A noncompact generalized quasi-variational inequality

Journal of Computational and Applied Mathematics 113 (2000) 309–315www.elsevier.nl/locate/cam

A noncompact generalized quasi-variational inequality(

Ming-Po Chena;†, Kok-Keong Tanb; ∗aInstitute of Mathematics, Academia Sinica, Taipei, Taiwan

bDepartment of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5

Received 19 March 1999

Abstract

By applying a �xed point theorem of Yannelis and Prabhakar, a generalized quasi-variational inequality is proved ona noncompact convex subset of a locally convex Hausdor� topological vector space. c© 2000 Elsevier Science B.V. Allrights reserved.

MSC: primary 47H04; 47H10; 49J40

Keywords: Fixed point; Paracompact; Totally bounded; Bounded; Quasi-variational inequality; Generalizedquasi-variational inequality; Topological vector space

1. Introduction

If X is a set, we shall denote by 2X the family of all nonempty subsets of X . If E is a vectorspace and X is a subset of E, we shall denote by co(X ) the convex hull of X . Let E be a topologicalvector space. We shall denote by E∗ the continuous dual of E, by 〈w; x〉 the pairing between E∗ andE for w ∈ E∗ and x ∈ E and by Re〈w; x〉 the real part of 〈w; x〉. Suppose X is a nonempty subsetof E, S :X → 2X is a set-valued map and T :X → E∗ is a single-valued map. The quasi-variationalinequality (QVI) problem is to �nd a point y ∈ X such that y ∈ S(y) and Re〈T (y); y − x〉60for all x ∈ S(y). The QVI was �rst introduced by Bensousson and Lions in 1973 (see, e.g., [2])in connection with impulse control. Other work concerning QVI can be found in Mosco [8]. If

( This work is partially supported by a research grant from NSERC of Canada under grant number A-8096.∗ Corresponding author.E-mail address: [email protected] (K.-K. Tan)†Deceased.

0377-0427/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0377-0427(99)00263-0

310 M.-P. Chen, K.-K. Tan / Journal of Computational and Applied Mathematics 113 (2000) 309–315

T :X → 2E∗is also a set-valued map, then the generalized quasi-variational inequality (GQVI)

problem is(1) to �nd y ∈ X such that y ∈ S(y) and

supw∈T (y)

Re〈w; y − x〉60 for all x ∈ S(y) (1.1)

or(2) to �nd y ∈ X and w ∈ T (y) such that y ∈ S(y) andRe〈w; y − x〉60 for all x ∈ S(y): (1.2)

Inequalities of the type (1.1) or (1.2) are called generalized quasi-variational inequalities. GQVIwas �rst introduced by Chan and Pang [3] for E = Rn, and by Shih and Tan [9] for E being any(in�nite-dimensional) locally convex space. Other works can be found in Kim [6] and Shih and Tan[10] and others.The purpose of this paper is to present a slight improvement of a �xed-point theorem of Yannelis

and Prabhakar in 1983 and a generalized quasi-variational inequality for mappings on a noncompactconvex set in a locally convex Hausdor� topological vector space.

2. Preliminaries

De�nition 2.1. Let X and Y be topological spaces. Then F :X → 2Y is said to be upper semicon-tinuous (respectively, lower semicontinuous) if, for each open (respectively, closed) subset U of Y ,the set {x ∈ X : F(x)⊂U} is open (respectively, closed) in X .

We shall need the following Lemma 1 of Ding et al. [4, p. 206].

Lemma 2.2. Let X be a nonempty compact subset of a topological vector space. Then co(X ) is�-compact and is hence paracompact.

We shall need the following simple fact; for completeness, we shall include its simple proof:

Lemma 2.3. Let E be a locally convex topological vector space and C be a nonempty compactsubset of E. Then co(C) is totally bounded; and hence bounded in E.

Proof. Let U be any open convex neighborhood of 0 in E. Let V be another open convex neighbor-hood of 0 in E such that V +V ⊂U . Since C is compact, there exists a �nite subset B={b1; : : : ; bn}of C such that C ⊂B+V . Suppose c1; : : : ; cm ∈ C; �1; : : : ; �m¿0 with ∑m

i=1 �i=1 are given. For eachi = 1; : : : ; m, let bji ∈ B and vi ∈ V be such that ci = bji + vi. Since V is convex, it follows that

m∑i=1

�ici =m∑i=1

�ibji +m∑i=1

�ivi ∈ co(B) + V:

Thus co(C)⊂ co(B) + V . But co(B) is also compact so that there exists another �nite subset D ofco(B) such that co(B)⊂D + V . Hence

co(C)⊂ co(B) + V ⊂D + V + V ⊂D + U:

M.-P. Chen, K.-K. Tan / Journal of Computational and Applied Mathematics 113 (2000) 309–315 311

Therefore co(C) is totally bounded in E.

Let E be a topological vector space. For each nonempty bounded subset A of E and �¿ 0, let

U (A; �) ={f ∈ E∗: sup

x∈A|f(x)|¡�

}:

Let B= {U (A; �): A is a nonempty bounded subset of E and �¿ 0}. Let �(E∗; E) be the topologyon E∗ generated by B as a base for the neighborhood system at 0. Then �(E∗; E) is called thestrong topology on E∗.

Lemma 2.4. Let E be a topological vector space; E∗ be the dual of E equipped with the strongtopology; X be a nonempty bounded subset of E and T :X → 2E

∗be upper semicontinuous such

that (a) T (x) is closed for each x ∈ X and (b) T (X ) is contained in a (strongly) compact subsetof E∗. De�ne f :X × X → R by f(x; y) = inf w∈T (y) Re〈w; y − x〉; for all x; y ∈ X; then f is lowersemicontinuous on X × X .

Proof. Let � ∈ R be arbitrarily given and set A�:={(x; y) ∈ X × X : f(x; y)6�}. Let {(x�; y�)}�∈�be a net in A� such that (x�; y�)→ (x0; y0) ∈ X × X . Let �¿ 0 be arbitrarily given. Then for each� ∈ �, there exists w� ∈ T (y�) such that Re〈w�; y� − x�〉6f(x�; y�) + �6� + �. Since {w�}�∈� is anet in T (X ) which is contained in a strongly compact subset of E∗, there is a subnet {w�′}�′∈�′ of{w�}�∈� and there is a point w0 ∈ E∗ such that w�′ → w0 in the strong topology. Since T is uppersemicontinuous from relative topology on X to the strong topology on E∗ and each T (x) is stronglyclosed, w0 ∈ T (y0). Since X is bounded,

f(x0; y0) = infw∈T (y0)

Re〈w; y0 − x0〉6Re〈w0; y0 − x0〉= lim

�′Re〈w�′ ; y�′ − x�′〉6�+ �:

Since �¿ 0 is arbitrary, f(x0; y0)6� so that (x0; y0) ∈ A�. Hence A� is closed in X × X . Thereforef is lower semicontinuous on X × X .

The following is Theorem 1.2.4 of Aubin and Cellina in [1, p. 51].

Lemma 2.5. Let X and Y be topological spaces; W :X ×Y → R be lower semicontinuous; G :Y →2X be lower semicontinuous at y0 ∈ Y . Then V :Y → R∪{+∞}; de�ned by V (y)=supx∈S(y)W (x; y)for each y ∈ Y; is lower semicontinuous at y0.

The statement of the following result below was due to Takahashi [11, Lemma 3, p. 177]; acomplete proof was given by Shih and Tan in [10, Lemma 3, pp. 71–72].

Lemma 2.6. Let X; Y be topological spaces; let f :X → R be nonnegative and continuous; and letg :Y → R be lower semicontinuous. Then the map F :X × Y → R; de�ned by F(x; y) = f(x)g(y)for all (x; y) ∈ X × Y; is lower semicontinuous.


The following is Theorem 3:2 of Yannelis and Prabhakar in [12, p. 236].

Lemma 2.7. Let X be a nonempty paracompact convex subset of a locally convex Hausdor�topological vector space; C be a nonempty compact subset of X and F :X → 2C be such that(a) F(y) is convex for each y ∈ X and (b) F−1(x) = {x ∈ X : y ∈ F(x)} is open in X for eachx ∈ C. Then there exists x ∈ X such that x ∈ F(x).

In view of Lemma 2.2, we see that Lemma 2.7 (i.e., Theorem 3:2 in [12]) can be slightly improvedby removing the assumption that X be paracompact:

Lemma 2.8. Let X be a nonempty convex subset of a locally convex Hausdor� topological vectorspace; C be a nonempty compact subset of X and F :X → 2C be such that (a) F(y) is convexfor each y ∈ X and (b) F−1(x) is open in X for each x ∈ C. Then there exists x ∈ X such thatx ∈ F(x).

Proof. The set co(C) is nonempty and paracompact by Lemma 2.2. By Lemma 2.7 (i.e., Theorem 3:2in [12]), the conclusion follows.

3. Main results

We shall now prove a generalized quasi-variational inequality on noncompact domain as follows.

Theorem 3.1. Let E be a locally convex Hausdor� topological vector space; X be a nonemptyconvex subset of E and C be a nonempty compact subset of X. Let S :X → 2C be upper semi-continuous such that (a) for each x ∈ X; S(x) is closed convex and (b) for each y ∈ C; S−1(y)is open in X. Let T :X → 2E

∗be upper semicontinuous from relative topology on X to the strong

topology on E∗ such that for each x ∈ X; T (x) is strongly compact and convex. Then there existy ∈ X and w ∈ T (y) such that(i) y ∈ S(y) and(ii) Re〈w; y − x〉60 for all x ∈ S(y).

