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International Journal of Differential Equations
Volume 2011 (2011), Article ID 808175, 12 pages
http://dx.doi.org/10.1155/2011/808175
Research Article

The Existence of Positive Solutions for Singular Impulse Periodic Boundary Value Problem

Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China

Received 15 May 2011; Accepted 1 July 2011

Academic Editor: Jian-Ping Sun

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.

Abstract

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.

1. Introduction

Because of wide interests in physics and engineering, periodic boundary value problems have been investigated by many authors (see [119]). 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: 𝑢||(𝑡)+𝑀𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡𝐽,Δ𝑢𝑡=𝑡𝑘=𝐼𝑘𝑢𝑡𝑘,Δ𝑢||𝑡=𝑡𝑘=𝐽𝑘𝑢𝑡𝑘,𝑘=1,2,𝑙,𝑢(0)=𝑢(2𝜋),𝑢(0)=𝑢(2𝜋).(1.1) Here, 𝐽=[0,2𝜋], 0<𝑡1<𝑡2<<𝑡𝑙<2𝜋, 𝐽=𝐽{𝑡1,𝑡2,,𝑡𝑙}, 𝑀>0, 𝑓𝐶(𝐽×𝑅+,𝑅+), 𝐼𝑘𝐶(𝑅+,𝑅), 𝐽𝑘𝐶(𝑅+,𝑅+), 𝑅+=[0,+), 𝑅+=(0,+) with (1/𝑚)𝐽𝑘(𝑢)<𝐼𝑘(𝑢)<(1/𝑚)𝐽𝑘(𝑢), 𝑢𝑅+, 𝑚=𝑀. Δ𝑢|𝑡=𝑡𝑘=𝑢(𝑡+𝑘)𝑢(𝑡𝑘), Δ𝑢|𝑡=𝑡𝑘=𝑢(𝑡+𝑘)𝑢(𝑡𝑘), where 𝑢(𝑖)(𝑡+𝑘) and 𝑢(𝑖)(𝑡𝑘), 𝑖=0,1, respectively, denote the right and left limit of 𝑢(𝑖)(𝑡) at 𝑡=𝑡𝑘.

In [7], 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 problems𝑥[]𝑡(𝑡)+𝑎(𝑡)𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),a.e.𝑡0,𝑇1,,𝑡𝑝,||Δ𝑥𝑡=𝑡𝑘=𝐼𝑘𝑥𝑡𝑘𝑥,𝑘=1,,𝑝,(0)=𝑥(𝑇).(1.2) Jiang [5] has applied Krasnoselskii's fixed point theorem to establish the existence of positive solutions of problem 𝑥[],(𝑡)+𝑀𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡0,2𝜋𝑥(0)=𝑥(2𝜋),𝑥(0)=𝑥(2𝜋).(1.3) The work [5] proved that periodic boundary value problem (PBVP in short) (1.3) without singularity have at least one positive solutions provided 𝑓(𝑡,𝑥) is superlinear or sublinear at 𝑥=0+ and 𝑥=+. In [14], 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 [4] with the following limit inequality condition:(𝐴1)2𝜋𝑓0+𝑙𝑖=1𝐽0(𝑖)𝜎>2𝜋𝑀,2𝜋𝑓+𝑙𝑖=1𝐽(𝑖)𝜎>2𝜋𝑀,(1.4)(𝐴2)2𝜋𝑓0+𝑙𝑖=1𝐽0(𝑖)𝜎<2𝜋𝑀,2𝜋𝑓+𝑙𝑖=1𝐽(𝑖)𝜎<2𝜋𝑀.(1.5)

