On the stability of functional equations. (English) Zbl 1207.39044

Moszner, Zenon

Aequationes Math. 77, No. 1-2, 33-88 (2009).

Summary: We give some theorems on the stability of the equation of homomorphism, of Lobacevski’s equation, of almost Jensen’s equation, of Jensen’s equation, of Pexider’s equation, of linear equations, of Schröder’s equation, of Sincov’s equation, of modified equations of homomorphism from a group (not necessarily commutative) into a \({\mathbb{Q}}\)-topological sequentially complete vector space or into a Banach space, of the quadratic equation, of the equation of a generalized involution, of the equation of idempotency and of the translation equation. We prove that the different definitions of stability are equivalent for the majority of these equations. The boundedness stability and the stability of differential equations and the anomalies of stability are considered and open problems are formulated too.

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
39B62	Functional inequalities, including subadditivity, convexity, etc.
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

stability; equation of homomorphism; Lobacevski’s equation; Jensen’s equation; Pexider’s equation; Schröder’s equation; Sincov’s equation; group; Banach space; quadratic equation; generalized involution; translation equation