42.9. AN EVOLUTION INCLUSION 1405

which implies−(y− xλ −Aλ xλ ,Aλ xλ )≤C (xλ ) . (42.8.56)

I claim {C (xλ )} and {|xλ |}are bounded independent of λ .By 42.8.54 and monotonicity of Aλ ,

Φ(ξ )−Φ(xλ )≥ (y− xλ −Aλ xλ ,ξ − xλ )

≥ (y− xλ ,ξ − xλ )− (Aλ xλ ,ξ − xλ )

≥ (y− xλ ,ξ − xλ )− (Aλ ξ ,ξ − xλ )

≥ (y−ξ ,ξ − xλ )+ |ξ − xλ |2− (Aλ ξ ,ξ − xλ )

≥ |ξ − xλ |2−Cξ y |ξ − xλ |where Cξ y depends on ξ and y but is independent of λ because of the assumption thatξ ∈D(A)∩D(Φ) and Lemma 42.1.3 which gives a bound on |Aλ ξ | in terms |y| for y ∈ Ax.Therefore, there exist constants, C1 and C2, depending on ξ and y but not on λ such that

Φ(ξ )≥Φ(xλ )+ |xλ |2−C1 |xλ |−C2.

Since Φ≥ 0,

2(

Φ(ξ )+C2 +C2

12

)≥Φ(xλ )+ |xλ |2 .

This shows |xλ | is bounded independent of λ . Therefore, by 42.8.53

K2 +2K1

(Φ(ξ )+C2 +

C21

2

)≥ K2 +K1

(Φ(xλ )+ |xλ |2

)≥C (xλ ) ,

showing that both |xλ | and C (xλ ) are bounded independent of λ . Therefore, from 42.8.56,it follows Aλ xλ is bounded independent of λ . By Theorem 42.8.1 this shows there existsx ∈ D(∂Φ)∩D(A) such that

y ∈ Ax+∂Φ(x)+ x

and so A+∂Φ is maximal monotone since y ∈ H was arbitrary. This proves the theorem.

42.9 An Evolution InclusionIn this section is a theorem on existence and uniqueness for the initial value problem

x′+∂2φ (t,x) ∋ f , x(0) = x0.

Suppose {φ (t, ·)}t∈[0,T ] is a family of functions mapping H to [0,∞] which satisfy thefollowing axioms.

φ (t, ·) is convex and lower semicontinuous, (42.9.57)

D(φ (t, ·)) = D, independent of t ∈ [0,T ] , (42.9.58)

There exists a constant, K, such that for all x ∈ D,

|φ (t,x)−φ (s,x)| ≤ K(

φ (r,x)+ |x|2 +1)|t− s| (42.9.59)

for all r ∈ [0,T ] .