Weighted Pseudo-almost Periodic Solutions of Neutral Integral and Di ff erential Equations

The research is financed by by the National Natural Science Foundation of China (11071001) and the Research Fund for the Doctoral Program of Higher Education of China(20093401110001) Abstract In this paper, we study weighted pseudo-almost periodic solutions to general neutral integral equations. By applying the properties of weighted pseudo-almost periodic functions and the fixed point theorem, we establish the conditions for the existence and uniqueness of weighted pseudo-almost periodic solutions of the equations.


Introduction
Qualitative analysis such as periodicity, almost periodicity and stability of neutral functional differential equations has been studied extensively by many authors, we can refer to [X.X. Chen, 2007; S. Abbas, 2008;X.X. Chen, 2011;et al] and references cited therein.Neutral functional differential equations are not only an extension of functional differential equation but also provide good models in many fields including Mechanics, Biology, Population Ecology, Neural network, and so on.The existence of almost periodic , pseudo-almost periodic ,weighted pseudo-almost periodic solutions is one of the most attractive topic in qualitative theory of differential equations due to their important applications ; see for instance [A.M. Fink, 1974;X.X Chen, 2010;C.Y. He, 1992;et al] and references therein.In [2008], Abbas and Bahugura have studied the almost periodic solution of a nonlinear neutral system of the form: In [M.Pinto, 2010], Mannel studied the existence and uniqueness of pseudo-almost periodic and almost periodic solutions to the equation: More specifically, that is the neutral delay integral equations of advanced and delayed decomposition type: In recent papers, in [2006], Diagana introduced the concept of weighted pseudo-almost periodic functions which is a new generalization of pseudo-almost periodic functions.Motivated by these papers, the purpose of this paper is to investigate the existence and uniqueness of the weighted pseudo-almost periodic solutionsor the following nonlinear neutral integral system:

Preliminaries
Let X, Y be complex Banach spaces endowed with the norm • X and • Y respectively and BC((R, Y), • ∞ ) be the Banach spaces of bounded continuous functions from R to Y endowed with the supremum norm φ ∞ = sup t∈R φ(t) Y .Define a function λ : R 2 → (0, +∞), BC λ (R 2 × X, Y)will denote the vectorial space of continuous functions f : R 2 × X → Y such that f /λ is bounded.Let U be the collection of all functions (weights) ρ : R → [0, +∞) which are locally integrable over R such that ρ(x) > 0 for almost each x ∈ R. For each r > 0 and ρ ∈ U , set About the definitions of almost periodic and pseudo-almost periodic functions, we refer to [C.Y.Zhang, 1994;Y.H. Wang, 2011;J.K. Hale, 1977;et al].
Let AP(R, X) be the set of all almost periodic functions from R to X and AP(R × Ω, X) be the set of all periodic functions in t ∈ R uniformly from R × ΩtoX, where Ω is a closed subset of X.Then (AP(R, X), • ∞ )and (AP(R × Ω, X), • ∞ )are all Banach spaces.We denote the set of all pseudo-almost functions by PAP(R, X).Similarly, we can also define the PAP(P × X, X).For a function λ : R 2 → (0, +∞) and any closed subset Ω ∈ X, a function We mean that, for each ε, there exists A ε > 0 such that for every rectangle R 1 × R 2 ⊂ R 2 of area A ε and a number τ ∈ R 1 ∩ R 2 with the following property: The number τ above will be called an ε-translation number with respect to λ of F and the class of such functions F will be denoted AP λ (R 2 × Ω, X).
Next, we need to introduce the weighted ergodic terms i.e.the spaces PAP 0 (R, X, ρ) and PAP The collection of such functions f will be denoted by WPAP(R × X, R, ρ).
The collection of such functions will be denoted by Lemma 2.1 ([T.Diagana, 2006]) Fix ρ ∈ U ∞ , the decomposition of a weighted pseudo-almost periodic function f is unique: Lemma 2.4 ([T.Diagana, 2006]) If f ∈ WPAP(R, X, ρ) satisfies the Lipschitz condition: Lemma 2.6 Assume that (i) The continuously differentiable functions h i : R → R satisfy the following assumptions: (ii) The weight ρ : R → (0, +∞) is continuous and is nondecreasing with . By Lemma 2.1 and consider the decomposition: On one hand, from condition (i), v(h(t)) ∈ AP(R, X) is obvious.On the other hand, for r > 0, we have which converges to zero when r → +∞.Thus w(h(t)) ∈ PAP 0 (R, X, ρ).The proof is complete.

Main results
In this section, we require the following assumptions: (H1) The function F : R × R 2n → R 2n is weighted pseudo-almost periodic satisfying for a constant L ∈ (0, 1) such that (H2) For i = 1, 2, there exists μ i = μ i (t, s) such that for t, s ∈ R, x i , y i ∈ R n , G i satisfies the Lipschitz condition: The conditions of the Lemma 2.6 about the functions h i (i = 0, 1, 2) and the weight function ρ hold.
Define the nonlinear operator: Lemma 3.1 If the conditions (H1)-(H5) hold, then the operator Φu is weighted pseudo-almost periodic for u weighted pseudo-almost periodic.
Remark 3: For u ∈ PAP(R) and ρ ≡ 1, the integral equation ( 1.3) has a unique pseudo-almost periodic solution, so the results of this paper extend some present works' results in [M.Pinto, 2010].Finally, we can consider a general neutral differential equation where dx dt = A(t)x admits exponential dichotomy and the function F(t, y(t), y(h 0 (t))) is differentiable.As we all know, if X(t) is the fundamental matrix of (C2) Q = Q(t, u, v) and F A (t, u, v) = A(t)F (t, u, v) are weighted pseudo-almost and for all t ∈ R, u i , v i ∈ R n , i = 1, 2 satisfy where L A > 0 constant.
Then the neutral differential equation (3.5) has a unique weighted pseudo-almost periodic solution.
Proof: From the solution's relation of between equation (3.5) and (3.6), we can easily have the result of the Theorem 3.3 at once from the Theorem 3.2.