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Digital Expansions Endowed with Fixed Point Theory

Kumari Jyoti, Asha Rani*

Department of Mathematics, SRM University, Haryana, Sonepat- 131001, India

Abstract: The area of fixed point theory is very active in many branches of mathematics and other related disciplines such as image processing, computer vision, applied mathematics, etc. The main goal in this theory is to solve many problems and to give some useful applications. The aim of this paper is to associate fixed point theory and digital images.

Keywords: expansive mappings, digital-α-ψ– expansive mappings, fixed point theorems, digital metric space

Received March 14, 2017; Revised June 20, 2017; Accepted July 11, 2017

Citation: Kumari Jyoti, Asha Rani, Digital Expansions Endowed with Fixed Point Theory. Turkish Journal of Analysis and Number Theory, 5 (5). 146-152. January 2017. doi:10.12691/tjant-5-5-1

1. Introduction

The dawn of the fixed point theory starts when in 1912 Brouwer [1] proved a fixed point result for continuous self maps on a closed ball. In the last few decades, fixed point theory has been one of the most interesting research fields in nonlinear functional analysis. In 1984 Wang [21] introduced the expansive mappings and proved fixed point results for them.

Digital topology is a developing area which is related to features of 2D and 3D digital images using general topology and functional analysis. Up to now, several developments have occurred in the study of digital topology. Digital topology was first studied by Rosenfield [19]. Kong [18], then introduced the digital fundamental group of a discrete object. Boxer [2] has given the digital versions of several notions from topology and [3] studied a variety of digital continuous functions. Some results and characteristic properties on the digital homology groups of 2D digital images are given in [8] and [17]. Ege and Karaca [9,10] give relative and reduced Lefschetz fixed point theorem for digital images. They also calculate degree of antipodal map for the sphere like digital images using fixed point properties. In [12,13,14,15] Ege and Karaca studied some more properties of related to digital topologies. Boxer et. al. [7] gave approximate digital fixed points and universal functions for digital metric spaces. Ege and Karaca [11] then defined a digital metric space and proved the famous Banach Contraction Principle for digital images.

2. Preliminaries

Let be a subset of for a positive integer where is the set of lattice points in the - dimensional Euclidean space and represent an adjacency relation for the members of . A digital image consists of .

Definition 2.1 [4]: Let be positive integers, and two distinct points

and are - adjacent if there are at most indices such that and for all other indices such that , .

There are some statements which can be obtained from definition 2.1:

• and are 2- adjacent if .

• and in are 8- adjacent if they are distinct and differ by at most 1 in each coordinate.

• and in are 26- adjacent if they are distinct and differ at most 1 in each coordinate.

• and in are 18- adjacent if are 26- adjacent and differ by at most two coordinates.

• and are 6- adjacent if they are 18- adjacent and differ in exactly one coordinate.

A -neighbour [4] of is a point of that is - adjacent to where and . The set

is called the - neighbourhood of . A digital interval [9] is defined by

where and .

A digital image is - connected [16] if and only if for every pair of different points , there is a set of points of digital image such that and and are - neighbours where .

Definition 2.2: Let , be digital images and be a function.

• is said to be - continuous [4], if for all - connected subset of , is a - connected subset of .

• For all - adjacent points of , either or and are a - adjacent in if and only if is - continuous [4].

• If is - continuous, bijective and is - continuous, then is called - isomorphism [5]and denoted by .

A - continuous function , is called a digital - path [4] from to in a digital image if such that and . A simple closed - curve of points [4] in a digital image is a sequence of images of the - path such that and are - adjacent if and only if .

Definition 2.5 [11]: A sequence of points of a digital metric space is a Cauchy sequence if for all , there exists such that for all , then

Definition 2.6 [11]: A sequence of points of a digital metric space converges to a limit if for all , there exists such that for all , then

Definition 2.7 [11]: A digital metric space is a digital metric space if any Cauchy sequence of points of converges to a point of .

Definition 2.8 [11]: Let be any digital image. A function is called right- continuous if where, .

Definition 2.9 [11]: Let, be any digital metric space and be a self digital map. If there exists such that for all ,

then is called a digital contraction map.

Proposition 2.10 [11]: Every digital contraction map is digitally continuous.

Theorem 2.11 [11]: (Banach Contraction principle) Let be a complete metric space which has a usual Euclidean metric in . Let, be a digital contraction map. Then has a unique fixed point, i.e. there exists a unique such that .

