JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS VOLUME 21, 2016

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Volume 21, Number 1 ISSN:1521-1398 PRINT,1572-9206 ONLINE

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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July 2016

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

28601, USA.

Journal of Computational Analysis and Applications(JoCAAA) is published by EUDOXUS PRESS,LLC,1424 Beaver Trail Drive,Cordova,TN38016,USA,[email protected] http://www.eudoxuspress.com. Annual Subscription Prices:For USA and Canada,Institutional:Print $700, Electronic OPEN ACCESS. Individual:Print $350. For any other part of the world add $130 more(handling and postages) to the above prices for Print. No credit card payments. Copyright©2016 by Eudoxus Press,LLC,all rights reserved.JoCAAA is printed in USA. JoCAAA is reviewed and abstracted by AMS Mathematical Reviews,MATHSCI,and Zentralblaat MATH. It is strictly prohibited the reproduction and transmission of any part of JoCAAA and in any form and by any means without the written permission of the publisher.It is only allowed to educators to Xerox articles for educational purposes.The publisher assumes no responsibility for the content of published papers.

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

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

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

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

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

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

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

Luis A. Caffarelli Department of Mathematics The University of Texas at Austin Austin, Texas 78712-1082 512-471-3160 e-mail: [email protected] Partial Differential Equations George Cybenko Thayer School of Engineering

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

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

Partial Differential Equations, Semigroups of Operators H. H. Gonska Department of Mathematics University of Duisburg Duisburg, D-47048 Germany 011-49-203-379-3542 e-mail: [email protected] Approximation Theory, Computer Aided Geometric Design

Sever S. Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001, AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities, Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics.

John R. Graef Department of Mathematics University of Tennessee at Chattanooga Chattanooga, TN 37304 USA [email protected] Ordinary and functional differential equations, difference equations, impulsive systems, differential inclusions, dynamic equations on time scales, control theory and their applications

Oktay Duman TOBB University of Economics and Technology, Department of Mathematics, TR06530, Ankara, Turkey, [email protected] Classical Approximation Theory, Summability Theory, Statistical Convergence and its Applications

Weimin Han Department of Mathematics University of Iowa Iowa City, IA 52242-1419 319-335-0770 e-mail: [email protected] Numerical analysis, Finite element method, Numerical PDE, Variational inequalities, Computational mechanics

Saber N. Elaydi Department Of Mathematics Trinity University 715 Stadium Dr. San Antonio, TX 78212-7200 210-736-8246 e-mail: [email protected] Ordinary Differential Equations, Difference Equations

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Christodoulos A. Floudas Department of Chemical Engineering Princeton University Princeton,NJ 08544-5263 609-258-4595(x4619 assistant) e-mail: [email protected] Optimization Theory&Applications, Global Optimization

Margareta Heilmann Faculty of Mathematics and Natural Sciences, University of Wuppertal Gaußstraße 20 D-42119 Wuppertal, Germany, [email protected]

J .A. Goldstein Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 901-678-3130 [email protected]

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Hrushikesh N. Mhaskar Department Of Mathematics California State University Los Angeles, CA 90032 626-914-7002 e-mail: [email protected] Orthogonal Polynomials, Approximation Theory, Splines, Wavelets, Neural Networks

Approximation Theory (Positive Linear Operators) Xing-Biao Hu Institute of Computational Mathematics AMSS, Chinese Academy of Sciences Beijing, 100190, CHINA [email protected] Computational Mathematics

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

Jong Kyu Kim Department of Mathematics Kyungnam University Masan Kyungnam,631-701,Korea Tel 82-(55)-249-2211 Fax 82-(55)-243-8609 [email protected] Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, ODE, PDE, Functional Equations.

Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] [email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations, Almost Periodicity & Almost Automorphy

Robert Kozma Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA [email protected] Neural Networks, Reproducing Kernel Hilbert Spaces, Neural Percolation Theory Mustafa Kulenovic Department of Mathematics University of Rhode Island Kingston, RI 02881,USA [email protected] Differential and Difference Equations

M.Zuhair Nashed Department Of Mathematics University of Central Florida PO Box 161364 Orlando, FL 32816-1364 e-mail: [email protected] Inverse and Ill-Posed problems, Numerical Functional Analysis, Integral Equations, Optimization, Signal Analysis

Irena Lasiecka Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional Analysis, [email protected]

Mubenga N. Nkashama Department OF Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170 205-934-2154 e-mail: [email protected] Ordinary Differential Equations, Partial Differential Equations

Burkhard Lenze Fachbereich Informatik Fachhochschule Dortmund University of Applied Sciences Postfach 105018 D-44047 Dortmund, Germany e-mail: [email protected] Real Networks, Fourier Analysis, Approximation Theory

Vassilis Papanicolaou Department of Mathematics

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National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability

Approximation Theory, Banach spaces, Classical Analysis T. E. Simos Department of Computer Science and Technology Faculty of Sciences and Technology University of Peloponnese GR-221 00 Tripolis, Greece Postal Address: 26 Menelaou St. Anfithea - Paleon Faliron GR-175 64 Athens, Greece [email protected] Numerical Analysis

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

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

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

I. P. Stavroulakis Department of Mathematics University of Ioannina 451-10 Ioannina, Greece [email protected] Differential Equations Phone +3-065-109-8283 Manfred Tasche Department of Mathematics University of Rostock D-18051 Rostock, Germany [email protected] Numerical Fourier Analysis, Fourier Analysis, Harmonic Analysis, Signal Analysis, Spectral Methods, Wavelets, Splines, Approximation Theory

Tomasz Rychlik Polish Academy of Sciences Instytut Matematyczny PAN 00-956 Warszawa, skr. poczt. 21 ul. Śniadeckich 8 Poland [email protected] Mathematical Statistics, Probabilistic Inequalities Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620, USA Tel 813-974-9710 [email protected]

Roberto Triggiani Department of Mathematical Sciences University of Memphis Memphis, TN 38152 PDE, Control Theory, Functional

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Analysis, [email protected]

USA. Tel 334-844-6557 office 678-642-8703 home Fax 334-844-6555 [email protected] Approximation Theory, Chebychev Systems, Wavelet Theory

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Xiang Ming Yu Department of Mathematical Sciences Southwest Missouri State University Springfield, MO 65804-0094 417-836-5931 [email protected] Classical Approximation Theory, Wavelets

Xin-long Zhou Fachbereich Mathematik, Fachgebiet Informatik Gerhard-Mercator-Universitat Duisburg Lotharstr.65, D-47048 Duisburg, Germany e-mail:[email protected] Fourier Analysis, Computer-Aided Geometric Design, Computational Complexity, Multivariate Approximation Theory, Approximation and Interpolation Theory

Lotfi A. Zadeh Professor in the Graduate School and Director, Computer Initiative, Soft Computing (BISC) Computer Science Division University of California at Berkeley Berkeley, CA 94720 Office: 510-642-4959 Sec: 510-642-8271 Home: 510-526-2569 FAX: 510-642-1712 [email protected] Fuzzyness, Artificial Intelligence, Natural language processing, Fuzzy logic Richard A. Zalik Department of Mathematics Auburn University Auburn University, AL 36849-5310

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Numerical simulation of an electro-thermal model for superconducting nanowire single-photon detectors 1 Wan Tanga , Jianguo Huanga,b

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and Hao Lic

a Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China b Division of Computational Science, E-Institute of Shanghai Universities, Shanghai 200235, China c State Key Laboratory of Functional Materials for Information, SIMIT, CAS, Shanghai 200050, China

Abstract The electro-thermal model for Superconducting Nanowire Single-Photon Detectors is a nonlinear free boundary problem involving the temperature and the current, which are coupled together by a nonlinear parabolic interface equation and a second order ordinary differential equation. In this paper, we propose a novel method to numerically solve the preceding electro-thermal model. A series of numerical experiments are provided to demonstrate the effectiveness of the method proposed. Keywords. Electro-thermal model, free boundary problem, finite difference method, shooting method

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Introduction

In recent years, superconducting nanowires single photon detection (SNSPD) has emerged as a new and promising single photon detection technology and has received wide attention in the field of applied superconductivity (cf. [1, 8]). The corresponding device structures nanometer zigzag line on the ultra-thin superconducting material, and uses the highly sensitive response of superconducting nanowire to realize single-photon detection. As shown in Figure 1 (see [1]), the key step of SNSPD is to discover the variation of the photon-induced hotspot. In 2007, some researchers in MIT (cf. [15]) proposed a relevant electro-thermal mechanism to account for the variation of the photon-induced hotspot in SNSPD, after a small resistive hotspot forms along the nanowire. In this model, the SNSPD is approximated as a one-dimensional structure, the thermal response is modeled by a one-dimensional nonlinear parabolic interface equation involving the current flowing through the nanowire, and the electrical response is modeled by a second order ordinary differential equation. The two equations are coupled together to form a free boundary problem (cf. [3]). To be more precise, let L denote the length of the superconducting nanowire under discussion, d the wire thickness, and W the width of nanowire. The domain occupying ˜ = (−L/2, L/2). Due to the symmetry of the physical the nanowire is simply written as Ω process, it suffices for us to discuss the variation of physical quantities in the half part Ω = (0, L/2). This domain is further split into two regions, Ωnorm (t) = (0, l(t)) and Ωsuper (t) = (l(t), L/2), corresponding to the normal/resistive and superconducting states, respectively. The interface x = l(t) is used to separate the two states at time t. Let T (x, t) represent the temperature of the material in the point x at time t. Then, as given in [15], T (x, t) is determined by the parabolic interface equation ∂2T α ∂Cn T − (T − Tsub ) = , 0 < x < l(t), t > 0, 2 ∂x d ∂t α ∂Cs T ∂2T κs 2 − (T − Tsub ) = , l(t) < x < L/2, t > 0, ∂x d ∂t

J 2 ρ + κn

(1.1) (1.2)

1 The work of this author was partly supported by NSFC (Grant no. 11171219) and E–Institutes of Shanghai Municipal Education Commission (E03004). 2 Corresponding author. E-mail address: [email protected].

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(a)

(f)

(b)

(c) (e)

(d)

Figure 1: The variation of the photon-induced hotspot in SNSPD. (a) Bias direct current close to (but less than) its critical current, and set the nanowire temperature well below its superconducting critical temperature. (b) Form a small resistive hotspot. (c) The hotspot region forces the supercurrent to flow around the periphery of the hotspot, since the hotspot itself is not large enough to span the width of the nanowire. (d) Form a resistive barrier across the width of the nanowire, results in an easily measurable voltage pulse. (e) Resistive region is increased, the bias current is shunted by the external circuit. (f) The NbN nanowire becomes fully superconducting again. with the initial condition T (x, 0) = T0 , 0 ≤ x ≤ L/2. Observe that T (x, t) is symmetric about x = 0 with respect to x, and L is taken large enough such that the temperature at x = L/2 almost coincides with the substrate temperature. Then we impose the following boundary conditions: ∂T (x, t) = 0, ∂x x=0

T (L/2, t) = Tsub , t > 0.

(1.3)

Moreover, at the interface point x = l(t) we impose the standard interface conditions: T (x, t)|l− = T (x, t)|l+ ,

∂T (x, t) ∂T (x, t) κn − = κs + , t > 0, ∂x ∂x l l

(1.4)

as well as a phase transition condition Ic (T ) = Ic (0) × (1 − (T /TC )2 )2 .

(1.5)

Here, J = I(t) W d is the current density through the nanowire, ρ is the electrical resistivity, κn and κs are the thermal conductivity coefficients, α is the thermal boundary conductance between the film and the substrate, Tsub is the substrate temperature (since the nanowire is thin enough), Cn and Cs are the heat capacity (per unit volume) of the superconducting film, Ic (0) is the initial critical current, and Tc is the critical temperature. We mention that the transition condition is an empirical relation (see [15, p. 582]), which was obtained from an excellent fit with experimental measurements. Using this expression, one can determine 2

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a segment to be resistive when I > Ic (T ), where T = T (x) is the temperature of a nanowire at the position x. The remaining part of the nanowire then belongs to the superconducting state. On the other hand, using Kirchhoff’s first law, we can find as in [3] that the current I(t) through the nanowire satisfies an ordinary differential equation ∫ t dI(t) 1 I(t)R(t) + Lk = (Ibias − I(s))ds + (Ibias − I(t))Z0 , t > 0, (1.6) dt Cbt 0 with the initial condition I(0) = I0 . Differentiating (1.6) once with respect to t gives Cbt

( d2 L I d(I(t)R(t)) dI(t) ) k + + Z = Ibias − I(t), t > 0, 0 dt2 dt dt

(1.7)

where Cbt is the capacitor, an inductor Lk and a resistor R(t) represent respectively the kinetic inductance of the superconducting nanowire and the time-dependent hotspot resisl(t) tance, the time-dependent hotspot resistance respectively is given as R(t) = 2ρ l(t) S = 2ρ W d , Z0 is the impedance of the transmission line connecting the probe to RF amplifiers (cf. [9]), Ibias is the bias current of the SNSPD. It is easy to see that the above electro-thermal model is a nonlinear free boundary problem with the interface x = l(t) to be determined. Observe that the quantity J appearing in (1.1) satisfies that J = I(t) W d , and the quantity R(t) appearing in (1.6) satisfies that l(t) R(t) = 2ρ W d . Hence, the temperature T (x, t) and the current I(t) are coupled together by the equations (1.1)-(1.2) and (1.7). Therefore, it is very challenging to devise an efficient method for numerically approximating the solution of this model. As far as we know, there is no work discussing numerical solution for the previous model systematically in the literature. The goal of this paper is intended to design some efficient algorithms for such a problem. Before designing our algorithm, let us review some typical methods for numerically solving free boundary problems. First of all, front-tracking methods which use an explicit representation of the interface has always been a common way of solving moving boundary problems. Juric and Tryggvason presented in [5] a front-tracking method which use a fixed grid in space and explicit tracking of the liquid-solid interface, the method performs well in approximating the exact solution. The moving grid method can also be used to solve free boundary problems, which focuses on increasing the order of accuracy in discretization. For example, Javierr (cf. [4]) located the interface in the rth node and the grid should be adapted at each time step. Compared to the level set method, the accuracy of first-order convergence in the interface position was slightly higher. The level set method (cf. [2,6]) is also a widely used method for moving boundary problems. The main idea behind the method is that the interface position is represented by the zero level set, and it captures the interface position implicitly. Compared to the moving grid method, the level set has a main advantage that a fixed grid can be used, which avoids the mesh generation at every time step. Phase-field methods (cf. [4, 7]) have become increasingly popular for phase transition models over the past decade. These methods are based on phase field models, a free boundary arising from a phase field transition is assumed to have finite thickness, which differ from the classical model of a sharp interface. Phase-field methods present an advantage over front-tracking methods, because Phase-field methods only have an approximate representation of the front location. The main difference between the level set and phase-field methods is that 3

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the level set method can capture the front on a fixed grid, in order to apply discretizations that depend on the exact interface location. In contrast, in the phase-field model, the front is not being explicitly tracked, and thus near the front the discretization of the diffusion field is less accurate. However, although there have developed many numerical methods for free boundary problems, it seems very difficult to simulate the above electro-thermal model effectively with these methods. Concretely speaking, since the moving grid method requires to introduce a transformation mapping to map a fictitious domain into the physical domain to form space grid points, it will lead to essential difficulty in discretization of the thermal equation, which is a nonlinear parabolic interface equation; for the level set method, it is inconvenient to establish a level set equation coupled with the original equations governing the variation of the temperature and current; for the phase-field method, it is very difficult to construct a relevant phase-field functional which involves very deep physical interpretation of the model. Hence, we develop a new approach to solve the electro-thermal model under discussion. The main novelty of our method proposed here is that we determine the interface x = l(t) at time t by means of the idea of the shooting method (cf. [11]) combined with the phase transition condition (1.5). We notice that the shooting method is often used in solving nonlinear two-point boundary value problems (cf. [11]). Our algorithm can be briefly described as follows. We use the finite difference method with fixed mesh to discretize the thermal equations (1.1)-(1.2) and the current equation (1.7). Assume the temperature T and the current I are available at time t = tn . We then select a grid position ˜l as the guess of the interface position x = l(tn+1 ) at t = tn+1 . Next, we compute the critical current Ic (T ) at t = tn+1 in view of (1.5) at all grid points. If there exists a grid point x = ˜l1 such that the numerical current In+1 is greater than the critical current at the left point of x = ˜l1 , and less at the right side point, then we update the guess interface position ˜l as ˜l1 . Repeat the above computation process until it converges. We present some numerical examples to show the computational performance of our method. The rest of this paper is organized as follows. In section 2, we describe the CrankNicholson finite difference method and implicit-explicit scheme for the discretization of the thermal equation, and the trapezoidal rule for the discretization of the current equation. The algorithm for determination of the interface positions is given in section 3. A series of numerical results are given in section 4. In the final section, we present a short conclusion about our investigation in this paper.

2

Discretization of the governing equations

In order to numerically solve the electro-thermal model, we first partition the space region [0, L/2] into N intervals with equal width ∆x, to get the spatial nodes 0 = x0 < x1 < · · · < xN = L/2 with xi = i∆x, and then construct the time nodes tn = nτ with τ > 0 as the time stepsize, n = 0, 1, · · · . We denote by Tin the approximate solution of the temperature T at a grid point (xi , tn ) and denote by In the approximate solution of the current I at a grid point tn . In this section, we will design effective finite difference methods for solving Tin and In , respectively.

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2.1 2.1.1

Discretization of the thermal equation The Crank-Nicholson method

Because the physical parameters rely on the temperature T itself, the thermal equations (1.1)-(1.2) are highly nonlinear. Hence, we use linearized schemes to carry out discretization, in order to avoid heavy cost in solving a nonlinear system of algebraic equations. Let x = l = l0n+1 = xj be the approximate interface position at the time t = tn+1 = (n + 1)τ . For a spatial point x = xi = i∆x in (0, l), we view the physical parameters to be constant in the time interval [tn , tn+1 ], equal to the ones corresponding to the temperature at t = tn . Then we use the standard Crank-Nicholson finite difference method to discretize the equation (1.1) (cf. [12]), to get the following difference equation: n+1 n+1 ) ( 1 n − 2T n + T n Ti−1 − 2Tin+1 + Ti+1 In+1 + In )2 1 ( Ti−1 i i+1 × ρ + κn (Tin ) × + Wd 2 2 ∆x2 ∆x2 (2.1) ( ) n+1 Ti + Tin Tin+1 − Tin 1 n n − Tsub = M (Ti ) × , − α(Ti ) × d 2 τ

where M (T ) := Cn (T ) + T Cn′ (T ) so that ∂C∂tn T = M (T ) ∂T ∂t . Similarly, for x = xi = i∆x in (l, L/2) we can derive the following difference equation from (1.2): n+1 n+1 ) n − 2T n + T n Ti−1 − 2Tin+1 + Ti+1 1 ( Ti−1 i i+1 + 2 ∆x2 ∆x2 ( ) n+1 n Ti + Ti T n+1 − Tin 1 − α(Tin ) × − Tsub = H(Tin ) × i , d 2 τ

κs (Tin ) ×

(2.2)

where H(T ) := Cs (T ) + T Cs′ (T ) so that ∂C∂ts T = H(T ) ∂T ∂t . Next, let us deal with discretization of the boundary conditions. The homogeneous Neumann boundary condition is imposed at the left boundary point x = x0 . To ensure second order accuracy of approximation, we use the ghost point method (cf. [12]). We m denote introduce a ghost point x−1 = −∆x outside the solution region [0, L/2] and let T−1 the approximate solution of T at the grid point (x−1 , tm ) fictitiously. Then using the central difference scheme we have from (1.3) that m T1m − T−1 = 0. 2∆x

(2.3)

On the other hand, we assume the difference scheme (2.1) holds at x = x0 to get n+1 ( 1 n − 2T n + T n T−1 − 2T0n+1 + T1n+1 ) In+1 + In )2 1 ( T−1 0 1 × ρ + κn (T0n ) × + Wd 2 2 ∆x2 ∆x2 ( ) n+1 1 T0 + T0n T n+1 − T0n − α(T0n ) × − Tsub = M (T0n ) × 0 . d 2 τ

(2.4)

n = T n and T n+1 = T n+1 , and plugging them into (2.4) we obtain From (2.3) we know T−1 1 −1 1

( 1 In+1 + In )2 1 ( −2T0n + 2T1n −2T0n+1 + 2T1n+1 ) × ρ + κn (T0n ) × + Wd 2 2 ∆x2 ∆x2 ( ) n+1 n 1 T0 + T0 T n+1 − T0n − α(T0n ) × − Tsub = H(T0n ) × 0 . d 2 τ

(2.5)

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The Dirichlet condition is imposed at the right boundary point x = L/2, so we directly have TN = Tsub . (2.6) To discretize the interface condition (1.4) at the interface point x = xj , we use a (x,t) backward (resp. forward) scheme to approximate ∂T∂x from the left (resp. right) at x = xj . So we have from (1.4) that κn

Tj+1 − Tj Tj − Tj−1 = κs . ∆x ∆x

(2.7)

The combination of the difference equations (2.1), (2.2), (2.5)-(2.7) can uniquely deter−1 mine the grid function {Tin+1 }N i=0 . Obviously the scheme is implicit, and can be expressed in matrix notation as a linear system with a tridiagonal coefficient matrix. So we can obtain −1 {Tin+1 }N i=0 in an efficient way. 2.1.2

The Implicit-Explicit (IMEX) method

In order to derive an efficient implicit-explicit scheme for solving the thermal model given before, we first make a reformulation for the equations (1.1) and (1.2). As a matter of fact, from some direct and routine manipulation, the two equations can be rewritten as follows. ∂2T F (T ) ∂T J 2ρ F (T ) + Gn (T ) 2 + T− Tsub = , (2.8) M (T ) ∂x d d ∂t Gs (T )

∂2T E(T ) E(T ) ∂T + T− Tsub = . ∂x2 d d ∂t

(2.9)

(T ) κs (T ) α(T ) α(T ) where Gn (T ) = κMn(T ) , Gs (T ) = H(T ) , F (T ) = − M (T ) , E(T ) = − H(T ) . Next, we choose a positive constant G0 , large enough, such that G0 is no less than Gn (T ) and Gs (T ) at least. The constant can be obtained by some additional calculation in terms of the explicit form of the underlying function. In our numerical experiments developed in section 4, G0 is taken such that

G0 =

max

Tsub ≤T ≤TC

{Gn (T ), Gs (T )}.

(2.10)

Therefore, the above equations can be reformulated further as G0

∂2T F (T ) ∂T ∂2T J 2ρ F (T ) + + [G (T ) − G ] + T− Tsub = , n 0 ∂x2 M (T ) ∂x2 d d ∂t

(2.11)

∂2T E(T ) E(T ) ∂T ∂2T + [G (T ) − G ] + T− Tsub = . (2.12) s 0 2 2 ∂x ∂x d d ∂t Hence, borrowing the same ideas to treat the variable coefficients as for the CrankNicholson method and using the technique that we discretize the partial derives of T with constant coefficients via implicit schemes and the other terms via explicit schemes (cf. [10]), we obtain from (2.11) that, at x = xi = i∆x, x ∈ (0, l) and t = tn+1 = (n + 1)τ , the difference equation for (1.1) reads G0

n+1 n+1 ( 1 Ti−1 − 2Tin+1 + Ti+1 1 I n+1 + I n )2 + × × ρ ∆x2 M (Tin ) Wd 2 n T n − 2Tin + Ti+1 Tin+1 − Tin F (Tin ) n + [Gn (Tin ) − G0 ] × i−1 + × (T − T ) = . sub i ∆x2 d τ

G0 ×

(2.13)

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Similarly, we have for x = xi = i∆x, x ∈ (l, L/2), the difference equation for (1.2) reads G0 ×

n+1 n+1 n − 2T n + T n Ti−1 − 2Tin+1 + Ti+1 Ti−1 i i+1 n + [G (T ) − G ] × s i 0 ∆x2 ∆x2 T n+1 − Tin E(Tin ) × (Tin − Tsub ) = i . + d τ

(2.14)

Following the same ideas for construction of the Crank-Nicholson method mentioned above, we can derive the difference equations corresponding to the boundary conditions and the interface condition. Compared to the Crank-Nicholson method, the present implicit-explicit (IMEX) scheme has an advantage. That is, if the generic constant G0 is chosen feasibly, we only require to solve a linear system with the same coefficient matrix at different time nodes t = tn . This will reduce the computational cost greatly, in particular, in high-dimensional case.

2.2

Discretization of the current equation

The current equation (1.7) is a second order ordinary differential equation, we rewrite it as a system ∫of first-order equations and then carry out discretization. To this end, t let K(t) = C1bt 0 (Ibias − I(s))ds + Ibias Z0 . Hence, by some direct manipulation, (1.7) is equivalent to Ibias − I(t) , Cbt Lk I ′ (t) = K(t) − (R(t) + Z0 )I(t), K ′ (t) =

(2.15) (2.16)

where

l(t) l(t) = 2ρ . S Wd The corresponding initial conditions are given by R(t) = 2ρ

I(0) = I0 ,

K(0) = Ibias Z0 .

Integrating both sides of the equation (2.15) in the domain [tn , tn+1 ] implies ∫ tn+1 ∫ tn+1 Ibias − I(t) ′ dt, K (t)dt = Cbt tn tn and using the trapezoid method for numerical integration to the right side term we further have τ Kn+1 = Kn + (2Ibias − In − In+1 ), (2.17) 2Cbt Similarly, integrating both sides of the equation (2.16) in the domain [tn , tn+1 ], we have ∫ tn+1 ∫ tn+1 ′ Lk I (t)dt = (K(t) − (R(t) + Z0 )I(t))dt, tn

tn

which, in conjunction with the trapezoid method, implies ( ( τ τ ) τ τ ) τ τ Lk + Rn+1 + Z0 In+1 = Lk − Rn − Z0 In + Kn + Kn+1 . 2 2 2 2 2 2

(2.18)

where Rn = Rn (t). 7

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Making use of (2.17) and (2.18) immediately gives aIn+1 = bIn + τ Kn +

τ2 Ibias , 2Cbt

(2.19)

where a = Lk +

τ τ τ2 τ τ τ2 + Rn+1 + Z0 , b = Lk − − Rn − Z0 , 4Cbt 2 2 4Cbt 2 2 ln ln+1 , Rn+1 = 2ρ . Rn = 2ρ Wd Wd

We remark that in the real applications, the interface positions ln and ln+1 at t = tn and t = tn+1 should be replaced by their approximate values l0n and l0n+1 , respectively. Therefore, it is clear that we can get the current In+1 whenever the unknowns at the time t = tn and the interface position x = l0n+1 are available. Observing that for the superconducting nanowire single-photon detector described in Figure 1, while the current drops below critical current and the resistive region subsides, the wire becomes fully superconducting again, the bias current through the wire returns to the original value. Thus, the time-dependent hotspot resistance Rn (t) = 0, and the current equation (1.7) becomes dI(t) d2 I(t) + a2 I(t) = a3 , (2.20) + a1 dt2 dt Z0 bias where a1 = L , a2 = Cbt1Lk , a3 = CIbt Lk . k Since the equation (2.20) is a constant second order inhomogeneous linear equation, we can easily derive its closed form of the solution:

I(t) = c1 eλ1 t + c2 eλ2 t +

a3 . a2

(2.21)

If the initial conditions are given by I(tˆ) = a4 and I ′ (tˆ) = a5 , then we know by a direct manipulation that the undetermined coefficients in (2.21) are √ √ −a1 + a21 − 4a2 −a1 − a21 − 4a2 λ1 = , λ2 = , 2 2 c1 =

a4 λ2 − a5 −

a3 a 2 λ2 , λ1 tˆ

c2 =

(λ2 − λ1 )e

a4 λ1 − a5 − (λ1 − λ2

a3 a 2 λ1 . )eλ2 tˆ

Therefore, if the nanowire returns to superconducting state again, we are able to get the current from the expression (2.21) explicitly, instead of the numerical solution. This will increase the computational efficiency greatly.

3

Determination of the interface position

Similar to the standard numerical method for solving evolutionary equations, we will conduct numerical simulation for the electro-thermal model along the time direction. That means, once the numerical results at t = tn are obtained, we will try to get the numerical results at t = tn+1 . From our discussion given in the above section, we easily know the key difficulty is to derive the interface position at this instant. Our key points to overcome the above obstacle are as follows. First of all, we make the partition of the region [0, L/2] fine enough, i.e. ∆x is taken small enough, so that we 8

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All variable values are known at t n

Select a position of n
1 interface l0

Compute current I (tn
1) temperature T(x,tn
1) critical current Ic (x,tn
1)

Determine new position of n
1 interface l1 . If satisfy termination rule ?

N

Y Get the new position of interface and the temperature

Figure 2: The flow chart of the iterative algorithm to form the interface position at different time nodes. can assume that the interface positions always lie in the spatial grid points, approximately with desired accuracy. Next, we will use the shooting method to determine the interface position at t = tn+1 , in view of the idea of the shooting method (cf. [11]) combined with the phase transition condition (1.5). To be more precise, we choose x = ˜l = ln as the initial guess of the interface position at t = tn+1 . Then, by means of the finite difference methods in section 2, we can derive the approximate temperature values Tin+1 at all grid points as well as the approximate current In+1 , and compute the critical current Ic (T ) at t = tn+1 in view of (1.5) at all grid points. If the initial guess x = ˜l satisfies that the numerical current In+1 is greater than the critical current at the left point of x = ˜l, and less at the right side point, then we take l0n+1 = ˜l. Otherwise, we try to find a grid x = ˜l1 such that the numerical current In+1 is greater than the critical current at the left point of x = ˜l, and less at the right side point. And then replace the guess interface position ˜l by ˜l1 . Repeat the above computation process until it converges, and choose the final result ˜l as l0n+1 . In all the calculations presented in the following section, the termination rule is taken as |˜l1 − ˜l| < tol, with tol = 1 × 10−6 . For preciseness, the above algorithm is shown in a flow chart, described in Figure 2.

4

Numerical results

In this section, we give some numerical experiments to illustrate the performance and accuracy of our method introduced in sections 2 and 3, from which we can observe the evolution of interface positions, namely the growth of the normal region along the wire, the change in current through the wire, the change of resistance along the wire after a 9

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Ic I

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16

I(µ A)

15 14

14

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10

0

12

0

0.1

I(µ A)

I(µ A)

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0.2

time(ns)

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time(ns)

time(ns)

(a)

(b)

0.6

0.7

0.8

0.9

Figure 3: The current variation in the electro-thermal model.(a) The calculated current through the wire vs. time. (b) The calculated current and critical current at x = 100nm vs time. small resistive hotspot is formed. Here the photon-induced resistive barrier forms at t = 0. Most of the physical parameters are taken from the monograph [13] about the theory of superconductivity, and the other ones are taken from the related literature. In particular, since the hotspot only forms and exists for several nanoseconds (cf. [1]), we choose in our numerical simulation the terminal time to be tend = 10ns. Then we choose the stepsize in t to be τ = 1ps, where 1ps = 10−3 ns. If τ is taken a little larger, say τ = 5ps, our algorithms will not converge. In the normal/resistive state, the electrical resistivity ρ = 2.4 × 10−6 Ωm. According to the Wiedemann-Franz law, the ratio of the electronic contribution of the thermal conductivity κn to the electrical conductivity ρ of a metal, is proportional to the temperature T (κn = L Tρ , where L = 2.45 × 10−8 W Ω/K 2 is the Lorenz number). The heat capacity (per unit volume) of the superconducting film Cn includes electron specific Cen and phonon specific heat Cpn , where Cen is proportional to the temperature T (Cen = γT , where γ = 240), and Cpn is proportional to T 3 such that Cpn = 9.8T 3 (cf. [8]). The thermal boundary conductivity α between NbN and sapphire we used is obtained from [15], and we only considered its cubic dependence on temperature (α = BT 3 , where B = 800). In the region of superconducting state, ρ is taken to be zero naturally. We express the T2 (cf. [14]), where Tc = 10K is the critical temperature. thermal conductivity as κs = L ρT c The heat capacity Cs also include two parts, the electron specific was calculated such 3.5Tc that Ces = Ae− T with A = 1.93 × 105 (cf. [13]), and the phonon specific heat is state independent such that Cps = 9.8T 3 . The thermal boundary conductivity is given as α = BT 3 . Some more data used in all the computations are given as follows. The length of superconducting nanowire L = 2000nm, the wire thickness d = 4nm, the width of nanowire W = 100nm, the substrate temperature Tsub = 2K, the initial critical current Ic (0) = 20µA, the capacitor Cbt = 20 × 10−9 F , the kinetic inductance of the superconducting nanowire Lk = 807.7nH, the impedance of the transmission line connecting the probe to RF amplifiers Z0 = 50Ω, the current of the SNSPD Ibias = 16.589µA, the initial interface position l0 = 15nm, the initial temperature T0 = 5K where the wire is normal and T0 = Tsub = 2K where the wire is superconducting. For the Crank-Nicholson method and the IMEX method, we take N = 1000, so the space step is ∆x = L/2N = 1nm. The time step is τ = 1ps, as introduced at the beginning

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6

500 450

6

3

350

4

position(nm)

R(KΩ)

R(KΩ)

4

3 2 1

2

0

500

400

5

position(nm)

5

300 250 200 150

0

0.1

0.2

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0.4

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time(ns)

(a)

(b)

Figure 4: The resistance variation in the electro-thermal model. (a) The calculated total normal state resistance vs. time, and the inset shows in greater detail the change of the resistance. (b) The interface position vs. time, and the inset shows in greater detail the change of the position. 11 10 9

T(K)

8 7 6 5 4 3 2

0

0.5

1

1.5

2

2.5

3

time(ns)

(a)

(b)

Figure 5: The temperature variation in the electro-thermal model. (a) The calculated temperature (shown using colors) at different positions along the wire and in time. (b) The calculated temperature history at x = 100nm. of this section. Furthermore, for the IMEX method, we choose the parameter G0 in view of the formulation (2.10) to get G0 = 2 ∗ 10−5 . We first use the Crank-Nicholson method for solving the thermal equations, combined with the numerical method for solving the current equation and the algorithm in section 3 to search for interface positions, to implement numerical simulation. It is shown in Figure 3(a) the calculated current through the SNSPD. We find the curve first forms a sharp decline within a short time period. Afterwards, the nanowire under consideration switches to superconducting state, and the calculated current increases at an exponential rate. It is shown in Figure 3(b) the calculated current and the critical current at x = 100nm vs. time. It is shown in Figure 4(a) the calculated total normal state resistance along the wire. It appears that the resistance increases gradually and then decreases to 0 sharply, and the inset shows in greater detail the change of the resistance. It is shown in Figure 4(b) the evolution of the interface position in the electro-thermal model. The initial interface position is taken as l0 = 15nm. The interface point moves forward to about the position x = 463nm at t = 170ps, where the resistance increases to a maximum value. Then the 11

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18

CN IMEX

CN IMEX

16

10 14

T(K)

I(µ A)

12 10 8 6

8

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4 2

2 0

1

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4

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7

8

9

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0

2

4

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time(ns)

time(ns)

(a)

(b)

Figure 6: Some numerical comparison between the Crank-Nicholson finite difference method and the IMEX method. (a) The calculated current through the wire vs. time. (b) The calculated temperature history at x = 100nm. interface point returns gradually to the central position with x = 0, and the nanowire becomes superconducting state. It is shown in Figure 5(a) the calculated temperature (shown using colors) at different positions along the wire and in time, the temperature at each segment show the segment under consideration switches into the normal state or remains superconducting. And it is shown in Figure 5(b) the calculated temperature history at x = 100nm. At that position, the initial temperature is T = 5K where the wire is normal, then it increases to a maximum value of about 10.7K, after that the temperature gradually returns to 2K and the position lies in superconducting state. All the numerical results given above coincide with the physical phenomenon observed by experiments (cf. [15]). We also compare the numerical results with the thermal equations numerically solved by the Crank-Nicholson method and the IMEX method, respectively. We observe from the numerical data in Figure 6 that the two methods which perform in the similar manners, can produce very similar numerical results.

5

Conclusions

In this paper, we propose two algorithms for numerically solving the electro-thermal model for Superconducting Nanowire Single-Photon Detectors. Such a model is governed by a nonlinear free boundary problem involving the temperature and the current, which are coupled together by a nonlinear parabolic interface equation and a second order ordinary differential equation (see the equations (1.1)-(1.2) and (1.7) for details). In our numerical experiments, for a fixed spatial size ∆x, only if the stepsize in time τ is taken small enough, our numerical methods are convergent. Therefore, we only develop in this paper some initial but interesting results for the coupled system of equations (1.1)-(1.2) and (1.7). Due to the complexity of the model, it is very challenging to establish mathematical theory for this model and discuss convergence analysis of the methods proposed in this paper. Acknowledgements. The authors would like to thank the three referees for their valuable comments and suggestions, which greatly improved the early version of the paper.

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References [1] Chandra M N, Michael G T, Robert H H, Superconducting nanowire single-photon detectors: physics and applications. Supercond. Sci. Technol., 2012, 25(6): 1-17. [2] Chen S, Merriman B, Osher S, Smereka P, A simple level set method for solving Stefan problems. J. Comput. Phys., 1997, 135: 8-29. [3] Friedman A, Variational Principles and Free Boundary Problems. John Wiley and Sons, New York, 1982. [4] Javierre E, Vuik C, Vermolen F J, A comparison of numerical models for one-dimensional Stefan problems. J. Comput. Appl. Math., 2006, 192(2): 445-459. [5] Juric D, Tryggvason G, A front tracking method for dendritic solidification. J. Comput. Phys., 1996, 123(1): 127-148. [6] Mackenzie J A, Robertson M L, A moving mesh method for the solution of the one-dimensional phase field equations. J. Comput. Phys., 2002, 181: 526-544. [7] Penrose O, Fife P C, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. Physica D, 1990, 43(1): 44-62. [8] Semenov A D, Nebosis R S, Gousev Y P, Analysis of the nonequilibrium photoresponse of superconducting films to pulsed radiation by use of a two-temperature model. Phys. Rev. B., 1995, 52: 581-590. [9] Semenov A D, Goltsman G N, Korneev A A, Quantum detection by current carrying superconducting film. Physica C., 2001, 351: 349-356. [10] Steven J R, Implicit-explicit methods for reaction diffusion problems in pattern formation. J. Math. Biol., 1995, 34: 148-176. [11] Stoer J, Bulirsch R, Introduction to Numerical Analysis. 3rd ed. Springer, New York, 2002. [12] Thomas J W, Numerical Partial Differential Equations: Finite Difference Methods. Springer, New York, 1995. [13] Tinkham M, Introduction to Superconductivity. 2nd ed. New York, McGraw Hill, 1996. [14] Tinkham M, Free J U, Lau C N, Markovic N, Hysteretic I-V curves of superconducting nanowires. Physical Review B, 2003, 68(134515): 1-7. [15] Yang J K W, Kerman A J, Dauler E A, et al, Modeling the electrical and thermal response of superconducting nanowire single-Photon detectors. IEEE Transactions on Applied Superconductivity., 2007, 17(2): 518-585.

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

On the Existence of Meromorphic Solutions of Some Nonlinear Differential-Difference Equations Xiu-Min Zheng1∗, Hong-Yan Xu2 1

Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, China 2 Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, 333403, China ([email protected], [email protected])

Abstract: In this paper, we investigate the conditions concerning the existence or non-existence of transcendental meromorphic or entire solutions of some kinds of differential-difference equations. We also give examples to illustrate the sharpness of our results. Key words: Differential-difference equation, meromorphic solution, entire solution. AMS Classification No. 39B32, 34M05, 30D35 1 Introduction and main results Throughout this paper, we assume that f (z) is a meromorphic function in the whole complex plane, and use standard notations, such as m(r, f ), T (r, f ), N (r, f ), in the Nevanlinna theory (see e.g. [3, 7, 8, 17]). And we also use σ(f ) and σ2 (f ) to denote respectively the order and the hyper order of f (z). Moreover, we say that a meromorphic function g(z) is small with respect to f (z), if T (r, g) = S(r, f ), where S(r, f ) means any real quantity satisfying S(r, f ) = o(T (r, f )) as r → ∞ outside of a possible exceptional set of finite logarithmic measure. Recently, with some establishments of difference analogues of the classic Nevanlinna theory (two typical and most important ones can be seen in [2, 4–6]), there has been a renewed interest in the properties of complex difference expressions and meromorphic solutions of complex difference equations (see e.g. [10–12, 18]). Further, Yang-Laine gave analogies between nonlinear difference and differential equations in [15] . From then on, some results concerning nonlinear differential-difference equations were found (see e.g. [13]). In what follows, we use the defintion of the differential-difference polynomial in [15, 19]. A differential-difference polynomial is a polynomial in f (z), its shifts, its derivatives and derivatives of its shifts, that is, an expression of the form X P (z, f ) = aλ (z)f (z)λ0,0 f 0 (z)λ0,1 · · · f (k) (z)λ0,k λ∈I

·f (z + c1 )λ1,0 f 0 (z + c1 )λ1,1 · · · f (k) (z + c1 )λ1,k · · · ·f (z + cl )λl,0 f 0 (z + cl )λl,1 · · · f (k) (z + cl )λl,k ∗

Corresponding author.

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=

X

aλ (z)

l Y k Y

f (j) (z + ci )λi,j ,

(1.1)

i=0 j=0

λ∈I

where I is a finite set of multi-indices λ = (λ0,0 , · · · , λ0,k , λ1,0 , · · · , λ1,k , · · · , λl,0 , · · · , λl,k ), and c0 (= 0), c1 , · · · , cl are distinct complex constants. And we assume that the meromorphic coefficients aλ (z), λ ∈ I of P (z, f ) are of growth S(r, f ). We denote the degree l Q k Q and the weight of the monomial f (j) (z + ci )λi,j of P (z, f ) respectively by i=0 j=0

d(λ) =

l X k X

λi,j

and w(λ) =

i=0 j=0

l X k X

(j + 1)λi,j .

i=0 j=0

Then we denote the degree and the weight of P (z, f ) respectively by d(P ) = max{d(λ)} and w(P ) = max{w(λ)}. λ∈I

λ∈I

In the following, we assume d(P ) ≥ 1. We recall the following result due to Wang-Li [13] by rewriting the original differentialdifference polynomial in [13] as the one of the form (1.1). Theorem A. Suppose that a nonlinear differential-difference equation is f n (z) + P (z, f ) = p(z),

(1.2)

where n ∈ N, p(z) is a polynomial, and P (z, f ) is a differential-difference polynomial of the form (1.1) with polynomial coefficients. If X n > (s + 1)d(P ) − d(λ), (1.3) λ∈I

where s is the number of components of I, then the equation (1.2) has no transcendental entire solutions of finite order. Remark 1.1 Obviously, (1.3) results in X n > (s + 1)d(P ) − d(λ) ≥ (s + 1)d(P ) − sd(P ) = d(P ) ≥ 1. λ∈I

Then, our first main purpose is to improve Theorem A. On the one hand, we improve the restrict on n by introducing an important lemma of our own. On the other hand, we also consider the non-existence of meromorphic solutions of the equation (1.2). Our result is as follows. Theorem 1.1 Consider the nonlinear differential-difference equation f n (z) + P (z, f ) = c(z),

n ∈ N,

(1.4)

where P (z, f ) is a differential-difference polynomial of the form (1.1) with meromorphic coefficients aλ (z), λ ∈ I, and c(z) is a meromorphic functions. (i) If n > d(P ), then the equation (1.4) has no admissible transcendental entire solutions with hyper order less than 1. (ii) If n > w(P ), then the equation (1.4) has no admissible transcendental meromorphic solutions with hyper order less than 1.

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Remark 1.2 Here, a meromorphic or entire solution f (z) of the equation (1.4) is called admissible, if aλ (z), λ ∈ I and c(z) are small with respect to f (z), that is, T (r, aλ ) = S(r, f ), λ ∈ I and T (r, c) = S(r, f ). Wang-Li also investigated another kind of nonlinear differential-difference equation in [13] as follows. Theorem B For two integers n ≥ 3, k > 0 and a nonlinear differential-difference equation f n (z) + q(z)f (k) (z + t) = aeibz + de−ibz , (1.5) where q(z) is a polynomial and t, a, b, d are complex numbers such that |a| + |d| 6= 0, bt 6= 0. (i) Let n = 3. If q(z) is nonconstant, then the equation (1.5) does not admit entire solutions of finite order. If q = q(z) is constant, then the equation (1.5) admits three distinct transcendental entire solutions of finite order, provided that bt = 3mπ (m 6= 0, if q 6= 0),

q 3 = (−1)m+1 (

3i 3k ) 27ad, b

when k is even, or bt =

3π + 3mπ (if q 6= 0), 2

q 3 = i(−1)m (

3i 3k ) 27ad, b

when k is odd, for an integer m. (ii) Let n > 3. If ad 6= 0, then the equation (1.5) does not admit entire solutions of finite order. If ad = 0, then the equation (1.5) admits n distinct transcentental entire solutions of finite order, provided that q = q(z) ≡ 0. Moreover, they proposed a question in [13]: for the differential-difference equation of the form f n (z) + L(z, f ) = aeibz + de−ibz , n ≥ 3, where L(z, f ) is some linear differential-difference polynomial of f (z) with polynomial coefficients, what can we say considering Theorem B. Then, our second main purpose is to give the following results, which answer the above question to some extent. Theorem 1.2 Consider the nonlinear differential-difference equation n

f (z) +

l X k X

As,t (z)f (t) (z + cs ) = aeibz + de−ibz ,

n ∈ N, n ≥ 3,

(1.6)

s=0 t=0

where c0 (= 0), c1 , · · · , cl are distinct complex constants, As,t (z), s = 0, 1, · · · , l, t = 0, 1, · · · , k are polynomials, and a, b, d ∈ C such that b 6= 0 and |a| + |d| 6= 0. (i) Let n = 3. If ad 6= 0 and

l X k X s=0 t=0

´ ³ ibcs ib −ibcs −ib )t 6≡ 0, As,t (z) e 3 ( )t − e 3 ( 3 3

then the equation (1.6) has no transcendental entire solutions of finite order. If l X k X s=0 t=0

As,t (z)e

ibcs 3

ib ( )t ≡ 0 = d, 3

or

l X k X s=0 t=0

26

As,t (z)e

−ibcs 3

(

−ib t ) ≡ 0 = a, 3

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

or l X k X

As,t (z)e

ibcs 3

s=0 t=0

l

k

XX −ibcs −ib ib ( )t ≡ As,t (z)e 3 ( )t ≡ −3d1 d2 , 3 3 s=0 t=0

d31 = a, d32 = d,

then the equation (1.6) has three transcendental entire solutions of finite order. (ii) Let n > 3. If ad = 6 0, or ad = 0 and

l X k X

As,t (z)e

ibcs n

s=0 t=0

ib ( )t 6≡ 0, n

then the equation (1.6) has no transcendental entire solutions of finite order. If ad = 0 and

l X k X

As,t (z)e

ibcs n

s=0 t=0

ib ( )t ≡ 0, n

then the equation (1.6) has n transcendental entire solutions of finite order. In particular, we obtain more concrete results for a special linear difference polynomial L(z, f ) as follows. Theorem 1.3 Consider the nonlinear difference equation f n (z) + q(z)4m f (z) = aeibz + de−ibz ,

n, m ∈ N, n ≥ 3,

(1.7)

where q(z) is a polynomial, a, b, d ∈ C such that b 6= 0 and |a| + |d| 6= 0. (i) Let n = 3. If ad 6= 0 and q(z) is a nonconstant, then the equation (1.7) has no transcendental entire solutions of finite order. If ad 6= 0 and q(z) is a constant q, then the equation (1.7) has three transcendental entire solutions of the form f (z) = −ibz ibz d1 e 3 + d2 e 3 , d31 = a, d32 = d, provided that bc = 6kπ + 3π +

27ad 6sπ (k ∈ Z, s ∈ {0, 1, · · · , m − 1}) and q 3 = (−1)m+1 2sπi . m (e m + 1)3m

If ad = 0, then the equation (1.7) has three transcendental entire solutions of the form −ibz ibz f (z) = d1 e 3 + d2 e 3 , d31 = a, d32 = d, provided that q(z) ≡ 0 or bc = 6kπ, k ∈ Z. (ii) Let n > 3. If ad 6= 0, then the equation (1.7) has no transcendental entire solutions of finite order. If ad = 0, then the equation (1.7) has n transcendental entire −ibz ibz solutions of the form f (z) = d1 e n + d2 e n , dn1 = a, dn2 = d, provided that q(z) ≡ 0 or bc = 2knπ, k ∈ Z. Remark 1.3 Here, the forward difference 4m f (z) for m ∈ N and c ∈ C\{0} is defined in the standard way [14, p. 52] by 4f (z) = f (z+c)−f (z), 4m f (z) = 4(4m−1 f (z)) = 4m−1 f (z+c)−4m−1 f (z), m ≥ 2. And it is shown as in [2] that 4m f (z) =

m X

j Cm (−1)m−j f (z + jc),

f (z + mc) =

j=0

m X

j Cm 4j f (z),

j=0

j where Cm , j = 0, 1, · · · , m are the binomial coefficients.

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2 Lemmas Lemma 2.1. ( [19]) Let f (z) be a transcendental meromorphic function of σ2 (f ) < 1, and P (z, f ) be a differential-difference polynomial of the form (1.1), then we have m(r, P (z, f )) ≤ d(P )m(r, f ) + S(r, f ). Furthermore, if f (z) also satisfies N (r, f ) = S(r, f ), then we have T (r, P (z, f )) ≤ d(P )T (r, f ) + S(r, f ). Lemma 2.2. ( [6]) Let T : [0, +∞) → [0, +∞) be a non-decreasing continuous function and let s ∈ (0, +∞). If the hyper order of T is strictly less than one, i.e. 2 T (r) lim loglog = ζ < 1, and δ ∈ (0, 1 − ζ), then r r→∞

T (r) ), rδ where r runs to infinity outside of a set of finite logarithmic measure. It is shown in [3, p.66] and [1, Lemma 1] that the inequality T (r + s) = T (r) + o(

(1 + o(1))T (r − |c|, f ) ≤ T (r, f (z + c)) ≤ (1 + o(1))T (r + |c|, f ) holds for c 6= 0 and r → ∞. And from its proof, the above relation is also true for counting function. By combining Lemma 2.2 and these inequalities, we immediately deduce the following lemma. Lemma 2.3. Let f (z) be a nonconstant meromorphic function of σ2 (f ) < 1, and c be a nonzero complex constant. Then we have T (r, f (z + c)) = T (r, f ) + S(r, f ), N (r, f (z + c)) = N (r, f ) + S(r, f ),

N (r,

1 1 ) = N (r, ) + S(r, f ). f (z + c) f

Laine-Yang [9] gave a difference analogue of Clunie lemma as follows. Lemma 2.4. ( [9]) Let f (z) be a transcendental finite order meromorphic solution of U (z, f )P (z, f ) = Q(z, f ), where U (z, f ), P (z, f ), Q(z, f ) are difference polynomials in f (z) with small meromorphic coefficients, degf U = n and degf Q ≤ n. Moreover, we assume that U (z, f ) contains just one term of maximal total degree. Then m(r, P (z, f )) = S(r, f ). Remark 2.1. Yang-Laine [15] also pointed out that Lemma 2.4 is also true if P (z, f ), Q(z, f ) are differential-difference polynomials in f (z). Further, by a careful inspection of the proof of Lemma 2.4, we see that the same conclusion holds for the differential-difference case, if the coefficients bµ (z) of P (z, f ), Q(z, f ) satisfy m(r, bµ ) = S(r, f ) instead of T (r, bµ ) = S(r, f ). Lemma 2.5. ( [16]) Suppose that c is a nonzero complex constant, α(z) is a nonconstant meromorphic function. Then the differential equation f 2 (z) + (cf (n) (z))2 = α(z)

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has no transcendental meromorphic solutions satisfying T (r, α) = S(r, f ). 3 Proofs of Theorems 1.1-1.3 Proof of Theorem 1.1. (i) Let f (z) be an admissible transcendental entire solution of (1.4) with σ2 (f ) < 1. By Lemma 2.1, we see that m(r, P (z, f )) ≤ d(P )m(r, f ) + S(r, f ).

(3.1)

By (1.4) and (3.1), we obtain that nT (r, f ) = T (r, P (z, f )) + S(r, f ) = m(r, P (z, f )) + S(r, f ) ≤ d(P )T (r, f ) + S(r, f ). (3.2) Since n > d(P ), (3.2) is a contradiction. Thus, (1.4) has no admissible transcendental entire solutions with hyper order less than 1. (ii) Let f (z) be an admissible transcendental meromorphic solution of (1.4) with σ2 (f ) < 1. We consider each pole of P (z, f ). Since each pole of P (z, f ) in |z| < r comes from the poles of f (z + ci ), i = 0, · · · , l and aλ (z), λ ∈ I in |z| < r, and each pole of f (z + ci ) with multiplicity pi is a pole of P (z, f ) with multiplicity at most pi λi,0 +(pi +1)λi,1 +· · ·+(pi +k)λi,k ≤ pi (λi,0 +2λi,1 +· · ·+(k +1)λi,k ) = pi

k X

(j +1)λi,j ,

j=0

we have by Lemma 2.3 that N (r, P (z, f )) ≤ max{ λ∈I

l X k X

(j + 1)λi,j N (r, f (z + ci ))} + S(r, f )

i=0 j=0

l X k X = max{ (j + 1)λi,j N (r, f )} + S(r, f ) λ∈I

i=0 j=0

= max w(λ)N (r, f ) + S(r, f ) = w(P )N (r, f ) + S(r, f ). λ∈I

(3.3)

Clearly, (3.1) holds by Lemma 2.1 again. By (1.4), (3.1) and (3.3), we obtain that nT (r, f ) = T (r, P (z, f )) + S(r, f ) = m(r, P (z, f )) + N (r, P (z, f )) + S(r, f ) ≤ d(P )m(r, f ) + w(P )N (r, f ) + S(r, f ) ≤ w(P )T (r, f ) + S(r, f ).

(3.4)

Since n > w(P ), (3.4) is a contradiction. Thus, (1.4) has no admissible transcendental meromorphic solutions with hyper order less than 1.¤ Proof of Theorem 1.3. Suppose that f (z) is a transcentental entire solution of finite order of (1.7). Differentiating (1.7), we have nf n−1 (z)f 0 (z) + q 0 (z)

m X

j Cm (−1)m−j f (z + jc) + q(z)

j=0

m X

j Cm (−1)m−j f 0 (z + jc)

j=0

= ib(aeibz − de−ibz ).

(3.5)

By combining (1.7) and (3.5), we have ³ nf

n−1

0

0

(z)f (z) + q (z)

m X

j Cm (−1)m−j f (z

+ jc) + q(z)

j=0

m X

´2 j Cm (−1)m−j f 0 (z + jc)

j=0

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2

³

n

+b f (z) + q(z)

m X

´2 j Cm (−1)m−j f (z + jc) = 4adb2 ,

j=0

consequently,

f 2n−2 (z)(b2 f 2 (z) + n2 f 02 (z)) = Q(z, f ),

(3.6)

where Q(z, f ) is a differential-difference polynomial of f (z) with the total degree at most n + 1. If b2 f 2 (z) + n2 f 02 (z) ≡ 0, we differentiate it and obtain that n2 f 00 (z) + b2 f (z) = 0,

(3.7)

which implies the solution must be f (z) = d1 e

ibz n

+ d2 e

−ibz n

,

(3.8)

where d1 and d2 are arbitrary complex constants. If b2 f 2 (z) + n2 f 02 (z) 6≡ 0, we may apply Lemma 2.4 and Remark 2.1 to (3.6) and obtain that T (r, b2 f 2 + n2 f 02 ) = m(r, b2 f 2 + n2 f 02 ) = S(r, f ). Thus, by Lemma 2.5, we see that b2 f 2 (z) + n2 f 02 (z) must be a constant M . Differentiating b2 f 2 (z) + n2 f 02 (z) = M , we obtain (3.7) and (3.8) again. ibz Substituting (3.8) into (1.7) and denoting w = w(z) = e n , we obtain that dn1 w2n + Cn1 dn−1 d2 w2n−2 + Cn2 dn−2 d22 w2n−4 + · · · + Cnn−2 d21 dn−2 w4 + Cnn−1 d1 dn−1 w2 + dn2 1 1 2 2 +d1 q(z)

m X

ijbc j Cm (−1)m−j e n wn+1 + d2 q(z)

m X

j Cm (−1)m−j e

−ijbc n

wn−1 = aw2n + d. (3.9)

j=0

j=0

(i) Let n = 3, then (3.9) reduces into a6 w6 + a4 w4 + a2 w2 + a0 = 0, where

    a6 = d31 − a,      m P  ijbc ibc  2 j  a = 3d d + d q(z) Cm (−1)m−j e 3 = 3d21 d2 + d1 q(z)(e 3 − 1)m , 1  4 1 2 j=0

m P  −ibc  2 2 j m−j −ijbc  3 3 = 3d d − 1)m , a = 3d d + d q(z) + d q(z)(e C (−1) e 1 2 2 1 2  2 2 m   j=0       a0 = d32 − d.

Since w(z) is transcendental, we have a6 = a4 = a2 = a0 = 0. If ad 6= 0, then d31 = a 6= 0, d32 = d 6= 0. It follows from a4 = a2 = 0 that 3d1 d2 + q(z)(e

ibc 3

− 1)m = 3d1 d2 + q(z)(e

30

−ibc 3

− 1)m = 0.

(3.10)

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ibc

−ibc

If q(z) is a nonconstant, then (3.10) results in e 3 − 1 = e 3 − 1 = 0, which implies a contradiction that d1 = d2 = 0. Thus, (1.7) has no transcendental entire solutions of ibc finite order for this case. If q(z) is a constant q, then (3.10) results in (e 3 − 1)m = −ibc ibc (e 3 − 1)m . Denoting v = e 3 , we have (v − 1)m = ( v1 − 1)m , consequently, 1 v − 1 = us ( − 1), v 2sπi

s = 0, · · · , m − 1,

2πi

where us = e m = εs , ε = e m , s = 0, · · · , m − 1. If s = 0 (that is, u0 = 1), then 2ibc v 2 = e 3 = 1, that is, bc = 3kπ, where k ∈ Z. Substituting it into (3.10), we deduce that 3d1 d2 + q((−1)k − 1)m = 0. Then k is odd, and q 3 = (−1)m+1 27ad . Thus, (1.7) has three distinct transcendental 8m entire solutions of finite order for this case. If s ∈ {1, · · · , m − 1} (that is, us = εs ), then v = 1 or −εs , that is, bc = 6kπ or bc = 6kπ + 3π + 6sπ , where k ∈ Z. Substituting m bc = 6kπ into (3.10), we deduce that d1 d2 = 0, which is a contradiction. Substituting into (3.10), we deduce that bc = 6kπ + 3π + 6sπ m 3d1 d2 + q(−εs − 1)m = 0. Then q 3 = (−1)m+1

27ad

(e

2sπi m +1)3m

. Thus, (1.7) has three distinct transcendental entire

solutions of finite order for this case. If a 6= 0 and d = 0, then d31 = a 6= 0 and d2 = 0. If q(z) ≡ 0, then (1.7) has three distinct transcendental entire solutions of finite order for this case. If q(z) 6≡ 0, it −ibc ibc follows from a4 = a2 = 0 that e 3 − 1 = e 3 − 1 = 0, that is, bc = 6kπ, where k ∈ Z. Substituting bc = 6kπ into (1.7), we see that (1.7) has three distinct transcendental entire solutions of finite order for this case. If a = 0 and d 6= 0, then d1 = 0 and d32 = d 6= 0. We can deduce similar results as the above. (ii) Let n > 3 (which implies 2n − 2 > n + 1 and 2 < n − 1), then we deduce from (3.9) that a2n w2n + a2n−2 w2n−2 + · · · + a2 w2 + a0 = 0, (3.11) where

    a2n        a2n−2     ···        a2        a0

= dn1 − a, = ndn−1 d2 , 1 ··· = nd1 dn−1 , 2 = dn2 − d.

Since w(z) is transcendental, we have a2n = a2n−2 = · · · = a2 = a0 = 0. If ad 6= 0, then dn1 = a 6= 0, dn2 = d = 6 0. It follows from a2n−2 = a2 = 0 that d1 d2 = 0, which is a contradiction. Thus, (1.7) has no transcendental entire solutions of finite order for this case.

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If a 6= 0 and d = 0, then dn1 = a 6= 0 and d2 = 0. If n is even, then n + 1 is odd. Hence, the coefficient of wn+1 in (3.11) is an+1 = d1 q(z)

m X

j Cm (−1)m−j e

ijbc n

ibc

= d1 q(z)(e n − 1)m .

j=0 ibc

Since an+1 = 0, we have q(z) ≡ 0 or e n − 1 = 0 (that is, bc = 2knπ, where k ∈ Z). If n is odd, then n + 1 is even. Hence, the coefficient of wn+1 in (3.11) is n−1 2

an+1 = Cn

n+1 2

d1

n−1 2

d2

+ d1 q(z)

m X

j Cm (−1)m−j e

ijbc n

ibc

= d1 q(z)(e n − 1)m .

j=0

Hence, we deduce the same result as the above, that is, q(z) ≡ 0 or bc = 2knπ, where k ∈ Z. Thus, (1.7) has n distinct transcendental entire solutions of finite order for this case. If a = 0 and d 6= 0, then d1 = 0 and dn2 = d 6= 0. We can deduce similar results as the above.¤ Proof of Theorem 1.2. The proof of Theorem 1.2 is similar as the one of Theorem 1.3.¤ 4 Examples Example 4.1. In the following, we give examples to show the sharpness of Theorem 1.1. Consider the nonlinear differential-difference equation f 2 (z) + P1 (z, f ) = 1 + 4(z − π)2 ,

(4.1)

where P1 (z, f ) =

√ √ √ 1 02 02 2 2 f (z)+f (z−π)+4(z−π) f (z−π)+f (z+ π)+f (z− π)+2 cos(2 πz)f (z). 4z 2

Clearly, n = 2 = d(P ), and f1 (z) = sin z 2 is an admissible transcendental entire solution of (4.1). This shows our assumption “n > d(P )” in Theorem 1.1(i) is sharp. Consider the nonlinear differential-difference equation f 4 (z) + P2 (z, f ) = 1 + z, where P2 (z, f ) = 2f 0 (z) − f 02 (z + π) − zf (z)f (z +

(4.2) π ). 2

Clearly, n = 4 = w(P ), and f2 (z) = tan z is an admissible transcendental meromorphic solution of (4.2). This shows our assumption “n > w(P )” in Theorem 1.1(ii) is sharp. Example 4.2. In the following, we give examples to illustrate the existence of entire solutions of finite√ order of (1.7) under the assumptions in Theorem 1.3. Denote ε = − 21 + 23 i, which is a cubit root of unity, and consider the nonlinear difference equation f 3 (z) + q4m f (z) = π 3 eibz − e−ibz . (4.3)

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If m = 2, q = 43 π, b = 32 π, c = 2, then (4.3) has three solutions as follows. f1 (z) = πe

iπz 2

f2 (z) = πεe

− e−

iπz 2

f3 (z) = πε2 e

iπz 2

,

− ε2 e−

iπz 2

− εe−

iπz 2

,

iπz 2

.

If m = 3, q = 3π, b = π, c = 1, then (4.3) has three solutions as follows. f1 (z) = πe

iπz 3

f2 (z) = πεe

− e−

iπz 3

f3 (z) = πε2 e

iπz 3

,

− ε2 e−

iπz 3

− εe−

iπz 3

,

iπz 3

.

If m = 4, q = − 34 π, b = −3π, c = 12 , then (4.3) has three solutions as follows. f1 (z) = πe−iπz − eiπz , f2 (z) = πεe−iπz − ε2 eiπz , f3 (z) = πε2 e−iπz − εeiπz . Consider the nonlinear difference equation f n (z) + q(z)4m f (z) = ie−3z ,

n, m ∈ N, n ≥ 3,

(4.4)

where p(z) is a polynomial. If q(z) ≡ 0 or c = − 2knπi , k ∈ Z, then (4.4) has n solutions 3 as follows. 3z fj (z) = dj e− n , j = 0, 1, · · · , n − 1, where dnj = i,

j = 0, 1, · · · , n − 1.

Acknowledgements This work was supported by the National Natural Science Foundation of China (11301233, 11171119), the Natural Science Foundation of Jiangxi Province in China (20151BAB201004, 20151BAB201008), and the Youth Science Foundation of Education Bureau of Jiangxi Province in China (GJJ14271).

References [1] Ablowitz M.J., Halburd R.G. and Herbst B., On the extension of the Painlev´e property to difference equations, Nonlinearity, 13, 2000, 889-905. [2] Chiang Y.M. and Feng S.J, On the Nevanlinna characteristic of f (z + η) and difference equations in the complex plane, Ramanujan J., 16, 2008, 105-129.

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[3] Gol’dberg A.A. and Ostrovskii I.V., The distribution of values of meromorphic functions, Nauka, Moscow, 1970. (in Russian) [4] Halburd R.G. and Korhonen R.J., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314, 2006, 477-487. [5] Halburd R.G. and Korhonen R.J., Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31, 2006, 463-478. [6] Halburd R.G., Korhonen R.J. and Tohge K., Holomorphic curves with shift-invariant hyper-plane preimages, Trans. Amer. Math. Soc. 366, 2014, 4267-4298. [7] Hayman W.K., Meromorphic functions, Clarendon Press, Oxford, 1964. [8] Laine I., Nevanlinna theory and complex differential equations, Walter de Gruyter, Berlin, 1993. [9] Laine I. and Yang C.C., Clunie theorems for difference and q-difference polynomials, J. London Math. Soc., 76(3), 2007, 556-566. [10] Liu N.N., L¨ u W.R., Shen T.T. and Yang C.C., Entire solutions of certain type of difference equations, J. Inequal. Appl., 2014(63), 2014, 1-9. [11] Peng C.W. and Chen Z.X., On a conjecture concerning some nonlinear difference equations, Bull. Malays. Math. Sci. Soc., (2)36(1), 2013, 221-227. [12] Qi J.M., Ding J. and Zhu T.Y., Some results about a special nonlinear difference equation and uniqueness of difference polynomial, J. Inequal. Appl., 2011(50), 2011, 1-10. [13] Wang S.M. and Li S., On entire solutions of nonlinear difference-differential equations, Bull. Korean Math. Soc. 50(5), 2013, 1471-1479. [14] Whittaker J.M., Interpolatory function theory, Cambridge Tracts in Mathematics and Mathematical Physics No.33, Cambridge University Press, 1935. [15] Yang, C.C. and Laine I., On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser.A Math. Sci., 86(1), 2010, 10-14. [16] Yang C.C. and Li P., On the transcendental solutions of a certain type of nonlinear differential equation, Arch. Math., 82(5), 2004, 442-448. [17] Yang C.C. and Yi H.X., Uniqueness theory of meromorphic functions, Kluwer Academic Publishers Group, Dordrecht, 2003. [18] Zheng X.M. and Tu J., Existence and growth of meromorphic solutions of some nonlinear qdifference equations, Adv. Difference Equ., 2013(33), 2013, 1-9. [19] Zheng X.M. and Xu H.Y., On the deficiencies of some differential-difference polynomials, Abstr. Appl. Anal., 2014, Article ID 378151, 12 pages.

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Asymptotic approximations of a stable and unstable manifolds of a two-dimensional quadratic map J. Bekteˇsevi´c† and M.R.S Kulenovi´c‡1 and E. Pilav§2 †

Division of Mathematics Faculty of Mechanical Engineering, University of Sarajevo, Bosnia and Herzegovina ‡ Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA §

Department of Mathematics University of Sarajevo, Sarajevo, Bosnia and Herzegovina

Abstract. We find the asymptotic approximations of the stable and unstable manifolds of the saddle equilibrium solutions and the saddle period-two solutions of the following difference equation xn+1 = cx2n−1 + dxn + 1, where the parameters c and d are positive numbers and initial conditions x−1 and x0 are arbitrary nonnegative numbers. These manifolds determine completely global dynamics of this equation. Keywords. Basin of attraction, cooperative, difference equation, local stable manifold, local unstable manifold, monotonicity, period-two solutions; AMS 2000 Mathematics Subject Classification: Primary: 37B25, 37D10 Secondary: 37M99, 65P40, 65Q30 .

1

Introduction

In this paper we consider the difference equation xn+1 = cx2n−1 + dxn + 1,

(1)

where the parameters c and d are positive numbers and initial conditions x−1 and x0 are arbitrary nonnegative numbers. Set un = xn−1 and vn = xn for n = 0, 1, . . .

(2)

and write Eq.(1) in the equivalent form: un+1 vn+1

=

vn

=

cu2n

(3) + dvn + 1.

Let T be the corresponding map defined by:     u v T = . v cu2 + dv + 1

(4)

It is easy to see that T2

       u u cu2 + dv
+ 1 =T T = . v v d cu2 + dv + 1 + cv 2 + 1

(5)

The local dynamics of the map T was derived in [1] where it was shown that the following holds: Theorem 1 If d < 1 and (d − 1)2 − 4c ≥ 0 1 2

Corresponding author, e-mail: [email protected] Partially supported by FMON Grant No. 05-39-3632–1/14

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then Eq.(1) has the equilibrium points x ¯1 and x ¯2 where p p 1 − d − (d − 1)2 − 4c 1 − d + (d − 1)2 − 4c x ¯1 = , x ¯2 = 2c 2c and the following holds: i) x ¯1 is locally asymptotically stable if c
1. Then, there are two unique invariant manifolds W s and W tangents to (1, 0) and (0, 1) at (0,0), which are graphs of the maps u

ϕ : E1 → E2 and ψ : E1 → E2 , such that ϕ(0) = ψ(0) = 0 and ϕ0 (0) = ψ 0 (0) = 0. See [4, 5, 10, 14]. Letting ηn = ϕ(ξn ) yields ηn+1 = ϕ(ξn+1 ) = ϕ(µ1 ξn + g1 (ξn , ϕ(ξn ))).

(10)

On the other hand by (9) ηn+1 = µ2 ϕ(ξn ) + g2 (ξn , ϕ(ξn )). Equating equations (10) and (11) yields ϕ(µ1 ξn + g1 (ξn , ϕ(ξn ))) = µ2 ϕ(ξn ) + g2 (ξn , ϕ(ξn )).

(11) (12)

Similarly, letting ξn = ψ(ηn ) yields ξn+1 = ψ(ηn+1 ) = ψ(µ2 ηn + g2 (ψ(ηn ), ηn )).

(13)

By using (9) we obtain ξn+1 = µ1 ψ(ηn ) + g1 (ψ(ηn ), ηn ). Equating equations (13) and (14) yields ψ(µ2 ηn + g2 (ψ(ηn ), ηn )) = µ1 ψ(ηn ) + g1 (ψ(ηn ), ηn ).

(14) (15)

Thus the functional equations (12) and (15), define the local stable manifold W s = {(ξ, η) ∈ R2 : η = ϕ(ξ)}, and the local unstable manifold W u = {(ξ, η) ∈ R2 : ξ = ψ(η)}. Without loss generality, we can assume that solutions of the functional equations (12) and (15) take the forms ψ(η) = α2 η 2 + β2 η 3 + O(|η|4 ) and ϕ(ξ) = α1 ξ 2 + β1 ξ 3 + O(|ξ|4 ), where αi , βi , i = 1, 2 are undetermined coefficients.

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3.1

Normal form of the map T at x¯2

Put yn = xn − x ¯2 . Then Eq(1) becomes yn+1 = c (¯ x2 + yn−1 ) 2 + d (¯ x2 + yn ) − x ¯2 + 1.

(16)

un = yn−1 and vn = yn for n = 0, 1, . . .

(17)

Set and write Eq(16) in the equivalent form: un+1

=

vn+1

vn

(18) 2

c (¯ x2 + un ) + d (¯ x2 + vn ) − x ¯2 + 1.

=

Let F be the function defined by:     u v F = . 2 v c (¯ x2 + u) + d (¯ x2 + v) − x ¯2 + 1

(19)

Then F has the fixed point (0, 0) and maps (−¯ x2 , ∞)2 into (−¯ x2 , ∞)2 . The Jacobian matrix of F is given by   0 1 JacF (u, v) = . 2c (u + x ¯2 ) d At (0, 0), JacF (u, v) has the form  J0 = JacF (0, 0) =

0 2c¯ x2

1 d

 .

(20)

The eigenvalues of (20) are µ1,2 where   p p 1 1 µ1 = d − 8c¯ x2 + d2 and µ2 = d + 8c¯ x2 + d 2 , 2 2 and the corresponding eigenvectors are given by √ √ T  T  x2 + d2 d − 8c¯ x2 + d 2 d + 8c¯ ,1 and v2 = − ,1 , v1 = − 4c¯ x2 4c¯ x2 respectively. Then we have that F

   u 0 = v 2c¯ x2

1 d

    u f1 (u, v) + , v g1 (u, v)

(21)

where f1 (u, v) = 0 g1 (u, v) = x ¯2 (c¯ x2 + d − 1) + cu2 + 1. Then, the system (16) is equivalent to    un+1 0 = vn+1 2c¯ x2 Let



1 d

un vn





un vn 

=P·



ξn ηn



 f1 (un , vn ) + . g1 (un , vn )

(22)



where √ P =

−

d+

d2 +8c¯ x2 4c¯ x2

1

√ −

d−

d2 +8c¯ x2 4c¯ x2



!

 and P −1 = 

1

−√

2c¯ x2

d2 +8c¯ x2

√

2c¯ x2

d2 +8c¯ x2

Then system (22) is equivalent to    ξn+1 µ1 = ηn+1 0

0 µ2



ξn ηn



√  d2 +8c¯ x2 −d √  2 √d2 +8c¯ x2 . d+ d2 +8c¯ x2 √

   ξn + P −1 · H1 P · , ηn

2

d2 +8c¯ x2

(23)

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where H1 Let G1

    f1 (u, v) u . := g1 (u, v) v

       u u f˜1 (u, v) . = P −1 · H1 P · := v v g˜1 (u, v)

By straightforward calculation we obtain that √
Υ1 (u, v) 8c¯ x2 + d2 − d √ f˜1 (u, v) = , 16c¯ x22 8c¯ x2 + d2 √
Υ1 (u, v) 8c¯ x2 + d 2 + d √ , g˜1 (u, v) = 16c¯ x22 8c¯ x2 + d2 where Υ1 (u, v) = 8c2 x ¯42 + d u2 − v 2

3.2


p



8c¯ x2 + d2 + 4c¯ x2 2(d − 1)¯ x22 + 2¯ x2 + (u − v)2 + d2 u2 + v 2 .

Stable and unstable manifolds corresponding to x¯2

Assume that d < 1 and (d − 1)2 − 4c ≥ 0. Then Eq.(1) has the equilibrium point x ¯2 where p 1 − d + (d − 1)2 − 4c x ¯2 = 2c which is a saddle point if (d − 1)2 (1 − 3d)(d + 1) 0, Γ(q) 0 (t − s)1−q provided the integral exists. Lemma 2.3 (see [1], [2]) (i)

If α > 0, β > 0, β > α, f ∈ L(0, 1) then

I α I β f (t) = I α+β f (t), Dα I α f (t) = f (t), Dα I β f (t) = I β−α f (t). (ii)

c

Dα tλ−1 =

Γ(λ) λ−α−1 t , λ > [α] and Γ(λ − α)

c

Dα tλ−1 = 0, λ < [α].

To define the solution for the problem (1)-(2), we use the following lemma. Lemma 2.4 Let a 6= 2 and Γ(2 − β) 6= b. For φ ∈ C([0, 1], R), the integral solution of the linear problem  c α D x(t) = φ(t), t ∈ [0, 1], 1 < q ≤ 2,   Z 1 (3)   x(0) + x(1) = a x(s)ds, x0 (0) = b c Dγ x(1), 0 < γ ≤ 1, 0

is given by Z b(2t − 1)Γ(2 − γ) 1 (1 − s)α−γ−1 (t − s)α−1 φ(s)ds + φ(s)ds x(t) = Γ(α) 2(Γ(2 − γ) − b) 0 Γ(α − γ) 0 Z 1 Z 1 1 (1 − s)α−1 a (1 − s)α − φ(s)ds + φ(s)ds. 2−a 0 Γ(α) 2 − a 0 Γ(α + 1) Z

t

(4)

Proof. As argued in [2], the general solution of the fractional differential equation in (3) can be written as Z t (t − s)α−1 x(t) = c0 + c1 t + φ(s)ds, (5) Γ(q) 0 where c0 , c1 ∈ R are arbitrary constants. Using the boundary condition x0 (0) = b c Dγ x(1) in (5), we find that Z 1 (1 − s)α−β−1 bΓ(2 − β) φ(s)ds. c1 = Γ(2 − β) − b 0 Γ(α − β)

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FRACTIONAL DIFFERENTIAL EQUATIONS R1 In view of the condition x(0) + x(1) = a 0 x(s)ds, (5) yields Z 1 Z 1Z s (1 − s)α−1 (s − u)α−1 ac1 2c0 + c1 + φ(s)ds = a φ(u)du + + ac0 , Γ(α) Γ(α) 2 0 0 0 which, on inserting the value of c1 and using the composition law of Riemann-Liouville integration, gives Z 1 (1 − s)α−β−1 1 bΓ(2 − β) c0 = − φ(s)ds 2 [Γ(2 − β) − b] 0 Γ(α − β) Z 1 Z 1 a (1 − s)α (1 − s)α−1 1 + φ(s)ds − φ(s)ds. 2 − a 0 Γ(α + 1) 2−a 0 Γ(α) Substituting the values of c0 , c1 in (5) yields (4). This completes the proof.

3



Main Results

Let us introduce the space Xi = {ui (t)|ui (t) ∈ C([0, 1])} endowed with the norm kui k = sup{|ui (t)|, t ∈ [0, 1]}, i = 1, 2. Obviously (Xi , k · k) is a Banach space. In consequence, the product space (X1 × X2 , k(u1 , u2 )k) is also a Banach space with norm k(u1 , u2 )k = ku1 k + ku2 k. In view of Lemma 2.4, we define an operator T : X1 × X2 → X1 × X2 by   T1 (u, v)(t) T (u, v)(t) = , T2 (u, v)(t) where Z (t − s)α−1 b(2t − 1)Γ(2 − γ) 1 (1 − s)α−γ−1 f (s, u(s), v(s))ds + f (s, u(s), v(s))ds = Γ(α) 2(Γ(2 − γ) − b) 0 Γ(α − γ) 0 Z 1 Z 1 1 a (1 − s)α−1 (1 − s)α + f (s, u(s), v(s))ds − f (s, u(s), v(s))ds, 2−a 0 Γ(α) 2 − a 0 Γ(α + 1) Z

T1 (u, v)(t)

t

and =

t

Z (t − s)β−1 b1 (2t − 1)Γ(2 − δ) 1 (1 − s)β−δ−1 h(s, u(s), v(s))ds + h(s, u(s), v(s))ds Γ(α) 2(Γ(2 − δ) − b1 ) 0 Γ(β − δ) 0 Z 1 Z 1 1 a1 (1 − s)β−1 (1 − s)β + h(s, u(s), v(s))ds − h(s, u(s), v(s))ds. 2 − a1 0 Γ(β) 2 − a1 0 Γ(β + 1) Z

T2 (u, v)(t)

For the sake of convenience, let us set 1 + |2 − a| |b|Γ(2 − γ) |a| + + , |2 − a|Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1) |2 − a|Γ(α + 2)

(6)

1 + |2 − a1 | |b1 |Γ(2 − δ) |a1 | + + . |2 − a1 |Γ(β + 1) 2|Γ(2 − δ) − b1 |Γ(β − δ + 1) |2 − a1 |Γ(β + 2)

(7)

M1 = M2 =

We need the following known theorems in the sequel. Lemma 3.1 (Schauder’s fixed point theorem) [24]. Let U be a closed, convex and nonempty subset of a Banach space X. Let P : U → U be a continuous mapping such that P (U ) is a relatively compact subset of X. Then P has at least one fixed point in U. Lemma 3.2 (Leray-Schauder alternative) ([24] p. 4.) Let F : E → E be a completely continuous operator (i.e., a map that restricted to any bounded set in E is compact). Let E(F ) = {x ∈ E : x = λF (x) for some 0 < λ < 1}. Then either the set E(F ) is unbounded, or F has at least one fixed point.

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B. AHMAD, S. K. NTOUYAS AND A. ALSAEDI

3.1

Existence results

Here we study the existence of solutions for the system (1)-(2) by means of Schauder’s fixed point theorem and Leray-Schauder alternative. Theorem 3.3 Assume that there exist positive constants ci , di , ei ∈ (0, ∞), i = 1, 2 such that the following condition holds: (H1 ) |f (t, x, y)| ≤ c1 |x|ρ1 + d1 |y|σ1 + e1 , and |h(t, x, y)| ≤ c2 |x|ρ2 + d2 |y|σ2 + e2 , 0 < ρi , σi < 1, i = 1, 2. Then the system (1)-(2) has at least one solution on [0, 1]. Proof. Define a ball in Banach space X1 × X2 as BR = {(u, v) : (u, v) ∈ X1 × X2 , k(u, v)k ≤ R}, where 1

1

R ≥ max{(6Mi ci ) 1−ρi , (6Mi di ) 1−σi , 6Mi ei , }, i = 1, 2.

(8)

Obviously BR is a closed, bounded and convex subset of the Banach space X1 × X2 . In the first step, we show that T : BR → BR . For (u, v) ∈ BR . For that we have |T1 (u, v)(t)|

t

(t − s)α−1 |f (s, u(s), v(s))|ds Γ(α) 0 Z |b(2t − 1)|Γ(2 − γ) 1 (1 − s)α−γ−1 + |f (s, u(s), v(s))|ds 2|Γ(2 − γ) − b| Γ(α − γ) 0 Z 1 Z 1 1 (1 − s)α−1 |a| (1 − s)α |f (s, u(s), v(s))|ds + |f (s, u(s), v(s))|ds + |2 − a| 0 Γ(α) |2 − a| 0 Γ(α + 1) ( 1 |b|Γ(2 − γ) ρ1 σ1 ≤ (c1 R + d1 R + e1 ) + Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1) ) 1 |a| + + , |2 − a|Γ(α + 1) |2 − a|Γ(α + 2) Z

≤

which implies that kT1 (u, v)k ≤ M1 (c1 Rρ1 + d1 Rσ1 + e1 ) ≤

R R R R + + = . 6 6 6 2

Similarly, we can obtain kT2 (u, v)k ≤ M2 (c2 Rρ2 + d2 Rσ2 + e2 ) ≤

R R R R + + = . 6 6 6 2

Clearly kT (u, v)k = kT1 (u, v)k + kT2 (u, v)k ≤ R, and in consequence we get T : BR → BR . Observe that continuity of f, h implies that T is continuous. Next, we shall show that for every bounded subset BR of X1 × X2 the family F (BR ) is equicontinuous. Since f, g are continuous, we can assume that |f (t, u(t), v(t)| ≤ N1 and |h(t, u(t), v(t)| ≤ N2 for any u, v ∈ BR and t ∈ [0, 1]. Now let 0 ≤ t1 < t2 ≤ 1. Then we have |T1 (u, v)(t2 ) − T1 (u, v)(t1 )| Z t2 Z t1 1 1 ≤ (t2 − s)α−1 f (s, u(s), v(s))ds − (t1 − s)α−1 f (s, u(s), v(s))ds Γ(α) 0 Γ(α) 0 Z 1 α−γ−1 (1 − s) 2|b|Γ(2 − γ)|t2 − t1 | |f (s, u(s), v(s))|ds + 2|Γ(2 − γ) − b| Γ(α − γ) 0

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FRACTIONAL DIFFERENTIAL EQUATIONS

≤

N1 Γ(α)

Z

t1 α−1

[(t2 − s)

α−1

− (t1 − s)

0

N1 ]ds + Γ(α)

Z

t2

(t2 − s)α−1 ds

t1

Z 2N1 |b|Γ(2 − γ)|t2 − t1 | 1 (1 − s)α−γ−1 ds 2|Γ(2 − γ) − b| Γ(α − γ) 0 2N1 |b|Γ(2 − γ)|t2 − t1 | N1 α (tα . 2 − t1 ) + Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1)

+ ≤

Analogously, we can have |T2 (u, v)(t2 ) − T2 (u, v)(t1 )| ≤

N2 2N2 |b|Γ(2 − δ)|t2 − t1 | α (tα . 2 − t1 ) + Γ(β + 1) 2|Γ(2 − δ) − b|Γ(β − δ + 1)

So kT1 (u, v)(t2 ) − T1 (u, v)(t1 )k → 0, kT2 (u, v)(t2 ) − T2 (u, v)(t1 )k → 0, as t1 → t2 . Therefore it follows that the operator T : BR → BR is equicontinuous and uniformly bounded. Hence, by Arzel´a-Ascoli theorem, T is completely continuous operator. Thus all the conditions of Theorem 3.1 are satisfied, which in turn, implies that the problem (1) has at least one solution. This completes the proof.  Remark 3.4 For ρi , σi > 1 (i = 1, 2) in the condition (H1 ), the conclusion of Theorem 3.6 remains true with a modified value of R given by (8). Theorem 3.5 Assume that: (H2 ) There exist real constants ki , λi ≥ 0 (i = 1, 2) and k0 > 0, λ0 > 0 such that ∀xi ∈ R, i = 1, 2, we have |f (t, x1 , x2 )| ≤ k0 + k1 |x1 | + k2 |x2 |, |h(t, x1 , x2 )| ≤ λ0 + λ1 |x1 | + λ2 |x2 |. Then the system (1)-(2) has at least one solution, provided M1 k1 + M2 λ1 < 1 and M1 k2 + M2 λ2 < 1, where M1 and M2 are given by (6) and (7) respectively. Proof. First we show that the operator T : X1 ×X2 → X1 ×X2 is completely continuous. By continuity of functions f and h, the operator T is continuous. Let Ω ⊂ X1 × X2 be bounded. Then there exist positive constants L1 and L2 such that |f (t, u(t), v(t)| ≤ L1 ,

|h(t, u(t), v(t)| ≤ L2 , ∀(u, v) ∈ Ω.

Then for any (u, v) ∈ Ω, we have Z t (t − s)α−1 |f (s, u(s), v(s))|ds |T1 (u, v)(t)| ≤ Γ(α) 0 Z |b(2t − 1)|Γ(2 − γ) 1 (1 − s)α−γ−1 + |f (s, u(s), v(s))|ds 2|Γ(2 − γ) − b| Γ(α − γ) 0 Z 1 Z 1 1 (1 − s)α−1 |a| (1 − s)α + |f (s, u(s), v(s))|ds + |f (s, u(s), v(s))|ds |2 − a| 0 Γ(α) |2 − a| 0 Γ(α + 1) ( ) 1 |b|Γ(2 − γ) 1 |a| ≤ L1 + + + , Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1) |2 − a|Γ(α + 1) |2 − a|Γ(α + 2) which implies that ( kT1 (u, v)k

≤ L1

1 |b|Γ(2 − γ) + Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1)

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1 |a| + + |2 − a|Γ(α + 1) |2 − a|Γ(α + 2)

) = L1 M1 .

Similarly, we can get (

kT2 (u, v)k

1 |b1 |Γ(2 − δ) + Γ(β + 1) 2|Γ(2 − δ) − b1 |Γ(β − δ + 1) ) 1 |a1 | = L2 M2 . + + |2 − a1 |Γ(β + 1) |2 − a1 |Γ(β + 2)

≤ L2

Thus, it follows from the above inequalities that the operator T is uniformly bounded. Next, we show that T is equicontinuous. Let t1 , t2 ∈ [0, 1] with t1 < t2 . Then we have |T1 (u(t2 ), v(t2 )) − T1 (u(t1 ), v(t1 ))| Z t2 Z t1 1 1 α−1 α−1 ≤ L1 (t2 − s) (t1 − s) ds − ds Γ(α) 0 Γ(α) 0 Z 2|b|Γ(2 − γ)|t2 − t1 | 1 (1 − s)α−γ−1 ds +L1 2|Γ(2 − γ) − b| Γ(α − γ) 0 Z t1 Z t2 L1 1 ≤ [(t2 − s)α−1 − (t1 − s)α−1 ]ds + (t2 − s)α−1 ds Γ(α) 0 Γ(α) t1 Z 2|b|L1 Γ(2 − γ)|t2 − t1 | 1 (1 − s)α−γ−1 ds + 2|Γ(2 − γ) − b| Γ(α − γ) 0 L1 2L1 |b|Γ(2 − γ)|t2 − t1 | (tα − tα . ≤ 1)+ Γ(α + 1) 2 2|Γ(2 − γ) − b|Γ(α − γ + 1) Analogously, we can obtain |T2 (u(t2 ), v(t2 )) − T2 (u(t1 ), v(t1 ))| ≤

L2 2L2 |b1 |Γ(2 − δ)|t2 − t1 | (tα − tα . 1)+ Γ(α + 1) 2 2|Γ(2 − δ) − b1 |Γ(β − δ + 1)

Therefore, the operator T (u, v) is equicontinuous, and thus the operator T (u, v) is completely continuous. Finally, it will be verified that the set E = {(u, v) ∈ X1 × X2 |(u, v) = λT (u, v), 0 ≤ λ ≤ 1} is bounded. Let (u, v) ∈ E, then (u, v) = λT (u, v). For any t ∈ [0, 1], we have u(t) = λT1 (u, v)(t),

v(t) = λT2 (u, v)(t).

Then (

) 1 + |2 − a| |b|Γ(2 − γ) |a| + + (k0 + k1 kuk + k2 kvk), |2 − a|Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1) |2 − a|Γ(α + 2)

(

) 1 + |2 − a1 | |b|Γ(2 − δ) |a| + + (λ0 + λ1 kuk + λ2 kvk). |2 − a1 |Γ(α + 1) 2|Γ(2 − δ) − b1 |Γ(β − δ + 1) |2 − a|Γ(β + 2)

|u(t)| ≤ and |v(t)| ≤

Hence we have kuk ≤ M1 (k0 + k1 kuk + k2 kvk), kvk ≤ M2 (λ0 + λ1 kuk + λ2 kvk), which imply that kuk + kvk = (M1 k0 + M2 λ0 ) + (M1 k1 + M2 λ1 )kuk + (M1 k2 + M2 λ2 )kvk.

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FRACTIONAL DIFFERENTIAL EQUATIONS

Consequently, k(u, v)k ≤

M1 k0 + M2 λ0 , M0

for any t ∈ [0, 1], where M0 = min{1 − (M1 k1 + M2 λ1 ), 1 − (M1 k2 + M2 λ2 )}, ki , λi ≥ 0 (i = 1, 2). This shows that E is bounded. Thus, by Lemma 3.2, the operator T has at least one fixed point. Hence the problem (1)-(2) has at least one solution. This completes the proof. 

3.2

Uniqueness of solutions

In this subsection, we prove the uniqueness of solutions for the system (1)-(2) via Banach’s contraction mapping principle. Theorem 3.6 Assume that (H3 ) f, h : [0, 1] × R2 → R are continuous functions and there exist constants mi , ni (i = 1, 2) such that for all t ∈ [0, 1] and ui , vi ∈ R, i = 1, 2, |f (t, u1 , u2 ) − f (t, v1 , v2 )| ≤ m1 |u1 − v1 | + m2 |u2 − v2 | and |h(t, u1 , u2 ) − h(t, v1 , v2 )| ≤ n1 |u1 − v1 | + n2 |u2 − v2 |. Then the system (1)-(2) has a unique solution if M1 (m1 + m2 ) + M2 (n1 + n2 ) < 1, where M1 and M2 are given by (6) and (7) respectively. Proof. Let us fix supt∈[0,1] f (t, 0, 0) = ζ1 < ∞ and supt∈[0,1] h(t, 0, 0) = ζ2 < ∞ such that r≥

ζ 1 M1 + ζ 2 M2 . 1 − M1 (m1 + m2 ) − M2 (n1 + n2 )

As a first step, we show that T Br ⊂ Br , where Br = {(u, v) ∈ X × Y : k(u, v)k ≤ r}. For (u, v) ∈ Br , we have (Z t (t − s)α−1 |T1 (u, v)(t)| ≤ sup |f (s, u(s), v(s))|ds Γ(α) t∈[0,1] 0 Z |b(2t − 1)|Γ(2 − γ) 1 (1 − s)α−γ−1 + |f (s, u(s), v(s))|ds 2|Γ(2 − γ) − b| Γ(α − γ) 0 ) Z 1 Z 1 1 (1 − s)α−1 |a| (1 − s)α + |f (s, u(s), v(s))|ds + |f (s, u(s), v(s))|ds |2 − a| 0 Γ(α) |2 − a| 0 Γ(α + 1) (Z t (t − s)α−1 ≤ sup (|f (s, u(s), v(s)) − f (s, 0, 0)| + |f (s, 0, 0)|)ds Γ(α) t∈[0,1] 0 Z 1 |b|Γ(2 − γ) (1 − s)α−γ−1 + (|f (s, u(s), v(s)) − f (s, 0, 0)| + |f (s, 0, 0)|)ds 2|Γ(2 − γ) − b| 0 Γ(α − γ) Z 1 1 (1 − s)α−1 + (|f (s, u(s), v(s)) − f (s, 0, 0)| + |f (s, 0, 0)|)ds |2 − a| 0 Γ(α) ) Z 1 |a| (1 − s)α + |(f (s, u(s), v(s)) − f (s, 0, 0)| + |f (s, 0, 0)|)ds |2 − a| 0 Γ(α + 1) ( 1 |b|Γ(2 − γ) ≤ + Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1)

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B. AHMAD, S. K. NTOUYAS AND A. ALSAEDI ) 1 |a| + (m1 kuk + m2 kvk + ζ1 ) + |2 − a|Γ(α + 1) |2 − a|Γ(α + 2) = M1 [(m1 + m2 )r + ζ1 ]. Hence kT1 (u, v)(t)k ≤ M1 [(m1 + m2 )r + ζ1 ]. In the same way, we can obtain that kT2 (u, v)(t)k ≤ M2 [(n1 + n2 )r + ζ2 ]. Consequently, it follows that kT (u, v)(t)k ≤ r. Now for (u2 , v2 ), (u1 , v1 ) ∈ X1 × X2 , and for any t ∈ [0, 1], we get |T1 (u2 , v2 )(t) − T1 (u1 , v1 )(t)| Z t (t − s)α−1 ≤ |f (s, u2 (s), v2 (s)) − f (s, u1 (s), v1 (s))|ds Γ(α) 0 Z |b(2t − 1)|Γ(2 − γ) 1 (1 − s)α−γ−1 + |f (s, u2 (s), v2 (s)) − f (s, u1 (s), v1 (s))|ds 2|Γ(2 − γ) − b| Γ(α − γ) 0 Z 1 1 (1 − s)α−1 + |f (s, u2 (s), v2 (s)) − f (s, u1 (s), v1 (s))|ds |2 − a| 0 Γ(α) Z 1 |a| (1 − s)α + |f (s, u2 (s), v2 (s)) − f (s, u1 (s), v1 (s))|ds |2 − a| 0 Γ(α + 1) ( 1 |b|Γ(2 − γ) ≤ + Γ(α + 1) 2|Γ(2 − γ) − b|Γ(α − γ + 1) ) 1 |a| + + (m1 ku2 − u1 k + m2 kv2 − v1 k) |2 − a|Γ(α + 1) |2 − a|Γ(α + 2) = M1 (m1 ku2 − u1 k + m2 kv2 − v1 k) ≤ M1 (m1 + m2 )(ku2 − u1 k + kv2 − v1 k), and consequently we obtain kT1 (u2 , v2 )(t) − T1 (u1 , v1 )k ≤ M1 (m1 + m2 )(ku2 − u1 k + kv2 − v1 k).

(9)

Similarly, we can get kT2 (u2 , v2 )(t) − T2 (u1 , v1 )k ≤ M2 (n1 + n2 )(ku2 − u1 k + kv2 − v1 k).

(10)

Clearly it follows from (9) and (10) that kT (u2 , v2 )(t) − T (u1 , v1 )(t)k ≤ [M1 (m1 + m2 ) + M2 (n1 + n2 )](ku2 − u1 k + kv2 − v1 k). Since M1 (m1 + m2 ) + M2 (n1 + n2 ) < 1, therefore T is a contraction operator. So, by Banach’s fixed point theorem, the operator T has a unique fixed point, which is the unique solution of problem (1)-(2). This completes the proof.  Example 3.7 Consider the following problem  1 1 |x(t)|  c 3/2  D x(t) = +1+ sin2 y(t), t ∈ [0, 1],   2  4(t + 2) 1 + |x(t)| 32      1 |y(t)| 1  c 5/3    D y(t) = 32π sin(2πx(t)) + 16(1 + |y(t)|) + 2 , t ∈ [0, 1], Z 1  1   x(0) + x(1) = 4 x(s)ds, x0 (0) = c D1/2 x(1),    2 0    Z  1    y(0) + y(1) = 1 y(s)ds, y 0 (0) = 3 c D1/3 y(1).  5 0

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FRACTIONAL DIFFERENTIAL EQUATIONS

Here α = 3/2, β = 5/3, γ = 1/2, δ = 1/3, a = 4, a1 = 1/5, b = 1/2, b1 = 3. Using the given data, it is found that M1 ≈ 5.6166715, M2 ≈ 1.6038591, |f (t, u1 , u2 ) − f (t, v1 , v2 )| ≤

1 1 |u1 − u2 | + |v1 − v2 |, 16 16

|h(t, u1 , u2 ) − h(t, v1 , v2 )| ≤

1 1 |u1 − u2 | + |v1 − v2 |, 16 16

and M1 (m1 + m2 ) + M2 (n1 + n2 ) ≈ 0.9025662 < 1. As all the conditions of Theorem 3.6 are satisfied, therefore its conclusion applies to the problem (11).

3.3

Special cases

We obtain some special cases of the results obtained in this paper by fixing the parameters involved in the problem (1)-(2) which are listed below. R1 • If b = b1 = 0, then our results correspond to the boundary conditions: x(0)+x(1) = a 0 x(s)ds, x0 (0) = R1 0; y(0) + y(1) = a1 0 y(s)ds, y 0 (0) = 0. • We can get the results for the boundary data: x(0)+x(1) = 0, x0 (0) = 0; y(0)+y(1) = 0, y 0 (0) = 0 by fixing a = 0, a1 = 0, b = 0, b1 = 0. • In case we choose a = 0, a1 = 0, b 6= 0, b1 6= 0, we get the results for the boundary conditions: x(0) + x(1) = 0, x0 (0) = b c Dγ x(1); y(0) + y(1) = 0, y 0 (0) = b1 c Dδ y(1), 0 < γ, δ ≤ 1. • By taking γ = δ = 1 with b 6= 1 6= b1 , our results reduce to the ones for Ra given system of 1 fractional differential equations with boundary conditions: x(0) + x(1) = a 0 x(s)ds, x0 (0) = R 1 bx0 (1); y(0) + y(1) = a1 0 y(s)ds, y 0 (0) = b1 y 0 (1). We emphasize that all the results obtained for different values of the parameters are new. Acknowledgment. The project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. 41-130-35-HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support.

References [1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [2] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. [3] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009. [4] J. Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007. [5] B. Ahmad, J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 2011, 2011:36, 9 pp. [6] Z.B. Bai, W. Sun, Existence and multiplicity of positive solutions for singular fractional boundary value problems, Comput. Math. Appl. 63, (2012) 1369-1381. [7] Y. Su, Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument, Acta Appl. Math. 118 (2012), 81-105.

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[8] B..Ahmad, S.K. Ntouyas, A boundary value problem of fractional differential equations with antiperiodic type integral boundary conditions, J. Comput. Anal. Appl. 15 (2013), 1372-1380. [9] J. Tariboon, T. Sitthiwirattham, S. K. Ntouyas, Existence results for fractional differential inclusions with multi–point and fractional integral boundary conditions, J. Comput. Anal. Appl. 17 (2014), 343-360. [10] B. Ahmad, S.K. Ntouyas, A. Alsaedi, Hybrid Boundary value problems of q-difference equations and inclusions, J. Comput. Appl. Anal. 19 (2015), 984-993. [11] B. Ahmad, S.K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng. (2013), Art. ID 320415, 9 pp. [12] X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Differ. Equ. 2013, 2013:126. [13] B. Ahmad, S.K. Ntouyas, On higher-order sequential fractional differential inclusions with nonlocal three-point boundary conditions, Abstr. Appl. Anal. 2014, Art. ID 659405, 10 pp. [14] G. Wang, S. Liu, L. Zhang, Eigenvalue problem for nonlinear fractional differential equations with integral boundary conditions, Abstr. Appl. Anal. 2014, Art. ID 916260, 6 pp. [15] L. Zhang, B. Ahmad, G. Wang, Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half line, Bull. Aust. Math. Soc. 91 (2015), 116-128. [16] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009), 64-69. [17] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009), 1838-1843. [18] J. Wang, H. Xiang, Z. Liu, Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, Int. J. Differ. Equ. Volume 2010 (2010), Article ID 186928, 12 pages. [19] S.K. Ntouyas and M. Obaid, A coupled system of fractional differential equations with nonlocal integral boundary conditions, Adv. Differ. Equ 2012, 2012:130. [20] J. Sun, Y. Liu, G. Liu, Existence of solutions for fractional differential systems with antiperiodic boundary conditions, Comput. Math. Appl. 64 (2012), 1557-1566. [21] M. Faieghi, S. Kuntanapreeda, H. Delavari, D. Baleanu, LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dynam. 72 (2013), 301-309. [22] J. Lin, Robust resilient controllers synthesis for uncertain fractional-order large-scale interconnected system, J. Franklin Inst. 351 (2014), 1630-1643. [23] B. Ahmad, S.K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal. 17 (2014), 348-360. [24] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005.

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Solutions of the nonlinear evolution equation via the generalized Riccati equation mapping together with the (G0/G)-expansion method Qazi Mahmood Ul Hassana , Hasibun Naherb , Farah Abdullahb and Syed Tauseef Mohyud-Dina∗ a

HITEC University Taxila Cantt., Pakistan School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia b

Abstract In this article, we investigate the combined KdV-MKdV equation to obtain new exact traveling wave solutions via the generalized Riccati equation mapping together with the(G0 /G)-expansion method. In this method, G0 (θ) = h + f G (ϕ) + g G2 (θ) is used with constant coefficients, as the auxiliary equation and called the generalized Riccati equation. By using this method, we obtain twenty seven exact traveling wave solutions including solitons and periodic solutions and solutions are expressed in the hyperbolic, the trigonometric and the rational functions. It is found that one of our solutions is in good agreement for a special case with the published results which validates our other results. Keywords: The generalized Riccati equation, (G0 /G)-expansion method, Expfunction method, traveling wave solutions, nonlinear evolution equations. Mathematics Subject Classification: 35K99, 35P99, 35P05. PACS: 02.30.Jr,05.45.Yv,02.30.Ik. 1. Introduction The enormous analysis of exact solutions of the nonlinear partial differential equations (PDEs) is one of the important and amazing research fields in all areas in science and engineering, such as, plasma physics, fluid mechanics, chemical ∗ [email protected],[email protected],[email protected],[email protected]

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physics, optical fibers, solid state physics, chemistry, biology, plasma physics and many others [1-41]. In recent years, many researchers used various methods to different nonlinear partial differential equations for constructing traveling wave solutions, for instance, the Backlund transformation [1], the inverse scattereing [2,3], the Jacobi elliptic function expansion [4,5], the tanh function [6,7], the variational iteration [8], the Hirota’s bilinear transformation [9], the direct algebraic [10], the Cole-Hopf transformation [11], the Exp-function [12-18] and others [19-25]. Recently, Wang et al. [26] presented a method, called the (G0 /G)-expansion method. By using this method, they constructed exact traveling wave solutions for the nonlinear evolution equations (NLEEs). In this method, the second order linear ordinary differential equation with constant coefficients G00 (θ)+λG0 (θ)+ µG (θ) = 0 is used, as an auxiliary equation. Afterwards, many researchers applied the (G0 /G)-expansion method to obtain exact traveling wave solutions for the NLEEs. For example, Ozis and Aslan [27] investigated the Kawahara type equations using symbolic computation via this method. In Ref. [28] Gepreel employed this method and found exact solutions for nonlinear PDEs with variable coefficients in mathematical physics whilst Zayed and Al-Joudi [29] studied nonlinear partial differential equations by applying the same method to construct solutions. Naher et al. [30] investigated the Caudrey-Dodd-Gibbon equation by using the useful (G0 /G)-expansion method and obtained abundant exact traveling wave solutions. Feng et al. [31] applied the method to the KolmogorovPetrovskii-Piskunov equation for constructing traveling wave solutions. In Ref. [32], Zhao et al. concerned about this method to obtain exact solutions for the variant Boussinesq equations while Nofel et al. [33] implemented the same method to the higher order KdV equation to get exact traveling wave solutions and so on. Zhu [34] introduced the generalized Riccati equation mapping to solve the (2+1)dimensional Boiti-Leon-Pempinelle equation. In this generalized Riccati equation mapping, he employed G0 (θ) = h + f G (ϕ) + g G2 (θ) with constants co-

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efficients, as the auxiliary equation. In Ref. [35], Li et al. used the Riccati equation expansion method to solve the higher dimensional NLEEs. Bekir and Cevikel [36] investigated nonlinear coupled equation in mathematical physics by applying the tanh-coth method combined with the Riccati equation. Guo et al. [37] studied the diffusion-reaction and the mKdV equation with variable coefficient via the extended Riccati equation mapping method whilst Li and Dai [38] implemented the generalized Riccati equation mapping with the (G0 /G)-expansion method to construct traveling wave solutions for the higher dimensional Jimbo-Miwa equation. In Ref. [39,40] Salas used the projective Riccati equation method to obtain some exact solutions for the Caudrey-DoddGibbon equation and the generalized Sawada-Kotera equations respectively. Many researchers utilized different methods for investigating the combined KdVMKdV equation to construct exact traveling wave solutions, such as, Liu et al. [41] studied the equation by applying the (G0 /G)-expansion method to obtain traveling wave solutions. In the (G0 /G)-expansion method, they used the second order linear ordinary differential equation (LODE) with constant coefficients, as an auxiliary equation. To the best of our knowledge, the combined KdV-MKdV equation is not examined by applying the generalized Riccati equation mapping together with the (G0 /G)-expansion method. In this article, we construct twenty seven exact traveling wave solutions including solitons, periodic, and rational solutions of the combined KdV-MKdV equation involving parameters via the generalized Riccati equation mapping together with the (G0 /G)-expansion method and Exp-function method. 2. The generalized Riccati equation mapping together with the(G0 /G)expansion method Suppose the general nonlinear partial differential equation:

H (v, vt , vx , vxt , vt t , vx x , ...) = 0,

(1)

where v = v (x, t) is an unknown function, H is a polynomial in v = v (x, t)and

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the subscripts indicate the partial derivatives. The most important steps of the generalized Riccati equation mapping together with the (G0 /G)-expansion method [26,34] are as follows: Step 1. Consider the traveling wave variable:

θ = x − B t,

v (x, t) = r (θ) ,

(2)

where Bis the wave speed. Now using Eq. (2), Eq. (1) is converted into an ordinary differential equation for r (θ) :

F (r, r0 , r00 , r000 , ...) = 0,

(3)

where the superscripts stand for the ordinary derivatives with respect to θ. Step 2.

Eq. (3) integrates term by term one or more times according to

possibility, yields constant(s) of integration. The integral constant(s) may be zero for simplicity. Step 3. Suppose that the traveling wave solution of Eq. (3) can be expressed in the form [26,34]: r (θ) =

n X j =0

 ej

G0 G

j (4)

where ej (j = 0, 1, 2, ..., n)and en 6= 0, with G = G (θ) is the solution of the generalized Riccati equation:

G0 = h + f G + g G2 ,

(5)

where f, g, h are arbitrary constants and g 6= 0. Step 4. To decide the positive integer n, consider the homogeneous balance between the nonlinear terms and the highest order derivatives appearing in Eq. (3). Step 5. Substitute Eq. (4) along with Eq. (5) into the Eq. (3), then collect all the coefficients with the same order, the left hand side of Eq. (3) converts

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into polynomials in G k (θ) and G − k (θ) , (k = 0, 1, 2, ...). Then equating each coefficient of the polynomials to zero, yield a set of algebraic equations for fj (j = 0, 1, 2, ..., n) , f, g, h and B. Step 6. Solve the system of algebraic equations which are found in Step 5 with the aid of algebraic software Maple and we obtain values for fj (j = 0, 1, 2, ..., n) and B. Then, substitute obtained values in Eq. (4) along with Eq. (5) with the value of n, we obtain exact solutions of Eq. (1). In the following, we have twenty seven solutions including four different families of Eq. (5). Family 1: When f 2 − 4gh > 0 and f g 6= 0 or g h 6= 0 , the solutions of Eq. (5) are:

G3 =

−1 G1 = 2g

p f + f 2 − 4gh tanh

−1 G2 = 2g

p f + f 2 − 4gh coth

p

f 2 − 4gh θ 2

!! ,

!! p f 2 − 4gh θ , 2

 p  p  p −1  f + f 2 − 4gh tanh f 2 − 4gh θ ± i sec h f 2 − 4gh θ , 2g

 p  p   p −1  f + f 2 − 4gh coth f 2 − 4gh θ ± csc h f 2 − 4gh θ , 2g ! !!! p p p −1 f 2 − 4gh f 2 − 4gh 2 G5 = 2f + f − 4gh tanh θ + cot h θ , 4g 4 4 G4 =

 G6 =

1  −f + 2g

p  p (X 2 + Y 2 ) (f 2 − 4gh) − X f 2 − 4gh cosh f 2 − 4gh θ , p  X sinh f 2 − 4gh θ + Y

p

p  p p 2 − X 2 ) (f 2 − 4gh) + X 2 − 4gh sinh 2 − 4gh θ (Y f f 1  , p  G7 = −f − 2g X cosh f 2 − 4gh θ + Y 

where X and Y are two non-zero real constants and satisfies Y 2 − X 2 > 0.  √ f 2 −4gh θ 2 h cosh 2 √  √ , G8 = p f 2 −4gh f 2 −4gh 2 f − 4gh sinh θ − f cosh θ 2 2

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√ G9 =

f 2 −4gh 2



−2 h sinh θ  , √ p f 2 −4gh f 2 −4gh 2 − 4gh cosh θ − θ f 2 2

√ f sinh

G10

p  2 h cosh f 2 − 4gh θ p  p  =p , p f 2 − 4gh sinh f 2 − 4gh θ − f cosh f 2 − 4gh θ ± i f 2 − 4gh

G11

p  2 h sinh f 2 − 4gh θ p  p p  , = p −f sinh f 2 − 4gh θ + f 2 − 4gh cosh f 2 − 4gh θ ± f 2 − 4gh √

f 2 −4gh 4

G12 =

√

f 2 −4gh 4



4 h sinh θ cosh θ √  √  , p p f 2 −4gh f 2 −4gh f 2 −4gh 2 − 4gh cosh2 2 − 4gh θ cosh θ + 2 f θ − f 4 4 4

√ −2f sinh





Family 2: When f 2 − 4gh < 0 and f g 6= 0 or gh 6= 0 , the solutions of Eq. (5) are: G13

G14

G15 =

1 = 2g −1 = 2g

p −f + 4gh − f 2 tan

p f + 4gh − f 2 cot

!! p 4gh − f 2 θ , 2 !! p 4gh − f 2 θ , 2

 p   p p 1  4gh − f 2 θ ± sec 4gh − f 2 θ , − f + 4gh − f 2 tan 2g

 p   p p −1  f + 4gh − f 2 cot 4gh − f 2 θ ± csc 4gh − f 2 θ , 2g ! !!! p p 2 2 p 1 4gh − f 4gh − f = −2f + 4gh − f 2 tan θ − cot θ , 4g 4 4

G16 =

G17

G18

p  p p 2 − Y 2 ) (4gh − f 2 ) − X 2 cos 2θ ± (X 4gh − f 4gh − f 1  , p  −f + = 2g X sin 4gh − f 2 θ + Y

G19

p   p p ± (X 2 − Y 2 ) (4gh − f 2 ) + X 4gh − f 2 cos 4gh − f 2 θ 1  , p  = −f − 2g X sin 4gh − f 2 θ + Y



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where X and Y are two non-zero real constants and satisfies X 2 − Y 2 > 0. √



4gh−f 2 2

−2 h cos θ  √ , √ 4gh−f 2 4gh−f 2 θ + f cos θ 4gh − f 2 sin 2 2

G20 = p

√

4gh−f 2 2

G21 =

√ − f sin



2 h sin θ  √ , p 4gh−f 2 4gh−f 2 2 cos θ + 4gh − f θ 2 2

G22

p  − 2 h cos 4gh − f 2 θ p  p  p , =p 4gh − f 2 sin 4gh − f 2 θ + f cos 4gh − f 2 θ ± 4gh − f 2 θ

G23

p  2 h sin 4gh − f 2 θ p  p p  p = , − f sin 4gh − f 2 θ + 4gh − f 2 cos 4gh − f 2 θ ± 4gh − f 2 √

4gh−f 2 4

G24 =

√ −2f sin

4gh−f 2 4

 θ



√

4gh−f 2 4



4 h sin θ cos θ √  √  , p p 4gh−f 2 4gh−f 2 2 2 2 cos θ + 2 4gh − f cos θ − 4gh − f 4 4

Family 3: when h = 0 and f g 6= 0, the solution Eq. (5) becomes:

G25 =

− f b1 , g (b1 + cosh (f θ) − sinh (f θ))

G26 =

− f (cosh (f θ) + sinh (f θ)) , g (b1 + cosh (f θ) + sinh (f θ))

where b1 is an arbitrary constant. Family 4: when g 6= 0 and h = f = 0, the solution of Eq. (5) becomes:

G27 =

−1 , g θ + u1

where u1 is an arbitrary constant. 2.1. Exp-function Method Consider the general nonlinear partial differential equation of the type (1) Using the transformation (2) in equation (1) we have equation of the type (3).

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According to the Exp-function method, developed by He and Wu [12-18],we assume that the wave solutions can be expressed in the following form Pd an exp(nη) u(η) = Pqn=−c m=−p bm exp(mη)

(6)

where p, q, c and d are positive integers which are to be further determined,an and bm are unknown constants. We can rewrite equation (6) in the following equivalent form

u (η) =

ac exp (cη) + ... + a−d exp (−dη) . bp exp (pη) + ... + b−q exp (−qη)

(7)

To determine the value of c andp, we balance the linear term of highest order of equation (3) with the highest order nonlinear term. Similarly, to determine the value of dandq, we balance the linear term of lowest order of equation (3) with lowest order non linear term. 3. Solution procedure By using Exp-function method and the generalized Riccati equation mapping together with the(G0 /G)-expansion method, we construct new exact traveling wave solutions for the combined KdV-MKdV equation (Gardner equation) in this section. 3.1 The combined KdV-MKdV equation (Gardner equation) We consider the combined KdV-MKdV equation with parameters followed by Liu et al. [41]: ut + p u ux + q u2 ux − s uxxx = 0,

(8)

where p, s are free parameters and q 6= 0. Now, we use the transformation Eq. (2) into the Eq. (8), which yields:

− B r0 + prr0 + qr2 r0 − sr000 = 0,

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Eq. (9) is integrable, therefore, integrating with respect θ once yields: p q −B r + r2 + r3 − sr00 + K = 0, 2 3

(10)

where Kis an integral constant which is to be determined later. Taking the homogeneous balance between r00 and r3 in Eq. (10), we obtain n = 1. Therefore, the solution of Eq. (10) is of the form:

r (θ) = e1 (G0 /G) + e0 , e1 6= 0.

(11)

Using Eq. (5), Eq. (11) can be re-written as:


r (θ) = e1 f + h G−1 + g G + e0 ,

(12)

where f, g and hare free parameters. By substituting Eq. (12) into Eq. (10), collecting all coefficients of Gk and G − k (k = 0, 1, 2, ...) and setting them equal to zero, we obtain a set of algebraic equations for e0 , e1 , f, g, h, K and B (algebraic equations are not shown, for simplicity). Solving the system of algebraic equations with the help of algebraic software Maple, we obtain

e0 =

q ∓p 6s q −6sf q , ± 2q 6s q

K=

q

2sqf 2 −p2 +16sqgh , 4q  q   q   q  −48psqgh ± 6s +6psqf 2 ± 6s −p3 ± 6s +288s2 qf gh q q q  q  , 24q 2 ± 6s q

e1 = ±

6s q ,

B=

where p, s are free parameters and q 6= 0. Family 1: The soliton and soliton-like solutions of Eq. (6) (when f 2 − 4gh > 0 and f g 6= 0 or g h 6= 0) are:

r1 = e1

∆2 sec h2

∆ 2

2 f + ∆ tanh

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θ




∆ 2

θ



+ e0 ,

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p

2

f2

2

where ∆ = − 4gh, ∆ = f − 4gh, e0 =   2 2 θ = x − 2sqf −p4q+16sqgh t. − ∆2 csc h2

r2 = e1

2 f + ∆ coth

q ∓p 6s q −6sf q , ± 2q 6s q

∆ 2 θ

∆ 2 θ

e1 = ±

q

6s q

and




+ e0 ,


∆2 sec h2 (∆ θ) ∓ i tanh (∆θ) sec h (∆ θ) p r3 = e1 + e0 , f + f 2 − 4gh (tanh (∆ θ ) ± i sec h (∆ θ))
− ∆2 csc h2 (∆ θ) ± coth (∆θ) csc h (∆ θ ) r4 = e1 + e0 , f + ∆ (coth (∆θ) ± csc h (∆ θ))


∆ 2 sec h2 ∆4 θ − csc h2 ∆4 θ


+ e0 , r5 = e1 8 f + 4∆ tanh ∆4 θ + coth ∆4 θ   p −X f 2 X − sinh (∆ θ) f 2 Y − 4ghX + 4ghY sinh (∆ θ) − ∆2 (X 2 + Y 2 ) cosh (∆ θ)   +e0 , r6 = e1 p (X sinh (∆ θ) + Y ) f X sinh (∆ θ) + f Y − ∆ (X 2 + Y 2 ) + X∆ cosh (∆ θ)   p X f 2 Y cosh (∆ θ) f 2 Y − 4ghY cosh (∆ θ) − ∆2 − (X 2 − Y 2 ) sinh (∆ θ) + f 2 X − 4ghX   r7 = e1 +e0 , p (X cosh (∆ θ) + Y ) f X cosh (∆ θ) + f Y + ∆ − (X 2 − Y 2 ) + X∆ sinh (∆ θ) where X and Y are two non-zero real constants and satisfies Y 2 − X 2 > 0.

r8 = e1

r9 = e1

2 sinh

r10 = e1

r11 = e1

r12 = e1

4 sinh

∆ 4

∆ 2

2 cosh

∆ 2

θ

θ




−∆2
∆ sinh ∆2 θ − f cosh

∆2
−f sinh ∆2 θ + ∆ cosh




∆ 2θ

∆ 2



+ e0 ,

θ



+ e0 ,

−∆2 + i f 2 sinh (∆ θ) − i4gh sinh (∆ θ) + e0 , ∆ sinh (∆ θ) − f cosh (∆ θ) + i∆ cosh (∆ θ)

∆2 + f 2 cosh (∆ θ) − 4gh cosh (∆ θ) + e0 , (−f sinh (∆ θ) + ∆ cosh (∆ θ) + ∆ ) sinh (∆ θ)




θ cosh

∆ 4

θ




∆2
−2f sinh ∆4 θ cosh

∆ 4


θ + 2∆ cosh2

∆ 4



+e0 , θ −∆

Family 2: The periodic form solutions of Eq. (8) (when f 2 − 4gh < 0 and

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f g 6= 0 or gh 6= 0) are:

r13 = e1

2 cos

p

Ω 2θ




Ω2
−f cos Ω2 θ + Ω sin

2

− f2

2

where Ω = + 4 gh, Ω = 4 gh − f ,e0 =   2 2 θ = x − 2sqf −p4q+16sqgh t.

r14 = e1

r15 = e1

r16 = e1

r17 = e1

r18 = e1

4 cos2

2 −1 + cos2

Ω2



Ω 2θ

Ω 2θ



+ e0 ,

q ∓p 6s q −6sf q , ± 2q 6s q

f + Ω cot

Ω 2θ

e1 = ±

q

6s q

and



+ e0 ,

Ω2 (1 + sin (Ωθ)) + e0 , cos (Ωθ) (−f cos (Ωθ) + Ω sin (Ωθ) + Ω)

Ω2 sin (Ωθ) + e0 , cos (Ωθ) f sin (Ωθ) + Ω cos2 (Ωθ) − f sin (Ωθ) − Ω

Ω 4θ




−1 + cos2

Ω 4θ





−Ω2 −2f + Ω tan

Ω 4θ




− cot

Ω 4θ




+ e0 ,

X.N1

  +e0 , √ Ω (X 2 −Y 2 ) (−X 2 + X 2 cos2 (Ωθ) − 2XY sin (Ωθ) − Y 2 ) −f + X sin(Ωθ)+Y − XΩ cos (Ωθ) (13)

where N1 = −4ghX − 4ghY sin (Ωθ) + f 2 X + f 2 Y sin (Ωθ) + p p 4gh (X 2 − Y 2 ) cos (Ωθ) − f 2 (X 2 − Y 2 ) cos (Ωθ) r19 =

  √ √ X −4ghX−4ghY sin(Ωθ)+f 2 X+f 2 Y sin(Ωθ)−4gh (X 2 −Y 2 ) cos(Ωθ)+f 2 (X 2 −Y 2 ) cos(Ωθ) ! √ 2 2 e1 + Ω (X −Y ) (−X 2 +X 2 cos2 (Ωθ)−2XY sin(Ωθ)−Y 2 ) −f − X sin(Ωθ)+Y +XΩ cos(Ωθ)

e0 , where X and Y are two non-zero real constants and satisfies X 2 − Y 2 > 0.

r20

r21

r22





−Ω2 sec Ω2 θ Ω sin Ω2 θ + f cos Ω2 θ

= e1 2 4gh − 4gh cos2 Ω2 θ − f 2 + 2f 2 cos2 Ω2 θ + 2f Ω sin

Ω 2θ




cos

Ω 2θ



+e0 ,




−Ω2 −f sin Ω2 θ + Ω cos Ω2 θ





+e0 , = e1 2 sin Ω2 θ −f 2 + 2f 2 cos2 Ω2 θ + 2f Ω sin Ω2 θ cos Ω2 θ − 4gh cos2 Ω2 θ
1 2 2 2 e1 sec (Ωθ) −Ω − 4gh sin (Ωθ) + f sin (Ωθ) (Ω sin (Ωθ) + f cos (Ωθ) + Ω) +e0 , = N2

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where N2 = 4gh − 2gh cos2 (Ωθ) − f 2 + f 2 cos2 (Ωθ) + Ωf sin (Ωθ) cos (Ωθ) + 4gh sin (Ωθ) − f 2 sin (Ωθ) + f Ω cos (Ωθ)

r23 = e1

q24 =

−Ω2 (−f sin (Ωθ) + Ω cos (Ωθ) + Ω) + e0 , 2 sin (Ωθ) (−2gh cos (Ωθ) + f 2 cos (Ωθ) + f Ω sin (Ωθ) − 2gh)

−Ω2 4 e1

csc

Ω 4θ




sec

Ω 4θ





−2f sin Ω4 θ cos N3

Ω 4θ




+ 2Ω cos2

Ω 4θ





−Ω

+e0 ,

where        Ω Ω Ω Ω 2 4 3 θ + 8f cos θ + 8Ωf cos θ sin θ − N3 = −8f cos 4 4 4 4         Ω Ω Ω Ω 4f Ω sin θ cos θ − 16gh cos4 θ + 16gh cos2 θ − Ω2 4 4 4 4 2

2



Family 3: The soliton and soliton-like solutions of Eq. (6) (when h = 0 and f g 6= 0) are:

r25 = e1

f (cosh (f θ) − sinh (f θ)) + e0 , b1 + cosh (f θ) − sinh (f θ)

r26 = e1

f b1 + e0 , b1 + cosh (f θ) + sinh (f θ)

where b1 is an arbitrary constant, e0 =   2 2 x − 2sqf −p4q+16sqgh t.

q ∓p 6s q −6sf q , ± 2q 6s q

e1 = ±

q

6s q

and θ =

Family 4: The rational function solution (when g 6= 0 and h = f = 0) is:

r27 =

−e1 g , g θ + u1

where u1 is an arbitrary constant , e1 = ±

q

6s q

and θ = x−



2sqf 2 −p2 +16sqgh 4q



t.

3.2 The combined KdV-MKdV equation (Gardner equation) using Exp-function method

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We consider the combined KdV-MKdV equation (8) with parameters followed by Liu et al. [41]: Now, we use the transformation Eq. (2) into the Eq. (8), which yields (9). Using Exp-function Method we have following solution sets satisfy the given combined KdV-MKdV equation (8) 1st Solution set:     B = B, b−1 =    a0 =

2 1 (p +4Bq+−2sq ) , b0 = 2 4 b1 (p  +4Bq+4sq)   √ 1 p+ p2 +4Bq+4sq b0 4s−2B− 2 p q

√

p2 +4Bq+4sq

− 81

  √ b20 p+ p2 +4Bq+4sq (4Bq+p2 −2sq )

b0 , b1 = b1 , a−1 = qb1 (p2 +4Bq+4sq)  √  p+ p2 +4Bq+4sq , a1 = − 21 b1 q

  ,     

We therefore, obtained the following generalized solitary solution   √   √  1 p+ p2 +4Bq+4sq b0 4s−2B− 2 p b20 p+ p2 +4Bq+4sq (4Bq+p2 −2sq ) q √ 2 e−η + − 81 qb1 (p2 +4Bq+4sq) p +4Bq+4sq  √  p+ p2 +4Bq+4sq 1 η −2 b1 e q U (η) = 1 (p2 +4Bq+−2sq) 4

− 81 − 12

b1 (p2 +4Bq+4sq )

e−η +b0 +b1 eη

  √ b20 p+ p2 +4Bq+4sq (4Bq+p2 −2sq )



√ p+

qb1 (p2 +4Bq+4sq)

p2 +4Bq+4sq q

U (x, t) =



e−(x−Bt) +

 √   1 p+ p2 +4Bq+4sq b0 4s−2B− 2 p q

√

p2 +4Bq+4sq

b1 e(x−Bt)

1 (p2 +4Bq+−2sq) −(x−Bt) +b0 +b1 e(x−Bt) 4 b1 (p2 +4Bq+4sq) e

2nd Solution set:     B = B, b−1 =    a0 = −

2 2 1 (p +4Bq+−2sq )b0 2 +4Bq+4sq) , b0 = 4 b (p 1   √  1 p+ p2 +4Bq+4sq b0 4s−2B− 2 p q

√

p2 +4Bq+4sq

  √ b20 −p+ p2 +4Bq+4sq (4Bq+p2 −2sq ) b0 , b1 = b1 , a−1 = − 81 , qb1 (p2 +4Bq+4sq)   √ −p+ p2 +4Bq+4sq , a1 = 21 b1 q

We therefore, obtained the following generalized solitary  solution √    √ 1 p+ p2 +4Bq+4sq b0 4s−2B− 2 p b20 −p+ p2 +4Bq+4sq (4Bq+p2 −2sq ) q √ 2 − 18 e−η − qb1 (p2 +4Bq+4sq) p +4Bq+4sq   √ 2 −p+ p +4Bq+4sq 1 η −2 b1 e q U (η) = 2 1 (p2 +4Bq+−2sq)b0 −η + b0 + b1 eη 4 b1 (p2 +4Bq+4sq) e

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− 18 − 21

  √ b20 −p+ p2 +4Bq+4sq (4Bq+p2 −2sq )



√

−p+

qb1 (p2 +4Bq+4sq) p2 +4Bq+4sq q

U (x, t) =



e−(x−Bt) −

   √ 1 p+ p2 +4Bq+4sq b0 4s−2B− 2 p q

√

p2 +4Bq+4sq

b1 e(x−Bt)

2 1 (p2 +4Bq+−2sq)b0 −(x−Bt) 4 b1 (p2 +4Bq+4sq) e

+ b0 + b1 e(x−Bt)

4. Results and discussion It is significance mentioning that our solution q27 is coincided with u3,4 (x, t) in example 1 of section 4 of Liu et al. [41] for s = 1, q = 1, p = 2 and u1 = 0. Moreover, it is showing that our solution q27 is coincided with u3,4 (x, t) in example 2 of section 4 of Liu et al. [41] for s = −1, q = 1, p = 2 and u1 = 0. In addition, we construct many new exact traveling wave solutions for the combined KdV-MKdV equation in this work, which have not been found in the previous literature. Furthermore, the graphical demonstrations of some of them are depicted in the following subsection in figures below. 4.1 Graphical representations of the solutions The graphical depictions of the solutions are shown in the figures with the help of Maple:

Fig. 1: Periodic solutions for f = 5, g = 4, h = 3, p = 3, s = 2, q = 5

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Fig. 2: Periodic solutions for f = 9, g = 8, h = 0, p = 8, s = 6, q = 7, b1 = 8

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Fig. 3: Periodic solutions for f = 5, g = 4, h = 3, p = 2, s = 5, q = 9

Fig. 4: Periodic solutions for f = 3, g = 4, h = 0, p = 1, s = 3, q = 4, b1 = 4

Fig. 5: Periodic solutions for f = 3, g = 4, h = 0, p = 2, s = 5, q = 7, b1 = 8

Fig. 6: Periodic solutions for f = 5, g = 7, h = 0, p = 5, s = 5, q = 7, b1 = 2

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Fig. 7: Periodic solutions for f = 2, g = 4, h = 0, p = 1, s = 3, q = 1, b1 = 2

Fig. 8: Periodic solutions for f = 5, g = 4, h = 0, p = 3, s = 4, q = 3, b1 = 2

Fig. 9: Periodic solutions for f = 7, g = 15, h = 0, p = 3, s = 4, q = 5, b1 = 4

Fig. 10: Periodic solutions for f = 0, g = 11, h = 0, p = 3, s = 5, q = 9, b1 = 2

5. Conclusions In this article, we apply the Exp-function method and generalized Riccati equa-

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tion mapping together with the(G0 /G)-expansion method to solve the combined KdV-MKdV equation. In (G0 /G)-expansion method, the generalized Riccati equation G0 (θ) = h + f G (ϕ) + g G2 (θ) is used with constant coefficients, as the auxiliary equation, instead of the second order linear ordinary differential equation with constant coefficients. By applying these methods, we obtain abundant exact traveling wave solutions including solitons and periodic solutions and solutions are expressed in terms of the hyperbolic, the trigonometric and the rational functions. The correctness of the obtained solutions is verified to compare with the published results. We hope that these useful and powerful methods can be effectively used to solve many nonlinear evolution equations which are arising in technical arena. References [1] C. Rogers and W. F. Shadwick, “Backlund transformations,” Aca. Press, New York, 1982. [2] M. J. Ablowitz and P. A. Clarkson, “Solitons, nonlinear evolution equations and inverse scattering transform,” Cambridge Univ. Press, Cambridge, 1991. [3] J. Zhang, D. Zhang and D. Chen, “Exact solutions to a mixed Toda lattice hierarchy through the inverse scattering transform,” Journal of Physics A: Mathematical and Theoritical, doi: 10.1088/1751-8113/44/11/115201, 2011. [4] S. Liu, Z. Fu, S. Liu and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, pp. 69–74, 2001. [5] Z. I. A. Al-Muhiameed and E. A. B. Abdel-Salam, “ Generalized Jacobi elliptic function solution to a class of nonlinear Schrodinger –type equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 575679, 11 pages, doi:10.1155/2010/575679, 2011. [6] W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” Am. J. Phys. vol. 60, pp. 650–654, 1992. [7] A. M. Wazwaz, “The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol.

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STABILITY OF A LATTICE PRESERVING FUNCTIONAL EQUATION ON RIESZ SPACE: FIXED POINT ALTERNATIVE EHSAN MOVAHEDNIA, SEYED MOHAMMAD SADEGH MODARRES MOSADEGH, CHOONKIL PARK, AND DONG YUN SHIN∗

Abstract. The aim of this paper is to investigate Hyers-Ulam stability of the following lattice preserving functional equation on Riesz space with fixed point method: kF (τ x ∨ ηy) − τ F (x) ∨ ηF (y)k ≤ ϕ(τ x ∨ ηy, τ x ∧ ηy), where X is a Banach lattice and ϕ : X × X → X is a mapping such that   α x y , ϕ(x, y) ≤ (τ η) 2 ϕ τ η for all τ, η ≥ 1 and α ∈ [0, 12 ).

1. Introduction In 1940 Ulam [1] proposed the famous Ulam stability problem: When is it true that a function which satisfies some functional equation approximately must be close to one satisfying the equation exactly?. If the answer is affirmative, we would say that the equation is stable. In 1941, Hyers [2] solved this stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. The result of Hyers was generalized by Rassias [3] for linear mapping by considering an unbounded Cauchy difference. In 1996, Isac and Rassias [4] were the first authors to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. Some authors have considered the Hyers-Ulam stability of quadratic functional equations in random normed spaces [5, 6, 7, 8, 9, 10, 11, 12, 13]. By the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [14, 15]). We generalize the Agbeko’s theorem [16] and prove it by fixed point method. A non-empty set M with a relation ”≤” is said to be an order set whenever the following conditions are satisfied: 1. x ≤ x for every x ∈ M; 2. x ≤ y and y ≤ x implies that x = y; 3. x ≤ y and y ≤ z implies that x ≤ z. If, in addition, for two elements x, y ∈ M either x ≤ y or y ≤ x, then M is called a totally ordered set. Let A be a subset of an ordered set M. x ∈ M is called an upper bound of A if y ≤ x for all y ∈ A. z ∈ M is called a lower bound of A if y ≥ z for all y ∈ A. Moreover, if there is an upper bound of A, then A is said to be bounded from above. If there is an lower bound of A, then A is said to be bounded from below. If A is bounded from above and from below, then we will briefly say that A is order bounded. An order set (M, ≤) is called a lattice if any two elements x, y ∈ M have a least upper bound denoted by x ∨ y = sup(x, y) and a greatest lower bound denoted by x ∧ y = inf(x, y). A real vector space E which is also an order set is called an order vector space if the order and the vector space structure are compatible in the following sense: 1. if x, y ∈ E such that x ≤ y then x + z ≤ y + z for all z ∈ E; 2010 Mathematics Subject Classification. 47H10, 46B42, 39B82. Key words and phrases. Hyers-Ulam stability; Riesz space; fixed point theory. ∗ The corresponding author.

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2. if x, y ∈ E such that x ≤ y then αx ≤ αy for all α ≥ 0. (E, ≤) is called a Riesz space if (E, ≤) is a lattice and order vector space. A norm ρ on Riesz space E, is called a lattice norm if ρ(x) ≤ ρ(y) whenever |x| ≤ |y|. In the latter case (E, k.k) is called a normed Riesz space. (E, k.k) is called a Banach lattice if (E, k.k) is a Banach space, E is Riesz space and |x| ≤ |y| implies that kxk ≤ kyk for all x, y ∈ E. Example 1.1. Suppose that X is a compact Hausdorff space. We denote by C(K) the Banach space of all real-valued continuous functions on X . Let “≤” be a point-wise order on C(K), and f ≤ g if and only if f (t) ≤ g(t) for all t ∈ K. It is easy to show that (C(K), ≤) is a Banach lattice. Let E be a Riesz space, and let the positive cone E + of E consist of all ∈ E such that x ≥ 0. For every x ∈ E let x+ = x ∨ 0 x− = −x ∨ 0 |x| = x ∨ −x. Let E be a Riesz space. For all x, y, z ∈ E and a ∈ R, the following assertions hold: 1. x + y = x ∨ y + x ∧ y , −(x ∨ y) = −x ∧ y. 2. x + (y ∨ z) = (x + y) ∨ (x + z) , x + (y ∧ z) = (x + y) ∧ (x + z). 3. |x| = x+ + x− , |x + y| ≤ |x| + |y|. 4. x ≤ y is equivalent to x+ ≤ y + and y − ≤ x− 5. (x ∨ y) ∧ z = (x ∧ y) ∨ (y ∧ z) , (x ∧ y) ∨ z = (x ∨ y) ∧ (y ∨ z) A Riesz space E is called Archimedean if x ≤ 0 holds whenever the set {nx : n ∈ N } is bounded from above. Theorem 1.1. Let E be a normed Riesz space. The following assertions hold: 1. the lattice operations is continuous; 2. the positive cone E + is closed; 3. limn→∞ xn = sup{xn : n ∈ N }. Definition 1.1. Let X , Y be Banach lattices. A mapping T : X → Y is called positive if T (X + ) = {T (|x|) : x ∈ X } ⊂ Y + . Definition 1.2. Let X , Y be Banach lattices and let T : X → Y be a positive mapping. We define P1 ) lattice homomorphism: T (|x| ∨ |y|) = T (|x|) ∨ T (|y|); P2 ) semi-homogeneity: for all x ∈ X and all α ∈ R+ T (α|x|) = αT (|x|); P3 ) continuity from below on the positive cone: for all increasing sequences xn ⊂ X + lim T (xn ) = T ( lim xn ).

n→∞

n→∞

Observe that every lattice homomorphism T : X → Y is necessarily a positive operator. Indeed, if x ∈ E + then T (x) = T (x ∨ 0) = T (x) ∨ T (0) = T (x)+ ≥ 0 holds in Y. Also it is important to note that the range of a lattice homomorphism is a Riesz subspace. Theorem 1.2. For an operator T : X → Y between two Riesz spaces, the following statements are equivalent: 1. T is a lattice homomorphism; 2. T (x+ ) = T (x)+ for all x ∈ X ; 3. T (x ∧ y) = T (x) ∧ T (y); 4. if x ∧ y = 0 in X , then T (x) ∧ T (y) = 0 holds in Y; 5. T (|x|) = |T (x)|.

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Definition 1.3. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies the following conditions: (a) d(x, y) = 0 if and only if x = y for all x, y ∈ X ; (b) d(x, y) = d(y, x) for all x, y ∈ X ; (c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X . Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. Theorem 1.3. Let (X ,d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x ∈ X , either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (a) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (b) the sequence {J n x} converges to a fixed point y ∗ of J ; (c) y ∗ is the unique fixed point of J in the set Y = {y ∈ X : d(J n0 x, y) < ∞}; 1 d(y, J y) for all y ∈ Y. (d) d(y, y ∗ ) ≤ 1−L 2. Main results Using the fixed point method, we prove the Hyers-Ulam stability of lattice homomorphisms in Banach lattices. Theorem 2.1. Let X , Y be Banach lattices. Consider a positive operator F : X → Y such that kF (τ x ∨ ηy) − τ F (x) ∨ ηF (y)k ≤ ϕ(τ x ∨ ηy, τ x ∧ ηy),

(2.1)

where ϕ : X × X → X is a mapping such that α

ϕ(x, y) ≤ (τ η) 2 ϕ



x y , τ η



for all x, y ∈ X , τ, η ≥ 1 and for which there is a real number α ∈ [0, 21 ) Then there is a unique positive operator T : X → Y satisfying the properties P1 , P2 and the inequality τα kT (x) − F (x)k ≤ τ − τα for all x ∈ X . Proof. Putting τ = η and x = y in (2.1), we get kF (τ x) − τ F (x)k ≤ ϕ(τ x, τ y). Then



1

F (τ x) − F (x) ≤

τ

1 τ ϕ(τ x, τ x)

≤ τ α−1 ϕ(x, x).

(2.2)

Consider the set ∆ = {g | g : X → Y g(0) = 0} and introduce the generalized metric on ∆  d(g, h) = inf c ∈ R+ , kg(x) − h(x)k ≤ c ϕ(x, x) for all x ∈ X , where as usual, inf ∅ = ∞. It is easy to show that (∆, d) is complete generalized metric space. Now we define the operator J : ∆ → ∆ by 1 Jg(x) = g(τ x) τ for all x ∈ X . Given g, h ∈ ∆, let c ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ c, that is, kg(x) − h(x)k ≤ c ϕ(x, x).

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So we have kJg(x) − Jh(x)k = ≤

1 1 kg(τ x) − h(τ x)k ≤ c ϕ(τ x, τ x) τ τ 1 α α−1 c τ ϕ(x, x) = τ c ϕ(x, x) τ

for all x ∈ X , that is, d(Jg, Jh) < τ α−1 c. Thus we have d(Jg, Jh) ≤ τ α−1 d(g, h) for all g, h ∈ ∆. So J is a strictly contractive mapping with constant τ α−1 < 1 on ∆, For all g, h ∈ ∆ and α ∈ [0, 12 ). By (2.2), we have d(JF, F ) ≤ τ α−1 < ∞. By Theorem 1.3, there exists a mapping T : X → Y satisfying the following: 1. T is a fixed point of J, i.e., T (τ x) = τ T (x) for all x ∈ X . Also the mapping T is a unique fixed point of J in the set M = {g ∈ ∆ : d(g, h < ∞)}. This implies that P2 holds. 2. d(J n F, T ) → 0 as n → ∞. This implies the equality lim

n→∞

1 F (2n x) = T (x) τn

for all x ∈ X . 1 3. d(F, T ) ≤ 1−L d(F, JF ), which implies the inequality kT (x) − F (x)k ≤

τ α−1 τα = , 1 − τ α−1 τ − τα

which implies that the inequality (2.1) holds. Now we show that T satisfies P1 . Putting τ = η = τ n in (2.1), we get kF (τ n (x ∨ y)) − τ n F (x) ∨ τ n F (y)k ≤ τ 2nα ϕ(x ∨ y, x ∧ y). n

(2.3)

n

Replacing x, y by τ x and τ y in (2.3), respectively, we get

F (τ 2n (x ∨ y)) − τ n F (τ n x) ∨ τ n F (τ n y) ≤ τ 2nα ϕ(τ n x ∨ τ n y, τ n x ∧ τ n y) = τ 4nα (ϕ(x ∨ y, x ∧ y). Then

 

1

1 1 2n(2α−1) 2n n n

.ϕ(x ∨ y, x ∧ y) .

τ 2n F (τ (x ∨ y)) − τ n F (τ x) ∨ τ n F (τ y) ≤ τ Since α ∈ [0, 21 ), when n → ∞, we have kT (x ∨ y) − T (x) ∨ T (y)k ≤ 0. and so T (x ∨ y) = T (x) ∨ T (y) for all x, y ∈ X . Note that the lattice operation is continuous.



Theorem 2.2. Let X , Y be Banach lattices and let a continuous function p : [0, ∞) → [0, ∞) be given. Consider a positive T : X → Y for which there are real numbers ν ∈ (0, ∞) and 0 ≤ r < 1 such that



T (α|x| ∨ β|y|) − αp(α)T (|x|) ∨ βp(β)T (|y|) ≤ ν (kxkr + kykr ) (2.4)

p(α) ∨ p(β)

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for all x, y ∈ X and α, β ∈ R+ . Then there exist a unique positive mapping F : X → Y which satisfies the properties P1 , P2 and the inequality 2ν kF (|x|) − T (|x|)k ≤ 2 − 2r for all x ∈ X . Proof. Putting α = β = 2 and x = y in (2.4), we get



T (2|x| ∨ 2|x|) − 2p(2)T (|x|) ∨ 2p(2)T (|x|) ≤ 2νkxkr

p(2) ∨ p(2) for all x ∈ X and r ∈ [0, 1). Thus kT (2|x|) − 2T (|x|)k ≤ 2νkxkr and so



1

T (2|x| − T (|x|)) ≤ νkxkr

2

for all x ∈ X and α ∈ [0, 1). Consider the set

(2.5)

∆ := {S : S : X → Y , S(0) = 0} and introduce the generalized metric on ∆ d(S, H) = inf{c ∈ R+ , kS(x) − H(x)k ≤ ckxkr , ∀x ∈ X }, where, as usual, inf ∅ = ∞. It is know that (∆, d) is complete. Now we define the mapping J : ∆ → ∆ by 1 JS(|x|) = S(2|x|) 2 for all x ∈ X . First we assert that J is strictly contractive with constant 2r−1 on ∆. Given S, H ∈ ∆, let c ∈ [0, ∞] be an arbitrary constant with d(S, H) < c, that is, kS(|x|) − H(|x|)k ≤ ckxkr . So we have 1 1 kS(2|x|) − H(2|x|)k ≤ ck2xkr = 2r−1 ckxkr 2 2 r−1 for all x ∈ X , that is, d(JS, JH) ≤ 2 c. Thus we have kJS(x) − JH(x)k =

d(JS, JH) ≤ 2r−1 d(S, H) for all S, H ∈ ∆ and so J is strictly contractive with constant 2r−1 < 1 on ∆. For all S, H ∈ ∆ and r ∈ [0, 1]. By (2.5) we have d(JF, F ) ≤ ν < ∞ By Theorem 1.3, there exists a mapping F : X → Y satisfying the following: 1. F is a fixed point J i. e. F (2|x|) = 2F (|x|) for all x ∈ X . Also the mapping F is a unique fixed point of J in the set M = {S ∈ ∆ : d(S, H) < ∞}. n

2. d(J T, F ) → 0 as n → ∞. This implies the equality 1 lim n T (2n x) = F (x) n→∞ 2 for all x ∈ X . 1 3. d(T, F ) ≤ d(T, JT ), which implies the inequality 1−L ν 2ν kF (|x|) − T (|x|)k ≤ = . 1 − 2r−1 2 − 2r

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(2.6)

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This implies that the inequality (2.2) holds. Now, we show that F is a lattice homomorphism. Putting α = β = 2n in (2.4), kT (2n (|x| ∨ |y|)) − 2n (T (|x|) ∨ T (|y|))k ≤ ν(kxkr + kykr ). x

(2.7)

n

Replacing x and y by 2 and 2 y in (2.7), respectively, we obtain kT (4n (|x| ∨ |y|)) − 2n (T (2n |x|) ∨ T (2n |y|))k ≤ 2nr ν(kxkr + kykr ) and so



1

T (4n (|x| ∨ |y|)) − 1 (T (2n |x|) ∨ T (2n |y|)) ≤ 2n(r−2) ν(kxkr + kykr ).

4n

n 2 As n → ∞, we have kF (|x| ∨ |y|) − F (|x|) ∨ F (|y|)k ≤ 0. and so F (|x| ∨ |y|) = F (|x|) ∨ F (|y|) for all x, y ∈ X . Next we show that T (α|x|) = αT (|x|) for all x ∈ X and all real numbers α ∈ [0, ∞). Letting α = β, y = 0 and replacing α by 2n α in (2.4), we get F (0) = 0 and so F satisfies P1 . So T (0) = 0 with (2.6) and kT (2n α|x|) − 2n αT (|x|)k ≤ νkxkr (2.8) for all x ∈ X and all real numbers α ∈ [0, ∞). Replacing x by 2n x in (2.8), kT (4n α|x|) − 2n αT (2n |x|)k ≤ ν 2nr kxkr and so



T (4n α|x|) T (2n (|x|))

≤ ν 2n(r−2) kxkr −α

4n 2n

for all x ∈ X . As n → ∞, we obtain kF (α|x| − αF (|x|))k ≤ 0 and so F (α|x| = αF (|x|). for all x ∈ X and α ∈ [0, ∞).



Corollary 2.1. Let X , Y be Banach lattices. Consider a positive operator T : X → Y for which there are real numbers ν ∈ (0, ∞) and 0 ≤ r < 1 such that kT (α|x| ∨ β|y|) − αT (|x|) ∨ βT (|y|)k ≤ ν(kxkr + kykr ) for all x, y ∈ X and α, β ∈ R+ . Then there exists a unique positive mapping F : X → Y which satisfies the properties P1 , P2 and the inequality 2ν kF (|x|) − T (|x|)k ≤ . 2 − 2r for all x ∈ X . Corollary 2.2. Let X , Y be Banach lattices. Consider a positive operator T : X → Y for which there are real numbers ν ∈ (0, ∞) and 0 ≤ r < 1 such that

2 2

T (α|x| ∨ β|y|) − α T (|x|) ∨ β T (|y|) ≤ ν(kxkr + kykr )

α∨β for all x, y ∈ X and α, β ∈ R+ . Then there exists a unique positive mapping F : X → Y which satisfies the properties P1 , P2 and the inequality 2ν kF (|x|) − T (|x|)k ≤ 2 − 2r for all x ∈ X .

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LATTICE PRESERVING FUNCTIONAL EQUATION ON RIESZ SPACE

Acknowledgments C. Park and D. Y. Shin were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF2012R1A1A2004299) and (NRF-2010-0021792), respectively. References [1] S. M. Ulam, Problems in Modern Mathematics, Science Editions, John Wiley and Sons, 1964. [2] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224. [3] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 123-130. [4] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268-273. [5] L. Cadariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43-52. [6] L. Cadariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [7] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097-1105. [8] L. C˘ adariu, L. G˘ avruta, P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60-67. [9] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50-59. [10] S. Shagholi, M. Eshaghi Gordji, M. Bavand Savadkouhi, Nearly ternaty cubic homomorphisms in ternary Fr´ echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106-1114. [11] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51-59. [12] D. Shin, C. Park, Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964-973. [13] D. Shin, C. Park, Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125-134. [14] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86. [15] J. M. Rassias, On the stability of a multi-dimensional problem of Ulam, in Geometry, Analysis and Mechanics, World Sci. Publ., River Edge, NJ, pp. 365-376, 1994. [16] N. K. Agbeko, Stability of maximum preserving functional equation on Banach lattice. Miskols Mathematical Notes 13 (2012), 187-196. Ehsan Movahednia Department of mathematics, University of Yazd, P. O. Box 89195-741, Yazd, Iran E-mail address: [email protected] Seyed Mohammad Sadegh Modarres Mosadegh Department of mathematics, University of Yazd, P. O. Box 89195-741, Yazd, Iran E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail address: [email protected]

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UNIQUENESS THEOREM OF MEROMORPHIC FUNCTIONS AND THEIR k-TH DERIVATIVES SHARING SET ¨ JUNFENG XU AND FENG LU Abstract. In this paper, due to the theories of normal family and complex differential equation, we consider a uniqueness problem of meromorphic functions share set S = {a, b} with their k-th derivatives.

1. Introduction and main results Let F be a family of meromorphic functions defined in D. F is said to be normal in D, in the sense of Montel, if for any sequence fn ∈ F , there exists a subsequence fnj , such that fnj converges spherically locally uniformly in D, to a meromorphic function or ∞ (see, [18]). Let f and g be two meromorphic functions in a domain D, and let a be a complex number. If g(z) = a whenever f (z) = a, we write f (z) = a ⇒ g(z) = a. If f (z) = a ⇒ g(z) = a and g(z) = a ⇒ f (z) = a, we write f (z) = a ⇔ g(z) = a and say that f and g share the value a IM (ignoring multiplicity). If f − a and g − a have the same zeros with the same multiplicities, we write f (z) = a g(z) = a and say that f and g share the value a CM (counting multiplicity). Let S be a set of complex numbers. Provide that f (z) ∈ S if and only if g(z) ∈ S in a domain D, then we say f and g share the set S in D. It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna theory, as found in [4, 21]. In the theory of normal family, it is meaningful to find sufficient conditions for normality(see. [1, 7, 8, 9, 10, 11, 15, 17, 20]). Recently, Y. Li [7] obtained a normal family of holomorphic functions share set with their k-th derivatives as follows. 2000 Mathematics Subject Classification. 30D35, 30D45. Key words and phrases. Share set, Nevanlinna theory, uniqueness, normal family, differential equation. This work was supported by NSF of Guangdong Province(No. 2015A030313644), the Natural Science Foundation of Shandong Province Youth Fund Project (ZR2012AQ021) and the training plan for the Outstanding Young Teachers in Higher Education of Guangdong (Nos. Yq2013159, SYq2014002). 1

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Theorem A. Let F be a family of holomorphic functions in a domain D, let k(≥ 2) be a positive integer, and let a, b be two distinct finite complex numbers. If for each f ∈ F, all the zeros of f are of multiplicity at least k, and f and f ′ share the set S = {a, b}, then F is normal in D. Remark 1. In fact, for the case ab ̸= 0, the conclusion of Theorem A still holds if the condition f and f (k) share the set S = {a, b } CM is replaced by f (z) ∈ S ⇒ f (k) (z) ∈ S. See Section 3. In the uniqueness theory, an important subtopic that a meromorphic function and it’s derivative share some values or functions or set is well investigated. Due to Theorem A, Y. Li [7] obtained a uniqueness theorem of entire functions. Theorem B. Let k(≥ 2) be a positive integer, and let a, b be two distinct finite complex numbers, and let f be a non-constant entire function. If all the zeros of f are of multiplicity at least k, and f and f (k) share the set S = {a, b } CM, then (1). f (z) = CeDz , where C ̸= 0 and D are two constants with Dk = ±1, (2). f = −f (k) + a + b. In [7], Y. Li also gave an example to show that the case (2) can not omitted. Example 1. Let f (z) = cos2 z2 . Then f and f ′′ share set {0, 12 } CM and all zeros of f are of multiplicity at least 2. Obviously, f = −f ′′ + 12 . After considering Theorem B and Example 1, we naturally ask the following questions. Question 1. What happens if f is a meromorphic function? Question 2. Note that k = 2 in Example 1. Naturally, we ask whether Case (2) occurs for k ̸= 2 or not? Question 3. What’s the specific form of f in Case (2)? In the work, we focus on the above questions. Basing on the idea of Y. Li in [7] and due to the theories of normal family and complex differential equation, we further study the uniqueness problem of meromorphic functions of finitely many poles sharing a set CM with their derivatives. Theorem 1.1. Let k(≥ 2) be a positive integer, and let a, b be two distinct finite complex numbers, and let f be a non-constant meromorphic function with finitely many poles. If all the zeros of f are of multiplicity at least k, and f and f (k) share the set S = {a, b} CM, then one of the following cases must occur:

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(1). f (z) = CeDz , where C ̸= 0 and D are two constants with Dk = 1, and f = f (k) ; (2). f (z) = CeDz , where C ̸= 0 and D are two constants with Dk = −1, f = −f (k) and S = {a, −a}; (3). f (z) = A1 eiz + A2 e−iz + a + b, where A1 and A2 are two nonzero ′′ constants with (a + b)2 = 4A1 A2 , f = −f + a + b, and k must be 2. Remark 2. For the special case that A1 = A2 = 41 , a = 0 and b = 12 , then Case (3) becomes Example 1. Remark 3. We answer the Questions 2 and find out the case (2) occurs only for k = 2 in Theorem B. We also answer the Question 3 and give the form of f . We partial answer the Question 1. In 2008, we considered the case of k = 1 and obtained a normal criteria theorem and a uniqueness theorem[10]. Theorem C. Let F be a family of functions holomorphic in a domain, let a and b be two distinct finite complex numbers with a + b ̸= 0. If for all f ∈ F , f and f ′ share S = {a, b} CM, then F is normal in D. Theorem D. Let a and b be two distinct complex numbers with a + b ̸= 0, and let f (z) be a nonconstant entire function. If f and f ′ share the set {a, b} CM, then one and only one of the following conclusions holds: (i) f = Aez or (ii) f = Ae−z + a + b, where A is a nonzero constant. By the same way to Theorem 1.1, we can obtain the following. Theorem 1.2. Let a and b be two distinct complex numbers with a + b ̸= 0, and let f (z) be a nonconstant meromorphic function with finite poles. If f and f ′ share the set {a, b} CM, then one and only one of the following conclusions holds: (i) f = Aez or (ii) f = Ae−z + a + b, where A is a nonzero constant. 2. Some Lemmas Lemma 2.1. [15] Let F be a family of functions holomorphic on a domain D, all of whose zeros have multiplicity at least k , and suppose that there exists A ≥ 1 such that |f (k) (z)| ≤ A whenever f (z) = 0. Then if F is not normal at z0 ∈ D, for each 0 ≤ α ≤ k, there exist, (a) a number 0 < r < 1; (b) points zn → z0 ; (c) functions fn ∈ ζ, and (d) positive number ρn → ∞ such that ρ−α n fn (zn + ρn ξ) = gn (ξ) → g(ξ) locally uniformly, where g is a nonconstant entire function on C with order at most 1, all of whose zeros have multiplicity at least k, such that g ♯ (ξ) ≤ g ♯ (0) = kA + 1.

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Here, as usual,

|g ′ (ξ)| 1 + |g(ξ)|2

g ♯ (ξ) = is the spherical derivative.

Lemma 2.2. [3, 13] Let f be an entire (resp. meromorphic) function, and let M be a positive number. If f ♯ (z) ≤ M for any z ∈ C, then f is of order at most 1 (resp. 2). It is well known that it is very important of the Wiman-Valiron theory[5, 6] to investigate the property of the entire solutions of differential equations. In 1999, Zong-Xuan Chen[2] has extented the Wiman-Valiron theory from entire functions to meromorphic functions with infinitely many poles. Here we show the following form given by Jun Wang and Wei-Ran L¨ u[19]. g(z) Lemma 2.3. Let f (z) = d(z) be a meromorphic function with ρ(f ) = ρ, where g, d are entire functions satisfying one of the following conditions: (i) g being transcendental and d being polynomial; (ii) g, d all being transcendental and λ(d) = ρ(d) = β < ρ(g) = ρ. Then there exists one sequence {rk }(rk → ∞) such that

f (n) (z) ν(rk , g) n =( ) (1 + o(1)), f (z) z

n∈N

holds for enough large rk as |z| = rk and |g(z)| = M (rk , g), where ν(rk , g) denotes the central index of g. Lemma 2.4. [14] Let f be an entire function of order at most 1 and k be a positive integer, then f (k) ) = o(log r), as r → ∞. f Lemma 2.5. [21] Let f be a nonconstant meromorphic function, and aj (j = 1, · · · q) be q (≥ 3) distinct constant (one of them may be ∞), then m(r,

(q − 2)T (r, f ) ≤

q ∑

N (r,

j=1

where

1 ) + S(r, f ), f − aj

∑ f′ f′ S(r, f ) = m(r, ) + m(r, ) + O(1). f f − aj q

j=1

Combining Lemmas 2.4 and 2.5, we have the following special case of the Nevanlinna’s second fundamental theorem. Lemma 2.6. Let f be a nonconstant entire function of order at most 1, and aj (j = 1, · · · q) be q (≥ 3) distinct constants (one of them may be ∞), then (q − 2)T (r, f ) ≤

q ∑

N (r,

j=1

93

1 ) + o(log r). f − aj

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3. Proof of Theorem 1.1 Firstly, we will prove that the meromorphic (resp. entire) function f is of order at most 2 (resp. 1). Suppose that the spherical derivative of f is bounded. Then by Lemma 2.2, we have meromorphic (resp. entire) function f is of order at most 2 (resp. 1). Now, we assume that the spherical derivative of f is unbounded. Then there exist a sequence {wn } such that wn → ∞, f ♯ (wn ) → +∞ as n → ∞. Define D = {z : |z| < 1} and Fn (z) = f (wn + z). Since f only has finitely many poles, we can assume that all Fn (z) are analytic in D. Furthermore, Fn♯ (0) = f ♯ (wn ) → ∞ as n → ∞. It follows from Marty’s criterion that (Fn )n is not normal at z = 0. Obviously, for each n, Fn has zeros with multiplicities at least k, Fn and share S CM. Thus, from Theorem A, we derive that (Fn )n is normal at z = 0, a contradiction. (k) Fn

Thus, we prove that the meromorphic (resp. entire) function f is of order at most 2 (resp. 1). Since f and f (k) share S CM and f has finitely many poles, we have (3.1)

(f (k) − a)(f (k) − b) eQ = , (f − a)(f − b) P

where P, Q are two polynomials. Rewrite (3.1) as follows. (k)

(3.2)

Q = log P

(k)

( f f − fa )( f f − fb ) (1 − fa )(1 − fb )

,

where log h is the principle branch of Log h. g(z) If f (z) = d(z) is a transcendental meromorphic function, where g(z) is a transcendental entire function and d(z) is a polynomial. Then by Lemma 2.3, we get (3.3)

ν(rk , g) k f (k) (z) =( ) (1 + o(1)), f (z) z

holds for enough large rk as |z| = rk and |g(z)| = M (rk , g). Note that f is transcendental, we have fa |zr → 0 and fb |zr → 0 as r → ∞. It follows from

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the fact g is of finite order that log ν(r, g) = O(log r). Then, we deduce that (k)

|Q(z)| = | log P

(k)

( f f − fa )( f f − fb ) (1 − fa )(1 − fb )

|zr = O(log r),

for enough large rk as |z| = rk and |g(z)| = M (rk , g). It implies that Q is a constant. If f (z) is a rational function, then by (3.1) we know that Q must be a constant. Without loss of generality, we rewrite (3.1) as 1 (f (k) − a)(f (k) − b) = . P (f − a)(f − b) Next, we will prove that P is also a constant. On the contrary, suppose that P is not a constant. We know any zero of P comes from the pole of f , so d = deg P ≥ 2k. From the above equation, we get (k)

1=P

(k)

( f f − fa )( f f − fb ) (1 − fa )(1 − fb )

.

In a similar way as the above, we get 1 = |P (zr )(

ν(r, g) 2k ) (1 + o(1))| = |ν(r, g)2 |zr |d−2k | = ν(r, g)2 rd−2k , zr

possibly outside a finite logarithmic measure set E, where |g(zr )| = M (r, g) and |z| = rk . Since d = deg P ≥ 2k, it implies that ν(r, g) is bound, a contradiction. Hence, P is also a constant. Thus, we prove that A=

(f (k) − a)(f (k) − b) , (f − a)(f − b)

where A is a nonzero constant. From the above equation, we see that f is an entire function, so the order of f is at most 1. Set F = f − and G = − Then √ √ G − AF and h2 = G + AF , then we have a+b 2

a+b 2 .

f (k)

h1 h2 =

(a−b)2 4 (a−b)2 F 2− 4

G2 −

= A. Set h1 =

(a − b)2 (1 − A). 4

We consider two cases. Case 1. A ̸= 1.

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Obviously, h1 , h2 has no zeros and poles. Then we set h1 (z) = A1 eBz and h2 (z) = A2 e−Bz , where A1 , A2 , B are constants. Furthermore, we have a+b 1 f (z) = + √ (−A1 eBz + A2 e−Bz ), 2 2 A a+b 1 f (k) (z) = + (A1 eBz + A2 e−Bz ), 2 2 B f ′ (z) = √ (−A1 eBz − A2 e−Bz ). 2 A The above part is based on the idea in [7]. Now, we consider two subcases again. Case 1.1. A1 A2 ̸= 0. It follows from the form of f that f has infinitely many zeros. Noting that the zeros of f has multiplicities at least k, we have f (s) (s = 0, · · · k − 2) has multiple zeros. Clearly, f ′ just has simple zeros. Then, k − 2 ≤ 0, so k must equal to 2. By differentiating f ′ one time, we have B2 f ′′ (z) = √ (−A1 eBz + A2 e−Bz ). 2 A Comparing it to the above form of f (k) , we have a+b 1 B2 + (A1 eBz + A2 e−Bz ) = √ (−A1 eBz + A2 e−Bz ), 2 2 2 A which means that either A1 or A2 is zero, a contradiction. Case 1.2. A1 A2 = 0. Without loss of generality, we assume that A2 = 0. Then we have a+b 1 f (z) = − √ A1 eBz . 2 2 A From the form of f , it is easy to see that if f has zeros, then f just has simple zeros. It contradicts with the fact f has zeros of multiplicities at least k. So, f has no zeros and a + b = 0. Thus, we can set f (z) = CeDz , where C, D are two constants. By differentiating f k-times, we have f (k) (z) = CDk eDz . From f and f (k) share S CM, we have Dk = ±1. If Dk = 1, then f = f (k) , and f and f (k) share a, b CM. If Dk = −1, then f = −f (k) , and b = −a. Case 2. A = 1.

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Then it is easy to see that f = f (k) or f (k) + f = a + b. Suppose that f = f (k) . Noting that f equals to f (k) , so they share 0 CM. Moreover, from the fact that all the zeros of f has multiplicities at least k, we derive f has no zeros. Then, by the same way in Case 1.2, we get the same results. Finally, by the similar way in [12], we will discuss the case of f (k) + f = a + b. Solving the differential equation, we have (3.4)

f (z) =

k−1 ∑

Cj expλj z +a + b,

j=0 2jπ+π

where λj = exp k i and Cj are constants. Since f is a non-constant, then there exist Cj ∈ {C0 , C1 , · · · , Ck−1 } such that Cj ̸= 0. Denote the non-zero constants in {Cj } by Cjm 0 ≤ jm ≤ k − 1 and m = 0, 1, · · · , s, s ≤ k − 1. Thus, rewrite (3.4) as (3.5)

f (z) =

s ∑

Cjm expλjm z +a + b.

m=0

Differentiating (3.5) t-times yields (3.6)

f (t) (z) =

s ∑

Cjm λtjm expλjm z , (t = 1, 2 · · · , k − 1).

m=0

Suppose that f has finitely many zeros, then we can set f (z) = P1 (z)eλz , where P1 is a polynomial. By differentiating it k times, we have f (k) (z) = [λk P1 + λk−1 P1′ + H(P1′′ , P1′′′ , · · · , P1 )]eλz , (k)

where H(P1′′ , P1′′′ , · · · , P1 ) is the linear combination of P1′′ , P1′′′ , · · · , P1 . Substituting the above forms of f and f (k) into f + f (k) = a + b, we derive that (k)

(k)

P1 + λk P1 + λk−1 P1′ + H(P1′′ , P1′′′ , · · · , P1 ) = 0, (k)

which implies that λk = −1 and P1′ = 0. Thus, P1 is a constant and f has no zeros. By the same way in Subcase 1.2, we derive the desired results. Thus, in what follows, we assume that f has infinitely many zeros, say zn = rn eθn , where 0 ≤ θn < 2π. Without loss of generality, we may assume that θn → θ0 and rn → ∞ as n → ∞. Substituting zn into (3.5) and (3.6), we have (3.7)

f (zn ) =

s ∑

Cjm expλjm zn = −(a + b)

m=0

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and (3.8)

f (t) (zn ) =

s ∑

Cjm (λjm )t expλjm zn = 0, (t = 1, 2 · · · , k − 1).

m=0

We consider two cases again. Subcase 2.1. s = k − 1. From (3.7) and (3.8), we have    Cj0 a+b  0   Cj0 λj0     . = ..  ..   . 0 Cj0 (λj0 )k−1

C j1 C j1 λ j1

··· ···

Cj1 (λj1 )k−1

···

 expλj0 zn λj1 zn    exp   .  ..   . λj zn k−1 Cjk−1 (λjk−1 )k−1 exp Cjk−1 Cjk−1 λjk−1



We know    det  

C j0 C j0 λ j0 .. . Cj0 (λj0 )k−1

··· ···

C j1 Cj1 λj1

Cjk−1 Cjk−1 λjk−1

    

Cj1 (λj1 )k−1 · · · Cjk−1 (λjk−1 )k−1   1 1 ··· 1  λj0 λ j1 ··· λjk−1    = Cj0 Cj1 · · · Cjk−1 det   ..   . k−1 k−1 k−1 (λj0 ) (λj1 ) · · · (λjk−1 ) ∏ (λjp − λjq ). = Cj0 Cj1 · · · Cjk−1 0≤q 0 or cos(θ0 +

2jp π+π ) k

< 0.

2j π+π

If cos(θ0 + p k ) > 0, then we can assume (for n large enough) cos(θn + 2jp π+π ) > δ, here δ is a small positive number. Thus, as n → ∞, by (3.9) k

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we have |Dp | = exprn cos(θn +

2jp π+π ) k

→ ∞,

a contradiction. 2j π+π

If cos(θ0 + p k ) < 0, then we can assume (for n large enough) cos(θn + < −δ, here δ is a small positive number. Thus, as n → ∞, by (3.9) we have 2jp π+π |Dp | = exprn cos(θn + k ) → 0, a contradiction. Observing that 0 ≤ jp , jq ≤ k − 1, by (3.10), we deduce 2jp π + π 2jq π + π (3.11) | − | = π, (0 ≤ p ̸= q ≤ k − 1). k k Let jp = 0 and jq = k − 1. Substitute them into (3.11), we have 2jp π+π ) k

2(k − 1) = k, that is k = 2. Thus, k must be 2. ′′

Now we discuss the equation f + f = a + b. From the above discussion, we can obtain λ0 = i, λ1 = −i. Then, we have f (z) = A1 eiz + A2 e−iz + a + b. Noting that f has zeros of multiplicity at leat 2, Then (a + b)2 = 4A1 A2 . Then, we finish the proof of this subcase. Subcase 2.2. s < k − 1. Then, by (3.8), we can choose t = 1, 2, · · · , s + 1. Then they form a system of linearly equation of expλj0 zn , expλj1 zn , · · · , expλjs zn . By solving it, we have (3.12)

expλjp zn = 0,

a contradiction. Hence, we complete the proof of this theorem. 4. Proof of Theorem 1.2 If Theorem A is replaced by Theorem C, by the same way to the proof of Theorem 1.1, we can also obtain the (f ′ − a)(f ′ − b) , A= (f − a)(f − b) where A is a nonzero constant. From the above equation, we see that f is an entire function. Hence we can get the conclusion by Theorem D.

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References [1] J. M. Chang and M. L. Fang, Normality and shared functions of holomorphic functions and their derivatives, Michigan Math. J. 53(2005), 625–645. [2] Z.X. Chen, On the rate of growth of meromorphic solutions of higher order linear differential equations, Acta Math. Sinica, 42(1999), 552-558. [3] J. Clunie and W.K. Hayman, The spherical derivative of integral and meromorphic functions, Comment. Math. Helv, 40(1966), 117–148. [4] W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [5] W. Hayman, The local growth of power series: A survey of Wiman-Valiron method, Canad. Math. Bull., 17(1974), 317-358. [6] I. Laine, Nevanlinna theory and complex differential equations. Studies in Math. vol. 15, de Gruyter, Berlin, 1993. [7] J.T. Li and H.X. Yi, Normal families and uniqueness of entire functions and their derivatives, Arch. Math(Basel). 87(2006), 52–59. [8] Y.T. Li, Sharing set and normal families of entire functions, Results. Math, 63(2013), 543–556. [9] X. J. Liu and X. C. Pang, Shared values and normal function, Acta Mathematica Sinca, Chinese Series. 50(2007), 409–412. [10] F. L¨ u and J.F. Xu, Sharing set and normal families of entire functions and their derivatives, Houston J. Math. 34(2008), 1213-1223. [11] F. L¨ u, J. F. Xu and H.X. Yi, A note on a famous theorem of Pang and Zalcman, Journal of Computational Analysis and Applications, 18(2015), 662-671. [12] F. L¨ u and H.X. Yi, The Br¨ uck conjecture and entire functions sharing polynomials with their k-th derivatives, J. Korean. Math. Soc, 48(2011), 499–512. [13] D. Minda, Yosida functions, Lectures on Complex Analysis (Xian, 1987), (Chi Tai Chuang, ed.), World Scientific Pub. Co., Singapore, 1988, pp. 197–213. [14] V. Ngoan and I. V. Ostrovskii, The logarithmic derivative of a meromorphic function, Akad. Nauk. Armjan. SSR. Dokl, 41(1965), 272–277. [15] X.C. Pang and L. Zalcman, Normal families and shared values, Bull. London Math. Soc, 32(2000), 325–331. [16] W. Schwick, Sharing values and normality, Arch. Math(Basel). 59(1992), 50–54. [17] J. F. Xu and X. B. Zhang, A note on the shared set and normal family, Journal of Computational Analysis and Applications, 15(2013), 977-984. [18] J. Schiff, Normal families, Springer (1993). [19] J. Wang and W.R. L¨ u, The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients, Acta Math. Appl. Sinica, 27(2004), 72-80. [20] X.B. Zhang and J. F. Xu, Further results on normal families of meromorphic functions concerning shared values, Journal of Computational Analysis and Applications, 19(2015), 310-318. [21] C.C. Yang and H.X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557. Kluwer Academic Publishers Group, Dordrecht, 2003. Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, P.R.China E-mail address: [email protected] or [email protected] College of Science, China University of Petroleum, Qingdao, 266580, P. R. China. E-mail address: [email protected]

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Compact adaptive aggregation multigrid method for Markov chains Ying Chen∗, Ting-Zhu Huang†, Chun Wen School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, P.R. China

Abstract A new adaptive aggregation-based multigrid scheme is presented for the calculation of the stationary probability vector of an irreducible Markov chain. By exploiting the experimental observation that components of vectors converge nonuniformly, we develop a new algorithm to speed up the on-the-fly adaptive multigrid method proposed by Treiter and Yavney [On-the-fly adaptive smoothed aggregation multigrid for Markov chains, SIAM J. Sci. Comput., 33(2011): 2927-2949]. In our algorithm, the converged components are collected and compacted into one aggregate on the finest level, which is able to cut down the cost of coarsen operators construction and the total amount of work. In addition, we present a technique to delete the possible weak-links introduced in the process of aggregation. Several types of test cases are calculated, and experiment results show that the new adaptive method can improve the on-the-fly algorithm in terms of total execution time. Key words: Adaptive aggregation multigrid; on-the-fly adaptive method; Markov chains; converged components

1

Introduction

This paper is concerned with a new adaptive multigrid method for the numerical calculation of the stationary probability vector of irreducible, large and sparse Markov matrices. Let B ∈ Rn×n be a sparse column-stochastic matrix, which means 1T B = 1T , where 1 is the column vector of all ones, and 0 ≤ bi j ≤ 1∀i, j. We seek a vector x ∈ Rn that satisfies Bx = x, ∥ x ∥1 = 1, xi ≥ 0∀i. ∗ †

(1.1)

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

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Matrix B represents the transition matrix of a Markov chain and x is a stationary probability vector of this Markov chain. If B is irreducible, that is, there exists a path from each vertex i to each vertex j in its directed graph, then according to the Perron-Frobenius theorem for nonnegative matrices [1], the equation (1.1) has a unique solution x, with xi > 0∀i. This problem (1.1) is equivalent to the singular linear system Ax = 0, ∥ x ∥1 = 1, xi ≥ 0∀i.

(1.2)

where A := I − B, by 1T B = 1T , we have 1T A = 0,which means the vector we seek, x, is the only left null-vector of the matrix A. Algebraic multigrid method (AMG) was developed and applied widely due to its efficiency for solving large problems arising from partial differential equations and Mmatrices. Compared with geometric multigrid methods, AMG constructs the multigrid hierarchy only using the information of the given matrix, which extends the application of multigrid methods. However, it leads to the inefficiency and the lack of robustness, because the operators of these multigrid methods are constructed based on the unsatisfied assumptions made on the near null spaces of the matrices. To overcome this disadvantage, several adaptive algebraic multigrid methods were developed in [4, 26, 5]. The basic idea of these adaptive approaches was of improving multigrid methods by updating interpolation and coarsen operators to fit the slow-to-converge components of the vector. The idea was further developed in adaptive AMG [23] and adaptive SA [24, 25], where slow-toconverge components were exposed through multiscale development instead of relaxation on finest-level. The Markov chains solver which was outlined in [13] was actually another form of adaptive AMG, because they share the same concept of updating operator to get more accurate approximation of the near null space of A. With the same idea, a multilevel adaptive aggregation [7] was suggested with aggregates updated in each step of the iteration. Based on this algorithm, a collection of Markov chains solvers were proposed recently: adaptive aggregation multigrid for Markov chains (AGG) [7], smoothed aggregation multigrid (SA) [6], AMG for Markov chains (MCAMG) [8]. Several accelerated methods were developed in [18, 10]. While all these adaptive approaches improved the algorithms robustness and accuracy by adapting coarsen operators in every cycle, they also suffered from considerable computation time for calculating the coarsen matrix [27]. The on-the-fly adaptive multigrid hierarchy for Markov chains which was developed in [19] significantly cut the cost of constructing the coarse-level operators. Here, the classical solution cycles are preferred over the adaptive cycles, under the assumption that the former is comparatively cheaper but it needs the operators provided by the latter. The algorithm presented in this paper is inspired by the following experimental observation: when applying aggregation multigrid V-cycle to obtain approximation of stationary probability vector, the elements of the stationary probability vector do not converge uniformly. Based on this observation, we propose a compressed on-the-fly adaptive scheme to save the cost on constructing coarsen operators. The main idea is to compact the converged components into a single aggregate and rescale the coarsen operators . Also 2

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we develop a new technique that deletes weak-links introduced by the above procedure. As the improvement of the on-the-fly adaptive aggregation method, the new algorithm adopts the same adaptive hierarchy as on-the-fly method does. It differs, however, in that the on-the-fly method uses operators supplied by SET cycles without any amendment, whereas in new algorithm, the coarsen operators are rescaled to smaller size to fit the notconverged-yet components. It is shown numerically that the new algorithm can reduce the total execution time of the on-the-fly adaptive multigrid method. New algorithm can also be applied to various adaptive multigrid Markov solvers. In this paper we apply it to the aggregation-based algebraic multigrid solver (AGG), with unsmoothed interpolation and prolongation operators. In the next section, we give a brief description of multilevel aggregation multigrid method for Markov problems. Then we recall the on-the-fly adaptive framework in Section 3, which the new algorithm is based on. In Section 4, we outline the experimental observation as the stage for the introduction of new algorithm, and we compare the proposed algorithm with compatible relaxation method as well. Numerical tests are presented in Section 5.

2

Classical aggregation multigrid for Markov chains

In this section, we briefly recall the aggregation-based multigrid methods for Markov chains from [13, 7, 6]. The interpolation operators of aggregation multigrid are often smoothed to overcome the instinct difficulties produced by aggregation [6, 14]. In our work, we stay with the unsmoothed coarsen operators. First, we define the multiplicative error ei by x = diag(xi )ei , where xi is the current approximate at ith iterate. Thus we have Adiag(xi )ei = 0.

(2.1)

It is necessary to assume that all components of xi are nonzero. At convergence, xi = x and the fine-level error ei = 1, where 1 is the column vector with all ones. Note that the aggregation technique used in this paper is the same as that used in [7], e = Adiag(xi ), the benefit which is based on strength of connection in the scaled matrix A e instead of original matrix A is that the former gives more of using the scaled matrix A appropriate notion of weak and strong links than the latter, more details are in [7]. We consider node i is strongly connected to node j if −˜ai j ≥ θ max{−˜aik }. k,i

(2.2)

where θ ∈ [0, 1] is a strength threshold parameter, we choose θ = 0.8. Aggregates based on the strength of connection are then constructed by the following procedure: choose point i with the largest value in current proximation xi from the unassigned points as the seed point of a new aggregate, then add all unassigned points j satisfies (4) to the new 3

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aggregates. Repeat this procedure until all points are assigned to aggregates. Assuming that the n fine-level points are aggregated into m groups, then the aggregation matrix Q ∈ Rn×m are formed, where qi j = 1 indicates that fine-level point i belongs to aggregate j and qi j = 0 the opposite[6]. Then the coarse version of (3) is given by QT Adiag(xi )Qec = 0,

(2.3)

where ec is the coarse-level approximation of the fine-level error ei , with ei ≈ Qec . The restriction and prolongation operators, R and P are defined as follows: R = QT , P = diag(xi )Q.

(2.4)

RAPec = 0.

(2.5)

Then (5) can be rewrited as Same as the definition of fine-level multiplicative error xi , the coarse-level error xc is given by xc = diag(Rxi )ec . (2.6) Notice that PT 1 = Rxi ,thus (3) can be rewrited as RAPdiag(PT 1)−1 xc = 0.

(2.7)

Then the coarse-level error equation (5) is equivalent to coarse-level probability equation Ac xc = 0, with coarsen matrix Ac defined by Ac = RAPdiag(PT 1)−1 .

(2.8)

When the coarsen solution xc is obtained, the next iterate, xi+1 can be calculated according to the coarse-level correction xi+1 = Pec = Pdiag(PT 1)−1 xc .

(2.9)

In this paper we use weighted Jacobi method for all relaxation procedure, at each coarser level we perform v1 pre-relaxation and v2 post-relaxations. One iteration of weighted Jacobi relaxation applied to problem Ax = b is given by x ← x + ωD−1 (b − Ax).

(2.10)

where D is the diagonal part of A, its relaxation parameter ω = 0.7. On coarsest level we perform direct solver described in [8]. The procedure above is described in Algorithm 1, which is originally presented in [13]. The multilevel aggregation method is obtained by recursively applying Algorithm 1 to step 5.

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Algorithm 1: Two-level aggregation for Markov chains x ← AGG(A,x,µ,v1 ,v2 ) Input:Initial vector: x ∈ Rn , operator: A ∈ Rn×n , cycle index: µ, number of pre-relaxations: v1 , number of post-relaxations: v2 . Output: New approximation to the solution of Ax = 0. Algorithm: if not at coarsest level 1. x ← Relax(A,x,0) v1 times 2. Build Q based on A and x 3. Set R ← QT , P ← diag(xi )Q 4. Set xc ← Rx, and repeat µ ≥ 1 times: xc ← AGG(RAPdiag(PT 1)−1 , xc , µ, v1 , v2 ) 5. Coarse-grid correction: x ← Pdiag(PT 1)−1 xc 6. x ← Relax(A,x,0) v2 times else 7. Direct solve of Ax = 0 end

3 On-the-fly aggregation multigrid for Markov chains In this section, we briefly describe the on-the-fly multigrid method developed recently in [19]. The main idea of this method is reducing the cost of expensive SET cycle such as Algorithm 1, which updating the whole multigrid hierarchy of operators in every cycle, by using classical algebraic multigrid cycles (Algorithm 2) instead, as the two algorithms are actually equivalent. In the approach, SET cycle provide classical cycle with improved operators, while classical cycle use them without adaptation and then offer SET cycle with better approximation of vector. It is obvious that the classical cycle with frozen operators are much more cheaper than the SET cycle, the advantage of this scheme is, by combining the two algorithms neatly, it speeds up the multigrid methods without sacrificing the convergence rate. Classical algebraic multigrid method for linear systems are generally based on the following basic idea. Given the linear system Ax = b,

(3.1)

where A ∈ Rn×n is a positive definite matrix. Traditional one-level iterative method for calculating x, such as Power method or weighted Jocobi relaxation, converge very slowly due to only a relatively small number of components in the error, known as algebraically smooth, that approximately satisfy Ae = 0. To eliminate the algebraic smoothed errors, classical multigrid methods solve this problem on a coarse level with smaller size, referred to as coarse-grid correction process. It is noted that on the coarse grid, the smooth error appears to be relatively higher in frequency, which means relaxations are more effective 5

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on coarser grid [3]. Algorithm 2 gives a typical two-level classical multigid cycle [5], a multilevel V-cycle is obtained by recursively applying the algorithm in step 4. Algorithm 2: Two-level additive cycle Input:Initial vector: x ∈ Rn , Right-hand-side vector: b ∈ Rn , operator: A ∈ Rn×n , P ∈ Rn×nc ,R ∈ Rnc ×n ,Ac ∈ Rnc ×nc . Output: New approximation to the solution of Ax = b. Algorithm: 1. Apply pre-relaxations: x ← Relax(A,x,b) 2. Define the residual: r ← b − Ax 3. Restrict the residual: rc ← Rr 4. Define ec as the solution of the coarse-grid problem: Ac ec = rc 5. Prolong ec and apply coarse-grid correction: x ← x + Pec 6. Apply post-relaxations: x ← Relax(A,x,b) The difference between Algorithm 1 and Algorithm 2 is that, on the coarse-grid, the correction scheme of two-level additive cycle approximates the error e rather than the exact solution x. Moreover, the classical algorithm requires the whole hierarchy of coarsen operators in advance, while the setup schemes calculate them in every cycle. In spite of that, Algorithm 2 can be written as the form of Algorithm 1 equivalently. For the problem (2) in which b = 0, the residual r in step 2 and rc in step 3 of algorithm 2 are given as r = −Ax and rc = −RAx, the coarse-grid problem then is given by Ac ec = RAPec = −RAx,

(3.2)

RA(Pec + x) = 0.

(3.3)

then we obtain Since the approximation x is in the range of P, there exists a vector xc satisfies x = Pxc . Then the equation above can be rewritten as RAP(ec + xc ) = 0. Note that the xc we mentioned above is not necessary the same as xc in Step 5 in Algorithm 1. We define zc = ec + xc , thus Ac zc = 0, which is equivalent to the coarse-grid problem in SET cycle of Algorithm 1. In the on-the-fly approach, an initial SET cycle is performed, followed by a SOL cycle which freezes the operators the SET cycle provided. If the convergence speed of SOL cycle is acceptable, another SOL cycle is performed. Conversely, a SET cycle is performed to yield more accurate operators. This procedure is described as follows. Procedure: try-SOL-else-SET(γ) 1. Try a solution cycle: y = V sol (x) 2. If q(y) > q(x) do x ← V set (x) and return 3. If q(y) > γq(x) then x = y, else x ← V set (y) In above procedure, V sol represents a SOL cycle (Algorithm 2), V set represents a SET cycle (Algorithm 1) and γ ∈ [0, 1] is the scalar threshold for acceptable convergence speed of the SOL cycles. We use ∥ Ax ∥1 , (3.4) q(x) = ∥ x ∥1 6

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which means the convergence factor is measured by the l1 residual norm. The criteria q(y) > q(x) indicates that the SOL cycle increases the error and should be abandoned. The criteria q(y) > γq(x) indicates that if the convergence factor of SOL cycle is better than the scalar threshold, then accept it, otherwise, perform a SET cycle instead. The on-the-fly adaptive algorithm is described as in Algorithm 3. Algorithm 3: On-the-fly adaptive multigrid method Input:Initial tolerance: εα , convergence parameter: γ, operator: A ∈ Rn×n , initial guess x0 . Output: New approximation to the solution of Ax = 0. Algorithm: 1. Initial Setup: Apply a few relaxations to smooth x0 Do an initial Setup cycle: x ← V set (x0 ) if ∥ Ax0 ∥1 < εα , goto Step 4 2. Improve Solution Cycle: while ∥ Ax0 ∥1 > εα do try-SOL-else-SET(γ) 3. Finalize Setup cycle: x ← V set (x) 4. Solution: Apply x ← V sol (x) until convergence

4 Compact adaptive aggregation multigrid In this section, we show how compacting the converged points into an aggregate, coupled with deleting the weak-links between them, can lead to better performance of on-the-fly method for Markov chains.

4.1 Experimental observation We define a point has already converged as in [15]: |xi(ν+1) − xi(ν) |/|xi(ν) | < τ p ,

(4.1)

where xi denotes the ith element of the vector, xi(ν) denotes ith element at νth iterate, and τ p is the convergence parameter. In [15], it is noted that the convergence patterns of the stationary probability vector of web matrix in the power method have a nonuniform distribution. Additional theoretical analysis in [17] has confirmed this conclusion recently. During the application of AGG on Markov problems, we have seen the similar convergence behavior that some points converge quickly while some others need more iterations before convergence. It is shown in Figure 1 that the number of the converged points increased gradually as iteration number increased. To exploit this observation, the method outlined in [15] is that the converged components won’t be recomputed so that computation cost can be reduced. The basic idea 7

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(b) Uniform 2D lattice with n=4096

Fig. 1: (a) Tandem queueing network with n=1600, (b) Uniform 2D lattice with n=4096, where x-axis represents iterations and y-axis represents the proportion of the points that satisfy the equation (4.1). developed there has three steps: splitting the vector into converged and not-yet-converged components, setting the submatrix AN ∈ Rm×n which corresponds to the not-yet-converged components as target matrix, and then applying the power method until convergence without recomputing converged components. More details can be seen in [15]. However, as AN is not a n × n matrix, many algorithms including AMG can not be applied to this method. For this reason, with the similar principle but different procedures, we propose a new algorithm in this paper.

4.2 Compact adaptive aggregation multigrid The main idea of our algorithm is reducing the computational cost by reducing the size of the coarse levels as well as the time spent on the coarse matrix construction. The new algorithm follows the same framework as the on-the-fly adaptive multigrid method does. Consider that we have executed a setup cycle, then an approximation x and the aggregation matrix Q are constructed in this cycle. Perform the try-SOL-else-SET procedure until the number of converged points meets m > ζn, where m is the number of the converged points, n is the size of the problem, ζ ∈ (0, 1) is the threshold parameter. The reason why we set this standard will be addressed in the following paragraphs. Let C as set of the converged points whose elements are positive integers between 1 and n, and N as set of the points have not converged yet. Partitioning the finest-level matrix as ) ( ANN ANC ˆ . (4.2) A= ACN ACC

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Similarly, the current approximation x and its multiplicative error, eˆ are reordered as ( ) x xˆ = N , (4.3) xC ( ) e eˆ = N , eC

(4.4)

respectively. To reduce the time cost of coarse matrix construction, on-the-fly method proposed that the SOL cycle use the aggregation matrix Q which is offered by SET cycle without any modification [19]. Whereas in our method, we modify the aggregation matrix Q before we perform a SOL cycle. As to modifications, we keep the non-converged points in their aggregates and collect the converged points into a new aggregate. Then a further standard solution cycle is performed with amended operators and smaller scales. The motivation is that we try to speed up the multigrid solvers by cutting down the cost on coarsen operators construction as well as reducing the size of coarse operators. Now we show how to construct the new aggregation matrix Qˆ by modifying the aggregation matrix Q from the setup cycle. We first delete the rows of Q which belongs to C, then check for those columns with all zero elements and delete them, finally, construct ˆ as given in Algorithm 3, where the length of 1 equals to that of C. The procedure is Q simple and inexpensive: Procedure: Construct compact aggregation matrix Qˆ 1. Delete Q(i, :), i ∈ C 2. Delete( Q(:, )j) if Q(:, j) = 0 Q 0 3. Qˆ ← , where 1 is the column vector of all ones, with length equals that of C 0 1 ˆ As the same Now we constructed coarse operators based on aggregation matrix Q. definition in the classical AMG, the restriction and prolongation operators, R and P, are given by Rˆ = Qˆ T , (4.5) ˆ Pˆ = diag(ˆxC )Q,

(4.6)

respectively. The coarse-level operator Aˆ c is given by ˆ Aˆ c = Rˆ Aˆ P.

(4.7)

Thus we obtain the complete hierarchy of multigrid operators the SOL cycle required, then we perform a standard SOL cycle as the final step to finish the new solution cycle, as described in Algorithm 4.

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Algorithm 4: Compact Solution Cycle(C-SOL) Input:Approximate vector: x ∈ Rn , operator: A ∈ Rn×n , the converged points set C. aggregation matrix Q ∈ Rnc ×n Output: New approximation to the solution of Ax = 0. Initial setup: ( ) ( ) A A x NN NC 1. Set Aˆ ← , xˆ ← N ACN ACC xC 2. Construct compactive aggregation matrix Qˆ based on Q and C. ˆ Set Aˆ c = Rˆ Aˆ Pˆ 3. Set Rˆ = Qˆ T , Pˆ = diag(ˆxC )Q, Apply solution cycle: 4. Do a standard solution cycle described in Algorithm 2 In new method we prefer C-SOL cycle over SOL cycle if the former’s error reduction is acceptable. The underlying assumption is that C-SOL cycles are considerably cheaper with satisfied convergence rate. However, if the ratio of m above n is too small or too big, this assumption will be ruined. On the one hand, for most of test cases, when we put a small number (m < 0.1n) of the converged points into an aggregate, the C-SOL cycle is more expensive than the SOL cycle. This is because the cost on SET process in Algorithm 4 cannot be balanced out by the time saved by cutting scales of coarse-levels. On the other hand, if a large number of the converged points are compacted into an aggregate, it may lead to quite inaccurate operators in coarse-levels. Numerical experiments confirm that the resulting algorithm performs worse than the original on-the-fly method or leads to divergence for most problems. For the above reasons, we introduce the restriction for the number of converged points m: if m < ζn we perform the procedure try-SOL-else-SET(γ), elsewhere we perform try-CSOL-else-SET(γ) instead. Similar with on-the-fly method, the goal of our method is to fall off the time cost on reaching the accuracy ∥ Ax0 ∥1 < εα . Whereas the most distinguished difference of the new algorithm from on-the-fly adaptive multigrid is in Step 2. At Step 2 in new algorithm we initially perform the procedure try-SOL-else-SET until the number of the converged points meets the compactive condition m ≥ ζn. With the converged points set C supplied by the process above and the aggregation matrix Q provided by SET cycle, we construct the C-SOL cycle, then we repeat the procedure try-CSOL-else-SET with the until the residual norm of the approximation reduced to εα . It is noted that in C-SOL cycle we frozen the coarsen operators as well as the converged points. The algorithm is described in Algorithm 5.

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Algorithm 5: Compactive On-the-fly adaptive multigrid method Input:Initial tolerance: εα , convergence criterion: τ p ,convergence factor: γ, size control parameter: ζ, operator: A ∈ Rn×n ,initial guess x0 , Output: New approximation to the solution of Ax = 0. Algorithm: 1. Initial Setup: Apply a few relaxations to smooth x0 Do an initial Setup cycle: x ← V set (x0 ) if ∥ Ax0 ∥1 < εα goto Step 5 2. Improve Solution Cycle: [N, C] ← Detect-converged-points(x(1) , x(0) , τ p ) While ∥ Ax0 ∥1 > εα do While m < ζn do try-SOL-else-SET(γ) [N, C] ← Detect-converged-points(x(v+1) , x(v) , τ p ) end try-CSOL-else-SOL(γ) end 4. Finalize Setup cycle: x ← V set (x) 5. Solution: Apply x ← V sol (x) until convergence As mentioned above, compacting the converged points into an aggregate may lead to a single aggregate with a large number of points that are not strongly connected to each other. As is shown in [6], the aggregate of points that are weakly connected may result in very poor convergence of the multilevel method. The reason is that if the link between two points is weak compared to the other links in the same aggregate, the differences in the error of these two points can neither be eliminated efficiently by relaxation, nor smoothed out by coarse-level correction. Thus, although we have made the restriction for the size of the aggregate, it may still induce unsatisfied convergence. Our next work is trying to define and delete the weak links in the aggregation of converged points to avoid the poor convergency.

4.3 Compactive on-the-fly method with correction We illustrate with a simple example. In C-SOL cycles, the converged points are compacted into a single aggregation. Figure 2 is an example of such an aggregation. Links between the converged points and not-converged-yet points are not presented in this figure. We assume that the converged points set as C=[4,5,9,10,14,17,18,38]. To capture the weak links in this aggregate, we need to determine what is meant by weak links. In the classical AMG, the strong connection is defined by formula (2.2), which indicates that if the size of the transition probability from i to j timed with the 11

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Figure 2: Single aggregate of converged points with fine-level transitions. The converged points are indicated by numbers in cycles, and the transitions are indicated by arrows with strength based on the scaled matrix ACC diag(xC ). Connections between the converged points and those have not converged yet are not presented in this figure.

probability of residing in i is comparative large, then it is a strong link. Rather than the connection strength between two points, our attention is turned to the overall connection strength between a point and the rest points in the same aggregate, which is used to measure the importance of a point in its aggregate. We define the connection strength of point i based on scaled matrix ACC diag(xC ) with elements a˜ il by ∑ S i = − (˜ail + a˜ li ). (4.8) l,i

This definition has an simple intuitive interpretation that the overall connection strength of a point is measured not only by the probability from other points to it but also by the probability from it to others. If a point’s overall connection strength is comparative small, it cannot contribute efficiently to the elimination of errors but may lead to poor convergence. In the view of the above, we define a point is weakly connected to the others if S i ≤ δS¯ i , (4.9) where S¯ i is the mean value of all S i (i ∈ C), and δ is a fixed threshold parameter, whose function is as the same as θ in (2.2). Choosing δ > 1 may set down all points as ”not important” points especially when the number of points strong connected to the others is large. Meanwhile, it should not be taken much smaller than 1 because this may leave weak-links staying in the aggregate. The numerical results indicate that choosing δ < 1 but close to 1 results in the best convergence properties for the new method. In generally we take δ = 0.8. It is easy to calculate and conclude that the points 14,17,18,38 in figure 1 are weakly connected to the other points in the aggregate, thus we have the new C¯ = [4, 5, 9, 10] to replace the original C.

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5 Numerical results and discussion In this section, we demonstrate the performance of the new algorithm for several test problems. The algorithm is applied to the two-level classical aggregation multigrid method, without smoothing operators. We compare the results of original on-the-fly adaptive aggregation multigrid algorithms (OTF) and the compactive on-the-fly adaptive aggregation multigrid algorithms (C-OTF). We start with an initial guess of unit vector with its elements all equal to 1/n, in which n is the length of the vector. All setup cycles employ (4,1) cycles, with four prerelaxations and one postrelaxation on each level, while all compactive solution cycles and original solution cycles use (2,1) cycles. We use the stopping criteria ∥ Axv ∥1 stop if v > maxit or < τ∥ Ax0 ∥1 . (5.1) ∥ xv ∥1 proposed in [9],where maxit is the upper limit of the number of iterations the algorithm will be allowed to perform, v is the current vth iteration. Here we use maxit = 200 and τ = 10−8 . We also say the problem has reached global convergence if this criterion has met. Several threshold value τ p are tested in the experiments. Through extensive simulations, we found that τ p = 10−3 achieved the best performance among any others for most of test cases. For OTF algorithm we use scalar threshold γ = 0.8 and εα = 10−5 , while for C-OTF algorithm we use γ = 0.8 and various choices of εα are presented in the following table. As to the AGG part in algorithms we use the aggregation strategy based on scaled matrix proposed in [7], with the strength threshold parameter θ = 0.25. In the following tables, we show the operator complexity COP and the work units WU which is defined as the cost of a single V sol (2, 1) [19]. WU is calculated as follows: for each problem and its size, averaging the execution time of a V sol (2, 1) by calculating the mean value of last five solution cycles in step 4 in Algorithm 3, the work units are the total execution time of the algorithm divided by this time. The motivation is that the execution time of the algorithm is susceptible to MATLAB’s compilation time. V set , V sol , Vcsol are the number of SET cycles, SOL cycles and C-SOL cycles, respectively. The experiments were performed using MATLAB R2010a with an Intel core i3 CPU with 4 GB of RAM memory.

5.1 Uniform chain The first three test problems are generated by graphs with weighted edges[6, 20]. Their transition probabilities are determined by weights of the edge: if node i transforms to j with p weights and then its probability p ji is obtained by p divided by the sum of the weights of all outgoing edges from node i. Our first test problem is the 1D uniform chain, generated by linear graphs in which each of two connected points has one outgoing edge with weight 1. The stencil of the matrix of uniform chain is given by ) ( (5.2) HUni f ormChain = 21 0 12 . 13

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Table 5.1 shows the results for the uniform chain problem using OTF-AGG algorithm (Algorithm 3) and C-OTF-AGG algorithm (Algorithm 5). When we set τ p = 10−3 the new algorithm achieves the much better performance compared with τ p = 10−4 and τ p = 10−2 . Various choices of weak-links parameter δ are tested and it does not make too much difference when δ ≤ 0.9. We set δ = 0.8 and the size control parameter ζ = 0.45 for this test case. The experiments show that the SET cycle is significantly more expensive than the SOL cycle, while the C-SOL cycle is cheaper than SOL when the number of converged points meets m > 0.1n. Comparing the C-OTF and the corrected C-OTF under the same parameters, we observe a decrease in work units and the number of cycles. The results also indicate that a sufficiently small εα enhance the opportunities of executing C-SOL cycles, as is shown in the table 5.1, so that reduce the total execution time. Table 5.1. Uniform chain results. t sol is the average timing of a single V sol (2, 1) solution cycle, εα is the threshold parameter for performing the on-the-fly procedure at step 3 in algorithm 5, τ p is the threshold parameter to explore the converged points in equation (12), COP is the operator complexity, V set , V sol , Vcsol are the number of SET cycles, SOL cycles and C-SOL cycles, respectively. WU is the work units defined as the cost of a single V sol (2, 1) solution cycle. Iter is the number of overall iterations.

Algorithm (εα ,τ p ) V set , V sol , Vcsol −4 OTF (10 , - ) 2,18,0 −3 −5 C-OTF (10 ,10 ) 2,16,4 961 0.05s −5 −3 C-OTF(-cor) (10 ,10 ) 2,15,3 C-OTF(-cor) (10−6 ,10−3 ) 2,12,6 −4 OTF (10 , - ) 2,18,0 C-OTF(10−8 ,10−3 ) 2,15,16 4096 2.67s C-OTF(-cor) (10−8 ,10−3 ) 2,9,10 C-OTF(-cor) (10−10 ,10−3 ) 2,3,16 −4 OTF (10 , - ) 2,18,0 C-OTF (10−8 ,10−3 ) 2,18,11, 13225 339.25s C-OTF(-cor) (10−8 ,10−3 ) 2,9,9 C-OTF(-cor) (10−10 ,10−3 ) 2,4,15 n

t sol

COP 1.50 1.22 1.27 1.05 1.50 0.77 0.79 0.35 1.54 0.82 0.82 0.43

WU Iter 81 20 80 > 20 80 20 73 20 45 20 42 > 21 36 21 30 21 54 20 56 > 20 44 20 37 21

5.2 Uniform chain with two weak links The next test problem is a chain with uniform weights, except for two weak links with weight ϵ in the middle of the chain [6]. The stencil matrix is given by ( ) 1 ϵ 1 0 1+ϵ HT woWeakLinks = 12 1+ϵ (5.3) 2 . 14

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where ϵ = 10−3 same as in [6]. As the same as the first case, we set the weak-links parameter δ = 0.8 and the size control parameter ζ = 0.45 here. The experiments show that the convergence criterion parameter τ p = 10−3 is the best choice for this case. Results in Table 5.2 show again that the corrected C-OTF method is competitive compared with OTF and C-OTF without corrections.

5.3 Uniform 2D lattice The next test problem is a 2D lattice with uniform weights [6, 20]. The stencil matrix is given by    1  1  HUni f orm2D = 1 0 1 . (5.4)   4 1 We set the weak-links parameter δ = 0.8 and the size control parameter ζ = 0.45 for this test case. Table 5.3 shows numerical results for this problem. For the small scale n = 4096 of this case, the choice of τ p = 10−2 performs better than τ p = 10−3 because the components of the prototype vector converge comparative slowly. In the larger case n = 13225, when we set τ p = 10−3 , the new algorithm fails to reduce the work units of OTF, largely due to the poor convergency of C-SOL cycles. To be specific, if the convergence rate of C-SOL cycle is unacceptable, we perform a SOL cycle instead. This procedure costs more time than a single SOL cycle and thus results in the worse performance than that of OTF.

Table 5.2. Uniform chain with two weak links results.

n 962

4096

13224

t sol

Algorithm (εα ,τ p ) OTF (10−4 , - ) C-OTF (10−5 ,10−3 ) 0.05 C-OTF(-cor) (10−5 ,10−3 ) C-OTF(-cor) (10−6 ,10−3 ) OTF (10−4 , - ) C-OTF(10−8 ,10−4 ) 2.63s C-OTF(-cor) (10−8 ,10−3 ) C-OTF(-cor) (10−10 ,10−3 ) OTF (10−4 , - ) C-OTF (10−8 ,10−3 ) 426.50 C-OTF(-cor) (10−8 ,10−3 ) C-OTF(-cor) (10−10 ,10−3 )

V set , V sol , Vcsol 2,18,0 2,17,2 2,17,2 2,13,5 2,20,0 2,15,11 2,8,11 2,14,17 2,19,0 2,19,12, 2,9,10, 2,3,16

COP 1.50 1.40 1.41 1.26 1.50 0.91 0.71 0.31 1.54 0.77 0.79 0.35

WU Iter 79 20 80 21 80 21 78 21 47 22 42 > 22 36 21 29 21 53 21 43 > 21 35 21 26 21

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Table 5.3. Uniform 2D lattice results.

n

t sol

961

0.05s

4096

2.48s

13225 522.11s

Algorithm (εα ,τ p ) V set , V sol , Vcsol −4 OTF (10 , - ) 2,31,0 −5 −2 C-OTF (10 ,10 ) 2,32,5 C-OTF(-cor) (10−5 ,10−2 ) 2,29,8 −6 −2 C-OTF(-cor) (10 ,10 ) 2,31,12 −4 OTF (10 , - ) 2,33,0 −8 −3 C-OTF(10 ,10 ) 2,41,15 C-OTF(-cor) (10−8 ,10−2 ) 2,28,24 −6 −2 C-OTF(-cor) (10 ,10 ) 2,22,16 −4 OTF (10 , - ) 2,24,0 −8 −3 C-OTF (10 ,10 ) 2,37,23 −8 −3 C-OTF(-cor) (10 ,10 ) 2,36,21 C-OTF(-cor) (10−10 ,10−3 ) 2,37,25

COP 1.63 1.35 1.30 1.15 1.66 0.99 0.82 1.06 1.70 0.96 0.97 0.92

WU cycles 97 33 98 > 33 94 > 33 98 > 33 61 35 60 > 35 51 > 35 53 > 35 44 26 49 > 26 48 > 26 48 > 26

COP 1.57 1.57 1.57 1.42 1.60 1.20 1.59 1.22 1.66 1.30 1.30 1.10

WU 97 99 99 97 52 50 51 50 49 47 44 36

Table 5.4. Tandem queueing network results.

Algorithm (εα ,τ p ) V set , V sol , Vcsol −4 OTF (10 , - ) 2,23,0 −5 −3 C-OTF (10 ,10 ) 2,22,0 961 0.04 C-OTF(-cor) (10−5 ,10−3 ) 2,22,0 −6 −3 C-OTF(-cor) (10 ,10 ) 2,22,2 −4 OTF (10 , - ) 2,23,0 −8 −3 C-OTF(10 ,10 ) 2,15,5 4096 2.40s C-OTF(-cor) (10−6 ,10−3 ) 2,21,1 −8 −3 C-OTF(-cor) (10 ,10 ) 2,15,5 −4 OTF (10 , - ) 2,23,0 −8 −3 C-OTF (10 ,10 ) 2,25,3, 13225 570.38s C-OTF(-cor) (10−8 ,10−3 ) 2,25,3 −10 −3 C-OTF(-cor) (10 ,10 ) 2,26,6 n

t sol

Iter 25 24 24 26 25 > 25 24 > 25 25 > 25 > 25 >25

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5.4 Tandem queueing network The next test problem is a tandem queueing network appeared in [2, 6, 9, 20], which has two finite single-server queues placed in tandem. Customers arrive in Poisson distribution with rate µ, and two server stations’ service time distribution is Poisson with rates µ1 and µ2 respectively. The stencil matrix of tandem queueing work is given by    µ1  1   HT andemQueue = (5.5) µ 0  , µ + µ1 + µ2 µ2 where we use µ = 10, µ1 = 11, µ2 = 10 as in [2, 6, 9, 20]. Table 5.4 shows numerical results for this problem. In this case we set the weak-links parameter δ = 0.8, the size control parameter ζ = 0.45 and convergence parameter τ p = 10−3 . We also try using more strict convergency parameter τ p = 10−4 . Results show that the algorithm fails to expose the converged points and the number of C-SOL cycle is equal to 0. Similar with the previous problems, several choices of εα are tested. Experiments show that with a sufficiently small εα , new algorithm improves the performance of OTF in terms of the total execution time, but suffers from an unsatisfied convergence rate, which increases the number of iterations. For the reason that the operators of C-SOL cycles are less accurate than that of SOL cycles, they have a probability to lead to poor convergence rate. To achieve the same accuracy εα , more C-SOL cycles are needed. Whereas, the total execution time is reduced because C-SOL cycles are comparative cheaper than SOL cycles. Table 5.5. Random walk on unstructured planar graph results.

n

t sol

961

2

4096

0.23s

13225 6.16s

Algorithm (εα ,τ p ) V set , V sol , Vcsol OTF (10−4 , - ) 2,27,0 C-OTF (10−5 ,10−3 ) 2,29,3 −5 −3 C-OTF(-cor) (10 ,10 ) 2,29,3 C-OTF(-cor) (10−6 ,10−3 ) 2,40,2 OTF (10−4 , - ) 2,29,0 C-OTF(10−6 ,10−3 ) 2,32,6 −6 −3 C-OTF(-cor) (10 ,10 ) 2,18,6 −8 −3 C-OTF(-cor) (10 ,10 ) 2,18,6 −4 OTF (10 , - ) 2,28,0 C-OTF (10−6 ,10−3 ) 2,29,8, C-OTF(-cor) (10−6 ,10−3 ) 2,30,5 −9 −3 C-OTF(-cor) (10 ,10 ) 2,41,20

COP 1.20 1.07 1.12 0.95 1.21 0.99 0.99 0.94 1.21 0.98 0.94 0.57

WU 182 186 186 183 162 162 157 158 122 130 120 116

Iter 29 > 29 > 29 > 29 31 > 31 > 31 > 31 30 > 30 > 30 > 30

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5.5 Random walk on unstructured planar graph The next test problem is random walks on graphs, which have significant applications in many fields, one of the well-known examples is Google’s pagerank algorithm. Here we consider an unstructured planar graph, which is generated by choosing n random points in the unit square, and triangulating them by Delaunay triangulation. The transition probability from point i to point j is given by the reciprocal of the number of egdes incident on point i. In this test case, when m < 0.6n, a single C-SOL cycle costs more time than SOL cycle does, thus we use the size control parameter ζ = 0.6 here. We set the weak-links parameter δ = 0.8. Experiments show that convergence parameter τ p = 10−3 is the best choice among any others. The performance of corrected C-OTF method is moderate. However, the work units, which indicates the total execution time, is still smaller than that of OTF and C-OTF without corrections. Table 5.5 shows numerical results for this problem.

6

Conclusions

This paper proposes a compact on-the-fly adaptive aggregation multigrid method for Markov chain problems. As is known, adaptive multigrid methods suffer from the common defect that considerable computation cost is spent on coarsen operators construction. The reason is that they update the entire multigrid hierarchy of operators in every cycles. We consider distributing the converged points into an aggregate and reducing the scale of the coarsen operators to decrease this cost. Meanwhile, a simple technique is proposed to delete the possible weak-links introduced by the procedure above. According to numerical results, for most of test cases, the corrected algorithm leads to better performance than on-the-fly adaptive aggregation multigrid algorithm in terms of total execution time. New algorithm can also be applied to various adaptive multigrid Markov solvers. One future work may be to study how to improve the convergence rate of compacted solution cycles. Acknowledgements. This research is supported by NSFC (61170309), the Fundamental Research Funds for the Central Universities (ZYGX2013Z005).

References [1] R. A. Horn, and C. R. Johnson, Matrix Analysis, Cambridge university press, New York, 2012. [2] W. J. Stewart, An Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994. [3] W. L. Briggs, S. F. McCormick, A Multigrid Tutorial, 2nd ed., SIAM, Philadelphia, 2000.

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[4] A. Brandt, S. McCoruick, J. Huge, Algebraic Multigrid (AMG) for Sparse Matrix Equations, Sparsity and its Applications, D. J. Evans, ed., Cambridge University Press, Cambridge, UK, 1985, pp. 257-284. [5] A. Brandt, D. Ron, Multigrid Solvers and Multilevel Optimization Strategies, Multilevel Optimization and VLSICAD, Kluwer Academic Publishers, Dordrecht, 2003, pp. 1-69. [6] H. De Sterck, T. A. Manteuffel, S. F, McCormick, et al, Smoothed aggregation multigrid for Markov chains, SIAM J. Sci. Comput., 32 (2010), pp. 40-61. [7] H. De Sterck, T. A. Manteuffel, S. F, McCormick, et al, Multilevel adaptive aggregation for Markov chains, with application to web ranking, SIAM J. Sci. Comput., 30 (2008), pp. 2235-2262. [8] H. De Sterck, T. A. Manteuffel, S. F. McCormick, et al, Algebraic multigrid for Markov chains, SIAM Journal on Scientific Computing, 32 (2010), pp. 544-562. [9] H. De Sterck, K. Miller, E. Treister, et al, Fast multilevel methods for Markov chains, Numerical Linear Algebra with Applications, 18 (2011), pp. 961-980. [10] H. De Sterck, K. Miller, T. A. Manteuffel, et al, Top-level acceleration of adaptive algebraic multilevel methods for steady-state solution to Markov chains, Adv. Comput. Math., 35 (2011), pp. 375-403. [11] L.-J. Deng, T.-Z. Huang, X.-L. Zhao, L. Zhao, and S. Wang, An economical aggregation algorithm for algebraic multigrid (AMG), J. Comput. Anal. Appl., Vol. 16, No.1, 2014, pp. 181-198 [12] M. Bolten, A. Brandt, J. Brannick, A. Frommer, et al, A bootstrap algebraic multilevel method for Markov chains, SIAM J. Sci. Comput., 33 (2011), pp. 3425-3446. [13] G. Horton, S. T. Leutenegger, A multi-level solution algorithm for steady-state Markov chains, Perform. Eval. Rev., 22 (1994), pp. 191-200. [14] P. Vank, J. Mandel, M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), pp. 179-196. [15] S. Kamvar, T. Haveliwala, G. Golub, Adaptive methods for the computation of PageRank, Lin. Alg. Appl., 386 (2004), pp. 51-65. [16] E. Virnik, An algebraic multigrid preconditioner for a class of singular M-matrices, SIAM J. Sci. Comput., 29 (2007), pp. 1982-1991. [17] A. Bourchtein, L. Bourchtein, On some analytical properties of a general PageRank algorithm, Mathematical and Computer Modelling, 2011. [18] H. De Sterck, K. Miller, G. Sanders, et al, Recursively accelerated multilevel aggregation for Markov chains, SIAM J. Sci. Comput., 32 (2010), pp. 1652-1671. [19] E. Treister, I. Yavneh, On-the-fly adaptive smoothed aggregation multigrid for Markov chains, SIAM J. Sci. Comput., 33 (2011), pp. 2927-2949.

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[20] E. Treister, I. Yavneh, Square and stretch multigrid for stochastic matrix eigenproblems, Numer. Lin. Alg. Appl., 17 (2010), pp. 229-251. [21] A. C. Muresan, Y. Notay, Analysis of aggregation-based multigrid, SIAM J. Sci. Comput., 30 (2008), pp. 1082-1103. [22] Y. Notay, An aggregation-based algebraic multigrid method, Electronic Trans. Numerical Anal., 37 (2010), pp. 123-146. [23] M. Brezina, R. Falgout, S. MacLachlan, et al, Adaptive algebraic multigrid, SIAM J. Sci. Comput., 27 (2006), pp. 1261-1286. [24] M. Brezina, R. Falgout, S. MacLachlan, et al, Adaptive smoothed aggregation (α SA), SIAM J. Sci. Comput., 32 (2010), pp. 1896-1920. [25] M. Brezina, R. Falgout, S. MacLachlan, et al, Towards adaptive smoothed aggregation (α SA) for nonsymmetric problems, SIAM J. Sci. Comput., 25 (2004), pp. 14-39. [26] J. W. Ruge, K. Stben, Algebraic multigrid (AMG), Multigrid methods, Frontiers in Applied and Computational Mathematics, SIAM, 1987, pp. 73-130. [27] A. J. Cleary, R. D. Falgout, V. E. Henson, et al, Robustness and scalability of algebraic multigrid, SIAM J. Sci. Comput., 21 (2000), pp. 1886-1980. [28] B.-Y. Pu, T.-Z. Huang, C. Wen, A preconditioned and extrapolation-accelerated GMRES method for PageRank, Appl. Math. Lett., Vol. 37, 2014, Pages 95-100 [29] Y. Xie and T.-Z. Huang, A model based on cocitation for web information retrieval, Math. Problems in Engineering, Volume 2014, Article ID 418605, 6 pages

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Travelling Solitary Wave Solutions for Stochastic Kadomtsev-Petviashvili Equation. Hossam A. Ghany1,2 and M. Al Qurashi2 1 Department of Mathematics, Helwan University, Cairo(11282), Egypt. 2 Department of Mathematics, Taif University, Taif(888), Saudi Arabia. [email protected]

Abstract. In this paper, generalized Wick-type stochastic Kadomtsev-Petviashvili equations are investigated. Abundant white noise functional solutions for Wick-type generalized stochastic Kadomtsev-Petviashvili equations are obtained. By using white noise analysis, Hermite transform, modified Riccati equation and modified tanh-coth method many exact travelling wave solutions are given. Detailed computations and implemented examples for the investigated model are explicitly provided . Keywords: White noise; Stochastic ; Wick product; Kadomtsev-Petviashvili equations. PACS numbers: 02.50.Fz; 05.10.Gr.

1

Introduction

In this paper we investigate the generalized variable coefficient Kadomtsev-Petviashvili (KP) equation: ∂ ∂u ∂3u ∂2u ut + (φ(t)u + ψ(t) 3 ) + θ(t) 2 = 0, (x, y, t) ∈ R2 × R+ (1.1) ∂x ∂x ∂x ∂y where u is a stochastic process on R2 × R+ and φ(t), ψ(t) and θ(t) are bounded measurable or integrable functions on R+ . Equation (1.1) plays a significant role in many scientific applications such as solid state physics, nonlinear optics, chemical kinetics, etc. The KP equations[1-2] are universal models(normal forms) for the propagation of long, dispersive, weakly nonlinear waves that travel predominantly in the x direction, with weak transverse effects. The notion of wellposed-ness will be the usual one in the context of nonlinear dispersive equations, that is, it includes existence, uniqueness, persistence property, and continuous dependence upon the data. Recently, many researchers pay more attention to study of random waves, which are important subjects of stochastic partial differential equation (SPDE). Wadati [3] first answered the interesting question, How does external noise affect the motion of solitons? and studied the diffusion of soliton of the KdV equation under Gaussian noise, which satisfies a diffusion equation in transformed coordinates.

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Wadati and Akutsu also studied the behaviors of solitons under the Gaussian white noise of the stochastic KdV equations with and without damping [4]. Wadati [3] first answered the interesting question, “How does external noise affect the motion of solitons?” and studied the diffusion of soliton of the KdV equation under Gaussian noise, which satisfies a diffusion equation in transformed coordinates. The stochastic PDEs was discussed by many authors, e.g., de Bouard and Debussche [6, 7], Debussche and Printems [8, 9], Printems [17] and Ghany and Hyder [13]. On the basis of white noise functional analysis [5], Ghany et al. [10-16] studied more intensely the white noise functional solutions for some nonlinear stochastic PDEs. This paper is mainly concerned to investigate the white noise functional solutions for the generalized Wick-type stochastic Kadomstev-Petviashvili (KP) equation: Ut + Φ(t)  Ux  Ux + Ψ(t)  U  Uxx + Ψ(t)  Uxxxx + Θ(t)  Uyy = 0.

(1.2)

where “  ” is the Wick product on the Kondratiev distribution space (S)−1 and Φ(t), Ψ(t) and Θ(t) are (S)−1 -valued functions [5]. It is well known that the solitons are stable against mutual collisions and behave like particles. In this sense, it is very important to study the nonlinear equations in random environment. However, variable coefficients nonlinear equations, as well as constant coefficients equations, cannot describe the realistic physical phenomena exactly. The rest of this paper is organized as follows: In Section 2, we recall the definition and some properties of white noise analysis. In Section 3, we apply some method to explore exact travelling wave solutions for Eq.(1.1). In Section 4, we use the Hermite transform and [5,Theorem 4.1.1] to obtain white noise functional solutions for Eq.(1.2). In Section 5, we give illustrative examples for the investigated model. The last section is devoted to summary and discussion.

2

Preliminaries

Suppose that S(Rd ) and S 0 (Rd ) are the Hida test function space and the Hida distribution space on Rd , respectively. Let hn (x) be Hermite polynomials and put √ 1 2 ζn = e−x hn ( 2x)/((n − 1)!π) 2 ,

n > 1.

(2.1)

then, the collection {ζn }n>1 constitutes an orthogonal basis for L2 (R) . Let α = (α1 , α2 , ..., αd ) denote d-dimensional multi-indices with α1 , α2 , ..., αd ∈ N . The family of tensor products ζα := ζ(α1 ,α2 ,...,αd ) = ζα1 ⊗ ζα2 ⊗ ... ⊗ ζαd

(2.2)

forms an orthogonal basis for L2 (Rd ) . (i) (i) (i) Suppose that α(i) = (α1 , α2 , ..., αd ) is the i-th multi-index number in some fixed ordering of all d-dimensional multi-indices α . We can, and will, assume that this ordering has the property that (i) (i) (i) (j) (j) (j) (2.3) i < j ⇒ α1 + α2 + ... + αd < α1 + α2 + ... + αd i.e., the {α(j) }∞ j=1 occurs in an increasing order. Now

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Define ηi := ζα(i) ⊗ ζα(i) ⊗ ... ⊗ ζα(i) , 1

2

d

i > 1.

(2.4)

We need to consider multi-indices of arbitrary length. For simplification of notation, we regard multi-indices as elements of the space (NN 0 )c of all sequences α = (α1 , α2 , ..., αd ) with elements αi ∈ N0 and with compact support, i.e., with only finitely many αi 6= 0 . We write J = (NN 0 )c , for α ∈ J , Define ∞ Y Hα (ω) := hαi (< ω, ηi >), ω = (ω1 , ω2 , ..., ωd ) ∈ S 0 (Rd ) (2.5) i=1

n ∈ N and for all k ∈ N , suppose the space (S)n1 consists of those f (ω) = P For a fixed L n n such that k=1 L2 (μ) with cα ∈ R α cα Hα (ω) ∈ X c2α (α!)2 (2N)kα < ∞ (2.6) ||f ||21,k = α

Pn (k) 2 (1) (2) (n) ifQcα = (cα , cα , ..., cα )Q∈ Rn and μ is the white noise where, c2α = |cα |2 = k=1 (cα ) α αj for α ∈ J . measure on (S 0 (R), B(S 0 (R))) , α! = ∞ k=1 αk ! and (2N) = j (2j) P n n such The space (S)P −1 consists of all formal expansions F (ω) = α bα Hα (ω) with bα ∈ R that ||f ||−1,−q = α b2α (2N)−qα < ∞ for some q ∈ N. The family of seminorms ||f ||1,k , k ∈ N gives rise to a topology on (S)n1 , and we can regard (S)n−1 as the dual of (S)n1 by the action < F, f >=

X (bα , cα )α!

(2.7)

α

where (bα , cα ) is the inner product in Rn . P P The Wick product f  F of two elements f = α aα Hα , F = β bβ Hβ ∈ (S)n−1 with aα , bβ ∈ Rn , is defined by X (aα , bβ )Hα+β (2.8) f F = α,β

n n d 0 d The spaces (S) closed under Wick products. P1 , (S)−1 , S(R n) and S (R ) are For F = α bα Hα ∈ (S)−1 , with bα ∈ Rn , the Hermite transformation of F , is defined by

HF (z) = Fe (z) =

X α

bα z α ∈ CN

(2.9)

where z = (z1 , z2 , ...) ∈ CN (the set of all sequences of complex numbers) and z α = z1α1 z2α2 ...znαn , if α ∈ J , where zj0 = 1. For F, G ∈ (S)n−1 we have e ^ F  G(z) = Fe(z).G(z) (2.10)

e for all z such that Fe (z) and G(z) exist. The product on the right-hand side of the above formula is the complex bilinear product between two elements of CN defined by (z11 , z21 , ..., zn1 ).(z12 , z22 , ..., zn2 ) = Pn 1 2 k=1 zk zk . P e ∈ RN is called the generalized expectation Let X = α aα Hα , then the vector c0 = X(0)

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of X which denoted by E(X) . Suppose that g : U −→ CM is an analytic function, where U is a neighborhood of E(X) . Assume that the Taylor series of g around E(X) have coefficients in  −1 M e RM . Then the Wick if g has the power series P version g (X)α= H (g◦ X) ∈M(S)−1 . In other words, P expansion g(z) = aα (z − E(X)) , with aα ∈ R , then g (z) = aα (z − E(X))α ∈ (S)M −1 .

