Equilibrium Problems Via the Palais-Smale Condition

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 105, No. 2, pp. 299–323, MAY 2000

Equilibrium Problems with GeneralizedMonotone Bifunctions and Applications

to Variational Inequalities1

O. CHADLI,2 Z. CHBANI,3 AND H. RIAHI4

Communicated by S. Schaible

Abstract. This paper attempts to generalize and unify several newresults that have been obtained in the ongoing research area of existenceof solutions for equilibrium problems. First, we propose sufficient con-ditions, which include generalized monotonicity and weak coercivityconditions, for existence of equilibrium points. As consequences, wegeneralize various recent theorems on the existence of such solutions.For applications, we treat some generalized variational inequalities andcomplementarity problems. In addition, considering penalty functions,we study the position of a selected solution by relying on the viscosityprinciple.

Key Words. Equilibrium problems, monotone bifunctions and oper-ators, variational inequalities, complementarity problems, viscosityprinciple.

1. Introduction

Equilibrium problems are among the most interesting and intensivelystudied classes of problems; they include fundamental mathematical prob-lems like optimization, variational inequalities, and complementarity

1The authors thank Professor W. Oettli for suggesting this work and the anonymous refereesfor their careful reading of the first version of this paper. The third author gratefully acknowl-edges the support by Deutsche Akademische Austauschdienst (DAAD) and Action Integree95y0849 between the Universities of Marrakech, Rabat, and Montpellier.

2Doctor in Mathematics, University Cadi Ayyad, Semlalia Faculty of Sciences, Mathematics,Marrakech, Morocco.

3Assistant Professor, University Cadi Ayyad, Semlalia Faculty of Sciences, Mathematics, Mar-rakech, Morocco.

4Professor, University Cadi Ayyad, Semlalia Faculty of Sciences, Mathematics, Marrakech,Morocco.

2990022-3239y00y0500-0299$18.00y0 2000 Plenum Publishing Corporation

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problems. Many problems of practical interest in optimization, economics,and engineering involve equilibrium in their description. In recent years,these facts motivated several researchers to establish general results on theexistence of equilibrium. There is a vast literature on equilibrium problemsand their treatment in optimization, variational and quasivariationalinequalities, and complementarity problems. A most general and extensivedevelopment can be found in Refs. 1–2 and bibliography therein.

The purpose of the present paper is to generalize and unify some newexistence theorems concerning solutions of equilibrium problems underweaker conditions. The paper organization is described below.

In Section 2, we consider equilibrium problems and discuss typical situ-ations for which examples are mentioned. Then, we recall some recentnotions of monotonicity of real bifunctions and set-valued operators.

Section 3 is devoted to existence results for general equilibrium prob-lems. We employ the Fan lemma to derive existence results. After a briefconstruction of the main results, we study some important special cases inSection 4. As consequences, first we generalize to Hausdorff topologicalvector space a minimax inequality of Ben-El-Mechaiekh, Deguire, andGranas (Ref. 3); then we find a recent result obtained by Blum and Oettli(Refs. 2, 4) and theorems given by Bianchi and Schaible in the case ofpseudomonotone bifunctions (see Refs. 5–8).

In Section 5, direct applications are focused on the existence of solu-tions of set-valued variational inequalities and complementarity problems.In particular, recent results for nonlinear variational inequalities and comp-lementarity problems [see Ding–Tarafdar (Ref. 9) and Cottle–Yao (Ref. 10)]are extended to generalized pseudomonotone set-valued operators in generallocally convex Hausdorff topological vector spaces.

In Section 6, we exhibit solutions of equilibrium problems, called vis-cosity solutions, which are obtained as limit of solutions of approximateproblems. By the viscosity principle, we extend to equilibrium some recentworks on optimization problems; see Giusti (Ref. 11), Attouch (Ref. 12),and the quoted bibliography.

2. Equilibrium Problems and Basic Concepts

Let X be a real topological vector space with topological dual X*;denote the duality pairing between X and X* by ⟨ · , · ⟩; denote by 2X and2X* the family of all subsets of X and X*, respectively. Let K be a nonemptysubset of X; clX (K) and conv(K ) mean the closure and convex hull of K. Iff is a real bifunction defined on KBK, the equilibrium problem to be dis-cussed in Section 3 is denoted by

(EP) find y∈K such that f (x, y)Y0, for each x∈K. (1)

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By relying on bifunctions of the generalized monotone type, we treat thefollowing equilibrium problem:

(EP)′ find x∈K such that f (x, y)X0, for each y∈K. (2)

To explain our interest in equilibrium problems, we consider a number ofparticular cases, both classical and recently discovered, that have been men-tioned in Refs. 1–2. Typical examples are given below.

(i) Minimization. Let ψ : K→R and consider the minimizationproblem

(M) find x∈K such that ψ (x)Yψ ( y), for all y∈K. (3)

If we set f (x, y)Gψ ( y)Aψ (x) for each x, y∈K, then we see thatproblems (M) and (EP)′ are equivalent.

(ii) Saddle Point. Let F: UBV→R be a real bifunction, and con-sider the saddle-point problem

(SP) find (u, v)∈UBV such that

F (u, ν)YF (u, v)YF (u, v), for each (u, ν)∈UBV. (4)

By taking f ((u, v), (u′, v′ ))GF (u′, v)AF (u, v′ ), for each (u, v),(u′, v′ )∈KGUBV, we see that problems (SP) and (EP)′ areequivalent.

(iii) Variational Inequalities. Let T: K→X* be an operator, and letus consider the following variational inequality problem:

(VIP) find x∈K such that ⟨T(x), yAx⟩X0, for all y∈K. (5)

Let f (x, y)G⟨T(x), yAx⟩; then we see that problems (VIP) and(EP)′ are equivalent.

