25.7. MAXIMAL MONOTONE OPERATORS 905

monotone. Now a repeat of the above reasoning which shows that  is maximal monotoneshows that the fact that Â+ B̂ is maximal monotone implies that A+B is also. You justshift with −x0 instead of x0. It amounts to nothing more than the observation that maximalgraphs don’t lose their maximality by shifting their ranges and domains.

Suppose B,A are maximal monotone. Does there always exist a solution x to

x∗ ∈ Fx+Bλ x+Ax? (25.7.63)

Consider the monotone hemicontinuous and bounded operator F +Bλ .Is F̂ + B̂λ definedby (

F̂ + B̂λ

)(x)≡

(F̂ + B̂λ

)(x+ x0)

also coercive for some x0 ∈ D(A)? If so, the existence of the desired solution to the aboveinclusion follows from Corollary 25.7.40. Then for all ∥x∥ large enough that ∥x+ x0∥ >∥x0∥ ,

⟨F (x+ x0)+Bλ (x+ x0) ,x⟩∥x∥

=⟨F (x+ x0) ,x⟩

∥x∥+

≥0⟨Bλ (x+ x0)−Bλ (x0) ,x⟩

∥x∥+⟨Bλ (x0) ,x⟩∥x∥

≥ 12⟨F (x+ x0) ,x⟩∥x+ x0∥

−∥Bλ (x0)∥

≥ 12⟨F (x+ x0) ,x+ x0⟩

∥x+ x0∥− 1

2⟨F (x+ x0) ,x0⟩∥x+ x0∥

−∥Bλ (x0)∥

≥ 12⟨F (x+ x0) ,x+ x0⟩

∥x+ x0∥− 1

2⟨F (x+ x0) ,x0⟩

∥x0∥−∥Bλ (x0)∥

≥ 12⟨F (x+ x0) ,x+ x0⟩

∥x+ x0∥− 1

2∥x+ x0∥−∥Bλ (x0)∥

=12∥x+ x0∥2− 1

2∥x+ x0∥−∥Bλ (x0)∥

which shows that

lim∥x∥→∞

⟨F (x+ x0)+Bλ (x+ x0) ,x⟩∥x∥

= ∞

and so by Corollary 25.7.40, there exists a solution to 25.7.63. This shows half of thefollowing interesting theorem which is another version of the above major result.

Theorem 25.7.43 Suppose A,B are maximal monotone operators. Then for each x∗ ∈ X ′,there exists a solution xλ to

x∗ ∈ Fxλ +Bλ xλ +Axλ , λ > 0 (25.7.64)

If for λ ∈ (0,δ ) ,{Bλ xλ} is bounded, then there exists a solution x to

x∗ ∈ Fx+Bx+Ax