Existence and uniqueness of almost periodic solutions for a class of nonlinear Duffing system with time-varying delays*

In this paper, we investigate the existence and uniqueness of almost periodic solutions for a class of nonlinear Duffing system with time-varying delays. By using theory of exponential dichotomies and contraction mapping principle, we establish some new results and give an example to illustrate the theoretical analysis in this work.


Introduction
In recent years, the dynamic behaviors of nonlinear Duffing equations have been widely investigated in [1][2][3][4] due to the application in many fields such as physics, mechanics, engineering, other scientific fields. In such applications, it is important to know the existence of the almost periodic solutions for nonlinear Duffing equations. Some results on existence of the almost periodic solutions were obtained in the literature. We refer the reader to [5−8] and the references cited therein.
Recently, L. Q. Peng and W. T. Wang [9] considered the following model for nonlinear Duffing equation with a deviating argument where τ (t) and p(t) are almost periodic functions on R, m > 1, a, b and c are constants. Let Q 1 (t) be a continuous and differentiable function on R. Define (1.2) where ξ > 1 is a constant, then L. Q. Peng and W. T. Wang [9] transformed (1.1) into the following system can be naturally extended to the following Duffing system with time-varying coefficients and where α(t), β(t), τ (t), δ 1 (t), δ 2 (t), Q 1 (t) and Q 2 (t) are almost periodic functions on R, m > 1 is an integer, α(t) > 0, β(t) = 0. However, to the best of our knowledge, few authors have considered the problem for almost periodic solutions of system (1.4). Motivated by the above arguments, the main purpose of this present paper is to give the conditions to guarantee the existence of almost periodic solutions of system (1.4).
For convenience, we introduce some notations. Throughout this paper adopt the following notations:X = (x 1 , x 2 ) T ∈ R 2 to denote a column vector, in which the symbol ( T ) denote the transpose of a vector. We let |X| denote the absolute-value vector given by |X| = (|x 1 |, |x 2 |) T , and define ||X|| = max 1≤i≤2 |x i |. A vector X ≥ 0 means that all x i are greater than or equal to zero. X > 0 is defined similarly. For vectors X and Y , X ≥ Y (resp. X > Y ) means that X − Y ≥ 0(resp. X − Y > 0 ). Let δ 1 , δ 2 , δ * , l, θ and q be defined as where ϕ is an almost periodic function on R. For ∀ϕ ∈ B, we define the induced modulus Definition 1. [10,11] Let u(t) : R → R n be continuous in t. u(t) is said to be almost relatively dense, i.e., for ε > 0, it is possible to find a real number L = L(ε) > 0 where, for any interval with length L(ε) , there exists a number λ = λ(ε) in this interval such that Definition 2. [10,11] Let z ∈ R n and Q(t) be an n × n continuous matrix defined on R.
The linear system is said to admit an exponential dichotomy on R if there exist positive constants k, α, projection P and the fundamental solution matrix X(t) of (1.8) satisfying Lemma 1.1. [10,11] Let Q(t) = (q ij ) be an n × n almost periodic matrix defined on R and let there exist a positive constant ν such that Then the linear system (1.8) admits an exponential dichotomy on R.
Lemma 1.2. [10,11] If the linear system (1.8) admits an exponential dichotomy, then the has a unique almost periodic solution z(t) and Theorem 2.1. Assume δ i (t) > 0, i = 1, 2, and let positive constants l, θ and q satisfy Then there exists a unique almost periodic solution of system (1.4) in the region Proof. For any ϕ ∈ B, we consider the almost periodic solution of the nonlinear almost periodic two-dimensional system with time-varying delays Then, notice that the linear system admits an exponential dichotomy on R. Define a projection P by setting Thus, by Lemma 1.2, we obtain that the system (2.2) has exactly one almost periodic solution: Define a mapping T : B → B by setting Since B * = {ϕ|ϕ ∈ B, ||ϕ − ϕ 0 || ≤ θl 1−θ }, it is easy to see that B * is a closed convex subset of B . According to the definition of the norm in Banach space B , we derive Therefore, for any ϕ ∈ B * , we have Now, we prove that the mapping T is a self-mapping from B * to B * . In fact, for any ϕ ∈ B * , from (2.4), we obtain which implies that T ϕ ∈ B * . So, the mapping T is a self-mapping from B * to B * .
Next, we prove that the mapping T is a contraction mapping of the B * . In deed, in view of (1.5), (1.6), (1.7), (2.1), (2.4) and differential mean-value theorem, for all ϕ, ψ ∈ B * , we where h(s) ∈ (0, 1) is the mean point in Lagrange's mean value theorem. Then, which yields that It follow from q < 1 that the mapping T is a contraction. Therefore, the mapping T possesses a unique fixed point z * = (x * (t), y * (t)) T ∈ B * ,T z * = z * . By (2.2), z * satisfies (1.4). So z * is an almost periodic solution of system (1.4) in B * . The proof of Theorem 2.1 is now completed.
If δ i (t) < 0, i = 1, 2, we define a projection P by setting then using similar arguments to those above, we can show the following EJQTDE, 2012 No. 80, p. 6 Theorem 2.2. If δ i (t) < 0, i = 1, 2 , and assume that (2.1) holds. Then, there exists a unique almost periodic solution of system (1.4) in the region
The proof of example 3.1 is now completed. it is clear that the results obtained in [9] are invalid for the above example. Moreover, one can find that main theorem in [8] is a special one of Theorem 2.1 when δ 1 (t) and δ 2 (t) are constants. Thus, the results of this paper substantially extend and improve the main results of refs. [8][9].