On the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity*

Journal of Mathematical Analysis and Applications 237, 30᎐42Ž. 1999 Article ID jmaa.1999.6437, available online at http:rrwww.idealibrary.com on

† ‡ View metadata, citation and similarAris papers Daniilidis at core.ac.ukand Nicolas Hadjisavvas brought to you by CORE

provided by Elsevier - Publisher Connector Department of Mathematics, Uni¨ersity of the Aegean, 83200 Karlo¨assi, Samos, Greece

Submitted by Arrigo Cellina

Received January 7, 1998

The notions of cyclic quasimonotonicity and cyclic pseudomonotonicity are introduced. A classical result of convex analysis concerning the cyclic monotonicity of theŽ. Fenchel᎐Moreau subdifferential of a convex function is extended to corresponding results for the Clarke᎐Rockafellar subdifferential of quasiconvex and pseudoconvex functions. The notion of proper quasimonotonicity is also introduced. It is shown that this new notion retains the characteristic property of quasimonotonicityŽ i.e., a lower semicontinuous function is quasiconvex if and only if its Clarke᎐Rockafellar subdifferential is properly quasimonotone. , while it is also related to the KKM property of multivalued maps; this makes it more useful in applications to variational inequalities. ᮊ 1999 Academic Press

1. INTRODUCTION

Let X be a Banach space and f: X ª R jqÄ4ϱ a lower semicontinu- ousŽ. lsc function. According to a relatively recent result of Correa, Joffre, and ThibaultŽ seewx 7 for reflexive and wx 8 for arbitrary Banach spaces. , the function f is convex if and only if its Clarke᎐Rockafellar subdifferential Ѩ f is monotone. In the same line of research, much work has been done to characterize the generalized convexity of lsc functions by a corresponding generalized monotonicity of the subdifferential. Thus Lucwx 15 and, inde-

* Work supported by a grant of the Greek Ministry of Industry and Technology. † E-mail: [email protected] ‡ E-mail: [email protected]

30 0022-247Xr99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. SUBDIFFERENTIALS AND QUASICONVEXITY 31 pendently, Aussel, Corvellec and Lassondewx 2 , showed that f is quasiconvex if and only if Ѩ f is quasimonotone. Similarly, Penot and Quangwx 16 showed that if the function f is also radially continuous, then f is pseudoconvex if and only if Ѩ f is pseudomonotoneŽ in the sense of Karamardian and Schaiblewx 14 , as generalized for multivalued operators by Yaowx 20. . In Section 2, we review these results, together with some notation and definitions, and show that in most cases the radial continuity assumption is not necessary. However, since the Clarke᎐Rockafellar subdifferential of a convex function coincides with the classical Fenchel᎐Moreau subdifferentialwx 19 , it is not only monotone, but also cyclically monotonewx 17 . In Section 3 of this work, we define analogous notions of cyclic quasimonotonicity and cyclic pseudomonotonicity and show that the subdifferential of quasimonotone and pseudomonotone functions have these properties, respectively. Cyclic generalized monotonicity is not just a stronger property than the corresponding generalized monotonicity, but it expresses a behavior of a specific kind. In particular, an operator can even be strongly monotone without being cyclically quasimonotone. CyclicŽ. generalized monotonicity describes the behavior of an operator around a ‘‘cycle’’ consisting of a finite number of points. In Section 4 we consider instead the convex hull of such a cycle. We show that the definitions of monotone and pseudomonotone operators can be equivalently stated in terms of this convex hull. This is not so for quasimonotone operators; this leads to the introduction of the new notion of a properly quasimonotone operator. We show that this new notion, while retaining the important characteristics of quasimonotonicityŽ in particular, f is quasiconvex if and only if Ѩ f is properly quasimonotone. is often easier to handle; in particular, it is closely related to the KKM property of multivalued maps. We show this by an application to Variational Inequalities. In addition, quasimonotonicity and proper quasimonotonicity are identical on one-dimensional spaces, which is probably the reason why the latter escaped attention.

