Analytic invariant curves for a planar mapping (Q943504)

The main object of this paper is the investigation of the single variable functional equation \[ f[p(z)+q(f(z))]=h(f(z)), \qquad z\in\mathbb{C}, \] where \(p,q\), and \(h\) are given analytic functions. This equation characterizes the invariance of the curve \(w=f(z)\) under the planar mapping \(T(w,z)=(p(z)+q(w),h(w))\). By constructing a convergent power series solution of an auxiliary functional equation, the existence of analytic solutions in a neighborhood of zero of the above equation is proved.