An Existence Result for Second-Order Impulsive Differential Equations with Nonlocal Conditions

Abstract

The Leray-Schauder alternative is used to investigate the existence of solutions for second-order impulsive differential equations with nonlocal conditions in Banach spaces. The results improve some recent results.

1. Introduction

The theory of impulsive differential equations is emerging as an important area of investigation since it is a lot richer than the corresponding theory of nonimpulsive differential equations. Many evolutionary processes in nature are characterized by the fact that at certain moments in time an abrupt change of state is experienced. That is the reason for the rapid development of the theory of impulsive differential equations; see the monographs [1, 2].

This paper is concerned with the study on existence of second-order impulsive differential equations with nonlocal conditions of the form 𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥^′(𝑡)),𝑡∈𝐽=[0,𝑏],𝑡≠𝑡_𝑘,Δ𝑥∣_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑥(𝑡_𝑘)),𝑘=1,…,𝑚,Δ𝑥^′∣_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘)),𝑘=1,…,𝑚,𝑥(0)+𝑔(𝑥)=𝑥₀,𝑥^′(0)=𝜂, (1.1) where the state 𝑥(⋅) takes values in Banach space 𝑋 with the norm ‖⋅‖,𝑥₀,𝜂∈𝑋, 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑏,Δ𝑥∣_{𝑡=𝑡𝑘}=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘),Δ𝑥^′∣_{𝑡=𝑡𝑘}=𝑥^′(𝑡⁺_𝑘)−𝑥^′(𝑡⁻_𝑘).  𝑓,𝑔, and 𝐼_𝑘,𝐼_𝑘(𝑘=1,2,…,𝑚) are given functions to be specified later.

The nonlocal condition is a generalization of the classical initial condition. The first results concerning the existence and uniqueness of mild solutions to Cauchy problems with nonlocal conditions were studied by Byszewski [3]. Recently, theorems about existence, uniqueness and continuous dependence of impulsive differential abstract evolution Cauchy problems with nonlocal conditions have been studied by Fu and Cao [4], Anguraj and Karthikeyan [5], Abada et al. [6], Li and Han [7], and in the references therein.

Up to now there have been very few papers in this direction dealing with the existence of solutions for second-order impulsive differential equations with nonlocal conditions. Our purpose here is to extend the results of first-order impulsive differential equations to second-order impulsive differential equations with nonlocal conditions.

Our main results are based on the following lemma [8].

Lemma 1.1 (Leray-Schauder alternative). Let 𝑆 be a convex subset of a normed linear space 𝐸 and assume that 0∈𝑆. Let 𝐺∶𝑆→𝑆 be a completely continuous operator, and let 𝜁(𝐺)={𝑥∈𝑆∶𝑥=𝜆𝐺𝑥,𝑓𝑜𝑟𝑠𝑜𝑚𝑒0<𝜆<1}. (1.2)

Then either 𝜁(𝐺) is unbounded or 𝐺 has a fixed point.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Denote 𝐽₀=[0,𝑡₁],𝐽_𝑘=(𝑡_𝑘,𝑡_𝑘+1],𝐽^′=𝐽⧵{𝑡_𝑘},𝑘=1,2,…,𝑚. We define the following classes of functions:

PC(𝐽,𝑋)={𝑥∶𝐽→𝑋∶𝑥_𝑘∈𝐶(𝐽_𝑘,𝑋),𝑘=0,1,…,𝑚, and there exist 𝑥(𝑡⁺_𝑘),𝑥(𝑡⁻_𝑘),𝑘=1,…,𝑚 with 𝑥(𝑡_𝑘)=𝑥(𝑡⁻_𝑘)},

PC¹(𝐽,𝑋)={𝑥∈PC(𝐽,𝑋)∶𝑥^′_𝑘∈𝐶(𝐽_𝑘,𝑋),𝑘=0,1,…,𝑚, and there exist 𝑥^′(𝑡⁺_𝑘),𝑥^′(𝑡⁻_𝑘),𝑘=1,…,𝑚 with 𝑥^′(𝑡_𝑘)=𝑥^′(𝑡⁻_𝑘)}, where 𝑥_𝑘 and 𝑥^′_𝑘 represent the restriction of 𝑥 and 𝑥^′ to 𝐽_𝑘, respectively (𝑘=0,…,𝑚), and ‖𝑥_𝑘‖_𝐽𝑘=sup_{𝑠∈𝐽𝑘}‖𝑥_𝑘(𝑠)‖.

