Existence theorems for the implicit complementarity problem

VOL. 16 NO. (1993) 67-74

EXISTENCE THEOREMS FOR THE IMPLICIT

COMPLEMENTARITY

PROBLEM

G. ISAC

Dpartement

de Mathmatiques

CollgeMilitaireRoyal

St-Jean,

Quebec, Canada

J0J1R0

and

o.

GOELEVEN

Dpartement

deMathmatiques Facult des Sciences Universit de

Namur

Rempart

de la Vierge 8 B-5000

Namur,

Belgique

(Received

August

20,1991 and in revised formApril 16,

1992)

ABSTRACT.

Some existence theoremsfor thegeneral implicit complementarityproblem inan

infinite dimensional spaceareconsidered.

KEY

WORDS AND PHRASES.

Complementarity

Problem,

ExistenceTheorems.

1991

AMS SUBJECT

CLASSIFICATION CODES.

49J40.

1.

INTRODUCTION.

The study of Complementarity Problemsisaninteresting and important domain ofapplied mathematics

[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 controlproblems

[2], [3], [4].

Now,

it iswellknown

that,

the Implicit Complementarity

Problemcanbeused tostudythe optimal stopping ofMarkovchains

[6].

The first existence results for the Implicit Complementarity are the results obtained by Dolcettaand

Mosco

[7],

[18],

[19].

As

numerical methods for solving the Implicit Complementarity Problem we remark the

iterativemethodsproposed by

Pang

[20], [21]

and

Uosco

[22].

In

this paper, we study someexistence theoremsfor thegeneral Implicit Complementarity Problem inaninfinitedimensional space. Thispaper canbeconsideredas acomplement ofour

paper

[13].

2. DEFINITION

OF PROBLEM AND

PRELIMINARIES.

Let

<E,E* > beadual systemofBanachspaces.

Denote

byKapointedconvex coneinE, that is,asubset ofEsatisfyingthefollowingproperties:

I)K+Kc_K

T):c_K, forallRand3")Kn(-K)={0}.

Given a subset DCEand themappings S:D-,Kand T:D---,E*, the Implict Complementarity Problemassociated to T,SandK is

ICP(T,S,K):

findz_oE D such that

T(zo)EK*and

<S(Xo), T(xo)> O.

Wefind applications andexamplesof thisproblemin

[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_zo Dsuchthat

<z-S(zo), T(zo)> _>0;VxeK.

PROPOSITION

1. TheproblemSVI(T,S,K)isequivalent totheproblemICP(T,S,K).

PROOF. Indeed,if

ro

isasolutionoftheproblemSVI(T,S,K) thenS(Zo) eKandwehave (1): <z-S(xo), T(Xo)> >_0; VzEK

Let uEK be an arbitrary element. Ifweput z u

+

_S(zo) in(1) weobtain <u,T(zo)> >_O,

foreveryueK, that is, wehaveT(zo)EK*.

Ifweput z 0in (1)wehave < S(zo), T(zo)> _<0and since <S(Zo), T(zo)> _>0 wededuce <S(zo),T(zo)> O.

Conversely, let

o

bea solutionof theproblem ICP(T,S,K). We have, S(Xo)EK, T(%)E K*

and <S(zo), T(to)> 0whichimply <

-

S(%), T(ro)> _>0, foreveryzeK.

Given a nonempty subset DCE and the mappings T:D--,E* and S:D-K we consider the followingproblem

SVI(T,S,D):[

find

o

EDsuch that

<z-S(zo), T(zo)> _>0; VzED

Tosolve theproblemSVI(T,S,D)we usethefollowingclassicalresult.

THEOREM 1.

A

mapping To:D--,2

D,

where DCX, have a fixed point if the following

conditionsaresatisfied:

1):

X isalocallyconvexspace and theset D isnonempty,compact, andconvex,

2):

theset To(X)isnonempty andconvexfor all eDandthe preimages

TO-

l(y)

{zED lyETo(z)} arerelativelyopen withrespect to D, for allV ED.

