JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS VOLUME 13 - 2015 0649913201, 0644701007

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VOLUME 13, NUMBERS 1-2 APRIL 2015

JANUARY-

ISSN:1548-5390 PRINT, 1559-176X ONLINE

JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS EUDOXUS PRESS,LLC

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SCOPE AND PRICES OF THE JOURNAL Journal of Concrete and Applicable Mathematics A quartely international publication of Eudoxus Press,LLC Editor in Chief: George Anastassiou Department of Mathematical Sciences, University of Memphis Memphis, TN 38152, U.S.A. [email protected] Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC 28601, USA. The main purpose of the "Journal of Concrete and Applicable Mathematics" is to publish high quality original research articles from all subareas of Non-Pure and/or Applicable Mathematics and its many real life applications, as well connections to other areas of Mathematical Sciences, as long as they are presented in a Concrete way. It welcomes also related research survey articles and book reviews.A sample list of connected mathematical areas with this publication includes and is not restricted to: Applied Analysis, Applied Functional Analysis, Probability theory, Stochastic Processes, Approximation Theory, O.D.E, P.D.E, Wavelet, Neural Networks,Difference Equations, Summability, Fractals, Special Functions, Splines, Asymptotic Analysis, Fractional Analysis, Inequalities, Moment Theory, Numerical Functional Analysis,Tomography, Asymptotic Expansions, Fourier Analysis, Applied Harmonic Analysis, Integral Equations, Signal Analysis, Numerical Analysis, Optimization, Operations Research, Linear Programming, Fuzzyness, Mathematical Finance, Stochastic Analysis, Game Theory, Math.Physics aspects, Applied Real and Complex Analysis, Computational Number Theory, Graph Theory, Combinatorics, Computer Science Math.related topics,combinations of the above, etc. In general any kind of Concretely presented Mathematics which is Applicable fits to the scope of this journal. Working Concretely and in Applicable Mathematics has become a main trend in many recent years,so we can understand better and deeper and solve the important problems of our real and scientific world. "Journal of Concrete and Applicable Mathematics" is a peer- reviewed International Quarterly Journal. We are calling for papers for possible publication. The contributor should send via email the contribution to the editor in-Chief: TEX or LATEX (typed double spaced) and PDF files. [ See: Instructions to Contributors]

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Editorial Board Associate Editors of Journal of Concrete and Applicable Mathematics

Editor in -Chief: George Anastassiou Department of Mathematical Sciences The University Of Memphis Memphis,TN 38152,USA tel.901-678-3144,fax 901-678-2480 e-mail [email protected] www.msci.memphis.edu/~ganastss Areas:Approximation Theory, Probability,Moments,Wavelet, Neural Networks,Inequalities,Fuzzyness. Associate Editors: 1)Ravi P. Agarwal Chairman Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 Office: 361-593-2600

Email: [email protected] Differential Equations,Difference Equations,Inequalities

2) Carlo Bardaro Dipartimento di Matematica & Informatica Universita' di Perugia Via Vanvitelli 1 06123 Perugia,ITALY tel.+390755855034, +390755853822, fax +390755855024 [email protected] , [email protected] Functional Analysis and Approximation Th., Summability,Signal Analysis,Integral Equations, Measure Th.,Real Analysis 3) Francoise Bastin Institute of Mathematics University of Liege 4000 Liege BELGIUM [email protected] Functional Analysis,Wavelets

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4) Yeol Je Cho Department of Mathematics Education College of Education Gyeongsang National University Chinju 660-701 KOREA tel.055-751-5673 Office, 055-755-3644 home, fax 055-751-6117 [email protected] Nonlinear operator Th.,Inequalities, Geometry of Banach Spaces

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tel.++56 2 354 5922 fax.++56 2 552 5916 [email protected] Partial Differential Equations, Mathematical Physics, Scattering and Spectral Theory 9) A.M.Fink Department of Mathematics Iowa State University Ames,IA 50011-0001,USA tel.515-294-8150 [email protected] Inequalities,Ordinary Differential Equations 10) Sorin Gal Department of Mathematics University of Oradea Str.Armatei Romane 5 3700 Oradea,Romania [email protected] Approximation Th.,Fuzzyness,Complex Analysis 11) Jerome A.Goldstein Department of Mathematical Sciences The University of Memphis, Memphis,TN 38152,USA tel.901-678-2484 [email protected] Partial Differential Equations, Semigroups of Operators 12) Heiner H.Gonska Department of Mathematics University of Duisburg Duisburg,D-47048 Germany tel.0049-203-379-3542 office [email protected] Approximation Th.,Computer Aided Geometric Design 13) Dmitry Khavinson Department of Mathematical Sciences University of Arkansas Fayetteville,AR 72701,USA tel.(479)575-6331,fax(479)575-8630 [email protected] Potential Th.,Complex Analysis,Holomorphic PDE, Approximation Th.,Function Th.

30) Bl. Sendov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia 1090,Bulgaria [email protected] Approximation Th.,Geometry of Polynomials, Image Compression 31) Igor Shevchuk Faculty of Mathematics and Mechanics National Taras Shevchenko University of Kyiv 252017 Kyiv UKRAINE [email protected] Approximation Theory 32) H.M.Srivastava Department of Mathematics and Statistics University of Victoria Victoria,British Columbia V8W 3P4 Canada tel.250-721-7455 office,250-477-6960 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. 33) Stevo Stevic Mathematical Institute of the Serbian Acad. of Science Knez Mihailova 35/I 11000 Beograd, Serbia [email protected]; [email protected] Complex Variables, Difference Equations, Approximation Th., Inequalities 34) Ferenc Szidarovszky Dept.Systems and Industrial Engineering The University of Arizona Engineering Building,111 PO.Box 210020 Tucson,AZ 85721-0020,USA [email protected] Numerical Methods,Game Th.,Dynamic Systems, Multicriteria Decision making, Conflict Resolution,Applications in Economics and Natural Resources Management

35) Gancho Tachev 14) Virginia S.Kiryakova Institute of Mathematics and Informatics Dept.of Mathematics

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Univ.of Architecture,Civil Eng. and Geodesy 1 Hr.Smirnenski blvd BG-1421 Sofia,Bulgaria [email protected] Approximation Theory

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J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2, 11-22 , 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 11

INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER AHMET ALTUNDAG

Abstract. The inverse problem under consideration is to reconstruct the conductive function of a coated dielectric infinite cylinder from the far field pattern for scattering of a time-harmonic E-polarized electromagnetic plane wave. We propose an inverse algorithm that extends the approach suggested by Akduman and Kress [1] for an impedance cylinder embedded in an homogeneous background medium. It is based on a system of nonlinear boundary integral equation associated with a single-layer potential approach to solve the forward scattering problem. We present the mathematical foundations of the method and exhibit its feasibility by numerical examples.

1. Introduction The problem is to determine the conductive function defined on the coated boundary from scattering of time-harmonic E-polarized electromagnetic plane waves. In the current paper we deal with dielectric scatterers covered by a thin boundary layer described by a conductive boundary condition and confine ourselves to the case of infinitely long cylinders. Let the simply connected bounded domain D ⊂ IR2 with C 2 boundary ∂D represents the cross section of an infinitely long homogeneous dielectric cylinder having constant wave number kd with Im{kd }, IRe{kd } > 0 embedded in a homogeneous background with positive wave number k0 . Denote by ν the outward unit normal vector to ∂D. Then, given an incident plane wave ui = eik0 x.d with incident direction given by the unit vector d, the direct scattering problem for E-polarized electromagnetic waves by a coated dielectric is modeled by the following conductive boundary 1 ¯ and v ∈ H 1 (D) to the (IR2 \ D) value problem for the Helmholtz equation: Find solutions u ∈ Hloc Helmholtz equations ¯ (1.1) ∆u + k 2 u = 0 in R2 \ D, ∆v + k 2 v = 0 in D 0

d

with the conductive boundary conditions ∂u ∂v = + iηv on ∂D ∂ν ∂ν for some complex valued function η ∈ C 1 (∂D) with Re{η} ≤ 0 and the total field is given by u = ui + us with the scattered wave us fulfilling the Sommerfeld radiation condition  s  ∂u (1.3) lim r1/2 − ik0 us = 0, r = |x|, r→∞ ∂r uniformly with respect to all directions. The latter is equivalent to an asymptotic behavior of the form      eik0 |x| x 1 (1.4) us (x) = p u∞ +O , |x| → ∞, |x| |x| |x| (1.2)

u = v,

Key words and phrases. inverse scattering; Helmholtz equation; transmission problem; singlelayer approach; nonlinear integral equations. 1

AHMET ALTUNDAG

12

2 INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER

uniformly in all directions, with the far field pattern u∞ defined on the unit circle S 1 in IR2 (see[10]). In the above, u and v represent the electric field that is parallel to the cylinder axis, (1.1) corresponds to the time-harmonic Maxwell equations and the conductive boundary conditions (1.2) model the continuity of the tangential components of the electric and magnetic field across the interface ∂D with the term iηv modelling the boundary layer. The inverse obstacle problem we are interested in is, given the conductive layer and the far field pattern u∞ for one incident plane wave with incident direction d ∈ S 1 to determine the conductive function η. More generally, we also consider the reconstruction of η from the far field patterns for a small finite number of incident plane waves with different incident directions. This inverse problem is nonlinear and ill-posed, since the solution of the scattering problem (1.1)–(1.3) is nonlinear with respect to the conductive function and since the mapping from the conductive function into the far field pattern is extremely smoothing. For a stable solution of the inverse problem we propose an algorithm that extends the approach suggested by Akduman and Kress[1] for the case of an infinitely long impedance cylinder with arbitrarily shaped cross section embedded in a homogeneous background. Representing the solution ¯ v and us to the forward scattering problem in terms of single-layer potentials in D and in IR2 \ D with densities ϕd and ϕ0 , respectively, the conductive boundary condition (1.2) provides a system of two boundary integral equations on ∂D for the corresponding densities, that in the sequel we will denote as field equations. For the inverse problem, the required coincidence of the far field of the single-layer potential representing us and the given far field u∞ provides a further equation that we denote as data equation. The system of the field and data equations can be viewed as three equations for three unknowns, i.e., the two densities and the conductive function η. They are linear with respect to the densities and nonlinear with respect to the conductive function. To some extend, the inverse problem consists in solving a certain Cauchy problem, i.e., extending a solution to the Helmholtz equation from knowing their Cauchy data on some boundary curve. With this respect we also mention the related work of Ben Hassen, Ivanyshyn and Sini [6], Cakoni and Colton [7], Cakoni, Colton and Monk [8], Eckel and Kress [11], Fang and Zeng [12], Ivanyshyn and Kress [14], Jakubik and Potthast [15]. For the simultaneous reconstruction of the shape and the impedance function in a homogeneous background we refer to Kress and Rundell [19], Liu, Nakamura and Sini[21], Nakamura and Sini [23], and Serranho [24]. The plan of the paper is as follows: In Section 2, as ingredient of our inverse algorithm we provide an existence proof for the solution of the forward scattering problem via a single-layer approach followed by a corresponding numerical solution method in Section 3. The details of the inverse algorithm are presented in Section 4 and in Section 5 we demonstrate the feasibility of the method by some numerical examples.

2. The direct problem The forward scattering problem (1.1)–(1.3) has at most one solution (see Gerlach and Kress [13]). Existence can be proven via boundary integral equations by a combined single- and double-layer approach (see Gerlach and Kress [13]). Here, as one of the ingredients of our inverse algorithm, we follow [4] and suggest a single-layer approach. For this we denote by Φk (x, y) :=

i (1) H (k|x − y|), 4 0

x 6= y,

the fundamental solution to the the Helmholtz equation with wave number k in IR2 in terms of (1) the Hankel function H0 of order zero and of the first kind. Adopting the notation of [10], in a

AHMET ALTUNDAG

13

INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER 3

Sobolev space setting, for k = kd and k = k0 , we introduce the single-layer potential operators Sk : H −1/2 (∂D) → H 1/2 (∂D) by Z (2.5)

(Sk ϕ)(x) := 2

Φk (x, y)ϕ(y) ds(y),

x ∈ ∂D,

∂D

and the normal derivative operators Kk0 : H −1/2 (∂D) → H −1/2 (∂D) by (2.6)

(Kk0 ϕ)(x)

Z := 2 ∂D

∂Φk (x, y) ϕ(y) ds(y), ∂ν(x)

x ∈ ∂D.

For the Sobolev spaces and the mapping properties of these operators we refer to [17, 22]. Then, from the jump relations it can be seen that the single-layer potentials Z v(x) = Φkd (x, y)ϕd (y) ds(y), x ∈ D, ∂D

(2.7) us (x) =

Z Φk0 (x, y)ϕ0 (y) ds(y),

¯ x ∈ IR2 \ D,

∂D

solve the scattering problem (1.1)–(1.3) provided the densities ϕd and ϕ0 satisfy the system of integral equations Skd ϕd − Sk0 ϕ0 = 2ui |∂D , (2.8) ϕd + ϕ0 + iηSkd ϕd +

Kk0 d ϕd

−

Kk0 0 ϕ0

∂ui = 2 . ∂ν ∂D

Theorem 2.1. Provided k0 is not a Dirichlet eigenvalue of the negative Laplacian for the domain D the system (2.8) has a unique solution in H −1/2 (∂D) × H −1/2 (∂D). Proof. We first establish that (2.8) has at most one solution. If ϕd and ϕ0 satisfy the homogeneous form of (2.8) then the single-layer potentials (2.7) solve the scattering problem with zero incident field. Consequently, since the forward scattering problem has at most one solution, we have us = 0 ¯ and v = 0 in D. Then us is also defined in D and has vanishing trace on ∂D. Therefore, in IR2 \ D the assumption on k0 implies that us = 0 also in D and consequently ϕ0 = 0 on ∂D by the jump ¯ also has vanishing trace on ∂D and consequently relations. Analogously, v considered in IR2 \ D using the radiation condition if kd is real valued or the exponential decay at infinity if Im kd > 0 ¯ Again the jump relations imply ϕd = 0 on ∂D and the uniqueness proof we have v = 0 in IR2 \ D. for the solution of the system (2.8) is completed. To establish existence of a solution, we note that due to the assumption on k0 the inverse operator Sk−1 : H 1/2 (∂D) → H −1/2 (∂D) exists and is bounded. With its aid, we can equivalently 0 transform (2.8) into ϕd + Sk−1 [Skd − Sk0 ]ϕd − ϕ0 = 2Sk−1 ui |∂D , 0 0 (2.9) ϕd + ϕ0 + iηSkd ϕd + Kk0 d ϕd − Kk0 0 ϕ0 = 2

∂ui . ∂ν ∂D

AHMET ALTUNDAG

14

4 INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER

Now (2.9) is of the form 

ϕd







ϕ0

  = 2 

 + B

A ϕ0



ϕd

Sk−1 ui |∂D 0



   ∂ui ∂ν ∂D

with the matrix operators A, B : H −1/2 (∂D) × H −1/2 (∂D) → H −1/2 (∂D) × H −1/2 (∂D) given by 

−I

I

A= I







and B = 

I

Sk−1 [Skd − Sk0 ] 0 iηSkd +

Kk0 d

0 −Kk0 0

 .

Clearly, A has a bounded inverse and B is compact since its components are compact. In particular, Skd − Sk0 : H −1/2 (∂D) → H 1/2 (∂D) is compact because of cancellation of singularities in the two single-layer operators. Therefore existence of a solution follows from uniqueness by the Riesz–Fredholm theory for compact operators (see [17]).  We note that instead of using Sobolev spaces the existence analysis can also be carried out in a classical H¨older space setting replacing H −1/2 (∂D) by C 0,α (∂D). 3. Numerical solution For the numerical solution of (2.8) and the presentation of our inverse algorithm we assume that the boundary curve ∂D is given by a regular 2π–periodic parameterization ∂D = {z(t) : 0 ≤ t ≤ 2π}.

(3.1)

Then, via ψ = ϕ ◦ z. emphasizing the dependence of the operators on the boundary curve, we introduce the parameterized single-layer operator Sek : H −1/2 [0, 2π] × C 2 [0, 2π] → H 1/2 [0, 2π] by Z i 2π (1) H0 (k|z(t) − z(τ )|) |z 0 (τ )| ψ(τ ) dτ Sek (ψ, z)(t) := 2 0 and the parameterized normal derivative operators e k0 : H −1/2 [0, 2π] × C 2 [0, 2π] → H −1/2 [0, 2π] K by e k0 (ψ, z)(t) := ik K 2

Z 0

2π

[z 0 (t)]⊥ · [z(τ ) − z(t)] (1) H1 (k|z(t) − z(τ )|) |z 0 (τ )| ψ(τ ) dτ |z 0 (t)| |z(t) − z(τ )| (1)0

(1)

(1)

for t ∈ [0, 2π]. Here we made use of H0 = −H1 with the Hankel function H1 of order zero and of the first kind. Furthermore, we write a⊥ = (a2 , −a1 ) for any vector a = (a1 , a2 ), that is, a⊥ is obtained by rotating a clockwise by 90 degrees. Then the parameterized form of (2.8) is given by Sekd (ψd , z) − Sek0 (ψ0 , z) = 2 ui ◦ z, (3.2) e 0 (ψd , z) − K e 0 (ψ0 , z) = ψd + ψ0 + η ◦ z Sekd (ψd , z) + K kd k0

2 [z 0 ]⊥ · grad ui ◦ z. |z 0 |

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INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER 5

The kernels M (t, τ ) :=

i (1) H (k|z(t) − z(τ )|) |z 0 (τ )| 2 0

and L(t, τ ) :=

ik [z 0 (t)]⊥ · [z(τ ) − z(t)] (1) H1 (k|z(t) − z(τ )|) |z 0 (τ )| 2 |z 0 (t)| |z(t) − z(τ )|

e 0 can be written in the form of the operators Sek and K k   2 t−τ + M2 (t, τ ), M (t, τ ) = M1 (t, τ ) ln 4 sin 2 (3.3)   2 t−τ L(t, τ ) = L1 (t, τ ) ln 4 sin + L2 (t, τ ), 2 where 1 J0 (k|z(t) − z(τ )|)|z 0 (τ )|, 2π   2 t−τ M2 (t, τ ) := M (t, τ ) − M1 (t, τ ) ln 4 sin , 2

M1 (t, τ ) := −

k [z 0 (t)]⊥ · [z(τ ) − z(t)] J1 (k|z(t) − z(τ )|) |z 0 (τ )|, 2π |z 0 (t)| |z(t) − z(τ )|   t−τ L2 (t, τ ) := L(t, τ ) − L1 (t, τ ) ln 4 sin2 . 2

L1 (t, τ ) := −

The functions M1 , M2 , L1 , and L2 turn out to be smooth with diagonal terms    i C 1 k 0 M2 (t, t) = − − ln |z (t)| |z 0 (t)| 2 π π 2 in terms of Euler’s constant C and L2 (t, t) = −

1 [z 0 (t)]⊥ · z 00 (t) . 2π |z10 (t)|2

For integral equations with kernels of the form (3.3) a combined collocation and quadrature methods based on trigonometric interpolation as described in Section 3.5 of [10] or in [20] is at our disposal. We refrain from repeating the details. For a related error analysis we refer to [17] and note that we have exponential convergence for smooth, i.e., analytic boundary curves ∂D. For a numerical example, we consider the scattering of a plane wave by a dielectric cylinder with a non-convex kite-shaped cross section with boundary ∂D described by the parametric representation z(t) = (cos t + 0.65 cos 2t − 0.65, 1.5 sin t),

(3.4)

0 ≤ t ≤ 2π.

The following conductive functions are chosen in our experiments. • η1 = − sin4 (0.5t) + i cos4 (0.5t)

(3.5) • (3.6)

η2 = −1.5 − sin3 t + i sin t

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6 INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER

• 2

η3 = −0.5e−(t−pi) + i(0.6 + 0.2 sin t)

(3.7)

From the asymptotics for the Hankel functions, it can be deduced that the far field pattern of the single-layer potential us with density ϕ0 is given by Z (3.8) u∞ (ˆ x) = γ e−ik0 xˆ·y ϕ0 (y) ds(y), x ˆ ∈ S1, ∂D

where

π

ei 4 γ=√ . 8πk0 The latter expression can be evaluated by the composite trapezoidal rule after solving the system of integral equations (2.8) for ϕ0 , i.e., after solving (3.2) for ψ0 . Table 3.1 gives some approximate values for the far field pattern u∞ (d) and u∞ (−d) in the forward direction d and the backward direction −d. The direction d of the incident wave is d = (1, 0) and the wave numbers are k0 = 2.8 and kd = 1 + 1i, and the conductive function η1 is chosen. Note that the exponential convergence is clearly exhibited. Table 3.1. Numerical results for direct scattering problem n

Re u∞ (d)

Im u∞ (d)

Re u∞ (−d)

Im u∞ (−d)

8 16 32 64

-2.5727739209 -2.6086999117 -2.6087198359 -2.6087198065

0.4381005402 0.5099897666 0.5099895695 0.5099895747

-2.4613551693 -2.4645038927 -2.4645127478 -2.4645127414

0.6118414535 0.5815194742 0.5815282913 0.5815282789

4. The inverse problem The inverse scattering problem that we are concerned with is, given the shape of the scatterer, to determine the conductive function η from a knowledge of the far field pattern for one or several incident plane waves. The inverse problem is ill-posed since the mapping taking conductive function η into the farfield pattern associated with the scattering problem (1.1) and (1.2) is highly smoothing since the far field is an analytic function. We will handle this issue of ill-posedness by using Tikhonov regularization. We note that the far field pattern for one incident plane wave uniquely determine the conductive function η. As a consequence of Rellich’s lemma (see [10]), the far field ¯ provided Imkd > 0. Then from the first condition in (1.2) pattern uniquely determine us in IR2 \ D using Imkd > 0 we observe that v is also uniquely determined in D. From (1.2) we can read off the uniqueness of conductive function η if we assume that ∂D is analytic since in this case v can not vanish on open intervals of ∂D. We now proceed describing an algorithm for approximately solving the inverse scattering problem by extending the method proposed by Akduman and Kress [1]. After introducing the far field operator S∞ : H −1/2 (∂D) → L2 (S 1 ) by Z (4.1) (S∞ ϕ)(ˆ x) := γ e−ik0 xˆ·y ϕ(y) ds(y), x ˆ ∈ S1, ∂D

from (2.7) and (3.8) we observe that the far field pattern for the solution to the scattering problem (1.1)–(1.3) is given by (4.2)

u∞ = S∞ ϕ0

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INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER 7

in terms of the solution to (2.8). Here S 1 denotes the unit circle in IR2 . Therefore we can state the following theorem as theoretical basis of our inverse algorithm. Theorem 4.1. For a given incident field ui and a given far field pattern u∞ , assume that ∂D and the densities ϕd and ϕ0 satisfy the system of three integral equations Skd ϕd − Sk0 ϕ0 = 2ui , (4.3)

ϕd + ϕ0 + iηSkd ϕd + Kk0 d ϕd − Kk0 0 ϕ0 = 2

∂ui , ∂ν

S∞ ϕ0 = u∞ . Then η solves the inverse problem. Given the far field pattern u∞ , the density ϕ0 is found by solving the third equation in (4.3), i.e., the data equation (4.4)

S∞ ϕ0 = u∞ . 2

2

1

Since the operator S∞ : L (∂D) → L (S ) is compact, it can not have bounded inverse. Therefore the equation (4.4) is ill-posed. We also require the parameterized version Se∞ : H −1/2 [0, 2π] × C 2 [0, 2π] → L2 (S 1 ) of the far field operator as given by Z (4.5)

Se∞ (ψ, z)(ˆ x) := γ

2π

e−ik0 xˆ·z(τ ) ψ(τ ) dτ,

x ˆ ∈ S1.

0

Then the parameterized form of (4.3) is given by Sekd (ψd , z) − Sek0 (ψ0 , z) = 2 ui ◦ z, (4.6)

e 0 (ψ0 , z) = e 0 (ψd , z) − K ψd + ψ0 + iη ◦ z Sekd (ψd , z) + K k0 kd

2 [z 0 ]⊥ · grad ui ◦ z, |z 0 |

Se∞ (ψ0 , z) = u∞ . The third equation of (4.6) requires stabilization and for this we use Tikhonov regularization, i.e., the ill-posed data equation is replaced by ∗ e ∗ αψ0 + Se∞ S∞ ψ0 = Se∞ u∞ ,

(4.7)

with some positive regularization parameter α and the adjoint operator Se∗ : L2 (S 1 ) → L2 [0, 2π] of Se∞ . After finding the density ψ0 from the (4.7) we can now find density ψd from the first equation of (4.6). ψd = Sek−1 (2ui ◦ z − Sekd (ψ0 , z)) d

(4.8)

Now it remains to find the conductive function η from the second equation of (4.6). i

(4.9)

η ◦ z = −i

e 0 (ψd , z) + K e 0 (ψ0 , z) 2 ∂u∂ν◦z − ψ0 − ψd − K kd k0 Sek (ψd , z) d

The reconstruction of the conductive function from equation (4.9) will be sensitive to errors due to the fact that it blows up in the vicinity of zeros of the Sekd (ψd , z). To obtain a more stable solution

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8 INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER

(see Akduman and Kress [1]), we express the unknown conductive function in terms of some basis functions µj , j = 0, ±1, ±2, . . . , ±N as a linear combination (4.10)

η=

N X

aj µj

on ∂D.

j=−N

A possible choice of basis functions consists of splines or trigonometric polynomials. We satisfy the second equation of (4.6) in a least square sense, penalized via Tikhonov regularization, for the unknown coefficient a−N , . . . , aN i.e., we determine the coefficients a−N , . . . , aN in (4.10) such that for a set of grid points z1 , . . . , zM on ∂D the least square sum 2 N M i X X ∂u 0 0 e e e ψd (zm )+ψ0 (zm )+i (z ) (4.11) a µ (z ) S ψ (z )+ K ψ (z )− K ψ (z )− m j j m kd d m kd d m k0 0 m ∂ν m=1 j=−N is minimized. The above algorithm has a straightforward extension for the case of more than one incident wave. Assume that ui1 , . . . , uiP are P incident waves with different incident directions and u∞,1 , . . . , u∞,P the corresponding far field patterns for scattering from ∂D. Then the inverse problem to determine the unknown conductive function η from these given far field patterns and incident fields is equivalent to solving Sekd (ψd,p , z) − Sek0 (ψ0,p , z) = 2 uip ◦ z, (4.12)

e 0 (ψd,p , z) + ψ0,p − K e 0 (ψ0,p , z) = ψd,p + iη ◦ z Sekd (ψd,p , z) + K kd k0

2 [z 0 ]⊥ · grad uip ◦ z, |z 0 |

Se∞ (ψ0,p , z) = u∞,p . for p = 1, . . . , P . We first solve the first field and data equations in (4.12) for p = 1, . . . , P to obtain 2P densities ψd,1 , . . . , ψd,P and ψ0,1 , . . . , ψ0,P . We satisfy second equation of (4.12) in a least square sense, penalized via Tikhonov regularization, for the unknown coefficient a−N , . . . , aN such that for a set of grid points z1 , . . . , zM on ∂D the least square sum (4.13) 2 P X M N i X X ∂u p 0 0 e e e ψd,p (zm )+ψ0,p (zm )+i aj µj (zm )Skd ψd,p (zm )+ Kkd ψd,p (zm )− Kk0 ψ0,p (zm )− (zm ) ∂ν p=1 m=1 j=−N is minimized. 5. Numerical examples To avoid an inverse crime, in our numerical examples the synthetic far field data were obtained by a numerical solution of the boundary integral equations based on a combined single- and doublelayer approach (see [9, 18]) using the numerical schemes as described in [10, 16, 17]. In each iteration step of the inverse algorithm for the solution of the field equations we used the numerical method described in Section 3 using 64 quadrature points. The equation (4.11) was solved by Tikhonov regularization with an H 2 Sobolev penalty term and with regularization parameter λ. The regularized equation is solved by Nystr¨om’s method with the composite trapezoidal rule again using 64 quadrature points. In addition, the regularized data was solved by Tikhonov regularization with an L2 penalty term and with regularization parameter α.

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INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER 9

In all our example we used N = 10 as degree for the approximating trigonometric polynomials in (4.10), and the wave numbers k0 = 4.5 and kd = 2 + 2i, and the regularization parameters α = 10−7 and λ = 1. In order to obtain noisy data, random errors are added point-wise to u∞ , ||u∞ || (5.1) u e∞ = u∞ + δξ |ξ| with the random variable ξ ∈ C and {Reξ, Imξ} ∈ (0, 1). Table 5.2. Parametric representation of boundary curves. Counter type

Parametric representation

Apple-shaped

:

cos t+0.1 sin 2t (cos t, sin t) : t ∈ [0, 2π]} z(t) = { 0.5+0.4 1+0.7 cos t

Kite-shaped

:

z(t) = {(cos t + 1.3 cos2 t − 1.3, 1.5 sin t) : t ∈ [0, 2π]}

Peanut-shaped

:

√ z(t) = { cos2 t + 0.25 sin t (cos t, sin t) : t ∈ [0, 2π]} z(t) = {(2 + 0.3 cos 3t)(cos t, sin t) : t ∈ [0, 2π]}

Rounded triangle :

In all the following examples, the green curve represents exact graph of conductive function η, the blue curve represents the reconstruction obtained from noisy data and the red curve represents the reconstruction obtained from noiseless data. In all examples we represent reconstructions from exact data and perturbed data with 1% relative error in the L2 norm. In the figures 1, 2,3, and 4, the scatterers are apple-shaped, kite-shaped, peanut-shaped, and rounded-triangle-shaped respectively. −0.5

0

1.5 reconstructed noise exact

reconstructed noise exact

1

reconstructed noise exact

−0.5

reconstructed noise exact

0.8

1

0.6

−1

0.4

−1 0.5

0.2

−1.5

−1.5

0

0

−0.2

−2

−0.4

−2

−0.5

−0.6

−2.5 −0.8 −2.5 0

1

2

3

4

5

6

7

−1 0

1

2

3

4

5

6

7

−3 0

1

2

3

4

5

6

7

−1 0

1

2

3

4

Figure 1. Reconstruction of η2 (3.6). Real part of η2 obtained for P=1 (the left figure), imaginary part of η2 obtained for P=1 (the middle left fig.), real part of η1 obtained for P=8 (the middle right fig.), imaginary part of η2 obtained for P=8 (the right fig.).

5

6

7

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INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER 10

0.2

1.2

0.4 reconstructed noise exact

0.2

reconstructed noise exact

1

1

reconstructed noise exact

0

reconstructed noise exact

0.9 0.8

0

0.8

−0.2

−0.2

0.6

−0.4

−0.4

0.4

−0.6

−0.6

0.2

−0.8

−0.8

0

−1

0.7 0.6 0.5 0.4 0.3 0.2

−1 0

1

2

3

4

5

6

7

−0.2 0

1

2

3

4

5

6

7

0.1 0

−1.2 0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

Figure 2. Reconstruction of η1 (3.5). Real part of η1 obtained for P=1 (the left figure), imaginary part of η1 obtained for P=1 (the middle left fig.), real part of η1 obtained for P=8 (the middle right fig.), imaginary part of η1 obtained for P=8 (the right fig.).

0.2

0.8

0.2

0.85

reconstructed noise exact

0.1

reconstructed noise exact

0.8

reconstructed noise exact

0.1

0.75 0

0.7 0

0.7

0.65

0.65

−0.1

reconstructed noise exact

0.75

−0.1

0.6

0.6 −0.2

0.55

−0.2 0.55

−0.3

0.5

0.5

−0.3

0.45

0.45 −0.4

−0.4 0.4

−0.5 0

1

2

3

4

5

6

0.35

7

0.4 0

1

2

3

4

5

6

7

−0.5 0

1

2

3

4

5

6

0.35

7

0

1

2

3

4

5

6

7

Figure 3. Reconstruction of η3 (3.7). Real part of η3 obtained for P=1 (the left figure), imaginary part of η3 obtained for P=1 (the middle left fig.), real part of η3 obtained for P=8 (the middle right fig.), imaginary part of η3 obtained for P=8 (the right fig.).

−0.5

−0.5

1.5 reconstructed noise exact

1.5

reconstructed noise exact

reconstructed noise exact

reconstructed noise exact

1

1

−1

−1

0.5

0.5

−1.5

−1.5 0

0

−2

−2 −0.5

−2.5 0

1

2

3

4

5

6

7

−1 0

−0.5

1

2

3

4

5

6

7

−2.5 0

1

2

3

4

5

6

7

−1 0

1

2

3

4

5

6

7

Figure 4. Reconstruction of η2 (3.6). Real part of η2 obtained for P=1 (the left figure), imaginary part of η2 obtained for P=1 (the middle left fig.), real part of η2 obtained for P=8 (the middle right fig.), imaginary part of η2 obtained for P=8 (the right fig.).

Our examples clearly indicate the feasibility of the proposed algorithm with a reasonable stability against noise. From our further numerical experiments it became obvious that using more than one incident wave improved on the accuracy of the reconstruction and the stability. Further research will be directed towards applying the algorithm to real data, to extend the numerics to the three dimensional case. One can also extend this inverse scattering problem to a simultaneous reconstruction of the conductive function and the shape of the scatterer. Similar

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INVERSE OBSTACLE SCATTERING WITH CONDUCTIVE BOUNDARY CONDITION FOR A COATED DIELECTRIC CYLINDER 11

problems have recently been considered by Kress and Rundell [19] and Serranho [24] for impenetrable scatterers. The shape reconstruction of transmision problem via single-layer potential approach was investigated by Altundag and Kress [4] and [5] and by Altundag [2], [3]. Uniqueness in inverse obstacle scattering with the conductive boundary condition was established by Gerlach and Kress [13]. However, there is no uniqueness result for one or finitely many incident fields for conductive transmission problem for shape reconstruction. Acknowledgments The author thanks his supervisor Professor Rainer Kress for the discussion on the topic of this paper. This research was supported by the German Research Foundation DFG through the Graduiertenkolleg Identification in Mathematical Models. References [1] Akduman, I. and Kress, R. : Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape. Radio Sci. 38, 1055–1064 (2003). [2] Altundag, A. : On a two-dimensional inverse scattering problem for a dielectric. Dissertation, G¨ ottingen, February 2012. [3] Altundag, A. : A hybrid method for inverse scattering problem for a dielectric. Advances in Applied Mathematics and Approximation Theory, Springer, 41, pp. 185–203 [4] Altundag, A. and Kress, R. : On a two dimensional inverse scattering problem for a dielectric. Applicable Analysis 91, pp. 757–771 (2012). [5] Altundag, A. and Kress, R. : An Iterative Method for a Two-DimensionalInverse Scattering Problem for a Dielectric. Jour. On Inverse and Ill-Posed Problem 20, pp. 575–590 (2012). [6] Ben Hassen, F., Ivanyshyn, O. and Sini, M. : The 3D acoustic scattering by complex obstacles. Inverse Problems 26 105–118 (2010). [7] Cakoni, F. and Colton, D. : The determination of the surface impedance of a partially coated obstacle from the far field data. SIAM J. Appl. Math. 64, 709–723 (2004). [8] Cakoni, F., Colton, D. and Monk, P. : The determination of boundary coefficients from far field measurements. J. Integral Equat. Appl. 22, 167–191 (2010). [9] Colton, D. and Kress, R.: Integral Equation Methods in Scattering Theory. Wiley-Interscience Publications, New York 1983. [10] Colton, D. and Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. 2nd. ed. Springer, Berlin 1998. [11] Eckel, H. and Kress, R.: Nonlinear integral equations for the inverse electrical impedance problem. Inverse Problems 23, 475–491 (2007). [12] Fang, W. and Zeng, S. : A direct solution of the Robin inverse problem. J. Integral Equat. Appl. 21, 545–557 (2009). [13] Gerlach, T. and Kress, R.: Uniqueness in inverse obstacle scattering with conductive boundary condition. Inverse Problems 12, 619-625 (1996). [14] Ivanyshyn, O. and Kress, R.: Inverse Scattering for surface impedance from phase-less far field data. J. Comput. Phys. 230, 3443-3452 (2011). [15] Jakubik, P. and Potthast, R. : Testing in integrity of some cavity-the Cauchy problem and the range test. Appl. Numer. Math.2007, doi: 10.1016/j.apnum.2007.04.007. [16] Kress, R.: On the numerical solution of a hypersingular integral equation in scattering theory. J. Comp. Appl. Math. 61, 345–360 (1995). [17] Kress, R.: Integral Equations. 2nd. ed Springer Verlag, Berlin 1998. [18] Kress, R. and Roach, G.F.: Transmission problems for the Helmholtz equation. J. Math. Phys. 19, 1433–1437 (1978) . [19] Kress, R. and Rundell, W.: Inverse Scattering for shape and impedance. Inverse Problems 17, 1075–1085 (2001). [20] Kress, R. and Sloan, I.H.: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. Numerische Mathematik 66, 199–214 (1993). [21] Liu, J., Nakamura, G. and Sini, M. : Reconstruction of the shape and surface impedance from acoustic scattering data for arbitrary cylinder. SIAM J.Appl. Math. 67, 1124–1146 (2007). [22] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press 2000.

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[23] Nakamura, G. and Sini, M. : Reconstruction of the shape and surface impedance from acoustic scattering data for arbitrary cylinder. SIAM J. Anal. 39, 819–837 (2007). [24] Serranho, P: A hybrid method for inverse scattering for shape and impedance. Inverse Problems 22, 663–680 (2006). Istanbul Sabahattin Zaim University, Mathematics Education, 34303, Istanbul/Turkey E-mail address: [email protected]

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2, 23-34, 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 23

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay∗ LI Yanfang,LIU Xianghu† School of Mathematics and Computer Science, Zunyi Normal College, Zunyi, 563002, Guizhou, P. R. China.

Abstract: In this paper, the authors study some impulsive fractional-order neural network with mixed delay. By the fractional integral and the definition of stability, the existence of solutions of the network is proved, the sufficient conditions for stability of the system are presented. Some examples are given to illustrate the main results. Keywords: fractional-order neural network,mixed delay,fixed point theorem Clc number: 0175.13

Introduction In this paper, we study some impulsive fractional-order neural network with mixed delay    c Dtα x(t) = −Cx(t) + AF (x(t)) + BG(x(t − τ ))    Rt   + D −∞ K(t − η)H(x(η))dη + J, 0 < t ≤ b, t 6= tk , (1) −   4x(tk ) = I(x(tk )), k = 1, 2, 3, · · · , m      x(η) = ϕ(η), η ∈ (−∞, 0] where c Dtα is the standard Caputo fractional derivative of order α, α ∈ (0, 1),x(t) = (x1 (t), x2 (t), · · · , xn (t))T is the neuron state vector of the neural network; C = diag(c1 , c2 , · · · , cn ) is a diagonal matrix and ci > 0,i ∈ N = 1, 2, · · · , n, A = (aij )n×n , B = (bij )n×n , D = (dij )n×n are the connection weight matrix, the delayed weight matrix and the distributively delayed connection weight matrix, respectively; J = (J1 , J2 , · · · , Jn )T is an external input; F (x(·)) = (f1 (x1 (·)), f2 (x2 (·)), · · · , fn (xn (·)))T , G(x(·)) = (g1 (x1 (·)), g2 (x2 (·)), · · · , gn (xn (·)))T , H(x(·)) = (h1 (x1 (·)), h2 (x2 (·)), · · · , hn (xn (·)))T represents the neuron activation function; τ = (τij )n×n ∗

Project supported by Zunyi science and technology mutual fund,Guizhou Province Education Science and planning issues † Corresponding author: Xianghu Liu; E-mail addresses: [email protected];

1

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

24

is the transmission delay of the neural network satisfies 0 ≤ τij ≤ δ, where δ is a positive constant; K(·) = diag(k1 (·), k2 (·), · · · , kn (·)) is the delay kernel function and satisR∞ fies 0 ki (t)dt = 1, the function Ik : X → X is continous, and 0 = t0 < t1 < t2 < − + − · · · < tk < · · · < tm = b, ∆x(tk ) = x(t+ k ) − x(tk ), x(tk ) and x(tk ) denote the right and

the left limits of x(t) at t = tk , ϕ(η) = (ϕ1 (t), ϕ2 (t), · · · , ϕn (t))T is the initial function and ϕi (η) ∈ C([−∞, 0], R), i ∈ N ,ϕi (0) = 0, the norm of C([−∞, 0], R) is denoted by P ||ϕ(t)|| = ni=1 sup{|ϕi (t)|}. It is well known that the delayed and impulsive neural networks exhibiting the rich and colorful dynamical behaviors are important part of the delayed neural systems. The delayed and impulsive neural networks can exhibit some complicated dynamics and even chaotic behaviors. Due to their important and potential applications in signal processing, image processing, artificial intelligence as well as optimizing problems and so on, the dynamical issues of delayed and impulsive neural networks have attracted worldwide attention , many interesting stability criteria for the equilibriums and periodic solutions of delayed or impulsive neural networks have been derived via Lyapunov-type function or functional approaches. For example, Wang and Zheng[1] investigated the global asymptotic stability of the equilibrium point of a class of mixed recurrent neural networks with time delay in the leakage by using the Lyapunov functional method, linear matrix inequality approach and general convex combination technique term under impulsive perturbations. Sebdani et al.[2] considered bifurcations and chaos in a discrete-time-delayed Hopfield neural network with ring structures and different internal decays. M.U. Akhmet, E. ylmaz[3] got a criteria for the global asymptotic stability of the impulsive Hopfield-type neural networks with piecewise constant arguments of generalized type by using linearization. For the last decades, fractional differential equations

[4−11]

have received intensive atten-

tion because they provide an excellent tool for the description of memory and hereditary properties of various materials and processes, such as physics, mechanics, chemistry, engineering, etc. Therefore, it may be more meaningful to model by fractional-order derivatives than integer- order ones. Recently, fractional calculus is introduced into artificial neural network . For example, Boroomand and Menhajas

[12]

investigated stability of fractional-

order Hopfield-type neural networks through energy-like function analysis, Chen et al.[13] studied uniform stability and the the existence,uniqueness and stability of its equilibrium point of a class of fractional-order neural networks with constant delay . The authors

[14−17]

analyzed the stability of some other neural networks with delay. We all know that the delay is not always a constant, it maybe change in the network. Time-verying delays and 2

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

25

distributed delays may occur in neural processing and signal transmission, which can cause instability, oscillations, there are few papers that consider the problems for fractional-order neural network with mixed delay and impulse. Thus, it is worth investigating some impulsive fractional-order neural network with mixed delay To the best of our knowledge, the system(1) is still untreated in the literature and it is the motivation of the present work. The rest of this paper is organized as follows: In section 2, some notations and preparations are given. In section 3, some main results of (1) are obtained. At last, some examples are given to demonstrate the main results.

1

Preliminaries In this section,we will give some definitions and preliminaries which will be used in the

paper. In order to define the solution of (1), the authors consider the following spaces P C(J, R) = − − {x : J → R : x(t) ∈ C(tk , tk+1 ], k = 0, · · · , m, there exist x(t+ k ) and x(tk ) with x(tk ) =

x(tk ), k = 1, · · · , m}. The norm ||x||P C = sup{||x(t)||c : t ∈ J}.P C 1 (J, X)={x : J → X, x ∈ C 1 ((tk , tk+1 ], X), k = 0, 1, 2, · · · , n, there exist x0 (t+ k ), k = 1, 2, · · · , n, }. The norm ||x||P C 1 = sup{||x(t)||pc , ||x0 (t)||pc : t ∈ J}. Obviously P C(J, X) and P C 1 (J, X) are Banach spaces. Let’s recall some known definitions of fractional calculus. For more details, one can see[4,5,6]. Definition 1 The integral Itα f (t)

1 = Γ(α)

Z

t

(t − s)α−1 f (s)ds,

α > 0,

0

is called Riemann-Liouville fractional integral of order α, where Γ is the gamma function. For a function f (t) given in the interval [0, ∞), the expression Z 1 d n t L α (t − s)n−α−1 f (s)dt, Dt f (t) = ( ) Γ(n − α) dt 0 where n = [α] + 1, [α] denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α > 0. Definition 2 Caputo’s derivative for a function f : [0, ∞) → R can be written as c

Dtα f (t)

=

L

Dtα [f (t)

−

n−1 k X t k=0

3

k!

f (k) (0)],

n = [α] + 1,

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

26

where [α] denotes the integer part of real number α. Theorem 1 According to the Lemma2.6[18] , one can get that if u(t) ∈ P C 1 (J, X), then   u(t) − u(0), t ∈ [0, t1 ], I q c Dq u(t) = P  u(t) − k I (u(t )) − u(0), t ∈ (t , t ], k ≥ 1. i=1 i

i

k

k+1

Proofµ µ If t ∈ [0, t1 ], then I

q c

Z s 1 (t − s) (s − τ )−q u0 (τ )dτ ds Γ(1 − q) 0 0 Z t Z t 1 = u0 (τ ) (t − s)q−1 (s − τ )−q dsdτ Γ(q)Γ(1 − q) 0 τ Z t = u0 (τ )dτ

1 D u(t) = Γ(q) q

Z

t

q−1

0

= u(t) − u(0). If t ∈ (tk , tk+1 ], k ≥ 1, then I

q c

Z s Z t 1 0 q−1 D u(t) = (s − τ )−q u (τ )dτ ds (t − s) Γ(q)Γ(2 − q) 0 0 Z t1 Z s 1 0 = (t − s)q−1 (s − τ )−q u (τ )dτ ds Γ(q)Γ(1 − q) 0 0 Z Z s k−1 t i+1 X 1 0 q−1 + (t − s) (s − τ )−q u (τ )dτ ds Γ(q)Γ(1 − q) ti 0 i=1 Z t Z s 1 0 + (t − s)q−1 (s − τ )−q u (τ )dτ ds Γ(q)Γ(1 − q) tk 0 Z t1 Z s 1 0 = (s − τ )−q u (τ )dτ ds (t − s)q−1 Γ(q)Γ(1 − q) 0 0 Z k−1 i−1 Z tj+1 t i+1 X X 1 0 q−1 + (t − s) [ (s − τ )−q u (τ )dτ Γ(q)Γ(1 − q) ti i=1 j=0 tj Z s Z t k−1 Z tj+1 X 1 0 −q 0 q−1 (s − τ )−q u (τ )dτ + (s − τ ) u (τ )dτ ]ds + (t − s) [ Γ(q)Γ(1 − q) tk ti j=0 tj Z s 0 + (s − τ )−q u (τ )dτ ]ds tk Z t1 Z t1 1 0 = u (τ )dτ (t − s)q−1 (s − τ )−q ds Γ(q)Γ(1 − q) 0 τ Z tj+1 Z ti+1 k−1 X i−1 X 1 0 + u (τ )dτ (t − s)q−1 (s − τ )−q ds Γ(q)Γ(1 − q) tj ti i=1 j=1 q

4

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

+

k−1 X i=1 k−1 X

1 Γ(q)Γ(1 − q)

Z

ti+1

ti+1

Z

0

u (τ )dτ ti

27

(t − s)q−1 (s − τ )−q ds

τ

tj+1 t 1 0 u (τ )dτ (t − s)q−1 (s − τ )−q ds Γ(q)Γ(1 − q) tj tk j=0 Z t Z t 1 + u0 (τ )dτ (t − s)q−1 (s − τ )−q ds Γ(q)Γ(1 − q) tk τ

+

= u(t) −

Z

k X

Z

Ii (u(ti )) − u(0),

i=1

with the help of the substitution s = z(t − τ ) + τ, Z

t q−1

(t − s)

Z

−q

(s − τ ) ds =

τ

1

(1 − z)q−1 z −q dz

0

= B(1 − q, q) = =

Γ(1 − q)Γ(q) Γ(1 − q + q)

Γ(1 − q)Γ(q) = Γ(1 − q)Γ(q). Γ(1)

The proof is completed. Let recollect the definition of stability which can be found in [13] and will be used in our main results. Definition 3 The solution of system (1.1) is said to be stable if for any ε > 0, there exist δ(t0 , ε) > 0, such that 0 ≤ t0 ≤ t, ||ϕ(t) − φ(t)|| < δ imply ||y(t, t0 , ϕ(t)) − x(t, t0 , φ(t))|| < ε for any two solutions y(t, t0 , ϕ(t)), x(t, t0 , φ(t)) associated with ϕ(t), φ(t) that are the initial function. It is uniformly stable if the above δ is independent of t0 .

2

Existence and uniqueness of solution In this section, we will investigate the existence and uniqueness of solution for im-

pulsive fractional-order neural network with mixed delay. Without loss of generality, let t ∈ (tk , tk+1 ], 1 ≤ k ≤ m − 1. For sake of convenience ,the authors adopt the following notations and assumptions: H(1): For j = 1, 2, · · · , n, the functions fj , gj , hj , Ik : X → X satisfy: There exist Lipschitz constants Lf j > 0,Lgj > 0,Lhj > 0, Ljk > 0, such that |fj (x) − fj (y)| ≤ Lf j |x − y|, |gj (x) − gj (y)| ≤ Lgj |x − y|, |hj (x) − hj (y)| ≤ Lhj |x − y|, |Ik (x) − Ik (y)| ≤ 5

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

28

Ljk |x − y|, for all x, y ∈ X. H(2): The delay kernel function K(·) = diag(k1 (·), k2 (·), · · · , kn (·)) satisfies Z ∞ Z ∞ ∗ ki (t)dt = 1, k = ki (t)e−t dt < ∞, i = 1, 2, · · · , n. 0

0

H(3):cj , aij , bij , dij and Lf j , Lgj , Lhj , Ljk satisfy the following conditions: Pn Pn Pn Pn ∗ ∗ ∗ (i)||A∗ || = i=1 sup∀j {|bij |Lgj }, i=1 sup∀j {|aij |Lf j }, ||B || = i=1 |bi | = i=1 |ai | = Pm Pn Pm Pn ∗ ∗ ∗ ∗ ||D || = i=1 |di | = i=1 sup∀j {|dij |Lhj }, ||I || = k=1 |Lk | = k=1 sup∀j {|Ljk |}, m ≤ n; (ii)Cmax = max{cj },Cmin = min{cj }; bα (Cmax Γ(α+1)

(iii)M = ||I ∗ || +

+ ||A∗ || + ||B ∗ || + ||D∗ ||) < 1.

Theorem 2 If the assumption H(1),H(2) and H(3) hold , for each x0 = (x01 , x02 , · · · , x0n ), the system(1) has a unique solution. Proofµ µ Consider the system (1), we will study the solvability and stability of it. (1)Existence. By Theorem 1, it is showed that the (1) is equivalent to the following integral equation

xi (t) = xi (0) +

k X

α I(xi (t− l )) + I [−ci xi (t) +

n X j=1

l=1

+

n X

Z

bij g(xj (t − τij ))

j=1

kj (t − η)hj (xj (η))dη + Ji ] −∞

= xi (0) +

k X

I(xi (t− l ))

l=1

+

n X

t

dij

j=1

n X

aij fj (xj (t)) +

Z

1 + Γ(α)

Z

t

(t − s)

α−1

[−ci xi (s) +

0

n X

aij fj (xj (s)) +

j=1

n X

bij g(xj (t − τij ))

j=1

s

kj (s − η)hj (xj (η))dη + Ji ]ds.

dij

(2)

−∞

j=1

Constract the following sequences zin , zi0 = x0i , zin+1 (t)

=

x0i +

+

k X

l=1 n X

I(zin (t− l ))

1 + Γ(α)

bij g(zjn (t − τij )) +

j=1

t

Z

n X

(t − s) 0

−∞

||zin+1 (t) − zin (t)|| = sup{|zin+1 (t) − zin (t)|} 6

+

n X

aij fj (zjn (s))

s

dij

we can calculate that

[−ci zin (s)

j=1

Z

j=1

α−1

kj (s − η)hj (zjn (η))dη + Ji ]ds,

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

≤ sup|

k X

29


n−1 − I(zin (t− (tl )) l )) − I(zi

l=1

1 + Γ(α) + +

n X j=1 n X

Z

t α−1

(t − s)

[ci |zin (s)

−

zin−1 (s)|

+

n X

0

|aij ||(fj (zin (s)) − fj (zin−1 (s)))|

j=1

|bij ||(gj (zin (s − τij )) − g(zin−1 (s − τij )))| Z

s

kj (s − η)|hj (zin (η)) − hj (zin−1 (η))|dη]ds|

|dij | −∞

j=1

bα (Cmax + ||A∗ || + ||B ∗ || + ||D∗ ||))||zin (t) − zin−1 (t)|| Γ(α + 1) ≤ M n ||zi1 (t) − zi0 (t)|| ≤ (||I ∗ || +

≤ M n (M − ||I ∗ ||)||zi0 ||, so the sequences zin are convergent and the limits satisfy (2), the existence is proved . (2) Stability. Assume that x(t) = (x1 (t), x2 (t), · · · , xn (t))T and y(t) = (y1 (t), y2 (t), · · · , yn (t))T are the two solutions of (1) with the different initial condition xi (η) = φi (η) ∈ C((−∞, 0], R), φi (0) = 0, yi (η) = ϕi (η) ∈ C((−∞, 0], R), ϕi (0) = 0, i ∈ N . We have c

Dtα (yi (t)

− xi (t)) = −ci (yi (t) − xi (t)) +

n X

aij (fj (yj (t)) − fj (xj (t))) +

j=1 n X

n X

bij (g(yj (t − τij )) − g(xj (t − τij ))) +

j=1

Z

t

kj (t − η)(hj (yj (η)) − hj (xj (η)))dη.

dij −∞

j=1

According to the Definition 2 and the initial function ϕi (0) = 0, if n = 1, 0 < q < 1, I q Dq u(t) = u(t) − u(0). So the solution of the (1) can be denoted by the following form yi (t) − xi (t) =

k X

I(yi (t− l )

−

xi (t− l ))

α

+ I [−ci (yi (t) − xi (t)) +

bij (gj (yj (t − τij )) − g(xj (t − τij ))) +

j=1

=

k X l=1

I(yi (t− l )

n X j=1

−

xi (t− l ))

aij (fj (yj (t)) − fj (xj (t))) +

j=1

l=1 n X

n X

1 + Γ(α)

Z

Z

t

kj (t − η)(hj (yj (η)) − hj (xj (η)))dη]

dij −∞

t α−1

(t − s)

[−ci (yi (s) − xi (s)) +

0

n X j=1

7

aij (fj (yj (s)) − fj (xj (s))) +

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

n X

bij (gj (yj (s − τij )) − g(xj (s − τij ))) +

j=1

n X

Z

s

kj (s − η)(hj (yj (η)) − hj (xj (η)))dη]ds.

dij

j=1

30

−∞

Then |yi (t) − xi (t)| ≤

k X

− I(yi (t− l ) − xi (tl ))

l=1

1 + Γ(α) +

n X

Z

t α−1

(t − s)

[ci |yi (s) − xi (s)| +

0

n X

|aij ||(fj (yj (s)) − fj (xj (s)))|

j=1

|bij ||(gj (yj (s − τij )) − g(xj (s − τij )))|

j=1

+

n X j=1

≤

m X

Z

s

|dij |

kj (s − η)|hj (yj (η)) − hj (xj (η))|dη]ds −∞

− Ljk |yi (t− l ) − xi (tl )|

l=1

Z t 1 ci (t − s)α−1 |yi (s) − xi (s)|ds + Γ(α) 0 Z t n X 1 (t − s)α−1 |yj (s) − xj (s)|ds + |aij |Lf j Γ(α) 0 j=1 Z n τij X 1 + |bij |Lgj (t − s)α−1 |ϕj (s − τij ) − φj (s − τij )|ds Γ(α) 0 j=1 Z n t X 1 + |bij |Lgj (t − s)α−1 |yj (s − τij ) − xj (s − τij )|ds Γ(α) τij j=1 Z n t X 1 + |dij |Lhj (t − s)α−1 |yj (η1 ) − xj (η1 )|ds 0 ≤ η1 ≤ t Γ(α) 0 j=1 Z n t X 1 + |dij |Lhj (t − s)α−1 |ϕj (η) − φj (η)|ds − ∞ ≤ η ≤ 0 Γ(α) 0 j=1 ≤

m X

− Ljk |yi (t− l ) − xi (tl )|

l=1

Z t 1 +ci sup{|yi (t) − xi (t)|} (t − s)α−1 ds Γ(α) 0 Z t n X 1 + |aij |Lf j sup{|yj (t) − xj (t)|} (t − s)α−1 ds Γ(α) 0 j=1 Z τij n X 1 + |bij |Lgj sup{|ϕj (t) − φj (t)|} (t − s)α−1 ds Γ(α) 0 j=1 8

(3)

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

n X

1 + |bij |Lgj sup{|yj (t) − xj (t)|} Γ(α) j=1

Z

31

t

(t − s)α−1 ds

τij

n X

Z t 1 (t − s)α−1 ds + |dij |Lhj sup{|yj (η1 ) − xj (η1 )|} Γ(α) 0 j=1 Z t n X 1 + |dij |Lhj sup{|ϕj (η) − φj (η)|} (t − s)α−1 ds, Γ(α) 0 j=1

0 ≤ η1 ≤ s − ∞ ≤ η ≤ 0.

