Integrability of weak distributions on Banach manifolds

This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroids and Banach Poisson manifold. The main results of this paper generalize the works presented in [4], [20] and [7].