The original stability problem was posed by S. M. Ulam in 1940 and concerned approximate homomorphisms. The pursuit of solutions to this problem, but also to its generalizations and modifications for various classes of (difference, functional, differential, and integral) equations and inequalities, is an expanding area of research and has led to the development of what is now quite often called
This special issue on
As usual, most of the authors use in their investigations direct and fixed point methods. Some open problems are also formulated.
The issue covers a wide variety of problems for different classes of functional equations both in a single variable and in several variables. Their stability is traditionally investigated in classical Banach spaces, but also in complete (probabilistic) metric spaces, complete probabilistic quasimetric spaces,
Several papers deal with the stability of several kinds of derivations, and, thus, derivations in Riesz algebras,
The issue contains a few papers on the phenomenon of superstability, an article on the stability of a functional inequality in
Moreover, general solutions of two conditional quadratic functional equations of Pexider type and the structure of the set of all regular points and the set of all irregular points for a Brouwer homeomorphism which is embeddable in a flow are also considered.
Finally, the survey presents some selected recent developments (results and methods) in the theory of Ulam’s type stability. In particular, some aspects of stability and nonstability of functional equations in a single variable, the effect "stability implies completeness," some methods of proofs applied in that theory (the Forti method and the methods of fixed points), stability in non-Archimedean spaces, selected results on functional congruences, the notion of hyperstability, and stability of composite functional equations (e.g., of the Gołąb-Schinzel equation and its generalizations) are discussed there.
We believe that this volume will have some influence on the further research in that area of mathematics.