By C. Caratheodory

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**Extra resources for Theory of Functions of a Complex Variable, 2 Volumes. **

**Sample text**

10, (i), we get \J\x -x\< \J\x - J\xn\ + \J\xn - xn\ + \xn - x\ < 2\x - xn\ + \J\xn - xn\, n = 1, 2 , . . , x £ Q. From closedness of Q and dom A C <3 C dom A we get dom A = {a; G X \ J\x -t x as A 4- 0}. In order to prove convexity of dom A we choose X\,X2 G dom A, 0 < a < 1 and set x = ax\ + (1 — a)a;2. 10, (i)) \J\X - Xi\ < \X - Xi\ + \JXX! \ < \X - X2\ + \J\X2 — X2\, A > 0. Each of these estimates implies that (JA^)A>O is bounded. As a uniformly convex space, X is also reflexive. n^,00 J\nx.

We choose y G X. 24, a), there exist, for any A > 0, elements [xx,y\] € A such that \xx\ < r, A > 0, for some r > 0. 10, (iii), that \Bxx\ < \\Bx\\ for all x G domB. 28 Chapter 1. 24. A > 0. A > 0. are bounded on A > 0. • The next results are concerned with continuous perturbations of dissipative operators. We shall need some preparatory results and also some results on nonlinear semigroups which are special cases of results on general evolution problems and are stated in Chapter 6. 27. Lemma. Assume that A is a dissipative operator on X and let x € dom A.

Imply [a:, y] G A. 16. Theorem. Assume that A is an m-dissipative operator on X. Then the following is true: (i) The operator A is closed. Q for a family [x, y] G A. (XA)O• oo. 7) (yn — y,xn - x)- < 0 for all [x,y] G A. Since {•, •)_ is lower semi-continuous, we obtain (y-V,x-x)- < \immi(yn-y,xn -x)_ <0 for all [x,y] G A. This shows that A — Au {(x,y)} is a dissipative extension of A. , [x, y] G A. 13)), we see that lim^o J\x\ = x. 10, (ii), we conclude that A\X\ G AJ\X\ (note that dom J\ = range(J - XA) = X by m-dissipativity of A).