From(3), one can get ||y(t) − x(t)|| =

n X

sup{|yi − xi |} ≤ ||I ∗ || +

i=1

+


bα (Cmax + ||A∗ || + ||B ∗ || + ||D∗ ||) ||y(t) − x(t)|| Γ(α + 1)

bα (||B ∗ || + ||D∗ ||)||ϕ(t) − φ(t)||, Γ(α + 1)

which implies that ||y(t) − x(t)|| ≤

bα (||B ∗ || + ||D∗ ||) ||ϕ(t) − φ(t)||. Γ(α + 1) − Γ(α + 1)||I ∗ || − bα (Cmax + ||A∗ || + ||B ∗ || + ||D∗ ||)

For ∀ > 0, there exists δ =

Γ(α+1)−Γ(α+1)||I ∗ ||−bα (Cmax +||A∗ ||+||B ∗ ||+||D∗ ||) , bα (||B ∗ ||+||D∗ ||)

such that ||y(t) −

x(t)|| < , when ||ϕ(t) − φ(t)|| < δ , so the solution x(t) is uniformly stable, which means that the network(1) is uniformly stable.

3

Some examples In this section, according to the impulsive fractional-order neural network (1), some ex-

amples are given to illustrate the main results. 





 2 −1 −10 1 , B =  , D = Example 3.1 Choose C = diag(−0.01, 0.01), A =  5 6 −5 1       −10 1 0.8 0.2 1  , I =  , τ =  , fj (xj (t)) = gj (xj (t)) = hj (xj (t)) = , let 1+e−xj (t) −5 1 0.8 0.1 1 Lf j = Lgj = Lhj = 1000 , m = 30 , for the convenience, the authors take the interval 1 , it is easily to prove that the assumption (H1)(H3) hold, so example 3.1 is uniformly stable . For the convenience of studying the local and the whole conditions, the authors take the time t=40(fig1) and t=400(fig2) and get the unique equilibrium point x∗ =(-2.6047,-1.3012).

9

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

32



     1 −1 −1 1 1 −1 , B =  , D =  , Example 3.2 Choose C = diag(−0.02, 0.01), A =  5 6 1 1 −1 1     0.4 0.1 −xj (t) 1 , let Lf j = I =  , τ =  , fj (xj (t)) = gj (xj (t)) = 1+e−xj (t) , hj (xj (t)) = 1+e 1+e−xj (t) 0.6 0.1 1 Lgj = Lhj = 1000 , m = 30, the interval is 1, it is proved that the assumption (H1)(H3) hold, so example 3.2 is uniformly stable . The authors also choose the time t=40(fig3) and t=4000(fig4) and get the unique equilibrium point x∗ =(1.4574,-0.8297).

4

Conclusions In this paper,by the fractional integral, the authors changed the derivative equation to

integral one, for the convergence of sequences and the definition of stabilty, the existence of solutions of the network has been proved, the sufficient conditions for stability of the system have been presented. The authors also gave two examples and designed the relevant experimental procedures, after some experiments, the results have been illustrated .I think 10

Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, LI Yanfang,LIU Xianghu

33

that the design of impulsive item is difficult.The finite item is proved to be feasible, but how the infinite one or the variable one , this can be our future work.

References [1] Wang Y, Zheng C D , Feng E M . Stability analysis of mixed recurrent neural networks with time delay in the leakage term under impulsive perturbations[J].Neurocomputing, 2013,119: 454-461. [2] R M Sebdani, S Farjami. Bifurcations and chaos in a discrete-time-delayed Hopfield neural network with ring structures and different internal decays[J].Neurocomputing ,2013, 99:154-162. [3] M U Akhmet, E Ylmaz .Impulsive Hopfield-type neural network system with piecewise constant argument[J].Nonlinear Analysis-Real World Applications,2010,11:2584-2593 . [4] K S Miller, B Ross. An Introduction to the Fractional Calculus and Differential Equations[M], New York, John Wiley, 1993. [5] I Podlubny. Fractional Differential Equations[M]. San Diego, Academic Press, 1999. [6] A A Kilbas, Hari M. Srivastava, J. Juan Trujillo. Theory and Applications of Fractional Differential Equations[M]. vol 204. Amsterdam, North-Holland Mathematics Studies, Elsevier Science B V , 2006. [7] Wang J R , M Fe˜ ckan, Zhou Y . Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations[J]. Journal of Optimization Theory and Applications, 2013, 156:1332. [8] Wang J R, Zhou Y , Mi Medved, On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay[J]. Journal of Optimization Theory and Applications, 2012, 152: 31-50. [9] Yang X J , Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations[J].Communications in Nonlinear Science and Numerical Simulation, 2014, 19(1): 37-48, . [10] Yang X J ,Advanced Local Fractional Calculus and Its Applications[M]. New York, NY, USA,World Science, 2012. 11

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[11] Yang X J , D Baleanu. Fractal heat conduction problem solved by local fractional variation iteration method[J],Thermal Science, 2013, 17(2):625-628. [12] A Boroomand, M Menhaj. Fractional-order Hopfield neural networks[J]. Lecture Notes in Computer Science, 2009, 5506: 883-890. [13] Chen L P , Chai Y , Wu R C ,et al . Dynamic analysis of a class of fractional-order neural networks with delay[J]. Neurocomputing, 2013,111:190-194. [14] Li X D , R Rakkiyappan. Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays[J]. Communications in Nonlinear Science and Numerical Simulation, 2013,18: 1515-1523. [15] H Delavari , D Baleanu, J Sadati. Stability analysis of Caputo fractional-order nonlinear systems revisited[J]. Nonlinear Dynamics,2012, 67:2433-2439. [16] S Lakshmanan , J Park , et al. Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays [J]. Applied Mathematics and Computation, 2013,219 :9408-9423. [17] Chen H B . New delay-dependent stability criteria fo runcertain stochastic neural networks with discrete interval and distributed delays[J]. Neurocomputing,2013, 101: 1-9. [18] Liu Z H , Li X W . Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations[J]. Communications in Nonlinear Science and Numerical Simulation, 2012,18,1362-1373.

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2,35-50, 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 35

Basins of Attraction of Certain Homogeneous Second Order Quadratic Fractional Difference Equation M. Gari´c-Demirovi´c Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina

M. R. S. Kulenovi´c∗ Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881-0816, USA,

M. Nurkanovi´c Department of Mathematics, University of Tuzla, 75 000 Tuzla, Bosnia and Herzegovina August 31, 2013

Abstract We investigate the basins of attraction of equilibrium points and period-two solution of the difference equation of the form Bxn xn−1 + Cx2n−1 , n = 0, 1, . . . , xn+1 = ax2n + bxn xn−1 where the parameters a, b, C, B are positive numbers and the initial conditions x−1 , x0 are arbitrary nonnegative numbers. We show that this equation exhibits global period-two bifurcation, as certain parameters are pasing through the critical value.

Keywords: attractivity, basin, difference equation, invariant sets, periodic solutions, stable set AMS 2000 Mathematics Subject Classification: 39A10, 39A11 ∗

Corresponding author

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

1

1 36

Introduction and preliminaries

We investigate global behavior of the equation xn+1 =

Bxn xn−1 + Cx2n−1 , n = 0, 1, 2, ..., ax2n + bxn xn−1

(1)

where the parameters a, b, C, B are positive numbers and the initial conditions x−1 , x0 , x−1 + x0 > 0 are arbitrary nonnegative numbers. It is a special case of the second order homogeneous quadratic fractional equation Ax2n + Bxn xn−1 + Cx2n−1 xn+1 = , n = 0, 1, 2, ... . (2) ax2n + bxn xn−1 + cx2n−1 and of the general second order quadratic fractional equation xn+1 =

Ax2n + Bxn xn−1 + Cx2n−1 + Dxn + Exn−1 + F , n = 0, 1, 2, ... ax2n + bxn xn−1 + cx2n−1 + dxn + exn−1 + f

(3)

which global dynamics is under investigation. Some special cases of Eq.(3) have been considered in the series of papers [3, 4, 9, 10, 17]. Some special second order quadratic fractional difference equations have appeared in analysis of the systems of linear fractional difference equations in the plane, see [5, 7, 8, 14, 15, 16]. Describing the global dynamics of Eq.(3) is formidable task as this equation contains as a special cases many equations with well known but complicated dynamics, such as Lynes’ equation. Equation (3) contains as a special case linear fractional equation which dynamics was investigated in great details in [11] and in many papers which solved some conjectures and open problems posed in [11]. Equation (2) can be brought to the form  2   n n A xxn−1 + B xxn−1 +C xn+1 = , n = 0, 1, 2, ... (4)  2   n n a xxn−1 +c + b xxn−1 and one can take the advantage of this auxiliary equation to describe the dynamics of Eq.(2). In this paper we take a different approach based on the monotonic properties of the right hand side of Eq.(2) and theory of monotone maps, and use it to describe precisely the basins of attraction of all attractors of that equation. The special cases of Eq.(1) when C = 0 or a = 0 are linear fractional difference equations which global dynamics is described in [11]. We show that Eq.(1) exhibits three types of global behavior characterized by the existence of a unique positive equilibrium solution and a unique minimal period-two solution, which stable manifold serves as the boundary of the basins of attraction of locally stable equilibrium and points at infinity (0, ∞) and (∞, 0). In fact, Eq.(1) exhibits period-two bifurcation studied in great details in [12]. Equation (1) is a special case of the general second order difference equation xn+1 = f (xn , xn−1 ) ,

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

(5)

where

Buv + Cv 2 . au2 + buv The next result is important for the general second order difference equation of the form (5), see [2]. f (u, v) =

Theorem 1 Let I be a set of real numbers and f : I × I → I be a function which is non-increasing in the first variable and non-decreasing in the second variable. Then, for ever solution {xn }∞ n=−1 of the equation xn+1 = f (xn , xn−1 ) ,

x−1 , x0 ∈ I, n = 0, 1, 2, ...

(6)

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

2 37

∞ the subsequences {x2n }∞ n=0 and {x2n−1 }n=0 of even and odd terms of the solution do exactly one of the following: (i) Eventually they are both monotonically increasing. (ii) Eventually they are both monotonically decreasing. (iii) One of them is monotonically increasing and the other is monotonically decreasing.

We now give some basic notions about monotone maps in the plane. Consider a partial ordering  on 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. Clearly, being related is invariant under iteration of a strongly monotone map. Throughout this paper we shall use the North-East ordering (NE) for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x1 , y1 ) ne (x2 , y2 ) if x1 ≤ x2 and y1 ≤ y2 and the South-East (SE) ordering defined as (x1 , y1 ) se (x2 , y2 ) if x1 ≤ x2 and y1 ≥ y2 . A map T on a nonempty set 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 " # + − sign (JT (x)) = , (7) − + 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, for example, Q1 (x) = {y ∈ R2 : x1 ≤ y1 , x2 ≤ y2 }. 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 [13, 12], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by H. L. Smith in [18, 19]. 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.

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

3 38

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 [20] 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} ,

(8)

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 : → x} and unstable set W u (x) of x is the set   0 x ∈ R : there exists {xn }n=−∞ ⊂ R s.t. T (xn ) = xn+1 , x0 = x, and lim xn = x

T n (x)

n→−∞

When T is non-invertible, the set W s (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 Eq.(6) 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 Eq.(6) is a strictly competitive map on I × I, see [13].

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

4 39

Set xn−1 = un and xn = vn in Eq.(6) 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 [13]. Remark 2 The characteristic equation of Eq.(6) at an equilibrium point (¯ x, x ¯): λ2 − D1 f (¯ x, x ¯)λ − D2 f (¯ x, x ¯) = 0,

(9)

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 nonexistence of minimal period-two solution. There are several attractivity results for Eq. (6). Some of these results give the sufficient conditions for all solutions to approach a unique equilibrium and they were used efficiently in [11]. See also [1, 6]. Next result is from [6]. Theorem 6 Consider Eq. (6) where f : I × I → I is a continuous function and f is decreasing in the first argument and increasing in the second argument. Assume that x is a unique equilibrium point which is locally asymptotically stable and assume that (φ, ψ) and (ψ, φ) are minimal period-two solutions which are saddle points such that (φ, ψ) se (x, x) se (ψ, φ) . Then, the basin of attraction B ((x, x)) of (x, x) is the region between the global stable sets W s ((φ, ψ)) and W s ((ψ, φ)) . More precisely B ((x, x)) = {(x, y) : ∃yu , yl : yu < y < yl , (x, yl ) ∈ W s ((φ, ψ)) , (x, yu ) ∈ W s ((ψ, φ))} . The basins of attraction B ((φ, ψ)) = W s ((φ, ψ)) and B ((ψ, φ)) = W s ((ψ, φ)) are exatly the global stable sets of (φ, ψ) and (ψ, φ). If (x−1 , x0 ) ∈ W+ ((ψ, φ)) or (x−1 , x0 ) ∈ W− ((φ, ψ)), then T n ((x−1 , x0 )) converges to the other equilibrium point or to the other minimal period-two solutions or to the boundary of the region I × I.

2

Local stability analysis

First, notice the function f (u, v) is decreasing in the first variable and increasing in the second variable. ∞ ∞ By Theorem 1, for every solutions {xn }∞ n=−1 of Eq. (1) the subsequences {x2n }n=0 and {x2n−1 }n=0 are eventualy monotonic. It is clear that Equation (1) has a unique positive equilibrium point x = B+C a+b . If we denote f (u, v) =

Buv + Cv 2 , au2 + buv

then a linearization of Equation (1) is of the form yn+1 = syn + tyn−1 ,

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

5 40

where ∂f s = −t = (x, x) = ∂u



−aBu2 v − 2aCuv 2 − bCv 3



(x, x) (au2 + buv)2 aBx3 + 2aCx3 + bCx3 1 aB + 2aC + bC = − =− ·
2 x (a + b)2 ax2 + bx2 aB + 2aC + bC = − . (a + b) (B + C)

Lemma 1 Equation (1) has a unique positive equilibrium point x = B+C a+b . i) If (3a + b) C < (b − a) B, then the equilibrium point x is locally asymptotically stable. ii) If b ≤ a or if b > a and (3a + b) C > (b − a) B, then the equilibrium point x is a saddle point. iii) If (3a + b) C = (b − a) B, then the equilibrium point x is non-hyperbolic (with eigenvalues λ1 = −1 and λ2 = 21 ). Proof. i) Equilibrium point x is locally asymptotically stable if 1 |s| < 1 − t < 2 ⇔ |s| < 1 + s < 2 ⇔ − < s < 1. 2 Since s < 0, we have s

1 aB + 2aC + bC 1 − ⇔− >− 2 (a + b) (B + C) 2 ⇔ 2 (aB + 2aC + bC) < (a + b) (B + C) = aB + aC + bB + bC >

⇔ aB + 3aC + bC < bB ⇔ (3a + b) C < (b − a) B, which implies that b − a > 0. Therefore, the equilibrium x is locally asymptotically stable if (3a + b) C < (b − a) B. ii) If |s| > |1 − t| and s2 + 4t > 0, then the equilibrium point x is a saddle point. We obtain s2 + 4t > 0 ⇔ t2 + 4t > 0, which is satisfied by t > 0, and |s|

>

|1 − t| ⇔ −s > |1 + s| ⇔ s < 1 + s < −s ⇒ s < −

⇔ −

1 2

aB + 2aC + bC 1 < − ⇔ (3a + b) C > (b − a) B. (a + b) (B + C) 2

Notice that, if b − a ≤ 0 then (3a + b) C > (b − a) B is satisfied, so equiilibrium point x is a saddle point if b ≤ a or if b > a and (3a + b) C > (b − a) B. iii) Equilibrium point x is non-hyperbolic point if 1 |s| = |1 − t| ⇔ s = − ⇔ (3a + b) C = (b − a) B. 2 Since s = −t = − 12 , then the characteristic equation at the equilibrium point is of the form 1 1 λ2 + λ − = 0, 2 2 with eigenvalues λ1 = −1 and λ2 = 21 .

2

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

3

6 41

Periodic solutions

In this section we present results about existence of minimal period-two solutions of Eq. (1). Theorem 7 If b ≤ a or B ≤ C or (3a + b) C ≥ (b − a) B, then Eq. (1) has no minimal period-two solutions. b) If b > a and (3a + b) C < (b − a) B, then Equation (1) has minimal period-two solution: φ, ψ, φ, ψ, . . . (φ 6= ψ and φ > 0 and ψ > 0) , where φ=

√  √  B−C  B−C  4aC 1− D , ψ = 1+ D , D =1− > 0. 2a 2a (b − a) (B − C)

Proof. Notice that b > a and (3a + b) C < (b − a) B imply that B > C. Suppose that there is a minimal period-two solution {φ, ψ, φ, ψ, . . .} of Eq.(1), where φ and ψ are distinct positive real numbers. Then, we have Bψφ+Cφ2 , aψ 2 +bψφ 2 Bφψ+Cψ , aφ2 +bφψ

φ= ψ=

(10)

from which we obtain (since that φ > 0 and ψ > 0) Bψ + Cφ = aψ 2 + bψφ,

(11)

Bφ + Cψ = aφ2 + bφψ.

(12)

Subtracting Eq.(12) from Eq.(11), we have
B (ψ − φ) + C (φ − ψ) = a ψ 2 − φ2 , i.e. ψ+φ=

B−C . a

(13)

Adding Eq.(11) to Eq.(12) we obtain
B (ψ + φ) + C (φ + ψ) = a ψ 2 + ϕ2 + 2bψφ i.e. (B + C) (ψ + φ) = a (ψ + φ)2 + 2 (b − a) ψφ.

(14)

Substituting (13) into (14), we have (B + C)

B−C =a a



i.e. ψφ =

B−C a

2 + 2 (b − a) ψφ

C B−C . a b−a

From (13) and (15) we see that Eq.(1) has no minimal period-two solutions if B − C ≤ 0 or b − a ≤ 0.

(15)

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

7 42

By (13) and (15) we have that positive φ and ψ satisfy the quadratic equation φ2 −

B−C C B−C φ+ = 0, (b > a and B > C) a a b−a

with solutions φ± = where D =1−

√  B−C  1± D , 2a

4aC B (b − a) − C (3a + b) = > 0 ⇔ (3a + b) C < (b − a) B. (b − a) (B − C) (b − a) (B − C)

Equation (13) implies that ψ± =

B−C − φ± = φ∓ . a 2

By substitution 

xn−1 = un , xn = vn .

Eq.(1) is transformed to the system of equations   un+1 = vn Bun vn + Cu2n .  vn+1 = avn2 + bun vn The map T corresponding to the system (16) is of the form     v u , = T h (u, v) v where h (u, v) =

Buv + Cu2 . The second iteration of the map T is av 2 + buv         h (u, v) h (u, v) v u 2 , = = =T T k (u, v) h (v, h (u, v)) h (u, v) v

where k (u, v) =

Bvh (u, v) + Cu2 , ah2 (u, v) + bvh (u, v)

and the map T 2 is competitive by Remark 2 . The Jacobian matrix of the map T 2 is !   ∂h ∂h u ∂u ∂v JT 2 = . ∂k ∂k v ∂u

∂v

Now we obtain that Jacobian matrix of the map T 2 at the point (φ, ψ) is of the form
∂h φ
!   ∂h φ φ ∂u ψ ∂v ψ JT 2 =
∂k φ
, ∂k φ ψ ∂u ψ

∂v ψ

(16)

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

where

  ∂h φ aBψ 2 + 2aCφψ + bCφ2 = , ∂u ψ ψ (aψ + bφ)2
  φ aBψ 2 + 2aCφψ + bCφ2 ∂h φ =− , ∂v ψ (aψ 2 + bφψ)2

  aBψ 2 + 2aCφψ + bCφ2 aBφ2 + 2aCφψ + bCψ 2 ∂k φ =− . ∂u ψ φ2 (aψ + bφ)2 (aφ + bψ)2     ∂k φ aBφ2 + 2aCφψ + bCψ 2 aBψ 2 + 2aCφψ + bCφ2 = 1+ . ∂v ψ φ (aφ + bψ)2 ψ (aψ + bφ)2

8 43

(17) (18) (19) (20)

The corresponding characteristic equation is λ2 − pλ + q = 0, where     ∂h φ ∂k φ aBψ 2 + 2aCφψ + bCφ2 p = + = ∂u ψ ∂v ψ ψ (aψ + bφ)2   aBφ2 + 2aCφψ + bCψ 2 aBψ 2 + 2aCφψ + bCφ2 + 1+ φ (aφ + bψ)2 ψ (aψ + bφ)2 i.e. p = a∗ + b∗ (1 + a∗ ) = a∗ + b∗ + a∗ b∗ , where a∗ = Notice that now is

aBψ 2 + 2aCφψ + bCφ2 aBφ2 + 2aCφψ + bCψ 2 ∗ , b = . ψ (aψ + bφ)2 φ (aφ + bψ)2 ∂h φ ∂u ψ




= a∗ ,

∂h φ ∂v ψ




= − ψφ a∗ ,

∂k φ ∂u ψ




= − ψφ a∗ b∗ ,

∂k φ ∂v ψ




= b∗ (1 + a∗ ) ,

so that   φ q = det JT 2 = a∗ b∗ (1 + a∗ ) − a∗2 b∗ = a∗ b∗ ψ p = a∗ + b∗ + a∗ b∗ .

Theorem 8 Suppose that Eq.(1) has the minimal period-two solution. Then, this solution is a saddle point. Proof. We need show that |p| > |1 + q| and p2 − 4q > 0. Indeed, (i) p2 − 4q > 0 ⇔ (a∗ + b∗ + a∗ b∗ )2 − 4a∗ b∗ > 0 ⇔ a∗2 + b∗2 + a∗2 b∗2 + 2a∗2 b∗ + 2a∗ b∗2 > 2a∗ b∗ , which is sasfied because of a∗2 + b∗2 ≥ 2a∗ b∗ .

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

9 44

(ii) |p| > |1 + q| ⇔ p > 1 + q ⇔ a∗ + b∗ > 1

⇔

aBψ 2 + 2aCφψ + bCφ2 aBφ2 + 2aCφψ + bCψ 2 + >1 ψ (aψ + bφ)2 φ (aφ + bψ)2

aBψ 2 + 2aCφψ + bCφ2 φ (aφ + bψ)2 + aBφ2 + 2aCφψ + bCψ 2 ψ (aψ + bφ)2

>

φψ (aφ + bψ)2 (aψ + bφ)2 .

⇔

By (11) and (12) we have |p| > |1 + q| ⇔ >



aBψ 2 + 2aCφψ + bCφ2 φ (aφ + bψ)2 + aBφ2 + 2aCφψ + bCψ 2 ψ (aψ + bφ)2 (Bψ + Cφ) (aψ + bφ) (Bφ + Cψ) (aφ + bψ)




⇔ L = abBC φ4 + ψ 4 + 2aC (aB + bC) φψ φ2 + ψ 2 + a2 B 2 + 3a2 C 2 + b2 C 2 + 2abBC − b2 B 2 φ2 ψ 2 > 0
2

⇔ L = abBC φ2 + ψ 2 + 2aC (aB + bC) φψ φ2 + ψ 2 + a2 B 2 + 3a2 C 2 + b2 C 2 − b2 B 2 φ2 ψ 2 > 0. Using (13) and (15) we have φ2 + ψ 2 = (φ + ψ)2 − 2φψ = (B − C)

bB − aB − aC − bC , a2 (b − a)

which implies 

bB − aB − aC − bC L = abBC (B − C) a2 (b − a) + a2 B 2 + 3a2 C 2 + b2 C 2 − b2 B 2

2

C (B − C)2 (bB − aB − aC − bC) + 2aC (aB + bC) a3 (b − a)2


C 2 (B − C)2

a2 (b − a)2 bB − aC = C (B − C)3 [(b − a) B − (3a + b) C] 3 > 0, a (b − a) which is satisfied because B−C > 0, b−a > 0 and (3a + b) C < (b − a) B . Therefore, L > 0 ⇔ |p| > |1 + q|, i.e. the minimal period-two solution of Eq.(1) is a saddle point. 2

4

Global results and basins of attraction

In this section we present results about basins of attraction of Eq.(1). Theorem 9 If (3a + b) C < (b − a) B, then Eq.(1) has a unique equilibrium point x which is locally asymptotically stable, and two minimal period-two points (φ, ψ)and (ψ, φ) which are saddle points. Then, the basin of attraction B ((x, x)) of (x, x) is the region between the global stable sets W s ((φ, ψ)) and W s ((ψ, φ)). The basins of attraction B ((φ, ψ)) = W s ((φ, ψ)) and B ((ψ, φ)) = W s ((ψ, φ)) are exatly the global stable sets of (φ, ψ) and (ψ, φ). Furthermore, i) if (x−1 , x0 ) ∈ W− ((φ, ψ)), then lim x2n = ∞ and lim x2n+1 = 0; n→∞

n→∞

ii) if (x−1 , x0 ) ∈ W+ (ψ, φ), then lim x2n = 0 and lim x2n+1 = ∞. n→∞

n→∞

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

10 45

Figure 1: The orbits of solutions with B = 4, C = 1, a = 1, b = 3.

B+C Proof. Using assumption (3a + b) C < (b − a) B and its consequence B−C · that (φ, ψ) se (x, x) se (ψ, φ), where √  √  B−C  B−C  1− D , ψ = 1+ D , φ = 2a 2a 4aC B (b − a) − C (3a + b) D = 1− = , (b − a) (B − C) (b − a) (B − C)

2a a+b ,

it is easy to check

i.e. φ ≤ x ≤ ψ. Since the equilibrium point (x, x) is locally asymptotically stable for T it is also locally asymptotically stable for T 2 . Equation (1) corresponds to the system of difference equations (16) which can be decomposed into the system of the even-indexed and odd-indexed terms as follows:  u2n = v2n−1 ,      u2n+1 = v2n ,     2 Bu2n−1 v2n−1 + Cv2n−1 (21) v2n = ,  au22n−1 + bu2n−1 v2n−1     2  Bu2n v2n + Cv2n   .  v2n+1 = au22n + bu2n v2n The conclusion folows from Lema 1 and from Theorems 6, 7 and 8 and using the facts: i) if (u0 , v0 ) ∈ W− ((φ, ψ)) , then (u2n , v2n ) = T 2n ((u0 , v0 )) → (0, ∞) and (u2n+1 , v2n+1 ) = T 2n+1 ((u0 , v0 )) → (∞, 0) ;

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

11 46

ii) if (u0 , v0 ) ∈ W+ ((ψ, φ)), then (u2n , v2n ) = T 2n ((u0 , v0 )) → (∞, 0) and (u2n+1 , v2n+1 ) = T 2n+1 ((u0 , v0 )) → (0, ∞) . Consequently, i) if (x−1 , x0 ) ∈ W− ((φ, ψ)), then T 2n ((x−1 , x0 )) → (0, ∞) and T 2n+1 ((x−1 , x0 )) → (∞, 0), i. e. lim x2n = ∞ and lim x2n+1 = 0;

n→∞

n→∞

ii) if (x−1 , x0 ) ∈ W+ ((ψ, φ)), then T 2n+1 ((x−1 , x0 )) → (0, ∞) and T 2n ((x−1 , x0 ))→ (∞, 0), i. e. lim x2n = 0 and lim x2n+1 = ∞.

n→∞

n→∞

2

(See Figure 1 and Figure 2.)

Figure 2: The orbits of solutions with B = 4, C = 1, a = 1, b = 3.

Theorem 10 If b ≤ a or if b > a and (3a + b) C > (b − a) B), then Eq.(1) has a unique equilibrium point x which is saddle point and has no minimal period-two solutions . There exists a set C which is an invariant subset of the basin of attraction of (x, x). The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separets R = (0, ∞) × (0, ∞) into two connected and invariant components W− ((x, x)) and W+ ((x, x)) which satisfy: i) if (x−1 , x0 ) ∈ W− ((x, x)), then lim x2n = ∞ and lim x2n+1 = 0; n→∞

n→∞

ii) if (x−1 , x0 ) ∈ W+ ((x, x)), then lim x2n = 0 and lim x2n+1 = ∞. n→∞

n→∞

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

12 47

Proof. It is easy to check that (x, x) is a saddle point for the strictly competitive map T 2 as well. The existence of the set C with stated properties follows from Lema 1 and Theorems 7, 2 and 4. Therefore, using (21), we obtain: i) if (u0 , v0 ) ∈ W− ((x, x)) , then (u2n , v2n ) = T 2n ((u0 , v0 )) → (0, ∞) and (u2n+1 , v2n+1 ) = T 2n+1 ((u0 , v0 )) → (∞, 0) ; ii) if (u0 , v0 ) ∈ W+ ((x, x)), then (u2n , v2n ) = T 2n ((u0 , v0 )) → (∞, 0) and (u2n+1 , v2n+1 ) = T 2n+1 ((u0 , v0 )) → (0, ∞) . Consequently, i) if (x−1 , x0 ) ∈ W− ((x, x)), then T 2n ((x−1 , x0 )) → (0, ∞) and T 2n+1 ((x−1 , x0 ))→ (∞, 0), that is lim x2n = ∞ and lim x2n+1 = 0;

n→∞

n→∞

ii) if (x−1 , x0 ) ∈ W+ ((x, x)), then T 2n+1 ((x−1 , x0 )) → (0, ∞) and T 2n ((x−1 , x0 ))→ (∞, 0), that is lim x2n = 0 and lim x2n+1 = ∞.

n→∞

n→∞

2

(See Figure 3.)

Figure 3: The orbits of solutions with B = 4, C = 1, a = 2, b = 1.

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

13 48

Theorem 11 If (3a + b) C = (b − a) B, then Eq.(1) has a unique equilibrium point x which is nonhyperbolic and has no minimal period-two solutions. There exists a set C which is an invariant subset of the basin of attraction of x. The set C is a graph of a strictly increasing continuous function of the first variable on an interval and separates R = (0, ∞) × (0, ∞) into two connected and invariant components W− ((x, x)) and W+ ((x, x)) which satisfy: i) if (x−1 , x0 ) ∈ W− ((x, x)), then lim x2n = ∞ and lim x2n+1 = 0; n→∞

n→∞

ii) if (x−1 , x0 ) ∈ W+ ((x, x)), lim x2n = 0 and lim x2n+1 = ∞. n→∞

n→∞

Figure 4: The orbits of solutions with B = 5, C = 1, a = 1, b = 2. Proof. By Lema 1, the eigenvalues of the map T at the equilibrium point (x, x) are λ1 = −1, λ2 = 21 , which means that µ1 = λ21 = 1 and µ2 = λ22 = 41 are eigenvalues of the map T 2 . Using (17), (18), (19) and (20) we obtain ! R −R JT 2 (x, x) = , −R2 R (1 + R) bB−aC where R = (a+b)(B+C) . The straight-forward calculation shows that the eigenvector corresponding the 1 eigenvalue µ2 = 4 is of the form   v1
v2 = , v1 ∈ R \ {0} , 1 1 − 4R v1

which shows that eigenvector v2 is not parallel to coordinate axes. Therefore all conditions of Theorem 2 are satisfied for the map T 2 with R = (0, ∞) × (0, ∞). As a consequence of this and using (21), we have: i) if (u0 , v0 ) ∈ W− ((x, x)) , then (u2n , v2n ) → (0, ∞) and (u2n+1 , v2n+1 ) → (∞, 0);

14

SECOND ORDER QUADRATIC FRACTIONAL DIFFERENCE EQUATION, M. KULENOVIC ET AL

49

ii) if (u0 , v0 ) ∈ W+ ((x, x)), then (u2n , v2n ) → (∞, 0) and (u2n+1 , v2n+1 ) → (0, ∞). It means that: i) if (x−1 , x0 ) ∈ W− ((x, x)), then T 2n ((x−1 , x0 )) → (0, ∞) and T 2n+1 ((x−1 , x0 ))→ (∞, 0), i.e. lim x2n = ∞ and lim x2n+1 = 0;

n→∞

n→∞

ii) if (x−1 , x0 ) ∈ W+ ((x, x)), then T 2n+1 ((x−1 , x0 )) → (0, ∞) and T 2n ((x−1 , x0 ))→ (∞, 0), i.e. lim x2n = 0 and lim x2n+1 = ∞.

n→∞

n→∞

2

(See Figure 4.)

Remark 3 As one may notice from the figures all stable manifolds of either saddle point equilibrium points or non-hyperbolic equilibrium points or saddle period-two solutions are asymptotic to the origin, which is the point where Eq.(1) is not defined. These manifolds can not end in any other point on the axes since the union of axes without the origin is an invariant set, Thus the limiting points of the global stable manifolds of either saddle point equilibrium points or saddle period-two solutions have end points at (0, 0) and (∞, ∞).

References [1] 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. [2] E. Camouzis and G. Ladas, When does local asymptotic stability imply global attractivity in rational equations? J. Difference Equ. Appl., 12(2006), 863885. [3] M. Dehghan, C. M. Kent, R. Mazrooei-Sebdani, N. L. Ortiz and H. Sedaghat, H. Dynamics of rational difference equations containing quadratic terms, J. Difference Equ. Appl. 14 (2008), 191208. [4] M. Dehghan, C. M. Kent, R. Mazrooei-Sebdani, N. L. Ortiz and H. Sedaghat, Monotone and oscillatory solutions of a rational difference equation containing quadratic terms, J. Difference Equ. Appl., 14 (2008), 10451058. [5] E. Drymonis and G. Ladas, On the global character of the rational system xn+1 = yn+1 =

α2 +β2 xn A2 +B2 xn +C2 yn ,

α1 A1 +B1 xn +yn

and

Sarajevo J. Math., Vol. 8 (21) (2012), 293-309.

[6] M. Gari´c-Demirovi´c, M.R.S. Kulenovi´c and M. Nurkanovi´c, Basins of attraction of equilibrium points of second order difference equations, Appl. Math. Lett., 25 (2012) 2110-2115. [7] M. Gari´c-Demirovi´c, M. R. S. Kulenovi´c and M. Nurkanovi´c, Global behavior of four competitive rational systems of difference equations on the plane, Discrete Dyn. Nat. Soc., 2009, Art. ID 153058, 34 pp. [8] E. A. Grove, D. Hadley, E. Lapierre and S. W. Schultz, On the global behavior of the rational system 1 and yn+1 = α2 +βy2 nxn +yn , Sarajevo J. Math., Vol. 8 (21) (2012), 283-292. xn+1 = xnα+y n [9] C. M. Kent and H. Sedaghat, Global attractivity in a quadratic-linear rational difference equation with delay. J. Difference Equ. Appl. 15 (2009), 913925.

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[10] C. M. Kent and H. Sedaghat, Global attractivity in a rational delay difference equation with quadratic terms, J. Difference Equ. Appl., 17 (2011), 457466. [11] 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. [12] M.R.S. Kulenovi´c and O. Merino, Global bifurcations for competitivesystem 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. Bifur. Chaos 20 (2010),2471-2486. [14] C. D. Lynd, The global character of solutions of ananti-competitive system of rational difference equations, Sarajevo J. Math., Vol. 8 (21) (2012), 323-336. [15] S. Moranjki´c and Z. Nurkanovi´c, Basins of attraction of certain rational anti-competitive system of difference equations in theplane, Advances in Difference Equations, 2012, 2012:153. [16] F. J. Palladino, On projective systems of rational difference equations, Sarajevo J. Math., Vol. 8 (21) (2012), 337-353. [17] H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms. J. Difference Equ. Appl. 15 (2009), 215224. [18] H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations64 (1986), 163-194. [19] H. L. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal. 17 (1986), 1289-1318. [20] H. L. Smith, Planar competitive and cooperative difference equations. J. Differ. Equations Appl. 3 (1998), 335357.

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2,51-59 , 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 51

Aspects of univalent holomorphic functions involving Sa˘la˘gean operator and Ruscheweyh derivative Alb Lupa¸s Alina Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract Making use S˘ al˘ agean operator and Ruscheweyh derivative,we introduce a new class of analytic functions L(γ, α, β) defined on the open unit disc, and investigate its various characteristics. Further we obtain distortion bounds, extreme points and radii of close-to-convexity, starlikeness and convexity and neighborhood property for functions belonging to the class L(γ, α, β).

Keywords: Analytic functions, univalent functions, radii of starlikeness and convexity, neighborhood property, Salagean operator, Ruscheweyh operator. 2000 Mathematical Subject Classification: 30C45, 30A20, 34A40.

1

Introduction

P j Let A denote the class of functions of the form f (z) = z + ∞ j=2 aj z , which are analytic and univalent in the openPunit disc U = {z : z ∈ C : |z| < 1}. T is a subclassPof A consisting the functions of the form P∞ ∞ ∞ f (z) = z − j=2 |aj | z j . For functions f, g ∈ A given by f (z) = z + j=2 aj z j , g(z) = z + j=2 bj z j , we define P∞ the Hadamard product (or convolution) of f and g by (f ∗ g)(z) = z + j=2 aj bj z j , z ∈ U. Definition 1.1 Sa˘la˘gean [5]) For f ∈ A, n ∈ N, the operator S n is defined by S n : A → A,

S 0 f (z) = f (z) S 1 f (z) = zf 0 (z), ... S n+1 f (z) = z (S n f (z))0 , z ∈ U. P∞ P∞ Remark 1.1 If f ∈ A, f (z) = z + j=2 aj z j , then S n f (z) = z + j=2 j n aj z j , z ∈ U . P∞ P n j If f ∈ T , f (z) = z − j=2 aj z j , then S n f (z) = z − ∞ j=t+1 j aj z , z ∈ U .

Definition 1.2 (Ruscheweyh [4]) For f ∈ A, n ∈ N, the operator Rn is defined by Rn : A → A, R0 f (z) = f (z) R1 f (z) = zf 0 (z) , ... (n + 1) Rn+1 f (z) = z (Rn f (z))0 + nRn f (z) ,

z ∈ U.

P∞ P∞ j Remark 1.2 If f ∈ A, f (z) = z + j=2 aj z j , then Rn f (z) = z + j=2 (n+j−1)! n!(j−1)! aj z , z ∈ U . P∞ P (n+j−1)! j n j If f ∈ T , f (z) = z − ∞ j=2 aj z , then R f (z) = z − j=t+1 n!(j−1)! aj z , z ∈ U . Definition 1.3 [1], [2] Let γ ≥ 0, n ∈ N. Denote by Lnγ the operator given by Lnγ : A → A, Lnγ f (z) = (1 − γ)Rn f (z) + γS n f (z),

z ∈ U. o P∞ n n P (n+j−1)! j n Remark 1.3 If f ∈ A, f (z) = z + ∞ aj z j , z ∈ U. j=2 aj z , then Lγ f (z) = z + j=2 γj + (1 − γ) n!(j−1)! n o P∞ P ∞ If f ∈ T , f (z) = z − j=2 aj z j , then Lnγ f (z) = z − j=t+1 γj n + (1 − γ) (n+j−1)! aj z j , z ∈ U. n!(j−1)! 1

UNIVALENT HOLOMORPHIC FUNCTIONS, ALINA LUPAS

52

Following the work of Sh.Najafzadeh and E.Pezeshki [3] we can define the class L(γ, α, β) as follows. Definition 1.4 For γ ≥ 0 , 0 ≤ α < 1 and 0 < β ≤ 1, let L(γ, α, β) be the subclass of T consisting of functions that satisfying the inequality ¯ ¯ ¯ ¯ Lµ,n γ f (z) − 1 ¯ 0 for z ∈ U , ζ ∈ U . Q(z,ζ) ¡ ¢ If p is analytic with p (0, ζ) = q (0, ζ), p U × U ⊆ D and θ (p (z, ζ)) + zp0z (z, ζ) φ (p (z, ζ)) ≺≺ θ (q (z, ζ)) + zqz0 (z, ζ) φ (q (z, ζ)) ,

then p (z, ζ) ≺≺ q (z, ζ) and q is the best dominant. Lemma 1.2 Let q be convex univalent in U × U and ν and φ be analytic in a domain ¢ ¡ the function D containing ´ U . Suppose that ³ 0 q U× ν z (q(z,ζ)) 1. Re φ(q(z,ζ)) > 0 for z ∈ U, ζ ∈ U and 2. ψ (z, ζ) = zqz0 (z, ζ) φ (q (z, ζ)) is starlike¡univalent ¢ in U × U . If p (z, ζ) ∈ H∗ [q (0, ζ) , 1, ζ] ∩ Q∗ , with p U × U ⊆ D and ν (p (z, ζ)) + zp0z (z) φ (p (z, ζ)) is univalent in U × U and ν (q (z, ζ)) + zqz0 (z, ζ) φ (q (z, ζ)) ≺≺ ν (p (z, ζ)) + zp0z (z, ζ) φ (p (z, ζ)) , then q (z, ζ) ≺≺ p (z, ζ) and q is the best subordinant.

3

Differential sandwich theorems, Andrei Loriana

2

88

Main results

First, our purpose is to find sufficient conditions for certain normalized analytic functions f such that z δ DRλm,n+1 f (z, ζ) q1 (z, ζ) ≺≺ ¡ ¢1+δ ≺≺ q2 (z, ζ) , z ∈ U, ζ ∈ U , 0 < δ ≤ 1, DRλm,n f (z, ζ)

where q1 and q2 are given univalent functions. Theorem 2.1 Let

z δ DRm,n+1 f (z,ζ) λ

1+δ m,n DRλ f (z,ζ)

¡ ¢ ∈ H∗ U × U , z ∈ U , ζ ∈ U, f ∈ A∗ζ , m, n ∈ N, λ ≥ 0,

( ) 0 < δ ≤ 1 and let the function q (z, ζ) be convex and univalent in U × U such that q (0, ζ) = 1. Assume that ¶ µ zqz0 (z, ζ) zqz002 (z, ζ) α + 0 > 0, z ∈ U, ζ ∈ U , Re 1 + q (z, ζ) − (2.1) β q (z, ζ) qz (z, ζ) for α, β ∈ C,β 6= 0, z ∈ U, ζ ∈ U , and (α, β, δ; z, ζ) ψ m,n λ "

z δ DRλm,n+1 f (z, ζ) := α ¡ ¢1+δ + DRλm,n f (z, ζ)

(2.2)

# DRλm,n+1 f (z, ζ) β δ (n + 1) − 1 + (n + 2) . − (1 + δ) (n + 1) DRλm,n f (z, ζ) DRλm,n+1 f (z, ζ) DRλm,n+2 f (z, ζ)

If q satisfies the following strong differential subordination (α, β, δ; z, ζ) ≺≺ αq (z, ζ) + ψ m,n λ

βzqz0 (z, ζ) , q (z, ζ)

(2.3)

z ∈ U, ζ ∈ U ,

(2.4)

for α, β ∈ C, β 6= 0 then z δ DRλm,n+1 f (z, ζ) ¡ ¢1+δ ≺≺ q (z, ζ) , DRλm,n f (z, ζ)

and q is the best dominant.

m,n+1 z δ DRλ f (z,ζ)

1+δ , z ∈ U , ζ ∈ U , z 6= 0, 0 < f (z,ζ)) (DRm,n λ δ ≤ 1, f ∈ A∗ζ . The function p is analytic in U × U and p (0, ζ) = 1 Differentiating this function, to z,we get ∙ with respect ¸ 0 0 m,n+1 m,n m,n+1 δ z DR f (z,ζ) z (DRλ f (z,ζ))z ( )z z DRλ f (z,ζ) λ 0 δ+ . − (1 + δ) DRm,n f (z,ζ) zpz (z, ζ) = m,n+1 1+δ DRλ f (z,ζ) λ (DRλm,n f (z,ζ)) By using the identity (1.2), we obtain

Proof. Let the function p be defined by p (z, ζ) :=

DRλm,n+2 f (z, ζ) zp0z (z, ζ) = δ (n + 1) − 1 + (n + 2) − p (z, ζ) DRλm,n+1 f (z, ζ) (1 + δ) (n + 1)

DRλm,n+1 f (z, ζ) . DRλm,n f (z, ζ)

(2.5)

β , α, β ∈ C, β 6= 0 it can be easily verified that θ is analytic By setting θ (w) := αw and φ (w) := w in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}.

4

Differential sandwich theorems, Andrei Loriana

Also, by letting Q (z, ζ) = zqz0 (z, ζ) φ (q (z, ζ)) = lent in U × U . Let h (z, ζ) = θ (q (z, ζ)) + Q (z, ζ) = αq (z, ζ) +

βzqz0 (z,ζ) q(z,ζ) ,we

βzqz0 (z,ζ) q(z,ζ) ,

89

find that Q (z, ζ) is starlike univa-

z ∈ U, ζ ∈ U .

If we derive the function Q, with respect to z, perform calculations, we have Re ¶ µ zq 002 (z,ζ) zqz0 (z,ζ) α z Re 1 + β q (z, ζ) − q(z,ζ) + q0 (z,ζ) > 0.

³

zh0z (z,ζ) Q(z,ζ)

´

=

z

0

z (z,ζ) = By using (2.5), we obtain αp (z, ζ) + β zpp(z,ζ) ¸ ∙ m,n+1 m,n+2 m,n+1 δ z DRλ f (z,ζ) DRλ f (z,ζ) DRλ f (z,ζ) α − (1 + δ) (n + 1) DRm,n f (z,ζ) . 1+δ + β δ (n + 1) − 1 + (n + 2) DRm,n+1 f (z,ζ) f (z,ζ)) λ (DRm,n λ λ 0 βzpz (z,ζ) βzqz0 (z,ζ) By using (2.3), we have αp (z, ζ) + p(z,ζ) ≺≺ αq (z, ζ) + q(z,ζ) .

i.e.

Therefore, the conditions of Lemma 1.1 are met, so we have p (z, ζ) ≺≺ q (z, ζ), z ∈ U, ζ ∈ U , z δ DRm,n+1 f (z,ζ) λ

1+δ

(DRλm,n f (z,ζ))

≺≺ q (z, ζ), z ∈ U, ζ ∈ U , and q is the best dominant.

Corollary 2.2 Let q (z, ζ) = (2.1) holds. If f ∈ A∗ζ and

ζ+Az ζ+Bz ,

−1 ≤ B < A ≤ 1, m, n ∈ N, λ ≥ 0, z ∈ U, ζ ∈ U . Assume that

(α, β, δ; z, ζ) ≺ α ψ m,n λ

(A − B) ζz ζ + Az +β , ζ + Bz (ζ + Az) (ζ + Bz)

for α, β ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, 0 < δ ≤ 1 where ψ m,n is defined in (2.2), then λ

and

ζ+Az ζ+Bz

is the best dominant.

Proof. For q (z, ζ) =

ζ+Az ζ+Bz ,

z δ DRλm,n+1 f (z, ζ) ζ + Az ¡ ¢1+δ ≺≺ m,n ζ + Bz DRλ f (z, ζ) −1 ≤ B < A ≤ 1, in Theorem 2.1 we get the corollary.

Theorem 2.3 Let q be convex and univalent in U × U , such that q (0, ζ) = 1, m, n ∈ N, λ ≥ 0. Assume that ¶ µ α q (z, ζ) > 0, for α, β ∈ C, β 6= 0, z ∈ U, ζ ∈ U . Re (2.6) β m,n+1 z δ DRλ f (z,ζ)

m,n ∗ ∗ (α, β, δ; z, ζ) is univalent in 1+δ ∈ H [q (0, ζ) , 1, ζ] ∩ Q and ψ λ f (z,ζ) (DRm,n ) λ (α, β, δ; z, ζ) is as defined in (2.2), then U × U , where ψ m,n λ

If f ∈ A∗ζ ,0 < δ ≤ 1,

αq (z, ζ) +

βzqz0 (z, ζ) (α, β, δ; z, ζ) , ≺≺ ψ m,n λ q (z, ζ)

z ∈ U, ζ ∈ U ,

(2.7)

implies z δ DRλm,n+1 f (z, ζ) q (z, ζ) ≺≺ ¡ ¢1+δ , DRλm,n f (z, ζ)

z ∈ U, ζ ∈ U ,

(2.8)

and q is the best subordinant.

Proof. Let the function p be defined by p (z, ζ) := 0 < δ ≤ 1, f ∈

A∗ζ . 5

m,n+1 z δ DRλ f (z,ζ)

1+δ

f (z,ζ)) (DRm,n λ

, z ∈ U , ζ ∈ U , z 6= 0,

Differential sandwich theorems, Andrei Loriana

90

β By setting ν (w) := αw and φ (w) := w it can be easily verified that ν is analytic in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. ´ ³ 0 ´ ³ z (q(z,ζ)) Since q is convex and univalent function, it follows that Re νφ(q(z,ζ)) = Re αβ q (z, ζ) > 0, for α, β ∈ C, β 6= 0. By using (2.7) we obtain

αq (z, ζ) + β

zqz0 (z, ζ) zp0 (z, ζ) ≺≺ αp (z, ζ) + β z . q (z, ζ) p (z, ζ)

Using Lemma 1.2, we have z δ DRλm,n+1 f (z, ζ) q (z, ζ) ≺≺ p (z, ζ) = ¡ ¢1+δ , DRλm,n f (z, ζ)

z ∈ U, ζ ∈ U ,

and q is the best subordinant. Corollary 2.4 Let q (z, ζ) = If f ∈ A∗ζ ,

m,n+1 z δ DRλ f (z,ζ) 1+δ m,n DRλ f (z,ζ)

(

)

α

ζ+Az ζ+Bz ,

−1 ≤ B < A ≤ 1, m, n ∈ N, λ ≥ 0. Assume that (2.6) holds.

∈ H∗ [q (0, ζ) , 1, ζ] ∩ Q∗ and

ζ + Az (A − B) ζz (α, β, δ; z, ζ) , +β ≺≺ ψ m,n λ ζ + Bz (ζ + Az) (ζ + Bz)

is defined in (2.2), then for α, β ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψ m,n λ

and

ζ+Az ζ+Bz

z δ DRλm,n+1 f (z, ζ) ζ + Az ≺≺ ¡ ¢1+δ ζ + Bz DRλm,n f (z, ζ)

is the best subordinant.

ζ+Az Proof. For q (z, ζ) = ζ+Bz , −1 ≤ B < A ≤ 1 in Theorem 2.3 we get the corollary. Combining Theorem 2.1 and Theorem 2.3, we state the following sandwich theorem.

Theorem 2.5 Let q1 and q2 be analytic and univalent in U ×U such that q1 (z, ζ) 6= 0 and q2 , ζ 6= 0, for all z ∈ U , ζ ∈ U , with z (q1 )0z (z, ζ) and z (q2 )0z (z, ζ) being starlike univalent. Suppose that q1 satisfies (2.1) and q2 satisfies (2.6). If f ∈ A∗ζ ,

m,n+1 z δ DRλ f (z,ζ)

∗ ∗ 1+δ ∈ H [q (0, ζ) , 1, ζ] ∩ Q , 0 < δ ≤ 1 (DRλm,n f (z,ζ)) (α, β, δ; z, ζ) is as defined in (2.2) univalent in U × U , then and ψ m,n λ

αq1 (z, ζ) +

βz (q1 )0z (z, ζ) βz (q2 )0z (z, ζ) (α, β, δ; z, ζ) ≺≺ αq (z, ζ) + ≺≺ ψ m,n , 2 λ q1 (z, ζ) q2 (z, ζ)

for α, β ∈ C, β 6= 0, implies z δ DRλm,n+1 f (z, ζ) q1 (z, ζ) ≺≺ ¡ ¢1+δ ≺≺ q2 (z, ζ) , DRλm,n f (z, ζ)

δ ∈ C, δ 6= 0,

and q1 and q2 are respectively the best subordinant and the best dominant. ζ+A1 z For q1 (z, ζ) = ζ+B , q2 (z, ζ) = 1z following corollary.

ζ+A2 z ζ+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the 6

Differential sandwich theorems, Andrei Loriana

91

Corollary 2.6 Let m, n ∈ N, λ ≥ 0. Assume that (2.1) and (2.6) hold for q1 (z, ζ) = q2 (z, ζ) =

ζ+A2 z ζ+B2 z ,

respectively. If f ∈ A∗ζ , 0 < δ ≤ 1, α

m,n+1 z δ DRλ f (z,ζ)

1+δ

(DRλm,n f (z,ζ))

ζ+A1 z ζ+B1 z

and

∈ H∗ [q (0, ζ) , 1, ζ] ∩ Q∗ and

(A1 − B1 ) ζz ζ + A1 z (α, β, δ; z, ζ) +β ≺≺ ψ m,n λ ζ + B1 z (ζ + A1 z) (ζ + B1 z) ≺≺ α

ζ + A2 z (A2 − B2 ) ζz +β , ζ + B2 z (ζ + A2 z) (ζ + B2 z)

is defined in (2.2), then for α, β ∈ C, β 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψ m,n λ

hence

ζ+A1 z ζ+B1 z

and

ζ+A2 z ζ+B2 z

z δ DRλm,n+1 f (z, ζ) ζ + A2 z ζ + A1 z ≺≺ ≺≺ ¡ , ¢ 1+δ ζ + B1 z ζ + B2 z DRλm,n f (z, ζ)

are the best subordinant and the best dominant, respectively.

Next, our purpose is to find sufficient conditions for certain normalized analytic functions f such that à !δ aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) q1 (z, ζ) ≺≺ ≺≺ q2 (z, ζ) , z ∈ U, ζ ∈ U , (a + b) z where q1 and q2 are given univalent functions Theorem 2.7 Let

µ

m,n aDRm+1,n f (z,ζ)+bDRλ f (z,ζ) λ (a+b)z

¶δ

¡ ¢ ∈ H∗ U × U , f ∈ A∗ζ , z ∈ U , ζ ∈ U , δ, a, b ∈

C, δ 6= 0, a + b 6= 0, m, n ∈ N, λ ≥ 0 and let the function q (z, ζ) be convex and univalent in U × U such that q (0, ζ) = 1, z ∈ U, ζ ∈ U . Assume that ¶ µ zqz0 (z, ζ) zqz002 (z, ζ) α + 0 > 0, (2.9) Re 1 + q (z, ζ) − β q (z, ζ) qz (z, ζ) for α, β ∈ C, β 6= 0, z ∈ U, ζ ∈ U , and (a, b, α, β, δ; z, ζ) ψ m,n λ

:= α

Ã

aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) (a + b) z

!δ

+

(2.10)

h i βδ aDRλm+2,n f (z, ζ) + (b − a) DRλm+1,n f (z, ζ) − bDRλm,n f (z, ζ) ³ ´ . λ aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ)

If q satisfies the following strong differential subordination

ψ m,n (a, b, α, β, δ; z, ζ) ≺≺ αq (z, ζ) + λ for α, β ∈ C, β 6= 0, z ∈ U, ζ ∈ U , then !δ Ã aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) ≺≺ q (z, ζ) , (a + b) z and q is the best dominant. 7

βzqz0 (z, ζ) , q (z, ζ)

z ∈ U, ζ ∈ U , δ ∈ C, δ 6= 0,

(2.11)

(2.12)

Differential sandwich theorems, Andrei Loriana

Proof. Let the function p be defined by p (z, ζ) :=

µ

m+1,n m,n aDRλ f (z,ζ)+bDRλ f (z,ζ) (a+b)z

92

¶δ

, z ∈ U ,ζ ∈ U ,

z 6= 0, δ, a, b ∈ C, δ 6= 0, a + b 6= 0, f ∈ A∗ζ . The function p is analytic in U × U and p (0, ζ) = 1. Differentiating this function, with respect to z, we get ¶δ−1 µ m+1,n aDRλ f (z,ζ)+bDRm,n f (z,ζ) 1 λ zp0z (z, ζ) = δ a+b · (a+b)z ∙ ¸ 0 0 m+1,n m,n m+1,n m,n a(DRλ f (z,ζ))z +b(DRλ f (z,ζ))z aDRλ f (z,ζ)+bDRλ f (z,ζ) − . z z2 We have

Ã

!δ aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) (z, ζ) = δ · (a + b) z ∙ ³ ´0 1 m+1,n az DR f (z, ζ) + λ z aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) i ¢0 ¡ m,n m+1,n m,n f (z, ζ) − bDRλ f (z, ζ) . bz DRλ f (z, ζ) z − aDRλ zp0z

(2.13)

By using the identity (1.1) we obtain h i m+2,n m+1,n m,n δ aDR 0 f (z, ζ) + (b − a) DR f (z, ζ) − bDR f (z, ζ) λ λ λ zpz (z, ζ) ³ ´ . = p (z, ζ) λ aDRm+1,n f (z, ζ) + bDRm,n f (z, ζ) λ

(2.14)

λ

β , α, β ∈ C, β 6= 0 it can be easily verified that θ is analytic By setting θ (w) := αw and φ (w) := w in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. 0 z (z,ζ) Also, by letting Q (z, ζ) = zqz0 (z, ζ) φ (q (z, ζ)) = βzq q(z,ζ) ,we find that Q (z, ζ) is starlike univa-

lent in U × U . Let h (z, ζ) = θ (q (z, ζ)) + Q (z, ζ) = αq (z, ζ) +

βzqz0 (z,ζ) q(z,ζ) ,

z ∈ U , ζ ∈ U.

