ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 626490, 20 pagesdoi:10.5402/2012/626490

Research Article

Pseudo Almost Automorphic Solutions for Differential Equations Involving Reflection of the Argument

Elhadi Ait Dads,1,2 Samir Fatajou,1,2 and Lahcen Khachimi1,2

¹Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, BP 2390, Marrakech, Morocco²UMMISCO UMI 209, UPMC, IRD Bondy France, Unité Associée au CNRST URAC 02, Morocco

Received 11 June 2012; Accepted 11 July 2012

Academic Editors: G. Ólafsson and A. Rhandi

Copyright © 2012 Elhadi Ait Dads 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.

Abstract.

By means of the fixed point methods and the properties of the pseudo almost automorphic functions, the existence and uniqueness of pseudo almost automorphic solutions are obtained for differential equations involving reflection of the argument. For the nonscalar, case we use the exponential dichotomy properties.

1. Introduction

The existence, uniqueness, and stability of periodic, almost periodic, almost automorphic, asymptotically almost automorphic, and pseudo almost automorphic solutions has been one of the most attractive topics in the qualitative theory of ordinary or functional differential equations for its significance in the physical sciences, mathematical biology, control theory, and others.

The differential equations involving reflection of argument have many applications in the study of stability of differential-difference equations, see Šarkovskiĭ [1], and such equations show very interesting properties by themselves, so many authors have worked on this category of equations. Wiener and Aftabizadeh [2] initiated to study boundary value problems involving reflection of the argument. Gupta in [3, 4] investigated two point boundary value problems for this kind of equations and Aftabizadeh and Huang [5] studied the existence of unique bounded solution of They proved that is almost periodic by assuming the existence of bounded solution. In [6], Piao considers the case of pseudo almost periodic solution. This work is motivated by the last reference and devoted to investigate the existence and uniqueness of pseudo almost automorphic solution in the scalar and vectorial case.

The concept of almost automorphic functions, which was introduced by Bochner as an extension of one of the almost periodic functions, has recently caught the attention of many mathematicians (see, e.g., [7–11]. In [12], Zhang has introduced an extension of the almost periodic functions, the so-called pseudo almost periodic functions. For more details on this notion, we can refer to [12–18]. Then the combination between pseudo almost periodic and almost automorphic leads to the pseudo almost automorphic functions, which is considered in this work.

The theory of exponential dichotomy has played a central role in the study of ordinary differential equations and diffeomorphisms for finite dimensional dynamic systems. This theory, which addresses the issue of strong transversality in dynamic systems, originated in the pioneering works of Lyapunov (1892) and Poincaré (1890). During the last few years, one finds an ever growing use of exponential dichotomies to study the dynamic structures of various partial differential delay equations, for more details, we refer to [19, 20].

This paper is organized as follows. In Section 2, we recall some preliminary results which is divided in two sections, in the first one we give some results on the exponential dichotomy theory, and in the second one, we give some definitions of pseudo almost automorphic functions. The main results are announced and discussed in Section 3. In the last section, we give some illustrated examples.

2. Preliminaries

Throughout the paper and for .

2.1. Exponential Dichotomy

In the sequel, denotes a continuous mapping from to , where is the space of square matrices with real coefficients.

Definition 2.1. Let be a continuous square matrix on an interval and let be a fundamental matrix of the following system: satisfying , where is the unit matrix.The system of differential equations (2.1) is said to possess an exponential dichotomy on the interval , if there exists a projection matrix (i.e., ) and constants , such that We denote by the triple of elements associated to an exponential dichotomy.

Remark 2.2. When is constant, the system (2.1) has an exponential dichotomy on an infinite interval, if and only if the eigenvalues of have a nonzero real part. When is periodic, (2.1) has an exponential dichotomy on an infinite interval, if and only if the Floquet multipliers lie off the unit circle. For the properties of exponential dichotomies, one may refer to [13, 19–22].

Remark 2.3. Putting . Then equation has as fundamental matrix . Let be one of the following intervals . Equation (2.3) admits an exponential dichotomy with parameters on , if and only if (2.1) has an exponential dichotomy on with parameters . On the other hand, is a solution of if and only if is a solution of where .

Theorem 2.4 (see [21]). Assume that the following differential equation: has an exponential dichotomy on  resp. with parameters (.Let be a bounded continuous function such that ,  resp.. Then the perturbed equation has an exponential dichotomy on resp. with parameters (, where is a projection with the same kernel range, resp. as the one of . Moreover, if is the fundamental matrix of (2.7) satisfying , then

Lemma 2.5 (see [21]). Let be real constants, (,   resp.). If (2.6) has an exponential dichotomy on respectively , then it has one on respectively , with the same exponents and the same projection .

