Open Access

Almost Automorphic Solutions of Difference Equations

Advances in Difference Equations20092009:591380

DOI: 10.1155/2009/591380

Received: 25 March 2009

Accepted: 13 May 2009

Published: 22 June 2009

Abstract

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 .

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 [1016] 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 [2225]. A unified and homogeneous exposition of the theory and its applications was first given by N'Guérékata in his book [21]. 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 [2633], in the books [20, 21, 32] (concerning almost automorphic functions with values in Banach spaces), and in [34] (concerning almost automorphy on groups). Also, the theory of almost automorphic functions with values in fuzzy-number-type spaces was developed in [35] (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 [20].

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

Let be a real or complex Banach space. We recall that a function is said to be discrete almost periodic if for any positive there exists a positive integer such that any set consisting of consecutive integers contains at least one integer with the property that
(2.1)

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 proof can be found in [38, Theorem  1.26, pages 45-46]. Observe that functions with the property (N) are also called normal in literature (cf. [7, page 72] or [38]).

The above characterization, as well as the definition of continuous almost automorphic functions (cf. [21]), motivates the following definition.

Definition 2.1.

Let be a (real or complex) Banach space. A function is said to be discrete almost automorphic if for every integer sequence , there exists a subsequence such that
(2.2)
is well defined for each and
(2.3)

for each

Remark 2.2.
  1. (i)

    If is a continuous almost automorphic function in then is discrete almost automorphic.

     
  2. (ii)

    If the convergence in Definition 2.1 is uniform on then we get discrete almost periodicity.

     
We denote by the set of discrete almost automorphic functions. Such as the continuous case we have that discrete almost automorphicity is a more general concept than discrete almost periodicity; that is,
(2.4)

Remark 2.3.

Examples of discrete almost automorphic functions which are not discrete almost periodic were first constructed by Veech [39]. In fact, note that the examples introduced in [39] are not on the additive group but on its discrete subgroup A concrete example, provided later in [25, Theorem  1] by Bochner, is
(2.5)

where is any nonrational real number.

Discrete almost automorphic functions have the following fundamental properties.

Theorem 2.4.

Let be discrete almost automorphic functions; then, the following assertions are valid:
  1. (i)

    is discrete almost automorphic;

     
  2. (ii)

    is discrete almost automorphic for every scalar ;

     
  3. (iii)

    for each fixed in the function defined by is discrete almost automorphic;

     
  4. (iv)

    the function defined by is discrete almost automorphic;

     
  5. (v)

    ; that is, is a bounded function;

     
  6. (vi)
    , where
    (2.6)
     

Proof.

The proof of all statements follows the same lines as in the continuous case (see [21, Theorem  2.1.3]), and therefore is omitted.

As a consequence of the above theorem, the space of discrete almost automorphic functions provided with the norm
(2.7)

becomes a Banach space. The proof is straightforward and therefore omitted.

Theorem 2.5.

Let be Banach spaces, and let a discrete almost automorphic function. If is a continuous function, then the composite function is discrete almost automorphic.

Proof.

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

Corollary 2.6.

If is a bounded linear operator on and is a discrete almost automorphic function, then , is also discrete almost automorphic.

Theorem 2.7.

Let and be discrete almost automorphic. Then defined by , is also discrete almost automorphic.

Proof.

Let be a sequence in . There exists a subsequence of such that is well defined for each and for each Also we have that is well defined for each and for each . The proof now follows from Theorem 2.4, and the identities
(2.8)

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.

Definition 2.8.

A function is said to be discrete almost automorphic in for each if for every sequence of integers numbers there exists a subsequence such that
(2.9)
is well defined for each , , and
(2.10)

for each and .

The proof of the following result is omitted (see [21, Section  2.2]).

Theorem 2.9.

If are discrete almost automorphic functions in for each in then the followings are true.
  1. (i)

    is discrete almost automorphic in for each in

     
  2. (ii)

    is discrete almost automorphic in for each in where is an arbitrary scalar.

     
  3. (iii)

    for each in .

     
  4. (iv)

    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.

Theorem 2.10.

Let be discrete almost automorphic in for each in X, and satisfy a Lipschitz condition in uniformly in ; that is,
(2.11)

Suppose is discrete almost automorphic, then the function defined by is discrete almost automorphic.

Proof.

Let be a sequence in . There exists a subsequence of such that for all , and for each , . Also we have is well defined for each and for each . Since the function is Lipschitz, using the identities
(2.12)

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 , .

Theorem 2.11.

Let be a discrete almost automorphic function, then is also discrete almost automorphic.

Proof.

Since then by (i) and (iii) in Theorem 2.4, we have that is discrete almost automorphic.

More important is the following converse result, due to Basit [40, Theorem  1] (see also [17, Lemma  2.8]). Recall that is defined as the space of all sequences converging to zero.

Theorem 2.12.