In order to introduce digital--expansive mappings, we use the following definition, given by Samet et. al. [20]:

Definition 2.16 [20] Let denote the family of all functions which satisfy the following :

(i) for each , where is the nth iterate of ;

(ii) is non-decreasing.

3. Main Results

3.1. Expansive Maps

Definition 3.1: Suppose that is a complete digital metric space and is any mapping. The mapping satisfy the condition holds for all and , then is called a digital expansive mapping.

Theorem 3.2: Let be a digital expansive mapping on a complete digital metric space and is onto. Then has a fixed point in .

Proof: Let, , since is onto, there exists an element satisfying . By the same way, we can choose, where .

If for some , then is a fixed point of . Without loss of generality, we can suppose for every . So,

Clearly, is a digital Cauchy sequence in . Since, is complete converges to some . Since is onto there exists such that and for infinitely many , , for such

On taking limit as , we get,

Thus, has a fixed point in .

Corollary 3.3: Let be a digital expansive mapping on a complete digital metric space and is bijective. Then has a unique fixed point in .

Proof: Since, is bijective therefore, there is a unique element for every such that . So, by obvious argument the fixed point is unique.

Theorem 3.4: Let, be a complete digital metric space and be an onto self map on . Let satisfy the condition where then has a fixed point.

Proof: Let, , since is onto, there exists an element satisfying . By the same way, we can choose, where .

If for some , then is a fixed point of . Without loss of generality, we can suppose for every . So,

Clearly, is a Cauchy sequence in . Since, is complete converges to some . Since is onto there exists such that and for infinitely many , , for such

On taking limit as , we get,

Thus, has a fixed point in .

Theorem 3.5: Let, be a complete digital metric space and be an onto self map which is continuous on . Let satisfy the condition

where , and

then has a fixed point.

Proof: Let, , since is onto, there exists an element satisfying . By the same way, we can choose, where .

If for some , then is a fixed point of . Without loss of generality, we can suppose for every . So,

where,

Case 1:

Case 2:

where

(since, if , then clearly and if then we get which is a contradiction)

Case 3: , which is same as Case2 since .

So, in each case for some .

Clearly, is a Cauchy sequence in . Since, is complete converges to some . Since, is onto there exists such that and for infinitely many , , for such

Now,

On taking limit as , we get,

Also, , which implies . So, or .

If , then .

If , then but and is continuous. So, .

(Since, by continuity of , we get, . So converges to . Then, there exist , such that, , by construction and hence by uniqueness of limit )

Thus, has a fixed point in .

Theorem 3.6: Let, be a complete digital metric space and be an onto self map which is continuous on . Let satisfy the condition

where , and

then has a fixed point.

Proof: Let, , since is onto, there exists an element satisfying . By the same way, we can choose, where .

If for some , then is a fixed point of . Without loss of generality, we can suppose for every . So,

where,

Case 1:

Case 2:

where

(since, if , then clearly and if then we get which is a contradiction)

Case 3: , which is same Case 2 as

So, in each case for some .

Clearly, is a Cauchy sequence in . Since, is complete converges to some . Since is onto there exists such that and for infinitely many , , for such

Now,

On taking limit as , we get,

Also, , which implies . So, or .

If , then .

If , then but and is continuous. So, .

Thus, has a fixed point in .

Corollary 3.7: If the map in the theorems (3.3-3.6) be bijective, then has a unique fixed point.

Example 3.8: Let us consider the digital metric space , with the digital metric defined by . Consider the self mapping given by, . Clearly, is an expansive mapping. Now, for every , there exists an , such that, . So, is onto. So, it satisfies the axioms of our theorem 3.2. Clearly, its fixed point is .

Example 3.9: Let us consider the digital metric space , with the digital metric defined by . Consider the self mapping given by,

Clearly, satisfies the condition

Now, for every , there exists an such that, . So, is onto. So, it satisfies the axioms of our theorem 3.3. Clearly, it has two fixed points and .

3.2. Digital-α-ψ-expansive Maps

We introduce the concept of digital-–expansive mappings in non Newtonian metric spaces as follows:

Definition 3.10: Let be a digital metric space and be a given mapping. We say that is a digital-–expansive mapping, if there exists two functions and such that for all in , we have

Remark 3.11: Clearly, any expansive mapping is a digital-–expansive mapping with for all and , for all and .

Definition 3.12: Let and . We say that is -admissible if for all in , we have .

