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The Range of a Monotone Operator
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
199, 176]201 Ž1996.
0135
The Range of a Monotone Operator S. SimonsU Department of Mathematics, Uni¨ ersity of California, Santa Barbara, California 93106-3080 Submitted by William F. Ames Received May 1, 1995
INTRODUCTION The analysis discussed in this paper stems from the following results of Rockafellar Žsee w11, Proposition 1, p. 77; 12, Theorem 2, p. 89; 13, Theorem 1, p. 398x.: THEOREM 1. Let S be a maximal monotone operator on a reflexi¨ e Banach space E and J be the duality map; then: Ža. S q J is surjecti¨ e. Žb. R Ž S . is con¨ ex. Žc. int RŽ S . is con¨ ex. Generalizations of Theorem 1Ža, b. to the nonreflexive case were given by Gossez, with the introduction of ‘‘maximal monotone operators of type ŽD.,’’ and of Theorem 1Žb. by Fitzpatrick and Phelps, with the introduction of ‘‘locally maximal monotone operators.’’ It is known that the subdifferential of a proper convex lower semicontinuous function on a Banach space is both maximal monotone of type ŽD. and locally maximal monotone. However, nothing seems to be known at the moment about the relationship between these two classes of operators. We hope that the analysis presented here will eventually give some insight into this question. Ultimately, Gossez’ proofs used Brouwer’s fixed-point theorem. In Section I we show how Gossez’ results can be proved and indeed generalized using purely functional analytic techniques Žas was, indeed, the case with the proofs of Fitzpatrick and Phelps in w4x.. Our main results in this direction are Theorems 12, 17, and 20. We have couched the analysis of * E-mail: [email protected]. 176 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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177
Section I in functional analytic rather than convex analytic terms; that is, we have not used the concepts of conjugate function or subdifferential. In Section II we show what additional information can be obtained using Brouwer’s fixed-point theorem. In this connection, we refer the reader to Lemmas 5Ža. and 22Ža., both results about continuous convex functions on a simplex and matrices, which have very similar statements but very different proofs. While our main interest is in the nonreflexive case, in Remark 7 we digress and give a complete proof of Theorem 1Ža, b., as stated above, for the reflexive case. As far as we know, the proof of Theorem 1Ža. given in Remark 7 is the simplest and least technical available. It is worth noting that the analysis in Remark 7 is valid for any reflexive space, not only one which has been renormed so that J and Jy1 are single-valued. Some of our later results give more precise information about R Ž S . and int RŽ S . than that given in Theorem 1Žb, c.. Specifically, in Theorem 13 we will exhibit proper, convex lower semicontinuous functions fS : EU ª R j `4 and cS : E ª R j `4 such that R Ž S . s dom fS , int RŽ S . s intŽdom fS ., D Ž S . s dom cS , and int DŽ S . s intŽdom cS .. These representations in terms of the domains of the convex functions were suggested by an analysis of a result of Reich Žsee w10, Lemma 2.1, p. 315x.. Theorem 32 contains some generalizations of Theorem 1Ža., still for reflexive spaces. In Section I, we introduce two new classes of multifunctions from a normed space into its dual, the operators of type ŽWD. ŽWeak D. Žsee Definition 14. and the operators of type ŽNI. Žnegative infimum. Žsee Definition 10.. Maximal monotone operators of type ŽD. are of type ŽWD., and operators of type ŽWD. are of type ŽNI.. Monotone operators of type ŽNI. enjoy the generalization to nonreflexive spaces of Theorem 1Ža. Žsee Theorem 12Ža.., and their natural extensions to the bidual are ‘‘nearly maximal monotone’’ Žsee Theorem 19 for this and equivalent conditions.. For monotone operators of type ŽWD. we can give a precise description of R Ž S . as the closure of the domain of a convex function Žsee Theorem 17.. For monotone operators of type ŽNI. we can give a precise description of R Ž S . and int RŽ S . as the closure and interior of the domain of a convex U function, where S is the operator from EUU into 2 E defined by Gossez Žsee Theorem 20.. In order to explain further the results in Section II, we must elaborate a little on Gossez’ surjectivity result. Gossez proves that if S is maximal monotone of type ŽD. then RŽ S q Ž jU .y1 . s EU , where j: E ª R is defined by jŽ x . [ 5 x 5 2r2. In Section II, we prove an analogous result for monotone operators of type ŽNI. for RŽ S q Ž f U .y1 ., where f is a real continuous convex function defined on some convex open subset of E Žsee Theorem 30..
178
S. SIMONS
There are two unusual features about the way in which we have presented our analysis. The first of these is the use of matrices in Lemmas 5Ža. and 22. We have done this not because we think that these two results have any particular merit as results about matrices, but rather because the concise matrix notation enables us to make the comparisons referred to above a little more easily. The second unusual feature is our use of minimax theorems. In fact, we could have used the Hahn]Banach theorem instead, but minimax theorems seem the most natural idiom for the description of the manipulations we perform. More specifically, we shall use the following classical minimax theorem, which can be deduced from more general results of Fan Žsee w3x. or Sion Žsee w14x.. Fan’s proof used a separation theorem for sets in finite dimensional spaces, and Sion’s proof used the KKM theorem, but Theorem 2 can easily be proved without any functional analysis or fixed-point related concepts. See, for instance, the proof of Sion’s theorem given by Komiya in w7x. THEOREM 2. Let X and Y be nonempty compact con¨ ex subsets of topological ¨ ector spaces. Let f : X = Y ª R be Ž separately. conca¨ e and upper semicontinuous on X and con¨ ex and lower semicontinuous on Y. Then max min f s min max f . X
Y
Y
X
Notation 3. Let E be a real normed space. If m G 1, let
sm [ a s Ž a1 , . . . , a m . : a1 , . . . , a m G 0, a1 q ??? qam s 1 4 ; R m . The author thanks Professor Robert Phelps for reading through two earlier versions of this paper and making a number of suggestions that have improved the exposition substantially. He also thanks Professors Jon Borwein and Simeon Reich for giving him some very helpful references.
I. WITHOUT FIXED-POINT THEOREMS DEFINITION 4. We say that an m = m matrix Q is copositi¨ e if a g sm « aT Qa G 0. Symmetric copositive matrices were introduced by Hall and Newman in w6x. Copositivity has, in fact, been treated in more general situations in optimization theory. See Borwein’s discussion in w1x of copositivity as it relates to the linear complementarity problem. LEMMA 5. Ža.
Let Q [ qi j 4 be a copositi¨ e m = m matrix.
Let g: sm ª R be continuous and con¨ ex. Then max min g Ž a . y g Ž b . q aT Qb G 0.
bg sm ag sm
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RANGE OF A MONOTONE OPERATOR
Žb.
Let y 1 , . . . , ym g E. Then there exist b g sm and xU g EU such
that 2
for all i s 1, . . . , m,
Ý bj y j
q 5 xU 5 2 F 2² yi , xU : q
j
Ý q i j bj . j
Proof. Ža. This follows from Theorem 2 with X [ Y [ sm since min max g Ž a . y g Ž b . q aT Qb G min g Ž a . y g Ž a . q aT Qa G 0.
ag sm bg sm
ag sm
Žb. Let M [ 5 y 1 5 k ??? k 5 ym 5 and B [ xU : xU g EU , 5 xU 5 F M 4 with the topology w Ž EU , E .. Define an affine map L: sm ª E by LŽ a. [ a1 y 1 q ??? qa m ym . From Ža. with g Ž a. [ 5 LŽ a.5 2 , max min 5 L Ž a . 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
bg sm ag sm
From the Hahn]Banach theorem, for all a g sm , there exists xU g EU such that ² LŽ a., xU : s 5 xU 5 2 s 5 LŽ a.5 2 , from which 5 LŽ a.5 2 s 2² LŽ a., xU : y 5 xU 5 2 . Since 5 LŽ a.5 F M, xU g B. Thus max min max 2² L Ž a . , xU : y 5 xU 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
bg sm ag sm x UgB
For each a g sm , the map xU ª 2² LŽ a., xU : y 5 xU 5 2 is concave and upper semicontinuous on B and, for all xU g B, the map a ª 2² LŽ a., xU : q aT Qb is affine and continuous on sm ; hence, from Theorem 2 with X [ B and Y [ sm , max max min 2² L Ž a . , xU : y 5 xU 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
bg sm x UgB ag sm
Thus there exist b g sm and xU g EU such that for all a g sm ,
2² L Ž a . , xU : y 5 xU 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
If we now let a run through the vertices of sm , we obtain for all i s 1, . . . , m,
2² yi , xU : q
Ý qi j bj y 5 L Ž b . 5 2 y 5 xU 5 2 G 0, j
as required. LEMMA 6.
Let F be a nonempty finite monotone subset of E = EU and NF [
min
Ž y , y U .gF
2 Ž 5 y 5 q 5 yU 5 . .
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S. SIMONS
Then there exists Ž x F , xUF . g E = EU such that 5 x F 5 F NF , 5 xUF 5 F NF and for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 F 2² y, xUF : q 2² y y x F , yU : .
Ž 6.1. U .4 . Define the matrix Q by qi j [ Proof. Let F s Ž y 1 , yU1 ., . . . , Ž ym , ym U 2² yi y y j , yi :. Then Q is copositive since, for all a g sm ,
aT Qa s
Ý ai qi j a j q Ý ai qi j a j s Ý ai qi j a j q Ý a j q ji ai i-j
s
j-i
i-j
i-j
Ý ai Ž qi j q q ji . a j s Ý 2 ai ² yi y yj , yiU y yUj : a j G 0. i-j
i-j
From Lemma 5Žb., there exist b g sm and xUF g EU such that 2
for all i s 1, . . . , m,
Ý bj y j
q 5 xUF 5 2
j
F 2² yi , xUF : q
Ý 2² yi y yj , yiU : bj , j
that is, 2
for all i s 1, . . . , m,
Ý bj y j
q 5 xUF 5 2
j
F 2² yi , xUF : q 2² yi y
Ý bj yj , yiU : . j
Thus Ž6.1. follows with x F [ Ý j bj y j . It remains to prove that 5 x F 5 F NF and 5 xUF 5 F NF . From Ž6.1., for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 y 2 5 y 5 5 xUF 5 y 2 5 x F 5 5 yU 5 F 2 5 y 5 5 yU 5 ,
from which for all Ž y, yU . g F ,
Ž 5 x F 5 y 5 yU 5 .
2
q Ž 5 xUF 5 y 5 y 5 . F Ž 5 y 5 q 5 yU 5 . . 2
2
It follows easily from this that 5 x F 5 F NF and 5 xUF 5 F NF . This completes the proof of Lemma 6. Remark 7 Ž Proof of Theorem 1Ža, b... As promised in the Introduction, we now digress and give a complete proof of Theorem 1Ža, b.. So let E be U reflexive, S: E ª 2 E be maximal monotone, and zU g EU . Define T :
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181
U
E ª 2 E by T [ S y zU . Then T is maximal monotone. From Lemma 6, for all finite subsets F of GŽT ., there exists Ž x F , xUF . g E = EU such that 5 x F 5 F NF , 5 xUF 5 F NF , and for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 F 2² y, xUF : q 2² y y x F , yU : .
Put another way, the set
F Ž x, xU . : Ž x, xU . g E = EU , 5 x 5 F NF , 5 xU 5 F NF ,
Ž y , y U .gF
5 x 5 2 q 5 xU 5 2 F 2² y, xU : q 2² y y x, yU : 4 is nonempty. As F runs, these sets are w Ž E, EU . = w Ž EU , E .-compact and directed downwards; hence their intersection is nonempty. It follows that there exists Ž x, xU . g E = EU such that, for all Ž y, yU . g GŽT ., 5 x 5 2 q 5 xU 5 2 F 2² y, xU : q 2² y y x, yU : . Thus, for all Ž y, yU . g GŽT ., 5 x 5 2 q 5 xU 5 2 y 2² x, xU : F 2² y y x, yU q xU : .
Ž 7.1.
Since 5 x 5 2 q 5 xU 5 2 y 2² x, xU : G 5 x 5 2 q 5 xU 5 2 y 2 5 x 5 5 xU 5 s Ž 5 x 5 y 5 xU 5 . G 0, 2
it follows from the definition of maximal monotonicity that yxU g Tx. Putting Ž y, yU . s Ž x, yxU . in Ž7.1., we find in particular that 5 x 5 2 q 5 xU 5 2 y 2² x, xU : F 0.
Ž 7.2.
Consequently, Ž5 x 5 y 5 xU 5. 2 F 0, from which 5 xU 5 s 5 x 5. Substituting this back into Ž7.2., ² x, xU : s 5 x 5 2 ; hence xU g Jx. Since yxU g Tx, zU y xU g Sx. Thus zU s Ž zU y xU . q xU g Ž S q J . x. Since zU was an arbitrary element of EU , S q J is surjective. This completes the proof of Theorem 1Ža.. In order to establish Theorem 1Žb., it suffices to prove that xU1 , xU2 g R Ž S . «
xU1 q xU2 2
g RŽ S . .
182
S. SIMONS
So let xU1 , xU2 g RŽ S .. Choose x 1 , x 2 g E such that Ž x 1 , xU1 ., Ž x 2 , xU2 . g U GŽ S .. Let l ) 0. Define T : E ª 2 E by Tx [
1
l
S xq
ž
x1 q x 2 2
/
y
xU1 q xU2 2l
.
