A new fixed point theorem in $T_{\alpha}$-metric spaces with application

Fixed Points in $T_{\alpha}$-Metric Spaces

Abstract

In this article, we introduce the concept of $T_{\alpha}$-metric space as a generalization that encompasses several existing generalized metric spaces, including $b$-metric, multiplicative metric and extended $b$-metric spaces. We establish necessary and sufficient conditions for a function defined on such spaces to satisfy Banach's contraction principle, thus proving the existence and uniqueness of a fixed point. Our approach unifies various extensions of Banach's fixed point theorem present in recent literature. We illustrate our theoretical results with examples and provide an application to solving nonlinear Fredholm integral equation.