Internat. J. Math. & Math. Scl.
VOL. 16 NO.
(1993) 67-74
67
EXISTENCE THEOREMS FOR THE IMPLICIT
COMPLEMENTARITY PROBLEM
G. ISAC
Dpartement de Mathmatiques
Collge Militaire Royal
St-Jean, Quebec, Canada
J0J 1R0
and
o. GOELEVEN
Dpartement de Mathmatiques
Facult des Sciences
Universit de Namur
Rempart de la Vierge 8
B-5000 Namur, Belgique
(Received August 20, 1991 and in revised form April 16, 1992)
ABSTRACT. Some existence theorems for the general implicit complementarity problem in
an
infinite dimensional space are considered.
KEY WORDS AND PHRASES. Complementarity Problem, Existence Theorems.
1991 AMS SUBJECT CLASSIFICATION CODES. 49J40.
1.
INTRODUCTION.
The study of Complementarity Problems is an interesting and important domain of applied
[1], [5], [8], [10] etc. In this domain, a special chapter is the Implicit
Complementarity Problem. It seems that the first Implicit Complementarity Problem was
defined in 1973 by Bensoussan and Lions [2], as the mathematical .model of some stochastic
optimal control problems [2], [3], [4]. Now, it is well known that, the Implicit Complementarity
Problem can be used to study the optimal stopping of Markov chains [6].
The first existence results for the Implicit Complementarity are the results obtained by
Dolcetta and Mosco [7], [18], [19].
As numerical methods for solving the Implicit Complementarity Problem we remark the
iterative methods proposed by Pang [20], [21] and Uosco [22].
In this paper, we study some existence theorems for the general Implicit Complementarity
Problem in an infinite dimensional space. This paper can be considered as a complement of our
mathematics
paper [13].
2. DEFINITION OF PROBLEM AND PRELIMINARIES.
Let < E, E* > be a dual system of Banach spaces. Denote by K a pointed convex cone in E,
that is, a subset of E satisfying the following properties:
I)K+Kc_K T):c_K, forallRand3")Kn(-K)={0}.
The closed convex cone K*
{y C* < z,y > _> 0; for all z K} is called the dual of K.
68
G. ISAC AND D. GOELEVEN
Given a subset D C E and the mappings S: D-,K and T: D---,E*, the Implict Complementarity
Problem associated to T,S and K is
find zo E D such that
ICP(T,S,K):
T(zo) E K* and
< S(Xo), T(xo) >
O.
We find applications and examples of this problem in [2], [3], [4], [6], [7], [18], [19], [20], [21].
When D=K and S(z)= z, for all EK, the problem ICP(T,S,K) is exactly the nonlinear
complementarity problem, which has interesting applications in: Optimization, Game Theory,
Economics, Mechanics, etc. [1], [5], [8-15].
If the problem ICP(T,S,K) is defined, we consider the following special variational
inequality:
SVI(T,S,K):[
find z o D such that
< z- S(zo), T(zo) > _> 0; Vx e K.
PROPOSITION 1. The problem SVI(T,S,K) is equivalent to the problem ICP(T,S,K).
PROOF. Indeed, if ro is a solution of the problem SVI(T,S,K) then S(Zo) e K and we have
(1): <z-S(xo), T(Xo)> >_0; VzEK
Let
u + S(zo) in (1) we obtain < u,T(zo) > >_ O,
we
that
for every e K,
have T(zo) E K*.
is,
If we put z 0 in (1) we have < S(zo), T(zo) > _< 0 and since < S(Zo), T(zo) > _> 0 we deduce
< S(zo),T(zo) > O.
Conversely, let o be a solution of the problem ICP(T,S,K). We have, S(Xo)E K, T(%)E K*
and < S(zo), T(to) > 0 which imply <
S(%), T(ro) > _> 0, for every z e K.
Given a nonempty subset D C E and the mappings T:D--,E* and S:D-K we consider the
following problem
uE
K be an arbitrary element. If we put z
u
-
SVI(T,S,D):[
find
o E D such that
<z-S(zo), T(zo)> _>0; VzED
To solve the problem SVI(T,S,D) we use the following classical result.
THEOREM 1. A mapping To:D--,2 D, where D C X, have a fixed point if the following
conditions are satisfied:
1): X is a locally convex space and the set D is nonempty, compact, and convex,
2): the set To(X) is nonempty and convex for all e D and the preimages TO- l(y)
{z E D ly E To(z)} are relatively open with respect to D, for all V E D.
PROOF. The proof is in [25][Proposition 9.9, p. 453].
