"The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects"

Abstract

"A very impressive research work has been devoted in the last two decades to obtaining fixed point theorems in quasimetric spaces (also called b-metric spaces). Some incorrect and incomplete references with respect to the early developments on fixed point theory in b-metric spaces are though perpetually taking over from the existing publications to the new ones. Starting from this fact, our main aim in this note is threefold: (1) to briefly survey the early developments in the fixed point theory on quasimetric spaces (b-metric spaces); (2) to collect some relevant bibliography related to this topic; (3) to discuss some other aspects of current interest in the fixed point theory on quasimetric spaces (b-metric spaces)."

The nonlinear contraction in probabilistic cone b-metric spaces with application to integral equation

“…Some mathematicians have developed this result by relaxing or modifying the contraction conditions [3,7,8,20] or by introducing new metrics spaces [12,14,15,23] with different structures. In this regard, by modifying the triangle inequality, Vulpe et al [23] were the first to propose a generalized form of a metric space (b-metric space) in 1981 due to Berinde and Pȃcurar [6] according to the current bibliographical knowledge. In the same direction, Hussain and Shah [10] generalized the notion of a b-metric space by changing the set of real numbers by an ordered Banach space, and they proved some fixed point theorems on this type of spaces called the cone b-metric space, which is also a generalization of the idea of a cone metric space [2,11,14,21].…”

Controlled v-generalized metric type space and some fixed point theorems with application

“…One of those is the concept of b-metric spaces which was introduced by Czerwik [6] in 1993. The concepts of metric and b-metric structures have been generalized in many ways (see [3,4,10,15,16,17,18,19,20,21,22,23]). Branciari [5] presented rectangular metric space in 2000, another such generalized metric structure, where the triangle inequality had been replaced by a so-called quadrilateral or rectangular inequality.…”

F-Contractions Endowed with Mann’s Iterative Scheme in Convex Gb-Metric Spaces

“…It would be a real benefit if this article is considered for further research. Czerwik [5] defined this concept formally in 1993 by introducing a condition that was weaker than the third property of MS. Czerwik and many other researchers generalized the Banach contraction principle using these spaces (see [6][7][8][9]). Mustafa and Sims [10] presented the idea of G-MS in 2006, which was further generalized by Aghajani et al [11] by presenting the concept of G b -MS.…”