If we derive the function Q, with respect to z, perform calculations, we have Re µ ¶ 0 zq 00 (z,ζ) z (z,ζ) Re 1 + αβ q (z, ζ) − zqq(z,ζ) + qz0 2(z,ζ) > 0.

³

zh0z (z,ζ) Q(z,ζ)

´

=

z

0

z (z,ζ) By using (2.5), we obtain αp (z, ζ) + β zpp(z,ζ) = µ ¶ δ m+2,n m+1,n m+1,n m,n βδ [aDRλ f (z,ζ)+(b−a)DRλ f (z,ζ)−bDRm,n f (z,ζ)] aDRλ f (z,ζ)+bDRλ f (z,ζ) λ . α + m+1,n (a+b)z λ(aDRλ f (z,ζ)+bDRm,n f (z,ζ)) λ 0 (z,ζ) 0 z z (z,ζ) By using (2.11), we have αp (z, ζ) + β zpp(z,ζ) ≺≺ αq (z, ζ) + β zqp(z,ζ) . ¶δ µ m+1,n aDRλ f (z,ζ)+bDRm,n f (z,ζ) λ From Lemma 1.1, we have p (z, ζ) ≺≺ q (z, ζ), z ∈ U, ζ ∈ U, i.e. (a+b)z

≺≺ q (z, ζ), z ∈ U, ζ ∈ U , δ ∈ C, δ 6= 0 and q is the best dominant. Corollary 2.8 Let q (z, ζ) = (2.9) holds. If f ∈ A∗ζ and

ζ+Az ζ+Bz ,

z ∈ U, ζ ∈ U , −1 ≤ B < A ≤ 1, m, n ∈ N, λ ≥ 0. Assume that

ψ m,n (a, b, α, β, δ; z, ζ) ≺≺ α λ

ζ + Az (A − B) ζz +β , ζ + Bz (ζ + Az) (ζ + Bz)

is defined in (2.10), then for α, β ∈ C, β = 6 0, −1 ≤ B < A ≤ 1, where ψ m,n λ !δ Ã aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) ζ + Az ≺≺ , δ ∈ C, δ 6= 0, (a + b) z ζ + Bz 8

Differential sandwich theorems, Andrei Loriana

and

ζ+Az ζ+Bz

93

is the best dominant.

Proof. For q (z, ζ) =

ζ+Az ζ+Bz ,

−1 ≤ B < A ≤ 1, in Theorem 2.7 we get the corollary.

Theorem 2.9 Let q be convex and univalent in U × U such that q (0, ζ) = 1. Assume that µ ¶ α Re q (z, ζ) > 0, for α, β ∈ C, β 6= 0. (2.15) β µ ¶δ m+1,n m,n aDRλ f (z,ζ)+bDRλ f (z,ζ) ∗ If f ∈ Aζ , δ, a, b ∈ C, δ 6= 0, a + b 6= 0, ∈ H∗ [q (0, ζ) , 1, ζ] ∩ Q∗ and (a+b)z

(a, b, α, β, δ; z, ζ) is univalent in U × U , where ψ m,n (a, b, α, β, δ; z, ζ) is as defined in (2.10), ψ m,n λ λ then βzqz0 (z, ζ) ≺≺ ψ m,n αq (z, ζ) + (a, b, α, β, δ; z, ζ) (2.16) λ q (z, ζ) implies à !δ aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) q (z, ζ) ≺≺ , δ ∈ C, δ 6= 0, z ∈ U, ζ ∈ U , (2.17) (a + b) z and q is the best subordinant. Proof. Let the function p be defined by p (z, ζ) :=

µ

m,n aDRm+1,n f (z,ζ)+bDRλ f (z,ζ) λ (a+b)z

¶δ

, z ∈ U,

ζ ∈ U , z 6= 0, a, b ∈ C, a + b 6= 0, δ ∈ C, δ 6= 0, f ∈ A∗ζ . The function p is analytic in U × U and p (0, ζ) = 1. β it can be easily verified that ν is analytic in C, φ is By setting ν (w) := αw and φ (w) := w analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. ³ 0 ´ ´ ³ z (q(z,ζ)) Since q is convex and univalent function, it follows that Re νφ(q(z,ζ)) = Re αβ q (z, ζ) > 0, for α, β ∈ C, β 6= 0. By using (2.7) we obtain αq (z, ζ) + β

zqz0 (z, ζ) zp0 (z, ζ) ≺≺ αp (z, ζ) + β z . q (z, ζ) p (z, ζ)

From Lemma 1.2, we have à !δ aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) q (z, ζ) ≺≺ p (z, ζ) = , (a + b) z

z ∈ U, ζ ∈ U , δ ∈ C, δ 6= 0,

and q is the best subordinant. ζ+Az Corollary 2.10 Let q (z, ζ) = ζ+Bz , −1 ≤ B < A ≤ 1, z ∈ U, ζ ∈ U , m, n ∈ N, λ ≥ 0. Assume ¶δ µ m+1,n aDRλ f (z,ζ)+bDRm,n f (z,ζ) ∗ λ ∈ H∗ [q (0, ζ) , 1, ζ] ∩ Q∗ , δ ∈ C, δ 6= 0 that (2.15) holds. If f ∈ Aζ , (a+b)z

and

α

ζ + Az (A − B) ζz (a, b, α, β, δ; z, ζ) , +β ≺≺ ψ m,n λ ζ + Bz (ζ + Az) (ζ + Bz)

for α, β ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψ m,n is defined in (2.10), then λ Ã !δ aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) ζ + Az ≺≺ , δ ∈ C, δ 6= 0, a, b ∈ C, a + b 6= 0 ζ + Bz (a + b) z and

ζ+Az ζ+Bz

is the best subordinant. 9

Differential sandwich theorems, Andrei Loriana

94

ζ+Az Proof. For q (z, ζ) = ζ+Bz , −1 ≤ B < A ≤ 1, in Theorem 2.9 we get the corollary. Combining Theorem 2.7 and Theorem 2.9, we state the following sandwich theorem.

Theorem 2.11 Let q1 and q2 be convex and univalent in U ×U such that q1 (z, ζ) 6= 0 and q2 (z, ζ) 6= 0, for all z ∈ U, ζ ∈ U . Suppose that q1 satisfies (2.9) and q2 satisfies (2.15). If f ∈ A∗ζ , µ ¶δ m,n aDRm+1,n f (z,ζ)+bDRλ f (z,ζ) λ ∈ H∗ [q (0, ζ) , 1, ζ] ∩ Q∗ , δ ∈ C, δ 6= 0, a, b ∈ C, a + b 6= 0 and (a+b)z

(a, b, α, β, δ; z, ζ) is as defined in (2.10) univalent in U × U , then ψ m,n λ αq1 (z, ζ) +

βz (q1 )0z (z, ζ) βz (q2 )0z (z, ζ) (a, b, α, β, δ; z, ζ) ≺≺ αq (z, ζ) + ≺≺ ψ m,n , 2 λ q1 (z, ζ) q2 (z, ζ)

for α, β ∈ C, β 6= 0, implies q1 (z, ζ) ≺≺

Ã

aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) (a + b) z

!δ

≺≺ q2 (z, ζ) ,

z ∈ U, ζ ∈ U , δ ∈ C, δ 6= 0,

and q1 and q2 are respectively the best subordinant and the best dominant. ζ+A1 z For q1 (z, ζ) = ζ+B , q2 (z, ζ) = 1z following corollary.

ζ+A2 z ζ+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the

Corollary 2.12 Let m, n ∈ N, λ ≥ 0. Assume that (2.9) and (2.15) hold for q1 (z, ζ) = µ m+1,n ¶δ DRλ f (z,ζ) ζ+A2 z ∗ q2 (z, ζ) = ζ+B2 z , respectively. If f ∈ Aζ , ∈ H∗ [q (0, ζ) , 1, ζ] ∩ Q∗ and DRm,n f (z,ζ)

ζ+A1 z ζ+B1 z

and

λ

α

(A1 − B1 ) ζz ζ + A1 z +β ≺≺ ψ m,n (a, b, α, β, δ; z, ζ) λ ζ + B1 z (ζ + A1 z) (ζ + B1 z) ≺≺ α

(A2 − B2 ) ζz ζ + A2 z +β , ζ + B2 z (ζ + A2 z) (ζ + B2 z)

is defined in (2.10), then for α, β ∈ C, β 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψ m,n λ ζ + A1 z ≺≺ ζ + B1 z

Ã

aDRλm+1,n f (z, ζ) + bDRλm,n f (z, ζ) (a + b) z

δ ∈ C, δ 6= 0, a, b ∈ C, a + b 6= 0, hence dominant, respectively.

ζ+A1 z ζ+B1 z

and

!δ

ζ+A2 z ζ+B2 z

≺≺

ζ + A2 z , ζ + B2 z

z ∈ U, ζ ∈ U ,

are the best subordinant and the best

References [1] A. Alb Lupa¸s, On special strong differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Journal of Concrete and Applicable Mathematics, Vol. 10, No.’s 1-2, 2012, 17-23. [2] A. Alb Lupa¸s, G.I. Oros, Gh. Oros, On special strong differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Journal of Computational Analysis and Applications, Vol. 14, No. 2, 2012, 266-270.

10

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[3] A. Alb Lupa¸s, A note on strong differential subordinations using a generalized Sa ˘la˘gean operator and Ruscheweyh operator, Acta Universitatis Apulensis No. 34/2013, 105-114. [4] A. Alb Lupa¸s, Certain strong differential superordinations using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 6268. [5] A. Alb Lupa¸s, A note on strong differential superordinations using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Stud. Univ. Babes-Bolyai Math. 57(2012), No. 2, 153—165. [6] A. Alb Lupa¸s, Certain strong differential subordinations using Sa˘la ˘gean and Ruscheweyh operators, Advances in Applied Mathematical Analysis, Volume 6, Number 1 (2011), 27—34. [7] A. Alb Lupa¸s, A note on strong differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Libertas Mathematica, tomus XXXI (2011), 15-21. [8] A. Alb Lupa¸s, Certain strong differential superordinations using Sa ˘la˘gean and Ruscheweyh operators, Acta Universitatis Apulensis No. 30/2012, 325-336. [9] A. Alb Lupa¸s, A note on strong differential superordinations using Sa˘la˘gean and Ruscheweyh operators, Journal of Applied Functional Analysis, Vol. 7, No.’s 1-2, 2012, 54-61. [10] L. Andrei, On certain differential sandwich theorems involving a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted 2014. [11] L. Andrei, Differential Sandwich Theorems using an extending generalized Sa˘la˘gean operator and extended Ruscheweyh operator, submitted 2014. [12] L. Andrei, Strong differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Journal of Computational Analysis and Applications (to appear). [13] L. Andrei, Some strong differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted International Journal of Modern Mathematical Sciences. [14] L. Andrei, Strong differential superordination results using a generalized Sa˘la ˘gean operator and Ruscheweyh operator, submitted Journal of Computational Mathematics. [15] L. Andrei, Some strong differential superordination results using a generalized Sa˘la ˘gean operator and Ruscheweyh operator, International Journal of Research and Reviews in Applied Sciences (to appear). [16] J.A. Antonino, S. Romaguera, Strong differential subordination to Briot-Bouquet differential equations, Journal of Differential Equations, 114 (1994), 101-105. [17] G.I. Oros, Gh. Oros, Strong differential subordination, Turkish Journal of Mathematics, 33 (2009), 249-257. [18] G.I. Oros, Strong differential superordination, Acta Universitatis Apulensis, 19 (2009), 101-106.

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J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2,96-107 , 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 96

HERMITE-HADAMARD TYPE INEQUALITIES FOR (h ( ; m))-CONVEX FUNCTIONS M.E. OZDEMIR, HAVVA KAVURMACI ÖNALAN F , AND MERVE AVCI ARDIÇ

Abstract. In this paper, we de…ne (h ( ; m))-convex functions that is a new class of convex functions as a generalization of ( ; m) convexity and h convexity. We also prove some Hadamard’s type inequalities.

1. INTRODUCTION Let f : I R ! R be a convex function on the interval I of real numbers and a; b 2 I with a < b: The inequality Z b a+b 1 f (a) + f (b) (1.1) f f (x)dx 2 b a a 2

is known as Hermite-Hadamard’s inequality for convex functions, [9]. In [6], Toader de…ned m convexity as the following.

De…nition 1. The function f : [0; b] ! R; b > 0, is said to be m convex where m 2 [0; 1]; if we have f (tx + m(1

t)y)

tf (x) + m(1

t)f (y)

for all x; y 2 [0; b] and t 2 [0; 1]: We say that f is m concave if ( f ) is m convex. In [3], Dragomir proved the following theorem. Theorem 1. Let f : [0; 1) ! R be an m convex function with m 0 a < b: If f 2 L1 [a; b] ; then the following inequalities hold: Z b x f (x) + mf m a+b 1 (1.2) f dx 2 b a a 2 2 b a b f m + mf f (a) + mf 16 m2 m +m 6 4 2 2 2

2 (0; 1] and

3

7 7: 5

In [10], de…nition of ( ; m) convexity was introduced by Mihe¸san as following.

De…nition 2. The function f : [0; b] ! R; b > 0, is said to be ( ; m) convex, where ( ; m) 2 [0; 1]2 ; if we have f (tx + m(1

t)y)

t f (x) + m(1

t )f (y)

for all x; y 2 [0; b] and t 2 [0; 1]: 1991 Mathematics Subject Classi…cation. 20D10;20D15. Key words and phrases. (h ( ; m))-convex functions,Hadamard’s inequality. F Corresponding Author. 1

97

M .E. OZDEM IR, HAVVA KAVURM ACI ÖNALAN F , AND M ERVE AVCI ARDIÇ

2

Denote by Km (b) the class of all ( ; m) convex functions on [0; b] for which f (0) 0: If we take ( ; m) = f(0; 0) ; ( ; 0) ; (1; 0) ; (1; m) ; (1; 1) ; ( ; 1)g ; it can be easily seen that ( ; m) convexity reduces to increasing, starshaped, starshaped, m convex, convex and convex functions, respectively. In [16], Set et al. proved the following Hadamard type inequalities for ( ; m) convex functions. Theorem 2. Let f : [0; 1) ! R be an ( ; m) convex function with ( ; m) 2 2 a b ; m ; then one has the inequality: (0; 1] : If 0 a < b < 1 and f 2 L1 [a; b]\L1 m # "Z x b f (x) + m (2 1) f m a+b 1 dx : (1.3) f 2 b a a 2 Theorem 3. Let f : [0; 1) ! R be an ( ; m) convex function with ( ; m) 2 2 (0; 1] : If 0 a < b < 1 and f 2 L1 [a; b] ; then one has the inequality: ) ( Z b a b f (b) + mf m f (a) + mf m 1 (1.4) ; : f (x) dx min b a a +1 +1 Theorem 4. Let f : [0; 1) ! R be an ( ; m) convex function with ( ; m) 2 2 (0; 1] : If f 2 L1 [ma; b] where 0 a < b; then one has the inequality: Z mb Z b 1 1 (1.5) f (x) dx + f (x) dx mb a a b ma ma m+1 [f (a) + f (b)] : +1 In [1], Bakula et al. proved the following theorem. Theorem 5. Let f; g : [0; 1) ! [0; 1) be such that f g is in L1 ([a; b]) ; where 0 a < b < 1: If f is ( 1 ; m1 ) convex and g is ( 2 ; m2 ) convex on [a; b] for some …xed 1 ; m1 ; 2 ; m2 2 (0; 1] ; then Z b 1 (1.6) f (x) g (x) dx min fN1 ; N2 g ; b a a where f (a)g(a) 1 1 b N1 = + m2 f (a)g m2 1+ 2+1 1+1 1+ 2+1 +m1 2

1

1 +1

+m1 m2 1

1

+

2

1 1+1

+1

b m1

f

1 + 2+1

g (a) 1

1+

f

b m1

f (b)g

a m2

2+1

g

b m2

g

a m2

and N2

=

f (b)g(b) + m2 1+ 2+1 +m1 2

1 +1

+m1 m2 1

1 1+1 1

1

+

1 1+1

2

+1

1 1

f

1 + 2+1

+

2

a m1

+1

g (b) 1

1+

2+1

f

a m1

:

98

( ; m))-CONVEX FUNCTIONS

(h

3

For the recent results based on the above de…nition see the papers [1], [2], [11]-[14] and [16]. In [8], Hudzik and Maligranda considered among others the class of functions which are s convex in the second sense. De…nition 3. A function f : R+ ! R; where R+ = [0; 1); is said to be s convex in the second sense if s

f ( x + y)

f (x) +

s

f (y)

for all x; y 2 [0; 1); ; 0 with + = 1 and for some …xed s 2 (0; 1]: This class of s convex functions in the second sense is usually denoted by Ks2 : It can be easily seen that for s = 1; s convexity reduces to ordinary convexity of functions de…ned on [0; 1). In [4], Dragomir and Fitzpatrick proved a variant of Hadamard’s inequality which holds for s convex functions in the second sense. Theorem 6. Suppose that f : [0; 1) ! [0; 1) is an s convex function in the second sense, where s 2 (0; 1); and let a; b 2 [0; 1); a < b: If f 2 L1 [a; b]; then the following inequalities hold: Z b 1 f (a) + f (b) a+b f (x)dx : (1.7) 2s 1 f 2 b a a s+1 1 The constant k = s+1 is the best possible in the second inequality in (1.7). In [7], Godunova and Levin introduced the following class of functions.

De…nition 4. A function f : I R ! R is said to belong to the class of Q(I) if it is non-negative and for all x; y 2 I and 2 (0; 1) satis…es the inequality; f ( x + (1

)y)

f (x)

+

f (y) : 1

In [5], Dragomir et al. de…ned the following new class of functions. De…nition 5. A function f : I R ! R is P function or that f belongs to the class of P (I); if it is non-negative and for all x; y 2 I and 2 [0; 1]; satis…es the following inequality; f ( x + (1 )y) f (x) + f (y): In [5], Dragomir et al. proved the following inequalities of Hadamard type for class of Q(I) functions and P functions. Theorem 7. Let f 2 Q(I); a; b 2 I with a < b and f 2 L1 [a; b]: Then the following inequalities hold: Z b a+b 4 f f (x)dx 2 b a a and

1 b where p (x) =

(b x)(x a) ; (b a)2

a

Z

b

p (x) f (x)dx

a

x 2 [a; b] :

f (a) + f (b) 2

99

4

M .E. OZDEM IR, HAVVA KAVURM ACI ÖNALAN F , AND M ERVE AVCI ARDIÇ

Theorem 8. Let f 2 P (I); a; b 2 I with a < b and f 2 L1 [a; b]: Then the following inequalities hold: Z b a+b 2 (1.8) f f (x)dx 2[f (a) + f (b)]: 2 b a a In [15], Varošanec de…ned the following class of functions. I and J are intervals in R; (0; 1) J and functions h and f are real non-negative functions de…ned on J and I, respectively. De…nition 6. Let h : J R ! R be a non-negative function, h 6= 0: We say that f : I ! R is an h convex function, or that f belongs to the class SX(h; I), if f is non-negative and for all x; y 2 I, 2 (0; 1) we have (1.9)

f ( x + (1

)y)

h( )f (x) + h(1

)f (y):

If inequality (1.9) is reversed, then f is said to be h concave, i.e. f 2 SV (h; I). Obviously, if h( ) = ; then all non-negative convex functions belong to SX(h; I) and all non-negative concave functions belong to SV (h; I); if h( ) = 1 ; then SX(h; I) = Q(I); if h( ) = 1; then SX(h; I) P (I); and if h( ) = s ; where s 2 (0; 1) ; then SX(h; I) Ks2 : In [17], Sar¬kaya et al. proved a variant of Hadamard inequality which holds for h convex functions. Theorem 9. Let f 2 SX (h; I) ; a; b 2 I; with a < b and f 2 L1 ([a; b]) : Then

(1.10)

1 2h

1 2

f

a+b 2

1 b

a

Z

b

f (x) dx

a

[f (a) + f (b)]

Z

1

h( )d :

0

In [12], Özdemir et al. de…ned (h; m) convexity and obtained Hermite-Hadamardtype inequalities as following . Theorem 10. Let f : [0; 1) ! R be an (h; m) convex function with m 2 (0; 1] ; t 2 [0; 1] : If 0 a < b < 1 and f 2 L1 [a; b] ; then the following inequality holds: Z b 1 (1.11) f (x)dx b a a Z 1 Z 1 b h (1 t) dt ; min f (a) h(t)dt + mf m 0 0 Z 1 Z 1 a f (b) h(t)dt + mf h (1 t) dt : m 0 0 Theorem 11. Let f : [0; 1) ! R be an (h; m) convex function with m 2 (0; 1] ; t 2 [0; 1] : If 0 a < b < 1 and f 2 L1 [ma; b] ; then the following inequality holds: " # Z mb Z b 1 1 1 (1.12) f (x) dx + f (x) dx m + 1 mb a a b ma ma [f (a) + f (b)]

Z

0

1

h (t) dt:

100

(h

( ; m))-CONVEX FUNCTIONS

5

Theorem 12. Let f : [0; 1) ! R be an (h; m) convex function with m 2 (0; 1] ; t 2 [0; 1] : If 0 a < b < 1 and f 2 L1 [a; b] ; then the following inequality holds: (1.13) f

a+b 2

h b h

1 2

a 1 2

Z

b

a

h

f (x) + mf

x i dx m

a + mf f (a) + mf m

b m

2

+m f

b m2

Z

1

h(t)dt:

0

The aim of this paper is to de…ne a new class of convex function and then establish new Hermite-Hadamard-type inequalities. 2. MAIN RESULTS In the beginning we give a new de…nition (h ( ; m)) convex function. I and J are intervals on R; (0; 1) J and functions h and f are real non-negative functions de…ned on J and I, respectively. De…nition 7. Let h : J R ! R be a non-negative function, h 6= 0: We say that f : [0; b] [0; 1) ! R is an (h ( ; m)) convex function, or that f belongs to the class SX((h ( ; m)) ; b), if f is non-negative, we have (2.1)

f (tx + m(1

t)y)

h(t )f (x) + mh(1

t )f (y)

2

for all x; y 2 [0; b], ( ; m) 2 [0; 1] and t 2 [0; 1] : If the inequality (2.1) is reversed, then f is said to be (h ( ; m); b) concave function on [a; b] : Obviously, if we choose h(t) = t, then we have non-negative ( ; m) convex functions. If we choose = m = 1; then we have h convex functions. If we choose = 1; then we have (h m) convex functions. If we choose = m = 1 and h(t) = t; 1; 1t ; ts ; then we obtain non-negative convex functions, p functions, Godunova-Levin functions and s convex functions in the second sense, respectively. The following theorems were obtained by using the (h ( ; m)) convex functions. Theorem 13. Let h : J R ! R be a non-negative function, h 6= 0 and f : [0; b] [0; 1) ! R be an (h ( ; m)) convex function with ( ; m) 2 [0; 1] (0; 1] and t 2 [0; 1] : If f 2 L1 [ma; b] ; h 2 L1 [0; 1] ; one has the following inequality: Z b 1 (2.2) f (x)dx b a a Z 1 Z 1 b h (1 t ) dt; min f (a) h(t )dt + mf m 0 0 Z 1 Z 1 a f (b) h(t )dt + mf h (1 t ) dt : m 0 0 Proof. Since f is (h ( ; m)) convex function, t 2 [0; 1] and ( ; m) 2 [0; 1] (0; 1] ; then f (tx + m(1 t)y) h(t )f (x) + mh(1 t )f (y)

101

6

M .E. OZDEM IR, HAVVA KAVURM ACI ÖNALAN F , AND M ERVE AVCI ARDIÇ

for all x; y

0: It follows that f (ta + (1

t)b)

h(t )f (a) + mh(1

t )f

b m

and

a : m Integrating the above inequalities with respect to t over [0; 1] ; we have Z 1 Z 1 Z 1 b h(t )dt + mf f (ta + (1 t)b)dt f (a) h(1 t )dt m 0 0 0 and Z 1 Z 1 Z 1 a h(t )dt + mf h(1 t )dt: f (tb + (1 t)a)dt f (b) m 0 0 0 However, Z 1 Z 1 Z b 1 f (ta + (1 t)b)dt = f (tb + (1 t)a)dt = f (x)dx; b a a 0 0 we get Z b 1 f (x)dx b a a Z 1 Z 1 b min f (a) h(t )dt + mf h (1 t ) dt; m 0 0 Z 1 Z 1 a h (1 t ) dt f (b) h(t )dt + mf m 0 0 which is the required result. The proof is completed. f (tb + (1

t)a)

h(t )f (b) + mh(1

t )f

Remark 1. In Theorem 13, if we take h (t) = t; then the inequality (2.2) reduces to inequality (1.4). Remark 2. In Theorem 13, if we take = m = 1; then the inequality (2.2) reduces to the right hand side of inequality (1.10). Remark 3. In Theorem 13, if we take inequality (1.11).

= 1; then the inequality (2.2) reduces to

Remark 4. In Theorem 13, if we take = m = 1 and h (t) = ft; 1; ts g, then the inequality (2.2) reduces to the right hand side of inequality in (1.1), (1.8) and (1.7) which are Hermite-Hadamard-type for non-negative convex functions, p functions and s convex functions in the second sense, respectively. Theorem 14. Let h : J R ! R be a non-negative function, h 6= 0 and f : [0; b] [0; 1) ! R be an (h ( ; m)) convex function with ( ; m) 2 [0; 1] (0; 1] and t 2 [0; 1] : If f 2 L1 [ma; b] ; h 2 L1 [0; 1] ; one has the following inequality:

(2.3)

Z

mb

1 b ma a Z 1 Z [f (a) + f (b)] h(t )dt + m 1 mb a

f (x) dx +

0

Z

0

b

f (x) dx

ma

1

h(1

t )dt

102

(h

for all 0

ma

a

7

mb < b < 1:

Proof. Since f is an (h

f ((1

( ; m))-CONVEX FUNCTIONS

( ; m)) convex function, we can write

f (ta + m(1

t)b)

h(t )f (a) + mh(1

t )f (b) ;

f (tb + m(1

t)a)

h(t )f (b) + mh(1

t )f (a) ;

t)a + m (1

(1

t)) b)

h ((1

t) ) f (a) + mh (1

(1

t) ) f (b)

and f ((1

t)b + m (1

(1

t)) a)

h ((1

t) ) f (b) + mh (1

(1

t) ) f (a)

for all t 2 [0; 1] and ( ; m) 2 [0; 1] (0; 1] : Adding the above inequalities with each other, we get f (ta + m(1

t)b) + f (tb + m(1

h(t ) [f (a) + f (b)] + mh(1 +h ((1

t)a) + f ((1

t)a + mtb) + f ((1

t)b + mta)

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

t) ) [f (a) + f (b)] + mh (1

(1

t) ) [f (a) + f (b)] :

Integrating the above inequality with respect to t over [0; 1] ; we get Z mb Z b 2 2 f (x)dx + f (x)dx mb a a b ma ma Z 1 Z 1 [f (a) + f (b)] (h(t ) + h ((1 t) )) dt + m (h(1 t ) + h(1 0

R1 R1 Since 0 h(t )dt = 0 h((1 obtain the desired result.

(1

t) )) dt :

0

t) )dt and

R1 0

h(1

t )dt =

R1 0

h(1

(1

t) )dt; we

Remark 5. In Theorem 14, if we take h (t) = t; then the inequality (2.3) reduces to inequality (1.5). Remark 6. In Theorem 14, if we take = m = 1; then the inequality (2.3) reduces to the right hand side of inequality (1.10). Remark 7. In Theorem 14, if we take inequality (1.12).

= 1; then the inequality (2.3) reduces to

Remark 8. In Theorem 14, if we take = m = 1 and h (t) = ft; 1; ts g, then the inequality (2.3) reduces to the right hand side of inequality in (1.1), (1.8) and (1.7) which are Hermite-Hadamard-type for non-negative convex functions, p functions and s convex functions in the second sense, respectively. Theorem 15. Let h : J R ! R be a non-negative function, h 6= 0 and f : [0; b] [0; 1) ! R be an (h ( ; m)) convex function with ( ; m) 2 [0; 1] (0; 1]

103

8

M .E. OZDEM IR, HAVVA KAVURM ACI ÖNALAN F , AND M ERVE AVCI ARDIÇ

and t 2 [0; 1] : If f 2 L1 [a; b] ; h 2 L1 [0; 1] ; one has the following inequalities: (2.4) a+b 2

f

h

b

1 2

a

Z

b

1 2

f

b m

1 2

h

a

f (a) + mh 1

+m h for all 0

1

1 2

1

h(t )dt

0

1 2

+ mh 1

dx

Z

b m2

f

1

h(1

t )dt

0

( ; m)) convex function, we can write

x+y 2

f

h

1 2

a+b 2

1 2

h

f (ta + (1

for all t 2 [0; 1] and ( ; m) 2 [0; 1] over [0; 1] ; we have Z 1 a+b 1 f h f (ta + (1 2 2 0 Taking into account that Z 1 f (ta + (1 Z

1

f

0

Then we get a+b 2

b

a

Z

b

h

a

1 2

t) b) + mh 1

t) a; we get tb + (1 t) a m

f

(0; 1] : Thus by integrating with respect to t

1

t) b) dt =

b

1 2

dt =

a 1

b

a

Z

1 2

t) b) dt+mh 1

tb + (1 t) a m 1

y m

f

t) b; y = tb + (1

0

and

1 2

f (x) + mh 1

for all x; y 2 [0; 1) : If we choose x = ta + (1

f

Z

a f m

x m

f

a < b < 1:

Proof. Since f is an (h

(2.5) f

1 2

f (x) + mh 1

Z

1

f

0

tb + (1 t) a m

b

f (x)dx

a

Z

b

f

a

f (x) + mh 1

x dx: m 1 2

f

x m

dx

which is the …rst inequality in (2.4). To prove the second inequality in (2.4), we use the right side of (2.5) with (h ( ; m)) convexity of f; we have h

1 2

h

1 2

+mh 1

f (ta + (1

t) b) + mh 1

h(t )f (a) + mh(1 1 2

h ((1

t )f

t) ) f (

1 2

f

(1

t) a + tb m

(1

t) ) f

b m

a ) + mh (1 m

b m2

dt:

104

( ; m))-CONVEX FUNCTIONS

(h

9

for all t 2 [0; 1] and ( ; m) 2 [0; 1] (0; 1] : Thus by integrating the resulting inequality with respect to t over [0; 1] ; we have Z 1 1 1 (1 t) a + tb h f (ta + (1 t) b) + mh 1 f 2 2 m 0 Z 1 Z 1 b 1 h(t )dt + mf h f (a) h(1 t )dt 2 m 0 0 Z 1 Z 1 1 a b h(1 (1 t) )dt +mh 1 f h ((1 t) ) dt + mf 2 m 0 m2 0 which is equal to second inequality in (2.4). Remark 9. In Theorem 15, if we take h (t) = t; then we have Z b x f (x) + m (2 1) f m a+b 1 f dx 2 b a a 2 " a 1) f m f b + m (2 1) f 1 f (a) + m (2 + m m 2 +1 +1

b m2

#

:

The left hand side of this inequality is in ( 1.3). Remark 10. In Theorem 15, if we take (2.4) reduces to inequality (1.2). Remark 11. In Theorem 15, if we take inequality (1.13).

= 1 and h (t) = t; then the inequality = 1; then the inequality (2.4) reduces to

Remark 12. In Theorem 15, if we take = m = 1 and h (t) = ft; 1; ts g, then the inequality (2.4) reduces to inequalities in (1.1), (1.8) and (1.7) which are HermiteHadamard-type for non-negative convex functions, p functions and s convex functions in the second sense, respectively. Theorem 16. Let h : J R ! R be a non-negative function, h 6= 0 and f; g : [0; b] [0; 1) ! R be such that f g 2 L1 [a; b] ; h1 h2 2 L1 [0; 1] where 0 a < b < 1: If f is (h1 ( 1 ; m1 )) convex and g is (h2 ( 2 ; m2 )) convex function on [0; b] ; one has the following inequality: Z b 1 (2.6) f (x) g (x) dx min fH1 ; H2 g b a a where H1

= f (a)g(a)

Z

1

h1 (t 1 )h2 (t 2 )dt + m2 f (a)g

0

+m1 f

b m1

+m1 m2 f

g (a)

Z

1

h2 (t 2 )h1 (1

0

b m1

g

b m2

b m2

Z

0

Z

1

h1 (t 1 )h2 (1

0

t 1 )dt

1

h1 (1

t 1 )h2 (1

t 2 )dt;

t 2 )dt

105

10

M .E. OZDEM IR, HAVVA KAVURM ACI ÖNALAN F , AND M ERVE AVCI ARDIÇ

and

H2

Z

= f (b)g(b)

1

h1 (t )h2 (t )dt + m2 f (b)g

0

a m1

+m1 f

Z

g (b)

1

h2 (t 2 )h1 (1

0

a m1

+m1 m2 f

for some …xed (

a m2

2

1

g

1 ; m1 ) ; ( 2 ; m2 )

Proof. Since f 2 SX (h1

(

Z

a m2

Z

1

t 2 )dt

h1 (t 1 )h2 (1

0

t 1 )dt

1

t 1 )h2 (1

h1 (1

t 2 )dt

0

2 [0; 1]

1 ; m1 ); b)

(0; 1] and t 2 [0; 1] :

and g 2 SX (h2

(

2 ; m2 ); b) ;

f (ta + (1

t)b)

h1 (t 1 )f (a) + m1 h1 (1

t 1 )f

b m1

g(ta + (1

t)b)

h2 (t 2 )g(a) + m2 h2 (1

t 2 )g

b m2

we have

and

for all t 2 [0; 1] : Since f and g are non-negative, f (ta + (1

t)b)g(ta + (1

t)b)

h1 (t 1 )h2 (t 2 )f (a)g(a) + m2 f (a)g b m1

+m1 f

g (a) h2 (t 2 )h1 (1

b m2

h1 (t 1 )h2 (1

t 1 ) + m1 m2 f

b m1

t 2) g

b m2

h1 (1

t 1 ))h2 (1

Then integrating the resulting inequality with respect to t over [0; 1] ; 1 b

a

Z

b

f (x) g (x) dx

a

f (a)g(a)

Z

1

h1 (t 1 )h2 (t 2 )dt + m2 f (a)g

0

+m1 f

b m1

+m1 m2 f

g (a)

Z

b m2

Z

1

h1 (t 1 )h2 (1

0

1

h2 (t 2 )h1 (1

t 1 )dt

0

b m1

g

b m2

Z

0

1

h1 (1

t 1 )h2 (1

t 2 )dt:

t 2 )dt

t 2 ):

106

( ; m))-CONVEX FUNCTIONS

(h

Analogously we have Z b 1 f (x) g (x) dx b a a Z 1 f (b)g(b) h1 (t 1 )h2 (t 2 )dt + m2 f (b)g 0

+m1 f

a m1

+m1 m2 f

g (b)

Z

a m2

11

Z

1

h1 (t 1 )h2 (1

t 2 )dt

0

1

h2 (t 2 )h1 (1

t 1 )dt

0

a m1

g

which completes the proof.

a m2

Z

1

h1 (1

t 1 )h2 (1

t 2 )dt

0

Remark 13. In Theorem 16, if we take h1 (t) = h2 (t) = t; then the inequality (2.6) reduces to inequality (1.6). References [1] M.K. Bakula, M. E Ozdemir, J. Peµcari´c, Hadamard type inequalities for m convex and ( ; m) convex functions, J. Inequal. Pure Appl. Math. 9 (2008), Article 96. [2] M.K. Bakula, J. Peµcari´c and M. Ribiµci´c, Companion inequalities to Jensen’s inequality for m convex and ( ; m) convex functions, J. Inequal. Pure Appl. Math. 7 (2006), Article 194. [3] S.S. Dragomir, On some new inequalities of Hermite-Hadamard type for m convex functions, Tamkang Journal of Mathematics, 33(1) 2002, 55-65. [4] S.S. Dragomir and S. Fitzpatrick, The Hadamard’s inequality for s convex functions in the second sense, Demonstratio Math. 32 (4) (1999) 687–696. [5] S.S. Dragomir, J. Peµcari´c and L.E. Persson, Some inequalities of Hadamard Type, Soochow Journal of Mathematics, Vol.21, No:3, pp. 335-341, July 1995. [6] G. Toader, Some generalizations of the convexity, Proc. Colloq. Approx. Optim., Univ. ClujNapoca, Cluj-Napoca, 1984, 329-338. [7] E.K. Godunova and V.I. Levin, Neravenstra dlja funccii sirokogo klassa soderzascego vypuklye, monotonnye i nekotorye drugie vidy funkaii, Vycislitel Mat. i Mt. Fiz., Mezvuzov Sb. Nauc. Trudov. MPGI, Moscow, 1985, 138-142. [8] H. Hudzik and L. Maligranda, Some remarks on s convex functions, Aequationes Math. 48 (1994) 100–111. [9] J. Peµcari´c, F. Proschan and Y.L. Tong, Convex Functions, Partial Ordering and Statistical Applications, Academic Press, New York, (1991). [10] V.G. Mihe¸san, A generalization of the convexity, Seminar of Functional Equations, Approx. and Convex, Cluj-Napoca (Romania) (1993). [11] M.E. Özdemir, M. Avc¬ and H. Kavurmac¬, Hermite–Hadamard-type inequalities via ( ; m) convexity, Comput. Math.Appl., 61 (2011) 2614–2620. [12] M.E. Özdemir, A.O. Akdemir and E. Set, On (h m) Convexity and Hadamard-Type Inequalities, arXiv:1103.6163v1 [math.CA] 31 Mar 20. [13] M.E. Ozdemir, E. Set and M.Z. Sarikaya, Some new Hadamard’s type inequalities for coordinated m convex and ( ; m) convex functions, Hacettepe J. of. Math. and Ist., 40, 219229, (2011). [14] M.E. Özdemir, H. Kavurmac¬ and E. Set, Ostrowski’s Type Inequalities for ( ; m) Convex Functions, Kyungpook Math. J. 50(2010), 371-378. [15] Sanja Varošanec, On h-convexity, J. Math. Anal. Appl. 326 (2007) 303–311. [16] E. Set et al., On generalizations of the Hadamard Inequality for ( ; m) convex functions., Kyungpook Math. J., 2011, Accepted. Article 96, online: htpp://jipam.vu.edu.au. [17] M.Z. Sar¬kaya, Aziz Sa¼ g lam and Hüseyin Y¬ld¬r¬m, On some Hadamard-type inequalities for h convex functions, Journal of Mathematical Inequalities, 2(3) 2008, 335-341.

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M .E. OZDEM IR, HAVVA KAVURM ACI ÖNALAN F , AND M ERVE AVCI ARDIÇ

Atatürk University, K.K. Education Faculty, Department of Mathematics, 25240 Campus, Erzurum, Turkey E-mail address : [email protected] Van Yüzüncü Y¬l University, Faculty of Education, Department of Mathematics Education,Van, Turkey E-mail address : [email protected] Ad¬yaman University, Faculty Of Science And Art, Department Of Mathematics, 02040, Ad¬yaman, Turkey E-mail address : [email protected]

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2,108-119 , 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 108

Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, U.S.A. [email protected] [email protected] Abstract. Here we give asymptotic expansions for the generalized discrete versions of unitary Picard, Gauss-Weierstrass, and PoissonCauchy singular operators. These are of Voronovskaya type expansions and they are connected to the approximation properties of these operators. AMS 2010 Mathematics Subject Classi…cation: 41A60. Key Words and Phrases: Asymptotic expansion, singular operator.

1

Background

In [2] p.307-310, the authors studied the smooth general singular integral operators r; (f ; x) de…ned as follows. For r 2 N and n 2 Z+ , they de…ned 8 ( 1)r j rj j n ; j = 1; :::; r; < r P = (1) j ( 1)r i ri i n ; j=0 : 1 i=1

that is

r P

j=0

j

= 1: Let f : R ! R be Borel measurable. For each

probability Borel measure on R: They de…ned for x 2 R the integrals 0 Z 1 X r @ r;n; (f ; x) := 1

j=0

1

j f (x

1

+ jt)A d

(t):

> 0;

is a

(2)

Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

109

and they assumed r;n; (f ; x) 2 R; 8x 2 R: They also stated the fact that the operators r;n; are not in general positive. They gave their main result as R n n 1 Theorem 1 Suppose jtj d (t) ; 8 2 (0; 1] ; > 0 and ck; := 1 R1 k t d (t); k = 1; : : : ; n 1: Let f : R ! R be such that f (n) exists; n 2 N; 1 and is bounded and let ! 0+; 0 < 1: Then 0 1 n r X1 f (k) (x) X kA @ f (x) = ck; + o( n ): (3) r;n; (f ; x) jj k! j=1 k=1

When n = 1 the sum collapses.

They also covered the following special cases: Corollary 2 (n=1 case) Let f Rsuch that f 0 exists and it is bounded: Let ! 1 0+; 0 < 1: Here suppose 1 1 jtj d (t) ; 8 2 (0; 1] ; > 0: Then r;1;

(f ; x)

f (x) = o

1

:

(4)

Corollary 3 (n=2 case) Let f such Rthat f 00 exists and it is bounded: Let 2 1 2 ! 0+; 0 < 1: Here suppose t d (t) ; 8 2 (0; 1] ; > 0: 1 Then 0 1 r X 0 @ A c1; + o 2 : (5) (f ; x) f (x) = f (x) r;2; jj j=1

Corollary 4 (n=3 case) Let that f (3) exists and it is bounded: Let R f such 3 3 1 ! 0+; 0 < 1; with jtj d (t) ; 8 2 (0; 1] ; > 0: Then 1 1 1 0 0 r r 00 X X 2A A c1; + f (x) @ c2; + o( 3 ): f (x) = f 0 (x) @ r;3; (f ; x) jj jj 2 j=1 j=1

(6)

(4) Corollary 5 (n=4 case) Let exists and it is bounded: Let R f such that f 4 1 4 ! 0+; 0 < 1; with t d (t) ; 8 2 (0; 1] ; > 0: Then 1 0 1 0 1 r r 00 X X f (x) 2A @ A c1; + f (x) = f 0 (x) @ c2; (7) r;4; (f ; x) jj jj 2 j=1 j=1 0 1 r f 000 (x) @X 3A + c3; + o( 4 ): jj 6 j=1

2

Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

110

On the other hand, in [3], the authors de…ned important special cases of operators for discrete probability measures as follows : Let f 2 C n (R), n 2 Z+ ; 0 < i) When

r;

1, x 2 R. j j

e 1 P

( )=

e

j j

,

(8)

= 1

they de…ned the generalized discrete Picard operators as ! 1 r P P j f (x + j ) e = 1

Pr;n; (f ; x) :=

j j

j=0

1 P

j j

e

.

(9)

= 1

ii) When

2

e 1 P

( )=

2

,

(10)

e

= 1

they de…ned the generalized discrete Gauss-Weierstrass operators as ! 2 1 r P P j f (x + j ) e Wr;n; (f ; x) :=

= 1

j=0

1 P

.

2

(11)

e

= 1

iii) Let

2 N, and

>

1

: When 2

( )=

1 P

+

2

2

+

;

(12)

2

= 1

they de…ned the generalized discrete Poisson-Cauchy operators as ! 1 r P P 2 + 2 j f (x + j ) r;n;

(f ; x) :=

= 1

j=0

1 P

2

+

.

(13)

2

= 1

They observed that for c constant they have Pr;n; (c; x) = Wr;n; (c; x) =

3

r;n;

(c; x) = c.

(14)

Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

They assumed that the operators Pr;n; (f ; x), Wr;n; (f ; x), and R; for x 2 R: This is the case when kf k1;R < 1: In [3], for k = 1; :::; n, the authors de…ned the sums

ck; :=

1 P

k

= 1 1 P

r;n;

(f ; x) 2

j j

e

j j

e

111

,

(15)

;

(16)

= 1

pk; :=

1 P

2

k

= 1 1 P

e 2

e

= 1

and for

2 N;

>

n+r+1 ; 2

they introduced

qk; :=

1 P

k

= 1 1 P

2

2

+

: 2

+

2

= 1

Furthermore, they proved that these sums ck; ; pk; , and qk; are …nite.

4

(17)

Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

2

112

Main Results

First, we present our results for the generalized discrete Picard operators. Proposition 6 Let n 2 N: Then, there exists K1 > 0 such that 1 P

n

j j

n

= 1 1 P

j j e

< K1 < 1

j j

e

(18)

= 1

for all

2 (0; 1] :

Proof. Since

1 P

1 2

2

=

6

=1

(Euler, 1741), we have that 1 X

1


= 1 2

=

+2

2

1 X

(44) 2

+

2

=1

2

:

Therefore, we have 1 1 P

2

2

:

(45)

2

+

= 1

Hence, we get n

1 P

= 1 1 P

n

j j

2

2

+

2 2

+

n

2

1 X

j j

1 X

j j

= 1

= 1 2

n

= 1

1 X

=

= 1 1 X

2

10

r;n;

2

n 2

n 2

=1

for all 2 (0; 1] since 2 n > 1: Now, we present our main result for

n

:

j j 1

2

n

< 1;

+

2

(46)

Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

118

Theorem 19 Let f : R ! R be such that f (n) exists, n 2 N; and is bounded. Let ! 0+ ; 0 < 1; and > n+r+1 : Then 2 0 1 r n X1 f (k) (x) X kA @ f (x) = qk; + o( n ): (47) jj r;n; (f ; x) k! j=1 k=1

When n = 1; the sum on R.H.S. collapses.

Proof. By (17); Theorem 1, and Proposition 18. For n = 1; we have Corollary 20 Let f : R ! R be such that f 0 exists, and is bounded. Let > r+2 ! 0+ ; and 0 < 1: Then 2 ; r;1;

(f ; x)

f (x) = o(

1

):

(48)

Proof. By Theorem 19. For n = 2; we get Corollary 21 Let f : R ! R be such that f 00 exists, and is bounded. Let ! 0+ ; and 0 < 1: Then > r+3 2 ; 0 1 r X A q1; + o( 2 ): f (x) = f 0 (x) @ (49) jj r;2; (f ; x) j=1

Proof. By Theorem 19. For n = 3; we obtain

Corollary 22 Let f : R ! R be such that f 000 exists, and is bounded. Let > r+4 ! 0+ ; and 0 < 1: Then 2 ; 0 1 0 1 r r 00 X X f (x) 2A A q1; + @ f (x) = f 0 (x) @ q2; + o( 3 ): jj jj r;3; (f ; x) 2 j=1 j=1

(50)

Proof. By Theorem 19. For n = 4; we derive Corollary 23 Let f : R ! R be such that f (4) exists, and is bounded. Let > r+5 ! 0+ ; and 0 < 1: Then 2 ; 0 1 r X A q1; f (x) = f 0 (x) @ jj r;4; (f ; x) j=1

0 r f (x) @X + 2 j=1 00

Proof. By Theorem 19.

jj

11

1

2A

q2; +

000

0

r X

f (x) @ 6 j=1

jj

1

3A

q3; + o(

(51)

4

):

Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

References [1] G.A. Anastassiou, Approximation by Discrete Singular Operators, Cubo, Vol.15, No.1 (2013), 97-112. [2] G.A. Anastassiou and R.A. Mezei, Approximation by Singular Integrals, Cambridge Scienti…c Publishers, Cambrige, UK, 2012. [3] G.A. Anastassiou and M. Kester, Quantitative Uniform Approximation by Generalized Discrete Singular Operators, Submitted, 2013. [4] F. Smarandache, A triple inequality with series and improper integrals, arxiv.org/ftp/mat/papers/0605/0605027.pdf, 2006.

12

119

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2,120-151 , 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 120

SIMPLE AQ AND SIMPLE CQ FUNCTIONAL EQUATIONS M. ARUNKUMAR1 , C. DEVI SHYAMALA MARY2 , G. SHOBANA3 ,

1,3

Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India. E-mail: [email protected] 2 Department of Mathematics, St.Joseph’s College of Arts and Science, Cuddalore - 607 001, TamilNadu, India. E-mail: [email protected]

Abstract. In this paper, the authors established the general solution and generalized Ulam - Hyers stability of the simple additive-quadratic and simple cubicquartic functional equations f (2x) = 3f (x) + f (−x), g(2x) = 12g(x) + 4g(−x), via Banach space using direct and fixed point method.

1. INTRODUCTION In mathematics, a functional equation is any equation that specifies a function in implicit form. Often, the equation relates the value of a function (or functions) at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional equations they satisfy. But the theory of functional equations is relatively young. The beginning of a theory of functional equations is connected with the work of an excellent specialist in this field, Hungarian mathematician J. Aczel. The stability problem for functional equations first was planed in 1940 by Ulam [41]: When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?. 2010 Mathematics Subject Classification. :39B52, 32B72, 32B82 . Key words and phrases. : Additive functional equation, quadratic functional equation, cubic functional equation, quartic functional equation, mixed additive-quadratic functional equations, mixed cubic-quartic functional equations, generalized Ulam - Hyers stability, Banach space, fixed point. 1

121

2

M. ARUNKUMAR et. al.,

If the problem accepts a solution, we say that the equation is stable. This phenomenon is called U lam − Hyers stability and has been extensively investigated for different functional equations. Let (G, .) be a groupoid and let (Y, .) be a groupoid with the metric ρ. The following definition of stability of the equation of additive homomorphism from G to Y f (x + y) = f (x) + f (y) (1.1) is formulated. Definition 1.1. Equation (1.1) is stable in Hyers-Ulam sense, if for every  > 0 there exists δ > 0 such that for every function f : G → Y fulfilling ρ (f (x + y), f (x) + f (y)) ≤ δ,

x, y ∈ G

there exists a solution g of (1.1) satisfying ρ (f (x), g(x)) ≤ ,

x ∈ G.

The study of stability problems for functional equations concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [22]. It was further generalized and excellent results were obtained by number of authors [2, 17, 33, 36, 39]. Its solutions via various forms of functional equations like additive, quadratic, cubic, quartic, mixed type functional equations which involves only these types of functional equations were discussed. We refer the interested readers for more information on such problems to the monographs [1, 13, 14, 15, 16, 23, 26, 27, 37]. The generalized Ulam-Hyers stability of various types of functions equations in various spaces were discussed in [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 20, 19, 24, 29, 30, 31, 32, 34, 35, 38, 40, 42, 43, 44]. In this paper, the authors established the general solution and generalized Ulam Hyers stability of the simple additive-quadratic and simple cubic-quartic functional equations f (2x) = 3f (x) + f (−x),

(1.2)

g(2x) = 12g(x) + 4g(−x),

(1.3)

f (x) = ax + bx2 ,

(1.4)

g(x) = cx3 + dx4 ,

(1.5)

and having solutions and respectively. In Section 2 the the general solution of (1.2) and (1.3) are respectively provided.

122

SIMPLE AQ AND SIMPLE CQ . . .

3

In Section 3 the generalized Ulam - Hyers stability of (1.2) and (1.3) are respectively proved via Banach spaces a direct method. In Section 4 the generalized Ulam - Hyers stability of (1.2) and (1.3) are respectively given via Banach spaces with the help of fixed point method. 2. GENERAL SOLUTION OF (1.2) AND (1.3) In this section, the general solution of the functional equations (1.2) and (1.3) are respectively given. For this, let us consider X and Y be real vector spaces. 2.1. GENERAL SOLUTION OF (1.2). Using oddness and evenness of f , the following lemmas are trivial. Hence, we omit the proofs. Lemma 2.1. An odd mapping f : X → Y satisfying (1.2), then f is additive. Lemma 2.2. An even mapping f : X → Y satisfying (1.2), then f is quadratic. 2.2. GENERAL SOLUTION OF (1.3). Using oddness and evenness of g, the following lemmas are trivial. Hence, we omit the proofs. Lemma 2.3. An odd mapping g : X → Y satisfying (1.3), then g is cubic. Lemma 2.4. An even mapping g : X → Y satisfying (1.3), then g is quartic. 3. STABILITY RESULTS IN BANACH SPACE: DIRECT METHOD In this section, the generalized Ulam - Hyers stability of the functional equations (1.2) and (1.3) are respectively is provided. Also throughout this section, let us consider X and Y to be a normed space and a Banach space, respectively. 3.1. STABILITY RESULTS OF (1.2). Theorem 3.1. Let j ∈ {−1, 1} and α : X → [0, ∞) be a function such that ∞ X α (2nj x) α (2nj x) converges in R and lim =0 n→∞ 2nj 2nj n=0

(3.1)

for all x ∈ X . Let fa : X → Y be an odd function satisfying the inequality kfa (2x) − 3fa (x) − fa (−x)k ≤ α (x)

(3.2)

for all x ∈ X . Then there exists a unique additive mapping A : X → Y which satisfying (1.2) such that ∞ 1 X α(2kj x) kfa (x) − A(x)k ≤ (3.3) 2 1−j 2kj k=

2

123

4

M. ARUNKUMAR et. al.,

for all x ∈ X . The mapping A(x) is defined by fa (2nj x) A(x) = lim n→∞ 2nj

(3.4)

for all x ∈ X . Proof. Assume j = 1. Using oddness of fa in (3.2), it follows that



α (x) f (2x) a

fa (x) −

≤

2 2 for all x ∈ X . Now replacing x by 2x and dividing by 2 in (3.9), we get

fa (2x) fa (22 x) α (2x)

2 − 22 ≤ 22 for all x ∈ X . From (3.9) and (3.6), we obtain





2 2





fa (x) − fa (2 x) ≤ fa (x) − fa (2x) + fa (2x) − fa (2 x)



2 2 2 2 2 2   1 α(2x) ≤ α(x) + 2 2 for all x ∈ X . In general for any positive integer n, we get

n−1 ∞ n X

1 X α(2k x) α(2k x)

fa (x) − fa (2 x) ≤ 1 ≤

2n 2 k=0 2k 2 k=0 2k

(3.5)

(3.6)

(3.7)

(3.8)

for all x ∈ X . In order to prove the convergence of the sequence   fa (2n x) , 2n replace x by 2m x and dividing by 2m in (3.8), for any m, n > 0 , we deduce



n m

fa (2m x) fa (2n+m x)

1 f (2 · 2 x) a m

=

fa (2 x) −

−

2m

2(n+m) 2m 2n ∞

1 X α(2k+m x) → 0 as m → ∞ 2 k=0 2k+m   fa (2n x) for all x ∈ X . Hence the sequence is a Cauchy sequence. Since Y is 2n complete, there exists a mapping A : X → Y such that ≤

A(x) = lim

n→∞

fa (2n x) , ∀ x ∈ X. 2n

124

SIMPLE AQ AND SIMPLE CQ . . .

