Superstability of the difference-form functional equations related to distance measures

Abstract

The present work extends the study on the stability of the functional equation f(pr,qs)+ f(ps,qr )= f(p,q) f(r,s), which arises in the characterization of symmetrically compositive sum-form distance measures, and as a products of some multiplicative functions. In this paper, we obtain the superstability of the functional equations f(pr,qs) − f(ps,qr )= f(p,q)g(r,s) f(pr,qs) − f(ps,qr )= g(p,q)f(r,s) f(pr,qs) − f(ps,qr )= g(p,q)g(r,s) f(pr,qs) − f(ps,qr )= g(p,q)h(r,s), for all p,q,r,s ∈ G ,w hereG is an Abelian group. These functional equations arise in the char- acterization of the nonsymmetrically compositive difference-form related to distance measures, products of some multiplicative functions. In reduction, they can be represented as exponential functional equations.