An alternative functional equation of Jensen type on groups

Abstract

ABSTRACT:Given an integer λ = 2, we establish the general solution of an alternative functional equation of Jensen type on certain groups. First, we give a criterion for the existence of the general solution for the functional equationwhere f is a mapping from a group (G, ·) to a uniquely divisible abelian group (H, +). Then we show that, for λ / ∈ {0, −1, −2}, the above alternative functional equation is equivalent to the classical Jensen's functional equation. We also find the general solution in the case when G is a cyclic group and λ = 2 is an integer.

Hyers-Ulam stability of an alternative functional equation of Jensen type

“…It should be remarked that Srisawat et al 8 proved that when λ / ∈ {−2, −1, 0}, the alternative Jensen's functional equation 4is equivalent to Jensen's functional equation 3. On the other hand, when λ ∈ {−2, −1, 0}, (4) is not necessarily equivalent to (3).…”

“…Nakmahachalasint 7 then investigated the Hyers-Ulam stability of the alternative Jensen's functional equation (2) in the class of mappings from 2-divisible abelian groups to Banach spaces. Given an integer λ = 2, Srisawat et al 8 solved the alternative Jensen's functional equation of the form…”

Hyperstability of an alternative equation of Jensen type

“…Regarding the study of alternative equations of Jensen type, Nakmahachalasint [6] studied the solutions of f (x) + 2 f (x + y) + f (x + 2 y) = 0 or f (x) − 2 f (x + y) + f (x + 2 y) = 0 when f is a function from a semigroup to a uniquely divisible commutative group. Srisawat et al [7] have proved that…”