Kenneth Kuttler

884 CHAPTER 25. NONLINEAR OPERATORS

Corollary 25.7.17 Let X be a Banach space and suppose that

xn→ x, ∥x∗n∥→ ∞

Then denoting by Dr the closed disk centered at x with radius r. It follows that for every Dr,there exists y0 ∈ Dr and a subsequence with index nk such that⟨

x∗nk,xnk − y0

⟩→−∞

Proof: It follows that xn− x→ 0. Therefore, from Lemma 25.7.16, for every r > 0,there exists ŷ0 ∈ B(0,r) and a subsequence xnk such that⟨

x∗nk,(xnk − x

)− ŷ0

⟩→−∞

Thus ⟨x∗nk

,xnk − (x+ ŷ0)⟩→−∞

Just let y0 = x+ ŷ0. Then y0 ∈ Dr and satisfies the desired conditions.

Definition 25.7.18 A set valued mapping A : D(A)→P (X) is locally bounded at x ∈D(A) if whenever xn→ x, xn ∈ D(A) it follows that

lim supn→∞

{∥x∗n∥ : x∗n ∈ Axn}< ∞.

Lemma 25.7.19 A set valued operator A is locally bounded at x∈D(A) if and only if thereexists r > 0 such that A is bounded on B(x,r)∩D(A) .

Proof: Say the limit condition holds. Then if no such r exists, it follows that A isunbounded on every B(x,r)∩D(A). Hence, you can let rn → 0 and pick xn ∈ B(x,rn)∩D(A) with x∗n ∈ Axn such that ∥x∗n∥> n, violating the limit condition. Hence some r existssuch that A is bounded on B(x,r)∩D(A). Conversely, suppose A is bounded on B(x,r)∩D(A) by M. Then if xn → x, it follows that for all n large enough, xn ∈ B(x,r) and so ifx∗n ∈ Axn, ∥x∗n∥ ≤M. Hence limsupn→∞ {∥x∗n∥ : x∗n ∈ Axn} ≤M < ∞ which verifies the limitcondition.

With this definition, here is a very interesting result.

Theorem 25.7.20 Let A : D(A)→ X ′ be monotone. Then if x is an interior point of D(A) ,it follows that A is locally bounded at x.

Proof: You could use Corollary 25.7.17. If x is an interior point of D(A) , and A isnot locally bounded, then there exists xn → x and x∗n ∈ Axn such that ∥x∗n∥ → ∞. Then byCorollary 25.7.17, there exists y0 close to x, in D(A) and a subsequence xnk such that⟨

x∗nk,xnk − y0

⟩→−∞

Letting y∗0 ∈ Ay0, ⟨x∗nk− y∗0,xnk − y0

⟩≥ 0

884 CHAPTER 25. NONLINEAR OPERATORSCorollary 25.7.17 Let X be a Banach space and suppose thatXn x, ||x7|| 3 00Then denoting by D, the closed disk centered at x with radius r. It follows that for every D,,there exists yo € D, and a subsequence with index nx such that(Xn s%nz — YO) > —20Proof: It follows that x, —x— 0. Therefore, from Lemma 25.7.16, for every r > 0,there exists fo € B(0,r) and a subsequence x, such that(ng (ng, —*) — Fo) + —2°Thus(Xing Xng — (x+$o0)) —> —ooJust let yp = x+ 9. Then yo € D, and satisfies the desired conditions. JjDefinition 25.7.18 A set valued mapping A: D(A) > A(X) is locally bounded at x €D(A) if whenever xn — x, Xn € D(A) it follows thatlim sup {|l2,|| +, € Axn} <0.n—-ooLemma 25.7.19 A set valued operator A is locally bounded at x € D(A) if and only if thereexists r > 0 such that A is bounded on B(x,r) D(A).Proof: Say the limit condition holds. Then if no such r exists, it follows that A isunbounded on every B(x,r) D(A). Hence, you can let r, + 0 and pick x, € B(x,rm)ND(A) with x* € Ax, such that ||x*|| >, violating the limit condition. Hence some r existssuch that A is bounded on B(x,r) D(A). Conversely, suppose A is bounded on B(x,r)MD(A) by M. Then if x, — x, it follows that for all n large enough, x, € B(x,r) and so ifx7 € AXn, ||x4|| <M. Hence limsup,,_,.. {]|x%|] 1%; € Axn} <M < which verifies the limitcondition. §jWith this definition, here is a very interesting result.Theorem 25.7.20 Let A: D(A) — X’ be monotone. Then if x is an interior point of D(A),it follows that A is locally bounded at x.Proof: You could use Corollary 25.7.17. If x is an interior point of D(A), and A isnot locally bounded, then there exists x, — x and x* € Ax, such that ||x*|| + o. Then byCorollary 25.7.17, there exists yo close to x, in D(A) and a subsequence Xn, such that(xn Ang — yo) — —o°Letting yg € Ayo,(ny —Yos%ny —yo) = 0