5

Letting n → ∞ in (3.8), we see that (3.3) holds for all x ∈ X . To prove that A satisfies (1.2), replacing x by 2n x and dividing by 2n in (3.2), we obtain

1 1

n n n f (2 · 2x) − 3f (2 x) − f (−2 x)

≤ n α(2n x) a a a n 2 2 for all x ∈ X . Letting n → ∞ in the above inequality and using the definition of A(x) and (3.1), we see that A(2x) = 3A(x) + A(−x). Hence A satisfies (1.2) for all x ∈ X . To prove that A is unique, let B(x) be another additive mapping satisfying (1.2) and (3.3), then 1 kA(2n x) − B(2n x)k 2n 1 ≤ n {kA(2n x) − fa (2n x)k + kfa (2n x) − B(2n x)k} 2 ∞ X α(2k+n x) ≤ → 0 as n → ∞ 2(k+n) k=0

kA(x) − B(x)k =

for all x ∈ X . Hence A is unique. Thus the theorem holds for j = 1. Replacing x by

x 2

in (3.9), we arrive

   

2fa 2x − fa (x) ≤ α x

2 2

(3.9)

for all x ∈ X . The rest of the proof is similar to that of case j = 1. Thus, for j = −1 also the theorem is true. Hence the proof is complete.  The following corollary is an immediate consequence of Theorem 3.1 concerning the stability of (1.2). Corollary 3.2. Let λ and r be nonnegative real numbers. Let an odd function fa : X → Y satisfies the inequality  λ, kfa (2x) − 3fa (x) − fa (−x)k ≤ (3.10) λ||x||r , r 6= 1; for all x ∈ X . Then there exists a unique additive function A : X → Y such that   |λ|, λ||x||r kfa (x) − A(x)k ≤ (3.11) ,  r |2 − 2 | for all x ∈ X . Now, we will provide an example to illustrate that the functional equation (1.2) is not stable for r = 1 in condition (ii) of Corollary 3.2.

125

6

M. ARUNKUMAR et. al.,

Example 3.3. Let α : R → R be a function defined by  µx, if |x| 0 is a constant, and define a function fa : R → R by fa (x) =

∞ X α(2n x) n=0

2n

,

f or all

x ∈ R.

Then fa satisfies the functional inequality |fa (2x) − 3fa (x) − fa (−x)| ≤ 10 µ|x|

(3.12)

for all x ∈ R. Then there do not exist a additive mapping A : R → R and a constant κ > 0 such that |fa (x) − A(x)| ≤ κ|x|,

f or all

x ∈ R.

(3.13)

Proof. Now |fa (x)| ≤

∞ X |α(2n x)| n=0

|2n |

∞ X µ = = 2 µ. 2n n=0

Therefore, we see that fa is bounded. We are going to prove that fa satisfies (3.12). 1 then the left hand side of (3.12) is less 2 1 than 10µ. Now suppose that 0 < |x| < . Then there exists a positive integer k 2 such that 1 1 ≤ |x| < k−1 , (3.14) k 2 2 1 so that 2k−1 x < and consequently 2 If x = 0 then (3.12) is trivial. If |x| ≥

2k−1 (x), 2k−1 (−x), 2k−1 (2x) ∈ (−1, 1). Therefore for each n = 0, 1, . . . , k − 1, we have 2n (x), 2n (−x), 2n (2x) ∈ (−1, 1) and α(2n (2x)) − 3α(2n (x)) − α(2n (−x)) = 0

126

SIMPLE AQ AND SIMPLE CQ . . .

7

for n = 0, 1, . . . , k − 1. From the definition of fa and (3.14), we obtain that ∞ X 1 n n n fa (2x) − 3fa (x) − fa (−x) ≤ α(2 (2x)) − 3α(2 (x)) − α(2 (−x)) n 2 n=0 ∞ X 1 n n n ≤ α(2 (2x)) − 3α(2 (x)) − α(2 (−x)) n 2 n=k ≤

∞ X 2 1 5µ = 5 µ × k = 10 µ|x|. n 2 2 n=k

1 Thus fa satisfies (3.12) for all x ∈ R with 0 < |x| < . 2 We claim that the additive functional equation (1.2) is not stable for r = 1 in condition (ii) of Corollary 3.2. Suppose on the contrary that there exist a additive mapping A : R → R and a constant κ > 0 satisfying (3.13). Since fa is bounded and continuous for all x ∈ R, A is bounded on any open interval containing the origin and continuous at the origin. In view of Theorem 3.1, A must have the form A(x) = cx for any x in R. Thus, we obtain that |fa (x)| ≤ (κ + |c|) |x|.

(3.15)

But we can choose a positive integer m with mµ > κ + |c|.
1 If x ∈ 0, 2m−1 , then 2n x ∈ (0, 1) for all n = 0, 1, . . . , m − 1 . For this x, we get fa (x) =

∞ X α(2n x) n=0

2n

≥

m−1 X n=0

µ(2n x) = mµx > (κ + |c|) x 2n

which contradicts (3.15). Therefore the additive functional equation (1.2) is not stable in sense of Ulam, Hyers and Rassias if r = 1, assumed in the inequality condition (ii) of (3.11).  Theorem 3.4. Let j ∈ {−1, 1} and α : X → [0, ∞) be a function such that ∞ X α (2nj x) n=0

4nj

converges in R

and

α (2nj x) =0 n→∞ 4nj lim

(3.16)

for all x ∈ X . Let fq : X → Y be an even function satisfying the inequality kfq (2x) − 3fq (x) − fq (−x)k ≤ α (x)

(3.17)

for all x ∈ X . Then there exists a unique quadratic mapping Q2 : X → Y which satisfying (1.2) such that ∞ 1 X α(2kj x) kfq (x) − Q2 (x)k ≤ 4 1−j 4kj k=

2

(3.18)

127

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M. ARUNKUMAR et. al.,

for all x ∈ X . The mapping Q2 (x) is defined by fq (2nj x) n→∞ 4nj

Q2 (x) = lim

(3.19)

for all x ∈ X . Proof. Assume j = 1. Using evenness of fq in (3.17), it follows that

α (x)

f (2x) q

≤

fq (x) −

4 4 for all x ∈ X . The rest of the proof is similar to that of Theorem 3.1.

(3.20) 

The following corollary is an immediate consequence of Theorem 3.4 concerning the stability of (1.2). Corollary 3.5. Let λ and r be nonnegative real numbers. Let an even function fq : X → Y satisfies the inequality  λ, (3.21) kfq (2x) − 3fq (x) − fq (−x)k ≤ λ||x||r , r 6= 2; for all x ∈ X . Then there exists a unique quadratic function Q2 : X → Y such that  λ   ,  |3| kfq (x) − Q2 (x)k ≤ (3.22) λ||x||r   ,  |4 − 2r | for all x ∈ X . Now, we will provide an example to illustrate that the functional equation (1.2) is not stable for r = 2 in condition (ii) of Corollary 3.5. Example 3.6. Let α : R → R be a function defined by  µx2 , if |x| 0 is a constant, and define a function fq : R → R by ∞ X α(2n x) fq (x) = , f or all x ∈ R. n 4 n=0 Then fq satisfies the functional inequality 20µ 2 |x| (3.23) 3 for all x ∈ R. Then there do not exist a quadratic mapping Q2 : R → R and a constant κ > 0 such that |fq (2x) − 3fq (x) − fq (−x)| ≤

|fq (x) − Q2 (x)| ≤ κ|x|2 ,

f or all

x ∈ R.

(3.24)

128

SIMPLE AQ AND SIMPLE CQ . . .

9

Proof. Now |fq (x)| ≤

∞ X |α(2n x)| n=0

|4n |

∞ X µ 4µ = . = n 4 3 n=0

Therefore, we see that fq is bounded. We are going to prove that fq satisfies (3.23). 1 If x = 0 then (3.23) is trivial. If |x|2 ≥ then the left hand side of (3.23) is less 24 20µ 1 than . Now suppose that 0 < |x|2 < . Then there exists a positive integer k 3 4 such that 1 1 ≤ |x|2 < k , (3.25) k+1 4 4 1 so that 4k−1 x2 < and consequently 4 2k−1 (x), 2k−1 (−x), 2k−1 (2x) ∈ (−1, 1). Therefore for each n = 0, 1, . . . , k − 1, we have 2n (x), 2n (−x), 2n (2x) ∈ (−1, 1) and α(2n (2x)) − 3α(2n (x)) − α(2n (−x)) = 0 for n = 0, 1, . . . , k − 1. From the definition of fq and (3.25), we obtain that ∞ X 1 n n n α(2 (2x)) − 3α(2 (x)) − α(2 (−x)) fq (2x) − 3fq (x) − fq (−x) ≤ n 2 n=0 ∞ X 1 n n n α(2 (2x)) − 3α(2 (x)) − α(2 (−x)) ≤ n 2 n=k ∞ X 1 4 20µ ≤ 5µ = 5 µ × = |x|. n k 4 3·2 3 n=k

1 Thus fq satisfies (3.23) for all x ∈ R with 0 < |x|2 < . 4 We claim that the quadratic functional equation (1.2) is not stable for r = 2 in condition (ii) of Corollary 3.5. Suppose on the contrary that there exist a quadratic mapping Q2 : R → R and a constant κ > 0 satisfying (3.24). Since fq is bounded and continuous for all x ∈ R, Q2 is bounded on any open interval containing the origin and continuous at the origin. In view of Theorem 3.4, Q2 must have the form Q2 (x) = cx2 for any x in R. Thus, we obtain that |fq (x)| ≤ (κ + |c|) |x|2 . But we can choose a positive integer m with mµ > κ + |c|.

(3.26)

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1 If x ∈ 0, 2m−1 , then 2n x ∈ (0, 1) for all n = 0, 1, . . . , m − 1 . For this x, we get fq (x) =

∞ X α(2n x) n=0

2n

≥

m−1 X n=0

µ(2n x) = mµx > (κ + |c|) x2 2n

which contradicts (3.26). Therefore the quadratic functional equation (1.2) is not stable in sense of Ulam, Hyers and Rassias if r = 2, assumed in the inequality condition (ii) of (3.22).  Now, we are ready to prove our main theorem. Theorem 3.7. Let j ∈ {−1, 1} and α : X → [0, ∞) be a function with conditions (3.1) and (3.16) for all x ∈ X . Let f : X → Y be a function satisfying the inequality kf (2x) − 3f (x) − f (−x)k ≤ α (x)

(3.27)

for all x ∈ X . Then there exists a unique additive mapping A : X → Y and a unique quadratic mapping Q : X → Y which satisfying (1.2) such that   ∞  1  1 X α(2kj x) α(−2kj x) + kf (x) − A(x) − Q2 (x)k ≤ 2 2 1−j 2kj 2kj k= 2    ∞ kj kj 1 X α(2 x) α(−2 x)  + + (3.28) 4 1−j 4kj 4kj k=

2

for all x ∈ X . The mapping A(x) and Q2 (x) are defined in (3.4) and (3.19) respectively for all x ∈ X . fa (x) − fa (−x) for all x ∈ X . Then fo (0) = 0 and fo (−x) = 2 −fo (x) for all x ∈ X . Hence Proof. Let fo (x) =

kfo (2x) − 3fo (x) − fo (−x)k ≤

α(x) α(−x) + 2 2

(3.29)

for all x ∈ X . By Theorem 3.1, we have  ∞  1 X α(2kj x) α(−2kj x) kfo (x) − A(x)k ≤ + 4 1−j 2kj 2kj k=

(3.30)

2

fq (x) + fq (−x) for all x ∈ X . Then fe (0) = 0 and 2 fe (−x) = fe (x) for all x ∈ X . Hence for all x ∈ X . Also, let fe (x) =

kfe (2x) − 3fe (x) − fe (−x)k ≤

α(x) α(−x) + 2 2

(3.31)

130

SIMPLE AQ AND SIMPLE CQ . . .

11

for all x ∈ X . By Theorem 3.4, we have  ∞  1 X α(2kj x) α(−2kj x) kfe (x) − Q2 (x)k ≤ + 8 1−j 4kj 4kj k=

(3.32)

2

for all x ∈ X . Define f (x) = fe (x) + fo (x)

(3.33)

for all x ∈ X . From (3.30),(3.32) and (3.33), we arrive kf (x) − A(x) − Q2 (x)k = kfe (x) + fo (x) − A(x) − Q2 (x)k ≤ kfo (x) − A(x)k + kfe (x) − Q2 (x)k   ∞  1  1 X α(2kj x) α(−2kj x) ≤ + 2 2 1−j 2kj 2kj k=

2

  ∞  kj kj X 1 α(2 x) α(−2 x)  + + 4 1−j 4kj 4kj k=

2

for all x ∈ X . Hence the theorem is proved.



Using Corollaries 3.2 and 3.5, we have the following corollary concerning the stability of (1.2). Corollary 3.8. Let λ and r be nonnegative real numbers. Let a function f : X → Y satisfies the inequality  λ, (3.34) kf (2x) − 3f (x) − f (−x)k ≤ λ||x||r , r 6= 1, 2; for all x ∈ X . Then there exists a unique additive function A : X → Y and a unique quadratic function Q2 : X → Y such that    1   ,  λ |1| + |3|   kf (x) − A(x) − Q2 (x)k ≤ (3.35) r r ||x|| ||x||   + ,  λ |2 − 2r | |4 − 2r | for all x ∈ X . 3.2. STABILITY RESULTS OF (1.3). Theorem 3.9. Let j ∈ {−1, 1} and β : X → [0, ∞) be a function such that ∞ X β (2nj x) n=0

8nj

converges in R

β (2nj x) and lim =0 n→∞ 8nj

(3.36)

131

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for all x ∈ X . Let gc : X → Y be an odd function satisfying the inequality kgc (2x) − 12gc (x) − 4gc (−x)k ≤ β (x)

(3.37)

for all x ∈ X . Then there exists a unique cubic mapping C : X → Y which satisfying (1.3) such that ∞ 1 X β(2kj x) kgc (x) − C(x)k ≤ (3.38) 8 1−j 8kj k=

2

for all x ∈ X . The mapping C(x) is defined by gc (2nj x) n→∞ 8nj

C(x) = lim

(3.39)

for all x ∈ X . Proof. Assume j = 1. Using oddness of gc in (3.37), it follows that



gc (x) − gc (2x) ≤ β (x)

8 8 for all x ∈ X . The rest of the proof is similar to that of Theorem 3.1.

(3.40) 

The following corollary is an immediate consequence of Theorem 3.9 concerning the stability of (1.3). Corollary 3.10. Let µ and r be nonnegative real numbers. Let an odd function gc : X → Y satisfies the inequality  µ, (3.41) kgc (2x) − 12gc (x) − 4gc (−x)k ≤ µ||x||r , r 6= 3; for all x ∈ X . Then there exists a unique cubic function C : X → Y such that  µ   |7| , kgc (x) − C(x)k ≤ (3.42) µ||x||r   , |8 − 2r | for all x ∈ X . Now, we will provide an example to illustrate that the functional equation (1.3) is not stable for r = 3 in condition (ii) of Corollary 3.10. Example 3.11. Let β : R → R be a function defined by  µx3 , if |x| 0 is a constant, and define a function gc : R → R by ∞ X β(2n x) gc (x) = , f or all x ∈ R. 8n n=0

132

SIMPLE AQ AND SIMPLE CQ . . .

13

Then gc satisfies the functional inequality 17µ × 83 3 |x| (3.43) 7 for all x ∈ R. Then there do not exist a cubic mapping C : R → R and a constant κ > 0 such that |gc (2x) − 12gc (x) − 4gc (−x)| ≤

|gc (x) − C(x)| ≤ κ|x|3 , Proof. Now |gc (x)| ≤

∞ X |β(2n x)| n=0

|8n |

f or all

=

x ∈ R.

(3.44)

∞ X µ 8µ . = n 8 7 n=0

Therefore, we see that gc is bounded. We are going to prove that gc satisfies (3.43). 1 If x = 0 then (3.43) is trivial. If |x|3 ≥ then the left hand side of (3.43) is less 8 8 × 17µ 1 3 than . Now suppose that 0 < |x| < . Then there exists a positive integer 7 8 k such that 1 1 ≤ |x| < k+1 , (3.45) k+2 8 8 1 so that 8k−1 x < and consequently 8 2k−1 (x), 2k−1 (−x), 2k−1 (2x) ∈ (−1, 1). Therefore for each n = 0, 1, . . . , k − 1, we have 2n (x), 2n (−x), 2n (2x) ∈ (−1, 1) and β(2n (2x)) − 12β(2n (x)) − 4β(2n (−x)) = 0 for n = 0, 1, . . . , k − 1. From the definition of gc and (3.45), we obtain that ∞ X 1 n n n β(2 (2x)) − 12β(2 (x)) − 4β(2 (−x)) g (2x) − 12g (x) − 4g (−x) ≤ c c c n 2 n=0 ∞ X 1 n n n ≤ β(2 (2x)) − 12β(2 (x)) − 4β(2 (−x)) n 2 n=k ∞ X 1 8 17µ × 83 3 ≤ 17µ = 17 µ × = |x| . 8n 7 · 8k 7 n=k

1 Thus gc satisfies (3.43) for all x ∈ R with 0 < |x|3 < . 8 We claim that the cubic functional equation (1.3) is not stable for r = 3 in condition (ii) of Corollary 3.10. Suppose on the contrary that there exist a cubic mapping C : R → R and a constant κ > 0 satisfying (3.44). Since gc is bounded

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and continuous for all x ∈ R, C is bounded on any open interval containing the origin and continuous at the origin. In view of Theorem 3.9, C must have the form C(x) = cx3 for any x in R. Thus, we obtain that |gc (x)| ≤ (κ + |c|) |x|3 .

(3.46)

But we can choose a positive integer m with mµ > κ + |c|.
1 If x ∈ 0, 2m−1 , then 2n x ∈ (0, 1) for all n = 0, 1, . . . , m − 1 . For this x, we get gc (x) =

∞ X β(2n x) n=0

2n

≥

m−1 X n=0

µ(2n x3 ) = mµx3 > (κ + |c|) x3 n 2

which contradicts (3.46). Therefore the cubic functional equation (1.3) is not stable in sense of Ulam, Hyers and Rassias if r = 3, assumed in the inequality condition (ii) of (3.42).  Theorem 3.12. Let j ∈ {−1, 1} and β : X → [0, ∞) be a function such that ∞ X β (2nj x) n=0

16nj

converges in R

and

β (2nj x) =0 n→∞ 16nj lim

(3.47)

for all x ∈ X . Let gq : X → Y be an even function satisfying the inequality kgq (2x) − 12gq (x) − 4gq (−x)k ≤ β (x)

(3.48)

for all x ∈ X . Then there exists a unique quartic mapping Q4 : X → Y which satisfying (1.3) such that ∞ 1 X β(2kj x) kgq (x) − Q4 (x)k ≤ 16 1−j 16kj k=

(3.49)

2

for all x ∈ X . The mapping Q4 (x) is defined by gq (2nj x) n→∞ 16nj

Q4 (x) = lim

(3.50)

for all x ∈ X . Proof. Assume j = 1. Using evenness of gq in (3.48), it follows that



gq (x) − gq (2x) ≤ β (x)

16 16 for all x ∈ X . The rest of the proof similar to the Theorem 3.1

(3.51) 

The following corollary is an immediate consequence of Theorem 3.12 concerning the stability of (1.3).

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SIMPLE AQ AND SIMPLE CQ . . .

15

Corollary 3.13. Let µ and r be nonnegative real numbers. Let an even function gq : X → Y satisfies the inequality  µ, kgq (2x) − 12gq (x) − 4gq (−x)k ≤ (3.52) µ||x||r , r 6= 4; for all x ∈ X . Then there exists a unique quartic function Q4 : X → Y such that  µ   |15| , kgq (x) − Q4 (x)k ≤ (3.53) µ||x||r   , |4 − 2r | for all x ∈ X . Now, we will provide an example to illustrate that the functional equation (1.3) is not stable for r = 4 in condition (ii) of Corollary 3.13. Example 3.14. Let β : R → R be a function defined by  µx4 , if |x| 0 is a constant, and define a function gq : R → R by gq (x) =

∞ X β(2n x) n=0

16n

,

f or all

x ∈ R.

Then gq satisfies the functional inequality 17 × 16µ |x|4 (3.54) 15 for all x ∈ R. Then there do not exist a quartic mapping Q4 : R → R and a constant κ > 0 such that |gq (2x) − 12gq (x) − 4gq (−x)| ≤

|gq (x) − Q4 (x)| ≤ κ|x|4 ,

f or all

x ∈ R.

(3.55)

Proof. Now |gq (x)| ≤

∞ X |β(2n x)| n=0

|16n |

=

∞ X µ 16µ = . n 16 15 n=0

Therefore, we see that gq is bounded. We are going to prove that gq satisfies (3.54). 1 If x = 0 then (3.54) is trivial. If |x|4 ≥ then the left hand side of (3.54) is 16 16 × 17µ 1 less than . Now suppose that 0 < |x|4 < . Then there exists a positive 15 16 integer k such that 1 1 ≤ |x|4 < k−1 , (3.56) k 16 16

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M. ARUNKUMAR et. al.,

so that 2k−1 x
0 satisfying (3.55). Since gq is bounded and continuous for all x ∈ R, Q4 is bounded on any open interval containing the origin and continuous at the origin. In view of Theorem 3.12, Q4 must have the form Q4 (x) = cx4 for any x in R. Thus, we obtain that Thus gq satisfies (3.54) for all x ∈ R with 0 < |x|4
κ + |c|.
1 If x ∈ 0, 2m−1 , then 2n x ∈ (0, 1) for all n = 0, 1, . . . , m − 1 . For this x, we get gq (x) =

∞ X β(2n x) n=0

2n

≥

m−1 X n=0

µ(2n x) = mµx4 > (κ + |c|) x4 2n

which contradicts (3.57). Therefore the quartic functional equation (1.3) is not stable in sense of Ulam, Hyers and Rassias if r = 4, assumed in the inequality condition (ii) of (3.53).  Now we are ready to prove our main theorem.

136

SIMPLE AQ AND SIMPLE CQ . . .

17

Theorem 3.15. Let j ∈ {−1, 1} and β : X → [0, ∞) be a function with conditions (3.36) and (3.47) for all x ∈ X . Let g : X → Y be a function satisfying the inequality kg(2x) − 12g(x) − 4g(−x)k ≤ β (x)

(3.58)

for all x ∈ X . Then there exists a unique cubic mapping C : X → Y and a unique quartic mapping Q4 : X → Y which satisfying (1.3) such that   ∞  1  1 X β(2kj x) β(−2kj x) kg(x) − C(x) − Q4 (x)k ≤ + 2 8 1−j 8kj 8kj k= 2   ∞  kj kj X 1 β(2 x) β(−2 x)  + + (3.59) 16 1−j 16kj 16kj k=

2

for all x ∈ X . The mapping C(x) and Q4 (x) are defined in (3.39) and (3.50) respectively for all x ∈ X . Proof. The proof of the Theorem is similar to the Theorem 3.7.



Using Corollaries 3.10 and 3.13, we have the following corollary concerning the stability of (1.3). Corollary 3.16. Let µ and r be nonnegative real numbers. Let a function g : X → Y satisfies the inequality  µ, (3.60) kg(2x) − 12g(x) − 4g(−x)k ≤ µ||x||r , r 6= 3, 4; for all x, y ∈ X . Then there exists a unique cubic function C : X → Y and a unique quartic function Q4 : X → Y such that    1 1   + ,  µ  |7| r|16|  kg(x) − C(x) − Q4 (x)k ≤ (3.61) ||x|| ||x||r   + ,  µ |8 − 2r | |16 − 2r | for all x ∈ X . 4. STABILITY RESULTS OF (1.2) AND (1.3): FIXED POINT METHOD In this section, we apply a fixed point method for achieving stability of the functional equations (1.2) and (1.3) are respectively present. Now, first we will recall the fundamental results in fixed point theory.

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Theorem 4.1. (Banach’s contraction principle) Let (X, d) be a complete metric space and consider a mapping T : X → X which is strictly contractive mapping, that is (A1) d(T x, T y) ≤ Ld(x, y) for some (Lipschitz constant) L < 1. Then, (i) The mapping T has one and only fixed point x∗ = T (x∗ ); (ii)The fixed point for each given element x∗ is globally attractive, that is (A2) limn→∞ T n x = x∗ , for any starting point x ∈ X; (iii) One has the following estimation inequalities: 1 (A3) d(T n x, x∗ ) ≤ 1−L d(T n x, T n+1 x), ∀ n ≥ 0, ∀ x ∈ X; (A4) d(x, x∗ ) ≤

1 1−L

d(x, x∗ ), ∀ x ∈ X.

Theorem 4.2. [28] Suppose that for a complete generalized metric space (Ω, δ) and a strictly contractive mapping T : Ω → Ω with Lipschitz constant L. Then, for each given x ∈ Ω , either d(T n x, T n+1 x) = ∞ ∀ n ≥ 0, or there exists a natural number n0 such that (FP1) d(T n x, T n+1 x) < ∞ for all n ≥ n0 ; (FP2) The sequence (T n x) is convergent to a fixed to a fixed point y ∗ of T (FP3) y ∗ is the unique fixed point of T in the set ∆ = {y ∈ Ω : d(T n0 x, y) < ∞}; 1 (FP4) d(y ∗ , y) ≤ 1−L d(y, T y) for all y ∈ ∆. Hereafter throughout this section, let us consider E and F to be a normed space and a Banach space, respectively. 4.1. FIXED POINT STABILITY RESULTS OF (1.2). Theorem 4.3. Let fa : E → F be a odd mapping for which there exist a function ζ : E → [0, ∞) with the condition 1 lim k ζ(κki x) = 0 (4.1) k→∞ κi where  2 if i = 0; κi = (4.2) 1 if i = 1, 2 such that the functional inequality kfa (2x) − 3fa (x) − fa (−x)k ≤ ζ(x) for all x ∈ E. If there exists L = L(i) < 1 such that the function x x → ψ(x) = ζ , 2 has the property L ψ(x) = ψ (κi x) . κi

(4.3)

(4.4)

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SIMPLE AQ AND SIMPLE CQ . . .

19

for all x ∈ E. Then there exists a unique additive mapping A : E → F satisfying the functional equation (1.2) and kfa (x) − A(x)k ≤

L1−i ψ(x) 1−L

(4.5)

for all x ∈ E. Proof. Consider the set Γ = {p/p : E → F, p(0) = 0} and introduce the generalized metric on Γ, d(p, q) = inf{K ∈ (0, ∞) :k p(x) − q(x) k≤ Kψ(x), x ∈ E}. It is easy to see that (Γ, d) is complete. Define Υ : Γ → Γ by Υp(x) =

1 p(κi x), κi

for all x ∈ E. Now p, q ∈ Γ, d(p, q) ≤ K ⇒ k p(x) − q(x) k≤ Kψ(x), x ∈ E,

1

1 1

⇒

κi p(κi x) − κi q(κi x) ≤ κi Kψ(κi x), x ∈ E,



1 1

≤ LKψ(x), x ∈ E, p(κ x) − q(κ x) ⇒ i i

κi κi ⇒ k Υp(x) − Υq(x) k≤ LKψ(x), x ∈ E, ⇒d(p, q) ≤ LK. This implies d(Υp, Υq) ≤ Ld(p, q), for all p, q ∈ Γ. i.e., T is a strictly contractive mapping on Γ with Lipschitz constant L. Using oddness of fa in (4.3), we arrive kfa (2x) − 2f (x)k ≤ ζ(x) for all x ∈ E. It follows from (4.6) that

fa (2x)

ζ(x)

≤ − f (x) a

2

2

(4.6)

(4.7)

for all x ∈ E. Using (4.4) for the case i = 0 it reduces to

fa (2x)

≤ Lψ(x) − f (x) a

2

for all x ∈ E, i.e., d(Υfa , fa ) ≤ L ⇒ d(Υfa , fa ) ≤ L = L1 < ∞.

(4.8)

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Again replacing x =

x 2

in (4.6), we get

 x  x

fa (x) − 2fa

≤ζ 2 2 for all x ∈ E. Using (4.4) for the case i = 1 it reduces to

 x 

f (x) − 2f

≤ ψ(x)

a a 2 for all x ∈ E, i.e., d(fa , Υfa ) ≤ 1 ⇒ d(fa , Υfa ) ≤ 1 = L0 < ∞.

(4.9)

(4.10)

From (4.8) and (4.10), we arrive d(fa , Υfa ) ≤ L1−i . Therefore (FP1) holds. By (FP2), it follows that there exists a fixed point A of Υ in Γ such that fa (κki x) A(x) = lim , k→∞ κki

∀ x ∈ E.

(4.11)

To order to prove A : E → F is additive. Replacing x by κki x in (4.3) and dividing by κki , it follows from (4.1) that

1 1

k k k f (κ 2x) − 3fa (κi x) − fa (−κi x) ≤ k ζ(κki x) k a i κi κi for all x ∈ E. Letting k → ∞ in the above inequality and using the definition of A(x), we see that A(2x) = 3A(x) + A(−x) i.e., A satisfies the functional equation (1.2) for all x ∈ E. By (FP3), A is the unique fixed point of Υ in the set ∆ = {A ∈ Γ : d(fa , A) < ∞}, such that kfa (x) − A(x)k ≤ Kψ(x) for all x ∈ E and K > 0. Finally by (FP4), we obtain 1 d(fa , A) ≤ d(fa , Υfa ) 1−L this implies L1−i d(fa , A) ≤ 1−L which yields L1−i kfa (x) − A(x)k ≤ ψ(x) 1−L this completes the proof of the theorem.



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SIMPLE AQ AND SIMPLE CQ . . .

21

The following corollary is an immediate consequence of Theorem 4.3 concerning the stability of (1.2). Corollary 4.4. Let fa : E → F be an odd mapping and there exists real numbers ρ and r such that  (i) ρ, kfa (2x) − 3fa (x) − fa (−x)k ≤ (4.12) (ii) ρ||x||r , r 6= 1; for all x ∈ E. Then there exists a unique additive function A : E → F such that   (i) |ρ|, ρ||x||r (4.13) kfa (x) − A(x)k ≤ (ii) ,  |2 − 2r | for all x ∈ E. Proof. Setting  ζ(x) =

ρ, ρ||x||r ,

for all x ∈ E. Now,  ρ    k, 1 → 0 as k → ∞, κi k ζ(κi x) = = ρ k k r → 0 as k → ∞.  κi  k ||κi xi || , κi Thus, (4.1) is holds. But we have ψ(x) = ζ

x 2




has the property ψ(x) =

Hence ψ(x) = ζ

x 2

( =

L ψ (κi x) for all x ∈ E. κi

ρ ρ ||x||r . 2r

Now,  ρ  ρ  ,  , 1 κ κi i = ψ(κi x) =  ρ ||κi x||r ,  ρ κri ||x||r , κi κi κi

 =

κ−1 i ρ, = κr−1 ρ||x||r , i



Hence the inequality (4.4) holds either, L = 2−1 if i = 0 and L = Now from (4.5), we prove the following cases for condition (i). Case:1 L = 2−1 if i = 0 1−0

(2−1 ) kfa (x) − A(x)k ≤ ψ(x) = ρ. 1 − 2−1

κ−1 i ψ(x), κr−1 ψ(x). i 1 2−1

if i = 1.

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M. ARUNKUMAR et. al.,

Case:2 L =

1 2−1

if i = 1 1

kfa (x) − A(x)k ≤

2−1


1−1

1−

1 2−1

ψ(x) = −ρ.

1 Also the inequality (4.4) holds either, L = 2r−1 for r < 1 if i = 0 and L = 2r−1 for r > 1 if i = 1. Now from (4.5), we prove the following cases for condition (ii). Case:3 L = 2r−1 for r < 1 if i = 0
1−0 2(r−1) 1 ρ||x||r r kfa (x) − A(x)k ≤ ψ(x) = ρ||x|| = . 1 − 2(r−1) 2 − 2r 2 − 2r 1 for r > 1 if i = 1 Case:4 L = 2r−1
1−1 1 ρ||x||r 1 (r−1) s kfa (x) − A(x)k ≤ 2 ρ||x|| = . ψ(x) = 1 2r − 2 2r − 2 1 − 2(r−1)

Hence the proof is complete.



Theorem 4.5. Let fq : E → F be a even mapping for which there exist a function ζ : E → [0, ∞) with the condition 1 lim 2k ζ(κki x) = 0 (4.14) k→∞ κi where κi is defined in (4.2) such that the functional inequality kfq (2x) − 3fq (x) − fq (−x)k ≤ ζ(x)

(4.15)

for all x ∈ E. If there exists L = L(i) < 1 such that the function x x → ψ(x) = ζ , 2 has the property L ψ(x) = 2 ψ (κi x) . (4.16) κi for all x ∈ E. Then there exists a unique quadratic mapping Q2 : E → F satisfying the functional equation (1.2) and kfq (x) − Q2 (x)k ≤

L1−i ψ(x) 1−L

for all x ∈ E. Proof. Consider the set Γ = {p/p : E → F, p(0) = 0} and introduce the generalized metric on Γ, d(p, q) = inf{K ∈ (0, ∞) :k p(x) − q(x) k≤ Kψ(x), x ∈ E}. It is easy to see that (Γ, d) is complete.

(4.17)

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SIMPLE AQ AND SIMPLE CQ . . .

23

Define Υ : Γ → Γ by Υp(x) =

1 p(κi x), κ2i

for all x ∈ E. Now p, q ∈ Γ, d(p, q) ≤ K ⇒ k p(x) − q(x) k≤ Kψ(x), x ∈ E,

1

1 1

⇒

κ2 p(κi x) − κ2 q(κi x) ≤ κ2 Kψ(κi x), x ∈ E, i i

i

1

1

⇒

κ2 p(κi x) − κ2 q(κi x) ≤ LKψ(x), x ∈ E, i i ⇒ k Υp(x) − Υq(x) k≤ LKψ(x), x ∈ E, ⇒d(p, q) ≤ LK. This implies d(Υp, Υq) ≤ Ld(p, q), for all p, q ∈ Γ. i.e., T is a strictly contractive mapping on Γ with Lipschitz constant L. Using evenness of fq in (4.15), we arrive kfq (2x) − 4f (x)k ≤ ζ(x) for all x ∈ E. It follows from (4.18) that

fq (2x)

ζ(x)

≤ − f (x) q

4 4

(4.18)

(4.19)

for all x ∈ E. Using (4.16) for the case i = 0 it reduces to

fq (2x)



4 − fq (x) ≤ Lψ(x) for all x ∈ E, i.e., d(Υfq , fq ) ≤ L ⇒ d(Υfq , fq ) ≤ L = L1 < ∞. Again replacing x =

x 2

in (4.18), we get

x  x 

≤ζ

fq (x) − 4fq 2 2 for all x ∈ E. Using (4.16) for the case i = 1 it reduces to

 x 

fq (x) − 4fq

≤ ψ(x) 2 for all x ∈ E,

(4.20)

i.e., d(fq , Υfq ) ≤ 1 ⇒ d(fq , Υfq ) ≤ 1 = L0 < ∞. From the above two cases, we arrive d(fq , Υfq ) ≤ L1−i Therefore (FP1) holds. The rest of the proof is similar to that of Theorem 4.3. 

143

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M. ARUNKUMAR et. al.,

The following corollary is an immediate consequence of Theorem 4.3 concerning the stability of (1.2). Corollary 4.6. Let fq : E → F be an even mapping and there exists real numbers ρ and r such that  (i) ρ, kf (2x) − 3f (x) − f (−x)k ≤ (4.21) (ii) ρ||x||r , r 6= 2; for all x ∈ E. Then there exists a unique quadratic function Q2 : E → F such that  ρ   (i) |3| , (4.22) kfq (x) − Q2 (x)k ≤ ρ||x||r   (ii) , |4 − 2r | for all x ∈ E. Proof. The proof of the corollary is similar lines to the of Corollary 4.4.



Now, we are ready to prove the main fixed point stability results. Theorem 4.7. Let f : E → F be a mapping for which there exist a function ζ : E → [0, ∞) with the conditions (4.1) and (4.14) where κi is defined (4.2) such that the functional inequality kf (2x) − 3f (x) − f (−x)k ≤ ζ(x)

(4.23)

for all x ∈ E. If there exists L = L(i) < 1 such that the function x x → ψ(x) = ζ , 2 with the properties (4.4) and (4.16) for all x ∈ E. Then there exists a unique additive mapping A : E → F satisfying the functional equation and a unique quadratic mapping Q : E → F satisfying the functional equation (1.2) and kf (x) − A(x) − Q(x)k ≤

L1−i (ψ(x) + ψ(−x)) 1−L

(4.24)

for all x ∈ E. Proof. It follows from (3.29) and Theorem 4.3, we have 1 L1−i (ψ(x) + ψ(−x)) 21−L for all x ∈ E. Also, it follows from (3.31) and Theorem 4.5, we have kfo (x) − A(x)k ≤

kfe (x) − Q(x)k ≤

1 L1−i (ψ(x) + ψ(−x)) 21−L

(4.25)

(4.26)

for all x ∈ E. Define f (x) = fe (x) + fo (x)

(4.27)

144

SIMPLE AQ AND SIMPLE CQ . . .

25

for all x ∈ E. From (4.25),(4.26) and (4.27), we arrive kf (x) − A(x) − Q(x)k = kfe (x) + fo (x) − A(x) − Q(x)k ≤ kfo (x) − A(x)k + kfe (x) − Q(x)k 1 L1−i [(ψ(x) + ψ(−x)) + (ψ(x) + ψ(−x))] 21−L L1−i (ψ(x) + ψ(−x)) ≤ 1−L for all x ∈ X. Hence the theorem is proved. ≤



The following corollary is an immediate consequence of Theorem 4.7, using Corollaries 4.4 and 4.6 concerning the stability of (1.2). Corollary 4.8. Let f : E → F be a mapping and there exists real numbers ρ and r such that  (i) ρ, kf (2x) − 3f (x) − f (−x)k ≤ (4.28) (ii) ρ||x||r , r 6= 1, 2; for all x ∈ E. Then there exists a unique additive function A : E → F and a unique quadratic function Q2 : E → F such that    1   ,  |ρ| 1 + 3   kf (x) − A(x) − Q(x)k ≤ (4.29) 1 1  r  + ,  ρ||x|| |2 − 2r | |4 − 2r | for all x ∈ E. 4.2. FIXED POINT STABILITY RESULTS OF (1.3). Theorem 4.9. Let gc : E → F be a odd mapping for which there exist a function ζ : E → [0, ∞) with the condition 1 ζ(κki x) = 0 k→∞ κ3k i lim

(4.30)

where  κi =

2 if i = 0; 1 if i = 1, 2

(4.31)

such that the functional inequality kgc (2x) − 12gc (x) − 4gc (−x)k ≤ ζ(x) for all x ∈ E. If there exists L = L(i) < 1 such that the function x x → ψ(x) = ζ , 2

(4.32)

145

26

M. ARUNKUMAR et. al.,

has the property 1 ψ (κi x) . (4.33) κ3i for all x ∈ E. Then there exists a unique cubic mapping C : E → F satisfying the functional equation (1.3) and ψ(x) =

kgc (x) − C(x)k ≤

L1−i ψ(x) 1−L

(4.34)

for all x ∈ E. Proof. Consider the set Γ = {p/p : E → F, p(0) = 0} and introduce the generalized metric on Γ, d(p, q) = inf{K ∈ (0, ∞) :k p(x) − q(x) k≤ Kψ(x), x ∈ E}. It is easy to see that (Γ, d) is complete. Define Υ : Γ → Γ by Υp(x) =

1 p(κi x), κ3i

for all x ∈ E. Now p, q ∈ Γ, d(p, q) ≤ K ⇒ k p(x) − q(x) k≤ Kψ(x), x ∈ E,



1 1

≤ 1 Kψ(κi x), x ∈ E, p(κ x) − q(κ x) ⇒ i i 3

κ3

κi κi i

1

1

⇒

κ3 p(κi x) − κ3 q(κi x) ≤ LKψ(x), x ∈ E, i i ⇒ k Υp(x) − Υq(x) k≤ LKψ(x), x ∈ E, ⇒d(p, q) ≤ LK. This implies d(Υp, Υq) ≤ Ld(p, q), for all p, q ∈ Γ. i.e., T is a strictly contractive mapping on Γ with Lipschitz constant L. Using oddness of gc in (4.32), we arrive kgc (2x) − 8f (x)k ≤ ζ(x) for all x ∈ E. It follows from (4.35) that

gc (2x)

ζ(x)

≤ − g (x) c

8 8 for all x ∈ E. Using (4.33) for the case i = 0 it reduces to



gc (2x)

≤ Lψ(x)

− g (x) c

8

(4.35)

(4.36)

146

SIMPLE AQ AND SIMPLE CQ . . .

27

for all x ∈ E, i.e., d(Υgc , gc ) ≤ L ⇒ d(Υgc , gc ) ≤ L = L1 < ∞. Again replacing x =

x 2

in (4.35), we get

 x  x

gc (x) − 8gc

≤ξ 2 2 for all x ∈ E. Using (4.33) for the case i = 1 it reduces to

 x 

gc (x) − 8gc

≤ ψ(x) 2 for all x ∈ E,

(4.37)

i.e., d(gc , Υgc ) ≤ 1 ⇒ d(gc , Υgc ) ≤ 1 = L0 < ∞. From the above two cases, we arrive d(gc , Υgc ) ≤ L1−i . Therefore (FP1) holds. The rest of the proof is similar to that of Theorem 4.3.  The following corollary is an immediate consequence of Theorem 4.9 concerning the stability of (1.3). Corollary 4.10. Let gc : E → F be an odd mapping and there exists real numbers ρ and r such that  (i) ρ, (4.38) kgc (2x) − 12gc (x) − 4gc (−x)k ≤ (ii) ρ||x||r , r 6= 3; for all x ∈ E. Then there exists a unique cubic function C : E → F such that  ρ   (i) |7| , kgc (x) − C(x)k ≤ (4.39) ρ||x||r   (ii) , |8 − 2r | for all x ∈ E. Proof. The proof of the corollary is similar lines to the of Corollary 4.4.



Theorem 4.11. Let gq : E → F be a even mapping for which there exist a function ζ : E → [0, ∞) with the condition 1 lim 4k ζ(κki x) = 0 (4.40) k→∞ κi where κi is defined in (4.31) such that the functional inequality kgq (2x) − 12gq (x) − 4gq (−x)k ≤ ζ(x) for all x ∈ E. If there exists L = L(i) < 1 such that the function x x → ψ(x) = ζ , 2

(4.41)

147

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M. ARUNKUMAR et. al.,

has the property L ψ (κi x) . (4.42) κ4i for all x ∈ E. Then there exists a unique quartic mapping Q4 : E → F satisfying the functional equation (1.3) and ψ(x) =

kgq (x) − Q4 (x)k ≤

L1−i ψ(x) 1−L

(4.43)

for all x ∈ E. Proof. Consider the set Γ = {p/p : E → F, p(0) = 0} and introduce the generalized metric on Γ, d(p, q) = inf{K ∈ (0, ∞) :k p(x) − q(x) k≤ Kψ(x), x ∈ E}. It is easy to see that (Γ, d) is complete. Define Υ : Γ → Γ by Υp(x) =

1 p(κi x), κ4i

for all x ∈ E. Now p, q ∈ Γ, d(p, q) ≤ K ⇒ k p(x) − q(x) k≤ Kψ(x), x ∈ E.

1

1

≤ 1 Kψ(κi x), x ∈ E, ⇒ p(κ x) − q(κ x) i i 4

κ4

κ4 κi i

i

1 1

⇒

κ4 p(κi x) − κ4 q(κi x) ≤ LKψ(x), x ∈ E, i i ⇒ k Υp(x) − Υq(x) k≤ LKψ(x), x ∈ E, ⇒d(p, q) ≤ LK. This implies d(Υp, Υq) ≤ Ld(p, q), for all p, q ∈ Γ. i.e., T is a strictly contractive mapping on Γ with Lipschitz constant L. Using evenness of gq in (4.41), we arrive kgq (2x) − 16f (x)k ≤ ζ(x) for all x ∈ E. It follows from (4.44) that

gq (2x)

ζ(x)

≤ − g (x) q

16

16 for all x ∈ E. Using (4.42) for the case i = 0 it reduces to



gq (2x)

≤ Lψ(x)

− g (x) q

16

(4.44)

(4.45)

148

SIMPLE AQ AND SIMPLE CQ . . .

29

for all x ∈ E, i.e., d(Υgq , gq ) ≤ L ⇒ d(Υgq , gq ) ≤ L = L1 < ∞. Again replacing x =

x 2

in (4.44), we get

 x  x

gq (x) − 16gq

≤ξ 2 2 for all x ∈ E. Using (4.42) for the case i = 1 it reduces to

 x 

gq (x) − 16gq

≤ ψ(x) 2 for all x ∈ E,

(4.46)

i.e., d(gq , Υgq ) ≤ 1 ⇒ d(gq , Υgq ) ≤ 1 = L0 < ∞. From the above two cases, we arrive d(gq , Υgq ) ≤ L1−i . Therefore (FP1) holds. The rest of the proof is similar to that of Theorem 4.3.  The following corollary is an immediate consequence of Theorem 4.11 concerning the stability of (1.3). Corollary 4.12. Let gq : E → F be an even mapping and there exists real numbers ρ and r such that  (i) ρ, (4.47) kgq (2x) − 12gq (x) − 4gq f (−x)k ≤ (ii) ρ||x||r , r 6= 4; for all x ∈ E. Then there exists a unique quartic function Q4 : E → F such that  ρ   (i) |15| , kgq (x) − Q2 (x)k ≤ (4.48) ρ||x||r   (ii) , |16 − 2r | for all x ∈ E. Proof. The proof of the corollary is similar lines to the of Corollary 4.4.



Now, we are ready to prove the main fixed point stability results. Theorem 4.13. Let g : E → F be a mapping for which there exist a function ζ : E → [0, ∞) with the conditions (4.30) and (4.40) where κi is defined (4.31) such that the functional inequality kg(2x) − 12g(x) − 4g(−x)k ≤ ζ(x) for all x ∈ E. If there exists L = L(i) < 1 such that the function x x → ψ(x) = ζ , 2

(4.49)

149

30

M. ARUNKUMAR et. al.,

with the properties (4.33) and (4.42) for all x ∈ E. Then there exists a unique cubic mapping C : E → F satisfying the functional equation and a unique quartic mapping Q4 : E → F satisfying the functional equation (1.3) and L1−i kg(x) − C(x) − Q4(x)k ≤ (ψ(x) + ψ(−x)) 1−L

(4.50)

for all x ∈ E. Proof. The proof of the Theorem is similar to the Theorem 4.7.



The following Corollary is an immediate consequence of Theorem 4.13, using Corollaries 4.10 and 4.12 concerning the stability of (1.3). Corollary 4.14. Let g : E → F be a mapping and there exists real numbers ρ and r such that  (i) ρ, (4.51) kg(2x) − 12g(x) − 4g(−x)k ≤ (ii) ρ||x||r , r 6= 2, 4; for all x ∈ E. Then there exists a unique cubic function C : E → F and a unique quartic function Q4 : E → F such that    1 1   |ρ| + ,  7  15  (4.52) kg(x) − C(x) − Q4 (x)k ≤ 1 1  r  + ,  ρ||x|| |8 − 2r | |16 − 2r | for all x ∈ E. References [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. [3] M. Arunkumar, S. Hema Latha, Orthogonal Stability Of 2 Dimensional Mixed Type Additive And Quartic Functional Equation, International Journal of pure and Applied Mathematics, 63 No.4, (2010), 461-470. [4] M. Arunkumar, John M. Rassias, On the generalized Ulam-Hyers stability of an AQmixed type functional equation with counter examples, Far East Journal of Applied Mathematics, Volume 71, No. 2, (2012), 279-305. [5] M. Arunkumar, Matina J. Rassias, Yanhui Zhang, Ulam - Hyers stability of a 2variable AC - mixed type functional equation: direct and fixed point methods, Journal of Modern Mathematics Frontier (JMMF), 1 (3), 2012, 10-26. [6] M. Arunkumar, Solution and stability of modified additive and quadratic functional equation in generalized 2-normed spaces, International Journal Mathematical Sciences and Engineering Applications, Vol. 7 No. I (January, 2013), 383-391. [7] M. Arunkumar, Generalized Ulam - Hyers stability of derivations of a AQ - functional equation, ”Cubo A Mathematical Journal” dedicated to Professor Gaston M. N’Gurkata on the occasion of his 60th Birthday Vol.15, No 01, (159-169), March 2013.

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[29] S. Murthy, M. Arunkumar, G. Ganapathy, P. Rajarethinam, Stability of mixed type additive quadratic functional equation in Random Normed space, International Journal of Applied Mathematics (IJAM), Vol. 26. No. 2 (2013), 123-136. [30] S. Murthy, M. Arunkumar, G. Ganapathy, Perturbation of n- dimensional quadratic functional equation: A fixed point approach, International Journal of Advanced Computer Research (IJACR), Volume-3, Number-3, Issue-11 September-2013, 271-276. [31] A. Najati, M.B. Moghimi, On the stability of a quadratic and additive functional equation, J. Math. Anal. Appl. 337 (2008), 399-415. [32] Choonkil Park, Orthogonal Stability of an Additive-Quadratic Functional Equation, Fixed Point Theory and Applications 2011 2011:66. [33] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130. [34] J.M. Rassias, K.Ravi, M.Arunkumar and B.V.Senthil Kumar, Ulam Stability of Mixed type Cubic and Additive functional equation, Functional Ulam Notions (F.U.N) Nova Science Publishers, 2010, Chapter 13, 149 - 175. [35] M.J. Rassias, M. Arunkumar, S. Ramamoorthi, Stability of the Leibniz additivequadratic functional equation in Quasi-Beta normed space: Direct and fixed point methods, Journal Of Concrete And Applicable Mathematics, Vol. 14 No. 1-2, (2014), 22 - 46. [36] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), 297-300. [37] Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003. [38] K.Ravi and M.Arunkumar, On a n- dimensional additive Functional Equation with fixed point Alternative, Proceedings of International Conference on Mathematical Sciences 2007, Malaysia. [39] K. Ravi, M. Arunkumar and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, Autumn 2008 Vol.3, No. 08, 36-47. [40] K. Ravi, J.M. Rassias, M. Arunkumar, R. Kodandan, Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Article 114, 29 pp. [41] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964. [42] T.Z. Xu, J.M. Rassias, W.X Xu, Generalized Ulam-Hyers stability of a general mixed AQCQ-functional equation in multi-Banach spaces: a fixed point approach, Eur. J. Pure Appl. Math. 3 (2010), no. 6, 1032-1047. [43] T.Z. Xu, J.M. Rassias, M.J. Rassias, W.X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces, J. Inequal. Appl. 2010, Art. ID 423231, 23 pp. [44] T.Z. Xu, J.M Rassias, W.X. Xu, A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-Archimedean normed spaces, Discrete Dyn. Nat. Soc. 2010, Art. ID 812545, 24 pp.

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 1-2, 152-170, 2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 152

- Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan. E-mail address: [email protected]. Abstract: Recently Salimi et al. [Fixed Point Theory Appl., 2013:151] modi…ed the notion of -admissible mappings. In this paper, the concept of dominated mappings is introduced and several general common …xed point results for two, three and four mappings in a closed ball in 0-complete partial metric spaces are established. ——————————————————— 2010 Mathematics Subject Classi…cation: 46S40; 47H10; 54H25. Keywords and Phrases: Unique common …xed point, Contractive mapping, Closed ball, - Dominated mapping, 0-complete partial metric spaces. _____________________

1

Introduction and Preliminaries

Let T : X ! X be a mapping. A point x 2 X is called a …xed point of T if x = T x: Many results appeared in literature related to the …xed point of mappings which are contractive on the whole domain. It is possible that T : X ! X is not a contraction but T : Y ! X is a contraction, where Y is a subset of X: One can obtain …xed point results for such mapping by using suitable conditions. Recently Arshad et al. [7] proved a result concerning the existence of …xed points of a mapping satisfying a contractive conditions in closed ball in a complete partial metric space( see also [5, 6, 16, 20, 27, 28, 29]). Matthews [17] introduced the concept of a partial metric space. In partial metric spaces, Matthews excluded an additional condition of metric spaces that is d(x; x) = 0 for all x: As it is better to prove a result with weaker conditions, so one should prove new results in partial metric spaces. Partial metric spaces have applications in theoretical computer science (see [15, 17]). Romaguera [23] intoduced 0-complete partial metric space which is a genralization of complete partial metric space. The existence of …xed points of -admissible mappings in complete metric spaces has been studied by several researchers (see [4, 14, 15, 25, 26] and references therein). In this paper we discuss common …xed point results for dominated mappings in a closed ball in 0-complete partial metric space. The results given in this paper improve and extend several recent results in [7, 29]. 1

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

Consistent with [12, 17, 23], the following de…nitions and results will be needed in the sequel. De…nition 1.1. [17] Let X be a nonempty set. If for any x; y; z 2 X, mapping p : X X ! R+ satis…es (P1 ) p(x; x) = p(y; y) = p(x; y) if and only if x = y; (P2 ) p(x; x)

p(x; y);

(P3 ) p(x; y) = p(y; x); (P4 ) p(x; z)

p(x; y) + p(y; z)

p(y; y).

Then it is said to be a partial metric on X and the pair (X; p) is called a partial metric space. Each partial metric p on X induces a T0 topology p on X which has as a base the family of open balls fBp (x; ") : x 2 X; " > 0g, where Bp (x; ") = fy 2 X : p(x; y) < p(x; x) + "g. Also Bp (x; r) = fy 2 X : p(x; y) p(x; x) + rg is a closed ball in (X; p): It is clear that if p(x; y) = 0, then from P1 and P2 ; x = y. But if x = y, then p(x; y) may not be 0. De…nition 1.2. [17] Let (X; p) be a partial metric space, then, (a) A sequence fxn g in (X; p) converges to a point x 2 X if and only if limn!1 p(x; xn ) = p(x; x). (b) [23] A sequence fxn g in (X; p) is called 0-Cauchy if

lim p(xn ; xm ) = 0.

n;m!1

The space (X; p) is called 0-complete if every 0-Cauchy sequence in X converges to a point x 2 X such that limn!1 p(x; xn ) = p(x; x) = 0. (c) [23] If (X; p) is complete, then it is 0-complete. Romaguera [23] has given an example which proves that converse assertion of (c) does not hold. It is easy to see that every closed subset of a 0-complete partial metric space is 0-complete. We require the following lemma for subsequent use: Lemma 1.3. [12] Let X be a non empty set and f : X ! X a function. Then there exists a subset E X such that f E = f X and f : E ! X is one to one.

