a b c x KEZ is uniformly Cauchy on t b t b verification of

Last updated: 9/24/2023

a b c x KEZ is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 to 8 B x b which is given by x t lim x 1 1E1 b t b Thus x 1 lim xo f f s X s ds x 1 EB xo b for each 1 to b t b MKk k 1 x 1 x 1 1 2 Finally the validity of this integral equation has the following two implications X x f s x s ds x 0 x 1 to b t b k20 x lim ff s x s ds x lim f s x s ds x f f s lim x s ds x f f s x s ds Thus the function tx 1 satisfying the integral equation x 1 x f f s x s ds for all r te b t b dx dt 0 df f s x s ds f t X t This now proves that the curve X 1 8 4 89 thus obtained is a solution of the initial value problem