ε-Enlargements of maximal monotone operators in Banach spaces Regina Sandra Burachik (
[email protected])∗ Engenharia de Sistemas e Computa¸c˜ ao, COPPE–UFRJ, CP 68511, Rio de Janeiro–RJ, 21945–970, Brazil.
B.F. Svaiter (
[email protected])† IMPA, Instituto de Matem´ atica Pura e Aplicada, Estrada Dona Castorina, 110. Rio de Janeiro, RJ, CEP 22460-320, Brazil. Abstract. Given a maximal monotone operator T in a Banach space, we consider an enlargement T ε , in which monotonicity is lost up to ε, in a very similar way to the ε-subdifferential of a convex function. We establish in this general framework some theoretical properties of T ε , like a transportation formula, local Lipschitz continuity, local boundedness, and a Brøndsted & Rockafellar property. Keywords: Banach spaces, maximal monotone operators, enlargement of an operator, Brøndsted & Rockafellar property, transportation formula, Lipschitz continuity, local boundedness. Mathematics Subject Classification (1991): 47H05, 46B99
1. Introduction and motivation Let A and B be arbitrary sets and F : A → 2B a multifunction. By an enlargement or extension of F we mean a multifunction E : R+ × A → 2B such that F (x) ⊆ E(b, x) ∀b ≥ 0, x ∈ A. A well known and most important example of extension of a multifunction is the ε-subdifferential. Given a proper convex function f on a Banach space X, f : X → R ∪ {+∞}, the subdifferential of f at x, i.e., the set of subgradients of f at x, denoted by ∂f (x), is given by ∂f (x) = {u ∈ X ∗ : f (y) − f (x) − hu, y − xi ≥ 0 for all y ∈ X} . The ε-subdifferential enlargement (of ∂f ) was introduced in [1]. It is defined as ∂ε f (x) := {u ∈ X ∗ : f (y) − f (x) − hu, y − xi ≥ −ε for all y ∈ X} , ∗
Partially supported by PRONEX–Optimization. Partially supported by CNPq Grant 301200/93-9(RN) and by PRONEX– Optimization. †
c 1999 Kluwer Academic Publishers. Printed in the Netherlands.
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2 for any ε ≥ 0, x ∈ X. Note that ∂f = ∂0 f and in general ( i.e., for ε ≥ 0) ∂f (x) ⊆ ∂ε f (x). If f is proper closed convex, then ∂f is a maximal monotone operator (see [12]). So, we have an example of an enlargement of certain maximal monotone operators (those which are subdifferentials). N For an arbitrary maximal monotone operator T : RN → 2(R ) , Burachik, Iusem and Svaiter [3] proposed the following enlargement: Given ε ≥ 0 and x ∈ RN T ε (x) = {u ∈ RN | hv − u, y − xi ≥ −ε, ∀y ∈ RN , v ∈ T (y)} . The enlargement defined above can be formulated in a straightforward manner for operators defined on Hilbert spaces. In the case of a Banach space X, the scalar product is replaced by the usual “dual product” on X × X ∗ . Hence, for an arbitrary maximal monotone operator T : X → ∗ 2X , we consider T ε (x) = {u ∈ X ∗ | hv − u, y − xi ≥ −ε, ∀y ∈ X, v ∈ T (y)}, for any ε ≥ 0, x ∈ X. The aim of this paper is to study the basic properties of this enlargement for maximal monotone operators defined in Banach spaces. The idea of relaxing monotonicity has been explored in [13]. According to this work, an operator W is ε-monotone if hu − v, x − yi ≥ −ε
∀ u ∈ W (x), v ∈ W (y).
Observe that in the above definition there is no maximal monotone operator to be extended. Furthermore, given T a maximal monotone operator, there are in general many maximal ε-monotone multifunctions whose graph contain the graph of T . An important property concerning any extension of a maximal monotone operator T is whether an element in the graph of the extension of T can be approximated by an element in the graph of the original operator. This question has been successfully solved for the extension ∂ε f by Brøndsted & Rockafellar in [1]: Given ε ≥ 0 and vε ∈ ∂ε f (xε ), for any η > 0, there exists x and v ∈ ∂f (x) such that kx − xε k ≤ ε/η,
kv − vε k ≤ η.
