Banach and Edelstein Fixed Point Theorems for Digital Images

The current paper generalizes the Edelstein fixed point theorem for digital (𝜀𝜀, 𝑘𝑘) -chainable metric spaces. In order to generalize Edelstein fixed point theorem, we study the digital topological properties of digital images. Further, we establish the Banach fixed point theorem for digital images. We give the notion of digital (𝜀𝜀, 𝜆𝜆, 𝑘𝑘) -uniformly locally contraction mapping on digital (𝜀𝜀, 𝑘𝑘) -chainable metric spaces . Finally, we generalize the Banach fixed point theorem to digital (𝜀𝜀, 𝑘𝑘) -chainable metric spaces which is known as the Edelstein fixed point theorem for digital images on digital (𝜀𝜀, 𝑘𝑘) -chainable metric spaces.


Introduction
Fixed point theory plays an essential role in various branches of mathematics. The knowledge of the existence of fixed points has relevant applications in many branches of analysis and topology. Besides, it has application to some areas of computer sciences such as computer graphics, image processing, mathematical morphology and so forth. It is very useful to find out a solution if an equation has any solution. Many mathematical problems, originating from various branches of mathematics can be equivalently formulated as fixed point problems meaning that one has to find a fixed point of some functions. In metric spaces, this theory begins with the Banach fixed point theorem (also known as the Banach contraction mapping principle) by Stefan Banach in 1922 [2]. It is an important tool for solution of some problems in mathematics and engineering. Up to now, there are many generalizations of Banach fixed point theorem have been established [12]. In 1961, Michael Edelstein generalized that theorem on -chainable metric spaces [10,12].
Digital topology deals with the digital topological properties of , ∈ ℕ digital images. It was first introduced by Resenfeld in 1979 [17]. To be specific, he developed the notion of digital continuity for studying 2 and 3 digital images in 1986 [16]. In 1994, Boxer [3,4] expanded the digital versions of several notions such as digital continuous functions including homeomorphisms retractions and homotopies. Few years ago, the concept of digital continuity was extended into the study of digital images [13]. The fixed point properties and fixed point theory for digital images were first given by Ege and Karaca [6] named as Lefschetz fixed point theorem. They developed some applications of the Lefschetz fixed point theorem and Nielsen fixed point theorem in digital images to count fixed points [7,9]. They also got some new results associating digital homotopy and fixed point theory [8]. In 2015, they studied the Banach fixed point theorem for digital images [5]. Han [12] refined and improved several notions of that paper such as digital versions of both Cauchy sequence and limit of a sequence in a digital metric spaces. Recently, approximate fixed points and the approximate fixed point property (AFPP) of digitally continuous functions are introduced [4]. This paper is organized as follows. In the first part, we give the required background about the digital images and digital topology. After that, we study the property of the completeness of digital metric spaces. In the next part, we state and prove the Banach fixed point theorem for digital images. Finally, we give the notion of digital ( , )chainable metric spaces and then state and prove the Edelstein fixed point theorem for digital images. Lastly, we give the conclusion.

Preliminaries
To study the Banach fixed point theorem and Edelstein fixed point theorem from the viewpoint of digital topology, we discuss some basic notions from digital topology: Let ℕ and ℤ represent the sets of natural numbers and integer numbers respectively. Let ℤ , where ∈ ℕ be the sets of lattice points in the -dimensional Euclidean space. It is useful to distinguish between a digital picture and a digital image. We say ℤ is a digital picture [17]. A digital image is a subset of ℤ with the adjacency relation of the Using above fact, we can obtain the -adjacency relations of ℤ as follows [13]: Mathematically, a set ⊂ ℤ with the above adjacency relation is called a digital image and denoted by ( , ). Definition 1.2 [16]: Let ( , 0 ) ⊂ ℤ 0 and ( , 1 ) ⊂ ℤ 1 be digital images. Let ∶ → be a function. Then is digitally continuous at 0 ∈ if and only if for every ≥ 1 there is a ≥ 1 such that ∈ and 0 ( 0 , ) ≤ implies 1 � ( 0 ), ( )� ≤ . Definition 1.3 [15]: Let , ∈ ℤ with ≤ , then the set Using the -adjacency relations of ℤ of (2.1), we define that a digital neighborhood of in ℤ is the set [16] ( ) = { | is adjacent to }. Furthermore, we often use the notation [15] [11]: A digital image ⊂ ℤ is -connected if and only if every pair of different points , ∈ , there is a sequence { } ∈[0, ] of points of a digital image such that = 0 and = with and +1 are -neighbors. Now we can say that there is a simple -path with elements whose length is the number , denoted by ( , ).

Completeness of the Digital Metric Spaces
Proof: Since the Definition 2.1 implies that any two distinct points , ∈ { } in the digital metric space ( , , ) has at most of their co-ordinates which differs by ±1 and all other coincide. Thus the Euclidean distance � , � ∈ {1, √2, ⋯ ⋯ , √ } depending on the position of that two points. For instance, in Figure 1 (a), the Euclidean distance of any two points 2-adjacent in ℤ is 1. In Figure 1 (b), the Euclidean distance of any two points 4adjacent in ℤ 2 is 1. In Figure 1 (c), the Euclidean distance of any two points 8-adjacent in ℤ 2 is either 1 or √2 depending on the position of the given two points. For instance, consider two 8-adjacent points of (0, 0) in ℤ 2 are 1 (0, 1) and 2 (1, 1). Now the Euclidean distance of these two points are ( , 1 ) = 1 and ( , 2 ) = √2 . Similarly, in three dimensional case, we have Euclidean distances � √ | ∈ [1, 3] ℤ � depending on the positions of the two points.
Thus by using the above Proposition 3.1, we obtain the following: . . = for all , > . By using the Proposition 3.1, we can define the convergency of a sequence of a digital metric space ( , , ) as follows: Definition 3.2 [12]: A sequence { } of points of a digital metric space ( , , ) converges to a limit ∈ if there is ∈ ℕ such that for all > , we have 1 2 , . . .  Motivated by the paper [5], here we give the Banach fixed point theorem in the following way:

Edelstein Fixed Point Theorem for Digital ( , ) Chainable Metric Spaces
An extension of Banach fixed point theorem was given by Edelstein to a class of mappings on -chainable metric spaces [10]. Based on that we discuss Edelstein fixed point theorem for digital ( , ) chainable metric spaces