2. RELATIONS BETWEEN GENERALIZED CONVEXITY AND GENERALIZED MONOTONICITY

We denote by X* the dual of X and by Ž.x*, x the value of x* g X*at x g X. For x, y g X we set wxx, y s Ätx q Ž.1 y ty:0F t F 14 and define Ž x, yxw, x, y.Ž., and x, y analogously. Given a lsc function f: X ª R j Ä4qϱ with domain domŽ.f [ Ä x g X : fxŽ.- qϱ4 / л,the 32 DANIILIDIS AND HADJISAVVAS

᎐ g Clarke Rockafellar generalized derivative of f at x0 domŽ.f in the direction d g X is given byŽ seewx 19 .

fxŽ.Ž.q tdЈ y fx s fxŽ.0 , d sup lim sup inf ,Ž. 2.1 ␧) ª dЈgBdŽ. t 0 x f x0 ␧ to0

55 where Bd␧Ž.s ÄdЈ g X : dЈ y d - ␧4, t o 0 indicates the fact that t ) 0 ª ª ª ª and t 0, and x f x000means that both x x and fxŽ.fx Ž .. ᎐ The Clarke Rockafellar subdifferential of f at x0 is defined by

Ѩ s g F ᭙ g fxŽ.00Ä4x* X : Žx*, d .f Žx , d ., d X .2.2Ž.

We recall that a function f is called quasiconvex, if for any x, y g X and z g wxx, y we have

fzŽ.F maxÄ4fx Ž., fy Ž..2.3Ž.

A lsc function f is called pseudoconvexwx 16 , if for every x, y g X, the following implication holds:

᭚ x* g Ѩ fxŽ.Ž: x*, y y x .G 0 « fxŽ.F fy Ž..2.4 Ž .

It is knownwx 16 that a lsc pseudoconvex function which is also radially continuousŽ. i.e., its restriction to line segments is continuous , is quasiconvex. Both quasiconvexity and pseudoconvexity of functions are often used in Optimization and other areas of applied mathematics when a convexity assumption would be too restrictivewx 5 . Let T: X ª 2X * be a multivalued operator with domain DTŽ.s Ä x g X : TxŽ./ л4. The operator T is called g Ž.i cyclically monotone, if for every x12, x ,..., xn X and every U g U g U g x1122TxŽ., x TxŽ.,..., xnnTxŽ.we have

n U y F Ý Ž.xii, x q1 xi 02.5 Ž. is1

[ Ž.where xnq11x . Ž.ii monotone, if for any x, y g X, x* g TxŽ., and y* g TyŽ.we have

Ž.y* y x*, y y x G 0. Ž2.6. SUBDIFFERENTIALS AND QUASICONVEXITY 33

Žiii . pseudomonotone, if for any x, y g X, x* g Tx Ž., and y* g Ty Ž. the following implication holds:

Ž.x*, y y x G 0 « Ž.y*, y y x G 02.7 Ž. or equivalently,

Ž.x*, y y x ) 0 « Ž.y*, y y x ) 0. Ž.2.8

Ž.iv quasimonotone, if for any x, y g X, x* g TxŽ., and y* g TyŽ. the following implication holds:

Ž.x*, y y x ) 0 « Ž.y*, y y x G 0. Ž.2.9

The above properties were listed from the strongest to the weakest. We recall the hitherto known results connecting generalized convexity with generalized monotonicity:

THEOREM 2.1. Let f: X ª R jqÄ4ϱ be a lower semicontinuous function. Then Ž.i f is conëx if and only if Ѩ f is monotone wx8. In this case Ѩ f is also cyclically monotoneŽ see for instance wx17. . Ž.ii f is quasiconëx if and only if Ѩ f is quasimonotone Ž see wx2 or w15 x. . Ž.iii Let f be also radially continuous. Then f is pseudoconëx if and only if Ѩ f is pseudomonotoneŽ see wx4 or w16 x. . We now show that pseudoconvexity of a function f implies quasiconvexity of f and pseudomonotonicity of Ѩ f, even without the radial continuity assumption:

PROPOSITION 2.2. Let f: X ª R j Ä4ϱ be a lsc, pseudoconëx function with conëx domain. Then Ž.i f is quasiconëx Ž.ii Ѩ f is pseudomonotone. g g Proof. Ž.i Suppose that for some x12, x domŽ.f and some y ) s Ž.x12, x we have fy Ž.maxÄ fxŽ.Ž.1, fx 24Ä. Set m max fxŽ.Ž.1, fx 24. Since f is lower semicontinuous, there exists some ␧ ) 0 such that wx fyŽ.Ј ) m, for all yЈ g By␧ Ž.. From Ž 2.4 . it follows Ž see also 4. that the sets of local and global minimizers of the function f coincide; hence the point y cannot be a local minimizer, so there exists w g By␧Ž.such that fwŽ.- fy Ž.. Applying Zagrodny’s Mean Value Theoremwx 21, Theorem 4.3 wxg w ª to the segment w, y , we obtain u w, y., a sequence un u, and 34 DANIILIDIS AND HADJISAVVAS

U g Ѩ U y ) g unnfuŽ., such that Žu nn, y u . 0. Since y coÄ4 x12, x it follows U y ) g that Žuni, x u n. 0, for some i Ä41, 2 . Using relationŽ. 2.4 we get G G G m fxŽ.infu Ž .and, since f is lower semicontinuous, m fuŽ.. This clearly contradicts the fact that u g By␧Ž.. Ž.ii Let x* g Ѩ fxŽ.be such that Ž.x*, y y x ) 0. By partŽ. i , f is quasiconvex, so applying Theorem 2.1Ž. ii we conclude that Ѩ f is quasimonotone. Hence Ž.y*, y y x G 0, for all y* g Ѩ fy Ž.. Suppose to the contrary that for some y* g Ѩ fyŽ.we have Žy*, y y x .s 0. From relation Ž2.4 . we obtain fx Ž.G fy Ž..

y ) ␧ ) On the other hand, since fxŽ.; y x 0, there exists 1 0, such ª o Ј g q Ј that for some xnnx, t 0 and for all y By␧ Ž., we have fx Žnnty Ž y ) Ј1 ) Ј g xnn..Ž.fx . Quasiconvexity of f implies fyŽ.fx Ž.n, for every y By␧ Ž.. In particular fyŽЈ .G fx Ž.Žsince f is lsc . , hence fy ŽЈ .G fy Ž.. The 1 latter shows that y is a local minimizer, hence a global one. This is a ) contradiction, since we have at least fyŽ.fx Žn ..

It is still an open question whether pseudomonotonicity of Ѩ f implies pseudoconvexity of f, without the radial continuity assumption. As a partial result, we have the following proposition, which will be of use in the next section.

PROPOSITION 2.3. Let f be a lsc function such that Ѩ f is pseudomonotone. Then f has the following properties:

Ž.i If 0 g Ѩ fx Ž ., then x is a global minimizer Ž.ii ᭚ x* g Ѩ fx Ž.Ž: x*, y y x .) 0 « fy Ž.) fx Ž..

Proof. Ž.i Suppose that fy Ž.- fx Ž.. Then using again Zagrodny’s ª g w Mean Value Theorem, we can find a sequence zn z y, x. and U g Ѩ U y ) y znnfzŽ., such that Žz nn, x z .Ž0. By pseudomonotonicity, x*, x ) g Ѩ f Ѩ zn.Ž0 for all x* fx.Ž, i.e., 0 fx.. Ž.ii Let us assume that for some x* g Ѩ fxŽ.Ž.we have x*, y y x ) 0. We may choose ␧ ) 0 such that Ž.x*, yЈ y x ) 0, for all yЈ g By␧ Ž.. Since Ѩ f is obviously quasimonotone, from Theorem 2.1Ž. ii we conclude that f is quasiconvex; it then follows that fyŽ.G fx Ž.Žsee for instance Theorem 2.1 inwx 4. . Suppose to the contrary that fx Ž.Ž.s fy. Then fy Ž.Ž.Ž.Ј G fxs fy, so f has a local minimum at y. It follows that 0 g Ѩ fyŽ.Žsee for instance w21, Theorem 2.2Ž. cx. . However, Ѩ f is pseudomonotone, hence we should haveŽŽ..Ž. see relation 2.8 y*, y y x ) 0, for all y* g Ѩ fy Ž., a contradiction. SUBDIFFERENTIALS AND QUASICONVEXITY 35

3. GENERALIZED CYCLIC MONOTONICITY

We first introduce cyclic quasimonotonicity.

