1 2 x lim k no xo f f s lim x s ds x f f s x s ds fo Thus the function tx 1 satisfying the integral equation x 1 x dx dr 500 Finally the validity of this integral equation has the following two implications X 1 x ff s x s ds 0 d dts x f s x s ds for all 1 t b t b f s x s ds f s x s ds x 0 x 1 S lim f s x s ds 11 f s x s ds f 1 X t This now proves that the curve X 1 8 82 thus obtained is a solution of the initial value problem 2 3 UNIQUENESS OF A SOLUTION 20 We prove an inequality which will lead us to the uniqueness of the solutions fi A e Proposition 2 Let Gronwall s Inequality f a b 0 g a b 0x be continuous functions and A 0 a constant satisfying PESA Ss s gs ds for all tea b Then Chapter 2 Systmes of First Order ODE for all te a b Proof First we assume A 0 and put ht A Sf a te a b Then ht 0 for all 1 a b and ht f t g t h t g t fsgs ds for all
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