THEOREM 2. Let D C E be a nonempty compact convex set, T:D---,E* and S:D.--,K two
continuous mappings.
If for every z D we have < S(z), T(z)> _< < T(x)>, then the problem SVI(T,S,D) has a
solution.
PROOF. If the problem SVI(T,S,D) does not have a solution then,
,
(2):(V= E D)(3u E D)( < u- S(z), T(z) > < O)
69
EXISTENCE THEOREMS FOR THE IMPLICIT COMPLEMENTARITY PROBLEM
Let To:D-,D be the point-to-set mapping defined by, To(x
for every
r fi
{u D < u-S(x), T(=)> < 0},
D.
We remark that To(r) is nonempty and convex for every r D.
Since T and S are continuous, the mapping v--< r-S(v), T(v)> is continuous too and we
have that To’- l(y) {z D ly To(z)} {z D < y-S(z),T(z) > < 0} is relatively open with
respect to D.
Hence, by Theorem 1 there is an element r, D such that r, To(z,), that is, < r, S(z,),
T(z,)> <0, which is impossible since for every zD we have (by assumption) <S(z),
T(x) > <_ <
, T(z) >.
Let K be a pointed convex cone in E. We say that a subset B of K is a base, if B is convex
and for every rK\{0} there is a unique br B and a unique number Ar R+ \{0} such that
A closed pointed
convex cone K C E is locally compact if and only if, it has a compact base
[Klee’s Theorem].
If r R + \{0} we denote Kr-<
< r} and Kr<
We say that a convex cone K C E is a Gnlerkin cone [10] if there exists a countable family of
convex subcones {Kn} n N of K such that:
i) K n is locally compact for every n N,
ii) if n _< m then K n _C Krn
{r K
z
KONK
iii)
n
A Galerkin cone will be denoted by K(Kn) n N"
We recall that if D C E is a closed convex set, we say that a continuous operator (not
D.
for every
necessary linear) P: E-,E is a projection onto D if P(E) O and P(r)
By the same proof as in our paper [12] we can prove that if K(Kn) n N is a Galerkin cone in
N there exists a projection Pn onto K n such that for every
K we have lrnooPn(r) r.
we say that an operator (not necessary linear)
and (F,
Given two Banach spaces (E,
T:E---,F is strongly continuous if for every sequence {rn} n N C E, weakly convergent to
we
have that {T(rn)}n N is norm convergent to T(,).
This class of operators is very important and was intensively studied by Vainberg [24] and
a Banach space, then for every n
z
,
Lipkin [17].
3. PRINCIPAL RESULTS.
The principal aim of this paper is to give some existence theorems for the problem
ICP(T,S,K).
In this sense, we suppose given a dual system < E,E*> of Banach spaces. We consider on
E* the strong topology.
THEOREM 3. Let K CE be a pointed locally compact cone and $:K.--,E, T:K-,E*
continuous mappings. If the following assumptions are satisfied:
1") there is a number ,- > 0 such that S(Kr--- C_ K,
2") there is an element uo K such that S(Uo) K, II S(Uo)II < r and < r- S(uo), T(r) > > O,
for all r K satisfying r < z -< mar(r, ro) where ro is a number such that
,
3 < S(), T() > _< < r(x) > VrKr-,
then the problem ICP(T,S,K) has a solution r.
Kr<-
such that
G. ISAC AND D. GOELEVEN
70
we have that Kr--< is a convex compact set. Applying
element z, gr-< such that
PROOF. Since K is locally compact
Theorem 2 with D
Kr-<
we obtain an
(3): < z- S(,), T(z,) > > 0;
K r-
We have that S(**) K. Two cases are possible:
I) S(,)][ < r. If K is an arbitrary element then there is a sufficiently small A ]0,1[ such
that w
+ (1 A)S(**) Kr---. If in (3) we put z w we have, < z S(z.),r(z.) > >_ 0, that is,
< z-S(z.),T(z.)> >_ 0 for all z K and by Proposition 1 we obtain that z. is a solution of the
.
problem ICP(T,S < K).
II) (.)II > In this case we have r _< S(z,)I[
.
-< rnaz(t, r0) and by assumption 2) we obtain,
(4): < S(z,)- S(Uo), T(z,) > >_ O,
and since for every t /Kr-< we have
(): <
we
-
s(,), T(,) > > 0
deduce (using (4) and (5)),<-S(%), T(.)>
< z S(z,), T(z,) > + < S(,)- S(uo),T(,) > _> 0, that is,
<-S(.)+S(.)-S(%),
we
(): < .- s(%), T(**) > > 0; W e
v
T(z.)>
have
,<-.