T is maximal monotone and if
Ž u, uU . [
ž
x1 y x 2 2
,
xU1 y xU2 2l
/
,
then "Ž u, uU . g GŽT .. Arguing as above, there exists Ž x, xU . g E = EU such that yxU g Tx and 5 xU 5 2 s ² x, xU :. Since "Ž u, uU . g GŽT ., ² " u y x, " uU q xU : G 0. Taking the average of the expressions with the two signs, ² u, uU : y ² x, xU : G 0, from which 5 xU 5 2 s ² x, xU : F ² u, uU :. Thus 5lx
U
U
5 F 'l ² u, u 2
: s
(
l² x 1 y x 2 , xU1 y xU2 : 4
.
Since xU1 q xU2 2
gS xq
ž
x1 q x 2 2
/
q l xU ; R Ž S . q l xU ,
it follows by letting l ª 0 that xU1 q xU2 2
g RŽ S . .
This completes the proof of Theorem 1Žb.. We point out a crucial technical fact in the existence proof in Remark 7: the term 2² x, xU : appears after the compactness argument. The reason for this is that the function
Ž x, xU . ª 5 x 5 2 q 5 xU 5 2 y 2² y, xU : y 2² y y x, yU : is lower semicontinuous with respect to w Ž E, EU . = w Ž EU , E ., while the function Ž x, xU . ª ² x, xU : is not. Similar comments can be made about the proofs of Lemmas 9 and 29. DEFINITION 8. If y g E, we write ˆ y for the canonical image of y in the U U bidual EUU of E. Let T : E ª 2 E . We define T : EUU ª 2 E by xU g TxUU
m
inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : G 0.
RANGE OF A MONOTONE OPERATOR
183
The definition of T given above is due to Gossez}see w9, Section 3x for an exposition. GŽT . consists of those elements of EUU = EU that are ‘‘monotonically related’’ to every element of GŽT .. U
LEMMA 9. Let T : E ª 2 E be monotone. Then there exists Ž xU , xUU . g E = EUU such that yxU g TxUU and U
5 xUU 5 2 q 5 xU 5 2 y 2² xU , xUU : F
inf
Ž y , y U .gG ŽT .
2² yU q xU , ˆ y y xUU : . Ž 9.1.
Proof. From Lemma 6, for all finite subsets F of GŽT ., there exists Ž x F , xUF . g E = EU such that 5 x F 5 F NF , 5 xUF 5 F NF , and for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 F 2² y, xUF : q 2² y y x F , yU : . $
UU 5 s 5 x F 5 F NF and We now define xUU . Then 5 xUU F [ x Fg E F
for all Ž y, yU . g F ,
5 xUU 5 2 q 5 xUF 5 2 F 2² xUF , ˆ :. y : q 2² yU , ˆ y y xUU F F
Put another way, the set
F U
Ž y , y .gF
½Ž x
U
, xUU . : Ž xU , xUU . g EU = EUU , 5 xU 5 F NF , 5 xUU 5 F NF , 5 xUU 5 2 q 5 xU 5 2 F 2² xU , ˆ y : q 2² yU , ˆ y y xUU :
5
is nonempty. As F runs, these sets are w Ž EU , E . = w Ž EUU , EU .-compact and directed downwards; hence their intersection is nonempty. It follows that there exists Ž xU , xUU . g EU = EUU such that for all Ž y, yU . g G Ž T . ,
5 xUU 5 2 q 5 xU 5 2 F 2² xU , ˆ y : q 2² yU , ˆ y y xUU : .
Inequality Ž9.1. is immediate from this. Since 5 xUU 5 2 q 5 xU 5 2 y 2² xU , xUU : G 0, it follows from Ž9.1. that yxU g TxUU . This completes the proof of Lemma 9. Motivated by the analysis in Remark 7, we now introduce a new class of multifunctions. U
DEFINITION 10. Let T : E ª 2 E . We shall say that T is of type Ž NI . if for all Ž xUU , xU . g EUU = EU ,
inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : F 0.
184
S. SIMONS
The concept of type ŽNI. is defined for operators that are not necessarily monotone though our main result ŽTheorem 12. is about monotone operators of type ŽNI.. A monotone operator of type ŽNI. is not necessarily maximal monotone }let E [ R, T Žy1. [ Žy`, 0x, T 1 [ w0, `. and Tx [ B otherwise. ŽT is of type ŽNI. because RŽT . s R s EU .. Before proving our main result about operators of type ŽNI., we introduce a function on EU that is defined for any operator S. U
DEFINITION 11. If S: E ª 2 E , we define fS : EU ª R j `4 by U
fS Ž w . [
² wU y ¨ U , ¨ UU :
sup
1 q 5¨U 5
Ž ¨ UU , ¨ U .gG Ž S .
.
Ž 11.1.
We see that fS is clearly convex and lower semicontinuous. The function fS is closely related to the function introduced by Borwein and Fitzpatrick in their proof of the local boundedness of monotone operators on Banach spaces Žsee w2, Theorem 2, p. 440x.. THEOREM 12.
U
Let S: E ª 2 E be of type Ž NI . and monotone. Then:
Ža. For all zU g EU and l ) 0, there exists Ž xU , xUU . g EU = EUU such that zU y l xU g SxUU Žb.
5 xUU 5 2 s 5 xU 5 2 s ² xU , xUU : . Ž 12.1.
and
RŽ S . ; dom fS .
Žc. dom fS ; R Ž S . . Žd. intŽdom fS . ; RŽ S .. U
Proof. Ža. Define T : E ª 2 E by T [ Ž S y zU .rl. Then T is of type ŽNI. and monotone. From Lemma 9, there exists Ž xU , xUU . g EU = EUU such that yxU g TxUU s
SxUU y zU
l
and
5 xUU 5 2 q 5 xU 5 2 y 2² xU , xUU : F 0.
Ž 12.2.
Consequently, Ž5 xUU 5 y 5 xU 5. 2 F 0, from which 5 xUU 5 s 5 xU 5. Substituting this back into Ž12.2., ² xU , xUU : s 5 xU 5 2 , as required. This completes the proof of Ža.. Žb. Let wU g RŽ S .. Choose w g E so that Ž w, wU . g GŽ S .. Let
b [ ² w, wU : k 5 w 5 .
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185
It follows from the definition of S that, for all Ž ¨ UU , ¨ U . g GŽ S ., ² wU y ¨ U , ¨ UU : F ² wU y ¨ U , w ˆ : F ² wU , w ˆ : q 5 w 5 5 ¨ U 5 F b Ž 1 q 5 ¨ U 5. . Thus fS Ž wU . F b , from which wU g dom fS . This completes the proof of Žb.. Žc. Let zU g dom fS . Put
g [ < fS Ž zU . <
d [ g Ž 1 q 5 zU 5 . .
and
Let l ) 0. From Ža., there exists Ž xU , xUU . g EU = EUU satisfying Ž12.1.. Putting
Ž ¨ UU , ¨ U . [ Ž xUU , zU y l xU . g G Ž S . and using Ž11.1., ² l xU , xUU : 1 q 5¨U 5
s
² zU y ¨ U , ¨ UU : 1 q 5¨U 5
F fS Ž zU . F g ;
hence
l2 ² xU , xUU : F gl Ž 1 q 5 ¨ U 5 . F glŽ 1 q 5 zU 5 q 5 ¨ U y zU 5 . F dlŽ 1 q 5 ¨ U y zU 5 . . Combining this with Ž12.1., 5 ¨ U y zU 5 2 s l2 5 xU 5 2 F dl Ž 1 q 5 ¨ U y zU 5 . . Completing the square, 5 ¨ U y zU 5 F
dl q 'd 2l2 q 4dl 2
.
Since ¨ U g RŽ S ., it follows by letting l ª 0 that zU g R Ž S . . This completes the proof of Žc.. Žd. Let zU g intŽdom fS .. From Žc., for all l ) 0, there exists UU Ž ¨l , ¨lU . g GŽ S . such that 5 ¨lU y zU 5 F
dl q 'd 2l2 q 4dl 2
² zU y ¨lU , ¨lUU : s l 5 ¨lUU 5 2 G 0.
and
Ž 12.3.
186
S. SIMONS
We first prove that lim sup 5 ¨lUU 5 - `.
Ž 12.4.
lª0
Since zU g intŽdom fS ., it follows from w8, Proposition 3.3, p. 39x that there exist h , P ) 0 such that tU g EU and 5 tU 5 F h
«
fS Ž zU q tU . F P.
ŽThis is the so-called ‘‘local boundedness property’’.. Let l ) 0, and tU g EU with 5 tU 5 F h. Then, since Ž ¨lUU , ¨lU . g GŽ S ., ² zU q tU y ¨lU , ¨lUU : 1 q 5 ¨lU 5
F fS Ž zU q t U . F P ;
hence ² tU , ¨lUU : q ² zU y ¨lU , ¨lUU : F P Ž 1 q 5 ¨lU 5 . , and thus, from Ž12.3., ² tU , ¨lUU : F P 1 q 5 zU 5 q
ž
dl q 'd 2l2 q 4dl
/
2
.
Taking the supremum over tU ,
h 5 ¨lUU 5 F P 1 q 5 zU 5 q
ž
dl q 'd 2l2 q 4dl 2
/
,
from which it follows that lim sup 5 ¨lUU 5 F lª0
P Ž 1 q 5 zU 5 .
h
,
which gives Ž12.4.. Let zUU be a w Ž EUU , EU .-cluster point of the net ¨lUU . Then, using the fact that Ž ¨lUU , ¨lU . g GŽ S ., for all Ž y, yU . g GŽ S ., ² yU y z U , ˆ y y zUU : s ² yU y ¨lU , ˆ y y ¨lUU : q ² ¨lU y zU , ˆ y y ¨lUU : q ² yU y zU , ¨lUU y zUU : G ² ¨lU y zU , ˆ y y ¨lUU : q ² yU y zU , ¨lUU y zUU : G y5 ¨lU y zU 5 Ž 5 y 5 q 5 ¨lUU 5 . q ² yU y zU , ¨lUU y zUU : .
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RANGE OF A MONOTONE OPERATOR
Letting l ª 0 and using Ž12.3. and Ž12.4., ² yU y z U , ˆ y y zUU : G 0. Since this holds for all Ž y, yU . g GŽ S ., Ž zUU , zU . g GŽ S .. Thus zU g RŽ S .. This completes the proof of Žd.. Specializing to reflexive spaces, we obtain the following result: THEOREM 13. Define
U
Let E be reflexi¨ e and S: E ª 2 E be maximal monotone.
fS : EU ª R j ` 4
cS : E ª R j ` 4
and
by U
fS Ž w . [
² ¨ , wU y ¨ U :
sup
Ž ¨ , ¨ U .gG Ž S .
cS Ž w . [
sup
and
1 q 5¨U 5 ²w y ¨ , ¨U :
Ž ¨ , ¨ U .gG Ž S .
1 q 5¨ 5
.
Then R Ž S . s dom fS ,
int R Ž S . s int Ž dom fS . ,
D Ž S . s dom cS ,
int D Ž S . s int Ž dom cS . .
In particular, R Ž S . , int RŽ S ., D Ž S . , and int DŽ S . are con¨ ex. Proof. Since S s S in this case, it follows from Theorem 12 that R Ž S . ; dom fS ,
dom fS ; R Ž S . ,
int Ž dom fS . ; R Ž S . .
This gives the first two assertions. The second two follow by replacing S by Sy1 . U
DEFINITION 14. Let T : E ª 2 E . We say that T is maximal monotone of type Ž D . if T is maximal monotone and, for all xU g TxUU , there exists a $ U bounded net Ž ya , ya . in GŽT . such that yaª xUU in w Ž EUU , EU . and 5 yaU y xU 5 ª 0. The definition of maximal monotone of type Ž D . is also due to Gossez}again, see w9, Section 3x for an exposition. We shall say that T is of type ŽWD . if, for all xU g RŽT ., there exists a bounded net Ž ya , yaU . in GŽT . such that 5 yaU y xU 5 ª 0. If T is maximal monotone of type ŽD. then T is clearly of type ŽWD. }we conjecture that the converse of this statement is false. LEMMA 15.
U
If T : E ª 2 E is of type ŽWD. then T is of type ŽNI..
188
S. SIMONS
Proof. Suppose that T is not of type ŽNI.. Then there exists Ž xUU , xU . g EUU = EU such that inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : ) 0.
Ž 15.1.
Consequently, xU g TxUU . By hypothesis, there exists a$ bounded net Ž ya , yaU . in GŽT . such that 5 yaU y xU 5 ª 0. Then ² yaU y xU , ya y xUU : ª 0, which would contradict Ž15.1.. We conjecture that there exists a maximal monotone operator of type ŽNI. that is not of type ŽWD.. Our next result generalizes a result of Gossez}see w9, Theorem 3.6, p. 21x for an exposition. ŽGossez’ original result, which has slightly stronger hypotheses, appears in w5, Theoreme ´ ` 4.1, p. 379x.. U
COROLLARY 16. Let S: E ª 2 E be of type ŽWD. and monotone, and U z g EU . Then there exists Ž xU , xUU . g EU = EUU such that zU y xU g SxUU and 5 xUU 5 2 s 5 xU 5 2 s ² xU , xUU :. Proof. This is immediate from Theorem 12 and Lemma 15. THEOREM 17.
U
Let S: E ª 2 E be of type ŽWD. and monotone. Then R Ž S . s dom fS .