2

Common …xed point results

De…nition 2.1: Let T : X ! X and ; : X X ! [0; +1) two functions. We say that T is - dominated mapping if x 2 X such that (x; T x) (x; T x). If (x; y) (x; y) and (y; z) (y; z) implies that (x; z) (x; z) then we say that T is triangle - dominated mapping: Also, if we take (x; y) = 1 then T is called -dominated mapping and triangle -dominated mapping respectively and if we take (x; y) = 1 then T is called -subdominated mapping and 2

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154

triangle -subdominated mapping respectively. In the following we present common …xed point theorems for the pair of - -dominated contractive mappings in a closed ball. Theorem 2.2. Let (X; p) be a 0-complete partial metric space. Suppose there exist two functions, ; : X X ! [0; +1) such that S and T are - dominated mapping. Let x0 ; x; y 2 X, r > 0: If there exist some k; t such that; k + 2t 2 [0; 1) and the following conditions hold: p(Sx; T y)

kp(x; y) + t[p(x; Sx) + p(y; T y)];

for all x; y 2 B(x0 ; r) such that (x; y) p(x0 ; Sx0 )

(x; y) or (y; x)

(1

(2.1) (y; x) and

)[r + p(x0 ; x0 )];

(2.2)

k+t . 1 t Then there exists a point x in Bp (x0 ; r) such that p(x ; x ) = 0: Moreover, if for any sequence fxn g in Bp (x0 ; r) such that (xn ; xn+1 ) (xn ; xn+1 ) for all n 2 N [ f0g and xn ! u 2 Bp (x0 ; r) as n ! +1 then (xn ; u) (xn ; u) for all n 2 N [ f0g; then x is a common …xed point of S and T: Proof. Let x1 in X be such that x1 = Sx0 and x2 = T x1 . Continuing this process, we construct a sequence xn of points in X such that, where

=

x2i+1 = Sx2i ; and x2i+2 = T x2i+1 , wherei = 0; 1; 2; : : : : As S is - dominated mapping then (x0 ; x1 ) (x0 ; x1 ): As T is - dominated mapping then (x1 ; x2 ) (x1 ; x2 ): Continuing in this way we obtain (xn ; xn+1 ) (xn ; xn+1 ) for all n 2 N [f0g: First, we show that xn 2 Bp (x0 ; r) k+t for all n 2 N . Using inequality (2.2) and the fact that = ; we have, 1 t p(x0 ; Sx0 )

r + p(x0 ; x0 ):

It implies that x1 2 Bp (x0 ; r): Let x2 ; ; xj 2 Bp (x0 ; r) for some j 2 N . If j = 2i + 1, then as x1 ; x2 ; :::; xj 2 Bp (x0 ; r) and (x2i ; x2i+1 ) (x2i ; x2i+1 );where i = 0; 1; 2; : : : j 2 1 : So using inequality (2.1), we obtain, p(x2i+1 ; x2i+2 )

= p(Sx2i ; T x2i+1 ) kp(x2i ; x2i+1 ) +t[p(x2i ; Sx2i ) + p(x2i+1 ; T x2i+1 )];

which implies that, p(x2i+1 ; x2i+2 )

p(x2i ; x2i+1 ) 2 p(x2i 1 ; x2i )

3

:::

2i+1

p(x0 ; x1 ):

(2.3)

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

155

If j = 2i+2, then as x1 ; x2 ; :::; xj 2 Bp (x0 ; r) and (x2i+1 ; x2i+2 ) where i = 0; 1; 2; : : : ; j 2 2 : We obtain, 2i+2

p(x2i+2 ; x2i+3 )

(x2i+1 ; x2i+2 );

p(x0 ; x1 ):

(2.4)

Thus from inequalities (2.3) and (2.4) , we have, j

p(xj ; xj+1 )

p(x0 ; x1 ) for some j 2 N:

(2.5)

Now p(x0 ; xj+1 )

p(x0 ; x1 ) + ::: + p(xj ; xj+1 ) p(x0 ; x1 ) + ::: + j p(x0 ; x1 ); p(x0 ; x1 )[1 + ::: + j 1 + j ]

p(x0 ; xj+1 )

(1

)[r + p(x0 ; x0 )]

j+1

(1

(by 2.5) )

1

r + p(x0 ; x0 ); gives xj+1 2 Bp (x0 ; r): Hence xn 2 Bp (x0 ; r): Also (xn ; xn+1 ) then n p(xn ; xn+1 ) p(x0 ; x1 ); for all n 2 N:

(xn ; xn+1 ); (2.6)

So we have, p(xn+k ; xn )

p(xn+k ; xn+k 1 ) + ::: + p(xn+1 ; xn ) n+k 1 p(x0 ; x1 ) + ::: + n p(x0 ; x1 ); n p(x0 ; x1 )[ k 1 + k 2 + ::: + 1]

p(xn+k ; xn )

n

p(x0 ; x1 )

(1 1

k

)

(by 2.6)

! 0 as n ! 1:

Notice that the sequence fxn g is a 0-Cauchy sequence in (Bp (x0 ; r); p): As (Bp (x0 ; r); p) is complete, therefore by De…nition 1.2, there exists a point x 2 Bp (x0 ; r) with lim p(xn ; x ) = p(x ; x ) = 0: (2.7) n!1

Now, p(x ; Sx )

p(x ; x2n+2 ) + p(x2n+2 ; Sx )

p(x2n+2 ; x2n+2 ):

On taking limit as n ! 1 and using the fact that (xn ; x ) (xn ; xn+1 ) (xn ; xn+1 ) and xn ! x ; we have, p(x ; Sx )

lim [p(x ; x2n+2 ) + kp(x2n+1 ; x )

n!1

+tfp(x2n+1 ; x2n+2 ) + p(x ; Sx )g]; by inequalities (2.6) and (2.7), we obtain, (1

t) p(x ; Sx ) 4

0

(xn ; x ) when

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

and x = Sx : Similarly, by using, p (x ; T x )

p(x ; x2n+1 ) + p(x2n+1 ; T x )

p(x2n+1 ; x2n+1 );

we can show that x = T x : Hence S and T have a common …xed point in Bp (x0 ; r). If we take T = S for all x; y 2 X in Theorem 2.2, we obtain following result. Corollary 2.3. Let (X; p) be a 0-complete partial metric space. Suppose there exist two functions, ; : X X ! [0; +1) such that S is - dominated mapping . Let x0 ; x; y 2 X, r > 0: If there exist some k; t such that; k+2t 2 [0; 1) and the following conditions hold: p(Sx; Sy)

kp(x; y) + t[p(x; Sx) + p(y; Sy)];

for all x; y 2 B(x0 ; r) such that (x; y) p(x0 ; Sx0 )

(x; y) and

(1

)[r + p(x0 ; x0 )];

k+t . 1 t If for any sequence fxn g in Bp (x0 ; r) such that (xn ; xn+1 ) (xn ; xn+1 ) for all n 2 N [f0g and xn ! u 2 Bp (x0 ; r) as n ! +1 then (xn ; u) (xn ; u) for all n 2 N [f0g: Then there exists a point x in Bp (x0 ; r) such that x = Sx and p(x ; x ) = 0: where

=

If (x; y) = 1 for all x; y 2 X in Theorem 2.2, we obtain following result. Corollary 2.4. Let (X; p) be a 0-complete partial metric space. Suppose there exists, : X X ! [0; +1) such that S and T are -dominated mapping. Let x0 ; x; y 2 X, r > 0: If there exist some k; t such that; k + 2t 2 [0; 1) and the following conditions hold: p(Sx; T y)

kp(x; y) + t[p(x; Sx) + p(y; T y)];

for all x; y 2 B(x0 ; r) such that (x; y) p(x0 ; Sx0 )

1 or (y; x)

(1

1 and

)[r + p(x0 ; x0 )];

k+t . 1 t If for any sequence fxn g in Bp (x0 ; r) such that (xn ; xn+1 ) 1 for all n 2 N [ f0g and xn ! u 2 Bp (x0 ; r) as n ! +1 then (xn ; u) 1 for all n 2 N [f0g: Then there exists a point x in Bp (x0 ; r) such that x = Sx = T x and p(x ; x ) = 0: where

=

If (x; y) = 1 for all x; y 2 X in Theorem 2.2, we obtain following result. Corollary 2.5. Let (X; p) be a 0-complete partial metric space. Suppose there 5

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157

exists, : X X ! [0; +1) such that S and T are -subdominated mapping. Let x0 ; x; y 2 X, r > 0: If there exist some k; t such that k + 2t 2 [0; 1) and (x; y) 1 or (y; x) 1 implies that p(Sx; T y)

kp(x; y) + t[p(x; Sx) + p(y; T y)];

for all x; y 2 B(x0 ; r) and p(x0 ; Sx0 )

(1

)[r + p(x0 ; x0 )];

k+t . 1 t 1 for all If for any sequence fxn g in Bp (x0 ; r) such that (xn ; xn+1 ) n 2 N [ f0g and xn ! u 2 Bp (x0 ; r) as n ! +1 then (xn ; u) 1 for all n 2 N [f0g: Then there exists a point x in Bp (x0 ; r) such that x = Sx = T x and p(x ; x ) = 0: where

=

Corollary 2.6. Let (X; d) be a complete metric space. Suppose there exists, : X X ! [0; +1) such that S and T are -dominated mapping: Let (x; y)d(Sx; T y)

kd(x; y) + t[d(x; Sx) + d(y; T y)]

holds for all x; y 2 X and k + 2t 2 [0; 1): If for any sequence fxn g in X with (xn ; xn+1 ) 1 for all n 2 N [ f0g and xn ! x as n ! +1, we have (xn ; x) 1 for all n 2 N [ f0g. Then S and T have a common …xed point: Theorem 2.7. Adding the following conditions to the hypotheses of Theorem 2.2. (i) Let S and T are triangle - dominated mapping. (ii) If for any two points x; y in Bp (x0 ; r) there exists a point z0 2 Bp (x0 ; r) such that (x; z0 ) (x; z0 ), (y; z0 ) (y; z0 ): (iii) For all z 2 Bp (x0 ; r) such that (z; Sx0 ) p(x0 ; Sx0 ) + p(z; T z)

(z; Sx0 ) implies

p(x0 ; z) + p(Sx0 ; T z):

Then S and T have a unique common …xed point x and p(x ; x ) = 0: Proof. Let y be another point in Bp (x0 ; r) such that y = Sy = T y: Now, p(y; y) (1

k

2t)p(y; y)

= p(Sy; T y) kp(y; y) + tfp(y; T y) + p(y; Sy)g 0:

This implies that, p(y; y) = 0: 6

(2.9)

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

Now if (x ; y)

(x ; y), then, p(x ; y)

(1

k)p(x ; y)

= p(Sx ; T y) kp(x ; y) + t[p(x ; Sx ) + p(y; T y)] 0. (by 2.7 and 2.9)

This shows that x = y: Now if (x ; y) (x ; y), then there exists a point z0 2 Bp (x0 ; r) such that (x ; z0 ) (x ; z0 ) and (y; z0 ) (y; z0 ): Choose a point z1 in X such that z1 = T z0 and z2 = Sz1 . Continuing this process, we construct a sequence zn of points in X such that, z2i+1 = T z2i ; and z2i+2 = Sz2i+1 , wherei = 0; 1; 2; : : : : As T is - dominated mapping then (z0 ; z1 ) (z0 ; z1 ): As S is - dominated mapping then (z1 ; z2 ) (z1 ; z2 ): Continuing in this way we obtain (zn ; zn+1 ) (zn ; zn+1 ) for all n 2 N [ f0g: As S and T are triangle dominated mapping , so (z0 ; z1 ) (z0 ; z1 ) and (z1 ; z2 ) (z1 ; z2 ) implies that (z0 ; z2 ) (z0 ; z2 ): Continuing in this way we obtain (z0 ; zn+1 ) (z0 ; zn+1 ): Now (x ; z0 ) (x ; z0 ) and (z0 ; zn+1 ) (z0 ; zn+1 ) implies that (x ; zn+1 ) (x ; zn+1 ): Also (xn ; x ) (xn ; x ) and (x ; zn+1 ) (x ; zn+1 ) implies that (xn ; zn+1 ) (xn ; zn+1 ): Now we will show that zn 2 Bp (x0 ; r) for all n 2 N . Now, p(Sx0 ; T z0 ) p(Sx0 ; T z0 )

kp(x0 ; z0 ) + t[p(x0 ; x1 ) + p(z0 ; T z0 )] kp(x0 ; z0 ) + t[p(x0 ; z0 ) + p(x1 ; T z0 )]; (by (iii)) p(x0 ; z0 ) and (2.10)

p(x0 ; z1 )

p(x0 ; x1 ) + p(x1 ; z1 ) p(x1 ; x1 ) (1 )[r + p(x0 ; x0 )] + p(x0 ; z0 ); (by 2.2 and 2.10) p(x0 ; z1 ) (1 )[r + p(x0 ; x0 )] + [r + p(x0 ; x0 )] (as z0 2 Bp (x0 ; r)) = r + p(x0 ; x0 );

implies that z1 2 Bp (x0 ; r): Let z2 ; z3 ; :::; zj 2 Bp (x0 ; r) for some j 2 N: If j is even, then, we have, p(x1 ; T zj ) p(x1 ; T zj )

= kp(x0 ; zj ) + t[p(x0 ; x1 ) + p(zj ; T zj )] kp(x0 ; zj ) + t[p(x0 ; zj ) + p(x1 ; T zj )]; (by (iii)) p(x0 ; zj )

[r + p(x0 ; x0 )]: (as zj 2 Bp (x0 ; r)) (2.11)

Now, p(x0 ; T zj ) p(x0 ; zj+1 )

p(x0 ; x1 ) + p(x1 ; T zj ) p(x1 ; x1 ) (1 )[r + p(x0 ; x0 )] + [r + p(x0 ; x0 )]; (by 2.11) r + p(x0 ; x0 ) (2.12) 7

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159

Now, if j is odd then, following similar arguments as we have used to prove inequality (2.5), we have, j

p(zj ; zj+1 )

p(z0 ; z1 ) for some j 2 N:

(2.13)

Now, we have, p(x2 ; zj+1 )

= p(T x1 ; Szj ) kp(x1 ; zj ) + t[p(x1 ; T x1 ) + p(zj ; Szj )] kp(x1 ; zj ) + t[ p(x0 ; x1 ) + p(zj 1 ; T zj 1 )], (by 2.6 and 2.13) p(x2 ; zj+1 ) kp(x1 ; zj ) + t [p(x0 ; zj 1 ) + p(x1 ; zj )], (by (iii))

p(x2 ; zj+1 ) p(x2 ; zj+1 ) p(x2 ; zj+1 )

(k + t )p(x1 ; T zj 1 ) + t [r + p(x0 ; x0 )]; (as zj 1 2 Bp (x0 ; r)) [(k + t ) + t ][r + p(x0 ; x0 )], (by 2.11, as j 1 is even) 2 [r + p(x0 ; x0 )] (2.14)

Now, p(x0 ; zj+1 )

p(x0 ; x1 ) + p(x1 ; x2 ) + p(x2 ; zj+1 ) p(x0 ; x1 ) + p(x0 ; x1 ) + 2 [r + p(x0 ; x0 )], (by 2.6 and 2.14) r + p(x0 ; x0 ): (2.15)

p(x0 ; zj+1 )

Therefore, from inequalities (2.12) and (2.15), zj+1 2 Bp (x0 ; r) in both cases: Hence zn 2 Bp (x0 ; r) for all n 2 N . Thus, inequality (2.13) becomes n

p(zn ; zn+1 )

p(z0 ; z1 )

! 0 as n ! 1:

As (y; z0 ) (y; z0 ) and (z0 ; zn+1 ) (z0 ; zn+1 ) implies that (y; zn+1 ): Also (x ; zn+1 ) (x ; zn+1 ): Then, for i 2 N; p(T x ; Sz2i p(x ; Sz2i

(2.16) (y; zn+1 )

1)

kp(x ; z2i 1 ) + t[p(x ; T x ) + p(z2i 1 ; Sz2i 1 )] = kp(Sx ; T z2i 2 ) + tp(z2i 1 ; z2i ); k 2 p(x ; z2i 2 ) + ktp(z2i 2 ; z2i 1 ) + tp(z2i 1 ; z2i ) 1) .. . k 2i p(x ; z0 ) + k 2i 1 tp(z0 ; z1 ) + +ktp(z2i 2 ; z2i 1 ) + tp(z2i 1 ; z2i ):

On taking limit as i ! 1 and by inequality (2.16), we have, p(x ; Sz2i

1)

= 0:

(2.17)

! 0 as i ! 1:

(2.18)

Similarly, p(Sz2i

1 ; y)

Now by using inequality (2.17) and (2.18), we have p(x ; y)

p(x ; Sz2i

1)

+ p(Sz2i 8

1 ; y)

! 0 as i ! 1:

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

160

So, x = y: Hence x is a unique common …xed point of T and S in Bp (x0 ; r): Example 2.8. Let X = [0; +1) \ Q and p : X X ! R+ be the 0-complete partial metric on X de…ned by p(x; y) = maxfx; yg: De…ne (x; y) = 2x y; (x; y) = x y and 8 8 x 5x > < < if x 2 [0; 1] \ Q if x 2 [0; 1] \ Q 16 17 Sx = and T x = : 1 1 : x > : x if x 2 (1; 1) \ Q if x 2 (1; 1) \ Q 6 7

1 1 Clearly, S and T are - dominated mappings. Take, k = ; t = ; x0 = 5 6 1 1 1 r = ; then Bp (x0 ; r) = [0; 1] \ Q: We have, p(x0 ; x0 ) = maxf ; g = 2 2 2 11 k+t = with = 1 t 25 14 (1 )[r + p(x0 ; x0 )] = 25 and 1 1 1 p(x0 ; Sx0 ) = p( ; ) = < (1 )[r + p(x0 ; x0 )]: 2 32 2 Also if, x = y = 2 2 (1; 1) \ Q; then, p(S2; T 2)

=

maxf2

1 ;2 6

1 ; 2 1 ; 2

1 13 g= 7 7

1 1 maxf2; 2g + [maxf2; 2 5 6

1 g + maxf2; 2 6

1 16 g] = 7 15

So the contractive condition does not hold on X: Now if, x; y

Also, (z; Sx0 )

x 5y ; g 16 17 1 1 x 5y maxfx; yg + [maxfx; g + maxfy; g] 5 6 16 17 = kp(x; y) + t[p(x; Sx) + p(y; T y)] 2

Bp (x0 ; r); then p(Sx; T y) = maxf

(z; Sx0 ) implies p(x0 ; Sx0 ) + p(z; T z)

p(x0 ; z) + p(Sx0 ; T z)

for all z 2 Bp (x0 ; r): Therefore, all the conditions of Theorem 2.7 are satis…ed. Moreover, 0 is the common …xed point of S and T and p(0; 0) = 0: In Theorem 2.7, the condition “If for any sequence fxn g in Bp (x0 ; r) such that (xn ; xn+1 ) (xn ; xn+1 ) for all n 2 N [ f0g and xn ! u 2 Bp (x0 ; r) as n ! +1 then (xn ; u) (xn ; u) for all n 2 N [ f0g”; the condition (i), (ii), (iii) are imposed to restrict the condition (2.1) only for - dominated mapping (x; y) or (y; x) (y; x): and for those x; y in Bp (x0 ; r) for which (x; y) 9

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

However, the following result relax these restrictions but impose the condition (2.1) for all elements in Bp (x0 ; r): Theorem 2.9. Let (X; p) be a 0-complete partial metric space, x0 2 X, r > 0 and S; T : X ! X be two mappings. Suppose for k + 2t 2 [0; 1), the following conditions hold: p(Sx; T y)

kp(x; y) + t[p(x; Sx) + p(y; T y)];

for all x; y in Bp (x0 ; r) and p(x0 ; Sx0 )

(1

)[r + p(x0 ; x0 )];

k+t . Then there exists a unique point x in Bp (x0 ; r) such that 1 t x = Sx = T x and p(x ; x ) = 0: Moreover; S and T have no …xed point other than x : Proof. By following similar arguments of Theorem 2.1, we can obtain a unique point x in Bp (x0 ; r) such that x = Sx = T x : Let y = T y: Then y is the …xed point of T and it will not be a …xed point of S: Now, where

=

p(x ; y) p(x ; y)

= p(Sx ; T y) kp(x ; y) + t[p(x ; Sx ) + p(y; T y)] t p(y; y) p(y; y): (by 2.7) 1 k

A contradiction, as p(y; y) p(x ; y): Hence x = y. Thus T has no …xed point other than x : Similarly S has no …xed point other than x : In Theorem 2.7, the conditions (iii) and (2.2) are imposed to restrict the condition (2.1) only for x; y in Bp (x0 ; r) and Example 2.8 explains the utility of these restrictions. However, the following result relax the conditions (iii) and (2.2) but impose the condition (2.1) for all elements x; y 2 X such that (x; y) (x; y) or (y; x) (y; x). Theorem 2.10. Let (X; p) be a 0-complete partial metric space. Suppose there exist two functions, ; : X X ! [0; +1) such that S and T are triangle dominated mapping. If there exist some k; t such that; k + 2t 2 [0; 1) and the following conditions hold: p(Sx; T y)

kp(x; y) + t[p(x; Sx) + p(y; T y)];

for all x; y 2 X such that (x; y) (x; y) or (y; x) (y; x). If for any sequence fxn g in X such that (xn ; xn+1 ) (xn ; xn+1 ) for all n 2 N [ f0g and xn ! u 2 X as n ! +1 then (xn ; u) (xn ; u) for all n 2 N [ f0g: Also for any two points x; y in X there exists a point z0 2 X such that (x; z0 ) (x; z0 ), (y; z0 ) (y; z0 ): Then there exists a point x in X such that x = Sx = T x and p(x ; x ) = 0:

10

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Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

162

Now we apply our Theorem 2.7 to obtain unique common …xed point of three mappings in closed ball in 0-complete partial metric space. Theorem 2.11. Let (X; p) be a partial metric space, S; T; f : X ! X such that SX [T X f X: Suppose there exist two functions, ; : X X ! [0; +1) such that S T and f are - dominated mapping and (f x; Sx) (f x; Sx) and (f x; T x) (f x; T x): Suppose for k + 2t 2 [0; 1), x0 2 X, r > 0; Bp (f x0 ; r) (f x; f y) implies that f X and for all f x; f y 2 B(f x0 ; r); (f x; f y) p (Sx; T y)

kp(f x; f y) + t[p(f x; Sx) + p(f y; T y)];

(2.19)

and p(f x0 ; Sx0 ) where

=

(1

)[r + p(x0 ; x0 )];

(2.20)

k+t and 1 t p(f x0 ; Sx0 ) + p(f y; T y)

p(f x0 ; f y) + p(Sx0 ; T y);

(2.21)

for all f y 2 Bp (f x0 ; r) such that (f y; Sx0 ) (f y; Sx0 ): If for any sequence fxn g in Bp (f x0 ; r) such that (xn ; xn+1 ) (xn ; xn+1 ) for all n 2 N [ f0g and xn ! u 2 Bp (f x0 ; r) as n ! +1 then (xn ; u) (xn ; u) for all n 2 N [ f0g and for any two points x; y in Bp (f x0 ; r) there exists a point z0 2 Bp (f x0 ; r) such that (x; z0 ) (x; z0 ), (y; z0 ) (y; z0 ): If f X is 0-complete subspace of X and (S; f ) and (T; f ) are weakly compatible, then S; T and f have a unique common …xed point f z in Bp (f x0 ; r). Also p(f z; f z) = 0: Proof. By Lemma 1.3, there exists E X such that f E = f X and f : E ! X is one-to-one. Now since SX [ T X f X; we de…ne two mappings g; h : f E ! f E by g(f x) = Sx and h(f x) = T x respectively. Since f is one-to-one on E, then g; h are well-de…ned. As (f x; Sx) (f x; Sx) implies that (f x; g(f x)) (f x; g(f x)) and (f x; T x) (f x; T x) implies that (f x; h(f x)) (f x; h(f x)) therefore g and h are dominated maps. Now f x0 2 Bp (f x0 ; r) f X; then f x0 2 f X: Let y0 = f x0 ; choose a point y1 in f X such that y1 = h(y0 ): As (y0 ; h(y0 )) (y0 ; h(y0 )) so (y0 ; y1 ) (y0 ; y1 ) and let y2 = g(y1 ). Now (y1 ; g(y1 )) (y1 ; g(y1 )) gives (y1 ; y2 ) (y1 ; y2 ). Continuing this process and having chosen yn in f X such that y2i+1 = h(y2i ) and y2i+2 = g(y2i+1 ); where i = 0; 1; 2; :::; (yn ; yn+1 ) (yn ; yn+1 ): Following similar arguments of Theorem 2.2, yn 2 Bp (f x0 ; r): Also by inequalities (2.20) and (2.21) we obtain, p(f x0 ; g(f x0 )) + p(f y; h(f y))

p(f x0 ; f y) + p(g(f x0 ); h(f y));

for all f y 2 Bp (f x0 ; r) such that (f y; Sx0 ) p(f x0 ; g(f x0 ))

(1 11

(f y; Sx0 ) and

)[r + p(x0 ; x0 )]:

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

By using inequality (2.19), for f x; f y 2 B(f x0 ; r); (f x; f y) that p (g(f x); h(f y))

163

(f x; f y) implies

kp(f x; f y) + t[p(f x; g(f x)) + p(f y; h(f y))]:

As f X is a 0-complete space; all conditions of Theorem 2.7 are satis…ed, we deduce that there exists a unique common …xed point f z 2 Bp (f x0 ; r) of g and h: Also p(f z; f z) = 0: The rest of the proof is similar to the proof given in Theorem 4 [7], so we leave it. Hence we obtain a unique common …xed point of S; T and f . Unique common …xed point results of three and four mappings in 0-complete partial metric space in a closed ball are given below which can be proved with the help of Theorem 2.5, by using the technique given in Theorem 7 [7] Theorem 2.12. Let (X; p) be a partial metric space, x0 2 X, r > 0 and S; T f X: and f are self mappings on X such that SX [ T X f X; Bp (f x0 ; r) Suppose for k + 2t 2 [0; 1), the following conditions hold: p (Sx; T y)

kp(f x; f y) + t[p(f x; Sx) + p(f y; T y)];

for all f x; f y 2 Bp (f x0 ; r) and p(f x0 ; Sx0 )

(1

)[r + p(x0 ; x0 )];

k+t : If f X is 0-complete subspace of X and (S; f ) and (T; f ) are 1 t weakly compatible, then S; T and f have a unique common …xed point f z in Bp (f x0 ; r). Also p(f z; f z) = 0: where

=

Theorem 2.13. Let (X; p) be a partial metric space, x0 2 X, r > 0 and S; T; g and f be self mappings on X such that SX; T X f X = gX and Bp (f x0 ; r) f X: Suppose for k + 2t 2 [0; 1), the following conditions hold: p (Sx; T y)

kp(f x; gy) + t[p(f x; Sx) + p(gy; T y)];

for all f x; f y 2 Bp (f x0 ; r) and p(f x0 ; Sx0 )

(1

)[r + p(x0 ; x0 )];

k+t : If f X is 0-complete subspace of X and (S; f ) and (T; g) are 1 t weakly compatible, then S; T; f and g have a unique common …xed point f z in Bp (f x0 ; r). Also p(f z; f z) = 0: where

=

Corollary 2.14. (Theorem 2.9 of [29]) Let (X; p) be a partial metric space and S; T; g and f be self mappings on X such that SX; T X f X = gX: Assume that for r > 0 and x0 be an arbitrary point in X; the following conditions hold: p (Sx; T y)

kp(f x; gy) 12

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

for all elements f x; gy 2 B(f x0 ; r) p(f x0 ; Sx0 )

f X; 0

(1

k < 1 and

k)[r + p(f x0 ; f x0 )]:

If f X is 0-complete subspace of X then there exists f z 2 X such that p(f z; f z) = 0: Also if (S; f ) and (T; g) are weakly compatible, then S; T; f and g have a unique common …xed point f z in B(f x0 ; r). Further S and T have no …xed point other than x : Corollary 2.15. (Theorem 7 of [7]) Let (X; p) be a partial metric space, x0 ; x; y 2 X, r > 0 and S; T; g and f be self mappings on X such that SX; T X f X = gX and B(f x0 ; r) f X: Assume that the following condition holds: p (Sx; T y)

k[p(f x; Sx) + p(gy; T y)]

for all f x; f y 2 B(f x0 ; r), where 0 p(f x0 ; Sx0 )

k < 1=2; and

(1

)[r + p(f x0 ; f x0 )]

where = 1 k k : If f X is complete subspace of X and (S; f ) and (T; g) are weakly compatible, then S; T; f and g have a unique common …xed point f z in B(f x0 ; r). Also p(f z; f z) = 0: The study of existence of …xed points in partially ordered sets has been initiated by Ran and Reurings [22] with applications to matrix equations. Agarwal et al. [2], Ciric et al. [10] and Nashine et al. [18] presented some new results for nonlinear contractions in partially ordered metric spaces and noted that their theorems can be used to investigate a large class of problems. [1, 3, 8, 9, 19, 21, 24] gave some …xed point theorems in ordered partial metric spaces. Recall that if (X; ) is a preordered set and T : X ! X is such that for x 2 X; with Sx x, then the mapping T is said to be dominated. De…ne the set r by r = f(x; y) 2 X X : x y or y xg: Let (X; ; p) be a preordered 0-complete partial metric space. From Theorem 2.2 to Theorem 2.7, Theorem 2.10 and Theorem 2.11, we derive following important results in preordered 0-complete partial metric spaces. Theorem 2.16. Let (X; ; p) be a preordered 0-complete partial metric space, x0 ; x; y 2 X, r > 0 and S; T : X ! X be two dominated mappings. Suppose for k + 2t 2 [0; 1), the following conditions hold: p(Sx; T y) for all (x; y) in (Bp (x0 ; r)

kp(x; y) + t[p(x; Sx) + p(y; T y)]; Bp (x0 ; r)) \ r and

p(x0 ; Sx0 )

(1

)[r + p(x0 ; x0 )]; 13

164

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

165

k+t . If, for a nonincreasing sequence fxn g in Bp (x0 ; r); fxn g ! u 1 t implies that u xn , then there exists a point x in Bp (x0 ; r) such that x = Sx = T x and p(x ; x ) = 0: Also, x is unique, if for any two points x; y in Bp (x0 ; r) there exists a point z0 2 Bp (x0 ; r) such that z0 x and z0 y and for all z 2 Bp (x0 ; r) such that z Sx0 ; we have where

=

p(x0 ; Sx0 ) + p(z; T z)

p(x0 ; z) + p(Sx0 ; T z)

Corollary 2.17. Let (X; ; p) be a preordered 0-complete partial metric space, x0 2 X and S; T : X ! X be two dominated mappings. Suppose for k + 2t 2 [0; 1), the following conditions hold: p(Sx; T y)

kp(x; y) + t[p(x; Sx) + p(y; T y)];

for all (x; y) in r. If, for a nonincreasing sequence fxn g ! u implies that u xn , then there exists a point x in X such that x = Sx = T x and p(x ; x ) = 0: Also, x is unique, if for any two points x; y in X there exists a point z0 2 X such that z0 x and z0 y. In Theorem 2.13, the condition “for all z 2 Bp (x0 ; r) such that z have p(x0 ; Sx0 ) + p(z; T z) p(x0 ; z) + p(Sx0 ; T z)”

Sx0 ; we

is imposed to obtain unique …xed point of a contractive mapping satisfying conditions (2.1). However, the following result relax this restriction but impose the conditions (2.1) and (2.2) for t = 0: Corollary 2.18. (Theorem 2.2 of [29]) Let (X; ; p) be an ordered 0-complete partial metric space, S; T : X ! X be dominated maps and x0 be an arbitrary point in X. Suppose there exists k 2 [0; 1) with p(Sx; T y)

kp(x; y) for all comparable elements x; y in Bp (x0 ; r) and p(x0 ; Sx0 )

(1

k)[r + p(x0 ; x0 )]:

If for a non-increasing sequence fxn g in Bp (x0 ; r); fxn g ! u implies that u xn : Then there exists x 2 Bp (x0 ; r) such that p(x ; x ) = 0 and x = Sx = T x : Also if, for any two points x; y in Bp (x0 ; r) there exists a point z 2 Bp (x0 ; r) such that z x and z y, then x is a unique common …xed point in Bp (x0 ; r): Corollary 2.19. (Theorem 1 of [7]) Let (X; ; p) be an ordered complete partial metric space, x0 ; x; y 2 X, r > 0 and S; T : X ! X be two dominated mappings. 14

Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

1 Suppose for t 2 [0; ), the following conditions hold: 2 p(Sx; T y) for all (x; y) in (Bp (x0 ; r)

t[p(x; Sx) + p(y; T y)];

Bp (x0 ; r)) \ r and

p(x0 ; Sx0 )

(1

)[r + p(x0 ; x0 )];

t

. If, for a nonincreasing sequence fxn g in Bp (x0 ; r); fxn g ! u 1 t implies that u xn , then there exists a point x in Bp (x0 ; r) such that x = Sx = T x and p(x ; x ) = 0: Also, x is unique, if for any two points x; y in Bp (x0 ; r) there exists a point z0 2 Bp (x0 ; r) such that z0 x and z0 y and where

=

p(x0 ; Sx0 ) + p(z; T z) for all z 2 Bp (x0 ; r) such that z

p(x0 ; z) + p(Sx0 ; T z)

Sx0 :

Theorem 2.20. Let (X; ; p) be a preordered partial metric space, x0 2 X, r > 0 and S; T be self mapping and f be a dominated mapping on X such that SX [ T X f X; T x f x; Sx f x and Bp (f x0 ; r) f X: Suppose for k + 2t 2 [0; 1), the following conditions hold: p (Sx; T y)

kp(f x; f y) + t[p(f x; Sx) + p(f y; T y)];

for all (f x; f y) 2 (Bp (f x0 ; r)

Bp (f x0 ; r)) \ r and

p(f x0 ; Sx0 ) where

=

(1

)[r + p(x0 ; x0 )];

k+t and 1 t p(f x0 ; Sx0 ) + p(f y; T y)

p(f x0 ; f y) + p(Sx0 ; T y);

for all f y 2 Bp (f x0 ; r) such that f y Sx0 : If, for a nonincreasing sequence fxn g in Bp (f x0 ; r); fxn g ! u implies that u xn and for any two points z and x in Bp (f x0 ; r) there exists a point y 2 Bp (f x0 ; r) such that y z and y x: If f X is 0-complete subspace of X and (S; f ) and (T; f ) are weakly compatible, then S; T and f have a unique common …xed point f z in Bp (f x0 ; r). Also p(f z; f z) = 0: Corollary 2.21. (Theorem 2.8 of [29]) Let (X; ; p) be an ordered partial metric space and S; T self mapping and f be a domonited mapping on X such that SX [ T X f X and T x; Sx f x: Assume that for r > 0 and x0 be an arbitrary point in X; the following conditions hold: p (Sx; T y)

kp(f x; f y)

15

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Dominated Mappings and Related Common Fixed Point Results in Closed Ball Abdullah Shoaib

for all comparable elements f x; f y 2 B(f x0 ; r) p(f x0 ; T x0 )

(1

f X; 0

k < 1 and

k)[r + p(f x0 ; f x0 )]:

If for a non-increasing sequence fxn g ! u implies that u xn , also for any two points z and x in B(f x0 ; r) there exists a point y 2 B(f x0 ; r) such that y z and y x that is every pair of elements in B(f x0 ; r) has a lower bound in B(f x0 ; r). If f X is 0-complete subspace of X and (S; f ) and (T; f ) are weakly compatible, then S; T and f have a unique common …xed point f z in B(f x0 ; r). Also p(f z; f z) = 0: Corollary 2.22. (Theorem 4 of [7]) Let (X; ; p) be a ordered partial metric space, x0 ; x; y 2 X, r > 0 and S; T self mapping and f be a dominated mapping on X such that SX [ T X f X; B(f x0 ; r) f X and (T x; f x); (Sx; f x) 2 r: Assume that the following conditions hold: p (Sx; T y) for all (f x; f y) 2 (B(f x0 ; r)

k[p(f x; Sx) + p(f y; T y)] B(f x0 ; r)) \ r; where 0

p(f x0 ; Sx0 ) + p(f y; T y) for all f y 2 B(f x0 ; r) such that f y p(f x0 ; T x0 )

k < 1=2;

p(f x0 ; f y) + p(Sx0 ; T y) Sx0 ;

(1

)[r + p(f x0 ; f x0 )]

where = 1 k k : If, for a nonincreasing sequence fxn g in B(f x0 ; r); fxn g ! u implies that u xn and for any two points z and x in B(f x0 ; r) there exists a point y 2 B(f x0 ; r) such that y z and y x: If f X is complete subspace of X and (S; f ) and (T; f ) are weakly compatible, then S; T and f have a unique common …xed point f z in B(f x0 ; r). Also p(f z; f z) = 0: Theorem 2.23. Let (X; ; p) be a preordered partial metric space, x0 2 X, r > 0 and S; T be self mapping and f be a dominated mapping on X such that SX [ T X f X and T x f x; Sx f x: Suppose for k + 2t 2 [0; 1), the following conditions hold: p (Sx; T y)

kp(f x; f y) + t[p(f x; Sx) + p(f y; T y)];

for all (f x; f y) 2 r: If, for a nonincreasing sequence fxn g in X; fxn g ! u implies that u xn and for any two points z and x in X there exists a point y 2 X such that y z and y x: If f X is 0-complete subspace of X and (S; f ) and (T; f ) are weakly compatible, then S; T and f have a unique common …xed point f z in Bp (f x0 ; r). Also p(f z; f z) = 0: Remark 2.24. We can obtain the preordered complete metric version of all theorems, which are still not present in the literature. Competing interests The author declares that he has no competing interests.

16

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[27] A. Shoaib, M. Arshad and J. Ahmad, Fixed point results of locally cotractive mappings in ordered quasi-partial metric spaces, The Scienti…c World Journal, 2013 (2013), Article ID 194897, 8 pages. [28] A. Shoaib, M. Arshad and M. A. Kutbi, Common …xed points of a pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, J. Comput. Anal. Appl., 17(2014), 255-264. [29] A. Shoaib, M. Arshad and A. Azam, Fixed Points of a pair of Locally Contractive Mappings in Ordered Partial Metric Spaces, Articles in press in Matematiµcki vesnik.

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TABLE OF CONTENTS, JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.’S 1-2, 2015

Inverse Obstacle Scattering With Conductive Boundary Condition for a Coated Dielectric Cylinder, Ahmet Altundag,.….………………………………………………………..11-22 Dynamic Analysis of Some Impulsive Fractional Neural Network with Mixed Delay, Li Yanfang, and Liu Xianghu,…………………………………………………………23-34 Basins of Attraction of Certain Homogeneous Second Order Quadratic Fractional Difference Equation, M. Garić-Demirović, M. R. S. Kulenović, and M. Nurkanović,……………35-50 Aspects of Univalent Holomorphic Functions Involving Sălăgean Operator and Ruscheweyh Derivative, Alb Lupaș Alina,…………………………………………………………..51-59 Properties on a Subclass of Univalent Functions Defined by Using a Generalized Sălăgean Operator and Ruscheweyh Derivative, Alina Alb Lupaș, and Loriana Andrei,……….60-68 On PPF Dependent Fixed Point Theorems and Applications, A. Farajzadeh, Salahuddin, and B. S. Lee,………………………………………………………………………………..69-75 Spline Right Fractional Monotone Approximation Involving Right Fractional Differential Operators, George A. Anastassiou,……………………………………………………...76-84 On Certain Differential Sandwich Theorems Involving an Extended Generalized Sălăgean Operator and Extended Ruscheweyh Operator, Andrei Loriana,……………………….85-95 Hermite-Hadamard Type Inequalities for (ℎ − (𝛼, 𝑚)) - Convex Functions, M. E. Ozdemir, Havva Kavurmaci Önalan, and Merve Avci Ardiç,……………………………………..96-107 Voronovskaya Type Asymptotic Expansions for Generalized Discrete Singular Operators, George A. Anastassiou, and Merve Kester,……………………………………………..108-119 Simple AQ and Simple CQ Functional Equations, M. Arunkumar, C. Devi Shyamala Mary, and G. Shobana,………………………………………………………………………………120-151 𝛼 − 𝜂 Dominated Mappings and Related Common Fixed Point Results in Closed Ball, Abdullah Shoaib,……………………………………………………………………………………152-170

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Editor in -Chief: George Anastassiou Department of Mathematical Sciences The University Of Memphis Memphis,TN 38152,USA tel.901-678-3144,fax 901-678-2480 e-mail [email protected] www.msci.memphis.edu/~ganastss Areas:Approximation Theory, Probability,Moments,Wavelet, Neural Networks,Inequalities,Fuzzyness. Associate Editors: 1)Ravi P. Agarwal Chairman Department of Mathematics Texas A&M University - Kingsville 700 University Blvd. Kingsville, TX 78363-8202 Office: 361-593-2600

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J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 3-4,183-197 ,2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 183

QUALITATIVE ANALYSIS OF A SOLOW MODEL ON TIME SCALES MARTIN BOHNER, JULIUS HEIM AND AILIAN LIU Abstract. In this paper, we further analyze the Solow model on time scales. This model was recently introduced by the authors and it combines the continuous and discrete Solow models and extends them to different time scales. Assuming constant labor force growth in the Solow model, we establish a comparison theorem. Then, under the more realistic assumption that the labor force growth rate is a monotonically decreasing function, we discuss the comparison theorem as well as stability and monotonicity of the solutions of the Solow model. The economic meanings are also indicated in some remarks.

1. Introduction The neoclassical growth model, developed by Solow [17] and Swan [18], had a great impact on how the economists think about economic growth. Since then, it has stimulated an enormous amount of work [2, 12, 20]. Since differential equation systems are usually more easily handled than difference systems from the analytical point of view, some of the economic models have used continuous timing [1, 9, 13–15] while others are given in difference models because some people think economic data are collected at discrete intervals and transformation of capital into investments depends on the length of time lag, etc. [8, 10, 21]. Hence, in economic modeling, either continuous timing or discrete timing is present, and there is not a common view among economists on which representation of time is better for economic models [14]. Meanwhile, many results concerning differential equations may carry over quite easily to corresponding results for difference equations, while other results seems to be completely different in nature from their continuous counterparts [6]. 2000 Mathematics Subject Classification. 91B62, 34C10, 39A10, 39A11, 39A12, 39A13. Key words and phrases. Solow model, time scales, labor growth rate, comparison of solutions, stability. 1

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The blanket assumption that economic processes are either solely continuous or solely discrete, while convenient for traditional mathematical approaches, may sometimes be inappropriate, because in reality many economic phenomena do feature both continuous and discrete elements. In biology, a familiar example is a “seasonal breeding population in which generations do not overlap” [6, 19]. A similar typical example in economics is the “seasonally changing investment and revenue in which seasons play an important effect on this kind of economic activity”. In addition, option pricing and stock dynamics in finance [11] and the frequency and duration of market trading in economics [19] also contain this hybrid continuous-discrete processes. Therefore, there is a great need to find a more flexible mathematical framework to accurately model the dynamical blend of such systems, so that they are precisely described and better understood. To meet this requirement, an emerging, progressive and modern area of mathematics, known as “dynamic equations on time scales”, has been introduced. This calculus has the capacity to act as the framework to effectively describe the above phenomena and to make advances in their associate fields, see e.g., [3–5, 19]. This theory was introduced by Stefan Hilger in 1988 in his Ph.D. thesis [16] in order to unify continuous and discrete analysis, and has been developed by many mathematicians. A time scale T is defined as any nonempty closed subset of R. In the time scales setting, once a result is established, special cases include the result for the differential equation when the time scale is the set of all real numbers R and the result for the difference equation when the time scale is the set of all integers Z. The induction principle plays an important rˆole in the proofs of some of our results, so we give it here. Theorem 1.1 (See [6, Theorem 1.7]). Let t0 ∈ T and assume that {S(t) : t ∈ [t0 , ∞)} is a family of statements satisfying: A. The statement S(t0 ) is true. B. If t ∈ [t0 , ∞) is right-scattered and S(t) is true, then S(σ(t)) is also true. C. If t ∈ [t0 , ∞) is right-dense and S(t) is true, then there is a neighborhood U of t such that S(r) is true for all r ∈ U ∩ (t, ∞). D. If t ∈ [t0 , ∞) is left-dense and S(r) is true for all r ∈ [t0 , t), then S(σ(t)) is true. Then S(t) is true for all t ∈ [t0 , ∞).

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3

For other notations and a systematic introduction to time scales theory, we refer the reader to [6, 7]. 2. The Solow Model on Time Scales In this section, we will first recall some elements of the Solow model on general time scales as introduced by the authors in [5]. In the original Solow model [1, 17], the key elements are the production function, i.e., how the inputs of capital K and labor L are transformed into outputs, and how capital and labor force change over time. Here we still assume the following: 1. The production function F satisfies: (a) F (λK, λL) = λF (K, L) for any λ, K, L ∈ R+ (constant return to scales); (b) F (K, 0) = F (0, L) = 0 for any K, L ∈ R+ ; ∂F ∂F ∂ 2F ∂ 2F (c) > 0, > 0, < 0, < 0; ∂K ∂L ∂K 2 ∂L2 ∂F ∂F ∂F ∂F (d) lim+ = lim+ = ∞, lim = lim = 0 (Inada K→∞ ∂K L→∞ ∂L K→0 ∂K L→0 ∂L conditions). 2. The capital stock changes are equal to the gross investment I = sF (K, L) minus the capital depreciation δK, where s and δ are the savings rate and the depreciation factor of goods, respectively. 3. The labor force L changes at a constant rate n. The three assumptions give, for any t ∈ T  Y (t) = F (K(t), L(t)),    K ∆ (t) = I(t) − δK(t), (2.1)  I(t) = sY (t),    ∆ L (t) = nL(t). From (2.1), we obtain (2.2)

K ∆ (t) = sY (t) − δK(t) = sF (K(t), L(t)) − δK(t).

Define K(t) Y (t) and y(t) := , L(t) L(t) which are regarded as the capital stock per worker and the production per worker, respectively. Let   K f (k) := F , 1 = F (k, 1) L k(t) :=

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be the production function in intensive form. Then condition 1 changes to   f (0) = 0; f 0 (k) > 0 and f 00 (k) < 0 for all k ∈ R+ (2.3)   lim f 0 (k) = ∞ and lim f 0 (k) = 0. + k→∞

k→0

Applying the time scales quotient rule [6, Theorem 1.20], we use (2.2) to find  ∆ K K ∆ (t)L(t) − K(t)L∆ (t) ∆ (t) = k (t) = L L(t)Lσ (t) K ∆ (t) K(t)L∆ (t) = σ − L (t) L(t)Lσ (t) K(t)n K ∆ (t) − = L(t)(1 + µ(t)n) L(t)(1 + µ(t)n) sF (K(t), L(t)) − δK(t) K(t)n = − L(t)(1 + µ(t)n) L(t)(1 + µ(t)n) s δ+n = f (k(t)) − k(t), 1 + µ(t)n 1 + µ(t)n i.e., (2.4)

k ∆ (t) =

δ+n s f (k(t)) − k(t), 1 + µ(t)n 1 + µ(t)n

which describes the Solow model on time scales. When T = R, equation (2.4) is the continuous Solow model in [1], whereas when T = Z, equation (2.4) is the discrete Solow model discussed in [10]. Equation (2.4) has a nontrivial equilibrium, denoted by kˆn , which is the unique positive solution of the equation sf (k) = (δ + n)k. For n = 0, we denote by kˆ0 the nontrivial steady state of equation (2.4). Obviously, we have lim kˆn = kˆ0 ,

n→0+

and kˆn increases to kˆ0 as n decreases to zero. Next we will discuss some sufficient conditions for the existence and uniqueness of solutions of initial value problems for equation (2.4). Some comparison theorems between two solutions with different initial conditions will be given.

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Theorem 2.1. Assume (2.3). For t0 ∈ T and k0 ∈ R+ , the initial value problem  s δ+n k ∆ (t) = f (k(t)) − k(t), 1 + µ(t)n 1 + µ(t)n (2.5)  k(t0 ) = k0 , has a unique solution on T+ t0 = {t ∈ T : t ≥ t0 }. Proof. Let s δ+n f (k) − k. 1 + µ(t)n 1 + µ(t)n Then u(·, k) is rd-continuous and regressive on T, and ∂u |u(t, k1 ) − u(t, k2 )| = (t, ξ) |k1 − k2 | ∂k δ + n s 0 |k1 − k2 | = f (ξ) − 1 + µ(t)n 1 + µ(t)n ≤ (sf 0 (t0 ) + δ + n)|k1 − k2 |, u(t, k) =

where ξ ∈ (k1 , k2 ). With the theorem of global existence and uniqueness in [6, Theorem 8.20], we can deduce that the solution of the problem (2.5) exists uniquely.  Hence in the following, with condition (2.3), we always have the existence and uniqueness of solutions for initial value problems (2.5). Theorem 2.2. Assume (2.3) and let δ > 0 be such that −δ ∈ R+ . Let k1 and k2 be solutions of equation (2.4) on T+ t0 with initial conditions k1 (t0 ) = k01 and k2 (t0 ) = k02 , respectively. If 0 < k01 < k02 , then k1 < k2

on

T+ t0 .

Proof. We use the induction principle Theorem 1.1. A. If t = t0 , then the result is obvious from the hypothesis. B. If t ∈ T+ t0 is right-scattered and k1 (t) < k2 (t), then k1 (σ(t)) − k2 (σ(t)) = k1 (t) − k2 (t) + µ(t)(k1∆ (t) − k2∆ (t)) µ(t) = k1 (t) − k2 (t) + [sf (k1 (t)) − (δ + n)k1 (t)] 1 + µ(t)n µ(t) − [sf (k2 (t)) − (δ + n)k2 (t)] 1 + µ(t)n sµ(t) = k1 (t) − k2 (t) + [f (k1 (t)) − f (k2 (t))] 1 + µ(t)n

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(δ + n)µ(t) [k1 (t) − k2 (t)] 1 + µ(t)n 1 − µ(t)δ µ(t)s = [k1 (t) − k2 (t)] + [f (k1 (t)) − f (k2 (t))] 1 + µ(t)n 1 + µ(t)n < 0, −

so k1 (σ(t)) < k2 (σ(t)). C. If t is right-dense and k1 (t) < k2 (t), then there exists a right neigh˚+ (t)∩T of t such that k1 (r) < k2 (r) for any r ∈ U ˚+ (t)∩T. borhood U For if such a neighborhood does not exist, then there must exist a ˚+ (t) ∩ T and lim tn = t, such that decreasing series {tn } ⊂ U n→∞

k1 (tn ) ≥ k2 (tn ) and taking limit on both sides, we obtain k1 (t) ≥ k2 (t), a contradiction. D. If t is left-dense and k1 (r) < k2 (r) for any r ∈ [t0 , t) ∩ T, then from the continuity of the solutions, k1 (t) ≤ k2 (t). Uniqueness of solutions of initial value problems yields k1 (t) < k2 (t). Now an application of Theorem 1.1 concludes the proof.  Note that Theorem 2.2 implies that the solution for the initial value problem (2.5) is always positive provided k(t0 ) > 0. Corollary 2.3. Assume (2.3) and let δ > 0 be such that −δ ∈ R+ . Then all solutions of equation (2.4) converge to the nontrivial steady state kˆn monotonically, and the equilibrium point kˆn is asymptotically stable and hence is a global attractor. Proof. If k(t0 ) < kˆn , then Theorem 2.2 implies that k(t) < kˆn

for any t ∈ T+ t0 .

Hence k ∆ (t) =

s δ+n f (k(t)) − k(t) > 0 for any t ∈ T+ t0 . 1 + µ(t)n 1 + µ(t)n

This means that k is increasing to the equilibrium point. Similar arguments apply to the case with k(t0 ) > kˆn , which concludes the proof.  Remark 2.4. Corollary 2.3 means that for two countries or districts with constant population growth rates, the one with the smaller population growth rate has a bigger capital per worker in the long run.

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3. Improved Solow Model on Time Scales In Section 2, we assumed that the labor force L grows at a constant rate n on the time scale, i.e., L∆ (t) = nL(t),

(3.1)

which implies that the labor force grows exponentially, that is, L(t) = L0 en (t, t0 ), where L0 is the initial labor level at t0 ∈ T. With the properties of the exponential function on time scales and the fact that n > 0, we have lim L(t) = ∞.

t→∞

This means the labor force approaches infinity when t goes to infinity, which is unrealistic, because in reality the environment has a carrying capacity. So the simple growth model of labor in equation (3.1) can provide an adequate approximation to such growth only for an initial period, but does not accommodate growth reductions due to competition for environmental resources such as food, habitat and the policy factor etc. [1]. Since the 1950s, developing countries have recognized that the high population growth rate has seriously hampered the economic growth and adopted the population control policy. As a result, the population growth rates of many countries decreased fast in the last 40 years, such as in China. Also due to the aging of the population and, consequently, a dramatic increase in the number of deaths, the population growth rate decreased below zero in some developed countries, and is projected to decrease to zero during the next few decades in the developing countries [1]. So to incorporate the numerical upper bound on the growth size, on the reference of [10], we revise Condition 3 from Section 2 as follows. 3.0 The labor force L satisfies the following properties: (a) The population is strictly increasing and bounded, i.e., L > 0, L∆ > 0 on T+ t0

and

lim L(t) = L∞ < ∞.

t→∞

(b) The population growth rate is decreasing to 0, i.e., L∆ , then lim n(t) = 0 and n∆ < 0 on T+ t0 . t→∞ L Hence equation (2.4) takes the form If n =

(3.2)

k ∆ (t) =

s δ + n(t) f (k(t)) − k(t). 1 + µ(t)n(t) 1 + µ(t)n(t)

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Note that this is a nonautonomous dynamic equation on a time scale. Next we give the theorem of existence and uniqueness for solutions of initial value problems for (3.2). Theorem 3.1. Assume (2.3). For t0 ∈ T and k0 ∈ R+ , the initial value problem  s δ + n(t) k ∆ (t) = f (k(t)) − k(t) 1 + µ(t)n(t) 1 + µ(t)n(t) (3.3)  k(t0 ) = k0 , has a unique solution on T+ t0 . Proof. Following the same way as in the proof of Theorem 2.1, we let u(t, k) =

δ + n(t) s f (k) − k. 1 + µ(t)n(t) 1 + µ(t)n(t)

So u(·, k) is rd-continuous and regressive, and ∂u (t, ξ) ≤ sf 0 (k0 ) + δ + n(t0 ). ∂k From the theorem of global existence and uniqueness in [6], the solution of the problem (3.3) exists uniquely.  Theorem 3.2. Assume (2.3) and let δ > 0 be such that −δ ∈ R+ . Let k1 and k2 be solutions of equation (3.2) with initial conditions k1 (t0 ) = k01 and k2 (t0 ) = k02 , respectively. If 0 < k01 < k02 , then k1 < k2

on

T+ t0 .

Proof. The proof is similar to the proof of Theorem 2.2.



Remark 3.3. Theorem 3.2 means that if two economies have the same fundamentals, then the one with the bigger initial capital per worker will always have the bigger capital per worker for ever on any time scale. The result in Theorem 3.2 includes the results in [1] and [10] as special cases. Theorem 3.4. Assume (2.3) and let δ > 0 be such that −δ ∈ R+ . Let k1 and k2 be solutions of the dynamic equations on the same time scale (3.4) k ∆ (t) =

s δ + n1 (t) f (k(t)) − k(t) =: u(k(t), t) 1 + µ(t)n1 (t) 1 + µ(t)n1 (t)

and (3.5) k ∆ (t) =

s δ + n2 (t) f (k(t)) − k(t) =: v(k(t), t), 1 + µ(t)n2 (t) 1 + µ(t)n2 (t)

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respectively, with the same initial condition k1 (t0 ) = k2 (t0 ). If n1 < n2 on T+ t0 , then k1 ≥ k2

on

T+ t0 .

Proof. From n1 (t) < n2 (t), we have u(k(t), t) > v(k(t), t) for all t ∈ T+ t0 . Let z := k1 − k2 . Obviously, we have z(t0 ) = k1 (t0 ) − k2 (t0 ) = 0 and z ∆ (t0 ) = k1∆ (t0 ) − k2∆ (t0 ) = u(k1 (t0 ), t0 ) − v(k2 (t0 ), t0 ) > 0. So z is right-increasing at t0 , i.e., if t0 is right-scattered, then we have z(σ(t0 )) > z(t0 ) = 0; if t0 is right-dense, then there exists a nonempty ˚+ (t0 )∩T of t0 such that z(t) > 0 for any t ∈ U ˚+ (t0 )∩T. neighborhood U + We now show that z ≥ 0 holds on Tt0 . If this is not the case, then there must be a point t1 > t0 , t1 ∈ T such that z(t1 ) < 0 and z(t) ≥ 0 when t ∈ (t0 , t1 ) ∩ T. If t1 is left-dense, then continuity of z gives that z(t1 ) ≥ 0, which contradicts the assumption. Hence t1 is left-scattered. Let ρ(t1 ) = t2 . Then z(t2 ) ≥ 0, i.e., k1 (t2 ) ≥ k2 (t2 ). Let k20 be the solution of equation (3.5) satisfying the initial condition k20 (t2 ) = k1 (t2 ). From the discussion in the beginning of this proof, we obtain that k1 −k20 is also right-increasing at t2 , i.e., (3.6)

˚+ (t2 ) ∩ T, k1 (t) > k20 (t) for t ∈ U

˚+ (t2 ) ∩ T is a nonempty right neighborhood of t2 (at least where U including t1 ). Taking into account that k2 (t2 ) ≤ k20 (t2 ), Theorem 3.2 gives (3.7)

k2 (t) ≤ k20 (t) for all t ∈ T+ t2 .

