Almost Automorphic Solutions of Difference Equations
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Research Article Almost Automorphic Solutions of Difference Equations Daniela Araya, Rodrigo Castro, and Carlos Lizama Departamento de Matem´atica, Universidad de Santiago, 9160000 Santiago, Chile Correspondence should be addressed to Carlos Lizama, [email protected] Received 25 March 2009; Accepted 13 May 2009 Recommended by Mouffak Benchohra We study discrete almost automorphic functions sequences defined on the set of integers with values in a Banach space X. Given a bounded linear operator T defined on X and a discrete almost automorphic function fn, we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation Δun T un fn. We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation Δun T un gn, un assuming that gn, x is discrete almost automorphic in n for each x ∈ X, satisfies a global Lipschitz type condition, and takes values on X. Copyright q 2009 Daniela Araya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction The theory of difference equations has grown at an accelerated pace in the last decades. It now occupies a central position in applicable analysis and plays an important role in mathematics as a whole. A very important aspect of the qualitative study of the solutions of difference equations is their periodicity. Periodic difference equations and systems have been treated, among others, by Agarwal and Popenda 1 , Corduneanu 2 , Halanay 3 , Pang and Agarwal 4 , Sugiyama 5 , Elaydi 6 , and Agarwal 7 . Almost periodicity of a discrete function was first introduced by Walther 8, 9 and then by Corduneanu 2 . Recently, several papers 10–16 are devoted to study existence of almost periodic solutions of difference equations. Discrete almost automorphic functions, a class of functions which are more general than discrete almost periodic ones, were recently introduced in 17, Definition 2.6 in connection with the study of continuous almost automorphic bounded mild solutions of differential equations. See also 18, 19 . However, the concept of discrete almost automorphic functions has not been explored in the theory of difference equations. In this paper, we first review their main properties, most of which are discrete versions of N’Gu´er´ekata’s work
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Advances in Difference Equations
in 20, 21 , and then we give an application in the study of existence of discrete almost automorphic solutions of linear and nonlinear difference equations. The theory of continuous almost automorphic functions was introduced by Bochner, in relation to some aspects of differential geometry 22–25 . A unified and homogeneous exposition of the theory and its applications was first given by N’Gu´er´ekata in his book 21 . After that, there has been a real resurgent interest in the study of almost automorphic functions.
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