



By Theorem 4.5, we can show that the following corollaries: Corollary 4.6. Let ε and p be real numbers with ε ≥ 0 and p > 1. Let f : X −→ Y be a mapping with f (0) = 0 and (4.21). Then there exists an unique additive-quadratic mapping F : X −→ Y such that (22p − 2)t N (f (x) − F (x), t) ≥ 2p (2 − 2)t + εkxk2p for all x ∈ X and all t > 0. Corollary 4.7. Assume that φ : X 3 −→ [0, ∞) is a function with (4.24). Let f : X −→ Y be a mapping with f (0) = 0 and (4.22). Then there exists an unique additive-quadratic mapping F : X −→ Y such that N (f (x) − F (x),

L t) ≥ min{N 0 (φ(x, 0), t), N 0 (φ(−x, 0), t)} 2(1 − L)

for all x ∈ X and all t > 0. Corollary 4.8. Let ε and p be real numbers with ε ≥ 0 and p > 1. Let f : X −→ Y be a mapping with f (0) = 0 and (4.23). Then there exists an unique additive-quadratic mapping F : X −→ Y such that (22p − 2)t N (f (x) − F (x), t) ≥ 2p (2 − 2)t + εkxk2p for all x ∈ X and all t > 0.

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[27] B. C. Tripathy and A. J. Dutta, On I-acceleration convergence of sequences of fuzzy real numbers, Math. Modell. Analysis, 17(4)(2012), 549-557. [28] S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley and Sons, Inc., New York, 1964. Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Suji-gu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: [email protected] Department of Mathematics Education, Dankook University, 152, Jukjeon-ro, Suji-gu, Yongin-si, Gyeonggi-do, 16890, Korea E-mail address: [email protected]

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NEW CHARACTERIZATIONS OF WEIGHTS IN HARDY’S TYPE INEQUALITIES VIA OPIAL’S DYNAMIC INEQUALITIES S. H. SAKER, M. M. OSMAN AND I. ABOHELA Abstract. In this paper, we prove some new characterizations of weights in some Hardy-type inequalities on time scales. The results as special cases contain the results due to Beesack and Heinig, Leindler and Bloom and Kerman. Some new integral and discrete inequalities related to Copson’s, Flett’s, Bliss’s and Bennett’s will be formulated. The main results will be proved by using new generalizations of Opial’s type inequalities, Hölder’s inequality, Minkowski’s inequality and the chain rule on time scales. Keywords: Hardy’s inequality, Opial’s inequality, time scales. AMS Classif: 26A15, 26D10, 26D15, 39A13, 34A40.

1. Introduction During the last decades the inequality !1=q Z b Z t q dt C (1.1) r (t) f ( )d a

Z

!1=p

b p

s (t) f (t)dt

a

a

; 1 0; k > 1; then "Z # p+1 Z Z p+1 k t

b

(1.6)

a

b

f ( )d

r (t)

dt

s (t) f k (t)dt

(p + 1) K1 (p; 1; k)

a

;

a

where K1 (p; 1; k) =

1 k

1 p+1

Z

b

(R (t; b)) k

k 1

(s (t))

a

1 k 1

Z

p

t

s

1 k 1

( )d

a

! kk 1

dt

;

Rb and R (t; b) = t r ( ) d : In the last decades the study of discrete results on lp analogues for Lp bounds has been proved by some authors. One of the reasons for this upsurge of interest in discrete cases is due to the fact that the discrete operators may even behave di¤erently from their continuous counterparts. So it was natural to look on the discrete results on lp analogues for the above Lp results. We mention here that in some special cases it is possible to translate or adapt almost straightforward the objects and results from the continuous setting to the discrete setting or vice versa, however, in some other cases that is far from be trivial. But lp bounds for discrete analogues of more complicated operators are not implied by results in the continuous setting, and moreover the discrete analogues are resistant to conventional methods. The main challenge here is that there are no general methods to study these questions and the methods should to be developed starting from the basic de…nitions in the discrete space. For example, Leindler [22] established the discrete versions of (1.3) and (1.4), and proved that if 0 < p 1; an 0 and n > 0; then !p !p 1 1 n 1 X X X X 1 p p apn ; ak p (1.7) k n n n=1

k=1

1 X

1 X

n=1

and

(1.8)

n=1

n

k=n

ak

!p

p

p

1 X

n=1

k=n

1 p n

n X

k=1

k

!p

apn :

