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

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.


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 Šarkovskiȋ 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 ISRN Mathematical Analysis boundary value problems for this kind of equations and Aftabizadeh and Huang 5 studied the existence of unique bounded solution of x t f t, x t , x −t . 1.1 They proved that x t 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.

Preliminaries
Throughout the paper C b : {f : R → R n , f continuous and bounded} and for f ∈ C b , |f| sup{|f t | : t ∈ R}.

Exponential Dichotomy
In the sequel, A denotes a continuous mapping from R to M n R , where M n R is the space of square matrices with real coefficients.Definition 2.1.Let A t be a continuous square matrix on an interval J and let X t be a fundamental matrix of the following system: satisfying X 0 I, where I is the unit matrix.
The system of differential equations 2.1 is said to possess an exponential dichotomy on the interval J, if there exists a projection matrix P i.e., P 2 P and constants k > 1, α > 0, such that

2.2
We denote by P, k, α the triple of elements associated to an exponential dichotomy.

Remark 2.2. When A t
A is constant, the system 2.1 has an exponential dichotomy on an infinite interval, if and only if the eigenvalues of A have a nonzero real part.When A t 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

2.9
Denote by AA R, E is the set of all such functions.
If g is almost automorphic, then its range is relatively compact, thus bounded in norm.By the pointwise convergence, the function k is just measurable and not necessarily continuous.
If the convergence in both limits is uniform, then g 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.f t sin 1/ 2 − sin t − sin πt is an almost automorphic function, which is not almost periodic, because it is not uniformly continuous.Definition 2.8 see 23 .A continuous function f : R × E → E is said to be almost automorphic in t uniformly with respect to x in E, if the following two conditions hold: ii f is uniformly continuous on each compact subset K ⊂ X with respect to the second variable x, namely, for each compact subset K in E, and for all ε > 0, there exists δ > 0, such that for all x 1 , x 2 ∈ K, one has Denote by AA U R × E, E is the set of all such functions.
With these definitions, we have the following inclusions:

Pseudo Almost Automorphic Functions
Set ,g and ϕ are, respectively, called the almost automorphic component and the ergodic perturbation of f.Denote the set of all such functions by PAA R, E PAA R × E, E , resp. .

It is easy to verify
Definition 2.11.A closed subset K of R is said to be an ergodic zero set if meas K ∩ −t, t /2t → 0 as t → ∞, where meas is the Lebesgue measure on R.

2.14
We conclude by using Remark 2.12.

Main Results
In this part of this work, we are concerned with the following differential equation: where A and B are two square matrices, and f t, x, y is almost automorphic in t uniformly with respect to x and y in any compact subset of R 2n .
In the first time, we consider the following scalar and linear differential equation: where g is continuous on R. Let y t x −t , 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.

Scalar Case
In the scalar case, our main results can be stated as follows.
For the proof of Theorem 3.1, we use the following lemmas.
Proof of Theorem 3.1.Uniqueness.If there is two pseudo almost automorphic solutions x 1 t and x 2 t of 3.2 , then the difference x 1 t − x 2 t should be a solution of the homogeneous equation as x t ax t bx −t , b / 0, t ∈ R.

3.5
According to Lemma 2 of 2 , one can derive that for some constant C. If C / 0, x 1 t − x 2 t will be unbounded.This is a contradiction to the boundedness of pseudo almost automorphic function.So x 1 t x 2 t .Existence.From Lemmas 2 and 3 of 5 that we can derive to the following solution: is a particular solution of 3.2 for any g ∈ PAA R, R .Now we show that x t ∈ PAA R, R .Let us go back to the rest of the proof.Now we show then x t H t Φ t .Similar to the proof of Theorem 2.2 in 6 , we have Φ ∈ PAP 0 R, R .Now, we prove that H t is almost automorphic indeed, let s n ⊂ R be an arbitrary sequence.Since h ∈ AA R, R , then t → h −t is also almost automorphic, consequently t → h t , h −t is also almost automorphic, which leads to the fact that we can found a same subsequence s n of s n and two functions k, k 1 such that

ISRN Mathematical Analysis
We define exp α s − t α a k s bk 1 −s ds .

3.10
Now, consider the following:

3.11
Note that

3.12
Then by the Lebesgue dominated convergence theorem, lim n → ∞ H t s n K t , for all t ∈ R. In similar, way we can show that lim n → ∞ K t − s n H t for all t ∈ R, which ends the proof.

