A monotone method for fourth order boundary value problems involving a factorizable linear operator

Abstract

We consider the nonlinear fourth order beam equation

\[ u^{\iv}=f(t,u,u''), \]

with boundary conditions corresponding to the periodic or the hinged beam problem. In presence of upper and lower solutions, we consider a monotone method to obtain solutions. The main idea is to write the equation in the form

\[ u^{\iv}-cu''+du=g(t,u,u''), \]

where $c$, $d$ are adequate constants, and use maximum principles and a suitable decomposition of the operator appearing in the left-hand side.