PROOF. Theproofis in

[25][Proposition

9.9, p.

453].

THEOREM

2. Let DCE be a nonempty compact convex set, T:D---,E* and S:D.--,K two continuousmappings.

If foreveryz Dwehave < S(z), T(z)> _< <

,

T(x)>, then theproblem SVI(T,S,D) hasa solution.

Let

To:D-,D

be the point-to-set mapping defined by, To(x {u D < u-S(x), T(=)> < 0}, foreveryr fiD.

Weremark thatTo(r)isnonempty andconvexfor everyr 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 respectto D.

Hence,

by Theorem 1 thereis anelement r, Dsuch 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 bea pointedconvexconein E. Wesaythatasubset BofK is abase, ifB isconvex

and for every rK\{0} there is a unique

br

B and a unique number Ar

R+

\{0} such that

A

closed pointedconvexcone KCE islocally compact if andonly if, it has a compact base

[Klee’s Theorem].

Ifr R

+

\{0}wedenote

Kr

-<

{r K z <r} and

Kr

<

Wesaythat aconvexconeKCEisaGnlerkincone

[10]

ifthereexistsacountablefamily of

convexsubcones{Kn}n Nof K such that:

i)

Knislocally compactfor everyn N,

ii)

ifn_<mthenK_n_C

Krn

iii)

KONK

n

A

Galerkinconewillbedenotedby K(Kn)n N"

We recall that if DCE is a closed convex set, we say that a continuous operator

(not

necessary

linear)

P:E-,Eisaprojection ontoDifP(E) OandP(r) forevery D.

By

thesameproofasin ourpaper

[12]

we canprove that ifK(Kn)n Nis aGalerkinconein a Banach space, then for every n N there exists a projection

Pn

onto K_n such that for every z K wehave

lrnooPn(r)

r.

GiventwoBanach spaces (E, and (F, wesaythatanoperator

(not

necessary

linear)

T:E---,F is strongly continuous if for every sequence

{rn}

_{n N}CE, weakly convergent to

,

we

havethat

{T(rn)}n

_{N is}normconvergent toT(,).

This class ofoperators is very important and was intensively studied by Vainberg

[24]

and 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, wesupposegivenadual system < E,E*> ofBanach spaces. Weconsideron

E*

thestrong topology.

THEOREM

3. Let KCE be a pointed locally compact cone and $:K.--,E, T:K-,E* continuousmappings. If thefollowingassumptionsaresatisfied:

1")

thereisanumber,-_>0such 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-,

PROOF.

Since K is locally compactwehave that

Kr

--<

is aconvex compact set. Applying Theorem2 with D

Kr

-<

weobtainanelement z,

gr

-<

such that

(3): <z-S(,), T(z,)> >0; K

r-We

havethatS(**)K.

Two

casesarepossible:

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---.

Ifin

(3)

we put z w wehave, <z S(z.),r(z.)> >_0, that is, <z-S(z.),T(z.)> >_0 for all zK and by Proposition 1 we obtain that

z.

is a solutionof the

problem ICP(T,S < K).

II)

(.)II >

.

In

thiscasewehaver_< S(z,)I[

-<

rnaz(t,

r0)

andbyassumption 2

)

weobtain,

(4): < S(z,)-S(Uo), T(z,)> >_O,

andsincefor everyt_/Kr

-<

wehave

(): <

-

s(,), T(,)> >0

we deduce (using (4) and

(5)),<-S(%),

T(.)> <-S(.)+S(.)-S(%), T(z.)>

<z S(z,), T(z,) >

+

< S(,)-S(uo),T(,)> _>0, that is, wehave (): <

.-

s(%), T(**)> >0; W

e

,<-.

If zK is an arbitrary element then there is a sufficiently small A ]0,1[ such that

v Xz

+

(I )S(uo)

Kr

-<

Now,

ifweputz v_{in (6)}weobtain, (7):<z-S(Uo) T(z,) > _>0; Vz K.

Since

S(uo)II

<rwe canputz _S(uo)in (3) andwededuce,

(8): <S(uo)- S(u,), T(a:,)> >0.

