Symmetric Positive Solutions for Nonlinear Singular Fourth-Order Eigenvalue Problems with Nonlocal Boundary Condition

Fuyi Xu,Jian Liu

doi:10.1155/2010/187827

Abstract

We investigate nonlinear singular fourth-order eigenvalue problems with nonlocal boundary conditionu(4)(t)-λh(t)f(t,u,u′′)=0,0<t<1,u(0)=u(1)=∫01a(s)u(s)ds,u′′(0)=u′′(1)=∫01b(s)u′′(s)ds, wherea,b∈L1[0,1],λ>0,hmay be singular att=0and/or1. Moreoverf(t,x,y)may also have singularity atx=0and/ory=0. By using fixed point theory in cones, an explicit interval forλis derived such that for anyλin this interval, the existence of at least one symmetric positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and nonsingular cases. The associated Green's function for the above problem is also given.

Highlights

We investigate nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition u 4 t − λh t f t, u, u
Boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics and so on, and the existence of positive solutions for such problems has become an important area of investigation in recent years
A class of boundary value problems with nonlocal boundary conditions appeared in heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics

Summary

Introduction

Boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics and so on, and the existence of positive solutions for such problems has become an important area of investigation in recent years. By using fixed point theory in cones, an explicit interval for λ is derived such that for any λ in this interval, the existence of at least one symmetric positive solution to the boundary value problem is guaranteed. Motivated by the works mentioned above, in this paper, we study the existence of symmetric positive solutions of the following fourth-order nonlocal boundary value problem BVP : u 4 t − λh t f t, u, u 0, 0 < t < 1, u0 u1 a s u s ds, 1.1

Results

Conclusion