In particular, R Ž S . is con¨ ex. Proof. It follows from Lemma 15 and Theorem 12Žb, c. that R Ž S . ; dom fS ; R Ž S . . However, since S is of type ŽWD., RŽ S . ; R Ž S . ; thus R Ž S . ; dom fS ; R Ž S . s R Ž S . . This completes the proof of Theorem 17. Let T : R ª 2 R be as in the comment following Definition 10, and S [ Ty1 . Then S is of type ŽNI. and monotone, and RUŽ S . s y1, 14 is not convex. It is attractive to conjecture that if S: E ª 2 E is of type ŽNI. and maximal monotone then R Ž S . is convex. From the argument given above, U it would be sufficient to prove that if S: E ª 2 E is of type ŽNI. and maximal monotone then RŽ S . ; R Ž S . . Theorem 17 is not a complete generalization of Theorem 13, since it does not give an exact formula for int RŽ S .. In this sense, the ‘‘correct’’ generalization of Theorem 13 to nonreflexive spaces is probably Theorem
189
RANGE OF A MONOTONE OPERATOR
20 below, which is valid for operators of type ŽNI., but treats RŽ S . rather than RŽ S .. In order to prove it, we will need a strengthening of Theorem 12Žb. that uses the implication Ža. « Žb. from Theorem 19. U
U
DEFINITION 18. If T : E ª 2 E , we define Tˆ: EUU ª 2 E so that G Ž Tˆ . [ Ž ˆ y, yU . : Ž y, yU . g G Ž T . 4 ; EUU = EU . It is easy to see that if T is monotone then G Ž Tˆ . ; G Ž T . . THEOREM 19. Ž d . m Ž e ..
Ž 18.1.
U
Let T : E ª 2 E be monotone. Then Ž a. « Ž b . m Ž c . m
Ža. T is of type ŽNI.. U Žb. T : EUU ª 2 E is monotone. U Žc. T : EUU ª 2 E is maximal monotone. Žd. GŽT . is the unique maximal monotone subset of EUU = EU containing GŽTˆ.. Že. There exists a unique maximal monotone subset of EUU = EU containing GŽTˆ.. Proof. Ža. « Žb. Let Ž ¨ UU , ¨ U ., Ž wUU , wU . g GŽT . and put
Žx
UU
U
,x .[
Ž ¨ UU , ¨ U . q Ž wUU , wU . 2
g EUU = EU .
If Ž y, yU . g GŽT . then, by direct computation, 4² yU y xU , ˆ y y xUU : q ² ¨ U y wU , ¨ UU y wUU : s 2² yU y ¨ U , ˆ y y ¨ UU : q 2² yU y wU , ˆ y y wUU : .
Ž 19.1.
Since ¨ U g T¨ UU and wU g TwUU , 4² yU y xU , ˆ y y xUU : q ² ¨ U y wU , ¨ UU y wUU : G 0. Taking the infimum over Ž y, yU . g GŽT . and using the fact that T is of type ŽNI., ² ¨ U y wU , ¨ UU y wUU : G 0. This establishes Žb..
190
S. SIMONS
Žb. « Žc. Let Ž zUU , zU . g EUU = EU and inf
² xU y zU , xUU y zUU : G 0.
inf
² xU y zU , xUU y zUU : G 0,
Ž x UU , x U .gG ŽT .
From Ž18.1., Ž x UU , x U .gG ŽTˆ.
that is to say, inf
Ž y , y U .gG ŽT .
² yU y z U , ˆ y y zUU : G 0.
Consequently, zU g TzUU . From Žb., T is monotone. Thus we have established that T is maximal monotone, giving Žc.. Žc. « Žd. We know from Žc. that GŽT . is a maximal monotone subset of EUU = EU and, from Ž18.1. again, GŽT . > GŽTˆ.. Let M be any maximal monotone subset of EUU = EU such that M > GŽTˆ.. Let Ž xUU , xU . g M. For all Ž y, yU . g GŽT ., Ž ˆ y, yU . g GŽTˆ. ; M. Since M is monotone, U U UU ²y yx , ˆ y y x : G 0. Thus inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : G 0;
in other words, Ž xUU , xU . g GŽT .. So we have proved that M ; GŽT .. Since M is maximal monotone and GŽT . is monotone, M s GŽT .. This gives Žd.. Žd. « Že. This is immediate. Že. « Žb. Suppose that Žb. fails. Then there exists Ž xUU , xU ., Ž yUU , yU . g GŽT . such that ² xU y yU , xUU y yUU : - 0. Since Ž xUU , xU . g GŽT ., GŽTˆ. j Ž xUU , xU .4 is a monotone set. From Zorn’s lemma, there is a maximal monotone subset B of EUU = EU such that B > GŽTˆ. j Ž xUU , xU .4 . Since B is monotone, Ž xUU , xU . g B and ² xU y yU , xUU y yUU : - 0, Ž yUU , yU . f B. We can perform exactly the same argument with Ž xUU , xU . replaced by Ž yUU , yU . and find a maximal monotone subset C of EUU = EU such that C > GŽTˆ. j Ž yUU , yU .4 . Since Ž yUU , yU . g C R B, C / B. This shows that Že. fails. We now give our promised generalization of Theorem 13 to nonreflexive spaces.
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RANGE OF A MONOTONE OPERATOR
THEOREM 20.
U
Let S: E ª 2 E be of type ŽNI. and monotone. Then
R Ž S . s dom fS
and
int R Ž S . s int Ž dom fS . .
In particular, R Ž S . and int RŽ S . are con¨ ex. Proof. Let wU g RŽ S .. Choose wUU g EUU so that Ž wUU , wU . g GŽ S .. Let
b [ ² wU , wUU : k 5 wUU 5 . It follows from Theorem 19Ža. « Žb. that, for all Ž ¨ UU , ¨ U . g GŽ S ., ² wU y ¨ U , ¨ UU : F ² wU y ¨ U , wUU : F ² wU , wUU : q 5 wUU 5 5 ¨ U 5 F b Ž 1 q 5 ¨ U 5. . Thus fS Ž wU . F b , from which wU g dom fS . So we have proved that R Ž S . ; dom fS .
Ž 20.1.
Combining Ž20.1. with Theorem 12Žc., we have R Ž S . ; dom fS ; R Ž S . and, combining Ž20.1. with Theorem 12Žd., we have int R Ž S . ; int Ž dom fS . ; int R Ž S . . This completes the proof of Theorem 20. It follows from Lemma 15 that Theorem 19 is a strengthening of Gossez’ result that if T is maximal monotone of type ŽD., then T is maximal monotone. The equivalence of Žb. ] Že. in Theorem 19 is actually a special case of more general situation. Let P / B and M ; P = P be a symmetric set such that M > Ž p, p . : p g P 4 . If G ; P, we write G M [ p : p g P, p4 = G ; M 4 and say that G is an M-set if G = G ; M . We can then prove, exactly as in the proof of Theorem 19 above, that the following four conditions are equivalent: G M is an M-set. G M is a maximal M-set. G M is the unique maximal M-set containing G . There is a unique maximal M-set containing G . ŽThe symbol ‘‘ P ’’ is meant to suggest the phrase ‘‘product space,’’ and the symbol ‘‘ M ’’ is meant to suggest the phrase ‘‘monotonically related.’’.
192
S. SIMONS
It is interesting to compare the techniques used in Remark 7 and Theorem 19. Suppose, for simplicity, that Ž wUU , wU . s yŽ ¨ UU , ¨ U . in the latter, so that Ž xUU , xU . s 0. Then Ž19.1. becomes 4² yU , ˆ y : q ²2 ¨ U , 2 ¨ UU : s 2² yU y ¨ U , ˆ y y ¨ UU : q 2² yU q ¨ U , ˆ y q ¨ UU : , that is, ² yU , ˆ y : q ² ¨ U , ¨ UU : s
² yU y ¨ U , ˆ y y ¨ UU : q ² yU q ¨ U , ˆ y q ¨ UU : 2
.
After some change of symbols, this is exactly the ‘‘averaging argument’’ used in the last part of Remark 7.
II. WITH FIXED-POINT THEOREMS Notation 21. Let C be a nonempty convex subset of a vector space, and f : C ª R be convex. If x, y g C then the directional deri¨ ati¨ e of f at y in the direction x y y is given by dq f Ž y . Ž x y y . [
inf
f Ž y q u Ž x y y. . y f Ž y.
u
u g Ž0, 1 x
s lim
f Ž y q u Ž x y y. . y f Ž y.
u
uª0q
.
Lemma 22Ža. generalizes Lemma 5Ža.. We do not know if the use of Brouwer’s fixed-point theorem in its proof can be avoided. If Q is symmetric and we choose b g sm to minimize g Ž b . q bT Qbr2 then, for all a g sm and u g Ž0, 1x, g Ž b q u Ž a y b. . y g Ž b. q
T Ž b q u Ž a y b . . Q Ž b q u Ž a y b . . y bT Qb
2
G 0.
On dividing by u and letting u ª 0, we obtain q
d g Ž b. Ž a y b. q
T Ž a y b . Qb q bT Q Ž a y b .
2
G 0.
Thus Lemma 22Žb. follows from the symmetry of Q. Unfortunately, in our application of Lemma 22Žb., Q is not generally symmetric. Theorem 2 cannot be used directly since the function b ª g Ž c . q c T Qb y g Ž b . y bT Qb may fail to be concave Žtake g [ 0 and Q [ yI ..
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RANGE OF A MONOTONE OPERATOR
LEMMA 22. Let Q be any m = m matrix. Let g: sm ª R be continuous and con¨ ex. Then: max min g Ž c . y g Ž b . q c T Qb y bT Qb s 0.
Ž a. Ž b.
bg sm cg sm
max inf
bg sm ag sm
dq g Ž b . Ž a y b . q aT Qb y bT Qb s 0.
Proof. Since max min g Ž c . y g Ž b . q c T Qb y bT Qb
bg sm cg sm
F max g Ž b . y g Ž b . q bT Qb y bT Qb s 0, bg sm
if Ža. fails then, for all b g sm , there exists c b g sm such that g Ž c b . y g Ž b . q c Tb Qb y bT Qb - 0, and hence there exists an open neighborhood Nb of b in sm such that a g Nb « g Ž c b . y g Ž a . q c Tb Qa y aT Qa - 0. Since sm is compact, there exist b1, . . . , b n g sm such that sm s Nb 1 j ??? j Nb n . Let p1 , . . . , pn be a partition of unity on sm subordinate to this open cover. The map a ª Ý k pk Ž a. c b k is a continuous self-map of sm ; hence, from Brouwer’s fixed point theorem, there exists a g sm such that as
Ý pk Ž a . c b
k
.
Ž 22.1.
k
Now k g 1, . . . , n4 and pk Ž a . ) 0
«
g Ž c b k . y g Ž a . q c Tb k Qa y aT Qa - 0;
consequently,
Ý pk Ž a. g Ž c b . y g Ž a. q cTb Qa y aT Qa k
k
- 0.
k
It follows from Ž22.1. and the convexity of g that g Ž a . y g Ž a . y aT Qa y aT Qa - 0, which is impossible. This contradiction completes the proof of Ža..
194
S. SIMONS
From Ža., there exists b g sm such that for all c g sm ,
g Ž c . y g Ž b . q c T Qb y bT Qb G 0.
Ž 22.2.
Let a g sm . For all u g Ž0, 1x, we substitute c [ b q u Ž a y b . into Ž22.2. and obtain for all u g Ž 0, 1 ,
g Ž b q u Ž a y b . . y g Ž b . q u Ž aT Qb y bT Qb . G 0.
Now Žb. follows by dividing by u and then letting u ª 0 since max inf
bg sm ag sm
dq g Ž b . Ž a y b . q aT Qb y bT Qb
F max dq g Ž b . Ž b y b . q bT Qb y bT Qb s 0. bg sm
Notation 23. We suppose for the rest of this paper that U is a nonempty convex open subset of E and that f : U ª R is continuous and convex. THEOREM 24. Let Q be an m = m matrix and y 1 , . . . , ym g U. Then there exist b g sm and xU g f ŽÝ j bj y j . such that for all i s 1, . . . , m,
¦Ý j
bj y j , xU q bT Qb F ² yi , xU : q
;
Ý q i j bj . j
Proof. As in the proof of Lemma 5Žb., define L: sm ª E by LŽ c . [ c1 y 1 q ??? qc m ym . From Lemma 22Žb. with g [ f ( L, max inf
bg sm ag sm
dq f Ž L Ž b . . Ž L Ž a . y L Ž b . . q aT Qb y bT Qb s 0.
Since f is continuous in the neighborhood of LŽ b ., it follows from Chapter 1 of w8x and the Hahn]Banach theorem that for all y g E,
dq f Ž L Ž b . . Ž y . s
max
z Ug f Ž L Ž b ..
² y, zU : .
Thus, max inf
max
bg sm ag sm z Ug f Ž L Ž b ..
² L Ž a . y L Ž b . , zU : q aT Qb y bT Qb s 0.
Since f Ž LŽ b .. is w Ž EU , E .-compact, it follows from Theorem 2 with X [ f Ž LŽ b .. and Y [ sm that max
max
min ² L Ž a . y L Ž b . , zU : q aT Qb y bT Qb s 0.
bg sm z Ug f Ž L Ž b .. ag sm
195
RANGE OF A MONOTONE OPERATOR
Thus there exist b g sm and xU g f Ž LŽ b .. such that for all a g sm ,
² L Ž a . y L Ž b . , xU : q aT Qb y bT Qb G 0.
