Dynamic programming and fixed point theorems in generalized metric spaces

In this study, several important aspects of research in the field of fixed point theory are highlighted. The generalization and modification of mappings, various generalizations of the underlying spaces, and the corresponding variations of fixed point theorems in these spaces are discussed with illustrative examples. By investigating the practical applications of metric spaces and fixed point theory in G-metric spaces introduced by Mustafa and Sims (2006), the evolving structure of fixed point theorems in other generalized metric spaces is emphasized, and various results concerning fixed point theorems for Lipschitz-type mappings are included. When these metric spaces are complete, new common fixed point theorems for certain mappings are established, providing extensions of previous results. The presented findings are further supported by applications, including common solutions for some systems of functional equations in dynamic programming, graph theory, and integral-type contractions. In summary, a concise analysis of the recent developments in fixed point theory within modular G-metric and modular b-metric spaces is provided.