3

Exact travelling wave solutions

In this section, we will give exact solutions of Eq.(1.1). Taking the Hermite transform of Eq.(1.2), we get: g U g U ft (t, x, y, z) + Φ(t). fx (t, x, y, z).U fx (t, x, y, z) + Ψ(t). e (t, x, y, z).U g U xx (t, x, y, z) g U g g ^ + Ψ(t). xxxx (t, x, y, z) + Θ(t).Uyy (t, x, y, z) = 0

(3.1)

where z = (z1 , z2 , ...) ∈ C N is a parameter. To look for the travelling wave solution of Eq.(3.1),we e (t, x, y, z) = ϕ(ξ(t, x, y, z)) with make the transformations u(t, x, y, z) := U ξ(t, x, y, z) := k1 x + +k2 y + s

Z

t

l(τ, z)dτ + c

0

where k1 , k2 , s, c are arbitrary constants which satisfy k1 k2 s 6= 0 , l(τ, z) is a non zero functions of indicated variables to be determined. So, Eq.(3.1) can be changing into the form: slu0 (t, x, z) + k12 Φu0 (t, x, z)u0 (t, x, z) + k12 Ψu(t, x, z)u00 (t, x, z)+ k14 Ψu0000 (t, x, z) + k22 Θu00 (t, x, z) = 0

(3.2)

The solution can be proposed by the tanh method as a finite power series in Y in the form: u(μζ) = S(Y ) =

M X

ak Y k ,

(3.3)

k=0

limiting them to solitary and shock wave profiles. However, the extended tanh method admits the use of the finite expansion u(μζ) = S(Y ) =

M X

ak Y k +

k=0

M X

bk Y −k ,

(3.4)

k=1

where M is a positive integer, in most cases, that will be determined. Expansion (3.4) reduces to the standard tanh method [4-6], where Y (ξ) satisfies the Riccati equation Y 0 = c0 + c1 Y + c2 Y 2 ,

(3.5)

and c0 , c1 , c2 are constant to be prescribed later. By virtue of (3.3) and (3.4) with observation of the linear independence of Y n (n = −6, −5, ..., 6) and using Mathematica Eqn.(3.2) implies the

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following nonlinear algebraic system of equations:  2 + 2α α slα1,0 + k14 Ψα4,0 + k22 Θα2,0 + k12 [Φ(α1,0  1,1 1,−1 + 2α1,2 α1,−2 + 2α1,3 α1,−3 )     +Ψ(a0 α2,0 + a1 α2,−1 + a2 α2,−2 + b1 α2,1 + b2 α2,2 )] = 0,      slα1,1 + k14 Ψα4,1 + k22 Θα2,1 + k12 [Φ(2α1,0 α1,1 + 2α1,−1 α1,2 + 2α1,−2 α1,3 )      +Ψ(a0 α2,1 + a1 α2,0 + a2 α2,−1 + b1 α2,2 + b2 α2,3 )] = 0,      slα1,−1 + k14 Ψα4,−1 + k22 Θα2,−1 + k12 [Φ(2α1,0 α1,−1 + 2α1,1 α1,−2 + 2α1,2 α1,−3 )      +Ψ(a0 α2,−1 + a1 α2,0−2 + a2 α2,−3 + b1 α2,0 + b2 α2,1 )] = 0,     2 + 2α α  slα1,2 + k14 Ψα4,2 + k22 Θα2,2 + k12 [Φ(α1,1  1,0 1,2 + 2α1,−1 α1,3 )     +Ψ(a0 α2,2 + a1 α2,1 + a2 α2,0 + b1 α2,3 + b2 α2,4 )] = 0,     2  slα1,−2 + k14 Ψα4,−2 + k22 Θα2,−2 + k12 [Φ(α1,−1 + 2α1,0 α1,−2 + 2α1,1 α1,−3 )    +Ψ(a α 0 2,−2 + a1 α2,−3 + a2 α2,−4 + b1 α2,−1 + b2 α2,0 )] = 0,  slα1,3 + k14 Ψα4,3 + k22 Θα2,3 + k12 [Φ(2α1,0 α1,3 + 2α1,1 α1,2 )      +Ψ(a0 α2,3 + a1 α2,2 + a2 α2,1 + b1 α2,4 )] = 0,      slα1,−3 + k14 Ψα4,−3 + k22 Θα2,−3 + k12 [Φ(2α1,0 α1,−3 + 2α1,−1 α1,−2 )      +Ψ(a0 α2,−3 + a1 α2,−4 + b1 α2,−2 + b2 α2,−1 )] = 0,     2 ) + Ψ(a α  k14 Ψα4,4 + k22 Θα2,4 + k12 [Φ(2α1,1 α1,3 + α1,2 0 2,4 + a1 α2,3 + a2 α2,2 )] = 0,     4 2 2 2  k1 Ψα4,−4 + k2 Θα2,−4 + k1 [Φ(2α1,−1 α1,−3 + α1,−2 ) + Ψ(a0 α2,−4 + b1 α2,−3 + b2 α2,−2 )] = 0,      k14 Ψα4,5 + 2k12 α1,2 α1,3 + k12 (a1 α2,4 + a2 α2,3 ) = 0,     k14 Ψα4,−5 + 2k12 α1,−2 α1,−3 + k12 (b1 α2,−4 + b2 α2,−3 ) = 0,     2 + k 2 Ψa α k14 Ψα4,6 + k12 Φα1,3 2 2,4 = 0,  1   4 2 2 2 k1 Ψα4,−6 + k1 Φα1,−3 + k1 Ψb2 α2,−4 = 0, (3.6)

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where  α1,0 = a1 c0 − b1 c2 , α1,1 = a1 c1 + 2a2 c0 , α1,2 = a1 c2 + 2a2 c1 , α1,3 = 2a2 c2 ,      α1,−2 = −(b1 c0 + 2b2 c1 ), α1,−1 = −(b1 c1 + 2b2 c2 ), α2,0 = α1,1 c0 − α1,−1 c2 ,      α1,−3 = −2b2 c0 , α2,1 = α1,1 c1 + 2α1,2 c0 , α2,2 = α1,1 c2 + 2α1,2 c1 + 3α1,3 c0 ,      α2,3 = 2α1,2 c2 + 3α1,3 c1 , α2,4 = 3α1,3 c2 , α2,−1 = −(α1,−1 c1 + 2α1,−2 c2 ),      α2,−2 = −(α1,−1 c0 + 2α1,−2 c1 + 3α1,−3 c2 ), α2,−3 = −(2α1,−2 c0 + 3α1,−3 c1 ),      α2,−4 = −3α1,−3 c0 , α3,0 = α2,1 c0 − α2,−1 c2 , α3,1 = α2,1 c1 + 2α2,2 c0 ,      α3,2 = α2,1 c0 + 2α2,2 c1 + 3α2,3 c2 , α3,3 = 2α2,2 c2 + 3α2,3 c1 + 4α2,4 c0 ,    α = 3α c + 4α c , α = 4α c , α 3,4 2,3 2 2,4 1 3,5 2,4 2 3,−1 = −(α2,−1 c1 + 2α2,−2 c2 ),  α3,−2 = −(α2,−1 c0 + 2α2,−2 c1 + α2,−3 c2 ), α3,−4 = −(3α2,−3 c0 + 4α2,−4 c1 ),      α3,−3 = −(2α2,−2 c0 + 3α2,−3 c1 + 4α2,−4 c2 ), α3,−5 = −(4α2,−4 c0 ),      α4,0 = α3,1 c0 − α3,−1 c2 , α4,1 = α3,1 c1 + 2α3,2 c0 , α4,6 = 5α3,5 c2 ,     α4,2 = α3,1 c2 + 2α3,2 c1 + 3α3,3 c0 , α4,3 = 2α3,2 c2 + 3α3,3 c1 + 4α3,4 c0 ,     α4,4 = 3α3,3 c2 + 4α3,4 c1 + 5α3,5 c0 , α4,5 = 4α3,4 c2 + 5α3,5 c0 ,      α4,−1 = −(α3,−1 c1 + 2α3,−2 c2 ), α4,−3 = −(2α3,−2 c0 + 3α3,−3 c1 + 4α3,−4 c2 ),      α4,−2 = −(α3,−1 c0 + 2α3,−2 c1 + 3α3,−3 c2 ), α4,−6 = −5α3,−5 c0 ,    α4,−4 = −(3α3,−3 c0 + 4α3,−4 c1 + 5α3,−5 c2 ), α4,−5 = −(4α3,−4 c0 + 5α3,−5 c2 ).

At the rest of this section we will discuss and solve our problem for some particular cases for the Riccati equation as follows:

A. c0 = c1 = 1, c2 = 0

.

For this choice of the constants, the Riccati equation has the solution: Y1 (ξ) = exp(ξ) − 1

(3.7)

By the aid of Mathematica, the above system of equations (3.6) can be solved for the following cases:

Case 1: a1 = a2 = 0, αi,j = 0 for all i, j > 0 ; a0 =

1 e {sl k12 Ψ

e − k 2 Θ} e ; b1 = 12k 2 3Ψ ; b2 = −12k 2 Ψ . − k14 Ψ 2 1 5Φ 1Φ

According to (3.2),(3.6) and (3.7), Eq.(3.1) has the solution u1 (t, x, y, z) = where,

2e 2e 1 e − k 2 Θ} e + 36k1 Ψ (exp(ξ) − 1)−1 − 12k1 Ψ (exp(ξ) − 1)−2 . {sl − k14 Ψ 2 e e e k12 Ψ 5Φ Φ

ξ = k1 x + k2 y −

11.4k14

Z

0

t

e z)dτ Ψ(τ,

(3.8)

(3.9)

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Case 2:

e

k2

e

e

e

e

Ψ 2 2 Ψ 2 Ψ . 1 2Θ a2 = b2 = 0; a 0 = 25k12 Φ+3 e Ψ e − 25k1 − ( k22 ) Ψ e ; b1 = −50k1 Φ+2 e Ψ e ; a1 = −2k1 Φ+ e Ψ e Φ+2 According to (3.2),(3.6) and (3.7), Eq.(3.1) has the solution

u2 (t, x, y, z) = 25k12 where,

e + 3Ψ e e e e 50k12 Ψ Φ k2 Θ 2k12 Ψ − 25k12 − ( 22 )2 − (exp(ξ) − 1) − (exp(ξ) − 1)−1(3.10) e e e e + 2Ψ e e +Ψ e + 2Ψ k1 Ψ Φ Φ Φ

ξ = k1 x + k2 y +

k14

Z

t 0

f2 (τ, z) − 12Φ f2 (τ, z) + 12Φ(τ, e z)Ψ(τ, e z) 11Ψ e z)dτ Ψ(τ, e z) + Ψ(τ, e z))(Φ(τ, e z) + 3Ψ(τ, e z)) (Φ(τ,

B. c0 = −c2 = 0.5, c1 = 0

(3.11)

.

For this choice of the constants, the Riccati equation has the solution: Y2 (ξ) = tanh(ξ) ± isech(ξ)

(3.12)

Y3 (ξ) = coth(ξ) ± csch(ξ)

(3.13)

or

By the aid of Mathematica, the above system of equations (3.6) can be solved for the following case:

Case 3: e

e

e

e

Φ Ψ Ψ 2 2 2 a2 = b1 = b2 = 0; a 0 = 1.25k12 − ( kk21 )2 Θ e − 7.5k1 2Φ+3 e Ψ e − 3.75k1 2Φ+3 e Ψ e ; a2 = −15k1 2Φ+3 e Ψ e . Ψ According to (3.2),(3.6) and (3.7), Eq.(3.1) has the solution

ui (t, x, y, z) = 1.25k12 − (

15k12 where,

e e e k2 2 Θ Φ Ψ ) − 7.5k12 − 3.75k12 − e + 3Ψ e + 3Ψ e e e k1 Ψ 2Φ 2Φ e Ψ

e + 3Ψ e 2Φ

2 (ξ), Yi−1

ξ = k1 x + k2 y.

C. c2 = 4c0 = 1, c1 = 0

i = 3, 4.

(3.14)

(3.15)

.

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For this choice of the constants, the Riccati equation has the solution: Y4 (ξ) = 0.5tan(2ξ)

(3.16)

Y5 (ξ) = 0.5cot(2ξ)

(3.17)

or

By the aid of Mathematica, the above system of equations (3.6) can be solved for the following case:

Case 4:

e

e

Ψ 2 a1 = a2 = b1 = 0; a 0 = −16k12 − ( kk21 )2 Θ e ; b2 = −120k1 4Φ+6 e Ψ e . Ψ According to (3.2),(3.6) and (3.7), Eq.(3.1) has the solution

ui (t, x, y, z) = −16k12 − ( where,

e e k2 2 Θ Ψ Y −2 (ξ), ) − 120k12 e + 6Ψ e i−1 e k1 Ψ 4Φ

i = 5, 6.

ξ = k1 x + k2 y.

(3.18)

(3.19)

At the end of this section we should remark that, there exists infinitely number of solutions for Eqn.(1.1) these solution coming from solving the system (3.6) with regarding the Riccati equation (3.5). The above mentioned cases are just to clarify how far my technique is applicable.

4

White noise functional solutions

The main aim of the rest of this paper is to obtain white noise functional solutions of Eqs.(1.2). As pointed out from Xie [16], we will use Theorem 2.1 of for d = 2 . The properties of hyperbolic functions yield that there exists a bounded open set S ⊂ R+ × R2 , m > 0 and n > 0 such that u(x, y, t, z), uxt (x, y, t, z) are uniformally bounded for all (t, x, y, z) ∈ S × Km (n) , continuous with respect to (t, x, y) ∈ S for all z ∈ Km (n) and analytic with respect to z ∈ Km (n) for all (t, x, y) ∈ S . Using Theorem 2.1 of Xie [16], there exists a stochastic process U (t, x, y) such that the Hermite transformation of U (t, x, y) is u(t, x, y, z) for all S × Km (n) , and U (t, x, y) is the solution of (1.2). This implies that U (t, x, y) is the inverse Hermite transformation of u(t, x, y, z) . Hence, for Φ(t)Ψ(t)Θ(t) 6= 0 the white noise functional solutions of Eqs.(1.2) can be written as follows: U1 (t, x, y) =

1 36k12 Ψ(t) 4 2 {sl − k Ψ(t) − k Θ(t)} + 1 2 5Φ(t)(exp (Ξ1 (t, x, y)) − 1) k12 Ψ(t) −

12k12 Ψ(t) 2 1 (t, x, y)) − 1)

Φ(t)(exp (Ξ

(4.1)

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where, Ξ1 = k1 x + k2 y −

11.4k14

Z

t

Ψ(τ )dτ

(4.2)

0

and, U2 (t, x, y) = 25k12

k2 Θ(t) Φ(t) + 3Ψ(t) Ψ(t) − 25k12 − ( )2 Y  (Ξ2 (t, x, y)) − 2k12 k1 Ψ(t) Φ(t) + 2Ψ(t) Φ(t) + Ψ(t) 1 − 50k12

Ψ(t) Y − (Ξ2 (t, x, y)) Φ(t) + 2Ψ(t) 1

(4.3)

where, Y1 (Ξ2 (t, x, y)) = exp (Ξ2 (t, x, y)) − 1 and, Ξ2 = k1 x + k2 y + k14

Ui (t, x, y) = 1.25k12 − (

− 15k12

Z

t 0

11Ψ2 (τ ) − 12Φ2 (τ ) + 12Φ(τ )Ψ(τ ) Ψ(τ )dτ (Φ(τ ) + Ψ(τ ))(Φ(τ ) + 3Ψ(τ ))

(4.4)

k2 2 Θ(t) Φ(t) Ψ(t) − 7.5k12 − 3.75k12 ) 2Φ(t) + 3Ψ(t) 2Φ(t) + 3Ψ(t) k1 Ψ(t) Ψ(t)  Y 2 (Ξ3 (x, y)), 2Φ(t) + 3Ψ(t) i−1

i = 3, 4.

(4.5)

where Y2 (Ξ3 (x, y)) = tanh (Ξ3 (x, y)) ± isech (Ξ3 (x, y)) or Y3 (Ξ3 (x, y)) = coth (Ξ3 (x, y)) ± csch (Ξ3 (x, y)) Ui (t, x, y) = −16k12 − (

k2 2 Θ(t) Ψ(t) , − 120k12 ) 2 k1 Ψ(t) (Ξ3 (x, y)) (4Φ(t) + 6Ψ(t))Yi−1

i = 5, 6.

(4.6)

where, Y4 (Ξ3 (x, y)) = 0.5tan (2Ξ3 (x, y)) or Y5 (Ξ3 (x, y)) = 0.5cot (2Ξ3 (x, y))

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and, Ξ3 (x, y) = k1 x + k2 y

5

(4.7)

Discussions

Our first interest in present work being in implementing the extended tanh-coth method, Hermite transform and white noise analysis to stress its power in handling nonlinear equations so that one can apply it to models of various types of nonlinearity. The next interest is in the determination of exact travelling wave solutions for modified KP equations. Also, we have presented Riccati equation expansion method and applied it to the modified KP equations. As a result, some new exact travelling wave solutions of the modified KP equation are obtained because of more special solutions of Eq.(2.1). The method which we have proposed in this letter is standard, direct and computerized method, which allow us to do complicated and tedious algebraic calculation. It is shown that the algorithm can be also applied to other NLPDEs in mathematical physics such as KdV-Burgers, Modified KdV-Burgers, Zhiber-Shabat equations (specially: Liouville equation, Sinh-Gordon equation, Dodd-Bullough-Mikhailov equation, Dodd-Bullough-Mikhailov equation and Tzitzeica-DoddBullough equation) and Benjamin-Bona-Mahony equations. Also, we remark that, since the Riccati equation has other solution if select other values of c0 , c1 and c2 , there are many other exact solutions of variable coefficient and wick-type stochastic modified KP equation.

References [1] J. Bourgain, Geom. Funct. Anal. 3 (1993), 315-341. [2] H. Takaoka and N. Tzvetkov, Internat. Math. Res. Notices. (2001), 77-114. [3] M. Wadati, J. Phys. Soc. Jpn. 52 (1983), 2642-2648. [4] M. Wadati, Y. Akutsu, J. Phys. Soc. Jpn. 53 (1984), 3342-3350. [5] H. Holden, B. Øsendal, J. Ubøe and T. Zhang, Stochastic partial differential equations, Birhk¨auser: Basel, 1996. [6] A. de Bouard and A. Debussche, J. Funct. Anal. 154 (1998), 215-251. [7] A. de Bouard and A. Debussche, J. Funct. Anal. 169 (1999), 532-558. [8] A. Debussche and J. Printems, Physica D 134 (1999), 200-226. [9] A. Debussche and J. Printems, J. Comput. Anal. Appl. 3 (2001), 183-206. [10] H. A. Ghany, Chin. J. Phys. 49 (2011), 926-940. [11] H. A. Ghany, Chin. J. Phys. 51 (2013), 875-881. [12] H. A. Ghany and A. Hyder, International Review of Physics 6 (2012), 153-157.

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[13] H. A. Ghany and A. Hyder, J. Comput. Anal. Appl. 15 (2013), 1332-1343. [14] H. A. Ghany and A. Hyder, Kuwait Journal of Science, 41 (2014), 1-14. [15] H. A. Ghany and A. Hyder, Chin. Phys. B. 23 (2014), 060503:1-7. [16] H. A. Ghany and M. S. Mohammed, Chin. J. Phys., 50 (2012), 619-627. [17] J. Printems, J. Differential Equations 153 (1999), 338-373.

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Global Dynamics and Bifurcations of Two Quadratic Fractional Second Order Difference Equations S. Kalabuˇsi´c†1 , M. R. S Kulenovi´c‡2 and M. Mehulji´c§ † Department

of Mathematics University of Sarajevo, Sarajevo, Bosnia and Herzegovina ‡ Department

of Mathematics University of Rhode Island, Kingston, Rhode Island 02881-0816, USA § Division of Mathematics Faculty of Mechanical Engineering, University of Sarajevo, Bosnia and Herzegovina

Abstract.We investigate the bifurcations and the global asymptotic stability of the following two difference equation xn+1

=

βxn xn−1 + γxn−1 , Ax2n + Bxn xn−1

xn+1

=

αx2n + βxn xn−1 + γxn−1 , Ax2n

x0 + x−1 > 0, x0 > 0,

A+B >0 A>0

where all parameters and initial conditions are positive. Keywords. asymptotic stability, attractivity, bifurcation, difference equation, global, local stability, period two; AMS 2000 Mathematics Subject Classification: 39A10, 39A28, 39A30.

1

Introduction and Preliminaries

We investigate global behavior of the equations: xn+1 =

βxn xn−1 + γxn−1 , Ax2n + Bxn xn−1

n = 0, 1, . . .

(1)

αx2n + βxn xn−1 + γxn−1 , n = 0, 1, . . . (2) Ax2n where the parameters α, β, γ, A, B and the initial conditions x−1 , x0 are positive numbers. Equations (1), (2)) are the special cases of equations αx2n + βxn xn−1 + γxn−1 xn+1 = , n = 0, 1, 2, ... (3) Ax2n + Bxn xn−1 + Cxn−1 and Ax2n + Bxn xn−1 + Cx2n−1 + Dxn + Exn−1 + F , n = 0, 1, 2, .... (4) xn+1 = ax2n + bxn xn−1 + cx2n−1 + dxn + exn−1 + f xn+1 =

Some special cases of equation (4) have been considered in the series of papers [3, 4, 12, 13, 20, 22]. Some special second order quadratic fractional difference equations have appeared in analysis of competitive and anti-competitive systems of linear fractional difference equations in the plane, see [5, 8, 7, 9, 18, 19]. Local stability analysis of the equilibrium solutions of equation (3) was performed in [11]. Describing the global dynamics of equation (4) is a formidable task as this equation contains as a special cases many equations with complicated dynamics, such as the linear fractional difference equation xn+1 =

Dxn + Exn−1 + F , dxn + exn−1 + f

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

(5)

The special cases considered so far shows that all kind of dynamics are possible including conservative and non-conservative chaos, Naimark-Sacker bifurcation, period-doubling bifurcation, exchange of stability bifurcation, etc. In this paper we use the theory of monotone maps developed in [16, 17] to describe precisely the basins of attraction of all attractors of this equation as well as bifurcations. Equations (1) and (2) exhibit essentially one period doubling bifurcation with different outcomes. Equation (1) allows the coexistence of the unique minimal period-two solution, which is a saddle point and the equilibrium but only the equilibrium solution and the degenerate period-two solution (0, ∞) and (∞, 0) have substantial basins of attraction. In one region of parameters, Equation (2) also allows the coexistence of the unique minimal periodtwo solution, which is locally asymptotically stable and the equilibrium, but the period-two solution attracts all solutions outside the global stable manifold of the equilibrium. In the complementary region of parameters every solution is either attracted to the equilibrium or to the degenerate period-two solution (1, ∞) and (∞, 1). 1 Partially

supported by FMON Grant No. 05-39-3632–1/14 author, e-mail: [email protected]

2 Corresponding

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Our results will be based on the following theorem for a general second order difference equation xn+1 = f (xn , xn−1 ) ,

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

(6)

see [2]. 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 x−1 , x0 ∈ I, n = 0, 1, 2, ...

xn+1 = f (xn , xn−1 ) ,

(7)

∞ 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.

The consequence of Theorem 1 is that every bounded solution of (7) converges to either equilibrium or period-two solution or to the point on the boundary, and most important question becomes determining the basins of attraction of these solutions as well as the unbounded solutions. The answer to this question follows from an application of theory of monotone maps in the plane which will be presented for the sake of completeness. We now give some basic notions about monotone maps in the plane. Consider a partial ordering  on R2 . Two points x, y ∈ R2 are said to be related if x  y or x  y. 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 # " + − , (8) sign (JT (x)) = − + provided that R is open and convex. For x ∈ R2 , define Q` (x) for ` = 1, . . . , 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction, 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 [17, 16], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by H. L. Smith in [21, 22]. 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. 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 ∈ ∆.

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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 [22] 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 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} ,

(9)

such that the following statements are true. (i) W− is invariant, and dist(T n (x), Q2 (x)) → 0 as n → ∞ for every x ∈ W− . (ii) W+ is invariant, and dist(T n (x), Q4 (x)) → 0 as n → ∞ for every x ∈ W+ . (B) If, in addition to the hypotheses of part (A), x is an interior point of R and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q1 (x) ∪ Q3 (x) except for x, and the following statements are true. (iii) For every x ∈ W− there exists n0 ∈ N such that T n (x) ∈ int Q2 (x) for n ≥ n0 . (iv) For every x ∈ W+ there exists n0 ∈ N such that T n (x) ∈ int Q4 (x) for n ≥ n0 . If T is a map on a set R and if x is a fixed point of T , the stable set W s (x) of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set



x ∈ R : there exists {xn }0n=−∞ ⊂ R s.t. T (xn ) = xn+1 , x0 = x, and

 lim xn = x

n→−∞

W s (x)

When T is non-invertible, the set may not be connected and made up of infinitely many curves, or W u (x) may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on R, the sets W s (x) and W u (x) are the stable and unstable manifolds of x. Theorem 5 In addition to the hypotheses of part (B) of Theorem 4, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis. If the curve C of Theorem 2 has endpoints in ∂R, then C is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in R that is tangential to E µ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W u (x) in R are fixed points of T. Remark 1 We say that f (u, v) is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative D1 f negative and first partial derivative D2 f positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of equation (7) follows from the fact that if f is strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to equation (7) is a strictly competitive map on I × I, see [17]. Set xn−1 = un and xn = vn in Eq.(7) 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 [17]. Remark 2 The characteristic equation of Eq.(7) at an equilibrium point (¯ x, x ¯): λ2 − D1 f (¯ x, x ¯)λ − D2 f (¯ x, x ¯) = 0,

(10)

has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ, whenever f is strictly decreasing 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 global attractivity results for Eq. (7). Some of these results give the sufficient conditions for all solutions to approach a unique equilibrium and they were used efficiently in [14]. The next result is from [6]. See also [1]. Theorem 6 Consider Eq. (7) 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 exactly 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.

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4

Equation xn+1 =

βx2n +γxn−1 Ax2n +Bxn xn−1

In this section we present the global dynamics of Eq. (11).

2.1

Local stability analysis

By substitution xn =

β y , A n

this equation is reduced to the equation γA y β 2 n−1 , B y y A n n−1

yn yn−1 + yn+1 =

2 + yn

n = 0, 1, ...

Thus we consider the following equation xn xn−1 + γxn−1 , n = 0, 1, ... x2n + Bxn xn−1

xn+1 =

(11)

−

Equation (11) has the unique positive equilibrium x given by √ − 1+ 1+4γ(1+B) x= . 2(1+B) −

The partial derivatives associated to the Eq(11) at equilibrium x are √ 2 −2(1+2(1+B)(2+B)γ 1+4(1+B)γ) y−2γxy−Bγy 2 √ fx0 = −x (x , − = 2 +Bxy)2 2 (1+B)(1+

x

1+4(1+B)γ)

Characteristic equation associated to the Eq.(11) at equilibrium is √ 2(1+2(1+B)(2+B)γ 1+4(1+B)γ) √ λ− λ2 + 2 (1+B)(1+

1+4(1+B)γ)

fy0 =



x+γ (x+By)2

=

− x

1 . 1+B

1 = 0. 1+B

By applying the linearized stability Theorem [14, 15] we obtain the following result. √

−

1+

Theorem 7 The unique positive equilibrium point x =

1+4γ(1+B) 2(1+B)

of equation (11) is:

i) locally asymptotically stable when B > 4γ + 1; ii) a saddle point when B < 4γ + 1; ii) a nonhyperbolic point (with eigenvalues λ1 = −1 and λ2 =

1 ) 2+4γ

when B = 4γ + 1.

Lemma 1 If B > 1 + 4γ then Eq.(11) possesses a unique minimal period-two solution {P (φ, ψ) , Q (ψ, φ)} where φ=

1 − 2

√ B−1−4γ √ 2 B−1

and ψ =

1 + 2

√

B−1−4γ √ . 2 B−1

The minimal period-two solution {P (φ, ψ) , Q (ψ, φ)} is a saddle point. Proof. Periodic solution φ, ψ, φ, ψ, . . . is the positive solution of the following system  (B − 1)y − γ = 0 −xy + y = 0.

(12)

where φ + ψ = x and φψ = y. We have that solution of system (12) is x = 1 and y =

γ . B−1

Since

B − 1 − 4γ >0 B−1 if and only if B > 1 + 4γ, we have a unique minimal period-two solution {P (φ, ψ) , Q (ψ, φ)} where x2 − 4y =

φ=

1 − 2

√

B−1−4γ √ 2 B−1

and ψ =

1 + 2

√ B−1−4γ √ . 2 B−1

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

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and write equation (11) in the equivalent form un+1

=

vn+1

=

vn un vn + γun , n = 0, 1, ... . 2 + Bu v vn n n

Let T be the function on (0, ∞) × (0, ∞) defined by   u T = v

!

v

.

uv+γu v 2 +Buv

By a straightforward calculation we find that T2



u v



where g(u, v) =

uv + γu , v 2 + Buv

We have

 JT 2

h(u, v) = φ ψ



 =

g(u, v) h(u, v)

v 2 (Bu + v)(v 2 γ + u(v + γ + Bvγ)) . u(v + γ)(Bv 3 + u(v + B 2 v 2 + γ)) gv0 (φ, ψ) h0v (φ, ψ)

0 (φ, ψ) gu h0u (φ, ψ)

 =



 ,

where

h0u = − h0v

0 gu

=

v+γ , (Bu+v)2

gv0

=

−

u(v 2 +Bγu+2γv) , v 2 (Bu+v)2

v 3 (Bγv 5 +2uγv 2 (v+γ+B 2 v 2 )+u2 (v 2 +v(2+Bv(2+B 2 v))γ+(1+2Bv)γ 2 )) , u2 (v+γ)(Bv 3 +u(v+B 2 v 2 +γ))2

v(B 4 u3 v 3 γ 2 +B 3 u2 v 4 γ(v+4γ)+B(v+2γ)(v 6 γ+4u2 v 2 γ(v+γ)+u3 (v+γ)2 ) u2 (v+γ)(Bv 3 +u(v+B 2 v 2 +γ))2

=

+

B 2 uvγ(u2 (v+γ)(v+3γ)+v 4 (2v+5γ))+uv(v+γ)(u(v+γ)(2v+3γ)+v 2 γ(3v+5γ))) . u2 (v+γ)(Bv 3 +u(v+B 2 v 2 +γ))2

Set 0 S = gu (φ, ψ) + h0v (φ, ψ),

0 D = gu (φ, ψ)h0v (φ, ψ) − gv0 (φ, ψ)h0u (φ, ψ).

After some lengthy calculation one can see that S=

1 + 6γ + B(−3 − 6γ + B(2 + γ)) (B − 1)(B + (B − 1)γ)

D=

and

γ . (B − 1)(B + (B − 1)γ)

We have that |S| > |1 + D|

if and only if

B > 1 + 4γ.

By applying the linearized stability Theorem we obtain that a unique prime period-two solution {P (φ, ψ) , Q (ψ, φ)} of Eq.(11) is a saddle point if and only if B > 1 + 4γ .

2.2

Global results and basins of attraction

In this section we present global dynamics results for equation (11). Theorem 8 If B > 4γ + 1 then equation (11) has a unique equilibrium point E(x, x) which is locally asymptotically stable and there exists the minimal period-two solution {P (φ, ψ), Q(ψ, φ)}, where φ=

1 − 2

√ B−1−4γ √ 2 B−1

and ψ =

1 + 2

√ B−1−4γ √ 2 B−1

which is a saddle point. Furthermore, the global stable manifold of the periodic solution {P, Q} is given by W s ({P, Q}) = W s (P ) ∪ W s (Q) where W s (P ) and W s (Q) are continuous increasing curves, that divide the first quadrant into two connected components, namely W1+ := {x ∈ R \ W s (P ) : ∃y ∈ W s (P ) with y se x}, W1− := {x ∈ R \ W s (P ) : ∃y ∈ W s (P ) with x se y}, and W2+ := {x ∈ R \ W s (Q) : ∃y ∈ W s (Q) with y se x}, W2− := {x ∈ R \ W s (Q) : ∃y ∈ W s (Q) with x se y} respectively such that the following statements are true. i) If (u0 , v0 ) ∈ W s (P ) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to Q. ii) If (u0 , v0 ) ∈ W s (Q) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to Q and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P .

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iii) If (u0 , v0 ) ∈ W1− (the region above W s (P )) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (0, ∞) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (∞, 0). iv) If (u0 , v0 ) ∈ W2+ (the region below W s (Q)) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (∞, 0) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (0, ∞). v) If (u0 , v0 ) ∈ W1+ ∩ W2− (the region between W s (P ) and W s (Q)) then the sequence {(un , vn )} is attracted to E(x, x). Proof. From Theorem 7 Eq.(11) has a unique equilibrium point E(x, x), which is locally asymptotically stable. Theorem 1 implies that the periodic solution {P, Q} is a saddle point. The map T 2 (u, v) = T (T (u, v)) is competitive on R = R2 \ {(0, 0)} and strongly competitive on int(R). It follows from the Perron-Frobenius Theorem and a change of variables that at each point the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively, see [16, 17]. Also, as is well known [16, 17] if the map is strongly competitive then no eigenvector is aligned with a coordinate axis. i) By Theorem 4 we have that if (u0 , v0 ) ∈ W s (P ) then (u2n , v2n ) = T 2n (u0 , v0 ) → P as n → ∞, which implies that (u2n+1 , v2n+1 ) = T (T 2n (u0 , v0 )) → T (P ) = Q as n → ∞, which implies the statement i). ii) The proof of the statement ii) is similar to the proof of the statement i) and will be ommitted. iii) A straightforward calculation shows that (φ, ψ) se (x, x) se (ψ, φ). Since Eq.(11) has no the other equilibrium point or the other minimal-period two solution from Theorem 6 we have if (x−1 , x0 ) ∈ W1− , then (u2n , v2n ) = T 2n ((u0 , v0 )) → (0, ∞) and (u2n+1 , v2n+1 ) = T 2n+1 ((u0 , v0 )) → (∞, 0). and hence if (x−1 , x0 ) ∈ W1− , then lim x2n = ∞ and lim x2n+1 = 0.

n→∞

iv) If (x−1 , x0 ) ∈

W2+ ,

n→∞

then

(u2n , v2n ) = T 2n ((u0 , v0 )) → (∞, 0) and (u2n+1 , v2n+1 ) = T 2n+1 ((u0 , v0 )) → (0, ∞). and hence if (x−1 , x0 ) ∈ W2+ , then lim x2n = 0 and lim x2n+1 = ∞.

n→∞

n→∞

v) If (x−1 , x0 ) ∈ W1+ ∩ W2− , then

√

lim xn =

1+

n→∞

1+4γ(1+B) . 2(1+B)

Theorem 9 If B < 4γ + 1 then equation (11) has a unique equilibrium point E(x, x) which is a saddle point. The global stable manifold W s (E) which is a continuous increasing curve divides the first quadrant such that the following holds: i) Every initial point (u0 , v0 ) in W s (E) is attracted to E. ii) If (u0 , v0 ) ∈ W + (E) (the region below W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (∞, 0) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (0, ∞). iii) If (u0 , v0 ) ∈ W − (E) (the region above W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (0, ∞) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (∞, 0). Proof. From Theorem 7 equation (11) has a unique equilibrium point E(x, x), which is a saddle point. The map T has −

−

no fixed points or periodic points of minimal period-two in ∆ = R ∩ int(Q1 ( x) ∪ Q3 ( x)). It is immediate to see that detJT (E) < 0 and T (x) = x only for x = x. Since the map T is anti-competitive, see [10] and T 2 is strongly competitive we have that all conditions of Theorem 10 in [10] are satisfied from which the proof follows. Theorem 10 If B = 4γ + 1 then Eq.(11) has a unique equilibrium point E(x, x) = ( 12 , 12 ) which is a nonhyperbolic point. There exists a continuous increasing curve CE which is a subset of the basin of attraction of E and it divides the first quadrant such that the following holds: i) Every initial point (u0 , v0 ) in CE is attracted to E. ii) If (u0 , v0 ) ∈ W − (E) (the region above CE ) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (0, ∞) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (∞, 0). iii) If (u0 , v0 ) ∈ W + (E) (the region below CE ) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (∞, 0) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (0, ∞).

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Proof. From Theorem 7 equation (11) has a unique equilibrium point E(x, x) = ( 12 , 21 ), which is nonhyperbolic. All conditions of Theorem 4 are satisfied, which yields the existence of a continuous increasing curve CE which is a subset of −

the basin of attraction of E and for every x ∈ W − (E) there exists n0 ∈ N such that T n (x) ∈ intQ2 ( x) for n ≥ n0 and −

for every x ∈ W + (E) there exists n0 ∈ N such that T n (x) ∈ intQ4 ( x) for n ≥ n0 . Set p 1 − (4γ + 1)t + (1 − (4γ + 1)t)2 + 4γ . U (t) = 2 It is easy to see that (t, U (t)) se E if t < x and E se (t, U (t)) if t > x. One can show that 2

T (t, U (t)) =

2γ(t + γ) p t, t(−t + t2 + 2γ + 4t2 γ + 8γ 2 + t 4γ + (−1 + t + 4tγ)2 )

! .

Now we have that T 2 (t, U (t)) se (t, U (t)) if t < x and By monotonicity if t < x we obtain that (∞, 0) as n → ∞.

(t, U (t)) se T 2 (t, U (t)) if t > x. → (0, ∞) as n → ∞ and if t > x then we have that T 2n (t, U (t)) →

T 2n (t, U (t))

−

If (u0 , v 0 ) ∈ intQ2 ( x) then there exists t1 such that (u0 , v 0 ) se (t1 , U (t1 )) se E. By monotonicity of the map we obtain that T 2n (u0 , v 0 ) se T 2n (t1 , U (t1 )) se E which implies that T 2n (u0 , v 0 ) → (0, ∞) and T 2n+1 (u0 , v 0 ) → T (0, ∞) = (∞, 0) as n → ∞ which proves the statement ii).

T2

−

If (u00 , v 00 ) ∈ intQ4 ( x) then there exists t2 such that E se (t2 , U (t2 )) se (u00 , v 00 ). By monotonicity of the map T 2 we obtain that E se T 2n (t2 , U (t2 )) se T 2n (u00 , v 00 ) which implies that T 2n (u00 , v 00 ) → (∞, 0) and T 2n+1 (u00 , v 00 ) → T (∞, 0) = (0, ∞) as n → ∞ which proves the statement iii). This completes the proof of Theorem. Remark 3 Theorems 8, 9 and 10 show new type of period doubling bifurcation. When B ≤ 4γ + 1 all solutions outside the global stable manifold are asymptotic to (0, ∞) or to (∞, 0), and when B > 4γ + 1 all solutions are either asymptotic to (0, ∞) or to (∞, 0) or to the minimal period-two solution {P, Q} or a unique equilibrium E. In the second case each attractor has a substantial basin of attraction.

Figure 1: Visual illustration of Theorems 8, 9 and 10 . Figures are generated by Dynamica 3, [15].

3

Equation xn+1 =

αx2n +βxn xn−1 +γxn−1 Ax2n

In this section we present the global dynamics and bifurcation analysis of Equation (13).

3.1

Local stability analysis

This equation is reduced to the equation xn+1 =

x2n + βxn xn−1 + γxn−1 , n = 0, 1, ... x2n

(13)

Equation (13) has the unique positive equilibrium x ¯ given by √ − 1+β+ (1+β)2 +4γ x= . 2

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The partial derivatives associated to equation (13) at equilibrium x are √ 2(4γ+β(1+β+ (1+β)2 +4γ)) √ , fy0 = βx+γ fx0 = −xyβ−2γy − = − 3 2 x

x

x

(1+β)2 +4γ)2

(1+β+

√ =

2(2γ+β(1+β+

− x

√

(1+β+

(1+β)2 +4γ))

(1+β)2 +4γ)2

.

Characteristic equation associated to the Eq.(13) at equilibrium is √ √ 2(4γ+β(1+β+ (1+β)2 +4γ)) 2(2γ+β(1+β+ (1+β)2 +4γ)) √ √ λ2 + λ− = 0. 2 2 2 2 (1+β+

(1+β) +4γ)

(1+β+

(1+β) +4γ)

By applying the linearized stability Theorem we obtain the following result. √

−

Theorem 11 The unique positive equilibrium point x = i) locally asymptotically stable when 4γ + 2β +

β2

1+β+

(1+β)2 +4γ 2

of equation (13) is

< 3;

ii) a saddle point when 4γ + 2β + β 2 > 3; β+1 ) β+3

iii) a nonhyperbolic point (with eigenvalues λ1 = −1 and λ2 =

when 4γ + 2β + β 2 = 3.

Lemma 2 Equation (13) has the minimal period-two solution {P (φ, ψ) , Q (ψ, φ)} where √ √ −γ+βγ+γ −3+2β+β 2 +4γ −γ+βγ−γ −3+2β+β 2 +4γ and ψ = φ= 2(−1+β+γ) 2(−1+β+γ) if and only if 3 − 2β − β 2 < γ < 1 − β. 4 The minimal period-two solution {P (φ, ψ) , Q (ψ, φ)} is locally asymptotically stable. β < 1 and

Proof. Period-two solution is a positive solution of the following systems  x−y−γ =0 x2 − xy + (β − 1)y = 0.

(14)

where φ + ψ = x and φψ = y. We have that only one solution of system (14) is x=

(β − 1)γ , β+γ−1

Since x2 − 4y =

y=

−γ 2 , β+γ−1

γ 2 (−3 + 2β + β 2 + 4γ) >0 (β + γ − 1)2

if and only if 3 − 2β − β 2 0 if and only if β < 1 and γ < 1 − β, we have that φ and ψ are solution of the equation (β − 1)γ −γ 2 t+ =0 β+γ−1 β+γ−1

t2 − if and only if

β < 1 and

3 − 2β − β 2 < γ < 1 − β. 4

The second iterate of the map T is T2



u v



 =

g(u, v) h(u, v)



where v 2 + βuv + γu g(u, v) = , v2 We have

 v 4 1 + v(β + γ) + h(u, v) = 

JT 2

u(2+vβ)(vβ+γ) v2 (v 2 + βuv + γu)2

φ ψ



 =

0 (φ, ψ) gu h0u (φ, ψ)

gv0 (φ, ψ) h0v (φ, ψ)

+

u2 (vβ+γ)2 v4

 .



where

h0u = −

0 gu

=

gv0

=

vβ+γ , v2 u(vβ+2γ) − , v3

v 3 (vβ+γ)(uvβ 2 +uβγ+v 2 (β+2γ)) , (v2 +βuv+γu)3

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h0v =

v 2 (5u2 vβ 2 γ+3u2 βγ 2 +v 4 (β+γ)+3uv 3 β(β+γ)+uv 2 (2uβ 3 +γ(uβ+5γ))) . (v2 +βuv+γu)3

Set 0 (φ, ψ) + h0v (φ, ψ) and S = gu After some lengthy calculation one can see that

S=

0 (φ, ψ)h0v (φ, ψ) − gv0 (φ, ψ)h0u (φ, ψ). D = gu

4+β(−6+β+β 2 )−9γ+β(7+β)γ+6γ 2 , γ2

D=

(−1+γ)(−1+β+γ) . γ2

Applying the linearized stability Theorem we obtain that a unique prime period-two solution {P (φ, ψ) , Q (ψ, φ)} of Eq(13) is locally asymptotically stable when 3 − 2β − β 2 β < 1 and < γ < 1 − β. 4

3.2

Global results and basins of attraction

In this section we present global dynamics results for Eq.(13). Theorem 12 If 4γ + 2β + β 2 < 3 then Eq. (13) has a unique equilibrium point E(x, x) which is globally asymptotically stable. Proof. From Theorem 11 equation (13) has a unique equilibrium point E(x, x), which is locally asymptotically stable. Every solution of equation (13) is bounded from above and from below by positive constants. If 4γ + 2β + β 2 < 3 then β + γ < 1 and we have x2 + βxn xn−1 + γxn−1 ≥1 xn+1 = n x2n and βxn−1 γxn−1 xn+1 = 1 + + ≤ 1 + βxn−1 + γxn−1 = 1 + (β + γ)xn−1 . xn x2n x2n ≤ 1 + (β + γ)[1 + (β + γ)x2n−4 ] ≤ ... ≤ < x2n−1

≤ ≤
0 and n ≥ N and so every solution is bounded. Equation (13) has no other equilibrium points or period two points and using Theorem 1 we have that equilibrium point E(x, x) is globally asymptotically stable.

Theorem 13 If 4γ + 2β + β 2 > 3 and β + γ < 1 then equation (13) has a unique equilibrium point E(x, x) which is a saddle point and the minimal period-two solution {P (φ, ψ), Q(ψ, φ)} which is locally asymptotically stable, where √ √ −γ+βγ−γ −3+2β+β 2 +4γ −γ+βγ+γ −3+2β+β 2 +4γ φ= , ψ= . 2(−1+β+γ) 2(−1+β+γ) The global stable manifold W s (E) which is a continuous increasing curve, divides the first quadrant such that the following holds: i) Every initial point (u0 , v0 ) in W s (E) is attracted to E. ii) If (u0 , v0 ) ∈ W + (E) (the region below W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to Q and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to P . iii) If (u0 , v0 ) ∈ W − (E) (the region above W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} is attracted to P and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} is attracted to Q. Proof. From Theorem 11 equation (13) has a unique equilibrium point E(x, x), which is a saddle point. The map T has −

−

no fixed points or periodic points of minimal period-two in ∆ = R ∩ int(Q1 ( x) ∪ Q3 ( x)). A straightforward calculation shows that detJT (E) < 0 and T (x) = x only for x = x. Since the map T is anti-competitive and T 2 is strongly competitive we have that all conditions of Theorem 10 in [10] are satisfied from which the proof follows. Theorem 14 If 4γ + 2β + β 2 > 3 and β + γ ≥ 1 then Eq. (13) has a unique equilibrium point E(x, x ) which is a saddle point. The global stable manifold W s (E), which is a continuous increasing curve divides the first quadrant such that the following holds:

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i) Every initial point (u0 , v0 ) in W s (E) is attracted to E. ii) If (u0 , v0 ) ∈ W + (E) (the region below W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (∞, 1) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (1, ∞). iii) If (u0 , v0 ) ∈ W − (E) (the region above W s (E)) then the subsequence of even-indexed terms {(u2n , v2n )} tends to (1, ∞) and the subsequence of odd-indexed terms {(u2n+1 , v2n+1 )} tends to (∞, 1). Proof. The proof is similar to the proof of the previous theorem using the fact that every solution of equation (13) is bounded from below by 1.

Figure 2: Visual illustration of Theorems 12, 13, 14 and 15 . Figures are generated by Dynamica 3, [15]. Theorem 15 If 4γ + 2β + β 2 = 3 then Eq. (13) has a unique equilibrium point E(x, x ) which is a nonhyperbolic point and a global attractor. Proof. From Theorem 11 Eq.(13) has a unique equilibrium point E(x, x), which is non-hyperbolic. All conditions of Theorem 4 are satisfied, which yields the existence a continuous increasing curve CE which is a subset of the basin of −

attraction of E and for every x ∈ W − (E) (the region above CE ) there exists n0 ∈ N such that T n (x) ∈ intQ2 ( x) for n ≥ n0 and for every x ∈ Set

W + (E)

(the region below CE ) there exists n0 ∈ N such that p βt + β 2 t2 + (3 − 2β − β 2 )(t2 − t) U (t) = . 2(t − 1)

141

T n (x)

−

∈ intQ4 ( x) for n ≥ n0 .

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It is easy to see that (t, U (t)) se E if t < x and E se (t, U (t)) if t > x. One can show that   tβ+(3+(−2+4t−β)β) (3+(−2+4t−β)β)s (tβ + s)4 (8t2 + + t−1 t−1 2 , T (t, U (t)) = t, 8t4 (−3 + t(3 + (−2 + 4t − β)β) + β(2 + β + 2s))2 where s=

p t(t(3 − 2β) + (β − 1)(β + 3).

Now we have that T 2 (t, U (t)) se (t, U (t)) if t > x and (t, U (t)) se T 2 (t, U (t)) if t < x since (tβ+s)4 (8t2 +

tβ+(3+(−2+4t−β)β) (3+(−2+4t−β)β)s + t−1 t−1

8t4 (−3+t(3+(−2+4t−β)β)+β(2+β+2s))2

if and only if t > x. By monotonicity if t < x then we obtain that that T 2n (t, U (t)) → E as n → ∞.

√ −

βt+

β 2 t2 +(3−2β−β 2 )(t2 −t) 2(t−1)

T 2n (t, U (t))

> 0,

→ E as n → ∞ and if t > x then we have

−

If (u0 , v 0 ) ∈ intQ2 ( x) then there exists t1 such that (t1 , U (t1 )) se (u0 , v 0 ) se E. By monotonicity of the map T 2 we obtain that T 2n (t1 , U (t1 )) se T 2n (u0 , v 0 ) se E which implies that T 2n (u0 , v 0 ) → E and T 2n+1 (u0 , v 0 ) → T (E) = E, as n → ∞ which proves the statement ii). −

If (u00 , v 00 ) ∈ intQ4 ( x) then there exists t2 such that E se (u00 , v 00 ) se (t2 , U (t2 )). By monotonicity of the map we obtain that E se T 2n (u00 , v 00 ) se T 2n (t2 , U (t2 )) which implies that T 2n (u00 , v 00 ) → E and T 2n+1 (u00 , v 00 ) → T (E) = E as n → ∞ which proves the statement iii), which completes the proof of the Theorem.

T2

Remark 4 Theorems 12, 13, 14 and 15 show another type of period doubling bifurcation. When 4γ + 2β + β 2 ≤ 3 all solutions are asymptotic to the unique equilibrium E. When 4γ + 2β + β 2 > 3 and β + γ < 1 all solutions which starts off the global stable manifold of the unique equilibrium E are asymptotic to the unique minimal period-two solution {P, Q}. Finally, when 4γ + 2β + β 2 > 3 and β + γ ≥ 1 all solutions which starts off the global stable manifold of the unique equilibrium E are asymptotic to (1, ∞) or to (∞, 1).

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), 863- 885. [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), 191-208. [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), 1045- 1058. [5] E. Drymonis and G. Ladas, On the global character of the rational system xn+1 = α2 +β2 xn , A2 +B2 xn +C2 yn

α1 A1 +B1 xn +yn

and yn+1 =

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 and M. Nurkanovi´ c, Dynamics of an anti-competitive two dimensional rational system of difference equations, Sarajevo J. Math., 7 (19) (2011), 39-56. [8] 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. [9] E. A. Grove, D. Hadley, E. Lapierre and S. W. Schultz, On the global behavior of the rational system xn+1 = and yn+1 =

α2 +β2 xn +yn , yn

α1 xn +yn

Sarajevo J. Math., Vol. 8 (21) (2012), 283-292.

[10] S. Kalabuˇsi´ c, M. R. S. Kulenovi´ c and E. Pilav, Global Dynamics of Anti-Competitive Systems in the Plane, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20(2013), 477–505. [11] S. Kalabuˇsi´ c, M. R. S. Kulenovi´ c and M. Mehulji´ c, Global Period-doubling Bifurcation of Quadratic Fractional Second Order Difference Equation, Discrete Dyn. Nat. Soc., (2014), 13p. [12] C. M. Kent and H. Sedaghat, Global attractivity in a quadratic-linear rational difference equation with delay. J. Difference Equ. Appl. 15 (2009), 913–925. [13] C. M. Kent and H. Sedaghat, Global attractivity in a rational delay difference equation with quadratic terms, J. Difference Equ. Appl., 17 (2011), 457–466.

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[14] 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. [15] M. R. S. Kulenovi´ c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman& Hall/CRC Press, Boca Raton, 2002. [16] 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. [17] 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. [18] G. Ladas, G. Lugo and F. J. Palladino, Open problems and conjectures on rational systems in three dimensions,Sarajevo J.Math., Vol. 8 (21) (2012), 311-321. [19] S. Moranjki´ c and Z. Nurkanovi´ c, Basins of attractionof certain rational anti-competitive system of difference equations in theplane, Advances in Difference Equations, 2012, 2012:153. [20] H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms. J. Difference Equ. Appl. 15 (2009), 215–224. [21] H. L. Smith, Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Differential Equations64 (1986), 163-194. [22] H. L. Smith, Planar competitive and cooperative difference equations. J. Differ. Equations Appl. 3 (1998), 335- 357.

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Some New Inequalities of Hermite–Hadamard Type for Geometrically Mean Convex Functions on the Co-ordinates Xu-Yang Guo1 1

Feng Qi2,∗

Bo-Yan Xi1

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China

2

Department of Mathematics, College of Science, Tianjin Polytechnic University, 300160, China ∗

Corresponding author. E-mail: [email protected], [email protected]

Received on January 17, 2015; accepted on August 20, 2015

Abstract In the paper, the authors introduce a new concept geometrically mean convex function on co-ordinates and establish some new integral inequalities of Hermite–Hadamard type for geometrically mean convex functions of two variables on the co-ordinates. 2010 Mathematics Subject Classification: 26A51, 26D15, 26D20, 26E60, 41A55. Key words and phrases: geometrically mean convex functions; integral inequality of Hermite– Hadamard type; H¨ older inequality.

1

Introduction

The following definitions are well known in the literature. Definition 1.1 ([3, 4]). A function f : ∆ = [a, b] × [c, d] ⊆ R2 → R is said to be convex on the co-ordinates on ∆ with a < b and c < d if the partial mappings fy : [a, b] → R,

fy (u) = f (u, y)

and fx : [c, d] → R,

fx (v) = f (x, v)

are convex for all x ∈ (a, b) and y ∈ (c, d). Definition 1.2 ([3, 4]). A function f : ∆ = [a, b]×[c, d] → R is said to be convex on the co-ordinates on ∆ with a < b and c < d if f (tx + (1 − t)z, λy + (1 − λ)w) ≤ tλf (x, y) + t(1 − λ)f (x, w) + (1 − t)λf (z, y) + (1 − t)(1 − λ)f (z, w) holds for all t, λ ∈ [0, 1], (x, y), (z, w) ∈ ∆. Definition 1.3 ([1]). A function f : ∆ = [a, b] × [c, d] ⊆ R2 → R+ is called co-ordinated log-convex on ∆ with a < b and c < d for all t, λ ∈ [0, 1] and (x, y), (z, w) ∈ ∆, if f (tx + (1 − t)z, λy + (1 − λ)w) ≤ [f (x, y)]tλ [f (x, w)]t(1−λ) [f (z, y)](1−t)λ [f (z, w)](1−t)(1−λ) .

1

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In recent years, the following integral inequalities of Hermite–Hadamard type for the above kinds of convex functions were published. Theorem 1.1 ([3, Theorem 2.2] and [4, Theorem 2.2]). Let f : ∆ = [a, b] × [c, d] → R be convex on the co-ordinates on ∆ with a < b and c < d. Then 

    Z b  Z d  c+d a+b 1 1 1 f x, f ≤ dx + ,y dy 2 b−a a 2 d−c c 2 Z dZ b 1 f (x, y) d x d y ≤ (b − a)(d − c) c a    Z b Z d Z b Z d 1 1 1 ≤ f (x, d) d x + f (b, y) d y f (x, c) d x + f (a, y) d y + 4 b−a a d−c c a c  1 ≤ f (a, c) + f (b, c) + f (a, d) + f (b, d) . 4

a+b c+d f , 2 2



Theorem 1.2 ([7, Theorem 2.3]). Let f : ∆ = [a, b] × [c, d] → R be a partial differentiable function ∂2f is convex on the co-ordinates on ∆, then on ∆. If ∂x∂y            1 c+d c+d a+b c+d a+b a+b f a, + f b, +4f , +f , c +f , d 9 2 2 2 2 2 2 Z dZ b  1 1 + f (a, c) + f (a, d) + f (b, c) + f (b, d) + f (x, y) d x d y − A 36 (b − a)(d − c) c a   2  2 ∂ f (a, c) ∂ 2 f (a, d) ∂ 2 f (b, c) ∂ 2 f (b, d) 5 , ≤ + + + (b − a)(d − c) 72 ∂x∂y ∂x∂y ∂x∂y ∂x∂y where 1 A= b−a

   Z b  1 c+d f (x, c) + 4f x, +f (x, d) d x 6 2 a    Z d  a+b 1 1 f (a, y) + 4f , y +f (b, y) d y. + d−c c 6 2

Theorem 1.3 ([6, Theorem 2]). Let f : ∆ = [a, b] × [c, d] → R be a partial differentiable function 2 on ∆. If ∂ f is convex on the co-ordinates on ∆, then ∂x∂y

  Z dZ b a+b c+d 1 f (x, y) d x d y + f , −A (b − a)(d − c) c a 2 2 " ∂ 2 f (a,c) ∂ 2 f (a,d) ∂ 2 f (b,c) ∂ 2 f (b,d) # (b − a)(d − c) ∂x∂y + ∂x∂y + ∂x∂y + ∂x∂y ≤ , 16 4 where 1 A= b−a

Z a

b

  Z d  c+d 1 a+b f x, dx + f , y d y. 2 d−c c 2 

2

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Theorem 1.4 ([1, Corollary 3.1]). Suppose that f : ∆ = [a, b] × [c, d] → R+ is log-convex on the co-ordinates on ∆. Then   Z dZ b a+b c+d 1 ln f (x, y) d x d y ln f , ≤ 2 2 (b − a)(d − c) c a ln f (a, c) + ln f (b, c) + ln f (a, d) + ln f (b, d) . ≤ 4 In the papers [2, 5, 8, 9, 10, 11], there are also some new results on this topic.

2

A definition and lemmas

In this section, we introduce the notion “geometrically mean convex function” and establish an integral identity. Definition 2.1. A function f : ∆ = [a, b] × [c, d] ⊆ R2+ → R+ is said to be geometrically mean convex on the co-ordinates on ∆ with a < b and c < d, if
f xt z 1−t , y λ w1−λ ≤ [f (x, y)][t+λ]/4 [f (x, w)][t+(1−λ)]/4 [f (z, y)][(1−t)+λ]/4 [f (z, w)][(1−t)+(1−λ)]/4 holds for all t, λ ∈ [0, 1] and (x, y), (z, w) ∈ ∆. In order to prove our main results, we need the following integral identity. Lemma 2.1. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R have partial derivatives of the second order with a < b and c < d. If on ∆, then

∂2f ∂x∂y

∈ L1 (∆), where L1 (∆) denotes the set of all Lebesgue integrable functions

 4 f (a, c) + f (b, c) + f (a, d) + f (b, d) S(f ) , (ln b − ln a)(ln d − ln c) 4  Z dZ b 1 f (x, y) −A+ dxdy (ln b − ln a)(ln d − ln c) c a xy Z 1Z 1 2
∂ = tλat b1−t cλ d1−λ f at b1−t , cλ d1−λ d t d λ ∂x∂y 0 0 Z 1Z 1
∂2 − f at b1−t , c1−λ dλ d t d λ tλat b1−t c1−λ dλ ∂x∂y 0 0 Z 1Z 1
∂2 − f a1−t bt , cλ d1−λ d t d λ tλa1−t bt cλ d1−λ ∂x∂y 0 0 Z 1Z 1
∂2 + tλa1−t bt c1−λ dλ f a1−t bt , c1−λ dλ d t d λ, ∂x∂y 0 0 where A=

1 2(ln b − ln a)

Z a

b



  Z d f (x, c) f (x, d) 1 f (a, y) f (b, y) + dx + + d y. x x 2(ln d − ln c) c y y

3

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Proof. Let x = at b1−t and y = cλ d1−λ for 0 ≤ t, λ ≤ 1. Using integration by parts, we have Z 1Z 1
∂2 tλat b1−t cλ d1−λ f at b1−t , cλ d1−λ d t d λ ∂x∂y 0 0   Z 1 Z 1
1
1 ∂ ∂ = λcλ d1−λ t f at b1−t , cλ d1−λ − f at b1−t , cλ d1−λ d t d λ ln a − ln b 0 ∂y 0 0 ∂y Z 1  Z 1Z 1

∂ ∂ 1 λcλ d1−λ f at b1−t , cλ d1−λ d t d λ λcλ d1−λ f a, cλ d1−λ d λ − = ln a − ln b 0 ∂y ∂y 0 0  Z 1
1 = f (a, c) − f a, cλ d1−λ d λ (ln b − ln a)(ln d − ln c) 0  Z 1 Z 1Z 1

t 1−t t 1−t λ 1−λ − f a b ,c dt + f a b ,c d dtdλ 0

0

0

 Z b 1 1 f (x, c) f (a, c) − dx = (ln b − ln a)(ln d − ln c) ln b − ln a a x  Z d Z dZ b 1 f (a, y) 1 f (x, y) − dy + dxdy . ln d − ln c c y (ln b − ln a)(ln d − ln c) c a xy Similarly, we have Z 1Z 1


∂2 f at b1−t , c1−λ dλ d t d λ ∂x∂y 0 0  Z b 1 1 f (x, d) =− f (a, d) − dx (ln b − ln a)(ln d − ln c) ln b − ln a a x  Z d Z dZ b 1 1 f (a, y) f (x, y) − dy + dxdy , ln d − ln c c y (ln b − ln a)(ln d − ln c) c a xy Z 1Z 1 2
∂ tλa1−t bt cλ d1−λ f a1−t bt , cλ d1−λ d t d λ ∂x∂y 0 0  Z b 1 1 f (x, c) =− f (b, c) − dx (ln b − ln a)(ln d − ln c) ln b − ln a a x  Z d Z dZ b 1 1 f (b, y) f (x, y) − dy + dxdy , ln d − ln c c y (ln b − ln a)(ln d − ln c) c a xy tλat b1−t c1−λ dλ

and 1

1


∂2 f a1−t bt , c1−λ dλ d t d λ ∂x∂y 0 0  Z b 1 1 f (x, d) = f (b, d) − dx (ln b − ln a)(ln d − ln c) ln b − ln a a x  Z d Z dZ b 1 f (b, y) 1 f (x, y) − dy + dxdy . ln d − ln c c y (ln b − ln a)(ln d − ln c) c a xy Z

Z

tλa1−t bt c1−λ dλ

Lemma 2.1 is proved. 4

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Lemma 2.2. Let u, v > 0. Then 1

Z F (u, v) , 0

   L(u, v) − u , t 1−t tu v d t = 1ln v − ln u   u, 2

u 6= v, (2.1) u = v,

where L(u, v) is logarithmic mean defined by Z L(u, v) ,

1

ut v 1−t d t =

0

v−u , ln v − ln u u,  

u 6= v, u = v.

Proof. The proof is straightforward.

3

Some integral inequalities of Hermite–Hadamard type

In this section, we prove some new inequalities of Hermite–Hadamard type for geometrically mean convex functions. Theorem 3.1. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R be a partial differentiable function on ∆ ∂ 2 f q ∂2f with a < b, c < d and ∂x∂y ∈ L1 (∆). If ∂x∂y is geometrically mean convex functions on the co-ordinates on ∆ for q ≥ 1, then  1−1/q 

1/q |S(f )| ≤ F (a, b)F (c, d) F Mq (a, a), Mq (b, b) F Nq (c, c), Nq (d, d)  1−1/q 

1/q + F (a, b)F (d, c) F Mq (a, a), Mq (b, b) F Nq (d, d), Nq (c, c) (3.1)  1−1/q 

1/q + F (b, a)F (c, d) F Mq (b, b), Mq (a, a) F Nq (c, c), Nq (d, d)  1−1/q 

1/q + F (b, a)F (d, c) F Mq (b, b), Mq (a, a) F Nq (d, d), Nq (c, c) , where F (u, v) is defined by (2.1),   2 ∂ f (u, c) ∂ 2 f (u, d) q/4 , Mq (ur , u) = ur ∂x∂y ∂x∂y

  2 ∂ f (a, v) ∂ 2 f (b, v) q/4 Nq (v r , v) = v r ∂x∂y ∂x∂y

(3.2)

for r ≥ 0. 2 q ∂ f Proof. Since ∂x∂y is geometrically mean convex on coordinates ∆, using Lemma 2.1 and by H¨older’s integral inequality, we have 2 Z 1Z 1
t 1−t λ 1−λ t 1−t λ 1−λ ∂ |S(f )| ≤ tλa b c d dtdλ ∂x∂y f a b , c d 0 0 2 Z 1Z 1 ∂
tλat b1−t c1−λ dλ + f at b1−t , c1−λ dλ d t d λ ∂x∂y 0 0 2 Z 1Z 1
∂ + tλa1−t bt cλ d1−λ f a1−t bt , cλ d1−λ d t d λ ∂x∂y 0 0 2 Z 1Z 1
1−t t 1−λ λ ∂ 1−t t 1−λ λ + tλa b c d f a b ,c d dtdλ ∂x∂y 0 0 5

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1−1/q Z

2 f (a, c) q[t+λ]/4 tλa b c d tλa b c d dtdλ ≤ ∂x∂y 0 0 0 0 2 1/q ∂ f (a, d) q[t+(1−λ)]/4 ∂ 2 f (b, c) q[(1−t)+λ]/4 ∂ 2 f (b, d) q[(1−t)+(1−λ)]/4 d t d λ × ∂x∂y ∂x∂y ∂x∂y 2 q[t+(1−λ)]/4 1−1/q Z 1 Z 1 Z 1 Z 1 t 1−t 1−λ λ ∂ f (a, c) t 1−t 1−λ λ tλa b c d tλa b c d dtdλ + ∂x∂y 0 0 0 0 2 q[t+λ]/4 2 q[(1−t)+(1−λ)]/4 2 q[(1−t)+λ]/4 1/q ∂ f (a, d) ∂ f (b, c) ∂ f (b, d) dtdλ × ∂x∂y ∂x∂y ∂x∂y 2 1−1/q Z 1 Z 1 Z 1 Z 1 ∂ f (a, c) q[(1−t)+λ]/4 tλa1−t bt cλ d1−λ tλa1−t bt cλ d1−λ d t d λ + ∂x∂y 0 0 0 0 2 1/q ∂ f (a, d) q[(1−t)+(1−λ)]/4 ∂ 2 f (b, c) q[t+λ]/4 ∂ 2 f (b, d) q[t+(1−λ)]/4 × dtdλ ∂x∂y ∂x∂y ∂x∂y q[(1−t)+(1−λ)]/4 2 Z 1 Z 1 1−1/q Z 1 Z 1 1−t t 1−λ λ 1−t t 1−λ λ ∂ f (a, c) + tλa b c d dtdλ tλa b c d ∂x∂y 0 0 0 0 2 1/q ∂ f (a, d) q[(1−t)+λ]/4 ∂ 2 f (b, c) q[t+(1−λ)]/4 ∂ 2 f (b, d) q[t+λ]/4 d t d λ . × ∂x∂y ∂x∂y ∂x∂y (3.3) Z

1

Z

1

t 1−t λ 1−λ

Also by Lemma 2.2, we have Z 1Z 0

1

1

Z

t 1−t λ 1−λ ∂

1

tλat b1−t cλ d1−λ d t d λ = F (a, b)F (c, d)

(3.4)

0

and 2 ∂ f (a, c) q[t+λ]/4 ∂ 2 f (a, d) q[t+(1−λ)]/4 tλat b1−t cλ d1−λ ∂x∂y ∂x∂y 0 0 2 ∂ f (b, c) q[(1−t)+λ]/4 ∂ 2 f (b, d) q[(1−t)+(1−λ)]/4 × dtdλ ∂x∂y ∂x∂y Z 1Z 1 = tλ[Mq (a, a)]t [Mq (b, b)]1−t [Nq (c, c)]λ [Nq (d, d)]1−λ d t d λ 0 0

= F Mq (a, a), Mq (b, b) F Nq (c, c), Nq (d, d) . Z

1

Z

1

By simple computation, Z 1Z 1 tλat b1−t c1−λ dλ d t d λ = F (a, b)F (d, c), 0

Z

0 1

0

Z

Z

0

1

Z

1

tλa1−t bt cλ d1−λ d t d λ = F (b, a)F (c, d),

0

1

tλa1−t bt c1−λ dλ d t d λ = F (b, a)F (d, c),

0 1

Z

1

0

2 f (a, c) q[t+(1−λ)]/4 ∂ 2 f (a, d) q[t+λ]/4 ∂ 2 f (b, c) q[(1−t)+(1−λ)]/4 d ∂x∂y ∂x∂y ∂x∂y

t 1−t 1−λ λ ∂

tλa b 0

Z

c

6

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2 ∂ f (b, d) q[(1−t)+λ]/4

d t d λ = F Mq (a, a), Mq (b, b) F Nq (d, d), Nq (c, c) , × ∂x∂y 2 Z 1Z 1 ∂ f (a, c) q[(1−t)+λ]/4 ∂ 2 f (a, d) q[(1−t)+(1−λ)]/4 ∂ 2 f (b, c) q[t+λ]/4 tλa1−t bt cλ d1−λ ∂x∂y ∂x∂y ∂x∂y 0 0 2 ∂ f (b, d) q[t+(1−λ)]/4

× d t d λ = F Mq (b, b), Mq (a, a) F Nq (c, c), Nq (d, d) ∂x∂y and Z

1

Z

1

tλa 0

0

2 f (a, c) q[(1−t)+(1−λ)]/4 ∂ 2 f (a, d) q[(1−t)+λ]/4 ∂ 2 f (b, c) q[t+(1−λ)]/4 bc d ∂x∂y ∂x∂y ∂x∂y 2 ∂ f (b, d) q[t+λ]/4

× d t d λ = F Mq (b, b), Mq (a, a) F Nq (d, d), Nq (c, c) . (3.5) ∂x∂y

1−t t 1−λ λ ∂

Substituting equalities (3.4) to (3.5) into the inequality (3.3) and rearranging yield the inequality (3.1). Theorem 3.1 is proved. Corollary 3.1.1. Under the conditions of Theorem 3.1, when q = 1, we have |S(f )| ≤ F (M1 (a, a), M1 (b, b))F (N1 (c, c), N1 (d, d)) + F (M1 (a, a), M1 (b, b))F (N1 (d, d), N1 (c, c))

+ F M1 (b, b), M1 (a, a) F (N1 (c, c), N1 (d, d)) + F M1 (b, b), M1 (a, a) F (N1 (d, d), N1 (c, c)), where F (u, v) is defined by (2.1), and Mq (ur , u) and Nq (v r , v) are defined by (3.2). Theorem 3.2. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R be a partial differentiable function on ∆ ∂ 2 f q ∂2f with a < b, c < d and ∂x∂y ∈ L1 (∆). If ∂x∂y is geometrically mean convex functions on the co-ordinates on ∆ for q > 1 and q ≥ r ≥ 0, then  1−1/q |S(f )| ≤ F (a(q−r)/(q−1) , b(q−r)/(q−1) )F (c(q−r)/(q−1) , d(q−r)/(q−1) ) 

1/q × F Mq (ar , a), Mq (br , b) F Nq (cr , c), Nq (dr , d)  1−1/q + F (a(q−r)/(q−1) , b(q−r)/(q−1) )F (d(q−r)/(q−1) , c(q−r)/(q−1) ) 

1/q × F Mq (ar , a), Mq (br , b) F Nq (dr , d), Nq (cr , c)  1−1/q + F (b(q−r)/(q−1) , a(q−r)/(q−1) )F (c(q−r)/(q−1) , d(q−r)/(q−1) ) 

1/q × F Mq (br , b), Mq (ar , a) F Nq (cr , c), Nq (dr , d)  1−1/q + F (b(q−r)/(q−1) , a(q−r)/(q−1) )F (d(q−r)/(q−1) , c(q−r)/(q−1) ) 

1/q × F Mq (br , b), Mq (ar , a) F Nq (dr , d), Nq (cr , c) , where F (u, v) is defined by (2.1), and Mq (ur , u) and Nq (v r , v) are defined by (3.2).

7

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Proof. From Lemma 2.1, we have 2 Z 1Z 1
t 1−t λ 1−λ ∂ t 1−t λ 1−λ tλa b c d |S(f )| ≤ ∂x∂y f a b , c d dtdλ 0 0 2 Z 1Z 1 ∂
tλat b1−t c1−λ dλ + f at b1−t , c1−λ dλ d t d λ ∂x∂y 0 0 (3.6) 2 Z 1Z 1
∂ tλa1−t bt cλ d1−λ + f a1−t bt , cλ d1−λ d t d λ ∂x∂y 0 0 2 Z 1Z 1
1−t t 1−λ λ ∂ 1−t t 1−λ λ tλa b c f a b ,c + d d d t d λ. ∂x∂y 0 0 2 q ∂ f Using H¨older’s integral inequality, and by the geometrically mean convexity of ∂x∂y on ∆ and Lemma 2.2, it is easy to observe that 2 Z 1Z 1
t 1−t λ 1−λ ∂ t 1−t λ 1−λ tλa b c d ∂x∂y f a b , c d dtdλ 0 0 Z 1 Z 1 1−1/q
(q−r)/(q−1) ≤ tλ at b1−t cλ d1−λ dtdλ 0

0

2 f (a, c) q[t+λ]/4 ∂ 2 f (a, d) q[t+(1−λ)]/4 × tλa b c d ∂x∂y ∂x∂y 0 0 2 1/q ∂ f (b, c) q[(1−t)+λ]/4 ∂ 2 f (b, d) q[(1−t)+(1−λ)]/4 × dtdλ ∂x∂y ∂x∂y 

1−1/q = F a(q−r)/(q−1) , b(q−r)/(q−1) F c(q−r)/(q−1) , d(q−r)/(q−1) 

1/q × F Mq (ar , a), Mq (br , b) F Nq (cr , c), Nq (dr , d) . Z

1

Z

1

rt r(1−t) rλ r(1−λ) ∂

(3.7)

Similarly, we can show that 2 Z 1Z 1 ∂
f at b1−t , c1−λ dλ d t d λ tλat b1−t c1−λ dλ ∂x∂y 0 0 

1−1/q (q−r)/(q−1) (q−r)/(q−1) ≤ F a ,b F d(q−r)/(q−1) , c(q−r)/(q−1) 

1/q × F Mq (ar , a), Mq (br , b) F Nq (dr , d), Nq (cr , c) , 2 Z 1Z 1 ∂
f a1−t bt , cλ d1−λ d t d λ tλa1−t bt cλ d1−λ ∂x∂y 0 0 

1−1/q = F b(q−r)/(q−1) , a(q−r)/(q−1) F c(q−r)/(q−1) , d(q−r)/(q−1) 

1/q × F Mq (br , b), Mq (ar , a) F Nq (cr , c), Nq (dr , d) , and

1

1

2 ∂
tλa1−t bt c1−λ dλ f a1−t bt , c1−λ dλ d t d λ ∂x∂y 0 0 

1−1/q (q−r)/(q−1) (q−r)/(q−1) ≤ F b ,a F d(q−r)/(q−1) , c(q−r)/(q−1) 

1/q × F Mq (br , b), Mq (ar , a) F Nq (dr , d), Nq (cr , c) . Z

Z

(3.8)

8

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Using the inequalities (3.7) to (3.8) in the inequality (3.6), we conclude the required inequality. The proof is completed. Corollary 3.2.1. Under the conditions of Theorem 3.2, 1. when r = 0, we deduce  1−1/q |S(f )| ≤ F (aq/(q−1) , bq/(q−1) )F (cq/(q−1) , dq/(q−1) ) 

1/q × F Mq (1, a), Mq (1, b) F Nq (1, c), Nq (1, d)  1−1/q + F (aq/(q−1) , bq/(q−1) )F (dq/(q−1) , cq/(q−1) ) 

1/q × F Mq (1, a), Mq (1, b) F Nq (1, d), Nq (1, c)  1−1/q + F (bq/(q−1) , aq/(q−1) )F (cq/(q−1) , dq/(q−1) ) 

1/q × F Mq (1, b), Mq (1, a) F Nq (1, c), Nq (1, d)  1−1/q + F (bq/(q−1) , aq/(q−1) )F (dq/(q−1) , cq/(q−1) ) 

1/q × F Mq (1, b), Mq (1, a) F Nq (1, d), Nq (1, c) ; 2. when r = q, we have  1−1/q n 

1/q 1 F Mq (aq , a), Mq (bq , b) F Nq (cq , c), Nq (dq , d) |S(f )| ≤ 4 

1/q + F Mq (aq , a), Mq (bq , b) F Nq (dq , d), Nq (cq , c) 

1/q + F Mq (bq , b), Mq (aq , a) F Nq (cq , c), Nq (dq , d) 

1/q o + F Mq (bq , b), Mq (aq , a) F Nq (dq , d), Nq (cq , c) , where F (u, v) is defined by (2.1), and Mq (ur , u) and Nq (v r , v) are defined by (3.2). Proof. This follows from letting r = 0 and r = q respectively in Theorem 3.2. Theorem 3.3. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R+ be integrable on ∆ with a < b, c < d. If f is geometrically mean convex on ∆, then Z d Z b f (x, y)f x, cd
f ab , y
f 1 y x (ln b − ln a)(ln d − ln c) c a xy  1/4 ≤ f (a, c)f (a, d)f (b, c)f (b, d) .

√ √
f ab , cd ≤

ab cd x , y


1/4 dxdy

Proof. Taking x = at b1−t and y = cλ d1−λ for 0 ≤ t, λ ≤ 1 and using the geometrically mean convexity of f , we have √ √
Z 1Z 1
f ab , cd = f [at b1−t ]1/2 [a1−t bt ]1/2 , [cλ d1−λ ]1/2 [c1−λ dλ ]1/2 d t d λ 0

0

9

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1

Z ≤

Z 1h





i1/4 f at b1−t , cλ d1−λ f at b1−t , c1−λ dλ f a1−t bt , cλ d1−λ f a1−t bt , c1−λ dλ dtdλ

0

0

1 = (ln b − ln a)(ln d − ln c)

Z c

d

Z

b


f (x, y)f x, cd y f

ab x ,y




f

ab cd x , y

xy

a


1/4 dxdy

and Z

1

1

Z

1

Z

0

0

Z





1/4 dtdλ f at b1−t , cλ d1−λ f at b1−t , c1−λ dλ f a1−t bt , cλ d1−λ f a1−t bt , c1−λ dλ

≤ 0

1

[f (a, c)]t+λ [f (a, d)]t+(1−λ) [f (b, c)](1−t)+λ [f (b, d)](1−t)+(1−λ)

0

× [f (a, d)]t+λ [f (a, c)]t+(1−λ) [f (b, d)](1−t)+λ [f (b, c)](1−t)+(1−λ) × [f (b, c)]t+λ [f (b, d)]t+(1−λ) [f (a, c)](1−t)+λ [f (a, d)](1−t)+(1−λ) 1/16 dtdλ × [f (b, d)]t+λ [f (b, c)]t+(1−λ) [f (a, d)](1−t)+λ [f (a, c)](1−t)+(1−λ) Z 1Z 1  1/4  1/4 = f (a, c)f (a, d)f (b, c)f (b, d) d t d λ = f (a, c)f (a, d)f (b, c)f (b, d) . 0

0

The proof of Theorem 3.3 is complete. Theorem 3.4. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R+ be integrable on ∆ with a < b, c < d. If f is geometrically mean convex on ∆, then Z dZ b 1 f (x, y) dxdy (ln b − ln a)(ln d − ln c) c a xy

≤ L [f (a, c)f (a, d)]1/4 , [f (b, c)f (b, d)]1/4 L [f (a, c)f (b, c)]1/4 , [f (a, d)f (b, d)]1/4 , where L(u, v) is the logarithmic mean. Proof. Putting x = at b1−t and y = cλ d1−λ for 0 ≤ t, λ ≤ 1, from the geometrically mean convexity of f , we obtain Z dZ b Z 1Z 1
f (x, y) 1 dxdy = f at b1−t , cλ d1−λ d t d λ (ln b − ln a)(ln d − ln c) c a xy 0 0 Z 1Z 1  1/4 ≤ [f (a, c)]t+λ [f (a, d)]t+(1−λ) [f (b, c)](1−t)+λ [f (b, d)](1−t)+(1−λ) dtdλ 0 0

= L [f (a, c)f (a, d)]1/4 , [f (b, c)f (b, d)]1/4 L [f (a, c)f (b, c)]1/4 , [f (a, d)f (b, d)]1/4 . The proof of Theorem 3.4 is complete. Theorem 3.5. Let f : ∆ = [a, b] × [c, d] ⊆ R2+ → R+ be integrable on ∆ with a < b, c < d. If f is co-ordinated geometrically mean convex on ∆, then Z dZ b 1 f (x, y) d x d y (ln b − ln a)(ln d − ln c) c a

≤ L a[f (a, c)f (a, d)]1/4 , b[f (b, c)f (b, d)]1/4 L c[f (a, c)f (b, c)]1/4 , d[f (a, d)f (b, d)]1/4 , where L(u, v) is the logarithmic mean. 10

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Proof. Similar to the proof of Theorem 3.4, by the geometrically mean convexity of f , we drive Z 1Z 1 Z dZ b
1 at b1−t cλ d1−λ f at b1−t , cλ d1−λ d t d λ f (x, y) d x d y = (ln b − ln a)(ln d − ln c) c a 0 0 Z 1Z 1  1/4 at b1−t cλ d1−λ [f (a, c)]t+λ [f (a, d)]t+(1−λ) [f (b, c)](1−t)+λ [f (b, d)](1−t)+(1−λ) dtdλ ≤ 0 0

= L a[f (a, c)f (a, d)]1/4 , b[f (b, c)f (b, d)]1/4 L c[f (a, c)f (b, c)]1/4 , d[f (a, d)f (b, d)]1/4 . The proof of Theorem 3.5 is complete. We proceed similarly as in the proof of Theorem 3.3 to Theorem 3.5, we can get Theorem 3.6. Let f, g : ∆ = [a, b] × [c, d] ⊆ R2+ → R+ be integrable on ∆ with a < b, c < d. If f and g are co-ordinated geometrically mean convex on ∆, then f

√
√ √
√ ab , cd g ab , cd ≤ Z

d

Z

× c

b

1 (ln b − ln a)(ln d − ln c) 

ab
cd f (x, y)g(x, y)f x, cd y g x, y f x , y g

ab x ,y


f

ab cd x , y


g

ab cd x , y


1/4 dxdy

xy

a

 1/4 ≤ f (a, c)g(a, c)f (a, d)g(a, d)f (b, c)g(b, c)f (b, d)g(b, d) . Theorem 3.7. Under the conditions of Theorem 3.6, we have 1 (ln b − ln a)(ln d − ln c)

Z

d

c

Z

b

a

f (x, y)g(x, y) dxdy xy

≤ L [f (a, c)g(a, c)f (a, d)g(a, d)]1/4 , [f (b, c)g(b, c)f (b, d)g(b, d)]1/4




× L [f (a, c)g(a, c)f (b, c)g(b, c)]1/4 , [f (a, d)g(a, d)f (b, d)g(b, d)]1/4




and 1 (ln b − ln a)(ln d − ln c)

Z

d

Z

b

f (x, y)g(x, y) d x d y c

a

≤ L a[f (a, c)g(a, c)f (a, d)g(a, d)]1/4 , b[f (b, c)g(b, c)f (b, d)g(b, d)]1/4





× L c[f (a, c)g(a, c)f (b, c)g(b, c)]1/4 , d[f (a, d)g(a, d)f (b, d)g(b, d)]1/4 .

Acknowledgements This work was partially supported by the National Natural Science Foundation of China under Grant No. 11361038, by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY14192, and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, China.

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References [1] M. Alomari and M. Darus, On the Hadamard’s inequality for log-convex functions on the co-ordinates, J. Inequal. Appl., 2009 (2009), Article ID 283147, 13 pages; Available online at http://dx.doi.org/doi:10.1155/2009/283147. [2] L. Chun, Some new inequalities of Hermite-Hadamard type for (α1 , m1 )-(α2 , m2 )-convex functions on coordinates, J. Funct. Spaces, 2014 (2014), Article ID 975950, 7 pages; Available online at http://dx.doi.org/10.1155/2014/975950. [3] S. S. Dragomir, On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., 5 (2001), no. 4, 775–788. [4] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University (2000); Available online at http:// rgmia.org/monographs/hermite_hadamard.html [5] A.-P. Ji, T.-Y. Zhang, and F. Qi, Integral inequalities of Hermite–Hadamard type for (α, m)GA-convex functions, J. Comput. Anal. Appl., 18 (2015), no. 2, 255–265. [6] M. A. Latif and S. S. Dragomir, On some new inequalities for dfferentiable co-ordinated convex functions, J. Inequal. Appl., 2012, 2012:28; Available online at http://dx.doi.org/10.1186/ 1029-242X-2012-28. ¨ [7] M. E. Ozdemir, A. O. Akdemir, H. Kavurmaci, and M. Avci, On the Simpson’s inequality for co-ordinated convex functions, available online at http://arxiv.org/abs/1101.0075. [8] B.-Y. Xi, R.-F. Bai, and F. Qi, Hermite-Hadamard type inequalities for the m- and (α, m)geometrically convex functions, Aequationes Math., 84 (2012), no. 3, 261–269; Available online at http://dx.doi.org/10.1007/s00010-011-0114-x. [9] B.-Y. Xi, J. Hua, and F. Qi, Hermite-Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle, J. Appl. Anal., 20 (2014), no. 1, 29–39; Available online at http://dx.doi.org/10.1515/jaa-2014-0004. [10] B.-Y. Xi and F. Qi, Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means, J. Funct. Spaces Appl., 2012 (2012), Article ID 980438, 14 pages; Available online at http://dx.doi.org/10.1155/2012/980438. [11] B.-Y. Xi and F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex functions and applications, Hacettepe J. Math. Statist., 42 (2013), no. 3, 243–257.

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STABILITY OF THE GENERALAIZED VERSION OF EULER-LAGRANGE TYPE QUADRATIC EQUATION CHANG IL KIM1 , GILJUN HAN2 , AND JEONGWOOK CHANG3*

1

Department of Mathematics Education, Dankook University, 126 Jukjeon, Yongin, Gyeonggi 448-701, Korea

2

Department of Mathematics Education, Dankook University, 126 Jukjeon, Yongin, Gyeonggi 448-701, Korea

3

e-mail : [email protected]

e-mail : [email protected]

Department of Mathematics Education, Dankook University, 126 Jukjeon, Yongin, Gyeonggi 448-701, Korea

e-mail : [email protected]

2010 Mathematics Subject Classification. 39B52, 39B82 Key words and phrases. Generalized Hyers-Ulam Stability, Quadratic functional equation with general terms, functional operator

Abstract. In this paper, we consider the generalized Hyers-Ulam stability for the following quadratic functional equation. f (ax + by) + f (ax − by) + Gf (x, y) = 2a2 f (x) + 2b2 f (y) Here Gf is a functional operator of f . We consider some sufficient conditions on Gf which can be applied easily for the generalized Hyers-Ulam stability, and illustrate some new functional equations by using them.

1. Introduction In 1940, Ulam proposed the following stability problem (See [17]):

“Let G1 be a group and G2 a metric group with the metric d. Given a constant δ > 0, does there exist a constant c > 0 such that if a mapping f : G1 −→ G2 satisfies d(f (xy), f (x)f (y)) < c for all x, y ∈ G1 , then there exists a unique homomorphism h : G1 −→ G2 with d(f (x), h(x)) < δ for all x ∈ G1 ?” * Corresponding

author. 1

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In 1941, Hyers [7] answered this problem under the assumption that the groups are Banach spaces. Aoki [1] and Rassias [13] generalized the result of Hyers. Th. M. Rassias [13] solved the generalized Hyers-Ulam stability of the functional inequality kf (x + y) − f (x) − f (y)k ≤ (kxkp + kykp ) for some  ≥ 0 and p with p < 1 and for all x, y ∈ X, where f : X −→ Y is a function between Banach spaces. The paper of Rassias [13] has provided a lot of influence in the development of what we call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias’ theorem was obtained by Gˇavruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)

(1.1)

is called a quadratic functional equation and a solution of a quadratic functional equation is called quadratic. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [16] for mappings f : X −→ Y , where X is a normed space and Y is a Banach space. Cholewa [2] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [3] proved the generalized Hyers-Ulam stability for the quadratic functional equation and Park [12] proved the generalized Hyers-Ulam stability of the quadratic functional eqution in Banach modules over a C ∗ -algebra. Rassias [14] investigated the following Euler-Lagrange functional equation

f (ax + by) + f (bx − ay) = 2(a2 + b2 )[f (x) + f (y)] and Gordji and Khodaei [6] investigated other Euler-Lagrange functional equations

f (ax + by) + f (ax − by) = (1.2) +

b(a + b) f (x + y) 2

b(a + b) f (x − y) + (2a2 − ab − b2 )f (x) + (b2 − ab)f (y) 2

for fixed integers a, b with b 6= a, −a, −3a and (1.3)

f (ax + by) + f (ax − by) = 2a2 f (x) + 2b2 f (y)

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for fixed integers a, b with a2 6= b2 and ab 6= 0. In this paper, we are interested in what kind of terms can be added to the quadratic functional equation f (ax + by) + f (ax − by) = 2a2 f (x) + 2b2 f (y) while the generalized Hyers-Ulam stability still holds for the new functional equation. We denote the added term by Gf (x, y) which can be regarded as a functional operator depending on the variables x, y, and function f . Then the new functional equation can be written as f (ax + by) + f (ax − by) + Gf (x, y) = 2a2 f (x) + 2b2 f (y) for some rational numbers a, b with ab 6= 0 and a2 6= b2 . The precise definition of Gf is given in section 2. In fact, the functional operator Gf (x, y) was introduced and considered in the case of the additive functional equations with somewhat different point of view by the authors([11]). The new observation in this article makes possible to prove many previous problems on quadratic functional equations more easily and provides methods to construct new ones. So we can have a larger class of functional equations related with quadratic functions for the generalized Hyers-Ulam stability. We illustrate some new functional equations in section 3 in order to see how our observation works for the generalized Hyers-Ulam stability.

2. Quadratic functional equations with general terms Let X be a real normed linear space and Y a real Banach space. For given l ∈ N and any i ∈ {1, 2, · · ·, l}, let σi : X × X −→ X be a binary operation such that σi (rx, ry) = rσi (x, y) for all x, y ∈ X and all r ∈ R. It is clear that σi (0, 0) = 0. Also let F : Y l −→ Y be a linear, continuous function. For a map f : X −→ Y , define Gf (x, y) = F (f (σ1 (x, y)), f (σ2 (x, y)), · · ·, f (σl (x, y))).

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Here, Gf is a functional operator on the function space {f |f : X −→ Y }. In this paper, for an appropriate function φ : X 2 −→ [0, ∞), we consider the functional inequalty (2.1)

kf (ax + by) + f (ax − by) + Gf (x, y) − 2a2 f (x) − 2b2 f (y)k ≤ φ(x, y)

for fixed non-zero rational numbers a, b with a2 6= b2 , where the functional operator Gf satisfies Gf (x, 0) ≡ λ[f (ax) − a2 f (x)]

(2.2)

for some λ(λ 6= −2). Here, ≡ means that Gf (x, 0) = λ[f (ax) − a2 f (x)] holds for all x ∈ X and all f : X −→ Y . In fact, as we shall see in Theorem 2.2, for a function f with f (0) = 0 satisfying the equation (2.3)

f (ax + by) + f (ax − by) + Gf (x, y) = 2a2 f (x) + 2b2 f (y),

f is quadratic if and only if Gf (x, 0) = λ[f (ax) − a2 f (x)] and Gf (x, y) = Gf (y, x). So the condition (2.2) is reasonable for the stability problem of (2.1). From now on, we assume that the functional operator Gf satisfies the condition (2.2) unless otherwise stated. We deonte Hf (x, y) = f (x + y) + f (x − y) − 2f (x) − 2f (y). The following lemma is proved in the authors’ previous paper [10]. Lemma 2.1. [10] Consider the following functional equation. (2.4)

f (ax + by) + f (ax − by) + cHf (x, y) = 2a2 f (x) + 2b2 f (y)

for fixed non-zero rational numbers a, b with a2 6= b2 and a real number c. Then if f : X −→ Y satisfies (2.4) and f (0) = 0, f is quadratic. By using Lemma 2.1, we can examine the properties of a solution function of the equation (2.3). Theorem 2.2. Suppose the equation (2.3) holds. Then the following coditions are equivalent : (1) f is quadratic. (2) Gf (x, y) = Gf (y, x) for all x, y ∈ X, and f (0) = 0

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(3) There are non-zero rational numbers m, n, δ such that a2 m2 6= b2 n2 and (2.5)

Gf (mx, ny) = δHf (x, y), f (mx) = m2 f (x), f (nx) = n2 f (x)

for all x, y ∈ X. Proof. We prove the theorem by showing (1) ⇒ (2) ⇒ (3) ⇒ (1) (1) ⇒ (2) Since f is quadratic, f (0) = 0 and Gf (x, y) = 0 for all x, y ∈ X. So Gf (x, y) satisfies (2) with λ = 0 in (2.2). (2) ⇒ (3) Putting y = 0 in (2.3) and by (2.2), we have

(2.6)

(2 + λ)[f (ax) − a2 f (x)] = 0


for all x ∈ X and since f (0) = 0, Gf (x, 0) = 2 a2 f (x) − f (ax) = 0. From the condition Gf (x, 0) = Gf (0, x), we have

f (bx) + f (−bx) = 2b2 f (x) for all x ∈ X and so we have

b2 f (x) = b2 f (−x)

(2.7)

for all x ∈ X. Since b 6= 0, by (2.7), f is even and hence f (bx) = b2 f (x) for all x ∈ X. Thus (2.3) becomes

Hf (ax, by) + Gf (x, y) = 0, and from the condition Gf (x, y) = Gf (y, x) we have (2.8)

Gf (x, y) = −Hf (ay, bx)

for all x, y ∈ X. Replacing x and y by ax and by respectively in (2.8), we have Gf (ax, by) = −Hf (aby, abx) = −a2 b2 Hf (y, x) = −a2 b2 Hf (x, y)

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for all x, y ∈ X. The last equality comes from the fact that f is even. Note that a4 6= b4 . So we have (3) with m = a, n = b, δ = −a2 b2 .

(3) ⇒ (1) By (2.3) and (3),

δHf (x, y) − Gf (mx, ny) = δHf (x, y) + f (amx + bny) + f (amx − bny) − 2a2 f (mx) − 2b2 f (ny) = δHf (x, y) + f (amx + bny) + f (amx − bny) − 2a2 m2 f (x) − 2b2 n2 f (y) =0 for all x, y ∈ X. Since a2 m2 6= b2 n2 , by Lemma 2.1, f is quadratic.



Now we prove the following stability theorem. Theorem 2.3. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

(2.9)

a−2n φ(an x, an y) < ∞

n=0

for all x, y ∈ X. Assume that Gf (x, y) satisfies one of the conditions in Thorem 2.2 when the equation (2.3) holds, and let f : X −→ Y be a mapping such that (2.10)

kf (ax + by) + f (ax − by) + Gf (x, y) − 2a2 f (x) − 2b2 f (y)k ≤ φ(x, y)

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X −→ Y such that

(2.11)

kQ(x) − f (x) − f (0)k ≤

∞ 1 X −2(n+1) a φ(an x, 0) |λ + 2| n=0

for all x ∈ X. Proof. By the standard argument, we may assume that f (0) = 0. Setting y = 0 in (2.10), we have

kf (ax) + 2−1 Gf (x, 0) − a2 f (x)k ≤ 2−1 φ(x, 0) for all x ∈ X and by (2.2), we have

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kf (x) − a−2 f (ax)k ≤

(2.12)

1 a−2 φ(x, 0) |λ + 2|

for all x ∈ X. Replacing x by an x in (2.12) and dividing (2.12) by a2n , we have

ka−2n f (an x) − a−2(n+1) f (an+1 x)k ≤

1 a−2(n+1) φ(an x, 0) |λ + 2|

for all x ∈ X and all non-negative integer n. For m, n ∈ N ∪ {0} with 0 ≤ m < n,

ka−2m f (am x) − a−2n f (an x)k = a−2m kf (am x) − a−2(n−m) f (an−m (am x))k

(2.13)

≤

n−1 X 1 a−2(k+1) φ(ak x, 0) |λ + 2| k=m

for all x ∈ X. By (2.13), {a−2n f (an x)} is a Cauchy sequence in Y and since Y is a Banach space, there exists a mapping Q : X −→ Y such that

Q(x) = lim a−2n f (an x) n−→∞

for all x ∈ X and

kQ(x) − f (x)k ≤

∞ 1 X −2(n+1) a φ(an x, 0) |λ + 2| n=0

for all x ∈ X. Replacing x and y by an x and an y respectively in (2.10) and dividing (2.10) by a2n , we have

ka−2n f (an (ax + by)) + a−2n f (an (ax − by)) + a−2n Gf (an x, an y) − 2 · a2 · a−2n f (an x) − 2 · b2 · a−2n f (an y)k ≤ a−2n φ(an x, an y) for all x, y ∈ X and letting n → ∞ in the above inequality, we have

Q(ax + by) + Q(ax − by) (2.14)

+ lim a−2n Gf (an x, an y) − 2a2 Q(x) − 2b2 Q(y) = 0 n−→∞

for all x, y ∈ X. Since F is continuous, we have

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lim a−2n Gf (an x, an y)

n−→∞

= lim F (a−2n f (an σ1 (x, y)), a−2n f (an σ2 (x, y)), · · · , a−2n f (an σl (x, y))) n−→∞

= F (Q(σ1 (x, y)), Q(σ2 (x, y)), · · · , Q(σl (x, y))) = GQ (x, y) for all x, y ∈ X. Hence by (2.14), we have

(2.15)

Q(ax + by) + Q(ax − by) + GQ (x, y) = 2a2 Q(x) + 2b2 Q(y)

for all x, y ∈ X. Since Q satisfies (2.3), Q is quadratic by Theorem 2.2. Now, we show the uniqueness of Q. Suppose that Q0 is a quadratic mapping with (2.11). Then we have

kQ(x) − Q0 (x)k = a−2k kQ(ak x) − Q0 (ak x)k ≤

∞ X 2 a−2(n+1) φ(an x, 0) |λ + 2| n=k

for all x ∈ X. Hence, letting k → ∞ in the above inequality, we have

Q(x) = Q0 (x) for all x ∈ X.



Theorem 2.4. Assume that Gf satisfies all of the conditions in Theorem 2.3. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

a2n φ(a−n x, a−n y) < ∞

n=0

for all x, y ∈ X. Let f : X −→ Y be a mapping such that kf (ax + by) + f (ax − by) + Gf (x, y) − 2a2 f (x) − 2b2 f (y)k ≤ φ(x, y) for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X −→ Y such that kQ(x) − f (x) − f (0)k ≤

∞ 1 X 2(n+1) a φ(a−n x, 0) |λ + 2| n=0

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for all x ∈ X. As examples of φ(x, y) in Theorem 2.3 and Theorem 2.4, we can take φ(x, y) = (kxkp kykp + kxk2p + kyk2p ) which is appeared in [11]. Then we can formulate the following corollary Corollary 2.5. Assume that all of the conditions in Theorem 2.3 are satisfied. Let p be a real number with p 6= 1. Let f : X −→ Y be a mapping such that kf (ax + by) + f (ax − by) − 2a2 f (x) − 2b2 f (y) + Gf (x, y)k ≤ (kxkp kykp + kxk2p + kxk2p ) for fixed non-zero rational numbers a, b with a2 6= b2 , a fixed positive real number , and all x, y ∈ X. Then there exists a unique quadratic mapping Q : X −→ Y such that

kQ(x) − f (x) − f (0)k ≤

kxk2p a2 |λ + 2|[1 − a2(p−1) ]

(p < 1 and |a| > 1, or p > 1 and |a| < 1) and kQ(x) − f (x) − f (0)k ≤

a2 kxk2p |λ + 2|[1 − a2(1−p) ]

(p > 1 and |a| > 1, or p < 1 and |a| < 1) for all x ∈ X. 3. Applications In this section we illustrate how the theorems in section 2 work well for the generalized Hyers-Ulam stability of various quadratic functional equations. By applying the results in this article, we can construct many concrete members in our calss of functional equations easily. First, we consider the following functional equation related with Theorem 2.3.

f (ax + by) + f (ax − by) + f (x + y) + f (x − y) (3.1) + f (y − x) − f (−x) − f (−y) = 2(a2 + 1)f (x) + 2(b2 + 1)f (y) for fixed non-zero rational numbers a, b with a2 6= b2 .

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Using Theorem 2.3, we can prove the stability for (3.1). Theorem 3.1. Let φ : X 2 −→ [0, ∞) be a function with (2.9) and f : X −→ Y a mapping such that

kf (ax + by) + f (ax − by) + f (x + y) + f (x − y) + f (y − x) (3.2) − f (−x) − f (−y) − 2(a2 + 1)f (x) − 2(b2 + 1)f (y)k ≤ φ(x, y). for fixed non-zero rational numbers a, b with a2 6= b2 , and all x, y ∈ X. Then there exists a unique quadratic mapping Q : X −→ Y such that Q satisfies (3.1) and kQ(x) − f (x) − f (0)k ≤

(3.3)

∞ 1 X −2(n+1) a φ(an x, 0) 2 n=0

for all x ∈ X. Proof. In this case, Gf (x, y) = f (x + y) + f (x − y) + f (y − x) − f (−x) − f (−y) − 2f (x) − 2f (y). So Gf (x, 0) = 0. Hence Gf and f satisfies all the conditions in Theorem 2.3. and the functional inequality (3.2) can be rewritten as the functional inequality

kf (ax + by) + f (ax − by) + Gf (x, y) − 2a2 f (x) − 2b2 f (y)k ≤ φ(x, y). By Theorem 2.3, we get the result.



When the equation (2.3) holds, Gf (x, y) can be represented as different forms. In some cases, these forms together help us to analyze a solution. Especially the following case happens often in some interesting equations. We will give an example later. Lemma 3.2. Suppose when the equation (2.3) with a2 6= b4 holds, Gf (x, y) can be represented as both of the followings.

(3.4)

Gf (0, y) = k[f (y) − f (−y)]

(3.5)

Gf (x, by) = k1 Hf (x, y)

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for all x, y ∈ X and a fixed positive real number k with k 6= b2 , and a fixed real number k1 . Then Gf satisfies the condition (3) in Theorem 2.2. Proof. Suppose that (2.3) holds. Then we have Gf (0, y) − Gf (0, −y) = 2b2 [f (y) − f (−y)] for all y ∈ X. Also by (3.4), we have Gf (0, y) − Gf (0, −y) = 2k[f (y) − f (−y)] Since k 6= b2 , f is even and so f (bx) = b2 f (x). By (3.5) and a2 6= b4 , we get the result with m = 1, n = b, and k1 = δ.



Note that Lemma 3.2 is still valid if we does not impose the condition (2.2) on Gf . By Lemma 3.2 and Theorem 2.3, we can formulate the following proposition. Proposition 3.3. Let φ be a function in Theorem 2.2 and suppose that Gf (x, y) satisfies the condition in Lemma 3.2 when the equation (2.3) holds. Then there exists a unique quadratic mapping Q : X −→ Y such that

kQ(x) − f (x) − f (0)k ≤

∞ 1 X −2(n+1) a φ(an x, 0) |λ + 2| n=0

for all x ∈ X. Now, we consider the following functional equation related with Proposition 3.3.

f (ax + by) + f (ax − by) − f (bx + y) − f (bx − y) + 2f (bx) (3.6) = 2a2 f (x) + 2(b2 − 1)f (y) for fixed non-zero rational numbers a, b with a2 6= b2 and a2 6= b4 . Theorem 3.4. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

a−2n φ((an x, an y) < ∞

n=0

for all x, y ∈ X. Let f : X −→ Y be a mapping such that

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CHANG IL KIM, GILJUN HAN, AND JEONGWOOK CHANG

kf (ax + by) + f (ax − by) − f (bx + y) − f (bx − y) (3.7) + 2f (bx) − 2a2 f (x) − 2(b2 − 1)f (y)k ≤ φ(x, y). Then there exists a unique quadratic mapping Q : X −→ Y such that Q satisfies (3.6) and kQ(x) − f (x) − f (0)k ≤

∞ 1 X −2(n+1) a φ(an x, 0) 2 n=0

for all x ∈ X. Proof. It is enough to show with the condition f (0) = 0. In this case, Gf (x, y) = −f (bx + y) − f (bx − y) + 2f (bx) + 2f (y). First we can check Gf (x, 0) = 0 as a functional operator. Now suppose that f satisfies (3.6). Then Gf (0, y) = f (y) − f (−y) for all y ∈ X and hence f (by) = b2 f (y). So Gf (x, by) = −b2 Hf (x, y). Since all the conditions in Proposition 3.3 are satisfied, we have the result.



Similar to Proposition 3.3, we have the following proposition : Proposition 3.5. Suppose that f (0) = 0 and Gf (x, y) satisfies (3.8)

Gf (ax, y) = k2 Hf (x, y)

for all x, y ∈ X and a fixed real number k2 when the equation (2.3) holds. Let φ be a function in Theorem 2.3. Then there exists a unique quadratic mapping Q : X −→ Y such that

kQ(x) − f (x) − f (0)k ≤

∞ 1 X −2(n+1) a φ(an x, 0) |λ + 2| n=0

for all x ∈ X. Finally, we consider the following functional equation related with Proposition 3.5.

(3.9)

2f (2x + y) + f (2x − y) + f (x − 2y) = 13f (x) + 6f (y) + f (−y)

for all x, y ∈ X.

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Theorem 3.6. Let φ : X 2 −→ [0, ∞) be a function such that ∞ X

a−2n φ(an x, an y) < ∞

n=0

for all x, y ∈ X. Let f : X −→ Y be a mapping such that

k2f (2x + y) + f (2x − y) + f (x − 2y) − 13f (x) − 6f (y) − (−y)k (3.10) ≤ φ(x, y) for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X −→ Y such that Q satisfies (3.6) and kQ(x) − f (x) − f (0)k ≤

∞ 1 X −2(n+1) a φ(an x, 0) 3 n=0

for all x ∈ X. Proof. In this case Gf (x, y) = f (2x + y) + f (x − 2y) − 5f (x) − 4f (y) − f (−y), so Gf (x, 0) = f (2x) − 4f (x) under the condition f (0) = 0. Now suppose f satisfies (3.9). Since Gf (0, y) = 3[f (−y) − f (y)] for all y ∈ X, by the argument in Lemma 3.2, f is even. Aso, the functional equation (3.9) implies (3.11)

¯ f (x, y) = 8f (x) + 2f (y), f (2x + y) + f (2x − y) + G

¯ f (x, y) = 1 f (x + 2y) + 1 f (x − 2y) − 2 f (x) − 1 f (y) − 7 f (−y). where G 3 3 3 3 3 ¯ f (2x, y) = 4 Hf (x, y) for all x, y ∈ X. By (3.10), we Since f is even, we have G 3 have

¯ f (x, y) − 8f (x) − 2f (y)k kf (2x + y) + f (2x − y) + G (3.12) ≤

1 [φ(x, y) + φ(x, −y)] 3

for all x, y ∈ X and so by Proposition 3.5, we have the result.



Acknowledgement The present research was conducted by the research fund of Dankook University in 2014

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References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), 64-66. [2] P.W.Cholewa, Remarkes on the stability of functional equations, Aequationes Math., 27(1984), 76-86. [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62(1992), 59-64. [4] M. E. Gordji, M. B. Savadkouhi, and C. Park Quadratic-quartic functional equations in RN-spaces, Journal of inequalities and Applications, 2009(2009), 1-14. [5] P. Gˇ avruta, A generalization of the Hyer-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431-436. [6] M. E. Gordji and H. Khodaei, On the Generalized Hyers-Ulam-Rassias Stability of Quadratic functional equations, Abstr. Appl. Anal. 2009(2009), 1-11. [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224. [8] K.W Jun, H.M.Kim and I.S. Chang, On the Hyers-Ulam Stability of an Euler-Lagrange type cubic functional equation, J. comput. anal. appl. 7(2005), 21-33. [9] K.W Jun, H.M.Kim and J. Son, Generalized Hyers-Ulam Stability of a Quadratic functional equation, Functional Equations in Mathematical Analysis 2012(2012),153-164. [10] C.I. Kim, G.J. HAN and S.A Shim, Hyers-Ulam stability for a class of quadratic functional equations via a typical form, Abstract and Applied Analysis, Volume 2013, Article ID 283173, 8 pages. [11] C.I. Kim, G.J. HAN and S.A Shim, Cauchy functional equation with an extra term and its stability, preprint [12] C. G. Park, On the stability of the quadratic mapping in Banach modules, J. Math. Anal. Appl. 276(2002), 135-144. [13] Th. M. Rassias, On the stability of the linear mapping in Banach sapces, Proc. Amer. Math. Sco. 72(1978), 297-300. [14] J. M. Rassias, Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings, Journal of mathematical analysis and applications, 220(1998), 613-639. [15] K. Ravi, M. Arunkumar and J. M. Rassias, Ulam stability for the orthogonally general EulerLagrange type functional equation, Int. J. Math. Stat. 3(2008), 36-46. [16] F. Skof, Propriet´ a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53(1983), 113-129. [17] S. M. Ulam, A collection of mathematical problems, Interscience Publisher, New York, 1964.

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A FIXED POINT APPROACH TO THE STABILITY OF EULER-LAGRANGE SEXTIC (a, b)-FUNCTIONAL EQUATIONS IN ARCHIMEDEAN AND NON-ARCHIMEDEAN BANACH SPACES MOHAMMAD BAGHER GHAEMI, MEHDI CHOUBIN, REZA SAADATI, CHOONKIL PARK AND DONG YUN SHIN

Abstract. In this paper, we present a fixed point method to prove the Hyers-Ulam stability of the system of Euler-Lagrange quadratic-quartic functional equations   f (ax1 + bx2 , y) + f (bx1 + ax2 , y) + abf (x1 − x2 , y)     2 , y), = (a2 + b2 )[f (x1 , y) + f (x2 , y)] + 4abf ( x1 +x 2 (0.1) 2 1  f (x, ay1 + by2 ) + f (x, by1 + ay2 ) + 2 ab(a − b) f (x, y1 − y2 )    2 = (a2 − b2 )2 [f (x, y1 ) + f (x, y2 )] + 8abf (x, y1 +y ) 2 for all numbers a and b with a + b ∈ / {0, ±1}, ab + 2 6= 2(a + b)2 and ab(a − b)2 + 4 6= 4(a + b)4 in Archimedean and non-Archimedean Banach spaces and we show that the approximation in nonArchimedean Banach spaces is better than the approximation in (Archimedean) Banach spaces.

1. Introduction The stability problem of functional equations started with the following question concerning stability of group homomorphisms proposed by Ulam [69] during a talk before a Mathematical Colloquium at the University of Wisconsin. In 1941, Hyers [32] gave a first affirmative answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Rassias [61] for linear mappings by considering an unbounded Cauchy difference, respectively. In 1994, a generalization of the Rassias theorem was obtained by Gˇavruta [28] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. For more details about the results concerning such problems, the reader refer to [2, 5, 8, 11, 14, 15, 24, 27, 29, 33, 34, 35, 36, 40, 41, 42, 44], [52]–[67] and [71, 72, 73]. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) (1.1) is related to a symmetric bi-additive mapping [1, 43]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is called a quadratic mapping. The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof [68]. In [14], Czerwik proved the Hyers-Ulam stability of the equation (1.1). Eshaghi Gordji and Khodaei [25] obtained the general solution and the Hyers-Ulam stability of the following quadratic functional equation: for all a, b ∈ Z\{0} with a 6= ±1, ±b, f (ax + by) + f (ax − by) = 2a2 f (x) + 2b2 f (y).

(1.2)

2010 Mathematics Subject Classification. 39B82, 46S10, 47H10. Key words and phrases. Hyers-Ulam stability, Euler-Lagrange functional equation, fixed point, non-Archimedean space. Corresponding author: Dong Yun Shin. 170

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Jun and Kim [38] introduced the following cubic functional equation: f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x)

(1.3)

and they established the general solution and the Hyers-Ulam stability for the functional equation (1.3). Jun et al. [39] investigated the solution and the Hyers-Ulam stability for the cubic functional equation f (ax + by) + f (ax − by) = ab2 (f (x + y) + f (x − y)) + 2a(a2 − b2 )f (x), (1.4) where a, b ∈ Z\{0} with a 6= ±1, ±b. For other cubic functional equations, see [50]. Lee et. al. [48] considered the following functional equation: f (2x + y) + f (2x − y) = 4f (x + y) + 4f (x − y) + 24f (x) − 6f (y)

(1.5)

In fact, they proved that a mapping f between two real vector spaces X and Y is a solution of the equation (1.5) if and only if there exists a unique symmetric bi-quadratic mapping B2 : X × X → Y such that f (x) = B2 (x, x) for all x ∈ X. The bi-quadratic mapping B2 is given by 1 (f (x + y) + f (x − y) − 2f (x) − 2f (y)). 12 Obviously, the function f (x) = cx4 satisfies the functional equation (1.5), which is called the quartic functional equation. For other quartic functional equations, see [13]. Ebadian et al. [16] proved the Hyers-Ulam stability of the following systems of the additive-quartic functional equation:    f (x1 + x2 , y) = f (x1 , y) + f (x2 , y), (1.6) f (x, 2y1 + y2 ) + f (x, 2y1 − y2 )    = 4f (x, y + y ) + 4f (x, y − y ) + 24f (x, y ) − 6f (x, y ) B2 (x, y) =

1

2

1

2

1

2

and the quadratic-cubic functional equation:    f (x, 2y1 + y2 ) + f (x, 2y1 − y2 )

(1.7)

= 2f (x, y1 + y2 ) + 2f (x, y1 − y2 ) + 12f (x, y1 ),   f (x, y + y ) + f (x, y − y ) = 2f (x, y ) + 2f (x, y ). 1 2 1 2 1 2

For more details about the results concerning mixed type functional equations, the readers refer to [18, 20, 21] and [23]. Recently, Ghaemi et. al. [30] and Cho et. al. [10] investigated the the stability of the following systems of the quadratic-cubic functional equation:  2 2   f (ax1 + bx2 , y) + f (ax1 − bx2 , y) = 2a f (x1 , y) + 2b f (x2 , y), f (x, ay1 + by2 ) + f (x, ay1 − by2 )    = ab2 (f (x, y1 + y2 ) + f (x, y1 − y2 )) + 2a(a2 − b2 )f (x, y1 )

and the additive-quadratic-cubic functional equation:   f (ax1 + bx2 , y, z) + f (ax1 − bx2 , y, z) = 2af (x1 , y, z),    f (x, ay + by , z) + f (x, ay − by , z) = 2a2 f (x, y , z) + 2b2 f (x, y , z), 1 2 1 2 1 2  f (x, y, az + bz ) + f (x, y, az − bz )  1 2 1 2    2 = ab (f (x, y, z1 + z2 ) + f (x, y, z1 − z2 )) + 2a(a2 − b2 )f (x, y, z1 ) 171

(1.8)

(1.9)

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in P N -spaces and P M -spaces, where a, b ∈ Z\{0} with a 6= ±1, ±b. The function f : R × R → R given by f (x, y) = cx2 y 3 is a solution of the system (1.8). In particular, letting y = x, we get a quintic function g : R → R in one variable given by g(x) := f (x, x) = cx5 . In this paper, we present a fixed point method to prove the Hyers-Ulam stability of the following system of the Euler-Lagrange quadratic-quartic (a, b)-functional equation:   f (ax1 + bx2 , y) + f (bx1 + ax2 , y) + abf (x1 − x2 , y)     2 = (a2 + b2 )[f (x1 , y) + f (x2 , y)] + 4abf ( x1 +x 2 , y), (1.10)  f (x, ay1 + by2 ) + f (x, by1 + ay2 ) + 21 ab(a − b)2 f (x, y1 − y2 )     2 = (a2 − b2 )2 [f (x, y1 ) + f (x, y2 )] + 8abf (x, y1 +y 2 ) for all numbers a and b with a + b ∈ / {0, ±1}, ab + 2 6= 2(a + b)2 and ab(a − b)2 + 4 6= 4(a + b)4 in Archimedean and non-Archimedean Banach spaces. For details about the results concerning such problems in non-Archimedean normed spaces, the reader refer to [9, 12, 17, 20, 26, 37, 46, 47, 55, 72]. It is easy to see that the function f : R × R → R defined by f (x, y) = cx2 y 4 is a solution of the system (1.10). In particular, letting x = y, we get a sextic function h : R → R in one variable given by h(x) := f (x, x) = cx6 . The proof of the following propositions is evident. Proposition 1.1. Let X and Y be real linear spaces. If a mapping f : X × X → Y satisfies the system (1.10), then f (λx, µy) = λ2 µ4 f (x, y) for all x, y ∈ X and rational numbers λ, µ. In this paper, we investigate the Hyers-Ulam stability of a sextic mapping from linear spaces into Archimedean and non-Archimedean Banach spaces. Hensel [31] has introduced a normed space which does not have the Archimedean property. During the last three decades theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p–adic strings and superstrings [45]. Although many results in the classical normed space theory have a non–Archimedean counterpart, their proofs are different and require a rather new kind of intuition [4, 22, 51, 54, 70]. One may note that |n| ≤ 1 in each valuation field, every triangle is isosceles and there may be no unit vector in a non-Archimedean normed space; cf. [51]. These facts show that the non-Archimedean framework is of special interest. Definition 1.2. Let K be a field. A valuation mapping on K is a function | · | : K → R such that for any a, b ∈ K we have (i) |a| ≥ 0 and equality holds if and only if a = 0, (ii) |ab| = |a||b|, (iii) |a + b| ≤ |a| + |b|. A field endowed with a valuation mapping will be called a valued field. If the condition (iii) in the definition of a valuation mapping is replaced with (iii)0 |a + b| ≤ max{|a|, |b|} then the valuation | · | is said to be non-Archimedean. The condition (iii)0 is called the strict triangle inequality. By (ii), we have |1| = | − 1| = 1. Thus, by induction, it follows from (iii)0 that |n| ≤ 1 for each integer n. We always assume in addition that | · | is non trivial, i.e., that there is an a0 ∈ K such that |a0 | 6∈ {0, 1}.The most important examples of non-Archimedean spaces are p-adic numbers. 172

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Example 1.3. Let p be a prime number. For any non–zero rational number a = pr m n such that m −r and n are coprime to the prime number p, define the p-adic absolute value |a|p = p . Then | · | is a non-Archimedean norm on Q. The completion of Q with respect to | · | is denoted by Qp and is called the p-adic number field. Definition 1.4. Let X be a linear space over a scalar field K with a non-Archimedean non-trivial valuation | · |. A function k · k : X → R is a non-Archimedean norm (valuation) if it satisfies the following conditions: (N A1) kxk = 0 if and only if x = 0; (N A2) krxk = |r|kxk for all r ∈ K and x ∈ X; (N A3) the strong triangle inequality (ultrametric); namely, kx + yk ≤ max{kxk, kyk} (x, y ∈ X). Then (X, k · k) is called a non-Archimedean normed space. Definition 1.5. (i) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called Cauchy if for a given ε > 0 there is a positive integer N such that kxn − xm k < ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that kxn − xk < ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. In 2003, Radu [60] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [6, 7]). Our aim is based on the following fixed point approach: Let (X, d) be a generalized metric space. An operator T : X → X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L ≥ 0 such that d(T x, T y) ≤ Ld(x, y) for all x, y ∈ X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz. Theorem 1.6. ([49, 60]) Suppose that we are given a complete generalized metric space (Ω, d) and a strictly contractive mapping T : Ω → Ω with Lipschitz constant L. Then for each given x ∈ Ω, either d(T m x, T m+1 x) = ∞ for all m ≥ 0, or there exists a natural number m0 such that • d(T m x, T m+1 x) < ∞ for all m ≥ m0 ; 173

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• the sequence {T m x} is convergent to a fixed point y ∗ of T ; • y ∗ is the unique fixed point of T in Λ = {y ∈ Ω : d(T m0 x, y) < ∞}; • d(y, y ∗ ) ≤

1 1−L d(y, T y)

for all y ∈ Λ.

2. Sextic functional inequalities in non-Archimedean Banach spaces Throughout this section, we will assume that X is a non-Archimedean Banach space. In this section, we establish the conditional stability of sextic functional equations in non–Archimedean Banach spaces. Theorem 2.1. Let s ∈ {−1, 1} be fixed. Let E be a real or complex linear space and let X be a non–Archimedean Banach space. Suppose f : E ×E → X satisfies the condition f (x, 0) = f (0, y) = 0 and inequalities of the form kf (ax1 + bx2 , y) + f (bx1 + ax2 , y) + abf (x1 − x2 , y) x1 + x2 −(a2 + b2 )[f (x1 , y) + f (x2 , y)] − 4abf ( , y)k ≤ φ(x1 , x2 , y), 2 1 kf (x, ay1 + by2 ) + f (x, by1 + ay2 ) + ab(a − b)2 f (x, y1 − y2 ) 2 y1 + y2 −(a2 − b2 )2 [f (x, y1 ) + f (x, y2 )] − 8abf (x, )k ≤ ψ(x, y1 , y2 ), 2 where φ, ψ : E × E × E → [0, ∞) is given functions such that φ((a + b)s x1 , (a + b)s x2 , (a + b)s y) ≤ |a + b|6s Lφ(x1 , x2 , y), ψ((a + b)s x, (a + b)s y1 , (a + b)s y2 ) ≤ |a + b|6s Lψ(x, y1 , y2 ), and have the properties
lim (a + b)−6sn φ (a + b)sn x1 , (a + b)sn x2 , (a + b)sn y = 0, n→∞
lim (a + b)−6sn ψ (a + b)sn x, (a + b)sn y1 , (a + b)sn y2 = 0,

(2.1)

(2.2)

(2.3)

(2.4)

n→∞

for all x, x1 , x2 , y, y1 , y2 ∈ E and a constant 0 < L < 1. Then there exists a unique sextic mapping T : E × E → X satisfying the system (1.10) and 1 Φ(x, y), (2.5) kT (x, y) − f (x, y)k ≤ 1−L where n  1  s−1 s−1 s−1 Φ(x, y) := max (a + b)−3s+1 φ (a + b) 2 x, (a + b) 2 x, (a + b) 2 y , 2 o  s−1 s−1 (a + b)−3s−3 ψ (a + b) s+1 2 x, (a + b) 2 y, (a + b) 2 y for all x, y ∈ E. Proof. We denote A := a + b. Putting x1 = x2 = x in (2.1), we get 1 2 kf (Ax, y) − A f (x, y)k ≤ φ(x, x, y) 2 for all x, y ∈ E. Putting y1 = y2 = y and replacing x by Ax in (2.2), we get 1 4 kf (Ax, Ay) − A f (Ax, y)k ≤ ψ(Ax, y, y) 2 174

(2.6)

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for all x, y ∈ E. Thus by (2.6) and (2.7) we have 1  6 kf (Ax, Ay) − A f (x, y)k ≤ max A4 φ(x, x, y), ψ(Ax, y, y) , 2 for all x, y ∈ E. By last inequality we get 1  kA f (Ax, Ay) − f (x, y)k ≤ max A−2 φ(x, x, y), A−6 ψ(Ax, y, y) , 2 x y n  x x y   y y o 1 kA6 f , − f (x, y)k ≤ max A4 φ , , , ψ x, , , A A 2 A A A A A −6

(2.8) (2.9)

for all x, y ∈ E. Therefore



1 s s

A6s f (A x, A y) − f (x, y) ≤ Φ(x, y),

(2.10)

for all x, y ∈ E. We now consider the set S = {h : E × E → X,

h(x, 0) = h(0, x) = 0 for all x ∈ E}

and introduce the generalized metric on S as follows: n o d(h, k) = inf α ∈ R+ : kh(x, y) − k(x, y)k ≤ αΦ(x, y), ∀x, y ∈ E where, as usual, inf ∅ = +∞. The proof of the fact that (S, d) is a complete generalized metric space, can be found in [6]. Now we consider the mapping J : S → S defined by Jh(x, y) := A−6s h(As x, As y) for all h ∈ S and x, y ∈ E. Let f, g ∈ S such that d(f, g) < ε. Then kJg(x, y) − Jf (x, y)k = kA−6s g(As x, As y) − A−6s f (As x, As y)k = |A−6s |kg(As x, As y) − f (As x, As y)k ≤ |A−6s |εΦ(As x, As y) ≤ LεΦ(x, y), that is, if d(f, g) < ε, then we have d(Jf, Jg) ≤ Lε. This means that d(Jf, Jg) ≤ Ld(f, g) for all f, g ∈ S, that is, J is a strictly contractive self-mapping on S with the Lipschitz constant L. It follows from (2.10) that kJf (x, y) − f (x, y)k ≤ Φ(x, y) for all x, y ∈ E which implies that d(Jf, f ) ≤ 1. Due to Theorem 1.6, there exists a unique mapping T : E × E → X such that T is a fixed point of J, i.e., T (As x, As y) = A6s T (x, y) for all x, y ∈ E. Also, d(J m f, T ) → 0 as m → ∞, which implies the equality lim A−6sm f (Asm x, Asm y) = T (x, y)

m→∞

for all x, y ∈ E. 175

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It follows from (2.1), (2.2) and (2.4) that kT (ax1 + bx2 , y) + T (bx1 + ax2 , y) + abT (x1 − x2 , y) − (a2 + b2 )[T (x1 , y) + T (x2 , y)] x1 + x2 − 4abT ( , y)k = 2 f (Ans (x1 − x2 ), Ans y) f (Ans (ax1 + bx2 ), Ans y) f (Ans (bx1 + ax2 ), Ans y) + + ab lim k n→∞ A6ns A6ns A6ns ns ns ns ns ns ns f (A x1 , A y) + f (A x2 , A y) f (A {x1 + x2 /2}, A y) (2.11) − (a2 + b2 ) − 4ab k= A6ns A6ns lim |A−6ns |kf (Ans (ax1 + bx2 ), Ans y) + f (Ans (bx1 + ax2 ), Ans y) + abf (Ans (x1 − x2 ), Ans y) n→∞

− (a2 + b2 )[f (Ans x1 , Ans y) + f (Ans x2 , Ans y)] − 4abf (Ans {x1 + x2 /2}, Ans y)k ≤ lim |A−6ns |φ(Ans x1 , Ans x2 , Ans y) = 0, n→∞

for all x1 , x2 , y ∈ E, and 1 kT (x, ay1 + by2 ) + T (x, by1 + ay2 ) + ab(a − b)2 T (x, y1 − y2 ) − (a2 − b2 )2 [T (x, y1 ) + T (x, y2 )] 2 y1 + y2 − 8abT (x, )k = 2 f (Ans x, Ans (ay1 + by2 )) f (Ans x, Ans (by1 + ay2 )) lim k + n→∞ A6ns A6ns (2.12) ns ns 1 2 f (A x, A (y1 − y2 )) + ab(a − b) k= 2 A6ns ns ns f (Ans x, Ans y1 ) + f (Ans x, Ans y2 ) 2 f (A x, A {y1 + y2 /2}) − (a2 − b2 )2 − 8ab(a + b) k A6ns A6ns ≤ lim |A6ns |ψ(Ans x, Ans y1 , Ans y2 ) = 0, n→∞

for all x, y1 , y2 ∈ E. It follows from (2.11)) and (2.12) that T satisfies (1.10), that is, T is sextic. According to the fixed point alternative, since T is the unique fixed point of J in the set Ω = {g ∈ S : d(f, g) < ∞}, T is the unique mapping such that kf (x, y) − T (x, y)k ≤ Φ(x, y) for all x, y ∈ E. Using the fixed point alternative, we obtain that 1 1 d(f, T ) ≤ d(f, Jf ) ≤ Φ(x, y), 1−L 1−L for all x, y ∈ E, which implies the inequality (2.5).



Corollary 2.2. Let s ∈ {−1, 1} be fixed. Let E be a normed space and let F be a is a nonArchimedean Banach space. Suppose f : E × E → F is a mapping with f (x, 0) = f (0, y) = 0 and there exist constants θ, ϑ ≥ 0 and non-negative real number p such that ps < 6s and kf (ax1 + bx2 , y) + f (bx1 + ax2 , y) + abf (x1 − x2 , y) x1 + x2 −(a2 + b2 )[f (x1 , y) + f (x2 , y)] − 4abf ( , y)k ≤ θ(kx1 kp + kx2 kp + kykp ), 2 1 kf (x, ay1 + by2 ) + f (x, by1 + ay2 ) + ab(a − b)2 f (x, y1 − y2 ) 2 y1 + y2 2 2 2 −(a − b ) [f (x, y1 ) + f (x, y2 )] − 8abf (x, )k ≤ ϑ(kxkp + ky1 kp + ky2 kp ), 2 176

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for all x, x1 , x2 , y, y1 , y2 ∈ E, where norms of the left-hand side of last inequalities is the non– Archimedean norm on F . Then there exists a unique sextic mapping T : E × E → F such that n θ(2kxkp + kykp ) ϑ(k|a + b|xkp + 2kykp ) o kf (x, y) − T (x, y)k ≤ max , , 2s|a + b|2 − 2s|a + b|p−4 2s|a + b|6 − 2s|a + b|p for all x, y ∈ E. Proof. Defining φ(x1 , x2 , y) = θ(kx1 kp + kx2 kp + kykp ),

ψ(x, y1 , y2 ) = ϑ(kxkp + ky1 kp + ky2 kp ),

and applying Theorem 2.1, we get the desired result.



3. Sextic functional inequalities in (Archimedean) Banach spaces Throughout this section, we will assume that X is a (Archimedean) Banach space. In this section, we establish the conditional stability of sextic functional equations. Theorem 3.1. Let s ∈ {−1, 1} be fixed. Let E be a real or complex linear space and let X be a (Archimedean) Banach space. Suppose f : E × E → X satisfies the condition f (x, 0) = f (0, y) = 0 and inequalities of (2.1) and (2.2), where φ, ψ : E ×E ×E → [0, ∞) are given functions which satisfy (2.3) and have the properties (2.4) for all x, x1 , x2 , y, y1 , y2 ∈ E and a constant 0 < L < 1. Then there exists a unique sextic mapping T : E × E → X satisfying the system (1.10) and kT (x, y) − f (x, y)k ≤

1 ˜ Φ(x, y), 1−L

(3.1)

where n   1 s−1 s−1 s−1 ˜ Φ(x, y) := (a + b)−3s+1 φ (a + b) 2 x, (a + b) 2 x, (a + b) 2 y 2 o  s+1 s−1 s−1 + (a + b)−3s−3 ψ (a + b) 2 x, (a + b) 2 y, (a + b) 2 y for all x, y ∈ E. Proof. We denote A := a + b. Putting x1 = x2 = x in (2.1), we get 1 2 kf (Ax, y) − A f (x, y)k ≤ φ(x, x, y) 2

(3.2)

for all x, y ∈ E. Putting y1 = y2 = y and replacing x by Ax in (2.2), we get 1 4 kf (Ax, Ay) − A f (Ax, y)k ≤ ψ(Ax, y, y) 2

(3.3)

for all x, y ∈ E. Thus by (3.2) and (3.3) we have 1  6 kf (Ax, Ay) − A f (x, y)k ≤ A4 φ(x, x, y) + ψ(Ax, y, y) , 2 for all x, y ∈ E. By last inequality we get 1  kA f (Ax, Ay) − f (x, y)k ≤ A−2 φ(x, x, y) + A−6 ψ(Ax, y, y) , 2 n x y  y y o 1 x x y kA6 f , − f (x, y)k ≤ A4 φ , , + ψ x, , , A A 2 A A A A A −6

177

(3.4) (3.5)

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for all x, y ∈ E. Therefore



1 s s

˜

A6s f (A x, A y) − f (x, y) ≤ Φ(x, y), for all x, y ∈ E. The rest of the proof is similar to the proof of Theorem 2.1.



Corollary 3.2. Let s ∈ {−1, 1} be fixed. Let E be a normed space and let F be a (Archimedean) Banach space. Suppose f : E × E → F is a mapping with f (x, 0) = f (0, y) = 0 and there exist constants θ, ϑ ≥ 0 and non-negative real number p such that ps < 6s and kf (ax1 + bx2 , y) + f (bx1 + ax2 , y) + abf (x1 − x2 , y) x1 + x2 , y)k ≤ θ(kx1 kp + kx2 kp + kykp ), −(a2 + b2 )[f (x1 , y) + f (x2 , y)] − 4abf ( 2 1 kf (x, ay1 + by2 ) + f (x, by1 + ay2 ) + ab(a − b)2 f (x, y1 − y2 ) 2 y1 + y2 2 2 2 −(a − b ) [f (x, y1 ) + f (x, y2 )] − 8abf (x, )k ≤ ϑ(kxkp + ky1 kp + ky2 kp ), 2 for all x, x1 , x2 , y, y1 , y2 ∈ E. Then there exists a unique sextic mapping T : E × E → F such that kf (x, y) − T (x, y)k ≤

ϑ(k|a + b|xkp + 2kykp ) θ(2kxkp + kykp ) , + 2s|a + b|6 − 2s|a + b|p 2s|a + b|2 − 2s|a + b|p−4

for all x, y ∈ E. Proof. Defining φ(x1 , x2 , y) = θ(kx1 kp + kx2 kp + kykp ),

ψ(x, y1 , y2 ) = ϑ(kxkp + ky1 kp + ky2 kp ),

and applying Theorem 3.1, we get the desired result.



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Dynamical behaviors of a nonlinear virus infection model with latently infected cells and immune response M. A. Obaid Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. Email: [email protected]

Abstract In this paper, we study the global stability of a mathematical model that describes the virus dynamics under the e¤ect of antibody immune response. The model is a modi…cation of some of the existing virus dynamics models by considering the latently infected cells and nonlinear incidence rate for virus infections. We show that the global dynamics of the model is completely determined by two threshold values R0 , the corresponding reproductive number of viral infection and R1 , the corresponding reproductive number of antibody immune response, respectively. Using Lyapunov method, we have proven that, if R0 stable (GAS), if R1

1, then the uninfected steady state is globally asymptotically

1 < R0 , then the infected steady state without antibody immune response

is GAS, and if R1 > 1, then the infected steady state with antibody immune response is GAS. Keywords: Virus infection; Global stability; Immune response; Lyapunov function; nonlinear infection rate.

1

Introduction

Recently, mathematical modeling and analysis of viral infections such as hepatitis C virus (HCV) [1]-[3], hepatitis B virus (HBV) [4]-[5], human immunode…ciency virus (HIV) [6]-[15] human T cell leukemia (HTLV) [16] have attracted the interest several researchers. In 1996, Nowak and Bangham [7] has proposed the basic viral infection model which contains three compartments, the uninfected target cells, infected cells and free virus particles. This model does not take into consideration the latently infected cells which is due to the delay between the moment of infection and the moment when the infected cell becomes active to produce infectious viruses. Latently infected cells have been incorporated into the basic viral infection model in several papers (see e.g. [18], [19] and [20]). The

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basic viral infection model which takes into account the latently infected cells is given by [20]: x_ =

dx

w_ = xv

xv;

(1)

(e + b)w;

(2)

y_ = bw

ay;

(3)

v_ = ky

cv;

(4)

where x, w; y and v representing the populations of the uninfected target cells, latently infected cells, actively infected cells and free virus particles, respectively. Parameters

and k represent,

respectively, the rate at which new uninfected cells are generated from the source within the body, and the generation rate constant of free viruses produced from the actively infected cells. Parameters d, e, a and c are the natural death rate constants of the uninfected cells, latently infected cells, actively infected cells and free virus particles, respectively. Parameter

is the infection rate constant. Eq. (2)

describes the population dynamics of the latently infected cells and show that they are converted to actively infected cells with rate constant b. All the parameters given in model (1)-(4) are positive. We observe that in model (1)-(4), the immune response has been neglected. To provide more accurate modelling for the viral infection, the e¤ect of immune response has to be considered. The antibody immune response which is based on the antibodies that are produced by the B cells plays an important role in controlling the disease [17]. In the literature, several mathematical models have been formulated to consider the antibody immune response into the viral infection models (see e.g., [21]-[24]). However, in [21]-[24], it was assumed that all the infected cells are active which is an unrealistic assumption. The aim of this paper is to propose a viral infection model with antibody immune response taking into consideration both latently and actively infected cells and investigate its basic and global properties. The incidence rate is given by nonlinear function which is more general than the bilinear incidence rate given in model (1)-(4). Using Lyapunov functions, we prove that the global dynamics of the model is determined by two threshold parameters, the basic reproductive number of viral infection R0 and the the basic reproductive number of antibody immune response R1 . If R0

1; then the infection-free steady state is globally asymptotically stable (GAS), if R1

1 < R0 ,

then the chronic-infection steady state without antibody immune response is GAS, and if R1 > 1, then the chronic-infection steady state with antibody immune response is GAS.

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2

The mathematical model

In this section, we propose a viral infection model with latently infected cells and antibody immune response. The incidence rate is given by a nonlinear infection rate. xn v ; ( n + xn )( + v) xn v w_ = (1 ) n (e + b)w; ( + xn )( + v) xn v + bw ay; y_ = ( n + xn )( + v) x_ =

dx

v_ = ky

cv

z_ = gvz

) and

with 0
0, w(t) 0 s1 x(0) + w(0) + y(0) L1 . On the other hand, let

X2 (t) = v(t) + gr z(t), then X_ 2 = ky

cv

where s2 = minfc; g. Hence X2 (t) z(t)

0, then 0

v(t)

L2 and 0

r z g

kL1

r s2 v + z g

L2 , if X2 (0) z(t)

L3 if 0

184

= kL1

s2 X2 ;

kL1 . Since v(t) s2 L2 , where L3 = gLr 2 .

L2 , where L2 = v(0) + gr z(0)

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2.2

Steady states

In this subsection, we calculate the steady states of model (5)-(9) and derive two thresholds parameters. The steady states of model (5)-(9) satisfy the following equations: xn v = 0; ( n + xn )( + v)

dx (1

)

(10)

xn v (e + b)w = 0; ( n + xn )( + v) xn v + bw ay = 0; ( n + xn )( + v) ky

(11) (12)

cv

rvz = 0;

(13)

(gv

)z = 0:

(14)

Equation (14) has two possible solutions, z = 0 or v = =g. If z = 0, then from Eqs. (11) and (12) we obtain w and y as: w=

(1 ) e+b (

n

xn v ; + xn )( + v)

(e + b) a(e + b) (

y=

n

xn v : + xn )( + v)

(15)

Substituting Eq. (15) into Eq. (13), we obtain k(e + b) a(e + b) (

xn v n + xn )( + v)

cv = 0:

(16)

Equation (16) has two possibilities, v = 0 or v 6= 0. If v = 0, then w = y = 0 and x =

d

which leads

to the uninfected steady state E0 = (x0 ; 0; 0; 0; 0), where x0 = d . If v 6= 0, then from Eqs. (10) and (16) we obtain v=

xn v k(e + b)( dx) = n + xn )( + v) ac(e + b) ac(e + b) v: dk(e + b)

k(e + b) ac(e + b) (

) x = x0

(17) (18)

From Eq. (18) into Eq. Eq. (16) we get k(e + b) a(e + b) Let us de…ne a function

1

1 (v)

It is clear that,

1 (0)

continuous for all v

ac(e+b) dk(e +b) v

x0 n

n n

ac(e+b) dk(e +b) v

+ x0

v ( + v)

cv = 0:

as =

k(e + b) a(e + b)

ac(e+b) dk(e +b) v

x0 n

+ x0

= 0, and when v = v =

n

ac(e+b) dk(e +b) v

x0 dk(e +b) ac(e+b)

n

v ( + v)

> 0, then

cv = 0:

1 (v)

=

cv < 0. Since

0; then we have 0 1 (0)

=c

k(e + b) ac (e + b)

185

xn0 n + xn 0

1 :

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1 (v)

is

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Therefore, if

> 0 i.e. k(e + b) ac (e + b)

then there exist a v1 2 (0; v) such that Eq. (10) we de…ne a function

2

2 (0)

=

> 0 and

function of x; for all n;

2 (x0 )

(e +b) a(e+b) (

xn 1 v1 n +xn )( +v ) 1 1

=

= 0. From Eq. (13) we obtain y1 = kc v1 > 0 and from

(

> 0; then

2

2 (x1 )

xn v1 = 0: ( n + xn )( + v1 )

dx

=

unique x1 2 (0; x0 ) such that

xn0 > 1; + xn0

as:

2 (x)

We have

1 (v1 )

n

xn 0 v1 n +xn )( +v ) 1 0

< 0. Since f (x) =

xn n +xn

is a strictly increasing

is a strictly decreasing function of x, and there exist a

= 0. It follows that, w1 =

(1 ) e+b (

xn 1 v1 n +xn )( +v ) 1 1

> 0 and y1 =

> 0. It means that, an infected steady state without antibody immune response

E1 (x1 ; w1 ; y1 ; v1 ; 0) exists when

k(e +b) ac (e+b)

xn 0 n +xn 0

> 1. Then we can de…ne the basic reproductive number

of viral infection as: R0 =

k(e + b) ac (e + b)

xn0 : n + xn 0

The parameter R0 determines whether a chronic-infection can be established. The other possibility of Eq. (14) is v2 =

g

. Inserting v2 in Eq. (10) and de…ning a function

3

as: 3 (x)

Note that,

3

=

dx

xn v2 = 0: ( n + xn )( + v2 )

is a strictly decreasing function of x.

Clearly,

3 (0)

=

xn 0 v2 n +xn ( +v ) 2 0

< 0. Thus, there exists a unique x2 2 (0; x0 ) such that ) ( from Eqs. (11)-(13) that, (1 ) xn2 v2 (e + b) ; y2 = n n e + b ( + x2 )( + v2 ) a(e + b) ( n c k(e + b) x2 z2 = 1 : r ac(e + b) ( n + xn2 )( + v2 )

w2 =

Thus w2 ; y2 > 0, and if

k(e +b) ac(e+b) (

xn 2 n +xn )( 2

+v2 )

> 0 and 3 (x2 )

3 (x0 )

=

= 0. It follows

xn2 v2 ; n + xn )( + v ) 2 2

> 1, then z2 > 0. Now we de…ne the basic reproductive

number of antibody immune response: R1 =

k(e + b) ac(e + b) (

xn2 ; n + xn )( + v ) 2 2

which determines whether a persistent antibody immune response can be established. Hence, z2 can be rewritten as z2 = rc (R1

1). It follows that, there exists an infected steady state with antibody

immune response E2 (x2 ; w2 ; y2 ; v2 ; z2 ) when R1 > 1. Since x1 < x0 and v2 > 0, then R1 =

k(e + b) ac(e + b) (

xn2 k(e + b) < n + xn )( + v ) ac (e + b) 2 2

xn0 = R0 : n + xn 0

From above we have the following result. Lemma 1 (i) if R0 (ii) if R1

1; then there exists only one positive steady state E0 ,

1 < R0 ; then there exist two positive steady states E0 and E1 , and

(iii) if R1 > 1; then there exist three positive steady states E0 , E1 and E2 .

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3

Main results

In this section, we investigate the global stability of steady states E0 , E1 and E2 employing the direct Lyapunov method and LaSalle’s invariance principle.

3.1

Global stability of the uninfected steady state E0

Theorem 1. If R0

1; then E0 is globally asymptotically stable (GAS).

Proof. De…ne a Lyapunov functional W0 as follows: W0 = x

x0

Zx

xn0 ( sn (

x0 dW0 dt

Calculating

n

+ sn ) b e+b a(e + b) ar(e + b) ds + w+ y+ v+ z: n + xn ) e +b e +b k(e + b) kg(e + b) 0

along the trajectories of (5)-(9) as:

xn v b xn v + xn ) dx + (1 ) (e + b)w n + xn ) ( n + xn )( + v) e +b ( n + xn )( + v) 0 xn v a(e + b) ar(e + b) e+b + bw ay + (ky cv rvz) + (gvz z) + e +b ( n + xn )( + v) k(e + b) kg(e + b) xn0 ( n + xn ) x ac(e + b) ar (e + b) xn0 v = 1 1 v z + n n n n n x ( + x0 ) x0 ( + x0 )( + v) k(e + b) kg(e + b) x ar (e + b) xn0 ( n + xn ) xn0 ac(e + b) k(e + b) 1 1 v = 1 z + xn ( n + xn0 ) x0 k(e + b) ac(e + b) ( n + xn0 )( + v) kg(e + b) xn0 ( n + xn ) x ar (e + b) ac(e + b) = 1 1 R0 1 v z + n n n x ( + x0 ) x0 k(e + b) +v kg(e + b) x ac(e + b) ar (e + b) xn0 ( n + xn ) 1 + R0 1 v z = 1 n n n x ( + x0 ) x0 k(e + b) +v kg(e + b) d n (xn xn0 ) (x0 x) ac(e + b) ac(e + b)R0 v 2 ar (e + b) = + (R 1)v z: (19) 0 xn ( n + xn0 ) k(e + b) k(e + b) + v kg(e + b)

dW0 = dt

xn0 ( xn (

1

We have (xn

n

xn0 ) (x0

x)

dW0 dt

1 then

dW0 dt

0 for all x; v; z > 0. Thus, n o 0 the solutions of system (5)-(9) limited to M , the largest invariant subset of dW = 0 [25]. Clearly, dt it follows from Eq. (19) that

0 for all x; n > 0. Thus if R0

= 0 if and only if x(t) = x0 , v(t) = 0 and z(t) = 0. The set M is

invariant and for any element belongs to M satis…es v(t) = 0 and z(t) = 0, then v(t) _ = 0. We can see from Eq. (8) that, 0 = v(t) _ = ky(t), and thus y(t) = 0. Moreover, from Eq. (7) we get w(t) = 0. Hence,

dW0 dt

= 0 if and only if x(t) = x0 , w(t) = 0, y(t) = 0; v(t) = 0 and z(t) = 0. From LaSalle’s

invariance principle, E0 is GAS.

3.2

Global stability of the infected steady state without antibody immune response E1

Theorem 2. If R1

1 < R0 ; then E1 is GAS.

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Proof. We construct the following Lyapunov functional W1 = x

x1

Zx

xn1 ( sn (

x1

+

e+b y1 H e +b

y y1

n

b + sn ) ds + w1 H n + xn ) e +b 1 +

a(e + b) v1 H k(e + b)

v v1

w w1 +

ar(e + b) z: kg(e + b)

The time derivative of W1 along the trajectories of (5)-(9) is given by dW1 = dt

1

xn1 ( xn (

b 1 e +b e+b 1 + e +b a(e + b) + k(e + b)

+

Applying

= dx1 +

(

xn 1 v1 n +xn )( +v ) 1 1

xn v + xn ) dx n n+x ) ( n + xn )( + v) 1 xn v w1 (1 ) n (e + b)w w ( + xn )( + v) y1 xn v + bw ay y ( n + xn )( + v) v1 ar(e + b) (gvz 1 (ky cv rvz) + v kg(e + b)

n

z) :

and collecting terms of Eq. (20) we get

xn1 ( n + xn ) xn1 v (dx dx) + 1 xn ( n + xn1 ) ( n + xn1 )( + v) xn1 ( n + xn ) xn1 v1 1 + n ( + xn1 )( + v1 ) xn ( n + xn1 ) n b(1 ) x v w1 b(e + b) (e + b) xn v y1 + w 1 n n n n e + b ( + x )( + v) w e +b e + b ( + x )( + v) y (e + b)b y1 w e+b ac(e + b) a(e + b) yv1 ac(e + b) + ay1 v + v1 e +b y e +b k(e + b) (e + b) v k(e + b) ar(e + b) ar (e + b) + v1 z z: k(e + b) kg(e + b)

dW1 = dt

1

Using the equilibrium conditions for E1 : (1

)

(e + b) xn1 v1 = (e + b)w1 ; y1 = n n ( + x1 )( + v1 ) a(e + b) (

xn1 v1 ; cv1 = ky1 ; n + xn )( + v ) 1 1

we obtain dW1 xn1 ( n + xn ) x v( + v1 ) v xn1 v1 = dx1 1 1 + n dt xn ( n + x1 ) x1 ( n + xn1 )( + v1 ) v1 ( + v) v1 xn1 v1 xn1 ( n + xn ) b(1 ) (e + b) + + 1 e +b e +b ( n + xn1 )( + v1 ) xn ( n + xn1 ) b(1 ) xn1 v1 xn ( n + xn1 )( + v1 )vw1 e + b ( n + xn1 )( + v1 ) xn1 ( n + xn )( + v)v1 w b(1 ) xn1 v1 (e + b) xn1 v1 xn ( n + xn1 )( + v1 )vy1 + e + b ( n + xn1 )( + v1 ) e + b ( n + xn1 )( + v1 ) xn1 ( n + xn )( + v)v1 y b(1 ) xn1 v1 y1 w b(1 ) (e + b) xn1 v1 + + e + b ( n + xn1 )( + v1 ) yw1 e +b e +b ( n + xn1 )( + v1 ) b(1 ) (e + b) xn1 v1 yv1 + e +b e +b ( n + xn1 )( + v1 ) y1 v b(1 ) (e + b) xn1 v1 ar(e + b) + + + v1 z: n n e +b e +b ( + x1 )( + v1 ) k(e + b) g

188

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(20)

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Collecting terms we get ar(e + b) xn1 (v v1 )2 d n (xn xn1 ) (x x1 ) + v1 z n n 2 n n n x ( + x1 ) ( + x1 )( + v)( + v1 ) k(e + b) g b(1 ) xn1 v1 xn1 ( n + xn ) xn ( n + xn1 )( + v1 )vw1 y1 w + 5 (e + b) ( n + xn1 )( + v1 ) xn ( n + xn1 ) xn1 ( n + xn )( + v)v1 w yw1 n n n n x1 ( + x ) xn ( n + xn1 )( + v1 )vy1 yv1 x1 v1 (e + b) 4 + (e + b) ( n + xn1 )( + v1 ) xn ( n + xn1 ) xn1 ( n + xn )( + v)v1 y y1 v

dW1 = dt

yv1 +v y1 v + v1 +v : + v1 (21)

Clearly, the …rst two terms of Eq. (21) are less than or equal zero. Because the geometrical mean is less than or equal to the arithmetical mean, then the last two terms of Eq. (21) are less than or equal zero. Now we show that if R1

1 then v1

sgn (x2

r

= v2 . This can be achieved if we show that

x1 ) = sgn (v1

v2 ) = sgn (R1

1) :

We have (xn2 Suppose that, sgn (x2

xn1 ) (x2

x1 ) = sgn (v2

x1 ) > 0;

for all n > 0

(22)

v1 ). Using the conditions of the steady states E1 and E2 we

have (

dx2 )

(

xn1 v1 xn2 v2 ( n + xn2 )( + v2 ) ( n + xn1 )( + v1 ) xn2 v2 xn2 v1 xn2 v1 = n + ( + xn2 )( + v2 ) ( n + xn2 )( + v1 ) ( n + xn2 )( + v1 ) n (xn xn (v2 v1 ) v1 xn1 ) 2 = n 2 n + + x2 ( + v2 )( + v1 ) + v1 ( n + xn2 )( n + xn1 )

dx1 ) =

xn1 v1 ( n + xn1 )( + v1 )

and from inequalities (22) we get: sgn (x1 which leads to contradiction. Thus, sgn (x2 for E1 we have R1

k(e +b) ac(e+b) (

xn 1 n +xn )( 1

+v1 )

k(e + b) ac(e + b) k(e + b) = ac(e + b) xn2 + n ( + xn2 )( k(e + b) = ac(e + b)

1=

x2 ) = sgn (x2

x1 ) ;

x1 ) = sgn (v1

v2 ) : Using the steady state conditions

= 1, then

xn2 ( n + xn2 )( + v2 ) xn2 n ( n + x2 )( + v2 )

xn1 ( n + xn1 )( + v1 ) xn2 n ( n + x2 )( + v1 )

xn1 + v1 ) ( n + xn1 )( + v1 ) xn2 (v1 v2 ) + ( n + xn2 )( + v1 )( + v2 ) (

n

+

n (xn 2 xn2 )( n +

xn1 ) : xn1 )( + v1 )

From inequality (22) we get: sgn (R1

1) = sgn (v1

189

v2 ) :

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.1, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

It follows that, if R1

1 then v1

r

= v2 . Therefore, if R1

1 then

dW1 dt

0 for all x; w; y; v; z > 0,

where the equality occurs at the steady state E1 . LaSalle’s invariance principle implies the global stability of E1 .

3.3

Global stability of the infected steady state with antibody immune response E2

Theorem 3. If R1 > 1, then E2 is GAS. Proof. We construct the following Lyapunov functional W2 = x

x2

Zx

xn2 ( sn (

x2

e+b + y2 H e +b

n

+ sn ) b ds + w2 H n + xn ) e +b 2

y y2

+

a(e + b) v2 H k(e + b)

v v2

w w2 +

ar(e + b) z2 H kg(e + b)

z z2

:

We calculate the time derivative of W2 along the trajectories of (5)-(9) as: dW2 = dt

1

xn2 ( xn (

b 1 e +b e+b + 1 e +b a(e + b) + k(e + b) +

Applying

= dx2 +

(

xn v + xn ) dx n + xn ) ( n + xn )( + v) 2 w2 xn v (1 ) n (e + b)w w ( + xn )( + v) y2 xn v + bw ay y ( n + xn )( + v) v2 ar(e + b) z2 1 (ky cv rvz) + 1 (gvz v kg(e + b) z n

xn 2 v2 n +xn )( +v ) 2 2

z) :

(23)

and collecting terms of Eq. (23) we get

xn2 ( n + xn ) xn2 v (dx dx) + 2 xn ( n + xn2 ) ( n + xn2 )( + v) xn2 ( n + xn ) b(1 ) xn v w2 b(e + b) xn2 v2 + n 1 + w2 n n n n n n ( + x2 )( + v2 ) x ( + x2 ) e + b ( + x )( + v) w e +b (e + b) xn v y2 (e + b)b y2 w e+b ac(e + b) a(e + b) yv2 + ay2 v n e + b ( + xn )( + v) y e +b y e +b k(e + b) (e + b) v ar(e + b) ar (e + b) ar(e + b) ar (e + b) ac(e + b) + v2 + v2 z z z2 v + z2 : k(e + b) k(e + b) kg(e + b) k(e + b) kg(e + b)

dW2 = dt

1

Using the steady state conditions for E2 (1

)

xn2 v2 = (e + b)w2 ; ( n + xn2 )( + v2 )

xn2 v2 + bw2 = ay2 ; ky2 = cv2 + rv2 z2 ; ( n + xn2 )( + v2 )

190

= gv2 ;

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we get x v( + v2 ) xn2 ( n + xn ) xn2 v2 dW2 v 1 + = dx2 1 n n n n n dt x ( + x2 ) x2 ( + x2 )( + v2 ) v2 ( + v) v2 n x2 v2 xn2 ( n + xn ) b(1 ) (e + b) 1 + + e +b e +b ( n + xn2 )( + v2 ) xn ( n + xn2 ) n n n n x ( + x2 )( + v2 )vw2 b(1 ) x2 v2 n n e + b ( + x2 )( + v2 ) xn2 ( n + xn )( + v)v2 w xn ( n + xn2 )( + v2 )vy2 xn2 v2 (e + b) xn2 v2 b(1 ) + e + b ( n + xn2 )( + v2 ) e + b ( n + xn2 )( + v2 ) xn2 ( n + xn )( + v)v2 y b(1 ) xn2 v2 y2 w b(1 ) (e + b) xn2 v2 + + n n n e + b ( + x2 )( + v2 ) yw2 e +b e +b ( + xn2 )( + v2 ) yv2 xn2 v2 xn2 v2 b(1 ) (e + b) b(1 ) (e + b) + + + e +b e +b ( n + xn2 )( + v2 ) y2 v e +b e +b ( n + xn2 )( + v2 ) d n (xn xn2 ) (x x2 ) xn2 (v v2 )2 = xn ( n + xn2 ) ( n + xn2 )( + v)( + v2 )2 xn2 ( n + xn ) xn ( n + xn2 )( + v2 )vw2 y2 w yv2 xn2 v2 b(1 ) +v 5 + n n (e + b) ( + x2 )( + v2 ) xn ( n + xn2 ) xn2 ( n + xn )( + v)v2 w yw2 y2 v + v2 (e + b) xn2 ( n + xn ) xn ( n + xn2 )( + v2 )vy2 yv2 +v xn2 v2 + 4 : n n n n n n n n (e + b) ( + x2 )( + v2 ) x ( + x2 ) x2 ( + x )( + v)v2 y y2 v + v2 (24) Thus, if R1 > 1 then x2 ; w2 ; y2 ; v2 and z2 > 0. Clearly,

dW2 dt

0 and

dW2 dt

= 0 if and only if

x(t) = x2 ; w(t) = w2 and v(t) = v2 . From Eq. (8), if v(t) = v2 and y(t) = y2 , then v(t) _ = 0 and 0 = ky2

cv2

rv2 z(t), which yields z(t) = z2 and hence

dW2 dt

equal to zero at E2 . LaSalle’s invariance

principle implies global stability of E2 .

4

Conclusion

In this paper, we have proposed and analyzed a virus dynamics model with antibody immune response. The model is a …ve dimensional that describe the interaction between the uninfected target cells, latently infected cells, actively infected cells, free virus particles and antibody immune cells. The incidence rate has been represented by nonlinear function. We have derived two threshold parameters, the basic reproductive number of viral infection R0 and the basic reproductive number of antibody immune response R1 which completely determined the basic and global properties of the virus dynamics model. Using Lyapunov method and applying LaSalle’s invariance principle we have proven that, if R0

1, then the uninfected steady state is GAS, if R1

1 < R0 , then the infected steady

state without antibody immune response is GAS, and if R1 > 1, then the infected steady state with antibody immune response is GAS.

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5

Acknowledgements

This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

References [1] R. Qesmi, J. Wu, J. Wu and J.M. He¤ernan, “In‡uence of backward bifurcation in a model of hepatitis B and C viruses,” Math. Biosci., 224 (2010), 118-125. [2] R. Qesmi, S. ElSaadany, J.M. He¤ernan and J. Wu, “A hepatitis B and C virus model with age since infection that exhibit backward bifurcation,”SIAM J. Appl. Math., 71(4) (2011) 1509-1530. [3] A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J, Layden and A. S. Perelson, “Hepatitis C viral dynamics in vivo and the antiviral e¢ cacy of interferon-alpha therapy,”Science, 282 (1998), 103-107. [4] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, “The dynamics of a delay model of HBV infection with logistic hepatocyte growth,” Math. Biosc. Eng., 6 (2009), 283-299. [5] J. Li, K. Wang, Y. Yang, “Dynamical behaviors of an HBV infection model with logistic hepatocyte growth,” Math. Comput. Modelling, 54 (2011), 704-711. [6] M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, 2000. [7] M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science, 272 (1996), 74-79. [8] A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,” SIAM Rev., 41 (1999), 3-44. [9] L. Wang, M.Y. Li, “Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells,” Math. Biosc., 200(1), (2006), 44-57. [10] A. S. Alsheri, A.M. Elaiw and M. A. Alghamdi, Global dynamics of two target cells HIV infection model with Beddington-DeAngelis functional response and delay-discrete or distributed, Journal of Computational Analysis and Applications, 17 (2014), 187-202. [11] A. M. Elaiw, A. S. Alsheri and M. A. Alghamdi, Global properties of HIV infection models with nonlinear incidence rate and delay-discrete or distributed, Journal of Computational Analysis and Applications, 17 (2014), 230-244.

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[12] A. M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 253703. [13] X. Wang, A. Elaiw, and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Math. Appl. Comput., 218 (2012), 9405-9414. [14] A. M. Elaiw, “Global properties of a class of virus infection models with multitarget cells,” Nonlinear Dynam., 69 (2012) 423-435. [15] A. M. Elaiw, “Global properties of a class of HIV models,” Nonlinear Anal. Real World Appl., 11 (2010), 2253–2263. [16] M. Y. Li and H. Shu, “Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response,” Nonlinear Anal. Real World Appl., 13 (2012), 1080-1092. [17] J. A. Deans and S. Cohen, “Immunology of malaria,” Ann. Rev. Microbiol. 37 (1983), 25-49. [18] A.S. Perelson, D. Kirschner, R. De Boer, “Dynamics of HIV infection of CD4+ T cells,” Mathematical Biosciences 114(1) (1993) 81-125. [19] A.S. Perelson, and P.W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,”SIAM Rev., 41 (1999), 3-44. [20] A. Korobeinikov, “Global properties of basic virus dynamics models,”Bull. Math. Biol. 66, (2004), 879-883. [21] A. Murase, T. Sasaki and T. Kajiwara, “Stability analysis of pathogen-immune interaction dynamics,” J. Math. Biol., 51 (2005), 247-267. [22] M. A . Obaid and A.M. Elaiw, “Stability of virus infection models with antibodies and chronically infected cells,” Abstr. Appl. Anal, 2014, (2014) Article ID 650371. [23] A. M. Elaiw, A. Alhejelan, Global dynamics of virus infection model with humoral immune response and distributed delays. Journal of Computational Analysis and Applications, 17 (2014), 515-523. [24] S. Wang and D. Zou, “Global stability of in host viral models with humoral immunity and intracellular delays,” J. Appl. Math. Mod., 36 (2012), 1313-1322. [25] J. K. Hale, and S. V. Lunel, Introduction to functional di¤ erential equations, Springer-Verlag, New York, 1993.

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On the symmetric properties for the generalized twisted (h, q)-tangent numbers and polynomials associated with p-adic integral on Zp C. S. Ryoo Department of Mathematics, Hannam University, Daejeon 306-791, Korea

Abstract : In this paper, we study the symmetry for the generalized twisted (h, q)-tangent numbers (h) (h) Tn,χ,q,ζ and polynomials Tn,χ,q,ζ (x). We obtain some interesting identities of the power sums and the (h)

generalized twisted polynomials Tn,χ,q,ζ (x) using the symmetric properties for the p-adic invariant integral on Zp . Key words : Symmetric properties, power sums, the tangent numbers and polynomials, the generalized twisted (h, q)-tangent numbers and polynomials, p-adic integral on Zp . 2000 Mathematics Subject Classification : 11B68, 11S40, 11S80. 1. Introduction Recently, many mathematicians have studies different kinds of the Euler, Bernoulli, Genocchi, Tangent numbers and polynomials(see [1-10]). These numbers and polynomials play important roles in many different areas of mathematics such as number theory, combinatorics, special function and analysis. The purpose of this paper is to obtain some interesting identities of the power sums and (h) generalized twisted (h, q)-tangent polynomials Tn,χ,q,ζ (x) using the symmetric properties for the p-adic invariant integral on Zp . Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers and Z+ = N ∪ {0} , C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p−1 . When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C one normally assume that |q| < 1. If q ∈ Cp , we normally assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for |x|p ≤ 1. Let U D(Zp ) be the space of uniformly differentiable function on Zp . For g ∈ U D(Zp ) the 1

fermionic p-adic invariant q-integral on Zp is defined by Kim as follows: ∫

p∑ −1 1 f (x)(−q)x , see [1, 2, 3] . I−q (f ) = f (x)dµ−q (x) = lim N →∞ [pN ]−q Zp x=0 N

∫

Note that lim I−q (g) = I−1 (g) =

q→1

Zp

g(x)dµ−1 (x).

(1.1)

If we take gn (x) = g(x + n) in (1.1), then we see that I−1 (gn ) = (−1)n I−1 (g) + 2

n−1 ∑

(−1)n−1−l g(l).

(1.2)

l=0

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Let a fixed positive integer d with (p, d) = 1, set X = Xd = lim(Z/dpN Z), ←− N ∪ ∗ X = a + dpZp ,

X1 = Zp ,

0 0, there exists a positive integer n0 such that for all m, n, l ≥ n0 , G(xn , xm , xl ) < , • Gb -convergent to a point x ∈ X if for each  > 0, there exists a positive integer n0 such that for all m, n ≥ n0 , G(xn , xm , x) < .

Proposition 2.4. [5]. Let X be a Gb -metric space. (1) The sequence {xn } is Gb -Cauchy. (2) For any  > 0, there exists n0 ∈ N such that G(xn , xm , xm ) < , for all m, n ≥ n0 .

Proposition 2.5. [5]. Let X be a Gb -metric space. The following are equivalent: (1) {xn } is Gb -convergent to x. (2) G(xn , xn , x) → 0 as n → ∞. (3) G(xn , x, x) → 0 as n → ∞.

Denition 2.6.

[5]. A Gb -metric space X is called complete if every Gb -Cauchy sequence is Gb -convergent in X .

The complex metric space was initiated by Azam et al. [4]. Let C be the set of complex numbers and z1 , z2 ∈ C. Dene a partial order - on C as follows: z1 - z2 if and only if Re(z1 ) ≤ Re(z2 ) and Im(z1 ) ≤ Im(z2 ).

It follows that z1 - z2 if one of the following conditions is satised: (C1 ) Re(z1 ) = Re(z2 ) and Im(z1 ) = Im(z2 ), (C2 ) Re(z1 ) < Re(z2 ) and Im(z1 ) = Im(z2 ), (C3 ) Re(z1 ) = Re(z2 ) and Im(z1 ) < Im(z2 ), (C4 ) Re(z1 ) < Re(z2 ) and Im(z1 ) < Im(z2 ). Particularly, we write z1  z2 if z1 6= z2 and one of (C2 ), (C3 ) and (C4 ) is satised and we write z1 ≺ z2 if only (C4 ) is satised. The following statements hold: (1) If a, b ∈ R with a ≤ b, then az ≺ bz for all z ∈ C. (2) If 0 - z1  z2 , then |z1 | < |z2 |. (3) If z1 - z2 and z2 ≺ z3 , then z1 ≺ z3 .

3

Complex Valued

Gb -Metric

Spaces

In this section, we dene the complex valued Gb -metric space.

Denition 3.1. satises:

Let X be a nonempty set and s ≥ 1 be a given real number. Suppose that a mapping G : X × X × X → C

(CGb 1) G(x, y, z) = 0 if x = y = z ; (CGb 2) 0 ≺ G(x, x, y) for all x, y ∈ X with x 6= y ; (CGb 3) G(x, x, y) - G(x, y, z) for all x, y, z ∈ X with y 6= z ; (CGb 4) G(x, y, z) = G(p{x, y, z}), where p is a permutation of x, y, z ; (CGb 5) G(x, y, z) - s(G(x, a, a) + G(a, y, z)) for all x, y, z, a ∈ X (rectangle inequality). Then, G is called a complex valued Gb -metric and (X, G) is called a complex valued Gb -metric space. From (CGb 5), we have the following proposition.

Proposition 3.2. Let (X, G) be a complex valued Gb -metric space. Then for any x, y, z ∈ X , • G(x, y, z) - s(G(x, x, y) + G(x, x, z)), • G(x, y, y) - 2sG(y, x, y). 364

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Denition 3.3. Let (X, G) be a complex valued Gb -metric space, let {xn } be a sequence in X . (i) {xn } is complex valued Gb -convergent to x if for every a ∈ C with 0 ≺ a, there exists k ∈ N such that G(x, xn , xm ) ≺ a

for all n, m ≥ k. (ii) A sequence {xn } is called complex valued Gb -Cauchy if for every a ∈ C with 0 ≺ a, there exists k ∈ N such that G(xn , xm , xl ) ≺ a for all n, m, l ≥ k. (iii) If every complex valued Gb -Cauchy sequence is complex valued Gb -convergent in (X, G), then (X, G) is said to be complex valued Gb -complete.

Proposition 3.4. Let

(X, G) be a complex valued Gb -metric space and {xn } be a sequence in X . Then {xn } is complex valued Gb -convergent to x if and only if |G(x, xn , xm )| → 0 as n, m → ∞.

Proof. (⇒) Assume that {xn } is complex valued Gb -convergent to x and let   a = √ + i√ 2 2

for a real number  > 0. Then we have 0 ≺ a ∈ C and there is a natural number k such that G(x, xn , xm ) ≺ a for all n, m ≥ k. Thus, |G(x, xn , xm )| < |a| =  for all n, m ≥ k and so |G(x, xn , xm )| → 0 as n, m → ∞. (⇐) Suppose that |G(x, xn , xm )| → 0 as n, m → ∞. For a given a ∈ C with 0 ≺ a, there exists a real number δ > 0 such that for z ∈ C |z| < δ ⇒ z ≺ a. Considering δ , we have a natural number k such that |G(x, xn , xm )| < δ for all n, m ≥ k. This means that G(x, xn , xm ) ≺ a for all n, m ≥ k, i.e., {xn } is complex valued Gb -convergent to x. From Propositions 3.2 and 3.4, we can prove the following theorem.

Theorem 3.5. Let (X, G) be a complex valued Gb -metric space, then for a sequence {xn } in X and point x ∈ X , the following are equivalent: (1) {xn } is complex valued Gb -convergent to x. (2) |G(xn , xn , x)| → 0 as n → ∞. (3) |G(xn , x, x)| → 0 as n → ∞. (4) |G(xm , xn , x)| → 0 as m, n → ∞. Proof. (1) ⇒ (2) It is clear from Proposition 3.4. (2) ⇒ (3) By Proposition 3.2, we have

G(xn , x, x) - s(G(xn , xn , x) + G(xn , xn , x))

and using (2), we get |G(xn , x, x)| → 0 as n → ∞. (3) ⇒ (4) If we use (CGb 4) and Proposition 3.2, then G(xm , xn , x) = G(x, xm , xn ) - s(G(x, x, xm ) + G(x, x, xn )) = s(G(xm , x, x) + G(xn , x, x))

and |G(xm , xn , x)| → 0 as m, n → ∞. (4) ⇒ (1) We will use the equivalence in Proposition 3.4, (CGb 3) and (CGb 4). Since G(x, xn , xm ) = G(xm , x, xn ) - s(G(xm , xm , x) + G(xm , xm , xn )) - s(G(xm , xn , x))

and |G(xm , xn , x)| → 0 as m, n → ∞, we obtain |G(x, xn , xm )| → 0 and this completes the proof.

Theorem 3.6. Let (X, G) be a complex valued Gb -metric space and {xn } be a sequence in X . Then {xn } is complex valued Gb -Cauchy sequence if and only if |G(xn , xm , xl )| → 0 as n, m, l → ∞.

Proof. (⇒) Let {xn } be complex valued Gb -Cauchy sequence and ε ε b = √ + i√ 2 2

where ε > 0 is a real number. Then 0 ≺ b ∈ C and there is a natural number k such that G(xn , xm , xl ) ≺ b for all n, m, l ≥ k. Therefore, we get |G(xn , xm , xl )| < |b| = ε for all n, m, l ≥ k and the required result. (⇐) Assume that |G(xn , xm , xl )| → 0 as n, m, l → ∞. Then there exists a real number γ > 0 such that for z ∈ C |z| < γ implies z ≺ b

where b ∈ C with 0 ≺ b. For this γ , there is a natural number k such that |G(xn , xm , xl )| < γ for all n, m, l ≥ k. This means that G(xn , xm , xl ) ≺ b for all n, m, l ≥ k. Hence {xn } is complex valued Gb -Cauchy sequence. 365

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We prove the contraction principle in complex valued Gb -metric spaces as follows:

Theorem 3.7. Let (X, G) be a complete complex valued Gb -metric space with coecient s > 1 and T : X → X be a mapping satisfying: G(T x, T y, T z) - kG(x, y, z)

(3.1)

for all x, y, z ∈ X , where k ∈ [0, 1s ). Then T has a unique xed point. Proof. Let T satisfy (3.1), x0 ∈ X be an arbitrary point and dene the sequence {xn } by xn = T n x0 . From (3.1), we obtain G(xn , xn+1 , xn+1 ) - kG(xn−1 , xn , xn ).

(3.2)

Using again (3.1), we have G(xn−1 , xn , xn ) - kG(xn−2 , xn−1 , xn−1 )

and by (3.2), we get

G(xn , xn+1 , xn+1 ) - k2 G(xn−2 , xn−1 , xn−1 ).

If we continue in this way, we nd

G(xn , xn+1 , xn+1 ) - kn G(x0 , x1 , x1 ).

(3.3)

Using (CGb 5) and (3.3) for all n, m ∈ N with n < m, G(xn , xm , xm ) - s[G(xn , xn+1 , xn+1 ) + G(xn+1 , xm , xm )] - s[G(xn , xn+1 , xn+1 )] + s2 [G(xn+1 , xn+2 , xn+2 ) + G(xn+2 , xm , xm )] - s[G(xn , xn+1 , xn+1 )] + s2 [G(xn+1 , xn+2 , xn+2 )]+ s3 [G(xn+2 , xn+3 , xn+3 )] + . . . + sm−n G(xm−1 , xm , xm )] - (skn + s2 kn+1 + s3 kn+2 + . . . + sm−n km−1 )G(x0 , x1 , x1 ) - skn [1 + sk + (sk)2 + (sk)3 + . . . + (sk)m−n−1 ]G(x0 , x1 , x1 ) -

skn G(x0 , x1 , x1 ). 1 − sk

Thus, we obtain |G(xn , xm , xm )| ≤

skn |G(x0 , x1 , x1 )|. 1 − sk

Since k ∈ [0, 1s ) where s > 1, taking limits as n → ∞, then skn |G(x0 , x1 , x1 )| → 0. 1 − sk

This means that |G(xn , xm , xm )| → 0.

By Proposition 3.2, we get G(xn , xm , xl ) - G(xn , xm , xm ) + G(xl , xm , xm )

for n, m, l ∈ N. Thus, |G(xn , xm , xl )| ≤ |G(xn , xm , xm )| + |G(xl , xm , xm )|.

If we take limit as n, m, l → ∞, we obtain |G(xn , xm , xl )| → 0. So {xn } is complex valued Gb -Cauchy sequence by Theorem 3.6. Completeness of (X, G) gives us that there is an element u ∈ X such that {xn } is complex valued Gb -convergent to u. To prove T u = u, we will assume the contrary. From (3.1), we obtain G(xn+1 , T u, T u) - kG(xn , u, u)

and |G(xn+1 , T u, T u)| ≤ k|G(xn , u, u)|.

If we take the limit as n ≥ ∞, we get |G(u, T u, T u)| ≤ k|G(u, u, u)|,

which is a contradiction since k ∈ [0, As a result, T u = u. Lastly, we prove the uniqueness. Let w 6= u be another xed point of T . Using (3.1), 1 ). s

G(z, w, w) = G(T z, T w, T w) - kG(z, w, w).

and |G(z, w, w)| ≤ k|G(z, w, w)|.

Since k ∈ [0,

1 ), s

we have |G(z, w, w)| ≤ 0. Thus, u = w and so u is a unique xed point of T . 366

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Example 3.8.

Let X = [−1, 1] and G : X × X × X → C be dened as follows: G(x, y, z) = |x − y| + |y − z| + |z − x|

for all x, y, z ∈ X . (X, G) is complex valued G-metric space [12]. Dene G∗ (x, y, z) = G(x, y, z)2 . G∗ is a complex valued Gb -metric with s = 2 (see [5]). If we dene T : X → X as T x = condition for all x, y, z ∈ X : 1 x y z G(T x, T y, T z) = G( , , ) = G(x, y, z) - kG(x, y, z) 3 3 3 3 where k ∈ [ 13 , 1s ), s > 1. Thus x = 0 is the unique xed point of T in X .

x , 3

then T satises the following

We will prove Kannan's xed point theorem for complex valued Gb -metric spaces.

Theorem 3.9. Let (X, G) be a complete complex valued Gb -metric space and the mapping T

: X → X satises for every

x, y ∈ X G(T x, T y, T y) - α[G(x, T x, T x) + G(y, T y, T y)]

where α ∈ [0,

(3.4)

. Then T has a unique xed point.

1 ) 2

Proof. Let x0 ∈ X be arbitrary. We dene a sequence {xn } by xn+1 = T xn for all n ≥ 0. We shall show that {xn } is Gb -Cauchy sequence. If xn = xn+1 , then xn is the xed point of T . Thus, suppose that xn 6= xn+1 for all n ≥ 0. Setting G(xn , xn+1 , xn+1 ) = Gn , it follows from (3.4) that G(xn , xn+1 , xn+1 ) = G(T xn−1 , T xn , T xn ) - α[G(xn−1 , T xn−1 , T xn−1 ) + G(xn , T xn , T xn )] - α[G(xn−1 , xn , xn ) + G(xn , xn+1 , xn+1 )] - α[Gn−1 + Gn ] α Gn Gn−1 = βGn−1 , 1−α

where β =

α 1−α

< 1 as α ∈ [0, 21 ). If we repeat this process, then we get Gn - β n G0 .

(3.5)

We can also suppose that x0 is not a periodic point. If xn = x0 , then we have G0 - β n G0 .

Since β < 1, then 1 − β n < 1 and

(1 − β n )|G0 | ≤ 0 ⇒ |G0 | = 0.

It follows that x0 is a xed point of T . Therefore in the sequel of proof we can assume T n x0 6= x0 for n = 1, 2, 3, . . . From inequality (3.4), we obtain G(T n x0 , T n+m x0 , T n+m x0 ) - α[G(T n−1 x0 , T n+m x0 , T n+m x0 ) + G(T n+m−1 x0 , T n+m x0 , T n+m x0 )] - α[β n−1 G(x0 , T x0 , T x0 ) + β n+m−1 G(x0 , T x0 , T x0 )].

So, |G(xn , xn+m , xn+m )| → 0 as n → ∞. It implies that {xn } is a Gb -Cauchy in X . By the completeness of X , there exists u ∈ X such that xn → u. From (CGb 5), we get G(T u, u, u) - s[G(T u, T n+1 x0 , T n+1 x0 ) + G(T n+1 x0 , u, u)] - s(α[G(u, T u, T u) + G(T n x0 , T n+1 x0 , T n+1 x0 )]) + sG(T n+1 x0 , u, u) - sα[G(u, T u, T u) + sαG(T n x0 , T n+1 x0 , T n+1 x0 )]) + sG(T n+1 x0 , u, u).

Letting n → ∞, since sα < 1 and xn → u, we have |G(T u, u, u)| → 0, i.e., u = T u. Now we show that T has a unique xed point. For this, assume that there exists another point v in X such that v = T v . Now, G(v, u, u) - G(T v, T u, T u) - α[G(v, T v, T v) + G(u, T u, T u)] - α[G(v, v, v) + G(u, u, u)] - 0.

Hence, we conclude that u = v . 367

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[14] Z. Mustafa and B. Sims, Fixed point theorems for contractive mappings in complete G-metric spaces, Fixed Point Theory Appl, 2009:917175, (2009). [15] Z. Mustafa, W. Shatanawi and M. Bataineh, Existence of xed point results in G-metric spaces, Int J Math Math Sci, 2009:283028, (2009). [16] Z. Mustafa, J.R. Roshan and V. Parvaneh, Coupled coincidence point results for (ψ, ϕ)-weakly contractive mappings in partially ordered Gb -metric spaces, Fixed Point Theory and Applications, 2013:206, (2013). [17] Z. Mustafa, J.R. Roshan and V. Parvaneh, Existence of tripled coincidence point in ordered Gb -metric spaces and applications to a system of integral equations, J. Inequalities Appl, 2013:453, (2013). [18] V. Parvaneh, J.R. Roshan and S. Radenovic, Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations, Fixed Point Theory and Applications, 2013:130, (2013). [19] V. Parvaneh, A. Razani and J.R. Roshan, Common xed points of six mappings in partially ordered G-metric spaces, Math Sci, 7:18, (2013). [20] K.P.R. Rao, K. BhanuLakshmi, Z. Mustafa and V.C.C. Raju, Fixed and Related Fixed Point Theorems for Three Maps in G-Metric Spaces, J Adv Studies Topology, 3(4), 1219 (2012). [21] K.P.R. Rao, P.R. Swamy and J.R. Prasad, A common xed point theorem in complex valued b-metric spaces, Bulletin of Mathematics and Statistics Research, 1(1), 18 (2013). [22] A. Razani and V. Parvaneh, On Generalized Weakly G-Contractive Mappings in Partially Ordered G-Metric Spaces, Abstr Appl Anal, 2012:701910, (2012). [23] R. Saadati, S.M. Vaezpour, P. Vetro and B.E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Math Comput Modelling, 52, 797801 (2010). [24] S. Sedghi, N. Shobkolaei, J.R. Roshan and W. Shatanawi, Coupled xed point theorems in Gb -metric spaces, Mat. Vesnik, 66(2), 190201 (2014). [25] W. Shatanawi, Fixed point theory for contractive mappings satisfying Φ-maps in G-metric spaces, Fixed Point Theory Appl, 2010:181650, (2010).

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Finite Difference approximations for the Two-side Space-time Fractional Advection-diffusion Equations∗ Yabin Shao1,2 †, Weiyuan Ma

2

1. Department of Applied Mathematics Chongqing University of Posts and Telecommunications, Chongqing, 400065, China 2. College of Mathematics and Computer Science Northwest University for Nationalities, Lanzhou, 730030, China

February 4, 2015

Abstract Fractional order advection-diffusion equation is viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we present a new weighted finite difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the weighted finite difference approximation is conditionally stable and convergent. Numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims. Keywords: Fractional order advection-diffusion equation; Weighted finite difference approximation; Stability; Convergence.

1

INTRODUCTION

In recent years, fractional differential equations have attracted much attention. Many important phenomena in physics [1, 2, 3], finance [4, 5], hydrology [6], engineering [7], mathematics [8] and material science are well described by differential equations of fractional order. These fractional order models tend to be more appropriate than the traditional integer-order models. So, the fractional derivatives are considered to be a very powerful and useful tool. The fractional advection-diffusion equation provides a useful description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation [9]. In this paper, we consider a special case ∗ The authors would like to thank National Natural Science Foundation of China (No.11161041). † Corresponding author. E-mail: [email protected](Y. Shao)

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of anomalous diffusion, the two-sided space-time fractional advection-diffusion equation can be written in the following way ∂ β u(x,t) ∂tβ

α

= −v(x) ∂u(x,t) + d+ (x) ∂ ∂u(x,t) α ∂x +x

α

u(x,t) + d− (x) ∂ ∂− xα + f (x, t),

x ∈ [L, R], t ∈ (0, T ],

u(L, t) = 0, u(R, t) = ϕ(t), t ∈ [0, T ], u(x, 0) = u0 (x), x ∈ (L, R],

(1) (2) (3)

where α and β are parameters describing the order of the space- and timefractional derivatives, respectively, physical considerations restrict 0 < β < 1, 1 < α < 2. The functions v(x, t), d+ (x, t) and d− (x, t) are all non-negative, bounded and d+ (x, t), d− (x, t) ≥ v(x, t). The left-sided (+) and the right-sided (−) Riemann-Liouville fractional derivatives of order α of a function u(x, t) are defined as follows Z x 1 ∂n u(ξ, t) ∂ α u(x, t) = dξ (4) ∂+ xα Γ(n − α) ∂xn L (x − ξ)α+1−n and ∂ α u(x, t) (−1)n ∂ n = ∂− xα Γ(n − α) ∂xn

Z

R

x

u(ξ, t) dξ, (x − ξ)α+1−n

where n is an integer such that n − 1 < α ≤ n. The time derivative given by a Caputo fractional derivative Z t 1 ∂u(x, η) ∂ β u(x, t) = (t − η)−β dη, ∂tβ Γ(1 − β) 0 ∂η

(5) ∂ β u(x,t) ∂tβ

is

(6)

where Γ(·) is the gamma function. As is well known, the fractional order differential operator is a nonlocal operator, which requires more involved computational schemes for its handling. Finite difference schemes for fractional partial differential equations are more complex than partial differential equations [1, 2, 4, 10, 11, 12, 13, 14]. It should note the following work for fractional advection-diffusion equation. Su et al. [13] presented a Crank-Nicolson type finite difference scheme for two-sided space fractional advection-diffusion equation. Liu et al. [14] considered a spacetime fractional advection-diffusion with Caputo time fractional derivative and Riemann-Liouville space fractional derivatives. In this paper, we present a new weighted finite difference approximation for the equation. The rest of the paper is as follows. In Section 2, we derive the new weighted finite difference approximation (NWFDM) for the fractional advection-diffusion equation. The convergence and stability of the finite difference scheme is given in Section 3, where we apply discrete energy method. In Section 4, numerical results are shown which confirm that the numerical method is effective.

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2

NEW WEIGHTED FINITE DIFFERENCE SCHEME

To present the numerical approximation scheme, we give some notations: τ is the time step, unj be the numerical solution at (xi , tn ) for xj = L + ih, tn = nτ, j = 0, 1, · · · , J, n = 0, 1, · · · , N . The shifted Gr¨ unwald formula is applied to discretize the left-handed fractional derivative and right-handed fractional derivative [15], i+1 1 X ∂ α u(xi , tn ) = gj u(xi − (j − 1)h, tn ) + o(h), ∂+ xα hα j=0

(7)

N −i+1 ∂ α u(xi , tn ) 1 X = gj u(xi + (j − 1)h, tn ) + o(h), ∂ − xα hα j=0

(8)

where the Gr¨ unwald coefficients are defined by g0 = 1, gj = (1 −

α+1 )gj−1 , j

j = 1, 2, 3, · · · .

Adopting the discrete scheme in [15], we discretize the Caputo time fractional derivative as, n

τ 1−β X u(xi , tn+1−j ) − u(xi , tn−j ) ∂ β u(xi , tn ) = σj + o(τ ), ∂tβ Γ(2 − β) j=0 τ where σj = (j + 1)1−β − j 1−β . Now we replace (1) with the following weighted finite difference approximation: n

un − uni−1 τ 1−β X un+1−j − un−j i i σj = −vi [θ i+1 Γ(2 − β) j=0 τ 2h +(1 − θ)

i+1 n+1 un+1 d+i X i+1 − ui−1 ] + α [θ gk uni−k+1 2h h k=0

+(1 − θ)

i+1 X

gk un+1 i−k+1 ] +

k=0

+(1 − θ)

NX −i+1

NX −i+1 d−i [θ gk uni+k−1 hα k=0

n+1 n gk un+1 , i+k−1 ] + θfi + (1 − θ)fi

(9)

k=0

for i = 1, 2, · · · , J − 1, n = 0, 1, · · · , N − 1, where θ is the weighting parameter subjected to 0 ≤ θ ≤ 1. When θ = 0, 1, 21 , we get the space-time fractional implicit, explicit, Crank-Nicolson type difference scheme, respectively.

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The above equation (9) can be simplified, for n = 0, −(1 − θ)(ξi + ηi g2 +

ζi )u1i−1

− (1 − θ)ηi

i+1 X

gk u1i−k+1

k=3

−(1 − θ)ζi

J−i+1 X

gk u1i+k−1 + (1 − θ)(ξi − ηi − ζi g2 )u1i+1

k=3

+[1 − (1 − θ)(ηi g1 + ζi g1 )]u1i = θ(ξi + ηi g2 + ζi )u0i−1 +[1 + θ(ηi g1 + ζi g1 )]u0i + θ(−ξi + ηi + ζi g2 )u0i+1 i+1 X

+θηi

J−i+1 X

gk u0i−k+1 + θζi

k=3

gk u0i+k−1

k=3

+Γ(1 − β)τ β (θfi0 + (1 − θ)fi1 ),

(10)

and for n > 0, −(1 − θ)(ξi + ηi g2 + ζi )un+1 i−1 − (1 − θ)ηi

i+1 X

gk un+1 i−k+1

k=3

−(1 − θ)ζi

J−i+1 X

n+1 gk un+1 i+k−1 + (1 − θ)(ξi − ηi − ζi g2 )ui+1

k=3

+[1 − (1 − θ)(ηi g1 + ζi g1 )]un+1 = θ(ξi + ηi g2 + ζi )uni−1 i +[2 − 21−β + θ(ηi g1 + ζi g1 )]uni + θ(−ξi + ηi + ζi g2 )uni+1 +θηi

i+1 X

gk uni−k+1 + θζi

J−i+1 X

k=3

+u0i σn

gk uni+k−1 +

+ Γ(1 − β)τ

(θfin

+ (1 −

dj un−j i

j=1

k=3 β

n−1 X

θ)fin+1 ),

(11)

and Dirichlet boundary conditions un0 = 0, unJ = ϕ(tn ),

n = 1, 2, · · · , N − 1,

and initial conditions u0i = u0 (xi ),

i = 0, 1, · · · , J,

β

β

β

, ηi = d+i τ hΓ(2−β) , ζi = d−i τ hΓ(2−β) and dj = σj+1 − where ξi = vi τ Γ(2−β) α α 2h σj , j = 1, 2, · · · , n − 1. The numerical method (10) and (11) can be written in the matrix form: AU n+1

AU 1 = B0 U 0 + Q0 , = BU n + d1 U n−1 + · · · + dn−1 U 1 + σn U 0 + Qn ,

where Un

=

(un1 , un2 , · · · , unJ−1 )T , 4

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U 0 = [u0 (x1 ), u0 (x2 ), · · · , u0 (xJ−1 )]T , b = (ηJ−1 + ζJ−1 g2 )[(1 − θ)un+1 + θunJ ], J n F n = (f1n , f2n , · · · , fJ−1 + b)T , E Qn

= (ζ1 gJ , ζ2 gJ−1 , · · · , ζJ−1 g2 )T , = Γ(2 − β)τ β (θF n + (1 − θ)F n+1 ) +(1 − θ)UJn+1 E + θUJn E,

and matrix A = (Aij )(J−1)×(J−1) is defined as follows:

Aij =

          

−(1 − θ)(ξi + ηi g2 + ζi ), j = i − 1, 1 − (1 − θ)(ηi g1 + ζi g1 ), j = i, (1 − θ)(ξ − ηi − ζi g2 ), j = i + 1, −(1 − θ)ηi gi+1−j , j = 1, 2, · · · , i − 2, −(1 − θ)ζi gj+1−i , j = i + 2, i + 3, · · · , J − 1.

It is obvious that matrix A is strictly dominant, the system defined by (10) and (11) has unique solution.

3

STABILITY AND CONVERGENCE

In this section, we investigate the stability and convergence of the numerical scheme (9). Theorem 1 For θαΓ(2 − β)τ β max (d+ (x) + d− (x)) ≤ 2 − 21−β , hα x∈[L,R]

(12)

the weighted finite difference scheme (9) for solving equation (1)-(3) is stable. Proof. Let uni , u ˜ni (i = 1, 2, · · · , J, n = 0, 1, 2, · · · , N − 1) be the numerical solutions of (9) corresponding to the initial data u0i and u ˜0i , respectively. Let n n n εi = u ˜i − ui , the stability condition is equivalent to kE n k∞ ≤ kE 0 k∞ ,

n = 0, 1, · · · , N − 1,

(13)

where E n = (εk1 , εk2 , · · · , εkJ−1 ). We will use mathematical induction to get the above result. For n = 0, we have −(1 − θ)[(ξi + ηi g2 + ζi )ε1i−1 + ηi

i+1 X

gk ε1i−k+1

k=3

+ζi

J−i+1 X

gk ε1i+k−1 − (ξi − ηi − ζi g2 )ε1i+1 ]

k=3

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+[1 − (1 − θ)(ηi g1 + ζi g1 )]ε1i = θ[(ξi + ηi g2 + ζi )ε0i−1 +ζi

J−i+1 X

gk ε0i+k−1 + (−ξi + ηi + ζi g2 )ε0i+1

k=3

+ηi

i+1 X

gk ε0i−k+1 ] + [1 + θ(ηi g1 + ζi g1 )]ε0i ,

(14)

k=3

for n > 0, −(1 − θ)[(ξi + ηi g2 + ζi )εn+1 i−1 + ηi

i+1 X

gk εn+1 i−k+1

k=3

+ζi

J−i+1 X

n+1 gk εn+1 i+k−1 − (ξi − ηi − ζi g2 )εi+1 ]

k=3

+[1 − (1 − θ)(ηi g1 + ζi g1 )]εn+1 = i

n−1 X

dj εn−j i

j=1

+σn ε0i + θ[(−ξi + ηi + ζi g2 )εni+1 + ηi

i+1 X

gk εni−k+1

k=3

+ζi

J−i+1 X

gk εni+k−1 + (ξi + ηi g2 + ζi )εni−1 ]

k=3 1−β

+[2 − 2

+ θ(ηi g1 + ζi g1 )]εni .

(15)

Note that d+ (x, t), d− (x, t) ≥ v(x, t), we have ξi − ηi − ζi g2 ≤ 0.

(16)

In fact, if n = 0, supposeP|ε1l | = max1≤i≤J−1 |ε1i |, note that ξi , ηi , ζi > 0 and m for any integer number m, j=0 gj < 0, from (12), (16), we derive kE 1 k∞ = |ε1l | ≤ −(1 − θ)ηl

l+1 X

gk |ε1l | + |ε1l | − (1 − θ)ζl

k=0

≤

J−l+1 X

|ε1l |

k=0

| − (1 − θ)[(ξl + ηl g2 + ζl )ε1l−1 + ζl

J−l+1 X

gl ε1l+k−1

k=3

+(ηl + ζl g2 − ξl )ε1l+1 + ηl

l+1 X

gk ε1l−k+1 ]

k=3

+[1 − (1 − θ)(ηl g1 + ζl g1 )]ε1l | ≤ θ[(ξl + ηl g2 + ζl )|ε0l−1 | + ζl

J−l+1 X

gk |ε0l+k−1 |

k=3

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+(ηl + ζl g2 )|ε0l+1 | + ηl

l+1 X

gk |ε0l−k+1 |]

k=3

+[1 − θ(ξl − ηl g1 − ζl g1 )]|ε0l | ≤ kE 0 k∞ , Suppose that kE n k∞ ≤ kE 0 k∞ , n = 1, 2, · · · , s, then when n = s + 1, let |εs+1 | = max1≤i≤J−1 |εs+1 |. Similar to former estimate, we obtain i l kE s+1 k∞ ≤ | − (1 − θ)[(ξl + ηl g2 + ζl )εn+1 l−1 + ηl

l+1 X

gk εn+1 l−k+1

k=3

+ζl

J−l+1 X

n+1 gk εn+1 l+k−1 − (ξl − ηl − ζl g2 )εl+1 ]

k=3

+[1 − (1 − θ)(ηl g1 + ζl g1 )]εn+1 | l s ≤ θ(ξl + ηl g2 + ζl )|εl−1 | + θ(−ξl + ηl + ζl g2 )|εsl+1 | +[2 − 21−β + θ(ηl g1 + ζl g1 )]|εsl | + θηl

l+1 X

gk |εsl−k+1 |

k=3

+θζl

J−l+1 X

gk |εsl+k−1 | +

dj |εs−j | + σs |ε0l | l

j=1

k=3

≤

s−1 X

0

kE k∞ .

Hence, kE s+1 k∞ ≤ kE 0 k∞ . The proof is completed. Theorem 2 Suppose that u(x, t) is the sufficiently smooth solution of (1)-(3) and uki is the difference solution of difference scheme (9). If the condition (12) is satisfied, then −1 ku(xi , tn ) − uni k∞ ≤ M σn−1 (τ 1+β + τ β h),

where M is a positive constant. Proof. Define eni = u(xi , tn ) − uni and en = (en1 , en2 , · · · , enJ−1 ). Notice that e0j = 0, we have: when n = 0, −(1 − θ)[(ξi + ηi g2 +

ζi )e1i−1

+ ηi

i+1 X

gk e1i−k+1

k=3

+ζi

J−i+1 X

gk e1i+k−1 − (ξi − ηi − ζi g2 )e1i+1 ]

k=3

+[1 − (1 − θ)(ηi g1 + ζi g1 )]e1i = Ri1 ,

(17)

when n > 0, −(1 − θ)[(ξi + ηi g2 + ζi )en+1 i−1 + ηi

i+1 X

gk en+1 i−k+1

k=3

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+ζi

J−i+1 X

n+1 gk en+1 i+k−1 − (ξi − ηi − ζi g2 )ei+1 ]

k=3

+[1 − (1 − θ)(ηi g1 + ζi g1 )]en+1 − θηi i

i+1 X

gk eni−k+1

k=3

−[2 − 21−β + θ(ηi g1 + ζi g1 )]eni − θζi

J−i+1 X

gk eni+k−1

k=3 n−1 X

−θ(ξi + ηi g2 + ζi )eni−1 −

dj en−j i

j=1

−θ(−ξi + ηi +

ζi g2 )eni+1

= Rin+1 ,

(18)

where Rin+1 is the truncation error of difference scheme (9). Furthermore, there exists a positive constant M independent of step sizes such that |Rin+1 | ≤ M (τ 1+β + τ β h). We will prove by inductive method. Let |e1l | = max1≤i≤J−1 |e1i |. If k = 1, subject to the condition (12), based on (17), we have ke1 k∞ ≤ | − (1 − θ)[(ξi + ηi g2 + ζi )e1i−1 + ηi

i+1 X

gk e1i−k+1

k=3

+ζi

J−i+1 X

gk e1i+k−1 − (ξi − ηi − ζi g2 )e1i+1 ]

k=3

+[1 − (1 − θ)(ηi g1 + ζi g1 )]e1i | ≤ M (τ 1+β + τ β h) = σ0−1 M (τ 1+β + τ β h). −1 Assume that ken k∞ ≤ M σn−1 (τ 1+β + τ β h), n = 1, 2, · · · , s, then when n = s + 1, let |es+1 | = max1≤i≤J−1 |es+1 |, notice that σj−1 < σk−1 , j = 0, 1, · · · , k − 1. i l Similarly, we obtain

kes+1 k∞ ≤ d1 kes k∞ +

n−1 X

dj kes−j k∞ + M (τ 1+β + τ β h)

j=1

≤ ≤

−1 −1 (d1 σs−1 + d2 σs−1 + ··· −1 1+β β σs M (τ + τ h).

+ ds σ0−1 + 1)M (τ 1+β + τ β h)

Thus, the proof is completed. In additional, since n−1 1 σn−1 = lim = . n→∞ (1 − β)n−1 n→∞ nβ 1−β lim

there is a constant C1 for which ken k∞ ≤ C1 nβ (τ 1+β + τ β h). and nτ ≤ T is finite, we obtain the following result. 8

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Theorem 3 Under the conditions of Theorem 2, then numerical solution converges to exact solution as h and τ tend to zero. Furthermore there exists positive constant C > 0, such that ku(xi , tn ) − uni k ≤ C(τ + h), where i = 1, 2, · · · , J − 1; n = 1, 2, · · · , N.

4

NUMERICAL RESULTS

In this section, the following two-sided space-time space-time fractional advectiondiffusion equation in a bounded domain is considered in [15]: ∂u(x, t) ∂ 1.6 u(x, t) ∂ 0.6 u(x, t) =− + d+ (x, t) 0.6 ∂t ∂x ∂+ x1.6 ∂ 1.6 u(x, t) +d− (x, t) + f (x, t), (x, t) ∈ [0, 1] × [0, 1] ∂− x1.6 u(0, t) = 0, u(1, t) = 1 + 4t2 , t ∈ [0, 1], u(x, 0) = x2 , x ∈ [0, 1], 100 x2 t1.4 + where d+ (x, t) = 52 Γ(0.4)x0.6 , d− (x, t) = 5Γ(0.4)(1−x)1.6 , and f (x, t) = 7Γ(0.4) 2 2 2 2 (1 + 4t )(−25x + 40x − 12). The exact solution is u(x, t) = (1 + 4t )x .

Table 1: The error max |uki − u(xi , tk )| for the IWFDMs with θ = 1 N

J

State

The error

10

10

Divergence

1.1305e+019

100

10

Divergence

2.3237e+163

10000

10

Divergence

Infinity

30000

10

Convergence

1.3230

Table 1 shows the maximum absolute numerical error between the exact solution and the numerical solution obtained by NWFDM with θ = 1. From Table 1, it can see that our scheme is conditionally stable. Table 2 and Table 3 show the maximum absolute error, at time t = 1.0, between the exact analytical solution and the numerical solution obtained by NWFDM with θ = 1/2 and θ = 0, respectively. Table 4 and Table 5 show the comparison of maximum absolute numerical error of the weighted finite difference scheme in [12] (WFDM) and new weighted finite difference (NWFDM). We can see that the NNWDM is more accurate than WFDM at θ = 0, but at θ = 0.4 is opposite. From the above five tables, it can seen that the numerical tests are in excellent agreement with theoretical analysis. 9

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Table 2: The error and convergence rate for the scheme with θ = 1/2 N

J

Maximum error

Convergence rate

200

200

0.0809

-

400

400

0.0486

1.6646

800

800

0.0298

1.6309

1600

1600

0.0055

1.6022

Table 3: The error and convergence rate for the scheme with θ = 0 N

J

Maximum error

Convergence rate

200

200

0.0415

-

400

400

0.0209

1.9378

800

800

0.0107

1.9533

1600

1600

0.0054

1.9815

References [1] E. Sousa, Finite difference approximations for a fractional advection diffusion problem, Journal of Computational Physics, 228, 4038-4054 (2009). [2] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Applied Mathematics and Computation, 191, 2-20 (2007). [3] P. Zhuang, F. Liu, V. Anh, I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM Journal on Numerical Analysis, 46, 1079-1095 (2008). [4] M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in highfrequency financial data: an empirical study, Physica A: Statistical Mechanics and its Applications, 314, 749-755 (2002). [5] L. Sabatelli, S. Keating, J. Dudley, P. Richmond, Waiting time distributions in financial markets, The European Physical Journal B, 27, 273-275 (2002). [6] L. Galue , S. L. Kalla, B. N. Al-Saqabi, Fractional extensions of the temperature field problems in oil strata, Applied Mathematics and Computation, 186, 35-44 (2007). [7] X. Li, M. Xu, J. Xiang, Homotopy perturbation method to time-fractional diffusion equation with a moving boundary, Applied Mathematics and Computation, 208, 434-439 (2009).

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Table 4: The comparison of two schemes with θ = 0 N

J

NWFDM

WFDM

50

50

0.1514

0.1522

100

100

0.0783

0.0797

150

150

0.0533

0.0545

200

200

0.0405

0.0415

Table 5: The comparison of two schemes with θ = 0.4 N

J

NWFDM

WFDM

50

50

0.2198

0.1498

100

100

0.1242

0.0785

150

150

0.0899

0.0537

200

200

0.0717

0.0409

[8] Z. Odibat, S. Momani, V. S. Erturk, Generalized differential transform method: application to differential equations of fractional order, Applied Mathematics and Computation, 197, 467-477 (2008). [9] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1-77 (2000). [10] Y. Zhang, A finite difference method for fractional partial differential equation, Applied Mathematics and Computation, 215, 524-529 (2009). [11] Z. Q. Ding, A. G. Xiao, M. Li, Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients, Journal of Computational and Applied Mathematics, 233, 1905-1914 (2010). [12] Y. M. Lin, C. J. Xu, Finite difference/spectral approximations for the timefractional diffusion equation, Journal of Computational Physics, 225, 15331552 (2007). [13] L. J. Su, W. Q. Wang, Z. X. Yang, Finite difference approximations for the fractional advection-diffusion equation, Physics Letters: A, 373, 4405-4408 (2009). [14] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of difference methods for the space-time fractional advection-diffusion equation, Applied Mathematics and Computation, 191, 12-20 (2007). [15] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.

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A modified Newton-Shamanskii method for a nonsymmetric algebraic Riccati equation∗ Jian-Lei Lia†, Li-Tao Zhangb , Xu-Dong Lic , Qing-Bin Lib a

College of Mathematics and Information Science, North China University of,

Water Resources and Electric Power, Zhengzhou, Henan, 450011, PR China. b

Department of Mathematics and Physics, Zhengzhou Institute of,

Aeronautical Industry Management, Zhengzhou, Henan, 450015, PR China. c

Investment Projects Construction Agent Center of Luohe City People’s, Government, Luohe, Henan, 462001, PR China.

Abstract The non-symmetric algebraic Riccati equation arising in transport theory can be rewritten as a vector equation and the minimal positive solution of the non-symmetric algebraic Riccati equation can be obtained by solving the vector equation. In this paper, based on the Newton-Shamanskii method, we propose a new iterative method called modified Newton-Shamanskii method for solving the vector equation. Some convergence results are presented. The convergence analysis shows that sequence of vectors generated by the modified Newton-Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Finally, numerical experiments are presented to illustrate the performance of the modified Newton-Shamanskii method. Key words: non-symmetric algebraic Riccati equation; M -matrix; transport theory; minimal positive solution; modified Newton-Shamanskii method. AMSC(2000): 49M15, 65H10, 15A24

1

Introduction

For convenience, firstly, we give some definitions and notations. For any matrices A = [ai,j ] and B = [bi,j ] ∈ Rm×n , we write A ≥ B(A > B) if ai,j ≥ bi,j (ai,j > bi,j ) ∗

This research was supported by the natural science foundation of Henan Province(14B110023), Growth Funds for Scientific Research team of NCWU(320009-00200), Youth Science and Technology Innovation talents of NCWU(70491), Doctoral Research Project of NCWU(201119), National Natural Science Foundation of China(11501525,11501200). † Corresponding author. E-mail:[email protected].

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation holds for all i, j. The Hadamard product of A and B is defined by A ◦ B = [ai,j · bi,j ]. I denotes the identity matrix with appropriate dimension. The superscript T denotes the transpose of a vector or a matrix. We denote the norm by k · k for a vector or a matrix. In this paper we are interested in iteratively solving the following nonsymmetric algebraic Riccati equation (NARE) arising in transport theory (see [3–5, 21] and the references cited therein): XCX − XE − AX + B = 0,

(1.1)

where A, B, C, E ∈ Rn×n have the following special form: A = ∆ − eq T , B = eeT , C = qq T , E = D − qeT .

(1.2)

Here and in the following, e = (1, 1, ..., 1)T , q = (q1 , q2 , ..., qn )T with qi = ci /2ωi ,  1    ∆ = diag(δ1 , δ2 , ..., δn ) with δi = cω (1 + α) , i (1.3) 1   ,  D = diag(d1 , d2 , ..., dn ) with di = cωi (1 − α) and Pn

0 < c ≤ 1, 0 ≤ α < 1,

0 < ωn < ... < ω2 < ω1 < 1

(1.4)

i=1 ci

= 1, ci > 0, i = 1, 2, ..., n. The form of the Riccati equation (1.1) arises in Markov models [22] and in nuclear physics [3, 24], and it has many positive solutions in the componentwise sense. There have been a lot of studies about algebraic properties [11, 21] and iterative methods for the nonnegative solution of the nonsymmetric algebraic Riccati equations (1.1), including the basic fixed-point iterations [5–8,19], the doubling algorithm [9], the Schur method [23,28], the Matrix Sign Function method [13,25] and the alternately linearized implicit iteration method [15], and so on; see related references therein. The existence of positive solutions of (1.1) has been shown in [3] and [4], but only the minimal positive solution is physically meaningful. So it is important to develop some effective and efficient procedures to compute the minimal positive solution of Equation (1.1). Recently, Lu [10] has shown that the matrix equation (1.1) is equivalent to a vector equation and has developed a simple and efficient iterative procedure to compute the minimal positive solution of (1.1). The fixed-point iteration methods were further studied in [14, 16] for solving the vector equation. In [14] Bai, Gao and Lu proposed two nonlinear splitting iteration methods: the nonlinear block Jacobi and the nonlinear block Gauss-Seidel iteration methods. In [16] Bao, Lin and Wei proposed a modified simple iteration method for solving the vector equation. Furthermore, the convergence rates of various fixed-point iterations [10,14,16] were determined and compared in [20].

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation The Newton method has been presented and analyzed by Lu for solving the vector equation in [12]. It has been shown that the Newton method for the vector equation is more simple and efficient than using the corresponding Newton method directly for the original Riccati equation (1.1). Li, Huang and Zhang present a relaxed Newton-like method [17] for solving the vector equation. Especially, in [18] Lin and Bao applied the Newton-Shamanskii method [2, 26] to solve the vector equation. Based on the Newton-Shamanskii method [18], in this paper, we propose a modified Newton-Shamanskii method to solve the vector equation. The convergence analysis shows that the sequence of vectors generated by the new iterative method is monotonically increasing and converges to the minimal positive solution of the vector equation, which can be used to obtain the minimal positive solution of the original Riccati equation. Our method extends the recent work done by Lu [12] and Lin and Bao [18]. Now, we give the definition of Z-matrix and M -matrix, and also give the following two Lemmas which will be used later. Definition 1 [1] A real square matrix A is called a Z-matrix if all its off-diagonal elements are non-positive. Any Z-matrix A can be written as A = sI − B with B ≥ 0, s > 0. Definition 2 [1] Any matrix A of the form A = sI − B for which s > ρ(B), the spectral radius of B, is called an M -matrix. Lemma 1.1 [1] For a Z-matrix A, the following statements are equivalent: (1) A is a nonsingular M -matrix; (2) A is nonsingular and A−1 ≥ 0; (3) Av > 0 for some vector v ≥ 0. Lemma 1.2 [1] Let A ∈ Rn×n be a nonsingular M -matrix. If B ∈ Rn×n is a Z-matrix and satisfies the relation B ≥ A, then B ∈ Rn×n is also a nonsingular M -matrix. The rest of the paper is organized as follows. In Section 2, we review the NewtonShamanskii method and some useful results, and present the modified Newton-Shamanskii method. Some convergence results are given in Section 3. Section 4 and 5 give numerical experiments and conclusions, respectively.

2

The modified Newton-Shamanskii method

It has been shown in [10, 12] that the solution of (1.1) must have the following form: X = T ◦ (uv T ) = (uv T ) ◦ T,

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation where T = [ti,j ] = [1/(δi + dj )] and u, v are two vectors, which satisfy the vector equations: ( u = u ◦ (P v) + e, (2.1) v = v ◦ (P˜ u) + e, where P = [pi,j ] = [qj /(δi + dj )], P˜ = [˜ pi,j ] = [qj /(δj + di )]. Define w = [uT , v T ]T . The equation (2.1) can be rewritten equivalently as f (w) = w − w ◦ Pw − e = 0, where

· P=

0 P P˜ 0

(2.2)

¸ .

The minimal positive solution of (1.1) can be obtained via computing the minimal positive solution of the vector equation (2.2). The Newton method presented by Lu in [12] for the vector equation (2.2) is the following: wk+1 = wk − f 0 (wk )−1 f (wk ), k = 0, 1, 2... where for any w ∈ R2n , the Jacobian matrix f 0 (w) of f (w) is given by · ¸ G1 (v) H1 (u) 0 f (w) = I2n − G(w), with G(w) = H2 (v) G2 (u)

(2.3)

where G1 (v) = diag(P v), G2 (u) = diag(P˜ u), H1 (u) = [u ◦ p1 , u ◦ p2 , ..., u ◦ pn ] and H2 (v) = [v ◦ p˜1 , v ◦ p˜2 , ..., v ◦ p˜n ]. For i = 1, 2, ..., n, pi and p˜i are the ith column of P and P˜ , respectively. Obviously, when w > 0, G(w) ≥ 0 and f 0 (w) is a Z-matrix. The Newton-Shamanskii method for solving the vector equation (2.2) is given in [18] as follows: Algorithm 2.1 (Newton-Shamanskii method) For a given m ≥ 1 and k = 0, 1, 2, ...,  0 −1   w˜k,1 = wk − f (wk ) f (wk ), w˜k,p+1 = w˜k,p − f 0 (wk )−1 f (w˜k,p ), 1 ≤ p ≤ m − 1, (2.4)   w ˜k,m . k+1 = w It has been shown in [18] that the Newton-Shamanskii method has a better convergence than the Newton method [12]. However, if the inversion of the Jacobian matrix f 0 (w) is difficult to compute, the Newton-Shamanskii method may converge slowly. Hence, based on the Newton-Shamanskii method, we propose the following modified Newton-Shamanskii method:

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation

Algorithm 2.2 (Modified Newton-Shamanskii method) For a given m ≥ 1 and k = 0, 1, 2, ..., the Modified Newton-Shamanskii method is defined as follows:  −1   w˜k,1 = wk − Tk f (wk ), w˜k,p+1 = w ˜k,p − Tk−1 f (w˜k,p ), 1 ≤ p ≤ m − 1, (2.5)   w ˜k,m . k+1 = w where Tk is a Z-matrix and Tk ≥ f 0 (wk ). Remark 2.1 When Tk = f 0 (wk ), the modified Newton-Shamanskii method becomes the Newton-Shamanskii method [18]. When m = 1 and Tk = f 0 (wk ), the modified Newton-Shamanskii method becomes the Newton method [12]. Before we give the convergence analysis of the Modified Newton-Shamanskii method, let us now state some results which are indispensable for our subsequent discussions. Lemma 2.1 [18] For any vectors w1 , w2 ∈ R2n , f 0 (w1 ) − f 0 (w2 ) = G(w2 − w1 ). Furthermore, if w2 > w1 , we have f 0 (w1 ) − f 0 (w2 ) = G(w2 − w1 ) ≥ 0. Here and in the subsequent section, for convenience, [f 00 (w)y]y is define as f 00 (w)y 2 . Let f 00 (w)y = [L1 y, L2 y, ..., L2n y]T ∈ R2n×2n , where Li ∈ R2n×2n , y ∈ R2n and for k = 1, 2, ..., n, · ¸ · ¸ 0 (−ek PkT ) 0 (−P˜k eTk ) Lk = , Ln+k = (−ek PkT )T 0 (−P˜k eTk )T 0 T with eTk = (0, ..., 0, 1, 0, ...), PkT and P˜k are the kth rows of the matrices P and P˜ , respectively.

Lemma 2.2 [12] For any vectors w+ , w ∈ R2n , we have 1 f (w+ ) = f (w) + f 0 (w)(w+ − w) + f 00 (w)(w+ − w, w+ − w). 2

(2.6)

In particular, if w+ = w∗ , the minimal positive solution of (2.2), then 1 0 = f (w) + f 0 (w)(w∗ − w) + f 00 (w)(w∗ − w, w∗ − w). 2

(2.7)

Furthermore, for any y > 0 or y < 0, f 00 (w)y 2 < 0

384

(2.8)

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation and f 00 (w)y 2 is independent of w. Because of the independence, in the following, we denote the operator f 00 (w) by L , i.e., L (y, y) = f 00 (w)(y, y) for any y ∈ R2n . By (2.7), we have 1 f (w) = f 0 (w)(w − w∗ ) − L (w − w∗ , w − w∗ ), 2

(2.9)

1 f 0 (w)(w − w∗ ) = f (w) + L (w − w∗ , w − w∗ ). 2

(2.10)

Lemma 2.3 [12] If 0 ≤ w < w∗ and f (w) < 0, then f 0 (w) is a nonsingular M -matrix.

3

Convergence analysis of the Modified NewtonShamanskii method

Now, we analyse convergence of the modified Newton-Shamanskii method (2.5). Theorem 3.1 Given a vector wk ∈ R2n . w˜k,1 , w˜k,2 , ..., w˜k,m , wk+1 are obtained by the modified Newton-Shamanskii method (2.5). If wk < w∗ and f (wk ) < 0, then, f 0 (wk ) is a nonsingular M -matrix, moreover, (1) wk < w˜k,1 < w˜k,2 < ... < w˜k,m = wk+1 < w∗ ; (2) f (w˜k,p ) < 0 for p = 1, 2, ..., m; (3) f 0 (w˜k,p ) is a nonsingular M -matrix for p = 1, 2, ..., m. Therefore, wk+1 < w∗ , f (wk+1 ) < 0 and f 0 (wk+1 ) is a nonsingular M -matrix. Proof. Since wk < w∗ and f (wk ) < 0, by Lemma 2.3, we can easily obtain that f (wk ) is a nonsingular M -matrix. By Lemma 1.2, we can conclude that Tk is also a nonsingular M -matrix. Now, we prove the theorem by mathematical induction. Define the error vectors e˜k,i = w˜k,i − w∗ and ek = wk − w∗ , then ek < 0. For p = 1, we have w˜k,1 = wk − Tk−1 f (wk ). Since f (wk ) < 0 and Tk is also a nonsingular M -matrix, then w˜k,1 > wk by Lemma 1.1. By Eqs. (2.5) and (2.9), we obtain 0

e˜k,1 = ek − Tk−1 f (wk ) 1 = ek − Tk−1 [f 0 (wk )ek − L (ek , ek )] 2 1 = Tk−1 [Tk − f 0 (wk )]ek + Tk−1 L (ek , ek ) < 0. 2

(3.1)

Thus, w ˜k,1 < w∗ .

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation By Eq. (2.6) and Lemma 1.1, we have f (w˜k,1 ) = f (wk − Tk−1 f (wk )) 1 = f (wk ) − f 0 (wk )Tk−1 f (wk ) + L (Tk−1 f (wk ), Tk−1 f (wk )) 2 1 = [Tk − f 0 (wk )]Tk−1 f (wk ) + L (Tk−1 f (wk ), Tk−1 f (wk )) < 0. 2

(3.2)

By Lemma 2.3, it can be concluded that f 0 (w˜k,1 ) is a nonsingular M -matrix. Therefore, the results hold for p = 1. Assume the results are true for 1 ≤ p ≤ t. Then, for p = t + 1, we have w˜k,t+1 = w˜k,t − Tk−1 f (w˜k,t ). Since f (w˜k,t ) < 0 and Tk is a nonsingular M -matrix, then w˜k,t+1 > w˜k,t . Since wk < w˜k,1 < w˜k,2 < ... < w˜k,t , by Lemma 2.1, we have f 0 (wk ) > f 0 (w˜k,1 ) > f 0 (w˜k,2 ) > ... > f 0 (w˜k,t ). Therefore, Tk − f 0 (w˜k,t ) > ... > Tk − f 0 (w˜k,1 ) > Tk − f 0 (wk ) ≥ 0. By Eqs. (2.5) and (2.9), we have the following error vectors equation e˜k,t+1 = e˜k,t − Tk−1 f (w˜k,t ) 1 = e˜k,t − Tk−1 [f 0 (w˜k,t )˜ ek,t − L (˜ ek,t , e˜k,t )] 2 1 ek,t , e˜k,t ) < 0. = Tk−1 [Tk − f 0 (w˜k,t )]˜ ek,t + Tk−1 L (˜ 2

(3.3)

Therefore, w ˜k,t+1 < w∗ . Similarly, by Eq. (2.6) and Lemma 1.1, we have f (w˜k,t+1 ) = f (w˜k,t − Tk−1 f (w˜k,t )) 1 = f (w˜k,t ) − f 0 (w˜k,t )Tk−1 f (w˜k,t ) + L (Tk−1 f (w˜k,t ), Tk−1 f (w˜k,t )) 2 1 −1 0 = [Tk − f (w˜k,t )]Tk f (w˜k,t ) + L (Tk−1 f (w˜k,t ), Tk−1 f (w˜k,t )) < 0. 2

(3.4)

By Lemma 2.3, we have that f 0 (w˜k,t+1 ) is a nonsingular M -matrix. Therefore, the results hold for p = t + 1. Hence, by the principle of mathematical induction, the proof of the theorem is completed. ¤ In practical computation, we should choose Tk such that the iteration step (2.5) is less expensive to implement. For any wk ∈ R2n , according to the structure of the Jacobian f 0 (wk ), Tk may be chosen as · ¸ G1 (vk ) 0 Tk = I2n − (3.5) 0 G2 (uk )

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation or

· Tk = I2n −

G1 (vk ) H1 (uk ) 0 G2 (uk )

¸ .

(3.6)

Another choice for Tk is · Tk = I2n −

G1 (vk ) 0 H2 (vk ) G2 (uk )

¸ .

Numerical experiments show that the performance for this choice is almost the same as that for Tk given by (3.6). The following theorem provides some results concerning the convergence of the modified Newton-Shamanskii method for the vector equation (2.2). Theorem 3.2 Let w∗ be the minimal positive solution of the vector equation (2.2). The sequence of the vector sets {wk , w˜k,1 , w˜k,2 , ..., w˜k,m } obtained by the modified NewtonShamanskii method (2.5) with the initial vector w0 = 0 is well defined. For all k ≥ 0 and 1 ≤ p ≤ m, we have (1) f (wk ) < 0 and f (w˜k,p ) < 0; (2) f 0 (wk ) and f 0 (w˜k,p ) are nonsingular M -matrices; (3) w0 < w˜0,1 < w˜0,2 < ... < w˜0,m = w1 < w˜1,1 < w˜1,2 < ... < w˜1,m = w2 < ... < w˜k−1,m = wk < w˜k,1 < ... < w˜k,m = wk+1 < ... < w∗ . Furthermore, we have lim wk = w∗ . k→∞

Proof. This theorem can also be proved by mathematical induction. The proof is similar to that of the Theorem 1 in [18]. Therefore, it is omitted. ¤

4

Numerical experiments

In this section, we give numerical experiments to illustrate the performance of the modified Newton-Shamanskii method presented in Section 3 with two different choices of the matrix Tk . Let NS denote the Newton-Shamanskii iterative method [18], MNS1 and MNS2 denote the modified Newton-Shamanskii iterative method (2.5) with Tk given by(3.5) and (3.6), respectively. In order to show numerically the performance of the modified Newton-Shamanskii iterative method, we list the number of iteration steps (denoted as IT), the CPU time in seconds (denoted as CPU), and relative residual error (denoted as ERR). The residual error is defined by ¾ ½ kuk+1 − uk k2 kvk+1 − vk k2 , , ERR = max kuk+1 k2 kvk+1 k2

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CPU time with (c, α)=(0.999,0.001)

IT numbers with (c, α)=(0.999,0.001)

7

400 350 NS MNS1 MNS2

NS MNS1 MNS2

300

5 250 IT numbers

CPU time (seconds)

6

4

200 150

3 100 2 50 1

0

5

10 m

15

0

20

0

5

10 m

15

20

Figure 1: CPU time and IT numbers for (c, α) = (0.999, 0.001) and n = 512 with different m. Left: CPU time; right: IT numbers CPU time with (c, α)=(0.5,0.5)

IT numbers with (c, α)=(0.5,0.5)

2

16

1.8 14 1.6

NS MNS1 MNS2

1.2 1

IT numbers

CPU time (seconds)

NS MNS1 MNS2

12

1.4

0.8 0.6

10

8

6

0.4 4 0.2 0

1

2

3

4

5 m

6

7

8

2

9

1

2

3

4

5 m

6

7

8

9

Figure 2: CPU time and IT numbers for (c, α) = (0.5, 0.5) and n = 512 with different m. Left: CPU time; right: IT numbers where k · k2 is the 2-norm for a vector. For comparison, every experiment is repeated 5 times, and the average of the 5 CPU times is shown here. All the experiments are run in MATLAB 7.0 on a personal computer with Intel(R) Pentium(R) D 3.00GHz CPU and 0.99 GB memory, and all iterations are terminated once the current iterate satisfies ERR ≤ n · eps, where eps = 1 × 10−16 . In the test example, the constants ci and wi , i = 1, 2, ...n, are given by the numerical quadrature formula on the interval [0, 1], which are obtained by dividing [0, 1] into n4 subintervals of equal length and applying a Gauss-Legendre quadrature [27] with 4 nodes to each subinterval; see the Example 5.2 in [6] We test several different values (c, α). In Table 1, for n = 512 with different m and pairs of (c, α), and in Table 2, for the fixed (c, α) = (0.99, 0.01) with different n, we list ITs, CPUs and ERRs for the NS method and MNS methods, respectively. Figure 1 and

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Table 1: Numerical results for n = 512 and different pairs of (c, α) m

method NS

1

MNS1

MNS2

NS

3

MNS1

MNS2

NS

6

MNS1

MNS2

NS

12

MNS1

MNS2

IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR

(0.999, 0.001) 10 2.9380 2.1776e-15 376 5.9370 4.7938e-14 195 3.7500 4.7717e-014 6 2.5310 5.3953e-15 132 2.5780 4.3397e-14 69 1.8440 4.4170e-14 5 2.9220 1.9497e-15 68 1.9060 4.7883e-14 36 1.3280 4.7025e-14 4 3.4530 1.9512e-15 36 1.2660 2.9584e-14 20 1.2190 1.1402e-14

389

(c, α) (0.99, 0.01) (0.9, 0.1) 9 7 2.6100 2.2810 1.5433e-15 1.4280e-15 130 43 2.0630 0.7820 4.3618e-14 2.7158e-14 69 24 1.3430 0.5150 3.3654e-14 1.6087e-14 5 5 2.0630 2.0780 4.6570e-14 1.2318e-15 46 16 0.9370 0.3130 3.7357e-14 1.0270e-14 25 10 0.6720 0.2810 2.9843e-14 8.7831e-16 4 4 2.2340 2.2810 1.7274e-15 1.3919e-15 24 9 0.6100 0.2340 4.0900e-14 3.3139e-15 14 6 0.5160 0.2340 8.5873e-15 5.5104e-16 4 3 3.5160 2.5630 1.6885e-15 1.3243e-15 13 5 0.4680 0.1880 2.6812e-14 1.9980e-14 8 4 0.4530 0.2180 4.8204e-15 5.5981e-16

(0.5, 0.5) 5 1.6090 1.5773e-14 16 0.2810 7.0829e-15 10 0.2190 1.7311e-15 4 1.7190 1.0553e-15 6 0.1410 2.6302e-14 5 0.1410 1.5640e-16 3 1.7350 1.0832e-15 4 0.1100 2.9604e-16 3 0.1250 2.1332e-15 3 2.6410 1.1225e-15 3 0.1250 1.64101e-16 3 0.2030 1.6410e-16

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Table 2: Numerical results for (c, α) = (0.99, 0.01) and different n, m m

method NS

1 MNS1

MNS2

NS 5 MNS1

MNS2

NS 10 MNS1

MNS2

IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR IT CPU ERR

64 9 0.0310 6.8597e-16 140 0.0630 5.3918e-15 73 0.0160 6.1723e-15 5 0.0160 8.1022e-16 31 0.0150 2.7024e-15 17 0.0160 2.4804e-15 4 0.0160 7.2604e-16 16 0.0150 6.0508e-15 10 0.0160 5.6442e-16

128 9 0.0620 9.5495e-16 136 0.0940 1.2247e-14 72 0.0630 9.44380e-15 5 0.0630 8.6594e-16 30 0.0310 7.6059e-15 17 0.0310 2.4814e-15 4 0.0780 8.1207e-16 16 0.0150 5.8374e-15 10 0.0320 4.2340e-16

390

n 256 9 0.4220 1.1845e-15 133 0.4850 2.3157e-14 70 0.2970 2.1990e-14 5 0.4220 1.2226e-15 29 0.1410 2.2028e-14 16 0.0930 2.0690e-14 4 0.5000 1.1571e-15 16 0.0780 6.0658e-15 9 0.0630 1.4346e-14

512 9 2.6100 1.5433e-15 130 2.0630 4.3618e-14 69 1.3430 3.3654e-14 5 2.3750 1.6581e-15 29 0.6410 2.1877e-14 16 0.5160 2.0656e-14 4 2.7500 1.5460e-15 15 0.4680 4.9045e-14 9 0.4380 1.3941e-14

1024 9 16.9060 2.7143e-15 126 7.4530 1.0212e-14 67 4.7180 7.8688e-14 5 14.6560 2.1565e-15 28 2.3590 6.3491e-14 16 1.9530 2.0698e-14 4 16.6090 2.2290e-15 15 1.7350 4.8935e-14 9 1.6400 1.3698e-14

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation Figure 2 describe the CPU time and IT numbers of those methods when n = 512 for (c, α) = (0.999, 0.001) and (c, α) = (0.5, 0.5). From these Tables and Figures, we can see that the optimal choice of m for the modified Newton-Shamanskii method is larger when (c, α) = (0.999, 0.001), compared with (c, α) = (0.5, 0.5). Obviously, compared with the Newton-Shamanskii iterative method, though the iterations number of the modified Newton-Shamanskii iterative method is more, according to the CPU time, we can find that the modified Newton-Shamanskii iterative method outperforms the Newton-Shamanskii iterative method. Among these methods, the MNS2 method is the best one.

