Existence and Uniqueness of Fixed Points in $MR-Metric$ Spaces and Their Applications

This paper investigates fixed-point theorems within \( MR \)-metric spaces, an extension of standard metric spaces, emphasizing the existence and uniqueness of fixed points for continuous mappings \( S: \mathbb{X} \to \mathbb{X} \), where \( \mathbb{X} \) is a closed, bounded, and convex subset of a Banach space \( (E, \|\cdot\|) \). The study establishes that if \( S \) satisfies a contractive condition involving the \( MR \)-metric with a constant \( k \in [0, 1) \), and a measure of noncompactness condition governed by a function \( \phi \) where \( \phi(t) < t \) for \( t > 0 \), then \( S \) possesses a unique fixed point \( \upsilon^* \). The findings have significant applications in solving nonlinear integral equations, ensuring stability of iterative processes, optimization, game theory, economic equilibria, and boundary value problems, showcasing the versatility of \( MR \)-metric spaces in addressing noncompact settings and fixed-point problems.