zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Stability of cubic and quartic functional equations in non-Archimedean spaces. (English) Zbl 1192.39018
Using some ideas of {\it M. S. Moslehian} and {\it Th. M. Rassias} [Appl. Anal. Discrete Math. 1, No. 2, 325--334 (2007; Zbl 1257.39019)], {\it K. W. Jun} and {\it H. M. Kim} [J. Math. Anal. Appl. 274, No. 2, 867--878 (2002; Zbl 1021.39014)] and {\it W. G. Park} and {\it 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 $k\in \mathbb N$, where $f:G\to X$ 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 $\Bbb R$, $\Bbb C$, $\Bbb H$or quaternions)
39B52Functional equations for functions with more general domains and/or ranges
WorldCat.org
Full Text: DOI arXiv
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) · Zbl 1099.39019
[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) · Zbl 1196.39016
[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 · Zbl 1160.39334
[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) · Zbl 1274.39058
[12] Hensel, K.: Über eine neue Begründung der Theorie der algebraischen Zahlen. Jahresber. Dtsch. Math. Ver. 6, 83--88 (1897) · Zbl 30.0096.03
[13] 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
[14] Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998) · Zbl 0907.39025
[15] Jung, S.M.: Hyers--Ulam--Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press Inc., Palm Harbor (2001) · Zbl 0980.39024
[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) · Zbl 0920.11087
[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) · Zbl 0398.47040 · 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) · Zbl 1009.39025
[25] Rassias, T.M. (ed.): Functional Equations and Inequalities. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0945.00010
[26] Robert, A.M.: A Course in p-adic Analysis. Springer, New York (2000) · Zbl 0947.11035
[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) · Zbl 0086.24101
[29] Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-adic Analysis and Mathematical Physics. World Scientific, Singapore (1994) · Zbl 0812.46076