On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

Abstract

LetGbe a uniquely2-divisible commutative group and letf,g:G→Candσ:G→Gbe an involution. In this paper, generalizing the superstability of Lobačevskiǐ’s functional equation, we considerf(x+σy)/22-g(x)f(y)≤ψ(x)orψ(y)for allx,y∈G, whereψ:G→R+. As a direct consequence, we find a weaker condition for the functionsfsatisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.