A New Product Integral Representation for Differential Equations in Separable Banach Spaces

Let X be a separable Banach space. Consider the problem x9sf t , x ., t)0 D. x 0. sx0gX, where the continuous function f : w0, `.=XªX is locally Lipschitz continuous in x, uniformly in t on bounded intervals, and continuous in t uniformly w.r.t. x. The product integral formula n T T x T . slimnª` Iq f i x0 , 0FT-tmax / is0 n n for the solution x t. of D. has been shown to converge. We also show that if f t, .. is Lipschitz continuous on X with constant L, then the mapping x0ªx T. is Lipschitz continuous with constant e LT for any T)0. This formula has been recently developed for differential inclusions in Rn by Wolenski, but the infinite dimensional case is considerably more involved.