Uniqueness of the maximal extension of a monotone operator

Nonlinear Anulys~~. Theory, Printed in Great Britain.

Methods & Applications,

Vol. 7. No. 4, pp. 325-332,

1983.

0362%546x/s3/040325~3,8 %03.00/O 0 1983 Pergamon Press Ltd.

UNIQUENESS OF THE MAXIMAL EXTENSION OF A MONOTONE OPERATOR? LIQUN QI$ Computer Sciences Department,

University of Wisconsin-Madison, 53706, U.S.A.

1210 West Dayton Street, Madison. Wisconsin

(Receiued 2 August 1982)

(Dedicated to the memory of Prof. Kuan Chao-Chin) Key words and phrases: Monotone operator,

maximal monotone operator, convexity, normal cone,

1. INTRODUCTION R. ROBERT in [ll]

proved

the following

result.

[Robert]. Let E be a Banach space and A a maximal monotone operator from E into its dual E* and which satisfies: Int(conv dom A) # 0 and cl dom A convex. Let A” be an operator from E into E* which satisfies:

THEOREM

(a) Vx E E, Aox C Ax (b) domA is dense in dom A Then,

for every x in int dom A, Ax = &(E*,E) conv{x*; (x,x*)

Using THEOREM

this result, [Benilan].

P. Benilan Under

gave the following

the above

{(x,x*)EcldomA

hypothesis,

E clExo(~*,E) GrAO}.

(1.1)

theorem: the graph of A is equal

x E*;vy*~A~y,(x*

-y*,x

-y)aO}.

to (1.2)

Benilan did not publish his result, but S. Menou quoted it with its proof in [lo]. H. Attouch [9] also proved these two results for the case when A. is the minimal section of A, and applied them to the study of measurable dependence for a family of maximal monotone operators. In Hilbert spaces, these results can be interpreted as saying that a monotone operator can be uniquely extended to a maximal monotone operator within the closure of the convex hull of its domain if the closure of the domain of the original operator is convex and the interior of the convex hull of its domain is nonempty. In section 2, we give a direct proof of such a theorem without using Robert’s theorem. It is interesting to generalize (1.1) to the whole domain of A. The difficulty is the unboundedness of a maximal monotone operator at its boundary points. In section 3, we prove that a monotone operator is bounded on any open segment connecting an interior point and a boundary point belonging to its domain. As a corollary of this result, we also prove that a maximal monotone operator is bounded in a bounded convex set Pin the interior of its domain t Sponsored by the National Science Foundation under Grant No. MCS7901066, Mod. 2. $ Permanent Address: Department of Applied Mathematics, Tsinghua University, Beijing, The People’s Republic of China. 325

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LIQUN Qr

if and only if its minimal section is defined and bounded in cl P. These results themselves extend the known result about the local boundedness of a monotone operator at an interior point of its domain [2,4]. In section 4, we give a formulation similar to (1.1) but in the whole domain. In section 5, we use the above results to get a new formulation of convergence in the graph sense [9] for sequences of maximal monotone operators. We confine our discussion to Hilbert spaces in most places and to finite-dimensional spaces in some places. We use H to denote a real Hilbert space. We use i and j to denote integers, t and s to denote real numbers, and x, y, z, U, U, w, h, k, p and 9 with or without subscripts to denote vectors in H or R”. We use conv D, cl D and int D to denote the convex hull. the closure and the interior of a set D respectively, dom F to denote the domain of a function F, and 8f to denote the subdifferential of a convex function f. 2. UNIQUENESS

Our first theorem directly. THEOREM

convex

follows.

