Stability of functional equations in single variable

Abstract

This paper discusses Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers-Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the HyersUlam stability of Böttcher's equation. We also prove a general result of Hyers-Ulam stability for iterative equations.

On functions that are approximate fixed points almost everywhere and Ulam’s type stability

“…Corollary 2.5 yields at once the following two outcomes that generalize (at least to some extent) several results in [1,2,5,7,8,13,18,19].…”

“…Let us remind that the interest in investigation of that type of stability has been stimulated by a problem of S. M. Ulam (cf. [14,20]) and several papers that appeared afterwards (for more details and the references concerning that subject, we refer the reader to [1,3,4,11,12,15,16,17]). …”

Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces

“…All authors read and approved the final manuscript. 1 School of Mathematics, Beijing Institute of Technology, Beijing, 100081, P.R. China.…”

On the stability of the linear functional equation in a single variable on complete metric groups

“…Pólya and G. Szegö [18] For more details on Hyers-Ulam stability of functional equations and optimization theory we refer the reader to [2,4,12,13,15,17,19].…”