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

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: Δ𝑢||_{𝑡=𝑡𝑘}−𝑢^(𝑡)+𝑀𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈𝐽′,=𝐼_𝑘𝑢𝑢(𝑡0_𝑘)=,𝑢(−2𝜋Δ)𝑢,^||_{𝑡=𝑡𝑘}𝑢=_𝐽_𝑘𝑢𝑡_𝑘,𝑘=1,2,…𝑙,(0)=𝑢^(2𝜋).(1.1) Here, 𝐽=[0,2𝜋], 0<𝑡₁<𝑡₂<⋯<𝑡_𝑙<2𝜋, 𝐽′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑙}, 𝑀>0, 𝑓∈𝐶(𝐽×𝑅₊,𝑅⁺), 𝐼_𝑘∈𝐶(𝑅⁺,𝑅), 𝐽_𝑘∈𝐶(𝑅⁺,𝑅⁺), 𝑅⁺=[0,+∞), 𝑅₊=(0,+∞) with −(1/𝑚)𝐽_𝑘(𝑢)<𝐼_𝑘(𝑢)<(1/𝑚)𝐽_𝑘(𝑢), 𝑢∈𝑅⁺, 𝑚=√𝑀. Δ𝑢|_{𝑡=𝑡𝑘}=𝑢(𝑡^+𝑘)−𝑢(𝑡^−𝑘), Δ𝑢^|_{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=𝑡_𝑘.

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)+𝑎(𝑡)𝑥Δ(𝑥𝑡)||=_{𝑡=𝑡𝑘}𝑓(𝑡,𝑥(𝑡)),a.e.𝑡∈[0,𝑇]⧵𝑡₁,…,𝑡_𝑝,=𝐼_𝑘𝑥(𝑥0)=𝑡_𝑘𝑥(,𝑇)𝑘.=1,…,𝑝,(1.2) Jiang [5] has applied Krasnoselskii's fixed point theorem to establish the existence of positive solutions of problem 𝑥^(𝑥𝑡)(+0)𝑀=𝑥𝑥((𝑡2)𝜋=)𝑓,(𝑡,𝑥𝑥_(𝑡)),𝑡∈[0,2𝜋],(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+^𝑙_𝑖=1𝐽₀(𝑖)𝜎>2𝜋𝑀,2𝜋𝑓_∞+^𝑙_𝑖=1𝐽_∞(𝑖)𝜎>2𝜋𝑀,(1.4)(𝐴₂)2𝜋𝑓⁰+^𝑙_𝑖=1𝐽⁰(𝑖)𝜎<2𝜋𝑀,2𝜋𝑓^∞+^𝑙_𝑖=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.(𝐴^1)There exist nonnegative valued 𝜉(𝑥),𝜂(𝑥)∈𝐶((0,∞)) and 𝑃(𝑡),𝑄(𝑡)∈𝐿¹[0,2𝜋] such that 0≤𝑓su(p𝑡,𝑥_{𝑥∈(0,∞)})≤𝑃(𝑡)𝜉(𝑥)+𝑄(𝑡)𝜂(𝑥),a.e.(𝑡,𝑥)∈[0,2𝜋]×(0,∞),⎧⎪⎨⎪⎩𝑥∫^02𝜋𝑃(𝑡)𝑑𝑡𝜉(𝑥)/𝜂(𝑥)+∫^02𝜋𝑄(𝑡)𝑑𝑡𝜂𝛿_𝑗𝑡⎫⎪⎬⎪⎭>𝐵^𝑗,(1.6)