5

Conclusion

In this paper, based on the Newton-Shamanskii method, we have proposed a modified Newton-Shamanskii method for solving the minimal positive solution of the nonsymmetric algebraic Riccati equation arising in transport theory and have given the convergence analysis. The convergence analysis shows that the iteration sequence generated by the modified Newton-Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the modified Newton-Shamanskii method has a better performance than the Newton-Shamanskii method for the nonsymmetric algebraic Riccati equation. We find that when Tk is chosen as the block triangular of the Jacobian matrix, the modified Newton-Shamanskii method has a better convergence rate. The choice of the matrix Tk impacts the convergence rate of the modified Newton-Shamanskii method, hence, the determination of the optimum matrix Tk such that the modified Newton-Shamanskii method has a better convergence rate needs further to be studied.

References [1] A. Berman, R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, PA, 1994. [2] V. E. Shamanskii. A modification of Newtons method. Ukrainian Mathematical Journal. 19 (1967), pp. 133–138. [3] J. Juang. Existence of algebraic matrix Riccati equations arising in transport theory. Linear Algebra Appl., 230 (1995), pp. 89–100. [4] J. Juang and W. -W. Lin. Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl., 20 (1) (1999), pp. 228–243. [5] J. Juang and I. D. Chen. Iterative solution for a certain class of algebraic matrix Riccati equations arising in transport theory. Transport Theory Statist. Phys., 22 (1993), pp. 65–80.

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation [6] C. -H. Guo and A. J. Laub. On the iterative solution of a class of nonsymmetric algebraicRiccati equations. SIAM J. Matrix Anal. Appl., 22 (2)(2000), pp. 376–391. [7] N. J. Nigham and C. -H. Guo. Iterative solution of a nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl., 29 (2) (2007), pp. 396–412. [8] C. -H. Guo. Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl., 23 (1), (2001), pp. 225–242. [9] C. -H. Guo, B. Iannazzo and B. Meini. On the doubling algorithm for a (shifted) nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl., 29 (2007), pp. 1083–1100. [10] L. -Z. Lu. Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory. SIAM J. Matrix Anal. Appl., 26 (3) (2005), pp. 679–685. [11] L. -Z. Lu and M. K. Ng. Effects of a parameter on a nonsymmetric algebraic Riccati equation. Appl. Math. Comput., 172 (2006), pp. 753–761. [12] L. -Z. Lu. Newton iterations for a non-symmetric algebraic Riccati equation. Numer. Linear Algebra Appl., 12 (2005), pp. 191–200. [13] L. -Z. Lu and C. E. M. Pearce On the mstrix-sign-function method for solving algebraic Riccati equations. Appl. Math. Comput., 86 (1997), pp. 157–170. [14] Z. -Z. Bai, Y. -H. Gao and L. -Z. Lu. Fast iterative schemes for nonsymmetric algebraic raccati equations arising from transport theory. SIAM J.Sci.Comput., 30 (2) (2008), pp. 804–818. [15] Z. -Z. Bai, X. -X. Guo and S. -F. Xu. Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations. Numer. Linear Algebra Appl., 13 (8) (2006), pp. 655–674. [16] L. Bao, Y. -Q. Lin and Y. M. -Wei. A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory. Appl. Math. Comput., 181 (2006), pp. 1499–1504. [17] J. -L. Li, T. -Z. Huang and Z. -J. Zhang. The relaxed Newton-like method for a nonsymmetric algebraic Riccati equation. Journal of Computational Analysis and Applications., 13 (2011), pp. 1132–1142. [18] Y. -Q. Lin and L. Bao. Convergence analysis of the Newton-Shamanskii method for a nonsymmetric algebraic Riccati equations. Numer. Linear Algebra Appl., 15 (2008), pp. 535–546. [19] D. A. Bini, B. Iannazzo and F. Poloni. A fast Newton’s method for a nonsymmetric algebraic Riccati equations. SIAM J. Matrix Anal. Appl., 30 (2008), pp. 276–290.

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Jian-Lei Li etal: A modified Newton-Shamanskii method for Riccati equation [20] C. -H. Guo and W. -W. Lin. Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory. Linear Algebra Appl., 432 (2010), pp. 283–291. [21] V. Mehrmann and H. -G. Xu. Explicit solutions for a Riccati equation from transport theory. SIAM J. Matrix Anal. Appl., 30 (4) (2008), pp. 1339–1357. [22] L. C. G. Rogers. Fluid models in queueing theory and Wiener-Hopf factorization of Markov Chains. Ann. Appl. Probab., 4 (1994), pp. 390–413. [23] C. Paige and C. V. Loan. A Schur decomposition for Hamitonian matrices. Linear Algebra Appl., 41 (1981), pp. 11–32. [24] B. D. Ganapol. An investigating of a simple transport model. Transport Theory Statist. Phys., 21 (1992), pp. 1–37. [25] X. -X. Guo and Z. -Z. Bai. On the minimal nonnegative solution of nonsymmetric algebraic Riccati equation. J. Comput. Math., 23 (2005), pp. 305–320. [26] C. T. Kelley. Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, PA, 1995. [27] G. W. Stewart. Afternotes on Numerical Analysis. SIAM, Philadelphia, 1996. [28] A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Transactions on automatic control., 24 (1979), pp. 913–921.

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Hesitant fuzzy filters and hesitant fuzzy G-filters in residuated lattices G. Muhiuddin Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Abstract. Characterizations of a hesitant fuzzy filter in a residuated lattice are considered. Given a hesitant fuzzy set, a new hesitant fuzzy filter of a residuated lattice is constructed. The notion of a hesitant fuzzy G-filter of a residuated lattice is introduced, and its characterizations are discussed. Conditions for a hesitant fuzzy filter to be a hesitant fuzzy G-filter are provided. Finally, the extension property of a hesitant fuzzy G-filter is established.

1. Introduction The notions of Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets and fuzzy multisets etc. are a generalization of fuzzy sets. As another generalization of fuzzy sets, Torra and Narukawa [5] and Torra [6] introduced the notion of hesitant fuzzy sets and discussed the relationship between hesitant fuzzy sets and intuitionistic fuzzy sets. Xia and Xu [11] studied hesitant fuzzy information aggregation techniques and their application in decision making. They developed some hesitant fuzzy operational rules based on the interconnection between the hesitant fuzzy set and the intuitionsitic fuzzy set. Xu and Xia [12] proposed a variety of distance measures for hesitant fuzzy sets, and investigated the connections of the aforementioned distance measures and further developed a number of hesitant ordered weighted distance measures and hesitant ordered weighted similarity measures. Xu and Xia [13] defined the distance and correlation measures for hesitant fuzzy information and then considered their properties in detail. Wei [9] investigated the hesitant fuzzy multiple attribute decision making problems in which the attributes are in different priority level. Residuated lattices are a non-classical logic system which is a formal and useful tool for computer science to deal with uncertain and fuzzy information. Filter theory, which is an important notion, in residuated lattices is studied by Shen and Zhang [4] and Zhu and Xu [15]. Wei [10] introduced the notion of hesitant fuzzy (implicative, regular and Boolean) filters in residuated lattice, and discussed its properties. 2010 Mathematics Subject Classification: 06F35, 03G25, 06D72. Keywords: Hesitant fuzzy filter, Hesitant fuzzy G-filter. E-mail: [email protected]

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In this paper, we deal with further properties of a hesitant fuzzy filter in a residuated lattice. We consider characterizations of a hesitant fuzzy filter in a residuated lattice. Given a hesitant fuzzy set, we construct a new hesitant fuzzy filter of a residuated lattice. We introduce the notion of a hesitant fuzzy G-filter of a residuated lattice, and discuss its characterizations. We provide conditions for a hesitant fuzzy filter to be a hesitant fuzzy G-filter. Finally, we establish the extension property of a hesitant fuzzy G-filter.

2. Preliminaries Definition 2.1 ([1, 2, 3]). A residuated lattice is an algebra (L, ∨, ∧, , →, 0, 1) of type (2, 2, 2, 2, 0, 0) such that (1) (L, ∨, ∧, 0, 1) is a bounded lattice. (2) (L, , 1) is a commutative monoid. (3) and → form an adjoint pair, that is, (∀x, y, z ∈ L) (x ≤ y → z ⇔ x y ≤ z) . In a residuated lattice L, the ordering ≤ and negation ¬ are defined as follows: (∀x, y ∈ L) (x ≤ y ⇔ x ∧ y = x ⇔ x ∨ y = y ⇔ x → y = 1) and ¬x = x → 0 for all x ∈ L. Proposition 2.2 ([1, 2, 3, 7, 8]). In a residuated lattice L, the following properties are valid. (2.1)

1 → x = x, x → 1 = 1, x → x = 1, 0 → x = 1, x → (y → x) = 1.

(2.2)

y ≤ (y → x) → x.

(2.3)

x ≤ y → z ⇔ y ≤ x → z.

(2.4)

x → (y → z) = (x y) → z = y → (x → z).

(2.5)

x ≤ y ⇒ z → x ≤ z → y, y → z ≤ x → z.

(2.6)

z → y ≤ (x → z) → (x → y), z → y ≤ (y → x) → (z → x).

(2.7)

(x → y) (y → z) ≤ x → z.

(2.8)

x y ≤ x ∧ y.

(2.9)

x ≤ y ⇒ x z ≤ y z.

(2.10)

y → z ≤ x ∨ y → x ∨ z.

(2.11)

(x ∨ y) → z = (x → z) ∧ (y → z).

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Definition 2.3 ([4]). A nonempty subset F of a residuated lattice L is called a filter of L if it satisfies the conditions: (2.12)

(∀x, y ∈ L) (x, y ∈ F ⇒ x y ∈ F ) .

(2.13)

(∀x, y ∈ L) (x ∈ F, x ≤ y ⇒ y ∈ F ) .

Proposition 2.4 ([4]). A nonempty subset F of a residuated lattice L is a filter of L if and only if it satisfies: (2.14)

1 ∈ F.

(2.15)

(∀x ∈ F ) (∀y ∈ L) (x → y ∈ F ⇒ y ∈ F ) .

3. Hesitant fuzzy filters Let E be a reference set. A hesitant fuzzy set on E (see [6]) is defined in terms of a function h that when applied to E returns a subset of [0, 1], that is, h : E → P([0, 1]). In what follows, we take a residuated lattice L as a reference set. Definition 3.1 ([10]). A hesitant fuzzy set h on L is called a hesitant fuzzy filter of L if it satisfies: (3.1)

(∀x, y ∈ L) (x ≤ y ⇒ h(x) ⊆ h(y)) ,

(3.2)

(∀x, y ∈ L) (h(x) ∩ h(y) ⊆ h(x y)) .

Example 3.2. Let L = [0, 1] be a subset of R. For any a, b ∈ L, define a ∨ b = max{a, b}, a ∧ b = min{a, b}, ( 1 if a ≤ b, a→b= (1 − a) ∨ b otherwise, and ( a b=

0

if a + b ≤ 1,

a ∧ b otherwise.

Then (L, ∨, ∧, , →, 0, 1) is a residuated lattice (see [15]). We define a hesitant fuzzy set ( h : L → P([0, 1]), x 7→

(0.2, 0.7)

if x ∈ (c, 1] where 0.5 ≤ c ≤ 1,

(0.3, 0.6]

otherwise.

It is routine to verify that h is a hesitant fuzzy filter of L. Example 3.3. Let L = {0, a, b, c, d, 1} be a set with the lattice diagram appears in Figure 1.

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1r  HH  H r  H a H r b HH   H rc d r r 0 Figure 1 Consider two operation ‘ ’ and ’→’ shown in Table 1 and Table 2, respectively.

Table 1. Cayley table for the binary operation ‘ ’



0

a

b

c

d

1

0

0

0

0

0

0

0

a

0

a

c

c

0

a

b

0

c

b

c

d

b

c

0

c

c

c

0

c

d

0

0

d

0

0

d

1

0

a

b

c

d

1

Table 2. Cayley table for the binary operation ‘→’

→

0

a

b

c

d

1

0

1

1

1

1

1

1

a

d

1

b

b

d

1

b

0

a

1

a

d

1

c

d

1

1

1

d

1

d

a

1

1

1

1

1

1

0

a

b

c

d

1

Then (L, ∨, ∧, , →, 0, 1) is a residuated lattice. We define a hesitant fuzzy set ( [0.2, 0.9) if x ∈ {1, a}, h : L → P([0, 1]), x 7→ (0.3, 0.8] otherwise. It is routine to verify that h is a hesitant fuzzy filter of L. Wei [10] provided a characterization of a hesitant fuzzy filter as follows.

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Lemma 3.4 ([10]). A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if it satisfies (3.3)

(∀x ∈ L) (h(x) ⊆ h(1)) .

(3.4)

(∀x, y ∈ L) (h(x) ∩ h(x → y) ⊆ h(y)) .

We provide other characterizations of a hesitant fuzzy filter. Theorem 3.5. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if it satisfies: (3.5)

(∀x, y, z ∈ L) (x ≤ y → z ⇒ h(x) ∩ h(y) ⊆ h(z)) .

Proof. Assume that h is a hesitant fuzzy filter of L. Let x, y, z ∈ L be such that x ≤ y → z. Then h(x) ⊆ h(y → z) by (3.1), and so h(z) ⊇ h(y) ∩ h(y → z) ⊇ h(x) ∩ h(y) by (3.4). Conversely let h be a hesitant fuzzy set on L satisfying (3.5). Since x ≤ x → 1 for all x ∈ L, it follows from (3.5) that h(1) ⊇ h(x) ∩ h(x) = h(x) for all x ∈ L. Since x → y ≤ x → y for all x, y ∈ L, we have h(y) ⊇ h(x) ∩ h(x → y) for all x, y ∈ L. Hence h is a hesitant fuzzy filter of L.



Theorem 3.6. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if h satisfies the condition (3.3) and (3.6)

(∀x, y, z ∈ L) (h(x → (y → z)) ∩ h(y) ⊆ h(x → z)) .

Proof. Assume that h is a hesitant fuzzy filter of L. Then the condition (3.3) is valid. Using (2.4) and (3.4), we have h(x → z) ⊇ h(y) ∩ h(y → (x → z)) = h(y) ∩ h(x → (y → z)) for all x, y, z ∈ L. Conversely, let h be a hesitant fuzzy set on L satisfying (3.3) and (3.6). Taking x := 1 in (3.6) and using (2.1), we get h(z) = h(1 → z) ⊇ h(1 → (y → z)) ∩ h(y) = h(y → z) ∩ h(y) for all y, z ∈ L. Thus h is a hesitant fuzzy filter of L by Lemma 3.4.

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Lemma 3.7. Every hesitant fuzzy filter h on L satisfies the following condition: (∀a, x ∈ L) (h(a) ⊆ h((a → x) → x)) .

(3.7)

Proof. If we take y = (a → x) → x and x = a in (3.4), then h((a → x) → x) ⊇ h(a) ∩ h(a → ((a → x) → x)) = h(a) ∩ h((a → x) → (a → x)) = h(a) ∩ h(1) = h(a). This completes the proof.



Theorem 3.8. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if it satisfies the following conditions: (3.8)

(∀x, y ∈ L) (h(x) ⊆ h(y → x)) ,

(3.9)

(∀x, a, b ∈ L) (h(a) ∩ h(b) ⊆ h((a → (b → x)) → x)) .

Proof. Assume that h is a hesitant fuzzy filter of L. Using (2.1), (3.3) and (3.4), we have h(y → x) ⊇ h(x) ∩ h(x → (y → x)) = h(x) ∩ h(1) = h(x) for all x, y ∈ L. Using (3.6) and (3.7), we get h((a → (b → x)) → x) ⊇ h((a → (b → x)) → (b → x)) ∩ h(b) ⊇ h(a) ∩ h(b) for all a, b, x ∈ L. Conversely, let h be a hesitant fuzzy set on L satisfying two conditions (3.8) and (3.9). If we take y := x in (3.8), then h(x) ⊆ h(x → x) = h(1) for all x ∈ L. Using (3.9) induces h(y) = h(1 → y) = h((x → y) → (x → y)) → y) ⊇ h(x → y) ∩ h(x) for all x, y ∈ L. Therefore h is a hesitant fuzzy filter of L by Lemma 3.4.



Theorem 3.9. A hesitant fuzzy set h on L is a hesitant fuzzy filter of L if and only if the set hτ := {x ∈ L | τ ⊆ h(x)} is a filter of L for all τ ∈ P([0, 1]) with hτ 6= ∅. Proof. Assume that h is a hesitant fuzzy filter of L. Let x, y ∈ L and τ ∈ P([0, 1]) be such that x ∈ hτ and x → y ∈ hτ . Then τ ⊆ h(x) and τ ⊆ h(x → y). It follows from (3.3) and (3.4) that h(1) ⊇ h(x) ⊇ τ and h(y) ⊇ h(x) ∩ h(x → y) ⊇ τ and so that 1 ∈ hτ and y ∈ hτ . Hence hτ is a filter of L by Proposition 2.4. Conversely, suppose that hτ is a filter of L for all τ ∈ P([0, 1]) with hτ 6= ∅. For any x ∈ L, let h(x) = δ. Then x ∈ hδ and hδ is a filter of L. Hence 1 ∈ hδ and so h(x) = δ ⊆ h(1). For

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any x, y ∈ L, let h(x) = δx and h(x → y) = δx→y . If we take δ = δx ∩ δx→y , then x ∈ hδ and x → y ∈ hδ which imply that y ∈ hδ . Thus h(x) ∩ h(x → y) = δx ∩ δx→y = δ ⊆ h(y). Therefore h is a hesitant fuzzy filter of L by Lemma 3.4.



˜ be a hesitant fuzzy set on L defined by Theorem 3.10. For a hesitant fuzzy set h on L, let h ( h(x) if x ∈ hτ , ˜ : L → P([0, 1]), x 7→ h ∅ otherwise, ˜ where τ ∈ P([0, 1]) \ {∅}. If h is a hesitant fuzzy filter of L, then so is h. Proof. Suppose that h is a hesitant fuzzy filter of L. Then hτ is a filter of L for all τ ∈ P([0, 1]) ˜ ˜ with hτ 6= ∅ by Theorem 3.9. Thus 1 ∈ hτ , and so h(1) = h(1) ⊇ h(x) ⊇ h(x) for all x ∈ L. Let x, y ∈ L. If x ∈ hτ and x → y ∈ hτ , then y ∈ hτ . Hence ˜ ˜ → y) = h(x) ∩ h(x → y) ⊆ h(y) = h(y). ˜ h(x) ∩ h(x ˜ ˜ → y) = ∅. Thus If x ∈ / hτ or x → y ∈ / hτ , then h(x) = ∅ or h(x ˜ ˜ → y) = ∅ ⊆ h(y). ˜ h(x) ∩ h(x ˜ is a hesitant fuzzy filter of L. Therefore h



Theorem 3.11. If h is a hesitant fuzzy filter of L, then the set Γa := {x ∈ L | h(a) ⊆ h(x)} is a filter of L for every a ∈ L. Proof. Since h(1) ⊇ h(a) for all a ∈ L, we have 1 ∈ Γa . Let x, y ∈ L be such that x ∈ Γa and x → y ∈ Γa . Then h(x) ⊇ h(a) and h(x → y) ⊇ h(a). Since h is a hesitant fuzzy filter of L, it follows from (3.4) that h(y) ⊇ h(x) ∩ h(x → y) ⊇ h(a) so that y ∈ Γa . Hence Γa is a filter of L by Proposition 2.4.



Theorem 3.12. Let a ∈ L and let h be a hesitant fuzzy set on L. Then (1) If Γa is a filter of L, then h satisfies the following condition: (3.10)

(∀x, y ∈ L) (h(a) ⊆ h(x) ∩ h(x → y) ⇒ h(a) ⊆ h(y)).

(2) If h satisfies (3.3) and (3.10), then Γa is a filter of L.

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Proof. (1) Assume that Γa is a filter of L. Let x, y ∈ L be such that h(a) ⊆ h(x) ∩ h(x → y). Then x → y ∈ Γa and x ∈ Γa . Using (2.15), we have y ∈ Γa and so h(y) ⊇ h(a). (2) Suppose that h satisfies (3.3) and (3.10). From (3.3) it follows that 1 ∈ Γa . Let x, y ∈ L be such that x ∈ Γa and x → y ∈ Γa . Then h(a) ⊆ h(x) and h(a) ⊆ h(x → y), which imply that h(a) ⊆ h(x) ∩ h(x → y). Thus h(a) ⊆ h(y) by (3.10), and so y ∈ Γa . Therefore Γa is a filter of L by Proposition 2.4.



Definition 3.13 ([14]). A nonempty subset F of L is called a G-filter of L if it is a filter of L that satisfies the following condition: (3.11)

(∀x, y ∈ L) ((x x) → y ∈ F ⇒ x → y ∈ F ) .

We consider the hesitant fuzzification of G-filters. Definition 3.14. A hesitant fuzzy set h on L is called a hesitant fuzzy G-filter of L if it is a hesitant fuzzy filter of L that satisfies: (∀x, y ∈ L) (h((x x) → y) ⊆ h(x → y)) .

(3.12)

Note that the condition (3.12) is equivalent to the following condition: (∀x, y ∈ L) (h(x → (x → y)) ⊆ h(x → y)) .

(3.13)

Example 3.15. The hesitant fuzzy filter h in Example 3.3 is a hesitant fuzzy G-filter of L. Lemma 3.16. Every hesitant fuzzy filter h of L satisfies the following condition: (3.14)

(∀x, y, z ∈ L) (h(x → (y → z)) ∩ h(x → y) ⊆ h(x → (x → z))) .

Proof. Let x, y, z ∈ L. Using (2.4) and (2.6), we have x → (y → z) = y → (x → z) ≤ (x → y) → (x → (x → z)). It follows from Theorem 3.5 that h(x → (y → z)) ∩ h(x → y) ⊆ h(x → (x → z)). This completes the proof.



Theorem 3.17. Let h be a hesitant fuzzy set on L. Then h is a hesitant fuzzy G-filter of L if and only if it is a hesitant fuzzy filter of L that satisfies the following condition: (3.15)

(∀x, y, z ∈ L) (h(x → (y → z)) ∩ h(x → y) ⊆ h(x → z)) .

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Proof. Assume that h is a hesitant fuzzy G-filter of L. Then h is a hesitant fuzzy filter of L. Note that x ≤ 1 = (x → y) → (x → y), and thus x → y ≤ x → (x → y) for all x, y ∈ L. It follows from (3.1) that h(x → y) ⊆ h(x → (x → y)). Combining this and (3.13), we have (3.16)

h(x → y) = h(x → (x → y))

for all x, y ∈ L. Using (3.14) and (3.16), we have h(x → (y → z)) ∩ h(x → y) ⊆ h(x → z) for all x, y, z ∈ L. Conversely, let h be a hesitant fuzzy filter of L that satisfies the condition (3.15). If we put y = x and z = y in (3.15) and use (2.1) and (3.3), then h(x → y) ⊇ h(x → (x → y)) ∩ h(x → x) = h(x → (x → y)) ∩ h(1) = h(x → (x → y)) for all x, y ∈ L. Therefore h is a hesitant fuzzy G-filter of L.



Theorem 3.18. Let h be a hesitant fuzzy filter of L. Then h is a hesitant fuzzy G-filter of L if and only if the following condition holds: (3.17)

(∀x ∈ L) (h(x → (x x)) = h(1)) .

Proof. Assume that h satisfies the condition (3.17) and let x, y ∈ L. Since x → (x → y) = (x x) → y ≤ (x → (x x)) → (x → y) by (2.4) and (2.6), it follows from (3.1) that h(x → (x → y)) ⊆ h((x → (x x)) → (x → y)). Hence, we have h(x → y) ⊇ h((x → (x x)) → (x → y)) ∩ h(x → (x x)) ⊇ h(x → (x → y)) ∩ h(x → (x x)) = h(x → (x → y)) ∩ h(1) = h(x → (x → y)) by using (3.4), (3.17) and (3.3). Hence h is a hesitant fuzzy G-filter of L.



Theorem 3.19. (Extension property) Let h and g be hesitant fuzzy filters of L such that h ⊆ g, i.e., h(x) ⊆ g(x) for all x ∈ L and h(1) = g(1). If h is a hesitant fuzzy G-filter of L, then so is g.

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Proof. Assume that h is a hesitant fuzzy G-filter of L. Using (2.4) and (2.1), we have x → (x → ((x → (x → y)) → y)) = (x → (x → y)) → (x → (x → y)) = 1 for all x, y ∈ L. Thus g(x → ((x → (x → y)) → y)) ⊇ h(x → ((x → (x → y)) → y)) = h(x → (x → ((x → (x → y)) → y))) = h(1) = g(1) by hypotheses and (3.16), and so g(x → ((x → (x → y)) → y)) = g(1) for all x, y ∈ L by (3.3). Since g is a hesitant fuzzy filter of L, it follows from (3.4), (2.4) and (3.3) that g(x → y) ⊇ g(x → (x → y)) ∩ g((x → (x → y)) → (x → y)) = g(x → (x → y)) ∩ g(x → ((x → (x → y)) → y)) = g(x → (x → y)) ∩ g(1) = g(x → (x → y)) for all x, y ∈ L. Therefore g is a hesitant fuzzy G-filter of L.



Acknowledgements The author wishes to thank the anonymous reviewers for their valuable suggestions. Also, the author would like to acknowledge financial support for this work, from the Deanship of Scientific Research (DRS), University of Tabuk, Tabuk, Saudi Arabia, under grant no. S/0123/1436. References [1] R. Belohlavek, Some properties of residuated lattices, Czechoslovak Math. J. 53(123) (2003) 161–171. [2] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271–288. [3] P. H´ajek, Metamathematics of Fuzzy Logic, Kluwer Academic Press, Dordrecht, 1998. [4] J. G. Shen and X. H. Zhang, Filters of residuated lattices, Chin. Quart. J. Math. 21 (2006) 443–447. [5] V. Torra and Y. Narukawa, On hesitant fuzzy sets and decision, in: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378. 1382. [6] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010), 529–539. [7] E. Turunen, BL-algebras of basic fuzzy logic, Mathware & Soft Computing 6 (1999), 49–61.

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Boolean deductive systems of BL-algebras, Arch. Math. Logic 40 (2001)

467–473. [9] G. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowledge-Based Systems 31 (2012) 176–182. [10] Y. Wei, Filters theory of residuated lattices based on hesitant fuzzy sets, (submitted). [11] M. Xia and Z. S. Xu, Hesitant fuzzy information aggregation in decision making, Internat. J. Approx. Reason. 52(3) (2011) 395–407. [12] Z. S. Xu and M. Xia, Distance and similarity measures for hesitant fuzzy sets, Inform. Sci. 181(11) (2011) 2128–2138. [13] Z. S. Xu and M. Xia, On distance and correlation measures of hesitant fuzzy information, Int. J. Intell. Syst. 26(5) (2011) 410–425. [14] X. H. Zhang and W. H. Li, On fuzzy logic algebraic system MTL, Adv. Syst. Sci. Appl. 5 (2005) 475–483. [15] Y. Q. Zhu and Y. Xu, On filter theory of residuated lattices, Inform. Sci. 180 (2010) 3614–3632.

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 2, 2016

Existence and Uniqueness of Fuzzy Solutions for the Nonlinear Second-Order Fuzzy Volterra Integrodifferential Equations, Shaher Momani, Omar Abu Arqub, Saleh Al-Mezel, and Marwan Kutbi,………………………………………………………………………………………213 αβ-Statistical Convergence and Strong αβ-Convergence of Order γ for a Sequence of Fuzzy Numbers, Zeng-Tai Gong, and Xue Feng,…………………………………………………228 IF Rough Approximations Based On Lattices, Gangqiang Zhang, Yu Han, and Zhaowen Li,237 Some Results on Approximating Spaces, Neiping Chen,……………………………………254 Divisible and Strong Fuzzy Filters of Residuated Lattices, Young Bae Jun, Xiaohong Zhang, and Sun Shin Ahn,……………………………………………………………………………….264 Frequent Hypercyclicity of Weighted Composition Operators on Classical Banach Spaces, Shi-An Han, and Liang Zhang,………………………………………………………………277 On The Special Twisted q-Polynomials, Jin-Woo Park,…………………………………….283 Equicontinuity of Maps on [0, 1), Kesong Yan, Fanping Zeng, and Bin Qin,………………293 On Mixed Type Riemann-Liouville and Hadamard Fractional Integral Inequalities, Weerawat Sudsutad, S.K. Ntouyas, and Jessada Tariboon,…………………………………………….299 Weighted Composition Operators From F(p, q, s) Spaces To nth Weighted-Orlicz Spaces, Haiying Li, and Zhitao Guo,…………………………………………………………………315 Modified q-Daehee Numbers and Polynomials, Dongkyu Lim,…………………………….324 A Class of BVPS for Second-Order Impulsive Integro-Differential Equations of Mixed Type In Banach Space, Jitai Liang, Liping Wang, and Xuhuan Wang,………………………………331 Distribution and Survival Functions With Applications in Intuitionistic Random Lie C*-Algebras, Afrah A. N. Abdou, Yeol Je Cho, and Reza Saadati,…………………………345 Cubic ρ-Functional Inequality and Quartic ρ-Functional Inequality, Choonkil Park, Jung Rye Lee, and Dong Yun Shin,…………………………………………………………………….355 Complex Valued Gb-Metric Spaces, Ozgur Ege,…………………………………………….363

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 2, 2016 (continued) Finite Difference approximations for the Two-side Space-time Fractional Advection-diffusion Equations, Yabin Shao, and Weiyuan Ma,…………………………………………………369 A Modified Newton-Shamanskii Method for a Nonsymmetric Algebraic Riccati Equation, Jian-Lei Li, Li-Tao Zhang, Xu-Dong Li, Qing-Bin Li,………………………………..……380 Hesitant Fuzzy Filters and Hesitant Fuzzy G-Filters in Residuated Lattices, G. Muhiuddin,394

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.3, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Some new Chebyshev type quantum integral inequalities on finite intervals Feixiang Chen1 and Wengui Yang2∗ 1

Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Gorges University, Wanzhou 844000, China 2

Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia 472000, China

Abstract: By using the two parameters of deformation q1 and q2 , we establish some new Chebyshev type quantum integral inequalities on finite intervals. Furthermore, we also consider their relevance with other related known results. Keywords: Chebyshev type inequalities; quantum integral inequalities; synchronous (asynchronous) functions 2010 Mathematics Subject Classification: 34A08; 26D10; 26D15

1

Introduction

Let us start by considering the following celebrated Chebyshev functional (see [1]): Z T (f, g, p, q) =

b

! Z q(x)dx

a

!

b

p(x)f (x)g(x)dx

Z +

a

Z − a

b

b

! Z p(x)dx

a

! Z q(x)f (x)dx

q(x)f (x)g(x)dx

a

!

b

!

b

p(x)g(x)dx

Z

b

−

a

! Z p(x)f (x)dx

a

b

! q(x)g(x)dx , (1.1)

a

where f, g : [a, b] → R are two integrable functions on [a, b] and p(x) and q(x) are positive integrable functions on [a, b]. If f and g are synchronous on [a, b], that is, (f (x) − f (y))(g(x) − g(y)) ≥ 0, for any x, y ∈ [a, b], then we have (see, e.g., [2, 3]) T (f, g, p, q) ≥ 0,

(1.2)

The inequality in (1.2) is reversed if f and g are asynchronous on [a, b], that is, (f (x) − f (y))(g(x) − g(y)) ≤ 0, for any x, y ∈ [a, b]. If p(x) = q(x) for any x, y ∈ [a, b], we get the Chebyshev inequality, see [1]. Here we should point out that the Chebyshev functional (1.1) has attracted many researchers attention mainly due to its distinguished applications in numerical quadrature, probability and statistical problems and transform theory. At the same time, the Chebyshev functional (1.1) has also been employed to yield a number of integral inequalities, see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and the references therein. The integral inequalities can be applied for the study of qualitative and quantitative properties of integrals, see [14, 15, 16, 17]. In order to generalize and spread the existing inequalities, we specify two ways to overcome the problems which ensue from the general definition of q-integral. In [18], Gauchman has introduced the restricted q-integral over [a, b]. In [19], Stankovi´c, Rajkovi´c and Marinkovi´c have introduced the definition of the q-integral of the Riemann type. In [18], Gauchman gave the q-analogues of the well-known inequalities in the integral calculus, as Chebyshev, Gr¨ uss, Hermite-Hadamard for all the types of the q-integrals. In [19], Stankovi´c, Rajkovi´c and Marinkovi´c obtained some new q-Chebyshev, q-Gr¨ uss, q-Hermite-Hadamard type inequalities. In [20, 21], by using the weighted q-integral Montgomery identity, Yang and Liu and Yang established the ˇ weighted q-Cebyˇ sev-Gr¨ uss type inequalities for single and double integrals, respectively. Recently, Tariboon ∗ Corresponding author. Email: [email protected] (F. Chen) and [email protected] (W. Yang).

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and Ntouyas [22] introduced the quantum calculus on finite intervals, they extended the H¨older, Hermiteˇ Hadamard, trapezoid, Ostrowski, Cauchy-Bunyakovsky-Schwarz, Gr¨ uss, and Gr¨ uss-Cebyˇ sev integral inequalities to quantum calculus on finite intervals in the paper [23]. Motivated by the results mentioned above, by using the two parameters of deformation q1 and q2 , we establish some new Chebyshev type quantum integral inequalities on finite intervals. Furthermore, we also obtain their relevance with other related known results.

2

Preliminaries

Let J := [a, b] ⊂ R, K := [c, d] ⊂ R, J0 := (a, b) be interval and 0 < q, q1 , q2 < 1 be a constant. We give the definition q-derivative of a function f : J → R at a point x ∈ J on [a, b] as follows. Definition 2.1 ([22]). Assume f : J → R is a continuous function and let x ∈ J. Then the expression a Dq f (x)

=

f (x) − f (qx + (1 − q)a) , (1 − q)(x − a)

x 6= a,

a Dq f (a)

= lim a Dq f (x), x→a

(2.1)

is called the q-derivative on J of function f at x. We say that f is q-differentiable on J provided a Dq f (x) exists for all x ∈ J. Note that if a = 0 in (2.1), then 0 Dq f = Dq f , where Dq is the well-known q-derivative of the function f (x) defined by Dq f (x) =

f (x) − f (qx) . (1 − q)x

For more details, see [24]. Definition 2.2 ([22]). Assume f : J → R is a continuous function. Then the q-integral on J is defined by Z x ∞ X Iqa f (x) = f (t)a dq t = (1 − q)(x − a) q n f (q n x + (1 − q n )a), (2.2) a

n=0

for x ∈ J. Moreover, if c ∈ (a, x) then the definite q-integral on J is defined by Z x Z x Z c f (t)a dq t = f (t)a dq t − f (t)a dq t c

a

a

= (1 − q)(x − a)

∞ X

n

n

n

q f (q x + (1 − q )a) − (1 − q)(c − a)

n=0

∞ X

q n f (q n c + (1 − q n )a).

n=0

Note that if a = 0, then (2.2) reduces to the classical q-integral of a function f (x) defined by (see [24]) Z x ∞ X f (t)0 dq t = (1 − q)x q n f (q n x), ∀x ∈ [0, ∞). 0

n=0

Lemma 2.3. Assume f, g : J → R are two continuous functions and f (t) ≤ g(t) for all t ∈ J. Then Z x Z x f (t)a dq t ≤ g(t)a dq t. a

(2.3)

a

Proof. For x ∈ J, then q n x + (1 − q n )a ∈ J. Because f, g : J → R are two continuous functions and f (t) ≤ g(t) for all t ∈ J. Then f (q n x + (1 − q n )a) ≤ g(q n x + (1 − q n )a). (2.4) Summing from 0 to ∞ with respect to n and multiplying both sides of (2.4) by (1 − q)(x − a) ≥ 0, then we get Z x ∞ X f (t)a dq t = (1 − q)(x − a) q n f (q n x + (1 − q n )a) a

≤ (1 − q)(x − a)

n=0 ∞ X

n

n

n

Z

q g(q x + (1 − q )a) =

x

g(t)a dq t, a

n=0

which implies (2.3). The proof is completed. 2

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3

Chebyshev type quantum integral inequalities

In this section, we establish some new Chebyshev type quantum integral inequalities on finite intervals. For the sake of simplicity, we always assume that in this paper all of quantum integral exist and Iqa (uf )(b) =

Z

b

u(t)f (t)a dq t and Iqa (uf g)(b) =

Z

a

b

u(t)f (t)g(t)a dq t. a

Lemma 3.1. Let f and g be two continuous and synchronous functions on J and let u, v : J → [0, ∞) be two continuous functions. Then the following inequality holds true Iqa u(b)Iqa (vf g)(b) + Iqa v(b)Iqa (uf g)(b) ≥ Iqa (uf )(b)Iqa (vg)(b) + Iqa (vf )(b)Iqa (ug)(b).

(3.1)

Proof. Since f and g be two continuous and synchronous functions on J, then for all τ, ρ ∈ J, we have (f (τ ) − f (ρ))(g(τ ) − g(ρ)) ≥ 0.

(3.2)

f (τ )g(τ ) + f (ρ)g(ρ) ≥ f (τ )g(ρ) + f (ρ)g(τ ).

(3.3)

By (3.2), we write Multiplying both sides of (3.3) by v(τ ) and integrating the resulting identity with respect to τ from a to b, then we obtain Iqa (vf g)(b) + f (ρ)g(ρ)Iqa v(b) ≥ g(ρ)Iqa (vf )(b) + f (ρ)Iqa (vg)(b). (3.4) Multiplying both side of (3.4) by u(ρ) and integrating the resulting identity with respect to ρ from a to b, then we get Iqa u(b)Iqa (vf g)(b) + Iqa v(b)Iqa (uf g)(b) ≥ Iqa (uf )(b)Iqa (vg)(b) + Iqa (vf )(b)Iqa (ug)(b), which implies (3.1). Theorem 3.2. Let f and g be two continuous and synchronous functions on J and let φ, ϕ, ψ : J → [0, ∞) be three continuous functions. Then the following inequality holds true
2Iqa φ(b) Iqa ϕ(b)Iqa (ψf g)(b) + Iqa ψ(b)Iqa (ϕf g)(b) + 2Iqa ϕ(b)Iqa ψ(b)Iqa (φf g)(b)
≥ Iqa φ(b) Iqa (ϕf )(b)Iqa (ψg)(b) + Iqa (ψf )(b)Iqa (ϕg)(b) + Iqa ϕ(b) Iqa (φf )(b)Iqa (ψg)(b)

+ Iqa (ψf )(b)Iqa (ϕg)(b) + Iqa ψ(b) Iqa (φf )(b)Iqa (ϕg)(b) + Iqa (ϕf )(b)Iqa (φg)(b) . (3.5) Proof. Putting u = ϕ, v = ψ and using Lemma 3.1, we can write Iqa ϕ(b)Iqa (ψf g)(b) + Iqa ψ(b)Iqa (ϕf g)(b) ≥ Iqa (ϕf )(b)Iqa (ψg)(b) + Iqa (ψf )(b)Iqa (ϕg)(b).

(3.6)

Multiplying both sides of (3.6) by Iqa φ(b), we obtain

Iqa φ(b) Iqa ϕ(b)Iqa (ψf g)(b) + Iqa ψ(b)Iqa (ϕf g)(b) ≥ Iqa φ(b) Iqa (ϕf )(b)Iqa (ψg)(b) + Iqa (ψf )(b)Iqa (ϕg)(b) .

(3.7)

Putting u = φ, v = ψ and using Lemma 3.1, we can write Iqa φ(b)Iqa (ψf g)(b) + Iqa ψ(b)Iqa (φf g)(b) ≥ Iqa (φf )(b)Iqa (ψg)(b) + Iqa (ψf )(b)Iqa (φg)(b).

(3.8)

Multiplying both sides of (3.8) by Iqa ϕ(b), we obtain

Iqa ϕ(b) Iqa φ(b)Iqa (ψf g)(b) + Iqa ψ(b)Iqa (φf g)(b) ≥ Iqa ϕ(b) Iqa (φf )(b)Iqa (ψg)(b) + Iqa (ψf )(b)Iqa (φg)(b) .

(3.9)

With the same arguments as before, we can get

Iqa ψ(b) Iqa φ(b)Iqa (ϕf g)(b) + Iqa ϕ(b)Iqa (φf g)(b) ≥ Iqa ψ(b) Iqa (φf )(b)Iqa (ϕg)(b) + Iqa (ϕf )(b)Iqa (φg)(b) . (3.10) The required inequality (3.5) follows on adding the inequalities (3.7), (3.9) and (3.10).

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Lemma 3.3. Let f and g be two continuous and synchronous functions on J ∪ K and let u, v : J ∪ K → [0, ∞) be two continuous functions. Then the following inequality holds true Iqa1 u(b)Iqc2 (vf g)(d) + Iqc2 v(d)Iqa1 (uf g)(b) ≥ Iqa1 (uf )(b)Iqc2 (vg)(d) + Iqc2 (vf )(d)Iqa1 (ug)(b).

(3.11)

Proof. Multiplying both sides of (3.3) by v(ρ) and q2 -integrating the resulting inequality obtained with respect to ρ from c to d, then we have f (τ )g(τ )Iqc2 v(d) + Iqc2 (vf g)(d) ≥ f (τ )Iqc2 (vg)(d) + g(τ )Iqc2 (vf )(d).

(3.12)

Multiplying both sides of (3.12) by u(τ ) and q1 -integrating the resulting identity with respect to τ from a to b, then we obtain Iqa1 u(b)Iqc2 (vf g)(d) + Iqc2 v(d)Iqa1 (uf g)(b) ≥ Iqa1 (uf )(b)Iqc2 (vg)(d) + Iqc2 (vf )(d)Iqa1 (ug)(b), which implies (3.11). Theorem 3.4. Let f and g be two continuous and synchronous functions on J ∪ K and let φ, ϕ, ψ : J ∪ K → [0, ∞) be three continuous functions. Then the following inequality holds true
Iqa1 φ(b) Iqa1 ψ(b)Iqc2 (ϕf g)(d) + 2Iqa1 ϕ(b)Iqc2 (ψf g)(d) + Iqc2 ψ(d)Iqa1 (ϕf g)(b)

+ Iqa1 ϕ(b)Iqc2 ψ(d) + Iqc2 ϕ(d)Iqa1 ψ(b) Iqa1 (φf g)(b) ≥ Iqa1 φ(b) Iqa1 (ϕf )(b)Iqc2 (ψg)(d) + Iqc2 (ϕf )(d)Iqa1 (ϕg)(b)

+Iqa1 ϕ(b) Iqa1 (φf )(b)Iqc2 (ψg)(d)+Iqc2 (ψf )(d)Iqa1 (φg)(b) +Iqa1 ψ(b) Iqa1 (φf )(b)Iqc2 (ϕg)(d)+Iqc2 (ϕf )(d)Iqa1 (φg)(b) . (3.13) Proof. Putting u = ϕ, v = ψ and using Lemma 3.3, we can write Iqa1 ϕ(b)Iqc2 (ψf g)(d) + Iqc2 ψ(d)Iqa1 (ϕf g)(b) ≥ Iqa1 (ϕf )(b)Iqc2 (ψg)(d) + Iqc2 (ψf )(d)Iqa1 (ϕg)(b).

(3.14)

Multiplying both sides of (3.14) by Iqa1 φ(b), we obtain

Iqa1 φ(b) Iqa1 ϕ(b)Iqc2 (ψf g)(d) + Iqc2 ψ(d)Iqa1 (ϕf g)(b) ≥ Iqa1 φ(b) Iqa1 (ϕf )(b)Iqc2 (ψg)(d) + Iqc2 (ψf )(d)Iqa1 (ϕg)(b) , (3.15) Putting u = φ, v = ψ and using Lemma 3.3, we can write Iqa1 φ(b)Iqc2 (ψf g)(d) + Iqc2 ψ(d)Iqa1 (φf g)(b) ≥ Iqa1 (φf )(b)Iqc2 (ψg)(d) + Iqc2 (ψf )(d)Iqa1 (φg)(b).

(3.16)

Multiplying both sides of (3.16) by Iqa1 ϕ(b), we obtain

Iqa1 ϕ(b) Iqa1 φ(b)Iqc2 (ψf g)(d) + Iqc2 ψ(d)Iqa1 (φf g)(b) ≥ Iqa1 ϕ(b) Iqa1 (φf )(b)Iqc2 (ψg)(d) + Iqc2 (ψf )(d)Iqa1 (φg)(b) , (3.17) With the same arguments as before, we can get

Iqa1 ψ(b) Iqa1 φ(b)Iqc2 (ϕf g)(d) + Iqc2 φ(d)Iqa1 (φf g)(b) ≥ Iqa1 ψ(b) Iqa1 (φf )(b)Iqc2 (ϕg)(d) + Iqc2 (ϕf )(d)Iqa1 (φg)(b) , (3.18) The required inequality (3.14) follows on adding the inequalities (3.15), (3.17) and (3.18). Remark 3.5. The inequalities (3.5) and (3.13) are reversed in the following cases: (a) The functions f and g are synchronous on J ∪ K. (b) The functions φ, ϕ and ψ are negative on J ∪ K. (c) Two of he functions φ, ϕ and ψ are positive and the third one is negative on J ∪ K. Theorem 3.6. Let f, g, h be three continuous and synchronous functions on J ∪ K and let u : J ∪ K → [0, ∞) be a continuous function. Then the following inequality holds true Iqa1 u(b)Iqc2 (uf gh)(d) + Iqa1 (uh)(b)Iqc2 (uf g)(d) + Iqa1 (uf g)(b)Iqc2 (uh)(d) + Iqa1 (uf gh)(b)Iqc2 u(d) ≥ Iqa1 (uf )(b)Iqc2 (ugh)(d) + Iqa1 (ug)(b)Iqc2 (uf h)(d) + Iqa1 (ugh)(b)Iqc2 (uf )(d) + Iqa1 (uf h)(b)Iqc2 (ug)(d). (3.19)

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Proof. Let f, g, h be three continuous and synchronous functions on J ∪ K, then for all τ, ρ ∈ J ∪ K, we have (f (τ ) − f (ρ))(g(τ ) − g(ρ))(h(τ ) + h(ρ)) ≥ 0, which implies that f (τ )g(τ )h(τ ) + f (ρ)g(ρ)h(ρ) + f (τ )g(τ )h(ρ) + f (ρ)g(ρ)h(τ ) ≥ f (τ )g(ρ)h(τ ) + f (τ )g(ρ)h(ρ) + f (ρ)g(τ )h(τ ) + f (ρ)g(τ )h(ρ). (3.20) Multiplying both sides of (3.20) by u(τ ) and q2 -integrating the resulting identity with respect to τ from c to d, then we obtain Iqc2 (uf gh)(d) + f (ρ)g(ρ)h(ρ)Iqc2 u(d) + h(ρ)Iqc2 (uf g)(d) + f (ρ)g(ρ)Iqc2 (uh)(d) ≥ g(ρ)Iqc2 (uf h)(d) + g(ρ)h(ρ)Iqc2 (uf )(d) + f (ρ)Iqc2 (ugh)(d) + f (ρ)h(ρ)Iqc2 (ug)(d). (3.21) Multiplying both sides of (3.21) by u(ρ) and q1 -integrating the resulting inequality obtained with respect to ρ from c to d, then we have Iqa1 u(b)Iqc2 (uf gh)(d) + Iqa1 (uh)(b)Iqc2 (uf g)(d) + Iqa1 (uf g)(b)Iqc2 (uh)(d) + Iqa1 (uf gh)(b)Iqc2 u(d) ≥ Iqa1 (uf )(b)Iqc2 (ugh)(d) + Iqa1 (ug)(b)Iqc2 (uf h)(d) + Iqa1 (ugh)(b)Iqc2 (uf )(d) + Iqa1 (uf h)(b)Iqc2 (ug)(d), which implies ((3.19). Theorem 3.7. Let f, g, h be three continuous and synchronous functions on J ∪ K and let u, v : J ∪ K → [0, ∞) be two continuous functions. Then the following inequality holds true Iqa1 u(b)Iqc2 (vf gh)(d) + Iqa1 (uh)(b)Iqc2 (vf g)(d) + Iqa1 (uf g)(b)Iqc2 (vh)(d) + Iqa1 (uf gh)(b)Iqc2 v(d) ≥ Iqa1 (uf )(b)Iqc2 (vgh)(d) + Iqa1 (ug)(b)Iqc2 (vf h)(d) + Iqa1 (ugh)(b)Iqc2 (vf )(d) + Iqa1 (uf h)(b)Iqc2 (vg)(d). (3.22) Proof. Multiplying both sides of (3.20) by v(τ ) and integrating the resulting identity with respect to τ from c to d, then we obtain Iqc2 (vf gh)(d) + f (ρ)g(ρ)h(ρ)Iqc2 u(d) + h(ρ)Iqc2 (vf g)(d) + f (ρ)g(ρ)Iqc2 (vh)(d) ≥ g(ρ)Iqc2 (vf h)(d) + g(ρ)h(ρ)Iqc2 (vf )(d) + f (ρ)Iqc2 (vgh)(d) + f (ρ)h(ρ)Iqc2 (vg)(d). (3.23) Multiplying both sides of (3.23) by u(ρ) and integrating the resulting inequality obtained with respect to ρ from a to b, then we have Iqa1 u(b)Iqc2 (vf gh)(d) + Iqa1 (uh)(b)Iqc2 (vf g)(d) + Iqa1 (uf g)(b)Iqc2 (vh)(d) + Iqa1 (uf gh)(b)Iqc2 v(d) ≥ Iqa1 (uf )(b)Iqc2 (vgh)(d) + Iqa1 (ug)(b)Iqc2 (vf h)(d) + Iqa1 (ugh)(b)Iqc2 (vf )(d) + Iqa1 (uf h)(b)Iqc2 (vg)(d), which implies (3.22). Remark 3.8. It may be noted that the inequalities in (3.19) and (3.22) are reversed if functions f, g and h are asynchronous. It is also easily seen that the special case u = v of (3.22) in Theorem 3.7 reduces to that in Theorem 3.6.

4

Other quantum integral inequalities

The first class are the inequalities related to Cauchy’s inequality. Theorem 4.1. Let φ, f and g be three continuous functions on J. Then the following inequality holds true 2

[T (φ, f, g)] ≤ T (φ, f, f )T (φ, g, g),

(4.1)

where T (φ, f, g) = Iqa φ(b)Iqa (φf g)(b) − Iqa (φf )(b)Iqa (φg)(b). 5

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Proof. By simple computation, we have the following fact that Z Z 1 b b φ(ρ)φ(τ )[f (ρ) − f (τ )][g(ρ) − g(τ )]a dq ρa dq τ. T (φ, f, g) = 2 a a

(4.2)

From (4.2) and weighted Cauchy’s inequality, we easily obtain (4.1). Lemma 4.2. Let f and h be two continuous functions on J and let φ : J → [0, ∞) be a continuous function. Then the following inequality holds true m[g(ρ) − g(τ )] ≤ f (ρ) − f (τ ) ≤ M [g(ρ) − g(τ )],

∀ρ, τ ∈ J,

(4.3)

where m and M are given real numbers. Then for all t > 0 and ν > 0, we have T (φ, f, f ) + mM T (φ, g, g) ≤ (m + M )T (φ, f, g),

(4.4)

where T (φ, f, g) is defined as in Theorem 4.1. Proof. If we use the condition (4.3), we get (M [g(ρ) − g(τ )] − [f (ρ) − f (τ )])([f (ρ) − f (τ )] − m[g(ρ) − g(τ )]) ≥ 0,

∀ρ, τ ∈ J.

(4.5)

From (4.5) and through simple computation, we have [f (ρ) − f (τ )]2 + mM [g(ρ) − g(τ )]2 ≤ (m + M )[f (ρ) − f (τ )][g(ρ) − g(τ )].

(4.6)

Multiplying both sides of (4.6) by φ(ρ)φ(τ ) and integrating the resulting identity with respect to ρ and τ from a to b, we deduce the required inequality (4.4). Theorem 4.3. Let f, g, φ be defined as in Lemma 4.2 and 0 < m ≤ M < ∞. Then the following inequalities hold true T (φ, f, f )T (φ, g, g) ≤

0≤

p

(m + M )2 [T (φ, f, g)]2 , 4mM

√ √ ( M − m)2 √ T (φ, f, f )T (φ, g, g) − T (φ, f, g) ≤ T (φ, f, g), 2 mM

(4.7)

(4.8)

and 0 ≤ T (φ, f, f )T (φ, g, g) − [T (φ, f, g)]2 ≤

(M − m)2 [T (φ, f, g)]2 , 4mM

(4.9)

where T (φ, f, g) is defined as in Theorem 4.1. Proof. We use the following elementary inequality 2xy ≤ αx2 +

1 2 y , α

∀x, y ≥ 0, α > 0,

to get, for the choices α=

√ mM > 0,

x=

p T (φ, g, g) ≥ 0,

y=

p T (φ, f, f ) ≥ 0

the following inequality p √ 1 2 T (φ, f, f )T (φ, g, g) ≤ mM T (φ, g, g) + √ T (φ, f, f ). mM

(4.10)

Using (4.4) and (4.10), we deduce p m+M 2 T (φ, f, f )T (φ, g, g) ≤ √ T (φ, f, g). mM which is clearly equivalent to (4.7). By a few transformations of (4.7), similarly, we obtain (4.8) and (4.9). 6

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The second class are the inequalities related to H¨older’s inequality. Theorem 4.4. Let φ : J → [0, ∞) be a continuous function on J and f, g : J → (0, ∞) be two continuous functions on J such that 0 < m ≤ f α (τ )/g β (τ ) ≤ M < ∞ on J. If 1/α + 1/β = 1 with α > 1, then the following inequality holds true 1
1
1 M
αβ Iqa (φf g)(b). Iqa (φf α )(b) α Iqa (φg β )(b) β ≤ m

(4.11)

Proof. Since f α (τ )/g β (τ ) ≤ M , then f α/β ≤ M 1/β g. Multiplying by φf > 0, it follows that 1

α

φf α = φf 1+ β ≤ M β φf g and integrating the above inequality from a to b, then we have
1 1 1 Iqa (φf α )(b) α ≤ M αβ (Iqa (φf g)(b)) α .

(4.12)

On the other hand, since m ≤ f α (τ )/g β (τ ), then f ≥ m1/α g β/α . Multiplying by φg > 0, it follows that 1

β

1

φf g ≥ m α φg 1+ α = m α φg β . Integrating the above inequality from a to b, we obtain that
1
1 1 Iqa (φf g)(b) β ≥ m αβ Iqa (φg α )(b) β .

(4.13)

Combining (4.12) and (4.13), we have the desired inequality (4.11). The proof is completed. Theorem 4.5. Suppose that 1/α + 1/β = 1/γ with α, β, γ > 0. Let φ : J → [0, ∞) be a continuous function on J and f, g : J → (0, ∞) be two continuous functions on J. If 0 < m ≤ f γ (τ )/g βγ/α (τ ) ≤ M < ∞ for any τ ∈ J, then the following inequalities hold true α α
α α
(M − m)Iqa (φf α )(b) + mM γ − M m γ Iqa (φg β )(b) ≤ M γ − m γ Iqa (φf γ g γ )(b), (4.14) and 1

1 α

1 β

α β γ

− γ1

α

α

(M − m) α mM γ − M m γ α α
1 M γ − mγ γ


β1 Iqa (φf α )


α1


1
1 Iqa (φg β )(b) β ≤ Iqa (φf γ g γ )(b) γ .

Proof. If 0 < m ≤ xγ /y βγ/α ≤ M < ∞, then the following inequality is valid (see [25]): α α
α α
(M − m)xα + mM γ − M m γ y β ≤ M γ − m γ xγ y γ .

(4.15)

(4.16)

Substituting in the inequality (4.16) x → f (τ ) and y → g(τ ), and multiplying both sides of the obtained result by φ(τ ) and integrating the resulting identity with respect to τ from a to b, we obtain (4.14). Now, rewrite (4.14) in the form        γ   α  α α
α α
γ β (M − m)Iqa (φf α )(b) + mM γ − M m γ Iqa (φg β )(b) ≤ M γ − m γ Iqa (φf γ g γ )(b), α γ β γ (4.17) and applying arithmetic-geometric inequality on the left-hand side of (4.17) we get (4.15). The next class are the inequalities related to Minkowsky’s inequality. Theorem 4.6. Let p ≥ 1 and φ : J → [0, ∞) be a continuous function on J and f, g : J → (0, ∞) be two continuous functions on J. If 0 < m ≤ f (τ )/g(τ ) ≤ M < ∞ for any τ ∈ J, then we have
1
1
1 1 + M (m + 2) a Iqa (φf p )(b) p + Iqa (φg p )(b) p ≤ Iq (φ(f + g)p )(b) p . (m + 1)(M + 1)

7

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(4.18)

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Proof. Using the condition f (τ )/g(τ ) ≤ M for any τ ∈ J, we can get (M + 1)p f p (τ ) ≤ M p (f + g)p (τ ).

(4.19)

Multiplying both sides of (4.19) by φ(τ ) and integrating the resulting inequalities with respect to τ from a to b, we obtain (M + 1)p Iqa (f p )(b) ≤ M p Iqa ((f + g)p )(b). Hence, we can write
1 Iqa (φf p )(b) p ≤


1 M Iqa (φ(f + g)p )(b) p . M +1

(4.20)

On the other hand, using the condition m ≤ f (τ )/g(τ ), we can get (m + 1)p g p (τ ) ≤ (f + g)p (τ ).

(4.21)

Multiplying both sides of (4.21) by φ(τ ) and integrating the resulting inequalities with respect to τ from a to b, we obtain (m + 1)p Iqa (φg p )(b) ≤ Iqa (φ(f + g)p )(b). Hence, we can write
1 Iqa (φg p )(b) p ≤


1 1 I a (φ(f + g)p )(b) p . m+1 q

(4.22)

Adding the inequalities (4.20) and (4.22), we obtain the inequality (4.19). Theorem 4.7. Let p ≥ 1 and φ : J → [0, ∞) be a continuous function on J and f, g : J → (0, ∞) be two continuous functions on J. If 0 < m ≤ f (τ )/g(τ ) ≤ M < ∞ for any τ ∈ J, then we have  
2
1
2
1 (m + 1)(M + 1) Iqa (φf p )(b) p + Iqa (φg p )(b) p ≥ − 2 Iqa (φf p )(b) p Iqa (φg p )(b) p . (4.23) M Proof. Multiplying the inequalities (4.20) and (4.22), we obtain
1
1 (m + 1)(M + 1) a Iq (φf p )(b) p Iqa (φg p )(b) p ≤ M


1
2 Iqa (φ(f + g)p )(b) p .

(4.24)

Applying Minkowski’s inequality to the right hand side of (4.24), we get
1
2 Iqa (φ(f + g)p )(b) p ≤


1
1
2 Iqa (φf p )(b) p + [Iqa (φg p )(b) p
2
2
1
1 = Iqa (φf p )(b) p + Iqa (φg p )(b) p + 2 Iqa (φf p )(b) p Iqa (φg p )(b) p . (4.25)

Combining (4.24) and (4.25), we obtain (4.23). Theorem 4.8. Suppose that 1/α + 1/β = 1 with α > 1. Let φ : J → [0, ∞) be a continuous function on J and f, g : J → (0, ∞) be two continuous functions on J. If 0 < m ≤ f (τ )/g(τ ) ≤ M < ∞ for any τ ∈ J, then the following inequality holds true !  α  a   β Iq (φf α )(b) + Iqa (φg α )(b) Iqa (φf β )(b) + Iqa (φg β )(b) 2α M 2β 1 a Iq (φf g)(b) ≤ + .(4.26) α M +1 2 β m+1 2 Proof. From m ≤ f (τ )/g(τ ) ≤ M for any τ ∈ J, we have f (τ ) ≤

M (f (τ ) + g(τ )), M +1

g(τ ) ≤

8

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1 (f (τ ) + g(τ )). m+1

(4.27)

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From (4.27) and the following Young-type inequality 1 α 1 β x + y , α β

xy ≤

∀x, y ≥ 0, α > 1,

1 1 + = 1, α β

we obtain 1 a 1 I (φf α )(b) + Iqa (φg β )(b) α q β  α  β 1 M 1 1 a α ≤ Iq (φ(f + g) )(b) + Iqa (φ(f + g)β )(b). (4.28) α M +1 β m+1

Iqa (φf g)(b) ≤

Using the elementary inequality (c + d)α ≤ 2α−1 (cα + dα ), (α > 1 and c, d > 0) in (4.28), we get α  β 1 M 1 2α−1 Iqa (φ(f α + g α ))(b) + 2β−1 Iqa (φ(f β + g β ))(b) M +1 β m+1 !  α  a   β Iq (φf α )(b) + Iqa (φg α )(b) Iqa (φf β )(b) + Iqa (φg β )(b) 2α M 2β 1 = + . α M +1 2 β m+1 2 1 α

Iqa (φf g)(b) ≤



This completes the proof of the inequality in (4.26). Theorem 4.9. Suppose that 1/α + 1/β = 1 with α, β > 0. Let φ : J → [0, ∞) be a continuous function on J and f, g : J → (0, ∞) be two continuous functions on J. If 0 < m ≤ f (τ )/(f (τ ) + g(τ )) ≤ M < ∞ and 0 < m ≤ g(τ )/(f (τ ) + g(τ )) ≤ M < ∞ for any τ ∈ J, then we have 1

Iqa (φ(f

α
1 − M mα 1 (M − m) α mM 1 + g) )(b) α ≥ α α β β M α − mα


β1

α


1
1
Iqa (φf α )(b) α + Iqa (φg α )(b) α .

(4.29)

Proof. Due to (4.15) with γ = 1 of Theorem 4.5, m ≤ f (τ )/g β/α (τ ) ≤ M for any τ ∈ J, we have 1

1

1

αα β β

1
1
1 (M − m) α (mM α − M mα ) β a Iq (φf α )(b) α Iqa (φg β )(b) β ≤ Iqa (φf g)(b). α α M −m

(4.30)

By simple computation, we have Iqa (φ(f + g)α )(b) = Iqa (φf (f + g)α−1 ) + Iqa (φg(f + g)α−1 )(b).

(4.31)

From m ≤ f (τ )/(f (τ ) + g(τ )) ≤ M and m ≤ g(τ )/(f (τ ) + g(τ )) ≤ M for any τ ∈ J, we have m ≤ f (τ )/ (f (τ ) +
β/α g(τ ))α−1 ≤ M and m ≤ g(τ )/ (f (τ ) + g(τ ))α−1 )β/α ≤ M for any τ ∈ J. Applying (4.30) on right hand of (4.31), we get 1

1

1

1

Iqa (φ(f (f + g)α−1 )(b) ≥ α α β β

1 1 (M − m) α (mM α − M mα ) β a [Iq (φf α )(b)] α [Iqa (φ(f + g)(α−1)β )(b)] β , α α M −m 1

1

1

1

Iqa (φ(g(f + g)α−1 )(b) ≥ α α β β

1 1 (M − m) α (mM α − M mα ) β a [Iq (φg α )(b)] α [Iqa (φ(f + g)(α−1)β )(b)] β . (4.32) M α − mα

Using (4.31) and adding two inequalities of (4.32), we obtain 1

Iqa (φ(f

1

(M − m) α (mM α − M mα ) β + g) )(b) ≥ α β M α − mα α

1 α


1
1

1 Iqa (φf α )(b) α + Iqa (φg α )(b) α Iqa (φ(f + g)α )(b) β .

1 β

(4.33)
1 Dividing both sides of (4.33) by Iqa (φ(f + g)α )(b) β , we get (4.29).

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References [1] P.L. Chebyshev, Sur les expressions approximati ves des integrales definies par les autres prises entre lesmemes limites, Proc. Math. Soc. Kharkov 2 (1882) 93-98. [2] J.C. Kuang, Applied Inequalities, Shandong Sciences and Technologie Press, Shandong, China, 2004. [3] D.S. Mitrinovi´c, Analytic Inequalities, Springer, Berlin, Germany, 1970. [4] J. Choi, P. Agarwal, Some new Saigo type fractional integral inequalities and their q-analogues, Abstr. Appl. Anal. 2014 (2014) Art. ID 579260, 11 pages. [5] W. Sudsutad, S.K. Ntouyas, J. Tariboon, Fractional integral inequalities via Hadamard’s fractional integral, Abstr. Appl. Anal. 2014 (2014) Art. ID 563096, 11 pages. [6] D. Baleanu, P. Agarwal, Certain inequalities involving the fractional q-integral operators, Abstr. Appl. Anal. 2014 (2014) Art. ID 371274, 10 pages. [7] S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math. 10 (3) (2009) Art. 86, 5, pages. [8] S. Purohit, R. Raina, Chebyshev type inequalities for the Saigo fractional integrals and their q-analogues, J. Math. Inequal. 7 (2) (2013) 239-249. [9] Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci. 9 (4) (2010) 493-497. [10] W. Yang, Some new fractional quantum integral inequalities, Appl. Math. Lett. 25 (6) (2012) 963-969. [11] K. Brahim, S. Taf, Some fractional integral inequalities in quantum calculus, J. Frac. Calc. Appl. 4 (2) (2013) 245-250. [12] V. Chinchane, D. Pachpatte, On some integral inequalities using Hadamard fractional integral, Malaya J. Mat. vol. 1 (1) (2012) 62-66. [13] D. Baleanu, S.D. Purohit, and P. Agarwal, On fractional integral inequalities involving Hypergeometric operators, Chinese J. Math. 2014 (2014) Art. ID 609476, 5 pages. [14] D.S. Mitrinovi´c, J.E. Peˇcari´c, A.M. Fink, Classical and new inequalities in analysis, Kluwer Academic Publishers, 1993. [15] G.A. Anastassiou, Intelligent mathematics: computational analysis. Springer, New York, 2011. [16] G.A. Anastassiou, Taylor Widder representation formulae and Ostrowski, Gr¨ uss, integral means and Csiszar type inequalities, Comput. Math. Appl. 54 (2007) 9-23. [17] M.A. Kutbi, N. Hussain, A. Rafiq, Generalized Chebyshev inequalities with applications, J. Comput. Anal. Appl. 16 (4) (2014) 763-776. [18] H. Cauciman, Integral inequalities in q-calculus, Comput. Math. Appl. 47 (2-3) (2004) 281-300. [19] S. Marinkovi´c, P. Rajkovi´c, M. Stankovi´c, The inequalities for some types of q-integrals, Comput. Math. Appl. 56 (2008) 2490-2498. ˇ [20] W. Yang, On weighted q-Cebyˇ sev-Gr¨ uss type inequalities, Comput. Math. Appl. 61 (5) (2011) 1342-1347. ˇ [21] Z. Liu, W. Yang, New weighted q-Cebyˇ sev-Gr¨ uss type inequalities for double integrals, J. Comput. Anal. Appl. (accepted). [22] J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Difference Equ. 2013 (2013) 282. [23] J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl. 2014 (2014) 121. [24] V. Kac, P. Cheung, Quantum calculus, Springer, New York, 2002. [25] J.E. Peˇcari´c, On Jessen’s inequality for convex functions, III, J. Math. Anal. Appl. 158 (1991) 349-351.

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A FIRST ORDER DIFFERENTIAL SUBORDINATION AND ITS APPLICATIONS M. A. KUTBI AND A. A. ATTIYA Abstract. In this paper we use the di¤erential subordinations techniques to obtain some properties of functions belonging to the class of analytic functions in the open unit disc U. Also, some properties of the class of two …xed points in U; are also discussed. Furthermore, some interesting results of Hurwitz Lerch Zeta function and Digamma function are obtained.

1. Introduction Let Ak denote the class of functions f of the form

(1.1)

f (z) = z +

1 X

am z m

(k 2 N = f1; 2; :::g);

m=k+1

which are analytic in the open unit disc U = fz : jzj < 1g. Also, let H[a; k] denote the class of analytic functions in U in the form (1.2)

r(z) = a +

1 X

am z m

m=k

(z 2 U) ;

for a 2 C (C is the complex plane). Usually the analytic functions with the normalization f (0) = 0 = 0 f (0) 1 is studied. Moreover, we can obtain interesting results by using the Montel’s normalization of f (cf. [16], [6]) as follows (1.3)

f (z)jz=0 = 0

and

f (z) z

= 1; z=

2010 Mathematics Subject Classi…cation. 30C45; 30C80; 11M35; 33B15. Key words and phrases. Analytic function, di¤erential subordination, HurwitzLerch Zeta function, …xed point, two …xed points, Digamma function, SrivastavaAttiya operator. 1

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2

M. A. KUTBI AND A. A. ATTIYA

where is a …xed point in U: We see that, when = 0; we get the classical normalization in U: We denote by Ak; the class of functions f in Ak with Montel’s normalization. The class Ak; will be called the class of functions f with two …xed points. A function f in the class Ak is said to be in the class Rk ( ) if it satis…es (1.4)

Re

f (z) z

(z 2 U);

>

for some (0 < 1) . The classes R1 ( ) = C( ); and R1 (0) = C(0); were earlier studied by Èzrohi [7] and MacGregor [19], respectively. Further; a function f in the class Ak is said to be in the class Pk ( ) if it satis…es 0

(1.5)

(z 2 U);

Re f (z) >

for some (0 < 1) . The class P1 (0) = B(0) ; was earlier studied by Yamaguchi [28]. For some (0 < 1), 6= 0 with Re( ) 0 and z 2 U we write : (1.6) Rk1 ( ; ) :=

f (z) 2 Ak : Re (1

)

f (z) 0 + f (z) z

>

and Rk2 ( ; ) :=

(1.7)

n o 00 0 f (z) 2 Ak : Re f (z) + z f (z) > :

We note that (i) Rk1 ( ; 1) = Pk ( ); (ii) f 2 Rk2 ( ; ) if and only if

0

z f 2 Rk1 ( ; ) :

Now, if f 2 Ak , we de…ne the function Gk ( ; ; z) by (1.8) 0

Gk ( ; ; z) := with

f (z)

z1 (f (z)) (f (z))1 0

(f (z))1 z1 6= 0; for

0

(

1) + (1

2 R ,

zf (z) ) + f (z)

6= 0 with Re( )

00

zf (z) 1+ 0 f (z)

!!

;

0 and

z 2 U: Let Hk ( ; ; ) denote the class of functions f satisfying the condition

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A FIRST ORDER DIFFERENTIAL SUBORDINATION...

(1.9)

3

(z 2 U);

Re (Gk ( ; ; z)) >

for some (0 < 1) and Gk ( ; ; z) de…ned by (1.8). Also, we note that (i) For 6= 0 with Re( ) 0 (1.10) Hk 0;

1

;

= Rk1 ( ; ) and Hk 1;

(ii) One can de…ne the Rk1 ( ; ) for following relations (1.11)

Rk ( ) = Rk1 ( ; 0) =

1

= Rk2 ( ; ) .

;

= 0: Therefore we may use the

lim

!0

Hk 0;

1

;

;

and (1.12)

Rk2 ( ; 0) = lim Hk 1; !0

1

;

:

A general Hurwitz- Lerch Zeta function (or Lerch transcendent) (z; s; b) (cf., e.g., [24, Section 2.5, P. 121]) is the analytic continuation of the series

(1.13)

(z; s; b) =

1 X k=0

zk ; (k + b)s

which converges for b (b 2 C n Z0 ; Z0 = f0; 1; 2; ::: g) if z and s 2 C are any complex numbers with either z 2 U, or jzj = 1 and Re(s) > 1: See also [2, Section 1.11]. Many authors obtained several properties of (z; s; b), for example, Attiya and Hakami [1], Cho et al. [3], Choi and et al. [5], Ferreira and López [9], Guillera and Sondow [10, Section 2 ], Gupta et al. [11], Kutbi and Attiya ([13],[14]), Luo and Srivastava [15], Owa and Attiya [21], Prajapat and Bulboaca [22], Srivastava and Attiya [23], Srivastava et al. [25] and Wang et al. [27]. Moreover, the Digamma function (or Psi) (cf., e.g., [24, Section 1.2, P. 13]) is the logarithmic derivative of the classical gamma function,

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M. A. KUTBI AND A. A. ATTIYA

de…ned by 1 X + z n=1 1

0

(1.14)

(z) =

(z) = (z)

C

1 n

1 z+n

;

with the Euler constant C = 0:57721566::: . See also [2, Section 1.7] and [18, Section 5.1]. Several properties of can be found in [17], [4], [8] and [26]. We shall also need the following de…nitions De…nition 1.1. Let f and F be analytic functions. The function f is said to be subordinate to F , written f (z) F (z); if there exists a function w analytic in U; with w(0) = 0 and jw(z)j 1; and such that f (z) = F (w(z)): If F is univalent, then f (z) F (z) if and only if f (0) = F (0) and f (U) F (U) : De…nition 1.2. Let : C3 U ! C and let h be univalent in U. If q 2 H[a; k] satis…es the di¤erential subordination (1.15)

0

00

(p(z); z p (z); z 2 p (z) ; z)

h(z)

(z 2 U) ;

then q will be called (a; k) solution : The univalent function s is called (a; k) domainant; if q(z) s(z) for all q satisfying (1.15) , (a; k) domainant s(z) s(z) for all (a; k) domainant s of (1.15) is said to be the best (a; k) domainant of (1.15). In this paper, using the technique of di¤erential subordination, some properties of functions in the class Hk ( ; ; ) are obtained. Furthermore, some properties of the class of two …xed points in U; are also introduced. Some applications to Analytic Number Theory are also discussed. 2. The class Hk ( ; ; ) with first order differential subordination To prove our results, we need the following theorem due to Hallenbeck and Ruscheweyh [12] (see also Miller and Mocanu [20, P. 71]). Theorem 2.1. Let h be convex univalent in U, with h(0) = a; 0 and Re( ) 0: If q 2 H[a; k] and

6=

0

(2.1)

q(z) +

z q (z)

h(z);

then

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A FIRST ORDER DIFFERENTIAL SUBORDINATION...

(2.2)

q(z)

S(z)

5

h(z);

where

(2.3)

Zz

S(z) = k zk

h(t) t k

1

dt :

0

The function S is a convex univalent and is the best (a; k) domainint: Now, we prove Theorem 2.2. Let be a complex number satisfying Re( ) 0: If q 2 H[1; k] and

6= 0 with

0

(2.4)

z q (z)

Re q(z) +

>

;

then (2.5) Re (q(z)) > (2

1)+2(1

j j2 j j2 1; ; + 1; 1 k Re( ) k Re( )

)2 F 1

!

for some (0 < 1) and 6= 0 with Re( ) > 0 : The constant(2 j j2 j j2 2(1 )2 F1 1; k Re( ) ; k Re( + 1; 1 is the best possible. )

(z 2 U); 1)+

Proof. If we take the convex univalent function h de…ned by (2.6)

h(z) =

1 + (2 1) z 1+z

(0

< 1) ;

then, we have 0

(2.7)

q(z) +

z q (z)

h(z) ;

where h is de…ned by (2.6) satisfying h(0) = 1 : Applying Theorem 2.1, then (2.8)

q(z)

S(z) ;

where the convex function S de…ned by Zz 1 + (2 1) t S(z) = tk 1+t k zk 0

431

1

dt;

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M. A. KUTBI AND A. A. ATTIYA

(2.9)

= (2

1) + 2 (1

)

Z1 0

since Re(h(z)) > 0 and S(z) Also, since

(2.10)

k

z2 U

;

k

1+t z

h(z) ; we have Re (S(z)) > 0 :

1

inf Re

dt

1+t z

!

1

= 1+

(0

k Re( ) 2 t j j

t

1) :

Hence inf Re (S(z)) = (2

1) + 2(1

z2 U

)

Z1 0

(2.11)

= (2

1) + 2(1

)2 F1

dt 1+t

k Re( ) j j2

! j j2 j j2 1; ; + 1; 1 : k Re( ) k Re( ) 2

2

j j ; j j + 1; 1 Therefore, the constant (2 1)+2(1 )2 F1 1; k Re( ) k Re( ) cannot be replaced by a larger one, which completes the proof of Theorem 2.2.

Theorem 2.3. Let the function f de…ned by (1.1) be in the class Hk ( ; ; ), then (2.12) 0

Re

f (z)

!

(f (z))1 z1

> (2

for some (0 < 1); constant (2 1) + 2(1 possible.

1)+2(1

)2 F1

! j j2 j j2 1; ; + 1; 1 (z 2 U); k Re( ) k Re( )

2 R and 6= 0 with Re( ) 0: The j j2 j j2 )2 F1 (1; k Re( ) ; k Re( ) + 1; 1) is the best

Proof. De…ning the function (f (z))1

0

(2.13)

q(z) =

f (z)

z1

(z 2 U);

then, we have q 2 H[1; k] :

Taking the logarithmic di¤erentiation in both sides of (2.13), we have 0

(2.14)

q(z) +

z q (z)

= Gk ( ; ; z) ;

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A FIRST ORDER DIFFERENTIAL SUBORDINATION...

since f 2 Hk ( ; ; ); then (2.15)

7

0

Re q(z) +

z q (z)

> :

Therefore, we have (2.12) by applying Theorem 2.2. Putting have

= 0 and

= 1= ( 6= 0; Re( )

0); in Theorem 2.3, we

Corollary 2.1. Let the function f de…ned by (1.1) be in the class ; ); then

Rk1 (

(2.16) f (z) Re z

> (2

for some (0 (2 1) + 2(1

1)+2(1

)2 F1 1;

1 1 ; + 1; 1 k Re( ) k Re( )

(z 2 U);

< 1) and 6= 0 with Re( ) 0 : The constant 1 1 )2 F1 1; k Re( ) ; k Re( + 1; 1 is the best possible. )

Assuming = 1 and = 1= ( 6= 0; Re( ) 0); in Theorem 2.3 0 or putting zf instead of f; in Corollary (2.1) , we have Corollary 2.2. Let the function f de…ned by (1.1) be in the class ; ) ; then

Rk2 (

(2.17) 0

Re f (z) > (2 for some (0 (2 1) + 2(1

1)+2(1

)2 F1 1;

1 1 ; + 1; 1 k Re( ) k Re( )

(z 2 U);

< 1) and 6= 0 with Re( ) 0: The constant 1 1 )2 F1 1; k Re( ) ; k Re( ) + 1; 1 is the best possible.

Corollary 2.3. Let the function f de…ned by (1.1) be in the class Ak; of functions f with two …xed points. Also, let f be in the class Rk1 ( ; ); then ! 1 X m k 1 (2.18) Re am z k z m k 1 > m=k+2

2(1

)

2 F1

1;

1 1 ; + 1; 1 k Re( ) k Re( )

1

(z 2 U);

for some (0 < 1); is a …xed point in U de…ned in (1.3) and = 6 1 0 with Re( ) 0: The constant 2(1 ) 2 F1 1; k Re( ; 1 + 1; 1 ) k Re( ) is the best possible.

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M. A. KUTBI AND A. A. ATTIYA

Proof. Since f 2 Ak; ; then we have (2.19)

1 X

ak+1 =

am

m k 1

;

m=k+2

therefore, the function f (z)=z; takes the form

f (z) =1+ z

(2.20)

1 X

am z k (z m

k 2

m k 1

m=k+2

!

;

Then, we have the Corollary by applying Corollary 2.1. By using the same technique in Corollary 2.3, we have Corollary 2.4. Let the function f de…ned by (1.1) be in the class Ak; of functions f with two …xed points. Also, let f be in the class Rk2 ( ; ); then ! 1 X m k 1 (2.21) Re am z k (m z m k 1 ) > m=k+2

2(1

)

2 F1

1;

1 1 ; + 1; 1 k Re( ) k Re( )

1

(z 2 U);

for some (0 < 1); is a …xed point in U de…ned in (1.3) and 1 6= 0 with Re( ) 0: The constant 2 F1 1; k Re( ; 1 + 1; 1 is ) k Re( ) the best possible. 3. Some applications in Analytic Number Theory In this section we need the following lemma due to Guillera and Sondow [10]. Lemma 3.1. For z 2 C (3.1)

Z1 Z1

(1

[1; 1) and

> 0; we have

(x y) 1 dx dy = xyz) ln xy

(z; 1; ) ;

0 0

and

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A FIRST ORDER DIFFERENTIAL SUBORDINATION...

(3.2)

Z1 Z1

(x y) 1 1 dx dy = (1 + xy) ln xy 2

2

+

1 2

9

2

;

0 0

where

(z; s; ) and

( ) de…ned by (1.13) and (1.14) respectively

Corollary 3.1. Let (z; s; b) be the Lerch transcendental function de…ned by (1.13), then (3.3) Re ( (z; 1; )) >

1 2

2

+

1 2

(jzj < 1;

2

> 0);

and this result is the best possible. Proof. We can show that the function (3.4)

0

g(z) = z @(2

1) +

2(1

)

Z1

1

t 1

0

is a member in the class

R11 (

1

d tA (0 tz

< 1; z 2 U);

; ): Using (3.4) and (1.13) we obtain

(3.5) g(z) = z

(2

1) +

2(1

)

1 (z; 1; )

(0

< 1 ; z 2 U);

which a member in the class R11 ( ; ), where (z; s; b) is the Lerch transcendental function de…ned by (1.13). Using (2.16) and (3.5), we readily obtain the following property with > 0; real (3.6)

Re

z; 1;

1

Z1

>

dt 1+t

(jzj < 1);

0

which is equivalent to (3.7)

Re ( (z; 1; )) >

the constant

( 1; 1; ) ; (jzj < 1;

> 0);

( 1; 1; ) ; cannot be replaced by a larger one .

Using (1.14) and (3.7), we have (3.8) Re ( (z; 1; )) >

1 2

2

+

1 2

435

2

( jzj < 1;

> 0);

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M. A. KUTBI AND A. A. ATTIYA

this result is the best possible in general, which completes the proof of Corollary 3.2 . Using (3.2) and (3.3), we have the following corollary Corollary 3.2. For Z1 Z1

(3.9)

> 0; we have

(x y) 1 dx dy < Re ( (z; 1; )) (1 + xy) ln xy

(jzj < 1) :

0 0

This result is the best possible. Using (3.1) and (3.3), we have the following corollary Corollary 3.4. For (3.10) 0 1 1 Z Z @ Re 0 0

> 0; we have 1

1

(1

1 (x y) dx dy A > xy z) ln xy 2

2

+

1 2

2

(jzj < 1):

This result is the best possible.

Acknowledgment. This research was funded by the Deanship of Scienti…c Research (DSR), King Abdul-Aziz University, Jeddah, under grant no. 130-104-D1434. The authors, therefore, acknowledge with thanks DSR technical and …nancial support. References [1] A.A. Attiya and A. H. Hakami, Some subordination results associated with generalized Srivastava-Attiya operator, Adv. Di¤erence Equ. 2013, 2013:105, 1-14. [2] H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 1, McGrawHill, New York, 1953. [3] Nak Eun Cho, In Hwa and H.M. Srivastava, Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator, Appl. Math. Comput. 217 (2010), no. 2, 918-928. [4] Chao-Ping Chen, H.M. Srivastava, Li Li and Seiichi Manyama, Inequalities and monotonicity properties for the psi (or digamma) function and estimates for the Euler-Mascheroni constant, Integral Transforms Spec. Funct. 22 (2011), no. 9, 681-693. [5] J. Choi, D.S. Jang and H.M. Srivastava, A generalization of the Hurwitz-Lerch Zeta function, Integral Transforms Spec. Funct. 19(2008), no. 1-2, 65-79.

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A FIRST ORDER DIFFERENTIAL SUBORDINATION...

11

[6] J. Dziok, Classes of multivalent analytic and meromorphic functions with two …xed points, Fixed Point Theory Appl. 2013, 2013:86, 18 pp. [7] T.G. Èzrohi, Certain estimates in special class of univalent functions in the unit circle jzj < 1; Dopovidi Akad. Nauk. Ukrain RSR(1965), 984-988. [8] Feng Qi, Shou-Xin Chen and Wing-Sum Cheung, Logarithmically completely monotonic functions concerning gamma and digamma functions, Integral Transforms Spec. Funct. 18 (2007), no. 5-6, 435-443. [9] C. Ferreira and J.L. López, Asymptotic expansions of the Hurwitz-Lerch zeta function, J. Math. Anal. Appl. 298(2004), 210-224. [10] J. Guillera and J. Sondow, Double integrals and in…nite products for some classical constants via analytic continuations of Lerch’s transcendent, Ramanujan J. 16 (2008), no. 3, 247— 270. [11] P.L. Gupta, R.C. Gupta, S. Ong and H.M. Srivastava, A class of Hurwitz-Lerch zeta distributions and their applications in reliability, Appl. Math. Comput. 196 (2008), no. 2, 521–531. [12] D.J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc. 52(1975), 191-195. [13] M.A. Kutbi and A.A. Attiya, Di¤erential subordination result with the Srivastava-Attiya integral operator, J. Inequal. Appl. 2010(2010), 1-10. [14] M.A. Kutbi and A.A. Attiya, Di¤erential subordination results for certain integrodi¤erential operator and its applications, Abstr. Appl. Anal. 2012, Art. ID 638234, 1-13. [15] Q.M. Luo, and H.M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308(2005), 290-302. [16] P. Montel, Leçons sur les Fonctions Univalentes ou Multivalentes, GauthierVillars, Paris 1933. [17] A. Prabhu and H.M. Srivastava, Some limit formulas for the Gamma and Psi (or Digamma) functions at their singularities, Integral Transforms Spec. Funct. 22 (2011), no. 8, 587-592. [18] S. Kanemitsu and H. Tsukada, Vistas of Special Functions, World Scienti…c Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. [19] T.H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104(1962), 532-537. [20] S.S. Miller and P.T. Mocanu, Di¤erential Subordinations: Theory and Applications, Series in Pure and Applied Mathematics, No. 225. Marcel Dekker, Inc., New York, 2000. [21] S. Owa and A.A. Attiya, An application of di¤erential subordinations to the class of certain analytic functions, Taiwanese J. Math., 13(2009), no. 2A, 369375. [22] J.K. Prajapat and Teodor Bulboac¼ a, Double subordination preserving properties for a new generalized Srivastava-Attiya integral operator, Chin. Ann. Math. Ser. B 33 (2012), no. 4, 569-582. [23] H.M. Srivastava and A.A. Attiya, An integral operator associated with the Hurwitz-Lerch zeta function and di¤erential subordination, Integral Transforms Spec. Funct. 18 (2007), no. 3-4, 207-216. [24] H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001.

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[25] H.M. Srivastava, S. Gaboury and F. Ghanim, A uni…ed class of analytic functions involving a generalization of the Srivastava–Attiya operator, Appl. Math. Comput. 251 (2015), 35-45. [26] Tsu-Chen Wu, Shih-Tong Tu, and H.M. Srivastava, Some combinatorial series identities associated with the digamma function and harmonic numbers, Appl. Math. Lett. 13 (2000), no. 3, 101-106. [27] Zhi-Gang Wang, Zhi-Hong Liu and Yong Sun, Some properties of the generalized Srivastava-Attiya operator, Integral Transforms Spec. Funct. 23 (2012), no. 3, 223-236. [28] K.Yamaguchi, On functions satisfying Re (f (z)=z) > ; Proc. Amer. Math. Soc. 17(1966), 588-591. Department of Mathematics, Faculty of Science, King AbdulAziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia E-mail address: [email protected] Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura, Egypt Current address: Department of Mathematics, Faculty of Science, University of Hail, Hail, Saudi Arabia E-mail address: [email protected]

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Approximation by a complex summation-integral type operators in compact disks Mei-Ying Ren1∗ , Xiao-Ming Zeng2∗ 1

Department of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China

2

Department of Mathematics, Xiamen University, Xiamen 361005, Chnia E-mail: [email protected],

[email protected]

Abstract. In this paper we introduce a kind of complex summation-integral type operators and study the approximation properties of these operators. We obtain a Voronovskaja-type result with quantitative estimate for these operators attached to analytic functions on compact disks. We also study the exact order of approximation. More important, our results show the overconvergence phenomenon for these complex operators. Keywords: Complex summation-integral type operators; Voronovskaja-type result; Exact order of approximation; Simultaneous approximation; Overconvergence Mathematical subject classification: 30E10, 41A25, 41A36

1. Introduction In 1986, some approximation properties of complex Bernstein polynomials in compact disks were initially studied by Lorentz [15]. Very recently, the problem of the approximation of complex operators has been causing great concern, which is becoming a hot topic of research. A Voronovskaja-type result with quantitative estimate for complex Bernstein polynomials in compact disks was obtained by Gal [2]. Also, in [1, 3-14, 16-19] similar results for complex Bernstein-Kantorovich polynomials, Bernstein-Stancu polynomials, Kantorovich-Schurer polynomials, Kantorovich-Stancu polynomials, complex Favard-Sz´ asz-Mirakjan operators, complex Beta operators of first kind, complex Baskajov-Stancu operators, complex Bernstein-Durrmeyer polynomials, complex Bernstein-Durrmeyer operators based on Jacobi weights, complex genuine Durrmeyer Stancu polynomials and complex q-Durrmeyer type operators were obtained. The aim of the present article is to obtain approximation results for a kind of complex summation-integral type operators (introduced and studied in the case of real variable by Ren [20]), which are defined for f : [0, 1] → C continuous ∗ Corresponding

authors: Mei-Ying Ren and Xiao-Ming Zeng.

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on [0, 1] by Mn (f ; z) := pn,0 (z)f (0) +

n−1 X

pn,k (z)Ln,k (f ) + pn,n (z)f (1),

(1)

k=1

µ ¶ n where z ∈ C, n = 1, 2, . . . , pn,k (z) = z k (1 − z)n−k is Bernstein basis k R 1 nk−1 1 t (1 − t)n(n−k)−1 f (t)dt, B(x, y) is function, and Ln,k (f ) = B(n(n−k),nk) 0 Beta function.

2. Auxiliary results In the sequel, we shall need the following auxiliary results. Lemma 1. Let m, n ∈ N, z ∈ C, we have Mn (tm ; z) is a polynomial of degree less than or equal to min (m, n) and m

Mn (tm ; z) =

(n2 − 1)! X cs (m)n2s Bn (ts ; z), 2 (n + m − 1)! s=1

where cs (m) ≥ 0 are constants depending on m and Bn (f ; z) =

n X k=0

k pn,k (z)f ( ). n

Proof. By the definition of Beta function, for all m, n ∈ N, z ∈ C, we have Mn (tm ; z) =

n−1 (nk + m − 1)! (n2 − 1)! X pn,k (z) + zn. 2 (n + m − 1)! (nk − 1)! k=1

Considering the definition of the Bn (f ; z), for any m ∈ N, applying the principle of mathematical induction, we immediately obtain the desired conclusion. Let m = 0, 1, 2, by Lemma 1, we have Mn (1; z) = 1; Mn (t; z) = z; Mn (t2 ; z) =

n(n − 1) 2 n+1 z + 2 z. n2 + 1 n +1

Lemma 2. For all m, n ∈ N we can get the equality m

(n2 − 1)! X cs (m)n2s = 1. 2 (n + m − 1)! s=1 Proof. For all m, n ∈ N, by Lemma 1 we have m

Mn (tm ; 1) =

(n2 − 1)! X cs (m)n2s . 2 (n + m − 1)! s=1 2 440

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On the other hand, we have pn,k (1) = 0, k = 0, 1, 2, . . . , n − 1, and pn,n (1) = 1. So, by the formula (1) and using these above value, we have Mn (tm ; 1) = 1, which implies that we get desired conclusion. Corollary 1. Let em (t) = tm , m ∈ N∪{0}, z ∈ C, n ∈ N, for all |z| ≤ r, r ≥ 1, we have |Mn (em ; z)| ≤ rm . Proof. Since Mn (e0 ; z) = 1, therefor this result is established for m = 0. When m ∈ N, by using the methods Gal [5], p. 61, proof of Theorem 1.5.6, we have |Bn (ts ; z)| ≤ rs . Thus, for all m ∈ N and |z| ≤ r, the proof follows directly by Lemma 1 and Lemma 2. Lemma 3. Let em (t) = tm , m ∈ N ∪ {0} and z ∈ C, we have Mn (em+1 ; z) =

nz(1 − z) m + n2 z 0 (M (e ; z)) + Mn (em ; z). n m n2 + m n2 + m

(2)

Proof. By Lemma 1, we have Mn (e0 ; z) = 1 and Mn (e1 ; z) = z, therefore, this result is established for m = 0. Now let m ∈ N, in view of 0

z(1 − z) [pn,k (z)] = (k − nz)pn,k (z), it follows that z(1 − z)(Mn (em ; z))0 =

n−1 X k=1

=

n−1 X· k=1

=

(k − nz)pn,k (z)Ln,k (tm ) + nz n (1 − z) ¸ (n2 + m)(nk + m) m − pn,k (z)Ln,k (tm ) + nz n − nzMn (em ; z) n(n2 + m) n

n−1 n−1 mX n2 + m X pn,k (z)Ln,k (tm+1 ) − pn,k (z)Ln,k (tm ) + nz n − nzMn (em ; z) n n k=1

k=1

n2 + m m = Mn (em+1 ; z) − Mn (em ; z) − nzMn (em ; z) n n n2 + m m + n2 z = Mn (em+1 ; z) − Mn (em ; z), n n which implies the recurrence in the statement. Lemma 4. Let m, n ∈ N, z ∈ C, em (z) = z m , Sn,m (z) := Mn (em ; z) − z m , we have Sn,m (z) =

m − 1 + n2 z nz(1 − z) 0 (M (e ; z)) + Sn,m−1 (z) n m−1 n2 + m − 1 n2 + m − 1 m − 1 + n2 z m−1 + 2 z − zm n +m−1

(3)

Proof. Using the recurrence formula (2), by simple calculation, we can easily get the recurrence (3), the proof is omitted.

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3. Main results The first main result is expressed by the following upper estimates. Theorem 1. Let 1 ≤ r ≤ R, DR = {z ∈ C : |z| < R}. Suppose that ∞ P f : DR → C is analytic in DR , i.e. f (z) = cm z m for all z ∈ DR . (i) for all |z| ≤ r and n ∈ N, we have

m=0

|Mn (f ; z) − f (z)| ≤ where Kr (f ) = (1 + r)

∞ P m=1

Kr (f ) , n

|cm |m(m − 1)rm−1 < ∞.

(ii) (Simultaneous approximation) If 1 ≤ r < r1 < R are arbitrary fired, then for all |z| ≤ r and n, p ∈ N we have |(Mn (f ; z))(p) − f (p) (z)| ≤

Kr1 (f )p!r1 , n(r1 − r)p+1

where Kr1 (f ) is defined as at the above point (i). Proof. Taking em (z) = z m , by hypothesis that f (z) is analytic in DR , i.e. ∞ P f (z) = cm z m for all z ∈ DR , it is easy for us to obtain m=0

Mn (f ; z) =

∞ X

cm Mn (em ; z),

m=0

therefore, we get |Mn (f ; z) − f (z)| ≤ =

∞ X m=0 ∞ X

|cm | · |Mn (em ; z) − em (z)| |cm | · |Mn (em ; z) − em (z)|,

m=1

as Mn (e0 ; z) = e0 (z) = 1. (i) For m ∈ N, taking into account that Mn (em−1 ; z) is a polynomial degree ≤ min(m − 1, n), by the well-known Bernstein inequality and Corollary 1, we get |(Mn (em−1 ; z))0 | ≤

m−1 max{|Mn (em−1 ; z)| : |z| ≤ r} ≤ (m − 1)rm−2 . r

On the one hand, when m = 1, for |z| ≤ r, by Lemma 1, we have |Mn (e1 ; z) − e1 (z)| = (1 + r)

m(m − 1) m−1 r . n

On the other hand, when m ≥ 2, for n ∈ N, |z| ≤ r, r ≥ 1, using the

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recurrence formula (3) and the above inequality, we have |Mn (em ; z) − em (z)| = |Sn,m (z)| r(1 + r) (m − 1)rm−2 + r|Sn,m−1 (z)| n m−1 + (1 + r)rm−1 n 2(m − 1) = (1 + r)rm−1 + r|Sn,m−1 (z)|. n ≤

By writing the last inequality, for m = 2, · · · , we easily obtain step by step the following µ ¶ 2(m − 2) |Mn (em ; z) − em (z)| ≤ r r|Sn,m−2 (z)| + (1 + r)rm−2 n 2(m − 1) + (1 + r)rm−1 n 2(m − 2) + 2(m − 1) (1 + r)rm−1 = r2 |Sn,m−2 (z))| + n m(m − 1) m−1 r . ≤ . . . ≤ (1 + r) n In conclusion, for any m, n ∈ N, |z| ≤ r, r ≥ 1, we have |Mn (em ; z) − em (z)| ≤ (1 + r)

m(m − 1) m−1 r , n

it follows that |Mn (f ; z) − f (z)| ≤

∞ 1+r X |cm |m(m − 1)rm−1 . n m=1

By assuming that f (z) is analytic in DR , we have f (2) (z) = 1)z m−2 and the series is absolutely convergent in |z| ≤ r, so we get 1)rm−2 < ∞, which implies Kr (f ) = (1 + r)

∞ P m=1

∞ P m=2 ∞ P m=2

cm m(m − |cm |m(m−

|cm |m(m − 1)rm−1 < ∞.

(ii) For the simultaneous approximation, denoting by Γ the circle of radius r1 > r and center 0, since for any |z| ≤ r and υ ∈ Γ, we have |υ − z| ≥ r1 − r, by Cauchy’s formulas it follows that for all |z| ≤ r and n ∈ N, we have ¯Z ¯ p! ¯¯ Mn (f ; υ) − f (υ) ¯¯ (p) (p) dυ ¯ |(Mn (f ; z)) − f (z)| = 2π ¯ Γ (υ − z)p+1 Kr1 (f ) p! 2πr1 ≤ n 2π (r1 − r)p+1 Kr1 (f ) p!r1 = · , n (r1 − r)p+1 which proves the theorem. Theorem 2. Let R > 1, DR = {z ∈ C : |z| < R}. Suppose that f : DR → C

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is analytic in DR , i.e. f (z) =

∞ P

ck z k for all z ∈ DR . For any fixed r ∈ [1, R]

k=0

and all n ∈ N, |z| ≤ r, we have ¯ ¯ 00 ¯ ¯ ¯Mn (f ; z) − f (z) − (n + 1)z(1 − z)f (z) ¯ ≤ Mr (f ) . ¯ ¯ 2 2(n + 1) n2 where Mr (f ) =

∞ P

(4)

|ck |(k − 1)Fk,r rk and Fk,r = 10k 3 − 30k 2 + 39k − 16 + 4(k −

k=2

2)(k − 1)2 (1 + r).

Proof. For all z ∈ DR , we have ¯ ¯ 00 ¯ ¯ ¯Mn (f ; z) − f (z) − (n + 1)z(1 − z)f (z) ¯ ¯ ¯ 2 2(n + 1) ¯ ¯ ∞ X ¯ (n + 1)k(k − 1)(1 − z)z k−1 ¯¯ ¯ ≤ |ck | ¯Mn (ek ; z) − ek (z) − ¯. 2(n2 + 1) k=2

Denoting Ek,n (z) = Mn (ek ; z) − ek (z) −

(n + 1)k(k − 1)(1 − z)z k−1 , 2(n2 + 1)

it is obvious that Ek,n (z) is a polynomial of degree less than or equal to k. For all k ≥ 2, by simple computation and the use of Lemma 3, we can get Ek,n (z) =

k − 1 + n2 z nz(1 − z) (Ek−1,n (z))0 + 2 Ek−1,n (z) + Gk,n (z), 2 n +k−1 n +k−1

(5)

z k−2 2 n2 +k−1 (z Ak,n + zBk,n + Ck,n ) and Ak,n = −n(k − 1) + 2 2 n(n+1)(k−1)2 (k−2) +k−1) + n2 − n (n+1)(k−1)(k−2) − (n2 + k − 1) + (n+1)k(k−1)(n , 2(n2 +1) 2(n2 +1) 2(n2 +1) 2 2 2 (k−2) (k−2) Bk,n = n(k−1)− n(n+1)(k−1) − n(n+1)(k−1)(k−2) +k−1− (n+1)(k−1) + 2(n2 +1) 2(n2 +1) 2(n2 +1) 2 2 2 n (n+1)(k−1)(k−2) (n+1)k(k−1)(n +k−1) n(n+1)(k−1)(k−2) (n+1)(k−1)2 (k−2) − , Ck,n = + . 2(n2 +1) 2(n2 +1) 2(n2 +1) 2(n2 +1)

where Gk,n (z) =

For all k ≥ 2, n ∈ N and |z| ≤ r, r ≥ 1, we easily obtain |Ck,n | ≤ (k − 1)(k − 2)(2k − 3), it follows that |

(2k 3 − 9k 2 + 13k − 6)rk z k−2 Ck,n | ≤ . n2 + k − 1 n2

By simple computation, we have Bk,n = 2(n21+1) {2n(k − 1) − n(n + 1)(k − 1) (k − 2) − n(n + 1)(k − 1)(k − 2)2 + 2(n2 + 1)(k − 1) − (n + 1)(k − 1)2 (k − 2) + n2 (k − 1)(k − 2) − nk(k − 1)2 − n2 k(k − 1) − k(k − 1)2 }, so, we can get 2

|

(5k 3 − 15k 2 + 18k − 8)rk z k−1 Bk,n | ≤ . n2 + k − 1 n2

By simple computation, we have Ak,n = 2(n21+1) {−2n(k − 1) + n(n + 1)(k − 1) (k − 2) + 2n2 − n2 (k − 1)(k − 2) − 2(n2 k + k − 1) + nk(k − 1)2 + n2 k(k − 1) + k(k − 1)2 }, so, we can get 2

|

z k Ak,n (3k 3 − 6k 2 + 8k − 2)rk | ≤ . n2 + k − 1 n2 6 444

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Thus, for all k ≥ 2, n ∈ N and |z| ≤ r, r ≥ 1, we can obtain |Gk,n (z)| ≤

rk Dk , n2

whereDk = 10k 3 − 30k 2 + 39k − 16. By formula (5), for all k ≥ 2, n ∈ N and |z| ≤ r, r ≥ 1, we have |Ek,n (z)| ≤

r(1 + r) |(Ek−1,n (z))0 | + r|Ek−1,n (z)| + |Gk,n (z)|. n

Using the estimate in the proof of Theorem 1 (i), we get |Mn (ek ; z) − ek (z)| ≤

1+r k(k − 1)rk−1 , n

for all k, n ∈ N , |z| ≤ r, r ≥ 1. So, denote kf kr = max{|f (z)|; |z| ≤ r}, we have |(Ek−1.n (z))0 | ≤

k−1 k Ek−1,n kr r

· ¸ k−1 (n + 1)(k − 1)(k − 2)(1 − e1 )ek−2 kr ≤ kMn (ek−1 ; ·) − ek−1 kr + k r 2(n2 + 1) · ¸ k − 1 (k − 1)(k − 2)(1 + r)rk−2 (k − 1)(k − 2)(1 + r)rk−2 ≤ + r n n 4(k − 2)(k − 1)2 rk−1 , n for all n ∈ N, k ≥ 2 and |z| ≤ r, r ≥ 1. It follows ≤

4(k − 2)(k − 1)2 (1 + r)rk rk + r|Ek−1,n (z)| + 2 Dk 2 n n rk := 2 Fk,r + r|Ek−1,n (z)|, n

|Ek,n (z)| ≤

where Fk,r is a polynomial of degree 3 in k defined as Fk,r = Dk + 4(k − 2)(k − 1)2 (1 + r), Dk is expressed in the above. Since E0.n (q; z) = E1.n (q; z) = 0 for any z ∈ C , therefore, by writing the last inequality for k = 2, 3, . . ., we easily obtain step by step the following |Ek,n (z)| ≤

k rk X (k − 1)Fk,r rk Fj,r ≤ . 2 n j=2 n2

As a conclusion, we have ¯ ¯ X ∞ 00 ¯ ¯ ¯Mn (f ; z) − f (z) − (n + 1)z(1 − z)f (z) ¯ ≤ |ck ||Ek,n (q; z)| ¯ ¯ 2(n2 + 1) k=2

∞ 1 X ≤ 2 |ck |(k − 1)Fk,r rk . n k=2

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As f (4) (z) =

∞ P

ck k(k −1)(k −2)(k −3)z k−4 and the series is absolutely con-

k=4

∞ P

vergent in |z| ≤ r, it easily follows that which implies that rem.

|ck |k(k − 1)(k − 2)(k − 3)rk−4 < ∞,

k=4

∞ P

|ck |(k − 1)Fk,r rk < ∞, this completes the proof of theo-

k=2

In the following theorem, we will obtain the exact order in approximation. Theorem 3. Let R > 1, DR = {z ∈ C : |z| < R}. Suppose that f : DR → C is analytic in DR . If f is not a polynomial of degree ≤ 1 , then for any r ∈ [1, R) we have Cr (f ) kMn (f ; ·) − f kr ≥ , n ∈ N, n where kf kr = max{|f (z)|; |z| ≤ r} and the constant Cr (f ) > 0 depends on f , r but it is independent of n. Proof. Denote e1 (z) = z and Hn (f ; z) = Mn (f ; z) − f (z) −

(n + 1)z(1 − z) 00 f (z). 2(n2 + 1)

For all z ∈ DR and n ∈ N, we have ½ ¾ ¤ n+1 2(n2 + 1) £ 2 00 z(1 − z)f (z) + 2 n Hn (f ; z) . Mn (f ; z) − f (z) = 2(n2 + 1) n (n + 1) In view of the property: kF +Gkr ≥ |kF kr −kGkr | ≥ kF kr −kGkr , it follows ½ ¾ ¤ n+1 2(n2 + 1) £ 2 00 n ||Hn (f ; ·)||r . kMn (f ; ·) − f kr ≥ ke1 (1 − e1 )f kr − 2 2(n2 + 1) n (n + 1) Considering the hypothesis that f is not a polynomial of degree ≤ 1 in DR , we have ke1 (1 − e1 )f 00 kr > 0. Indeed, supposing the contrary, it follows that z(1 − z)f 00 (z) = 0, for all z ∈ Dr . By hypothesis that f (z) is analytic in DR , we can denote f (z) =

∞ P

ck z k , the

k=0

identification of the coefficients method immediately leads to ck = 0, k = 2, 3, ···. This implies that f is a polynomial of degree ≤ 1 on Dr , a contradiction with the hypothesis. Using the inequality (4), we get n2 kHn (f ; ·)kr ≤ Mr (f ), therefore, there exists an index n0 depending only on f , r, such that for all n ≥ n0 , we have ke1 (1 − e1 )f 00 kr −

¤ 1 2(n2 + 1) £ 2 n ||Hn (f ; ·)||r ≥ ke1 (1 − e1 )f 00 kr , n2 (n + 1) 2 8 446

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which implies kMn (f ; ·)−f kr ≥

n+1 1 ke1 (1 − e1 )f 00 kr ≥ ke1 (1 − e1 )f 00 kr , for all n ≥ n0 . 4(n2 + 1) 4n

For n ∈ {1, 2, · · ·, n0 − 1}, we have kMn (f ; ·) − f kr ≥

Wr,n (f ) , n

where Wr,n (f ) = nkMn (f ; ·) − f kr > 0. As a conclusion, we have kMn (f ; ·) − f kr ≥

Cr (f ) , for all n ∈ N, n

where ¾ ½ 1 Cr (f ) =min Wr,1 (f ), Wr,2 (f ), . . . , Wr,n0 −1 (f ), ke1 (1 − e1 )f 00 kr , 4 this complete the proof. Combining Theorem 3 with Theorem 1, we get the following result. Corollary 2. Let R > 1, DR = {z ∈ C : |z| < R}. Suppose that f : DR → C is analytic in DR . If f is not a polynomial of degree ≤ 1 , then for any r ∈ [1, R) we have 1 kMn (f ; ·) − f kr ³ , n ∈ N, n where kf kr = max{|f (z)|; |z| ≤ r} and the constants in the equivalence depend on f , r but it is independent of n. Theorem 4. Let R > 1, DR = {z ∈ C : |z| < R}. Suppose that f : DR → C is analytic in DR . Also, let 1 ≤ r < r1 < R and p ∈ N be fixed. If f is not a polynomial of degree ≤ max{1, p − 1}, then we have k(Mn (f ; ·))(p) − f (p) kr ³

1 , n ∈ N, n

where kf kr = max{|f (z)|; |z| ≤ r} and the constants in the equivalence depend on f , r, r1 , p, but it is independent of n. Proof. Taking into account the upper estimate in Theorem 1, it remains to prove the lower estimate only. Denoting by Γ the circle of radius r1 > r and center 0 , by the Cauchy’s formula it follows that for all |z| ≤ r and n ∈ N we have Z Mn (f ; v) − f (v) p! (p) (p) dv. Mn (f ; z) − f (z) = 2πi Γ (v − z)p+1 Keeping the notation there for Hn (f ; z), for all n ∈ N, we have ½ ¾ ¤ n+1 2(n2 + 1) £ 2 00 Mn (f ; z) − f (z) = z(1 − z)f (z) + 2 n Hn (f ; z) . 2(n2 + 1) n (n + 1)

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By using Cauchy’s formula, for all v ∈ Γ, we get n+1 {[z(1 − z)f 00 (z)]p 2(n2 + 1) ¾ Z 2 2(n2 + 1) p! n Hn (f ; v) + 2 dv , n (n + 1) 2πi Γ (v − z)p+1

Mn(p) (f ; z) − f (p) (z) =

passing now to k · kr and denoting e1 (z) = z, it follows ° n + 1 n° ° 00 (p) ° (1 − e )f ] kMn(p) (f ; ·) − f (p) kr ≥ °[e ° 1 1 2(n2 + 1) r ° ° ¾ Z 2 ° p! 2(n2 + 1) ° n H (f ; v) n ° − 2 dv ° ° ° , p+1 n (n + 1) 2πi Γ (v − ·) r Since for any |z| ≤ r and υ ∈ Γ we have |υ − z| ≥ r1 − r, so, by using Theorem 2, we get ° ° Z 2 2 ° p! n Hn (f ; v) ° ° ≤ p! 2πr1 n kHn (f ; ·)kr1 ≤ Mr1 (f )p!r1 . ° dv ° ° 2πi p+1 2π (r1 − r)p+1 (r1 − r)p+1 Γ (v − ·) r By hypothesis on f , we have k[e1 (1 − e1 )f 00 ](p) kr > 0. Indeed, supposing the contrary, it follows that k[e1 (1 − e1 )f 00 ](p) kr = 0, that is z(1 − z)f 00 (z) is a polynomial of degree ≤ p − 1. let p = 1 and p = 2, then the analyticity of f obviously implies that f is a polynomial of degree ≤ 1 = max(1, p − 1), a contradiction. Now let p ≥ 3, then the analyticity of f obviously implies that f is a polynomial of degree ≤ p − 1 = max(1, p − 1), a contradiction with the hypothesis. In conclusion, k[e1 (1−e1 )f 00 ](p) kr > 0 and in continuation, reasoning exactly as in the proof of Theorem 3, we can get the desired conclusion. Remark 1. If we use King’s approach to consider King type modification of the complex extension of the operators which was given by (1), we will obtain better approximation (cf. [21-23]).

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant no. 61572020), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant no. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant no. 2014J01021 and 2013J01017).

References 1. Anastassiou, G.A., Gal, S.G.: Approximation by complex Bernstein-Schurer and Kantorovich-Schurer polynomials in compact disks. Comput. Math. Appl. 58(4), 734-743 (2009) 2. Gal, S.G.: Voronovskaja’s theorem and iterations for complex Bernstein polynomials in compact disks. Mediterr. J. Math. 5(3), 253-272 (2008) 10 448

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3. Gal, S.G.: Approximation by complex Bernstein-Kantorovich and StancuKantorovich polynomials and their iterates in compact disks. Rev. Anal. Num´er. Th´eor. Approx. (Cluj) 37(2), 159-168 (2008) 4. Gal, S.G.: Exact orders in simultaneous approximation by complex Bernstein Stancu polynomials. Rev. Anal. Num´er. Th´eor. Approx. (Cluj) 37(1), 47-52 (2008) 5. Gal, S.G.: Approximation by Complex Bernstein and Convolution Type Operators. World Scientific Publ. Co., Singapore (2009) 6. Gal, S.G.: Exact orders in simultaneous approximation by complex Bernstein polynomials. J. Concr. Appl. Math. 7(3), 215-220 (2009) 7. Gal, S.G.: Approximation by complex Bernstein-Stancu polynomials in compact disks. Results Math. 53(3-4), 245-256 (2009) 8. Gal, S.G.: Approximation by complex genuine Durrmeyer type polynomials in compact disks. Appl. Math. Comput. 217, 1913-1920 (2010) 9. Gal, S.G.: Approximation by complex Bernstein-Durrmeyer polynomials with Jacobl weights in conpact disks. Math. Balkanica (N.S.). 24(1-2), 103-110 (2010) 10. Gal, S.G., Gupta, V.: Approximation by a Durrmeyer-type operator in compact disks. Ann. Univ. Ferrara. 57, 261-274 (2011) 11. Gal, S.G., Gupta, V.: Approximation by complex Beta operators of first kind in strips of compact disks. Mediterr. J. Math. 10 (1), 31-39 (2013) 12. Gupta, V.: Approximation properties by Bernstein-Durrmeyer type operators. Complex Anal. Oper. Theory. (2011) doi: 10.1007/s11785-0110167-9 13. Gal, S.G., Gupta, V., etc.: Approximation by complex Baskakov-Stancu operators in compact disks. Rend. Circ. Mat. Palermo. 61(2), 153-165 (2012) 14. Gal, S.G., Mahmudov, N.I. and Kara, M.: Approximation by complex q-Sz´ asz-Kantorovich operators in compact disks, q > 1. Complex Anal. Oper. Theory. (2012) doi: 10.1007/s11785-012-0257-3 15. Lorentz, G.G.: Bernstein Polynomials. 2nd ed., Chelsea Publ, New York (1986) 16. Mahmudov, N.I.: Approximation properties of complex q-Sz´ asz-Mirakjan operators in compact disks. Comput. Math. Appl. 60, 1784-1791 (2010) 17. Mahmudov, N.I., Gupta, V.: Approximation by genuine Durrmeyer-Stancu polynomials in compact disks. Math. Comput. Model. 55, 278-285 (2012) 18. Ren M.Y., Zeng X.M.: Approximation by Complex Schurer-Stancu Operators in Compact Disks. J. Comput. Amal. Appl. 15(5), 833-843 (2013) 19. Ren M.Y., Zeng X.M.: Exact orders in simultaneous approximation by complex q-Durrmeyer type operators. J. Comput. Amal. Appl. 16(5), 895-905 (2014) 11 449

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20. Ren, M.Y.: Approximation properties of a kind of Bernstein type operators. J. Wuyi Univ. 2, 1-4 (2012) 21. King, J.P.: Positive linear operators which preserve x2 . Acta Math. Hungar. 99(3), 203-208 (2003) 22. Mahmudov, N.I.: q-Sz´asz-Mirakjan operators which preserve x2 , J. Comput. Appl. Math. 235, 4621-4628 (2011) ¨ u, M.: King type modification of Meyer-K¨onig and Zeller 23. Do˘ gru, O., Orkc¨ operators based on the q-integers. Math. Comput. Model. 50, 1245-1251 (2009)

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On the Convergence of Mann and Ishikawa Type Iterations in the Class of Quasi Contractive Operators Shin Min Kang1 , Faisal Ali2, Arif Rafiq3, Young Chel Kwun4,∗ and Shaista Jabeen5 1

2

Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected]

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan e-mail: [email protected] 3

Department of Mathematics and Statistics, Virtual University of Pakistan, Lahore 54000, Pakistan e-mail: [email protected] 4

5

Department of Mathematics, Dong-A University, Busan 604-714, Korea e-mail: [email protected]

Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan e-mail: [email protected] Abstract In this paper, we introduce two new iteration schemes, namely modified Mann and modified Ishikawa to approximate the fixed points of quasi contractive operators on a normed space. Various test problems are presented to reveal the validity and high efficiency of these iterative schemes. AMS Subject Classification: 47J25. Key Words: Quasi contraction, fixed point, strong convergence.

1

Introduction and preliminaries

In the last few decades, various researchers have explored the fixed points of contractive type operators in metric spaces, Hilbert spaces and different classes of Banach spaces, see [1] and references there in. To approximate unique fixed point of strict contractive type operators, Picard iterative scheme can be used effectively [1, 10, 15, 16]. But this scheme does not generally converge for the operators with slightly weaker contractive ∗

Corresponding author

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conditions. For such operators, Mann iteration [13] (cf. [6, 14]), Ishikawa iteration [7] and Krasnosel’okiˇı iteration [11] (cf. [3]) are much useful. Let E be a normed space and C a nonempty convex subset of E. Let T : C → C be an operator and {αn } and {βn } sequences of real numbers in [0, 1]. The Mann iteration [13] is defined by the sequence {xn }∞ n=0 as xn+1 = (1 − αn ) xn + αn T xn ,

n ≥ 0.

(1.1)

n ≥ 0,

(1.2)

The sequence {xn }∞ n=0 defined by xn+1 = (1 − αn ) xn + αn T yn , yn = (1 − βn ) xn + βn T xn ,

n≥0

is called Ishikawa iteration [7]. It is noticeable that for αn = λ (constant), the iterative procedure (1.1) turn into Krasnosel’okiˇı iteration. Also for βn = 0, Ishikawa iteration(1.2) reduces to Mann iteration (1.1). Definition 1.1. Let (X, d) be a metric space and a ∈ (0, 1). A mapping T : X → X satisfying d (T x, T y) ≤ ad (x, y) for all x, y ∈ X (1.3) is called a contraction. The following theorem is the classical Banach’s contraction principle and of fundamental importance in the study of Fixed Point Theory. Theorem 1.2. Let (X, d) be a complete metric space and T : X → X be a contraction. Then T has a unique fixed point p and the Picard iteration {xn }∞ n=0 defined by xn+1 = T xn ,

n≥0

(1.4)

converges to p for any x0 ∈ X. The contraction in the above theorem forces T to be continuous. Despite this condition, Theorem 1.2 has many applications in solving the nonlinear equation f (x) = 0. Kannan [9] developed a fixed point theorem by
relaxing the condition of continuity of T . He produced the following by taking b in 0, 21 : d (T x, T y) ≤ b [d (x, T x) + d (y, T y)]

for all x, y ∈ X.

(1.5)


Chatterjea [4] obtained a similar result by considering c ∈ 0, 21 as follows: d (T x, T y) ≤ c [d (x, T y) + d (y, T x)]

for all x, y ∈ X.

(1.6)

In 1972, Zamferescu [17] proved the following very interesting and important fixed point theorem by taking into account (1.3), (1.5) and (1.6). 2 452

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Theorem 1.3. Let (X, d) be a metric space and T : X → X be a mapping for which there exist real numbers a, b and c satisfying 0 < a < 1, 0 < b and c < 21 such that for each x, y ∈ X, at least one of the following is true: (z1 ) d (T x, T y) ≤ ad (x, y) , (z2 ) d (T x, T y) ≤ b [d (x, T x) + d (y, T y)] , (z3 ) d (T x, T y) ≤ c [d (x, T y) + d (y, T x)] . Then T has a unique fixed point p and the Picard iteration {xn }∞ n=0 defined by xn+1 = T xn ,

n≥0

converges to p for any x0 ∈ X. An operator T : X → X satisfying the contractive conditions (z1 ), (z2 ) and (z3 ) is called Zamferescu operator. ´ c [5] obtained a more general contraction to approximate unique fixed In 1974, Ciri´ point with the help of Picard iteration: there exists 0 < h < 1 such that for all x, y ∈ X d(T x, T y) ≤ h max{d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x)}.

(1.7)

Definition 1.4. Let (X, d) be a metric space and T : X → X a mapping satisfying (1.7). Then T is called quasi contraction. A new class of operators on an arbitrary Banach space E, satisfying kT x − T yk ≤ δ kx − yk + 2δ kx − T xk

for all x, y ∈ E, 0 ≤ δ < 1,

(1.8)

was established by Berinde [2] in 2004. He approximated fixed points of this class of operators via Ishikawa iteration. It is well known that a nonlinear equation f (x) = 0 can be expressed in terms of fixed point iteration method as follows: x = T x. (1.9) Taking up the technique of [8], if T 0 x 6= 1, θ 6= −1, it can easily be seen by adding θx to both sides of (1.9) that θx + T x x= = Tθ x. (1.10) 1+θ So as to make (1.10) to be efficient, we can choose Tθ0 x = 0, which gives θ = −T 0 x.

(1.11)

Now we are in a position to define modified Mann and modified Ishikawa iterative schemes. Replacing T xn and T yn in (1.1) and (1.2) with Tθ xn and Tθ yn , respectively, we get xn+1 = (1 − αn ) xn + αn Tθ xn

(1.12)

xn+1 = (1 − αn ) xn + αn Tθ yn ,

(1.13)

and

yn = (1 − βn ) xn + βn Tθ xn . 3 453

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Using (1.10) in (1.12) and (1.13) and also the error term, we obtain   1 1 xn+1 = 1 − αn xn + αn T xn + µn 1+θ 1+θ

(1.14)

and     θβn αn θβn αn T xn + T yn + µn , 1+ xn+1 = 1 − xn + 1+θ 1+θ 1+θ 1+θ   1 1 β n xn + βn T xn + νn . yn = 1 − 1+θ 1+θ 

(1.15)

We call the procedures defined in (1.14) and (1.15), the modified Mann and modified Ishikawa iterative procedures. It is obvious that (1.14) and (1.15) without error term reduce to (1.1) and (1.2), respectively for θ = 0. In this paper, we have proved the strong convergence of quasi contractive operator T satisfying (1.14) and (1.15) in the setting of normed space. We also present some test problems to compare the iterative procedures defined in (1.1), (1.2), (1.14) and (1.15). The numerical results obtained demonstrate the high performance and efficiency of modified Mann and modified Ishikawa iterative processes. We use the following lemma in the sequal. Lemma 1.5. ([12]) Let {rn }, {sn }, {tn } and {kn } be the sequences of nonnegative numbers satisfying rn+1 ≤ (1 − sn ) rn + sn tn + kn , n ≥ 0. P∞ P If n=0 sn = ∞ and limn→∞ tn = 0 and ∞ n=0 kn < ∞ hold, then limn→∞ rn = 0.

2

Main results

Assuming that the operator T has at least one fixed point, we prove the covergence theorems for iterative procedures (1.14) and (1.15). Theorem 2.1. Let C be a nonempty closed convex subset of a normed space E and T : C → C be an operator satisfying (1.8). For arbitrary x0 ∈ C, let {xn }∞ n=0 be the P∞ sequence defined by the iterative process (1.14) satisfying θ > −1, n=0 αn = ∞ and kµn k = 0(αn ). Then the sequence {xn }∞ converges strongly to a fixed point of T . n=0 Proof. Let p be the fixed point of the operator T . We consider kxn+1 − pk





1 1

= 1− αn xn + αn T xn + µn − p

1+θ 1+θ

   

1 1 1 1

= 1− αn xn + αn T xn + µn − 1 − αn + αn p

1+θ 1+θ 1+θ 1+θ

 

1 1

=

1 − 1 + θ αn (xn − p) + 1 + θ αn (T xn − p) + µn   1 1 ≤ 1− αn kxn − pk + αn kT xn − pk + kµn k . 1+θ 1+θ

(2.1)

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Substituting y = xn and x = p in (1.8), we get kT xn − pk ≤ δ kxn − pk . Thus (2.1) implies   1 δ kxn+1 − pk ≤ 1 − αn kxn − pk + αn kxn − pk + kµn k 1+θ 1+θ   1 δ = 1− αn + αn kxn − pk + kµn k 1+θ 1+θ   1−δ αn kxn − pk + kµn k . = 1− 1+θ Using Lemma 1.5 and the fact that 0 ≤ δ < 1, 0 ≤ αn ≤ 1, θ > −1, kµn k = 0(αn ) and P∞ n=0 αn = ∞, we obtain lim kxn − pk = 0. n→∞

Hence xn → p. This completes the proof. Taking θ = 0 in the setting of normed space and the contraction condition (1.8), we obtain the following corollary. Corollary 2.2. Let C be a nonempty closed convex subset of a normed space E and T : C → C be an operator satisfying (1.8). For arbitrary x0 ∈ C, let {xn }∞ n=0 be the P sequence defined by the iterative process (1.1) satisfying ∞ α = ∞. Then the sequence n=0 n ∞ {xn }n=0 converges strongly to a fixed point of T . Now we prove the convergence of modified Ishikawa iterative process in the form of the following theorem. Theorem 2.3. Let C a nonempty closed convex subset of a normed space E and T : C → C be an operator satisfying (1.8). For arbitrary x0 ∈ C, let {xn }∞ n=0 be the sequence P defined by the iterative process (1.15) satisfying θ > −1, ∞ α = ∞, kνn k = 0(αn ) and n=0 n ∞ kµn k = 0(αn ). Then the sequence {xn }n=0 converges strongly to a fixed point of T . Proof. Let p be the fixed point of the operator T . We consider kxn+1 − pk = k(1 − αn ) xn + αn Tθ yn + µn − pk

 

θyn + T yn

= (1 − αn ) xn + αn + µn − p

1+θ

T yn + θyn

≤ (1 − αn ) kxn − pk + αn

1 + θ − p + kµn k

(T yn − p) + θ (yn − p)

+ kµn k = (1 − αn ) kxn − pk + αn

1+θ αn θαn ≤ (1 − αn ) kxn − pk + kT yn − pk + kyn − pk + kµn k . 1+θ 1+θ

(2.2)

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Substituting x = p and y = yn in (1.8), we get kT yn − pk ≤ δ kyn − pk .

(2.3)

Thus (2.2) implies kxn+1 − pk δαn θαn kyn − pk + kyn − pk + kµn k . 1+θ 1+θ δ+θ αn kyn − pk + kµn k . = (1 − αn ) kxn − pk + 1+θ ≤ (1 − αn ) kxn − pk +

(2.4)

Consider





1 1

kyn − pk = 1 − β n xn + βn T xn + νn − p

1+θ 1+θ

 

1 1

βn (xn − p) + βn (T xn − p) + νn = 1−

1+θ 1+θ   1 1 ≤ 1− βn kxn − pk + βn kT xn − pk + kνn k . 1+θ 1+θ

(2.5)

Substituting x = p and y = xn in (1.8), we get kT xn − pk ≤ δ kxn − pk .

(2.6)

Thus (2.5) implies 

 1 δ kyn − pk ≤ 1 − βn kxn − pk + βn kxn − pk + kνn k 1+θ 1+θ   1−δ = 1− βn kxn − pk + kνn k . 1+θ

(2.7)

Using (2.7) in (2.4), we get kxn+1 − pk    δ+θ 1−δ ≤ (1 − αn ) kxn − pk + αn 1 − βn kxn − pk + kνn k + kµn k 1+θ 1+θ    δ+θ 1−δ δ+θ = 1 − αn + αn 1 − βn kxn − pk + αn kνn k + kµn k 1+θ 1+θ 1+θ     δ+θ 1−δ kxn − pk = 1 − αn 1 − 1− βn 1+θ 1+θ δ+θ + αn kνn k + kµn k . 1+θ

(2.8)

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Let 

  δ+θ 1−δ An = 1 − αn 1 − βn 1− 1+θ 1+θ   δ + θ (δ + θ) (1 − δ) βn = 1 − αn 1 − + 1+θ (1 + θ)2   (1 + θ) (δ + θ) − (δ + θ) (1 − δ) βn = 1 − αn 1 − (1 + θ)2   (1 + θ)2 − (1 + θ) (δ + θ) + (δ + θ) (1 − δ) βn = 1 − αn (1 + θ)2   (1 − δ) (1 + θ) + (δ + θ) (1 − δ) βn = 1 − αn (1 + θ)2   (1 + θ) + (δ + θ) βn (1 − δ) αn =1− (1 + θ) (1 + θ)   (1 − δ) δ+θ =1− αn 1 + βn . (1 + θ) 1+θ

(2.9)

δ+θ δ+θ Since βn ≥ 0, 0 ≤ δ < 1 and θ > −1, therefore 1+θ βn ≥ 0 and 1 + 1+θ βn ≥ 1. (2.9) gives Hence   1−δ δ+θ 1−δ An = 1 − αn 1 + βn ≤ 1 − αn . 1+θ 1+θ 1+θ

Thus from (2.8), we get kxn+1 − pk ≤



 1−δ δ+θ 1− αn kxn − pk + αn kνn k + kµn k . 1+θ 1+θ

With the help of Lemma 1.5 and using the fact that 0 ≤ δ < 1, 0 < αn < 1, θ > −1, P kνn k = 0(αn ), kµn k = 0(αn ) and ∞ n=0 αn = ∞, we get lim kxn − pk = 0.

n→∞

Consequently, xn → p ∈ F and this completes the proof. Corollary 2.4. Let C a nonempty closed convex subset of a normed space E and T : C → C be an operator satisfying (1.8). For arbitrary x0 ∈ C, let {xn }∞ n=0 be the sequence P∞ defined by the iterative process (1.2) satisfying n=0 αn = ∞. Then the sequence {xn }∞ n=0 converges strongly to a fixed point of T . The above corollary in fact is the generalization of Theorem 2 of Berinde [2] in the context of a normed space and the contraction condition (1.8).

3

Applications

In this section, we consider various test problems to apply Mann (M), modified Mann (MM), Ishikawa (I) and modified Ishikawa (MI) iterative procedures for the estimation 7 457

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of fixed points. The data in the following table indicates the rapidness of convergence in each problem. We make use of Maple software and 10−3 tolerence for the purpose. Here we denote the number of iterations (NI). Tx

θ

αn

βn

x0

3 − x2

2x

√1 n+1

√1 n+1

1

3(1−x) − cos x

´ ` ln 3 3(1−x) − sin x

1 1+n

1 1+n

0.5

1 − x − cos x

1 − sin x

√1 n+1

√1 n+1

0.1

cos x − ex + 1

sin x + ex

√1 n+1

√1 n+1

0.4

1− x 2

−1 2

√1 n+1

√1 n+1

0.5

x 2

−1 2

√1 n+1

√1 n+1

0.5

1 1+n

1 1+n

0.5

e(1−x)

4

2

−1

2 (1 − x) e(1−x)

2

M ethod M MM I MI M MM I MI M MM I MI M MM I MI M MM I MI M MM I MI M MM I MI

NI 9 4 22 2 4 4 11 1 4 1 6 1 5 1 12 1 2 1 3 1 12 1 6 1 4 2 13 1

x [k] 1.3044 1.3047 1.3009 1.3009 0.6657 0.6576 0.6570 0.6588 0.0000 −0.0026 0.0037 0.0001 0.4120 0.4101 0.4076 0.4101 0.6616 0.6667 0.6616 0.6667 0.0181 0.0000 0.0154 0.0000 0.4160 0.4089 0.4159 0.4136

Tx 1.2985 1.2977 1.3076 1.3077 0.6572 0.6652 0.6658 0.6641 −0.0000 0.0026 −0.0036 −0.0001 0.4066 0.4100 0.4150 0.4101 0.6692 0.6667 0.6692 0.6667 0.0091 0.0000 0.0077 0.0000 0.4065 0.4182 0.4067 0.4104

|x [k] − T x| 0.0059 0.0070 0.0067 0.0068 0.0085 0.0076 0.0088 0.0053 0.0000 0.0052 0.0073 0.0002 0.0054 0.0001 0.0074 0.0000 0.0076 0.0000 0.0076 0.0000 0.0091 0.0000 0.0077 0.0000 0.0095 0.0093 0.0092 0.0032

Conclusion

We have developed two new iterative schemes, namely modified Maan and modified Ishikawa. The convergence theorems for our proposed schemes have been proved. In Section 2, the table provides comparison between Mann, modified Mann, Ishikawa and modified Ishikawa iterative procedures. Our results clearly indicate that how rapidly our proposed methods converge to the fixed points. In some given test problems, due to large difference in number of iterations, it is obvious that modified Mann and modified Ishikawa iterative schemes require very little time to produce fixed point.

Acknowledgment This study was supported by research funds from Dong-A University.

References [1] V. Berinde, Iterative Approximation of Fixed Points, Editura Efemeride, Baia Mare, 2002. [2] V. Berinde, On the convergence of the Ishikawa iteration in the class of quasi contractive operators, Acta Math. Univ. Comenian. (N.S.) 73 (2004), 119–126. 8 458

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[3] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228. [4] S. K. Chatterjea, Fixed-point theorems, C.R. Acad. Bulgare Sci. 25 (1972), 727–730. ´ c, A generalization of Banach’s contraction principle, Proc. Amer. Math. [5] Lj. B. Ciri´ Soc. 45 (1974), 267–273. [6] C. W. Groetsch, A note on segmenting Mann iterates, J. Math. Anal. Appl. 40 (1972), 369–372. [7] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. [8] S. M. Kang, A. Rafiq, and Y. C. Kwun, A new second-order iteration method for solving nonlinear equations, Abstr. Appl. Anal. 2013 (2013), Article ID 487062, 4 pages. [9] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 10 (1968), 71–76. [10] W. A. Kirk and B. Sims, Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers, 2001. [11] M. A. Krasnosel’skiˇı, Two remarks on the method of successive approximations, (Russian) Uspehi Mat. Nauk (N.S.) 10 (1955), 123–127. [12] L. Liu, Ishikawa and Mann iterative process with erros for non linear strongly accreive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), 114–125. [13] W. R. Mann, Mean value methods in iteration Proc. Amer. Math. Soc. 44 (1953), 506–510. [14] B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196 (1974), 161–176. [15] I. A. Rus, Principles and Applications of the Fixed Point Theory, (Romanian) Editura Dacia. Cluj-Napoca, 1979. [16] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, ClujNapoca, 2001. [17] T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math. (Basel) 23 (1972), 292–298.

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MEAN ERGODIC THEOREMS FOR SEMIGROUPS OF LINEAR OPERATORS IN P-BANACH SPACES H. M. KENARI, REZA SAADATI, AND CHOONKIL PARK∗

Abstract. In this paper, by using the Rode’s method, we extend Yosida’s theorem to semigroups of linear operators in p-Banach spaces. Our paper is motivated from ideas in [7].

1. Introduction In 1938, Yosida [14] Proved the following mean ergodic theorem for linear operators: Let E be a real Banach space and T be a linear operator of E into itself such that there exists a constant C with kT n k ≤ C for n = 1, 2, 3, ..., and T is weakly completely continuous,i.e., T maps the closed unite ball of E into a weakly compact subset of E. Then, the Cesaro mean n 1X k T x Sn x = n k=1

converges strongly as n → +∞ to a fixed point of T for each x ∈ E. On the other hand, in 1975, Baillon [1] proved the following nonlinear ergodic theorem: Let X be a Banach space and C a closed convex subset of X. The mapping T : C → C is called nonexpansive on C if kT x − T yk ≤ kx − yk ∀x, y ∈ C. Let F (T ) be the set of fixed point of T . If X is stricly convex, F (T ) is closed and convex. In [1, 4], Baillon proved the first nonlinear ergodic theorem such that if X is a real Hilbert space and F (T ) 6= ∅, then for each x ∈ C, the sequence {Sn x} defined by 1 Sn x = ( )(x + T x + ... + T n−1 x) n converges weakly to a fiexd point of T . It was also shown by Pazy [8] thet if X a real Hilbert space and Sn x converges weakly to y ∈ C, then y ∈ F (T ). Recently, Rode [10] and Takahashi [13] tried to extend this nonlinear ergodic theorem to semigroup, generalizing the Cesaro means on N = {1, 2, ...}, such that the corresponding sequence of mappings converges to a projection onto the set of common fixed points. In this paper, by using the Rode’s method, we extend Yosida’s theorem to semigroups of linear operators in P-Banach spaces. The proofs employ the methods of Yosida[14], Greenleaf [5], Rode [10] and Takahashi [6, 12] . Our paper is motivated from ideas in [7] 2. p-Norm Definition 2.1. ([3, 11]) Let X be a real linear space. A function k.k : X → R is a quasi– norm ( valuation ) if it satisfies the following conditions : (1) kxk ≥ 0 for all x ∈ X and kxk = 0 if and only if x = 0; MSC(2010): Primary 39A10, 39B72; Secondary 47H10, 46B03. Keywords: ergodic theorem, semigroup, p-Banach space. ∗ Corresponding author: [email protected] (Choonkil Park).

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(2) kλxk = |λ|.kxk for all λ ∈ R and all x ∈ X; (3) There is a constant M ≥ 1 such that kx + yk ≤ M (kxk + kyk) for all x, y ∈ X. Then (X, k.k) is called a quasi-normed space. The smallest possible M is called the modulus of concavity of k.k. A quasi-Banach space is a complete quas-normed space. A quasi-norm k.k is called a p-norm 0 < p < 1 if kx + ykp ≤ kxkp + kykp for all x, y ∈ X. In this case, a quasi-Banach space is called a p-Banach space. By the Aoki-Rolewicz [11], each quasi-norm is equivalent to some p-norm (see also [9]). Since it is much easier to work with p-norm, henceforth we restrict our attention mainly to p-norms. 3. Preliminaries and lemmas Let E a real p-Banach space and let E ∗ be the conjugate space of E, that is, the space of all continuous linear functionals on E. The value of x∗ ∈ E ∗ at x ∈ E will be denoted by < x, x∗ >. We denote by coD the convex hull of D, coD the closure of coD. Let U be a linear continuous operator of E into itself. Then, we denote by U ∗ the conjugate operator of U . Assumption (A). Let (E, k.kp ) be a p-Banach space and {Tt : t ∈ G}, be a family of linear continuous operators of a real Banach space E into itself such that there exist a real number C with kTt kp ≤ C for all t ∈ G and the weak closure of {Tt x : t ∈ G} is weakly compact, for each x ∈ E. The index set G is a topological semigroup such that Tst = Ts .Tt for all s, t ∈ G and T is continuous with respect to the weak operator topology : < Ts x, x∗ >→< Tt x, x∗ > for all x ∈ E and x∗ ∈ E ∗ if s → t in G. We denote by m(G) the p-Banach space of all bounded continuous real valued functions on the topological semigroup G with the p-norm. For each s ∈ G and f ∈ m(G), we define elements ls f and rs f in m(G) given by ls f (t) = f (st) and rs f (t) = f (ts) for all t ∈ G. An element µ ∈ m(G)∗ (the conjugate space of m(G)) is called a mean on G if kµkp = µ(1) = 1. A mean µ on G is called left (right) invariant if µ(ls f ) = µ(f ) (µ(rs f ) = µ(f )) for all f ∈ m(G) and s ∈ G. An invariant mean is a left and right invariant mean. We know that µ ∈ m(G)∗ is a mean on G if and only if inf{f (t) : t ∈ G} ≤ µ(f ) ≤ sup{f (t) : t ∈ G} for every f ∈ m(G); see [4, 5, 9]. Let {Tt : t ∈ G} be a family of linear continuous operators of E into itself satisfying the assumption (A) and µ be a mean on G. Fix x ∈ E. Then, for x∗ ∈ E ∗ , the real valued function t →< Tt x, x∗ > is in m(G). Denote by µt < Tt x, x∗ > the valued of µ at this function. By linearity of µ and of < ., . >, this is linear in x∗ ; moreover, since |µt < Tt x, x∗ > | ≤ kµkp · sup | < Tt x, x∗ > | ≤ sup kTt xkp · kx∗ kp ≤ C · kxkp · kx∗ kp , t

t

it is continuous in x∗ . Hence µt < Tt x, . > is an element of E ∗∗ . So it follows from weak compactness of co{Tt x : t ∈ G} that µt < Tt x, x∗ >=< Tµ x, x∗ > for every x∗ ∈ E ∗ . Put K = co{Tt x : t ∈ G} and suppose that the element µt < Tt x, . > is not contained in the n(K), where n is the natural embedding of the p-Banach space E into its second conjugate space E ∗∗ . Since the convex set n(K) is compact in the weak ∗ topology of E ∗∗ , there exists an element y ∗ ∈ E ∗ such that µt < Tt x, y ∗ >< inf{< y ∗ , z ∗∗ >: z ∗∗ ∈ n(k)}

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Hence we have µt < Tt x, y ∗ > < inf{< y ∗ , z ∗∗ >: z ∗∗ ∈ n(k)} ≤ inf{< Tt x, y ∗ >: t ∈ G} ≤ µt < Tt x, y ∗ > . This is a contradiction. Thus, for a mean µ on G, we can define a linear continuous operator Tµ of E into itself such that kTµ kp ≤ C, T µx ∈ co{Tt x : t ∈ G} for all x ∈ E, and µt < Tt x, x∗ >=< Tµ x, x∗ > for all x ∈ E and x∗ ∈ E ∗ . we denote by F (G) the set all common fixed points of the mappings Tt , t ∈ G. Lemma 3.1. Assume that a left invariant mean µ exists on G. Then Tµ (E) ⊂ F (G). Especially, F (G) is not empty. Proof. Let x ∈ E and µ be a left invariant mean on G. Then since, for s ∈ G and x∗ , < Ts Tµ x, x∗ > = < Tµ x, Ts∗ x∗ >= µt < Tt x, Ts∗ x∗ >= µt < Ts Tt x, x∗ > = µt < Tst x, x∗ >= µt < Tt x, x∗ >=< Tµ x, x∗ >, we have Ts Tµ x = Tµ x. Hence Tµ (E) ⊂ F (G).



Lemma 3.2. Let λ be an invariant mean on G. Then Tλ Ts = Ts Tλ = Tλ for each s ∈ G and Tλ Tµ = Tµ Tλ = Tλ for each mean µ on G. Especially, Tλ is a projection of E onto F (G) . Proof. Let s ∈ G. Since < Tλ Ts x, x∗ >= λt < Tt Ts x, x∗ >= λt < Tts x, x∗ >= λt < Tt x, x∗ >=< Tλ x, x∗ > for x ∈ E and x∗ ∈ E ∗ , we have Tλ Ts = Tλ . It follows from Lemma 3.1 that Ts T λ = T λ for each s ∈ G. Let µj be a mean on G. Then, since < Tµ Tλ x, x∗ >= µt < Tt Tλ x, x∗ >= µt Tλ x, x∗ >=< Tλ x, x∗ > and < Tλ Tµ x, x∗ > = < Tµ x, Tλ∗ x∗ >= µt < Tt x, Tλ∗ x∗ >= µt < Tλ Tt x, x∗ > = µt < Tλ x, x∗ >=< Tλ x, x∗ > for x ∈ E and x∗ ∈ E ∗ , we have Tµ Tλ = Tλ Tµ = Tλ , Putting µ = λ, we have Tλ2 = Tλ and hence Tλ is a projection of E onto F (G).  As a direct consequence of Lemma 3.2, we have the following. Lemma 3.3. Let µ and λ be invariant means on G. Then Tµ = Tλ . Lemma 3.4. Assume that an invariant mean exists on G. Then, for each x ∈ E, the set co{Tt x : t ∈ G} ∩ F (G) consists of a single point. Proof. Let x ∈ E and µ be an invariant mean on G. Then, we know that Tµ x ∈ F (G) and Tµ x ∈ co{Tt x : t ∈ G}. So, we show that co{Tt x : t ∈ G} ∩ F (G) = {Tµ x}. Let ∗ ∗ x P0 n∈ co{Tt x : t ∈ G} ∩ F (G) and  > 0. Then, for x ∗ ∈ EP, nthere exists an element i=1 αi Tti x in the set co{Tt x : t ∈ G} such that  > C · kx kp .k i=1 αi Tti x − x0 kp . Hence we have n n X X  > C · kx∗ kp · k αi Tti x − x0 kp ≥ sup kTt kp · k αi Tti x − x0 kp · kx∗ kp i=1

≥ sup k t

= |

n X

n X

t

αi Tj,t Tj,ti x − x0 kj · kx∗ kj ≥ |
|

i=1

αi µt < Ttti x − x0 , x∗ > | = |µt < Tt x − x0 , x∗ > | = | < Tµ x − x0 , x∗ > |.

i=1

Since  is arbitrary, we have < Tµ x, x∗ >=< x0 , x∗ > for every x∗ ∈ E ∗ and hence Tµ x = x0 . 

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4. Ergodic Theorems Now we can prove mean ergodic theorems for semigroups of linear continuous operators in p-Banach space. Theorem 4.1. Let {Tt : t ∈ G} be a family of linear continuous operators in a real p-Banach space E satisfying Assumption (A). If a net {µα : α ∈ I} of means on G is asymptotically invariant, i.e., µα − rs∗ µα

and

µα − ls∗ µα

converge to 0 in the weak ∗ topology of m(G)∗ for each s ∈ G, then there exists a projection Q of E onto F (G) such that kQkp ≤ C, Tµα x converges weakly to Qx for each x ∈ E, QTt = Tt Q = Q for each t ∈ G, and Qx ∈ co{Tt x : t ∈ G} for each x ∈ E. Furthermore, the projection Q onto F (G) is the same for all asymptotically invariant nets. Proof. Let µ be a cluster point of net {µα : α ∈ I} in the weak ∗ topology of m(G)∗ . Then µ is an invariant mean on G. Hence, by Lemma 3.2, Tµ is a projection of E onto F (G) such that kTµ kp ≤ C, Tµ Tt = Tt Tµ = Tµ for each t ∈ G and Tµ x ∈ co{Tt x : t ∈ G} for each x ∈ E. Setting Q = Tµ , we show that Tµα x converges weakly to Qx for each x ∈ E. Since Tµα x ∈ co{Tt x : t ∈ G} for all α ∈ I and co{Tt x : t ∈ G} is weakly compact, there exists a subnet {Tµβ x : β ∈ J} of {Tµα x : α ∈ I} such that Tµβ x converges weakly to an element x0 ∈ co{Tt x : t ∈ G}. To show that Tµα x converges weakly to Qx, it is sufficient to show x0 = Qx. Let x∗ ∈ E ∗ and s ∈ G. since Tµβ x → x0 weakly, we have µβt < Tt x, x∗ >→< x0 , x∗ > and µβt < Tt x, Ts∗ x∗ >→< x0 , Ts∗ x∗ >=< Ts x0 , x∗ >. On the other hand, since µβ − ls∗ µβ → 0 in the weak ∗ topology, we have µβt < Tt x, x∗ > −ls∗ µβt < Tt x, x∗ >

=

µβt < Tt x, x∗ > −µβt < Tst x, x∗ >

= µβt < Tt x, x∗ > −µβt < Tt x, Ts∗ x∗ > → 0. Hence, we have < x0 , x∗ >=< Ts x0 , x∗ > and hence x0 ∈ F (G). So, we obtain Qx = T µx = x0 by Lemma 3.4. That the projection Q is the same for all asymptotically invariant nets is obvious from Lemma 3.3.  As a direct consequence of Theorem 4.1, we have the following. Corollary 4.2. Let {Tt : t ∈ G} be as in Theorem 4.1 and assume that an invariant mean exists on G. Then, there exists a projection Q of E onto F such that kQkp ≤ C, QTt = Tt Q = Q for each t ∈ G and Qx ∈ co{Tt x : t ∈ G} for each x ∈ E Theorem 4.3. Let {Tt : t ∈ G} be as in Theorem 4.1. If a net {µα : α ∈ I} of means on G is asymptotically invariant and further µα −rs∗ µα converges to 0 in the strong topology of m(G)∗ , then exists a projection Q of E onto F (G) such that kQkp ≤ C, Tµα x converges strongly to Qx for each x ∈ E, QTt = Tt Q = Q for each t ∈ G, and Qx ∈ co{Tt x : t ∈ G} for each x ∈ E. Proof. As in the proof of Theorem 4.1, let Q = Tµ , where µ is a cluster point of the net {µα : α ∈ I} in the weak ∗ topology of m(G)∗ . Then we show that Tµα x converges strongly to Qx for each x ∈ E. Let E0 = co{y − Tt y : y ∈ E, t ∈ G}. Then, for any z ∈ E0 , Tµα z converges strongly to 0. In fact, if z = y − Ts y, then since, for any y ∗ ∈ E ∗ , | < Tµα z, y ∗ > | = |µαt < Tt (y − Ts y), y ∗ > | = |µαt < Tt y, y ∗ > −µαt < Tts y, y ∗ > | = |(µαt − rs∗ µαt ) < Tt y, y ∗ > | ≤ kµα − rs∗ µα kp · sup | < Tt y, y ∗ > | t

α

≤ kµ −

rs∗ µα kp

∗

· C · kykp · ky kp ,

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we have kTµα zkp ≤ C · kµα − rs∗ µα kp · kykp . Using this inequality, we show that Tµα z converges strongly to 0 for any z ∈ E0 . Let z be anyPelement of E0 and  be any positive number. By the definition of E0 , there existsPan element ni=1 ai (yi − Tsi yi ) in the set co{y − Ts y : y ∈ E, s ∈ G} such that  > 2C · kz − ni=1 ai (yi − Tsi yi )kp . On the other hand, from kµα − rs∗ µα kp → 0 for all s ∈ G, there exists a0 ∈ I such that, for all α ≥ α0 and i = 1, 2, ..., n,  > kµα − rs∗i µα kp · 2Ckyi kp . This implies kT zkp µα

n n X X α ≤ kT z − T ( ai (yi − Tsi yi ))kp + kTµ ( ai (yi − Tsi yi ))kp µα

µα

≤ kTµα kp · kz −

i=1 n X

i=1

ai (yi − Tsi yi )kp + |

i=1

≤ C · kz −

n X

n X

ai |P kTµα (yi − Tsi yi )kp

i=1

ai (yi − Tsi yi )kP + sup kµα − rs∗i µα kp · C · kyi kp i

i=1

  + = . 2 2 Hence Tµα z converges strongly to 0 for each z ∈ E0 . Next, assume that x − Tµ x for some x ∈ E is not contained in the set E0 . Then, by the Hahn– Banach theorem, there exists a linear continuous functional y ∗ such that < x − Tµ x, y ∗ >= 1 and < z, y ∗ >= 0 for all z ∈ E0 . and so since x − Tt x ∈ E0 for all t ∈ G, we have
= µt < x − Tt x, y ∗ >= 0. This is a contradiction. Hence x − Tµ for all x ∈ E are contained in E0 . Therefore, we have Tµα x − Tµ x = Tµα (x − Tµ ) converges strongly to 0 for all x ∈ E. This completes the proof.  By using Theorem 4.3, we can obtain the following corollary. Corollary 4.4. Let E be a real p-Banach space and T be a linear operator of E into itself such that exists a constant C with kT n kp ≤ C for n = 1, 2, .... Assume that T is weakly completely continuous, i.e., T maps the closed unit ball of E into a weakly compact subset of E. Then there exists a projection Q of PE onto the set F (T ) of all fixed points of T such that kQkp ≤ C, the Cesaro means Sn = n1 nk=1 T k x converges strongly to Qx for each x ∈ E, and T Q = QT = Q. Proof. Let x ∈ E. Then, since {T n x : n = 1, 2, ...} = T ({T n−1 x : n = 1, 2, ...}) ⊂ T (B(0, kx · (c + 1))), where B(x, r) means the closed ball with center x and radius r, the weak closure of {T n x : n = 1, 2, ...} is weakly compact. On the other hand, P let G = {1, 2, 3, ...} with the discrete topology and µn be a mean on G such that µn (f ) = ni=1 ( n1 )f (i) for each f ∈ m(G). Then, it is obvious that kµn − rk∗ µn kp ≤ 2k n → 0 for all k ∈ G. So, it follows from Theorem 4.3 that the corollary is true.  If G = [0, ∞) whit the natural topology, then we obtain the corresponding result. Corollary 4.5. Let E be a real p-Banach space and {Tt : t ∈ [0, ∞)} be a family of linear operators of E into itself satisfying Assumption (A). Then there exists a projection Q of E RT R onto F (G) such that kQkp ≤ C, T1 0 T t xdt converges strongly to Qx for each x ∈ E, and Tt Q = QTt = Q for each t ∈ [0, ∞). RT R Remark. T1 0 T t xdt is a weak vector valued integral with respect to means on G = [0, ∞). As in Section IV of Rode [10], we can also obtain the strong convergence of the

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sequences (1 − r)

∞ X

rk T k x,

r → 1−

k=1

and

Z λ

∞

e−λt Tt xdt,

λ→0+.

0

References [1] J. B. Baillon, Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Ser. A-B 280 (1975), A1511–A1514. [2] J. B. Baillon, Comportement asymptotique des iteres de contractions non lineaires dans les espace Lp , C. R. Acad. Sci. Paris Ser. A-B 286 (1978), 157–159. [3] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis. vol. 1, colloq. publ. vol. 48, Amer. Math. Soc, Providence, RI, 2000. [4] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. [5] F. P. Greenleaf, Invariant Mean on Topological Groups and Their Applications van Nostrand Mathematical Studies, Vol. 16, van Nostrand-Reinhold, New York, 1969. [6] N. Hirano, W. Takahashi, Nonlinear ergodic theorems for an amenable semigroup of nonexpansive mappings in a Banach space, Pacific J. Math. 112 (1984), 333–346. [7] K. Kido, W. Takahashi, Mean ergodic theorems for semigroup of linear operators, J. Math. Anal. Appl. 103 (1984), 387–394. [8] A. Pazy, On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space, Israel J. Math. 26 (1977), 197–204. [9] Y. Purtas, H. Kiziltunc, Weak and strong convergence of an explicit iteration process for an asymptotically quasi-nonexpansive mapping in Banach space, J. Nonlinear Sci. Appl, 5 (2012), 403–411. [10] G. Rode, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl., 85 (1982), 172–178. [11] S. Rolewicz, Metric Linear Space, Pwn-Polish Sci. Publ/Reidel, Warzawa and Pordrecht, 1984. [12] W. Takahashi, Invariant functions for amenable semigroups of positive contractions on L1 , Kodai Math. Sem. Rep. 23 (1971), 131–143. [13] W. Takahashi, A nonlinear ergodic theorems for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981), 253–256. [14] K. Yosida, Mean ergodic theorem in Banach space, Proc. Imp. Acad. Tokyo. 14 (1938), 292–294. H. M. Kenari Department of Mathematics, Science and Research Branch, Islamic Azad University, Post Code 14778, Ashrafi Esfahani Ave, Tehran, I.R. Iran. Email: hmkenari@@hush.ai Reza Saadati Department of Mathematics, Iran University of Science and Technology, Tehran, Iran Email: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea Email: [email protected]

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´ c type Fixed Point Results for Ciri´ - -GF -Contractions Marwan Amin Kutbia Muhammad Arshadb , Aftab Hussainc , (a)

Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia and (b;c)

Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan

E-mail address: a [email protected], b [email protected], c

[email protected]

Abstract: The aim of this paper is to establish some new …xed point results for ´ c type - -GF -contraction in a complete metric space. We extend the conCiri´ ´ c type - -GF -contraction. cept of F -contraction and introduce the notion Ciri´ An example is given to demonstrate the novelty of our work. 2000 Mathematics Subject Classi…cation: 46S40; 47H10; 54H25, ´ c type Keywords and Phrases: Metric space; …xed point; F contraction; Ciri´ - -GF -contraction.

1

Introduction

In metric …xed point theory the contractive conditions on underlying functions play an important role for …nding solutions of …xed point problems. Banach contraction principle [4] is a fundamental result in metric …xed point theory. Due to its importance and simplicity, several authors have generalized/extended it in di¤erent directions. In 1973, Geraghty [9] studied a generalization of Banach ´ c [5], introduced quasi contraction theorem, which contraction principle. Ciri´ generalized Banach contraction principle. Over the years, Banach contraction theorem has been generalized in di¤erent ways by several mathematicians (see [1-24]). In 2012, Samet et al. [22], introduced a concept of

- contractive type

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mappings and established various …xed point theorems for mappings in complete metric spaces. Afterwards Karapinar et al. [16], re…ned the notion and obtained various …xed point results. Hussain et al. [12], extended the concept of -admissible mappings and obtained useful …xed point theorems. Subsequently, Abdeljawad [1] introduced pair of

admissible mappings satisfying new suf-

…cient contractive conditions di¤erent from those in [12, 22], and proved …xed point and common …xed point theorems. Lately, Salimi et al. [21], modi…ed the concept of

contractive mapping and established …xed point results.

De…nition 1 ([22]). Let T : X ! X and T is

-admissible if x; y 2 X;

(x; y)

:X

X ! [0; +1). We say that

1 implies that

De…nition 2 ([21]). Let T : X ! X and

;

: X

(T x; T y)

X ! [0; +1) be two

functions. We say that T is -admissible mapping with respect to (x; y)

(x; y) implies that

(T x; T y)

1:

if x; y 2 X;

(T x; T y):

If (x; y) = 1; then above de…nition reduces to de…nition 1. If

(x; y) = 1;

then T is called an -subadmissible mapping. De…nition 3 [11] Let (X; d) be a metric space. Let T : X ! X and X

X ! [0; +1) be two functions. We say that T is

;

:

-continuous mapping

on (X; d) if for given x 2 X; and sequence fxn g with xn ! x as n ! 1;

(xn ; xn+1 )

(xn ; xn+1 ) for all n 2 N ) T xn ! T x:

In 1962, Edelstein proved the following version of the Banach contraction principle. Theorem 4 [7]. Let (X; d) be a metric space and T : X ! X be a self mapping. Assume that d(T x; T y) < d(x; y); holds for all x; y 2 X with x 6= y: Then T has a unique …xed point in X.

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In 2012, Wardowski [24] introduce a new type of contractions called F contraction and proved new …xed point theorems concerning F -contraction. He generalized the Banach contraction principle in a di¤erent way than as it was done by di¤erent investigators. Piri et al. [19] de…ned the F -contraction as follows. De…nition 5 [19] Let (X; d) be a metric space. A mapping T : X ! X is said to be an F contraction if there exists 8x; y 2 X; d(T x; T y) > 0 )

> 0 such that

+ F (d(T x; T y))

F (d(x; y)) ;

(1.1)

where F : R+ ! R is a mapping satisfying the following conditions: (F1) F is strictly increasing, i.e. for all x; y 2 R+ such that x < y, F (x) < F (y); (F2) For each sequence f only if limn!1 F (

n)

=

1 n gn=1

of positive numbers, limn!1

1;

(F3) There exists k 2 (0; 1) such that lim We denote by

F,

! 0+

k

n

= 0 if and

F ( ) = 0.

the set of all functions satisfying the conditions (F1)-(F3).

Example 6 [24] Let F : R+ ! R be given by the formula F ( ) = ln : It is clear that F satis…ed (F1)-(F2)-(F3) for any k 2 (0; 1): Each mapping T : X ! X satis…ying (1:1) is an F -contraction such that d(T x; T y)

e

d(x; y); for all x; y 2 X; T x 6= T y:

It is clear that for x; y 2 X such that T x = T y the inequality d(T x; T y) e

d(x; y); also holds, i.e. T is a Banach contraction.

Example 7 [24] If F ( ) = ln

+ ;

> 0 then F satis…es (F1)-(F3) and the

condition (1:1) is of the form d(T x; T y) d(x; y)

ed(T x;T y)

d(x;y)

e

; for all x; y 2 X; T x 6= T y:

Remark 8 From (F1) and (1:1) it is easy to conclude that every F -contraction is necessarily continuous.

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Wardowski [24] stated a modi…ed version of the Banach contraction principle as follows. Theorem 9 [24] Let (X; d) be a complete metric space and let T : X ! X be an F contraction. Then T has a unique …xed point x 2 X and for every x 2 X the sequence fT n xgn2N converges to x .

Hussain et al. [11] introduced the following family of new functions. Let

G

denotes the set of all functions G : R+4 ! R+ satisfying:

(G) for all t1 ; t2 ; t3 ; t4 2 R+ with t1 t2 t3 t4 = 0 there exists

> 0 such that

G(t1 ; t2 ; t3 ; t4 ) = . De…nition 10 [11] Let (X; d) be a metric space and T be a self mapping on X: Also suppose that ; : X

X ! [0; +1) be two function. We say that T is -

-GF -contraction if for x; y 2 X; with (x; T x)

(x; y) and d(T x; T y) > 0 we

have G(d(x; T x); d(y; T y); d(x; T y); d(y; T x)) + F (d(T x; T y)) where G 2

2

G

and F 2

F (d(x; y)) ;

F:

Main Result

´ c type - -GF -contraction In this section, we de…ne a new contraction called Ciri´ and obtained some new …xed point theorems for such contraction in the setting ´ c type - -GF -contraction as follows: of complete metric spaces. We de…ne Ciri´ De…nition 11 Let (X; d) be a metric space and T be a self mapping on X: Also suppose that ´ Ciri´c type

;

: X

X ! [0; +1) two functions. We say that T is

- -GF -contraction if for all x; y 2 X; with (x; T x)

(x; y) and

d(T x; T y) > 0; we have G(d(x; T x); d(y; T y); d(x; T y); d(y; T x)) + F (d(T x; T y))

F (M (x; y)) (2.1)

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where M (x; y) = max d(x; y); d(x; T x); d(y; T y); G2

G

and F 2

d(x; T y) + d(y; T x) 2

F:

Now we state our main result. ´ Theorem 12 Let (X; d) be a complete metric space. Let T be a Ciri´c type - -GF -contraction satisfying the following assertions: (i) T is an -admissible mapping with respect to ; (ii) there exists x0 2 X such that (x0 ; T x0 ) (iii) T is

(x0 ; T x0 );

-continuous.

Then T has a …xed point in X: Moreover, T has a unique …xed point when (x; y)

(x; x) for all x; y 2 F ix(T ).

Proof. Let x0 in X such that struct a sequence

1 fxn gn=1

(x0 ; T x0 )

(x0 ; T x0 ): For x0 2 X; we con-

such that x1 = T x0 , x2 = T x1 = T 2 x0 . Con-

tinuing this process, xn+1 = T xn = T n+1 x0 , for all n 2 N: Now since, T is an

-admissible mapping with respect to

then

(x0 ; x1 ) =

(x0 ; T x0 )

(x0 ; T x0 ) = (x0 ; x1 ). By continuing in this process we have, (xn

1 ; T xn 1 )

= (xn

1 ; xn )

(xn

1 ; xn );

for all n 2 N:

(2.2)

If there exists n 2 N such that d(xn ; T xn ) = 0, there is nothing to prove. So, we assume that xn 6= xn+1 with d(T xn

1 ; T xn )

= d(xn ; T xn ) > 0; 8n 2 N:

´ c type - -GF -contraction, for any n 2 N; we have Since T is Ciri´ G(d(xn +F (d(T xn

1 ; T xn ))

1 ; T xn 1 ); d(xn ; T xn ); d(xn 1 ; T xn ); d(xn ; T xn 1 ))

F (M (xn

1 ; xn ))

which implies G(d(xn +F (d(T xn

1 ; T xn ))

1 ; xn ); d(xn ; xn+1 ); d(xn 1 ; xn+1 ); 0)

F (M (xn

1 ; xn ))

(2.3)

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Now by de…nition of G, d(xn exists

1 ; xn ):d(xn ; xn+1 ):d(xn 1 ; xn+1 ):0

= 0, so there

> 0 such that, G(d(xn

1 ; xn ); d(xn ; xn+1 ); d(xn 1 ; xn+1 ); 0)

= :

Therefore F (d(xn ; xn+1 )) = F (d(T xn

1 ; T xn ))

F (M (xn

1 ; xn ))

:

(2.4)

Now M (xn

8 < d(x n 1 ; xn ) = max : 8 < d(x n = max : =

9 = ; x ); d(x ; x ); 1 n n n+1 d(xn 1 ;xn+1 )+d(xn ;xn ) ; ; 2 9 ; x ); d(x ; x ); d(x ; x ); =

1 ; xn ); d(xn

1

n

n 1

d(xn

n

n

1 ;xn+1 )

2

;

d(xn

max d(xn

1 ; xn ); d(xn ; xn+1 );

max fd(xn

1 ; xn ); d(xn ; xn+1 )g :

n+1

1 ; xn )

;

+ d(xn ; xn+1 ) 2

So, we have F (d(xn ; xn+1 )) = F (d(T xn In this case M (xn

1 ; xn )

1 ; T xn ))

F (max fd(xn

= max fd(xn

1 ; xn ); d(xn ; xn+1 )g)

1 ; xn ); d(xn ; xn+1 )g

:

= d(xn ; xn+1 ) is

impossible, because F (d(xn ; xn+1 )) = F (d(T xn

1 ; T xn ))

F (d(xn ; xn+1 ))

< F (d(xn ; xn+1 )) :

Which is a contradiction. So M (xn

1 ; xn )

= max fd(xn

1 ; xn ); d(xn ; xn+1 )g

= d(xn

1 ; xn ):

Thus from (2:4); we have F (d(xn ; xn+1 ))

F (d(xn

1 ; xn ))

:

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Continuing this process, we get F (d(xn ; xn+1 ))

F (d(xn

1 ; xn ))

= F (d(T xn F (d(xn

2 ; T xn 1 ))

2 ; xn 1 ))

= F (d(T xn F (d(xn .. .

2

3 ; T xn 2 ))

3 ; xn 2 ))

F (d(x0 ; x1 ))

2

3

n :

This implies that F (d(xn ; xn+1 ))

F (d(x0 ; x1 ))

From (2:5), we obtain limn!1 F (d(xn ; xn+1 )) =

n :

1. Since F 2

(2.5) F;

we have

lim d(xn ; xn+1 ) = 0:

(2.6)

n!1

From (F 3), there exists k 2 (0; 1) such that k

lim

(d(xn ; xn+1 )) F (d(xn ; xn+1 )) = 0:

n!1

(2.7)

From (2:5), for all n 2 N; we obtain k

(d(xn ; xn+1 )) (F (d(xn ; xn+1 ))

k

F (d(x0 ; x1 )))

(d(xn ; xn+1 )) n

0: (2.8)

By using (2:6), (2:7) and letting n ! 1; in (2:8), we have lim

n!1

k

n (d(xn ; xn+1 ))

= 0:

(2.9) k

We observe that from (2:9); then there exists n1 2 N; such that n (d(xn ; xn+1 )) 1 for all n

n1 , we get d(xn ; xn+1 )

1 1

nk

for all n

n1 :

(2.10)

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Now, m; n 2 N such that m > n

n1 . Then, by the triangle inequality and

from (2:10) we have d(xn ; xm )

d(xn ; xn+1 ) + d(xn+1 ; xn+2 ) + d(xn+2 ; xn+3 ) + ::: + d(xm 1(2.11) ; xm ) m X1

=

i=n 1 X i=n 1 X i=n

The series

P1

i=n

d(xi ; xi+1 )

d(xi ; xi+1 ) 1 1

ik

1 1

ik

:

is convergent: By taking limit as n ! 1; in (2:11); we

have limn;m!1 d(xn ; xm ) = 0: Hence fxn g is a Cauchy sequence. Since X is a complete metric space there exists x 2 X such that xn ! x as n ! 1: T is an

- -continuous and

(xn

1 ; xn )

(xn

1 ; xn );

for all n 2 N then

xn+1 = T xn ! T x as n ! 1: That is, x = T x . Hence x is a …xed point of T . To prove uniqueness, let x 6= y be any two …xed point of T , then from (2:1); we have G(d(x; T x); d(y; T y); d(x; T y); d(y; T x)) + F (d(T x; T y))

F (M (x; y))

we obtain + F (d(x; y))

F (d(x; y)) :

which is a contradiction. Hence, x = y. Therefore, T has a unique …xed point.

Theorem 13 Let (X; d) be a complete metric space. Let T be a self mapping satisfying the following assertions: (i) T is an -admissible mapping with respect to ; ´ c type - -GF -contraction; (ii) T is Ciri´ (iii) there exists x0 2 X such that (x0 ; T x0 ) (iv) if fxn g is a sequence in X such that

(x0 ; T x0 );

(xn ; xn+1 )

(xn ; xn+1 ) with

xn ! x as n ! 1 then either (T xn ; x)

(T xn ; T 2 xn ) or (T 2 xn ; x)

(T 2 xn ; T 3 xn )

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holds for all n 2 N. Then T has a …xed point in X: Moreover, T has a unique …xed point when (x; y)

(x; x) for all x; y 2 F ix(T ).

Proof. As similar lines of the Theorem 12, we can conclude that (xn ; xn+1 )

(xn ; xn+1 ) and xn ! x as n ! 1:

Since, by (iv), either (T xn ; x )

(T xn ; T 2 xn ) or (T 2 xn ; x )

(T 2 xn ; T 3 xn );

holds for all n 2 N. This implies (xn+1 ; x )

(xn+1 ; xn+2 ) or (xn+2 ; x )

(xn+2 ; xn+3 ); for all n 2 N:

Then there exists a subsequencefxnk g of fxn g such that (xnk ; T xnk ) = (xnk ; xnk +1 )

(xnk ; x ):

From (2:1), we have G(d(xnk ; T xnk ); d(x ; T x ); d(xnk ; T x ); d(x ; T xnk )) + F (d(T xnk ; T x )) F (M (xnk ; x )) 8 91 0 < d(x ; x ); d(x ; T x ); d(x ; T x ); = nk nk nk A = F @max d(xnk ;T x )+d(x ;T xnk ) ; : ; 2 8 91 0 < d(x ; x ); d(x ; x = nk nk nk +1 ); d(x ; T x ); A: = F @max d(xnk ;T x )+d(x ;xnk +1 ) : ; 2

Using the continuity of F and the fact that

lim d(xnk ; x ) = 0 = lim d(xnk +1 ; x )

k!1

k!1

(2.12)

we obtain + F (d(x ; T x ))

F (d(x ; T x )) :

(2.13)

Which is a contradiction. Therefore, d(x ; T x ) = 0, implies x is a …xed point of T . Uniqueness follows similar lines as in Theorem 12. In the following we extend the Wardowski type …xed point theorem. 9

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Theorem 14 Let T be a continuous self mapping on a complete metric space X. If for x; y 2 X with d(x; T x)

d(x; y) and d(T x; T y) > 0; we have

G(d(x; T x); d(y; T y); d(x; T y); d(y; T x)) + F (d(T x; T y))

F (M (x; y)) ;

where M (x; y) = max d(x; y); d(x; T x); d(y; T y); G2

G

and F 2

F:

d(x; T y) + d(y; T x) 2

:

Then T has a …xed point in X:

Proof. Let us de…ne ; : X

X ! [0; +1) by

(x; y) = d(x; y) and (x; y) = d(x; y) for all x; y 2 X: Now, d(x; y)

d(x; y) for all x; y 2 X; so

(x; y)

(x; y) for all x; y 2 X:

That is, conditions (i) and (iii) of Theorem 12 hold true. Since T is continuous, so T is d(x; T x)

- -continuous. Let (x; T x)

(x; y) and d(T x; T y) > 0; we have

d(x; y) with d(T x; T y) > 0; then

G(d(x; T x); d(y; T y); d(x; T y); d(y; T x)) + F (d(T x; T y))

F (M (x; y)) :

´ c type - -GF -contraction mapping. Hence, all conditions of That is, T is Ciri´ Theorem 12 satis…ed and T has a …xed point. Corollary 15 Let T be a continuous selfmapping on a complete metric space X. If for x; y 2 X with d(x; T x)

d(x; y) and d(T x; T y) > 0; we have

+ F (d(T x; T y)) where

> 0; and F 2

F:

F (M (x; y)) ;

Then T has a …xed point in X:

Corollary 16 Let T be a continuous selfmapping on a complete metric space X. If for x; y 2 X with d(x; T x)

d(x; y) and d(T x; T y) > 0; we have

+ F (d(T x; T y)) where

> 0; and F 2

F:

F (d(x; y)) ;

Then T has a …xed point in X: 10

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Corollary 17 [11] Let (X; d) be a complete metric space. Let T : X ! X be a self-mapping satisfying the following assertions: (i) T is an -admissible mapping with respect to ; (ii) T is an - -GF -contraction (iii) there exists x0 2 X such that (x0 ; T x0 ) (iv) T is

(x0 ; T x0 );

-continuous.

Then T has a …xed point in X: Moreover, T has a unique …xed point when (x; y)

(x; x) for all x; y 2 F ix(T ).

Corollary 18 [11] Let (X; d) be a complete metric space. Let T : X ! X be a self-mapping satisfying the following assertions: (i) T is an -admissible mapping with respect to ; (ii) T is an - -GF -contraction (iii) there exists x0 2 X such that (x0 ; T x0 ) (iv)if fxn g is a sequence in X such that

(x0 ; T x0 );

(xn ; xn+1 )

(xn ; xn+1 ) with

xn ! x as n ! 1 then either (T xn ; T 2 xn ) or (T 2 xn ; x)

(T xn ; x)

(T 2 xn ; T 3 xn )

holds for all n 2 N. Then T has a …xed point in X: Moreover, T has a unique …xed point when (x; y)

(x; x) for all x; y 2 F ix(T ).

Example 19 Consider the sequence, S1 = 1

3

S2 = 1

3+2

5

S3 = 1

3+2

5+3

7

Sn = 1

3+2

5+3

7::: + n

(2n + 1) =

Let X=fSn : n 2 Ng and d (x; y) = jx space. If F ( ) =

+ ln ;

n(n+1)(4n+5) : 6

yj : Then (X; d) is a complete metric

> 0 and G(t1 ; t2 ; t3 ; t4 ) =

where

= 1: De…ne the

11

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mapping T : X ! X by, T (S1 ) = S1 and T (Sn ) = Sn if x 2 X; (x; T x) = lim

n!1

1 2

1; n

1 and (x; y) = 1

for all x 2 X: we have

d (T (Sn ) ; T (S1 )) Sn 1 3 (n 1) n(4n + 1) = lim = n!1 Sn d (Sn ; S1 ) 3 n(n + 1)(4n + 5)

18 = 1: 18

So we conclude the following two cases: Case 1: we observe that for every m 2 N; m > 2; n = 1 or n = 1 and m > 1 then (Sm ; Sn )

(Sm ; T (Sm )), we have

d (T (Sm ) ; T (S1 )) d(T (Sm );T (S1 )) e M (Sm ; S1 )

M (Sm ;S1 )

= =
n > 1, then (Sm ; Sn )

= =

(Sm ; T (Sm )), we have

d (T (Sm ) ; T (Sn )) d(T (Sm );T (Sn )) M (Sm ;Sn ) e M (Sm ; Sn ) Sm 1 Sn 1 Sn Sn 1 +Sm 1 Sm e Sm Sn (m 1) m(4m + 1) (n 1) n(4n + 1) n(n+1)(4n+5) 6 e m(m + 1)(4m + 5) n(n + 1)(4n + 5)

m(m+1)(4m+5) 6

e

1

:

So all condition of theorems are satis…ed, T has a …xed point in X: Let (X; d; ) be a partially ordered metric space. Let T : X ! X is such that for x; y 2 X; with x

y implies T x

T y, then the mapping T is said

to be non-decreasing. We derive following important result in partially ordered metric spaces. Theorem 20 Let (X; d; ) be a complete partially ordered metric space. Assume that the following assertions hold true: (i) T is nondecreasing and ordered GF -contraction; 12

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(ii) there exists x0 2 X such that x0

T x0 ;

(iii) either for a given x 2 X and sequence fxn g in X such that xn ! x as n ! 1 and xn

xn+1 for all n 2 N we have T xn ! T x

or if fxn g is a sequence in X such that xn

xn+1 with xn ! x as n ! 1

then either T xn

x or T 2 xn

x

holds for all n 2 N. Then T has a …xed point in X: : R+

De…ne z = f

! R+ :

is a Lebesgue integral mapping which is R summable, nonnegative and satis…es (t)dt > 0, for each > 0g: 0

We can easily deduce following result involving integral type inequalities.

Theorem 21 Let T be a continuous selfmapping on a complete metric space X. If for x; y 2 X with d(x;T Z x)

(t)dt

0

d(x;y) Z

(t)dt and

0

d(TZx;T y)

(t)dt > 0;

0

we have G(

d(x;T Z x)

0

where

B F@

(t)dt;

0

MZ(x;y) 0

d(y;T Z y)

1

0

(t)dt;

d(x;T Z y) 0

(t)dt;

d(y;T Z x)

G

B (t)dt) + F @

d(TZx;T y) 0

1

C (t)dtA

C (t)dtA ;

M (x; y) = max d(x; y); d(x; T x); d(y; T y); 2 z; G 2

0

0

and F 2

F:

d(x; T y) + d(y; T x) 2

:

Then T has a …xed point in X:

Con‡ict of Interests The authors declare that they have no competing interests. Acknowledgement: The …rst author gratefully acknowledges the support from the Deanship of Scienti…c Research (DSR) at King Abdulaziz University (KAU) during this research. 13

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References [1] T. Abdeljawad, Meir-Keeler

contractive …xed and common …xed point

theorems, Fixed PoinTheory Appl. 2013 doi:10.1186/1687-1812-2013-19. [2] Ö. Acar and I. Altun, A Fixed Point Theorem for Multivalued Mappings with -Distance, Abstr. Appl. Anal., Volume 2014, Article ID 497092, 5 pages. [3] M. Arshad , Fahimuddin, A. Shoaib and A. Hussain, Fixed point results for - -locally graphic contraction in dislocated qusai metric spaces, Math Sci., (2014) doi 10.1007/s40096-014-0132, 7 pages. [4] S.Banach, Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales, Fund. Math., 3 (1922) 133–181. ´ c, A generalization of Banach’s contraction principle. Proc. Am. [5] LB. Ciri´ Math. Soc., 45, (1974) 267-273 [6] M.

Cosentino,

mappings

of

P.

Vetro,

Fixed

Hardy-Rogers-Type,

point

results

Filomat

for

F-contractive

28:4(2014),

715-722.

doi:10.2298/FIL1404715C [7] M. Edelstein, On …xed and periodic points under contractive mappings. J. Lond. Math. Soc., 37, 74-79 (1962). [8] B. Fisher, Set-valued mappings on metric spaces, Fundamenta Mathematicae, 112 (2) (1981) 141–145. [9] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973) 604-608. [10] N. Hussain, M. Arshad, A. Shoaib and Fahimuddin, Common …xed point results for

-contractions on a metric space endowed with graph, J.

Inequal. Appl., (2014) 2014:136.

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[11] N. Hussain and P. Salimi, suzuki-wardowski type …xed point theorems for

-GF -contractions, Taiwanese J. Math., 20 (20) (2014), doi:

10.11650/tjm.18.2014.4462 [12] N. Hussain, E. Karap¬nar, P. Salimi and F. Akbar, -admissible mappings and related …xed point theorems, J. Inequal. Appl., 114 (2013) 1-11. [13] N. Hussain, P Salimi and A. Latif, Fixed point results for single and set-valued

- - -contractive mappings, Fixed Point Theory Appl. 2013,

2013:212. [14] N. Hussain, E. Karapinar, P. Salimi, P. Vetro, Fixed point results for Gm Meir-Keeler contractive and G-( ; )-Meir-Keeler contractive mappings, Fixed Point Theory Appl. 2013, 2013:34. [15] N. Hussain, S. Al-Mezel and P. Salimi, Fixed points for

- -graphic con-

tractions with application to integral equations, Abstr. Appl. Anal., (2013) Article 575869. [16] E. Karapinar and B. Samet, Generalized (

) contractive type mappings

and related …xed point theorems with applications, Abstr. Appl. Anal., (2012) Article id:793486. [17] MA. Kutbi, M. Arshad and A. Hussain, On Modi…ed

Contractive

mappings, Abstr. Appl. Anal., (2014) Article ID 657858, 7 pages. [18] SB. Nadler, Multivalued contraction mappings, Pac. J. Math., 30 (1969), 475-488. [19] H. Piri and P. Kumam, Some …xed point theorems concerning F contraction in complete metric spaces, Fixed Poin Theory Appl. (2014) 2014:210. [20] M. Sgroi and C. Vetro, Multi-valued F -contractions and the solution of certain functional and integral equations, Filomat 27:7 (2013), 1259–1268.

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[21] P. Salimi, A. Latif and N. Hussain, Modi…ed

-Contractive mappings

with applications, Fixed Point Theory Appl. (2013) 2013:151. [22] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for

-contractive

type mappings, Nonlinear Anal. 75 (2012) 2154–2165. [23] NA. Secelean, Iterated function systems consisting of F -contractions, Fixed Point Theory Appl. 2013, Article ID 277 (2013). doi:10.1186/1687-18122013-277 [24] D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed PoinTheory Appl. (2012) Article ID 94.

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FIXED POINT AND QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN NON-ARCHIMEDEAN BANACH SPACES JUNG RYE LEE AND DONG YUN SHIN∗ Abstract. In this paper, we solve the following quadratic ρ-functional inequalities

        x−y−z y−x−z z−x−y

x+y+z +f +f +f

f 2 2 2 2 −f (x) − f (y) − f (z)k ≤ kρ(f (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y)

(0.1)

−4f (x) − 4f (y) − 4f (z))k, where ρ is a fixed non-Archimedean number with |ρ|
0 there is a positive integer N such that kxn − xm k ≤ ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that kxn − xk ≤ ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. The stability problem of functional equations originated from a question of Ulam [27] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [12] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Rassias [24] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [9] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y)

(1.1)

is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [26] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [2] noticed that the theorem of Skof is still true if the relevant domain E1 is replaced by an Abelian See  group.   [7, 15, 16] for more functional equations.  x−y x+y The functional equation 2f 2 + 2 2 = f (x) + f (y) is called a Jensen type quadratic equation. In [10], Gil´anyi showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k

(1.2)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (xy) + f (xy −1 ). See also [25]. Gil´anyi [11] and Fechner [8] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [22] proved the Hyers-Ulam stability of additive functional inequalities. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. We recall a fundamental result in fixed point theory.

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FIXED POINT AND QUADRATIC ρ-FUNCTIONAL INEQUALITIES

Theorem 1.3. [3, 6] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−α d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [13] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [4, 5, 14, 17, 20, 21, 23]). In Section 2, we deal with quadratic functional equations. In Section 3, we solve the quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in non-Archimedean Banach spaces. In Section 4, we solve the quadratic ρfunctional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in non-Archimedean Banach spaces. Throughout this paper, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| 6= 1. 2. Quadratic functional equations Theorem 2.1. Let X and Y be vector spaces. A mapping f : X → Y satisfies 

f

x+y+z x−y−z y−x−z z−x−y + + + 2 2 2 2



= f (x) + f (y) + f (z)

(2.1)

if and only if the mapping f : X → Y is a quadratic mapping. Proof. Assume that f : X → Y satisfies (2.1) Letting x = y = z = 0 in (2.1), we have 4f (0) = 3f (0). So f (0) = 0. Letting y = z = 0 in (2.1), we get x 2

 

2f

x + 2f − 2 

x 2f − 2





= f (x)

&



x 2

 

+ 2f

= f (−x)

(2.2)

for all x ∈ X, which imply that f (x) =
f (−x) for all x ∈ X. From this and (2.2), we obtain 4f x2 = f (x) or f (2x) = 4f (x) for all x ∈ X. Putting z = 0 in (2.1), we obtain 12 f (x + y) + 21 f (x − y) = f (x) + f (y) for all x, y ∈ X, which means that f : X → Y is a quadratic mapping. The converse is obviously true.  Corollary 2.2. Let X and Y be vector spaces. An even mapping f : X → Y satisfies f (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) = 4f (x) + 4f (y) + 4f (z)

(2.3)

for all x, y, z ∈ X. Then the mapping f : X → Y is a quadratic mapping. Proof. Assume that f : X → Y satisfies (2.3). Letting x = y = z = 0 in (2.3), we have 4f (0) = 12f (0). So f (0) = 0. Letting z = 0 in (2.3), we get 2f (x+y)+2f (x−y) = 4f (x)+4f (y) and so f (x+y)+f (x−y) = 2f (x) + 2f (y) for all x, y ∈ X. 

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J. LEE AND D. SHIN

3. Quadratic ρ-functional inequality (0.1) 1 Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |4| . In this section, we solve and investigate the quadratic ρ-functional inequality (0.1) in nonArchimedean normed spaces.

Lemma 3.1. An even mapping f : X → Y satisfies

       

f x + y + z + f x − y − z + f y − x − z + f z − x − y

2 2 2 2 −f (x) − f (y) − f (z)k ≤ kρ(f (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) −4f (x) − 4f (y) − 4f (z))k

(3.1)

for all x, y, z ∈ X if and only if f : X → Y is quadratic. Proof. Assume that f : X → Y satisfies (3.1). Letting x = y = z = 0 in (3.1), we get kf
(0)k ≤ |ρ|k8f (0)k. So f (0) = 0.

Letting y = z = 0 in (3.1), we get 4f x2 − f (x) ≤ 0 and so x 2

 

f

1 = f (x) 4

(3.2)

for all x ∈ X. By (3.1) and (3.2), we have         x+y+z x−y−z y−x−z z−x−y f +f +f +f = f (x) + f (y) + f (z) 2 2 2 2 1 for all x, y, z ∈ X, since |ρ| < |4| . The converse is obviously true.



Now we prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (3.1) in non-Archimedean Banach spaces. Theorem 3.2. Let ϕ : X 3 → [0, ∞) be a function with ϕ(0, 0, 0) = 0 such that there exists an L < 1 with   x y z L ϕ , , ≤ ϕ (x, y, z) (3.3) 2 2 2 |4| for all x, y, z ∈ X. Let f : X → Y be an even mapping such that

       

f x + y + z + f x − y − z + f y − x − z + f z − x − y

2 2 2 2 −f (x) − f (y) − f (z)k ≤ kρ(f (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) (3.4) −4f (x) − 4f (y) − 4f (z))k + ϕ(x, y, z) for all x, y, z ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 1 kf (x) − Q(x)k ≤ ϕ (x, 0, 0) 1−L for all x ∈ X. Proof. Letting x = y = z = 0 in (3.4), we get kf (0)k ≤ |ρ|k8f (0)k. So f (0) = 0. Letting y = z = 0 in (3.4), we get

 



4f x − f (x) ≤ ϕ(x, 0, 0)

2

485

(3.5)

(3.6)

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FIXED POINT AND QUADRATIC ρ-FUNCTIONAL INEQUALITIES

for all x ∈ X. Consider the set S := {h : X → Y, h(0) = 0} and introduce the generalized metric on S: d(g, h) = inf {µ ∈ R+ : kg(x) − h(x)k ≤ µϕ (x, 0, 0) , ∀x ∈ X} , where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see
[18]). Now we consider the linear mapping J : S → S such that Jg(x) := 4g x2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then kg(x) − h(x)k ≤ εϕ (x, 0, 0) for all x ∈ X. Hence kJg(x) − Jh(x)k

     

x x x

≤ |4|εϕ − 4h , 0, 0 = 4g 2 2 2

≤ |4|ε

L ϕ (x, 0, 0) ≤ Lεϕ (x, 0, 0) |4|

for all x ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (3.6) that d(f, Jf ) ≤ 1. By Theorem 1.3, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., x 2

 

Q (x) = 4Q

(3.7)

for all x ∈ X. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (3.7) such that there exists a µ ∈ (0, ∞) satisfying kf (x) − Q(x)k ≤ µϕ (x, 0, 0) for all x ∈ X;
(2) d(J l f, Q) → 0 as l → ∞. This implies the equality liml→∞ 4n f 2xn = Q(x) for all x ∈ X; 1 1 (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤ 1−L . So kf (x) − Q(x)k ≤ 1 1−L ϕ(x, 0, 0) for all x ∈ X. It follows from (3.3) and (3.4) that

       

Q x + y + z + Q x − y − z + Q y − x − z + Q z − x − y

2 2 2 2

−Q(x) − Q(y) − Q(z)k

       

x+y+z x−y−z y−x−z z−x−y n = lim |4| f +f +f +f n→∞ 2n+1 2n+1 2n+1 2n+1       x y z

−f −f −f n n 2 2 2n

       

x+y+z x−y−z y−x−z z−x−y n

+f +f +f ≤ lim |4| |ρ| f n→∞ 2n 2n 2n 2n         y z 1 x y z x

−4f − 4f − 4f + lim ϕ n, n, n 2n 2n 2n n→∞ |4|n 2 2 2

= kρ(Q(x + y + z) + Q(x − y − z) + Q(y − x − z) + Q(z − x − y) −4Q(x) − 4Q(y) − 4Q(z))k

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for all x, y, z ∈ X. So



      

Q x + y + z + Q x − y − z + Q y − x − z + Q z − x − y − Q(x) − Q(y) − Q(z)

2 2 2 2 ≤ kρ(Q(x + y + z) + Q(x − y − z) + Q(y − x − z) + Q(z − x − y) − 4Q(x) − 4Q(y) − 4Q(z))k for all x, y, z ∈ X. By Lemma 3.1, the mapping Q : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (3.5). Then we have

   

q x x q

kQ(x) − T (x)k = 4 Q q − 4 T 2 2q            

q

q x x 1 x x x q q



≤ max 4 Q q − 4 f , 4 T ≤ lim −4 f ϕ n , 0, 0 , n→∞ |4|n 2 2q 2q 2q 2

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mapping satisfying (3.5).  Corollary 3.3. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be an even mapping such that

       

f x + y + z + f x − y − z + f y − x − z + f z − x − y

2 2 2 2 −f (x) − f (y) − f (z)k ≤ kρ(f (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) (3.8) r r r −4f (x) − 4f (y) − 4f (z))k + θ(kxk + kyk + kzk ) for all x, y, z ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

|2|r θ kxkr |2|r − |2|2

for all x ∈ X. Proof. The proof follows from Theorem 3.2 by takig ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Then we can choose L = |2|2−r and we get desired result.  Theorem 3.4. Let ϕ : X 3 → [0, ∞) be a function with ϕ(0, 0, 0) = 0 such that there exists an L < 1 with   x y z ϕ (x, y, z) ≤ |4|Lϕ , , 2 2 2 for all x, y, z ∈ X Let f : X → Y be an even mapping satisfying (3.4). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

L ϕ (x, 0, 0) 1−L

for all x ∈ X. Proof. It follows from (3.6) that



f (x) − 1 f (2x) ≤ 1 ϕ(2x, 0, 0) ≤ Lϕ(x, 0, 0)

4 |4|

(3.9)

for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. Now we consider the linear mapping J : S → S such that Jg(x) := 14 g (2x) for all x ∈ X.

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FIXED POINT AND QUADRATIC ρ-FUNCTIONAL INEQUALITIES L It follows from (3.9) that d(f, Jf ) ≤ L. So d(f, Q) ≤ 1−L . So kf (x) − Q(x)k ≤ for all x ∈ X. The rest of the proof is similar to the proof of Theorem 3.2.

L 1−L ϕ(x, 0, 0)



Corollary 3.5. Let r > 2 and θ be positive real numbers, and let f : X → Y be an even mapping satisfying (3.8). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

|2|r θ kxkr |2|2 − |2|r

for all x ∈ X. Proof. The proof follows from Theorem 3.4 by takig ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Then we can choose L = |2|r−2 and we get desired result.  4. Quadratic ρ-functional inequality (0.2) Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < |8|. In this section, we solve and investigate the quadratic ρ-functional inequality (0.2) in nonArchimedean normed spaces. Lemma 4.1. An even mapping f : X → Y satisfies kf (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) − 4f (x) − 4f (y) − 4f (z)k x−y−z y−x−z +f 2 2 2   

z−x−y − f (x) − f (y) − f (z) +f

2 for all x, y, z ∈ X if and only if f : X → Y is quadratic.

  

x+y+z

≤

ρ f









+f

(4.1)

Proof. Assume that f : X → Y satisfies (4.1). Letting x = y = z = 0 in (4.1), we get k8f (0)k ≤ |ρ|kf (0)k. So f (0) = 0. Letting x = y, z = 0 in (4.1), we get k2f (2x) − 8f (x)k ≤ 0 x 2




(4.2)

1 4 f (x)

and so f = for all x ∈ X. By (4.1) and (4.2), we have f (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) = 4f (x) + 4f (y) + 4f (z) for all x, y, z ∈ X, since |ρ| < |8| ≤ |4|. The converse is obviously true.



We prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (4.1) in nonArchimedean Banach spaces. Theorem 4.2. Let ϕ : X 3 → [0, ∞) be a function with ϕ(0, 0, 0) = 0 such that there exists an L < 1 with   x y z L ϕ , , ≤ ϕ (x, y, z) (4.3) 2 2 2 |4| for all x, y, z ∈ X. Let f : X → Y be an even mapping satisfying kf (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) − 4f (x) − 4f (y) − 4f (z)k

        

x+y+z x−y−z y−x−z z−x−y

≤ ρ f +f +f +f 2 2 2 2

−f (x) − f (y) − f (z))k + ϕ(x, y, z)

488

(4.4)

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for all x, y, z ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

L ϕ (x, x, 0) |4|(1 − L)

(4.5)

for all x ∈ X. Proof. Letting x = y = z = 0 in (4.4), we get k8f (0)k ≤ |ρ|kf (0)k. So f (0) = 0. Letting x = y, z = 0 in (4.4), we get

   

4f x − f (x) ≤ ϕ x , x , 0 ≤ L ϕ (x, x, 0)

2 2 2 |4|

(4.6)

for all x ∈ X. Consider the set S := {h : X → Y, h(0) = 0} and introduce the generalized metric on S: d(g, h) = inf {µ ∈ R+ : kg(x) − h(x)k ≤ µϕ (x, x, 0) , ∀x ∈ X} , where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see
[18]). Now we consider the linear mapping J : S → S such that Jg(x) := 4g x2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then kg(x) − h(x)k ≤ εϕ (x, x, 0) for all x ∈ X. Hence

   

x x

≤ Lεϕ (x, x, 0) kJg(x) − Jh(x)k = 4g − 4h 2 2

for all a ∈ X. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. L It follows from (4.6) that d(f, Jf ) ≤ |4| . By Theorem 1.3, there exists a mapping Q : X → Y satisfying the following: (1) Q is a fixed point of J, i.e., x 2

 

Q (x) = 4Q

(4.7)

for all x ∈ X. The mapping Q is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that Q is a unique mapping satisfying (4.7) such that there exists a µ ∈ (0, ∞) satisfying kf (x) − Q(x)k ≤ µϕ (x, x, 0) for all x ∈ X;
(2) d(J l f, Q) → 0 as l → ∞. This implies the equality liml→∞ 4n f 2xn = Q(x) for all x ∈ X; 1 L (3) d(f, Q) ≤ 1−L d(f, Jf ), which implies the inequality d(f, Q) ≤ |4|(1−L) . So kf (x) − Q(x)k ≤

L ϕ(x, x, 0) |4|(1 − L)

for all x ∈ X.

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It follows from (4.3) and (4.4) that

       

Q x + y + z + Q x − y − z + Q y − x − z + Q z − x − y

2 2 2 2

−Q(x) − Q(y) − Q(z)k

       

x+y+z x−y−z y−x−z z−x−y f = lim |4|n + f + f + f

n→∞ 2n+1 2n+1 2n+1 2n+1       y z x

−f −f −f n n 2 2 2n

       

x+y+z x−y−z y−x−z z−x−y f ≤ lim |4|n |ρ| + f + f + f

n→∞ 2n 2n 2n 2n         x y z

+ lim |4|n ϕ x , y , z −4f − 4f − 4f n n n 2 2 2 n→∞ 2n 2n 2n

= kρ(Q(x + y + z) + Q(x − y − z) + Q(y − x − z) + Q(z − x − y) −4Q(x) − 4Q(y) − 4Q(z))k for all x, y, z ∈ X. So

       

Q x + y + z + Q x − y − z + Q y − x − z + Q z − x − y

2 2 2 2

−Q(x) − Q(y) − Q(z)k ≤ kρ(Q(x + y + z) + Q(x − y − z) + Q(y − x − z) + Q(z − x − y) −4Q(x) − 4Q(y) − 4Q(z))k for all x, y, z ∈ X. By Lemma 4.1, the mapping Q : X → Y is quadratic. The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 4.3. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be an even mapping such that kf (x + y + z) + f (x − y − z) + f (y − x − z) + f (z − x − y) − 4f (x) − 4f (y) − 4f (z)k x−y−z y−x−z +f (4.8) 2 2 2   

z−x−y r r r +f − f (x) − f (y) − f (z)

+ θ(kxk + kyk + kzk ) 2

  

x+y+z

≤

ρ f









+f

for all x, y, z ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

|2|r

2θ kxkr − |2|2

for all x ∈ X. Proof. The proof follows from Theorem 4.2 by takig ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Then we can choose L = |2|2−r and we get desired result.  Theorem 4.4. Let ϕ : X 3 → [0, ∞) be a function with ϕ(0, 0, 0) = 0 such that there exists an L < 1 with x y z ϕ (x, y, z) ≤ |4|Lϕ , , 2 2 2 

490



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for all x, y, z ∈ X Let f : X → Y be an even mapping satisfying (4.4). Then there exists a unique quadratic mapping Q : X → Y such that 1 kf (x) − Q(x)k ≤ ϕ (x, x, 0) |4|(1 − L) for all x ∈ X. Proof. It follows from (4.6) that



f (x) − 1 f (2x) ≤ 1 ϕ(x, x, 0)

4 |4|

(4.9)

for all x ∈ X. Let (S, d) be the generalized metric space defined in the proof of Theorem 4.2. Now we consider the linear mapping J : S → S such that 1 Jg(x) := g (2x) 4 for all x ∈ X. 1 1 . So d(f, Q) ≤ |4|(1−L) d(f, Jf ), which implies the It follows from (4.9) that d(f, Jf ) ≤ |4| inequality 1 d(f, Q) ≤ . 1−L So 1 ϕ(x, x, 0) kf (x) − Q(x)k ≤ |4|(1 − L) for all x ∈ X. The rest of the proof is similar to the proof of Theorem 3.2.



Corollary 4.5. Let r > 2 and θ be nonnegative real numbers, and let f : X → Y be an even mapping satisfying (4.8). Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

2θ kxkr |2|2 − |2|r

for all x ∈ X. Proof. The proof follows from Theorem 4.4 by takig ϕ(x, y, z) = θ(kxkr + kykr + kzkr ) for all x, y, z ∈ X. Then we can choose L = |2|r−2 and we get desired result.  References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [3] L. C˘ adariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). [4] L. C˘ adariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [5] L. C˘ adariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [6] J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [7] G. Z. Eskandani, J. M. Rassias, Approximation of a general cubic functional equation in Felbin’s type fuzzy normed linear spaces, Results Math. 66 (2014), 113–123. [8] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161.

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[9] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. [10] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [11] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [12] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [13] G. Isac, Th. M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [14] Y. Jung, I. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl. 306 (2005), 752–760. [15] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), 368–372. [16] J. Lee, Stability of functional equations in matrix random normed spaces: a fixed point approach, Results Math. 66 (2014), 99–112. [17] M. Mirzavaziri, M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [18] D. Mihet¸, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [19] M. S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.–TMA 69 (2008), 3405–3408. [20] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [21] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [22] C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. 2007, Article ID 41820 (2007). [23] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [24] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [25] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. [26] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [27] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Jung Rye Lee Department of Mathematics Daejin University Kyunggido 487-711 Republic of Korea E-mail address: [email protected] Dong Yun Shin Department of Mathematics University of Seoul Seoul 130-743 Republic of Korea E-mail address: [email protected]

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Dynamics and Global Stability of Higher Order Nonlinear Di erence Equation E. M. Elsayed1;2 and Asma Alghamdi1;3 1 Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 3 Mathematics Department, University College of Umluj, Tabuk University, P. O. Box 741, Umluj 71491, Saudi Arabia. E-mails: [email protected], [email protected]. ABSTRACT In this paper, we study the behavior of the solutions of the following rational di¤erence equation with big order xn+1 = axn

l

+ bxn

k

+

cxn exn

+ dxn s + f xn s

t

:

t

where the parameters a; b; c; d; e and f are positive real numbers and the initial conditions x and x0 are positive real numbers where r = maxfl; k; s; tg.

r;

x

r+1

:::; x

1

Keywords: recursive sequence, periodicity, boundedness, stability, di¤erence equations. Mathematics Subject Classi…cation: 39A10 ——————————————————————

1. INTRODUCTION Di¤erence equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and as such these equations are in their own right important mathematical models. Di¤erence equations related to di¤erential equations as discrete mathematics related to continuous mathematics. In recent years nonlinear di¤erence equations have attracted the attention of many researchers, for example: Agarwal and Elsayed [1] studied the global stability, periodicity character and gave the solution form of some special cases of the recursive sequence xn+1 = axn +

bxn xn 3 cxn 2 + dxn

: 3

Cinar [5] obtained the solutions of the following di¤erence equation xn 1 1 + axn xn

xn+1 =

: 1

El-Metwally et al.[10] dealed with the following di¤erence equation yn+1 =

yn

+p (2k+1) + qyn

yn

(2k+1)

: 2l

Elsayed [12] studied the global stability, and periodicity character of the following recursive sequence xn+1 = axn

l

+

bxn cxn

493

l

l

dxn

: k

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Elsayed et al. [18] investigated the behavior of the following second order rational di¤erence equation xn+1 = axn +

b + cxn d + exn

1

:

1

Elsayed and El-Dessoky [16] investigated the global convergence, boundedness, and periodicity of solutions of the di¤erence equation bxn l + cxn k : xn+1 = axn s + dxn l + exn k Karatas et al. [21] got the solution of the di¤erence equation xn+1 =

xn 5 1 + xn 2 xn

: 5

Obaid et al. [24] studied the global attractivity and periodic character of the following fourth order di¤erence equation bxn 1 + cxn 2 + dxn 3 xn+1 = axn + : xn 1 + xn 2 + xn 3 Yalcinkaya [29] dealed with the following di¤erence equation xn+1 =

+

xn m : xkn

Zayed and El-Moeam [31], [32] studied the global asymptotic properties of the solutions of the following di¤erence equations xn+1 xn+1

= axn

bxn cxn dxn

= Axn + Bxn

k

+

: k

pxn + xn k : q + xn k

For some related work see [1-33]. Our goal in this article is to investigate the global stability character and the periodicity of solutions of the recursive sequence cxn s + dxn t xn+1 = axn l + bxn k + : (1) exn s + f xn t where the parameters a; b; c; d; e and f are positive real numbers and the initial conditions x and x0 are positive real numbers where r = maxfl; k; s; tg.

r;

x

r+1

:::; x

1

2. SOME BASIC PROPERTIES AND DEFINITIONS Here, we recall some basic de…nitions and some theorems that we need in the sequel. Let I be some interval of real numbers and let F : I k+1 ! I; be a continuously di¤erentiable function. Then for every set of initial conditions x k ; x k+1 ; ::: ; x0 2 I; the di¤erence equation xn+1 = F (xn ; xn 1

has a unique solution fxn gn=

k

1 ; :::; xn k );

n = 0; 1; :::;

(2)

:

De…nition 1. (Equilibrium point) A point x 2 I is called an equilibrium point of Equation (2) if x = F (x; x; :::; x): That is, xn = x f or n

0, is a solution of Equation (2), or equivalently, x is a …xed point of F .

De…nition 2. (Periodicity)

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1

A sequence fxn gn=

k

is said to be periodic with period p if xn+p = xn for all n

k.

De…nition 3. (Stability)

(i) The equilibrium point x of Equation (2) is locally stable if for every for all x k ; x k+1 ; :::; x 1 ; x0 2 I with jx

k

xj + jx

xj + ::: + jx0

k+1

> 0, there exists

> 0 such that

xj < ;

we have jxn

xj
0, such that for all x k ; x k+1 ; :::; x 1 ; x0 2 I with jx

k

xj + jx

xj + ::: + jx0

k+1

xj < ;

we have lim xn = x:

n!1

(iii) The equilibrium point x of Equation (2) is global attractor if for all x

k ; x k+1 ; :::; x 1 ; x0

2 I; we have

lim xn = x:

n!1

(iv) The equilibrium point x of Equation (2) is globally asymptotically stable if x is locally stable, and x is also a global attractor of Equation (2). (v) The equilibrium point x of Equation (2) is unstable if is not locally stable. The linearized equation of Equation (2) about the equilibrium point x is the linear di¤erence equation yn+1 =

k X @F (x; x; :::; x)

@xn

i=0

yn i :

(3)

i

Theorem A [22] Assume that pi 2 R; i = 1; 2; ::: and k 2 f0; 1; 2; :::g: Then k X i=1

jpi j < 1;

(4)

is a su¢ cient condition for the asymptotic stability of the di¤ erence equation yn+k + p1 yn+k

1

+ ::: + pk yn = 0;

n = 0; 1; ::: :

(5)

Theorem B [23] Let g : [a; b]k+1 ! [a; b], be a continuous function, where k is a positive integer, and where [a; b] is an interval of real numbers. Consider the di¤erence equation xn+1 = g(xn ; xn

1 ; :::; xn k );

n = 0; 1; ::: :

(6)

Suppose that g satis…es the following conditions. (1) For each integer i with 1 z1 ; z2 ; :::; zi 1 ; zi+1 ; :::; zk+1 :

i

k + 1; the function g(z1 ; z2 ; :::; zk+1 ) is weakly monotonic in zi for …xed

(2) If m; M is a solution of the system m = g(m1 ; m2 ; :::; mk+1 );

M = g(M1 ; M2 ; :::; Mk+1 );

then m = M , where for each i = 1; 2; :::; k + 1; we set mi =

m; M;

if g is non-decreasing in zi ; ; if g is non-increasing in zi ;

Mi =

M; m;

if g is non-decreasing in zi ; : if g is non-increasing in zi .

Then there exists exactly one equilibrium point x of Equation.(6), and every solution of Equation (6) converges to x.

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3. LOCAL STABILITY OF EQUATION (1) In this section, we investigate the local stability character of the solutions of Equation (1). Equation (1) has a unique positive equilibrium point and is given by x = ax + bx +

cx + dx : ex + f x

If (a + b) < 1;then the unique positive equilibrium point is x=

[1

c+d : (a + b)](e + f )

Let f : (0; 1)4 ! (0; 1) be a function de…ned by f (u0 ; u1 ; u2 ; u3 ) = au0 + bu1 +

cu2 + du3 : eu2 + f u3

Therefore it follows that @f (u0 ; u1 ; u2 ; u3 ) @u0 @f (u0 ; u1 ; u2 ; u3 ) @u2

@f (u0 ; u1 ; u2 ; u3 ) = b; @u1 (cf de)u3 @f (u0 ; u1 ; u2 ; u3 ) (de cf )u2 ; = : 2 (eu2 + f u3 ) @u3 (eu2 + f u3 )2

= a; =

Then, we see that @f (x; x; x; x) @u0 @f (x; x; x; x) @u2

@f (x; x; x; x) = b; @u1 (cf de)[1 (a + b)] @f (x; x; x; x) (de cf )[1 (a + b)] ; : = (e + f )(c + d) @u3 (e + f )(c + d)

= a; =

Then, the linearized equation of Equation (1) about x is yn+1 + ayn

l

+ byn

k

+

(cf

de)[1 (a + b)] yn (e + f )(c + d)

s

+

(de

cf )[1 (a + b)] yn (e + f )(c + d)

p

= 0:

(7)

Theorem 1. Assume that 2 jcf

dej < (e + f ) (c + d) :

Then the equilibrium point of Equation (1) is locally asymptotically stable. Proof. It follows by Theorem A that Equation (7) is asymptotically stable if jaj + jbj +

(cf

or 2

de)[1 (a + b)] (de cf )[1 (a + b)] + < 1; (e + f )(c + d) (e + f )(c + d) (cf

de)[1 (a + b)] < [1 (e + f )(c + d)

(a + b)];

and so 2 jcf

dej < (e + f ) (c + d) :

This completes the proof.

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4. GLOBAL ATTRACTIVITY OF THE EQUILIBRIUM POINT OF EQUATION (1) In this section we deals the global attractivity character of solutions of Equation (1). Theorem 2. The equilibrium point x is a global attractor of equation (1) if one of the following conditions holds: (i) cf (ii) de

de cf

0; d 0; c

c: d:

Proof. Let r; s be nonnegative real numbers and assume that f : [r; s]4 ! [r; s] be a function de…ned by f (u0 ; u1 ; u2 ; u3 ) = au0 + bu1 +

cu2 + du3 : eu2 + f u3

Then @f (u0 ; u1 ; u2 ; u3 ) @u0 @f (u0 ; u1 ; u2 ; u3 ) @u2

@f (u0 ; u1 ; u2 ; u3 ) = b; @u1 @f (u0 ; u1 ; u2 ; u3 ) (cf de)u3 (de cf )u2 ; = : (eu2 + f u3 )2 @u3 (eu2 + f u3 )2

= a; =

We consider two cases: Case1: Assume that cf de 0 is true, then we can easily see that the function f (u0 ; u1 ; u2 ; u3 ) is increasing in u0 ; u1 ; u2 and decreasing in u3 . Suppose that (m; M ) is a solution of the system M = f (M; M; M; m)

and

m = f (m; m; m; M ):

Then from Equation (1), we see that M M [1

= aM + bM +

(a + b)]

=

cM + dm ; eM + f m

cM + dm ; m[1 eM + f m

m = am + bm +

(a + b)] =

cm + dM ; em + f M

cm + dM ; em + f M

then M 2 e[1 m2 e[1

(a + b)] + mM f [1 (a + b)] + mM f [1

(a + b)] = cM + dm; (a + b)] = cm + dM:

Subtracting this two equations, we obtain (M

m) fe(M + m)[1

(a + b)] + (d

c)g = 0;

under the condition (a + b) < 1; d > c; we see that M = m: It follows from Theorem B that x is a global attractor of Equation (1). Case 2: Similar to Case 1.

5. BOUNDEDNESS OF SOLUTIONS OF EQUATION (1) In this section we study the boundedness nature of the solutions of Equation (1). Theorem 3. Every solution of Equation (1) is bounded if a + b < 1: 1

r

l

+ bxn

Proof. Let fxn gn= xn+1 = axn

be a solution of Equation (1). It follows from Equation (1) that k

+

cxn exn

+ dxn + f xn s s

t

= axn

l

t

497

+ bxn

k

+

exn

cxn s s + f xn

+ t

exn

dxn t s + f xn

: t

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Then xn+1 6 axn

l

+ bxn

k

+

cxn exn

s

+

s

dxn f xn

t

= axn

l

+ bxn

k

t

+

c d + f or all n e f

1:

By using a comparison, we can right hand side as follows zn+1 = azn

l

+ bzn

k

+

c d + : e f

and this equation is locally asymptotically stable if a + b < 1; and converges to the equilibrium point z = cf +de ef [1 (a+b)] : Therefore lim sup xn 6

n!1

cf +de ef [1 (a+b)] :

Thus the solution is bounded. Theorem 4. Every solution of Equation (1) is unbounded if a > 1or b > 1: 1

Proof. Let fxn gn=

r

be a solution of Equation (1).Then from Equation (1) we see that

xn+1 = axn

l

+ bxn

k

+

cxn exn

+ dxn + f xn s s

p

> axn

l

f or all n

1:

p

We see that the right hand side can be written as follows zn+1 = azn l : Then zln+i = an zl+i + const:;

i = 0; 1; :::; l; 1

and this equation is unstable because a > 1, and lim zn = 1:Then by using ratio test fxn gn= n!1 from above. When b > 1 is similar.

r

is unbounded

6. EXISTENCE OF PERIODIC SOLUTIONS Here we study the existence of periodic solutions of Equation (1). The following theorem states the necessary and su¢ cient conditions that this equation has periodic solution of prime period two. Theorem 5. Equation (1) has a prime period two solutions if and only if (i) (ii) (iii) (iv)

(d (c (c (d

(vii) (d (viii) (c

c)(e d)(f d)(f c)(e (v) (vi) c)(e d)(f

f )(a + b + 1) e)(a + b + 1) e)(1 + a b) f )(1 + a b) (c d)(f e) (d c)(e f ) f )(1 a + b) e)(1 a + b)

> > > > > > > >

4[cf + de(a + b)]; 4[cf (a + b) + de]; 4[de(1 b) + caf ]; 4[cf (1 b) + dae]; 4de; c > d; f > e; 4cf; d > c; e > f; 4[cf + dbe daf ]; 4[cbf dae + de];

l; k; s even and t l; k; t even and s l; t even and k; s l; s even and k; t l; k; s odd; and t l; k; t odd and s l; t odd and k; s l; s odd and k; t

odd: odd: odd: odd: even: even: even: even:

Proof. We prove …rst case when l; k and s are even, and t is odd ( the other cases are similar and will be left to readers). First suppose that there exists a prime period two solution :::p; q; p; q; :::; of Equation (1).We will prove that Inequality (i) holds. We see from Equation (1) when l; k; s are even, and t is odd that p = aq + bq +

cq + dp ; eq + f p

q = ap + bp +

cp + dq : ep + f q

Then epq + f p2 epq + f q 2

= =

(a + b)eq 2 + (a + b)f pq + cq + dp; (a + b)ep2 + (a + b)f pq + cp + dq:

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Subtracting (8) from (9) gives f (p2 q 2 ) = f (p q)(p + q) = Since p 6= q , it follows that

(a + b)e(p2 q 2 ) c(p q) + d(p q); (a + b)e(p q)(p + q) c(p q) + d(p q);

f (p + q) =

(a + b)e(p + q)

p+q =

c + d;

d c : f + (a + b)e

(10)

Again, adding (8) and (9) yields 2epq + f (p2 + q 2 ) = (a + b)e(p2 + q 2 ) + 2(a + b)f pq + (c + d)(p + q); (p2 + q 2 )[f (a + b)e] = (c + d)(p + q) + 2pq[(a + b)f e];

(11)

It follows by (10), (11) and the relation p2 + q 2 = (p + q)2

2pq

f or all

p; q 2 R;

that [(p + q)2 2pq][f (a + b)e] = (c + d)(p + q) + 2pq[(a + b)f e]; 2pq[(a + b)f (a + b)e + f e] = (p + q)2 [f (a + b)e] (c + d)(p + q); pq =

(d c)[cf + de(a + b)] : (a + b + 1)(e f )[f + (a + b)e]2

(12)

Now it is clear from Equations (10) and (12) that p and q are the two distinct roots of the quadratic equation r2

d c f + (a + b)e

(f + (a + b)e)r2 and so

(d

(d

(d c)[cf + de(a + b)] (a + b + 1)(e f )[f + (a + b)e]2 (d c)[cf + de(a + b)] c)r + (a + b + 1)(e f )[f + (a + b)e] r+

c)2 2

[f + (a + b)e]

>

=

0;

=

0;

(13)

4(d c)[cf + de(a + b)] : (a + b + 1)(e f )[f + (a + b)e]2

Thus (d

c)(e

f )(a + b + 1) > 4[cf + de(a + b)]:

Therefore Inequality (i) holds. Second suppose that Inequality (i) is true. We will show that Equation (1) has a prime period two solution. Assume that d c+ d c p= ; q= ; 2(f + Ae) 2(f + Ae) where =

s

4(d c)(cf + Ade) ; (A + 1)(e f )

2

(d

c)

(d

c)(e

(d

c)2

and

A = (a + b):

We see from Inequality (1) that f )(a + b + 1) > 4[cf + de(a + b)]:

which equivalents to 2

[f + (a + b)e]

>4

(d c)[cf + de(a + b)] ; (a + b + 1)(e f )[f + (a + b)e]2

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Therefore p and q are distinct real numbers. Set x l = p; x k = p; x q; x0 = p: We wish to show that x1 = x 1 = q and x2 = x0 = p:

s

= p; x

t

= q; :::; x

2

= p; x

1

It follows from Equation (1) that x1 = A

d c+ d c c[ 2(f d c+ +Ae) ] + d[ 2(f +Ae) ] + ; d c+ d c 2(f + Ae) e[ 2(f +Ae) ] + f [ 2(f +Ae) ]

Dividing the denominator and numerator by 2(f + Ae) gives x1 = A

d c+ c[d + 2(f + Ae) e[d

d 2(f d = A 2(f

= A

c + ] + d[d c + ] + f [d

] ; ]

c+ d2 c2 + (c d) + ; + Ae) e(d c) + f (d c) + (e f ) c+ (d c)(d + c ) + ; + Ae) (d c)(e + f ) + (e f )

Multiplying the denominator and numerator of the right side by (d x1

c c

c)(e + f )

(e

f ) gives

(d c)(d + c )[(d c)(e + f ) (e f )] c+ ; + + Ae) [(d c)(e + f ) + (e f )][(d c)(e + f ) (e f )] c+ (d c)(d + c )[(d c)(e + f ) (e f )] + ; 2 2 2 2 + Ae) (d c) (e + f ) (e f ) d c+ (d c)(d + c )[de + df ce cf e f] h i; = A + 2 4(d c)(cf +Ade) 2(f + Ae) (d c)2 (e + f )2 (e f )2 (d c)

d 2(f d = A 2(f = A

(A+1)(e f )

2

2

2

= A

(d c)[e(d c ) + f (d d c+ + 2 2 2(f + Ae) (d c) [e + 2ef + f 2 (e2

= A

(d d c+ + 2(f + Ae)

= A

d c+ + 2(f + Ae)

c)[(d2

4(d

d c+ + 2(f + Ae)

= A

d c+ 2(f + Ae)

d c+ 2(f + Ae) d c+ = A 2(f + Ae) d c+ = A 2(f + Ae) Ad Ac + A =

= A

2ef +

c)

de) +

+ 2

c)(

f )]

f )]

2(d c)(A 1)(cf A+1

3Ade

de)

+ 2 (cf

;

;

3cf +Af c Af d Aec+ed (A+1) Ac+d c)]+[ecf +Ae2 d cf 2 Adef ] (A+1)

de)+

(d

(e

4(d c)(e f )(cf +Ade) (A+1)

(e

c)(e f )(cf +Ade) (A+1)

c)[ef (Ad

h

f 2 )]

2

de) +

c)2 +4(d

(d c)[(d2 c2 )(e+f )+2 (cf

= A

c ) + 2 (cf

c2 )(e + f ) + 2 (cf

4ef (A+1)(d

(d

2

ec

f d)

]

;

i de)

; Acef + Ae2 d cf 2 ] (A + 1) h i c)(A 1) 2(d c)(cf de) (d A+1 + + ; 4(d c)[ef d Acef + Ae2 d cf 2 ] (A + 1) (cf de)f(d c)(A 1) + (A + 1)g + ; 2[ef d Acef + Ae2 d cf 2 ] (cf de)f(d c)(A 1) + (A + 1)g + ; 2(f + Ae)(de cf ) f(d c)(A 1) + (A + 1)g + ; 2(f + Ae) Ad + d + Ac c A d c = = q: 2(f + Ae) 2(f + Ae) 4(d

c)[ef d

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Similarly as before we can easily show that x2 = p: Then it follows by induction that x2n = p and x2n+1 = q for all n 1: Thus Equation (1) has the prime period two solution :::; p; q; p; q; :::; where p and q are the distinct roots of the quadratic equation (13) and the proof is complete.

7. NUMERICAL EXAMPLES For con…rming the results of this article, we consider numerical examples which represent di¤erent types of solutions to Equation (1). Example 1. We consider numerical example for the di¤erence equation (1) when we take the constants and the initial conditions as follows: l = 3; k = 2; s = 1 ; t = 3; x 3 = 5; x 2 = 12; x 1 = 6; x0 = 8; a = 0:4; b = 0:3; c = 2; d = 4; e = 6; f = 8: See Figure 1. Example 2. See Figure (2) when we take Equation (1) with l = 1; k = 3; s = 2 ; t = 3; x 9; x 1 = 7; x0 = 5; a = 0:6; b = 0:4; c = 3; d = 2; e = 5; f = 8.

3

= 13; x

2

=

Example 3. Figure (3) shows the behavior of the solution of the di¤erence equation (1) when we put l = 2; k = 1; s = 3 ; t = 3; x 3 = 15; x 2 = 11; x 1 = 9; x0 = 5; a = 0:6; b = 1:4; c = 2; d = 4; e = 6; f = 9. Example 4. We assume l = 2; k = 3; s = 1 ; t = 2; x 0:2; c = 2; d = 0; e = 6; f = 7: See Figure 4.

3

= 15; x

2

= 11; x

1

=

9; x0 = 5; a = 1:5; b =

Example 5. Figure (5) shows the period two solution of Equation (1) when l = 0; k = 2; s = 2 ; t = 3; x 3 = p; x 2 = q; x 1 = p; x0 = q; a = 0:06; b = 0:03; c = 1; d = 5; e = 7; f = 2, since p and q as in the previous theorem.

X(n+1)=a*X(n-l)+b*X(n-k)+((c*X(n-s)+d*X(n-t))/(e*X(n-s)+f*X(n-t)))

X(n+1)=a*X(n-l)+b*X(n-k)+((c*X(n-s)+d*X(n-t))/(e*X(n-s)+f*X(n-t))) 20

10

15

5 10

0 x(n)

x(n)

5 0

-5

-5

-10 -10

-15

0

10

20

30

40

50

60

70

80

-15

90

0

10

20

30

40

n

Figure 1.

60

70

80

90

Figure 2.

X(n+1)=a*X(n-l)+b*X(n-k)+((c*X(n-s)+d*X(n-t))/(e*X(n-s)+f*X(n-t))) x 10

6 X(n+1)=a*X(n-l)+b*X(n-k)+((c*X(n-s)+d*X(n-t))/(e*X(n-s)+f*X(n-t))) x 10

5

16

50 n

10

14

8

12

6

8

x(n)

x( n)

10

6 4

4 2

2

0 0 -2

0

10

20

30

40

50

60

70

80

-2

90

0

n

10

20

30

40

50

60

70

80

90

n

Figure 3.

Figure 4.

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x(n+1)=ax(n-l)+bx(n-k)+((cx(n-s)+dx(n-t))/(ex(n-s)+fx(n-t))) 1 0.9

x(n)

0.8 0.7 0.6 0.5

0

5

10

15 n

20

25

30

Figure 5.

Acknowledgements This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

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503

k

+

k

, Comm.

pxn +xn k q+xn k ;

xn + xn k + Cxn +Dxn

k

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A fra tional derivative in lusion problem via an integral boundary

ondition 1,2 , Mehdi Moghaddam3 , Hakimeh Mohammadi4 , Shahram Rezapour5

Dumitru Baleanu

1

Department of Mathemati s, Cankaya University,

Ogretmenler Cad. 14 06530, Balgat, Ankara, Turkey 2 3

Institute of Spa e S ien es, Magurele, Bu harest, Romania

Department of Mathemati s, Miandoab Bran h, Islami Azad University, Miandoab, Iran 4

Department of Mathemati s, Bonab Bran h, Islami Azad University, Bonab, Iran 5

Department of Mathemati s, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran

Abstra t.

We investigate the existen e of solutions for the fra tionalZdierential in lusion c Dα x(t) ∈ F (t, x(t)) η

equipped with the boundary value problems x(0) = 0 and x(1) =

x(s)ds, where 0 < η < 1, 1 < α ≤ 2,

0

c

Dα is the standard Caputo dierentiation and F : [0, 1] × R → 2R is a ompa t valued multifun tion. An

illustrative example is also dis ussed. Keywords: Fixed point, Fra tional dierential in lusion, Integral boundary value problem.