Obviously, PC(𝐽,𝑋) is a Banach space with the norm ‖𝑥‖_PC=max{‖𝑥_𝑘‖_𝐽𝑘,𝑘=0,…,𝑚}, and PC¹(𝐽,𝑋) is also a Banach space with the norm ‖𝑥‖_PC¹=max{‖𝑥‖_PC,‖𝑥^′‖_PC}.

Definition 2.1. A map 𝑓∶𝐽×𝑋×𝑋→𝑋 is said to be an 𝐿¹-Carathéodory if(i)𝑓∶(⋅,𝑤,𝑣)∶𝐽→𝑋 is measurable for every 𝑤,𝑣∈𝑋,(ii)𝑓∶(𝑡,⋅,⋅)∶𝑋×𝑋→𝑋 is continuous for almost all 𝑡∈𝐽,(iii)for each 𝑖>0, there exists 𝛼_𝑖∈𝐿¹(𝐽,𝑅₊) such that for almost all 𝑡∈𝐽sup_{‖𝑤‖,‖𝑣‖≤𝑖}‖𝑓(𝑡,𝑤,𝑣)‖≤𝛼_𝑖(𝑡). (2.1)

Definition 2.2. A function 𝑥∈PC¹(𝐽,𝑋)∩𝐶²(𝐽^′,𝑋) is said to be a solution of (1.1) if 𝑥 satisfies the equation 𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥^′(𝑡)) a.e. on 𝐽^′, the conditions Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑥(𝑡_𝑘)),Δ𝑥^′|_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘)),𝑘=1,…,𝑚, and 𝑥(0)+𝑔(𝑥)=𝑥₀,𝑥^′(0)=𝜂.

Lemma 2.3. If 𝑥∈PC¹(𝐽,𝑋)∩𝐶²(𝐽^′,𝑋) satisfies 𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥^′(𝑡)),𝑡≠𝑡_𝑘(𝑘=1,2,…,𝑚), (2.2) then 𝑥^′(𝑡)=𝑥^′(0)+^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}[𝑥^′(𝑡⁺_𝑘)−𝑥^′(𝑡_𝑘)],∀𝑡∈𝐽,𝑥(𝑡)=𝑥(0)+𝑥^′(0)𝑡+_{0<𝑡𝑘<𝑡}[𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘)]+_{0<𝑡𝑘<𝑡}[𝑥^′(𝑡⁺_𝑘)−𝑥^′(𝑡_𝑘)](𝑡−𝑡_𝑘)+^𝑡₀(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠,∀𝑡∈𝐽. (2.3)(2.4)

Proof. Assume that 𝑡_𝑘<𝑡≤𝑡_𝑘+1 ( here 𝑡₀=0,𝑡_𝑚+1=𝑏 ). Then 𝑥^′(𝑡₁)−𝑥^′(0)=^𝑡1₀𝑓(𝑠,𝑥(𝑠),𝑥^′𝑥(𝑠))𝑑𝑠,_′(𝑡₂)−𝑥^′(𝑡⁺₁)=^𝑡2_𝑡1𝑓(𝑠,𝑥(𝑠),𝑥^′⋮𝑥(𝑠))𝑑𝑠,_′(𝑡_𝑘)−𝑥^′(𝑡⁺_𝑘−1)=^𝑡𝑘_𝑡𝑘−1𝑓(𝑠,𝑥(𝑠),𝑥^′𝑥(𝑠))𝑑𝑠,_′(𝑡)−𝑥^′(𝑡⁺_𝑘)=^𝑡_𝑡+𝑘𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠. (2.5)
Adding these together, we get 𝑥^′(𝑡)=𝑥^′(0)+^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}[𝑥^′(𝑡⁺_𝑘)−𝑥^′(𝑡_𝑘)], (2.6) that is, (2.3) holds.
Similarly, we have 𝑥(𝑡)=𝑥(0)+^𝑡₀𝑥^′(s)𝑑𝑠+_{0<𝑡𝑘<𝑡}[𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘)]. (2.7)
Substitution of (2.3) in (2.7) gives 𝑥(𝑡)=𝑥(0)+𝑥^′(0)𝑡+_{0<𝑡𝑘<𝑡}[𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘)]+_{0<𝑡𝑘<𝑡}[𝑥^′(𝑡⁺_𝑘)−𝑥^′(𝑡_𝑘)](𝑡−𝑡_𝑘)+^𝑡₀(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠,∀𝑡∈𝐽, (2.8) that is, (2.4) holds.

