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The stability of the quartic functional equation in various spaces. (English) Zbl 1205.39029
Summary: The purpose of this paper is first to introduce the notation of intuitionistic random normed spaces, and then by virtue of this notation to study the stability of a quartic functional equation in the setting of these spaces under arbitrary triangle norms. Then we prove the stability of above quartic functional equation in non-Archimedean random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the theory of intuitionistic spaces and the theory of functional equations are also presented in the paper.
MSC:
39B82Stability, separation, extension, and related topics
28E99Miscellaneous topics of measure theory
39B52Functional equations for functions with more general domains and/or ranges
46S10Functional analysis over fields (not , , or quaternions)
References:
[1]Ulam, S. M.: Problems in modern mathematics, (1964)
[2]Hyers, D. H.: On the stability of the linear functional equation, Proc. natl. Acad. sci. USA 27, 222-224 (1941) · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3]Aoki, T.: On the stability of the linear transformation in Banach spaces, J. math. Soc. Japan 2, 64-66 (1950) · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[4]Rassias, Th.M.: On the stability of the linear mapping in Banach spaces, Proc. amer. Math. soc. 72, 297-300 (1978) · Zbl 0398.47040 · doi:10.2307/2042795
[5]Cho, Y. J.; Park, C.; Saadati, R.: Functional inequalities in non-Archimedean Banach spaces, Appl. math. Lett. 10, 1238-1242 (2010) · Zbl 1203.39015 · doi:10.1016/j.aml.2010.06.005
[6]Czerwik, S.: Functional equations and inequalities in several variables, (2002) · Zbl 1011.39019
[7]Hyers, D. H.; Isac, G.; Rassias, Th.M.: Stability of functional equations in several variables, (1998)
[8]Miheţ, D.: Fuzzy stability of additive mappings in non-Archimedean fuzzy normed spaces, Fuzzy sets systems 161, 2206-2212 (2010)
[9]Rassias, Th.M.: Functional equations, inequalities and applications, (2003)
[10]Baak, C.; Moslehian, M. S.: On the stability of J*-homomorphisms, Nonlinear anal. TMA 63, 42-48 (2005) · Zbl 1085.39026 · doi:10.1016/j.na.2005.04.004
[11]Jun, K. W.; Kim, H. M.: The generalized Hyers-Ulam-rassias stability of a cubic functional equation, J. math. Anal. appl. 274, 867-878 (2002) · Zbl 1021.39014 · doi:10.1016/S0022-247X(02)00415-8
[12]Rassias, Th.M.: Solution of the Ulam stability problem for quartic mappings, Glas. mat. Ser. III 34, 243-252 (1999) · Zbl 0951.39008
[13]Alsina, C.: On the stability of a functional equation arising in probabilistic normed spaces, General inequalities 5, 263-271 (1987) · Zbl 0633.60029
[14]Miheţ, 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 · doi:10.1016/j.jmaa.2008.01.100
[15]Miheţ, D.; Saadati, R.; Vaezpour, S. M.: The stability of the quartic functional equation in random normed spaces, Acta appl. Math. 110, 797-803 (2010) · Zbl 1195.46081 · doi:10.1007/s10440-009-9476-7
[16]D. Miheţ, R. Saadati, S.M. Vaezpour, The stability of an additive functional equation in Menger probabilistic φ-normed spaces, Math. Slovak, (in press).
[17]Mirmostafaee, A. K.; Moslehian, M. S.: Fuzzy versions of Hyers–Ulam–rassias theorem, Fuzzy sets systems 159, 720-729 (2008) · Zbl 1178.46075 · doi:10.1016/j.fss.2007.09.016
[18]Mirmostafaee, A. K.; Mirzavaziri, M.; Moslehian, M. S.: Fuzzy stability of the Jensen functional equation, Fuzzy sets systems 159, 730-738 (2008) · Zbl 1179.46060 · doi:10.1016/j.fss.2007.07.011
[19]Mirmostafaee, A. K.; Moslehian, M. S.: Fuzzy stability of the Jensen functional equation, fuzzy approximately cubic mappings, Inform. sci. 178, 3791-3798 (2008)
[20]Saadati, R.; Vaezpour, S. M.; Cho, Y.: A note on the on the stability of cubic mappings and quadratic mappings in random normed spaces”, J. inequal. Appl. 2009 (2009) · Zbl 1176.39024 · doi:10.1155/2009/214530
[21]R. Saadati, C. Park, J.M. Rassias, Gh. Sadeghi, Stability of a quartic functional equation in various random normed spaces, Abstr. Appl. Anal., (in press).