(iv) Complementarity Problems. Let T: K→X* be an operator. Thecomplementarity problem is

(CP) find x∈K such that T(x)∈K* and ⟨T(x), x⟩G0, (6)

where

K*G{x*∈X*: ⟨x*, y⟩X0, for all y∈K}

is the polar cone to K. If K is a closed convex cone and f (x, y)G⟨T(x), yAx⟩, we see that problems (CP), (EP)′, (VIP) are equival-ent. For a different approach to problems equivalent to comp-lementarity problems, see Refs. 13–14.

These examples explain why we suppose that f (x, x)X0 for each x∈Kthroughout this paper.

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Before embarking on the development of existence of solution of equi-librium problems, we give some notations and definitions that will be neededin the sequel.

Definition 2.1. Let f: KBK→R be a real bifunction.

(i) f is said to be monotone [resp. strictly monotone] if, for each x,y∈K, one has f (x, y)Cf ( y, x)Y0 [resp.F0 if x≠y].

(ii) f is said to be pseudomonotone [resp. strictly pseudomonotone]if, for each x, y∈K, one has

f (x, y)X0 implies f ( y, x)Y0 [resp.F0 if x≠y]. (7)

(iii) f is said to be quasimonotone if, for each x, y∈K, one has that

f (x, y)H0 implies f ( y, x)Y0. (8)

(iv) f is said to be maximal if, for every convex real function θ definedon K, one has that x∈Kθ (x)G0 and f ( y, x)Yθ( y), for all y∈K,implies 0Yf (x, y)Cθ ( y), for all y∈K.

Clearly, each monotone [resp. strictly monotone] bifunction is pseudo-monotone [resp. strictly pseudomonotone]; each pseudomonotone bifunc-tion is quasimonotone; the converse statements are not true. We note thatthe pseudomonotonicity notion is taken from Blum–Oettli (Ref. 4), Brezis–Nirenberg–Stampacchia (Ref. 15), Karamardian–Schaible (Ref. 16), andSchaible (Refs. 7–8); quasimonotonicity and strict pseudomonotonicity aretaken from Ref. 6.

Definition 2.2. Let an operator T: K→2X * be given. We recall that Tis:

(i) monotone if

x, y∈K, x*∈T(x), y*∈T( y) ⇒ ⟨y*Ax*, yAx⟩X0; (9)

(ii) maximal monotone if it is maximal in the family of monotoneoperators in KBX* ordered by inclusion;

(iii) quasimonotone if

x, y∈K, x*∈T(x), y*∈T( y) and

⟨x*, yAx⟩H0 ⇒ ⟨y*, yAx⟩X0; (10)

(iv) pseudomonotone if

x, y∈K, x*∈T(x), y*∈T( y) and

⟨x*, yAx⟩X0 ⇒ ⟨y*, yAx⟩X0. (11)

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The definitions of monotonicity and maximal monotonicity are wellknown. The definitions of pseudomonotonicity and quasimonotonicity aretaken from Refs. 17–21 and Ref. 8.

For X a real Banach space, Correa–Jofre–Thibault (Ref. 22) and Aus-sel–Corvellec–Lassonde (Ref. 23) have derived in recent years equivalencebetween convexity [resp. quasiconvexity] of lower semicontinuous real func-tions and monotonicity [resp. quasimonotonicity] of their generalizedClarke subdifferentials; see also Refs. 24–25.

One can see that a pseudomonotone operator is quasimonotone. Thefollowing example (see Ref. 26) shows that the converse is not true: KG

[0, 1]B[0, 1] and

T(x1 , x2)G{(−ty(tC1), −1y(tC1))},

where tG(x1C√x21C4x2)y2.

Exploiting the special structure of pseudomonotonicity and quasi-monotonicity of a real bifunction, we define this notion for set-valuedoperators.

Definition 2.3. Let T: K→2X* be a set-valued operator. T is called w-pseudomonotone [resp. w-quasimonotone] if, for every x, y∈K, v∈T( y),and ⟨v, yAx⟩Y0 [resp. infv∈T(y) ⟨v, yAx⟩F0] implies infu∈T(x) ⟨u, yAx⟩Y0.

Remark 2.1.

(i) These definitions coincide with those introduced by Karamardian–Schaible (Ref. 16) and Schaible (Ref. 8) when only single-valued operatorsare considered. In the set-valued case, the next definitions generalize thesame named opertors by Schaible.

(ii) Our definition of w-pseudomonotonicity coincides with the notionof quasimonotonicity introduced by Ding (Ref. 27) and Ding–Tarafdar(Ref. 9).

(iii) For other notions of pseudomonotone and quasimonotone set-valued operators, one can consult Refs. 21, 28, 29.

(iv) Let T: K→2X* be a set-valued operator. Then, T quasimonotone[resp. pseudomonotone] implies T w-quasimonotone [resp. w-pseudo-monotone].

Notice that the relation between the last two definitions on real bifunc-tions and operators is most tight. In fact, if T: K→X* is a single-valuedoperator and if f (x, y)G⟨T(x), yAx⟩, for each x, y∈K, it is easy to see thatthe notions of monotonicity, pseudomonotonicity, and quasimonotonicityof T and f are equivalent.

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By repeating a similar construction for a set-valued operator T: K→2X*

[i.e., f (x, y)Gsupζ∈T(x) ⟨ζ , yAx⟩], we adopt the following definition of maxi-mality of a bifunction:

for each (ζ , x)∈X*BK, if f ( y, x)Y⟨−ζ , yAx⟩ for all y∈K,

then f (x, y)C⟨−ζ , yAx⟩X0.