DEFINITION 3.1. An operator T: X ª 2 X * is called cyclically quasi- g g monotone, if for every x12, x ,..., xn X, there exists an i Ä41, 2, . . . , n such that U y F ᭙ U g Ž.xii, x q1 xiii0, x Tx Ž.Ž.3.1

[ Ž.where xnq11x . It is easy to see that a cyclically monotone operator is cyclically quasimonotone, while a cyclically quasimonotone operator is quasimonotone. Cyclic quasimonotonicity is considerably more restrictive than quasimono- tonicityŽ. see Example 3.5 below . However, this property characterizes subdifferentials of quasiconvex functions, as shown by the next theorem.

THEOREM 3.2. Let f: X ª R jqÄ4ϱ be a lower semicontinuous function. Then f is quasicon¨ex if and only if Ѩ f is cyclically quasimonotone. Proof. In view of Theorem 2.1Ž. ii , we have only to prove that if f is quasiconvex then Ѩ f is cyclically quasimonotone. g Ѩ Assume to the contrary that there exist x12, x ,..., xn DŽ.f and U g Ѩ U y ) s xiifxŽ.such that Žx ii, x q1 xi.Ž0, for i 1, 2, . . . , n where as s y ) usual xnq11x .. It follows that fxŽ.ii, x q1 xi 0. In particular, for ␧ ) ␦ ) every i there exists ii0, 0 such that

X q y X fxŽ.Ž.iitd fx lim sup inf ) ␦ ) 0.Ž. 3.2 X i ª dgBx␧ Ž.q yx t x ifix i i 1 i to0

␧ s ␧ ␦ s ␦ g We set min is1, 2 иииniand min is1, 2 иииni. For any y X g y X g y Bx␧ r2Ž.iiand x q1 Bx␧ r2Ž.iq1 we have y xiq1 Bx␧ Ž.iq1 xi ; hence g g we can choose xi Bx␧ r2Ž.iiand t Ž0, 1 . such that

q X y y fxŽ.iiitxŽ.Ž.q1 xiifx inf ) ␦ ) 03.3Ž. X g x iq1 Bx␧ r2Ž.iq1 ti or equivalently

q XXy ) q ␦ ᭙ g fxŽ.iiitxŽ.Ž.q1 xiiiifx t , x q1 Bx␧ r2 Ž.Ž.iq1 3.4 for i s 1, 2, . . . , n. 36 DANIILIDIS AND HADJISAVVAS

X s Now for every i we choose xiq1 xiq1, henceŽ. 3.4 becomes q y ) q ␦ fxŽ.iiitxŽ.q1 xiiifxŽ. t Ž.3.5 for i s 1, 2, . . . , n. Since f is quasiconvex,Ž. 3.5 implies that G q y fxŽ.iq1 fxŽ.iiitxŽ.q1 xi Ž.3.6 for i s 1, 2, . . . , n. Combining withŽ. 3.5 and adding for i s 1, 2, . . . , n,we ) ␦ n get 0 Ž.Ýis1 ti , a contradiction. Inwx 18 it was proved that the subdifferential of a convex function is a maximal monotone and maximal cyclically monotone operator. An analogous property does not hold for quasiconvex functions, since, for the quasiconvex function fxŽ.s sgn Ž.xx'<<, x g R, it is knownŽ seewx 15. that Ѩ f is not maximal quasimonotone. The following proposition shows that it is neither maximal cyclically quasimonotone:

PROPOSITION 3.3. E¨ery quasimonotone operator T: R ª 2 R is cyclically quasimonotone. Proof. We assume to the contrary that the operator T is quasimono- g U g tone and there exist x12, x ,..., xniiR, x TxŽ., such that U y ) Ž.xii, x q1 xi 03.7Ž. s s s for i 1, 2, . . . , n Ž.where xnq11x . Set xMimax s1,2,...,nix . Then U - - relationŽ. 3.7 implies that xMM0. On the other hand, since x y1 xM , U ) U y ) we conclude fromŽ. 3.7 that xMy1 0. Thus Ž xMy1, xMMx y1. 0, U y - while Ž xMM, x x My1. 0, which contradicts the definition of quasimonotonicity. We now introduce cyclic pseudomonotonicity:

DEFINITION 3.4. An operator T: X ª 2 X * is called cyclically pseu- g domonotone, if for every x12, x ,..., xn X, the following implication holds: ᭚ g ᭚ UUg y ) i Ä41,2,...,n , xiiiiTxŽ.Ž: x , x q1 xi . 0 « ᭚ g ᭙ UUg y - j Ä41,2,..., n , x jjjjTxŽ.Ž: x , x q1 x j .03.8 Ž . [ Ž.where xnq11x . One can easily check that every cyclically monotone operator is cyclically pseudomonotone, while every cyclically pseudomonotone operator is pseudomonotone and cyclically quasimonotone. On the other hand, the SUBDIFFERENTIALS AND QUASICONVEXITY 37 following example shows that cyclic generalized monotonicity differs essen- tially from generalized monotonicity: 22ª syabq EXAMPLE 3.5. Let T: R R be defined by TaŽ.Ž, b 22b, a .. Then the operator T is monotoneŽ and even strongly monotone, i.e., satisfies ŽŽ.Txy Ty Ž., x y y .G kx55y y 2 for all x, y g R2 where k is a positive constant. . In particular, T is pseudomonotone and quasimonotone. However, it is not cyclically quasimonotone, as one sees by consider- s s sy s y ing the points x123Ž.1, 0 , x Ž.0, 1 , x Ž1, 0 . , and x 4 Ž0, 1. . We now show the following strengthening of Theorem 2.1Ž. iii .

THEOREM 3.6. Let f: X ª R jqÄ4ϱ be a lsc function. If f is pseudo- conëx, then Ѩ f is cyclically pseudomonotone. Conërsely, if Ѩ f is pseudomonotone and f is radially continuous, then f is pseudoconëx. Proof. Again we have only to show that if f is pseudoconvex then Ѩ f is cyclically pseudomonotone. Assume to the contrary that there exist g Ѩ U g Ѩ U y G x12, x ,..., xniiiiDŽ.f and x fxŽ.such that Žx , x q1 xi. 0, for U i s 1, 2, . . . , n Ž.where x q s x , while for some i and some x g Ѩ fxŽ. n 11 oiioo we have

U x , x q y x ) 0. 3.9 Ž.iioo1 i o Ž.

G By the definition of pseudoconvexityŽŽ..Ž. relation 2.4 we have fxiq1 fxŽ., for i s 1, 2, . . . , n, hence all fxŽ.are equal. In particular, fxŽ.q iiio 1 s fxŽ., which contradictsŽ. 3.9 in view of Proposition 2.3. i o

4. PROPER QUASIMONOTONICITY

The definitions of monotonicity and pseudomonotonicity have an equivalent formulation, which involves a finite cycle of points and its convex hull:

PROPOSITION 4.1.Ž. i An operator T is monotone, if and only if for any g ¨ s n ␭ n ␭ s ␭ ) x12, x ,..., xniX and e ery y Ý s1 iix , with Ý is1 ii1 and 0, one has

n ␭ U y F Ý iiisup Ž.x , y x 0. Ž. 4.1 s Ug i 1 x iiTxŽ.

Ž.ii An operator T with con¨ex domain DŽ. T is pseudomonotone, g ¨ s n ␭ if and only if for any x12, x ,..., xniX and e ery y Ý s1 iix , with 38 DANIILIDIS AND HADJISAVVAS

n ␭ s ␭ ) Ýis1 ii1 and 0, the following implication holds: ᭚ g ᭚ UUg y ) i Ä41,2,..., n , xiiiiTxŽ.Ž: x , y x . 0 « ᭚ g ᭙ UUg y - j Ä41,2,...,n , x jjjjTxŽ.Ž: x , y x .0. Ž 4.2 .