From(7) and (8)weobtain

(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 ofthe problem

ICP(T,S,K) and theproofisfinished.

Theorem 3 can be extended to Galerkin cones. To obtain this extension we need to

introduceanewconcept.

We

say thatS:K-E issubordinate tothe approximation(Kn)n Nif thereexistsno Nsuch

that for everyn_>n_owehaveS(Kn)C_K_n.

In

[13]

weindicatedsomeexamplesof mappings with thisproperty.

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-mappingcanbe approximatedby an F-mapping while the class of DC-mappingsis veryreach.

We

say that S:K-E is r-subordinate to the approximation (Kn)_{n N} ifthere exist r_>0 and

no Nsuch that foreveryn>_nowehave

S(gr

CKn, where

Kiln

{z gnl

11

z

II

-<

r}.

REMARK.

IfS:K--.E is continuous and r-subordinate to the approximation

(Kn)

n N then

S(K

c

.

THE IMPLICIT COMPLEMENTARITY PROBLEM 7!

a)

t _<r. Since K isa Galerkincone there is asequence {tn}n(5N such that t

timn_.oot.,,

and

foreverynEN, t_nEK_n.

There exists eN such that

n-

<r-

I1

II,

for every n n_1, which implies,

n

n

+

<

-Since, foreveryn _maz(no,

nl)

wehaveS(n)KnCK,weobtMn bycontinuitythat S(z)eK.

b)

r. If forevery neN, z_neK_n

z

_n zd r<

n

then consideringthe sequence

Un

II

,

*n Xn, where 0<

en

<

,

II;

Vn N d

en

0 we_{have that Yn}

Kn,

Vn < d

tim,v.,.

z,whichimply that S(z)

S(Vn)

K.

THEOREM 4. Let (E, II) bea reflexive Bh spaceand K(Kn)n Na GMerkin conein E. LetS:KEdT:KE*bestronglycontinuousmappings.

If thefollowingsumptionsesatisfied:

10)

Sis r-subordinate tothe approximation K(Kn)neN,

2

)

there exist meN d uo Km such that S(Uo)II <

,

S(Uo)

Km

d <z-S(Uo),

T(z)> 0, for M1 zKn satisfying

r

Ileal

mat(r,rn) where rn is a number such that

p{ S()II u

K

n

dforeveryn _mar(no,m), 3

)

< S(),T() > < ,,T(,) > W

,

then theproblemICP(T,S,K) hasolution

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

boundedsequence.

BecauseEis reflexive

{t}

_neNh a_{wetly convergent subsequence}

{zk}k

eN" Wedenote

agMn

this subsequence by

{Z}ne

_N d we put

z.

(w)-z.

We have that

.

eK d

.

r, since

K

isclosed dconvex. Hence S(r.) eK.

LetreKbe bitry element. Foreveryn mat(no,m wehave, (0): <P.()-

s(.;),

r()

> _>0,

where

{Pn}n

6N iS asequence of projections. SinceSand T arestrongly continuous, computing

the limit in (10)weobtain,

(11):<t-S(t.), T(t,) > >0; foralltK.

The proofis finished since from (11) by Proposition 1 we have that _{t. is} a solutionof the problem ICP(T,S,K).

We

considernowthecasewhen S(K)c_K.

THEOREM

5. Let (E, beaBanachspace, KCEapointedlocally compact convexcone

and S:K-,K, T:K-E*continuousmappings. If thefollowingassumptionsaresatisfied:

10)

<S(t), T(z)> < <t, T(t)> YtK,

2*)

thereis r_>0suchthat for everyteKwithr< t thereis anelement

v

EKsuch 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 D_n {t6

KIll

t _<n}. Since K is locally compact we have that for

everyn N, D_nis&convexcompactset.

So

wehave:

foreveryneNthere_{is Xn}eDsuchthat (12):

<S(zt)-v,

T(x)>

<0; VvCDn

The sequence

{X}n

_N is bounded. >O)(3n N)(

x,

_>).

