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.
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.
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 5and 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.
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 
.
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
 | (1) |
Recalling that 
is 
-admissible, we have, 
.
Using mathematical induction, we obtain
 | (2) |
From (1) and (2), it follows that for all 
, we have
 |
Since 
is non-decreasing, by induction, we have
 | (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
 | (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
 | (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
 | (6) |
Recalling the 
-admissible property of 
, we obtain from (6)
 | (7) |
Therefore, by repeatedly applying the 
-admissible property of 
, we get
 | (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 
.
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 StudyThe 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.
The Authors declare that they have no competing interest.
| [1] | L. E. S. Brouwer, Uber Abbildungen Von Mannigfaltigkeiten, Math. Ann., 77(1912), 97-115. | ||
| In article | |||
| [2] | L. Boxer, Digitally Continuous Functions, Pattern Recognition Letters, 15 (1994), 833-839. | ||
| In article | View Article | ||
| [3] | L. Boxer, Properties of Digital Homotopy, J. Math. Imaging Vis., 22(2005), 19-26. | ||
| In article | View Article | ||
| [4] | L. Boxer, A Classical Constructions for The Digital Fundamental Group, J. Math. Imaging Vis., 10(1999), 51-62. | ||
| In article | View Article | ||
| [5] | L. Boxer, Digital Products, Wedges and Covering Spaces, J. Math. Imaging Vis., 25(2006), 159-171. | ||
| In article | View Article | ||
| [6] | L. Boxer, Continuous Maps on Digital Simple Closed Curves, Appl. Math., 1(2010), 377-386. | ||
| In article | View Article | ||
| [7] | L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology, 17(2), 159-172 (2016). | ||
| In article | View Article | ||
| [8] | O. Ege, I. Karaca, Fundamental Properties of Simplicial Homology Groups for Digital Images, American Journal of Computer Technology and Application, 1(2013), 25-42. | ||
| In article | View Article | ||
| [9] | O. Ege, I. Karaca, Lefschetz Fixed Point Theorem for Digital Images, Fixed Point Theory Appl., 2013(2013), 13 pages. | ||
| In article | View Article | ||
| [10] | O. Ege, I. Karaca, Applications of The Lefschetz Number to Digital Images, Bull. Belg. Math. Soc. Simon Stevin, 21(2014), 823-839. | ||
| In article | View Article | ||
| [11] | O. Ege, I.karaca, Banach Fixed Point Theorem for Digital Images, J. Nonlinear Sci. Appl., 8(2015), 237-245. | ||
| In article | View Article | ||
| [12] | O. Ege and I. Karaca, Digital homotopy fixed point theory, Comptes Rendus Mathematique, 353(11), 1029-1033 (2015). | ||
| In article | View Article | ||
| [13] | O. Ege, Complex valued Gb -metric spaces, Journal of Computational Analysis and Applications, 21(2), 363-368 (2016). | ||
| In article | |||
| [14] | O. Ege and I. Karaca, Digital fibrations, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 87(1), 109-114 (2017). | ||
| In article | View Article | ||
| [15] | O. Ege and I. Karaca, Nielsen fixed point theory for digital images, Journal of 2 Computational Analysis and Applications, 22(5), 874-880 (2017). | ||
| In article | View Article | ||
| [16] | G. T. Herman, Oriented Surfaces in Digital Spaces, CVGIP: Graphical Models and Image Processing, 55(1993), 381-396. | ||
| In article | View Article | ||
| [17] | I. Karaca, O. Ege, Some Results on Simplicial Homology Groups of 2D Digital Images, Int. J. Inform. Computer Sci., 1(2012), 198-203. | ||
| In article | View Article | ||
| [18] | T. Y. Kong, A Digital Fundamental Group, Computers and Graphics, 13(1989), 159-166. | ||