We say that an enlargement E of T satisfies the Brøndsted & Rockafellar property if any element in the graph of E can be approximated in a similar way, i.e.: Given b ≥ 0, vb ∈ E(b, xb ), for any η > 0 there exists x and v ∈ T (x) such that kx − xb k ≤ ε/η,
kv − vb k ≤ η.
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3
When X is a Hilbert space, it has been proved in [4], that the enlargement defined above satisfies the Brøndsted & Rockafellar property. In this work, we prove that T ε has this property for any reflexive Banach space. Another key question concerning an extension of T is how to construct an element of the extension using elements of T . For the extension ∂ε f , the tool for doing this is the so-called “transportation formula” (see Proposition 4.2.2, Vol.2 [8]). For the extension T ε , a transportation formula was found in [4] for operators defined on Hilbert spaces. In this work, we will show that this formula still holds for operators defined on Banach spaces. This “alternative” transportation formula was already obtained for the extension ∂ε f in [10] (Proposition 1.2.10). Finally, we prove that (ε, x) ,→ T ε (x) is locally bounded by an affine function of ε on the interior of the domain of T , as well as Lipschitz continuous, for T defined in a Banach space. We point out that this result was proved in [4] for X = RN . Algorithmic applications of this enlargement can be found in [3], [4] and [5]. The paper is organized as follows. In Section 2 we give the theoretical preliminaries. The main results are established in Section 3: The Brøndsted & Rockafellar property, the “transportation formula”, the “ε-affine” local boundedness and the Lipschitz-continuity of the multifunction (ε, x) ,→ T ε (x).
2. Basic definitions ¿From now on X is a real Banach space, X ∗ is its dual. When X is reflexive, X ∗∗ (the dual of X ∗ ) is identified with X. Given x ∈ X and v ∈ X ∗ , v(x) will be denoted indifferently by p hx, vi and hv, xi. The norm in X × X ∗ will be taken as k(x, v)k = kxk2 + kvk2 . We need first some notation. Let R+ := {α ∈ R | α ≥ 0}. For ρ ≥ 0, we denote by B(x, ρ) the open ball with center at x and radius ρ, i.e., B(x, ρ) := {x ∈ X : kxk < ρ}. ∗ Given a set A ⊆ X and a multifunction S : X → 2X , − we define the set S(A) :=
S
a∈A S(a) .
− The domain, image and graph of S are respectively denoted by D(S) := {x ∈ X : S(x) 6= ∅} , R(S) := S(X) and G(S) := {(x, v) : x ∈ X and v ∈ S(x)} .
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4 − The inverse of S is the multifunction S −1 : X ∗ → 2X defined by S −1 (v) = {x ∈ X : v ∈ S(x)} . − S is locally bounded at x if there exists a neighborhood U of x such that the set S(U ) is bounded. − S is monotone if hu − v, x − yi ≥ 0 for all u ∈ S(x) and v ∈ S(y), for all x, y ∈ X. ∗
− S: X → 2X is maximal monotone on X if it is monotone and its graph is maximal with respect to this property, i.e., its graph is not properly contained in the graph of any other monotone operator. Based on [3], we define the enlargement of monotone operators on a Banach space: ∗
DEFINITION 2.1. Let T : X → 2X be monotone. The ε-enlargement of T is the multifunction T (·) (·) : R+ × X → 2X
∗
defined by T ε (x) = {u ∈ X ∗ : hv − u, y − xi ≥ −ε , ∀y ∈ X , v ∈ T (y)}
(1)
for x ∈ X and ε ≥ 0. We will also use the notation ∗
T ε (·) = T (ε, ·) : X → 2X . Although some results below hold for the above defined enlargement of arbitrary monotone multifunctions, we are concerned in this paper with maximal monotone operators. Therefore, from now on, T : X → 2X
∗
is a maximal monotone operator. Observe that if T = ∂f , for some proper closed and convex function f , then ∂ε f (x) ⊆ T ε (x). This result, as well as examples showing that the inclusion can be strict, can be found in [3]. Note that ε is used to define different concepts and objects: εenlargement, ε-subdifferential, ε-monotonicity, ε solutions etc. Note also that ε is also a real variable. To avoid confusion, from now on we will try not to use ε as a variable.
