EXISTENCE OF ALMOST PERIODIC SOLUTIONS OF A NONLINEAR SYSTEM

This paper considers the existence of almost periodic solutions of a N -dimension non-autonomous Lotka-Volterra model with delays. The method is based on exponential dichotomy and Schauder fixed point theorem. The obtained results generalize some previously known ones.


Introduction
The existence of almost periodic solutions of ordinary differential equations has been discussed extensively in theory and in practice. Many useful methods have been developed for the study of almost periodic differential equations, we refer the reader to classical references such as Hale [5], Fink [3], Yoshizawa [10]. These methods are useful in studying the existence of almost periodic solutions of differential equations, and thus have many applications in specific systems arising form biology, neural networks, physics, chemistry and engineering. One of the powerful approaches are the combination between exponential dichotomy and fixed point theory. The use of different fixed point theorems will yield different sufficient (necessary) conditions for the existence of almost periodic solutions. For example, if Banach fixed point theorem is employed, we need r < 1 (where ∥T x − T y∥ ≤ r∥x − y∥). However, this restriction would not be necessary if one uses the Schauder fixed point theorem (e.g. see [2]). There are many fixed point theorems including Banach fixed point, Brower fixed point, generalized fixed point, Leray-Schauder fixed point, Horn fixed point, Schauder fixed point and so on. In this paper, we shall employ Schauder fixed point theorem to study a nonlinear system arising from the biological system -Lotka Volterra model.
In this paper, we consider the almost periodic Lotka-Voterra system as follows: ] . (1.1) For system (1.1), many results have been obtained in regard to permanence, extinction, bifurcation, chaos, asymptotic stability and periodic solutions. Some authors have employed the hull system theory and asymptotic stability theory to obtain the existence of almost periodic solutions of (1.1). However, by such methods, they have obtained relatively strict conditions on the parameters in order to guarantee the asymptotic stability. The reader may refer to [1,4,[6][7][8][9]. To obtain weaker sufficient conditions for the existence of almost periodic solutions of (1.1), we shall employ the exponential dichotomy theory and Schauder fixed point theorem to system (1.1). The proof and results are much different from the previous literature.
The structure of this paper is as follows. In Section 2, we shall introduce some definitions and lemmas. In Section 3, the main results are presented.

Some Definitions and Lemmas
In this section, we recall some definitions and lemmas which can be found in [3,5,10]. Suppose that f : R−→R is an almost periodic function. Define the mean value of f (t) by The notation mod(f ) is used to denote the modulus of the almost periodic function is also almost periodic and mod(g/f ) ⊂ mod(f, g). For the continuous and bounded function f (t), denote

Lemma 2.1. Suppose that {f n (t)} is an almost periodic function sequence and f (t) is an almost periodic function.
If mod(f n ) ⊂ mod(f ), n = 1, 2, · · · , and f n is locally uniformly convergent on R, then {f n } is uniformly convergent on R.

Lemma 2.3. (Schauder fixed point theorem) Suppose that B is a Banach Space, C is a closed convex subset of B.
If T : C → C is a continuous and compact operator, then T has a fixed point in C.
Lemma 2.4. Suppose that the linear systemẋ = A(t)x has an exponential dichotomy on R, i.e., there exist constants K, α > 0 and a fundamental matrix , which can be represented as

Results and Proofs
Throughout the paper, we always take i = 1, 2, · · · , N and j = 1, 2, · · · , N , unless otherwise stated. Throughout the paper, we assume the following: Making the change of variables , For any φ(t) ∈ S, consider the following auxiliary equatioṅ has an exponential dichotomy. From Lemma 2.4, system (3.2) has a unique almost periodic solution satisfying To prove that system (3.2) has a unique almost periodic solution, we shall apply Lemma 2.3 (Schauder fixed point theorem). To this end, we proceed in three steps.
Step 1: We shall show that for any φ ∈ S with |φ i (t)| < n 0 , there exists a sufficiently large n 0 such that system (3.3) has a unique almost periodic solution By way of contradiction, suppose the above does not hold, then there exists a sequence φ It follows from (3.3) and (3.4) that Let n → ∞, in view of (H 3 ), we get which is a contradiction. Hence, there exists n 0 (we can choose n 0 > m) such that for any φ(t) ∈ S, system (3.2) has a unique almost periodic solution u φ (t) with Step 2: To apply Lemma 2.3, we need to construct the sets C and B and the then C is a closed convex subset of B. Define a operator T : C → B as (3.5) Step 3: By employing Schauder fixed point theorem, we shall show that T has a fixed point in C.