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):
Throughout this paper, we assume
(A1) there exist two constants m and M 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:
(L1) there exist two constants m and M such that
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.
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 using (2.2). Integrating the first terms by parts, we derive
Thus, we complete the proof of Theorem 2. □
has a unique solution.
By Theorem 3, PIBVP (2.1) has only a trivial solution, which shows
From the boundary value conditions, we have
3 Nonlinear equations
Let us prove Theorem 1. Rewrite (1.1) as follows:
To prove the main result, we need the following Lemma 2.
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
By the Arzela-Ascoli theorem, passing to a subsequence, we may assume that
From PIBVP (3.2), we obtain
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. □
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. □
Employing (A2), we have
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).
Lazer, AC: Application of a lemma on bilinear forms to a problem in nonlinear oscillations. Proc. Am. Math. Soc.. 33, 89–94 (1972). Publisher Full Text
Wang, H, Li, Y: Existence and uniqueness of periodic solution for Duffing equations across many points of resonance. J. Differ. Equ.. 108, 152–169 (1994). Publisher Full Text
Asakawa, H: Landesman-Lazer type problems for Fucik spectrum. Nonlinear Anal.. 26, 407–414 (1996). Publisher Full Text
Jankowski, T: Differential equations with integral boundary conditions. J. Comput. Appl. Math.. 147, 1–8 (2002). Publisher Full Text
Yang, Z: Positive solutions to a system of second-order nonlocal boundary value problems. Nonlinear Anal.. 62, 1251–1265 (2005). Publisher Full Text
Yang, Z: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal.. 65, 1489–1511 (2006). Publisher Full Text
Yang, Z: Positive solutions of a second-order integral boundary value problem. J. Math. Anal. Appl.. 321, 751–765 (2006). Publisher Full Text