In Definition 2.1(iv), if we suppose that the family of real convex func-tions is Gateaux-differentiable, then the two maximality notions are equival-ent. In this case, it is easy to see that, if T is maximal monotone, then fdefined for all x, y∈K by

f (x, y)G supζ∈T(x)

⟨ζ , yAx⟩

is maximal and monotone. For details concerning these properties, the inter-ested reader may consult Refs. 2, 30, 31.

Definition 2.4. Let K be a closed convex subset of X. A real bifunctionf: KBK→R is said to be upper hemicontinuous if, for all x, y, z∈K, thefunction t∈[0, 1] > f (tyC(1At)x, z) is upper semicontinuous.

As an important characterization of the maximality of real bifunctions,we have the following lemma.

Lemma 2.1. Let X be a topological vector space, let K be a closedconvex nonempty subset of K, and let f: KBK→R be a real bifunctionsatisfying f (x, x)X0 for each x∈K. If f is upper hemicontinuous and convexin the second argument, then f is maximal.

Proof. For x∈K, consider a convex function θ : K→R such thatθ (x)G0 and f ( y, x)Yθ ( y) for each y∈K. Fix t∈(0, 1], y∈K, and set xtG

tyC(1At)x. By the convexity of f (xt , · ), one has

0Yf (xt , xt)

Ytf (xt , y)C(1At) f (xt , x)

Ytf (xt , y)C(1At)θ (xt)

Ytf (xt , y)C(1At)tθ ( y)C(1At)2θ (x)

Yt[ f (xt , y)C(1At)θ ( y)]. (12)

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Dividing both sides of the last inequality and letting t90, by using the upperhemicontinuity of f it follows that

f (x, y)Cθ ( y)Xlim supt→0+

( f (xt , y)C(1At)θ ( y))X0. (13)

The proof is complete. h

To close this section, we recall the following definition.

Definition 2.5. Let X, Y be two topological spaces, and let T: X→2Y

be a set-valued mapping. T is said to be upper semicontinuous at x0 in X if,for every open set V in Y containing T(x0), there is an open neighborhoodU of x0 in X such that T(x)⊂V for all x∈U.

3. Main Results on Existence of Solutions

The main aim of this section is to present results on the existence ofsolutions for equilibrium problems. We start with a well-known theoremwhich characterizes the compactness of a general topological space.

Lemma 3.1. Compactness Lemma. Let E be a compact topologicalspace, and let F G{Fi: i∈I} be a family of closed subsets of E. If everyfinite subfamily of F has a nonempty intersection, then )i∈I Fi≠∅.

For the existence of solutions of equilibrium problems, we shall needthe following Fan lemma (Ref. 32, Lemma 1), which is an infinite-dimen-sional generalization of the classical result of Knaster, Kuratowski, andMazurkiewicz (KKM lemma).

Lemma 3.2. Fan Lemma. Let X be a Hausdorff topological vectorspace,5 and let B be a nonempty subset. Consider a set-valued operatorT: B→2X such that:

(i) for every x∈B, T(x) is closed and nonempty in X;(ii) there exists x0∈B such that T(x0) is compact;(iii) for every finite subset A of B, conv(A)⊂*x∈A T(x).

Then, )x∈B T(x)≠∅.

5Following a remark quoted in Ref. 33, one can suppose X to be a topological vector spacewithout the Hausdorff property.

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Our first main existence theorem is the following.

Theorem 3.1. Let X be a Hausdorff topological vector space, let K bea nonempty closed convex subset. Consider two real bifunctions ϕ and ψdefined on KBK such that:

(H1) For each x, y∈K, if ψ (x, y)Y0, then ϕ(x, y)Y0.(H2) For each fixed x∈X, the function ϕ(x, · ) is lower semicontinu-

ous on every compact subset of K.(H3) For each finite subset A of K, one has

supy∈conv(A)

minx∈A

ψ (x, y)Y0.

(H4) Compactness Assumption. There exists a compact convex sub-set C of K such that either (i) or (ii) below holds:

(i) for all y∈K \C, there exists x∈C such that

ϕ(x, y)H0; (14a)(ii) there exists x0∈C such that, for all y∈K \C,

ψ (x0 , y)H0. (14b)

Then, there exists an equilibrium point y∈C; i.e., ϕ(x, y)Y0 for each x∈K.Furthermore, the set of solutions is compact.

Proof. Let AG{x1 , . . . , xn} be a finite subset of K, and consider thesubset BGconv(A∪C ) of K. B is convex and compact, since C is convexand compact; see Ref. 34, Theorem 15. Consider now the operator S: B→2B

defined for each x∈B by

S(x)G{y∈B: ψ (x, y)Y0}.

Let {z1 , . . . , zm} be a finite subset of B; we contend that

conv{z1 , . . . , zm}⊂*m

iG1S(zi).

Suppose the contrary; one can find y∈conv{z1 , . . . , zm} such that

ψ (zi , y)H0, for iG1 to m.

We obtain

miniG1,...,m

ψ (zi , y)H0,

which contradicts (H3). Now, from (H3) and the compactness of B, for eachx∈B we have x∈S(x); hence, cl(S(x)) is nonempty and compact.

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From the Fan lemma, we deduce that

)x∈B

cl(S(x))≠∅.

