By SOREN PRIP BEIER

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**Example text**

7 2 . This example 2 In 2 is also treated in Rail [69]. The operator Tu is syntone for u(t) >_ 0 and therefore the theory of MDO, No. 6 is applicable too. For m one can use v 0 0, v, = 1, w - 2, w < 1 + In 2 < 2 a "A one has immediately existence of a solution and the bounds 1 £ u(t) £ 1 + In 2, which can be improved e a s i l y . 4) Newton's method is convenient, if the derivatives T and T'~l are e a s y to get. ,x ) = 0 J = i, n or if one is discretizing the given (nonlinear) differential or integral equations.

Proposition 2. 6. Suppose f: R -*R and Df(x) e x i s t s . Then Df(x) is represented by the Jacobian matrix at x if and only if Df (x) is a linear operator. Proof: Observe that in Example 2. 6 we used only the fact that f'(x) was linear. Remark. A point worth mentioning is that the derivative (Gateaux or Frechet) always has its domain in the same space as the original operator, i . e . , if f: X ->Y then Df is also defined in X . However, since Df: X -*[X,Y], we could consider Df: XXX -*Y . With this interpretation f and Df will have the same range s p a c e .

46 DIFFERENTIATION AND INTEGRATION Let C [a,b] be the vector space of all real-valued functions which are continuously differentiable on the in terval [a,b] and vanish at the end points a and b . Sup pose f: R3 -*R has continuous second partial derivatives with respect to all three variables. 1) . a The simplest problem in the calculus of variations is e s s e n tially that of finding y € C:: [a,b] which minimizes J, i . e . 2) minimize J(y); y € C [a,b] . 2), then for each r\ € C p j a , b ] , a(t) = J(y + trj) is a real-valued function of the real variable t which has a minimum at t = 0; hence a'(0) = 0 .

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