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On the different definitions of the stability of functional equations. (Sur les définitions différentes de la stabilité des équations fonctionnelles.) (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 $\left(*\right)$ $L\left(f\right)=R\left(f\right)$ be a functional equation with an unknown function $f$ and let $\rho$ be a metric in the target space. The equation $\left(*\right)$ is (uniquely) stable if for each $\epsilon >0$ there exists $\delta >0$ such that for each $g$ satisfying $\rho \left(L\left(g\right),R\left(g\right)\right)<\delta$, for all the variables of $g$, there exists (a unique) solution $f$ of the equation such that $\rho \left(g,f\right)<\epsilon$ (for all the variables of $f$ and $g$).

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

One says that $\left(*\right)$ is (uniquely) uniformly $b$-stable if for each $\delta >0$ there exists $\epsilon >0$ such that for each $g$ satisfying $\rho \left(L\left(g\right),R\left(g\right)\right)<\delta$ there is a (unique) solution $f$ of $\left(*\right)$ such that $\rho \left(f,g\right)<\epsilon$. In particular, if for some $\alpha >0$, $\epsilon =\alpha \delta$, $\left(*\right)$ 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 $\left(*\right)$ is superstable if for each $g$ for which $\rho \left(L\left(g\right),R\left(g\right)\right)$ is bounded, $g$ is bounded or it is a solution of the equation $\left(*\right)$; if functions $g$, in the case considered, are bounded by the same constant, $\left(*\right)$ is called strongly superstable. If for each $g$ for which $\rho \left(L\left(g\right),R\left(g\right)\right)$ is bounded, $g$ a solution of $\left(*\right)$, then we call $\left(*\right)$ 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