An alternative Jensen's functional equation on semigroups

Abstract

We study the alternative Jensen's functional equation f (x) ± 2f (xy) + f (xy 2 ) = 0 when f is a function from a semigroup or a group to a uniquely divisible abelian group.

An alternative functional equation of Jensen type on groups

“…Note that when λ = −2, we recover the results in Ref. 10 in the case when the domain is a group. Furthermore, if λ / ∈ {0, −1, −2}, then we show that (3) is equivalent to the classical Jensen's functional equation…”

“…For instance, Kannappan and Kuczma 1 studied the solutions of the alternative Cauchy functional equation Motivated by the work on the alternative Cauchy functional equation and many extensive studies on Jensen's functional equation on different kinds of groups (see, e.g., Refs. 5-9) Nakmahachalasint 10 has investigated the alternative Jensen's functional equation of the form…”

“…Note that one of the results in Ref. 10 is that if the domain of the alternative Jensen's functional equation (2) is a 2-divisible group, then (2) is equivalent to the classical Jensen's functional equation in the sense that the solution f satisfies Jensen's functional equation…”

“…Inspired by the work of Nakmahachalasint 10 and Forti 4 , we will investigate the alternative Jensen's functional equation in an even more general setting. Namely, given an integer λ = 2, we find a criterion of the existence of the general solution of the alternative Jensen's functional equation…”

Stability of an alternative functional equation related to Jensen's functional equation

“…In this section, we will prove the Hyers-Ulam stability of the alternative Jensen's functional equation (4). The following lemma is crucial for the main theorem.…”