where 𝜂(𝑥) is nonincreasing and 𝜉(𝑥)/𝜂(𝑥) is nondecreasing on (0,∞),(𝐴^2)lim_𝑡→0+infmin∫^02𝜋𝑓(𝑥,𝑤)𝑑𝑥∶𝛿_𝑗𝑡≤𝑤≤𝑡𝑡>𝐴1_𝑗,(1.7)(𝐴^3)lim_𝑡→+∞infmin∫^02𝜋𝑓(𝑥,𝑤)𝑑𝑥∶𝛿_𝑗𝑡≤𝑤≤𝑡𝑡>𝐴1_𝑗.(1.8) Here, 𝛿_𝑗,𝐴_𝑗,𝐵_𝑗 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≠𝑡_𝑘, left continuous at 𝑡=𝑡_𝑘, 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=sup_𝑡∈𝐽|𝑢(𝑡)|. Let 𝑃𝐶¹[𝐽,𝑅]={𝑢∈𝑃𝐶[𝐽,𝑅]∶𝑢′(𝑡) exists at 𝑡≠𝑡_𝑘 and is continuous at 𝑡≠𝑡_𝑘, and 𝑢^(𝑡^+𝑘), 𝑢^(𝑡^−𝑘) exist and 𝑢^(𝑡) is left continuous at 𝑡=𝑡_𝑘, 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,𝑡_𝑘𝐽_𝑘𝑢𝑡_𝑘+^𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢𝑡_𝑘,(2.1) where 𝐺(𝑡,𝑠) is the Green's function to the following periodic boundary value problem: 𝑢(0)=𝑢(−2𝜋𝑢)^,+𝑀𝑢𝑢_=0,𝐺(𝑡,𝑠)∶=1(0)=𝑢^(2𝜋),Γ⎧⎪⎨⎪⎩⎧⎪⎨⎪⎩𝑒^{𝑚(𝑡−𝑠)}𝑒+𝑒^{𝑚(2𝜋−𝑡+𝑠)},0≤𝑠≤𝑡≤2𝜋,_{𝑚(𝑠−𝑡)}+𝑒^{𝑚(2𝜋−𝑠+𝑡)},0≤𝑡≤𝑠≤2𝜋,(2.2) here, Γ=2𝑚(𝑒^2𝑚𝜋−1). It is clear that 2𝑒^𝑚𝜋Γ=𝐺(𝜋)≤𝐺(𝑡,𝑠)≤𝐺(0)=𝑒^2𝑚𝜋+1Γ.(2.3) Define 𝐾=𝑢∈𝑃𝐶[𝐽,𝑅]∶𝑢(𝑡)≥𝜎‖𝑢‖_𝑝𝑐,𝑡∈𝐽,(2.4) where 𝜎=1𝑒^2𝑚𝜋.(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),𝑞(𝑡)∈𝐿¹([0,2𝜋]) such that𝑓(𝑡,𝑢m)a≤x_{1≤𝑘≤𝑙}𝑝(𝑡)𝜉(𝑢)+𝑞(𝑡)𝜂(𝑢),(𝑡,𝑢)∈[0,2𝜋]×(0,∞),(3.1)𝐽_𝑘(𝑢)≤𝛾(𝑢),(𝑡,𝑢)∈[0,2𝜋]×(0,+∞),(3.2)(𝐻₂)there exists a positive number 𝑟>0 such that 𝐴2max^{[]𝑥∈𝜎𝑟,𝑟}𝜉(𝑥)^02𝜋𝑝(𝑠)𝑑𝑠+max_{[]𝑥∈𝜎𝑟,𝑟}𝜂(𝑥)^02𝜋𝑞(𝑠)𝑑𝑠+𝐴𝑙𝛾(𝑟)<𝑟,(3.3)(𝐻₃)for the constant 𝑟 in (H₂), there exists a function Φ_𝑟>0 such that 𝑓(𝑡,𝑢)>Φ_𝑟(𝑡),(𝑡,𝑢)∈[0,2𝜋]×(0,𝑟],^02𝜋Φ_𝑟(𝑠)𝑑𝑠>0.