Superstability of Pexiderized functional equations arising from distance measures

In this paper, we obtain the superstability of the functional equation f(pr, qs) + g(ps, qr) = θ(pq, rs)h(p, q)k(r, s) for all p, q, r, s ∈ G, where G is an Abelian group, f, g, h, k are functionals on G2, and θ is a cocycle on G2. This functional equation is a generalized form 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 the information measures, and also they can be represented as products of some multiplicative functions and the exponential functional equations. As corollaries, we obtain the superstability of the many functional equations (combination of three variables functions, for example: f(pr, qs) + g(ps, qr) = θ(pq, rs)h(p, q)g(r, s)). c ©2016 All rights reserved.