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VII.1 Hille-Yosida Theorem

Explore the Hille-Yosida Theorem and elementary properties of maximal monotone operators in real Hilbert spaces, including Yosida regularization. Learn about Cauchy, Lipschitz, and Picard solutions in Banach spaces, as well as the Hille-Yosida and Riesz Lemmas.

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VII.1 Hille-Yosida Theorem

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  1. VII.1 Hille-Yosida Theorem

  2. VII.1 Definition and Elementary Properties of maximal monotone operators

  3. Maximal Monotone Let H be a real Hilbert space and let be an unbounded linear operator . A is called monotone if A is called maximal monotone if furthermore i.e.

  4. Proposition VII.1 Let A be maximal monotone. Then (a) D(A) is dense in H (b) A is closed. (c) For every is a bijection from D(A) onto H is a bounded operator with

  5. Yosida Regularization of A Let A be maximal monotone, for each let (by Prop.VII.1 ) is called a resolvent of A and is called Yosida regularization of A

  6. Proposition VII. 2 p.1 Let A be maximal monotone, Then (a1) (a2) (b) (c)

  7. Proposition VII. 2 p.2 (d) (e) (f)

  8. VII.2 Solution of problem of evolution

  9. Theorem VII.3 Cauchy, Lipschitz. Picard Let E be a Banach space and F be a mapping From E to E such that there is a unique then for all such that

  10. Lemma VII.1 If is a function satisfing , then the functions and are decreasing on

  11. Theorem VII.4 (Hille-Yosida) p.1 Let A be a maximal monotone operator in a Hilbert space H then for all there is a unique s.t.

  12. Theorem VII.4(Hille-Yosida) where D(A) is equipped with graph norm i.e. for Furthermore, and

  13. Lemma VI.1 (Riesz-Lemma) Let For any fixed , apply Green’s second identity to u and in the domain we have and then let

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