If zK is an arbitrary element then there is a sufficiently small A ]0,1[ such that
)S(uo) Kr-< Now, if we put z v in (6) we obtain,
Xz + (I
(7): < z- S(Uo) T(z,) > _> 0; Vz K.
Since
S(uo)II < r we can put
z
S(uo) in (3) and
we
deduce,
(8): < S(uo)- S(u,), T(a:,) > > 0.
From (7) and (8)
we obtain
(9): < a:- S(z,), T(a:,) > > 0; Ya: K.
Since s(x,) K, from (9) and Proposition 1 we obtain that z, is a solution of the problem
ICP(T,S,K) and the proof is finished.
Theorem 3 can be extended to Galerkin cones. To obtain this extension we need to
introduce a new concept.
We say that S:K-E is subordinate to the approximation (Kn)n N if there exists n o N such
that for every n _> no we have S(Kn)C_ K n.
In [13] we indicated some examples of mappings with this property.
Independent of us in [16] is defined the concept of F-mapping which is similar to our
concept.
In [16] we showed that every DC-mapping can be approximated by an F-mapping while the
class of DC-mappings is very reach.
We say that S:K-E is r-subordinate to the approximation (Kn) n N if there exist r > 0 and
no N such that for every n >_ no we have S(gr C K n, where Kiln
{z gnl 11 z II -< r}.
REMARK. If S:K--.E is continuous and r-subordinate to the approximation (Kn) n N then
S(K c
Indeed, if Kr-< then we have two cases:
.
EXISTENCE THEOREMS FOR THE IMPLICIT COMPLEMENTARITY PROBLEM
7!
t
< r. Since K is a Galerkin cone there is a sequence {tn} n (5 N such that t timn_.oot.,, and
a)
for every n E N, t n E K n.
There exists
e N such that
n- < r- I1 II, for every n n 1, which implies,
<
+
n
n
Since, for every n maz(no, nl) we have S(n)K n CK, we obtMn by continuity that S(z) e K.
r. If for every n e N, z n e K n
then considering the sequence
b)
n z d r< n
0 we have that Yn Kn, Vn <
*n Xn, where 0 < en <
Un
II; Vn N d
d II
-
,
tim,v.,.
,
z, which imply that S(z)
z
S(Vn)
en
K.
THEOREM 4. Let (E, II) be a reflexive Bh space and K(Kn)n N
E. Let S:KE d T:KE* be strongly continuous mappings.
a
GMerkin cone in
If the following sumptions e satisfied:
10) S is r-subordinate to the approximation K(Kn)n e N,
2) there exist m e N d uo K m such that S(Uo)II <
S(Uo) Km d < z- S(Uo),
T(z)> 0, for M1 zK n satisfying r Ileal mat(r, rn) where r n is a number such that
p{ S()II u
n d for every n mar(no, m),
3)
< S(),T() > < ,,T(,) > W
then the problem ICP(T,S,K) h a solution r. such that
r.
PROOF. We remark that for every n ma(no, m the all sumptions of Threm 3 e
satisfied for every problem ICP(T,S, Kn) d hence we have a solutionr for each of these
problems.
Since for ever t (with nmar(no, m)) we have I111 r we have that {Zn}neN is a
bounded sequence.
Because E is reflexive {t} n e N h a wetly convergent subsequence {zk}k e N" We denote
agMn this subsequence by {Z}ne N d we put z. (w)-z. We have that e K d
is closed d convex. Hence S(r.) e K.
r, since
Let r e K be bitry element. For every n mat(no, m we have,
,
,
K
.
.
.
K
(0): < P.()- s(.;), r() > _> 0,
where {Pn}n 6 N iS a sequence of projections. Since S and T are strongly continuous, computing
the limit in (10) we obtain,
(11): <t-S(t.), T(t,) > >0; for all tK.
The proof is finished since from (11) by Proposition 1 we have that t. is a solution of the
problem ICP(T,S,K).
We consider now the case when S(K)c_ K.
THEOREM 5. Let (E,
be a Banach space, K C E a pointed locally compact convex cone
and S:K-,K, T:K-E* continuous mappings.
If the following assumptions are satisfied:
10) < S(t), T(z) > < < t, T(t) > Yt K,
2*) there is r > 0 such that for every t e K with r < t there is an element v E K such that
vt < r and < S(t)-vt, T(t)> > 0, then the problem ICP(T,S,K) has a solution z. such that
PROOF. We denote Dn {t 6 KIll t _< n}. Since K is locally compact
every
n
N, D n is
&
we have that for
convex compact set.
