Abstract
In this paper we deal with one kind of second order periodicintegrable 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; periodicintegrable boundary value problems1 Introduction and main results
In this paper, we consider the solutions to the following periodicintegrable boundary value problem (for short, PIBVP):
where
Throughout this paper, we assume
(A1) there exist two constants m and M such that
for all
(A2) there exists
Recently, boundary value problems with integral conditions have been studied extensively [610]. As we all know Lazer type conditions are essential for the existence and uniqueness of periodic solutions of equations [14]. In [5] the existence of periodic solutions has been considered for the following second order equation:
Motivated by the above works, we will consider periodicintegrable 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
(L1) there exist two constants m and M such that
for all
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
It is clear that
Define a bilinear form on
for any
where N suits assumption (L1), and
From
for all
for all
for all
For every x on
The following lemma is very useful in our proofs.
Lemma 1If
has only a trivial solution.
Proof It is clear that 0 is a solution of two points boundary value problem (2.2). If
by using (2.2). Integrating the first terms by parts, we derive
By assumption (L1),
Proof of Theorem 2 It is clear that PIBVP (2.1) has at least one solution, for example,
Case 1:
Case 2:
Take
From
only has a trivial solution. Hence we obtain
From
This contradicts
Case 3:
Thus, we complete the proof of Theorem 2. □
Theorem 3If
has a unique solution.
Proof Let
Assume that
By Theorem 3, PIBVP (2.1) has only a trivial solution, which shows
Let
From the boundary value conditions, we have
From (2.5) constants
3 Nonlinear equations
Let us prove Theorem 1. Rewrite (1.1) as follows:
where
Define
Fix
To prove the main result, we need the following Lemma 2.
Lemma 2Iffsatisfies (A1) and (A2), then for any given
Proof From condition (A2), it follows that
By Theorem 3, PIBVP (3.2) has only one solution
Because the set
is bounded convex in
By the ArzelaAscoli theorem, passing to a subsequence, we may assume that
and
By
one has
which implies
From PIBVP (3.2), we obtain
This shows that
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. □
Set
Define an operator
Lemma 3OperatorFis completely continuous on
Proof We first prove that F is continuous. Given any
We would prove that
Utilizing Lemma 2 and ArzelaAscoli theorem, passing to a subsequence, we may assume
that
Moreover,
Hence, from Theorem 2,
Proof of Theorem 1 By Lemma 2, Lemma 3 and Schauder’s fixed point theorem, F has a fixed point in
The following is to prove uniqueness. Let
Employing (A2), we have
Hence by Theorem 3,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
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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