(3.4) Then PBVP (1.1) has at least one positive periodic solution with 0<‖𝑢‖<𝑟, where 𝐴=𝑒^2𝑚𝜋+1𝑚𝑒^2𝑚𝜋−1=𝑒^√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^{[]𝑥∈𝜎𝑟,𝑟}𝜉(𝑥)^02𝜋𝑝(𝑠)𝑑𝑠+max_{[]𝑥∈𝜎𝑟,𝑟}𝜂(𝑥)^02𝜋𝑞(𝑠)𝑑𝑠+𝐴𝑙𝛾(𝑟)+1𝑛₀<𝑟.(3.6) Let 𝑁₀={𝑛₀,𝑛₀+1,…}. For 𝑛∈𝑁₀. We consider the family of equations −𝑢^(𝑡)+𝑀𝑢(𝑡)=𝜆𝑓_𝑛(𝑡,𝑢(𝑡))+𝑀𝑛,𝑡∈𝐽^Δ𝑢||_{𝑡=𝑡𝑘},=𝐼_𝑘𝑢𝑢(𝑡0_𝑘)=,𝑢(−2𝜋Δ)𝑢,^||_{𝑡=𝑡𝑘}𝑢=_𝐽_𝑘𝑢𝑡_𝑘,𝑘=1,2,…𝑙,(0)=𝑢^(2𝜋),(3.7) where 𝜆∈[0,1] and 𝑓_𝑛(𝑡,𝑢)=𝑓𝑡,max𝑢,1𝑛,(𝑡,𝑢)∈𝐽×[0,+∞).(3.8) For every 𝜆 and 𝑛∈𝑁₀, define an operator as follows: 𝑇_𝜆,𝑛𝑢(𝑡)=𝜆^02𝜋𝐺(𝑡,𝑠)𝑓_𝑛(𝑠,𝑢(𝑠))𝑑𝑠+^𝑙_𝑘=1+𝐺𝑡,𝑡_𝑘𝐽_𝑘𝑢𝑡_𝑘_𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢𝑡_𝑘,𝑢∈𝐾.(3.9) Then, we may verify that 𝑇_𝜆,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=𝑇_𝜆,𝑛𝑢+1𝑛.(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 𝑓_𝑛(𝑡,𝑢)≥0. 𝑢(𝑡)≥1/𝑛 for all 𝑡∈𝐽 and 𝑟≥𝑢(𝑡)≥(1/𝑛)+𝜎‖𝑢−1/𝑛‖. By the choice of 𝑛₀, 1/𝑛≤1/𝑛₀<𝑟. Hence, for all 𝑡∈𝐽, we get 𝑟≥𝑢(𝑡)≥1𝑛+𝜎‖‖‖𝑢−1𝑛‖‖‖≥1𝑛+𝜎|||‖𝑢‖−1𝑛|||≥1𝑛+𝜎𝑟−1𝑛>𝜎𝑟.(3.13) From (3.2), we have 𝐽_𝑘𝑢𝑡_𝑘≤max_{1≤𝑘≤𝑙}𝐽_𝑘𝑢𝑡_𝑘≤𝛾𝑢𝑡_𝑘≤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)=𝜆^02𝜋𝐺(𝑡,𝑠)𝑓_𝑛(𝑠,𝑢(𝑠))𝑑𝑠+𝑘=1^𝑙_𝑘=1𝐺𝑡,𝑡_𝑘𝐽_𝑘𝑢𝑡_𝑘+^𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢𝑡_𝑘+1≤_02𝜋𝑛𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑙_𝑘=1𝐺𝑡,𝑡_𝑘𝐽_𝑘𝑢𝑡_𝑘+^𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢𝑡_𝑘+1≤_02𝜋𝑛+1𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠Γ_{𝑡𝑘≤𝑡}𝑒^{𝑚(𝑡−𝑡𝑘)}++𝑒^{𝑚(2𝜋−𝑡+𝑡𝑘)}𝐽_𝑘𝑢𝑡_𝑘_{𝑡𝑘>𝑡}+𝑒^{𝑚(𝑡𝑘−𝑡)}+𝑒^{𝑚(2𝜋−𝑡𝑘+𝑡)}𝐽_𝑘𝑢𝑡_𝑘+_{𝑡𝑘≤𝑡}−𝑒^{𝑚(𝑡−𝑡𝑘)}+𝑒^{𝑚(2𝜋−𝑡+𝑡𝑘)}𝑚𝐼_𝑘𝑢𝑡_𝑘_{𝑡𝑘>𝑡}=_02𝜋𝑒^{𝑚(𝑡𝑘−𝑡)}−𝑒^{𝑚(2𝜋−𝑡𝑘+𝑡)}𝑚𝐼_𝑘𝑢𝑡_𝑘+1𝑛+1𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠Γ_{𝑡𝑘≤𝑡}+𝑒^{𝑚(𝑡−𝑡𝑘)}𝐽_𝑘𝑢𝑡_𝑘−𝑚𝐼_𝑘𝑢𝑡_𝑘_{𝑡𝑘≤𝑡}+𝑒^{𝑚(2𝜋−𝑡+𝑡𝑘)}𝐽_𝑘𝑢𝑡_𝑘+𝑚𝐼_𝑘𝑢𝑡_𝑘+_{𝑡𝑘>𝑡}𝑒^{𝑚(𝑡𝑘−𝑡)}𝐽_𝑘𝑢𝑡_𝑘+𝑚𝐼_𝑘𝑢𝑡_𝑘_{𝑡𝑘>𝑡}𝑒^{𝑚(2𝜋−𝑡𝑘+𝑡)}𝐽_𝑘𝑢𝑡_𝑘−𝑚𝐼_𝑘𝑢𝑡_𝑘+1𝑛.(3.15) It follows from −(1/𝑚)𝐽_𝑘(𝑢)<𝐼_𝑘(𝑢)<(1/𝑚)𝐽_𝑘(𝑢) that 𝐽_𝑘𝑢𝑡_𝑘−𝑚𝐼_𝑘𝑢𝑡_𝑘>0,𝐽_𝑘𝑢𝑡_𝑘+𝑚𝐼_𝑘𝑢𝑡_𝑘>0.(3.16) So, we get from (3.1), (3.2), and (3.3) that 𝑢(𝑡)≤^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+2𝑒^2𝑚𝜋+1Γ^𝑙_𝑘=1𝐽_𝑘𝑢𝑡_𝑘+1≤_02𝜋𝑛𝐺(𝑡,𝑠)[𝑝(𝑠)𝜉(𝑢(𝑠))+𝑞(𝑠)𝜂(𝑢(𝑠))]𝑑𝑠+𝐴𝑙𝛾(𝑟)+1𝑛₀≤𝐴2^02𝜋𝑝(𝑠)𝑑𝑠max_{[]𝑥∈𝜎𝑟,𝑟}𝜉(𝑥)+^02𝜋𝑞(𝑠)𝑑𝑠max_{[]𝑥∈𝜎𝑟,𝑟}𝜉(𝑥)+𝐴𝑙𝛾(𝑟)+1𝑛₀.(3.17) Therefore, 𝑟=‖𝑢‖≤𝐴2^02𝜋𝑝(𝑠)𝑑𝑠max_{[]𝑥∈𝜎𝑟,𝑟}𝜉(𝑥)+^02𝜋𝑞(𝑠)𝑑𝑠max_{[]𝑥∈𝜎𝑟,𝑟}𝜉(𝑥)+𝐴𝑙𝛾(𝑟)+1𝑛₀<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 𝑢_𝑛 in Ω. So, (3.7) (with 𝜆=1) has a positive solution 𝑢_𝑛 with ‖‖𝑢_𝑛‖‖<𝑟,𝑢_𝑛(𝑡)≥1𝑛,𝑡∈𝐽.(3.19)Step 4. We show that {𝑢_𝑛} have a uniform positive lower bound; that is, there exists a constant 𝛿>0, independent of 𝑛∈𝑁₀, such that min_𝑡𝑢_𝑛(𝑡)≥𝛿.