# Differential Equations: An Introduction with Applications by Lothar Collatz

By Lothar Collatz

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We next check whether T is a contraction operator. 56) z°(E) - OI <_ P( ) II Z_ 211. Further we can now choose p(x) to be a positive continuous function on J. For p(x)=e-"I x-xoI with a>Ls we have P(x) di J XoP( ) Hence it follows altogether that II TZ - TZII <- KIIZ - ZII with K=Ls< 1. a Therefore the operator T is indeed a contraction in the domain considered. 22. BANACH'S FIXED-POINT THEOREM AND THE EXISTENCE THEOREM FOR ORDINARY DIFFERENTIAL EQUATIONS We have now introduced the concepts needed for formulating a general fixed- point theorem, from which the existence theorem for ordinary differential equations can easily be deduced.

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)+ . , that the series of its absolute values converges IR(x)I < IYo(x)I+IY1-YoI+IY2-Y1I+....

N - 1), Y(n) = f = r(x) - Z PU(x)Y("11), V=0 then the partial derivatives of f with respect to the y(q) are precisely the coefficient functionspq _ i (x) for q = 1, 2, ... , n. If we assume these to be continuous and bounded in an interval [A, B] , which might extend to infinity in one or both directions, I pq(x)I <- P for q=0,1, ... , n - 1 and I r(x) I _< P with P? 1, then P can be used as the Lipschitz constant k. Bounds M for I y(q+ 1) I and I f I are also required. ,n-l. where, of course, I x - xo s a must be contained in [A, B] and b must be x - xoI <-a, >--1.