From (3.6) and (3.7), we have ˚+ (t2 ) ∩ T, k1 (t) > k2 (t) for t ∈ U and thus k1 (t1 ) > k2 (t1 ), which contradicts the fact z(t1 ) < 0. This concludes the proof.  Remark 3.5. Theorem 3.4 implies that, on any economic domain, for two economies with the same initial capital per worker, the economy with the smaller population growth rate will always have the bigger capital per worker on any time scale. The result here also includes the results in [1, 13] and [10] as special cases. Theorem 3.6. Assume (2.3) and let δ > 0 be such that −δ ∈ R+ . If k solves (3.2), then lim k(t) = kˆ0 . t→∞

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Proof. We want to prove that for any ε > 0, there exists T > 0, such that if t > T , t ∈ T, we have |k(t) − kˆ0 | < ε. Now let ε > 0. Since lim kˆn = kˆ0 ,

n→0+

we know that there exists n > 0 such that ε |kˆn − kˆ0 | < for all n ∈ (0, n). 3 Let t1 ∈ T+ t0 such that nt1 = n(t1 ) < n, i and let knt1 and k0 be the solutions of δ + nt1 s f (k(t)) − k(t) k ∆ (t) = 1 + µ(t)nt1 1 + µ(t)nt1 and k ∆ (t) = sf (k(t)) − δk(t), respectively, with the initial conditions knt1 (t1 ) = k0 (t1 ) = k(t1 ). Then Theorem 3.4 implies that knt1 (t) ≤ k(t) ≤ k0 (t) for all t ∈ T+ t1 . Since lim k0 (t) = kˆ0 , there exists T1 > 0 such that t→∞

ε |k0 (t) − kˆ0 | < for all t > T1 . 3 Moreover, since lim knt1 (t) = kˆnt1 , there exists T2 > 0 such that t→∞

ε |knt1 (t) − kˆnt1 | < for all t > T2 . 3 Hence for t > T := max{T1 , T2 , t1 }, we have 2 ε ε kˆ0 − ε < kˆnt1 − < knt1 (t) ≤ k(t) ≤ k0 (t) < kˆ0 + , 3 3 3 + which implies that |k(t) − kˆ0 | < ε for any t ∈ T . T



Remark 3.7. Theorem 3.6 says that for any economic domain T, the population growth rate n(t) has no influence on the level of per worker output in the long run. That is, provided that the economy possesses a population growth rate strictly decreasing to zero, the capital per worker always converges to the positive steady state of the Solow model on a time scale with a population growth rate of zero. Theorem 3.8. Assume (2.3) and let δ > 0 be such that −δ ∈ R+ . Then the solution k of (3.2) with k(t0 ) = k0 is asymptotically stable.

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Proof. To prove the Lyapunov stability of k in equation (3.2) with initial condition k(t0 ) = k0 , we have to show that for any ε > 0, there exists η > 0 such that for any solution q of equation (3.2) with initial condition q(t0 ) = q0 and such that |k(t0 ) − q(t0 )| < η, we have |k(t) − q(t)| < ε for any t ∈ T+ t0 . Let ϕ1 and ϕ2 be the solutions of equation (3.2) with initial conditions 3 1 ϕ1 (t0 ) = k(t0 ) and ϕ2 (t0 ) = k(t0 ), respectively. From Theorem 3.6, 2 2 we have lim ϕ1 (t) = lim ϕ2 (t) = kˆ0 = lim k(t). t→∞

t→∞

t→∞ T+ t0 ,

Thus, for any ε > 0, there exists t1 > t0 , t1 ∈ such that ε ε |ϕ1 (t) − k(t)| < and |ϕ2 (t) − k(t)| < for all t ∈ T+ t1 . 2 2   3 1 k(t0 ), k(t0 ) . From Let q solve (3.2) with the initial condition q0 ∈ 2 2 Theorem 3.2, we have ϕ1 (t) < q(t) < ϕ2 (t) for any t ∈ T+ t0 . Thus |q(t) − k(t)| < ε for any t ∈ T+ t1 . Next we choose η such that for any solution q with initial value q0 , |q0 − k0 | < η implies |k − q| < ε on [t0 , t1 ] ∩ T. Following the proof of the theorem of continuous dependence on initial conditions, making use of the finite covering theorem, we can obtain that for any ε > 0, there exists η < k0 /2 such that |q0 − k0 | < η implies |k(t) − q(t)| < ε for all t ∈ [t0 , t1 ] ∩ T. From Theorem 3.6, for any solutions k and q of equation (3.2), we have that lim k(t) = lim q(t) = kˆ0 ,

t→∞

t→∞

and then lim |q(t) − k(t)| = 0.

t→∞

So the solution of equation (3.2) is asymptotically stable.



Remark 3.9. Theorem 3.8 says that under the same fundamentals, if two economies operating on the same time domain have nearly the same initial capital per worker, the following capitals per worker will take on similar behavior. Next we will present the monotonicity of the solutions of (3.2).

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Theorem 3.10. Assume (2.3) and let δ > 0 be such that −δ ∈ R+ . Let t0 ∈ T and k, knt0 , k0 be solutions of the dynamic equation (3.2), k ∆ (t) =

(3.8)

s δ + nt0 f (k(t)) − k(t), 1 + µ(t0 )nt0 1 + µ(t0 )nt0

and k ∆ (t) = sf (k(t)) − δk(t),

(3.9)

respectively, with the initial values k(t0 ) = knt0 (t0 ) = k0 (t0 ). Then 1. knt0 ≤ k ≤ k0 on T+ t0 ; 2. If k(t0 ) ≤ kˆn0 , then k is strictly increasing on T+ t0 ; 3. If kˆn0 < k(t0 ) ≤ kˆ0 , then there exists e t ∈ T such that k is decreasing on [t0 , e t] ∩ T and is increasing on Te+ ; t e 4. If kˆ0 < k(t0 ), then k is increasing on T+ t0 , or there exists t ∈ T such that k is decreasing on [t0 , e t] ∩ T and is increasing on Te+ . t Here kˆn and kˆ0 are the equilibria of (3.8) and (3.9), respectively. 0

Proof. 1. For t > t0 , t ∈ T, we have n(t0 ) > n(t) > 0. So from Theorem 3.4, we obtain the result easily. 2. We want to prove the statement S(t) given by k ∆ (t) > 0 is true for any t ∈ T+ t0 . To do this, we use Theorem 1.1. A. Since k(t0 ) < kˆn0 , we have k ∆ (t0 ) > 0. So S(t) holds at t = t0 . B. If t is right-scattered and k ∆ (t) > 0, then k(σ(t)) = k(t) + µ(t)k ∆ (t) sf (k(t)) − (δ + n(t))k(t) = k(t) + µ(t) 1 + µ(t)n(t) (1 − µ(t)δ)k(t) + sµ(t)f (k(t)) = 1 + µ(t)n(t) (1 − µ(t)δ)k(σ(t)) + sµ(t)f (k(σ(t))) < 1 + µ(t)n(σ(t)) [1 + µ(t)n(σ(t))]k(σ(t)) = 1 + µ(t)n(σ(t)) µ(t)[sf (k(σ(t))) − (δ + n(σ(t)))k(σ(t))] + 1 + µ(t)n(σ(t)) µ(t) = k(σ(t)) + [1 + µ(σ(t))n(σ(t))]k ∆ (σ(t)), 1 + µ(t)n(σ(t))

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so k ∆ (σ(t)) > 0. C. If t is right-dense and k ∆ (t) > 0, then there exists a neighbor˚+ (t) ∩ T such that k ∆ (r) > 0 for any r ∈ U ˚+ (t) ∩ T. hood U To prove this, we assume that there does not exist such a neighborhood. Then there must exist a decreasing sequence ˚+ (t) ∩ T such that lim tn = t and k ∆ (tn ) ≤ 0. {tn } ⊂ U n→∞ From the properties of f , taking limit on both sides, we obtain k ∆ (t) ≤ 0, which is a contradiction. D. Assume that t is left-dense and k ∆ (r) > 0 for any r ∈ [t0 , t)∩T. From continuity, we can get k ∆ (t) ≥ 0. If k ∆ (t) = 0, then for any r ∈ [t0 , t) ∩ T, from the chain rule in [6], we have [(1 + µn)k ∆ ]∆ (r) = [s(f ◦ k) − (δ + n)k]∆ (r) = sf 0 (k(r))k ∆ (r) − n∆ (r)k(r) −(δ + nσ (r))k ∆ (r). Taking limit on both sides when r → t− , we obtain [(1 + µn)k ∆ )]∆ (t) = −n∆ (t)k(t) > 0. So since t is left-dense and from the continuity, we have (1 + µ(t)n(t))k ∆ (t) > (1 + µ(r)n(r))k ∆ (r) > 0 ˚− (t) ∩ T. Hence k ∆ (t) > 0. for all r ∈ U 3. If kˆn0 < k(t0 ) ≤ kˆ0 , then s δ + n(t0 ) f (k(t0 )) − k(t0 ) 1 + µ(t0 )n(t0 ) 1 + µ(t0 )n(t0 ) s δ + n(t0 ) = f (knt0 (t0 )) − kn (t0 ) 1 + µ(t0 )n(t0 ) 1 + µ(t0 )n(t0 ) t0 = kn∆t0 (t0 ) < 0.

k ∆ (t0 ) =

Hence k is right-decreasing at t0 , i.e., if t0 is right-scattered, then k(σ(t0 )) < k(t0 ); if t0 is right-dense, then there exists a nonempty ˚+ (t0 ) ∩ T of t0 such that k(t) < k(t0 ) for any t ∈ neighborhood U + ˚+ (t0 ) ∩ T. If k ∆ ≤ 0 is true on T+ U t0 , then k is decreasing on Tt0 . Considering lim k(t) = kˆ0 in Theorem 3.6, we have t→∞

kˆ0 ≤ k(t) < k(t0 ) ≤ kˆ0

for t ∈ T+ t0 ,

which is a contradiction. So there must exist e t ∈ T+ t0 such that ∆ e e k (t) > 0, and for simplicity we assume t is the first point that verifies the inequality. So it must be proved that k ∆ (t) > 0 for all t ∈ Te+ , which is similar to the proof of Statement 2. t

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4. Following the same proof as in Statement 3, we can obtain the monotonicity. This completes the proof.  Acknowledgements This work was financially supported by Grants ZR2012AQ024 and ZR2010AL014 from NSF of Shandong Province. References [1] Elvio Accinelli and Juan Gabriel Brida. Population growth and the Solow-Swan model. Int. J. Ecol. Econ. Stat., 8(S07):54–63, 2007. [2] Robert J. Barro and Xavier Sala-i Martin. Economic Growth. McGraw-Hill, Inc., 1994. [3] Martin Bohner and Gregory Gelles. Risk aversion and risk vulnerability in the continuous and discrete case. A unified treatment with extensions. Decis. Econ. Finance, 35:1–28, 2012. [4] Martin Bohner, Gregory Gelles, and Julius Heim. Multiplier-accelerator models on time scales. Int. J. Stat. Econ., 4(S10):1–12, 2010. [5] Martin Bohner, Julius Heim, and Ailian Liu. Solow models on time scales. Cubo, 15(1):13–32, 2013. [6] Martin Bohner and Allan Peterson. Dynamic equations on time scales: an introduction with applications. Birkh¨auser Boston Inc., Boston, MA, 2001. [7] Martin Bohner and Allan Peterson. Advances in dynamic equations on time scales. Birkh¨ auser Boston Inc., Boston, MA, 2003. [8] Juan Gabriel Brida. Poblaci´on y crecimiento econ´omico. Una versi´o mejorada del modelo de Solow. El Trimestre Econ´ omico, LXXV:7–24, 2008. [9] Juan Gabriel Brida and Erick Jos´e Limas Maldonado. Closed form solutions to a generalization of the Solow growth model. Appl. Math. Sci. (Ruse), 1(3740):1991–2000, 2007. [10] Juan Gabriel Brida and Juan Sebasti´an Pereyra. The Solow model in discrete time and decreasing population growth rate. Economics Bulletin, 3(41):1–14, 2008. [11] D. Brigo and F. Mercurio. Discrete time vs continuous time stock-price dynamics and implications for option pricing. Finance Stoch., 4:147–159, 2000. [12] E. Burmeister and A. R. Dobell. Mathematical theories of economic growth. The Macmillan Company, New York, 1970. [13] Donghan Cai. An improved Solow-Swan model. Chinese Quart. J. Math., 13(2):72–78, 1998. [14] Giancarlo Gandolfo. Economic dynamics. North-Holland Publishing Co., 1997. [15] Luca Guerrini. The Solow-Swan model with a bounded population growth rate. J. Math. Econom., 42(1):14–21, 2006. [16] Stefan Hilger. Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math., 18(1-2):18–56, 1990. [17] Robert M. Solow. A contribution to the theory of economic growth. The Quarterly Journal of Economics, 70(1):65–94, February 1956. [18] T. W. Swan. Economic growth and capital accumulation. Economic Record, 32(2):334–361, 1956.

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[19] Christopher C. Tisdell and Atiya Zaidi. Basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling. Nonlinear Anal., 68(11):3504–3524, 2008. [20] Henry Y. Wan Jr. Economic Growth. Harcourt Brace Jovanivich., New York, 1971. [21] Wei-Bin Zhang. A discrete economic growth model with endogenous labor. Discrete Dyn. Nat. Soc., (2):101–109, 2005. Missouri University of Science and Technology, Department of Mathematics and Statistics, Rolla, MO 65409-0020, USA E-mail address: [email protected], [email protected] Shandong University of Finance and Economics, School of Mathematics and Quantitative Economics, Jinan 250014, China E-mail address: [email protected]

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 3-4,198-208 ,2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 198

SINGULARITIES OF THE CONTINUOUS WAVELET TRANSFORM IN Lp (Rn ) JAIME NAVARRO AND SALVADOR ARELLANO-BALDERAS

Jaime Navarro Universidad Aut´onoma Metropolitana Departamento de Ciencias B´asicas P. O. Box 16-306, M´exico City, 02000 M´exico [email protected] Salvador Arellano-Balderas Universidad Aut´onoma Metropolitana Departamento de Ciencias B´asicas P. O. Box 16-306, M´exico City, 02000 M´exico [email protected] Abstract. The continuous wavelet transform for functions f in Lp (Rn ) where 1 ≤ p < ∞ is defined with respect to a radially symmetric admissible function so that the singularities of f are the singularities of the continuous wavelet transform. Key words and phrases: The space Lp (Rn ), the continuous wavelet transform, inversion formula, admissibility condition. 2010 AMS classification:

42A38, 44A05, 46F10

1. Introduction The local regularity of functions f in the Hilbert space L2 (Rn ) by means of the continuous wavelet transform has been studied in [3], where the main point of this result is the use of the inversion formula given for functions in L2 (Rn ), and where the continuous wavelet transform is defined with respect to a radially symmetric admissible function h in C0∞ (Rn ). 1

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In this paper we extend this local property for functions f in the space Lp (Rn ) with 1 ≤ p < ∞. In this case we will use the inversion formula for the continuous wavelet transform for functions in Lp (Rn ) given in [5]. For this purpose, we will consider two symmetric functions h1 and h2 instead of one h, one for the decomposition and the other one for the inversion formula, in such a way that the admissibility condition will depend of h1 and h2 . 2. Notations and definitions In this section we will give the definition of and admissible function in order to define the continuous wavelet transform for a function in the space Lp (Rn ). Definition 1. For h in L2 (Rn ), the dilation operator Ja : L2 (Rn ) → L2 (Rn ) and the translation operator Tb : L2 (Rn ) → L2 (Rn ) are defined respectively as: n

1) (Ja h)(x) = a− 2 h(a−1 x), where a > 0 and x ∈ Rn . 2) (Tb h)(x) = h(x − b), where x, b ∈ Rn . Now, the admissibility condition is given. Definition 2. A radially symmetric function h in L2 (Rn ) is admissible if Z 1 Ch ≡ |η(k)|2 dk < ∞, where b h(y) = η(|y|). k R+

In this case, b h is the Fourier transform of h.

Thus, the continuous wavelet transform with respect to an admissible function now is given. Definition 3. Let f be in Lp (Rn ) with 1 ≤ p < ∞, and let h be a radially symmetric admissible function in L1 (Rn )∩L2 (Rn ). The continuous wavelet transform of f with respect to h is defined as (Lh f)(a, b) = [(Ja h)∼ ∗ f] (b), where ∗ means convolution and h∼ (x) = h(−x).

(1)

200

SINGULARITIES OF THE CONTINUOUS WAVELET TRANSFORM IN Lp (Rn )

3

Remark 1. According to Definition 3, and since Ja h ∈ L1(Rn ) and f ∈ Lp (Rn ), it follows from Young’s Inequality that (Lh f)(a, b) ∈ Lp (Rn ) and |(Lh f)(a, b)|p ≤ kfkp kJa hk1 Remark 2. The inversion formula of the continuous wavelet transform for f in Lp (Rn ) with 1 < p < ∞ can be obtained from Theorem 1 given by Weisz in [5]. In this case, f=

lim

S→0,U →∞

1 Ch1 ,h2

Z

S

T

Z

(Lh1 f)(a, b)TbJa h2 Rn

1 an+1

da db,

(2)

where the convergence is in the Lp (Rn ) norm, and where h1 and h2 are radially symmetric admissible functions in L1 (Rn ) ∩ L2 (Rn ), and where Z 1 0 < Ch1 ,h2 ≡ (3) η1 (k)η2 (k) dk < ∞. k R+ In this case, b h1 (y) = η1 (|y|j ) and b h2(y) = η2 (|y|j ), where j = 1 or j = ∞.

We should remark also that for the special case p = 2, the inversion formula (2) and the condition (3) are given in [1]. Also, for the case that f ∈ Lp (R) the inversion formula for the continuous wavelet transform is given in [4]. Lemma 1. If f ∈ Lp (Rn ) with 1 ≤ p < ∞ and h ∈ C0∞ (Rn ) is a radially symmetric admissible function, then for a > 0, (Lh f)(a, b) is of class C ∞ , and Z

1 (−1)|α| α = f(x) n (∂ h) a 2 a|α| Rn for any multi-index α ∈ Rn . ∂bα(Lh f)(a, b)



 x−b dx. a

(4)

Proof. Since Ja h ∈ C0∞ (Rn ), it follows that (Jah)∼ ∗ f ∈ C ∞ and ∂bα [(Ja h)∼ ∗ f] (b) = [∂bα (Jah)∼ ∗ f] (b) for any multi-index α ∈ Rn . Then from (1), (Lh f)(a, b) is of class C ∞ , and    Z 1 x−b α α ∼ α ∂b (Lh f)(a, b) = [∂b (Ja h) ∗ f] (b) = ∂b f(x)dx n h a a2 Rn   Z (−1)|α| 1 α x − b = f(x)dx. n h a|α| a 2 a Rn

(5)

201

4

JAIME NAVARRO AND SALVADOR ARELLANO-BALDERAS

This proves Lemma 1.



Note that from (4) and (1), ∂bα(Lh f)(a, b) =

(−1)|α| (L∂ α h f)(a, b) a|α|

(6)

3. Partial result In this section we will prove first that if f in Lp (Rn ) is of class C ∞ in a neighborhood of b0, then we have the existence of the limit of 1 1 α n ∂ (Lh f)(a, b) a a2 b as (a, b) → (0, b1 ) for any b1 in a neighborhood of b0 . That is, we have the following result. Theorem 1. Suppose that h ∈ C0∞ (R) is a non-zero radially symmetric admissible function where b h(0) = 0. If f in Lp (Rn ) is of class C ∞ in a neighborhood of x = b0 in Rn where 1 ≤ p < ∞, then for each multi-index α ∈ Rn , 1 1 α n ∂b (Lh f)(a, b) (a,b)→(0,b1) a a 2 lim

exists for each b1 in a neighborhood of b0 ∈ Rn . Proof. First, note that from Lemma 1 the function Whα f is continuous at any (a1, b1 ) ∈ R+ × Rn for any multi-index α ∈ Rn . Suppose now that f in Lp (Rn ) is of class C ∞ in a neighborhood of x = b0 ∈ Rn containing the closed ball B∆ (b0 ), where ∆ > 0. Take b1 in the open ball B ∆ (b0), 2 and choose b ∈ B ∆ (b0 ). 2

Now, since f ∈ Lp (Rn ) and h ∈ C0∞(R), it follows from (4) that ∂bα(Lh f)(a, b)

   1 x−b |α| α = f(x) n (−1) ∂x h dx. a a2 Rn Z

Moreover, since h ∈ C0∞ (Rn ), there exists L > 0 such that supph ⊂ BL (0). Then supp hα ⊂ BL (0) also. Then

202

SINGULARITIES OF THE CONTINUOUS WAVELET TRANSFORM IN Lp (Rn )

∂bα(Lh f)(a, b)

5

   1 x−b |α| α = f(x) n (−1) ∂x h dx. a a2 BaL(b) Z

∆ ). Then for b ∈ B ∆ (b0 ) Now consider a such that a ∈ (0, 2L 2 we have BaL (b) ⊂ B∆ (b0). Then by hypothesis f is of class C ∞ on BaL(b). Therefore we may integrate by parts to obtain

∂bα (Lh f)(a, b)

=

Z

∂xα f(x) BaL (b)

1 n h a2



 x−b dx. a

Hence, for x = b + ay, ∂bα(Lh f)(a, b)

=

Z

n

a 2 ∂bαf(b + ay)h(y) dy. BL(0)

∞

Since f is C at the points in the region of integration, for y ∈ BL (0) we have from Taylor’s formula with integral remainder given by ∂bαf(b

α

+ ay) = ∂ f(b) +

Z

1

X 1 β+α ∂ f(b + tay) a y β dt, β! b

0 |β|=1

that Z 1 1 α 1 1 n a 2 ∂bαf(b + ay)h(y)dy n ∂b (Lh f)(a, b) = n aa2 a a 2 BL (0)   Z Z 1X 1 ∂ αf(b) + = ∂bβ+αf(b + tay) ay β dt h(y) dy a BL(0) 0 |β|=1 Z Z Z 1X 1 α h(y)dy + ∂bβ+αf(b + tay) y β h(y) dt dy. = ∂ f(b) a BL (0) BL (0) 0 |β|=1

Since b h(0) = 0, then

Z

h(y)dy = 0, and since f is of class C ∞ in a neighBL (0)

borhood of x = b0 , and since ∂ β+αf is continuous near b1 , it follows that for b and b1 in B ∆ (b0 ), 2

203

6

JAIME NAVARRO AND SALVADOR ARELLANO-BALDERAS

1 1 α lim n ∂b (Lh f)(a, b) = (a,b)→(0,b1) a a 2

Z

BL(0)

Z

1

X

0 |β|=1

∂bβ+αf(b1 ) y β h(y) dt dy

Z X  β+α = ∂b f(b1 ) |β|=1

β



y h(y) dy . BL (0)

Note that since h ∈ C0∞ (Rn ), we have Z y β h(y) dy < ∞. BL(0)

Thus,

1 1 α n n ∂b (Lh f)(a, b) exists for each multi-index α ∈ R . (a,b)→(0,b1) a a 2 lim



Now, let us prove the converse of Theorem 1, which is our main result. 4. Main result Theorem 2. Suppose h1 ∈ C0∞ (Rn ) and h2 ∈ C0(Rn ) are non-zero radially symmetric admissible functions and satisfy condition (3). Consider f in Lp (Rn ) with 1 < p < ∞. If for each multi-index α ∈ Rn we have the existence of the limit of (Whα1 f)(a, b) ≡ a1 1n2 ∂bα(Lh f)(a, b) as (a, b) → (0, b1) for each b1 in an a open neighborhood of x = b0 ∈ Rn , then f is of class C ∞ in an open neighborhood of b0 ∈ Rn for any multi-index α ∈ Rn . Proof. Suppose then that for any multi-index α ∈ Rn Fhα1 (b1) ≡

lim

(a,b)→(0,b1)

(Whα1 f)(a, b)

exists for each b1 in an open neighborhood containing the closed ball Br (b0), where r > 0. Now, for fixed x in Br (b0), let  h (−y) (W α f)(a, x + ay) if a > 0 2 h1 (Ihα1 ,h2 f)(a, x, y) = h2 (−y) F α (x) if a = 0 h1

Then we have the following notes.

204

SINGULARITIES OF THE CONTINUOUS WAVELET TRANSFORM IN Lp (Rn )

7

Note 1. For x in Br (b0), the function Ihα1 ,h2 f is well-defined for all a ≥ 0 and all y in Rn . Note 2. For a > 0, the function Ihα1 ,h2 f is given by 1 1 (−1)|α| ∼ (Ihα1 ,h2 f)(a, x, y) = h2 (−y) √ n [(Ja hα1 ) ∗ f] (x + ay), a a a|α|

(7)

and since h1 ∈ C0∞, it comes from the definition of (Whα1 f)(a, b) and from (6), that for fixed y ∈ Rn and a > 0, the function Ihα1 ,h2 f is infinitely differentiable in the variable x. Then we have the following three results, where the proofs are given in the Appendix. Lemma 2. The function Ihα1 ,h2 f is continuous on [0, ∞) × Br (b0 ) × Rn . Lemma 3. For fixed x in Br (b0), the function Ihα1 ,h2 f is in L1([0, ∞) × Rn ). Lemma 4. For x in the open ball Br (b0), let Z ∞Z w(x) = (Ih1 ,h2 f)(a, x, y)dyda, Rn

0

and let (Ihα1 ,h2 f)(x)

=

Z

0

∞Z

Rn

(Ihα1 ,h2 f)(a, x, y)dyda.

Then ∂ αw(x) = (Ihα1 ,h2 f)(x)

for each multi-index α ∈ Rn .

(8)

Back to the proof of Theorem 2, note that from Lemma 4, the function w is of class C ∞ on Br (b0 ). Thus if we define uc (x) =

Z cZ 1 c

1 1 h2 (−y) √ n (Lh1 f)(a, x + ay)dyda a a Rn

for any x in Rn and c > 0, we have from Lemma 4 that for x ∈ Br (b0), lim uc (x) = w(x).

c→+∞

That is, uc → w pointwise on Br (b0) as c → +∞.

205

8

JAIME NAVARRO AND SALVADOR ARELLANO-BALDERAS

On the other hand, by (2), we have uc → Ch1 ,h2 f in the Lp (Rn ) norm. Then f = (Ch1 ,h2 )−1 w almost everywhere on Br (b0 ). Finally, since from (8) the function w is C ∞ on Br (b0 ), it follows that f is of class C ∞ on Br (b0 ). This completes the proof of Theorem 2. 

5. Appendix Proof of Lemma 2. Let (a1 , x1, y1) be any point in [0, ∞) × Br (b0) × Rn . Note that if a1 > 0, then from (7), the function Ihα1 ,h2 f is continuous at (a1, x1 , y1). Now, if a1 = 0, then by hypothesis of Theorem 2, (Ihα1 ,h2 f)(a, x, y) =

lim

(a,x,y)→(0,x1,y1 )

= h2 (−y1)

lim

(a,x,y)→(0,x1 ,y1 )

lim

(a,x,y)→(0,x1,y1 )



h2 (−y)(Whα1 f)(a, x + ay)

(Whα1 f)(a, x + ay) = h2 (−y1)

lim

(a,b)→(0,x1)



(Whα1 f)(a, b)

= h2 (−y1)Fhα1 (x1) = (Ihα1 ,h2 f)(0, x1 , y1). This completes the proof of Lemma 2.



Proof of Lemma 3. Note that for a > 0, and from the definition of (Whα1 f)(a, b) and then from (6) that (Ihα1 ,h2 f)(a, x, y) = h2 (−y)(Whα1 f)(a, x + ay)

= h2 (−y)

1 1 α n ∂ (Lh1 f)(a, x + ay) aa2 x

= h2 (−y)


1 1 (−1)|α| Lhα1 f (a, x + ay). n |α| aa2 a

(9)

206

SINGULARITIES OF THE CONTINUOUS WAVELET TRANSFORM IN Lp (Rn )

1 p

9

Now, since f ∈ Lp (Rn ) and h1 ∈ C0∞(Rn ), we can choose 1 ≤ q < ∞ so that + q1 = 1 and hence, h ∈ Lq (Rn ). Thus, from H¨older’s inequality, α (I

h1 ,h2 f)(a, x, y)

Now, let

n ≤ |h2 (−y)| a−n−1−|α|+ q kfkp khαkq

(10)

1

  (I α f)(a, x, y) if 0 ≤ a ≤ 1 h1 ,h2 α (Gh1 ,h2 f)(a, y) = |h2(−y)| a−n−1−|α|+ nq kfkp khαkq if a > 1 1 Then

α (Ih ,h f)(a, x, y) ≤ (Ghα ,h f)(a, y) 1 2 1 2

(11)

for all (a, x, y) ∈ [0, ∞) × Br (b0) × Rn . Z

Hence, ∞Z

0

= =

Z

Z

0

0

Rn 1Z

α (Gh ,h f)(a, y) dyda 1 2

Rn

1Z

Rn

α (G

h1 ,h2

α (I

f)(a, y) dyda +

h1 ,h2 f)(a, x, y)

Z

∞ 1

dyda +

Z

1

Z

Rn ∞Z

α (G

dyda f)(a, y) h1 ,h2

Rn

n

|h2 (−y)| a−n−1−|α|+ q kfkp khα1 kq dyda.

Suppose now that supp h2 ⊂ Bd (0) for some d > 0. Then Z

∞ 0

=

Z

0

+

Rn

Z 1Z

α (G

h1 ,h2 f)(a, y)

Bd (0)

α (I

kfkp khα1 kq

dyda

h1 ,h2 f)(a, x, y)

Z

Bd (0)

dyda

|h2(−y)|dy

 Z

(12) ∞

a 1

−n−1−|α|+ n q



da .

207

10

JAIME NAVARRO AND SALVADOR ARELLANO-BALDERAS

from Lemma 2 the function Ihα1 ,h2 f is continuous on [0, ∞) × Bd (0), and Z Since ∞ n a−n−1−|α|+ q da < ∞ for any multi-index α ∈ Rn , where 1p + 1q = 1 with 1

1 ≤ q < ∞, it follows that

Ghα1 ,h2 f ∈ L1 ([0, ∞) × Rn ).

(13)

Hence, (Ihα1 ,h2 f)(·, x, ·) ∈ L1 ([0, ∞) × Rn ). This completes the proof of Lemma 3.



Proof of Lemma 4. First note that from (9), ∂xα(Ih1 ,h2 f)(a, x, y) = (Ihα1 ,h2 f)(a, x, y)

(14)

for any multi-index α ∈ Rn . Now note that since from Note 1 for α = 0 the function Ih1 ,h2 f is well-defined on [0, ∞)×Br (b0)×Rn , from Lemma 3 the function (Ih1 ,h2 f)(·, x, ·) is integrable for each x ∈ Br (b0 ), from Note 2 we have that ∂x (Ih1 ,h2 f) exists, and since from (11) and (13) there is Gh1 ,h2 f ∈ L1([0, ∞) × Rn ) such that |∂xα (Ih1 ,h2 f)(a, x, y)| ≤ (Gh1 ,h2 f)(a, y) for all [0, ∞) × Br (b0) × Rn , it follows from (Theorem 2.27, [2]) that w(x) =

Z

0

∞Z

(Ih1 ,h2 f)(a, x, y)dyda Rn

is differentiable and ∂w(x) =

Z

∞ 0

Z

∂x (Ih1 ,h2 f)(a, x, y)dyda

Rn

Hence, for any multi-index α ∈ Rn and from (14)

208

SINGULARITIES OF THE CONTINUOUS WAVELET TRANSFORM IN Lp (Rn )

α

∂ w(x) =

Z

0

∞Z

Rn

∂xα(Ih1 ,h2 f)(a, x, y)dyda

=

Z

∞ 0

Z

Rn

11

(Ihα1 ,h2 f)(a, x, y)dyda

= (Ihα1 ,h2 f)(x) This completes the proof of Lemma 4.



References [1] I. Daubechies, Ten Lectures on Wavelets, Siam, Philadelphia, 1992. [2] G. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd. Ed., A WileyInterscience Publication, 1999. [3] J. Navarro, Wavelet Transform and singularities of L2 -functions in Rn , Revista Colombiana de Matem´ aticas, vol. 32, No. 2, pp. 93-99, (1998). ˘ [4] T. Rao, H. Siki´c and R. Song, Application of Carleson’s Theorem to wavelet inversion, Control Cybernetics, 23, (1994) pp. 761-771. [5] F. Weisz, Inversion formulas for the continuous wavelet transform, Acta Math. Hungar., 138(3), (2012), 237-258.

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 3-4,209-218 ,2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 209

SOME ESTIMATION OF THE STRUVE TRANSFORM IN A QUOTIENT SPACE OF BOEHMIANS S.K.Q. AL-OMARI Department of Applied Sciences; Faculty of Engineering Technology Al-Balqa’Applied University; Amman 11134; Jordan [email protected] Abstract In this paper, we investigate the Struve transformation on certain space of Boehmians. The transform under consideration is de…ned and some of its properties are also illustrated . Keywords: Struve transform; Modi…ed Struve, Generalized function; Boehmian.

1

Introduction

In some physical situations, di¤erential equations are sometimes governed by boundary conditions that are not enough smooth but are generalized functions. It is of importance to extend the classical integral transforms to generalized functions especially that possess a convolution property of Fourier convolution type. As a youngest space of generalized functions, and more particularly of distributions, the space of Boehmians was initiated by the concept of regular operators [6]. Regular operators form a subalgebra of the …eld of Mikusinski operators and include only such functions whose support is bounded from the left. The space of Boehmians was initiated to contain all regular operators, all distributions and some objects which are neither operators nor distributions. In literature, several integral transforms have been extended to various soaces of Boehmians by many authors. We reacall, Roopkumar in [19; 20] ; Karunakaran and Vembu in [18] ; Karunakaran and Roopkumar in [17] ; Mikusinski and Zayed in [16] ; Al-Omari in [6] ; Al-Omari and Kilicman in [5; 9; 15] ; Al-Omari et. al. in [11] ; Loonker et. al. in [25; 26] and many others. The Struve Hv - transform as an example of a symmetric Watson transform is de…ned by [1; 2] Z 1 p tr Sv g (x) = xyHv (xy) g (y) dy; x 2 R+ ; (1) 0

where Hv denotes the Struve function of order v with a power series form Hv (x) =

1 X 0

( 1)m m + 23 m+d+

3 2

x 2

2m+ +1

;

(2)

210

2

S.K.Q. Al-Omari

is the gamma function. When Re v > 12 ; the Struve function can be written in an integral representation; giving x v Z 1 2 2 sin (x cos t) sin2v (t) dt: (3) Hv (x) = p 1 v+v+ 2 0 Struve functions have many applications in physics and applied mathematics. In optics, they occur as the normalized line spread function and, in ‡uid dynamics they occur as acoustics for impedance calculations [3]. The Struve transform have been investigated on the space lv;q (R+ ) of those measurable functions de…ned on R+ such that [1] kf k

;q

Z

=

1 0

dx jx f (x)j x q

1 q

< 1:

(4)

In the strip, 2 < Re v < 0; v = 21 ; q = 2; we have l 1 ;2 = l2 (R+ ) : Hence, the 2 Struve transform is bounded on l2 (R+ ) and for Re v 6= 1, we have [1; p.p.209] Svtr g

l2 (R+ )

C kgkl2 (R+ ) :

(5)

The Bessel-Struve transform was de…ned in [24] and the modi…ed Struve transform was de…ned in [4] :

2

De…nitions and Notations

The construction of Boehmians is similar to the construction of the …eld of quotients and in some cases, it just gives the …eld of quotients. On the other hand, the construction is possible where there are zero divisors, such as space C (the space of continous functions) with the operations of pointwise additions and convolution. Let G be a linear space and S be a subspace of G: We assume to each pair of elements f 2 G and ! 2 S; is assigned the product f g such that the following conditions are satis…ed : (1) If !; 2 S; then ! ? 2 S and ! ? = ? !: (2) If f 2 G and !; 2 S; then (f !) = f (! ? ) : (3) If f; g 2 G; ! 2 S and 2 R; then (f + g) ! = f

! + g ! and

(f

!) = ( f ) !:

Let

be a family of sequences from S; such that : and f 1 If f; g 2 G; f n g 2 n =g n ; then f = g; 8n 2 N: ; then f! n ? n g 2 : 2 If f! n g ; f n g 2 Elements of will be called delta sequences. Consider the class A of pair of sequences de…ned by A = (ffn g ; f! n g) : ffn g

G N ; f! n g 2

;

211

3

on some de…nition of a struve transform in the space ...

for each n 2 N: An element (ffn g ; f! n g) 2 A is called a quotient of sequences, denoted by; ffn g if fn ! m = fm ! n ; 8n; m 2 N. f! n g ffn g fgn g ffn g Two quotients of sequences and are said to be equivalent, f! n g f ng f! n g fgn g ; if fn m = gm ! n ; 8n; m 2 N: f ng The relation is an equivalent relation on A and hence, splits A into equivaffn g ffn g lence classes. The equivalence class containing : These is denoted by f! n g f! n g equivalence classes are called Boehmians; or usual Boehmians; and the space of all Boehmians is denoted by B: The sum of two Boehmians and multiplication by a scalar can be de…ned in a natffn g fgn g ffn ffn g ffn g n + gn ! n g ural way : + = and = = f! n g f ng f! n f! n g f! n g ng f fn g ; 2 C, space of complex numbers. f! n g fgn g ffn g = The operation and the di¤ erentiation are de…ned by : f! n g f ng ffn gn g ffn g fD fn g and D = : f! n g f! g f! n g n n Many a time, G is equipped with a notion of convergence. The relationship between the notion of convergence and are given by: (4) If fn ! f as n ! 1 in G and, ! 2 S is any …xed element, then fn ! ! f

! in G as n ! 1:

(5) If fn ! f as n ! 1 in G and f! n g 2

; then

fn ! n ! f in G as n ! 1: The operation

can be extended to B ffn g f! n g

In B; two types of convergence, as follows:

S by : If !=

ffn !g : f! n g

- convergence and

- convergence: A sequence of Boehmians f to a Boehmian f! n g such that

in B; denoted by (

n

ffn g 2 B and ! 2 S, then f! n g

!k ) ; (

n

!

ng

- convergence, are de…ned

in B is said to be

; if there exists a delta sequence

! k ) 2 G; 8k; n 2 N;

and (

n

!k ) ! (

convergent

! k ) as n ! 1; in G; for every k 2 N:

The following is equivalent for the statement of

- convergence :

212

4

S.K.Q. Al-Omari

n

!

such that G:

as n ! 1 in B if and only if there is ffn;k g ; ffk g 2 G and f! k g 2 ffn;k g ffk g ; = and for each k 2 N; fn;k ! fk as n ! 1 in n = f! k g f! k g

- convergence: A sequence of Boehmians f

ng

in B is said to be

- convergent

to a Boehmian in B; denoted by n ! ; if there exists a f! n g 2 such that ( n ) ! n 2 G; 8n 2 N; and ( n ) ! n ! 0 as n ! 1 in G:

See; [5] [9] ; [11] ; [13] ; [27] ; [28] ; [29] for further investigation of the abstract construction of Boehmians.

3

Basic Theorem and Notations

Let (R+ ) denote the space of test functions of compact support de…ned on R+ [14; 10] : Then, the following de…nitions are needful tour next investigations. De…nition 1 Let 2 l2 (R+ ) ; ' 2 (R+ ); then denote by the Mellin - type convolution product of …rst kind given by [14; 12] Z 1 1 (6) ( ') (y) = y 1 '( )d : 0

Properties of may be introduced as : (i) ( ) (t) = ( ) (t) ; (ii) (( + ) ') (t) = ( ') (t) + ( ') (t) ; (iii) ( ) (t) = ( ) (t) ; is complex number ; (vi) (( ) ') (t) = ( ( ')) (t) : De…nition 2 Let and '; by

2 l2 (R+ ) and ' 2 (

') (x) =

(R+ ); then we de…ne a product Z

Theorem 3 Let 2 < Re v < 0 and ' 2 every 2 l2 (R+ ) :

1

(x ) ' ( ) d :

; between (7)

0

(R+ ); then Svtr (

') = Svtr

'; for

Proof Under the Hypothesis of the theorem we by De…nition 1.2. and (1) write Z 1 p tr Sv ( ') (x) = ( ') (y) xyHv (xy) dy 0 Z 1 Z 1 p 1 = y 1 '( )d xyHv (xy) dy 0 Z 1 Z 01 p 1 = '( ) y 1 xyHv (xy) dyd (8) 0

0

213

5

on some de…nition of a struve transform in the space ...

By change of variables y Svtr (

') (x) = = =

1

=

Z

Z

1 0 1

0 Svtr

and Fubini’s theorem, (8) then gives Z 1 p ( ) x Hv (x ) d ' ( ) d 0

Svtr

(x ) ' ( ) d

' (x) :

This completes the proof of the theorem.

4

Space of Boehmians

Let us now construct spaces of Boehmians where our Struve transform is de…ned. Let be the set of delta sequences (approximating identities) satisfying the following properties : (1) : fZ n g 2 (R+ ) ; 1

(2) : (3) :

Z

0 1 0

n (x) dx

j

(4) : supp

= 1; n 2 N;

n (x)j dx n (x)

< 1; n 2 N;

[a; bn ] ; 0 < an < bn and an ; bn ! 0 as n ! 1; where supp

n (x)

= fx 2 R+ :

n (x)

6= 0; 8n 2 Ng :

We …rst consider the space B (l2 ; ( ; ) ; ) for our construction. Theorem 4 Let tities

1;

2

2 l2 (R+ ) ; '1 ; '2 2

(R+ ); then we have the following iden-

(i) ( 1 + 2 ) '1 = 1 '1 + 2 '1 : (ii) '1 '2 = '2 '1 in (R+ ) : (iii) Let n ! in l2 (R+ ); then for every ' 2 (R+ ) ; n n ! 1: Proof of part (i) and (iii) follows from simple integration. Part (ii) follows from the properties of enumerated in [14] : Theorem 5 Let 2 l2 (R+ ) and '1 ; '2 2 ( '1 ) ' 2 :

(R+ ); then '

' !

('1 '2 ) = '

Proof Under the hypothesis of the theorem and Fubini’s theorem we write Z 1 ( ('1 '2 )) (x) = (x ) ('1 '2 ) ( ) d 0 Z 1 Z 1 = (x ) y 1 '1 y 1 '2 (y) dyd 0 0 Z 1 Z 1 1 = '2 (y) y (x ) '1 y 1 dyd : 0

0

' as

(9)

214

6

S.K.Q. Al-Omari

Change of variables on (9) implies Z ( ('1 '2 )) (x) = Z =

1Z 1 0 1

(xy ) '1 ( ) d '2 (y) dy

0

(

'1 ) (xy) '2 (y) dy

0

= ((

'1 )

'2 ) (x) :

This completes the proof of the theorem. As …nal in this construction of B (l2 ; ( ; ) ; ), we prove the following theorem. Theorem 6 Let n ! 1:

2 l2 (R+ ) and f n g be a delta sequence; then

Proof By properties delta sequences we have Z 1 2 k kl2 (R+ ) = j( n n ) (x) =

Z

Z

0 1 0 bn

an

Z

Z

1

0 1 0

= M (bn

(x )

j (x )

an )

Z

n

k

n kl2 (R+ )

k kl2 (R+ )

k

as

(x)j2 dx n(

)

Z

(x)j2 j

1

2

(x)

n(

)d

dx

0 n(

)j dxd

(By Jensen’s inequality) 1 0

j (x )

(x)j2 dx ! 0 as n ! 1;

where Kn = [an ; bn ] ; is a compact set containing the support of But 0

!

n

n ; 8n

2 N:

kl2 (R+ ) ! 0 as n ! 1:

Hence, we have obtained k

n kl2 (R+ )

k kl2 (R+ ) ! 0 as n ! 1:

This completes the proof of the theorem. This our space is constructed and regarded as a space of Boehmians. The sum of two Boehmians and multiplication by a scalar can be de…ned in a natural way : ffn g fgn g ffn !ng n + gn + = (10) f! n g f ng f! n ng and

ffn g = f! n g

2 C, the space of complex numbers.

ffn g f fn g = ; f! n g f! n g

215

7

on some de…nition of a struve transform in the space ...

The operation ffn g f! n g

and the di¤erentiation are de…ned by : fgn g ffn = f ng f! n

gn g ng

and D

ffn g fD fn g = : f! n g f! n g

(11)

Similarly, the space B (l2 ; ( ; ) ; ) can be established: Sum and multiplication by a scalar in B (l2 ; ( ; ) ; ) can be de…ned as : fgn g ffn ffn g n + gn ! n g + = f! n g f ng f! n ng and

ffn g = f! n g

(12)

ffn g f fn g = ; f! n g f! n g

2 C, the space of complex numbers. The product and the di¤erentiation in B (l2 ; ( ; ) ; ) are de…ned as : ffn g f! n g

fgn g ffn gn g = f ng f! n ng

and D

ffn g fD fn g = : f! n g f! n g

(13)

f ng 2 B (l2 ; ( ; ) ; ); then, in view of Theorem 3, for f ng f ng Re v < 0; we de…ne the Struve transform of as f ng

De…nition 7 Let

Svtr

f ng Svtr f n g = f ng f ng

2
0 such that γ(t) := (u(t), v(t)) maps (−c, c) into γ ∩ (I1 × J1 ). We observe that if u(c/2) = 0, then v(c/2) also vanishes since f is a function passing through (0, 0). Then we have γ(c/2) = γ(0) which contradicts the assumption that γ has no self-intersections. Thus, neither u(c/2) nor u(−c/2) vanish. Without loss of generality we can assume u(c/2) > 0. Then u(−c/2) must be negative. Otherwise either γ has a selfintersection or f is not a function. Since γ(t) is continuous, its trace from t = −c/2 to t = c/2 must coincide with the graph of f from u(−c/2) < 0 to
u(c/2) > 0. Thus, for I := u(−c/2), u(c/2) , and J := J1 , the function f satisfies Definition 1. As a corollary we obtain the promised result on supersmoothness: Theorem 3 Let γ ⊂ R2 be the trace of a Jordan arc that divides the open disk Ω into two subsets Ω1 and Ω2 as in Figure 3. Further assume that γ is not smooth at P ∈ γ. Let f1 , f2 be C 1 functions on Ω continuously glued along γ, that is, let  f1 (x, y) if (x, y) ∈ Ω1 (3) F (x, y) := f2 (x, y) if (x, y) ∈ Ω2 be a continuous function on Ω. Then the piecewise function F is differentiable at P , that is, (4)

∇f1 (P ) = ∇f2 (P ). 4

235

Intrinsic Supersmoothness Boris Shekhtman, Tatyana Sorokina

Proof. Consider a C 1 function h = f1 − f2 . The fact that f1 and f2 are continuously glued along γ means that h(γ) = 0 and by Theorem 2 0 = ∇h(P ) = ∇f1 (P ) − ∇f2 (P ). Thus, ∇f1 (P ) = ∇f2 (P ), and the proof is complete. A partial converse of Theorem 3 holds true in the following sense: Theorem 4 Let γ ⊂ R2 be the trace of a Jordan arc that divides the open disk Ω into two subsets Ω1 and Ω2 . Assume that γ is smooth at a point P ∈ γ. Then there exists a neighborhood U of P and and two differentiable functions f1 , f2 ∈ C 1 (U ) such that the function  f1 (x, y) if (x, y) ∈ Ω1 ∩ U (5) F (x, y) := f2 (x, y) if (x, y) ∈ Ω2 ∩ U is not differentiable at P . Proof. Let U and h be chosen as in Theorem 2, i.e., satisfying conditions (2). Let f1 (x, y) := h(x, y) and f2 (x, y) ≡ 0. Then, since h(γ ∩ U ) = 0, the function F defined by (5) is continuous and not differentiable at P because ∇f2 (P ) = 0 6= ∇f1 (P ). Theorem 4 provides only a partial converse of Theorem 3 because the function F is defined locally, in a neighborhood U of P , and not on all of Ω. We believe that the global version of this theorem also holds and end this section with a conjecture. Conjecture 5 Let γ ⊂ R2 be a continuous curve that divides an open disk Ω centered at P into two subsets Ω1 and Ω2 . Then γ is smooth at P if and only if we can glue two continuously differentiable functions along the curve as in (5) so that the resulting piecewise function F is not differentiable at P .

3

Supersmoothness of higher derivatives.

Consider two non-collinear rays v1 and v2 emanating from the origin in R2 . The curve formed by these two rays is not smooth and partitions the open unit disk Ω into two sectors ∆1 and ∆2 . It follows from the results of the previous section that two differentiable functions f1 and f2 continuously glued along the boundary of the sectors as in (5) produce a piecewise function F2 differentiable at the origin: (6)

F2 ∈ C(Ω) ⇒ F2 ∈ C 1 (0). 5

236

Intrinsic Supersmoothness Boris Shekhtman, Tatyana Sorokina

Farin’s observation (1) shows that for three pairwise non-collinear rays emanating from the origin and a piecewise function F3 consisting of three differentiable pieces as in Figure 1 the following holds: F3 ∈ C 1 (Ω) ⇒ F3 ∈ C 2 (0). However, as it was pointed out in the introduction, for three non-collinear rays amplification (6) may not hold, that is, in general F3 ∈ C(Ω) 6⇒ F3 ∈ C 1 (0). In this section we extend this pattern. For a fixed n ≥ 0, we partition the open disk Ω into n+2 sectors ∆1 , . . . , ∆n+2 , by pairwise non-collinear vectors (rays) v1 , . . . , vn+2 , positioned clockwise as in Figure 4. Then we create a piecewise function Fn+2 by gluing n+2 functions f1 , . . . , fn+2 ∈ C n (Ω) along the rays. Thus for 1 ≤ j ≤ n + 1, the sector ∆j is formed by vj and vj+1 , and the sector ∆n+2 is formed by vn+2 and v1 . We will show that similarly to (1) the following holds: Fn+2 ∈ C n (Ω) ⇒ Fn+2 ∈ C n+1 (0);

(7)

yet the weaker assumption Fn+2 ∈ C n−1 (Ω) may not imply the associated conclusion that Fn+2 ∈ C n (0). We start with a simple lemma that shows that two differentiable functions continuously glued along a ray v must be differentiable in the direction of v. We use Dv to denote the directional derivative in the direction of v. Lemma 6 Let v = (a, b) be a unit vector in R2 . Let f and g be continuously differentiable functions in an ε-neighborhood of the origin in R2 such that (8)

f (ta, tb) = g(ta, tb), for all t ∈ [0, ε).

Then Dv f (ta, tb) = Dv g(ta, tb), for all t ∈ [0, ε). Proof. It suffices to prove the result for t = 0. We obtain f (ta, tb) − f (0) f (ta, tb) − f (0) = lim t→0 t→0+ t t g(ta, tb) − g(0) g(ta, tb) − g(0) by (8) = lim = lim = Dv g(0), t→0+ t→0 t t

Dv f (0) = lim

where the second and the fourth equalities follow from the continuity of Dv f and Dv g, respectively. We are now ready to prove statement (7). For brevity, we use F := Fn+2 . 6

237

Intrinsic Supersmoothness Boris Shekhtman, Tatyana Sorokina

238

Theorem 7 Let functions f1 , . . . , fn+2 , be n times continuously differentiable on Ω and let F be defined piecewise on each sector ∆j by F |∆j := fj , j = 1, . . . , n + 2. If F ∈ C n (Ω) then F has all derivatives of order n + 1 at the origin; that is, F ∈ C n+1 (0), n ≥ 0. Proof. If n = 0, the proof is given in Theorem 3. Let n ≥ 1. We will show that for two neighboring functions, say fj and fj+1 , all partial derivatives of order n + 1 coincide at the origin. Then for every k = 0, . . . , n, Dxk Dyn−k f1 (0) = Dxk Dyn−k f2 (0) = . . . = Dxk Dyn−k fn+2 (0), which would prove the theorem. Without loss of generality we consider the neighboring functions f1 and f2 . It is clearly enough to prove that (9)

Dvk2 Dvn−k f1 (0) = Dvk2 Dvn−k f2 (0), for every k = 0, . . . , n. 1 1

Observe that for k ≥ 1, the assumption F ∈ C n (Ω) implies that the functions Dvk−1 Dvn−k f1 and Dvk−1 Dvn−k f2 are continuously glued along the ray v2 . 2 1 2 1 Hence, by Lemma 6 we obtain

Dv2 Dvk−1 Dvn−k f1 (0) = Dv2 Dvk−1 Dvn−k f2 (0) 2 1 2 1 which implies (9) for k ≥ 1. Hence it remains to prove that Dvn1 f1 (0) = Dvn1 f2 (0).

(10)

Since all the vectors vj are pairwise non-collinear we can find constants αj and βj such that v1 = αj v2 + βj vj for all j = 3, . . . , n + 2. Then (11)

Dvn1 =(α3 Dv2 + β3 Dv3 ) . . . (αn+2 Dv2 + βn+2 Dvn+2 ) = Dv2 p(Dv2 , . . . , Dvn+2 ) + γ

n+2 Y

Dvj

j=3

for some constant γ and some homogeneous polynomial p of order n − 1. Since, by the assumption, p(Dv2 , . . . , Dvn+2 )f1 and p(Dv2 , . . . , Dvn+2 )f2 coincide along the ray v2 , by Lemma 6 (12)

Dv2 p(Dv2 , . . . , Dvn+2 )f1 (0) = Dv2 p(Dv2 , . . . , Dvn+2 )f2 (0).

Similarly, for every k = 3, . . . , n + 2, the functions n+2 Y

Dvj fk−1

and

j=3,j6=k

n+2 Y

j=3,j6=k

7

Dvj fk

Intrinsic Supersmoothness Boris Shekhtman, Tatyana Sorokina

239

coincide along the ray vk . Hence, by Lemma 6, for every k = 3, . . . , n + 2, n+2 Y

Dvj fk−1 (0) =Dvk

j=3

n+2 Y

Dvj fk−1 (0) = Dvk

j=3 j6=k

n+2 Y

Dvj fk (0) =

j=3 j6=k

n+2 Y

Dvj fk (0).

j=3

Thus, we obtain the following chain of equalities (13)

n+2 Y

Dvj f2 (0) =

j=3

n+2 Y

Dvj f3 (0) = · · · =

j=3

n+2 Y

Dvj fn+2 (0) =

j=3

n+2 Y

Dvj f1 (0).

j=3

The last equality follows from Lemma 6 since f1 and fn+2 share a common edge v1 . Thus by (11) Dvn1 f2 (0) =

Dv2 p(Dv2 , . . . , Dvn+2 )f2 (0) + γ

n+2 Y

Dvj f2 (0)

j=3

by (12,13)

=

Dv2 p(Dv2 , . . . , Dvn+2 )f1 (0) + γ

n+2 Y

Dvj f1 (0)

by (11)

=

Dvn1 f1 (0).

j=3

which completes the proof of (9), and consequently proves the theorem. The next result is a direct consequence of applying Theorem 7 to the derivatives of the piecewise function. Corollary 8 Let functions f1 , . . . , fn+2 , be m times continuously differentiable on Ω, with m ≥ n, and let Fn+2 be defined piecewise on each sector ∆j by Fn+2 |∆j := fj , j = 1, . . . , n + 2. If Fn+2 ∈ C m (Ω) then Fn+2 has all derivatives of order m+1 at the origin, that is, Fn+2 ∈ C m+1 (0), m ≥ n ≥ 0. We finish this section and this article by constructing polynomials (hence smooth functions) f1 , . . . , fn+2 , n ≥ 1, such that the spline Fn+2 defined by Fn+2 |∆j = fj is in C n−1 (Ω) yet Fn+2 6∈ C n (0). We note that if n = 0, it is immediately obvious that f1 ≡ 0 and f2 ≡ 1 do not join continuously at the origin. The following observation is the key to the construction: Lemma 9 Given n ≥ 1, consider the polynomial g(x, y) :=

n+1 X

ci (y + ai x)n .

i=1

Then the system of equations with the unknowns (c1 , . . . , cn+1 ): ∂k g(x, 0) = 0, ∂xj ∂y k−j

for all 0 ≤ j ≤ k ≤ n − 1,

has a non-trivial solution. 8

Intrinsic Supersmoothness Boris Shekhtman, Tatyana Sorokina

240

Proof. Indeed for 0 ≤ k ≤ n − 1 and 0 ≤ j ≤ k we have n+1 X i=1

n+1 X j ∂k n! n ci j k−j (y + ai x) ci ai (y + ai x)n−k = ∂x ∂y (n − k)! y=0 y=0 i=1

n+1

X n−k+j n! xn−k ci ai = 0. (n − k)!

