On approximate solutions of the linear functional equation of higher order (Q711039)

The authors consider linear functional equations of the type \[ 0=\Delta(\phi)(x):=\phi(f^m(x))-\sum_{i=1}^m a_i(x)\phi(f^{m-i}(x))+F(x) \] where \(\phi\) is the unknown function and \(f^m\) denotes the \(m\)th iterate of \(f\). They prove that under suitable conditions for every approximate solution \(\phi_s\), satisfying \(\|\Delta(\phi_s)\|\leq \epsilon_0(x)\), there exists a solution \(\phi\) of the original problem \(\Delta(\phi)=0\) close to \(\phi_s\). In the simplest case the conditions are that \(f\) is bijective, the \(a_j\) and \(\epsilon_0\) are \(f\)-invariant (that is, \(a_j(f(x))=a_j(x)\) for all \(x\)), and that \(l_j>0\), where \(l_j=\inf_x|1-|r_j(x)||\) and \(r_j(x)\) are the complex roots of the polynomial \(\sum_{j=1}^m a_j(x)z^{m-j}\).