Linear functional equations and their Baire category properties (Q1592821)

The author considers the functional equation \[ \varphi(f(x))=g(x)\varphi(x)+h(x) \tag{*} \] to show that only for functions \(f\) having a very simple dynamical behaviour equation (*) has continuous solutions. Given \(f:I \to I\) (\(I\) non degenerate real interval) and a positive integer \(n\), Per\((f,n)\) denotes the set of all periodic points of \(f\) of order \(n\). The main results of the paper are the following. Theorem 1: Let \(Y\) be a metric space, \(f \in C(I)\) and \(g:I \to \mathbb R\) continuous with the following properties: \[ |g(x)g(f(x))|<1 \quad \text{for} \quad x\in \text{Per}(f^2,1) \] and \[ g(x)\neq 0 \quad \text{for} \quad x\in \Cup_{i=1}^{\infty} (f^i)^{-1}(\text{Per}(f,2)). \] If for each \(h \in C(I,Y)\) equation (*) has a solution \(\varphi \in C(I,Y)\), then \(f\) can have only cycles of orders \(1\) and \(2\). Theorem 2: Assume that \(Y\) is a Banach space, let \(f\) and \(g\) be as in Theorem 1. If the set of functions \(h \in C(I,Y)\) for which equation (*) has a solution \(\varphi \in C(I,Y)\) is of second Baire category in \(C(I,Y)\), then \(f\) can have only cycles of orders \(1\) and \(2\).