International Journal of Nonlinear Analysis and Applications
Fixed point theory in digital topology
Document Type : Research Paper
Authors
Mathematics Division, SASL, VIT Bhopal University 466114(M.P), India
Abstract
In this paper, we review some research works on exploring image processing in digital spaces using fixed point theorems. The basic concepts of digital images are mentioned. Moreover, we prove some theorems on digital metric spaces by replacing the conditions in the previously established theorem with a suitable condition.
Keywords
[1] R.P. Agarwall, M. Meehan and D. O’Regan, Fixed point theory and applications, Cambridge University Press,
2001.
[2] A.G. Aksoy and M.A. Khamsi, Nonstandard methods in fixed point theory, Springer, New York, Berlin, 1990.
[3] S. Banach, Sur less opeerations dans les ensembles abstraits et leur applications aux equations integrates, Fund.
Math. 3(1922), 133–181.
[4] L.E.S. Brouwer, Uber abbildungen von mannigfaltigkeiten, Math. Ann. 77 (1912), 97–115.
[5] L. Boxer, Digitally continuous function, Pattern Recog. Lett. 15 (1994), 833–839.
[6] L. Boxer, A classical constructions for the digital fundamental group, J. Math. Imag. Vis. 10 (1999), 51–62.
[7] L. Boxer, Properties of digital homotopy, J. Math. Imag. Vis. 22 (2005), 19–26.
[8] L. Boxer, Digital products, wedges and covering spaces, J. Math. Imag, Vis. 25 (2006), no. 2, 159–171.
[9] O. Ege and I. Karaca, Lefschetz fixed point theorem for digital images, Fixed Point Theory, Appl. 2013 (2013),
13 pages.
[10] O. Ege and I. Karaca, Applications of the Lefschetz number to digital images, Bull. Belg. Math. Soc. Simon Stevin
21 (2014), 823–839.
[11] O. Ege and I. Karaca, Banach fixed point theorem for digital images, J. Nonlinear Sci. Appl. 8 (2015), 237–245.
[12] G.T. Herman, Oriented surfaces in digital spaces, CVGIP Graph. Mod. Image Process. 55 (1993) 381–396.
[13] S. Jain, S. Jain and L.B. Jain, On Banach contraction principle in a cone metric space, J. Nonlinear Sci. Appl.
5 (2012), 198–203.
[14] M. Jleli and B. Samet, A new generalisation of the Banach contraction principle, J. Inequal. Appl. 2014 (2014),
8 pages.
[15] K. Jyoti and A. Rani, Digital expansions endowed with fixed point theory, Turk. J. Anal. Number Theory 5 (2017),
no. 5, 146–152.
[16] R. Kannan, Some results on fixed point, Bull. Cal. Math. Soc. 60 (1968), 71–76.
[17] R. Kannan, Some results on fixed point II, Amer. Math. Month. 76 (1969), no. 4, 405–408.
[18] T.Y. Kong, A digital fundamental group, Comput. Graph. 13 (1989), 159–166.
[19] A. Rosenfeld, Digital topology, Amer. Math. Month. 86 (1979), 76–87.[20] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl.
Math. 24 (1993), no. 11, 691–703.
[21] V.M. Sehgal and A.T. Reid Bharucha, Fixed points of contraction mappings on PM-spaces, Math. Syst. Theory
6 (1972), 97–102.
[22] W. Shatanawi and H.K. Nashine, A generalisation of Banach’s contraction principle for nonlinear contraction in
a partial metric space, J. Nonlinear Sci. Appl. 5 (2012), 37–43.
[23] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23 (1972), 292–298.
2001.
[2] A.G. Aksoy and M.A. Khamsi, Nonstandard methods in fixed point theory, Springer, New York, Berlin, 1990.
[3] S. Banach, Sur less opeerations dans les ensembles abstraits et leur applications aux equations integrates, Fund.
Math. 3(1922), 133–181.
[4] L.E.S. Brouwer, Uber abbildungen von mannigfaltigkeiten, Math. Ann. 77 (1912), 97–115.
[5] L. Boxer, Digitally continuous function, Pattern Recog. Lett. 15 (1994), 833–839.
[6] L. Boxer, A classical constructions for the digital fundamental group, J. Math. Imag. Vis. 10 (1999), 51–62.
[7] L. Boxer, Properties of digital homotopy, J. Math. Imag. Vis. 22 (2005), 19–26.
[8] L. Boxer, Digital products, wedges and covering spaces, J. Math. Imag, Vis. 25 (2006), no. 2, 159–171.
[9] O. Ege and I. Karaca, Lefschetz fixed point theorem for digital images, Fixed Point Theory, Appl. 2013 (2013),
13 pages.
[10] O. Ege and I. Karaca, Applications of the Lefschetz number to digital images, Bull. Belg. Math. Soc. Simon Stevin
21 (2014), 823–839.
[11] O. Ege and I. Karaca, Banach fixed point theorem for digital images, J. Nonlinear Sci. Appl. 8 (2015), 237–245.
[12] G.T. Herman, Oriented surfaces in digital spaces, CVGIP Graph. Mod. Image Process. 55 (1993) 381–396.
[13] S. Jain, S. Jain and L.B. Jain, On Banach contraction principle in a cone metric space, J. Nonlinear Sci. Appl.
5 (2012), 198–203.
[14] M. Jleli and B. Samet, A new generalisation of the Banach contraction principle, J. Inequal. Appl. 2014 (2014),
8 pages.
[15] K. Jyoti and A. Rani, Digital expansions endowed with fixed point theory, Turk. J. Anal. Number Theory 5 (2017),
no. 5, 146–152.
[16] R. Kannan, Some results on fixed point, Bull. Cal. Math. Soc. 60 (1968), 71–76.
[17] R. Kannan, Some results on fixed point II, Amer. Math. Month. 76 (1969), no. 4, 405–408.
[18] T.Y. Kong, A digital fundamental group, Comput. Graph. 13 (1989), 159–166.
[19] A. Rosenfeld, Digital topology, Amer. Math. Month. 86 (1979), 76–87.[20] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl.
Math. 24 (1993), no. 11, 691–703.
[21] V.M. Sehgal and A.T. Reid Bharucha, Fixed points of contraction mappings on PM-spaces, Math. Syst. Theory
6 (1972), 97–102.
[22] W. Shatanawi and H.K. Nashine, A generalisation of Banach’s contraction principle for nonlinear contraction in
a partial metric space, J. Nonlinear Sci. Appl. 5 (2012), 37–43.
[23] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. 23 (1972), 292–298.
)
Volume 13, Special Issue for selected papers of ICDACT-2021
The link to the conference website is
https://vitbhopal.ac.in/event/icdact_dec_21/March 2022Pages 157-163
- Receive Date: 13 August 2021
- Revise Date: 18 November 2021
- Accept Date: 11 January 2022