- Open Access
Existence and uniqueness of solutions for periodic-integrable boundary value problem of second order differential equation
© Hua et al.; licensee Springer 2012
Received: 26 March 2012
Accepted: 16 July 2012
Published: 7 August 2012
In this paper we deal with one kind of second order periodic-integrable boundary value problem. Using the lemma on bilinear form and Schauder’s fixed point theorem, we give the existence and uniqueness of solutions for the problem under Lazer type nonresonant condition.
MSC:34B15, 34B16, 37J40.
1 Introduction and main results
where is a given T-periodic function in , and ; is T-periodic in t.
Throughout this paper, we assume
for all and ;
Motivated by the above works, we will consider periodic-integrable boundary value problem (1.1). The main result obtained by us is the following theorem.
Theorem 1 Assume that (A 1) and (A 2) are satisfied. Then PIBVP (1.1) has a unique solution.
This paper is organized as follows. Section 2 deals with a linear problem. There, using the bilinear lemma developed by Lazer, one proves the uniqueness of solutions for linear equations. In Section 3, applying the result in Section 2 and Schauder’s fixed point theorem, we complete the proof of Theorem 1.
2 Linear equation
here is a given T-periodic function in , and ; is a T-periodic function. Assume that
for all . Moreover, m and M suit (A2).
Theorem 2 Assume that (L 1) and (A 2) are satisfied, then PIBVP (2.1) has only a trivial solution.
In order to prove Theorem 2, let us give some following concepts.
where N suits assumption (L1), and , , , and are some constants. Then .
for all . Thus, is positive definite on and negative definite on . By the lemma in , we assert that if for all , then .
The following lemma is very useful in our proofs.
has only a trivial solution.
By assumption (L1), is positive definite on and negative definite on . These show for , that is, for . The proof of Lemma 1 is ended. □
Proof of Theorem 2 It is clear that PIBVP (2.1) has at least one solution, for example, . Assume that PIBVP (2.1) possesses a nontrivial solution . The proof is divided into three parts.
Case 1: . By Lemma 1 ( and ), PIBVP (2.1) has only a trivial solution. This contradicts .
This contradicts and .
Case 3: . This case is similar to Case 2.
Thus, we complete the proof of Theorem 2. □
has a unique solution.
From (2.5) constants , are unique. Thus, PIBVP (2.4) has only one solution. □
3 Nonlinear equations
To prove the main result, we need the following Lemma 2.
Lemma 2 If f satisfies (A 1) and (A 2), then for any given , PIBVP (3.2) has only one solution, denoted as and .
By Theorem 3, PIBVP (3.2) has only one solution . If does not hold, there would exist a sequence such that , . Choose a subsequence of , without loss of generality, express as itself, such that the sequences are weakly convergent in . Denote the limit as . It is obvious that .
and in . Thus, and .
which implies , for any . Hence, .
On the other hand, by Theorem 2, PIBVP (3.4) has only zero, which leads to a contradiction. The proof of Lemma 2 is completed. □
Define an operator by . Applying Lemma 2, .
Lemma 3 Operator F is completely continuous on .
Hence, from Theorem 2, . This implies F is continuous. By Lemma 2, for any bounded subset , is also bounded. Hence, applying the continuity of F and Arzela-Ascoli theorem, is relatively compact. This shows F is completely continuous on . The proof of Lemma 3 is completed. □
Proof of Theorem 1 By Lemma 2, Lemma 3 and Schauder’s fixed point theorem, F has a fixed point in , that is, PIBVP (1.1) has a solution .
Hence by Theorem 3, . The uniqueness is proved. □
The authors are grateful to the referees for their useful comments. The research of F. Cong was partially supported by NSFC Grant (11171350) and Natural Science Foundation of Jilin Province of China (201115133).
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