Nieto [10] 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 𝑓 involved in above papers [5, 7, 10, 14] does not have singularity. Xiao et al. [16] investigate the multiple positive solutions of singular boundary value problem for second-order impulsive singular differential equations on the halfline, where the function 𝑓(𝑡,𝑢) is singular only at 𝑡=0 and/or 𝑡=1. Reference [19] studied PBVP (1.3), where the function 𝑓 has singularity at 𝑥=0. The authors present the existence of multiple positive solutions via the Krasnoselskii's fixed point theorem under the following conditions.(𝐴1)There exist nonnegative valued 𝜉(𝑥),𝜂(𝑥)𝐶((0,)) and 𝑃(𝑡),𝑄(𝑡)𝐿1[0,2𝜋] such that []0𝑓(𝑡,𝑥)𝑃(𝑡)𝜉(𝑥)+𝑄(𝑡)𝜂(𝑥),a.e.(𝑡,𝑥)0,2𝜋×(0,),sup𝑥(0,)𝑥02𝜋𝑃(𝑡)𝑑𝑡𝜉(𝑥)/𝜂(𝑥)+02𝜋𝜂𝛿𝑄(𝑡)𝑑𝑡𝑗𝑡>𝐵𝑗,(1.6)

where 𝜂(𝑥) is nonincreasing and 𝜉(𝑥)/𝜂(𝑥) is nondecreasing on (0,),(𝐴2)lim𝑡0+infmin02𝜋𝑓(𝑥,𝑤)𝑑𝑥𝛿𝑗𝑡𝑤𝑡𝑡>1𝐴𝑗,(1.7)(𝐴3)lim𝑡+infmin02𝜋𝑓(𝑥,𝑤)𝑑𝑥𝛿𝑗𝑡𝑤𝑡𝑡>1𝐴𝑗.(1.8) Here, 𝛿𝑗,𝐴𝑗,𝐵𝑗 are some constants.

In this paper, the nonlinear term 𝑓(𝑡,𝑢) is singular at 𝑢=0, 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 𝜂, 𝜉/𝜂 used in [19]. 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.

2. Preliminaries

Consider the space 𝑃𝐶[𝐽,𝑅]={𝑢𝑢 is a map from 𝐽 into 𝑅 such that 𝑢(𝑡) is continuous at 𝑡𝑡𝑘, left continuous at 𝑡=𝑡𝑘, and 𝑢(𝑡+𝑘) exists, for 𝑘=1,2,𝑙.}. It is easy to say that 𝑃𝐶[𝐽,𝑅] is a Banach space with the norm 𝑢𝑝𝑐=sup𝑡𝐽|𝑢(𝑡)|. Let 𝑃𝐶1[𝐽,𝑅]={𝑢𝑃𝐶[𝐽,𝑅]𝑢(𝑡) exists at 𝑡𝑡𝑘 and is continuous at 𝑡𝑡𝑘, and 𝑢(𝑡+𝑘), 𝑢(𝑡𝑘) exist and 𝑢(𝑡) is left continuous at 𝑡=𝑡𝑘, for 𝑘=1,2,𝑙.} with the norm 𝑢𝑝𝑐1=max{𝑢𝑝𝑐,𝑢𝑝𝑐}. Then, 𝑃𝐶1[𝐽,𝑅] is also a Banach space.

Lemma 2.1 (see [15]). 𝑢𝑃𝐶1(𝐽,𝑅)𝐶2(𝐽,𝑅) is a solution of PBVP (1.1) if and only if 𝑢𝑃𝐶(𝐽) is a fixed point of the following operator 𝑇: 𝑇𝑢(𝑡)=02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘,(2.1) where 𝐺(𝑡,𝑠) is the Green's function to the following periodic boundary value problem: 𝑢+𝑀𝑢=0,𝑢(0)=𝑢(2𝜋),𝑢(0)=𝑢1(2𝜋),𝐺(𝑡,𝑠)=Γ𝑒𝑚(𝑡𝑠)+𝑒𝑚(2𝜋𝑡+𝑠)𝑒,0𝑠𝑡2𝜋,𝑚(𝑠𝑡)+𝑒𝑚(2𝜋𝑠+𝑡),0𝑡𝑠2𝜋,(2.2) here, Γ=2𝑚(𝑒2𝑚𝜋1). It is clear that 2𝑒𝑚𝜋Γ𝑒=𝐺(𝜋)𝐺(𝑡,𝑠)𝐺(0)=2𝑚𝜋+1Γ.(2.3) Define []𝐾=𝑢𝑃𝐶𝐽,𝑅𝑢(𝑡)𝜎𝑢𝑝𝑐,𝑡𝐽,(2.4) where 1𝜎=𝑒2𝑚𝜋.(2.5)

The following nonlinear alternative principle of Leray-Schauder type in cone is very important for us.