1

Introdu tion

During the last de ade the fra tional dierential equations were investigated from theoreti al and applied viewpoints (see for example, [1℄-[6℄, [8℄-[15℄, and [32℄). A spe ial attention was given to the real world appli ations where the power law ee t is present and where the fra tional models a give better results than the lassi al ones. We re all that the Riemann-Liouville fra tional integral of order α > 0 of f : (0, ∞) → R Z

is given by I α f (t) =

t

1 Γ(α)

(t − s)α−1 f (s)ds provided the right side is pointwise dened on

0

(0, ∞) (see [26℄, [29℄, [31℄, [34℄ and [35℄). Also, the Caputo fra tional derivative of order α of Z t f (n) (s) 1 c α f is dened by D f (t) = Γ(n−α) ds, where n = [α] + 1 (see [26℄, [29℄, [31℄, [34℄ α−n+1 0 (t − s) and [35℄). We re all that the basi theory for fra tional dierential in lusions is represented by the xed point theory of multivalued mappings whi h was intensively investigated during last years (the reader an nd more details in [18℄-[25℄, [30℄ and the related referen es). Thus, many papers about ordinary and fra tional dierential in lusions were written (e.g. [1℄-[2℄, 1

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[7℄, [16℄, [17℄ and [33℄). Let (X, d) be a metri spa e. Let us denote by P (X) and 2X the lass of all subsets and the lass of all nonempty subsets of X respe tively. As a result, Pcl (X), Pbd (X), Pcv (X) and Pcp (X) denote the lass of all losed, bounded, onvex and ompa t subsets of X respe tively. A mapping Q : X → 2X is alled a multifun tion on X and u ∈ X is alled a xed point of Q whenever u ∈ Qu ([24℄). Also, we say that Q is onvex whenever Qx is onvex for all x ∈ X ([24℄). A multifun tion G : [0, 1] → Pcl (R) is said to be measurable whenever the fun tion t 7→ d(y, G(t)) = inf{|y − z| : z ∈ G(t)} is measurable for all y ∈ R. Put J = [0, 1]. The aim of this manus ript is to investigate the existen e of solutions for the fra tional dierential in lusion c α D x(t) ∈ F (t, x(t)) (∗) Z η via the boundary value problems x(0) = 0 and x(1) = x(s)ds, where c D α is the standard 0

Caputo dierentiation, 0 < η < 1, 1 < α ≤ 2 and F : J × R → 2R is a ompa t valued multifun tion. We say that F : J × R → 2R is a Caratheodory multifun tion whenever t 7→ F (t, x) is measurable for all x ∈ R and x 7→ F (t, x) is upper semi- ontinuous for almost all t ∈ J . Also, a Caratheodory multifun tion F : J ×R → 2R is alled L1 -Caratheodory whenever for ea h ρ > 0 there exists φρ ∈ L1 (J, R+ ) su h that k F (t, x) k= sup{|v| : v ∈ F (t, x)} ≤ φρ (t) for all kxk∞ ≤ ρ and for almost all t ∈ J . For ea h x ∈ C(J, R), dene the set of sele tions of F by SF,x := {v ∈ L1 (J, R) : v(t) ∈ F (t, x(t)) for almost all t ∈ J}. Let E be a nonempty losed subset of a Bana h spa e X and G : E → 2X a multifun tion with nonempty losed values. We say that the multifun tion G is lower semi- ontinuous whenever the set {y ∈ E : G(y) ∩ B 6= ∅} is open for all open set B in X . It has been proved that ea h ompletely ontinuous multifun tion is lower semi- ontinuous (see [24℄). We shall use the following xed point results.

([30℄) Let X be a Bana h spa e, F : J → Pcp,cv (X) an L1 -Caratheodory multifun tion and Θ a linear ontinuous mapping from L1 (J, X) to C(J, X). Then the operator ΘoSF : C(J, X) → Pcp,cv (C(J), X) dened by (ΘoSF )(x) = Θ(SF,x) is a losed graph operator in C(J, X) × C(J, X). Lemma 1.1.

It has been proved that if dimX < ∞, then SF (x) 6= ∅ for all x ∈ C(J, X) ([30℄). Lemma 1.2. ([24℄) Let E be a Bana h spa e, C a losed onvex subset of E , U an open subset of C and 0 ∈ U . Suppose that F : U → Pcp,cv (C) is a upper semi- ontinuous ompa t map, where Pcp,cv (C) denotes the family of nonempty, ompa t onvex subsets of C . Then either F has a xed point in U or there exist u ∈ ∂U and λ ∈ (0, 1) su h that u ∈ λF (u).

Let (X, k.k) be a normed spa e. Dene the Hausdor metri Hd : 2X × 2X → [0, ∞]} by

Hd (A, B) = max{sup d(a, B), sup d(A, b)}, a∈A

b∈B

2

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where d(A, b) = inf a∈A d(a; b). Then (Pb,cl (X), Hd) is a metri spa e and (Pcl (X), Hd ) is a generalized metri spa e ([27℄). A multifun tion N : X → Pcl (X) is alled a ontra tion whenever there exists γ > 0 su h that Hd (N(x), N(y)) ≤ γd(x, y) for all x, y ∈ X .

([19℄) Let (X, d) be a omplete metri spa e. If N : X → Pcl (X) is a ontra tion, then N has a xed point. Lemma 1.3.

[11℄ Let 0 < η < 1. Then x is a solution for the dierential equation cZDα x(t) = η v(t) (t ∈ J and 1 < α ≤ 2) via the boundary value onditions x(0) = 0 and x(1) = x(s)ds 0 if and only if x is a solution of the integral equation Lemma 1.4.

Z 1 2t (t − s) v(s)ds − (1 − s)α−1 v(s)ds 2 (2 − η )Γ(α) 0 0 Z η Z s 2t + ( (s − m)α−1 v(m)dm)ds (t ∈ J). (2 − η 2 )Γ(α) 0 0

1 x(t) = Γ(α)

2

Z

t

α−1

Main results

Here, we give our results about the existen e of solutions for the in lusion problem (∗).

Suppose that F : J ×R → 2R is a Caratheodory multifun tion with ompa t and

onvex values and there exist a bounded ontinuous non-de reasing map ψ : [0, ∞) → (0, ∞) and a ontinuous fun tion p : J → (0, ∞) su h that Theorem 2.1.

kF (t, x(t))k = sup{|v| : v ∈ F (t, x(t))} ≤ p(t)ψ(kxk∞ )

for all t ∈ J and x ∈ C(J, R). Then the problem (∗) has at least one solution. Proof. By using Lemma 1.4, we know that the existen e of solution for the problem (∗) is

equivalent to the existen e of solution for the integral equation Z t Z 1 1 2t α−1 x(t) ∈ (t − s) v(s)ds − (1 − s)α−1 v(s)ds Γ(α) 0 (2 − η 2 )Γ(α) 0 Z η Z s 2t ( (s − m)α−1 v(m)dm)ds (t ∈ J). + (2 − η 2 )Γ(α) 0 0 Put E = C(J, R). Dene the operator N : E → 2E by

Z t Z 1 1 2t α−1 N(x) = {h ∈ E : h(t) = (t − s) v(s)ds − (1 − s)α−1 v(s)ds Γ(α) 0 (2 − η 2 )Γ(α) 0 Z η Z s 2t + ( (s − m)α−1 v(m)dm)ds, for some v ∈ SF,x }. (2 − η 2 )Γ(α) 0 0 3

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We show that the operator N satises the assumptions of Lemma 1.2. First, we show that N(x) is onvex for all x ∈ C(J, R). Let h1 , h2 ∈ N(x). Choose v1 , v2 ∈ SF,x su h that

Z 1 2t (t − s) vi (s)ds − (1 − s)α−1 vi (s)ds 2 (2 − η )Γ(α) 0 0 Z η Z s 2t + ( (s − m)α−1 vi (m)dm)ds 2 (2 − η )Γ(α) 0 0

1 hi (t) = Γ(α)

Z

t

α−1

for all t ∈ J and i = 1, 2. Let 0 ≤ w ≤ 1. Then, we have Z t 1 [wh1 + (1 − w)h2 ](t) = (t − s)α−1 [wv1 (s) + (1 − w)v2 (s)]ds Γ(α) 0

Z 1 2t − (1 − s)α−1 [wv1 (s) + (1 − w)v2 (s)]ds (2 − η 2 )Γ(α) 0 Z η Z s 2t + ( (s − m)α−1 [wv1 (m) + (1 − w)v2 (m)]dm)ds (2 − η 2 )Γ(α) 0 0 for all t ∈ J . Sin e SF,x is onvex (be ause F has onvex values), wh1 + (1 − w)h2 ∈ N(x). Now, we show that N(x) maps bounded sets of C(J, R) into bounded sets. Let r > 0 and Br = {x ∈ C(J, R) : kxk∞ ≤ r}. For ea h h ∈ N(x) and x ∈ Br hoose v ∈ SF,x su h that

Z 1 2t (t − s) v(s)ds − (1 − s)α−1 v(s)ds 2 )Γ(α) (2 − η 0 0 Z η Z s 2t + ( (s − m)α−1 v(m)dm)ds (2 − η 2 )Γ(α) 0 0

1 h(t) = Γ(α)

and

Z

t

α−1

Z 1 2t (t − s) v(s)ds − (1 − s)α−1 v(s)ds 2 (2 − η )Γ(α) 0 0 Z η Z s 2t + ( (s − m)α−1 v(m)dm)ds | 2 (2 − η )Γ(α) 0 0 Z t Z 1 1 2t α−1 ≤ sup (t − s) | v(s) | ds + sup | | (1 − s)α−1 | v(s) | ds 2 t∈J Γ(α) 0 t∈[0,1] (2 − η )Γ(α) 0 Z η Z s 2t + sup | | ( (s − m)α−1 | v(m) | dm)ds ≤ kpk∞ ψ(kxk∞ )A 2 )Γ(α) (2 − η t∈J 0 0 1 |h(t)| ≤ sup | Γ(α) t∈J

Z

t

α−1

for all t ∈ J , where kpk∞ = supt∈J p(t) and A =

(α+1)(2−η2 )+2(α+1)+2ηα+1 . (2−η2 )Γ(α+2)

Thus,

kh(t)k∞ = sup |h(t)| ≤ Akpk∞ ψ(kxk∞ ). t∈J

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Now, we show that N maps bounded sets into equi- ontinuous sets of C(J, R). Let t1 , t2 ∈ J with t1 < t2 and x ∈ Br . Then, Z t2 Z t1 1 1 α−1 |h(t2 ) − h(t1 )| =| (t2 − s) v(s)ds − (t1 − s)α−1 v(s)ds Γ(α) 0 Γ(α) 0

2t2 − (2 − η 2 )Γ(α)

Z

1

(1 − s)

Z 1 2t1 v(s)ds + (1 − s)α−1 v(s)ds (2 − η 2 )Γ(α) 0 Z η Z s ( (s − m)α−1 v(m)dm)ds

α−1

0

2t2 (2 − η 2 )Γ(α) 0 0 Z η Z s 2t1 − ( (s − m)α−1 v(m)dm)ds | (2 − η 2 )Γ(α) 0 0 +

(2 − η 2 )(tα2 − tα1 ) + 2(t1 − t2 ) 2(t2 − t1 )η α+1 ≤ kpk∞ ψ(kxk∞ )[ + ] (2 − η 2 )Γ(α + 1) (2 − η 2 )Γ(α + 2) For all h ∈ N(x). Thus, limt2 →t1 |h(t2 ) − h(t1 )| = 0 for all x ∈ Br . Hen e by using the Arzela-As oli theorem, N is ompletely ontinuous. Here, we show that N has a losed graph. Let xn → x0 , hn ∈ N(xn ) for all n and hn → h0 . We have to show that h0 ∈ N(x0 ). For ea h n hoose vn ∈ SF,xn su h that

Z 1 2t (t − s) vn (s)ds − (1 − s)α−1 vn (s)ds 2 )Γ(α) (2 − η 0 0 Z η Z s 2t ( (s − m)α−1 vn (m)dm)ds + (2 − η 2 )Γ(α) 0 0

1 hn (t) = Γ(α)

Z

t

α−1

for all t ∈ J . Dene the ontinuous linear operator θ : L1 (J, R) → C(J, R) by

Z 1 2t (t − s) v(s)ds − (1 − s)α−1 v(s)ds 2 (2 − η )Γ(α) 0 0 Z η Z s 2t + ( (s − m)α−1 v(m)dm)ds. (2 − η 2 )Γ(α) 0 0

1 θ(v) = Γ(α)

Note that,

Z

t

α−1

Z t 1 khn (t) − h0 (t)k = k (t − s)α−1 (vn (s) − v0 (s))ds Γ(α) 0 Z 1 2t (1 − s)α−1 (vn (s) − v0 (s))ds − (2 − η 2 )Γ(α) 0 Z η Z s 2t + ( (s − m)α−1 (vn (m) − v0 (m))dm)dsk (2 − η 2 )Γ(α) 0 0

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for all n and so lim n → ∞khn (t) − h0 (t)k = 0. By using Lemma 1.1, θoSF is a losed graph operator. Sin e hn (t) ∈ θ(SF,xn ) for all n and xn → x0 , there exists v0 ∈ SF,x0 su h that

Z 1 2t (t − s) v0 (s)ds − (1 − s)α−1 v0 (s)ds 2 (2 − η )Γ(α) 0 0 Z η Z s 2t + ( (s − m)α−1 v0 (m)dm)ds. 2 (2 − η )Γ(α) 0 0

1 h0 (t) = Γ(α)

Z

t

α−1

Thus, N has a losed graph. If there exists λ ∈ (0, 1) su h that x ∈ λN(x), then there exists v ∈ SF,x su h that

Z 1 2λt (t − s) v(s)ds − (1 − s)α−1 v(s)ds (2 − η 2 )Γ(α) 0 0 Z η Z s 2λt + ( (s − m)α−1 v(m)dm)ds (2 − η 2 )Γ(α) 0 0

λ x(t) = Γ(α)

Z

t

α−1

2

)+2(α+1)+2η for all t ∈ J . Now, hoose M > 0 su h that kpk∞ ψ(kxk∞ )([α+1)(2−η (2−η2 )Γ(α+2) x ∈ E . This is possible be ause ψ is bounded. Thus,

kxk∞ ≤

α+1 ]

< M for all

kpk∞ ψ(kxk∞ )([α + 1)(2 − η 2 ) + 2(α + 1) + 2η α+1 ] < M. (2 − η 2 )Γ(α + 2)

Now, put U = {x ∈ C(J, R) : kxk∞ < M + 1}. Thus, there are not x ∈ ∂U and λ ∈ (0, 1) su h that x ∈ λN(x). Note that, the operator N : U → Pcp,cv (U ) is upper semi- ontinuous be ause it is ompletely ontinuous. Now by using Lemma 1.2, N has a xed point in U whi h is a solution of the problem (∗). This ompletes the proof. Now, we present our next result about the existen e of solutions for the problem (∗) with non- onvex valued assumption. 2η Let m ∈ C(J, R+ ) be su h that kmk∞ ( (2−η4−η 2 )Γ(α+1) + (2−η 2 )Γ(α+2) ) < 1. Suppose that F : J × R → Pcp(R) is a multifun tion su h that Hd (F (t, x), F (t, y)) ≤ m(t)|x − y| and d(x, F (t, x)) ≤ m(t) for almost all t ∈ J and x, y ∈ R. Then the boundary value in lusion problem (∗) has a solution. 2

α+1

Theorem 2.2.

Proof. Note that, SF,x is nonempty for all x ∈ C(J, R). By using Theorem III.6 in [18℄, we get

F has a measurable sele tion. Now, similar to the proof of Theorem 2.1, onsider the operator N : E → 2E by Z t Z 1 1 2t α−1 N(x) = {h ∈ E : h(t) = (t − s) v(s)ds − (1 − s)α−1 v(s)ds Γ(α) 0 (2 − η 2 )Γ(α) 0 Z η Z s 2t + ( (s − m)α−1 v(m)dm)ds, for some v ∈ SF,x }, (2 − η 2 )Γ(α) 0 0 6

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where E = C(J, R). First, we show that N(x) is a losed subset of E for all x ∈ E . Let x ∈ E and {un }n≥1 be a sequen e in N(x) with un → u. For ea h n, hoose vn ∈ SF,x su h that

Z 1 2t (t − s) vn (s)ds − (1 − s)α−1 vn (s)ds 2 (2 − η )Γ(α) 0 0 Z η Z s 2t + ( (s − m)α−1 vn (m)dm)ds 2 (2 − η )Γ(α) 0 0

1 un (t) = Γ(α)

Z

t

α−1

for all t ∈ J . Sin e F has ompa t values, {vn }n≥1 has a subsequen e whi h onverges to some v ∈ L1 (J, R). We denote this subsequen e again by {vn }n≥1 . It is easy to he k that v ∈ SF,x and Z t Z 1 1 2t α−1 un (t) → u(t) = (t − s) v(s)ds − (1 − s)α−1 v(s)ds 2 Γ(α) 0 (2 − η )Γ(α) 0 Z η Z s 2t + ( (s − m)α−1 v(m)dm)ds 2 (2 − η )Γ(α) 0 0 for all t ∈ J . This implies that u ∈ N(x) and so the multifun tion N has losed values. Now, we show that N is a ontra tive multifun tion with onstant

γ = kmk∞ (

4 − η2 2η α+1 + ) < 1. (2 − η 2 )Γ(α + 1) (2 − η 2 )Γ(α + 2)

Let x, y ∈ E and h1 ∈ N(x). Choose v1 ∈ SF,x su h that

Z 1 2t (t − s) v1 (s)ds − (1 − s)α−1 v1 (s)ds 2 (2 − η )Γ(α) 0 0 Z η Z s 2t + ( (s − m)α−1 v1 (m)dm)ds 2 (2 − η )Γ(α) 0 0

1 h1 (t) = Γ(α)

Z

t

α−1

for all t ∈ J . Sin e Hd (F (t, x), F (t, y)) ≤ m(t)|x(t) − y(t)| for almost all t ∈ J , there exists w0 ∈ F (t, y(t)) su h that |v1 − w0 | ≤ m(t)|x(t) − y(t)| for almost all t ∈ [0, 1]. Dene the multifun tion U : J → 2R by

U(t) = {w ∈ R : |v1 (t) − w| ≤ m(t)|x(t) − y(t)|} for almost all t ∈ J}. By using Proposition III.4 in [18℄, we get the multifun tion U(t) ∩ F (t, y(t)) is measurable. It is easy to see that there exists v2 ∈ SF,y su h that |v1 (t) − v2 (t)| ≤ m(t)|x(t) − y(t)| For all t ∈ J . Now, dene Z t Z 1 1 2t α−1 h2 (t) = (t − s) v2 (s)ds − (1 − s)α−1 v2 (s)ds 2 Γ(α) 0 (2 − η )Γ(α) 0 Z η Z s 2t + ( (s − m)α−1 v2 (m)dm)ds (2 − η 2 )Γ(α) 0 0 7

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for all t ∈ J . Thus,

Z t 1 | h1 (t) − h2 (t) |≤ (t − s)α−1 | v1 (s) − v2 (s) | ds Γ(α) 0 Z 1 2t +| | (1 − s)α−1 | v1 (s) − v2 (s) | ds (2 − η 2 )Γ(α) 0 Z η Z s 2t +| | ( (s − m)α−1 | v1 (m) − v2 (m) | dm)ds 2 (2 − η )Γ(α) 0 0 ≤ kmk∞ (

4 − η2 2η α+1 + )kx − yk∞ = γkx − yk∞ . (2 − η 2 )Γ(α + 1) (2 − η 2 )Γ(α + 2)

Therefore, the multifun tion N is a ontra tion with losed values. By using Lemma 1.3, N has a xed point whi h is a solution of the in lusion problem (∗).

3

Appli ation

Consider the problem

c

D 3/2 x(t) ∈ F (t, x(t)) (t ∈ [0, 1]) Z 3/4 via the boundary value onditions x(0) = 0 and x(1) = x(s)ds, where F : [0, 1] × R → 2R 0

is the multifun tion dened by

x5 t+1 1 1 F (t, x) = [ + , sin x + (t + 1)]. 5 4(x + 3) 8 4 4 5

Sin e max[ 4(xx5 +3) +

t+1 1 ,4 8

sin x + 14 (t + 1)] ≤ 43 , it is easy to he k that sup{|γ| : γ ∈ F (t, x)} ≤ p(t)ψ(kxk∞ )

for all x ∈ C([0, 1], R), where p(t) = 1 and ψ(t) = 34 for all t ∈ [0, 1]. Thus by using Theorem 2.1, this in lusion problem has at least one solution.

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Stability and hyperstability of generalized orthogonally quadratic ternary homomorphisms in non-Archimedean ternary Banach algebras: a fixed point approach M. Eshaghi Gordji1 , G. Askari2 , N. Ansari3 , G. A. Anastassiou4 , C. Park5∗ 1,2,3

4

Department of Mathematics, Semnan University, P.O Box 35195-363, Semnan, Iran Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA 5 Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea

Abstract: Using the fixed point method, we prove the stability and the hyperstability of generalized orthogonally quadratic ternary homomorphisms in non-Archimedean ternary Banach algebras. 1. Introduction and preliminaries The stability problem of functional equations had been first raised by Ulam [29]. This problem solved by Hyers [16] in the framework of Banach spaces. For more details about the result concerning such problems, we refer the reader to ([1, 3, 11, 17, 22, 25, 26, 27, 28, 31, 32]). The stability of homomorphisms and derivations in Banach algebras, Banach ternary algebras, C ∗ -algebras, Lie C ∗ -algebras, C ∗ -ternary algebras has been studied by many authors (see [9, 25, 26, 27, 28]). Let A, B be two ternary algebras. A mapping f : A → B is called a quadratic ternary homomorphism if f is a quadratic mapping (i.e. f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ A) and satisfies f ([a, b, c]) = [f (a), f (b), f (c)] for all a, b, c ∈ A. A mapping g : A → B is called a generalized quadratic ternary homomorphism if there exists a quadratic ternary homomorphism f : A → B such that g([a, b, c]) = [g(a), f (b), f (c)] for all a, b, c ∈ A. In 2003, C˘ adariu and Radu applied the fixed point methods to the investigation of Jensen functional equations [4] (see also [5, 6, 12, 21, 24]). Arriola and Beyer [2] initiated the stability of functional equations in non-Archimedean spaces. In fact they established the stability of the Cauchy functional equation over p-adic fields. After their results some papers (see, for instance, ([7, 8, 9, 10]) on the stability of other equations in such spaces have been published. In 1897, Hensel [15] discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. During the last three decades p-adic numbers have gained the interest in of physicists for their research, in particular, in the problems coming from quantum physics, p-adic strings and hyperstrings [18, 19]. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: For any x, y > 0, there exists n ∈ N such that x < ny (see [13, 30]). Let K denote a field and function (valuation absolute) |.| from K into [0, ∞). A non-Archimedean valuation is a function |.| that satisfies the strong triangle inequality; namely |x + y| ≤ max{|x|, |y|} ≤ |x| + |y| for all x, y ∈ K. The associated field K is referred to as a non-Archimedean filed. Clearly, |1| = | − 1| = 1 and |n| ≤ 1 for all n ≥ 1. A trivial example of a non-Archimedean valuation is the function |.| taking everything except 0 into 1 and |0| = 0. We always assume in addition that |.| is non trivial, i.e., there is a z ∈ K such that |z| = 6 0, 1. 0

2010 Mathematics Subject Classification: 17A40, 39B52, 39B55, 39B82, 47H10, 47S10. Keywords: non-Archimedean ternary Banach algebra; generalized orthogonally quadratic ternary homomorphism; quadratic functional equation; fixed point approach; stability and hyperstability. ∗ Corresponding author (Choonkil Park). 0 E-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] 0

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Let X be a linear space over a field K with a non-Archimedean nontrivial valuation |.|. A function k.k : X → [0, ∞) is said to be a non-Archimedean norm if it is a norm over K with the strong triangle inequality (ultrametric); namely, kx + yk ≤ max{kxk, kyk} for all x, y ∈ K. Then (X, k.k) is called a non-Archimedean space. In any such a space a sequence {xn }n∈N is a Cauchy sequence if and only if {xn+1 , xn }n∈N converges to zero. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. A non-Archimedean ternary Banach algebra is a complete non-Archimedean space A equipped with a ternary product (x, y, z) → [x, y, z] of A3 into A which is K-linear in each variables and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v] and satisfies the following: k[x, y, z]k ≤ kxk · kyk · kzk (see [14]). Let X be a nonempty set and let d : X × X → [0, ∞] satisfy: d(x, y) = 0 if and only if x = y, d(x, y) = d(y, x) and d(x, z) ≤ max{d(x, y), d(y, z)} (strong triangle inequality), for all x, y, z ∈ X. Then (X, d) is called a non-Archimedean generalized metric space. (X, d) is called complete if every d-Cauchy sequence in X is d-convergent. Suppose that X is a real vector space (or an algebra) with dim X ≥ 2 and ⊥ is a binary relation on X with the following properties: (O1 ) totality of ⊥ for zero: x ⊥ 0, 0 ⊥ x for all x ∈ X; (O2 ) independence: if x, y ∈ X − {0}, x ⊥ y, then x, y are linearly independent; (O3 ) homogeneity: if x, y ∈ X, x ⊥ y, then αx ⊥ βy for all α, β ∈ R; (O4 ) the Thalesian property: if P is a 2−dimensional subspace (subalgebra) of X, x ∈ P and λ ∈ R+ , then there exists ux ∈ P such that x ⊥ ux and x + ux ⊥ λx − ux . The pair (X, ⊥) is called an orthogonality space (algebra). By an orthogonality normed space (normed algebra) we mean an orthogonality space (algebra) having a normed structure (see [23]). 2. Main results Using the strong triangle inequality in the proof of the main result of [20], we get to the following result: Theorem 2.1. (Non-Archimedean Alternative Contraction Principle) Suppose that (Ω, d) is a non-Archimedean generalized complete metric space and T : Ω → Ω is a strictly contractive mapping with the Lipschitz constant L. Let x ∈ Ω. If either (i) d(T m (x), T m+1 (x)) = ∞ for all m ≥ 0, or (ii) there exists some m0 such that d(T m (x), T m+1 (x)) < ∞ for all m ≥ m0 , then the sequence {T m (x)} is convergent to a fixed point x∗ of T ; x∗ is the unique fixed point of T in the set Λ = {y ∈ Ω : d(T m0 (x), y) < ∞}; and d(y, x∗ ) ≤ d(y, T (y)) for all y in this set. ´ S{(x, αx) : x ∈ A, α ∈ R}, In this section, we suppose that A is a non-Archimedean ternary Banach algebra with ⊥:= ⊥ S where ⊥ is an orthogonality on A, and B is a non-Archimedean ternary Banach algebra and l ∈ {1, −1} is fixed. Also, let |4| < 1 and we assume that 4 6= 0 in K (i.e., the characteristic of K is not 4). Theorem 2.2. Let g, f : A → B be two mappings with g(0) = f (0) = 0 for which there exists a function ϕ : A8 → [0, ∞] such that kη(ax + by) + η(ax − by) − 2a2 η(x) − 2b2 η(y)k + kf ([u, v, w]) − [f (u), f (v), f (w)]k (2.1) + kg([r, s, t]) − [g(r), f (s), f (t)]k ≤ ϕ(x, y, u, v, w, r, s, t) for all η ∈ {f, g}, x, y ∈ A with x ⊥ y and for all u, v, w, r, s, t, ∈ A, that are mutually orthogonal and nonzero fixed integers a, b. Suppose that there exists L < 1 such that x y u v w r s t (2.2) ϕ(x, y, u, v, w, r, s, t) ≤ |4|l(l+2) Lϕ( l , l , l , l , l , l , l , l ) 2 2 2 2 2 2 2 2 516

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for all x, y ∈ A with x ⊥ y and for all u, v, w, r, s, t, ∈ A, that are mutually orthogonal. Then there exist a unique orthogonally quadratic ternary homomorphism h : A → B and a unique generalized orthogonally quadratic ternary homomorphism H : A → B (respect to h) such that 1−l

max{kg(x) − H(x)k, kf (x) − h(x)k} ≤

L 2 ψ(x) |4|

(2.3)

for all x ∈ A, where x x x 1 ψ(x) := max{ϕ( , , 0, 0, 0, 0, 0, 0), ϕ( , 0, 0, 0, 0, 0, 0, 0), ϕ(x, x, 0, 0, 0, 0, 0, 0), a b a |2b2 | 1 x ϕ(x, −x, 0, 0, 0, 0, 0, 0), ϕ(0, , 0, 0, 0, 0, 0, 0)}. |2b2 | b for all x ∈ A. Proof. By (2.2), one can show that lim

n→∞

1 ϕ(2ln x, 2ln y, 2ln u, 2ln v, 2ln w, 2ln r, 2ln s, 2ln t) = 0 |4|l(l+2)n

(2.4)

for all x, y ∈ A with x ⊥ y and for all u, v, w, r, s, t, ∈ A, that are mutually orthogonal. Putting η = g in (2.1) and u = v = w = r = s = t = 0 in (2.1), we get kg(ax + by) + g(ax − by) − 2a2 g(x) − 2b2 g(y)k ≤ ϕ(x, y, 0, 0, 0, 0, 0, 0)

(2.5)

for all x, y ∈ A with x ⊥ y. Putting y = 0 in (2.5). Since x ⊥ 0, we get k2g(ax) − 2a2 g(x)k ≤ ϕ(x, 0, 0, 0, 0, 0, 0, 0)

(2.6)

for all x ∈ A. Setting y = −y in (2.5), by the definition of ⊥, we get kg(ax − by) + g(ax + by) − 2a2 g(x) − 2b2 g(−y)k ≤ ϕ(x, −y, 0, 0, 0, 0, 0, 0)

(2.7)

for all x, y ∈ A with x ⊥ y. It follows from (2.5) and (2.7) that k2b2 g(y) − 2b2 g(−y)k ≤ max{ϕ(x, y, 0, 0, 0, 0, 0, 0), ϕ(x, −y, 0, 0, 0, 0, 0, 0)}

(2.8)

for all x, y ∈ A with x ⊥ y. Putting y = by in (2.8), by the definition of ⊥, we get kg(by) − g(−by)k ≤ max{

1 1 ϕ(x, by, 0, 0, 0, 0, 0, 0), ϕ(x, −by, 0, 0, 0, 0, 0, 0)} |2b2 | |2b2 |

(2.9)

for all x, y ∈ A with x ⊥ y. Let x = 0 in (2.5). Since 0 ⊥ x, we get kg(by) + g(−by) − 2b2 g(y)k ≤ ϕ(0, y, 0, 0, 0, 0, 0, 0)

(2.10)

for all y ∈ A. It follows from (2.9) and (2.10) that k2g(by) − 2b2 g(y)k ≤ max{

1 1 ϕ(x, by, 0, 0, 0, 0, 0, 0), ϕ(x, −by, 0, 0, 0, 0, 0, 0), ϕ(0, y, 0, 0, 0, 0, 0, 0)} |2b2 | |2b2 |

(2.11)

for all x, y ∈ A with x ⊥ y. Replacing x and y by xa and yb in (2.5), respectively, and by the definition of ⊥, we get x x x x kg(2x) − 2a2 g( ) − 2b2 g( )k ≤ ϕ( , , 0, 0, 0, 0, 0, 0) (2.12) a b a b for all x ∈ A. Setting x = xa in (2.6), by the definition of ⊥, we get x x k2a2 g( ) − 2g(x)k ≤ ϕ( , 0, 0, 0, 0, 0, 0, 0) (2.13) a a for all x ∈ A. Putting y = xb in (2.9), by the definition of ⊥, we get x 1 1 x k2b2 g( ) − 2g(x)k ≤ max{ 2 ϕ(x, x, 0, 0, 0, 0, 0, 0), ϕ(x, −x, 0, 0, 0, 0, 0, 0), ϕ(0, , 0, 0, 0, 0, 0, 0)} b |2b | |2b2 | b

(2.14)

for all x ∈ A. It follows from (2.12), (2.13) and (2.14) that kg(2x) − 4g(x)k ≤ ψ(x)

(2.15)

for all x ∈ A. Consider the set X := {´ g : g´ : A → B g´(0) = 0}. 517

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´ ∈ X, define For every g´, h ´ := inf{K ∈ (0, ∞) : k´ ´ d(´ g , h) g (x) − h(x)k ≤ Kψ(x), ∀x ∈ A}. It is easy to show that (X, d) is a complete generalized non-Archimedean metric space. Now, we consider the J : X → X such that 1 J (´ g )(x) := l g´(2l x) 4 ´ ∈ X, it follows that for all x ∈ A for all x ∈ A. For any g´, h ´ < K ⇒ k´ ´ d(´ g , h) g (x) − h(x)k ≤ Kψ(x) ´ l x) g´(2l x) h(2 ψ(2l x) − k≤K 4l 4l |4|l ´ ⇒ kJ g´(x) − J h(x)k ≤ LKψ(x). ⇒ k

Hence we have ´ ≤ Ld(´ ´ d(J (´ g ), J (h)) g , h). 1−l

By applying the inequality (2.15), we see that d(J(f ), f ) ≤ L |4|2 . It follows from Theorem 2.1 that J has a unique fixed point H : A → B in the set Λ : {´ g ∈ X : d(´ g , g) < ∞}, where H is defined by H(x) = lim J n g(x) = lim n→∞

n→∞

1 g(2ln x) 4ln

(2.16)

for all x ∈ A. It follows from (2.4), (2.5) and (2.16) that kH(ax + by) + H(ax − by) − 2a2 H(x) − 2b2 H(y)k 1 kg(2ln ax + 2ln by) + g(2ln ax − 2ln by) − 2a2 g(2ln x) − 2b2 g(2ln y)k |4|ln 1 ≤ lim ϕ(2ln x, 2ln y, 0, 0, 0, 0, 0, 0) n→∞ |4|ln 1 ϕ(2ln x, 2ln y, 0, 0, 0, 0, 0, 0) = 0 ≤ lim n→∞ |4|ln(l+2) = lim

n→∞

for all x, y ∈ A with x ⊥ y. This shows that H is an orthogonally quadratic. Putting η = f , u = v = w = r = s = t = 0 in (2.1), we get kf (ax + by) + f (ax − by) − 2a2 f (x) − 2b2 f (y)k ≤ ϕ(x, y, 0, 0, 0, 0, 0, 0) for all x, y ∈ A with x ⊥ y. By the same reasoning as above, we can show that the limit 1 h(x) =: lim ln f (2ln x) n→∞ 4 exists for all x ∈ A. Moreover, we can show that h is an orthogonally quadratic mapping on A satisfying (2.3). On the other hand, we have 1 kh([u, v, w]) − [h(u), h(v), h(w)]k = lim kf (4ln [u, v, w]) − [f (2ln u), f (2ln v), f (2ln w)]k n→∞ |4|2ln 1 ≤ lim ϕ(0, 0, 2ln u, 2ln v, 2ln w, 0, 0, 0) n→∞ |4|2ln 1 ≤ lim ϕ(0, 0, 2ln u, 2ln v, 2ln w, 0, 0, 0) = 0 n→∞ |4|l(l+2)n for all u, v, w ∈ A, that are mutually orthogonal. Therefore, h is an orthogonally quadratic ternary homomorphism on A. Also, we have 1 kH([r, s, t]) − [H(r), h(s), h(t)]k = lim kg(4ln [r, s, t]) − [g(2ln r), f (2ln s), f (2ln t)]k n→∞ |4|2ln 1 ≤ lim ϕ(0, 0, 0, 0, 0, 2ln r, 2ln s, 2ln t) n→∞ |4|2ln 1 ≤ lim ϕ(0, 0, 0, 0, 0, 2ln r, 2ln s, 2ln t) = 0 n→∞ |4|l(l+2)n 518

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for all r, s, t ∈ A, that are mutually orthogonal. It follows that H is a generalized orthogonally quadratic ternary homomorphism (respect to h) on A. This completes the proof.  From now on, we use the following abbreviation for any mappings g, f : A → B: ∆(g, f )(z1 , ..., z8 ) :=kf (az1 + bz2 ) + f (az1 − bz2 ) − 2a2 f (z1 ) − 2b2 f (z2 )k + kg(az1 + bz2 ) + g(az1 − bz2 ) − 2a2 g(z1 ) − 2b2 g(z2 )k + kf ([z3 , z4 , z5 ]) − [f (z3 ), f (z4 ), f (z5 )]k + kg([z6 , z7 , z8 ]) − [g(z6 ), f (z7 ), f (z8 )]k. Corollary 2.3. Let K = Q2 be the 2-adic number field. Let A be a non-Archimedean ternary Banach algebra on K with ´ S{(x, αx) : x ∈ X, α ∈ R} and B be a non-Archimedean ternary Banach algebra on K. Let  be a nonnegative real ⊥= ⊥ number and let p be a real number such that p > 6 if l = 1 and 0 < p < 2 if l = −1. Suppose that mappings g, f : A → B satisfy f (0) = g(0) = 0 and ∆(g, f )(z1 , ..., z8 ) ≤  max{kzi kp : 1 ≤ i ≤ 8} for all z1 , z2 ∈ A with z1 ⊥ z2 and for all z3 , ..., z8 ∈ A, that are mutually orthogonal. Then there exist a unique orthogonally quadratic ternary homomorphism h : A → B and a unique generalized orthogonally quadratic ternary homomorphism H : A → B (respect to h) such that max{kg(z) − H(z)k, kf (z) − h(z)k}   2,    ip l(4−p)+p max{2 , 2}, 2 kzkp ≤ |2| jp 2j+1  max{2 , 2 },    max{2jp , 22j+1 },

gcd(a, 2) = gcd(b, 2) = 1; a = k2i , gcd(b, 2) = 1; gcd(a, 2) = 1, b = m2j ∨ a = k2i , b = m2j (j ≥ i); a = k2i , b = m2j (i ≥ j)

for all x ∈ A, where i, j, k, m ≥ 1 are integers and gcd(k, 2) = gcd(m, 2) = 1. Now, we have the following result on hyperstability of generalized orthogonally quadratic ternary homomorphisms in non-Archimedean ternary Banach algebras. Corollary 2.4. Let p > 0 be a nonnegative real number such that |2|(2l+4)p ≥ 1 and let j ∈ {3, 4, ..., 8} be fixed. Suppose that mappings g, f : A → B satisfy f (0) = g(0) = 0 and ∆(g, f )(z1 , ..., z8 ) ≤ (

8 X

kzi kp )kzj kp

i=1

for all z1 , z2 ∈ A with z1 ⊥ z2 and for all z3 , ..., z8 ∈ A, that are mutually orthogonal, where a, b are positive fixed integers. Then f is an orthogonally quadratic ternary homomorphism and g is a generalized orthogonally quadratic ternary homomorphism related to f . Proof. It follows from Theorem 2.2 by taking ϕ(z1 , ..., z8 ) = (

8 X

kzi kp )kzj kp

i=1

for all z1 , z2 ∈ A with z1 ⊥ z2 and for all z3 , ..., z8 ∈ A, that are mutually orthogonal and putting L = |2|−(2l+4)p .



References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] L. M. Arriola and W. A. Beyer, Stability of the Cauchy functional over p-adic fields, Real Anal. Exch. 31 (2005-2006), 125–132. [3] L. C˘ adariu, L. G˘ avruta and P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. 519

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[4] L. Cˇ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Ineq. Pure Appl. Math. 4 (2003), Art. No. 4. [5] L. Cˇ adariu and V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [6] L. Cˇ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 2008, Art. ID 749392 (2008). [7] Y. Cho and R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett. 23 (2010), 1238–1242. [8] Y. Cho and R. Saadati, Lattice non-Archimedean random stability of ACQ-functional equation, Adv. Difference Equ. 2011, Art. ID 2011:31 (2011). [9] Y. Cho, R. Saadati and J. Vahidi, Approximation of homomorphism and non-Archimedean Lie C ∗ -algebra via fixed point method, Discrete Dyn. Nat. Soc. 2012, Art. ID 373904 (2012). [10] M. Eshaghi Gordji, R, Khodabakhsh, H. Khodaei, C. Park and D. Shin, A function equation related to inner product space in non-Archimedaen normed space, Adv. Difference Equ. 2011, Art. ID 2011:37 (2011). [11] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [12] P. Gˇ avruta and L. Gˇ avruta, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl. 1 (2010), 11–18. [13] F. Q. Gouvea, A p-Adic Number, Springer, Berlin, 1997. [14] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5 (1964), 848–861. [15] K. Hensel, Uber eine neue Begrundung der Theorie der algebraischen Zahlen, Jahresber. Dtsch. Math. Ver. 6 (1897), 83–88. [16] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. [17] H. Khodaei and Th.M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 22–41. [18] A. Khrennikov, p-Adic Valued Distribution in Mathematical Physics, Kluwer Academic, Dordrecht, 1994. [19] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic, Dordrecht, 1997. [20] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on the generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968), 305–309. [21] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [22] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [23] J. Ratz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35–49. [24] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [25] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivations on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [26] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Nearly ternaty cubic homomorphisms in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [27] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [28] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [29] S.M. Ulam, Problems in Modern Mathematics, Chapter V I, Science Editions., Wiley, New York, 1964. [30] V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics, Worid Scientific, Singapore, 1994. [31] C. Zaharia, On the probabilistic stability of the monomial functional equation, J. Nonlinear Sci. Appl. 6 (2013), 51–59. [32] S. Zolfaghari, Approximation of mixed type functional equations in p-Banach spaces, J. Nonlinear Sci. Appl. 3 (2010), 110–122.

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SYMMETRY IDENTITIES OF HIGHER-ORDER q-EULER POLYNOMIALS UNDER THE SYMMETRIC GROUP OF DEGREE FOUR DAE SAN KIM AND TAEKYUN KIM

Abstract. In this paper, we give some new identities of symmetry for the higher-order q-Euler polynomials under the symmetric group of degree four which are derived from the fermionic p-adic q-integrals on Zp .

1. Introduction Let p be a fiixed odd prime number. Throughout this paper, Zp , Qp , and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp . Let νp be the normalized exponential valuation of Cp with |p|p = p−νp (p) = p1 . Let us assume that q is an indeterminate x

1

in Cp such that |1 − q|p < p− p−1 . The q-number of x is defined as [x]q = 1−q 1−q . Note that limq→1 [x]q = x. Let C (Zp ) be the space of all Cp -valued continuous functions on Zp . For f ∈ C (Zp ), the fermionic p-adic q-integral on Zp is defined by Kim as (1.1)

I−q (f ) ˆ = f (x) dµ−q (x) Zp N

pX −1 1 x f (x) (−q) , N →∞ [pN ]−q x=0

= lim

(see [9, 10, 12, 13]) .

Thus, by (1.1), we get (1.2)

qI−q (f1 ) + I−q (f ) = [2]q f (0) ,

(see [9]) ,

where f1 (x) = f (x + 1). The Carlitz-type q-Euler numbers are defined by (1.3)

n

q (Eq + 1) + En,q = [2]q δ0,n ,

E0,q = 1,

(see [9, 10]) ,

Eqn

with the usual convention about replacing by En,q . The q-Euler polynomials are given by n   X n n−l (1.4) En,q (x) = [x]q q lx El,q , (see [9]) . l l=0

2000 Mathematics Subject Classification. 11B68, 11S80, 05A19, 05A30. Key words and phrases. identities of symmetry, higher-order q-Euler polynomials, symmetric group of degree four. 1

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DAE SAN KIM AND TAEKYUN KIM

From (1.1) and (1.4), we have ˆ n (1.5) [x + y]q dµ−q (y) = En,q (x) ,

(n ≥ 0) ,

(see [9, 10, 12]) .

Zp

For r ∈ N, we consider the higher-order q-Euler polynomials as follows: ˆ ˆ ∞ X tn (r) (1.6) ··· et[x1 +···+xr +x]q dµ−q (x1 ) · · · dµ−q (xr ) = En,q (x) . n! Zp Zp n=0 Thus, by (1.3), we get (r) En,q (x) ˆ ˆ = ···

(1.7)

Zp

n

[x1 + · · · + xr + x]q dµ−q (x1 ) · · · dµ−q (xr ) ,

(see [9]) .

Zp (r)

(r)

When x = 0, En,q = En,q (0) are called the higher-order q-Euler numbers. In this paper, we give some new identities of symmetry for the higher-order q-Euler polynomials under the symmetric group S4 of degree four. Recently, several authors have studied q-extensions of Euler numbers and polynomials in the several different areas (see [1–23]). (r)

2. Symmetry identities of En,q (x) under S4 Let w1 , w2 , w3 , w4 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), w3 ≡ 1 (mod 2), w4 ≡ 1 (mod 2). Then we have (2.1) ˆ

ˆ ···

e

[w1 w2 w3

Pr

l=1

xl +w1 w2 w3 w4 x+w4 w2 w3

Pr

l=1 il +w4 w1 w3

Pr

l=1

jl +w4 w1 w2

Pr

l=1

kl ] t q

Zp

Zp

× dµ−qw1 w2 w3 (x1 ) · · · dµ−qw1 w2 w3 (xr ) N pX −1

1

Pr x (−q w1 w2 w3 ) l=1 l r N N →∞ [p ] w1 w2 w3 −q x1 ,...,xr =0 P P P Pr [w1 w2 w3 l=1 xl +w1 w2 w3 w4 x+w4 w2 w3 rl=1 il +w4 w1 w3 rl=1 jl +w4 w1 w2 rl=1 kl ] t

= lim ×e

q

wX 4 −1

1

N pX −1

Pr Pr l x +···+xr (−1) i=1 i q w1 w2 w3 i=1 (li +w4 xi ) (−1) 1 r N N →∞ [w4 p ] w1 w2 w3 −q l1 ,...,lr =0 x1 ,...,xr =0 Pr P P P w w w (l +w x [ 1 2 3 i=1 i 4 i )+w1 w2 w3 w4 x+w4 w2 w3 rl=1 il +w4 w1 w3 rl=1 jl +w4 w1 w2 rl=1 kl ] t

= lim ×e

q

.

Now, we observe that (2.2) wX wX wX 1 −1 2 −1 3 −1 Pr 1 (i +j +k ) (−1) l=1 l l l r [2]qw1 w2 w3 i ,...,i =0 j ,...,j =0 k1 ,...,kr =0 1 r 1 r P P P w4 w2 w3 rl=1 il +w4 w1 w3 rl=1 jl +w4 w1 w2 rl=1 kl

×q ˆ ×

Zp

ˆ

···

e

[w1 w2 w3

Pr

l=1

xl +w1 w2 w3 w4 x+w4 w2 w3

Pr

l=1 il +w4 w1 w3

Pr

l=1

jl +w4 w1 w2

Pr

l=1

kl ] t q

Zp

× dµ−qw1 w2 w3 (x1 ) · · · dµ−qw1 w2 w3 (xr )

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SYMMETRY IDENTITIES OF HIGHER-ORDER q-EULER POLYNOMIALS

 = lim

1 + q w1 w2 w3 w4 pN

N →∞

×q

w4 w2 w3

l=1 il +w4 w1 w3

q w1 w2 w3

Pr

i=1

wX 1 −1

wX 2 −1

wX 3 −1

wX 4 −1

Pr

(−1)

n=1 (ln +jn +in +kn )

i1 ,...,ir =0 j1 ,...,jr =0 k1 ,...,kr =0 l1 ,...,lr =0 Pr Pr Pr j=1 jl +w4 w1 w2 l=1 kl +w1 w2 w3 i=1 li

Pr

N pX −1

×

r

1

3

xi

x1 +···+xr

(−1)

x1 ...,xr =0

×e

[w1 w2 w3

Pr

i=1 (li +xi w4 )+w1 w2 w3 w4 x+w4 w2 w3

Pr

Pr

l=1 il +w4 w1 w3

jl +w4 w1 w2

l=1

Pr

l=1

kl ] t q

.

As this expression is invariant under S4 , we have the following theorem. Theorem 2.1. For w1 , w2 , w3 , w4 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), w3 ≡ 1 (mod 2), w4 ≡ 1 (mod 2), the following expression 1

wσ(1) −1

wσ(2) −1

wσ(3) −1

X

X

X

Pr

(−1) r [2]qwσ(1) wσ(2) wσ(3) i ,...,i =0 j ,...,j =0 k1 ,...,kr =0 1 r 1 r × q wσ(4) wσ(2) wσ(3)

Pr

l=1 il +wσ(4) wσ(1) wσ(3)

Pr

l=1

l=1 (il +jl +kl )

jl +wσ(4) wσ(1) wσ(2)

Pr

ˆ

l=1

ˆ

kl

e[A]q t

··· Zp

Zp

× dµ−qwσ(1) wσ(2) wσ(3) (x1 ) · · · dµ−qwσ(1) wσ(2) wσ(3) (xr ) are the same for any σ ∈ SP 4, Pr r where A = wσ(1) wσ(2) wσ(3) l=1 il +wσ(1) wσ(2) wσ(3) wσ(4) x+wσ(4) wσ(2) wσ(3) l=1 il Pr Pr +wσ(4) wσ(1) wσ(3) l=1 jl + wσ(4) wσ(1) wσ(2) l=1 kl . From (1.7), we have (2.3) ˆ

ˆ ···

Zp

e

[w1 w2 w3

Pr

l=1

xl +w1 w2 w3 w4 x+w4 w2 w3

Pr

l=1 il +w4 w1 w3

Pr

l=1

jl +w4 w1 w2

Pr

l=1

kl ] t q

Zp

× dµ−qw1 w2 w3 (x1 ) · · · dµ−qw1 w2 w3 (xr ) #n ˆ ˆ "X ∞ r r r r X w4 X w4 X w4 X n = [w1 w2 w3 ]q ··· xl + w4 x + il + jl + kl w1 w2 w3 Zp Zp w n=0 l=1

l=1

l=1

l=1

q

1 w2 w3

n

t n! ! r r X w4 w4 X w4 tn w4 x + il + jl + kl . w1 w2 w3 n!

× dµ−qw1 w2 w3 (x1 ) · · · dµ−qw1 w2 w3 (xr ) =

∞ X

n

(r)

[w1 w2 w3 ]q En,qw1 w2 w3

n=0

l=1

l=1

Thus, by (2.3), we get (2.4) ˆ

ˆ ···

Zp

" w1 w2 w3

Zp

r X

xl + w4 w2 w3

r X

l=1

il + w4 w1 w3

l=1

r X

jl + w4 w1 w2

l=1

r X l=1

#n kl q

× dµ−qw1 w2 w3 (x1 ) · · · dµ−qw1 w2 w3 (xr ) n = [w1 w2 w3 ]q

(r) En,qw1 w2 w3

r r r w4 X w4 X w4 X w4 x + il + jl + kl w1 w2 w3 l=1

l=1

! .

l=1

Therefore, by (2.4) and Theorem 2.1, we obtain the following theorem.

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Theorem 2.2. For n ≥ 0, w1 , w2 , w3 , w4 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), w3 ≡ 1 (mod 2), w4 ≡ 1 (mod 2), the following expression  n Pr X−1 wσ(2) X−1 wσ(3) X−1 wσ(1) wσ(2) wσ(3) q wσ(1) l=1 (il +jl +kl ) (−1) r [2]qwσ(1) wσ(2) wσ(3) i ,...,i =0 j ,...,j =0 1

×q ×

wσ(4) wσ(2) wσ(3)

r

1

k1 ,...,kr =0 Pr Pr l=1 jl +wσ(4) wσ(1) wσ(2) l=1 kl

r

Pr

l=1 il +wσ(4) wσ(1) wσ(3)

r r r wσ(4) X wσ(4) X wσ(4) X il + jl + kl wσ(4) x + wσ(1) wσ(2) wσ(3)

(r) En,qwσ(1) wσ(2) wσ(3)

l=1

l=1

!

l=1

are the same for any σ ∈ S4 . Now, we observe that " r # r r r X w4 X w4 X w4 X (2.5) xl + w 4 x + il + jl + kl w1 w2 w3 l=1 l=1 l=1 l=1 q w 1 w2 w3 " # r r r X X X [w4 ]q w2 w3 il + w1 w3 jl + w1 w2 kl = [w1 w2 w3 ]q w l=1

+q

w2 w3 w4

Pr

l=1

l=1 il +w1 w3 w4

Pr

l=1

l=1

jl +w1 w2 w4

Pr

l=1

kl

q

4

.

By (2.5), we get " r #n r r r X w4 X w4 X w4 X xl + w4 x + il + jl + kl (2.6) w1 w2 w3 l=1 l=1 l=1 l=1 q w1 w2 w3 !n−m " #n−m   n r r r X n X X X [w4 ]q w2 w3 il + w1 w3 jl + w1 w2 kl = m [w1 w2 w3 ]q m=0 l=1 l=1 l=1 q w4 " r #m Pr Pr Pr X x +w x . × q m(w2 w3 w4 l=1 il +w1 w3 w4 l=1 jl +w1 w2 w4 l=1 kl ) l

4

l=1

q w1 w 2 w3

From (2.6), we can derive the following equation: #n ˆ ˆ "X r r r r w4 X w4 X w4 X ··· xl + w 4 x + il + jl + kl (2.7) w1 w2 w3 Zp Zp w l=1

l=1

l=1

l=1

q

1 w2

× dµ−qw1 w2 w3 (x1 ) · · · dµ−qw1 w2 w3 (xr ) !n−m " #n−m r r r n   X X X X [w4 ]q n w2 w3 il + w1 w3 jl + w1 w2 kl = m [w1 w2 w3 ]q w m=0 l=1

×

l=1

(r) Em,qw1 w2 w3 (w4 x) P P P m(w2 w3 w4 rl=1 il +w1 w3 w4 rl=1 jl +w1 w2 w4 rl=1 kl )

×q

l=1

q

4

.

Thus, by (2.7), we get (2.8) n

[w1 w2 w3 ]q

wX 1 −1

wX 2 −1

wX 3 −1

Pr

(−1) r [2]qw1 w2 w3 i ,...,i =0 j ,...,j =0 k ,...,k =0 1 r 1 r 1 r

524

l=1 (il +jl +kl )

q w2 w3 w4

Pr

l=1 il +w4 w1 w3

Pr

l=1

jl

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SYMMETRY IDENTITIES OF HIGHER-ORDER q-EULER POLYNOMIALS

×q

w1 w2 w4

ˆ

Pr

l=1 kl

ˆ

" r X

··· Zp

Zp

l=1

r r r w4 X w4 X w4 X il + jl + kl xl + w4 x + w1 w2 w3 l=1

l=1

l=1

× dµ (x1 ) · · · dµ (xr )   m wX wX n 1 −1 2 −1 X n [w1 w2 w3 ]q n−m (r) [w ] E (w x) = w w w 4 4 m,q 1 2 3 q m [2]rqw1 w2 w3 m=0 i ,...,i =0 j ,...,j =0 −q w1 w2 w3

1

l=1 (il +jl +kl ) (m+1)(w2 w3 w4

× w2 w3

r X

#n q w1 w2 w3

−q w1 w2 w3

Pr

× (−1) "

5

q

il + w1 w3

l=1

r X

Pr

l=1 il +w4 w1 w3

jl + w1 w2

l=1

r X

r

1

r

Pr

l=1 jl +w4 w1 w2

wX 3 −1

k1 ,...,kr =0 Pr l=1 kl )

#n−m kl

l=1

q w4

  m n [w1 w2 w3 ]q (r) n−m (r) [w4 ]q Em,qw1 w2 w3 (w4 x) Tn,qw4 (w1 , w2 , w3 | m) , = r m [2] w w w 1 2 3 q m=0 n X

where (2.9)

(r) Tn,q (w1 , w2 , w3 | m)

=

wX 1 −1

wX 2 −1

wX 3 −1

Pr

(−1)

l=1 (il +jl +kl )

i1 ,...,ir =0 j1 ,...,jr =0 k1 ,...,kr =0 P P P (m+1)(w2 w3 rl=1 jl +w1 w3 rl=1 jl +w1 w2 rl=1 kl )

×q "

× w2 w3

r X l=1

il + w1 w3

r X

jl + w1 w2

l=1

r X l=1

#n−m kl

. q

As this expression is invariant under S4 , we have the following theorem. Theorem 2.3. For n ≥ 0, w1 , w2 , w3 , w4 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), w3 ≡ 1 (mod 2), w4 ≡ 1 (mod 2), the following expression m  n   w X n−m σ(1) wσ(2) wσ(3) q  n wσ(4) q r w w w m [2]q σ(1) σ(2) σ(3) m=0

(r) (r) × Em,qwσ(1) wσ(2) wσ(3) wσ(4) x Tn,qwσ(4) wσ(1) , wσ(2) , wσ(3) | m are the same for any σ ∈ S4 . References 1. L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954), 332–350. MR 0060538 (15,686a) 2. M. Chester, Is symmetry identity?, Int. Stud. Philos. Sci. 16 (2002), no. 2, 111–124. MR 2022083 (2004i:00005) 3. A. M. Fu, H. Pan, and I. F. Zhang, Symmetric identities on Bernoulli polynomials, J. Number Theory 129 (2009), no. 11, 2696–2701. MR 2549525 (2010j:11039) 4. S. Gaboury, R. Tremblay, and B.-J. Fug`ere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115–123. MR 3184467

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SYMMETRY IDENTITIES OF HIGHER-ORDER q-EULER POLYNOMIALS

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21. Y. Simsek, Interpolation functions of the Eulerian type polynomials and numbers, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 2, 301–307. MR 3088760 22. S. L. Yang and Z. K. Qiao, Some symmetry identities for the Euler polynomials, J. Math. Res. Exposition 30 (2010), no. 3, 457–464. MR 2680613 (2011d:11045) 23. Y. Zhang, Z.-W. Sun, and H. Pan, Symmetric identities for Euler polynomials, Graphs Combin. 26 (2010), no. 5, 745–753. MR 2679944 (2011k:05032) Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected]

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Soft saturated and dried values with applications in BCK/BCI-algebras Seok Zun Songa , Hee Sik Kimb,∗ , and Young Bae Junc a

Department of Mathematics Jeju National University, Jeju 690-756, Korea e-mail: [email protected] b

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea e-mail: [email protected] c

Department of Mathematics Education, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected] Abstract The notions of soft saturated values and soft dried values are introduced, and their applications in BCK/BCI-algebras are discussed. Using these notions, properties of energetic subsets are investigated. Using the concepts of intersectional (union) ideals, properties of right vanished (stable) subsets are explored. Keywords: Energetic subset, Right vanished subset, Right stable subset, Saturated value, Dried value. 2010 Mathematics Subject Classification. 06F35, 03G25, 06D72.

1

Introduction

The real world is inherently uncertain, imprecise and vague. Various problems in system identification involve characteristics which are essentially non-probabilistic in nature [32]. In response to this situation Zadeh [33] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a ∗

Correspondence: Tel.: +82 2 2220 0897, Fax: +82 2 2281 0019 (H. S. Kim).

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more general framework, the approach to uncertainty is outlined by Zadeh [34]. To solve complicated problem in economics, engineering, and environment, we can’t successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties can’t be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [29]. Maji et al. [26] and Molodtsov [29] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [29] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years. Evidence of this can be found in the increasing number of high-quality articles on soft sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences in recent years. Maji et al. [26] described the application of soft set theory to a decision making problem. Maji et al. [25] also studied several operations on the theory of soft sets. Akta¸s and C ¸ a˘gman [2] studied the basic concepts of soft set theory, and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. BCK and BCI-algebras are two classes of logical algebras which are introduced by Imai and Is´eki (see [9, 10]). This notion originated from two different ways: (1) set theory, and (2) classical and non-classical propositional calculi. In set theory, we have the following simple relations: (A − B) − (A − C) ⊆ C − B and A − (A − B) ⊆ B. Several properties on BCK/BCI-algebras are investigated in the papers [11, 12, 13, 14] and [27]. There is a deep relation between BCK/BCI-algebras and posets. Today BCK/BCI-algebras have been studied by many authors and they have been applied to many branches of mathematics, such as group, functional analysis, probability theory, topology, fuzzy set theory, and so on. Jun and Park [24] studied applications of soft sets in ideal theory of BCK/BCI-algebras. Jun et al. [20, 22] introduced the notion of 2

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intersectional soft sets, and considered its applications to BCK/BCI-algebras. Also, Jun [16] discussed the union soft sets with applications in BCK/BCI-algebras. We refer the reader to the papers [1, 3, 5, 6, 7, 15, 18, 19, 21, 23, 30, 31, 35] for further information regarding algebraic structures/properties of soft set theory. In this paper, we introduce the notions of soft saturated values and soft dried values, and discuss their applications in BCK/BCI-algebras. Using these notions, we investigate several properties of energetic subsets. Using the concepts of intersectional (union) ideals, we explore some properties of right vanished (stable) subsets.

2

Preliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. Is´eki and was extensively investigated by several researchers. An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions: (I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), (II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0), (III) (∀x ∈ X) (x ∗ x = 0), (IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y). If a BCI-algebra X satisfies the following identity: (V) (∀x ∈ X) (0 ∗ x = 0), then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following axioms: (∀x ∈ X) (x ∗ 0 = x) ,

(2.1)

(∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) ,

(2.2)

(∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) ,

(2.3)

(∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y)

(2.4)

where x ≤ y if and only if x ∗ y = 0. A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. A subset I of a BCK/BCI-algebra 3

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X is called an ideal of X if it satisfies: 0 ∈ I,

(2.5)

(∀x ∈ X) (∀y ∈ I) (x ∗ y ∈ I ⇒ x ∈ I) .

(2.6)

We refer the reader to the books [8, 28] for further information regarding BCK/BCIalgebras. A soft set theory is introduced by Molodtsov [29], and C ¸ aˇgman et al. [4] provided new definitions and various results on soft set theory. In what follows, let U be an initial universe set and E be a set of parameters. We say that the pair (U, E) is a soft universe. Let P(U ) denote the power set of U and A, B, C, · · · ⊆ E. Definition 2.1 ([4, 29]). A soft set (f, A) over U is defined to be the set of ordered pairs (f, A) := {(x, f (x)) : x ∈ E, f (x) ∈ P(U )} , where f : E → P(U ) such that f (x) = ∅ if x ∈ / A. The function f is called an approximate function of the soft set (f, A). The subscript A in the notation f indicates that f is the approximate function of (f, A). Definition 2.2 ([16]). Let (U, E) = (U, X) where X is a BCK/BCI-algebra. A soft set (f, X) over U is called a union soft subalgebra over U if the following condition holds: (∀x, y ∈ X) (f (x ∗ y) ⊆ f (x) ∪ f (y)) .

(2.7)

Definition 2.3 ([16]). Let (U, E) = (U, X) where X is a BCK/BCI-algebra. A soft set (f, X) over U is called a union soft ideal over U if it satisfies: (∀x, y ∈ X) (f (0) ⊆ f (x) ⊆ f (x ∗ y) ∪ f (y)) .

(2.8)

Proposition 2.4 ([16]). Let (U, E) = (U, X) where X is a BCK/BCI-algebra. Every union soft ideal (f, X) over U satisfies the following condition: (∀x, y ∈ X) (x ≤ y ⇒ f (x) ⊆ f (y)) .

(2.9)

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3

Energetic subsets and soft saturated (dried) values (

In what follows, let

Q(U ) R(U )

)

( (∀A, B, C ∈ P(U ))

be the class of all subsets of U such that A ∩ B ⊆ C ⇒ A ⊆ C or B ⊆ C A ⊆ B ∪ C ⇒ A ⊆ B or A ⊆ C

) ,

and let (U, E) = (U, X) where X is a BCK/BCI-algebra unless otherwise specified. Definition 3.1 ([17]). A non-empty subset G of X is said to be S-energetic if it satisfies: (∀a, b ∈ X) (a ∗ b ∈ G ⇒ {a, b} ∩ G 6= ∅) .

(3.1)

Example 3.2 ([17]). Let X = {0, a, b, c, d} be a BCK-algebra with the following Cayley table: ∗ 0 a b c d

0 0 a b c d

a 0 0 a b a

b 0 0 0 a a

c 0 0 0 0 a

d 0 0 a b 0

The set G := {a, b, c} is an S-energetic subset of X, but H := {a, b} is not an S-energetic subset of X since d ∗ c = a ∈ H but {d, c} ∩ H = ∅. Definition 3.3 ([22]). A soft set (f, X) over U is called an int-soft subalgebra over U if it satisfies: (∀x, y ∈ X) (f (x ∗ y) ⊇ f (x) ∩ f (y)) .

(3.2)

Definition 3.4 ([22]). A soft set (f, X) over U is called an int-soft ideal over U if it satisfies: (∀x ∈ X) (f (x) ⊆ f (0)) ,

(3.3)

(∀x, y ∈ X) (f (x ∗ y) ∩ f (y) ⊆ f (x)) .

(3.4)

Lemma 3.5 ([22]). Every int-soft ideal (f, X) over U satisfies the following conditions: (1) (∀x, y ∈ X) (x ≤ y ⇒ f (y) ⊆ f (x)) . 5

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(2) (∀x, y, z ∈ X) (x ∗ y ≤ z ⇒ f (y) ∩ f (z) ⊆ f (x)) . Given a soft set (f, X) over U and α ∈ P(U ), we define useful subsets of X. fα⊆ := {x ∈ X | f (x) ⊆ α}, fα⊂ := {x ∈ X | f (x) ⊂ α}, fα⊇ := {x ∈ X | f (x) ⊇ α}, fα⊃ := {x ∈ X | f (x) ⊃ α}. Proposition 3.6. If (f, X) is an int-soft subalgebra over U with f : X → Q(U ), then
(∀α ∈ Q(U )) fα⊆ 6= ∅ ⇒ fα⊆ is an S-energetic subset of X . Proof. Let x, y ∈ X be such that x ∗ y ∈ fα⊆ . Then f (x) ∩ f (y) ⊆ f (x ∗ y) ⊆ α, and so f (x) ⊆ α or f (y) ⊆ α, that is, x ∈ fα⊆ or y ∈ fα⊆ . Hence {x, y} ∩ fα⊆ 6= ∅. Therefore fα⊆ is an S-energetic subset of X. Corollary 3.7. If (f, X) is an int-soft subalgebra over U with f : X → Q(U ), then (∀α ∈ Q(U )) (fα⊂ 6= ∅ ⇒ fα⊂ is an S-energetic subset of X) . Proof. Straightforward. The following example shows that the converse of Proposition 3.6 is not true. Example 3.8. Let (U, E) = (U, X) where X = {0, a, b, c, d} is a BCK-algebra as in Example 3.2. Let (f, X) be a soft set over U in which f is given as follows:    γ2 if x = 0, f : X → Q(U ), x 7→ γ3 if x = d,   γ1 if x ∈ {a, b, c}, where γ1 , γ2 , γ3 ∈ Q(U ) with γ1 ( γ2 ( γ3 . For any α ∈ Q(U ), if γ1 ⊆ α ( γ2 then fα⊆ = {a, b, c} is an S-energetic subset of X. But (f, X) is not an int-soft subalgebra over U since f (d ∗ d) = f (0) = γ2 + γ3 = f (d) ∩ f (d). Let (U, E) = (U, X) where X is a BCK-algebra. Then every int-soft ideal over U is an int-soft subalgebra over U (see [22]). Hence we have the following corollary.

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Corollary 3.9. Let (U, E) = (U, X) where X is a BCK-algebra. If (f, X) is an int-soft ideal over U with f : X → Q(U ), then
(∀α ∈ Q(U )) fα⊆ 6= ∅ ⇒ fα⊆ is an S-energetic subset of X . The following example shows that the converse of Corollary 3.9 is not true. Example 3.10. Consider the soft set (f, X) over U as in Example 3.8. For any α ∈ Q(U ), if γ1 ⊆ α ( γ2 then fα⊆ = {a, b, c} is an S-energetic subset of X. But (f, X) is not an int-soft ideal over U since f (d) = γ3 * γ2 = f (0). Definition 3.11. Let (f, X) be a soft set over U and α ∈ P(U ) with fα⊇ 6= ∅. Then α is called a soft saturated S-value for (f, X) if the following assertion is valid: (∀a, b ∈ X) (f (a ∗ b) ⊇ α ⇒ f (a) ∪ f (b) ⊇ α) .

(3.5)

Example 3.12. Let (U, E) = (U, X) where X = {0, 1, 2, 3} is a BCK-algebra with the following Cayley table: ∗ 0 1 2 3

0 0 1 2 3

1 0 0 2 1

2 0 1 0 3

3 0 0 2 0

Consider a soft set (f, X) over U in which f    γ1 f : X → P(U ), x 7→ γ2   γ3

is given as follows: if x = 0, if x = 1, if x ∈ {2, 3},

where γ1 , γ2 and γ3 are subsets of U with γ1 ( γ2 ( γ3 . Take α ∈ P(U ) with γ2 ( α ⊆ γ3 . Then fα⊇ = {2, 3}, and it is easy to check that α is a soft saturated S-value for (f, X). Example 3.13. Let (U, E) = (N, X) where N is the set of all natural numbers and X = {0, 1, 2, a, b} is a BCI-algebra with the following Cayley table: ∗ 0 1 2 a b

0 0 1 2 a b

1 0 0 2 a b

2 0 1 0 a b

a b b b 0 a

b a a a b 0 7

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Consider a soft set (f, X) over U in which f is given as follows:  N if x = 0,     2N if x ∈ {1, a}, f : X → P(U ), x 7→  2N − {2, 4, 6} if x = 2,    2N − {4, 6, 8} if x = b. If α = 2N − {4}, then fα⊇ = {0, 1, a} 6= ∅, f (2 ∗ b) = f (a) = 2N ⊇ α, and f (2) ∪ f (b) = 2N − {4, 6} + α. Hence α is not a soft saturated S-value for (f, X). Proposition 3.14. Let (f, X) be an int-soft subalgebra over U with f : X → R(U ). If α ∈ R(U ) is a soft saturated S-value for (f, X), then fα⊇ 6= ∅ ⇒ fα⊇ is an S-energetic subset of X. Proof. Let a, b ∈ X be such that a ∗ b ∈ fα⊇ . Then f (a ∗ b) ⊇ α, which implies from (3.5) that f (a) ∪ f (b) ⊇ α. Thus f (a) ⊇ α or f (b) ⊇ α, that is, a ∈ fα⊇ or b ∈ fα⊇ . Hence {a, b} ∩ fα⊇ 6= ∅. Therefore fα⊇ is an S-energetic subset of X. Theorem 3.15. Let (f, X) be a soft set over U and α ∈ P(U ) be such that fα⊇ 6= ∅. If (f, X) is a union soft subalgebra over U, then α is a soft saturated S-value for (f, X). Proof. Let x, y ∈ X be such that f (x ∗ y) ⊇ α. Then α ⊆ f (x ∗ y) ⊆ f (x) ∪ f (y), and so α is a soft saturated S-value for (f, X). Corollary 3.16. Let (U, E) = (U, X) where X is a BCK-algebra. Let (f, X) be a soft set over U and let α ∈ P(U ) be such that fα⊇ 6= ∅. If (f, X) is a union soft ideal over U, then α is a soft saturated S-value for (f, X). Definition 3.17. Let (f, X) be a soft set over U and α ∈ P(U ) with fα⊆ 6= ∅. Then α is called a soft dried S-value for (f, X) if the following assertion is valid: (∀a, b ∈ X) (f (a ∗ b) ⊆ α ⇒ f (a) ∩ f (b) ⊆ α) .

(3.6)

Example 3.18. Let (U, E) = (U, X) where X = {0, 1, 2, 3} is a BCK-algebra as in Example 3.12. Consider a soft set (f, X) over U in which f is given as follows:    γ2 if x = 0, f : X → P(U ), x 7→ γ1 if x = 1,   γ3 if x ∈ {2, 3}, 8

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where γ1 , γ2 and γ3 are subsets of U with γ1 ( γ2 ( γ3 . For any α ∈ P(U ) with γ1 ⊆ α ( γ2 , fα⊆ = {1} and α is a soft dried S-value for (f, X). Theorem 3.19. Let (f, X) be a union soft subalgebra over U with f : X → Q(U ). For any soft dried S-value α ∈ Q(U ) for (f, X), we have fα⊆ 6= ∅ ⇒ fα⊆ is an S-energetic subset of X. Proof. Let a, b ∈ X be such that a ∗ b ∈ fα⊆ . Then f (a ∗ b) ⊆ α, and so f (a) ∩ f (b) ⊆ α by (3.6). Thus f (a) ⊆ α or f (b) ⊆ α, i.e., a ∈ fα⊆ or b ∈ fα⊆ . Hence {a, b} ∩ fα⊆ 6= ∅. Therefore fα⊆ is an S-energetic subset of X. Corollary 3.20. Let (U, E) = (U, X) where X is a BCK-algebra. Let (f, X) be a union soft ideal over U with f : X → Q(U ). For any soft dried S-value α ∈ Q(U ) for (f, X), we have fα⊆ 6= ∅ ⇒ fα⊆ is an S-energetic subset of X. Theorem 3.21. Let (f, X) be an int-soft subalgebra over U and let α ∈ P(U ) be such that fα⊆ 6= ∅. Then α is a soft dried S-value for (f, X). Proof. Let a, b ∈ X be such that f (a ∗ b) ⊆ α. Then α ⊇ f (a ∗ b) ⊇ f (a) ∩ f (b), which shows that α is a soft dried S-value for (f, X). Corollary 3.22. Let (U, E) = (U, X) where X is a BCK-algebra. Let (f, X) be an intsoft ideal over U and let α ∈ P(U ) be such that fα⊆ 6= ∅. Then α is a soft dried S-value for (f, X). Definition 3.23 ([17]). Let X be a BCK/BCI-algebra. A non-empty subset G of X is said to be I-energetic if it satisfies: (∀x, y ∈ X) (y ∈ G ⇒ {x, y ∗ x} ∩ G 6= ∅) .

(3.7)

Example 3.24 ([17]). Let X = {0, 1, 2, a, b} be a BCI-algebra with the following Cayley table: ∗ 0 1 2 a b

0 0 1 2 a b

1 0 0 2 a b

2 0 1 0 a b

a b b b 0 a

b a a a b 0

It is routine to verify that G := {a, b} is an I-energetic subset of X. 9

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Example 3.25 ([17]). Let X = {0, 1, 2, 3, 4} be a BCK-algebra with the following Cayley table: ∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 1 1 4

2 0 0 0 1 4

3 0 0 1 0 4

4 0 1 2 3 0

It is routine to verify that G := {0, 1, 4} is an I-energetic subset of X. The notion of I-energetic subsets is independent to the notion of S-energetic subsets. In fact, the S-energetic subset G := {a, b, c} in Example 3.2 is not an I-energetic subset of X since {d, a ∗ d} ∩ G = ∅. Also, in Example 3.25, the I-energetic subset G := {0, 1, 4} is not an S-energetic subset of X since 3 ∗ 2 = 1 ∈ G and {3, 2} ∩ G = ∅ (see [17]). Definition 3.26. Let (f, X) be a soft set over U and α ∈ P(U ) with fα⊇ 6= ∅. Then α is called a soft saturated I-value for (f, X) if the following assertion is valid: (∀x, y ∈ X) (f (y) ⊇ α ⇒ f (y ∗ x) ∪ f (x) ⊇ α) .

(3.8)

Example 3.27. Let (U, E) = (U, X) where X = {0, 1, 2, 3} is a BCK-algebra as in Example 3.12. Consider a soft set (f, X) over U in which f is given as follows:    γ3 if x = 0, f : X → P(U ), x 7→ γ2 if x ∈ {1, 3},   γ1 if x = 2, where γ1 , γ2 and γ3 are subsets of U with γ1 ( γ2 ( γ3 . Put α ∈ P(U ) with γ1 ( α ⊆ γ2 . Then fα⊇ = {0, 1, 3}. It is easy to check that α is a soft saturated I-value for (f, X). Theorem 3.28. Let (U, E) = (U, X) where X is a BCK-algebra. If (f, X) is a union soft subalgebra over U, then every soft saturated I-value for (f, X) is a soft saturated S-value for (f, X). Proof. Since (f, X) is a union soft subalgebra over U, f (0) ⊆ f (x) for all x ∈ X. Let α ∈ P(U ) be a soft saturated I-value for (f, X). Assume that f (a ∗ b) ⊇ α for all a, b ∈ X. Using (3.8), (2.3), (III) and (V), we have α ⊆ f ((a ∗ b) ∗ a) ∪ f (a) = f ((a ∗ a) ∗ b) ∪ f (a) = f (0 ∗ b) ∪ f (a) = f (0) ∪ f (a) = f (a). Thus f (a) ∪ f (b) ⊇ f (a) ⊇ α and therefore α is a soft saturated S-value for (f, X). 10

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Corollary 3.29. Let (U, E) = (U, X) where X is a BCK-algebra. If (f, X) is a union soft ideal over U, then every soft saturated I-value for (f, X) is a soft saturated S-value for (f, X). Proof. Straightforward. The converse of Theorem 3.28 is not true as seen in the following example. Example 3.30. Let (U, E) = (U, X) where X = {0, a, b, c} is a BCK-algebra with the following Cayley table: ∗ 0 a b c

0 0 a b c

a 0 0 a c

b 0 0 0 c

c 0 a b 0

Consider a soft set (f, X) over U in which f    γ1 f : X → P(U ), x 7→ γ2   γ3

is given as follows: if x = 0, if x = a, if x ∈ {b, c},

where γ1 , γ2 and γ3 are subsets of U with γ1 ( γ2 ( γ3 . Take α ∈ P(U ) with γ2 ( α ⊆ γ3 . Then fα⊇ = {b, c}. It is easy to check that α is a soft saturated S-value for (f, X), but not a soft saturated I-value for (f, X) since f (b) ⊇ γ3 and f (b ∗ a) ∪ f (a) = f (a) = γ2 + γ3 . Theorem 3.31. Let (f, X) be an int-soft ideal over U with f : X → Q(U ). Then
(∀α ∈ Q(U )) fα⊆ 6= ∅ ⇒ fα⊆ is an I-energetic subset of X . Proof. Let x, y ∈ X be such that y ∈ fα⊆ . Then f (y) ⊆ α. It follows from (3.4) that α ⊇ f (y) ⊇ f (y ∗ x) ∩ f (x). Thus f (y ∗ x) ⊆ α or f (x) ⊆ α, i.e., y ∗ x ∈ fα⊆ or x ∈ fα⊆ . Hence {x, y ∗ x} ∩ fα⊆ 6= ∅, and so fα⊆ is an I-energetic subset of X. Theorem 3.32. Let (f, X) be an int-soft ideal over U with f : X → R(U ). If α ∈ R(U ) is a soft saturated I-value for (f, X), then fα⊇ 6= ∅ ⇒ fα⊇ is an I-energetic subset of X. 11

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Proof. Let x, y ∈ X be such that y ∈ fα⊇ . Then f (y) ⊇ α, which implies from (3.8) that f (y ∗ x) ∪ f (x) ⊇ α. Hence f (y ∗ x) ⊇ α or f (x) ⊇ α, that is, y ∗ x ∈ fα⊇ or x ∈ fα⊇ . Thus {x, y ∗ x} ∩ fα⊇ 6= ∅, and therefore fα⊇ is an I-energetic subset of X. Theorem 3.33. Let α ∈ P(U ) be such that fα⊇ 6= ∅. If (f, X) is a union soft ideal over U, then α is a soft saturated I-value for (f, X). Proof. Let x, y ∈ X be such that f (y) ⊇ α. Then α ⊆ f (y) ⊆ f (y ∗ x) ∪ f (x) by (2.8). Hence α is a soft saturated I-value for (f, X). Theorem 3.34. If (f, X) is a union soft ideal over U with f : X → R(U ), then
(∀α ∈ R(U )) fα⊇ 6= ∅ ⇒ fα⊇ is an I-energetic subset of X . Proof. Let x, y ∈ X be such that y ∈ fα⊇ . Then f (y) ⊇ α, and so α ⊆ f (y) ⊆ f (y ∗ x) ∪ f (x) by (2.8). Thus f (y∗x) ⊇ α or f (x) ⊇ α, i.e., y∗x ∈ fα⊇ or x ∈ fα⊇ . Hence {x, y∗x}∩fα⊇ 6= ∅, and so fα⊇ is an I-energetic subset of X. Definition 3.35. Let (f, X) be a soft set over U and α ∈ P(U ) with fα⊆ 6= ∅. Then α is called a soft dried I-value for (f, X) if the following assertion is valid: (∀x, y ∈ X) (f (y) ⊆ α ⇒ f (y ∗ x) ∩ f (x) ⊆ α) .

(3.9)

Example 3.36. Let (U, E) = (U, X) where X = {0, a, b, c} is a BCK-algebra as in Example 3.30. Consider the soft set (f, X) over U in Example 3.30. Take α ∈ P(U ) with γ2 ⊆ α ( γ3 . Then fα⊆ = {0, a}. It is easy to check that α is a soft dried I-value for (f, X). Theorem 3.37. Let (U, E) = (U, X) where X is a BCK-algebra. If (f, X) is an int-soft subalgebra over U, then every soft dried I-value for (f, X) is a soft dried S-value for (f, X). Proof. Since (f, X) is an int-soft subalgebra over U, f (0) ⊇ f (x) for all x ∈ X. Let α ∈ P(U ) be a soft dried I-value for (f, X). Assume that f (a ∗ b) ⊆ α for all a, b ∈ X. Using (3.9), (2.3), (III) and (V), we have α ⊇ f ((a ∗ b) ∗ a) ∩ f (a) = f ((a ∗ a) ∗ b) ∩ f (a) = f (0 ∗ b) ∩ f (a) = f (0) ∩ f (a) = f (a). Thus f (a) ∩ f (b) ⊆ f (a) ⊆ α and therefore α is a soft dried S-value for (f, X). 12

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Theorem 3.38. If (f, X) is a union soft ideal over U with f : X → R(U ), then
(∀α ∈ R(U )) fα⊇ 6= ∅ ⇒ fα⊇ is an I-energetic subset of X . Proof. Let x, y ∈ X be such that y ∈ fα⊇ . Then f (y) ⊃ α. It follows from (2.8) that α ⊆ f (y) ⊆ f (y ∗ x) ∪ f (x). Thus f (y ∗ x) ⊇ α or f (x) ⊇ α, i.e., y ∗ x ∈ fα⊇ or x ∈ fα⊇ . Hence {x, y ∗ x} ∩ fα⊇ 6= ∅, and so fα⊇ is an I-energetic subset of X. Theorem 3.39. Let (f, X) be a union soft ideal over U with f : X → Q(U ). If α ∈ Q(U ) is a soft dried I-value for (f, X), then fα⊆ 6= ∅ ⇒ fα⊆ is an I-energetic subset of X. Proof. Let x, y ∈ X be such that y ∈ fα⊆ . Then f (y) ⊆ α, which implies from (3.9) that f (y ∗ x) ∩ f (x) ⊆ α. Hence f (y ∗ x) ⊆ α or f (x) ⊆ α, that is, y ∗ x ∈ fα⊆ or x ∈ fα⊆ . Thus {x, y ∗ x} ∩ fα⊆ 6= ∅, and therefore fα⊆ is an I-energetic subset of X. Definition 3.40 ([17]). Let Q be a non-empty subset of a BCK/BCI-algebra X. Then Q is said to be right vanished if it satisfies: (∀a, b ∈ X) (a ∗ b ∈ Q ⇒ a ∈ Q) .

(3.10)

Q is said to be right stable if Q ∗ X := {a ∗ x | a ∈ Q, x ∈ X} ⊆ Q. Theorem 3.41. Let (U, E) = (U, X) where X is a BCK-algebra and let (f, X) be an int-soft ideal over U. Then fα⊇ and fα⊃ are right stable subsets of X for any α ∈ P(U ) with fα⊇ 6= ∅ 6= fα⊃ . Proof. Let x ∈ X and a ∈ fα⊇ . Then f (a) ⊇ α. Since a ∗ x ≤ a and (f, X) is an int-soft ideal over U, it follows from Lemma 3.5(1) that f (a ∗ x) ⊇ f (a) ⊇ α, i.e., a ∗ x ∈ fα⊇ . Hence fα⊇ is a right stable subset of X. Similarly, fα⊃ is a right stable subset of X. Theorem 3.42. Let (U, E) = (U, X) where X is a BCK-algebra. If (f, X) is a union soft ideal over U, then fα⊆ and fα⊂ are right stable subsets of X for any α ∈ P(U ) with fα⊆ 6= ∅ 6= fα⊂ .

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Proof. Let α ∈ P(U ) and x, a ∈ X be such that a ∈ fα⊆ . Then f (a) ⊆ α. Note that a ∗ x ≤ a, i.e., (a ∗ x) ∗ a = 0. Since (f, X) is a union soft ideal of X, it follows that f (a ∗ x) ⊆ f ((a ∗ x) ∗ a) ∪ f (a) = f (0) ∪ f (a) = f (a) ⊆ α. Hence a ∗ x ∈ fα⊆ , and so fα⊆ is a right stable subset of X. Similarly, fα⊂ is a right stable subset of X. Theorem 3.43. Let (U, E) = (U, X) where X is a BCK-algebra. If (f, X) is a union soft ideal over U, then fα⊇ and fα⊃ are right vanished subsets of X for any α ∈ P(U ) with fα⊇ 6= ∅ 6= fα⊃ . Proof. Let α ∈ P(U ) and a, b ∈ X be such that a ∗ b ∈ fα⊇ . Then f (a ∗ b) ⊇ α. Note that a ∗ b ≤ a, i.e., (a ∗ b) ∗ a = 0. Since (f, X) is a union soft ideal of X, it follows from (2.8), (2.3), (III) and (V) that α ⊆ f (a ∗ b) ⊆ f ((a ∗ b) ∗ a) ∪ f (a) = f ((a ∗ a) ∗ b) ∪ f (a) = f (0 ∗ b) ∪ f (a) = f (0) ∪ f (a) = f (a), and so a ∈ fα⊇ . Therefore fα⊇ is a right vanished subset of X. Similarly, fα⊃ is a right vanished subset of X.

4

Conclusions

We have introduced the notions of soft saturated values and soft dried values, and discussed their applications in BCK/BCI-algebras. Using these notions, we have investigated several properties of energetic subsets. Using the concepts of int-soft ideals (union ideals), we have explored some properties of right vanished (stable) subsets. Work is on going. Some important issues for further work are: 1. To develop strategies for obtaining more valuable results, 2. To apply these notions and results for studying related notions in other (soft) algebraic structures such as soft (semi-, near-, Γ-) rings, soft lattices, soft BL-algebras, soft R0 -algebras, soft MV-algebras and soft MTL-algebras, etc., 3. To study (fuzzy) rough set theoretical aspects based on this article. 14

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Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this article.

References [1] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl. 59 (2010) 3458–3463. [2] H. Akta¸s and N. C ¸ a˘gman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726– 2735. [3] A. O. Atag¨ un and A. Sezgin, Soft substructures of rings, fields and modules, Comput. Math. Appl. 61 (2011) 592–601. [4] N. C ¸ aˇgman, F. C ¸ itak and S. Engino˘glu, Soft set theory and uni-int decision making, Eur. J. Oper. Res. 207 (2010) 848–855. [5] N. C ¸ aˇgman and S. Engino˘glu, FP-soft set theory and its applications, Ann. Fuzzy Math. Inform. 2 (2011) 219–226. [6] F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math. Inform. 2 (2011) 69–80. [7] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621–2628. [8] Y. Huang, BCI-algebra, Science Press, Beijing, 2006. [9] Y. Imai and K. Is´eki, On axiom systems of propositional calculi, Proc. Jpn. Acad. 42 (1966), 19–21. [10] K. Is´eki, An algebra related with a propositional calculus, Proc. Japan Acad. 42 (1966), 26–29. [11] K. Is´eki, Some examples of BCI-algebras, Math. Seminar Notes 8 (1980), 237–240. [12] K. Is´eki, On ideals in BCK-algebras, Math. Seminar Notes 3 (1975), 1–12.

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[13] K. Is´eki, On some ideals in BCK-algebras, Math. Seminar Notes 3 (1975), 65–70. [14] K. Is´eki and S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), 1–26. [15] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408–1413. [16] Y. B. Jun, Union soft sets with applications in BCK/BCI-algebras, Bull. Korean Math. Soc. 50(6) (2013) 1937–1956. [17] Y. B. Jun, S. S. Ahn and E. H. Roh, Energetic subsets and permeable values with applications in BCK/BCI-algebras, Appl. Math. Sci. (Ruse) 7(89-92) (2013) 4425– 4438. [18] Y. B. Jun, H. S. Kim and J. Neggers, Pseudo d-algebras, Inform. Sci. 179 (2009) 1751–1759. [19] Y. B. Jun, K. J. Lee and A. Khan, Soft ordered semigroups, Math. Logic Q. 56 (2010) 42–50. [20] Y. B. Jun, M. S. Kang and K. J. Lee, Intersectional soft sets and applications to BCK/BCI-algebras, Comm. Korean Math. Soc. 28(1) (2013) 11–24. [21] Y. B. Jun, K. J. Lee and C. H. Park, Soft set theory applied to ideals in d-algebras, Comput. Math. Appl. 57 (2009) 367–378. [22] Y. B. Jun, K. J. Lee and E. H. Roh, Intersectional soft BCK/BCI-ideals, Ann. Fuzzy Math. Inform. 4(1) (2012) 1–7. [23] Y. B. Jun, K. J. Lee and J. Zhan, Soft p-ideals of soft BCI-algebras, Comput. Math. Appl. 58 (2009) 2060–2068. [24] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCIalgebras, Inform. Sci. 178 (2008) 2466–2475. [25] P. K. Maji, R. Biswas and A. R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555–562. [26] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077–1083.

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[27] J. Meng, Commutative ideals in BCK-algerbas, Pure Appl. Math. (in China) 9 (1991), 49–53. [28] J. Meng and Y. B. Jun, BCK-algebras, Kyungmoon Sa Co. Seoul, 1994. [29] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31. ¨ urk, Soft WS-algebras, Commun. Korean Math. [30] C. H. Park, Y. B. Jun and M. A. Ozt¨ Soc. 23 (2008) 313–324. [31] S. Z. Song, K. J. Lee and Y. B. Jun, Intersectional soft sets applied to subalgebras/ideals in BCK/BCI-algebras, Algeb. Represent. Theory (submitted). [32] L. A. Zadeh, From circuit theory to system theory, Proc. Inst. Radio Eng. 50 (1962) 856–865. [33] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–353. [34] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) - an outline, Inform. Sci. 172 (2005) 1–40. [35] J. Zhan and Y. B. Jun, Soft BL-algebras based on fuzzy sets, Comput. Math. Appl. 59 (2010) 2037–2046.

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Boundedness from Below of Composition Followed by Differentiation on Bloch-type Spaces Xiaosong Liu and Songxiao Li Abstract. Let φ be an analytic self-map of the unit disk D. The composition followed by differentiation operator, denoted by DCφ , is defined by DCφ f (z) = (f ◦ φ)′ = f ′ (φ)φ′ , f ∈ H(D). In this paper, under some assumption conditions, we give a necessary and sufficient condition for the operator DCφ : B α → B β to be bounded below. MSC 2000: 47B33, 30H30. Keywords: Bloch-type space, composition operator, differentiation operator, bounded below.

1

Introduction

Let D denote the open unit disk in the complex plane C and ∂D be its boundary. Let H(D) be the space of analytic functions on D. For 0 < α < ∞, an f ∈ H(D) is said to belong to Bloch-type space( or α-Bloch space), denoted by Bα = Bα (D), if ∥f ∥α = sup(1 − |z|2 )α |f ′ (z)| < ∞. z∈D

It is easy to check that Bα is a Banach space with the norm ∥f ∥Bα = |f (0)| + ∥f ∥α . When α = 1, B 1 = B is the well-known Bloch space. Throughout the paper, S(D) denotes the set of all analytic self-maps of D. Associated with φ ∈ S(D) is the composition operator Cφ defined by Cφ f = f ◦ φ for f ∈ H(D). The main subject in the study of composition operators is to describe operator theoretic properties of Cφ in terms of function theoretic properties of φ. See [4] and the references therein for the study of the composition operator. See [7, 8, 9, 10, 11, 12, 13, 14, 15] for the study of composition operators on Bloch-type spaces. Let D be the differentiation operator and φ ∈ S(D). The composition followed by differentiation operator, denoted by DCφ , is defined as follows. DCφ f (z) = (f ◦ φ)′ = f ′ (φ)φ′ , f ∈ H(D).

1

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In [7], the authors studied the boundedness and compactness of DCφ between Bloch-type spaces. For example, they obtained the following results: Theorem A. [7] Let α, β > 0 and φ ∈ S(D). Then DCφ : B α → B β is bounded if and only if M1 := sup z∈D

|φ′ (z)|2 (1 − |z|2 )β < ∞, (1 − |φ(z)|2 )α+1

M2 := sup z∈D

|φ′′ (z)|(1 − |z|2 )β < ∞. (1 − |φ(z)|2 )α

Theorem B. [7] Let α, β > 0, φ ∈ S(D) such that DCφ : Bα → B β is bounded. Then DCφ : B α → B β is compact if and only if |φ′ (z)|2 (1 − |z|2 )β =0 |φ(z)|→1 (1 − |φ(z)|2 )α+1

(1)

|φ′′ (z)|(1 − |z|2 )β = 0. (1 − |φ(z)|2 )α |φ(z)|→1

(2)

lim

and lim

Recall that the operator DCφ : B α → B β is said to be bounded, if there exists a C > 0, such that ∥DCφ f ∥Bβ ≤ C∥f ∥Bα for all f ∈ B α . A bounded operator DCφ : B α → B β is said to be bounded below, if there exists a δ > 0, such that ∥DCφ f ∥Bβ ≥ δ∥f ∥Bα for all f ∈ B α . We notice that DCφ : Bα → B β is bounded below if and only if DCφ has closed range. The boundedness from below of composition operator Cφ on B was studied by Gathage, Zheng and Zorboska in terms of sampling sets, see [6]. More precisely, they proved that Cφ is bounded below on B if and only if there exists ε > 0, such that Gε = φ(Ωε ) is a sampling set for B, where Ωε = {z ∈ D :

(1 − |z|2 )|φ′ (z)| ≥ ε}. 1 − |φ(z)|2

See [1, 2, 5, 6] for other characterizations of the boundedness from below of composition operator on B. The boundedness from below of multiplication operator on Bloch-type spaces was studied in [3]. In this paper, we give a necessary and sufficient condition for the boundedness from below of the operator DCφ : B α → B β , i.e., we obtain the following results. Theorem 1. Let 0 < α, β < ∞. Let φ ∈ S(D) such that φ′ (z) ̸≡ 0 and (2) hold′ (z)|2 (1−|z|2 )β s. Suppose that DCφ is bounded from B α to Bβ and limφ(z)→∂D |φ(1−|φ(z)| 2 )α+1 exists. Then DCφ : B α → B β is bounded below if and only if |φ′ (z)|2 (1 − |z|2 )β > 0. φ(z)→∂D (1 − |φ(z)|2 )α+1 lim

(3)

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Theorem 2. Let 0 < α, β < ∞. Let φ ∈ S(D) such that φ′ (z) ̸≡ 0 and (1) hold′′ (z)|(1−|z|2 )β s. Suppose that DCφ is bounded from B α to B β and limφ(z)→∂D |φ(1−|φ(z)| 2 )α β α exists. Then DCφ : B → B is bounded below if and only if |φ′′ (z)|(1 − |z|2 )β > 0. (1 − |φ(z)|2 )α φ(z)→∂D lim

(4)

Throughout the paper, we denote by C a positive constant which may differ from one occurrence to the next. We say that P ≼ Q if there exists a constant C such that P ≤ CQ. The symbol P ≈ Q means that P ≼ Q ≼ P .

2

Proof of main results

In this section, we prove the main results in this paper. For this purpose, we need the following lemma. Lemma 1. Let φ ∈ S(D) such that φ′ (z) ̸≡ 0. Suppose that β > 0 and fn ∈ H(D) for n = 1, 2, · · · . If ∥DCφ fn ∥Bβ → 0 as n → ∞, then fn′ ◦ φ → 0 as n → ∞, locally uniformly in D. Proof. The proof is similar to the proof of Lemma 2.9 in [3]. For the convenience of the readers, we give the detail of the proof. Since φ′ (z) ̸≡ 0, then for any r0 ∈ (0, 1), there exists an r′ such that r0 < r′ < 1 and φ′ (z) ̸= 0 for |z| = r′ . By Lemma 2.2 of [3], |φ′ (z)fn′ (φ(z))| ≤ Cβ,r′ ∥φ′ fn′ ◦ φ∥Bβ for n = 1, 2, · · · , and |z| = r′ . Let δ = min|z|=r′ |φ′ (z)| > 0. Then we have |fn′ (φ(z))| ≤ (Cβ,r′ /δ)∥φ′ fn′ ◦ φ∥Bβ , for n = 1, 2, · · · , and |z| = r′ . By Maximum principle and the assumption that ∥φ′ fn′ ◦ φ∥Bβ → 0 as n → ∞, we have fn′ ◦ φ → 0 as n → ∞, uniformly for |z| ≤ r′ . The proof of the lemma is finished. Lemma 2. [16] Let m be a positive integer and α > 0. Then f ∈ B α if and only if sup(1 − |z|2 )m+α−1 |f (m) (z)| < ∞. z∈D

Moreover, ∥f ∥Bα ≈

m−1 ∑

|f (j) (0)| + sup(1 − |z|2 )m+α−1 |f (m) (z)|. z∈D

j=0

Proof of Theorem 1. Necessity. By the assumption that DCφ is bounded from B α to B β , from Theorem A, we have M1 := sup z∈D

|φ′ (z)|2 (1 − |z|2 )β < ∞. (1 − |φ(z)|2 )α+1 3

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Hence, M3 := sup(1 − |z|2 )β |φ′ (z)|2 < sup z∈D

z∈D

|φ′ (z)|2 (1 − |z|2 )β < ∞. (1 − |φ(z)|2 )α+1

Assume that (3) does not hold, i.e., for any η1 > 0, there exists a δ1 > 0 such that |φ′ (z)|2 (1 − |z|2 )β < η1 (1 − |φ(z)|2 )α+1

(5)

for |φ(z)| > δ1 . Let an ∈ D such that φ(an ) → ∂D as n → ∞(n = 1, 2, · · · ). Set fn (z) =

1 − |φ(an )|2

1

αφ(an ) (1 − φ(an )z)α

.

It is easy to check that 1 ≤ ∥fn ∥α ≤ 2α+1 . Since DCφ from Bα to B β is bounded below, then, there exists a δ > 0 such that ∥DCφ fn ∥Bβ ≥ δ∥fn ∥Bα ≥ δ∥fn ∥α ≥ δ.

(6)

On the other hand, we obtain ( )′ ∥DCφ fn ∥Bβ = |φ′ (0)fn′ (φ(0))| + sup(1 − |z|2 )β | φ′ · fn′ ◦ φ (z)| = |φ ≤

′

(0)fn′ (φ(0))|

z∈D ′2

+ sup(1 − |z| ) |φ (z)fn′′ (φ(z)) + φ′′ (z)fn′ (φ(z))| 2 β

z∈D

|φ′ (0)fn′ (φ(0))| + sup(1 − |z|2 )β |φ′2 (z)fn′′ (φ(z))| + sup(1 − |z|2 )β |φ′′ (z)fn′ (φ(z))| z∈D

= |φ

′

(0)fn′ (φ(0))|

z∈D

+ E1 + E2 ,

where E1 = sup(1 − |z|2 )β |φ′2 (z)fn′′ (φ(z))| and E2 = sup(1 − |z|2 )β |φ′′ (z)fn′ (φ(z))|. z∈D

z∈D

First we estimate E1 . For any z ∈ D such that |φ(z)| > δ1 , by (5), we have (1 − |z|2 )β |φ′2 (z)fn′′ (φ(z))| = ≤

(α + 1)(1 − |z|2 )β |φ′ (z)|2 |φ(an )|

1 − |φ(an )|2

|1 − φ(an )φ(z)|α+2 |φ′ (z)|2 (1 − |z|2 )β (1 − |φ(an )|2 )(1 − |φ(z)|2 )α+1 (α + 1) (1 − |φ(z)|2 )α+1 |1 − φ(an )φ(z)|α+2 |φ′ (z)|2 (1 − |z|2 )β (1 + |φ(an )|)(1 + |φ(z)|)α+1 (1 − |φ(z)|2 )α+1

≤

(α + 1)

≤

(α + 1)2α+2 η1 .

(7)

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For any η2 > 0, there exists a positive integer N , 1 − |φ(an )|2 < η2 holds for all n > N . For any z ∈ D such that |φ(z)| ≤ δ1 and n > N , we deduce (1 − |z|2 )β |φ′2 (z)fn′′ (φ(z))| = (α + 1)(1 − |z|2 )β |φ′ (z)|2 |φ(an )|

1 − |φ(an )|2

|1 − φ(an )φ(z)|α+2 1 − |φ(an )|2 ≤ (α + 1)(1 − |z|2 )β |φ′ (z)|2 (1 − |φ(z)|)α+2 η2 ≤ (α + 1)(1 − |z|2 )β |φ′ (z)|2 (1 − δ1 )α+2 M3 η2 . ≤ (α + 1) (1 − δ1 )α+2

(8)

From (7) and (8), we have E1

≤

sup (1 − |z|2 )|φ′ (z)|2 |fn′′ (φ(z))|

|φ(z)|>δ1

+

sup (1 − |z|2 )|φ′ (z)|2 |fn′′ (φ(z))|

|φ(z)|≤δ1

< (α + 1)2α+2 η1 + (α + 1)

M3 η2 , (1 − δ1 )α+2

as

n > N.

By the arbitrary of η1 and η2 , we see that E1 → 0 as n → ∞. Next we estimate E2 . From (2), for any η3 > 0, there exists a δ2 > 0 such that |φ′′ (z)|(1 − |z|2 )β < η3 , when |φ(z)| > δ2 . (1 − |φ(z)|2 )α For any z ∈ D such that |φ(z)| > δ2 , (1 − |z|2 )β |φ′′ (z)fn′ (φ(z))|

=

(1 − |z|2 )β |φ′′ (z)|

1 − |φ(an )|2

|1 − φ(an )φ(z)|α+1 |φ′′ (z)|(1 − |z|2 )β ≤ 2α (1 + |φ(an )|) (1 − |φ(z)|2 )α < 2α+1 η3 .

(9)

For any z ∈ D such that |φ(z)| ≤ δ2 and n > N , we have (1 − |z|2 )β |φ′′ (z)fn′ (φ(z))|

=

(1 − |z|2 )β |φ′′ (z)|

1 − |φ(an )|2

|1 − φ(an )φ(z)|α+1 |φ (z)|(1 − |z| ) 1 − |φ(an )|2 ≤ 2α (1 − |φ(z)|2 )α 1 − |φ(z)| M η 2 2 ≤ 2α . (10) 1 − δ2 ′′

2 β

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Then, E2

≤

sup (1 − |z|2 )|φ′′ (z)fn′ (φ(z))| +

|φ(z)|>δ2

≤ 2α+1 η3 + 2α

M2 η2 , 1 − δ2

sup (1 − |z|2 )|φ′′ (z)fn′ (φ(z))|

|φ(z)|≤δ2

as n > N.

Since η2 and η3 are arbitrary, then E2 → 0 as n → ∞. In addition, |φ′ (0)fn′ (φ(0))| → 0 as n → ∞. Therefore, ∥DCφ fn ∥Bβ → 0 as n → ∞, which contradicts (6). Therefore (3) holds. Sufficiency. Now assume that (3) holds. Denoted ϵ=

|φ′ (z)|2 (1 − |z|2 )β > 0. φ(z)→∂D (1 − |φ(z)|2 )α+1 lim

(11)

Suppose on the contrary that DCφ is not bounded below from B α to B β . Then, there exists a sequence {fn } ⊂ B α such that ∥fn ∥Bα = 1 for n = 1, 2, · · · , and sup(1 − |z|2 )β |(fn′ (φ)φ′ )′ (z)| ≤ ∥DCφ fn ∥Bβ → 0 z∈D

as n → ∞. By Lemma 1, fn′ ◦ φ → 0 and hence fn′ → 0 as n → ∞, locally uniformly in D. By Cauchy’s estimate we see that fn′′ → 0 as n → ∞, locally uniformly in D. Let zn ∈ D be a sequence such that (1 − |φ(zn )|2 )α+1 |fn′′ (φ(zn ))| ≥

1 . 2

(12)

Since for every n = 1, 2, · · · , ∥fn ∥Bα = 1, we see that the above {zn } exist by Lemma 2. Then φ(zn ) → ∂D as n → ∞. Hence by (2) and (11), we get |φ′′ (zn )|(1 − |zn |2 )β →0 (1 − |φ(zn )|2 )α

(13)

|φ′ (zn )|2 (1 − |zn |2 )β ≥ ϵ/2 (1 − |φ(zn )|2 )α+1

(14)

and

for sufficiently large n, respectively. Therefore, by (12), (13), (14) and Lemma 2, we obtain ∥DCφ fn ∥Bβ

≥ (1 − |z|2 )β |(φ′ · fn′ ◦ φ)′ (z)| |φ′ (zn )|2 (1 − |zn |2 )β (1 − |φ(zn )|2 )α+1 |f ′′ (φ(zn ))| (1 − |φ(zn )|2 )α+1 |φ′′ (zn )|(1 − |zn |2 )β − (1 − |φ(zn )|2 )α |f ′ (φ(zn ))| (1 − |φ(zn )|2 )α ϵ ≥ , as n → ∞. 4

≥

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We arrive at a contradiction. Therefore DCφ is bounded below from Bα to B β . This completes the proof of this theorem. Proof of Theorem 2. The proof of Theorem 2 is similar to the proof of Theorem 1. Hence we omit the details. Acknowledgement. This work was supported by NSF of China (No.11471143) and NSF of Guangdong, China (No.S2013010011978).

References [1] H. Chen, Boundness from below of composition operators on the Bloch spaces, Sci. China Ser. A 46 (2003), 838–846. [2] H. Chen and P. Gauthier, Boundness from below of composition operators on the α-Bloch spaces, Canad. Math. Bull Ser. 51 (2008), 195–204. [3] H. Chen and M. Zhang, Boundness from below of multiplication operators between αBloch spaces, Canad. Math. Bull Ser. 53 (2010), 23–36. [4] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Math., CRC Press, Boca Raton, 1995. [5] P. Ghatage, J. Yan and D. Zheng, Composition operators with close range on the Bloch spaces, Proc. Amer. Math. Soc. 129 (2001), 2309–2044. [6] P. Ghatage, D. Zheng and N. Zorboska, Sampling sets and close range composition operators on the Bloch space, Proc. Amer. Math. Soc. 133 (2004), 1371–1377. [7] S. Li and S. Stevi´ c, Composition followed by differentiation between Bloch type spaces, J. Comput. Anal. Appl. 9 (2007), 195–205. [8] S. Li and S. Stevi´ c, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338 (2008), 1282–1295. [9] S. Li and S. Stevi´ c, Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput. 206 (2008), 825–831. [10] Z. Lou, Composition operators on Bloch type spaces, Analysis (Munich), 23 (2003), 81–95. [11] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679–2687. [12] S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch-type spaces, Rocky. Mountain J. Math. 33 (2003), 191–215. [13] M. Tjani, Compact composition operators on some M¨ obius invariant Banach space, PhD dissertation, Michigan State University, 1996. [14] H. Wulan, D. Zheng and K. Zhu, Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc. 137 (2009), 3861–3868. [15] W. Yang, Products of composition differentiation operators from QK (p, q) spaces to Bloch-type spaces, Abstr. Appl. Anal. Volume 2009, Article ID 741920, 14 pages. [16] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. Xiaosong Liu: Department of Mathematics, Jiaying University, Meizhou 514015, China. Email: [email protected] Songxiao Li: Department of Mathematics, Jiaying University, Meizhou 514015, China. Email: [email protected] ⋆Corresponding author: Songxiao Li

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Uni-soft filters of BE-algebras Young Bae Juna and N. O. Alshehrib,∗ a

Department of Mathematics Education Gyeongsang National University, Jinju 660-701, Korea b

Department of Mathematics, Faculty of science for girls King Abdulaziz University, Jeddah, KSA Abstract Further properties of uni-soft filters in a BE-algebra are investigated. The problem of classifying uni-soft filters by their τ -exclusive filter is solved. New uni-soft filter from old one is established. Keywords: (Self distributive) BE-algebra, Filter, Uni-soft filter, 2010 Mathematics Subject Classification. 06F35, 03G25, 06D72.

1

Introduction

To solve complicated problem in economics, engineering, and environment, we can’t successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties can’t be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [8]. Maji et al. [7] and Molodtsov [8] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [8] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from *Corresponding author. e-mail: [email protected] (Y. B. Jun), [email protected] (N. O. Alshehri)

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the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [7] described the application of soft set theory to a decision making problem. Maji et al. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. C ¸ aˇgman et al. [3] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision making method based on FP-soft set theory, and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Feng [4] considered the application of soft rough approximations in multicriteria group decision making problems. Akta¸s and C ¸ a˘gman [2] studied the basic concepts of soft set theory, and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. As a generalization of a BCK-algebra, Kim and Kim [6] introduced the notion of a BE-algebra, and investigated several properties. In [1], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras. Jun et al. [5] introduced the notion of uni-soft filter of a BE-algebra, and investigated their properties. They considered characterizations of a uni-soft filter, and provided conditions for a soft set to be a uni-soft filter. In this paper, we investigate further properties of a uni-soft filter. We solve the problem of classifying uni-soft filters by their τ -exclusive filters. We make a new uni-soft filter from old one.

2

Preliminaries

Let K(τ ) be the class of all algebras of type τ = (2, 0). By a BE-algebra (see [6]) we mean a system (X; ∗, 1) ∈ K(τ ) in which the following axioms hold: (∀x ∈ X) (x ∗ x = 1),

(2.1)

(∀x ∈ X) (x ∗ 1 = 1),

(2.2)

(∀x ∈ X) (1 ∗ x = x),

(2.3)

(∀x, y, z ∈ X) (x ∗ (y ∗ z) = y ∗ (x ∗ z)). (exchange)

(2.4)

A relation “≤” on a BE-algebra X is defined by (∀x, y ∈ X) (x ≤ y ⇐⇒ x ∗ y = 1).

(2.5)

A BE-algebra (X; ∗, 1) is said to be transitive (see [1]) if it satisfies: (∀x, y, z ∈ X) (y ∗ z ≤ (x ∗ y) ∗ (x ∗ z)).

(2.6)

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A BE-algebra (X; ∗, 1) is said to be self distributive (see [6]) if it satisfies: (∀x, y, z ∈ X) (x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z)).

(2.7)

Every self distributive BE-algebra (X; ∗, 1) satisfies the following properties: (∀x, y, z ∈ X) (x ≤ y ⇒ z ∗ x ≤ z ∗ y, y ∗ z ≤ x ∗ z) .

(2.8)

(∀x, y ∈ X) (x ∗ (x ∗ y) = x ∗ y) ,

(2.9)

(∀x, y, z ∈ X) (x ∗ y ≤ (z ∗ x) ∗ (z ∗ y)) ,

(2.10)

(∀x, y, z ∈ X) ((x ∗ y) ∗ (x ∗ z) ≤ x ∗ (y ∗ z)) .

(2.11)

Note that every self distributive BE-algebra is transitive, but the converse is not true in general (see [1]). Let (X; ∗, 1) be a BE-algebra and let F be a non-empty subset of X. Then F is a filter of X (see [6]) if (F1) 1 ∈ F ; (F2) (∀x, y ∈ X)(x ∗ y, x ∈ F ⇒ y ∈ F ). A soft set theory is introduced by Molodtsov [8]. In what follows, let U be an initial universe set and X be a set of parameters. Let P(U ) denotes the power set of U and A, B, C, · · · ⊆ ³X. ´ A soft set F˜ , A of X over U is defined to be the set of ordered pairs ³ ´ n³ ´ o F˜ , A := x, F˜ (x) : x ∈ X, F˜ (x) ∈ P(U ) , ˜ where F˜ : X → P(U / A. ³ ) such ´ that F (x) = ∅ if x ∈ ³ ´ ˜ ˜ For a soft set F , A of X and a subset τ of U, the τ -exclusive set of F , A , denoted ³ ´ by eA F˜ ; τ , is defined to be the set ³ ´ n o eA F˜ ; τ := x ∈ A | τ ⊇ F˜ (x) . ³ ´ ³ ´ ³ ´ ³ ´ ˜ ˜ ˜ ˜ For any soft sets F , X and G , X of X, we call F , X a soft subset of G , X , ³ ´ ³ ´ ³ ´ ˜ G˜, X , if F˜ (x) ⊆ G˜(x) for all x ∈ X. The soft union of F˜ , X denoted by F˜ , X ⊆ ³ ´ ³ ´ ³ ´ ³ ´ ˜ G˜, X , is defined to be the soft set F˜ ∪ ˜ G˜, X of and G˜, X , denoted by F˜ , X ∪ ˜ G˜ is defined by X over U in which F˜ ∪ ³ ´ ˜ ˜ ˜ F ∪ G (x) = F˜ (x) ∪ G˜(x) for all x ∈ M. 3

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³ ´ ³ ´ ³ ´ ³ ´ ˜ G˜, X , is defined The soft intersection of F˜ , X and G˜, X , denoted by F˜ , X ∩ ³ ´ ˜ ˜ ˜ ˜ G˜ is defined by to be the soft set F ∩ G , M of X over U in which F˜ ∩ ³

3

´ ˜ G˜ (x) = F˜ (x) ∩ G˜(x) for all x ∈ S. F˜ ∩

Uni-soft filters

In what follows, we take a BE-algebra X, as a set of parameters unless otherwise specified. ³ ´ ˜ Definition 3.1 ([5]). A soft set F , X of X over U is called a uni-soft filter of X if it satisfies: ³ ´ (∀x ∈ X) F˜ (1) ⊆ F˜ (x) , (3.1) ³ ´ (∀x, y ∈ X) F˜ (y) ⊆ F˜ (x ∗ y) ∪ F˜ (x) . (3.2) We make a new uni-soft filter from old one. ³ ´ ˜ Lemma 3.2 ([5]). For a soft set F , X over U , the following are equivalent. ³ (i)

F˜ , X

´ is a uni-soft filter of X over U .

³ ´ ³ ´ (ii) The τ -exclusive set eX F˜ ; τ is a filter of X for all τ ∈ P(U ) with eX F˜ ; τ 6= ∅. ³ ´ Theorem 3.3. For a soft set F˜ , X over U , define a soft set (F˜ ∗ , X) over U by ( F˜ ∗ : X → P(U ), x 7→

F˜ (x) U

if x ∈ eX (F˜ ; τ ), otherwise

³

´ ˜ where τ is a nonempty subset of U . If F , X is a uni-soft filter of X over U , then so is (F˜ ∗ , X). ³ ´ Proof. Assume that F˜ , X is a uni-soft filter of X over U . Then eX (F˜ ; τ )(6= ∅) is a filter of X over U for all τ ⊆ U by Lemma 3.2. Hence 1 ∈ eX (F˜ ; τ ), and so F˜ ∗ (1) = F˜ (1) ⊆ F˜ (x) ⊆ F˜ ∗ (x) for all x ∈ X. Let x, y ∈ X. If x ∗ y ∈ eX (F˜ ; τ ) and x ∈ eX (F˜ ; τ ), then y ∈ eX (F˜ ; τ ). Hence F˜ ∗ (y) = F˜ (y) ⊆ F˜ (x ∗ y) ∪ F˜ (x) = F˜ ∗ (x ∗ y) ∪ F˜ ∗ (x). 4

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If x ∗ y ∈ / eX (F˜ ; τ ) or x ∈ / eX (F˜ ; τ ), then F˜ ∗ (x ∗ y) = U or F˜ ∗ (x) = U. Thus F˜ ∗ (y) ⊆ U = F˜ ∗ (x ∗ y) ∪ F˜ ∗ (x). Therefore (F˜ ∗ , X) is a uni-soft filter of X over U . ³ ´ ³ ´ Theorem 3.4. If F˜ , X and G˜, X are uni-soft filters of X over U , then the soft ³ ´ ³ ´ ³ ´ ³ ´ ˜ G˜, X of F˜ , X and G˜, X is a uni-soft filter of X over U . union F˜ , X ∪ Proof. For any x ∈ X, we have ³ ´ ˜ G˜ (1) = F˜ (1) ∪ G˜(1) ⊆ F˜ (x) ∪ G˜(x) = (F˜ ∪ ˜ G˜)(x). F˜ ∪ Let x, y ∈ X. Then ³ ´ ˜ G˜ (y) = F˜ (y) ∪ G˜(y) F˜ ∪ ³ ´ ³ ´ ˜ ˜ ˜ ˜ ⊆ F (x ∗ y) ∪ F (x) ∪ G (x ∗ y) ∪ G (x) ³ ´ ³ ´ = F˜ (x ∗ y) ∪ G˜(x ∗ y) ∪ F˜ (x) ∪ G˜(x) ³ ´ ³ ´ ˜ G˜ (x ∗ y) ∪ F˜ ∪ ˜ G˜ (x). = F˜ ∪ ³ ´ ³ ´ ˜ G˜, X is a uni-soft filter of X over U . Hence F˜ , X ∪ The following example shows that the soft intersection of uni-soft filters of X over U may not be a uni-soft filter of X over U Example 3.5. Consider a BE-algebra X = {1, a, b, c, d, 0} with the Cayley table which is given in Table 1 (see [1]). Let E = ³X be the and U = Z be the initial universe set. Define two ´ set³of parameters ´ soft sets F˜ , X and G˜, X over U as follows: (

4N if x ∈ {1, c} 2Z if x ∈ {a, b, d, 0}

F˜ : X → P(U ), x 7→ and ( G˜ : X → P(U ), x 7→

8N if x ∈ {1, a, b} 4Z if x ∈ {c, d, 0}

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Table 1: Cayley table for the “∗”-operation

∗ 1 a b c d 0

1 1 1 1 1 1 1

a a 1 1 a 1 1

b b a 1 b a 1

c c c c 1 1 1

d d c c a 1 1

0 0 d c b a 1

³

´ ³ ´ ˜ ˜ respectively. It is easy to check that F , X and G , X are uni-soft filters of X over ³ ´ ³ ´ ˜ G˜, X = (F˜ ∩ ˜ G˜, X) is not a uni-soft filter of X over U , since U . But F˜ , X ∩ ˜ G˜)(d) = F˜ (d) ∩ G˜(d) = 2Z ∩ 4Z (F˜ ∩ = 4Z * 4N = 8N ∪ 4N = (F˜ (a) ∩ G˜(a)) ∪ (F˜ (c) ∩ G˜(c)) ˜ G˜)(a) ∪ (F˜ ∩ ˜ G˜)(c) = (F˜ ∩ ˜ G˜)(c ∗ d) ∪ (F˜ ∩ ˜ G˜)(c). = (F˜ ∩ ³ ´ Theorem 3.6. Let F˜ , X be a uni-soft filter of X. Let τ1 and τ2 be subsets of U such ³ ´ ³ ´ that τ1 ) τ2 . If the τ1 -exclusive set of F˜ , X is equal to the τ2 -exclusive set of F˜ , X , then there is no x ∈ X such that τ1 ) F˜ (x) ) τ2 . Proof. Straightforward. The converse of Theorem 3.6 is not true in general as seen in the following example. Example 3.7. Consider a BE-algebra X = {1, a, b, c} with the Cayley table which is given in Table 2. ³ ´ ˜ Given U = X, consider a soft set F , X of X over U which is given by   if x = 1,  ∅ ˜ F : X → P(U ), x 7→ {1, a, c} if x ∈ {a, b},   {1, a} if x = c. 6

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Table 2: Cayley table for the “∗”-operation

∗ 1 a b c

1 1 1 1 1

a a 1 1 a

b b a 1 a

c c a a 1

³ ³ ´ ´ Then F˜ , X is a uni-soft filter of X. The τ -exclusive sets of F˜ , X are described as follows:   if τ ∈ {X, {1, a, c}}  X ˜ eX (F ; τ ) = {1, c} if {1, a} ⊆ τ ( {1, a, c},   {1} otherwise. If we take τ1 = X and τ2 = {1, b}, then τ1 ) τ2 and there is no x ∈ X such that τ1 ) F˜ (x) ) τ2 . But eX (F˜ ; τ1 ) = X 6= {1} = eX (F˜ ; τ2 ). ³ ´ Theorem 3.8. Let F˜ , X be a uni-soft filter of X. Let τ1 and τ2 be subsets of U such that τ1 ) τ2 and {τ1 , τ2 , F˜ (x)} is totally ordered by set inclusion for ³all x ∈´X. If there is no x ∈ X such that τ1 ⊇ F˜ (x) ) τ2 , then the τ1 -exclusive set of F˜ , X is equal to ³ ´ the τ2 -exclusive set of F˜ , X . Proof. Since τ1 ) τ2 , we have eX (F˜ ; τ2 ) ⊆ eX (F˜ ; τ1 ). If x ∈ eX (F˜ ; τ1 ), then τ1 ⊇ F˜ (x). Since {τ1 , τ2 , F˜ (x) | x ∈ X} is totally ordered by set inclusion and there is no x ∈ X such ˜ that τ1 ⊇ F˜ (x) )³τ2 , it ´ follows that τ2 ⊇ F˜ (x), that is, x ³ ∈ eX (F ´ ; τ2 ). Therefore the τ1 -exclusive set of F˜ , X is equal to the τ2 -exclusive set of F˜ , X . We have the following question. ³ ´ Question. Given a uni-soft filter F˜ , X of X, does any filter can be represented as a ³ ´ ˜ τ -exclusive set of F , X ? The following example shows that the answer to the question above is false.

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Example 3.9. Let³X = {1, ´ a, b, c} be the BE-algebra as in Example 3.7. Given U = X, ˜ consider a soft set F , X of X over U which is given by (

{c} if x = 1, {1, c} if x ∈ {a, b, c}. ³ ´ ³ ´ Then F˜ , X is a uni-soft filter of X. The τ -exclusive sets of F˜ , X are described as follows:   if τ ⊇ {1, c},  X ˜ eX (F ; τ ) = {1} if {c} ⊆ τ ( {1, c},   ∅ otherwise. F˜ : X → P(U ), x 7→

The filter {1, b} cannot be a τ -exclusive set eX (F˜ ; τ ), since there is no τ ⊆ U such that eX (F˜ ; τ ) = {1, b}. However, we have the following theorem. Theorem 3.10. Every filter of a BE-algebra can be represented as a τ -exclusive set of a X, there ³ uni-soft ´ filter, that is, given a filter F of a BE-algebra ³ ´ exists a uni-soft filter ˜ ˜ F , X of X over U such that F is the τ -exclusive set of F , X for a nonempty subset τ of U . Proof. ³ Let´F be a filter of a BE-algebra X. For a nonempty subset τ of U, define a soft set F˜ , X over U by (

τ if x ∈ F, U if x ∈ / F. ³ ´ ˜ ˜ Obviously, F = eX (F ; τ ). We now prove that F , X is a uni-soft filter of X. Since 1 ∈ F = eX (F˜ ; τ ), we have F˜ (1) ⊆ τ ⊆ F˜ (x) for all x ∈ X. Let x, y ∈ X. If x ∗ y ∈ F and x ∈ F, then y ∈ F because F is a filter of X. Hence F˜ (x ∗ y) = F˜ (x) = F˜ (y) = τ, ˜ ˜ and so F˜ (y) ⊆ F˜ (x ∗ y) ∪ F˜ (x). If x ∗ y ∈ / F or x³∈ / F, then ´ F (x ∗ y) = U or F (x) = U . Hence F˜ (y) ⊆ U = F˜ (x ∗ y) ∪ F˜ (x). Therefore F˜ , X is a uni-soft filter of X. F˜ : X → P(U ), x 7→

Note that if E = X is a finite BE-algebra, then the number of filters of X is finite whereas the number of τ -exclusive sets of a uni-soft filter of X over U = Z appears to be infinite. But, since every τ -exclusive set is indeed a filter of X, not all these τ -exclusive sets are distinct. The next theorem characterizes this aspect. 8

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³ ´ Theorem 3.11. Let F˜ , X be a uni-soft filter of X over U = Z and let τ1 ( τ2 ⊆ U be such that {τ1 , τ2 , F˜ (x)} is a chain for all x ∈ X. Two τ -exclusive sets eX (F˜ ; τ1 ) and eX (F˜ ; τ2 ) are equal if and only if there is no x ∈ X such that τ1 ( F˜ (x) ( τ2 . Proof. Let τ1 and τ2 be subsets of U such that eX (F˜ ; τ1 ) = eX (F˜ ; τ2 ). Assume that there exists x ∈ X such that τ1 ( F˜ (x) ( τ2 . Then eX (F˜ ; τ2 ) is a proper superset of eX (F˜ ; τ1 ), which contradicts the hypothesis. Conversely, suppose that there is no x ∈ X such that τ1 ( F˜ (x) ( τ2 . Obviously, eX (F˜ ; τ2 ) ⊇ eX (F˜ ; τ1 ). If x ∈ eX (F˜ ; τ2 ), then τ2 ⊇ F˜ (x). It follows from the hypothesis that τ1 ⊇ F˜ (x), i.e., x ∈ eX (F˜ ; τ1 ). Therefore eX (F˜ ; τ1 ) = eX (F˜ ; τ2 ). ³ ´ Let F˜ , X be a soft set of X over U . For any a, b ∈ X and k ∈ N, consider the set n o ¡ ¢ F˜ [ak ; b] := x ∈ X | F˜ ak ∗ (b ∗ x) = F˜ (1) where F˜ (ak ∗ x) = F˜ (a ∗ (a ∗ (· · · ∗ (a ∗ (a ∗ x)) · · · ))) in which a appears k-times. Note that a, b, 1 ∈ F˜ [ak ; b] for all a, b ∈ X and k ∈ N. ³ ´ ˜ Proposition 3.12. Let F , X be a soft set of X over U satisfying the condition (3.1) and F˜ (x∗y) = F˜ (x)∩ F˜ (y) for all x, y ∈ X. For any a, b ∈ X and k ∈ N, if x ∈ F˜ [ak ; b], then y ∗ x ∈ F˜ [ak ; b] for all y ∈ X. Proof. Assume that x ∈ F˜ [ak ; b]. Then F˜ (ak ∗ (b ∗ x)) = F˜ (1), and so F˜ (ak ∗ (b ∗ (y ∗ x))) = F˜ (ak ∗ (y ∗ (b ∗ x))) = F˜ (y ∗ (ak ∗ (b ∗ x))) = F˜ (y) ∩ F˜ (ak ∗ (b ∗ x)) = F˜ (y) ∩ F˜ (1) = F˜ (1) for all y ∈ X by the exchange property of the operation ∗ and (3.1). Hence y ∗x ∈ F˜ [ak ; b] for all y ∈ X. ³ ´ Proposition 3.13. For any soft set F˜ , X of X over U , if an element a ∈ X satisfies a ∗ x = 1 for all x ∈ X then F˜ [ak ; b] = X = F˜ [bk ; a] for all b ∈ X and k ∈ N. Proof. For any x ∈ X, we have F˜ (ak ∗ (b ∗ x)) = F˜ (ak−1 ∗ (a ∗ (b ∗ x))) = F˜ (ak−1 ∗ (b ∗ (a ∗ x))) = F˜ (ak−1 ∗ (b ∗ 1)) = F˜ (1), 9

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and so x ∈ F˜ [ak ; b]. Similarly, x ∈ F˜ [bk ; a]. ³

Proposition 3.14. Let X be a self distributive BE-algebra and let F˜ , X

´

be an orderreversing soft set of X over U with the property (3.1). If b ≤ c in X, then F˜ [ak ; c] ⊆ F˜ [ak ; b] for all a ∈ X and k ∈ N. Proof. Let a, b, c, ∈ X be such that b ≤ c. For any k ∈ N, if x ∈ F˜ [ak ; c], then F˜ (1) = F˜ (ak ∗ (c ∗ x)) = F˜ (c ∗ (ak ∗ x)) ⊇ F˜ (b ∗ (ak ∗ x)) = F˜ (ak ∗ (b ∗ x)) by (2.4) and (2.8), and so F˜ (ak ∗ (b ∗ x)) = F˜ (1). Thus x ∈ F˜ [ak ; b], which completes the proof. ³ ´ The following example shows that there exists a soft set F˜ , X of X over U , a, b ∈ X and k ∈ N such that F˜ [ak ; b] is not a filter of X. ³ ´ Example 3.15. Consider the BE-algebra X = {1, a, b, c} in Example 3.7. Let F˜ , X be a soft set of X over U = N which is given by ( 6N if x = 1, F˜ : X → P(U ), x 7→ 3N if x ∈ {a, b, c}. Then it is a soft set of X over U . But F˜ [c; b] = {x ∈ X|F˜ (c ∗ (b ∗ x)) = F˜ (1)} = {1, a, b} is not a filter, since a ∗ c = a ∈ F˜ [c; b] and c ∈ / F˜ [c; b]. We provide conditions for a set F˜ [ak ; b] to be a filter. ³ ´ Theorem 3.16. Let F˜ , X be a soft set over X. If X is a self distributive BE-algebra and F˜ is injective, then F˜ [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N. Proof. Assume that X is a self distributive BE-algebra and F˜ is injective. Obviously, 1 ∈ F˜ [ak ; b]. Let a, b, x, y ∈ X and k ∈ N be such that x ∗ y ∈ F˜ [ak ; b] and x ∈ F˜ [ak ; b]. Then F˜ (ak ∗ (b ∗ x)) = F˜ (1) which implies that ak ∗ (b ∗ x) = 1 since F˜ is injective.

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Using (2.7), we have F˜ (1) = F˜ (ak ∗ (b ∗ (x ∗ y))) = F˜ (ak−1 ∗ (a ∗ (b ∗ (x ∗ y)))) = F˜ (ak−1 ∗ (a ∗ ((b ∗ x) ∗ (b ∗ y)))) = ··· = F˜ ((ak ∗ (b ∗ x)) ∗ (ak ∗ (b ∗ y))) = F˜ (1 ∗ (ak ∗ (b ∗ y))) = F˜ (ak ∗ (b ∗ y)), which implies that y ∈ F˜ [ak ; b]. Therefore F˜ [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N. ³ ´ ˜ Theorem 3.17. Let X be a self distributive BE-algebra. Let F , X be a soft set of X over U satisfying the condition (3.1) and ³ ´ ˜ ˜ ˜ (∀x, y ∈ X) F (x ∗ y) = F (x) ∪ F (y) . (3.3) Then F˜ [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N. Proof. Let a, b ∈ X and k ∈ N. Obviously, 1 ∈ F˜ [ak ; b]. Let x, y ∈ X be such that ¡ ¢ x ∗ y ∈ F˜ [ak ; b] and x ∈ F˜ [ak ; b]. Then F˜ ak ∗ (b ∗ x) = F˜ (1), which implies from (3.3) and (3.1) that F˜ (1) = F˜ (ak ∗ (b ∗ (x ∗ y))) = F˜ (ak−1 ∗ (a ∗ (b ∗ (x ∗ y)))) = F˜ (ak−1 ∗ (a ∗ ((b ∗ x) ∗ (b ∗ y)))) = ··· = F˜ ((ak ∗ (b ∗ x)) ∗ (ak ∗ (b ∗ y))) = F˜ (ak ∗ (b ∗ x)) ∪ F˜ (ak ∗ (b ∗ y)) = F˜ (1) ∪ F˜ (ak ∗ (b ∗ y)) = F˜ (ak ∗ (b ∗ y)). Hence y ∈ F˜ [ak ; b] and therefore F˜ [ak ; b] is a filter of X for all a, b ∈ X and k ∈ N.

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References [1] S. S. Ahn and K. S. So, On ideals and upper sets in BE-algerbas, Sci. Math. Jpn. 68 (2008), 279–285. [2] H. Akta¸s and N. C ¸ a˘gman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726– 2735. [3] N. C ¸ aˇgman and S. Engino˘glu, FP-soft set theory and its applications, Ann. Fuzzy Math. Inform. 2 (2011) 219–226. [4] F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math. Inform. 2 (2011) 69–80. [5] Y. B. Jun, K. J. Lee and S. Z. Song, Filters of BE-algebras associated with uni-soft set theory, Commun. Korean Math. Soc. (submitted). [6] H. S. Kim and Y. H. Kim, 113–116.

On BE-algerbas, Sci. Math. Jpn. 66 (2007), no. 1,

[7] P. K. Maji, A. R. Roy and R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl. 44 (2002) 1077–1083. [8] D. Molodtsov, Soft set theory - First results, Comput. Math. Appl. 37 (1999) 19–31.

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On q-analogue of Stancu-Schurer-Kantorovich operators based on q-Riemann integral Shin Min Kang1 , Ana Maria Acu2, Arif Rafiq3 and Young Chel Kwun4,∗

1

Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected]

2

Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Sibiu 550012, Romania e-mail: [email protected] 3

Department of Mathematics and Statistics, Virtual University of Pakistan, Lahore 54000, Pakistan e-mail: [email protected] 4

Department of Mathematics, Dong-A University, Busan 604-714, Korea e-mail: [email protected] Abstract

In the present paper we introduce the Kantorovich type generalization of StancuSchurer operators based on q-Riemann integral. A convergence theorem using the well known Bohman-Korovkin criterion is proven and the rate of convergence involving the modulus of continuity is established. Also, we obtain a Voronovskaja type theorem for these operators. 2010 Mathematics Subject Classification: 41A10, 41A25, 41A36. Key words and phrases: q-Stancu-Kantorovich operator, modulus of continuity, rate of convergence, Voronovskaja theorem.

1

Introduction

In recent years, the applications of q-calculus have played an important role in the area of approximation theory, generalizations of many well-known linear and positive operators based on the q-integers were studied by numbers of authors ([2, 5, 7, 10–12, 14, 16, 18–20]). In 1987, Lupa¸s [9] introduced and studied q-analogue of Bernstein operators and in 1996 another generalization of these operators were introduced by Philips [17]. More results on q-Bernstein polynomials were obtained by Ostrowska in [15]. In [1], Agratini introduced a new class of q-Bernstein type operators which fix certain polynomials and studied their approximation properties. Very recently, Muraru [14] proposed the q-Bernstein-Schurer ∗

Corresponding author

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operators. Agrawal et al. [3] introduced the Stancu type generalization of BernsteinSchurer operators. Aral et al. [4] also presented many results on convergence of different q-operators recently in their book. First, we present some basic definitions and notations from q-calculus. Let q > 0. For each nonnegative integer k, the q-integer [k]q and q-factorial [k]q ! are respectively defined by ( k ( 1−q , q 6= 1, [k]q [k − 1]q · · · [1]q , k ≥ 1, 1−q [k]q := [k]q ! := 1, k = 0. k, q = 1, For the integers n, k satisfying n ≥ k ≥ 0, the q-binomial coefficients are defined by   [n]q ! n := . k q [k]q ![n − k]q ! Q j We denote (a + b)kq = k−1 j=0 (a + bq ). The q-Jackson integral on the interval [0, b] is defined as Z

b

0

f (t)dq t = (1 − q)b

∞ X

f (q j b)q j ,

0 < q < 1,

j=0

provided that sums converge absolutely. Suppose 0 < a < b. The q-Jackson integral on the interval [a, b] is defined as Z b Z b Z a f (t)dq t = f (t)dq t − f (t)dq t, 0 < q < 1. a

0

0

Dalmano˜glu [5], Mahmudov and Sabancigil [12] defined some q-type generalizations of Bernstein-Kantorovich operators using q-Jackson integral. In [18], Ren and Zeng were introduced two kinds of Kantorovich-type q-Bernstein-Stancu operators. The first version is defined using q-Jackson integral as follows (α,β) Sn,q (f, x)

= ([n + 1]q + β)

n X

q

−k

pn,k (q; x)

k=0

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

f (t)dq t,

(1.1)

h n i xk (1 − x)n−k . q k q To guarantee the positivity of q-Bernstein-Stancu-Kantorovich operators, in [18] is considered the Riemann-type q-integral (see [13]) defined by

where 0 ≤ α ≤ β, f ∈ C[0, 1] and pn,k (q; x) =

Z

b a

f (t)dR q t = (1 − q)(b − a)

∞ X j=0


f a + (b − a)q j q j .

(1.2)

(α,β)

Ren and Zeng [18] redefine Sn,q by putting the Riemann-type q-integral into the operators instead of the q-Jackson integral as (α,β) S˜n,q (f, x)

= ([n + 1]q + β)

n X

q

−k

k=0

pn,k (q; x)

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

f (t)dR q t.

(1.3)

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Very recently, the q-Kantorovich extension of the Bernstein-Schurer operators have been considered by Kumar et al. [8] as follows: Z [k+1]q /[n+1]q n+p X −k Kn,p(f, q; x) = [n + 1]q bn,k (q; x)q f (t)dR (1.4) q t, [k]q /[n+1]q

k=0

i n+p where x ∈ [0, 1], f ∈ C[0, 1+p], p ∈ N = N∪{0} and bn,k (q; x) = xk (1−x)n+p−k . q k q In the present paper, inspired by the new Kantorovich type generalization of the qBernstein-Schurer operators we introduce the Kantorovich type of Stancu-Schurer operators involving the Riemann-type q-integral. h

0

2

Construction of the operators

In this section we construct the Stancu-Schurer-Kantorovich operators based on q-integers. Let α, β ∈ R be such that 0 ≤ α ≤ β and p ∈ N0 = N ∪ {0}, then for any f ∈ C[0, 1 + p], q ∈ (0, 1), the Stancu-Schurer-Kantorovich operators are defined using q-Riemann integral as follows Z [k+1]q +α n+p X [n+1]q +β (α,β) −k Kn,p (f, q; x) = ([n + 1]q + β) q bn,k (q; x) [k] +α f (t)dR (2.1) q t. k=0

q [n+1]q +β

(α,β)

Lemma 2.1. Let Kn,p (f, q; x) be given by (2.1). Then the following equalities hold: (α,β) (i) Kn,p (1, q; x) = 1; o n (α,β) 2q (ii) Kn,p (t, q; x) = [n+1]1q +β [2]1 q + α + [2] [n + p] x ; q q n    2 (α,β) 2 (q−1) 2 + 3q+1 + (iii) Kn,p (t , q; x) = ([n+1]1q +β)2 q 1 + 2(q−1) + [n + p] [n + p − 1] x q q [2]q [3]q [2]q  o 4αq q 2 −1 1 2α 2 + [3]q [n + p]q x + [3]q + [2]q + α . [2]q

Proof. By definition of q-Riemann integral (1.2), we have Z Z Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β [k+1]q +α [n+1]q +β [k]q +α [n+1]q +β [k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

dR qt= tdR qt=

qk ; [n + 1]q + β 1 ([n + 1]q + β)2

t2 dR qt=

(2.2)   q 2k q k ([k]q + α) + ; [2]q

1 ([n + 1]q + β)3

(2.3)

  2q 2k q 3k q k ([k]q + α)2 + ([k]q + α) + . [2] [3]q

(2.4)

The following identities hold n+p X

k=0 n+p X k=0

bn,k (q; x)q k = 1 − (1 − q)[n + p]q x;

(2.5)

bn,k (q; x)q 2k = 1 − (1 − q 2 )[n + p]q x + q(1 − q)2 [n + p]q [n + p − 1]q x2 .

(2.6)

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Hence, by using equality

n+p X

bn,k (q; x) = 1 and equation (2.2), we get

k=0

(α,β) Kn,p (1, q; x) = 1.

By using relations (2.3) and (2.5) we have (α,β) Kn,p (t, q; x)

  n+p X 1 q 2k −k k q bn,k (q; x) q ([k]q + α) + = [n + 1]q + β [2]q k=0   1 2q 1 = +α+ [n + p]q x . [n + 1]q + β [2]q [2]q

(2.7)

Now, from the equations (2.4) and (2.6), we get (α,β) 2 Kn,p (t , q; x)

  n+p X 1 2q 2k q 3k −k k 2 = q bn,k (q; x) q ([k]q + α) + ([k]q + α) + ([n + 1]q + β)2 [2] [3]q k=0 (  2 n+p X 1 1 bn,k (q; x) = +α ([n + 1]q + β)2 1−q k=0         1 1 1 1 1 2 1 k 2k + 2q +α − +α +q − + [2]q 1 − q 1−q 1−q (1 − q)2 (1 − q)[2]q [3]q    2(q − 1) (q − 1)2 1 = q 1+ + [n + p]q [n + p − 1]q x2 ([n + 1]q + β)2 [2]q [3]q    3q + 1 4αq q 2 − 1 1 2α 2 + + + [n + p]q x + + +α . [2]q [2]q [3]q [3]q [2]q

Remark 2.2. From Lemma 2.1, we get (α,β) Kn,p (t − x, q; x) =

(α,β) Kn,p (t − x)2 , q; x

1 + [2]q α + [2]q ([n + 1]q + β)



 2q [n + p]q − 1 x; [2]q [n + 1]q + β




(α,β) 2 (α,β) (α,β) = Kn,p (t ; x) − 2xKn,p (t; x) + x2 Kn,p (1; x)  2 2 1 q (4q + q + 1) = [n + p]q [n + p − 1]q x2 2 ([n + 1]q + β) [2]q [3]q  q(4q 2 + 5q + 3) + 4αq(q 2 + q + 1) 1 2α 2 + [n + p]q x + + +α [2]q [3]q [3]q [2]q   2x 1 2q − +α+ [n + p]q x + x2 [n + 1]q + β [2]q [2]q

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 4q [n + p]q q 2 (4q 2 + q + 1) [n + p]q [n + p − 1]q − + 1 x2 = [2]q [3]q ([n + 1]q + β)2 [2]q [n + 1]q + β   q(4q 2 + 5q + 3) + 4αq(q 2 + q + 1) 2(1 + [2]q α) [n + p]q + x − [2]q [3]q ([n + 1]q + β)2 [2]q ([n + 1]q + β)   2α 1 1 2 + +α . + ([n + 1]q + β)2 [3]q [2]q 

Lemma 2.3. For f ∈ C[0, p + 1], we have (α,β) kKn,p (f, q; ·)kC[0,1] ≤ kf kC[0,p+1],

where k · kC[0,p+1] is the sup-norm on [0, p + 1]. Proof. We have n+p X (α,β) K ≤ ([n + 1]q + β) (f, q; ·) q −k bn,k (q; x) n,p

≤

k=0 (α,β) kf kC[0,p+1]Kn,p (1, q; x)

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

|f (t)|dR qt

= kf kC[0,p+1].

Lemma 2.4. For each x ∈ [0, 1] and 0 < q < 1, we have
8(α2 + 3β 2 + 3[p]2q + 4) ϕ2 (x) (α,β) Kn,p (t − x)2 , q; x ≤ 4C + , [n + p]q ([n + 1]q + β)2 (α,β) Kn,p

83 (α4 + 27[p]4q + 27β 4 + 28) ϕ2 (x) ˜ (t − x) , q; x ≤ 64C + , [n + p]2q ([n + 1]q + β)4 4




(2.8) (2.9)

˜ are some constants. where ϕ2 (x) = x(1 − x) and C, C Proof. We have (α,β) Kn,p (t − x)2 , q; x

= ([n + 1]q + β)




n+p X

q

−k

bn,k (q; x)

k=0

n+p X

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

(t − x)2 dR qt

2 ∞  X [k]q + α qk j = (1 − q) bn,k (q; x) + q − x qj [n + 1]q + β [n + 1]q + β j=0 k=0   2 n+p ∞  ∞ 2k q 3j X X X [k] + α q q  bn,k (q; x)  ≤ 2(1 − q) − x qj + [n + 1]q + β ([n + 1]q + β)2 j=0

k=0

≤2

n+p X k=0

j=0





[k]q + α [k]q [k]q bn,k (q; x) − − x− [n + 1]q + β [n + p]q [n + p]q

2

n+p q 2k 2 X + bn,k (q; x) [3]q ([n + 1]q + β)2 k=0

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≤4

n+p X k=0



[k]q [k]q +α − bn,k (q; x) [n + 1]q +β [n + p]q

2

+4

n+p X k=0

 bn,k (q; x) x−

[k]q [n + p]q

2

2 + [3]q ([n + 1]q +β)2  2 n+p X [k]q + α [k]q ≤8 bn,k (q; x) − [n + 1]q + β [n + 1]q + β k=0 2  n+p X [k]q [k]q bn,k (q; x) − +8 [n + 1]q + β [n + p]q k=0


2 1 + 4Bn+p (t − x)2 , q; x + , [3]q ([n + 1]q + β)2
P f ([k]q /[n]q ) is the q-Bernstein operators. On where Bn (f, q; x) = nk=0 nk q xk (1 − x)n−k q the other hand by [10], we have |Bn ((t − x)m, q; x)| ≤ Km

x(1 − x)

b(m+1)/2c

[n]q

,

for some constant Km > 0, where bac is the integer part of a ≥ 0. We find that
(α,β) Kn,p (t − x)2 , q; x n+p

≤

n+1 X 8α2 [p]q − q n+p − β)2 2 (q 8 b (q; x)[k] + n,k q ([n + 1]q + β)2 [n + p]2q ([n + 1]q + β)2 k=0

ϕ2 (x)

2 [n + p]q [3]q ([n + 1]q + β)2 8α2 ϕ2 (x) 24([p]2 + 1 + β 2 ) 2 ≤ 4C + + + 2 2 ([n + 1]q + β) ([n + 1]q + β) [n + p]q [3]q ([n + 1]q + β)2 8(α2 + 3β 2 + 3[p]2q + 4) ϕ2 (x) ≤ 4C + . [n + p]q ([n + 1]q + β)2 + 4C

+

Also, we obtain (α,β) Kn,p (t − x)4 , q; x

= ([n + 1]q + β)




n+p X

q

−k

bn,k (q; x)

k=0

= (1 − q)

n+p X

bn,k (q; x)

j=0

k=0

≤ 8(1 − q)

n+p X k=0

+ 8(1 − q)

∞  X

Z

[k+1]q +α [n+1]q +β [k]q +α [n+1]q +β

(t − x)4 dR qt

[k]q + α qk + qj − x [n + 1]q + β [n + 1]q + β

4

qj

4 ∞  X [k]q + α bn,k (q; x) −x q j [n + 1]q +β

n+p X k=0

j=0

bn,k (q; x)

∞  X j=0

qk [n + 1]q +β

4

q 5j

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n+p X

4 4  n+p [k]q + α 8 X qk −x + bn,k (q; x) =8 bn,k (q; x) [n + 1]q + β [5]q [n + 1]q + β k=0 k=0   4 n+p X [k]q [k]q + α [k]q − =8 bn,k (q; x) − x− [n + 1]q +β [n + p]q [n + p]q k=0   n+p 4 8 X qk + bn,k (q; k) [5]q [n + 1]q +β k=0  4 n+p X
[k]q + α [k]q 4 ≤ 64Bn+p (t − x) , q; x + 64 bn,k (q; x) − [n + 1]q + β [n + p]q 

k=0

8 + [5]q ([n + 1]q + β)4

n+p X



[k]q + α [k]q ≤ 64Bn+p (t − x) , q; x + 8 bn,k (q; x) − [n + 1]q + β [n + 1]q + β k=0  4 n+p X [k]q [k]q 8 3 bn,k (q; x) +8 − + [n + 1]q + β [n + p]q [5]q ([n + 1]q + β)4 4




3

4

k=0

n+p

n+1 [p] − q n+p −β)4 X ϕ2 (x) 83 α4 q 3 4 (q ≤ 64C˜ 8 b (q; x)[k] + + n,k q [n + p]2q ([n + 1]q +β)4 [n + p]4q ([n + 1]q +β)4 k=0

8 + [5]q ([n + 1]q +β)4

≤ 64C˜

[p]4q + 1 + β 4 ϕ2 (x) 83 α4 8 3 24 + + + [n + p]2q ([n + 1]q + β)4 ([n + 1]q + β)4 [5]q ([n + 1]q + β)4

83 (α4 + 27[p]4q + 27β 4 + 28) ϕ2 (x) ≤ 64C˜ + . [n + p]2q ([n + 1]q + β)4

3

Direct theorems

In this section we propose to study some approximation properties of the Stancu-SchurerKantorovich operators defined in (2.1). First, we prove the basic convergence theorem of (α,β) Kn,p and then obtain the rate convergence of these operators in term of the modulus of continuity. Further, we study local direct results for the q-analogue of Stancu-SchurerKantorovich operators. Theorem 3.1. Let (qn )n , 0 < qn < 1 be a sequence satisfying the following conditions lim qn = 1,

n→∞

lim q n n→∞ n

= a,

a ∈ [0, 1).

(3.1)

(α,β)

Then for any f ∈ C[0, p + 1], the sequence Kn,p (f, qn ; x) converges to f uniformly on [0, 1]. 7 570

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Proof. From (3.1) we obtain [n + 1]qn → ∞ as n → ∞. Further (α,β) Kn,p (1, qn; x)

(α,β) Kn,p (t, qn ; x)

(α,β) Kn,p (t2 , qn ; x)

[n+p]qn [n+1]qn

→ 1, hence

→ 1, → x and → x2 uniformly on [0, 1] as n → ∞. Therefore, using the Bohman-Korovkin theorem implies that Kn,p (f, qn ; ·) converges to f uniformly on [0, 1]. Let us consider the following K-functional  K2 (f, δ) := inf kf − gk + δkg 00k, g ∈ C 2 [0, p + 1] ,

where δ ≥ 0.

It is known (see [6]) there exist an absolute constant C > 0 such that √ K2 (f, δ) ≤ Cω2 (f, δ),

(3.2)

(3.3)

where the second order modulus of smoothness for f ∈ C[0, p + 1] is defined as ω2 (f, δ) :=

sup 0 0 for all u > 0, 'j (0) = 0; j = 1; 2; 3; (ii) '0j (u) > 0; for all u > 0; j = 1; 3, '02 (u) > 0; for all u 0, (iii) there are j 0; j = 1; 2; 3 such that 'j (u) 0: j u; for all u (x; y; v) Assumption A4. is decreasing with respect to v for all v > 0: '2 (v)

1.1

Properties of solutions

In this subsection, we study some properties of the solution of the model such as the non-negativity and boundedness of solutions. Proposition. Assume that Assumptions A1-A3 are satis…ed. Then there exist positive numbers Li , i = 1; 2; 3, such that the compact set = (x; y; v; z) 2 R4 0 : 0

x; y

L1 ; 0

v

L2 ; 0

z

L3

is positively invariant. Proof. We have x_ jx=0 = n(0) > 0, v_ jv=0 = k'1 (y)

y_ jy=0 = (x; 0; v) 0 for all y

0,

0 for all x

0; v

0;

z_ jz=0 = 0:

Hence, the orthant R4 0 is positively invariant for system (5)-(8). Next, we show that the solutions of the system a v(t) + rq z(t), then are bounded. Let T (t) = x(t) + y(t) + 2k a ac q a ac q T_ (t) = n(x) '1 (y) '2 (v) '3 (z) s sx 1y 2v 3z 2 2k r 2 2k r a q s x+y+ v+ z =s T (t); 2k r where = minfs; a2 1 ; c 2 ; 3 g. Then, s s s T (t) T (0)e t + 1 e t = e t T (0) + : Hence, 0 and 0

T (t)

z(t)

L1 if T (0)

L3 for all t

0 if

L1 for t

0 where L1 =

s

: It follows that, 0

a x(0)+y(0)+ 2k v(0)+ rq z(0)

x(t); y(t) L1 , 0 v(t) L2 2kL1 rL1 L1 , where L2 = and L3 = . Therefore, a q

x(t); y(t); v(t) and z(t) are all bounded.

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1.2

The equilibria and threshold parameters

In this subsection we calculate the equilibria of the model and derive two threshold parameters. Lemma. Assume that Assumptions A1-A4 are satis…ed, then there exist two threshold parameters R0 > 0 and R1 > 0 with R1 < R0 such that (i) if R0 1; then there exists only one positive equilibrium E0 2 , (ii) if R1 1 < R0 ; then there exist only two positive equilibria E0 2 and E1 2 , and (iii) if R1 > 1; then there exist three positive equilibria E0 2 , E1 2 Proof. At any equilibrium we have n(x) (x; y; v)

a'1 (y)

and E2 2 .

(x; y; v) = 0;

(9)

q'1 (y)'3 (z) = 0;

(10)

k'1 (y) (r'1 (y)

c'2 (v) = 0;

(11)

) '3 (z) = 0:

(12)

From Eq. (12), either '3 (z) = 0 or '3 (z) 6= 0. If '3 (z) = 0, then from Assumption A3 we get, z = 0 and from Eqs. (9)-(11) we have ac'2 (v) n(x) = (x; y; v) = a'1 (y) = : (13) k kn(x) From Eq. (13), we have '1 (y) = n(x) a ; '2 (v) = ac : Since '1 ; '2 are continuous and strictly increasing 1 1 functions with '1 (0) = '2 (0) = 0, then '1 ; '2 exist and they are also continuous and strictly increasing [21]. Let {1 (x) = '1 1 n(x) and {2 (x) = '2 1 kn(x) , then a ac

y = {1 (x); v = {2 (x):

(14)

Obviously from Assumption A1, {1 (x); {2 (x) > 0 for x 2 [0; x0 ) and {1 (x0 ) = {2 (x0 ) = 0. Substituting from Eq. (14) into Eq. (13) we get ac '2 ({2 (x)) = 0: (15) (x; {1 (x); {2 (x)) k We note that, x = x0 is a solution of Eq. (15). Then, from Eq. (14) we have y = v = 0, and this leads to the infection-free equilibrium E0 = (x0 ; 0; 0; 0). Let 1

(x) =

(x; {1 (x); {2 (x))

ac '2 ({2 (x)) = 0: k

Then from Assumptions A1-A3, we have 1 (0)

=

1 (x0 )

=

ac '2 ({2 (0)) < 0; k ac '2 (0) = 0: (x0 ; 0; 0) k

Moreover, 0 1

(x0 ) =

@ (x0 ; 0; 0) @ (x0 ; 0; 0) @ (x0 ; 0; 0) + {10 (x0 ) + {20 (x0 ) @x @y @v

Assumption A2 implies that

@ (x0 ;0;0) @x 0 1

(x0 ) =

=

@ (x0 ;0;0) @y

ac 0 ' (0){20 (x0 ): k 2

= 0. Also, from Assumption A3, we have '02 (0) > 0; then

ac 0 { (x0 )'02 (0) k 2

@ (x0 ; 0; 0) k ac'02 (0) @v

1 :

From Eq. (14), we get 0 1

(x0 ) = n0 (x0 )

k @ (x0 ; 0; 0) 0 ac'2 (0) @v

1 :

k @ (x0 ; 0; 0) > 1; then 01 (x0 ) < 0 and there ac'02 (0) @v 1 (x1 ) = 0. From Eq. (14), we have y1 = {1 (x1 ) > 0 and v1 = {2 (x1 ) > 0.

From Assumption A1, we have n0 (x0 ) < 0. Therefore, if exists a x1 2 (0; x0 ) such that

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It follows that, a chronic-infection equilibrium without CTL immune response E1 = (x1 ; y1 ; v1 ; 0) exists when k @ (x0 ; 0; 0) > 1. Let us de…ne ac'02 (0) @v R0 =

k @ (x0 ; 0; 0) ; 0 ac'2 (0) @v

which describes the average number of newly infected cells generated from one infected cell at the beginning of > 0 and the infectious process. The other possibility of Eq. (12) is '3 (z) 6= 0 which leads to y2 = '1 1 r k v2 = '2 1 > 0. Substituting y = y2 and v = v2 in Eq. (9), we letting cr 2 (x)

= n(x)

(x; y2 ; v2 ) = 0:

Clearly, 2 (0)

= n(0) > 0 and

2 (x0 )

=

(x0 ; y2 ; v2 ) < 0:

According to Assumptions A1 and A2, 2 (x) is a strictly decreasing function of x. Thus, there exists a unique x2 2 (0; x0 ) such that 2 (x2 ) = 0. Now from Eq. (10) we have z2 = ' 3 1 From Assumption A3, we have if

a q

k (x2 ; y2 ; v2 ) ac'2 (v2 )

1

:

k (x2 ; y2 ; v2 ) > 1; then z2 > 0. Now we de…ne ac'2 (v2 ) R1 =

k (x2 ; y2 ; v2 ) ; ac'2 (v2 )

which represents the immune response reproduction ratio which expresses the CTL load during the lifespan of a (R1 1) . It follows that, there is a chronic-infection a CTL cell. Hence, z2 can be rewritten as z2 = '3 1 q equilibrium with CTL immune response E2 = (x2 ; y2 ; v2 ; z2 ) if R1 > 1. Now we show that E0 ; E1 2

and E2 2 . Clearly, E0 2 . We have x1 < x0 , then from Assumption A1 0 = n(x0 ) < n(x1 )

It follows that

s s

0 < x1
0 and H has the global minimum H(1) = 0. Theorem 1. Let Assumptions A1-A4 hold true and R0 1; then the infection-free equilibrium E0 is globally asymptotically stable (GAS) in : Proof. We construct a Lyapunov functional as: Z x (x0 ; 0; v) a aq U0 (x; y; v; z) = x x0 lim d + y + v + z: (16) + ( ; 0; v) k rk x0 v!0 It is seen that, U0 (x; y; v; z) > 0 for all x; y; v; z > 0 while U0 (x; y; v; z) reaches its global minimum at E0 . We 0 calculate dU dt along the solutions of model (5)-(8) as: dU0 = dt +

1

(x0 ; 0; v) (x; 0; v)

lim

v!0+

a (k'1 (y) k

= n(x) 1

c'2 (v)

(n(x)

a'1 (y)

aq (r'3 (z)'2 (v) '3 (z)) rk (x0 ; 0; v) ac '2 (v) + (x; y; v) lim+ (x; 0; v) k v!0

q'2 (v)'3 (z)) +

(x0 ; 0; v) (x; 0; v)

lim

v!0+

(x; y; v)) + (x; y; v)

aq '3 (z): rk

(17)

Since n(x0 ) = 0; then we get dU0 = (n(x) dt

n(x0 )) 1

lim

v!0+

(x0 ; 0; v) (x; 0; v)

+

ac k

k (x; y; v) lim ac '2 (v) v!0+

(x0 ; 0; v) (x; 0; v)

1 '2 (v)

aq '3 (z): rk (18)

From Assumptions A2 and A4 we have (x; y; v) < '2 (v)

(x; 0; v) '2 (v)

lim

v!0+

(x; 0; v) 1 @ (x; 0; 0) = 0 : '2 (v) '2 (0) @v

Then, dU0 dt

(n(x)

n(x0 )) 1

= (n(x)

n(x0 )) 1

(@ (@ (@ (@

(x0 ; 0; 0)=@v) (x; 0; 0)=@v) (x0 ; 0; 0)=@v) (x; 0; 0)=@v)

ac k @ (x0 ; 0; 0) 1 '2 (v) k ac'02 (0) @v aq ac (R0 1) '2 (v) '3 (z): + k rk +

aq '3 (z) rk (19)

From Assumptions A1 and A2, we have (n(x)

n(x0 )) 1

(@ (x0 ; 0; 0)=@v) (@ (x; 0; 0)=@v)

0:

0 Therefore, if R0 1, then dU 0 for all x; v; z > 0. We note that solutions of system (5)-(8) limited to , the dt dU0 0 largest invariant subset of dt = 0 [22]. We see that, dU dt = 0 if and only if x(t) = x0 , v(t) = 0 and z(t) = 0 for all t. By the above discussion, each element of satis…es v(t) = 0 and z(t) = 0. Then from Eq. (7) we get

v(t) _ = 0 = k'1 (y(t)): It follows from Assumption A3 that, y(t) = 0 for all t. Using LaSalle’s invariance principle, we derive that E0 is GAS. To prove the global stability of the two equilibria E1 and E2 , we need the following condition on the incidence rate function.

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Assumption A5. (x; y; v) (x; yi ; vi )

'2 (v) '2 (vi )

(x; yi ; vi ) (x; y; v)

1

0,

x; y; v > 0; i = 1; 2

Theorem 2. Assume that Assumptions A1-A5 are satis…ed and R1 equilibrium without CTL immune response E1 is GAS in : Proof. We de…ne the following Lyapunov functional U1 (x; y; v; z) = x

+

x1 0

a@ v k

Z

x

(x1 ; y1 ; v1 ) d + y y1 ( ; y1 ; v1 ) x1 1 Zv '2 (v1 ) A q v1 d + z: '2 ( ) r

1 < R0 , then the chronic-infection

Zy

y1

'1 (y1 ) d '1 ( )

(20)

v1

It is seen that, U1 (x; y; v; z) > 0 for all x; y; v; z > 0 while U1 (x; y; v; z) reaches its global minimum at E1 . The time derivative of U1 along the trajectories of (5)-(8) is given by (x1 ; y1 ; v1 ) '1 (y1 ) (n(x) (x; y; v)) + 1 ( (x; y; v) a'1 (y) (x; y1 ; v1 ) '1 (y) a '2 (v1 ) q + 1 (k'1 (y) c'2 (v)) + (r'1 (y)'3 (z) '3 (z)) k '2 (v) r (x; y; v) '1 (y1 ) (x1 ; y1 ; v1 ) n(x) + (x1 ; y1 ; v1 ) (x; y; v) = 1 (x; y1 ; v1 ) (x; y1 ; v1 ) '1 (y) ac '2 (v1 ) ac q + a'1 (y1 ) + q'1 (y1 )'3 (z) '2 (v) a'1 (y) + '2 (v1 ) '3 (z): k '2 (v) k r

dU1 = dt

1

q'1 (y)'3 (z))

(21)

Using the equilibrium conditions for E1 : n(x1 ) = (x1 ; y1 ; v1 ) = a'1 (y1 ) =

ac '2 (v1 ); k

we obtain dU1 = (n(x) dt

(x1 ; y1 ; v1 ) (x; y; v) (x1 ; y1 ; v1 ) + a'1 (y1 ) + 3a'1 (y1 ) a'1 (y1 ) (x; y1 ; v1 ) (x; y1 ; v1 ) (x; y1 ; v1 ) '1 (y1 ) (x; y; v) '2 (v) '2 (v1 )'1 (y) a'1 (y1 ) a'1 (y1 ) a'1 (y1 ) + q '1 (y1 ) '3 (z): '1 (y) (x1 ; y1 ; v1 ) '2 (v1 ) '2 (v)'1 (y1 ) r n(x1 )) 1

(22)

Collecting terms to get dU1 = (n(x) dt

(x1 ; y1 ; v1 ) + a'1 (y1 ) (x; y1 ; v1 ) (x1 ; y1 ; v1 ) '1 (y1 ) (x; y; v) (x; y1 ; v1 ) '1 (y) (x1 ; y1 ; v1 )

n(x1 )) 1

+ a'1 (y1 ) 4 + q ('1 (y1 )

(x; y; v) '2 (v) '2 (v) (x; y1 ; v1 ) 1+ (x; y1 ; v1 ) '2 (v1 ) '2 (v1 ) (x; y; v) '2 (v1 )'1 (y) '2 (v) (x; y1 ; v1 ) '2 (v)'1 (y1 ) '2 (v1 ) (x; y; v)

'1 (y2 )) '3 (z):

(23)

Eq. (23) can be rewritten as: dU1 = (n(x) dt

(x1 ; y1 ; v1 ) + a'1 (y1 ) (x; y1 ; v1 ) (x1 ; y1 ; v1 ) '1 (y1 ) (x; y; v) (x; y1 ; v1 ) '1 (y) (x1 ; y1 ; v1 )

n(x1 )) 1

+ a'1 (y1 ) 4 + q ('1 (y1 )

(x; y; v) '2 (v) (x; y1 ; v1 ) 1 (x; y1 ; v1 ) '2 (v1 ) (x; y; v) '2 (v1 )'1 (y) '2 (v) (x; y1 ; v1 ) '2 (v)'1 (y1 ) '2 (v1 ) (x; y; v)

'1 (y2 )) '3 (z):

(24)

From Assumptions A1-A5, we get that the …rst and second terms of Eq. (24) are less than or equal zero. Since the geometrical mean is less than or equal to the arithmetical mean, the third term of Eq. (24) is also

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a (R1 1) 0. It follows that, q : 1 0 for all x; y; v; z > 0. z(t) = (r'1 (y) ) '3 (z) 0, i.e. '1 (y1 ) '1 (y2 ). It follows from above that, dU dt dU1 The solutions of system (5)-(8) limited to , the largest invariant subset of (x; y; v; z) : dt = 0 [22]. We have dU1 contains a unique point, that is dt = 0 if and only if x(t) = x1 , y(t) = y1 , v(t) = v1 and z(t) = 0. So, E1 . Thus, the global asymptotic stability of the chronic-infection equilibrium without CTL immune response E1 follows from LaSalle’s invariance principle. Theorem 3. Let Assumptions A1-A5 hold true and R1 > 1, then the chronic-infection equilibrium with less than or equal zero. Now, if R1

1, then E2 does not exist since z2 =

CTL immune response E2 is GAS in : Proof. We construct a Lyapunov functional as follows: Z

Zy (x2 ; y2 ; v2 ) '1 (y2 ) U2 (x; y; v; z) = x x2 d + y y2 d ( ; y ; v ) '1 ( ) 2 2 x2 y2 0 0 1 Zv '2 (v2 ) A q @ a + q'3 (z2 ) @ z z2 v v2 d + + k '2 ( ) r x

Zz

z2

v2

1 '3 (z2 ) A d : '3 ( )

(25)

We have U2 (x; y; v; z) > 0 for all x; y; v; z > 0 and U2 (x2 ; y2 ; v2 ; z2 ) = 0. Calculating the derivative of U2 along positive solutions of (5)-(8) gives us the following dU2 = dt +

(x2 ; y2 ; v2 ) '1 (y2 ) (n(x) (x; y; v)) + 1 ( (x; y; v) a'1 (y) q'1 (y)'3 (z)) (x; y2 ; v2 ) '1 (y) '2 (v2 ) q '3 (z2 ) a + q'3 (z2 ) 1 (k'1 (y) c'2 (v)) + 1 (r'1 (y)'3 (z) '3 (z)): k '2 (v) r '3 (z)

1

(26)

Collecting terms of Eq. (26) and using n(x2 ) = (x2 ; y2 ; v2 ) we obtain dU2 = (n(x) dt

(x2 ; y2 ; v2 ) (x2 ; y2 ; v2 ) + (x2 ; y2 ; v2 ) (x2 ; y2 ; v2 ) (x; y2 ; v2 ) (x; y2 ; v2 ) (x2 ; y2 ; v2 ) '1 (y2 ) (x; y; v) ac '2 (v2 ) + (x; y; v) + a'1 (y2 ) + q'1 (y2 )'3 (z) '2 (v) a'1 (y) (x; y2 ; v2 ) '1 (y) k '2 (v) ac qc '2 (v2 ) qc q q + '2 (v2 ) '3 (z2 )'2 (v) q'3 (z2 )'1 (y) + '3 (z2 )'2 (v2 ) '3 (z) + '3 (z2 ): (27) k k '2 (v) k r r n(x2 )) 1

By using the equilibrium conditions of E2 (x2 ; y2 ; v2 ) = a'1 (y2 ) + q'1 (y2 )'3 (z2 ) =

qc ac '2 (v2 ) + '3 (z2 )'2 (v2 ); k k

'1 (y2 ) =

r

;

we obtain dU2 = (n(x) dt

n(x2 )) 1

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

+ 3 (x2 ; y2 ; v2 )

(x2 ; y2 ; v2 )

(x2 ; y2 ; v2 ) (x; y2 ; v2 )

(x; y; v) '1 (y2 ) (x; y; v) (x2 ; y2 ; v2 ) (x; y2 ; v2 ) '1 (y) (x2 ; y2 ; v2 ) '2 (v2 )'1 (y) '2 (v2 )'1 (y) '2 (v) a'1 (y2 ) q'1 (y2 )'3 (z2 ) : (x2 ; y2 ; v2 ) '2 (v2 ) '2 (v)'1 (y2 ) '2 (v)'1 (y2 )

+ (x2 ; y2 ; v2 )

(28)

Collecting terms of Eq. (28), we get dU2 = (n(x) dt

n(x2 )) 1

+ (x2 ; y2 ; v2 ) 4

(x2 ; y2 ; v2 ) + (x2 ; y2 ; v2 ) (x; y2 ; v2 ) (x2 ; y2 ; v2 ) '1 (y2 ) (x; y; v) (x; y2 ; v2 ) '1 (y) (x2 ; y2 ; v2 )

(x; y; v) '2 (v) '2 (v) (x; y2 ; v2 ) 1+ (x; y2 ; v2 ) '2 (v2 ) '2 (v2 ) (x; y; v) '2 (v2 )'1 (y) '2 (v) (x; y2 ; v2 ) : (29) '2 (v)'1 (y2 ) '2 (v2 ) (x; y; v)

We can rewrite (29) as dU2 = (n(x) dt

n(x2 )) 1

+ (x2 ; y2 ; v2 ) 4

(x2 ; y2 ; v2 ) + (x2 ; y2 ; v2 ) (x; y2 ; v2 ) (x2 ; y2 ; v2 ) '1 (y2 ) (x; y; v) (x; y2 ; v2 ) '1 (y) (x2 ; y2 ; v2 )

584

(x; y; v) '2 (v) (x; y2 ; v2 ) 1 (x; y2 ; v2 ) '2 (v2 ) (x; y; v) '2 (v2 )'1 (y) '2 (v) (x; y2 ; v2 ) : '2 (v)'1 (y2 ) '2 (v2 ) (x; y; v)

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We note that from Assumptions A1-A5 and the relationship between the arithmetical and geometrical means, 2 we have dU 0 for all x; y; v; z > 0: The solutions of model (5)-(8) limited to , the largest invariant subset dt dU2 2 of (x; y; v; z) : dU dt = 0 [22]. We have dt = 0 if and only if x(t) = x2 , y(t) = y2 and v(t) = v2 for all t: Therefore, if v(t) = v2 and y(t) = y2 , then from Eq. (6), (x2 ; y2 ; v2 ) a'1 (y2 ) q'1 (y2 )'3 (z(t)) = 0, which 2 gives z(t) = z2 for all t. Thus, dU dt = 0 occurs at E2 . The global asymptotic stability of the chronic-infection equilibrium with CTL immune response E2 follows from LaSalle’s invariance principle.

3

Conclusion

In this paper, we have proposed and analyzed a nonlinear viral infection model with CTL immune response. We have considered more general nonlinear functions for the: (i) intrinsic growth rate of uninfected cells; (ii) incidence rate of infection; (iii) natural death rate of infected cells; (iv) rate at which the infected cells are killed by CTL cells; (v) production and removal rates of viruses; (vi) activation and natural death rates of CTLs. We have derived a set of conditions on these general functions and have determined two threshold parameters to prove the existence and the global stability of the model’s equilibria. The global asymptotic stability of the three equilibria, infection-free, chronic-infection without CTL immune response and chronic-infection with CTL immune response has been proven by using direct Lyapunov method and LaSalle’s invariance principle.

4

Acknowledgements

This article was funded by the Deanship of Scienti…c Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and …nancial support.

References [1] M.A. Nowak and C.R.M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [2] M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford Uni., Oxford, (2000). [3] P. K. Roy, A. N. Chatterjee, D. Greenhalgh and Q. J.A. Khan, Long term dynamics in a mathematical model of HIV-1 infection with delay in di¤ erent variants of the basic drug therapy model, Nonlinear Anal. Real World Appl., 14 (2013), 1621-1633. [4] A. S. Alsheri, A.M. Elaiw and M. A. Alghamdi, Global dynamics of two target cells HIV infection model with Beddington-DeAngelis functional response and delay-discrete or distributed, J. Comput. Anal. Appl, 17 (2014), 187-202. [5] A. M. Elaiw, A. S. Alsheri and M. A. Alghamdi, Global properties of HIV infection models with nonlinear incidence rate and delay-discrete or distributed, J. Comput. Anal. Appl, 17 (2014), 230-244. [6] A. Alhejelan and A. M. Elaiw, Global dynamics of virus infection model with humoral immune response and distributed delays, J. Comput. Anal. Appl, 17 (2014), 515-523. [7] A.M. Elaiw and M. A. Alghamdi, Global analysis for delay virus infection model with multitarget cells, J. Comput. Anal. Appl, 17 (2014), 187-202. [8] A.M. Elaiw and S.A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Method Appl. Sci., 36 (2013), 383-394.

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[9] A.M. Elaiw, I. A. Hassanien and S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49 (2012), 779-794. [10] A.M. Elaiw, Global dynamics of an HIV infection model with two classes of target cells and distributed delays, Discrete Dyn. Nat. Soc., (2012), Article ID 253703. [11] A.M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263. [12] M.Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J Appl Math, 70 (2010), 2434-2448. [13] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynam., 69 (2012) 423-435. [14] A. M. Elaiw, A. Alhejelan and M. A. Alghamdi, Mathematical analysis of a humoral immunity virus infection model with Crowley-Martin functional response and distributed delays, J. Comput. Anal. Appl, 18 (3) (2015), 524-535. [15] P. Georgescu and Y.H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J Appl Math, 67 (2006), 337-353. [16] A. M. Elaiw, R. M. Abukwaik and E. O. Alzahrani, Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays, Int. J. Biomath., 7 (5) (2014) 1450055, 25 pages. [17] X. Shi, X. Zhou, and X. Son, Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Anal. Real World Appl., 11 (2010), 1795-1809. [18] C. Lv, L. Huang, and Z. Yuan, Global stability for an HIV-1 infection model with Beddington–DeAngelis incidence rate and CTL immune response, Communications in Nonlinear Science and Numerical Simulation, 19, (2014) 121-127. [19] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with in…nitely distributed intracellular delays and CTL imune responses, SIAM J. Appl. Math., 73 (3) (2013), 1280-1302. [20] X. Wang, A. M. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Appl. Math. Comput., 218 (18) (2012), 9405-9414. [21] R. Larson and B. H. Edwards, Calculus of a single variable, Cengage Learning, Inc., USA, (2010). [22] J.K. Hale and S. Verduyn Lunel, Introduction to functional di¤ erential equations, Springer-Verlag, New York, (1993).

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ON THE STABILITY OF CUBIC LIE ∗-DERIVATIONS DONGSEUNG KANG Abstract. We will show the general solution of the functional equation f (sx + y) + f (x − sy) − s2 f (x + y) − sf (x − y) = (s − 1)(s2 − 1)f (x) − (s+1)(s2 −1)f (y) and investigate the stability of cubic Lie ∗-derivations associated with the given functional equation.

1. Introduction The concept of stability problem of a functional equation was first posed by Ulam [14] concerning the stability of group homomorphisms as follows: Let G1 be a group and let G2 be a metric group with the metric d(·, ·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1 ? In other words, we are looking for situations when the homomorphisms are stable, i.e., if a mapping is almost a homomorphism then there exists a true homomorphism near it. By the problem raised by Ulam, several stability problems of a large variety of functional equations have been extensively studied and generalized by a number of authors and many interesting results have been obtained for the last nearly fifty years. For further information about the topic, we refer the reader to [9], [5], [1] and [2]. Recall that a Banach ∗-algebra is a Banach algebra (complete normed algebra) which has an isometric involution. Jang and Park [6] investigated the stability of ∗-derivations and of quadratic ∗-derivations with Cauchy functional equation and the Jensen functional equation on Banach ∗-algebra. The stability of ∗-derivations on Banach ∗-algebra by using fixed point alternative was proved by Park and Bodaghi and also Yang et al.; see [12] and [15], respectively. Also, the stability of cubic Lie derivations was introduced by Foˇsner and Foˇsner; see [4]. Jun and Kim [8] introduced the following cubic functional equation: f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x) and established a general solution. Najati [11] investigated the following generalized cubic functional equation: (1.1)

f (sx + y) + f (sx − y) = sf (x + y) + sf (x − y) + 2(s3 − s)f (x)

2000 Mathematics Subject Classification. 39B55; 47B47; 39B72. Keywords : Hyers-Ulam-Rassias Stability, Cubic Mapping, Lie ∗-Derivation, Fixed Point Alternative, Banach ∗-Algebra. 1

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for a positive integer s ≥ 2 . Also, Jun and Kim [7] proved the Hyers-UlamRassias stability of a Euler-Lagrange type cubic mapping as follows: (1.2)

f (sx + y) + f (x + sy)

= (s + 1)(s − 1)2 [f (x) + f (y)] + s(s + 1)f (x + y) , where s ∈ Z (s 6= 0 , ±1) . In this paper, we deal with the following the functional equation: (1.3)

f (sx + y) + f (x − sy) − s2 f (x + y) − sf (x − y)

= (s − 1)(s2 − 1)f (x) − (s + 1)(s2 − 1)f (y) for all s ∈ Z (s 6= 0 , ±1) . We will show the general solution of the functional equation (1.3) and investigate the stability of cubic Lie ∗-derivations associated with the given functional equation on normed algebras. 2. Cubic Functional Equations In this section let X and Y be vector spaces and we investigate the general solution of the functional equation (1.3). Theorem 2.1. A function f : X → Y satisfies the functional equation (1.3) if and only if it satisfies the functional equation (1.1). Proof. Suppose that f satisfies the equation (1.3). It is easy to show that f (0) = 0 , f (sx) = s3 f (x) for all x ∈ X and all s ∈ Z (s 6= 0 , ±1) . Letting x = −x in the equation (1.3), we have (2.1)

−f (sx − y) − f (x + sy) + (s + 1)(s2 − 1)f (y)

= −s2 f (x − y) − sf (x + y) − (s − 1)(s2 − 1)f (x) for all x , y ∈ X . Replacing x and y in the equation (2.1), we get (2.2)

f (x − sy) − f (sx + y) + (s + 1)(s2 − 1)f (x)

= s2 f (x − y) − sf (x + y) − (s − 1)(s2 − 1)f (y) for all x , y ∈ X . Subtracting the equation (2.2) from the equation (1.3), we obtain (2.3)

2f (sx + y) + 2(s2 − 1)f (y)

= (s2 + s)f (x + y) + (s − s2 )f (x − y) + 2s(s2 − 1)f (x) for all x , y ∈ X . Now, letting y = −y in the equation (2.3) (2.4)

2f (sx − y) − 2(s2 − 1)f (y)

= (s2 + s)f (x − y) + (s − s2 )f (x + y) + 2s(s2 − 1)f (x) for all x , y ∈ X . Adding two equations (2.3) and (2.4), we have (2.5) 2f (sx + y) + 2f (sx − y) = 2sf (x + y) + 2sf (x − y) + 4s(s2 − 1)f (x) for all x , y ∈ X . Thus we have the equation (1.1). Conversely, suppose that f satisfies the equation (1.1). It is easy to see that f (0) = 0 , f (sx) = s3 f (x)

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for all x ∈ X and all s ∈ Z(s 6= 0) . Letting y = sy in the equation (1.1), we have (2.6)

f (x + sy) + f (x − sy) = s2 f (x + y) + s2 f (x − y) − 2(s2 − 1)f (x)

for all x , y ∈ X . Replacing x and y in the equation (2.6), we get (2.7)

f (sx + y) − f (sx − y) = s2 f (x + y) − s2 f (x − y) − 2(s2 − 1)f (y)

for all x , y ∈ X . By adding two equations (1.1) and (2.7), we obtain (2.8) 2f (sx+y) = (s2 +s)f (x+y)+(s−s2 )f (x−y)+2s(s2 −1)f (x)−2(s2 −1)f (y) for all x , y ∈ X . Now, letting y = sy in the equation (2.7), we have (2.9)

f (x + sy) − f (x − sy) = sf (x + y) − sf (x − y) + 2s(s2 − 1)f (y)

for all x , y ∈ X . Subtracting the equation (2.9) from the equation (2.6), we know that (2.10) 2f (x−sy) = (s2 −s)f (x+y)+(s2 +s)f (x−y)−2(s2 −1)f (x)−2s(s2 −1)f (y) for all x , y ∈ X . By adding two equations (2.8) and (2.10), we have the desired equation (1.3).  3. Cubic Lie ∗-Derivations Throughout this section, we assume that A is a complex normed ∗-algebra and M is a Banach A-bimodule. We will use the same symbol || · || as norms on a normed algebra A and a normed A-bimodule M . A mapping f : A → M is a cubic homogeneous mapping if f (µa) = µ3 f (a) , for all a ∈ A and µ ∈ C . A cubic homogeneous mapping f : A → M is called a cubic derivation if f (xy) = f (x)y 3 + x3 f (y) holds for all x , y ∈ A . For all x , y ∈ A , the symbol [x, y] will denote the commutator xy−yx . We say that a cubic homogeneous mapping f : A → M is a cubic Lie derivation if f ([x, y]) = [f (x), y 3 ] + [x3 , f (y)] for all x, y ∈ A . In addition, if f satisfies in condition f (x∗ ) = f (x)∗ for all x ∈ A , then it is called the cubic Lie ∗-derivation. Example 3.1. Let A = C be a complex field endowed with the map z 7→ z ∗ = z¯ (where z¯ is the complex conjugate of z). We define f : A → A by f (a) = a3 for all a ∈ A . Then f is cubic and f ([a, b]) = [f (a), b3 ] + [a3 , f (b)] = 0 for all a ∈ A . Also, ¯ = f (a)∗ f (a∗ ) = f (¯ a) = a ¯3 = f (a) for all a ∈ A . Thus f is a cubic Lie ∗-derivation.

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In the following, T1 will stand for the set of all complex units, that is, T1 = {µ ∈ C | |µ| = 1} . For the given mapping f : A → M , we consider (3.1) ∆µ f (x, y) := f (sµx + µy) + f (µx − sµy) − s2 µ3 f (x + y) − sµ3 f (x − y) −µ3 (s − 1)(s2 − 1)f (x) + µ3 (s + 1)(s2 − 1)f (y) , ∆f (x, y) := f ([x, y]) − [f (x), y 3 ] − [x3 , f (y)] for all x, y ∈ A , µ ∈ C and s ∈ Z (s 6= 0 , ±1) . Theorem 3.2. Suppose that f : A → M is a mapping with f (0) = 0 for which there exists a function φ : A5 → [0, ∞) such that ∞ X 1 e b, x, y, z) := (3.2) φ(a, φ(sj a, sj b, sj x, sj y, sj z) < ∞ |s|3j j=0

(3.3)

||∆µ f (a, b)|| ≤ φ(a, b, 0, 0, 0)

(3.4)

||∆f (x, y) + f (z ∗ ) − f (z)∗ || ≤ φ(0, 0, x, y, z)

for all µ ∈ T11 = {eiθ | 0 ≤ θ ≤ n0

2π n0 }

and all a, b, x, y, z ∈ A in which

n0 ∈ N . Also, if for each fixed a ∈ A the mapping r 7→ f (ra) from R to M is continuous, then there exists a unique cubic Lie ∗-derivation L : A → M satisfying 1 e (3.5) ||f (a) − L(a)|| ≤ 3 φ(a, 0, 0, 0, 0) . |s| Proof. Let b = 0 and µ = 1 in the inequality (3.3), we have 1 1 (3.6) ||f (a) − 3 f (sa)|| ≤ 3 φ(a, 0, 0, 0, 0) s |s| for all a ∈ A . Using the induction, it is easy to show that (3.7)

||

t−1 1 1 1 X φ(sj a, 0, 0, 0, 0) t k f (s a) − f (s a)|| ≤ s3t |s|3 |s|3j s3k j=k

for t > k ≥ 0 and a ∈ A . The inequalities (3.2) and (3.7) imply that the 1 sequence { s3n f (sn a)}∞ n=0 is a Cauchy sequence. Since M is complete, the sequence is convergent. Hence we can define a mapping L : A → M as 1 (3.8) L(a) = lim 3n f (sn a) n→∞ s for a ∈ A . By letting t = n and k = 0 in the inequality (3.7), we have (3.9)

||

n−1 1 1 X φ(sj a, 0, 0, 0, 0) n f (s a) − f (a)|| ≤ s3n |s|3 |s|3j j=0

for n > 0 and a ∈ A . By taking n → ∞ in the inequality (3.9), the inequalities (3.2) implies that the inequality (3.5) holds.

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Now, we will show that the mapping L is a unique cubic Lie ∗-derivation such that the inequality (3.5) holds for all a ∈ A . We note that (3.10)

||∆µ L(a, b)|| = lim

n→∞

1 ||∆µ f (sn a, sn b)|| |s|3n

φ(sn a, sn b, 0, 0, 0) = 0, n→∞ |s|3n

≤ lim

for all a, b ∈ A and µ ∈ T11 . By taking µ = 1 in the inequality (3.10), it n0

follows that the mapping L is a Euler-Lagrange cubic mapping. Also, the inequality (3.10) implies that ∆µ L(a, 0) = 0 . Hence L(µa) = µ3 L(a) for all a ∈ A and µ ∈ T11 . Let µ ∈ T1 = {λ ∈ C | |λ| = 1} . Then µ = eiθ , n0

iθ

1

where 0 ≤ θ ≤ 2π . Let µ1 = µ n0 = e n0 . Hence we have µ1 ∈ T11 . Then n0

3 0 L(µa) = L(µn1 0 a) = µ3n 1 L(a) = µ L(a)

for all µ ∈ T1 and a ∈ A . Suppose that ρ is any continuous linear functional on A and a is a fixed element in A . Then we can define a function g : R → R by g(r) = ρ(L(ra)) for all r ∈ R . It is easy to check that g is cubic. Let  f (sk ra)  gk (r) = ρ s3k for all k ∈ N and r ∈ R . Note that g as the pointwise limit of the sequence of measurable functions gk is measurable. Hence g as a measurable cubic function is continuous (see [3]) and g(r) = r3 g(1) for all r ∈ R . Thus ρ(L(ra)) = g(r) = r3 g(1) = r3 ρ(L(a)) = ρ(r3 L(a)) for all r ∈ R . Since ρ was an arbitrary continuous linear functional on A we may conclude that L(ra) = r3 L(a) µ for all r ∈ R . Let µ ∈ C (µ 6= 0) . Then |µ| ∈ T1 . Hence µ   µ 3  µ 3 L(µa) = L |µ|a = L(|µ|a) = |µ|3 L(a) = µ3 L(a) |µ| |µ| |µ|

for all a ∈ A and µ ∈ C (µ 6= 0) . Since a was an arbitrary element in A , we may conclude that L is cubic homogeneous.

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Next, replacing x, y by sk x, sk y , respectively, and z = 0 in the inequality (3.4), we have ∆f (sn x, sn y) || n→∞ s3n 1 ≤ lim φ(0, 0, sn x, sn y, 0) = 0 n→∞ |s|3n

||∆L(x, y)|| =

lim ||

for all x, y ∈ A . Hence we have ∆L(x, y) = 0 for all x, y ∈ A . That is, L is a cubic Lie derivation. Letting x = y = 0 and replacing z by sk z in the inequality (3.4), we get f (sn z ∗ ) f (sn z)∗ φ(0, 0, 0, 0, sn z) (3.11) − ≤ 3n 3n s s |s|3n for all z ∈ A . As n → ∞ in the inequality (3.11), we have L(z ∗ ) = L(z)∗ for all z ∈ A . This means that L is a cubic Lie ∗-derivation. Now, assume L0 : A → A is another cubic ∗-derivation satisfying the inequality (3.5). Then 1 ||L(a) − L0 (a)|| = ||L(sn a) − L0 (sn a)|| |s|3n  1  n n n 0 n ≤ ||L(s a) − f (s a)|| + ||f (s a) − L (s a)|| |s|3n ∞ 1 X 1 ≤ φ(sj+n a, 0, 0, 0, 0) |s|3n+1 |s|3j j=0

=

1 |s|3

∞ X j=n

1 φ(sj a, 0, 0, 0, 0) , |s|3j

which tends to zero as k → ∞ , for all a ∈ A . Thus L(a) = L0 (a) for all a ∈ A . This proves the uniqueness of L .  Corollary 3.3. Let θ , r be positive real numbers with r < 3 and let f : A → M be a mapping with f (0) = 0 such that ||∆µ f (a, b)|| ≤ θ(||a||r + ||b||r ) ||∆f (x, y) + f (z ∗ ) − f (z)∗ || ≤ θ(||x||r + ||y||r + ||z||r ) for all µ ∈ T11 and a, b, x, y, z ∈ A . Then there exists a unique cubic Lie n0

∗-derivation L : A → M satisfying ||f (a) − L(a)|| ≤

θ||a||r |s|3 − |s|r

for all a ∈ A . Proof. The proof follows from Theorem 3.2 by taking φ(a, b, x, y, z) = θ(||a||r + ||b||r + ||x||r + ||y||r + ||z||r ) for all a, b, x, y, z ∈ A . 

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Now, we will investigate the stability of the given functional equation (3.1) using the alternative fixed point method. Before proceeding the proof, we will state the theorem, the alternative of fixed point; see [10] and [13]. Definition 3.4. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y ; (2) d(x, y) = d(y, x) for all x, y ∈ X ; (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X . Theorem 3.5 ( The alternative of fixed point [10], [13] ). Suppose that we are given a complete generalized metric space (Ω, d) 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) = ∞ for all n ≥ 0 , or there exists a natural number n0 such that (1) d(T n x, T n+1 x) < ∞ for all n ≥ n0 ; (2) The sequence (T n x) is convergent to a fixed point y ∗ of T ; (3) y ∗ is the unique fixed point of T in the set 4 = {y ∈ Ω|d(T n0 x, y) < ∞} ; (4) d(y, y ∗ ) ≤

1 1−l

d(y, T y) for all y ∈ 4 .

Theorem 3.6. Let f : A → M be a continuous mapping with f (0) = 0 and let φ : A5 → [0, ∞) be a continuous mapping such that (3.12)

||∆µ f (a, b)|| ≤ φ(a, b, 0, 0, 0)

(3.13)

||∆f (x, y) + f (z ∗ ) − f (z)∗ || ≤ φ(0, 0, x, y, z)

for all µ ∈ T11 and a, b, x, y, z ∈ A . If there exists a constant l ∈ (0, 1) such that

n0

(3.14)

φ(sa, sb, sx, sy, sz) ≤ |s|3 lφ(a, b, x, y, z)

for all a, b, x, y, z ∈ A , then there exists a cubic Lie ∗-derivation L : A → M satisfying (3.15)

||f (a) − L(a)|| ≤

1 |s|3 (1

− l)

φ(a, 0, 0, 0, 0)

for all a ∈ A . Proof. Consider the set Ω = {g | g : A → A , g(0) = 0} and introduce the generalized metric on Ω , d(g, h) = inf {c ∈ (0, ∞) | k g(a) − h(a) k≤ cφ(a, 0, 0, 0, 0) , for all a ∈ A} .

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It is easy to show that (Ω, d) is complete. Now we define a function T : Ω → Ω by 1 (3.16) T (g)(a) = 3 g(sa) s for all a ∈ A . Note that for all g, h ∈ Ω , let c ∈ (0, ∞) be an arbitrary constant with d(g, h) ≤ c . Then ||g(a) − h(a)|| ≤ cφ(a, 0, 0, 0, 0)

(3.17)

for all a ∈ A . Letting a = sa in the inequality (3.17) and using (3.14) and (3.16), we have 1 ||g(sa) − h(sa)|| |s|3 1 c φ(sa, 0, 0, 0, 0) ≤ c l φ(a, 0, 0, 0, 0) , |s|3

||T (g)(a) − T (h)(a)|| = ≤ that is,

d(T g, T h) ≤ c l . Hence we have that d(T g, T h) ≤ l d(g, h) , for all g, h ∈ Ω , that is, T is a strictly self-mapping of Ω with the Lipschitz constant l . Letting µ = 1 , b = 0 in the inequality (3.12), we get ||

1 1 f (sa) − f (a)|| ≤ 3 φ(a, 0, 0, 0, 0) 3 s |s|

for all a ∈ A . This means that d(T f, f ) ≤

1 . |s|3

We can apply the alternative of fixed point and since limn→∞ d(T n f, L) = 0 , there exists a fixed point L of T in Ω such that f (sn a) , n→∞ s3n

(3.18)

L(a) = lim

for all a ∈ A . Hence d(f, L) ≤

1 1 1 d(T f, f ) ≤ 3 . 1−l |s| 1 − l

This implies that the inequality (3.15) holds for all a ∈ A . Since l ∈ (0, 1) , the inequality (3.14) shows that (3.19)

φ(sn a, sn b, sn x, sn y, sn z) = 0. n→∞ |s|3n lim

Replacing a , b by sn a , sn b , respectively, in the inequality (3.12), we have 1 φ(sn a, sn b, 0, 0, 0) ||∆µ f (sn a, sn b)|| ≤ . 3n |s| |s|3n

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9

Taking the limit as k tend to infinity, we have ∆µ f (a, b) = 0 for all a , b ∈ A and all µ ∈ T11 . The remains are similar to the proof of Theorem 3.2.  n0

Corollary 3.7. Let θ , r be positive real numbers with r < 3 and let f : A → M be a mapping with f (0) = 0 such that ||∆µ f (a, b)|| ≤ θ(||a||r + ||b||r ) ||∆f (x, y) + f (z ∗ ) − f (z)∗ || ≤ θ(||x||r + ||y||r + ||z||r ) for all µ ∈ T11 and a, b, x, y, z ∈ A . Then there exists a unique cubic Lie n0

∗-derivation L : A → M satisfying ||f (a) − L(a)|| ≤

θ||a||r |s|3 (1 − l)

for all a ∈ A . Proof. The proof follows from Theorem 3.6 by taking φ(a, b, x, y, z) = θ(||a||r + ||b||r + ||x||r + ||y||r + ||z||r ) for all a, b, x, y, z ∈ A .  Acknowledgement The present research was conducted by the research fund of Dankook University in 2015. References [1] N. Brillou¨et-Belluot, J. Brzd¸ek and K. Ciepli´ nski, Fixed Point Theory and the Ulam Stability, Abstract and Applied Analysis 2014, Article ID 829419, 16 pages, (2014). [2] J. Brzd¸ek, L. Cˇ adariu and K. Ciepli´ nski, On Some Recent Developments in Ulam’s Type Stability, Abstract and Applied Analysis 2012, Article ID 716936, 41 pages, (2012). [3] St. Czerwik, On the Stability of the Quadratic Mapping in Normed Spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. [4] A. Foˇsner and M. Foˇsner, Approximate cubic Lie derivations, Abstract and Applied Analysis 2013, Article ID 425784, 5 pages, (2013). [5] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Mass, USA, (1998). [6] S. Jang and C. Park, Approximate ∗-derivations and approximate quadratic ∗derivations on C ∗ -algebra, J. Inequal. Appl. 2011 Articla ID 55 (2011). [7] K.-W. Jun and H.-M. Kim, On the stability of Euler-Lagrange type cubic functional equations in quasi-Banach spaces, J. Math. Anal. Appl. 332 (2007), 1335–1350. [8] K.-W. Jun and H.-M. Kim, The generalized Hyer-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878. [9] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, (2011). [10] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126, 74(1968), 305–309. [11] A. Najati, The Generalized Hyers-Ulam-Rassias Stability of a Cubic Functional Equation, Turk. J. Math. 31 (2007) 395–408.

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DONGSEUNG KANG

[12] C. Park and A. Bodaghi, On the stability of ∗-derivations on Banach ∗-algebras, Adv. Diff. Equat. 2012 2012:138(2012). [13] I.A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, ClujNapoca, 1979 (in Romanian). [14] S. M. Ulam, Problems in Morden Mathematics, Wiley, New York (1960). [15] S.Y. Yang, A. Bodaghi, K.A.M. Atan,Approximate cubic *-derivations on Banach *-algebra, Abstract and Applied Analysis, 2012, Article ID 684179, 12 pages, (2012). Department of Mathematical Education, Dankook University, 152, Jukjeon, Suji, Yongin, Gyeonggi, South Korea 448-701 E-mail address: [email protected] (D. Kang)

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Fuzzy share functions for cooperative fuzzy games† Zeng-Tai Gonga,∗ , Qian Wanga,b a College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China b College of Mathematics and Computer Science, Northwest University for Nationalities,Lanzhou 730030,China

Abstract In this paper, the concept of fuzzy share functions of cooperative fuzzy games with fuzzy characteristic functions is proposed. Players in the proposed cooperative fuzzy game do not need to know precise information about the payoff value. We generalize the axiom of additivity by introducing a positive fuzzy value function µ ˜ on the class of cooperative fuzzy games in fuzzy characteristic function form. The so-called axiom of µ ˜-additivity generalizes the classical axiom of additivity by putting the weight µ ˜(˜ v ) on the value of the game v˜. We show that any additive function µ ˜ determines a unique fuzzy share function satisfying the axioms of efficient shares, null player property, symmetry and µ ˜-additivity on the subclass of games on which µ ˜ is positive and which contains all positively scaled unanimity games. Finally, we introduce the fuzzy Shapley share functions and fuzzy Banzhaf share functions for the cooperative fuzzy games with fuzzy characteristic functions. Keywords: Cooperative fuzzy game; Fuzzy share functions; Characteristic functions; Fuzzy numbers. 1. Introduction A cooperative game with transferable utility, or simply a TU-game, is a finite set of players N and for any subset (coalition) of players a worth representing the total payoff that the coalition can obtain by cooperating. A value function for TU-games is a function that assigns to every TU-game with n players an n-dimensional vector representing a distribution of payoffs among the players. A value function is efficient if for every game it distributes exactly the worth of the ’grand coalition’, N , over all players. The most famous efficient value function is the Shapley value[16]. An example of a value function that is not efficient is the Banzhaf value[3, 8, 14]. Since the Banzhaf value is not efficient, it is not adequate in allocating the worth v(N ). In order to allocate v(N ) and according to the Banzhaf value, Van der Laan et al. in [18] characterize the normalized Banzhaf value, which distributes the worth v(N ) proportional to the Banzhaf values of the players. A different approach to efficiently allocate the worth v(N ) is described in [19], who introduce share functions as an alternative type of solution for TU-games. A share vector for an n-player game is an n-dimensional real vector whose components add up to one. The ith component is player i’s share in the total payoff that is to be distributed among the players. A share function assigns such a share vector to every game. The share function corresponding to the Shapley value is the Shapley share function, which is obtained by dividing the Shapley value of each player by v(N ), i.e., by the sum of the Shapley values of all players. Similarly, the Banzhaf share function is obtained by dividing the Banzhaf-value or normalized Banzhaf-value by the corresponding sum of payoffs over all players. One advantage of share functions over value functions is that share functions avoid the ”efficiency issue”, i.e., they avoid the question of what is the final worth to be distributed over the players. This yields some major simplifications. For example, although the Banzhaf and normalized Banzhaf value are very different value functions (e.g. the Banzhaf value satisfies linearity and the dummy player property which are not satisfied by the normalized Banzhaf value), they correspond to the same Banzhaf share function. Another main advantage of share functions has been discovered by [15], who shows that on a ratio scale meaningful statements can be made for a certain class of share functions, whereas all statements with respect to value functions are meaningless. Besides the advantages of share functions for general TU-games, in [2, 20] they study share † ∗

Supported by the Natural Scientific Fund of China (61262022, 11461062). Corresponding Author: Zeng-Tai Gong. Tel.:+86 09317971430. E-mail addresses: [email protected], [email protected](Zeng-Tai Gong) and [email protected](Qian Wang). 597

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Zeng-Tai Gong and Qian Wang: Fuzzy share functions for cooperative fuzzy games

functions for so-called games in coalition structure, for which an extra advantage is that they provide a natural method to define solutions for such games. Share functions, when multiplied by the worth of the grand coalition N , yield a distribution of the worth of the grand coalition reflecting the individual bargaining position of the players. Mares and Vlach [12, 13] were concerned about the uncertainty in the value of the characteristic function associated with a game. In their models, the domain of the characteristic function of a game remains to be the class of crisp (deterministic) coalitions but the values assigned to them are fuzzy quantities. However, the implicit assumption that all players and coalitions know the expected payoffs even before the negotiation process, is evidently unrealistic. In fact, during the process of negotiation and coalition forming, the players can have only vague idea about the real outcome of the situation, and this vague expectation can be modeled by mathematical tools (see [12]). In this paper, we consider the fuzzy share functions of a cooperative fuzzy game with fuzzy characteristic function. The paper will be organized as follows. In Section 2, we introduce the concepts of fuzzy numbers and the Hukuhara difference on fuzzy numbers. Then, the model of cooperative fuzzy games is introduced. Moreover, some basic concepts of crisp games will be discussed. In Section 3, the fuzzy share functions of cooperative fuzzy games with fuzzy characteristic function is proposed, we generalize the axiom of additivity by introducing a positive fuzzy valued function µ ˜ on the class of cooperative fuzzy games in fuzzy characteristic function form. The so-called axiom of µ ˜-additivity generalizes the classical axiom of additivity by putting the weight µ ˜(˜ v ) on the value of the game v˜. We show that any additive function µ ˜ determines a unique fuzzy share function satisfying the axioms of efficient shares, null player property, symmetry and µ ˜-additivity on the subclass of games on which µ ˜ is positive and which contains all positively scaled unanimity games. In Section 4, we introduce fuzzy Shapley share functions and fuzzy Banzhaf share functions, furthermore, an applicable example is given. Finally, some conclusions will be discussed in Section 5. 2. Preliminaries In this section, we first recall the concept of fuzzy number, and then introduce some basic concepts and notations in cooperative games with fuzzy characteristic functions. 2.1 A review of fuzzy numbers Let us start by recalling the most general definition of a fuzzy number. Let R be (−∞, +∞), i.e., the set of all real numbers. Definition 2.1. A fuzzy number, denoted by a ˜, is a fuzzy subset of R with membership function ua˜ : R → [0, 1] satisfying the following conditions: (1) there exists at least one number a0 ∈ R such that ua˜ (a0 ) = 1; (2) ua˜ (x) is nondecreasing on (−∞, a0 ) and nonincreasing on (a0 , +∞); (3) ua˜ (x) is upper semi-continuous, i.e., limx→x+ ua˜ (x) = ua˜ (x0 ) if x0 < a0 ; and limx→x− ua˜ (x) = 0 0 ua˜ (x0 ) if x0 > a0 ; (4) Supp(ua˜ ), the support set of a ˜, is compact, where Supp(ua˜ ) = cl{x ∈ (R)|ua˜ (x) > 0}. We denote the set of all fuzzy numbers by 0, then Jaα+ is bounded from Cγ,ln [a, b] into Cγ−α,ln [a, b] : φ (t) = δ n g (t) , dk =

(



α

Ja+ g

Cγ−α,ln

(

≤ k ∥g∥Cγ,ln , k =

b ln a

)Re(α)

Γ (1 − γ) . Γ (1 + α − γ)

)

In particular Jaα+ is bounded in Cγ,ln [a, b] . ( ) b. If γ ≤ α, then Jaα+ is bounded from Cγ,ln [a, b] into C [a, b] : )Re(α)−γ (

Γ (1 − γ) b

α .

Ja+ g ≤ k ∥g∥Cγ,ln , k = ln a Γ (1 + α − γ) C ( ) In particular Jaα+ is bounded in Cγ,ln [a, b] . ( ) Lemma 2.7 ([10]). The fractional operator Jaα+ represents a mapping from C [a, b] to C [a, b] and



α

Ja+ g ≤ C

1 Re (α) Γ (α)

(

b ln a

)Re(α) ∥g∥C .

Theorem 2.8 (Banach fixed point Theorem, [10]). Let (X, d) be a nonempty complete metric space, let 0 ≤ w < 1, and let T : X −→ X be a map such that for every x, x ˜ ∈ X, the relation d (T x, T x ˜) ≤ wd (x, x ˜) , holds. Then the operator T has a uniquely defined fixed point x∗ ∈ X. Furthermore, if T k (k ∈ N) is the sequence defined by T 1 = T, T k = T T k−1 (k ∈ N − {1}) , { }k=∞ then, for any x0 ∈ X T k x0 k=1 converges to the above fixed point x∗ . Definition 2.9 ([10]). Let l ∈ N, G ⊂ Rl , [a, b] ⊂ R, g : [a, b] × G −→ R be a function such that, for any (x1 , ..., xl ) , (˜ x1 , ..., x ˜l ) ∈ G, g satisfies generalized Lipschitizian condition: |g [t, x1 , ..., xl ] − g [t, x ˜, x ˜1 , ..., x ˜l ]| ≤ A1 |x1 − x ˜1 | + ... + Al |xl − x ˜l | , Aj ≥ 0, j = 1, ..., l. (6) In particular,g satisfies the Lipschitzian condition with respect to the second variable if for all t ∈ (a, b] and for any x, x ˜ ∈ G one has |g [t, x] − g [t, x ˜]| ≤ A |x − x ˜| , A > 0.

(7)

4

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3

Nonlinear Cauchy problem

α,r In this section, we present the existence and uniqueness results in the space Cδ,γ,ln [a, b] of the Cauchy problem for the nonlinear fractional differential equation in the frame of Caputo Hadamarad fractional derivative. That is we consider the equation ( ) C α Da+ x (t) = h [t, x (t)] , Re (α) > 0, t > a > 0, (8)

subject to the initial conditions ( k ) δ x (a+ ) = dk , dk ∈ R, k = 0, ..., n − 1, n = [Re(α)] + 1.

(9)

The Volterra type integral equation corresponding to problem (8)-(9) is : x(t) =

n−1 ∑ j=0

dj j!

( )j )α−1 ∫ t( t 1 t dτ ln + ln h [τ, x (τ )] , a ≤ t ≤ b. a Γ (α) a τ τ

In partuclar, if α = n ∈ N then the problem (8)-(9) is as follows: ( ) (δ n x) (t) = h [t, x (t)] , a ≤ t ≤ b, δ k x (a+ ) = dk ∈ R, k = 0, 1, ...n − 1.

(10)

(11)

The corresponding integral equation to the problem (11) has the form: x(t) =

n−1 ∑ j=0

dj j!

(

t ln a

)j

( ) + Jan+ h (t) , a ≤ t ≤ b.

(12)

Firstly, we we have to prove the equivalence of the Cauchy problem to the Volterra type integral equation in the sense that, if x ∈ Cδr [a, b] satisfies one of them, then it also satisfies the other one. Theorem 3.1. Let Re (α) > 0, n = [Re(α)] + 1, (0 < a < b < +∞), and 0 ≤ γ < 1 be such that α ≥ γ. Let G be an open set in R and let h : [a, b] × G −→ R be a function such that h [t, x] ∈ Cγ,ln [a, b] for any x ∈ Cγ,ln [a, b]. (i) Let r = n − 1 for α ∈ / N, if x ∈ Cδn−1 [a, b] then x satisfies the relations (8) and (9) iff x satisfies equation (10) . (ii) Let r = n for α ∈ N, if x ∈ Cδn [a, b] then x satisfies the relation (11) if and only if, x satisfies equation (12) . Proof. (i) Let α ∈ / N, n − 1 < α < n and x ∈ Cδn−1 [a, b] . (i.a) Here we prove the necessity. From definition of C Daα+ and (3) we obtain    n−1 ∑ δ j x (a) ( τ )j C α x (τ ) −  (t) . ln Da+ x (t) = (δ n ) Jan−α + j! a j=0 By hypothesis, h [t, x] ∈ Cγ,ln [a, b] and it follows from (8) that C Daα+ x (t) ∈ Cγ,ln [a, b] , and hence, by applying Lemma 2.5, we have    n−1 ∑ δ j x (a) ( t )j n Jan−α x (τ ) −  (t) ∈ Cδ,γ,ln ln [a, b] . + j! τ j=0 5

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By using Lemma 2.4, we obtain Jaα+

(

C

)

Daα+ x

(t) = x (t) −

n−1 ∑ j=0

δ j x (a) j!

( )j t ln . a

(13)

( ) In view of Lemma 2.6-(b) , Jaα+ h [t, x] belongs to the C [a, b] space, Applying Jaα+ to the both sides of (8) and utilizing (13), with respect to the initial conditions (9), we deduce that there exists a unique solution x ∈ Cδn−1 [a, b] to equation (10). (i.b) Let x ∈ Cδn−1 [a, b] satisfies the equation (10).

( ) – We want to show that x satisfies equation (8) . Applying Daα+ to both sides of (10) , and taking into account (4) , (9) , Property 2.2 and Property 2.3, we get   ) ( )α−1 ∫ t( n−1 ∑ δ j x (a) ( t )j t dτ 1 α  α  Da+ x (t) − ln ln , = Da+ h [τ, x (τ )] j! a Γ (α) a τ τ j=0 (

then

C

) ( )( ) Daα+ x (t) = Daα+ Jaα+ h (t) ≡ h [t, x (t)] .

– Now, we show that x satisfies the initial relations (9). We obtain by differentiation both sides of (10) that, ( )j−k )α−k−1 ∫ t( n−1 ∑ dj t 1 t δ k x (t) = ln + ln h [τ, x (τ )] dτ. (j − k)! a Γ (α − k) a τ j=k

Changing the variable τ = a

k

δ x (t) =

=

( )s t , yieldys a

)α−k−1 ( )j−k ∫ 1( dj t 1 t ln + ln ( t )s (j − k)! a Γ (α − k) 0 a a j=k [ ( )s ( ( )s )] ( ) ( )s t t t t ×h a ,x a a ln ds a a a a ( )α−k t ( ) [ ( )s ( ( )s )] ∫ 1 ln n−1 j−k ∑ dj t t t a α−k−1 ln + (1 − s) h a ,x a ds. (j − k)! a Γ (α − k) 0 a a n−1 ∑

j=k

for k = 0, ..., n − 1. Because α − k > n − 1 − k ≥ 0, using the continuity of h, Property 2.3 and Lemma 2.7 we get Jaα+ h [t, x] ∈ C [a, b], and taking a limit as t −→ a+ , we obtain δ k x (a+ ) = dk . (ii) For α ∈ N and x (t) ∈ Cδn [a, b] be the solution to the Cauchy problem (11). ( ) (ii.a) Firstly, we prove the necessity. Applying Jan+ to both sides of equation (11), using (4) and Lemma 2.4, we have n−1 ∑ δ k x (a) ( t )k ln = Jan+ h (t) , Jan+ δ n x (t) = x (t) − k! a k=0

since δ k x (a+ ) = dk , we arrive at equation (12) and hence the necessity is proved. 6

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(ii.b) If x ∈ Cδn [a, b] satisfies the equation (12), in addition, by term-by-term differentiation of (12) in the usual sense k times, we get δ k x (t) =

n−1 ∑ j=k

dj (j − k)!

( )j−k )n−k−1 ∫ t( t 1 t dτ ln + ln h [τ, x (τ )] , a (n − k − 1)! a τ τ

for k = 0, ..., n. Using Property 2.3 , taking the limit as t −→ a+ , we obtain δ k x (a+ ) = dk , and δ n x (t) = h [t, x (t)] . Thus the Theorem 3.1 is proved for α ∈ N. This completes the proof of the theorem. Corollary 3.2. Under the hypotheses of Theorem 3.1, with 0 < x ∈ Cδ [a, b] then x (t) satisfies the relation ( ) C α Da+ x (t) = h [t, x (t)] , t > a > 0, x (a) = d0 ,

Re (α) < 1, if

if and only if, x satisfies the equation ( ) x(t) = d0 + Jaα+ h (t) , a ≤ t ≤ b. The next step is to prove the existence of a unique solution to the Cauchy problem α,r (8)-(9) in the space of functions Cδ,γ,ln [a, b] by using the Banach’s fixed point theorem. Theorem 3.3. Let α > 0, ad n = [ℜ(α)] + 1, 0 ≤ γ < 1 be such that α ≥ γ. Let G be an open set in R and h : ]a, b] × G −→ C be a function such that, for any x ∈ G, h [t, x] ∈ Cγ,ln [a, b], x ∈ Cγ,ln [a, b], and the Lipshitz condition (7) holds with respect to the second variable. α,n−1 (i) If n−1 < α < n, then there exists a unique solution x to (8)-(9) in the space Cδ,γ,ln [a, b] . n (ii) If α = n, then there exists a unique solution x ∈ Cδ,γ,ln [a, b].

Since the problem (8)-(9) and the equation (10) are equivalent, it is enough to prove that there exists only one solution to (10). Proof. Here we prove (i) only as (ii) can be proved similarly. Step 1. First we show that there exists a unique solution x ∈ Cδn−1 [a, b]. Divide the interval [a, b] into M subdivisions [a, t1 ] , [t1 , t2 ] , ..., [tM −1 , b] such that a < t1 < t2 < ... < tM −1 < b. (a) Choose t1 ∈ ]a, b] such that the inequality w1 = A

n−1 ∑ k=0

Γ (1 − γ) Γ (α − k − γ + 1)

( )Re(α)−k t1 ln < 1, A > 0, a

(14)

holds. Now we prove that there exists a unique solution x (t) ∈ Cδn−1 [a, t1 ] to equation (10) in the interval [a, t1 ].

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It is easy to see that Cδn−1 [a, t1 ] is a complete metric space equipped with the distance d (x1 , x2 ) = ∥x1 − x2 ∥C n−1 [a,t1 ] =

n−1 ∑

δ

( k )

δ x1 − δ k x2

C[a,t1 ]

.

k=0

Now, for any x ∈ Cδn−1 [a, t1 ] , define operator T as follows )α−1 ∫ t( t dτ 1 ln ) h [τ, x (τ )] , (T x) (t) ≡ T x (t) = x0 (t) + Γ (α) a τ τ with x0 (t) =

n−1 ∑ j=0

dj j!

( ln

t a

(15)

)j .

(16)

Transforming the problem (10) into a fixed point problem, x (t) = T x (t) , where T is defined by (15). One can see that the fixed points of T are nothing but solutions to problem (8)-(9) . Applying the Banach contraction mapping, we shall prove that T has a unique fixed point. Firstly, we have to show that: (a.i) if x (t) ∈ Cδn−1 [a, t1 ], then (T x) (t) ∈ Cδn−1 [a, t1 ] . (a.ii) ∀x1 , x2 ∈ Cδn−1 [a, t1 ] the following inequality holds: ∥T x1 − T x2 ∥C n−1 [a,t1 ] ≤ w1 ∥x1 − x2 ∥C n−1 [a,t1 ] , 0 < w1 < 1. δ

δ

(a.i) Let us prove that T x : [a, t1 ] −→ [a, t1 ] is a continuous operator. Differentiating (15) k (k = 0, ..., n − 1) times, we arrive at the equality )α−1−k ∫ t( ( k ) 1 t dτ k δ T x (t) = δ x0 (t) + ln( h [τ, x (τ )] , Γ (α − k) a τ τ Cδn−1

Cδn−1

with δ k x0 (t) =

n−1 ∑ j=k

dj (j − k)!

( )j−k t ln . a

It follows that δ k x0 (t) ∈ Cδ [a, t1 ] because x0 (t) might be further decomposed as a finite sum of functions in Cδn−1 [a, t1 ] . When x0 (t) ∈ Cδn−1 [a, t1 ] then ∥x0 (t)∥C[a,t1 ] ≤ ∥x0 (t)∥C n−1 [a,t1 ] =

n−1 ∑

δ

( k )

δ x0 (t)

C[a,t1 ]

+ ∥x0 (t)∥C[a,t1 ] .

k=1

On the other hand, we can apply Lemma 2.6-(b) with α ≥ γ, and α being replaced by (α − k) , we have Jaα−k h [τ, x (τ )] (t) ∈ Cδ [a, t1 ] . + In view of Lemma 2.6 and (7) , for all k = 0, ..., n − 1, we have ( )Re(α)−k−γ

t1 Γ (1 − γ)

α−k

ln ∥h [t, x (t)]∥Cγ,ln [a,t1 ] ≤

Ja+ h [τ, x (τ )] Γ (1 + α − k − γ) a C[a,t1 ] ( )Re(α)−k−γ Γ (1 − γ) t1 ∥x (t)∥Cγ,ln [a,t1 ] ≤ A ln Γ (1 + α − k − γ) a ( )Re(α)−k Γ (1 − γ) t1 ≤ A ln ∥x (t)∥C[a,t1 ] . Γ (1 + α − k − γ) a 8

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As fractional integrals are bounded in the space of functions continuous in interval [a, t1 ]. The above implies that T x (t) belongs to the Cδn−1 [a, t1 ] space. (a.ii) Next, we let x1 , x2 ∈ Cδn−1 [a, t1 ] the following estimate holds:



∥T x1 − T x2 ∥C n−1 [a,t1 ] = Jaα+ (h [τ, x1 (τ )] − h [τ, x2 (τ )]) (t) n−1 δ Cδ [a,t1 ]

∑n−1

= k=0 Jaα−k (h [τ, x (τ )] − h [τ, x (τ )]) (t)

1 2 + ≤

∑n−1

Γ(1−γ) k=0 Γ(α−k−γ+1)

≤A ≤A ≤A

(

∑n−1

ln (

∑n−1

(

∑n−1

(

Γ(1−γ) k=0 Γ(α−k−γ+1)

Γ(1−γ) k=0 Γ(α−k−γ+1) Γ(1−γ) k=0 Γ(α−k−γ+1)

) t1 Re(α)−k−γ a ln ta1 ln ta1 ln ta1

C[a,t1 ]

∥h [τ, x1 (τ )] − h [τ, x2 (τ )]∥Cγ,ln [a,t1 ]

)Re(α)−k−γ )Re(α)−k )Re(α)−k

∥x1 (t) − x2 (t)∥Cγ,ln [a,t1 ]

∥x1 (t) − x2 (t)∥C[a,t1 ] ∥x1 (t) − x2 (t)∥C n−1 [a,t1 ] . δ

Thus ∥T x1 − T x2 ∥C n−1 [a,t1 ] ≤ A δ

∑n−1

Γ(1−γ) k=0 Γ(α−k−γ+1)

(

ln ta1

)Re(α)−k

∥x1 (t) − x2 (t)∥C n−1 [a,t1 ] . δ

˙ The last estimate shows that the operator T is a contraction mapping from Cδn−1 [a, t1 ]Thus, ∗ the Banach fixed point theorem implies that there exists a unique function (solution) x0 ∈ Cδn−1 [a, t1 ] and this given as: x∗0 =

lim T m x∗00 , (m ∈ N∗ ) ,

m→+∞

where (T m x∗00 ) (t) = x0 (t) +

1 Γ (α)

)α−1 ∫ t( [ ( ) ] dτ t ln h τ, T m−1 x∗00 (τ ) , τ τ a

with x∗00 ∈ Cδn−1 [a, t1 ] is an arbitrary starting function. Let us take x∗00 (t) = x0 (t) when dk ̸= 0 with x0 (t) defined by (16), if we denote by xm (t) = (T m x∗00 ) (t) , (m ∈ N∗ ) , then

lim ∥xm (t) − x∗0 (t)∥C n−1 [a,t1 ] = 0.

m→+∞

δ

x∗0

Now we show that this solution (t) is unique. Suppose that there exist two solutions x∗0 (t) , x ˜∗0 (t) of equation (10) on [a, t1 ]. Using Lemma 2.6 and substituting them into (10), we get ∥x∗0 (t) − x ˜∗0 (t)∥C n−1 [a,t1 ] ≤ A

n−1 ∑

δ

k=0

Γ (1 − γ) Γ (α − k − γ + 1)

( )Re(α)−k t1 ln ∥x∗0 (t) − x ˜∗0 (t)∥C n−1 [a,t1 ] . δ a

This relation yields A

n−1 ∑ k=0

Γ (1 − γ) Γ (α − k − γ + 1)

)Re(α)−k ( t1 ln ≥ 1, a

which contradicts the assumption (14). Thus there is a unique solution x∗0 (t) ∈ Cδn−1 [a, t1 ]. 9

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(b) We prove the existence of an unique solution x (t) ∈ Cδn−1 [t1 , b] . analogously Further, if we consider the closed interval [t1, b], we can rewrite equation (10) in the form x (t) = (T x) (t) where (T x) (t) = x01 (t) +

1 Γ (α)

)α−1 ∫ t( t h [τ, x (τ )] ln dτ, τ τ t1

(17)

where x01 (t) defined by 1 x01 (t) = x0 (t) + Γ (α)

∫

t1

(

a

)α−1

t ln τ

h [τ, x (τ )] dτ, τ

is a known function. We note that x01 (t) ∈ Cδn−1 [t1 , b] . Differentiating (17) k (k = 0, ..., n − 1) times, we arrive at the equality ( k ) δ T x (t) = δ k x01 (t) +

1 Γ (α − k)

∫ t( a

t ln τ

)α−k−1 h [τ, x (τ )]

dτ . τ

It follows that δ k x01 (t) ∈ Cδ [t1 , b] and Jaα−k h [τ, x (τ )] ∈ Cδ [t1 , b] thus (T x) (t) ∈ + n−1 Cδ [t1 , b] . (b.i) Choose t2 ∈ ]t1 , b] such that the inequality w2 = A

n−1 ∑ k=1

Γ (1 − γ) Γ (α − k − γ + 1)

)Re(α)−k ( t2 < 1, ln t1

hold. Let x1 , x2 ∈ Cδn−1 [t1 , t2 ] the following estimate holds: ∥T x1 − T x2 ∥C n−1 [t1 ,t2 ] δ

≤

n−1 ∑

α−k

Ja+ (h [τ, x1 (τ )] − h [τ, x2 (τ )]) (t)

k=0 n ∑

≤ A

k=0

Γ (1 − γ) Γ (α − k + 1)

( ln

t2 t1

C[t1 ,t2 ]

)Re(α)−k ∥x1 (t) − x2 (t)∥C n−1 [t1 ,t2 ] . δ

Hence T x is a contraction in Cδn−1 [t1 , t2 ] . By Lemma 2.6-(b) and α being replaced by α−k, we obtain that Jtα−k (h [τ, x1 (τ )] − h [τ, x2 (τ )]) is 1+ continuous in [t1 , t2 ]. Then, the Banach fixed point theorem implies that there exists a unique solution x∗1 ∈ Cδn−1 [t1 , t2 ] to the equation (10) on the interval [t1 , t2 ] . Notice that x∗1 (t1 ) = x∗0 (t1 ) = x01 (t1 ) . Further, Theorem 2.8 guarantees that this solution x∗1 (t) is the limit of the convergent sequence T m x∗01 . Thus, we have lim ∥T m x∗01 − x∗1 ∥C n−1 [t1 ,t2 ] = 0,

m→+∞

δ

with (T m x∗01 ) (t)

1 = x01 (t) + Γ (α)

)α−1 ∫ t( [ ( ) ] dτ t ln( ) h τ, T m−1 x∗01 (τ ) , (m ∈ N∗ ) . τ τ t1

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If x0 (t) ̸= 0 then we can take x∗01 (t) = x0 (t), therefore, lim ∥xm (t) − x∗1 (t)∥C n−1 [t1 ,t2 ] = 0, xm (t) = (T m x∗01 ) (t) .

m→+∞

δ

{

Now let ∗

x (t) =

x∗0 (t) x∗1 (t)

t ∈ [t1 , t2 ] , t ∈ [a, t1 ] .

Moreover, since x∗ ∈ Cδn−1 [a, t1 ] and x∗ ∈ Cδn−1 [t1 , t2 ], we have x∗ ∈ Cδn−1 [a, t2 ] , and hence there is a unique solution x∗ ∈ Cδn−1 [a, t2 ] to the equation (10) on the interval [a, t2 ]. (b.ii) Finally, we prove that a unique solution x (t) ∈ Cδn−1 [t2 , b] exists. If t2 ̸= b, we choose ti+1 ∈ ]ti , b] such that the relation ( )Re(α)−k n−1 ∑ Γ (1 − γ) ti+1 wi+1 = A ln < 1, i = 2, 3, ..., M, b = tM . Γ (α − k − γ + 1) ti k=0

Repeating the above process i times, we also deduce that there exists a unique solution x∗i ∈ Cδn−1 [ti , ti+1 ] given as a limit of a convergent sequence T m x∗0i i.e., lim

m−→+∞

∥T m x∗0i − x∗i ∥C n−1 [ti ,ti+1 ] = 0, i = 2, 3, ..., M. δ

Consequently, the previous relation can be rewritten as lim ∥xm (t) − x∗ (t)∥C n−1 [a,b] = 0,

m→+∞

with

(18)

δ

xm (t) = T m x∗0i , x∗0i (t) = x0 (t) , x∗ (t) = x∗i (t) , i = 0, 1, ..., M,

and

x∗i (ti+1 ) = x∗i+1 (ti+1 ) , [a, b] = ∪ [ti , ti+1 ] , a = t0 < ... < tM = b. ( ) Step 2. Now we show that C Daα+ x∗ (t) ∈ Cγ,ln [a, b] . By (8) , (18) and the Lipschitzian condition (7), we have that

( ) ( )

lim C Daα+ xm (t) − C Daα+ x∗ (t) = lim ∥h [t, xm (t)] − h [t, x∗ (t)]∥Cγ,ln [a,b]

m→+∞

Cγ,ln [a,b]

m→+∞

≤ A lim ∥xm (t) − x∗ (t)∥Cγ,ln [a,b] m→+∞

( )γ ≤ A ln ab ( )γ ≤ A ln ab

lim ∥xm (t) − x∗ (t)∥C[a,b]

m→+∞

lim ∥xm (t) − x∗ (t)∥C n−1 [a,b] .

m→+∞

δ

It is obvious that the right hand side of the above inequality approaches to zero independently, thus

( ) ( )

lim C Daα+ xm (t) − C Daα+ x∗ (t) = 0. m→+∞

(

Cγ,ln [a,b]

)

By hypothesis, C Daα+ xm (t) = h [t, xm (t)] and h [t, x (t)] ∈ Cγ,ln [a, b] for x ∈ Cδn−1 [a, b] , ( ) we have C Daα+ x∗ (t) ∈ Cγ,ln [a, b] . α,n−1 Consequently, x∗ ∈ Cδ,γ,ln [a, b] is the unique solution to the problem (8)-(9) .

