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

AbstractLetXbe a separable Banach space. Consider the problem[formula]where the continuous functionf: [0,∞)×X→Xis locally Lipschitz continuous inx, uniformly inton bounded intervals, and continuous intuniformly w.r.t.x. The product integral formulax(T)=limn→∞∏i=0n[I+Tnf(iTn)]x0,0≤T<tmaxfor the solutionx(t) of (D) has been shown to converge. We also show that iff(t,.) is Lipschitz continuous onXwith constantL, then the mappingx0→x(T) is Lipschitz continuous with constanteLTfor anyT>0. This formula has been recently developed for differential inclusions inRnby Wolenski, but the infinite dimensional case is considerably more involved