42.2. EVOLUTION INCLUSIONS 1377

It follows from this and the equation 42.2.6, that

|Aλ yλ (t)| ≤C ≡max{| f (t)|H : t ∈ [0,T ]}+

+(

eT(

min{|a| : a ∈ Ay0}+ | f (0)|+1+∥∥ f ′∥∥2

L2(0,T ;H)

))1/2(42.2.14)

I am using C as a generic constant which is sufficiently large.This is a very nice estimate. The next task is to show uniform convergence of the yλ as

λ → 0. From 42.2.8 (Dh(yλ (t)− yµ (t)

),yλ (t +h)− yµ (t +h)

)+(

1h

∫ t+h

t

(Aλ yλ (s)−Aµ yµ (s)

)ds,yλ (t +h)− yµ (t +h)

)= 0

Then from the argument in 42.2.10,

1h

12

(∣∣yλ (t +h)− yµ (t +h)∣∣2− ∣∣yλ (t)− yµ (t)

∣∣2)+

(1h

∫ t+h

t

(Aλ yλ (s)−Aµ yµ (s)

)ds,yλ (t +h)− yµ (t +h)

)≤ 0

Now take limsuph→0 to obtain

12

D+∣∣yλ (t)− yµ (t)

∣∣2 + (Aλ yλ (t)−Aµ yµ (t) ,yλ (t)− yµ (t))≤ 0

Using the definition of Aλ Aλ x≡ 1λ

x− 1λ

Jλ x, this equals

12

D+∣∣yλ (t)− yµ (t)

∣∣2+(Aλ yλ (t)−Aµ yµ (t) ,λAλ yλ (t)+ Jλ yλ (t)−

(µAµ yµ (t)+ Jµ yµ (t)

))≤ 0

Now this last term splits into the following sum(Aλ yλ (t)−Aµ yµ (t) ,λAλ yλ (t)−µAµ yµ (t)

)+(Aλ yλ (t)−Aµ yµ (t) ,Jλ yλ (t)− Jµ yµ (t)

)By Lemma 42.1.3 the second of these terms is nonnegative. Also from the estimate 42.2.14,the first term converges to 0 uniformly in t as λ ,µ → 0. Then by Lemma 42.2.1 it followsthat if λ is any sequence converging to 0, yλ (t) is uniformly Cauchy. Let

y(t)≡ limλ→0

yλ (t) .

Thus y is continuous because it is the uniform limit of continuous functions. Since Aλ yλ (t)is uniformly bounded and Aλ x≡ 1

λx− 1

λJλ x, it also follows

y(t) = limλ→0

Jλ yλ (t) uniformly in t. (42.2.15)