Fixed Point Methods for the Generalized Stability of Functional Equations in a Single Variable

Abstract

We discuss on the generalized Ulam-Hyers stability for functional equations in a single variable, including the nonlinear functional equations, the linear functional equations, and a generalization of functional equation for the square root spiral. The stability results have been obtained by a fixed point method. This method introduces a metrical context and shows that the stability is related to some fixed point of a suitable operator.

On the Fuzzy Stability of Cubic Mappings Using Fixed Point Method

“…Hence we have d(f, Jf ) ≤ L 2a 4 . By Theorem 1.6, there exists a mapping C : X → Y such that (1) C is a fixed point of J , that is,…”

“…By using fixed point methods the stability problems of several functional equations have been extensively investigated by a number of authors; see [4], [5], [21] and [24].…”

A fixed point approach to the stability of an AQCQ-functional equation in RN-spaces

“…In 1996, Isac and Rassias [27] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [6,7,8,33,38,40,41,43]). …”

On the probabilistic stability of the monomial functional equation

“…The (generalized) Ulam -Hyers stability for the monomial functional equation was previously studied in [1], [7], [8] and [3]. We also mention the recent papers [18] and [19].…”