Applications of Fixed Point Theorems in Uniform Spaces

Authors: Vikrant Panchal, Vinit Kumar Sharma

Abstract: Fixed point theory is a cornerstone of modern mathematical analysis with applications spanning topology, functional analysis, and applied sciences. Uniform spaces generalize metric spaces by replacing distance with a uniform structure, thereby allowing broader applicability. This paper investigates extensions of classical fixed point theorems—particularly Banach and Schauder types—within uniform spaces. We establish conditions for existence and uniqueness of fixed points and demonstrate applications in functional equations, integral equations, and optimization problems. We studied certain popular fixed-point theorems for pairs of weakly and semi-compatible mappings. For a pair of self-mappings in Hausdorff uniform spaces, we discussed various stability results for a few common fixed points. Fixed point theorems in uniform spaces provide powerful, generalized tools to ensure the existence of solutions for equations where standard metric space distances (like Banach’s) are insufficient, such as in topological vector spaces. These applications include solving systems of operator equations, finding fixed points for contractive mappings on gauge spaces, and analyzing stability in Hausdorff uniform spaces.

DOI: https://doi.org/10.5281/zenodo.19677564