×

A fixed point approach to almost quartic mappings in quasi fuzzy normed spaces. (English) Zbl 1187.46067

Summary: We define a notion for a quasi fuzzy \(p\)-normed space, then we use the fixed point alternative theorem to establish Hyers-Ulam-Rassias stability of the quartic functional equation where functions map a linear space into a complete quasi fuzzy \(p\)-normed space. Later, we show that there exists a close relationship between the fuzzy continuity behavior of a fuzzy almost quartic function, control function and the unique quartic mapping which approximates the almost quartic map. Finally, some applications of our results in the stability of quartic mappings from a linear space into a quasi \(p\)-norm space will be exhibited.

MSC:

46S40 Fuzzy functional analysis
39B82 Stability, separation, extension, and related topics for functional equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aoki, T., Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 18, 588-594 (1942) · Zbl 0060.26503
[2] Bag, T.; Samanta, S. K., Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11, 3, 687-705 (2003) · Zbl 1045.46048
[3] Bag, T.; Samanta, S. K., Fuzzy bounded linear operators, Fuzzy Sets and Systems, 151, 513-547 (2005) · Zbl 1077.46059
[4] Cheng, S. C.; Mordeson, J. N., Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 86, 429-436 (1994) · Zbl 0829.47063
[5] Diaz, J. B.; Margolis, B., A fixed point theorem of the alternative for the contractions on generalized complete metric space, Bull. Amer. Math. Soc., 74, 305-309 (1968) · Zbl 0157.29904
[6] Felbin, C., Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48, 239-248 (1992) · Zbl 0770.46038
[7] Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403
[8] Katsaras, A. K., Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12, 143-154 (1984) · Zbl 0555.46006
[9] Kramosil, I.; Michalek, J., Fuzzy metric and statistical metric spaces, Kybernetica, 11, 326-334 (1975)
[10] Krishna, S. V.; Sarma, K. K.M., Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems, 63, 207-217 (1994) · Zbl 0849.46058
[11] D. Mihet, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems (2008), doi:10.1016/j.fss.2008.06.014; D. Mihet, The fixed point method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems (2008), doi:10.1016/j.fss.2008.06.014
[12] Mihet, D.; Radu, V., On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343, 567-572 (2008) · Zbl 1139.39040
[13] Mirmostafaee, A. K.; Mirzavaziri, M.; Moslehian, M. S., Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159, 730-738 (2008) · Zbl 1179.46060
[14] Mirmostafaee, A. K.; Moslehian, M. S., Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159, 720-729 (2008) · Zbl 1178.46075
[15] Mirmostafaee, A. K.; Moslehian, M. S., Fuzzy almost quadratic functions, Results Math., 52, 161-177 (2008) · Zbl 1157.46048
[16] Mirmostafaee, A. K.; Moslehian, M. S., Fuzzy approximately cubic mappings, Inform. Sci., 78, 19, 3791-3798 (2008) · Zbl 1160.46336
[17] V. Radu, The fixed point alternative and the stability of functional equations, in: Seminar on Fixed Point Theory, Cluj-Napoca, Vol. 4, 2003.; V. Radu, The fixed point alternative and the stability of functional equations, in: Seminar on Fixed Point Theory, Cluj-Napoca, Vol. 4, 2003. · Zbl 1051.39031
[18] Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040
[19] S.M. Ulam, Some questions in analysis: §1, stability, Problems in Modern Mathematics, Science eds., Wiley, New York, 1964 (Chapter VI).; S.M. Ulam, Some questions in analysis: §1, stability, Problems in Modern Mathematics, Science eds., Wiley, New York, 1964 (Chapter VI).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.