SC-90-09.pdf - ZIB

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and the first part of the theorem is proved by a simple inductive argument. In order to prove uniqueness in S(y*,2/uj) let y° :'= y** for some y** ^ y* with F(y**) = 0) which implies y 1 = y** as well. Upon insertion into (1.19), this yields ||y«-y'||< w /2||y"-y*|| a
Indeed (1.6) follows from (1.21) by taking z — y and u = x + s(y — x), while (1.12) follows from (1.21) for z = x and u = x + s(y — x). Let u, x € D and denote by L the line containing u and x, and by L\ the half line of origin u, contained in L but not containing x. Assumption (1.6) says that (1.21) is satisfied only for z € L\ and not necessarily for all z £ L. However, we will prove that in case F is twice Frechet differentiate assumptions (1.6) and (1.21) are equivalent. First note that if F is twice Frechet differentiate then, by dividing both sides of (1.6) by s, and then setting s —* 0, we obtain \\F'( y )- 1 F"(x)(y-xY\\(u) - F'(x))(u -x)\\< »J llZ {~* { ly 2 dt which proves that (1.21) is satisfied. l = u 1 \\u - x\\ 2 dt = u\\u -x\\ 2 , o 6
Page 1 and 2: Konrad-Zuse-Zentrum für Informatio
Page 3 and 4: The authors wish to thank Erlinda C
Page 5 and 6: 1. A Refined Newton-Mysovskii Theor
Page 7: We note that under the hypothesis o
Page 11 and 12: 2. Asymptotic Mesh Independence of
Page 13 and 14: C\: if Wj G Xj and w € X are solu
Page 15 and 16: and by using property C 0 we obtain
Page 17 and 18: then for k > k(x°,e), where \x —
Page 19 and 20: Condition (2.30) is generally not s
Page 21 and 22: implies (2.12) — without any quas
Page 23: [12] L. B. Rail: A Note on the Conv

SC-90-09.pdf - ZIB

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