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About the sequence x kezt we have the following a x 1 B x b...

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Algebra

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About the sequence x kezt we have the following a x 1 B x b for each t to b t b MK b 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 1 8 1 8 B x b which is given by x t lim x 1 1 l b to b Thus x 1 lim x f s 1 x 0 1 2 te t b t b k20 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 lim ff s x s ds x lim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function tx t satisfying the integral equation x 1 x f f s x s ds for all t te b 1 b dx 0 f f s x s ds f t X 1 dt This now proves that the curve X 1 8 4 82 thus obtained is a solution of the initial value problem

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