Using (H1) and (H2), we have that

)x∈B

cl(S(x))⊂)x∈B

cl({y∈B: ϕ(x, y)Y0})

G)x∈B

{y∈B: ϕ(x, y)Y0}. (15)

Let

y∈)x∈B

cl (S(x))

and invoke the compactness assumption (H4) to obtain y∈C. If (H4)(i) issatisfied, y∉C leads to the existence of x0∈C such that ϕ(x0 , y)H0, whichcontradicts (15). Therefore, y∈C. If (H4)(ii) is satisfied, we obtain S(x0)⊂Cand, by closedness of C, we have y∈C.

Consequently, for each finite subset A of K, we have that

)x∈A

{y∈C: ϕ(x, y)Y0}

is nonempty. Let us denote for each x∈K the subset {y∈C: ϕ(x, y)Y0} byT(x). Observe that (H2) justifies that T(x) is compact for each x∈K, sinceC is compact. Thus, by virtue of Lemma 3.1, we deduce that

)x∈K

T(x)≠0.

This means that (EP) has a solution. The set of solutions of (EP) is equalto

)x∈K

{y∈C: ϕ(x, y)Y0};

thus, it is compact. h

Remark 3.1.

(i) Convexity of K is imposed only to justify that the convex hull ofall finite subsets is included in K.

(ii) If K is compact, then condition (H4) of Theorem 3.1 can bedropped.

Now we turn our attention to modifying assumption (H1) of Theorem3.1. In fact, this condition is closely tied to pseudomonotonicity. For this

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reason, by relying on the scheme of the above proof, the next theorem indi-cates a condition which is like quasimonotonicity. In this context, we makethe following assumptions:

(H5) For each x, y∈K, if ψ (x, y)F0, then ϕ(x, y)Y0.(H6) For each x∈K, ψ (x, x)Y0 and ϕ(x, x)Y0.(H7) For each finite subset {xi , 1YiYn} of K with nX2, for each y∈

conv{xi , 1YiYn}, with y≠xi for all iG1, . . . , n, one hasmin1YiYn ψ (xi , y)F0.

Theorem 3.2. Suppose that (H2) and (H4)–(H7) hold. Then, the con-clusion of Theorem 3.1 holds true.

Proof. By the scheme of proof of Theorem 3.1, for each x∈K, weconsider

T(x)G{y∈C: ϕ(x, y)Y0},

and for each x∈B,

S1(x)G{x}∪{y∈B: ψ (x, y)F0}.

Using the same argument, we shall prove only that, for each arbitrary finitefamily {z1 , . . . , zm}⊂B, we have

conv{z1 , . . . , zm}⊂*m

iG1S1(zi ).

Suppose on the contrary the existence of some y0∈conv{z1 , . . . , zm} suchthat y0≠zi and ψ (zi , y0)X0 for each iG1, . . . , m. This means thatmin1YiYm ψ (zi , y0)X0, a contradiction. h

4. Particular Cases

It is desirable to compare Theorem 3.1 with existing results in the litera-ture. In the following, we present some concrete particular cases and explainthe role of Assumptions (H1)–(H4).

First, we treat Assumption (H3) as in a similar result by Tan and Yan(Ref. 33).

Lemma 4.1. Suppose that:

(i) ψ (x, x)Y0, for each x∈K;(ii) for each y∈K, {x∈K: ψ (x, y)H0} is convex.

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Then, Assumption (H3) is satisfied.

Proof. On the contrary, if (H3) does not hold, we have the existenceof x1 , . . . , xn in K and λ1 , . . . , λnX0 such that ∑n

iG1 λ iG1 andψ (xi , ∑n

iG1 λ i xi)H0 for each iG1, . . . , n. By assumption (ii), ifyG∑n

iG1 λ ixi , we deduce that ψ ( y, y)H0; but this contradicts (i). h

Now, we can state the well-known Ky Fan result; see Ref. 35, Theorem8.6 when X is a Hilbert space.

Theorem 4.1. Let K be a convex compact subset of a Hausdorff topo-logical vector space, and let f: KBK→R be a real bifunction such that:

(i) for each x∈K, f (x, · ) is lower semicontinuous;(ii) for each y∈K, f ( · , y) is quasiconcave.

Then, there exists y∈K such that supx∈K f (x, y)Ysupx∈K f (x, x).

Proof. We introduce the bifunctions ϕ and ψ defined on KBK by

ϕ(x, y)Gψ (x, y)Gf (x, y)Asupx∈K

f (x, x). (16)

Theorem 3.1 implies that there exists y∈K such that ϕ(x, y)Y0 for eachx∈K. h

Theorem 4.2. Let X be a topological vector space, let K be a closedconvex subset of X, and let f, g: KBK→R. Suppose that the followingconditions hold:

(i) for each x, y∈K, g(x, y)Yf (x, y);(ii) for each y∈K, the function f ( · , y) is quasiconcave;(iii) for each x∈K, g(x, · ) is lower semicontinuous;(iv) for a given λ∈R, there exists a convex compact subset C of K

such that either (a) or (b) below holds:

(a) for all y∈K \C, there exists x∈C such that g(x, y)Hλ ;(b) there exists x∈C such that, for all y∈K \C, f (x, y)Hλ .

Then, either (A) there exists y∈C such that g(x, y)Yλ for each x∈K or (B)there exists x∈K such that f (x, x)Hλ .

Proof. For x, y∈K, choose ϕ(x, y)Gg(x, y)Aλ and also ψ (x, y)Gf (x, y)Aλ . Assumptions (H1), (H2), (H4)(i) of Theorem 3.1 follow immedi-ately from (i), (iii), (iv) respectively.

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Suppose that (B) of Theorem 4.2 does not hold. Then, ψ (x, x)Y0 forall x in K. Hence, according to Lemma 4.1 and (ii) of Theorem 4.2, Assump-tion (H3) is satisfied. Therefore, Theorem 3.1 yields (A) of Theorem 4.2. h

Remark 4.1.

