Lower and upper solutions method to the fully elastic cantilever beam equation with support

Abstract The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is, { u ( 4 ) ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) , u ‴ ( t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ′ ( 0 ) = 0 , u ″ ( 1 ) = 0 , u ‴ ( 1 ) = g ( u ( 1 ) ) , $$ \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u(0)=u'(0)=0, \\ u''(1)=0,\qquad u'''(1)=g(u(1)), \end{cases} $$ where f : [ 0 , 1 ] × R 4 → R $f:[0,1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}$ , g : R → R $g: \mathbb{R}\rightarrow \mathbb{R}$ are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.