Document Type: Research Paper

Author

Department of Mathematics, Faculty of Science, University of Bu-Ali Sina, P.O.Box 6517838695, Hamedan, Iiran.

Abstract

In 2006, Espinola and Kirk made a useful contribution on combining fixed point theory
and graph theory. Recently, Reich and Zaslavski studied a new inexact iterative scheme for fixed points of contractive and nonexpansive multifunctions. In this paper, by using  the main idea of their work and the idea of combining fixed point theory on intuitionistic fuzzy metric spaces and graph theory, we present some iterative scheme results for $G$-fuzzy contractive and $G$-fuzzy nonexpansive mappings on graphs.

Keywords

Main Subjects

[1] R.P. Agarwal, M.A. El-Gebeily, and D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Analysis, 87 (2008), pp. 109-116.

[2] C. Alaca, D. Turkoghlu, and C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 29 (2006), pp. 1073-1078.

[3] S.M.A. Aleomraninejad, Sh. Rezapour, and N. Shahzad, Some fixed point results on metric space witha graph, Topology Appl., 159 (2012), pp. 659-663.

[4] I. Altun and G. Durmaz, Some fixed point results in cone metric spaces, Rend Circ. Math. Palermo, 58 (2009), pp. 319-325.

[5] K. Atanassov, Intuitionistic fuzzy Sets, Fuzzy Sets and Systems, 20 (1986), pp. 87-96.

[6] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3 (1922), pp. 133-181.

[7] I. Beg, A.R. Butt, and S. Radojevic, The contraction principle for set valued mappings on a metric space with a graph, Comput. Math. Appl., 60 (2010), pp. 1214-1219.

[8] V. Berinde, Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Analysis, 74 (2011), pp. 7347-7355.

[9] J. Caristi, Fixed point theorems for mapping satisfying inwardness conditions, Trans. Amer. Math. Soc., 215 (1976), pp. 241-251.

[10] D. Coker, An introduction to intuitionistic fuzzy metric spaces, Fuzzy Sets and Systems, 88 (1997), pp. 81-89.

[11] C. Di Bari and, C. Vetro, A fixed point theorem for a family of mappings in a fuzzy metric space, Rend Circ. Math. Palermo, 52 (2003), pp. 315-321.

[12] F. Echenique, A short and constructive proof of Tarski's fixed point theorem, Internat, J. Game Theory, 33(2) (2005), pp. 215-218.

[13] M.S. El Naschie, A review of E-infinity theory and the mass spectrum of high energy particle physics, Chaos, solitons and Fractals, 19 (2004), pp. 209-236.

[14] M.S. El Naschie, On a fuzzy Kahler-like manifold which is consistent with two-slit experiment, Int. J. Nonlinear Science and Numerical Simul., 6 (2005), pp. 95-98.

[15] M.S. El Naschie, On the uncertainty of Contorian geometry and two-slit experiment, Chaos, Soliton and Fractals, 9 (1998), pp. 517-529.

[16] M.S. El Naschie, On the verification of heterotic strings theory and ε(∞) theory, Chaos, Soliton and Fractals, 11 (2000), pp. 2397-2407.

[17] M.S. El Naschie, On two new fuzzy Kahler manifolds, Klein modular space and Hooft holographic principles, Chaos, Solitons and Fractals, 29 (2006), pp. 876-881.

[18] M.S. El Naschie, The idealized quantum two-slit Gedanken experiment revisited criticism and reinterpretation, Chaos, Solitons and Fractals, 27 (2006), pp. 843-849.

[19] R. Espinola and W.A. Kirk, Fixed point theorems in R-trees with applications to graph theory, Topology Appl., 153 (2006), pp. 1046-1055.

[20] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27 (1988), pp. 385-389.

[21] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994), pp. 395-399.

[22] V. Gregori and A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 125 (2002), pp. 245-252.

[23] G. Gwozdz-Lukawska and J. Jachymski, IFS on metric space with a graph structure and extentions of the Kelisky-Rivilin theorem, J. Math. Anal. Appl., 356 (2009), pp. 453-463.

[24] T.L. Hicks, Fixed point theorems for quasi-metric spaces, Math. Japon., 33 (1988), pp. 231-236.

[25] J. Jachymski, The contraction principle for mappings on metric space with a graph, Proc, Amer. Math. Soc., 136 (2008), pp. 1359-1373.

[26] S.G. Jeong and B.E. Rhoades, More maps for which F(T)=F(Tn), Demonestraio Math., 40 (2007), pp. 671-680.

[27] H. Karayilan and M. Telci, Common fixed point theorem for contractive type mappings in fuzzy metric spaces, Rend. Circ. Mat. Palermo., 60 (2011), pp. 145-152.

[28] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), pp. 326-334.

[29] K. Menger, Statistical metrices, Proc. Natl. Acad. Sci., 28 (1942), pp. 535-537.

[30] D. Mihet, A banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems, 144 (2004), pp. 431-439.

[31] J.H. Park, Intiutionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004), pp. 1039-1046.

[32] J.S. Park, Y.C. Kwun, and J.H. Park, A fixed point theorem in the intiutionistic fuzzy metric spaces, Far East J. Math. Sci., 16 (2005), pp. 137-149.

[33] M. Rafi and M.S.M. Noorani, Fixed point theorem on intuitionistic fuzzy metric spaces, Iranian J. of Fuzzy Systems, 3 (2006), pp. 23-29.

[34] Sh. Rezapour and P. Amiri, Some fixed point results for multivalued operators in generalized metric spaces, Computers and Mathematics with Applications, 61 (2011), pp. 2661-2666.

[35] Sh. Rezapour and R. Hamlbarani, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Analysis Appl., 345 (2008), pp. 719-724.

[36] J. Rodriquez-Lopez and S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets and Systems, 147 (2004), pp. 173-176.

[37] B. Samet, C. Vetro, and P. Vetro, Fixed point theorem for α-ψ-contractive type mappings, Nonlinear Analysis, 75 (2012), pp. 2154-2165.

[38] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math., 10 (1960), pp. 314-334.

[39] L.D.J. Sigalotte and A. Mejias, On El Naschie's conjugate complex time, fractal E(∞) space-time and faster-than-light particles, Int. J. Nonlinear Sci. Number Simul., 7 (2006), pp. 467-472.

[40] P. Veeramani, Best approximation in fuzzy metricspaces, Journal of fuzzy mathematics, 9 (2001), pp. 75-80.

[41] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), pp. 338-353.