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

• Main Authors: Yongxiang Li, Yabing Gao
• Format: Article
• Language: English
• Published: SpringerOpen 2019-05-01
• Series: Journal of Inequalities and Applications
• Subjects: Fully fourth-order boundary value problem; Cantilever beam equation; Lower and upper solution; Existence
• Online Access: http://link.springer.com/article/10.1186/s13660-019-2088-5

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.