Existence and location result for the bending of a single elastic beam

Abstract

In the present work we prove an existence and location result for the fourth order fully nonlinear equation

u^{(iv)}=f(t,u,u′,u′′,u′′′), 0<t<1,

with the Lidstone boundary conditions

u(0)=u′′(0)=u(1)=u′′(1)=0,

where f is a continuous function satisfying a Nagumo-type condition. We remark that f must only verify some assumptions of bound type and no monotonicity restrictions are considered, as it is usual. The existence of at least a solution lying between a pair of well ordered lower and upper solutions is obtained using an a priori estimates, lower and upper solutions method and degree theory.