We apply Theorem 2 with D Dn and
we obtain a solution
tr for the problem SVI(T,S, Dn).
G. ISAC AND D. GOELEVEN
72
So we have:
(12):
The sequence
> O)(3n N)( x,
{X}n N
for every
n
e N there is Xn e D such that
<S(zt)-v, T(x)>
is
bounded.
<0; VvCDn
Indeed, supposing the contrary
we
have
_> ).
If k > then there is a natural number n such that, n > [I Xn* _> _> r.
assumption 2) there is an element vx. fi K such that
< r and,
For this n,* by
v. n
n
(13): < S(x) I/. T(x) > > 0.
?1
But, since
V+.n
< < +n _< n, from (12) we have < $(r+)
vz. n, T(z) > _< 0,
which is a
contradiction of (13).
Hence, {zr}n e/v is bounded and because K is locally compact the sequence
{zr}n e/v has a
convergent subsequence Ix*n k k N"
Let x. =/mzr We show now that z, is a solution of the problem ICP(T,S,K).
k.
Indeed, if v K is an arbitrary element, then there is m N such that for every n > m we
and < S(zk v, T(zk > < O.
have, v Dn and for every n k > m, v
Using the continuity of S and T we obtain,
< S(x,)-v, T(z.)> < 0, YveK, that is z. is a solution of the problem SVI(T,S,K) which, by
Proposition 1 is equivMent to the problem ICP(T,S,K). Obviously, by assumption 2*) we must
norm
Dnk
have
x.
< r.
From Theorem 5 we deduce two important corollaries.
COROLLARY 1. Let K C E be a pointed locally compact
cone and S:K-K, T:K-,E*
continuous mappings. If the following assumptions are satisfied:
1 ) < S(,),T() > >_ < ,T() > W e K,
2*) there is a number r>0 such that for every xK with r<
< S(,),T() > > 0,
,
.
.
.
IIll
we
have
then the problem ICP(T,S,K) has a solution such that
_< r.
PROOF. We apply Theorem 5 with v, 0 for every K satisfying
>COROLLARY 2. Let K C E be a pointed locally compact cone and S:K---,K,
continuous mappings. If the following assumptions are satisfied:
) < S(),T() > _< < T(x) > V K,
we have
2*) there is a number ro > 0 and uo K such that for every z K with r <
< S(x)- uo, T(x) > > O.
then the problem ICP(T,S,K) has a solution such that
< + maz(ro, Uo )PROOF. If we denote, r max(ro, Uo I1) + we have r > ro and r > uo IINow, we can apply Theorem 5 since assumption 2*) of this theorem is satisfied with vx uo,
for every z K with
>- r.
REMARK. Condition 2) of Corollary 2 is satisfied if T is semicoercive with respect to S in
the following sense:
,
(quK)(
lim
<S(x)-u’T(z)>
EXISTENCE THEOREMS FOR THE IMPLICIT COMPLEMENTARITY PROBLEM
73
If S(z)= z, for every z K, this notion is similar to the semicoercivity used by Stampacchia
and Lions [22], [23].
Obviously, condition 2 is satisfied if there is a number a>0 such that <S(x),
2, for every z e K.
T(x) > > a
Finally, we give an extension of Theorem 5 to Galerkin cone.
THEOREM 6. Let (E, II) be a reflexive Banach space and K(Kn) n N a Galerkin cone in
E. Let S:K--.K and T:KE* be strongly continuous mappings.
If the following assumptions are satisfied:
) S is subordinate to the approximation (K)n n e N’
2 <S(z), T(z)> _< <z, T(z)>; VzEK,
3) there is a number r > 0 such that for every n >_ no and every z K n with _< z there is
an element v. E K n such that vr < r and < S(.) vr, T(r > > O,
then the problem ICP(T,S,K) has a solution such that r. -< r.
**
PROOF. Since, for every n >_ no we have S(Kn)c_ K n and the all assumptions of Theorem 5
are satisfied, we have that for every n >_ no the problem ICP(T,S, Kr, has a solution znBecause for every n >_ no we have
< r and E is reflexive the sequence {z} n 6 N has a
weakly convergent subsequence denoted again by {z}n N" We put
is a solution of the problem
Now, as in the proof of Theorem 4 we conclude that
_< r.
t.CP(T,S,K). Obviously,
z
.
ACKNOWLEDGEMENT.
.
The authors thank Professor V.H. Nguyen for several helpful
comments on the first version of the paper.
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