1. Suppose and nonempty,

OF THE

MAXIMAL

It may be derived

from

EXTENSION

Benilan’s

but we prove

V(u, U) E Graph F}

for all x E cl D. Then F* is the unique Graph F* and dom F* C cl D.

maximal

monotone

it

in H. that D = int dom F is

that F is a set-valued monotone operator and that cl D = cl dom F. Define

F*(x):={yI(y-U,x-u)aO,

theorem,

operator

satisfying

(2.1) Graph

F C

Proof. It is already known that there exists a maximal monotone operator F’ satisfying Graph F C Graph F’ and dom F’ C cl D [2]. By (2. l), any such maximal extension must be contained in F*. Therefore, if we know that F* is monotone, we can conclude that F* is the unique maximal extension of F, satisfying dom F* C cl D. We now prove that F* is monotone. Suppose that (xi, yl), (x2, ~2) E Graph F*. Write u = (xi + x2)/2; then u E cl D. Choose any ~0 E int dom F. Let Ui = u + ti(uo - u), 0 < t; < 1, ti + 0, i 7 1,2,3, . . . Since uo E D, uEclDandD=intdomF-int(clD),weknowthatuiEDCdomF,i= 1,2,3,. . .Therefore, we can choose u; E F(u,), i = 0, 1,2,3, . , and we have

~Yl~Y2~x1~X2~~~YI~U~~xI~x2~+~Y2~Ui~X2~XxI~ =

2(y, - U,,Xr - U) + 2(y2 - uj,x2-

= 2(Yi -

Ul,X]

-

Ui) +

+

2(Yi - 01, ui - u) +

=

2(Yi -

Ui,Xl

-

Uj) +

2(Yzvy2

U)

Uj,X2_Uu,)

-

UI, u, -

2(Y2 -

Ul,X2-

u) U,)

+4(UO-Ui,U,-U)+2(y~+y2-2UO,U,-U) = 2(yl - Ul>Xl - U,) +

2(y2

-

Ui3 X2 -

Ut)

Uniqueness of the maximal extension of a monotone operator

+ 4(fi/(l

327

- ti)) (UO- Ui, UO- Uj) + 2ti(yl + y2 - 2U(),UO- U)

3 2QYl +

y2

-

200,

uo

i = 1) 2,3, . . .

4,

of F and to (2.1). Letting i+

The last inequality is due to the monotonicity (Y1 -y2>x1

-

+ CC,we have

n

-x2)20.

COROLLARY1.1. If the interiors of the domains of two maximal monotone operators are identical and nonempty, and if they have a common single-valued selection in this interior, then these two maximal monotone operators are identical. COROLLARY1.2. If a maximal monotone operator has a cyclically monotone single-valued selection over its domain, then this maximal monotone operator is also cyclically monotone. The proof of corollary 2 follows a line similar to the proof of corollary 2.8, p. 39 of [2]. These two corollaries extend corollary 2.2, p. 29 and corollary 2.8, p. 39 of [2] to more general selections. 3. A BOUNDEDNESS

PROPERTY

It is known that a monotone operator in H is locally bounded at every interior point of its domain, and that a maximal monotone operator must be unbounded at every boundary point of its domain [4]. In the following theorem, we show that a monotone operator is bounded on an open segment between an interior point and a boundary point of its domain. This makes it easier to investigate some properties at a boundary point by approximating it from the interior. THEOREM 2. Suppose that F is a set-valued monotone dom F. Then {Y IY E F(tx + (1 - f)z>,

operator

in H, z E int dom F, x E

O
(3.1)

is bounded. Proof. Suppose that u E F(z), F(tx + (1 - t)z), we have

u E F(x).

Write

h = x - z.

For

0
(u - y, h) = (l/(1 - t))(u - y, x - (r~ + (1 - t)x)) 3 0,

(y - U, h) = (l/t) (y - U, (tx + (1 - t)z) - z) 2 0, (u, h) c (Y, h) 6 h, h). Given any w E H, let k = Aw, A > 0, such that z+kEdomF,z-kEdomF. Choose p E F(z + k), q E F(z - k).

(3.2)

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LIQUN

QI

For 0 < r < 1 and y E F(fx + (1 - t)z), we have Oc(p-y,z+k-@x+(1

-t)z))=(p-y,k-A), (Y,

4 s G

(~3

W - Q> h) + GY>h)

(3.3)

(p, W - t(p, W + lb, h)

s (p>k) + I(P> W

+ I(u, h)l;

0 =z (q - y, z - k - (tx + (1 - t)z)) = (q - y, -k - th), (Y, W 3 (4. k) + f(q, h) - +J, h) 2

(3.4)

(4, 4 + f(q, h) - G> h)

3 (49 k) - lb,

W - l(u, M.

