*(French)*Zbl 1060.39031

In a great number of papers and monographs dealing with the stability of functional equations, various types of this concept are considered. The paper under review is a kind of survey; the main types of the stability are defined and compared. Here are the list of them:

Let $(*)$ $L\left(f\right)=R\left(f\right)$ be a functional equation with an unknown function $f$ and let $\rho $ be a metric in the target space. The equation $(*)$ is (uniquely) stable if for each $\epsilon >0$ there exists $\delta >0$ such that for each $g$ satisfying $\rho \left(L\right(g),R(g\left)\right)<\delta $, for all the variables of $g$, there exists (a unique) solution $f$ of the equation such that $\rho (g,f)<\epsilon $ (for all the variables of $f$ and $g$).

The equation $(*)$ is (uniquely) $b$-stable if for each $g$ for which $\rho \left(L\right(g),R(g\left)\right)$ is bounded, there exists (a unique) solution $f$ of $(*)$ such that $\rho (f,g)$ is bounded.

One says that $(*)$ is (uniquely) uniformly $b$-stable if for each $\delta >0$ there exists $\epsilon >0$ such that for each $g$ satisfying $\rho \left(L\right(g),R(g\left)\right)<\delta $ there is a (unique) solution $f$ of $(*)$ such that $\rho (f,g)<\epsilon $. In particular, if for some $\alpha >0$, $\epsilon =\alpha \delta $, $(*)$ is said to be strongly stable (or strongly and uniquely stable).

There are also considered definitions of not uniquely and totally not uniquely stability as well as (uniquely/not uniquely/totally not uniquely) iterative stability.

The equation $(*)$ is superstable if for each $g$ for which $\rho \left(L\right(g),R(g\left)\right)$ is bounded, $g$ is bounded or it is a solution of the equation $(*)$; if functions $g$, in the case considered, are bounded by the same constant, $(*)$ is called strongly superstable. If for each $g$ for which $\rho \left(L\right(g),R(g\left)\right)$ is bounded, $g$ a solution of $(*)$, then we call $(*)$ completely superstable.

There are suitable examples for the above definitions and comparisons. Also some properties and related results are proved. The so-called Hyers’ operator and the stability of conditional functional equations are also mentioned.

##### MSC:

39B82 | Stability, separation, extension, and related topics |

39-02 | Research monographs (functional equations) |

39B52 | Functional equations for functions with more general domains and/or ranges |