Uniqueness of Positive Solutions for a Class of Fourth-Order Boundary Value Problems

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fourth-order boundary value problem: , , . Moreover, under certain assumptions, we will prove that the above boundary value problem has a unique symmetric positive solution. Finally, we present some examples and we compare our results with the ones obtained in recent papers. Our analysis relies on a fixed point theorem in partially ordered metric spaces.

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

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.