Proof. By replacing X with co(C), we may assume, without loss of generality, that X is paracompact(by Lemma 2.2) and bounded (by Lemma 2.3).By Lemma 2.4, the function (x; y)→ inf w∈T (y) Re〈w; y− x〉 for x; y ∈ X is lower semicontinuous

on X × X . It follows from Lemma 2.5 that the function � :X → R de�ned by�(y) = sup

x∈S(y)inf

w∈T (y)Re〈w; y − x〉 for y ∈ X

is also lower semicontinuous on X . Thus the set V (p0) = {y ∈ X : �(y)¿ 0} is open in X .We shall continue the proof in two steps:Step 1: There exists a point y ∈ S such that y ∈ S(y) and �(y)60.Suppose the assertion were false. Then for each y∈X , either y 6∈ S(y) or �(y)¿ 0. If �(y)¿ 0,

then y∈V (p0). If y 6∈ S(y), then since S(y) is compact convex, by Hahn–Banach separation theorem,


there exists p∈E∗ with Re〈p; y〉 − supx∈S(y) Re〈p; x〉¿ 0 so that y∈V (p):={z ∈X : Re〈p; z〉 −supx∈S(z) Re〈p; x〉¿ 0}. Since S is upper semicontinuous, V (p) is open in X for each p∈E∗. ThusV:={V0}∪{V (p): p∈E∗} is an open cover of X . Since X is paracompact, by [5, Theorem VIII.1.4,p. 162], V has an open precise neighborhood-�nite re�nement V′:={V ′(p0)}∪{V ′(p): p∈E∗}. Let{�0} ∪ {�p: p∈E∗} be a continuous partition of unity on X subordinated by V′; i.e., �0; �p :X →[0; 1] are continuous for each p∈E∗ such that (a) for each x∈X , there is an open neighborhoodNx of x in X and there is a �nite subset A of E∗ such that �p(x′) = 0 for all x′ ∈Nx and for allp∈E∗ \ A, (b) for each y∈X; �0(y) +∑

p∈E∗ �p(y) = 1 and (c) for each x ∈ X , if �0(x)¿ 0, thenx ∈ V ′(p0) and if �p(x)¿ 0 for p ∈ E∗, then x ∈ V ′(p).Now de�ne F :X → 2C ∪ {∅} by

F(y) =

x ∈ S(y): �0(y) infw∈T (y)

Re〈w; y − x〉+∑p∈E∗

�p(y)Re〈p; y − x〉¿ 0

for each y ∈ X . Since S(y)⊂C for each y ∈ X , F(y)⊂C for each y ∈ X so that F is wellde�ned. We shall now show that F(y) 6= ∅ for all y∈X . Suppose the contrary that F(y) = ∅ forsome y∈X . Then

�0(y) infw∈T (y)

Re〈w; y − x〉+∑p∈E∗

�p(y)Re〈p; y − x〉60 for all x ∈ S(y): (∗)

If �(y)¿ 0, choose any x∈ S(y) with inf w∈T (y) Re〈w; y − x〉¿�(y)=2; if �(y)60, simply chooseany x ∈ S(y).Now if �0(y)¿ 0, then y ∈ V ′(p0)⊂V (p0) so that inf w∈T (y) Re〈w; y − x〉¿�(y)=2¿ 0 and

hence �0(y)inf w∈T (y) Re〈w; y − x〉¿ 0. If p ∈ E∗ and �p(y)¿ 0, then p ∈ V ′(p)⊂V (p) so thatRe〈p; y〉 − supx∈S(y) Re〈p; x〉¿ 0; it follows that Re〈p; y〉¿supx∈S(y) Re〈p; x〉¿Re〈p; x〉 which im-plies Re〈p; y − x〉¿ 0 and hence �p(y)Re〈p; y − x〉¿ 0. Therefore �0(y)inf w∈T (y) Re〈w; y − x〉 +∑

p∈E∗ �p(y)Re〈p; y − x〉¿ 0 which contradicts (∗) as x ∈ S(y). This shows that F(y) 6= ∅ for ally ∈ X . Hence F :X → 2C .Next we shall show that F(y) is convex for each y ∈ X . Indeed, suppose y ∈ X is given and

let x1; x2 ∈ F(y) and �1; �2¿0 with �1 + �2 = 1. Then x1; x2 ∈ S(y) so that �1x1 + �2x2 ∈ S(y) sinceS(y) is convex. Since for each i = 1; 2, �0(y)inf w∈T (y) Re〈w; y − xi〉+∑

p∈E∗ �p(y)〈p; y − xi〉¿ 0,


Re〈w; y − (�1x1 + �2x2)〉+∑p∈E∗

�p(y)Re〈p; y − (�1x1 + �2x2)〉

¿�1�0(y) infw∈T (y)

