Differential equations driven by rough signals (I): an extension of an inequality of L. C. Young

If one fixes x one may ask about the existence and uniqueness of y with finite p-variation where to avoid triviality we assume d > 1.We prove that if each f i is (1+α)-Lipschitz in the sense of [7] then a unique solution exists and that it can be recovered as a limit of Picard iterations; in consequence it varies continuously with x.If each f i is α-Lipschitz, one still has existence of solutions, but examples of A.M. Davie show that they are not, in general, unique.