We present a generalized FKKM Theorem and it's application to the existence of solution for the variationals inequalities using a generalized coercivity type condition for correspondences defined in L-space.L-structure, L-spaces, L-KKM correspondences, L-coercing family and variational inequality.

Hakim Hammami

English

Research Papers in Economics

Documents de Travail du
Centre d’Economie de la Sorbonne
Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647  Paris Cedex 13
http://ces.univ-paris1.fr/cesdp/CES-docs.htm
ISSN : 1955-611X
A Generalized FKKM Theorem and Variational Inequality
Hakim HAMMAMI
2007.76
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8A Generalized FKKM Theorem and Variational
Inequality
H. HAMMAMI1
Centre d' Economie de la Sorbonne, UMR 8174, CNRS-Universit e Paris 1
Abstract
We present a generalized FKKM Theorem and it's application to
the existence of solution for the variationals inequalities using a gener-
alized coercivity type condition for correspondences dened in L-space.
Key words and phrases: L-structures, L-spaces, L-KKM correspon-
dences, L-coercing family and variational inequality.
Classication-JEL: C02, C69, C72.
The purpose of this article is to give a generalization of FKKM Theo-
rem [KKM] and it's application in variational inequalities. All these results
extend classical results obtained in topological vector spaces by Fan [F1]
[F2], Dugundji and Granas [DG], Ding and Tan [DT] and Yen [Y] as well
as results obtained in H-spaces by Bardaro and Ceppitelli [BC1], [BC2], in
convex spaces in the sense of Lassonde [L] and in L-spaces by Chebbi, Gour-
del and Hammami [CGH], [GH].
In this paper, we will use the same notation as in [CGH]. We remind
the denition given in [CGH] of L-KKM correspondences, which extend the
notion of KKM correspondences to L-spaces, and the concept of L-coercing
family for correspondences dened in L-spaces. Let A be a subset of a vector
space X. We denote by hAi the family of all nonempty nite subsets of A
and convA the convex hull of A. Since topological spaces in this paper are
not supposed to be Hausdor, following the terminology used in [B], a set is
called quasi-compact if it satises the Finite Intersection Property while a
Hausdor quasi-compact is called compact. In what follows, the correspon-
dences are represented by capital letters F, G, Q, S,  , ..., and the single
valued functions will be represented by small letters. We denote by graphF
the graph of the correspondence F. If X and Y are two topological spaces,
(X;Y ) denotes the set of all continuous functions from X to Y .
If n is any integer, n denotes the unit-simplex of Rn+1 and for every
J  f0;1;:::;ng, J denotes the face of n corresponding to J. Let X be
1 Ecole Polytechnique de Tunisie B.P. 743, 2078 La Marsa, Tunis, Tunisia and Paris
School of Economics, University of Paris 1 Panth eon-Sorbonne, CNRS, CES, M.S.E. 106
Boulevard de l'H^ opital, 75647 Paris cedex 13, France.
E-mail address: hakim.hammami@malix.univ-paris1.fr
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8a topological space. An L-structure (also called L-convexity) on X is given
by a correspondence   : hXi ! X with nonempty valued such that for every
A = fx0;:::;xng 2 hXi, there exists a continuous function fA : n !  (A)
such that for all J  f0;:::;ng, fA(J)   (fxj;j 2 Jg). Such a pair
(X; ) is called an L-space. A subset C  X is said to be L-convex if for
every A 2 hCi,  (A)  C. A subset P  X is said to be L-quasi-compact
if for every A 2 hXi, there exists a quasi-compact L-convex set D such that
A [ P  D. Clearly, if C is an L-convex subset of an L-space (X; ), then
the pair (C; jhCi) is an L-space.
1 A Generalized FKKM Theorem
In this section, we rst remind some known denitions of L-KKM correspon-
dences and L-coercing family quoted in [CGH], then we give a generalized
FKKM Theorem and we deduce a more adapted theorem to study the va-
riational inequality.
Denition 1.1 Let (X; ) be an L-space and Z  X an arbitrary subset.
A correspondence F : Z ! X is called L-KKM if and only if:
8A 2 hZi;  (A) 
[
x2A
F(x):
Denition 1.2 Let Z be an arbitrary set of an L-space (X; ), Y a topolo-
gical space and s 2 (X;Y ). A family f(Ca;K)ga2X is said to be L-coercing
for a correspondence F : Z ! Y with respect to s if and only if:
(i) K is a quasi-compact subset of Y ,
(ii) for each A 2 hZi, there exists a quasi-compact L-convex set DA in X
containing A such that:
x 2 DA ) Cx \ Z  DA \ Z;
(iii)
8
<
:
y 2 Y j y 2
[
z2s 1(y)
\
x2Cz\Z
F(x)
9
=
;
 K.
For more explanation of the L-coercivity and to see that this coercivi-
ty can't be compared to the coercivity in the sense of Ben-El-Mechaiekh,
Chebbi and Florenzano in [BCF], see [CGH].
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8Denition 1.3 If X is a topological space, a subset B of X is called strongly
compactly closed (open respectively) if for every quasi-compact subset K of
X, B \ K is closed (open, respectively) in K.
We remind the following result given in [CGH], which is an extension of
a lemma in [F1] to L-spaces.
Lemma 1.1 Let (X; ) be an L-space, Z a nonempty subset of X and F :
Z ! X an L-KKM correspondence with strongly compactly closed values.
Suppose that for some z 2 Z, the correspondence F(z) is quasi-compact,
then
\
x2Z
F(x) 6= ;.
Proof: see [CGH].
The main result of this paper is the following generalized FKKM Theo-
rem (see for example Theorem 4 in [F2] and Theorem 1 in [CGH]:
Theorem 1.1 Let Z be an arbitrary set in the L-space (X; ), Y an arbi-
trary topological space and F,G : Z ! Y two correspondences such that:
(a) for every x 2 Z, F(x) is strongly compactly closed,
(b) for every x 2 Z, G(x)  F(x),
(c) there is a function s 2 (X;Y ) satisfying :
1. the correspondence R : Z ! X dened by R(x) = s 1(F(x)) is
L-KKM,
2. there exists an L-coercing family f(Ca;K)ga2X for G with respect
to s,
3. for each quasi-compact L-convex set C in X:
\
x2C\Z
G(x) \ s(C) 6= ; ,
\
x2C\Z
F(x) \ s(C) 6= ;:
Then
\
x2Z
F(x) 6= ; more precisely K
T
 