The result now follows as in Lemma 5Žb. by letting a run through the vertices of sm . Theorem 24 is a considerable generalization of Lemma 5Žb.. ŽTake f [ 5 5 2r2 and replace Q by Qr2.. The point we want to make is that the additional power of Theorem 24 is not used in any of our results up to and including Theorem 20. The rest of this paper contains results for which this additional power seems to be required. Notation 25. We extend f to be a function from E into R j `4 by defining f Ž x . [ ` if x g E R U. f U : EU ª R j `4 is defined, as usual, by f U Ž xU . [ sup E Ž xU y f . s supU Ž xU y f . . f U is proper and convex and w Ž EU , E .-lower semicontinuous. If g: E ª R j `4 Žresp. g: EU ª R j `4., we define dom g [ x : x g E, g Ž x . g R4 Žresp. xU : xU g EU , g Ž xU . g R4.. The first order of business is to find an analog for the constant 2Ž5 y 5 q 5 yU 5. used in Lemma 6. To this end, we introduce two constants determined by the rates of growth of f and f U at `. We define
rŽ f . [
lim inf
f Ž x.
xgU , 5 x 5ª`
and
5 x5
U
r Žf. [
f U Ž xU .
lim inf
x Ugdom f U , 5 x U 5ª`
5 xU 5
,
with the understanding that r Ž f . [ ` if U is bounded and r U Ž f . [ ` if dom f U is bounded. We note that f Žresp. f U . is said to be coerci¨ e if r Ž f . [ ` Žresp. r Ž f . [ `.. U
LEMMA 26. Ža. Let y g U, 5 y 5 - r U Ž f ., yU g ydom f U and 5 yU 5 Ž r f .. Then there exists K Ž y, yU . g R such that K Ž y, yU . G 0, x g U and 5 x 5 ) K Ž y, yU .
«
f Ž x. 5 x5
) 5 yU 5 q
f Ž y . q ² y, yU : 5 x5
and xU g dom f U and 5 xU 5 ) K Ž y, yU . «
f U Ž xU . 5 xU 5
) 5 y5 q
f U Ž yyU . q ² y, yU : 5 xU 5
.
196
S. SIMONS
Žb.
Let x g U and xU g f Ž x .. Then
² y y x, yU q xU : G 0
5 x 5 F K Ž y, yU . and 5 xU 5 F K Ž y, yU . .
«
Proof. If U is unbounded then lim inf
f Ž x.
xgU , 5 x 5ª`
5 x5
s r Ž f . ) 5 yU 5 s lim
5 x 5ª`
5 yU 5 q
f Ž y . q ² y, yU : 5 x5
;
hence there exists K 1 g R such that x g U and 5 x 5 ) K 1
f Ž x.
«
5 x5
) 5 yU 5 q
f Ž y . q ² y, yU : 5 x5
. Ž 26.1.
If U is bounded we take K 1 [ sup 5 U 5, and Ž26.1. is vacuously true. In a similar fashion, there exists K 2 g R such that xU g dom f U and 5 xU 5 ) K 2 «
f U Ž xU . 5 xU 5
) 5 y5 q
f U Ž yyU . q ² y, yU : 5 xU 5
.
Now Ža. follows with K Ž y, yU . [ K 1 k K 2 k 0. From Ža., if 5 x 5 ) K Ž y, yU . then ² y y x, yU q xU : s ² y y x, yU : q ² y y x, xU : F ² y y x, yU : q f Ž y . y f Ž x . F 5 x 5 5 yU 5 q f Ž y . q ² y, yU : y f Ž x . - 0, while if 5 xU 5 ) K Ž y, yU . then ² y y x, yU y xU : s ² y, yU q xU : q ² x, yyU y xU : F ² y, yU q xU : q f U Ž yyU . y f U Ž xU . F 5 y 5 5 xU 5 q f U Ž yyU . q ² y, yU : y f U Ž xU . - 0. This completes the proof of Žb.. THEOREM 27. and
Let F be a nonempty finite monotone subset of U = EU
R F [ min K Ž y, yU . : Ž y, yU . g F , 5 y 5 - r U Ž f . , yU g ydom f U , 5 yU 5 - r Ž f . 4 . Then there exist x F g U and xUF g f Ž x F . such that 5 x F 5 F R F , 5 xUF 5 F R F and for all Ž y, yU . g F ,
f Ž x F . q f U Ž xUF . F ² y, xUF : q ² y y x F , yU : .
Ž 27.1.
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RANGE OF A MONOTONE OPERATOR
U .4 . Define the matrix Q by qi j [ Proof. Let F s Ž y 1 , yU1 ., . . . , Ž ym , ym U ² yi y y j , yi :. Arguing as in Lemma 6, Q is copositive. Thus, from Theorem 24, there exist b g sm and xUF g f ŽÝ j bj y j . such that
for all i s 1, . . . , m,
¦
yi y
Ý bj yj , xUF
;
j
q
Ý qi j bj G bT Qb G 0. j
It follows from this that for all i s 1, . . . , m,
¦
Ý bj yj , xUF
¦
Ý bj yj , xUF
yi y
;
j
q
Ý ² yi y yj , yUi : bj G 0, j
that is, for all i s 1, . . . , m,
yi y
;¦
j
q yi y
Ý bj yj , yUi j
;
G 0.
Hence, writing x F [ Ý j bj y j , for all Ž y, yU . g F ,
² y y x F , yU q xUF : G 0.
Then Ž27.1. follows since f Ž x F . q f U Ž xUF . s ² x F , xUF :. Finally, from Lemma 26Žb., 5 x F 5 F R F and 5 xUF 5 F R F . Notation 28. f UU : EUU ª R j `4 is defined, as usual, by f UU Ž xUU . [ sup EU Ž xUU y f U .. f UU is proper and convex and w Ž EUU , EU .-lower semicontinuous. Furthermore, for all x g U,
f UU Ž ˆ x. F f Ž x. .
Ž 28.1.
We write V [ y : y g E, 5 y 5 - r U Ž f .4 and V U [ yU : yU g EU , 5 yU 5 - r Ž f .4 . We do not generally have equality in Ž28.1., since we are not assuming that f is lower semicontinuous on E. U
LEMMA 29. Let T : E ª 2 E be monotone, DŽT . ; U, and T Ž V . q dom f U l V U 2 0. Then there exists Ž xU , xUU . g EU = EUU such that yxU g TxUU and f UU Ž xUU . q f U Ž xU . y ² xU , xUU : F
inf
Ž y , y U .gG ŽT .
² yU q x U , ˆ y y xUU : .
Ž 29.1. Proof. We choose y 0 g U l V and yU0 g Ty 0 such that yyU0 g dom f U l V U . From Theorem 27, for all finite subsets F of GŽT . such that
198
S. SIMONS
F 2 Ž y 0 , yU0 ., there exist x F g U and xUF g f Ž x F . such that 5 x F 5 F K Ž y 0 , yU0 ., 5 xUF 5 F K Ž y 0 , yU0 . and for all Ž y, yU . g F ,
f Ž x F . q f U Ž xUF . F ² y, xUF : q ² y y x F , yU : . $
UU . F f Ž x F . and We now define xUU . Then f UU Ž xUU F [ x Fg E F
for all Ž y, yU . g F ,
U U ² U y : q ² yU , ˆ :. f UU Ž xUU y y xUU F . q f Ž xF . F xF , ˆ F
It now follows from the w Ž EU , E . = w Ž EUU , EU .-compactness of the sets
Ž xU , xUU . : Ž xU , xUU . g EU = EUU , 5 xU 5 F K Ž y 0 , yU0 . , 5 xUU 5 F K Ž y 0 , yU0 . 4 and the w Ž EU , E .- and w Ž EUU , EU .-lower semicontinuity of f U and f UU that there exists Ž xU , xUU . g EU = EUU such that for all Ž y, yU . g G Ž T . ,
f UU Ž xUU . q f U Ž xU . F ² xU , ˆ y : q ² yU , ˆ y y xUU : .
Inequality Ž29.1. is immediate from this. Since f UU Ž xUU . q f U Ž xU . y ² xU , xUU : G 0, it follows from Ž29.1. that yxU g TxUU . This completes the proof of Lemma 29. THEOREM 30. DŽ S . ; U. Then
Let S: E ª 2 E
RŽ S q Ž f U .
y1
U
be of type ŽNI. and monotone and
. > S Ž V . q dom f U l V U . U
Proof. Let xU g SŽ V . q dom f U l V U . Define T : E ª 2 E by T [ S y zU . Then T is of type ŽNI. and monotone, DŽT . s DŽ S . ; U, and T Ž V . q dom f U l V U s S Ž V . q dom f U l V U y zU 2 0. From Lemma 29, there exists Ž xU , xUU . g EU = EUU such that yxU g TxUU s SxUU y zU and f UU Ž xUU . q f U Ž xU . y ² xU , xUU : F 0, i.e., xUU g f U Ž xU .. Thus zU s Ž zU y xU . q zU g SxUU q Ž f U . This completes the proof of Theorem 30.
y1
Ž xUU . g R Ž S q Ž f U .
y1
..
199
RANGE OF A MONOTONE OPERATOR
Remark 31. We now give some special cases of Theorem 30. U
Ža. If f is coercive Žthat is to say, r Ž f . s `., S: E ª 2 E is of type ŽNI. and monotone, and DŽ S . ; U then RŽ S q Ž f U .
y1
. > S Ž V . q dom f U .
ŽSee Remark 33 for an improvement of this result.. U Žb. If f U is coercive Žthat is, r U Ž f . s `., S: E ª 2 E is of type ŽNI. and monotone, and DŽ S . ; U then RŽ S q Ž f U .
y1
. > R Ž S . q dom f U l V U .
Ž 31.1. U
We take some time now to discuss a special case of Ž31.1.. LetU S: E ª 2 E be of type ŽNI. and monotone. Let l ) 0, and L: EUU ª 2 E be defined by LxUU [ xU : xU g EU , 5 xU 5 F l , ² xU , xUU : s l 5 xUU 5 4 . Then R Ž S q L . > R Ž S . q x U : 5 x U 5 - l4 .
Ž 31.2.
In order to prove this, we take U [ E and f [ l 5 5, for which f U is coercive. Then Ž31.2. follows from Ž31.1., since Ž f U .y1 s L, V U s xU : 5 xU 5 - l4 and dom f U s V U . U Žc. If both f and f U are coercive, S: E ª 2 E is of type ŽNI. and monotone, and DŽ S . ; U then RŽ S q Ž f U .
y1
. > R Ž S . q dom f U .
ŽSee Remark 33 for an improvement of this result. THEOREM 32. Let E be reflexi¨ e, f : E ª R j `4 be proper, con¨ ex, and U lower semicontinuous, and dom f be open. Let S: E ª 2 E be monotone. Ža.
If S is of type ŽNI. and DŽ S . ; dom f then R Ž S q f . > S Ž V . q dom f U l V U .
Žb.
If S is maximal monotone and DŽ S . ; dom f then R Ž S q f . > S Ž V . q dom f U l V U .
Žc.
If S is of type ŽNI. and DŽ S . ; dom f then R Ž S q f . > S Ž V . q dom f U l V U .
200
S. SIMONS
Proof. We write U [ dom f. From w8, Proposition 3.3, p. 39x, f is continuous on U. Further, Ž f U .y1 s f. Thus Ža. follows from Theorem 30. Žb. is immediate from Ža. since if S is maximal monotone then S is of type ŽNI. and S s S. Žc. follows by applying Žb. with S replaced by S since, from Theorem 19ŽŽa. « Žc.., S is maximal monotone. If E is reflexive and f [ 5 5 2r2 then f s J, r Ž f . s r U Ž f . s `, V s E, V s EU , and dom f U s EU ; hence Theorem 32Žb. generalizes Theorem 1Ža.. Finally, still supposing that E is reflexive, we conjecture that the conclusion of Theorem 30 cannot be strengthened to read U
R Ž S q f . > S Ž V . q dom f U l V U Žthat is, unless f is lower semicontinuous on E and Theorem 32Žb. can be applied.. Remark 33. The author is grateful to Professors Heinz Bauschke and Jon Borwein for pointing out to him that
rŽ f . s `
« dom f U s EU .
Ž 33.1.
Indeed, suppose that r Ž f . s `. Let z g U and zU g f Ž z . be fixed. If now xU is any element of EU , then there exists K g R such that K G 5 z 5 and x g U and 5 x 5 ) K
«
f Ž x.
) 5 xU 5 q
5 x5
f Ž z . y ² z, xU : 5 x5
,
from which it follows that x g U and 5 x 5 ) K
« ² x, xU : y f Ž x . F ² z, xU : y f Ž z . .
Thus f U Ž xU . s sup Ž xU y f . s sup ² x, xU : y f Ž x . 5 x 5FK
U
F sup w ² x, xU y zU : x q sup ² x, zU : y f Ž x . 5 x 5FK U
5 x 5FK
U
U
U
F K 5 x y z 5 q f Ž z . - `. This establishes Ž33.1.. Consequently, in Remark 31Žc., S q Ž f U .y1 is surjective. The same is true in Remark 31Ža. provided that DŽ S . l V / B.
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201
REFERENCES 1. J. M. Borwein, Generalized linear complementarity problems treated without fixed-point theory, J. Optim. Theory Appl. 43 Ž1984., 343]356. 2. J. Borwein and S. Fitzpatrick, Local boundedness of monotone operators under minimal hypotheses, Bull. Austral. Math. Soc. 39 Ž1988., 439]441. 3. K. Fan, Minimax theorems, Proc. Nat. Acad. Sci. U.S. A. 39 Ž1953., 42]47. 4. S. P. Fitzpatrick and R. R. Phelps, Bounded approximants to monotone operators on Banach spaces, Ann. Inst. H. Poincare, 9 Ž1992., 573]595. ´ Anal. Non Lineaire ´ 5. J.-P. Gossez, Operateurs monotones non lineaires dans les espaces de Banach non ´ ´ reflexifs, J. Math. Anal. Appl. 34 Ž1971., 371]395. ´ 6. M. Hall, Jr., and M. Newman, Copositive and completely positive quadratic forms, Proc. Cambridge Philos. Soc. 59 Ž1963., 329]339. 7. H. Komiya, Elementary proof for Sion’s minimax theorem, Kodai Math. J. 11 Ž1988., 5]7. 8. R. R. Phelps, ‘‘Convex Functions, Monotone Operators and Differentiability,’’ 2nd ed., Lecture Notes in Mathematics, Vol. 1364 Springer-Verlag, BerlinrNew York, 1993. 9. R. R. Phelps, Lectures on maximal monotone operators Ž2nd summer school on Banach spaces, related areas and applications, Prague and Paseky, August 15]28, 1993., preprint, 30, pp. 10. S. Reich, The range of sums of accretive and monotone operators, J. Math. Anal. Appl. 69 Ž1979., 310]317. 11. R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 Ž1970., 75]88. 12. R. T. Rockafellar, On the virtual convexity of the domain and range of a nonlinear maximal monotone operator, Math. Ann. 185 Ž1970., 81]90. 13. R. T. Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J. 16 Ž1969., 397]407. 14. M. Sion, On general minimax theorems, Pacific J. Math. 8 Ž1958., 171]176.