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Corollary 3.4. Under the hypotheses of Theorem 3.3, with γ = 0, there exists a unique solution x to the problem (8)-(9) in the space Cδα,n−1 [a, b] and to the problem (11) in the space Cδn [a, b] . Proof. The above Corollary can be demonstrated in a similar way to that of Theorem 3.3, using the following inequality ( )Re(α)−k n−1 ∑ 1 ti+1 wi+1 = A < 1, i = 0, ..., M, a = t0 , b = tM , ln Re (α − k) Γ (α − k + 1) ti k=0

where ti ∈ [a, b] and we observe that T is a contractive mapping when the following inequality holds, indeed, for any x1 , x2 ∈ Cδn−1 [ti , ti+1 ]

∑n−1

∥T x1 − T x2 ∥C n−1 [ti ,ti+1 ] = k=0 Jtα−k (h [τ, x1 (τ )] − h [τ, x2 (τ )]) (t) i+ δ

C[ti ,ti+1 ]

≤ ≤

∑n−1

ln

ti+1 ti

)Re(α)−k

k=0 Re(α−k)Γ(α−k+1) ∥h [t, x1 (t)] − ( )Re(α)−k ∑n−1 ln ti+1 ti A k=0 Re(α−k)Γ(α−k+1) ∥x1 (t) − x2

≤A

4

(

∑n−1

( ln

ti+1 ti

h [t, x2 (t)]∥C[ti ,ti+1 ] (t)∥C[ti ,ti+1 ]

)Re(α)−k

k=0 Re(α−k)Γ(α−k+1)

∥x1 (t) − x2 (t)∥C n−1 [ti ,ti+1 ] . δ

The Generalized Cauchy type problem

The results in the previous section can be extended to the following equation, which is more general than (8) : ( ) [ ( ) ( ) ] c α Da+ x (t) = h t, x (t) , c Daα+1 x (t) , ..., c Daα+l x (t) , (19) ( α ) with αj ∈ (j − 1, j], j = 1, 2, ..., l, α0 = 0, and c Da+j denotes the Caputo Hadamard operator of order αj . The initial conditions for (19) are ( k ) δ x (a+ ) = dk , dk ∈ R (k = 0, ..., n − 1) . (20) [ ( ) ( ) ] For simplicity, we denote by h [t, φ (t, x)] instead of h t, x (t) , c Daα+1 x (t) , ..., c Daα+l x (t) . Similar to the things discussed in the previous, our investigations are based on reducing the problem (19)-(20) to the Volterra equation x(t) =

n−1 ∑ j=0

dj j!

(

t ln a

)j

1 + Γ (α)

∫t ( ln

t τ

)α−1 h [τ, φ (τ, x)]

dτ , (t > a) . τ

(21)

a

Theorem 4.1. Let α > 0, n = [Re(α)] + 1 and αj ∈ C (j = 0, ..., l) be such that 0 = Re (α0 ) < Re (α1 ) < ... < Re (αl ) < n − 1.

(22)

Let G ∈ Rl+1 be open subsets and let h : (a, b] × G −→ R be a function such that h [t, x, x1 , ..., xl ] ∈ Cγ,ln [a, b] for arbitrary x, x1 , ..., xl ∈ Cγ,ln [a, b] and the Lipschitz condition (6) is fulfilled. 12

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α,n−1 (i) If x ∈ Cδ,γ,ln [a, b], then x holds the relations (19)-(20) if and only if x holds the equation (21) . α (ii) If 0 < α < 1, then x ∈ Cδ,γ,ln [a, b] satisfies the relations

(

c

) Daα+ x (t) = h [t, φ (t, x)] , x (a+ ) = d0 , d0 ∈ R,

(23)

iff x satisfies the equation ( ) x(t) = d0 + Jaα+ h [τ, φ (τ, x)] (t) , (t > a) .

(24)

Proof. Let α ∈ (n − 1, n] and x ∈ Cδn−1 [a, b] satisfies the relations (19)-(20) . (i.a) According to (4) and (19) , (

c

)

Daα+ x

( (t) =

Daα+

)

[ x (τ ) −

n−1 ∑ k=0

( We have

c

] δ k x (a) ( τ )k ln (t) . k! a

) Daα+ x (t) ∈ Cγ,ln [a, b] and hence  x (τ ) − δ n Jan−α +

n−1 ∑ j=0

Thus,

 x (τ ) − Jan−α +

n−1 ∑ j=0

 δ j x (a) ( τ )j  ln ∈ Cγ,ln [a, b] . j! a

 ( ) j δ x (a) τ  n ln ∈ Cδ,γ,ln [a, b] , j! a j

and by Lemma 2.4 (

Jaα+

)(

c

n−1 ) ∑ δ j x (a) ( t )j−1 Daα+ x (t) = x (t) − ln , (j − 1)! a j=1

Then, from (19) , (20) and the last relation, we obtain x(t) =

n−1 ∑ j=0

dj j!

( ln

t a

)j

( ) + Jaα+ h [τ, φ (τ, x)] (t) , (t > a) .

That is x ∈ Cδn−1 [a, b] satisfy the equation (21) . (i.b) Now we prove the sufficiency. Let x ∈ Cδn−1 [a, b] satisfies equation (21) . – From (21) we have x(t) −

n−1 ∑ j=0

dj j!

( )j ( ) [ ( ) ( ) ] t ln = Jaα+ h τ, x (τ ) , c Daα+1 x (τ ) , ..., c Daα+l x (τ ) (t) . a

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( ) Applying Daα+ on both sides of this relation, taking into acount the conditions for h and using Property 2.2, we get   n−1 ( ) ( )( ) ∑ dj ( t ) j  = ln Daα+ x(t) − Daα+ Jaα+ h [τ, φ (τ, x)] (t) j! a j=0 = h [t, φ (t, x)] . ( By (4), the left hand of the above expression is (

c

c

Daα+

) and thus

) [ ( ) ( ) ] Daα+ x (t) = h t, x (t) , c Daα+1 x (t) , ..., c Daα+l x (t) .

Hence x ∈ Cδn−1 [a, b] satisfies (19) . – Applying δ k (k = 0, ..., n − 1) to both sides of (21), we have k

δ x(t) =

n−1 ∑ j=k

dj (j − k)!

( )j−k ) ( )( t ln + δ k Jaα+ h [τ, φ (τ, x)] (t) , (t > a) , a

(25)

(( ) ( )) Since x ∈ Cδn−1 [a, b] for any c Daα+1 x , ..., c Daα+l x ∈ Rn−1 and α −k > γ −(n − 1) > 0, we have ( ) [ ( ) ( ) ] c α1 c αl Jaα−k h τ, x (τ ) , D x (τ ) , ..., D x (τ ) ∈ C [a, b] . (26) a+ a+ + On the other hand, by Lemma 2.3, we let τ −→ a+ on the both sides of (25) , then we obtain δ k x(τ ) τ =a+ = dk , k = 0, ..., n − 1. Hence, x satisfying (21) satisfies the initial condition (20). That is x ∈ Cδn−1 [a, b] satisfies the Cauchy problem (19)-(20). Similarly, we prove the second part of the Theorem. Theorem 4.2. Let α ∈ C, n = [Re(α)] + 1, 0 ≤ γ < 1 be such that γ ≤ α. Let αj > 0 (j = 1, ..., l) be such that conditions in (22) are satisfied. Let G be an open set in Rl+1 and let h : (a, b] × G −→ R be a function such that h [t, x, x1 , ..., xl ] ∈ Cγ,ln [a, b] for any x, x1 , ..., xl ∈ Cγ,ln [a, b] and the Lipschitz condition (6) is fulfilled. (i) If n − 1 < α < n, then there is a unique solution x to the problem (19)-(20) in the space α,n−1 Cδ,γ,ln [a, b] . α [a, b] to (19) with the condition (ii) If 0 < α < 1, then there is a unique solution x ∈ Cδ,γ,ln

x (a+ ) = d0 ∈ R. Proof. By Theorem 4.1 it is sufficient to establish the existence of a unique solution x α,n−1 ∈ Cδ,γ,ln [a, b] to the integral equation (21) . Step 1. First we show that there exists a unique solution x ∈ Cδn−1 [a, b].

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(a) We choose t1 ∈ ]a, b] , we prove the existence of a unique solution x ∈ Cδn−1 [a, t1 ] , so that the conditions n−1 l )Re(α−αj )−k ( ∑ ∑ Γ(1−γ) w1 = Aj ln ta1 Γ(1−γ+α−αj −k) < 1 if γ ≤ α, k=0 j=0

holds, and apply the Banach fixed point theorem to prove the existence of a unique solution x ∈ Cδn−1 [a, t1 ] of the integral equation (21) . We rewrite the equation (21) in the form x (t) = (T x) (t) , where )α−1 ∫ t( t dτ 1 ln h [τ, φ (τ, x)] , (T x) (t) = x0 (t) + Γ (α) a τ τ with x0 (t) =

n−1 ∑ j=0

dj j!

( ln

t a

)j .

It follows that x0 (t) ∈ Cδn−1 [a, t1 ] because x0 (t) my be further decomposed as a finite sum of functions in Cδn−1 [a, t1 ] , h [τ, φ (τ, x)] ∈ Cγ,ln [a, b] =⇒ h [τ, φ (τ, x)] ∈ Cγ,ln [a, t1 ] , and, from Lemma 2.6-(b) , we have, using the fact that α > 0 and 0 ≤ γ < 1, Jaα+ h [τ, φ (τ, x)] ∈ C [a, t1 ] if γ ≤ α. Let x ∈ Cδn−1 [a, t1 ], by Lemma 2.7, the integral in the right-hand side of (21) also belongs to Cδn−1 [a, t1 ] i.e., Jaα+ h [τ, φ (τ, x)] ∈ Cδn−1 [a, t1 ] , and hence T x ∈ Cδn−1 [a, t1 ] , this proves T is continuous on Cδn−1 [a, t1 ]. To show that T is a contraction we have to prove that, for any x1 , x2 ∈ Cδn−1 [a, t1 ] there exists w1 > 0 such that ∥T x1 − T x2 ∥C n−1 [a,t1 ] ≤ w1 ∥x1 − x2 ∥C n−1 [a,t1 ] . δ

δ

By Lipschitzian condition (6) , Property 2.2 and Lemma 2.4, thus

( ( [ ] [ ]))

α

(t)

Ja+ h τ, x1 ,c Daα+1 x1 , ...,c Daα+l x1 − h τ, x2 ,c Daα+1 x2 , ...,c Daα+l x2 ( [ ] [ ] )

≤ Jaα+ h τ, x1 ,c Daα+1 x1 , ...,c Daα+l x1 − h τ, x2 ,c Daα+1 x2 , ...,c Daα+l x2 (t) ≤ = =

(

)

α−αj

αj ( c αj ) A J J D (x − x )

(t) a a a j 1 2 + + + j=0

∑l

(∑

l j=0

[(∑

α−αj

Aj Ja+

l j=0

αj ( c αj )

)

Ja+ Da+ (x1 − x2 ) (t)

α−αj

Aj Ja+

) ∑n −1 ∥x1 − x2 ∥ (τ ) − kjj=0

δ kj (x1 −x2 )(a+ ) kj !

(

ln at

)kj ]

.

By the hypothesis and Lemma 2.4, δ kj x1 (a+ ) = δ kj (x2 ) (a+ ), kj = 0, ..., nj − 1, nj = Re (αj ) + 1, thus

nj −1 k

( ) ∑ δ j (x1 − x2 ) (a+ ) ( t )kj

αj c αj

ln Da+ (x1 − x2 ) (t) =

Ja+

(x1 − x2 ) (t) − kj ! a

kj =0 = ∥(x1 − x2 ) (t)∥ 15

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for arbitrary t ∈ [a, t1 ] . Thus we may continue our estimation above according to l

( ) ∑ ( )

α

j Aj Jaα−α (∥x1 − x2 ∥) (t) .

Ja+ {h [τ, φ (τ, x1 )] − h [τ, φ (τ, x2 )]} (t) ≤ +

(27)

j=0

Moreover by Lemma 2.6-(b), (27) and by (a.ii) in Theorem 3.3 the following holds, indeed, for x1 , x2 ∈ Cδn−1 [a, t1 ]

n−1



∑ α−k

α

≤ Ja+ (h [τ, φ (τ, x1 )] − h [τ, φ (τ, x2 )]) (t)

Ja+ (h [τ, φ (τ, x1 )] − h [τ, φ (τ, x2 )]) (t) n−1

Cδ

≤

n−1 ∑

l ∑

k=0 j=0

[a,t1 ]

k=0

( )Re(α−αj )−k Aj ln ta1

Cδ [a,t1 ]

Γ(1−γ) Γ(1−γ+α−αj −k)

∥x1 (t) − x2 (t)∥C n−1 [a,t1 ] . δ

We conclude that mapping T satisfies ′

∥T x1 − T x2 ∥C n−1 [a,t1 ] ≤ w1 ∥x1 − x2 ∥C n−1 [a,t1 ] δ

δ

for any functions x1 , x2 ∈ [a, t1 ] . Hence, a unique fixed point in space Cδn−1 [a, t1 ] exists and it is explicitly given as a limit of iterations of the mapping T i.e., ∃x∗0 ∈ Cδn−1 [a, t1 ] such that lim ∥xm (t) − x∗0 (t)∥C n−1 [a,t1 ] = 0, Cδn−1

m→+∞

δ

Thus we deduce that a unique solution x∗ (t) ∈ Cδn−1 [a, b] xists such that lim ∥xm (t) − x∗ (t)∥C n−1 [a,b] = 0,

m→+∞

where and

δ

xm (t) = T m x∗0i , x∗0i (t) = x0 (t) , x∗ (t) = x∗i (t) , i = 0, 1, ..., M, x∗i (ti+1 ) = x∗i+1 (ti+1 ) , [a, b] = ∪ [ti , ti+1 ] , a = t0 < ... < tM = b.

Step 2. To complete the proof of Theorem 4.2, we show that this unique solution x (t) = α,n−1 x∗ (t) ∈ Cδn−1 [a, b] belongs to the space Cδ,γ,ln [a, b]. It is sufficient to prove that ( ) c α α Da+ x (t) ∈ Cδ,γ,ln [a, b]. Using the estimate (27) , we have

( ) ( )

c α

Da+ xm (t) − c Daα+ x∗ (t)

Cγ,ln [a,b]

= ∥h [t, φ (t, xm )] − h [t, φ (t, x∗ )]∥Cγ,ln [a,b]

≤

l ∑ j=0

≤ ≤

l ∑



n−1−αj n−1

Aj Ja+ δ (xm (t) − x∗ (t))

Cγ,ln [a,b]

l ∑

)γ (

n−1−αj n−1

δ (xm (t) − x∗ (t)) Aj ln ab Ja+

l ∑ j=0

≤

γ,ln [a,b]

j=0

j=0

≤

α

Aj c Da+j (xm (t) − x∗ (t)) C

l ∑ j=0

C[a,b]

(ln ) Aj Re(n−1−αj )Γ(n−1−αj ) δ n−1 (xm (t) − x∗ (t)) C[a,b] b a

γ

γ

(ln ab ) ∥xm (t) − x∗ (t)∥C n−1 [a,b] , Aj Re(n−1−αj )Γ(n−1−α j) 16

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It is clear that the right hand side of the above inequality approaches to zero independently. Hence,

( ) ( )

lim c Daα+ xm (t) − c Daα+ x∗ (t) = 0. m→+∞

Cγ,ln [a,b]

α,n−1 Consequently, a unique solution x∗ ∈ Cδ,γ,ln [a, b] of equation (21) exists. The second part of the theorem can be proved analogously.

Corollary 4.3. Under the hypotheses of Theorem 4.2 with γ = 0. Then there exists a unique solution x∗ (t) ∈ Cδn−1 [a, b] to the Cauchy problem (19)-(20) . Proof. The above Corollary can be demonstrated in a similar way to that of Theorem 4.2, using the following inequality

α

Ja+ (h [τ, φ (τ, x1 )] − h [τ, φ (τ, x2 )]) (t) ≤

C[ti ,ti+1 ] ( )Re(α−α )−k j t n−1 l ln t i ∑ ∑ i+1 Aj ℜ(α−αj −k)Γ(α−αj −k) k=0 j=0

∥x1 (t) − x2 (t)∥C[ti ,ti+1 ] ,

for i = 0, 1, ..., M, a =t0 , b = tM , and

( ) ( )

c α

Da+ xm (t) − c Daα+ x∗ (t)

Cγ,ln [a,b]

l ∑ j=0

≤

( b )γ ln a ∥xm (t) − x∗ (t)∥C n−1 [a,b] . Aj Re (n − 1 − αj ) Γ (n − 1 − αj )

We can derive the corresponding results for the Cauchy problems for linear fractional equations. Corollary 4.4. Let α > 0, n = [Re(α)] + 1 and 0 ≤ γ < 1 be such that α ≥ γ. Let l ∈ N, αj > 0 (j = 1, ..., l) be such that conditions in (22) are satisfied and let dj (t) ∈ C [a, b] (j = 1, ..., l) and f (t) ∈ Cγ,ln [a, b] . Then the Cauchy problem for the following linear differential equation of order α (

c

l ) ( ) ∑ Daα+ x (t) + dj (t) c Daα+j x (t) + d0 (t) x (t) = f (t) (t > a) , j=1

α,n−1 with the initial conditions (9) has a unique solution x (t) in the space Cδ,γ,ln [a, b] . α,n−1 In particular, there exists a unique solution x (t) in the space Cδ,γ,ln [a, b] to the Cauchy problem for the equation with λj ∈ R and βj = 0 (j = 1, ..., l) :

(

c

)β ( )β ( l ) ∑ t 0 t j (c α j ) Da+ x (t) + λ0 ln Daα+ x (t) + λj ln x (t) = f (t) (t > a) . a a j=1

Proof. The proof is a direct consequence of Theorem 4.2.

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5

Illustrative Examples

We give here some applications of the above results to Cauchy problems with the Caputo Hadamard derivative. Example 5.1. We consider the fractional differential equation of the form (

c

Daα+ x

)

( )β t m (t) = λ ln [x (t)] ; t > a > 0; Re(α) > 0, m > 0; m ̸= 1, a

with λ, β ∈ R (λ ̸= 0) , with the initial conditions ( k ) δ x (a+ ) = 0, k = 0, ..., n − 1.

(28)

(29)

(a) Suppose that the solution has the folowing form: ( )ν t , x (t) = c ln a then, this equation has the explicit solution [

Γ (γ − α + 1) x (t) = λΓ (γ + 1)

1 ] (m−1) (

ln

t a

)α−γ , γ=

(β + mα) . (m − 1)

(30)

Moreover, the condition (29) is satisfied. Hence x (t) is an eigenfunction if both of γ +1 and γ −α+1 are not equal to 0 or negative integer. also using Property 2.3 it is easily verified that if the condition (β + α) ≥ −1, (m − 1)

(31)

holds, this solution x (t) belongs to Cγ [a, b] and to C [a, b] in the respective cases 0 ≤ α and γ − α ≤ 0. x (t) ∈ Cγ [a, b] if 0 ≤ γ < 1 and 0 ≤ α, (32) x (t) ∈ C [a, b] if γ − α ≤ 0. The right-hand side of the equation (28) takes the form [

Γ (γ − α + 1) h [t, x (t)] = λΓ (γ + 1)

m ] (m−1) (

t ln a

)−γ .

(33)

The function h [t, x (t)] ∈ Cγ [a, b] when 0 ≤ γ < 1 and h [t, x (t)] ∈ C [a, b] when γ ≤ 0 h [t, x (t)] ∈ Cγ [a, b]

if

0 ≤ γ < 1,

h [t, x (t)] ∈ C [a, b]

if

γ ≤ 0.

(34)

In accordance with (31) , the following case is possible for the space of the right-hand side (33) and of the solution (30) :

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1. When α > 0 and m > 1, −mα ≤ β < m − 1 − mα, β ≤ −α, or 0 < m < 1, m − 1 − mα < β ≤ −mα, β ≥ −α. 2. If 0 < α < 1 these conditions take the following forms m > 1, −mα ≤ β ≤ −α or 0 < m < 1, −α ≤ β ≤ −mα.

(35)

3. If α ≥ 1 then m > 1, −mα ≤ β < m − 1 − mα or 0 < m < 1, m − 1 − mα < β ≤ −mα.

(36)

(b) Now we establish the conditions for the uniqueness of the solution (30) to the above problem (28)-(29). For this we have to choose the domain G and check when the Lipschitz condition (7) with the right-hand side of (28) is valid. We choose the following domain: { ( } )q t 2 G = (t, x) ∈ R : 0 < a < t ≤ b, 0 < x < p ln , q ∈ R, p > 0 . a To prove the Lipschitz condition (7) with ( )β t m h [t, x (t)] = λ ln (x (t)) , a

(37)

(38)

we have, for any (t, x1 ) (t, x2 ) ∈ G : ( )β t m |h [t, x1 ] − h [t, x2 ]| ≤ |λ| ln |xm 1 − x2 | . a

(39)

By definition (37) , we have ( )q t m |xm − x | < mK ln |x1 − x2 | , m > 0. 1 2 a Substituting this estimate into (39) , we obtain ( )β+(m−1)q t |h [t, x1 ] − h [t, x2 ]| ≤ |λ| mK ln |x1 − x2 | . a Then the functions h [t, x (t)] fulfil the Lipschitizian condition provided that β+(m − 1) q ≥ 0. Proposition 5.2. Let λ, β ∈ R (λ ̸= 0) and m > 0 (m ̸= 1), γ = (β + mα) \ (m − 1) . Let G be the domain (37), where q ∈ R is such that β + (m − 1) q ≥ 0. (i) Let 0 < α < 1, if either of the conditions (35) holds, then the Cauchy problem )β ( ( ) t m c α [x (t)] and x (a+ ) = 0, Da+ x (t) = λ ln a

(40)

α has a unique solution x (t) ∈ Cδ,γ,ln [a, b] and this solution is given by (30).

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(ii) Let n−1 < α < n (n ∈ N \ {1}) , if either of the conditions (36) holds, then the problem (

c

( )β ) ( ) t m Daα+ x (t) = λ ln [x (t)] and δ k x (a+ ) = 0, k = 0, ..., n − 1, a

(41)

α,n−1 has a unique solution x (t) ∈ Cδ,γ,ln [a, b] and this solution is given by (30).

Remark 5.3. If β = 0, 0 < Re(α) < 1 then the Lipschitz condition is violated in the domain (37) . The Cauchy problem (41) admits of two continuous solutions x = 0 and [ x (t) =

Γ (γ + 1) Γ (γ + 1 − α)

1 ] (m−1) (

ln

t a

)γ , γ=

α . (1 − m)

Example 5.4. Let us consider the following problem of order α (Re (α) > 0) (

c

( )β )ν ( ) t t m Daα+ x (t) = λ ln [x (t)] + c ln , λ, c ∈ R (λ ̸= 0) and ν, β ∈ R. a a

Then it is verified that the equation (42) has the solution of the form ( )γ t x (t) = µ ln , γ = (β + α) \ (1 − m) . a

(42)

(43)

In this case the right-hand side of 42 takes the form (

t h [t, x (t)] = (λ + c) ln a

)(β+αm)\(1−m) .

(44)

Using the same arguments as in the proof of Proposition 5.2 we derive the uniqueness result for the Cauchy problem 42. Proposition 5.5. Let λ, β ∈ R (λ ̸= 0) and m > 0 (m ̸= 1), γ = (β + mα) \ (m − 1). Let G be the domain (37), where q ∈ R is such that β + (m − 1) q ≥ 0. Let ν = −γ and let the transcendental equation ( ) ( ) α+β α+β Γ + 1 − α [λy m + c] − Γ + 1 y = 0, 1−m 1−m have a unique solution y = µ. (i) Let 0 < α < 1, if either of the conditions (35) holds, then the Cauchy problem (

c

)β ( )ν ( ) t t m [x (t)] + c ln , x (a+ ) = 0, Daα+ x (t) = λ ln a a

α has a unique solution x (t) ∈ Cδ,γ,ln [a, b] and this solution is given by (43).

(ii) Let n − 1 < α < n, if either of the conditions (36) holds, then the problem (42)-(29) α,n−1 has a unique solution x (t) ∈ Cδ,γ,ln [a, b] and this solution is given by (43).

Acknowledgments This work is partially supported by the Scientific and Technical Research Council of Turkey. 20

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References [1] K. Diethelm, “The analysis of fractional differential equations, Lecture Notes in Mathematics”, Springer (2004) . [2] G. A. Anastassiou, “Fractional Differentiation Inequalities”, Springer (2009) . [3] G. A. Anastassiou, “Advances on Fractional Inequalities”, Springer (2011) . [4] F. Jarad, D. Baleanu, T. Abdeljawad, “Caputo-type modification of the Hadamard fractional derivatives”, Adv. Differ. Equ. 2012, 142, (2012). [5] C. Kou, J. Liu, and Y. Ye, “Existence and uniqueness of solutions for the cauchy- type problems of fractional differential equations”, Discrete Dynamics in Nature and Society, vol. 2010, Article ID142175, 15 pages, (2010). [6] J. Hadamard,“Essai sur l’etude des fonctions donnes par leur developpment de Taylor”, J. Pure Appl. Math. 4 (8), 101 − 186, (1892). [7] R. Hilfer,“Fractional time evolution”, in Applications of Fractional Calculus in Physics, WorldScientiic, London, UK, (2000) . [8] A. Kilbas, “Hadamard-type fractional calculus”, Journal of Korean Mathematical Society (38) (6) , 1191 − 1204, (2001). [9] A. Kilbas, S. A. Marzan, “Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions”, Differential’nye Uravneniya, 41 (1), 82 − 86, (2005) . [10] A. Kilbas, H. M. Srivastava, J.J. Trujillo, “Theory and Applications of Fractional Differential Equations”, Elsevier, Amsterdam (2006). [11] S, G. Samko, A. A. Kilbas, O. I. Marichev, “Fractional Integrals and Derivatives: Theory and Applications”, Gordon and Breach Science Publishers, Switzerland, (1993). [12] I. Podlubny. “Fractional Differential Equations”. Academic Press, San Diego, California,(1999). [13] J. J. Trujillo, M. Rivero, B. Bonilla, “On a Riemann–Liouville generalized Taylor’s Formula”, J. Math. Anal. Appl. 231, 255 − 265, (1999). [14] J. Tariboon, S. K. Ntouyas, W. Sudsutad, C. Thaiprayoon, “Nonlinear Langevin Equation of Hadamard-Caputo Type Fractional Derivatives with Nonlocal Fractional Integral Conditions ”, Adv. Math. Phys. 2014, doi : 10.1155/2014/372749, (2014).

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Multivalued Generalized Contractive Maps and Fixed Point Results Marwan A. Kutbi Department of Mathematics, King Abdulaziz University, P.O.Box 80203, Jeddah 21589, Saudi Arabia E-mail: [email protected]

Abstract In this paper, we prove some fixed point results for generalized contractive multimaps with respect to generalized distance. Consequently, several known fixed point results either generalized or improved including the corresponding recent fixed point results of Ciric, BinDehaish-Latif, Latif-Albar, Klim-Wardowski, Feng-Liu.

1

Introduction and Preliminaries

Let (X, d) be a metric space, 2X a collection of nonempty subsets of X, and CB(X) a collection of nonempty closed bounded subsets of X, Cl(X) a collection of nonempty closed subsets of X, K(X) a collection of nonempty compact subsets of X and H the Hausdorff metric induced by d. Then for any A, B ∈ CB(X), H(A, B) = max{sup d(x, B), sup d(y, A)}, x∈A

y∈B

where d(x, B) = inf y∈B d(x, y). An element x ∈ X is called a fixed point of a multivalued map T : X → 2X if x ∈ T (x). We denote F ix(T ) = {x ∈ X : x ∈ T (x)}. A sequence {xn } in X is called an orbit of T at x0 ∈ X if xn ∈ T (xn−1 ) for all n ≥ 1. A map f : X → R 0

2000 Mathematics Subject Classification: 47H09, 54H25. Keywords: Fixed point, contractive multimap, w-distance, metric space.

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is called T-orbitally lower semicontinuous if for any orbit {xn } of T and x ∈ X, xn → x imply that f (x) ≤ lim inf f (xn ). n→∞

Using the concept of Hausdorff metric, Nadler [13] introduced a notion of multivalued contraction maps and proved a multivalued version of the wellknown Banach contraction principle, which states that each closed bounded valued contraction map on a complete metric space has a fixed point. Since then various fixed point results concerning multivalued contractions have appeared. Feng and Liu [4] extended Nadler’s fixed point theorem without using the concept of Hausdorff metric. While in [7] Klim and Wardowski generalized their result. Ciric [3] obtained some interesting fixed point results which extend and generalize these cited results. In [6], Kada et al. introduced the concept of w-distance on a metric space and studied the properties, examples and some classical results with respect to w-distance. Using this generalized distance, Suzuki and Takahashi [14] have introduced notions of single-valued and multivalued weakly contractive maps and proved fixed point results for such maps. Consequently, they generalized the Banach contraction principle and Nadler’s fixed point result. Some other fixed point results concerning w-distance can be found in [8, 9, 10, 16, 18]. In [15], Susuki generalized the concept of w-distance by introducing the notion of τ -distance on metric space (X, d). In [15], Suzuki improved several classical results including the Caristi’s fixed point theorem for single-valued maps with respect to τ -distance. In the literature, several other kinds of distances and various versions of known results are appeared. Most recently, Ume [17] generalized the notion of τ -distance by introducing a concept of u-distance as follows: A function p : X × X → R+ is called u-distance on X if there exists a function θ : X × X × R+ × R+ → R+ such that the following hold for each x, y, z ∈ X: (u1 ) p(x, z) ≤ p(x, y) + p(y, z). (u2 ) θ(x, y, 0, 0) = 0 and θ(x, y, s, t) ≥ min{s, t} for each s, t ∈ R+ , and for every  > 0, there exists δ > 0 such that | s − s0 |< δ, | t − t0 |< δ, s, s0 , t, t0 ∈ R+ and y ∈ X imply | θ(x, y, s, t) − θ(x, y, s0 , t0 ) |< . 2

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(u3 )

limn→∞ xn = x lim sup{θ(wn , zn , p(wn , xm ), p(zn , xm )) : m ≥ n} = 0

n→∞

imply p(y, x) ≤ lim inf p(y, xn ) n→∞

(u4 ) lim sup{p(xn , wm )) : m ≥ n} = 0,

n→∞

lim sup{p(yn , zm )) : m ≥ n} = 0,

n→∞

lim θ(xn , wn , sn , tn ) = 0,

n→∞

lim θ(yn , zn , sn , tn ) = 0

n→∞

imply lim θ(wn , zn , sn , tn ) = 0

n→∞

or lim sup{p(wn , xm )) : m ≥ n} = 0,

n→∞

lim sup{p(zm , yn )) : m ≥ n} = 0,

n→∞

lim θ(xn , wn , sn , tn ) = 0,

n→∞

lim θ(yn , zn , sn , tn ) = 0

n→∞

imply lim θ(wn , zn , sn , tn ) = 0;

n→∞

(u5 ) lim θ(wn , zn , p(wn , xn ), p(zn , xn )) = 0,

n→∞

lim θ(wn , zn , p(wn , yn ), p(zn , yn )) = 0

n→∞

imply lim d(xn , yn ) = 0

n→∞

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or lim θ(an , bn , p(xn , an ), p(xn , bn )) = 0,

n→∞

lim θ(an , bn , p(yn , an ), p(yn , bn )) = 0

n→∞

imply lim d(xn , yn ) = 0.

n→∞

Remark 1.1 [17] (a) Suppose that θ from X × X × R+ × R+ into R+ is a mapping satisfying (u2) to (u5). Then there exists a mapping η from X × X × R+ × R+ into R+ such that η is nondecreasing in its third and fourth variable, respectively, satisfying (u2)η to (u5)η, where (u2)η to (u5)η stand for substituting η for θ in (u2) to (u5), respectively. (b) In the light of (a), we may assume that θ is nondecreasing in its third and fourth variables, respectively, for a function θ from X × X × R+ × R+ into R+ satisfying (u2) to (u5). (c) Each τ -distance p on a metric space (X, d) is also a u-distance on X. Here we present some examples of u-distance which are not τ -distance. (For the detail, see [17]). Example 1.2. Let X = R+ with the usual metric. Define p : X × X → R+ by p(x, y) = ( 41 )x2 . Then p is a u-distance on X but not a τ distance on X. Example 1.3. Let X be a normed space with norm k.k. Then a function p : X × X → R+ defined by p(x, y) = kxk for every x, y ∈ X is a u-distance on X but not a τ -distance. It follows from the above examples and Remark 1.1(c) that u-distance is a proper extension of τ -distance. Other useful examples on u-distance are also given in [17]. Let (X, d) be a metric space and let p be a u-distance on X. A sequence {xn } in X is called p-Cauchy [17] if there exists a function θ from X × X × R+ ×R+ into R+ satisfying (u2)∼(u5) and a sequence {zn } of X such that lim sup{θ(zn , zn , p(zn , xm ), p(zn , xm )) : m ≥ n} = 0,

n→∞

or 4

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lim sup{θ(zn , zn , p(xm , zn ), p(xm , zn )) : m ≥ n} = 0.

n→∞

The following lemmas concerning u-distance are crucial for the proofs of our results. Lemma 1.4 [17] Let (X, d) be a metric space and let p be a u-distance on X. If {xn } is a p-Cauchy sequence in X, then {xn } is a Cauchy sequence. Lemma 1.5 [5] Let (X, d) be a metric space and let p be a u-distance on X. If {xn } is a p-Cauchy sequence and {yn } is a sequence satisfying lim sup{p(xn , ym )) : m ≥ n} = 0,

n→∞

then {yn } is also a p-Cauchy sequence and lim d(xn , yn ) = 0. n→∞

Lemma 1.6 [17] Let (X, d) be a metric space and let p be a u-distance on X. Suppose that a sequence {xn } of X satisfies

lim sup{p(xn , xm )) : m > n} = 0,

n→∞

or lim sup{p(xm , xn )) : m > n} = 0.

n→∞

Then {xn } is a p-Cauchy sequence. The aim of this paper is to present some more general fixed point results with respect to u-distance for multivalued maps satisfying certain conditions. Our results unify and generalize the corresponding results of Mizoguchi and Takahashi [12], Klim and Wardowski [7], Latif and Abdou [10], BinDehaish and Latif [2], Ciric [3], Feng and Liu [4], and several others.

2

The Results

Using the u-distance, we prove a general result on the existence of fixed points for multivalued maps. Theorem 2.1 Let (X, d) be a complete metric space. Let T : X → Cl(X) be a multivalued map and let ϕ : [0, ∞) → [0, 1) be such that lim sup ϕ(r) < 1 r→t+

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for each t ∈ [0, ∞). Let p be a u-distance on X and assume that the following conditions hold : (I) for any x ∈ X, there exists y ∈ T (x) satisfying p(x, y) ≤ (2 − ϕ(p(x, y)))p(x, T (x)), and p(y, T (y)) ≤ ϕ(p(x, y))p(x, y) (II) the map f : X → R, defined by f (x) = p(x, T (x)) is T-orbitally lower semicontinuous. Then there exists v0 ∈ X such that f (v0 ) = 0. Further if p(v0 , v0 ) = 0, then v0 ∈ T (v0 ). Proof. let x0 ∈ X be an arbitrary but fixed element in X. Then there exists x1 ∈ T (x0 ) such that p(x0 , x1 ) ≤ (2 − ϕ(p(x0 , x1 )))p(x0 , T (x0 )),

(1)

p(x1 , T (x1 )) ≤ ϕ(p(x0 , x1 ))p(x0 , x1 ).

(2)

and

From (1) and (2), we get p(x1 , T (x1 )) ≤ ϕ(p(x0 , x1 ))(2 − ϕ(p(x0 , x1 )))p(x0 , T (x0 )).

(3)

Define a function ψ : [0, ∞) → [0, ∞) by ψ(t) = ϕ(t)(2 − ϕ(t)) = 1 − (1 − ϕ(t))2 .

(4)

Using the facts that for each t ∈ [0, ∞), ϕ(t) < 1 and limr→t+ sup ϕ(r) < 1, we have ψ(t) < 1 (5) and lim sup ψ(r) < 1, for all t ∈ [0, ∞)

(6)

r→t+

From (3) and (4), we have p(x1 , T (x1 )) ≤ ψ(p(x0 , x1 ))p(x0 , T (x0 )).

(7)

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Similarly, for x1 ∈ X, there exists x2 ∈ T (x1 ) such that p(x1 , x2 ) ≤ (2 − ϕ(p(x1 , x2 )))p(x1 , T (x1 )), and p(x2 , T (x2 )) ≤ ϕ(p(x1 , x2 ))p(x1 , x2 ). Thus p(x2 , T (x2 )) ≤ ψ(p(x1 , x2 ))p(x1 , T (x1 )). Continuing this process we can get an orbit {xn } of T in X satisfying the following p(xn , xn+1 ) ≤ (2 − ϕ(p(xn , xn+1 )))p(xn , T (xn )) (8) and p(xn+1 , T (xn+1 )) ≤ ψ(p(xn , xn+1 ))p(xn , T (xn )),

(9)

for each integer n ≥ 0. Since ψ(t) < 1 for each t ∈ [0, ∞) and from (9), we have for all n ≥ 0 p(xn+1 , T (xn+1 )) < p(xn , T (xn )).

(10)

Thus the sequence of non-negative real numbers {p(xn , T (xn ))} is decreasing and bounded below, thus convergent. Therefore, there is some δ ≥ 0 such that lim p(xn , T (xn )) = δ.

n→∞

(11)

From (8), as ϕ(t) < 1 for all t ≥ 0, we get p(xn , T (xn )) ≤ p(xn , xn+1 ) < 2p(xn , T (xn )),

(12)

Thus, we conclude that the sequence of non-negative reals {ω(xn , xn+1 )} is bounded. Therefore, there is some θ ≥ 0 such that lim inf p(xn , xn+1 ) = θ. n→∞

(13)

Note that p(xn , xn+1 ) ≥ p(xn , T (xn )) for each n ≥ 0, so we have θ ≥ δ. Now we shall show that θ = δ. If δ = 0. Then we get lim p(xn , xn+1 ) = 0.

n→∞

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Now consider δ > 0. Suppose to the contrary, that θ > δ. Then θ − δ > 0 and so from (11) and (13) there is a positive integer n0 such that p(xn , T (xn )) < δ +

θ−δ 4

for all n ≥ n0

(14)

and

θ−δ < p(xn , xn+1 ) for all n ≥ n0 4 Then from (15), (8) and (14), we get θ−

θ−

θ−δ 4




3θ + δ , 3δ + θ

that is; 1 + (1 − ϕ(p(xn , xn+1 ))) > 1 +

2(θ − δ) , 3δ + θ

and we get −(1 − ϕ(p(xn , xn+1 )))2 < −



2(θ − δ) 3δ + θ

2 .

Thus for all n ≥ n0 , ψ(p(xn , xn+1 ))

= 1 − (1 − ϕ(p(xn , xn+1 )))2 2  2(θ − δ) . < 1− 3δ + θ

(16)

Thus, from (9) and (16) , we get p(xn+1 , T (xn+1 )) ≤ hp(xn , T (xn )) for all n ≥ n0 ,

(17)

h i2 where h = 1 − 2(θ−δ) . Clearly h < 1 as θ > δ. From (14) and (17), we 3δ+θ have for any k ≥ 1 p(xn0 +k , T (xn0 +k )) ≤ hk p(xn0 , T (xn0 )).

(18)

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Since δ > 0 and h < 1, there is a positive integer k0 such that hk0 p(xn0 , T (xn0 )) < δ. Now, since δ ≤ p(xn , T (xn )) for each n ≥ 0, by (18) we have δ ≤ p(xn0 +k0 , T (xn0 +k0 )) ≤ hk0 p(xn0 , T (xn0 )) < δ. a contradiction. Hence, our assumption θ > δ is wrong. Thus δ = θ. Now we show that θ = 0. Since θ = δ ≤ p(xn , T (xn )) ≤ p(xn , xn+1 ), then from (13) we can read as lim inf p(xn , xn+1 ) = θ+, n→∞

so, there exists a subsequence {p(xnk , xnk +1 )} of {p(xn , xn+1 )} such that lim p(xnk , xnk +1 ) = θ + .

k→∞

Now from (6) we have lim sup

ψ(p(xnk , xnk +1 )) < 1,

(19)

p(xnk ,xnk +1 )→θ+

and from (9), we have p(xnk , T (xnk +1 )) ≤ ψ(p(xnk , xnk +1 ))p(xnk , T (xnk )). Taking the limit as k → ∞ and using (11), we get δ

=

lim sup p(xnk+1 , T (xnk +1 )) k→∞

≤ (lim sup ψ(p(xnk+1 , xnk +1 )))(lim sup p(xnk , T (xnk )) k→∞

=

(

k→∞

lim sup

ψ(p(xnk , xnk +1 )))δ.

p(xnk ,xnk +1 )→θ+

If we suppose that δ > 0, then from last inequality, we have ψ(p(xnk , xnk +1 )) ≥ 1,

lim sup p(xnk ,xnk +1 )→θ+

which contradicts with (19). Thus δ = 0. Then from (11) and (12), we have lim p(xn , T (xn )) = 0+,

(20)

lim p(xn , xn+1 ) = 0 + .

(21)

n→∞

and thus n→∞

9

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Now, let α=

lim

p(xnk ,xnk +1 )→0+

sup ψ(p(xnk , xnk +1 )).

Then by (6), α < 1. Let q be such that α < q < 1. Then there is some n0 ∈ N such that ψ(p(xn , xn+1 )) < q,

for all n ≥ n0 .

Thus it follows from (9) p(xn+1 , T (xn+1 )) ≤ qp(xn , T (xn ))

for all n ≥ n0 .

By induction we get p(xn+1 , T (xn+1 )) ≤ q n+1−n0 p(xn0 , T (xn0 ))

for all n ≥ n0 .

(22)

Now, using (12) and (22), we have p(xn , xn+1 ) ≤ 2q n−n0 p(xn0 , T (xn0 )) for all n ≥ n0 .

(23)

Now, we show that {xn } is a Cauchy sequence, for all m ≥ n ≥ n0 , we get p(xn , xm ) ≤

m−1 X

p(xk , xk+1 )

k=n

≤ 2

m−1 X

q k−n0 p(xn0 , T (xn0 ))

k=n n−n0

≤ 2(

q )p(xn0 , T (xn0 )). 1−q

(24)

and hence lim sup{p(xn , xm ) : m ≥ n} = 0.

n→∞

Thus, by Lemma 1.6, {xn } is a p- Cauchy sequence and hence by Lemma 1.4, {xn } is a Cauchy sequence. Due to the completeness of X, there exists some v0 ∈ X such that limn→∞ xn = v0 . Since f is T-orbitally lower semicontinuous and from (20), we have 0 ≤ f (v0 ) ≤ lim inf f (xn ) = lim inf p(xn , T (xn )) = 0, n→∞

n→∞

and thus, f (v0 ) = p (v0 , T (v0 )) = 0. Thus there exists a sequence {vn } ⊂ T (v0 ) such that limn→∞ p(v0 , vn ) = 0. It follows that 0 ≤ lim sup{p(xn , vm )) : m ≥ n} ≤ lim sup{p(xn , v0 ) + p(v0 , vm ) : m ≥ n} = 0. (25) n→∞

n→∞

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Since {xn } is a p-Cauchy sequence, thus it follows from (25) and Lemma 1.5 that {vn } is also a p-Cauchy sequence and limn→∞ d(xn , vn ) = 0. Thus, by Lemma 1.4, {vn } is a Cauchy sequence in the complete space. Due to closedness of T (v0 ), there exists z0 ∈ X such that limn→∞ vn = z0 ∈ T (v0 ). Consequently, using (u3 ) we get p(v0 , z0 ) ≤ lim inf p(v0 , vn ) = 0, n→∞

and thus p(v0 , z0 ) = 0. But, since limn→∞ xn = v0 , limn→∞ vn = z0 and limn→∞ d(xn , vn ) = 0, we have v0 = z0 . Hence v0 ∈ F ix(T ) and p(v0 , v0 ) = 0. Remarks 2.2 Theorem 2.1 generalizes fixed point theorems of Latif and Abdou [10, Theorem 2.1], Ciric [3, Theorem 5], Bin Dehaish and Latif [2, Theorem 2.2], Latif and Abdou [8, Theorem 2.2], Susuki [15, Theorem 2], Bin Dehaish and Latif [1, Theorem 2.2], Suzuki and Takahashi [14, Theorem 1], Klim and Wardowski [7, Theorem 2.1] and Feng and Liu [4, Theorem 3.1]which contains Nadler’ fixed point theorem. We also have the following interesting result by replacing the hypothesis (II) of Theorem 2.1 with another suitable condition. Theorem 2.3 Suppose that all the hypotheses of Theorem 2.1 except (II) hold. Assume that inf{p(x, v)) + p(x, T (x))} : x ∈ X} > 0, for every v ∈ X with v ∈ / T (v). Then F ix(T ) 6= ∅. Proof. Following the proof of Theorem 2.1, there exists there exists an orbit {xn } of T , which is Cauchy sequence in a complete metric space X. Thus, there exists v0 ∈ X such that lim xn = v0 . Thus, using (u3 ) and (24) we have for n→∞ all n ≥ n0 p(xn , v0 ) ≤ lim inf p(xn , xm ) ≤ ( m→∞

2q n−n0 )p(xn0 , T (xn0 )), 1−q

and p(xn , T (xn )) ≤ p(xn , xn+1 ) ≤ 2q n−n0 p(xn0 , T (xn0 )). Assume that v0 ∈ / T (v0 ). Then, we have 0 < inf{p(x, v0 ) + p(x, T (x)) : x ∈ X} 11

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≤ inf{p(xn , v0 ) + p(xn , T (xn )) : n ≥ n0 } 2q n−n0 )p(xn0 , T (xn0 )) + 2q n−n0 p(xn0 , T (xn0 )) : n ≥ n0 } ≤ inf{( 1−q 2(2 − q) = p(xn0 , T (xn0 )) inf{q n : n ≥ n0 } = 0, (1 − q)q n0 which is impossible and hence v0 ∈ F ix(T ). Remarks 2.4 Theorem 2.3 generalizes [8, Theorem 2.4], [10, Theorem 3.3] and [2, Theorem 2.5]. Competing interests The author declares that he has no competing interests. Acknowledgement This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No.130-104D1434 The author, therefore, acknowledges with thanks DSR technical and financial support. The author also thanks the referees for their valuable comments and suggestions.

References [1] B.A. BinDehaish and A. Latif, Fixed point results for multivalued contractive maps, Fixed Point Theory and Applications, 2012, 2012:61. [2] B.A. BinDehaish and A. Latif, Fixed point theorems for generalized contractive type multivalued maps, Fixed Point Theory and Applications, 2012, 2012:135. [3] L. Ciric, Fixed point theorems for multi-valued contractions in metric spaces, J. Math. Anal. Appl. 348 (2008) 499-507. [4] Y. Feng and S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi Type mappings, J. Math. Anal. Appl. 317 (2006) 103-112. [5] S Hirunworkakit and N. Petrot, Some fixed point theorem for contractive multi-valued mappings induced by generalized distance in metric spaces, 12

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Fixed Point Theory and Applications 2011, 2011:78, doi:10. 116/16871812-2011-78. [6] O. Kada, T. Susuki and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Jpn. 44 (1996) 381-391. [7] D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132-139. [8] A. Latif and A.A.N. Abdou, Fixed points of generalized contractive maps, Fixed Point Theory and Applications,(2009), Article ID 487161, 9 pages. [9] A. Latif and A.A.N. Abdou, Fixed points of generalized contractive multimaps in metric spaces, Fixed Point Theory and Applications,(2009), Article ID 432130, 16 pages. [10] A. Latif and W.A. Albar, Fixed point results in complete metric spaces, Demonstratio Mathematica 41 (2008) 145-150. [11] L.J. Lin and W.S. Du, Some equivalent formulations of the generalized Ekland’s variational principle and their applications, Nonlinear Anal. 67 (2007) 187-199. [12] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177188. [13] S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475-488. [14] T. Suzuki and W. Takahashi, Fixed point Theorems and characterizations of metric completeness, Topol. Methods Nonlinear Anal., 8 (1996), 371-382. [15] T. Suzuki, Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl., 253 (2001), 440-458. [16] W. Takahashi, Nonlinear Functional Analysis: Fixed point theory and its applications, Yokohama Publishers, 2000.

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[17] J. S. Ume, Existence theorems for generalized distance on complete metric space, Fixed Point Theory and Applications, Vol. 2010, Article ID 397150, 21 pages. [18] J. S. Ume, B. S. Lee and S. J. Cho, Some results on fixed point theorems for multivalued mappings in complete metric spaces, IJMMS, 30 (2002), 319-325.

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ESSENTIAL COMMUTATIVITY AND ISOMETRY OF COMPOSITION OPERATOR AND DIFFERENTIATION OPERATOR GENG-LEI LI

Abstract. In this paper, we characterize the essential commutativity and isometry of composition operator and differentiation operator from the Bloch type space to space of all weighted bounded analytic functions in the disk.

1. Introduction Let D be the unit disk of the complex plane, and S(D) be the set of analytic self-maps of D. The algebra of all holomorphic functions with domain D will be denoted by H(D). A positive continuous function v on [0, 1) is called normal (see, e.g., [17]), if there exist three constants 0 ≤ δ < 1, and 0 < a < b < ∞, such that for r ∈ [δ, 1) v(r) ↓ 0, (1 − r)a

v(r) ↑∞ (1 − r)b

as r → 1. Assume v is normal, the weighted-type space Hv∞ consists of all f ∈ H(D) such that kf kHv∞ = sup v(z)|f (z)| < ∞. z∈D

When v(z) = 1, we know that Hv∞ = H ∞ , that is H ∞ = {f ∈ H(D), sup |f (z)| < ∞}. z∈D

We recall that the Bloch type space B α (α > 0) consists of all f ∈ H (D) such that 2 kf kBα = sup(1 − |z| )α |f 0 (z)| < ∞, z∈D

then k·kBα is a complete semi-norm on B α , which is M¨obius invariant. It is well known that B α is a Banach space under the norm kf k = |f (0)| + kf kBα . Let ϕ be an analytic self-map of D, the composition operator Cϕ induced by ϕ is defined by (Cϕ f )(z) = f (ϕ(z)) 2010 Mathematics Subject Classification. Primary: 47B38; Secondary: 30H30, 30H05, 47B33. Key words and phrases. Composition operator, differentiation operator, Bloch type space, essential commutativity. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11371276), and by the Research Programs Financed by Tianjin Collegiate Fund for Science and Technology Development(Grand No. 20131002). 1

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2

G.L. LI

for z ∈ D and f ∈ H(D). Let D be the differentiation operator on H(D), that is Df (z) = f 0 (z). For f ∈ H(D), the products of composition and differentiation operators DCϕ and Cϕ D are defined by Cϕ D(f ) = f 0 (ϕ) DCϕ (f ) = (f ◦ ϕ)0 = f 0 (ϕ)ϕ0 The boundedness and compactness of DCϕ on the Hardy space were discussed by Hibschweiler and Portnoy in [7] and by Ohno in [14]. We write Tϕ for the operators DCϕ − Cϕ D, which is from the Bloch type space B α to Hv∞ . Generally speaking, it is clear that DCϕ 6= Cϕ D, but it is interesting to study when DCϕ (B α → Hv∞ ) ≡ Cϕ D(B α → Hv∞ ), modK where K denotes the collection of all compact operators from Bloch type space B α to Hv∞ . If the upper properties is satisfied, we say they are essential commutative. In the past few decades, boundedness, compactness, isometries and essential norms of composition and closely related operators between various spaces of holomorphic functions have been studied by many authors, see, e.g., [1, 3, 5, 9, 12, 15, 16, 21, 22]; the results about difference and other properties can be seen [?, 4, 6, 10, 11, 13, 18, 20] and the related references therein. Recently, many interests focused on studying the essential commutativity of virous different composition operators. In [23], Zhou and Zhang studied the essential commutativity of the integral operators and composition operators from a mixed-norm space to Bloch type space. In [19, ?], Tong and Zhou characterized the intertwining relations for Volterra operators on the Bergman space, and compact intertwining relations for composition operators between the weighted Bergman spaces and the weighted Bloch spaces, respectively. The paper continues this line of research, and discusses the essential commutativity of composition operator and differentiation operator from the Bloch type space to the space of all weighted bounded analytic functions in the disk. 2. Notation and Lemmas To begin the discussion, let us introduce some notations and state a couple of lemmas. For a ∈ D, the involution ϕa which interchanges the origin and point a, is defined by a−z ϕa (z) = . 1 − az For z, w in D, the pseudo-hyperbolic distance between z and w is given by z−w , ρ(z, w) = |ϕz (w)| = 1 − zw and the hyperbolic metric is given by Z |dξ| 1 1 + ρ(z, w) β (z, w) = inf 2 = 2 log 1 − ρ(z, w) , γ 1 − |ξ| γ

where γ is any piecewise smooth curve in D from z to w.

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ESSENTIAL COMMUTATIVITY AND ISOMETRY

3

The following lemma is well known [24]. Lemma 1. For all z, w ∈ D, we have 1 − ρ2 (z, w) =

(1 − |z|2 )(1 − |w|2 ) |1 − zw|

2

.

For ϕ ∈ S(D), the Schwarz-Pick lemma shows that ρ (ϕ(z), ϕ(w)) ≤ ρ(z, w), and if equality holds for some z 6= w, then ϕ is an automorphism of the disk. It is also well known that for ϕ ∈ S(D), Cϕ is always bounded on B. Lemma 2. [8, Lemma 3] Assume that f ∈ H ∞ (D), then for each n ∈ N , there is a positive constant C independent of f such that sup (|1 − |z|)n f (n) (z) < C||f ||∞ . z∈D

A little modification of Lemma 1 in [?] shows the following lemma. Lemma 3. There exists a constant C > 0 such that  α  α 2 2 f 0 (z) − 1 − |w| f 0 (w) ≤ C kf kBα · ρ(z, w) 1 − |z| for all z, w ∈ D and f ∈ B α . The following lemma is an easy modification of Proposition 3.11 in [2]. Lemma 4. Let 0 < α < ∞, g ∈ H(B) and ϕ be a holomorphic self-map of B. Then Pϕg : H ∞ → B α is compact if and only if Pϕg : H ∞ → B α is bounded and for any bounded sequence (fk )k∈N in H ∞ which converges to zero uniformly on B as g ) k → ∞, we have ||(Pϕg11 − Pϕ22 fk ||Bα → 0 as k → ∞. Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one appearance to the next. 3. Main theorems Theorem 1. Let 0 < α < ∞ and ϕ be a analytic self map of the unit disk. Then Tϕ = DCϕ − Cϕ D is a bounded operator from B α to Hv∞ if and only if sup z∈D

υ(z) |ϕ0 (z) − 1| 2

(1 − |ϕ(z)| )α

< ∞.

(1)

Proof. We prove the sufficiency first. Assume that (1) is true, for every f ∈ B α , we have ||Tϕ f ||Hv∞

=

sup υ(z) |f 0 (ϕ(z))ϕ0 (z) − f 0 (ϕ(z))| z∈D

=

sup

υ(z) |ϕ0 (z) − 1| 2

(1 − |ϕ(z)| )α ≤ Ckf kBα . z∈D

2

(1 − |ϕ(z)| )α |f 0 (ϕ(z))|

This means that Tϕ = DCϕ − Cϕ D is a bounded operator from B α to Hv∞ . Now we turn to the necessity. Suppose that Tϕ : B α → Hv∞ is a bounded operator, that is, there exists a constant C such that ||Tϕ f ||Hv∞ ≤ Ckf kBα , for any f ∈ B α .

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For any a ∈ D, we begin by taking test function Z z (1 − |a|2 )α fa (z) = dt. ¯t)2α 0 (1 − a It is clear that fa0 (z) =

(1−|a|2 )α (1−¯ az)2α .

(1 − |z|2 )α |fa0 (z)| =

Using Lemma 1, we have

(1 − |z|2 )α (1 − |a|2 )α = (1 − ρ2 (a, z))α . |1 − a ¯z|2α

So kfa kBα = sup(1 − |z|2 )α |fa0 (z)| ≤ 1. z∈D

α

that is fa (z) ∈ B . Therefore ∞ > =

C||Tϕ ||Bα →Hυ∞ > kTϕ fϕ(a) kHυ∞ sup

υ(z) |ϕ0 (z) − 1| 2

(1 − |ϕ(z)| )α υ(z) |ϕ0 (a) − 1| . 2 (1 − |ϕ(a)| )α

z∈D

≥

0 2 (1 − |ϕ(z)| )α fϕ(a) (ϕ(z))

So (1) follows by noticing a is arbitrary. This completes the proof of this theorem.



Theorem 2. Let 0 < α < ∞ and ϕ be a analytic self map of the unit disk. Then Tϕ = DCϕ − Cϕ D is operator from B α to Hv∞ . Then Cϕ and D are essential commutative if and only if Tϕ is bounded and lim

|ϕ(z)|→1

υ(z) |ϕ0 (z) − 1| 2

(1 − |ϕ(z)| )α

= 0.

(2)

Proof. We prove the sufficiency first. Assume that Tϕ is bounded and condition (2) holds. By the Theorem 1, we have sup υ(z) |ϕ0 (z) − 1| < ∞

(3)

z∈D

for any z ∈ D. Let {fk }k∈N be a arbitrary sequence in B α which converges to zero uniformly on compact subset of D as k → ∞,and its norm ||fk ||Bα ≤ C. Then, it follows from (2) that for any ε > 0, there is a δ > 0, with δ < |ϕ(z)| < 1, such that υ(z) |ϕ0 (z) − 1| ε (4) sup 2 α < C. z∈D (1 − |ϕ(z)| ) Let A = {z ∈ D : |ϕ(z)| ≤ δ} and B = {w : |w| ≤ δ}, then B is a compact subset of D.

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The boundedness of Tϕ implies (1) is true by the Theorem 1. Combining (3) and(4), it follows from Lemma 2 that ||Tϕ fk ||Hv∞

sup υ(z) |fk0 (ϕ(z))ϕ0 (z) − fk0 (ϕ(z))|

=

z∈D

υ(z) |ϕ0 (z) − 1|

2

(1 − |ϕ(z)| )α |fk0 (ϕ(z))| 2 (1 − |ϕ(z)| )α sup υ(z) |ϕ0 (z) − 1| |fk0 (ϕ(z))|

=

sup z∈D

≤

z∈A

+

υ(z) |ϕ0 (z) − 1|

sup

2

(1 − |ϕ(z)| )α ≤ C sup |fk (w)| + ε. z∈D\A

2

(1 − |ϕ(z)| )α |fk0 (ϕ(z))|

w∈B

As we are assume that fk → 0 on compact subset of D as k → ∞ , and ε is an arbitrary positive number. Letting k → ∞, we have ||Tϕ fk ||Hv∞ → 0. Therefore, the operator Tϕ is a compact operator by Lemma 3, so the operators Cϕ and D are essentially commutative. Now we turn to the necessity. Assume that Cϕ and D are essentially commutative. Then Tϕ = DCϕ − Cϕ D is obvious bounded since it is a compact operator. Nest, let {zk }k∈N is a arbitrary sequence in D such that |ϕ(zk )| → 1 as k → ∞. we will show (2) holds. For any zk , we begin by taking test function Z z (1 − |ϕ(zk )|2 )α dt. fk (z) = ¯ 2α 0 (1 − ϕ(zk )t) It is clear that fk0 (z) =

(1−|ϕ(zk )|2 )α ¯ k )z)2α . (1−ϕ(z

(1 − |z|2 )α |fk0 (z)| =

Using Lemma 1, we have

(1 − |z|2 )α (1 − |ϕ(zk )|2 )α = (1 − ρ2 (ϕ(zk ), z))α . ¯ k )z|2α |1 − ϕ(z

So kfk kBα = sup(1 − |z|2 )α |fk0 (z)| ≤ 1. z∈D

α

that is fa (z) ∈ B , and the sequence {fk } converges to 0 uniformly on compact subset of D as k → ∞. As the operator Tϕ = DCϕ − Cϕ D is a compact operator, it follows from Lemma 3 that lim kTϕ fk kHυ∞ = 0.

k→∞

(5)

So, we have ||Tϕ fk ||Hv∞

=

sup(υ(z) |fk0 (ϕ(z))ϕ0 (z) − fk0 (ϕ(z))| z∈D

≥ υ(zk ) |fk0 (ϕ(zk ))ϕ0 (zk ) − fk0 (ϕ(zk ))| = υ(zk ) |ϕ0 (zk ) − 1| |fk0 (ϕ(zk ))| 1 = υ(zk ) |ϕ0 (zk ) − 1| 2 (1 − |ϕ(zk )| )α So, the condition (2) is followed by combining (5) and the above result. This completes the proof of this theorem.

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G.L. LI

Remark If α = 1, υ(z) = 1 then the space B α and Hυ∞ will be Bloch space B and H ∞ . The similar results from Bloch space B to the H ∞ corresponding to Theorems 1 and 2 also hold. In the next, we study the isometry of the operator Tϕ = DCϕ − Cϕ , which is from B α to space Hβ∞ , and give the following theorem. Theorem 3. Let 0 < α < ∞ and ϕ be a analytic self maps of the unit disk . Then the operator Tϕ = DCϕ − Cϕ D : B α → Hυ∞ is an isometry in the semi-norm if and only if the following conditions hold: υ(z)|ϕ0 (z)−1| (a) sup (1−|ϕ(z)|2 )α ≤ 1; z∈D

For every a ∈ D, there at least exists a sequence {zn } in D, such that (1−|zn |2 )β |ϕ0 (zn )−1| lim ρ(ϕ(zn ), a) = 0 and lim = 1. (1−|ϕ(z )|2 )α (b)

n→∞

n→∞

n

Proof. We prove the sufficiency first. As condition (a), for every f ∈ B α , we have ||Tϕ f ||Hv∞

=

sup υ(z) |f 0 (ϕ(z))ϕ0 (z) − f 0 (ϕ(z)))| z∈D

=

sup

(1 − kf kBα .

z∈D

≤

υ(z) |ϕ0 (z) − 1| 2 |ϕ(z)| )α

2

(1 − |ϕ(z)| )α |f 0 (ϕ(z))|

Next we show that the property (b) implies ||Tϕ f ||Hv∞ ≥ ||f ||Bα . Given any f ∈ B α , then ||f ||Bα = lim (1 − |am |2 )α |f 0 (am )| for some sequence m→∞

{am } ⊂ D. For any fixed m, by property (b), there is a sequence {zkm } ⊂ D such that υ(zkm ) |ϕ0 (zkm ) − 1| →1 ρ(ϕ(zkm ), am ) → 0 and 2 (1 − |ϕ(zkm )| )α as k → ∞. By Lemma 3, for all m and k, (1 − |ϕ(zkm )|2 )α f 0 (ϕ(zkm )) − (1 − |am |2 )α f 0 (am ) ≤ C||f ||Bα · ρ(ϕ(zkm ), am ). Hence (1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))| ≥ (1 − |am |2 )α |f 0 (am )| − C||f ||Bα · ρ(ϕ(zkm ), am ). Therefore, ||Tϕ f ||Hv∞

=

sup υ(z) |f 0 (ϕ(z))ϕ0 (z) − f 0 (ϕ(z)))| z∈D

υ(z) |ϕ0 (z) − 1|

2

(1 − |ϕ(z)| )α |f 0 (ϕ(z))| 2 (1 − |ϕ(z)| )α 0 m υ(z(m k )) |ϕ (zk ) − 1| ≥ lim sup (1 − |ϕ(zkm )|2 )α |f 0 (ϕ(zkm ))| 2 α m (1 − |ϕ(zk )| ) k→∞ =

sup z∈D

=

(1 − |am |2 )α |f 0 (am )|.

The inequality ||Tϕ f ||Hv∞ ≥ ||f ||Bα follows by letting m → ∞. From the above discussions, we have kTϕ f kHv∞ = kf kBα , which means that Tϕ is an isometry operator in the semi-norm from B α to Hυ∞ . Now we turn to the necessity.

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ESSENTIAL COMMUTATIVITY AND ISOMETRY

7

For any a ∈ D, we begin by taking test function Z z (1 − |a|2 )α fa (z) = dt. ¯t)2α 0 (1 − a It is clear that fa0 (z) =

(1−|a|2 )α (1−¯ az)2α .

(6)

Using Lemma 1, we have

(1 − |z|2 )α (1 − |a|2 )α = (1 − ρ2 (a, z))α . |1 − a ¯z|2α

(1 − |z|2 )α |fa0 (z)| =

(7)

So kfa kBα = sup(1 − |z|2 )α |fa0 (z)| ≤ 1. z∈D

2 2α

(1−|a| ) On the other hand, since (1 − |a| ) |fa0 (a)| = (1−|a| 2 )2α = 1, we have kfa kB α = 1. By isometry assumption, for any a ∈ D, we have 2 α

1

= ||fϕ(a) ||Bα = kTϕ fϕ(a) kHv∞ 0 0 (ϕ(z))ϕ0 (z) − fϕ(a) (ϕ(z))) = sup υ(z) fϕ(a) z∈D

=

sup

υ(z) |ϕ0 (z) − 1| 2 |ϕ(z)| )α

(1 − υ(a) |ϕ0 (a) − 1|

z∈D

≥

2

(1 − |ϕ(a)| )α

0 2 (1 − |ϕ(z)| )α fϕ(a) (ϕ(z))

.

So (a) follows by noticing a is arbitrary. Since ||Tϕ fa ||Hv∞ = ||fa ||Bα = 1, there exists a sequence {zm } ⊂ D such that υ(zm ) |(Tϕ fa )(zm )| = υ(zm )|fa0 (ϕ(zm ))||ϕ0 (zm ) − 1| → 1

(8)

as m → ∞. It follows from (a) that υ(zm )|fa0 (ϕ(zm ))||ϕ0 (zm ) − 1| υ(zm ) |ϕ0 (zm ) − 1| 2 (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| = 2 (1 − |ϕ(zm )| )α

(9)

2

≤ (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| .

(10)

Combining (8) and (10), it follows that 2

1 ≤ lim inf (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| m→∞

2

≤ lim sup(1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| ≤ 1. m→∞

The last inequality follows by (7) since ϕ(zm ) ∈ D. Consequently, 2

lim (1 − |ϕ(zm )| )α |fa0 (ϕ(zm ))| = lim (1 − ρ2 (ϕ(zm ), a))α = 1.

m→∞

m→∞

(11)

That is, lim ρ(ϕ(zm ), a) = 0. m→∞

Combining (8), (9) and (11), we know lim

m→∞

(1 − |zm |2 )β |ϕ0 (zm ) − 1| 2

(1 − |ϕ(zm )| )α

This completes the proof of this theorem.

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References [1] J. Bonet, M. Lindstr¨ om and E. Wolf, Isometric weighted composition operators on weighted Banach spaces of type H ∞ , Proc. Amer. Math. Soc. 136(12) (2008), 4267-4273 . [2] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Math., CRC Press, Boca Raton, 1995. [3] E. Berkson, Composition operators isolated in the uniform operator topology, Proc. Amer. Math Soc. 81 (1981), 230-232. [4] Z.S. Fang and Z.H. Zhou,Differences of composition operators on the space of bounded analytic functions in the polydisc, Bull. Aust. Math. Soc. 79 (2009), 465-471. [5] T. Hosokawa, K. Izuchi and D. Zheng, Isolated points and essential components of composition operators on H ∞ , Proc. Amer. Math. Soc. 130 (2002), 1765-1773. [6] T. Hosokawa, S. Ohno, Differences of weighted composition operators acting from Bloch space to the space H ∞ , Trans. Amer. Math. Soc. 363(10) (2011), 5321-5340. [7] R.A. Hibschweiler and N. Portony, Composition followed by differtition between Bergman and Hardy spaces, Rocky Mountain J. Math. 35 (2005), 843-855. [8] S. Li and S. Stevi´ c, Products of Volterra type operators from H ∞ and the Bloch type spaces to Zygmund spaces, J. Math. Anal. Appl. 345 (2008), 40-52. [9] G.L. Li and Z.H. Zhou, Isometries on Products of composition and integral operators on Bloch type space, J. Inequal. Appl. 2010 (2010), Article ID 184957, 9 pages. [10] G.L. Li and Z.H. Zhou, Isometries of Composition and Differentiation Operators from Bloch Type Space to , J. Comput. Anal. Appl. 15(3) (2013), 526-533. [11] J. Moorhouse, Compact difference of composition operators, J. Funct. Anal. 219 (2005), 70-92. [12] A. Montes-Rodr´ıguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London Math. Soc., 61(3) (2000), 872-884. [13] B. MacCluer, S. Ohno and R. Zhao, Topological structure of the space of composition operators on H ∞ , Integr. Equ. Oper. Theory 40(4) (2001), 481-494. [14] Ohno, Product of composition and differtition between Hardy spaces, Bull. Austral. Math. Soc. 73 (2006), 235-243. [15] J.H. Shapiro, Composition operators and classical function theory, Spriger-Verlag, 1993. [16] J.H. Shi and L. Luo, Composition operators on the Bloch space, Acta Math. Sinica 16 (2000), 85-98. [17] S. Stevic´ and E. Wolf, Differences of composition operators between Cn , Appl. Math. Comput. 215(5) (2009), 1752-1760. [18] C. Toews, Topological components of the set of composition operators on H ∞ (BN ), Integr. Equ. Oper. Theory 48 (2004), 265-280. [19] C.Z. Tong and Z.H. Zhou, Intertwining relations for Volterra operators on the Bergman space, Illinois J. Math. 57 (1)(2013), 195-211. [20] E. Wolf. Differences of composition operators between weighted Banach spaces of holomorphic functions on the unit polydisk, Result. Math. 51 (2008), 361-372. [21] Z.H. Zhou and R.Y. Chen, Weighted composition operators fom F (p, q, s) to Bloch type spaces, Internat. J. Math. 19(8) (2008), 899-926. [22] Z.H. Zhou and J.H. Shi, Compactness of composition operators on the Bloch space in classical bounded symmetric domains, Michigan Math. J. 50 (2002), 381-405. [23] Z.H. Zhou and L. Zhang, Essential commutativity of some integral and composition operators, Bull. Aust. Math. Soc., 85 (2012), 143-153. [24] K.H. Zhu, Spaces of holomorphic functions in the unit ball, Springer-Verlag (GTM 226), 2004.

Geng-Lei Li Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, P.R. China. E-mail address: [email protected]

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APPROXIMATION OF JENSEN TYPE QUADRATIC-ADDITIVE MAPPINGS VIA THE FIXED POINT THEORY ∗ YANG-HI LEE, JOHN MICHAEL RASSIAS, AND HARK-MAHN KIM

†

Abstract. In this article, we investigate the stability results of a Jensen type quadratic-additive functional equation f (x + y) + f (x − y) + 2f (z) = 2f (x) + f (z + y) + f (z − y) via the fixed point theory. And then, we present two counter-examples which do not satisfy the stability results.

1. Introduction A classical question in the theory of functional equations is “when is it true that a mapping, which satisfies approximately a functional equation, must be somehow close to an exact solution of the equation?” Such a problem, called a stability problem of functional equations, was formulated by S. M. Ulam [31] in 1940 as follows: Let G1 be a group and G2 a metric group with metric ρ(·, ·). Given ϵ > 0, does there exist a δ > 0 such that if f : G1 → G2 satisfies ρ(f (xy), f (x)f (y)) < δ for all x, y ∈ G1 , then a homomorphism h : G1 → G2 exists with ρ(f (x), h(x)) < ϵ for all x ∈ G1 ? When this problem has a solution, we say that the homomorphisms from G1 to G2 are stable. In 1941, D. H. Hyers [16] considered the case of approximately additive mappings between Banach spaces and proved the following result. Suppose that E1 and E2 are Banach spaces and f : E1 → E2 satisfies the following condition: there is a constant ϵ ≥ 0 such that ∥f (x + y) − f (x) − f (y)∥ ≤ ϵ n

for all x, y ∈ E1 . Then the limit h(x) = limn→∞ f (22n x) exists for all x ∈ E1 and it is a unique additive mapping h : E1 → E2 such that ∥f (x) − h(x)∥ ≤ ϵ. The method which was provided by Hyers, and which produces the additive mapping h, was called a direct method. This method is the most important and most powerful tool for studying the stability of various functional equations. Hyers’ Theorem was generalized by T. Aoki [1] and D.G. Bourgin [3] for additive mappings by considering an unbounded Cauchy difference. In 1978, Th.M. Rassias [26] also provided a generalization of Hyers’ Theorem for linear mappings which allows the Cauchy difference to be unbounded like this ∥x∥p + ∥y∥p . A generalized result of 2000 Mathematics Subject Classification. 39B82, 39B72, 47L05. Key words and phrases. Fixed point method; Generalized Hyers–Ulam stability; Jensen type quadratic-additive mapping. ∗ This work was supported by research fund of Chungnam National University. † Corresponding author:[email protected]. 1

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Th.M. Rassias’ theorem was obtained by P. Gˇavruta in [10] and S. Jung in [18]. In 1990, Th.M. Rassias [27] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. In 1991, Z. Gajda [9] following the same approach as in [26], gave an affirmative solution to this question for p > 1. It was shown by Z. Gajda [9], as well as by Th.M. ˘ Rassias and P. Semrl [28], that one cannot prove a Th.M. Rassias’ type theorem when p = 1. In 2003-2007 J.M. Rassias and M.J. Rassias [23, 24] and J.M. Rassias [25] solved the above Ulam problem for Jensen and Jensen type mappings. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians, see [6, 14, 17, 29, 22, 15]. Almost all subsequent proofs, in this very active area, have used Hyers’ direct method, namely, the mapping F , which is a solution of he functional equation, is explicitly constructed by the limit function of a Cauchy sequence starting from the given approximate mapping f . The first result of the generalized Hyers–Ulam stability for Jensen equation was given in the paper [19] by the direct method. In 2003, L. C˘adariu and V. Radu [4] observed that the existence of the solution F for a Cauchy functional equation and the estimation of the mapping F with the approximate mapping f of the equation can be obtained from the alternative fixed point theorem. This method is called a fixed point method. In addition, they applied this method to prove stability theorems of the Jensen’s functional equation: ( ) x+y (1.1) 2f − f (x) − f (y) = 0 ⇔ 2f (x) − f (x + y) − f (x − y) = 0. 2 On the other hand, some properties of generalized Hyers-Ulam stability for a functional equation of Jensen type were obtained in [7] by the fixed point method. Further, the authors [5] obtained the stability of the quadratic functional equation: (1.2)

f (x + y) + f (x − y) − 2f (x) − 2f (y) = 0

by using the fixed point method. Notice that if we consider the functions f1 , f2 : R → R defined by f1 (x) = ax + b and f2 (x) = cx2 , respectively, where a, b and c are real constants, then f1 satisfies the equation (1.1) and f2 is a solution of the equation (1.2), respectively. Associating the equation (1.1) with the equation (1.2), we see the following well known Drygas functional equation: (1.3)

f (x + y) + f (x − y) = 2f (x) + f (y) + f (−y),

which has quadratic solutions Q of equation (1.2) in the class of even functions, and has additive solutions A of equation (1.1) in the class of odd functions. Hence the general solution f of (1.3) is given by f (x) = Q(x) + A(x) [30]. Now, adding the equation (1.3) and the following Drygas functional equation (1.4)

2f (z) + f (y) + f (−y) = f (z + y) + f (z − y),

we get the Jensen type quadratic-additive functional equation: (1.5)

f (x + y) + f (x − y) + 2f (z) = f (z + y) + f (z − y) + 2f (x),

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of which the general solution function f (x) − f (0) has of the form f (x) − f (0) = (−x) Q(x) + A(x), where Q(x) := f (x)+f − f (0) is a quadratic mapping satisfying the 2 f (x)−f (−x) equation (1.2) and A(x) := is a Jensen mapping satisfying the equation 2 (1.1). In the paper, without splitting the given approximate mapping f : X → Y of the equation (1.5) into two approximate even and odd parts, we are going to derive the desired approximate solution F near f at once. Precisely, we introduce a strictly contractive mapping with Lipschitz constant 0 < L < 1, and then, we show that the contractive mapping has the fixed point F in a generalized metric function space by using the fixed point method in the sense of L. C˘adariu and V. Radu, where, the fixed point F yields the precise solution of the equation (1.5) near f . In Section 2, we prove several stability results of the functional equation (1.5) using the fixed point method under suitable conditions. In Section 3, we use the results in the previous section to get stability results of the Jensen’s functional equation (1.1) and to get that of the quadratic functional equation (1.2), respectively. 2. Generalized Hyers–Ulam stability of (1.5) In this section, we prove the generalized Hyers–Ulam stability of the Jensen type quadratic-additive functional equation (1.5). We recall the following fundamental result of the fixed point theory by Margolis and Diaz [20]. Theorem 2.1. Suppose that a complete generalized metric space (X, d), which means that the metric d may assume infinite values, and a strictly contractive mapping Λ : X → X with the Lipschitz constant 0 < L < 1 are given. Then, for each given element x ∈ X, either d(Λn x, Λn+1 x) = +∞, ∀n ∈ N ∪ {0}, or there exists a nonnegative integer k such that • • • •

d(Λn x, Λn+1 x) < +∞ for all n ≥ k; the sequence {Λn x} is convergent to a fixed point y ∗ of Λ; y ∗ is the unique fixed point of Λ in X1 := {y ∈ X, d(Λk x, y) < +∞}; 1 d(y, y ∗ ) ≤ 1−L d(y, Λy) for all y ∈ X1 .

Throughout this paper, let V be a (real or complex) linear space and Y a Banach space. For a given mapping f : V → Y , we use the following abbreviation Df (x, y, z) := f (x + y) + f (x − y) + 2f (z) − f (z + y) − f (z − y) − 2f (x) for all x, y, z ∈ V . In the following theorem, we prove the stability of the Jensen type quadraticadditive functional equation (1.5) using the fixed point method. Theorem 2.2. Let f : V → Y be a mapping for which there exists a mapping φ : V 3 → R+ := [0, ∞) such that (2.1)

∥Df (x, y, z)∥ ≤ φ(x, y, z)

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for all x, y, z ∈ V . If φ(x, y, z) = φ(−x, −y, −z) for all x, y, z ∈ V and there exists a constant 0 < L < 1 such that φ(2x, 2y, 2z) ≤ 2Lφ(x, y, z),

(2.2)

for all x, y, z ∈ V , then there exists a unique Jensen type quadratic-additive mapping F : V → Y such that DF (x, y, z) = 0 for all x, y, z ∈ V and ∥f (x) − F (x)∥ ≤

(2.3)

φ(x, x, 0) 2(1 − L)

for all x ∈ V . In particular, F is represented by ] [ f (2n x) + f (−2n x) f (2n x) − f (−2n x) (2.4) + F (x) = f (0) + lim n→∞ 2 · 4n 2n+1 for all x ∈ V . Proof. If we consider the mapping f˜ := f − f (0), then f˜ : V → Y satisfies f˜(0) = 0 and ∥Df˜(x, y, z)∥ = ∥Df (x, y, z)∥ ≤ φ(x, y, z)

(2.5)

for all x, y, z ∈ V . Let S be the set of all mappings g : V → Y with g(0) = 0, and then we introduce a generalized metric d on S by { } (2.6) d(g, h) := inf K ∈ R+ : ∥g(x) − h(x)∥ ≤ Kφ(x, x, 0) ∀x ∈ V . It is easy to show that (S, d) is a generalized complete metric space. Now, we consider an operator Λ : S → S defined by (2.7)

Λg(x) :=

g(2x) − g(−2x) g(2x) + g(−2x) + 4 8

for all g ∈ S and all x ∈ V. Then we notice that (2.8)

g(2n x) − g(−2n x) g(2n x) + g(−2n x) Λ g(x) = + 2n+1 2 · 4n n

for all n ∈ N and x ∈ V . First, we show that Λ is a strictly contractive self-mapping of S with the Lipschitz constant L. Let g, h ∈ S and let K ∈ [0, ∞] be an arbitrary constant with d(g, h) ≤ K. From the definition of d, we have (2.9)

3 1 ∥g(2x) − h(2x)∥ + ∥g(−2x) − h(−2x)∥ 8 8 1 Kφ(2x, 2x, 0) ≤ LKφ(x, x, 0) ≤ 2

∥Λg(x) − Λh(x)∥ =

for all x ∈ V , which implies that (2.10)

d(Λg, Λh) ≤ Ld(g, h)

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for any g, h ∈ S. That is, Λ is a strictly contractive self-mapping of S with the Lipschitz constant L. Moreover, by (2.5) we see that 1 ∥ − 3Df˜(x, x, 0) + Df˜(−x, −x, 0)∥ ∥f˜(x) − Λf˜(x)∥ = 8 1 ≤ φ(x, x, 0) 2 for all x ∈ V . It means that d(f˜, Λf˜) ≤ 12 < ∞ by the definition of d. Therefore, according to Theorem 2.1, the sequence {Λn f˜} converges to the unique fixed point F˜ : V → Y of Λ in the set S1 = {g ∈ S|d(f˜, g) < ∞}, which is represented by

(2.11)

[ f˜(2n x) − f˜(−2n x) f˜(2n x) + f˜(−2n x) ] ˜ (2.12) + F (x) = lim n→∞ 2n+1 2 · 4n for all x ∈ V . Putting F := F˜ + f (0), then we have the equality ∥f (x) − F (x)∥ = ∥f˜(x) − F˜ (x)∥ for all x ∈ V . Thus one notes that (2.13)

d(f˜, F˜ ) ≤

1 1 d(f˜, Λf˜) ≤ , 1−L 2(1 − L)

which implies (2.3) and (2.4). By the definitions of F and F˜ , together with (2.5) and (2.2), we have that ∥DF (x, y, z)∥ = ∥DF˜ (x, y, z)∥

Df (2n x, 2n y, 2n z) − f (−2n x, −2n y, −2n z)

= lim n→∞ 2n+1 n Df (2 x, 2n y, 2n z) + Df (−2n x, −2n y, −2n z)

+

2 · 4n ) 2n + 1 ( n n n n n n ≤ lim φ(2 x, 2 y, 2 z) + φ(−2 x, −2 y, −2 z) n→∞ 2 · 4n = 0 for all x, y, z ∈ V . Thus, the mapping F satisfies the Jensen type quadratic-additive  functional equation (2.3). This completes the proof of this theorem. Theorem 2.3. Let f : V → Y be a mapping for which there exists a mapping φ : V 3 → [0, ∞) such that (2.14)

∥Df (x, y, z)∥ ≤ φ(x, y, z)

for all x, y, z ∈ V . If φ(x, y, z) = φ(−x, −y, −z) for all x, y, z ∈ V and there exists a constant 0 < L < 1 such that the mapping φ has the property L φ(x, y, z) ≤ φ(2x, 2y, 2z) (2.15) 4 for all x, y, z ∈ V , then there exists a unique Jensen type quadratic-additive mapping F : V → Y such that DF (x, y, z) = 0 for all x, y, z ∈ V and (2.16)

∥f (x) − F (x)∥ ≤

708

Lφ(x, x, 0) 4(1 − L)

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for all x ∈ V. In particular, F is represented by (2.17) F (x) := f (0) [ 4n ( ( x ) ( −x ) ) ( (x) ( −x ))] + lim f n + f n − 2f (0) + 2n−1 f n − f n n→∞ 2 2 2 2 2 for all x ∈ V . Proof. The proof is similarly verified by the same argument as that of Theorem 2.2.  3. Applications For a given mapping f : V → Y , we use the following abbreviations ( ) x+y Jf (x, y) := 2f − f (x) − f (y), 2 Qf (x, y) := f (x + y) + f (x − y) − 2f (x) − 2f (y) for all x, y ∈ V . Using Theorem 2.2 and Theorem 2.3, we will obtain the stability results of the quadratic functional equation Qf ≡ 0 and the Jensen’s functional equation Jf ≡ 0 in the following corollaries. Corollary 3.1. Suppose that each fi : V → Y, i = 1, 2, and ϕi : V 2 → [0, ∞), i = 1, 2, are given functions satisfying ∥Qfi (x, y)∥ ≤ ϕi (x, y) and ϕi (x, y) = ϕi (−x, −y) for all x, y ∈ V , respectively. If there exists 0 < L < 1 such that ϕ1 (2x, 2y) ≤ 2Lϕ1 (x, y), L (3.2) ϕ2 (x, y) ≤ ϕ2 (2x, 2y) 4 for all x, y ∈ V , then we have unique quadratic mappings F1 , F2 : V → Y such that

(3.1)

(3.3) (3.4)

ϕ1 (0, x) + ϕ1 (x, x) , 2(1 − L) L[ϕ2 (0, x) + ϕ2 (x, x)] ∥f2 (x) − F2 (x)∥ ≤ 4(1 − L)

∥f1 (x) − f1 (0) − F1 (x)∥ ≤

for all x ∈ V . In particular, F1 and F2 are represented by (3.5) (3.6)

f1 (2n x) , n→∞ 4n( ) x F2 (x) = lim 4n f2 n n→∞ 2

F1 (x) =

lim

for all x ∈ V . Moreover, if 0 < L < 12 and ϕ1 is continuous, then f1 − f1 (0) is itself a Jensen type quadratic-additive mapping.

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Proof. Notice that ∥Dfi (x, y, z)∥ = ∥Qfi (x, y) − Qfi (z, y)∥ ≤ ϕi (x, y) + ϕi (z, y) for all x, y, z ∈ V and i = 1, 2. Put φi (x, y, z) := ϕi (x, y) + ϕi (z, y) for all x, y, z ∈ V . Then φ1 satisfies (2.2) and φ2 satisfies (2.15). According to Theorem 2.2, there exists a unique mapping F1 : V → Y satisfying (3.3) which is represented by ( ) f1 (2n x) + f1 (−2n x) f1 (2n x) − f1 (−2n x) + . F1 (x) = lim n→∞ 2 · 4n 2n+1 Observe that

f1 (2n x) − f1 (−2n x) 1 n

= lim lim

n→∞ 2n+1 ∥Qf1 (0, 2 x)∥ n→∞ 2n+1 1 ≤ lim ϕ (0, 2n x) 2n+1 1 n→∞

≤ lim

n→∞

as well as



f1 (2n x) − f1 (−2n x)

≤ lim lim

n→∞ n→∞ 2 · 4n

Ln ϕ (0, x) 2 1

=0

Ln ϕ (0, x) 2n+1 1

=0

for all x ∈ V . From these two properties, we lead to the mapping (3.5) for all x ∈ V . Moreover, we have

Qf1 (2n x, 2n y) ϕ1 (2n x, 2n y) Ln

≤ ≤ ϕ1 (x, y)

4n 4n 2n for all x, y ∈ V . Taking the limit as n → ∞ in the above inequality, we get QF1 (x, y) = 0 for all x, y ∈ V and so F1 : V → Y is a quadratic mapping. On the other hand, since Lϕ2 (0, 0) ≥ 4ϕ2 (0, 0) and ∥2f2 (0)∥ = ∥Qf2 (0, 0)∥ ≤ ϕ2 (0, 0) we can show that ϕ2 (0, 0) = 0 and f2 (0) = 0. According to Theorem 2.3, there exists a unique mapping F2 : V → Y satisfying (3.4), which is represented by (2.17). We have (x) ( x ) ( x ) 4n 4n



lim

− f2 n + f2 − n = lim

Qf2 0, n n→∞ 2 n→∞ 2 2 2 2 4n ( x ) ≤ lim ϕ2 0, n n→∞ 2 2 Ln ≤ lim ϕ2 (0, x) = 0 n→∞ 2 as well as

(x) ( x )

lim 2n−1 f2 n − f2 − n = 0 n→∞ 2 2

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for all x ∈ V . From these and (2.8), we get (3.6). Notice that

(x y) ( x y )

n

4 Qf2 n , n ≤ 4n ϕ2 n , n ≤ Ln ϕ2 (x, y) 2 2 2 2 for all x, y ∈ V . Taking the limit as n → ∞, then we have shown that QF2 (x, y) = 0 for all x, y ∈ V and so F2 : V → Y is a quadratic mapping. This completes the corollary.  Now, we obtain a stability result of Jensen functional equations. Corollary 3.2. Let fi : V → Y, i = 1, 2, be mappings for which there exist functions ϕi : V 2 → [0, ∞), i = 1, 2, such that ∥Jfi (x, y)∥ ≤ ϕi (x, y)

(3.7)

and ϕi (x, y) = ϕi (−x, −y) for all x, y ∈ V , respectively. If there exists 0 < L < 1 such that the mapping ϕ1 has the property (3.1) and ϕ2 holds (3.2) for all x, y ∈ V, then there exist unique Jensen mappings Fi : V → Y, i = 1, 2, such that (3.8) (3.9)

ϕ1 (0, 2x) + ϕ1 (x, −x) , 2(1 − L) L(ϕ2 (0, 2x) + ϕ2 (x, −x)) ∥f2 (x) − F2 (x)∥ ≤ 4(1 − L)

∥f1 (x) − F1 (x)∥ ≤

for all x ∈ V . In particular, the mappings F1 , F2 are represented by (3.10) (3.11)

f1 (2n x) + f1 (0), n n→∞ 2 ( (x) ) F2 (x) = lim 2n f2 n − f2 (0) + f2 (0) n→∞ 2

F1 (x) =

lim

for all x ∈ V . Proof. The proof is similar to that of Theorem 3.1.  Now, we obtain generalized Hyers-Ulam stability results in the framework of normed spaces using Theorem 2.2 and Theorem 2.3. Corollary 3.3. Let X be a normed space, θ ≥ 0, and p ∈ (0, 1) ∪ (2, ∞). Suppose that a mapping f : X → Y satisfies the inequality ∥Df (x, y, z)∥ ≤ θ(∥x∥p + ∥y∥p + ∥z∥p ) for all x, y, z ∈ X. Then there exists a unique Jensen type quadratic-additive mapping F : X → Y such that { 2θ ∥x∥p , if 0 < p < 1; 2−2p ∥f (x) − F (x)∥ ≤ 2θ ∥x∥p , if p > 2, 2p −4 for all x ∈ X.

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Proof. It follows from Theorem 2.2 and Theorem 2.3, by putting L := 2p−1 < 1 if 0 < p < 1,and L := 22−p < 1 if p > 2.  In the following, we present counter-examples for the singular cases p = 1 and p = 2 in Corollary 3.3. Example 3.1. We remark that if we consider an odd function f : R → R defined by  ∞  µx, if − 1 < x < 1; i ∑ ϕ(2 x) µ, if x ≥ 1; , ϕ(x) = f (x) =  2i −µ, if x ≤ −1, (µ > 0), i=1 which is the same type as that in the paper [9], then it follows from [28] that |f (x + y) + f (x − y) − 2f (x)| ≤ θ(|x| + |y|), |f (z + y) + f (z − y) − 2f (z)| ≤ θ(|z| + |y|), and so |Df (x, y, z)| ≤ 2θ(|x| + |y| + |z|), for all x, y, z and for some constant θ > 0. However, there doesn’t exist Jensen type quadratic-additive function F : R → R such that |f (x) − F (x)| ≤ K(θ)|x| for all x and for some constant K(θ). Hence, there exists a counter-example for the case p = 1 in Corollary 3.3. Also, we remark that if we consider an even function f : R → R defined by { ∞ ∑ ϕ(2i x) µx2 , if |x| < 1; f (x) = = , ϕ(x) µ, if |x| ≥ 1, (µ > 0), 4i i=1

which is the same type as that in the paper [8], then it is well known that |f (x + y) + f (x − y) − 2f (x) − 2f (y)| ≤ θ(|x|2 + |y|2 ), |f (z + y) + f (z − y) − 2f (z) − 2f (y)| ≤ θ(|z|2 + |y|2 ), and so |Df (x, y, z)| ≤ 2θ(|x|2 + |y|2 + |z|2 ), for all x, y, z and for some constant θ > 0. However, there doesn’t exist Jensen type quadratic-additive function F : R → R such that |f (x) − F (x)| ≤ K(θ)|x|2 for all x and for some constant K(θ). Hence, there exists a counter-example for the case p = 2 in Corollary 3.3. Corollary 3.4. Let X be a normd space, θ ≥ 0 and p, q, r > 0 be reals with p+q+r ∈ (−∞, 1) ∪ (2, ∞). If a mapping f : X → Y satisfies ∥Df (x, y, z)∥ ≤ θ∥x∥p ∥y∥q ∥z∥r for all x, y, z ∈ X, then f is itself a Jensen type quadratic-additive mapping. Proof. It follows from Theorem 2.2 and Theorem 2.3, by putting L := 2p+q+r−1 < 1 if p + q + r < 1,and L := 22−p−q−r < 1 if p + q + r > 2. 

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Corollary 3.5. Let X be a normd space, θi ≥ 0, (i = 1, 2, 3) and p, q, r > 0 be reals such that either max{p + q, q + r, p + r} < 1 or min{p + q, q + r, p + r} > 2. If a mapping f : X → Y satisfies ∥Df (x, y, z)∥ ≤ θ1 ∥x∥p ∥y∥q + θ2 ∥y∥q ∥z∥r + θ3 ∥x∥p ∥z∥r for all x, y, z ∈ X, Then there exists a unique Jensen type quadratic-additive mapping F : X → Y such that { θ1 ∥x∥p+q , if max{p + q, q + r, p + r} < 1; 2−2max{p+q,q+r,p+r} ∥f (x) − F (x)∥ ≤ θ1 ∥x∥p+q , if min{p + q, q + r, p + r} > 2, 2min{p+q,q+r,p+r} −4 for all x ∈ X. Proof. It follows from Theorem 2.2 and Theorem 2.3, by putting L := 2max{p+q,q+r,p+r}−1 < 1, if max{p + q, q + r, p + r} < 1, and L := 22−min{p+q,q+r,p+r} < 1, if min{p + q, q + r, p + r} > 2.  In the following, we present counter-examples for the singular cases max{p+q, q + r, p + r} = 1 and min{p + q, q + r, p + r} = 2 in Corollary 3.5. Example 3.2. We remark that if we consider an odd function f : R → R defined by { xln|x|, if x ̸= 0; f (x) = 0, if x = 0, then for any p with 0 < p < 1 it follows from [11, 12] that there exists a constant c > 0 such that |f (x + y) − f (x) − f (y)| ≤ c|x|p |y|1−p , and so |f (x − y) − f (x) + f (y)| ≤ c|x|p |y|1−p , |f (x + y) + f (x − y) − 2f (x)| ≤ 2c|x|p |y|1−p , |f (z + y) + f (z − y) − 2f (z)| ≤ 2c|z|p |y|1−p , which yield |Df (x, y, z)| ≤ 2c(|x|p |y|1−p + |y|1−p |z|p ), for all x, y, z. However, there doesn’t exist Jensen type quadratic-additive function F : R → R such that |f (x) − F (x)| ≤ K(c, p)|x| for all x and for some constant K(c, p). Hence, there exists a counter-example for the case max{p + q, q + r, p + r} = 1 in Corollary 3.5. Also, we remark that if we consider an even function f : R → R defined by { 2 x ln|x|, if x ̸= 0; f (x) = 0, if x = 0,

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then for any p with 0 < p < 2 it follows from [13] that there exists a constant k > 0 such that |f (x + y) + f (x − y) − 2f (x) − 2f (y)| ≤ k|x|p |y|2−p , |f (z + y) + f (z − y) − 2f (z) − 2f (y)| ≤ k|z|p |y|2−p , which yield |Df (x, y, z)| ≤ k(|x|p |y|2−p + |y|2−p |z|p ), for all x, y, z. However, there doesn’t exist Jensen type quadratic-additive function F : R → R such that |f (x) − F (x)| ≤ K(k, p)|x|2 for all x and for some constant K(k, p). Hence, there exists a counter-example for the case min{p + q, q + r, p + r} = 2 in Corollary 3.5. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. [2] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ. vol. 48, Amer. Math. Soc., Providence, RI, 2000. [3] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237. [4] L. C˘adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4(1), (2003), Art.4. [5] L. C˘adariu and V. Radu Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform. 41 (2003), 25–48. [6] L. C˘adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications, Volume 2008 (2008), Article ID 749392, 15 pages. [7] L. C˘adariu and V. Radu, The fixed Point method to stability properties of a functional equation of Jensen type, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si, Ser. Nou˘a, Mat. 54 (2008), No. 2, 307–318. [8] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Hamburg, 62 (1992), 59–64. [9] Z. Gajda, On the stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431–434. [10] P. Gˇavruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [11] P. Gˇavruta, An answer to a question of John M. Rassias concerning the stability of Cauchy equation, in: Advances in Equations and Inequalities, Hadronic Math. Series, USA, 1999, 67–71. [12] P. Gˇavruta, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings, J. Math. Anal. Appl. 261 (2001), 543–553. [13] L. Gˇavruta and P. Gˇavruta, On a problem of John M. Rassias concerning the stability in Ulam sense of Euler–Lagrange equation, in: Functional Equations, Difference Inequalities, Nova Sci. Publishers, Inc. 2010, 47–53. [14] Z.X. Gao, H.X. Cao, W.T. Zheng and L. Xu, Generalized Hyers–Ulam–Rassias stability of functional inequalities and functional equations, J. Math. Inequal. 3 (1)(2009), 63–77. [15] M.E. Gordji, A. Divandari, M. Rostamian, C. Park, and D.S. Shin, Hyers–Ulam Stability of a Tribonacci Functional Equation in 2-Normed Spaces, J. Comput. Anal. Appl. 16(3), 2014, 503–508.

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[16] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [17] S. Jin and Y. Lee, Fuzzy stability of a functional equation deriving from quadratic and additive mappings, Abstract and Applied Analysis, Volume 2011 (2011), Article ID 534120, 15 pages. [18] S. Jung, On the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 204 (1996), 221–226. [19] Y. Lee and K. Jun, A generalization of the Hyers–Ulam–Rassias stability of Jensens equation, J. Math. Anal. Appl. 238 (1999), 303–315. [20] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [21] M.S. Moslehian and A. Najati, An application of a fixed point theorem to a functional inequality, Fixed Point Theory, 10 (2009) No. 1, 141–149. [22] C. Park, S. Shagholi, A. Javadian, M.B. Savadkouhi, and M.E. Gordji, Quadratic Derivations on non-Archimedean Banach Algebras, J. Comput. Anal. Appl. 16(3), 2014, 565–570. [23] J.M. Rassias and M.J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281(2003), 516–524. [24] J.M. Rassias and M.J. Rassias, Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. 129(2005), 545–558. [25] J.M. Rassias, Refined Hyers–Ulam approximation of approximately Jensen type mappings, Bull. Sci. Math. 131(2007), 89–98. [26] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [27] Th.M. Rassias, The stability of mappings and related topics, In ‘Report on the 27th ISFE’, Aequationes Math. 39 (1990), 292–293. ˇ [28] Th.M. Rassias and P. Semrl, On the behaviour of mappings which do not satisfy Hyers–Ulam– Rassias stability, Proc. Amer. Math. Soc. 114 (1992), 989–993. [29] D.Y. Shin, C. Park, and S. Farhadabadi, On the Superstability of Ternary Jordan C*homomorphisms, J. Comput. Anal. Appl. 16(5), 2014, 964–973. [30] Gy. Szab´o, Some functional equations related to quadratic functions, Glasnik Math. 38 (1983), 107–118. [31] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Korea, E-mail address: [email protected] Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4, Agamemnonos St., Aghia Paraskevi, Athens, 15342, Greece E-mail address: [email protected] Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon, 305-764, Korea E-mail address: [email protected]

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On certain subclasses of p-valent analytic functions involving Saitoh operator J. Patel1 and N.E. Cho2,∗ 1 Department

2 Department

of Mathematics, Utkal University, Vani Vihar, Bhubaneswar-751004, India E-mail:[email protected]

of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea E-Mail:[email protected] ∗ Corresponding Author

Abstract: The object of the present investigation is to solve Fekete-Szegö problem for a new class Vλp (a, c, A, B) of p-valent analytic functions involving the Saitoh operator in the unit disk. We also obtain subordination results and some interesting corollaries for functions in Ap involving this operator. Relevant connections of the results obtained here with those given by earlier workers on the subject are also mentioned. 2010 Mathematics Subject Classification: 30C45. Key words and phrases: Analytic function, Subordination, Hadamard product, Fekete-Szegö problem, Saitoh operator. 1. Introduction and preliminaries Let Ap be the class of functions of the form p

f (z) = z +

∞ X

ap+k z p+k

(1.1)

k=1

analytic in the unit disk U = {z ∈ C : |z| < 1} with p ∈ N = {1, 2, 3, · · · }. Let S be the subclass of A1 = A consisting of univalent functions. A function f ∈ Ap is said to be p-valent starlike of order α, denoted by S∗p (α), if and only if zf 0 (z) Re f (z) 



>α

(0 ≤ α < p; z ∈ U).

(1.2)

Similarly, a function f ∈ Ap is said to be p-valent convex of order α, denoted by Cp (α), if and only if   zf 00 (z) Re 1 + 0 > α (0 ≤ α < p; z ∈ U). (1.3) f (z) From (1.2) and (1.3), it follows that f (z) ∈ Cp (α) ⇐⇒

zf 0 (z) ∈ S∗p (α). p

Furthermore, we say that a function f ∈ Ap is said to be in the class Rp (α), if and only if 

Re

f 0 (z) z p−1



>α

(0 ≤ α < p; z ∈ U).

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For functions f and g, analytic in the unit disk U, we say the f is said to be subordinate to g, written as f ≺ g or f (z) ≺ g(z) (z ∈ U), if there exists an analytic function ω in U with ω(0) = 0, |ω(z)| ≤ |z| (z ∈ U) and f (z) = g(ω(z)) for all z ∈ U. In particular, if g is univalent in U, then we have the following equivalence (cf., e.g., [14]): f (z) ≺ g(z) ⇐⇒ f (0) = g(0) and f (U) ⊂ g(U). For the functions f and g given by the power series f (z) =

∞ X

an z n ,

g(z) =

n=0

∞ X

bn z n ,

(z ∈ U)

n=0

their Hadamard product (or convolution), denoted by f ? g is defined as (f ? g)(z) =

∞ X

an bn z n = (g ? f )(z)

(z ∈ U).

n=0

Note that f ? g is analytic in U. By making use of the Hadamard product, Saitoh [18] defined a linear operator Lp (a, c) : Ap −→ Ap in terms of the function ϕp as Lp (a, c)f (z) = ϕp (a, c; z) ? f (z) where ϕp (a, c; z) =

∞ X (a)k k=0

(c)k

z p+k

(f ∈ Ap ; z ∈ U),

(1.5)

(a ∈ C, c ∈ C \ {· · · , −2, −1, 0}; z ∈ U) .

(1.6)

and (x)k is the Pochhammer symbol (or shifted factorial) given by (x)k =

 1,

n = 0,

x(x + 1) · · · (x + k − 1),

k ∈ N.

For p = 1, the operator Lp (a, c) reduces to the Carlson-Shaffer operator L(a, c) [1]. If f ∈ Ap is given by (1.1), then it follows from (1.5) and (1.6) that Lp (a, c)f (z) = z p +

∞ X (a)k k=1

(c)k

ap+k z p+k

(z ∈ U)

(1.7)

and z (Lp (a, c)f )0 (z) = aLp (a + 1, c)f (z) − (a − p)Lp (a, c)f (z)

(z ∈ U).

(1.8)

It is easily seen that for f ∈ Ap (i) Lp (a, a)f (z) = f (z), zf 0 (z) (ii) Lp (p + 1, p)f (z) = , p (iii) Lp (n + p, p)f (z) = Dn+p−1 f (z) (n ∈ Z; n > −p), the operator studied by Goel and Sohi [5]. For the case p = 1, Dn is the Ruscheweyh derivative operator [17]. (iv) Lp (p + 1, n + p)f (z) = In,p f (z) (n ∈ Z; n > −p), the extended Noor integral operator considered by Liu and Noor [10].

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(v) Lp (p+1, p+1−µ)f (z) = Ωz f (z) (−∞ < µ < p+1), the extended fractional differintegral operator studied by Patel and Mishra [16]. Note that Ω(0,p) f (z) = f (z), Ω(1,p) f (z) = z z

zf 0 (z) z 2 f 00 (z) and Ω(2,p) f (z) = (p ≥ 2). z p p(p − 1)

As a special case, we get the operator Ωµz f (z) (0 ≤ µ < 1) for p = 1 introduced and studied by Owa-Srivastava [15]. With the aid of the operator Lp (a, c), we introduce a subclass of Ap as follows. Definition 1.1. For the fixed parameters A, B (−1 ≤ B < A ≤ 1), a > 0 and c > 0, we say that a function f ∈ Ap is said to be in the class Vλp (a, c, A, B), if it satisfies the following subordination relation (1 − λ)

1 + Az Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) + ≺ zp p Lp (a, c)f (z) 1 + Bz

(0 ≤ λ ≤ 1; z ∈ U).

(1.9)

We note that the class Vλp (a, c, A, B) includes many known subclasses of Ap as mentioned below. 

(i) V1p a, c, 1 −

2α p , −1



= Sp (a, c; α) (0 ≤ α < p)

z (Lp (a, c)f )0 (z) f ∈ Ap : Re Lp (a, c)f (z)

(

=

!

)

> α, z ∈ U .

Note that Sp (a, a; α) = S∗p (α), the class of p-valent starlike functions of order α and Sp (p + 1, p; α) = Cp (α), the class of p-valent convex functions of order α. 

(ii) V0p a, c, 1 −



2α p , −1

= Rp (a, c; α) (0 ≤ α < p)

Lp (a, c)f (z) α > ,z ∈ U zp p which, in turn yields the class Rp (α) for a = p + 1 and c = p. 

= f ∈ Ap : Re







For 0 ≤ α < 1, the functions in the class R1 (α) = R(α) are called functions of bounded turning. By the Nashiro-Warschowski Theorem [3], the functions in R(α) are univalent and also close-to-convex in U. It is well-known that R(α) S∗1 (0) = S∗ and S∗ R(α). For more information on the class R(0) = R (cf., e.g., [12]). Fekete and Szegö [4] proved a remarkable result that the estimate |a3 −

γa22 |

2γ ≤ 1 + 2 exp − 1−γ 



is sharp and holds for each γ ∈ [0, 1] over the class S consisting of functions f ∈ A of the form f (z) = z +

∞ X

an z n

(z ∈ U).

(1.10)

n=2

The coefficient functional Φγ (f ) = a3 − γa22 =

1 3γ 00
2 f (0) f 000 (0) − 6 2 

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on the functions in A represents various geometrical properties, for example, when γ = 1, Φγ (f ) = a3 − a22 becomes Sf (0)/6, where Sf denote the Schwarzian derivative (

Sf (z) =

f 00 (z) f 0 (z)

0

1 − 2



f 00 (z) f 0 (z)

2 )

f 000 (z) 3 = 0 − f (z) 2



f 00 (z) f 0 (z)

2

of locally univalent functions in U. For a family F of functions in A of the form (1.10), the more general problem of maximizing the absolute value for the functional Φγ (f ) for some γ (real as well as complex) is popularly known as Fekete-Szegö problem for the class F. In literature, there exists a large number of results about the inequalities for |Φγ (f )| corresponding to various subclasses of S (see, e.g., [4, 7, 8, 9]). The object of the present study is to solve Fekete-Szegö problem for a new class Vλp (a, c, A, B) of p-valent analytic functions in U involving the Saitoh operator. We also obtain some subordination results along with some interesting corollaries for functions in Ap involving this operator. Relevant connections of the results obtained here with those given by earlier workers on the subject are pointed out. 2. Preliminaries Let P denote the family of all functions of the form ϕ(z) = 1 + q1 z + q2 z 2 + · · ·

(2.1)

analytic in U and satisfying the condition Re{ϕ(z)} > 0 in U. To establish our main results, we need the following lemmas. Lemma 2.1. If the function ϕ, given by (2.1) belongs to the class P, then for any complex number γ, |qk | ≤ 2

(2.2)

q2 − γq12 ≤ 2 max{1, |2γ − 1|}.

(2.3)

and The result in (2.2) is sharp for the function ϕ1 (z) = (1 + z)/(1 − z) (z ∈ U), where as, the result in (2.3) is sharp for the functions ϕ2 (z) = (1 + z 2 )/(1 − z 2 ) (z ∈ U) and ϕ1 (z). We note that the estimate (2.2) is contained in [3], the estimate (2.3) is due to Ma and Minda [11]. The following lemma is due to Miller and Mocanu [14]. Lemma 2.2. Let q be univalent in U and let θ and φ be analytic in a domain Ω containing q(U) with φ(w) 6= 0, when w ∈ q(U). Set Q(z) = zq 0 (z)φ(q(z)), h(z) = θ(q(z)) + Q(z) and suppose that either (i) h is convex, or (ii) Q is starlike. In addition, assume that

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 zh0 (z)

> 0 (z ∈ U). Q(z) If ψ is analytic in U with ψ(0) = q(0), ψ(U) ⊂ Ω and

(iii) Re

θ(ψ(z)) + zψ 0 (z)φ(ψ(z)) ≺ θ(q(z)) + zq 0 (z)φ(q(z))

(z ∈ U),

then ψ(z) ≺ q(z) (z ∈ U) and the function q is the best dominant. 3. Main results Unless otherwise mentioned, we assume throughout the sequel that a > 0, c > 0, 0 ≤ λ ≤ 1 and −1 ≤ B < A ≤ 1. We first solve the Fekete-Szegö problem for the class Vλp (a, c, A, B). Theorem 3.1. If γ ∈ R and the function f , given by (1.1) belongs then  −p(A − B)c Q    ,   (p + λ(1 − p))2 (p + λ(2 − p))a(a)2   p(A − B)(c)  2 , ap+2 − γa2p+1 ≤  {p + (2 − p)λ}(a) 2     p(A − B)c Q   ,  (p + λ(1 − p))2 (p + λ(2 − p))a(a)2

to the class Vλp (a, c, A, B), γ < ρ1 ρ1 ≤ γ ≤ ρ2

(3.1)

γ > ρ2 ,

where h

i

Q = γp(p + λ(2 − p))(A − B)(a + 1)c + {B(p + λ(1 − p))2 − λp(A − B)}a(c + 1) , λp(A − B) − (1 + B){p + λ(1 − p)}2 a(c + 1) , ρ1 = p{p + λ(2 − p)}(A − B)(a + 1)c 



and λp(A − B) + (1 − B){p + λ(1 − p)}2 a(c + 1) ρ2 = . p{p + λ(2 − p)}(A − B)(a + 1)c 



All these results are sharp. Proof. From (1.9), it follows that Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) 1 − A + (1 + A)ϕ(z) (1 − λ) + = zp p Lp (a, c)f (z) 1 − B + (1 + B)ϕ(z)

(z ∈ U),

where the function ϕ defined by (2.1) belongs to the class P. Substituting the power series expansion of Lp (a, c)f and ϕ in the above expression, we deduce that ap+1 =

p(A − B)c q1 , 2(p + λ(1 − p))a

(3.2)

and ap+2

p(A − B)(c)2 = 2(p + λ(2 − p))(a)2 = q2 +



q2 −



1+B 2 λp(A − B) q1 + q2 2 2(p + λ(1 − p))2 1 

λp(A − B) − (1 + B)(p + λ(1 − p))2 2 q1 . 2(p + λ(1 − p))2

720



(3.3)

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With the aid of (3.2) and (3.3), we get p(A − B)(c)2 ap+2 − γ a2p+1 = 2 (p + λ(2 − p)) (a)2 γ p (p + λ(2 − p)) (A − B)(a + 1)c + {(1 + B) (p + λ(1 − p))2 − λ p(A − B)}a(c + 1) 2 × q2 − q1 , 2 (p + λ(1 − p))2 a(c + 1)

which in view of Lemma 2.1 yields p(A − B)(c)2 ap+2 − γ a2p+1 = (p + λ(2 − p)) (a)2    γ p (p + λ(2 − p)) (A − B)(a + 1)c + {B (p + λ(1 − p))2 − λ p(A − B)}a(c + 1) 

× max 1,

(p + λ(1 − p))2 a(c + 1)



.



(3.4) Now, we consider the following cases. (i) If 2 γ p (p + λ(2 − p)) (A − B)(a + 1)c + {B (p + λ(1 − p)) − λ p(A − B)}a(c + 1)

(p + λ(1 − p))2 a(c + 1)

≤ 1,

then it is easily seen that ρ1 ≤ γ ≤ ρ2 and (3.4) gives the second estimate in (3.1). (ii) For 2 γ p (p + λ(2 − p)) (A − B)(a + 1)c + {B (p + λ(1 − p)) − λ p(A − B)}a(c + 1)

(p + λ(1 − p))2 a(c + 1)

> 1,

we have either γ p (p + λ(2 − p)) (A − B)(a + 1)c + {B (p + λ(1 − p))2 − λ p(A − B)}a(c + 1) < −1 (p + λ(1 − p))2 a(c + 1) or γ p (p + λ(2 − p)) (A − B)(a + 1)c + {B (p + λ(1 − p))2 − λ p(A − B)}a(c + 1) > 1. (p + λ(1 − p))2 a(c + 1) The above inequalities implies that either γ < ρ1 or γ > ρ2 . Thus, again by use of (3.4), we get the first and the third estimate in (3.1). We note that the results are sharp for the function f defined in U by (1 − λ)

 1 + Az   , if γ < ρ1 or γ > ρ2 

Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) + = 1 + Bz2  1 + Az zp p Lp (a, c)f (z)   , if ρ1 ≤ γ ≤ ρ2 , 1 + Bz 2

where 0 ≤ λ ≤ 1, a > 0, c > 0 and −1 ≤ B < A ≤ 1. This completes the proof of Theorem 3.1.  Taking λ = 1, A = 1 − (2α/p) (0 ≤ α < p) and B = −1 in Theorem 3.1, we obtain

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Corollary 3.1. If γ ∈ R and the function f , given by (1.1) belongs to the class Sp (a, c; α), then ap+2 − γa2p+1  (p − α){(2(p − α) + 1)a(c)2 − 4γ(p − α)(a + 1)c2 }    ,   a(a)2    (p − α)(c)

≤

2

,

 (a)2     (p − α){4γ(p − α)(a + 1)c2 − (2(p − α) + 1)a(c)2 }   , 

a(a)2

a(c + 1) 2(a + 1)c a(c + 1) (p + 1 − α)a(c + 1) ≤γ≤ 2(a + 1)c 2(p − α)(a + 1)c (p + 1 − α)a(c + 1) γ> . 2(p − α)(a + 1)c γ
, 2(a + 1)c 2(p − α)(a + 1)c a(c + 1) (p + 1 − α)a(c + 1) ≤γ≤ . 2(a + 1)c 2(p − α)(a + 1)c

γ


α p

(0 ≤ α < p; z ∈ U),

then

ap+2 − γa2p+1 ≤

      α α    −2 1 − 2γ 1 − −1 ,   p p     

α ,  p         α α   2γ 1 − −1 , 2 1 − p p 2 1−

γ 1−

α p

.

These results are sharp for the function f ∈ A defined in U by      2α p     z 1+ 1− z   α −1 p   , γ < 0 or γ > 1 −  1−z p     f (z) = 2α  2 p z 1 + 1 −   z    α −1 p   , 0≤γ ≤ 1− . 1 − z2 p For the choice a = p + 1 and c = p in Corollary 3.2, we obtain Corollary 3.4. If γ ∈ R and the function f ∈ A, given by (1.1) satisfies the condition 

Re

f 0 (z) z p−1



>α

(0 ≤ α < p; z ∈ U),

then

ap+2 − γa2p+1 ≤

 2(p − α){2γ(p + 2)(p − α) − (p + 1)2 }    , −   (p + 1)2 (p + 2)    2(p − α)

,

 p+2     2(p − α){2γ(p + 2)(p − α) − (p + 1)2 }   ,  2

(p + 1) (p + 2)

γ . (p + 2)(p − α) 0≤γ≤

These results are sharp for the function f ∈ A defined in U by      p 1 + 1 − 2α z     z   α −1 p   , γ < 0 or γ > 1 − ϕp (p, p + 1; z) ∗ 1−z p     f (z) = 2α     zp 1 + 1 − z2    α −1 p  ϕp (p, p + 1; z) ∗ , 0 ≤ γ ≤ 1 − . 1 − z2 p

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Next, we prove the following subordination result. Theorem 3.2. If a function f ∈ Ap satisfies the subordination relation Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) + zp p Lp (a, c)f (z) (A − B)z λ(A − B)z ≺ 1 + (1 − λ) + 1 + Bz p(1 + Az)(1 + Bz)

(1 − λ)

(0 < λ ≤ 1, −1 ≤ B < A ≤ 1; z ∈ U),

(3.5)

then

1 + Az Lp (a, c)f (z) ≺ = qe(z) (say) p z 1 + Bz and the function qe is the best dominant of (3.6).

(z ∈ U)

(3.6)

Proof. Setting q(z) =

1 + Az (z ∈ U), θ(w) = λ + (1 − λ)w (w ∈ C) 1 + Bz

we see that Q(z) =

φ(w) =

and

λ (0 6= w ∈ C), pw

λ zq 0 (z) λ (A − B)z = p q(z) p(1 + Az)(1 + Bz)

and

zQ0 (z) 1 Bz Re − = Re > 0, Q(z) 1 + Az 1 + Bz so that Q is starlike in U. Further, letting h(z) = θ(q(z)) + Q(z), we get 



Re

zh0 (z) Q(z)









(1 − λ)p zQ0 (z) Re{q(z)} + Re λ Q(z) 

=



>0

(z ∈ U).

Suppose that ψ(z) =

Lp (a, c)f (z) zp

(z ∈ U).

Then the hypothesis (3.5) implies that θ(ψ(z)) + zψ 0 (z)φ(ψ(z)) ≺ θ(q(z)) + zq 0 (z)φ(q(z))

(z ∈ U),

which in view of Lemma 2.2 gives the required assertion (3.6) and the function qe is the best dominant. The proof of Theorem 3.2 is thus completed.  Taking λ = 1, A = −α/p and B = −1 in Theorem 3.2, we get Corollary 3.5. If a function f ∈ Ap satisfies the subordination relation z (Lp (a, c)f )0 (z) (p − α)z ≺p+ Lp (a, c)f (z) (p − αz)(1 − z) then 

Re

Lp (a, c)f (z) zp



>

p+α 2p

(0 ≤ α < p; z ∈ U),

(z ∈ U)

and the result is the best possible. Putting A = 1 − (2α/p) (0 ≤ α < p) and B = −1 in Theorem 3.2, we obtain

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Corollary 3.6. If a function f ∈ Ap satisfies the subordination relation Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) + zp p Lp (a, c)f (z) 2(1 − λ)(p − α) z 2λ(p − α)z ≺1+ + p 1 − z p{p + (p − 2α)z}(1 − z)

(1 − λ)

then 

Re

Lp (a, c)f (z) zp



>

α p

(0 < λ ≤ 1; z ∈ U),

(z ∈ U)

and the result is the best possible. For the choice c = a (a = p + 1 and c = p, respectively), Corollary 3.5 yields the following result. Corollary 3.7. For 0 < λ ≤ 1 and 0 ≤ α < p, let Φp (λ, α; z) = 1 +

2λ(p − α)z 2(1 − λ)(p − α) z + p 1 − z p{p + (p − 2α)z}(1 − z)

(z ∈ U).

(i) If a function f ∈ Ap satisfies (1 − λ) then

f (z) λ zf 0 (z) + ≺ Φp (λ, α; z) zp p f (z) f (z) Re zp 



>

α p

(z ∈ U),

(z ∈ U).

(ii) If a function f ∈ Ap satisfies (1 − λ)

f 0 (z) +λ z p−1



1+

zf 00 (z) f 0 (z)



≺ p Φp (λ, α; z)

(z ∈ U),

then

f 0 (z) Re >α z p−1 The results in (i) and (ii) are the best possible. 



(z ∈ U).

Remark 3.2. 1. Letting a = p + 1, c = p in Corollary 3.5 and noting that 

p + Re

(p − α)z (p − αz)(1 − z)



>

(2p − 1)(p + α) + 2α 2(p + α)

(0 ≤ α < p; z ∈ U),

we get the corresponding result obtained by Deniz [2, Theorem 2.1]. 2. Setting p = 1 and α = 0(p = 1 and α = 1/2, respectively) in Corollary 3.6, we get the following the following results due to Singh et al. [19, Theorem 1 and Theorem 2]. (i) If f ∈ A satisfies 

0

zf 00 (z) 1+ 0 f (z)



Re (1 − λ)f (z) + λ



>λ

(0 < λ < 1; z ∈ U),

then Re{f 0 (z)} > 0

(z ∈ U)

and the result is sharp for the function f (z) = −z − 2 log(1 − z), z ∈ U.

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(ii) If f ∈ A satisfies 

zf 00 (z) 1+ 0 f (z)



0

Re (1 − λ)f (z) + λ



>

1 2

(0 < λ ≤ 1; z ∈ U),

then 1 (z ∈ U) 2 and the result is sharp for the function f (z) = − log(1 − z), z ∈ U. Re{f 0 (z)} >

Theorem 3.3. If 0 < λ < 1 and a function f ∈ Ap satisfies Lp (a, c)f (z) Re zp 



>

α p

(0 ≤ α < p; z ∈ U),

(3.7)

then (1 − λ)

α Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) + > (1 − λ) + λ p z p Lp (a, c)f (z) p

(|z| < Rp (λ, α)) ,

(3.8)

where

Rp (λ, α) =

 p  {λ + (1 − λ)(p − α)} − λ2 + 2λ(1 − λ)(p − α)   , 

(1 − λ)(p − 2α)

   

(1 − λ)p , 2λ + (1 − λ)p

p 2 p α= . 2

α 6=

The result is the best possible. Proof. From (3.7), it follows that Lp (a, c)f (z) α α = + 1− ϕ(z) p z p p 



(z ∈ U),

where ϕ ∈ P. Differentiating the above expression logarithmically followed by a simple calculations, we deduce that (1 − λ)

Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) α + − (1 − λ) − λ p z p Lp (a, c)f (z) p α = (1 − λ) 1 − p 

λ zϕ0 (z) ϕ(z) + (1 − λ){α + (p − α)ϕ(z)}





so that Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) Re (1 − λ) + zp p Lp (a, c)f (z) (

α ≥ (1 − λ) 1 − p 



)

− (1 − λ)

α −λ p

λ |zϕ0 (z)| Re{ϕ(z)} − . (1 − λ){|α + (p − α)ϕ(z)|} 

(3.9)

Using the estimates [13] |zϕ0 (z)| 2r ≤ Re{ϕ(z)} 1 − r2

and

Re{ϕ(z)} ≥

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in (3.9), we get Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) + Re (1 − λ) zp p Lp (a, c)f (z) (

)

− (1 − λ)

α −λ p

α 2λ r ≥ (1 − λ) 1 − Re{ϕ(z)} 1 − . 2 p (1 − λ) {α(1 − r ) + (p − α)(1 − r)2 } 







(3.10)

We note that the right hand side of (3.10) is positive, provided r < Rp (λ, α), where Rp (λ, α) is defined as in the theorem. To show that the bound Rp (λ, α) is the best possible, we consider the function f ∈ Ap defined by zp

2α 1+ 1− z p 1−z



f (z) = ϕp (c, a; z) ?



 

(0 ≤ α < p; z ∈ U).

It follows that 2α 1+ 1− z Lp (a, c)f (z) p = (0 ≤ α < p; z ∈ U), zp 1−z which on differentiating logarithmically followed by a routine calculation yields 

(1 − λ)



 

Lp (a, c)f (z) λ z (Lp (a, c)f )0 (z) α + − (1 − λ) − λ p z p Lp (a, c)f (z) p     α 1+z 2λ z = (1 − λ) 1 − 1+ p 1−z α(1 − z 2 ) + (p − α)(1 − z)2 z → −Rp (λ, α).

= 0 as



This completes the proof of Theorem 3.3.

For the choice c = a, p = 1 and α = 0 (a = 2, c = p = 1 and α = 0, respectively), Theorem 3.3 yields the following result. Corollary 3.8. Let 0 < λ < 1. If a function f ∈ A satisfies 

Re then 

Re (1 − λ)

f (z) z



(z ∈ U),

>0

f (z) zf 0 (z) +λ z f (z)



>λ





e |z| < R(λ) ,

and if it satisfies Re{f 0 (z)} > 0 then 

0

zf 00 (z) 1+ 0 f (z)



Re (1 − λ)f (z) + λ

(z ∈ U), 

>λ





e |z| < R(λ) ,

where 1− e R(λ) =

p

λ(2 − λ) . 1−λ

The results are the best possible.

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4. Acknowledgements This work was supported by a Research Grant of Pukyong National University(2014 year) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037). References [1] B.C. Carlson and D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737-745. [2] E. Deniz, On p-valently close-to-convex, starlike and convex functions, Hacet. J. Math. Stat., 41(5) (2012), 635-642. [3] P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, USA (1983). [4] M. Fekete and G. Szegö, Eine Bemerkung über ungerede schlichte funktionen, J. London Math. Soc., 8 (1933), 85-89. [5] R.M. Goel and N.S. Sohi: A new criterion for p-valent functions, Proc. Amer. Math. Soc., 78 (1980), 353-357. [6] T. Hayami and S.Owa, Hankel determinant for p-valently starlike and convex functions of order α, General Math., 17(4) (2009), 29-44. [7] F. R.Keogh and E. P.Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8-12. [8] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions-II, Arch. Math.(Basel), 49 (1987), 420-433. [9] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc., 101 (1987), 89-95. [10] J.-L. Liu and K.I. Noor, Some properties of Noor integral operator, J. Natur. Geom., 21 (2002), 81-90. [11] W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Z. Li, F. Ren, L. Yang and S. Zhang (Eds.), Int. Press, Cambridge, MA, (1994) 157-169. [12] T.H. MacGregor, Functions whose derivative have a positive real part. Trans. Amer. Math. Soc. 104(3) (1962), 532-537. [13] T.H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., 14(3) (1963), 514-520. [14] S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and Applications in: Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York, 2000. [15] S. Owa and H.M. Srivastava, Univalent and starlike generalized hyper- geometric functions. Canad. J. Math. 39 (1987), 1057-1077. [16] J. Patel and A.K. Mishra, On certain multivalent functions associated with an extended fractional differintegral operator, J. Math. Anal. Appl., 332 (2007), 109-122. [17] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115. [18] H. Saitoh, A linear operator and its application of first order differential subordinations, Math. Japonica, 44 (1996), 31-38. [19] V. Singh, S. Singh and S. Gupta, A problem in the theory of univalent functions, Integral Transforms Spec. Funct., 16(2) (2005), 179-186.

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GENERALIZED φ-WEAK CONTRACTIVE FUZZY MAPPINGS AND RELATED FIXED POINT RESULTS ON COMPLETE METRIC SPACE AFSHAN BATOOL, TAYYAB KAMRAN, SUN YOUNG JANG∗ AND CHOONKIL PARK∗ Abstract. In this paper, we discuss the existence and uniqueness of a (common) fixed point of generalized φ-weak contractive fuzzy mappings on complete metric spaces. We present some examples to illustrate the obtained results.

1. Introduction and preliminaries 1.1. Fuzzy fixed points of fuzzy mappings. In fixed point theory, the importance of various contractive inequalities cannot be overemphasized. Existence theorems of fixed points have been established for mappings defined on various types of spaces and satisfying different types of contractive inequalities. The notion of fuzzy sets was introduced by Zadeh [27] in 1965. Following this initial result, Weiss [24] and Butnariu [9] studied on the characterization of several notion in the sense of fuzzy numbers. Heilpern [14] introduced the fuzzy mapping and further he established fuzzy Banach contraction principle on a complete metric space. Subsequently several other researchers studied the existence of fixed points and common fixed points of fuzzy mappings satisfying a contractive type condition on a metric space (see [1, 3, 4, 7, 8, 10, 16, 19, 20, 22, 25]). The following are some definitions and concepts required for our discussion in the paper. In fact most of these are discussed in [13, 14, 17] in metric linear spaces. We discuss them in metric spaces. Suppose that (X, d) is a metric space. A fuzzy set A over X is defined by a function µA , µA : X → [0, 1], where µA is called a membership function of A, and the value µA (x) is called the grade of membership of x in X. The value represents the degree of x belonging to the fuzzy set X. The α-level set of A is denoted by [A]α , and is defined as follows: [A]α = {x : A(x) ≥ α} if α ∈ (0, 1], [A]0 = {x : A(x) > 0}, where B denotes the closure of the set B. Let F(X) be the collection of all fuzzy sets in a metric space X. For A, B ∈ F(X), A ⊂ B means A(x) ≤ B(x) for each x ∈ X. A fuzzy set A in a metric linear space V is said to be an approximate quantity if and only if [A]α is compact and convex in V for each α ∈ [0, 1] and supx∈V A(x) = 1. We 2010 Mathematics Subject Classification: Primary 47H10, 54E50, 54E40, 46S50. Key words and phrases: contractive fuzzy mapping; complete metric space; fixed point. ∗ Corresponding author.

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A. BATOOL, T. KAMRAN, S. Y. JANG, C. PARK

denote the collection of all approximate quantities in a metric linear space V by W (V ). Clearly when X is a metric linear space W (X) ⊂ F(X). Let X be an arbitrary set and (Y, d) be a metric space . A mapping G is called fuzzy mapping if G is a mapping from X into F(Y ). A fuzzy mapping G is a fuzzy subset on X × Y with membership function G(x)(y). The function G(x)(y) is the grade of membership of y in G(x).For convenience, we denote α-level set of G(x) by [Gx]α instead of [G(x)]α . Definition 1. Let G, H be fuzzy mappings from X into F(X). A point z in X is called an α-fuzzy fixed point of H if z ∈ [Hz]α . The point z is called a common α-fuzzy fixed point of G and H if z ∈ [Gz]α ∩ [Hz]α . 1.2. Fixed point theory on metric spaces. Let (X, d) be a metric space, B(X) and CB(X) be the sets of all nonempty bounded and closed subsets of X, respectively. For P, Q ∈ B(X) we define δ(P, Q) = sup{d(p, q) : p ∈ P, q ∈ Q} and D(P, Q) = inf{d(p, q) : p ∈ P, q ∈ Q}. If P = {p}, we write δ(P, Q) = δ(p, Q), and if Q = {q}, then δ(p, Q) = d(p, q). For P, Q, R in B(X) one can easily prove the following properties. δ(P, Q) = δ(P, Q) ≥ 0, δ(P, Q) ≤ δ(P, R) + δ(R, Q), δ(P, P ) = sup{d(p, r) : p, r ∈ P } = diam P δ(P, Q) = 0 implies that P = Q = {p}. Let {An } be a sequence in B(X). Then the sequence {An } converges to A if and only if (i) a ∈ A implies that an → a for some sequence {an } with an ∈ An for n ∈ N , and (ii) for any ε > 0, there exist n, m ∈ N with n > m such that An ⊆ Aε = {x ∈ X : d(x, a) < ε f or some a ∈ A}. See [10, 11]. The following results will be useful in the proof of our main result. Lemma 1. [11] Let {An } and {Bn } be sequences in B(X) and (X, d) be a complete metric space. If An → A ∈ B(X) and Bn → B ∈ B(X), then δ(An , Bn ) → δ(A, B). Lemma 2. [15] Let (X, d) be a complete metric space. If {An } is a sequence of nonempty bounded subsets in (X, d) and if δ(An , y) → 0 for some y ∈ X, then An → {y}. Theorem 1. [21] Let(X, d) be a complete metric space and T be a φ-weak contraction on X; that is, for each x, y ∈ X, there exists a function φ : [0, ∞) → [0, ∞) such that φ is positive on (0, ∞) and φ(0) = 0, and d(T x, T y) ≤ d(x, y) − φ(d(x, y)) (1) Also if φ is a continuous and nondecreasing function, then T has a unique fixed point.

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A weakly contractive mapping is a map satisfying the inequality (1) which was first defined by Alber and Guerre-Delabriere [2]. For more results on these mappings, see [5, 6, 12, 18, 23] and the related references therein. Zhang and Song [26] gave the following theorem. Theorem 2. [26] Let (X, d) be a complete metric space and T, S : X → X be two mappings such that for each x, y ∈ X, d(T x, Sy) ≤ m(x, y) − φ(m(x, y)), where φ : [0, ∞) → [0, ∞) is a lower semi-continuous function with φ(t) > 0 for t > 0 and φ(0) = 0, and } { 1 m(x, y) = max d(x, y), d(x, T x), d(y, Sy), [d(y, T x) + d(x, Sy)] 2 Then there exists a unique point u ∈ X such that u = T u = Su. 2. Main results This section includes the main theorem of the paper. More precisely, we find out a common fixed point of fuzzy mappings which is also unique. Let (X, d) be a complete metric space. Then we define and use the following notations: ξX

= {A : A is the subset of X},

B(ξ X ) = {A ∈ ξ X : A is nonempty bounded}, CB(ξ X ) = {A ∈ ξ X : A is nonempty closed and bounded}. Theorem 3. Let (X, d) be a complete metric space and S, T : X → F(X) and for x ∈ X, there exist αS (x), αT (x) ∈ (0, 1] such that [Sx]αS (x) , [T x]αT (x) ∈ B(ξ x ), such that for all x, y ∈ X. ( ) δ [Sx]αS (x) , [T y]αT (y) ≤ M (x, y) − φ (M (x, y)) (2) where, φ : [0, ∞) → [0, ∞) is a lower semicontinous function with φ(t) > 0 for t ∈ (0, ∞) and φ (0) = 0 and { ]} 1[ M (x, y) = max d(x, y), D(x, [Sx]αS (x) ), D(y, [T y]αT (y) ), D(y, [Sx]αS (x) ) + D(x, [T y]αT (y) ) (3) 2 Then there exists a unique z ∈ [Sx]αS (x) and z ∈ [T x]αT (x) . Proof. Take a0 ∈ X. According to the given condition, there exists α(a0 ) ∈ (0, 1] such that [Sa0 ]α(a0 ) ∈ CB(ξ X ). Let us denote α(x0 ) by α1 . We set a1 ∈ [Sa0 ]α(a0 ) , for this a1 there exists α2 ∈ (0, 1] such that, [T a1 ]α2 ∈ CB(ξ x ). Iteratively, we shall construct a sequence {an } in X in a way that a2k+1 ∈ [Sa2k ]α2k+1 , a2k+2 ∈ [T a2k+1 ]α2k+2 It is clear that if M (an , an+1 ) = 0, then the proof is completed. Consequently, throughout the proof, we suppose that M (an , an+1 ) > 0 for all n ≥ 0. (4) We shall prove that d(a2n+1 , a2n+2 ) ≤ d(a2n , a2n+1 ) f or all n ≥ 0.

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A. BATOOL, T. KAMRAN, S. Y. JANG, C. PARK

Suppose, on the contrary, that there exists n ˇ ≥ 0 such that d(a2ˇn+1 , a2ˇn+2 ) > d(a2ˇn , a2ˇn+1 ), which yields that M (a2ˇn , a2ˇn+1 ) ≤ d(a2ˇn+1 , a2ˇn+2 ). Regarding (2), we derive that d(a2ˇn0 +1 , a2ˇn+2 ) ≤ δ([Sa2ˇn ]α(a2ˇn ) , [T a2ˇn+1 ]α(a2ˇn+1 ) ) ≤ M (a2ˇn , a2ˇn+1 ) − φ(M (a2ˇn , a2ˇn+1 )) ≤ d(a2ˇn , a2ˇn+1 ) − φ(M (a2ˇn , a2ˇn+1 )). Consequently, we obtain that φ(M (a2ˇn , a2ˇn+1 )) = 0 and so we have M (a2ˇn , a2ˇn+1 ) = 0. This contradicts the observation (4). Hence we have the inequality (5). In an analogous way, one can conclude that d(a2n+2 , a2n+3 ) ≤ d(a2n+1 , a2n+2 ) f or all n ≥ 0.

(6)

By combining (5) and (6), we get that d(an+1 , an+2 ) ≤ d(an , an+1 ) f or all n ≥ 0. Hence we derive that the sequence {d(an , an+1 )} is non-increasing and bounded below. Since (X, d) is complete, there exists l ≥ 0 such that lim d(an , an+1 ) = l.

(7)

n→∞

Due to hypothesis, we observe that d(a2n , a2n+1 ) ≤ M (a2n , a2n+1 ) { } d(a2n , [a2n+1 ), D(a2n , [Sa2n ]α(a2n ) ), D(a2n+1 , [T a2n+1 ]α(a2n+1 ), ) ] = max 1 2 D(a2n+1 , [Sa2n ]α(a2n ) ) + D(a2n , [T a2n+1 ]α(a2n+1 ) ) } { 1 ≤ max d(a2n , a2n+1 ), d(a2n+1 , a2n+2 ), [d(a2n , a2n+1 ) + d(a2n+1 , a2n+2 )] . 2 Thus we have l ≤ lim M (a2n , a2n+1 ) ≤ l. n→∞

Hence we get lim M (a2n , a2n+1 ) = l.

(8)

lim M (a2n+1 , a2n+2 ) = l.

(9)

n→∞

Analogously, we have n→∞

By combining (7), (8) and (9), we derive that lim d(an , an+1 ) = lim M (an , an+1 ) = l.

n→∞

n→∞

By the lower semi-continuity of φ, we find φ(l) ≤ lim inf φM (an , an+1 )). n→∞

Now we claim that l = 0. From (2), we have d(a2n+1 , a2n+2 ) ≤ δ([Sa2n ]α(a2n ) , [T a2n+1 ]α(a2n+1 ) ) ≤ M (a2n , a2n+1 ) − φ(M (a2n , a2n+1 )

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GENERALIZED φ-WEAK CONTRACTIVE FUZZY MAPPINGS

By letting the upper limit as n → ∞ in the inequality above, we obtain l ≤ l − lim inf φ(M (a2n , a2n+1 )) n→∞

≤ l − φ(l), that is, φ(l) = 0. Regarding the property of φ, we conclude that l = 0. As a next step, we shall show that {an } is Cauchy. For this purpose, it is sufficient to get that {a2n } is Cauchy. Suppose, on the contrary, that {a2n } is not Cauchy. Then there is an ϵ > 0 such that for an even integer 2k there exist even integers 2m(k) > 2n(k) > 2k such that d(a2n(k) , a2m(k) ) > ϵ.

(10)

For every even integer 2k, let 2m(k) be the least positive integer exceeding 2n(k), satisfying (10), and such that d(a2n(k) , a2m(k)−2) < ϵ. (11) Now ϵ ≤ d(a2n(k) , a2m(k) ) ≤ d(a2n(k) , a2m(k)−2 ) + d(a2m(k)−2 , a2m(k)−1 ) +d(a2m(k)−1 , a2m(k) ). By (10) and (11), we get lim d(a2n(k) , a2m(k) ) = ϵ.

(12)

k→∞

Due to the triangle inequality, we have d(a2n(k) , a2m(k)−1 ) − d(a2n(k) , a2m(k) ) < d(a2m(k)−1 , a2m(k) ). By (12), we get d(a2n(k) , a2m(k)−1 ) = ϵ.

(13)

Now by (3) we observe that d(a2n(k) , a2m(k)−1 ) ≤ M (a2n(k) , a2m(k)−1 )  d(a2n(k) , a2m(k)−1 ), D(a2n(k) , [Sa2n(k) ]α(a2n(k) ) ),   D(a2m(k)−1 , [T a2m(k)−1 ]α(a2m(k)−1 )) , = max   1 D(a 2m(k)−1 , [Sa2n(k) ]α(a2n(k) ) ) + D(a2n(k) , [T a2m(k)−1 ]α(a2m(k)−1 ) )} 2 { } d(a2n(k) , a2m(k)−1 ), d(a2n(k) , a2n(k)+1 ), d(a2m(k)−1 , a2m(k) ) ≤ max 1 2 [d(a2m(k)−1 , a2n(k)+1) + d(a2n(k) , a2m(k) )] } { d(a2n(k) , a2m(k)−1 ), d(a2n(k) , a2n(k)+1 ), d(a2m(k)−1 , a2m(k) ) ] [ . ≤ max 1 2 [d(a2m(k)−1 , a2n(k) ) + d(a2n(k) , a2n(k)+1 ) + d(a2n(k) , a2m(k) )

    

By letting k → ∞ in the inequality above and taking (12) and (13) into account, we conclude that ε ≤ lim M (x2n(k) , x2m(k)−1 ) ≤ ε. k→∞

Consequently, we have lim M (x2n(k) , x2m(k)−1 ) = ε.

k→∞

By the lower semi-continuity of φ, we derive that φ(ε) ≤ lim inf φ(M (x2n(k) , x2m(k)−1 )). k→∞

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A. BATOOL, T. KAMRAN, S. Y. JANG, C. PARK

Now by (2), we get d(x2n(k) , x2m(k) ) ≤ d(x2n(k) , x2n(k)+1 ) + δ([Sx2n(k) ]α(x2n(k) ) , [T x2m(k)−1 ]α(x2m(k)−1 ) ) ≤ d(x2n(k) , x2n(k)+1 ) + M (x2n(k) , x2m(k)−1 ) − φ(M (x2n(k) , x2m(k)−1 )). Letting the upper limit k → ∞ in the inequality above, we have ϵ ≤ ϵ − lim inf φ(M (a2n(k) , a2m(k)−1 )) k→∞

≤ ϵ − φ(ϵ), which is a contradiction. Hence we conclude that {a2n } is a Cauchy sequence. It follows from the completeness of X that there exists c ∈ X such that an → c as n → ∞. Furthermore, a2n → c and a2n+1 → c. We shall prove that c ∈ [Sc]αS (c) . D(c, [Sc]αS (c) ) ≤ M (c, a2n−1 ) } { d(c, a2n−1 ), D(c, [Sc]αS (c) ), D(a2n−1 , [T a2n−1 ]αT (a2n−1 ) ), = max 1 2 [D(a2n−1 , , [Sc]αS (c) ) + D(c, [T a2n−1 ]αT (a2n−1 ) )] { } d(c, a2n−1 ), D(c, [Sc]αS (c) ), d(a2n−1 , a2n ) ≤ max 1 2 [D(a2n−1 , , [Sc]αS (c) ) + d(c, a2n )] Letting n → ∞, we have lim M (c, a2n−1 ) = D(c, [Sc]αS (c) ). Due to the lower semi-continuity of φ, we n→∞ have φ(D(c, [Sc]αS (c) )) ≤ lim φ(M (c, a2n−1 )).

(14)

n→∞

On the other hand, from (2) δ([Sc]αS (c) , a2n ) ≤ δ([Sc]αS (c) , [T a2n−1 ]αT (a2n−1 ) ) ≤ M (c, a2n−1 ) − φ(M (c, a2n−1 )) and letting n → ∞, we have δ(([Sc]αS (c) , c) ≤ D(c, ([Sc]αS (c) ) − lim φ(M (c, a2n−1 )).

(15)

n→∞

This shows that lim φ(M (c, a2n−1 )) = 0 and so from (14), we have φ(D(c, [Sc]αS (c) )) = 0, that is, n→∞

D(c, [Sc]αS (c) ) = 0. This implies, from (15), that {c} = [Sc]αS (c) . Now, from (3) it is easy to see that M (c, c) = D(c, [T c]αT (c) ), and so from (2) we have δ(c, [T c]αT (c) ) ≤ δ([Sc]αS (c) , [T c]αT (c) ) ≤ M (c, c) − φ(M (c, c)) = D(c, [T c]αT (c) ) − φ(D(c, [T c]αT (c) )). Therefore, we have c ∈ [T c]αT (c) and so {c} = [T c]αT (c) . As a consequence, we have {c} = [Sc]αS (c) = [T c]αT (c) , that is, c is a common fixed point of S and T . Now we will show that this common fixed point is unique. Assume that a and b are two common fixed points of S and T . Then a ∈ [Sa]αS (a) , a

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GENERALIZED φ-WEAK CONTRACTIVE FUZZY MAPPINGS

∈ [T a]αT (a) and b ∈ [Sb]αS (b) , b ∈ [T b]αT (b) . Therefore, from (3) we have M (a, b) ≤ d(a, b) and so from (2) we have d(a, b) ≤ δ([Sa]αS (a) , [T b]αT (b) ) ≤ M (a, b) − φ(M (a, b)) ≤ d(a, b) − φ(M (a, b)). □

This shows that M (a, b) = 0 and so a = b.

Example 1. Let X = [0, 1], d (a, b) = |a − b|, when a, b ∈ X and let G, H : X → F(X) be fuzzy mappings defined as:  1 if 0 ≤ t < a6         1 a a   2 if 6 ≤ t ≤ 4 G(a)(t) =

         

1 3

if

0 if

a 4 a 3

≤t
e; the only positive equilibrium point of Eq.(1) is given by d x= : (®¡ e)(1 ¡ a ¡ b ¡ c) Theorem 1 The equilibrium x of Eq. (1) is locally asymptotically stable if ®¡ e > 2d: Proof: Let f : (0; 1)5 ¡! (0; 1) be a continuous function de…ned by f (u1 ; u2 ; u3 ; u4 ; u5 ) = au1 + bu2 + cu3 +

du4 : eu4 ¡ ®u5

(6)

Therefore, it follows that @f(u1 ; u2 ; u3 ; @u1 @f(u1 ; u2 ; u3 ; @u2 @f(u1 ; u2 ; u3 ; @u3 @f(u1 ; u2 ; u3 ; @u4 @f(u1 ; u2 ; u3 ; @u5

u4 ; u5 ) u4 ; u5 ) u4 ; u5 ) u4 ; u5 )

= a; = b; = c;

®du5 : (eu4 ¡ ®u5 )2 u4 ; u5 ) ®du4 = ¡ : (eu4 ¡ ®u5 )2 =

So, we can write @f(x; x; x; x; x) = a = p1 ; @u1 @f(x; x; x; x; x) = b = p2 ; @u2 @f(x; x; x; x; x) = c = p3 ; @u3

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@f (x; x; x; x; x) d(1 ¡ a ¡ b ¡ c) = = p4 ; @u4 (e ¡ ®) @f (x; x; x; x; x) d(1 ¡ a ¡ b ¡ c) = ¡ = p5 : @u5 (e ¡ ®) Then the linearized equation of Eq.(1) about x is yn+1 ¡ p1 yn ¡ p2 yn¡k ¡ p3 yn¡l ¡ p4 yn¡s ¡ p5 yn¡t = 0: It follows by Theorem A that, Eq.(1) is asymptotically stable if and only if jp1 j + jp2 j + jp3 j + jp4 j + jp5 j < 1: Thus,

and so

¯ ¯ ¯ ¯ ¯ d(1 ¡ a ¡ b ¡ c) ¯ ¯ d(1 ¡ a ¡ b ¡ c) ¯ ¯ ¯ ¯ ¯ < 1; jaj + jbj + jcj + ¯ ¯+¯ ¯ (®¡ e) (®¡ e) ¯ ¯ ¯ d(1 ¡ a ¡ b ¡ c) ¯ ¯ < 1 ¡ b ¡ a ¡ c; 2 ¯¯ ¯ (®¡ e)

or

2d < ®¡ e:

The proof is complete. Example 1. The solution of the di¤erence equation (1) is local stability if k = 2; l = 1; s = 3; t = 2; a = 0:23; b = 0:12; c = 0:3; d = 0:1; e = 0:6 and ® = 0:9 and the initial conditions x¡3 = 11:1; x¡2 = 1:1; x¡1 = 1:4 and x0 = 1:9 (See Fig. 1). plot of x(n+1)=ax(n)+bx(n-k)+cx(n-l)-dx(n-s)/(ex(n-s)-alfax(n-t)) 80

70

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x(n)

40

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20

10

0

-10

0

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50 n

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Figure 1. Plot the behavior of the solution of equation (1).

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Example 2. The solution of the di¤erence equation (1) if k = 2; l = 1; s = 3; t = 2; a = 0:4; b = 0:2; c = 0:5; d = 0:1; e = 0:6 and ® = 0:9 and the initial conditions x¡3 = 11:1; x¡2 = 1:1; x¡1 = 1:4 and x0 = 1:9 (See Fig. 2). plot of x(n+1)=ax(n)+bx(n-k)+cx(n-l)-dx(n-s)/(ex(n-s)-alfax(n-t)) 1600

1400

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x(n)

1000

800

600

400

200

0

0

10

20

30

40

50 n

60

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Figure 2. Plot the behavior of the solution of equation (1).

3

Global Attractivity of the Equilibrium Point of Eq.(1)

In this section, the global asymptotic stability of Eq.(1) will be studied. Theorem 2 The equilibrium point x is a global attractor of Eq.(1) if a + b + c < 1: Proof: Suppose that ³and ´ are real numbers and assume that g : [³; ´]5 ¡! [³; ´] is a function de…ned by g(u1 ; u2 ; u3 ; u4 ; u5 ) = au1 + bu2 + cu3 ¡

du4 : eu4 ¡ ®u5

Then @g(u1 ; u2 ; u3 ; @u1 @g(u1 ; u2 ; u3 ; @u3 @g(u1 ; u2 ; u3 ; @u4 @g(u1 ; u2 ; u3 ; @u5

u4 ; u5 ) u4 ; u5 ) u4 ; u5 )

= a;

@g(u1 ; u2 ; u3 ; u4 ; u5 ) = b; @u2

= c;

®du5 ; (eu4 ¡ ®u5 )2 u4 ; u5 ) ®du4 = ¡ : (eu4 ¡ ®u5 )2 =

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Now, we can see that the function g(u1 ; u2 ; u3 ; u4 ; u5 ) increasing in u1 ; u2 ; u3 ; u4 and decreasing in u5 : Let (m; M) be a solution of the system M = g(M; M; M; M; m) and m = g(m; m; m; m; M). Then from Eq.(1), we see that dM dm M = aM + bM + cM ¡ and m = am + bm + cm ¡ ; eM ¡ ®m em ¡ ®M and then dM dm M(1 ¡ a ¡ b ¡ c) = ¡ and m(1 ¡ a ¡ b ¡ c) = ¡ ; eM ¡ ®m em ¡ ®M thus e(1 ¡ a ¡ b ¡ c)M 2 ¡ ®(1 ¡ a ¡ b ¡ c)Mm = ¡dM and e(1 ¡ a ¡ b ¡ c)m2 ¡ ®(1 ¡ a ¡ b ¡ c)Mm = ¡dm: Subtracting we obtain e(1 ¡ a ¡ b ¡ c)(M 2 ¡ m2 ) + d(M ¡ m) = 0;

then

(M ¡ m)fe(1 ¡ a ¡ b ¡ c)(M + m) + dg = 0

under the condition a + b + c < 1; we see that

M = m: It follows by Theorem B that x is a global attractor of Eq.(1). This completes the proof. Example 3. The solution of the di¤erence equation (1) is global stability if k = 2, l = 1; s = 3; t = 2; a = 0:2; b = 0:2; c = 0:5; d = 0:12; e = 0:6 and ® = 0:9 and the initial conditions x¡3 = 11:1; x¡2 = 1:1; x¡1 = 1:4 and x0 = 1:9 (See Fig. 3). plot of x(n+1)=ax(n)+bx(n-k)+cx(n-l)-dx(n-s)/(ex(n-s)-alfax(n-t)) 12

10

x(n)

8

6

4

2

0

0

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20

30

40

50 n

60

70

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90

100

Figure 3. Plot the behavior of the solution of equation (1).

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4

Existence of Periodic Solutions

In this section, we investigate the existence of periodic solutions of Eq.(1). The following theorem states the necessary and su¢cient conditions for the Eq.(1) to be periodic solutions of prime period two. Theorem 3 Equation (1) has a prime period two solutions if and only if one of the following conditions satis…es (i) (e ¡ 3®)(a + b + c) + e + ® > 0; l; k; t ¡ even and s ¡ odd: (ii) (e + ®)(a + b + c + 1) ¡ 4® > 0; l; k; s ¡ even and t ¡ odd: (iii) (®+ e)(a + c ¡ b + 1) ¡ 4®(a + c) > 0; l; t ¡ even and k; s ¡ odd: (iv) (®+ e)(b ¡ a ¡ c ¡ 1) ¡ 4®(b ¡ 1) > 0; l; s ¡ even and k; t ¡ odd: (v) (® + e)(b ¡ a ¡ c ¡ 1) ¡ 4®a > 0; l; k; s ¡ odd; and t ¡ even: (vi) (®+ e)(b + c ¡ a ¡ 1) ¡ 4®(b + c ¡ 1) > 0; l; k; t ¡ odd and s ¡ even: (vii) (®+ e)(b + a + c ¡ 1) ¡ 4®(c ¡ 1) > 0; l; t ¡ odd and k; s ¡ even: (viii) (® + e)(b + a ¡ c + 1) ¡ 4®(a + b) > 0; l; s ¡ odd and k; t ¡ even: Proof: We prove …rst case when l; k and t are even and s odd ( the other cases are similar and will be left to readers). First suppose that there exists a prime period two solution :::p; q; p; q; :::; of Equation (1).We will prove that Inequality (i) holds. We see from Equation (1) when l; k and t are even and s odd that p = aq + bq + cq ¡

dp ; ep ¡ ®q

q = ap + bp + cp ¡

dq : eq ¡ ®p

and

Therefore,

ep2 ¡ ®pq = e(a + b + c)pq ¡ ®(a + b + c)q2 ¡ dp;

(7)

eq2 ¡ ®pq = e(a + b + c)pq ¡ ®(a + b + c)p2 ¡ dq:

(8)

and Subtracting (8) from (7) gives e(p2 ¡ q2 ) ¡ ®(a + b + c)(p2 ¡ q2 ) + d(p ¡ q) = 0; then (p ¡ q) [e ¡ ®(a + b + c))(p + q) + d] = 0

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Since p 6= q, then

(p + q) =

Again; adding (7) and (8) yields

d : ®(a + b + c) ¡ e

(9)

e(q 2 + p2 ) ¡ 2®pq = 2e(a + b + c)pq ¡ ®(a + b + c)(q2 + p2 ) ¡ d(q + p); then 2(e(a + b + c) + ®)pq = (e + ®(a + b + c))(q2 + p2 ) + d(q + p):

(10)

By using (9); (10) and the relation p2 + q 2 = (p + q)2 ¡ 2pq for all p; q 2 R; we obtain (e + ®(a + b + c))((p + q)2 ¡ 2pq) + d(q + p) = 2(e(a + b + c) ¡ ®)pq; 2 [e(a + b + c) + ®+ e + ®(a + b + c)] pq = (e + ®(a + b + c))(p + q)2 + d(q + p); ³

d ®(a+b+c)¡e

´2

(e + ®(a + b + c) + ®(a + b + c) ¡ e) ; ³ ´2 d 2(e + ®)(a + b + c + 1)pq = 2®(a + b + c) ®(a+b+c)¡e : 2(e + ®)(a + b + c + 1)pq =

Then,

pq =

µ

®(a + b + c) (a + b + c + 1)(e + ®)

¶µ

d ®(a + b + c) ¡ e

¶2

:

(11)

Now it is obvious from Eq.(9) and Eq.(11) that p and q are the two distinct roots of the quadratic equation µ ¶µ ¶2 d ®(a + b + c) d 2 t ¡ t+ = 0; ®(a + b + c) ¡ e (a + b + c + 1)(e + ®) ®(a + b + c) ¡ e d2 ®(a + b + c) = 0; (12) (®(a + b + c) ¡ e) t2 ¡ dt + (a + b + c + 1)(e + ®) (®(a + b + c) ¡ e) and so d2 ¡

4d2 ®(a + b + c) (®(a + b + c) ¡ e) > 0; (a + b + c + 1)(e + ®) (®(a + b + c) ¡ e) (a + b + c + 1)(e + ®) ¡ 4®(a + b + c) > 0; e(a + b + c + 1) + ®¡ 3®(a + b + c) > 0;

or (e ¡ 3®)(a + b + c) + e + ® > 0:

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For ®(a + b + c) > e and e > 3® then the Inequality (i) holds. Second suppose that Inequality (i) is true. We will show that Equation (1) has a prime period two solution. Suppose that d(1 ¡ ³) d(1 + ³) p= and q = ; 2(®A ¡ e) 2(®A ¡ e) r 4®A where ³= 1 ¡ and A = a + b + c: (A + 1)(e + ®) We see from the inequality (i) that (e ¡ 3®)(a + b + c) + e + ® > 0; (a + b + c + 1)(e + ®) ¡ 4®(a + b + c) > 0; which equivalents to (A + 1)(e + ®) ¡ 4®A > 0:

Therefore p and q are distinct real numbers. Set

x¡l = q; x¡k = q; x¡s = p; x¡t = q; :::; x¡3 = p; x¡2 = q; x¡1 = p; x0 = q: We would like to show that x1 = x¡1 = p and x2 = x0 = q: It follows from Eq.(1) that dp ; ep ¡ ®q ³ ´ d(1+³) µ ¶ d 2(®A¡e) d(1 ¡ ³) ´ ³ ´: = (a + b + c) ¡ ³ d(1+³) d(1¡³) 2(®A ¡ e) e 2(®A¡e) ¡ ® 2(®A¡e)

x1 = aq + bq + cq ¡

Dividing the denominator and numerator by 2(®A ¡ e) we get µ ¶ d(1 ¡ ³) d(1 + ³) x1 = (a + c + b) ¡ : 2(®A ¡ e) (e ¡ ®) + (e + ®)³

Multiplying the denominator and numerator of the right side by (e ¡ ®) ¡ (e + ®)³ ¶ µ d(1 ¡ ³) d(1 + ³) ((e ¡ ®) ¡ (e + ®)³) x1 = (a + c + b) ¡ ; 2(®A ¡ e) ((e ¡ ®) + (e + ®)³) ((e ¡ ®) ¡ (e + ®)³) ¡ ¢ dA(1 ¡ ³) d (e ¡ ®) ¡ 2®³¡ (e + ®)³2 = ¡ 2(®A ¡ e) (e ¡ ®)2 ¡ (e + ®)2 ³2 Ad(1 ¡ ³) d ((A ¡ 1) ¡ ³(A + 1)) = ¡ ; 2(®A ¡ e) 2(A® ¡ e)

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x1 =

d (A ¡ A³¡ A + 1 + A³+ ³) d (1 + ³) = = p: 2(®A ¡ e) 2(®A ¡ e)

Similarly as before, it is easy to show that x2 = q: Then by induction we get x2n = q

and

x2n+1 = p

for all

n ¸ ¡2.

Thus Eq.(1) has the prime period two solution :::;p;q;p;q;:::; where p and q are the distinct roots of the quadratic equation (12) and the proof is complete. Example 4. The solution of the di¤erence equation (1) has a prime period two solution when k = 4; l = 2; s = 3; t = 2; a = 0:3; b = 0:02; c = 0:01; d = 9; e = 3 and ® = 1:1 and the initial conditions x¡5 = p; x¡4 = q; x¡3 = p; x¡2 = q; x¡1 = p and x0 = q since p and q as in the previous theorem (See Fig. 4). plot of x(n+1)=ax(n)+bx(n-k)+cx(n-l)-dx(n-s)/(ex(n-s)-alfax(n-t)) 0

-0.5

-1

x(n)

-1.5

-2

-2.5

-3

-3.5

0

5

10

15

20

25 n

30

35

40

45

50

Figure 4. Plot the periodicity of the solution of equation (1).

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Theorem 4 Equation (1) has no prime period two solutions if one 6= 6 = 6= 6= 6= 6= 6= 6=

(i) 1 + a + b + c (ii) 1 + a ¡ b ¡ c (iii) 1 + a + c ¡ b (iv) 1 + a + c + b (v) 1 + a + c ¡ b (vi)1 + a + b ¡ c (vii) 1 + a ¡ b ¡ c (viii) 1 + a + b ¡ c

0; 0; 0; 0; 0; 0; 0; 0;

l; k; s; t ¡ even: l; k; s; t ¡ odd: l; s; t ¡ even and k ¡ odd: l; k ¡ even and t; s ¡ odd: l ¡ even and k; s; t ¡ odd: l; s; t ¡ odd and k ¡ even: l; k ¡ odd and s; t ¡ even: l ¡ odd and k; s; t ¡ even:

Proof: We prove …rst case when l; k; s and t are both even positive integers ( the other cases are similar and will be left to readers). First suppose that there exists a prime period two solution :::p; q; p; q; :::; of Equation (1).We will prove that Inequality (i) holds. We see from Equation (1) when l; k; s andt are both even positive integers that p = aq + bq + cq ¡

dq ; eq ¡ ®q

q = ap + bp + cp ¡

dp : ep ¡ ®p

and

Therefore,

p ¡ (¡a ¡ b ¡ c)q = ¡

d ; e¡®

(13)

q ¡ (¡a ¡ b ¡ c)p = ¡

d : e¡®

(14)

and

Subtracting (14) from (13) gives

(1 ¡ a ¡ b ¡ c)(p ¡ q) = 0: Since a + b + c 6= 1, then p = q. This is a contradiction. Thus, the proof of (i) is now completed. Example 5. Figure (5) shows the di¤erence equation (1) has no period two solution when k = 4; l = 2; s = 2; t = 4; a = 0:09; b = 0:2; c = 1; d = 9; e = 3 and ® = 2:1

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and the initial conditions x¡4 = 2; x¡3 = 5; x¡2 = 8; x¡1 = 1:2 and x0 = 5. plot of x(n+1)=ax(n)+bx(n-k)+cx(n-l)-dx(n-s)/(ex(n-s)-alfax(n-t)) 350 300 250

x(n)

200 150 100 50 0 -50

0

10

20

30 n

40

50

60

Figure 5. Plot of the solution of equation (1) has no periodic.

5

Existence of Bounded and Unbounded Solutions of Eq.(1)

In this section, we investigate the boundedness nature of the positive solutions of Eq.(1). Theorem 5 Suppose fxn g be a solution of Eq.(1). Then the following statements are true: (i) Let d < e and for some N ¸ 0; the initial conditions xN ¡¾+1 ; xN ¡¾+2 ; :::; xN ¡1 ; xN 2 [ de ; 1], are valid, then for d 6= ® and e2 6= d®, we have the inequality d d d2 (a + b + c) ¡ · xn · a + b + c ¡ 2 ; e d¡® e ¡ ®d

for all n ¸ N:

(15)

(ii) Let d > e and for some N ¸ 0; the initial conditions xN ¡¾+1 ; xN ¡¾+2 ; :::; xN¡1 ; xN 2 [1; de ], are valid, then for d 6= ®; e2 6= d® and exn¡s 6= ®xn¡t , we have the inequality a+b+c¡

d2 d d · xn · (a + b + c) ¡ ; for all n ¸ N: 2 e ¡ ®d e d¡®

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Proof: Let fxn g be a solution of Eq.(1). If for some N ¸ 0; we have

d e

· xn · 1 and d 6= ®;

dxn¡s dxn¡s ·a+b+c¡ : exn¡s ¡ ®xn¡t exn¡s ¡ ®xn¡t ¡ ¢ 2 , But, we can see that exn¡s ¡ ®xn¡t · e ¡ ® de ; exn¡s ¡ ®xn¡t · e ¡®d e d de( e ) 2 dxn¡s dxn¡s dxn¡s 1 e ¸ e2 ¡®d ; exn¡s ¸ e2 ¡®d ; exn¡s ¸ e2 d¡®d ; ¡ exn¡s · exn¡s ¡®xn¡t ¡®xn¡t ¡®xn¡t ¡®xn¡t xn+1 = axn + bxn¡k + cxn¡l ¡

2

¡ e2 d¡®d : Then for ®d 6= e2 ; we get

xn+1 · a + b + c ¡

d2 : e2 ¡ ®d

(17)

Similarly, we can show that xn+1 = axn + bxn¡k + cxn¡l ¡ But, exn¡s ¡®xn¡t ¸ d¡®, d : Then for d 6= ®; we get ¡ d¡®

dxn¡s d dxn¡s ¸ (a + b + c) ¡ : exn¡s ¡ ®xn¡t e exn¡s ¡ ®xn¡t

1 exn¡s ¡®xn¡t

·

dxn¡s 1 ; d¡® exn¡s ¡®xn¡t

·

d ; d¡®

dxn¡s ¡ exn¡s ¸ ¡®xn¡t

d d xn+1 ¸ (a + b + c) ¡ : e d¡®

(18)

From (17) and (18), we get d d2 d (a + b + c) ¡ · xn+1 · a + b + c ¡ 2 ; e d¡® e ¡ ®d

for all n ¸ N:

The proof of part (i) is completed. Similarly, for some N ¸ 0; 1 · xn · de ; d 6= ® and e2 6= d® we can prove part (ii) which is omitted here for convenience. Thus, the proof is now completed.

References [1] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Di¤erence Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993. [2] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Di¤erence Equations with Open Problems and Conjectures, Chapman & Hall / CRC Press, 2001. [3] R. P. Agarwal and E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv. Stud. Contemp. Math., 20 (4), (2010), 525–545.

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[4] Taixiang Sun, Xin Wu, Qiuli He, Hongjian Xi, On boundedness of solutions of the di¤erence equation xn+1 = p + xxn¡1 for p < 1, J. Appl. Math. Comput., n 44(1-2), (2014), 61-68. [5] E. M. Elsayed and M. M. El-Dessoky, Dynamics and behavior of a higher order rational recursive sequence, Adv. Di¤erence Equ., Vol. 2012, (2012), 69. [6] E. M. E. Zayed, Dynamics of the nonlinear rational di¤erence equation xn+1 = n +xn¡k Axn + Bxn¡k + pxq+x ; Eur. J. Pure Appl. Math., 3 (2), (2010), 254-268. n¡k [7] E. M. Elsayed, On the global attractivity and the periodic character of a recursive sequence, Opuscula Math., 30, (2010), 431-446. [8] M. A. El-Moneam, On the Dynamics of the Higher Order Nonlinear Rational Di¤erence Equation, Math. Sci. Lett., 3(2), (2014), 121-129. [9] I. Yalç¬nkaya, On the di¤erence equation xn+1 = ® + Soc., Vol. 2008, (2008), Article ID 805460, 8 pages.

xn¡m , xkn

Discrete Dyn. Nat.

[10] E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the di¤erence equation bxn xn+1 = axn ¡ cxn ¡dx ; Adv. Di¤erence Equ., Vol. 2006, (2006), Article ID n¡1 82579, 1–10. [11] Kenneth S. Berenhaut, Katherine M. Donadio and John D. Foley, On the rational n¡1 for small A, Appl. Math. Lett., Vol. 21(9), recursive sequence yn+1 = A + yyn¡k (2008), 906-909. [12] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, On the nonautonomous xp di¤erence equation xn+1 = An + n¡1 , Appl. Math. Comput., Vol. 217(12), (2011), xqn 5573-5580. [13] Taixiang Sun, Hongjian Xi and Qiuli He, On boundedness of the di¤erence equation xn+1 = pn + xxn¡2s+1 source with period-k coe¢cients, Appl. Math. Comput., n¡s+1 Vol. 217(12), (2011), 5994–5997. [14] Kenneth S. Berenhaut, John D. Foley and Stevo Stevi´c, Quantitative bounds for yn the recursive sequence yn+1 = A + yn¡k , Appl. Math. Lett., Vol., 19(9), (2006), 983–989. [15] Y. Yazlik, D. T. Tollu, N. Taskara, On the Behaviour of Solutions for Some Systems of Di¤erence Equations, J. Comput. Anal. Appl., 18 (1), (2015), 166178. ½ ½ [16] Mehmet G½um½u¸s and Ozkan Ocalan, Some Notes on the Di¤erence Equation xn¡1 xn+1 = ®+ xkn , Discrete Dyn. Nat. Soc., Vol., 2012, (2012), Article ID 258502, 12 pages.

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[17] A. E. Hamza and A. Morsy, On the recursive sequence xn+1 = ® + Math. Lett., Vol., 22(1), (2009), 91-95.

xn¡1 , xkn

Appl.

[18] H. M.EL-Owaidy, A. M. Ahmed and M. S. Mousa, On asymptotic behavior of xp the di¤erence equation xn+1 = ® + n¡1 , J. Appl. Math. Comput., 12, (2003), xpn 31-37. [19] M. R. S. Kulenovi´c and M. Pilling, Global Dynamics of a Certain Twodimensional Competitive System of Rational Di¤erence Equations with Quadratic Terms, J. Comput. Anal. Appl., 19 (1), (2015), 156-166. [20] Mehmet G½um½u¸s, The Periodicity of Positive Solutions of the Nonlinear Di¤erence xp Equation xn+1 = ® + n¡k p , Discrete Dyn. Nat. Soc., Vol. 2013, (2013), Article xn ID 742912, 3 pages. ½ ½ [21] Mehmet Gümü¸s, Ozkan Ocalan and Nilüfer B. Felah, On the Dynamics of the p xn¡k Recursive Sequence xn+1 = ®+ xqn , Discrete Dyn. Nat. Soc., Vol. 2012, (2012), Article ID 241303, 11 pages. [22] E. M. E. Zayed, M. A. El-Moneam, On the global asymptotic stability for a rational recursive sequence, Iran. J. Sci. Technol. Trans. A Sci., Vol. 35, (2012), 333-339. [23] Fangkuan Sun, Xiaofan Yang, and Chunming Zhang, On the Recursive Sequence xp , Discrete Dyn. Nat. Soc., Vol., 2009, (2009), Article ID 608976, xn+1 = A + xn¡k r n¡1 8 pages. [24] E. M. Elsayed and M. M. El-Dessoky, Dynamics and global behavior for a fourthorder rational di¤erence equation, Hacet. J. Math. Stat., 42 (5), (2013), 479–494. [25] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1 = bxn axn ¡ cxn ¡dx , Comm. Appl. Nonlinear Anal., 15 (2), (2008), 47-57. n¡k [26] Taixiang Sun, Hongjian Xi and Hui Wu, On boundedness of the solutions of 1 , Discrete Dyn. Nat. Soc., Vol., 2006, the di¤erence equation xn+1 = xn ¡ p+x n (2006), Article ID 20652, 7 pages. [27] C. J. Schinas, G. Papaschinopoulos, and G. Stefanidou, On the Recursive Sexp quence xn+1 = A + xn¡1 q , Adv. Di¤er. Equ., Vol., 2009 (2009), Article ID 327649, n 11 page. [28] Awad A. Bakery, Some Discussions on the Di¤erence Equation, J. Funct. Spaces, Vol., 2015, (2015), Article ID 737420, 9 pages

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A QUADRATIC FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES EHSAN MOVAHEDNIA, MADJID ESHAGHI GORDJI, CHOONKIL PARK, AND DONG YUN SHIN∗ Abstract. In this paper, we define an intuitionistic fuzzy 2-normed space. Using the fixed point alternative approach, we investigate the Hyers-Ulam stability of the following quadratic functional equation a a f (ax + by) + f (ax − by) = f (x + y) + f (x − y) + (2a2 − a)f (x) + (2b2 − a)f (y) 2 2 in intuitionistic fuzzy 2-Banach spaces.

1. Introduction In 1940, Ulam [1] proposed the famous Ulam stability problem for a metric group homomorphism. In 1941, Hyers [2] solved this stability problem for additive mappings subject to the Hyers condition on approximately additive mappings in Banach spaces. In 1951, Bourgin [3] treated the Ulam stability problem for additive mappings. Subsequently the result of Hyers was generalized by Rassias [4] for linear mapping by considering an unbounded Cauchy difference. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings f : X → Y, where X is a normed space and Y is a Banach space. In 1984, Katrasas [6] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a linear space from various points of view [7, 8]. In particular, in 2003, Bag and Samanta [9], following Cheng and Mordeson [10], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [11]. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces. Recently, considerable attention has been increasing to the problem of fuzzy stability of functional equations. Several various fuzzy stability results concerning Cauchy, Jensen, simple quadratic, and cubic functional equations have been investigated [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Quite recently, the stability results in the setting of intuitionistic fuzzy normed space were studied in [23, 24, 25, 26]; respectively, while the idea of intuitionistic fuzzy normed space was introduced in [27].

2010 Mathematics Subject Classification. 47S40, 54A40, 46S40, 39B52, 47H10. Key words and phrases. Intuitionistic fuzzy 2-normed space; Fixed point; Hyers-Ulam stability; Quadratic functional equation. ∗ The corresponding author.

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2. Preliminaries Definition 2.1. Let X be a real linear space of dimension greater than one and let ∥·, ·∥ be a real-valued function on X × X satisfying the following condition: 1. ∥x, y∥=∥y, x∥ for all x, y ∈ X 2. ∥x, y∥ = 0 if and only if x, y are linearly dependent. 3. ∥αx, y∥ = |α|∥x, y∥ for all x, y ∈ X and α ∈ R. 4. ∥x, y + z∥ ≤ ∥x, y∥ + ∥x, z∥ for all x, y, z ∈ X . Then the function ∥., .∥ is called a 2-norm on X and pair (X, ∥., .∥) is called a 2-normed linear space. Definition 2.2. A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if ∗ satisfies the following conditions: 1. ∗ is commutative and associative; 2. ∗ is continuous; 3 a ∗ 1 = a for all a ∈ [0, 1]; 4 a ∗ b ≤ c ∗ d, whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Example 2.1. An example of continuous t-norm is a ∗ b = min{a, b} Definition 2.3. A binary operation ⋄ : [0, 1] × [0, 1] → [0, 1] is a continuous t-conorm if ⋄ satisfies the following conditions: 1. ⋄ is commutative and associative; 2. ⋄ is continuous; 3 a ⋄ 0 = a for all a ∈ [0, 1]; 4 a ⋄ b ≤ c ⋄ d, whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. Example 2.2. An example of continuous t-conorm is a ⋄ b = max{a, b} Definition 2.4. Let X be a real linear space. A fuzzy subset µ of X × X × R is called a fuzzy 2-norm on X if and only if for x, y, z ∈ X , and t, s, c ∈ R: 1. µ(x, y, t) = 0 if t ≤ 0. 2. µ(x, y, t) = 1 if and only if x, y are linearly dependent, for all t > 0. 3. µ(x, y, t) is invariant under any permutation of x, y. t 4. µ(x, cy, t) = µ(x, y, |c| ) for all t > 0 and c ̸= 0. 5. µ(x + z, y, t + s) ≥ µ(x, y, t) ∗ µ(z, y, s) for all t, s > 0. 6. µ(x, y, .) is a non-decreasing function on R and lim µ(x, y, t) = 1.

t→∞

Then µ is said to be a fuzzy 2-norm on a linear space X , and the pair (X , µ) is called a fuzzy 2-normed linear space. Example 2.3. Let (X , ∥., .∥) be a 2-normed linear space. Define  t  t+∥x,y∥ if t > 0 µ(x, y, t) =  0 if t ≤ 0 where x, y ∈ X and t ∈ R. Then (X , µ) is a fuzzy 2-normed linear space.

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FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES

Definition 2.5. Let (X , µ) be a fuzzy 2-normed linear space. Let {xn } be a sequence in X . Then {xn } is said to be convergent if there exists x ∈ X such that lim µ(xn − x, y, t) = 1

n→∞

for all t > 0. Definition 2.6. Let (X , µ) be a fuzzy 2-normed linear space. Let {xn } be a sequence in X . Then {xn } is said to be a Cauchy sequence if lim µ(xn+p − xn , y, t) = 1

n→∞

for all t > 0 and p = 1, 2, 3, · · · . Let (X , µ) be a fuzzy 2-normed linear space and {xn } be a Cauchy sequence in X . If {xn } is convergent in X then (X , µ) is said to be a fuzzy 2-Banach space. Definition 2.7. Let X be a real linear space. A fuzzy subset ν of X × X × R such that for all x, y, z ∈ X , and t, s, c ∈ R 1. ν(x, y, t) = 1, for all t ≤ 0. 2. ν(x, y, t) = 0 if and only if x, y are linearly dependent, for all t > 0. 3. ν(x, y, t) is invariant under any permutation of x, y. t 4. ν(x, cy, t) = ν(x, y, |c| ) for all t > 0, c ̸= 0. 5. ν(x, y + z, t + s) ≤ ν(x, y, t) ⋄ ν(x, z, s) for all s, t > 0 6. ν(x, y, .) is a nonincreasing function and lim ν(x, y, t) = 0

t→∞

Then ν is said to be an anti fuzzy 2-norm on a linear space X and the pair (X , ν) is called an anti fuzzy 2-normed linear space. Definition 2.8. Let (X , ν) be an anti fuzzy 2-normed linear space and {xn } be a sequence in X . Then {xn } is said to be convergent if there exists x ∈ X such that lim ν(xn − x, y, t) = 0

n→∞

for all t > 0. Definition 2.9. Let (X , ν) be an anti fuzzy 2-normed linear space and {xn } be a sequence in X . Then {xn } is said to be a Cauchy sequence if lim ν(xn+p − xn , y, t) = 0

n→∞

for all t > 0 and p = 1, 2, 3, · · · . Let (X , ν) be an anti fuzzy 2-normed linear space and {xn } be a Cauchy sequence in X . If {xn } is convergent in X then (X , ν) is said to be an anti fuzzy 2-Banach space. The following lemma is easy to prove and we will omit it. Lemma 2.1. Consider the set L∗ and operation ≤L∗ defined by L∗ = {(x1 , x2 ) : (x1 , x2 ) ∈ [0, 1]2 and x1 + x2 ≤ 1} (x1 , x2 ) ≤L∗ (y1 , y2 ) ⇐⇒ x1 ≤ y1 , x2 ≥ y2 for all (x1 , x2 ), (y1 , y2 ) ∈ L∗ . Then (L∗ , ≤L∗ ) is a complete lattice.

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Definition 2.10. A continuous t-norm τ on L = [0, 1]2 is said to be continuous t-representable if there exist a continuous t-norm ∗ and a continuous t-conorm ⋄ on [0, 1] such that, for all x = (x1 , x2 ), y = (y1 , y2 ) ∈ L τ (x, y) = (x1 ∗ y1 , x2 ⋄ y2 ). Definition 2.11. Let X be a set. A function d : X × X −→ [0, ∞] is called a generalized metric on X if and only if d satisfies: (M1 ) d(x, y) = 0 ⇔ x = y ∀x, y ∈ X (M2 ) d(x, y) = d(y, x) ∀x, y ∈ X (M3 ) d(x, z) ≤ d(x, y) + d(y, z) ∀x, y, z ∈ X Theorem 2.1. ([28]) Let (X ,d) be a complete generalized metric space and J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x ∈ X , either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (a) d(J n x, J n+1 x) < ∞ for all n ≥ n0 ; (b) the sequence {J n x} converges to a fixed point y ∗ of J ; (c) y ∗ is the unique fixed point of J in the set Y = {y ∈ X : d(J n0 x, y) < ∞}; 1 (d) d(y, y ∗ ) ≤ 1−L d(y, J y) for all y ∈ Y. 3. Main results 3.1. Intuitionistic fuzzy 2-normed spaces. In this subsection we define an intuitionistic fuzzy 2-normed space. Then in next subsection by the fixed point technique we investigate the Hyers-Ulam stability of a generalized quadratic functional equation in intuitionistic fuzzy 2-normed spaces. Definition 3.1. A 3-tuple (X , ρµ,ν , τ ) is said to be an intuitionistic fuzzy 2-normed space(for short, IF2NS) if X is a real linear space, and µ and ν are a fuzzy 2-norm and an anti fuzzy 2-norm, respectively, such that ν(x, y, t) + µ(x, y, t) ≤ 1. τ is continuous t-representable, and ρµ,ν : X × X × R → L∗ ρµ,ν (x, y, t) = (µ(x, y, t), ν(x, y, t)) is a function satisfying the following conditions, for all x, y, z ∈ X , and t, s, α ∈ R (1) ρµ,ν (x, y, t) = (0, 1) = 0L∗ for all t ≤ 0. (2) ρµ,ν (x, y, t) = (1, 0) = 1L∗ if and only if x, y are linearly dependent, for all t > 0. t (3) ρµ,ν (αx, y, t) = ρµ,ν (x, y, |α| ) for all t > 0 and α ̸= 0 (4) ρµ,ν (x, y, t) is invariant under any permutation of x, y. (5) ρµ,ν (x + z, y, t + s) ≥L∗ τ (ρµ,ν (x, y, t), ρµ,ν (z, y, s)) for all t, s > 0. (6) ρµ,ν (x, y, .) is continuous and lim ρµ,ν (x, y, t) = 0L∗ and lim ρµ,ν (x, y, t) = 1L∗ t→∞

t→0

Then ρµ,ν is said to be an intuitionstic fuzzy 2-norm on a real linear space X . Example 3.1. Let (X , ∥·, ·∥) be a 2-normed space, τ (a, b) = (a1 b1 , min(a2 + b2 , 1)) be continuous t-representable for all a = (a1 , a2 ), b = (b1 , b2 ) ∈ L∗ and µ, ν be a fuzzy and an anti fuzzy 2-norm, respectively. We define

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FUNCTIONAL EQUATION IN INTUITIONISTIC FUZZY 2-BANACH SPACES

( ρµ,ν (x, y, t) =

t ∥x, y∥ , t + m∥x, y∥ t + m∥x, y∥

)

for all t ∈ R+ and m > 1. Then (X , ρµ,ν , τ ) is an IF2NS. Definition 3.2. A sequence {xn } in an IF2NS (X , ρµ,ν , τ ) is said to be convergent to a point x ∈ X if lim ρµ,ν (xn − x, y, t) = 1L∗ n→∞

for every t > 0. Definition 3.3. A sequence {xn } in an IF2NS (X , ρµ,ν , τ ) is said to be a Cauchy sequence if for any 0 < ϵ < 1 and t > 0, there exists n0 ∈ N such that ρµ,ν (xn − xm , y, t) ≥L∗ (1 − ϵ, ϵ) for all n, m ≥ n0 . Definition 3.4. An IF2NS space (X , ρµ,ν , τ ) is said to be complete if every Cauchy sequence in (X , ρµ,ν , τ ) is convergent. A complete intuitionistic fuzzy 2-normed space is called an intuitionistic fuzzy 2-Banach space. 3.2. Hyers-Ulam stability of a generalized quadratic functional equation in IF2NS. In this subsection, using the fixed point alternative approach, we prove the Hyers-Ulam stability of a generalized quadratic functional equation in intuitionistic fuzzy 2-Banach spaces. Definition 3.5. Let X , Y be real linear spaces. For a given mapping f : X → Y, we define a Df (x, y) := f (ax + by) + f (ax − by) − f (x + y) 2 a 2 − f (x − y) − (2a − a)f (x) − (2b2 − a)f (y) 2 2 where a, b ≥ 1 , a ̸= 2b and x, y ∈ X . Theorem 3.1. Let X be a real linear space, (Z, ρ′µ,ν , τ ′ ) an intuitionistic fuzzy 2-normed space and let ϕ : X × X → Z, φ : X × X → Z be mappings such that for some 0 < |α| < a ) (α (3.1) ρ′µ,ν (ϕ(ax, ay), φ(ax, ay), t) ≥L∗ ρ′µ,ν 2 ϕ(x, y), φ(x, y), t a for all x, y ∈ X and t ∈ R+ . Let (Y, ρµ,ν , τ ) be a complete intuitionistic fuzzy 2-normed space. If ξ : X × X → Y is a mapping such that ξ(ax, ay) = αa1 2 ξ(x, y) for all x, y ∈ X and f : X → Y is a mapping satisfying f (0) = 0 and ρµ,ν (Df (x, y), ξ(x, y), t) ≥L∗ ρ′µ,ν (ϕ(x, y), φ(x, y), t)

(3.2)

for all x, y ∈ X , t > 0, then there is a unique quadratic mapping Q : X → Y such that ( ) ρµ,ν (f (x) − Q(x), ξ(x, 0), t) ≥L∗ ρ′µ,ν ϕ(x, 0), φ(x, 0), (2(a2 − α2 )t

(3.3)

Proof. Putting y = 0 in (3.2), we have ( ) ρµ,ν 2f (ax) − a2 f (x), ξ(x, 0), t ≥L∗ ρ′µ,ν (ϕ(x, 0), φ(x, 0), t) and so

( ρµ,ν

1 f (ax) − f (x), ξ(x, 0), t a2

) ≥

L∗

765

ρ′µ,ν

(

1 ϕ(x, 0), φ(x, 0), t 2a2

) (3.4)

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for all x ∈ X and t > 0. Consider the set E = {g : X → Y} and define a generalized metric d on E by { } d(g, h) = inf c ∈ R+ : ρµ,ν (g(x) − h(x), ξ(x, 0), t) ≥L∗ (cϕ(x, 0), φ(x, 0), t) for all x ∈ X and t > 0 with inf ∅ = ∞. It is easy to show that (E, d) is complete (see [29]). Define J : X → X by Jg(x) = a12 g(ax) for all x ∈ X . Now, we prove that J is strictly contractive 2 mapping of E with the Lipschitz constant αa2 . Let g, h ∈ E be given such that d(g, h) < ϵ. Then ρµ,ν (g(x) − h(x), ξ(x, 0), t) ≥L∗ ρ′µ,ν (ϵϕ(x, 0), φ(x, 0), t) for all x ∈ X and t > 0. So ρµ,ν (Jg(x) − Jh(x), ξ(x, 0), t)

= ≥L∗ =L∗

( ) ρµ,ν g(ax) − h(ax), ξ(x, 0), a2 t ) ( t ′ ρµ,ν cϕ(ax, 0), φ(ax, 0), α ) ( 2 α ′ cϕ(x, 0), φ(x, 0), t . ρµ,ν a2

α2 d(g, h) for all g, h ∈ E. It follows from (3.4) that a2 1 d(f, Jf ) ≤ 2 < ∞ 2a It follows from Theorem 2.1 that there exists a mapping Q : X → Y satisfying the following Then d(Jg, Jh)
0,

x ∈ Γ0 ,

t ≥ 0,

x ∈ Γ1 ,

t > 0,

(1.1)

x ∈ Ω.

Here Ω is a regular and bounded domain in RN (N ≥ 1) and ∂Ω = Γ0 ∪ Γ1 , mes(Γ0 ) > 0, Γ0 ∩ Γ1 = ∅. We denote ∆ the Laplacian operator with respect to the x variable and

∂ ∂ν

the

unit outer normal derivative, m ≥ 2, p > 2, α, r are positive constants and u0 and u1 are given functions. From the mathematical point of view, the boundary conditions that do not neglect the acceleration terms are usually called dynamic boundary conditions. Researches on these problems are very important in practical problems as well as in the theoretical fields. For the cases of one dimension space, many results have been established (see [1, 2, 3, 11, 12, 13, 15, 24, 23, 35]). For example, Grobbelaar-van Dalsen [12] studied the following problem:  x ∈ (0, L), t > 0,   utt − uxx − utxx = 0,    u(0, t) = 0, t > 0,    utt (L, t) = − [ux + utx ] (L, t), t > 0, (1.2)    u(x, 0) = u (x), u (x, 0) = v (x), x ∈ (0, L), 0 t 0      u(L, 0) = η, ut (L, 0) = µ t > 0. By using the theory of B-evolutions and the theory of fractional powers, the author proved that problem (1.2) gives rise to an analytic semigroup in an appropriate functional space and obtained the existence and the uniqueness of solutions. For a problem related to (1.2), an exponential 1

Corresponding author. E-mail: [email protected] 1

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decay result was obtained in [13], which describes the weakly damped vibrations of an extensible beam. Later, Zhang and Hu [35] considered (1.2) in a more general form and an exponential and polynomial decay rates for the energy were obtained by using the Nakao inequality. Pellicer and Sol`a-Morales [24] considered the linear wave equation with strong damping and dynamical boundary conditions as an alternative model for the classical spring-mass-damper ODE: m1 u′′ (t) + d1 u′ (t) + k1 (t) = 0.

(1.3)

Based on the semigroup theory, spectral perturbation analysis and dominant eigenvalues, they compared analytically these two approaches to the same physical system. )Then, Pellicer [23] ( (1,t) considered the same problem with a control acceleration εf u(1, t), ut√ as a model for a ε controlled spring-mass-damper system and established some results concerning its large time behavior. By applying invariant manifold theory, the author proved that the infinite dimensional system admits a two-dimensional attracting manifold where the equation is well represented by a classical nonlinear oscillations ODE, which can be exhibited explicitly. For the multi-dimensional cases, we can cite [5, 6, 14, 21, 22, 30] for problems with the Dirichlet boundary conditions and [27, 28, 29] for the Cauchy problems. Recently, Gerbi and Said-Houari [7, 8] studied problem (1.1), in which the strong damping term −∆ut is involved. They showed in [7] that if the initial data are large enough then the energy and the Lp norm of the solution of problem (1.1) is unbounded and grows up exponentially as time goes to infinity. Later, they established in [8] the global existence and asymptotic stability of solutions starting in a stable set by combining the potential well method and the energy method. A blow-up result for the case m = 2 with initial data in the unstable set was also obtained. However, as indicated in [8], the blow-up of solutions in the presence of a strong damping and a nonlinear boundary damping (i.e., m > 2) at the same time is still an open problem. For other related works, we refer the readers to [4, 10, 9, 17, 18, 19, 20, 25, 26, 31, 32, 33, 34] and the references therein. Motivated by the above works, in this article, we intend to extend the exponentially growth result in [7] to a blow-up result with positive initial energy. The main difficulty here is the simultaneous appearance of the strong damping term ∆ut , the nonlinear boundary damping term r|ut |m−2 ut , and the nonlinear source term |u|p−2 u. For our purpose, the functional like L(t) = H(t) + εF (t) in [7] is modified to L(t) = H 1−α (t) + εF (t) for some α > 0 in this paper. ∫ m−2 ut udσ so that the appearance We also give a modified manner to estimate the term Γ1 |ut |

of the form like : γ = RH −σ (t) (for constants γ, R and σ) which has been used in many earlier

works (for example in [10, 16, 21, 22]) can be avoided. The paper is organized as follows. In Section 2 we present some notations and assumptions and state the main result. Section 3 is devoted to proof of the blow-up result - Theorem 2.2.

2. Preliminaries and main result In this section, we first recall some notations and assumptions given in [7]. We denote { } HΓ10 (Ω) = u ∈ H 1 (Ω) | uΓ0 = 0

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with the scalar product (·, ·) in L2 (Ω) and we also mean by ∥ · ∥q the Lq (Ω) norm for 1 ≤ q ≤ ∞ and by ∥ · ∥q,Γ1 the Lq (Γ1 ) norm. We will use the following embedding 2 ≤ q ≤ q¯,

HΓ10 (Ω) ,→ Lq (Γ1 ),   2(N − 1) , q¯ = N −2  + ∞,

where

if N ≥ 3,

(2.1)

if N = 1, 2.

We state the following local existence and uniqueness theorem established in [7]. { } q¯ Theorem 2.1. ([7, theorem 2.1]) Let 2 ≤ p ≤ q¯ and max 2, q¯+1−q ≤ m ≤ q¯. Then given u0 ∈ HΓ10 (Ω) and u1 ∈ L2 (Ω), there exists T > 0 and a unique solution u(t) of the problem (1.1) on [0, T ) such that u ∈ C(0, T ; HΓ10 (Ω)) ∩ C 1 (0, T ; L2 (Ω)), ut ∈ L2 (0, T ; HΓ10 (Ω)) ∩ Lm (Γ1 × [0, T )). We define the energy functional 1 1 E(t) = ∥ut ∥22 + ∥∇u∥22 − 2 2 and set

1 1 ∥u∥pp + ∥ut ∥22,Γ1 p 2

(2.2)

) 1 1 α1 = B , E1 = − α12 , 2 p where B is the best constant of the embedding H01 (Ω) ,→ Lp (Ω). We can easily get −p/(p−2)

(

E ′ (t) = −α∥∇ut ∥22 − r∥ut ∥m m,Γ1 ≤ 0.

(2.3)

(2.4)

Our main result reads as follows. Theorem 2.2. Suppose that m < p with 2 < p ≤ q¯ and that { } N −1 p−2 2(p − m) N − ≤ min , 0< 2 m p mp

(2.5)

holds. Assume that E(0) < E1 ,

∥∇u0 ∥2 > α1 .

(2.6)

Then the solution of problem (1.1) blows up in a finite time T0 , in the sense that [ ] lim ∥ut ∥22 + ∥ut ∥22,Γ1 + ∥∇u∥22 = +∞. t→T0−

(2.7)

3. Blow-up of solutions In this section, we prove our main result and use C to denote a generic positive constant. To this end, we need the following lemmas. Lemma 3.1. ([7, Lemma 3.1]) Let u be the solution of problem (1.1). Assume that 2 < p ≤ q¯ and (2.6) holds. Then there exists a constant α2 > α1 such that ∥∇u(·, t)∥2 ≥ α2 ,

∀ t ≥ 0,

(3.1)

and ∥u∥p ≥ Bα2 ,

779

∀ t ≥ 0.

(3.2)

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Lemma 3.2. Let u be the solution of problem (1.1). Assume that 2 < p ≤ q¯ and (2.6) holds. Then we have E1
α1 and combining the definition of E1 , we have [ ( )p ] 2 p−2 α1 −p −p 1 − − 2E1 B α2 = 1− > 0. p p α2 So, we can choose β small enough so that s

2 rCC12 β m −θ1 1− − H (0) − 2E1 B −p α2−p > 0. p m Once β is fixed, we choose ε small enough such that ∫ ∫ εα ∥∇u0 ∥22 > 0 L(0) = H 1−θ (0) + ε u0 u1 dx + ε u0 u1 dσ + 2 Ω Γ1 and

s

εCC12 (m − 1) −(θ1 −θ) 1−θ− H (0) > 0. mβ m Hence, we have ( ) L′ (t) ≥Λε ∥ut ∥22 + ∥ut ∥22,Γ1 + ∥u∥pp + H(t)

(3.15)

for some positive constant Λ. On the other hand, we have ( ∫ ∫ 1 L 1−θ (t) = H 1−θ (t) + ε ut udx + ε

εα ut udσ + ∥∇u∥22 2 Γ1

Ω

781

)

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( ≤C

∫ ∫ 1 1−θ + H(t) + ut udx

Γ1

Ω

) 1 2 1−θ ut udσ + ∥∇u∥21−θ .

Using H¨older and Young inequalities, (3.7), (3.11) and Lemma 3.3, we get ∫ 1 ( ) 1 1 2 1−θ 1 1−θ 1−θ 1−2θ 2 ut udx 1−θ ≤C(∥u∥2 ∥ut ∥2 ) ≤ C∥u∥p ∥ut ∥2 ≤ C ∥u∥p + ∥ut ∥2 Ω ( ) ≤C H(t) + ∥ut ∥22 + ∥ut ∥22,Γ1 + ∥u∥pp , 1 1−s 1 s 1−θ 1 1−θ 1−θ 1−θ ut udx ≤ C(∥u∥2,Γ1 ∥ut ∥2,Γ1 ) 1−θ ≤ C∥ut ∥2,Γ ∥u∥ ∥∇u∥ p 2 1 Γ1 ( ) 2(1−s) s ( ) ≤CC12(1−θ) ∥u∥p1−2θ + ∥ut ∥22,Γ1 ≤ C H(t) + ∥ut ∥22 + ∥ut ∥22,Γ1 + ∥u∥pp ,

(3.16)

(3.17)

∫

and

2

(3.18)

1

∥∇u∥21−θ ≤ C11−θ .

(3.19)

Using the Poincar´e’s inequality and (3.7), we have p

∥u∥pp ≤ B p ∥∇u∥p2 ≤ B p C12 .

(3.20)

By virtue of (3.5) and (3.20), we know that H(t) is bounded. There exists a positive constant C2 such that

1

H(t) + C11−θ ≤ C2 H(t). Therefore, we obtain ( ) 1 L 1−θ (t) ≤ C ∥ut ∥22 + ∥ut ∥22,Γ1 + ∥u∥pp + H(t) .

(3.21)

A combining of (3.15) and (3.21) leads to L′ (t) ≥

1 εΛ 1−θ L (t). C

(3.22)

A simple integration of (3.22) over [0, t] gives 1

θ

L 1−θ (t) ≥

L

θ − 1−θ

(0) −

θΛε C(1−θ) t

,

∀ t ≥ 0.

(3.23)

This shows that L(t) blows up in a finite time T0 , where T0 ≤ If we choose T ∗ ≥

(1−θ)C , Λεθ[L(0)]θ/(1−θ)

(1 − θ)C . Λεθ[L(0)]θ/(1−θ)

then we obtain T0 ≤ T ∗ , which contradicts to our assumption.

This completes the proof. References [1] K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl. 197 (1996), no. 3, 781–795. ¨ Morg¨ [2] F. Conrad and O. ul, On the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim. 36 (1998), no. 6, 1962–1986 (electronic). [3] G. G. Doronin and N. A. Larkin, Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions, Nonlinear Anal. 50 (2002), no. 8, Ser. A: Theory Methods, 1119–1134. [4] C. G. Gal, G. R. Goldstein and J. A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evol. Equ. 3 (2003), no. 4, 623–635. [5] F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 23 (2006), no. 2, 185–207.

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Grobbelaar-Van Dalsen and A. van der Merwe, Boundary stabilization for the extensible beam with attached load, Math. Models Methods Appl. Sci. 9 (1999), no. 3, 379–394. [12] M. Grobbelaar-Van Dalsen, On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions, Appl. Anal. 53 (1994), no. 1-2, 41–54. [13] M. Grobbelaar-Van Dalsen, On the initial-boundary-value problem for the extensible beam with attached load, Math. Methods Appl. Sci. 19 (1996), no. 12, 943–957. [14] J.-M. Jeong, J. Y. Park, Y. H. Kang, Energy decay rates for viscoelastic wave equation with dynamic boundary conditions, J. Comput. Anal. Appl. 19 (2015), no. 3, 500–517. [15] M. Kirane, Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type, Hokkaido Math. J. 21 (1992), no. 2, 221–229. [16] G. Li, Y. N. Sun and W. J. Liu, Global existence and blow-up of solutions for a strongly damped Petrovsky system with nonlinear damping, Appl. Anal. 91 (2012), no. 3, 575–586. [17] W. J. Liu, General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term, Taiwanese J. Math. 17 (2013), no. 6, 2101–2115. [18] W. J. Liu, Arbitrary rate of decay for a viscoelastic equation with acoustic boundary conditions, Appl. Math. Lett. 38 (2014), 155–161. [19] W. J. Liu and K. W. Chen, Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions, Z. Angew. Math. Phys., 66 (2015), DOI: 10.1007/s00033-014-0489-3. [20] W. J. Liu and Y. Sun, General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions, Z. Angew. Math. Phys. 65 (2014), no. 1, 125–134. [21] S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr. 260 (2003), 58–66. [22] S. A. Messaoudi, Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl. 320 (2006), no. 2, 902–915. [23] M. Pellicer, Large time dynamics of a nonlinear spring-mass-damper model, Nonlinear Anal. 69 (2008), no. 9, 3110–3127. [24] M. Pellicer and J. Sol` a-Morales, Analysis of a viscoelastic spring-mass model, J. Math. Anal. Appl. 294 (2004), no. 2, 687–698. [25] F. Q. Sun and M. X. Wang, Non-existence of global solutions for nonlinear strongly damped hyperbolic systems, Discrete Contin. Dyn. Syst. 12 (2005), no. 5, 949–958. [26] F. Tahamtani, Blow-up results for a nonlinear hyperbolic equation with Lewis function, Bound. Value Probl. 2009, Art. ID 691496, 9 pp. [27] G. Todorova, Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, C. R. Acad. Sci. Paris S´er. I Math. 326 (1998), no. 2, 191–196. [28] G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl. 239 (1999), no. 2, 213–226. [29] G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in Rn , J. Math. Anal. Appl. 303 (2005), no. 1, 242–257. [30] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999), no. 2, 155–182. [31] E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste 31 (2000), suppl. 2, 245–275. [32] S.-T. Wu, Non-existence of global solutions for a class of wave equations with nonlinear damping and source terms, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 4, 865–880. [33] R. Z. Xu, Global existence, blow up and asymptotic behaviour of solutions for nonlinear Klein-Gordon equation with dissipative term, Math. Methods Appl. 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UNIQUENESS OF MEROMORPHIC FUNCTIONS WITH THEIR DIFFERENCE OPERATORS XIAOGUANG QI, YONG LIU AND LIANZHONG YANG Abstract. This paper is devoted to considering sharing value problems for a meromorphic function f (z) with its difference operator ∆c f = f (z + c) − f (z), which improve some recent results in Chen and Yi in [2].

1. Introduction In this paper a meromorphic function will mean meromorphic in the whole complex plane. We assume that the reader is familiar with the elementary Nevanlinna Theory, see, e.g. [8, 19]. In particular, we denote the order, exponent of convergence of zeros and poles of a meromorphic function f (z) by σ(f ), λ(f ) and λ( f1 ), respectively. As usual, the abbreviation CM stands for ”counting multiplicities”, while IM means ”ignoring multiplicities”. The classical results in the uniqueness theory of meromorphic functions are the five-point, resp. four-point, theorems due to Nevanlinna [17]: Theorem A. If two meromorphic functions f (z) and g(z) share five distinct values a1 , a2 , a3 , a4 , a5 ∈ C ∪ {∞} IM, then f (z) = g(z). Theorem B. If two meromorphic functions f (z) and g(z) share four distinct values a1 , a2 , a3 , a4 ∈ C ∪ {∞} CM, then f (z) = g(z) or f (z) = T ◦ g(z), where T is a M¨obius transformation. It is well-known that 4 CM can not be improved to 4 IM, see [4]. Further, Gundersen [6, Theorem 1] has improved the assumption 4 CM to 2 CM+2 IM, while 1 CM+3 IM remains an open problem. Meanwhile, Gundersen [7], Mues and Stinmetz [16] got some uniqueness results on the case when g(z) is the derivative of f (z): Theorem C. If a meromorphic functions f (z) and its derivative f 0 (z) share two distinct values a1 , a2 CM, then f (z) = f 0 (z). 2010 Mathematics Subject Classification. 30D35, 39A05. Key words and phrases. Difference operator; Meromorphic functions; Value sharing; Nevanlinna theory. This work was supported by the National Natural Science Foundation of China (No. 11301220, No. 11371225 and No. 11401387), the NSF of Shandong Province, China (No. ZR2012AQ020) and the Fund of Doctoral Program Research of University of Jinan (XBS1211). 1

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Gundersen [7] has given a counterexample to show that the conclusion of Theorem C is not valid if 2 CM is replaced by 1 CM +1 IM. However, 2 CM can be replaced by 3 IM, see [5, 15]. In recent papers [9, 10], Heittokangas et al. started to consider the uniqueness of a finite order meromorphic function sharing values with its shift. They concluded that: Theorem D. Let f (z) be a meromorphic function of finite order, let c ∈ C, and let a1 , a2 , a3 ∈ S(f ) ∪ {∞} be three distinct periodic functions with period c. If f (z) and f (z + c) share a1 , a2 CM and a3 IM, then f (z) = f (z + c) for all z ∈ C. Some improvements of Theorem D can be found in [2, 11, 12, 18]. The difference operator ∆c f = f (z + c) − f (z) can be regarded as the difference counterpart of f 0 (z). Therefore, some research results [9, 13] have been obtained for the problem that ∆c f and f (z) share one value a CM, which can be seen as difference analogues of Br¨ uck conjecture in [1]. Here, we just recall the following result in [2] as an example: Theorem E. Let f (z) be a finite order transcendental entire function which has a finite Borel exceptional value a, and let f (z) be not periodic of period c. If ∆c f and f (z) share a CM, then a = 0 and ∆c f = τ f (z), where τ is a non-zero constant. Zhang et al. gave some improvements of Theorem E, the reader is referred to [14, 20]. A natural question is: what is the uniqueness result on the case when f (z) is meromorphic and a(z) is a small function of f (z) in Theorem E. Corresponding to this question, we get the following results: Theorem 1.1. Let f (z) be a transcendental meromorphic function of finite order which has two Borel exceptional values a and ∞, and let f (z) be not periodic of period c. If ∆c f and f (z) share values a and ∞ CM, then a = 0 and f (z) = AeBz , where A, B are non-zero constants. Theorem 1.2. Let f (z) be a transcendental meromorphic function of finite order which has a Borel exceptional value ∞, and let a(z) be a non-constant meromorphic function such that σ(a) < σ(f ) and λ(f − a) < σ(f ). If ∆c f and f (z) share values a(z) and ∞ CM, then f (z) = a(z) + CeDz and σ(a) < 1, where C, D are non-zero constants, . 2. Some Lemmas Lemma 2.1. [3, Theorem 2.1] Let f (z) be a non-constant meromorphic function with finite order σ, and let c be a non-zero constant. Then, for each ε > 0, we have T (r, f (z + c)) = T (r, f (z)) + O(rσ−1+ε ) + O(log r).

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3

Lemma 2.2. [3, Theorem 8.2] Let f (z) be a meromorphic function of finite order σ, c be a non-zero constant. Let ε > 0 be a given real constant, then there exits a subset E ⊂ (1, ∞) of finite logarithmic measure such that for all |z| = r ∈ / [0, 1] ∪ E, we have ¯ ¯ ¯ f (z + η) ¯ σ−1+ε ¯ ≤ exp(rσ−1+ε ). exp(−r ) ≤ ¯¯ f (z) ¯ 3. Proof of Theorem 1.2 It follows by the assumption that f (z) = a(z) +

u(z) h(z) e , v(z)

(3.1)

where u(z), v(z) are two non-zero entire functions, h(z) is a nonconstant polynomial of degree m. Furthermore, we know f (z) is of normal growth, and a(z), u(z), v(z) satisfy: 1 λ(f − a) = λ(u) = σ(u) < σ(f ) = m, λ( ) = λ(v) = σ(v) < σ(f ), f and T (r, a) = S(r, f ),

T (r, u) = S(r, eh(z) ),

T (r, v) = S(r, eh(z) ) = S(r, f ).

From (3.1), we have ¶ µ u(z + c) h(z+c)−h(z) u(z) h(z) e − e + a(z + c) − a(z) ∆c f = v(z + c) v(z) h(z)

= H(z)e

(3.2)

+ a(z + c) − a(z).

Applying Lemma 2.1 to equation (3.2), we conclude σ(H) < m = σ(f ), which means that T (r, H) = S(r, f ). By the sharing assumption, we obtain that ∆c f − a(z) = ep(z) , f (z) − a(z)

(3.3)

where p(z) is a polynomial. By combining Lemma 2.1 and (3.3), it follows that deg p(z) ≤ σ(f ) = m. From (3.1), (3.2) and (3.3), we deduce that u(z) h(z)+p(z) e . (3.4) H(z)eh(z) + a(z + c) − 2a(z) = v(z) Case 1. Suppose that a(z + c) − 2a(z) 6≡ 0. Then by (3.4) we get ! Ã µ ¶ 1 1 = S(r, eh(z) ), ≤ N r, N r, a(z+c)−2a(z) h(z) u(z) e + H(z) when H(z) 6≡ 0, which is a contradiction.

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If H(z) ≡ 0, then it follows from (3.2) that u(z + c) h(z+c)−h(z) u(z) e − ≡ 0, v(z + c) v(z) this gives u(z + c)v(z) h(z+c)−h(z) e ≡ 1. u(z)v(z + c)

(3.5)

Denote

u(z + c)v(z) . (3.6) u(z)v(z + c) From equation (3.5), we know that G(z) is a non-zero entire function. By Lemma 2.2, we see ¯ ¯ ¯ ¯ ¯ ¯ ¯ u(z + c) ¯ ¯ ≤ exp(rσ(u)−1+ε ), ¯ v(z) ¯ ≤ exp(rσ(v)−1+ε ), ¯ ¯ v(z + c) ¯ ¯ u(z) ¯ G(z) =

which implies that

¯ ¯ ¯ u(z + c)v(z) ¯ ¯ ≤ exp(2rσ−1+ε ), ¯ |G(z)| = ¯ u(z)v(z + c) ¯

where σ = max{σ(u), σ(v)} < σ(f ) = m, and 0 < ε
0 there is a positive integer N such that kxn − xm k ≤ ε for all n, m ≥ N . (ii) Let {xn } be a sequence in a non-Archimedean normed space X. Then the sequence {xn } is called convergent if for a given ε > 0 there are a positive integer N and an x ∈ X such that kxn − xk ≤ ε for all n ≥ N . Then we call x ∈ X a limit of the sequence {xn }, and denote by limn→∞ xn = x. (iii) If every Cauchy sequence in X converges, then the non-Archimedean normed space X is called a non-Archimedean Banach space. The stability problem of functional equations originated from a question of Ulam [25] concerning the stability of group homomorphisms. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [17] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [8] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [24] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [5] noticed that the theorem of Skof isstill true if the relevant domain E1 is replaced by an Abelian group. The +2 x−y = f (x)+f (y) is called a Jensen type quadratic equation. The functional equation 2f x+y 2 2 stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 3, 4, 15, 16, 19, 20, 21, 22, 23, 26, 27]). In [9], Gil´anyi showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (xy) + f (xy −1 ). See also [18]. Gil´anyi [10] and Fechner [7] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [14] proved the Hyers-Ulam stability of additive functional inequalities. The stability problems of functional equations and inequalities have also been treated by many authors ([6, 13]). In Section 2, we solve the quadratic ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.1) in non-Archimedean Banach spaces. We moreover prove the Hyers-Ulam stability of a quadratic ρ-functional equation associated with the quadratic ρ-functional inequality (0.1) in non-Archimedean Banach spaces. In Section 3, we solve the quadratic ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (0.2) in non-Archimedean Banach spaces. We moreover prove the Hyers-Ulam stability of a quadratic ρ-functional equation associated with the quadratic ρ-functional inequality (0.2) in non-Archimedean Banach spaces. Throughout this paper, assume that X is a non-Archimedean normed space and that Y is a non-Archimedean Banach space. Let |2| 6= 1. 2. Quadratic ρ-functional inequality (0.1) in non-Archimedean normed spaces Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < 1.

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In this section, we solve the quadratic ρ-functional inequality (0.1) in non-Archimedean normed spaces. Lemma 2.1. A mapping f : X → Y satisfies

     

x+y x−y

kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤ ρ 2f + 2f − f (x) − f (y)

(2.1) 2 2

for all x, y ∈ X if and only if f : X → Y is quadratic. Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get k2f (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = x in (2.1), we get kf (2x) − 4f (x)k ≤ 0 and so f (2x) = 4f (x) for all x ∈ X. Thus x 2

 

f

1 = f (x) 4

(2.2)

for all x ∈ X. It follows from (2.1) and (2.2) that

     

ρ 2f x + y + 2f x − y − f (x) − f (y)

2 2

kf (x + y) + f (x − y) − 2f (x) − 2f (y)k ≤

|ρ| kf (x + y) + f (x − y) − 2f (x) − 2f (y)k 2

= and so

f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X. The convesre is obviously true.



Corollary 2.2. A mapping f : X → Y satisfies x+y f (x + y) + f (x − y) − 2f (x) − 2f (y) = ρ 2f 2 for all x, y ∈ X if and only if f : X → Y is quadratic. 







+ 2f

x−y 2





− f (x) − f (y)

(2.3)

The functional equation (2.3) is called a quadratic ρ-functional equation. We prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (2.1) in non-Archimedean Banach spaces. Theorem 2.3. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)k

     

x−y x+y r r

≤ ρ 2f + 2f − f (x) − f (y)

+ θ(kxk + kyk ) 2 2 for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤

2θ kxkr |2|r

(2.4)

(2.5)

for all x ∈ X. Proof. Letting x = y = 0 in (2.4), we get k2f (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = x in (2.4), we get kf (2x) − 4f (x)k ≤ 2θkxkr

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(2.6)

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≤ 2r θkxkr for all x ∈ X. Hence |2|

   

l x

4 f x − 4m f

l 2 2m

         

m−1

l x x x x l+1 m



f , · · · , 4 ≤ max 4 f −4 f −4 f 2l 2l+1 2m−1 2m

 

       

x x x x l m−1

= max |4| f , · · · , |4| − 4f

f 2m−1 − 4f 2m 2l 2l+1 ( )

for all x ∈ X. So f (x) − 4f

≤ max

x 2

|4|l |4|m−1 , · · · , |2|rl+1 |2|r(m−1)+1

2θkxkr =

2θ

|2|(r−2)l+1

(2.7)

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping h : X → Y by x h(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5). It follows from (2.4) that kh(x + y) + h(x − y) − 2h(x) − 2h(y)k

       

x−y x y x+y

+ f − 2f − 2f = lim |4|n f

n→∞ 2n 2n 2n 2n

        n

x+y x−y x y n

+ lim |4| θ (kxkr + kykr ) ≤ lim |4| |ρ| 2f + 2f − f − f n→∞ 2n+1 2n+1 2n 2n n→∞ |2|nr

   

x−y x+y + 2h − h(x) − h(y) = |ρ|

2h 2 2 for all x, y ∈ X. So

     

x+y x−y

kh(x + y) + h(x − y) − 2h(x) − 2h(y)k ≤ ρ 2h + 2h − h(x) − h(y)

2 2

for all x, y ∈ X. By Lemma 2.1, the mapping h : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (2.5). Then we have

   

q x x q

kh(x) − T (x)k = 4 h q − 4 T 2 2q          

q

q x 2θ x x x q q



≤ ≤ max 4 h q − 4 f , 4 T −4 f kxkr ,

q q q (r−2)q+1 2 2 2 2 |2|

which tends to zero as q → ∞ for all x ∈ X. So we can conclude that h(x) = T (x) for all x ∈ X. This proves the uniqueness of h. Thus the mapping h : X → Y is a unique quadratic mapping satisfying (2.5).  Theorem 2.4. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.4). Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤

2θ kxkr |4|

(2.8)

for all x ∈ X.

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Proof. It follows from (2.6) that



f (x) − 1 f (2x) ≤ 2θ kxkr

4 |4|

for all x ∈ X. Hence



1

f (2l x) − 1 f (2m x)

4l 4m   

1  l 

1 l+1

≤ max l f 2 x − l+1 f 2 x

,··· 4 4

 

   1

f 2l x − 1 f 2l+1 x , · · · , = max

|4|l 4 ( )

≤ max

|2|rl |2|r(m−1) , · · · , |4|l+1 |4|(m−1)+1

2θkxkr =

(2.9)

  

1

1 m−1 m

, m−1 f 2 x − m f (2 x) 4 4



 

1 1 m−1 m f 2 x − f (2 x)

|4|m−1 4

2θ |2|(2−r)l+2

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.9) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping h : X → Y by 1 f (2n x) n→∞ 4n

h(x) := lim

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.9), we get (2.8). The rest of the proof is similar to the proof of Theorem 2.3.



Let A(x, y) := f (x + y) + f (x − y) − 2f (x) − 2f (y) and 

B(x, y) := ρ 2f



x+y 2





+ 2f

x−y 2





− f (x) − f (y)

for all x, y ∈ X. For x, y ∈ X with kA(x, y)k ≤ kB(x, y)k, kA(x, y)k − kB(x, y)k ≤ kA(x, y) − B(x, y)k. For x, y ∈ X with kA(x, y)k > kB(x, y)k, kA(x, y)k = ≤ = ≤

kA(x, y) − B(x, y) + B(x, y)k max{kA(x, y) − B(x, y)k, kB(x, y)k} kA(x, y) − B(x, y)k kA(x, y) − B(x, y)k + kB(x, y)k,

since kA(x, y)k > kB(x, y)k. So we have

     

x+y x−y

kf (x + y) + f (x − y) − 2f (x) − 2f (y)k − ρ 2f + 2f − f (x) − f (y)

2 2

     

x−y x+y ≤ + 2f − f (x) − f (y)

.

f (x + y) + f (x − y) − 2f (x) − 2f (y) − ρ 2f 2 2

As corollaries of Theorems 2.3 and 2.4, we obtain the Hyers-Ulam stability results for the quadratic ρ-functional equation (2.3) in non-Archimedean Banach spaces.

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Corollary 2.5. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that kf (x + y) + f (x − y) − 2f (x) − 2f (y)      

x−y x+y r r + 2f − f (x) − f (y) −ρ 2f

≤ θ(kxk + kyk ) 2 2

(2.10)

for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y satisfying (2.5). Corollary 2.6. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (2.10). Then there exists a unique quadratic mapping h : X → Y satisfying (2.8). 3. Quadratic ρ-functional inequality (0.2) Throughout this section, assume that ρ is a fixed non-Archimedean number with |ρ| < 12 . In this section, we solve the quadratic ρ-functional inequality (0.2) in non-Archimedean normed spaces. Lemma 3.1. A mapping f : X → Y satisfies

   

2f x + y + 2f x − y − f (x) − f (y) ≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k

2 2

(3.1)

for all x, y ∈ X if and only if f : X → Y is quadratic. Proof. Assume that f : X → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get k2f (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = 0 in (3.1), we get

 



4f x − f (x) ≤ 0

2

(3.2)

and so f x2 = 14 f (x) for all x ∈ X. It follows from (3.1) and (3.2) that


x+y x−y 1 kf (x + y) + f (x − y) − 2f (x) − 2f (y)k = 2f + 2f − f (x) − f (y)

2 2 2 ≤ |ρ|kf (x + y) + f (x − y) − 2f (x) − 2f (y)k









and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X. The converse is obviously true.



Corollary 3.2. A mapping f : X → Y satisfies 

2f

x+y 2





+ 2f

x−y 2



− f (x) − f (y) = ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y))

(3.3)

for all x, y ∈ X and only if f : X → Y is quadratic. The functional equation (3.3) is called a quadratic ρ-functional equation. We prove the Hyers-Ulam stability of the quadratic ρ-functional inequality (3.1) in non-Archimedean Banach spaces.

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES

Theorem 3.3. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that x−y x+y + 2f − f (x) − f (y)k 2 2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − 2f (y))k + θ(kxkr + kykr ) 







k2f

(3.4)

for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤ θkxkr

(3.5)

for all x ∈ X. Proof. Letting x = y = 0 in (3.4), we get k2f (0)k ≤ |ρ|k2f (0)k. So f (0) = 0. Letting y = 0 in (3.4), we get

 

4f x − f (x) ≤ θkxkr

2

(3.6)

for all x ∈ X. So

   

l x

4 f x − 4m f

2l 2m

         

l

m−1 x x x x l+1 m



≤ max 4 f , · · · , 4 f −4 f −4 f 2l 2l+1 2m−1 2m

 

       

x x x x l m−1

= max |4| f − 4f , · · · , |4|

f 2m−1 − 4f 2m 2l 2l+1 ( )

≤ max

|4|l |4|m−1 , · · · , |2|rl |2|r(m−1)

θkxkr =

θ

|2|(r−2)l

(3.7)

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.7) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping h : X → Y by h(x) := lim 4n f ( n→∞

x ) 2n

for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.7), we get (3.5). The rest of the proof is similar to the proof of Theorem 2.3.



Theorem 3.4. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (3.4). Then there exists a unique quadratic mapping h : X → Y such that kf (x) − h(x)k ≤

|2|r θ kxkr |4|

(3.8)

for all x ∈ X. Proof. It follows from (3.6) that

r

f (x) − 1 f (2x) ≤ |2| θ kxkr

4 |4|

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S. YUN, C. PARK

for all x ∈ X. Hence



1

f (2l x) − 1 f (2m x)

4l 4m

     

1  l 

1

1 1 l+1 m−1 m

x − m f (2 x) ≤ max l f 2 x − l+1 f 2 x , · · · , m−1 f 2 4 4 4 4

 



   1  

1 1 1 l m−1 l+1 m

= max f 2 x − f 2 x , · · · , m−1 f 2 x − f (2 x)

l |4| 4 |4| 4 (

≤ max

|2|rl |2|r(m−1) , · · · , |4|l+1 |4|(m−1)+1

)

|2|r θkxkr =

|2|r θ |2|(2−r)l+2

(3.9)

kxkr

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping h : X → Y by 1 h(x) := lim n f (2n x) n→∞ 4 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.9), we get (3.8). The rest of the proof is similar to the proofs of Theorems 2.3 and 3.3.  Let A(x, y) := 2f



x+y 2



+ 2f



x−y 2



− f (x) − f (y) and

B(x, y) := ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y)) for all x, y ∈ X. For x, y ∈ X with kA(x, y)k ≤ kB(x, y)k, kA(x, y)k − kB(x, y)k ≤ kA(x, y) − B(x, y)k. For x, y ∈ X with kA(x, y)k > kB(x, y)k, kA(x, y)k = ≤ = ≤

kA(x, y) − B(x, y) + B(x, y)k max{kA(x, y) − B(x, y)k, kB(x, y)k} kA(x, y) − B(x, y)k kA(x, y) − B(x, y)k + kB(x, y)k,

since kA(x, y)k > kB(x, y)k. So we have



  

2f x + y + 2f x − y − f (x) − f (y) − kρ (f (x + y) + f (x − y) − 2f (x) − 2f (y))k

2 2



  

x+y x−y

≤ 2f + 2f − f (x) − f (y) − ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y))

. 2 2 As corollaries of Theorems 3.3 and 3.4, we obtain the Hyers-Ulam stability results for the quadratic ρ-functional equation (3.3) in non-Archimedean Banach spaces. Corollary 3.5. Let r < 2 and θ be nonnegative real numbers, and let f : X → Y be a mapping such that

   

2f x + y + 2f x − y − f (x) − f (y) (3.10)

2 2 −ρ (f (x + y) + f (x − y) − 2f (x) − 2f (y))k ≤ θ(kxkr + kykr ) for all x, y ∈ X. Then there exists a unique quadratic mapping h : X → Y satisfying (3.5). Corollary 3.6. Let r > 2 and θ be positive real numbers, and let f : X → Y be a mapping satisfying (3.10). Then there exists a unique quadratic mapping h : X → Y satisfying (3.8).

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QUADRATIC ρ-FUNCTIONAL INEQUALITIES

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 4, 2016

On Structures of IVF Approximation Spaces, Ningxin Xie,………………………………621 Stability of Ternary m-Derivations on Ternary Banach Algebras, Madjid Eshaghi Gordji, Vahid Keshavarz, Jung Rye Lee, and Dong Yun Shin,……………………………………………640 On IF Approximating Spaces, Bin Qin, Fanping Zeng, and Kesong Yan,…………………645 On Cauchy Problems with Caputo Hadamard Fractional Derivatives, Y. Adjabi, F. Jarad, D. Baleanu, and T. Abdeljawad,……………………………………………………………….661 Multivalued Generalized Contractive Maps and Fixed Point Results, Marwan A. Kutbi,…682 Essential Commutativity and Isometry of Composition Operator and Differentiation Operator, Geng-Lei Li,…………………………………………………………………………………696 Approximation of Jensen Type Quadratic-Additive Mappings via the Fixed Point Theory, Yang-Hi Lee, John Michael Rassias, and Hark-Mahn Kim,……………………………….704 On Certain Subclasses of p-Valent Analytic Functions Involving Saitoh Operator, J. Patel, and N.E. Cho,……………………………………………………………………………………716 Generalized φ -Weak Contractive Fuzzy Mappings and Related Fixed Point Results on Complete Metric Space, Afshan Batool, Tayyab Kamran, Sun Young Jang, and Choonkil Park,…….729 On Carlitz’s Degenerate Euler Numbers and Polynomials, Dae San Kim, Taekyun Kim, and Dmitry V. Dolgy,……………………………………………………………………………738 Dynamics and Behavior of the Higher Order Rational Difference Equation, M. El-Dessoky,743 A Quadratic Functional Equation In Intuitionistic Fuzzy 2-Banach Spaces, Ehsan Movahednia, Madjid Eshaghi Gordji, Choonkil Park, and Dong Yun Shin,………………………………761 On A q-Analogue of (h,q)-Daehee Numbers and Polynomials of Higher.., Jin-Woo Park,…769 On Blow-Up of Solutions For A Semilinear Damped Wave Equation with Nonlinear Dynamic Boundary Conditions, Gang Li, Biqing Zhu, and Danhua Wang,…………………………...777 Uniqueness of Meromorphic Functions with Their Difference Operators, Xiaoguang Qi, Yong Liu, and Lianzhong Yang,……………………………………………………………………784

TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 4, 2016 (continued) Quadratic ρ-Functional Inequalities in Non-Archimedean Normed Spaces, Sungsik Yun, and Choonkil Park,………………………………………………………………………………791

Volume 21, Number 5 ISSN:1521-1398 PRINT,1572-9206 ONLINE

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Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Mixed problems of fractional coupled systems of Riemann-Liouville differential equations and Hadamard integral conditions S.K. Ntouyas1,2 Jessada Tariboon3 and Phollakrit Thiramanus3 1

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece e-mail: [email protected] 2 Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3

Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800 Thailand e-mail: [email protected], [email protected] Abstract IIn this paper we study existence and uniqueness of solutions for mixed problems consisting nonlocal Hadamard fractional integrals for coupled systems of Riemann-Liouville fractional differential equations. The existence and uniqueness of solutions is established by using the Banach’s contraction principle, while the existence of solutions is derived by applying Leray-Schauder’s alternative. Examples illustrating our results are also presented.

Key words and phrases: Riemann-Liouville fractional derivative; Hadamard fractional integral; coupled system; existence; uniqueness; fixed point theorems. AMS (MOS) Subject Classifications: 34A08; 34A12; 34B15.

