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

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.