In recent years the study of dynamic equations and inequalities on time scales has received a lot of attention in the literature and has become a major …eld in pure and applied mathematics. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function is a so-called time scale T, which may be an arbitrary closed subset of the real numbers R, to avoid proving results twice, once for di¤erential inequality and once again for di¤erence inequality. This idea goes back to its founder Stefan Hilger [19] who started the study of dynamic equations on time scales. Since the integral and discrete inequalities are important in the analysis of qualitative properties of solutions of di¤erential and di¤erence equations, we also believe that the dynamic Hardy type inequalities with weights on time scales will play the same e¤ective role in the analysis of qualitative properties of dynamic equations with boundary conditions like oscillation, nonoscillation and distribution of zeros of solutions. For related dynamic inequalities on time scales, we refer the reader to the papers [26, 27, 32, 33] and the books [2,3]. Our technique in this paper will overcame the lack of calculus in the discrete

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DYN AM IC IN EQ U ALITIES O F H ARDY-TYPE

space where there is no power rules and also there is no chain rule which are the main tools used in the proofs of the continuous case. The aim of this paper is to prove some new dynamic inequalities by employing some Opial’s type inequalities on an arbitrary time scale T which contain the integral and discrete inequalities (1.3)–(1.6) as special cases. For applications of the main results we get some well-known dynamic inequalities as special cases. The paper is divided into two sections. In Section 2, we introduce some preliminaries on time scales and establish some basic lemmas that will be needed in the proofs. In Section 3, we prove the main results and formulate some discrete results to show the application of the new results. 2. Preliminaries and Some Basic Lemmas In this section, we present some basic de…nitions and results concerning the delta calculus on time scales; for more details we refer the reader to the book [14]. A time scale T is an arbitrary nonempty closed subset of the real numbers R. The forward jump operator and the backward jump operator are de…ned by (t) := inffs 2 T : s > tg; and (t) := supfs 2 T : s < tg; where sup ; = inf T. A point t 2 T; is said to be left–dense if (t) = t and t > inf T; is right-dense if (t) = t; is left–scattered if (t) < t and right–scattered if (t) > t: A function f : T ! R is said to be right–dense continuous (rd–continuous) provided f is continuous at right–dense points and at left–dense points in T; left hand limits exist and are …nite. The set of all such rd–continuous functions is denoted by Crd (T): Also, the set of functions that are di¤erentiable and whose derivative is rd–continuous is 1 1 denoted by Crd (T) = Crd (T; R): The graininess function for a time scale T is de…ned by (t) := (t) t, and for any function f : T ! R the notation f (t) denotes f ( (t)): Without loss of generality, we assume that sup T = 1, and de…ne the time scale interval [a; b]T by [a; b]T := [a; b] \ T: Recall of the following product and quotient rules for the derivative of the product f g and the quotient f =g (where gg 6= 0, here g = g ) of two di¤erentiable functions f and g (2.1)

(f g) = f g + f g

= f g + f g ; and

f g

=

f g fg : gg

The …rst chain rule that we will use in this paper is (2.2)

(f (t)) =

Z1

[hf + (1

h)f ]

1

dhf (t);

2 R;

0

which is a simple consequence of Keller’s chain rule [14, Theorem 1.90]. The second chain rule that we will use in this paper is given in the following. Let f : R ! R be continuously di¤erentiable and suppose g : T ! R is delta di¤erentiable, then f g : T ! R is delta di¤erentiable and (2.3)

0

f (g (t)) = f (g(d)) g (t) ;

for d 2 [t; (t)]:

In this paper we will refer to the (delta) integral which we can de…ne as follows. If F (t) = Rt f (t), then the Cauchy (delta) integral of f is de…ned by t0 f (s) s := F (t) F (t0 ): It Rt can be shown (see [14]) that if f 2 Crd (T); then the Cauchy integral F (t) := t0 f (s) s exists, t0 2 T, and satis…es F (t) = f (t), t 2 T: An in…nite integral is de…ned as R1 Rb f (t) t = limb!1 a f (t) t: Integration on discrete time scales is de…ned by a Z b X f (t) t = (t)f (t): a

t2[a;b)

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S. H. SAKER, M . M . O SM AN AND I. ABOHELA

The integration by parts formula on time scales reads Z b Z b b (2.4) u(t) (t) t = [u(t) (t)]a u (t) a

