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 of quadratic functional equations. (English) Zbl 1146.39045
Summary: Let $X,Y$ be vector spaces and $k$ a fixed positive integer. It is shown that a mapping $f(kx+y)+f(kx-y)=2k^{2}f(x)+2f(y)$ for all $x,y\in X$ if and only if the mapping $f:X\rightarrow Y$ satisfies $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\in X$. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
WorldCat.org
Full Text: DOI EuDML
References:
[1] S. M. Ulam, “Problems in Modern Mathematics,” p. xvii+150, John Wiley & Sons, New York, NY, USA, 1960. · Zbl 0137.24201
[2] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[4] Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 · doi:10.2307/2042795
[5] Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae, vol. 39, no. 2-3, pp. 292-293, 309, 1990.
[6] Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431-434, 1991. · Zbl 0739.39013 · doi:10.1155/S016117129100056X · eudml:46657
[7] Th. M. Rassias and P. \vSemrl, “On the Hyers-Ulam stability of linear mappings,” Journal of Mathematical Analysis and Applications, vol. 173, no. 2, pp. 325-338, 1993. · Zbl 0789.46037 · doi:10.1006/jmaa.1993.1070
[8] J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445-446, 1984. · Zbl 0599.47106
[9] F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113-129, 1983. · Zbl 0599.39007 · doi:10.1007/BF02924890
[10] P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1, pp. 76-86, 1984. · Zbl 0549.39006 · doi:10.1007/BF02192660 · eudml:137013
[11] St. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59-64, 1992. · Zbl 0779.39003 · doi:10.1007/BF02941618
[12] K.-W. Jun and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality,” Mathematical Inequalities & Applications, vol. 4, no. 1, pp. 93-118, 2001. · Zbl 0976.39031
[13] S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126-137, 1998. · Zbl 0928.39013 · doi:10.1006/jmaa.1998.5916
[14] J. M. Rassias, “Solution of a quadratic stability Hyers-Ulam type problem,” Ricerche di Matematica, vol. 50, no. 1, pp. 9-17, 2001. · Zbl 1221.39039
[15] J. M. Rassias, “On approximation of approximately quadratic mappings by quadratic mappings,” Annales Mathematicae Silesianae, no. 15, pp. 67-78, 2001. · Zbl 1087.39518
[16] M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361-376, 2006. · Zbl 1118.39015 · doi:10.1007/s00574-006-0016-z
[17] Th. M. Rassias, “On the stability of the quadratic functional equation and its applications,” Studia Universitatis Babe\cs-Bolyai, vol. 43, no. 3, pp. 89-124, 1998. · Zbl 1009.39025