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

10. 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

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

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

19. VII.2 Solution of problem of evolution

20. 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

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

29. 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.

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

34. 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