Abstract
In this paper, we obtain the superstability of the functional equation f(pr,qs)+f(ps,qr)=θ(pq,rs)f(p,q)f(r,s) for all p,q,r,s∈G, where G is an Abelian group, f a functional on G 2 , and θ 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)=θ(pq,rs)f(p,q)g(r,s), f(pr,qs)+f(ps,qr)=θ(pq,rs)g(p,q)f(r,s), f(pr,qs)+f(ps,qr)=θ(pq,rs)g(p,q)g(r,s), f(pr,qs)+f(ps,qr)=θ(pq,rs)g(p,q)h(r,s).MSC:39B82, 39B52.
Highlights
We investigate the superstability with following functional equations: f + f = θf (p, q)g(r, s), f + f = θg(p, q)f (r, s), f + f = θg(p, q)g(r, s), f + f = θg(p, q)h(r, s)
Denote the set of all n-ary discrete complete probability distributions, that is, o n is the class of discrete distributions on a finite set of cardinality n with n ≥
Let us consider the generalized characterization of a symmetrically compositive sum form related to distance measures with a cocycle: (CDM) f + f = θ(pq, rs)f (p, q)f (r, s) for all p, q, r, s ∈ G and where f, θ are functionals on G, which can be represented as exponential functional equation in reduction
Summary
The following functional equation: (FE) f (pr, qs) + f (ps, qr) = f (p, q)f (r, s) holding for all p, q, r, s ∈ I was instrumental in the characterization of symmetrically compositive sum-form distance measures. These functional equations arise in the characterization of symmetrically compositive sum-form distance measures, products of some multiplicative functions.
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