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5
3. Properties of T ε The following properties, which also can be found in [3], still hold in Banach spaces. LEMMA 3.1. 1. T 0 (·) = T (·) 2. For any b1 , b2 ∈ R+ , if b1 ≤ b2 then T (b1 , x) ⊆ T (b2 , x),
∀x ∈ X.
3. If I ⊆ R+ is nonempty then ∩b∈I T (b, x) = T (¯b, x),
∀x ∈ X.
where ¯b = inf I. 4. T b (x) is weak∗closed and convex for any b ≥ 0, x ∈ X. Proof. The proof follows directly from Definition 2.1. We point out that only item 1 depends on the maximal monotonicity of T . ∗ Since T : X → 2X is maximal monotone, T −1 : X ∗ → 2X is monotone. If X is reflexive then T −1 is maximal monotone. In this case the ε-enlargements of T and T −1 are quite connected. LEMMA 3.2. If X is a reflexive Banach space, then for any b ≥ 0 (T −1 )b = (T b )−1 . Proof. The proof is straightforward from Definition 2.1. Given a closed and convex function f and a positive α, it follows easily from the definition of ε-subdifferential that ∂ε (αf ) = α ∂(ε/α) (f ). This property can be easily established also for the multifunction T b . More specifically: LEMMA 3.3. For any α > 0, (αT )b = α T b/α . It is well-known (see [7], Proposition 1.3) that the following relation between the epsilon subdifferential of a sum of two proper convex lower-semicontinuous functions f1 , f2 and the sum of the corresponding
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6 epsilon subdifferentials holds: If x0 ∈ dom f1 ∩ dom f2 and one of the functions is continuous at this point then, given ε > 0,
∂ε (f1 + f2 )(x0 ) = ∪θ∈[0,1] ∂θε f1 (x0 ) + ∂(1−θ)ε f2 (x0 ) . For the T ε enlargement, however, we recover only one of the inclusions. Suppose T1 , T2 and T1 + T2 maximal monotone. Direct application of Definition 2.1 yields: for any x ∈ D(T1 ) ∩ D(T2 ) and b ≥ 0,
(1−θ)b
(T1 + T2 )b (x) ⊃ ∪θ∈[0,1] T1θb (x) + T2
(x) .
It is well-known that the graph of a maximal monotone operator is demiclosed. This property is shared by the graph of T (·, ·). ∗
PROPOSITION 3.4. The graph of T (·, ·) : R+ × X → 2X is demiclosed, i.e., the conditions below hold. (a) If {xk } ⊂ X converges strongly to x0 , {uk ∈ T (bk , xk )} converges weak ∗ to u0 in X ∗ and {bk } ⊂ R+ converges to b, then u0 ∈ T (b, x0 ). (b) If {xk } ⊂ X converges weakly to x0 , {uk ∈ T (bk , xk )} converges strongly to u0 in X ∗ and {bk } ⊂ R+ converges to b, then u0 ∈ T (b, x0 ). Proof. To prove (a), observe that since {xk } converges strongly and {uk } converges weak∗, they are bounded sequences. So, let M < +∞ be a bound for kxk k and kuk k. Take any (y, v) ∈ G(T ). From Definition 2.1, it follows that −bk ≤ = = ≤
hxk − y, uk − vi hxk − y, u0 − vi + hxk − y, uk − u0 i hxk − y, u0 − vi + hxk − x0 , uk − u0 i + hx0 − y, uk − u0 i hxk − y, u0 − vi + kxk − x0 k(M + ku0 k) + hx0 − y, uk − u0 i
Taking the limit k → ∞, we obtain −b ≤ hx0 − y, u0 − vi. Since (y, v) is an arbitrary element in the graph of T , the conclusion follows. For proving item (b), recall that weak convergent sequences are also bounded and use a similar reasoning.