(t) t:

a

Hölder’s inequality [5, Theorem 6.2] states that for f; g 2 Crd ([a; b]T , R); we have "Z #1=p "Z #1=q Z b b b p q jf (t)j t jg(t)j t ; (2.5) jf (t)g(t)j t a

a

a

where p > 1; 1=p + 1=q = 1 and a; b 2 T: This inequality is reversed if 0 < p < 1 and Rb Rb q p jg(t)j t > 0; and it is also reversed if p < 0 and a jf (t)j t > 0: a Throughout this paper, we will assume that r (t) ; s (t) and f (t) are nonnegative rdcontinuous functions and the integrals considered are assumed to exist. In order to prove our main results in Section 3, we need the following lemmas. Lemma 2.1. Assume F : T ! R is di¤ erentiable and positive. If F then (2.6)

F

F

1

(F (t))

;

if

is always positive,

1;

and (2.7)

F

F

(F (t))

1

;

if

0

1;

Proof. If F is increasing and 1; then F 1 is increasing and thus F that 1 F = F F 1 = F (F (t)) +F F 1 0: This shows (2.6), and (2.7) follows similarly. The proof is complete. Lemma 2.2. Let T be a time scale with a; b 2 T: If p > 0; then !p+1 Z b Z (t) Z b p (2.8) r (t) f( ) t (p + 1) R (t; b) (F (t)) F a

a

> 0 so

(t)

t;

a

where (2.9)

1

R (t; b) =

Z

b

r( )

;

and

F (t) =

t

Z

t

f( )

:

a

Proof. From (2.9) and applying integration by parts (2.4) with u (t) = R (t; b) and p+1 (t) = (F (t)) ; we obtain !p+1 Z b Z (t) Z b p+1 r (t) f( ) t = R (t; b) (F (t)) t a

a

a

R (t; b) F p+1 (t)

=

b a

+

Z

b

R (t; b) F p+1 (t)

Using the fact that R (b; b) = 0 and F (a) = 0; we have !p+1 Z b Z (t) Z b (2.10) r (t) f( ) t= R (t; b) F p+1 (t) a

a

t:

a

t:

a

By the chain rule (2.2) and the fact that F (t) = f (t) Z 1 F p+1 (t) = (p + 1) [hF (t) + (1

0 yields p

h) F (t)] F

(t)

0

(p + 1)

Z

1

[hF (t) + (1

p

h) F (t)] F

(t)

0

=

p

(p + 1) (F (t)) F

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SAKER ET AL 1072-1085

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

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DYN AM IC IN EQ U ALITIES O F H ARDY-TYPE

Substituting into (2.10), we get (2.8). The proof is complete. Lemma 2.3. Let T be a time scale with a; b 2 T: If p > 0; then !p+1 Z b Z b Z b R (a; (t)) F p (t) f (t) f( ) t (p + 1) r (t) (2.11)

t;

a

t

a

where (2.12)

R (a; t) =

Z

t

r( )

;

and

F (t) =

Z

b

f( )

Proof. From (2.12) and applying integration by parts (2.4) with u (t) = F p+1 (t) ; we obtain !p+1 Z b Z b Z b f( ) r (t) t = R (a; t) F p+1 (t) t t

a

:

t

a

(t) = R (a; t) and

a

= R (a; t) F

p+1

Z

b a

(t)

b

R (a; (t)) F p+1 (t)

Using the fact that R (a; a) = 0 and F (b) = 0; we have !p+1 Z b Z b Z b (2.13) r (t) f( ) t= R (a; (t)) F p+1 (t) a

t

t:

a

By the chain rule (2.3) and the fact that F F p+1 (t)

t:

a

(t) =

= (p + 1) F p (d) F

f (t)

0 and t

(p + 1) F p (t) F

(t)

d; we see that (t) :

Substituting into (2.13), we get (2.11). The proof is complete. 3. Main Results In this section, we prove the main results. Theorem 3.1. Let T be a time scale with a 2 [0; 1)T ; 0 < p < 1. If !p Z 1 Z (t) r (t) f( ) t < 1; a

a

then (3.1)

Z

1

Z

r (t)

a

(t)

a

f( )

!p

p

t

p

Z

1

a

r

1 p

(t)

