Existence and localization of solutions for fourth-order boundary-value problems

Abstract

In this paper, we study the existence of solutions for the differential equation $$ u^{(4)}(t)=fig(t,u(t),u''(t)ig), $$ where $f$ satisfies one-sided Lipschitz conditions with respect to $u$ and $u''$, with periodic conditions or boundary conditions from ``simply supported'' beam theory. We assume the existence of lower and upper solutions (well-ordered and in some cases reversely ordered) and we make use of a fourth-order linear differential operator factorization.

Keywords

• Beam equation
• fourth order boundary value problem
• maximum principles
• lower and upper solutions
• reversed order