About the sequence x keZ we have the following a 1 EBx b for

Last updated: 9/25/2023

About the sequence x keZ we have the following a 1 EBx b for each t to b te b MKk b x 0 x 0 te t b t b k 20 k 1 c xk Z 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 to 8 to 8 B x b C which is given by x t lim x 1 1 t b to b Thus x t lim x f s x s ds 5 8 8 de 1 2 x lim ff s x s ds x j Finally the validity of this integral equation has the following two implications X f x f s x s ds x 0 xo x f f s lim x s ds x ff s x s ds fo Thus the function tx 1 satisfying the integral equation x 1 x f f s x s ds for all 1 t b t b dt lim f s x s ds 0 ff s x s ds f t X t This now proves that the curve X 1 8 4 82 thus obtained is a solution of the initial value problem