Document Type : Research Paper


Department of Mathematics, Siksha-Bhavana, Visva-Bharati, Santiniketan-731235, Birbhum, West-Bengal, India.


Following the definition of fuzzy normed linear space which was introduced by Bag and Samanta in general t-norm settings, in this paper,  definition of fuzzy strong $\phi$-b-normed linear space is given. Here the scalar function $|c| $ is replaced by a general function $ \phi(c) $ where $ \phi $ satisfies some properties. Some basic results on finite dimensional  fuzzy strong $\phi$-b-normed linear space are studied.


[1] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, The J. Fuzzy Math., 11 (2003), pp. 687-705.
[2] T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst., 151 (2005), pp. 513-547.
[3] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Ann. Fuzzy Math. Inform., 6 (2013), pp. 271-283. 
[4] T. Bag, Finite dimensional fuzzy cone normed linear spaces, Int. J. Math. Scientific Computing, 3 (2013), pp. 9-14.
[5] S. Chatterjee, T. Bag and S.K. Samanta, Some results on G-fuzzy normed linear space, Int. J. Pure Appl. Math, 5 (2018), pp. 1295-1320.
[6] S.C. Cheng and J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Cal. Math. Soc., 86 (1994), pp. 429-436. 
[7] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math Inf Univ Ostraviensis, 1 (1993), pp. 5-11.
[8] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst., 48 (1992), pp. 239-248.
[9] S. Gahler, 2-metrische Raume und ihre topologische Struktur, Mathe-matische Nachrichten, 26 (1963), pp. 115-118. 
[10] S. Gahler, Lineare 2-normierte raume, Math. Nachr., 28 (1964), pp. 1–43.
[11] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), pp. 395-399. 
[12] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst., 12 (1984), pp. 215-229. 
[13] A.K. Katsaras, Fuzzy topological vector spaces I, Fuzzy Sets Syst., 12 (1984), pp. 143-154.
[14] K.A. Khan, Generalized normed spaces and fixed point theorems, J. Math. Computer Sci., 13 (2014), pp. 157-167.
[15] W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014. 
[16] G.J. Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic, Printice-Hall of India Private Limited, New Delhi-110001, 1997.
[17] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), pp. 326-334. 
[18] F. Mehmood, R. Ali, C.Ionescu and T. Kamran, Extended fuzzy b-Metric Spaces, J. Math. Anal., 8 (2017), pp. 124-131.
[19] Z. Mustafa, H. Obiedat and F. Awawdeh, Some Fixed Point Theorem for Mapping on Complete G-Metric Spaces, Fixed Point Theory Appl., 2008, 12 pages.
[20] S. Nădăban, Fuzzy b-Metric Spaces, Int. J. Computers Communications and Control, 11 (2016), pp. 273-281.
[21] T. Oner, M.B. Kandemir and B. Tanay, Fuzzy cone metric spaces, J. Nonlinear Sci. Appl., 8 (2015), pp. 610-616. 
[21] T. Oner, On topology of fuzzy strong b-metric spaces, J. New Theory, 21 (2018), pp. 59-67.
[22] L.A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), pp. 338-353.