1

Introduction

The aim of this paper is to investigate the existence and uniqueness of solutions for nonlocal Hadamard fractional integrals for a coupled system of Riemann-Liouville fractional differential equations of the form:  p t ∈ [0, T ], 1 < p ≤ 2,  RL D x(t) = f (t, x(t), y(t)),   q  D y(t) = g(t, x(t), y(t)), t ∈ [0, T ], 1 < q ≤ 2,  RL   m1 n1  ∑ ∑  x(0) = 0, µiH I αi x(ηi ) = δj H I βj y(ξj ) + λ1 , (1)  i=1 j=1   n m 2 2  ∑ ∑    ωlH I νl y(θl ) + λ2 , τkH I σk x(γk ) =  y(0) = 0,  l=1

k=1

where RL Dq , RL Dp are the standard Riemann-Liouville fractional derivative of orders q, p, two continuous functions f, g : [0, T ] × R2 → R, H I αi , H I βj , H I σk and H I νl are the Hadamard fractional integral of orders αi , βj , σk , νl > 0, λ1 , λ2 ∈ R are given constants, ηi , ξj , γk , θl ∈ (0, T ), and µi , δj , τk , ωl ∈ R, for m1 , m2 , n1 , n2 ∈ N, i = 1, 2, . . . , m1 , j = 1, 2, . . . , n1 , k = 1, 2, . . . , m2 , l = 1, 2, . . . , n2 are real constants such that ( (m )( n ) n ) q−1 m2 q−1 p−1 1 2 1 p−1 ∑ ∑ ∑ ∑ δ ξ ωl θ l τ γ µi ηi j j k k  6=  . (p − 1)αi (q − 1)νl (q − 1)βj (p − 1)σk i=1 j=1 l=1

k=1

Fractional calculus has a long history with more than three hundred years. Up to now, it has been proved that fractional calculus is very useful. Many mathematical models of real problems arising

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S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS

in various fields of science and engineering were established with the help of fractional calculus, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, and electromagnetic waves. For examples and recent development of the topic, see ([1, 2, 3, 4, 5, 6, 7, 14, 16, 17, 18, 19, 20, 21]). However, it has been observed that most of the work on the topic involves either RiemannLiouville or Caputo type fractional derivative. Besides these derivatives, Hadamard fractional derivative is another kind of fractional derivatives that was introduced by Hadamard in 1892 [12]. This fractional derivative differs from the other ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains logarithmic function of arbitrary exponent. For background material of Hadamard fractional derivative and integral, we refer to the papers [8, 9, 10, 13, 14, 15]. The paper is organized as follows: In Section 2 we will present some useful preliminaries and lemmas. The main results are given in Section 3, where existence and uniqueness results are obtained by using Banach’s contraction principle and Leray-Schauder’s alternative. Finally the uncoupled integral boundary conditions case is studied in Section 4. Examples illustrating our results are also presented.

2

Preliminaries

In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later [18, 14]. Definition 2.1 The Riemann-Liouville fractional derivative of order q > 0 of a continuous function f : (0, ∞) → R is defined by ( )n ∫ t 1 d q (t − s)n−q−1 f (s)ds, n − 1 < q < n, RL D f (t) = Γ(n − q) dt 0 where n = [q]+1, [q] denotes the integer part of a real number q, provided ∫ ∞the right-hand side is point-wise defined on (0, ∞), where Γ is the gamma function defined by Γ(q) = 0 e−s sq−1 ds. Definition 2.2 The Riemann-Liouville fractional integral of order q > 0 of a continuous function f : (0, ∞) → R is defined by q RL I f (t) =

1 Γ(q)

∫

t

(t − s)q−1 f (s)ds, 0

provided the right-hand side is point-wise defined on (0, ∞). Definition 2.3 The Hadamard derivative of fractional order q for a function f : (0, ∞) → R is defined as ( )n ∫ t ( )n−q−1 d t f (s) 1 q t log ds, n − 1 < q < n, n = [q] + 1, D f (t) = H Γ(n − q) dt s s 0 where log(·) = loge (·). Definition 2.4 The Hadamard fractional integral of order q ∈ R+ of a function f (t), for all t > 0, is defined as q H I f (t) =

1 Γ(q)

)q−1 ∫ t( t ds log f (s) , s s 0

provided the integral exists. Lemma 2.5 ([14], page 113) Let q > 0 and β > 0. Then the following formulas HI

t = β −q tβ

q β

and

HD

q β

t = β q tβ

hold.

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MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS Lemma 2.6 Let q > 0 and x ∈ C(0, T ) ∩ L(0, T ). Then the fractional differential equation RL Dq x(t) = 0 has a unique solution x(t) = c1 tq−1 + c2 tq−2 + . . . + cn tq−n , where ci ∈ R, i = 1, 2, . . . , n, and n − 1 < q < n. Lemma 2.7 Let q > 0. Then for x ∈ C(0, T ) ∩ L(0, T ) it holds q q RL I RL D x(t)

= x(t) + c1 tq−1 + c2 tq−2 + . . . + cn tq−n ,

where ci ∈ R, i = 1, 2, . . . , n, and n − 1 < q < n. Lemma 2.8 Given φ, ψ ∈ C([0, T ], R), the unique solution of the problem  p t ∈ [0, T ], 1 < p ≤ 2,  RL D x(t) = φ(t),   q  D y(t) = ψ(t), t ∈ [0, T ], 1 < q ≤ 2,  RL   m1 n1  ∑ ∑  x(0) = 0, µiH I αi x(ηi ) = δj H I βj y(ξj ) + λ1 ,  i=1 j=1   m2 n2  ∑ ∑   σk  y(0) = 0, τ I x(γ ) = ωlH I νl y(θl ) + λ2 ,  kH k  k=1

is

(2)

l=1

{ n ( n ) m1 2 1 ∑ ∑ ωl θlq−1 tp−1 ∑ βj q αi p δj H I RL I ψ(ξj ) − µiH I RL I φ(ηi ) + λ1 x(t) = RL I φ(t) + Ω (q − 1)νl j=1 i=1 l=1 ( )} n2 m2 n1 ∑ ∑ ∑ δj ξjq−1 ωlH I νl RL I q ψ(θl ) − τkH I σk RL I p φ(γk ) + λ2 , − βj (q − 1) j=1 p

l=1

and y(t)

k=1

{m ( n ) m1 p−1 1 2 q−1 ∑ ∑ ∑ τ γ t k k = RL I q ψ(t) + δj H I βj RL I q ψ(ξj ) − µiH I αi RL I p φ(ηi ) + λ1 Ω (p − 1)σk j=1 i=1 ( n k=1 )} m2 m1 2 ∑ ∑ ∑ µi ηip−1 ωlH I νl RL I q ψ(θl ) − τkH I σk RL I p φ(γk ) + λ2 , − αi (p − 1) i=1 l=1

where Ω=

(3)

(4)

k=1

m2 m1 n2 n1 ∑ ∑ δj ξjq−1 ∑ ωl θlq−1 τk γkp−1 µi ηip−1 ∑ − 6= 0. (p − 1)αi (q − 1)νl j=1 (q − 1)βj (p − 1)σk i=1

(5)

k=1

l=1

Proof. Using Lemmas 2.6-2.7, the equations in (2) can be expressed as equivalent integral equations x(t) = RL I p φ(t) + c1 tp−1 + c2 tp−2 ,

(6)

y(t) = RL I q ψ(t) + d1 tq−1 + d2 tq−2 ,

(7)

for c1 , c2 , d1 , d2 ∈ R. The conditions x(0) = 0, y(0) = 0 imply that c2 = 0, d2 = 0. Taking the Hadamard fractional integral of order αi > 0, σk > 0 for (6) and βj > 0, νl > 0 for (7) and using the property of the Hadamard fractional integral given in Lemma 2.5 we get the system m1 ∑ i=1 m2 ∑ k=1

µiH I αi RL I p φ(ηi ) + c1

n1 n1 m1 ∑ ∑ ∑ δj ξjq−1 µi ηip−1 βj q = δ I I ψ(ξ ) + λ1 , + d jH RL j 1 (p − 1)αi (q − 1)βj j=1 j=1 i=1

τkH I σk RL I p φ(γk ) + c1

m2 n2 n2 ∑ ∑ ∑ τk γkp−1 ωl θlq−1 νl q = ω I I ψ(θ ) + d + λ2 , lH RL l 1 (p − 1)σk (q − 1)νl

k=1

l=1

815

l=1

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S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS

from which we have c1

( n ) n2 m1 1 ∑ ∑ ∑ ωl θlq−1 αi p βj q = µiH I RL I φ(ηi ) + λ1 δj H I RL I ψ(ξj ) − (q − 1)νl j=1 i=1 l=1 ( n )} n1 m2 2 ∑ ∑ ∑ δj ξjq−1 q νl σk p − τkH I RL I φ(γk ) + λ2 ωlH I RL I ψ(θl ) − (q − 1)βj j=1 1 Ω

{

k=1

l=1

and d1

=

( n ) m2 m1 1 ∑ ∑ ∑ τk γkp−1 αi p βj q δj H I RL I ψ(ξj ) − µiH I RL I φ(ηi ) + λ1 (p − 1)σk j=1 i=1 k=1 ( n )} m2 m1 2 ∑ ∑ ∑ µi ηip−1 q νl σk p τkH I RL I φ(γk ) + λ2 ωlH I RL I ψ(θl ) − − . (p − 1)αi i=1 1 Ω

{

k=1

l=1

Substituting the values of c1 , c2 , d1 and d2 in (6) and (7), we obtain the solutions (3) and (4).

3

¤

Main Results

Throughout this paper, for convenience, we use the following expressions ∫ v 1 w I h(s, x(s), y(s))(v) = (v − s)w−1 h(s, x(s), y(s))ds, RL Γ(w) 0 and 1 H I RL I h(s, x(s), y(s))(v) = Γ(u)Γ(w) u

∫

w

v

∫ t( log

0

0

h(s, x(s), y(s)) v )u−1 (t − s)w−1 dsdt, t t

where u ∈ {ρi , γj }, v ∈ {t, T, ηi , θj }, w = {p, q} and h = {f, g}, i = 1, 2, . . . , n, j = 1, 2, . . . , m. Let C = C([0, T ], R) denotes the Banach space of all continuous functions from [0, T ] to R. Let us introduce the space X = {x(t)|x(t) ∈ C([0, T ])} endowed with the norm kxk = max{|x(t)|, t ∈ [0, T ]}. Obviously (X, k · k) is a Banach space. Also let Y = {y(t)|y(t) ∈ C([0, T ])} be endowed with the norm kyk = max{|y(t)|, t ∈ [0, T ]}. Obviously the product space (X × Y, k(x, y)k) is a Banach space with norm k(x, y)k = kxk + kyk. ( ) T1 (x, y)(t) In view of Lemma 2.8, we define an operator T : X × Y → X × Y by T (x, y)(t) = , T2 (x, y)(t) where T1 (x, y)(t)

=

tp−1 RL I f (s, x(s), y(s))(t) + Ω

{

p

−

m1 ∑

µiH I

αi

RL I

p

n2 ∑ ωl θlq−1 (q − 1)νl l=1 )

f (s, x(s), y(s))(ηi ) + λ1

i=1

−

m2 ∑

(

τk H I

RL I

p

δj H I βj RL I q g(s, x(s), y(s))(ξj )

j=1

− )}

σk

n1 ∑

n1 ∑ δj ξjq−1 j=1

f (s, x(s), y(s))(γk ) + λ2

,

{

(

(

(q − 1)βj

n2 ∑

ωlH I νl RL I q g(s, x(s), y(s))(θl )

l=1

k=1

and T2 (x, y)(t)

=

tq−1 RL I g(s, x(s), y(s))(t) + Ω q

−

m1 ∑

µiH I

αi

RL I

p

m2 ∑ τk γkp−1 (p − 1)σk k=1 )

f (s, x(s), y(s))(ηi ) + λ1

i=1

816

n1 ∑

δj H I βj RL I q g(s, x(s), y(s))(ξj )

j=1

m1 ∑ µi ηip−1 − (p − 1)αi i=1

(

n2 ∑

ωlH I νl RL I q g(s, x(s), y(s))(θl )

l=1

Ntouyas et al 813-828

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS

−

m2 ∑

)} τk H I

σk

RL I

p

f (s, x(s), y(s))(γk ) + λ2

.

k=1

Let us introduce the following assumptions which are used hereafter. (H1 ) Assume that f, g : [0, T ] × R2 → R are continuous functions and there exist constants mi , ni , i = 1, 2 such that for all t ∈ [0, T ] and ui , vi ∈ R, i = 1, 2, |f (t, u1 , u2 ) − f (t, v1 , v2 )| ≤ K1 |u1 − v1 | + K2 |u2 − v2 | and |g(t, u1 , u2 ) − g(t, v1 , v2 )| ≤ L1 |u1 − v1 | + L2 |u2 − v2 |. (H2 ) Assume that there exist real constants ki , li ≥ 0 (i = 1, 2) and k0 > 0, l0 > 0 such that ∀xi ∈ R, (i = 1, 2) we have |f (t, x1 , x2 )| ≤ k0 + k1 |x1 | + k2 |x2 |,

|g(t, x1 , x2 )| ≤ l0 + l1 |x1 | + l2 |x2 |.

For the sake of convenience, we set ( ) m2 n2 m1 n1 |δj |ξjq−1 ∑ |τk |γkp |ωl |θlq−1 ∑ T p−1 ∑ |µi |ηip T p−1 ∑ 1 p T + + , M1 = Γ(p + 1) |Ω| (q − 1)νl i=1 pαi |Ω| j=1 (q − 1)βj pσk k=1 l=1 ( n ) n1 n2 n1 2 ∑ |δj |ξjq ∑ |δj |ξjq−1 ∑ |ωl |θlq |ωl |θlq−1 ∑ T p−1 + M2 = , |Ω|Γ(q + 1) (q − 1)νl j=1 q βj (q − 1)βj q νl j=1 l=1 l=1 ( ) n2 n1 ∑ ∑ |δj |ξjq−1 |ωl |θlq−1 T p−1 M3 = |λ1 | + |λ2 | , |Ω| (q − 1)νl (q − 1)βj j=1 l=1 ( ) m2 n1 m1 n2 |δj |ξjq |τk |γkp−1 ∑ |ωl |θlq T q−1 ∑ T q−1 ∑ 1 |µi |ηip−1 ∑ q + T + , M4 = Γ(q + 1) |Ω| (p − 1)σk j=1 q βj |Ω| i=1 (p − 1)αi q νl k=1 l=1 (m ) m1 m1 m2 2 ∑ |τk |γkp−1 ∑ |τk |γkp T q−1 |µi |ηip ∑ |µi |ηip−1 ∑ M5 = + , |Ω|Γ(p + 1) (p − 1)σk i=1 pαi (p − 1)αi pσk i=1 k=1 k=1 ( ) m2 m1 ∑ ∑ |τk |γkp−1 T q−1 |µi |ηip−1 M6 = |λ1 | + |λ2 | , |Ω| (p − 1)σk (p − 1)αi i=1

(8)

(9)

(10)

(11)

(12)

(13)

k=1

and M0 = min{1 − (M1 + M5 )k1 − (M2 + M4 )l1 , 1 − (M1 + M5 )k2 − (M2 + M4 )l2 },

(14)

ki , li ≥ 0 (i = 1, 2). The first result is concerned with the existence and uniqueness of solutions for the problem (1) and is based on Banach’s contraction mapping principle. Theorem 3.1 Assume that (H1 ) holds. In addition, suppose that (M1 + M5 )(K1 + K2 ) + (M2 + M4 )(L1 + L2 ) < 1, where Mi , i = 1, 2, 4, 5 are given by (3.1)-(3.2) and (3.4)-(3.5). Then the boundary value problem (1) has a unique solution. Proof. Define supt∈[0,T ] f (t, 0, 0) = N1 < ∞ and supt∈[0,T ] g(t, 0, 0) = N2 < ∞ such that { r ≥ max

} M1 N1 + M2 N2 + M3 M4 N2 + M5 N1 + M6 , , 1 − (M1 K1 + M2 L1 + M1 K2 + M2 L2 ) 1 − (M4 L1 + M5 K1 + M4 L2 + M5 K2 )

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S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS

where M3 and M6 are defined by (3.3) and (3.6), respectively. We show that T Br ⊂ Br , where Br = {(x, y) ∈ X × Y : k(x, y)k ≤ r}.

For (x, y) ∈ Br , we have { |T1 (x, y)(t)|

= max t∈[0,T ]

( ×

n1 ∑

[ n 2 ωl θlq−1 tp−1 ∑ RL I f (s, x(s), y(s))(t) + Ω (q − 1)νl p

l=1

δj H I βj RL I q g(s, x(s), y(s))(ξj ) −

j=1 n1 ∑ δj ξjq−1 − (q − 1)βj j=1

−

m2 ∑

τkH I

σk

(

m1 ∑

µiH I αi RL I p f (s, x(s), y(s))(ηi ) + λ1

i=1 n2 ∑

ωlH I νl RL I q g(s, x(s), y(s))(θl )

l=1

RL I

)

p

)]}

f (s, x(s), y(s))(γk ) + λ2

k=1

[ n 2 |ωl |θlq−1 T p−1 ∑ ≤ RL I (|f (s, x(s), y(s)) − f (s, 0, 0)| + |f (s, 0, 0)|)(T ) + |Ω| (q − 1)νl l=1 ( n 1 ∑ × |δj |H I βj RL I q (|g(s, x(s), y(s)) − g(s, 0, 0)| + |g(s, 0, 0)|)(ξj ) p

j=1

+

+

m1 ∑ i=1 n1 ∑ j=1

+

m2 ∑

|µi |H I

) αi

RL I

|δj |ξjq−1 (q − 1)βj |τk |H I

σk

(

p

(|f (s, x(s), y(s)) − f (s, 0, 0)| + |f (s, 0, 0)|)(ηi ) + |λ1 |

n2 ∑

|ωl |H I νl RL I q (|g(s, x(s), y(s)) − g(s, 0, 0)| + |g(s, 0, 0)|)(θl )

l=1

RL I

p

)]

(|f (s, x(s), y(s)) − f (s, 0, 0)| + |f (s, 0, 0)|)(γk ) + |λ2 |

k=1

[ n 2 |ωl |θlq−1 T p−1 ∑ ≤ RL I (K1 kxk + K2 kyk + N1 )(T ) + |Ω| (q − 1)νl l=1 ( n 1 ∑ × |δj |H I βj RL I q (L1 kxk + L2 kyk + N2 )(ξj ) p

j=1

+

+

m1 ∑ i=1 n1 ∑ j=1

+

m2 ∑

|µi |H I

) αi

RL I

|δj |ξjq−1 (q − 1)βj |τk |H I

σk

(

p

(K1 kxk + K2 kyk + N1 )(ηi ) + |λ1 |

n2 ∑ l=1

RL I

p

)]

(K1 kxk + K2 kyk + N1 )(γk ) + |λ2 |

k=1

=

|ωl |H I νl RL I q (L1 kxk + L2 kyk + N2 )(θl )

{

n2 m1 |ωl |θlq−1 ∑ T p−1 ∑ |µi |H I αi RL I p (1)(ηi ) |Ω| (q − 1)νl i=1 l=1 } { m2 ∑ T p−1 |τk |H I σk RL I p (1)(γk ) + (L1 kxk + L2 kyk + N2 ) |Ω|

(K1 kxk + K2 kyk + N1 ) n1 |δj |ξjq−1 T p−1 ∑ + |Ω| j=1 (q − 1)βj

RL I

p

(1)(T ) +

k=1

} n2 n1 n1 n2 ∑ |δj |ξjq−1 ∑ |ωl |θlq−1 ∑ T p−1 ∑ βj q νl q × |δj |H I RL I (1)(ξj ) + |ωl |H I RL I (1)(θl ) (q − 1)νl j=1 |Ω| j=1 (q − 1)βj l=1

l=1

818

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MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS n2 n1 |δj |ξjq−1 |ωl |θlq−1 T p−1 ∑ T p−1 ∑ + |λ | 2 |Ω| (q − 1)νl |Ω| j=1 (q − 1)βj l=1 { m1 n2 ∑ |ωl |θlq−1 ∑ Tp T p−1 |µi |ηip = (K1 kxk + K2 kyk + N1 ) + Γ(p + 1) |Ω|Γ(p + 1) (q − 1)νl i=1 pαi l=1 } { n1 m2 ∑ |δj |ξjq−1 ∑ |τk |γkp T p−1 T p−1 + (L kxk + L kyk + N ) + 1 2 2 β σ |Ω|Γ(p + 1) j=1 (q − 1) j p k |Ω|Γ(q + 1) k=1 } n1 n1 n2 n2 ∑ ∑ |δj |ξjq |δj |ξjq−1 ∑ |ωl |θlq |ωl |θlq−1 ∑ T p−1 × + (q − 1)νl j=1 q βj |Ω|Γ(q + 1) j=1 (q − 1)βj q νl

+|λ1 |

l=1

l=1

+|λ1 |

n2 p−1 ∑

T |Ω|

l=1

|ωl |θlq−1 (q − 1)νl

+ |λ2 |

n1 p−1 ∑

T |Ω|

j=1

|δj |ξjq−1 (q − 1)βj

= (K1 kxk + K2 kyk + N1 )M1 + (L1 kxk + L2 kyk + N2 )M2 + M3 = (M1 K1 + M2 L1 )kxk + (M1 K2 + M2 L2 )kyk + M1 N1 + M2 N2 + M3 ≤ (M1 K1 + M2 L1 + M1 K2 + M2 L2 )r + M1 N1 + M2 N2 + M3 ≤ r. In the same way, we can obtain that

{

m2 n1 ∑ |δj |ξjq |τk |γkp−1 ∑ T q−1 Tq + Γ(q + 1) |Ω|Γ(q + 1) (p − 1)σk j=1 q βj k=1 } { m1 n2 ∑ |ωl |θlq T q−1 |µi |ηip−1 ∑ T q−1 + + (K kxk + K kyk + N ) 1 2 1 |Ω|Γ(q + 1) i=1 (p − 1)αi q νl |Ω|Γ(p + 1) l=1 } m2 m1 m1 m2 ∑ ∑ |τk |γkp−1 ∑ |τk |γkp |µi |ηip T q−1 |µi |ηip−1 ∑ × + (p − 1)σk i=1 pαi |Ω|Γ(p + 1) i=1 (p − 1)αi pσk

|T2 (x, y)(t)| ≤

(L1 kxk + L2 kyk + N2 )

k=1

+|λ1 |

k=1

m2 q−1 ∑

T |Ω|

k=1

|τk |γkp−1 (p − 1)σk

+ |λ2 |

m1 q−1 ∑

T |Ω|

i=1

|µi |ηip−1 (p − 1)αi

= (L1 kxk + L2 kyk + N2 )M4 + (K1 kxk + K2 kyk + N1 )M5 + M6 = (M4 L1 + M5 K1 )kxk + (M4 L2 + M5 K2 )kyk + M4 N2 + M5 N1 + M6 ≤ (M4 L1 + M5 K1 + M4 L2 + M5 K2 )r + M4 N2 + M5 N1 + M6 ≤ r. Consequently, kT (x, y)(t)k ≤ r. Now for (x2 , y2 ), (x1 , y1 ) ∈ X × Y, and for any t ∈ [0, T ], we get |T1 (x2 , y2 )(t) − T1 (x1 , y1 )(t)|

[ n 2 |ωl |θlq−1 T p−1 ∑ ≤ RL I |f (s, x2 (s), y2 (s)) − f (s, x1 (s), y1 (s))|(T ) + |Ω| (q − 1)νl l=1 ( n 1 ∑ × |δj |H I βj RL I q (|g(s, x2 (s), y2 (s)) − g(s, x1 (s), y1 (s))|)(ξj ) p

j=1

+

+

m1 ∑ i=1 n1 ∑ j=1

+

m2 ∑

|µi |H I

) αi

RL I

|δj |ξjq−1 (q − 1)βj

(

p

(|f (s, x2 (s), y2 (s)) − f (s, x1 (s), y1 (s))|)(ηi )

n2 ∑

|ωl |H I νl RL I q (|g(s, x2 (s), y2 (s)) − g(s, x1 (s), y1 (s))|)(θl )

l=1

)]

|τk |H I σk RL I p (|f (s, x2 (s), y2 (s)) − f (s, x1 (s), y1 (s))|)(γk )

k=1

819

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S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS {

m1 n2 ∑ |ωl |θlq−1 ∑ T p−1 |µi |ηip Tp + Γ(p + 1) |Ω|Γ(p + 1) (q − 1)νl i=1 pαi l=1 } n1 m2 ∑ |δj |ξjq−1 ∑ |τk |γkp T p−1 + + (L1 kx2 − x1 k + L2 ky2 − y1 k) |Ω|Γ(p + 1) j=1 (q − 1)βj pσk k=1 { } n1 n2 n1 n2 ∑ ∑ |δj |ξjq |δj |ξjq−1 ∑ |ωl |θlq−1 ∑ |ωl |θlq T p−1 T p−1 × + |Ω|Γ(q + 1) (q − 1)νl j=1 q βj |Ω|Γ(q + 1) j=1 (q − 1)βj q νl

≤ (K1 kx2 − x1 k + K2 ky2 − y1 k)

l=1

l=1

= (K1 kx2 − x1 k + K2 ky2 − y1 k)M1 + (L1 kx2 − x1 k + L2 ky2 − y1 k)M2 = (M1 K1 + M2 L1 )kx2 − x1 k + (M1 K2 + M2 L2 )ky2 − y1 k, and consequently we obtain kT1 (x2 , y2 )(t) − T1 (x1 , y1 )k ≤ (M1 K1 + M2 L1 + M1 K2 + M2 L2 )[kx2 − x1 k + ky2 − y1 k].

(15)

kT2 (x2 , y2 )(t) − T2 (x1 , y1 )k ≤ (M4 L1 + M5 K1 + M4 L2 + M5 K2 )[kx2 − x1 k + ky2 − y1 k].

(16)

Similarly,

It follows from (15) and (16) that kT (x2 , y2 )(t) − T (x1 , y1 )(t)k ≤ [(M1 + M5 )(K1 + K2 ) + (M2 + M4 )(L1 + L2 )](kx2 − x1 k + ky2 − y1 k).

Since (M1 + M5 )(K1 + K2 ) + (M2 + M4 )(L1 + L2 ) < 1, therefore, T is a contraction operator. So, By Banach’s fixed point theorem, the operator T has a unique fixed point, which is the unique solution of problem (1). This completes the proof. ¤ In the next result, we prove the existence of solutions for the problem (1) by applying Leray-Schauder alternative. Lemma 3.2 (Leray-Schauder alternative) ([11], page.4.) Let F : E → E be a completely continuous operator (i.e., a map that restricted to any bounded set in E is compact). Let E(F ) = {x ∈ E : x = λF (x) for some 0 < λ < 1}. Then either the set E(F ) is unbounded, or F has at least one fixed point. Theorem 3.3 Assume that (H2 ) holds. In addition it is assumed that (M1 + M5 )k1 + (M2 + M4 )l1 < 1 and (M1 + M5 )k2 + (M2 + M4 )l2 < 1, where M1 , M2 , M4 , M5 are given by (3.1)-(3.2) and (3.4)-(3.5). Then there exists at least one solution for the boundary value problem (1). Proof. First we show that the operator T : X × Y → X × Y is completely continuous. By continuity of functions f and g, the operator T is continuous. Let Θ ⊂ X × Y be bounded. Then there exist positive constants P1 and P2 such that |f (t, x(t), y(t))| ≤ P1 ,

|g(t, x(t), y(t))| ≤ P2 , ∀(x, y) ∈ Θ.

Then for any (x, y) ∈ Θ, we have kT1 (x, y)k

≤

[ n ( n 2 1 |ωl |θlq−1 ∑ T p−1 ∑ |δj |H I βj RL I q |g(s, x(s), y(s))|(ξj ) RL I |f (s, x(s), y(s))|(T ) + |Ω| (q − 1)νl j=1 l=1 ) m1 ∑ αi p + |µi |H I RL I |f (s, x(s), y(s))|(ηi ) + |λ1 | p

i=1

820

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS n1 ∑ |δj |ξjq−1 + (q − 1)βj j=1

+ (

m2 ∑

|τk |H I

σk

(

n2 ∑

|ωl |H I νl RL I q |g(s, x(s), y(s))|(θl )

l=1

RL I

p

)]

|f (s, x(s), y(s))|(γk ) + |λ2 |

k=1

1 2 ∑ |ωl |θlq−1 ∑ T p−1 |µi |ηip Tp + Γ(p + 1) |Ω|Γ(p + 1) (q − 1)νl i=1 pαi l=1 ) ( n1 m2 ∑ |δj |ξjq−1 ∑ |τk |γkp T p−1 T p−1 P1 + + β σ j k |Ω|Γ(p + 1) j=1 (q − 1) p |Ω|Γ(q + 1)

≤

m

n

k=1

×

n1 |ωl |θlq−1 ∑ (q − 1)νl j=1 l=1

n2 ∑

+|λ1 | =

n2 p−1 ∑

T |Ω|

l=1

|δj |ξjq q βj

|ωl |θlq−1 (q − 1)νl

) n1 n2 ∑ |δj |ξjq−1 ∑ |ωl |θlq T p−1 P2 + |Ω|Γ(q + 1) j=1 (q − 1)βj q νl l=1

+ |λ2 |

n1 p−1 ∑

T |Ω|

j=1

|δj |ξjq−1 (q − 1)βj

M1 P1 + M2 P2 + M3 .

Similarly, we get kT2 (x, y)k

(

2 1 ∑ |δj |ξj |τk |γkp−1 ∑ Tq T q−1 + Γ(q + 1) |Ω|Γ(q + 1) (p − 1)σk j=1 q βj k=1 ) ( m1 n2 p−1 ∑ q−1 ∑ |ωl |θlq T |µi |ηi T q−1 + P + 2 α ν |Ω|Γ(q + 1) i=1 (p − 1) i q l |Ω|Γ(p + 1)

≤

m

n

q

l=1 m1 m1 m2 p−1 ∑ p ∑ |τk |γk |µi |ηi T q−1 |µi |ηip−1 ∑ × + (p − 1)σk i=1 pαi |Ω|Γ(p + 1) i=1 (p − 1)αi k=1 k=1 m2 m1 p−1 p−1 q−1 ∑ q−1 ∑ |τk |γk T T |µi |ηi +|λ1 | + |λ2 | |Ω| (p − 1)σk |Ω| i=1 (p − 1)αi k=1 m2 ∑

) |τk |γkp P1 pσk

= M4 P2 + M5 P1 + M6 . Thus, it follows from the above inequalities that the operator T is uniformly bounded. Next, we show that T is equicontinuous. Let t1 , t2 ∈ [0, T ] with t1 < t2 . Then we have |T1 (x(t2 ), y(t2 )) − T1 (x(t1 ), y(t1 ))| ∫ t1 1 [(t2 − s)p−1 − (t1 − s)p−1 ]|f (s, x(s), y(s))|ds ≤ Γ(p) 0 [ n ∫ t2 2 ∑ |ωl |θlq−1 1 tp−1 − tp−1 p−1 2 1 (t2 − s) |f (s, x(s), y(s))|ds + + Γ(p) t1 |Ω| (q − 1)νl l=1 ( n ) m1 1 ∑ ∑ × |δj |H I βj RL I q |g(s, x(s), y(s))|(ξj ) + |µi |H I αi RL I p |f (s, x(s), y(s))|(ηi ) + |λ1 | j=1 n1 ∑ |δj |ξjq−1 + (q − 1)βj j=1

+

m2 ∑

|τk |H I

σk

(

i=1 n2 ∑

|ωl |H I νl RL I q |g(s, x(s), y(s))|(θl )

l=1

RL I

p

)]

|f (s, x(s), y(s))|(γk ) + |λ2 |

k=1

≤

P1 Γ(p)

∫

t1

[(t2 − s)p−1 − (t1 − s)p−1 ]ds +

0

821

P1 Γ(p)

∫

t2

(t2 − s)p−1 ds

t1

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS [ n ( ) n1 m1 2 ∑ ∑ ∑ |δj |ξjq |ωl |θlq−1 tp−1 − tp−1 |µi |ηip 2 1 P2 ) + P1 + |λ1 | + |Ω| (q − 1)νl q βj Γ(q + 1) pαi Γ(p + 1) j=1 i=1 l=1 ( )] n2 m2 n1 ∑ ∑ ∑ |δj |ξjq−1 |ωl |θlq |τk |γkp P2 + P1 + |λ2 | . + (q − 1)βj q νl Γ(q + 1) pσk Γ(p + 1) j=1 l=1

k=1

Analogously, we can obtain |T2 (x(t2 ), y(t2 )) − T2 (x(t1 ), y(t1 ))| ∫ t1 ∫ t2 P2 P2 [(t2 − s)q−1 − (t1 − s)q−1 ]ds + (t2 − s)q−1 ds ≤ Γ(q) 0 Γ(q) t1 [m ( ) n1 m1 2 ∑ ∑ ∑ |δj |ξjq |τk |γkp−1 tq−1 − tq−1 |µi |ηip 2 1 + P2 ) + P1 + |λ1 | |Ω| (p − 1)σk q βj Γ(q + 1) pαi Γ(p + 1) j=1 i=1 k=1 ( )] n2 m2 m1 ∑ ∑ ∑ |ωl |θlq |τk |γkp |µi |ηip−1 P2 + P1 + |λ2 | . + (p − 1)αi q νl Γ(q + 1) pσk Γ(p + 1) i=1 l=1

k=1

Therefore, the operator T (x, y) is equicontinuous, and thus the operator T (x, y) is completely continuous. Finally, it will be verified that the set E = {(x, y) ∈ X × Y |(x, y) = λT (x, y), 0 ≤ λ ≤ 1} is bounded. Let (x, y) ∈ E, then (x, y) = λT (x, y). For any t ∈ [0, T ], we have x(t) = λT1 (x, y)(t), Then

y(t) = λT2 (x, y)(t).

(

|x(t)|

2 1 ∑ |ωl |θlq−1 ∑ Tp T p−1 |µi |ηip + Γ(p + 1) |Ω|Γ(p + 1) (q − 1)νl i=1 pαi l=1 ) m2 n1 ∑ |δj |ξjq−1 ∑ |τk |γkp T p−1 + + (l0 + l1 kxk + l2 kyk) |Ω|Γ(p + 1) j=1 (q − 1)βj pσk k=1 ( ) n2 n2 n1 n1 ∑ ∑ |δj |ξjq |δj |ξjq−1 ∑ |ωl |θlq−1 ∑ |ωl |θlq T p−1 T p−1 + × |Ω|Γ(q + 1) (q − 1)νl j=1 q βj |Ω|Γ(q + 1) j=1 (q − 1)βj q νl

n

m

≤ (k0 + k1 kxk + k2 kyk)

l=1

l=1

+|λ1 |

n2 p−1 ∑

T |Ω|

l=1

|ωl |θlq−1 (q − 1)νl

and

+ |λ2 |

n1 p−1 ∑

T |Ω|

j=1

|δj |ξjq−1 (q − 1)βj

(

|y(t)| ≤

m2 n1 ∑ |δj |ξjq |τk |γkp−1 ∑ T q−1 Tq + Γ(q + 1) |Ω|Γ(q + 1) (p − 1)σk j=1 q βj k=1 ) m1 n2 ∑ |ωl |θlq T q−1 |µi |ηip−1 ∑ + + (k0 + k1 kxk + k2 kyk) |Ω|Γ(q + 1) i=1 (p − 1)αi q νl l=1 ( ) m2 m1 m1 m2 ∑ ∑ |τk |γkp−1 ∑ |τk |γkp T q−1 |µi |ηip T q−1 |µi |ηip−1 ∑ × + |Ω|Γ(p + 1) (p − 1)σk i=1 pαi |Ω|Γ(p + 1) i=1 (p − 1)αi pσk

(l0 + l1 kxk + l2 kyk)

k=1

k=1

m2 m1 |τk |γkp−1 T q−1 ∑ T q−1 ∑ |µi |ηip−1 +|λ1 | + |λ | . 2 |Ω| (p − 1)σk |Ω| i=1 (p − 1)αi k=1

Hence we have kxk ≤ (k0 + k1 kxk + k2 kyk)M1 + (l0 + l1 kxk + l2 kyk)M2 + M3

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MIXED PROBLEMS OF FRACTIONAL COUPLED SYSTEMS

and kyk ≤ (l0 + l1 kxk + l2 kyk)M4 + (k0 + k1 kxk + k2 kyk)M5 + M6 , which imply that kxk + kyk

≤

(M1 + M5 )k0 + (M2 + M4 )l0 + [(M1 + M5 )k1 + (M2 + M4 )l1 ]kxk +[(M1 + M5 )k2 + (M2 + M4 )l2 ]kyk + M3 + M6 .

Consequently, k(x, y)k ≤

(M1 + M5 )k0 + (M2 + M4 )l0 + M3 + M6 , M0

for any t ∈ [0, T ], where M0 is defined by (14), which proves that E is bounded. Thus, by Lemma 3.2, the operator T has at least one fixed point. Hence the boundary value problem (1) has at least one solution. The proof is complete. ¤

3.1

Examples

Example 3.4 Consider the following system of coupled Riemann-Liouville fractional differential equations with Hadamard type fractional integral boundary conditions                

|x(t)| |y(t)| e−t 3 t + + , t ∈ [0, 2], (t + 6)2 (1 + |x(t)|) (t2 + 3)3 (1 + |y(t)|) 4 1 1 5 3/2 y(t) = sin x(t) + 2t cos y(t) + , t ∈ [0, 2], RL D 18 2 + 19 4 √ √ 2/3 7/5 x(0) = 0, 2H I x(3/5) + π H I x(1) = 2H I 3/2 y(1/3) + e2 H I 5/4 y( 3) + 4,      √ 2   y(0) = 0, −3H I 9/5 x(2/3) + 4H I 7/4 x(9/7) + H I 1/3 x( 2)    5    e  11/6 12/11  = HI y(8/5) − 2H I y(1/4) − 10. 2 RL D

4/3

x(t) =

(17)

Here p = 4/3, q = 3/2, T = 2, λ1 = 4, λ2 = −10, m √1 = 2, n1 = 2, m2 = 3, n2 = 2, µ1 = 2, µ2 = √ π, α1 = 2/3, α2 = 7/5, η1 = 3/5, η2 = 1, δ1 = 2, δ2 = e2 , β1 = 3/2, β2 = 5/4, ξ1 = 1/3, √ ξ2 = 3, τ1 = −3, τ2 = 4, τ3 = 2/5, σ1 = 9/5, σ2 = 7/4, σ3 = 1/3, γ1 = 2/3, γ2 = 9/7, γ3 = 2, ω1 = e/2, ω2 = −2, ν1 = 11/6, ν2 = 12/11, θ1 = 8/5, θ2 = 1/4 and f (t, x, y) = (t|x|)/(((t + 6)2 )(1 + |x|)) + (e−t |y|)/(((t2 + 3)3 )(1 + |y|)) + (3/4) and g(t, x, y) = (sin x/18) + (cos y)/(22t + 19) + (5/4). Since |f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ ((1/18)|x1 − x2 | + (1/27)|y1 − y2 |) and |g(t, x1 , y1 ) − g(t, x2 , y2 )| ≤ ((1/18)|x1 − x2 | + (1/20)|y1 − y2 |). By using the Maple program, we can find n2 n1 m2 m1 ∑ ∑ δj ξjq−1 ∑ ωl θlq−1 τk γkp−1 µi ηip−1 ∑ − ≈ −218.9954766 6= 0. Ω= (p − 1)αi (q − 1)νl j=1 (q − 1)βj (p − 1)σk i=1 l=1

k=1

With the given values, it is found that K1 = 1/18, K2 = 1/27, L1 = 1/18, L2 = 1/20, M1 ' 2.847852451, M2 ' 0.5295490231, M4 ' 4.723846069, M5 ' 1.276954854, and (M1 + M5 )(K1 + K2 ) + (M2 + M4 )(L1 + L2 ) ' 0.9364516398 < 1. Thus all the conditions of Theorem 3.1 are satisfied. Therefore, by the conclusion of Theorem 3.1, the problem (17) has a unique solution on [0, 2]. Example 3.5 Consider the following system of coupled Riemann-Liouville fractional differential equa-

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S. K. NTOUYAS, J. TARIBOON AND P. THIRAMANUS

tions with Hadamard type fractional integral boundary conditions  1 1 2 π/2  tan−1 x(t) + y(t), t ∈ [0, 3], x(t) = +  RL D  2  5 (t + 6) 20e   √   1 1 π   + sin x(t) + y(t) cos x(t), t ∈ [0, 3],  RL D7/4 y(t) =   2 42 t + 20   √ √  √  1/4 2 3    x(0) = 0, 3H I x(5/2) + 5H I x(7/8) + tan(4)H I x(9/4) √  8π 5/3   y(5/4) − 2H I 6/11 y(π/3) + 2, =  HI  3   √    2 2/3 2 1/3 √  6/5  x( 2)  y(0) = 0, − H I x(π/2) + 3H I x(5/3) + HI   3 π    √  7  + H I 11/9 x( 5) = eH I 7/6 y(π/6) − log(9)H I 3/4 y(7/4) − 1. 9

(18)

√ Here p = π/2, q = 7/4, T = 3, √ λ1 = 2, λ2√= −1, m1 = 3, n1 = 2, m2 = 4, n2 = 2, µ √1 = 3, µ2 = 5, µ3 = tan(4), α1 = 1/4, α2 = 2, α3 = 3, η1 = 5/2, η2 = 7/8, η3 = √ 9/4, δ1 = 8π/3, δ2 = −2, β1 = 5/3, β2 = 6/11, ξ1 = 5/4, ξ2 = π/3, τ1 = −2/3, τ2 =√3, τ3 = √2/π, τ4 = 7/9, σ1 = 2/3, σ2 = 6/5, σ3 = 1/3, σ4 = 11/9, γ1 = π/2, γ2 = 5/3, γ3 = 2, γ4 = 5, ω1 = e, ω2 = − log(9), −1 ν1 = 7/6, ν√ x)/((t + 6)2 ) + (y)/(20e) and 2 = 3/4, θ1 = π/6, θ2 = 7/4, f (t, x, y) = (2/5) + (tan g(t, x, y) = ( π/2) + (sin x)/(42) + (y cos x)/(t + 20). By using the Maple program, we get Ω=

m2 n2 n1 m1 ∑ ∑ δj ξjq−1 ∑ τk γkp−1 ωl θlq−1 µi ηip−1 ∑ − ≈ −59.01857601 6= 0. (p − 1)αi (q − 1)νl j=1 (q − 1)βj (p − 1)σk i=1 l=1

k=1

Since |f (t, x, y)| ≤ √ k0 + k1 |x| + k2 |y| and |g(t, x, y)| ≤ l0 + l1 |x| + l2 |y|, where k0 = 2/5, k1 = 1/36, k2 = 1/(20e), l0 = π/2, l1 = 1/42, l2 = 1/20, it is found that M1 ' 7.406711671, M2 ' 1.110132269, M4 ' 6.802999724, M5 ' 7.790182643. Furthermore, we can find that (M1 + M5 )k1 + (M2 + M4 )l1 ≈ 0.6105438577 < 1, and (M1 + M5 )k2 + (M2 + M4 )l2 ≈ 0.6751878489 < 1. Thus all the conditions of Theorem 3.3 holds true and consequently the conclusion of Theorem 3.3, the problem (18) has at least one solution on [0, 3].

4

Uncoupled integral boundary conditions case

In this section we consider the following system  p t ∈ [0, T ], 1 < p ≤ 2,  RL D x(t) = f (t, x(t), y(t)),   q  D y(t) = g(t, x(t), y(t)), t ∈ [0, T ], 1 < q ≤ 2,  RL   m1 n1  ∑ ∑  αi x(0) = 0, µiH I x(ηi ) = δj H I βj x(ξj ) + λ1 ,  i=1 j=1   n2 m2  ∑ ∑   σk  y(γ ) = ωlH I νl y(θl ) + λ2 . y(0) = 0, τ I  k kH 

(19)

l=1

k=1

Lemma 4.1 (Auxiliary Lemma) For h ∈ C([0, T ], R), the unique solution of the problem  p 1 < p ≤ 2, t ∈ [0, T ]   RL D x(t) = mh(t), n1 1 ∑ ∑ αi x(0) = 0, µ I x(η ) = δj H I βj x(ξj ) + λ1 , iH i   i=1

(20)

j=1

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is given by   n1 m1 p−1 ∑ ∑ t  δj H I βj RL I p h(ξj ) − x(t) = RL I p h(t) + µiH I αi RL I p h(ηi ) + λ1  , Λ j=1 i=1 where Λ :=

4.1

n1 m1 ∑ ∑ δj ξjp−1 µi ηip−1 − 6= 0. (p − 1)αi j=1 (p − 1)βj i=1

(21)

(22)

Existence results for uncoupled case

(

In view of Lemma 4.1, we define an operator T : X × Y → X × Y by T(u, v)(t) =

T1 (u, v)(t) T2 (u, v)(t)

)

where T1 (u, v)(t)

tp−1 = RL I f (s, u(s), v(s))(t) + Λ

(

p

−

m1 ∑

µiH I

αi

RL I

p

n1 ∑ j=1

δj H I βj RL I p f (s, u(s), v(s))(ξj ) )

f (s, u(s), v(s))(ηi ) + λ1 ,

i=1

and T2 (u, v)(t)

tq−1 = RL I g(s, u(s), v(s))(t) + Φ q

−

m2 ∑

τkH I

σk

RL I

q

(

n2 ∑ l=1

ωlH I νl RL I q g(s, u(s), v(s))(θl ) )

g(s, u(s), v(s))(γk ) + λ2 ,

k=1

where Φ=

m2 n2 ∑ ∑ τk γkq−1 ωl θlq−1 − 6= 0. (q − 1)σk (q − 1)νl

k=1

l=1

In the sequel, we set M1

=

M2

=

M3 M4

= =

  p n1 m1 p p−1 ∑ p−1 ∑ |δ |ξ 1 T T |µ |η j i i  j T p + + , Γ(p + 1) |Λ| j=1 pβj |Λ| i=1 pαi T p−1 λ1 , |Λ| 1 Γ(q + 1)

(23)

(24) ( Tq +

n2 m2 |ωl |θlq |τk |γkq T q−1 ∑ T q−1 ∑ + |Φ| q νl |Φ| q σk l=1

) ,

(25)

k=1

T q−1 λ2 . |Φ|

(26)

Now we present the existence and uniqueness result for the problem (19). We do not provide the proof of this result as it is similar to the one for Theorem 3.1. Theorem 4.2 Assume that f, g : [0, T ] × R2 → R are continuous functions and there exist constants K i , Li , i = 1, 2 such that for all t ∈ [0, T ] and ui , vi ∈ R, i = 1, 2, |f (t, u1 , u2 ) − f (t, v1 , v2 )| ≤ K 1 |u1 − v1 | + K 2 |u2 − v2 |

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and |g(t, u1 , u2 ) − g(t, v1 , v2 )| ≤ L1 |u1 − v1 | + L2 |u2 − v2 |. In addition, assume that M 1 (K 1 + K 2 ) + M 3 (L1 + L2 ) < 1, where M 1 and M 3 are given by (23) and (25) respectively. Then the boundary value problem (19) has a unique solution. Example 4.3 Consider the following system of coupled Riemann-Liouville fractional differential equations with uncoupled Hadamard type fractional integral boundary conditions  |x(t)| 3et/2 |y(t)| 2 cos(πt)  e/2  + + , t ∈ [0, 4], D x(t) =  RL t 2 3  (π + 4) |x(t)| + 2 (t + 5) |y(t)| + 3 e   √   √  2 |y(t)| + 1 sin x(t)  3   + + 5, t ∈ [0, 4], y(t) = RL D   15(et + 3) 7π(t + 3)      √ tan2 (5) 10/3 5   x(0) = 0, 11H I 5/2 x(2/3) + x(π) = H I 3/7 x(e) HI 20 e (27)  √ √  7 π  5 2/5  − H I x( 2) + H I x(12/7) + 11,   6 2     √  log(15) 7/4   y(1/4) + 2H I 5/6 y( 7) y(0) = 0, HI   9     √ √  π 2 4/3  = y(1/e) + 5H I 9/7 y(7/2) + 8/3. HI 15 √ √ √ Here p = e/2, q = 3, T = 4, λ1 = 11, λ2 = 8/3, m1 = 2, n1 = 3, m2 = 2, n2 = 2, µ1 = 11, 2 µ2 = tan √ (5)/20, α1 = 5/2, α2 = 10/3, √ η1 = 2/3, η2 = π, δ1 = 5/e, δ2 = −7/6, δ3 = π/2, β1 = 3/7, 2/5, ξ1 = e, ξ2 = 2, √ ξ3 = 12/7, τ1 = log(15)/9, τ2 = 2, σ1 = 7/4, σ2 = 5/6, β2 = 5, β3 = √ γ1 = 1/4, γ2 = 7, ω1 = π 2 /15, ω2 = 5, ν1 = 4/3, ν2 = 9/7, θ1 = 1/e, θ2 = 7/2, f (t, x, y) = t (cos(πt)|x|)/(((π +4)2 )(|x|+2))+(3et/2 |y|)/(((t+5)3 )(|y|+3))+(2/e) and g(t, x, y) = (sin x(t))/(15(et + √ 3)) + (2 |y| + 1)/(7π(t + 3)) + 5. Since |f (t, x1 , y1 ) − f (t, x2 , y2 )| ≤ ((1/50)|x1 − x2 | + (e2 /125)|y1 − y2 |) and |g(t, x1 , y1 ) − g(t, x2 , y2 )| ≤ ((1/60)|x1 − x2 | + (1/(21π))|y1 − y2 |). By using the Maple program, we can find m1 n1 ∑ ∑ δj ξjp−1 µi ηip−1 Λ := − ≈ 69.35947949 6= 0 (p − 1)αi j=1 (p − 1)βj i=1 and Φ=

m2 n2 ∑ ∑ τk γkq−1 ωl θlq−1 − ≈ −3.358717154 6= 0. (q − 1)σk (q − 1)νl

k=1

l=1

With the given values, it is found that K 1 = 1/50, K 2 = e2 /125, L1 = 1/60, L2 = 1/(21π), M 1 ' 5.673444294, M 3 ' 15.54186374. In consequence, M 1 (K 1 + K 2 ) + M 3 (L1 + L2 ) ≈ 0.9434486991 < 1. Thus all the conditions of Theorem 4.2 are satisfied. Therefore, there exists a unique solution for the problem (27) on [0, 4]. The second result dealing with the existence of solutions for the problem (19) is analogous to Theorem 3.3 and is given below. Theorem 4.4 Assume that there exist real constants k¯i , ¯li ≥ 0 (i = 1, 2) and k¯0 > 0, ¯l0 > 0 such that ∀xi ∈ R, (i = 1, 2) we have |f (t, x1 , x2 )| ≤ k¯0 + k¯1 |x1 | + k¯2 |x2 |, |g(t, x1 , x2 )| ≤ ¯l0 + ¯l1 |x1 | + ¯l2 |x2 |.

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In addition it is assumed that k¯1 M 1 + ¯l1 M 3 < 1 and k¯2 M 1 + ¯l2 M 3 < 1, where M 1 and M 3 are given by (23) and (25) respectively. Then the boundary value problem (19) has at least one solution. Proof. Setting M 0 = min{1 − k¯1 M 1 − ¯l1 M 3 , 1 − k¯2 M 1 − ¯l2 M 3 }, k¯i , ¯li ≥ 0 (i = 1, 2), ¤

the proof is similar to that of Theorem 3.3. So we omit it.

References [1] R.P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl. 59 (2010), 1095-1100. [2] B. Ahmad, J.J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl. 2013, Art. ID 149659, 8 pp. [3] B. Ahmad, J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 2011, 2011:36, 9 pp. [4] B. Ahmad, S.K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, Math. Probl. Eng. (2013), Art. ID 320415, 9 pp. [5] B. Ahmad, S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ. (2011) Art. ID 107384, 11pp. [6] D. Baleanu, K. Diethelm, E. Scalas, J.J.Trujillo, Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, 2012. [7] D. Baleanu, O.G. Mustafa, R.P. Agarwal, On Lp -solutions for a class of sequential fractional differential equations, Appl. Math. Comput. 218 (2011), 2074-2081. [8] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), 387-400. [9] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1-27. [10] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), 1-15. [11] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. [12] J. Hadamard, Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Mat. Pure Appl. Ser. 8 (1892) 101-186. [13] A.A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), 1191-1204. [14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [15] A.A. Kilbas, J.J. Trujillo, Hadamard-type integrals as G-transforms, Integral Transform. Spec. Funct.14 (2003), 413-427.

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[16] X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Differ. Equ. 2013, 2013:126. [17] D. O’Regan, S. Stanek, Fractional boundary value problems with singularities in space variables, Nonlinear Dynam. 71 (2013), 641-652. [18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [19] J. Tariboon, T. Sitthiwirattham, S. K. Ntouyas, Existence results for fractional differential inclusions with multi–point and fractional integral boundary conditions, J. Comput. Anal. Appl. 17 (2014), 343-360. [20] P. Thiramanus, J. Tariboon, S. K. Ntouyas, Average value problems for nonlinear second-order impulsive q-difference equations, J. Comput. Appl. Anal. 18 (2015), 590-611. [21] L. Zhang, B. Ahmad, G. Wang, R.P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math. 249 (2013), 51–56.

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Ternary Jordan ring derivations on Banach ternary algebras: A fixed point approach Madjid Eshaghi Gordji1 , Shayan Bazeghi1 , Choonkil Park2∗ and Sun Young Jang3∗ 1

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran 2 Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea 3 Department of Mathematics, University of Ulsan, Ulsan 680-749, Korea

e-mail: [email protected], [email protected], [email protected], [email protected] Abstract. Let A be a Banach ternary algebra. An additive mapping D : (A, [ ]) → (A, [ ]) is called a ternary Jordan ring derivation if D([xxx]) = [D(x)xx] + [xD(x)x] + [xxD(x)] for all x ∈ A. In this paper, we prove the Hyers-Ulam stability of ternary Jordan ring derivations on Banach ternary algebras.

1. Introduction We say that a functional equation (Q) is stable if any function g satisfying the equation (Q) approximately is near to true solution of (Q). Also, we say that a functional equation is superstable if every approximately solution is an exact solution of it. Recently, Bavand Savadkouhi et al. [4] investigate the stability of ternary Jordan derivations on Banach ternary algebras by direct methods. Ternary algebraic operations were considered in the 19th century by several mathematicians. Cayley [7] introduced the notion of cubic matrix, which in turn was generalized by Kapranov, Gelfand and Zelevinskii [17]. The comments on physical applications of ternary structures can be found in [3, 12, 13, 14, 22, 23, 26, 28, 31, 32]. Let A be a Banach ternary algebra. An additive mapping D : (A, [ ]) → (A, [ ]) is called a ternary ring derivation if D([xyz]) = [D(x)yz] + [xD(y)z] + [xyD(z)] for all x, y, z ∈ A. An additive mapping D : (A, [ ]) → (A, [ ]) is called a ternary Jordan ring derivation if D([xxx]) = [D(x)xx] + [xD(x)x] + [xxD(x)] for all x ∈ A. Theorem 1.1. ([11]) Suppose that (Ω, d) is a complete generalized metric space and T : Ω → Ω is a strictly contractive mapping with the Lipschitz constant L. Then, for any x ∈ Ω, either d(T n x, T n+1 x) = ∞,

∀n ≥ 0,

or there exists a positive integer n0 such that (1) d(T n x, T n+1 x) < ∞ for all n ≥ n0 ; (2) the sequence {T n x} is convergent to a fixed point y ∗ of T ; (3) y ∗ is the unique fixed point of T in Λ = {y ∈ Ω : d(T n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, T y) for all y ∈ Λ. The study of stability problems originated from a famous talk given by Ulam [30] in 1940: “Under what condition does there exist a homomorphism near an approximate homomorphism?” In the next year 1941, Hyers [15] answered affirmatively the question of Ulam for additive mappings between Banach spaces. A generalized version of the theorem of Hyers for approximately additive mappings was given by Rassias [24] in 1978. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [1, 5, 8, 10, 18, 19, 20, 21, 25, 27, 29, 33, 34]). In this paper, we prove the Hyers-Ulam stability and superstability of ternary Jordan ring derivations on Banach ternary algebras by the fixed point method. 0

2010 Mathematics Subject Classification. Primary 39B52; 39B82; 47H10; 46B99; 17A40. Keywords: Hyers-Ulam stability; ternary ring derivation; Banach ternary algebra; fixed point method; ternary Jordan ring derivation. 0∗ Corresponding author. 0

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M. Eshaghi Gordji, Sh. Bazeghi, C. Park, S. Y. Jang 2. Hyers-Ulam stability of ternary Jordan ring derivations In this section, we prove the Hyers-Ulam stability of ternary Jordan ring derivations on Banach ternary algebras. Throughout this section, assume that A is a Banach ternary algebra. Lemma 2.1. Let f : A → A be an additive mapping. Then the following assertions are equivalent. f ([a, a, a]) = [f (a), a, a] + [a, f (a), a] + [a, a, f (a)]

(2.1)

for all a ∈ A, and f ([a, b, c] + [b, c, a] + [c, a, b]) =[f (a), b, c] + [a, f (b), c] + [a, b, f (c)] + [f (b), c, a] + [b, f (c), a] + [b, c, f (a)] + [f (c), a, b] + [c, f (a), b] + [c, a, f (b)]

(2.2)

for all a, b, c ∈ A. Proof. Replacing a by a + b + c in (2.1), we have f ([(a + b + c), (a + b + c), (a + b + c)]) =[f (a + b + c), (a + b + c), (a + b + c)] + [(a + b + c), f (a + b + c), (a + b + c)] + [(a + b + c), (a + b + c), f (a + b + c)] and so f ([(a + b + c), (a + b + c), (a + b + c)]) = f ([a, a, a] + [a, b, a] + [a, c, a] + [b, a, a] + [b, b, a] + [b, c, a] + [c, a, a] + [c, b, a] + [c, c, a] + [a, a, b] + [a, b, b] + [a, c, b] + [b, a, b] + [b, b, b] + [b, c, b] + [c, a, b] + [c, b, b] + [c, c, b] + [a, a, c] + [a, b, c] + [a, c, c] + [b, a, c] + [b, b, c] + [b, c, c] + [c, a, c] + [c, b, c] + [c, c, c]) = f ([a, a, a]) + f ([a, b, a]) + f ([a, c, a]) + f ([b, a, a]) + f ([b, b, a]) + f ([b, c, a]) + f ([c, a, a]) + f ([c, b, a]) + f ([c, c, a]) + f ([a, a, b]) + f ([a, b, b]) + f ([a, c, b]) + f ([b, a, b]) + f ([b, b, b]) + f ([b, c, b]) + f ([c, a, b]) + f ([c, b, b]) + f ([c, c, b]) + f ([a, a, c]) + f ([a, b, c]) + f ([a, c, c]) + f ([b, a, c]) + f ([b, b, c]) + f ([b, c, c]) + f ([c, a, c]) + f ([c, b, c]) + f ([c, c, c]) = [f (a), a, a] + [a, f (a), a] + [a, a, f (a)] + [f (a), b, a] + [a, f (b), a] + [a, b, f (a)] + [f (a), c, a] + [a, f (c), a] + [a, c, f (a)] + [f (b), a, a] + [b, f (a), a] + [b, a, f (a)] + [f (b), b, a] + [b, f (b), a] + [b, b, f (a)] + [f (b), c, a] + [b, f (c), a] + [b, c, f (a)] + [f (c), a, a] + [c, f (a), a] + [c, a, f (a)] + [f (c), b, a] + [c, f (b), a] + [c, b, f (a)] + [f (c), c, a] + [c, f (c), a] + [c, c, f (a)] + [f (a), a, b] + [a, f (a), b] + [a, a, f (b)] + [f (a), b, b] + [a, f (b), b] + [a, b, f (b)] + [f (a), c, b] + [a, f (c), b] + [a, c, f (b)] + [f (b), a, b] + [b, f (a), b] + [b, a, f (b)] + [f (b), b, b] + [b, f (b), b] + [b, b, f (b)] + [f (b), c, b] + [b, f (c), b] + [b, c, f (b)] + [f (c), a, b] + [c, f (a), b] + [c, a, f (b)] + [f (c), b, b] + [c, f (b), b] + [c, b, f (b)] + [f (c), c, b] + [c, f (c), b] + [c, c, f (b)] + [f (a), a, c] + [a, f (a), c] + [a, a, f (c)] + [f (a), b, c] + [a, f (b), c] + [a, b, f (c)] + [f (a), c, c] + [a, f (c), c] + [a, c, f (c)] + [f (b), a, c] + [b, f (a), c] + [b, a, f (c)] + [f (b), b, c] + [b, f (b), c] + [b, b, f (c)] + [f (b), c, c] + [b, f (c), c] + [b, c, f (c)] + [f (c), a, c] + [c, f (a), c] + [c, a, f (c)] + [f (c), b, c] + [c, f (b), c] + [c, b, f (c)] + [f (c), c, c] + [c, f (c), c] + [c, c, f (c)] for all a, b, c ∈ A. On the other hand, we have f ([(a + b + c), (a + b + c), (a + b + c)]) = [f (a), a, a] + [f (a), a, b] + [f (a), a, c] + [f (a), b, a] + [f (a), b, b] + [f (a), b, c] + [f (a), c, a] + [f (a), c, b] + [f (a), c, c] + [f (b), a, a] + [f (b), a, b] + [f (b), a, c] + [f (b), b, a] + [f (b), b, b] + [f (b), b, c] + [f (b), c, a] + [f (b), c, b] + [f (b), c, c] + [f (c), a, a] + [f (c), a, b] + [f (c), a, c] + [f (c), b, a] + [f (c), b, b] + [f (c), b, c] + [f (c), c, a] + [f (c), c, b] + [f (c), c, c] + [a, f (a), a] + [a, f (a), b] + [a, f (a), c] + [b, f (a), a] + [b, f (a), b] + [b, f (a), c] + [c, f (a), a] + [c, f (a), b] + [c, f (a), c] + [a, f (b), a] + [a, f (b), b] + [a, f (b), c] + [b, f (b), a] + [b, f (b), b] + [b, f (b), c] + [c, f (b), a] + [c, f (b), b] + [c, f (b), c] + [a, f (c), a] + [a, f (c), b] + [a, f (c), c] + [b, f (c), a] + [b, f (c), b] + [b, f (c), c] + [c, f (c), a] + [c, f (c), b] + [c, f (c), c] + [a, a, f (a)] + [a, b, f (a)] + [a, c, f (a)] + [b, a, f (a)] + [b, b, f (a)] + [b, c, f (a)] + [c, a, f (a)] + [c, b, f (a)] + [c, c, f (a)] + [a, a, f (b)] + [a, b, f (b)] + [a, c, f (b)] + [b, a, f (b)] + [b, b, f (b)] + [b, c, f (b)] + [c, a, f (b)] + [c, b, f (b)] + [c, c, f (b)] + [a, a, f (c)] + [a, b, f (c)] + [a, c, f (c)] + [b, a, f (c)] + [b, b, f (c)] + [b, c, f (c)] + [c, a, f (c)] + [c, b, f (c)] + [c, c, f (c)] for all a, b, c ∈ A.

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Ternary Jordan ring derivations on Banach ternary algebras It follows that ([f (b), c, a] + [b, f (c), a] + [b, c, f (a)]) + ([f (c), a, b] + [c, f (a), b] + [c, a, f (b)]) + ([f (a), b, c] + [a, f (b), c] + [a, b, f (c)]) = f ([b, c, a]) + f ([c, a, b]) + f ([a, b, c]) = f ([b, c, a] + [c, a, b] + [a, b, c]) = f ([a, b, c] + [b, c, a] + [c, a, b]) and [f (a), b, c] + [f (b), c, a] + [f (c), a, b] + [c, f (a), b] + [a, f (b), c] + [b, f (c), a] + [b, c, f (a)] + [c, a, f (b)] + [a, b, f (c)] = ([f (a), b, c] + [a, f (b), c] + [a, b, f (c)]) + ([f (b), c, a] + [b, f (c), a] + [b, c, f (a)]) + ([f (c), a, b] + [c, f (a), b] + [c, a, f (b)]) for all a, b, c ∈ A. Then f ([a, b, c] + [b, c, a] + [c, a, b]) =([f (a), b, c] + [a, f (b), c] + [a, b, f (c)]) + ([f (b), c, a] + [b, f (c), a] + [b, c, f (a)]) + ([f (c), a, b] + [c, f (a), b] + [c, a, f (b)]) for all a, b, c ∈ A. Hence (2.2) holds true. For the converse, replacing b and c by a in (2.2), we have f ([a, a, a] + [a, a, a] + [a, a, a]) =[f (a), a, a] + [a, f (a), a] + [a, a, f (a)] + [f (a), a, a] + [a, f (a), a] + [a, a, f (a)] + [f (a), a, a] + [a, f (a), a] + [a, a, f (a)] and so f (3[a, a, a]) = 3([f (a), a, a] + [a, f (a), a] + [a, a, f (a)]) for all a ∈ A. Thus f ([a, a, a]) = [f (a), a, a] + [a, f (a), a] + [a, a, f (a)] for all a ∈ A. This completes the proof.



Theorem 2.2. Let f : A → A be a mapping for which there exists function ϕ : A × A × A → [0, ∞) such that kf (x + y) − f (x) − f (y)k ≤ ϕ(x, y, 0),

(2.3)

kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), y, z] − [x, f (y), z] − [x, y, f (z)] − [f (y), z, x] −[y, f (z), x] − [y, z, f (x)] − [f (z), x, y] − [z, f (x), y] − [z, x, f (y)]k ≤ ϕ(x, y, z) for all x, y, z ∈ A. If there exists a constant 0 < L < 1 such that x y z  L ϕ , , ≤ ϕ(x, y, z) 2 2 2 8 for all x, y, z ∈ A, then there exists a unique ternary Jordan ring derivation D : A → A such that kf (x) − D(x)k ≤

L ϕ(x, x, 0) 8 − 2L

(2.4)

(2.5)

(2.6)

for all x ∈ A. Proof. It follows from (2.5) that x y z  =0 (2.7) , , n→∞ 2n 2n 2n for all x, y, z ∈ A. By (2.5), ϕ(0, 0, 0) = 0. Letting x = y = 0 in (2.3), we get kf (0)k ≤ ϕ(0, 0, 0) = 0 and so f (0) = 0. Let Ω = {g : A → X, g(0) = 0}. We introduce a generalized metric on Ω as follows: lim 23n ϕ

d(g, h) = dϕ (g, h) = inf{C ∈ (0, ∞) : kg(x) − h(x)k ≤ Cϕ(x, x, 0), ∀x ∈ A}

.

It is easy to show that (Ω, d) is a generalized complete metric space [16]. Now, we consider the mapping T : Ω → Ω defined by T g(x) = 2g( x2 ) for all x ∈ A and g ∈ Ω. Note that, for all g, h ∈ Ω and x ∈ A, d(g, h) < C ⇒ kg(x) − h(x)k ≤ Cϕ(x, x, 0) x x x x ⇒ k2g( ) − 2h( )k ≤ 2 C ϕ( , , 0) 2 2 2 2 x x L C ϕ(x, x, 0) ⇒ k2g( ) − 2h( )k ≤ 2 2 4 L ⇒ d(T g, T h) ≤ C. 4 Hence we obtain that d(T g, T h) ≤

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M. Eshaghi Gordji, Sh. Bazeghi, C. Park, S. Y. Jang for all g, h ∈ Ω, that is, T is a strictly contractive mapping of Ω with the Lipschitz constant L. Putting y = x in (2.3), we have kf (2x) − 2f (x)k ≤ ϕ(x, x, 0), (2.8) and so

 x  x x  L

, , 0 ≤ ϕ(x, x, 0)

f (x) − 2f

≤ϕ 2 2 2 8 for all x ∈ A. Let us denote x (2.9) D(x) = lim 2n f n→∞ 2n for all x ∈ A. By the result in ([2, 6]), D is an additive mapping and so it follows from the definition of D, (2.4) and (2.7) that kD([x, y, z] + [y, z, x] + [z, x, y]) − [D(x), y, z] − [x, D(y), z] − [x, y, D(z)] − [D(y), z, x] − [y, D(z), x] − [y, z, D(x)] − [D(z), x, y] − [z, D(x), y] − [z, x, D(y)]k x, y, z y, z, x z, x, y x y z x y z x y z = lim 8n kf ([ 3n ] + [ 3n ] + [ 3n ]) − [f ( n ), n , n ] − [ n , f ( n ), n ] − [ n , n , f ( n )] n→∞ 2 2 2 2 2 2 2 2 2 2 2 2 z x y z x y z x z x y z x y z x y y − [f ( n ), n , n ] − [ n , f ( n ), n ] − [ n , n , f ( n )] − [f ( n ), n , n ] − [ n , f ( n ), n ] − [ n , n , f ( n )]k 2 2 2 2  2 2 2 2 2 2 2 2 2 2 2 2 2 2 x y z ≤ lim 8n ϕ n , n , n = 0 n→∞ 2 2 2 for all x, y, z ∈ A and so D([x, y, z] + [y, z, x] + [z, x, y]) = [D(x), y, z] + [x, D(y), z] + [x, y, D(z)] + [D(y), z, x] + [y, D(z), x] + [y, z, D(x)] + [D(z), x, y] + [z, D(x), y] + [z, x, D(y)], which implies that D is a ternary Jordan ring derivation, by Lemma 2.1. According to Theorem 1.1, since D is the unique fixed point of T in the set Λ = {g ∈ Ω : d(f, g) < ∞}, D is the unique mapping such that kf (x) − D(x)k ≤ C ϕ(x, x, 0) for all x ∈ A and C > 0. By Theorem 1.1, we have 1 4L d(f, D) ≤ d(f, T f ) ≤ 8(4 − L) 1 − L4 and so kf (x) − D(x)k ≤

L ϕ(x, x, 0) 8 − 2L

for all x ∈ A. This completes the proof.



Corollary 2.3. Let θ, r be nonnegative real numbers with r > 1. Suppose that f : A → A is a mapping such that kf (x + y) − f (x) − f (y)k ≤ θ(kxkr + kykr ),

(2.10)

kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), y, z] − [x, f (y), z] − [x, y, f (z)] − [f (y), z, x] −[y, f (z), x] − [y, z, f (x)] − [f (z), x, y] − [z, f (x), y] − [z, x, f (y)]k ≤ θ(kxkr + kykr + kzkr )

(2.11)

for all x, y, z ∈ A. Then there exists a unique ternary Jordan ring derivation D : A → A satisfying θ kf (x) − D(x)k ≤ r+1 kxkr 2 −1 for all x ∈ A. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y, z) := θ(kxkr + kykr + kzkr ) for all x, y, z ∈ A. Then we can choose L = 21−r and so we obtain the desired conclusion.



Remark 2.4. Let f : A → A be a mapping with f (0) = 0 such that there exists a function ϕ : A × A × A → [0, ∞) satisfying (2.3) and (2.4). Let 0 < L < 1 be a constant such that ϕ(2x, 2y, 2z) ≤ 2Lϕ(x, y, z) for all x, y, z ∈ A. By a similar method as in the proof of Theorem 2.2, one can show that there exists a unique ternary Jordan ring derivation D : A → A satisfying 2 kf (x) − D(x)k ≤ ϕ(x, x, 0) 4−L for all x ∈ A. For the case ϕ(x, y, z) := δ + θ(kxkr + kykr + kzkr ),

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Ternary Jordan ring derivations on Banach ternary algebras (where θ, δ are nonnegative real numbers and 0 < r < 1, there exists a unique ternary Jordan ring derivation D : A → X satisfying 4δ 8θ kf (x) − D(x)k ≤ + kxkr 8 − 2r 8 − 2r for all x ∈ A. Now, we formulate a theorem for the superstability of ternary Jordan ring derivations. Theorem 2.5. Suppose that there exist a function ϕ : A × A × A → [0, ∞) and a constant 0 < L < 1 such that x y z  L ≤ ϕ(x, y, z) ψ , , 2 2 2 8 for all x, y, z ∈ A. Moreover, if f : A → A is an additive mapping such that kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), y, z] − [x, f (y), z] − [x, y, f (z)] − [f (y), z, x] −[y, f (z), x] − [y, z, f (x)] − [f (z), x, y] − [z, f (x), y] − [z, x, f (y)]k ≤ ϕ(x, y, z) for all x, y, z ∈ A, then f is a ternary Jordan ring derivation. Proof. The proof is similar to the proof of Theorem 2.2. We will omit it.



Corollary 2.6. Let θ, s be nonnegative real numbers and s > 3. If f : A → A is an additive mapping such that kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), y, z] − [x, f (y), z] − [x, y, f (z)] − [f (y), z, x] −[y, f (z), x] − [y, z, f (x)] − [f (z), x, y] − [z, f (x), y] − [z, x, f (y)]k ≤ θ(kxks + kyks + kzks ) for all x, y, z ∈ A, then f is a ternary Jordan ring derivation. Remark 2.7. Suppose that there exist a function ψ : A × A × A → [0, ∞) and a constant 0 < L < 1 such that ϕ(2x, 2y, 2z) ≤ 2Lϕ(x, y, z) for all x, y, z ∈ A. Moreover, if f : A → A is an additive mapping such that kf ([x, y, z] + [y, z, x] + [z, x, y]) − [f (x), y, z] − [x, f (y), z] − [x, y, f (z)] − [f (y), z, x] −[y, f (z), x] − [y, z, f (x)] − [f (z), x, y] − [z, f (x), y] − [z, x, f (y)]k ≤ ϕ(x, y, z) for all x, y, z ∈ A, then f is a ternary Jordan ring derivation. Acknowledgments S. Y. Jang was supported by the Research Fund, University of Ulsan, 2014.

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Initial value problems for a nonlinear integro-differential equation of mixed type in Banach spaces∗ Xiong-Jun Zheng†, Jin-Ming Wang College of Mathematics and Information Science, Jiangxi Normal University Nanchang, Jiangxi 330022, People’s Republic of China

Abstract In this paper, we discuss the following initial value problem for first order nonlinear integro-differential equations of mixed type in a Banach space:  0 u = f (t, u, T u, Su) u(t0 ) = u0 . Rt In the case of the integral kernel k(t, s) of the operator (T u)(t) = t0 k(t, s)u(s)ds being unbounded, we obtain the existence of maximal and minimal solutions for the above problem by establishing a new comparison theorem. Keywords: noncompactness measure, unbounded integral kernel, maximal and minimal solutions, integro-differential equations.

1

Introduction and Preliminaries

Suppose that E is a Banach space. In this paper, We consider the following initial value problem for first order nonlinear integro-differential equations of mixed type in E:  u = f (t, u, T u, Su) (1.1) u(t0 ) = u0 , where f ∈ C[J × E × E × E, E], J = [t0 , t0 + a](a > 0), u0 ∈ E, and Z

t

(T u)(t) =

Z

t0 +a

k(t, s)u(s)ds, (Su)(t) = t0

h(t, s)u(s)ds.

(1.2)

t0

∗ The work was supported by the Natural Science Foundation of Jiangxi Province (No. 20122BAB201008, 20143ACB21001) and Science and Technology Plan of Education Department of Jiangxi Province (No. GJJ08169). † Corresponding author. E-mail address: [email protected], [email protected].

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ρ(t,s) + + In (1.2), k(t, s) = (t−s) α (0 < α < 1), ρ(t, s) ∈ C[D, R ], and h(t, s) ∈ C[D0 , R ], where   R+ = [0, +∞), D = (t, s) ∈ R2 t0 ≤ s ≤ t ≤ t0 + a , D0 = (t, s) ∈ R2 (t, s) ∈ J × J . Here, k(t, s) isunbounded on D, on D, and h(t, s) is bounded on D0 . ρ(t, s) is bounded  Set R0 = max ρ(t, s) (t, s) ∈ D , h0 = max h(t, s) (t, s) ∈ D0 . The study of initial value problems for nonlinear integro-differential equations has been of great interest for many researchers for its physical backgrounds and applications in mathematical models. We refer the reader to [1, 5–12] and references therein for some recent results on equation (1.1). However, in many earlier results, the kernel k(t, s) of the operator T is bounded. In this paper, we will make further study on the initial value problem (1.1) in the case of k(t, s) being unbounded. By establishing a comparison theorem, we achieve an existence theorem about minimal and maximal solutions for equation (1.1). Throughout the rest of this paper, let (E, k · k) be a real Banach space and P be a cone in E which defines a partial ordering in E denoted by ”≤”. Suppose that E ∗ is the dual space of E, the dual cone of the cone P is P ∗ = {ϕ ∈ ∗ E |ϕ(x) ≥ 0, ∀x ∈ P }. A cone P ⊂ E is said to be normal there exists a constant γ > 0 such that θ ≤ x ≤ y =⇒ kxk ≤ γkyk, ∀x, y ∈ E.

The cone P is normal if and only if any order interval [x, y] = {z ∈ E|x ≤ z ≤ y} is bounded in norm(see [3]). Set n o C[J, E] = u(t) : J → E u(t) is continuous on J , C 1 [J, E]

n o 0 = u(t) : J → E u(t) and u (t) are continuous on J .

Let kukc = max ku(t)k be a norm for u ∈ C[J, E], then C[J, E] will be a Banach space t∈J  with norm k·kc . It is easy to know Pc = u ∈ C[J, E] u(t) ≥ θ, ∀t ∈ J is a cone in C[J, E]. The cone Pc defines an ordering in C[J, E] which also denoted by ”≤” here. Obviously, when the cone P is normal, Pc is a normal cone in C[J, E]. Assume that V is a bounded set in E. The Kuratowski measure of noncompactness α(V ) and the Hausdorff measure of noncompactness β(V ) are defined respectively as follow: m S

α(V ) = inf{δ > 0|V can be expressed as the union S =

Vi of a finite number of sets Vi

i=1

with diameter diam(Vi ) ≤ δ}, n β(V ) = inf δ > 0 V can be covered by a finite number of closed balls Vi with diameter o diam(Vi ) ≤ δ . The relationship of the two noncompactness measures is β(V ) ≤ α(V ) ≤ 2β(V ).

(1.3)

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For the basic properties of cones and noncompactness measures, we refer the reader to [2–4]. For convenience, the Kuratowski measure of noncompactness for bounded sets in E and C[J, E] are all denoted by α(·). In the sequel, we denote B(t) = {u(t)|u ∈ B}, (T B)(t) = {(T u)(t)|u ∈ B}, (SB)(t) = {(Su)(t)|u ∈ B} for all B ⊂ C[J, E] with t ∈ J. Lemma 1.1. Let m ∈ C 1 [J, R1 ] be such that t

Z

0

m (t) ≥ −M m(t) − N

k(t, s)m(s)ds, m(t0 ) ≥ 0, t ∈ J,

(1.4)

t0

where M ≥ 0 and N ≥ 0 are two constants satisfying one of the following conditions: (i) a2−α N R0 eM a ≤ 1; (1.5) 1−α (ii) N R0 a2−α ≤ 1. 1−α

aM +

(1.6)

Then m(t) ≥ 0 for all t ∈ J. Proof. Case 1. If the condition (i) is established, let v(t) = m(t)eM t . From (1.4), we have Z t 0 v (t) ≥ −N k ∗ (t, s)v(s)ds, ∀t ∈ J, v(t0 ) ≥ 0, (1.7) t0

where

k ∗ (t, s)

=

k(t, s)eM (t−s) .

Now, we prove that v(t) ≥ 0, ∀t ∈ J.

(1.8)

In fact, if there exists t0 ≤ t1 ≤ t0 + a such that v(t1 ) < 0 and let max{v(t) : t0 ≤ t ≤ t1 } = b, then b ≥ 0. If b = 0, then v(t) ≤ 0 for all t0 ≤ t ≤ t1 and so (1.7) implies that v 0 (t) ≥ 0, ∀t0 ≤ t ≤ t1 . Hence we have v(t1 ) ≥ v(t0 ) = m(t0 )eM t0 ≥ 0, which contradicts v(t1 ) < 0. If b > 0, then there exists t0 ≤ t2 < t1 such that v(t2 ) = b > 0 and so there exists t2 < t3 < t1 such that v(t3 ) = 0. Then, by the mean value theorem, there exists t2 < t4 < t3 such that v 0 (t4 ) =

v(t3 ) − v(t2 ) −v(t2 ) −b b = = 1 which contradicts (1.5). Therefore, (1.8) Then from (1.9), we have N R0 eM a a1−α is true and so m(t) ≥ 0 for all t ∈ J. Case 2. If the assumption (ii) holds, but the conclusion does not hold, then there exists t1 ∈ (t0 , t0 + a] such that

m(t1 ) = min m(t) < 0, t∈J

and so

m0 (t

1)

≤ 0. If max m(t) ≤ 0, from (1.4), we have t0 ≤t≤t1

Z

0

t1

0 ≥ m (t1 ) ≥ −M m(t1 ) − N

k(t1 , s)m(s)ds ≥ −M m(t1 ) > 0, t0

which is a contradictory statement. Therefore, there exists t2 ∈ [t0 , t1 ) such that m(t2 ) = max m(t) = µ > 0. Then, by the mean value theorem, there exists t3 ∈ (t2 , t1 ) such that t0 ≤t≤t1

m0 (t3 ) =

m(t1 ) − m(t2 ) µ m0 (t3 ) ≥ −M m(t3 ) − N a

Z Z

t3

t0 t3

≥ −M µ − N R0 µ t0

ρ(t3 , s) m(s)ds (t3 − s)α 1 ds (t3 − s)α

(t3 − t0 )1−α = −M µ − N R0 µ 1−α 1−α a ≥ −M µ − N R0 µ , 1−α 2−α

i.e. aM + N R0 a1−α > 1 which contradicts (1.6). The Lemma is proved. 4

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Lemma 1.2. Let m ∈ C[J, R+ ] be such that Z

t

Z

t0 +a

m(s)ds + M2 (t − t0 )

m(t) ≤ M1

m(s)ds, t ∈ J

t0

(1.10)

t0

where M1 > 0, M2 ≥ 0, are constants for satisfying one of the following conditions: (i)aM2 (eaM1 − 1) < M1 , (ii)a(2M1 + aM2 ) < 2. Then m(t) ≡ 0, t ∈ J. Proof. Case 1. If the condition (i) holds, letting v(t) = m(t)eM t , then m1 (t0 ) = 0, m01 (t) = m(t), t ∈ J. If m1 (t0 + a) 6= 0, it follows from (1.10) that m01 (t) ≤ M1 m1 (t) + aM2 m1 (t0 + a), t ∈ J and from e−M1 (t−t0 ) > 0 we have  0 m1 (t)e−M1 (t−t0 ) ≤ aM2 m1 (t0 + a)e−M1 (t−t0 ) , t ∈ J. Now, we integrate the above inequality between t0 and t with noticing m1 (t0 ) = 0, we can obtain Z t −M1 (t−t0 ) m1 (t)e ≤ aM2 m1 (t0 + a) e−M1 (s−t0 ) ds t0

  aM2 m1 (t0 + a) 1 − e−M1 (t−t0 ) , t ∈ J. M1

≤ By choosing t = t0 + a, we can get


aM2 eaM1 − 1 ≥ M1 R t +a which contradicts (i). Consequently, m1 (t0 + a) = t00 m(s)ds = 0 which implies m(t) ≡ 0, t ∈ J. Case 2. If the condition (ii) is established, it follows from (1.10) that Z

t0 +a

m(t) ≤ [M1 + M2 (t − t0 )]

m(s)ds. t0

Integrating the above inequality between t0 and t0 + a, we get   Z t0 +a Z t0 +a a 2 M2 m(t)dt ≤ aM1 + m(s)ds. 2 t0 t0 From the above inequality and conditio (ii), it follows that 0, t ∈ J. This completes the proof.

R t0 +a t0

m(t)dt = 0, so m(t) ≡

Lemma 1.3. If B is a equicontinuous bounded set in ⊂ C[J, E], then α(B) = max α(B(t)). t∈J

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Lemma 1.4. If B is a equicontinuous bounded set in ⊂ C[J, E] with J = [a, b], then α({u(t)|u ∈ B}) is continuous with respect to t ∈ J and Z b  Z b  α u(t) u ∈ B dt. α u(t)dt u ∈ B ≤ a

a

Lemma 1.5. (see [2]) Let E be a separable Banach space, J = [a, b] and {un } : J → E be continuous abstract function sequences. If there φ ∈ L[a, b] such that a function
 exists is integrable on J kun (t)k ≤ φ(t), t ∈ J, n = 1, 2, 3, · · · , then β un (t) n = 1, 2, 3, · · · and Z b  Z b 
β un (t)dt n = 1, 2, 3, · · · β un (t) n = 1, 2, 3, · · · dt. ≤ a

a

Now, we give our assumptions: (H1 ) There exist v0 , ω0 ∈ C 1 [J, E] such that v0 (t) ≤ ω0 (t)(t ∈ J) and v0 , ω0 are a lower solution and an upper solution respectively for the initial value problem (1.1), that is v00 ≤ f (t, v0 , T v0 , Sv0 ), ∀t ∈ J; v0 (t0 ) ≤ u0 , ω00 ≥ f (t, ω0 , T ω0 , Sω0 ), ∀t ∈ J; ω0 (t0 ) ≥ u0 . (H2 ) For any t ∈ J, any u, v ∈ [v0 , ω0 ] = {u ∈ C[J, E]|v0 ≤ u ≤ ω0 } and u ≤ v, we have f (t, v, T v, Sv) − f (t, u, T u, Su) ≥ −M (v − u) − N T (v − u), where M > 0, N ≥ 0 are constants satisfying the condition (i) or (ii) in Lemma 1.1. (H3 ) For any t ∈ J and equicontinuous bounded monotone sequences B = {un } ⊂ [v0 , ω0 ], we have α(f (t, B(t), (T B)(t), (SB)(t)) ≤ c1 α(B(t)) + c2 α((T B)(t)) + c3 α((SB)(t)), where ci (i =1, 2, 3) are constants satisfying one  of the following two conditions: c R0 a1−α 2N R0 a1−α 1−α 1−α 2a(c1 +M + 2 1−α + ) 0a 0a 1−α (i)ah0 c3 e − 1 < c1 + M + c2 R1−α + 2N R ; 1−α   1−α 1−α 0a 0a (ii)a 2c1 + 2M + 2c2 R + 4N R + ah0 c3 < 1. 1−α 1−α

2

Main results

Theorem 2.1. Let E be a real Banach space, P ⊂ E be a normal cone and the conditions (H1 ), (H2 ), (H3 ) be satisfied. Then the initial value problem (1.1) has a minimal solution and a maximal solution u, u∗ ∈ C 1 [J, E] in [v0 , ω0 ], and for the initial value v0 and ω0 , the iterative sequences {vn (t)} and {ωn (t)} defined by the following formulas converge uniformly to u(t), u∗ (t) on J according to the norm in E respectively: Z t 
eM (s−t) f s, vn−1 (s), (T vn−1 )(s), (Svn−1 )(s) vn (t) = u0 e−M (t−t0 ) + t0

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 +M vn−1 (s) − N T (vn − vn−1 )(s) ds, ∀t ∈ J,

ωn (t) = u0 e−M (t−t0 ) +

Z

(2.1)

t


eM (s−t) f s, ωn−1 (s), (T ωn−1 )(s), (Sωn−1 )(s) t0  +M ωn−1 (s) − N T (ωn − ωn−1 )(s) ds, ∀t ∈ J (n = 1, 2, 3, · · · ).

(2.2)

Moreover, there holds v0 ≤ v1 ≤ · · · ≤ vn ≤ · · · ≤ u ≤ u∗ ≤ · · · ≤ ω1 ≤ ω0 .

(2.3)

Proof. For any η ∈ [v0 , ω0 ], we consider the initial value problem of linear integro-differential equation in Banach space E: u0 = g(t) − M u − N T u, u(t0 ) = u0 , (2.4)
where g(t) = f t, η(t), (T η)(t), (Sη)(t) + M η(t) + N (T η)(t). It is easy to show that u is a solution of the linear initial value problem (2.4) if and only if u is the fixed point in C[J, E] of the following operator Z t   −M (t−t0 ) (Au)(t) = u0 e + eM (s−t) g(s) − N (T u)(s) ds. (2.5) t0

In the following, we will prove there exists n0 such that An0 is a contraction operator. For any u, v ∈ C[J, E], t ∈ J, it follows from (2.5) that Z k(Au)(t) − (Av)(t)k ≤ N

t

kT (u − v)(s)kds t0 Z t Z s



≤ N

k(s, τ )ku(τ ) − v(τ )kdτ ds  ρ(s, τ ) = N ku(τ ) − v(τ )kdτ ds α t0 t0 (s − τ ) Z tZ s 1 ≤ N R0 ku − vkc dτ ds (s − τ )α t0 t0 t0 t0 Z t Z s

=

N R0 (t − t0 )2−α ku − vkc . (1 − α)(2 − α)

(2.6)

In the same way, by (2.5) and (2.6), we have Z t

2

(A u)(t) − (A2 v)(t) ≤ N kT (Au − Av)(s)kds t0  Z t Z s k(s, τ )k(Au)(τ ) − (Av)(τ )kdτ ds ≤ N t0

t0

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Z t Z

≤ = ≤ = =

s

 1 N R0 (τ − t0 )2−α N R0 ku − vkc dτ ds α (1 − α)(2 − α) t0 t0 (s − τ )  Z t Z s (τ − t0 )2−α (N R0 )2 dτ ds ku − vkc (1 − α)(2 − α) (s − τ )α t0 t0 Z Z (N R0 )2 ku − vkc t s (s − τ )−α (s − t0 )2−α dτ ds (1 − α)(2 − α) t0 t0 Z (N R0 )2 ku − vkc t (s − t0 )3−2α ds (1 − α)(2 − α) t0 1−α (N R0 )2 ku − vkc (t − t0 )4−2α . (1 − α)2 2(2 − α)2

It is easy to prove that by mathematical induction

n

(A u)(t) − (An v)(t) ≤

(N R0 )n (t − t0 )n(2−α) ku − vkc , t ∈ J, n = 1, 2, 3, · · · . n![(1 − α)(2 − α)]n

Thus kAn u − An vkc ≤

(N R0 a2−α )n ku − vkc , n = 1, 2, 3, · · · . n![(1 − α)(2 − α)]n 2−α n

(N R0 a ) n0 a contraction We can choose n0 ∈ {1, 2, 3, · · · } such that n![(1−α)(2−α)] n < 1, and so A operator in C[J, E]. Therefore, it follows from the principle of contraction mapping that An0 , that is, A has a unique fixed point uη in C[J, E] which implies the linear initial value problem (2.4) has a unique solution uη in C[J, E]. Now, we define a operator

Bη = uη

(2.7)

where uη is a unique solution for η of the linear initial value problem (2.4), and satisfies u0η = f (t, η(t), (T η)(t), (Sη)(t)) − M (uη (t) − η(t)) − N T (uη − η)(t), uη (t0 ) = u0 . Then B : [v0 , ω0 ] −→ C[J, E], and the iterative sequences (2.1)(2.2) can be written vn = Bvn−1 , ωn = Bωn−1 , n = 1, 2, 3, · · · . Moreover, we claim that the operator B defined by (2.7) satisfies i) v0 ≤ Bv0 , Bω0 ≤ ω0 ;

(2.8)

(2.9)

ii) Bη1 ≤ Bη2 , ∀η1 , η2 ∈ [v0 , ω0 ], η1 ≤ η2 .

(2.10)

Next, we will prove i) and ii). Firstly, we prove the result i). Set v1 = Bv0 , it follows from the definition of B that

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v10 = f (t, v0 , T v0 , Sv0 ) − M (v1 − v0 ) − N T (v1 − v0 ), v1 (t0 ) = u0 .

(2.11)

For any ϕ ∈ P ∗ , let m(t) = ϕ(v1 (t) − v0 (t)), it follows from (2.11) and the assumption (H1 ) that Z t

m0 (t) ≥ −M m(t) − N

k(t, s)m(s)ds, m(t0 ) ≥ 0. t0

Thus, by lemma 1.1, it follows that m(t) ≥ 0 for all t ∈ J, which implies v1 (t) − v0 (t) ≥ 0 for all t ∈ J. It follows theorem 2.4.3 in [3] that v0 ≤ Bv0 . Similarly, we can prove that Bω0 ≤ ω0 . Consequently, the result i) is proved. Next, we prove ii). Let uη1 = Bη1 , uη2 = Bη2 , it follows from the hypothesis (H2 ) and the definition of B that u0η1 − u0η2

= f (t, η2 , T η2 , Sη2 ) − M (uη2 − η2 ) − N T (uη2 − η2 ) −f (t, η1 , T η1 , Sη1 ) + M (uη1 − η1 ) + N T (uη1 − η1 ) ≥ −M (uη2 − uη1 ) − N T (uη2 − uη1 )

(2.12)

uη2 (t0 ) − uη1 (t0 ) = u0 − u0 = θ.

(2.13)

and For any ϕ ∈ P ∗ , let m(t) = ϕ(uη2 (t) − uη1 (t)). From (2.12) and (2.13), it follows that 0

Z

t

m (t) ≥ −M m(t) − N

k(t, s)m(s)ds, m(t0 ) = 0 t0

Thus, by lemma 1.1, it follows that m(t) ≥ 0 for all t ∈ J, which implies uη2 (t)−uη1 (t) ≥ θ, t ∈ J, that is, Bη1 ≤ Bη2 . So the result ii) is proved. Form (2.8)-(2.10) and observing that v0 ≤ ω0 , it follows that v0 ≤ v1 ≤ · · · ≤ vn ≤ · · · ≤ ωn ≤ · · · ≤ ω1 ≤ ω0 .

(2.14)

and B is a mapping with [v0 , ω0 ] into [v0 , ω0 ]. In the following, we prove that {vn (t)} converges uniformly to some element u ∈ C[J, E] in J. By the normality of P , the cone Pc is normal in C[J, E] which implies the order interval [v0 , ω0 ] is a bounded set in C[J, E]. Then, it follows from (2.14) that {vn } is a bounded set in C[J, E]. On the one hand, for any η ∈ [v0 , ω0 ], by the conditions (H1 ) and (H2 ), we have v00 + M v0 + N T v0 ≤ f (t, v0 , T v0 , Sv0 ) + M v0 + N T v0 ≤ f (t, η, T η, Sη) + M η + N T η ≤ f (t, ω0 , T ω0 , Sω0 ) + M ω0 + N T ω0 ≤ ω00 + M ω0 + N T ω0 .

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 Then, by the normality of Pc , the set f (t, η, Tη, Sη) N T η η ∈ [v0 , ω0 ] is a + Mη + bounded set in C[J, E]. On the other hand, the set T η η ∈ [v0 , ω0 ] is also a bounded set in C[J, E], because it follows from the boundedness of [v0 , ω0 ] that for any η ∈ [v0 , ω0 ], Z t k(t, s)kη(s)kds kT η(t)k ≤ t0

Z

t

ρ(t, s) ds α t0 (t − s) Z t 1 ≤ R0 kηkc ds α t0 (t − s) ≤ kηkc

= R0 kηkc

(t − t0 )1−α . 1−α

Therefore, {f (t, η, T η, Sη)|η ∈ [v0 , ω0 ]} is a bounded set in C[J, E]. Thus, from vn0 = f (t, vn−1 , T vn−1 , Svn−1 )−M (vn −vn−1 )−N T (vn −vn−1 ), t ∈ J, n = 1, 2, 3, · · · , (2.15) it follows that {vn0 |n = 1, 2, 3, · · · } is a bounded set in C[J, E]. Applying the mean value theorem, we see that all the functions {vn (t)|n = 1, 2, 3, · · · } is equicontinuous on J. From Lemma 1.3, we have α({vn |n = 1, 2, 3, · · · }) = max α({vn (t)|n = 1, 2, 3, · · · }). t∈J

(2.16)

Now, we prove α({vn |n = 1, 2, 3, · · · }) = 0. From (2.4), (2.5), (2.7) and (2.8), it follows that Z t  −M (t−t0 ) vn (t) = u0 e + eM (s−t) f (s, vn−1 (s), (T vn−1 ), (Svn−1 )(s)) t0  +M vn−1 (s) − N T (vn − vn−1 )(s) ds. (2.17) Let m(t) = α{vn (t)|n = 1, 2, 3, · · · }, then m(t0 ) = α({u0 }) = 0, m ∈ C[J, R+ ]. For every n, by the continuity of vn (t), {vn (t)|t ∈ J} is a separable set in E, so {vn (t)|t ∈ J, n = 1, 2, 3, · · · } is a separable set in E. Thus, we can assume that E is a separable Banach space without loss of generality otherwise, the closed subspace in E is spanned
by {vn (t)|t ∈ J, n = 1, 2, 3, · · · } can be used in place of E . By (2.17), (1.3) and Lemma 1.5 and observing 0 < eM (s−t) ≤ 1, (t, s) ∈ D, we can obtain  Z t   m(t) ≤ α eM (s−t) f (s, B(s), (T B)(s), (SB)(s)) + M B(s) − N T (B1 − B)(s) ds t Z0 t    M (s−t) ≤ 2β e f (s, B(s), (T B)(s), (SB)(s)) + M B(s) − N T (B1 − B)(s) ds t0

Z

t

≤ 2

  β f (s, B(s), (T B)(s), (SB)(s)) + M B(s) − N T (B1 − B)(s) ds

t0

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Z ≤ 2

t

β (f (s, B(s), (T B)(s), (SB)(s)))  +M β(B(s)) + N β(T (B1 − B)(s)) ds. t0

(2.18)

where B(s) = {vn (s)|n = 0, 1, 2, · · · }, B1 (s) = {vn (s)|n = 1, 2, 3, · · · }. By the condition (H3 ) and (1.3), we have β (f (s, B(s), (T B)(s), (SB)(s))) ≤ α (f (s, B(s), (T B)(s), (SB)(s))) ≤ c1 α(B(s)) + c2 α(((T B))(s)) + c3 α((SB)(s)).

(2.19)

From the uniform boundedness of B(s) and uniform continuity of h(t, s), it easy to prove (SB)(s) is a equicontinuous bounded set, so it follows from Lemma 1.4 that Z t0 +a  Z t0 +a h(s, τ )B(τ )dτ ≤ h0 m(τ )dτ. (2.20) α((SB)(s)) = α t0

t0


Now, we consider dealing with α (T B)(s) . Firstly, Z s Z s Z s ρ(s, τ ) 1 R0 a1−α k(s, τ )dτ = dτ ≤ R dτ ≤ . 0 α α 1−α t0 t0 (s − τ ) t0 (s − t) Since B(s) is equicontinuous bounded sequences and α(B(s)) = m(s), there exists a parl l S S tition B(s) = Bi such that the partition (T B)(s) = T Bi exists, where T Bi = i=1 i=1 o nR s t0 k(s, τ )vi (τ )dτ vi ∈ Bi , so we have diam(T Bi ) =

sup ∀vi1 ,vi2 ∈Bi

≤ =
≥ 0 ∀F 2 ∈ K2 (H 1 , H 2 ),

(3.1)

Proof. First assume that (3.1) holds and (2.3) does not hold. Then there exist ω ∈ Ω and q, s ∈ R(ω) such that Cqi (H 1 , H 2 ) < Csi (H 1 , H 2 ), Hqi < µis , Hqi > λis , i = 1, 2.

(3.2)

Let δi = min{µiq − Hqi , his − λis }, i = 1, 2. Then δi > 0, i = 1, 2. We define a vector Fi ∈ Ki (H 1 , H 2 ), i = 1, 2, whose components are Fqi (t) = Hqi + δi , Fsi (t) = Hsi − δi ,

Fri = Hri ,

(3.3)

when r 6= q, s. Thus, < C i (H 1 , H 2 ), F i − H i >=

n X

Cji (H 1 , H 2 )(Fji − Hji ) = δi (Cqi (H 1 , H 2 ) − Csi (H 1 , H 2 )) < 0,

(3.4)

j=1

and so (3.1) is not satisfied. Therefore, it is proved that (3.1) implies (2.4). Next, assume that (2.4) holds. That is Cqi (H 1 , H 2 ) < Csi (H 1 , H 2 ) ⇒ Hqi = µiq , or

Hsi = µis , i = 1, 2.

(3.5)

Let F i ∈ Ki (H 1 , H 2 ) for i = 1, 2. Then (3.1) holds from Definition 2.1. The proof is completed. Furthermore, we discuss the existence and uniqueness of the solution for the dynamic traffic equilibrium system (3.1). In order to get our main results, the following definitions will be employed. Definition 3.2. C i (x, y)(i = 1, 2) is said to be θ-strictly monotone with respect to x on K1 (H 1 , H 2 )× K2 (H 1 , H 2 ) if there exists θ > 0such that < C i (x1 , y) − C i (x2 , y), x1 − x2 >≥ θkx1 − x2 k22 ,

(3.6)

∀x1 , x2 ∈ K1 (H 1 , H 2 ), ∀y ∈ K2 (H 1 , H 2 ).

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Definition 3.3. C i (x, y)(i = 1, 2) is said to be L-Lipschitz continuous with respect to x on 1 2 1 K1 (H , H ) × K2 (H , H 2 ) if there exists θ > 0such that kC i (x1 , y) − C i (x2 , y)k2 ≤ Lkx1 − x2 k2 ,

(3.7)

∀x1 , x2 ∈ K1 (H 1 , H 2 ), ∀y ∈ K2 (H 1 , H 2 ). Remark 3.4. Based on Definitions 3.2 and 3.3, we can similarly define the θ -strict monotonicity and L-Lipschitz continuity of C i (x, y) with respect to y on K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ), for i = 1, 2. Theorem 3.5. (H 1 , H 2 ) ∈ K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ) is an equilibrium flow if and only if there exist α > 0 and β > 0 such that H 1 = PK1 (H 2 − αC 1 (H 1 , H 2 )), H 2 = PK2 (H 2 − βC 1 (H 1 , H 2 )),

(3.8)

where Pki : Rn → Ki (H 1 , H 2 ) is a projection operator for i = 1, 2. Proof. The proof is analogous to that of Theorem 5.2.4 of [18]. Let k(x, y)1 k be the norm on space K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ) defined as follows: k(x, y)k1 = kxk2 + kyk2 , ∀x ∈ K1 (H 1 , H 2 ), y ∈ K2 (H 1 , H 2 ).

(3.9)

It is easy to see that (K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ), k.k1 )is a Banach space. Similar to Theorem 3.9 in He et. al. [1], one can easily obtain the following theorem, the proof is omitted. Theorem 3.6. Suppose that C 1 (H 1 , H 2 ) is θ1 -strictly monotone and L11 -Lipschitz continuous with respect to H 1 , and L12 -Lipschitz continuous with respect to H 2 on K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ). Suppose that C 2 (H 1 , H 2 ) is L21 -Lipschitz continuous with respect to H 1 , θ2 -strictly monotone, and L22 -Lipschitz continuous with respect to H 2 on K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ). If there exist α > 0 and β > 0 such that q 1 − 2γθ1 + α2 L211 + βL21 < 1, (3.10) q 1 − 2ηθ2 + β 2 L222 + αL12 < 1, then problem (3.1) admits unique solution. Remark 3.7. If fj1 (H 1 , H 2 ) is θˆj1 -strictly monotone with respect to H 1 and gj1 ◦ strictly monotone with respect to H 1 , then θ1 =

n X

Pn

rs 1 j=1 δpj tj

1

is θj -

1 rs ˆ1 (θj + δpj θj ).

j=1

In fact, ˆ 1 , H 2 ), H 1 − H ˆ1 > < Cj1 (H 1 , H 2 ) − Cj2 (H n n n n X X X X rs 1 rs rs 1 rs 1 ˆ 1 =< gr ( δpj tj (H 1 , H 2 )) + δpj τj + δpj fj (H 1 , H 2 ) − gr ( δpj tj (H , H 2 )) j=1

j=1

+

n X

j=1

rs δpj τj +

j=1

n X

(3.11)

j=1

rs 1 ˆ 1 ˆ1 > δpj fj (H , H 2 ), H 1 − H

j=1

≥

n X

1

rs ˆ1 ˆ 1 k2 θj )kH 1 − H (θj + δpj 2

j=1

So,

ˆ 1 , H 2 ), H 1 − H ˆ 1 >≥ θ1 kH 1 − H ˆ 1 k. < Cj (H 1 , H 2 ) − Cj (H 4

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4

Algorithms for solving the multicriteria transportation equilibrium system problem

Here, we describe an iterative algorithm with fixed step-sizes, and also describe a self-adaptive algorithm, which uses a self-adaptive stratedgy of step-size choice. Algorithm 4.1 Iterative Method with fixed step-sizes Step 1. Given ² > 0, α, β ∈ [0, 1), and (H10 , H20 ) ∈ K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ), set k = 0. Step 2. Get the next iterate: H 1,k+1 = PK1 (H 2,k − αC 1 (H 1,k , H 2,k ), H 2,k+1 = PK2 (H 2,k − βC 1 (H 1,k , H 2,k ), (k+1)

Step 3. Compute r1 = kH1 go to step 2.

(k)

(k+1)

−H1 k, r2 = kH2

(k)

−H2 k , if r1 , r2 < ², then stop; Otherwise,k = k+1,

Algorithm 4.2 SI method Step 1. Given ² > 0, γ ∈ [1, 2),µ ∈ (0, 1),ρ > 0 ,δ ∈ (0, 1) ,δ0 ∈ (0, 1) , and µ0 ∈ H, set k = 0. Step 2. Setρk = ρ , if kr1 (H 1K , ρ)k < ² and kr1 (H 1K , ρ)k < ², then stop; otherwise, find the smallest nonnegative integer mk , such that ρk = ρµmk satisfying kρk (C 1 (H 1k , H 2k ) − C 1 (wk , H 2k )k ≤ δkr(xk , ρk )k,

(4.1)

where wk = PK [H 1k − ρk C 1 (H 1k , H 2k )]. Step 3. Compute d(H 1k , ρk ) = r(H 1k , ρk ) − ρk C 2 (H 1k , H 2k ) + ρk C 2 (PK [H 1k − ρk C(H 1k , H 2k )], H 2k ),

(4.2)

where r(H 1k , ρ) = H 1k − PK [H 1k − ρC 2 (H 1k , H 2k )] . Step 4. Get the next iterate: H 2k = PK [H 1k − γd(H 1k , ρk ) − γC 2 (H 1k , H 2k )]; H 1,k+1 = PK [H 1k − ρC 1 (H 1k , H 2k )]

(4.3)

Step 5. If kρk (C(H 1k , H 2k ) − C(wk , H 2k )k ≤ δ0 kr(xk , ρk )k , then set ρ = ρk /µ, else set ρ = ρk . Set k = k + 1, and go to Step 2. Remark 4.2. Note that Algorithm 4.2 is obviously a modification of the standard procedure. In Algorithm 4.2, the searching direction is taken as H 1k − γd(H 1k , ρk ) − γC(H 1k , H 2k ) , which is closely related to the projection residue, and differs from the standard procedure. In addition, the self-adaptive strategy of step-size choice is used. The numerical results show that these modifications can introduce computational efficiency substantially. Theorem 4.3. Suppose that C 1 (H 1 , H 2 ) is θ1 -strictly monotone and L11 -Lipschitz continuous with respect to H 1 , and L12 -Lipschitz continuous with respect to H2 on K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ). Suppose that C 2 (H 1 , H 2 ) is L21 -Lipschitz continuous with respect to H 1 ,θ2 -strictly monotone, and L22 -Lipschitz continuous with respect to H 2 on K1 (H 1 , H 2 ) × K2 (H 1 , H 2 ). Let H 1∗ , H 2∗ ∈ K form a 1k solution setpfor the SNVI (2.1) and let the sequences {Hp } and {H 2k } be generated by Algorithm 4.2. 2 2 If 0 < θ < 1 − 2ρθ1 + 2ρ L12 (1 + γL21 )/(1 − γL22 ) + 2ρL211 + 2ρL11 < 1 , then the sequence {H 1k } converges to H 1∗ and the sequence {H 2k } converges to H 2∗ , for 0 < ρ < 2r/s2 . Proof. Since (H 1∗ , H 2∗ ) is a solution of transportation equilibrium system (3.2), it follows from Theorem 3.5 that H 1∗ = PK1 [H 2∗ − ρC 1 (H 1∗ , H 2∗ )], H 2∗ = PK2 [H 1∗ − γC 2 (H 1∗ , H 2∗ )]

(4.4)

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Applying Algorithm 4.2, we know kH 1,k+1 − H 1∗ k = kPK1 [H 2k − ρC 1 (H 1k , H 2k )] − PK1 [H 2∗ − ρC 1 (H 1∗ , H 2∗ )] ≤ kH 2k − H 2∗ − ρC 1 (H 1k , H 2k ) + ρC 1 (H 1∗ , H 2∗ )k Since T is r-strongly monotone and s-Lipschitz continuous, we know kH 2k − H 2∗ − ρC 1 (H 1∗ , H 2∗ )k2 ≤ kH 2k − H 2∗ k2 − 2ρ < C 1 (H 1k − H 2k − C 1 (H 1∗ , H 2∗ , H 2k − H 2∗ > +ρ2 kC 1 (H 1k , H 2k ) − C 1 (H 1∗ , H 2∗ )k ≤ kH 2k − H 2∗ k2 − 2ρθ1 kH 2k − H 2∗ k2 + 2ρL11 kH 1k − H 1∗ k2 + ρ2 kC 1 (H 1k , H 2k ) − C 1 (H 1∗ , H 2∗ )k ≤ kH 2k − H 2∗ k2 − 2ρθ1 kH 2k − H 2∗ k2 + 2ρL11 kH 1k − H 1∗ k2 + 2ρ2 L211 kH 1k − H 1∗ k2 + 2ρ2 L212 kH 2k − H 2∗ k2 ≤ (1 − 2ρθ1 + 2ρ2 L212 )kH 2k − H 2∗ k2 + (2ρ2 L211 + 2ρL11 )kH 1k − H 1∗ k2 It follows that q kH 1,k+1 − H 1∗ k ≤

1 − 2ρθ1 + 2ρ2 L212 kH 2k − H 2∗ k +

q 2ρ2 L211 + 2ρL11 kH 1k − H 1∗ k.

(4.5)

Next, we consider kH 2k − H 2∗ k = kPK2 [H 1k − γd(H 1k , ρk ) − γC 2 (H 1k , H 2k )] − PK2 [H 1∗ − γC 2 (H 1∗ , H 2∗ )k ≤ kH 1k − γd(H 1k , ρk ) − γC 2 (H 1k , H 2k ) − H 1∗ + γC 2 (H 1∗ , H 2∗ )k ≤ kH 1k − γd(H 1k , ρk ) − H 1∗ k + γkC 2 (H 1k , H 2k ) − C 2 (H 1∗ , H 2∗ )k

(4.6)

where we use the property of the operator PK . Now, we consider kH 1k − H 1∗ − γd(H 1k , ρk )k2 ≤ kH 1k − H 1∗ k2 − 2γ < H 1k − H 1∗ , d(H 1k , ρk ) > +γ 2 kd(H 1k , ρk )k2 ≤ kH 1k − H 1∗ k2 ,

(4.7)

where we use the definition of d(H 2k , ρk ). It follows that kH 2,k − H 2∗ k ≤ (1 + γL21 )kH 1k − H 1∗ k + γL22 kH 2k − H 2∗ k.

(4.8)

From (4.5) to (4.8), we know q q kH 1,k+1 − H 1∗ k ≤ ( 1 − 2ρθ1 + 2ρ2 L212 (1 + γL21 )/(1 − γL22 ) + 2ρL211 + 2ρL11 )kH 1k − H 1∗ k. (4.9) Since 0 < θ < 1 , from (4.9), we know H 1k → H 1∗ . Thus from (4.8), we know H 2k → H 2∗ .

5

Numerical results

In this section, we presented some numerical results for the proposed method. we consider a simple traffic network consisting of two nodes, only a origin-destination (O/D) pair, and a set R of routes. Each route r ∈ R links the origin-destination pair in parallel. Assume that n is the number of the route in R. Let C 1 (H1 , H2 ) = DH1 (t) + cT1 H2 (t), C 2 (H1 , H2 ) = DH1 (t) + cT2 H2 (t) , where   4 −2 · · · · · ·  1 4 ··· ···   D=  · · · · · · 4 −2  , ··· ··· 1 4 c1 = (−1, −1, · · · , −1)T , c2 = (1, 1, · · · , 1)T ,H1 (t) = H1 ∈ Rn ,H1 (t) = H2 ∈ Rn . let K1 (H1 , H2 ) = {H1 |H1 ∈ [l, u], H1i + H2i ≤ 2000, i = 1, 2, · · · , n}, 6

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K2 (H1 , H2 ) = {H1 |H1 ∈ [l, u], H1i + H2i ≤ 2000, i = 1, 2, · · · , n}. where l = (0, 0, · · · , 0)T , u = (1000, 1000, · · · , 1000)T . The calculations are started with vectors H1 = (0, 0, · · · , 0)T , H2 = (5, 5, · · · , 5)T and stopped whenever r1 , r2 < 10−5 . Table 1 gives the numerical results of Algorithms 4. 1. Table 2 gives the numerical results of Algorithms 4. 2. Comparing Table 2 and Table 1, it show that Algorithm 4.2 is very effective for the problem tested. In addition, it seems that the computational time and the iteration numbers are not very sensitive to the problem size. Table 1: Computation performance with different scales by Algorithm 4.1 n 50 100 200 300

Iteration 366 183 93 63

CPU(s) 29.5469 28.5469 27.2813 30.4375

Table 2: Computation performance with different scales by Algorithm 4.2 n 50 100 200 300

Iteration 220 110 56 38

CPU(s) 17.7282 16.1281 15.3687 18.2625

References [1] Y. P. He, J. P. Xu, N. J. Huang, and M. Wu, Dynamic traffic network equilibrium system, Fixed Point Theory and Application, vol. 2010, Art. 873025. [2] S. C. Dafermos, The general multinodal network equilibrium problem with elastic demand, Networks, vol. 12, pp. 57-72, 1982. [3] M. Fukushima, On the dual approach to the traffic assignment problem, Transportation Res. Part B, vol. 18, pp. 235-245, 1984. [4] M. Fukushima and T. Itoh, A dual approach to asymmetric traffic equilibrium problems, Math. Japon. vol. 32 , pp. 701-721, 1987. [5] A. Maugeri, Stability results for variational inequalities and applications to traffic equilibrium problem, Supplemento Rendiconti Circolo Matematico di Palermo vol. 8, pp. 269-280, 1985. [6] S. Gabriel, D. Bernstein, The traffic equilibrium problem with nonadditive path costs, Transportation Science, Vol 31, pp. 337-348, 1997. [7] K. Scott, D. Bernstein, Solving a best path problem when the value of time function is nonlinear, in: The 77th Annual Meeting of Transportation Research Board, 1998. [8] K. Scott, D. Bernstein, Solving a traffic equilibrium problem when paths are not additive, in: The 78th Annual Meeting of Transportation Research Board, 1999. [9] F. Facchinei, J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM Journal of Optimization, Vol. 7, pp. 225-247 , 1997. [10] H.K. Lo, A. Chen, Traffic equilibrium problem with route-specific costs: formulation and algorithms, Transportation Research Part B, Vol. 34 , pp. 493-513 , 2000. 7

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[11] A. Chen, H.K. Lo, H. Yang, A self-adaptive projection and contraction algorithm for the traffic assignment problem with path-specific costs, European Journal of Operational Research, Vol. 135, pp. 27-41 , 2001. [12] T. Larsson, P.O. Lindberg, J. Lundgren, M. Patriksson, C. Rydergren, On traffic equilibrium models with a nonlinear time/money relation, in: M. Patriksson, M. Labbe (Eds.), Transportation Planning: State of the Art, Kluwer Academic Publishers, 2002. [13] D.H. Bernstein, L. Wynter, Issues of uniqueness and convexity in non-additive bi-criteria equilibrium models, in: Proceedings of the EURO Working Group on Transportation, Rome, Italy, 2000. [14] E. Altman, L. Wynter, Equilibrium, games, and pricing in transportation and telecommunication networks, Networks and Spatial Economics , Vol. 4, pp. 7-21, 2004. [15] R. U. Verma, Projection methods, algorithms, and a new system of nonlinear variational inequalities, Computers and Mathematics with Applications, vol. 41, no. 7-8, pp. 1025-1031, 2001. [16] A. Bnouhachem, A self-adaptive method for solving general mixed variational inequalities, Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 136-150, 2005. [17] C. F. Shi, A self-adaptive method for solving a system of nonlinear variational inequalities, Mathematical Problems in Engineering, Art. 23795, 2007. [18] P. Daniele, Dynamic Networks and Evolutionary Variational Inequalities, New Dimensions in Networks, Edward Elgar, Cheltenham, UK, 2006.

Chaofeng Shi Department of Financial and Economics Chongqing Jiaotong University Chongqing, 400074, P. R. China and Department of Economics University of California Riverside, CA 92507, USA. Correspondence should be addressed to Chaofeng Shi, E-mail: [email protected] Yingrui Wang Department of Financial and Economics Chongqing Jiaotong University Chongqing, 400074, P. R. China

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BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM

Abstract. In this paper, we consider the Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomials. Using the umbral calculus, we derive several explicit formulas and recurrence relations for these polynomials. Also, we establish connections between our polynomials and several known families of polynomials.

1. Introduction For λ ̸= 1, s ∈ N, the Frobenius-Euler polynomials of order s are defined by the generating function ( )s ∞ ∑ 1−λ tn xt (1.1) e = , (see [7, 12, 19]) . H(s) n (x | λ) t e −λ n! n=0 Let a1 , a2 , . . . , ar , λ1 , λ2 , . . . , λr ∈ C with a1 , . . . , ar ̸= 0, λ1 , . . . , λr ̸= 1. Then the Barnes’ multiple Frobenius-Euler polynomials Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) are given by the generating funciton (1.2) ) r ( ∞ ∏ ∑ 1 − λj tn xt . , a ; λ , . . . , λ ) e = , (see [13, 15]) . H (x | a , . . r 1 r n 1 eaj t − λj n! n=0 j=1 When x = 0, Hn (a1 , . . . , ar ; λ1 , . . . , λr ) = Hn (0 | a1 , . . . , ar ; λ1 , . . . , λr ) are called the Barnes’ multiple Frobenius-Euler numbers (see [13]). For a1 = a2 = · · · = ar = 1, λ1 = λ2 = · · · = λr = λ, we have Hn (x | (r) (r) (r) 1, 1, . . . , 1; λ, λ, . . . , λ) = Hn (x | λ). When x = 0, Hn (λ) = Hn (0 | λ) are called | {z } | {z } r−times

r−times

the Frobenius-Euler numbers of order r. (ν) The Hermite polynomials Hn (x) of variance ν (0 ̸= ν ∈ R) are given by the generating function (1.3)

e−νt

2

/2 xt

e

=

∞ ∑

Hn(ν) (x)

n=0 (ν)

tn , n!

(see [24]) .

(ν)

When x = 0, Hn = Hn (0) are called the Hermite numbers of variance ν. It is well known that the Bernoulli polynomials of order r (∈ N) are defined by the generating function 2010 Mathematics Subject Classification. 05A19, 05A40, 11B75, 11B83. Key words and phrases. Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomials, umbral calculus. 1

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2

DAE SAN KIM, DMITRY V. DOLGY, AND TAEKYUN KIM

( (1.4)

t et − 1 (r)

)r ext =

∞ ∑

Bn(r) (x)

n=0

tn , n!

(see [1–24, 26]) .

(r)

When x = 0, Bn = Bn (0) are called the Bernoulli numbers of order r. For n ≥ 0, the Stirling numbers of the first kind are given by n ∑ (1.5) (x)n = x (x − 1) · · · (x − n + 1) = S1 (n, l) xl , (see [24]) . l=0

The Stirling numbers of the second kind are defined by the generating function ∞ ∑ ( t )n xl (1.6) e − 1 = n! S2 (l, n) , (see [24]) . l! l=n

Let C be the complex field and let F be the set of all formal power series in the variable t: { } ∞ ∑ tk (1.7) F = f (t) = ak ak ∈ C . k! k=0

∗

Let P = C [x] and let P be the vector space of all linear functionals on P. We use the notation ⟨L | p (x)⟩ to denote the action of the linear functional L on the polynomial p (x), and we recall that the vector space operations on P∗ are defined by ⟨L + M | p (x)⟩ = ⟨L | p (x)⟩ + ⟨M | p (x)⟩, and ⟨cL | p (x)⟩ = c ⟨ L| p (x)⟩, where c is a complex constant in C. The linear functional ⟨ f (t)| ·⟩ on P is defined by (1.8)

⟨ f (t)| xn ⟩ = an ,

(n ≥ 0) ,

By (1.8), we easily get ⟨ k n⟩ (1.9) t x = n!δn,k ,

where f (t) ∈ F ,

(n, k ≥ 0) ,

(see [17, 21, 24]) .

(see [8, 21, 24]) ,

where δn,k is the Kronecker’s symbol. ∑∞ ⟨ L|xk ⟩ Let fL (t) = k=0 k! tk . Then, by (1.9), we get ⟨ fL (t)| xn ⟩ = ⟨L | xn ⟩ . So, the map L 7→ fL (t) is a vector space isomorphism from P∗ onto F. Henceforth, F denotes both the algebra of formal power series in t and the vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional. We call F the umbral algebra and the umbral calculus is the study of umbral algebra. The order o (f (t)) of a power series f (t) ̸= 0 is the smallest integer k for which the coefficient of tk does not vanish. If the order of f (t) is 1, then f (t) is called a delta series; if the order g (t) is 0, then g (t) is called an invertible series. Let f (t) , g (t) ∈ F with o (f (t)) = 1 and o⟨(g (t)) = 0. Then there exists a unique sequence sn (x) (deg sn (x) = n) such ⟩ k

that g (t) f (t) | sn (x) = n!δn,k for n, k ≥ 0. Such a sequence sn (x) is called the Sheffer sequence for (g (t) , f (t)) which is denoted by sn (x) ∼ (g (t) , f (t)) (see [21, 24]). In particular, if sn (x) ∼ (g (t) , t), then sn (x) is called an Appell sequence for g (t). For f (t) , g (t) ∈ F , we have (1.10) ⟨ f (t) g (t)| p (x)⟩ = ⟨ f (t)| g (t) p (x)⟩ = ⟨ g (t)| f (t) p (x)⟩ = ⟨ 1| f (t) g (t) p (x)⟩ , (1.11)

f (t) =

∞ ∑ ⟨ k=0

f (t)| xk

⟩ tk , k!

p (x) =

∞ ∑ ⟨ k ⟩ xk , t p (x) k!

(see [24]) .

k=0

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BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS 3

Thus, by (1.11), we get (1.12) ⟨ ⟩ dk p (x) tk p (x) = p(k) (x) = , eyt p (x) = p (x + y) , and eyt p (x) = p (y) . dxk The sequence sn (x) is Sheffer for (g (t) , f (t)) if and only if ∞

(1.13)

∑ sk (x) 1 ) eyf (t) = tk , k! g f (t) k=0 (

(y ∈ C) ,

(see [17, 21, 24]) ,

( ) where f (t) is the compositional inverse of f (t) with f (f (t)) = f f (t) = t. It is well known that the Sheffer identity is given by (1.14) ∞ ( ) ∑ n sn (x + y) = sj (x) pn−j (y) , where pn (x) = g (t) sn (x) , (see [17, 24]) . j j=0 For sn (x) ∼ (g (t) , f (t)), we have ) ( 1 g ′ (x) sn (x) , (1.15) sn+1 (x) = x − g (x) f ′ (x) (1.16)

sn (x) =

(n ≥ 0) ,

n ⟩ ∑ )−1 1 ⟨ ( j f (t) xn xj , g f (t) j! j=0

and (1.17)

⟨ f (t)| xp (x)⟩ = ⟨ ∂t f (t)| p (x)⟩ ,

f (t) sn (x) = nsn−1 (x) ,

(n ≥ 1) .

Let sn (x) ∼ (g (t) , f (t)) and rn (x) ∼ (h (t) , l (t)), (n ≥ 0). Then we have (1.18)

sn (x) =

n ∑

Cn,m rm (x) ,

(n ≥ 0) ,

m=0

where (1.19)

Cn,m

1 = m!

⟩ ⟨ ( ) )m n h f (t) ( ( ) l f (t) x , g f (t)

(see [17, 21, 24]) .

(ν)

In this paper, we consider the polynomials F Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) whose generating function is given by ) ) r ( r ( ∏ ∏ 2 1 − λj 1 − λj −νt2 /2 xt (1.20) e e = ext−νt /2 aj t − λ aj t − λ e e j j j=1 j=1 =

∞ ∑ n=0

F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr )

tn , n!

where r ∈ Z>0 , a1 , . . . , ar , λ1 , . . . , λr ∈ C with a1 , . . . , ar ̸= 0, λ1 , . . . , λr ̸= 1, and (ν) ν ∈ R with ν ̸= 0. F Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) are called Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomials. (ν) (ν) When x = 0, F Hn (a1 , . . . , ar ; λ1 , . . . , λr ) = F Hn (0 | a1 , . . . , ar ; λ1 , . . . , λr ) are called the Barnes’ multi[ple Frobenius-Euler and Hermite mixed-type numbers. (ν) We observe here that F Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ), Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ),

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and Hn (x) are respectively Appell sequences for and e

νt2 /2

(1.21)

. That is,



F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) ∼ 

j=1

(1.22)

Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) ∼ 

)

1 − λj

) r ( aj t ∏ e − λj j=1

and

eaj t −λj 1−λj

1 − λj

2

eνt

) r ( aj t ∏ e − λj j=1



(

∏r

/2

,

∏r

(

j=1

n

eaj t −λj 1−λj

) ,

 eνt

2

/2

, t ,

 , t ,

( 2 ) Hn(ν) (x) ∼ eνt /2 , t .

(1.23)

From the Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomials, we investigate some properties of those polynomials. Finally, we give some new and interesting identities which are derived from umbral calculus. 2. Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomials From (1.21), (1.22) and (1.23), we note that d (2.1) tF Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) = F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) dx (ν) = nF Hn−1 (x | a1 , . . . , ar ; λ1 , . . . , λr ) , (2.2)

d Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) dx = nHn−1 (x | a1 , . . . , ar ; λ1 , . . . , λr ) ,

tHn (x | a1 , . . . , ar ; λ1 , . . . , λr ) =

and d (ν) (ν) H (x) = nHn−1 (x) . dx n Now, we give explicit expressions related to the Barnes’ multiple Frobenius-Euler and Hermite mixed-type polynomials. From (1.13), we note that ) r ( ∏ 1 − λj (ν) −νt2 /2 (2.4) F Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) = e xn aj t − λ e j j=1 tHn(ν) (x) =

(2.3)

=e−νt

2

Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) 1 ( ν )m 2m = − t Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) m! 2 m=0 /2

∞ ∑

[ n2 ] ∑ 1 ( ν )m = − (n)2m Hn−2m (x | a1 , . . . , ar ; λ1 , . . . , λr ) m! 2 m=0 [ n2 ] ( ) ∑ n (2m)! ( ν )m = − Hn−2m (x | a1 , . . . , ar ; λ1 , . . . , λr ) . 2m m! 2 m=0

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By (1.9), we get (2.5)

F Hn(ν) (y | a1 , . . . , ar ; λ1 , . . . , λr ) ⟩ ⟨ ∞ ∑ ti n (ν) = F Hi (y | a1 , . . . , ar ; λ1 , . . . , λr ) x i! i=0 ⟨ r ( ⟩ ) ∏ 1 − λj −νt2 /2 yt n e e x = a t j e − λj j=1 ⟨ r ( ⟩ ) ∏ −νt2 /2 yt n 1 − λj = e e x eaj t − λj j=1 ⟨ r ( ⟩ ) ∑ ∏ ∞ (ν) 1 − λj tl n = Hl (y) x eaj t − λj l! j=1 l=0 ⟨ r ( ⟩ ) n ( ) ∑ ∏ n−l n 1 − λ j (ν) = Hl (y) x l eaj t − λj j=1 l=0 ⟨ ∞ ⟩ n ( ) ∑ ∑ ti n−l n (ν) Hi (a1 , . . . , ar ; λ1 , . . . λr ) x = Hl (y) i! l i=0 l=0 ( ) n ∑ n (ν) = Hn−l (a1 , . . . , ar ; λ1 , . . . , λr ) Hl (y) . l l=0

Thus, by (2.5), we get (2.6) F Hn(ν)

n ( ) ∑ n (ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) = Hn−l (a1 , . . . , ar ; λ1 , . . . , λr ) Hl (x) . l l=0

Therefore, by (2.4) and (2.6), we obtain the following theorem. Theorem 2.1. For n ≥ 0, we have F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) [ n2 ] ( ) ∑ n (2m)! ( ν )m = − Hn−2m (x | a1 , . . . , ar ; λ1 , . . . , λr ) 2m m! 2 m=0 n ( ) ∑ n (ν) = Hn−l (a1 , . . . , ar ; λ1 , . . . , λr ) Hl (x) . l l=0

From (1.9), we have (2.7)

F Hn(ν) (y | a1 , . . . , ar ; λ1 , . . . , λr ) ⟩ ⟨ ∞ i ∑ t (ν) = F Hi (y | a1 , . . . , ar ; λ1 , . . . , λr ) xn i! i=0 ⟨ r ( ⟩ ) ∏ 1 − λj −νt2 /2 yt n = e e x a t j e − λj j=1

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⟨ = e−νt ⟨

2

⟩ ∏ ) r ( 1 − λ j /2 eyt xn a t j=1 e j − λj

⟩ ∞ ∑ tl n = e Hl (y | a1 , . . . , ar ; λ1 , . . . , λr ) x l! l=0 ⟨ ∞ ⟩ ( ) n ∑ n ∑ (ν) ti n−l = Hl (y | a1 , . . . , ar ; λ1 , . . . , λr ) Hi x l i! i=0 l=0 n ( ) ∑ n (ν) Hl (y | a1 , . . . , ar ; λ1 , . . . , λr ) Hn−l . = l −νt2 /2

l=0

Thus, by (2.7), we get (2.8) F Hn(ν)

n ( ) ∑ n (ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) = Hl (x | a1 , . . . , ar ; λ1 , . . . , λr ) Hn−l . l l=0

(ν)

will use the conjugation representation in (1.16). For F Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) ∼ ) ( Now, we ∏r ( eaj t −λ ) 2 g (t) = j=1 1−λj j eνt /2 , f (t) = t , we observe that ⟩ ⟨ ( )−1 j g f (t) f (t) xn (2.9) ⟩ ⟨ r ( ) ∏ 1 − λj −νt2 /2 j n e t = x a t e j − λj j=1 ⟩ ⟨ r ( ) ∏ 1 − λj −νt2 /2 j n e = t x eaj t − λj j=1 ⟨ r ( ⟩ ) ∏ 1 − λj −νt2 /2 n−j = (n)j e x eaj t − λj j=1 ⟨ ⟩ 2 = (n)j e−νt /2 Hn−j (x | a1 , . . . , ar ; λ1 , . . . , λr ) ⟩ ⟨ ∞ ∑ 1 ( ν )m 2m = (n)j − t Hn−j (x | a1 , . . . , ar ; λ1 , . . . , λr ) m! 2 m=0

= (n)j

n−j [∑ 2 ]

m=0

1 ( ν )m − (n − j)2m Hn−j−2m (a1 , . . . , ar ; λ1 , . . . , λr ) . m! 2

From (1.16) and (2.9), we can derive the following equation: (2.10)

F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) n ( ) [∑ 2 ] ∑ 1 ( ν )m n − (n − j)2m Hn−j−2m (a1 , . . . , ar ; λ1 , . . . , λr ) xj . = m! 2 j m=0 j=0 n−j

Therefore, by (2.8) and (2.10), we obtain the following theorem.

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BARNES’ MULTIPLE FROBENIUS-EULER AND HERMITE MIXED-TYPE POLYNOMIALS 7

Theorem 2.2. For n ≥ 0, we have F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) n ( ) ∑ n (ν) = Hn−l Hl (x | a1 , . . . , ar ; λ1 , . . . , λr ) l l=0

n ( ) [∑ 2 ] ∑ n 1 ( ν )m − = (n − j)2m Hn−j−2m (a1 , . . . , ar ; λ1 , . . . , λr ) xj . j m! 2 m=0 j=0 n−j

Remark. From (1.14), we have F Hn(ν) (x + y | a1 , . . . , ar ; λ1 , . . . , λr ) n ( ) ∑ n (ν) = F Hj (x | a1 , . . . , ar ; λ1 , . . . , λr ) y n−j . j j=0

(2.11)

By (1.15) and (1.21), we get (ν)

F Hn+1 (x | a1 , . . . , ar ; λ1 , . . . , λr ) ) ( g ′ (t) F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) , = x− g (t) ∏r ( eaj t −λ ) 2 where g (t) = j=1 1−λj j eνt /2 . Now, we compute that (2.12)

g ′ (t) ′ = (log g (t)) g (t)  ′ r r ∑ ( aj t ) ∑ 1 log e − λj − log (1 − λj ) + νt2  = 2 j=1 j=1

(2.13)

=

r ∑ aj eaj t + νt. a e j t − λj j=1

So (2.14)

g ′ (t) F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) g (t) ) r r ( ∑ 2 1 − λj ∏ 1 − λi aj eaj t = · aj t e−νt /2 xn ai t − λ 1 − λ e − λ e j j i=1 i j=1 + νtF Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) =

r ∑ j=1

aj F Hn(ν) (x + aj | a1 , . . . , ar , aj ; λ1 , . . . , λr , λj ) 1 − λj (ν)

+ nνF Hn−1 (x | a1 , . . . , ar ; λ1 , . . . , λr ) =

r ∑ j=1

aj F Hn(ν) (x + aj | a1 , . . . , ar , aj ; λ1 , . . . , λr , λj ) 1 − λj (ν)

+ nνF Hn−1 (x | a1 , . . . , ar , aj ; λ1 , . . . , λr ) .

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By (2.12) and (2.14), we get (2.15)

(ν)

F Hn+1 (x | a1 , . . . , ar ; λ1 , . . . , λr ) =xF Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) −

r ∑ j=1

aj F Hn(ν) (x + aj | a1 , . . . , ar , aj ; λ1 , . . . , λr , λj ) 1 − λj (ν)

− nνF Hn−1 (x | a1 , . . . , ar ; λ1 , . . . , λr ) . For n ≥ 2, by (1.9), we get (2.16)

F Hn(ν) (y | a1 , . . . , ar ; λ1 , . . . , λr ) ⟩ ⟨ ∞ ∑ ti n (ν) = F Hi (y | a1 , . . . , ar ; λ1 , . . . , λr ) x i! i=0 ⟨ r ( ⟩ ) ∏ 1 − λj −νt2 /2 yt n = e e x a t j e − λj j=1   ⟨ ⟩ ) r ( ∏ 2 1 − λ j −νt /2 yt  n−1  = ∂t e e x eaj t − λj j=1   ⟨ ⟩ ) r ( ∏ 1 − λj −νt2 /2 yt n−1   ∂t = e e x eaj t − λj j=1 ⟨ r ( ⟩ )( ) ∏ 2 1 − λj −νt /2 yt n−1 + ∂t e e x eaj t − λj j=1 ⟩ ⟨ r ( ) ∏ ( yt ) n−1 1 − λj −νt2 /2 . + e ∂t e x eaj t − λj j=1

The third term is

⟨

(2.17)

y

⟩ ) r ( ∏ 1 − λj −νt2 /2 yt n−1 e e x eaj t − λj j=1 (ν)

=yF Hn−1 (y | a1 , . . . , ar ; λ1 , . . . , λr ) . The second term is ⟩ ) r ( ∏ n−1 1 − λj −νt2 /2 yt −ν e e tx eaj t − λj j=1 ⟨ r ( ⟩ ) ∏ 1 − λj −νt2 /2 yt n−2 = − ν (n − 1) e e x eaj t − λj j=1 ⟨

(2.18)

(ν)

= − ν (n − 1) F Hn−2 (y | a1 , . . . , ar ; λ1 , . . . , λr ) .

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We observe that

 ) r ( ∏ 1 − λj  ∂t  aj t − λ e j j=1 

(2.19)

)′ ∏ ( ) r ( ∑ 1 − λi 1 − λj eai t − λi eaj t − λj i=1 j̸=i )∑ r ( r ∏ 1 − λj ai eai t , =− eaj t − λj i=1 eai t − λi j=1 =

where (2.20) r r ∞ r ∑ ∑ ∑ ∑ ai ai am a i ea i t i m ai t 1 − λi ai t = e = e H (λ ) t . m i ai t − λ ai t − λ e 1 − λ e 1 − λ m! i i i i m=0 i=1 i=1 i=1 So, by (2.19) and (2.20), we get (2.21)   )∑ r ( r r ∞ ∏ ∏ ∑ 1 − λ 1 − λj ai am j i m ai t   = − ∂t e t . H (λ ) m i aj t − λ aj t − λ e e 1 − λ m! j j i m=0 j=1 j=1 i=1 Now, the first term is (2.22) −

r ∑ i=1

ai 1 − λi

⟨

⟩ n−1 ) r ( m ∑ ∏ 2 1 − λ a j e(y+ai )t e−νt /2 Hm (λi ) i tm xn−1 aj t − λ e m! j m=0 j=1

) n−1 ( ai ∑ n − 1 =− Hm (λi ) am i 1 − λ m i m=0 i=1 ⟨ ⟩ ) r ( ∏ 2 1 − λj (y+ai )t −νt /2 n−1−m e × e x eaj t − λj j=1 r ∑

=−

r ∑

) n−1 ( ai ∑ n − 1 Hm (λi ) am i 1 − λi m=0 m

i=1 ∞ ∑

⟩ tl n−1−m × (y + ai | a1 , . . . , ar ; λ1 , . . . , λr ) x l! l=0 ( ) r n−1 ∑ ∑ n − 1 am+1 (ν) i =− Hm (λi ) F Hn−1−m (y + ai | a1 , . . . , ar ; λ1 , . . . , λr ) . m 1 − λ i i=1 m=0 ⟨

(ν) F Hl

Therefore, by (2.16), (2.17), (2.18) and (2.22), we obtain the following theorem. Theorem 2.3. For n ≥ 2, we have F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) (ν)

(ν)

=xF Hn−1 (x | a1 , . . . , ar ; λ1 , . . . , λr ) − ν (n − 1) F Hn−2 (x | a1 , . . . , ar ; λ1 , . . . , λr )

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−

r n−1 ∑ ∑ (n − 1) am+1 i

i=1 m=0

1 − λi

m

(ν)

Hm (λi ) F Hn−1−m (x + ai | a1 , . . . , ar ; λ1 , . . . , λr ) .

Remark. We compute the following in two different ways in order to derive an identity: ⟩ ⟨ r ( ) ∏ 2 1 − λj −νt /2 m n e t x , (m, n ≥ 0) . a t j e − λj j=1 On one hand, it is

⟩ ) r ( ∏ 1 − λj −νt2 /2 m n e t x a t e j − λj j=1 ⟩ ⟨ r ( ) ∏ 1 − λj −νt2 /2 n−m = (n)m e x eaj t − λj j=1

(2.23)

⟨

(ν)

= (n)m F Hn−m (a1 , . . . , ar ; λ1 , . . . , λr ) . On the other hand, it is  ⟨  r ( ⟩ ) ∏ n−1 2 1 − λ j −νt /2 m (2.24) ∂t  e t  x eaj t − λj j=1  ⟨ ⟩ ) r ( ∏ n−1 2 1 − λ j −νt /2 m e t x = ∂t eaj t − λj j=1 ⟨ r ( ⟩ )( ) ∏ 2 1 − λj −νt /2 m n−1 + ∂t e t x eaj t − λj j=1 ⟨ r ( ⟩ ) ∏ 1 − λj −νt2 m n−1 e (∂t t ) x + . eaj t − λj j=1 From (2.23) and (2.24), we can derive the following equation: for n ≥ m + 2, (2.25) (ν)

F Hn−m (a1 , . . . , ar ; λ1 , . . . , λr ) (ν)

= − ν (n − m − 1) F Hn−m−2 (a1 , . . . , ar ; λ1 , . . . , λr ) r n−m−1 ∑ ∑ (n − m − 1) al+1 (ν) i − Hl (λi ) F Hn−1−l−m (ai ; a1 , . . . , ar ; λ1 , . . . , λr ) . 1 − λ l i i=1 l=0 ) 2 ) (∏ ( aj t (ν) e −λj r νt /2 e , For F Hn (x | a1 , . . . , ar ; λ1 , . . . , λr ) ∼ t , (x)n ∼ (1, et − 1), j=1 1−λj we have n ∑ (2.26) F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) = Cn,m (x)m , m=0

(2.27)

Cn,m

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⟩ ) r ( ∏ ( ) 2 1 − λj m −νt /2 t e e − 1 xn aj t − λ e j j=1 ⟨ r ( ⟩ ) ∏ ( ) 2 1 − λj 1 m = e−νt /2 et − 1 xn aj t − λ e m! j j=1 ⟩ ⟨ r ( ∑ ) ∞ l ∏ 2 t 1 − λj e−νt /2 = S2 (l, m) xn eaj t − λj l! l=m j=1 ⟨ r ( ⟩ ) n ( ) ∑ ∏ n 1 − λj −νt2 /2 n−l = S2 (l, m) e x l eaj t − λj j=1 1 = m!

⟨

l=m

=

n ( ) ∑ n (ν) S2 (l, m) F Hn−l (a1 , . . . , ar ; λ1 , . . . , λr ) . l

l=m

Therefore, by (2.26) and (2.27), we obtain the following theorem. Theorem 2.4. For n ≥ 0, we have F Hn(ν) (x | a1 , . . . , ar ; λ1 . . . , λr ) =

n ∑ n ( ) ∑ n (ν) S2 (l, m) F Hn−l (a1 , . . . , ar ; λ1 , . . . , λr ) (x)m . l m=0 l=m

It is easy to show that

( ) x(n) = x (x + 1) · · · (x + n − 1) ∼ 1, 1 − e−t .

From (1.18) and (1.19), we have (2.28)

F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) =

n ∑

Cn,m x(m) ,

m=0

where (2.29)

Cn,m ⟩ ⟨ r ( ) ∏ ( ) 2 1 − λj 1 −νt /2 −t m n e 1 − e = x a t j m! j=1 e − λj ⟩ ⟨ r ( ) ∏ ( ) 2 1 − λj 1 m n −νt /2 −mt t x e e e − 1 = m! j=1 eaj t − λj ⟨ r ( ⟩ ) ∏ ( t )m n 1 − λj −νt2 /2 −mt 1 = e e m! e − 1 x eaj t − λj j=1 ⟨ r ( ⟩ ∞ ) ∏ ∑ 2 1 − λj tl n −νt /2 −mt = S2 (l, m) x e e eaj t − λj l! l=m j=1 ⟨ r ( ⟩ ) n ( ) ∑ ∏ 2 n 1 − λj −νt /2 −mt n−l = S2 (l, m) e e x l eaj t − λj j=1 l=m

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⟨ ∞ ⟩ n ( ) ∑ ∑ n ti n−l (ν) = S2 (l, m) F Hi (−m | a1 , . . . , ar ; λ1 , . . . , λr ) x l i! i=0 l=m n ( ) ∑ n (ν) = S2 (l, m) F Hn−l (−m | a1 , . . . , ar ; λ1 , . . . , λr ) . l l=m

Therefore, by (2.28) and (2.29), we obtain the following theorem. Theorem 2.5. For n ≥ 0, we have F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) n ∑ n ( ) ∑ n (ν) = S2 (l, m) F Hn−l (−m | a1 , . . . , ar ; λ1 , . . . , λr ) x(m) . l m=0 l=m

From (1.4), (1.13), (1.18), (1.19) and (1.21), we have (2.30)

F Hn(ν)

(x | a1 , . . . , ar ; λ1 , . . . , λr ) =

n ∑

(s) Cn,m Bm (x) ,

(s ∈ N) ,

m=0

where (2.31) Cn,m ⟨ r ( ) ( t )s ⟩ ∏ 1 1 − λj e −1 −νt2 /2 m n = e t x a t m! j=1 e j − λj t ⟩ ⟨ ( t ) )s ( ) ∏ r ( n 1 − λj −νt2 /2 e − 1 n−m e x = eaj t − λj t m j=1 ⟩ ∑ ( )⟨∏ ) r ( ∞ 1 − λj n s! −νt2 /2 l n−m = e S2 (l + s, s) t x m eaj t − λj l=0 (l + s)! j=1 ⟨ r ( ⟩ ( ) n−m ) ∑ ∏ s! n 1 − λj −νt2 /2 n−m−l = S2 (l + s, s) (n − m)l e x m (l + s)! eaj t − λj j=1 l=0 (n−m) ( ) n−m n ∑ (ν) l = (l+s ) S2 (l + s, s) F Hn−m−l (a1 , . . . , ar ; λ1 , . . . , λr ) . m s l=0

Therefore, by (2.30) and (2.31), we obtain the following theorem. Theorem 2.6. For n ≥ 0, and s ∈ N, we have F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) (n−m) n ( ) n−m ∑ n ∑ (ν) (s) l = (x) . (l+s ) S2 (l + s, s) F Hn−m−l (a1 , . . . , ar ; λ1 , . . . , λr ) Bm m s m=0 l=0

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From (1.1), (1.18), (1.19) and (1.21), we have (2.32)

F Hn(ν) (x | a1 , . . . , ar ; λ1 , , . . . , λr ) =

n ∑

Cn,m H(s) m (x | λ) ,

(s ∈ N) ,

m=0

where (2.33)

Cn,m ⟨ r ( ) ( t )s ⟩ ∏ 2 e −λ 1 1 − λj e−νt /2 tm xn = a t j m! j=1 e − λj 1−λ ⟨ r ( ⟩ ) ∏ ( ) 2 1 1 − λj s m n −νt /2 t = e e − λ t x s eaj t − λj m! (1 − λ) j=1 ⟩ (n) ⟨ r ( ∑ ) ( ) ∏ s s 1 − λj s−j jt n−m −νt2 /2 m e x e (−λ) = s eaj t − λj (1 − λ) j=0 j j=1 ⟨ r ( ⟩ (n) ) s ( ) ∑ ∏ 2 s 1 − λ j s−j −νt /2 jt n−m m = (−λ) e e x s eaj t − λj (1 − λ) j=0 j j=1 (n) s ( ) ∑ s (ν) s−j m (−λ) F Hn−m (j | a1 , . . . , ar ; λ1 , . . . , λr ) . = s (1 − λ) j=0 j

Therefore, by (2.32) and (2.33), we obtain the following theorem. Theorem 2.7. For n ≥ 0, we have F Hn(ν) (x | a1 , . . . , ar ; λ1 , . . . , λr ) n ( )∑ s ( ) ∑ 1 n s (ν) s−j = (−λ) F Hn−m (j | a1 , . . . , ar ; λ1 , . . . , λr ) H(s) s m (x|λ) . (1 − λ) m=0 m j=0 j References 1. T. Agoh and M. Yamanaka, A study of Frobenius-Euler numbers and polynomials, Ann. Sci. Math. Qu´ebec 34 (2010), no. 1, 1–14. MR 2744192 (2012a:11024) 2. A. Bayad and T. Kim, Results on values of Barnes polynomials, Rocky Mountain J. Math. 43 (2013), no. 6, 1857–1869. MR 3178446 3. A. Bayad, T. Kim, W.J. Kim, S. H. Lee, Arithmetic properties of q-Barnes polynomials, J. Comput. Anal. Appl. 15 (2013), no. 1, 111–117. MR 3076723 4. M. Can, M. Cenkci, V. Kurt, and Y. Simsek, Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler l-functions, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 2, 135–160. MR 2508979 (2010a:11072) ˙ N. Cang¨ 5. I. ul, V. Kurt, H. Ozden, and Y. Simsek, higher-order w-q-Genocchi numbers, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2009), no. 1, 39–57. MR 2542124 (2011b:05010) 6. L. Carlitz, Some polynomials related to the Bernoulli and Euler polynomials, Utilitas Math. 19 (1981), 81–127. MR 624049 (82j:10023)

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13. 14.

15. 16.

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, Some remarks on the multiplication theorems for the Bernoulli and Euler polynomials, Glas. Mat. Ser. III 16(36) (1981), no. 1, 3–23. MR 634291 (83b:10009) R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 433–438. MR 2976601 S. Gaboury, R. Tremblay, and B.-J. Fug`ere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115–123. MR 3184467 H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79 (1972), 44–51. MR 0306102 (46 #5229) D. S. Kim and T. Kim, Higher-order Frobenius-Euler and poly-Bernoulli mixedtype polynomials, Adv. Difference Equ. (2013), 2013:251, 13. MR 3108262 D. S. Kim, T. Kim, S.-H. Lee, and S.-H. Rim, A note on the higherorder Frobenius-Euler polynomials and Sheffer sequences, Adv. Difference Equ. (2013), 2013:41, 12. MR 3032702 D. S. Kim, T. Kim, q-Bernoulli polynomials and q-umbral calculus, Sci. China Math. no. 9, 57 (2014), no. 9, 1867–1874. MR 3249396 D.S. Kim, T. Kim, T. Komatsu, S.-H. Lee Barnes-type Daehee of the first kind and poly-Cauchy of the first kind mixed-type polynomials, Adv. Difference Equ. 2014 (2014), 2014:140, 22 pp. MR 3259855 T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), no. 3, 261–267. MR 2012900 (2004j:11106) , An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Zp , Rocky Mountain J. Math. 41 (2011), no. 1, 239–247. MR 2845943 (2012k:11027) , Identities involving Laguerre polynomials derived from umbral calculus, Russ. J. Math. Phys. 21 (2014), no. 1, 36–45. MR 3182545 T. Kim and J. Choi, A note on the product of Frobenius-Euler polynomials arising from the p-adic integral on Zp , Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 2, 215–223. MR 2961614 T. Kim and B. Lee, Some identities of the Frobenius-Euler polynomials, Abstr. Appl. Anal. (2009), Art. ID 639439, 7. MR 2487361 (2010a:11231) T. Kim, A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials, Russ. J. Math. Phys. 22 (2015), no. 1, 26–33. T. Kim and T. Mansour, Umbral calculus associated with Frobenius-type Eulerian polynomials, Russ. J. Math. Phys. 21 (2014), no. 4, 484–493. MR 3284958 Q.-M. Luo and F. Qi, Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 7 (2003), no. 1, 11–18. MR 1981601 H. Ozden, I. N. Cangul, and Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 1, 41–48. MR 2479746 (2009k:11037) S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984. MR 741185 (87c:05015) K. Shiratani, On Euler numbers, Mem. Fac. Sci. Kyushu Univ. Ser. A 27 (1973), 1–5. MR 0314755 (47 #3307)

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26. Y. Simsek, O. Yurekli, and V. Kurt, On interpolation functions of the twisted generalized Frobenius-Euler numbers, Adv. Stud. Contemp. Math. (Kyungshang) 15 (2007), no. 2, 187–194. MR 2356176 (2008g:11193) Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected] Institute of Mathematics and Computer Science, Far Eastern Federal University, 690950 Vladivostok, Russia E-mail address: d− [email protected] Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected]

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Robust stability and stabilization of linear uncertain stochastic systems with Markovian switching

Yifan Wu1 Department of Basic Courses Jiangsu Food & Pharmaceutical Science College, HuaiAn, Jiangsu, 223003, China

Abstract.

This paper is concerned with robust stability and stabilization problem for a class of

linear uncertain stochastic systems with Markovian switching. The uncertain system under consideration involves parameter uncertainties both in the drift part and in the diffusion part. New criteria for testing the robust stability of such systems are established in terms of bi-linear matrix inequalities (BLMIs), and sufficient conditions are proposed for the design of robust state-feedback controllers. An example illustrates the proposed techniques. Keywords: Bi-linear matrix inequalities (BLMIs); Robust stabilization; Stochastic system with Markovian switching; Uncertainty

1

Introduction Stochastic systems with Markovian switching have been used to model many practical systems

where they may experience abrupt changes in their structure and parameters. Such systems have played a crucial role in many applications, such as hierarchical control of manufacturing systems([4, 5, 16]), financial engineering ([19]) and wireless communications ( [6]). In the past decades, the stability and control of Markovian jump systems have recently received a lot of attention. For example, [3] and [15] systematically studied stochastic stability properties of jump linear systems. [1] discussed the stability of a semi-linear stochastic differential equation with Markovian switching. [7, 9, 10, 12] discussed the exponential stability of general nonlinear stochastic systems with Markovian switching of the form dx(t) = f (x(t), t, r(t))dt + g(x(t), t, r(t))dB(t). 1 E-mail

(1.1)

address: [email protected]

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Over the last decade, stochastic control problems governed by stochastic differential equation with Markovian switching have attracted considerable research interest, and we here mention[2, 11, 20, 23, 24]. It is well known that uncertainty occurs in many dynamic systems and is frequently a cause of instability and performance degradation. In the past few years, considerable attention has been given to the problem of designing robust controllers for linear systems with parameter uncertainty, such as[8, 13, 17, 21, 22]. However, a literature search reveals that the issue of stabilization of uncertain system under consideration involves parameter uncertainties both in the drift part and in the diffusion part has not been fully investigated and remains important and challenging. This situation motivates the present study on the robust stabilization of linear uncertain stochastic systems with Markovian switching. We aim at designing a robust state-feedback controller such that, for all admissible uncertainties , the closed-loop system is exponentially stable in mean square. The structure of this paper is as follows. In Section 2, we introduce notations, definitions and results required from the literature. In Section 3, we shall discuss the problem of mean square exponential stabilization for a linear jump stochastic system. In Section 4, sufficient conditions are proposed for the design of robust state-feedback controllers. An example is discussed for illustrating our main results in Section 5.

2

Preliminaries

In this paper, we will employ the following notation. Let |.| be the Euclidean norm in Rn . The interval [0, ∞) be denoted by R+ . If A is a vector or matrix, its transpose is denoted by AT . In denotes the n × n identity matrix. If A is a symmetric matrix λmin (A) and λmax (A) mean the smallest and largest eigenvalue, respectively. If A and B are symmetric matrices, by A > B and A ≥ B we mean that A − B is positive definite and nonnegative definite, respectively. And C 2,1 (Rn × R+ × S; R+ ) denotes the family of all R+ -valued functions on Rn × R+ × S which are continuously twice differentiable in x and once differentiable in t. We write diag(a1 , ..., an ) for a diagonal matrix whose diagonal entries starting in the upper left corner are a1 , ..., an . Let (Ω, F, (F)t , P ) be a complete probability space with a filtration (F)t satisfying the usual conditions. Let r(t), t ≥ 0, be a right-continuous Markov chain on the probability space taking values in a finite state space S = {1, 2, ..., N } with generator Q = (qij )N ×N given by    qij ∆ + o(∆), if i 6= j P (r(t + ∆) = j | r(t) = i) =   1 + qii ∆ + o(∆), if i = j where ∆ > 0, and qij ≥ 0 denotes the switching rate from i to j if i 6= j while qii = −

X

qij .

i6=j

2

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Definition 1 ([9]) The trivial solution of system (1), or simply system (1) is said to be exponentially stable in mean square if 1 lim sup log(E|x(t; t0 , x0 , r0 )|2 ) < 0, t→∞ t

for all (t0 , x0 , r0 ) ∈ R+ × Rn × S. If V ∈ C 2,1 (Rn × R+ × S; R+ ) , define operator LV (x, t, i) associated with system (1) by ∂V (x, t, i) ∂V (x, t, i) + f (x, t, i) ∂t ∂x N X 1 ∂ 2 V (x, t, i) + tr[g T (x, t, i) g(x, t, i)] + qij V (x, t, i). 2 ∂x2 j=1

LV (x, t, i) =

We have the following lemma. Lemma 2.1([9]) Let λ, c1 , c2 be positive numbers. Assume that there exists a function V (x, t, i) ∈ C 2,1 (Rn × R+ × S; R+ ) such that c1 |x(t)|2 ≤ V (x, t, i) ≤ c2 |x(t)|2 and LV (x, t, i) ≤ −λ|x(t)|2 for all (x, t, i) ∈ Rn × R+ × S, then system (1) is exponentially stable in mean square. In this note, we consider the following linear uncertain stochastic systems with Markovian switching: ˜ dx(t) = A(r(t))x(t)dt +

d X

˜k (r(t))x(t)dwk (t), B (2.2)

k=1

x(t0 ) = x0 ∈ Rn , t ≥ t0 , where w(t) = (w1 (t), w2 (t), · · · , wd (t))T denotes a d-dimensional Brownian motion or Wiener process, x(t) ∈ Rn is the system state, we assume that w(t) and r(t) are independent. For any i ∈ S, 1 ≤ ˜ ˜ki = B ˜k (r(t) = i) are not precisely known a priori, but belong to the k ≤ d, A˜i = A(r(t) = i) and B following admissible uncertainty domains: Da = {Ai + D0i F0i (t)E0i : F0i (t)T F0i (t) ≤ I, i ∈ S}, Dbk = {Bki + Dki Fki (t)Eki : Fki (t)T Fki (t) ≤ I, i ∈ S}, where Ai , Bki , D0i , E0i , Dki , Eki are known constant real matrices with appropriate dimensions, while F0i (t) and Fki (t) denotes the uncertainties in the system matrices, for all i ∈ S.

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Lemma 2.2 ([14, 18]) Let A, D, E, W and F (t) be real matrices of appropriate dimensions such that F T (t)F (t) ≤ I and W > 0, then , 1) For scalar ε > 0,

DF (t)E + (DF (t)E)T ≤ εDDT + 1ε E T E

2) For any scalar ε > 0 such that W − εDDT > 0, (A + DF (t)E)T W −1 (A + DF (t)E) ≤ AT (W − εDDT )−1 A + 1ε E T E.

3

Robust stability analysis

This section, we discuss the robust stability for system (2). For convenience, we will let the initial values x0 and r0 be non-random, namely x0 ∈ Rn and r0 ∈ S, but the theory developed in this paper can be generalized without any difficulty to cope with the case of random initial values, and we write x(t; t0 , x0 , r0 ) = x(t) simply. Theorem 3.1 Suppose that there exist N symmetric positive-definite matrices Pi and positive scalars εi , γi , and λi , such that ∀ i ∈ S, the following BLMIs hold:

  Π11  E P  0i i    Π31    Π41   Π51

 ∗

∗

∗

−γi I

∗

∗

0

Π33

∗

0

0

−εi I

0

0

0

∗   ∗     < 0, ∗    ∗    Π55

(3.3)

where the symbol ‘∗’ denotes the transposed element at the symmetric position, and T Π11 = Ai Pi + Pi ATi + qii Pi + λi Pi + γi D0i D0i , T T T T Π31 = [Pi B1i , Pi B2i , . . . , Pi Bdi ] , T T T T Π41 = [Pi E1i , Pi E2i , . . . , Pi Edi ] , T T Π33 = diag[εi D1i D1i − Pi , . . . , εi Ddi Ddi − Pi ],

Π51 = [Pi , Pi , . . . , Pi ]T , | {z } N −1

Π55 = diag[

−1 −1 −1 −1 P1 , . . . , Pi−1 , Pi+1 , . . . , PN ], qi1 qi(i−1) qi(i+1) qiN

then system (2) is exponentially stable in mean square. Proof Let Xi = Pi−1 and define V (x, i) = xT Xi x for all i ∈ S. And let c1 = min{λmin (Xi ) : i ∈ S}, c2 = max{λmax (Xi ) : i ∈ S}, it is clear that c1 |x(t)|2 ≤ V (x, i) ≤ c2 |x(t)|2 .

(3.4) 4

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On the other hand, a calculation shows that N X  LV (x, i) = x(t)T Xi (Ai + D0i F0i E0i ) + (Ai + D0i F0i E0i )T Xi + qij Xj j=1

+

d X

 (Bki + Dki Fki Eki )T Xi (Bki + Dki Fki Eki ) x(t),

k=1 T by Lemma 2.2, for all i ∈ S, if there exist positive scalars εi and γi such that εi Dki Dki − Pi < 0,

1 ≤ k ≤ d, then we have  T T E0i LV (x, i) ≤ x(t)T Xi Ai + ATi Xi + γi Xi D0i D0i Xi + γ1i E0i d d N X X 1 X  T T −1 T + Bki (Pi − εi Dki Dki ) Bki + Eki Eki + qij Xj x(t). εi j=1 k=1

k=1

Thus, there exists a λ > 0 such that LV (x, i) ≤ −λ|x(t)|2 will hold if for any i ∈ S there exists a λi > 0 such that T T E0i Xi Ai + ATi Xi + γi Xi D0i D0i Xi + γ1i E0i d d N X X 1 X T T −1 T + Bki (Pi − εi Dki Dki ) Bki + E Eki + qij Xj + λi Xi εi ki j=1 k=1

(3.5)

k=1

< 0. Pre- and post-multiplying (5) by Pi yields T T E0i Pi Ai Pi + Pi ATi + γi D0i D0i + γ1i Pi E0i d X T T −1 + Pi Bki (Pi − εi Dki Dki ) Bki Pi

+

k=1 d X k=1

X 1 T Pi Eki Eki Pi + qij Pi Pj−1 Pi + qii Pi + λi Pi εi j6=i

< 0, which is equivalent to inequality (3) in view of Schur complement equivalence. The assertion of this theorem follows from Lemma 2.1 immediately. Remark 1 Theorem 3.1 provides the analysis results for the exponential stability of the system (2). It can be seen from (3) that we need to check whether there exist N symmetric positive-definite matrices Pi and positive scalars εi , γi , and λi meeting the N coupled matrix inequalities. It is clear that inequality (3) is BLMIs, and it is LMIs for a prescribed λi , then we are able to determine exponential stability of the system (3) readily by checking the solvability of the LMIs.

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4

Robust stabilization synthesis

This section deals with the robust stabilization problem for linear uncertain stochastic systems with Markovian switching. Let us consider the uncertain stochastic control system of the form d X     ˜ ˜k (r(t))x(t) + Ck (r(t))u(t) dwk (t), dx(t) = A(r(t))x(t) + C(r(t))u(t) dt + B

(4.6)

k=1

x(t0 ) = x0 ∈ Rn , t ≥ t0 . We aim to design a state-feedback controller u(t) = K(r(t))x(t) such that the resulting closed-loop system d X     ˜ ˜k (r(t)) + Ck (r(t))K(r(t)) x(t)dwk (t), dx(t) = A(r(t)) + C(r(t))K(r(t)) x(t)dt + B k=1

(4.7)

x(t0 ) = x0 ∈ Rn , t ≥ t0 . is exponentially stable in mean square over all admissible uncertainty domains Da and Dbk , where Ki = K(r(t) = i) (i ∈ S) is the controller to be determined. The following results solve the robust stabilization problem for system (6). Theorem 4.1 The closed-loop system (7) is exponentially stable in mean square with respect to state-feedback gain Ki = Yi Pi−1 , if there exist N symmetric positive-definite matrices Pi , N matrices Yi and positive scalars εi , γi , and λi , such that∀ i ∈ S, the following BLMIs hold:

  Π11  E P  0i i    Π31    Π41   Π51

 ∗

∗

∗

−γi I

∗

∗

0

Π33

∗

0

0

−εi I

0

0

0

∗   ∗     < 0, ∗    ∗    Π55

(4.8)

where Π11 = (Ai Pi + Ci Yi ) + (Ai Pi + Ci Yi )T + qii Pi T + λi Pi + γi D0i D0i ,

Π31 = [(B1i Pi + C1i Yi )T , . . . , (Bdi Pi + Cdi Yi )T ]T , T T T T Π41 = [Pi E1i , Pi E2i , . . . , Pi Edi ] , T T Π33 = diag[εi D1i D1i − Pi , . . . , εi Ddi Ddi − Pi ],

Π51 = [Pi , Pi , . . . , Pi ]T , {z } | N −1

Π55 = diag[

−1 −1 −1 −1 P1 , . . . , Pi−1 , Pi+1 , . . . , PN ]. qi1 qi(i−1) qi(i+1) qiN 6

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Proof The proof is similar to that of Theorem 3.1, so we only give an outlined one. Let Xi = Pi−1 and define V (x, i) = xT Xi x . There exists a λ > 0 such that LV (x, i) ≤ −λ|x(t)|2 will hold if for T any i ∈ S there exist positive scalars εi , γi and λi , where εi Dki Dki − Pi < 0, 1 ≤ k ≤ d, such that T T Xi + γ1i E0i E0i Xi (Ai + Ci Ki ) + (Ai + Ci Ki )T Xi + γi Xi D0i D0i d X T −1 + (Bki + Cki Ki )T (Pi − εi Dki Dki ) (Bki + Cki Ki )

+

k=1 d X k=1

(4.9)

N X

1 T E Eki + qij Xj + λi Xi < 0. εi ki j=1

Noting that Yi = Ki Pi , and Pre- and post-multiplying (9) by Pi yields T T E0i Pi + γ1i Pi E0i (Ai Pi + Ci Yi ) + (Ai Pi + Ci Yi )T + γi D0i D0i d X T −1 + (Bki Pi + Cki Yi )T (Pi − εi Dki Dki ) (Bki Pi + Cki Yi )

+

k=1 d X k=1

X 1 T Pi Eki Eki Pi + qij Pi Pj−1 Pi + qii Pi + λi Pi < 0, εi j6=i

which is equivalent to (8) in view of Schur complement equivalence. The assertion of this theorem follows from Lemma 2.1 immediately. Remark 2 It is shown in Theorem 4.1 that the robust exponentially stabilization of system (6)(7) is guaranteed if the inequalities (8) are valid. And the inequality (8) is linear in Yi and Pi for a prescribed λi , thus the standard LMI techniques can be exploited to check the exponential stability of the closed-loop system (7).

5

Example Let w(t) be a one-dimensional Brownian motion, let r(t) be a right-continuous Markov chain 

 −1 1   taking values in S = {1, 2} with generator Q =  , consider a two-dimensional stochastic 1 −1 systems with Markovian switching of the form  
dx(t) = A(r(t)) + D0 (r(t))F0 (r(t), t)E0 (r(t)) x(t) + C(r(t))u(t) dt   (5.10)
+ B(r(t)) + D1 (r(t))F1 (r(t), t)E1 (r(t)) x(t) + C1 (r(t))u(t) dw(t), where   0.5 A1 =  0.3





0.2   1 , A2 =  0.8 0.2









0.1   −1 0.5   −2 , B1 =  , B2 =  2 0.5 −1 0.1

 0.1  , 1

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D01 = diag(−1, −2), D02 = diag(0.2, 0.3), D11 = diag(−1, −1), D12 = diag(5, −0.5), E01 = diag(0.2, 0.2), E02 = diag(−3, −5), E11 = diag(−0.9, −0.9), E12 = diag(0.5, 1), 











0  0.1   −1  −20  −8 C1 =  , C11 =  , C2 =  2 0 −30 0.05 −10





0.5   −2 1  , , C12 =  0.5 −4 3

for i = 1, 2, F0i (t) and F1i (t) denote the uncertainties of system (10). Let λ1 = 1, λ2 = 2, by solving LMIs (8), we find the feasible solution:

  98.708 P1 =   4.383











4.383   233.108 −0.786   93.468 −16.376  , P2 =  , Y1 =  ,      85.385 −0.786 180.327 −20.698 70.862





 171.947 −64.056  , γ1 = 0.082, γ2 = 1.170, ε1 = 0.034, ε2 = 0.004, Y2 =    75.520 82.513 therefore, by Theorem 4.1, closed-loop system (10) is exponentially stable in mean square with respect to state-feedback gain Ki = Yi Pi−1 .

6

Conclusions Based on the exponential stability theory, we have investigated the robust stochastic stability

of the uncertain stochastic system with Markovian switching, sufficient stability conditions were developed. The robust stability of such systems can be tested based on the feasibility of bi-linear matrix inequalities An example has been presented to illustrate the effectiveness of the main results. It is believed that this approach is one step further toward the descriptions of the uncertain stochastic systems.

References [1] G.K. Basak, A. Bisi, M.K. Ghosh, Stability of a random diffusion with linear drift. J. Math. Anal. Appl. 202, 604-622 (1996) [2] Y. Dong, J. Sun, On hybrid control of a class of stochastic non-linear Markovian switching systems. Automatica. 44, 990-995 (2008)

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[3] X. Feng, K.A. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properties of jump linear systems. IEEE Trans. Automat. Contr. 37, 38-52 (1992) [4] M.K. Ghosh, A. Arapostathis, S.I. Marcus, Ergodic control of switching diffusions. SIAM J. Control Optim. 35, 1952-1988 (1997) [5] M.K. Ghosh, A. Arapostathis, S.I. Marcus, Optimal control of switching diffusions with application to flexible manufacturing systems. SIAM J. Contr. Optim. 35, 1183-204 (1993) [6] V. Krishnamurthy, X. Wang, G. Yin,Spreading code optimization and adaptation in CDMA via discrete stochastic approximation. IEEE Trans. Inform. Theory. 50, 1927-1949 (2004) [7] R. Khasminskii, C. Zhu, G. Yin, Stability of regime-switching diffusions. Stoch. Proc. Appl. 117, 1037-1051 (2007) [8] J. Lian, F. Zhang, P. Shi, Sliding mode control of uncertain stochastic hybrid delay systems with average dwell time. Circuits Syst. Signal Process. 31, 539-553 (2012) [9] X. Mao, Stability of stochastic differential equations with Markovian switching. Stoch. Proc. Appl. 79, 45-67(1999) [10] X. Mao, G. Yin, C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations. Automatica. 43, 264-273(2007) [11] X. Mao, J. Lam, L. Huang, Stabilisation of hybrid stochas- tic differential equations by delay feedback control. Syst. Control Lett. 57, 927-935 (2008) [12] S. Pang, F. Deng, X. Mao, Almost sure and moment exponential stability of eulercmaruyama discretizations for hybrid stochastic differential equations. J. Comput. Appl. Math. 213, 127-141 (2008) [13] P.V. Pakshin, Robust stability and stabilization of family of jumping stochastic systems. Nonlinear Analysis. 30, 2855-2866 (1997) [14] I.R. Petersen, A stabilization algorithm for a class of uncertain linear systems. Syst. Control Lett., 8:351-357,1987. [15] M.A. Rami, L.E. Ghaoui, LMI optimization for nonstandard Riccati equations arising in stochastic control. IEEE Trans. Automat. Contr. 41, 1666-1671(1996) [16] S.P. Sethi, Q. Zhang. Hierarchical decision making in stochastic manufacturing systems( Birkh¨ auser, Boston, 1994) 9

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[17] Z. Wang, H. Qiao, K.J. Burnham, On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters. IEEE Trans. Automat. Contr. 47, 640-646 (2002) [18] S. Xu, T. Chen. Robust H∞ control for uncertain stochastic systems with state delay, IEEE Trans. Automat. Control. 47, 2089-2094 (2002) [19] G. Yin, R.H. Liu, Q. Zhang, Recursive algorithms for stock liquidation: A stochastic optimization approach. SIAM J. Contr. Optim. 13, 240-263(2002) [20] C. Yuan, J. Lygeros, On the exponential stability of switching diffusion processes, IEEE Trans. Automat. Control. 50, 1422-1426 (2005) [21] C. Yuan, J. Lygeros, Stabilization of a class of stochastic differential equations with Markovian switching. Syst. Control Lett. 54, 819-833 (2005) [22] C. Yuan, X. Mao, Robust stability and controllability of stochastic differential delay equations with Markovian switching. Automatica. 40, 343-354(2004) [23] C. Zhu, Optimal control of the risk process in a regime-switching environment. Automatica. 47, 1570-1579 (2011) [24] F.Zhu, Z.Han, J.Zhang, Stability analysis of stochastic differential equations with Markovian switching. Syst. Control Lett. 61, 1209-1214(2012)

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On interval valued functions and Mangasarian type duality involving Hukuhara derivative Izhar Ahmad1,∗ Deepak Singh2 , Bilal Ahmad Dar3 , S. Al-Homidan4 1,4

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. 2 Department of Applied Sciences, NITTTR (under Ministry of HRD, Govt. of India), Bhopal, M.P., India. 3 Department of Applied Mathematics, Rajiv Gandhi Proudyogiki Vishwavidyalaya (State Technological University of M.P.), Bhopal, M.P., India. Abstract In this paper, we introduce twice weakly differentiable and twice Hdifferentiable interval valued functions. The existence of twice H-differentiable interval-valued function and its relation with twice weakly differentiable functions are presented. Interval valued bonvex and generalized bonvex functions involving twice H-differentiability are proposed. Under the proposed settings, necessary conditions are elicited naturally in order to achieve LU efficient solution. Mangasarian type dual is discussed for a nondifferentiable multiobjective programming problem and appropriate duality results are also derived. The theoretical developments are illustrated through non-trivial numerical examples.

Keyword: Interval valued functions; twice weak differentiability; twice H- differentiability, LU -efficient solution; generalized bonvexity; duality. Mathematics Subject Classification: 90C25, 90C29, 90C30.

1

Introduction

The study of uncertain programming is always challenging in its modern face. Several attempts to achieve optimal in the same have been made in several directions. However optimality conditions still needs to be optimized. In this direction interval valued programming is one of the several techniques which has got attention of researchers in the recent past. Existing literature [2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 17, 19, 20, 21, 22] contains many interesting results on the study of interval valued programming involving different types of differentiability concepts and various types of convexity concepts of interval valued functions. Second order duality gives tighter bounds for the value of the objective function when approximations are used. For more information, authors may see ([11], pp ∗

Corresponding author.

E-mail addresses: [email protected] (Izhar Ahmad), [email protected] (Deepak Singh), [email protected] (Bilal Ahmad Dar), [email protected] (S.Al-Homidan)

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93). One more advantage is that if a feasible point in the primal problem is given and first order duality does not use, then we can apply second order duality to provide a lower bound of the value of the primal problem. Note that the study of nondifferentiable interval valued programming problems has not been studied extensively as quoted in Sun and Wang [18] therefore to study the second order duals of the aforesaid problem is an interesting move, we consider the following nondifferentiable vector programming problem with interval valued objective functions and constraint conditions and study its second order dual of Mangasarian type. (IP )   1 1 1 T T T 2 2 2 min f (x) + (x Bx) = f1 (x) + (x Bx) , ..., fk (x) + (x Bx) subject to gj (x) LU [0, 0], j ∈ Λm U L where fi = [fi , fi ], i ∈ Λk and gj = [gjL , gjU ], j ∈ Λm are interval valued functions with fiL , fiU , gjL , gjU : Rn → R, i ∈ Λk , j ∈ Λm be twice differentiable functions. The remaining paper is designed as: section 2 is devoted to preliminaries. Section 3 represents the differentiation of interval valued functions with the introduction of twice weakly differentiable and twice H-differentiable interval valued functions. Some properties of these functions are also presented. Section 4 highlights the concept of so-called bonvexity and its quasi and pseudo forms of interval valued functions and their properties. In section 5, the necessary conditions for proposed solution concept are elicited naturally by considering above settings. Finally with the proposed settings the section 6 is devoted to study the Mangasarian type dual of primal problem (IP ). Lastly we conclude in section 7.

2

Preliminaries

Let Ic denote the class of all closed and bounded intervals in R. i.e., Ic = {[a, b] : a, b ∈ R and a ≤ b}. And b − a is the width of the interval [a, b] ∈ Ic . Then for A ∈ Ic we adopt the notation A = [aL , aU ], where aL and aU are respectively the lower and upper bounds of A. Let A = [aL , aU ], B = [bL , bU ] ∈ Ic and λ ∈ R, we have the following operations. (i) A + B = {a + b : a ∈ A and b ∈ B} = [aL + bL , aU + bU ] (ii)  [λaL , λaU ] if λ ≥ 0 L U λA = λ[a , a ] = [λaU , λaL ] if λ < 0; (iii) A × B = [min, max], ab

ab

where min = min{aL bL , aL bU , aU bL , aU bU } ab

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and max = max{aL bL , aL bU , aU bL , aU bU } ab

In view of (i) and (ii) we see that −B = −[bL , bU ] = [−bU , −bL ] and A − B = A + (−B) = [aL − bL , aU − bL ]. Also the real number a ∈ R can be regarded as a closed interval Aa = [a, a], then we have for B ∈ Ic a + B = Aa + B = [a + bL , a + bU ]. Note that the space Ic is not a linear space with respect to the operations (i) and (ii), since it does not contain inverse elements.

3

Differentiation of interval valued functions

Definition 1. [20] Let X be open set in R. An interval-valued function f : X → Ic is called weakly differentiable at x∗ if the real-valued functions f L and f U are differentiable at x∗ (in the usual sense). Given A, B ∈ Ic , if there exists C ∈ Ic such that A = B +C, then C is called the Hukuhara difference of A and B. We also write C = A H B when the Hukuhara difference C exists, which means that aL − bL ≤ aU − bU and C = [aL − bL , aU − bU ]. Proposition 1. [20] Let A = [aL , aU ] and B = [bL , bU ] be two closed intervals in R. If aL −bL ≤ aU −bU , then the Hukuhara difference C exists and C = [aL −bL , aU −bU ]. Definition 2. [20] Let X be an open set in R. An interval-valued function f : X → Ic is called H-differentiable at x∗ if there exists a closed interval A(x∗ ) ∈ Ic such that f (x∗ ) H f (x∗ + h) f (x∗ + h) H f (x∗ ) and lim h→0+ h→0+ h h lim

both exist and are equal to A(x∗ ). In this case, A(x∗ ) is called the H-derivative of f at x∗ . Proposition 2. [20] Let f be an interval-valued function defined on X ⊆ Rn . If f is H-differentiable at x∗ ∈ X, then f is weakly differentiable at x∗ . Next we introduce twice differentiable interval valued functions and study some properties. Definition 3. Let X be an open set in Rn , and let x∗ = (x∗1 , ..., x∗n ) ∈ X be fixed. Then we say that f is twice weakly differentiable interval valued function at x∗ if

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f L and f U are twice differentiable functions at x∗ (in usual sense). We denote by ∇2 f the second differential of f , then we have ∇2 f (x∗ ) = ∇(∇f (x))x=x∗ = ∇(∇[f L (x), f U (x)])x=x∗ = ∇([∇f L (x), ∇f U (x)])x=x∗ = [∇2 f L (x), ∇2 f U (x)]x=x∗ " #   2 U  ∂ 2f L ∂ f = (x), (x) ∂xi ∂xj i,j ∂xi ∂xj i,j

. x=x∗

Definition 3 is illustrated by the following example. Example 1. Consider the interval valued function f (x1 , x2 ) = [f L = 2x1 + x22 , f U = x21 + x22 + 1].

(1)

Therefore we have ∇f (x) = [(2, 2x2 ), (2x1 , 2x2 )]T and 2

∇ f (x) =



0 0 0 2

   2 0 , . 0 2

Definition 4. Let X be an open set in Rn , and let x∗ = (x∗1 , ..., x∗n ) ∈ X be fixed. Then we say that f is twice H-differentiable interval valued function if f 0 is Hdifferentiable at x∗ , where f 0 is H-derivative of f . We denote by ∇2H f the second order H-differential of f , then we have ∇2H f (x∗ ) = ∇H (∇H f (x))x=x∗  T ∂f ∂f = ∇H (x), ..., (x) ∂x1 ∂xn x=x∗   T  L  L U ∂f ∂f U ∂f ∂f = ∇H (x), (x) , ..., ∇H (x), (x) ∂x1 ∂x1 ∂xn ∂xn ∗  h i h 2 L i  x=x 2 U 2 U 2 L ∂ f (x), ∂∂ 2fx1 (x) ... ∂x∂ 1f∂xn (x), ∂x∂ 1f∂xn (x)  ∂ 2 x1    .. .. .. . =  . . . i h 2 L i   h 2 L 2 U 2 U ∂ f (x), ∂x∂ nf∂x1 (x) ... ∂∂ 2fxn (x), ∂∂ 2fxn (x) ∂xn ∂x1 n×n,x=x∗

(2) Following example justifies the existence of twice H-differentiable interval valued function. Example 2. Consider the interval valued function (1), then by definition we have ∇H f (x) = ([2, 2x1 ], [2x2 , 2x2 ])T 4 884

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which exist for x1 ≥ 1. Therefore we have ∇2H f (x) = ∇H ([2, 2x1 ], [2x2 , 2x2 ])T   [0, 2] [0, 0] = . [0, 0] [2, 2] The relation between twice weakly differentiable and twice H-differentiable interval valued functions is furnished as follows. Proposition 3. Let f be an interval-valued function defined on X ⊆ Rn . If f is twice H-differentiable at x∗ ∈ X, then f is twice weakly differentiable at x∗ . Proof. From (2) we have  2 L   ∂2f L ∂ f 2 x (x) ... ∂x ∂x (x) ∂ n 1 1    .. .. .. , ∇2H f (x∗ ) =    . . .  2 L ∂2f L (x) ... ∂∂ 2fxn (x) n×n ∂xn ∂x1 2 L

  2 U ... ∂x∂ 1f∂xn (x)   .. .. ..   . . .  ∂2f U ∂2f U (x) ... ∂ 2 xn (x) n×n ∂xn ∂x1 ∂2f U (x) ∂ 2 x1

x=x∗

2 U

= [∇ f (x), ∇ f (x)]x=x∗ = ∇2 f (x∗ ).

We authenticate Proposition 3 by following example. Example 3. From Example 2 we have ∇2H f (x)



 [0, 2] [0, 0] = [0, 0] [2, 2]     0 0 2 0 = , 0 2 0 2  2 L  = ∇ f (x), ∇2 f U (x) = ∇2 f (x). (see Example 1).

The converse of Proposition 3 is not true in general, however we have the following result. Proposition 4. Let f ∈ T , be twice weakly differentiable function at x∗ , with 0 0 (f L )00 (x∗ ) = aL (x∗ ) and (f U )00 (x∗ ) = aU (x∗ ). 1. if (f L )0 (x∗ +h)−(f L )0 (x∗ ) ≤ (f U )0 (x∗ +h)−(f U )0 (x∗ ) and (f L )0 (x∗ )−(f L )0 (x∗ − h) ≤ (f U )0 (x∗ )−(f U )0 (x∗ −h) for every h > 0, then f is twice H-differentiable 0 0 with second H-derivative [aL (x∗ ), aU (x∗ )]. 0

0

2. if aL (x∗ ) > aU (x∗ ), then f is not twice H-differentiable at x∗ . Proof. The proof is similar as that of Proposition 4.3 of [20]. The existence of twice weakly differentiable interval valued functions which are not twice H-differentiable is proved by following example. Example 4. Consider f : [0, 2] → [x3 + x2 + 1, x3 + 2x + 2] be an interval valued function defined on [0, 2]. Then f is twice weakly differentiable on (0, 2) but f is 0 0 not twice H-differentiable on (0, 2) as aL (x∗ ) > aU (x∗ ). 5 885

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4

Interval valued bonvex functions

Convexity is an important concept in studying the theory and methods of mathematical programming, which has been generalized in several ways. For differentiable functions numerous generalizations of convexity exist in the literature. An important concept so-called second order convexity for twice differentiable real valued functions was introduced in Mond [14], however Bector and Chandra [6] named them as bonvex functions. Now consider f to be real valued twice differentiable function, then for the definitions of (strictly) bonvexity, (strictly) pseudobonvexity and (strictly) quasibonvexity, one is refered to [3]. In this section, we introduce LU -bonvex, LU -pseudobonvex and LU -quasibonvex interval valued functions and their strict conditions. We consider T to be the set of all interval valued functions defined on X ⊆ Rn . Definition 5. Let f ∈ T be twice H-differentiable function at x∗ ∈ X. If we have for every x ∈ X and P = (P1 , ..., Pn ) with Pi ∈ Ic such that PiL ≥ 0, i ∈ Λk . 1.  1 f (x) H f (x∗ ) + P T ∇2H f (x∗ )P LU ∇H f (x∗ ) + ∇2H f (x∗ )P (x − x∗ ) 2 then we say that f is LU -bonvex at x∗ . We also say that f is strictly LU bonvex at x∗ (6= x) if the inequality is strict. 2. If 1 f (x) H f (x∗ ) + P T ∇2H f (x∗ )P LU [0, 0], 2  ∗ 2 ⇒ ∇H f (x ) + ∇H f (x∗ )P (x − x∗ ) LU [0, 0] then we say that f is LU -quasibonvex at x∗ . We also say that f is strictly LU -quasibonvex x∗ (6= x) if the inequality is strict. 3. If 

∇H f (x∗ ) + ∇2H f (x∗ )P (x − x∗ ) LU [0, 0],

1 ⇒ f (x) H f (x∗ ) + P T ∇2H f (x∗ )P LU [0, 0] 2 then we say that f is LU -pseudobonvex at x∗ . We also say that f is strictly LU pseudobonvex at x∗ (6= x) if the inequality is strict. Now we present some non-trivial examples which authenticates that the class of interval valued functions introduced in this section is non-empty. Example 5. Consider an interval valued function f (x) = [x2 + 3x + 2, x2 + 3x + 5], x ≥ 0. Then we have ∇H f (x) = ([2x + 3, 2x + 3]) 6 886

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= ([3, 3])x=0 and ∇2H f (x) = ([2, 2]) we have 1 [x2 + 3x + 2, x2 + 3x + 5] H [2, 5] + ([0, 1])T [2, 2][0, 1] = [x2 + 3x + 2, x2 + 3x + 2] 2 LU ([3, 3] + [2, 2][0, 1])(x) = [3x, 5x] therefore f is LU -bonvex at x = 0. Next consider another interval valued functions defined as f (x1 , x2 ) = [x21 + x22 + 3, x21 + x22 + 5], x ≥ 0. Then we have ∇H f (x1 , x2 ) = ([2x1 , 2x1 ], [2x2 , 2x2 ])T
T = [4, 4], [4, 4]T (x1 ,x2 )=(2,2) and ∇2H f (x)



[2, 2] [0, 0] [0, 0] [2, 2]

=



Now let [x21

+

x22

+

3, x21

+

x22

1 + 5] H [11, 13] + 2



[1, 1] [1, 1]

T 

[2, 2] [0, 0] [0, 0] [2, 2]



[1, 1] [1, 1]



LU [0, 0] then 

[4, 4] [4, 4]



 +

[2, 2] [0, 0] [0, 0] [2, 2]



[1, 1] [1, 1]

 

x1 − 2 x2 − 2

 LU [0, 0].

this shows that f is LU -quasibonvex at (2, 2). However if        [4, 4] [2, 2] [0, 0] [1, 1] x1 − 2 + LU [0, 0]. [4, 4] [0, 0] [2, 2] [1, 1] x2 − 2 then [x21

+

x22

+

3, x21

+

x22

1 + 5] H [11, 13] + 2



[1, 1] [1, 1]

T 

[2, 2] [0, 0] [0, 0] [2, 2]



[1, 1] [1, 1]



LU [0, 0] this shows that f is LU -pseudobonvex at (2, 2). 7 887

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Proposition 5. Let f ∈ T be twice H-differentiable function at x∗ and P = (P1 , ..., Pn ) with Pi ∈ Ic such that PiL ≥ 0, i ∈ Λn . 1. if f is LU -bonvex at x∗ then f L and f U are bonvex functions at x∗ . 2. if f is LU -quasibonvex at x∗ then f L and f U are quasibonvex functions at x∗ . 3. if f is LU -pseudobonvex at x∗ then f L and f U are pseudobonvex functions at x∗ . Proof. (i) Let f is LU -bonvex at x∗ , then by definition we have  1 f (x) H f (x∗ ) + P T ∇2H f (x∗ )P LU ∇H f (x∗ ) + ∇2H f (x∗ )P (x − x∗ ) 2 Since f is twice H-differentiable at x∗ , then by Proposition 3 and Definition 3 f L and f U are twice differentiable at x∗ . Also since PiL ≥ 0, therefore we have  1 T f L (x) − f L (x∗ ) + P L ∇2 f L (x∗ )P L ≥ ∇f L (x∗ ) + ∇2 f L (x∗ )P L (x − x∗ ), 2 and  1 T f U (x) − f U (x∗ ) + P U ∇2 f U (x∗ )P U ≥ ∇f U (x∗ ) + ∇2 f U (x∗ )P U (x − x∗ ). 2 Therefore f L and f U are bonvex functions at x∗ . (ii) and (iii) follows by similar treatment. Note that the converse of Proposition 5 follows in the light of Proposition 4. Proposition 6. Let f ∈ T be twice H-differentiable function at x∗ and P = (P1 , ..., Pn ) with Pi ∈ Ic such that PiL ≥ 0, i ∈ Λn . 1. if f is strictly LU -bonvex at x∗ then either f L or f U or both are strictly bonvex functions at x∗ . 2. if f is strictly LU -quasibonvex at x∗ then either f L or f U or both are strictly quasibonvex functions at x∗ . 3. if f is strictly LU -pseudobonvex at x∗ then either f L or f U or both are strictly pseudobonvex functions at x∗ . Proof. Proof is same as that of Proposition 5. Remark 1. If we assume that f L = f U , then bonvexity comes as a sub-case of LU -bonvexity, and similarly for quasi and pseudobonvexity.

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5

Solution concept and necessary conditions

In this section we shall propose solution concept and derive some necessary conditions for problem (IP ). We define by SIP the set of feasible solutions of (IP ). Definition 6. Let x∗ ∈ SIP . We say that x∗ is an efficient solution of (IP ) if there exist no xˆ ∈ SIP , such that fi (ˆ x) LU fi (x∗ ), i ∈ Λk and fh (ˆ x) ≺LU fh (x∗ ), for at least one index h. An efficient solution x∗ is said to be properly efficient solution of (IP ) if there exist scalar M > 0, such that for all i ∈ Λk , fi (x) ≺LU fi (x∗ ) and x ∈ SIP imply that fi (x∗ ) H fi (x) LU M {fh (x) H fh (x∗ )} for atleast one index h ∈ Λk − i such that fh (x∗ ) ≺LU fh (x). Theorem 1. (Mond et al. [16]) Let x∗ be a properly efficient solution of (P ) (see, [3]) at which constraint qualification [15] is satisfied. Then there exist λ∗ ∈ Rk , u∗ ∈ Rm and vi∗ ∈ Rn , i ∈ ΛK such that k X

λ∗i (fi (x∗ ) + Bi vi∗ ) + ∇u∗ T g(x∗ ) = 0,

i=1

u∗ T g(x∗ ) = 0, 1

(x∗ T Bi x∗ ) 2 = x∗ T Bi vi∗ , i ∈ Λk , vi∗ T Bi vi∗ ≤ 1, i ∈ Λk , ∗

λ > 0,

k X

λ∗i = 1, u∗ ≥ 0.

i=1

Now we present the necessary conditions for problem(IP ). Consider the following constraint qualification CQ1 dT ∇H gj (x∗ ) LU [0, 0], j ∈ J0 (x∗ ) 1

dT ∇H fi (x∗ ) + dT Bi x∗ /(x∗ T Bi x∗ ) 2 LU [0, 0], if x∗ T Bi x∗ > 0 1

dT ∇H fi (x∗ ) + (dT Bi d) 2 LU [0, 0], if x∗ T Bi x∗ = 0 Theorem 2. Let x∗ be a properly efficient solution of (IP ) at which a constraint qualification CQ1 is satisfied. Then there exist λ∗ ∈ Rk , u∗ ∈ Rm and vi∗ ∈ Rn , i ∈ ΛK such that k X

λ∗i (∇H fi (x∗ ) + Bi vi∗ ) + ∇H u∗ T g(x∗ ) = [0, 0],

i=1

u∗ T g(x∗ ) = [0, 0], 1

(x∗ T Bi x∗ ) 2 = x∗ T Bi vi∗ , i ∈ Λk , vi∗ t Bi vi∗ ≤ 1, i ∈ Λk , ∗

λ > 0,

k X

λ∗i = 1, u∗ ≥ 0.

i=1

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Proof. Since x∗ is properly efficient solution of (IP ) at which a constraint qualification CQ1 is satisfied. Then using the property of intervals and twice H-derivative, for 0 < ξiL , ξiU ∈ R, i ∈ Λk with ξiL + ξiU = 1, i ∈ Λk , we have CQ2 dT ∇gjL (x∗ ) > 0, j ∈ J0 (x∗ ) dT ∇gjU (x∗ ) > 0, j ∈ J0 (x∗ ) 1

dT (ξiL ∇fiL (x∗ ) + ξiU ∇fiU (x∗ )) + dT Bi x∗ /(x∗ T Bi x∗ ) 2 < 0, if x∗ T Bi x∗ > 0 1

dT (ξiL ∇fiL (x∗ ) + ξiL ∇fiU (x∗ )) + (dT Bi d) 2 < 0, if x∗ T Bi x∗ = 0 Further using the property of intervals and twice H-derivative, for 0 < ξiL , ξiU ∈ R, i ∈ Λk with ξiL + ξiU = 1, i ∈ Λk we have new conditions as k X

λ∗i ((ξiL ∇fiL (x∗ ) + ξiL ∇fiU (x∗ )) + Bi vi∗ ) + ∇u∗ T (g L (x∗ ) + g U (x∗ )) = 0,

i=1

u∗ T g L (x∗ ) = 0, u∗ T g U (x∗ ) = 0, 1

(x∗ T Bi x∗ ) 2 = x∗ T Bi vi∗ , i ∈ Λk , vi∗ T Bi vi∗ ≤ 1, i ∈ Λk , ∗

λ > 0,

k X

λ∗i = 1, u∗ ≥ 0.

i=1

Now using constraint qualification CQ2 the above conditions are justified by Theorem 1 for the problem (say (IP 1)) heaving objective function (ξ1L f1L (x) + ξ1L f1U (x), ..., ξkL fkL (x)+ξkL fkU (x)) and constraint functions gjL (x), gjU (x) ≤ 0, j ∈ Λm . Now it is easy to see that the optimal solutions of (IP ) and (IP 1) are same. This completes the proof.

6

Mangasarian type duality

In this section, we propose the following Mangasarian type dual of primal problem (IP ). (M SD) V-maximize  1 f1 (y) + uT g(y) + y T B1 v1 H P T ∇2H {f1 (y) + uT g(y)}P, ..., 2  1 T 2 T T T fk (y) + u g(y) + y Bk vk H P ∇H {fk (y) + u g(y)}P 2 subject to k X

λi (∇H fi (y) + ∇2H fi (y)P + Bi vi ) + ∇H uT g(y) + ∇2H uT g(y)P = [0, 0]

(3)

i=1

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viT Bi vi ≤ 1, i ∈ Λk r X λ > 0, λi = 1

(4) (5)

i=1

u = (u1 , ..., um ) ≥ 0, g = (g1 , ..., gm ) such that gj = [gjL , gjU ], j = 1, ..., m, P = (P1 , ..., Pn ) with Pi ∈ Ic such that PiL ≥ 0, i ∈ Λk . and y, vi ∈ Rn . T

We define by SM SD the set of all feasible solutions of (M SD), therefore if z ∈ SM SD then z = (y, u, v, λ, P ), such that v ∈ Rk with vi ∈ Rn , and Pi ∈ Ic such that PiL ≥ 0, i ∈ Λk . We shall use the following generalized Schwartz inequality: xT Az ≤ (xT Ax)1/2 (z T Az)1/2 , where x, z ∈ Rn and A is positive semidefinite symmetric matrix of order n. Theorem 3. (weak duality) Let x ∈ SIP and z ∈ SM SD . Assume that fi (.) + (.)T Bi vi , i ∈ Λk and gj (.), j ∈ Λm are LU -bonvex at y, then the following can not hold. 1 1 fi (x)+(xT Bi x) 2 LU fi (y)+uT g(y)+y T Bi vi H P T {∇2H fi (y) + uT g(y)}P, i ∈ Λk . 2 (6) and 1 1 fh (x)+(xT Bh x) 2 ≺LU fh (y)+uT g(y)+y T Bh vh H P T {∇2H fh (y) + uT g(y)}P, (7) 2 for at least one index h. Proof. From (3) we have k X


λi ∇fiL (y) + ∇2 fiL (y)P L + Bi vi + ∇uT g L (y) + ∇2 uT g L (y)P L = 0.

i=1 k X


λi ∇fiU (y) + ∇2 fiU (y)P U + Bi vi + ∇uT g U (y) + ∇2 uT g U (y)P U = 0.

i=1

Adding we get, k X


λi ∇fiL (y) + ∇fiU (y) + ∇2 fiL (y)P L + ∇2 fiU (y)P U + 2Bi vi + ∇uT g L (y)

i=1

+∇uT g U (y) + ∇2 uT g L (y)P L + ∇2 uT g U (y)P U = 0.

(8)

If possible let (6) and (7) holds then by definition we have ( 1 T fiL (x) + (xT Bi x) 2 ≤ fiL (y) + uT g L (y) + y T Bi vi − 21 P L ∇2 {fiL (y) + uT g L (y)}P L . 1 T fiU (x) + (xT Bi x) 2 ≤ fiU (y) + uT g U (y) + y T Bi vi − 21 P U ∇2 {fiU (y) + uT g U (y)}P U . for i ∈ Λk , and ( 1 T fhL (x) + (xT Bh x) 2 < fhL (y) + uT g L (y) + y T Bh vh − 21 P L ∇2 {fhL (y) + uT g L (y)}P L . 1 T fhU (x) + (xT Bh x) 2 ≤ fhU (y) + uT g U (y) + y T Bh vh − 21 P U ∇2 {fhU (y) + uT g U (y)}P U . 11 891

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or (

or (

1

T

1

T

fhL (x) + (xT Bh x) 2 ≤ fhL (y) + uT g L (y) + y T Bh vh − 21 P L ∇2 {fhL (y) + uT g L (y)}P L . 1 T fhU (x) + (xT Bh x) 2 < fhU (y) + uT g U (y) + y T Bh vh − 12 P U ∇2 {fhU (y) + uT g U (y)}P U .

fhL (x) + (xT Bh x) 2 < fhL (y) + uT g L (y) + y T Bh vh − 21 P L ∇2 {fhL (y) + uT g L (y)}P L . 1 T fhU (x) + (xT Bh x) 2 < fhU (y) + uT g U (y) + y T Bh vh − 12 P U ∇2 {fhU (y) + uT g U (y)}P U .

for atleast one index h. This yields for λ = (λ1 , ..., λr ); λi > 0 k X

λi

n

  o 1 1 fiL (x) + (xT Bi x) 2 + fiU (x) + (xT Bi x) 2
0,

k X

λ∗i = 1, u∗ ≥ 0.

i=1

Which yields that (x∗ , u∗ , vi∗ , λ∗ , P ∗ T = ([0, 0], ..., [0, 0])) ∈ SM SD and the corresponding objective values of (IP ) and (M SD) are equal. Now let (x∗ , u∗ , vi∗ , λ∗ , P ∗ T = ([0, 0], ..., [0, 0])) is not efficient solution of dual problem (M SD), then by Definition there exist (y ∗ , u∗ , vi∗ , λ∗ , P ∗ ) ∈ SM SD , such that fi (x∗ ) + x∗ T Bi vi∗ + u∗ T g(x∗ ) LU fi (y ∗ ) + u∗ T g(y ∗ ) + y ∗ T Bi vi∗ 1 H P ∗ T ∇2H {fi (y ∗ ) + u∗ T g(y ∗ )}P ∗ , i ∈ Λk 2 and fi (x∗ ) + x∗ T Bi vi∗ + u∗ T g(x∗ ) ≺LU fi (y ∗ ) + u∗ T g(y ∗ ) + y ∗ T Bi vi∗ 1 H P ∗ T ∇2H {fi (y ∗ ) + u∗ T g(y ∗ )}P ∗ , 2 for atleast one index h. 1

Now using (x∗ T Bi x∗ ) 2 = x∗ T Bi vi∗ , i ∈ Λk and u∗ T g(y ∗ ) = [0, 0], we get a contradiction to weak duality theorem. Therefore (x∗ , u∗ , vi∗ , λ∗ , P ∗ T = ([0, 0], ..., [0, 0])) is an efficient solution of dual problem (M SD). 13 893

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Theorem 5. (Strict converse duality) Let x∗ ∈ SIP and z ∗ ∈ SM SP such that k X

λ∗i {fi (x∗ )

+x

∗T

Bi vi∗ }

k X

LU

i=1

λ∗i

i=1

  1 ∗T 2 ∗ ∗T ∗ ∗ ∗ fi (y ) + y Bi vi H P ∇H fi (y )P 2

1 ∗T 2 ∗T P ∇H u g(y ∗ )P ∗ . (12) 2 Assume that fi (.) + (.)T Bi vi∗ , i ∈ Λk are strictly LU -bonvex at y ∗ and gj (.), j ∈ Λm is LU -bonvex at y ∗ then x∗ = y ∗ . +u∗ T g(y ∗ ) H

Proof. If possible let x∗ 6= y ∗ . Now since fi (.) + (.)T Bi vi∗ , i ∈ Λk are strictly LU bonvex at y ∗ , we have 1 fi (x∗ ) + x∗ T Bi vi∗ H (fi (y ∗ ) + y ∗ T Bi vi∗ ) + P ∗ T ∇2H fi (y ∗ )P ∗ LU 2
∇H fi (y ∗ ) + ∇2H fi (y ∗ )P ∗ + Bi vi ∗ (x∗ − y ∗ ), i ∈ Λk .

(13)

and 1 gj (x∗ ) H gj (y ∗ )+ P ∗ T ∇2H gj (y ∗ )P ∗ LU (∇H gj (y ∗ )+∇2H gj (y ∗ )P ∗ )(x∗ −y ∗ ), j ∈ Λm . 2 (14) Now multiplying (13) by λ∗i , i ∈ Λk and (14) by u∗j , j ∈ Λm and then summing up we get k X

λ∗i



∗

fi (x ) + x

∗T

Bi vi∗



∗T

∗

+ u g(x ) H

i=1

k X

n λ∗i fi (y ∗ ) + y ∗ T Bi vi H

i=1

1 1 ∗T 2 P ∇H fi (y ∗ )P ∗ H u∗ T g(y ∗ ) + P ∗ T ∇2H u∗ T gj (y ∗ )P ∗ LU 2 2 ) o

(

k X

λ∗j (∇H fi (y ∗ )+∇2H fi (y ∗ )P ∗ +Bi vi∗ )+∇H u∗ T g(y ∗ )+∇2H u∗ T g(y ∗ )P ∗ (x∗ −y ∗ ).

i=1

The above inequality on using (3) and u∗ T g(x∗ ) LU [0, 0] gives k X

λ∗i {fi (x∗ )

+x

∗T

Bi vi∗ }

LU

i=1

k X

n λ∗i fi (y ∗ ) + y ∗ T Bi vi∗ H

i=1

1 1 ∗T 2 P ∇H fi (y ∗ )P ∗ + u∗ T g(y ∗ ) H P ∗ T ∇2H u∗ T g(y ∗ )P ∗ . 2 2 ∗ ∗ which is a contradiction to (12). Hence x = y o

7

Conclusions

This paper represents the study of nondifferentiable vector problem in which objective functions and constraints are interval valued. Firstly the twice H- differentiable interval valued functions are introduced, secondly the concepts of LU -bonvexity, LU -quasibonvexity and LU -pseudobonvexity are introduced, thirdly the necessary conditions for proposed solution concept are obtained. And lastly the Mangasarian 14 894

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type dual is proposed and the corresponding duality results are obtained. Although the interval valued equality constraints are not considered in this paper, the similar methodology proposed in this paper can also be used to handle the interval valued equality constraints. However it will be interesting to study the Mond-Weir type duality results [1] for the problem (IP ). Future research is oriented to consider the uncertain environment in order to study the optimality conditions involving Fuzzy parameters. Acknowledgements The research of the first and fourth author is financially supported by King Fahd University of Petroleum and Minerals, Saudi Arabia under the Internal Research Project No. IN131026.

References [1] I. Ahmad, Z. Husain, Second order (F, α, ρ, d)-convexity and duality in multiobjective programming, Inform Sci 176 (2006) 3094-3103. [2] I. Ahmad, A. Jayswal, J. Banerjee: On interval-valued optimization problems with generalized invex functions, J Inequal Appl (2013) 1-14. [3] I. Ahmad, S. Sharma, Second order duality for nondifferentiable multiobjective programming problems, Num Func Anal Optim 28 (9-10) (2007) 975-988. [4] I. Ahmad, D. Singh, B. A. Dar, Optimality conditions for invex interval-valued nonlinear programming problems involving generalized H-derivative, Filomat (2015) (Accepted). [5] A. Jayswal, I. Stancu-Minasian, J. Banerjee, A.M. Stancu, Sufficieny and duality for optimization problems involving interval-valued invex functions in parametric form, Oper Res Int J 15(2015) 137-161. [6] C. R. Bector, S. Chandra, Generalized-bonvexity and higher order duality for fractional programming, Opsearch 24 (1987)143-154. [7] A. Bhurjee, G. Panda, Efficient solution of interval optimization problem, Math Meth Oper Res 76 (2012) 273-288. [8] A. Bhurjee, G. Panda, Multiobjective optimization problem with bounded parameters, Rairo-Oper. Res. 48(2014), 545-558. [9] Y. Chalco-Cano, H. Roman-Flores, MD. Jimenez-Gamero, Generalized derivative and π-derivative for set valued functions, Inform Sci 181 (2011) 2177-2188. [10] Y. Chalco-Cano, W.A. Lodwick, A. Rufian-Lizana, Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative, Fuzzy Optim Dec Making 12 (2013) 305-322.

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[11] Y. Chalco-Cano, A. Rufian-Lizana, H. Roman-Flores, M.D. Jimenez-Gamero, Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and Systems 219 (2013) 49-67. [12] H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of interval valued objective functions, Eur J Oper Res 48 (1990) 219-225. [13] L. Li, S. Liu, J. Zhang, Univex interval-valued mapping with differentiability and its application in nonlinear programming, J Appl Math Art. ID 383692, (2013). http://dx.doi.org/10.1155/2013/383692. [14] B. Mond, Second order duality for nonlinear programs, Opsearch 11 (1974) 90-99. [15] B. Mond, A class of nondifferentiable mathematical programming problems. J. Math Anal Appl 46 (1974) 169-174. [16] B. Mond, I. Husain, M.V. Durgaprasad, Duality for a class of nondifferentiable multiple objective programming problems, J Inform Optim Sci 9 (1988) 331-341. [17] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal 71 (2009) 13111328. [18] Y. Sun, L. Wang, Optimality conditions and duality in nondifferentiable interval valued progra-mming, J Indust Manag Optim 9 no. 1 (2013) 131-142. [19] D. Singh, B. A. Dar, A. Goyal, KKT optimality conditions for interval valued optimization problems, J Nonl Anal Optim 5 no. 2 (2014) 91-103. [20] H. C. Wu, The karush Kuhn tuker optimality conditions in an optimization problem with interval valued objective functions, Eur J Oper Res 176 (2007) 46-59. [21] H. C. Wu, The karush Kuhn tuker optimality conditions in multiobjective programming problems with interval valued objective functions, Eur J Oper Res 196 (2009) 49-60. [22] J Zhang, S. Liu, L. Li, Q. Feng, The KKT optimality conditions in a class of generalized convex optimization problems with an interval-valued objective function, Optim Lett 8 (2014) 607-631.

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN β-HOMOGENEOUS NORMED SPACES SUNGSIK YUN, GEORGE A. ANASTASSIOU AND CHOONKIL PARK∗ Abstract. In this paper, we solve the following additive-quadratic ρ-functional inequalities kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k

      x+y x−y 3 1 1 1

≤ ρ 2f + 2f − f (x) + f (−x) − f (y) − f (−y) , 2 2 2 2 2 2 where ρ is a fixed complex number with |ρ| < 1, and



   x+y x−y 3 1 1 1

+ 2f − f (x) + f (−x) − f (y) − f (−y)

2f 2 2 2 2 2 2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y))k,

(0.1)

(0.2)

1 , 2

where ρ is a fixed complex number with |ρ| < and prove the Hyers-Ulam stability of the additivequadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive-quadratic ρ-functional equations associated with the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces.

1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [24] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [11] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [15] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by G˘avruta [8] by replacing the unbounded Cauchy difference   by x+y a general control function in the spirit of Rassias’ approach. The functional equation f 2 = + 12 f (y) is called the Jensen equation. The functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) is called the quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. The stability of quadratic functional equation was proved by Skof [23] for mappings f : E1 → E2 , where E1 is a normed space and E2 is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true ifthe relevant domain E1 is replaced by an Abelian group. The  x−y functional equation 2f x+y + 2 = f (x) + f (y) is called a Jensen type quadratic equation. 2 2 The stability problems of various functional equations have been extensively investigated by a number of authors (see [1, 4, 5, 13, 14, 18, 19, 20, 21, 22]). 1 2 f (x)

2010 Mathematics Subject Classification. Primary 39B62, 39B72, 39B52, 39B82. Key words and phrases. Hyers-Ulam stability; β-homogeneous space; additive-quadratic ρ-functional equation; additive-quadratic ρ-functional inequality. ∗ Corresponding author: Choonkil Park (email: [email protected]).

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S. YUN, G. A. ANASTASSIOU, C. PARK

In [9], Gil´anyi showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (xy −1 )k ≤ kf (xy)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (xy) + f (xy −1 ). See also [16]. Gil´anyi [10] and Fechner [7] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [12] proved the Hyers-Ulam stability of additive functional inequalities. Definition 1.1. Let X be a linear space. A nonnegative valued function k · k is an F -norm if it satisfies the following conditions: (FN1 ) kxk = 0 if and only if x = 0; (FN2 ) kλxk = kxk for all x ∈ X and all λ with |λ| = 1; (FN3 ) kx + yk ≤ kxk + kyk for all x, y ∈ X; (FN4 ) kλn xk → 0 provided λn → 0; (FN5 ) kλxn k → 0 provided xn → 0. Then (X, k · k) is called an F ∗ -space. An F -space is a complete F ∗ -space. An F -norm is called β-homogeneous (β > 0) if ktxk = |t|β kxk for all x ∈ X and all t ∈ C (see [17]). In Section 2, we solve the additive-quadratic ρ-functional inequality (0.1) and prove the HyersUlam stability of the additive-quadratic ρ-functional inequality (0.1) in β-homogeneous complex Banach spaces. We moreover prove the Hyers-Ulam stability of an additive-quadratic ρ-functional equation associated with the additive-quadratic ρ-functional inequality (0.1) in β-homogeneous complex Banach spaces. In Section 3, we solve the additive-quadratic ρ-functional inequality (0.2) and prove the HyersUlam stability of the additive-quadratic ρ-functional inequality (0.2) in β-homogeneous complex Banach spaces. We moreover prove the Hyers-Ulam stability of an additive-quadratic ρ-functional equation associated with the additive-quadratic ρ-functional inequality (0.2) in β-homogeneous complex Banach spaces. Throughout this paper, let β1 , β2 be positive real numbers with β1 ≤ 1 and β2 ≤ 1. Assume that X is a β1 -homogeneous real or complex normed space with norm k · k and that Y is a β2 homogeneous complex Banach space with norm k · k. 2. Additive-quadratic ρ-functional inequality (0.1) Throughout this section, assume that ρ is a fixed complex number with |ρ| < 1. In this section, we investigate the additive-quadratic ρ-functional inequality (0.1) in β-homogeneous complex Banach spaces. Lemma 2.1. An even mapping f : X → Y satisfies kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k

     

x−y 3 1 1 1 x+y

+ 2f − f (x) + f (−x) − f (y) − f (−y) ≤ ρ 2f

2 2 2 2 2 2 for all x, y ∈ X if and only if f : X → Y is quadratic.

(2.1)

Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get k2f (0)k ≤ |ρ|β2 k2f (0)k. So f (0) = 0. Letting y = x in (2.1), we get kf (2x) − 4f (x)k ≤ 0 and so f (2x) = 4f (x) for all x ∈ X. Thus   x 1 f = f (x) (2.2) 2 4

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES

for all x ∈ X. It follows from (2.1) and (2.2) that kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k

     

x+y x−y 3 1 1 1

+ 2f − f (x) + f (−x) − f (y) − f (−y) ≤ ρ 2f

2 2 2 2 2 2 |ρ|β2 = β2 kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k 2 and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X. The converse is obviously true.



Corollary 2.2. An even mapping f : X → Y satisfies f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)       x+y x−y 3 1 1 1 = ρ 2f + 2f − f (x) + f (−x) − f (y) − f (−y) 2 2 2 2 2 2 for all x, y ∈ X if and only if f : X → Y is quadratic.

(2.3)

The functional equation (2.3) is called an additive-quadratic ρ-functional equation. We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (2.1) in β-homogeneous complex Banach spaces for an even mapping case. Theorem 2.3. Let r > mapping such that

2β2 β1

and θ be nonnegative real numbers, and let f : X → Y be an even

kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k (2.4)

     

x − y 3 1 1 1 x + y r r + 2f − f (x) + f (−x) − f (y) − f (−y) ≤

+ θ(kxk + kyk )

ρ 2f 2 2 2 2 2 2 for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that 2θ kf (x) − Q(x)k ≤ β1 r kxkr (2.5) 2 − 4β2 for all x ∈ X. Proof. Letting x = y = 0 in (2.4), we get k2f (0)k ≤ |ρ|β2 k2f (0)k. So f (0) = 0. Letting y = x in (2.4), we get kf (2x) − 4f (x)k ≤ 2θkxkr

(2.6)


for all x ∈ X. So f (x) − 4f x2 ≤ 2β21 r θkxkr for all x ∈ X. Hence

    m−1     m−1 X X 4β2 j

l

j x x

4 f x − 4m f

≤

4 f x − 4j+1 f

≤ 2 θkxkr



j j+1 β1 r β1 rj 2l 2m 2 2 2 2 j=l j=l

(2.7)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.7) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.7), we get (2.5). Since f : X → Y is even, the mapping Q : X → Y is even.

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It follows from (2.4) that kQ(x + y) + Q(x − y) − 2Q(x) − Q(y) − Q(−y)k

         

x+y x−y x y −y β2 n

= lim 4 f +f − 2f −f −f n→∞ 2n 2n 2n 2n 2n         

x+y x−y 3 1 x −x β2 n β2 ≤ lim 4 |ρ|

2f 2n+1 + 2f 2n+1 − 2 f 2n + 2 f 2n n→∞      β2 n 1 1 y −y

+ lim 4 θ (kxkr + kykr ) − − f f n→∞ 2β1 nr 2 2n 2 2n

   

x+y x−y 3 1 1 1

2Q = |ρ|β2 + 2Q − Q(x) + Q(−x) − Q(y) − Q(−y)

2 2 2 2 2 2 for all x, y ∈ X. So kQ(x + y) + Q(x − y) − 2Q(x) − Q(y) − Q(−y)k

     

x+y x−y 3 1 1 1

≤ ρ 2Q + 2Q − Q(x) + Q(−x) − Q(y) − Q(−y)

2 2 2 2 2 2 for all x, y ∈ X. By Lemma 2.1, the mapping Q : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (2.5). Then we have

   

x x β2 n

kQ(x) − T (x)k = 4 Q n − T 2 2n          

x x x x β2 n

≤ 4

Q 2n − f 2n + T 2n − f 2n 4 · 4β2 n ≤ θkxkr , (2β1 r − 4β2 )2β1 nr which tends to zero as n → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mapping satisfying (2.5).  2 Theorem 2.4. Let r < 2β β1 and θ be nonnegative real numbers, and let f : X → Y be an even mapping satisfying (2.4). Then there exists a unique quadratic mapping Q : X → Y such that

kf (x) − Q(x)k ≤

2θ kxkr 4β2 − 2β1 r

(2.8)

for all x ∈ X.



Proof. It follows from (2.6) that f (x) − 14 f (2x) ≤

2θ kxkr 4β2

for all x ∈ X. Hence

m−1

m−1 β1 rj X 1 X 2

1

2θ

f (2l x) − 1 f (2m x) ≤

f (2j x) − 1 f (2j+1 x) ≤ kxkr

4l

m j j+1 β j β2 2 4 4 4 4 4 j=l j=l

(2.9)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.9) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 f (2n x) 4n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.9), we get (2.8). The rest of the proof is similar to the proof of Theorem 2.3. Q(x) := lim

n→∞

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES

Lemma 2.5. An odd mapping f : X → Y satisfies (2.1) if and only if f : X → Y is additive. Proof. Since f : X → Y is an odd mapping, f (0) = 0. Assume that f : X → Y satisfies (2.1). Letting y = x in (2.1), we get kf (2x) − 2f (x)k ≤ 0 and so f (2x) = 2f (x) for all x ∈ X. Thus   x 1 f = f (x) (2.10) 2 2 for all x ∈ X. It follows from (2.1) and (2.10) that kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k

     

x−y 3 1 1 1 x+y

+ 2f − f (x) + f (−x) − f (y) − f (−y) ≤ ρ 2f

2 2 2 2 2 2 = |ρ|β2 kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k and so f (x + y) + f (x − y) = 2f (x)

(2.11)

for all x, y ∈ X. Letting z = x + y and w = z − y in (2.11), we get   z+w f (z) + f (w) = 2f = f (z + w) 2 for all z, w ∈ X. The converse is obviously true.



Corollary 2.6. An odd mapping f : X → Y satisfies (2.3) if and only if f : X → Y is additive. We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (2.1) in β-homogeneous complex Banach spaces for an odd mapping case. Theorem 2.7. Let r > ββ21 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (2.4). Then there exists a unique additive mapping A : X → Y such that 2θ kf (x) − A(x)k ≤ β1 r kxkr (2.12) 2 − 2β2 for all x ∈ X. Proof. Letting x = y = 0 in (2.4), we get k2f (0)k ≤ |ρ|β2 k2f (0)k. So f (0) = 0. Letting y = x in (2.4), we get kf (2x) − 2f (x)k ≤ 2θkxkr

(2.13)


for all x ∈ X. So f (x) − 2f x2 ≤ 2β21 r θkxkr for all x ∈ X. Hence

    m−1     m−1 X X 2β2 j

j

l x x

2 f x − 2m f

≤

2 f x − 2j+1 f

≤ 2 θkxkr



l m j j+1 β r β1 rj 1 2 2 2 2 2 2 j=l j=l

(2.14)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.14) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.14), we get (2.12). Since f : X → Y is odd, the mapping A : X → Y is odd.

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It follows from (2.4) that kA(x + y) + A(x − y) − 2A(x) − A(y) − A(−y)k

         

x+y x−y x y −y β2 n

+f − 2f −f −f = lim 2 f n→∞ 2n 2n 2n 2n 2n         

x+y x−y 3 1 x −x β2 n β2 ≤ lim 2 |ρ|

2f 2n+1 + 2f 2n+1 − 2 f 2n + 2 f 2n n→∞      β2 n 1 1 y −y

+ lim 2 θ (kxkr + kykr ) − f − f n→∞ 2β1 nr 2 2n 2 2n

   

x−y 3 x+y 1 1 1

+ 2A − 2A A(x) + A(−x) − A(y) − A(−y) = |ρ|β2

2 2 2 2 2 2 for all x, y ∈ X. So kA(x + y) + A(x − y) − 2A(x) − A(y) − A(−y)k

    

x+y x−y 3 1

≤ ρ 2A + 2A − A(x) + A(−x) − 2 2 2 2 for all x, y ∈ X. By Lemma 2.5, the mapping A : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (2.12).

1 1 A(y) − A(−y)

2 2 

Then we have

   

x x β2 n

kA(x) − T (x)k = 2 A n − T 2 2n          

x x x x β2 n

≤ 2

A 2n − f 2n + T 2n − f 2n

≤

4 · 2β2 n θkxkr , (2β1 r − 2β2 )2β1 nr

which tends to zero as n → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A. Thus the mapping A : X → Y is a unique additive mapping satisfying (2.12).  Theorem 2.8. Let r < ββ21 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (2.4). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

2β2

2θ kxkr − 2 β1 r

(2.15)

for all x ∈ X.



Proof. It follows from (2.13) that f (x) − 12 f (2x) ≤

2θ kxkr 2β2

for all x ∈ X. Hence

m−1

m−1 X X 2β1 rj 2θ

1

1

f (2l x) − 1 f (2m x) ≤

f (2j x) − 1 f (2j+1 x) ≤ kxkr

2l

2j

j+1 β2 j 2 β2 2m 2 2 j=l j=l

(2.16)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (2.16) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 f (2n x) n→∞ 2n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (2.16), we get (2.15). The rest of the proof is similar to the proof of Theorem 2.7. A(x) := lim

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES

By the triangle inequality, we have kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k

     

x+y x−y 3 1 1 1

ρ 2f + 2f − − f (x) + f (−x) − f (y) − f (−y)

2 2 2 2 2 2 ≤ kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)      

x+y x−y 3 1 1 1 −ρ 2f + 2f − f (x) + f (−x) − f (y) − f (−y)

. 2 2 2 2 2 2 As corollaries of Theorems 2.3, 2.4, 2.7 and 2.8, we obtain the Hyers-Ulam stability results for the additive-quadratic ρ-functional equation (2.3) in β-homogeneous complex Banach spaces. Corollary 2.9. Let r > mapping such that

2β2 β1

and θ be nonnegative real numbers, and let f : X → Y be an even

kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y) (2.17)      

x+y x−y 3 1 1 1 r r −ρ 2f + 2f − f (x) + f (−x) − f (y) − f (−y)

≤ θ(kxk + kyk ) 2 2 2 2 2 2 for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y satisfying (2.5). 2 Corollary 2.10. Let r < 2β β1 and θ be nonnegative real numbers, and let f : X → Y be an even mapping satisfying (2.17). Then there exists a unique quadratic mapping Q : X → Y satisfying (2.8).

Corollary 2.11. Let r > ββ21 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (2.17). Then there exists a unique additive mapping A : X → Y satisfying (2.12). Corollary 2.12. Let r < ββ21 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (2.17). Then there exists a unique additive mapping A : X → Y satisfying (2.15). Remark 2.13. If ρ is a real number such that −1 < ρ < 1 and Y is a β2 -homogeneous real Banach space, then all the assertions in this section remain valid. 3. Additive-quadratic ρ-functional inequality (0.2) Throughout this section, assume that ρ is a fixed complex number with |ρ| < 21 . In this section, we investigate the additive-quadratic ρ-functional inequality (0.2) in β-homogeneous complex Banach spaces. Lemma 3.1. An even mapping f : X → Y satisfies



  

2f x + y + 2f x − y − 3 f (x) + 1 f (−x) − 1 f (y) − 1 f (−y)

2 2 2 2 2 2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y))k

(3.1)

for all x, y ∈ X if and onlt if f : X → Y is quadratic. Proof. Assume that f : X → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get k2f (0)k ≤ |ρ|β2 k2f (0)k. So f (0) = 0. Letting y = 0 in (3.1), we get

 



4f x − f (x) ≤ 0

2

903

(3.2)

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J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.5, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

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for all x ∈ X. So f x2 = 14 f (x) for all x ∈ X. It follows from (3.1) and (3.2) that 1 kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k 2β2 

  

x+y x−y 3 1 1 1

2f = + 2f − f (x) + f (−x) − f (y) − f (−y)

2 2 2 2 2 2 ≤ |ρ|β2 kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k


and so f (x + y) + f (x − y) = 2f (x) + 2f (y) for all x, y ∈ X. The converse is obviously true.



Corollary 3.2. An even mapping f : X → Y satisfies     x+y x−y 3 1 1 1 2f + 2f − f (x) + f (−x) − f (y) − f (−y) 2 2 2 2 2 2 = ρ (f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y))

(3.3)

for all x, y ∈ X if and only if f : X → Y is quadratic. The functional equation (3.3) is called an additive-quadratic ρ-functional equation. We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (3.1) in β-homogeneous complex Banach spaces for an even mapping case. 2 Theorem 3.3. Let r > 2β β1 and θ be nonnegative real numbers, and let f : X → Y be an even mapping such that     x+y x−y 3 1 1 1 k2f + 2f − f (x) + f (−x) − f (y) − f (−y)k (3.4) 2 2 2 2 2 2 ≤ kρ(f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y))k + θ(kxkr + kykr )

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that kf (x) − Q(x)k ≤

2β1 r θ kxkr 2β1 r − 4β2

(3.5)

for all x ∈ X. Proof. Letting x = y = 0 in (3.4), we get k2f (0)k ≤ |ρ|β2 k2f (0)k. So f (0) = 0. Letting y = 0 in (3.4), we get

 



4f x − f (x) ≤ θkxkr

2 for all x ∈ X. So

  m−1   m−1     X X 4β2 j

l

j x x

≤

≤

4 f x − 4m f

4 f x − 4j+1 f θkxkr



2l 2m 2j 2j+1 2β1 rj j=l

(3.6)

(3.7)

j=l

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.7) that the sequence {4n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {4n f ( 2xn )} converges. So one can define the mapping Q : X → Y by x Q(x) := lim 4n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.7), we get (3.5).

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Since f : X → Y is even, the mapping Q : X → Y is even. It follows from (3.4) that

   

2Q x + y + 2Q x − y − 3 Q(x) + 1 Q(−x) − 1 Q(y) − 1 Q(−y)

2 2 2 2 2 2              

x+y x−y 3 x 1 −x 1 y 1 −y β2 n

= lim 4

2f 2n+1 + 2f 2n+1 − 2 f 2n + 2 f 2n − 2 f 2n − 2 f 2n n→∞

          

x+y x−y x y −y β2 n

≤ lim 4 ρ f +f − 2f −f −f

n→∞ 2n 2n 2n 2n 2n β n 4 2 θ + lim β1 nr (kxkr + kykr ) n→∞ 2 = kρ(Q(x + y) + Q(x − y) − 2Q(x) − Q(y) − Q(−y))k for all x, y ∈ X. So

   

2Q x + y + 2Q x − y − 3 Q(x) + 1 Q(−x) − 1 Q(y) − 1 Q(−y)

2 2 2 2 2 2

≤ kρ(Q(x + y) + Q(x − y) − 2Q(x) − Q(y) − Q(−y))k for all x, y ∈ X. By Lemma 3.1, the mapping Q : X → Y is quadratic. Now, let T : X → Y be another quadratic mapping satisfying (3.5). Then we have

   

x x β2 n

kQ(x) − T (x)k = 4 Q n − T 2 2n          

x x x x β2 n

≤ 4

Q 2n − f 2n + T 2n − f 2n 2 · 4β2 n · 2β1 r ≤ θkxkr , (2β1 r − 4β2 )2β1 nr which tends to zero as n → ∞ for all x ∈ X. So we can conclude that Q(x) = T (x) for all x ∈ X. This proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quadratic mapping satisfying (3.5).  2 Theorem 3.4. Let r < 2β β1 and θ be nonnegative real numbers, and let f : X → Y be an even mapping satisfying (3.4). Then there exists a unique quadratic mapping Q : X → Y such that

kf (x) − Q(x)k ≤

2β1 r θ kxkr 4β2 − 2β1 r

(3.8)

for all x ∈ X.



