zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
The generalized Hyers--Ulam--Rassias stability of a cubic functional equation. (English) Zbl 1021.39014
The authors consider the functional equation $$f(2x+y) +f(2x-y)=2f(x+y)+ 2f(x-y)+12f(x).$$ They determine the general solution, which is of the form $f(x)= B(x,x,x)$ where $B$ is symmetric and additive in each variable. Moreover they investigate the stability properties of this equation.

39B82Stability, separation, extension, and related topics
Full Text: DOI
[1] Aczél, J.; Dhombres, J.: Functional equations in several variables. (1989) · Zbl 0685.39006
[2] Baker, J.: The stability of the cosine equation. Proc. amer. Math. soc. 80, 411-416 (1980) · Zbl 0448.39003
[3] Cholewa, P. W.: Remarks on the stability of functional equations. Aequationes math. 27, 76-86 (1984) · Zbl 0549.39006
[4] Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. math. Sem. univ. Hamburg 62, 59-64 (1992) · Zbl 0779.39003
[5] Amir, D.: Characterizations of inner product spaces. (1986) · Zbl 0617.46030
[6] Ga\check{}vruta, P.: A generalization of the Hyers--Ulam--rassias stability of approximately additive mappings. J. math. Anal. appl. 184, 431-436 (1994) · Zbl 0818.46043
[7] Grabiec, A.: The generalized Hyers--Ulam stability of a class of functional equations. Publ. math. Debrecen 48, 217-235 (1996) · Zbl 1274.39058
[8] Hyers, D. H.: On the stability of the linear functional equation. Proc. nat. Acad. sci. 27, 222-224 (1941) · Zbl 0061.26403
[9] Hyers, D. H.; Isac, G.; Rassias, Th.M.: Stability of functional equations in several variables. (1998) · Zbl 0907.39025
[10] Hyers, D. H.; Isac, G.; Rassias, Th.M.: On the asymptoticity aspect of Hyers--Ulam stability of mappings. Proc. amer. Math. soc. 126, 425-430 (1998) · Zbl 0894.39012
[11] Hyers, D. H.; Rassias, Th.M.: Approximate homomorphisms. Aequationes math. 44, 125-153 (1992) · Zbl 0806.47056
[12] Jun, K. W.; Kim, H. M.: Remarks on the stability of additive functional equation. Bull. korean math. Soc. 38, 679-687 (2001) · Zbl 0991.39020
[13] Jordan, P.; Von Neumann, J.: On inner products in linear metric spaces. Ann. of math. 36, 719-723 (1935) · Zbl 61.0435.05
[14] Jun, K. W.; Lee, Y. H.: On the Hyers--Ulam--rassias stability of a pexiderized quadratic inequality. Math. ineq. Appl. 4, 93-118 (2001) · Zbl 0976.39031
[15] Jung, S. M.: On the Hyers--Ulam stability of the functional equations that have the quadratic property. J. math. Anal. appl. 222, 126-137 (1998) · Zbl 0928.39013
[16] Jung, S. M.: On the Hyers--Ulam--rassias stability of a quadratic functional equation. J. math. Anal. appl. 232, 384-393 (1999) · Zbl 0926.39013
[17] Kannappan, Pl.: Quadratic functional equation and inner product spaces. Results math. 27, 368-372 (1995) · Zbl 0836.39006
[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] Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. math. Anal. appl. 251, 264-284 (2000) · Zbl 0964.39026
[20] Skof, F.: Proprietà locali e approssimazione di operatori. Rend. sem. Mat. fis. Milano 53, 113-129 (1983)
[21] Ulam, S. M.: Problems in modern mathematics. (1960) · Zbl 0086.24101