Abstract
In this paper, we prove common fixed point results for quadruple self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize many relevant results existing in literature. Also, we define the concept of compatible maps and its variants in the setting of digital metric space and establish some common fixed point results for these maps. Also, an application of the proposed results is quoted in this note.
Highlights
We prove common fixed point results for quadruple self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions
We define the concept of compatible maps and its variants in the setting of digital metric space and establish some common fixed point results for these maps
Digital topology is an emerging area based on general topology and functional analysis and focuses on studying digital topological properties of n-dimensional digital spaces, where as Euclidean topology deals with topological properties of subspaces of the n-dimensional real space, which has contributed to the study of some areas of computer sciences such as computer graphics, image processing, approximation theory, mathematical morphology, optimization theory etc
Summary
Digital topology is an emerging area based on general topology and functional analysis and focuses on studying digital topological properties of n-dimensional digital spaces, where as Euclidean topology deals with topological properties of subspaces of the n-dimensional real space, which has contributed to the study of some areas of computer sciences such as computer graphics, image processing, approximation theory, mathematical morphology, optimization theory etc. Rosenfield [1] was the first to consider digital topology as a tool to study digital images. Ege and Karaca [4] [5] established relative and reduced Lefschetz fixed point theorem for digital images and proposed the notion of a digital metric space and proved the famous Banach Contraction Principle for digital images. In metric spaces, this theory begins with the Banach fixed-point theorem which provides a constructive method of finding fixed points and an essential tool for solution of some problems in mathematics and engineering and has been generalized in many ways. A major shift in the arena of fixed point theory came in 1976 when Jungck [6] [7] [8], defined the concept of commutative and compatible maps and proved the common fixed point results for such maps
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