Proof. It follows from (3.6) that f (x) − 14 f (2x) ≤

2β1 r θ kxkr 4β2

for all x ∈ X. Hence

m−1

β1 r m−1 X 2β1 rj X

1

1

f (2l x) − 1 f (2m x) ≤

f (2j x) − 1 f (2j+1 x) ≤ 2 θ kxkr

4l

m j j+1 β β2 j 2 4 4 4 4 4 j=l j=l

(3.9)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.9) that the sequence { 41n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 41n f (2n x)} converges. So one can define the mapping Q : X → Y by 1 Q(x) := lim n f (2n x) n→∞ 4 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.9), we get (3.8). The rest of the proof is similar to the proof of Theorem 3.3. 

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Lemma 3.5. An odd mapping f : X → Y satisfies (3.1) if and only if f : X → Y is additive. Proof. Assume that f : X → Y satisfies (3.1). Letting x = y = 0 in (3.1), we get k2f (0)k ≤ |ρ|β2 k2f (0)k. So f (0) = 0. Letting y = 0 in (3.1), we get

 

4f x − 2f (x) ≤ 0

2

(3.10)

for all x ∈ X. So f x2 = 12 f (x) for all x ∈ X. It follows from (3.1) and (3.10) that


1 kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k 2β2 

  

x+y x−y 3 1 1 1

= 2f + 2f − f (x) + f (−x) − f (y) − f (−y)

2 2 2 2 2 2 ≤ |ρ|β2 kf (x + y) + f (x − y) − 2f (x) − f (y) − f (−y)k and so f (x + y) + f (x − y) = 2f (x) for all x, y ∈ X. The converse is obviously true.



Corollary 3.6. An odd mapping f : X → Y satisfies (3.3) if and only if f : X → Y is additive. We prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequality (3.1) in β-homogeneous complex Banach spaces for an odd mapping case. Theorem 3.7. Let r > ββ12 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (3.4). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

2β1 r θ kxkr (2β1 r − 2β2 )2β2

(3.11)

for all x ∈ X. Proof. Letting x = y = 0 in (3.4), we get k2f (0)k ≤ |ρ|β2 k2f (0)k. So f (0) = 0. Letting y = 0 in (3.4), we get

 

4f x − 2f (x) ≤ θkxkr

2

(3.12)

  m−1     m−1   X X 2β2 j θ

j

l x x

≤

2 f x − 2j+1 f

≤

2 f x − 2m f kxkr

j j+1 β1 rj 2β2 2l 2m 2 2 2 j=l j=l

(3.13)

for all x ∈ X. So

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.13) that the sequence {2n f ( 2xn )} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2n f ( 2xn )} converges. So one can define the mapping A : X → Y by x A(x) := lim 2n f ( n ) n→∞ 2 for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.13), we get (3.11). Since f : X → Y is odd, the mapping A : X → Y is odd.

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It follows from (3.4) that

   

2A x + y + 2A x − y − 3 A(x) + 1 A(−x) − 1 A(y) − 1 A(−y)

2 2 2 2 2 2              

x+y x−y 3 1 1 1 x −x y −y β2 n

= lim 2

2f 2n+1 + 2f 2n+1 − 2 f 2n + 2 f 2n − 2 f 2n − 2 f 2n n→∞

          

x+y x−y x y −y β2 n

≤ lim 2 ρ f +f − 2f −f −f

n→∞ 2n 2n 2n 2n 2n β n 2 2 θ + lim β1 nr (kxkr + kykr ) n→∞ 2 = kρ(A(x + y) + A(x − y) − 2A(x) − A(y) − A(−y))k for all x, y ∈ X. So

   

2A x + y + 2A x − y − 3 A(x) + 1 A(−x) − 1 A(y) − 1 A(−y)

2 2 2 2 2 2

≤ kρ(A(x + y) + A(x − y) − 2A(x) − A(y) − A(−y))k for all x, y ∈ X. By Lemma 3.5, the mapping A : X → Y is additive. Now, let T : X → Y be another additive mapping satisfying (3.11). Then we have

   

x x

kA(x) − T (x)k = 2β2 n A − T

n 2 2n          

x x

+ T x − f x ≤ 2β2 n A − f

n n n n 2 2 2 2 2 · 2β2 n · 2β1 r θ ≤ kxkr , β r β β nr 1 2 1 (2 − 2 )2 2β2 which tends to zero as n → ∞ for all x ∈ X. So we can conclude that A(x) = T (x) for all x ∈ X. This proves the uniqueness of A. Thus the mapping A : X → Y is a unique additive mapping satisfying (3.11).  Theorem 3.8. Let r < ββ12 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (3.4). Then there exists a unique additive mapping A : X → Y such that kf (x) − A(x)k ≤

(2β2

2β1 r θ kxkr − 2β1 r )2β2

(3.14)

for all x ∈ X.



Proof. It follows from (3.12) that f (x) − 21 f (2x) ≤

2β1 r θ kxkr 4β2

for all x ∈ X. Hence

m−1

β1 r m−1 X 2β1 rj X 1

1

f (2l x) − 1 f (2m x) ≤

f (2j x) − 1 f (2j+1 x) ≤ 2 θ kxkr

2l

2j j+1 β2 β2 j 2m 2 4 2 j=l j=l

(3.15)

for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (3.15) that the sequence { 21n f (2n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence { 21n f (2n x)} converges. So one can define the mapping A : X → Y by 1 f (2n x) n→∞ 2n for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.15), we get (3.14). The rest of the proof is similar to the proof of Theorem 3.7. A(x) := lim

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S. YUN, G. A. ANASTASSIOU, C. PARK

By the triangle inequality, we have



  

2f x + y + 2f x − y − 3 f (x) + 1 f (−x) − 1 f (y) − 1 f (−y)

2 2 2 2 2 2 − kρ (f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y))k

   

x+y 1 1 1 x−y 3 2f + 2f − f (x) + f (−x) − f (y) − f (−y) ≤

2 2 2 2 2 2 −ρ (f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y))k . As corollaries of Theorems 3.3, 3.4, 3.7 and 3.8, we obtain the Hyers-Ulam stability results for the additive-quadratic ρ-functional equation (3.3) in β-homogeneous complex Banach spaces. 2 Corollary 3.9. Let r > 2β β1 and θ be nonnegative real numbers, and let f : X → Y be an even mapping such that

   

2f x + y + 2f x − y − 3 f (x) + 1 f (−x) − 1 f (y) − 1 f (−y) (3.16)

2 2 2 2 2 2 −ρ (f (x + y) + f (x − y) − 2f (x) − f (y) − f (−y))k ≤ θ(kxkr + kykr )

for all x, y ∈ X. Then there exists a unique quadratic mapping Q : X → Y satisfying (3.5). 2 Corollary 3.10. Let r < 2β β1 and θ be nonnegative real numbers, and let f : X → Y be an even mapping satisfying (3.16). Then there exists a unique quadratic mapping Q : X → Y satisfying (3.8).

Corollary 3.11. Let r > ββ21 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (3.16). Then there exists a unique additive mapping A : X → Y satisfying (3.11). Corollary 3.12. Let r < ββ21 and θ be nonnegative real numbers, and let f : X → Y be an odd mapping satisfying (3.16). Then there exists a unique additive mapping A : X → Y satisfying (3.14). Remark 3.13. If ρ is a real number such that − 21 < ρ < 21 and Y is a β2 -homogeneous real Banach space, then all the assertions in this section remain valid. References [1] M. Adam, On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl. 4 (2011), 50–59. [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [3] J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in C ∗ -ternary algebras, Bull. Korean Math. Soc. 47 (2010), 195–209. [4] L. C˘ adariu, L. G˘ avruta and P. G˘ avruta, On the stability of an affine functional equation, J. Nonlinear Sci. Appl. 6 (2013), 60–67. [5] A. Chahbi and N. Bounader, On the generalized stability of d’Alembert functional equation, J. Nonlinear Sci. Appl. 6 (2013), 198–204. [6] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. [7] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [8] P. Gˇ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [9] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Math. 62 (2001), 303–309. [10] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [11] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224. [12] C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. 2007 (2007), Article ID 41820, 13 pages.

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ADDITIVE-QUADRATIC ρ-FUNCTIONAL INEQUALITIES

[13] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [14] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [15] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [16] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. [17] S. Rolewicz, Metric Linear Spaces, PWN-Polish Scientific Publishers, Warsaw, 1972. [18] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [19] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [20] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [21] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [22] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [23] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. [24] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [25] L. G. Wang and B. Liu, The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in quasi-β-normed spaces, Acta Math. Sin., Engl. Ser. 26 (2010), 2335–2348. [26] Z. H. Wang and W. X. Zhang, Fuzzy stability of quadratic-cubic functional equations, Acta Math. Sin., Engl. Ser. 27 (2011), 2191–2204. [27] T. Z. Xu, J. M. Rassias and W. X. Xu, Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces, Acta Math. Sin., Engl. Ser. 28 (2012), 529–560. Sungsik Yun Department of Financial Mathematics, Hanshin University, Gyeonggi-do 447-791, Korea E-mail address: [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected]

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A note on stochastic functional differential equations driven by G-Brownian motion with discontinuous drift coefficients ∗

Faiz Faizullah , Aamir Mukhtar1 , M. A. Rana1 ∗

Department of BS and H, College of E and ME, National University of Sciences and Technology (NUST) Pakistan. 1 Department of Basic Sciences, Riphah International University, Islamabad, Pakistan. March 27, 2015

Abstract In the fields of sciences and engineering, the role of discontinuous functions is of immense importance. Heaviside function, for instance, describes the switching process of voltage in an electrical circuit through mathematical process. The current paper aims at exploring the existence theory for stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) whose drift coefficients may not be continuous. It is ascertain that G-SFDEs with discontinuous drift coefficients have more than one bounded and continuous solutions. Key words: Stochastic functional differential equations, discontinuous drift coefficints, G-Brownian motion, existence.

1

Introduction

For the purpose of analysis and formulation of systems pertaining to engineering, economics and social sciences, stochastic dynamical systems play an important role. Through these equations, while considering the present status, one reconstructs the history and predicts the future of the dynamical systems. On the other hand, in several applications, analysis of the modeling system predicts that the change rate of the system’s existing status depends not only on the state that is prevalent but also on the precedent record of the system. This leads to stochastic functional differential equations. The stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) with Lipschitz continuous coefficients was initiated by Ren et.al. [12]. Afterwards, Faizullah used the Caratheodory approximation scheme for developing the existence and uniqueness of solution for G-SFDEs with continuous coefficients [3]. On the other hand, in this case, we study ∗

Corresponding author, E-mail: faiz [email protected] ¯

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the existence theory for G-SFDEs with discontinuous drift coefficients, such as in the following G-SFDE dX(t) = H(Xt )dt + dhBi(t) + dB(t), where H : R → R is the Heaviside function defined by ( 0, if x < 0 ; H(x) = 1, if x ≥ 0.

The above mentioned equations arise, when we take into account the effects of background noise switching systems with delays [5]. For more details on SDEs with discontinuous drift coefficients see [4, 7]. The following stochastic functional differential equation driven by G-Brownian motion (G-SFDE) with finite delay is considered dX(t) = α(t, Xt )dt + β(t, Xt )dhB, Bi(t) + σ(t, Xt )dB(t), 0 ≤ t ≤ T,

(1.1)

where X(t) is the value of stochastic process at time t and Xt = {X(t + θ) : −τ ≤ θ ≤ 0} is a BC([−τ, 0]; R)-valued stochastic process, which represents the family of bounded continuous R-valued functions ϕ defined on [−τ, 0] having norm kϕk = sup | ϕ(θ) | . Let α : [0, T ] × −τ ≤θ≤0

BC([−τ, 0]; R) → R, β : [0, T ] × BC([−τ, 0]; R) → R and σ : [0, T ] × BC([−τ, 0]; R) → R are Borel measurable. The condition ξ(0) ∈ R is given , {hB, Bi(t), t ≥ 0} is the quadratic variation process of G-Brownian motion {B(t), t ≥ 0} and α, β, σ ∈ MG2 ([−τ, T ]; R). Let L2 denote the space of all Ft -adapted process X(t), 0 ≤ t ≤ T , such that k X kL2 = sup |X(t)| < ∞. We define the −τ ≤t≤T

initial condition of equation (1.1) as follows; Xt0 =ξ = {ξ(θ) : −τ < θ ≤ 0} is F0 − measurable, BC([−τ, 0]; R) − valued random variable such that ξ ∈ MG2 ([−τ, 0]; R) .

(1.2)

G-SFDEs (1.1) with initial condition (1.2) can be written in the following integral form; Z t Z t Z t X(t) = ξ(0) + α(s, Xs )ds + β(s, Xs )dhB, Bi(s) + σ(s, Xs )dB(s). 0

0

0

Consider the following linear growth and Lipschitz conditions respectively. (i) For any t ∈ [0, T ], |α(t, x)|2 + |β(t, x)|2 + |σ(t, x)|2 ≤ K(1 + |x|2 ), K > 0. (ii) For all x, y ∈ (BC[−τ, 0]; R) and t ∈ [0, T ], |α(t, x) − α(t, y)|2 + |β(t, x) − |β(t, y)|2 + |σ(t, x) − σ(t, y)|2 ≤ K(x − y)2 , K > 0. The above G-SFDE has a unique solution X(t) ∈ MG2 ([−τ, T ]; R) if all the coefficients α, β and σ satisfy the Linear growth and Lipschitz conditions [3, 12]. However, we suppose that the drift coefficient α does not need to be continuous. The solution of equation 1.1 with initial condition 1.2 is an R valued stochastic processes X(t), t ∈ [−τ, T ] if 2

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(i) X(t) is path-wise continuous and Ft -adapted for all t ∈ [0, T ]; (ii) α(t, Xt ) ∈ L1 ([o, T ]; R) and β(t, Xt ), σ(t, Xt ) ∈ L2 ([o, T ]; R); (iii) X0 = ξ and for each t ∈ [0, T ], dX(t) = α(t, Xt )dt + β(t, Xt )dhB, Bi(t) + σ(t, Xt )dB(t) q.s. In the subsequent section, some preliminaries are given whereas in section 3, the comparison theorem is developed. The last section, shows that under some suitable conditions, the G-SFDE (1.1), provides more than one solutions.

2

Basic concepts and notions

In this section, we give some notions and basic definitions of the sublinear expectation [1, 2, 10, 11, 13]. Let Ω be a (non-empty) basic space and H be a linear space of real valued functions defined on Ω such that any arbitrary constant c ∈ H and if X ∈ H then |X| ∈ H. We consider that H is the space of random variables. Definition 2.1. A functional E : H → R is called sub-linear expectation, if ∀ X, Y ∈ H, c ∈ R and λ ≥ 0 it satisfies the following properties (1) (Monotonicity): If X ≥ Y

then E[X] ≥ E[Y ].

(2) (Constant preserving): E[c] = c. (3) (Sub-additivity): E[X + Y ] ≤ E[X] + E[Y ]. (4) (Positive homogeneity): E[λX] = λE[X]. The triple (Ω, H, E) is called a sublinear expectation space. Consider the space of random variables H such that if X1 , X2 , ..., Xn ∈ H then ϕ(X1 , X2 , ..., Xn ) ∈ H for each ϕ ∈ Cl.Lip (Rn ), where Cl.Lip (Rn ) is the space of linear functions ϕ defined as the following Cl.Lip (Rn ) ={ϕ : Rn → R | ∃ C ∈ [0, ∞) : ∀ x, y ∈ Rn , |ϕ(x) − ϕ(y)| ≤ C(1 + |x|C + |y|C )|x − y|}. G-expectation and G-Brownian Motion. Let Ω = C0 ([0, ∞)), that is, the space of all R-valued continuous paths (wt )t∈[0,∞) with w0 = 0 equipped with the distance ∞ X 1 ρ(w , w ) = ( max |w1 − wt2 | ∧ 1), 2k t∈[0,k] t 1

2

k=1

and consider the canonical process Bt (w) = wt for t ∈ [0, ∞), w ∈ Ω then for each fixed T ∈ [0, ∞) we have Lip (ΩT ) = {ϕ(Bt1 , Bt2 , ..., Btn ) : t1 , ..., tn ∈ [0, T ], ϕ ∈ Cl.Lip (Rn ), n ∈ N}, 3

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where Lip (Ωt ) ⊆ Lip (ΩT ) for t ≤ T and Lip (Ω) = ∪∞ m=1 Lip (Ωm ). ˆ ˆ ˆ Consider a sequence {ξi }∞ i=1 of random variables on a sublinear expectation space (Ω, Hp , E) such that ξi+1 is independent of (ξ1 , ξ2 , ..., ξi ) for each i = 1, 2, ... and ξi is G-normally distributed for each i ∈ {1, 2, ...}. Then a sublinear expectation E[.] defined on Lip (Ω) is introduced as follows. For 0 = t0 < t1 < ... < tn < ∞ (t0 , t1 , ..., tn ∈ [t, ∞)) [13], ϕ ∈ Cl.Lip (Rn ) and each X = ϕ(Bt1 − Bt0 , Bt2 − Bt1 , ..., Btn − Btn−1 ) ∈ Lip (Ω),

E[ϕ(Bt1 − Bt0 , Bt2 − Bt1 , ..., Btn − Btn−1 )] p √ ˆ = E[ϕ( t1 − t0 ξ1 , ..., tn − tn−1 ξn )]. The conditional sublinear expectation of X ∈ Lip (Ωt ) is defined by E[X|Ωt ] = E[ϕ(Bt1 , Bt2 − Bt1 , ..., Btm − Btm−1 )|Ωt ] = ψ(Bt1 , Bt2 − Bt1 , ..., Btj − Btj−1 ), where ˆ ψ(x1 , ..., xj ) = E[ϕ(x 1 , ..., xj ,

p

tj+1 − tj ξj+1 , ...,

p

tn − tn−1 ξn )].

Definition 2.2. The sublinear expectation E : Lip (Ω) → R defined above is called a G-expectation and the corresponding canonical process {Bt , t ≥ 0} is called a G-Brownian motion. The completion of Lip (Ω) under the norm kXkp = (E[|X|p ])1/p [11, 13] for p ≥ 1 is denoted by LpG (Ω) and LpG (Ωt ) ⊆ LpG (ΩT ) ⊆ LpG (Ω) for 0 ≤ t ≤ T < ∞. The filtration generated by the canonical process (Bt )t≥0 is denoted by Ft = σ{Bs , 0 ≤ s ≤ t}, F = {Ft }t≥0 . Itˆ o’s Integral of G-Brownian motion. For any T ∈ [0, ∞), a finite ordered subset πT = {t0 , t1 , ..., tN } such that 0 = t0 < t1 < ... < tN = T is a partition of [0, T ] and µ(πT ) = max{|ti+1 − ti | : i = 0, 1, ..., N − 1}. N N N A sequence of partitions of [0, T ] is denoted by πTN = {tN 0 , t1 , ..., tN } such that lim µ(πT ) = 0. N →∞

Consider the following simple process: Let p ≥ 1 be fixed. For a given partition πT = {t0 , t1 , ..., tN } of [0, T ], ηt (w) =

N −1 X

ξi (w)I[ti ,ti+1 ] (t),

(2.1)

i=0

where ξi ∈ LpG (Ωti ), i = 0, 1, ..., N − 1. The collection containing the above type of processes, that is, containing ηt (w) is denoted by MGp,0 (0, T ). The completion of MGp,0 (0, T ) under the norm RT kηk = { 0 E[|ηu |p ]du}1/p is denoted by MGp (0, T ) and for 1 ≤ p ≤ q, MGp (0, T ) ⊃ MGq (0, T ).

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Definition 2.3. For each ηt ∈ MG2,0 (0, T ), the Itˆo’s integral of G-Brownian motion is defined as T

Z

ηu dBu =

I(η) = 0

N −1 X

ξi (Bti+1 − Bti ).

i=0

Definition 2.4. An increasing continuous process {hBit : t ≥ 0} with hBi0 = 0, defined by hBit =

Bt2

Z −2

t

Bu dBu , 0

is called the quadratic variation process of G-Brownian motion.

3

Comparison theorem for G-SFDEs

The purpose of this section is to establish comparison result for problem (1.1) with initial data (1.2). Consider the following two stochastic functional differential equations X (t) = ξ 1 (0) +

Z

t

Z

t0 2

t

α1 (s, Xs )ds + Z

t0

t

X (t) = ξ (0) +

Z

Z

σ(s, Xs )dB(s),

t ∈ [0, T ], (3.1)

σ(s, Xs )dB(s),

t ∈ [0, T ]. (3.2)

t0

t

α2 (s, Xs )ds + t0

t

β(s, Xs )dhB, Bi(s) + Z

t

β(s, Xs )dhB, Bi(s) + t0

t0

Theorem 3.1. Assume that: (i) X 1 and X 2 are unique strong solutions of problems (3.1) and (3.2) respectively. (ii) α1 (s, Xs ) ≤ α2 (s, Xs ) componentwise for all t ∈ [t0 , T ], x ∈ BC([−τ, 0]; Rd ) and ξ 1 ≤ ξ 2 . (iii) α1 or α2 is increasing such that f (t, x) ≤ f (t, y) when x ≤ y for all x, y ∈ C([−τ, 0]; R). Then for all t > 0 we have X 1 ≤ X 2 q.s. Proof. First, we define an operator q(., .) : C([−τ, 0]; R) × C([−τ, 0]; R) → C([−τ, 0]; R) such that q(x, y) = max[x, y]. Obviously, y → q(x, y) satisfies the linear growth and Lipschitz conditions. Now we suppose that α2 is increasing and consider the following equation Z t Z t 2 1 Y (t) = ξ (0) + α2 (s, q(Xs , Ys ))ds + β(s, q(Xs 1 , Ys )dhB, Bi(s) t0 t0 (3.3) Z t 1 + σ(s, q(Xs , Ys )dB(s), t0 ≤ t ≤ T. t0

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Thus it is easy to see that the coefficients satisfy the linear growth and Lipschitz conditions, so 3.3 has a unique solution Y (t). We shall prove that Y (t) ≥ Xs1 q.s. We define the following two stopping times. For more details on stopping times we refere the reader to [7, 8]. τ1 = inf{t ∈ [t0 , T ] : Xs1 − Y (t) > 0} where τ1 < T, τ2 = inf{t ∈ [τ1 , T ] : Xs1 − Y (t) < 0}. Contrary suppose that there exist an interval (τ1 , τ2 ) ⊂ [t0 , T ] such that Y (τ1 ) = X 1 (τ1 ) = ξ ∗ (0) and Y (t) ≤ X 1 (t) for all t ∈ (τ1 , τ2 ). Then, Z t Z t 1 1 ∗ β(s, q(Xs 1 , Ys ))dhB, Bi(s) α2 (s, q(Xs , Ys ))ds + Y (t) − X (t) = ξ (0) + τ1

τ1

Z

t

∗

1

Z

t

α1 (s, Xs 1 )ds

σ(s, q(Xs , Ys ))dB(s) − ξ (0) −

+ τ1 t

Z −

τ1

β(s, Xs 1 )dhB, Bi(s) −

τ1

Z

t

σ(s, Xs 1 )dB(s), t ∈ (τ1 , τ2 ).

τ1

Y (t) − X 1 (t) =

Z Z

t

[α2 (s, q(Xs 1 , Ys )) − α1 (s, Xs 1 )]ds

τ1 t

+ τ1 Z t

+

[β(s, q(Xs 1 , Ys )) − β(s, Xs 1 )]dhB, Bi(s) [σ(s, q(Xs 1 , Ys )) − σ(s, Xs 1 )]dB(s), t ∈ (τ1 , τ2 ).

τ1

But our supposition Y (t) ≤ X 1 (t) yields q(X 1 , Y ) = max[X 1 , Y ] = X 1 . So, we have Z t 1 Y (t) − X (t) = [α2 (s, Xs 1 ) − α1 (s, Xs 1 )]ds Z

τ1 t

[β(s, Xs 1 ) − β(s, Xs 1 )]dhB, Bi(s)

+ τ1 Z t

+

[σ(s, Xs 1 ) − σ(s, Xs 1 )]dB(s)

τ1

Y (t) − X 1 (t) =

Z

t

[α2 (s, Xs 1 ) − α1 (s, Xs 1 )]ds ≥ 0,

τ1

because α2 (t, x) ≥ α1 (t, x). Which gives contradiction. So, our supposition Y (t) ≤ X 1 (t) for all t ∈ (τ1 , τ2 ) is wrong. Thus Y (t) ≥ X 1 (t) q.s. and so p(X 1 , Y ) = Y . It means that Y = X 2 ≥ X 1 because G-SFDE (3.3) has a unique solution X 2 .

4

G-SFDEs with discontinuous drift coefficients

We now suppose that α is left continuous, increasing and α(t, x) ≥ 0 for all (t, x) ∈ [0, T ] × BC([−τ, 0]; R) but not continuous. Consider the following sequence of problems. Z t Z t Z t n n−1 n X (t) = ξ(0) + α(s, Xs )ds + β(s, Xs )dhB, Bi(s) + σ(s, Xsn )dB(s), t ∈ [0, T ], (4.1) 0

0

0

6

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where X 0 = Lt , Lt is the unique solution of the following problem Z t Z t σ(s, Ls )dB(s), t ∈ [0, T ]. β(s, Ls )dhB, Bi(s) + Lt = ξ +

(4.2)

0

0

Thus using the comparison theorem and the fact that α(t, x) ≥ 0, we have X 1 ≥ Lt . So, we can see that X n is an increasing sequence. Now we shall prove that X n is bounded in L2 norm. Lemma 4.1. Suppose X n (t) be a solution of problem (4.1) then there exists a positive constant C independent of n such that, ! E

sup |X n (s)|2

≤ C.

−τ ≤s≤T

Proof. For any n ≥ 1 we define the following stopping time in a similar way as given in [9] τm = T ∧ inf {t ∈ [t0 , T ] : kXtn k ≥ m}. We have τm ↑ T and define X n,m (t) = X n (t ∧ τm ) for t ∈ (−τ, T ). Then for t ∈ [0, T ], Z t Z t Z t n−1,m n,m n,m X (t) = ξ(0) + α(t, Xt )I[o,τm ] dt + β(t, Xt )I[o,τm ] dhB, Bit + σ(t, Xtn,m )I[o,τm ] dBt . 0

0

Z

0

t

Z

t

|X n,m (t)|2 = |ξ(0) + α(t, Xtn−1,m )I[0,τm ] dt + β(t, Xtn,m )I[0,τm ] dhB, Bit 0 0 Z t σ(t, Xtn,m )I[0,τm ] dBt |2 + 0 Z t Z t n−1,m 2 2 ≤ 4|ξ(0)| + 4| α(t, Xt )I[0,τm ] dt| + 4| β(t, Xtn,m )I[0,τm ] dhB, Bit |2 0 0 Z t + 4| σ(t, Xtn,m )I[0,τm ] dBt |2 0

Taking G-expectation, using properties of G-integral, G-quadratic variation process [10, 11] and linear growth condition we get Z t Z t n−1,m 2 n,m 2 2 E[|X (t)| ] ≤ 4E|ξ(0)| + 4C1 [1 + E|Xt | ]dt + 4C2 [1 + E|Xtn,m |2 ]dt| 0 0 Z t + 4C3 [1 + E|Xtn,m |2 ]dt 0 Z t Z t Z t Z t n−1,m 2 2 ≤ 4E|ξ(0)| + 4C1 dt + 4C1 E|Xt | dt + 4C2 dt + 4C2 E|Xtn,m |2 dt 0 0 0 0 Z t Z t n,m 2 + 4C3 dt + 4C3 E|Xt | dt 0 0 Z t Z t = 4E|ξ(0)|2 + 4C1 T + 4C1 E|Xtn−1,m |2 dt + 4C2 T + 4C2 E|Xtn,m |2 dt 0 0 Z t + 4C3 T + 4C3 E|Xtn,m |2 dt. 0

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Then for any k ∈ N we have, max E[|X

1≤n≤k

n,m

Z

2

(t)| ] ≤ C4 +4C1

t

max

0 1≤n≤k

E|Xtn−1,m |2 dt+4C2

t

Z

max

0 1≤n≤k

E|Xtn,m |2 dt+4C3

Z

t

max E|Xtn,m |2 dt,

0 1≤n≤k

where C4 = 4[E|ξ|2 + C1 T + C2 T + C3 T ] and thus using Doob’s martingale inequality for any n, m ∈ N we have, Z t n,m 2 E|Xsn,m |2 dt, (4.3) E[ sup |X (s)| ] ≤ C4 + C5 0≤s≤t

0

where C5 = 4(C1 + C2 + C3 ). One can observe the fact [9], sup |X n,m (s)|2 ≤ kξk + sup |X n,m (s)|2 , −τ ≤s≤t

0≤s≤t

and thus 4.3 yields E[ sup |X n,m (s)|2 ] ≤ E[kξk] + C4 + C5 −τ ≤s≤t

Z

t

E|Xsn,m |2 dt

0

Z

t

≤ C6 + C5

E[ sup |X n,m (r)|2 ]dt,

0

−τ ≤r≤s

where C6 = E[kξk] + C4 . So, using the Gronwall inequality and taking m → ∞ we have, E[ sup |X n (s)|2 ] ≤ C6 eC4 t . −τ ≤s≤t

Letting t = T we get the desired result, E[ sup |X n (s)|2 ] ≤ C ∗ ,

C ∗ = C4 eCT .

−τ ≤s≤T

Theorem 4.2. Suppose that: (i) The coefficient α be left continuous and increasing in the second variable x. (ii) For all (t, x) ∈ [0, T ] × BC([−τ, 0]; R), α(t, x) ≥ 0. Then the G-SFDE (1.1) has more than one solution X(t) ∈ MG2 ([−τ, T ]; R). Proof. By theorem 3.1 we know that {X n } is increasing and by Lemma 4.1 it is bounded in L2 . Then by Dominated Convergence theorem we can deduce that X n converges in L2 . Denoting the limit of X n by X and thus for almost all w, we get α(t, X n (t)) → α(t, X(t)) as n → ∞, and |α(t, X n (t))| ≤ K(1 + sup |X n (t)|) ∈ L1 ([t0 , T ]). n

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Thus, for almost all w and uniformly in t Z t Z t n α(s, X(s))ds, n → ∞. α(s, X (s))ds → 0

0

By the properties of β, σ and by the continuity properties of G-integral and its quadratic variation process we have, Z t Z t n sup β(s, X (s))dhB, Bis − β(s, X(s))dhB, Bis → 0 (q.s), n → ∞. 0≤t≤T

0

0

Z t Z t n sup σ(s, X (s))dB(s) − σ(s, X(s))dB(s) → 0 (q.s), n → ∞.

0≤t≤T

0

0

It is easy to conclude that X n converges uniformly to X in t, hence X is continuous. Taking limit in equation (4.1), we get that X is the desired solution for stochastic functional differential equation (1.1) with initial condition (1.2).

5

Acknowledgments

We are very grateful to Dr. Ali Anwar for his careful reading and some useful suggestions. First author acknowledge the financial support of National University of Sciences and Technology (NUST) Pakistan for this research.

References [1] Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths. Potential Anal., 2010; 34: 139-161. [2] Faizullah F. A note on the Caratheodory approximation scheme for stochastic differential equations under G-Brownian motion. Zeitschrift fr Naturforschung A. 2012; 67a: 699-704. [3] Faizullah F. Existence of solutions for G-SFDEs with Caratheodory Approximation Scheme. Abstract and Applied Analysis. http://dx.doi.org/10.1155/2014/809431, 2014; volume 2014: pages 8. [4] Faizullah F, Piao D. Existence of solutions for G-SDEs with upper and lower solu- tions in the reverse order. International Journal of the Physical Sciences. 2012; 16(12): 1820-1829. [5] Halidias N, Ren Y. An existence theorem for stochastic functional differential equations with delays under weak conditions. Statistics and Probability Letters. 2008; 78: 2864-2867. [6] Halidias N, Kloeden P. A note on strong solutions for stochastic differential equations with discontinuous drift coefficient. J. Appl. Math. Stoch. Anal. doi:10.1155/JAMSA/2006/73257. Article ID 73257., 2006; 78: 1-6. 9

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[7] Hu M, Peng S. Extended conditional G-expectations and related stopping times. arXiv:1309.3829v1[math.PR] 16 Sep 2013. [8] Li X, Peng S. Stopping times and related Ito’s calculus with G-Brownian motion. Stochastic Processes and thier Applications. 2011; 121: 1492-1508. [9] Mao X. Stochastic differential equations and their applications. Horwood Publishing Chichester 1997. [10] Peng S. G-expectation, G-Brownian motion and related stochastic calculus of Ito’s type. The abel symposium 2005, Abel symposia 2, edit. benth et. al., Springer-vertag. 2006; 541-567. [11] Peng S. Multi-dimentional G-Brownian motion and related stochastic calculus under Gexpectation. Stochastic Processes and thier Applications. 2008; 12: 2223-2253. [12] Ren Y, Bi Q, Sakthivel R. Stochastic functional differential equations with infinite delay driven by G-Brownian motion. Mathematical Methods in the Applied Sciences, 2013; 36(13): 17461759. [13] Song Y. Properties of hitting times for G-martingale and their applications. Stochastic Processes and thier Applications. 2011; 8(121): 1770-1784.

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SUBCLASSES OF JANOWSKI-TYPE FUNCTIONS DEFINED BY CHO-KWON-SRIVASTAVA OPERATOR ˘ AND BADR S. ALKAHTANI SAIMA MUSTAFA, TEODOR BULBOACA, Abstract. We introduce a new subclass of analytic functions in the unit disk U defined by using Cho-Kwon Srivastava integral operator. Inclusion results radius problem and integral preserving properties are investigated.

1. Introduction Let Ap be the class of analytic functions in U of the form (1.1)

p

f (z) = z +

∞ X

ap+n z p+n , z ∈ U,

(p ∈ N) ,

n=1

where N := {1, 2, . . . }. For p = 1 we denotes A := A1 . Note that the class Ap is closed under the convolution (or Hadamard) product, that is ∞ X p ap+n bp+n z p+n , z ∈ U, (p ∈ N) , f (z) ∗ g(z) := z + n=1

P p+n , z ∈ U. where f is given by (1.1) and g(z) = z p + ∞ n=1 bp+n z p The operator L (d, e) : Ap → Ap is defined by using the Hadamard (convolution) product, that is (1.2)

Lp (d, e)f (z) := f (z) ∗ ϕp (d, e; z),

where p

ϕp (d, e; z) := z +

∞ X (d)n n=1

(e)n

z p+n ,


d ∈ C, e ∈ C \ Z− 0 ,

and (d)n = d(d + 1) . . . (d + n − 1), with (d)0 = 1, represents the well-known Pochhammer symbol. From (1.2) it follows immediately that z (Lp (d, e)f (z))0 = dLp (d + 1, e)f (z) − (d − p)Lp (d, e)f (z). The operator Lp (d, e) was introduced by Saitoh [16] and this is an extension of the operator L(d, e) which was defined by Carlson and Shaffer [2]. 2010 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. Analytic functions, univalent functions, Cho-KwonSrivastava operator. 1

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2

Analogous to the Lp (d, e) operator, Cho et. al. [4] introduced the operator Iµp (d, e) : Ap → Ap defined by Iµp (d, e)f (z) := f (z) ∗ ϕ†p (d, e; z),

(1.3) where ϕ†p (d, e; z) := z p +

∞ X (µ + p)n (e)n n=1

n!(d)n

z p+n ,


d, e ∈ C \ Z− 0 , µ > −p .

We notice that ϕ†p (d, e; z) ∗ ϕp (d, e; z) =

zp , z ∈ U. (1 − z)µ+p

From (1.3), the following identities can be easily obtained [4]:
0 (1.4) z Iµp (d + 1, e)f (z) = dIµp (d, e)f (z) − (d − p) Iµp (d + 1, e)f (z),
0 p (1.5) z Iµp (d, e)f (z) = (µ + p)Iµ+1 (d, e)f (z) − µIµp (d, e)f (z). We may easily remark the following relations I1p (p + 1, 1)f (z) = f (z),

I1p (p, 1)f (z) =

zf 0 (z) , p

and remark that the operator Iµ1 (a + 2, 1), with µ > −1 and a > −2, was studied in [5]. If f and g are two analytic functions in U, we say that f is subordinate to g, written symbolically as f (z) ≺ g(z), if there exists a Schwarz function w, which (by definition) is analytic in U, with w(0) = 0, and |w(z)| < 1, z ∈ U, such that f (z) = g(w(z)), for all z ∈ U. Furthermore, if the function g is univalent in U, then we have the following equivalence, (cf., e.g., [10], see also [11, p. 4]): f (z) ≺ g(z) ⇔ f (0) = g(0) and f (U) ⊂ g(U). Definition 1.1. 1. Like in [3], for arbitrary fixed numbers A, B and β, with −1 ≤ B < A ≤ 1 and 0 ≤ β < 1, let P [A, B, β] denote the family of functions p that are analytic in U, with p(0) = 1, and such that 1 + [(1 − β)A + βB] z p(z) ≺ . 1 + Bz We will use the notations P [A, B] := P [A, B, 0] and P (0) := P [1, −1, 0]. 2. Let Pl [A, B, β] denote the class of functions p that are analytic in U, with p(0) = 1, that are represented by     l 1 l 1 (1.6) p(z) = + K1 (z) − − K2 (z), 4 2 4 2 where K1 , K2 ∈ P [A, B, β], −1 ≤ B < A ≤ 1, 0 ≤ β < 1, and l ≥ 2.

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Remarks 1.1. (i) Remark that the class Pl (β) := Pl [1, −1, β] was defined and studied in [12], while for l = 2 and β = 0 the above class was introduced by Janowski [8]. Moreover, the class Pl := Pl [1, −1, 0] is the well-known class of Pinchuk [15]. e where βe = 1 − A1 and A1 = Also, we see that Pl [A, B, β] ⊂ Pl (β), 1−B (1 − β)A + βB. (ii) Notice that, if g is analytic in U with g(0) = 1, then there exit functions g1 and g2 analytic in U with g1 (z) = g2 (z) = 1, such that the function g could be written in the form (1.6). For example, taking g(z) + 1 g(z) − 1 g(z) − 1 g(z) + 1 + and g1 (z) = − , g1 (z) = k 2 2 k then g1 and g2 are analytic in U, and g1 (z) = g2 (z) = 1. We will assume throughout our discussion, unless otherwise stated, that λ > 0, d, e ∈ R \ Z− 0 , µ > −p, −1 ≤ B < A ≤ 1, ϑ ≥ 0, and p ∈ N. Moreover, all the powers are the principal ones. Using the Cho-Kwon-Srivastava integral operator Iµp (d, e) defined by (1.4), we will define the following subclasses of Ap . Definition 1.2. Let d, e ∈ R \ Z− 0 , λ > 0, µ > −p, 0 ≤ β < 1, and ϑ ≥ 0. For the function f ∈ Ap , p ∈ N, we say that f ∈ λ,ϑ Nl,p (d, e; µ; β, A, B), with l ≥ 2, if and only if  λ  λ p Iµ+1 (d, e)f (z) zp zp (1 + ϑ) −ϑ p ∈ Pl [A, B, β]. Iµp (d, e)f (z) Iµ (d, e)f (z) Iµp (d, e)f (z) We need to notice that, since the left-hand side function from the above definition need to be analytic in U, we implicitly assumed that ˙ Iµp (d, e)f (z) 6= 0 for all z ∈ U. Remarks 1.2. We remark the following special cases of the above classes: (i) for β = 0 and l = 2 we obtain the subclass of non-Bazilevi´c functions defined by [18]; (ii) for µ = 0, l = 2, ϑ = B = −1, A = 1 and λ > 0, the above class reduces to the class Q(λ) of p–valent non-Bazilevi´c functions (see [14]). 2. Preliminaries The following definitions and lemmas will be required in our present investigation. Lemma 2.1. [7] Let h be a convex function in U with h(0) = 1. Suppose also that the function p given by p(z) = 1 + cn z n + cn+1 z n+1 + . . . , z ∈ U, is analytic in U. Then zp0 (z) p(z) + ≺ h(z) γ

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4

implies (2.1)

γ γ p(z) ≺ q(z) = z − n n

z

Z

γ

t n −1 h(t) d t ≺ h(z),

0

and q is the best dominant of (2.1). For real or complex numbers a, b and c, the Gauss hypergeometric function is defined by

(2.2)

a(a + 1) · b(b + 1) z 2 a·b z + + ... 2 F1 (a, b, c; z) = 1 + c 1! c(c + 1) 2! ∞ X (a)k (b)k z k = , a, b ∈ C, c ∈ C \ {0, −1, −2, . . . }, (c) k! k k=0

where (d)k is the previously recalled Pochhammer symbol. The series (2.2) converges absolutely for z ∈ U, hence it represents an analytic function in U (see [19, Chapter 14]). Each of the following identities are fairly well-known: Lemma 2.2. [19, Chapter 14] For all real or complex numbers a, b and c, with c 6= 0, −1, −2, . . . , , the next equalities hold: Z 1 Γ(b)Γ(c − b) tb−1 (1 − t)c−b−1 (1 − tz)−a d t = (2.3) 2 F1 (a, b, c; z) Γ(c) 0 where Re c > Re b > 0,   z −a (2.4) , 2 F1 (a, b, c; z) = (1 − z) 2 F1 a, c − b, c; z−1 and (2.5)

2 F1 (a, b, c; z)

Lemma 2.3. [17] Let f (z) = ∞ X

=2 F1 (b, a, c; z).

∞ X

ak z k be analytic in U and g(z) =

k=0

bk z k be analytic and convex in U. If f (z) ≺ g(z), then

k=0

|ak | ≤ |b1 | , k ∈ N. λ,ϑ 3. Main results for the class Nl,p (d, e; µ; β, A, B) λ,ϑ In this section, some properties of the class Nl,p (d, e; µ; β, A, B) such as inclusion results, integral preserving property, radius problem, coefficient bound will be discussed. λ,ϑ Theorem 3.1. 1. If f ∈ Nl,p (d, e; µ; β, A, B), then  λ zp ∈ Pl [A, B, β]. Iµp (d, e)f (z)

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λ,ϑ 2. Moreover, if f ∈ Nl,p (d, e; µ; β, γ) with ϑ 6= 0, then  λ zp ∈ Pl (β1 ), Iµp (d, e)f (z)

where β1 := β + (1 − β)ϑ1 and

(

A B

+ 1−

1−


A

ϑ1 := ϑ1 (p, λ, ϑ, µ; A, B) =   λ(µ+p) B −1 (1 − B) 2 F1 1, 1, ϑ + 1, B−1 , B 6= 0,

B λ(µ+p) A, λ(µ+p)+ϑ

B = 0.

(All the powers are the principal ones). Proof. Since the implication is obvious for ϑ = 0, suppose that ϑ > 0. Letting  λ zp (3.1) K(z) = . Iµp (d, e)f (z) It follows that K is analytic in U, with K(0) = 1, and according to the part (ii) of Remarks 1.1 the function K could be written in the form     k 1 k 1 (3.2) K(z) = + − K1 (z) − K2 (z), 4 2 4 2 where K1 and K2 are analytic in U, with K1 (z) = K2 (z) = 1. From the part 2. of Definition 1.1 we have that K ∈ Pl [A, B, β], if and only if the function K has the representation given by the above relation, where K1 , K2 ∈ P [A, B, β]. Consequently, supposing that K is of the form (3.2), we will prove that K1 , K2 ∈ P [A, B, β]. Differentiating the relation (3.1) and using the identity (1.5), we have  λ p Iµ+1 (d, e)f (z) zK 0 (z) zp = K(z) − p , λ(µ + p) Iµ (d, e)f (z) Iµp (d, e)f (z) and from this relation we deduce that  λ  λ p Iµ+1 (d, e)f (z) zp zp (1 + ϑ) −ϑ p = Iµp (d, e)f (z) Iµ (d, e)f (z) Iµp (d, e)f (z) ϑ K(z) + zK 0 (z). λ(µ + p) λ,ϑ Since f ∈ Nl,p (d, e; µ; β, A, B), from the above relation it follows that

K(z) +

ϑ zK 0 (z) ∈ Pl [A, B, β], λ(µ + p)

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˘ AND B. S. ALKAHTANI S. MUSTAFA, T. BULBOACA,

and according to the second part of the Definition 1.1, this is equivalent to ϑ Ki (z) + zKi0 (z) ∈ P [A, B, β], (i = 1, 2), λ(µ + p) that is   1 ϑ 0 Ki (z) + zK (z) − β ∈ P [A, B], (i = 1, 2). 1−β λ(µ + p) i Writing (3.3)

Ki (z) = (1 − β)pi (z) + β, (i = 1, 2),

from the previous relation we have ϑ pi (z) + zp0 (z) ∈ P [A, B], (i = 1, 2). λ(µ + p) i By using Lemma 2.1 for γ = relation we deduce that

λ(µ + p) and n = 1, from the above ϑ

1 + Az , (i = 1, 2), 1 + Bz where Z λ(µ + p) − λ(µ+p) z λ(µ+p) −1 1 + At ϑ z dt q(z) = t ϑ ϑ 1 + Bt 0 is the best dominant for pi , i = 1, 2. 1 + Az Since pi (z) ≺ , i = 1, 2, from (3.3) it follows that Ki ∈ 1 + Bz P [A, B, β], i = 1, 2, and according to (3.1) we conclude that K ∈ Pl [A, B, β], which proves the first part of the theorem. For the second part of our result, we distinguish the following two cases: (i) For B = 0, a simple computation shows that pi (z) ≺ q(z) ≺

pi (z) ≺ q(z) = 1 +

λ(µ + p) Az, (i = 1, 2). λ(µ + p) + ϑ

(ii) For B 6= 0, making the change of variables s = zt, followed by the use of the identities (2.3), (2.4) and (2.5) of Lemma 2.2, we obtain Z λ(µ + p) 1 λ(µ+p) −1 1 + Asz pi (z) ≺ q(z) = s ϑ ds = ϑ 1 + Bsz 0     A A λ(µ + p) Bz −1 + 1− (1 + Bz) 2 F1 1, 1, + 1, , (i = 1, 2). B B ϑ Bz + 1 Now, it is sufficient to show that (3.4)

inf {Re q(z) : z ∈ U} = q(−1).

We may easily show that 1 + Az 1 − Ar Re ≥ , 1 + Bz 1 − Br

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1 + Atz λ(µ + p) λ(µ+p) −1 and d µ(t) = t ϑ d t, which 1 + Btz ϑ is a positive measure on [0, 1], we have Z 1 G(t, z) d µ(t), q(z) = Denoting G(t, z) =

0

hence it follows Z Re q(z) ≥ 0

1

1 − Atr d µ(t) = q(−r), 1 − Btr

for |z| ≤ r < 1.

By letting r → 1− we obtain (3.4), and from (3.3) and (3.1) we conclude that K ∈ Pl (β1 ), which completes our proof.  Theorem 3.2. If 0 ≤ ϑ1 < ϑ2 , then λ,ϑ1 λ,ϑ2 (d, e; µ; β, A, B) (d, e; µ; β, A, B) ⊂ Nl,p Nl,p

Proof. The first part of Theorem 3.1 shows that the above inclusion holds whenever ϑ1 = 0. λ,ϑ2 If 0 < ϑ1 < ϑ2 , for an arbitrary f ∈ Nl,p (d, e; µ; β, A, B) let denote  λ  λ p Iµ+1 (d, e)f (z) zp zp U1 (z) = (1 + ϑ1 ) − ϑ1 p Iµp (d, e)f (z) Iµ (d, e)f (z) Iµp (d, e)f (z) and λ zp U0 (z) = . Iµp (d, e)f (z) A simple computation shows that λ  λ  p Iµ+1 (d, e)f (z) zp zp (1 + ϑ1 ) − ϑ1 p = Iµp (d, e)f (z) Iµ (d, e)f (z) Iµp (d, e)f (z)   ϑ1 ϑ1 U0 (z) + U2 (z). 1− ϑ2 ϑ2 

Since Pl [A, B, β] is a convex set, from the first part of Theorem 3.1, according to the above notations it follows that   ϑ1 ϑ1 1− U0 (z) + U2 (z) ∈ Pl [A, B, β], ϑ2 ϑ2 λ,ϑ1 that is f ∈ Nl,p (d, e; µ; β, A, B).



λ,0 λ,ϑ Theorem 3.3. If f ∈ Nl,p (d, e; µ; β, 1, −1), then f (ρz) ∈ Nl,p (d, e; µ; β, 1, −1), where ρ is given by   r 2 ϑ ϑ − β + λ(µ+p) + β + λ(µ+p) + 1 − β2 (3.5) ρ= . 1+β

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8

λ,0 Proof. For an arbitrary f ∈ Nl,p (d, e; µ; β, 1, −1), let denote  λ     zp l 1 l 1 = K(z) = + K1 (z) − − K2 (z), Iµp (d, e)f (z) 4 2 4 2

where K1 , K2 ∈ P [1, −1, β], which in equivalent to K1 (0) = K2 (0) = 1 and Re K1 (z) > β, Re K2 (z) > β, z ∈ U . With this notation, like in the proof of Theorem 3.1 we obtain  λ  λ p Iµ+1 (d, e)f (z) zp zp (1 + ϑ) −ϑ p = Iµp (d, e)f (z) Iµ (d, e)f (z) Iµp (d, e)f (z) ϑ K(z) + zK 0 (z) = λ(µ + p)       l l 1 ϑ 1 ϑ 0 0 + K1 (z) + zK (z) − − K2 (z) + zK (z) . 4 2 λ(µ + p) 1 4 2 λ(µ + p) 2 λ,ϑ In order to have f (ρz) ∈ Nl,p (d, e; µ; β, 1, −1), according to the above formula, we need to find the (bigger) value of ρ, such that   ϑ 0 Re Ki (z) + zKi (z) > β, |z| < ρ, (i = 1, 2). λ(µ + p)

From the well-known estimates for the class P (0) (see, eq., [6]) we have 2 Re Ki (z) , |z| ≤ r < 1, (i = 1, 2), |Ki0 (z)| ≤ 1 − r2 1−r Re Ki (z) ≥ , |z| ≤ r < 1, (i = 1, 2), 1+r thus, we deduce that   ϑ ϑ 0 Re Ki (z) + zKi (z) ≥ Re Ki (z) − |zKi0 (z)| ≥ λ(µ + p) λ(µ + p)   ϑ 2r (3.6) Re Ki (z) 1 − , |z| ≤ r < 1, (i = 1, 2). λ(µ + p) 1 − r2 A simple computation shows that 1−

(3.7)

ϑ 2r ≥ 0, λ(µ + p) 1 − r2

for 0 ≤ r ≤ r0 , where r0 :=

−ϑ +

p

ϑ2 + λ2 (µ + p)2 ∈ (0, 1). λ(µ + p)

Now, from the inequality (3.6) we have     ϑ 1−r ϑ 2r 0 Re Ki (z) + zKi (z) ≥ 1− , |z| ≤ r0 < 1, λ(µ + p) 1+r λ(µ + p) 1 − r2

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for i = 1, 2. It is easy to check that   1−r ϑ 2r 1− >β 1+r λ(µ + p) 1 − r2 for 0 ≤ r < ρ, where ρ is given by (3.5). Moreover, since the above inequality is equivalent to 1−

ϑ 2r 1+r β > 2 λ(µ + p) 1 − r 1−r

for r ∈ [0, 1), if follows that (3.7) holds for r ∈ [0, ρ), and our theorem is completely proved.  Next we will consider some properties of generalized p–valent Bernardi integral operator. Thus, for f ∈ Ap , let Fη,p : Ap → Ap be defined by Z η+p z (3.8) Fη,p f (z) = f (t)tη−1 d t, (η > −p). η z 0 We will give a short proof that this operator is well-defined, as follows. If the function f ∈ Ap is of the form (1.1), then the definition relation (3.8) could be written as Z η+p z f (t)tη−1 d t = (η + p)Iη,p f (z), Fη,p f (z) = zη 0 where

Z 1 z f (t)tη−1 d t. Iη,p f (z) = η z 0 We see that integral operator Iη,p defined above is similar to that of Lemma 1.2c. of [11]. According to this lemma, it follows that Iη,p is an analytic integral operator for any function f of the form (1.1) whenever Re η > −p, and Fη,p f ∈ Ap has the form p

Fη,p f (z) = z + (η + p)

∞ X n=1

ap+n z p+n , z ∈ U. p+n+η

The operator defined in (3.8) is called the generalized p–valent Bernardi integral operator, and for special case p = 1 we get the generalized Bernardi integral operator. Thus, for p = 1 and η ∈ N, the operator Fη := Fη,1 was introduced by Bernardi [1], and in particular, if η = 1 it reduces to the operator F1 that was earlier introduced by Livingston [9]. Theorem 3.4. Let f ∈ Ap and F = Fη,p f , where Fη,p is given by (3.8). If (3.9)  λ  λ Iµp (d, e)f (z) zp zp (1 + ϑ) −ϑ p ∈ Pl [A, B, β], Iµp (d, e)F (z) Iµ (d, e)F (z) Iµp (d, e)F (z)

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10

where d, e ∈ R \ Z− 0 , λ > 0, µ > −p, 0 ≤ β < 1, ϑ ≥ 0 and l ≥ 2, then  λ zp ∈ Pl [A, B, β]. Iµp (d, e)F (z) (All the powers are the principal ones). Proof. Like we mentioned after the Definition 1.2, since the left-hand side function from the relation (3.9) need to be analytic in U, we im˙ plicitly assumed that Iµp (d, e)F (z) 6= 0 for all z ∈ U. The implication is obvious for ϑ = 0, hence suppose that ϑ > 0. Letting  λ zp (3.10) = K(z), Iµp (d, e)F (z) from the assumption (3.9) it follows that K is analytic in U, with K(0) = 1. It is easy to check that, if f, g ∈ Ap , then  0 z z 0 0 (3.11) (f (z) ∗ g(z)) = f (z) ∗ g(z). p p Moreover, since F = Fη,p f , where Fη,p is given by (3.8), a simple differentiation shows that
0 (3.12) z Iµp (d, e)F (z) = (η + p)Iµp (d, e)f (z) − ηIµp (d, e)F (z). Taking the logarithmical differentiation of (3.10), we have "
0 # z Iµp (d, e)F (z) zK 0 (z) , = λ p− Iµp (d, e)F (z) K(z) and using the relations (3.11) and (3.12), it follows that Iµp (d, e)f (z) 1 zK 0 (z) = 1 − , Iµp (d, e)F (z) λ(η + p) K(z) and thus  (1 + ϑ)

 λ Iµp (d, e)f (z) zp −ϑ p = Iµ (d, e)F (z) Iµp (d, e)F (z) ϑ K(z) + zK 0 (z). λ(p + η)

zp Iµp (d, e)F (z)

λ

From the assumption (3.9), the above relation gives that K(z) +

ϑ zK 0 (z) ∈ Pl [A, B, β], λ(p + η)

and using a similar proof with those of the first part of Theorem 3.1 we obtain that K ∈ Pl [A, B, β], which proves our result. 

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Theorem 3.5. If f (z) = z p +

(3.13)

∞ X

λ,ϑ (d, e; µ; β, A, B) , ap+n z p+n ∈ Nl,p

n=1

where d, e ∈ R \ (3.14)

Z− 0,

λ > 0, µ > −p, 0 ≤ β < 1, ϑ ≥ 0 and l ≥ 2, then d (1 − β)(A − B) |ap+1 | ≤ . e |ϑ + λ(µ + p)|

The inequality (3.14) is sharp. Proof. If we let (3.15) λ  λ  p Iµ+1 (d, e)f (z) zp zp −ϑ p = p(z), (1 + ϑ) Iµp (d, e)f (z) Iµ (d, e)f (z) Iµp (d, e) f (z) using the fact that ∞ X (µ + p)n (e)n p p Iµ (d, e)f (z) = z + ap+n z p+n , n!(d) n n=1 we have λ  λ p Iµ+1 (d, e)f (z) zp zp (1 + ϑ) −ϑ p = Iµp (d, e)f (z) Iµ (d, e)f (z) Iµp (d, e)f (z)   ϑ (µ + p)1 (e)1 λ ap+1 z + · · · = 1− 1+ λ(µ + p) 1!(d)1 e 1 − [ϑ + λ(µ + p)] ap+1 z + . . . , z ∈ U. (3.16) d λ,ϑ Since f ∈ Nl,p (d, e; µ; β, A, B), it follows that the function p defined by (3.15) is of the form     l 1 l 1 + − p(z) = p1 (z) − p2 (z), 4 2 4 2 

where p1 , p2 ∈ P [A, B, β]. It follows that 1 + [(1 − β)A + βB] z (i = 1, 2), 1 + Bz and from the above relation we deduce that 1 + [(1 − β)A + βB] z (3.17) p(z) ≺ . 1 + Bz According to (3.16), from the subordination (3.17) we obtain pi (z) ≺

p(z) − β 1 + Az e ϑ + λ(µ + p) ap+1 z + · · · = ≺ , d 1−β 1−β 1 + Bz and from Lemma 2.3 we conclude that e ϑ + λ(µ + p) − |ap+1 | ≤ |A − B|, d 1−β 1−

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˘ AND B. S. ALKAHTANI S. MUSTAFA, T. BULBOACA,

which proves our result. To prove that the inequality (3.14) is sharp we need to show that λ,ϑ (d, e; µ; β, A, B) of the form (3.13), there exists a function f ∈ Nl,p such that for this function we have equality in (3.14). λ,ϑ Thus, we will prove that there exists f ∈ Nl,p (d, e; µ; β, A, B), such that the identity (3.15) holds for the special case p(z) =

1 + [(1 − β)A + βB] z . 1 + Bz

Setting  (3.18)

K(z) =

zp Iµp (d, e)f (z)

λ ,

like in the proof of Theorem 3.1 we deduce that the relation (3.15) is equivalent to (3.19)

K(z) +

1 zK 0 (z) = p(z), γ

where γ :=

λ(µ + p) . ϑ

(i) If ϑ = 0, the above differential equation has the solution K = p. (ii) If ϑ > 0, then γ > 0 whenever λ > 0 and µ > −p. Since the function p is convex in the unit disk U, according to Lemma 2.1 it follows that this differential equation has the solution Z z γ e K(z) = γ tγ−1 p(t) d t ≺ p(z). z 0 It is easy to check that p(z) 6= 0 for all z ∈ U, and from the above e subordination we get that K(z) 6= 0, z ∈ U. Now, if we define the function K0 by  p(z), if ϑ = 0, K0 (z) = e K(z), if ϑ > 0, then K0 is the analytic solution of the differential equation (3.19), and moreover K0 (z) 6= 0, z ∈ U. Thus, for K = K0 the relation (3.18) is equivalent to −1/λ

Iµp (d, e)f (z) = z p K0

(z),

and this equation has the solution   −1/λ f0 (z) := ψp (d, e; z) ∗ z p K0 (z) ,

(3.20) where p

ψp (d, e; z) := z +

∞ X n=1

n!(d)n z p+n . (µ + p)n (e)n


d, e ∈ C \ Z− , µ > −p , 0

Consequently, for the function f0 defined by (3.20) we get equality in (3.14), hence the sharpness of our result is proved. 

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As a special case, for l = 2 and β = 0 we obtain the corresponding λ,ϑ result for the class N2,p (d, e; µ; 0, A, B) (see [18] for n = 1). Acknowledgement. The work of the second author (T. Bulboac˘a) was entirely supported by the grant given by Babe¸s-Bolyai University, dedicated for Supporting the Excellence Research 2015. The third author B. S. Alkahtani is grateful to King Saud University, Deanship of Scientific Research, College of Science Research Center, for supporting this project. References [1] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135(1969), 429–446. [2] B. C. Carlson and D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 159(1984), 737–745. [3] M. C ¸ a˘ glar, Y. Polato˘ glu, E. Yavuz, λ–Fractional properties of generalized Janowski functions in the unit disc, Mat. Vesnik, 50(2008), 165–171. [4] N. E. Cho, O. S. Kwon and H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl., 292(2004), 470–483. [5] J. H. Choi, M. Saigo and H. M. Srivastava, Some inclusion properties of certain family of integral operators, J. Math. Anal. Appl., 276(2002) 432–444. [6] A. W. Goodman, Univalent Functions, Vol. I, II, Polygonal Publishing House, Washington, New Jersey, 1983. [7] D. J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc., 52(1975), 191–195. [8] W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math., 28(1973), 297–327. [9] A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc., 17(1966), 352–357. [10] S. S. Miller and P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., 65(1978), 289–305. [11] S. S. Miller and P. T. Mocanu, Differential Subordinations. Theory and Applications, Marcel Dekker Inc., New York Basel, 2000. [12] K. I. Noor and M. A. Noor, On certain classes of analytic functions defined by Noor integral operator, J. Math. Anal. Appl., 281(2003), 244-252. [13] K. I. Noor, Applications of certain operators to the classes related with generalized Janowski functions, Integral Transforms Spec. Funct., 2(2010), 557–567. [14] K. I. Noor, M. Ali, M. Arif, On a class of p–valent non-Bazilevi´c functions, Gen. Math., 18(2)(2010), 31–46 [15] B. Pinchuk, Functions with bounded boundary rotation, Israel J. Math., 10(1971), 7–16. [16] H. Saitoh, A linear operator and its applications of first order differential subordinations, Math. Japon., 44(1996), 31–38. [17] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (Ser. 2), 48(1943), 48–82. [18] Z.-G. Wang, H.-T. Wang and Y. Sun, A class of multivalent non-Bazilevi´c functions involving the Cho-Kwon-Srivastava operator, Tamsui Oxf. J. Inf. Math. Sci., 26(1)(2010), 1–19. [19] E. T. Whittaker and G. N. Watson, A Course on Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions;

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With an Account of the Principal Transcendental Functions, Fourth Edition, Cambridige Univ. Press, Cambridge, 1927. Department of Mathematics, Pir Mehr Ali Shah Arid Agriculture University, Rawalpindi, Pakistan E-mail address: [email protected] Faculty of Mathematics and Computer Science, Babes¸-Bolyai University, 400084 Cluj-Napoca, Romania E-mail address: [email protected] Mathematics Department, College of Science, King Saud University, P.O.Box 1142, Riyadh 11989, Saudi Arabia E-mail address: [email protected]

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ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES HAIYING LI AND ZHITAO GUO

A BSTRACT. The boundedness and compactness of a product-type operator DMu Cψ from mixed-norm spaces to Bloch-Orlicz spaces are characterized in this paper.

1. I NTRODUCTION Let D denote the unit disk in the complex plane C, H(D) the class of all analytic functions on D and N the set of nonnegative integers. A positive continuous function φ on [0,1) is called normal if there exist two positive numbers s and t with 0 < s < t, and δ ∈ [0, 1) such that (see [19]) φ(r) φ(r) is decreasing on [δ, 1), lim = 0; r→1 (1 − r)s (1 − r)s φ(r) φ(r) is increasing on [δ, 1), lim = ∞. t r→1 (1 − r)t (1 − r) For p, q ∈ (0, ∞) and φ normal, the mixed-norm space H(p, q, φ)(D) = H(p, q, φ) is the space of all functions f ∈ H(D) such that Z 1  p1 φp (r) p kf kH(p,q,φ) = Mq (f, r) dr < ∞, 1−r 0 where

 q1 Z 2π 1 iθ q Mq (f, r) = |f (re )| dθ . 2π 0 For 1 ≤ p, q < ∞, H(p, q, φ), equipped with the norm kf kH(p,q,φ) , is a Banach space, while for the other vales of p and q, k · kH(p,q,φ) is a quasinorm on H(p, q, φ), H(p, q, φ) 

α+1

is a Fr´echet space but not a Banach space. Note that if φ(r) = (1 − r) p , then H(p, q, φ) is equivalent to the weighted Bergman space Apα (D) = Apα defined for 0 < p < ∞ and α > −1, as the spaces of all f ∈ H(D) such that Z kf kpApα = (α + 1) |f (z)|p (1 − |z|2 )α dm(z) < ∞, D

where dm(z) = π1 rdrdθ is the normalized Lebesgue area measure on D ([8, 12, 18, 25, 27, 33, 35, 48, 51]). For more details on the mixed-norm space on various domains and operators on them, see, e.g., [1, 7, 10, 20, 22, 23, 24, 28, 29, 34, 36, 37, 38, 41, 42, 43, 44, 46, 47, 54]. 2000 Mathematics Subject Classification. Primary 47B33. Key words and phrases. A product-type operator, mixed-norm spaces, Bloch-Orlicz spaces, boundedness, compactness. 1

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For every 0 < α < ∞, the α-Bloch space, denoted by B α , consists of all functions f ∈ H(D) such that sup(1 − |z|2 )α |f 0 (z)| < ∞. z∈D

B α is a Banach space under the norm kf kBα = |f (0)| + sup(1 − |z|2 )α |f 0 (z)|. z∈D

For α = 1 is obtained the Bloch space. α-Bloch space is introduced and studied by numerous authors. Recently, many authors studied different classes of Bloch-type spaces, where the typical weight function, ω(z) = 1 − |z|2 (z ∈ D) is replaced by a bounded continuous positive function µ defined on D. More precisely, a function f ∈ H(D) is called a µ-Bloch function, denoted by f ∈ B µ , if kf kµ = sup µ(z)|f 0 (z)| < ∞. z∈D α

Clearly, if µ(z) = ω(z) with α > 0, B µ is just the α-Bloch space B α . It is readily seen that B µ is a Banach space with the norm kf kBµ = |f (0)| + kf kµ . For some information on the Bloch, α-Bloch and Bloch-type spaces, as well as some operators on them see, e.g., [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 23, 25, 26, 27, 29, 30, 31, 32, 34, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55]. Recently, Fern´andez in [17] used Young’s functions to define the Bloch-Orlicz space. More precisely, let ϕ : [0, ∞) → [0, ∞) be a strictly increasing convex function such that ϕ(0) = 0 and limt→∞ ϕ(t) = ∞. The Bloch-Orlicz space associated with the function ϕ, denoted by B ϕ , is the class of all analytic functions f in D such that sup(1 − |z|2 )ϕ(λ|f 0 (z)|) < ∞ z∈D

for some λ > 0 depending on f . Also, since ϕ is convex, it is not hard to see that the Minkowski’s functional    0 f kf kϕ = inf k > 0 : Sϕ ≤1 k define a seminorm for B ϕ , which, in this case, is known as Luxemburg’s seminorm, where Sϕ (f ) = sup(1 − |z|2 )ϕ(|f (z)|) z∈D ϕ

We know that B is a Banach space with the norm kf kBϕ = |f (0)| + kf kϕ . We also have that the Bloch- Orlicz space is isometrically equal to µ-Bloch space, where 1 , z ∈ D. µ(z) = −1 1 ϕ ( 1−|z| 2) Thus for any f ∈ B ϕ , we have kf kBϕ = |f (0)| + sup µ(z)|f 0 (z)|. z∈D

It is well known that the differentiation operator D is defined by (Df )(z) = f 0 (z), f ∈ H(D). Let u ∈ H(D), then the multiplication operator Mu is defined by (Mu f )(z) = u(z)f (z), f ∈ H(D).

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Let ψ be an analytic self-map of D. The composition operator Cψ is defined by (Cψ f )(z) = f (ψ(z)), f ∈ H(D). Investigation of products of these and integral-type operators attracted a lot of attention recently (see, e.g., [2]-[49], [51]-[55]). For example, in [3] and [17], the authors investigated bounded superposition operators between Bloch-Orlicz and α-Bloch spaces and composition operators on Bloch-Orlicz type spaces. In [37] and [38], S. Stevi´c investigated extended Ces`aro operators between mixed-norm spaces and Bloch-type spaces and an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball. In [36] and [41], S. Stevi´c investigated an integral-type operator from logarithmic Bloch-type and mixed-norm spaces to Bloch-type spaces and weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. In [42] and [46], S. Stevi´c investigated an integral-type operator from Zygmund-type spaces to mixed-norm spaces on the unit ball and weighted differentiation composition operators from the mixed-norm space to the nth weighted-type space on the unit disk. S. Stevi´c in [34] gave the properties of products of integral-type operators and composition operators from the mixed norm space to Bloch-type spaces. In [47], S. Stevi´c investigated weighted radial operator from the mixed-norm space to the nth weighted-type space on the unit ball. In [54], X. Zhu studied extended Ces`aro operators from mixed-norm spaces to Zygmund type spaces. Motivated, among others, by these papers, we will study here the boundedness and compactness of the following operator, which is also a product-type one, (DMu Cψ f )(z) = u0 (z)f (ψ(z)) + u(z)ψ 0 (z)f 0 (ψ(z)), f ∈ H(D), from H(p, q, φ) to B ϕ . In what follows, µ(z) =

1

ϕ−1 (

, 1 1−|z|2 )

and we use the letter C to denote a positive constant whose value may change at each occurrence. 2. T HE B OUNDEDNESS AND COMPACTNESS OF DMu Cφ : H(p, q, φ) → Bϕ In this section, we will give our main results and proofs. In order to prove our main results, we need some auxiliary results. Our first lemma characterizes compactness in terms of sequential convergence. Since the proof is standard, it is omitted here (see, Proposition 3.11 in [4]). Lemma 1. Suppose u ∈ H(D), ψ is an analytic self-map of D, 0 < p, q < ∞ and φ is normal. Then the operator DMu Cψ : H(p, q, φ) → B ϕ is compact if and only if it is bounded and for each sequence {fn }n∈N which is bounded in H(p, q, φ) and converges to zero uniformly on compact subsets of D as n →∞, we have kDMu Cψ fn kBϕ → 0 as n → ∞. The following lemma can be found in [36]. Lemma 2. Assume 0 < p, q < ∞, ψ is normal and f ∈ H(p, q, φ). Then for every n ∈ N, there is a positive constant C independent of f such that |f (n) (z)| ≤

Ckf kH(p,q,φ) 1

φ(|z|)(1 − |z|2 ) q +n

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Theorem 3. Let u ∈ H(D), ψ be an analytic self-map of D, 0 < p, q < ∞ and φ be normal. Then DMu Cψ : H(p, q, φ) → Bϕ is bounded if and only if k1 = sup z∈D

µ(z)|u00 (z)|

1 < ∞, φ(|ψ(z)|)(1 − |ψ(z)|2 ) q µ(z)|2u0 (z)ψ 0 (z) + u(z)ψ 00 (z)|

(1)

< ∞, 1 φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +1 µ(z)|u(z)||ψ 0 (z)|2 k3 = sup < ∞. 1 +2 z∈D φ(|ψ(z)|)(1 − |ψ(z)|2 ) q

k2 = sup

(2)

z∈D

(3)

Proof. Assume that (1), (2) and (3) hold. By Lemma 2, then we get C1 kf kH(p,q,φ)

|f (ψ(z))| ≤

,

1

|f 0 (ψ(z))| ≤

φ(|ψ(z)|)(1 − |ψ(z)|2 ) q C2 kf kH(p,q,φ)

|f 00 (ψ(z))| ≤

1

φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +1 C3 kf kH(p,q,φ) 1

,

φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +2

.

Then for each f ∈ H(p, q, φ) \ {0}, we have:  Sϕ

(DMu Cψ f )0 (z) Ckf kH(p,q,φ)



1

k1 φ(|ψ(z)|)(1 − |ψ(z)|2 ) q |f (ψ(z))| Cµ(z)kf kH(p,q,φ) z∈D   1 2 q +1 0 k2 φ(|ψ(z)|)(1 − |ψ(z)| ) |f (ψ(z))| + Cµ(z)kf kH(p,q,φ)   1 k3 φ(|ψ(z)|)(1 − |ψ(z)|2 ) q +2 |f 00 (ψ(z))| + Cµ(z)kf kH(p,q,φ)   k1 C1 + k2 C2 + k3 C3 ≤ sup(1 − |z|2 )ϕ Cµ(z) z∈D    1 ≤ sup(1 − |z|2 )ϕ ϕ−1 =1 1 − |z|2 z∈D ≤ sup(1 − |z|2 )ϕ





where C is a constant such that C ≥ k1 C1 + k2 C2 + k3 C3 . Now, we can conclude that there exists a constant C such that kDMu Cψ f kBϕ ≤ Ckf kH(p,q,φ) for all f ∈ H(p, q, φ), so the product-type operator DMu Cψ : H(p, q, φ) → Bϕ is bounded. Conversely, suppose that DMu Cψ : H(p, q, φ) → B ϕ is bounded, i.e., there exists C > 0 such that kDMu Cψ f kB ϕ ≤ Ckf kH(p,q,φ) for all f ∈ H(p, q, φ). Taking the function f (z) = 1 ∈ H(p, q, φ), and kf kH(p,q,φ) ≤ C, then    00   00  u (z) |u (z)| (DMu Cψ f )0 (z) 2 = Sϕ = sup(1 − |z| )ϕ ≤ 1. Sϕ C C C z∈D It follows that sup µ(z)|u00 (z)| < ∞.

(4)

z∈D

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ON A PRODUCT-TYPE OPERATOR FROM MIXED-NORM SPACES TO BLOCH-ORLICZ SPACES

5

Taking the function f (z) = z ∈ H(p, q, φ), and kf kH(p,q,φ) ≤ C, then   (DMu Cψ f )0 (z) Sϕ C   00 |u (z)ψ(z) + (2u0 (z)ψ 0 (z) + u(z)ψ 00 (z))| 2 = sup(1 − |z| )ϕ ≤ 1. C z∈D Hence sup µ(z)|u00 (z)ψ(z) + 2u0 (z)ψ 0 (z) + u(z)ψ 00 (z)| < ∞. z∈D

By (4) and the boundedness of ψ(z), we can see that sup µ(z)|2u0 (z)ψ 0 (z) + u(z)ψ 00 (z)| < ∞.

(5)

z∈D

Taking the function f (z) =

z2 2

∈ H(p, q, φ), similarly, we can get

sup µ(z)|u(z)||ψ 0 (z)|2 < ∞.

(6)

z∈D

For a fixed ω ∈ D, set fψ(ω) (z)

=

A(1 − |ψ(ω)|2 )t+1 1 q +t+1

+

B(1 − |ψ(ω)|2 )t+2 1

φ(|ψ(ω)|)(1 − ψ(ω)z) φ(|ψ(ω)|)(1 − ψ(ω)z) q +t+2 (1 − |ψ(ω)|2 )t+3 + , 1 φ(|ψ(ω)|)(1 − ψ(ω)z) q +t+3 where the constant t is from the definition of the normality of the function φ. Then supω∈D kfψ(ω) kH(p,q,φ) < ∞, and we have fψ(ω) (ψ(ω)) = 0 fψ(ω) (ψ(ω)) =

A+B+1 1

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q

,

(AM1 + BM2 + M3 )ψ(ω) 1

(7)

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q +1

, 2

00 fψ(ω) (ψ(ω)) =

(AM1 M2 + BM2 M3 + M3 M4 )ψ(ω) 1

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q +2

.

where Mi = 1q + t + i, i = 1, 2, 3, 4. To prove (1), we choose the corresponding function in (7) with M3 2M3 A= , B=− , M1 M2 and denote it by fψ(ω) , then we have fψ(ω) (ψ(ω)) =

P 1

φ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q

0 00 , fψ(ω) (ψ(ω)) = fψ(ω) (ψ(ω)) = 0,

(8)

M3 3 where P = M − 2M M2 + 1. 1 By the boundedness of DMu Cψ : H(p, q, φ) → B ϕ , we have kDMu Cψ fψ(ω) kBϕ ≤ C, then   (DMu Cψ fψ(ω) )0 (z) 1 ≥ Sϕ C   P |u00 (ω)| ≥ sup (1 − |ω|2 )ϕ , 1 w∈D Cφ(|ψ(ω)|)(1 − |ψ(ω)|2 ) q

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from which we can get (1). To prove (2), we choose the corresponding function in (7) with A=

−2M2 − M1 M2 + M3 M4 M1 M2 − M3 M4 , B= , 2M2 2M2

and denote it by gψ(ω) , then we have 0 gψ(ω) (ψ(ω)) =

Eψ(ω) φ(|ψ(ω)|)(1 −

1

|ψ(ω)|2 ) q +1

00 , gψ(ω) (ψ(ω)) = gψ(ω) (ψ(ω)) = 0,

(9)

where E=

−2M1 M2 − M12 M2 + M1 M3 M4 M1 M2 − M3 M4 + + M3 . 2M2 2

By the boundedness of DMu Cψ : H(p, q, φ) → B ϕ , we have kDMu Cψ gψ(ω) kBϕ ≤ C, then   (DMu Cψ gψ(ω) )0 (z) 1 ≥ Sϕ C   |(DMu Cψ gψ(ω) )0 (ω)| ≥ sup (1 − |ω|2 )ϕ 1 C 2 0, for all z ∈ U, f (z) and f ∈ A is convex if and only if "

# 00 zf (z) > 0, for all z ∈ U. Re 1 + 0 f (z) The classes consisting of starlike and convex functions are denoted by S ∗ and K, respectively. Further, we denote by S ∗ (δ) and K(δ) the class of starlike functions of order δ and the class of convex functions of order δ (0 ≤ δ < 1), respectively, where " 0 # " # 00 zf (z) zf (z) Re > δ and Re 1 + 0 > δ. f (z) f (z) 1

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Recently, Frasin and Jahangiri [4] defined the family B(µ, λ), µ ≥ 0, 0 ≤ λ < 1 consisting of functions f ∈ A satisfying the condition  µ 0 z f (z) − 1 < 1 − λ, for all z ∈ U. f (z) We note that B(1, λ) = S ∗ (λ), B(2, λ) = B(λ) (see [3]) and B(2, 0) = S. For the functions f, g ∈ A and α, ζ ∈ C we define the integral operator Iαζ (f, g) given by " Z !α # ζ1 0 z f (t) ζ α+ζ−1 Iα (f, g)(z) = ζ t dt . (2) g(t) 0 Note that the integral operator Iαζ (f, g)(z) generalizes the integral operator Iα (f, g)(z) introduced in [2]. In this paper our purpose is to derive univalence conditions, starlikeness properties and the order of convexity for the integral operator introduced in (2). Recently, many authors studied the problem of integral operators which preserve the class S (see [5], [9]). In order to prove our results, we have to recall here the following: Lemma 1.1 (Mocanu and S ¸ erb [7]) Let M0 = 1, 5936..., the positive solution of equation (2 − M )eM = 2.

(3)

If f ∈ A and 00 f (z) 0 ≤ M0 , for all z ∈ U, f (z) then

0 zf (z) − 1 < 1, for all z ∈ U. f (z)

The edge M0 is sharp. Lemma 1.2 (Pascu [8]) Let γ be a complex number, Reγ > 0 and let the function f ∈ A. If 00 1 − |z|2Reγ zf (z) · 0 ≤ 1, f (z) Reγ for all z ∈ U, then for any complex number ζ, Reζ ≥ Reγ, the function  Z z  ζ1 0 Fζ (z) = ζ tζ−1 f (t)dt 0

is regular and univalent in U . Lemma 1.3 (General Schwarz Lemma [6]) Let f be regular function in the disk UR = {z ∈ C : |z| < R} with |f (z)| < M, M fixed. If f has in z = 0 one zero with multiply bigger than m, then |f (z)| ≤ The equality case hold only if f (z) = eiθ ·

M Rm

M |z|m , Rm

z ∈ UR .

· z m , where θ is constant.

Lemma 1.4 (Ready and Padmanabhan [10]) Let the functions p, q be analytic in U with p(0) = q(0) = 0, and let δ be a real number. If the function q maps the unit disk U onto a region which is starlike with respect to the origin, the inequality " 0 # p (z) Re 0 > δ, for all z ∈ U q (z) implies that  Re

 p(z) > δ, for all z ∈ U. q(z) 2

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Lemma 1.5 (Wilken and Feng [11]) If 0 ≤ δ < 1 and f ∈ K(δ), then f ∈ S ∗ (ν(δ)), where ( 1−2δ , if δ 6= 12 , 2(1−δ) ν(δ) = 2 1 −2 if δ = 12 . 2 log 2 ,

2

(4)

Main results

The univalence condition for the operator Iαζ (f, g) defined in (2) is proved in the next theorem, by using Pascu univalence criterion. Theorem 2.1 Let α, γ be complex numbers, Reγ 1, 5936..., and f, g ∈ A. If 00 f (z) 0 ≤ M0 , f (z)

> 0, M0 the positive solution of the equation (3), M0 = 00 g (z) 0 ≤ M0 , g (z)

z∈U

(5)

and 2M0 Reγ + (2Reγ + 1)

2Reγ+1 2Reγ

Reγ · (2Reγ + 1) ≤ |α|

2Reγ+1 2Reγ

,

(6)

then for any complex number ζ, Reζ ≥ Reγ, the integral operator " Z ζ Iα (f, g)(z) = ζ

0

z

tα+ζ−1

0

f (t) g(t)

# ζ1

!α

dt

is in the class S. Proof. Let the function Z

z

"

h(z) = 0

0

tf (t) g(t)

#α dt.

(7)

0

The function h is regular in U and h(0) = h (0) − 1 = 0. From (7) we have #α " 0 0 zf (z) h (z) = g(z) and 0

00

h (z) = α

zf (z) g(z)

!α−1 "

# 0 00 0 0 f (z) zf (z) f (z) g (z) · + −z· · . g(z) g(z) g(z) g(z)

We get " # " 00 00 0 00 zh (z) zf (z) zg (z) zf (z) =α 1+ 0 − =α − h0 (z) f (z) g(z) f 0 (z)

!# 0 zg (z) −1 . g(z)

From (8) we obtain 1 − |z|2Reγ Reγ

00 00 0 zh (z) 1 − |z|2Reγ f (z) 1 − |z|2Reγ zg (z) · 0 · |z| · |α| · 0 · |α| · − 1 . ≤ + h (z) f (z) g(z) Reγ Reγ

From (5) and applying Lemma 1.1 we obtain 0 zg (z) − 1 < 1, for all z ∈ U, g(z) 3

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which implies that 1 − |z|2Reγ Reγ Since

00 zh (z) 1 − |z|2Reγ 1 − |z|2Reγ · 0 · |z| · |α| · M0 + · |α|. ≤ h (z) Reγ Reγ 1 − |z|2Reγ 2 · |z| = 2Reγ+1 , Reγ |z|≤1 (2Reγ + 1) 2Reγ max

we have 1 − |z|2Reγ Reγ

00 zh (z) 2 |α| · 0 · |α| · M0 + . ≤ h (z) (2Reγ + 1) 2Reγ+1 Reγ 2Reγ

(9)

Using (6) in (9) we obtain 1 − |z|2Reγ Reγ

00 zh (z) · 0 ≤ 1, h (z)

z ∈ U,

(10)

and by applying Lemma 1.2, we obtain that the function Iαζ (f, g)(z) is in the class S. If we put ζ = 1 in Theorem 2.1, we obtain Corollary 2.2 Let α, γ be complex numbers, 0 < Reγ ≤ 1, M0 the positive solution of the equation (3), M0 = 1, 5936..., and f, g ∈ A. If 00 00 f (z) g (z) 0 ≤ M0 , 0 ≤ M0 , z ∈ U, f (z) g (z) and 2M0 Reγ + (2Reγ + 1)

2Reγ+1 2Reγ

≤

Reγ · (2Reγ + 1) |α|

then the integral operator Z

z

"

Iα (f, g)(z) = 0

0

tf (t) g(t)

2Reγ+1 2Reγ

,

#α dt,

is in the class S. Putting Reγ = 1 in Corrolary 2.2, we obtain Corollary 2.3 Let α, γ be complex numbers, 0 < Reγ ≤ 1, M0 the positive solution of the equation (3), M0 = 1, 5936..., and f, g ∈ A. If 00 00 f (z) g (z) 0 ≤ M0 , 0 ≤ M0 , z ∈ U, f (z) g (z) and

√ 3 3 √ |α| ≤ 2M0 + 3 3

then the integral operator Z Iα (f, g)(z) = 0

z

"

0

tf (t) g(t)

#α dt,

is in the class S. This result was also obtained in [2]. In the following theorem we give sufficient conditions such that the integral operator Iαζ (f, g)(z) ∈ S ∗ .

4

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Theorem 2.4 Let α, ζ be complex numbers, M ≥ 1, f ∈ A and g ∈ B(µ, λ) such that 00 zf (z) < 1 and |g(z)| < M, z ∈ U. 0 f (z) If |α| ≤

|ζ| , 2 + (2 − λ)M µ−1

then the integral operator Iαζ (f, g)(z) is in the class S ∗ . Proof. Let’s consider the function ϕ given by ϕ(z) = Iαζ (f, g)(z),

z ∈ U.

(11)

Then, by differentiating ϕ with respect to z, we obtain z α+ζ

0



0

f (z) g(z)

α

zϕ (z)  0 α . = R z (t) ϕ(z) ζ 0 tα+ζ−1 fg(t) dt Letting 0

p(z) = zϕ (z) and q(z) = ϕ(z), we find that

" # 00 0 0 α zf (z) zg (z) p (z) =1+ 1+ 0 − . q 0 (z) ζ f (z) g(z)

Thus, 00 # 0 " |α| zf (z) zg 0 (z) p (z) − 1 ≤ 1+ 0 + 0 f (z) g(z) q (z) |ζ| 00 " #   µ zf (z)  0 g(z) µ−1 |α| z ≤ 1+ 0 . − 1 + 1 + g (z) · f (z) |ζ| g(z) z Since |g(z)| < M , z ∈ U , by applying the Schwarz Lemma, we have g(z) z ≤ M , for all z ∈ U.

(13)

By using the hypothesis and (13) we obtain 0 p (z) |α|   − 1 ≤ 2 + (2 − λ) · M µ−1 ≤ 1, 0 q (z) |ζ| that is

# 0 p (z) Re 0 > 0, q (z)

"

z ∈ U.

Therefore, applying Lemma 1.4, we find that  p(z) Re > 0, q(z)



z ∈ U.

This completes the proof. of the theorem. Taking µ = 1 in Theorem 2.4, we have

5

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Corollary 2.5 Let α, ζ be complex numbers, M ≥ 1, f ∈ A and g ∈ S ∗ (λ) such that 00 zf (z) < 1 and |g(z)| < M, z ∈ U. 0 f (z) If |α| ≤

|ζ| , 4−λ

then the integral operator Iα (f, g) is in the class S ∗ . Letting λ = 0 in Corollary 2.5, we obtain ∗ Corollary 2.6 Let α, ζ be complex numbers with |α| = |ζ| 4 and M ≥ 1. If f ∈ A and g ∈ S satisfies 00 zf (z) 0 < 1 and |g(z)| < M, z ∈ U, f (z)

then the integral operator Iαζ (f, g) is in the class S ∗ . Next, we find sufficient conditions such that Iαζ (f, g)(z) ∈ K(δ). Theorem 2.7 Let α, ζ be complex numbers, M, N ≥ 1, f ∈ A and g ∈ B(µ, λ). If 00 f (z) |g(z)| < M and 0 < N, f (z) for all z ∈ U then, the integral operator Iαζ (f, g) is in the class K(δ), where α α µ−1 ] and 0 < [1 + N + (2 − λ)M µ−1 ] ≤ 1. δ = 1 − [1 + N + (2 − λ)M ζ ζ Proof. Letting the function ϕ be given by (11), we have " # 00 00 0 zϕ (z) α zf (z) zg (z) = 1+ 0 − . ϕ0 (z) ζ f (z) g(z) Therefore, using the hypothesis of the theorem and applying the Schwarz Lemma, we obtain 00 0 00 # " # " zf (z) zg 0 (z) zg (z)  z µ  g(z) µ−1 zϕ (z) α α 1+ 0 · 0 ≤ + ≤ 1 + N + · ϕ (z) ζ f (z) g(z) g(z) ζ g(z) z      µ α α 0 z ≤ 1 + N + g (z) − 1 + 1 · M µ−1 ≤ [1 + N + (2 − λ) · M µ−1 ] = 1 − δ. ζ g(z) ζ This evidently completes the proof. Letting µ = 1 in Theorem 2.7, we have Corollary 2.8 Let α, ζ be complex numbers, M, N ≥ 1, f ∈ A and g ∈ S ∗ (λ). If 00 f (z) |g(z)| < M and 0 < N, f (z) for all z ∈ U then, the integral operator Iαζ (f, g) is in the class K(δ), where α α δ = 1 − (3 + N − λ) and 0 < (3 + N − λ) ≤ 1. ζ ζ Letting δ = λ = 0 in Corollary 2.8, we obtain 6

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Corollary 2.9 Let α, ζ be complex numbers, M, N ≥ 1, f ∈ A and g ∈ S ∗ . If 00 f (z) |g(z)| < M and 0 0, if f and g share five ¯ (i = 1, 2, 3, 4, 5) IM in Ω(θ − ε, θ + ε), then f = g. distinct values ai ∈ C In order to state the next result, we also need the following notation. Let f be a non-constant meromorphic function in C, and let a be an arbitrary ¯ D, f ) to denote the zeros set of f − a in complex number. We use E(a, D ⊆ C, in which each zero is counted only once. Clearly, we say that f and ¯ D, f ) = E(a, ¯ D, g). We use E(a, ¯ f ) to denote g share a IM in D, if E(a, the zeros set of f − a in D = C. In [18, Theorem 3.2], C.C.Yang improved Theorem A by proving Theorem D. Let f and g be two non-constant meromorphic functions in ¯ (i = 1, 2, 3, 4, 5) be five distinct values. If C and ai ∈ C (1.2)

¯ i , f ) ⊆ E(a ¯ i , g), E(a

i = 1, 2, 3, 4, 5,

and (1.3)

lim inf r→∞

5 ∑ i=1

∑ 1 ¯ (r, 1 ) > 1 , )/ N f − ai i=1 g − ai 2 5

¯ (r, N

then f = g. Now, it is natural to ask the following question.

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Question 1. Do f and g coincide if they satisfy the conditions of Theorem D in an angular domain? In the present paper, we answer to Question 1 is affirmative for some class of meromorphic functions by using Nevanlinna theory in an angular domain which is recalled in Lemma 2.1 below. The first result is stated as follows. Theorem 1.1. Let ρ(r) be a proximate order of meromorphic function f ¯ (i = 1, 2, 3, 4, 5) be of infinite order in C and let g ∈ M (ρ(r)). Let ai ∈ C five distinct values. Suppose that arg z = θ ∈ [0, 2π) is a Brole direction of order ρ(r) of f . For any given ε > 0, if ¯ i , Ω(θ − ε, θ + ε), f ) ⊆ E(a ¯ i , Ω(θ − ε, θ + ε), g), (1.4) E(a

i = 1, 2, 3, 4, 5,

and (1.5)

lim inf r→∞

5 ∑ i=1

∑ 1 1 1 C¯θ−ε,θ+ε (r, )/ )> , f − ai i=1 g − ai 2 5

C¯θ−ε,θ+ε (r,

then f = g. Before stating the following result, we need some notation concerning Ahlfors theory in an angular domain Ω(α, β) which can be found [14, pp. 258-259], or for reference [26, pp. 66-76]. ∫ ∫ 1 r β |f ′ (teiφ )| 2 ) tdtdφ, SA (r, Ω(α, β), f ) = ( π 0 α 1 + |f (teiφ )|2 ∫

r

T (r, Ω(α, β), f ) = 0

SA (t, Ω(α, β), f ) dt. t

Especially the corresponding notation in the whole complex plane are denoted by ∫ ∫ 1 r 2π |f ′ (teiφ )| 2 ( ) tdtdφ, SA (r, f ) = π 0 0 1 + |f (teiφ )|2 ∫

r

T (r, f ) = 0

SA (t, f ) dt. t

By using the relationship between Ahlfors characteristic function in an angular domain and sectorial Nevanlinna characteristic function which is introduced in Lemma 2.7 of Section 2, we can prove the following result. Theorem 1.2. Let f and g be two non-constant meromorphic functions of ¯ (i = 1, 2, 3, 4, 5) be five distinct values. Suppose finite order in C and ai ∈ C

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BOREL DIRECTIONS AND UNIQUENESS OF MEROMORPHIC FUNCTIONS

5

that Ω(α, β) is an angular domain such that f satisfies (1.6)

lim sup lim sup ε→0+

r→∞

log T (r, Ω(α + ε, β − ε), f ) > ω, log r

where ω =

π . β−α

(1.7)

¯ i , Ω(α, β), f ) ⊆ E(a ¯ i , Ω(α, β), g), E(a

If i = 1, 2, 3, 4, 5,

and (1.8)

lim inf r→∞

5 ∑

∑ 1 1 1 )/ C¯α,β (r, )> , f − ai i=1 g − ai 2 5

C¯α,β (r,

i=1

then f = g. Theorem 1.3. Let f and g be two non-constant meromorphic functions of ¯ (i = 1, 2, 3, 4, 5) be five distinct values. Suppose finite order in C and ai ∈ C that Ω(α, β) is an angular domain such that for any ε > 0 and for some ¯ a∈C (1.9)

lim sup r→∞

where ω =

π . β−α

1 ) log n(r, Ω(α + ε, β − ε), f −a

log r

> ω,

If f and g satisfy (1.7) and (1.8), then f = g.

Remark 1.4. It is well know that every meromorphic function of order ρ ∈ (0, ∞) must have at least one direction arg z = θ ∈ [0, 2π) such that for sufficiently small ε > 0, lim sup r→∞

1 log n(r, Ω(α + ε, β − ε), f −a )

log r

=ρ

¯ with at most two exceptional values, which can be found holds for all a ∈ C in [20, Chapter 3]. So the angular domain satisfying (1.9) must exist when f is of order ρ ∈ ( 12 , ∞). This paper is organized as follows. In Section 2, we recall the properties of sectorial Nevanlinna characteristic and state some Lemmas which are needed in proving our results. The proof of Theorem 1.1 is given in Section 3. Finally, we prove Theorem 1.2 and 1.3 in Section 4. 2. Auxiliary results Let f be a meromorphic function in the angular domain Ω(α, β) = {z : α ≤ arg z ≤ β}, where α < β and β − α < 2π. We recall the following

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definitions that were found in [6, Chapter 1]. ∫ tω ω r 1 dt ( ω − 2ω ){log+ |f (teiα )| + log+ |f (teiβ )|} , Aα,β (r, f ) = π 1 t r t ∫ β 2ω Bα,β (r, f ) = ω log+ |f (teiθ )| sin ω(θ − α)dθ, πr α ∑ |bn |ω 1 − ) sin ω(θn − α), Cα,β (r, f ) = 2 ( |bn |ω r2ω 1 0 and for some a ∈ C, (2.2)

lim sup

1 log n(r, Ω(α + ε, β − ε), f −a )

log r

r→∞

where ω =

π , β−α

> ω,

then

lim sup lim sup r→∞

ε→0+

log T (r, Ω(α + ε, β − ε), f ) > ω. log r

Proof. For any given ε > 0, from (2.2), there exists a sequence {rn }, rn → ∞ as n → ∞, such that lim

1 log n(rn , Ω(α + ε, β − ε), f −a )

= λ > ω. log rn Let σ be a real number such that ω < σ < λ, we have 1 ) > rnσ > rnω , n ≥ n0 . n(rn , Ω(α + ε, β − ε), f −a By this and n→∞

1 ωε N (r, Ω(α + ε, β − ε), f −a ) ) ≥ 2ω sin( ) f −a 2 rω ∫ r N (t, Ω(α + ε, β − ε), 1 ) ωε f −a , + 2ω 2 sin( ) ω+1 2 t 1 1

Cα+ 2ε ,β− 2ε (r,

which can be found in [26, Lemma 2.2.2], we have 1 (2.3) Cα+ 2ε ,β− 2ε (rn , ) > rnσ−ω . f −a By using Lemma 2.1 and (2.3), we get Sα+ 2ε ,β− 2ε (rn , f ) > rnσ−ω . It follows from Lemmas 2.6 and 2.7 that ε ε T (rn , Ω(α + , β − ), f ) > rnσ . 2 2 Thus, log T (r, Ω(α + 2ε , β − 2ε ), f ) > σ > ω. log r r→∞ Noting ε is arbitrary small, hence lemma holds. lim sup

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3. Proof of Theorem 1.1 Suppose that ρ(r) is a proximate order of meromorphic function f of infinite order, g ∈ M (ρ(r)) and that arg z = θ ∈ [0, 2π) is a Borel direction of order ρ(r) of f . For any given ε > 0, f and g satisfy (1.4) and (1.5) in the angular domain Ω(θ − ε, θ + ε) = {z : θ − ε ≤ arg z ≤ θ + ε}. Firstly, we claim that arg z = θ is also a Borel direction of order ρ(r) of g. Since arg z = θ is a Borel direction of order ρ(r) of f , for above given ε, by using Lemmas 2.1, 2.3 and 2.4, then there exists a value a such that lim sup

1 log C¯θ−ε,θ+ε (r, f −a )

ρ(r) log r

r→∞

≥ 1.

Without loss of generality, we may assume that a = a1 . Thus, lim sup

1 log C¯θ−ε,θ+ε (r, f −a ) 1

ρ(r) log r

r→∞

≥ 1.

It follows from (1.4) that lim sup

1 log C¯θ−ε,θ+ε (r, g−a ) 1

r→∞

ρ(r) log r

≥ 1.

Therefore, we get lim sup r→∞

log Sθ−ε,θ+ε (r, g) ≥ 1. ρ(r) log r

Combining this and g ∈ M (ρ(r)), we have lim sup r→∞

log Sθ−ε,θ+ε (r, g) = 1. ρ(r) log r

By using Lemma 2.4, we know that arg z = θ is a Borel direction of order ρ(r) of g. In order to prove that f = g, we assume on the contrary to the assertion that f ̸= g. Now we use the similar method of [23] to complete the proof. To this end, we consider two cases. Case 1. We may assume that all ai (i = 1, 2, 3, 4, 5) are finite. By using Lemma 2.1, we can obtain (3.1)

3Sθ−ε,θ+ε (r, f ) ≤

5 ∑

C¯θ−ε,θ+ε (r,

1 ) + Rθ−ε,θ+ε (r, f ), f − ai

C¯θ−ε,θ+ε (r,

1 ) + Rθ−ε,θ+ε (r, g). g − ai

i=1

and (3.2)

3Sθ−ε,θ+ε (r, g) ≤

5 ∑ i=1

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From (1.4), we have 5 ∑

C¯θ−ε,θ+ε (r,

i=1

1 1 ) ≤ Cθ−ε,θ+ε (r, ) f − ai f −g 1 ) f −g ≤ Sθ−ε,θ+ε (r, f ) + Sθ−ε,θ+ε (r, g) + O(1). ≤ Sθ−ε,θ+ε (r,

(3.3)

Since arg z = θ is a Borel direction of order ρ(r) of f , by using Lemma 2.4, then we have log Sθ−ε,θ+ε (r, f ) lim sup = 1. ρ(r) log r r→∞ It follows from this and Lemma 2.3, we have (3.4)

lim sup r→∞

Rθ−ε,θ+ε (r, f ) = 0. Sθ−ε,θ+ε (r, f )

Similarly, we have (3.5)

lim sup r→∞

Rθ−ε,θ+ε (r, g) = 0. Sθ−ε,θ+ε (r, g)

Combining (3.1)-(3.5), for sufficiently large r, we have 5 ∑

∑ 1 1 1 C¯θ−ε,θ+ε (r, ) ≤ ( + o(1)) ) f − ai 3 f − ai i=1 5

C¯θ−ε,θ+ε (r,

i=1

∑ 1 1 C¯θ−ε,θ+ε (r, ). + ( + o(1)) 3 g − ai i=1 5

Therefore, ∑ ∑ 1 1 2 1 C¯θ−ε,θ+ε (r, C¯θ−ε,θ+ε (r, ( + o(1)) ) ≤ ( + o(1)) ). 3 f − a 3 g − a i i i=1 i=1 5

5

It follows that lim inf r→∞

5 ∑

∑ 1 1 1 C¯θ−ε,θ+ε (r, )/ )≤ . f − ai i=1 g − ai 2 5

C¯θ−ε,θ+ε (r,

i=1

This contradicts to (1.5), and hence f = g. Case 2. If one of the values ai (i = 1, 2, 3, 4, 5) is ∞, without loss of generality, we may assume that a5 = ∞. Take a finite value c such that 1 1 c ̸= ai (i = 1, 2, 3, 4) and set F = f −c , G = g−c , bi = ai1−c (i = 1, 2, 3, 4) and ¯ i , Ω(θ−ε, θ+ε), F ) ⊆ E(b ¯ i , Ω(θ−ε, θ+ε), G) b5 = 0, then F and G satisfy E(b (i = 1, 2, 3, 4, 5), and lim inf r→∞

5 ∑ i=1

∑ 1 1 1 )/ C¯θ−ε,θ+ε (r, )> . F − bi i=1 G − bi 2 5

C¯θ−ε,θ+ε (r,

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From Lemma 2.1, we also know that Sθ−ε,θ+ε (r, F ) = Sθ−ε,θ+ε (r, f ) + O(1) and Sθ−ε,θ+ε (r, G) = Sθ−ε,θ+ε (r, g)+O(1). From the previous proof, we know F = G. Therefore f = g. The proof is completed. 4. Proofs of Theorems 1.2 and 1.3 Proof of Theorem 1.2. Suppose that f and g be two non-constant meromorphic functions of finite order in C satisfying (1.6)-(1.8), Ω(α, β) = {z : π α ≤ arg z ≤ β} is an angular domain and ω = β−α . Set (4.1)

lim sup lim sup r→∞

ε→0+

log T (r, Ω(α + ε, β − ε), f ) = λ. log r

Firstly, we claim that (4.2)

lim sup lim sup r→∞

ε→0+

log Sα+ε,β−ε (r, g) ≥ λ − ω. log r

From (4.1), for any given ε1 ∈ (0, λ−ω ), there exists at least some ε2 ∈ (0, ε1 ) 2 such that log T (r, Ω(α + ε2 , β − ε2 ), f ) lim sup = λ′ ≥ λ − ε1 , log r r→∞ where λ′ (≤ λ) is a constant. It follows from Lemma 2.5 and (1.6) that lim sup r→∞

log Sα+ε2 ,β−ε2 (r, f ) = λ′ − ω ≥ λ − ω − ε1 . log r

By using Lemmas 2.1 and 2.3, then there exists a value a such that 1 ) log C¯α+ε2 ,β−ε2 (r, f −a ≥ λ − ω − ε1 . lim sup log r r→∞ Without loss of generality, we may assume that a = a1 . Thus, 1 ) log C¯α+ε2 ,β−ε2 (r, f −a 1 lim sup ≥ λ − ω − ε1 . log r r→∞ It follows from (1.7) that lim sup

1 log C¯α+ε2 ,β−ε2 (r, g−a ) 1

r→∞

log r

≥ λ − ω − ε1 .

Therefore, we get lim sup r→∞

log Sα+ε2 ,β−ε2 (r, g) ≥ λ − ω − ε1 . log r

Noting ε1 is arbitrary and ε2 < ε1 , so (4.2) holds. We assume on the contrary to the assertion that f ̸= g. We consider two cases. Case 1. We may assume that all ai (i = 1, 2, 3, 4, 5) are finite.

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By arguing similar to that proof of Theorem 1.1, we can obtain the following inequalities, 3Sα+ε,β−ε (r, f ) ≤

(4.3)

5 ∑

C¯α+ε,β−ε (r,

1 ) + Rα+ε,β−ε (r, f ), f − ai

C¯α+ε,β−ε (r,

1 ) + Rα+ε,β−ε (r, g), g − ai

i=1

3Sα+ε,β−ε (r, g) ≤

(4.4)

5 ∑ i=1

5 ∑

C¯α+ε,β−ε (r,

i=1

1 1 ) ≤ Cα+ε,β−ε (r, ) f − ai f −g ≤ Sα+ε,β−ε (r, f ) + Sα+ε,β−ε (r, g) + O(1).

(4.5)

By using (1.6), Lemmas 2.3 and 2.5, we get (4.6)

lim sup r→∞

Rα+ε,β−ε (r, f ) = 0. Sα+ε,β−ε (r, f )

Similarly, it follows from (4.2) that lim sup

(4.7)

r→∞

Rα+ε,β−ε (r, g) = 0. Sα+ε,β−ε (r, g)

Combining (4.3)-(4.7), for sufficiently large r, we have ∑ ∑ 2 1 1 1 ( + o(1)) C¯α+ε,β−ε (r, C¯α+ε,β−ε (r, ) ≤ ( + o(1)) ). 3 f − a 3 g − a i i i=1 i=1 5

5

It follows that lim inf r→∞

5 ∑ i=1

∑ 1 1 1 C¯α+ε,β−ε (r, )/ )≤ . f − ai i=1 g − ai 2 5

C¯α+ε,β−ε (r,

Noting ε → 0, this contradicts to (1.8), and hence f = g. Case 2. If one of the values ai (i = 1, 2, 3, 4, 5) is ∞, without loss of generality, we may assume that a5 = ∞. By using similar way of the proof of Theorem 1.1, we can easily obtain f = g. The proof is completed.  Proof of Theorem 1.3. By Lemma 2.8, (1.9) implies (1.6). So combining Theorem 1.2 we get the conclusion of Theorem 1.3.  Acknowledgements. This research was partly supported by the United Technology Foundation of Science and Technology Department of Guizhou Province and Guizhou Normal University (Grant No. LKS[2012]12), the National Natural Science Foundation of China (Grant No. 11171277).

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References [1] T. B. Cao and H. X. Yi, On the uniqueness of meromorphic functions that share four values in one angular domain, J. Math. Anal. Appl. 358 (2009), 81-97. [2] C. T. Chuang, Sur les fonctions-types, Sci. Sinica 10 (1961), 171-181. [3] C. T. Chuang, On Borel directions of meromorphic functions of infinite order(⨿), Bull. HongKong Math. Soc. 2 (1999), no. 2, 305-323. [4] C. T. Chuang, Singular directions of meromorphic functions, Science Press, Beijing, 1982. (In Chinese) [5] D. Drasin and A. Weitsman, On the Julia Directions and Borel Directions of Entire Functions, Proc. London Math. Soc. 32 (1976), no. 3, 199-212. [6] A. A. Gol’dberg and I. V. Ostrovshii, Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. [7] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [8] K. L. Hiong, Sur les fonctions enti´eres et les fonctions m´eromorphes d’ordre infini, J. Math. Pures Appl. 14 (1935), 233-308. [9] W. C. Lin, S. Mori and H. X. Yi, Uniqueness theorems of entire functions with shared-set in an angular domain, Acta Math. Sinica 24 (2008), 1925-1934. [10] W. C. Lin, S. Mori and K. Tohge, Uniqueness theorems in an angular domain, Tohoku Math. J. 58 (2006), 509-527. [11] J. R. Long and P. C. Wu, Borel directions and Uniqueness of meromorphic functions, Chinese Ann. Math. 33A (2012), no. 3, 261-266. [12] R. Nevanlinna, Le th´ eor´ eme de Picard-Borel et la th´ eorie de fonctions m´ eromorphes, Paris, 1929. [13] M. Tsuji, On Borel directions of meromorphic functions of finite order, I, Tohoku Math. J. 2 (1950), no. 2, 97-112. [14] M. Tsuji, Potential theoty in modern function theory, Maruzen, Tokyo, 1959. [15] S. J. Wu, Further results on Borel removable set s of entire functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 19 (1994), 67-81. [16] S. J. Wu, On the distribution of Borel directions of entire function, Chinese Ann. Math. 14A (1993), no. 4, 400-406.

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[17] J. F. Xu and H. X. Yi, On uniqueness of meromorphic functions with shared four values in some angular domains, Bull. Malays Sci. Soc. (2) 31 (2008), no. 1, 57-65. [18] C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Publ., Dordrecht, 2003. [19] L. Yang and C. C. Yang, Angular distribution of f f ′ , Sci. China Ser. A 37 (1994), no. 3, 284- 294. [20] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993. [21] L. Yang, Borel directions of meromorphic functions in an angular domain, Sci. China Math. I (1979), 149-164. [22] L. Yang and G. H. Zhuang, The distribution of Borel directions of entire functions, Sci. China Ser. A 3 (1976), 157-168. [23] Q. C. Zhang, Meromorphic functions sharing values in an angular domain, J. Math. Anal. Appl. 349 (2009), 100-112. [24] J. H. Zheng, On uniqueness of meromorphic functions shared values in some angular domains, Canad. Math. Bull. 47 (2004), no. 1, 152-160. [25] J. H. Zheng, On uniqueness of meromorphic functions shared values in one angular domain, Complex Variables 48 (2003), no. 9, 777-785. [26] J. H. Zheng, Value Distribution of Meromorphic Functions, Springer and Tsinghua University Publishing House, 2010. Jianren Long School of Mathematics and Computer Science, Guizhou Normal University, 550001, Guiyang, P.R. China. The current address. School of Mathematical Sciences, Xiamen University, 361005, Xiamen, P.R. China. E-mail address: [email protected], [email protected] Chunhui Qiu School of Mathematical Sciences, Xiamen University, 361005, Xiamen, P.R. China. E-mail address: [email protected]

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ON AN INTERVAL-REPRESENTABLE GENERALIZED PSEUDO-CONVOLUTION BY MEANS OF THE INTERVAL-VALUED GENERALIZED FUZZY INTEGRAL AND THEIR PROPERTIES JEONG GON LEE AND LEE-CHAE JANG

Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Republic of Korea E-mail : [email protected] General Education Institute, Konkuk University, Chungju 138-701, Korea E-mail : [email protected]

Abstract. In this paper, we consider the generalized pseudo-convolution in the theory of probabilistic metric space and their properties which was introduced by Pap-Stajner (1999). Wu-Wang-Ma(1993) and Wu-Ma-Song(1995) studied the generalized fuzzy integral and their properties. Recently, Jang(2013) defined the interval-valued generalized fuzzy integral by using an interval-representable pseudo-multiplication. From the generalized fuzzy integral, we define a generalized pseudo-convolution by means of the generalized fuzzy integral and investigate their properties. In particular, we also define an interval-representable generalized pseudo-convolution of interval-valued functions by means of the interval-valued generalized fuzzy integral and investigate their properties.

1. Introduction Fang [8-10], Wu-Wang-Ma [35], Wu-Ma-Song [36], Xie-Fang [37] have studied the generalized fuzzy integral(for short, the (G) fuzzy integral) by using a pseudo-multiplication which is a generalization of fuzzy integrals in [5, 25, 26, 29, 31, 33, 39]. Pap-Stajner [28] introduced a notion of the generalized pseudo-convolution of functions based on pseudo-operations and proved their mathematical theories such as optimization, probabilistic metric spaces, and information theory Many researchers [1,2,7,13-19, 21, 30, 34, 38, 40] have been studying various integrals of measurable multi-valued functions which are used for representing uncertain functions, for examples, the Aumann integral, the fuzzy integral, and the Choquet integral of measurable interval-valued functions in many different mathematical theories and their applications. Recently, Jang [20] defined the interval-valued generalized fuzzy integral (for short, the (IG) fuzzy integral) with respect to a fuzzy measure by using an interval-representable pseudo-multiplication of measurable interval-valued functions and investigated some convergence properties of them. 1991 Mathematics Subject Classification. 28E10, 28E20, 03E72, 26E50 11B68. Key words and phrases. fuzzy measure, generalized fuzzy integral, interval-representable pseudomultiplication, interval-valued function, generalized pseudo-convolution. 1

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The purpose of this study is to define the generalized pseudo-convolution of functions by means of the (G) fuzzy integral and to investigate some properties of them. In particular, we also define the interval-valued generalized pseudo-convolution of interval-valued functions by means of the (IG) fuzzy integral and to investigate some properties of them. The paper is organized in five sections. In section 2, we list definitions and some properties of the generalized fuzzy integral with respect to a fuzzy measure by using generalized pseudo-multiplication and the interval-valued generalized fuzzy integral with respect to a fuzzy measure by using interval-representable generalized pseudo-multiplication. In section 3, we define the generalized pseudo-convolution of integrable nonnegative functions by means of the (G) fuzzy integral and investigate their properties. Furthermore, we give an example of the generalized pseudo-convolution of integrable nonnegative functions. In section 4, we define a interval-representable semigroup and the interval-valued generalized pseudoconvolution of integrable interval-valued functions by means of the (IG) fuzzy integral and investigate their properties. Furthermore, we give an example of the interval-valued generalized pseudo-convolution of integrable interval-valued functions. In section 5, we give a brief summary results and some conclusions.

2. Definitions and Preliminaries In this section, we introduce some definitions and properties of a fuzzy measure, a pseudomultiplication, a pseudo-addition, the (G) fuzzy integral with respect to a fuzzy measure by using a pseudo-multiplication of a measurable functions. Let X be a set and (X, A) be a measurable space. Denote by F(X) the set of all measurable nonnegative functions on X. Definition 2.1. ([25, 26]) (1) A fuzzy measure µ : A −→ [0, ∞] is a set function satisfying (i) (ii)

µ(∅) = 0 µ(A) ≤ µ(B) whenever A, B ∈ A and A ⊂ B.

(2) A fuzzy measure µ is said to be finite if µ(X) < ∞.

Definition 2.2. ([10, 33, 37]) (1) A binary operation ⊕ : [0, ∞]2 −→ [0, ∞] is called a pseudo-addition if it is non-decreasing in both components, associative, and 0 is its neutral element. (2) A binary operation : [0, ∞]2 −→ [0, ∞] is called a pseudo-multiplication corresponding to ⊕ if it satisfies the following axioms: (i) a b = b a, (ii) a (x ⊕ y) = (a x) ⊕ (a y), (iii) a ≤ b =⇒ a x ≤ b x, (vi) a x = 0 ⇐⇒ a = 0 or x = 0, (v) there exists a unit element, that is, ∃e ∈ (0, ∞] such that e x = x for all x ∈ [0, ∞], (vi) an −→ a ∈ (0, ∞) and xn −→ x ∈ [0, ∞] =⇒ an xn −→ a x and lima→∞ a x = ∞ x for all x ∈ (0, ∞].

Definition 2.3. ([20, 33, 37]) (1) Let (X, A, µ) be a fuzzy measure space,f ∈ F(X), and A ∈ A. The (G) fuzzy integral with respect to a fuzzy measure µ by using a pseudomultiplication corresponding to the pseudo-addition ⊕ = max(maximum) of f on A is

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defined by Z



f dµ = sup α µA,f (α),

(G) A

(1)

α>0

where µA,f (α) = µ({x ∈ A|f (x) ≥ α}) for all α ∈ (0, ∞). R (2) f is said to be integrable if (G) A f dµ is finite. Let F(X)∗ be the set of all nonnegative integrable functions on X. We consider the intervals, a standard interval-valued pseudo-multiplication, and an extended interval-valued pseudo-multiplication. Let I(Y ) be the set of all bounded closed intervals (intervals, for short) in Y as follows: I(Y ) = {a = [al , ar ] | al , ar ∈ Y and al ≤ ar },

(2)

where Y is [0, ∞) or [0, ∞]. For any a ∈ Y , we define a = [a, a]. Obviously, a ∈ I(Y ) (see[4, 7, 16-21, 30, 34, 38-40]). Denote by IF(X) the set of all measurable interval-valued functions on X. Definition 2.4. ([20]) If a = [al , ar ], b = [bl , br ], an = [aln , arn ], aα = [alα , arα ] ∈ I(Y ) for all n ∈ N and α ∈ [0, ∞), and k ∈ [0, ∞), then we define arithmetic, maximum, minimum, order, inclusion, superior, inferior operations as follows: (1) a + b = [al + bl , ar + br ], (2) ka = [kal , kar ], (3) ab = [al bl , ar br ], (4) a ∨ b = [al ∨ bl , ar ∨ br ], (5) a ∧ b = [al ∧ bl , ar ∧ br ], (6) a ≤ b if and only if al ≤ bl and ar ≤ br , (7) a < b if and only if al ≤ bl and al 6= bl , (8) a ⊂ b if and only if bl ≤ al and ar ≤ br ], (9) supn an = [supn anl , supn anr ], (10) inf n an = [inf n anl , inf n anr ], (11) supα aα = [supα aαl , supα aαr ], and (12) inf α aα = [inf α aαl , inf α aαr ].

J 2 Definition 2.5. ([20]) (1) A mapping I : I([0, ∞]) −→ I([0, ∞]) is called a standard interval-valued pseudo-multiplication if there exist pseudo-multiplications l and r such that x l y ≤ x r y for all x, y ∈ [0, ∞], and such that for all a = [al , ar ], b = [bl , br ] ∈ I([0, ∞]), K a b = [al l bl , ar r br ] . (3) I J Then l and r are J called the representants of I . (2) A mapping II : I([0, ∞])2 −→ I([0, ∞]) is called an extended interval-valued pseudomultiplication if there exists a pseudo-multiplication such that for any a = [al , ar ], b = [bl , br ] ∈ I([0, ∞]), K a b = [al bl , max{al br , ar bl }] . (4) II J Then is called the representant of II .

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We also introduce the (IG) fuzzy integral with respect to a fuzzy measure by using two interval-representable pseudo-multiplications which are used to define the interval-valued generalized pseudo-convolution in the next section 4. Definition 2.6. ([20]) Let (X, A, µ) be a fuzzy measure space. (1) An interval-valued function f : X → I([0, ∞) \ {∅} is said to be measurable if for any open set O ⊂ [0, ∞), f

−1

(O) = {x ∈ X | f ∩ O 6= ∅} ∈ A.

(5)

(2) If : I([0, ∞])2 −→ I([0, ∞]) is an interval-representable pseudo-multiplication and J f ∈ IF(X) and A ∈ A, then the (IG) fuzzy integral with respect to µ by using of f on A is defined by Z J K (IG) f dµ = sup α µA,f (α), (6) J

A

α>0

where µA,f (α) = [µA,fl (α), µA,fr (α)] for all α ∈ [0, ∞). (3) f is said to be integrable on A if Z J (IG) f dµ ∈ P([0, ∞)) \ {∅},

(7)

A

where P(R+ ) is the set of all subsets of [0, ∞). Let IF(X)∗ be the set of all integrable interval-valued functions. We consider the following theorem which is used to investigate some characterizations of the interval-valued generalized pseudo-convolution by means of the (IG) fuzzy integral. Theorem 2.1. (1) Let l and J r be pseudo-multiplications on [0, ∞] corresponding to a pseudo-addition ⊕ = max. If I is a standard interval-valued pseudo-multiplication, A ∈ A, and f ∈ IF(X)∗ , then we have   Z l Z r Z JI f dµ = (G) fl dµ, (G) fr dµ . (8) (IG) A

A

A

(2) Let : [0, ∞] −→ [0, ∞] be a pseudo-multiplication, f = [fl , fr ] ∈ IF(X)∗ , and A ∈ A. J If II is an extended interval-valued pseudo-multiplication, then we have   Z Z JII Z (IG) f dµ = (G) fl dµ, (G) fr dµ . (9) 2

A

A

A

3. The generalized pseudo-convolution on F(X)∗ In this section, we consider a semigroup ([0, ∞), ⊗) and define the generalized pseudoconvolution on F(X)∗ .

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Definition 3.1. Let f, h ∈ F(X)∗ and t ∈ [0, ∞). The generalized pseudo-convolution of f and h by means of the (G) fuzzy integral is defined by Z (f ∗ h)(t) = (G) f (t − u) ⊗ h(u)dµ(u). (10) [0,t]

Then we obtain the following basic properties and examples of the generalized pseudoconvolution of nonnegative measurable functions. Theorem 3.1. (1) If f, h ∈ F(X)∗ and t ∈ [0, ∞) and ⊗ is a minimum operation(min) and f (t − u) ≤ h(u) for all u ∈ [0, t], then we have (f ∗ h)(t) = sup α µ[0,t],f (α).

(11)

α∈[0,t]

(2) If f, h ∈ F(X)∗ and t ∈ [0, ∞) and ⊗ is a multiplication operation(·) and f (x) = c for all x ∈ [0, ∞), then we have α . (12) (f ∗ h)(t) = sup α µ[0,t],h c α∈[0,t] (3) If f, h ∈ F(X)∗ and t ∈ [0, ∞) and {t − x|f (x) > 0} ∩ {x|h(x) > 0} = ∅ and a ⊗ 0 = 0 for all a ∈ [0, t], then we have (f ∗ h)(t) = 0.

(13)

(4) If f, h ∈ F(X)∗ and t ∈ [0, ∞) and is a minimum operation(min) and µ({u ∈ [0, t]|f (t − u) ⊗ h(u) > α}) = g(α) ≥ α for all α ∈ [0, ∞), then we have (f ∗ h)(t) = t.

(14)

Proof.(1) Suppose that ⊗ is a minimum operation(min) and f (t − u) ≤ h(u) for all u ∈ [0, t]. Then we have µ[0,t],f (t−·)⊗h(·) (α)

= µ({u ∈ [0, t]| min{f (t − u), h(u)} > α}) = µ({u ∈ [0, t]|h(u) > α}) = µ[0,t],h (α).

(15)

By (15), we have Z (f ∗ h)(t)

=



f (t − u) ⊗ h(u)dµ(u)

(G) [0,t]

=

sup α µ[0,t],f (t−·)⊗h(·) (α) α∈[0,t]

=

sup α µ[0,t],h (α).

(16)

α∈[0,t]

(2) Suppose that ⊗ is a multiplication operation(·) and f (x) = c for all x ∈ [0, ∞). Then we have µ[0,t],f h (α)

= µ({u ∈ [0, t]|f (t − u)h(u) > α}) = µ({u n ∈ [0, t]|ch(u) > α}) α α o = µ u ∈ [0, t]|h(u) > = µ[0,t],h . c c

(17)

By (17), we have Z (f ∗ h)(t)

=



f (t − u) ⊗ h(u)dµ(u)

(G) [0,t]

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Z =



ch(u)dµ(u)

(G) [0,t]

sup α µ[0,t],ch (α) α = sup α µ[0,t],h . c α∈[0,t] =

α∈[0,t]

(18)

(3) Suppose that {t − x|f (x) > 0} ∩ {x|h(x) > 0} = ∅ and u ⊗ 0 = 0 for all u ∈ [0, t]. Then we have = µ({u ∈ [0, t]|f (t − u) ⊗ h(u) > α}) = µ(∅) = 0.

µ[0,t],f (t−·)⊗h(·) (α)

(19)

By (19) and Definition 2.2 (2)(vi), we have (f ∗ h)(t)

sup α µ[0,t],f (t−·)⊗h(·) (α)

=

α∈[0,t]

sup α 0 = 0.

=

(20)

α∈[0,t]

(4) Suppose that is a minimum operation(min) and µ({u ∈ [0, t]|f (t − u) ⊗ h(u) > α}) = g(α) ≥ α for all α ∈ [0, ∞). Then we have (f ∗ h)(t)

sup α µ[0,t],f (t−·)⊗h(·) (α)

=

α∈[0,t]

=

sup min{α, g(α)} α∈[0,t]

=

sup α = t.

(21)

α∈[0,t]

Theorem 3.2. Let ([0, ∞), ⊗) be a semigroup and e be a unit element with respect to ⊗, that is, e ⊗ u = u for all u ∈ [0, ∞). If f ∈ F(X)∗ , then we have Z (e ∗ f )(t) = (G) f dµ. (22) [0,t]

Proof. Since (e ⊗ f )(u) = e ⊗ f (u) = f (u) for all u ∈ [0, ∞), we have (e ∗ f )(t)

=

sup α µ[0,t],e⊗f (α) α∈[0,t]

=

sup α µ[0,t],f (α) Z (G) f dµ.

α∈[0,t]

=

(23)

[0,t]

Remark 3.3. A function f : X −→ [0, ∞) is an idempotent with respect to the generalized pseudo-convolution ∗ induced by semigroup ([0, ∞), ⊗) if and only if f ∗ f = f . It is easy to see that if e is a unit element as in Theorem 3.3, that is, f ∗ e = f for all f ∈ F(X), then we also have e ∗ e = e. Therefore, e is an idempotent with respect to ∗.

Example 3.1. Let u v = min{u, v} and u ⊗ v = u · v for all u, v ∈ [0, ∞), and f (x) = 1 and h(x) = x2 for all x ∈ [0, ∞), and m be the Lebesgue measure on [0, ∞). If µ = m2 , then clearly µ is a fuzzy measure. Thus, we have µ[0,t],f (t−·)⊗h(·) (α)

= µ({u ∈ [0, t]|1 ⊗ u2 > α})

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√ √ = µ([ α, t]) = (t − α)2 . By (24), we have (f ∗ h)(t)

sup min{α, (t −

=

(24)

√ 2 α) }

α∈[0,t] 2

t . 4

=

(25)

4. The interval-valued generalized pseudo-convolution on IF(X)∗ N In this section, we define a standard interval-valued semigroup (I([0, ∞), ) and the interval-representable generalized pseudo-convolution of interval-valued functions by means of the (IG) fuzzy integral on IF(X)∗ . N Definition 4.1. A pair (I([0, ∞), ) is called a standard interval-valued semigroup if there exist two semigroups ([0, ∞), ⊗l ) and ([0, ∞), ⊗r ) such that O u v = [ul ⊗l vl , ur ⊗r vr ], (26) for all u = [ul , ur ], v = [vl , vr ] ∈ I([0, ∞)). Definition 4.2. Let f , h ∈ IF(X)∗ and t ∈ [0, ∞). The interval-valued generalized pseudoconvolution of f and h by means of the (IG) fuzzy integral is defined by Z J (f ∗ h)(t) = (IG) f (t − u) ⊗ h(u)dµ(u) (27) [0,t]

where

J

is an interval-representable pseudo-multiplication.

Then we obtain the following basic properties and examples of the interval-valued generalized pseudo-convolution of measurable interval-valued functions. Theorem 4.1. (1) Let l andJ r be pseudo-multiplications on [0, ∞] corresponding to a pseudo-addition ⊕ = max. If I is a standard interval-valued pseudo-multiplication and f = [fl , fr ], h = [hl , hr ] ∈ IF∗ (X), t ∈ [0, ∞] and Z JI O (f ∗1 f )(t) = (IG) f (t − u) h(u)dµ(u), (28) A

then we have (29) (f ∗1 f )(t) = [(fl ∗1l hl , fr ∗1r hr ], R l R r where (fl ∗1l hl )(t) = (G) [0,t] fl (t − u) ⊗l hl (u)dµ(u) and (fr ∗1r hr )(t) = (G) [0,t] fr (t − u) ⊗r hr (u)dµ(u). (2) Let J be a pseudo-multiplications on [0, ∞] corresponding to a pseudo-addition ⊕ = max. If II is an extended interval-valued pseudo-multiplication and f = [fl , fr ], h = [hl , hr ] ∈ IF∗ (X), t ∈ [0, ∞] and Z JII O (f ∗2 f )(t) = (IG) f (t − u) h(u)dµ(u), (30) A

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then we have (f ∗2 f )(t) = [(fl ∗2l hl , fr ∗2r hr ], (31) R R where (fl ∗2l hl )(t) = (G) [0,t] fl (t − u) ⊗l hl (u)dµ(u) and (fr ∗2r hr )(t) = (G) [0,t] fr (t − u) ⊗r hr (u)dµ(u). N h = [fl ⊗l hl , fr ⊗r hr ], by Theorem 2.7 (1), we have Proof. (1) Since f Z JI O (f ∗1 f )(t) = (IG) f (t − u) h(u)dµ(u) A # " Z l Z r fr ⊗r hr dµ = (G) fl ⊗l hl dµ, (G) [0,∞]

[0,∞]

= (2) Since f

[(fl ∗1l hl , fr ∗1r hr ].

N

h = [fl ⊗l hl , fr ⊗r hr ], by Theorem 2.7 (2), we have Z JII O (f ∗2 f )(t) = (IG) f (t − u) h(u)dµ(u) " # ZA Z = (G) fl ⊗l hl dµ, (G) fr ⊗r hr dµ [0,∞]

=

[0,∞]

[(fl ∗2l hl , fr ∗2r hr ].

Theorem 4.2. (1) If f = [fl , fr ], h = [hl , hr ] ∈ IF(X)∗ and t ∈ [0, ∞), and ⊗l = ⊗r are minimum operation(min) and f (t − u) ≤ h(u) for all u ∈ [0, t], then we have " # (f ∗1 h)(t) =

sup α l µ[0,t],fl (α), sup α r µ[0,t],fr (α) α∈[0,t]

(32)

α∈[0,t]

and #

" (f ∗2 h)(t) =

sup α µ[0,t],fl (α), sup α µ[0,t],fr (α) . α∈[0,t]

(33)

α∈[0,t]

(2) If f = [fl , fr ], h = [hl , hr ] ∈ IF(X)∗ and t ∈ [0, ∞) and ⊗l = ⊗r is multiplication operation(·) and f (x) = [c, d] ∈ I([0, ∞)) for all x ∈ [0, ∞), then we have " # α α , sup α r µ[0,t],fr (34) (f ∗1 h)(t) = sup α l µ[0,t],hl c α∈[0,t] d α∈[0,t] and " (f ∗2 h)(t) =

sup α µ[0,t],hl α∈[0,t]

α c

, sup α µ[0,t],fr α∈[0,t]

α d

# .

(35)

∗ (3) If f = [fl , fN r ], h = [hl , hr ] ∈ IF(X) and t ∈ [0, ∞) and {t − x|f (x) > [0, 0]} ∩ {x|h(x) > [0, 0]} = ∅ and a [0, 0] = [0, 0] for all a ∈ I([0, t]), then we have

(f ∗1 h)(t) = 0

(36)

(f ∗2 h)(t) = 0.

(37)

and

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(4) If f = [fl , fr ], h = [hl , hr ] ∈ IF(X)∗ and t ∈ [0, ∞) and µ({u ∈ [0, t]|fl (t − u) ⊗l hl (u) > α}) = gl (α) and µ({u ∈ [0, t]|fr (t − u) ⊗r hr (u) > α}) = gr (α) for all α ∈ [0, ∞), then we have (f ∗1 h)(t) = [ sup α l gl (α), sup α r gr (α)] α∈[0,t]

(38)

α∈[0,t]

and (f ∗2 h)(t) = [ sup α gl (α), sup α gr (α)]. α∈[0,t]

(39)

α∈[0,t]

Proof.(1) Suppose that ⊗l = ⊗r are minimum operation(min) and f (t − u) ≤ h(u) for all u ∈ [0, t]. Then we have fl (t − u) ≤ hl (u) and fr (t − u) ≤ hr (u) for all u ∈ [0, t]. Thus, by Theorem 4.1(1) and Theorem 3.1 (1), we have (f ∗1 h)(t)

= " [(fl ∗1l hl )(t), (fr ∗1l hr )(t)] Z Z l

=

fr ⊗r hr dµ

fl ⊗l hl dµ, (G)

(G)

#

r

[0,t]

[0,t]

"

# sup α l µ[0,t],fl (α), sup α r µ[0,t],fr (α) .

=

α∈[0,t]

(40)

α∈[0,t]

By Theorem 4.1(2) and Theorem 3.1 (1), we have (f ∗2 h)(t)

= " [(fl ∗2l hl )(t), (fr ∗2l hr )(t)] Z Z

=

#



fl ⊗l hl dµ, (G)

(G) [0,t]

fr ⊗r hr dµ [0,t]

#

"

sup α µ[0,t],fl (α), sup α µ[0,t],fr (α) .

=

α∈[0,t]

(41)

α∈[0,t]

(2) Suppose that f (x) = [c, d] ∈ I([0, ∞)) for all x ∈ [0, ∞). By Theorem 3.1 (2) and Theorem 4.1 (1), we have (f ∗1 h)(t)

= " [(fl ∗1l hl )(t), (fr ∗1l hr )(t)] Z l

=

fl (t − u) · hl (u)dµ(u), (G)

(G) [0,t]

" =

[0,t]

l

Z

fr (t − u) · hr (u)dµ(u) #

d · hr (u)dµ(u)

[0,t]

[0,t]

" sup α l µ[0,t],hl

=

r

Z c · hl (u)dµ(u), (G)

(G)

#

r

Z

α∈[0,t]

α c

, sup α r µ[0,t],hr

α

# .

d

α∈[0,t]

(42)

By Theorem 3.1 (2) and Theorem 4.1 (2), we have (f ∗2 h)(t)

= " [(fl ∗2l hl )(t), (fr ∗2l hr )(t)] Z

=

Z

fl (t − u) · hl (u)dµ(u), (G)

(G) [0,t]

" =

Z

[0,t]



Z

[0,t]

" =



c · hl (u)dµ(u), (G)

(G)

sup α µ[0,t],hl α∈[0,t]

#



fr (t − u) · hr (u)dµ(u) #

d · hr (u)dµ(u) [0,t]

α c

, sup α µ[0,t],hr α∈[0,t]

1068

α d

# .

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N (3) Suppose that {t − x|f (x) > [0, 0]} ∩ {x|h(x) > [0, 0]} = ∅ and a [0, 0] = [0, 0] for all a ∈ I([0, t]). Then we have that {t − x|fl (x) > 0} ∩ {x|hl (x) > 0} = ∅ and al ⊗ 0 = 0 for all al ∈ [0, t], and {t − x|fr (x) > 0} ∩ {x|hr (x) > 0} = ∅ and ar ⊗ 0 = 0 for all ar ∈ [0, t]. By Theorem 3.1(3), we have (fl ∗1l hl )(t) = 0 and (fr ∗1r hr )(t) = 0

(44)

(fl ∗2l hl )(t) = 0 and (fr ∗2r hr )(t) = 0.

(45)

and

By (44) and Theorem 4.1(1), we have (f ∗1 h)(t)

=

[(fl ∗1l hl )(t), (fr ∗1r hr )(t)] = 0.

(46)

[(fl ∗2l hl )(t), (fr ∗2r hr )(t)] = 0.

(47)

By (45) and Theorem 4.1(2), we have (f ∗2 h)(t)

=

(4) Suppose that f f = [fl , fr ], h = [hl , hr ] ∈ IF(X) and t ∈ [0, ∞) and µ({u ∈ [0, t]|fl (t − u) ⊗l hl (u) > α}) = gl (α) and µ({u ∈ [0, t]|fr (t − u) ⊗r hr (u) > α}) = gr (α) for all α ∈ [0, ∞). By Theorem 3.1 (4), we have (fl ∗1l hl )(t) = sup α l gl (α) and (fr ∗1r hr )(t) = sup α r gr (α), α∈[0,t]

(48)

α∈[0,t]

and (fl ∗2l hl )(t) = sup α gl (α) and (fr ∗2r hr )(t) = sup α gr (α). α∈[0,t]

(49)

α∈[0,t]

By (48) and Theorem 4.1(1), we have (f ∗1 h)(t)

= [(fl ∗1l hl )(t), (fr ∗1r hr )(t)] = [ sup α l gl (α), sup α r gr (α)]. α∈[0,t]

(50)

α∈[0,t]

By (49) and Theorem 4.1(2), we have (f ∗2 h)(t)

= [(fl ∗2l hl )(t), (fr ∗2r hr )(t)] = [ sup α gl (α), sup α gr (α)]. α∈[0,t]

(51)

α∈[0,t]

N Theorem 4.3. Let (I([0, ∞)), = [⊗l , ⊗r ]) be a standard interval-valued semigroup and el be a unit element with respect to ⊗l and and er be a unit element with respect to ⊗r . If f ∈ IF(X)∗ , then we have Z NI (e ∗1 f )(t) = (IG) f dµ (52) [0,t]

and Z

N

II

(e ∗2 f )(t) = (IG)

f dµ

(53)

[0,t]

where e = [el , er ]. Proof. By Theorem 3.2, we have Z Z l fl dµ and (er ∗1r fr )(t) = (G) (el ∗1l fl )(t) = (G) [0,t]

r

fr dµ

(54)

[0,t]

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and Z



(el ∗2l fl )(t) = (G)

Z



fl dµ and (er ∗2r fr )(t) = (G) [0,t]

fr dµ.

(55)

[0,t]

By Theorem 4.1(1) and (54), we have (e ∗1 f )(t)

= " [el ∗1l fl , er ∗1r fr ] Z l

=

(G) el ⊗l fl dµ, (G) [0,t] " Z Z l

=

(G)

er ⊗r fr dµ #

[0,t]

r

fl dµ, (G) [0,t] Z N

#

r

Z

fr dµ [0,t]

I

=

(IG)

f dµ.

(56)

[0,t]

By (55) and Theorem 4.1(2), we have (e ∗2 f )(t)

= " [el ∗2l fl , er ∗2r fr ] Z

=

Z

el ⊗l fl dµ, (G)

(G) [0,t]

" =

Z

er ⊗r fr dµ #

[0,t]



Z



fr dµ

fl dµ, (G)

(G)

#



[0,t]

[0,t] Z N

II

=

(IG)

f dµ.

(57)

[0,t]

Remark 4.4. A function f : X −→ I([0, ∞)) is an interval-valued idempotent with respect to the standard interval-valued generalized pseudo-convolution ∗i (for i = 1, 2) induced by a N standard interval-valued semigroup (I([0, ∞)), ) if and only if f ∗i f = f for i = 1, 2. It is easy to see that if e = [el , er ] is a unit element as in Theorem 4.2, that is, f ∗i E = f for all f ∈ IF(X)∗ , then we also have e ∗i e = e for i = 1, 2. Therefore, e is an interval-valued idempotent with respect to ∗i for i = 1, 2.

Example 4.1. Suppose that l = r = and u v = min{u, v} and u ⊗l v = u ⊗r v = u · v for all u, v ∈ [0, ∞), and f (x) = [1, 2] and h(x) = [x2 , 2x2 ] for all x ∈ [0, ∞), and m be the Lebesgue measure on [0, ∞). If µ = m2 , then clearly µ is a fuzzy measure. Thus, we have µ[0,t],fl (t−·)⊗l hl (·) (α)

= µ({u ∈ [0, t]|1 ⊗ u2 > α}) √ √ = µ([ α, t]) = (t − α)2

(58)

and µ[0,t],fr (t−·)⊗r hr (·) (α)

= µ({u ∈ [0, t]|2 ⊗ 2u2 > α})    √ √ 2 α α ,t = t− . = µ 2 2

(59)

By (58) and Theorem 4.1(1), we have (f " ∗1 h)(t) =

#   sup min α, µ[0,t],fl (t−·)⊗l hl (·) (α) , sup min α, µ[0,t],fr (t−·)⊗r hr (·) (α)

α∈[0,t]

α∈[0,t]

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JEONG GON LEE AND LEE-CHAE JANG

(  √ 2 )#  √ 2 α = sup min α, (t − α) , sup min α, t − 2 α∈[0,t] α∈[0,t]   2 t , 4t2 . = 4 "

5. Conclusions This study was to define the generalized pseudo-convolution of integrable functions by means of the (G) fuzzy intgeral(see Definition 3.1) and to investigate some properties and an example of the generalized pseudo-convolution on F(X)∗ in Theorems 3.2, 3.3 and Example 3.1. By using the concept of an interval-representable pseudo-multiplication(see Definitions 2.5 and 2.6), we can define a standard interval-valued semigroup (see Definition 4.1) and the interval-representable generalized pseudo-convolution on IF(X)∗ (see definition 4.2). From Theorems 4.3, 4.4, and 4.5, we investigate some characterizations of the interval-representable generalized pseudo-convolution of integrable interval-valued functions. Furthermore, some applications of the interval-representable generalized pseudo-convolution are focused on various transform operations including pseudo-Laplace transform. For this reason, the future work can also be directed to interval-representable generalized pseudotransform operations by means of the (IG) fuzzy integral. Acknowledgement: This paper was supported by Wonkwang University in 2014.

References [1] J.P. Aubin, Set-valued Analysis, Birkhauser Boston, (1990). [2] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl., 12 (1965), 1-12. [3] F. Baccelli, G. Cohen, G.J. Olsder, Qudrat, synechronization and linearity: an algebra for discrete event systems, Wiley, NewYork, 1992. [4] G. Beliakov, H. Bustince, D.P. Goswami, U.K. Mukherjee, On averaging operators for Atanassov’s intuitionistic fuzzy sets, Information Sciences, 181 (2011), 1116-1124. [5] P. Benvenuti, R. Mesiar, D. Vivona, Monotone set functions-based integrals Handbook of measure theory, Volume II, 2011, 1116-1124. [6] P. Benvenuti, R. Mesiar Pseudo-arithmetical operations as a basis for the general measure and integration theory, Information Sciences, 160 (2004), 1-11. [7] G. Deschrijver, Generalized arithemetic operators and their relatioship to t-norms in interval-valued fuzzy set theory, Fuzzy Sets and Systems, 160 (2009), 3080-3102. [8] J. Fang, A note on the convergence theorem of generalized fuzzy integrals, Fuzzy Sets and Systems, 127 (2002), 377-381. [9] J. Fang, On the convergence theorems of generalized fuzzy integral sequence, Fuzzy Sets and Systems, 124 (2001), 117-123. [10] J. Fang, Some properties of sequences of generalized fuzzy integrable functions, Fuzzy Sets and Systems, 158 (2007), 1832-1842. [11] M. Grabisch, Fuzzy integral in multicriteria decision making, Fuzzy Sets and Systems, 69(1995), 279-298. [12] M. Ha, C. Wu, Fuzzy measurs and integral theory, Science Press, Beijing, 1998. [13] L.C. Jang, B.M. Kil, Y.K. Kim, J.S. Kwon, Some properties of Choquet integrals of set-valued functions, Fuzzy Sets and Systems, 91 (1997), 61-67. [14] L.C. Jang, J.S. Kwon, On the representation of Choquet integrals of set-valued functions and null sets, Fuzzy Sets and Systems, 112 (2000), 233-239. [15] L.C. Jang, T. Kim, J.D. Jeon,On the set-valued Choquet integrals and convergence theorems(II), Bull. Korean Math. Soc. 40(1) (2003), 139-147. [16] L.C. Jang, Interval-valued Choquet integrals and their apllications, J. Appl. Math. and Computing, 16(12) (2004), 429-445.

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[17] L.C. Jang, A note on the monotone interval-valued set function defined by the interval-valued Choquet integral, Commun. Korean Math. Soc., 22 (2007), 227-234. [18] L.C. Jang, On properties of the Choquet integral of interval-valued functions, Journal of Applied Mathematics, 2011 (2011), Article ID 492149, 10pages. [19] L.C. Jang, A note on convergence properties of interval-valued capacity functionals and Choquet integrals, Information Sciences, 183 (2012), 151-158. [20] L.C. Jang, A note on the interval-valued generalized fuzzy integral by means of an interval-representable pseudo-multiplication and their convergence properties, Fuzzy Sets and Systems, 222 (2013), 45-57. [21] L.C. Jang, Some characterizations of the Choquet integral with respect to a monotone interval-valued set function, International Journal of Fuzzy Logic and Intelligent Systems, 13(1) (2013), 75-81. [22] E.P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000. [23] V.P. Maslov, S.N. Samborskij, Idempotent Analysis, Advances in Soviet Mathematics 13, Amer. Math. Soc. Providence, Rhode Island, 1992. [24] R. Mesiar, J. Rybarisk, PAN-operations structure, Fuzzy Sets and Systems, 74 (1995), 365-369. [25] T. Murofushi, M. Sugeno, A theory of fuzzy measures: representations, the Choquet integral, and null sets, J. Math. Anal. Appl., 159 (1991), 532-549. [26] T. Murofushi, M. Sugeno, M. Suzaki, Autocontinuity, convergence in measure, and convergence indistribution, Fuzzy Sets and Systems, 92(1997) 197-203. [27] E. Pap, Null-additive set functions, Kluwer Academic Publishers,Dordrecht,1995. [28] E. Pap, I. Stajner, Generalized pseudo-convolution in the theory of probabilistic metric spaces, information, fuzzy numbers, optimatation, system theory , Fuzzy Sets and Systems, 102(1999),393-415. [29] D.A. Ralescu, G. Adams, The fuzzy integral, J. Math. Anal. Appl., 75(2) (1980), 562-570. [30] H. Schjear-Jacobsen, Representation and calculation of economic uncertains: intervals, fuzzy numbers and probabilities, Int. J. of Production Economics, 78(2002), 91-98. [31] N. Shilkret, Maxitive measures and integration, Indag Math, 33(1971), 109-116. [32] M. Sugeno, Theory of fuzzy integrals and its applications, Doctorial Thesis, Tokyo Institute of Techonology, Tokyo,(1974). [33] M. Sugeno, T. Murofushi, Pseudo-additive measures and integrals, J. Math. Anal. Appl., 122(1987), 197-222. [34] K. Wechselberger, The theory of interval-probability as a unifying concept for uncertainty, Int. J. Approximate Reasoning, 24(2000), 149-170. [35] C. Wu, S. Wang, M. Ma, Generalized fuzzy integrals: Part1. Fundamental concept, Fuzzy Sets and Systems, 57(1993), 219-226. [36] C. Wu, M. Ma, S. Song, S. Zhang Generalized fuzzy integrals: Part3. convergence theorems, Fuzzy Sets and Systems, Fuzzy Sets and Systems, 70 (1995), 75-87. [37] Q. Xie, J. Fang, Corrections and remarks to the paper in Fuzzy Sets and Systems 124(2001) 117-123, Fuzzy Sets and Systems, Fuzzy Sets and Systems, 157(2006), 699-704. [38] D. Zhang, C. Guo, D. Lin, Set-valued Choquet integrals revisited, Fuzzy Sets and Systems, 147(2004), 475-485. [39] D. Zhang, C. Guo, On the convergence of sequences of fuzzy measures and generalized convergences theorems of fuzzy integral, Fuzzy Sets and Systems, 72(1995), 349-356. [40] D. Zhang, Z. Wang, Fuzzy integrals of fuzzy-valued functions, Fuzzy Sets and Systems, 54 (1993), 63-67.

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Fixed point and coupled fixed point theorems for generalized cyclic weak contractions in partially ordered probabilistic metric spaces Chuanxi Zhu, Wenqing Xu† Department of Mathematics, Nanchang University, Nanchang, 330031, P. R. China

Abstract. In this paper, we introduce the concept of new generalized cyclic weak contraction mappings and prove a class of fixed point theorems for such mappings in partially ordered probabilistic metric spaces. In addition, we also establish a coupled fixed point for mixed monotone mappings under contractive conditions in partially ordered probabilistic metric spaces. Our results extend and generalize Harjani et al. (Nonlinear anal. 71(2009)3403-3410) and Wu (Fixed Point Theory Appl. 2014(2014)49). Also, we introduce an example to support the validity of our results. Finally, an application of our results extends fixed point theorems for generalized weak contraction mappings in ordered metric spaces. Keywords: Menger probabilistic metric space; partially ordered; cyclic weak contractions; fixed point MR Subject Classification: 47H10, 34B15, 46S50

1

Introduction and preliminaries Fixed point theory in metric spaces is an important banch of nonlinear analysis, which is closely related

to the existence and uniqueness of solutions of differential and integral equations. The celebrated Banach’s contraction mapping principle is one of the cornerstones in development of nonlinear analysis. In the past years, Kirk and Srinvasan [1] presented fixed point theorems for mappings satisfying cyclical contractive conditions. Ran and Reurings [2] introduced fixed point theorems of Banach contraction operator in partially ordered metric spaces. Agarwal et al. [3] proved fixed point results of generalized contractive operators in partially ordered metric spaces; Harjani and Sadarangani [4] presented some fixed point theorems for weakly contractive mappings in complete metric spaces endowed with a partial order. Shatanwi [5] introduced nonlinear weakly C-contractive mappings in ordered metric spaces and proved some fixed point theorems. For more detail on fixed point theory and related results, we refer to [6-12] and the references therein. In 1942, Menger [13] introduced the concept of probabilistic metric spaces, a number of authors have done considerable works on probabilistic metric spaces [14-19]. Recently, the extension of fixed point theory to generalized structures as partially ordered probabilistic metric spaces has received much attention (see, [20-22]). † To

whom correspondence should be addressed. E-mail:wen qing [email protected](W. Xu). work has been supported by the National Natural Science Foundation of China (11361042,11071108), the Provincial

† This

Natural Science Foundation of Jiangxi, China (20132BAB201001,2010GZS0147).

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However, we rarely see any work about fixed point theorems for mappings under weakly contractive conditions in partially ordered probabilistic metric spaces. The aim of this paper is to determine some fixed point theorems for generalized cyclic weak contractions in the framework of partially ordered probabilistic metric spaces. Also, we introduce an example to support the validity of our results. Our results extend and generalize the main results of [3-8,11-12]. We introduce some useful concepts and lemmas for the development of our results. Let R denote the set of reals and R+ the nonnegative reals. A mapping F : R → R+ is called a distribution function if it is nondecreasing and left continuous with inf F (t) = 0 and sup F (t) = 1. We will denote by D the t∈R

t∈R

set of all distribution functions and D+ = {F ∈ D : F (t) = 0, t ≤ 0}. Let H denote the specific distribution function defined by   0, x ≤ 0; H(x) =  1, x > 0.

Definition 1.1 ([14]). The mapping ∆ : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (for short, a t-norm) if the following conditions are satisfied: (∆ − 1) ∆(a, 1) = a, for all a ∈ [0, 1]; (∆ − 2) ∆(a, b) = ∆(b, a); (∆ − 3) ∆(a, b) ≤ ∆(c, d), for c ≥ a, d ≥ b; (∆ − 4) ∆(a, ∆(b, c)) = ∆(∆(a, b), c). Two typical examples of continuous t-norm are ∆1 (a, b) = max{a + b − 1, 0} and ∆2 (a, b) = ab, for all a, b ∈ [0, 1]. Definition 1.2 ([14]). A triplet (X, F, ∆) is called a Menger probabilistic metric space (for short, Menger PMspace), if X is a nonempty set, ∆ is a t-norm and F is a mapping from X × X → D+ satisfying the following conditions (for x, y ∈ X, we denote F (x, y) by Fx,y ): (MS-1) Fx,y (t) = H(t), for all t ∈ R, if and only if x = y; (MS-2) Fx,y (t) = Fy,x (t), for all x, y ∈ X and t ∈ R; (MS-3) Fx,z (s + t) ≥ ∆(Fx,y (s), Fy,z (t)), for all x, y, z ∈ X and s, t ≥ 0. Definition 1.3 ([15]). (X, F, ∆) is called a non-Archimedean Menger PM-space (shortly, a N.A Menger PMspace), if (X, F, ∆) is a Menger PM-space and ∆ satisfies the following condition: for all x, y, z ∈ X and t1 , t2 ≥ 0, Fx,z (max{t1 , t2 }) ≥ ∆(Fx,y (t1 ), Fy,z (t2 )).

(1.1)

Definition 1.4 ([15]). A non-Archimedean Menger PM-space (X, F, ∆) is said to be type (D)g if there exists a g ∈ Ω such that g(∆(s, t)) ≤ g(s) + g(t), for all s, t ∈ [0, 1], where Ω = {g : g : [0, 1] → [0, ∞) is continuous, strictly decreasing, g(1) = 0}. 2 1074

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Example 1.1 . (X, F, ∆) is a N.A Menger PM-space, and ∆ ≥ ∆1 , where ∆1 (s, t) = max{s + t − 1, 0}, then (X, F, ∆) is of (D)g -type for g ∈ Ω defined by g(t) = 1 − t. Remark 1.1 Schweizer and Sklar [14] point out that if (X, F, ∆) is a Menger probabilistic metric space and ∆ is continuous, then (X, F, ∆) is a Hausdorff topological space in the (ε, λ)-topology T , i.e., the family of sets {Ux (ε, λ) : ε > 0, λ ∈ (0, 1]} (x ∈ X) is a basis of neighborhoods of a point x for T , where Ux (ε, λ) = {y ∈ X : Fx,y (ε) > 1 − λ}. Lemma 1.1 ([15]). Let {xn } be a sequence in X such that lim Fxn ,xn+1 (t) = 1 for all t > 0. If the sequence n→∞

{xn } is not a Cauchy sequence in X, then there exist ε0 > 0, t0 > 0 and two sequences {k(i)}, {m(i)} of positive integers such that (1) m(i) > k(i), and m(i) → ∞ as i → ∞; (2) Fxm(i) ,xk(i) (t0 ) < 1 − ε0 and Fxm(i)−1 ,xk(i) (t0 ) ≥ 1 − ε0 , for i = 1, 2, · · · . Definition 1.5 ([1]). Let X be a non-empty set, m be a positive integer, A1 , A2 , . . . , Am be subsets of X, Y = ∪m i=1 Ai and a mapping f : Y → Y . Then Y is said to be a cyclic representation of Y with respect to f , if (i) Ai , i = 1, 2, . . . , m, are nonempty closed sets; (ii) f (A1 ) ⊆ A2 , . . ., f (Am−1 ) ⊆ Am , f (Am ) ⊆ A1 . Example 1.2 Let X = R+ . Let A1 = [0, 2], A2 = [ 21 , 32 ], A3 = [ 34 , 45 ], and Y = fx =

1 2

S3

i=1

Ai . Defined f : Y → Y by

+ 12 x, for all x ∈ Y .

Clearly Y =

S3

i=1

Ai is a cyclic representation of Y with respect to f .

Definition 1.6 ([9]). The function h : [0, ∞) → [0, ∞) is called an altering distance function, if the following properties are satisfied: (a) h is continuous and nondecreasing; (b) h(t) = 0 if and only if t = 0. In [10], Bhasker and Lakshmikantham introduced the concepts of mixed monotone mappings and coupled fixed point. Definition 1.7 ([10]). Let (X, ≤) be a partially ordered set and A : X × X → X. The mapping A is said to have the mixed monotone property if A is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument, that is, for any x, y ∈ X, x1 , x2 ∈ X,

x1 ≤ x2

=⇒

A(x1 , y) ≤ A(x2 , y),

y1 , y2 ∈ X,

y2 ≤ y1

=⇒

A(x, y1 ) ≤ A(x, y2 ).

Definition 1.8 ([10]). An element (x, y) ∈ X 2 is said to be a coupled fixed point of the mapping A : X 2 → X if A(x, y) = x and A(y, x) = y. For a ˜ = (x, y), ˜b = (u, v) ∈ X 2 , we introduce a distribution function F˜ from X 2 into D+ defined by F˜a˜,˜b (t) = min{Fx,u (t), Fy,v (t)}, for all t > 0. In [20], Wu proved the following results: 3 1075

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Lemma 1.2 ([20]). If (X, F, ∆) is a complete Menger PM space, then (X 2 , F˜ , ∆) is also a complete Menger PM space. In the section 3 of this paper, we establish some coupled point theorems under contractive conditions in partially ordered probabilistic metric spaces. The obtained results extend and generalized the main results of [20-22]. Finally, we also obtain the corresponding fixed point theorems for generalized weak contraction mapping in ordered metric spaces.

2

Fixed point theorems for generalized cyclic weak contractions We start with the definition of generalized cyclic weak contraction mappings in probabilistic metric spaces.

Definition 2.1 Let (X, ≤) be a partially ordered set and (X, F, ∆) be a N.A Menger PM-space of type (D)g . Let m be a positive integer, A1 , A2 , . . . , Am be subsets of X, Y = ∪m i=1 Ai . A mapping T : X → X is said to be a generalized cyclic weak contraction, if Y is a cyclic representation of Y with respect to T , Am+1 = A1 and for k ∈ {1, 2, . . . , m}, and for all x, y ∈ X, x ∈ Ak and y ∈ Ak+1 are comparable with h(g(FT x,T y (t))) ≤ h(Mt (x, y)) − φ(Mt (x, y)),

for all t > 0,

(2.1)

where Mt (x, y) = max{g(Fx,y (t)), g(Fx,T x (t)), g(Fy,T y (t)), 12 [g(Fx,T y (t)) + g(Fy,T x (t))]}, h is a altering distance function, φ : [0, ∞) → [0, ∞) is a continuous function such that φ(s) = 0 if and only if s = 0. Theorem 2.1 Let (X, ≤) be a partially ordered set and (X, F, ∆) be a complete N.A Menger PM-space of type (D)g . Let m be a positive integer, A1 , A2 , . . . , Am be subsets of X, Y = ∪m i=1 Ai , T : Y → Y be a generalized cyclic weak contraction, and T be nondecreasing. Also assume that either (a) T is continuous or, (b) if a nondecreasing sequence xn → x, then xn ≤ x, for all n ∈ N . If there exists x0 ∈ A1 such that x0 ≤ T x0 , then T has a fixed point. Furthermore, the set of fixed points of T is well ordered if and only if T has a unique fixed point. Proof. Since T (A1 ) ⊆ A2 , there exists an x1 ∈ A2 , such that x1 = T x0 . Since T (A2 ) ⊆ A3 , there exists an x2 ∈ A3 , such that x2 = T x1 . Continuing this process, we can construct a sequence {xn } such that xn+1 = T xn , for all n ∈ N , and there exists in ∈ {1, 2, . . . , m} such that xn ∈ Ain and xn+1 ∈ Ain +1 . Since x0 ≤ T x0 = x1 and T is nondecreasing, we have T x0 ≤ T x1 , that is, x1 ≤ x2 . By induction, we get that x0 ≤ x1 ≤ · · · ≤ xn ≤ · · · , for all n ∈ N . Without loss of generality, assume that xn+1 6= xn , for all n ∈ N (otherwise, xn+1 = T xn = xn , then the conclusion holds). Since xn ∈ Ain and xn+1 ∈ Ain +1 are comparable, for in ∈ {1, 2, . . . , m}, by inequality (2.1), we get h[g(Fxn+1 ,xn (t))] ≤ h[Mt (xn , xn−1 )] − φ(Mt (xn , xn−1 )),

for all t > 0,

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(2.2)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

where 1 Mt (xn , xn−1 ) = max{g(Fxn ,xn−1 (t)), g(Fxn ,xn−1 (t)), g(Fxn ,xn+1 (t)), g(Fxn−1 ,xn+1 (t))} 2 1 ≤ max{g(Fxn ,xn−1 (t)), g(Fxn ,xn+1 (t)), g(∆(Fxn−1 ,xn (t), Fxn ,xn+1 (t)))} 2 1 ≤ max{g(Fxn ,xn−1 (t)), g(Fxn ,xn+1 (t)), [g((Fxn−1 ,xn (t)) + g(Fxn ,xn+1 (t))]} 2 = max{g(Fxn ,xn−1 (t)), g(Fxn ,xn+1 (t))} = Mt (xn , xn−1 ). Suppose that Mt (xn , xn−1 ) = g(Fxn ,xn+1 (t)), by (2.2), we have h[g(Fxn+1 ,xn (t))] ≤ h[g(Fxn ,xn+1 (t))] − φ(g(Fxn ,xn+1 (t))),

for all t > 0,

which implies that φ(g(Fxn ,xn+1 (t))) = 0. Thus, g(Fxn ,xn+1 (t)) = 0, that is, Fxn ,xn+1 (t) = 1 for all t > 0. Then xn = xn+1 , which is in contradiction to xn 6= xn+1 , for any n ∈ N . Hence, Mt (xn , xn−1 ) = g(Fxn ,xn−1 (t)), it follows from (2.2) that h[g(Fxn+1 ,xn (t))] ≤ h[g(Fxn ,xn−1 (t))] − φ(g(Fxn ,xn−1 (t))) ≤ h[g(Fxn ,xn−1 (t))],

∀t > 0,

(2.3)

Since h is nondecreasing, it follows from (2.3) that {g(Fxn+1 ,xn (t))} is a decreasing sequence, for every t > 0. Hence, there exists rt ≥ 0 such that lim g(Fxn+1 ,xn (t)) = rt . n→∞

By using the continuities of h and φ, letting n → ∞ in (2.3), we get h(rt ) ≤ h(rt ) − φ(rt ), which implies that φ(rt ) = 0. Then rt = 0, that is, lim g(Fxn+1 ,xn (t)) = 0 and lim Fxn+1 ,xn (t) = 1, for all t > 0. n→∞

n→∞

In the sequel, we will prove that {xn } is Cauchy sequence. To prove this fact, we first prove the following claim. Claim: for every t > 0, ε > 0, there exists n0 ∈ N , such that p, q ≥ n0 with p − q ≡ 1 mod m then Fxp ,xq (t) > 1 − ε, that is, g(Fxp ,xq (t)) < g(1 − ε). In fact, suppose to the contrary, there exist t0 > 0 and ε0 > 0, such that for any n ∈ N , we can find p(n) > q(n) ≥ n with p(n) − q(n) ≡ 1 mod m satisfying Fxp(n) ,xq(n) (t0 ) ≤ 1 − ε0 , that is, g(Fxp(n) ,xq(n) (t0 )) ≥ g(1 − ε0 ). Now, we take n > 2m. Then corresponding to q(n) ≥ n, we can choose p(n) in such a way that it is the smallest integer with p(n) > q(n) satisfying p(n) − q(n) ≡ 1 mod m and g(Fxp(n) ,xq(n) (t0 )) ≥ g(1 − ε0 ). Therefore, g(Fxp(n)−m ,xq(n) (t0 )) < g(1 − ε0 ). Using the non-Archimedean Menger triangular inequality and Definition 1.5, we have g(1 − ε0 ) ≤ g(Fxq(n) ,xp(n) (t0 )) ≤ g(∆(Fxq(n) ,xq(n)+1 (t0 ), Fxq(n)+1 ,xp(n) (t0 ))) ≤ g(Fxq(n) ,xq(n)+1 (t0 )) + g(Fxq(n)+1 ,xp(n) (t0 )) ≤ g(Fxq(n) ,xq(n)+1 (t0 )) + g(Fxq(n)+1 ,xp(n)+1 (t0 )) + g(Fxp(n)+1 ,xp(n) (t0 )) ≤ 2g(Fxq(n) ,xq(n)+1 (t0 )) + g(Fxq(n) ,xp(n)+1 (t0 )) + g(Fxp(n)+1 ,xp(n) (t0 )) ≤ 2g(Fxq(n) ,xq(n)+1 (t0 )) + g(Fxq(n) ,xp(n) (t0 )) + 2g(Fxp(n)+1 ,xp(n) (t0 )) ≤ 2g(Fxq(n) ,xq(n)+1 (t0 )) + g(Fxq(n) ,xp(n)−m (t0 )) + g(Fxp(n)−m ,xp(n) (t0 )) + 2g(Fxp(n)+1 ,xp(n) (t0 )) ≤ 2g(Fxq(n) ,xq(n)+1 (t0 )) + g(1 − ε0 ) +

m X

g(Fxp(n)−i ,xp(n)−i+1 (t0 )) + 2g(Fxp(n)+1 ,xp(n) (t0 )).

i=1

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(2.4)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

Since lim g(Fxn+1 ,xn (t)) = 0 for all t > 0, letting n → ∞ in (2.4), we have n→∞

g(1 − ε0 ) = lim g(Fxq(n) ,xp(n) (t0 )) = lim g(Fxq(n)+1 ,xp(n) (t0 )) n→∞

n→∞

(2.5)

= lim g(Fxq(n)+1 ,xp(n)+1 (t0 )) = lim g(Fxq(n) ,xp(n)+1 (t0 )). n→∞

n→∞

By p(n) − q(n) ≡ 1 mod m, we know that xp(n) and xq(n) lie in different adjacently labeled sets Ai and Ai+1 , for 1 ≤ i ≤ m. Using the fact that T is a generalized cyclic weak contraction, we have h[g(Fxq(n)+1 ,xp(n)+1 (t0 ))] = h[g(FT xq(n) ,T xp(n) (t0 ))] ≤ h[Mt0 (xq(n) , xp(n) )] − φ(Mt0 (xq(n) , xp(n) )),

(2.6)

where Mt0 (xq(n) , xp(n) ) = max{g(Fxq(n) ,xp(n) (t0 )), g(Fxq(n) ,xq(n)+1 (t0 )), g(Fxp(n) ,xp(n)+1 (t0 )), 1 [g(Fxq(n) ,xp(n)+1 (t0 )) + g(Fxp(n) ,xq(n)+1 (t0 ))]}. 2 By (2.5), we have lim Mt0 (xq(n) , xp(n) ) = max{g(1 − ε0 ), 0, 0, 21 [g(1 − ε0 ) + g(1 − ε0 )]} = g(1 − ε0 ). According n→∞

to the continuities of h and φ, letting n → ∞ in (2.6), we get h[g(1 − ε0 )] ≤ h[g(1 − ε0 )] − φ(g(1 − ε0 )). Thus, φ(g(1 − ε)) = 0, that is g(1 − ε0 ) = 0. Then ε0 = 0, which is in contradiction to ε0 > 0. Therefore, our claim is proved. In the sequel, we will prove that {xn } is Cauchy sequence. By the continuity of g and g(1) = 0, we have lim g(1 − a) = 0, for any given ε > 0. Since g is strictly a→0+

decreasing, then there exists a > 0 such that g(1 − aε) ≤

g(1−ε) . 2

For any given t > 0, ε > 0, there exists a > 0 such that g(1 − aε) ≤

g(1−ε) . 2

By the claim, we find n0 ∈ N

such that if p, q ≥ n0 with p − q ≡ 1 mod m, then Fxp ,xq (t) > 1 − aε,

and

g(Fxp ,xq (t)) < g(1 − aε) ≤

g(1 − ε) . 2

(2.7)

Since lim g(Fxn+1 ,xn (t)) = 0, we also find n1 ∈ N such that ree n→∞

g(Fxn+1 ,xn (t)) ≤

g(1 − ε) , 2m

(2.8)

for any n > n1 . Suppose that r, s ≥ max{n0 , n1 } and s > r. Then there exists k ∈ {1, 2, . . . , m} such that s − r ≡ k mod m. Therefore, s − r + j ≡ 1 mod m, for j = m − k + 1, j ∈ {0, 1, . . . , m − 1}. So, we have g(Fxr ,xs (t)) ≤ g(Fxr ,xs+j (t)) + g(Fxs+j ,xs+j−1 (t)) + · · · + g(Fxs+1 ,xs (t)). From (2.7), (2.8) and the last inequality, we get g(Fxr ,xs (t))
1 − ε. Therefore {xn } is Cauchy sequence. 6 1078

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Since X is a complete PM-space, Y = ∪m i=0 Ai is closed, then Y also is a complete space. Thus there exists x∗ ∈ Y such that xn → x∗ . As Y = ∪m i=1 Ai is a cyclic representation of Y with respect to T , then the sequence {xn } has infinite terms in each Ai for i ∈ {1, 2, . . . , m}. First, suppose that x∗ ∈ Ai , then T x∗ ∈ Ai+1 , and we take a subsequence {xnk } of {xn } with xnk ∈ Ai−1 (the existence of this subsequence is guaranteed by above mentioned comment). Case (a): If T is continuous. Since lim xn = x∗ , we have T x∗ = x∗ . n→∞

Case (b): If it satisfies a nondecreasing sequence xn → x∗ , such that xn ≤ x∗ , then xnk ∈ Ai−1 and x∗ ∈ Ai are comparable. By (2.1), we have h[g(Fxnk +1 ,T x∗ (t))] = h[g(FT xnk ,T x∗ (t))] ≤ h[Mt (xnk , x∗ )] − φ(Mt (xnk , x∗ )),

(2.10)

where Mt (xnk , x∗ ) = max{g(Fxnk ,x∗ (t)), g(Fxnk ,xnk+1 (t)), g(Fx∗ ,T x∗ (t)), 1 [g(Fxnk ,T x∗ (t)) + g(Fxnk+1 ,x∗ (t))]}. 2 Let G0 be the set of all the discontinuous points of Fx∗ ,T x∗ (t). Since g, h, and φ are continuous, we obtain that G0 also is the set of all the discontinuous points of g(Fx∗ ,T x∗ (t)), h[g(Fx∗ ,T x∗ (t))] and φ(g(Fx∗ ,T x∗ (t))). Moreover, we know that G0 is a countable set. Let G = R+ \G0 . When t ∈ G\{0} (t is a continuity point of Fx∗ ,T x∗ (t)), we have 1 lim Mt (xnk , x∗ ) = max{0, 0, g(Fx∗ ,T x∗ (t)), [g(Fx∗ ,T x∗ (t)) + 0]} = g(Fx∗ ,T x∗ (t)). k→∞ 2 Letting n → ∞ in (2.10), we get h[g(Fx∗ ,T x∗ (t))] ≤ h[g(Fx∗ ,T x∗ (t))] − φ(g(Fx∗ ,T x∗ (t))). Thus, φ(g(Fx∗ ,T x∗ (t))) = 0, that is, g(Fx∗ ,T x∗ (t)) = 0. Then Fx∗ ,T x∗ (t) = H(t),

for all t ∈ G.

(2.11)

When t ∈ G0 with t > 0, by the density of real numbers, there exist t1 , t2 ∈ G such that 0 < t1 < t < t2 . Since the distribution is nondecreasing, we have 1 = H(t1 ) = Fx∗ ,T x∗ (t1 ) ≤ Fx∗ ,T x∗ (t) ≤ Fx∗ ,T x∗ (t2 ) = 1. This shows that, for all t ∈ G0 with t > 0, Fx∗ ,T x∗ (t) = H(t).

(2.12)

Combing (2.11) with (2.12), we have Fx∗ ,T x∗ (t) = H(t), for all t > 0, that is, T x∗ = x∗ . Hence, in all case, we have T x∗ = x∗ . Finally, we prove the uniqueness of the fixed point under the additional conditions. In fact, suppose that there exist x∗ , y ∗ ∈ Y such that T x∗ = x∗ , T y ∗ = y ∗ , then we have x∗ , y ∗ ∈ ∩m i=1 Ai . 7 1079

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Since the set of fixed points of T is well ordered, we have x∗ ∈ Ai and y ∗ ∈ Ai+1 are comparable. By (2, 1), we have h[g(Fx∗ ,y∗ (t))] ≤ h[Mt (x∗ , y ∗ )] − φ(Mt (x∗ , y ∗ )),

for all t > 0,

where 1 Mt (x∗ , y ∗ ) = max{g(Fx∗ ,y∗ (t)), g(Fx∗ ,x∗ (t)), g(Fy∗ ,y∗ (t)), [g(Fx∗ ,y∗ (t)) + g(Fx∗ ,y∗ (t))]} = g(Fx∗ ,y∗ (t)). 2 Thus, φ(g(Fx∗ ,y∗ (t))) = 0, that is, g(Fx∗ ,y∗ (t)) = 0. Hence, Fx∗ ,y∗ (t) = 1, for all t > 0. Then x∗ = y ∗ . Remark 2.1 Theorem 2.1 generalizes and extends Theorem 2.1 in [6] and Theorem 2.4 in [7]. Corollary 2.1 Let (X, ≤) be a partially ordered set and (X, F, ∆) be a complete N.A Menger PM-space, T : X → X be a nondecreasing mapping. Suppose that for comparable x, y ∈ X, we have 1 g(FT x,T y (t)) ≤ Φ(max{g(Fx,y (t)), g(Fx,T x (t)), g(Fy,T y (t)), [g(Fx,T y (t)) + g(Fy,T x (t))]}), 2 for all t > 0, where Φ : [0, ∞) → [0, ∞) is a continuous function, Φ(t) < t, for t > 0 and Φ(0) = 0. Also assume that either (a) T is continuous or, (b) if a nondecreasing sequence xn → x, then xn ≤ x, for all n ∈ N . If there exists x0 ∈ X such that x0 ≤ T x0 , then T has a fixed point. Furthermore, the set of fixed points of T is well ordered if and only if T has a unique fixed point. Proof. Taking h(x) = x and Φ(t) = t − φ(t) in Theorem 2.1, we can easily obtain the above corollary. Corollary 2.2 Let (X, ≤) be a partially ordered set and (X, F, ∆) be a complete N.A Menger PM-space, T : X → X be a nondecreasing mapping. Suppose that for comparable x, y ∈ X, we have 1 FT x,T y (t) ≥ ψ(min{Fx,y (t), Fx,T x (t), Fy,T y (t), [Fx,T y (t) + Fy,T x (t)]}), 2

for all t > 0,

where ϕ : [0, 1] → [0, 1] is a continuous function, t < ψ(t) < 1 for t ∈ [0, 1), ψ(t) = 1 if and only if t = 1. Also assume that either (a) T is continuous or, (b) if a nondecreasing sequence xn → x, then xn ≤ x, for all n ∈ N . If there exists x0 ∈ X such that x0 ≤ T x0 , then T has a fixed point. Furthermore, the set of fixed points of T is well ordered if and only if T has a unique fixed point. Proof. Taking h(x) = x and g(t) = 1 − t, ψ(t) = t + φ(1 − t) in Theorem 2.1, we can easily obtain the above corollary. Remark 2.2 Corollary 2.2 generalizes and extends Theorem 2.1 in [22]. Now, we give an example to demonstrate Theorem 2.1. Example 2.1 . Let X = R+ , ∆1 (a, b) = max{a + b − 1, 0}, F be defined by   t ≤ 0,   0, min{x,y} Fx,y (t) = 0 < t ≤ 1, max{x,y} ,    1, t > 1. 8 1080

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for all x, y ∈ X. Then, for every given x, y ∈ X, it is easy to verify that Fx,y is a distribution function and (X, F, ∆) is a complete N.A Menger PM-space. In fact, (M S − 1) and (M S − 2) are easy to check. To prove inequality (1.1). We consider the case: Case 1. If t1 > 1 or t2 > 1, then Fx,z (max{t1 , t2 }) ≥ ∆1 (Fx,y (t1 ), Fy,z (t2 )), for any x, y, z ∈ X. Case 2. If 0 < t1 , t2 ≤ 1 and x ≤ y ≤ z, for x, y, z ∈ R+ , then Fx,z (max{t1 , t2 }) − ∆1 (Fx,y (t1 ), Fy,z (t2 )) =

x y (y − x)(z − y) x +1−( + )= ≥ 0. z y z yz

Case 3. If 0 < t1 , t2 ≤ 1 and y ≤ x ≤ z, for x, y, z ∈ R+ , then Fx,z (max{t1 , t2 }) − ∆1 (Fx,y (t1 ), Fy,z (t2 )) =

x y y (x + z)(x − y) +1−( + )= ≥ 0. z x z xz

Case 4. If 0 < t1 , t2 ≤ 1 and x ≤ z ≤ y, for x, y, z ∈ R+ , then Fx,z (max{t1 , t2 }) − ∆1 (Fx,y (t1 ), Fy,z (t2 )) =

x x z (x + z)(y − z) +1−( + )= ≥ 0. z y y yz

Hence, in all case, we have Fx,z (max{t1 , t2 }) ≥ ∆1 (Fx,y (t1 ), Fy,z (t2 )), for all t1 , t2 ∈ R+ , that is, 1.1 holds. Suppose that A1 = [0, 1], A2 = [ 21 , 1], A3 = [ 34 , 1], and Y =

S3

i=1

Ai . Let f : Y → Y and f x =

1 2

+ 21 x, for

all x ∈ Y , Clearly Y =

S3

i=1

Ai is a cyclic representation of Y with respect to f .

We next prove that it satisfies the conditions of Theorem 2.1, where h(x) = 21 x, φ(x) = 16 x, and g(t) = 1 − t. By the definitions of F , g, h and φ, we only need to prove that 1 2 1 Ff x,f y (t) ≥ Qt (x, y) + (1 − Qt (x, y)) = Qt (x, y) + , 3 3 3

(2.13)

where Qt (x, y) = min{Fx,y (t), Fx,T x (t), Fy,T y (t), 21 [Fx,T y (t) + Fy,T x (t)]}. Since f x =

1 2

+ 12 x. If 0 ≤ x ≤ y, for x, y ∈ [0, 1], then we have

1 x Qt (x, y) = min{Fx,y (t), Fx,T x (t), Fy,T y (t), [Fx,T y (t) + Fy,T x (t)]} ≤ Fx,y (t) = . 2 y Hence, we consider the following two cases: Case 1. If 0 < t ≤ 1, we have 2 1 x + 1 2x 1 (2 − y)(y − x) Ff x,f y (t) − Qt (x, y) − ≥ − − = ≥ 0, 3 3 y+1 3y 3 3(y + 1)y which implies that (2.13) holds. Case 2. If t > 1, by the definition of F , we have 2 1 Ff x,f y (t) − Qt (x, y) − = 0, 3 3 which implies that (2.13) holds. Hence, in all case, we obtain that (2.13) holds. Thus, all hypotheses of Theorem 2.1 are satisfied, and we deduce that f has a unique fixed point in Y . Here, x = 1 is the unique fixed point of f . 9 1081

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3

Coupled fixed point theorems in partially ordered probabilistic metric spaces In the section, we will apply the Corollary 2.2 in the Section 2 to prove the coupled fixed point theorems

under contractive conditions in partially ordered probabilistic metric spaces. Lemma 3.1 If (X, F, ∆) is a N.A Menger PM space, then (X 2 , F˜ , ∆) is also a N.A Menger PM space. Proof. It is sufficient to prove that, for a ˜ = (x, y), ˜b = (u, v), c˜ = (p, q) ∈ X 2 , F˜a˜,˜c (max{t1 , t2 }) ≥ ∆(F˜a˜,˜b (t1 ), F˜˜b,˜c (t2 )), for all t1 , t2 ≥ 0. In fact, for all a ˜ = (x, y), ˜b = (u, v), c˜ = (p, q) ∈ X 2 and t1 , t2 ≥ 0 we have F˜a˜,˜c (max{t1 , t2 }) = min{Fx,p (max{t1 , t2 }), Fy,q (max{t1 , t2 })} ≥ min{∆(Fx,u (t1 ), Fu,p (t2 )), ∆(Fy,v (t1 ), Fv,q (t2 ))} ≥ ∆(min{Fx,u (t1 ), Fy,v (t1 )}, min{Fu,p (t2 ), Fv,q (t2 )}) = ∆(F˜a˜,˜b (t1 ), F˜˜b,˜c (t2 )). The proof is complete. Theorem 3.1 Let (X, ≤) be a partially ordered set and (X, F, ∆) be a complete N.A Menger PM-space, A : X ×X → X be a mapping satisfying the mixed monotone property on X. Suppose that for all x, y, u, v ∈ X, x ≤ u and v ≤ y, we have FA(x,y),A(u,v) (t) ≥ ψ(min{Fx,u (t), Fy,v (t), Fx,A(x,y) (t), Fu,A(u,v) (t), Fy,A(y,x) (t), Fv,A(v,u) (t), 1 [min{Fx,A(u,v) (t), Fy,A(v,u) (t)} + min{Fu,A(x,y) (t), Fv,A(y,x) (t)}]}), 2 for all t > 0, where ψ : [0, 1] → [0, 1] is a continuous function, t < ψ(t) < 1 for t ∈ [0, 1), ψ(t) = 1 if and only if t = 1. Also assume that either (a) A is continuous or, (b) if a nondecreasing sequence xn → x, then xn ≤ x, for all n ∈ N ; If a nonincreasing sequence xn → x, then y ≤ yn , for all n ∈ N . If there exist x0 , y0 ∈ X such that x0 ≤ A(x0 , y0 ) and A(y0 , x0 ) ≤ y0 , then A has a coupled fixed point, that is, there exist p, q ∈ X such that A(p, q) = p and A(q, p) = q. ˜ = X × X, for a ˜ we introduce the order  as Proof. Let X ˜ = (x, y), ˜b = (u, v) ∈ X, a ˜  ˜b if and only if x ≤ u, v ≤ y. It follows from Lemma 1.2 and Lemma 3.1 that (X, , F˜ , ∆) is also a complete partially ordered N.A Menger PM-space, where F˜a˜,˜b (t) = min{Fx,u (t), Fy,v (t)}. ˜ →X ˜ is given by The self-mapping T : X ˜ Ta ˜ = (A(x, y), A(y, x)) for all a ˜ = (x, y) ∈ X. 10 1082

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Then a coupled point of A is a fixed point of T and vice versa. If a ˜  ˜b, then x ≤ u and v ≤ y. Noting the mixed monotone property of A, we see that A(x, y) ≤ A(u, v) ˜ and A(v, u) ≤ A(y, x), then T a ˜  T ˜b. Thus T is a nondecreasing mapping with respect to the order  on X. ˜ with a On the other hand, for all t > 0 and a ˜ = (x, y), ˜b = (u, v) ∈ X ˜  ˜b, we have FA(x,y),A(u,v) (t) ≥ ψ(min{Fx,u (t), Fy,v (t), Fx,A(x,y) (t), Fu,A(u,v) (t), Fy,A(y,x) (t), Fv,A(v,u) (t) 1 [min{Fx,A(u,v) (t), Fy,A(v,u) (t)} + min{Fu,A(x,y) (t), Fv,A(y,x) (t)}) 2 = ψ(min{min{Fx,u (t), Fy,v (t)}, min{Fx,A(x,y) (t), Fy,A(y,x) (t)}, min{Fu,A(u,v) (t), Fv,A(v,u) (t)}, 1 [min{Fx,A(u,v) (t), Fy,A(v,u) (t)} + min{Fu,A(x,y) (t), Fv,A(y,x) (t)}) 2 1 = ψ(min{F˜a˜,˜b (t), F˜a˜,T a˜ (t), F˜b,T ˜b (t), [F˜a˜,T ˜b (t) + F˜T a˜,˜b (t)]}) 2 Similarly, FA(y,x),A(v,u) (t) ≥ ψ(min{F˜a˜,˜b (t), F˜a˜,T a˜ (t), F˜b,T ˜b (t), 12 [F˜a˜,T ˜b (t) + F˜T a˜,˜b (t)]}). Thus, 1 FT a˜,T ˜b (t) ≥ ψ(min{F˜a˜,˜b (t), F˜a˜,T a˜ (t), F˜b,T ˜b (t), [F˜a˜,T ˜b (t) + F˜T a˜,˜b (t)]}). 2 ˜ such that x Also, there exists an x ˜0 = (x0 , y0 ) ∈ X ˜0  T x ˜0 = (A(x0 , y0 ), A(y0 , x0 )). ˜ tends to x If a nondecreasing monotone sequence {˜ xn } = {(xn , yn )} in X ˜ = (x, y), then x ˜n = (xn , yn )  (xn+1 , yn+1 ) = x ˜n+1 , that is, xn ≤ xn+1 and yn+1 ≤ yn . Thus {xn } is nondecreasing sequence tending to x and {yn } a nonincreasing sequence tending to y. Thus xn ≤ x and y ≤ yn for all n ∈ N . This implies x ˜n  x. Obviously, the continuity of A implies the continuity of T . Therefore, all hypotheses of Corollary 2.2 are satisfied. Following Corollary 2.2, we deduce that A has a ˜ such that A(p, q) = p and A(q, p) = q. coupled point, that is, there exist p, q ∈ X Remark 3.1 Theorem 3.1 generalizes and extends Theorem 71 in [21] and Corollary 2.1 in [22].

4

An application In this section, using the Theorem 2.1, we establish some fixed results for generalized weak contractions in

partially ordered metric spaces. Theorem 4.1 Let (X, d, ≤) be an ordered complete metric space, T : X → X be a nondecreasing mapping. Suppose that for comparable x, y ∈ X, we have d(T x, T y) ≤ M (x, y) − ϕ(M (x, y)),

∀t > 0,

(4.1)

where M (x, y) = max{d(x, y), d(x, T x), d(y, T y), 21 [d(x, T y) + d(y, T x)]}, ϕ : [0, ∞) → [0, ∞) is a continuous function,

ϕ(s) t

≥ ϕ( st ), for all t > 0, and ϕ(s) = 0 if and only if s = 0. Also assume that either

(a) T is continuous or, (b) if a nondecreasing sequence xn → x, then xn ≤ x, for all n ∈ N . If there exists x0 ∈ X such that x0 ≤ T x0 , then T has a fixed point. Furthermore, the set of fixed points of T is well ordered if and only if T has a unique fixed point. 11 1083

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Proof. Let (X, F, ∆2 ) be the induced N.A Menger PM-space, where F is defined by Fx,y (t) = e−

d(x,y) t

, for

t > 0, x, y ∈ X. We can easily prove that a sequence {xn } in X converges in the metric d to a point x∗ ∈ X if and if only {xn } in (X, F, ∆2 ) τ -converges to x∗ . Let g ∈ Ω, where g(t) = 1 − t. Since (X, d) is a complete metric space, then (X, F, ∆2 ) is a τ -complete N.A Menger PM-space of type (D)g . For x, y ∈ X, x and y are comparable, by (4.1), for t > 0, we have 1 − e−

d(T x,T y) t

≤ 1 − e−

M (x,y) ϕ(M (x,y)) + t t

≤ 1 − e−

M (x,y) M (x,y) +ϕ( ) t t

=1−e

M (x,y) − t

−e

Let φ : [0, 1) → [0, +∞), where φ(u) = [1 − u][eϕ(ln −1

ϕ

(0) = 0, then φ also is continuous and φ

Since φ(1 − e−

M (x,y) t

) = e−

M (x,y) t

[eϕ(

−1

M (x,y) ) t

M (x,y) − t

1 1−u

)

(4.2) [e

M (x,y) ϕ( ) t

− 1].

− 1], for u ∈ [0, 1]. Since ϕ is continuous and

(0) = 0. − 1], g(s) = 1 − s, and Fx,y (t) = e−

d(x,y) t

, by (4.2), we get

g(FT x,T y (t)) ≤ Mt (x, y) − φ(Mt (x, y)), for t > 0, where Mt (x, y) = max{g(Fx,y (t)), g(Fx,T x (t)), g(Fy,T y (t)), 21 [g(Fx,T y (t)) + g(Fy,T x (t))]}. Thus, all hypotheses of Theorem 2.1 are satisfied, when h(s) = s and m = 1. Then the conclusion holds.

References [1] W.A. Kirk, P.S. Srinivasan and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, 79-89 (2003). [2] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132, 1435-1443 (2004). [3] R.P. Agarwal, M.A. El-Gebeily and D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87, 1-8 (2008). [4] J. Harjani and K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71, 3403-3410 (2009). [5] J. Esmaily, S.M. Vaezpour, B.E. Rhoades, Coincidence point theorem for generalized weakly contractions in ordered metric spaces, Appl. Math. Comput. 219, 5684-5692 (2013). [6] J. Harjani, B. Lopez and K. Sadarangani, Fixed point theorems for weakly C-contractive mappings in ordered metric spaces, Comput. Math. Appl. 61, 790-796 (2011). [7] H.K. Nashine and B. Samet, Fixed point results for mappings satisfying (ψ, ϕ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal. 74, 2201-2209 (2011). [8] J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 74, 768-774 (2011). 12 1084

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[9] M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30(1), 1-9 (1984). [10] T.G Bhaskar and V. Lakshmikantham, Fixed point theory in partially ordered metric spaces and applications, Nonlinear Anal. 65, 1379-1393 (2006). [11] M. Abbas, T. Nazir and S. Radenovi´c, Common fixed points of four maps in partially ordered metric spaces, Appl. Math. Lett. 24, 1520-1526 (2011). [12] W. Shatanawi and M. Postolache, Common fixed point results for mappings under nonlinear contraction of cyclic form in ordered metric spaces, Fixed Point Theory Appl. 2013, 60 (2013). [13] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA. 28, 535-537 (1942). [14] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier/North-Holland, New York, 1983. [15] S.S. Chang, Y.J. Cho and S.M. Kang, Probabilistic Metric Spaces and Nonlinear Operator Theory. Chengdu: Si chuan University Press. 1994. [16] J. Jachymski, On probilistic ϕ−contraction on Menger spaces. Nonlinear Anal. 73, 2199-2203 (2010). [17] C.X. Zhu and J.D. Yin, Calculations of a random fixed point index of a random sem-cloosed 1-setcontractive operator. Math. Comput. Model. 51, 1135-1139 (2010). [18] C.X. Zhu, Research on some problems for nonlinear operators. Nonlinear Anal. 71, 4568-4571 (2009). [19] C.X. Zhu, Several nonlinear operator problems in the Menger PN space. Nonlinear Anal. 65, 1281-1284 (2006). [20] X.Q. Hu and X.Y. Ma, Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces, Nonlinear Anal. 74, 6451-6458 (2011). ´ c, R.P. Agarwal and B. Samet, Mixed monotone generalized contractions in partially ordered [21] Lj. B. Ciri´ probabilistic metric spaces, Fixed Point Theory Appl. 2011, 56 (2011). [22] J. Wu, Some fixed-point theorems for mixed monotone operators in partially ordered probabilistic metric spaces, Fixed Point Theory Appl. 2014, 49 (2014).

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Weak Galerkin finite element method for time dependent reaction-diffusion equation Fuzheng Gao1 School of Materials Science and Engineering, Shandong University School of Mathematics, Shandong University, Jinan, Shandong 250100, China

Guoqun Zhao2 School of Materials Science and Engineering, Shandong University, Jinan, Shandong 250061, China

Abstract We propose a weak Galerkin finite element procedure for time dependent reaction-diffusion equation by using weakly defined gradient operators over discontinuous functions with heterogeneous properties, in which the classical gradient operator is replaced by the discrete weak gradient. Numerical analysis and numerical experiments illustrate and confirm that our new method has effective numerical performances. Mathematics subject classifications: 65M15, 65M60. Keywords: Galerkin finite element methods, parabolic equation, weak gradient, error estimate, numerical experiment. 1. Introduction. Time dependent reaction-diffusion equations are a large important class of equations. In this paper, we consider the following time dependent reactiondiffusion equation: ut + Au = f (x, t), x ∈ Ω, 0 < t ≤ T, u = u0 (x), x ∈ Ω, t = 0, 1 2

(1a) (1b)

Corresponding author, e-mail: [email protected] (Fuzheng Gao). e-mail:[email protected].

Preprint submitted to JOCAA

April 28, 2015

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with homogenous Dirichlet boundary condition, where Ω is a bounded region ∂u in R2 , with a Lipschitz continuous boundary; ut = ; and A is a second ∂t order elliptic differential operator: Au ≡ −∇ · (a∇u) + cu, where a and c are sufficiently smooth functions of x and satisfy 0 < a∗ ≤ a(x) ≤ a∗ and c(x) ≥ 0 for fixed a∗ , a∗ . We define the following bilinear form Z a(u, v) := (a∇u · ∇v + cuv)dx. (2) Ω

It is obvious that there is a constant α0 > 0 such that a(u, u) ≥ α0 kuk21 , ∀u ∈ H01 (Ω).

(3)

The variational weak form to (1) is: find u = u(x, t) ∈ L2 (0, T ; H01(Ω)), such that (ut , v) + a(u, v) = (f, v), ∀v ∈ H01 (Ω), t > 0, u(x, 0) = u0 (x), x ∈ Ω,

(4a) (4b)

where (·, ·) denotes the inner product of L2 (Ω). Many numerical methods for solving such problems have been developed, please see [3, 6, 7, 11, 12, 13, 16] and references in. In [5], a weak Galerkin finite element method (WG-FEM) was introduced and analyzed for parabolic equation based on a discrete weak gradient arising from local Raviart-Thomas (RT ) elements [10]. Due to the use of RT elements, the WG finite element formulation of [5] was limited to finite element partitions of triangles for two dimensional problem. To overcome this, we presented a WG-FEM in [4] with a stabilization term for a diffusion equation without reaction term and derived optimal convergence rate in L2 norm based on a dual argument technique for the solution of the WG-FEM. The WG-FEM was first introduced in [14] for solving second order elliptic problems. Later, the WG-FEMs were studied from implementation point of view in [8] and applied to solve the Helmholtz problem with high wave numbers in [9]. The purpose of this paper is to present a weak Galerkin (WG) finite element procedures using more flexible elements in arbitrary unstructured meshes for time dependent reaction-diffusion problem, and derive optimal convergence rate in the H 1 norm. 2

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The outline of this article is as follows. In Section 3, we define the weak gradient and present semi-discrete and fully-discrete WG-FEMs for problem (1). In Section 4, we establish the optimal order error estimates in H 1 -norm to the WG-FEMs for the parabolic problem. Finally in Section 5 we give some numerical examples to verify the theory. Throughout this paper, the notations of standard Sobolev spaces L2 (Ω), H k (Ω) and associated norms k · k = k · kL2 (Ω) , k · kk = k · kH k (Ω) are adopted. 2. A weak gradient operator and its discrete approximation Let T be any polygonal domain with interior T 0 and boundary ∂T . A weak function on the region T refers to a function v = {v0 , vb } such that 1 v0 ∈ L2 (T ) and vb ∈ H 2 (∂T ). v0 represents the value of v on T 0 and vb represents that of v on ∂T . Note that vb may not necessarily be related to the trace of v0 on ∂T . Denote by W (T ) the space of weak function associated with T ; i.e., n o 1 2 2 (5) W (T ) = v = {v0 , vb } : v0 ∈ L (T ), vb ∈ H (∂T ) .

Definition 2.1. [14] The dual of L2 (T ) can be identified with itself by using the standard L2 inner product as action of linear functional. With a similar interpretation, for any v ∈ W (T ), the weak gradient of v is defined as a linear functional ∇w v in the dual space of H(div, T ) whose action on each q ∈ H(div, T ) is given by Z Z (∇w v, q)T := − v0 ∇ · qdT + vb q · nds, (6) T

∂T

where n is the outer normal direction to ∂T .

Next, we introduce a discrete weak gradient operator by defining ∇w in a polynomial subspace of H(div, T ). To this end, for any non-negative integer r ≥ 0, denote by Pr (T ) the set of polynomials on T with degree no more than r. Let V (K, r) ⊂ [Pr (T )]2 be a subspace of the space of vectorvalued polynomials of degree r. A discrete weak gradient operator, denoted by ∇w,r , is defined so that ∇w,r v ∈ V (T, r) is the unique solution of the following equation Z Z (∇w,r v, q)T := − v0 ∇ · qdT + vb q · nds, ∀q ∈ V (T, r). (7) T

∂T

It is easy to know that ∇w,r is a Galerkin-type approximation of the weak gradient operator ∇w by using the polynomial space V (T, r). 3

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3. Weak Galerkin finite element methods Let Th be a regular finite element grid on Ω with mesh size h. Assume that the partition Th is shape regular so that the routine inverse inequality in the finite element analysis holds true (see [2]). In the general spirit of the Galerkin procedure, we shall design a weak Galerkin finite element method for (4) by the following two basic principles: (1) replace H 1 (Ω) by a space of discrete weak functions defined on the finite element partition Th and the boundary of triangular elements; and (2) replace the classical gradient operator by a discrete weak gradient operator ∇w for weak functions on each triangle T . For each T ∈ Th , denote by Pj (T 0 ) the set of polynomials on T 0 , which is the interior of triangle T , with degree no more than j, and Pl (∂T ) the set of polynomials on ∂T with degree no more than l (i.e., polynomials of degree l on each line segment of ∂T ). A discrete weak function v = {v0 , vb } on T refers to a weak function v = {v0 , vb } such that v0 ∈ Pj (T 0 ) and vb ∈ Pl (∂T ) with j ≥ 0 and l ≥ 0. Denote this space by W (T, j, l), i.e., W (T, j, l) := {v = {v0 , vb } : v0 ∈ Pj (T 0 ), vb ∈ Pl (∂T )}. The corresponding FE space would be defined by matching W (T, j, l) over all the triangles T ∈ Th as Vh := {v = {v0 , vb } : {v0 , vb }|T ∈ W (T, j, l), ∀T ∈ Th }.

(8)

Denote by Vh0 the subspace of Vh with zero boundary values on ∂Ω; i.e., Vh0 := {v = {v0 , vb } ∈ Vh , vb |∂T T ∂Ω = 0, ∀T ∈ Th }.

(9)

According to (7), for each v = {v0 , vb } ∈ Vh0 , the discrete weak gradient ∇w,r v of v on each element T is given by the following equation: Z Z Z ∇w,r v · qdx = − v0 ∇ · qdx + vb q · nds, ∀q ∈ V (T, r). (10) T

T

∂T

For simplicity of notation, we shall drop the subscript r in the discrete weak gradient operator ∇w,r from now on. Now, we define the semi-discrete weak Galerkin finite element scheme for (1) as: find uh = {u0 , ub }(·, t) ∈ Vh0 (0 ≤ t ≤ T ) such that (uh,t, v) + aw (uh , v) = (f, v0 ),∀ v = {v0 , vb } ∈ Vh0 , t > 0, uh (x, 0) = Qh u0(x), x ∈ Ω,

(11a) (11b)

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where the bilinear form aw (·, ·) is defined as P R aw (v, w) = (a∇w v · ∇w w + cv0 w0 )dx T T ∈Th P −1 + hT < v0 − vb , w0 − wb >∂T ,

(12)

T ∈Th

and Qh u = {Q0 u, Qb u} is the L2 projection onto Pj (T 0 ) × Pl (∂T ). In other words, on each element T , the function Q0 u is defined as the L2 projection of u on Pj (T ) and on ∂T , Qb u is the L2 projection in Pl (∂T ). Hereafter, we choose l = j. Let {ϕi (x) : i = 1, 2, · · · , N}, where N = dim(Vh0 ), be the bases of Vh0 . For example, when j = 0 in Pj (T ), ϕi is a function which takes value one in the interior of triangle T of Th and zero everywhere else; and ϕi is a function that takes value one on the edge e ∈ ∂T and zero everywhere else. Then (11) can be expressed as: find a solution of the form uh = {u0 , ub} =

N X

µj (t)ϕi (x),

i=1

such that its coefficients µ1 (t), µ2 (t), · · · , µN satisfy N X dµi (t) (ϕi , ϕj ) + µi aw (ϕi , ϕj )] = (f, ϕj ), [ dt i=1

t > 0.

(13)

By Introducing the following matrix and vector notations: M = [mij ] = [(ϕi , ϕj )],

K = [kij ] = [aw (ϕi , ϕj )],

µ = [µ1 , µ2, · · · , µN ]T , F = [(f, ϕ1 ), (f, ϕ2 ), · · · , (f, ϕN )]T , then (13) can be rewritten as dµ + Kµ = F. (14) dt M and K are positive definite matrix. The ordinary differential equation (ODE) theory tells us that the semi-discrete WG scheme has a unique solution for any f ∈ L2 (Ω). Define a norm |k · k|w,1 as sX 2 (k∇w vk20,T + kvk20,T + h−1 |kvk|w,1 := (15) T kv0 − vb k0,∂T ), M

T ∈Th

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which is a H 1 -equivalent norm for conventional finite element functions, since the presence of the L2 (T ) term renders the norm to be an equivalent H 1 norm for any H 1 function, regardless the value of their zeroth order traces on ∂T ; R R where kvk20,T = T v 2 dx and kv0 − vb k20,∂T = ∂T (v0 − vb )2 ds. Moreover, the following P oincar´ e inequality holds true for functions in Vh0 . Lemma 3.1. Assume that the finite element partition Th is shape regular. Then there exists a constant C independent of the mesh size h such that kvk ≤ |kvk|w,1, ∀ v = {v0 , vb } ∈ Vh0 .

(16)

Let us now return to our semi-discrete problem in the formulation (11). A basic stability inequality for problem (1) with f = 0, for simplicity, is as follows: Theorem 3.1. For the numerical solution to scheme (11) with initial setting (11b), there is a L2 -stability as follows Z d u2 (x, t)dx ≤ 0. (17) dt Ω h Proof. Taking v = uh in (11a), with f = 0, we get (uh,t (t), uh (t)) + aw (uh (t), uh (t)) = 0. From the definition of bilinear form aw (·, ·) in (12), we know that aw (uh (t), uh (t)) ≥ 0. Based on this fact, Z 1d 1d u2h (t)dx = (uh (t), uh (t)) = (uh,t (t), uh (t)) ≤ 0. 2 dt Ω 2 dt This completes the proof.  n Let τ denote the time step size, and tn = nτ (n = 0, 1, · · · ), uh := uh (tn ) = {un0 , unb }. At time t = tn , using backward difference quotient ∂¯t unh = (unh − un−1 h )/τ

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to approximate the differential quotient uh,t in the semi-discrete scheme (11), we get the fully-discrete WG-FE scheme: find unh = {un0 , unb } ∈ Vh0 for n = 1, 2, · · · , such that X X (∂¯t unh , vh )T + aw (unh , vh ) = (f n , v0 )T ,∀vh = {v0 , vb } ∈ Vh0 , (18a) T ∈Th u0h =

T ∈Th

0

Qh u (x).

(18b)

From (12), for v, w ∈ Vh0 , we get aw (v, v) ≥ α0 |kvk|2w,1, ∀v ∈ Vh0 , and aw (v, w) ≤ C ∗ |kvk|w,1k|wk|w,1, which guarantees the existence and uniqueness of the solution unh = {un0 , unb } to (18) for a given uhn−1 = {un−1 , un−1 }. 0 b 4. Error estimate In this section we will present a priori error estimates in H 1 -norm for the semi-discrete scheme (11) and fully-discrete scheme (18) for smooth solutions of (1). For simplicity, we assume that diffusion coefficient a is piecewise constant with respect to the finite element partition Th . The corresponding results can be extended to the case of variable coefficients provided that the coefficient function a is sufficiently smooth. Below we denote C (maybe with indicates) as a positive constant depending solely on the exact solution, which may have different values in each occurrence. 4.1. Preliminaries 4.1.1. Sobolev space definitions and notations Let Ω be any domain in R2 . In this paper, we adopt the standard definition for the Sobolev space W s,r (Ω), which consists of functions with (distributional) derivatives of order less than or equal to s in Lr (Ω) for 1 ≤ r ≤ +∞ and integer s. And their associated inner products (·, ·)s,r,Ω, norms k · ks,r,Ω , and seminorms | · |s,r,Ω. Further, k · k∞,Ω represents the norm on L∞ (Ω), and k · kL∞ ([0,T ];W s,r (Ω)) the norm on L∞ ([0, T ]; W s,r (Ω)). See Adams [1] for more details. 7

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4.1.2. Properties of finite element space In our analysis, we shall use two kinds of polynomial finite element spaces associated with each element T ∈ Th . one is a scalar polynomial space Pk (T ), in which the degree of polynomial is no more than k on T 0 and ∂T , and the other is the vector value polynomial space [Pk−1 (T )]2 which is used to define the discrete weak gradient ∇w in (10). For convenience, we denote [Pk−1(T )]d by Gk−1 (T ), which is called a local discrete gradient space. In addition, we define the local L2 -projection of the vector value function w(x) in this paper by Qh w(x). It is defined in each element T ∈ Th as the unique vector value function in Gk−1 (T ) such that Z Z Qh w(x) · q(x)dx = w(x) · q(x)dx, ∀q(x) ∈ Gk−1 (T ). (19) T

T

The following three lemmas are listed without any proof. Their proofs can be found in [14]. Lemma 4.1. Let Qh be the L2 projection operator. Then, on each element T ∈ Th , we have the following relation ∇w (Qh φ) = Qh (∇φ),

∀φ ∈ H 1 (Ω).

(20)

Lemma 4.2. Let T be an element with e ∈ ∂T is a portion of its boundary. For any function φ ∈ H 1 (T ), the following trace inequality is valid for general meshes (see [14] for details): 2 2 kφk2e ≤ C(h−1 T kφkT + hT k∇φkT ).

(21)

Lemma 4.3. Let Th be a finite element partition of domain Ω satisfying corresponding shape regularity assumptions as specified in [15]. Then, for any φ ∈ H k+1(Ω), we have X X kφ − Q0 φk2T + h2T k∇(φ − Q0 φ)k2T ≤ Ch2(k+1) kφk2k+1 . (22) T ∈Th

T ∈Th

X

ka(∇φ − Qh (∇φ))k2T ≤ Ch2k kφk2k+1 .

(23)

T ∈Th

Lemma 4.4. Assume that Th is shape regular. We have the following relation X k | h−1 (24) T < Q0 w − Qb w, v0 − vb >∂T | ≤ Ch kwkk+1 |kvk|w,1, T ∈Th

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and |

X

< a(∇w − Qh ∇w) · n, v0 − vb >∂T | ≤ Chk kwkk+1|kvk|w,1

(25)

T ∈Th

for ∀w ∈ H k+1(Ω) and v = {v0 , vb } ∈ Vh0 . 4.2. Error estimate for semi-discrete WG scheme In this section, we analyze semi-discrete WG scheme (11) first. Theorem 4.1. Let u(x, t) and uh (x, t) be the solutions to the problem (1) and the semi-discrete WG scheme (11), respectively. Assume that the exact solution has a regularity such that u, ut ∈ H k+1 (Ω). Then, there exists a constant C such that RT |ku − uh k|2w,1 ≤ C[|ku0 − u0h k|2w,1 + h2k 0 (h2 kut k2k+1 + kuk2k+1)dt]. (26) Proof Let

ρ = u − Qh u, e = Qh u − uh .

(27)

where Qh is the local L2 -projection operator and e = {e0 , eb } = {Q0 u − u0 , Qb u − ub }. Then we have u − uh = ρ + e.

(28)

To estimate ρ, we apply Lemma 4.3 and 4.4. We start by estimating e. Since u and uh satisfy (4) and (11) respectively, we have (ut − uh,t , v) + a(u, v) − aw (uh , v) = 0,

∀v ∈ Vh0 .

Further, (ut − Qh ut + Qh ut − uh,t, v) + a(u, v) − aw (uh , v) = 0,

∀v ∈ Vh0 ,

i.e., (et , v) + a(u, v) − aw (uh , v) = −(ρt , v),

∀v ∈ Vh0 .

(29)

In the following, we analyze the term a(u, v) − awP (uh , v). Recalling the definitions of a(u, v) and aw (uh , v) and noting that < a∇u · n, vb >∂T = 0, T ∈Th

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we derive a(u,P v) − aw (uh , v) P = [(a∇u, ∇v)T + (cu, v)T ] − < a∇u · n, v >∂T T ∈ThP T ∈Th P −1 hT < u0 − ub , v0 − vb >∂T − [(a∇w uh , ∇w v)T + (cuh , v)T ] − . T ∈Th PT ∈Th P = [(a∇u, ∇v)T + (cu, v)T ] − < a∇u · n, v0 − vb >∂T T ∈ThP T ∈Th P −1 − [(a∇w uh , ∇w v)T + (cuh , v)T ] − hT < u0 − ub , v0 − vb >∂T T ∈Th

T ∈Th

Further, we have

a(u,P v) − aw (uh , v) = [(a∇u, ∇v)T − (a∇w uh , ∇w v)T ] T ∈ThP P −1 + [(cu, v)T − (cuh , v)T ] − hT < u0 − ub, v0 − vb >∂T T ∈ThP T ∈Th − < a∇u · n, v0 − vb >∂T T ∈T h P = [(a∇u, ∇v)T − (a∇w Qh u, ∇w v)T

(30)

T ∈Th

+

+(a∇w Qh u,P∇w v)T − (a∇w uh , ∇w v)T ] [(cu, v)T − (cuh , v)T ] − h−1 T < u0 − ub , v0 − vb >∂T T ∈ThP T ∈Th − < a∇u · n, v0 − vb >∂T . P

T ∈Th

From the definitions of the weak discrete gradient ∇w and the projection Qh , as well as the expressions in (10) and (20), we get (a∇w Qh u, ∇w v)T = (aQh (∇u), ∇w v)T = (∇w v, aQh (∇u))T = −(v0 , ∇ · (aQh (∇u)))T + < vb , aQh (∇u) · n) >∂T = (∇v0 , aQh (∇u))T − < v0 − vb , aQh (∇u) · n >∂T = (a∇u, ∇v0 )T − < v0 − vb , aQh (∇u) · n >∂T . Substituting (31) into (30) arrives at P a(u, v) − aw (uh , v) = [(a∇w e, ∇w v)T + (cρ, v)T + (ce, v)T ] P −1T ∈Th − hT < u0 − ub , v0 − vb >∂T T ∈Th P + < a(Qh (∇u) − ∇u) · n, v0 − vb >∂T .

(31)

(32)

T ∈Th

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Combining (29) with (32) gives P (et , v) + [(a∇w e, ∇w v)T + (cρ, v)T + (ce, v)T ] T ∈Th P − h−1 T < u0 − ub , v0 − vb >∂T T ∈Th P + < a(Qh (∇u) − ∇u) · n, v0 − vb >∂T = −(ρt , v).

(33)

T ∈Th

Adding the term

P

T ∈Th

have

h−1 T < Q0 u − Qb u, v0 − vb >∂T to both sides of (33), we

(et , v) + aw (e, v) P P −1 = −(ρt , v) − (cρ, v)T + hT < Q0 u − Qb u, v0 − vb >∂T T ∈T T ∈T h h P + < a(∇u − Qh (∇u)) · n, v0 − vb >∂T .

(34)

T ∈Th

Choosing the test function v = et in (34), we have

1d aw (e, e) 2 dt P P −1 = −(ρt , et ) − (cρ, et )T + hT < Q0 u − Qb u, e0,t − eb,t >∂T T ∈T T ∈T h h P + < a(∇u − Qh (∇u)) · n, e0,t − eb,t >∂T T ∈Th P −1 = −(ρt , et ) − (cρ, et ) + hT < Q0 u − Qb u, e0,t − eb,t >∂T T ∈T h P + < a(∇u − Qh (∇u)) · n, e0,t − eb,t >∂T

ket k20 +

T ∈Th

≡ R1 + R2 + R3 + R4 .

(35) We estimate each term of R1 , R2 , R3 and R4 , separately. For R3 and R4 , we use Lemma 4.4, yielding: |R3 | ≤ Chk kukk+1k|et k|w,1,

|R4 | ≤ Chk kukk+1k|et k|w,1.

(36)

The other two terms R1 and R2 can be bound by applying the H o¨lder inequality and Lemma 3.1, i.e., 1 |R1 | = | − (ρt , et )| ≤ Ckρt kket k ≤ Ckρt k20 + ket k20 . 2

(37)

1 |R2 | = | − (cρ, et )| ≤ Ckρkket k ≤ Ckρk20 + ket k20 . 2

(38)

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Substituting (37), (38) and (36) into (35) leads to: 1d aw (e, e) ≤ C(kρk2 + kρt k2 + hk kukk+1k|et k|w,1) 2 dt

(39)

Integrating (39) with respect to t from 0 to T , we have aw (e(T ), e(T )) − aw (e(0), e(0)) RT RT RT ≤ C[ 0 kρk2 dt + 0 kρt k2 dt + hk 0 kukk+1k|et k|w,1dt)] RT RT RT RT ≤ C[ 0 kρk2 dt + 0 kρt k2 dt + h2k 0 kuk2k+1dt + 0 k|et k|2w,1dt].

(40)

By virtue of Lemma 4.3,

kρt k0 = kut − Qh ut k0 ≤ Chk+1kut kk+1 .

(41)

A combination of (22) and (40)-(41) with Gronwall lemma leads to (26).  4.3. Error estimate for fully discrete WG scheme Theorem 4.2. Let u and {unh } be the solutions to the parabolic equation (1) and the fully discrete WG scheme (18), respectively. Then ku(tn ) − unh k2w,1 ≤ C{ku0 − u0h k2w,1 + h2k [(ku0 k2k+1 + Rt +τ 2 0 n kutt k20 dt}.

R tn 0

kut k2k+1dt) + τ

n P

i=1

kui k2k+1]

(42)

Proof Set

ρn = u(tn ) − Qh u(tn ),

en = Qh u(tn ) − unh ,

then u(tn ) − unh = ρn + en

(43)

It follows from Lemma 4.3 that n

n

k

k

0

|kρ k|w,1 ≤ Ckρ k1 ≤ Ch ku(tn )kk+1 ≤ Ch [ku kk+1 +

Z

tn

kuτ kk+1 dτ ] (44)

0

In (4a), we set t = tn we have (unt , v) + a(un , v) = (f n , v),

(45)

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∂u(x, t) at t = tn , and similar ∂t n n definitions to u and f . Subtracting (18a) from (45), then we have where unt denotes the value of derivative

(unt − ∂¯t unh , v) + a(un , v) − aw (unh , v) = 0,

(46)

(∂¯t en , v) + a(un , v) − aw (unh , v) = (∂¯t Qh u(tn ) − unt , v).

(47)

further

For the term a(un , v) − aw (unh , v), taking the same measures used in the analysis course of semi-discrete case, we have (∂¯t en , v) + aw (en , v) P = (∂¯t Qh u(tn ) − unt , v) − (cρn , v)T T ∈Th P −1 + hT < Q0 un − Qb un , v0 − vb >∂T T ∈ThP + < a(∇un − Qh (∇un )) · n, v0 − vb >∂T .

(48)

|LL1 | = |(∂¯t en , ∂¯t en )| = k∂¯t en k20 .

(49)

T ∈Th

Let LL1 , LL2 be the two terms of the left hand side (LHS) of the equation (48) and RR1 , RR2 , RR3 , RR4 be the four terms of the right hand side (RHS) of (48), respectively. Nest, we choose the test function v = ∂¯t en in (48), and estimate these six terms consecutively. For the two terms LL1 , LL2 of the LHS of the error equation, we have

Note that aw (en , en ) ≥ α0 |ken k|2w,1, and aw (en , en−1 ) ≤ C ∗ k|en k|w,1k|en−1k|w,1. using the weighted H o¨lder inequality and choosing a suitable weight ǫ , such that ǫ < α0 and aw (en , en−1 ) ≤ ǫk|en k|2w,1 + Ck|en−1 k|2w,1. |LL2 | ≥

1 [(α0 − ǫ)k|en k|2w,1 − Ck|en−1k|2w,1]. τ

(50)

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The terms RR1 through RR4 in the RHS of (48) are estimated as follows: 1 1 |RR1 | ≤ Ck∂¯t Qh u(tn ) − unt k2 + k∂¯t en k20 , |RR2 | ≤ Ckρn k2 + k∂¯t en k20 . (51) 2 2 k n n |RR3 | ≤ Ch ku kk+1 k|∂¯t e k|w,1 C1 (k|en k|2w,1 + k|en−1 k|2w,1), ≤ Ch2k kun k2k+1 + τ (52) |RR4 | ≤ Chk kun kk+1 k|∂¯t en k|w,1 C1 (k|en k|2w,1 + k|en−1 k|2w,1), ≤ Ch2k kun k2k+1 + τ 1 where C1 has to be less than (α0 − ǫ). A combination of (48)-(52) leads to 2 n 2 |ke k|w,1 ≤ βk|en−1k|2w,1 + Cτ (k∂¯t Qh u(tn ) − unt k20 + kρn k20 + h2k kun k2k+1) (53) n P 0 2 i i 2 i 2 2k i 2 ¯ ≤ β|ke k| + Cτ (k∂t Qh u(t ) − u k + kρ k + h ku k ), w,1

t 0

i=1

0

k+1

2C1 + C . α0 − ǫ − 2C1 Introducing z i = ∂¯t Qh u(ti ) − uit , and writing z i = z1i + z2i , where Z 1 ti i i i ¯ ¯ (Qh − I)ut dt, z1 = ∂t Qh u(t ) − ∂t u(t ) = τ ti−1

where β =

and z2i

1 = ∂¯t u(ti ) − ut (ti ) = − τ

From Lemma 4.3, n P

i=1

Similarly

n X

Z

ti

(t − ti−1 )utt dt. ti−1

n R P ti ( ti−1 Chk+1 kut kk+1 dt)2 i=1 R t ≤ Cτ −1 hk+1 0 n kut k2k+1dt.

kz1i k20 ≤ Cτ −2

kz2i k20

i=1

Again by Lemma 4.3,

n Z X ≤ ( i=1

ti

2

kutt k0 dt) = τ

Z

0

ti−1

(54)

tn

kutt k20 dt

(55)

|ke0 k|2w,1 = |kQh u0 − u0h k|2w,1 = |kQh u0 − u0 + u0 − u0h k|2w,1 ≤ Ch2k ku0 k2k+1 + k|u0 − u0h k|2w,1. A combination of (53), (44) and (54)-(56) leads to (42).

(56) 

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5. Numerical Experiment In this section, we give three numerical examples using scheme (18) and consider the following parabolic problem [11] ut − div(D∇u) = f,

in Ω × J,

(57)

with proper Dirichlet boundary and initial conditions. For simplicity, we let D = 1, 10; Ω = (0, 1) × (0, 1) be unit square; and the time interval J = (0, T ) be (0, 1), in all three numerical examples. One can determine the initial and boundary conditions and source term f (x, t) according to the corresponding analytical solution of each example. We construct triangular mesh as follows. Firstly, we partition the square domain Ω = (0, 1) × (0, 1) into N × N sub-squares uniformly to obtain the square mesh. Secondly, we divide each square element into two triangles by the diagonal line with a negative slope so that we complete the constructing of triangular mesh. In the first example, the analytical solution is u = sin(πx) sin(πy) exp(−t).

(58)

For a set of simulations, different mesh sizes h = 1/N(N = 4, 8, 16, 32, 64) and different diffusion coefficients D = 1 and D = 10 are taken, and their corresponding discrete norms errors and convergence rates (CR) are listed in Table 1 for D = 1 and D = 10. Here |k · k|w,1 is defined as discrete version of the definition of (15) without the term kvk20,T . Table 1: Numerical results of the fist example for D = 1 and D = 10.

D=1 h |||u − uh |||w,1 2.5000e-01 1.6044e-01 1.2500e-01 8.0594e-02 6.2500e-02 4.0329e-02 3.1250e-02 2.0165e-02 1.5625e-02 1.0082e-02

CR 0.99 1.00 1.00 1.00

D = 10 |||u − uh |||w,1 1.2252e+00 6.0921e-01 3.0412e-01 1.5200e-01 7.5990e-02

CR 1.01 1.00 1.00 1.00

In the second example, the analytical solution is u = x(1 − x)y(1 − y) exp(x − y − t).

(59)

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Numerical error results and CRs are listed in Table 2 for D = 1 and D = 10 based on the same triangular mesh as those of the first example. Table 2: Numerical results of the second example for D = 1 and D = 10.

h 2.5000e-01 1.2500e-01 6.2500e-02 3.1250e-02 1.5625e-02

D=1 |||u − uh |||w,1 1.5858e-02 7.9786e-03 3.9940e-03 1.9973e-03 9.9864e-04

CR 0.99 1.00 1.00 1.00

D = 10 |||u − uh |||w,1 1.2052e-01 5.9967e-02 2.9945e-02 1.4967e-02 7.4829e-03

CR 1.01 1.00 1.00 1.00

In the third example, the analytical solution is u = x(1 − x)y(1 − y) exp(x + y + t).

(60)

Numerical error results and CRs are listed in Table 3 for D = 1 and D = 10 based on the same triangular mesh as those of the first example. Table 3: Numerical results of the third example for D = 1 and D = 10.

h 2.5000e-01 1.2500e-01 6.2500e-02 3.1250e-02 1.5625e-02

D=1 |||u − uh |||w,1 3.1035e-01 1.5916e-01 8.0078e-02 4.0099e-02 2.0056e-02

CR 0.96 0.99 1.00 1.00

D = 10 |||u − uh |||w,1 2.3672e+00 1.1977e+00 6.0062e-01 3.0052e-01 1.5029e-01

CR 0.98 1.00 1.00 1.00

All three numerical examples show good agreement with the theoretical results in Section 4, which show that the WG-FEM (18) is stable and first order convergent in H 1 norm. Acknowldgements The first author’s research is partially supported by the Natural Science Foundation of Shandong Province of China grant ZR2013AM023, China Postdoctoral Science Foundation no. 2014M560547, and the fundamental research funds of Shandong university grant 2015JC019.

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References [1] R. A. Adams. Sobolev spaces. New York: Academic Press, 1975. [2] P. G. Ciarlet, The finite element method for elliptic problems. Amsterdam: NorthHolland, 1978. [3] M. A. T. Elshebli, Discrete maximum principle for the finite element solution of linear non-stationary diffusion-reaction problems, Appl Math Model, 32, 2008: 1530-1541. [4] F. Gao and L. Mu, L2 error estimate for weak Galerkin finite element methods for parabolic problems, J. Comp. Math. 32(2), 2014: 195-204. [5] Q. H. Li, J. Wang, Weak Galerkin finite element methods for parabolic equations, Numer Methods Partial Differential Equations 29(6), 2013: 2004-2024. [6] M. Majidi, G. Starke,Least-square Galerkin methods for parabolic problems: Semidiscretization in time, SIAM J. Numer. Anal. 39, 2001: 1302-1323. [7] A. K. Pani and G. Fairweather, H1 -Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA J. Numer. Anal. 22, 2002: 231252. [8] L. Mu, J. Wang, Y. Wang and X. Ye, A computational study of the weak Galerkin method for the second order elliptic equations, Numer Algor 63, 2013: 753-777. [9] L. Mu, J. Wang, X. Ye and S. Zhao, A numerical study on the weak Galerkin method for the Helmholtz equation, Commun. Comput. Phys. 15(5), 2014: 1461-1479. [10] P. Raviart, J. Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method, I.Galligani, E.Magenes, eds., Lectures Notes in Math. 606, Springer-Verlag, New York, 1977. [11] H. X. Rui, S. D. Kim, S. Kim, Split least-squares finite element methods for linear and nonlinear parabolic problems, J. Comput. Appl. Math., 223, 2009: 938-952. [12] M. Tabata, Uniform convergence of the upwind finite element approximation for semilinear parabolic problems, J.Math.Kypto Univ. (JMKYAZ), 18(2), 1978: 327-351. [13] T. Vejchodsk´ y , S. Korotov, A. Hannukainen, Discrete maximum principle for parabolic problems solved by prismatic finite elements, Math Comput Simul, 80, 2010: 1758-1770. [14] J. Wang and X. Ye, A weak Galerkin method for second order elliptic problems, J. Comput. Appl. Math., 241, 2013: 103-115. [15] J. Wang and X. Ye, A Weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp. 83, 2014: 2101C2126. [16] W. Wu, Error estimates of generalized difference methods for nonlinear parabolic equations, Math. Numer. Sinica, 2, 1987: 119-132.

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Some fixed point theorems for generalized expansive mappings in cone metric spaces over Banach algebras Binghua Jiang1 , Shaoyuan Xu2∗, Huaping Huang1 , Zelin Cai1 1. School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China 2. School of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China

Abstract: In this paper, we prove some fixed point theorems for expansive mappings in cone metric spaces over Banach algebras without the assumption of normality of cones. Moreover, we give some examples to support our results. Our results improve and generalize the recent results of Aage and Salunke(2011). MSC: 54H25, 47H10, 54E50 Keywords: Generalized expansive mapping, Cone metric space over Banach algebra, Spectral radius

1

Introduction and Preliminaries In 2007 Huang and Zhang[1] introduced cone metric space and proved some fixed point

theorems of contractive mappings in such spaces. Since then, some authors proved lots of fixed point theorems for contractive or expansive mappings in cone metric spaces that expanded certain fixed point results in metric spaces (see [2-14]). However, recently, it is not an attractive topic since some authors have appealed to the equivalence of some metric and cone metric fixed point results (see [21-24]). Recently [13] introduced the concept of cone metric space with Banach algebra and obtained some fixed point theorems in such spaces. Moreover, the authors of [13] gave an example to illustrate that the non-equivalence of fixed point theorems between cone metric spaces over Banach algebras and metric spaces ∗

Corresponding author: E-mail: [email protected]

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(in usual sense). As a result, it is necessary to further investigate fixed point theorems in cone metric spaces over Banach algebras. In this paper, we generalize the famous Banach expansive mapping theorems as follows: Let (X, d) be a complete cone metric space over Banach algebra A and K be a cone in A. Suppose the mapping T : X → X is onto and satisfies the generalized expansive condition: d(T x, T y) º kd(x, y), for all x, y ∈ X, where k, k −1 ∈ K are generalized constants with ρ(k −1 ) < 1. Then T has an unique fixed point in X. Further, we give some other fixed point theorems for expansive mappings with generalized constants in cone metric spaces over Banach algebras. In addition, all cones are not necessarily normal ones. In these cases, our main results are not equivalent to those in metric spaces (see [7]). For the sake of completeness, we introduce some basic concepts as follows: Let A be a Banach algebras with a unit e, and θ the zero element of A. A nonempty closed convex subset K of A is called a cone if and only if (i) {θ, e} ⊂ K; T (ii) K 2 = KK ⊂ K, K (−K) = {θ}; (iii) λK + µK ⊂ K for all λ, µ ≥ 0. On this basis, we define a partial ordering ¹ with respect to K by x ¹ y if and only if y − x ∈ K, we shall write x ≺ y to indicate that x ¹ y but x 6= y, while x ¿ y will indicate that y − x ∈ intK, where intK stands for the interior of K. If intK 6= ∅, then K is called a solid cone. Write k · k as the norm on A. A cone K is called normal if there is a number M > 0 such that for all x, y ∈ A, θ ¹ x ¹ y ⇒ kxk ≤ M kyk. The least positive number satisfying above is called the normal constant of K. An element x ∈ A is said to be invertible if there is an element y ∈ A such that yx = xy = e. The inverse of x is denoted by x−1 . For more details, we refer to [10, 13]. In the following we always suppose that A is a real Banach algebra with a unit e, K is a solid cone in A and ¹ is a partial ordering with respect to K. Definition 1.1([13]) Let X be a nonempty set and A a Banach algebra. Suppose that the mapping d : X × X → A satisfies: 2

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(i) θ ≺ d(x, y) for all x, y ∈ X with x 6= y and d(x, y) = θ if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) d(x, y) ¹ d(x, z) + d(z, y) for all x, y, z ∈ X. Then d is called a cone metric on X, and (X, d) is called a cone metric space over Banach algebra A. Definition 1.2([2]) Let (X, d) be a cone metric space, x ∈ X and {xn } a sequence in X. Then (i) {xn } converges to x whenever for every c ∈ A with θ ¿ c there is a natural number N such that d(xn , x) ¿ c for all n ≥ N, we denote this by lim xn = x or xn → x (as n→∞

n → ∞). (ii) {xn } is a Cauchy sequence whenever for every c ∈ A with θ ¿ c there is a natural number N such that d(xn , xm ) ¿ c for all n, m ≥ N . (iii) (X, d) is a complete cone metric space if every Cauchy sequence is convergent. Lemma 1.3 ([7]) Let u, v, w ∈ A. If u ¹ v and v ¿ w, then u ¿ w. Lemma 1.4 ([7]) Let A be a Banach algebra and {an } a sequence in A. If an → θ (n → ∞), then for any c À θ, there exists N such that for all n > N , one has an ¿ c. 1

Lemma 1.5 ([10]) Let A be a Banach algebra with a unit e, x ∈ A, then lim kxn k n n→∞

exists and the spectral radius ρ(x) satisfies 1

1

ρ(x) = lim kxn k n = inf kxn k n . n→∞

If ρ(x) < |λ|, then λe − x invertible in A, moreover, (λe − x)

−1

∞ X xi = , λi+1 i=0

where λ is a complex constant. Lemma 1.6([10]) Let A be a Banach algebra with a unit e, a, b ∈ A. If a commutes with b, then ρ(a + b) ≤ ρ(a) + ρ(b),

ρ(ab) ≤ ρ(a)ρ(b).