2.2. Almost Automorphic Functions

Definition 2.6. A continuous function is said to be almost automorphic if for every sequence of real numbers , there exists a subsequence of , denoted such that for each Denote by is the set of all such functions.

If is almost automorphic, then its range is relatively compact, thus bounded in norm.

By the pointwise convergence, the function is just measurable and not necessarily continuous.

If the convergence in both limits is uniform, then is almost periodic. The concept of almost automorphy is then larger than the one of almost periodicity. It was introduced in the literature by Bochner and recently studied by several authors. A complete description of their properties and further applications to evolution equations can be found in the monographs by N’Guérékata [10, 11].

Example 2.7. is an almost automorphic function, which is not almost periodic, because it is not uniformly continuous.

Definition 2.8 (see [23]). A continuous function is said to be almost automorphic in uniformly with respect to in , if the following two conditions hold:(i)for all ;(ii) is uniformly continuous on each compact subset with respect to the second variable , namely, for each compact subset in , and for all , there exists , such that for all , one has

Denote by is the set of all such functions.

With these definitions, we have the following inclusions:

Theorem 2.9 (see [23]). Let and . Then .

2.3. Pseudo Almost Automorphic Functions

Set

Definition 2.10. is called pseudo almost-automorphic, if with and , and are, respectively, called the almost automorphic component and the ergodic perturbation of . Denote the set of all such functions by .

It is easy to verify that is a translation invariant closed subspace of containing the constant functions. Furthermore,

Definition 2.11. A closed subset of is said to be an ergodic zero set if as , where is the Lebesgue measure on .

Remark 2.12. One sees that is in if and only if for , the set is an ergodic zero set in .

Example 2.13. (a) is a pseudo almost automorphic function.(b) A continuous function satisfying is an ergodic function. Indeed by hypothesis, for all , there exists such that , then for all , We conclude by using Remark 2.12.

3. Main Results

In this part of this work, we are concerned with the following differential equation: where and are two square matrices, and is almost automorphic in uniformly with respect to and in any compact subset of .

In the first time, we consider the following scalar and linear differential equation: where is continuous on . Let , then (3.2) is changed into the following system: which is in a formally Hamilton system with Hamiltonian function as So, one may say that some first order scalar differential equations can also generate Hamilton systems.

3.1. Scalar Case

In the scalar case, our main results can be stated as follows.

Theorem 3.1. For any , with , (3.2) has an unique pseudo almost automorphic solution .

For the proof of Theorem 3.1, we use the following lemmas.

Lemma 3.2 (see [10]). If , then .

Lemma 3.3 (see [15]). If , then .

Lemma 3.4 (see [8]). If , and , then .

Proof of Theorem 3.1. Uniqueness. If there is two pseudo almost automorphic solutions and of (3.2), then the difference should be a solution of the homogeneous equation as According to Lemma 2 of [2], one can derive that for some constant . If ,   will be unbounded. This is a contradiction to the boundedness of pseudo almost automorphic function. So .Existence. From Lemmas and of [5] that we can derive to the following solution: is a particular solution of (3.2) for any . Now we show that .Let us go back to the rest of the proof. Now we show . Assume that ,  ,  . Let then . Similar to the proof of Theorem  2.2 in [6], we have . Now, we prove that is almost automorphic indeed, let be an arbitrary sequence. Since , then is also almost automorphic, consequently is also almost automorphic, which leads to the fact that we can found a same subsequence of and two functions such that We define Now, consider the following: Note that Then by the Lebesgue dominated convergence theorem, , for all . In similar, way we can show that for all , which ends the proof.

3.2. The Vectorial Case

Let us consider the following equation with reflection: where ,    , and bounded continuous functions.

Putting , one has

If we put , then is a solution of the following system: where

Theorem 3.5. If the system has a fundamental matrix and has an exponential dichotomy with parameters , then the following system: where , has a fundamental matrix and admits an exponential dichotomy with parameters , where

Proof. If is a fundamental matrix of the system (3.17), Consequently, Furthermore, since (3.17) has an exponential dichotomy, then there exist a projection and positive constants such that If we put , then it is easy to see that is a projection, and that which ends the proof.

Lemma 3.6. If the system has an exponential dichotomy with parameters and if is continuous and uniformly bounded in such that , then the system has an exponential dichotomy.

Proof. The proof is a direct application of Theorems 2.4 and 3.5.

Corollary 3.7. If the system has an exponential dichotomy with parameters ,  and if is continuous and uniformly bounded in , such that ( then the system has an exponential dichotomy too.

Proof. The proof is a direct consequence of Corollary in Ait Dads and Arino [13] and Theorem 3.5.

Theorem 3.8. Under the hypothesis ( or (, if moreover is pseudo almost automorphic, then (3.13) has a unique pseudo almost automorphic solution.

