An Existence Result of Positive Solutions for the Bending Elastic Beam Equations

Abstract

This paper is concerned with the existence of positive solutions to the fourth-order boundary value problem u(4)(x)=f(x,u(x),u″(x)) on the interval [0, 1] with the boundary condition u(0)=u(1)=u″(0)=u″(1)=0, which models a statically bending elastic beam whose two ends are simply supported. Without assuming that the nonlinearity f(x, u, v) is nonnegative, an existence result of positive solutions is obtained under the inequality conditions that |(u, v)| is small or large enough. The discussion is based on the method of lower and upper solutions.