\
x2Z
F(x)
!
6= ;.
Proof: The correspondence F has strongly compactly closed values, then
in order to prove that:
K
\
 
\
x2Z
F(x)
!
6= ;;
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8it suces to prove that for each nite subset A of Z,
 
\
x2A
F(x)
!
\ K 6= ;.
Let A 2 hZi, by condition (ii) of Denition 1.2, there exists a quasi-
compact L-convex set DA containing A such that for all y 2 DA, Cy \ Z 
DA \ Z. Consider now the correspondence RA : DA \ Z ! DA dened by
RA(x) = R(x) \ DA. By Hypothesis (c:1) and the L-convexity of DA, it is
immediate that the correspondence RA is L-KKM. Next, by the continuity of
s, F(x)\s(DA) is closed in s(DA) then RA(x) = s 1
0 (F(x)\s(DA)), where
s0 is the restriction of s to DA, is closed in DA and consequently RA(x) is
quasicompact. Since (DA; jhDAi) is also an L-space, we deduce by Lemma
1.1 that
\
x2DA
RA(x) 6= ;, then
\
x2DA\Z
RA(x) 6= ;. Since for all x 2 DA \ Z,
s(RA(x))  F(x) \ s(DA), we have:
\
x2DA\Z
 
F(x) \ s(DA)

6= ; then by
(c:3),
\
x2DA\Z
 
G(x) \ s(DA)

