A Natural Extension of the Banach Fixed Point Theorem in a b-Metric Space with an Orthogonal Direct Sum Structure

This study aims to develop new versions of the Banach fixed point theorem in generalized metric spaces endowed with a direct sum structure. Specifically, we assume a diagonal matrix \ (A\) in \ (R^d d\) and establish more appropriate contraction conditions to improve the applicability of fixed point results within this framework. Since the condition that the matrix \ (A\) must converge to zero is unnecessary, our approach yields stronger results than the Perov one. As an application of our findings, we examine the existence and uniqueness of solutions for a system of matrix equations. This version is more powerful than the Perov version. We introduced some examples and applications to illustrate our result.