(3.20) In fact, from (3.4), (3.8), (3.16), and (3.19), we get 𝑢_𝑛(𝑡)=^02𝜋𝐺(𝑡,𝑠)𝑓_𝑛𝑠,𝑢_𝑛(𝑠)𝑑𝑠+^𝑙_𝑘=1+𝐺𝑡,𝑡_𝑘𝐽_𝑘𝑢_𝑛𝑡_𝑘_𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢_𝑛𝑡_𝑘+1=_02𝜋𝑛𝐺(𝑡,𝑠)𝑓𝑠,𝑢_𝑛(𝑠)𝑑𝑠+^𝑙_𝑘=1+𝐺𝑡,𝑡_𝑘𝐽_𝑘𝑢_𝑛𝑡_𝑘_𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢_𝑛𝑡_𝑘+1≥_02𝜋𝑛+1𝐺(𝑡,𝑠)Φ_𝑟(𝑠)𝑑𝑠Γ_{𝑡𝑘≤𝑡}+𝑒^{𝑚(𝑡−𝑡𝑘)}𝐽_𝑘𝑢_𝑛𝑡_𝑘−𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘_{𝑡𝑘≤𝑡}+𝑒^{𝑚(2𝜋−𝑡+𝑡𝑘)}𝐽_𝑘𝑢_𝑛𝑡_𝑘+𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘_{𝑡𝑘>𝑡}+𝑒^{𝑚(𝑡𝑘−𝑡)}𝐽_𝑘𝑢_𝑛𝑡_𝑘+𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘_{𝑡𝑘>𝑡}𝑒^{𝑚(2𝜋−𝑡𝑘+𝑡)}𝐽_𝑘𝑢_𝑛𝑡_𝑘−𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘+1≥_02𝜋𝑛≥2𝑒_𝑚𝜋𝐺(𝑡,𝑠)Φ_𝑟(𝑠)𝑑𝑠Γ^02𝜋Φ_𝑟(𝑠)𝑑𝑠∶=𝛿>0.(3.21)Step 5. We prove that ‖‖𝑢^𝑛‖‖<𝐻,𝑛≥𝑛₀(3.22) for some constant 𝐻>0. Equations (3.19) and (3.20) tell us that 𝛿≤𝑢_𝑛(𝑡)≤𝑟, so we may let 𝑀₁=max_{[]𝑡∈𝐽,𝑢∈𝛿,𝑟}𝑓(𝑡,𝑢),𝑀₂=max_{𝑡,𝑠∈𝐽}||𝐺^𝑡(𝑡,𝑠)||,𝑀₃=max_{[]𝑙𝑢∈𝛿,𝑟}_𝑘=1𝐽_𝑘(𝑢).(3.23) Then, ‖‖𝑢^𝑛‖‖==ssuupp_{𝑡∈𝐽𝑡∈𝐽}||𝑢^𝑛(𝑡)|||||||_02𝜋𝐺^𝑡(𝑡,𝑠)𝑓𝑠,𝑢_𝑛(𝑠)𝑑𝑠+^𝑙_𝑘=1+𝐺^𝑡𝑡,𝑡_𝑘𝐽_𝑘𝑢_𝑛𝑡_𝑘_𝑙_𝑘=1𝜕=sup_𝑡∈𝐽𝜕𝑡𝜕𝐺𝜕(𝑠𝑡,𝑠)|||_{𝑠=𝑡𝑘}𝐼^𝑘𝑢_𝑛𝑡_𝑘||||||||^02𝜋+𝑚𝐺^𝑡(𝑡,𝑠)𝑓𝑠,𝑢_𝑛(𝑠)𝑑𝑠Γ_{𝑡𝑘≤𝑡}−𝑒^{𝑚(𝑡−𝑡𝑘)}𝐽_𝑘𝑢_𝑛𝑡_𝑘−𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘_{𝑡𝑘≤𝑡}−𝑒^{𝑚(2𝜋−𝑡+𝑡𝑘)}𝐽_𝑘𝑢_𝑛𝑡_𝑘+𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘_{𝑡𝑘>𝑡}+𝑒^{𝑚(𝑡𝑘−𝑡)}𝐽_𝑘𝑢_𝑛𝑡_𝑘+𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘_{𝑡𝑘>𝑡}≤sup_𝑡∈𝐽𝑒^{𝑚(2𝜋−𝑡𝑘+𝑡)}𝐽_𝑘𝑢_𝑛𝑡_𝑘−𝑚𝐼_𝑘𝑢_𝑛𝑡_𝑘|||||^02𝜋||𝐺^𝑡(𝑡,𝑠)||𝑓𝑠,𝑢_𝑛(𝑠)𝑑𝑠+2𝑚𝑒^2𝑚𝜋+1Γ^𝑙_𝑘=1≤2𝜋𝑀₁𝐽_𝑘𝑢_𝑛𝑡_𝑘𝑀₂+2𝑚𝑒^2𝑚𝜋+1Γ𝑀³∶=𝐻.