Bounded solutions and stability of functional equations for two variable functions (Q1895202)

This paper generalizes some recent results of \textit{L. Széhelyhidi} [Publ. Math. 47, No. 1-2, 95-100 (1995)] concerning the bounded solutions and stability of two functional equations for two-variable functions. The paper at hand deals with the family of functional equations \[ F(x, y)+ {1\over n} \sum^n_{i= 1} F(x\phi_i(y), z)= {1\over n} \sum^n_{i= 1} F(x, y\phi_i(z))+ F(y, z), \] where \(F: S\times S\to \mathbb{C}\) is unknown, \(S\) is an amenable semigroup, \(n\) is a positive integer, and the maps \(\phi_i: S\to S\) \((i= 1,\dots, n)\) form a group of homomorphisms of \(S\). Széhelyhidi (op. cit.) dealt with the cases \(n= 1\), which has a long history and has become known as the cocycle equation, and \(n= 2\). Generalizing, the author proves two main results for general \(n\). First, the bounded solutions are of the form \[ F(x, y)= f(x)+ f(y)- {1\over n} \sum^n_{i= 1} f(x\phi_i(y)), \] for arbitrary bounded \(f: S\to \mathbb{C}\); second, an approximate solution must be close to a solution. A number of interesting examples are given.