The Drygas functional equation on abelian semigroups with endomorphisms (Q2038376)

For an abelian semigroup \((S,+)\), a uniquely 2-divisible group \((H,+)\), endomorphisms \(\sigma,\tau\colon S\to S\) and an unknown mapping \(f\colon S\to H\) the authors consider a {Drygas-type functional equation} \[ f(x+\sigma(y))+f(x+\tau(y))=2f(x)+f(\sigma(y))+f(\tau(y)),\qquad x,y\in S. \] The solutions of this equation are determined in the form \[ f(x)=A(x)+Q(x,x),\qquad x\in S, \] where \(A\colon S\to H\) and \(Q\colon S\times S\to H\) are additive and bi-additive mappings, respectively. Further, continuous solutions, if \(S\) is a topological group and \(H\) is a topological vector space, are constructed as well.