Banach fixed point theorem for digital images

Ege, Ozgur¹ ; Karaca, Ismet²

¹ Department of Mathematics, Celal Bayar University, Muradiye, 45140, Manisa, Turkey
² Departments of Mathematics, Ege University, Bornova, 35100, Izmir, Turkey

Journal of nonlinear sciences and its applications, Tome 8 (2015) no. 3, p. 237-245.

Résumé

In this paper, we prove Banach fixed point theorem for digital images. We also give the proof of a theorem which is a generalization of the Banach contraction principle. Finally, we deal with an application of Banach fixed point theorem to image processing.

Détail

DOI : 10.22436/jnsa.008.03.08

Classification : 47H10, 54E35, 68U10
Keywords: Digital image, fixed point, Banach contraction principle, digital contraction.

Bibliographie

[1] Banach, S. Sur les operations dans les ensembles abstraits et leurs applications aux equations integrales, Fund. Math., Volume 3 (1922), pp. 133-181

[2] Bertrand, G. Simple points, topological numbers and geodesic neighborhoods in cubic grids, Pattern Recognition Letters, Volume 15 (1994), pp. 1003-1011

[3] Bertrand, G.; Malgouyres, R. Some topological properties of discrete surfaces, J. Math. Imaging Vis., Volume 20 (1999), pp. 207-221

[4] Boxer, L. Digitally continuous functions, Pattern Recognition Letters, Volume 15 (1994), pp. 833-839

[5] Boxer, L. A classical construction for the digital fundamental group, J. Math. Imaging Vis., Volume 10 (1999), pp. 51-62

[6] Boxer, L. Properties of digital homotopy, J. Math. Imaging Vis., Volume 22 (2005), pp. 19-26

[7] Boxer, L. Digital products, wedges and covering spaces, J. Math. Imaging Vis., Volume 25 (2006), pp. 159-171

[8] Boxer, L. Continuous maps on digital simple closed curves, Appl. Math., Volume 1 (2010), pp. 377-386

[9] Ege, O.; I. Karaca Fundamental properties of simplicial homology groups for digital images, American Journal of Computer Technology and Application, Volume 1 (2013), pp. 25-42

[10] Ege, O.; Karaca, I. Lefschetz Fixed Point Theorem for Digital Images, Fixed Point Theory Appl., Volume 2013 (2013), pp. 1-13

[11] Ege, O.; Karaca, I. Applications of the Lefschetz Number to Digital Images, Bull. Belg. Math. Soc. Simon Stevin, Volume 21 (2014), pp. 823-839

[12] S. E. Han An extended digital \((k_0; k_1)\)-continuity, J. Appl. Math. Comput., Volume 16 (2004), pp. 445-452

[13] Han, S. E. Minimal simple closed 18-surfaces and a topological preservation of 3D surfaces, Inform. Sci., Volume 176 (2006), pp. 120-134

[14] Han, S. E. Connected sum of digital closed surfaces, Inform. Sci., Volume 176 (2006), pp. 332-348

[15] Herman, G. T. Oriented surfaces in digital spaces , CVGIP: Graphical Models and Image Processing, Volume 55 (1993), pp. 381-396

[16] Jain, Sh.; Jain, Sh.; Jain, L. B. On Banach contraction principle in a cone metric space, J. Nonlinear Sci. Appl., Volume 5 (2012), pp. 252-258

[17] Jleli, M.; Samet, B. A new generalization of the Banach contraction principle, J. Inequal. Appl., Volume 2014 (2014), pp. 1-8

[18] Karaca, I.; Ege, O. Some results on simplicial homology groups of 2D digital images, Int. J. Inform. Computer Sci., Volume 1 (2012), pp. 198-203

[19] Kong, T. Y. A digital fundamental group, Computers and Graphics, Volume 13 (1989), pp. 159-166

[20] Malgouyres, R.; Bertrand, G. A new local property of strong n-surfaces, Pattern Recognition Letters, Volume 20 (1999), pp. 417-428

[21] Rosenfeld, A. Digital topology, Amer. Math. Monthly, Volume 86 (1979), pp. 76-87

[22] Samet, B.; Vetro, C.; Vetro, P. Fixed point theorems for \(\alpha-\psi\)-contractive type mappings , Nonlinear Anal., Volume 75 (2012), pp. 2154-2165

[23] Shatanawi, W.; Nashine, H. K. A generalization of Banach's contraction principle for nonlinear contraction in a partial metric space, J. Nonlinear Sci. Appl., Volume 5 (2012), pp. 37-43