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 [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: âˆ’ğ‘¢î…žî…ž||(𝑡)+𝑀𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈𝐽′,Δ𝑢𝑡=𝑡𝑘=𝐼𝑘𝑢𝑡𝑘,âˆ’Î”ğ‘¢î…ž||𝑡=𝑡𝑘=𝐽𝑘𝑢𝑡𝑘,𝑘=1,2,…𝑙,𝑢(0)=𝑢(2𝜋),ğ‘¢î…ž(0)=ğ‘¢î…ž(2𝜋).(1.1) Here, 𝐽=[0,2𝜋], 0<𝑡1<𝑡2<⋯<𝑡𝑙<2𝜋, 𝐽′=𝐽⧵{𝑡1,𝑡2,…,𝑡𝑙}, 𝑀>0, 𝑓∈𝐶(𝐽×𝑅+,𝑅+), 𝐼𝑘∈𝐶(𝑅+,𝑅), 𝐽𝑘∈𝐶(𝑅+,𝑅+), 𝑅+=[0,+∞), 𝑅+=(0,+∞) with −(1/𝑚)𝐽𝑘(𝑢)<𝐼𝑘(𝑢)<(1/𝑚)𝐽𝑘(𝑢), 𝑢∈𝑅+, √𝑚=𝑀. Δ𝑢|𝑡=𝑡𝑘=𝑢(𝑡+𝑘)−𝑢(𝑡−𝑘), Î”ğ‘¢î…ž|𝑡=𝑡𝑘=𝑢′(𝑡+𝑘)−𝑢′(𝑡−𝑘), where 𝑢(𝑖)(𝑡+𝑘) and 𝑢(𝑖)(𝑡−𝑘), 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 problemsğ‘¥î…ž[]⧵𝑡(𝑡)+ğ‘Ž(𝑡)𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),a.e.𝑡∈0,𝑇1,…,𝑡𝑝,||Δ𝑥𝑡=𝑡𝑘=𝐼𝑘𝑥𝑡𝑘𝑥,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 ğ‘¥î…žî…ž[],(𝑡)+𝑀𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈0,2𝜋𝑥(0)=𝑥(2𝜋),ğ‘¥î…ž(0)=ğ‘¥î…ž(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+𝑙𝑖=1𝐽0(𝑖)ğœŽ>2𝜋𝑀,2ğœ‹ğ‘“âˆž+𝑙𝑖=1ğ½âˆžîƒ­(𝑖)ğœŽ>2𝜋𝑀,(1.4)(𝐴2)2𝜋𝑓0+𝑙𝑖=1𝐽0(𝑖)ğœŽ<2𝜋𝑀,2ğœ‹ğ‘“âˆž+𝑙𝑖=1ğ½âˆžîƒ­(𝑖)ğœŽ<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.(ğ´î…ž1)There exist nonnegative valued 𝜉(𝑥),𝜂(𝑥)∈𝐶((0,∞)) and 𝑃(𝑡),𝑄(𝑡)∈𝐿1[0,2𝜋] such that []0≤𝑓(𝑡,𝑥)≤𝑃(𝑡)𝜉(𝑥)+𝑄(𝑡)𝜂(𝑥),a.e.(𝑡,𝑥)∈0,2𝜋×(0,∞),sup𝑥∈(0,∞)âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¥î‚€âˆ«02𝜋∫𝑃(𝑡)𝑑𝑡𝜉(𝑥)/𝜂(𝑥)+02𝜋𝜂𝛿𝑄(𝑡)ğ‘‘ğ‘¡ğ‘—ğ‘¡î€¸âŽ«âŽªâŽ¬âŽªâŽ­>𝐵𝑗,(1.6)

where 𝜂(𝑥) is nonincreasing and 𝜉(𝑥)/𝜂(𝑥) is nondecreasing on (0,∞),(ğ´î…ž2)lim𝑡→0+∫infmin02𝜋𝑓(𝑥,𝑤)𝑑𝑥∶𝛿𝑗𝑡≤𝑤≤𝑡𝑡>1𝐴𝑗,(1.7)(ğ´î…ž3)lim𝑡→+∞∫infmin02𝜋𝑓(𝑥,𝑤)𝑑𝑥∶𝛿𝑗𝑡≤𝑤≤𝑡𝑡>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]={𝑢∶𝑢 is a map from 𝐽 into 𝑅 such that 𝑢(𝑡) is continuous at 𝑡≠𝑡𝑘, 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 ‖𝑢‖𝑝𝑐=sup𝑡∈𝐽|𝑢(𝑡)|. Let 𝑃𝐶1[𝐽,𝑅]={𝑢∈𝑃𝐶[𝐽,𝑅]∶𝑢′(𝑡) exists at 𝑡≠𝑡𝑘 and is continuous at 𝑡≠𝑡𝑘, and ğ‘¢î…ž(𝑡+𝑘), ğ‘¢î…ž(𝑡−𝑘) exist and ğ‘¢î…ž(𝑡) is left continuous at 𝑡=𝑡𝑘, for 𝑘=1,2,…𝑙.} with the norm ‖𝑢‖𝑝𝑐1=max{‖𝑢‖𝑝𝑐,‖𝑢′‖𝑝𝑐}. Then, 𝑃𝐶1[𝐽,𝑅] is also a Banach space.