Re〈w; y − x1〉+ �2�0(y) infw∈T (y)

Re〈w; y − x2〉

+ �1∑p∈E∗

�p(y)Re〈p; y − x1〉+ �2∑p∈E∗

�p(y)Re〈p; y − x2〉

= �1


Re〈w; y − x1〉+∑p∈E∗

�p(y)Re〈p; y − x1〉

+ �2


Re〈w; y − x2〉+∑p∈E∗

�p(y)Re〈p; y − x2〉

¿ 0:


Thus �1x1 + �2x2 ∈ F(y). Hence F(y) is convex for each y ∈ X .Finally, we shall show that for each x ∈ C, F−1(x) is open in X . De�ne f :X × X → R by

f(x; y) = �0(y) infw∈T (y)

Re〈w; y − x〉+∑p∈E∗

�p(y)Re〈p; y − x〉 for all x; y ∈ X:

Note that (x; y) → ∑p∈E∗ �p(y)Re〈p; y − x〉 is continuous on X × X . Moreover, by Lemma 2.4,

(x; y) → inf w∈T (y) Re〈w; y − x〉 is lower semicontinuous and (x; y) → �0(y) is continuous andnonnegative, it follows from Lemma 2.6 that f is lower semicontinuous. Thus for each x ∈ C,

F−1(x) = {y ∈ X : x ∈ F(y)}= {y ∈ X : x ∈ S(y) and f(x; y)¿ 0}= S−1(x) ∩ {y ∈ X : f(x; y)¿ 0}

is open in X .By Lemma 2.7 (i.e., Theorem 3:2 in [12]), there exists x ∈ X such that x ∈ F(x). Thus0¡�0(x) inf

w∈T (x)Re〈w; x − x〉+

∑p∈E∗

�p(x)Re〈p; x − x〉= 0

which is a contradiction. This proves Step 1.Step 2: There exists a point w ∈ T (y) such that Re〈w; y − x〉60 for all x ∈ S(y).Indeed, de�ne g : S(y)× T (y)→ R byg(x; z):=Re〈z; y − x〉 for all (x; z) ∈ S(y)× T (y):

Note that (a) for each �xed x ∈ S(y), z → g(x; z) is a�ne and continuous on T (y) equipped withthe relative strong topology, and (b) for each z ∈ T (y), x → g(x; z) is a�ne. Thus by Kneser’sminimax theorem [7], we have

minz∈T (y)

maxx∈S(y)

g(x; z) = maxx∈S(y)

minz∈T (y)

g(x; z):

Thus

minz∈T (y)

maxx∈S(y)

Re〈z; y − x〉60 by Step 1:

Since T (y) is compact, there exists w ∈ T (y) such thatRe〈w; y − x〉60 for all x ∈ S(y):

References

[1] J.P. Aubin, A. Cellina, Di�erential Inclusions, Springer, Berlin, 1984.[2] A. Bensousson, J.L. Lions, Nouvelle formulation des probl�emes de control impulsionnel et applications, C. R. Acad.

Sci. 29 (1973) 1189.[3] D. Chan, J.S. Pang, The generalized quasi-variational inequality problem, Math. Oper. Res. 7 (1982) 211.[4] X.P. Ding, W.K. Kim, K.K. Tan, A selection theorem and its applications, Bull. Austral. Math. Soc. 46 (1992) 205.[5] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.[6] W.K. Kim, Remark on a generalized quasi-variational inequality, Proc. Amer. Math. Soc. 103 (1988) 667.[7] H. Kneser, Sur un theoreme fondamental de la theorie des jeux, C. R. Acad. Sci. 234 (1952) 2418.[8] U. Mosco, Implicit variational problems and quasi-variational inequalities, in: J.P. Gossez, E.J. Lami Dozo,

J. Mawhin, L. Waelbroek (Eds.), Lecture Notes in Mathematics No. 453, Springer, New York, 1976, p. 83.


[9] M.H. Shih, K.K. Tan, Generalized quasi-variational inequalities in locally convex topological vector spaces, J. Math.Anal. Appl. 108 (1985) 333.

[10] M.H. Shih, K.K. Tan, Generalized bi-quasi-variational inequalities, J. Math. Anal. Appl. 143 (1989) 66.[11] W. Takahashi, Nonlinear variational inequalities and �xed point theorems, J. Math. Soc. Japan 28 (1976) 168.[12] N.C. Yannelis, N.D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math.

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A noncompact generalized quasi-variational inequality