By S. Schwabik, M. Tvrdý, O. Vejvoda

ISBN-10: 9027708029

ISBN-13: 9789027708021

Booklet via Schwabik, S., Tvrdý, M., Vejvoda, O.

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**Extra info for Differential and Integral Equations: Boundary Value Problems and Adjoints**

**Example text**

I j , x(l) = 0 and extending the definition of x to the whole [0,1] in such a way that x is linear on the rest of [0,1], we obtain m-í rxj I 1=0 Jtj m-1 x(s)d[j(s)] *(sKMI + < Z Varrj / + Var L / < £ ' J 1=0 Hence x(s)d[j(s)] m- 1 i ю-ňъ-ä + iлo -л^-л +1 =10 1=i m-l Пj x(s)d[л-)] + X(5)d[/( )] S > Z IЛ^) - ЛVi)l + ІЛO -Л^m-i)| - в > ЦAfl - 2г > a - Зв > 0, 1=i where a = {0 = r0 < T 0 < tx < zx < ... < tm-x < xm-x < xm < tm = 1}. Since obviously xeWp and x(0) = x(l) = 0, this completes the proof.

Theorem (Substitution Theorem). If h e BV[a, ft], g: [a, ft] -> R and b f: [a, ft] -* K, the integral \ a g dh exists a/irf / is bounded on [a,ft],then the integral d T d T J« /(0 (Ja g( ) M )) exfefs if and only if the integral J* f(t) g(t) dh(t) exists and in this case the two integrals are equal. Proof. Let us show that the following statement holds. If \bagdh exists then for every n > 0 there is an Sle9)[a,ft] such that for every AeA(S1)9 A: a = a 0 < tj < ... < Tk < ock = ft we have k I (4,20) Ja,- i <ч.

If AeB(X, Y), then the unique operator _4# e B(Y, X) satisfying (3,12) is called the principal generalized inverse operator of A. 31. Remark. 2, then the vector u0 = A*beRn satisfies the conditions (3,13) and (3,7). 4#b = d*(_4# - _4#_4_4#) b = 0. Furthermore, A*AA* = A* by (2,17). |e, _4#b is for any beRm a unique best approximate solution of (2,1). 4. Perron-Stieltjes integral This section contains the definition of the Perron-Stieltjes integral based on the work of J. Kurzweil [1], [2]. Some facts concerning this integral are collected here.

### Differential and Integral Equations: Boundary Value Problems and Adjoints by S. Schwabik, M. Tvrdý, O. Vejvoda

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