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
In the above definition is called an -almost period of or an -translation number. We denote by the set of discrete almost periodic functions.
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.
If is a continuous almost automorphic function in then is discrete almost automorphic.
If the convergence in Definition 2.1 is uniform on then we get discrete almost periodicity.
where is any nonrational real number.
Discrete almost automorphic functions have the following fundamental properties.
is discrete almost automorphic;
is discrete almost automorphic for every scalar ;
for each fixed in the function defined by is discrete almost automorphic;
the function defined by is discrete almost automorphic;
; that is, is a bounded function;
- (vi), where(2.6)
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.
Let be Banach spaces, and let a discrete almost automorphic function. If is a continuous function, then the composite function is discrete almost automorphic.
Let be a sequence in , and since there exists a subsequence of such that is well defined for each and for each Since is continuous, we have In similar way, we have therefore is in
If is a bounded linear operator on and is a discrete almost automorphic function, then , is also discrete almost automorphic.
Let and be discrete almost automorphic. Then defined by , is also discrete almost automorphic.
which are valid for all .
For applications to nonlinear difference equations the following definition, of discrete almost automorphic function depending on one parameter, will be useful.
for each and .
The proof of the following result is omitted (see [21, Section 2.2]).
is discrete almost automorphic in for each in
is discrete almost automorphic in for each in where is an arbitrary scalar.
for each in .
for each in where is the function in Definition 2.8.
The following result will be used to study almost automorphy of solution of nonlinear difference equations.
Suppose is discrete almost automorphic, then the function defined by is discrete almost automorphic.
valid for all we get the desired proof.
We will denote the space of the discrete almost automorphics functions in for each in .
Let denote the forward difference operator of the first-order, that is, for each and , .
Let be a discrete almost automorphic function, then is also discrete almost automorphic.
Since then by (i) and (iii) in Theorem 2.4, we have that is discrete almost automorphic.
is discrete almost automorphic. Then is also discrete almost automorphic.
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.
and then is discrete almost automorphic.
- (i)The same conclusions of the previous results holds in case of the finite convolution(2.21)
- (ii)The results are true if we consider an operator valued function such that(2.23)
A typical example is , where satisfies .
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 We begin studying the scalar case. We denote .
- (i)Define . Then and hence, by Theorem 2.13, we obtain Next, we note that is solution of (3.2) because(3.3)
- (ii)Define and since we have . It follows, by Theorem 2.13, that . Finally, we check that is solution of (3.2) as follows:(3.4)
As a consequence of the previous theorem, we obtain the following result in case of a matrix .
Suppose is a constant matrix with eigenvalues . Then for any function there is a discrete almost automorphic solution of (3.2).
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.
for some .
We can replace in Theorem 3.1 by a general bounded operator and use (ii) of Remark 2.14 in the proof of the first part of Theorem 3.1, to obtain the following result.
Let be a Banach space, and let such that . Let . Then there is a discrete almost automorphic solution of (3.2).
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.
By Corollary 2.6 we have since is bounded. Therefore, by Theorem 3.1, we get . We conclude that as a finite sum of discrete almost periodic functions.
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 .
Let be a Banach space that does not contain any subspace isomorphic to . Assume that is countable, and let . Then each bounded solution of (3.2) is discrete almost automorphic.
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.
Let . Then a solution of (3.2) is discrete almost automorphic if and only if it is bounded.
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.
Let and be a bounded solution of (3.10) on . Then is almost automorphic if and only if the sequence is almost automorphic.
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
where is a bounded linear operator defined on a Banach space and .
Our main result in this section is the following theorem for the scalar case.
in case , and
Since we obtain that the function is a contraction. Then there exists a unique function in such that . That is, satisfies and hence is solution of (4.1) (cf. the proof of (i) in Theorem 3.1).
and hence is a solution of (4.1) (cf. the proof of (ii) in Theorem 3.1).
In the particular case we obtain the following corollary.
Then for each , (4.1) have a unique discrete almost automorphic solution whenever or .
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.
Then (4.1) have a unique discrete almost automorphic solution whenever .
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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