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7
3.1. Brøndsted & Rockafellar Property For a closed proper convex function f , the theorem of Brøndsted & Rockafellar (see [1]), states that any ε-subgradient of f at a point xε can be approximated by some exact subgradient, computed at some x, possibly different from xε . The T ε enlargement also satisfies this property. This fact has been proved for a Hilbert space in [4]. We extend below this result to a reflexive Banach space. DEFINITION 3.5. Consider g : X → R defined by 1 g(x) = kxk2 . 2 ∗
The duality mapping J : X → 2X is defined by J(x) := ∂g(x). As a subdifferential of a convex function, J is maximal monotone. We recall below two important results concerning the duality mapping J. PROPOSITION 3.6. (i) J(x) = {v ∈ X ∗ | hx, vi = kxk kvk, kxk = kvk}, (ii) Let X be reflexive, and take S any maximal monotone operator on X. Then S + λJ is onto for any λ > 0. The proof of (i) can be found in [6]. Item (ii) follows from a result of Browder [2], Theorem 7.2. THEOREM 3.7. Assume X is a reflexive Banach space. Let b ≥ 0 and (xb , vb ) ∈ G(T b ). Then for all η > 0 there exists (x, v) ∈ G(T ) such that kv − vb k ≤
b η
and
kx − xb k ≤ η .
(2)
Proof. If b = 0, then (2) holds with (x, v) = (xb , vb ) ∈ G(T ). Suppose now b > 0. For an arbitrary positive coefficient β define the operator ∗
G β : X → 2X y 7→ βT (y) + {J(y − xb )} , where J is the duality operator of X. Since βT is maximal monotone, by Proposition 3.6(ii), Gβ is onto. In particular βvb is in the image of this operator and there exist x ∈ X and v ∈ T (x) such that βvb ∈ βv + J(x − xb ) .
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8 Therefore β(vb − v) ∈ J(x − xb ). This, together with Proposition 3.6(i) and Definition 2.1, yields 1 hvb − v, xb − xi = − kx − xb k2 β = −βkv − vb k2 ≥ −b . Choosing β := η 2 /b, the result follows. Observe that the proof above only uses the inequality which characterizes elements in T b . This means that the Brøndsted & Rockafellar property holds for any extension ET of T which satisfies ET (b, ·) ⊂ T b (·). As ∂b f (·) ⊂ (∂f )b (·), we recover Brøndsted & Rockafellar’s theorem for b-subdifferentials in a reflexive Banach space. The following corollary, which extends slightly Proposition 2 in [3], establishes a relation between the image, domain and graph of an operator and its extension T ε . COROLLARY 3.8. Let X be a reflexive Banach space. The following inclusions hold. (i) R(T ) ⊂ R(T b ) ⊂ R(T ), (ii) D(T ) ⊂ D(T b ) ⊂ D(T ), (iii) If d(· ; ·) denotes the point-to-set distance, then √ d((xb , vb ); G(T )) ≤ 2b , whenever (xb , vb ) ∈ G(T b ). Proof. The leftmost inclusions in (i) and (ii) are straightforward from Definition 2.1. As for the right ones, they follow from Theorem 3.7, making η → +∞ and η → 0√in (i) and (ii) respectively. To prove (iii), take η = b in (2), write d((xb , vb ); G(T ))2 ≤ kx − xb k2 + kv − vb k2 ≤ 2b , and take square roots. 3.2.
“ε-affine” local boundedness
It is a well-known fact (see [9],[11]) that any maximal monotone operator in an arbitrary Banach space is locally bounded on the interior of its domain. The following result establishes that also the extended operator T ε is locally bounded on the interior of the domain of T , by an affine function of ε. In order to prove this, we start with a technical result.
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PROPOSITION 3.9. Let U ⊂ D(T ). Define M := sup{kuk | u ∈ T (U )}.
(3)
If V ⊂ X and ρ > 0 are such that V + B(0, ρ) ⊂ U, then sup{kvk | v ∈ T (b, V )} ≤
(4) b + M, ρ
(5)
for any b ≥ 0. Proof. Observe that (4) is equivalent to the following inclusion: {x ∈ X | d(x, V ) < ρ} ⊂ U.
(6)
Take x ˜ ∈ V and v˜ ∈ T (b, x ˜). Recall that the norm in X ∗ is given by k˜ v k := sup{hz, v˜i | z ∈ X , kzk = 1}. Hence there exists a sequence {z k } ⊂ X such that kz k k = 1 and lim hz k , v˜i = k˜ v k.
k→∞
Define y k := x ˜ + σz k , where σ ∈ (0, ρ). Clearly d(y k , V ) ≤ ky k − x ˜k = σ < ρ. Hence, by our assumption, {y k } ⊂ U . Using this fact and the definition of M , we conclude that for any wk ∈ T (y k ), kwk k ≤ M . By definition of T b and y k , −b ≤ h˜ v − wk , x ˜ − y k i = h˜ v − wk , −σz k i. Dividing the expression above by σ, we get −
b ≤ −h˜ v , z k i + hwk , z k i ≤ σ −h˜ v , z k i + kwk kkz k k = −h˜ v , z k i + kwk k ≤ −h˜ v , z k i + M,
(7)
where we used the definitions of {z k } and M . Rearranging (7), we obtain h˜ v, zk i ≤
b + M. σ
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10 Taking now limit for k tending to infinity, we obtain a bound for v˜: k˜ vk ≤
b + M. σ
Since the inequality above is true for any σ < ρ, we conclude that k˜ vk ≤
b + M. ρ
Now we are able to prove a property of T ε that is slightly stronger than local boundedness. This property will be useful in the sequel (for proving Lipschitz continuity). COROLLARY 3.10 (“ε-affine” local boundedness). For any x in the interior of D(T ) there exist a neighborhood of x, V ⊂ int D(T ) and constants L, M > 0, such that sup{kvk | v ∈ T ε (V )} ≤ εL + M for any ε ≥ 0. Proof. Take x ∈ int D(T ). By [11], Theorem 1, T is locally bounded on int D(T ). Hence, for some R > 0, B(x, R) ⊆ D(T ) and T is bounded on B(x, R). Take ρ := R/2; V := B(x, ρ) and apply Proposition 3.9. 3.3. Transportation Formula We already mentioned that the set T (b, x) approximates T (x), but this fact is of no use as long as there is no way of computing elements of T (b, x). The question is then how to construct an element in T (b, x) with the help of some elements (xi , v i ) ∈ G(T ). The answer is given by the “transportation formula” stated below. Therein we use the notation ∆m := {α ∈ Rm | αi ≥ 0 ,
m X
αi = 1}
i=1
for the unit-simplex in Rm . THEOREM 3.11. Consider a set of m triplets {(bi , xi , v i ∈ T (bi , xi ))}i=1,...,m .
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For any α ∈ ∆m define i x ˆ := Pm 1 αi x m i vˆ := 1 αi v ˆb := Pm αi bi + Pm αi hxi − x ˆ, v i − vˆi . 1 1
P
(8)
Then ˆb ≥ 0 and vˆ ∈ T (ˆb, x ˆ). Proof. Recalling Definition 2.1, we need to show that hˆ x − y, vˆ − vi ≥ −ˆb , for any (y, v) ∈ G(T ). Combine (8) and (1), with (b, x, u) replaced by (bi , xi , v i ), to obtain i hˆ x − y, vˆ − vi = Pm ˆ − vi 1 αi hx − y, v m i hxi − y, v i − vi] = P1 αi [hx − y, vˆ − v i i +P m i i ˆ−v i− m ≥ 1 αi bi . 1 αi hx − y, v
P
(9)
Since Pm 1
m i ˆ, vˆ − v i i + hˆ x − y, vˆ − v i i] αi hxi − y, vˆ − v i i = P 1 αi [hx − x m i i ˆ, v − vˆi + 0 = − P1 αi hx − x m i ˆ, v i − vˆi , = − 1 αi hx − x
P
with (9) and (8) we get hˆ x − y, vˆ − vi ≥ −ˆb .
(10)
For contradiction purposes, suppose that ˆb < 0. Then hˆ x −y, vˆ −vi > 0 for any (y, v) ∈ G(T ) and the maximality of T implies that (ˆ x, vˆ) ∈ G(T ). In particular, the pair (y, v) = (ˆ x, vˆ) yields 0 > 0. Therefore ˆb must be nonnegative. Since (10) holds for any (y, v) ∈ G(T ), we conclude from (1) that vˆ ∈ T (ˆb, x ˆ). Observe that when bi = 0, for all i = 1, . . . , m, this theorem shows how to construct vˆ ∈ T (ˆb, x ˆ), using (xi , v i ) ∈ G(T ). ˆ The formula above holds also when replacing T b by ∂ˆb f , with f a proper closed and convex function. This is Proposition 1.2.10 in [10], where an equivalent expression is given for ˆb: ˆb =
m X
m 1 X αi bi + αi αj hxi − xj , v i − v j i . 2 i=1 i,j=1
Observe also that, when compared to the standard transportation formula for ε-subdifferentials, Theorem 3.11 is a weak transportation
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12 formula, in the sense that it only allows us to express some selected ε-subgradients in terms of subgradients. The transportation formula can also be used for the ε-subdifferential to obtain the lower bound: If {(v i ∈ ∂bi f (xi ))}i=1,2
then hx1 − x2 , v 1 − v 2 i ≥ −(b1 + b2 ) .
For the enlargement T ε , we have a weaker bound: COROLLARY 3.12. Take v 1 ∈ T (b1 , x1 ) and v 2 ∈ T (b2 , x2 ). Then hx1 − x2 , v 1 − v 2 i ≥ −( b1 + p
b2 )2 .
p
(11)
Proof. If b1 or b2 are zero the result holds trivially. Otherwise choose α ∈ ∆2 as follows √ √ b2 b1 √ √ α1 := √ α2 := 1 − α1 = √ (12) b1 + b 2 b1 + b2 and define the convex sums x ˆ, vˆ and ˆb as in (8). Because ˆb ≥ 0, we can write 0 ≤ ˆb = α1 b1 + α2 b2 + α1 hx1 − x ˆ, v 1 − vˆi + α2 hx2 − x ˆ, v 2 − vˆi 2 1 2 1 = α1 b1 + α2 b2 + α1 α2 hx − x , v − v i , where we have used the identities x1 − x ˆ = α2 (x1 − x2 ), x2 − x ˆ = 2 1 1 1 2 2 α1 (x − x ), v − vˆ = α2 (v − v ) and v − vˆ = α1 (v 2 − v 1 ) first, and then α1 α22 + α12 α2 = α1 α2 . Now, combine the expression above with (8) and (12) to obtain √ p b1 b2 √ b1 b 2 + √ hx1 − x2 , v 1 − v 2 i ≥ 0 . ( b1 + b2 )2 Rearranging terms and simplifying the resulting expression, (11) is proved. Observe that the result above implies that T ε is 4ε-monotone in the sense of [13]. 3.4. Lipschitz Continuity For the sake of generality, we will work with any ET , extension of T , which satisfies: (E1 ) For any (b, x) ∈ R+ × X , T (x) ⊂ ET (b, x) ⊂ T (b, x), (E2 ) If 0 ≤ b1 ≤ b2 , then ET (b1 , x) ⊂ ET (b2 , x) for any x ∈ X.
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13
(E3 ) The transportation formula holds for ET (·, ·). More precisely, let {(bi , xi , v i ∈ ET (bi , xi ))}i=1,...,m , be any set of m triplets on the graph of ET (·, ·) and take α ∈ ∆m . Consider P i x ˆ := Pm 1 αi x m i vˆ := (13) 1 αi v i ˆb := Pm αi bi + Pm αi hxi − x ˆ, v − vˆi . 1 1 Then ˆb ≥ 0 and vˆ ∈ ET (ˆb, x ˆ). Our aim is to prove that any such extension is Lipschitz continuous. REMARK 3.13. Observe that for T = ∂f , the extension ET (b, x) = ∂b f (x) satisfies (E1 ) − (E3 ). This fact will allow us to give an alternative proof of the Lipschitz continuity of ∂b f (x) through Theorem 3.14. For an arbitrary maximal monotone operator T , the extension ET (b, x) = T b (x) satisfies (E1 ) − (E3 ) by definition of T ε , Lemma 3.1 and Theorem 3.11. For the extension ∂ε f (·) the Lipschitz continuity was proved in [7]. For the extension T ε in a finite dimensional space, this property was established in [4]. We recall that a closed-valued locally bounded multifunction S is continuous at x ¯ if for any positive there exists δ > 0 such that
kx − x ¯k ≤ δ
=⇒
S(x) ⊂ S(¯ x) + B(0, ) , S(¯ x) ⊂ S(x) + B(0, ) .
Furthermore, S is Lipschitz continuous on V if there exists a nonnegative constant L such that for any y 1 , y 2 ∈ V and s1 ∈ S(y 1 ) there exists s2 ∈ S(y 2 ) satisfying ks1 − s2 k ≤ Lky 1 − y 2 k. For simplicity of notation, we write D for the interior of D(T ). THEOREM 3.14. Let ET be an extension of T which satisfies (E1 ) − (E3 ). Take V ⊂ X and ρ > 0. Suppose that T is bounded on U ⊂ D and that {x ∈ X | d(x, V ) < ρ} ⊂ U. (14) Then ET is Lipschitz continuous on [b, b]×V . In other words, if b, b ≥ 0 are such that 0 < b ≤ b < +∞, then there exist nonnegative constants A and B such that for any (b1 , x1 ), (b2 , x2 ) ∈ [b, b] × V and v 1 ∈ ET (b1 , x1 ), there exists v 2 ∈ ET (b2 , x2 ) satisfying kv 1 − v 2 k ≤ Akx1 − x2 k + B|b1 − b2 | .
(15)
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14 Proof. Define M := sup{kuk | u ∈ T (U )},
(16)
which is finite by assumption. We claim that (15) holds for the following choice of A and B:
A :=
1 2M + ρ b
!
b + 2M , ρ
B :=
1 2M + ρ b
.
(17)
To see this, take x1 , x2 , b1 , b2 and v 1 as above. Take l := kx1 − x2 k and let x3 be in the line containing x1 and x2 such that kx3 − x2 k = ρ ,
kx3 − x1 k = ρ + l ,
(18)
as shown in Figure 1. r
l
- r
x1
ρ
r
x2
x3
Figure 1.
