Document Type: Research Paper

Author

Department of Mathematics, Payame Noor University, p.o.box.19395-3697, Tehran, Iran.

Abstract

Proving  fixed point theorem in a fuzzy metric space is not possible for  Meir-Keeler contractive mapping. For this, we introduce the notion of $c_0$-triangular fuzzy metric space. This new space allows us to prove some fixed point theorems for  Meir-Keeler contractive mapping. As some pattern we introduce the class of $\alpha\Delta$-Meir-Keeler contractive and we establish some results of fixed point for such a mapping in the setting of $c_0$-triangular fuzzy metric space. An example is furnished to demonstrate the validity of these obtained results.

Keywords

Main Subjects

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

[2] C. Di Bari and C. Vetro,  Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space, J. Fuzzy Math., 13 (2005), pp. 973–-982.

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

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

[5] J. Jachymski,  Equivalent condition and the Meir-Keeler type theorems , J. Math. Anal. Appl., 194 (1995), pp. 293-303.

[6] E. Karapinar, P. Kumam, and P. Salimi,  On $alpha$$-psi$-Meir-Keeler contractive mappings, Fixed Point Theory Appl., 1 (2013), pp. 1-12.

[7] I. Kramosil and J. Michálek,  Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975), pp. 336-344.

[8] A. Meir and E. Keeler,  A theorem on contraction mapping, J. Math. Anal. Appl., 28 (1969), pp. 326-329.

[9] S. Park and B.E. Rhoades,  Meir-Keeler type contractive condition, Math. Japon., 26 (1981), pp. 13-20.

[10] A.C.M. Ran and M.C. Reurings,  A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2004), pp. 1435-1443.

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

[12] Y.Shen, D.Qiu, and W.Chenc, Fixed point theorems in fuzzy metric spaces, Appl. Math. Letters, 25 (2012), pp. 138-141.

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