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Ulam-Hyers stability of functional equations in quasi-\(\beta\)-Banach spaces. (English) Zbl 1447.39024

Brzdęk, Janusz (ed.) et al., Ulam type stability. Based on the conferences on Ulam type stability (CUTS), Cluj-Napoca, Romania, July 4–9, 2016 and Timisoara, Romania, October 8–13, 2018. Cham: Springer, 97-130 (2019).
The main purpose of this paper is to give a survey on Ulam-Hyers stability of functional equations in quasi-\(\beta\)-Banach spaces. Special attention is given to \(p\)-Banach spaces, quasi-Banach spaces and \((\beta, p)\)-Banach spaces.
The first section of the paper is an introduction about quasi-normed and quasi-\(\beta\)-normed spaces.
Section two contains general results and facts about the stability theory of functional equations. Here, the main emphasis is given to the Cauchy equation \[ f(x+y)= f(x)+f(y), \] the quadratic equation \[ f (x + y) + f (x - y) = 2f (x) + 2f (y) \] and some rather special linear functional equations.
The methods presented here are based on the so-called Hyers’ method.
For the entire collection see [Zbl 1431.39001].

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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