From (3.3) and (3.4), we know that for all y E F(tx + (1 - t)z), 0
FORM OF THE MAXIMAL

EXTENSION

In this section, we confine our discussion to R”. To give another form of the maximal extension of a monotone operator, we use theorem 2 and the fact that the recession cone of the function value set of a maximal monotone operator at each point of its domain is exactly the normal cone of the closure of its domain at that point [6]. THEOREM 3. Suppose that F is a set valued monotone operator in R”, that D = int dom F is nonempty and convex, that cl D = cl dom F and that F* is the unique maximal extension

Uniqueness of the maximal extension of a monotone operator

329

defined by (2.1). Then for all x E cl D, F*(x)

= cl(conv S(x)) + N(x),

(4.1)

where N(x) is the normal cone to cl D at x, and S(x) is the set of all limits of sequences {YilYiE F(tix + (1 - ti>z),

1,

0
ti+1)

for every z E D. Proof. Since F* is closed, we know that S(x) C F*(x). we know that F*(x)

Since F*(x)

3 cl(conv S(x)).

From [6], we know that N(x) is the recession cone of F*(x). F*(x)

is closed and convex,

Therefore,

3 cl(conv S(x)) + N(x).

(4.2)

The opposite inclusion must now be proved. If F*(x) is empty, that is trivial. Suppose F*(x) is nonempty. Just as in the proof of theorem 25.6 of [3], it follows that F*(x)

C cl(conv E) + N(x),

where E is the set of all exposed points of F*(x). F*(x)

To prove

C cl(conv S(x)) + N(x),

(4.3)

it suffices to prove E c S(x).

(4.4)

Given any exposed point w of F*(x), there exists, by definition, a supporting hyperplane to F*(x), which meets F*(x) only at W. Thus there exists a vector h with llhll = 1 such that h is normal to F*(x) at w but not normal to F*(x) at any other points, i.e. (h, w> ’ (h, UA Since N(x) is the recession cone of F*(x), (h, W < 0,

v + w.

Vu E F*(x),

(4.5)

the latter condition on v implies in particular that Vk E N(x),

k # 0.

Hence there does not exist a vector k # 0 such that (u, k) G (x, k) s (x + th, k) for every u E dom F” and every nonnegative number t 2 0. In other words, the half line {x + fh ) t > 0} cannot be separated from dom F * . It follows from theorem 11.3 of [3] that this half line must meet the interior of dom F*. There is a positive number s > 0 such that z=x+shEintdomF*

=intdomF.

From theorem 2, {Y\Y E F(tx + (1 - Qz),

O
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LIQUNQI

is bounded.

There

is thus a convergent

subsequence

{Yjly,EF(tjX+(l-ti)Z),

O
ti+l}

Yi+y*.

According

of F*,

to the closedness

y* E F*(x). From the monotonicity

of F*,

(y; - w, h) = (l/s) (YL - w, = (l/(S(l

Letting

i+

(4.6)

2 - x)

- ti)))(yl-

W, (tjX + (1 -ti)Z)

-X)aO.

CD,tl-+ 1, we have (y* - W, h) 3 0,

i.e.

(4.7)

(w, A) d (y*, h). Comparing

(4.7) with (4.5) and (4.6), we know that y* = w. n

i.e. w E S(x), so (4.4) holds. COROLLARY 3.1. Theorem

3 is still true if S(x) is replaced

by the set s(x):

{X1(X,,y,)~(X,y),(Xiy,)EGraphF,i=1,2,3,...}. Proof. Since F* is closed, we know that

we know that s(x) F*(x)

Again,

since N(x) is the recession

C F*(x).

(4.8) Since F*(X) is closed

3 cl conv s(x).

cone of F*(x), F*(x)

3 cl conv s(X) + N(x).

