On Multi-Metric Spaces

Journal of Nonlinear and Convex Analysis, 2014

"Gahler ([3] ,[4]) introduced the concept of 2-metric as a possible generalization of usual notion of a metric space. In many cases the results obtained in the usual metric spaces and 2-metric spaces are found to be unrelated (see [5]). Mustafa and Sims [8] took a different approach and introduced the notion of G-metric. The author [6] generalized the notion of G-metric to more than three variables and introduced the concept of K-metric as a function K: X^n---->R^+, (n> or = 3). In this paper, We improve the definition of K-metric by making symmetry condition more general. This improved metric denoted by G_n is called the Generalized n-metric. We develop the theory for generalized n-metric spaces and obtain some fixed point theorems."

Arxiv, 2021

Mustafa and Sims [12] introduced the notion of G-metric as a possible generalization of usual notion of a metric space. The author generalized the notion of G-metric to more than three variables and introduced the concept of Generalized n-metric spaces [10]. In this paper, We prove Banach fixed point theorem and a Suzuki-type fixed point theorem in Generalized n-metric spaces. We also discuss applications to certain functional equations arising in dynamic programming.

We introduce a new concept of generalized metric spaces and extend some well-known related fixed point theorems including Banach contraction principle, Ćirić's fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and Rodríguez-López. This new concept of generalized metric spaces and extend some well-known related fixed point theorems recover various topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces.

We introduce the notion of -metric as a generalization of a metric by replacing the triangle inequality with a more generalized inequality. We investigate the topology of the spaces induced by a -metric and present some essential properties of it. Further, we give characterization of well-known fixed point theorems, such as the Banach and Caristi types in the context of such spaces.

2016

Our main aim in this paper is to introduce a general concept of multidimensional fixed point of a mapping in spaces with distance and establish various multidimensional fixed point results. This new concept simplifies the similar notion from [A. Roldan, J. Martinez-Moreno, C. Roldan, {\it Multidimensional fixed point theorems in partially ordered complete metric spaces}, J. Math. Anal. Appl. 396 (2012), 536--545]. The obtained multiple fixed point theorems extend, generalise and unify many related results in literature.

Journal of Mathematics, 2022

The notion of Δ -metric spaces has been proposed in this study as a generalization of b -metric spaces, extended b -metric spaces, and p -metric spaces. A number of topological characteristics of such spaces have been investigated in this paper. On such spaces, a noncompactness measure has been established, and some results in the framework of noncompactness measure have been achieved. We prove an analogous of the Banach contraction principle in such spaces based on this approach. In order to investigate the validity of the underlying space and our proven fixed-point theorems, supporting examples have been presented. Furthermore, the well-posedness of the fixed-point problem has been tested using our fixed-point result.

We present a survey of fixed point results in generalized metric spaces (g.m.s.) in the sense of Branciari , A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen, 57, 31-37]. Since it may happen that the topology of such space is not Hausdorff, several authors added Hausdorfness (or some other condition) as an additional assumption in order to obtain their results. We show here that such assumptions are usually superfluous. Finally, we state some open questions on the topic.

arXiv: General Topology, 2017

In this paper we make some observations concerning m-metric spaces and point out some discrepancies in the proofs found in the literature. To remedy this, we propose a new topological construction and prove that it is in fact a generalization of a partial metric space. Then, using this construction, we present our main theorem having as its corollaries the fixed point theorems found in previous publications.

In this paper, the Banach contraction principle theorem is proved in rectangular metric-like spaces and rectangular metric spaces. The extension of fixed point theorems in rectangular metric type spaces is generalized by various contraction mappings to obtain new fixed point results of rectangular metric spaces, which are extended to rectangular metric-like spaces.

2015

In this paper we combine the notions of partial metric spaces with negative distances, $G_p$-metric spaces and n-metric spaces together into one structure called the partial n-metric spaces. These are generalizations of all the said structures, and also generalize the notions of $G$-metric and $G_p$-metric spaces to arbitrary finite dimension. We prove Cauchy mapping theorems and other fixed point theorems for such spaces.