Now, we prove our main results.

Theorem 3.13: Let be a complete digital metric space and be a bijective, digital-–expansive mapping and satisfies the following conditions:

(i) is -admissible;

(ii) There exists such that ;

(iii) is digitally continuous.

Then has a fixed point, that is, there exists in such that .

Proof. Let us define the sequence in by for all , where is such that . If for some , then is a fixed point of . So, we can assume that for all .

It is given that

[picture] | (1)

Recalling that is -admissible, we have, .

Using mathematical induction, we obtain

[picture] | (2)

From (1) and (2), it follows that for all , we have

Since is non-decreasing, by induction, we have

[picture] | (3)

Using (3), we have

Since, and , we get .

Thus, we have .

This implies that is a digital Cauchy sequence in digital metric space . But is complete, so there exists in such that as . From the continuity of , it follows that as . By the uniqueness of the limit, we get , that is, is a fixed point of .

In what follows, we prove that Theorem 3.13 still holds for not necessarily continuous, assuming the following condition:

(M) If is a sequence in such that for all and as , then

[picture] | (4)

Theorem 3.14: If in Theorem 3.13, we replace the continuity of by the condition (M), then the result holds true.

Proof. Following the proof of Theorem 3.11, we know that is a digital Cauchy sequence in such that for all and as . Now, from the hypothesis (4), we have

[picture] | (5)

Using (1) and (5), we get

Continuity of at implies that as That is, . Consider, , which implies that, is a fixed point of .

We now present some examples in support of our results and show that the hypotheses in Theorems 3.11 and 3.12 do not guarantee uniqueness.

To ensure the uniqueness of the fixed point in Theorems 3.11 and 3.12, we consider the condition:

(S): For all , there exists such that and .

Theorem 3.15: Adding the condition (S) to the hypotheses of Theorem 3.13 (resp. Theorem 3.14), we obtain the uniqueness of the fixed point of .

Proof. From Theorem 3.13 and 3.14, the set of fixed points is non-empty. We shall show that if and are two fixed points of , that is, and , then .

From the condition (S), there exists such that

[picture] | (6)

Recalling the -admissible property of , we obtain from (6)

[picture] | (7)

Therefore, by repeatedly applying the -admissible property of , we get

[picture] | (8)

From the inequalities (1) and (8), we get

Repetition of the above inequality implies that , for all . Thus, we have as .

Using the similar steps as above, we obtain as . Now, the uniqueness of the limit of gives . This completes the proof.

Example 3.16: Let us consider the digital metric space , with the digital metric defined by . Consider the self mapping given by

and by

Clearly, is a digital-–expansive mapping with for all . In fact for all , we have .

Moreover, there exists such that . In fact, for , we have . Obviously, is non Newtonian continuous, and so it remains to show that is -admissible. For this, let such that . This implies that and , and by the definitions of and , we have , and . Then is -admissible. Now, all the hypothesis of Theorem 3.13 are satisfied. Consequently, has a fixed point. Clearly, and are two fixed points of .

Now, we give an example involving a function that is digitally discontinuous.

Example 3.17: Let us consider the digital metric space , with the digital metric defined by . Consider the self mapping given by

and by

Due to the discontinuity of at 1, Theorem 3.14 is not applicable in this case. Clearly, is a digital-– expansive mapping with for all . In fact for all in , we have Moreover, there exists such that . In fact, for , we have . Now, let such that . This implies that and , and by the definitions of and , we have , and . Then is -admissible. Finally, let be a sequence in such that for all and as . Since for all , by the definition of , we have . Therefore, all the required hypothesis of Theorem 3.14 are satisfied and so has a fixed point. Here, and are two fixed points of .

3.3. Application of Expansive Mappings to Digital Images

The zooming in and zooming out of the digital images may be very important in certain circumstances. But when we zoom out an image due to the use of pixels the image quality is distorted and the image becomes blurred. By assigning an expansive map to the points of the image the procedure may become more fruitful, i.e., the image can be zoomed out without disturbing the quality of the image.

3.4. Scope of the Study

The aim of this paper is to give the digital version of some fixed point theorems where the map is expansive. We hope the results will be useful in digital topology and fixed point theory. These results can therefore be used for the expansion or Zooming out of digital images and the array of digital images. In future, some other properties of digital images can be discussed with the viewpoint of fixed point theory.

Competing Interests

The Authors declare that they have no competing interest.

References

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