Let $C$ be a subset of a Hilbert space, and let $f$ and $g$ be self-maps on $C$ such that the range of $f$ is a convex, closed, and bounded subset of the range of $g$. If $f$ does not increase distances more than $g$, we demonstrate that $f$ and $g$ have coincidence points. This result generalizes a fixed point theorem of Browder-Petryshyn. As applications, we establish the existence of solutions to both matrix and integral equations.