Lemma 2.1 (see [15]). 𝑢∈𝑃𝐶1(𝐽,𝑅)∩𝐶2(𝐽′,𝑅) is a solution of PBVP (1.1) if and only if 𝑢∈𝑃𝐶(𝐽) is a fixed point of the following operator 𝑇: 𝑇𝑢(𝑡)=02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘,(2.1) where 𝐺(𝑡,𝑠) is the Green's function to the following periodic boundary value problem: âˆ’ğ‘¢î…žî…ž+𝑀𝑢=0,𝑢(0)=𝑢(2𝜋),ğ‘¢î…ž(0)=ğ‘¢î…ž1(2𝜋),𝐺(𝑡,𝑠)∶=Î“âŽ§âŽªâŽ¨âŽªâŽ©âŽ§âŽªâŽ¨âŽªâŽ©ğ‘’ğ‘š(𝑡−𝑠)+𝑒𝑚(2𝜋−𝑡+𝑠)𝑒,0≤𝑠≤𝑡≤2𝜋,𝑚(𝑠−𝑡)+𝑒𝑚(2𝜋−𝑠+𝑡),0≤𝑡≤𝑠≤2𝜋,(2.2) here, Γ=2𝑚(𝑒2𝑚𝜋−1). It is clear that 2𝑒𝑚𝜋Γ𝑒=𝐺(𝜋)≤𝐺(𝑡,𝑠)≤𝐺(0)=2𝑚𝜋+1Γ.(2.3) Define []𝐾=𝑢∈𝑃𝐶𝐽,𝑅∶𝑢(𝑡)â‰¥ğœŽâ€–ğ‘¢â€–ğ‘ğ‘î€¾,𝑡∈𝐽,(2.4) where 1ğœŽ=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 𝑇∶Ω→𝐾 be a compact map with 0∈Ω. 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),ğ‘ž(𝑡)∈𝐿1([0,2𝜋]) such that[]𝑓(𝑡,𝑢)≤𝑝(𝑡)𝜉(𝑢)+ğ‘ž(𝑡)𝜂(𝑢),(𝑡,𝑢)∈0,2𝜋×(0,∞),(3.1)max1≤𝑘≤𝑙𝐽𝑘[](𝑢)≤𝛾(𝑢),(𝑡,𝑢)∈0,2𝜋×(0,+∞),(3.2)(𝐻2)there exists a positive number 𝑟>0 such that 𝐴2max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟𝜉(𝑥)02𝜋𝑝(𝑠)𝑑𝑠+max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟𝜂(𝑥)02ğœ‹î‚¼ğ‘ž(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+1𝑚𝑒2𝑚𝜋=𝑒−1√2𝜋𝑀+1√𝑀𝑒√2𝜋𝑀−1.(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 𝐴2max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟𝜉(𝑥)02𝜋𝑝(𝑠)𝑑𝑠+max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟𝜂(𝑥)02𝜋1ğ‘ž(𝑠)𝑑𝑠+𝐴𝑙𝛾(𝑟)+𝑛0<𝑟.(3.6) Let 𝑁0={𝑛0,𝑛0+1,…}. For 𝑛∈𝑁0. We consider the family of equations âˆ’ğ‘¢î…žî…ž(𝑡)+𝑀𝑢(𝑡)=𝜆𝑓𝑛𝑀(𝑡,𝑢(𝑡))+𝑛,ğ‘¡âˆˆğ½î…ž,||Δ𝑢𝑡=𝑡𝑘=𝐼𝑘𝑢𝑡𝑘,âˆ’Î”ğ‘¢î…ž||𝑡=𝑡𝑘=𝐽𝑘𝑢𝑡𝑘𝑢,𝑘=1,2,…𝑙,(0)=𝑢(2𝜋),ğ‘¢î…ž(0)=ğ‘¢î…ž(2𝜋),(3.7) where 𝜆∈[0,1] and 𝑓𝑛1(𝑡,𝑢)=𝑓𝑡,max𝑢,𝑛[,(𝑡,𝑢)∈𝐽×0,+∞).