- Research Article
- Open Access
Almost Automorphic Solutions of Difference Equations
© Daniela Araya et al. 2009
Received: 25 March 2009
Accepted: 13 May 2009
Published: 22 June 2009
We study discrete almost automorphic functions (sequences) defined on the set of integers with values in a Banach space . Given a bounded linear operator defined on and a discrete almost automorphic function , we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation . We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation assuming that is discrete almost automorphic in for each , satisfies a global Lipschitz type condition, and takes values on .
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 , Corduneanu , Halanay , Pang and Agarwal , Sugiyama , Elaydi , and Agarwal . Almost periodicity of a discrete function was first introduced by Walther [8, 9] and then by Corduneanu . 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'Guérékata's work 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'Guérékata in his book . After that, there has been a real resurgent interest in the study of almost automorphic functions.
Important contributions to the theory of almost automorphic functions have been obtained, for example, in the papers [26–33], in the books [20, 21, 32] (concerning almost automorphic functions with values in Banach spaces), and in  (concerning almost automorphy on groups). Also, the theory of almost automorphic functions with values in fuzzy-number-type spaces was developed in  (see also [20, Chapter 4]). Recently, in [36, 37], the theory of almost automorphic functions with values in a locally convex space (Fréchet space) and a -Fréchet space has been developed.
The range of applications of almost automorphic functions includes at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, and dynamical systems. A recent reference is the book .
This paper is organized as follows. In Section 2, we present the definition of discrete almost automorphic functions (sequences) as a natural generalization of discrete almost periodic functions, and then we give some basic and related properties for our purposes. In Section 3, we discuss the existence of almost automorphic solutions of first-order linear difference equations. In Section 4, we discuss the existence of almost automorphic solutions of nonlinear difference equations of the form , where is a bounded operator defined on a Banach space .
2. The Basic Theory
Bochner's criterion: is a discrete almost periodic function if and only if (N) for any integer sequence , there exists a subsequence such that converges uniformly on as . Furthermore, the limit sequence is also a discrete almost periodic function.
The above characterization, as well as the definition of continuous almost automorphic functions (cf. ), motivates the following definition.
Discrete almost automorphic functions have the following fundamental properties.
The proof of all statements follows the same lines as in the continuous case (see [21, Theorem 2.1.3]), and therefore is omitted.
becomes a Banach space. The proof is straightforward and therefore omitted.
For applications to nonlinear difference equations the following definition, of discrete almost automorphic function depending on one parameter, will be useful.
The proof of the following result is omitted (see [21, Section 2.2]).
The following result will be used to study almost automorphy of solution of nonlinear difference equations.
As is well known a uniformly convex Banach space does not contain any subspace isomorphic to . In particular, every finite-dimensional space does not contain any subspace isomorphic to . The following result will be the key in the study of discrete almost automorphic solutions of linear and nonlinear difference equations.
is also discrete almost automorphic.
3. Almost Automorphic Solutions of First-Order Linear Difference Equations
Difference equations usually describe the evolution of certain phenomena over the course of time. In this section we deal with those equations known as the first-order linear difference equations. These equations naturally apply to various fields, like biology (the study of competitive species in population dynamics), physics (the study of motions of interacting bodies), the study of control systems, neurology, and electricity; see [6, Chapter 3].
where is a complex number and is a discrete almost automorphic function. Hence, all we need to show is that any solution of (3.6) is discrete almost automorphic. But this is the content of Theorem 3.1. It then implies that the th component of the solution of (3.5) is discrete almost automorphic. Then substituting in the th equation of (3.5) we obtain again an equation of the form (3.6) for and so on. The proof is complete.
The procedure in the Proof of Theorem 3.2 is called "Method of Reduction" and introduced, in the continuous case, by N'Guérékata [20, Remark 6.2.2]. See also [41, 42]. In the discrete case, it was used earlier by Agarwal (cf. [7, Theorem 2.10.1]).
As an application of the above Theorem and [7, Theorem 5.2.4] we obtain the following Corollary.
We can also prove the following result.
Let be a Banach space. Suppose and where the complex numbers are mutually distinct with , and forms a complex system of mutually disjoint projections on . Then (3.2) admits a discrete almost automorphic solution.
The following important related result corresponds to the general Banach space setting. It is due to Minh et al. [17, Theorem 2.14]. We denote by the part of the spectrum of on .
We point out that in the finite dimensional case, the above result extend Corduneanu's Theorem on discrete almost periodic functions (see [7, Theorem 2.10.1, page 73]) to discrete almost automorphic functions. We state here the result for future reference.
where is a bounded linear operator on a Banach space and is the largest integer function. These results are based in the following connection between discrete and continuous almost automorphic functions.
We finish this section with the following simple example concerning the heat equation (cf. [6, page 157]).
we obtain for all eigenvalues of . For each , Theorem 3.5 then implies that, for , there is a discrete almost automorphic solution of (3.11). On the other hand, Theorem 3.7 implies that, without restriction on , each bounded solution of (3.11) is discrete almost automorphic.
4. Almost Automorphic Solutions of Semilinear Difference Equations
Our main result in this section is the following theorem for the scalar case.
and hence is a solution of (4.1) (cf. the proof of (ii) in Theorem 3.1).
The case of a bounded operator can be treated assuming extra conditions on the operator. The proof of the next result follows the same lines of the first part in the proof of Theorem 4.1, using (ii) of Remark 2.14.
5. Conclusion and Future Directions
The aim of the present paper is to present for the first time a brief exposition of the theory of discrete almost automorphic funtions and its application to the field of difference equations in abstract spaces. We first state, for future reference, several results which can be directly deduced from the continuous case, and then we analyze the existence of discrete almost automorphic solutions of linear and nonlinear difference equations in the scalar and in the abstract setting. Many questions remain open, as for example to prove the converse of (i) in Remark 2.2, that is, assuming that is a discrete almost automorphic function, to find an almost automorphic function such that for all (see [38, Theorem 1.27] in the almost periodic case). Concerning almost automorphic solutions of difference equations, it remains to study discrete almost automorphic solutions of Volterra difference equations as well as discrete almost automorphic solutions of functional difference equations with infinite delay. This topic should be handled by looking at the recent papers of Song [13, 14].
Carlos Lizama is partially supported by Laboratorio de Análisis Estocástico, Proyecto Anillo PBCT-ACT-13.
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