Existence theorem for nonlinear difference equations

We prove an asymptotic existence theorem for locally analytic, nonlinear difference equations possessing a formal power series solution at ∞. The proof is based on a well-known contraction argument applied to Banach spaces of holomorphic functions on suitable unbounded domains of $\[$\mathbb{C}$$ . The main difficulty lies in the definition of appropriate domains for the case that the equation possesses a level “1⁺” in combination with lower levels.