Anti-Periodic Solutions for a Class of Third-Order Nonlinear Differential Equations with a Deviating Argument ∗

In this paper, we study a class of third-order nonlinear differential equations with a deviating argument and establish some sufficient conditions for the existence and exponential stability of anti-periodic solutions of the equation. These conditions are new and complement to previously known results.

Just as above, in the past few decades, the study for third order differential equation has been paid attention to by many scholars.Many results relative to the stability, instability of solutions, boundedness of solutions, convergence of solutions and existence of periodic solutions for equation (1.1) and its analogue equations have been obtained (see [7,8] and references therein).However, as pointed out in [8], the results about the existence of antiperiodic solutions for nonlinear differential equations whose orders are more than two are relatively scarce.Moreover, it is well known that the existence of anti-periodic solutions play a key role in characterizing the behavior of nonlinear differential equations (See [9−12]).
Thus, it is worthwhile to continue to investigate the existence and stability of anti-periodic solutions of Eq. (1.1).
A primary purpose of this paper is to study the problem of anti-periodic solutions of (1.1).
We will establish some sufficient conditions for the existence and exponential stability of the anti-periodic solutions of (1.1).Our results are new and complement to previously known results.In particular, an example is also provided to illustrate the effectiveness of the new results.
Let d 1 and d 2 be constants.Define then we can transform (1.1) into the following equivalent system Throughout this article, it will be assumed that there exists a constant T > 0 such that We suppose that there exists a constant L + such that It is known in [14−16] that for g 1 , g 2 , a, b, τ and p continuous, given a continuous initial function ϕ ∈ C([−τ , 0], R) and a vector (y 0 , z 0 ) ∈ R 2 , then there exists a solution of (1.3) on an interval [0, T ) satisfying the initial condition and satisfying (1. If there exist constants λ > 0 and M > 1 such that for every solution Z(t) = (x(t), y(t), z(t)) of system (1.3) with any We also assume that the following condition holds.
(C 1 ) There exist constants The paper is organized as follows.In Section 2, we establish some preliminary results, which are important in the proofs of our main results.Based on the preparations in Section 2, we state and prove our main results in Section 3.Moreover, an illustrative example is given in Section 4. EJQTDE, 2010 No. 8, p. 3

Preliminary Results
The following lemmas will be useful to prove our main results in Section 3.
Lemma 2.1.Let (C 1 ) hold.Suppose that ( x(t), y(t), z(t)) is a solution of system (1.3) with initial conditions where Proof.Assume, by way of contradiction, that (2.2) does not hold.Then, one of the following cases must occur.
If Case 1 holds, calculating the upper left derivative of | x(t)|, together with (C 1 ), (1.3) and (2.3) imply that which is a contradiction and implies that (2.2) holds.The proof of Lemma 2.1 is now complete.
Remark 2.1.In view of the boundedness of this solution, from the theory of functional differential equations in [15], it follows that ( x(t), y(t), z(t)) can be defined on [0, ∞).
Case I: There exists

13)
Case II: There exists T 2 > 0 such that

Main Results
In this section, we establish some results for the existence, uniqueness and exponential stability of the T-anti-periodic solution of (1.3).
In view of (3.4), we can choose a sufficiently large constant N > 0 and a positive constant α such that Next, we prove that Z * (t) is a solution of (1.1).In fact, together with the continuity of the right side of (1.3), (3.1), (3.2) and (3.3) imply that{((−1) m+1 v(t + (m + 1)T )) ′ } uniformly converges to a continuous function on any compact set of R. Thus, letting m −→ ∞, we obtain 2 )y * (t) + p(t).
(  Remark 4.1.Since Eq. (4.1) is a form of third-order nonlinear differential equation with varying time delays.One can observe that all the results in [8][9][10][11][12][13] and the references cited therein can not be applicable to prove that Eq. (4.1) has a unique anti-periodic periodic solution which is globally exponentially stable.Moreover, we propose a totally new approach to proving the existence of anti-periodic solutions of third-order nonlinear differential equation, which is different from [8] and the references therein.This implies that the results of this paper are essentially new.