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

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

One says that (*) is (uniquely) uniformly b-stable if for each δ>0 there exists ε>0 such that for each g satisfying ρ(L(g),R(g))<δ there is a (unique) solution f of (*) such that ρ(f,g)<ε. In particular, if for some α>0, ε=αδ, (*) 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 ρ(L(g),R(g)) 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 ρ(L(g),R(g)) 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:
39B82Stability, separation, extension, and related topics
39-02Research monographs (functional equations)
39B52Functional equations for functions with more general domains and/or ranges