Positive solutions of some nonlocal fourth-order boundary value problem

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Abstract

By the use of the Krasnosel’skii’s fixed point theorem, the existence of one or two positive solutions for the nonlocal fourth-order boundary value problemu(4)(t)+βu(t)=λf(t,u(t),u(t)),0<t<1,u(0)=u(1)=01p(s)u(s)ds,u(0)=u(1)=01q(s)u(s)ds,is considered, where p,qL[0,1],λ>0,fC([0,1]×[0,)×(-,0],[0,)).

Introduction

The deformation of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by a fourth-order ordinary equation boundary value problem. Owing to its significance in physics, it has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point index theory and the method of upper and lower solutions, see for example, [1], [2], [3], [4], [6], [7], [8], [9], [10], [9], [10], [11], [12], [13], [14], [15], [16].

Recently, there has been much attention focused on the question of positive solution of fourth-order differential equation with one or two parameters. For example, Li [6] investigated the existence of positive solutions for the fourth-order boundary value problemu(4)(t)+βu(t)-αu(t)=f(t,u(t)),0<t<1,u(0)=u(1)=u(0)=u(1)=0,under the assumptions:

  • (J1)

    f:[0,1]×[0,+)[0,+) is continuous;

  • (J2)

    β<2π2,α-β2/4,α/π4+β/π2<1.

Chai [3] studied the above generalizing form with variable parameters. All of the results consider only an equation with two-point boundary value condition. More recently, Ma [9], [10] studied the existence of symmetric positive solutions of the nonlocal fourth-order boundary value problemu(t)=h(t)f(t,u),0<t<1,u(0)=u(1)=01p(s)u(s)ds,u(0)=u(1)=01q(s)u(s)ds.Motivated by above papers we consider the following nonlocal boundary value problemu(4)(t)+βu(t)=λf(t,u(t),u(t)),0<t<1,u(0)=u(1)=01p(s)u(s)ds,u(0)=u(1)=01q(s)u(s)ds.

We assume the following conditions throughout

  • (A1)

    λ>0 and 0<β<π2.

  • (A2)

    fC([0,1]×[0,)×(-,0],[0,)),p,qL[0,1],p(s)0,q(s)0,

0p(s)ds<1,01q(s)sinβsds+01q(s)sinβ(1-s)ds<sinβ.

Section snippets

The preliminary lemmas

Set λ1=0,0>λ2=-β>-π2. Given mL[0,1], letδ1=1-01m(x)dx,δ2=sinβ-01m(s)sinβsds-01m(s)sinβ(1-s)ds.By (A1),(A2), there hold δi0,i=1,2. Denote by Hi(t,s),i=1,2 the Green’s functions of the following problems-u(t)+λiu(t)=0,0<t<1,u(0)=u(1)=01m(s)u(s)ds.Then, carefully calculation yieldH1(t,s)=G1(t,s)+1δ101G1(s,x)m(x)dx;G1(t,s)=t(1-s),0ts1;s(1-t),0st1;H2(t,s)=G2(t,s)+sinβt+sinβ(1-t)δ201G2(s,x)m(x)dx;G2(t,s)=sinβtsinβ(1-s)βsinβ,0ts1;sinβssinβ(1-t)βsinβ,0st1.Denoteρ1=11-01p(x)dx,ρ2(t)

The main results

Suppose K1,K2,G2,ρ2,C2,θare defined as in Section 2, we introduce some notations as follows:A=0101K1(s,s)K2(s,τ)dτds,B=01G2(s,s)+ρ21201G2(s,x)q(x)dxds,η0=1A+C2B,η1=1θ1434K2(12,τ)dτ,f¯0=limsup|u|+|v|0maxt[0,1]f(t,u,v)|u|+|v|,f̲0=liminf|u|+|v|0mint[1/4,3/4]f(t,u,v)|u|+|v|,f¯=limsup|u|+|v|+maxt[0,1]f(t,u,v)|u|+|v|,f̲=liminf|u|+|v|+mint[1/4,3/4]f(t,u,v)|u|+|v|,

Theorem 3.1

Assume (A1), (A2) hold. Problem (1.3) has at least one positive solution, if one of the following cases holds

  • (i)

    f¯0<1λη0,f̲>1λ

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