We assume the following hypotheses:(𝐻₁)𝑓∶𝐽×𝑋×𝑋→𝑋 is an 𝐿¹-Carathéodory map;(𝐻₂)𝐼_𝑘∈𝐶(,𝑋),𝐼_𝑘∈𝐶(𝑋×𝑋,𝑋), and there exist constants 𝑑_𝑘,𝑑_𝑘 such that ‖𝐼_𝑘(𝑤)‖≤𝑑_𝑘,‖𝐼_𝑘(𝑤,𝑣)‖≤𝑑_𝑘(𝑘=1,…,𝑚) for every 𝑤,𝑣∈𝑋;(𝐻₃)𝑔∶PC(𝐽,𝑋)→𝑋 is a continuous function and there exists a constant 𝑀 such that ‖𝑔(𝑥)‖≤𝑀,foreach𝑥∈PC(𝐽,𝑋); (2.9)(𝐻₄) there exists a function 𝑝∈𝐿¹(𝐽,𝑅₊) such that ‖𝑓(𝑡,𝑤,𝑣)‖≤𝑝(𝑡)𝜓(‖𝑤‖+‖𝑣‖),fora.e.𝑡∈𝐽andevery𝑤,𝑣∈𝑋, (2.10) where 𝜓∶[0,∞)→(0,∞) is a continuous nondecreasing function with (𝑏+1)^𝑏₀𝑝(𝑠)𝑑𝑠<^∞_𝑐𝑑𝑠,𝜓(𝑠) (2.11) where 𝑐=‖𝑥₀‖+𝑀+(𝑏+1)‖𝜂‖+^𝑚_𝑘=1[𝑑_𝑘+(𝑏+1−𝑡_𝑘)𝑑_k]; (2.12)(𝐻₅) for each bounded 𝐵⊆PC¹(𝐽,𝑋) and 𝑡∈𝐽 the set {𝑥₀−𝑔(𝑥)+𝑡𝜂+_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘))+_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘))(𝑡−𝑡_𝑘)+^𝑡₀(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠∶𝑥∈𝐵} (2.13) is relatively compact in 𝑋.

3. Main Results

Theorem 3.1. If the hypotheses (𝐻₁)-(𝐻₅) are satisfied, then the second-order impulsive nonlocal initial value problem (1.1) has at least one solution on 𝐽.

