Metric Space: A Generalization

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.

Publicationes Mathematicae Debrecen

We give a fixed point theorem related to the contraction mapping principle of Banach and Caccioppoli; here we have considered generalized metric spaces, that is metric spaces with the triangular inequality replaced by similar ones which involve four or more points instead of three. At the end of the paper an example is provided to show the improvement of our result with respect to the classical one.

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."

Several fixed point and common fixed point theorems are obtained in the setting of metric-type spaces introduced by M. A. Khamsi in 2010.

2017

The subject of this paper belongs to the theory of approximate metrics [23]. An approximate metric on X is a real application defined on X × X that satisfies only a part of the metric axioms. In a recent paper [23], we introduced a new type of approximate metric, named C-metric, that is an application which satisfies only two metric axioms: symmetry and triangular inequality. The remarkable fact in a C-metric space is that a topological structure induced by the C-metric can be defined. The innovative idea of this paper is that we obtain some convergence properties of a C-metric space in the absence of a metric. In this paper we investigate C-metric spaces. The paper is divided into four sections. Section 1 is for Introduction. In Section 2 we recall some concepts and preliminary results. In Section 3 we present some properties of C-metric spaces, such as convergence properties, a canonical decomposition and a C-fixed point theorem. Finally, in Section 4 some conclusions are highlighted.

International Journal of ADVANCED AND APPLIED SCIENCES

The Banach contraction principle, which shows that every contractive mapping has a unique fixed point in a complete metric space, has been extended in many directions. One of the branches of this theory is devoted to the study of fixed points. Especially, Fixed point theory in C *-algebra valued metric spaces has greatly developed in recent times. Also, we study on generalized C *-algebra valued metric space and give some examples, the idea of this metric is to replace the set of real numbers by the positive cone C *-algebras, the set of positive elements on the C *-algebras the notation introduced recently. Also, we prove certain fixed-point theorem for a singlevalued mapping in such spaces. The mapping we consider here is assumed to satisfy certain D-metric conditions with generalized fixed-point theorem. Moreover, the paper provides an application to prove the existence and uniqueness of fixed points.

Filomat, 2021

Non-triangular metric spaces have been introduced by Khojasteh and Khandani [12] in 2020. The concept of non-triangular metric spaces is fresh and therefore, requires quite some analysis on its topology. In this article, we have made an attempt to study the topology of non-triangular metric spaces by giving a natural definition of open sets in this context. The introduction of non-triangular metric spaces has shown that there is no inherent need for the triangle inequality to prove various fixed point results. Keeping this in mind, we give a fixed point result for Suzuki type Z-contractions in the context of non-triangular metric spaces by introducing a new property of maps.

Transactions of the American Mathematical Society, 1986

Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of Zp-cubes into the ¡i-cube (Hamming cube) are discussed.

2014

Gähler ([4], [5]) introduced and investigated the notion of 2-metric spaces and 2-normed spaces in sixties. These concepts are inspired by the notion of area in two dimensional Euclidean space. In this paper, we choose a fundamentally different approach and introduce a possible generalization of usual norm retaining the distance analogue properties. This generalized norm will be called as G-norm. We show that every G-normed space is a G-metric space and therefore, a topological space and develop the theory for G-normed spaces. We also introduce G-Banach spaces and obtain some fixed point theorems

The main purpose of this paper is to introduce several concepts of the metric-like spaces. For instance, we define concepts such as equal-like points, cluster points and completely separate points. Furthermore, this paper is an attempt to present compatibility definitions for the distance between a point and a subset of a metric-like space and also for the distance between two subsets of a metric-like space. In this study, we define the diameter of a subset of a metric-like space, and then we provide a definition for bounded subsets of a metric-like space. In line with the aforemen-tioned issues, various examples are provided to better understand this space.