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Stability of cubic and quartic functional equations in non-Archimedean spaces. (English) Zbl 1192.39018

Using some ideas of M. S. Moslehian and Th. M. Rassias [Appl. Anal. Discrete Math. 1, No. 2, 325–334 (2007; Zbl 1257.39019)], K. W. Jun and H. M. Kim [J. Math. Anal. Appl. 274, No. 2, 867–878 (2002; Zbl 1021.39014)] and W. G. Park and J. H. Bae [Nonlinear Anal., Theory Methods Appl. 62, No. 4 (A), 643–654 (2005; Zbl 1076.39027)], the authors investigate the generalized Hyers-Ulam-Rassias stability of the cubic functional equation

f(kx+y)+f(kx-y)=k[f(x+y)+f(x-y)]+2(k 3 -k)f(x),

and the quartic functional equation

f(kx+y)+f(kx-y)=k 2 [f(x+y)+f(x-y)]+2k 2 (k 2 -1)f(x)-2(k 2 -1)f(y)

for all k, where f:GX is a mapping, G is an additive group and X is a complete non-Archimedean space.

MSC:
39B82Stability, separation, extension, and related topics
46S10Functional analysis over fields (not , , or quaternions)
39B52Functional equations for functions with more general domains and/or ranges
References:
[1]Arriola, L.M., Beyer, W.A.: Stability of the Cauchy functional equation over p-adic fields. Real Anal. Exch. 31, 125–132 (2005/2006)
[2]Borelli, C., Forti, G.L.: On a general Hyers–Ulam stability result. Int. J. Math. Math. Sci. 18, 229–236 (1995) · Zbl 0826.39009 · doi:10.1155/S0161171295000287
[3]Cholewa, P.W.: Remarks on the stability of functional equations. Aequ. Math. 27, 76–86 (1984) · Zbl 0549.39006 · doi:10.1007/BF02192660
[4]Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hambg. 62, 59–64 (1992) · Zbl 0779.39003 · doi:10.1007/BF02941618
[5]Eshaghi Gordji, M.: Stability of a functional equation deriving from quartic and additive functions. Bull. Korean Math. Soc. (to appear)
[6]Eshaghi Gordji, M., Park, C., Savadkouhi, M.B.: Stability of a quartic type functional equation. Fixed Point Theory (to appear)
[7]Eshaghi Gordji, M., Ebadian, A., Zolfaghari, S.: Stability of a functional equation deriving from cubic and quartic functions. Abstr. Appl. Anal. 2008 (2008), Article ID 801904, 17 pages
[8]Gajda, Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991) · Zbl 0739.39013 · doi:10.1155/S016117129100056X
[9]Gǎvruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994) · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[10]Gouvêa, F.Q.: p-adic Numbers. Springer, Berlin (1997)
[11]Grabiec, A.: The generalized Hyers–Ulam stability of a class of functional equations. Publ. Math. Debr. 48, 217–235 (1996)
[12]Hensel, K.: Über eine neue Begründung der Theorie der algebraischen Zahlen. Jahresber. Dtsch. Math. Ver. 6, 83–88 (1897)
[13]Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941) · doi:10.1073/pnas.27.4.222
[14]Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)
[15]Jung, S.M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press Inc., Palm Harbor (2001)
[16]Jung, K.W., Kim, H.M.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274(2), 267–278 (2002) · Zbl 1012.39025 · doi:10.1016/S0022-247X(02)00328-1
[17]Khrennikov, A.: Non-Archimedean Analysis, Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic Publishers, Dordrecht (1997)
[18]Moslehian, M.S., Rassias, Th.M.: Stability of functional equations in non-Archimedean spaces. Appl. Anal. Discrete Math. 1, 325–334 (2007) · Zbl 1257.39019 · doi:10.2298/AADM0702325M
[19]Najati, A.: On the stability of a quartic functional equation. J. Math. Anal. Appl. 340(1), 569–574 (2008) · Zbl 1133.39030 · doi:10.1016/j.jmaa.2007.08.048
[20]Park, C.: On the stability of the quadratic mapping in Banach modules. J. Math. Anal. Appl. 276, 135–144 (2002) · Zbl 1017.39010 · doi:10.1016/S0022-247X(02)00387-6
[21]Park, C.: Generalized quadratic mappings in several variables. Nonlinear Anal. 57, 713–722 (2004) · Zbl 1058.39024 · doi:10.1016/j.na.2004.03.013
[22]Park, W.G., Bae, J.H.: On the stability a bi-quartic functional equation. Nonlinear Anal. 62(4), 643–654 (2005) · Zbl 1076.39027 · doi:10.1016/j.na.2005.03.075
[23]Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) · doi:10.1090/S0002-9939-1978-0507327-1
[24]Rassias, Th.M.: On the stability of the quadratic functional equation and its applications. Stud. Univ. Babes-Bolyai 43, 89–124 (1998)
[25]Rassias, T.M. (ed.): Functional Equations and Inequalities. Kluwer Academic Publishers, Dordrecht (2000)
[26]Robert, A.M.: A Course in p-adic Analysis. Springer, New York (2000)
[27]Skof, F.: Proprietá localie approssimazione dioperatori. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983) · Zbl 0599.39007 · doi:10.1007/BF02924890
[28]Ulam, S.M.: A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics. Interscience, New York (1960)
[29]Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-adic Analysis and Mathematical Physics. World Scientific, Singapore (1994)