Three-point boundary value problems in Banach spaces

Abstract

Using degree for $$\alpha $$ -condensing maps, we obtain the existence of at least one solution for nonlinear boundary value problems $$\begin{aligned} \left\{ \begin{array}{lll} (\varphi (u' ))' = f(t,u,u') &{} &{} u(0)=u(1)=u'(0), &{} &{} \quad \quad \end{array}\right. \end{aligned}$$ where $$\varphi : X\rightarrow X $$ is a linear homeomorphism, $$f:\left[ 0, 1\right] \times X \times X \rightarrow X $$ is a continuous function and X is a real Banach space.