The method of lower and upper solutions for the cantilever beam equations with fully nonlinear terms

In this paper we discuss the existence of solutions of the fully fourth-order boundary value problem {u(4)=f(t,u,u′,u″,u‴),t∈[0,1],u(0)=u′(0)=u″(1)=u‴(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} u^{(4)}=f(t, u, u', u'', u'''), \quad t\in [0, 1], \\ u(0)=u'(0)=u''(1)=u'''(1)=0 , \end{cases} $$\end{document} which models the deformations of an elastic cantilever beam in equilibrium state, where f:[0,1]×R4→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:[0, 1]\times {\mathbb{R}}^{4}\to \mathbb{R}$\end{document} is continuous. Using the method of lower and upper solutions and the monotone iterative technique, we obtain some existence results under monotonicity assumptions on nonlinearity.