Z

1

p

r( )

f p (t) t:

t

Rt Proof. De…ne F (t) = a f ( ) : Integrating the left hand side of (3.1) by parts (2.4) p with u (t) = r (t) and (t) = (F (t)) ; we obtain Z 1 Z 1 p 1 p r (t) (F (t)) t = u (t) F (t)ja u (t) (F p (t)) t a a Z 1 (3.2) = ( u (t)) (F p (t)) t; where u (t) = d (t)) (3.3)

R1 t

a

r( )

: From (2.3), we have (note that F (t) = f (t)

(F p (t)) = pF p

1

(d)F (t)

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p 1

p (F (t))

0 and

f (t):

SAKER ET AL 1072-1085

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO.6, 2020, COPYRIGHT 2020 EUDOXUS PRESS, LLC

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S. H. SAKER, M . M . O SM AN AND I. ABOHELA

Substitute (3.3) into (3.2) and applying Hölder’s inequality (2.5) to get Z 1 Z 1 Z 1 p p 1 r( ) t r (t) (F (t)) t p f (t) (F (t)) t a a Z 1 Z 1 0 0 p r( ) r1=p (t) (F (t)) f (t) r 1=p (t) = p a

Z

p

1

r

1 p

Z

(t)

a

Z

1

t

t

1

1=p

p

f p (t) t

r( )

t

1

1=p p

r (t) (F (t))

0

;

t

a

and consequently, we obtain Z 1 p r (t) (F (t)) t

Z

1=p

p

1

r1

p

Z

(t)

1=p

p

f p (t) t

r( )

;

t

a

a

1

which is (3.1). The proof is complete. Remark 3.1. If T = R, then inequality (3.1) reduces to the Beesack and Heinig integral inequality (1.3). Remark 3.2. If T = N, then inequality (3.1) reduces to the Leindler discrete inequality (1.7). Here, we state the Minkowski inequality [29, Lemma 2.6] on time scales which is needed in the proof of our next main result. Lemma 3.1. Let T be a time scale with a; b 2 T and let f; g be nonnegative rd-continuous functions on [a; b]T : If 1; then Z

(3.4)

Z

b

f (x)

a

(x)

g (t)

a

!

!1=

t

Z

x

Z

b

g (t)

a

b

f (x)

t

!1=

x

t:

Theorem 3.2. Let T be a time scale with a 2 [0; 1)T ; 0 < p < 1. If Z 1 Z 1 p r (t) f( ) t < 1; a

t

then (3.5)

Z

a

1

r (t)

Z

p

f( )

t

p

p

t

Proof. De…ne F (t) := (3.6)

1

Z

1

r

1 p

(t)

a

R1 t

: Since Z F p (t) =

Z

(t)

r( )

a

!p

f p (t) t:

f( )

1

Fp ( )

;

f( )

0; and d

t

so, from (2.3), we have ( note that F ( ) = (3.7)

Fp ( )

= pF p

1

(d)F ( )

pF p

1

)

( )f ( ):

Substitute (3.7) into (3.6) gives F p (t)

p

Z

1

Fp

1

( )f ( )

:

t

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DYN AM IC IN EQ U ALITIES O F H ARDY-TYPE

Applying Minkowski’s inequality and Hölder’s inequality to get Z 1 Z 1 Z 1 F p 1 ( )f ( ) t r (t) F p (t) r(t) t p t a a ! Z Z 1

p

f ( )r

1=p

( )

0

( )

r (t)

1

0

( )r1=p ( )

a

a

= p

t Fp

(Z

1

r

1 p

Z

( )

a

!p

f ( )

r (t)

!p

( )

r (t)

t

a

Z

1

1=p

)1=p

p

0

F p ( )r ( )

;

a

and consequently, we obtain Z

1

Z

1=p

F p (t) r(t) t

p

1

r1

p

Z

( )

a

a

( )

f p( )

t

a

!1=p

;

which is the desired inequality (3.5). The proof is complete. Remark 3.3. If T = R, then inequality (3.5) reduces to the Beesack and Heinig integral inequality (1.4). Remark 3.4. If T = N, then inequality (3.5) reduces to the Leindler discrete inequality (1.8). (See also [28, Remark 3.5]) Theorem 3.3. Let T be a time scale with a 2 [0; 1)T , 1 < p < 1. If Z 1 p (s (t) f (t)) t < 1; a

and

Z

(3.8)

1

s

1

( )

t

Z

1

p p

r (x)

(3.9)

1

r (t)

a

Z

x

C

1

rp ( )