Let be a Banach space that does not contain any subspace isomorphic to Let and assume that
(2.13)

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.

Theorem 2.13.

Let be a summable function, that is,
(2.14)
Then for any discrete almost automorphic function the function defined by
(2.15)

is also discrete almost automorphic.

Proof.

Let be a arbitrary sequence of integers numbers. Since is discrete almost automorphic there exists a subsequence of such that
(2.16)
is well defined for each and
(2.17)
for each . Note that
(2.18)
then, by Lebesgue's dominated convergence theorem, we obtain
(2.19)
In similar way, we prove
(2.20)

and then is discrete almost automorphic.

Remark 2.14.
  1. (i)
    The same conclusions of the previous results holds in case of the finite convolution
    (2.21)
     
and the convolution
(2.22)
  1. (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].

We are interested in finding discrete almost automorphic solutions of the following system of first-order linear difference equations, written in vector form
(3.1)
where is a matrix or, more generally, a bounded linear operator defined on a Banach space and is in . Note that (3.1) is equivalent to
(3.2)

where We begin studying the scalar case. We denote .

Theorem 3.1.

Let be a Banach space. If and is discrete almost automorphic, then there is a discrete almost automorphic solution of (3.2) given by
  1. (i)

    in case

     
  2. (ii)

    in case

     
Proof.
  1. (i)
    Define . Then and hence, by Theorem 2.13, we obtain Next, we note that is solution of (3.2) because
    (3.3)
     
  2. (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 .

Theorem 3.2.

Suppose is a constant matrix with eigenvalues . Then for any function there is a discrete almost automorphic solution of (3.2).

Proof.

It is well known that there exists a nonsingular matrix such that is an upper triangular matrix. In (3.2) we use now the substitution to obtain
(3.5)
Obviously, the system (3.5) is of the form as (3.2) with a discrete almost automorphic function. The general case of an arbitrary matrix can now be reduced to the scalar case. Indeed, the last equation of the system (3.5) is of the form
(3.6)

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.

Remark 3.3.

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.

Corollary 3.4.

Assume that is a constant matrix with eigenvalues , and suppose that is such that
(3.7)
for all large , where and . Then there is a discrete almost automorphic solution of (3.2), which satisfies
(3.8)

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.

Theorem 3.5.

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.

Theorem 3.6.

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.

Proof.

Let be fixed. Applying the projection to (3.2) we obtain
(3.9)

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 .

Theorem 3.7.

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.

Theorem 3.8.

Let . Then a solution of (3.2) is discrete almost automorphic if and only if it is bounded.

Interesting examples of application of Theorem 3.7 are given in [19, Theorems  3.4 and  3.7], concerning the existence of almost automorphic solutions of differential equations with piecewise constant arguments of the form
(3.10)

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.

Theorem 3.9.

Let and be a bounded solution of (3.10) on . Then is almost automorphic if and only if the sequence is almost automorphic.

For a proof, see [19, Lemma  3.3]. A corresponding result for compact almost automorphic functions is also true (see [19, Lemma  3.6]).

We finish this section with the following simple example concerning the heat equation (cf. [6, page 157]).

Example 3.10.

Consider the distribution of heat through a thin bar composed by a homogeneous material. Let be equidistant points on the bar. Let be the temperature at time at the point , Under certain conditions one may derive the equation
(3.11)
where the vector consists of the components , , and is a tridiagonal Toeplitz matrix. Its eigenvalues may be found by the formula
(3.12)
where is a constant of proportionality concerning the difference of temperature between the point and the nearby points and (see [6]). Assuming
(3.13)

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

We want to find conditions under which it is possible to find discrete almost automorphic solutions to the equation
(4.1)

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.

Theorem 4.1.

Let and be discrete almost automorphic in for each . Suppose that satisfies the following Lipschitz type condition
(4.2)
Then (4.1) have a unique discrete almost automorphic solution satisfying
  1. (i)

    in case , and

     
  2. (ii)

    in case

     

Proof.

Case : we define the operator by
(4.3)
Since and satisfies (4.2), we obtain by Theorem 2.10 that is in . So is well-defined thanks to Theorem 2.13. Now, given we have
(4.4)

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).

Case : we define by
(4.5)
and with similar arguments as in the previous case we obtain that is well-defined. Now, given we have
(4.6)
Therefore is a contraction, and then there exists a unique function such that . The function satisfies
(4.7)

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.

Corollary 4.2.

Let . Suppose that satisfies a Lipschitz condition
(4.8)

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.

Theorem 4.3.

Let and suppose that is such that
(4.9)

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].

Declarations

Acknowledgment

Carlos Lizama is partially supported by Laboratorio de Análisis Estocástico, Proyecto Anillo PBCT-ACT-13.

Authors’ Affiliations

(1)
Departamento de Matemática, Universidad de Santiago

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© Daniela Araya et al. 2009

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