Indeed, supposing the contrary we have

*

If k> then there is a natural number n such that, n> [I_Xn _> _>r.

assumption 2

)

thereis anelement

vx.

nfiKsuchthat

v.

n <rand,

* by For this n,

(13): <

S(x)

_I/.

T(x)

> >0.

?1

But,

since

V+.n

< <

+n

_<n, from (12) wehave <$(r+)

vz.

n,

T(z)

> _<0, whichis a contradictionof (13).

Hence,

{zr}n

_e/v is bounded and because K is locally compact the sequence

{zr}n

_e/v hasa

Ix*

normconvergent subsequence n

k k N"

Let

x.

=/mzr

k. Weshownowthat z,isasolutionofthe problemICP(T,S,K).

Indeed, if v K is an arbitrary element, then thereis m N such that for every n>m we

have, v D_nandforeveryn

k>m, v

Dnk

and <

S(zk

v,

T(zk

> <O. Using the continuity ofSandTweobtain,

<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 have

x.

<r.

From

Theorem 5wededucetwoimportant corollaries.

COROLLARY

1. Let KCE be a pointed locally compact cone and S:K-K, T:K-,E* continuousmappings. If thefollowingassumptionsaresatisfied:

1

)

< S(,),T()> >_ <,T()> WeK,

2*)

there is a number r>0 such that for every xK with r<

IIll

we have < S(,),T() > >0,

thentheproblemICP(T,S,K) hasasolution

,

such that

.

_<r.

PROOF. We apply Theorem5with v, 0forevery Ksatisfying

>-COROLLARY

2. Let K CE be a pointed locally compact cone and S:K---,K, continuousmappings. If thefollowingassumptionsaresatisfied:

)

< S(),T() > _< <

,

T(x) > V K,

2*)

there is a number _ro>0 and u_o K such that for every z K with r < wehave < S(x)-uo, T(x) > >O.

then theproblemICP(T,S,K) hasasolution

.

such that

.

<

+

maz(ro, Uo

)-PROOF. Ifwedenote, r max(ro, _UoI1)

+

wehaver>roandr>

uo

II-Now,

we canapply Theorem 5 sinceassumption

2*)

ofthistheorem issatisfied with v_x u_o,

foreveryz Kwith

>-

r.

REMARK.

Condition 2

)

ofCorollary2issatisfiedif T is semicoercivewith respectto S in thefollowingsense:

If S(z)=z,for every z K, this notion is similar tothe semicoercivity used by Stampacchia

and Lions

[22],

[23].

Obviously, condition 2 is satisfied if there is a number a>0 such that <S(x),

T(x) > >a

2,

for everyzeK.

Finally, wegiveanextensionof Theorem 5to Galerkincone.

THEOREM 6. Let (E, II) be areflexive Banach space and K(Kn)n N a Galerkinconein E. LetS:K--.K andT:KE*bestronglycontinuousmappings.

If thefollowingassumptionsaresatisfied:

)

S is subordinate tothe approximation

(K)n

_neN’ 2 <S(z), T(z)> _< <z, T(z)>; VzEK,

3

)

thereis anumberr_>0 such that for everyn>_noandeveryz K_nwith _< z thereis anelement

v.

EK_nsuch that

vr

<r_{and < S(.) vr, T(r > >}O,

then theproblemICP(T,S,K) hasasolution

**

such that

r.

-<

r.

PROOF. Since, forevery n>_nowehave S(Kn)c_K_n and the all assumptions of Theorem 5

aresatisfied,wehave thatfor everyn>_n_otheproblemICP(T,S,

Kr,

hasasolution

zn-Because for everyn>_no we have

z

<rand E is reflexivethe sequence

{z}

_n6N has a weaklyconvergentsubsequence denoted againby

{z}n

_N" Weput

Now,

as in the proof of Theorem 4 we conclude that

.

is a solution of the problem

t.CP(T,S,K). Obviously,

.

_<r.

ACKNOWLEDGEMENT. The authors thank Professor V.H.

Nguyen

for several helpful

commentson thefirst versionof thepaper.

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