Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

Abstract

This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation , , where the nonlinearity is continuous and is -periodic in . Under certain inequality conditions that may be superlinear growth on , an existence result of odd -periodic solutions is obtained via Leray-Schauder fixed point theorem.

1. Introduction and Main Result

In this paper, we discuss the existence of odd -periodic solutions for the fully second-order ordinary differential equation where the nonlinearity is continuous and is -periodic with respect to .

The existence of periodic solutions for nonlinear second-order ordinary differential equations is an important topic in ordinary differential equation qualitative analysis. It has attracted many authors’ attention and concern, and the most works are on the special equation which does not contain explicitly first-order derivative term in nonlinearity. Many theorems and methods of nonlinear functional analysis have been applied to the periodic problems of (2), such as the upper and lower solutions method and monotone iterative technique [1–4], the continuation method of topological degree [5–9], variational method and critical point theory [10–14], method of phase-plane analysis [15–19], the Krasnoselskii’s type fixed point theorem in cone [20–23], and the theory of fixed point index [24–26].

However, there are not so many existence results for the second-order periodic problem that nonlinearity is dependent on the derivative. See [27–30]. In [27, 28], the authors discussed some special cases that nonlinearity is with a separated derivative term . By finding the fixed point of the Poincaré mapping, they obtained several existence results. In [29], Hakl et al. considered the second-order periodic problem with linear derivative term by the method of lower and upper solutions. In [30], Li and Jiang researched the existence of positive -periodic solution of the general second-order differential equation (1) by employing fixed point index theory in cones under the nonlinearity satisfying a sign condition. Contrary to the results in [30], whether BVP (1) has a sign-changing periodic solution, especially an odd periodic solution, is an interesting problem. For the special second-order ordinary differential equation (2), the existence of odd periodic solutions has been discussed by the present author in [31]. But no one has discussed the general second-order equation (1). The main purpose of this paper is to obtain the existence of odd -periodic solutions for the general second-order equation (1). Our main result is as follows.

Theorem 1. Assume that is continuous; is -periodic with respect to and satisfies the following conditions:(F1), .(F2)there exist nonnegative constants and satisfying and a positive constant , such that (F3)For any given , there is a positive continuous function on satisfying  such that Then (1) has at least one odd -periodic solution.

In Theorem 1, condition (F1) means that is an odd function on . Condition (F2) allows that may be superlinear growth on and . Condition (F3) is a Nagumo type growth condition on , which requests the growth of on which cannot be hyperquadric.

The proof of Theorem 1 is based on the Leray-Schauder fixed point theorem, which will be given in the next section. Two examples to illustrate the applicability of our main result are presented at the end of Section 2. Our result and method are different from those in the references mentioned above.

2. Proof of the Main Result

Let denote the Banach space of all continuous -periodic function in with norm and the Banach space of all th-order continuously differentiable -periodic functions in with the norm , where is a positive integer. Let be the Hilbert space of locally square integrable -periodic functions in with the interior product and the norm . Let be the Sobolev space of -periodic functions with the norm . means that and is absolutely continuous on any finite interval of and . Let be the subspace of odd functions in . Denote , , and . Clearly, is a closed subspace of and hence it is a Banach space by the norm of , is a closed subspace of , and hence it is a Banach space by the norm of .

Given , we consider the existence of odd -periodic solution for the linear second-order differential equation

Lemma 2. For every , the linear equation (7) has a unique odd -periodic solution , and it satisfies Moreover, when the solution and is a completely continuous linear operator.

Proof. Let . Since is an odd function, it can be expressed by the Fourier sine series expansion where , , and the Parseval equality holds. It is easy to verify that belongs to and is a unique odd -periodic solution of the linear equation (7) in Carathéodory sense. If , the solution is a classical solution. From (11) we can easily see that is a linearly bounded operator. By the compactness of the Sobolev embedding , we see that the embedding is compact. Hence, by the boundedness of the embedding , maps into and is completely continuous.
On the other hand, since is an even function, it can be expressed by the cosine series expansion whereHence we obtain the cosine series expansion of : By the expansions (11) and (14), using Parseval equality, we obtain that Hence (8) holds.

Proof of Theorem 1. For every , set By the continuity of and Assumption (F1), is continuous and it maps every bounded set of into a bounded set of . Define a mapping by By Lemma 2, is a completely continuous linear operator. Hence the composite mapping is completely continuous. By the definition of the operator , the odd -periodic solution of (1) is equivalent to the fixed point of . We will use the Leray-Schauder fixed point theorem [32] to show that has a fixed point. To do this, we consider the homotopic family of the operator equations We need to prove that the set of the solutions of the equations (18) is bounded in the space .
Let be a solution of an equation of (18) for . Then . Set . Since , by the definition of , is the unique odd periodic solution of the linear equation (7). Hence satisfies the differential equation Multiplying this equation by and using Assumption (F2), we haveIntegrating this inequality on , using integration by parts for the left side and (8), we haveFrom this inequality it follows that By this and (8) we obtain that Hence, by the continuity of the Sobolev embedding , we have where is the Sobolev embedding constant.
For this , by Assumption (F3), there is a positive continuous function on satisfying (5) such that (6) holds. Hence by (6) and (24), By (5), there exists such that We use (25) and (26) to show that Let . By the periodicity of , there exist and , , such that Clearly, one of the following cases holds.
Case  1. , .
Case  2. , .
Case  3. , .
Case  4. , .
We only consider Case  1; the other cases can be dealt with by a similar method. Let Case  1 hold. Set Then , and by the definition of supremum, Hence, for every , by (19) and (25), we haveso we obtain that Integrating both sides of this inequality on and making the variable transformation for the left side, we have From this inequality and (26) it follows that . Hence, ; namely, (27) holds.
Now by (24) and (27), we have This means that the set of the solutions of the equations (18) is bounded in . By the Leray-Schauder fixed point theorem, has a fixed point in , which is an odd -periodic solution of (1).
The proof of Theorem 1 is completed.

In the above proof, applying the Leray-Schauder fixed point theorem and the technique of prior estimation we proved Theorem 1. Since the nonlinearity may be superlinear growth on and , Theorem 1 cannot be proved by the simple Schauder fixed point theorem.

Example 3. Consider the superlinear second-order ordinary differential equationWe verify that the corresponding nonlinearity satisfies the conditions of Theorem 1. Clearly, satisfies conditions (F1) and (F3). For any , by definition (36) we havethat is, satisfies condition (F2) for , , and . Hence satisfies the conditions of Theorem 1. By Theorem 1, (35) has at least one odd -periodic solution.

Example 4. Consider the sublinear second-order ordinary differential equation Evidently, the corresponding nonlinearity satisfies conditions (F1) and (F3). We show that it also satisfies condition (F2). For any , by definition (39),which is derived by using the Young inequality to the terms of , , , and . From this inequality it follows that Hence, satisfies condition (F2). By Theorem 1, (38) has at least one odd -periodic solution.

Example 3 shows that Theorem 1 is applicable to the superlinear fully second-order ordinary differential equation, and Example 4 shows that Theorem 1 is also applicable to the sublinear fully second-order ordinary differential equation. It should be pointed out that, in Theorem 1 since the nonlinearities contain derivative terms and do not have monotonicity, the conclusions of Examples 3 and 4 cannot be obtained from the known results of [1–31].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by NNSFs of China (11661071, 11261053, and 11361055).