Positive Almost Periodic Solutions for a Class of Nonlinear Duffing Equations with a Deviating Argument

In this paper, we study a class of nonlinear Duffing equations with a deviating argument and establish some sufficient conditions for the existence of positive almost periodic solutions of the equation. These conditions are new and complement to previously known results.


Introduction
Consider 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.
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 [5−7] and the references cited therein.Suppose that the following condition holds: (H 0 ) a = b = 1, c = 0, τ : R → R is a constant function, m > 1 is an integer, and ). (1. 2) The authors of [6] and [7] obtained some sufficient conditions ensuring the existence of almost periodic solutions for Eq.(1.1).However, to the best of our knowledge, few authors have considered the problem of almost periodic solutions for Eq.(1.1) without the assumption (H 0 ).Thus, it is worthwhile to continue to investigate the existence of almost periodic solutions Eq. (1.1) in this case.
A primary purpose of this paper is to study the problem of positive almost periodic solutions of (1.1).Without assuming (H 0 ), we derive some sufficient conditions ensuring the existence of positive almost periodic solutions for Eq.(1.1), which are new and complement to previously known results.Moreover, an example is also provided to illustrate the effectiveness of our results.
Let Q 1 (t) be a continuous and differentiable function on R. Define where ξ > 1 is a constant, then we can transform (1.1) into the following system (1.4) is relatively dense, i.e., for any ε > 0, it is possible to find a real number l = l(ε) > 0, for any interval with length l(ε), there exists a number δ = δ(ε) in this interval such that Throughout this paper, it will be assumed that τ, Q 1 , Q 2 : R → [0, +∞) are almost periodic functions.From the theory of almost periodic functions in [8,9], it follows that for any ǫ > 0, it is possible to find a real number l = l(ǫ) > 0, for any interval with length l(ǫ), there exists a number δ = δ(ǫ) in this interval such that for all t ∈ R. We suppose that there exist constants L, L + and τ such that ) and a number y 0 , then there exists a solution of (1.4) on an interval [0, T ) satisfying the initial condition and satisfying (1.4) on [0, T ).If the solution remains bounded, then T = +∞.We denote such a solution by (x(t), y(t)) = (x(t, ϕ, y 0 ), y(t, ϕ, y 0 )).Let y(s) = y(0) for all s ∈ (−∞, 0] and x(s) = x(−τ ) for all s ∈ (−∞, −τ ].Then (x(t), y(t)) can be defined on R.
We also assume that the following conditions hold.

Preliminary Results
The following lemmas will be useful to prove our main results in Section 3.
If Case 2 holds, calculating the upper right derivative of | y(t)|, together with (C 1 ), (1.4), (1.6) and (2.4) imply that which is a contradiction and implies that (2.2) holds.The proof of Lemma 2.1 is now completed.
Lemma 2.2.Suppose that (C 1 ) and (C 2 ) hold.Moreover, we choose a sufficiently large constant θ > 0 such that for all t > 0, ζ = L + ηθ < L + η , and ) is a solution of system (1.4) with initial conditions which is a contradiction and implies that (2.7) holds.
If Case II holds, together with (C 2 ) , (1.4), (2.5) and (2.9) imply that which is a contradiction and implies that (2.7) holds.The proof of Lemma 2.2 is now completed.
The proof of Lemma 2.3 is now completed.

Main Results
EJQTDE, 2010 No. 6, p. 8 In this section, we establish some results for the existence of the positive almost periodic solution of system (1.4).

Proof.
Let (x(t), y(t)) be a solution of system (1.4) with initial conditions (2.10).Set Now, we prove that Z * (t) is a positive solution of (1.4).In fact, for any t > 0 and ∆t ∈ R, we have Therefore, Z * (t) is a positive solution of (1.4).
Secondly, we prove that Z * (t) is a positive almost periodic solution of (1.4).From Lemma 2.3, for any ǫ > 0, there exists l = l(ǫ) > 0, such that every interval [α, α + l] contains at least one number δ for which there exists N > 0 satisfies Then, for any fixed s ∈ R, we can find a sufficient large positive integer N 0 > N such that for any k > N 0  Remark 4.1.Since τ (t) = sin 2 t, p(t) = 12(1 + 0.9 sin t) + 1.8 cos t + 1 + 0.01 sin √ 2t, it is clear that the condition (H 0 ) is not satisfied.Therefore, all the results in [1][2][3][4][5][6][7] and the references therein can not be applicable to prove that the existence of positive almost periodic solutions for nonlinear Duffing equation (4.1).Moreover, we propose a totally new approach to proving the existence of positive almost periodic solutions of nonlinear Duffing equation, which is different from [1][2][3][4][5][6][7][8][9] and the references therein.This implies that the results of this paper are essentially new.