6= ;. To nish the proof, we will show that:
\
x2DA\Z
 
G(x) \ s(DA)


\
x2A
F(x) \ K.
Indeed, it is clear by (b) that
\
x2DA\Z
 
G(x) \ s(DA)


\
x2A
F(x), then
it only remains to show that:
\
x2DA\Z
 
G(x) \ s(DA)

 K. Let y 2
\
x2DA\Z
 
G(x) \ s(DA)

, then y 2 s(DA) which implies that there exists
z 2 s 1(y) \ DA. By condition (ii) of Denition 1.2, Cz \ Z  DA \ Z,
it follows that y 2
[
z2s 1(y)
\
x2Cz\Z
G(x). Hence, by hypothesis (c:2), y 2 K
and the theorem is proved.
Remark 1.1 (1) The main result of [CGH] (Theorem 1) becomes an im-
mediate corollary of Theorem 1.1: it suces to take F = G.
(2) In view of our approach, it is possible to state a weakened version of
Theorem 1 in [CGH] by replacing the coercivity on F by a coercivity
on G together with condition (c:3) of our Theorem 1.1.
Corollary 1.1 Under the conditions of Theorem 1.1, if we assume in ad-
dition that X is a quasi-compact set and s is a surjective function, then we
can reinforce the conclusion:
\
x2Z
G(x) 6= ;:
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8Proof: All the requirement of Theorem 1.1 are satised then
\
x2Z
F(x) 6=
;. By the denition of L-space, it is clear that X is an L-convex set. In
addition, X is a quasi-compact set and s(X) = Y , then by assumption (c:3), \
x2Z
G(x) 6= ;.
Remark 1.2 It is obvious as in [DG] that if we add the following condition:
\
x2Z
F(x) 6= ; ,
\
x2Z
G(x) 6= ;
in Theorem 1.1 then in addition to
\
x2Z
F(x) 6= ; we have
\
x2Z
G(x) 6= ;.
The next theorem is more specially adapted to the study of variational
inequality. It can be seen as a corollary of Theorem 1.1 and it is a generali-
zation of Theorem II [L] and Corollary 1.4 [DG].
Theorem 1.2 Let Z be an arbitrary set in the L-space (X; ), Y an arbi-
trary topological space and F, G : Z ! Y two correspondences such that:
(a) for every x 2 Z, F(x) is strongly compactly closed,
(b) for every x 2 Z, G(x)  F(x),
(c) there is a surjective function s 2 (X;Y ) satisfying :
1. he correspondence R : Z ! X dened by R(x) = s 1(F(x)) is
L-KKM,
2. there exists an L-coercing family f(Ca;K)ga2X for G with respect
to s,
3. for each L-convex set C in X:
\
x2C\Z
G(x) \ s(C) 6= ; ,
\
x2C\Z
F(x) \ s(C) 6= ;:
Then
\
x2Z
G(x) 6= ;.
Proof: It is obvious to see that Assumption (c:3) of Theorem 1.2 imply
Assumption (c:3) of Theorem 1.1, then
\
x2Z
F(x) 6= ;. By the denition of
L-space, it is clear that X is an L-convex set and hence, for the particular
case where C = X, Assumption (c:3) implies that
\
x2Z
G(x) 6= ; and the
theorem is proved.
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82 Application to variational inequalities
In this section we will prove the existence of solutions of variational inequa-
lities using Theorem 1.2.
Let E and P denote two real topological vector space, X a nonempty
convex set in E and h;i a bilinear form on PE whose for each xed v 2 P,
the restriction of hv;i on any quasi-compact subset Q of X is continuous2
(the natural example is between a normed topological vector space E and
its dual space equipped with the strong topology).
Denition 2.4 A non empty valued correspondence T : X ! P is said to
be monotone if for each (x;u) and (y;v) in the graph of T, hu v;x yi  0.
Remark 2.3 One checks easily that if a correspondence T is upper hemi-
continuous in the sense of Cornet [C1] (see for example [C2] and [F]) then
