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 be a functional equation with an unknown function and let be a metric in the target space. The equation is (uniquely) stable if for each there exists such that for each satisfying , for all the variables of , there exists (a unique) solution of the equation such that (for all the variables of and ).
The equation is (uniquely) -stable if for each for which is bounded, there exists (a unique) solution of such that is bounded.
One says that is (uniquely) uniformly -stable if for each there exists such that for each satisfying there is a (unique) solution of such that . In particular, if for some , , 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 for which is bounded, is bounded or it is a solution of the equation ; if functions , in the case considered, are bounded by the same constant, is called strongly superstable. If for each for which is bounded, 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.