On a functional equation concerning affine transformations
Let U, V be real vector spaces and X, P be nonempty subsets of U such that \(X+P\subset X\), \(\lambda\) \(P\subset P\) for \(\lambda\geq 0\) and R be a subset of L(U) such that \(I\in R\), R(X)\(\subset X\) for \(R\in {\mathfrak R}\) and \(\lambda\) \({\mathfrak R}\subset {\mathfrak R}\) for \(\lambda >0\). The author proves that, if the functional equation \(u(Rx+p)=\alpha (R,p)u(x)+\beta (R,P),\) where \(x\in X\), \(p\in P\), \(\alpha\) : \({\mathfrak R}\times P\to R\setminus \{0\}\), \(\beta\) : \({\mathfrak R}\times P\to V\) are given functions, has a nonconstant solution \(u: X\to V\), then \(\alpha\) and \(\beta\) have to be a very special form. Moreover the author obtains, under the additional hypotheses that \({\mathfrak R}\) is a group and one of the supplementary spaces of Lin\(\{\) Rp; \(R\in {\mathfrak R}\), \(p\in P\}\) is R-invariant, the general solution of this equation.
scientific article; zbMATH DE number 4038396