Jensen's functional equation on semigroups (Q6111803)

The author considers the functional equation \[(1) \;\;\; f(x\varphi(y))+f(\psi(y)x)=2f(x), \quad x,y\in S,\] with \(s\colon S\to H\), \(S\) a semigroup, \(H\) a 2-torsion free abelian group and \(\varphi,\psi\colon S\to S\) endomorphisms.\par It is shown that under the assumption that \(\varphi\) or \(\psi\) is surjective the solutions of (1) are of the form \(f=A+c\) with some constant \(c\) and \(A\colon S\to H\) additive with the additional property that \(A\circ\psi=-A\circ\varphi\).\par Many examples are given showing known functional equations to be special cases of (1). Moreover examples are given that \(f\) need not be of the above form if neither \(\varphi\) nor \(\psi\) is surjective.