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

Abstract

Abstract 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, $$ \textstyle\begin{cases} u^{(4)}=f(t, u, u', u'', u'''), \quad t\in [0, 1], \\ u(0)=u'(0)=u''(1)=u'''(1)=0 , \end{cases} $$ which models the deformations of an elastic cantilever beam in equilibrium state, where f:[0,1]×R4→R $f:[0, 1]\times {\mathbb{R}}^{4}\to \mathbb{R}$ 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.

Keywords

• Fully fourth-order boundary value problem
• Cantilever beam equation
• Lower and upper solution
• Existence