Existence and Uniqueness on Periodic Solutions of Fourth-order Nonlinear Differential Equations ∗

In this paper, we study the problem of periodic solutions for fourth-order nonlinear differential equations. Under proper conditions, we employ a novel proof to establish some criteria to ensure the existence and uniqueness of T -periodic solutions. Moreover, we give two examples to illustrate the effectiveness of our main results.


Introduction
The existence of periodic solutions for fourth-order nonlinear differential equations have been widely investigated and are still being investigated due to their applications in many fields such as nonlinear oscillations [1][2], fluid mechanical and nonlinear elastic mechanical phenomena [3][4][5][6][7][8].Recently, C. Bereanu [9], C. Zhao et al. [10] and Q. Fan et al. [11] discussed the existence of T -periodic solutions of fourth-order nonlinear differential equation of the type where a, p, q ∈ R, g : R 2 → R is continuous, T -periodic in t, f : R → R is continuous, and e : R → R is a continuous T -periodic function.
Moreover, the authors in [9−11] also provided some sufficient conditions for the existence of T -periodic solutions of equation (1.1), which generalized and improved the known results in references [1,4,6,7].However, to the best of our knowledge, most authors of the bibliographies listed above only consider the existence of periodic solutions of equation (1.1), and there exist few results for the existence and uniqueness of periodic solutions of this equation.Thus, in this case, it is worth to study the problem of existence and uniqueness of periodic solutions of fourth-order nonlinear differential equation (1.1).
The purpose of this article is to investigate the existence and uniqueness of T -periodic solutions of (1.1).By using some inequality techniques and Mawhin's continuation theorem, we establish some sufficient conditions for the existence and uniqueness of T -periodic solutions of (1.1).Moreover, two illustrative examples are given in Section 4.

Preliminary Results
Let us introduce some notations.For m ∈ N, we denote by C m T the Banach space where, for a function v ∈ C 0 T , we have that |v|.
For x ∈ C 0 T , we will denote Now, let f : R 5 → R be a continuous function, T -periodic with respect to the first variable and consider the fourth-order differential equation Lemma 2.1(see [12]) Assume that the following conditions hold.
(i) There exists ρ > 0 such that, for each λ ∈ (0, 1], one has that any possible T -periodic solution u of the problem satisfies the a priori estimation u (3) < ρ.
(ii) The continuous function F : R → R defined by Then, (2.1) has a least one T -periodic solution u such that u (3) < ρ.
From Lemma 2.2 in [13] and the proof of inequality (10) in [11, pp 124], we obtain Lemma 2.2.Let x(t) ∈ C 1 T .Suppose that there exists a constant D ≥ 0 such that Then Proof.Lemma 2.3 is a direct consequence of the Wirtinger inequality, and see [14,15] for its proof.
Lemma 2.4.(see[11]) For any u ∈ C 4 T one has that Lemma 2.5.Assume that that one of the following conditions is satisfied: Then (1.1) has at most one T -periodic solution.
Proof.Suppose that u 1 (t) and u 2 (t) are two T -periodic solutions of (1.1).Set Z(t) = u 1 (t) − u 2 (t).Then, we obtain Therefore, in view of integral mean value theorem, it follows that there exists a constant Since g(t, x) is a strictly monotone function in x, (2.7) implies that Then, from (2.3), we have (2.9) Multiplying Z(t) and (2.6) and then integrating it from 0 to T , it follows that (2.10) Now suppose that (A 1 )(or (A 2 )) holds, we shall consider two cases as follows.

Main Results
Theorem 3.1.Assume that that one of the following conditions is satisfied: (A 1 ) * Let (A 1 ) hold, and there exists a nonnegative constant d 0 such that (g(t, u) + e(t))u < 0, for all t ∈ R, |u| ≥ d 0 ; (A 2 ) * There exist nonnegative constants d 0 and B such that (A 2 ) holds, and (g(t, u) + e(t))u > 0, for all t ∈ R, |u| ≥ d 0 .
Then equation (1.1) has a unique T -periodic solution.
Proof.From Lemma 2.5, together with (A 1 ) * (or (A 2 ) * ), it is easy to see that equation For λ ∈ (0, 1], we consider the fourth-order differential equation Let us show that (i) in Lemma 2.1 is satisfied, that means, there exists ρ > 0 such that any possible T -periodic solution u of (3.1) is such that It follows from (2.3) that In view of ( implies that It follows that and On the other hand, multiplying equation (3.1) by u and integrating it from 0 to T , it follows that Now suppose that (A 1 ) * (or (A 2 ) * ) holds, we shall consider two cases as follows.