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.
Keywords:lemma on bilinear forms; Schauder’s fixed point theorem; existence and uniqueness; periodic-integrable boundary value problems
1 Introduction and main results
In this paper, we consider the solutions to the following periodic-integrable boundary value problem (for short, PIBVP):
where is a given T-periodic function in , and ; is T-periodic in t.
Throughout this paper, we assume
(A1) there exist two constants m and M such that
for all and ;
(A2) there exists such that
Recently, boundary value problems with integral conditions have been studied extensively [6-10]. As we all know Lazer type conditions are essential for the existence and uniqueness of periodic solutions of equations [1-4]. In  the existence of periodic solutions has been considered for the following second order equation:
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 1Assume that (A1) and (A2) 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
Consider the following linear PIBVP:
here is a given T-periodic function in , and ; is a T-periodic function. Assume that
(L1) there exist two constants m and M such that
for all . Moreover, m and M suit (A2).
Theorem 2Assume that (L1) and (A2) are satisfied, then PIBVP (2.1) has only a trivial solution.
In order to prove Theorem 2, let us give some following concepts.
First, for any interval , define
It is clear that is a linear space with the norm as follows:
Define a bilinear form on as follows:
for any and . Let
where N suits assumption (L1), and , , , and are some constants. Then .
From and , we can obtain that there exist two constants and such that
for all . Then from assumptions (L1) and (A2), we have
for all , and
for all . Thus, is positive definite on and negative definite on . By the lemma in , we assert that if for all , then .
For every x on with , we introduce an auxiliary function
The following lemma is very useful in our proofs.
Lemma 1If , are continuous and satisfy (L1) and (A2), then the following two points boundary value problem
has only a trivial solution.
Proof It is clear that 0 is a solution of two points boundary value problem (2.2). If is a solution of problem (2.2), then . For any , we have
by using (2.2). Integrating the first terms by parts, we derive
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 .
Case 2: . Denote
From , there are at least two points in the set S, which implies that and . By Lemma 1 ( and ) the two points boundary value problem
only has a trivial solution. Hence we obtain , . By the definitions of a and b, one has
From , we get
This contradicts and .
Case 3: . This case is similar to Case 2.
Thus, we complete the proof of Theorem 2. □
Theorem 3If , are continuous and satisfy (L1) and (A2), then the following PIBVP
has a unique solution.
Proof Let and be two linear independent solutions of the following linear equation:
Assume that is a solution of PIBVP (2.1), where and are constants. Then by the boundary value conditions of (2.1),
By Theorem 3, PIBVP (2.1) has only a trivial solution, which shows
Let be a solution of PIBVP (2.4), where is a solution of the equation
From the boundary value conditions, we have
From (2.5) constants , are unique. Thus, PIBVP (2.4) has only one solution. □
3 Nonlinear equations
Let us prove Theorem 1. Rewrite (1.1) as follows:
Fix , introduce an auxiliary PIBVP
To prove the main result, we need the following Lemma 2.
Lemma 2Iffsatisfies (A1) and (A2), then for any given , PIBVP (3.2) has only one solution, denoted as and .
Proof From condition (A2), it follows that
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 .
Because the set
is bounded convex in , by the Mazur theorem, we have . Hence,
By the Arzela-Ascoli theorem, passing to a subsequence, we may assume that
and in . Thus, and .
which implies , for any . Hence, .
From PIBVP (3.2), we obtain
This shows that is a nontrivial solution of the following PIBVP:
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 3OperatorFis completely continuous on .
Proof We first prove that F is continuous. Given any such that . Put . From the definition
We would prove that in . If not, then there would be a such that
Utilizing Lemma 2 and Arzela-Ascoli theorem, passing to a subsequence, we may assume that . Similar to the proof of Lemma 2, we have . Then
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 .
The following is to prove uniqueness. Let and be any two solutions of equation (1.1). Then is a solution of the equation
Employing (A2), we have
Hence by Theorem 3, . The uniqueness is proved. □
The authors declare that they have no competing interests.
Each of the authors, HH, FC and YC contributed to each part of this work equally and read and approved the final version of the manuscript.
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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