Fixed Points and Generalized Hyers‐Ulam Stability

Abstract

In this paper we prove a fixed-point theorem for a class of operators with suitable properties, in very general conditions. Also, we show that some recent fixed-points results in Brzdęk et al., (2011) and Brzdęk and Ciepliński (2011) can be obtained directly from our theorem. Moreover, an affirmative answer to the open problem of Brzdęk and Ciepliński (2011) is given. Several corollaries, obtained directly from our main result, show that this is a useful tool for proving properties of generalized Hyers-Ulam stability for some functional equations in a single variable.

Stability of a generalization of the Fréchet functional equation

“…For related outcomes we refer to [7,9]; a similar approach to stability of functional equations has been already applied in [5,24].…”

Hyperstability of the Fréchet Equation and a Characterization of Inner Product Spaces

“…The main tool in the proof of the main theorem is a fixed point result for function spaces from [17] (for related outcomes see [18,19]). Similar method of the proof has been already applied in [11,20].…”

Hyers-Ulam stability of derivations in fuzzy Banach space

“…By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [1,9,12,13,17,19,28,33,35,39]). …”