Yet another variant of the Drygas functional equation on groups

Let G be a group and C the field of complex numbers. Suppose σ1,σ 2 : G → G are endomorphisms satisfying the condition σi(σi(x)) = x for all x in G and for i = 1, 2. In this paper, we find the central solution f : G → C of the equation f (xy) + f (σi(y)x) =2f (x) + f (y) + f (σ2(y)) for all x,y ∈ G which is a variant of the Drygas functional equation with two involutions. Further, we present a generalization the above functional equation and determine its central solutions. As an application, using the solutions ofthe generalized equation, we determine the solutions f, g, h, k : GxG → C ofthefunc-tional equation f (pr, qs) + g(sp, rq) = 2f (p, q) + h(r, s) + k(s, r) when f satisfies the condition f (pr, qs) = f (rp, sq) for all p, q, r, s ∈ G.