Existence of Periodic Solutions for Nonlinear Fully Third-Order Differential Equations

In this paper, we study the existence of periodic solutions to nonlinear fully third-order differential equation x′′′ + f(t, x, x′, x′′) � 0, t ∈ R ≔ (−∞,∞), where f: R4⟶ R is continuous and T-periodic in t. By using the topological transversality method together with the barrier strip technique, we obtain new existence results of periodic solutions to the above equation without growth restrictions on the nonlinearity. Meanwhile, as applications, an example is given to demonstrate our results.


Introduction
In this paper, we consider the existence of periodic solutions for nonlinear fully third-order differential equation where f: R 4 ⟶ R is continuous and T-periodic with respect to t. e third-order periodic problem arises in many areas of applied mathematics and physics, and so it has been extensively studied by many authors via various methods, for instance, see  and the references therein. Among a substantial number of works, we mention that the upper and lower solutions method is used in [5,6,15,16], Leray-Schauder continuation theorem is used in [1,10,20,21], Leray-Schauder degree theory or the Schauder-fixed-point theorem is used in [10,12,13,18], Mawhin coincidence degree theory is used in [2,3,8,17], and fixed-point theorem in cone or fixed-point index theory is used in [7,9,11,19,22]. However, to the best of our knowledge, there is no work that refers to periodic solutions of equation (1) using the topological transversality method.
Recently, Kelevedjiev and Todorov [23] have used the topological transversality method and barrier strip technique to study various third-order two-point boundary value problems. But, they did not consider the third-order periodic boundary value problem.
Motivated and inspired by the aforementioned works, the aim of this paper is to establish new existence results of periodic solutions to equation (1) by using the topological transversality method together with barrier strip technique. It is worth mentioning that our results do not need any growth restrictions on the nonlinearity. In addition, compared with the corresponding ones in the known literature, the barrier strip technique we use to estimate a prior bounds of periodic solutions is essentially new.
is work is organized as follows. In Section 2, we first introduce some notations and lemmas and then estimate a prior bounds of periodic solutions of equation (1) by using barrier strip technique. Finally, by using the topological transversality method, we establish the existence results of periodic solutions to equation (1). As applications of our main results, an example is given in the last section.
(H 0 ′ ): there exists M > 0 such that, for any T-periodic function x ∈ C 2 (R), x, y, z) does not change its sign for (t, x, y, z) ∈ D 1 and for (t, x, y, z) ∈ D 2 , respectively, here and M 0 � M + TM 1 . Let us introduce some notations of the topological transversality method for the convenience of the reader. Let U be a convex subset of a Banach space Y and Ω ⊂ U be open in U. Denote by H zΩ (Ω, U) the set of compact operators F: Ω ⟶ U which are fixed-point-free on zΩ. We say that F ∈ H zΩ (Ω, U) is essential if every operator in H zΩ (Ω, U) which agrees with F on zΩ has a fixed point in Ω. e next two lemmas can be found in [24].
en, H(·, 1) is essential, and therefore, it has a fixed point in Ω.
We note that the existence of T-periodic solutions for equation (1) is equivalent to the solvability of the following third-order periodic boundary value problem (for short PBVP): We first consider the family of following PBVPs: where λ ∈ (0, 1]. A priori bounds for the solutions of PBVP (7), (8) are presented in the following lemmas. (7) and (8) en, Proof. At first, we show that (10) holds. Indeed, suppose on the contrary that there exist Without loss of generality, we assume that t 2 ∈ (0, T); which is a contradiction. is means that (10) holds.
Next, we prove that (9) holds. Indeed, integrating the equation in (7) from 0 to T, we obtain that is together with the condition (H 0 )(or (H 0 ′ )) implies that there exists ξ ∈ [0, T] such that |x(ξ)| ≤ M. It follows from (10) that, for t ∈ [0, T], Journal of Function Spaces which means that (9) holds. is completes the proof of the lemma.  (7) and (8) for some λ ∈ (0, 1]. en, Proof. We estimate x ″ (t) by using the barrier strip technique. From condition (H 2 ), f(t, x, y, z) does not change its sign for (t, x, y, z) ∈ D 1 and for (t, x, y, z) ∈ D 2 , respectively. For the sake of certainty, without loss of generality, we assume that Notice that x ′ (0) � x ′ (T), and from Rolle's mean value theorem, there exists η ∈ (0, T) such that x ″ (η) � 0. Let We now assert that the sets S 0 and S 1 are empty. We shall complete the proof in two steps.
(36) en, U is a closed and convex subset of Y, and Ω is an open subset of U.
We now give two lemmas which will be used in the proof of our main theorem.
Proof. At first, we define operator A: where It is easy to check that, for each (x, r, λ) ∈ Ω × [0, 1], A(x, r, λ) is the unique solution of the following boundary value problem: Furthermore, by a standard argument, it is easy to show that the operator A(x, r, λ) is completely continuous.