Pseudo almost periodic solutions for a class of di erential equation with delays depending on state

Delay di erential equation of the form x′(t) = f (t, x(t), x(t − τ1(t)), . . . , x(t − τk(t))) has been discussed in [1, 2]. In particularly, the delay functions τj(t), j = 0, 1, . . . , k depend not only on unknown function, but also state, the delay functions τj(t, x(t)), j = 0, 1, . . . , k have been studied in many literatures. In [3], Cooke pointed out that it is highly desirable to establish the existence and stability properties of periodic solutions for equations of the form x′(t) + ax(t − h(t, x(t))) = F(t), in which the lag h(t, x(t)) implicitly involves x(t). Si andWang [4] discussed the smooth solutions of equation of x′(t) = λ1x(t) + λ2x(t) + . . . + λnx(t) + f (t), where x[1](t) = x(t), x[2](t) = x(x(t)), x[3](t) = x(x(x(t))), . . ., i.e., x[i](t) denotes ith iterate of x(t), i = 1, 2, . . . , n. Later, by Schröder transformation, Liu and Si [5] considered the analytic solutions of the form


Introduction
Delay di erential equation of the form x (t) = f (t, x(t), x(t − τ (t)), . . . , x(t − τ k (t))) has been discussed in [1,2]. In particularly, the delay functions τ j (t), j = , , . . . , k depend not only on unknown function, but also state, the delay functions τ j (t, x(t)), j = , , . . . , k have been studied in many literatures. In [3], Cooke pointed out that it is highly desirable to establish the existence and stability properties of periodic solutions for equations of the form in which the lag h(t, x(t)) implicitly involves x(t). Si and Wang [4] discussed the smooth solutions of equation of (t))), . . ., i.e., x [i] (t) denotes ith iterate of x(t), i = , , . . . , n. Later, by Schröder transformation, Liu and Si [5] considered the analytic solutions of the form where x [n] (t) denotes nth iterate of x(t), n = , , , . . . , k. Recently, In [6], Zhao and Fečkan studied the periodic solution of For some various properties of solutions for several iterative functional di erential equations, we refer the interested reader to [7]- [11].
On the other hand, the existence of pseudo almost periodic solutions is among the most attractive topics in qualitative theory of di erential equations due to their applications, especially in biology, economics and physics [12]- [16]. As pointed out by Ait Das and Ezzinbi [13], it is an interesting thing to study the pseudo almost periodic systems with delays. It is obviously that iterative functional di erential equations are special type state-dependent delay di erential equations. In [17], Liu pointed that the properties of the almost periodic functions do not always hold in the set of pseudo almost periodic functions and given an example: when x(t) is a pseudo almost periodic function, x(x(t)) may not be a pseudo almost periodic function. To the best of our knowledge, there are few results about pseudo almost periodic solutions for iterative functional di erential equations except [17], [18] and [19].
In the present work, we propose an existence result for pseudo almost periodic solutions of Eq (1.1) by using Tikhonov xed theorem. Uniqueness of the solution is achieved by Banach xed point theorem.
This paper is organized as follows. In Section 2 we give some notes and establish the main existence result. In Section 3, we show that (1.1) has a unique pseudo almost periodic solution under some suitable conditions. Furthermore, we prove the stability depend on C l,n and G. In Section 4, we present an example to illustrate the theory. Related problems are also studied in [20].

Existence result
In this section, the existence of pseudo almost periodic solutions of equation (1.1) will be studied. Throughout this paper, it will be assumed that For convenience, we will use C(R, R) to denote the set of all continuous functions from R to R endowed with the usual metric is the uniform convergence on each compact intervals of R. We also consider the set BC(R, R) of all bounded and continuous functions from R to R with the norm f = sup t∈R |f (t)|, so the topology on BC(R, R) is the uniform convergence on R. Denote by AP(R, R) the set of all almost periodic functions from A function φ ∈ BC(R, R) is called pseudo almost periodic if it can be expressed as φ = h + g, where h ∈ AP(R, R) and g ∈ PAP (R, R). The collection of such functions will be denoted by PAP(R, R). In particular, (PAP(R, R), · ) is a Banach space [21]. For M, L > , de ne From [17], it is easy to see that B where G ∈ PAP(R, R) and C , (t) < . It is easy to see that the linear equation admits an exponential dichotomy on R, by Theorem 2.3 in [15], we know that (2.2) has exactly one solution Integrating the both sides of (2.5) from −m to t, we have for any t ∈ [−m, m]. Then Gronwall's inequality implies

Uniqueness and stability
In this section, uniqueness and stability of (1.1) will be proved.  Then i.e.,  This completes the proof.

Example
In this section, an example is provided to illustrate that the assumptions of Theorem 2.3 do not self-contradict.