Proof. Consider the space 𝐵=PC¹(𝐽,𝑋) with norm ‖𝑥‖_PC¹=max{‖𝑥‖_PC,‖𝑥^′‖_PC}. (3.1)
We will now show that the operator 𝐺 defined by 𝐺𝑥(𝑡)=𝑥₀−𝑔(𝑥)+𝑡𝜂+_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘))+_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘))(𝑡−𝑡_𝑘)+^𝑡₀(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠,𝑡∈𝐽 (3.2) has a fixed point. This fixed point is then a solution of (1.1).
First we obtain a priori bounds for the following equation: 𝑥(𝑡)=𝜆[𝑥₀−𝑔(𝑥)+𝑡𝜂]+𝜆_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘))+𝜆_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘))(𝑡−𝑡_𝑘)+𝜆^𝑡₀(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠,𝑡∈𝐽. (3.3)
We have ‖𝑥(𝑡)‖≤‖𝑥₀‖+𝑀+𝑏‖𝜂‖+^𝑚_𝑘=1𝑑_𝑘+^𝑚_𝑘=1(𝑏−𝑡_𝑘)𝑑_𝑘+𝑏^𝑡₀𝑝(𝑠)𝜓(‖𝑥(𝑠)‖+‖𝑥^′(𝑠)‖)𝑑𝑠,𝑡∈𝐽. (3.4)
Denoting by 𝜇(𝑡) the right-hand side of the above inequality, we have 𝜇(0)=‖𝑥₀‖+𝑀+𝑏‖𝜂‖+^𝑚_𝑘=1[𝑑_𝑘+(𝑏−𝑡_𝑘)𝑑_𝑘𝜇],‖𝑥(𝑡)‖≤𝜇(𝑡),𝑡∈𝐽,_′(𝑡)=𝑏𝑝(𝑡)𝜓(‖𝑥(t)‖+‖𝑥^′(𝑡)‖),𝑡∈𝐽. (3.5) But 𝑥^′(𝑡)=𝜆𝜂+𝜆^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠+𝜆_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘)),𝑡∈𝐽. (3.6) Thus we have ‖𝑥^′(𝑡)‖≤‖𝜂‖+^𝑡₀𝑝(𝑠)𝜓(‖𝑥(𝑠)‖+‖𝑥^′(𝑠)‖)𝑑𝑠+^𝑚_k=1𝑑_𝑘,𝑡∈𝐽. (3.7) Denoting by 𝑟(𝑡) the right-hand side of the above inequality, we have 𝑟(0)=‖𝜂‖+^𝑚_𝑘=1𝑑_𝑘,‖𝑥^′𝑟(𝑡)‖≤𝑟(𝑡),𝑡∈𝐽,_′(𝑡)=𝑝(𝑡)𝜓(‖𝑥(𝑡)‖+‖𝑥^′(𝑡)‖),𝑡∈𝐽. (3.8) Let 𝑤(𝑡)=𝜇(𝑡)+𝑟(𝑡),𝑡∈𝐽. (3.9) Then 𝑤𝑤(0)=𝜇(0)+𝑟(0)=𝑐,_′(𝑡)=𝜇^′(𝑡)+𝑟^′(𝑡)≤𝑏𝑝(𝑡)𝜓(𝑤(𝑡))+𝑝(𝑡)𝜓(𝑤(𝑡))=(𝑏+1)𝑝(𝑡)𝜓(𝑤(𝑡)),𝑡∈𝐽. (3.10)
This implies that ^𝑤(𝑡)_𝑤(0)𝑑𝑠𝜓(𝑠)≤(𝑏+1)^𝑏₀𝑝(𝑠)𝑑𝑠<^∞_𝑐𝑑𝑠𝜓(𝑠),𝑡∈𝐽. (3.11)
This inequality implies that there is a constant 𝐾 such that 𝑤(𝑡)=𝜇(𝑡)+𝑟(𝑡)≤𝐾,𝑡∈𝐽. (3.12) Then ‖𝑥(𝑡)‖≤𝜇(𝑡),𝑡∈𝐽,‖𝑥^′(𝑡)‖≤𝑟(𝑡),𝑡∈𝐽, (3.13) and hence ‖𝑥‖_PC¹=max{‖𝑥‖_PC,‖𝑥^′‖_PC}≤𝐾. (3.14)
Second, we must prove that the operator 𝐺∶𝐵→𝐵 is a completely continuous operator.
Let 𝐵_𝑖={𝑥∈𝐵∶‖𝑥‖_PC¹≤𝑖} for some 𝑖≥1. We first show that 𝐺 maps 𝐵_𝑖 into an equicontinuous family. Let 𝑥∈𝐵_𝑖 and 𝑡,𝑡∈𝐽. Then for 0<𝑡<𝑡≤𝑏, we have ‖(𝐺𝑥)(𝑡)−(𝐺𝑥)(𝑡)‖≤‖𝑡𝜂−𝑡𝜂‖+‖_{𝑡≤𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘))‖+‖_{0<𝑡𝑘<𝑡}[(𝑡−𝑡_𝑘)−(𝑡−𝑡_𝑘)]𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘))‖+‖_{𝑡≤𝑡𝑘<𝑡}(𝑡−𝑡_𝑘)𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘))‖+‖^𝑡₀[(𝑡−𝑠)−(𝑡−𝑠)]𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠‖+‖^𝑡_𝑡(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠‖≤(𝑡−𝑡)‖𝜂‖+_{𝑡≤𝑡𝑘<𝑡}𝑑_𝑘+_{0<𝑡𝑘<𝑡}(𝑡−𝑡)𝑑_𝑘+_{𝑡≤𝑡𝑘<𝑡}(𝑡−𝑡_𝑘)𝑑_𝑘+^𝑡₀(𝑡−𝑡)𝛼_𝑖(𝑠)𝑑𝑠+^𝑡_𝑡(𝑡−𝑠)𝛼_𝑖(𝑠)𝑑𝑠, (3.15) and