JID:FSS AID:7377 /FLA [m3SC+; v1.279; Prn:21/02/2018; 9:18] P.1(1-10) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 46 46 47 47 48 48 49 49 50 50 51 51 52 52 A new class of fuzzy contractive mappings and fixed point theorems Satish Shukla a , Dhananjay Gopal b , Wutiphol Sintunavarat c a Department of Applied Mathematics, Shri Vaishnav Institute of Technology & Science, Gram Baroli, Sanwer Road, Indore, 453331, (M.P.) India b Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Surat, 395-007, Gujarat, India c Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand Received 26 July 2016; received in revised form 14 February 2018; accepted 15 February 2018 Abstract The main aim of this work is to unify different classes of fuzzy contractive mappings by introducing a new class of fuzzy contractive mappings called fuzzy Z-contractive mappings. For this new class of mappings, suitable conditions are framed to ensure the existence of fixed point in M-complete fuzzy metric spaces (in the sense of George and Veeramani). A comprehensive set of examples are presented to support the claim.  2018 Elsevier B.V. All rights reserved. Keywords: Fuzzy metric space; M-completeness; Fuzzy Z-contractive mapping; Fixed point; Edelstein’s mapping 1. Introduction Kramosil and Michálek [6] first introduced the notion of fuzzy metric spaces. George and Veeramani [1] modi- fied the notion of Kramosil and Michálek so that they obtained a Hausdorff topology in fuzzy metric spaces. The fixed point theory in fuzzy metric spaces was first introduced by Grabiec [2]. He extended the famous Banach con- traction principle and the fixed point result of Edelstein [5] in fuzzy metric spaces in the sense of Kramosil and Michálek [6]. However, the Grabiec’s fixed point result was based on a strong condition associated with completeness of fuzzy metric spaces called G-completeness. George and Veeramani [1] weakened this condition and introduced the M -completeness of fuzzy metric spaces. A basic and natural contractive condition in probabilistic metric spaces was used by Sherwood [13] (see also Tirado [16]). Gregori and Sapena [3] introduced the notion of fuzzy contractive map- pings and proved the fixed point results for such mappings. The class of fuzzy contractive mappings of Gregori and Sapena [3] includes the class of Sherwood’s contraction. Radu [10] rewrote the fuzzy contractive condition of Gregori and Sapena in fuzzy Menger spaces. Mihe¸ t [7] introduced the notion of fuzzy ψ -contractive mappings and unified and generalized the definitions used in [13] and [3]. Mihe¸ t also proved a fixed point result for fuzzy ψ -contractive E-mail addresses: satishmathematics@yahoo.co.in (S. Shukla), gopaldhananjay@yahoo.in (D. Gopal), wutiphol@mathstat.sci.tu.ac.th (W. Sintunavarat). https://doi.org/10.1016/j.fss.2018.02.010 0165-0114/ 2018 Elsevier B.V. All rights reserved.