the following condition used by Lassonde [L] for monotone correspondences
is satised 3:
For any (x;y) 2 X  X, the function hxy : [0;1] ! R dened, for all
t 2 [0;1], by: hxy(t) = inf
u2T((1 t)y+tx)
hu;y   xi is lower semi-continuous at
point t = 0, (resp. the function ~ hxy : [0;1] ! R dened by for all t 2 [0;1],
~ hxy(t) = sup
u2T((1 t)y+tx)
hu;x   yi is upper semi-continuous at point t = 0).
The following theorem is a general version of one of the basic facts in
the theory of variational inequalities (see for example [HS], [DG] and [L]).
Theorem 2.3 Let T : X ! P be a non empty monotone correspondence,
' : X ! R [ f+1g a convex function lower semi-continuous on any quasi-
compact subset of X4. Let us suppose that there exists a family f(Cx;K)gx2X
of pairs of sets satisfying:
(a) K is a quasi-compact subset of X,
(b) for each A 2 hXi, there exists a quasi-compact convex set DA contai-
ning A such that:
x 2 DA ) Cx  DA;
2Which is equivalent, if we suppose that for all x 2 Z, 'v(x) = hv;xi, to : for every
closed subset F of R, '
 1(F) is a strongly compactly closed subset.
3It suces to consider p equal to the (continuous) linear form h;y xi in the following
denition given by Cornet: a correspondence F : X ! P is said upper hemi-continuous in
a point x0 2 X in the sense of Cornet if for any continuous linear function p, the function
h : x ! sup
y2'(x)
p(y) (resp. ~ h : x ! inf
y2'(x)
p(y)) is upper semi-continuous (resp. lower
semi-continuous) at the point x0.
4Or equivalently: for every  2 R, '
 1(]   1;]) is a strongly compactly closed set.
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8(c)
(
y 2 X j '(y)  '(x) + sup
v2T(y)
hv;x   yi for all x 2 Cy
)
 K,
(d) for each (x;y) 2 XX, the function hxy : [0;1] ! R given for t 2 [0;1]
by hxy(t) = sup
u2T((1 t)y+tx)
hu;x   yi is upper semi-continuous at t = 0.
Then there is a point y0 2 X such that,
'(y0)  '(x) + sup
v2T(y0)
hv;x   y0i 8x 2 X:
Proof: The proof is similar to the proof of [DG] and [L]. For each x 2 X,
let
G(x) = fy 2 X j '(y)  '(x) + sup
v2T(y)
hv;x   yig;
we have to show that Theorem 1.2 can be applied in order to get
\
x2X
G(x) 6=
;. Let us now consider the correspondence
F(x) = fy 2 X j '(x)  '(y) + sup
u2T(x)
hu;y   xig:
We will verify that G and F satises requirements of Theorem 1.2 (with
Z = X and s = the identity function).
(i.1) From the l.s.c assumption of ' and the \regularity" assumption of the
bilinear form hu;:i, it follows that F(x) is strongly compactly closed
in X for each x 2 X.
(i.2) Let us prove that for every x 2 X, G(x)  F(x) : Let y 2 G(x), then
'(y)  '(x)+ sup
v2T(y)
hv;x yi. By the monotonicity of T, we have: for
all u 2 T(x) and v 2 T(y), hu;x   yi  hv;x   yi then
inf
u2T(x)
hu;x   yi  sup
v2T(y)
hv;x   yi
consequently
  sup
u2T(x)
hu;y   xi  sup
v2T(y)
hv;x   yi
which implies sup
u2T(x)
hu;y   xi + '(y)  '(x), i.e. y 2 F(x).
(ii.1) We will prove that F is KKM. Let y 2 convfx1;:::;xng, then there
exists i 2 [0;1] for i = 1;:::;n such that y = n
i=1ixi and n
i=1i =
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81. By the monotonicity of T, for all u 2 T(xi) and v 2 T(y), hu;xi  
yi  hv;xi   yi, then
inf
u2T(xi)
hu;xi   yi  hv;xi   yi
consequently
sup
u2T(xi)
hu;y   xii   hv;xi   yi
which implies n
i=1i sup
u2T(xi)
hu;y xii  0. It follows from the convex-
ity of ' that '(y)  n
i=1i'(xi). The two previous inequalities allows
us to deduce that: n
i=1i
 
