Research Article | Open Access
The Existence of Positive Solutions for Singular Impulse Periodic Boundary Value Problem
We obtain new result of the existence of positive solutions of a class of singular impulse periodic boundary value problem via a nonlinear alternative principle of Leray-Schauder. We do not require the monotonicity of functions in paper (Zhang and Wang, 2003). An example is also given to illustrate our result.
Because of wide interests in physics and engineering, periodic boundary value problems have been investigated by many authors (see [1–19]). In most real problems, only the positive solution is significant.
In this paper, we consider the following periodic boundary value problem (PBVP in short) with impulse effects: Here, , , , , , , , , with , , . , , where and , , respectively, denote the right and left limit of at .
In , Liu applied Krasnoselskii's and Leggett-Williams fixed-point theorem to establish the existence of at least one, two, or three positive solutions to the first-order periodic boundary value problems Jiang  has applied Krasnoselskii's fixed point theorem to establish the existence of positive solutions of problem The work  proved that periodic boundary value problem (PBVP in short) (1.3) without singularity have at least one positive solutions provided is superlinear or sublinear at and . In , Tian et al. researched PBVP (1.1) without singularity. They obtained the existence of multiple positive solutions of PBVP (1.1) by replacing the suplinear condition or sublinear condition of  with the following limit inequality condition:
Nieto  introduced the concept of a weak solution for a damped linear equation with Dirichlet boundary conditions and impulses. These results will allow us in the future to deal with the corresponding nonlinear problems and look for solutions as critical points of weakly lower semicontinuous functionals.
We note that the function involved in above papers [5, 7, 10, 14] does not have singularity. Xiao et al.  investigate the multiple positive solutions of singular boundary value problem for second-order impulsive singular differential equations on the halfline, where the function is singular only at and/or . Reference  studied PBVP (1.3), where the function has singularity at . The authors present the existence of multiple positive solutions via the Krasnoselskii's fixed point theorem under the following conditions.There exist nonnegative valued and such that
where is nonincreasing and is nondecreasing on , Here, are some constants.
In this paper, the nonlinear term is singular at , and positive solution of PBVP (1.1) is obtained by a nonlinear alternative principle of Leray-Schauder type in cone. We do not require the monotonicity of functions , used in . An example is also given to illustrate our result.
This paper is organized as follows. In Section 1, we give a brief overview of recent results on impulsive and periodic boundary value problems. In Section 2, we present some preliminaries such as definitions and lemmas. In Section 3, the existence of one positive solution for PBVP (1.1) will be established by using a nonlinear alternative principle of Leray-Schauder type in cone. An example is given in Section 4.
Consider the space is a map from into such that is continuous at , left continuous at , and exists, for . It is easy to say that is a Banach space with the norm . Let exists at and is continuous at , and , exist and is left continuous at , for with the norm . Then, is also a Banach space.
Lemma 2.1 (see ). is a solution of PBVP (1.1) if and only if is a fixed point of the following operator : where is the Green's function to the following periodic boundary value problem: here, . It is clear that Define where
The following nonlinear alternative principle of Leray-Schauder type in cone is very important for us.
Lemma 2.2 (see ). Assume that is a relatively open subset of a convex set in a Banach space . Let be a compact map with . Then, either (i) has a fixed point in , or,(ii)there is a and a such that .
3. Main Results
In this section, we establish the existence of positive solutions of PBVP (1.1).
Theorem 3.1. Assume that the following three hypothesis hold: ()there exists nonnegative functions and such that()there exists a positive number such that ()for the constant in (H2), there exists a function such that Then PBVP (1.1) has at least one positive periodic solution with , where
Proof. The existence of positive solutions is proved by using the Leray-Schauder alternative principle given in Lemma 2.2. We divide the rather long proof into six steps. Step 1. From (3.3), we may choose such that
Let . For . We consider the family of equations
For every and , define an operator as follows:
Then, we may verify that
To find a positive solution of (3.7) is equivalent to solve the following fixed point problem in :
then is a relatively open subset of the convex set .Step 2. We claim that any fixed point of (3.11) for any must satisfies .
Otherwise, we assume that is a solution of (3.11) for some such that . Note that . for all and . By the choice of , . Hence, for all , we get From (3.2), we have Consequently, for any fixed point of (3.11), by (3.8), (3.13), and (3.14), we have It follows from that So, we get from (3.1), (3.2), and (3.3) that Therefore, This is a contraction, and so the claim is proved.Step 3. From the above claim and the Leray-Schauder alternative principle, we know that operator (3.9) (with ) has a fixed point denoted by in . So, (3.7) (with ) has a positive solution with Step 4. We show that have a uniform positive lower bound; that is, there exists a constant , independent of , such that In fact, from (3.4), (3.8), (3.16), and (3.19), we get Step 5. We prove that for some constant . Equations (3.19) and (3.20) tell us that , so we may let Then, Step 6. Now, we pass the solution of the truncation equation (3.7) (with ) to that of the original equation (1.1). The fact that and (3.22) show that is a bounded and equi-continuous family on . Then, the Arzela-Ascoli Theorem guarantees that has a subsequence , converging uniformly on . From the fact and (3.20), satisfies for all . Moreover, also satisfies the following integral equation: Let , and we get where the uniform continuity of on is used. Therefore, is a positive solution of PBVP (1.1). This ends the proof.
4. An Example
Consider the following impulsive PBVP: where are constants. Then, PBVP (4.1) has at least one positive solution with .
To see this, we will apply Theorem 3.1.
Let then has a repulsive singularity at Denote Then, it is easy to say that (3.1), (3.2), and (3.3) hold. From (3.5), we know So, we may choose large enough to guarantee that (3.3) holds. Then, the result follows from Theorem 3.1.
The authors are grateful to the anonymous referees for their helpful suggestions and comments. Zhaocai Hao acknowledges support from NSFC (10771117), Ph.D. Programs Foundation of Ministry of Education of China (20093705120002), NSF of Shandong Province of China (Y2008A24), China Postdoctoral Science Foundation (20090451290), ShanDong Province Postdoctoral Foundation (200801001), and Foundation of Qufu Normal University (BSQD07026).
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Copyright © 2011 Zhaocai Hao and Tanggui Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.