Proof. If the operator T satisfies conditionŽ.Ž 4.1 respectively Ž.. 4.2 , s q r then by choosing y Ž.x12x 2, we conclude that it is monotone Ž.respectively, pseudomonotone . Hence it remains to show the two oppo- site directions. g Let us first suppose that T is monotone. Then for any x12, x ,..., xn X, U g s s nn␭ ␭ s any xiiTxŽ.Žfor i 1, 2, . . . , n.and any y Ý js1 jjx , with Ý js1 j ␭ ) 1 and j 0, we have nnn ␭ UUy s ␭␭ y ÝÝiiŽ.x , y x i Ýiji Žx , x jx i . is1 is1 js1 s ␭␭ UUy q y Ý ijŽ.Ž.x i, x jx ix j, x ix j i)j s ␭␭ U y U y F Ý ijŽ.x ix j, x jx i 0, i)j where the last inequality is a consequence of the monotonicity of T. Hence T satisfies relationŽ. 4.1 . We now suppose that the operator T is pseudomonotone. If relation g U g Ž.4.2 does not hold, then there exist x12, x ,..., xniiX, x TxŽ.for s s n ␭ n ␭ s ␭ ) i 1, 2, . . . , n, and some y Ý js1 jjx with Ý js1 jj1 and 0, such that U y G Ž.xii, y x 04.3Ž. while for at least one i Ž.say i s 1, U y ) Ž.x11, y x 0.Ž. 4.4 g / л In particular we have x12, x ,..., xn DTŽ., hence Ty Ž. . Choose any y* g TyŽ.. Relations Ž. 2.7 and Ž. 4.3 show that y G Ž.y*, y xi 04.5Ž. g ␭ y s for all y* TyŽ.and all i’s. Since ÝiiŽ.y*, y xi 0, relationsŽ. 4.5 y s show that Ž.y*, y xi 0 for all i’s. On the other hand, relationŽ. 4.4 y ) together with relationŽ. 2.8 imply that Žy*, y x1 . 0, a contradiction. In view of Proposition 4.1, one could seek an equivalent formulation for the definition of quasimonotonicity, which would involve again the convex SUBDIFFERENTIALS AND QUASICONVEXITY 39 hull of a finite cycle. However, in contrast to monotone and pseudomonotone operators, this leads to a different, more restrictive definition:

DEFINITION 4.2. An operator T: X ª 2 X * is called properly quasi- g s n ␭ monotone, if for every x12, x ,..., xniX and every y Ý s1 iix , with n ␭ s ␭ ) Ýis1 ii1 and 0, there exists i such that ᭙ U g U y F xiiiiTxŽ.Ž: x , y x . 0.Ž. 4.6 s q r Choosing y Ž.x12x 2, we see that a properly quasimonotone operator is quasimonotone. As in Proposition 3.3, it is easy to show that the converse is true whenever X s R; however, it is not true in general, as the following example shows. s 2 s s s EXAMPLE 4.3. Let X R , x123Ž.0, 1 , x Ž.0, 0 , x Ž.1, 0 . We 22ª sy y s s define T: R R by TxŽ.Ž1231, 1, .Ž.Ž.Ž.Ž.Tx 1, 0 , Tx 0, 1 , and TxŽ.s 0 otherwise. It is easy to check that T is quasimonotone but s q q r not properly quasimonotoneŽŽ.. it suffices to consider y x123x x 3. The class of properly quasimonotone operators, though strictly smaller than the class of quasimonotone operators, is in a sense not much smaller. This is shown in the next proposition.

PROPOSITION 4.4.Ž. i Eëry pseudomonotone operator with conëx domain is properly quasimonotone. Ž.ii Eëry cyclically quasimonotone operator is properly quasimonotone. Proof. Ž.i This is an obvious consequence of Proposition 4.1Ž. ii . Ž.ii Suppose that the operator T is not properly quasimonotone. g U g s Then there would exist x12, x ,..., xniiDTŽ., x TxŽ., and y n ␭ ␭ ) Ýis1 iix with i 0, such that U y ) Ž.xii, y x 04.7Ž. s s ␭ U for i 1, 2, . . . , n. Set xiŽ1. x1. RelationŽ. 4.7 implies that Ý jjŽx iŽ1., x j y ) / U y ) xiŽ1..Ž0. It follows that for some x j x1 we have xiŽ1., x jix Ž1.. 0. s We set xiŽ2. x j and apply relationŽ. 4.7 again. Continuing in this way, we define a sequence xiŽ1., xiŽ2., . . . such that U y ) Ž.xiŽk. , xiŽkq1. xiŽk. 04.8 Ž. for all k g N. g - Since the set Ä4x12, x ,..., xn is finite, there exist m, k N, m k such s that xiŽkq1. xiŽm.. Thus, for the finite sequence of points xiŽm., xiŽmq1.,..., xiŽk., relationŽ. 4.8 holds. This means that T is not cyclically quasimonotone. 40 DANIILIDIS AND HADJISAVVAS

