International Journal of Nonlinear Analysis and Applications

A study on interval-valued generalized fuzzy n-normed linear space

Document Type : Research Paper

Authors

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

Abstract

Following the definition of interval-valued fuzzy n-normed linear space given by S. Vijayabalaji et al., in this paper, the notion of interval-valued generalized fuzzy n-normed linear space is introduced. The notion of convergent sequence, Cauchy sequence and their relation are studied. Some basic results are established on finite-dimensional interval-valued generalized fuzzy n-normed linear space.

Keywords

[1] T. Bag, Finite dimensional fuzzy cone normed linear spaces, Int. J. Math. Sci. Comput. 3 (2013), 9–14.
[2] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (2003), 687–705.
[3] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Ann. Fuzzy Math. Inf. 6 (2013), 271–283.
[4] S. Chatterjee, T. Bag, and S.K. Samanta, Some results on G-fuzzy normed linear space, Int. J. Pure Appl. Math. 5 (2018), 1295–1320.
[5] S.C. Cheng and J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436.
[6] F.C. Company and P. Tirado, A note on interval-valued fuzzy metric spaces, Ann. Fuzzy Math. Inf. 12 (2016), 585–590.
[7] P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl. 63 (2012), 708–715.
[8] P. Debnath, Results on lacunary difference ideal convergence in intuitionistic fuzzy normed linear spaces, J. Intell. Fuzzy Syst. 28 (2015), 1299–1306.
[9] P. Debnath, A generalized statistical convergence in intuitionistic fuzzy n-normed linear spaces, Ann. Fuzzy Math. Inf. 12 (2016), 559–572.
[10] P. Debnath and M. Sen, Some completeness results in terms of infinite series and quotient spaces in intuitionistic fuzzy n-normed linear spaces, J. Intell. Fuzzy Syst. 26 (2014), 975–982.
[11] P. Debnath and M. Sen, Some results of calculus for functions having values in an intuitionistic fuzzy n-normed linear space, J. Intell. Fuzzy Syst. 26 (2014), 2983–2991.
[12] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48 (1992) 239–248.
[13] S. Gahler, Lineare 2-normierte raume, Math. Nach. 28 (1964), 1—43.
[14] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994), 395–399.
[15] M.E. Gordji, M. Ramezani, H. Khodaei and H. Baghani, Cone normed spaces, Caspian J. Math. Sci. 1 (2012), no. 1, 7—12.
[16] H. Gunawan and M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27 (2001), 631–639.
[17] A.K. Katsaras, Fuzzy topological vector spaces I, Fuzzy Sets Syst. 12 (1984), 143–154.
[18] K.A. Khan, Generalized normed spaces and fixed point theorems, J. Math. Comput. Sci. 13 (2014), 157–167.
[19] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Printice-Hall of India Private Limited, 1997.
[20] N. Konwar and P. Debnath, Iλ-convergence in intuitionistic fuzzy n-normed linear space, Ann. Fuzzy Math. Inf. 13 (2017), 91–107.
[21] C. Li, Distances between interval-valued fuzzy sets, Proc. 28th North Amer. Fuzzy Inf. Process. Soc. Ann. Conf. (NAFIPS2009), Cincinnati, USA, 2009.
[22] R. Moore, Interval Analysis, Pretince-Hall, Englewood Clis, 1996.
[23] Y. Shen, H. Li, and F. Wang, On interval-valued fuzzy metric spaces, Int. J. Fuzzy Syst. 14 (2012), 35–44.
[24] U. Samanta and T. Bag, Completeness and compactness of finite dimensional fuzzy n-normed linear spaces, Ann. Fuzzy Math. Inf. 7 (2014), 837–850.
[25] M. Sen and P. Debnath, Lacunary statistical convergence in intuitionistic fuzzy n-normed linear spaces, Math. Comput. Modell. 54 (2011), 2978–2985.
[26] R.M. Somasundaram and T. Beaula, Some aspects of 2-fuzzy 2-normed linear spaces, Bull. Malays. Math. Sci. Soc. Second Ser. 32 (2009), 211–221.
[27] S. Vijayabalaji, S.A. Shanthi, and N. Thillaigovindan, Interval valued fuzzy n-normed linear space, Malays. J. Fund. Appl. Sci. 4 (2008), 287–297.
[28] S. Vijayabalaji and N. Thillaigovindan, Complete fuzzy n-normed linear space, J. Fund. Appl. Sci. 3 (2007), 119–126.
[29] L.A. Zadeh, The concept of a linguistic variable and its application to approximation reasoning I, Inf. Sci. 8 (1975), 199–249.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 10
October 2024
Pages 311-321
  • Receive Date: 14 March 2023
  • Revise Date: 30 August 2023
  • Accept Date: 20 October 2023