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Structure for $g$-Metric Spaces and Related Fixed Point Theorems

Hayoung Choi, Sejong Kim, Seung Yeop Yang

Abstract

In this paper, we propose a generalized notion of a distance function, called a $g$-metric. The $g$-metric with degree $n$ is a distance of $n+1$ points, generalizing the ordinary distance between two points and $G$-metric between three points. Indeed, it is shown that the $g$-metric with degree 1 (resp. degree 2) is equivalent to the ordinary metric (resp. the $G$-metric). Fundamental properties and several examples for the $g$-metric are also given. Moreover, topological properties on the $g$-metric space including the convergence of sequences and the continuity of mappings on the $g$-metric space are studied. Finally, we generalize some well-known fixed point theorems including Banach contraction mapping principle and Ćirić fixed point theorem in the $g$-metric space.

Structure for $g$-Metric Spaces and Related Fixed Point Theorems

Abstract

In this paper, we propose a generalized notion of a distance function, called a -metric. The -metric with degree is a distance of points, generalizing the ordinary distance between two points and -metric between three points. Indeed, it is shown that the -metric with degree 1 (resp. degree 2) is equivalent to the ordinary metric (resp. the -metric). Fundamental properties and several examples for the -metric are also given. Moreover, topological properties on the -metric space including the convergence of sequences and the continuity of mappings on the -metric space are studied. Finally, we generalize some well-known fixed point theorems including Banach contraction mapping principle and Ćirić fixed point theorem in the -metric space.

Paper Structure

This paper contains 5 sections, 21 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction and preliminaries
  2. Theory of a $g$-metric
  3. Topology on a $g$-metric space
  4. Convergence and continuity in $g$-metric spaces
  5. Fixed point theorems

Key Result

Theorem 2.4

Let $\Omega$ be a given nonempty set. The following are true.

Figures (1)

  • Figure 1: While a convergent sequence is defined by the distance between $x_k$ and $x$ (left), a $g$-convergent sequence is defined by the distance (i.e., $g$-metric) between $x_{i_1},\ldots, x_{i_n}$ and $x$ (right).

Theorems & Definitions (72)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Example 1.5
  • Definition 1.6
  • Example 1.7
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 62 more