On the stability of functional equations and a problem of Ulam.

*(English)*Zbl 0981.39014In 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 \[ \bigl\|f(x+y)-f(x)-f(y)\bigr\|\leq\delta\tag{\(\ast\)} \] for all \(x,y\in X\), then there exists the unique additive function \(A:X\to Y\) such that \(\bigl\|f(x)-A(x)\bigr\|\leq\delta\) for all \(x\in X\).

The stability of functional equations may be considered from some points of view. In (\(\ast\)) 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.

\smallskip 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.

\smallskip 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)].

Let \(X,Y\) be Banach spaces and \(\delta>0\). If the function \(f:X\to Y\) satisfies the inequality \[ \bigl\|f(x+y)-f(x)-f(y)\bigr\|\leq\delta\tag{\(\ast\)} \] for all \(x,y\in X\), then there exists the unique additive function \(A:X\to Y\) such that \(\bigl\|f(x)-A(x)\bigr\|\leq\delta\) for all \(x\in X\).

The stability of functional equations may be considered from some points of view. In (\(\ast\)) 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.

\smallskip 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.

\smallskip 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)].

Reviewer: Szymon Wasowicz (Bielsko-Biala)

##### MSC:

39B82 | Stability, separation, extension, and related topics for functional equations |

39B72 | Systems of functional equations and inequalities |

47H10 | Fixed-point theorems |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |