25.7. MAXIMAL MONOTONE OPERATORS 893

and so ε ∥xε − x0∥ = ε ∥F (xε − x0)∥ ≤ ∥y− y0∥ < r/2. Thus yε stays in B(y0,r). This isbecause y is closer to y0 than r/2 while yε is within r/2 of y. It follows that the xε arebounded and so xε − x0 is bounded and so εF (xε − x0)→ 0. Thus yε → y strongly. Sincethe xε are bounded, there exists a further subsequence, still denoted as xε such that xε → x,some point of X . Then if [u,v] ∈ G (A) ,

⟨yε − v,xε −u⟩ ≥ 0

and letting ε → 0 using the strong convergence of yε one obtains

⟨y− v,x−u⟩ ≥ 0

which shows that y ∈ Ax. Thus B(y0,

r2

)⊆ A(D(A)) ≡ D

(A−1

)and so A(D(A)) is open.

The proof featured the usual duality map.Note that as part of the proof A(D(A)) was shown to be closed so although it was

assumed at the outset that A−1 was locally bounded on A(D(A)), this is the same as sayingthat A−1 is locally bounded on A(D(A)).

Corollary 25.7.32 Suppose A : D(A)→P (X ′) is maximal monotone and coercive. ThenA is onto.

Proof: From Theorem 25.7.31 it suffics to show that A−1 is locally bounded at y∗ ∈A(D(A)). The case of an interior point follows from Theorem 25.7.20. Assume then thaty∗ is a limit point of A(D(A)). Of course this includes the case of interior points. Thenthere exists y∗n→ y∗ where y∗n ∈ Axn. Then

⟨y∗n,xn⟩∥xn∥

≤ ∥y∗n∥

and the right side is bounded. Hence by coercivity, so is ∥xn∥. Therefore, there is a furthersubsequence, still denoted as xn such that xn → x weakly while y∗n → y∗ strongly. Thenletting [u,v∗] ∈ G (A) ,

⟨y∗− v∗,x−u⟩= limn→∞⟨y∗n− v∗,xn−u⟩ ≥ 0

Hence y∗ ∈ Ax and y∗ ∈ A(D(A)). Thus A−1 is locally bounded on A(D(A)) and so A isonto from the above theorem.

25.7.4 Approximation TheoremsThis section continues following Barbu [13]. Always it is assumed that the situation is ofa real reflexive Banach space X having strictly convex norm and its dual X ′. As observedearlier, there exists a solution xλ to the inclusion

0 ∈ F (xλ − x)+λA(xλ )