Hyperstability of general linear functional equation

Abstract

Abstract. Our purpose is to investigate criteria for hyperstability of linear type functional equations. We prove that a function satisfying the equation approximately in some sense, must be a solution of it. We give some conditions on coefficients of the functional equation and a control function which guarantee hyperstability. Moreover, we show how our outcomes may be used to check whether the particular functional equation is hyperstable. Some relevant examples of applications are presented.Mathematics Subject Classification. Primary 39B82, 47H14, 47J20; Secondary 39B62, 47H10.

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].…”

Hyperstability of Some Functional Equations on Restricted Domain

“…We can now state a counterpart (on restricted domain) of Theorem 2.1 in [8], concerning -hyperstability of (5). …”

“…The above notion of hyperstability for functions defined on subsets of a linear space (not necessarily on the whole space but on the restricted domain) is a counterpart of hyperstability considered in [8]. In this paper we obtain, applying the fixed point theorem, some results concerning stability and hyperstability for (4) …”

“…The stability and hyperstability of the above functional equation and its particular cases were studied by many mathematicians (cf., e.g., [5][6][7][8][9][10][11][12][13][14][15][16]). Let us recall (see [9]) that the theory of Hyers-Ulam stability, in general, was motivated by a problem raised in 1940 by Ulam [17] and a partial answer to it was provided by Hyers [18].…”

On the Stability of a Generalized Fréchet Functional Equation with Respect to Hyperplanes in the Parameter Space

“…Equation (1) can be treated as a special case of the general linear equation. The stability problem of the general linear equation was studied in [29][30][31][32]. In this article, we want to look at Equation (1) in order to get estimates of the difference between approximate and exact solutions more closely connected to the values of the coefficients of the equation and the form of the control function.…”