< 1;

t

then Z

Z

0

(t)

f( )

a

!p

t

C

Z

1

p

(s (t) f (t))

t;

a

Rt Proof. Assume …rst that (3.8) holds and de…ne F (t) = a f ( ) : Integrating the left p hand side of (3.9) by parts (2.4) with u (t) = rp (t) and (t) = (F (t)) ; we obtain Z 1 Z 1 p 1 p p r (t) (F (t)) t = u (t) F (t)ja u (t) (F p (t)) t a a Z 1 = ( u (t)) (F p (t)) t; where u (t) =

and so Z 1

R1 t

a

rp ( )

: From (2.3), we have (F p (t))

p

rp (t) (F (t))

t

p

a

Z

1

p 1

p (F (t))

Z

p 1

f (t) (F (t))

Z

1

rp ( )

t

a

= p

f (t);

1

p 1

s (t) f (t) (F (t))

a

s

1

(t)

t Z

1

rp ( )

t:

t

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S. H. SAKER, M . M . O SM AN AND I. ABOHELA

R1 p If we assume that a (s (t) f (t)) t = 1; then Hölder’s inequality (2.5) gives Z 1 Z 1 1=p p p p r (t) (F (t)) t (s (t) f (t)) t p a

a

8 + 1; we have that (Z ) +k Z b

b

(3.12)

q(t) (y ) (t)(y (t))

t

G1 ( ; ; k)

p(t) y (t)

a

k

t

;

a

where G1 ( ; ; k) := c

(Z

b

(q(t)) k

k 1

k 1)(k

(p(t)) (k

R

1)

k k

)k

1 k

;

(t) t

1

a

with k

k k

c=

1

1

+1 +

k

+1 k

;

and

R(t) =

Z

t

a

(p( )) k

:

1 1

From (2.6), inequality (3.12) becomes as follow: If p (t) ; q (t) 2 Crd ([a; b]T ; R) are 1 positive functions and y 2 Crd ([a; b]T ; R) with y > 0 satis…es y (a) = 0; then for 1; 0 and k > + 1 (Z ) +k Z b

(3.13)

a

1

q(t) jy (t)j

y (t)

b

+1

t

G1 ( ; ; k)

k

p(t) y (t)

;

t

a

where G1 ( ; ; k) is de…ned as in (3.12). Theorem 3.6. Let T be a time scale with a; b 2 T: If p > 0 and k > 1; then (3.14) !p+1 "Z # p+1 Z b Z (t) k b k r (t) f( ) t (p + 1) G1 (p + 1; k) s(t) (f (t)) t ; a

a

a

where 2 Z 4 G1 (p + 1; k) :=

b

(R (t; b)) k

Z

k 1

p+1

t

s

1 k 1

( )

a

a

3 kk 1

!

t5

and R (t; b) is de…ned as in (2.9).

;

Proof. Applying Opial’s inequality (3.13) with y (t) = F (t) ; q (t) = R (t; b) ; p (t) = s (t) ; = p + 1 and = 0; we obtain "Z # p+1 Z k b

(3.15)

b

p

R (t; b) (F (t)) F

(t)

t

G1 (p + 1; k)

a

k

s(t) (f (t))

t

:

a

The result follows from (2.8) and (3.15). The proof is complete. Theorem 3.7. Let T be a time scale with a; b 2 T: If p > 0 and k > 1; then !p+1 "Z Z Z b

(3.16)

b

r (t)

a

b

f( )

t

(p + 1) K2 (p; 1; k)

k

s(t) (f (t))

t

a

t

# p+1 k

;

where K2 (p; 1; k) :=

1 p+1

1 k

"Z

b

(R (a; (t))) k

a

k 1

s

1 k 1

(t)

Z

t

b

s

1 k 1

( )

!p

t

# kk 1

;

and R (a; t) is de…ned as in (2.12).