=

i=1

With s := n − k + j, the system of n equations with n + 1 unknowns n+1 X

ci asi = 0,

s = 0, . . . , n − 1,

i=1

has a non-trivial solution. Now we can proceed with our construction. As in Figure 4, choose n + 2 consecutive positioned clockwise rays vi emanating from the origin whose equations are given by the following lines li l1 : y = 0, l2 : y + a2 x = 0,

. . . , ln+2 : y + an+2 x = 0.

Note that without loss of generality we assume that v1 goes along the positive direction of the x-axis. Define f1 :≡ 0 to be the function between v1 and v2 . Let the function between vk and vk+1 be defined as follows: fk :=

k X

ci lin ,

for each 2 ≤ k ≤ n + 2,

i=2

with the convention vn+3 := v1 . We next define: gk (x, y) := fk+1 (x, y) − fk (x, y) = ck+1 (y + ak+1 x)n ,

for all 2 ≤ k ≤ n + 1.

All partial derivatives of gk of order n − 1 or less vanish for y = −ak+1 x, that is, at the line lk+1 . It remains to choose the coefficients c2 , . . . , cn+2 in such a way that fn+2 is glued smoothly to f1 ≡ 0 at l1 , that is, so that all derivatives of order n − 1 or less of the polynomial fn+2 =

n+2 X

ci lin

i=2

vanish at y = 0. By Lemma 9 this leads to a system of n equations with n + 1 unknowns (c2 , . . . , cn+2 ) that has a nontrivial solution. 9

Intrinsic Supersmoothness Boris Shekhtman, Tatyana Sorokina

Hence there exists a non-zero homogeneous polynomial fn+2 of order n between ln+2 and l1 which is C n−1 -smoothly glued to fn+1 across ln+1 and C n−1 -smoothly glued to f1 ≡ 0 across l1 . Finally fn+2 is a nonzero homogeneous polynomial of order n. Thus there exists a partial derivative of fn+2 of order n which is a non-zero constant. In particular, its value at the origin is not zero, yet the same derivative of f1 ≡ 0 is zero. The resulting piecewise function Fn+2 does not have a derivative of order n at the origin. Remark 10 The existence of the spline Fn+2 implicitly constructed above also follows from Theorem 9.3 in [3]. Indeed this theorem shows that the dimension of polynomial splines of degree n and smoothness
n − 1 defined over the union of n + 2 sectors is strictly greater than n+2 2 . The latter is the dimension of bivariate polynomials. Thus, there exists a spline that does not have a derivative of order n at the origin. We decided to provide a development here that would be accessible to an audience not familiar with spline theory.

References [1] P. Alfeld, A trivariate Clough-Tocher scheme for tetrahedral data. Comput. Aided Geom. Design 1 (1984), 169–181. [2] G. Farin, B`ezier polynomials over triangles. Report TR/91, Dept. of Mathematics, Brunel University, Uxbridge, UK (1980). [3] M.-J. Lai and L. L. Schumaker, Spline functions on triangulations, Cambridge University Press, Cambridge, 2007. [4] T. Sorokina, Intribsic supersmoothness of multivariate splines. Numerische Mathematik 116 (2010), 421–434. [5] T. Sorokina, Redundancy of smoothness conditions and supersmoothness of bivariate splines, IMA Journal of Numerical Analysis, to appear.

10

241

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 3-4,242-251 ,2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 242

Approximation by Kantorovich and Quadrature type quasi-interpolation neural network operators George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here we present multivariate basic approximation by Kantorovich and Quadrature type quasi-interpolation neural network operators with respect to supremum norm. This is done with rates using the …rst multivariate modulus of continuity. We approximate continuous and bounded functions on RN . When they are also uniformly continuous we have pointwise and uniform convergences.

2010 AMS Mathematics Subject Classi…cation: 41A17, 41A25, 41A30, 41A36. Keywords and Phrases: sigmoidal and hyperbolic tangent activation functions, multivariate quasi-interpolation neural network approximation, Kantorovich and quadrature type operators.

1

Background

We consider here the sigmoidal function of logarithmic type si (xi ) =

1 1+e

xi

each has the properties

;

xi 2 R, i = 1; :::; N ; x := (x1 ; :::; xN ) 2 RN ; lim si (xi ) = 1 and

xi !+1

lim si (xi ) = 0, i = 1; :::; N:

xi ! 1

These functions play the role of activation functions in the hidden layer of neural networks, also have applications in biology, demography, etc. As in [7], we consider i

(xi ) :=

1 (si (xi + 1) 2

si (xi 1

1)) , xi 2 R, i = 1; :::; N:

Approximation by Kantorovich and Quadrature type quasi-interpolation neural networks George A. Anastassiou

243

We notice the following properties: i) ii) iii)

(xi ) > 0, 8 xi 2 R;

i

P1

ki = 1

i

(xi

ki = 1

i

(nxi

P1

ki ) = 1, 8 xi 2 R, ki ) = 1, 8 xi 2 R; n 2 N;

v)

R1 i

is a density function,

vi)

i

is even:

iv)

i

1

(xi ) dxi = 1,

i

( xi ) =

i

(xi ), xi

0, for i = 1; :::; N:

We see that ([5]) i

vii)

i

(xi ) =

e2 1 2e2

1 (1 +

exi 1 ) (1

+e

xi 1 )

; i = 1; :::; N:

is decreasing on R+ , and increasing on R , i = 1; :::; N:

Let 0
0, 8 x 2 RN ;

We see that 1 X

1 X

k1 = 1 k2 = 1 1 X

1 X

k1 = 1 k2 = 1

That is

:::

1 X

:::

1 X

(x1

k1 ; x2

k2 ; :::; xN

kN = 1 N Y

i

(xi

ki ) =

kN = 1 i=1

N Y

i=1

2

1 X

ki = 1

i

(xi

kN ) = !

ki )

= 1:

(2)

Approximation by Kantorovich and Quadrature type quasi-interpolation neural networks George A. Anastassiou

244

(ii)’ 1 X

(x

1 X

k) :=

1 X

:::

k1 = 1 k2 = 1

k= 1

1 X

(x1

k1 ; :::; xN

kN ) = 1;

kN = 1

(3)

k := (k1 ; :::; kn ), 8 x 2 RN : (iii)’ 1 X

(nx

k) :=

k= 1 1 X

1 X

1 X

:::

k1 = 1 k2 = 1

(nx1

k1 ; :::; nxN

kN ) = 1;

(4)

kN = 1

8 x 2 RN ; n 2 N. (iv)’

Z

(x) dx = 1;

(5)

RN

that is

is a multivariate density function.

Here kxk1 := max fjx1 j ; :::; jxN jg, x 2 RN , also set 1 := (1; :::; 1), 1 := ( 1; :::; 1) upon the multivariate context. For 0 < < 1 and n 2 N, …xed x 2 RN , we have proved ([4]) (v)’ bnbc 8 < :

X

k n

(nx

k = dnae x 1>

k)

3:1992e

n(1

)

.

(6)

1 n

Let f 2 CB RN (bounded and continuous functions on RN , N 2 N). We de…ne the multivariate Kantorovich type neural network operators (n 2 N, 8 x 2 RN ) ! Z k+1 1 X n Kn (f; x) := Kn (f; x1 ; :::; xN ) := nN f (t) dt (nx k) := (7) k= 1

1 X

k1 = 1

:::

1 X

kN = 1

We observe that Z k+1 n k n

n

N

Z

k1 +1 n

:::

k1 n

f (t) dt =

Z

Z

kN +1 n

f (t1 ; :::; tN ) dt1 :::dtN

kN n

k1 +1 n k1 n

k n

:::

Z

!

N Y

i=1

kN +1 n kN n

3

f (t1 ; :::; tN ) dt1 :::dtN =

i

(nxi

!

ki ) :

Approximation by Kantorovich and Quadrature type quasi-interpolation neural networks George A. Anastassiou

Z

0

1 n

:::

Z

1 n

f

t1 +

0

k1 kN ; :::; tN + n n

dt1 :::dtN =

Kn (f; x) =

nN

Z

1 n

k t+ n

f

0

k= 1

1 n

f

t+

0

Thus it holds

1 X

Z

245

dt

!

(nx

k n

dt:

k) :

(8)

(9)

Again for f 2 CB RN , N 2 N, we de…ne the multivariate neural network operators of quadrature type Qn (f; x), n 2 N, as follows. Let = ( 1 ; :::; N ) 2 NN , r = (r1 ; :::; rN ) 2 ZN 0, such that + , wr = wr1 ;:::;rN X

1 X

wr =

r=0

r1 =0

:::

N X

rN =0

wr1 ;:::;rN = 1; k 2 ZN

and nk

(f ) :=

n;k1 ;:::;kN

(f ) :=

X

wr f

r=0

:=

1 X

:::

r1 =0

where r = r11 ; :::; We de…ne

N X

N

1 X

nk

k= 1

:=

:::

k1 = 1

1 X

;

(11)

k)

(12)

:

Qn (f; x) := Qn (f; x1 ; :::; xN ) := 1 X

(10)

k1 r1 rN kN + + ; :::; n n 1 n n N

wr1 ;:::;rN f

rN =0

rN

k r + n n

n;k1 ;:::;kN

(f )

N Y

i=1

kN = 1

i

(nxi

(f )

(nx !

ki ) ;

8 x 2 RN :

We consider also here the hyperbolic tangent function tanh x, x 2 R (see also [2]) ex e x tanh x := x . (13) e +e x It has the properties tanh 0 = 0; 1 < tanh x < 1, 8 x 2 R, and tanh ( x) = tanh x. Furthermore tanh x ! 1 as x ! 1, and tanh x ! 1, as x ! 1, and it is strictly increasing on R. This function plays the role of an activation function in the hidden layer of neural networks. We further consider ([2]) (x) :=

1 (tanh (x + 1) 4

tanh (x

We easily see that ( x) = (x), that is di¤erentiable, thus continuous. 4

1)) > 0, 8 x 2 R. is even on R. Obviously

(14) is

Approximation by Kantorovich and Quadrature type quasi-interpolation neural networks George A. Anastassiou

Proposition 1 ([2]) Obviously

(x) for x

246

0 is strictly decreasing.

(x) is strictly increasing for x

0. Also it holds lim

x! 1

(x) =

0 = lim (x) : x!1 Infact has the bell shape with horizontal asymptote the x-axis. So the maximum of is zero, (0) = 0:3809297: Theorem 2 ([2]) We have that Thus

1 X

(nx

P1

(x

i= 1

i) = 1;

8 n 2 N; 8 x 2 R:

i= 1

Also it holds

1 X

(x + i) = 1;

8x 2 R:

i= 1

Theorem 3 ([2]) It holds So

R1

1

i) = 1, 8 x 2 R.

(x) dx = 1.

(x) is a density function on R.

Theorem 4 ([2]) Let 0
0, 9 neighborhood V (x0 ) : (x) (x0 ) + ", 8 x 2 V (x0 ) : 1

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

280

(iv) the set supp ( ) is compact in R (where supp( ) := fx 2 R; (x) > 0g). We call a fuzzy real number. Denote the set of all with RF : E.g., fx0 g 2 RF , for any x0 2 R; where fx0 g is the characteristic function at x0 . r For 0 < r 1 and 2 RF de…ne [ ] := fx 2 R : (x) rg and 0 [ ] := fx 2 R : (x) > 0g: r

Then it is well known that for each r 2 [0; 1], [ ] is a closed and bounded interval of R. For u; v 2 RF and 2 R, we de…ne uniquely the sum u v and the product u by r

[u

r

r

v] = [u] + [v] ;

r

r

[

u] =

r

[u] ; 8 r 2 [0; 1] ;

r

where [u] + [v] means the usual addition of two intervals (as subsets of R) and r [u] means the usual product between a scalar and a subset of R (see, e.g., [16]). Notice 1 u = u and it holds uh v = v iu, u = u . If 0 r1 r2 1 (r) (r) (r) (r) (r) (r) r2 r1 r then [u] [u] . Actually [u] = u ; u+ , where u < u+ , u ; u+ 2 R, 8 r 2 [0; 1] : De…ne D : RF RF ! R+ [ f0g by D (u; v) := sup max r2[0;1]

n

u

(r)

v

(r)

(r)

; u+

(r)

v+

o

;

h i (r) (r) r where [v] = v ; v+ ; u; v 2 RF . We have that D is a metric on RF . Then (RF ; D) is a complete metric space, see [16], with the properties D (u

w; v

w) = D (u; v) ;

D (k

u; k

D (u

v; w

8 u; v; w 2 RF ;

(1)

v) = jkj D (u; v) , 8 u; v 2 RF , 8 k 2 R; e)

D (u; w) + D (v; e) , 8 u; v; w; e 2 RF :

Let f; g : R ! RF be fuzzy real number valued functions. The distance between f; g is de…ned by D (f; g) := supD (f (x) ; g (x)) : x2R

On RF we de…ne a partial order by " ": u; v 2 RF , u (r) (r) u+ v+ , 8 r 2 [0; 1] : We need Lemma 2 ([5]) For any a; b 2 R : a b D (a

u; b

where oe 2 RF is de…ned by oe :=

u)

2

(r)

v

(r)

and

0 and any u 2 RF we have ja

f0g :

v i¤ u

bj D (u; oe) ;

(2)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

281

Lemma 3 ([5]) (i) If we denote oe := f0g , then oe 2 RF is the neutral element with respect to , i.e., u oe = oe u = u, 8 u 2 RF : (ii) With respect to oe, none of u 2 RF , u 6= oe has opposite in RF . (iii) Let a; b 2 R : a b 0, and any u 2 RF , we have (a + b) u = a u b u. For general a; b 2 R, the above property is false. (iv) For any 2 R and any u; v 2 RF , we have (u v) = u v: (v) For any ; 2 R and u 2 RF , we have ( u) = ( ) u: (vi) If we denote kukF := D (u; oe), 8 u 2 RF , then k kF has the properties of a usual norm on RF , i.e., ku

kukF = 0 i¤ u = oe, k vkF

ukF = j j kukF ;

kukF + kvkF ;

kukF

kvkF

D (u; v) :

(3)

Notice that (RF ; ; ) is not a linear space over R; and consequently (RF ; k kF ) is not a normed space. As in Remark 4.4 ([5]) one can show easily that a sequence of operators of the form n X Ln (f ) (x) := f (xkn ) wn;k (x) , n 2 N, (4) k=0

X ( denotes the fuzzy summation) where f : Rd ! RF , xkn 2 Rd , d 2 N, wn;k (x) real valued weights, are linear over Rd , i.e., Ln ( 8 ;

f

g) (x) =

Ln (f ) (x)

Ln (g) (x) ;

(5)

2 R, any x 2 Rd ; f; g : Rd ! RF . (Proof based on Lemma 3 (iv).) We further need

De…nition 4 (see also [14], De…nition 13.16, p. 654) Let (X; B; P ) be a probability space. A fuzzy-random variable is a B-measurable mapping g : X ! RF (i.e., for any open set U RF ; in the topology of RF generated by the metric D, we have g 1 (U ) = fs 2 X; g (s) 2 U g 2 B). (6) The set of all fuzzy-random variables is denoted by LF (X; B; P ). Let gn ; g 2 LF (X; B; P ), n 2 N and 0 < q < +1. We say gn (s) lim

n!+1

Z

q

”q-m ean”

!

n!+1

D (gn (s) ; g (s)) P (ds) = 0:

g (s) if

(7)

X

Remark 5 (see [14], p. 654) If f; g 2 LF (X; B; P ), let us denote F : X ! R+ [ f0g by F (s) = D (f (s) ; g (s)), s 2 X. Here, F is B-measurable, because 3

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

282

F = G H, where G (u; v) = D (u; v) is continuous on RF RF , and H : X ! RF RF , H (s) = (f (s) ; g (s)), s 2 X, is B-measurable. This shows that the above convergence in q-mean makes sense. De…nition 6 (see [14], p. 654, De…nition 13.17) Let (T; T ) be a topological space. A mapping f : T ! LF (X; B; P ) will be called fuzzy-random function (or fuzzy-stochastic process) on T . We denote f (t) (s) = f (t; s), t 2 T , s 2 X. Remark 7 (see [14], p. 655) Any usual fuzzy real function f : T ! RF can be identi…ed with the degenerate fuzzy-random function f (t; s) = f (t), 8 t 2 T , s 2 X. Remark 8 (see [14], p. 655) Fuzzy-random functions that coincide with probability one for each t 2 T will be consider equivalent. Remark 9 (see [14], p. 655) Let f; g : T ! LF (X; B; P ). Then f k f are de…ned pointwise, i.e., (f

g) (t; s) = f (t; s)

(k

f ) (t; s) = k

g and

g (t; s) ;

f (t; s) , t 2 T; s 2 X:

De…nition 10 (see also De…nition 13.18, pp. 655-656, [14]) For a fuzzyrandom function f : Rd ! LF (X; B; P ), d 2 N, we de…ne the (…rst) fuzzyrandom modulus of continuity (F ) 1

sup

( Z

(f; )Lq = 1 q

q

D (f (x; s) ; f (y; s)) P (ds)

X

0< ;1

d

: x; y 2 R ; kx

ykl1

)

;

(8)

q < 1:

De…nition 11 ([9]) Here 1 q < +1. Let f : Rd ! LF (X; B; P ), d 2 N, be a fuzzy random function. We call f a (q-mean) uniformly continuous fuzzy random function over Rd , i¤ 8 " > 0 9 > 0 :whenever kx ykl1 ; x; y 2 Rd ; implies that Z q

(D (f (x; s) ; f (y; s))) P (ds)

":

(9)

X

U

q We denote it as f 2 CF R Rd :

U

q Proposition 12 ([9]) Let f 2 CF R Rd . Then

(F ) 1

(f; )Lq < 1, any

> 0:

Proposition 13 ([9]) Let f; g : Rd ! LF (X; B; P ), d 2 N, be fuzzy random functions. It holds (F ) (i) 1 (f; )Lq is nonnegative and nondecreasing in > 0: 4

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

283

Uq (F ) (F ) (f; )Lq = 1 (f; 0)Lq = 0, i¤ f 2 CF R Rd : 1 #0 (F ) (F ) (F ) (f; 1 + 2 )Lq (f; 1 )Lq + 1 (f; 2 )Lq , 1 ; 2 1 1 (F ) (F ) (f; n )Lq n 1 (f; )Lq , > 0; n 2 N. 1 (F ) (F ) (F ) (f; )Lq d e 1 (f; )Lq ( + 1) 1 (f; )Lq , 1

(ii) lim

(iii) > 0: (iv) (v) > 0, > 0, where d e is the ceilling of the number. (F ) (F ) (F ) (vi) 1 (f g; )Lq (f; )Lq + 1 (g; )Lq , > 0. Here f g is a 1 fuzzy random function. Uq (F ) (vii) 1 (f; )Lq is continuous on R+ , for f 2 CF R Rd : We give De…nition 14 ([7]) Let f (t; s) be a stochastic process from Rd (X; B; P ) into R, d 2 N, where (X; B; P ) is a probability space. We de…ne the q-mean multivariate …rst moduli of continuity of f by 1

sup

( Z

X

> 0; 1

jf (x; s)

(f; )Lq := 1 q

q

: x; y 2 Rd ; kx

f (y; s)j P (ds)

)

ykl1

;

(10)

q < 1.

For more see [7]. We mention (F ) 1

Proposition 15 ([9]) Assume that Then n (F ) (f; ) sup max q 1 L

1

(f; )Lq is …nite, f

(r)

;

r2[0;1]

; q

L

> 0; 1

(r)

1

f+ ;

Lq

o

q < 1. :

(11)

The reverse direction " " "is not possible. Remark 16 ([9]) For each s 2 X we de…ne the usual …rst modulus of continuity of f ( ; s) by (F )

!1

(f ( ; s) ; ) :=

sup

D (f (x; s) ; f (y; s)) ;

> 0:

(12)

x;y2Rd kx ykl 1

Therefore (F )

Dq (f (x; s) ; f (y; s)) 8 s 2 X and x; y 2 Rd : kx

ykl1

!1

,

> 0.

5

q

(f ( ; s) ; )

;

(13)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

Hence it holds Z Dq (f (x; s) ; f (y; s)) P (ds)

Z

1 q

X

8 x; y 2 Rd : kx ykl1 We have that (F ) 1

(F )

!1

284

1 q

q

(f ( ; s) ; )

P (ds)

;

X

(14)

: Z

(f; )Lq

(F )

!1

1 q

q

(f ( ; s) ; )

P (ds)

;

(15)

X

under the assumption that the right hand side of (15) is …nite. The reverse " " of the last (15) is not true. Also we have Proposition 17 ([6]) (i) Let Y (t; !) be a real valued stochastic process such that Y is continuous in t 2 [a; b]. Then Y is jointly measurable in (t; !) : (ii) Further assume that the expectation (E jY j) (t) 2 C ([a; b]), or more Rb generally a (E jY j) (t) dt makes sense and is …nite. Then ! Z Z b

E

b

Y (t; !) dt

=

a

(EY ) (t) dt:

(16)

a

According to [13], p. 94 we have the following De…nition 18 Let (Y; T ) be a topological space, with its -algebra of Borel sets B := B (Y; T ) generated by T . If (X; S) is a measurable space, a function f : X ! Y is called measurable i¤ f 1 (B) 2 S for all B 2 B. By Theorem 4.1.6 of [13], p. 89 f as above is measurable i¤ f

1

(C) 2 S for all C 2 T .

We would need Theorem 19 (see [13], p. 95) Let (X; S) be a measurable space and (Y; d) be a metric space. Let fn be measurable functions from X into Y such that for all x 2 X, fn (x) ! f (x) in Y . Then f is measurable. I.e., lim fn = f is n!1 measurable. We need also Proposition 20 Let f; g be fuzzy random variables from S into RF . Then (i) Let c 2 R, then c f is a fuzzy random variable. (ii) f g is a fuzzy random variable. 6

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

285

For the de…nition of general fuzzy integral we follow [15] next. De…nition 21 Let ( ; ; ) be a complete -…nite measure space. We call F : ! RF measurable i¤ 8 closed B R the function F 1 (B) : ! [0; 1] de…ned by F 1 (B) (w) := sup F (w) (x) , all w 2 (17) x2B

is measurable, see [15]. Theorem 22 ([15]) For F : ! RF , n (r) (r) F (w) = F (w) ; F+ (w) j0

r

the following are equivalent (1) F is measurable, (r) (r) (2) 8 r 2 [0; 1], F ; F+ are measurable.

o 1 ;

(r)

(r)

Following [15], given that for each r 2 [0; 1], F ; F+ have that the parametrized representation Z Z (r) (r) F d ; F+ d j0 r 1 A

(18)

are integrable we

(19)

A

is a fuzzy real number for each A 2 : The last fact leads to De…nition 23 ([15]) A measurable function F : ! RF , n o (r) (r) F (w) = F (w) ; F+ (w) j0 r 1 is called integrable if for each r 2 [0; 1], F (0) F are integrable.

(r)

are integrable, or equivalently, if

In this case, the fuzzy integral of F over A 2 Z Z Z (r) (r) F d := F d ; F+ d A

A

A

is de…ned by j0

r

1 :

By [15], F is integrable i¤ w ! kF (w)kF is real-valued integrable. Here denote kukF := D u; e 0 , 8 u 2 RF :

We need also

7

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

Theorem 24 ([15]) Let F; G : ! RF be integrable. Then (1) Let a; b 2 R, then aF + bG is integrable and for each A 2 , Z Z Z (aF + bG) d = a Fd + b Gd ; A

A

In particular,

A

Z

(21)

A

Z

Fd

A

(20)

A

(2) D (F; G) is a real- valued integrable function and for each A 2 , Z Z Z D Fd ; Gd D (F; G) d : A

286

A

F

kF kF d :

(22)

Above could be the Lebesgue measure, in this article the multivariate Lebesgue measure, with all the basic properties valid here too. Basically here we have Z Z Z r (r) (r) Fd = F d ; F+ d ; (23) A

i.e.

Z

A

(r)

Fd

=

A

Z

A

F

(r)

A

d ; 8 r 2 [0; 1] :

(24)

Next we state Fubini’s theorem for fuzzy number-valued functions and fuzzy number-valued integrals, see [15]. Theorem 25 Let ( 1 ; 1 ; 1 ) and ( 2 ; 2 ; 2 ) be two complete -…nite measure spaces, and let ( 1 2; 1 2; 1 2 ) be their product measure space. If a fuzzy number-valued function F : 1 2 ! RF is 1 2 -integrable, then (1) the fuzzy-number-valued function F ( ; ! 2 ) : ! 1 ! F (! 1 ; ! 2 ) is 1 integrable for ! 2 2 2 , 2 -a.e., R (2) the fuzzy-number-valued function ! 2 ! F (! 1 ; ! 2 ) d 1 (! 1 ) is 2 1 integrable, and (3) Z Z Z Fd( 1 F (! 1 ; ! 2 ) d 1 (! 1 ) d 2 (! 2 ) = (25) 2) = 1

2

2

Z

1

Z

1

F (! 1 ; ! 2 ) d 2

We further mention

8

2

(! 2 ) d

1

(! 1 ) :

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

Theorem 26 Let f :

N Q

i=1

287

[ai ; bi ] ! LF (X; B; P ), be a fuzzy-random function.

N ! ! Q We assume that f t ; s is fuzzy continuous in t 2 [ai ; bi ], for any s 2 X. i=1 R ! ! Then Q f t ; s d t exists in RF and it is a fuzzy-random variable in N [ai ;bi ]

i=1

s 2 X.

Proof. By de…nition of fuzzy integral we notice here that Z ! ! f t ;s d t = N Q [ai ;bi ]

i=1

80 1 Z < Z ! ! ! ! (r) (r) @ N t ;s d t ; Q f+ t ; s d t A j0 f N Q : [ai ;bi ] [ai ;bi ]

r

i=1

i=1

9 = 1 ; ;

(26)

N Q ! ! where d t is the multivariate Lebesgue measure on [ai ; bi ]. Because f t ; s i=1

! (r) ! is fuzzy continuous in t , we get that f t ; s , 0 r 1, are real valued conR ! ! (r) ! tinuous in t , for each s 2 X. Hence the real integrals Q f t ;s d t N [ai ;bi ]

i=1

are multivariate Riemann integrals that exist, for each s 2 X. R ! ! f t ; s d t 2 RF , i.e. it exists. Thus Q N [ai ;bi ]

i=1

By Theorem 19 and the de…nition of multivariate Riemann integral we get R ! (r) ! t ; s d t are P -measurable functions in s 2 X. f that Q N [ai ;bi ]

i=1

Taking into account (24) and Theorem 22 we derive that

is a fuzzy-random variable in s 2 X.

2

R

N Q

f

[ai ;bi ]

! ! t ;s d t

i=1

Main Results

We are morivated by [9], [11], [12]. Here the activation function b : Rd ! R+ , d 2 N; is of compact support d Q B := [ Tj ; Tj ], Tj > 0; j = 1; :::; d. That is b (x) > 0 for any x 2 B, and j=1

clearly b may have jump discontinuities. Also the shape of the graph of b is immaterial. Typically in neural networks approximation we take b to be a d-dimensional bell-shaped function (i.e. per coordinate is a centered bell-shaped function), or a product of univariate centered bell-shaped functions, or a product of sigmoid functions, in our case all them of compact support B. 9

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

Example 27 Take b (x) = (x1 ) (x2 ) ::: (xd ), where functions, i = 1; :::; d : (i) (xj ) is the characteristic function on [ 1; 1] ; (ii) (xj ) is the hat function over [ 1; 1], that is,

288

is any of the following

8 1 xj 0; < 1 + xj , (xj ) = 1 xj ; 0 < xj 1; : 0, elsewhere,

(27)

(iii) the truncated sigmoids ( 1 or tanh xj or erf (xj ) , for xj 2 [ Tj ; Tj ] , with large Tj > 0; x (xj ) = 1+e j 0, xj 2 R [ Tj ; Tj ] , (iv) the truncated Gompertz function ( xj e e , xj 2 [ Tj ; Tj ] ; (xj ) = 0, xj 2 R [ Tj ; Tj ] ,

;

> 0; large Tj > 0;

The Gompertz functions are also sigmoid functions, with wide applications to many applied …elds, e.g. demography and tumor growth modeling, etc. Thus the general activation function b we will be using here includes all kinds of activation functions in neural network approximations. Uq Here we consider functions f 2 CF R Rd : Let here the parameters: 0 < < 1; x = (x1 ; :::; xd ) 2 Rd , n 2 N; r = (r1 ; :::; rd ) 2 Nd ; i = (i1 ; :::; id ) 2 Nd ; with ij = 1; 2; :::; rj , j = 1; :::; d; also let rd r1 P r2 P P wi = wi1 ;:::;id 0, such that ::: wi1 ;:::;id = 1, in brief written as i1 =1 i2 =1

r P

i=1

id =1

wi = 1. We further consider the parameters k = (k1 ; :::; kd ) 2 Zd ;

i1 ; :::; id Call min = i

Rd+ ,

i

=

Rd+ ;

2 and i = i1 ;:::;id , i = i1 ;:::;id 0: i = ( i1 ; :::; id ) 2 minf i1 ; :::; id g: In this article we study in q-mean (1 q < 1) the pointwise and uniform convergences with rates over Rd , to the fuzzy-random unit operator, of the following fuzzy-random normalized one hidden layer multivariate perturbed neural network operators, where s 2 X, (X; B; P ) a probability space, n 2 N; (i) the Stancu type HnF R (f ) (x; s) = HnF R (f ) (x1 ; :::; xd ; s) = Pn2

k= n2

r P

wi

f

i=1

Pn2

k= n2

k+ n+

i i

b n1

;s

b n1

10

x

k n

x

(28)

k n

=

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

Pn2

:::

k1 = n2

Pn2

Pn2

rP 1

kd = n2

k1 = n2

:::

i1 =1

Pn2

:::

wi1 ;:::;id

k1 + n+

f

id =1

k d = n2

b n1

rP d

k1 n

b n1

x1

k1 n

; :::; n1

x1

i1 i1

289

; :::;

; :::; n1

id id

;s

!

kd n

xd

kd n

xd

kd + n+

;

(ii) the Kantorovich type KnF R (f ) (x; s) = Pn2

n2

k=

r P

wi (n +

i=1

Pn2

2

2

R

1 n+ i

:::

1 n+ i ;:::;i 1 d

0

Pn2

kd = n2

:::

k1 = n2

R

:::

f

f t+

0

b n1

r1 X

n X

k1 = n2

:::

R

k= n2

n X

R

d

i)

:::

i1 =1

t1 +

Pn2

k d = n2

i i

x

k n

=

k n

x

rd X

b n1

; s dt

d

wi1 ;:::;id n +

i1 ;:::;id

id =1

k1 + n+

k+ n+

(29)

i1;:::;i d i1;:::;i d

b n1

; :::; td + k1 n

x1

kd + n+

i1 ;:::;id i1 ;:::;id

; :::; n1

; s dt1 :::dtd

xd

kd n

(30)

b n1

x1

k1 n

; :::; n1

kd n

xd

;

and (iii) the quadrature type Pn2

k=

MnF R Pn2

(f ) (x; s) =

k 1 = n2

:::

Pn2

Pn2

k d = n2

k1 = n2

:::

rP 1

:::

i1 =1

Pn2

b n1

n2

k d = n2

x1

r P

i=1

rP d

wi

f

k n

Pn2

k= n2

wi1 ;:::;id

+

b n1

x1

k1 n

; :::; n1

b n1

k1 n

; :::; n1 xd

k n

=

kd i1 nr1 ; :::; n

k1 n

+

x

k n

x

f

id =1

b n1

i nr ; s

kd n

xd

+

id nrd ; s

!(31)

kd n

:

Similar operators de…ned for d-dimensional bell-shaped activation functions and sample coe¢ cients f nk = f kn1 ; :::; knd were studied initially in [9]. In this article we assume that n o 1 n max Tj + jxj j ; Tj ; (32) j2f1;:::;dg

11

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

290

see also [3], p. 91. So, by (32) we can rewrite the operators as follows ([ ] is the integral part and d e the ceiling of a number). We denote by T = (T1 ; :::; Td ), [nx + T n ] = ([nx1 + T1 n ] ; :::; [nxd + Td n ]), dnx T n e = (dnx1 T1 n e ; :::; dnxd Td n e). It holds (i) HnF R (f ) (x; s) = HnF R (f ) (x1 ; :::; xd ; s) = (33) P[nx+T n

] k=dnx T n e

r P

wi

k+ n+

f

i=1

P[nx+T n

] k=dnx T n e

i i

b n1

;s

b n1

=

k n

x

rd rP 1 k1 + P[nxd +Td n ] P ::: wi1 ;:::;id f n+ k1 =dnx1 T1 n kd =dnxd Td n e i1 =1 id =1 P[nxd +Td n ] P[nx1 +T1 n ] k ::: b n1 (x1 n1 );:::;n1 k1 =dnx1 T1 n e kd =dnxd Td n e

P[nx1 +T1 n ]

::: e

b n1

k1 n

x1

k n

x

; :::; n1

i1 i1

;:::;

kd n

xd

xd

kd + i d n+ i d

!

;s

kd n

;

(ii) KnF R (f ) (x; s) = P[nx+T n

] k=dnx T n e

r P

wi (n +

i=1

P[nx+T n

[nxd +Td n ]

X

R

1 n+ i ;:::;i 1 d

0

P[nx1 +T1 n

X

:::

k1 =dnx1 T1 n e

:::

1 n+ i

0

] k=dnx T n e

[nx1 +T1 n ]

R

R

d i)

kd =dnxd Td n e

R ::: f

] k1 =dnx1 T1 n e

:::

t1 +

k1 + n+

P[nxd +Td n

b n

1

b n1

i i

b n1

; s dt

x

k n

=

k n

x

:::

i1 =1

i1;:::;i d i1;:::;i d

k1 n

k+ n+

(34) r1 X

] kd =dnxd Td n e

x1

f t+

rd X

d

wi1 ;:::;id n +

i1 ;:::;id

id =1

; :::; td +

b n1

kd + n+

x1

i1 ;:::;id i1 ;:::;id

k1 n

; s dt1 :::dtd

; :::; n1

kd n

xd

(35)

; :::; n

1

xd

kd n

;

and (iii)

MnF R

(f ) (x; s) =

P[nx+T n

] k=dnx T n e

r P

i wi f nk + nr ;s i=1 P[nx+T n ] 1 k=dnx T n e b n

12

b n1 x

x

k n

=

k n

(36)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

291

rd rP 1 P[nxd +Td n ] P k i k i ::: wi1 ;:::;id f n1 + nr1 ;:::; nd + nrd ;s kd =dnxd Td n e 1 d i1 =1 id =1 P[nx1 +T1 n ] P[nxd +Td n ] kd k1 1 1 ::: b n xd n (x1 n );:::;n k1 =dnx1 T1 n e kd =dnxd Td n e

P[nx1 +T1 n ] k1 =dnx1

T1 n

e

:::

b n1 So if n1

kj n

xj

k1 n

x1

; :::; n1

kd n

xd

!

:

Tj , all j = 1; :::; d; we get that k n

x

kT kl1 : n1

l1

(37)

For convinience we call [nx+T n ]

X

V (x) =

b n1

x

k n

k1 n

; :::; n1

k=dnx T n e [nxd +Td n ]

[nx1 +T1 n ]

X

X

:::

k1 =dnx1 T1 n e

b n1

x1

kd =dnxd Td n e

=

xd

kd n

: (38)

We make Remark 28 Here always k is as in n2

nxj

Tj n

kj

n2 ;

nxj + Tj n

(39)

I) We observe that k+ n+

i

k n+

x

i

l1

k n+

x i

+

x i

l1

+ l1

i

n+

i

l1

k i kl 1 : n + min i

(40)

Next see that k n+

x i

l1

k n+

i

k n

+ l1

kkkl2 k i kl2

n n+

min i

k n +

kk i kl1

x l1

kT kl1 =: ( ) : n1

We notice for j = 1; :::; d we get jkj j

n jxj j + Tj n : 13

n n+

min i

+

kT kl1 (41) n1 (42)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

292

Hence kkkl2

kn jxj + T n kl2

n kxkl2 + kT kl2 n ;

where jxj = (jx1 j ; :::; jxd j) : Thus n kxkl2 + kT kl2 n ( ) n n + min i

k i kl 2

(43)

kT kl1 : n1

+

(44)

So we get k n+

n kxkl2 + kT kl2 n

x i

n n+

l1

k i kl2 kxkl2 n + min i

k i kl2 kT kl2

+

Hence it holds

n1

k+ n+

k i kl2 kxkl2 + k i kl1 n + min i

k i kl2

min i

n+

i

(45)

x

i

l1

1

+

kT kl1 = n1

kT kl1 : n1

+

min i

+

kT kl1 +

n1

k i kl2 kT kl2 n+

min i

!

:

(46)

II) We also have for 0 that t+

k+ n+

d n+

i

x l1

d n+

+ i

+ i

i

+

i

d i n+ i)

i

l1

+ i

i

k n+

(47)

k n

i

i

x

i

l1

(48)

x l1

k n+

k n+

k n+

+

k+ n+

i

+

n+

i

i)

k+ n+

i

d (1 + n+ d (1 + n+

, j = 1; :::; d; i

ktkl1 +

i

+

d n+

1 n+

tj

=

x i

l1

x i

= l1

x i

+ l1

(49) l1

k n

x

kT k d (1 + i ) i + kkkl1 + 1 l1 n+ i n (n + i ) n

14

l1

(50)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

d (1 + n+

i)

i

+

n (n +

i

d (1 + n+ i

i)

+

i

kxkl1 + (n + i ) n1 i

kxkl1 + d ( n+ i

+ 1)

i

+

n kxkl1 + kT kl1 n

i)

+

i

293

kT kl1 = n1

kT kl1 kT k + 1 l1 = (n + i ) n

kT kl1 n1

1+

i

n+

(51)

:

(52)

:

(53)

i

We have found that t+ i

kxkl1 + d ( n+ i

k+ n+

i

+ 1)

i

x

i

l1

+

kT kl1 n1

1+

i

n+

i

III) We observe that k i + n nr

k n

x l1

+

x l1

1 i n r

l1

kT kl1 d + : 1 n n

(54)

Convergence results follow. U

q Theorem 29 Let f 2 CF R Rd , 1

q < +1, x 2 Rd and n 2 N such that

1

n

max

j2f1;:::;dg

Tj + jxj j ; Tj Z

D

,0
0. Then

1 q

(f ) (x; s) ; f (x; s) P (ds)

X

r X

wi

(F ) 1

i=1

f;

k i kl2 kxkl2 + k i kl1 n + min i

+

1

kT kl1 +

n1

k i kl2 kT kl2 n+

min i

!!

=

Lq

(55)

r1 X

i1 =1 (F ) 1

f;

k i kl2 kxkl2 + k i kl1 n + min i

:::

rd X

wi1 ;:::;id

id =1

+

1

kT kl1 +

n1

k i kl2 kT kl2

where i = (i1 ; :::; id ) : As n ! 1, we get that HnF R (f ) (x; s) with rates at the speed

1 n1

"q-m ean"

:

15

!

f (x; s)

n+

min i

!!

Lq

; (56)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

294

Proof. We may write [nx+T n ]

HnF R

r X

X

(f ) (x; s) =

wi

k+ n+

f

i=1

k=dnx T n e

i

!

;s

i

b n1

x V (x)

k n

:

(57)

We observe that HnF R (f ) (x; s) ; f (x; s) = ! b n1 x k+ i f ;s n+ i V (x)

D 0

[nx+T n ]

r X

X

D@

0

wi

i=1

k=dnx T n e

[nx+T n ]

r X

X

D@

D@

k+ n+

f

i=1

k=dnx T n e

0

wi

r X r X

X

wi

f (x; s)

wi

k+ n+

[nx+T n ]

r X

X

X

b n1

k=dnx T n e

x V (x)

i

wi

f (x; s) r X

k n

;s

i

wi D f

i=1

b n !

!

b n1

1

1A =

; f (x; s)

(58)

!

V (x) V (x) i

!

i=1

k=dnx T n e [nx+T n ]

f

i=1

k=dnx T n e

;s

!

i=1

[nx+T n ]

i

k n

1

;

=

b n1

x V (x)

k+ n+

x V (x)

k n

i

(59) x V (x) k n

1 A

k n

;

(1)

; s ; f (x; s)

!

:

(60)

; s ; f (x; s)

!

:

(61)

i

Thus it holds so far that HnF R (f ) (x; s) ; f (x; s)

D [nx+T n ]

X

b n1

k=dnx T n e

Hence

Z

x V (x)

D

q

r X

k n

wi D f

i=1

HnF R

k+ n+

i i

1 q

(f ) (x; s) ; f (x; s) P (ds)

X [nx+T n ]

r X i=1

wi

Z

X

X

b n1

k=dnx T n e

Dq f

k+ n+

i

x V (x)

k n

1 q

; s ; f (x; s) P (ds)

i

16

!

(62)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

[nx+T n ]

X

b n1

k=dnx T n e r X

wi

(F ) 1

r X

k n

x V (x)

wi

i=1

k i kl2 kxkl2 + k i kl1 n + min i

f;

i=1

k+ f; n+

(F ) 1

1

+

295

i

x

i

l1

(46)

(63)

Lq

k i kl2 kT kl2

kT kl1 +

n1

! !

n+

min i

!!

;

Lq

proving the claim. d Q

Corollary 30 (to Theorem 29) Let x 2 (

1 ; :::;

d)

and n 2 N such that n Z

D

HnF R

q

j;

j=1

wi

(F ) 1

f;

i=1

> 0,

j

:=

1

max fTj +

j2f1;:::;dg

j ; Tj

g. Then

1 q

(f ) (x; s) ; f (x; s) P (ds)

X

r X

Rd ,

j

1;

k i kl2 k kl2 + k i kl1 n + min i

1

+

kT kl1 +

n1

d Q

[

j; j

j=1

]

k i kl2 kT kl2 n+

min i

!!

:

Lq

(64)

We continue with Theorem 31 All assumptions as in Theorem 29. Additionally assume that f (t; s) is fuzzy continuous in t 2 Rd , for each s 2 X. Then Z

D

KnF R

q

1 q

(f ) (x; s) ; f (x; s) P (ds)

X

r X

wi

(F ) 1

f;

i

i=1

r1 P

:::

i1 =1

rd P

id =1

wi1 ;:::;id

kxkl1 + d ( n+ i

(F ) 1

i

+ 1)

+

i kxkl1 +d( i +1)

f;

n+

i

kT kl1 n1 +

kT kl 1 n1

1+

i

n+

1+

= i

i

n+

Lq

i

(65)

Lq

:

As n ! 1, we get KnF R (f ) (x; s)

with rates at the speed

1 n1

"q-m ean "

!

f (x; s)

:

Proof. By Remark 5 the function F (t; s) := D f t +

k+ n+

i i

; s ; f (x; s) is

B-measurable over X with respect to s. Also F is continuous in t 2 Rd , similar reasoning as in explanation of Remark 5, for each s 2 X.

17

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

296

Thus for F (t; s), by joint measurability theorem of Caratheodory, see [1], p. 156, we get that F (t; s) is jointly measurable over Rd X. Hence in what follow we can apply Fubini;s theorem. We may rewrite [nx+T n ]

KnF R

r X

X

(f ) (x; s) =

Z

d

wi (n +

i)

k+ t+ n+

f

0

i=1

k=dnx T n e

1 n+ i

b n1

k n

x V (x)

i

; s dt

i

:

!

(67)

We observe that KnF R (f ) (x; s) ; f (x; s) =

D 0

[nx+T n ]

X

D@

k=dnx T n e

r X

wi (n +

i)

r X

X

Z

d i)

wi (n +

1 n+ i

[nx+T n ]

wi (n +

d

i)

D

Z

X

1 n+ i

b n

1

f

[nx+T n ]

X

b n1

k=dnx T n e r X

wi (n +

d

i)

Z

1 n+ i

D f

0

i=1

x V (x)

k+ t+ n+

0

i=1

k+ t+ n+

f

i

!

b n

1

x V (x)

Z

1 n+ i

k n

f (x; s) dt

0

i

k+ t+ n+

; s dt

i

!

1 A

(1)

(68)

k n

; s dt;

x V (x)

i

;

f (x; s) dt

k=dnx T n e

r X

k n

x V (x)

0

i=1

k=dnx T n e

1 n+ i

0

i=1

b n1

[nx+T n ]

Z

d

!!

(21)

k n

i

!

(69)

!

(70)

; s ; f (x; s) dt :

i

That is it holds [nx+T n ]

D

X

KnF R (f ) (x; s) ; f (x; s)

b n1

k=dnx T n e r X i=1

wi (n +

d

i)

Z

1 n+ i

D f

0

18

k+ t+ n+

i i

x V (x)

k n

; s ; f (x; s) dt :

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

Hence (let p; q > 1 : Z

D

+

1 q

= 1) [nx+T n ]

1 q

KnF R

q

1 p

X

(f ) (x; s) ; f (x; s) P (ds)

X

0 @

r X

297

b n1

x V (x)

k=dnx T n e

Z

d i)

wi (n +

Z

k+ t+ n+

D f

0

X

i=1

1 n+ i

i

; s ; f (x; s) dt

i

!q

k n

! q1 1 P (ds) A (71)

(by Hölder’s inequality) [nx+T n ]

X

b n1

k=dnx T n e

Z

Z

1

X

(n +

i)

dq p

1 n+ i

Dq f

[nx+T n ]

X

(by Fubini’s theorem )

=

b n1

Z

1 n+ i

0

Dq f

X

[nx+T n ]

X

1 n+ i

b n1

(F ) 1

0 [nx+T n ]

X

1 (n + r X i=1

i)

wi

d p

(F ) 1

(F ) 1

f;

i

f;

i

r X

k n

(n +

d i)

(n +

i)

wi

i=1

kxkl1 + d ( n+ i

i

i

wi

i=1

i

x

i

l1

+ 1)

+

d i)

(n +

i)

Lq

wi

i=1

+ 1)

(n +

!q

r X

k n

x V (x)

kxkl1 + d ( n+ i

r X

k n

x V (x)

k+ f; t + n+ b n1

k=dnx T n e

i

d p

! q1 1 (8) k+ i t+ ; s ; f (x; s) P (ds) dt A n+ i

k=dnx T n e

Z

! q1 1 i ; s ; f (x; s) dt P (ds) A !

x V (x)

k=dnx T n e

Z

d i)

wi (n +

i=1

k+ t+ n+

0

r X

k n

x V (x)

! q1 1 (53) dt A

(n +

d i)

(n +

i)

kT k + 1 l1 n kT kl1 n1

d p

1+

1+

d p

i

n+

i

Lq

i

n+

; i

proving the claim. The case of q = 1 goes through in a similar and simpler way.

19

(73)

Lq

!

=

(74)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

d Q

Corollary 32 (to Theorem 31) Let x 2 (

1 ; :::;

d)

and n 2 N such that n Z

D

Rd ,

j

max fTj +

j ; Tj

j2f1;:::;dg

(75) 1;

(F ) 1

wi

i

f;

i=1

:=

g. Then

(f ) (x; s) ; f (x; s) P (ds)

X

r X

> 0,

j

1

1 q

KnF R

q

j;

j=1

298

k kl1 + d ( n+ i

i

+ 1)

+

kT kl1 n1

d Q

[

j;

j=1

1+

j

]

i

n+

: i

Lq

We …nish with Theorem 33 All as in Theorem 29. Then Z

MnF R (f ) (x; s) ; f (x; s) P (ds)

Dq

1 q

X (F ) 1

f;

kT kl1 d + 1 n n

:

(76)

Lq

As n ! 1, we get that MnF R (f ) (x; s) with rates at the speed

1 n1

"q-m ean"

!

f (x; s)

:

Proof. We may rewrite [nx+T n ]

MnF R

X

(f ) (x; s) =

k=dnx T n e

r X

wi

f

i=1

k i + ;s n nr

!

b n1

x V (x)

k n

:

(77)

We observe that MnF R (f ) (x; s) ; f (x; s) = ! r X b n1 x k i wi f + ;s n nr V (x) i=1

D 0

D@

[nx+T n ]

X

k=dnx T n e [nx+T n ]

X

k=dnx T n e [nx+T n ]

X

k=dnx T n e

b n1

r X

wi

!

f (x; s)

i=1

x V (x)

k n

r X

wi D f

i=1

20

b n1

x V (x)

k n

1 A

k n

;

(1)

k i + ; s ; f (x; s) n nr

!

:

(78)

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

299

Hence it holds Z

1 q

MnF R (f ) (x; s) ; f (x; s) P (ds)

Dq

X [nx+T n ]

r X

Z

wi

X

b n1

k=dnx T n e

b n1

k=dnx T n e

D

q

f

X

i=1

[nx+T n ]

X

x V (x)

k n

x V (x)

1 q

k i + ; s ; f (x; s) P (ds) n nr r X

wi

(F ) 1

i=1

(F ) 1

k n

f;

kT kl1 d + n1 n

i k + f; n nr

!

x l1

(79) ! !

(54)

Lq

;

(80)

Lq

proving the claim. Corollary 34 (to Theorem 33) Here all as in Corollary 30. Then Z

Dq

MnF R (f ) (x; s) ; f (x; s) P (ds)

X

1 q

1;

(F ) 1

f;

kT kl1 d + 1 n n

:

d Q

[

j=1

j; j

] (81)

Lq

Comment 35 All the convergence fuzzy random results of this article can have real analogous results for real valued random functions (stochastic processes) using 1 (f; )Lq , see (10).

References [1] C.D. Aliprantis, O. Burkinshaw, Priciples of Real Analysis, Third edition, Academic Press, Inc., San Diego, CA, 1998. [2] G.A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, Journal of Mathematical Analysis and Application, Vol. 212 (1997), 237-262. [3] G.A. Anastassiou, Quantitative Approximation, Chapman & Hall/CRC, 2001, Boca Raton, New York.

21

Multivariate Fuzzy-Random Perturbed Neural Network Approximation George A. Anastassiou

[4] G.A. Anastassiou, Rate of convergence of Fuzzy neural network operators, univariate case, The Journal of Fuzzy Mathematics, 10, No. 3 (2002), 755780. [5] G.A. Anastassiou, S. Gal, On a fuzzy trigonometric approximation theorem of Weierstrass-type, the Journal of Fuzzy Mathematics, 9, No. 3 (2001), 701-708. [6] G.A. Anastassiou, Univariate fuzzy-random neural network approximation operators, Computers and Mathematics with Applications, Special issue/ Proceedings edited by G. Anastassiou of special session "Computational Methods in Analysis", AMS meeting in Orlando, Florida, November 2002, Vol. 48 (2004), 1263-1283. [7] G.A. Anastassiou, Multivariate Stochastic Korovkin Theory given quantitatively, Mathematics and Computer Modelling, 48 (2008), 558-580. [8] G.A. Anastassiou, Fuzzy Mathematics: Approximation Theory, Springer, Heidelberg, New York, 2010. [9] G.A. Anastassiou, Multivariate Fuzzy-Random normalized Neural Network Approximation Operators, Annals of Fuzzy Mathematics and Informatics, Vol. 6, No. 1, 2013, 191-212. [10] G.A. Anastassiou, Higher order multivariate fuzzy approximation by basic neural network operators, Cubo, accepted 2013. [11] G.A. Anastassiou, Approximation by Multivariate Perturbed Neural Network Operators, submitted 2014. [12] P. Cardaliaguet, G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Networks 5(1992), 207-220. [13] R.M. Dudley, Real Analysis and Probability, Wadsworth & Brooks / Cole Mathematics Series, Paci…c Grove, California, 1989. [14] S. Gal, Approximation Theory in Fuzzy Setting, Chapter 13 in Handbook of Analytic-Computational Methods in Applied Mathematics, pp. 617-666, edited by G. Anastassiou, Chapman & Hall/CRC, Boca Raton, New York, 2000. [15] Yun Kyong Kim, Byung Moon Ghil, Integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems, 86(1997), 213-222. [16] Wu Congxin, Gong Zengtai, On Henstock integrals of fuzzy number valued functions (I), Fuzzy Sets and Systems, Vol. 115, No. 3, 2000, 377-391.

22

300

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 3-4,301-311 ,2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 301

NOTES ON THE GENERALIZED HANKEL INTEGRAL TRANSFORM AND ITS EXTENSION TO A CLASS OF BOEHMIANS S. K. Q. AL - OMARI Department of Applied Sciences; Faculty of Engineering Technology Al-Balqa’Applied University; Amman 11134; Jordan [email protected] Abstract In this paper we establish certain spaces of Boehmians for the generalized Hankel integral transform l ; ;w;k; . We estimate the generalization lge; ;w;k; of l ; ;w;k; and explore some of its properties. Continuity of lge; ;w;k; with respect to and convergence is also discussed in some detail. Keywords: l Space.

1

; ;w;k;

Transform; lge;

;w;k;

Transform; Generalized Function; Boehmian

INTRODUCTION

Classical integral transforms have found their application in various areas of mathematics, mathematical physics, engineering and some other di¤erential equations that arise in certain physical problems. As some physical situations are governed by di¤erential equations whose boundary conditions are not enough smooth but are generalized functions; it was of great importance to extend some classical integral transforms to generalized functions. Among those integral transforms that have applications in the space of Boehmians we recall, but are not limited, some such as : Fourier transform [17] ; Radon transform [16; 19] ; Stieltjes tranform [20] ; Hartley Hilbert and Fourier - Hilbert transforms [15] ; Hartley transform [6; 13] ; di¤raction Fresnel transform [5] ; Fresnel wavelet transform [9] ; Hilbert transform [18] ; Fourier sine (cosine) transform [11] ; Ridgelet transform [1] ; Hankel [24] ; Wavelet [25] ; and many others to mention but a few. In addition to this sequence of integrals, this paper investigates the integral transform [2] (l

; ;w;k;

f ) (x) = x

Z

1

|

1

(xt) k tw f (t) dt; (x > 0) ;

(1)

0

with 2 C (Re ( ) > 1) ; 2 C; w 2 C; k > 0 and > 0; | is the Bessel function of the …rst kind of order ; f 2 C0 is a continuous function of compact support on (0; 1) for the range of parameters that indicated.

