Three-Point Boundary Value Problem for Second-OrderDifferential Equations with Impulses

YANG Xu-xin,SHEN Jian-hua

(1. Department of Mathematics and Physics, Hunan First Normal University， Changsha 410205,China;2. College of Mathematics and Computer Science, Hunan Normal University， Changsha 410081,China;3. Department of Mathematics, Hangzhou Normal University，Hangzhou 310036,China)

1 Introduction

This paper, we prove existence results for the following second-order three-point boundary value problem with impulses(BVP):

(1)

whereη∈(0,1),η≠tk,α∈,J=(0,1),f∈C(J×,),0=t0

LetJ-=J{tk,k=1,2,…,m}. Denote byPC(J,) the set of functionsx:J→, such thatx(t) is continuous everywhere except for sometkat whichandexist andx(tk),k=1,2,…,m.PC1(J,)= {x∈PC(J,);x′(t) is continuous differentiable everywhere except for sometkat whichandexist, andx′(tk), fork=1,2,...,m}. ThenPC(J,),PC1(J,) are Banach spaces with the norm

respectively. LetE=PC1(J,)∩C2(J-,). A functionx∈Esatisfying the BVP (1) is called nontrivial solution ifx(t)≢0.

The study of three-point BVP for certain nonlinear ordinary differential equations was initiated by Gupta[1]. Over the past ten years, three-point boundary value problems have been extensively studied by many authors[2-10]. But in the existing literature on the BVP(1) is few. In a recent paper, Infante[6]investigated the BVP (1) without impulses for the first time. Boundary value problems with impulses have been studied by various authors[11-13]. The aim of the present paper is to establish some criteria of the existence of nontrivial solution for the BVP(1). Note that we do not require any monotonicity and nonnegativity conditions onf. The results we have obtained are new, and they generalize work of Sun[8]to the situation with impulses.

The paper is organized as follows. In Section 2, we present two lemmas that will be used to prove the main results. In Section 3, we obtain some existence results for nontrivial solution of the BVP(1). Finally,in Section 4, as an application, we give some examples to illustrate the main results we obtain.

2 Preliminaries

We will start with two preliminary results. A solution of the boundary value problem (1) is called a nontrivial solution if it does not identically vanish on the unit interval. The proof of the next lemma is elementary, and we will omit it.

Lemma1Lety(t) ∈PC(J,) be any solution of the boundary value problem (2) with boundary value condition. Then the three point boundary value problem

(2)

has a unique solution

(3)

Let the integral operatorT:PC(J,)→PC(J,) be defined by

(4)

wheret∈[0,1].

Then,by Lemma 1,the boundary value problem (1) has a solution if and only if the operator (4) has a fixed point inPC(J,). Hence,it is necessary for us only to find a fixed point ofTinPC(J,). It follows from the Arzel-Ascoli theorem thatTis a completely continuous operator. This approach makes use of following Leray-Schauder nonlinear alternative[14].

3 Main Results

In this section, we prove our main results. Namely, we shall show that the boundary value problem with impulses (1) has at least one nontrivial solution under certain growth conditions.

Theorem1Suppose thatf(t,0) is not identically zero, and there exist nonnegative functionsk,h∈L1[0,1] such that

whereLk,lk,Rk,rk≥0, |f(t,x)|≤k(t)|x|+h(t) for a.e. (t,x)∈[0,1]×, and

Then the BVP(1) has at least one nontrivial solutionx*∈E.

Proof Write

ThenA< 1. Becausef(t,0) is continuous everywhere except at {tk}×and not identically zero, there exists an interval[σ,τ]⊂[0,1] such that|f(t,0)|>0. Similarly, ash(t)≥|f(t,0)|a.e.t∈[0,1], we haveB>0.

WriteM=B/(1-A),and letΩ= {x∈PC(J,):‖x‖1 such thatTx=λx. Then

Theorem2Suppose thatf(t,0) is not identically zero. Then the boundary value problem(1) has a nontrivial solutionx*∈Eif the conditions below hold:

(i) There exists a positive constantrsuch that

Proof The condition (i) follows there exists a nonnegative functionsh∈L1[0,1] such that

|f(t,x)|≤r|x|+h(t)

for a.e. (t,x)∈[0,1]×andk(s)≤rfors∈[0,1].From the definition ofρ(k),ξ(k) andIk,Jk∈C(,), we obtain that there existsM1>0 such that

|Jk(x)|≤ρ(k)|x|+M1, |Ik(x)|≤ξ(k)|x|+M1

forx∈.

From condition (ii),letM> 0 be a sufficiently large constant such that

andΩ= {x∈PC(J,):‖x‖ 1 withTx=λx. Then ‖x‖=Mand

Hence,

4 Examples

In this section, we give some examples to illustrate the applications of our main results.

Example1Consider the three point boundary value problem with impulses

(5)

Example2Consider the three point boundary value problem with impulses

(6)

whereη≠tiandη∈(0,1). We note that

Since

it follows by Theorem 2 that the boundary value problem (6) has at least one nontrivial solutionx*inE.

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