  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.
