Analytic solutions of difference equations with small step size

On Banach spaces of functions analytic on bounded domains, we construct a right inverse of the difference operator Delta (epsilon) defined by Delta (epsilon)y(x) = (1/epsilon)(y(x + epsilon) - y(x)), epsilon a small parameter. We derive the existence of analytic solutions of difference equations of the form Delta (epsilon)y = f(epsilon, x, y) that have an asymptotic expansions as epsilon --> 0. We study the analytic continuation of the Borel transforms of the corresponding formal solutions. As applications we derive explicit error bounds for Euler's method and the midpoint rule (Nystrom's method) in the theory of the numerical solution of initial value problems and a (known) theorem on the inclusion of analytic diffeomorphisms close to the identity into analytic flows.