Solvability of an elastic beam equation in presence of a sign-type Nagumo control

Abstract

The aim of this work is to establish an existence and localization result for the fourth order two-point boundary value problem\begin{equation*}\left\{\begin{array}{l}u^{(iv)}=f\left( x,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime}\right) ,\quad 0<x<1,\medskip \\u\left( 0\right) =u^{\prime }\left( 1\right) =u^{\prime \prime }\left(0\right) =u^{\prime \prime \prime }\left( 1\right) =0,\end{array}\right.\end{equation*} related to elastic beam models. It is assumed that $f:\left[ 0,1 \right] \times \mathbb{R}^{4}\rightarrow \mathbb{R}$ is a continuous function\ satisfying a sign-type Nagumo growth condition which allows an asymmetric unbounded behavior\ on the nonlinearity. The arguments make use of the lower and upper solution method and degree theory.