(3.8) For every 𝜆 and 𝑛∈𝑁0, define an operator as follows: 𝑇𝜆,𝑛𝑢(𝑡)=𝜆02𝜋𝐺(𝑡,𝑠)𝑓𝑛(𝑠,𝑢(𝑠))𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘,𝑢∈𝐾.(3.9) Then, we may verify that 𝑇𝜆,𝑛∶𝐾⟶𝐾iscompletelycontinuous.(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={𝑥∈𝐾∶‖𝑥‖<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 ‖𝑢‖≠𝑟.
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 𝑡∈𝐽 and 𝑟≥𝑢(𝑡)≥(1/𝑛)+ğœŽâ€–ğ‘¢âˆ’1/𝑛‖. By the choice of 𝑛0, 1/𝑛≤1/𝑛0<𝑟. Hence, for all 𝑡∈𝐽, we get 1𝑟≥𝑢(𝑡)≥𝑛‖‖‖1+ğœŽğ‘¢âˆ’ğ‘›â€–â€–â€–â‰¥1𝑛|||1+ğœŽâ€–ğ‘¢â€–âˆ’ğ‘›|||≥1𝑛1+ğœŽğ‘Ÿâˆ’ğ‘›î‚>ğœŽğ‘Ÿ.(3.13) From (3.2), we have 𝐽𝑘𝑢𝑡𝑘≤max1≤𝑘≤𝑙𝐽𝑘𝑢𝑡𝑘𝑢𝑡≤𝛾𝑘≤𝛾(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=1𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘+1𝑛≤02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘+1𝑛≤02𝜋+1𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠Γ𝑡𝑘≤𝑡𝑒𝑚(𝑡−𝑡𝑘)+𝑒𝑚(2𝜋−𝑡+𝑡𝑘)𝐽𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘−𝑡)+𝑒𝑚(2𝜋−𝑡𝑘+𝑡)𝐽𝑘𝑢𝑡𝑘+𝑡𝑘≤𝑡−𝑒𝑚(𝑡−𝑡𝑘)+𝑒𝑚(2𝜋−𝑡+𝑡𝑘)𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘−𝑡)−𝑒𝑚(2𝜋−𝑡𝑘+𝑡)𝑚𝐼𝑘𝑢𝑡𝑘+1𝑛=02𝜋+1𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠Γ𝑡𝑘≤𝑡𝑒𝑚(𝑡−𝑡𝑘)𝐽𝑘𝑢𝑡𝑘−𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘≤𝑡𝑒𝑚(2𝜋−𝑡+𝑡𝑘)𝐽𝑘𝑢𝑡𝑘+𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘−𝑡)𝐽𝑘𝑢𝑡𝑘+𝑚𝐼𝑘𝑢𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(2𝜋−𝑡𝑘+𝑡)𝐽𝑘𝑢𝑡𝑘−𝑚𝐼𝑘𝑢𝑡𝑘+1𝑛.(3.15) It follows from −(1/𝑚)𝐽𝑘(𝑢)<𝐼𝑘(𝑢)<(1/𝑚)𝐽𝑘(𝑢) that 𝐽𝑘𝑢𝑡𝑘−𝑚𝐼𝑘𝑢𝑡𝑘>0,𝐽𝑘𝑢𝑡𝑘+𝑚𝐼𝑘𝑢𝑡𝑘>0.