About the sequence x kEZ we have the following 1 EB x b for

Last updated: 10/8/2023

About the sequence x kEZ we have the following 1 EB x b for each t to b t b a b x 1 1 5 0 MK te to b te bk20 k 1 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 8B x b C which is given by x t lim x 1 1E1 b to b Thus X 1 lim x f s x s ds 6 X 1 1 da kix x lim ff s x s ds x flim f s x s ds Finally the validity of this integral equation has the following two implications X t x s x s ds x 0 x 1 x f f s lim x s ds x ff s x s ds Thus the function 1 x 1 satisfying the integral equation x 1 x ff s x s ds for all t t b t b 2 dx 0 aff s x s ds f 1 X 1 dt d This now proves that the curve X To 8 to 85 thus obtained is a solution of the initial value problem