Journal of Mathematical Sciences and Applications
ISSN (Print): 2333-8784 ISSN (Online): 2333-8792 Website: http://www.sciepub.com/journal/jmsa Editor-in-chief: Prof. (Dr.) Vishwa Nath Maurya, Cenap ozel
Open Access
Journal Browser
Go
Journal of Mathematical Sciences and Applications. 2017, 5(2), 36-39
DOI: 10.12691/jmsa-5-2-2
Open AccessArticle

Banach and Edelstein Fixed Point Theorems for Digital Images

Akram Hossain1, , Razina Ferdausi2, Samiran Mondal1 and Harun Rashid1

1Department of Mathematics, Jessore University of Science & Technology, Jessore, Bangladesh

2Department of Mathematics, University of Dhaka, Dhaka, Bangladesh

Pub. Date: November 29, 2017

Cite this paper:
Akram Hossain, Razina Ferdausi, Samiran Mondal and Harun Rashid. Banach and Edelstein Fixed Point Theorems for Digital Images. Journal of Mathematical Sciences and Applications. 2017; 5(2):36-39. doi: 10.12691/jmsa-5-2-2

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.

Keywords:
digital image digital continuity digital metric space digital (ελk)-uniformly locally contraction digital (εk)-chainable metric space Banach fixed point theorem Edelstein fixed point theorem

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  R. P. Agarwall, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001.
 
[2]  S. Banach, Sur less opérations dans les ensembles abstraits et leur applications aux equations integrates, Fund. Math. 3(1922), 133-181.
 
[3]  L. Boxer, Digitally Continuous Function, Pattern Recognition Letter, 15(1994), 833-839.
 
[4]  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).
 
[5]  O. Ege, I. Karaca, Banach fixed point theorem for digital images, J. Nonlinear. Sci. Appl., 8(2015), 237-245.
 
[6]  O. Ege, I. Karaca, Lefschetz fixed point theorem for digital images, Fixed Point Theory and Applications, 2013, 2013:253.
 
[7]  O. Ege and I. Karaca, Applications of the Lefschetz number to digital images, Bulletin of the Belgian Mathematical Society - Simon Stevin, 21(5), 823-839 (2014).
 
[8]  O. Ege and I. Karaca, Digital homotopy fixed point theory, Comptes Rendus Mathematique, 353(11), 1029-1033 (2015).
 
[9]  O. Ege and I. Karaca, Nielsen fixed point theory for digital images, Journal of Computational Analysis and Applications, 22(5), 874-880 (2017).
 
[10]  M. Edelstein, An Extention of Banach’s Contraction Principle, Amer. Math. Soc., 12(1961), 7-10.
 
[11]  G.T. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing, 55(1993).
 
[12]  S. E. Han, Banach fixed point theorem from the viewpoint of digital topology, J. Nonlinear. Sci. Appl. 9(2016), 895-905.
 
[13]  S. E. Han, Non-product property of the digital fundamental group, Inform. Sci., 171(2005), 73-91.
 
[14]  M. C. Joshi, R.K. Bose, Some Topics in Non-linear Functional Analysis, Wiley Eastern Limited, New Delhi 110 002, 1985.
 
[15]  T. Y. Kong, A. Rosenfeld, Topological Algorithms for the Digital Image Processing, Elsevier Science, Amsterdam, 1996.
 
[16]  A. Rosenfeld, Continuous functions on digital pictures, Pattern Recognition Letter, 4(1986), 177-184.
 
[17]  A. Rosenfeld, Digital topology, Amer. Math. Soc., 86(1979), 621-630.