Lemma 2.2 (see [4]). Assume that Ω is a relatively open subset of a convex set 𝐾 in a Banach space 𝑃𝐶[𝐽,𝑅]. Let 𝑇Ω𝐾 be a compact map with 0Ω. Then, either (i)𝑇 has a fixed point in Ω, or,(ii)there is a 𝑢𝜕Ω and a 𝜆<1 such that 𝑢=𝜆𝑇𝑢.

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: (𝐻1)there exists nonnegative functions 𝜉(𝑢),𝜂(𝑢),𝛾(𝑢)𝐶(0,+) and 𝑝(𝑡),𝑞(𝑡)𝐿1([0,2𝜋]) such that[]𝑓(𝑡,𝑢)𝑝(𝑡)𝜉(𝑢)+𝑞(𝑡)𝜂(𝑢),(𝑡,𝑢)0,2𝜋×(0,),(3.1)max1𝑘𝑙𝐽𝑘[](𝑢)𝛾(𝑢),(𝑡,𝑢)0,2𝜋×(0,+),(3.2)(𝐻2)there exists a positive number 𝑟>0 such that 𝐴2max[]𝑥𝜎𝑟,𝑟𝜉(𝑥)02𝜋𝑝(𝑠)𝑑𝑠+max[]𝑥𝜎𝑟,𝑟𝜂(𝑥)02𝜋𝑞(𝑠)𝑑𝑠+𝐴𝑙𝛾(𝑟)<𝑟,(3.3)(𝐻3)for the constant 𝑟 in (H2), there exists a function Φ𝑟>0 such that 𝑓(𝑡,𝑢)>Φ𝑟[]],(𝑡),(𝑡,𝑢)0,2𝜋×(0,𝑟02𝜋Φ𝑟(𝑠)𝑑𝑠>0.(3.4) Then PBVP (1.1) has at least one positive periodic solution with 0<𝑢<𝑟, where 𝑒𝐴=2𝑚𝜋+1𝑚𝑒2𝑚𝜋=𝑒12𝜋𝑀+1𝑀𝑒2𝜋𝑀1.(3.5)

