Superstability of the functional equation with a cocycle related to distance measures

In this paper, we obtain the superstability of the functional equation f (pr, qs) + f (ps, qr) = theta(pq, rs) f (p, q) f (r, s) for all p, q, r, s is an element of G, where G is an Abelian group, f a functional on G(2), and theta a cocycle on G(2). 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 as products of some multiplicative functions. In reduction, they can be represented as exponential functional equations. Also we investigate the superstability with following functional equations: f (pr, qs) + f (ps, qr) = theta(pq, rs) f (p, q) g(r, s), f (pr, qs) + f (ps, qr) = theta(pq, rs) g(p, q) f (r, s), f (pr, qs) + f (ps, qr) = theta(pq, rs) g(p, q) g(r, s), f (pr, qs) + f (ps, qr) = theta (pq, rs) g(p, q) h(r, s).