Category of a map and nonlinear analysis

A classic problem in analysis is to solve nonlinear equations of the form \begin{equation*} F(x)=0, \end{equation*} where $F:D^n\to {\mathbb{R}}^m$ is a continuous map of the closed unit disk $D^n\subset {\mathbb{R}}^n$ in ${\mathbb{R}}^m$. A topological technique, which exists in the literature, for the existence of solutions of nonlinear equations is the topological degree theory. In this work, we will use the category of a map theory to solve the problem of existence of solutions of nonlinear equations. This theory, as we will show in this work, provides an alternative topological technique to study nonlinear equations.