Common Fixed Point Results for Compatible Map in Digital Metric Space

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.


Introduction
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.Boxer [2] produced the digital versions of the topological concepts and later studied digital continuous functions [3].Ege and Karaca [4]   science, engineering, game theory, fuzzy theory, image processing and so forth.
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 consequently has been generalized in many ways.Up to now, several developments have occurred in this area.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.
Later on, Sessa [9] gave the concept of weakly compatible, and proved results for set valued maps.Certain altercations of commutativity and compatibility can also be found in [10] [11] [12] [13].
In 2014, He et al. [14] proved the common fixed points for pair of weak commutative mappings on a complete multiplicative metric spaces as follow: Theorem 1.Let , , S T A and B be mappings of a complete multiplicative metric space ( ) , X d into itself satisfying the following conditions: 3) one of the mappings , , S T A and B is continuous.Assume that the pairs The objective of this paper is to give digital version of above theorem using an implicit function which is general enough to cover several linear as well as some nonlinear contractions.Our results generalize and extend the many results existing in literature.
This paper is organized as follows.In the first part, we give the required background about the digital topology and fixed point theory.In the next section, we state and prove main results for compatible mappings and compatible mappings of types (A) and (P) in digital metric spaces.Our results improve and generalize many other results existing in literature.Finally, we give an important application of fixed point theorem to digital images.Lastly, we make some conclusions.

Preliminaries
Let X be subset of n Z for a positive integer n where n Z is the set of lattice points in the n-dimensional Euclidean space and ρ represent an adjacency relation for the members of X.A digital image consists of ( ) , X ρ .Definition 2. [15] Let , l n be positive integers, 1 l n ≤ ≤ and two distinct points ( ) , , , , a and b are l k -adjacent if there are at most l indices i such that and for all other indices j such that 1, .
Z that is ρ-adjacent to a where { } , where , and .
and i u and ρ ⊂ be digital images and : T X Y → be a function, then 1) T is said to be ( ) 2) For all , ρ ρ -isomorphism and denoted by X d ρ is a Cauchy sequence if for all 0 ε > , there exists N δ ∈ such that for all , n m δ > , then ( ) of points of a digital metric space ( ) , , X d ρ converges to a limit p X ∈ if for all 0 ε > , there exists δ ∈  such that for all n δ > , then ( ) Proposition 8. [16] Every digital contraction map is digitally continuous.
Theorem 9. [16] (Banach Contraction principle) Let ( ) , , X d ρ be a complete digital metric space which has a usual Euclidean metric in n Z .Let, : T X X → be a digital contraction map.Then T has a unique fixed point, i.e. there exists a unique p X ∈ such that ( ) T p p = .

Implicit Relation and Related Concepts
In recent years, Popa [21] have used implicit functions rather than contraction conditions to prove fixed point theorems in metric spaces whose strength lie in their unifying power, as an implicit function can cover several contraction conditions at the same time, which include known as well as some unknown contraction conditions.This fact is evident from examples furnished in Popa [21].In this section, in order to prove our results, we define a set of suitable implicit functions involving six real non-negative arguments that was given in [22].
In the literature, there are several types of implicit contraction mappings where many nice consequences of fixed point theorems could be derived.First, denote Φ the set of functions where n φ is the nth iterate of φ .Remark 6.It is easy to see that if φ ∈Φ , then ( ) [22] Let R + denote the set of non-negative real numbers and let Γ be the set of all continuous functions 6 : R R Γ → satisfying the following conditions: 1) For each , 0 , , , , ,0 0 2) ( ) ,0,0, , ,0 0 T u u u > or ( ) ) ) [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.

Fixed
point theory leads to lots of applications in mathematics, computer S. Dalal et al.
weakly commuting.Then , , S T A and B have a unique common fixed point.
pairs ( ) , A S and ( ) , B T are compatible; c) one of , , S T A and B is continuous; d Definition 11. [17]The self maps S and T of a digital metric space ( ) , , X d ρ is a complete digital metric space DOI: 10.4236/apm.2018.83019→∞ = 4) O -compatible if for any sequence { } as n → ∞ and hence the proof.
, F d Ax By d Sx Ty d Ax Sx d By Ty d Ax Ty d By Sx , ASxconverge to Az as n → ∞ .Since the mappings A and S are compatible on X, it follows from Proposition 17 that { }