Combining Proposition 4.4Ž. ii and Theorem 3.2, we get the following corollary.

COROLLARY 4.5. A lower semicontinuous function f is quasicon¨ex if and only if Ѩ f is properly quasimonotone. The converse of Proposition 4.4 does not hold. For instance, the operator T defined in Example 3.5 is properly quasimonotoneŽ since it is monotone, hence pseudomonotone. , but not cyclically quasimonotone. On the other hand, any subdifferential of a continuous quasiconvex function f is properly quasimonotone, but not pseudomonotone unless f is also pseudoconvex. Thus, between the various generalized monotonicity properties we considered, the following strict implications hold, and none other: cyclically monotone ª monotone xx cyclically pseudomonotone ª pseudomonotone xx cyclically quasimonotone ªproperly quasimonotone x quasimonotone

Note that the implicationŽ pseudomonotone ª properly quasimonotone. holds under the assumption that the domain of the operator is convex. We recall that a multivalued mapping G: X ª 2X * is called KKMwx 11 , g g g if for any x12, x ,..., xn X and any y coÄ4 x12, x ,..., xn one has y D ª X * iiGxŽ.. It is easy to see that an operator T: X 2 is properly quasimonotone if and only if the multivalued mapping G: X ª 2 X * defined by GxŽ.s ½5y g K : sup Ž.x*, y y x F 04.9Ž. x*gTxŽ. is KKM. This suggests an obvious application to variational inequalities. All known theorems of existence of solutions for quasimonotone variational inequality problems require extra assumptions on the domain of the operatorŽ seewx 12. and, in the case of a multivalued operator, on its values Žseewx 9. . As the following theorem shows, existence of solutions for properly quasimonotone operators requires very weak assumptions. We first recall fromwx 1 the following definition.

DEFINITION 4.6. The operator T: X ª 2 X * is called upper hemicontinuous, if its restriction to line segments of its domain is upper semicontinuous, when X* is equipped to the weak-) topology. SUBDIFFERENTIALS AND QUASICONVEXITY 41

We now have:

THEOREM 4.7. Let K be a nonempty, con¨ex and w-compact subset of X. If T is a properly quasimonotone, upper hemicontinuous operator with K : g ¨ g DTŽ., then there exists an x0 K, such that for e ery x K, there exists g x* TŽ. x0 such that

y G Ž.x*, x x0 0.Ž. 4.10

Proof. Since the multivalued map G defined byŽ. 4.9 is KKM, and the sets GxŽ.are obviously weakly closed, by Ky Fan’s Lemmawx 10 one has F / л g F x g K GxŽ.. Take any x0 x g K Gx Ž.. We shall show that x0 is actually a solution ofŽ. 4.10 . g g We assume, to the contrary, that for some x K and all x* TxŽ.0 we y - s g y - have Ž.x*, x x000. The set V Ä x* X*:Ž.x*, x x 0isa4 w*- s q y neighborhood of TxŽ.0 ; hence, if we set xt tx Ž.1 tx0 , by the upper : hemicontinuity assumption, we have TxŽ.t V for all t sufficiently small. y s y y - g Since xt x00txŽ.x , this means that Žx*, xt x0 .0 for all x* f TxŽ.t , i.e., x0 Gx Ž.t . This contradicts the definition of x0. We conclude with a final remark. The notion of a quasimonotone operator was introduced to describe a property that characterizes the subdifferential of a lsc quasiconvex function. Since proper quasimonotonicity does exactly the same thing and is directly related to the KKM property, it is possibly a good candidate to replace quasimonotonicity in most theoretical and practical applications.

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