# 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 functional equations and a problem of Ulam. (English) Zbl 0981.39014

In 1940 S. M. Ulam posed the problem concerning the stability of homomorphisms. In 1941 D. H. Hyers gave the first significant partial solution:

Let $X,Y$ be Banach spaces and $\delta >0$. If the function $f:X\to Y$ satisfies the inequality

$∥f\left(x+y\right)-f\left(x\right)-f\left(y\right)∥\le \delta \phantom{\rule{2.em}{0ex}}\left(*\right)$

for all $x,y\in X$, then there exists the unique additive function $A:X\to Y$ such that $∥f\left(x\right)-A\left(x\right)∥\le \delta$ for all $x\in X$.

The stability of functional equations may be considered from some points of view. In ($*$) the left-hand side of the inequality is bounded. Many results concerning the stability were proved with the assumption that the left-hand sides of the appropriate inequalities may be unbounded. The stability may be also considered on restricted domains.

The stability of functional equations is extensively investigated by many researchers. The reviewed paper contains the wide range survey of both classical results and current research concerning the stability. Many results are presented with proofs, so the paper is self-contained. It is of interest to researchers in the field and it is accessible to graduate students as well. The related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

The paper consists of nine sections: Introduction, Additive functional equation, Jensen’s functional equation, Quadratic functional equations, Exponential functional equations, Multiplicative functional equation, Logarithmic functional equation, Trigonometric functional equations, Other functional equations. The bibliography contains 139 items.

For other surveys devoted to the stability of functional equations cf. G. L. Forti, [Aequationes Math. 50, No. 1-2, 143-190 (1995; Zbl 0836.39007)] and D. H. Hyers and T. M. Rassias [ibid. 44, No. 2/3, 125-153 (1992; Zbl 0806.47056)].

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B72 Systems of functional equations and inequalities 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)