Problems with mixed boundary conditions in Banach spaces

TL;DR

This paper establishes the existence of solutions for certain boundary value problems in Banach spaces using degree theory, addressing issues with mixed boundary conditions.

Contribution

It introduces a method employing Leray-Schauder degree for b1-condensing maps to prove solution existence in Banach space boundary value problems.

Findings

Existence of solutions proven for specific boundary value problems.

Applicable to problems with mixed boundary conditions in Banach spaces.

Utilizes degree theory for b1-condensing maps.

Abstract

Using Leray-Schauder degree or degree for $\alpha$-condensing maps we obtain the existence of at least one solution for the boundary value problem of the type \[ \left\{\begin{array}{lll} (\varphi(u' ))' = f(t,u,u') & & \\ u(T)=0=u'(0), & & \quad \quad \end{array}\right. \] where $\varphi: X\rightarrow X $ is a homeomorphism with reverse Lipschitz such that $\varphi(0)=0$, $f:\left[0, T\right]\times X \times X \rightarrow X $ is a continuous function, $T$ a positive real number and $X$ is a real Banach space.