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S. H. SAKER, M . M . OSM AN AND I. ABOHELA

Proof. Applying Opial’s inequality (3.11) with y (t) = F (t) ; q (t) = R (a; (t)) ; p (t) = s (t) ; = p and = 1; we obtain "Z # p+1 Z k b

b

R (a; (t)) F p (t) f (t)

(3.17)

t

k

s(t) (f (t))

K2 (p; 1; k)

t

:

a

a

The result follows from (2.11) and (3.17). The proof is complete. The next result follows from Theorems 3.6 and 3.7 by choosing k = p + 1. Corollary 3.1. Let T be a time scale with a; b 2 T: If k > 1, then !k Z b Z (t) Z b k (3.18) r (t) f( ) t kG1 (k) s(t) (f (t)) a

where

a

2 Z G1 (k) := 4

b

(R (t; b)) k

Z

k 1

Z

(3.19)

Z

b

r (t)

a

k

t

s

1 k 1

( )

a

a

and

t;

a

!k

b

f( )

t

t

kK2 (k)

Z

!

b

3 kk 1

t5 k

s(t) (f (t))

;

t;

a

where 1 k

1 k

K2 (k) :=

2 Z 4

b

(R (a; (t))) k

k 1

s

1 k 1

Z

(t)

b

s

1 k 1

( )

t

a

!k

3 kk 1

1

t5

:

Remark 3.5. Note that Theorems 3.6 and 3.7 are consequences of the weighted Hardytype inequality due to Saker et al. [29, 36] with p + 1 = q and k = p: As special cases of Theorems 3.6 and 3.7 when T = N; we have the following new discrete results Corollary 3.2. Let fxn g ; f n g and fwn g be nonnegative sequences. If p > 0 and k > 1; then !p+1 ! p+1 k N n N X X X k rn xi (p + 1) G1 (p + 1; k) sn xn ; n=1

n=1

i=1

where

2

G1 (p + 1; k) := 4

N X

(R (n; N )) k

n X

k 1

n=1

with R (n; N ) =

PN

i=1

i=n ri :

!p+1 3 k k 1 1 5 (si ) k 1 ;

Corollary 3.3. Let fxn g ; f n g and fwn g be nonnegative sequences. If p > 0 and k > 1; then !p+1 ! p+1 k N N N X X X k rn xi (p + 1) K2 (p; 1; k) sn xn ; n=1

n=1

i=n

where

1 K2 (p; 1; k) := p+1 Pn with R (1; n + 1) = i=1 ri :

1 k

"

N X

(R (1; n + 1)) k

n=1

k 1

(sn )

1 k 1

N X i=n

1081

(si )

1 k 1

!p # k k 1

;

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DYN AM IC IN EQ U ALITIES O F H ARDY-TYPE

By making suitable substitutions for the two weighted functions r (t) and s (t) ; we µ get some extensions related to the dynamic inequalities due to Rehák [25] and Saker et al. [30, 31] respectively. Also when T = N and T = R; we get consequences due to Bennett [6], Bliss [9] and Flett [15]. For illustrations, we will present these special cases in the following examples. k

Example 3.1. If r (t) = ( (t) a) and s (t) = 1; then inequality (3.18) reduces to the following extension of the Hardy-type inequality due to µRehák [25, Theorem 2.1] !k Z (t) Z 1 Z 1 1 f( ) t kR1 f k (t) t; (t) a a a a where R1 :=

Z

a

1

k

(R (t; 1))

k k

1 k

k

(t

1

a)

t

:

Example 3.2. If we choose r (t) = 1=t and s (t) = 1=t k ; > 1 in Corollary 3.1, we get the inequality !k Z (t) Z 1 Z 1 1 1 k f( ) t kR2 f (t) t; k t t a a a which is related to the inequality due to Saker and O’Regan [30, Theorem 2.2], where 2 3 kk 1 0 !k 1 1 Z 1 Z t k 1 k 1 6 7 A (R (t; 1)) k 1 @ R2 := 4 t5 : k a

a

Example 3.3. If we choose r (t) = 1= have the inequality Z 1 Z 1 1 f( ) (t) a t

k

(t) and s (t) = 1=

k

t

kR3

Z

1

(t) in Corollary 3.1, we

1 k

a

(t)

f k (t) t;

which is related to the inequality due to Saker and O’Regan [30, Theorem 2.1], where # kk 1 1 "Z Z 1 k 1 1 k k k 1 k t R3 := (R (a; (t))) k 1 ( (t)) k 1 ( ( )) k 1 : k a t Example 3.4. If we take f (t) = r (t) =

(t) g (t) ;

(t) ; s (t) = (t))

(

1 k

k

(t) (

(t))

t

Z

; k

> 1;

in Corollary 3.1, we have the inequality Z

a

b

(

(t) (t))

Z

!k

(t)

( )g( )

a

kR4

b

(t) (

k

g k (t)

(t))

which is related to the inequality due to Saker et al. [31, Theorem 2.1], where Rt ( ) and a 2 Z 4 R4 :=

a

b

(R (t; b)) k

k 1

Z

a

1082

t;

a

k

t

sk

1 1

( )

!

