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)
On the stability problem for a mixed type of quartic and quadratic functional equation. (English) Zbl 1106.39027
The problem “If we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality must be close to the solutions of the given equation?” is the essence of Hyers-Ulam-Rassias stability theory; cf. {\it Th. M. Rassias} [Acta Appl. Math. 62, No. 1, 23--130 (2000; Zbl 0981.39014)]. For a mapping $f: E_1 \to E_2$ between real vector spaces, let us define $\biguplus_{x_2}f(x_1)$ to be $f(x_1+x_2)+f(x_1-x_2)$ and $\biguplus_{x_2, \dots, x_{n+1}}^nf(x_1)= \biguplus_{x_{n+1}} (\biguplus_{x_2, \dots, x_n}^{n-1}f(x_1))$ $(n \in \Bbb N)$. In the paper under review, the author determines the general solution for the mixed type functional equation $$\biguplus_{x_2, \dots, x_n}^{n-1}f(x_1)+2^{n-1}(n-2)\sum_{i=1}^nf(x_i)=2^{n-2}\sum_{1\leq i < j \leq n}\left(\biguplus_{x_j}f(x_j)\right),$$ and proves its Hyers-Ulam-Rassias stability by using the Hyers type sequences; see {\it Th. M. Rassias} [J. Math. Anal. Appl. 158, No. 1, 106--113 (1991; Zbl 0746.46038)].

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
WorldCat.org
Full Text: DOI
References:
[1] Aczél, J.; Dhombres, J.: Functional equations in several variables. (1989) · Zbl 0685.39006
[2] Amir, D.: Characterizations of inner product spaces. (1986) · Zbl 0617.46030
[3] Bourgin, D. G.: Classes of transformations and bordering transformations. Bull. amer. Math. soc. 57, 223-237 (1951) · Zbl 0043.32902
[4] Chung, J.; Sahoo, P. K.: On the general solution of a quartic functional equation. Bull. korean math. Soc. 40, 565-576 (2003) · Zbl 1048.39017
[5] Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. math. Sem. univ. Hamburg 62, 59-64 (1992) · Zbl 0779.39003
[6] Fischer, P.; Mokanski, J. P.: A class of symmetric biadditive functionals. Aequationes math. 23, 169-174 (1981) · Zbl 0515.15008
[7] Gruber, P. M.: Stability of isometries. Trans. amer. Math. soc. 245, 263-277 (1978) · Zbl 0393.41020
[8] Hyers, D. H.: On the stability of the linear functional equation. Proc. natl. Acad. sci. 27, 222-224 (1941) · Zbl 0061.26403
[9] Jordan, P.; Von Neumann, J.: On inner products in linear metric spaces. Ann. of math. 36, 719-723 (1935) · Zbl 61.0435.05
[10] K. Jun, H. Kim, Solution of Ulam stability problem for approximately biquadratic mappings and functional inequalities, J. Inequal. Appl. (edited by Th.M. Rassias), in press
[11] Kannappan, Pl.: Quadratic functional equation and inner product spaces. Results math. 27, 368-372 (1995) · Zbl 0836.39006
[12] Lee, S. H.; Im, S. M.; Hwang, I. S.: Quartic functional equations. J. math. Anal. appl. 307, No. 2, 387-394 (2005) · Zbl 1072.39024
[13] Rassias, J. M.: On approximation of approximately linear mappings by linear mappings. J. funct. Anal. 46, 126-130 (1982) · Zbl 0482.47033
[14] Rassias, J. M.: Solution of a problem of Ulam. J. approx. Theory 57, 268-273 (1989) · Zbl 0672.41027
[15] Rassias, J. M.: Solution of the Ulam stability problem for quartic mappings. Glas. mat. 34, 243-252 (1999) · Zbl 0951.39008
[16] Rassias, Th.M.: Inner product spaces and applications. (1997) · Zbl 0882.00018
[17] Rassias, Th.M.: On the stability of the quadratic functional equation and its applications. Studia univ. Babeş-bolyai math. 43, No. 3, 89-124 (1998)
[18] Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. math. Anal. appl. 251, 264-284 (2000) · Zbl 0964.39026
[19] Rassias, Th.M.: On the stability of functional equations originated by a problem of Ulam. Mathematica (67) 44, No. 1, 33-36 (2002)
[20] Röhmel, J.: Eine charakterisierung quadratischer formen durch eine funktionalgleichung. Aequationes math. 15, 163-168 (1977)
[21] Senechalle, D. A.: A characterization of inner product spaces. Proc. amer. Math. soc. 19, 1306-1312 (1968) · Zbl 0172.39802
[22] Ulam, S. M.: A collection of the mathematical problems. (1960) · Zbl 0086.24101
[23] Zhou, D. -X.: On a conjecture of Z. Ditzian. J. approx. Theory 69, 167-172 (1992) · Zbl 0755.41029