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C-Cosine Functions and the Abstract Cauchy Problem, II

✍ Scribed by Chung-Cheng Kuo; Sen-Yen Shaw

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
246 KB
Volume
210
Category
Article
ISSN
0022-247X

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✦ Synopsis

For a bounded linear injection C on a Banach space X and a closed linear Ε½ . Ε½ . operator A : D A ; X Βͺ X which commutes with C we prove that 1 the Ε½ . Ε½ . Ε½ . Ε½ . abstract Cauchy problem, uΠ‰ t s Au t , t g R, u 0 s Cx, uЈ 0 s Cy, has a Ε½ . Ε½ . Ε½ 2 . unique strong solution for every x, y g D A if and only if 2

. Ε½ generates a C -cosine function on X D A with the graph norm , if and only if, 1 1

. Ε½ . in case A has nonempty resolvent set 3 A generates a C-cosine function on X.

Here C s C N X . Under the assumption that A is densely defined and C y1 AC s 1 1

Ε½ . Ε½ .

A, statement 3 is also equivalent to each of the following statements: 4 the

Ε½ . problem Β¨Π‰ t s AΒ¨t q C x q ty q H Cg r dr, t g R, Β¨0 s ¨Ј 0 s 0, has a 0 1

Ε½ .

Ε½ . unique strong solution for every g g L and x, y g X; 5 the problem wΠ‰ t s l o c Ε½ . Ε½ . Ε½ . Ε½ . Aw t q Cg t , t g R, w 0 s Cx, wЈ 0 s Cy, has a unique weak solution for every g g L 1 and x, y g X. Finally, as an application, it is shown that for any bounded l o c operator B which commutes with C and has range contained in the range of C, A q B is also a generator.

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