3 kk 1

t5

(t) =

:

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S. H. SAKER, M . M . OSM AN AND I. ABOHELA

Example 3.5. If we choose r (t) = t 1 (p+1) =tp+1 and s (t) = t 1 k ; > 1 in Theorem 3.6, we obtain the inequality !p+1 # p+1 "Z R (t) Z b k b f ( ) a t 1 (p+1) ; t (p + 1) R5 t 1 k f k (t) t t a a which is related to the inequalities due to Flett [15] and Bliss [9], Hardy and Littlewood [18] (with = 1=k), where 2 3 kk 1 ! Z t Z b p+1 k 1 R5 := 4 (R (t; b)) k 1 sk 1 ( ) t5 : a

a

Example 3.6. If we take rn =

n 1 n

(p+1) (1 k

c)

; sn =

1 k n

k c n ;

c > 1 and xn =

n yn ;

in Corollary 3.2, we get the inequality N X

(p+1)(1 k

c)

1

n

n

n=1

n X

i yi

i=1

!p+1

(p + 1) R6

N X

n=1

n

k c k n yn

! p+1 k

Pn which is related to Bennett’s inequality [6, Corollary 7], where n = i=1 2 !p+1 3 k k 1 N n X X 1 k 5 (R (n; N )) k 1 (si ) k 1 ; R6 := 4 n=1

with R (n; N ) =

PN

i

i=n

(p+1)(1 k

i

i

; and

i=1

c)

1

:

Remark 3.6. As an application, we can apply Opial’s inequalities together with a Hardytype inequality (3.16) on time scales to establish some lower bounds of the distance between zeros of a solution and/or its derivatives for the fourth-order dynamic equation (see [13, Theorem 5.1]) (3.20)

r(t)y

3

(t)

p(t)y (t)

+ q(t)y (t) = 0;

t 2 [a; b]T :

Availability of supporting data: The authors declare that all data and materials in the article are available and veritable. Competing interests: The authors declare that they have no competing interests. Funding: Not applicable Authors’contributions: The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the …nal manuscript. Acknowledgements: The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. Authors’information: S. H. Saker: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, E-mails:[email protected] M. Osman: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, E-mails: [email protected] Islam Abohela: College of Engineering, Applied Science University, Kingdom of Bahrain, E-mail: [email protected]

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13

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 28, NO. 6, 2020

Sharp Inequalities Between Toader and Neuman Means, Wei-Mao Qian, Zai-Yin He, and Yu-Ming Chu,………………………………………………………………………………929 On Strongly Starlikeness of Strongly Convex Functions, Adel A. Attiya, Nak Eun Cho, and M. F. Yassen,……………………………………………………………………………………941 Invariance Analysis of a Four-Dimensional System of Fourth-Order Difference Equations with Variable Coefficients, Mensah Folly-Gbetoula,…………………………………………….949 Dynamics of an Anti-Competitive System of Difference Equations, J. Ma and A. Q. Khan,962 An Iterative Scheme for Solving Split System of Minimization Problems, Anteneh Getachew Gebrie and Rabian Wangkeeree,…………………………………………………………….968 Complex Korovkin Theory, George A. Anastassiou,………………………………………..981 Additive 𝜌𝜌-Functional Inequalities in Non-Archimedean Banach Spaces, Inho Hwang,……997

Square Root and 3rd Root Functional Equations in C*-Algebras, Choonkil Park, Sun Young Jang, and, Jieun Ahn,………………………………………………………………………1004 Approximation by Multivariate Sublinear and Max-product Operators, Revisited, George A. Anastassiou,………………………………………………………………………………..1011 New Dynamic Inequalities on Time Scales by Using the Sneak-Out Principle, S. H. Saker, M. M. Osman, and I. Abohela,……………………………………………………………………1047 Additive-Quadratic Functional Inequalities in Fuzzy Normed Spaces and Its Stability, Chang Il Kim and Giljun Han,………………………………………………………………………1057 New Characterizations of Weights in Hardy’s Type Inequalities via Opial’s Dynamic Inequalities, S. H. Saker, M. M. Osman, and I. Abohela,…………………………………1072