Existence of positive periodic solutions for third-order nonlinear delay differential equations with variable coefficients

In this paper, the following third-order nonlinear delay differential equation with periodic coefficients x(t) + p(t)x(t) + q(t)x(t) + r(t)x(t) = f ( t, x(t), x(t− τ(t)) ) + d dt g ( t, x(t− τ(t)) ) , is considered. By employing Green’s function and Krasnoselskii’s fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order nonlinear delay differential equation.


Introduction
Third order differential equations arise from in a variety of different areas of applied mathematics and physics, as the deflection of a curved beam having a constant or varying cross section, three layer beam, electromagnetic waves or gravity driven flows and so on [24,29].
Delay differential equations have received increasing attention during recent years since these equations have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering, see the monograph [12,26] and the papers [1]- [23], [25] [27]- [29], [31]- [34] and the references therein.
The second order nonlinear delay differential equation with periodic coefficients has been investigated in [9]. By using Krasnoselskii's fixed point theorem and the contraction mapping principle, Ardjouni and Djoudi obtained existence and uniqueness of periodic solutions. In [29], Ren, Siegmund and Chen discussed the existence of positive periodic solutions for the third-order differential equation By employing the fixed point index, the authors obtained existence results for positive periodic solutions.
Inspired and motivated by the works mentioned above and the papers [1]- [23], [25], [27]- [29], [31]- [34] and the references therein, we concentrate on the existence of positive periodic solutions for the third-order nonlinear delay differential equation where p, q, r, τ are continuous real-valued functions. The functions g : The obtained integral equation splits in the sum of two mappings, one is a contraction and the other is compact. The organization of this paper is as follows. In section 2, we introduce some notations and lemmas, and state some preliminary results needed in later section, then we give the Green's function of (1), which plays an important role in this paper. In section 3, we present our main results on existence of positive periodic solutions of (1).
We state Krasnoselskii's fixed point theorem which enables us to prove the existence of positive periodic solutions to (1). For its proof we refer the reader to [30]. Then there exists z ∈ M with z = H 1 z + H 2 z.
In this paper, we give the assumptions as follows that will be used in the main results.
(h2) p, q, r, τ ∈ C (R, R + ) are T -periodic functions with τ (t) ≥ τ * > 0 and (h3) The functions g (t, x) and f (t, x, y) are continuous T -periodic in t and continuous in x and in x and y, respectively.

Green's function of third-order differential equation
For T > 0, let P T be the set of all continuous scalar functions x, periodic in t of period T . Then (P T , . ) is a Banach space with the supremum norm We consider (2) x where h is a continuous T -periodic function. Obviously, by the condition (h1), (2) is transformed into ). If y, h ∈ P T , then y is a solution of equation Corollary 2.1. Green function G 1 satisfies the following properties and 28]). Suppose that (h1) and (h2) hold and (5) Then there are continuous T -periodic functions a and b such that and . Suppose the conditions of Lemma 2.2 hold and y ∈ P T . Then the equation has a T periodic solution. Moreover, the periodic solution can be expressed by Corollary 2.2. Green's function G 2 satisfies the following proprieties G 2 (t + T, s + T ) = G 2 (t, s), G 2 (t, t + T ) = G 2 (t, t), where Lemma 2.5 ( [15]). Suppose the conditions of Lemma 2.2 hold and h ∈ P T . Then the equation Third-order nonlinear delay differential equations has a T -periodic solution. Moreover, the periodic solution can be expressed by Corollary 2.4. Green's function G satisfies the following properties .

Main Results
In this section we will study the existence of positive periodic solutions of (1). Lemma 3.1. Suppose (h1) − (h3) and (5) hold. The function x ∈ P T is a solution of (1) if and only if Proof. Let x ∈ P T be a solution of (1). From Lemma 2.5, we have Performing an integration by parts, we get t+T t G (t, s) ∂ ∂s g (s, x (s − τ (s))) ds We obtain (11) by substituting (13) in (12). Since each step is reversible, the converse follows easily. This completes the proof.
Define the mapping H : Note that to apply Krasnoselskii's fixed point theorem we need to construct two mappings, one is a contraction and the other is compact. Therefore, we express (14) as where H 1 , H 2 : P T → P T are given by and (16) ( To simplify notations, we introduce the constants In this section we obtain the existence of a positive periodic solution of (1) by considering the two cases; g (t, x) ≥ 0 and g (t, x) ≤ 0 for all t ∈ R. For a non-negative constant K and a positive constant L we define the set which is a closed convex and bounded subset of the Banach space P T . We assume that the function g (t, x) is locally Lipschitz continuous in x. That Third-order nonlinear delay differential equations is, there exists a positive constant k such that (18) |g (t, x) − g (t, y)| ≤ k x − y , for all t ∈ [0, T ] , x, y ∈ D.
In case g (t, x) ≥ 0, we assume that there exist a nonnegative constant k 1 and a positive constant k 1 such that (19) k and for all t ∈ [0, T ] , x, y ∈ D Proof. Let H 1 be defined by (15). Obviously, H 1 ϕ is continuous and it is easy to show that (H 1 ϕ) (t + T ) = (H 1 ϕ) (t). For t ∈ [0, T ] and for ϕ ∈ D, we have Thus from the estimation of |(H 1 ϕ) (t)| we have This shows that H 1 (D) is uniformly bounded.
Consequently, by invoking (17) and (21), we obtain for some positive constant D. Hence the sequence (H 1 ϕ n ) is equicontinuous. The Ascoli-Arzela theorem implies that a subsequence (H 1 ϕ n k ) of (H 1 ϕ n ) converges uniformly to a continuous T -periodic function. Thus H 1 is continuous and H 1 (D) is contained in a compact subset of D.
Lemma 3.3. Suppose that (18) holds. If H 2 is given by (16) with then H 2 : D → P T is a contraction.
This shows that H 2 ψ + H 1 ϕ ∈ D. Clearly, all the Hypotheses of Theorem 1.1, are satisfied. Thus there exists a fixed point x ∈ D such that x = H 1 ψ + H 2 ϕ. By Lemma 3.1 this fixed point is a solution of (1) and the proof is complete.
In the case g (t, x) ≤ 0, we substitute conditions (19)- (21) with the following conditions respectively. We assume that there exist a negative constant k 3 and a non-positive constant k 4 such that (23) k 3 x ≤ g (t, x) ≤ k 4 x, for all t ∈ [0, T ] , x ∈ D, The proof follows along the lines of Theorem 3.1, and hence we omit it.