A quadratic functional equation with involutive automorphisms on semigroups
The authors consider a generalization of the classical quadratic functional equation in the following form: \[ f(x\sigma(y))+f(\tau(y)x)=2f(x)+2f(y),\qquad x,y\in S, \] where \(S\) is a semigroup (not necessarily abelian), \(H\) is an abelian uniquely 2-divisible group, \(\sigma,\tau\colon S\to S\) are involutive automorphisms of \(S\) and \(f\colon S\to H\) is the unknown function. This equation has been considered in many previous works, for particular cases or under additional assumptions, e.g., for mappings defined on abelian semigroups. Some related functional equations have been also widely studied. The main result in this paper is the general solution of the above functional equation. Precisely, it is proved that \(f\) satisfies the above equation if and only if it is of the form \[ f(x)=Q_1(x,\sigma(x))+Q_2(x,x)+\psi(x)+a(x),\qquad x\in S, \] where \(Q_1,Q_2\colon S\times S\to H\) are bi-additive and invariant with respect to \(\sigma\) and \(\tau\) mappings such that \(Q_i(\tau(y),x)=-Q_i(x,\sigma(x))\) for \(x,y\in S\) (\(i=1,2\)), \(a\colon S\to H\) is an additive mapping such that \(a\circ \sigma=a\circ\tau=-a\) and \(\psi\colon S\to H\) satisfies \[ \psi(xy)+\psi(yx)=2\psi(x)+2\psi(y),\qquad x,y\in S \] and \[ \psi(\tau(x)y)-\psi(\sigma(x)y)=4a(x),\qquad x\in S. \] Many of the known previous results about special cases of the considered functional equation are thus particular cases of the general description given in this paper.
scientific article; zbMATH DE number 7493122
scientific article