| In article | View Article | ||
| [19] | A. Rosenfeld, Digital Topology, Amer. Math. Monthly, 86(1979), 76-87. | ||
| In article | View Article | ||
| [20] | B. Samet, C. Vetro, P. Vetro, Fixed point theorem for α-ψ-contractive type mappings, Nonlinear Analysis 75 (2012), 2154-2165. | ||
| In article | View Article | ||
| [21] | S. Z. Wang, B. Y. Li, Z. M. Gao, K. Iseki, Some fixed point theorems on expansion mappings, Math. Jpn. 29, 631-636, 1984. | ||
| In article | View Article | ||
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
https://creativecommons.org/licenses/by/4.0/
| [1] | L. E. S. Brouwer, Uber Abbildungen Von Mannigfaltigkeiten, Math. Ann., 77(1912), 97-115. | ||
| In article | |||
| [2] | L. Boxer, Digitally Continuous Functions, Pattern Recognition Letters, 15 (1994), 833-839. | ||
| In article | View Article | ||
| [3] | L. Boxer, Properties of Digital Homotopy, J. Math. Imaging Vis., 22(2005), 19-26. | ||
| In article | View Article | ||
| [4] | L. Boxer, A Classical Constructions for The Digital Fundamental Group, J. Math. Imaging Vis., 10(1999), 51-62. | ||
| In article | View Article | ||
| [5] | L. Boxer, Digital Products, Wedges and Covering Spaces, J. Math. Imaging Vis., 25(2006), 159-171. | ||
| In article | View Article | ||
| [6] | L. Boxer, Continuous Maps on Digital Simple Closed Curves, Appl. Math., 1(2010), 377-386. | ||
| In article | View Article | ||
| [7] | L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology, 17(2), 159-172 (2016). | ||
| In article | View Article | ||
| [8] | O. Ege, I. Karaca, Fundamental Properties of Simplicial Homology Groups for Digital Images, American Journal of Computer Technology and Application, 1(2013), 25-42. | ||
| In article | View Article | ||
| [9] | O. Ege, I. Karaca, Lefschetz Fixed Point Theorem for Digital Images, Fixed Point Theory Appl., 2013(2013), 13 pages. | ||
| In article | View Article | ||
| [10] | O. Ege, I. Karaca, Applications of The Lefschetz Number to Digital Images, Bull. Belg. Math. Soc. Simon Stevin, 21(2014), 823-839. | ||
| In article | View Article | ||
| [11] | O. Ege, I.karaca, Banach Fixed Point Theorem for Digital Images, J. Nonlinear Sci. Appl., 8(2015), 237-245. | ||
| In article | View Article | ||
| [12] | O. Ege and I. Karaca, Digital homotopy fixed point theory, Comptes Rendus Mathematique, 353(11), 1029-1033 (2015). | ||
| In article | View Article | ||
| [13] | O. Ege, Complex valued Gb -metric spaces, Journal of Computational Analysis and Applications, 21(2), 363-368 (2016). | ||
| In article | |||
| [14] | O. Ege and I. Karaca, Digital fibrations, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 87(1), 109-114 (2017). | ||
| In article | View Article | ||
| [15] | O. Ege and I. Karaca, Nielsen fixed point theory for digital images, Journal of 2 Computational Analysis and Applications, 22(5), 874-880 (2017). | ||
| In article | View Article | ||
| [16] | G. T. Herman, Oriented Surfaces in Digital Spaces, CVGIP: Graphical Models and Image Processing, 55(1993), 381-396. | ||
| In article | View Article | ||
| [17] | I. Karaca, O. Ege, Some Results on Simplicial Homology Groups of 2D Digital Images, Int. J. Inform. Computer Sci., 1(2012), 198-203. | ||
| In article | View Article | ||
| [18] | T. Y. Kong, A Digital Fundamental Group, Computers and Graphics, 13(1989), 159-166. | ||
| In article | View Article | ||
| [19] | A. Rosenfeld, Digital Topology, Amer. Math. Monthly, 86(1979), 76-87. | ||
| In article | View Article | ||
| [20] | B. Samet, C. Vetro, P. Vetro, Fixed point theorem for α-ψ-contractive type mappings, Nonlinear Analysis 75 (2012), 2154-2165. | ||
| In article | View Article | ||
| [21] | S. Z. Wang, B. Y. Li, Z. M. Gao, K. Iseki, Some fixed point theorems on expansion mappings, Math. Jpn. 29, 631-636, 1984. | ||
| In article | View Article | ||