Since S(x) C s(x), we get the opposite direction from (4.1). n Notice that corollary 3.1 is the same as Robert’s theorem applied 5. SEQUENCES

and convex,

OF MAXIMAL

MONOTONE

to R” if x E int dom F.

OPERATORS

The following results were suggested by H. Attouch. The definition about convergence the graph sense (or, equivalently, in the resolvent sense) may be seen in [9].

in

THEOREM 4. Let A be a maximal monotone operator in H, int(conv dom A) # 0 and cl dom A be convex. Let {A,, i = 0, 1,2, . . .} be a sequence of maximal monotone operators in H, with A, -->A in the graph sense of [9]. Denote their minimal sections by A, and A respectively.

Uniqueness

of the maximal

extension

of a monotone

operator

331

Then for every x in int dom A, Ax = cl conv T(X).

(5.1)

T(x) = {weak-limit Ai

lyi*X,

yi E domAi},

(5.2)

where the closure is also in the weak sense. Proof. According to Robert’s theorem, Ax =

cl conv S(X),

where the closure is in the weak sense (we will not mention this further in the proof) and S(X) =

{weak-limit A(xj)

1xj *

x}.

From theorem 1.1 of [9], we know that Ai + A and therefore {A,(Xj)lLii(Xj)+Ld(Xj)

asi+

there exists a double sequence m}

According to lemma 1.6 of [9], we know that there exists {yiIyl+ x, yi E dom Ai} such that weak-limit Ai (yi) = weak-limit A(Xj). Thus, we have proved Ax C

cl conv T(x).

However, from (5.2) and the fact that Ai+ A, Ai E Ai( is convex and closed in the weak sense, we know that

we see

that

T(X) C Ax.

Since

Ax

Ax 3

Thus, we have proved the theorem. THEOREM

cl conv T(x).

n

5. If H = R” in theorem 4, then for every x in domA, Ax = cl conv

T(X) + N(x),

(5.3)

where T(x) is defined by (5.2) and N(x) is the normal cone to cl domA at x. Proof. This time we use corollary 3.1 instead of Robert’s theorem and follow the same argument. Notice that the conditions of that corollary can be obtained from the hypothesis of theorem 4 by using theorem 0.3 of [lo]. n Acknowledgements-I am grateful to Professor S. Robinson for his guidance. I am also grateful to Professors Benilan, M. Crandall and R. Robert, and to Mr. Zhou Shuzi for their help. I am especially grateful to Professor Brezis and H. Attouch for their valuable comments.

P. H.

REFERENCES 1. ATTOUCH H., On the maximality of the sum of two maximal monotone operators, Nonlinear Analysis, Theory, Method & Applications 5, 143-147 (1981). 2. BREZIS H., OpCrateurs Maximaux Monotones et Semi-Groupes de Contractions dam les &paces de Hilbert, North-Holland/American Elsevier, Amsterdam/N.Y. (1973). 3. ROCKAFELLAR R. T., Convex Analysis, Princeton University Press, Princeton, N.J. (1970). 4. ROCKAFELLAR R. T., Local boundedness of nonlinear, monotone operators, Mich. math. J. 16, 397-407 (1969).

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5. ROCKAFELLAR R. T., On the virtual convexity of the domain and range of a nonlinear maximal monotone operator, Mufh. Ann. 185, 81-90 (1970). Computer Sciences Technical Report No 443, 6. QI L., Complete closedness of maximal monotone operators, University of Wisconsin-Madison (1981), to appear in: Mathematics of Operations Research. 7. REED M. & SIMON B., Functional Analysis, Academic Press, New York (1972). 8. YOSIDA K., Functional Analysis, Springer-Verlag, Berlin (1980). 9. ATTOUCH H., Familles d’operateurs maximaux monotones et mesurabilite, Ann. math. IV, CXX, 35-111 (1979). 10. MENOU S.,Famille mesurable d’operateurs maximaux monotones, These, Besancon (1979). 11. ROBERT R., Contributions a l’analyse non lineaire, These d’Etat, Grenoble (1976).