Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-22T20:40:02.802Z Has data issue: false hasContentIssue false
  • English
  • Français

The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph

Published online by Cambridge University Press:  20 November 2018

Monther Rashed Alfuraidan
Monther Rashed Alfuraidan*
Affiliation:
Department of Mathematics & Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler and Edelstein’s fixed point theorems to modular metric spaces endowed with a graph.

Keywords

47H0946B2047H1047E10fixed point theorymodularmetric spacesmultivalued contractionmappingconnected digraph
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 1 , 01 March 2016 , pp. 3 - 12
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Abdou, A. N. and Khamsi, M. A., Fixed points results ofpointwise contractions in modular metric spaces. Fixed Point Theory Appl. 2013, 2013:163.Google Scholar
[2] Abdou, A. N. and Khamsi, M. A., Fixed points of multivalued contraction mappings in modular metric spaces. Fixed Point Theory Appli. 2014, 2014:249 http://dx.doi.Org/10.1186/1687-1812-2014-249 Google Scholar
[3] Alfuraidan, M. R., Fixed points of multivalued mappings in modular function spaces with a graph. Fixed Point Theory Appli. 2015, 2015:42.Google Scholar
[4] Alfuraidan, M. R., Remarks on monotone multivalued mappings on a metric space with a graph. J. Inequal. Appl., to appear.Google Scholar
[5] Beg, I., Fixed points of fuzzy multivalued mappings with values in fuzzy ordered sets. J. Fuzzy Math. 6(1998), no. 1,127-131.Google Scholar
[6] Beg, I. and Butt, A. R., Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal. 71(2009), no. 9, 3699–3704. http://dx.doi.Org/10.1016/j.na.2009.02.027 Google Scholar
[7] Beg, I., Butt, A. R., and S. Radojevic, The contraction principle for set valued mappings on a metric space with a graph. Comput. Math. Appl. 60(2010), no. 5,1214-1219. http://dx.doi.Org/1 0.101 6/j.camwa.2O10.06.003 Google Scholar
[8] Chartrand, G., Lesniak, L., and Zhang, P., Graphs & digraphs. Fifth éd., CRC Press, Boca Raton, FL, 2011.Google Scholar
[9] Chistyakov, V. V., Modular metric spaces. I. Basic concepts. Nonlinear Anal. 72(2010), no. 1.1-14. http://dx.doi.Org/10.1016/j.na.2009.04.057 Google Scholar
[10] Chistyakov, V. V., Modular metric spaces. II. Application to superposition operators. Nonlinear Anal. 72(2010), no. 1, 15–30. http://dx.doi.Org/10.1016/j.na.2009.04.018 Google Scholar
[11] Drici, Z., McRae, F. A., and Vasundhara Devi, J., Fixed point theorems in partially ordered metric space for operators with PPF dependence. Nonlinear Anal. 67(2007), no. 2, 641–647. http://dx.doi.Org/10.1016/j.na.2006.06.022 Google Scholar
[12] Edelstein, M., An extension ofBanach's contraction principle. Proc. Amer. Math. Soc. 12(1961), 7–10.Google Scholar
[13] Feng, Y. and Liu, S., Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings. J. Math. Anal. Appl. 317(2006), no. 1,103-112. http://dx.doi.Org/10.1016/j.jmaa.2OO5.12.004 Google Scholar
[14] Fujimoto, T., An extension of Tarski's fixed point theorem and its application to isotone complementarity problems. Math. Programming 28(1984), no. 1,116-118. http://dx.doi.Org/10.1007/BF02612716 Google Scholar
[15] Granas, A. and Dugundji, J., Fixed point theory. Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.Google Scholar
[16] Jachymski, J., The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc. 136(2008), no. 4,1359-1373. http://dx.doi.Org/10.1090/S0002-9939-07-09110-1 Google Scholar
[17] Johnsonbaugh, R., Discrete mathematics. Prentice-Hall, Inc., New Jersey, 1997.Google Scholar
[18] Harjani, J. and Sadarangani, K., Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72(2010), no. 3-4, 1188-1197. http://dx.doi.Org/10.1016/j.na.2009.08.003 Google Scholar
[19] Kirk, W. A. and Goebel, K., Topics in metric fixed point theory. Cambridge Studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990.Google Scholar
[20] Klim, D. and Wardowski, D., Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 334(2007), no. 1, 132–139. http://dx.doi.Org/10.1016/j.jmaa.2006.12.012 Google Scholar
[21] Kozlowski, W. M., Modular function spaces. Monographs and Textbooks in Pure and Applied Mathematics, 122, Dekker, New York, 1988.Google Scholar
[22] Lukawska, G. G. and Jachymski, J., IFS on a metric space with a graph structure and extension of the Kelisky-Rivlin theorem. J. Math. Anal. Appl. 356(2009), no. 2, 453–463. http://dx.doi.Org/10.1016/j.jmaa.2009.03.023 Google Scholar
[23] Mizoguchi, N. and Takahashi, W., Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1989), no. 1,177-188. http://dx.doi.Org/10.1016/0022-247X(89)90214-X Google Scholar
[24] Musielak, J., Orlicz spaces and modular spaces. Lecture Notes in Mathematics, 1034, Springer-Verlag, Berlin, 1983.Google Scholar
[25] Nadler, S. B., Jr., Multi-valued contraction mappings. Pacific J. Math. 30(1969), 475–488. http://dx.doi.Org/10.2140/pjm.1969.30.475 Google Scholar
[26] Nakano, H., Modulared semi-ordered linear spaces. Maruzen Co., Ltd., Tokyo, 1950.Google Scholar
[27] Nieto, J. J., Pouso, R. L., and Rodriguez-Lopez, R., Fixed point theorems in ordered abstract spaces. Proc. Amer. Math. Soc. 135(2007), no. 8, 2505–2517. http://dx.doi.Org/10.1090/S0002-9939-07-08729-1 Google Scholar
[28] Orlicz, W., Collected Papers, part I, II. PWN-Polish Scientific Publishers, Warsaw, 1988.Google Scholar
[29] O'Regan, D. and Petrusel, A., Fixed point theorems for generalized contraction in ordered metric spaces. J. Math. Anal. Appl. 341(2008), no. 2,1241-1252. http://dx.doi.Org/1 0.101 6/j.jmaa.2007.11.02 6 Google Scholar
[30] Petrusel, A. and Rus, I. A., Fixed point theorems in ordered L-spaces. Proc. Amer. Math. Soc. 134(2006), no. 2, 411–418.Google Scholar
[31] M. Ran, A. C. and Reurings, M. C. B., A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132(2004), no. 5,1435-1443. http://dx.doi.Org/10.1090/S0002-9939-03-07220-4 Google Scholar
[32] Reich, S., Some fixed point problems. Atti. Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 57(1974), no. 3-4, 194–198.Google Scholar
[33] Tarski, A., A lattice theoretical fixed point and its application. Pacific J. Math. 5(1955), 285–309. http://dx.doi.Org/10.2140/pjm.1955.5.285 Google Scholar