[22]Shakeri, S.: Intuitionistic fuzzy stability of Jensen type mapping, J. nonlinear sci. Appl. 2, 105-112 (2009) · Zbl 1167.54004 · doi:emis:journals/TJNSA/no6.htm
[23]El Naschie, M. S.: Remarks on superstring, fractal gravity, nagaswas diffusion and Cantorian space–time, Chaos, solitons and fractals 8, 1873-1886 (1997) · Zbl 0934.83049 · doi:10.1016/S0960-0779(97)00124-0
[24]El Naschie, M. S.: On the uncertainty of Cantorian geometry and two-slit experiment, Chaos solitons fractals 9, 517-529 (1998) · Zbl 0935.81009 · doi:10.1016/S0960-0779(97)00150-1
[25]Li, M.: Fuzzy gravitions from uncertain space–time, Phys. rev. D. 63, 63-76 (2001)
[26]Nozari, K.; Fazlpour, B.: Some consequences of space–time fuzziness, Chaos solitons fractals 34, 224-234 (2007) · Zbl 1132.83306 · doi:10.1016/j.chaos.2006.03.066
[27]Sidharth, B. G.: Fuzzy non-commutative space–time: A new paradigm for A new century, frontiers of fundamental physics, Fuzzy non-commutative space–time: A new paradigm for A new century, frontiers of fundamental physics 4 (2001)
[28]Sigalotti, L. G.; Mejias, A.: On el naschie’s conjugate complex time, fractal E() space–time and faster-than-light particles, Int. J. Nonlinear sci. Numer. simul. 7, 467-472 (2006)
[29]Chang, S. S.; Rassias, J. M.; Saadati, R.: The stability of the cubic functional equation in intuitionistic random normed spaces, Appl. math. Mech. 31, 1-7 (2010) · Zbl 1198.46057 · doi:10.1007/s10483-010-0103-6
[30]Hadžić, O.; Pap, E.: Fixed point theory in PM-spaces, (2001)
[31]Hadžić, O.; Pap, E.; Budincević, M.: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces, Kybernetica 38, No. 3, 363-381 (2002)
[32]Hensel, K.: Uber eine neue begründung der theorie der algebraischen zahlen, Jahresber. deutsch. Math. -verein 6, 83-88 (1897) · Zbl 30.0096.03
[33]Schweizer, B.; Sklar, A.: Probabilistic metric spaces, (1983)
[34]Chang, S. S.; Cho, Y. J.; Kang, S. M.: Nonlinear operator theory in probabilistic metric spaces, (2001)
[35]Deschrijver, G.; O’regan, D.; Saadati, R.; Vaezpour, S. M.: L-fuzzy Euclidean normed spaces and compactness, Chaos solitons fractals 42, 40-45 (2009) · Zbl 1200.46065 · doi:10.1016/j.chaos.2008.10.026
[36]Goudarzi, M.; Vaezpour, S. M.; Saadati, R.: On the intuitionistic fuzzy inner product spaces, Chaos, solitons fractals 41, 1105-1112 (2009) · Zbl 1200.46066 · doi:10.1016/j.chaos.2008.04.040
[37]Kutukcu, S.; Tuna, A.; Yakut, A. T.: Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations, Appl. math. Mech. 28, 799-809 (2007) · Zbl 1231.46021 · doi:10.1007/s10483-007-0610-z
[38]Saadati, R.: On the L-fuzzy topological spaces, Chaos solitons fractals 37, 1419-1426 (2008) · Zbl 1142.54318 · doi:10.1016/j.chaos.2006.10.033
[39]Saadati, R.; Park, J.: On the intuitionistic fuzzy topological spaces, Chaos solitons fractals 27, 331-344 (2006) · Zbl 1083.54514 · doi:10.1016/j.chaos.2005.03.019
[40]Šerstnev, A. N.: On the notion of a random normed space, Dokl. akad. Nauk SSSR 149, 280-283 (1963) · Zbl 0127.34902
[41]Atanassov, K. T.: Intuitionistic fuzzy sets, Fuzzy sets systems 20, 87-96 (1986) · Zbl 0631.03040 · doi:10.1016/S0165-0114(86)80034-3
[42]Deschrijver, G.; Kerre, E. E.: On the relationship between some extensions of fuzzy set theory, Fuzzy sets systems 23, 227-235 (2003) · Zbl 1013.03065 · doi:10.1016/S0165-0114(02)00127-6