A variant of the quadratic functional equation on groups and an application

摘要

Let G be a group and C the field of complex numbers. Suppose σ:G→G is an endomorphism satisfying σ(σ(x))=x for all x in G. In this paper, we first determine the central solution, f:G or G×G→C, of the functional equationf(xy)+f(σ(y)x)=2f(x)+2f(y)for all x,y∈G,which is a variant of the quadratic functional equation. Using the central solution of this functional equation, we determine the general solution of the functional equation f(pr,qs)+f(sp,rq)=2f(p,q)+2f(r,s) for all p,q,r,s∈G, which is a variant of the equation f(pr,qs)+f(ps,qr)=2f(p,q)+2f(r,s)studied by Chung, Kannappan, Ng and Sahoo in [3](see also [16]). Finally, we determine the solutions of this equation on thefree groups generated by one element, the cyclic groups of order m, thesymmetric groups of order m, and the dihedral groups of order 2m form ≥ 2.