42.2. EVOLUTION INCLUSIONS 1373

Therefore, letting λ denote a sequence converging to 0 it follows

limλ→0

xλ = x1 ∈ H

for some x, the convergence being strong convergence. Also taking a further subsequenceand using weak compactness it can be assumed

Bλ xλ ⇀ z1

where this time the convergence is weak. Taking another subsequence, it can also be as-sumed

y− xλ −Bλ xλ ⇀ z2 (42.1.4)

the convergence being weak convergence. Recall Bλ xλ ∈ BJλ (B)xλ and also note that byassumption there is a constant C independent of λ such that

C ≥ |Bλ xλ | ≥1λ(xλ − Jλ (B)x)

which showsJλ (B)xλ → x1

also. Now it follows from Lemma 42.1.5 that x1 ∈ D(B) and z1 ∈ Bx1. Recall

y− xλ −Bλ xλ ∈ Axλ

and so by the same lemma again,

x1 ∈ D(A) , z2 ∈ Ax1

By 42.1.4 it followsy− x1− z1 = z2 ∈ Ax1

Thusy = x1 + z1 + z2 ∈ x1 +Bx1 +Ax1

and this proves the theorem.

42.2 Evolution InclusionsOne of the interesting things about maximal monotone operators is the concept of evolutioninclusions. To facilitate this, here is a little lemma.

Lemma 42.2.1 Let f : [0,T ]→ R be continuous and suppose

D+ f (t)≡ lim suph→0+

f (t +h)− f (t)h

< g(t)

where g is a continuous function. Then

f (t)− f (0)≤∫ t

0g(s)ds.