Stability of Generalized Quadratic Functional Equation in Non-Archimedean Fuzzy Normed Spaces

Abstract:

In this paper, we obtain the general solution and investigate the Hyers-Ulam-Rassias stability of the Generalized Quadratic functional equation in non-Archimedean fuzzy normed spaces.

• Periodical: Advanced Materials Research (Volumes 403-408)
• Pages: 879-887
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.403-408.879
• Online since: November 2011
• Authors: K. Ravi, P. Narasimman
• Keywords: Fuzzy Metric Spaces, Fuzzy Normed Spaces, Hyers-Ulam-Rassias Stability, Intuitionistic Fuzzy Metric Spaces, Intuitionistic Fuzzy Normed Spaces, Non-Archimedean Fuzzy Normed Spaces, Quadratic Functional Equation

References

[1] J. Aczel and J. Dhombres, Functional Equation in Several Variables, Cambridge Univ. Press, (1989).

[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math . Soc. Japan, 2 (1950), 64-66.

[3] T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11(3) (2003), 687-705.

[4] V. Balopoulos, A. G. Hatzimichailidis and Basil K. Papadopoulos, Distance and similarity measures for fuzzy operators, Inform. Sci. 177 (2007), 2336-2348.

DOI: 10.1016/j.ins.2007.01.005

[5] R. Biswas, Fuzzy inner product space and fuzzy norm functions, Inform. Sci. 53 (1991), 185-190.

DOI: 10.1016/0020-0255(91)90063-z

[6] S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcuta Math. Soc. 86 (1994), 429-436.

[7] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86.

DOI: 10.1007/bf02192660

[8] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64.

DOI: 10.1007/bf02941618

[9] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets Syst. 48 (1992), 239-248.

DOI: 10.1016/0165-0114(92)90338-5

[10] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.

DOI: 10.1006/jmaa.1994.1211

[11] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), 222-224.

DOI: 10.1073/pnas.27.4.222

[12] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst. 12 (1984), 143-154.

DOI: 10.1016/0165-0114(84)90034-4

[13] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica11 (1975), 326-334.

[14] S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets Syst. 63 (1994), 207-217.

DOI: 10.1016/0165-0114(94)90351-4

[15] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets Syst. 159 (2008), 720-729.

DOI: 10.1016/j.fss.2007.09.016

[16] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results Math. doi: 10. 1007/s00025-007-0278-9.

[17] Th. M. Rassias, On the stability of the linear mapping in Banacb spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

DOI: 10.1090/s0002-9939-1978-0507327-1

[18] K. Ravi, P. Narasimman and R. Kishore Kumar, Generalized Hyers-Ulam- Rassias stability and J. M. Rassias stability of a quadratic functional equation, IJMSEA, 3(2)(2009), 79-94.

DOI: 10.9734/bjmcs/2016/21874

[19] R. Saadati, J. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals 27 (2006) 331–344.

DOI: 10.1016/j.chaos.2005.03.019

[20] Shieh, Infinite fuzzy relation equations with continuous t-norms, Inform. Sci. 178(2008), 1961-(1967).

DOI: 10.1016/j.ins.2007.12.006

[21] F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis Milano 53 (1983), 113-129.

[22] S. M. Ulam, A Colloection of the Mathematical Problems, Interscience Publ., NewYork, (1960).

[23] Cpmgxin Wu and Jinxuan Fang, Fuzzy generalization of Klomogoroffs theorem, J. Harbin Inst. Technol. 1 (1984), 1-7.