(3.24)Step 6. Now, we pass the solution 𝑢_𝑛 of the truncation equation (3.7) (with 𝜆=1) to that of the original equation (1.1). The fact that ‖𝑢_𝑛‖<𝑟 and (3.22) show that {𝑢_𝑛}_𝑛∈𝑁0 is a bounded and equi-continuous family on [0,2𝜋]. Then, the Arzela-Ascoli Theorem guarantees that {𝑢_𝑛}_𝑛∈𝑁0 has a subsequence {𝑢_𝑛𝑗}_𝑗∈𝑁, converging uniformly on [0,2𝜋]. From the fact ‖𝑢_𝑛‖<𝑟 and (3.20), 𝑢 satisfies 𝛿≤𝑢(𝑡)≤𝑟 for all 𝑡∈𝐽. Moreover, 𝑢_𝑛𝑗 also satisfies the following integral equation: 𝑢_𝑛𝑗(𝑡)=+^02𝜋𝐺(𝑡,𝑠)𝑓𝑠,𝑢_𝑛𝑗(𝑠)𝑑𝑠_𝑙_𝑘=1𝐺𝑡,𝑡_𝑘𝐽_𝑘𝑢^𝑛_𝑗𝑡_𝑘+^𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢^𝑛_𝑗𝑡_𝑘+1𝑛_𝑗.(3.25) Let 𝑗→+∞, and we get 𝑢(𝑡)=+^02𝜋𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝑙_𝑘=1𝐺𝑡,𝑡_𝑘𝐽_𝑘𝑢𝑡_𝑘+^𝑙_𝑘=1𝜕𝐺(𝑡,𝑠)𝜕𝑠|||_{𝑠=𝑡𝑘}𝐼_𝑘𝑢𝑡_𝑘,(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+||s𝑢in^3/2𝑢||+𝑡(1+|cos𝑢|),𝑡∈𝐽^Δ𝑢||_{𝑡=𝑡𝑘},=min𝑐¹,𝑐₂,…,𝑐_𝑙2√𝑀𝑢𝑡^𝑘𝑢(0)=𝑢(2𝜋),,−𝑢Δ_𝑢^||_{𝑡=𝑡𝑘}=𝑐_𝑘𝑢𝑡_𝑘,𝑘=1,2,…,𝑙,(0)=𝑢^(2𝜋),(4.1) where 𝑐_𝑘>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+||s𝑢in^3/2𝑢||+𝑡(1+|cos𝑢|),(4.2) then 𝑓(𝑡,𝑢) has a repulsive singularity at 𝑢=0lim_𝑢→0+𝑓(𝑡,𝑢)=+∞,uniformalyin𝑡.(4.3) Denote 𝑝(𝑡)=𝑡²,𝑞(𝑡)=𝑡,𝜉(𝑢)=1+||sin𝑢||𝑢^3/2𝛾(𝑢)=max𝑐₁,𝜂(𝑢)=1+|cos𝑢|,𝑟=1,Φ_𝑟,𝑐₂,…,𝑐_𝑙𝑢,(𝑡)=𝑡+𝑡².(4.4) Then, it is easy to say that (3.1), (3.2), and (3.3) hold. From (3.5), we knowlim_𝑀→+∞𝐴=lim_𝑀→+∞𝑒^√2𝜋𝑀+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).