199, 176]201 Ž1996.
0135
The Range of a Monotone Operator S. SimonsU Department of Mathematics, Uni¨ ersity of California, Santa Barbara, California 93106-3080 Submitted by William F. Ames Received May 1, 1995
INTRODUCTION The analysis discussed in this paper stems from the following results of Rockafellar Žsee w11, Proposition 1, p. 77; 12, Theorem 2, p. 89; 13, Theorem 1, p. 398x.: THEOREM 1. Let S be a maximal monotone operator on a reflexi¨ e Banach space E and J be the duality map; then: Ža. S q J is surjecti¨ e. Žb. R Ž S . is con¨ ex. Žc. int RŽ S . is con¨ ex. Generalizations of Theorem 1Ža, b. to the nonreflexive case were given by Gossez, with the introduction of ‘‘maximal monotone operators of type ŽD.,’’ and of Theorem 1Žb. by Fitzpatrick and Phelps, with the introduction of ‘‘locally maximal monotone operators.’’ It is known that the subdifferential of a proper convex lower semicontinuous function on a Banach space is both maximal monotone of type ŽD. and locally maximal monotone. However, nothing seems to be known at the moment about the relationship between these two classes of operators. We hope that the analysis presented here will eventually give some insight into this question. Ultimately, Gossez’ proofs used Brouwer’s fixed-point theorem. In Section I we show how Gossez’ results can be proved and indeed generalized using purely functional analytic techniques Žas was, indeed, the case with the proofs of Fitzpatrick and Phelps in w4x.. Our main results in this direction are Theorems 12, 17, and 20. We have couched the analysis of * E-mail: [email protected]. 176 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
RANGE OF A MONOTONE OPERATOR
177
Section I in functional analytic rather than convex analytic terms; that is, we have not used the concepts of conjugate function or subdifferential. In Section II we show what additional information can be obtained using Brouwer’s fixed-point theorem. In this connection, we refer the reader to Lemmas 5Ža. and 22Ža., both results about continuous convex functions on a simplex and matrices, which have very similar statements but very different proofs. While our main interest is in the nonreflexive case, in Remark 7 we digress and give a complete proof of Theorem 1Ža, b., as stated above, for the reflexive case. As far as we know, the proof of Theorem 1Ža. given in Remark 7 is the simplest and least technical available. It is worth noting that the analysis in Remark 7 is valid for any reflexive space, not only one which has been renormed so that J and Jy1 are single-valued. Some of our later results give more precise information about R Ž S . and int RŽ S . than that given in Theorem 1Žb, c.. Specifically, in Theorem 13 we will exhibit proper, convex lower semicontinuous functions fS : EU ª R j `4 and cS : E ª R j `4 such that R Ž S . s dom fS , int RŽ S . s intŽdom fS ., D Ž S . s dom cS , and int DŽ S . s intŽdom cS .. These representations in terms of the domains of the convex functions were suggested by an analysis of a result of Reich Žsee w10, Lemma 2.1, p. 315x.. Theorem 32 contains some generalizations of Theorem 1Ža., still for reflexive spaces. In Section I, we introduce two new classes of multifunctions from a normed space into its dual, the operators of type ŽWD. ŽWeak D. Žsee Definition 14. and the operators of type ŽNI. Žnegative infimum. Žsee Definition 10.. Maximal monotone operators of type ŽD. are of type ŽWD., and operators of type ŽWD. are of type ŽNI.. Monotone operators of type ŽNI. enjoy the generalization to nonreflexive spaces of Theorem 1Ža. Žsee Theorem 12Ža.., and their natural extensions to the bidual are ‘‘nearly maximal monotone’’ Žsee Theorem 19 for this and equivalent conditions.. For monotone operators of type ŽWD. we can give a precise description of R Ž S . as the closure of the domain of a convex function Žsee Theorem 17.. For monotone operators of type ŽNI. we can give a precise description of R Ž S . and int RŽ S . as the closure and interior of the domain of a convex U function, where S is the operator from EUU into 2 E defined by Gossez Žsee Theorem 20.. In order to explain further the results in Section II, we must elaborate a little on Gossez’ surjectivity result. Gossez proves that if S is maximal monotone of type ŽD. then RŽ S q Ž jU .y1 . s EU , where j: E ª R is defined by jŽ x . [ 5 x 5 2r2. In Section II, we prove an analogous result for monotone operators of type ŽNI. for RŽ S q Ž f U .y1 ., where f is a real continuous convex function defined on some convex open subset of E Žsee Theorem 30..
178
S. SIMONS
There are two unusual features about the way in which we have presented our analysis. The first of these is the use of matrices in Lemmas 5Ža. and 22. We have done this not because we think that these two results have any particular merit as results about matrices, but rather because the concise matrix notation enables us to make the comparisons referred to above a little more easily. The second unusual feature is our use of minimax theorems. In fact, we could have used the Hahn]Banach theorem instead, but minimax theorems seem the most natural idiom for the description of the manipulations we perform. More specifically, we shall use the following classical minimax theorem, which can be deduced from more general results of Fan Žsee w3x. or Sion Žsee w14x.. Fan’s proof used a separation theorem for sets in finite dimensional spaces, and Sion’s proof used the KKM theorem, but Theorem 2 can easily be proved without any functional analysis or fixed-point related concepts. See, for instance, the proof of Sion’s theorem given by Komiya in w7x. THEOREM 2. Let X and Y be nonempty compact con¨ ex subsets of topological ¨ ector spaces. Let f : X = Y ª R be Ž separately. conca¨ e and upper semicontinuous on X and con¨ ex and lower semicontinuous on Y. Then max min f s min max f . X
Y
Y
X
Notation 3. Let E be a real normed space. If m G 1, let
sm [ a s Ž a1 , . . . , a m . : a1 , . . . , a m G 0, a1 q ??? qam s 1 4 ; R m . The author thanks Professor Robert Phelps for reading through two earlier versions of this paper and making a number of suggestions that have improved the exposition substantially. He also thanks Professors Jon Borwein and Simeon Reich for giving him some very helpful references.
I. WITHOUT FIXED-POINT THEOREMS DEFINITION 4. We say that an m = m matrix Q is copositi¨ e if a g sm « aT Qa G 0. Symmetric copositive matrices were introduced by Hall and Newman in w6x. Copositivity has, in fact, been treated in more general situations in optimization theory. See Borwein’s discussion in w1x of copositivity as it relates to the linear complementarity problem. LEMMA 5. Ža.
Let Q [ qi j 4 be a copositi¨ e m = m matrix.
Let g: sm ª R be continuous and con¨ ex. Then max min g Ž a . y g Ž b . q aT Qb G 0.
bg sm ag sm
179
RANGE OF A MONOTONE OPERATOR
Žb.
Let y 1 , . . . , ym g E. Then there exist b g sm and xU g EU such
that 2
for all i s 1, . . . , m,
Ý bj y j
q 5 xU 5 2 F 2² yi , xU : q
j
Ý q i j bj . j
Proof. Ža. This follows from Theorem 2 with X [ Y [ sm since min max g Ž a . y g Ž b . q aT Qb G min g Ž a . y g Ž a . q aT Qa G 0.
ag sm bg sm
ag sm
Žb. Let M [ 5 y 1 5 k ??? k 5 ym 5 and B [ xU : xU g EU , 5 xU 5 F M 4 with the topology w Ž EU , E .. Define an affine map L: sm ª E by LŽ a. [ a1 y 1 q ??? qa m ym . From Ža. with g Ž a. [ 5 LŽ a.5 2 , max min 5 L Ž a . 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
bg sm ag sm
From the Hahn]Banach theorem, for all a g sm , there exists xU g EU such that ² LŽ a., xU : s 5 xU 5 2 s 5 LŽ a.5 2 , from which 5 LŽ a.5 2 s 2² LŽ a., xU : y 5 xU 5 2 . Since 5 LŽ a.5 F M, xU g B. Thus max min max 2² L Ž a . , xU : y 5 xU 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
bg sm ag sm x UgB
For each a g sm , the map xU ª 2² LŽ a., xU : y 5 xU 5 2 is concave and upper semicontinuous on B and, for all xU g B, the map a ª 2² LŽ a., xU : q aT Qb is affine and continuous on sm ; hence, from Theorem 2 with X [ B and Y [ sm , max max min 2² L Ž a . , xU : y 5 xU 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
bg sm x UgB ag sm
Thus there exist b g sm and xU g EU such that for all a g sm ,
2² L Ž a . , xU : y 5 xU 5 2 y 5 L Ž b . 5 2 q aT Qb G 0.
If we now let a run through the vertices of sm , we obtain for all i s 1, . . . , m,
2² yi , xU : q
Ý qi j bj y 5 L Ž b . 5 2 y 5 xU 5 2 G 0, j
as required. LEMMA 6.
Let F be a nonempty finite monotone subset of E = EU and NF [
min
Ž y , y U .gF
2 Ž 5 y 5 q 5 yU 5 . .
180
S. SIMONS
Then there exists Ž x F , xUF . g E = EU such that 5 x F 5 F NF , 5 xUF 5 F NF and for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 F 2² y, xUF : q 2² y y x F , yU : .
Ž 6.1. U .4 . Define the matrix Q by qi j [ Proof. Let F s Ž y 1 , yU1 ., . . . , Ž ym , ym U 2² yi y y j , yi :. Then Q is copositive since, for all a g sm ,
aT Qa s
Ý ai qi j a j q Ý ai qi j a j s Ý ai qi j a j q Ý a j q ji ai i-j
s
j-i
i-j
i-j
Ý ai Ž qi j q q ji . a j s Ý 2 ai ² yi y yj , yiU y yUj : a j G 0. i-j
i-j
From Lemma 5Žb., there exist b g sm and xUF g EU such that 2
for all i s 1, . . . , m,
Ý bj y j
q 5 xUF 5 2
j
F 2² yi , xUF : q
Ý 2² yi y yj , yiU : bj , j
that is, 2
for all i s 1, . . . , m,
Ý bj y j
q 5 xUF 5 2
j
F 2² yi , xUF : q 2² yi y
Ý bj yj , yiU : . j
Thus Ž6.1. follows with x F [ Ý j bj y j . It remains to prove that 5 x F 5 F NF and 5 xUF 5 F NF . From Ž6.1., for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 y 2 5 y 5 5 xUF 5 y 2 5 x F 5 5 yU 5 F 2 5 y 5 5 yU 5 ,
from which for all Ž y, yU . g F ,
Ž 5 x F 5 y 5 yU 5 .
2
q Ž 5 xUF 5 y 5 y 5 . F Ž 5 y 5 q 5 yU 5 . . 2
2
It follows easily from this that 5 x F 5 F NF and 5 xUF 5 F NF . This completes the proof of Lemma 6. Remark 7 Ž Proof of Theorem 1Ža, b... As promised in the Introduction, we now digress and give a complete proof of Theorem 1Ža, b.. So let E be U reflexive, S: E ª 2 E be maximal monotone, and zU g EU . Define T :
RANGE OF A MONOTONE OPERATOR
181
U
E ª 2 E by T [ S y zU . Then T is maximal monotone. From Lemma 6, for all finite subsets F of GŽT ., there exists Ž x F , xUF . g E = EU such that 5 x F 5 F NF , 5 xUF 5 F NF , and for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 F 2² y, xUF : q 2² y y x F , yU : .
Put another way, the set
F Ž x, xU . : Ž x, xU . g E = EU , 5 x 5 F NF , 5 xU 5 F NF ,
Ž y , y U .gF
5 x 5 2 q 5 xU 5 2 F 2² y, xU : q 2² y y x, yU : 4 is nonempty. As F runs, these sets are w Ž E, EU . = w Ž EU , E .-compact and directed downwards; hence their intersection is nonempty. It follows that there exists Ž x, xU . g E = EU such that, for all Ž y, yU . g GŽT ., 5 x 5 2 q 5 xU 5 2 F 2² y, xU : q 2² y y x, yU : . Thus, for all Ž y, yU . g GŽT ., 5 x 5 2 q 5 xU 5 2 y 2² x, xU : F 2² y y x, yU q xU : .
Ž 7.1.
Since 5 x 5 2 q 5 xU 5 2 y 2² x, xU : G 5 x 5 2 q 5 xU 5 2 y 2 5 x 5 5 xU 5 s Ž 5 x 5 y 5 xU 5 . G 0, 2
it follows from the definition of maximal monotonicity that yxU g Tx. Putting Ž y, yU . s Ž x, yxU . in Ž7.1., we find in particular that 5 x 5 2 q 5 xU 5 2 y 2² x, xU : F 0.
Ž 7.2.
