Hyperstability of Some Functional Equations on Restricted Domain

Abstract

The paper concerns functions which approximately satisfy, not necessarily on the whole linear space, a generalization of linear functional equation. A Hyers-Ulam stability result is proved and next applied to give conditions implying the hyperstability of the equation. The results may be used as tools in stability studies on restricted domains for various functional equations. We use the main theorem to obtain a few hyperstability results of Fréchet equation on restricted domain for different control functions.

Generalizations of an asymptotic stability theorem of Bahyrycz, Páles and Piszczek on Cauchy differences to generalized cocycles

“…Hyperstability results, for the Cauchy equation and its generalizations, have also been proved by Maksa and Páles [53], Najati and Rassias [59], Alimohammady and Sadeghi [3], and Brzdek [14,15,16], Piszczek [61,62], Almahalebi, Charifi and Kabbaj [4], Bahyrycz and Olko [7,8], Aiemsomboon and Sintunavarat [1,2], Molaei, Najati and Park [55,58].…”

General criteria for hyperstability of Cauchy-type equations

“…(for p = q and r = 1, X = S and D = S 2 , in [2]). In [5], φ-hyperstability of the Cauchy equation was proved for r = 1 and p = q < 0 on a domain of type (k 1 ) (see case (3) in the next result).…”

“…A synthesis of some results on the hyperstability of homomorphism equation on restricted domains -in the wider context of generalized Ulam stability -can be found in [8]. In [3], the hyperstability of some generalizations of Fréchet's equation on domains in linear spaces is analyzed. The hyperstability of arithmetically homogeneous functions is used in [10] to demonstrate the hyperstability of the monomial equation (therefore also of equation (1.1)) on Abelian semigroups.…”

Fixed Point Approach: Ulam Stability Results of Functional Equation in Non-Archimedean Fuzzy φ-2-Normed Spaces and Non-Archimedean Banach Spaces

“…From that point forward, numerous stability problems for different functional equations have been explored in [6][7][8][9][10][11][12][13][14][15]. Later, the stability issues for different types of functional equations were investigated in [16,17].…”