Recursive procedure in the stability of Fréchet polynomials

Abstract

By means of a new stability result, established for symmetric and multi-additive mappings, and using the concepts of stability couple and of stability chain, we prove, by a recursive procedure, the generalized stability of two of Fréchet's polynomial equations. We also give a new functional characterization of generalized polynomials and a new approach to solving the generalized stability of the monomial equation. MSC: 39B82; 39B52; 20M15; 65Q30

Stability Threshold for Multiadditive and Symmetric Mappings

“…In the following lines we complete that result using control functions which verifies (3). For convenience, we reproduce from [3] the proof of the result mentioned above also. Theorem 2.2.…”

“…Using the following elementary lemma, in [3] we have shown that the functions which verifie (2) constitute a class of control functions that provide stability for n-additive and symmetric functions.…”

“…[3] Let (b k ) k∈N be a sequence in B, (α k ) k∈N be a sequence of positive numbers, and c > 0 such that β :=…”

Certain properties of difference operator and stability of Fréchet functional equation

“…The Hyers-Ulam stability of (2) and (3) has been studied in [6][7][8][9]. The Gǎvruţa-type stability of (1) has been studied by Dăianu [10]. Dăianu used an equivalence theorem, which is more general than that presented by Kuczma [11], to obtain a stability result for (2) under the assumption that M is (n + 1)!-divisible.…”

Hyperstability of an alternative equation of Jensen type

“…where f is a function on groups. Following the stability in the sense of Hyers and Ulam [9] on Cauchy functional equation, many researchers have studied and have extended the concept of stability of related functional equations [10][11][12]. Hyers-Ulam type stability of (1) has been investigated in [13].…”