(i) In the particular case where K is convex and compact, this exist-ence result was established by Ben-El-Mechaiekh, Deguire, and Granas(Ref. 3).

(ii) In the case where X is a reflexive Banach space or the dual of aBanach space, condition (iv) of Theorem 4.2 can be replaced by

there exists a∈K such that g(a, x)→+S,

when uuxuu→+S and x∈K. (17)

Theorem 4.3. Consider a real bifunction f: KBK→R such that:

(i) for each x∈K, f (x, x)G0;(ii) f is pseudomonotone and upper hemicontinuous;(iii) for each x∈K, f (x, · ) is lower semicontinuous;(iv) if x, y, z∈K, f ( y, x)F0, and f ( y, z)Y0, then f ( y, txC(1At)z)F0

for every 0FtF1;(v) there exists a convex compact subset B of K such that, for all

y∈K \B, there exists x∈B such that f (x, y)H0.

Then, there exists y∈B such that f ( y, y)X0 for each y∈K. The solution isunique when furthermore we suppose that f is strictly pseudomonotone.

Proof. Existence. For each x, y∈K, choose ϕ(x, y)Gf (x, y) and alsoψ (x, y)G−f (y, x). By the pseudomonotonicity of f, (H1) is satisfied. (H2)and (H4) are direct consequences of (iii) and (v). Assumptions (i) and (iv)permit to realize (i) and (ii) of Lemma 4.1, and consequently (H3).

All conditions of Theorem 3.1 are verified; thus, there exists y∈B suchthat f (x, y)Y0 for every x∈K. The assertion of Theorem 4.3 is not yetproved. For this, let y∈K and consider, for t∈(0, 1), ytGtyC(1At)y. SinceK is convex, then for each t∈(0, 1), f ( yt , y)Y0. Assume that, for some t0∈(0, 1), we have f (yt0 , y)F0. According to (iv), we obtain f (yt0 , yt0)F0, a con-tradiction. It follows that, for every t∈(0, 1), f (yt , y)X0. Letting t90, upperhemicontinuity of f yields f ( y, y)X0; thus, y is an equilibrium point.

Uniqueness. Suppose the existence of two solutions y1 and y2 suchthat y1≠y2; then, f (y1 , y2)X0. By strict pseudomonotonicity of f, we deducethat f (y2 , y1)F0, which contradicts y2 being a solution. h

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Remark 4.2.

(i) Note that the conclusion of the next theorem is similar to sayingthat ( y, y) is a saddle point of f in KBK. Indeed, since f ( y, y)G0, combin-ing the first and the second parts of the existence proof, we obtain that, foreach y∈K,

f ( y, y)Yf ( y, y )Y f ( y, y). (18)

(ii) Theorem 4.3 is similar to recent existence results established, forpseudomonotone and quasimonotone bifunctions by Bianchi and Schaiblein Ref. 6, p. 36, Theorem 3.1; more precisely, the coercivity condition (iv)in Ref. 6 implies condition (v) in Theorem 4.3.

We mention that, as in Ref. 6, this theorem generalizes also results inBrezis, Nirenberg, and Stampacchia (Ref. 15).

It is also possible to establish existence results for perturbed problems.

Theorem 4.4. Suppose that the real bifunctions f, g: KBK→R satisfythe conditions below:

(i) g is maximal and monotone;(ii) for each x∈K, f (x, x)Gg(x, x)G0;(iii) the functions f (x, · ) and g(x, · ) are convex for each x∈K;(iv) the function g(x, · ) is lower semicontinuous for each x∈K;(v) the function f ( · , y) is upper semicontinuous for each y∈K;(vi) there exists a convex compact subset B of K such that either, for

all y∈K \B, there exists x∈B such that f ( y, x)Ag(x, y)F0 or thereexists x∈B such that, for all y∈K \B, f ( y, x)Cg( y, x)F0.

Then, there exists x∈B such that f (x, y)Cg(x, y)X0 for every y∈K.

Proof. For each x, y∈K, consider

ϕ(x, y)Gg(x, y)Af ( y, x) and ψ (x, y)G−f ( y, x)Ag( y, x).

According to assumption (i), we have

ϕ(x, y)Gg(x, y)Af ( y, x)

Gg(x, y)Cg( y, x)Ag( y, x)Af ( y, x)

Yψ (x, y). (19)

It follows that (H1) is satisfied. Assumptions (H2) and (H4) arise from (iv)–(v) and (vi), respectively. For (H3), it suffices to use Lemma 4.1 and assump-tions (ii) and (iii). We conclude from Theorem 3.1 that there exists x∈B

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such that

ϕ( y, x)Gg(y, x)Af (x, y)Y0, for every y∈K.

Set θ ( y)Gf (x, y), which is a convex function such that θ (x)G0. As g ismaximal, we have that, for every y∈K,

0Yg(x, y)Cθ ( y)Gg(x, y)Cf (x, y). h

Theorem 4.5. Let f, g: KBK→R be two bifunctions satisfying theconditions below:

(i) g is upper hemicontinuous;(ii) for each x∈K, f (x, x)Gg(x, x)G0;(iii) for each x∈K, f (x, · ) and g(x, · ) are convex;(iv) for each x∈K, g(x, · ) is lower semicontinuous on every compact

subset of K;(v) for each y∈K, f ( · , y) is upper semicontinuous on every compact

subset of K and g( · , y) is concave;(vi) there exists B⊂K convex compact such that, for all y∈K \B, there

exists x∈B such that

f ( y, x)Ag(x, y)F0. (20)

Then, there exists x∈B such that g(x, y)Cf (x, y)X0 for every y∈K.

