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Stability of the general form of quadratic-quartic functional equations in non-Archimedean \(\mathcal{L}\)-fuzzy normed spaces. (English) Zbl 1390.39084
Summary: In this paper, we introduce and obtain the general solution of a new generalized mixed quadratic and quartic functional equation and investigate its stability in non-Archimedean \(\mathcal{L}\)-fuzzy normed spaces.

MSC:
39B52 Functional equations for functions with more general domains and/or ranges
39B72 Systems of functional equations and inequalities
39B82 Stability, separation, extension, and related topics for functional equations
46B03 Isomorphic theory (including renorming) of Banach spaces
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[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math . Soc. Japan, 2 (1950), 64-66. · Zbl 0040.35501
[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), No. 1, 87-96. · Zbl 0631.03040
[3] K. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Sci.-Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulg.)
[4] T. Bag and S . K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11 (3) (2003) 687-705. · Zbl 1045.46048
[5] R. Biswas, Fuzzy inner product space and fuzzy norm functions, Inform. Sci., 53 (1991), 185-190. · Zbl 0716.46061
[6] A. Bodaghi, Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, J. Intel. Fuzzy Syst., 30 (2016), 2309-2317. · Zbl 1361.39013
[7] A. Bodaghi, Approximate mixed type additive and quartic functional equation, Bol. Soc. Paran. Mat., 35 (1) (2017), 43-56.
[8] A. Bodaghi, Stability of a mixed type additive and quartic function equation, Filomat, 28 (8) (2014), 1629-1640. · Zbl 1390.39098
[9] A. Bodaghi, D. Kang and J. M. Rassias, The mixed cubic-quartic functional equation, An. Stiint. Univ. Al. I. Cuza Iasi Mat. (N.S.), 63, No 1 (2017), 215-227. · Zbl 1399.39062
[10] A. Bodaghi and S. O. Kim, Ulam’s type stability of a functional equation deriving from quadratic and additive functions, J. Math. Ineq., 9, No. 1 (2015), 73-84. · Zbl 1314.39028
[11] A. Bodaghi and S. O. Kim, Stability of a functional equation deriving from quadratic and ad- ditive functions in non-Archimedean normed spaces, Abst. Appl. Anal., 2013, Art. ID 198018, 10 pages, doi: · Zbl 1291.39039
[12] A. Bodaghi, C. Park and J. M. Rassias, Fundamental stabilities of the nonic functional equation in intuitionistic fuzzy normed spaces, Commun. Korean Math. Soc., 31 (2016), No. 4, 729-743. · Zbl 1372.39025
[13] S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcuta Math. Soc., 86 (1994), 429-436. · Zbl 0829.47063
[14] G. Deschrijver, E. E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Set. Syst., 133 (2003), 227-235. · Zbl 1013.03065
[15] G. Deschrijver, D. O’Regan, R. Saadati, S. M. Vaezpour, L-Fuzzy Euclidean normed spaces and compactness, Chaos Sol. Fract., 42 (2009), 40-45. · Zbl 1200.46065
[16] M. Eshaghi Gordji, S. Abbaszadeh and C. Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces, J. Inequ. Appl., 2009, Art. ID 153084, 26 pages, doi: · Zbl 1187.39035
[17] M. Eshaghi Gordji, H. Khodaei and Hark-Mahn Kim, Approximate quartic and quadratic mappings in quasi-Banach spaces, Int. J. Math. Math., Sci., 2011, Art. ID 734567, 18 pages, doi: · Zbl 1223.39016
[18] C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Set. Syst., 48 (1992), 239-248. · Zbl 0770.46038
[19] P. Găavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436. · Zbl 0818.46043
[20] K. Hensel, Uber eine neue Begrndung der Theorie der algebraischen Zahlen, Jahresber, Deutsche Mathematiker-Vereinigung, 6 (1897), 83-88.
[21] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222-224. · JFM 67.0424.01
[22] A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Set. Syst., 12 (1984), 143-154. · Zbl 0555.46006
[23] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), 326-334. · Zbl 0319.54002
[24] Y. S. Lee, S. Kim and C. Kim, A stability for the mixed type of quartic and quadratic functional equations, J. Funct. Spaces, 2014, Art. ID 853743, 10 pages. · Zbl 1297.39033
[25] M. S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal., 69 (2008), 3405-3408. · Zbl 1160.46049
[26] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Sol. Fract., 22 (2004), 1039-1046. · Zbl 1060.54010
[27] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. · Zbl 0398.47040
[28] R. Saadati, A note on \Some results on the IF-normed spaces”, Chaos Sol. Fract., 41 (2009), 206-213. · Zbl 1198.54023
[29] R. Saadati, Y. J. Cho and J. Vahidi, The stability of the quartic functional equation in various spaces, Compu. Math. Appl., 60 (2010), 1994-2002. · Zbl 1205.39029
[30] R. Saadati and C. Park, Non-archimedean L-fuzzy normed spaces and stability of functional equations, Comput. Math. Appl., 60 (2010), 2488-2496. · Zbl 1205.39023
[31] R. Saadati, A. Razani and H. Adibi, A common fixed point theorem in L-fuzzy metric spaces”, Chaos Sol. Fract., 33 (2007), 354-363.
[32] R. Saadati and J. Park, On the intuitionistic fuzzy topological spaces, Chaos Sol. Fract., 27 (2006), 331-344. · Zbl 1083.54514
[33] R. Saadati, On the L-Fuzzy topological spaces, Chaos Sol. Fract., 37 (2008), 1419-1426. · Zbl 1142.54318
[34] S. M. Ulam, A Colloection of the Mathematical Problems, Interscience Publ., New York, 1960.
[35] C. Wu and J. Fang, Fuzzy generalization of Klomogoroffs theorem, J. Harbin Inst. Tech., 1 (1984), 1-7.
[36] T. Z. Xu, M. J. Rassias and W. X. Xu, Stability of a general mixed additive-cubic functional equation in non-archimedean fuzzy normed spaces, J. Math. Physics. Art., 093508, 19 pages, 51 (2010). · Zbl 1309.30029
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