Lower and upper solutions for a fully nonlinear beam equation

In this paper the two point fourth order boundary value problem is considered { u ( i v ) = f ( t , u , u ′ , u ″ , u ‴ ) , 0 < t < 1 , u ( 0 ) = u ′ ( 1 ) = u ″ ( 0 ) = u ‴ ( 1 ) = 0 , where f : [ 0 , 1 ] × R 4 → R 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.