(3.16) So, we get from (3.1), (3.2), and (3.3) that 𝑢(𝑡)≤02𝜋2𝑒𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+2𝑚𝜋+1Γ𝑙𝑘=1𝐽𝑘𝑢𝑡𝑘+1𝑛≤02𝜋[]1𝐺(𝑡,𝑠)𝑝(𝑠)𝜉(𝑢(𝑠))+ğ‘ž(𝑠)𝜂(𝑢(𝑠))𝑑𝑠+𝐴𝑙𝛾(𝑟)+𝑛0≤𝐴202𝜋𝑝(𝑠)𝑑𝑠max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟𝜉(𝑥)+02ğœ‹ğ‘ž(𝑠)𝑑𝑠max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟1𝜉(𝑥)+𝐴𝑙𝛾(𝑟)+𝑛0.(3.17) Therefore, 𝐴𝑟=‖𝑢‖≤202𝜋𝑝(𝑠)𝑑𝑠max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟𝜉(𝑥)+02ğœ‹ğ‘ž(𝑠)𝑑𝑠max[]ğ‘¥âˆˆğœŽğ‘Ÿ,𝑟1𝜉(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,𝑡∈𝐽.(3.19)Step 4. We show that {𝑢𝑛} have a uniform positive lower bound; that is, there exists a constant 𝛿>0, independent of 𝑛∈𝑁0, such that min𝑡𝑢𝑛(𝑡)≥𝛿.(3.20) In fact, from (3.4), (3.8), (3.16), and (3.19), we get 𝑢𝑛(𝑡)=02𝜋𝐺(𝑡,𝑠)𝑓𝑛𝑠,𝑢𝑛(𝑠)𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑛𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑡𝑘+1𝑛=02𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢𝑛(𝑠)𝑑𝑠+𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑛𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑡𝑘+1𝑛≥02𝜋𝐺(𝑡,𝑠)Φ𝑟+1(𝑠)𝑑𝑠Γ𝑡𝑘≤𝑡𝑒𝑚(𝑡−𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘−𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘≤𝑡𝑒𝑚(2𝜋−𝑡+𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘−𝑡)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(2𝜋−𝑡𝑘+𝑡)𝐽𝑘𝑢𝑛𝑡𝑘−𝑚𝐼𝑘𝑢𝑛𝑡𝑘+1𝑛≥02𝜋𝐺(𝑡,𝑠)Φ𝑟≥(𝑠)𝑑𝑠2𝑒𝑚𝜋Γ02𝜋Φ𝑟(𝑠)𝑑𝑠∶=𝛿>0.(3.21)Step 5. We prove that â€–â€–ğ‘¢î…žğ‘›â€–â€–<𝐻,𝑛≥𝑛0(3.22) for some constant 𝐻>0. Equations (3.19) and (3.20) tell us that 𝛿≤𝑢𝑛(𝑡)≤𝑟, so we may let 𝑀1=max[]𝑡∈𝐽,𝑢∈𝛿,𝑟𝑓(𝑡,𝑢),𝑀2=max𝑡,𝑠∈𝐽||ğºî…žğ‘¡||(𝑡,𝑠),𝑀3=max[]𝑙𝑢∈𝛿,𝑟𝑘=1𝐽𝑘(𝑢).