1410 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

and [J1 (·)s] is in L2 (0,T ;H) . If [ f ] ∈ L2 (0,T ;H) is arbitrary, take a sequence of simplefunctions, sn converging to f pointwise and [sn]→ [ f ] in L2 (0,T ;H). Then

|J1 (t)sn (t)− J1 (t) f (t)| ≤ |sn (t)− f (t)|

and it follows J1 (t)sn (t) converges pointwise to J1 (t) f (t) showing that t → J1 (t) f (t) ismeasurable. Now the equivalence class of functions equal to this one a.e. is in L2 (0,T ;H)by Fatou’s lemma and the assumption that the simple functions, sn converge in L2 (0,T ;H) .This shows A is maximal and proves the lemma.

Conditions 42.9.70 and 42.9.71 are just what is needed to obtain the conclusion ofLemma 42.9.6 but it may not be clear how to verify these conditions easily. The follow-ing lemma gives sufficient conditions which are easy to verify which imply 42.9.70 and42.9.71.

Lemma 42.9.7 Suppose there exists [ξ ] ∈ L2 (0,T ;H) such that

J1 (t)ξ (t) ,φ (t,J1 (t)ξ (t))

are bounded independent of t ∈ [0,T ] and t → J1 (t)ξ (t) is measurable. Then the conclu-sion of Lemma 42.9.6 holds.

Proof: Let y(t) = J1 (t)ξ (t) . Thus

y(t)+∂2φ (t,y(t)) ∋ ξ (t) .

Now suppose x ∈ H and letx(s)+ z(s) = x (42.9.72)

where z(s) ∈ ∂2φ (s,x(s)) , so x(s) = J1 (s)x. Take the inner product of both sides withx(s)− y(s) to obtain

(x(s) ,x(s)− y(s))H +(z(s) ,x(s)− y(s))H = (x,x(s)− y(s))H

and therefore,12|x(s)|2H −

12|y(s)|2H ≤ φ (s,y(s))−φ (s,x(s))

+ |x|H |x(s)|H + |x|H |y(s)|H ≤14|x(s)|2H + c |x|2H +

12|y(s)|2H

+φ (s,y(s))−φ (s,x(s)) .

Consequently,

φ (s,x(s))+14|x(s)|2H ≤ |y(s)|

2H + c |x|2H +φ (s,y(s))<C (42.9.73)

a constant depending on x. Replacing s with t in 42.9.72 and subtracting yields

x(t)− x(s)+ z(t)− z(s) = 0.