Consequently, Ž5 x 5 y 5 xU 5. 2 F 0, from which 5 xU 5 s 5 x 5. Substituting this back into Ž7.2., ² x, xU : s 5 x 5 2 ; hence xU g Jx. Since yxU g Tx, zU y xU g Sx. Thus zU s Ž zU y xU . q xU g Ž S q J . x. Since zU was an arbitrary element of EU , S q J is surjective. This completes the proof of Theorem 1Ža.. In order to establish Theorem 1Žb., it suffices to prove that xU1 , xU2 g R Ž S . «
xU1 q xU2 2
g RŽ S . .
182
S. SIMONS
So let xU1 , xU2 g RŽ S .. Choose x 1 , x 2 g E such that Ž x 1 , xU1 ., Ž x 2 , xU2 . g U GŽ S .. Let l ) 0. Define T : E ª 2 E by Tx [
1
l
S xq
ž
x1 q x 2 2
/
y
xU1 q xU2 2l
.
T is maximal monotone and if
Ž u, uU . [
ž
x1 y x 2 2
,
xU1 y xU2 2l
/
,
then "Ž u, uU . g GŽT .. Arguing as above, there exists Ž x, xU . g E = EU such that yxU g Tx and 5 xU 5 2 s ² x, xU :. Since "Ž u, uU . g GŽT ., ² " u y x, " uU q xU : G 0. Taking the average of the expressions with the two signs, ² u, uU : y ² x, xU : G 0, from which 5 xU 5 2 s ² x, xU : F ² u, uU :. Thus 5lx
U
U
5 F 'l ² u, u 2
: s
(
l² x 1 y x 2 , xU1 y xU2 : 4
.
Since xU1 q xU2 2
gS xq
ž
x1 q x 2 2
/
q l xU ; R Ž S . q l xU ,
it follows by letting l ª 0 that xU1 q xU2 2
g RŽ S . .
This completes the proof of Theorem 1Žb.. We point out a crucial technical fact in the existence proof in Remark 7: the term 2² x, xU : appears after the compactness argument. The reason for this is that the function
Ž x, xU . ª 5 x 5 2 q 5 xU 5 2 y 2² y, xU : y 2² y y x, yU : is lower semicontinuous with respect to w Ž E, EU . = w Ž EU , E ., while the function Ž x, xU . ª ² x, xU : is not. Similar comments can be made about the proofs of Lemmas 9 and 29. DEFINITION 8. If y g E, we write ˆ y for the canonical image of y in the U U bidual EUU of E. Let T : E ª 2 E . We define T : EUU ª 2 E by xU g TxUU
m
inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : G 0.
RANGE OF A MONOTONE OPERATOR
183
The definition of T given above is due to Gossez}see w9, Section 3x for an exposition. GŽT . consists of those elements of EUU = EU that are ‘‘monotonically related’’ to every element of GŽT .. U
LEMMA 9. Let T : E ª 2 E be monotone. Then there exists Ž xU , xUU . g E = EUU such that yxU g TxUU and U
5 xUU 5 2 q 5 xU 5 2 y 2² xU , xUU : F
inf
Ž y , y U .gG ŽT .
2² yU q xU , ˆ y y xUU : . Ž 9.1.
Proof. From Lemma 6, for all finite subsets F of GŽT ., there exists Ž x F , xUF . g E = EU such that 5 x F 5 F NF , 5 xUF 5 F NF , and for all Ž y, yU . g F ,
5 x F 5 2 q 5 xUF 5 2 F 2² y, xUF : q 2² y y x F , yU : . $
UU 5 s 5 x F 5 F NF and We now define xUU . Then 5 xUU F [ x Fg E F
for all Ž y, yU . g F ,
5 xUU 5 2 q 5 xUF 5 2 F 2² xUF , ˆ :. y : q 2² yU , ˆ y y xUU F F
Put another way, the set
F U
Ž y , y .gF
½Ž x
U
, xUU . : Ž xU , xUU . g EU = EUU , 5 xU 5 F NF , 5 xUU 5 F NF , 5 xUU 5 2 q 5 xU 5 2 F 2² xU , ˆ y : q 2² yU , ˆ y y xUU :
5
is nonempty. As F runs, these sets are w Ž EU , E . = w Ž EUU , EU .-compact and directed downwards; hence their intersection is nonempty. It follows that there exists Ž xU , xUU . g EU = EUU such that for all Ž y, yU . g G Ž T . ,
5 xUU 5 2 q 5 xU 5 2 F 2² xU , ˆ y : q 2² yU , ˆ y y xUU : .
Inequality Ž9.1. is immediate from this. Since 5 xUU 5 2 q 5 xU 5 2 y 2² xU , xUU : G 0, it follows from Ž9.1. that yxU g TxUU . This completes the proof of Lemma 9. Motivated by the analysis in Remark 7, we now introduce a new class of multifunctions. U
DEFINITION 10. Let T : E ª 2 E . We shall say that T is of type Ž NI . if for all Ž xUU , xU . g EUU = EU ,
inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : F 0.
184
S. SIMONS
The concept of type ŽNI. is defined for operators that are not necessarily monotone though our main result ŽTheorem 12. is about monotone operators of type ŽNI.. A monotone operator of type ŽNI. is not necessarily maximal monotone }let E [ R, T Žy1. [ Žy`, 0x, T 1 [ w0, `. and Tx [ B otherwise. ŽT is of type ŽNI. because RŽT . s R s EU .. Before proving our main result about operators of type ŽNI., we introduce a function on EU that is defined for any operator S. U
DEFINITION 11. If S: E ª 2 E , we define fS : EU ª R j `4 by U
fS Ž w . [
² wU y ¨ U , ¨ UU :
sup
1 q 5¨U 5
Ž ¨ UU , ¨ U .gG Ž S .
.
Ž 11.1.
We see that fS is clearly convex and lower semicontinuous. The function fS is closely related to the function introduced by Borwein and Fitzpatrick in their proof of the local boundedness of monotone operators on Banach spaces Žsee w2, Theorem 2, p. 440x.. THEOREM 12.
U
Let S: E ª 2 E be of type Ž NI . and monotone. Then:
Ža. For all zU g EU and l ) 0, there exists Ž xU , xUU . g EU = EUU such that zU y l xU g SxUU Žb.
5 xUU 5 2 s 5 xU 5 2 s ² xU , xUU : . Ž 12.1.
and
RŽ S . ; dom fS .
Žc. dom fS ; R Ž S . . Žd. intŽdom fS . ; RŽ S .. U
Proof. Ža. Define T : E ª 2 E by T [ Ž S y zU .rl. Then T is of type ŽNI. and monotone. From Lemma 9, there exists Ž xU , xUU . g EU = EUU such that yxU g TxUU s
SxUU y zU
l
and
5 xUU 5 2 q 5 xU 5 2 y 2² xU , xUU : F 0.
Ž 12.2.
Consequently, Ž5 xUU 5 y 5 xU 5. 2 F 0, from which 5 xUU 5 s 5 xU 5. Substituting this back into Ž12.2., ² xU , xUU : s 5 xU 5 2 , as required. This completes the proof of Ža.. Žb. Let wU g RŽ S .. Choose w g E so that Ž w, wU . g GŽ S .. Let
b [ ² w, wU : k 5 w 5 .
RANGE OF A MONOTONE OPERATOR
185
It follows from the definition of S that, for all Ž ¨ UU , ¨ U . g GŽ S ., ² wU y ¨ U , ¨ UU : F ² wU y ¨ U , w ˆ : F ² wU , w ˆ : q 5 w 5 5 ¨ U 5 F b Ž 1 q 5 ¨ U 5. . Thus fS Ž wU . F b , from which wU g dom fS . This completes the proof of Žb.. Žc. Let zU g dom fS . Put
g [ < fS Ž zU . <
d [ g Ž 1 q 5 zU 5 . .
and
Let l ) 0. From Ža., there exists Ž xU , xUU . g EU = EUU satisfying Ž12.1.. Putting
Ž ¨ UU , ¨ U . [ Ž xUU , zU y l xU . g G Ž S . and using Ž11.1., ² l xU , xUU : 1 q 5¨U 5
s
² zU y ¨ U , ¨ UU : 1 q 5¨U 5
F fS Ž zU . F g ;
hence
l2 ² xU , xUU : F gl Ž 1 q 5 ¨ U 5 . F glŽ 1 q 5 zU 5 q 5 ¨ U y zU 5 . F dlŽ 1 q 5 ¨ U y zU 5 . . Combining this with Ž12.1., 5 ¨ U y zU 5 2 s l2 5 xU 5 2 F dl Ž 1 q 5 ¨ U y zU 5 . . Completing the square, 5 ¨ U y zU 5 F
dl q 'd 2l2 q 4dl 2
.
Since ¨ U g RŽ S ., it follows by letting l ª 0 that zU g R Ž S . . This completes the proof of Žc.. Žd. Let zU g intŽdom fS .. From Žc., for all l ) 0, there exists UU Ž ¨l , ¨lU . g GŽ S . such that 5 ¨lU y zU 5 F
dl q 'd 2l2 q 4dl 2
² zU y ¨lU , ¨lUU : s l 5 ¨lUU 5 2 G 0.
and
Ž 12.3.
186
S. SIMONS
We first prove that lim sup 5 ¨lUU 5 - `.
Ž 12.4.
lª0
Since zU g intŽdom fS ., it follows from w8, Proposition 3.3, p. 39x that there exist h , P ) 0 such that tU g EU and 5 tU 5 F h
«
fS Ž zU q tU . F P.
ŽThis is the so-called ‘‘local boundedness property’’.. Let l ) 0, and tU g EU with 5 tU 5 F h. Then, since Ž ¨lUU , ¨lU . g GŽ S ., ² zU q tU y ¨lU , ¨lUU : 1 q 5 ¨lU 5
F fS Ž zU q t U . F P ;
hence ² tU , ¨lUU : q ² zU y ¨lU , ¨lUU : F P Ž 1 q 5 ¨lU 5 . , and thus, from Ž12.3., ² tU , ¨lUU : F P 1 q 5 zU 5 q
ž
dl q 'd 2l2 q 4dl
/
2
.
Taking the supremum over tU ,
h 5 ¨lUU 5 F P 1 q 5 zU 5 q
ž
dl q 'd 2l2 q 4dl 2
/
,
from which it follows that lim sup 5 ¨lUU 5 F lª0
P Ž 1 q 5 zU 5 .
h
,
which gives Ž12.4.. Let zUU be a w Ž EUU , EU .-cluster point of the net ¨lUU . Then, using the fact that Ž ¨lUU , ¨lU . g GŽ S ., for all Ž y, yU . g GŽ S ., ² yU y z U , ˆ y y zUU : s ² yU y ¨lU , ˆ y y ¨lUU : q ² ¨lU y zU , ˆ y y ¨lUU : q ² yU y zU , ¨lUU y zUU : G ² ¨lU y zU , ˆ y y ¨lUU : q ² yU y zU , ¨lUU y zUU : G y5 ¨lU y zU 5 Ž 5 y 5 q 5 ¨lUU 5 . q ² yU y zU , ¨lUU y zUU : .
187
RANGE OF A MONOTONE OPERATOR
Letting l ª 0 and using Ž12.3. and Ž12.4., ² yU y z U , ˆ y y zUU : G 0. Since this holds for all Ž y, yU . g GŽ S ., Ž zUU , zU . g GŽ S .. Thus zU g RŽ S .. This completes the proof of Žd.. Specializing to reflexive spaces, we obtain the following result: THEOREM 13. Define
U
Let E be reflexi¨ e and S: E ª 2 E be maximal monotone.
fS : EU ª R j ` 4
cS : E ª R j ` 4
and
by U
fS Ž w . [
² ¨ , wU y ¨ U :
sup
Ž ¨ , ¨ U .gG Ž S .
cS Ž w . [
sup
and
1 q 5¨U 5 ²w y ¨ , ¨U :
Ž ¨ , ¨ U .gG Ž S .
1 q 5¨ 5
.
Then R Ž S . s dom fS ,
int R Ž S . s int Ž dom fS . ,
D Ž S . s dom cS ,
int D Ž S . s int Ž dom cS . .
In particular, R Ž S . , int RŽ S ., D Ž S . , and int DŽ S . are con¨ ex. Proof. Since S s S in this case, it follows from Theorem 12 that R Ž S . ; dom fS ,
dom fS ; R Ž S . ,
int Ž dom fS . ; R Ž S . .
This gives the first two assertions. The second two follow by replacing S by Sy1 . U
DEFINITION 14. Let T : E ª 2 E . We say that T is maximal monotone of type Ž D . if T is maximal monotone and, for all xU g TxUU , there exists a $ U bounded net Ž ya , ya . in GŽT . such that yaª xUU in w Ž EUU , EU . and 5 yaU y xU 5 ª 0. The definition of maximal monotone of type Ž D . is also due to Gossez}again, see w9, Section 3x for an exposition. We shall say that T is of type ŽWD . if, for all xU g RŽT ., there exists a bounded net Ž ya , yaU . in GŽT . such that 5 yaU y xU 5 ª 0. If T is maximal monotone of type ŽD. then T is clearly of type ŽWD. }we conjecture that the converse of this statement is false. LEMMA 15.
U
If T : E ª 2 E is of type ŽWD. then T is of type ŽNI..
188
S. SIMONS
Proof. Suppose that T is not of type ŽNI.. Then there exists Ž xUU , xU . g EUU = EU such that inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : ) 0.
Ž 15.1.
Consequently, xU g TxUU . By hypothesis, there exists a$ bounded net Ž ya , yaU . in GŽT . such that 5 yaU y xU 5 ª 0. Then ² yaU y xU , ya y xUU : ª 0, which would contradict Ž15.1.. We conjecture that there exists a maximal monotone operator of type ŽNI. that is not of type ŽWD.. Our next result generalizes a result of Gossez}see w9, Theorem 3.6, p. 21x for an exposition. ŽGossez’ original result, which has slightly stronger hypotheses, appears in w5, Theoreme ´ ` 4.1, p. 379x.. U
COROLLARY 16. Let S: E ª 2 E be of type ŽWD. and monotone, and U z g EU . Then there exists Ž xU , xUU . g EU = EUU such that zU y xU g SxUU and 5 xUU 5 2 s 5 xU 5 2 s ² xU , xUU :. Proof. This is immediate from Theorem 12 and Lemma 15. THEOREM 17.