Proof. The proof is a direct application of the two following results.

Lemma 3.9 (see [7]). If is pseudo almost automorphic, then is also pseudo almost automorphic.

Lemma 3.10. If the system (2.6) has an exponential dichotomy and if is pseudo almost automorphic, then the system has an unique pseudo almost automorphic solution.

Proof (The unique bounded solution, when we consider bounded). A solution is represented as follows (see [24]): where is a piecewise continuous function on the plane. If , where is almost automorphic and is an ergodic perturbation, then Moreover, it is known that is an ergodic perturbation [13]. It remains to be prove that is almost automorphic. For this, we use the following result.

Proposition 3.11 (see [9]). Let be continuous function and assume that the equation has an exponential dichotomy on , then for , the unique bounded solution of is almost automorphic.

Corollary 3.12. If and are -periodic with the same period, such that the Floquet multipliers of lie of the unit circle and verifies the condition ( or (, then the system (3.27) has an exponential dichotomy. Moreover, if is periodic, then (3.13) has an unique periodic solution.

3.3. Autonomous Case

Definition 3.13. For , the spetrum of denoted by

Proposition 3.14. In the autonomous case, If , and is pseudo almost automorphic, then (3.13) has an unique pseudo almost automorphic solution.

Remark 3.15. If the matrices and are constant, the system (3.15) has an exponential dichotomy if and only if the eigenvalues of the matrix have nonzero real part. One has with and . Let . One has and On the other hand, and , so, we have then In the sequel, we suppose that Then, (3.40) has an unique bounded solution denoted by .One has Putting one has

Remark 3.16. is a bounded solution of (3.40), by the uniqueness of bounded solution, we have that in the sequel , consequently, is a bounded solution of (3.13). Finally (3.38) has a unique bounded solution, and the application is a bijective from the set of bounded solutions of (3.13) to the set of bounded solutions of (3.40).

3.4. Goal Result of Nonlinear Case

Consider the following equation involving reflection of the argument: If we put , then (3.41) is changed into system where , and is defined as in (3.16) and

We assume that there exists , with such that

In what follows, let us put .

Remark 3.17. If satisfies (3.44), then satisfies

Theorem 3.18. Assume that and satisfies the Lipschitz condition (3.44), if the system has an exponential dichotomy with parameters and if is continuous and uniformly bounded in , such that or holds. If then (3.41) has an unique pseudo almost automorphic solution.

For the proof, we need the following preliminary result.

Lemma 3.19. Let , and put Then is an equivalent norm to the uniform convergence norm.

Proof. In fact,

Proof of Theorem 3.18. is a Banach space with the supremum norm. If , then for any is also pseudo almost automorphic. For , the following differential equation: has a unique pseudo almost automorphic solution, denoted by , then we define a mapping as Thus, , so that is well defined.Now for ,, we have Let us put then one has On the other hand, which leads to Hence, therefore, If we choose such that then, Then, will be a contraction, which proves that is continuous. So by the Banach fixed point theorem, there exists a unique , such that , that is, The proof is complet.

Proposition 3.20. Assume that and satisfies the Lipschitz condition as If the system has an exponential dichotomy with parameters and if is continuous and uniformly bounded in , such that or holds, then (3.41) has an unique pseudo almost automorphic solution provided that .

Corollary 3.21. In the scalar case, if , and and satisfies the lipschitz condition as for any , where , and , then (3.41) has an unique pseudo almost automorphic solution.

Proof. Thanks to Lemmas 3.2, 3.3, and 3.6, and from Theorem 3.1, for any , we see that the following equation: has a unique pseudo almost automorphic solution, which we denote by . Then, if we consider the operator . Now we can show that is a contraction. Indeed, for and , the following equation: has a unique pseudo almost automorphic solution . Moreover, So Since so is a contraction mapping, and so has a unique fixed point in , which proves that (3.41) has a unique pseudo almost automorphic solution.

3.5. Examples

3.5.1. Scalar Case

Consider the following equation: In this situation, admits exponential dichotomy and the function satisfies that so the condition (ii) in Theorem 3.18 is satisfied, and the function is a pseudo almost automorphic function, so all the hypotheses of Theorem 3.18 hold, and so (3.70) has an unique pseudo almost automorphic solution.

3.6. Vectorial Case

Let us consider the following example of Markus and Yamabe: where The matrix is -periodic and the eigenvalues , of are and, in particular, the real parts of the eigenvalues are negative. If ,   are the characteristic multipliers of , where is the minimal period and the eigenvalues of , then, we have One of the characteristic multipliers is . The other multiplier is , since the product of the multipliers is . So, the system has an exponential dichotomy. The matrix is an ergodic function. is pseudo almost automorphic, hence, (3.71) has an unique pseudo almost automorphic solution.

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