1368 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

which yields

|J1x− J1y|2 ≤ (x− y,J1x− J1y)

≤ |x− y| |J1x− J1y|

which yields the result.Next consider the claim that λA is maximal monotone. The monotone part is immedi-

ate. The only thing in question is whether I +λA is onto. Let r ∈ (−1,1) and pick f ∈ H.Consider solving the equation for u

(1+ r)u+Au ∋ (1+ r) f (42.1.1)

This is equivalent to finding u such that

(I +A)u ∋ (1+ r) f − ru

or in other words finding u such that

u = J1 ((1+ r) f − ru)

However, ifTu≡ J1 ((1+ r) f − ru) ,

then since |r|< 1, T is a contraction mapping and so there exists a unique solution to 42.1.1.Thus

u+1

1+ rAu ∋ f

It follows for any |r| < 1,(1+ r)−1 A is maximal monotone. This takes care of all λ ∈( 1

2 ,∞). Now do the same thing for (2/3)A to get the result for all λ ∈(( 2

3

)( 12

),∞). Now

apply the same argument to (2/3)2 A to get the result for all λ ∈(( 2

3

)2 ( 12

),∞). Next

consider the same argument to (2/3)3 A to get the desired result for all λ ∈(( 2

3

)3 ( 12

),∞).

Continuing this way shows λA is maximal monotone for all λ > 0. Also from the first partof the proof (I +λA)−1 is Lipschitz continuous with Lipschitz constant 1.

A maximal monotone operator can be approximated with a Lipschitz continuous oper-ator which is also monotone and has certain salubrious properties. This operator is calledthe Yosida approximation and as in the case of linear operators it is obtained by formallyconsidering

A1+λA

If you do the division formally you get the definition for Aλ ,

Aλ x≡ 1λ

x− 1λ

Jλ x (42.1.2)

where Jλ = (I +λA)−1 as above. It is obvious that Aλ is Lipschitz continuous with Lip-schitz constant no more than 2/λ . Actually you can show 1/λ also works but this is notimportant here.