An alternative Cauchy functional equation on a semigroup

Hyperstability of an alternative equation of Jensen type

“…From this point onward, we will assume that ϕ(x, e) = 0 for every x ∈ G. This always holds without loss of generality since (2) e (x) = 0.…”

“…Proof : Let x, y ∈ G. Suppose that (λ) y (x). Case I: (λ) y (x − y) and (2) y (x + y). Then one of the following linear combinations is applicable.…”

“…The study of alternative functional equations can be viewed as a generalization of an original equation. For instance, Cauchy (additive) functional equation f (x + y) = f (x) + f ( y) can be generalized to ( f (x + y)) 2 = ( f (x)+ f ( y)) 2 , which may be rewritten as…”

“…This kind of generalizations is later recognized as alternative functional equations, in the sense that for each point in the designated domain, either the original equation or a modified counterpart is satisfied. Some recent studies on alternative functional equations can be found in [2][3][4][5].…”