The Vectorial Case
Let us consider the following equation with reflection: Proof.If X t is a fundamental matrix of the system 3.17 ,

ISRN Mathematical Analysis
Furthermore, since 3.17 has an exponential dichotomy, then there exist a projection P and positive constants α, k such that

3.22
If we put Q P 0 0 I−P , then it is easy to see that Q is a projection, and that

ISRN Mathematical Analysis
Proof (The unique bounded solution, when we consider f bounded).A solution x t is represented as follows see 24 : 3.29 G t, s is a piecewise continuous function on the t, s plane.If f t g t ϕ t , where g is almost automorphic and ϕ is an ergodic perturbation, then

3.30
Moreover, it is known that ∞ −∞ G t, s ϕ s ds is an ergodic perturbation 13 .It remains to be prove that ∞ −∞ G t, s g s ds is almost automorphic.For this, we use the following result.Proposition 3.11 see 9 .Let A : R → M n R be continuous function and assume that the equation dx/dt A t x t has an exponential dichotomy on R, then for f ∈ AA R, R n , the unique bounded solution of dx/dt A t x t f t is almost automorphic.Corollary 3.12.If A and B are ω-periodic with the same period, such that the Floquet multipliers of A t lie of the unit circle and B verifies the condition (H 1 or (H 2 , then the system 3.27 has an exponential dichotomy.Moreover, if g is ω periodic, then 3.13 has an unique ω periodic solution.

3.33
On the other hand,

3.36
In the sequel, we suppose that

3.40
Remark 3.16.Z t is a bounded solution of 3.40 , by the uniqueness of bounded solution, we have that Z t X t in the sequel y t x −t , consequently, x is a bounded solution of 3.13 .Finally 3.38 has a unique bounded solution, and the application x t → x t x −t is a bijective from the set of bounded solutions of 3.13 to the set of bounded solutions of 3.40 .

Goal Result of Nonlinear Case
Consider the following equation involving reflection of the argument:

3.41
If we put y t x −t , then 3.41 is changed into system where X t x t y t , and M t is defined as in 3.16 and

3.43
We assume that there exists
Remark 3.17.If f satisfies 3.44 , then F satisfies For the proof, we need the following preliminary result.

3.48
Then • c is an equivalent norm to the uniform convergence norm.
Proof.In fact,

3.49
Proof of Theorem 3.18.PAA R × R 2n , R n is a Banach space with the supremum norm.If f t, x, y ∈ PAA R × R 2n , R n , then for any ϕ ∈ PAA R, R n , f t, ϕ t , ϕ −t is also pseudo almost automorphic.For ϕ ∈ PAA R, R n , the following differential equation: has a unique pseudo almost automorphic solution, denoted by Tϕ t , then we define a mapping as exp pcμ s 2 p p s ds exp pcμ s μ s ds

3.60
Then, K will be a contraction, which proves that K is continuous.So by the Banach fixed point theorem, there exists a unique u The proof is complet.
Proposition 3.20.Assume that f ∈ PAA R × R 2n , R n and satisfies the Lipschitz condition as

Scalar Case
Consider the following equation:

3.70
In this situation, x t −x t admits exponential dichotomy and the function t → 1/ 1 t 2 satisfies that lim t → ∞ 1/ 1 t 2 0, so the condition ii in Theorem 3.18 is satisfied, and the function t → sin 1/ 2 − sin t − sin πt 1/ √ 1 t 2 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.

Definition 3 . 13 . 31 Proposition 3 . 14 .Remark 3 . 15 . 2 − 1 / 2 I
For A ∈ M n R , the spetrum of A denoted by sp A {λ ∈ C, such that there exists x ∈ M n,1 C , x / 0 with Ax λx}.3.In the autonomous case, If sp A − B A B ∩ R − ∅, and g is pseudo almost automorphic, then 3.13 has an unique pseudo almost automorphic solution.If the matrices A and B are constant, the system 3.15 has an exponential dichotomy if and only if the eigenvalues of the matrix M have nonzero real part.One has B 2 and D AB − BA.Let P I n I n I n −I n .One has P −1
K 0 is a contraction mapping, and so K 0 has a unique fixed point in PAA R, R , which proves that 3.41 has a unique pseudo almost automorphic solution.

2.2. Almost Automorphic Functions
Equation 2.3 admits an exponential dichotomy with parameters P, k, α on J, if and only if 2.1 has an exponential dichotomy on −J with parameters I − P, k, α .On the other hand, x 13, 19-22 .Remark 2.3.Putting A t −A −t .Then equation y A t y 2.3 has as fundamental matrix Y t X −t .Let J be one of the following intervals R , R − .
Under the hypothesis (H 1 or (H 2 , if moreover g 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 3.10.If the system 2.6 has an exponential dichotomy and if f is pseudo almost automorphic, then the system dx/dt t A t x t f t has an unique pseudo almost automorphic solution.
Let us consider the following example of Markus and Yamabe: particular, the real parts of the eigenvalues are negative.If ρ j exp λ j ω , j 1, 2 . . ., n are the characteristic multipliers of A t , where ω is the minimal period and λ j the eigenvalues of A t , then, we have One of the characteristic multipliers is exp π/2 .The other multiplier is exp −π , since the product of the multipliers is exp −π/2 .So, the system dx/dt A t x t has an exponential dichotomy.The matrix B t is an ergodic function.G is pseudo almost automorphic, hence, 3.71 has an unique pseudo almost automorphic solution.
The matrix A t is π-periodic and the eigenvalues λ 1 t , λ 2 t of A t are