Lemma 1.7([20])) Let K be a cone in a Banach algebra A and k ∈ K be a given vector. Let {un } be a sequence in K. If for each c1 À θ, there exists N1 such that un ¿ c1 for all n > N1 , then for each c2 À θ, there exists N2 such that kun ¿ c2 for all n > N2 . Lemma 1.8([20]) If A is a Banach algebra with a solid cone K and kxn k → 0(n → ∞), then for any θ ¿ c, there exists N such that for all n > N , we have xn ¿ c. Remark 1.9 Let A be a Banach algebra and k ∈ A. If ρ(k) < 1, then lim kk n k = 0. n→∞

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2

Main results

In this section, we shall prove some fixed point theorems for expansive mappings in the setting of non-normal cone metric spaces over Banach algebras. Furthermore, we display two examples to support our main conclusions. Theorem 2.1 Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X is onto and satisfies the expansive expansive condition: d(T x, T y) º kd(x, y) + ld(T x, y),

(2.1)

for all x, y ∈ X, where k, l, k −1 ∈ K are two generalized constants. If e − l ∈ K and ρ(k −1 ) < 1, then T has a fixed point in X. Proof

Since T is an onto mapping, for each x0 ∈ X, there exists x1 ∈ X such that

T x1 = x0 . Continuing this process, we can define {xn } by xn = T xn+1 (n = 0, 1, 2, . . .). Without loss of generality, we assume xn−1 6= xn for all n ≥ 1. According to (2.1), we have d(xn , xn−1 ) = d(T xn+1 , T xn ) º kd(xn+1 , xn ) + ld(T xn+1 , xn ) = kd(xn+1 , xn ) + ld(xn , xn ) = kd(xn+1 , xn ), then d(xn+1 , xn ) ¹ k −1 d(xn , xn−1 ). Letting k −1 = h we get d(xn+1 , xn ) ¹ hd(xn , xn−1 ) ¹ · · · ¹ hn d(x1 , x0 ).

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So by the triangle inequality and ρ(h) < 1, for all m > n, we see d(xm , xn ) ¹ d(xm , xm−1 ) + d(xm−1 , xm−2 ) + · · · + d(xn+1 , xn ) ¹ (hm−1 + hm−2 + · · · + hn )d(x1 , x0 ) = (e + h + · · · + hm−n−1 )hn d(x1 , x0 ) ∞ ³X ´ i ¹ h hn d(x1 , x0 ) i=0

= (e − h)−1 hn d(x1 , x0 ). By Lemma 1.8 and the fact that k(e − h)−1 hn d(x1 , x0 )k → 0(n → ∞) (Because of Remark 1.9, khn k → 0 (n → ∞)), it follows that for any c ∈ A with θ ¿ c, there exists N such that for all m > n > N , we have d(xm , xn ) ¹ (e − h)−1 hn d(x1 , x0 ) ¿ c, which implies that {xn } is a Cauchy sequence. By the completeness of X, there exists x∗ ∈ X such that xn → x∗ (n → ∞). Consequently, we can find an x∗∗ ∈ X such that T x∗∗ = x∗ . Now we show that x∗∗ = x∗ . In fact, d(x∗ , xn ) = d(T x∗∗ , T xn+1 ) º kd(x∗∗ , xn+1 ) + ld(T x∗∗ , xn+1 ) = kd(x∗∗ , xn+1 ) + ld(x∗ , xn+1 ). Since d(x∗ , xn ) ¹ d(x∗ , xn+1 ) + d(xn+1 , xn ), it follows that kd(x∗∗ , xn+1 ) ¹ (e − l)d(x∗ , xn+1 ) + d(xn+1 , xn ). Now, we have d(x∗∗ , xn+1 ) ¹ k −1 ((e − l)d(x∗ , xn+1 ) + d(xn+1 , xn )). Note that xn → x∗ (n → ∞), by Lemma 1.7, it follows that for any c ∈ A with θ ¿ c, there exists N such that for any n > N , we have k −1 ((e − l)d(x∗ , xn+1 ) + d(xn+1 , xn )) ¿ c. 5

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Thus d(x∗∗ , xn+1 ) ¿ c. Since the limit of a convergent sequence in cone metric space over Banach algebra is unique, we get x∗∗ = x∗ , i.e., x∗ is a fixed point of T . Theorem 2.2 Let (X, d) be a complete cone metric space over Banach algebra A and K be a solid cone in A. Suppose that the mapping T : X → X is onto and satisfies the generalized expansive condition: d(T x, T y) º kd(x, y) + ld(x, T x) + pd(y, T y),

(2.2)

for all x, y ∈ X, where k, l, p, e − p ∈ K are generalized constants with (k + l)−1 ∈ K and £ ¤ ρ (k + l)−1 (e − p) < 1. Then T has a fixed point in X. Proof

Since T is an onto mapping, for each x0 ∈ X, there exists x1 ∈ X such that

T x1 = x0 . Continuing this process, we can define {xn } by xn = T xn+1 (n = 0, 1, 2, . . .). Without loss of generality, we suppose xn−1 6= xn for all n ≥ 1. According to (2.2), we have d(xn , xn−1 ) = d(T xn+1 , T xn ) º kd(xn+1 , xn ) + ld(xn+1 , T xn+1 ) + pd(xn , T xn ) = kd(xn+1 , xn ) + ld(xn+1 , xn ) + pd(xn , xn−1 ), which implies that (k + l)d(xn , xn+1 ) ¹ (e − p)d(xn , xn−1 ). Put k + l = r, then rd(xn , xn+1 ) ¹ (e − p)d(xn , xn−1 ).

(2.3)

Since r is invertible, to multiply r−1 in both sides of (2.3), we have d(xn , xn+1 ) ¹ hd(xn , xn−1 ),

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where h = (k + l)−1 (e − p). Note that ρ(h) < 1 and for all m > n, d(xm , xn ) ¹ d(xm , xm−1 ) + d(xm−1 , xm−2 ) + · · · + d(xn+1 , xn ) ¹ (hm−1 + hm−2 + · · · + hn )d(x1 , x0 ) = (e + h + · · · + hm−n−1 )hn d(x1 , x0 ) ∞ ³X ´ ¹ hi hn d(x1 , x0 ) i=0

= (e − h)−1 hn d(x1 , x0 ). As is shown in the proof of Theorem 2.1, it follows that {xn } is a Cauchy sequence. Then by the completeness of X, there exists x∗ ∈ X such that xn → x∗ (n → ∞). Consequently, we can find a x∗∗ ∈ X such that T x∗∗ = x∗ . Now we show that x∗∗ = x∗ . Indeed, Since d(x∗ , xn ) = d(T x∗∗ , T xn+1 ) º kd(x∗∗ , xn+1 ) + ld(x∗∗ , T x∗∗ ) + pd(xn+1 , T xn+1 ) = kd(x∗∗ , xn+1 ) + ld(x∗∗ , x∗ ) + pd(xn+1 , xn ). Then d(x∗ , xn ) º kd(x∗∗ , xn+1 ) + ld(x∗∗ , xn+1 ) − ld(x∗ , xn+1 ) + pd(xn+1 , xn ). Note that d(x∗ , xn ) ¹ d(x∗ , xn+1 ) + d(xn+1 , xn ), thus d(x∗ , xn+1 ) + d(xn+1 , xn ) º (k + l)d(x∗∗ , xn+1 ) − ld(x∗ , xn+1 ) + pd(xn+1 , xn ), which implies that (k + l)d(x∗∗ , xn+1 ) ¹ (e + l)d(x∗ , xn+1 ) + (e − p)d(xn+1 , xn ). Since k + l = r is invertible, we have ¡ ¢ d(x∗∗ , xn+1 ) ¹ r−1 (e + l)d(x∗ , xn+1 ) + (e − p)d(xn+1 , xn ) . Owing to xn → x∗ (n → ∞), it follows by Lemma 1.7 that for any c ∈ A with θ ¿ c there exists N such that for any n > N , ¡ ¢ r−1 (e + l)d(x∗ , xn+1 ) + (e − p)d(xn+1 , xn ) ¿ c, 7

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hence d(x∗∗ , xn+1 ) ¿ c. Since the limit of a convergent sequence in cone metric space over Banach algebra is unique, we have x∗∗ = x∗ , i.e., x∗ is a fixed point of T . Corollary 2.3 Let (X, d) be a complete cone metric space over Banach algebra A and K be a cone in A. Suppose the mapping T : X → X is onto and satisfies the generalized expansive condition: d(T x, T y) º kd(x, y),

(2.4)

for all x, y ∈ X, where k, k −1 ∈ K are generalized constants with ρ(k −1 ) < 1. Then T has an unique fixed point in X. Proof

By using Theorem 2.1 and Theorem 2.2, letting l = p = θ, we need to only prove

the fixed point is unique. Indeed, if y ∗ is another fixed point of T , then d(x∗ , y ∗ ) = d(T x∗ , T y ∗ ) º kd(x∗ , y ∗ ), that is, d(x∗ , y ∗ ) ¹ k −1 d(x∗ , y ∗ ) = hd(x∗ , y ∗ ). Thus d(x∗ , y ∗ ) ¹ hd(x∗ , y ∗ ) ¹ h2 d(x∗ , y ∗ ) ¹ · · · ¹ hn d(x∗ , y ∗ ). In view of khn d(x∗ , y ∗ )k → 0(n → ∞), it establishes that for any c ∈ A with θ ¿ c, there exists N2 such that for all n > N2 , we have d(x∗ , y ∗ ) ¹ hn d(x∗ , y ∗ ) ¿ c, so d(x∗ , y ∗ ) = θ, which implies that x∗ = y ∗ . Hence, the fixed point is unique. Remark 2.4 Note that Corollary 2.3 only assumes that ρ(k −1 ) < 1, which implies ρ(k) > 1, neither k  e nor kkk > 1. This is a vital improvement. Remark 2.5 Since we get the fixed point theorems in the setting of non-normal cone metric spaces over Banach algebras, our results are never equivalent to the fixed point 8

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versions in metric spaces (see [7, 13]). The following examples illustrate our conclusions. Example 2.6 Let A = CR1 [0, 14 ] and define a norm on A by kxk = kxk∞ + kx0 k∞ for x ∈ A. Define multiplication in A as just pointwise multiplication. Then A is a Banach algebra with a unit e = 1. The set K = {x ∈ A : x ≥ 0} is a non-normal cone in A (see [7]). Let X = R. Define d : X × X → A by d(x, y)(t) = |x − y|et , for all t ∈ [0, 41 ]. Further, 4 . let T : X → X be a mapping defined by T x = 2x and let k ∈ K define by k(t) = 2t+3 By careful calculations one sees that all the conditions of Corollary 2.3 are fulfilled. The point x = 0 is the unique fixed point of the mapping T . Example 2.7 Let A = {a = (aij )3×3 | aij ∈ R, 1 ≤ i, j ≤ 3} and kak =

1 3

P

| aij |.

1≤i,j≤3

Then the set K = {a ∈ A | aij ≥ 0, 1 ≤ i, j ≤ 3} is a normal cone in A. Let X = {1, 2, 3}. Define d : X × X → A by d(1, 1) = d(2, 2) = d(3, 3) = θ and   1 2 3 d(1, 2) = d(2, 1) =  0 0 0  , 4 5 6 

 2 4 6 d(1, 3) = d(3, 1) =  0 0 0  , 3 4 5   2 4 6 d(2, 3) = d(3, 2) =  0 0 0  . 3 4 5 We find that (X, d) is a solid cone metric space over Banach algebra A. Let T : X → X be a mapping defined by T 1 = 2, T 2 = 1, T 3 = 3, and let k, l, p ∈ K be defined by   4 0 0 5 k =  0 45 0  , 0 0 45 

0  0 p=l= 0

0 1 10

0

 0 0 . 0

Then d(T x, T y) º kd(x, y) + ld(x, T x) + pd(y, T y), where k, l, p, e − p ∈ K are generalized constants. It is easy to prove that ke − k − lk < 1 and k(k + l)−1 (e − p)k < 1, which imply £ ¤ ρ(e − k − l) < 1 and ρ (k + l)−1 (e − p) < 1. Clearly, all conditions of Theorem 2.2 are 9

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fulfilled. Hence T has a fixed point x = 3 in X. Remark 2.8 It needs to emphasis that according to the expansive condition of [11, Theorem 2.1], we are easy to see that the mapping discussed is an injection, and the authors attempt to use [11, Example 2.7] to support this theorem. But unfortunately, this is impossible, since the mapping appearing in this example is not an injection at all. Therefore, it is unreasonable. Basing on the facts above, we may verify that Example 2.7 in this paper is reasonable. It is also interesting, since here we use matrixes as generalized constants.

Competing interests The authors declare that there have no competing interests.

Acknowledgements The research is partially supported by the Foundation of Education Ministry, Hubei Province, China (no. Q20122203). The research is also partially supported by the foundation of the research item of Strong Department of Engineering Innovation, Research on Fixed Point Theory with Banach Algebras in Abstract Spaces and Applications, which is sponsored by the Strong School of Engineering Innovation of Hanshan Normal University, China (2013).

References [1] L.-G. Huang, X. Zhang, “Cone metric space and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, Vol. 332, pp. 14681476, 2007. [2] Sh. Rezapour, R. Hamlbarani, “Some notes on the paper ‘Cone metric spaces and fixed point theorems of contractive mappings’,” Journal of Mathematical Analysis and Applications, Vol. 345, pp 719-724, 2008. [3] Y. Han, S.-Y. Xu, “New common fixed point results for four maps on cone metric spaces,” Applied Mathematics, Vol. 2, pp. 1114-1118, 2011. 10

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[4] C.D. Bari, P. Vetro, “ϕ-pairs and common fixed points in cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, Vol. 57, pp. 279-285, 2008. [5] X. Zhang, “Common fixed point theorems of Lipschitz type mappings in cone metric spaces,” Acta Mathematica Sinica(in Chinese), Vol. 53, no. 1, pp. 1139-1148, 2010. [6] M. Abbas, G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metic spaces,”Journal of Mathematical Analysis and Applications, Vol. 341, pp. 416-420, 2008. [7] S. Jankovi´c, Z. Kadelburg, S. Radenovi´c, “On cone metric spaces: A survey,” Nonlinear Analysis, Vol. 74, pp. 2591-2601, 2011. [8] S. Radenovi´c, “Common fixed points under contractive conditions in cone metric spaces,” Computers and Mathematics with Applications, Vol. 58, pp. 1273-1278, 2009. [9] I. Altun, B. Damjanovi´c, D. Djori´c, “Fixed point and common fixed point theorems on ordered cone metric spaces,” Applied Mathematics Letter, Vol. 23, pp. 310-316, 2010. [10] W. Rudin, Functional Analysis, McGraw-Hill, New York, NY, USA, 2nd edition, 1991. [11] C. T. Aage, J. N. Salunke, “Some fixed point theorems for expansion onto mappings on cone metric spaces,” Acta Mathematica Sinica(English series), Vol. 27, no. 6, pp. 1101-1106, 2011. [12] Y. Han, S.Y. Xu, “Some new theorems of expanding mappings without continuity in cone metric spaces,” Fixed point Theory and Applications, Vol. 3, pp. 1-9, 2013. [13] H. Liu, S. Xu, “Cone metric space with Banach algebras and fixed point theorems of generalized Lipschitz mapping,” Fixed point Theory and Applications, Vol. 320, pp. 1-10, 2013. [14] W. Shatanawi, F. Awawdeh, “Some fixed point and coincidence point theorems for expansive maps in cone metric space,” Fixed point Theory and Applications, Vol. 19, pp. 1-10, 2012. 11

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[15] S. Chouhan, N. Malviya, “A fixed point theorem for expansive type mappings in cone metric spaces,” International Mathematical Forum, Vol. 6, no. 18, pp. 891-897, 2011. [16] W. Shatanawi, F. Awawdeh, “Some fixed and coincidence point theorems for expansive maps in cone metric spaces,” Fixed point Theory and Applications, Vol. 19, pp. 1-10, 2012. [17] Z. Kadelburg, P. P. Murthy, S. Radenovi´c, “Common fixed points for expansive mappings in cone metric spaces,” International Journal of Mathemathical Analysis, Vol. 5, no. 27, 1309-1319, 2011. [18] B.-H. Jiang, S.-Y. Xu, L. Shi, “Coupled coincidence points for mixed monotone random operators in partially ordered metric spaces,” Abstract and Applied Analysis, Vol. 2014, Article ID 484857, 9 pages, http://dx. doi. org/10. 1155/2014/484857. [19] A. G. B Ahmad, Z. M. Fadail, M. Abbas, Z. Kadelburg, Stojan Radenovi´c, “Some fixed and periodic points in abstract metric spaces,” Abstract and Applied Analysis, Vol. 2012, Article ID 908423, 15 pages, doi: 10.1155/2012/908423 62, 1677-1684. [20] S. -Y. Xu, S. Radenovi´c, “Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality,” Fixed point Theory and Applications, Vol. 102, pp. 1-12, 2014. [21] H. Cakalli, A. Sonmez, C. Genc, “On an equivalence of topological vector space valued cone metric spaces and metric spaces,” Applied Mathematics Letter, Vol. 25, pp. 429-433, 2012. [22] W.-S. Du, “A note on cone metric fixed point theory and its equivalence,” Nonlinear Analysis, Vol. 72, no. 5, pp. 2259-2261, 2010. [23] Z. Kadelburg, S. Radenovi´c, V. Rakocevi´c, “A note on the equivalence of some metric and cone metric fixed point results,” Applied Mathematics Letter, Vol. 24, pp. 370374, 2011. [24] Y.-Q Feng, W. Mao, “The equivalence of cone metric spaces and metric spaces,” Fixed Point Theory, Vol. 11, no. 2, pp. 259-263, 2010.

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ADDITIVE ρ-FUNCTIONAL INEQUALITIES IN FUZZY NORMED SPACES JI-HYE KIM, GEORGE A. ANASTASSIOU AND CHOONKIL PARK∗ Abstract. In this paper, we solve the following additive ρ-functional inequalities       x+y − f (x) − f (y) , t (0.1) N (f (x + y) − f (x) − f (y), t) ≤ N ρ 2f 2 and     x+y N 2f − f (x) − f (y), t ≤ N (ρ (f (x + y) − f (x) − f (y)) , t) (0.2) 2 in fuzzy normed spaces, where ρ is a fixed real number with |ρ| < 1. Using the fixed point method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in fuzzy Banach spaces.

1. Introduction and preliminaries Katsaras [19] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [11, 23, 48]. In particular, Bag and Samanta [2], following Cheng and Mordeson [8], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [22]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [3]. We use the definition of fuzzy normed spaces given in [2, 27, 28] to investigate the Hyers-Ulam stability of additive ρ-functional inequalities in fuzzy Banach spaces. Definition 1.1. [2, 27, 28, 29] Let X be a real vector space. A function N : X × R → [0, 1] is called a fuzzy norm on X if for all x, y ∈ X and all s, t ∈ R, (N1 ) N (x, t) = 0 for t ≤ 0; (N2 ) x = 0 if and only if N (x, t) = 1 for all t > 0; t (N3 ) N (cx, t) = N (x, |c| ) if c 6= 0; (N4 ) N (x + y, s + t) ≥ min{N (x, s), N (y, t)}; (N5 ) N (x, ·) is a non-decreasing function of R and limt→∞ N (x, t) = 1. (N6 ) for x 6= 0, N (x, ·) is continuous on R. The pair (X, N ) is called a fuzzy normed vector space. The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [26, 27]. Definition 1.2. [2, 27, 28, 29] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to be convergent or converge if there exists an x ∈ X such that limn→∞ N (xn −x, t) = 1 2010 Mathematics Subject Classification. Primary 46S40, 39B52, 47H10, 39B62, 26E50, 47S40. Key words and phrases. fuzzy Banach space; additive ρ-functional inequality; fixed point method; HyersUlam stability. ∗ Corresponding author.

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for all t > 0. In this case, x is called the limit of the sequence {xn } and we denote it by N limn→∞ xn = x. Definition 1.3. [2, 27, 28, 29] Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called Cauchy if for each ε > 0 and each t > 0 there exists an n0 ∈ N such that for all n ≥ n0 and all p > 0, we have N (xn+p − xn , t) > 1 − ε. It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn } converging to x0 in X, then the sequence {f (xn )} converges to f (x0 ). If f : X → Y is continuous at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). The stability problem of functional equations originated from a question of Ulam [47] concerning the stability of group homomorphisms. The functional equation f (x + y) = f (x) + f (y) is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping. Hyers [15] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [1] for additive mappings and by Th.M. Rassias [39] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Th.M. Rassias theorem was obtained by G˘avruta [12] by replacing the unbounded Cauchy difference by a general control function in the spiritof Th.M. Rassias’ approach.  x+y 1 The functional equation f 2 = 2 f (x)+ 12 f (y) is called the Jensen equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [7, 16, 18, 20, 21, 24, 35, 36, 37, 41, 42, 43, 44, 45, 46]). Gil´anyi [13] showed that if f satisfies the functional inequality k2f (x) + 2f (y) − f (x − y)k ≤ kf (x + y)k

(1.1)

then f satisfies the Jordan-von Neumann functional equation 2f (x) + 2f (y) = f (x + y) + f (x − y). See also [40]. Fechner [10] and Gil´anyi [14] proved the Hyers-Ulam stability of the functional inequality (1.1). Park, Cho and Han [34] investigated the Cauchy additive functional inequality kf (x) + f (y) + f (z)k ≤ kf (x + y + z)k

(1.2)

and the Cauchy-Jensen additive functional inequality

 

x+y

+ z kf (x) + f (y) + 2f (z)k ≤ 2f

2

(1.3)

and proved the Hyers-Ulam stability of the functional inequalities (1.2) and (1.3) in Banach spaces. Park [32, 33] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. We recall a fundamental result in fixed point theory. Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x) for all x, y ∈ X;

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(3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 1.4. [4, 9] Let (X, d) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ X, either d(J n x, J n+1 x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(J n x, J n+1 x) < ∞, ∀n ≥ n0 ; (2) the sequence {J n x} converges to a fixed point y ∗ of J; (3) y ∗ is the unique fixed point of J in the set Y = {y ∈ X | d(J n0 x, y) < ∞}; 1 (4) d(y, y ∗ ) ≤ 1−L d(y, Jy) for all y ∈ Y . In 1996, G. Isac and Th.M. Rassias [17] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [5, 6, 26, 30, 31, 37, 38]). In Section 2, we solve the additive ρ-functional inequality (0.1) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in fuzzy Banach spaces by using the fixed point method. In Section 3, we solve the additive ρ-functional inequality (0.2) and prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.2) in fuzzy Banach spaces by using the fixed point method. 2. Additive ρ-functional inequality (0.1) In this section, we prove the Hyers-Ulam stability of the additive ρ-functional inequality (0.1) in fuzzy Banach spaces. Let ρ be a real number with |ρ| < 1. We need the following lemma to prove the main results. Lemma 2.1. Let (Y, N ) be a fuzzy normed vector spaces. Let f : X → Y be a mapping such that       x+y − f (x) − f (y) , t (2.1) N (f (x + y) − f (x) − f (y), t) ≥ N ρ 2f 2 for all x, y ∈ X and all t > 0. Then f is Cauchy additive, i.e., f (x + y) = f (x) + f (y) for all x, y ∈ X. Proof. Assume that f : X → Y satisfies (2.1). Letting x = y = 0 in (2.1), we get N (f (0), t) = N (0, t) = 1. So f (0) = 0. Letting y = x in (2.1), we get N (f (2x) − 2f (x), t) ≥ N (0, t) = 1 and so f (2x) = 2f (x) for all x ∈ X. Thus   x 1 = f (x) (2.2) f 2 2 for all x ∈ X. It follows from (2.1) and (2.2) that       x+y N (f (x + y) − f (x) − f (y), t) ≥ N ρ 2f − f (x) − f (y) , t 2 = N (ρ(f (x + y) − f (x) − f (y)), t)   t = N f (x + y) − f (x) − f (y), |ρ|

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for all t > 0. By (N5 ) and (N6 ), N (f (x + y) − f (x) − f (y), t) = 1 for all t > 0. It follows from (N2 ) that f (x + y) = f (x) + f (y) for all x, y ∈ X.  Theorem 2.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ(x, y) ≤ ϕ (2x, 2y) 2 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying N (f (x + y) − f (x) − f (y), t)         t x+y − f (x) − f (y) , t , ≥ min N ρ 2f 2 t + ϕ(x, y)

(2.3)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f 2xn exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2L)t N (f (x) − A(x), t) ≥ (2.4) (2 − 2L)t + Lϕ(x, x) for all x ∈ X and all t > 0.


Proof. Letting y = x in (2.3), we get N (f (2x) − 2f (x), t) ≥

t t + ϕ(x, x)

(2.5)

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



where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [25, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, x) for all x ∈ X and all t > 0. Hence       x x N (Jg(x) − Jh(x), Lεt) = N 2g − 2h , Lεt 2 2       x x L = N g −h , εt 2 2 2 ≥ =

Lt 2


≥ x

+ ϕ x2 , 2 t t + ϕ(x, x) Lt 2

1118

Lt 2

+

Lt 2 L 2 ϕ(x, x)

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for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (2.5) that 

x L t , t ≥ 2 2 t + ϕ(x, x)

 

N f (x) − 2f



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ L2 . By Theorem 1.4, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e.,   1 x = A(x) A (2.6) 2 2 for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}. This implies that A is a unique mapping satisfying (2.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + ϕ(x, x) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality   x n N - lim 2 f = A(x) n→∞ 2n for all x ∈ X; 1 d(f, Jf ), which implies the inequality (3) d(f, A) ≤ 1−L d(f, A) ≤

L . 2 − 2L

This implies that the inequality (2.4) holds. By (2.3),          x y x+y n − f − f , 2 t N 2n f 2n 2n 2n (   )        x+y x y t n+1 n n n
≥ min N ρ 2 f −2 f −2 f ,2 t , 2n+1 2n 2n t + ϕ 2xn , 2yn for all x, y ∈ X, all t > 0 and all n ∈ N. So          x+y x y N 2n f − f − f ,t n n 2 2 2n (          x+y x y n+1 n n −2 f −2 f ,t , ≥ min N ρ 2 f n+1 n 2 2 2n for all x, y ∈ X, all t > 0 and all n ∈ N. Since limn→∞ t > 0,  

N (A(x + y) − A(x) − A(y), t) ≥ N ρ 2A

1119



t 2n t Ln + 2n ϕ(x,y) 2n

x+y 2



t 2n

+

t 2n n L 2n ϕ (x, y)

)

= 1 for all x, y ∈ X and all 



− A(x) − A(y) , t

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for all x, y ∈ X and all t > 0. By Lemma 2.1, the mapping A : X → Y is Cauchy additive, as desired.  Corollary 2.3. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with the norm k · k. Let f : X → Y be an odd mapping satisfying N (f (x + y) − f (x) − f (y), t)         x+y t ≥ min N ρ 2f − f (x) − f (y) , t , 2 t + θ(kxkp + kykp ) for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f ( 2xn ) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2p − 2)t N (f (x) − A(x), t) ≥ p (2 − 2)t + 2θkxkp for all x ∈ X and all t > 0. Proof. The proof follows from Theorem 2.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 21−p , and we get the desired result.  Theorem 2.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with   x y ϕ(x, y) ≤ 2Lϕ , 2 2 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying (2.3). Then A(x) := N limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (2 − 2L)t N (f (x) − A(x), t) ≥ (2.7) (2 − 2L)t + ϕ(x, x) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 2.2. It follows from (2.5) that   1 1 t N f (x) − f (2x), t ≥ 2 2 t + ϕ(x, x) for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 12 . Hence 1 d(f, A) ≤ , 2 − 2L which implies that the inequality (2.7) holds. The rest of the proof is similar to the proof of Theorem 2.2.



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

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

Letting y = 0 in (3.1), we get N 2f x2 − f (x), t ≥ N (0, t) = 1 and so x 2

 

f

1 = f (x) 2

(3.2)

for all x ∈ X. It follows from (3.1) and (3.2) that x+y N (f (x + y) − f (x) − f (y), t) = N 2f − f (x) − f (y) , t 2 ≥ N (ρ(f (x + y) − f (x) − f (y)), t)   t = N f (x + y) − f (x) − f (y), |ρ| 







for all t > 0. By (N5 ) and (N6 ), N (f (x + y) − f (x) − f (y), t) = 1 for all t > 0. It follows from (N2 ) that f (x + y) = f (x) + f (y) for all x, y ∈ X.  Theorem 3.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with L ϕ(x, y) ≤ ϕ (2x, 2y) 2 for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying     x+y N 2f − f (x) − f (y), t 2   t ≥ min N (ρ (f (x + y) − f (x) − f (y)) , t) , t + ϕ(x, y)

(3.3)

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ 2n f 2xn exists for each x ∈ X and defines an additive mapping A : X → Y such that (1 − L)t N (f (x) − A(x), t) ≥ (3.4) (1 − L)t + ϕ(x, 0) for all x ∈ X and all t > 0.


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Proof. Since f is odd, f (0) = 0. Letting y = 0 in (3.3), we get         x x t N f (x) − 2f , t = N 2f − f (x), t ≥ 2 2 t + ϕ(x, 0)

(3.5)

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

t , ∀x ∈ X, ∀t > 0 , t + ϕ(x, 0) 

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

where, as usual, inf φ = +∞. It is easy to show that (S, d) is complete (see [25, Lemma 2.1]). Now we consider the linear mapping J : S → S such that   x Jg(x) := 2g 2 for all x ∈ X. Let g, h ∈ S be given such that d(g, h) = ε. Then t N (g(x) − h(x), εt) ≥ t + ϕ(x, 0) for all x ∈ X and all t > 0. Hence x x N (Jg(x) − Jh(x), Lεt) = N 2g − 2h , Lεt 2 2       x L x −h , εt = N g 2 2 2 

≥ =

 

Lt 2 Lt 2

 


≥

+ ϕ x2 , 0 t t + ϕ(x, 0)

Lt 2

+



Lt 2 L 2 ϕ(x, 0)

for all x ∈ X and all t > 0. So d(g, h) = ε implies that d(Jg, Jh) ≤ Lε. This means that d(Jg, Jh) ≤ Ld(g, h) for all g, h ∈ S. It follows from (3.5) that 

x t ,t ≥ 2 t + ϕ(x, 0)

 

N f (x) − 2f



for all x ∈ X and all t > 0. So d(f, Jf ) ≤ 1. By Theorem 1.4, there exists a mapping A : X → Y satisfying the following: (1) A is a fixed point of J, i.e.,   x 1 A = A(x) (3.6) 2 2 for all x ∈ X. Since f : X → Y is odd, A : X → Y is an odd mapping. The mapping A is a unique fixed point of J in the set M = {g ∈ S : d(f, g) < ∞}.

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This implies that A is a unique mapping satisfying (3.6) such that there exists a µ ∈ (0, ∞) satisfying t N (f (x) − A(x), µt) ≥ t + ϕ(x, 0) for all x ∈ X; (2) d(J n f, A) → 0 as n → ∞. This implies the equality N - lim 2n f



n→∞

for all x ∈ X; (3) d(f, A) ≤

1 1−L d(f, Jf ),

x 2n



= A(x)

which implies the inequality d(f, A) ≤

1 . 1−L

This implies that the inequality (3.4) holds. By (3.3), x y −2 f − 2n f , 2n t N 2 f n 2 2n (     )       x + y x y t
≥ min N ρ 2n f −f −f , 2n t , 2n 2n 2n t + ϕ 2xn , 2yn 

n+1



x+y 2n+1



n











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

t 2n Ln t + ϕ(x,y) 2n 2n

t 2n

+

t 2n Ln 2n ϕ (x, y)

)

= 1 for all x, y ∈ X and all

x+y − A(x) − A(y), t ≥ N (ρ (A(x + y) − A(x) − A(y)) , t) 2 for all x, y ∈ X and all t > 0. By Lemma 3.1, the mapping A : X → Y is Cauchy additive, as desired.  







N 2A

Corollary 3.3. Let θ ≥ 0 and let p be a real number with p > 1. Let X be a normed vector space with the norm k · k. Let f : X → Y be an odd mapping satisfying 



N 2f

x+y 2





− f (x) − f (y), t

t ≥ min N (ρ (f (x + y) − f (x) − f (y)) , t) , t + θ(kxkp + kykp ) 



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

(2p − 2)t (2p − 2)t + 2p θkxkp

for all x ∈ X and all t > 0.

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Proof. The proof follows from Theorem 3.2 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 21−p , and we get the desired result.  Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an L < 1 with x y ϕ(x, y) ≤ 2Lϕ , 2 2 



for all x, y ∈ X. Let f : X → Y be an odd mapping satisfying (3.3). Then A(x) := N limn→∞ 21n f (2n x) exists for each x ∈ X and defines an additive mapping A : X → Y such that (1 − L)t (3.7) N (f (x) − A(x), t) ≥ (1 − L)t + Lϕ(x, 0) for all x ∈ X and all t > 0. Proof. Let (S, d) be the generalized metric space defined in the proof of Theorem 3.2. It follows from (3.5) that 1 t N f (x) − f (2x), Lt ≥ 2 t + ϕ(x, 0) 



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

L , 1−L

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



Corollary 3.5. Let θ ≥ 0 and let p be a real number with 0 < p < 1. Let X be a normed vector space with the norm k · k. Let f : X → Y be an odd mapping satisfying 



N 2f

x+y 2





− f (x) − f (y), t

t ≥ min N (ρ (f (x + y) − f (x) − f (y)) , t) , t + θ(kxkp + kykp ) 

for all x, y ∈ X and all t > 0. Then A(x) := N -limn→∞ defines an additive mapping A : X → Y such that N (f (x) − A(x), t) ≥

1 n 2n f (2 x)



exists for each x ∈ X and

(2 − 2p )t (2 − 2p )t + 2p θkxkp

for all x ∈ X. Proof. The proof follows from Theorem 3.4 by taking ϕ(x, y) := θ(kxkp + kykp ) for all x, y ∈ X. Then we can choose L = 2p−1 , and we get the desired result.  References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. [2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687– 705. [3] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005), 513–547. [4] L. C˘ adariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).

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[5] L. C˘ adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. [6] L. C˘ adariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). [7] I. Chang and Y. Lee, Additive and quadratic type functional equation and its fuzzy stability, Results Math. 63 (2013), 717–730. [8] S.C. Cheng and J.M. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436. [9] J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309. [10] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. [11] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48 (1992), 239–248. [12] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436. [13] A. Gil´ anyi, Eine zur Parallelogrammgleichung a ¨quivalente Ungleichung, Aequationes Math. 62 (2001), 303– 309. [14] A. Gil´ anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707–710. [15] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. [16] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨ auser, Basel, 1998. [17] G. Isac and Th.M. Rassias, Stability of ψ-additive mappings: Appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. [18] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, Florida, 2001. [19] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984), 143–154. [20] H. Kim, M. Eshaghi Gordji, A. Javadian and I. Chang, Homomorphisms and derivations on unital C ∗ algebras related to Cauchy-Jensen functional inequality, J. Math. Inequal. 6 (2012), 557–565. [21] H. Kim, J. Lee and E. Son, Approximate functional inequalities by additive mappings, J. Math. Inequal. 6 (2012), 461–471. [22] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11 (1975), 326–334. [23] S.V. Krishna and K.K.M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63 (1994), 207–217. [24] J. Lee, C. Park and D. Shin, An AQCQ-functional equation in matrix normed spaces, Results Math. 27 (2013), 305–318. [25] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. [26] M. Mirzavaziri and M.S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. 37 (2006), 361–376. [27] A.K. Mirmostafaee, M. Mirzavaziri and M.S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159 (2008), 730–738. [28] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159 (2008), 720–729. [29] A.K. Mirmostafaee and M.S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci. 178 (2008), 3791–3798. [30] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). [31] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). [32] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17–26. [33] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397–407. [34] C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820 (2007).

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[35] C. Park, K. Ghasemi, S. G. Ghaleh and S. Jang, Approximate n-Jordan ∗-homomorphisms in C ∗ -algebras, J. Comput. Anal. Appl. 15 (2013), 365-368. [36] C. Park, A. Najati and S. Jang, Fixed points and fuzzy stability of an additive-quadratic functional equation, J. Comput. Anal. Appl. 15 (2013), 452–462. [37] C. Park and Th.M. Rassias, Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal. 1 (2007), 515–528. [38] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. [39] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. [40] J. R¨ atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. ˇ ankov´ [41] L. Reich, J. Sm´ıtal and M. Stef´ a, Singular solutions of the generalized Dhombres functional equation, Results Math. 65 (2014), 251–261. [42] S. Schin, D. Ki, J. Chang and M. Kim, Random stability of quadratic functional equations: a fixed point approach, J. Nonlinear Sci. Appl. 4 (2011), 37–49. [43] S. Shagholi, M. Bavand Savadkouhi and M. Eshaghi Gordji, Nearly ternary cubic homomorphism in ternary Fr´echet algebras, J. Comput. Anal. Appl. 13 (2011), 1106–1114. [44] S. Shagholi, M. Eshaghi Gordji and M. Bavand Savadkouhi, Stability of ternary quadratic derivation on ternary Banach algebras, J. Comput. Anal. Appl. 13 (2011), 1097–1105. [45] D. Shin, C. Park and Sh. Farhadabadi, On the superstability of ternary Jordan C ∗ -homomorphisms, J. Comput. Anal. Appl. 16 (2014), 964–973. [46] D. Shin, C. Park and Sh. Farhadabadi, Stability and superstability of J ∗ -homomorphisms and J ∗ -derivations for a generalized Cauchy-Jensen equation, J. Comput. Anal. Appl. 17 (2014), 125–134. [47] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960. [48] J.Z. Xiao and X.H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003), 389–399. Ji-hye Kim Department of Mathematics, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected] George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA E-mail address: [email protected] Choonkil Park Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea E-mail address: [email protected]

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A NOTE ON BARNES-TYPE BOOLE POLYNOMIALS WITH λ-PARAMETER TAEKYUN KIM, DMITRY V. DOLGY, AND DAE SAN KIM

Abstract. In this paper, we consider Barnes-type Boole polynomials and give some formulae related to these polynomials.

1. Introduction Let p be a fixed odd prime number. Throughout this paper, Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp . The p-adic norm is normalized as |p|p = p1 . Let C (Zp ) be the space of continuous functions on Zp . For f ∈ C (Zp ), the fermionic p-adic integral on Zp is defined by Kim as (1.1) N Z pX −1 x f (x) (−1) , (see [1–19, 21, 22]) . I−1 (f ) = f (x) dµ−1 (x) = lim N →∞

Zp

x=0

From (1.1), we have (1.2)

I−1 (fn ) + (−1)

n−1

I−1 (f ) = 2

n−1 X

n−1−l

(−1)

f (l) ,

(see [14]) .

l=0

As is well known, the Boole polynomials are given by the generating function (1.3)

1

x

λ

(1 + t) + 1

(1 + t) =

∞ X

Bln (x | λ)

n=0

tn , n!

(see [10]) .

When x = 0, Bln (λ) = Bln (0 | λ) are called the Boole numbers. For a1 , a2 , . . . , ar ∈ Cp , the Barnes-type Euler polynomials are given by the generating function (1.4)

∞ X 2r tn xt e = E (x | a , . . . , a ) . n 1 r a t a t a t (e 1 + 1) (e 2 + 1) · · · (e r + 1) n! n=0

When x = 0, En (a1 , . . . , ar ) = En (0 | a1 , . . . , ar ) are called the Barnes-type Euler numbers (see [12, 20]). From (1.1), we can derive the following equation: Z (1.5) f (x) dµ−1 (x) Zp

2010 Mathematics Subject Classification. 11B75, 11B83, 11S80. Key words and phrases. Barnes-type Boole polynomial, Barnes-type Euler polynomial, fermionic p-adic integral. 1

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TAEKYUN KIM, DMITRY V. DOLGY, AND DAE SAN KIM

= lim

N →∞

N dpX −1

x

f (x) (−1)

x=0 N

= lim

N →∞

=

d−1 X

−1 d−1 pX X

a+x

f (a + dx) (−1)

a=0 x=0 a

Z

(−1)

a=0

f (a + dx) µ−1 (x) , Zp

where d ∈ N with d ≡ 1 (mod 2). In [10], Kim-Kim derived the Witt-type formula for Boole polynomials which are given by Z 1 x+λy (1.6) (1 + t) dµ0 (y) 2 Zp 1 x = (1 + t) λ (1 + t) + 1 ∞ X tn = Bln (x | λ) . n! n=0 In this paper, we consider Barnes-type Boole polynomials and give some formulae related to these polynomials. 2. Barnes-type Boole polynomials with λ-parameter Let a1 , a2 , . . . , ar ∈ Cp . Then, we consider the Barnes-type Boole polynomials which are given by the multivariate fermionic p-adic integral on Zp as follows: Z Z 1 λa y +λa2 y2 +···+λar yr +x (2.1) · · · (1 + t) 1 1 dµ−1 (y1 ) · · · dµ−1 (yr ) 2r Zp Zp ! r Y 1 x = (1 + t) λal 1 + (1 + t) l=1 =

∞ X

Bln,λ (x | a1 , . . . , ar )

n=0

tn . n!

(1)

Note that Bln,λ (x | 1) = Bln (x | λ), (n ≥ 0). When x = 0, Bln,λ (a1 , . . . , ar ) = Bln,λ (0 | a1 , . . . , ar ) are called the Barnes-type Boole numbers. From (2.1), we have Z Z 1 (2.2) · · · (λa1 y1 + · · · + λar yr + x)n dµ−1 (y1 ) · · · dµ−1 (yr ) 2r Zp Zp = Bln,λ (x | a1 , . . . , ar ) ,

(n ≥ 0) ,

where (x)n = x (x − 1) · · · (x − n + 1). We observe that (2.3) Z Z 1 · · · (λa1 y1 + · · · + λar yr + x)n dµ−1 (y1 ) · · · dµ−1 (yr ) 2r Zp Zp

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3

Z Z n 1 X l = r S1 (n, l) ··· (λa1 y1 + · · · + λar yr + x) dµ−1 (y1 ) · · · dµ−1 (yr ) 2 Zp Zp l=0 Z Z  n x l 1 X S1 (n, l) λl ··· a1 y1 + · · · + ar yr + dµ−1 (y1 ) · · · dµ−1 (yr ) , = r 2 λ Zp Zp l=0

where S1 (n, l) is the Stirling number of the first kind. From (1.2), we have Z Z ··· e(a1 x1 +···+ar xr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) (2.4) Zp

Zp

 2r ext (ea1 t + 1) · · · (ear t + 1) ∞ X tn = En (x | a1 , . . . , ar ) . n! n=0 

=

Thus, by (2.4), we get Z Z n ··· (a1 x1 + · · · + ar xr + x) dµ−1 (x1 ) · · · dµ−1 (xr ) (2.5) Zp

Zp

= En (x | a1 , . . . , ar ) ,

(n ≥ 0) .

From (2.2) and (2.5), we obtain the following theorem. Theorem 1. For n ≥ 0, we have n  x  1 X l S (n, l) λ E a , . . . , a 1 l 1 r = Bln,λ (x | a1 , . . . , ar ) . 2r λ l=0

By (2.1), we get  r  x 1 Y 2 eλt r a t l 2 e +1

(2.6)

l=1

=

=

∞ X

 Bln,λ (x | a1 , . . . , ar )

n=0 ∞ X m=0

λ

−m

m X

n 1 eλt − 1 n! !

Bln,λ (x | a1 , . . . , ar ) S2 (m, n)

n=0

tm , m!

where S2 (m, n) is the Stirling number of the second kind. By (1.4), we get  r  ∞  x  tm Y X x 2 t λ (2.7) e = E a , . . . , a . m 1 r eal t + 1 λ m! m=0 l=1

Therefore, by (2.6) and (2.7), we obtain the following theorem. Theorem 2. For m ≥ 0, we have  x  λm Em a1 , . . . , ar λ m X = 2r Bln,λ (x | a1 , . . . , ar ) S2 (m, n) . n=0

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TAEKYUN KIM, DMITRY V. DOLGY, AND DAE SAN KIM

From (1.5), we have (2.8) Z

Z

λa1 y1 +···+λar yr +x

···

(1 + t)

Zp d−1 X

=

dµ−1 (y1 ) · · · dµ−1 (yr )

Zp k1 +···+kr

(−1)

k1 ,...,kr =0

Z

Z

×

(1 + t)

Zp

dµ−1 (y1 ) · · · dµ−1 (yr )

Zp

d−1 X

=

λa1 (k1 +dy1 )+···+λar (kr +dyr )+x

···

Z

k1 +···+kr

  a1 k1 +···+ar kr + x λ +a y +···+a y λd 1 1 r r d

Z ···

(−1)

(1 + t)

Zp

k1 ,...,kr =0

Zp

×dµ−1 (y1 ) · · · dµ−1 (yr ) d−1 X

= 2r



∞ X

d−1 X

k1 +···+kr

(−1)

 n=0

tn n! 

Bln,λd (λa1 k1 + · · · + λar kr + x | a1 , . . . , ar )

n=0

k1 ,...,kr =0

= 2r

∞ X

k1 +···+kr

(−1)

Bln,λd (λa1 k1 + · · · + λar kr + x | a1 , . . . , ar )

k1 ,...,kr =0

tn , n!

where d ∈ N with d ≡ 1 (mod 2). From (2.8), we have (2.9) ∞ X





d−1 X

k1 +···+kr

(−1)

 n=0

1 = r 2 =

k1 ,...,kr =0

Z

Z

λa1 y1 +···+λar yr +x

···

(1 + t)

Zp

∞ X

Bln,dλ (λa1 k1 + · · · + λar kr + x | a1 , . . . , ar )

tn n!

dµ−1 (y1 ) · · · dµ−1 (yr )

Zp

Bln,λ (x | a1 , . . . , ar )

n=0

tn . n!

By comparing the coefficients on the both sides of (2.9), we obtain the following equation: Theorem 3. For d ∈ N with d ≡ 1 (mod 2), n ≥ 0, we have Bln,λ (x | a1 , . . . , ar ) =

d−1 X

k1 +···+kr

(−1)

Bln,λd (λa1 k1 + · · · + λar kr + x | a1 , . . . , ar ) .

k1 ,...,kr =0

Let d ∈ N with d ≡ 1 (mod 2). From (1.2), we have Z (2.10)

e Zp

a1 (y1 +d)t

Z dµ−1 (y1 ) + Zp

1130

ea1 y1 t dµ−1 (y) = 2

d−1 X

l

(−1) ea1 lt .

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A NOTE ON BARNES-TYPE BOOLE POLYNOMIALS WITH λ-PARAMETER

5

Thus, by (2.10), we get Z

ea1 y1 t dµ−1 (y) =

(2.11) Zp

d−1 X

2 ea1 dt + 1

l

(−1) ea1 lt .

l=0

From (2.11), we can derive Z Z (2.12) ··· e(a1 y1 +a2 y2 +···+ar yr +x)t dµ−1 (x1 ) · · · dµ−1 (xr ) Zp

Zp

d−1 X

=

l +···+lr

(−1) 1

l1 ,...,lr =0

=

∞ X

dn



2 eal dt + 1

l=1 d−1 X

n=0

r  Y



l +···+lr

(−1) 1

En

l1 ,...,lr =0

e(a1 l1 +···+ar lr +x)t

 n t a1 l1 + · · · + ar lr + x a , . . . , a . r 1 d n!

From (2.12) and (2.4), we get En (x | a1 , . . . , ar ) d−1 X

= dn



l +···+lr

(−1) 1

En

l1 ,...,lr =0

 a1 l1 + · · · + ar lr + x a , . . . , a r , 1 d

where d ∈ N with d ≡ 1 (mod 2). On the other hand, Z Z λa (d+y1 ) (2.13) (1 + t) 1 dµ−1 (y) + Zp d−1 X

=2

λa1 y1

(1 + t)

dµ−1 (y)

Zp l

λa1 l1

(−1) 1 (1 + t)

,

l1 =0

where d ∈ N such that d ≡ 1 (mod 2). By (2.13), we get (2.14) Z

Z

λa1 y1 +···+λar yr +x

···

(1 + t)

Zp

=

r Y

d−1 X

2

l +···+lr

(−1) 1

λal d

l=1 1 + (1 + t)

= 2r

∞ X

Blm,λd (a1 , . . . , ar )

×



= 2r

∞ X

l +···+lr

(−1) 1

(λa1 l1 + · · · + λar lr + x)k 

l1 ,...,lr =0



n   X n

 n=0

tm m! 

d−1 X

 k=0

k=0

λa1 l1 +···+λar lr +x

(1 + t)

l1 ,...,lr =0

m=0 ∞ X

dµ−1 (y1 ) · · · dµ−1 (yr )

Zp

k

d−1 X

tk k! 

l +···+lr

(−1) 1

(λa1 l1 + · · · + λar lr + x)k Bln−k,λd (a1 , . . . , ar )

l1 ,...,lr =0

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tn . n!

TAEKYUN KIM et al 1127-1134

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

6

TAEKYUN KIM, DMITRY V. DOLGY, AND DAE SAN KIM

From (2.9) and (2.14), we note that Bln (x | a1 , . . . , ar ) n   d−1 X X n l +···+lr = (−1) 1 (x + λa1 l1 + · · · + λar lr )k k

(2.15)

k=0

l1 ,...,lr =0

×Bln−k,λd (a1 , . . . , ar ) , where n ≥ 0 and d ∈ N with d ≡ 1 (mod 2). Therefore, by (2.15), we obtain the following theorem. Theorem 4. For n ≥ 0 and d ∈ N with d ≡ 1 (mod 2), we have Bln (x | a1 , . . . , ar ) n   d−1 X X n l +···+lr = (x + λa1 l1 + · · · + λar lr )k (−1) 1 k k=0

l1 ,...,lr =0

×Bln−k,λd (a1 , . . . , ar ) . From (2.14), we have (2.16) 1 2r =

Z

Z

λa1 y1 +···+λar yr

··· d−1 X

l1 +···+lr

(−1)

=



!

1

λa1 l1 +···+λar lr

λal d

(1 + t)



d−1 X

l +···+lr

(−1) 1

 n=0

r Y

l=1 1 + (1 + t)

l1 ,...,lr =0 ∞ X

dµ−1 (y1 ) · · · dµ−1 (yr )

(1 + t) Zp

Zp

l1 ,...,lr =0

n

t Bln,λd (λa1 l1 + · · · + λar lr | a1 , . . . , ar ) , n!

and Z

Z

λa1 y1 +···+λar yr

···

(2.17)

(1 + t)

= 2r

dµ−1 (y1 ) · · · dµ−1 (yr )

Zp

Zp ∞ X

Bln,λ (a1 , . . . , ar )

n=0

tn . n!

Therefore, by (2.16) and (2.17), we obtain the following theorem. Theorem 5. For n ≥ 0, d ∈ N with d ≡ 1 (mod 2), we have d−1 X

Bln,λ (a1 , . . . , ar ) =

l +···+lr

(−1) 1

Bln,λd (λa1 l1 + · · · + λar lr | a1 , . . . , ar ) .

l1 ,...,lr =0 1

By replacing t by e λ t − 1 in (2.14), we get Z

Z ···

(2.18) Zp

=

∞ X

x

Zp



n  X

 n=0

e(a1 y1 +···+ar yr + λ )t dµ−1 (y1 ) · · · dµ−1 (yr )

k=0

n k



d−1 X

l +···+lr

(−1) 1

(λa1 l1 + · · · + λar lr + x)k

l1 ,...,lr =0

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TAEKYUN KIM et al 1127-1134

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.6, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

A NOTE ON BARNES-TYPE BOOLE POLYNOMIALS WITH λ-PARAMETER

7

n 1  1t × Bln−k,λd (a1 , . . . , ar )) eλ − 1 n!    ∞ m X n d−1 X X X n l +···+lr −m  = λ S2 (m, n) (−1) 1 k m=0 n=0 k=0 l1 ,...,lr =0

× (λa1 l1 + · · · + λar lr + x)k Bln−k,λd (a1 , . . . , ar ))

tm m!

Thus, by (2.18), we get   x (2.19) λm Em a1 , . . . , ar λ   m X n d−1 X X n l +···+lr = S2 (m, n) (−1) 1 k n=0 k=0 l1 ,...,lr =0

× (λa1 l1 + · · · + λar lr + x)k Bln−k,λd (a1 , . . . , ar ) , where m ≥ 0 and d ∈ N with d ≡ 1 (mod 2). Therefore, by (2.19), we obtain the following theorem. Theorem 6. For m ≥ 0, d ∈ N with d ≡ 1 (mod 2), we have  x  λm Em a1 , . . . , ar λ   m n d−1 XX X n l +···+lr = S2 (m, n) (−1) 1 k n=0 k=0 l1 ,...,lr =0

× (λa1 l1 + · · · + λar lr + x)k Bln−k,λd (a1 , . . . , ar ) . ACKNOWLEDGEMENTS. This paper is supported by grant NO 14-11-00022 of Russian Scientific Fund References 1. S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 399–406. 2. , A note on the values of weighted q-Bernstein polynomials and weighted q-Genocchi numbers, Adv. Difference Equ. (2015), 2015:30, 9 pp. 3. A. Bayad and J. Chikhi, Apostol-Euler polynomials and asymtotics for negative binomial reciprocals, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2014), no. 1, 39–57. 4. A. Bayad and T. Kim, Results on values of Barnes polynomials, Rocky Mountain J. Math. 24 (2013), no. 1, 33–37. 5. I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, On the higher-order w-qGenocchi numbers, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2009), no. 1, 39–57. 6. D. Ding and J. Yang, Some identities related to the Apostol-Euler and ApostolBernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 7–21. 7. S. Gaboury, R. Tremblay, and B.-J. Fug´ere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Adv. Difference Equ. (2013), 2013:246, 10 pp.

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TAEKYUN KIM, DMITRY V. DOLGY, AND DAE SAN KIM

8. Y. He, Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials, Adv. Difference Equ. (2013), 2013:246, 10 pp. 9. J.-H. Jeong, J.-H. Jin, J.-W. Park, and S.-H. Rim, On the twisted weak q-Euler numbers and polynomials with weight 0, Proc. Jangjeon Math. Soc. 16 (2013), no. 2, 157–163. 10. D. S. Kim and T. Kim, A note on Boole polynomials, Integral Transform Spec. Funct. 25 (2014), no. 8, 627–633. 11. D. S. Kim, T. Kim, and J. J. Seo, A note on q-analogue of Boole polynomials, Appl. Math. Inf. Sci. 9 (2015), no. 6, 1–6. 12. T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys 10 (2003), no. 3, 261–267. , A note on p-adic invariant integral in the rings of p-adic integers, Adv. 13. Stud. Contemp. Math. (Kyungshang) 13 (2006), no. 1, 95–99. 14. , Symmetric p-adic invariant integral on Zp for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), no. 12, 1267–1277. , Some identities on the q-Euler polynomials of higher order and q15. Stirling numbers by fermionic p-adic integral on Zp , Russ. J. Math. Phys 16 (2009), no. 4, 484–491. , Barnes-type multiple q-zeta functions and q-euler polynomials, J. Phys. 16. A. 43 (2010), no. 25, 255201, 11 pp. 17. , Barnes’ type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. 258 (2015), 556–564. 18. T. Kim, D. V. Dolgy, Y. S. Jang, and J. J. Seo, A note on symmetric identities for the generalized q-Euler polynomials of the second kind, Proc. Jangjeon Math. Soc. 17 (2014), no. 3, 375–381. 19. Q.-M. Luo, q-analogue of some results for the Apostol-Euler polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 2010. 20. F. Qi, M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications , Appl. Math. Comput. 258 (2015), 597–607. 21. E. Sen, Theorems on Apostol-Euler polynomials of higher order arising from Euler basis, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 2, 337– 345. 22. Y. Simsek, Interpolation functions of the Eulerian type polynomials and numbers, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 2, 337–345. 23. Z. Zhang and H. Yang, Some closed formulas for generalized Bernoulli-Euler numbers and polynomials, Proc. Jangjeon Math. Soc. 11 (2008), no. 2, 191–198. Department of Mathematics, Tianjin Polytechnic University, Tianjin, China Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea E-mail address: [email protected], [email protected] School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia E-mail address: d− [email protected] Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea E-mail address: [email protected]

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TAEKYUN KIM et al 1127-1134

 

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TABLE OF CONTENTS, JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO. 6, 2016

Some Results of a New Integral Operator, Roberta Bucur, Loriana Andrei, Daniel Breaz,1017 A New Generalization Of The Ostrowski Inequality and Ostrowski Type Inequality for Double Integrals on Time Scales, A. Tuna, and S. Kutukcu,………………………………………1024 Approximate Ternary Jordan Bi-Homomorphisms in Banach Lie Triple Systems, Madjid Eshaghi Gordji, Amin Rahimi, Choonkil Park, and Dong Yun Shin,…………………….1040 Borel Directions and Uniqueness of Meromorphic Functions Sharing Five Values, Jianren Long, and Chunhui Qiu,…………………………………………………………………………..1046 On An Interval-Representable Generalized Pseudo-Convolution By Means Of the IntervalValued Generalized Fuzzy Integral and Their Properties, Jeong Gon Lee, Lee-Chae Jang,1060 Fixed Point and Coupled Fixed Point Theorems for Generalized Cyclic Weak Contractions in Partially Ordered Probabilistic Metric Spaces, Chuanxi Zhu, and Wenqing Xu,…………1073 Weak Galerkin Finite Element Method for Time Dependent Reaction-Diffusion Equation, Fuzheng Gao, and Guoqun Zhao,…………………………………………………………1086 Some Fixed Point Theorems for Generalized Expansive Mappings in Cone Metric Spaces over Banach Algebras, Binghua Jiang, Shaoyuan Xu, Huaping Huang, and Zelin Cai,………1103 Additive ρ-Functional Inequalities in Fuzzy Normed Spaces, Ji-Hye Kim, George A. Anastassiou, and Choonkil Park,…………………………………………………………1115 A Note on Barnes-Type Boole Polynomials with λ-Parameter, Taekyun Kim, Dmitry V. Dolgy, and Dae San Kim,………………………………………………………………………..1127

Volume 21, Number 7 ISSN:1521-1398 PRINT,1572-9206 ONLINE

December 15, 2016

Journal of Computational Analysis and Applications EUDOXUS PRESS,LLC

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

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Book 2. G.G.Lorentz, (title of book in italics) Bernstein Polynomials (2nd ed.), Chelsea,New York,1986.

Contribution to a Book 3. M.K.Khan, Approximation properties of beta operators,in(title of book in italics) Progress in Approximation Theory (P.Nevai and A.Pinkus,eds.), Academic Press, New York,1991,pp.483-495.

11. All acknowledgements (including those for a grant and financial support) should occur in one paragraph that directly precedes the References section. 12. Footnotes should be avoided. When their use is absolutely necessary, footnotes should be numbered consecutively using Arabic numerals and should be typed at the bottom of the page to which they refer. Place a line above the footnote, so that it is set off from the text. Use the appropriate superscript numeral for citation in the text. 13. After each revision is made please again submit via email Latex and PDF files of the revised manuscript, including the final one. 14. Effective 1 Nov. 2009 for current journal page charges, contact the Editor in Chief. Upon acceptance of the paper an invoice will be sent to the contact author. The fee payment will be due one month from the invoice date. The article will proceed to publication only after the fee is paid. The charges are to be sent, by money order or certified check, in US dollars, payable to Eudoxus Press, LLC, to the address shown on the Eudoxus homepage. No galleys will be sent and the contact author will receive one (1) electronic copy of the journal issue in which the article appears.

15. This journal will consider for publication only papers that contain proofs for their listed results.

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PRODUCT-TYPE OPERATORS FROM WEIGHTED BERGMAN-ORLICZ SPACES TO BLOCH-ORLICZ SPACES HONG-BIN BAI AND ZHI-JIE JIANG

Abstract. Let D = {z ∈ C : |z| < 1} be the open unit disk, ϕ an analytic self-map of D and ψ an analytic function on D. Let D be the differentiation operator and Wϕ,ψ the weighted composition operator. The boundedness and compactness of the product-type operators DWϕ,ψ from the weighted Bergman-Orlicz spaces to the Bloch-Orlicz spaces on D are characterized.

1. Introduction Let C be the complex plane, D = {z ∈ C : |z| < 1} the open unit disk and H(D) the class of all analytic functions on D. Let ϕ be an analytic self-map of D and ψ ∈ H(D). Weighted composition operator Wϕ,ψ on H(D) is defined by Wϕ,ψ f (z) = ψ(z)f (ϕ(z)), z ∈ D. If ψ ≡ 1, the operator is reduced to, so called, composition operator and usually denoted by Cϕ . If ϕ(z) = z, it is reduced to, so called, multiplication operator and usually denoted by Mψ . A standard problem is to provide function theoretic characterizations when ϕ and ψ induce a bounded or compact weighted composition operator. Composition operators and weighted composition operators between various spaces of holomorphic functions on different domains have been studied in many papers, see, for example, [1,3,8,11–15,17,19,22,26,27,31,33–36,40,42,48,53,55,60] and the references therein. Let D be the differentiation operator on H(D), that is, Df (z) = f 0 (z), z ∈ D. Operator DCϕ has been studied, for example, in [6, 16, 18, 24, 25, 28, 41, 45, 50]. In [32] Sharma studied the operators DMψ Cϕ and DCϕ Mψ from Bergman spaces to Bloch type spaces. These operators on weighted Bergman spaces were also studied in [58] and [59] by Stevi´c, Sharma and Bhat. If we consider the producttype operator DWϕ,ψ , it is clear that DMψ Cϕ = DWϕ,ψ and DCϕ Mψ = DWϕ,ψ◦ϕ . Quite recently, the present author has considered this operator in [7, 9]. For some other product-type operators, see, for example, [10,20,21,23,37–39,43,44,46,47,51, 52,54,56,61] and the references therein. This paper is devoted to characterizing the boundedness and compactness of the operators DWϕ,ψ from the weighted BergmanOrlicz spaces to the Bloch-Orlicz spaces. Next we are ready to introduce the needed spaces and some facts in [30]. The function Φ 6≡ 0 is called a growth function, if it is a continuous and nondecreasing 2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B33, 47B37. Key words and phrases. Weighted Bergman-Orlicz spaces, product-type operators, BlochOrlicz spaces, boundedness, compactness. E-mail address: [email protected], [email protected]. 1

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function from the interval [0, ∞) onto itself. It is clear that these conditions imply that Φ(0) = 0. It is said that the function Φ is of positive upper type (respectively, negative upper type), if there are q > 0 (respectively, q < 0) and C > 0 such that Φ(st) ≤ Ctq Φ(s) for every s > 0 and t ≥ 1. By Uq we denote the family of all growth functions Φ of positive upper type q (q ≥ 1), such that the function t 7→ Φ(t)/t is nondecreasing on [0, ∞). It is said that function Φ is of positive lower type (respectively, negative upper type), if there are r > 0 (respectively, r < 0) and C > 0 such that Φ(st) ≤ Ctr Φ(s) for every s > 0 and 0 < t ≤ 1. By Lr we denote the family of all growth functions Φ of positive lower type r (0 < r ≤ 1), such that the function t 7→ Φ(t)/t is nonincreasing on [0, ∞). If f ∈ Uq , we will also assume that it is convex. Let dA(z) = π1 dxdy be the normalized Lebesgue measure on D. For α > −1, let dAα (z) = (α + 1)(1 − |z|2 )α dA(z) be the weighted Lebesgue measure on D. Let Φ Φ be a growth function. The weighted Bergman-Orlicz space AΦ α (D) := Aα is the space of all f ∈ H(D) such that Z kf kAΦα = Φ(|f (z)|)dAα (z) < ∞. D

On

AΦ α

is defined the following quasi-norm Z  o n |f (z)|  dAα (z) ≤ 1 . kf klux = inf λ > 0 : Φ Φ Aα λ D

If Φ ∈ Uq or Φ ∈ Lr , then the quasi-norm on AΦ α is finite and called the Luxembourg norm. The classical weighted Bergman space Apα , p > 0, corresponds to Φ(t) = tp , consisting of all f ∈ H(D) such that Z kf kpApα = |f (z)|p dAα (z) < ∞. D

It is well known that for p ≥ 1 it is a Banach space, while for 0 < p < 1 it is a translation-invariant metric space with d(f, g) = kf − gkpApα . Moreover, if Φ ∈ Us , Φ

then Aα p , where Φp (t) = Φ(tp ), is a subspace of Apα ( [30]). Recently, the Bloch-Orlicz space was introduced in [29] by Ramos Fern´andez. More precisely, let Ψ be a strictly increasing convex function such that Ψ(0) = 0. From these conditions it follows that limt→∞ Ψ(t) = ∞. The Bloch-Orlicz space associated with the function Ψ, denoted by B Ψ , is the class of all f ∈ H(D) such that sup(1 − |z|2 )Ψ(λ|f 0 (z)|) < ∞ z∈D

for some λ > 0 depending on f . On B Ψ Minkowski’s functional n f0  o kf kΨ = inf k > 0 : SΨ ≤1 k defines a seminorm, where SΨ (f ) = sup(1 − |z|2 )Ψ(f (z)). z∈D

Moreover, B

Ψ

is a Banach space with the norm kf kBΨ = |f (0)| + kf kΨ .

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In fact, Ramos Fern´ andez in [29] proved that B Ψ is isometrically equal to µ-Bloch space, where 1 µ(z) = −1 , z ∈ D. 1 Ψ ( 1−|z| 2) Thus, for f ∈ B Ψ , we have kf kBΨ = |f (0)| + sup µ(z)|f 0 (z)|. z∈D Φ Aα p

We can study the operator DWϕ,ψ : → B Ψ with the help of this equivalent norm. It is obviously seen that if Ψ(t) = tp with p > 0, then the space B Ψ coincides with the weighted Bloch space B α (see [62]), where α = 1/p. Also, if Ψ(t) = t log(1 + t) then B Ψ coincides with the Log-Bloch space (see [2]). For the generalization of Log-Bloch spaces, see, for example, [49, 57]. Let X and Y be topological vector spaces whose topologies are given by translation invariant metrics dX and dY , respectively. It is said that a linear operator L : X → Y is metrically bounded if there exists a positive constant K such that dY (Lf, 0) ≤ KdX (f, 0) for all f ∈ X. When X and Y are Banach spaces, the metrical boundedness coincides with the usual definition of bounded operators between Banach spaces. Operator L : X → Y is said to be metrically compact if it maps bounded sets into relatively compact sets. When X and Y are Banach spaces, the metrical compactness coincides with the usual definition of compact operators between Banach spaces. Let X = AΦ α and Y a Banach space. The norm of operator L : X → Y is defined by kLkAΦα →Y = sup kLf kY kf kAΦ ≤1 α

and often written by kLk. Throughout this paper, an operator is bounded (respectively, compact), if it is metrically bounded (respectively, metrically compact). C will be a constant not necessary the same at each occurrence. The notation a . b means that a ≤ Cb for some positive constant C. When a . b and b . a, we write a ' b. 2. Auxiliary results In order to prove the compactness of the product-type operators, we need the following result which is similar to Proposition 3.11 in [4]. The details of the proof are omitted. Lemma 2.1. Let p ≥ 1, α > −1, and Φ ∈ Us such that Φp ∈ Lr . Then the Φ bounded operator DWϕ,ψ : Aα p → B Ψ is compact if and only if for every bounded Φp sequence {fn }n∈N in Aα such that fn → 0 uniformly on every compact subset of D as n → ∞, it follows that lim kDWϕ,ψ fn kBΨ = 0.

n→∞

We formulate the following two useful point estimates. For the first, see Lemma 2.4 in [30], and for the second, see Lemma 2.3 in [9].

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Lemma 2.2. Let p ≥ 1, α > −1 and Φ ∈ Us . Then for every f ∈ Aα p and z ∈ D we have  4 α+2  kf klux |f (z)| ≤ Φ−1 Φ . p Aα p 1 − |z|2 Lemma 2.3. Let p ≥ 1, α > −1, Φ ∈ Us and n ∈ N. Then there are two positive Φ constants Cn = C(α, p, n) and Dn = D(α, p, n) independent of f ∈ Aα p and z ∈ D such that   Cn Dn (n) −1 |f (z)| ≤ Φ kf klux Φ . Aα p (1 − |z|2 )n p (1 − |z|2 )α+2 We also need the following lemma which provides a class of useful test functions Φ in space Aα p (see [9]). Lemma 2.4. Let p > 0, α > −1 and Φ ∈ Us . Then for every t ≥ 0 and w ∈ D the Φ following function is in Aα p  C α+2  1 − |w|2  2(α+2) +t p fw,t (z) = Φ−1 , p 2 1 − |w| 1 − wz where C is an arbitrary positive constant. Moreover, sup kfw,t klux Φp . 1. Aα

w∈D

Φ

3. The operator DWϕ,ψ : Aα p → B Ψ Φ

First we characterize the boundedness of operator DWϕ,ψ : Aα p → B Ψ . We Φ assume that Φ ∈ Us such that Φp ∈ Lr . Under this assumption, Aα p is a complete metric space (see [30]). Theorem 3.1. Let p ≥ 1, α > −1, and Φ ∈ Us such that Φp ∈ Lr . Then the following conditions are equivalent: Φ

(i) The operator DWϕ,ψ : Aα p → B Ψ is bounded. (ii) Functions ϕ and ψ satisfy the following conditions:  α+2  4 M1 := sup µ(z) ψ 00 (z) Φ−1 < ∞, p 2 1 − |ϕ(z)| z∈D M2 := sup z∈D

 α+2  D1 µ(z) ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) Φ−1 < ∞, p 1 − |ϕ(z)|2 1 − |ϕ(z)|2

and M3 := sup z∈D

α+2   µ(z) D2 0 2 −1 < ∞. |ψ(z)||ϕ (z)| Φ p (1 − |ϕ(z)|2 )2 1 − |ϕ(z)|2 Φ

Moreover, if the operator DWϕ,ψ : Aα p → B Ψ is nonzero and bounded, then

DWϕ,ψ ' 1 + M1 + M2 + M3 .

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Φ

Proof. (i) ⇒ (ii). Suppose that DWϕ,ψ : Aα p → B Ψ is bounded. For w ∈ D, we choose the function  1 − |ϕ(w)|2  2(α+2)  1 − |ϕ(w)|2  2(α+2) +1 p p + c1 f1,ϕ(w) (z) = c0 1 − ϕ(w)z 1 − ϕ(w)z  1 − |ϕ(w)|2  2(α+2)  1 − |ϕ(w)|2  2(α+2) +2 +3 p p + c2 − , 1 − ϕ(w)z 1 − ϕ(w)z where 2(α + 2) + 3p 6(α + 2) + 9p 6(α + 2) + 9p , c1 = − , c2 = . 2(α + 2) 2(α + 2) + p 2(α + 2) + 2p

c0 =

By a direct calculation, we have 0 00 (ϕ(w)) = f1,ϕ(w) (ϕ(w)) = 0. f1,ϕ(w)

(1)

Using the function f1,ϕ(w) , we define the function  α+2  4 f1,ϕ(w) (z). f (z) = Φ−1 p 1 − |ϕ(w)|2 Applying (1) to f 0 and f 00 , we obtain f 0 (ϕ(w)) = f 00 (ϕ(w)) = 0.

(2)

It is clear that f (ϕ(w)) = CΦ−1 p



α+2  4 , 2 1 − |ϕ(w)|

(3)

where C=

2(α + 2) + 3p 6(α + 2) + 9p 6(α + 2) + 9p − + − 1 6= 0. 2(α + 2) 2(α + 2) + p 2(α + 2) + 2p Φ

By Lemma 2.4, f ∈ Aα p and kf kAΦp ≤ C. By (2), (3) and the boundedness of α

Φ

DWϕ,ψ : Aα p → B Ψ ,  µ(w) ψ 00 (w) Φ−1 p

α+2  4 ≤ CkDWϕ,ψ k, 2 1 − |ϕ(w)|

(4)

which means that M1 = sup µ(z)|ψ 00 (z)|Φ−1 p z∈D



α+2  4 ≤ CkDWϕ,ψ k < ∞. 1 − |ϕ(z)|2

(5)

Next we will prove M2 < ∞. For this we consider the functions f1 (z) = z and Φ f2 (z) ≡ 1, respectively. Since the operator DWϕ,ψ : Aα p → B Ψ is bounded, we have sup µ(z) ψ 00 (z)ϕ(z) + 2ψ 0 (z)ϕ0 (z) + ψ(z)ϕ00 (z) z∈D

≤ kDWϕ,ψ f1 kBΨ ≤ CkDWϕ,ψ k

(6)

and sup µ(z)|ψ 00 (z)| ≤ kDWϕ,ψ f2 kBΨ ≤ CkDWϕ,ψ k.

(7)

z∈D

By (6), (7) and the boundedness of ϕ, J1 := sup µ(z) ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) ≤ CkDWϕ,ψ k.

(8)

z∈D

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For w ∈ D, choose the function  1 − |ϕ(w)|2  2(α+2)  1 − |ϕ(w)|2  2(α+2) +1 p p f2,ϕ(w) (z) = c0 + c1 1 − ϕ(w)z 1 − ϕ(w)z  1 − |ϕ(w)|2  2(α+2)  1 − |ϕ(w)|2  2(α+2) +2 +3 p p − , + c2 1 − ϕ(w)z 1 − ϕ(w)z where c1 =

36p(α + 2)2 + 78p2 (α + 2) + 36p3 , [4(α + 2) + 2p][2(α + 2) + 2p][2(α + 2) + 3p] c2 =

4(α + 2)2 + 42p(α + 2) + 36p2 , [2(α + 2) + 2p][4(α + 2) + 6p]

and c0 = 1 − c1 − c2 . From a calculation, we obtain 00 (ϕ(w)) = 0. f2,ϕ(w) (ϕ(w)) = f2,ϕ(w)

(9)

Define the function g(z) = Φ−1 p



α+2  D1 f2,ϕ(w) (z). 1 − |ϕ(w)|2

Then by (9), g(ϕ(w)) = g 00 (ϕ(w)) = 0,

(10)

and by a direct calculation, g 0 (ϕ(w)) = C

 α+2  D1 ϕ(w) Φ−1 , p 2 2 1 − |ϕ(w)| 1 − |ϕ(w)|

(11)

Φ

where C = c1 + 2c2 − 3. Also by Lemma 2.4, g ∈ Aα p and kgkAΦp ≤ C. Since α

Φ

DWϕ,ψ : Aα p → B Ψ is bounded, we have µ(z)|(DWϕ,ψ g)0 (z)| ≤ CkDWϕ,ψ k,

(12)

for all z ∈ D. By (10) and (11), letting z = w in (12) gives α+2  −1  µ(w)|ϕ(w)| D1 00 0 0 Φp J(w) : = ψ(w)ϕ (w) + 2ψ (w)ϕ (w) 1 − |ϕ(w)|2 1 − |ϕ(w)|2 ≤ CkDWϕ,ψ k. (13) Hence sup J(z) ≤ CkDWϕ,ψ k.

(14)

z∈D

For the fixed δ ∈ (0, 1), by (8)  α+2  µ(z) D1 ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) Φ−1 sup p 2 1 − |ϕ(z)|2 {z∈D:|ϕ(z)|≤δ} 1 − |ϕ(z)|    J1 D1 α+2 ≤ Φ−1 ≤ CkDWϕ,ψ k, (15) p 2 1−δ 1 − δ2

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and by (14)  α+2  µ(z) D1 ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) Φ−1 p 2 1 − |ϕ(z)|2 {z∈D:|ϕ(z)|>δ} 1 − |ϕ(z)| sup

≤

1 sup J(z) ≤ CkDWϕ,ψ k. δ z∈D

(16)

Consequently, it follows from (15) and (16) that M2 ≤ CkDWϕ,ψ k < ∞.

(17)

Now we prove that M3 < ∞. First taking the function f (z) = z 2 , we have sup µ(z) ψ 00 (z)ϕ(z)2 + 4ψ 0 (z)ϕ0 (z)ϕ(z) + 2ψ(z)ϕ00 (z)ϕ(z) + 2ψ(z)ϕ0 (z)2 z∈D

≤ kDWϕ,ψ z 2 kBΨ ≤ CkDWϕ,ψ k

(18)

By (7) and the boundedness of ϕ, we obtain sup µ(z)|ψ 00 (z)||ϕ(z)|2 ≤ CkDWϕ,ψ k.

(19)

z∈D

From (8), (18), (19) and the boundedness of ϕ, it follows that J2 := sup µ(z)|ψ(z)||ϕ0 (z)|2 ≤ CkDWϕ,ψ k.

(20)

z∈D

For w ∈ D, consider the function  1 − |ϕ(w)|2  2(α+2)  1 − |ϕ(w)|2  2(α+2) +1 p p f3,ϕ(w) (z) = c0 + c1 1 − ϕ(w)z 1 − ϕ(w)z  1 − |ϕ(w)|2  2(α+2)  1 − |ϕ(w)|2  2(α+2) +2 +3 p p + c2 − , 1 − ϕ(w)z 1 − ϕ(w)z where c0 =

2(α + 2) + p 3(α + 2) + 4p 6(α + 2) + 7p , c1 = − , c2 = . 2(α + 2) + 2p α+2+p 2(α + 2) + 2p

For the function f3,ϕ(w) , we have 0 f3,ϕ(w) (ϕ(w)) = f3,ϕ(w) (ϕ(w)) = 0.

(21)

For the function h(z) = Φ−1 p



α+2  D2 f3,ϕ(w) (z), 1 − |ϕ(w)|2

it follows from (21) that h(ϕ(w)) = h0 (ϕ(w)) = 0.

(22) Φ

By (21) and (22), the boundedness of the operator DWϕ,ψ : Aα p → B Ψ gives  α+2  µ(w)|ϕ(w)|2 D2 |ψ(w)||ϕ0 (w)|2 Φ−1 ≤ CkDWϕ,ψ k. K(w) : = p 2 2 2 (1 − |ϕ(w)| ) 1 − |ϕ(w)| This yields sup K(z) ≤ CkDWϕ,ψ k < ∞.

(23)

z∈D

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For the fixed δ ∈ (0, 1), by (20) and (23) we respectively obtain  α+2  D2 µ(z) 0 2 −1 sup |ψ(z)||ϕ (z)| Φ p 2 2 1 − |ϕ(z)|2 {z∈D:|ϕ(z)|≤δ} (1 − |ϕ(z)| )    D2 α+2 J2 ≤ Φ−1 ≤ CkDWϕ,ψ k p 2 2 (1 − δ ) 1 − δ2

(24)

and  α+2  µ(z) D2 0 2 −1 |ψ(z)||ϕ (z)| Φ p 2 2 1 − |ϕ(z)|2 {z∈D:|ϕ(z)|>δ} (1 − |ϕ(z)| ) sup

≤

1 sup K(z) ≤ CkDWϕ,ψ k. δ 2 z∈D

(25)

So, by (24) and (25) we have M3 ≤ CkDWϕ,ψ k < ∞.

(26)

Φ

(ii) ⇒ (i). By Lemmas 2.2 and 2.3, for all f ∈ Aα p we have kDWϕ,ψ f kBΨ = (ψ · f ◦ ϕ)0 (0) + sup µ(z) (ψ · f ◦ ϕ)00 (z) z∈D

≤ (ψ · f ◦ ϕ)0 (0) + + sup µ(z) ψ 00 (z) f (ϕ(z)) z∈D 00 + sup µ(z) ψ(z)ϕ (z) + 2ψ 0 (z)ϕ0 (z) f 0 (ϕ(z)) z∈D + sup µ(z)|ψ(z)||ϕ0 (z)|2 f 00 (ϕ(z)) z∈D


≤ C 1 + M1 + M2 + M3 kf kAΦp .

(27)

α

Φ

From condition (ii) and (27), it follows that DWϕ,ψ : Aα p → B Ψ is bounded. Φ Suppose that the operator DWϕ,ψ : Aα p → B Ψ is nonzero and bounded. Then from the preceding inequalities (5), (17) and (26), we obtain M1 + M2 + M3 . kDWϕ,ψ k.

(28)

Φ Aα p

Since the operator DWϕ,ψ : → B Ψ is nonzero, we have kDWϕ,ψ k > 0. From this, we can find a positive constant C such that 1 ≤ CkDWϕ,ψ k. This means that 1 . kDWϕ,ψ k.

(29)

Hence, combing (28) and (29) gives 1 + M1 + M2 + M3 . kDWϕ,ψ k.

(30)

From (27), it is clear that kDWϕ,ψ k . 1 + M1 + M2 + M3 .

(31)

So, from (30) and (31), we obtain the asymptotic expression of kDWϕ,ψ k. The proof is finished.  Φ

Remark 3.1. If DWϕ,ψ : Aα p → B Ψ is a zero operator, then is obviously kDWϕ,ψ k = 0. Hence, the case is usually excluded from such considerations. Φ

Now we characterize the compactness of operator DWϕ,ψ : Aα p → B Ψ . Theorem 3.2. Let p ≥ 1, α > −1, and Φ ∈ Us such that Φp ∈ Lr . Then the following conditions are equivalent:

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Φ

(i) The operator DWϕ,ψ : Aα p → B Ψ is compact. (ii) Functions ϕ and ψ are such that ψ 0 ∈ B Ψ , J1 := sup µ(z) ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) < ∞, z∈D

J2 := sup µ(z)|ψ(z)||ϕ0 (z)|2 < ∞, z∈D

lim

00 −1  ψ (z) Φp µ(z) −

|ϕ(z)|→1

lim

|ϕ(z)|→1−

α+2  4 = 0, 1 − |ϕ(z)|2

 α+2  µ(z) D1 ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) Φ−1 = 0, p 2 2 1 − |ϕ(z)| 1 − |ϕ(z)|

and lim

|ϕ(z)|→1−

 α+2  D2 µ(z) 0 2 −1 |ψ(z)||ϕ (z)| Φ = 0. p (1 − |ϕ(z)|2 )2 1 − |ϕ(z)|2 Φ

Proof. (i) ⇒ (ii). Suppose that (i) holds. Then the operator DWϕ,ψ : Aα p → B Ψ is bounded. In the proof of Theorem 3.1, we have obtained that ψ 0 ∈ B Ψ and J1 , J2 < ∞. Next consider a sequence {ϕ(zn )}n∈N in D such that |ϕ(zn )| → 1− as n → ∞. If such sequence does not exist, then condition (ii) obviously holds. Using this sequence, we define the functions  α+2  4 fn (z) = Φ−1 f1,ϕ(zn ) (z). p 2 1 − |ϕ(zn )| Φ

By Lemma 2.4, we know that the sequence {fn }n∈N is uniformly bounded in Aα p . From the proof of Theorem 3.6 in [30], it follows that the sequence {fn }n∈N uniformly converges to zero on any compact subset of D as n → ∞. Hence by Lemma 2.1, lim kDWϕ,ψ fn kBΨ = 0. n→∞

From this, (2) and (3), we have  lim µ(zn ) ψ 00 (zn ) Φ−1 p n→∞

α+2  4 = 0. 1 − |ϕ(zn )|2

By using the sequence of functions α+2   D1 f2,ϕ(zn ) (z), gn (z) = Φ−1 p 2 1 − |ϕ(zn )| similar to the above, we obtain α+2   µ(zn ) D1 lim = 0. ψ(zn )ϕ00 (zn ) + 2ψ 0 (zn )ϕ0 (zn ) Φ−1 p 2 2 n→∞ 1 − |ϕ(zn )| 1 − |ϕ(zn )| Also, by using sequence of functions  α+2  D2 hn (z) = Φ−1 f3,ϕ(zn ) (z), p 1 − |ϕ(zn )|2 we obtain  α+2  D2 µ(zn ) 0 2 −1 |ψ(z )||ϕ (z )| Φ = 0. n n p n→∞ (1 − |ϕ(zn )|2 )2 1 − |ϕ(zn )|2 lim

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The proof of the implication is finished. Φ (ii) ⇒ (i). We first check that DWϕ,ψ : Aα p → B Ψ is bounded. For this we observe that condition (ii) implies that for every ε > 0, there is an η ∈ (0, 1) such that  α+2  4 L1 (z) := µ(z) ψ 00 (z) Φ−1 < ε, (32) p 1 − |ϕ(z)|2 L2 (z) :=

 α+2  D1 µ(z) ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) Φ−1 < ε, (33) p 1 − |ϕ(z)|2 1 − |ϕ(z)|2

and L3 (z) :=

 α+2  D2 µ(z) 0 2 −1 |ψ(z)||ϕ (z)| Φ < ε, p (1 − |ϕ(z)|2 )2 1 − |ϕ(z)|2

(34)

for any z ∈ K = {z ∈ D : |ϕ(z)| > η}. Then since ψ 0 ∈ B Ψ and by (32), we have  4 α+2  + ε. M1 = sup L1 (z) ≤ sup L1 (z) + sup L1 (z) ≤ kψ 0 kBΨ Φ−1 p 1 − η2 z∈K z∈D z∈D\K By (33) and J1 < ∞, we obtain M2 = sup L2 (z) ≤ sup L2 (z) + sup L2 (z) ≤ z∈D

z∈K

z∈D\K

 D α+2  J1 1 −1 Φ + ε. p 1 − η2 1 − η2 Φ

By (34) and J2 < ∞, it follows that M3 < ∞. So by Theorem 3.1, DWϕ,ψ : Aα p → B Ψ is bounded. Φ To prove that the operator DWϕ,ψ : Aα p → BΨ is compact, by Lemma 2.1 we Φ just need to prove that, if {fn }n∈N is a sequence in Aα p such that kfn kAΦp ≤ M α and fn → 0 uniformly on any compact subset of D as n → ∞, then lim kDWϕ,ψ fn kBΨ = 0.

n→∞

For any ε > 0 and the above η, we have, by using again the condition (ii), Lemma 2.2 and Lemma 2.3, sup µ(z) (DWϕ,ψ fn )0 (z) = sup µ(z) (ψ · fn ◦ ϕ)00 (z) ≤ sup µ(z) ψ 00 (z) fn (ϕ(z)) z∈D

z∈D

z∈D

+ sup µ(z) ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) fn0 (ϕ(z)) + sup µ(z)|ψ(z)||ϕ0 (z)|2 fn00 (ϕ(z)) z∈D

z∈D

≤ sup µ(z) ψ 00 (z) fn (ϕ(z)) + sup µ(z) ψ 00 (z) fn (ϕ(z)) z∈K

z∈D\K

+ sup µ(z) ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) fn0 (ϕ(z)) z∈D\K

+ sup µ(z) ψ(z)ϕ00 (z) + 2ψ 0 (z)ϕ0 (z) fn0 (ϕ(z)) z∈K + sup µ(z)|ψ(z)||ϕ0 (z)|2 fn00 (ϕ(z)) + sup µ(z)|ψ(z)||ϕ0 (z)|2 fn00 (ϕ(z)) z∈K

z∈D\K

≤ Kn + M sup L1 (z) + M sup L2 (z) + M sup L3 (z) z∈K

z∈K

z∈K

≤ Kn + 3M ε, where Kn = kψ 0 kBΨ

sup

2 X fn (z) + Ji

{z:|z|≤η}

i=1

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Hence kDWϕ,ψ fn kBΨ ≤ Kn + 3M ε + (ψ · fn ◦ ϕ)0 (0) = Kn + 3M ε + ψ 0 (0)fn (ϕ(0)) + ψ(0)fn0 (ϕ(0))ϕ0 (0) .

(35)

It is easy to see that, when {fn }n∈N uniformly converges to zero on any compact subset of D, {fn0 }n∈N and {fn00 }n∈N also do as n → ∞. From this, we obtain Kn → 0 as n → ∞. Since {z : |z| ≤ η} and {ϕ(0)} are compact subsets of D, letting n → ∞ in (35) gives lim kDWϕ,ψ fn kBΨ = 0. n→∞

Φ

From Lemma 2.1, it follows that the operator DWϕ,ψ : Aα p → B Ψ is compact. The proof is finished.  Acknowledgments. This work is supported by the National Natural Science Foundation of China (No.11201323), the Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No.2013QZJ01, No.2013YY01) and the Fund Project of Sichuan Provincial Department of Education (No.12ZB288).

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Lyapunov inequalities of linear Hamiltonian systems on time scales Jing Liu1 1

Taixiang Sun1

Xin Kong1

Qiuli He2,∗

Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications, Nanning, 530004, China 2

College of Electrical Engineering, Guangxi University, Nanning, Guangxi 530004, China

Abstract In this paper, we establish several Lyapunov-type inequalities for the following linear Hamiltonian systems x∆ (t) = −A(t)x(σ(t)) − B(t)y(t), y ∆ (t) = C(t)x(σ(t)) + AT (t)y(t) on the time scale interval [a, b]T ≡ [a, b] ∩ T for some a, b ∈ T, where B and C are real n × n symmetric matrix-valued functions on [a, b]T with B being semi-positive definite, A is real n × n matrix-valued function on [a, b]T with I + µ(t)A being invertible, and x, y are real vector-valued functions on [a, b]T . AMS Subject Classification: 34K11, 34N05, 39A10. Keywords: Lyapunov inequality; Hamiltonian system; Time scale

1. Introduction In 1990, Hilger introduced in [9] the theory of time scales with one goal being the unified treatment of differential equations (the continuous case) and difference equations (the discrete case). A time scale T is an arbitrary nonempty closed subset of the real numbers R, which has the topology that it inherits from the standard topology on R. The two most popular examples are R and the integers Z. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice-once for differential equations and once again for difference equations. Not only can the theory of dynamic equations unify the theories of differential equations and difference equations, but also extends these classical cases to cases “in between”, e.g., to the so-called q-difference equations when T = {1, q, q 2 , · · · , q n , · · · }, which has important applications in quantum theory (see [11]). For the time scale calculus, and some related basic concepts, and the basic notions connected to time scales, we refer the readers to the books by Bohner and Peterson [2,3] for further details. In this paper, we study Lyapunov-type inequalities for the following linear Hamiltonian Project Supported by NNSF of China (11461003) and NSF of Guangxi (2012GXNSFDA276040). ∗ Corresponding author: E-mail: [email protected]

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systems x∆ (t) = −A(t)x(σ(t)) − B(t)y(t), y ∆ (t) = C(t)x(σ(t)) + AT (t)y(t),

(1.1)

on the time scale interval [a, b]T ≡ [a, b] ∩ T for some a, b ∈ T, where B and C are real n × n symmetric matrix-valued functions on [a, b]T with B being semi-positive definite, A is real n × n matrix-valued function on [a, b]T with I + µ(t)A being invertible, and x, y are real vector-valued functions on [a, b]T . When n = 1, (1.1) reduces to x∆ (t) = α(t)x(σ(t)) + β(t)y(t), y ∆ (t) = −γ(t)x(σ(t)) − α(t)y(t)

(1.2)

on an arbitrary time scale T, where α(t), β(t) and γ(t) are real-valued rd-continuous functions defined on T with β(t) ≥ 0 for any t ∈ T. In [10], Jiang and Zhou obtained some interesting Lyapunov-type inequalities. Theorem 1.1[10] Suppose that for any t ∈ T, 1 − µ(t)α(t) > 0, β(t) > 0, γ(t) > 0, and let a, b ∈ Tk with σ(a) < b. Assume that (1.2) has a real solution (x(t), y(t)) such that x(a)x(σ(a)) < 0, and x(b)x(σ(b)) < 0. Then the inequality Z

hZ

b

|α(t)| 4 (t) + a

Z

σ(b)

b

β(t) 4 (t) a

i1/2 γ(t) 4 (t) >1

(1.3)

a

holds. Theorem 1.2[10] Suppose that for any t ∈ T, 1 − µ(t)α(t) > 0, β(t) > 0, and let a, b ∈ Tk with σ(a) < b. Assume that (1.2) has a real solution (x(t), y(t)) such that x(a)x(σ(a)) < 0, and x(σ(b)) = 0. Then the inequality Z

hZ

b

|α(t)| 4 (t) + σ(a)

Z

σ(b)

β(t) 4 (t) σ(a)

b

i1/2 γ + (t) 4 (t) >1

(1.4)

a

holds, where γ + (t) = max{γ(t), 0}. In [8], He et al. obtained the following Lyapunov-type inequality. Theorem 1.3[8] Suppose for any t ∈ T, 1 − µ(t)α(t) > 0, and let a, b ∈ Tk with σ(a) ≤ b. Assume that (1.2) has a real solution (x(t), y(t)) such that x(t) has generalized zeros at end-points a and b and x(t) is not identically zero on [a, b]T ≡ {t ∈T : a ≤ t ≤ b}, i.e., x(a) = 0 or x(a)x(σ(a)) < 0; x(b) = 0 or x(b)x(σ(b)) < 0; max |x(t)| > 0. t∈[a,b]T

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Then the inequality Z

hZ

b

a

Z

σ(b)

|α(t)| 4 (t) +

β(t) 4 (t)

b

i1/2 γ + (t) 4 (t) ≥2

(1.5)

a

a

holds. For some other related results on Lyapunov-type inequality, see, for example, [1,4-6,8,10,1216].

2. Preliminaries and some lemmas For any x ∈ Rn and any A ∈ Rn×n (the space of real n × n matrices), denote by |x| =

√

xT x

and

|A| =

max

x∈Rn ,|x|=1

|Ax|

the Euclidean norm of x and the matrix norm of A respectively, where C T is the transpose of a n × m matrix C. It is easy to show |Ax| ≤ |A||x| the space of all symmetric real n × n for any x ∈ Rn and any A ∈ Rn×n . Denote by Rn×n s matrices. For A ∈ Rn×n , we say that A is semi-positive definite (resp. positive definite), written s as A ≥ 0 (resp. A > 0), if xT Ax ≥ 0 (resp. xT Ax > 0) for all x ∈ Rn . If A is semi-positive definite (resp. positive definite), then there exists a unique semi-positive definite matrix (resp. √ √ positive definite matrix), written as A, such that [ A]2 = A. In this paper, we study Lyapunov-type inequalities of (1.1) which admits some solution (x(t), y(t)) satisfying x(a) = x(b) = 0 and max |x(t)| > 0, t∈[a,b]T

(2.1)

where a, b ∈ T with σ(a) < b, A, B, C ∈ Crd (T, Rn×n ) are n × n-matrix-valued functions on T and B ≥ 0. we first introduce the following notions with I + µ(t)A being invertible, B,C ∈ Rn×n s and lemmas. A partition of [a, b)T is any finite ordered subset P = {t0 , t1 , · · · , tn } ⊂ [a, b]T with a = t0 < t1 < · · · < tn = b. For given δ > 0, we denote by Pδ ([a, b)T ) the set of all partitions P : a = t0 < t1 < · · · < tn = b that possess the property: for every i ∈ {1, 2, · · · , n}, either ti − ti−1 ≤ δ or ti − ti−1 > δ and σ(ti ) = ti−1 . Definition 2.1[7] Let f be a bounded function on [a, b)T , and let P : a = t0 < t1 < · · · < tn = b be a partition of [a, b)T . In each interval [ti−1 , ti )T (1 ≤ i ≤ n), choose an arbitrary point ξi and form the sum S(P, f ) = Σni=1 f (ξi )(ti − ti−1 ). We say that f is ∆-integrable from a to b (or on [a, b)T ) if there exists a constant number I with the following property: for each ε > 0 there exists δ > 0 such that |S(P, f ) − I| < ε

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for every P ∈ Pδ ([a, b)T ) independent of the way in which we choose ξi ∈ [ti−1 , ti )T (1 ≤ i ≤ n). Rb It is easily seen that such a constant number I is unique. The number I, written as a f (t)∆t, is called the ∆-integral of f from a to b. Remark 2.2 In [7], Guseinov showed that if there exists F : T → R such that F 4 (t) = f (t) holds for all t ∈ Tk , then Z

b

f (t) 4 t = F (b) − F (a), for any a, b ∈ T. a

Lemma 2.3 Let ai , bi , ci ∈ R (i ∈ {1, 2, · · · , n}) with ci ≥ 0. Then n ³X

ai ci

´2

n ³X

+

i=1

bi ci

´2

≤

i=1

n q hX

i2 a2i + b2i ci .

(2.2)

i=1

Proof. Since 2ai bi aj bj ≤ b2i a2j + b2j a2i for any i, j ∈ {1, 2, · · · , n}, we have q a2i + b2i ci

ai ci aj cj + bi ci bj cj ≤

q

a2j + b2j cj ,

which implies n X n X

n X n q X

(ai ci aj cj + bi ci bj cj ) ≤

i=1 j=1

That is

a2i + b2i ci

q a2j + b2j cj .

i=1 j=1

n ³X

ai ci

´2

n ³X

+

i=1

bi ci

´2

≤

i=1

n q hX

i2 a2i + b2i ci .

i=1

This completes the proof of Lemma 2.3 Lemma 2.4 Let f, g, f 2 + g 2 be ∆-integrable from a to b. Then hZ

b

f (t)∆t a

i2

hZ +

b

g(t)∆t

i2 hZ bp f 2 (t) + g 2 (t)∆t . ≤

i2

a

(2.3)

a

Proof. By Definition 2.1, for any ε > 0 there exists δi > 0 (i = 1, 2, 3) such that Z |S(P1 , f ) −

b

Z |S(P2 , g) − and

f (t)∆t| < ε,

(2.4)

g(t)∆t| < ε

(2.5)

a b

a

Z bp p 2 2 |S(P3 , f (t) + g (t)) − f 2 (t) + g 2 (t)∆t| < ε

(2.6)

a

for every Pi ∈ Pδi ([a, b)T ). Let P = P1 ∪ P2 ∪ P3 (∈ ∩3i=1 Pδi ([a, b)T )) : a = t0 < t1 < · · · < tn = b

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and choose an arbitrary point ξi ∈ [ti−1 , ti ). Then from (2.4)-(2.6) and Lemma 2.3 we have hZ b i2 h Z b i2 g(t)∆t ≤ [|S(P, f )| + ε]2 + [|S(P, g)| + ε]2 f (t)∆t + a

a

= [|Σni=1 f (ξi )(ti − ti−1 )| + ε]2 + [|Σni=1 g(ξi )(ti − ti−1 )| + ε]2 ≤ [Σni=1 f (ξi )(ti − ti−1 )]2 + [Σni=1 g(ξi )(ti − ti−1 )]2 Z b h Z b i +2ε | f (t)∆t| + | g(t)∆t| + 3ε a pa n 2 2 ≤ [Σi=1 f (ξi ) + g (ξi )(ti − ti−1 )]2 Z b h Z b i +2ε | f (t)∆t| + | g(t)∆t| + 3ε a a Z bp f 2 (t) + g 2 (t)∆t + ε]2 ≤ [ a Z b h Z b i +2ε | f (t)∆t| + | g(t)∆t| + 3ε . a

a

Let ε −→ 0, we obtain (2.3). This completes the proof of Lemma 2.4. Corollary 2.5 Let a, b ∈ T with a < b and f1 (t), f2 (t), · · · , fn (t) be ∆-integrable on [a, b]T . write x(t) = (f1 (t), f2 (t), · · · , fn (t)). Then Z b Z b n ³Z b n nX ´2 o 1 Z b n X o1 2 2 2 | x(t)∆t| = fi (t)∆t ≤ |x(t)|∆t. fi (t) ∆t = a

a

i=1

a

(2.7)

a

i=1

Proof. By Lemma 2.4, we know that (2.7) holds when n = 2. Assume that (2.7) holds when n = k ≥ 2, that is

k ³Z X

b

a

i=1

fi (t)∆t

´2

≤

k hZ bnX a

o 1 i2 2 fi2 (t) ∆t .

i=1

Then k+1 hZ bnX a

o1

fi2 (t)

2

∆t

i2

nZ

b

= a

i=1

³Z ≥ a

≥

k o2 X 1 1 2 {fk+1 (t) + [( fi2 (t)) 2 ]2 } 2 ∆t i=1

b

fk+1 (t)∆t

k+1 ³ Z b X i=1

a

´2

fi (t)∆t

hZ

+ a

´2

b

k i2 X 1 { fi2 (t)} 2 ∆t i=1

.

This completes the proof of Corollary 2.5. Lemma 2.6[2] (Cauchy-Schwarz inequality) Let a, b ∈ T and f, g ∈ Crd (T, R). Then Z b Z b o1 nZ b 2 2 g 2 (t) 4 (t) . f (t) 4 (t) · |f (t)g(t)| 4 (t) ≤ a

Lemma 2.7[2]

(2.8)

a

a

Suppose that A ∈ Crd (T, Rn×n ) with I + µ(t)A being invertible and f ∈

Crd (T, Rn ). Let t0 ∈ T and xo ∈ Rn . Then the initial value problem x∆ (t) = −A(t)x(σ(t)) + f (t), x(t0 ) = x0 5 1164

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has a unique solution x : T → Rn . Moreover, this solution is given by Z t x(t) = eΘA (t, t0 )x0 + eΘA (t, τ )f (τ )∆τ.

(2.9)

t0

Lemma 2.8 Let C ∈ Rn×n . Then for any C1 ∈ Rn×n with C1 ≥ C (i.e.,C1 − C ≥ 0), we have s s (xσ )T Cxσ ≤ |C1 ||xσ |2 , x ∈ Rn .

(2.10)

Proof. For C, C1 ∈ Rn×n with C1 ≥ C, we have C1 − C ≥ 0. Then for all x ∈ Rn , we obtain s (xσ )T (C1 − C)xσ ≥ 0. Thus (xσ )T Cxσ ≤ (xσ )T C1 xσ ≤ |xσ ||C1 xσ | ≤ |xσ ||C1 ||xσ | = |C1 ||xσ |2 . This completes the proof of Lemma 2.8.

3. Main results and proofs Denote

Z

σ(t)

ξ(σ(t)) = a

and

Z

b

η(σ(t)) = σ(t)

|B(s)||eΘA (σ(t), s)|2 ∆s

(3.1)

|B(s)||eΘA (σ(t), s)|2 ∆s.

(3.2)

Theorem 3.1 Let a, b ∈ T with σ(a) < b. If (1.1) has a solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then for any C1 ∈ Rn×n with C1 (t) ≥ C(t), one has the following inequality s Z b ξ(σ(t))η(σ(t)) |C1 (t)| 4 t ≥ 1. (3.3) ξ(σ(t)) + η(σ(t)) a Proof. At first let us notice that any solution (x(t), y(t)) of (1.1) satisfies the following equality (y T (t)x(t))∆ = (y T (t))∆ xσ (t) + y T (t)x∆ (t) = (xσ (t))T y ∆ (t) + y T (t)x∆ (t) = (xσ (t))T C(t)xσ (t) − y T (t)B(t)y(t).

(3.4)

By integrating (3.4) from a to b and taking into account that x(a) = x(b) = 0, one has Z b Z b T y (t)B(t)y(t) 4 t = (xσ (t))T C(t)xσ (t) 4 t. a

a

Moreover, since B(t) is semi-positive definite, we have y T (t)B(t)y(t) ≥ 0, t ∈ [a, b]T . If y T (t)B(t)y(t) ≡ 0, t ∈ [a, b]T 6 1165

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then B(t)y(t) = 0. Thus the first equation of (1.1) would read as x∆ (t) = −A(t)x(σ(t)), x(a) = 0. By Lemma 2.7, it follows x(t) = eΘA (t, a) · 0 = 0, a contradiction with (2.1). Hence we have that Z b Z b T y (t)B(t)y(t) 4 t = (xσ )T (t)C(t)xσ (t) 4 t > 0, a

(3.5)

a

and for t ∈ [a, b]T , let t0 = a and t0 = b, from Lemma 2.7, we obtain Z t Z t x(t) = − eΘA (t, τ )B(τ )y(τ )∆τ = − eΘA (t, τ )B(τ )y(τ )∆τ.

(3.6)

Which follows that for t ∈ [a, b)T , Z σ(t) Z σ x (t) = − eΘA (σ(t), τ )B(τ )y(τ )∆τ = +

(3.7)

a

b

a

b

σ(t)

eΘA (σ(t), τ )B(τ )y(τ )∆τ.

Note that for a ≤ τ ≤ σ(t) ≤ b, |eΘA (σ(t), τ )B(τ )y(τ )| ≤ |eΘA (σ(t), τ )||B(τ )y(τ )| 1

= |eΘA (σ(t), τ )|{y T (τ )B T (τ )B(τ )y(τ )} 2 p p 1 = |eΘA (σ(t), τ )|{( B(τ )y(τ ))T B(τ ) B(τ )y(τ )} 2 p p 1 ≤ |eΘA (σ(t), τ )|{| B(τ )y(τ )||B(τ )|| B(τ )y(τ )|} 2 1

1

= |eΘA (σ(t), τ )||B(τ )| 2 (y T (τ )B(τ )y(τ )) 2 . Then from Corollary 2.5 and Lemma 2.6 we obtain Z σ(t) σ |x (t)| = | eΘA (σ(t), τ )B(τ )y(τ )∆τ | Z

a σ(t)

|eΘA (σ(t), τ )B(τ )y(τ )|∆τ

≤ a

Z

σ(t)

≤ a

³Z

1

1

|eΘA (σ(t), τ )||B(τ )| 2 (y T (τ )B(τ )y(τ )) 2 ∆τ

σ(t)

≤

´1 ³ Z

2

|eΘA (σ(t), τ )| |B(τ )|4τ

a

that is

Z σ

σ(t)

2

|x (t)| ≤ ξ(σ(t))

2

σ(t)

y T (τ )B(τ )y(τ )∆τ

´1

2

,

a

y T (τ )B(τ )y(τ )∆τ.

(3.8)

a

Similarly, by letting η(σ(t)) be as in (3.2), for a ≤ σ(t) ≤ τ ≤ b, we have Z b y T (τ )B(τ )y(τ )∆τ. |xσ (t)|2 ≤ η(σ(t))

(3.9)

σ(t)

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It follows from (3.8) and (3.9) that Z

σ(t)

η(σ(t))ξ(σ(t))

y T (τ )B(τ )y(τ )∆τ ≥ |xσ (t)|2 η(σ(t))

a

and

Z

b

η(σ(t))ξ(σ(t))

y T (τ )B(τ )y(τ )∆τ ≥ |xσ (t)|2 ξ(σ(t)).

σ(t)

Thus ξ(σ(t))η(σ(t)) |x (t)| ≤ ξ(σ(t)) + η(σ(t)) σ

Z

2

b

y T (τ )B(τ )y(τ )∆τ.

a

By Lemma 2.8 we see Z a

Z

b

σ

2

|C1 (t)||x (t)| ∆t ≤

ξ(σ(t))η(σ(t)) (|C1 (t)| ξ(σ(t)) + η(σ(t))

b

ξ(σ(t))η(σ(t)) ∆t |C1 (t)| ξ(σ(t)) + η(σ(t))

a

Z =

a

Z

b

= a

Z

Z

b

b

≤ a

b

a

ξ(σ(t))η(σ(t)) |C1 (t)| ∆t ξ(σ(t)) + η(σ(t)) ξ(σ(t))η(σ(t)) ∆t |C1 (t)| ξ(σ(t)) + η(σ(t))

y T (τ )B(τ )y(τ )∆τ )∆t

Z

b

y T (τ )B(τ )y(τ )∆τ

a

Z

b

(xσ (t))T C(t)xσ (t) 4 t

a

Z

b

a

|C1 (t)||xσ (t)|2 ∆t.

Since Z a

b

Z |C1 (t)||xσ (t)|2 ∆t ≥

we get

Z

b

a

b

Z (xσ )T (t)C(t)xσ (t) 4 t =

a

b

y T (t)B(t)y(t)∆t > 0,

a

ξ(σ(t))η(σ(t)) |C1 (t)| 4 t ≥ 1. ξ(σ(t)) + η(σ(t))

This completes the proof of Theorem 3.1. Theorem 3.2 Let a, b ∈ T with σ(a) < b. If (1.1) has a solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then for any C1 ∈ Rn×n with C1 (t) ≥ C(t), one has the following inequality s Z

b

a

Proof. Note

o nZ b |B(s)||eΘA (σ(t), s)|2 ∆s 4 t ≥ 4. |C1 (t)|

(3.10)

a

ξ(σ(t))η(σ(t)) ξ(σ(t)) + η(σ(t)) ≤ . ξ(σ(t)) + η(σ(t)) 4

It follows from (3.3) that Z

b

a

ξ(σ(t)) + η(σ(t)) |C1 (t)| 4 t ≥ 1. 4

Combining (3.1) and (3.2), we obtain Z b³Z a

a

b

´ |B(s)||eΘA (σ(t), s)|2 ∆s|C1 (t)| 4 t ≥ 4.

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That is

Z

b

a

nZ b o |C1 (t)| |B(s)||eΘA (σ(t), s)|2 ∆s 4 t ≥ 4. a

This completes the proof of Theorem 3.2. Theorem 3.3 Let a, b ∈ T with σ(a) < b. If (1.1) has a solution (x(t), y(t)) satisfying (2.1) on the interval [a, b]T , then for any C1 ∈ Rn×n with C1 (t) ≥ C(t), one has the following inequality s Z

³Z

b

b

|A(t)| 4 t + a

´1/2 ´1/2 ³ Z b p 2 |C1 (t)| 4 t | B(t)| 4 t ≥ 2.

(3.11)

a

a

Proof. From the proof of Theorem 3.1, we have Z

b

Z y T (t)B(t)y(t) 4 t =

a

b

(xσ (t))T C(t)xσ (t) 4 t.

a

It follows from the first equation of (1.1) that for all a ≤ t ≤ b, we get Z

t

(−A(τ )xσ (τ ) − B(τ )y(τ )) 4 τ

x(t) = a

Z

b

x(t) =

(A(τ )xσ (τ ) + B(τ )y(τ )) 4 τ.

t

Thus, from Corollary 2.5, Lemma 2.6 and Lemma 2.8 we obtain |x(t)| = ≤ ≤ ≤ ≤ = = ≤

Z Z b i 1h t | (A(τ )xσ (τ ) + B(τ )y(τ )) 4 τ | + | (A(τ )xσ (τ ) + B(τ )y(τ )) 4 τ | 2 a t Z Z b i 1h t |A(τ )xσ (τ ) + B(τ )y(τ )| 4 τ + |A(τ )xσ (τ ) + B(τ )y(τ )| 4 τ 2 a t Z i 1h b (|A(τ )xσ (τ )| + |B(τ )y(τ )|) 4 τ 2 a Z Z b p i p 1h b σ |A(τ )||x (τ )| 4 τ + | B(τ )|| B(τ )y(τ )| 4 τ 2 a a Z b Z b p h ³ ´1/2 ³ Z b p ´1/2 i 1 σ 2 |A(t)||x (t)| 4 t + | B(t)| 4 t | B(t)y(t)|2 4 t 2 a a a Z b Z b p Z b p h ³ ´ ³ ´1/2 i p 1/2 1 |A(t)||xσ (t)| 4 t + | B(t)|2 4 t ( B(t)y(t))T B(t)y(t) 4 t 2 a a a Z ³Z b p ´1/2 ³ Z b ´1/2 i 1h b |A(t)||xσ (t)| 4 t + | B(t)|2 4 t (xσ )T (t)C(t)(xσ (t)) 4 t 2 a a a Z ³Z b p ´1/2 ³ Z b ´1/2 i 1h b |A(t)||xσ (t)| 4 t + | B(t)|2 4 t |C1 (t)||xσ (t)|2 4 t . 2 a a a

Denote M = maxa≤t≤b |x(t)| > 0, then M≤

1h 2

Z

³Z

b

|A(t)|M 4 t + a

a

b

´1/2 ³ Z b ´1/2 i p | B(t)|2 4 t |C1 (t)|M 2 4 t .

(3.12)

a

Thus inequality (3.11) follows from (3.12).This completes the proof of Theorem 3.3.

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REFERENCES [1] R.Agarwal, M.Bohner, P.Rehak, Half-linear dynamic equations, Nonlinear Anal.Appl.1(2003):156. [2] M.Bohner, A.Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001. [3] M.Bohner, A.Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. [4] S.S.Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math.J.12(1983):10 5-112. [5] S.Cheng, Lyapunov inequalities for differential and difference equations, Fasc.Math.23(19 91):25-41. [6] G.Sh.Guseinov, B.Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput.Math.Appl.45(2003):1399-1416. [7] G.Sh.Guseinov, Integration on time scales, J.Math.Anal.Appl.285(2003):107-127. [8] X.He, Q.Zhang, X.Tang, On inequalities of Lyapunov for linear Hamiltonian systems on time scales, J.Math.Anal.Appl.381(2011):695-705. [9] S.Hilger, Analysis on measure chains − a unified approach to continuous and discrete calculus, Results Math., 18(1990):18-56. [10] L.Jiang, Z.Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math.Anal.Appl. 310(2005):579-593. [11] V.Kac, P.Chueng, Quantum Calculus, Universitext,2002. [12] X.Liu, M.Tang, Lyapunov-type inequality for linear Hamiltonian systems on time scales, J.Math.Anal.Appl. 310(2005):579-593. [13] W.Reid, A generalized Lyapunov inequality, J.Differential equations, 13(1973):182-196. [14] X.Tang, M.Zhang, Lyapunov inequalities and stability for linear Hamiltonian systems, J.Differential Equations. 252(2012):358-381. [15] F.Wong, S.Yu, C.Yeh, W.Lian, Lyapunov’s inequality on time scales,Appl.Math.Lett.19(20 06):1293-1299. [16] Q.Zhang, X.He, J.Jiang, On Lyapunov-type inequalities for nonlinear dynamic systems on time scales, Comput.Math.Appl. 62(2011):4028-4038.

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Error analysis of distributed algorithm for large scale data classification ∗ Cheng Wang

Feilong Cao

†

Department of Applied Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, P R China

Abstract The distributed algorithm is an important and basic approach, and it is usually used for large scale data processing. This paper aims to error analysis of distributed algorithm for large scale data classification generated from Tikhonov regularization schemes associated with varying Gaussian kernels and convex loss functions. The main goal is to provide fast convergence rates for the excess misclassification error. The number of subsets randomly divided from a large scale datasets is determined to guarantee that the distributed algorithm have lower time complexity and memory complexity. Keywords: Distributed algorithm; Classification; Large scale data; Generalization error Mathematics Subject Classification: 68T05, 68P30.

1

Introduction

In [11], a binary classification problem, which is generated from Tikhonov regularization schemes with general convex loss functions and varying Gaussian kernels, was studied well. This paper addresses error analysis of distributed algorithm for the classification with large scale datasets. For ease of description, we first introduce some concepts and notations. Most of them are the same as that of [11]. We denote the input space by a compact subset X of Rp . To represent the two classes, we write the output space Y = {−1, 1}. Clearly, classification algorithms produce binary classifiers C : X → Y , and the prediction power of such classifier C can be measured by using its misclassification error defined by ∫ R(C) = Prob(C(x) ̸= y) = P (y ̸= C(x)|x) dρX , X

where ρ is a probability distribution on Z := X × Y , ρX is the marginal distribution of ρ on X, and P (y|x) is the conditional distribution at x ∈ X. So-called Bayes rule is the classifier minimizing R(C), and is given by { 1, if P (y = 1|x) ≥ P (y = −1|x), fc (x) = −1, otherwise. So the excess misclassification error R(C) − R(fc ) of a classifier C can be used to measure the performance of the classifier C. In this paper we consider classifiers Cf induced by real-valued functions f : X → R, which is defined by { 1, if f (x) ≥ 0, Cf = sgn(f )(x) = −1, otherwise. The real-valued functions are generated from Tikhonov regularization schemes associated with general convex loss functions and varying Gaussian kernels. Now we give a definition for loss function [11]. ∗ This work was supported by the National Natural Science Foundation of China (Grant Nos. 10901137, 91330118, and 61272023) and Zhejiang Provincial Natural Science Foundation of China (No. LY14A010026) † Corresponding author. E-mail: [email protected]

1

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C. Wang & F. L. Cao: Distributed algorithm for large scale data classification

Definition 1.1. (see [11]) We say φ : R → R+ is a classifying loss (function) if it is convex, differentiable at 0 with φ′ (0) < 0, and the smallest zero of φ is 1. For details of such loss function, we refer reader to { Cucker }and Zhou [4]. ′ | is called the Gaussian kernel with The function on X × X given by K σ (x, x′ ) = exp − |x−x 2σ 2 variance σ > 0. From [1], this function can be used to define a reproducing kernel Hilbert space (RKHS). We denote the RKHS by Hσ . From [10] and [5], the Tikhonov regularization scheme with the loss φ, Gaussian kernel K σ , and a sample z = {(xi , yi )}ni=1 ∈ Z n can be defined as the solution fz of the following minimization problem { } n 1 ∑ 2 fz = argmin φ(yi f (xi )) + λ∥f ∥Hσ , (1.1) m i=1 f ∈Hσ where λ > 0 is called the regularization parameter. The regularizing function in terms of the generalization error E φ is defined as ∫ φ 2 φ ˜ fσ,λ := arg min {E (f ) + λ∥f ∥Hσ }, where E (f ) = φ(yf (x)) dρ. f ∈Hσ

Z

This function was used in Zhang [13], De Vito et al. [6], and Yao [12]. Zhou and Xiang [11] constructed a function (denoted by fσ,λ ) which works better than f˜σ,λ due to the special approximation ability of varying Gaussian kernels. The construction of fσ,λ is done under a Sobolev smoothness condition of a measurable function fρφ minimizing E φ , i.e., for almost everywhere x ∈ X, ∫ fρφ (x) = argmin φ(yt) dρ(y|x) = argmin{φ(t)P (y = 1|x) + φ(−t)P (y = −1|x)}. t∈R

Y

t∈R

The constructed function fσ,λ was used to estimate the excess misclassification error in [11]. The following Lemma 2.2 is a key result in [11], which will be employed as a base of our proof. We will use the concept of Sobolev space with index s > 0 and denote the space by H s (Rp ). In fact, the space is consisted by all functions in L2 (Rp ) with the finite semi-norm ∫ 1 |f |H s (Rp ) = {(2π)−n |ξ|2s |fˆ(ξ)|2 dξ} 2 , Rp

where fˆ is the Fourier transform of f defined for f ∈ L1 (Rp ) as fˆ(ξ) = It was proved in Chen et al. [3] and Bartlett et al. [2] that √ R(sgn(f )) − R(fc ) ≤ cφ E φ (f ) − E φ (fρφ )

∫ Rp

f (x)e−ixξ dx.

(1.2)

holds for some cφ > 0. Although the statistical aspects of (1.1) are well investigated, the computation of (1.1) can be complicated for large data with size N . For example, in a standard implementation [9], it requires costs O(N 3 ) and O(N 2 ) in time and memory, respectively. Such scaling are prohibitive when the sample size is large. In this work, we study a decomposition-based learning approach for large datasets, which is also called distributed algorithm for large datasets. Recently, the approach has attacked more attentions of researchers, and more results have been explored, such as McDonald et al. [8] for perceptron-based algorithms, Kleiner et al. [7] for bootstrap, and Zhang et al. [14] for parametric smooth convex optimization problems. The aim of this paper is to study the binary classification error of the distributed algorithm with varying λ and σ for general loss functions. For this purpose, we first describe the distributed algorithm [15]. We are given N samples (x1 , y1 ), . . . , (xN , yN ) drawn independent identically distributed (i.i.d.) according to the distribution ρ on Z = X × Y . Rather than solving the problem (1.1) on all N samples, we execute the following three steps: (1) Divide the set of samples {(x1 , y1 ), . . . , (xN , yN )} randomly and evenly into m disjoint subsets S1 , . . . , Sm ⊂ Z, and each

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Si has n =

N m

samples; (2) For each i = 1, 2, . . . , m, compute the local estimate   1 ∑  fˆi := argmin φ(yf (x)) + λ∥f ∥2Hσ ;  f ∈Hσ  n (x,y)∈Si

∑m ˆ 1 (3) Average together the local estimates and output f¯ = m i=1 fi . Our aim is to estimate the error R(sgn(f¯)) − R(fc ). However, from (1.2), we only need to estimate E φ (f¯) − E φ (fρφ ). The following section presents some results to bound E φ (f¯) − E φ (fρφ ) and R(sgn(f¯)) − R(fc ). When solving each fˆi , similarly to [11], we take λ = λ(n) = n−γ , σ = σ(n) = λζ = n−γζ , for some γ, ζ > 0.

2

Main results Lemma 2.1. We have E φ (f¯) − E φ (fρφ ) ≤

1 m

∑m ( i=1

) E φ (fˆi ) − E φ (fρφ ) .

Proof. Due to the convexity of φ, we have ∫

∫

E φ (f¯) = Z

Z

So E φ (f¯) − E φ (fρφ ) ≤

1 m

∑m i=1

1 ∑ 1 ∑ φ(y fˆi (x)) dρ = m i=1 m i=1 m

φ(y f¯(x)) dρ ≤ (

m

∫

1 ∑ φ ˆ E (fi ). m i=1 m

φ(y fˆi (x)) dρ = Z

) E φ (fˆi ) − E φ (fρφ ) .

Now in order to bound E φ (f¯) − E φ (fρφ ), we only need to estimate E φ (fˆi ) − E φ (fρφ ) for each i. In fact, the results for each i are the same because fˆi (i = 1, 2, . . . , m) are i.i.d., and share the same properties. We take Xiang and Zhou’s approach [11] and make some modifications. Lemma 2.2. (see [11]) Assume that for some s > 0, dρX fρφ = f˜ρφ |X for some f˜ρφ ∈ H s (Rp ) ∩ L∞ (Rp ) and ∈ L2 (X). dx

(2.1)

Then we can find functions {fσ,λ ∈ Hσ : 0 < σ ≤ 1, λ > 0} such that ˜ ∥fσ,λ ∥L∞ (X) ≤ A,

(2.2)

˜ s + λσ −p ) D(σ, λ) := E φ (fσ,λ ) − E φ (fρφ ) + λ∥fσ,λ ∥2Hσ ≤ A(σ for 0 < σ ≤ 1, λ > 0, where A˜ ≥ 1 is a constant independent of σ and λ. Using the method of error decomposition of [11], we easily obtain the following Lemma 2.3. Lemma 2.3. Let φ be a classifying loss function, we have E φ (fi ) − E φ (fρφ ) ≤ D(σ, λ) + Sz (fσ,λ ) − Sz (fˆi ),

(2.3)

where Sz (f ) is defined for any f by Sz (f ) = [Ezφ (f ) − Ezφ (fρφ )] − [E φ (f ) − E φ (fρφ )], and Ezφ (f ) = ∑ 1 (x,y)∈Si φ(yf (x)). n We also need the following Definition 2.1. Definition 2.1. (see [11]) A variancing power τ = τφ,ρ of the pair (φ, ρ) is the maximal ˜ ≥ 1, there exists C1 = C1 (B) ˜ satisfying number τ in [0, 1] such that for any B E[φ(yf (x)) − φ(yfρφ (x))]2 ≤ C1 [E φ (f ) − E φ (fρφ )]τ

˜ B], ˜ ∀f : X → [−B,

(2.4)

where Eξ denotes the expected value of ξ. The following Lemma 2.4 is to bound the second term of (2.3).

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˜ Lemma 2.4. (see [11]) Suppose A˜ and fσ,λ are as in Lemma 2.2, τ = τφ,ρ and C1 = C1 (A) δ are as in Definition 2.1. Then for any 0 < δ < 1, with confidence 1 − 2 , we have ( ) 2 ( φ ) − 1 φ φ Sz (fσ,λ ) ≤ 2 ∥φ∥C[−A, ˜ A] ˜ + C1 ln n 2−τ + E (fσ,λ ) − E (fρ ) . δ To bound the third term of (2.3), −Sz (fˆi ), we need the following Lemma 2.5, Lemma 2.6, and Lemma 2.7. √ Lemma 2.5. For any λ > 0 , we have ∥fˆi ∥Hσ ≤ φ(0)/λ. The proof is easy by taking f = 0 in the definition of fˆi , referring to De Vito et al. [6]. The next Lemma 2.6 is from Cucker and Zhou [4]. Lemma 2.6. (see [4]) Let 0 ≤ τ ≤ 1, c, B ≥ 0, and G be a set of functions on Z such that for every g ∈ G, E(g) ≥ 0, ∥g − E(g)∥∞ ≤ B and E(g 2 ) ≤ c(E(g))τ . Then for all ε > 0, { } ∑n { } E(g) − n1 i=1 f (zi ) nε2−τ 1− τ2 Probn sup √ > 4ε ≤ N (G, ε) exp − , z∈Z 2(c + 13 Bε1−τ ) (E(g))τ + ετ g∈G where N (G, ε) denotes the covering number to be the minimal ℓ ∈ N such that there exist ℓ disks in G with radius ε covering G. √ √ Note that if ∥f ∥Hσ ≤ φ(0)/λ, then ∥f ∥∞ ≤ Cσ φ(0)/λ. From the above Lemma 2.6, we obtain the following Lemma 2.7. √ ˜ = Cσ φ(0)/λ and C1 = C1 (B) ˜ in Definition 2.1. For Lemma 2.7. Let τ = τφ,ρ with B any ε > 0, there holds     [E φ (f ) − E φ (fρφ )] − [Ezφ (f ) − Ezφ (fρφ )] 1− τ2 √ sup Probn ≤ 4ε ≥ z∈Z ∥f ∥ ≤√φ(0)/λ  (E φ (f ) − E φ (fρφ ))τ + ετ Hσ ( ) √ { } ε λ nε2−τ √ 1 − N B1 , exp − , 2C1 + 34 D2 ε1−τ D1 φ(0) ˜ |φ′− (B)|}, ˜ where D1 = max{|φ′+ (−B)|, and D2 = max{φ(−1), ∥φ∥C[−B, ˜ B] ˜ }. Proof. We apply the above Lemma 2.6 to the function set { } √ G = φ(yf (x)) − φ(yfρφ (x)) : ∥f ∥Hσ ≤ φ(0)/λ , and see that each function g ∈ G satisfies E(g 2 ) ≤ c(E(g))τ for c = C1 . Obviously ∥g∥∞ ≤ D2 := max{φ(−1), ∥φ∥C[−B, ˜ B] ˜ }, so ∥g − E(g)∥∞ ≤ B := 2D2 . To draw our conclusion, we only need√ to bound the covering number N (G, ε). To do so, note that for f1 and f2 satisfying ∥f ∥Hσ ≤ φ(0)/λ and (x, y) ∈ Z, we have |{φ(yf1 (x)) − φ(yfρφ (x))} − {φ(yf2 (x)) − φ(yfρφ (x))}| = |φ(yf1 (x)) − φ(yf2 (x))| ≤ D1 ∥f1 − f2 ∥∞ . ( ) √ Therefore, N (G, ε) ≤ N B√φ(0)/λ , Dε1 = N (B1 , ε√ λ ), where B√φ(0)/λ denotes the ball D1 φ(0) √ with radius φ(0)/λ in Hσ . The statement is proved. Let ε∗ (n, λ, σ, δ) denote the smallest positive number ε satisfying ) ( √ { } nε2−τ δ ε λ √ exp − ≥1− . 1 − N B1 , 4 1−τ 2 2C1 + 3 D2 ε D1 φ(0) Then we have the following proposition.

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Proposition 2.1. Let σ = λζ with 0 < ζ < p1 (Noting p is the dimension of X), s be as in Lemma 2.2, and fσ,λ ∈ Hσ satisfy (2.2). For any 0 < δ < 1, with confidence at least 1 − δ, we have 2 − 1 ˜ min{sζ,1−pζ} + 40ε∗ (n, λ, σ, δ) + 4(∥φ∥ E φ (fˆi ) − E φ (fρφ ) ≤ 8Aλ ˜ A] ˜ + C1 ) ln n 2−τ . (2.5) C[−A, δ Proof. Xiang and Zhou [11] (see Proposition 2 in [11]) have proved that for any 0 < δ < 1, with confidence at least 1 − δ, 2 − 1 E φ (fˆi ) − E φ (fρφ ) ≤ 4D(σ, λ) + 40ε∗ (n, λ, σ, δ) + 4(∥φ∥C[−A, ˜ A] ˜ + C1 ) ln n 2−τ . δ ζ sζ 1−pζ min{sζ,1−pζ} ˜ ˜ +λ ) ≤ 2Aλ . So Proposition With Lemma 2.2 and σ = λ , we have D ≤ A(λ 2.1 is proved. To get the more explicit bound, we need the following Lemma 2.8 to bound ε∗ (m, λ, σ, δ). It can be proved via the same method as in [11]. Lemma 2.8. Let 0 ≤ τ ≤ 1, λ = n−γ and σ = λζ with γ > 0 and 0 < ζ < we have ε∗ (m, λ, σ, δ) ≤ C2 n−

1−2γζ(p+1) 2−τ

ln

2 δ

1 2γ(p+1) .

.

Then (2.6)

From Proposition 2.1 and Lemma 2.8, we have the following Proposition 2.2. Proposition 2.2. Let σ = λζ and λ = n−γ for some 0 < ζ < p1 and 0 < γ < (2.1) is valid for some s > 0, then for any 0 < δ < 1, with confidence 1 − δ we have

where

1 2ζ(p+1) .

If

˜ −θ ln 2 , E φ (fˆi ) − E φ (fρφ ) ≤ Cn δ

(2.7)

{ } 1 − 2γζ(p + 1) θ = min sζγ, γ(1 − pζ), , 2−τ

(2.8)

and C˜ is a constant independent of n and δ. Proof. Applying the bound for ε∗ from Lemma 2.8 on Proposition 2.1, with confidence at least 1 − δ, we have 2 − 1 + 4(∥φ∥C[−A, ˜ A] ˜ + C1 ) ln n 2−τ . δ ˜ −θ ln 2 . Here Putting λ = n−γ into the above formula, we easily see that E φ (fˆi ) − E φ (fρφ ) ≤ Cn δ θ is given by (2.8) and C˜ is the constant independent of n and δ given by C˜ = 8A˜ + 40C2 + 4(∥φ∥C[−A, ˜ A] ˜ + C1 ). ˜ min{sζ,1−pζ} + 40C2 n− E φ (fˆi ) − E φ (fρφ ) ≤ 8Aλ

1−2γζ(p+1) 2−τ

ln

2 δ

Now we can obtain our main result to bound E φ (f ) − E φ (fρφ ). Theorem 2.1. Under the condition of Proposition 2.2, for any 0 < δ < 1, with confidence 1 − δ we have ˜ −θ ln E φ (f ) − E φ (fρφ ) ≤ Cn

2m , δ

(2.9)

where θ and C˜ are as in Proposition 2.2. δ ˜ −θ ln 2m . Proof. From Proposition 2.2, for any δ > 0, with confidence 1− m , E φ (fˆi )−E φ (fρφ ) ≤ Cn δ From Lemma 2.1, { } } { m ( ) ∑ 2m 1 2m φ φ −θ φ φ φ −θ φ ˜ ˜ ln ≥ Prob E (fˆi ) − E (fρ ) ≤ Cn Prob E (f ) − E (fρ ) ≤ Cn ln δ m i=1 δ {m { } } ∩ δ ˜ −θ ln 2m ≥ Prob E φ (fˆi ) − E φ (fρφ ) ≤ Cn ≥1−m× = 1 − δ. δ m i=1

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C. Wang & F. L. Cao: Distributed algorithm for large scale data classification

1

a

Remark 2.1. Given N , we take n = ma , i.e. m = N a+1 and n = N a+1 . We easily see that the above bound m1aθ ln 2m δ → 0(m → ∞) for all a > 0. As mentioned in Introduction, the Tikhonov regularization scheme for all N samples have time complexity O(N 3 ) and memory complexity O(N 2 ). Now we can determine m (also n) to guarantee that the distributed algorithm have lower time complexity and memory complexity. Corollary 2.1. For any k < 3, the time complexity of the distributed algorithm is less than 3−k O(N k ) if and only if m > N 2 . 1

Proof. Let n = ma , i.e. m = N a+1 . The time complexity is m · O(n3 ) = O(m3a+1 ) = O(N 1 3−k k−1 a+1 > N 2 . For k < 3, to ensure 3a+1 a+1 < k, it only needs a < 3−k . So m = N

3a+1 a+1

).

For memory complexity, we have a similar result as follows. Corollary 2.2. For any k < 2, the memory complexity of the distributed algorithm is less than O(N k ) if and only if m > N 2−k . Due to (1.2), we have Theorem 2.2. R(sgn(f¯)) − R(fc ) ≤ cφ

√ ˜ −θ ln 2m . Cn δ

References [1] N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68: 337-404, 1950. [2] P. L. Bartlett, M. I. Jordan, J. D. McAuliffe. Convexity, classification, and risk bounds. J. Amer. Statis. Assoc., 101: 138-156, 2006. [3] D. R. Chen, Q. Wu, Y. Ying, D. X. Zhou. Support vector machine soft margin classifiers: error analysis. J. Mach. Lear. Res., 5: 1143-1175, 2004. [4] F. Cucker, D. X. Zhou. Learning Theory: An Approximation Theory Viewpoint. Cambridge University Press, 2007. [5] N. Cristianini, J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge University Press, 2000. [6] E. De Vito, A. Caponnetto, L. Rosasco. Model selection for regularized least-squares algorithm in learning theory. Found. Comput. Math., 5: 59-85, 2006. [7] A. Kleiner, A. Talwalkar, P. Sarkar, M. Jordan. Bootstrapping big data. in: Proc. 29th Inter. Conf. Mach. Lear., Edinburgh, Scotland, UK, 2012. http://www.cs.ucla.edu/ ameet/blb-icml2012-final.pdf [8] R. McDonald, K. Hall, G. Mann. Distributed training strategies for the structured perceptron. in: Proc. 2010 Annual Conference North American Chapter of the Association for Computational Linguistics (NAACL), pp. 456-464, Association for Computational Linguistics Stroudsburg, PA, USA, 2010. [9] C. Saunders, A. Gammerman, V. Vovk. Ridge regression learning algorithm in dual variables. in Proc. 15th Inter. Conf. Mach. Lear., pp. 515-521, Morgan Kaufmann, 1998. [10] G. Wahba. Spline Models for Observatianal Data. SIAM, 1990. [11] D. H. Xiang, D. X. Zhou. Classification with Gaussians and convex loss. J. Mach. Lear. Res., 10: 1147-1468, 2009. [12] Y. Yao. On complexity issue of online learning algorithms. IEEE Trans. Inform. Theory, 56(12): 6470-6481, 2010. [13] T. Zhang. Statistical behavior and consistency of classification methods based on convex risk minimization. Annals of Statis., 32: 56-85, 2004. [14] Y. Zhang, J. C. Duchi, M.J. Wainwright. Communication-efficient algorithms for statistical optimization. J. Mach. Lear. Res., 14: 3321-3363, 2013. [15] Y. Zhang, J. C. Duchi, M.J. Wainwright. Divide and Conquer Kernel Ridge Regression. JMLR: Workshop and Conference Proceedings, 30: 1-26, 2013.

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Korovkin type statistical approximation theorem for a function of two variables G. A. Anastassiou1) and M. Arsalan Khan2) 1)

Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA [email protected]

2)

Department of Civil Engineering, Aligarh Muslim University, Aligarh 202002, India [email protected]

Abstract. In this paper, we prove a Korovkin type approximation theorem for a function of two variables by using the notion of convergence in the Pringsheim’s sense and statistical convergence of double sequences. We also display an example in support of our results.

Keywords and phrases: Double sequence; statistical convergence; positive linear operator; Korovkin type approximation theorem. AMS subject classification (2000): 41A10, 41A25, 41A36, 40A30, 40G15.

1. Introduction and preliminaries The concept of statistical convergence for sequences of real numbers was introduced by Fast [8] and further studied Fridy [9] and many others. Let K ⊆ N and Kn = {k ≤ n : k ∈ K} .Then the natural density of K is defined by δ(K) = limn n−1 |Kn | if the limit exists, where |Kn | denotes the cardinality of Kn . A sequence x = (xk ) of real numbers is said to be statistically convergent to L provided that for every  > 0 the set K := {k ∈ N : |xk − L| ≥ } has natural density zero, i.e. for each  > 0, 1 lim |{j ≤ n : |xj − L| ≥ }| = 0. n n By the convergence of a double sequence we mean the convergence in the Pringsheim’s sense [20]. A double sequence x = (xjk ) is said to be Pringsheim’s convergent (or P -convergent) if for given  > 0 there exists an integer N such that |xjk − `| <  whenever j, k > N . In this case, ` is called the Pringsheim limit of x = (xjk ) and it is written as P − lim x = `. A double sequence x = (xjk ) is said to be bounded if there exists a positive number M such that |xjk | < M for all j, k. 1

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Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. The idea of statistical convergence for double sequences was introduced and studied by Moricz [17] and Mursaleen and Edely [18], independently in the same year and further studied in [15]. Let K ⊆ N × N be a two-dimensional set of positive integers and let K(m, n) = {(j, k) : j ≤ m, k ≤ n}. Then the double natural density of the set K is defined as P − lim m,n

| K(m, n) | = δ2 (K) mn

provided that the sequence (| K(m, n) | /mn) has a limit in Pringsheim’s sense. For example, let K = {(i2 , j 2 ) : i, j ∈ N}. Then | K(m, n) | δ2 (K) = P − lim ≤ P − lim m,n m,n mn

√ √ m n = 0, mn

i.e. the set K has double natural density zero, while the set {(i, 2j) : i, j ∈ N} has double natural density 21 . A real double sequence x = (xjk ) is said to be statistically convergent to the number L if for each  > 0, the set {(j, k), j ≤ m and k ≤ n :| xjk − L |≥ } has double natural density zero. In this case we write st2 - lim xjk = L. j,k→∞

Remark 1.1. Note that if x = (xjk ) is P -convergent then it is statisically convergent but not conversely. See the following example. Example 1.1. The double sequence w = (wjk ) defined by ( 1 , if j and k are squares; wjk = 0 , otherwise .

(1.1.1)

Then w is statistically convergent to zero but not P -convergent. Let C[a, b] be the space of all functions f continuous on [a, b] equipped with the norm kf kC[a,b] := sup |f (x)|, f ∈ C[a, b]. x∈[a,b]

The classical Korovkin approximation theorem states as follows (cf. [10], [13]):

2

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Let (Tn ) be a sequence of positive linear operators from C[a, b] into C[a, b]. Then limn kTn (f, x) − f (x)kC[a,b] = 0, for all f ∈ C[a, b] if and only if limn kTn (fi , x) − fi (x)kC[a,b] = 0, for i = 0, 1, 2, where f0 (x) = 1, f1 (x) = x and f2 (x) = x2 . Korovkin type approximation theorems are also proved for different summability methods to replace the ordinary convergence, e.g. [4], [7], [11], [14], [16] etc.. Quite recently, such type of approximation theorems are proved in [1], [2], [3], [6] and [19] for functions of two variables by using almost convergence and statistical convergence of double sequences, respectively. For single sequences, Boyanov and Veselinov [2] have proved the Korovkin theorem on C[0, ∞) by using the test functions 1, e−x , e−2x . In this paper, we extend the result of Boyanov and Veselinov for functions of two variables by using the notion of Pringsheim’s convergence and statistical convergence of double sequences. 2. Main result Let C(I 2 ) be the Banach space with the uniform norm k . k of all real-valued two dimensional continuous functions on I × I, where I = [0, ∞); provided that lim(x,y)→(∞,∞) f (x, y) is finite. Suppose that Tm,n : C(I 2 ) → C(I 2 ). We write Tm,n (f ; x, y) for Tm,n (f (s, t); x, y); and we say that T is a positive operator if T (f ; x, y) ≥ 0 for all f (x, y) ≥ 0. The following result is an extension of Boyanov and Veselinov theorem [5] for functions of two variables. Theorem 2.1. Let (Tj,k ) be a double sequence of positive linear operators from C(I 2 ) into C(I 2 ). Then for all f ∈ C(I 2 )



P - lim Tj,k (f ; x, y) − f (x, y)

= 0.

(2.1.0)

j,k→∞

if and only if



= 0, P - lim T (1; x, y) − 1 j,k

(2.1.1)



−s −x

= 0, P - lim T (e ; x, y) − e j,k

(2.1.2)



−t −y

= 0, P - lim T (e ; x, y) − e j,k

(2.1.3)



−2s −2t −2x −2y

= 0. P - lim T (e + e ; x, y) − (e + e ) j,k

(2.1.4)

j,k→∞

j,k→∞

j,k→∞

j,k→∞

3

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Proof. Since each 1, e−x , e−y , e−2x + e−2y belongs to C(I 2 ), conditions (2.1.1)-(2.1.4) follow immediately from (2.1.0). Let f ∈ C(I 2 ). There exist aconstant M > 0 such that |f (x, y)| ≤ M for each (x, y) ∈ I 2 . Therefore, |f (s, t) − f (x, y)| ≤ 2M, − ∞ < s, t, x, y < ∞.

(2.1.5)

It is easy to prove that for a given ε > 0 there is a δ > 0 such that |f (s, t) − f (x, y)| < ε,

(2.1.6)

whenever |e−s − e−x | < δ and |e−t − e−y | < δ for all (x, y) ∈ I 2 . Using (2.1.5), (2.1.6), putting ψ1 = ψ1 (s, x) = (e−s − e−x )2 and ψ2 = ψ2 (t, y) = (e−t − e−y )2 , we get |f (s, t) − f (x, y)| < ε +

2M (ψ1 + ψ2 ), ∀ |s − x| < δ and |t − y| < δ. δ2

This is, 2M 2M (ψ + ψ ) < f (s, t) − f (x, y) < ε + (ψ1 + ψ2 ). 1 2 δ2 δ2 Now, operating Tj,k (1; x, y) to this inequality since Tj,k (f ; x, y) is monotone and linear. −ε −

We obtain   2M < Tj,k (1; x, y)(f (s, t) − f (x, y)) Tj,k (1; x, y) −ε − 2 (ψ1 + ψ2 ) δ   2M < Tj,k (1; x, y) ε + 2 (ψ1 + ψ2 ) . δ Note that x and y are fixed and so f (x, y) is constant number. Therefore −εTj,k (1; x, y) −

2M Tj,k (ψ1 + ψ2 ; x, y) < Tj,k (f ; x, y) − f (x, y)Tj,k (1; x, y) δ2 < εTj,k (1; x, y) +

2M Tj,k (ψ1 + ψ2 ; x, y). δ2

(2.1.7)

But Tj,k (f ; x, y) − f (x, y) = Tj,k (f ; x, y) − f (x, y)Tj,k (1; x, y) + f (x, y)Tj,k (1; x, y) − f (x, y) = [Tj,k (f ; x, y) − f (x, y)Tj,k (1; x, y)] + f (x, y)[Tj,k (1; x, y) − 1].

(2.1.8)

Using (2.1.7) and (2.1.8), we have Tj,k (f ; x, y) − f (x, y) < εTj,k (1; x, y) +

2M Tj,k (ψ1 + ψ2 ; x, y) + f (x, y)(Tj,k (1; x, y) − 1). δ2 (2.1.9)

Now Tj,k (ψ1 + ψ2 ; x, y) = Tj,k ((e−s − e−x )2 + (e−t − e−y )2 ; x, y) 4

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= Tj,k (e−2s − 2e−s e−x + e−2x + e−2t − 2e−t e−y + e−2y ; x, y) = Tj,k (e−2s + e−2t ; x, y) − 2e−x Tj,k (s; x, y) − 2e−y Tj,k (t; x, y) +(e−2x + e−2y )Tj,k (1; x, y) = [Tj,k (e−2s + e−2t ; x, y) − (e−2x + e−2y )] − 2e−x [Tj,k (e−s ; x, y) − e−x ] − 2e−y [Tj,k (e−t ; x, y) − e−y ] + (e−2x + e−2y )[Tj,k (1; x, y) − 1]. Using (2.1.9), we obtain Tj,k (f ; x, y) − f (x, y) < εTj,k (1; x, y) +

2M {[Tj,k ((e−2s + e−2t ); x, y) − (e−2x + e−2y )] δ2

− 2e−x [Tj,k (e−s ; x, y) − e−x ] − 2e−y [Tj,k (e−t ; x, y) − e−y ] + (e−2x + e−2y )[Tj,k (1; x, y) − 1]} + f (x, y)(Tj,k (1; x, y) − 1) 2M {[Tj,k ((e−2s +e−2t ); x, y)−(e−2x +e−2y )] δ2 − 2e−x [Tj,k (e−s ; x, y) − e−x ] − 2e−y [Tj,k (e−t ; x, y) − e−y ]

= ε[Tj,k (1; x, y)−1]+ε+

+ (e−2x + e−2y )[Tj,k (1; x, y) − 1]} + f (x, y)(Tj,k (1; x, y) − 1). Since ε is arbitrary, we can write Tj,k (f ; x, y) − f (x, y) ≤ ε[Tj,k (1; x, y) − 1] +

2M {[Tj,k ((e−2s + e−2t ); x, y) − (e−2x + e−2y )] δ2

− 2e−x [Tj,k (e−s ; x, y) − e−x ] − 2e−y [Tj,k (e−t ; x, y) − e−y ] + (e−2x + e−2y )[Tj,k (1; x, y) − 1]} + f (x, y)(Tj,k (1; x, y) − 1). Therefore | Tj,k (f ; x, y)−f (x, y) |≤ ε+(ε+M ) | Tj,k (1; x, y)−1 | +

2M −2x −2y | e +e || Tj,k (1; x, y)−1 | δ2

2M | Tj,k (e−2s + e−2t ; x, y) | −(e−2x + e−2y ) | δ2 4M 4M + 2 | e−x || Tj,k (e−s ; x, y) − e−x | + 2 | e−y || Tj,k (e−t ; x, y) − e−y | δ δ 2M 4M ≤ ε + (ε + M + 2 ) | Tj,k (1; x, y) − 1 | + 2 | e−2x + e−2y || Tj,k (1; x, y) − 1 | δ δ 2M + 2 | Tj,k (e−2s + e−2t ; x, y) − (e−2x + e−2y ) | δ 4M 4M + 2 | Tj,k (e−s ; x, y) − e−x | + 2 | Tj,k (e−t ; x, y) − e−y | . (2.1.10) δ δ since | e−x |, | e−y |≤ 1 for all x, y ∈ I. Now, taking sup(x,y)∈I 2 , we get





Tj,k (f ; x, y) − f (x, y) ≤ ε + K Tj,k (1; x, t) − 1



+

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−s −x −t −y

+ Tj,k (e ; x, y) − e + Tj,k (e ; x, y) − e



−2s −2t −2x −2y

+ Tj,k (e + e ; x, y) − (e + e ) , where where K = max{ε + M +

4M 4M 2M , δ2 , δ2 }. δ2

(2.1.11)

Taking P -lim as j, k → ∞ and using

(2.1.1), (2.1.2), (2.1.3), (2.1.4), we get



P − lim Tj,k (f ; x, y) − f (x, y)

= 0, uniformly in m, n. p,q→∞

This completes the proof of the theorem. 3. Statistical version In the following theorem we use the notion of statistical convergence of double sequences to generalize the above theorem. We also display an interesting example to show its importance. Theorem 3.1. Let (Tj,k ) be a double sequence of positive linear operators from C(I 2 ) into C(I 2 ). Then for all f ∈ C(I 2 )



st2 - lim

Tj,k (f ; x, y) − f (x, y) = 0.

(3.1.0)

j,k→∞

if and only if



= 0, T (1; x, y) − 1 st2 - lim j,k

(3.1.1)



−s −x

= 0, T (e ; x, y) − e st2 - lim j,k

(3.1.2)



−t −y

= 0, T (e ; x, y) − e st2 - lim j,k

(3.1.3)



−2s −2t −2x −2y

= 0. st2 - lim T (e + e ; x, y) − (e + e ) j,k

(3.1.4)

j,k→∞

j,k→∞

j,k→∞

j,k→∞

Proof. For a given r > 0 choose ε > 0 such that ε < r . Define the following sets



D := {(j, k), j ≤ m and k ≤ n :

Tj,k (f ; x, y) − f (x, y) ≥ r},



r−ε

≥ D1 := {(j, k), j ≤ m and k ≤ n : T (1; x, y) − 1 }, j,k

4K



r−ε −s −x

≥ D2 := {(j, k), j ≤ m and k ≤ n : T (e ; x, y) − e }, j,k

4K 6

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r−ε −t −y

}. D3 := {(j, k), j ≤ m and k ≤ n : Tj,k (e ; x, y) − e ≥ 4K



r−ε −2s −2t −2x −2y

D4 := {(j, k), j ≤ m and k ≤ n : Tj,k (e + e ; x, y) − (e + e ) ≥ }. 4K Then from (2.1.11), we see that D ⊂ D1 ∪ D2 ∪ D3 ∪ D4 and therefore δ2 (D) ≤ δ2 (D1 ) + δ2 (D2 ) + δ2 (D3 ) + δ2 (D4 ). Hence conditions (3.1.1)–(3.1.4) imply the condition (3.1.0). This completes the proof of the theorem. We show that the following double sequence of positive linear operators satisfies the conditions of Theorem 3.1 but does not satisfy the conditions of Theorem 2.1. Example 3.2. Consider the sequence of classical Baskakov operators of two variables [12]     ∞ X ∞ X j k m−1+j n−1+k j f , Bm,n (f ; x, y) := x (1+x)−m−j y k (1+y)−n−k ; m n j k j=0 k=0 where 0 ≤ x, y < ∞ . Let Lm,n : C(I 2 ) → C(I 2 ) be defined by Lm,n (f ; x, y) = (1 + wmn )Bm,n (f ; x, y), where the sequence (wmn ) is defined by (1.1.1). Since Bm,n (1; x, y) = 1, 1

Bm,n (e−s ; x, y) = (1 + x − xe− m )−m , 1

Bm,n (e−t ; x, y) = (1 + y − ye− n )−n , 1

1

Bm,n (e−2s + e−2t ; x, y) = (1 + x2 − x2 e− m )−m + (1 + y 2 − y 2 e− n )−n , we have that the sequence (Lm,n ) satisfies the conditions (3.1.1), (3.2.2), (3.1.3) and (3.1.4). Hence by Theorem 3.1, we have st2 - lim kLm,n (f ; x, y) − f (x, y)k = 0. m,n→∞

On the other hand, we get Lm,n (f ; 0, 0) = (1+wmn )f (0, 0), since Bm,n (f ; 0, 0) = f (0, 0), and hence kLm,n (f ; x, y) − f (x, y)k ≥ |Lm,n (f ; 0, 0) − f (0, 0)| = wmn |f (0, 0)|. We see that (Lm,n ) does not satisfy the conditions of Theorem 2.1, since P - lim wmn m,n→∞

does not exist. 7

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References [1] A. Alotaibi, M. Mursaleen and S.A. Mohiuddine, Statistical approximation for periodic functions of two variables, Jour. Function Spaces Appl., Volume 2013, Article ID 491768, 5 pages http://dx.doi.org/10.1155/2013/491768. [2] G. A. Anastassiou, M. Mursaleen and S. A. Mohiuddine, Some approximation theorems for functions of two variables through almost convergence of double sequences, Jour. Comput. Analy. Appl., 13(1)(2011) 37-40. [3] C. Belen, M. Mursaleen and M. Yildirim, Statistical A-summability of double sequences and a Korovkin type approximation theorem, Bull. Korean Math. Soc., 49(4) (2012) 851-861. [4] C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application,” Appl. Math. Comput., 219 (2013) 9821-9826. [5] B. D. Boyanov and V. M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. Roum., 14(62) (1970) 9-13. [6] F. Dirik and K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense, Turk. J. Math. 33(2009)1-11. [7] O.H.H. Edely, S.A. Mohiuddine and A. K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett., 23(2010)1382-1387. [8] H. Fast, Sur la convergence statistique, Colloq. Math, 2 (1951), 241-244. [9] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313. [10] A. D. Gad˘ziev, The convergence problems for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P.Korovkin, Soviet Math. Dokl., 15(1974) 1433-1436. [11] A.D. Gadjiv and C. Orhan, Sme approximzation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002) 129-138. [12] M. Gurdek, L. Rempulska and M. Skorupka, The Baskakov operators for functions of two variables, Collect. Math. 50, 3 (1999) 289–302. [13] P. P. Korovkin, Linear operators and approximation theory, Hindustan Publ. Co., Delhi,1960. 8

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[14] S.A. Mohiuddine, An application of almost convergence in approximation theorems, Appl. Math. Lett., 24(2011)1856-1860. [15] S.A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces, Abstract Appl. Anal., Volume 2012, Article ID 719729, 9 pages, doi:10.1155/2012/719729. [16] S.A. Mohiuddine and A. Alotaibi, Statistical convergence and approximation theorems for functions of two variables, J. Comput. Anal. Appl., 15 (2) (2013) 218-223. [17] F. Moricz, Statistical convergence of multiple sequences, Arch. Math. 81 (2003) 82-89. [18] M. Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223-231. [19] M. Mursaleen and A. Alotaib, Korovkin type approximation theorem for statistical A-summability of double sequences, Jour. Comput. Anal. Appl., 15(6) (2013) 1036-1045. [20] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Z., 53(1900) 289-321.

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Advanced Fractional Taylor’s formulae George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] Abstract Here are presented …ve new advanced fractional Taylor’s formulae under as weak as possible assumptions.

2010 AMS Subject Classi…cation: 26A33. Key Words and Phrases: fractional integral, fractional derivative, fractional Taylor’s formula.

1

Introduction

In [3] we proved Theorem 1 Let f; f 0 ; :::; f (n) ; g; g 0 be continuous functions from [a; b] (or [b; a]) (k) into R, n 2 N. Assume that g 1 , k = 0; 1; :::; n; are continuous functions. Then it holds f (b) = f (a) +

n X1

f

g

1 (k)

(g (a))

k!

k=1

(g (b)

k

g (a)) + Rn (a; b) ;

(1)

where Rn (a; b) :=

1 (n

=

1)!

Z

1 (n

1)!

b

(g (b)

n 1

f

g

n 1

f

g

g (s))

1 (n)

(g (s)) g 0 (s) ds

(2)

a

Z

g(b)

(g (b)

t)

1 (n)

(t) dt:

g(a)

Remark 2 Let g be strictly increasing and g 2 AC ([a; b]) (absolutely continuous functions). Set g ([a; b]) = [c; d], where c; d 2 R, i.e. g (a) = c, g (b) = d, and call l := f g 1 . 1

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Assume that l 2 AC n ([c; d]) (i.e. l(n 1) 2 AC ([c; d])). [Obviously here it is implied that f 2 C ([a; b]) :] (n) Furthermore assume that f g 1 2 L1 ([c; d]). [By this very last as(n)

n 1

sumption, the function (g (b) t) f g 1 (t) is integrable over [c; d]. Since g 2 AC ([a; b]) and it is increasing, by [9] the function (n) n 1 (g (b) g (s)) f g 1 (g (s)) g 0 (s) is integrable on [a; b], and again by [9], (2) is valid in this general setting.] Clearly (1) is now valid under these general assumptions.

2

Results

We need Lemma 3 Let g be strictly increasing and g 2 AC ([a; b]). Assume that f is Lebesgue measurable function over [c; d]. Then f where f

g

1 (m)

g

1 (m)

f

1;[c;d]

g

1 (m)

g

1;[a;b]

;

g

1 (m)

(3)

g 2 L1 ([a; b]) :

Proof. We observe by de…nition of k k1 that: f

g

n n inf M : m t 2 [a; b] :

1 (m)

f

g

1;[a;b]

1 (m)

g

=

(4)

g (t) > M

o

o =0 ;

where m is the Lebesgue measure. Because g is absolutely continuous and strictly increasing function on [a; b], by [11], p. 108, exercise 14, we get that n o (m) m z 2 [c; d] : f g 1 (z) > M =

given that

n m g (t) 2 [c; d] : f g n m g t 2 [a; b] : f g 1 n m t 2 [a; b] :

f

g

Therefore each M of (4) ful…lls n n M 2 L : m z 2 [c; d] :

1 (m)

(g (t)) > M o (m) (g (t)) > M

1 (m)

f

The last implies (3). We give

g

g (t) > M

1 (m)

o

o

= = 0;

= 0:

o o (z) > L = 0 :

(5)

2

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De…nition 4 (see also [10, p. 99]) The left and right fractional integrals, respectively, of a function f with respect to given function g are de…ned as follows: Let a; b 2 R, a < b, > 0. Here g 2 AC ([a; b]) and is strictly increasing, f 2 L1 ([a; b]). We set Z x 1 1 0 (g (x) g (t)) g (t) f (t) dt; x a; (6) Ia+;g f (x) = ( ) a where

0 is the gamma function, clearly Ia+;g f (a) = 0, Ia+;g f := f and

Ib

;g f

(x) =

1 ( )

Z

b

(g (t)

1

g (x))

g 0 (t) f (t) dt; x

b;

(7)

x

clearly Ib ;g f (b) = 0, Ib0 ;g f := f: When g is the identity function id, we get that Ia+;id = Ia+ , and Ib ;id = Ib , the ordinary left and right Riemann-Liouville fractional integrals, where Z x 1 1 (x t) f (t) dt; x a; (8) Ia+ f (x) = ( ) a Ia+ f (a) = 0 and Ib f (x) =

1 ( )

Z

b

(t

1

x)

f (t) dt;

x

b;

(9)

x

Ib f (b) = 0: In [5], we proved Lemma 5 Let g 2 AC ([a; b]) which is strictly increasing and f Borel measurable in L1 ([a; b]). Then f g 1 is Lebesgue measurable, and kf k1;[a;b]

f

g

1 1;[g(a);g(b)]

;

(10)

:

(11)

i.e. f g 1 2 L1 ([g (a) ; g (b)]). If additionally g 1 2 AC ([g (a) ; g (b)]), then kf k1;[a;b] = f

g

1 1;[g(a);g(b)]

Remark 6 We proved ([5]) that Ia+;g f (x) = Ig(a)+ f

g

1

(g (x)) , x

a

(12)

Ib

g

1

(g (x)) , x

b:

(13)

and ;g f

(x) = Ig(b)

f

3

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It is well known that, if f is a Lebesgue measurable function, then there exists f a Borel measurable function, such that f = f , a.e. Also it holds kf k1 = R R kf k1 , and :::f:::dx = :::f :::dx: Of course a Borel measurable function is a Lebesgue measurable function. Thus, by Lemma 5, we get kf k1;[a;b] = kf k1;[a;b]

f

1

g

:

1;[g(a);g(b)]

(14)

We observe the following: Let ; > 0, then Ia+;g Ia+;g f Ig(a)+

Ia+;g f

1

g

(x) = Ia+;g Ia+;g f

(g (x)) = Ig(a)+ Ig(a)+ f

Ig(a)+ Ig(a)+ f + Ig(a)+ f

(g (x))

+ (g (x)) = Ia+;g f

1

g

1

g

(x) = 1

g

g g

1

(g (x)) = (15)

(by [8], p. 14)

=

+ (x) = Ia+;g f (x) a.e.

The last is true for all x, if + 1 or f 2 C ([a; b]). We have proved the semigroup composition property + Ia+;g Ia+;g f (x) = Ia+;g f (x) = Ia+;g Ia+;g f (x) ;

a.e., which is true for all x, if Similarly we get Ib Ig(b)

Ib

;g f

;g

g

Ib

1

f

g

1

Ig(b)

a,

(16)

1 or f 2 C ([a; b]) :

(x) = Ib

(g (x)) = Ig(b)

Ig(b) + Ig(b)

;g f

+

x

f

Ib

;g

Ig(b) g

1

;g f

f

(g (x))

(g (x)) = Ib +;g f

(x) = g

1

g g

1

(g (x)) = (17)

(by [1])

=

(x) = Ib +;g f (x) a.e.,

true for all x 2 [a; b], if + 1 or f 2 C ([a; b]). We have proved the semigroup property that Ib

;g Ib ;g f

(x) = Ib +;g f (x) = Ib

which is true for all x 2 [a; b], if

+

;g Ib ;g f

(x) ; a.e., x

b,

(18)

1 or f 2 C ([a; b]) :

From now on without loss of generality, within integrals we may assume that f = f , and we mean that f = f , a.e. We make 4

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De…nition 7 Let > 0, d e = n, d e the ceiling of the number. Again here g 2 (n) AC ([a; b]) and strictly increasing. We assume that f g 1 g 2 L1 ([a; b]). We de…ne the left generalized g-fractional derivative of f of order as follows: Z x 1 (n) n 1 0 (g (t)) dt, (g (x) g (t)) g (t) f g 1 Da+;g f (x) := (n ) a (19) x a: If 2 = N, by [6], we have that Da+;g f 2 C ([a; b]). We see that n Ia+;g

f

g

1 (n)

g

(x) = Da+;g f (x) , x

We set n Da+;g f (x) :=

f

g

1 (n)

a:

(20)

g (x) ;

(21)

0 Da+;g f (x) = f (x) , 8 x 2 [a; b] :

(22)

Da+;g f = Da+;id f = D a f;

(23)

When g = id, then the usual left Caputo fractional derivative. We make (n)

g 2 L1 ([a; b]), which could Remark 8 Under the assumption that f g 1 be considered as Borel measurable within integrals, we obtain n Ia+;g Da+;g f (x) = Ia+;g Ia+;g +n Ia+;g

f 1

(n

1)!

g Z

1 (n)

x

n 1

g (t))

g 0 (t)

1 (n)

g

n (x) = Ia+;g

g

(g (x)

f

f f

g g

g

1 (n)

1 (n)

(x) = g (x) =

(24)

g (t) dt:

a

We have proved that Ia+;g Da+;g f (x) =

1 (n

1)!

Z

x

(g (x)

n 1

g (t))

g 0 (t) f

g

1 (n)

(g (t)) dt

a

(25)

= Rn (a; x) , 8 x

a;

see (2). But also it holds Rn (a; x) = Ia+;g Da+;g f (x) = 1 ( )

Z

x

(g (x)

g (t))

1

g 0 (t) Da+;g f (t) dt; x

(26) a:

a

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We have proved the following g-left fractional generalized Taylor’s formula: Theorem 9 Let g be strictly increasing function and g 2 AC ([a; b]). We assume that f g 1 2 AC n ([g (a) ; g (b)]), where N 3 n = d e, > 0. Also we assume that f

1 (n)

g

g 2 L1 ([a; b]). Then

f (x) = f (a) +

n X1

f

g

1 (k)

k=1

1 ( )

Z

x

(g (x)

(g (a))

k! 1

g (t))

a

k

(g (x)

g (a)) +

g 0 (t) Da+;g f (t) dt; 8 x 2 [a; b] :

(27)

Calling Rn (a; x) the remainder of (27), we get that 1 ( )

Rn (a; x) =

Z

g(x)

(g (x)

1

z)

Da+;g f

g

1

(z) dz; 8 x 2 [a; b] :

g(a)

(28)

Remark 10 By [6], Rn (a; x) is a continuous function in x 2 [a; b]. Also, by [9], change of variable in Lebesgue integrals, (28) is valid. By [3] we have Theorem 11 Let f; f 0 ; :::; f (n) ; g; g 0 be continuous from [a; b] into R, n 2 N. (k) Assume that g 1 , k = 0; 1; :::; n; are continuous. Then f (x) = f (b) +

n X1

f

g

1 (k)

(g (b))

k!

k=1

k

(g (x)

g (b)) + Rn (b; x) ;

(29)

where Rn (b; x) :=

=

1 (n

1 (n

1)!

1)!

Z

Z

x

n 1

(g (x)

g (s))

f

g

1 (n)

(g (s)) g 0 (s) ds

(30)

b

g(x)

n 1

(g (x)

t)

f

g(b)

g

1 (n)

(t) dt; 8 x 2 [a; b] :

(31)

Notice that (29), (30) and (31) are valid under more general weaker assumptions, as follows: g is strictly increasing and g 2 AC ([a; b]), f g 1 2 AC n ([g (a) ; g (b)]), and f

g

1 (n)

2 L1 ([g (a) ; g (b)]) :

We make

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(n)

De…nition 12 Here we assume that f g 1 g 2 L1 ([a; b]), where N 3 n = d e, > 0. We de…ne the right generalized g-fractional derivative of f of order as follows: Z b n ( 1) (n) n 1 0 (g (t) g (x)) g (t) f g 1 (g (t)) dt, Db ;g f (x) := (n ) x (32) all x 2 [a; b] : If 2 = N, by [7], we get that Db ;g f 2 C ([a; b]). We see that Ibn

n

( 1)

;g

f

g

1 (n)

g (x) = Db

We set Dbn

;g f

n

(x) = ( 1)

Db0

;g f

f

;g f

1 (n)

g

(x) , a

x

b:

g (x) ;

(33)

(34)

(x) = f (x) , 8 x 2 [a; b] :

When g = id, then Db

;g f

(x) = Db

;id f

(x) = Db f;

(35)

the usual right Caputo fractional derivative. We make Remark 13 Furthermore it holds Ib Ibn

;g Db ;g f n

;g

( 1)

f n

( 1) (n 1)!

g Z

1 (n)

g

( 1)

f

n

Ibn

(x) = ( 1)

b

(g (t)

n

n ;g Ib ;g

(x) = Ib

n 1

g (x))

g 0 (t)

g ;g

f

g

f

g

1 (n)

f 1 (n)

g

g

(x) =

1 (n)

g

(x) = (36)

g (t) dt =

x

2n Z x ( 1) n 1 0 (g (x) g (t)) g (t) (n 1)! b Z x 1 n 1 0 (g (x) g (t)) g (t) f g (n 1)! b

1 (n)

1 (n)

g (t) dt =

g (t) dt = Rn (b; x) ; (37)

as in (30). That is Rn (b; x) = Ib 1 ( )

Z

b

(g (t)

g (x))

1

;g Db ;g f

g 0 (t) Db

;g f

(x) =

(t) dt; all a

x

b:

(38)

x

7

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We have proved the g-right generalized fractional Taylor’s formula: Theorem 14 Let g be strictly increasing function and g 2 AC ([a; b]). We assume that f g 1 2 AC n ([g (a) ; g (b)]), where N 3 n = d e, > 0. Also we assume that f

g

1 (n)

f (x) = f (b) +

g 2 L1 ([a; b]). Then

n X1

f

1 (k)

g

k=1

1 ( )

Z

b

(g (t)

1

g (x))

(g (b))

k! g 0 (t) Db

;g f

(g (x)

k

g (b)) +

(t) dt; all a

x

b:

(39)

x

Calling Rn (b; x) the remainder in (39), we get that Rn (b; x) =

1 ( )

Z

g(b)

(z

1

g (x))

Db

;g f

g

1

g(x)

(z) dz; 8 x 2 [a; b] : (40)

Remark 15 By [7], Rn (b; x) is a continuous function in x 2 [a; b]. Also, by [9], change of variable in Lebesgue integrals, (40) is valid. Basics 16 The right Riemann-Liouville fractional integral of order L1 ([a; b]), a < b, is de…ned as follows: Z

1 ( )

Ib f (x) :=

b

(z

x)

1

x

f (z) dz, 8 x 2 [a; b] :

> 0, f 2

(41)

Ib0 := I (the identity operator). Let

;

0, f 2 L1 ([a; b]). Then, by [1], we have Ib Ib f = Ib + f = Ib Ib f;

(42)

valid a.e. on [a; b]. If f 2 C ([a; b]) or + 1, then the last identity is true on all of [a; b] : The right Caputo fractional derivative of order > 0, m = d e, f 2 m AC ([a; b]) is de…ned as follows: m

Db f (x) := ( 1) Ibm

f (m) (x) ;

(43)

that is m

Db f (x) =

( 1) (m

)

Z

b

(z

m

x)

x

1

f (m) (z) dz; 8 x 2 [a; b] ;

(44)

m

with Dbm f (x) := ( 1) f (m) (x) : 8

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By [1], we have the following right fractional Taylor’s formula: Let f 2 AC m ([a; b]), x 2 [a; b], > 0, m = d e, then m X1

f (x)

k=0

f (k) (b) (x k! m

1

m

( 1)

(m

1)! m

Z

Z

b

(z

1

x)

Db f (z) dz =

f (m) (x) = ( 1)

b

m 1

(z

x)

Ibm f (m) (x) =

f (m) (z) dz =

x

x ( 1) m 1 (m) (x z) f (z) dz = (m 1)! b Z x 1 m 1 (m) (x z) f (z) dz: (m 1)! b m

( 1)

(45)

x m

Ib Ibm

Ib Db f (x) = ( 1)

Z

1 ( )

k

b) =

(46)

That is m

Ib Db f (x) = ( 1) f (x)

m X1 k=0

f (k) (b) (x k!

1

k

b) =

(m

1)!

Ibm f (m) (x) = Z

x

(x

m 1

z)

f (m) (z) dz:

(47)

b

We make

Remark 17 If 0
0, over (x; b), and Z

b

(z

x)

1

dz =

(b

x)

x

thus (z

x)

1

< 1; for any 0
0, Re 1 + q (z) + β β q (z) q (z) 1221

4

(2.7)

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for α, ξ, µ, β ∈ C, β 6= 0, z ∈ U, and ψ m,n λ,l

(δ, α, ξ, µ, β; z) := α + ξ

βδ (n + 2)

Ã

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

m,n+2 f (z) IRλ,l m,n+1 IRλ,l f (z)

!δ

− βδ (n + 1)

+µ

Ã

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

m,n+1 f (z) IRλ,l m,n IRλ,l f (z)

!2δ

+

(2.8)

− βδ.

If q satisfies the following subordination 2 ψ m,n λ,l (δ, α, ξ, µ, β; z) ≺ α + ξq (z) + µq (z) +

for α, ξ, µ, β ∈ C, β 6= 0, z ∈ U, then

µ

IRm,n+1 f (z) λ,l IRm,n λ,l f (z)

¶δ

βzq 0 (z) , q (z)

(2.9)

≺ q (z), z ∈ U, δ ∈ C, δ 6= 0, and q is the best dominant.

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

µ

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

analytic in U and p (0) = 1 ¶δ ∙ µ m,n+1 0 IRλ,l f (z) z (IRm,n+1 f (z)) λ,l We have zp0 (z) = δ − z IRm,n f (z) λ,l

¶δ

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

·

, z ∈ U , z 6= 0, f ∈ A. The function p is m,n z (IRλ,l f (z)) IRm,n λ,l f (z)

By using the identity (1.3), we obtain

0

¸

.

m,n+2 m,n+1 IRλ,l f (z) f (z) IRλ,l zp0 (z) = δ (n + 2) . − δ (n + 1) m,n m,n+1 p (z) IR f (z) IRλ,l f (z) λ,l

(2.10)

β , it can be easily verified that θ is analytic in C, φ is analytic in By setting θ (w) := α + ξw + µw2 and φ (w) := w C\{0} and that φ (w) 6= 0, w ∈ C\{0}. 0 (z) Also, by letting Q (z) = zq 0 (z) φ (q (z)) = βzq q(z) , we find that Q (z) is starlike univalent in U. 0

(z) Let h (z) = θ (q (z)) + Q (z) = α + ξq (z) + µq 2 (z) + βzq q(z) . ³ 0 ´ ³ ´ (z) q 0 (z) q 00 (z) ξ 2µ 2 We have Re zh q (z) + q (z) − z + z = Re 1 + > 0. 0 Q(z) β β q(z) q (z) 2

By using (2.10), we obtain α + ξp (z) + µ (p (z)) + +βδ (n + 2)

m,n+2 IRλ,l f (z) m,n+1 IRλ,l f (z)

− βδ (n + 1)

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

− βδ. 2

0 (z) β zpp(z)

= α+ξ

µ

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

0

¶δ

+µ

µ

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

¶2δ

0

(z) (z) By using (2.9), we have α + ξp (z) + µ (p (z)) + β zpp(z) ≺ α + ξq (z) + µq 2 (z) + βzq q(z) . µ m,n+1 ¶δ IRλ,l f (z) From Lemma 1.1, we have p (z) ≺ q (z), z ∈ U, i.e. ≺ q (z), z ∈ U, δ ∈ C, δ 6= 0 and q is the m,n IR f (z) λ,l

best dominant.

Corollary 2.11 Let q (z) =

1+Az 1+Bz ,

z ∈ U, −1 ≤ B < A ≤ 1, m, n ∈ N, λ, l ≥ 0. Assume that (2.7) holds. If f ∈ A ³ ´2 (A−B)z 1+Az 1+Az and ψ m,n (δ, α, ξ, µ, β; z) ≺ α+ξ +µ +β (1+Az)(1+Bz) , for α, ξ, µ, β, δ ∈ C, β, δ 6= 0, −1 ≤ B < A ≤ 1, λ,l 1+Bz 1+Bz ¶δ µ m,n+1 IRλ,l f (z) 1+Az 1+Az where ψ m,n ≺ 1+Bz , and 1+Bz is the best dominant. λ,l is defined in (2.8), then IRm,n f (z) λ,l

Proof. For q (z) =

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, in Theorem 2.10 we get the corollary.

³ ´γ 1+z Corollary 2.12 Let q (z) = 1−z , m, n ∈ N, λ, l ≥ 0. Assume that (2.7) holds. If f ∈ A and ψ m,n λ,l (δ, α, ξ, µ, β; z) ≺ ³ ³ ´γ ´2γ m,n 2βγz 1+z 1+z + µ 1−z + 1−z is defined in (2.8), then α + ξ 1−z 2 , for α, ξ, µ, β, δ ∈ C, 0 < γ ≤ 1, β, δ 6= 0, where ψ λ,l ¶δ ³ µ m,n+1 ³ ´ ´ γ γ IRλ,l f (z) 1+z 1+z ≺ 1−z , and 1−z is the best dominant. IRm,n f (z) λ,l

1222

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Proof. Corollary follows by using Theorem 2.10 for q (z) =

³

1+z 1−z

´γ

, 0 < γ ≤ 1.

Theorem 2.13 Let q be convex and univalent in U such that q (0) = 1. Assume that ¶ µ 2µ 2 ξ q (z) q 0 (z) + q (z) q 0 (z) > 0, for α, ξ, µ, β ∈ C, β 6= 0. (2.11) Re β β ¶δ µ m,n+1 IR f (z) m,n ∈ H [q (0) , 1] ∩ Q and ψ m,n If f ∈ A, IRλ,lm,n f (z) λ,l (δ, α, ξ, µ, β; z) is univalent in U , where ψ λ,l (δ, α, ξ, µ, β; z) λ,l

is as defined in (2.8), then

α + ξq (z) + µq 2 (z) +

implies q (z) ≺

µ

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

¶δ

βzq 0 (z) ≺ ψ m,n λ,l (δ, α, ξ, µ, β; z) q (z)

(2.12)

, δ ∈ C, δ 6= 0, z ∈ U, and q is the best subordinant.

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

µ

m,n+1 IRλ,l f (z) m,n IRλ,l f (z)

¶δ

, z ∈ U , z 6= 0, δ ∈ C, δ 6= 0, f ∈ A. The

function p is analytic in U and p (0) = 1. β By setting ν (w) := α + ξw + µw2 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 0 (q(z)) ν (q(z)) ξ 2µ 2 2 0 0 0 = βξ q (z) q 0 (z) + 2µ q (z) q (z), it follows that Re q (z) q (z) + q (z) q (z) > = Re Since νφ(q(z)) β φ(q(z)) β β 0, for α, ξ, µ, β ∈ C, β 6= 0. 0 (z) βzp0 (z) 2 Now, by using (2.12) we obtain α + ξq (z) + µq 2 (z) + βzq q(z) ≺ α + ξp (z) + µp (z) + p(z) , z ∈ U. From Lemma µ m,n+1 ¶δ IRλ,l f (z) 1.2, we have q (z) ≺ p (z) = , z ∈ U, δ ∈ C, δ 6= 0, and q is the best subordinant. IRm,n f (z) λ,l

1+Az Corollary 2.14 Let q (z) = 1+Bz , −1 ≤ B < A ≤ 1, z ∈ U, m, n ∈ N, λ, l ≥ 0. Assume that (2.11) holds. If f ∈ A, µ m,n+1 ¶δ ³ ´2 IRλ,l f (z) (A−B)z 1+Az 1+Az ∈ H [q (0) , 1]∩Q, δ ∈ C, δ 6= 0 and α+ξ 1+Bz +µ 1+Bz +β (1+Az)(1+Bz) ≺ ψ m,n m,n λ,l (δ, α, ξ, µ, β; z) , IRλ,l f (z) ¶δ µ m,n+1 IRλ,l f (z) 1+Az is defined in (2.8), then ≺ , δ ∈ C, for α, ξ, µ, β ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψ m,n m,n λ,l 1+Bz IR f (z) λ,l

δ 6= 0, and

1+Az 1+Bz

is the best subordinant.

Proof. For q (z) =

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, in Theorem 2.13 we get the corollary.

¶δ µ m,n+1 ³ ´γ IR f (z) 1+z Corollary 2.15 Let q (z) = 1−z , m, n ∈ N, λ, l ≥ 0. Assume that (2.11) holds. If f ∈ A, IRλ,lm,n f (z) λ,l ³ ³ ´γ ´2γ m,n 2βγz 1+z 1+z ∈ H [q (0) , 1] ∩ Q and α + ξ 1−z + µ 1−z + 1−z2 ≺ ψ λ,l (δ, α, ξ, µ, β; z) , for α, ξ, µ, β, δ ∈ C, 0 < γ ≤ 1, β ³ ³ ´γ µ IRm,n+1 f (z) ¶δ ´γ λ,l 1+z 1+z ,δ 6= 0, where ψ m,n is defined in (2.8), then ≺ , and is the best subordinant. m,n λ,l 1−z 1−z IR f (z) λ,l

³ ´γ , 0 < γ ≤ 1. Proof. Corollary follows by using Theorem 2.13 for q (z) = 1+z 1−z Combining Theorem 2.10 and Theorem 2.13, we state the following sandwich theorem.

Theorem 2.16 Let q1 and q2 be convex and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all z ∈ U . ¶δ µ m,n+1 IRλ,l f (z) ∈ H [q (0) , 1] ∩ Q , δ ∈ C, δ 6= 0 and Suppose that q1 satisfies (2.7) and q2 satisfies (2.11). If f ∈ A, IRm,n f (z) λ,l

βzq 0 (z)

ψ m,n λ,l

1 (δ, α, ξ, µ, β; z) is as defined in (2.8) univalent in U , then α+ξq1 (z)+µq12 (z)+ q1 (z) ≺ ψ m,n λ,l (δ, α, ξ, µ, β; z) ≺ ¶δ µ m,n+1 IRλ,l f (z) βzq20 (z) α + ξq2 (z) + µq22 (z) + q2 (z) , for α, ξ, µ, β ∈ C, β 6= 0, implies q1 (z) ≺ ≺ q2 (z), z ∈ U, δ ∈ C, IRm,n f (z) λ,l

δ 6= 0, and q1 and q2 are respectively the best subordinant and the best dominant. 1223

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For q1 (z) =

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

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

1+A2 z 1z Corollary 2.17 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.7) and (2.11) hold for q1 (z) = 1+A 1+B1 z and q2 (z) = 1+B2 z , ¶δ µ m,n+1 ³ ´2 IRλ,l f (z) (A1 −B1 )z 1+A1 z 1+A1 z ∈ H [q (0) , 1] ∩ Q and α + ξ + µ + β (1+A ≺ respectively. If f ∈ A, m,n 1+B1 z 1+B1 z IRλ,l f (z) 1 z)(1+B1 z) ³ ´2 (A2 −B2 )z 1+A2 z 1+A2 z ψ m,n + β (1+A , z ∈ U, for α, ξ, µ, β ∈ C, β 6= 0, −1 ≤ B2 ≤ λ,l (δ, α, ξ, µ, β; z) ≺ α + ξ 1+B2 z + µ 1+B2 z 2 z)(1+B2 z) ¶δ µ m,n+1 IRλ,l f (z) m,n 1+A1 z 2z B1 < A1 ≤ A2 ≤ 1, where ψ λ,l is defined in (2.2), then 1+B1 z ≺ ≺ 1+A 1+B2 z , z ∈ U, δ ∈ C, δ 6= 0, IRm,n f (z) λ,l

hence

1+A1 z 1+B1 z

and

For q1 (z) =

³

1+A2 z 1+B2 z

1+z 1−z

´γ 1

are the best subordinant and the best dominant, respectively. , q2 (z) =

³

1+z 1−z

´γ 2

, where 0 < γ 1 < γ 2 ≤ 1, we have the following corollary.

³ ´γ 1 1+z Corollary 2.18 Let m, n ∈ N, λ, l ≥ 0. Assume that (2.7) and (2.11) hold for q1 (z) = 1−z and q2 (z) = ¶δ µ m,n+1 ³ ³ ³ ´γ 2 ´ ´ γ1 2γ 1 IRλ,l f (z) 1+z 1+z 1+z 1z , respectively. If f ∈ A, ∈ H [q (0) , 1] ∩ Q and α + ξ 1−z + µ 1−z + 2βγ m,n 1−z 1−z 2 ≺ IRλ,l f (z) ³ ³ ´γ 2 ´2γ 2 1+z 1+z 2z ψ m,n (δ, α, ξ, µ, β; z) ≺ α + ξ + µ + 2βγ λ,l 1−z 1−z 1−z 2 , z ∈ U, for α, ξ, µ, β ∈ C, β 6= 0, 0 < γ 1 < γ 2 ≤ 1, ¶δ ³ µ ³ ³ ´γ 1 ´γ 2 ´γ 1 m,n+1 IRλ,l f (z) m,n 1+z 1+z 1+z where ψ λ,l is defined in (2.2), then 1−z ≺ ≺ , z ∈ U, δ ∈ C, δ = 6 0, hence m,n 1−z 1−z IRλ,l f (z) ³ ´γ 2 1+z and 1−z are the best subordinant and the best dominant, respectively.

References [1] A. Alb Lupas, Differential Sandwich Theorems using a multiplier transformation and Ruscheweyh derivative, submitted GFTA 2015. [2] A. Alb Lupas, Some differential sandwich theorems using a multiplier transformation and Ruscheweyh derivative, submitted 2015. [3] T. Bulboaca˘, Classes of first order differential superordinations, Demonstratio Math., Vol. 35, No. 2, 287-292. [4] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 10, 815-826, October, 2003. [5] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker Inc., New York, 2000. [6] C. Selvaraj, K.T. Karthikeyan, Differential Subordination and Superordination for Analytic Functions Defined Using a Family of Generalized Differential Operators, An. St. Univ. Ovidius Constanta, Vol. 17 (1) 2009, 201-210. [7] T.N. Shanmugan, C. Ramachandran, M. Darus, S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions involving a linear operator, Acta Math. Univ. Comenianae, 16 (2007), no. 2, 287-294. [8] H.M. Srivastava, A.Y. Lashin, Some applications of the Briot-Bouquet differential subordination, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), no. 2, Article 41, 7 pp. (electronic).

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Approximating fixed points with applications in fractional calculus George A. Anastassiou1 and Ioannis K. Argyros2 1 Department of Mathematical Sciences University of Memphis Memphis, TN 38152, U.S.A. [email protected] 2 Department of Mathematical Sciences Cameron University Lawton, Ok 73505, USA [email protected] Abstract We approximate fixed points of some iterative methods on a generalized Banach space setting. Earlier studies such as [5, 6, 7, 12] require that the operator involved is Fréchet-differentiable. In the present study we assume that the operator is only continuous. This way we extend the applicability of these methods to include generalized fractional calculus and problems from other areas. Some applications include generalized fractional calculus involving the Riemann-Liouville fractional integral and the Caputo fractional derivative. Fractional calculus is very important for its applications in many applied sciences.

2010 AMS Subject Classification: 26A33, 65G99, 47J25. Key Words and phrases: Generalized Banach space, Fixed point, semilocal convergence, Riemann-Liouville fractional integral, Caputo fractional derivative.

1

Introduction

Many problems in Computational sciences can be formulated as an operator equation using Mathematical Modelling [7, 10, 13, 14, 15]. The fixed points of these operators can rarely be found in closed form. That is why most solution methods are usually iterative.

1

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The semilocal convergence is, based on the information around an initial point, to give conditions ensuring the convergence of the method. We present a semilocal convergence analysis for some iterative methods on a generalized Banach space setting to approximate fixed point or a zero of an operator. A generalized norm is defined to be an operator from a linear space into a partially order Banach space (to be precised in section 2). Earlier studies such as [5, 6, 7, 12] for Newton’s method have shown that a more precise convergence analysis is obtained when compared to the real norm theory. However, the main assumption is that the operator involved is Fréchet-differentiable. This hypothesis limits the applicability of Newton’s method. In the present study we only assume the continuity of the operator. This may be expand the applicability of these methods. The rest of the paper is organized as follows: section 2 contains the basic concepts on generalized Banach spaces and auxiliary results on inequalities and fixed points. In section 3 we present the semilocal convergence analysis of these methods. Finally, in the concluding sections 4-5, we present special cases and applications in generalized fractional calculus.

2

Generalized Banach spaces

We present some standard concepts that are needed in what follows to make the paper as self contained as possible. More details on generalized Banach spaces can be found in [5, 6, 7, 12], and the references there in. Definition 2.1 A generalized Banach space is a triplet (x, E, /·/) such that (i) X is a linear space over R (C) . (ii) E = (E, K, k·k) is a partially ordered Banach space, i.e. (ii1 ) (E, k·k) is a real Banach space, (ii2 ) E is partially ordered by a closed convex cone K, (iii3 ) The norm k·k is monotone on K. (iii) The operator /·/ : X → K satisfies /x/ = 0 ⇔ x = 0, /θx/ = |θ| /x/ , /x + y/ ≤ /x/ + /y/ for each x, y ∈ X, θ ∈ R(C). (iv) X is a Banach space with respect to the induced norm k·ki := k·k · /·/ . Remark 2.2 The operator /·/ is called a generalized norm. In view of (iii) and (ii3 ) k·ki , is a real norm. In the rest of this paper all topological concepts will be understood with respect to this norm. ¡ ¢ Let L X j , Y stand for the space of j-linear symmetric and bounded operators from X j to Y , where X and Y are Banach spaces. For X, Y partially

2

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¡ ¢ ordered L+ X j , Y stands for the subset of monotone operators P such that 0 ≤ ai ≤ bi ⇒ P (a1 , ..., aj ) ≤ P (b1 , ..., bj ) .

(2.1)

Definition 2.3 The set of bounds for an operator Q ∈ L (X, X) on a generalized Banach space (X, E, /·/) the set of bounds is defined to be: B (Q) := {P ∈ L+ (E, E) , /Qx/ ≤ P /x/ for each x ∈ X} .

(2.2)

Let D ⊂ X and T : D → D be an operator. If x0 ∈ D the sequence {xn } given by xn+1 := T (xn ) = T n+1 (x0 ) (2.3) is well defined. We write in case of convergence T ∞ (x0 ) := lim (T n (x0 )) = lim xn . n→∞

(2.4)

We need some auxiliary results on inequations. Lemma 2.4 Let (E, K, k·k) be a partially ordered Banach space, ξ ∈ K and M, N ∈ L+ (E, E). (i) Suppose there exists r ∈ K such that R (r) := (M + N ) r + ξ ≤ r

(2.5)

and k

(M + N ) r → 0 as k → ∞.

(2.6)

Then, b := R∞ (0) is well defined satisfies the equation t = R (t) and is the smaller than any solution of the inequality R (s) ≤ s. (ii) Suppose there exists q ∈ K and θ ∈ (0, 1) such that R (q) ≤ θq, then there exists r ≤ q satisfying (i). Proof. (i) Define sequence {bn } by bn = Rn (0). Then, we have by (2.5) that b1 = R (0) = ξ ≤ r ⇒ b1 ≤ r. Suppose that bk ≤ r for each k = 1, 2, ..., n. Then, we have by (2.5) and the inductive hypothesis that bn+1 = Rn+1 (0) = R (Rn (0)) = R (bn ) = (M + N ) bn +ξ ≤ (M + N ) r +ξ ≤ r ⇒ bn+1 ≤ r. Hence, sequence {bn } is bounded above by r. Set Pn = bn+1 − bn . We shall show that Pn ≤ (M + N )n r for each n = 1, 2, ...

(2.7)

We have by the definition of Pn and (2.6) that P1 = R2 (0) − R (0) = R (R (0)) − R (0) = R (ξ) − R (0) =

Z

0

1 0

R (tξ) ξdt ≤

Z

1

R0 (ξ) ξdt

0

3

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≤

Z

1

0

R0 (r) rdt ≤ (M + N ) r,

which shows (2.7) for n = 1. Suppose that (2.7) is true for k = 1, 2, ..., n. Then, we have in turn by (2.6) and the inductive hypothesis that Pk+1 = Rk+2 (0) − Rk+1 (0) = Rk+1 (R (0)) − Rk+1 (0) = Z

1 0

¡ ¢ ¡ ¢ Rk+1 (ξ) − Rk+1 (0) = R Rk (ξ) − R Rk (0) =

¡ ¡ ¢¢ ¡ k ¢ R0 Rk (0) + t Rk (ξ) − Rk (0) R (ξ) − Rk (0) dt ≤

¡ ¡ ¢¡ ¢ ¢¡ ¢ R Rk (ξ) Rk (ξ) − Rk (0) = R0 Rk (ξ) Rk+1 (0) − Rk (0) ≤ ¡ ¢ R0 (r) Rk+1 (0) − Rk (0) ≤ (M + N ) (M + N )k r = (M + N )k+1 r, 0

which completes the induction for (2.7). It follows that {bn } is a complete sequence in ³ a Banach ´space and as such it converges to some b. Notice that R (b) = R lim Rn (0) = lim Rn+1 (0) = b ⇒ b solves the equation R (t) = t. n→∞

n→∞

We have that bn ≤ r ⇒ b ≤ r, where r a solution of R (r) ≤ r. Hence, b is smaller than any solution of R (s) ≤ s. (ii) Define sequences {vn }, {wn } by v0 = 0, vn+1 = R (vn ), w0 = q, wn+1 = R (wn ). Then, we have that 0 ≤ vn ≤ vn+1 ≤ wn+1 ≤ wn ≤ q,

(2.8)

n

wn − vn ≤ θ (q − vn )

and sequence {vn } is bounded above by q. Hence, it converges to some r with r ≤ q. We also get by (2.8) that wn − vn → 0 as n → ∞ ⇒ wn → r as n → ∞. We also need the auxiliary result for computing solutions of fixed point problems. Lemma 2.5 Let (X, (E, K, k·k) , /·/) be a generalized Banach space, and P ∈ B (Q) be a bound for Q ∈ L (X, X) . Suppose there exists y ∈ X and q ∈ K such that P q + /y/ ≤ q and P k q → 0 as k → ∞. (2.9) Then, z = T ∞ (0), T (x) := Qx + y is well defined and satisfies: z = Qz + y and /z/ ≤ P /z/ + /y/ ≤ q. Moreover, z is the unique solution in the subspace {x ∈ X|∃ θ ∈ R : {x} ≤ θq} . The proof can be found in [12, Lemma 3.2].

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3

Semilocal convergence

Let (X, (E, K, k·k) , /·/) and Y be generalized Banach spaces, D ⊂ X an open subset, G : D → Y a continuous operator and A (·) : D → L (X, Y ). A zero of operator G is to be determined by a method starting at a point x0 ∈ D. The results are presented for an operator F = JG, where J ∈ L (Y, X). The iterates are determined through a fixed point problem: xn+1 = xn + yn , A (xn ) yn + F (xn ) = 0

(3.1)

⇔ yn = T (yn ) := (I − A (xn )) yn − F (xn ) . Let U (x0 , r) stand for the ball defined by U (x0 , r) := {x ∈ X : /x − x0 / ≤ r} for some r ∈ K. Next, we present the semilocal convergence analysis of method (3.1) using the preceding notation. Theorem 3.1 Let F : D ⊂ X, A (·) : D → L (X, Y ) and x0 ∈ D be as defined previously. Suppose: (H1 ) There exists an operator M ∈ B (I − A (x)) for each x ∈ D. (H2 ) There exists an operator N ∈ L+ (E, E) satisfying for each x, y ∈ D /F (y) − F (x) − A (x) (y − x)/ ≤ N /y − x/ . (H3 ) There exists a solution r ∈ K of R0 (t) := (M + N ) t + /F (x0 )/ ≤ t. (H4 ) U (x0 , r) ⊆ D. k (H5 ) (M + N ) r → 0 as k → ∞. Then, the following hold: (C1 ) The sequence {xn } defined by xn+1 = xn + Tn∞ (0) , Tn (y) := (I − A (xn )) y − F (xn )

(3.2)

is well defined, remains in U (x0 , r) for each n = 0, 1, 2, ... and converges to the unique zero of operator F in U (x0 , r) . (C2 ) An apriori bound is given by the null-sequence {rn } defined by r0 := r and for each n = 1, 2, ... rn = Pn∞ (0) , Pn (t) = M t + N rn−1 . (C3 ) An aposteriori bound is given by the sequence {sn } defined by sn := Rn∞ (0) , Rn (t) = (M + N ) t + N an−1 , 5

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bn := /xn − x0 / ≤ r − rn ≤ r, where an−1 := /xn − xn−1 / for each n = 1, 2, ... Proof. Let us define for each n ∈ N the statement: (In ) xn ∈ X and rn ∈ K are well defined and satisfy rn + an−1 ≤ rn−1 . We use induction to show (In ). The statement (I1 ) is true: By Lemma 2.4 and (H3 ), (H5 ) there exists q ≤ r such that: M q + /F (x0 )/ = q and M k q ≤ M k r → 0 as k → ∞. Hence, by Lemma 2.5 x1 is well defined and we have a0 ≤ q. Then, we get the estimate P1 (r − q) = M (r − q) + N r0 ≤ M r − M q + N r = R0 (r) − q ≤ R0 (r) − q = r − q. It follows with Lemma 2.4 that r1 is well defined and r1 + a0 ≤ r − q + q = r = r0 . Suppose that (Ij ) is true for each j = 1, 2, ..., n. We need to show the existence of xn+1 and to obtain a bound q for an . To achieve this notice that: M rn + N (rn−1 − rn ) = M rn + N rn−1 − N rn = Pn (rn ) − N rn ≤ rn . Then, it follows from Lemma 2.4 that there exists q ≤ rn such that k

q = M q + N (rn−1 − rn ) and (M + N ) q → 0, as k → ∞.

(3.3)

By (Ij ) it follows that bn = /xn − x0 / ≤

n−1 X j=0

aj ≤

n−1 X j=0

(rj − rj+1 ) = r − rn ≤ r.

Hence, xn ∈ U (x0 , r) ⊂ D and by (H1 ) M is a bound for I − A (xn ) . We can write by (H2 ) that /F (xn )/ = /F (xn ) − F (xn−1 ) − A (xn−1 ) (xn − xn−1 )/ ≤ N an−1 ≤ N (rn−1 − rn ) .

(3.4)

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It follows from (3.3) and (3.4) that M q + /F (xn )/ ≤ q. By Lemma 2.5, xn+1 is well defined and an ≤ q ≤ rn . In view of the definition of rn+1 we have that Pn+1 (rn − q) = Pn (rn ) − q = rn − q, so that by Lemma 2.4, rn+1 is well defined and rn+1 + an ≤ rn − q + q = rn , which proves (In+1 ). The induction for (In ) is complete. Let m ≥ n, then we obtain in turn that /xm+1 − xn / ≤

m X

j=n

aj ≤

m X

j=n

(rj − rj+1 ) = rn − rm+1 ≤ rn .

(3.5)

Moreover, we get inductively the estimate n+1

rn+1 = Pn+1 (rn+1 ) ≤ Pn+1 (rn ) ≤ (M + N ) rn ≤ ... ≤ (M + N )

r.

It follows from (H5 ) that {rn } is a null-sequence. Hence, {xn } is a complete sequence in a Banach space X by (3.5) and as such it converges to some x∗ ∈ X. By letting m → ∞ in (3.5) we deduce that x∗ ∈ U (xn , rn ). Furthermore, (3.4) shows that x∗ is a zero of F . Hence, (C1 ) and (C2 ) are proved. In view of the estimate Rn (rn ) ≤ Pn (rn ) ≤ rn the apriori, bound of (C3 ) is well defined by Lemma 2.4. That is sn is smaller in general than rn . The conditions of Theorem 3.1 are satisfied for xn replacing x0 . A solution of the inequality of (C2 ) is given by sn (see (3.4)). It follows from (3.5) that the conditions of Theorem 3.1 are easily verified. Then, it follows from (C1 ) that x∗ ∈ U (xn , sn ) which proves (C3 ). In general the aposterior, estimate is of interest. Then, condition (H5 ) can be avoided as follows: Proposition 3.2 Suppose: condition (H1 ) of Theorem 3.1 is true. (H03 ) There exists s ∈ K, θ ∈ (0, 1) such that R0 (s) = (M + N ) s + /F (x0 )/ ≤ θs. (H04 ) U (x0 , s) ⊂ D. Then, there exists r ≤ s satisfying the conditions of Theorem 3.1. Moreover, the zero x∗ of F is unique in U (x0 , s) . 7

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Remark 3.3 (i) Notice that by Lemma 2.4 Rn∞ (0) is the smallest solution of Rn (s) ≤ s. Hence any solution of this inequality yields on upper estimate for Rn∞ (0). Similar inequalities appear in (H2 ) and (H02 ). (ii) The weak assumptions of Theorem 3.1 do not imply the existence of −1 A (xn ) . In practice the computation of Tn∞ (0) as a solution of a linear equation is no problem and the computation of the expensive or impossible to compute in general A (xn )−1 is not needed. (iii) We can used the following result for the computation of the aposteriori estimates. The proof can be found in [12, Lemma 4.2] by simply exchanging the definitions of R. Lemma 3.4 Suppose that the conditions of Theorem 3.1 are satisfied. If s ∈ K is a solution of Rn (s) ≤ s, then q := s − an ∈ K and solves Rn+1 (q) ≤ q. This k solution might be improved by Rn+1 (q) ≤ q for each k = 1, 2, ... .

4

Special cases and applications

Application 4.1 The results obtained in earlier studies such as [5, 6, 7, 12] require that operator F (i.e. G) is Fréchet-differentiable. This assumption limits the applicability of the earlier results. In the present study we only require that F is a continuous operator. Hence, we have extended the applicability of these methods to include classes of operators that are only continuous. Example 4.2 The j-dimensional space Rj is a classical example of a generalized Banach space. The generalized norm is defined by componentwise absolute values. Then, as ordered Banach space we set E = Rj with componentwise ordering with e.g. the maximum norm. A bound for a linear operator (a matrix) is given by the corresponding matrix with absolute values. Similarly, we can define the ”N ” operators. Let E = R. That is we consider the case of a real normed space with norm denoted by k·k. Let us see how the conditions of Theorem 3.1 look like. Theorem 4.3 (H1 ) kI − A (x)k ≤ M for some M ≥ 0. (H2 ) kF (y) − F (x) − A (x) (y − x)k ≤ N ky − xk for some N ≥ 0. (H3 ) M + N < 1, kF (x0 )k r= . 1 − (M + N )

(4.1)

(H4 ) U (x0 , r) ⊆ D. k (H5 ) (M + N ) r → 0 as k → ∞, where r is given by (4.1). Then, the conclusions of Theorem 3.1 hold.

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5

Applications to k-Fractional Calculus

Background We apply Theorem 4.3 in this section. Let f ∈ L∞ ([a, b]), the k-left Riemann-Liouville fractional integral ([15]) of order α > 0 is defined as follows: Z x α 1 α J f (x) = (x − t) k −1 f (t) dt, (5.1) k a+ kΓk (α) a all x ∈ [a, b], where k > 0, and Γk (a) is the k-gamma function given by Γk (α) = R ∞ α−1 − tk t e k dt. 0 α It holds ([4]) Γk (α + k) = αΓk (α), Γ (α) = lim Γk (α), and we set k Ja+ f (x) = k→1

f (x) . Similarly, we define the k-right Riemann-Liouville fractional integral as α k Jb− f

1 (x) = kΓk (α)

Z

b

x

α

(t − x) k −1 f (t) dt,

(5.2)

α f (x) = f (x) . for all x ∈ [a, b], and we set k Jb− Results α I) Here we work with k Ja+ f (x). We observe that Z x ¯ α ¯ α 1 −1 ¯k Ja+ f (x)¯ ≤ (x − t) k |f (t)| dt kΓk (α) a

kf k∞ ≤ kΓk (α) =

Z

a

α

x

(x − t)

α k −1

kf k∞ (x − a) k dt = α kΓk (α) k

(5.3)

α α kf k∞ kf k∞ (x − a) k ≤ (b − a) k . Γk (α + k) Γk (α + k)

We have proved that α k Ja+ f

and

° α ° °k Ja+ f °

∞

(a) = 0,

(5.4)

α

(b − a) k kf k∞ , ≤ Γk (α + k)

(5.5)

α proving that k Ja+ is a bounded linear operator. ¡ α ¢ By [3], p. 388, we get that k Ja+ f is a continuous function over [a, b] and in particular continuous over [a∗ , b], where a < a∗ < b. Thus, there exist x1 , x2 ∈ [a∗ , b] such that ¡ α ¢ ¡ α ¢ (5.6) k Ja+ f (x1 ) = min k Ja+ f (x) , ¡ α ¢ ¡ α ¢ ∗ (5.7) k Ja+ f (x2 ) = max k Ja+ f (x) , x ∈ [a , b] .

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We assume that

¡

α k Ja+ f

Hence

° α ° °k Ja+ f °

Here it is

∞,[a∗ ,b]

¢

=

(x1 ) > 0. ¡

α k Ja+ f

¢

(5.8) (x2 ) > 0.

(5.9)

J (x) = mx, m 6= 0.

(5.10)

Jf (x) = 0, x ∈ [a∗ , b] ,

(5.11)

Therefore the equation has the same solutions as the equation F (x) := Notice that à α k Ja+

Call

f ¡ α ¢ 2 k Ja+ f (x2 )

We notice that

Hence it holds

!

2

¡

Jf (x) ¢ = 0, x ∈ [a∗ , b] . (x2 )

(5.12)

α k Ja+ f

(x) =

¡ α ¢ f (x) 1 kJ ¡ a+ ¢ ≤ < 1, x ∈ [a∗ , b] . α f (x ) 2 2 k Ja+ 2

¡ α ¢ f (x) kJ ¢ , ∀ x ∈ [a∗ , b] . A (x) := ¡ a+ α 2 k Ja+ f (x2 )

¡ α ¢ f (x1 ) 1 kJ ¢ ≤ A (x) ≤ , ∀ x ∈ [a∗ , b] . 0 < ¡ a+ α 2 2 k Ja+ f (x2 )

¡ α ¢ f (x1 ) kJ ¢ =: γ 0 , ∀ x ∈ [a∗ , b] . |1 − A (x)| = 1 − A (x) ≤ 1 − ¡ a+ α 2 k Ja+ f (x2 )

(5.13)

(5.14)

(5.15)

(5.16)

Clearly γ 0 ∈ (0, 1) . We have proved that

|1 − A (x)| ≤ γ 0 , ∀ x ∈ [a∗ , b] .

(5.17)

Next we assume that F (x) is a contraction, i.e. |F (x) − F (y)| ≤ λ |x − y| ; ∀ x, y ∈ [a∗ , b] ,

(5.18)

and 0 < λ < 12 . Equivalently we have |Jf (x) − Jf (y)| ≤ 2λ

¡

α k Ja+ f

¢

(x2 ) |x − y| , all x, y ∈ [a∗ , b] .

(5.19)

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We observe that |F (y) − F (x) − A (x) (y − x)| ≤ |F (y) − F (x)| + |A (x)| |y − x| ≤ λ |y − x| + |A (x)| |y − x| = (λ + |A (x)|) |y − x| =: (ψ 1 ) , ∀ x, y ∈ [a∗ , b] . (5.20) We have that α

Hence

k ¯¡ α ¢ ¯ ¯ k Ja+ f (x)¯ ≤ (b − a) kf k < ∞, ∀ x ∈ [a∗ , b] . ∞ Γk (α + k)

(5.21)

¯¡ α ¢ ¯ α ¯ k Ja+ f (x)¯ (b − a) k kf k∞ ¡ α ¢ ≤ < ∞, ∀ x ∈ [a∗ , b] . |A (x)| = ¡ α ¢ 2 k Ja+ f (x2 ) 2Γk (α + k) k Ja+ f (x2 ) (5.22) Therefore we get à ! α (b − a) k kf k∞ ¡ α ¢ (ψ 1 ) ≤ λ + (5.23) |y − x| , ∀ x, y ∈ [a∗ , b] . 2Γk (α + k) k Ja+ f (x2 )

Call

α

0 < γ 1 := λ +

(b − a) k kf k∞ ¡ α ¢ , 2Γk (α + k) k Ja+ f (x2 )

(5.24)

choosing (b − a) small enough we can make γ 1 ∈ (0, 1). We have proved that

|F (y) − F (x) − A (x) (y − x)| ≤ γ 1 |y − x| , ∀ x, y ∈ [a∗ , b] , γ 1 ∈ (0, 1) . (5.25) Next we call and we need that ¡ α ¢ a (b − a) k kf k∞ k Ja+ f (x1 ) ¡ ¢ ¡ ¢ 0 < γ := γ 0 + γ 1 = 1 − +λ+ < 1, α f (x ) α f (x ) 2 k Ja+ 2Γk (α + k) k Ja+ 2 2 (5.26) equivalently, ¡ α ¢ a f (x1 ) (b − a) k kf k∞ kJ ¡ α ¢ ¢ λ+ < ¡ a+ , (5.27) α 2Γk (α + k) k Ja+ f (x2 ) 2 k Ja+ f (x2 ) equivalently,

2λ

¡

α k Ja+ f

¢

α

(b − a) k kf k∞ ¡ α ¢ < k Ja+ f (x1 ) , (x2 ) + Γk (α + k)

(5.28)

which is possible for small λ, (b − a). That is γ ∈ (0, 1). So our numerical method converges and solves (5.11). 11

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α II) Here we act on k Jb− f (x), see (5.2). Let f ∈ L∞ ([a, b]) . We have that

¯ α ¯ ¯k Jb− f (x)¯ ≤

kf k∞ ≤ kΓk (α) =

Z

b

x

α

(t − x) k

−1

|f (t)| dt α

b

x

Z

1 kΓk (α)

(t − x)

α k −1

kf k∞ (b − x) k dt = α kΓk (α) k

α α kf k∞ kf k∞ (b − x) k ≤ (b − a) k . Γk (α + k) Γk (α + k)

(5.29)

We observe that α k Jb− f

and

° α ° °k Jb− f °

∞

(b) = 0,

(5.30)

α

(b − a) k kf k∞ . ≤ Γk (α + k)

(5.31)

α That is k Jb− is a bounded linear operator. Let here a < b∗ < b. α By [4] we get that k Jb− f is continuous over [a, b] , and in particular it is ∗ continuous over [a, b ]. Thus, there exist x1 , x2 ∈ [a, b∗ ] such that ¡ α ¢ ¡ α ¢ (5.32) k Jb− f (x1 ) = min k Jb− f (x) , ¡ α ¢ ¡ α ¢ ∗ k Jb− f (x2 ) = max k Jb− f (x) , x ∈ [a, b ] .

We assume that

¡

α k Jb− f

Hence

° α ° °k Jb− f °

Here it is

∞,[a∗ ,b]

¢

(x1 ) > 0.

=

¡

α k Jb− f

¢

(5.33) (x2 ) > 0.

(5.34)

J (x) = mx, m 6= 0.

(5.35)

Jf (x) = 0, x ∈ [a, b∗ ] ,

(5.36)

Therefore the equation has the same solutions as the equation F (x) := Notice that à α k Jb−

f ¡ α ¢ 2 k Jb− f (x2 )

!

Jf (x) ¢ = 0, x ∈ [a, b∗ ] . α f (x ) 2 k Jb− 2 ¡

(5.37)

¡ α ¢ 1 k J f (x) ¢ ≤ < 1, x ∈ [a, b∗ ] . (x) = ¡ b− α 2 2 k Jb− f (x2 )

(5.38)

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Call A (x) := We notice that

Hence we have

¡ α ¢ k J f (x) ¡ b− ¢ , ∀ x ∈ [a, b∗ ] . α f (x ) 2 k Jb− 2

(5.39)

¡ α ¢ 1 k J f (x1 ) ¢ ≤ A (x) ≤ , ∀ x ∈ [a, b∗ ] . 0 < ¡ b− α f (x ) 2 2 k Jb− 2

(5.40)

¡ α ¢ k J f (x1 ) ¢ |1 − A (x)| = 1 − A (x) ≤ 1 − ¡ b− =: γ 0 , ∀ x ∈ [a, b∗ ] . α f (x ) 2 k Jb− 2

(5.41)

Clearly γ 0 ∈ (0, 1) . We have proved that

|1 − A (x)| ≤ γ 0 , ∀ x ∈ [a, b∗ ] , γ 0 ∈ (0, 1) .

(5.42)

Next we assume that F (x) is a contraction, i.e. |F (x) − F (y)| ≤ λ |x − y| ; ∀ x, y ∈ [a, b∗ ] ,

(5.43)

and 0 < λ < 12 . Equivalently we have |Jf (x) − Jf (y)| ≤ 2λ We observe that

¡

α k Jb− f

¢

(x2 ) |x − y| , all x, y ∈ [a, b∗ ] .

(5.44)

|F (y) − F (x) − A (x) (y − x)| ≤ |F (y) − F (x)| + |A (x)| |y − x| ≤ λ |y − x| + |A (x)| |y − x| = (λ + |A (x)|) |y − x| =: (ψ 1 ) , ∀ x, y ∈ [a, b∗ ] . (5.45) We have that α

Hence

k ¯¡ α ¢ ¯ ¯ k Jb− f (x)¯ ≤ (b − a) kf k < ∞, ∀ x ∈ [a, b∗ ] . ∞ Γk (α + k)

(5.46)

¯ ¯¡ α ¢ α ¯ k J f (x)¯ (b − a) k kf k∞ b− ¡ α ¢ ≤ < ∞, ∀ x ∈ [a, b∗ ] . |A (x)| = ¡ α ¢ 2 k Jb− f (x2 ) 2Γk (α + k) k Jb− f (x2 ) (5.47) Therefore we get à ! α (b − a) k kf k∞ ¡ α ¢ (ψ 1 ) ≤ λ + (5.48) |y − x| , ∀ x, y ∈ [a, b∗ ] . 2Γk (α + k) k Jb− f (x2 ) 13

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Call

α

(b − a) k kf k∞ ¡ α ¢ 0 < γ 1 := λ + , 2Γk (α + k) k Jb− f (x2 )

(5.49)

choosing (b − a) small enough we can make γ 1 ∈ (0, 1). We have proved that

|F (y) − F (x) − A (x) (y − x)| ≤ γ 1 |y − x| , ∀ x, y ∈ [a, b∗ ] , γ 1 ∈ (0, 1) . (5.50) Next we call and we need that ¡ α ¢ a (b − a) k kf k∞ k Jb− f (x1 ) ¡ α ¢ 0 < γ := γ 0 + γ 1 = 1 − ¡ α ¢ +λ+ < 1, 2 k Jb− f (x2 ) 2Γk (α + k) k Jb− f (x2 ) (5.51) equivalently, ¡ α ¢ a (b − a) k kf k∞ k J f (x1 ) ¡ α ¢ ¢ λ+ < ¡ b− , (5.52) α f (x ) 2Γk (α + k) k Jb− f (x2 ) 2 k Jb− 2 equivalently,

2λ

¡

α k Jb− f

¢

α

(b − a) k kf k∞ ¡ α ¢ < k Jb− f (x1 ) , (x2 ) + Γk (α + k)

(5.53)

which is possible for small λ, (b − a). That is γ ∈ (0, 1). So our numerical method converges and solves (5.36). III) Here we deal with the fractional M. Caputo-Fabrizio derivative defined as follows (see [9]): let 0 < α < 1, f ∈ C 1 ([0, b]), ¶ µ Z t 1 α CF α (t − s) f 0 (s) ds, D∗ f (t) = exp − (5.54) 1−α 0 1−α for all 0 ≤ t ≤ b. Call γ := I.e. CF

We notice that

D∗α f (t) =

1 1−α

¯ ¯CF α ¯ D∗ f (t)¯ ≤

Z

α > 0. 1−α

(5.55)

e−γ(t−s) f 0 (s) ds, 0 ≤ t ≤ b.

(5.56)

t 0

1 1−α

µZ

0

t

¶ e−γ(t−s) ds kf 0 k∞

¢ ¢ e−γt ¡ γt 1¡ e − 1 kf 0 k∞ = 1 − e−γt kf 0 k∞ ≤ = α α

µ

1 − e−γb α

¶

kf 0 k∞ . (5.57)

14

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That is and

¯CF α ¯ ¯ D∗ f (t)¯ ≤

¡CF µ

¢ D∗α f (0) = 0,

1 − e−γb α

¶

kf 0 k∞ , ∀ t ∈ [0, b] .

(5.58)

(5.59)

Notice here that 1 − e−γt , t ≥ 0 is an increasing function. Thus the smaller the t, the smaller it is 1 − e−γt . We rewrite Z t e−γt CF α D∗ f (t) = eγs f 0 (s) ds, (5.60) 1−α 0 ¢ ¡ proving that CF D∗α f is a continuous function over [0, b], in particular it is continuous over [a, b], where 0 < a < b. Therefore there exist x1 , x2 ∈ [a, b] such that CF

D∗α f (x1 ) = min

CF

D∗α f (x) ,

(5.61)

and CF

D∗α f (x2 ) = max

CF

D∗α f (x) , for x ∈ [a, b] .

We assume that CF

D∗α f (x1 ) > 0.

(i.e. CF D∗α f (x) > 0, ∀ x ∈ [a, b]). Furthermore °CF α ° ° D∗ f G°

∞,[a,b]

=CF D∗α f (x2 ) .

(5.62)

(5.63)

Here it is

J (x) = mx, m 6= 0.

(5.64)

Jf (x) = 0, x ∈ [a, b] ,

(5.65)

The equation has the same set of solutions as the equation Jf (x) = 0, x ∈ [a, b] . (x2 )

F (x) := Notice that µ CF α D∗

f (x) 2CF D∗α f (x2 )

¶

CF D α f ∗

1 D∗α f (x) ≤ < 1, ∀ x ∈ [a, b] . 2CF D∗α f (x2 ) 2

(5.66)

CF

=

We call

CF

A (x) := We notice that

D∗α f (x) , ∀ x ∈ [a, b] . 2CF D∗α f (x2 )

(5.67)

CF

0
0 (z ∈ U). f (z) In recent years, many authors (see, for example, [1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18]) have investigated and derived sufficient conditions for Carath´eodory functions and some of their results have been applied to find some sufficient conditions for starlikeness or convexity of analytic functions (see, for example, [5, 11, 13, 14, 15, 17]). Following the principle of differential subordination, we say that a function f is subordinate to F in U, written as f ≺ F , if and only if
f (z) = F w(z) (z ∈ U) for some Schwarz function w(z), with w(0) = 0

|w(z)| < 1

and

(z ∈ U).

If F (z) is univalent in U, then the subordination f ≺ F is equivalent to f (0) = F (0)

and

f (U) ⊂ F (U).

We denote by Q the class of functions q that are analytic and injective on U \ E(q), where   E(q) = ζ : ζ ∈ ∂U and lim q(z) = ∞ , z→ζ

and are such that q 0 (ζ) 6= 0


ζ ∈ ∂U \ E(q) .

Furthermore, let the subclass of Q for which q(0) = a be denoted by Q(a). The main object of this paper is to investigate and present several sets of sufficient conditions for Carath´eodory functions in the open unit disk U. The main results proven here are shown to lead to some conditions for starlike functions in U. We also consider the relevant connections of our results with various known results.

2. A Set of Main Results In order to prove our main results, we need the following lemma due to Miller and Mocanu [7, p. 24]. Lemma 1. Let q ∈ Q(a) and let the function p(z) given by p(z) = a + an z n + · · ·

(n = 1)

be analytic in U with p(0) = a. If p is not subordinate to q, then there exist points z0 ∈ U and ζ0 ∈ ∂U \ E(q) for which (i) p(z0 ) = q(ζ0 ) and 2

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(ii) z0 p0 (z0 ) = mζ0 q 0 (ζ0 )

(m = n = 1).

Applying Lemma 1, we can obtain the following results. Theorem 1. Let P : U → C with R {P (z)} = I {P (z)} tan α = 0



05α
4A cos2 α 2 cos α where A = cos 2α +

(1)

B 2 cos α

(2)

and B = R {P (z)} cos α − I {P (z)} sin α,

(3)

then

  π π −α 05α< ; z∈U . 2 2 Proof. Let us define two functions q(z) and h1 (z) by   π iα iα q(z) = e p(z) q(z) 6≡ e ; 0 5 α < ; z ∈ U 2 and   eiα + eiα z π h1 (z) = 05α< ; z∈U , 1−z 2 respectively. Then the functions q(z) and h1 (z) are analytic in U with | arg {p(z)} |
0} .

We now suppose that the function q is not subordinate to h1 . Then, by Lemma 1, there exist points z1 ∈ U and ζ1 ∈ ∂U \ {1} such that q(z1 ) = h1 (ζ1 ) = iρ (ρ ∈ R)

and

z1 q 0 (z1 ) = mζ1 h01 (ζ1 ) = mσ1

(m = 1),

(6)

where

ρ2 − 2ρ sin α + 1 . 2 cos α Using the equations (4), (5), (6) and (7), we obtain  R [p(z1 )]2 + P (z1 )z1 p0 (z1 ) n o 2 = R e−iα q(z1 ) + P (z1 )e−iα z1 q 0 (z1 )  = R e−2iα [h1 (ζ1 )]2 + P (z1 )e−iα mζ1 h01 (ζ1 )  = R e−2iα (iρ)2 + P (z1 )e−iα mσ1 σ1 = −

= −ρ2 cos 2α + mσ1 B1     B1 sin α B1 B1 2 ρ + ρ− 5 − cos 2α + 2 cos α cos α 2 cos α   B1 sin α B1 2 = −A1 ρ + ρ− cos α 2 cos α =: g(ρ), 3

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(7)

(8)

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 21, NO.7, 2016, COPYRIGHT 2016 EUDOXUS PRESS, LLC

where B1 and A1 are given by B1 = R {P (z1 )} cos α − I {P (z1 )} sin α and

B1 , 2 cos α respectively. By a simple calculation, we see that the function g1 (ρ) in (8) takes on the maximum value at ρ∗ given by B1 sin α ρ∗ = . 2A1 cos α A1 = cos 2α +

Hence we have  R [p(z1 )]2 + P (z1 )z1 p0 (z1 ) 5 g1 (ρ∗ ) B1 B12 sin2 α − 4A1 cos2 α 2 cos α B 2 sin2 α B 5 − , 2 4A cos α 2 cos α =

where A and B are given by (2) and (3), respectively. Moreover, this inequality is a contradiction to (1). Therefore, we obtain    π R eiα p(z) > 0 05α< ; z∈U . (9) 2 Next, let us define two analytic functions by −iα

r(z) = e



p(z)

 π 05α< ; z∈U 2

(10)

and

  e−iα + e−iα z π 05α< ; z∈U . 1−z 2 Then the functions r and h2 are analytic in U with h2 (z) =

r(0) = h2 (0) = e−iα ∈ C

and h2 (U) = {w : w ∈ C

(11)

and R {w} > 0} = h1 (U).

Suppose that r is not subordinate to h2 . Then, by Lemma 1, there exist points z2 ∈ U and ζ2 ∈ ∂U \ {1} such that r(z2 ) = h2 (ζ2 ) = iρ

(ρ ∈ R)

and

where σ2 = −

z2 r0 (z2 ) = mζ2 h02 (ζ2 ) = mσ2 (m = 1),

ρ2 + 2ρ sin α + 1 . 2 cos α

(13)

4

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From the equations (10), (11), (12) and (13), we get  R [p(z2 )]2 + P (z2 )z2 p0 (z2 )  = R e2iα [h2 (ζ2 )]2 + P (z2 )eiα mζ2 h02 (ζ2 )  = R e2iα (iρ)2 + P (z2 )eiα mσ2 = −ρ2 cos 2α + mσ2 B 5 −ρ2 cos 2α + σ2 B   B B sin α 2 ρ− = −Aρ − cos α 2 cos α = g2 (ρ)   B sin α 5 g2 − 2A cos α 2 B sin2 α B = − , 2 4A cos α 2 cos α which is a contradiction to (1). Therefore, we have    π R e−iα p(z) > 0 05α< ; z∈U . 2

(14)

Hence, by applying the inequalities (9) and (14), we find that   π π 05α< ; z∈U . | arg {p(z)} | < − α 2 2 This evidently complete the proof of Theorem 1. If we take P (z) ≡ β (β > 0) in Theorem 1, then we have the following corollary. Corollary 1. Let the function p be analytic in U with p(0) = 1. If  2 1 (β + 4β) sin2 α − β 2 − 2β 2β + 4 cos 2α   π β > 0; 0 5 α < ; z ∈ U , 2   π π | arg {p(z)} | < − α 05α< ; z∈U . 2 2

 R [p(z)]2 + βzp0 (z) >

then

More specially, if we take P (z) ≡ 1 in Theorem 1 or set β = 1 in Corollary 1, we obtain the following corollary. Corollary 2. Let the function p be analytic in U with p(0) = 1. If  5 sin2 α − 3 R [p(z)]2 + zp0 (z) > 6 − 8 sin2 α then | arg {p(z)} |
− R f (z) f (z) 2

(z ∈ U)

implies that f ∈ S ∗ .

3. Further Sufficient Conditions We now find another another set of sufficient conditions for Carath´eodory functions. Theorem 2. Let p(z) be a nonzero analytic function in U with p(0) = 1 and 0   zp (z) 1 π < cos α 0 5 α < ; z ∈ U . [p(z)]2 2 2 Then | arg {p(z)} |
0 (z ∈ U) (17) for the case when α 6= 0. Thus, from (16) and (17), we have   π π | arg {p(z)} | < − α 0 0. Let p be a nonzero analytic function with p(0) = 1 and     π zp0 (z) < δ2 (α) 05α< ; z∈U , δ1 (α) < I p(z) + β (18) p(z) 2 where

p δ1 (α) = −

and

(2 cos2 α + u)u + u sin α cos α

p

(2 cos2 α + u)u − u sin α . cos α

δ2 (α) = Then | arg {p(z)} |
0. 8

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Then, by using the same method as the above, we obtain ρ cos α −

mσ1 u ρ

σ1 u 5 −˜ ρ cos α + ρ˜   1 u 2 =− (2 cos α + u)˜ ρ + + 2u sin α 2 cos α ρ˜ n o p 1 2 (2 cos2 α + u)u + 2u sin α 5− 2 cos α = δ1 (α). Moreover, this last inequality yields   z1 p0 (z1 ) I p(z1 ) + β 5 δ1 (α), p(z1 ) which is a contradiction to (18). Hence we have    π R eiα p(z) > 0 05α< ; z∈U . 2

(19)

We next define the functions r and h2 by (10) and (11), respectively. Then, by using a similar method as the above, we obtain    π (20) R e−iα p(z) > 0 05α< ; z∈U . 2 Thus, from (19) and (20), we have | arg {p(z)} |
0. Let p be a nonzero analytic function with p(0) = 1 and     0 π p(z) + γ zp (z) − 1 < γ + 1 |p(z)| cos α 0 5 α < ; z ∈ U . (21) p(z) 2 2 Then | arg {p(z)} |