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 𝑛0{1,2,} such that 𝐴2max[]𝑥𝜎𝑟,𝑟𝜉(𝑥)02𝜋𝑝(𝑠)𝑑𝑠+max[]𝑥𝜎𝑟,𝑟𝜂(𝑥)02𝜋1𝑞(𝑠)𝑑𝑠+𝐴𝑙𝛾(𝑟)+𝑛0<𝑟.(3.6) Let 𝑁0={𝑛0,𝑛0+1,}. For 𝑛𝑁0. We consider the family of equations 𝑢(𝑡)+𝑀𝑢(𝑡)=𝜆𝑓𝑛𝑀(𝑡,𝑢(𝑡))+𝑛,𝑡𝐽,||Δ𝑢𝑡=𝑡𝑘=𝐼𝑘𝑢𝑡𝑘,Δ𝑢||𝑡=𝑡𝑘=𝐽𝑘𝑢𝑡𝑘𝑢,𝑘=1,2,𝑙,(0)=𝑢(2𝜋),𝑢(0)=𝑢(2𝜋),(3.7) where 𝜆[0,1] and 𝑓𝑛1(𝑡,𝑢)=𝑓𝑡,max𝑢,𝑛[,(𝑡,𝑢)𝐽×0,+).(3.8) For every 𝜆 and 𝑛𝑁0, define an operator as follows: 𝑇𝜆,𝑛𝑢(𝑡)=𝜆02𝜋𝐺(𝑡,𝑠)𝑓𝑛(𝑠,𝑢(𝑠))𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘,𝑢𝐾.(3.9) Then, we may verify that 𝑇𝜆,𝑛𝐾𝐾iscompletelycontinuous.(3.10) To find a positive solution of (3.7) is equivalent to solve the following fixed point problem in 𝑃𝐶[𝐽,𝑅]: 𝑢=𝑇𝜆,𝑛1𝑢+𝑛.(3.11) Let Ω={𝑥𝐾𝑥<𝑟},(3.12) then Ω is a relatively open subset of the convex set 𝐾.Step 2. We claim that any fixed point 𝑢 of (3.11) for any 𝜆[0,1) must satisfies 𝑢𝑟.
Otherwise, we assume that 𝑢 is a solution of (3.11) for some 𝜆[0,1) such that 𝑢=𝑟. Note that 𝑓𝑛(𝑡,𝑢)0. 𝑢(𝑡)1/𝑛 for all 𝑡𝐽 and 𝑟𝑢(𝑡)(1/𝑛)+𝜎𝑢1/𝑛. By the choice of 𝑛0, 1/𝑛1/𝑛0<𝑟. Hence, for all 𝑡𝐽, we get 1𝑟𝑢(𝑡)𝑛1+𝜎𝑢𝑛1𝑛|||1+𝜎𝑢𝑛|||1𝑛1+𝜎𝑟𝑛>𝜎𝑟.(3.13) From (3.2), we have 𝐽𝑘𝑢𝑡𝑘max1𝑘𝑙𝐽𝑘𝑢𝑡𝑘𝑢𝑡𝛾𝑘𝛾(𝑟).(3.14) Consequently, for any fixed point 𝑢 of (3.11), by (3.8), (3.13), and (3.14), we have 𝑢(𝑡)=𝜆02𝜋𝐺(𝑡,𝑠)𝑓𝑛(𝑠,𝑢(𝑠))𝑑𝑠+𝑘=1𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘+1𝑛02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘+1𝑛02𝜋+1𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠Γ𝑡𝑘𝑡𝑒𝑚(𝑡𝑡𝑘)+𝑒𝑚(2𝜋𝑡+𝑡𝑘)𝐽𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘𝑡)+𝑒𝑚(2𝜋𝑡𝑘+𝑡)𝐽𝑘𝑢𝑡𝑘+𝑡𝑘𝑡𝑒𝑚(𝑡𝑡𝑘)+𝑒𝑚(2𝜋𝑡+𝑡𝑘)𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘𝑡)𝑒𝑚(2𝜋𝑡𝑘+𝑡)𝑚𝐼𝑘𝑢𝑡𝑘+1𝑛=02𝜋+1𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠Γ𝑡𝑘𝑡𝑒𝑚(𝑡𝑡𝑘)𝐽𝑘𝑢𝑡𝑘𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘𝑡𝑒𝑚(2𝜋𝑡+𝑡𝑘)𝐽𝑘𝑢𝑡𝑘+𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘𝑡)𝐽𝑘𝑢𝑡𝑘+𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(2𝜋𝑡𝑘+𝑡)𝐽𝑘𝑢𝑡𝑘𝑚𝐼𝑘𝑢𝑡𝑘+1𝑛.(3.15) It follows from (1/𝑚)𝐽𝑘(𝑢)<𝐼𝑘(𝑢)<(1/𝑚)𝐽𝑘(𝑢) that 𝐽𝑘𝑢𝑡𝑘𝑚𝐼𝑘𝑢𝑡𝑘>0,𝐽𝑘𝑢𝑡𝑘+𝑚𝐼𝑘𝑢𝑡𝑘>0.(3.16) So, we get from (3.1), (3.2), and (3.3) that 𝑢(𝑡)02𝜋2𝑒𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+2𝑚𝜋+1Γ𝑙𝑘=1𝐽𝑘𝑢𝑡𝑘+1𝑛02𝜋[]1𝐺(𝑡,𝑠)𝑝(𝑠)𝜉(𝑢(𝑠))+𝑞(𝑠)𝜂(𝑢(𝑠))𝑑𝑠+𝐴𝑙𝛾(𝑟)+𝑛0𝐴202𝜋𝑝(𝑠)𝑑𝑠max[]𝑥𝜎𝑟,𝑟𝜉(𝑥)+02𝜋𝑞(𝑠)𝑑𝑠max[]𝑥𝜎𝑟,𝑟1𝜉(𝑥)+𝐴𝑙𝛾(𝑟)+𝑛0.(3.17) Therefore, 𝐴𝑟=𝑢202𝜋𝑝(𝑠)𝑑𝑠max[]𝑥𝜎𝑟,𝑟𝜉(𝑥)+02𝜋𝑞(𝑠)𝑑𝑠max[]𝑥𝜎𝑟,𝑟1𝜉(𝑥)+𝐴𝑙𝛾(𝑟)+𝑛0<𝑟.(3.18) 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 𝜆=1) has a fixed point denoted by 𝑢𝑛 in Ω. So, (3.7) (with 𝜆=1) has a positive solution 𝑢𝑛 with 𝑢𝑛<𝑟,𝑢𝑛1(𝑡)𝑛,𝑡𝐽.