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By Lothar Collatz

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The pair y(x),z(x) is a `fixed point' of the operator T. (L32) 29 §6. THE GENERAL EXISTENCE AND UNIQUENESS THEOREM 18. THE EXISTENCE THEOREM For the system of differential equations y' = f(x, Y, z); z' = g(x, Y, Z) `Peano's theorem'-which we quote here without proof-states that the existence of a solutions system y(x), z(x) is assured provided that the functions f(x, y, z) and g(x, y, z) are continuous and bounded. But to prove the uniqueness we require rather more, for example, the `Lipschitz conditions', which in our case can be written If(x,y,z)-f(x,y*,z*)I

31.

Xo; the same bound holds for x < xo. 37) I (n + 1)! k (n + 1)! which can easily be proved by induction, as follows. 36) that Sn+1 <- 2k x Sn(E) dE c 2 M ( 2 k ) xO rX I S - Xo I"+1 dE Jxo (n+1)! Ix-xoln+2 =2M(2k)"+1 (n + 2)! again first for x>- xo; but the same bound is also obtained for x < X. 37) holds for n = 1, it must therefore hold for all n. We now consider a series R(x), the partial sums of which are precisely the y, (x), and which will turn out to be the required solution function y (x): R(x) = yo(x)+(y1 -Yo)+(y2-y1)+ .

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