Abstract

The current paper generalizes the Edelstein fixed point theorem for digital (ε,k)-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 (ε,λ,k)-uniformly locally contraction mapping on digital (ε,k) -chainable metric spaces Finally, we generalize the Banach fixed point theorem to digital (ε,k)-chainable metric spaces which is known as the Edelstein fixed point theorem for digital images on digital (ε,k)-chainable metric spaces.

1. 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 digital images. It was first introduced by Resenfeld in 1979 ¹⁷. To be specific, he developed the notion of digital continuity for studying and 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.

2. 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 and represent the sets of natural numbers and integer numbers respectively. Let where 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 is a digital picture ¹⁷. A digital image is a subset of with the adjacency relation of the members of that image.

Definition 1.1: We say that two distinct points are -(or ) adjacent for digital images if they satisfy the followings ¹³:

For a natural number two distinct points

are or -adjacent if at most of their co-ordinates differ by and all other points coincide.

Using above fact, we can obtain the -adjacency relations of as follows ¹³:

(2.1)

where

Mathematically, a set with the above adjacency relation is called a digital image and denoted by

Definition 1.2 ¹⁶: Let and be digital images. Let be a function. Then is digitally continuous at if and only if for every there is a such that and implies

Definition 1.3 ¹⁵: Let with , then the set

with 2-adjacency is called a digital interval.

Using the -adjacency relations of of (2.1), we define that a digital neighborhood of in is the set ¹⁶ is adjacent to .

Furthermore, we often use the notation ¹⁵

Definition 1.4 ¹¹: A digital image is -connected if and only if every pair of different points there is a sequence of points of a digital image such that and with and are -neighbors. Now we can say that there is a simple -path with elements whose length is the number denoted by

For a digital image as a generalization of the digital -neighborhood of with radius is defined in to be the following subset ¹³ of

where is the length of a shortest simple -path from to and Concretely, for we obtain

Proposition 2.1 ¹³: Let and be digital images in and respectively. A function is digitally -continuous if and only if for every

According to Proposition 2.1, we see that the points is mapped into the points which implies that for the points which are -adjacent a -continuous map has the property

Definition 2.1 ^{3, 12}: Let and be digital images in and respectively. Then a map is called a -isomorphism if is a digital -continuous, bijective and further is -continuous which can be denoted by

3. Completeness of the Digital Metric Spaces

Definition 3.1 ⁵: Let denote a digital metric space with -adjacency relation, where is the standard Euclidean function on

In relation to the study of Banach fixed point theorem and Edelstein fixed point theorem, we recall the following:

Definition 3.2 ⁵: A sequence of points of a digital metric space is a Cauchy sequence if for all there exists such that for all then

Since the sequence is defined in the digital metric space Han ¹² observed that the Euclidean distance between any two distinct points is greater than or equal to 1 as follows:

Proposition 3.1 ¹²: In a digital metric space consider two points in a sequence of such that they are -adjacent or and Then they have the Euclidean distance

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 and all other coincide. Thus the Euclidean distance depending on the position of that two points. For instance, in Figure 1 (a), the Euclidean distance of any two points 2-adjacent in is 1. In Figure 1 (b), the Euclidean distance of any two points 4-adjacent in is 1. In Figure 1 (c), the Euclidean distance of any two points 8-adjacent in is either 1 or depending on the position of the given two points. For instance, consider two 8-adjacent points of in are and Now the Euclidean distance of these two points are and . Similarly, in three dimensional case, we have Euclidean distances depending on the positions of the two points.

Thus by using the above Proposition 3.1, we obtain the following:

Proposition 3.2 ¹²: A sequence of points of a digital metric space is a Cauchy sequence if and only if there is such that for all we have

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 such that for all we have

Definition 3.3 ¹²: A digital metric space is a complete digital metric space if any Cauchy sequence of points of converges to a point

4. Banach Fixed Point Theorem for Digital Contraction Mapping

Definition 4.1 ⁵: Let be a digital metric space and be a self-mapping. If there exists such that for all , we have then is called a digital contraction mapping.

Proposition 4.1: Every digital contraction mapping is digitally continuous.

Proof: Let be a digital metric space and be a digital contraction mapping. Let and by the Proposition 3.1, we have

where for all Then is a digitally -continuous.

Motivated by the paper ⁵, here we give the Banach fixed point theorem in the following way:

Theorem 4.1 (Banach contraction principle for digital images): Let be a complete digital metric space with Euclidean metric on Let be a digital contraction mapping. Then has a unique fixed point.

Proof: For notational purposes, we define and inductively by and To show existence, we select We first show that is a Cauchy sequence.

Notice for that

Thus for we have

By Proposition 3.1, we have which implies . This shows that is a Cauchy sequence and since is complete, so there exists such that

Moreover, the continuity of yields

Therefore, is a fixed point of

For uniqueness, suppose that is a another fixed point of such that and then

But so Therefore, and our proof is completed.

5. Edelstein Fixed Point Theorem for Digital (ε,k) 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 from the viewpoint of digital topology. In this regard, we discuss digital -chain, -uniformly locally contractive mapping and -chainable metric spaces for digital topology.

Definition 5.1: Let be a sequence of points of a digital metric space and suppose Then the sequence is said to be chain joining and if and are neighbors,

Definition 5.2: The digital metric space is said to be digitally -chainable metric space if each pair of its points, there exists an -chain joining and

Definition 5.3: A mapping is called digitally -uniformly locally contractive if there exists and such that which implies for each

Theorem 5.1 (Edelstein fixed point theorem for digital images): Let be a complete digital -chainable metric space and be a digital - uniformly locally contractive mapping. Then has a unique fixed point in

Proof: Given is a complete digital -chainable metric space. Now we define a new metric by

where infimum is taken over all chains joining and . Then the metric on satisfies

(i)

(ii) for

Then the metric is complete whenever is complete. With given and any -chains joining and we have

where for .

Now we can write

Hence, by the Definition 5.3, we have

So is an -chain joining and

Then

Now being an arbitrary -chain, we have

Since has a unique fixed point given by for any [By Banach fixed point theorem].

But by we have Therefore, our proof is completed.

6. Conclusion

We have reviewed some notions of digital topology for image processing technique. Further, we have studied the Banach fixed point theorem for digital images. At last, we have discussed the digital version of Edelstein fixed point theorem for digital images which is the generalization of Banach fixed point theorem in metric spaces. We hope that all results in this paper help us to understand better structure of digital images.

In future, we will study the other fixed point theorem for digital images in digital topology.

Conflicts of Interest

The authors declare that there is no conflicts of interest regarding the publication of this paper.

References