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 \(\mathbb{R}^{d\times d}\) 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.