Lower and upper solutions for a fully nonlinear beam equation

Abstract

In this paper the two point fourth order boundary value problem is considered u^{(iv)}=f(t,u,u',u'',u'''), 0<t<1, u(0)=u'(1)=u''(0)=u'''(1)=0, where is a continuous function satisfying a Nagumo-type condition. We prove the existence of a solution lying between lower and upper solutions using an a priori estimation, lower and upper solutions method and degree theory. The same arguments can be used, with adequate modifications, for any type of two-point boundary value problem, including all derivatives until order three, with the second and the third derivatives given in different end-points. An application to the extended Fisher-Kolmogorov problem will be obtained.