Proof. For each x, y∈K, choose

ϕ(x, y)Gψ(x, y)Gg(x, y)Af (y, x).

According to Lemma 2.1, we have that g is maximal. The conclusion followsby the same method as in the proof of Theorem 4.4. h

5. Application to Multivalued Variational Inequalities

In this section, we investigate classes of variational inequalities associ-ated to general set-valued operators. The investigations are based on themain theorems of this paper. Let X be a topological vector space, let K bea nonempty closed convex subset of X, and let T: K→2X* be an everywheredefined operator. Let us consider the set-valued variational inequalityproblem

(SVIP) find x∈K and u∈T(x) such that⟨u, xAx⟩Y0, for every x∈K. (21)

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Theorem 5.1. Let X be a locally convex Hausdorff topological vectorspace, and let K be a closed convex subset of X. Suppose that the set-valuedoperator T: K→2X* satisfies the conditions below:

(i) for each x∈K, T(x) is a strongly compact subset of X*;(ii) T is w-pseudomonotone;(iii) T is weak*-upper semicontinuous on every line segment in K;(iv) there exists a convex weakly compact subset C of K such that

either (a) or (b) below holds:

(a) for all y∈K \C, there exists x∈C such that

supu∈T(x)

⟨u, xAy⟩F0; (22a)

(b) there exists x0∈C such that, for all y∈K \C,

supu∈T(y)

⟨u, x0Ay⟩F0. (22b)

Then, there exists x∈K such that supξ∈T(x) ⟨ξ, yAx⟩X0 for each y∈K. If inaddition T(x) is convex, then x is the solution of (SVIP).

Proof. For each x, y∈K, consider

ϕ(x, y)G infu∈T(x)

⟨u, yAx⟩ and ψ (x, y)G infv∈T(y)

⟨v, yAx⟩, (23)

Our objective is to verify Assumptions (H1)–(H4) of Theorem 3.1.

(H1) This follows immediately from (ii).(H2) Let A be a weakly compact subset of K. Fix x∈K and show

that, for each λ∈R, Aλ (x)_{y∈A: ϕ(x, y)Yλ} is weakly closed. To thisaim, let (yβ )β be a net of Aλ (x) weakly converging to y. Choose uβ∈T(x)such that

⟨uβ , yβAx⟩Yλ . (24)

As T(x) is strongly compact, there exists a strongly converging subnet(uβ i)i of (uβ )β to an element u of T(x). By passing to the limit in (24), seeLemma 7.1 in the Appendix, one has ⟨u, yAx⟩Yλ . Hence,

ϕ(x, y)G infu∈T(x)

⟨u, yAx⟩Yλ ;

i.e., y∈Aλ (x).(H3) This follows from the definition of ψ and Lemma 4.1.(H4) This compactness assumption is similar to (iv).

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By Theorem 3.1, one can find x∈C such that, for all y∈K, ϕ( y, x)Y0;therefore,

infv∈T(y)

⟨v, xAy⟩Y0, for all y∈K. (25)

Fix y∈K, and set

ytGtyC(1At)x∈K, for t∈[0, 1].

We have that, for each t∈(0, 1]

α (t)G infv∈T(yt)

⟨v, xAy⟩G(1yt) infv∈T(yt)

⟨v, xAyt ⟩Y0. (26)

Since T is weak*-upper semicontinuous on the segment [x, y]⊂K, we obtainfrom the Berge theorem (Ref. 36, p. 122) that α is lower semicontinuous.Therefore,

α (0)G infv∈T(x)

⟨v, xAy⟩Y0,

hence

supy∈K

infv∈T(x)

⟨v, xAy⟩Y0,

which is our first assertion. Suppose furthermore that T(x) is convex andcompact; then, according to the Sion minimax theorem (see Refs. 37 or 38),we lead to the existence of some u∈T(x) such that ⟨u, xAy⟩Y0 for ally∈K. h

Theorem 5.2. Let X be a Banach space, and let K be a closed convexsubset of X. Suppose that the set-valued operator T: K→2X* satisfies theconditions below:

(i) for each x∈K, T(x) is a weak*-compact subset of X*;(ii) T is w-pseudomonotone;(iii) T is weak*-upper semicontinuous on every line segment in K;(iv) there exists a convex strongly compact subset C of K such that

either (a) or (b) below holds:

(a) for all y∈K \C, there exists x∈C such that

supu∈T(x)

⟨u, xAy⟩F0; (27a)

(b) there exists x0∈C such that, for all y∈K \C,

supu∈T(y)

⟨u, x0Ay⟩F0. (27b)

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Then, there exists x∈K such that supξ∈T(x) ⟨ξ, yAx⟩X0, for each y∈K. Inaddition, if T(x) is convex, then there exists u∈T(x) such that

⟨u, yAx⟩X0, for each y∈K. (28)

To prove this theorem, we can proceed analogously to the proof ofTheorem 5.1.

Remark 5.1.

(i) Theorem 5.1 extends slightly the results given in Ding (Ref. 27,Theorem 2.2), since the space X here is not supposed to be of secondcategory.

(ii) In comparison with Theorem 5.2, Yao gives a similar result (seeRef. 39, Theorem 2.3), where X is a reflexive Banach space, T is pseudo-monotone, and C is a bounded subset of K.

Corollary 5.1. Let X be a reflexive Banach space, let K be a closedconvex subset, and let T: K→X* be a pseudomonotone operator which isweak*-upper semicontinuous on every line segment in K. Suppose that thereexists rH0 such that, for each y∈K, with uuyuuHr, there exists x∈K such thatuuxuuYr and ⟨T( y), yAx⟩H0. Then, there exists x∈K satisfying⟨T(x), yAx⟩X0 for each y∈K.

