Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem

Abstract

This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem u(4)=f(t,u,u′,u″,u‴) on [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, where f:[0,1]×R+×R×R−×R→R+ is continuous. Some precise inequality conditions on f guaranteeing the existence of positive solutions are presented. The inequality conditions allow that f(t,u,v,w,z) may be asymptotically linear, superlinear, or sublinear on u,v,w, and z as |(u,v,w,z)|→0 and |(u,v,w,z)|→∞. Our discussion is based on the fixed point index theory in cones.

Keywords

• fully fourth-order boundary value problem
• simply supported beam equation
• positive solution
• cone
• fixed point index