(3.19)Step 4. We show that {𝑢𝑛} have a uniform positive lower bound; that is, there exists a constant 𝛿>0, independent of 𝑛𝑁0, such that min𝑡𝑢𝑛(𝑡)𝛿.(3.20) In fact, from (3.4), (3.8), (3.16), and (3.19), we get 𝑢𝑛(𝑡)=02𝜋𝐺(𝑡,𝑠)𝑓𝑛𝑠,𝑢𝑛(𝑠)𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑛𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑡𝑘+1𝑛=02𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢𝑛(𝑠)𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑛𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑡𝑘+1𝑛02𝜋𝐺(𝑡,𝑠)Φ𝑟+1(𝑠)𝑑𝑠Γ𝑡𝑘𝑡𝑒𝑚(𝑡𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘𝑡𝑒𝑚(2𝜋𝑡+𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘𝑡)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(2𝜋𝑡𝑘+𝑡)𝐽𝑘𝑢𝑛𝑡𝑘𝑚𝐼𝑘𝑢𝑛𝑡𝑘+1𝑛02𝜋𝐺(𝑡,𝑠)Φ𝑟(𝑠)𝑑𝑠2𝑒𝑚𝜋Γ02𝜋Φ𝑟(𝑠)𝑑𝑠=𝛿>0.(3.21)Step 5. We prove that 𝑢𝑛<𝐻,𝑛𝑛0(3.22) for some constant 𝐻>0. Equations (3.19) and (3.20) tell us that 𝛿𝑢𝑛(𝑡)𝑟, so we may let 𝑀1=max[]𝑡𝐽,𝑢𝛿,𝑟𝑓(𝑡,𝑢),𝑀2=max𝑡,𝑠𝐽||𝐺𝑡||(𝑡,𝑠),𝑀3=max[]𝑙𝑢𝛿,𝑟𝑘=1𝐽𝑘(𝑢).(3.23) Then, 𝑢𝑛=sup𝑡𝐽||𝑢𝑛||(𝑡)=sup𝑡𝐽|||||02𝜋𝐺𝑡(𝑡,𝑠)𝑓𝑠,𝑢𝑛(𝑠)𝑑𝑠+𝑙𝑘=1𝐺𝑡𝑡,𝑡𝑘𝐽𝑘𝑢𝑛𝑡𝑘+𝑙𝑘=1𝜕𝜕𝑡𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑡𝑘||||=sup𝑡𝐽||||02𝜋𝐺𝑡(𝑡,𝑠)𝑓𝑠,𝑢𝑛+𝑚(𝑠)𝑑𝑠Γ𝑡𝑘𝑡𝑒𝑚(𝑡𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘𝑚𝐼𝑘𝑢𝑛𝑡𝑘𝑡𝑘𝑡𝑒𝑚(2𝜋𝑡+𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘𝑡)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(2𝜋𝑡𝑘+𝑡)𝐽𝑘𝑢𝑛𝑡𝑘𝑚𝐼𝑘𝑢𝑛𝑡𝑘|||||sup𝑡𝐽02𝜋||𝐺𝑡||𝑓(𝑡,𝑠)𝑠,𝑢𝑛𝑒(𝑠)𝑑𝑠+2𝑚2𝑚𝜋+1Γ𝑙𝑘=1𝐽𝑘𝑢𝑛𝑡𝑘2𝜋𝑀1𝑀2+𝑒2𝑚2𝑚𝜋+1Γ𝑀3=𝐻.(3.24)Step 6. Now, we pass the solution 𝑢𝑛 of the truncation equation (3.7) (with 𝜆=1) to that of the original equation (1.1). The fact that 𝑢𝑛<𝑟 and (3.22) show that {𝑢𝑛}𝑛𝑁0 is a bounded and equi-continuous family on [0,2𝜋]. Then, the Arzela-Ascoli Theorem guarantees that {𝑢𝑛}𝑛𝑁0 has a subsequence {𝑢𝑛𝑗}𝑗𝑁, converging uniformly on [0,2𝜋]. From the fact 𝑢𝑛<𝑟 and (3.20), 𝑢 satisfies 𝛿𝑢(𝑡)𝑟 for all 𝑡𝐽. Moreover, 𝑢𝑛𝑗 also satisfies the following integral equation: 𝑢𝑛𝑗(𝑡)=02𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢𝑛𝑗+(𝑠)𝑑𝑠𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑛𝑗𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑗𝑡𝑘+1𝑛𝑗.(3.25) Let 𝑗+, and we get 𝑢(𝑡)=02𝜋+𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘,(3.26) where the uniform continuity of 𝑓(𝑡,𝑢) on 𝐽×[𝛿,𝑟] is used. Therefore, 𝑢 is a positive solution of PBVP (1.1). This ends the proof.

