In this paper, by introducing the concept of a fuzzy rectangular-b-metric space, the notion of a fuzzy metric space and a fuzzy b-metric space are generalized. The well known metric fixed point theorems are established in the setting of fuzzy rectangular b-metric spaces and illustrated by examples. To show the significance of our result an application is presented to establish the existence of a solution of integral equation. Our results generalize many existing theorems in the literature.
R.P.Agarwal, N.Hussain and M.A.Taoudi, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, Abstr Appl Anal2012 (2012), 15. Art. ID 245872.
2.
I.A.Bakhtin, The contraction mapping principle in quasi metric spaces, Funct Anal Unianowsk Gos Ped Inst30 (1989), 26–37.
3.
A.Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces, Publ Math Debrecen57 (2000), 31–37.
4.
R.Chugh and S.Kumar, Weakly compatible maps in generalized fuzzy metric spaces, J Analysis10 (2002), 65–74.
5.
S.Czerwik, Contraction mappings in b-metric space, Acta Math InfUniv Ostraviensis1 (1993), 5–11.
6.
A.George and P.Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems64 (1994), 395–399.
7.
R.George, S.Radenovic, K.P.Reshma and S.Shukla, Rectangular b-metric space and contraction principles, J Nonlinear Sci Appl8 (2015), 1005–1013.
8.
D.Gopal and C.Vetro, Some new fixed point theorems in fuzzy metric spaces, Iranian Journal of Fuzzy Systems11(3) (2014), 95–107.
9.
M.Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems27 (1988), 385–389.
10.
V.Gupta, N.Mani and A.Saini, Fixed point theorems and its applications in Fuzzy metric spaces, Conference Paper, 2013.
11.
T.L.Hicks and B.E.Rhoades, A Banach type fixed point theorem, Math Jpn24 (1979), 327–330.
12.
S.Heilpern, Fuzzy maps and fixed point theorems, Anal Appl83(2) (1981), 566–569.
13.
A.-F.Roldan-Lopez-de-Hierro, E.Karapinar and S.Manro, Some new fixed point theorems in fuzzy metric spaces, Journal of Intelligent and Fuzzy Systems27(5) (2014), 2257–2264.
14.
N.Hussain, P.Salimi and V.Parvaneh, Fixed point results for various contractions in parametric and fuzzy b-metric spaces, J Nonlinear Sci Appl8(5) (2015), 719–739.
15.
H.Jiang, J.Zhan and D.Chen, Covering based variable precision (I, T)-fuzzy rough sets with applications to multiattribute decision-making, IEEE Transactions on Fuzzy Systems (2018).
16.
T.Kamran, Common fixed points theorems for fuzzy mappings, Chaos, Solitons and Fractals38(5) (2008), 1378–1382.
17.
M.A.Khamsi and N.Hussain, KKM mappings in metric type spaces, Nonlinear Anal73 (2010), 3123–3129.
18.
I.Kramosil and J.Michalek, Fuzzy metric and statistical metric spaces, Kybernetica11 (1975), 326–334.
19.
U.Mishra, C.Vetro and P.Kumam, On modified a-0-fuzzy contractive mappings and an application to integral equations, Journal of Inequalities and Applications1 (2016), 67.
20.
S.Nǎdǎban, Fuzzy b-metric spaces, International Journal of Computers Communications and Control11(2) (2016), 273–281.
21.
J.J.Nieto and R.Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order22 (2005), 223–239.
22.
J.R.Roshan, V.Parvaneh, Z.Kadelburg and N.Hussain, New fixed point results in b-rectangular metric spaces, Nonlinear Anal, Model Control21(5) (2016), 614–634.
D.Turkoglu and M.Sangurlu, Fixed point theorems for fuzzy ψ-contractive mappings in fuzzy metric spaces, Journal of Intelligent and Fuzzy Systems26(1) (2014), 137–142.
25.
R.Vasuki and P.Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets and Systems135 (2003), 415–417.
26.
C.Vetro, Fixed points in a weak non-Archemedean fuzzy metric spaces, Fuzzy Sets and Systems162 (2011), 84–90.
27.
L.A.Zadeh, Fuzzy sets, Information and Control8 (1965), 338–353.
28.
J.Zhan, B.Sun and J.C.R.Alcantud, Covering based multigranulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making, Information Sciences476 (2019), 290–318.
29.
L.Zhang, J.Zhan and Z.Xu, Covering-based generalized IF rough sets with applications to multi-attribute decision-making, Information Sciences478 (2019), 275–302.
30.
K.Zhang, J.Zhan, W.Wu and J.C.R.Alcantud, Fuzzy-covering Covering-based (I, T) fuzzy rough set models and applications to multi-attribute decision-making, Computers and Industrial Engineering128 (2019), 605–621.