U
Let S: E ª 2 E be of type ŽWD. and monotone. Then R Ž S . s dom fS .
In particular, R Ž S . is con¨ ex. Proof. It follows from Lemma 15 and Theorem 12Žb, c. that R Ž S . ; dom fS ; R Ž S . . However, since S is of type ŽWD., RŽ S . ; R Ž S . ; thus R Ž S . ; dom fS ; R Ž S . s R Ž S . . This completes the proof of Theorem 17. Let T : R ª 2 R be as in the comment following Definition 10, and S [ Ty1 . Then S is of type ŽNI. and monotone, and RUŽ S . s y1, 14 is not convex. It is attractive to conjecture that if S: E ª 2 E is of type ŽNI. and maximal monotone then R Ž S . is convex. From the argument given above, U it would be sufficient to prove that if S: E ª 2 E is of type ŽNI. and maximal monotone then RŽ S . ; R Ž S . . Theorem 17 is not a complete generalization of Theorem 13, since it does not give an exact formula for int RŽ S .. In this sense, the ‘‘correct’’ generalization of Theorem 13 to nonreflexive spaces is probably Theorem
189
RANGE OF A MONOTONE OPERATOR
20 below, which is valid for operators of type ŽNI., but treats RŽ S . rather than RŽ S .. In order to prove it, we will need a strengthening of Theorem 12Žb. that uses the implication Ža. « Žb. from Theorem 19. U
U
DEFINITION 18. If T : E ª 2 E , we define Tˆ: EUU ª 2 E so that G Ž Tˆ . [ Ž ˆ y, yU . : Ž y, yU . g G Ž T . 4 ; EUU = EU . It is easy to see that if T is monotone then G Ž Tˆ . ; G Ž T . . THEOREM 19. Ž d . m Ž e ..
Ž 18.1.
U
Let T : E ª 2 E be monotone. Then Ž a. « Ž b . m Ž c . m
Ža. T is of type ŽNI.. U Žb. T : EUU ª 2 E is monotone. U Žc. T : EUU ª 2 E is maximal monotone. Žd. GŽT . is the unique maximal monotone subset of EUU = EU containing GŽTˆ.. Že. There exists a unique maximal monotone subset of EUU = EU containing GŽTˆ.. Proof. Ža. « Žb. Let Ž ¨ UU , ¨ U ., Ž wUU , wU . g GŽT . and put
Žx
UU
U
,x .[
Ž ¨ UU , ¨ U . q Ž wUU , wU . 2
g EUU = EU .
If Ž y, yU . g GŽT . then, by direct computation, 4² yU y xU , ˆ y y xUU : q ² ¨ U y wU , ¨ UU y wUU : s 2² yU y ¨ U , ˆ y y ¨ UU : q 2² yU y wU , ˆ y y wUU : .
Ž 19.1.
Since ¨ U g T¨ UU and wU g TwUU , 4² yU y xU , ˆ y y xUU : q ² ¨ U y wU , ¨ UU y wUU : G 0. Taking the infimum over Ž y, yU . g GŽT . and using the fact that T is of type ŽNI., ² ¨ U y wU , ¨ UU y wUU : G 0. This establishes Žb..
190
S. SIMONS
Žb. « Žc. Let Ž zUU , zU . g EUU = EU and inf
² xU y zU , xUU y zUU : G 0.
inf
² xU y zU , xUU y zUU : G 0,
Ž x UU , x U .gG ŽT .
From Ž18.1., Ž x UU , x U .gG ŽTˆ.
that is to say, inf
Ž y , y U .gG ŽT .
² yU y z U , ˆ y y zUU : G 0.
Consequently, zU g TzUU . From Žb., T is monotone. Thus we have established that T is maximal monotone, giving Žc.. Žc. « Žd. We know from Žc. that GŽT . is a maximal monotone subset of EUU = EU and, from Ž18.1. again, GŽT . > GŽTˆ.. Let M be any maximal monotone subset of EUU = EU such that M > GŽTˆ.. Let Ž xUU , xU . g M. For all Ž y, yU . g GŽT ., Ž ˆ y, yU . g GŽTˆ. ; M. Since M is monotone, U U UU ²y yx , ˆ y y x : G 0. Thus inf
Ž y , y U .gG ŽT .
² yU y x U , ˆ y y xUU : G 0;
in other words, Ž xUU , xU . g GŽT .. So we have proved that M ; GŽT .. Since M is maximal monotone and GŽT . is monotone, M s GŽT .. This gives Žd.. Žd. « Že. This is immediate. Že. « Žb. Suppose that Žb. fails. Then there exists Ž xUU , xU ., Ž yUU , yU . g GŽT . such that ² xU y yU , xUU y yUU : - 0. Since Ž xUU , xU . g GŽT ., GŽTˆ. j Ž xUU , xU .4 is a monotone set. From Zorn’s lemma, there is a maximal monotone subset B of EUU = EU such that B > GŽTˆ. j Ž xUU , xU .4 . Since B is monotone, Ž xUU , xU . g B and ² xU y yU , xUU y yUU : - 0, Ž yUU , yU . f B. We can perform exactly the same argument with Ž xUU , xU . replaced by Ž yUU , yU . and find a maximal monotone subset C of EUU = EU such that C > GŽTˆ. j Ž yUU , yU .4 . Since Ž yUU , yU . g C R B, C / B. This shows that Že. fails. We now give our promised generalization of Theorem 13 to nonreflexive spaces.
191
RANGE OF A MONOTONE OPERATOR
THEOREM 20.
U
Let S: E ª 2 E be of type ŽNI. and monotone. Then
R Ž S . s dom fS
and
int R Ž S . s int Ž dom fS . .
In particular, R Ž S . and int RŽ S . are con¨ ex. Proof. Let wU g RŽ S .. Choose wUU g EUU so that Ž wUU , wU . g GŽ S .. Let
b [ ² wU , wUU : k 5 wUU 5 . It follows from Theorem 19Ža. « Žb. that, for all Ž ¨ UU , ¨ U . g GŽ S ., ² wU y ¨ U , ¨ UU : F ² wU y ¨ U , wUU : F ² wU , wUU : q 5 wUU 5 5 ¨ U 5 F b Ž 1 q 5 ¨ U 5. . Thus fS Ž wU . F b , from which wU g dom fS . So we have proved that R Ž S . ; dom fS .
Ž 20.1.
Combining Ž20.1. with Theorem 12Žc., we have R Ž S . ; dom fS ; R Ž S . and, combining Ž20.1. with Theorem 12Žd., we have int R Ž S . ; int Ž dom fS . ; int R Ž S . . This completes the proof of Theorem 20. It follows from Lemma 15 that Theorem 19 is a strengthening of Gossez’ result that if T is maximal monotone of type ŽD., then T is maximal monotone. The equivalence of Žb. ] Že. in Theorem 19 is actually a special case of more general situation. Let P / B and M ; P = P be a symmetric set such that M > Ž p, p . : p g P 4 . If G ; P, we write G M [ p : p g P, p4 = G ; M 4 and say that G is an M-set if G = G ; M . We can then prove, exactly as in the proof of Theorem 19 above, that the following four conditions are equivalent: G M is an M-set. G M is a maximal M-set. G M is the unique maximal M-set containing G . There is a unique maximal M-set containing G . ŽThe symbol ‘‘ P ’’ is meant to suggest the phrase ‘‘product space,’’ and the symbol ‘‘ M ’’ is meant to suggest the phrase ‘‘monotonically related.’’.
192
S. SIMONS
It is interesting to compare the techniques used in Remark 7 and Theorem 19. Suppose, for simplicity, that Ž wUU , wU . s yŽ ¨ UU , ¨ U . in the latter, so that Ž xUU , xU . s 0. Then Ž19.1. becomes 4² yU , ˆ y : q ²2 ¨ U , 2 ¨ UU : s 2² yU y ¨ U , ˆ y y ¨ UU : q 2² yU q ¨ U , ˆ y q ¨ UU : , that is, ² yU , ˆ y : q ² ¨ U , ¨ UU : s
² yU y ¨ U , ˆ y y ¨ UU : q ² yU q ¨ U , ˆ y q ¨ UU : 2
.
After some change of symbols, this is exactly the ‘‘averaging argument’’ used in the last part of Remark 7.
II. WITH FIXED-POINT THEOREMS Notation 21. Let C be a nonempty convex subset of a vector space, and f : C ª R be convex. If x, y g C then the directional deri¨ ati¨ e of f at y in the direction x y y is given by dq f Ž y . Ž x y y . [
inf
f Ž y q u Ž x y y. . y f Ž y.
u
u g Ž0, 1 x
s lim
f Ž y q u Ž x y y. . y f Ž y.
u
uª0q
.
Lemma 22Ža. generalizes Lemma 5Ža.. We do not know if the use of Brouwer’s fixed-point theorem in its proof can be avoided. If Q is symmetric and we choose b g sm to minimize g Ž b . q bT Qbr2 then, for all a g sm and u g Ž0, 1x, g Ž b q u Ž a y b. . y g Ž b. q
T Ž b q u Ž a y b . . Q Ž b q u Ž a y b . . y bT Qb
2
G 0.
On dividing by u and letting u ª 0, we obtain q
d g Ž b. Ž a y b. q
T Ž a y b . Qb q bT Q Ž a y b .
2
G 0.
Thus Lemma 22Žb. follows from the symmetry of Q. Unfortunately, in our application of Lemma 22Žb., Q is not generally symmetric. Theorem 2 cannot be used directly since the function b ª g Ž c . q c T Qb y g Ž b . y bT Qb may fail to be concave Žtake g [ 0 and Q [ yI ..
193
RANGE OF A MONOTONE OPERATOR
LEMMA 22. Let Q be any m = m matrix. Let g: sm ª R be continuous and con¨ ex. Then: max min g Ž c . y g Ž b . q c T Qb y bT Qb s 0.
Ž a. Ž b.
bg sm cg sm
max inf
bg sm ag sm
dq g Ž b . Ž a y b . q aT Qb y bT Qb s 0.
Proof. Since max min g Ž c . y g Ž b . q c T Qb y bT Qb
bg sm cg sm
F max g Ž b . y g Ž b . q bT Qb y bT Qb s 0, bg sm
if Ža. fails then, for all b g sm , there exists c b g sm such that g Ž c b . y g Ž b . q c Tb Qb y bT Qb - 0, and hence there exists an open neighborhood Nb of b in sm such that a g Nb « g Ž c b . y g Ž a . q c Tb Qa y aT Qa - 0. Since sm is compact, there exist b1, . . . , b n g sm such that sm s Nb 1 j ??? j Nb n . Let p1 , . . . , pn be a partition of unity on sm subordinate to this open cover. The map a ª Ý k pk Ž a. c b k is a continuous self-map of sm ; hence, from Brouwer’s fixed point theorem, there exists a g sm such that as
Ý pk Ž a . c b
k
.
Ž 22.1.
k
Now k g 1, . . . , n4 and pk Ž a . ) 0
«
g Ž c b k . y g Ž a . q c Tb k Qa y aT Qa - 0;
consequently,
Ý pk Ž a. g Ž c b . y g Ž a. q cTb Qa y aT Qa k
k
- 0.
k
It follows from Ž22.1. and the convexity of g that g Ž a . y g Ž a . y aT Qa y aT Qa - 0, which is impossible. This contradiction completes the proof of Ža..
194
S. SIMONS
From Ža., there exists b g sm such that for all c g sm ,
g Ž c . y g Ž b . q c T Qb y bT Qb G 0.
Ž 22.2.
Let a g sm . For all u g Ž0, 1x, we substitute c [ b q u Ž a y b . into Ž22.2. and obtain for all u g Ž 0, 1 ,
g Ž b q u Ž a y b . . y g Ž b . q u Ž aT Qb y bT Qb . G 0.
Now Žb. follows by dividing by u and then letting u ª 0 since max inf
bg sm ag sm
dq g Ž b . Ž a y b . q aT Qb y bT Qb
F max dq g Ž b . Ž b y b . q bT Qb y bT Qb s 0. bg sm
Notation 23. We suppose for the rest of this paper that U is a nonempty convex open subset of E and that f : U ª R is continuous and convex. THEOREM 24. Let Q be an m = m matrix and y 1 , . . . , ym g U. Then there exist b g sm and xU g f ŽÝ j bj y j . such that for all i s 1, . . . , m,
¦Ý j
bj y j , xU q bT Qb F ² yi , xU : q
;
Ý q i j bj . j
Proof. As in the proof of Lemma 5Žb., define L: sm ª E by LŽ c . [ c1 y 1 q ??? qc m ym . From Lemma 22Žb. with g [ f ( L, max inf
bg sm ag sm
dq f Ž L Ž b . . Ž L Ž a . y L Ž b . . q aT Qb y bT Qb s 0.
Since f is continuous in the neighborhood of LŽ b ., it follows from Chapter 1 of w8x and the Hahn]Banach theorem that for all y g E,
dq f Ž L Ž b . . Ž y . s
max
z Ug f Ž L Ž b ..
² y, zU : .
Thus, max inf
max
bg sm ag sm z Ug f Ž L Ž b ..
² L Ž a . y L Ž b . , zU : q aT Qb y bT Qb s 0.