This result is an easy consequence of Theorem 5.1 and is similar toTheorem 3.1 in Hadjisavvas–Schaible (Ref. 26).

Corollary 5.2. Assume that X and T satisfy the assumptions ofTheorem 5.1 or Theorem 5.2, and suppose that K is a closed convex cone.Then, the complementarity problem (CP) has at least one solution; i.e.,there exists x∈K and u∈T(x) such that u∈K* and ⟨u, x⟩G0.

Proof. Since K is a closed convex cone, the following variationalinequality and complementarity problem are equivalent:

(VI) x∈K, u∈T(x) such that ⟨u, yAx⟩X0, for each y∈K, (29)

(CP) x∈K, u∈T(x) such that u∈K* and ⟨u, x⟩G0; (30)

see Blum–Oettli (Ref. 2) and Ding (Ref. 27). By virtue of Theorem 5.1, theconclusion of the corollary follows. h

Remark 5.2. See Refs. 9–10 for other interesting nonlinear pseudo-monotone complementarity problems.

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Next, we derive an existence theorem for a perturbed variational prob-lem by relying on Theorems 4.4 and 4.5.

Theorem 5.3. Let X be a Hausdorff topological vector space, let K bea closed convex subset of X, let T, B: K→2X* be two set-valued operators,and let ϕ: K→R be a convex lower semicontinuous function. Suppose that:

(i) B is monotone, upper semicontinuous and, for each x∈K, B (x)is compact;

(ii) T is upper semicontinuous and for each x∈K, T(x) is compact;(iii) there exists a convex compact subset C of K such that either (a)

or (b) below holds;

(a) ∀y∈K \C, there exists x∈C, ζ∈B (x) such that, ∀η ∈T( y),⟨ζCη, yAx⟩Hϕ(x)Aϕ( y);

(b) there exists x∈C such that, ∀y∈K \C, ζ∈B ( y), η ∈T( y),⟨ζCη, yAx⟩Hϕ(x)Aϕ( y).

Then, there exist x∈C, η ∈T(x), and θr ∈B (x) such that:

for all y∈K, ⟨η, yAx⟩C⟨θr , yAx⟩Cϕ( y)Aϕ(x)X0. (31)

Proof. Let us consider in Theorem 4.4

g(x, y)G supθ∈B (x)

⟨θ , yAx⟩Cϕ( y)Aϕ(x) and f (x, y)G supη ∈T(x)

⟨η, yAx⟩. (32)

For each x∈K, one has f (x, x)Gg(x, x)G0. By the monotonicity of B, wededuce that g is monotone. Since B is upper semicontinuous and, for eachx∈K, B (x) is compact and ϕ is lower semicontinuous, then g(x, · ) is lowersemicontinuous and g( · , y) is upper semicontinuous. Hence, g is upper hemi-continuous. By the convexity of ϕ, we deduce that g(x, · ) is convex. Thus,by Lemma 2.1, g is maximal.

On the other hand, as T is upper semicontinuous and, for each x∈K,T(x) is compact, then f ( · , y) is upper semicontinuous. Also, the condition(iii) implies (vi) in Theorem 4.4. This finishes the proof. h

Remark 5.3.

(a) Using the convex subdifferential operator, the conclusion of thelast theorem can be written as

there exists x∈C such that 0∈T(x)CB (x)C∂(ϕCδK)(x), (33)

where ϕ is defined by

ϕ(x)Gϕ(x), if x∈K,

ϕ(x)G+S, otherwise.

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Taking into account the qualification condition of Attouch and Brezis(Ref. 40), *λH0 λ (KAK ) is a closed vector subspace of X, one can write(33) as

there exists x∈C such that 0∈T(x)CB (x)C∂ϕ(x)CNK(x), (34)

where NK(x) is the normal cone to K at x.(b) If X is a Banach space, B is maximal monotone, and the domain

of B is the hull space X [i.e., B (x)≠∅, for each x∈X ], assumption (i) on Bis satisfied; see Phelps (Ref. 41, pp. 31 and 102).

6. Viscosity Principle for Equilibrium Problems

Let X be a topological vector space, let K be a closed convex subset ofX, and let g: KBK→R be a bifunction such that g(x, x)G0 for each x∈K.Let us consider the following equilibrium problem

(EP) find x∈K such that g(x, y)X0, for each y∈K. (35)

Let us denote by S the set of solutions of (EP).Let h: KBK→R be a bifunction such that h(x, x)G0 for each x∈K,

and let us consider the following approximate problem:

(EP( ) find x(∈K such that

g(x( , y)C(h(x( , y)X0, for each y∈K. (36)

Motivated and inspired by the research work of Attouch (Ref. 12) we intro-duce and study the perturbed problem (EP( ). If S is reduced to one pointonly, this perturbation scheme can be seen as a penalty method for optimiz-ation problems with parameter (H0 and penalty function h. The followingtheorem indicates the conditions under which the viscosity principle gener-ates a family of solutions of the perturbed problems (EP( ) accumulating topoints on the solutions set S of (EP).

Theorem 6.1. Suppose that:

(i) g is monotone and upper hemicontinuous;(ii) g(x, · ) is convex lower semicontinuous for each x∈K;(iii) h( · , y) is upper semicontinuous for each y∈K;(iv) the approximate problem (EP() has a solution x( for each (H0

sufficiently small.

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Then, every cluster point x of {x(}(H0 is a solution of (EP) and verifies thefollowing selection principle of viscosity:

h(x, z)X0, for each z∈S. (37)

Proof.

