Advancements in Fixed Point Results of Generalized Metric Spaces: A Survey

Abstract

The increasing significance of metric spaces and their applications in sciences and engineering has manifested over the years. This has led to the emergence of fixed point theory in metric spaces which in turn, has many practical usefulness in inequalities, approximation theory, optimization theory, image restoration and filtering, to mention but a few. Following up this development, in this paper, various results on G-metric spaces (also called generalized metric spaces) introduced by Mustafa and Sims are reviewed. Extensions of fixed point theorems for Lipschitzian-type mappings on G-metric spaces are compiled and a concise report on the transition in fixed point theorems on G-metric spaces are established. The aim of this survey is therefore, to examine and provide an up-to-date analysis of the important advancements in the fixed point theory of G-metric spaces. Consequently, this note is handy for researchers in the domain of metric and pseudo-metric spaces as they can easily appreciate how new results are delineated from the subsequent ones.

Advancements in Hybrid Fixed Point Results and F-Contractive Operators

“…In 2023, Hammad et al [11] defined a new class of control functions and discussed fixed point results which properly refined some known ones in the literature. The idea of [11] is to take care of some peculiarities of a G-MS. For a recent survey of the advancements in fixed point results in G-MS, the reader can consult Jiddah et al [12].…”

On Two-Point Boundary Value Problems and Fractional Differential Equations via New Quasi-Contractions

“…One of the extensions of the principle by replacing the contractive constant with a family of functions was established by Rakotch [2]. For some recent refinements of the Banach fixed point theorem, one can consult [3][4][5] and the references therein. A few of these earlier improvements that are of interest to us in this current project include the work of Ciric [6], Geraghty [7], Jaggi [8], and Dass-Gupta [9].…”