Then d(x3 , V ) ≤ d(x3 , x2 ) = ρ and hence x3 ∈ U by (14). By definition of x3 , it is straightforward that x2 = (1 − θ)x1 + θx3 ,
with
θ=
l ∈ [0, 1) . ρ+l
Now, take u3 ∈ T (x3 ) and define v˜2 := (1 − θ)v 1 + θu3 . By property (E3 ), v˜2 ∈ ET (˜b2 , x2 ), with ˜b2 = (1 − θ)b1 + (1 − θ)hx1 − x2 , v 1 − v˜2 i + θ x3 − x2 , u3 − v˜2
= (1 − θ)b1 + θ(1 − θ) x1 − x3 , v 1 − u3 . Use Proposition 3.9 with v 1 ∈ ET (b1 , x1 ) ⊂ ET (b1 , V ) ⊂ T (b1 , V ), together with the definition of M , to obtain kv 1 − u3 k ≤ kv 1 k + ku3 k ≤ (
b1 b1 + M) + M ≤ + 2M , ρ ρ
(19)
where we are using also that u3 ∈ T (U ). Using that the dual product on X × X ∗ satisfies hx, vi ≤ kxkkvk, (19), (18), and the definition of θ, we get ˜b2 ≤ (1 − θ)b1 + θ(1 − θ)kx1 − x3 k kv 1 − u3 k b1 ≤ (1 − θ)b1 + θ(1 − θ)(ρ + l) + 2M (20) ρ ρl = b1 + 2M. ρ+l
teps-banach.tex; 1/01/1999; 22:11; p.14
15
ε-Enlargements in Banach spaces
The definition of v˜2 combined with (19) yields kv 1
−
v˜2 k
b1 = − ≤θ + 2M ρ 1 2 1 b1 + 2M , ≤ kx − x k ρ ρ θkv 1
u3 k
(21)
where we used the definition of θ in the last inequality. Now consider two cases: (i) ˜b2 ≤ b2 , (ii) ˜b2 > b2 . If (i) holds, by (E2 ), v˜2 ∈ ET (˜b2 , x2 ) ⊆ ET (b2 , x2 ). Then, choosing := v˜2 and using (21) together with (17), (15) follows. b2 < 1 and v 2 := (1 − β)u2 + β˜ v 2 , with In case (ii), define β := ˜b2 u2 ∈ T (x2 ). Because of (E3 ), v 2 ∈ ET (b2 , x2 ) ⊂ ET (b2 , V ). On the other hand, as u2 ∈ T (x2 ), and x2 ∈ V ⊂ U , we obtain, in a similar way as in (19): b1 + 2M . (22) ku2 − v 1 k ≤ ρ v2
Combining the definition of v 2 and (22), we obtain kv 2 − v 1 k ≤ (1 − β)ku2 − v 1 k + βk˜ v2 − v1k b1 b1 ≤ (1 − β) + 2M + βθ + 2M ρ ρ b1 = (1 − β(1 − θ)) + 2M . ρ
(23)
Using (20) we have that β=
b2 b2 ≥ . ˜b2 ρl b1 + 2M ρ+l
Some elementary algebra, the inequality above and the definitions of θ and l, yield ρ(b1 − b2 ) b1 + ρ2M + (ρ + l)b1 + ρl2M (ρ + l)b1 + ρl2M 1 2M 1 ≤ kx1 − x2 k + + |b1 − b2 | . ρ b1 b1
1 − β(1 − θ) ≤ l
(24)
Altogether, with (23), (24) and our assumptions on b1 , b2 , b, b, the conclusion follows.
teps-banach.tex; 1/01/1999; 22:11; p.15
16 As a result from Theorem 3.14, we obtain Lipschitz continuity on any compact set contained in the interior of the domain of T . COROLLARY 3.15. Let ET be an extension of T which satisfies (E1 )− (E3 ). Let K ⊂ D be a compact set. Take b, b ≥ 0 such that 0 < b ≤ b < +∞. Then there exists an open set V , K⊂V ⊂D such that ET is Lipschitz continuous on [b, ¯b] × V , i.e., there exist nonnegative constants A and B such that for any (b1 , x1 ), (b2 , x2 ) ∈ [b, b] × V and v 1 ∈ ET (b1 , x1 ), there exists v 2 ∈ ET (b2 , x2 ) satisfying kv 1 − v 2 k ≤ Akx1 − x2 k + B|b1 − b2 | . Proof. By a compactness argument, the local boundedness of T implies that there exists an open set U such that K ⊂ U ⊂ D and T (U ) is bounded. Using again the compactness of K it follows that for some R > 0, {x ∈ X | d(x, K) < R} ⊂ U. (25) Take ρ = R/2 , V = {x ∈ X | d(x, K) < ρ} . Then, K ⊂ V ⊂ D. Furthermore, {x ∈ X | d(x, V ) < ρ} ⊂ U and we can apply Theorem 3.14.
Acknowledgements We are indebted to the anonymous referees for the corrections they made on the original version of this paper.
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