4. An Example

Consider the following impulsive PBVP:𝑢(𝑡)+𝑀𝑢(𝑡)=𝑡2||||1+sin𝑢𝑢3/2+𝑡(1+|cos𝑢|),𝑡𝐽,||Δ𝑢𝑡=𝑡𝑘=𝑐min1,𝑐2,,𝑐𝑙2𝑀𝑢𝑡𝑘,Δ𝑢||𝑡=𝑡𝑘=𝑐𝑘𝑢𝑡𝑘,𝑘=1,2,,𝑙,𝑢(0)=𝑢(2𝜋),𝑢(0)=𝑢(2𝜋),(4.1) where 𝑐𝑘>0 are constants. Then, PBVP (4.1) has at least one positive solution 𝑢 with 0<𝑢<1.

To see this, we will apply Theorem 3.1.

Let 𝑓(𝑡,𝑢)=𝑡2||||1+sin𝑢𝑢3/2+𝑡(1+|cos𝑢|),(4.2) then 𝑓(𝑡,𝑢) has a repulsive singularity at 𝑢=0lim𝑢0+𝑓(𝑡,𝑢)=+,uniformalyin𝑡.(4.3) Denote 𝑝(𝑡)=𝑡2||||,𝑞(𝑡)=𝑡,𝜉(𝑢)=1+sin𝑢𝑢3/2𝑐,𝜂(𝑢)=1+|cos𝑢|,𝛾(𝑢)=max1,𝑐2,,𝑐𝑙𝑢,𝑟=1,Φ𝑟(𝑡)=𝑡+𝑡2.(4.4) Then, it is easy to say that (3.1), (3.2), and (3.3) hold. From (3.5), we knowlim𝑀+𝐴=lim𝑀+𝑒2𝜋𝑀+1𝑀𝑒2𝜋𝑀1=0.(4.5) So, we may choose 𝑀 large enough to guarantee that (3.3) holds. Then, the result follows from Theorem 3.1.

