On a theorem by Browder and its application to nonlinear boundary value problems

TL;DR

This paper extends Browder's 1960 theorem on fixed points of parameter-dependent functions to more general spaces and applies it to nonlinear boundary value problems, offering new insights, results, and open questions.

Contribution

It generalizes Browder's theorem to arbitrary normed spaces and introduces new applications and perspectives for nonlinear boundary value problems.

Findings

Extended Browder's theorem to normed spaces

Presented new viewpoints on nonlinear boundary value problems

Identified open problems in the field

Abstract

In a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions $f_t:X\to X$ depending continuously on a parameter $t\in [0,1]$, where $X$ is a convex and compact subset of $\R^n$. Here, the result is presented for a compact mapping $f:A\times X\to X$ where $X$ is a convex, closed and bounded subset of an arbitrary normed space and $A$ is an arcwise connected topological space. Applications to nonlinear boundary value problems are given; specifically, we shall present new viewpoints of known results, introduce some novel results and exhibit some open problems.