(3.23) Then, â€–â€–ğ‘¢î…žğ‘›â€–â€–=sup𝑡∈𝐽||ğ‘¢î…žğ‘›||(𝑡)=sup𝑡∈𝐽|||||02ğœ‹ğºî…žğ‘¡î€·(𝑡,𝑠)𝑓𝑠,𝑢𝑛(𝑠)𝑑𝑠+𝑙𝑘=1ğºî…žğ‘¡î€·ğ‘¡,𝑡𝑘𝐽𝑘𝑢𝑛𝑡𝑘+𝑙𝑘=1𝜕𝜕𝑡𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑡𝑘||||=sup𝑡∈𝐽||||02ğœ‹ğºî…žğ‘¡î€·(𝑡,𝑠)𝑓𝑠,𝑢𝑛+𝑚(𝑠)𝑑𝑠Γ𝑡𝑘≤𝑡𝑒𝑚(𝑡−𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘−𝑚𝐼𝑘𝑢𝑛𝑡𝑘−𝑡𝑘≤𝑡𝑒𝑚(2𝜋−𝑡+𝑡𝑘)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘−𝑡𝑘>𝑡𝑒𝑚(𝑡𝑘−𝑡)𝐽𝑘𝑢𝑛𝑡𝑘+𝑚𝐼𝑘𝑢𝑛𝑡𝑘+𝑡𝑘>𝑡𝑒𝑚(2𝜋−𝑡𝑘+𝑡)𝐽𝑘𝑢𝑛𝑡𝑘−𝑚𝐼𝑘𝑢𝑛𝑡𝑘|||||≤sup𝑡∈𝐽02𝜋||ğºî…žğ‘¡||𝑓(𝑡,𝑠)𝑠,𝑢𝑛𝑒(𝑠)𝑑𝑠+2𝑚2𝑚𝜋+1Γ𝑙𝑘=1𝐽𝑘𝑢𝑛𝑡𝑘≤2𝜋𝑀1𝑀2+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}𝑛∈𝑁0 is a bounded and equi-continuous family on [0,2𝜋]. Then, the Arzela-Ascoli Theorem guarantees that {𝑢𝑛}𝑛∈𝑁0 has a subsequence {𝑢𝑛𝑗}𝑗∈𝑁, converging uniformly on [0,2𝜋]. From the fact ‖𝑢𝑛‖<𝑟 and (3.20), 𝑢 satisfies 𝛿≤𝑢(𝑡)≤𝑟 for all 𝑡∈𝐽. Moreover, 𝑢𝑛𝑗 also satisfies the following integral equation: 𝑢𝑛𝑗(𝑡)=02𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢𝑛𝑗+(𝑠)𝑑𝑠𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑛𝑗𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑛𝑗𝑡𝑘+1𝑛𝑗.(3.25) Let 𝑗→+∞, and we get 𝑢(𝑡)=02𝜋+𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠𝑙𝑘=1𝐺𝑡,𝑡𝑘𝐽𝑘𝑢𝑡𝑘+𝑙𝑘=1𝜕𝐺(𝑡,𝑠)|||𝜕𝑠𝑠=𝑡𝑘𝐼𝑘𝑢𝑡𝑘,(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:âˆ’ğ‘¢î…žî…ž(𝑡)+𝑀𝑢(𝑡)=𝑡2||||1+sin𝑢𝑢3/2+𝑡(1+|cos𝑢|),ğ‘¡âˆˆğ½î…ž,||Δ𝑢𝑡=𝑡𝑘=𝑐min1,𝑐2,…,𝑐𝑙2√𝑀𝑢𝑡𝑘,âˆ’Î”ğ‘¢î…ž||𝑡=𝑡𝑘=𝑐𝑘𝑢𝑡𝑘,𝑘=1,2,…,𝑙,𝑢(0)=𝑢(2𝜋),ğ‘¢î…ž(0)=ğ‘¢î…ž(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=0lim𝑢→0+𝑓(𝑡,𝑢)=+∞,uniformalyin𝑡.(4.3) Denote 𝑝(𝑡)=𝑡2||||,ğ‘ž(𝑡)=𝑡,𝜉(𝑢)=1+sin𝑢𝑢3/2𝑐,𝜂(𝑢)=1+|cos𝑢|,𝛾(𝑢)=max1,𝑐2,…,𝑐𝑙𝑢,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→+âˆžğ´=lim𝑀→+âˆžğ‘’âˆš2𝜋𝑀+1√𝑀𝑒√2𝜋𝑀−1=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).