(i) Since x( is a solution of (EP(), then for each y∈K one has

g(x( , y)C(h(x( , y)X0.

By the monotonicity of g, one has

(h(x( , y)Xg( y, x( ).

Let x be a cluster point of the sequence {x(}(H0; by the lower semicontinuityof g( y, · ) and upper semicontinuity of h( · , y), one has

g( y, x)Y0, for each y∈K. (38)

Let t∈(0, 1], y∈K, and set

xtGtyC(1At)x∈K;

then,

0Gg(xt , xt )Ytg(xt , y)C(1At)g(xt , x). (39)

It follows that g(xt , y)X0. Since g is upper hemicontinuous, then

g(x, y)X0, for each y∈K. (40)

(ii) Let z∈S, and let x(∈K be a solution of (EP( ); i.e.,

g(x( , z)C(h(x( , z)X0.

Then, by the monotonicity of g, one has

(h(x( , z)Xg(z, x( )X0.

Hence, h(x(, z)X0 and, by the upper semicontinuity of h( · , z), we deducethat

h(x, z)X0, for each z∈S. (41)

The proof is complete. h

Attempts to solve (EP) by the viscosity principle are motivated by theexistence of solutions of the approximate problems (EP() and also by thecompactness property. These are expressed in the following theorems.

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Theorem 6.2. Let assumptions (i), (ii), (iii) of Theorem 3.1 be satisfied.In addition, we suppose that:

(C1) there exists B⊂X convex and compact, and there exists a∈B∩Ksuch that

for all x∈K \B, g(x, a)C(h(x, a)F0. (42)

Then, for each (H0, (EP( ) has a solution in B.

This theorem is an immediate consequence of Theorem 4.4.

Remark 6.1. When X is assumed to be a reflexive Banach space,Assumption (C1) in the preceding theorem can be replaced by

(C2) there exists a∈K such that limuuxAauu→+S h(x, a)yuuxAauuG−S.

Indeed, fix r1H0 and consider

CG{x∈K: uuxAauuYr1},

a convex σ (X, X*)-compact subset of K. As X is a reflexive Banach space,g(a, · ) is weakly lower semicontinuous, and C is weakly compact, we deducethat there exists α0∈R, such that α0Fg(a, u) for each u∈C. Let x∈K \C andset

yG[r1yuuxAauu]xC[1Ar1yuuxAauu]a, y∈C.

Since g(a, · ) is convex and g(a, a)G0, we have

α0(uuxAauuyr1)Fg(a, x), for each x∉C.

Since g is monotone, it follows that, for each x∉C,

g(x, a)C(h(x, a)F(h(x, a)Aα0(uuxAα uuyr1).

Using Assumption (C2), we have the existence of r2H0 such that, for allx∈K with uuxAauuHr2 , one has

g(x, a)C(h(x, a)F0.

Then, condition (vi) of Theorem 4.4 is realized with

BG{x∈X: uuxAauuYr}, where rGmax(r1 , r2).

The following result gives a characterization of solutions {x(}(H0 of theapproximate problems (EP( )(H0 in the particular case where X is a reflexiveBanach space.

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Theorem 6.3. Let X be a reflexive Banach space; let assumptions (i)and (ii) of Theorem 6.1 be satisfied; let g be maximal. In addition, supposethat the following coercivity hypothesis holds:

(C3) for x∈K and uuxAauuXR, h(x, a)YMuuxAauu andg(x, a)yuuxAauu→−S when uuxAauu→+S.

Then, the approximate problem (EP( ) has a solution x( , and the sequence{x(}(H0 is bounded.

Proof.

(i) Let us show that Hypothesis (C3) realizes Hypothesis (C1). Indeed,as

g(x, a)yuuxAauu→−S, when uuxAauu→+S,

then there exists R′H0 such that

g(x, a)yuuxAauuF−M(, for each x with uuxAauuXR ′.

Let us set rGmax(R, R′ ); thus, if uuxAauuHr, one has

g(x, a)yuuxAauuC([h(x, a)yuuxAauu]F(MC(−(M )G0; (43)

hence,

g(x, a)C(h(x, a)F0, for each uuxAauuHr.

(ii) Let us suppose that the family {x(}(H0 is not bounded; then, asequence {x(n}n∈N can be found such that uux(nAauu→+S. On the otherhand,

g(x(n , a)C(nh(x(n , a)X0;

then,

M(n uux(nAauuCg(x(n , a)X0.

Hence,

M(nCg(x(n , a)yuux(nAauuX0,

which is, by passing to the limit, in contradiction with Assumption (C3). h

7. Appendix

The following lemma seems to be a standard well-known result in func-tional analysis, but we have not found any classical reference for generallocally convex Hausdorff topological vector space.

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Lemma 7.1. Let X be locally convex Hausdorff topological vectorspace. Suppose that a net {xα∈X: α∈I} [resp. {yα∈X*: α∈I}] is weaklybounded in X and converges weakly to 0 [resp. converges strongly to 0].Then, ⟨yα , xα⟩ converges to 0.

Proof. The set BG{xα : α∈I} is weakly bounded; hence, by Theorem3.18 of Rudin (Ref. 42), B is strongly bounded. Then, for each (H0, the set

VG5y∈X*: supx∈B

u⟨y, x⟩uF(6is a strongly open neighborhood of 0 in X*. Since yα converges strongly to0, then there exists α0∈I, such that yα∈V for αXα0 . It follows that

u⟨yα , xα⟩uF(,

and so

limα∈I

⟨yα , xα⟩G0,

which ends the proof of the lemma. h

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