Remark 4.1. Functions 𝜉, 𝜂 in example (4.1) do not have the monotonicity required as in [19]. So, the results of [19] cannot be applied to PBVP (4.1).

Acknowledgments

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).

References

  1. M. Benchohra, J. Henderson, and S. Ntouyas, “Impulsive differential equations and inclusions,” Contemporary Mathematics and Its Applications, vol. 2, pp. 1–380, 2006. View at Google Scholar · View at Scopus
  2. J. Chu and Z. Zhou, “Positive solutions for singular non-linear third-order periodic boundary value problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 64, no. 7, pp. 1528–1542, 2006. View at Publisher · View at Google Scholar · View at Scopus
  3. W. Ding and M. Han, “Periodic boundary value problem for the second order impulsive functional differential equations,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 709–726, 2004. View at Publisher · View at Google Scholar · View at Scopus
  4. D. J. Guo, J. Sun, and Z. Liu, Nonlinear Ordinary Differential Equations Functional Technologies, Shan-Dong Science Technology, 1995.
  5. D. Jiang, “On the existence of positive solutions to second order periodic BVPs,” Acta Mathematica Scientia, vol. 18, pp. 31–35, 1998. View at Google Scholar
  6. Y. H. Lee and X. Liu, “Study of singular boundary value problems for second order impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 159–176, 2007. View at Publisher · View at Google Scholar · View at Scopus
  7. Y. Liu, “Positive solutions of periodic boundary value problems for nonlinear first-order impulsive differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 5, pp. 2106–2122, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. Y. Liu, “Multiple solutions of periodic boundary value problems for first order differential equations,” Computers and Mathematics with Applications, vol. 54, no. 1, pp. 1–8, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. J. J. Nieto and R. Rodríguez-López, “Boundary value problems for a class of impulsive functional equations,” Computers and Mathematics with Applications, vol. 55, no. 12, pp. 2715–2731, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. J. J. Nieto, “Variational formulation of a damped Dirichlet impulsive problem,” Applied Mathematics Letters, vol. 23, pp. 940–942, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. J. J. Nieto and D. O’Regan, “Singular boundary value problems for ordinary differential equations,” Boundary Value Problems, vol. 2009, Article ID 895290, 2 pages, 2009. View at Publisher · View at Google Scholar
  12. S. Peng, “Positive solutions for first order periodic boundary value problem,” Applied Mathematics and Computation, vol. 158, no. 2, pp. 345–351, 2004. View at Publisher · View at Google Scholar · View at Scopus
  13. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
  14. Y. Tian, D. Jiang, and W. Ge, “Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 123–132, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. Z. L. Wei, “Periodic boundary value problem for second order impulsive integro differential equations of mixed type in Banach space,” Journal of Mathematical Analysis and Applications, vol. 195, pp. 214–229, 1995. View at Google Scholar
  16. J. Xiao, J. J. Nieto, and Z. Luo, “Multiple positive solutions of the singular boundary value problem for second-order impulsive differential equations on the half-line,” Boundary Value Problems, vol. 2010, Article ID 281908, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. X. Yang and J. Shen, “Periodic boundary value problems for second-order impulsive integro-differential equations,” Journal of Computational and Applied Mathematics, vol. 209, no. 2, pp. 176–186, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. Q. Yao, “Positive solutions of nonlinear second-order periodic boundary value problems,” Applied Mathematics Letters, vol. 20, no. 5, pp. 583–590, 2007. View at Publisher · View at Google Scholar · View at Scopus
  19. Z. Zhang and J. Wang, “On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,” Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 99–107, 2003. View at Google Scholar · View at Scopus