Ordinary differential equations on infinite dimensional manifolds

By using the Tangent bundle of a finite or infinite dimensional manifold we provide an alternative way to study the first and second order ordinary differential equations on M. We apply the vector fields with their integral curves, autoparallel curves and a new technique in the sense of [2], [6], [7], to introduce a new way to study ordinary differential equations on Banach manifolds and also a certain type of Frechet manifolds obtained as projective limits of Banach manifolds. Moreover we extend the concept of completeness for Banach and Frechet manifolds. In the other words we will prove that if M is a compact Banach (Frechet) manifold then it is complete ie it's autoparallel curves are defined on the whole of real line R.