Since f Ž LŽ b .. is w Ž EU , E .-compact, it follows from Theorem 2 with X [ f Ž LŽ b .. and Y [ sm that max
max
min ² L Ž a . y L Ž b . , zU : q aT Qb y bT Qb s 0.
bg sm z Ug f Ž L Ž b .. ag sm
195
RANGE OF A MONOTONE OPERATOR
Thus there exist b g sm and xU g f Ž LŽ b .. such that for all a g sm ,
² L Ž a . y L Ž b . , xU : q aT Qb y bT Qb G 0.
The result now follows as in Lemma 5Žb. by letting a run through the vertices of sm . Theorem 24 is a considerable generalization of Lemma 5Žb.. ŽTake f [ 5 5 2r2 and replace Q by Qr2.. The point we want to make is that the additional power of Theorem 24 is not used in any of our results up to and including Theorem 20. The rest of this paper contains results for which this additional power seems to be required. Notation 25. We extend f to be a function from E into R j `4 by defining f Ž x . [ ` if x g E R U. f U : EU ª R j `4 is defined, as usual, by f U Ž xU . [ sup E Ž xU y f . s supU Ž xU y f . . f U is proper and convex and w Ž EU , E .-lower semicontinuous. If g: E ª R j `4 Žresp. g: EU ª R j `4., we define dom g [ x : x g E, g Ž x . g R4 Žresp. xU : xU g EU , g Ž xU . g R4.. The first order of business is to find an analog for the constant 2Ž5 y 5 q 5 yU 5. used in Lemma 6. To this end, we introduce two constants determined by the rates of growth of f and f U at `. We define
rŽ f . [
lim inf
f Ž x.
xgU , 5 x 5ª`
and
5 x5
U
r Žf. [
f U Ž xU .
lim inf
x Ugdom f U , 5 x U 5ª`
5 xU 5
,
with the understanding that r Ž f . [ ` if U is bounded and r U Ž f . [ ` if dom f U is bounded. We note that f Žresp. f U . is said to be coerci¨ e if r Ž f . [ ` Žresp. r Ž f . [ `.. U
LEMMA 26. Ža. Let y g U, 5 y 5 - r U Ž f ., yU g ydom f U and 5 yU 5 Ž r f .. Then there exists K Ž y, yU . g R such that K Ž y, yU . G 0, x g U and 5 x 5 ) K Ž y, yU .
«
f Ž x. 5 x5
) 5 yU 5 q
f Ž y . q ² y, yU : 5 x5
and xU g dom f U and 5 xU 5 ) K Ž y, yU . «
f U Ž xU . 5 xU 5
) 5 y5 q
f U Ž yyU . q ² y, yU : 5 xU 5
.
196
S. SIMONS
Žb.
Let x g U and xU g f Ž x .. Then
² y y x, yU q xU : G 0
5 x 5 F K Ž y, yU . and 5 xU 5 F K Ž y, yU . .
«
Proof. If U is unbounded then lim inf
f Ž x.
xgU , 5 x 5ª`
5 x5
s r Ž f . ) 5 yU 5 s lim
5 x 5ª`
5 yU 5 q
f Ž y . q ² y, yU : 5 x5
;
hence there exists K 1 g R such that x g U and 5 x 5 ) K 1
f Ž x.
«
5 x5
) 5 yU 5 q
f Ž y . q ² y, yU : 5 x5
. Ž 26.1.
If U is bounded we take K 1 [ sup 5 U 5, and Ž26.1. is vacuously true. In a similar fashion, there exists K 2 g R such that xU g dom f U and 5 xU 5 ) K 2 «
f U Ž xU . 5 xU 5
) 5 y5 q
f U Ž yyU . q ² y, yU : 5 xU 5
.
Now Ža. follows with K Ž y, yU . [ K 1 k K 2 k 0. From Ža., if 5 x 5 ) K Ž y, yU . then ² y y x, yU q xU : s ² y y x, yU : q ² y y x, xU : F ² y y x, yU : q f Ž y . y f Ž x . F 5 x 5 5 yU 5 q f Ž y . q ² y, yU : y f Ž x . - 0, while if 5 xU 5 ) K Ž y, yU . then ² y y x, yU y xU : s ² y, yU q xU : q ² x, yyU y xU : F ² y, yU q xU : q f U Ž yyU . y f U Ž xU . F 5 y 5 5 xU 5 q f U Ž yyU . q ² y, yU : y f U Ž xU . - 0. This completes the proof of Žb.. THEOREM 27. and
Let F be a nonempty finite monotone subset of U = EU
R F [ min K Ž y, yU . : Ž y, yU . g F , 5 y 5 - r U Ž f . , yU g ydom f U , 5 yU 5 - r Ž f . 4 . Then there exist x F g U and xUF g f Ž x F . such that 5 x F 5 F R F , 5 xUF 5 F R F and for all Ž y, yU . g F ,
f Ž x F . q f U Ž xUF . F ² y, xUF : q ² y y x F , yU : .
Ž 27.1.
197
RANGE OF A MONOTONE OPERATOR
U .4 . Define the matrix Q by qi j [ Proof. Let F s Ž y 1 , yU1 ., . . . , Ž ym , ym U ² yi y y j , yi :. Arguing as in Lemma 6, Q is copositive. Thus, from Theorem 24, there exist b g sm and xUF g f ŽÝ j bj y j . such that
for all i s 1, . . . , m,
¦
yi y
Ý bj yj , xUF
;
j
q
Ý qi j bj G bT Qb G 0. j
It follows from this that for all i s 1, . . . , m,
¦
Ý bj yj , xUF
¦
Ý bj yj , xUF
yi y
;
j
q
Ý ² yi y yj , yUi : bj G 0, j
that is, for all i s 1, . . . , m,
yi y
;¦
j
q yi y
Ý bj yj , yUi j
;
G 0.
Hence, writing x F [ Ý j bj y j , for all Ž y, yU . g F ,
² y y x F , yU q xUF : G 0.
Then Ž27.1. follows since f Ž x F . q f U Ž xUF . s ² x F , xUF :. Finally, from Lemma 26Žb., 5 x F 5 F R F and 5 xUF 5 F R F . Notation 28. f UU : EUU ª R j `4 is defined, as usual, by f UU Ž xUU . [ sup EU Ž xUU y f U .. f UU is proper and convex and w Ž EUU , EU .-lower semicontinuous. Furthermore, for all x g U,
f UU Ž ˆ x. F f Ž x. .
Ž 28.1.
We write V [ y : y g E, 5 y 5 - r U Ž f .4 and V U [ yU : yU g EU , 5 yU 5 - r Ž f .4 . We do not generally have equality in Ž28.1., since we are not assuming that f is lower semicontinuous on E. U
LEMMA 29. Let T : E ª 2 E be monotone, DŽT . ; U, and T Ž V . q dom f U l V U 2 0. Then there exists Ž xU , xUU . g EU = EUU such that yxU g TxUU and f UU Ž xUU . q f U Ž xU . y ² xU , xUU : F
inf
Ž y , y U .gG ŽT .
² yU q x U , ˆ y y xUU : .
Ž 29.1. Proof. We choose y 0 g U l V and yU0 g Ty 0 such that yyU0 g dom f U l V U . From Theorem 27, for all finite subsets F of GŽT . such that
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S. SIMONS
F 2 Ž y 0 , yU0 ., there exist x F g U and xUF g f Ž x F . such that 5 x F 5 F K Ž y 0 , yU0 ., 5 xUF 5 F K Ž y 0 , yU0 . and for all Ž y, yU . g F ,
f Ž x F . q f U Ž xUF . F ² y, xUF : q ² y y x F , yU : . $
UU . F f Ž x F . and We now define xUU . Then f UU Ž xUU F [ x Fg E F
for all Ž y, yU . g F ,
U U ² U y : q ² yU , ˆ :. f UU Ž xUU y y xUU F . q f Ž xF . F xF , ˆ F
It now follows from the w Ž EU , E . = w Ž EUU , EU .-compactness of the sets
Ž xU , xUU . : Ž xU , xUU . g EU = EUU , 5 xU 5 F K Ž y 0 , yU0 . , 5 xUU 5 F K Ž y 0 , yU0 . 4 and the w Ž EU , E .- and w Ž EUU , EU .-lower semicontinuity of f U and f UU that there exists Ž xU , xUU . g EU = EUU such that for all Ž y, yU . g G Ž T . ,
f UU Ž xUU . q f U Ž xU . F ² xU , ˆ y : q ² yU , ˆ y y xUU : .
Inequality Ž29.1. is immediate from this. Since f UU Ž xUU . q f U Ž xU . y ² xU , xUU : G 0, it follows from Ž29.1. that yxU g TxUU . This completes the proof of Lemma 29. THEOREM 30. DŽ S . ; U. Then
Let S: E ª 2 E
RŽ S q Ž f U .
y1
U
be of type ŽNI. and monotone and
. > S Ž V . q dom f U l V U . U
Proof. Let xU g SŽ V . q dom f U l V U . Define T : E ª 2 E by T [ S y zU . Then T is of type ŽNI. and monotone, DŽT . s DŽ S . ; U, and T Ž V . q dom f U l V U s S Ž V . q dom f U l V U y zU 2 0. From Lemma 29, there exists Ž xU , xUU . g EU = EUU such that yxU g TxUU s SxUU y zU and f UU Ž xUU . q f U Ž xU . y ² xU , xUU : F 0, i.e., xUU g f U Ž xU .. Thus zU s Ž zU y xU . q zU g SxUU q Ž f U . This completes the proof of Theorem 30.
y1
Ž xUU . g R Ž S q Ž f U .
y1
..
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RANGE OF A MONOTONE OPERATOR
Remark 31. We now give some special cases of Theorem 30. U
Ža. If f is coercive Žthat is to say, r Ž f . s `., S: E ª 2 E is of type ŽNI. and monotone, and DŽ S . ; U then RŽ S q Ž f U .
y1
. > S Ž V . q dom f U .
ŽSee Remark 33 for an improvement of this result.. U Žb. If f U is coercive Žthat is, r U Ž f . s `., S: E ª 2 E is of type ŽNI. and monotone, and DŽ S . ; U then RŽ S q Ž f U .
y1
. > R Ž S . q dom f U l V U .
Ž 31.1. U
We take some time now to discuss a special case of Ž31.1.. LetU S: E ª 2 E be of type ŽNI. and monotone. Let l ) 0, and L: EUU ª 2 E be defined by LxUU [ xU : xU g EU , 5 xU 5 F l , ² xU , xUU : s l 5 xUU 5 4 . Then R Ž S q L . > R Ž S . q x U : 5 x U 5 - l4 .
Ž 31.2.
In order to prove this, we take U [ E and f [ l 5 5, for which f U is coercive. Then Ž31.2. follows from Ž31.1., since Ž f U .y1 s L, V U s xU : 5 xU 5 - l4 and dom f U s V U . U Žc. If both f and f U are coercive, S: E ª 2 E is of type ŽNI. and monotone, and DŽ S . ; U then RŽ S q Ž f U .
y1
. > R Ž S . q dom f U .
ŽSee Remark 33 for an improvement of this result. THEOREM 32. Let E be reflexi¨ e, f : E ª R j `4 be proper, con¨ ex, and U lower semicontinuous, and dom f be open. Let S: E ª 2 E be monotone. Ža.
If S is of type ŽNI. and DŽ S . ; dom f then R Ž S q f . > S Ž V . q dom f U l V U .
Žb.
If S is maximal monotone and DŽ S . ; dom f then R Ž S q f . > S Ž V . q dom f U l V U .
Žc.
If S is of type ŽNI. and DŽ S . ; dom f then R Ž S q f . > S Ž V . q dom f U l V U .
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S. SIMONS
Proof. We write U [ dom f. From w8, Proposition 3.3, p. 39x, f is continuous on U. Further, Ž f U .y1 s f. Thus Ža. follows from Theorem 30. Žb. is immediate from Ža. since if S is maximal monotone then S is of type ŽNI. and S s S. Žc. follows by applying Žb. with S replaced by S since, from Theorem 19ŽŽa. « Žc.., S is maximal monotone. If E is reflexive and f [ 5 5 2r2 then f s J, r Ž f . s r U Ž f . s `, V s E, V s EU , and dom f U s EU ; hence Theorem 32Žb. generalizes Theorem 1Ža.. Finally, still supposing that E is reflexive, we conjecture that the conclusion of Theorem 30 cannot be strengthened to read U
R Ž S q f . > S Ž V . q dom f U l V U Žthat is, unless f is lower semicontinuous on E and Theorem 32Žb. can be applied.. Remark 33. The author is grateful to Professors Heinz Bauschke and Jon Borwein for pointing out to him that
rŽ f . s `
« dom f U s EU .
Ž 33.1.
Indeed, suppose that r Ž f . s `. Let z g U and zU g f Ž z . be fixed. If now xU is any element of EU , then there exists K g R such that K G 5 z 5 and x g U and 5 x 5 ) K
«
f Ž x.
) 5 xU 5 q
5 x5
f Ž z . y ² z, xU : 5 x5
,
from which it follows that x g U and 5 x 5 ) K
« ² x, xU : y f Ž x . F ² z, xU : y f Ž z . .
Thus f U Ž xU . s sup Ž xU y f . s sup ² x, xU : y f Ž x . 5 x 5FK
U
F sup w ² x, xU y zU : x q sup ² x, zU : y f Ž x . 5 x 5FK U
5 x 5FK
U
U
U
F K 5 x y z 5 q f Ž z . - `. This establishes Ž33.1.. Consequently, in Remark 31Žc., S q Ž f U .y1 is surjective. The same is true in Remark 31Ža. provided that DŽ S . l V / B.
RANGE OF A MONOTONE OPERATOR
201
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