On the different definitions of the stability of functional equations (Q704874)

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 \(\rho\) be a metric in the target space. The equation \((*)\) is (uniquely) stable if for each \(\varepsilon>0\) there exists \(\delta>0\) such that for each \(g\) satisfying \(\rho(L(g),R(g))<\delta\), for all the variables of \(g\), there exists (a unique) solution \(f\) of the equation such that \(\rho(g,f)<\varepsilon\) (for all the variables of \(f\) and \(g\)). The equation \((*)\) is (uniquely) \(b\)-stable if for each \(g\) for which \(\rho(L(g),R(g))\) 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 \(\varepsilon>0\) such that for each \(g\) satisfying \(\rho(L(g),R(g))<\delta\) there is a (unique) solution \(f\) of \((*)\) such that \(\rho(f,g)<\varepsilon\). In particular, if for some \(\alpha>0\), \(\varepsilon=\alpha\delta\), \((*)\) 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 \(\rho(L(g),R(g))\) 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 \(\rho(L(g),R(g))\) 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.