Abstract

We investigate the solvability of a fully fourth-order periodic boundary value problem of the form where satisfies Carathéodory conditions. By using the coincidence degree theory, the existence of nontrivial solutions is obtained. Meanwhile, as applications, some examples are given to illustrate our results.

1. Introduction

In this paper, we consider a fully nonlinear fourth-order periodic boundary value problem of the form subject to the boundary conditions where satisfies Carathéodory conditions; that is, (i)for a.e. , the function is continuous;(ii)for every , the function is measurable;(iii)for each , there is a real valued function such that  for a.e. and .

It is well known that fourth-order periodic boundary value problems are important research topics which arise in a variety of different areas, such as nonlinear oscillations, fluid mechanical, and nonlinear elastic mechanical phenomena, and thus have been extensively studied; for instance, see [1–30] and references therein. However, most of the works in the above-mentioned references allow only having , or , , in the right-hand side nonlinear function ; see [2–11, 13, 15–18, 20–30]. The works on the fully nonlinear cases of which contains explicitly and every derivative of up to order three have been quite rarely seen; see [1, 12, 14, 19].

The aim of this paper is to establish the existence of solutions and nontrivial solutions for the fully nonlinear fourth-order PBVP (1), (2). Our main tool is the coincidence degree theory. The paper [31] motivated our study.

2. Preliminary

In this section, we present some lemmas which are needed for our main results.

At first, we will briefly recall some notations that are needed for our discussion.

Let , be real Banach spaces. A linear mapping will be called a Fredholm mapping of index zero if the following two conditions hold:(i) is a closed subspace of ;(ii).

Let be a Fredholm mapping of index zero; then there exist continuous projectors and such that so that It follows that is invertible. We denote the inverse of that map by . Let be an open bounded subset of such that ; the map will be called -compact on , if and are compact.

Lemma 1 (see [32]). Let be a linear Fredholm mapping of index zero and let be an open bounded set. Let be -compact on and let be -completely continuous such that(i);(ii)for every ,  and assume that . Then equation  has at least one solution in .

Lemma 2 (see [33]). Let be a linear Fredholm mapping of index zero and let be an open bounded set. Let be -compact on and the coincidence degree is well defined. If there exists with such that then .

In the following, we take Banach space with the norm , and . Define a linear map by where and is the usual Sobolev space. It is easy to see that is a Fredholm mapping of index zero. Also define a nonlinear map by

Define two projects and as follows:

Let be Green function for the homogeneous BVP Then can be given by Hence the map is continuous. We note that if satisfying Carathéodory conditions, then is bounded and continuous by Lebesgue's dominated convergence theorem. Furthermore, is -compact on every bounded set .

3. Main Results

For and we put

In order to introduce our main theorem, we need some lemmas.

Lemma 3. Let be a nonnegative function and . Let , , , , and fulfil (14). Then for any , the inequalities imply

Proof. Since , there exists such that We will show that By contradiction, assume that there exists such that Then there exists such that There are two cases to consider.
Case 1. Consider on . In this case, integrating (16) from to and using (15) and (21) we infer that Thus , which contradicts (20).
Case 2. Consider on . Similar to Case 1, we have Thus, , which contradicts (20). Therefore (19) is true. Furthermore, from the fact it follows that .
Finally, we show that Suppose on the contrary that there exists satisfying Then there exists such that There are two cases to consider.
Case  1 Consider on . Similar to Case 1, we have , which contradicts (25).
Case  2 Consider on . Similar to Case 2, one has , which also contradicts (25). Hence (24) is true.
In summary, from (19) and (24) it follows that estimate (17) holds. This completes the proof of the lemma.

Lemma 4. Let and let be a nonnegative function. Let , , , , and fulfil (14). Then for any function the inequalities imply

Proof. For every , from (28), there exist such that Integrating (27) by (14) and (30) we get This completes the proof of the lemma.

Now, we apply Lemma 1 to establish the existence results of solutions for the fourth-order PBVP (1), (2).

Theorem 5. Assume that there exist , , and a nonnegative function . Suppose further that (H₀) satisfies the Carathéodory conditions;(H₁)if , , , , then (H₂)if , ,  ,  , then  where , , , fulfil (14). Then PBVP (1), (2) has at least one solution such that

Proof. Let Then iff there exist some such that
Now, we show that where , . To do this, we assume that is the solution of the following periodic boundary value problem: Integrating the equation as above on , we obtain Thus, by Wirtinger inequality, Hence from (38) it follows that If , then, from , the following contradiction holds: Therefore, .
Finally, we show that, for every , To do this, let and let be a solution of the following PBVP: Then . In fact, let Then, by (33), for a.e. . Applying Lemma 3, we obtain Thus according to (32), we have for a.e. . It follows from Lemma 4 that Thus . This implies that condition (ii) of Lemma 1 is valid.
In summary, all conditions of Lemma 1 are satisfied. Therefore the conclusion of Theorem 5 holds. This completes the proof of the theorem.