302

2

S.K.Q. Al-Omari

The substitution

= w = 12 ;

= k = 1 gives the classical Hankel integral [10] Z 1 1 (xt) 2 | (xt) f (t) dt; (2) (l f ) (x) = 0

1 2 C; Re ( ) > 1; x > 0; whereas, the substitution = k; = w = k a modi…ed Hankel transform Z 1 1 1 1 (xt) k 2 | k (xt) k f (t) dt; (l ;k f ) (x) =

1 is in fact 2 (3)

0

2 C; Re ( ) > 1; k > 0; x > 0 : Let f be a complex-valued function de…ned on (0; 1) : Then, the generalized Hankel transform (1) has thoroughly been investigated on the space lv;r of those complex-valued Lebesgue measurable functions de…ned on (0; 1) such that 1 Z 1 r r dx v kf kv;r = jx f (x)j ; x 0 where 1

r < 1 and kf kv;1 = esssupxv jf (x)j ; v 2 ( 1; 1) : x>0

By

( ) we denote the maximum value function given by [1; (16)] ( ) = max

1 1 ; r r0

;

(4)

1 1 + 0 = 1 and 1 1: r r By g we denote the Mellin - type convolution product of …rst kind [10; 12] Z 1 y 1 f ty 1 g (y) dy; (5) (f g g) (t) =

where

0

and, by D we denote the Schwartz’space of test functions of compact support de…ned on (0; 1) : Properties of the product g are enumerated as [3; 4] (i) (f g g) (t) = (g g f ) (t) ; (ii) ((f + g) g h) (t) = (f g h) (t) + (g g h) (t) ; (iii) ( f g g) (t) = (g g f ) (t) ; is complex number ; (vi) ((f g g) g h) (t) = (f g (g g h)) (t) : For a complete investigation, we recall the following theorem [2; Theorem 4 (9)] . Theorem 1 Let 1 r 1 and ( ) be given as in (4) : If 1 < r < 1 and (r) k (v Re (w) 1) + 32 < Re ( ) + 32 ; then for all s r such that 0

s

k (v

Re (w)

3 1) + 2

1

1 1 and + 0 = 1, the operator l ; ;w;k; belongs to lv;r ; l1 v+Re(w s s further a one-to-one mapping from lv;r onto l1 v+Re(w );s :

);s

and is

303

3

on a generalized Hankel integral transform and its extension to ...

2

NECESSARY THEOREMS

Theorem 2 Let f; g be l1 functions de…ned on (0; 1) and x > 0: Let product de…ned by the integral equation Z 1 y Re(w ) f (xy) g (y) dy: (f g) (x) =

denote the

(6)

0

Then, we have (l

; ;w;k;

(f g g)) (x) = (l

; ;w;k;

f

g) (x) ;

whenever the integrals exist. Proof Let f; g 2 l1 : Then, the Fubini’s theorem gives Z 1 1 | (xt) k tw (f g g) (t) dt (l ; ;w;k; (f g g)) (x) = x Z 01 Z 1 1 w k i.e = x | f ty 1 g (y) y (xt) t 0

1

dydt:

0

Hence, change of variables yields Z 1 Z 1 1 (l ; ;w;k; (f g g)) (x) = (xy) | ((xy) z) k z w f (z) dzy w 0 Z 01 i.e = y (w ) (l ; ;w;k; f ) (xy) g (y) dy:

g (y) dy

0

Hence, the theorem is completely proved. Theorem 3 Let f; g and h be l1 functions de…ned on (0; 1); then we have f (g g h) = (f g) h: Proof By using de…nitions (5) and (6) ; we obtain Z 1 (f (g g h)) (x) = y Re(w ) f (xy) (g g h) (y) dy 0 Z 1 Z 1 Re(w ) i.e = y t 1 g yt 1 h (t) dt dy 0 Z 1 Z 1 0 1 i.e = t h (t) y Re(w ) f (xy) g yt 1 dydt: 0

0

As earlier, change of variables and Fubini’s theorem yield Z 1 Z 1 (f (g g h)) (x) = h (t) (tz)Re(w ) f (x (tz)) g (z) dzdt 0 Z 01 Z 1 w i.e = t h (t) z Re(w ) f ((xt) z) g (z) dzdt 0 0 Z 1 Re(w ) i.e = t (f g) (xt) h (t) dt: 0

304

4

S.K.Q. Al-Omari

That is, (g g h)) (x) = ((f

(f

g)

h) (x) :

This completes the proof of the theorem. For more convenience, let us consider the case where r = s and v = Therefore, we state and prove the requirements of construction. Theorem 4 Let f 2 lv;r and g 2 D: Then, f

Re (w

)+1 z

:

g 2 lv;r :

Proof Let f 2 lv;r and g 2 D be given; then we have Z

1 0

v

jx (f

dx g) (x)j = x r

By Jensen’s inequality, we get Z 1 Z r dx v jx (f g) (x)j x 0 Z i.e

Z

1

v

x

0

1Z 1

0 0 1Z 1 0

Z

1

y

Re(w

)

r

f (xy) g (y) dy

0

xv y

Re(w

xv y

Re(w

0

dx : x

r

dx x r dx ) f (xy) jg (y)j dy x

)

f (xy) g (y) dy

dx (since is a measure on x (0; 1) we can apply Jensen’s inequality) Z 1 Z 1 (w ) vr dx y Re g (y) j(xy)v f (xy)jr x 0 0

i.e

dy: (7)

Applying change of variables for (7) and using the fact that f 2 lv;r imply Z 1 Z 1 Z 1 dz r dx v (Re(w ) v)r+1 jx (f g) (x)j y g (y) dy jz v f (z)jr x z 0 0 0 Z 1 dz M jz v f (z)jr ; z 0 where M= Therefore,

Z kf

y (Re(w

) v)r+1

g (y) dy:

K

gkv;r < M kf kv;r .

(8)

The proof of the theorem is therefore completed.

2.1

CONSTRUCTED SPACES OF BOEHMIANS

Let us denote by

the set of all sequences ( n ) 2 D such that Z 1 n (x) dx = 1; 0

(9)

305

5

on a generalized Hankel integral transform and its extension to ...

Z

1 0

supp

j

n (x)j dx n (x)

< A; A 2 R; A > 0;

(10)

! 0 as n ! 1;

(11)

then every sequence ( n ) satisfying the equations (9) (11) is called a delta sequence. ;g Following theorems are needful for establishing the space D; of Boehmians: lv;r ; Theorem 5 Let f 2 lv;r and ;

( g ) = (f

2 D; then f

)

:

Similar proof is already given to Theorem 3. Hence, we prefer we omit the details. Theorem 6 Let f1 ; f2 2 lv;r and

1;

2

2 D; then we have

(i) (f1 + f2 ) 1 = f1 1 + f2 1; (ii)Let fn ! f as n ! 1; then for every 2 D we have fn !f as n!1: Proof of this theorem is straightforward. It follows from simple integration. Theorem 7 Let f 2 lv;r and ( n ) 2 Proof Let f 2 lv;r and ( n ) 2 choose 2 D such that

; then f

n

! f as n ! 1 in lv;r :

: Since the space D is dense in lv;r we can easily kf

k< ;

(12)

> 0: By using (8) and (12) we get that k(f

)

nk

M kf

k 0 such that jg (y) g (x)j < whenever jy xj : Since supp n ! 0 as n ! 1 there is an integer n1 2 N such that supp

n

[

; ] ; 8n

n1 :

In addition, let [a; b] be a bounded set such that supp 0; 8x 2 = [a ; b + ].

[a; b] : Hence,

(x) =

306

6

S.K.Q. Al-Omari

Moreover, the fact that j(g (y) g (1))j < and Jensen’s inequality imply Z 1 Z 1 Re(w ) r v k( ) (x)kv;r x y (xy) n (y) dy (x) n (t) dy n 0 0 Z 1 Z 1 j(g (y) g (1))jr j n (y)j dydx i.e xv 0 0 Z 1 Z b+ r j n (y)j dy dx i.e a

r

i.e

r

dx x

0

(b

a + 2 ) N;

where

Z

N=

1 0

j

(14)

n (y)j dy:

Hence, by using (12) ; (13) and (14) ; we, for large values of n; write kf

n

f krv;r

k(f

r n kv;r

)

< M +

r

=

r 1

M+

(b

+k

krv;r + k

n

a + 2 )N +

(b

a + 2 )N + 1

f krv;r

:

Hence, f n ! f as n ! 1: ;g is therefore well de…ned. The Boehmian space D; lv;r ; A typical element in

D; ;g lv;r ;

is denoted by the equivalence class of quotients as ffn g ; f! n g

where ffn g 2 lv;r and f! n g 2

:

De…nition 8 (i)The sum of two Boehmians by

fgn g ffn g ; 2 f! n g f ng

D; ;g lv;r ;

is de…ned

ffn g fgn g ffn !ng n + gn + = ; f! n g f ng f! n g n g (ii)The multiplication of a Boehmian by a scalar in ffn g = f! n g

fgn g ffn gn g = f ng f! n g n g

and D

Similarly, the reader can establish the space De…nitions for

D; ;g lv;r ;g

is de…ned by

ffn g f fn g = ; f! n g f! n g

2 C, the space of complex numbers. (iii)The operation and the di¤erentiation in ffn g f! n g

D; ;g lv;r ;

D; ;g lv;r ;

ffn g fD fn g = : f! n g f! n g

D; ;g lv;r ;g

can be stated as follows.

are de…ned by

.

307

7

on a generalized Hankel integral transform and its extension to ... D; ;g lv;r ;g

De…nition 9 (i) The sum and scalar multiplication in

are de…ned as

ffn g fgn g ffn g n + gn g ! n g + = f! n g f ng f! n g n g and ffn g = f! n g

ffn g f fn g = ; f! n g f! n g

(ii)The operation g and the di¤erentiation in ffn g ffn g gn g fgn g = g f! n g f ng f! n g n g De…nition 10 Let

ffn g 2 f! n g

D; ;g lv;r ;g ;

generalized Hankel transform lge; lge;

are also given as

ffn g fD fn g = : f! n g f! n g

and D

then, by vertue of Theorem 2, we de…ne the ffn g as f! n g

of

;w;k;

ffn g f! n g

;w;k;

D; ;g lv;r ;g

fl

=

; ;w;k;

f! n g

fn g

(15)

D; ;g lv;r ; :

which belongs to the space

Theorem 11 The mapping in (15) is well - de…ned. Let

ffn g fgn g = 2 f! n g f ng

D; ;g lv;r ;g ;

fn g

then we have m

= gm g ! n :

(16)

Employing the mapping (15) and Theorem 2 imply that l Hence,

fl

; ;w;k;

f! n g Therefore,

fn g

; ;w;k;

fl

fn

; ;w;k;

fl

f ng

m

gn g

; ;w;k;

f! n g

=l

gm

!n:

(17)

D; ;g lv;r ; :

in

fn g

; ;w;k;

=

fl

; ;w;k;

f ng

gn g

:

This completes the proof of the theorem. Lemma 12 lge;

;w;k;

is an isomorphism from

Proof Let us …rst establish that lge;

;w;k;

D; ;g lv;r ;g

into

is one-to-one.

D; ;g lv;r ; :

308

8

S.K.Q. Al-Omari

Given lge; m

=l

; ;w;k;

ffn g fgn g = lge; ;w;k; : Then, it follows that l f! n g f ng ! n : Therefore, Theorem 2 implies

;w;k;

gm

l

; ;w;k;

Employing the transform l

(fn g

m)

=l

; ;w;k;

implies fn g

; ;w;k;

m

; ;w;k;

fn

(gm g ! n ) : = gm g ! n : That is,

ffn g fgn g = : f! n g f ng Now we establish that lge; ;w;k; is onto. fl ; ;w;k; fn g ;g Let be arbitrary; then l ; ;w;k; fn ! m = l ; ;w;k; fm 2 D; lv;r ; f! n g ! n for every choice of m; n 2 N: Hence, l ; ;w;k; (fn g ! m ) = l ; ;w;k; (fm g ! n ). That is, fl ; ;w;k; fn g ffn g lge; ;w;k; = : f! n g f! n g

This complete the proof of the lemma. The product g can also be extended to lge;

;w;k;

ffn g g f! n g

D; ;g lv;r ;g

ffn g g f! n g l ; ;w;k; (fn g ) f! n g ((By Equation 15)) fl ; ;w;k; fn g f! n g ((By Theorem 2)) fl ; ;w;k; fn g f! n g

= lge;

i.e

=

i.e

=

i.e

=

in the sense that

;w;k;

Hence, by Equation 15, we obtain lge;

;w;k;

ffn g g f! n g

= lge;

Theorem 13 The mapping lge; ;w;k; : to and - convergence. Proof We show …rst that lge; to - convergence. Let

n

!

in

D; ;g lv;r ;g

;w;k;

:

;w;k;

D; ;g lv;r ;g D; ;g lv;r ;g

!

!

ffn g f! n g D; ;g lv;r ; D; ;g lv;r ;

:

is continuous with respect is continuous with respect

as n ! 1: Then, there are fn;k and fk in lv;r such that n

=

ffn;k g f! k g

and

=

ffk g f! k g

309

9

on a generalized Hankel integral transform and its extension to ...

and fn;k ! fk as n ! 1 for every k 2 N: Therefore l n ! 1 in lv;r : Therefore, fl as n ! 1 in That is lge;

fn;k ! l

; ;w;k;

fk as

fn;k g fl ; ;w;k; fk g ! f! k g f! k g

; ;w;k;

D; ;g lv;r ; .

ffn;k g f! k g

;w;k;

! lge;

ng ;

2

D; ;g lv;r ;g

ffk g f! k g

;w;k;

Now, we establish the continuity of lge; Let f

; ;w;k;

;w;k;

with respect to

be given such that

we …nd ffn g 2 lv;r and f! n g 2

such that (

n n

fn ! 0 as n ! 1: Therefore ;w;k;

((

((

n

) g !n) =

n

l

) g !n) =

lge;

as n ! 1 in

!

- convergence.

D; ;g lv;r ;g

as n ! 1: Then, ffn g ! k g ) g !n = ' fn and f! k g

; ;w;k;

in

D; ;g lv;r ; :

(fn g ! k ) : !k

Hence, we have lge;

;w;k;

fl

; ;w;k;

;w;k;

n

fn

!k

!k g

'l

; ;w;k;

fn ! 0

as n ! 1 in lv;r : Therefore lge;

;w;k;

((

n

) g ! n ) = lge;

lge;

;w;k;

! n ! 0 as n ! 1:

Hence, lge; ;w;k; n ! lge; ;w;k; as n ! 1: The proof of this theorem is completed. Con‡ict of Interests The author declares that there is no con‡ict of interests regarding the publication of this paper.

References [1] R. Roopkumar, Ridgelet transform on square integrable Boehmians, Bull. Korean Math. Soc. 46(5) (2009), 835-844 [2] A. A. Kilbas and J. J. Trujillo, Generalized Hankel transform on `v;r space, Integ. Transf. Spec. Funct., 9(4), (2000), 271-286. [3] R. Roopkumar, Mellin transform for Boehmians, Bulletin of the Institute of Mathematics, Academia Sinica (New Series)4(1) (2009), pp. 75-96

310

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[4] S. K. Q. Al-Omari, On the distributional Mellin transformation and its extension to Boehmian spaces, Int. J. Contemp. Math. Sciences, 6(17) (2011), 801 810. [5] S. K. Q. Al-Omari and A. Kilicman , On di¤raction Fresnel transforms for Boehmians, Abstract and Applied Analysis, Volume 2011(2012), Article ID 712746. [6] S. K. Q. Al-Omari , Notes for Hartley transforms of generalized functions , Italian J. Pure Appl. Math. N.28(2011), 21-30. [7] P. Mikusinski, Tempered Boehmians and ultradistributions, Proc. Amer. Math. Soc. 123(3)(1995), 813-817. [8] T. K. Boehme, (1973), The support of Mikusinski operators, Tran.Amer. Math. Soc.176,319-334. [9] S. K. Q. Al-Omari and Kilicman, A. Note on Boehmians for class of optical Fresnel wavelet transforms, Journal of Function Spaces and Applications, Volume 2012(2012), Article ID 405368, doi:10.1155/2012/405368. [10] A. H. Zemanian, Generalized integral transformation, Dover Publications, Inc., New York. First published by interscience publishers (1987), New York. [11] S. K. Q. Al-Omari., Loonker D., Banerji P.K. and Kalla, S.L. Fourier sine(cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces, Integ. Trans. Spl. Funct. 19(6) (2008), 453 –462. [12] R. S. Pathak, Integral transforms of generalized functions and their applications, Gordon and Breach Science Publishers (1997), Australia , Canada, India, Japan. [13] S. K. Q. Al-Omari, Hartley transforms on certain space of generalized functions, Georgian Mathematical Journal, 20(3) (2013),415–426. [14] A. H. Zemanian, Distribution theory and transform analysis. Dover Publications (1987), Inc. New York. [15] S. K. Q. Al-Omari and A. Kilicman, On generalized Hartley - Hilbert and Fourier - Hilbert transforms, Advances in Di¤erence Equations, Vol. 2012, 2012:232 doi:10.1186/1687-1847-2012-232(2012), 1-12. [16] P. Mikusinski and A. Zayed , The Radon transform of Boehmians, Amer. Math. Soc.,118(2) (1993), 561-570. [17] V. Karunakaran and R. Roopkumar, Operational calculus and Fourier transform on Boehmians, Colloq. Math. 102 (2005), 21–32. [18] V. Karunakaran and R. Vembu, Hilbert transform on periodic Boehmians, Houston J. Math. 29 (2003), 439–454.

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on a generalized Hankel integral transform and its extension to ...

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[19] R. Roopkumar, Generalized Radon transform, Rocky Mount. J. Math., 36(4)(2006), 1375-1390. [20] R. Roopkumar, Stieltjes transform for Boehmians, Integral Transf. Spl. Funct. 18(11)(2007),845-853. [21] D. Loonker, P. K. Banerji, L. Debnath , On the Hankel transform for Boehmians, Integral Trasforms Spec. Funct., 21(7)(2010) , 479–486. [22] D. Loonker and P. K. Banerji , Wavelet transform of fractional integrals for integrable Boehmians, Applications and Applied Mathematics, 5(1)(2010),1 – 10

J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 3-4,312-321 ,2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 312

ELASTIC STRING WITH INTERNAL BODY FORCES IGOR NEYGEBAUER

Abstract. The internal body forces are included into the model of an elastic string. That allows to consider the steady state problems for infinite string. The body forces are considered as nonlocal forces. Variational formulation of the string problem is given using the principle of virtual work. Some analytical solutions of the stated boundary value problems are presented.

1. Introduction The models of an elastic string are considered in many papers and books [1], [2], [3],[4], [5], [6], [7], [8], [9], [14], [16]. These models do not use any constitutive law for the internal body forces and they cannot give a solution of the steady state problem for the infinite string. There are exist MAC models in mechanics [10], [11], [12]. The MAC model for an elastic string is given in [13], where an additional term in form of a local internal body force was added to the string equation. That string model can consider infinite strings with zero displacements at infinity. This paper consider the nonlocal internal body forces and that allows to use arbitrary boundary conditions at infinity. 2. Internal body forces The internal body forces could be considered from two points of view. The first one is that these forces are physical forces with their specific law. The second one is that the additional body forces are some corrections to the stated physical model to include the specific solutions into the MAC model. We will not go into detailed physical considerations of the deformations of a string like [13]. The constitutive law for the internal body forces of a string is taken in the following form: (2.1)

f = f1 + f2 ,

where 0 < x < L, L is the finite length of the string, L < ∞, the first force f1 is the resultant force of interactions between a given internal point of a string and other internal points.  2  Z  1 L ∂ u ∂2u 0 (2.2) f1 = α1 [u(x) − u(x0 )] + α2 (x) − (x ) dx0 + L 0 ∂x2 ∂x2 (2.3)

1 + L

Z 0

L

  4  ∂ u ∂4u 0 α3 (x) − (x ) dx0 = ∂x4 ∂x4

Key words and phrases. Mechanical models, elastic string, internal body forces. 2010 AMS Math. Subject Classification. Primary 74A99, 76A99, 78A99; Secondary 80A99, 81P99. 1

ELASTIC STRING WITH INTERNAL BODY FORCES

2

I. NEYGEBAUER

∂2u ∂4u (x) + α3 4 (x) − C1 , 2 ∂x ∂x where α1 , α2 , α3 are the material constants and the constant C1 equals  Z  1 L ∂2u ∂4u α1 u(x0 ) + α2 2 (x0 ) + α3 4 (x0 ) dx0 . (2.5) C1 = L 0 ∂x ∂x (2.4)

= α1 u(x) + α2

The local internal body force f1 was considered in [13] in the form ∂4u ∂2u (x) + α (x). 3 ∂x2 ∂x4 The second force f2 is the resultant force of interactions between a given internal point of a string and the boundary points.   2 ∂2u ∂ u (x) − (0) + (2.7) f2 = β1 [u(x) − u(0)] + β2 ∂x2 ∂x2  4  ∂ u ∂4u (2.8) +β3 (x) − (0) + β1 [u(x) − u(L)] + ∂x4 ∂x4  2    4 ∂ u ∂2u ∂4u ∂ u (2.9) +β2 (x) − (L) + β3 (x) − (L) = ∂x2 ∂x2 ∂x4 ∂x4 (2.6)

f1 = α1 u(x) + α2

∂2u ∂4u (x) + 2β3 4 (x) − C2 , 2 ∂x ∂x where β1 , β2 , β3 are the material constants and the constant C2 equals    2  4 ∂2u ∂4u ∂ u ∂ u (2.11) C2 = β1 [u(0) + u(L)] + β2 (0) + (L) + β (0) + (L) . 3 ∂x2 ∂x2 ∂x4 ∂x4

(2.10)

= 2β1 u(x) + 2β2

The force f2 was taken as f2 = 0 in [13]. Then the equation 2.1 will take the form (2.12)

f = γ1 u(x) + γ2

∂2u ∂4u (x) + γ (x) − C, 3 ∂x2 ∂x4

where C = C1 + C2 and (2.13)

γ1 = α1 + 2β1 , γ2 = α2 + 2β2 , γ3 = α3 + 2β3 .

If the length of the string is infinite then u(∞) in the equation (2.11) will be replaced by the parameter p which could obtained satisfying the boundary condition. 3. Statement of the problem Many books and papers consider the statement of the string problem, for example [3], [14], [15], [17]. The equation of one-dimensional motion of the string is taken in the form ∂2u ∂2u (3.1) T0 2 − f = ρ 2 − q(x, t), ∂x ∂t where T0 is the constant tension applied to the string, x− is a Cartesian coordinate of a cross-section, 0 ≤ x ≤ L, L− is the length of the string, ρ− is the density of mass per unit length, u− is the transversal displacement of a cross-section, t− is time, q(x, t)− is the density of the transversal external body forces per unit length. The density of the transversal internal body forces per unit length is taken in the form of Eq. (2.12).

313

ELASTIC STRING WITH INTERNAL BODY FORCES

I. NEYGEBAUER

314

3

The following boundary conditions could be taken into consideration: (3.2)

u(0) = u0 , u(L) = uL ,

∂2u ∂2u (0) = 0, (L) = 0. ∂x2 ∂x2 The transversal forces at the ends of the string is given by

(3.3)

P (0) = −T0 u0 (0), P (L) = T0 u0 (L).

(3.4)

4. Principle of virtual work and boundary conditions Consider a string with β3 = 0. That means that the non-local internal body forces are included into the string equation. 4.1. Case α3 = 0. Consider the steady state problem given by the eq. (3.1) at α3 = 0 ∂2u ∂2u (4.1) T0 2 − α1 u − α2 2 + C + q = 0. ∂x ∂x Let δu(x) is a virtual displacement of the string and P (0), P (L) are the transversal forces acting at the end points of the string. Then multiplying the Eq. (4.1) by δu and integrating the result from 0 to L we will get  Z L ∂2u ∂2u (4.2) T0 2 − α1 u − α2 2 + C + q δu(x)dx = 0. ∂x ∂x 0 We can use the integration by parts and transform the integral Z

L 00

0

(T0 − α2 )u δudx = (T0 − α2 )u

(4.3)

δu|L 0

0

Z −

L

(T0 − α2 )u0 δu0 dx.

0

If we substitute the eq. (4.3) into the eq. (4.2) then we obtain Z L (4.4) [(T0 − α2 ) u0 δu0 + α1 uδu − Cδu − qδu] dx − (T0 − α2 )u0 δu|L 0 0

or Z (4.5) 0

L

 1  δ (T0 − α2 )u02 + α1 u2 dx = 2

Z

L

(C + q)δudx + (T0 − α2 )u0 δu|L 0.

0

The last term in the eq. (4.5) equals zero according to the prescribed boundary conditions eq. (3.2). Then the following variational problem is obtained: Z L Z L  1 (T0 − α2 )u02 + α1 u2 − 2qu dx − 2 (4.6) δ Cδudx = 0. 0 0 2 The external transversal forces at the ends of a string should be calculated as (4.7)

P (L) = T0 u0 (L), P (0) = −T0 u0 (0).

If we accept the existence of the potential energy and the potential energy of a string is introduced in the form Z  1 L (T0 − α2 )u02 + α1 u2 dx (4.8) Π= 2 0 then the principle of virtual work could be applied: (4.9)

δΠ = δ 0 W,

ELASTIC STRING WITH INTERNAL BODY FORCES

4

I. NEYGEBAUER

where the variation of the potential energy is Z L   1 (4.10) δΠ = δ α1 u2 + (T0 − α2 )u02 dx. 2 0 The virtual work of the external forces q, P and of a part of internal forces is Z L (4.11) δ0 W = (C + q)δudx + P δu|L 0, 0

where P is the transversal force applied at the ends of the string. If we substitute the eqs. (4.10), (4.11) into the eq. (4.9) then the following equation will be available Z L Z  1 α1 u2 + (T0 − α2 )u02 dx = (C + q)δudx + P δu|L (4.12) δ 0. 2 0 Subtracting eq. (4.12) from the eq. (4.5) we will get [(T0 − α2 )u0 − P ] δu|L 0 = 0.

(4.13)

The eq. (4.13) shows that the following boundary conditions at the ends of the string could be applied: P (L) = (T0 − α2 )u0 |L , P (0) = −(T0 − α2 )u0 |0

(4.14) or (4.15)

u

is prescribed. 4.2. Case α3 6= 0. Consider the steady state problem given by the eq. (3.1) ∂2u ∂2u ∂4u − α1 u − α2 2 − α3 4 + C + q = 0. 2 ∂x ∂x ∂x Let δu(x) is a virtual displacement of the string and P (0), P (L) are the transversal forces acting at the end points of the string. Then multiplying the Eq. (4.16) by δu and integrating the result from 0 to L we will get  Z L ∂2u ∂4u ∂2u (4.17) T0 2 − α1 u − α2 2 − α3 4 + C + q δu(x)dx = 0. ∂x ∂x ∂x 0 (4.16)

T0

We can use the integration by parts and transform the integrals Z L Z L L 0000 000 (4.18) α3 u δudx = [α3 u δu]0 − α3 u000 δu0 dx = 0

0

00 0 L = α3 u000 δu|L 0 − α3 u δu |0 +

(4.19)

Z

L

α3 u00 δu00 dx,

0

Z (4.20) 0

L

(T0 − α2 )u00 δudx = (T0 − α2 )u0 δu|L 0 −

Z

L

(T0 − α2 )u0 δu0 dx.

0

If we substitute the eqs. (4.18), (4.19), (4.20) into the eq. (4.17) then we obtain Z L (4.21) [(T0 − α2 ) u0 δu0 + α1 uδu + α3 u00 δu00 − (C + q)δu] dx− 0

(4.22)

000 L 00 0 L −(T0 − α2 )u0 δu|L 0 + α3 u δu|0 − α3 u δu |0 = 0

315

ELASTIC STRING WITH INTERNAL BODY FORCES

316

I. NEYGEBAUER

5

or Z (4.23) 0

Z (4.24)

=

L

 1  δ (T0 − α2 )u02 + α1 u2 + α3 u002 dx = 2

L 00 0 L (C + q)δudx + [(T0 − α2 )u0 − α3 u000 ] δu|L 0 + α3 u δu |0 .

0

The last two terms of the eq. (4.24) equal zero because of the boundary conditions eqs. (3.2) and (3.3). Then the following variational problem is obtained: Z (4.25)

δ 0

L

 1 (T0 − α2 )u02 + α1 u2 + α3 u002 − 2qu dx − 2 2

Z

L

Cδudx = 0. 0

The external transversal forces at the ends of a string should be calculated according to the eqs. (4.7). If we accept the existence of the potential energy and the potential energy of a string is introduced in the form Z  1 L (4.26) Π= (T0 − α2 )u02 + α1 u2 + α3 u002 dx 2 0 then the principle of virtual work could be applied: δΠ = δ 0 W,

(4.27)

where the variation of the potential energy is Z L   1 α1 u2 + (T0 − α2 )u02 + α3 u002 dx. (4.28) δΠ = δ 2 0 The virtual work of the external forces q, P and of the part of the internal forces is Z L (4.29) δ0 W = (C + q)δudx + P δu|L 0, 0

where P is the transversal force applied at the ends of the string. If we substitute the eqs. (4.28), (4.29) into the eq. (4.27) then the following equation will be available Z L Z  1 2 02 002 α1 u + (T0 − α2 )u + α3 u dx = (C + q)δudx + P δu|L (4.30) δ 0. 2 0 Subtracting eq. (4.30) from the eq. (4.23), (4.24) we will get (4.31)

00 0 L [(T0 − α2 )u0 − α3 u000 − P ] δu|L 0 + α3 u δu |0 = 0.

The eq. (4.31) shows that the following boundary conditions at the ends of the string could be applied • either u00 = 0 or u0 is prescribed, • either P = −α3 u000 + (T0 − α2 )u0 or u is prescribed.

ELASTIC STRING WITH INTERNAL BODY FORCES

6

I. NEYGEBAUER

5. Examples of string with local internal body forces 5.1. Example of string without internal body forces. Consider a simple particular example to show that the theory of the string with internal body forces has solution, where the classical problem does not have any one. Let us take the steady state problem without any given distributed external forces and the length of the string is infinite. Then the classical equation is d2 u = 0. dx2

(5.1) If the boundary conditions are

u(0) = u0 6= 0, u(∞) = 0,

(5.2)

then it is easy to see, that the solution of the stated problem Eqs. (5.1), (5.2) does not exist. 5.2. Example 1 of string with internal body forces. Consider now the same steady state problem for the string with the internal body forces. The differential equation of the problem at α3 = 0 is d2 u d2 u − α1 u − α2 2 = 0. 2 dx dx The boundary conditions are the Eq. (5.2). The solution of the problem (5.2), (5.3) with internal body force exists and equals (5.3)

T0

(5.4)

u = u0 exp(λx),

where r λ=−

(5.5)

α1 . T0 − α2

The above solution Eq. (5.4) exists if (5.6)

T0 > α2 .

The first of the eqs. (3.4) gives the following value of the transversal force applied at the origin √ T0 u0 α1 (5.7) P (0) = −T0 u0 (0) = −T0 u0 λ = √ . T0 − α2 The second of the eqs. (4.14) gives the external transversal force applied at the origin p (5.8) P = −(T0 − α2 )u0 (0) = u0 α1 (T0 − α2 ). 5.3. Example 2 of string with internal body forces. If we consider more general problem with internal body force and α3 6= 0, then the differential equation of the problem will take the form d2 u d2 u d4 u − α1 u − α2 2 − α3 4 = 0. 2 dx dx dx The boundary conditions are taken the Eqs. (5.1), (5.2): (5.9)

T0

(5.10)

u(0) = u0 6= 0, u(∞) = 0,

(5.11)

d2 u d2 u (0) = 0, 2 (∞) = 0. 2 dx dx

317

ELASTIC STRING WITH INTERNAL BODY FORCES

I. NEYGEBAUER

318

7

The boundary conditions Eqs. (5.11) are obtained as follows - we require that the equation Eq. (5.1) without body forces should be satisfied at the boundary or we can use the variational conditions. The solution of the problem with internal body forces Eqs. (5.9), (5.10), (5.11) exists and equals (5.12)

u=

u0 (λ2 exp λ1 x − λ21 exp λ2 x), λ22 − λ21 2

where s (5.13)

λ1,2 = −

T0 − α2 ±

p (T0 − α2 )2 − 4α1 α3 , 2α3

where two inequalities should be fulfilled. The first inequality is the Eq. (5.6) and the second one is (T0 − α2 )2 − 4α1 α3 > 0.

(5.14)

The external transversal force applied at the origin according to the variational approach is
√ √ α1 α3 + T0 − α2 u0 α1 0 000 p (5.15) P (0) = −(T0 − α2 )u (0) + α3 u (0) = . √ T0 − α2 + 2 α1 α3 The eq. (3.4) gives another expression of P : (5.16)

√ T0 u0 α1 . √ T0 − α2 + 2 α1 α3

P (0) = −T0 u0 (0) = p

If the left hand-side of the Eq. (5.14) equals to zero, then the solution will take the form   λx (5.17) u = u0 1 + exp(−λx), 2 where r (5.18)

λ=

2α1 . T0 − α2

Then the transversal end forces are r 0

000

(5.19) P (0) = −(T0 − α2 )u (0) + α3 u (0) = u0

  α1 (T0 − α2 ) 2α1 α3 1+ 2 (T0 − α2 )2

and (5.20)

√ T0 u0 α1 P (0) = −T0 u (0) = 2(T0 − α2 ) 0

correspondingly. The considered example of the string problem shows that the introduced internal body forces allow to obtain solutions in the cases, where the classical problem does not have any solution.

ELASTIC STRING WITH INTERNAL BODY FORCES

8

I. NEYGEBAUER

6. Examples of the string with nonlocal internal body forces 6.1. Example 1 of string with finite length. Consider the problem Z 1 L (6.1) T0 u00 − (α1 + 2β1 )u + α1 udx + β1 (u0 + uL ) = 0, L 0 (6.2)

u(0) = u0 , u(L) = uL .

Let (6.3)

γ1 = α1 + 2β1 ,

and the constant is (6.4)

1 C= L

L

Z

α1 udx + β1 (u0 + uL ). 0

Then the equation(6.1) will take the form (6.5)

T0 u00 − γ1 u + C = 0.

The general solution of the equation (6.5) is (6.6)

u = A1 ekx + A2 e−kx +

C , γ1

where A1 , A2 are arbitrary constants and r γ1 (6.7) k= . T0 If the equation (6.6) is substituted into the equations (6.2) and (6.4) then the following system of the three linear algebraic equations with respect to C, A1 , A2 will be obtained. (6.8) (6.9)

α1 (ekL − 1)A1 − α1 (e−kL − 1)A2 − 2β1 kLC = −β1 kL(u0 + uL ), A1 + A2 +

1 C = u0 , γ1

1 C = uL . γ1 This system of equations could be easily solved and the solution of the problem will be the equation (6.6). (6.10)

ekL A1 + e−kL A2 +

6.2. Example 2 of the string with infinite length. Consider the problem given in example 1. Let L → 0. It is supposed that the solution should be bounded. Then the equation (6.6) shows that (6.11)

A1 = 0.

Then the equations (6.8), (6.9), (6.10) will take the form γ1 (6.12) C = (u0 + u∞ ), 2 (6.13)

u0 = A2 +

(6.14)

u∞ =

C , γ1

C . γ1

319

ELASTIC STRING WITH INTERNAL BODY FORCES

I. NEYGEBAUER

320

9

Substituting the equation (6.12) into the equation (6.14) and simplifying the result the following consequence follows (6.15)

u0 = u∞ .

The equation (6.15) the contradicts to the real situation when we can apply the boundary conditions independent on both ends of a string. Therefore the statement of the problem should be improved. Let (6.16)

T0 u00 − γ1 u + C = 0,

(6.17)

u(0) = u0 , u(∞) = u∞ ,

where Z 1 L α1 udx + β1 (u0 + p), L → ∞. L 0 p is a parameter which will be defined later. Following the solution of the examples 1 and 2 the equations (6.12), (6.13), (6.14) will take the form γ1 (6.19) C = (u0 + p), 2

(6.18)

C=

(6.20)

u0 = A2 +

(6.21)

u∞ =

C , γ1

C . γ1

The equations (6.19), (6.20), (6.21) give (6.22)

A2 = u0 − u∞ ,

(6.23)

p = 2u∞ − u0 .

Then the solution of the given problem is (6.24)

u = (u0 − u∞ )e−kx + u∞ . References

[1] S.S. Antman, The equations for large vibrations of string, American Mathematical Monthly, 87, 359–380, (1980). [2] S.S. Antman, Non-linear problems in elasticity, Springer, Applied Mathematical Sciences 107, 1991. [3] S. Antman, Nonlinear Problems of Elasticity, Springer, 2005. [4] J.E. Bolwell, The flexible string’s neglected term, Journal of Sound and Vibrations, 206, 618–623, (1997). [5] R. Haberman, Applied partial differential equations, Pearson Prentice Hall, 2004. [6] J.B. Keller, Large amplitude motion of a string, American Journal of Physics, 27, 584–586, (1959). [7] H. Leckar, R. Sampaio, E. Cataldo, Validation of the D’Alembert’s equation for the vibrating string problem, TEMA Tend. Mat. Apl. Comput., 7, No. 1, 75–84, (2006). [8] L.A. Medeiros, J. Limaco, S.B. Menezes, Vibrations of elastic strings: mathematical aspects, J. of Computational Analysis and Applications, Vol. 4, No. 2, Part One, 91–127, (April 2002). [9] L.A. Medeiros, J. Limaco, S.B. Menezes, Vibrations of elastic strings: mathematical aspects, J. of Computational Analysis and Applications, Vol. 4, No. 3, Part Two, 211–263, (July 2002). [10] I. Neygebauer, MAC solution for a rectangular membrane, Journal of Concrete and Applicable Mathematics, Vol. 8, No. 2, 344–352, (2010).

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[11] I.N. Neygebauer, MAC model for the linear thermoelasticity, Journal of Materials Science and Engineering, Vol.1, No.4, 576–585, (2011). [12] I. Neygebauer, Differential MAC models in continuum mechanics and physics, Journal of Applied Functional Analysis, Vol.8, No.1, 100–124, (2013). [13] I. Neygebauer, Mechanical models with internal body forces, Journal of Concrete and Applicable Mathematics, Vol. 12, No.’s 3-4, 181–200, (2014). [14] I.G. Petrovsky, Lectures on Partial Differential Equations, Dover, 1991. [15] A.P.S.Selvadurai, Partial Differential Equations in Mechanics, Springer, 2010. [16] M.D.G. da Silva, L.A. Medeiros, A.C. Biazutti, Vibration of elastic strings: unilateral problem, J. of Compt. Analysis and Applications, Vol. 8, No. 1, 53–73, (2006). [17] P.Villaggio, Mathematical Models for Elastic Structures, Cambridge University Press, 1997. (I. Neygebauer) University of Dodoma, Dodoma, Tanzania E-mail address: [email protected]

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J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.'S 3-4,322-329 ,2015, COPYRIGHT 2015 EUDOXUS PRESS LLC 322

On a Characterization of Hilbert Spaces through Minimality of Orthogonal Projections and Related Topics. Boris Shekhtman Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave., CMC 114, Tampa, FL 33620-5700 [email protected]

Leslaw Skrzypek Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave., CMC 114, Tampa, FL 33620-5700 [email protected]

October 1, 2014 Abstract In the first part we present a new characterization of finite dimensional Hilbert spaces in terms of minimal projections. If a finite dimensional Banach space X = (Rn , || · ||) has the property that for every subspace V in X, the orthogonal projection from X onto V has the minimal norm, then the space X has to be isometric to the Hilbert space. In the second part we estimate the norms of minimal projections onto subspaces of X as operators from lpn to l2n . In the case of hyperplanes the obtained bound is proved to be optimal.

1

Introduction

In this paper we will consider a finite dimensional Banach spaces X = (Rn , ||·||). The symbols S(X) and B(X) will, respectively, denote the unit sphere and the unit ball of X. We will use h·, ·i to denote the standard inner product on Rn and identify the dual space X ∗ with (Rn , ||·||∗ ) where kxk∗ := sup{hx, yi : y ∈ S(X)}.

(1)

If V is a linear subspace of X then the set of all projections from X onto V is denoted by P (X, V ). A projection Q from X onto V is called minimal

1

Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, L. Skrzypek

323

if ||Q|| = λ(V, X) = inf{kP k : P ∈ P (X, V )}. (2) p If X is a Hilbert space (i.e., kxk = kxk2 := hx, xi) then, for any subspace V of X, the orthogonal projection PV onto V is the unique projection of norm one. In particular, the projection PV is minimal, i.e., it has the least norm among all projections onto V . The main result of section 1 is the reverse statement: Theorem 1. If for every subspace V ⊂ X = (Rn , || · ||) the orthogonal projection is minimal then Y is isometric to a Hilbert space. Since the minimal projection onto every one-dimensional subspace of any Banach space has norm 1, the theorem will directly follow from the following statement: Theorem 2. If for every one-dimensional subspace V ⊂ X = (Rn , || · ||) the orthogonal projection onto Y has norm 1 then X is isometric to a Hilbert space. Is is worth mentioning that there are many other characterizations of Hilbert spaces through projections (see a survey paper [7] for detailed information) and the relation between minimal projections and other class of projections (orthogonal or radial) is also being investigated (see [2], [6] and [10]). The results of Section 1 can be viewed in terms of characterizing minimal projections in the original norm || · || through minimality in `n 2 norm. This leads to section 2 where we investigate the minimal projections equipped with p → 2 norm (that is we will consider the projections as operators from lpn to l2n .). We can define a p → 2 norm of a linear operator L : Rn → Rn as follows ||L||p→2 = sup ||Lx||2 .

(3)

||x||p =1

We obtain the theorem (see Theorem 11 and Theorem 12): Theorem 3. For p > 2, the norms of all minimal projections as operators 1−1 from lpn to l2n are bounded by n 2 p . Additionally, for the class of minimal projections onto hyperplanes, this estimate is optimal (whenPn is even) and the bound is attained for the hyperplane T := ker1 = {x : n i=1 xi = 0}. It is interesting to note that obtaining the sharp estimate for the norms of minimal projections onto hyperplanes in `n p is an open problem (except for the cases p = 1, 2, ∞, see [10]) although it is conjectured that the Pn hyperplane T := ker1 = {x : x = 0} is the worst possible case i i=1 ([11]).

2

Characterization of Hilbert Spaces.

The purpose of this section is to establish a characterization of finite dimensional Hilbert spaces in terms of minimal projections. Before we prove the Theorem 2 we will need to define a few preliminaries (see Chapter 2 in [1]). By Hahn-Banach theorem, for every x ∈ S(X) there exists a functional f ∈ S(X ∗ ) such that f (x) = 1. A point x ∈ S(X)

2

Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, L. Skrzypek

324

is called smooth if such functional is unique. The space X is called smooth if every point in S(X) is smooth. A well-known characterization of smooth points is the following Theorem 4. A point x ∈ S(X) is smooth if and only if the norm || · || is Gateaux differentiable at x, i.e., for every y ∈ S(x) the following limit exists kx + λyk − kxk lim := ρ0 (x, y). (4) λ→0 λ A point x ∈ S(X) is called an exposed point if there exists a functional f ∈ S(X ∗ ) such that f (x) = 1 and for every y ∈ S(X), y 6= x, f (y) < 1. Finally, the space X is called strictly convex if, for any x, y ∈ S(X), x 6= y and all t ∈ (0, 1) we have ktx + (1 − t)yk < 1.

(5)

In a strictly convex space every point x ∈ S(x) is an exposed point (see remark after Theorem 1, Chapter 2 in [1]). We are now ready to address the proof of Theorem 2. We will start with the following Lemma 5. If X = (Rn , || · ||) the orthogonal projection onto every onedimensional subspace of X is minimal then X is strictly convex and smooth. Proof. If every orthogonal projection is minimal then, in particular, every orthogonal projection onto a one-dimensional subspace is minimal hence is of norm one. Now, assume that X is not strictly convex. Then there exists two distinct vectors u, v ∈ S(X) such that w(t) = tu + (1 − t)v ∈ S(X) for all t ∈ [0, 1]. Since hw(t), u − vi = t ku − vk22 + (hu, vi − kvk22 ),

(6)

we can choose t ∈ (0, 1) such that hw(t), u − vi 6= 0.

(7)

w(s) = su + (1 − s)v = w(t) + (s − t)(u − v) ∈ S(X),

(8)

w(t) + α(u − v) ∈ S(X)

(9)

Since we obtain for all α in some interval (−ε, ε). We have   w(t) w(t) , w(t) + α(u − v) Pw(t) (w(t) + α(u − v)) = kw(t)k2 kw(t)k2 =

kw(t)k22 + α hw(t), u − vi kw(t)k22

w(t)

(10) (11)

and since w(t), w(t)+α(u−v) ∈ S(X) we conclude that for all α ∈ (−ε, ε) 2



Pw(t) ≥ Pw(t) (w(t) + α(u − v)) = kw(t)k2 + α hw(t), u − vi . (12) kw(t)k22

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Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, L. Skrzypek

325

By (7) we can choose α ∈ (−ε, ε) such that α hw(t), u − vi > 0, hence

Pw(t) > 1 contradicting the assumptions of the lemma. To prove that X is uniformly smooth, we observe that for every orthogonal projection P on X, the adjoint P ∗ is an orthogonal projection on X ∗ = (Rn , || · ||∗ ) and kP k = kP ∗ k. Thus, in particular, the orthogonal projection onto every one-dimensional space in X ∗ has norm 1 and X ∗ is strictly convex. This implies that X is smooth (see Theorem 2, Chapter 2 in [1]). Theorem 6. Let X = (Rn , || · ||) be a normed space. The following are equivalent: (i) X is a Hilbert space, i.e., k·k = k·k2 . (ii) For every subspace V ⊂ X, the orthogonal projection onto V is minimal. (iii) For every one-dimensional subspace V ⊂ X, the orthogonal projection onto V is minimal, i.e., has norm one. Proof. The implications (i) ⇒ (ii) ⇒ (iii) are obvious. We will prove that (iii) implies (i). Observe that if V is a one-dimensional subspace of X spanned by a vector u ∈ S(x) then every projection Q onto V has the form Qx = f (x)u for some f ∈ X ∗ with f (u) = 1. Additionally kQk = kf k. If Q has norm one then f ∈ S(X ∗ ). Since, by the previous lemma, every point u ∈ S(X) is smooth there exists the unique functional fu ∈ S(X ∗ ) such that fu (u) = 1 and thus there exists the unique projection onto V of norm one: Qx = fu (x)u. By the previous lemma, u is an exposed point: (u + ker fu ) ∩ S(X) = {u},

(13)

i.e., u + ker fu is the unique hyperplane tangent to the sphere S(X). Finally, since Q is the orthogonal projection, we have * +   u u u fu (x)u = ,x u (14) ,x = kuk2 kuk2 kuk22 and fu =

u . kuk22

(15)

The above arguments means that ker fu = u⊥ := {x ∈ X : hu, xi = 0} and hence at every point u on the sphere S(X) the (unique) tangent hyperplane at this point is perpendicular to the vector u. This, combined with the smoothness of S(X), gives a convincing geometric argument that S(X) must be the Euclidean sphere. In a sequel we present a formal proof of this. Since the norm || · || : Rn → R is smooth, it is Gateaux differentiable at every x 6= 0 and for every u ∈ S(X) lim

λ→0

hx, yi kxk kx + λyk − kxk x (y) = = f kxk λ kxk22

(16)

Since the norm is a Gateaux differentiable Lipschitz function on a finitedimensional space, it is Frechet differentiable away from zero (see [5])

4

Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, L. Skrzypek

326

and its Gateuax derivative in the directions of unit basis vectors give the gradient of || · ||, i.e.,   ||x|| ∇(||x||) = · x. (17) ||x||22 On the other hand   ||x|| ||x||2 · ∇(||x||) − ||x|| · ∇(||x||2 ) ∇ = ||x||2 ||x||22 x ||x||2 · ∇(||x||) − ||x|| · ||x|| 2 = 2 ||x||2 =

||x||22 · ∇(||x||) − ||x|| · x . ||x||32

As a result, using (17) we obtain   ||x|| ∇ = 0. ||x||2

(18) (19) (20)

(21)

Hence kxk = C kxk2

(22)

which is what we set out to prove. Remark 7. It is interesting to note that the part (iii) in the statement of the theorem 6 cannot be replaced by (iii’) For every two-dimensional subspace U ⊂ X the orthogonal projection onto U is minimal; or by (iii”) For every subspace of U ⊂ X of codimension 1, the orthogonal projection onto U is minimal. Indeed, A. Komisarski [4], constructed an example of a three dimensional Banach space X, not isometric to a Hilbert space, such that all minimal projections onto every two-dimensional supspaces are unique, orthogonal and have the same norm λ > 1.

3 Minimizing the norms of projections as operators from lpn to l2n . In this section we will investigate the projections as operators from lpn to l2n . For a fixed subspace V , we will minimize the norm of all such projections onto V. The goal will be to find the maximal norms of all minimal projections. Definition 8. For any linear subspace V of Rn we define λp→2 (V, Rn ) = inf{||P ||p→2 : P ∈ P (Rn , V )},

(23)

n λN p→2 = sup{λp→2 (V, R ) : dim V = N }.

(24)

and

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Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, L. Skrzypek

327

Observe that Remark 9. Geometrically speaking, the norm of a projection P : (Rn , || · ||1 ) → (V, || · ||2 )

(25)

is the smallest constant C such that
P (B||·||1 ) ⊂ C · B||·||2 ∩ V .

(26)

As a result, Remark 10. Fix p ≤ 2. For any linear subspace V 6= 0 of Rn , the orthogonal projection PV : `n p → (V, || · ||2 ) has norm one. Hence λp→2 (V, Rn ) = 1

(27)

λN p→2 = 1

(28)

and The main result of this section is the following theorem Theorem 11. Fix p > 2 and consider Rn . For N ≥ 1 we have 1−1 p

2 λN p→2 ≤ n

1 p

Proof. Take q such that can see that

+

1 q

.

(29)

= 1. Considering the dual projections we

λp→2 (V, Rn ) = inf{||P ||p→2 : P ∈ P (Rn , V )}

(30)

= inf{||P ∗ ||2→q : P ∗ ∈ P (Rn , V )} = λ2→q (V, Rn ).

(31)

Hence λp→2 (V, Rn ) ≤ ||PV ||2→q ,

(32)

where PV is the orthogonal projection onto V . Noting that PV (B(`n p )) = B(`n p ) ∩ V and using Remark 9 we can observe that ||PV ||2→q = ||Id/V ||2→q

(33)

λp→2 (V, Rn ) ≤ ||Id/V ||2→q .

(34)

and For a fixed N, the union of all subspaces of dimension N gives the whole space Rn . As a result λN p→2 =

λp→2 (V, Rn ) ≤

sup dim V =N

||Id/V ||2→q = ||Id||2→q .

sup

(35)

dim V =N

Once can easily compute the last quantity using the classical inequality between power means (see [3]). For q < 2 we have  Pn

i=1

n

|xi |q

1/q

 Pn ≤

6

i=1

n

|xi |2

1/2 ,

(36)

Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, L. Skrzypek

328

with ”=” if and only if |xi | = c, for all i. Using the above inequality we get ||Id(x)||q = ||x||q =

n X

!1/q |xi |

q

(37)

i=1

≤ n1/q−1/2

n X

!1/2 |xi |2

1−1 p

= n2

||x||2 .

(38)

i=1

We will show now that the estimate in the above theorem is essentially the optimal. Theorem 12. Fix p > 2 and consider Rn , where n is an even integer. Then 1−1 2 p. λn−1 (39) p→2 = n Pn Proof. Consider the hyperplane T := {x : i=1 xi = 0}. Any projection onto T is given by n X Pz x = x − ( xi ) · z, (40) i=1

for some z such that

Pn

i=1

zi = 1. Let

Qx = x −

n 1 X xi ) · (1, ..., 1). ( n i=1

(41)

Using standard averaging argument (see [9] and [12]) we will show that, for any z, we have ||Q||p→2 ≤ ||Pz ||p→2 . Fix any permutation σ ∈ Sn , we will denote zσ = (zσ(1) , ..., zσ(n) ). One can easily see that ||Pz ||p→2 = ||Pzσ ||p→2 and 1 X Q= Pzσ . (42) n! σ∈S n

Hence ||Q||p→2 ≤ ||Pz ||p→2 and λp→2 (T, Rn ) = ||Q||p→2 . Assume n = 2k, 1 1 take x ∈ S(`n p ) such that x2i = − n1/p and x2i−1 = n1/p , for i = 1, ..., k. 1−1 p

Then ||Q(x)||2 = n 2 n

1−1 2 p

. As a result 1−1 p

2 = ||Q||p→2 = λp→2 (T, Rn ) ≤ λn−1 p→2 ≤ n

,

(43)

which completes the proof. P Essentially the above theorem shows that T := ker1 = {x : n i=1 xi = 0} is the maximal hyperplane for minimal projections considered as a n operators between `n p and `2 spaces.

7

Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, L. Skrzypek

References [1] J. Diestel. Geometry of Banach spacesselected topics, Lecture Notes in Mathematics, Vol. 485. Springer-Verlag, Berlin-New York, 1975 MR 0461094 (57 #1079) [2] C. Franchetti, Projections onto Hyperplanes in Banach Spaces, J. Approx. Theory 38 (1983), 319–333. MR 0711458 (84h:46023) [3] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, Cambridge, at the University Press, 1934. [4] A. Komisarski, On the minimal projections in the finite dimensional Banach Spaces, Ph.D. Thesis, University of Lodz, 2004. In polish. [5] J. Lindenstrauss and D. Preiss, On Frechet differentiability of Lipschitz maps between Banach spaces, Ann. of Math. (2) 157 (2003), no. 1, 257288. MR 1954267 (2003k:46058) [6] W. Odyniec and G. Lewicki, Minimal projections in Banach spaces, Lecture Notes in Mathematics, vol. 1449, Springer-Verlag, Berlin, 1990, Problems of existence and uniqueness and their application. MR 1079547 (92a:41021) [7] B. Randrianantoanina, Norm-one projections in Banach spaces, International Conference on Mathematical Analysis and its Applications (Kaohsiung, 2000). Taiwanese J. Math. 5 (2001), no. 1, 3595. MR 1816130 (2003d:46001). [8] S. Rolewicz, On projections on subspaces of codimension one, Studia Math. 96 (1990), no. 1, 17–19. MR 1055074 (91i:46017) [9] B. Shekhtman and L. Skrzypek Uniqueness of Minimal Projections onto Two-Dimensional Subspaces, Studia Math. 168 (2005), no. 3, 273284. MR 2146127 (2006b:41054) [10] B. Shekhtman and L. Skrzypek Minimal Versus Orthogonal Projecn tions onto Hyperplanes in `n 1 and `∞ , Approximation Theory XIV: San Antonio 2013 Springer Proceedings in Mathematics & Statistics Volume 83, 2014, 343–349. [11] L. Skrzypek, On the Lp norm of the Rademacher projection and related inequalities, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2661– 2669. MR 2497479 (2010d:41043) [12] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics 25, Cambridge University Press, Cambridge, 1991. MR 1144277 (93d:46001)

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TABLE OF CONTENTS, JOURNAL OF CONCRETE AND APPLICABLE MATHEMATICS, VOL. 13, NO.’S 3-4, 2015

Qualitative Analysis of a Solow Model on Time Scales, Martin Bohner, Julius Heim, and Ailian Liu,….…………………………………………………….................................................183-197 Singularities of the Continuous Wavelet Transform in 𝐿𝑝 (ℝ𝑛 ), Jaime Navarro, and Salvador Arellano-Balderas,………………………………………………………………………...198-208 Some Estimation of the Struve Transform in a Quotient Space of Boehmians, S.K.Q. Al-Omari,.. …209-218 On the Extended Fourier Transform of Generalized Functions, S.K.Q. Al-Omari,……...219-231 Intrinsic Supermoothness, Boris Shekhtman, and Tatyana Sorokina,……………………232-241 Approximation by Kantorovich and Quadrature Type Quasi-Interpolation Neural Network Operators, George A. Anastassiou,……………………………………………………….242-251 Asymptotics for Szegő Polynomials of Polya Type, Michael Arciero, and Lewis Pakula,252-265 On One Nonclassical Problem for a Multidimensional Parabolic Equation, Olga Danilkina,…… …….266-278 Multivariate Fuzzy-Random Perturbed Neural Network Approximation, George A. Anastassiou,……………………………………………………………………………….279-300 Notes on the Generalized Hankel Integral Transform and its Extension to a Class of Boehmians, S. K. Q. Al – Omari,……………………………………………………………………….301-311 Elastic String with Internal Body Forces, Igor Neygebauer,………………………………312-321 On a Characterization of Hilbert Spaces through Minimality of Orthogonal Projections and Related Topics, Boris Shekhtman, and Lesław Skrzypek,………………………………...322-329