Common fixed point theorems for discontinuous mappings in M-metric space and numerical approximations

Generalized metric spaces have emerged as efficient extensions to the traditional metric space and provided novel insights into the generalized fixed point results. In this manuscript, we present a novel approach for fixed point results within the framework of $M$-metric space, characterized by the contraction mappings. The condition of continuity of these Banach contraction mappings is not essential in this structure, as it is in the usual metric space. We identify the scenarios in which traditional contraction mappings in metric space do not hold. But their extended versions within $M$-metric space establish the desired results. Additionally, we study the behavior of contraction mappings graphically in both metric space and $M$-metric space to highlight the difference and importance of $M$-metric space. Further, some results on the existence of common fixed points for pairs of self-mappings in incomplete space are established. Also, we discuss the result on the existence of a common fixed point for three self-mappings (not necessarily continuous), which is an extension of the work of Petrov (2023). To support our findings, various examples are discussed and some numerical iterations with the graphs for approximating the common fixed point are presented.