Some common fixed point results of multivalued mappings on fuzzy metric space

Document Type : Research Paper


1 Department of Mathematics, School of Computing, University of Engineering and technology Roorkee, Haridwar, Uttarakhand-247667, India

2 Department of Mathematics, Kanya Gurukula Campus, Gurukula Kangri Vishwavidyalaya, Haridwar, Uttarakhand-249404, India

3 Department of Mathematics, Shaheed Srimati Hansa Dhanai Government Degree College, Agrora (Dharmandal), Tehri Garhwal, Uttarakhand -249127, India


The aim of this paper is to establish new fixed point theorems for single-valued and multivalued maps which satisfy α-ψ-contraction conditions in the complete fuzzy metric space. In this paper, we extend the results of Hussain et al. and Samet et al. Some comparative examples are also given which demonstrate the superiority of our results from the exiting results in the literature.


[1] A. George and P. Veeramani, On some result in fixed point theorems in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399.
[2] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α − ψ−contractive type mappings, Nonlinear Anal. 75 (2012) 2154–2165.
[3] D. Gopal and C. Vetro, Some new fixed point theorems in fuzzy metric spaces, Iran. J. Fuzzy Syst. 11(3) (2014) 95–107.
[4] D. Gopal, M. Imded, C. Vetro and M. Hasan, Fixed point theory for cyclic weak ϕ−contraction in fuzzy metric spaces, J. Nonlinear Anal. Appl. Article ID jnaa- pages,doi: 10.5899/2012/jnaa-0110, 2012 (2012).
[5] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets Syst. 144 (3) (2004) 431–439.
[6] D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Syst. 158 (8) (2007) 915–921.
[7] I. Kramosil and J. Michalek, Fuzzy metric and Statistical metric spaces, Ky-Bernetica 11 (1975) 336–344.
[8] J. Hasanzade Asl, S. Rezapour and N. Shahzad, On fixed points of α−ψ−contractive multifunctions, Fixed Point Theory Appl. (2012) 2012 : 212.
[9] J. Rodriguez-Lopez and S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets Syst. 147 (2004) 273–283.
[10] L.A. Zadeh, Fuzzy sets Information and Control, Fuzzy Sets and Syst. 8 (1965) 338–353.
[11] M. Grabiec, Fixed point in fuzzy metric space, Fuzzy Sets Syst. 27 (1988) 385–389.
[12] M.U. Ali and T. Kamran, On (α⋆−ψ−)contractive multi-valued mappings, Fixed Point Theory Appl. 2013 (2013) Article ID 137.
[13] N. Hussain, J. Ahmad and A. Azam, Generalized fixed point theorems for multi-valued α − ψ− contractive mappings, J. Inequal. Appl. 2014 (2014) 348.
[14] R. Arora and M. Kumar, Unique fixed point theorems for α−ψ−contractive type mappings in fuzzy metric space, Cogent. Math. Statis. 3(1) (2016) 1–8.
[15] S.B.Jr. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–478.
[16] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960) 385–389.
[17] S. Phiangsungnoen, W. Sintunavarat and P. Kumam, Fuzzy fixed point theorems in Hausdorff fuzzy metric spaces, J. Inequal. Appl. 2014 (2014) 201.
[18] V. Gregori and A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets Syst. 125(2) (2002) 245–253.
[19] V.L. Lazar, Fixed point theory for multivalued ϕ−contractions, Fixed Point Theory Appl. 2011 (2011) 50.
Volume 13, Issue 1
March 2022
Pages 653-662
  • Receive Date: 30 May 2019
  • Revise Date: 08 December 2019
  • Accept Date: 13 December 2019