sup
u2T(xi)
hu;y   xii + '(y)   '(xi)
!
 0.
Therefore, there exists i 2 f1;:::;ng such that sup
u2T(xi)
hu;y   xii +
'(y)  '(xi), then y 2
n [
i=1
F(xi) and F is KKM.
(ii.2) The assumptions (a), (b) and (c), mean exactly that f(Cx;K)gx2X is
a coercing family of the correspondence G.
(b.3) Let C be any non-empty convex subset of X. Due to the inclusion be-
tween F and G, it is enough to show
\
x2C
(F(x) \ C) 
\
x2C
(G(x) \ C).
Let y 2
\
x2C
(F(x) \ C). Let us x z in C and prove that y 2 G(z).
Obviously, we may assume '(z) < +1. Since y 2 F(z), this implies
that '(y) is also nite. For each 0 < t < 1, let zt = (1 t)y+tz. Since
C is convex, then zt 2 C and recalling that y 2
\
x2C
F(x), we can de-
duce that y 2 F(zt) or equivalently, sup
ut2T(zt)
hut;y  zti+'(y)  '(zt),
8t 2 ]0;1[.
Using the convexity of ', it implies:
sup
ut2T(zt)
hut;y   zti  t('(z)   '(y)) 8t 2 ]0;1[:
By the convexity of the function y ! sup
u2T(zt)
hu;y   zti, it follows that
0 = sup
ut2T(zt)
hut;zt zti  (1 t) sup
ut2T(zt)
hut;y zti+t sup
ut2T(zt)
hut;z zti:
Consequently
0  t(1   t)('(z)   '(y)) + t sup
ut2T(zt)
hut;z   zti;
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8or
0  t(1   t)
 
'(z)   '(y) + sup
ut2T(zt)
hut;z   yi
!
:
Let us rst simplify by t(1 t) and let t tend to 0, then from Assump-
tion (d), it follows that,
0  '(z)   '(y) + sup
v2T(y)
hv;z   yi
and the theorem is proved.
Remark that together with the monotonicity of the correspondence T,
Assumption (c) of Corollary 3.1 in [GH] implies assumption (c) of the pre-
vious theorem. Then, Corollary 3.1 of [GH] is an immediate corollary of the
previous theorem.
References
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[C1] B. Cornet, Fixed point and surjectivity theorems for correspon-
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[C2] B. Cornet, Th ese de doctorat d' etat : Contributions  a la
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A Generalization of Fan's Matching Theorem,

A generalization of Tychono's  point theorem,

A generalized KKMF principle,

A minimax inequality and its applications to variational inequalities,

Applications of A Generalized Ky Fan's Matching Theorem in Minimax and Variational Inequality,

Ein Beweis des Fixpunktsatsez f ur n-dimensionale simplexe, Fundamental Mathematics XIV

Fixed point and surjectivity theorems for correspondences; applications,

Fixed point theorems and vector valued minimax theorems,

General Equilibrium Analysis Existence and Optimality Properties of Equilibria,

General Topology:

KKM Maps and Variational Inequalities,

On some nonlinear elliptic dierential equations,

On the Use of KKM correspondences in  point theory and related topics,

Some further generalizations of Knaster-Kuratowski-Mazurkiewicz Theorem and Minimax Inequalities,

Some properties of convex sets related to  point theorems,

Th ese de doctorat d' etat : Contributions  a la th eorie math ematique des m ecanismes dynamiques d'allocation des ressources,

A generalized FKKM theorem and variational inequality

http://halshs.archives-ouvertes.fr/docs/00/20/46/01/PDF/B07076.pdf

A generalized FKKM theorem and variational inequality

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