similarly ‖(𝐺𝑥)^′(𝑡)−(𝐺𝑥)^′(𝑡)‖≤‖^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠−^𝑡₀𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠‖+‖_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘))−_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘))‖≤^𝑡_𝑡𝛼_𝑖(𝑠)𝑑𝑠+_{𝑡≤𝑡𝑘<𝑡}𝑑_𝑘. (3.16)
The right-hand sides are independent of 𝑥∈𝐵_𝑖 and tend to zero as 𝑡−𝑡→0. Thus 𝐺 maps 𝐵_𝑖 into an equicontinuous family of functions. It is easy to see that the family 𝐺𝐵_𝑖 is uniformly bounded. And from (𝐻₅), we know that 𝐺𝐵_𝑖 is compact. Then by Arzela-Ascoli theorem, we can conclude that the map 𝐺∶𝐵→𝐵 is compact.
Next, we show that 𝐺∶𝐵→𝐵 is continuous. Let {𝑢_𝑛}^∞₀⊑𝐵 with 𝑢_𝑛→𝑢 in 𝐵. Then there is an integer 𝑞 such that ‖𝑢_𝑛(𝑡)‖,‖𝑢^′_𝑛(𝑡)‖≤𝑞 for all 𝑛 and 𝑡∈𝐽, so 𝑢_𝑛∈𝐵_𝑞 and 𝑢∈𝐵_𝑞. By (𝐻₁), 𝑓(𝑡,𝑢_𝑛(𝑡),𝑢^′_𝑛(𝑡))→𝑓(𝑡,𝑢(𝑡),𝑢^′(𝑡))(𝑛→∞) for almost all 𝑡∈𝐽, and since ‖𝑓(𝑡,𝑢_𝑛(𝑡),𝑢^′_𝑛(𝑡))−𝑓(𝑡,𝑢(𝑡),𝑢^′(𝑡))‖<2𝛼_𝑞(𝑡), we have by the dominated convergence theorem that ‖𝐺𝑢_𝑛−𝐺𝑢‖_PC=sup_𝑡∈𝐽‖[𝑔(𝑢_𝑛)−𝑔(𝑢)]+^𝑡₀(𝑡−𝑠)[𝑓(𝑠,𝑢_𝑛(𝑠),𝑢^′_𝑛(𝑠))−𝑓(𝑠,𝑢(𝑠),𝑢^′(𝑠))]𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑢_𝑛(𝑡_𝑘))−_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑢(𝑡_𝑘))+_{0<𝑡𝑘<𝑡}(𝑡−𝑡_𝑘)𝐼_𝑘(𝑢_𝑛(𝑡_𝑘),𝑢^′_𝑛(𝑡_𝑘))−_{0<𝑡𝑘<𝑡}(𝑡−𝑡_𝑘)𝐼_𝑘(𝑢(𝑡_𝑘),𝑢^′(𝑡_𝑘))‖≤‖𝑔(𝑢_𝑛+)−𝑔(𝑢)‖_𝑏0(𝑏−𝑠)‖𝑓(𝑠,𝑢_𝑛(𝑠),𝑢^′_𝑛(𝑠))−𝑓(𝑠,𝑢(𝑠),𝑢^′+(𝑠))‖𝑑𝑠_𝑚𝑘=1‖𝐼_𝑘(𝑢_𝑛(𝑡_𝑘))−𝐼_𝑘(𝑢(𝑡_𝑘+))‖_𝑚𝑘=1(𝑏−𝑡_𝑘)‖𝐼_𝑘(𝑢_𝑛(𝑡_𝑘),𝑢^′_𝑛(𝑡_𝑘))−𝐼_𝑘(𝑢(𝑡_𝑘),𝑢^′(𝑡_𝑘))‖→0,‖(𝐺𝑢_𝑛)^′−(𝐺𝑢)^′‖_PC=sup_𝑡∈𝐽‖^𝑡₀[𝑓(𝑠,𝑢_𝑛(𝑠),𝑢^′_𝑛(𝑠))−𝑓(𝑠,𝑢(𝑠),𝑢^′(𝑠))]𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑢_𝑛(𝑡_𝑘),𝑢^′_𝑛(𝑡_𝑘))−_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑢(𝑡_𝑘),𝑢^′(𝑡_𝑘≤))‖_𝑏0‖[𝑓(𝑠,𝑢_𝑛(𝑠),𝑢^′_𝑛(𝑠))−𝑓(𝑠,𝑢(𝑠),𝑢^′+(𝑠))]‖𝑑𝑠_𝑚𝑘=1‖𝐼_𝑘(𝑢_𝑛(𝑡_𝑘),𝑢^′_𝑛(𝑡_𝑘))−𝐼_𝑘(𝑢(𝑡_𝑘),𝑢^′(𝑡_𝑘))‖→0. (3.17)(3.18) Thus 𝐺 is continuous. This completes the proof that 𝐺 is completely continuous.
Finally, the set 𝜁(𝐺)={𝑥∈𝐵∶𝑥=𝜆𝐺𝑥,𝜆∈(0,1)} is bounded, as we proved in the first step. As a consequence of Lemma 1.1, we deduce that 𝐺 has a fixed point 𝑥∈𝐵 which is a solution of (1.1).

Acknowledgments

This work is supported by NNSF of China (no. 10971139) and Chinese Universities Scientific Fund (no. B08-1).