Vietnam Journal of Mathematics 2g:4 (2000)2g$'g10
Vi.etham lournal
of
MATHEMATICS
@ Springer-Verlag
2000
some Sufficient conditions for the Existenee
of Equilibrium Points
Concerning Multivalued Mappings
Nguyen Ba Minh and Nguyen Xuan Thn
p.o. Box 6s1,Bo Ho 10000,
Instituteof Mathematics,
Hanoi, vietnom
Received April 9, lggg
Revised April g, 2000
Abstract.
Some results on the continuity with respect to a cone
of multivalued
mappings are shown and applied to consider sufrcient conditions
for the existence of
weak equilibrium points concerning multivalued mappings.
Some applications of these
results to the existence of efficient points of subsets in reflexive
Banach space and of
solutions of vector optimization problems are also discussed.
1. Introduction
LeJ X' Y be topological locally convex Hausdorff spaces,D
c X a closed convex
subset, C c Y a closed convex pointed cone. Given a
multivalued mapping
F : D x D -+ 2v with F(*,il
* a for all x,y € D, we are interested in
considering the problem of finding
I e D suchthat F(i,A) I -intC, for all y e D.
(1)
The point f is called a weak equilibrium point (or, a solution)
of the weak
equilibriumproblemwith respectto c, denotedby (wEp,c).
in. proulemof
finding
fr e D suchthat F(n,il g -Q \ {0}), for all € D,
(2)
U
is called an equilibrium problem with respectto c and denoted
by (Ep,c),
Sucha point r is said to be an equilibriumpoint (or, a solutionj of
tfr. problem
(EP'C)' It is clearthat if int C:0, then everypoint r
€ D is an equilibrium
point of (WEP,C). In this caseit is not interestingto consider
(1). ifrur"foru,
in this paper, we only study (1) with int C I 0. Frrrther,if u is
Ln equilibrium
point of (EP,c), then it is alsoa weakequilibriumpoint of (wEp,c).
And, if
296
Ngugen Ba Minh and Nguyen Xuon To'
y with e c C and x is an (a weak
d is another closed convex point cone in
equilibrium point of (EP,C) ((WEP,O)), then n is also an (a weak) equilibriur
point of (EP,d) ((WEP,C), respectively). F\rther, if C satisfies condition: l
there is aclosed convex pointed cone C such that C\{0} C intf and r e D i
a solution of (WEP,C), then i is a solution of (EP,C). Such a condition on C i
satisfied, for example, if it has a convex compact base (see the proof in [12]). I
this paper we consider (WEP,C) only'
Before presenting the main results we generalize the well-known Banact
Steinhaus theorem [3] for a family of convex functions. We prove necessEryan
sufficient conditions for a multivalued mapping to become a lower C-continuor:
multivalued mapping (Theorem 2.3). Then, using the new Banach-Steinha
theorem for convex functions we show that if C has the polar C', which is
polyhedral cone and F ; D + 2v is C-convex and lower C-continuous wit
f (;) * C convex for all x € D, then it is weak lower C-continuous (Theorer
3.4).
we extend the result in [12] to the multivaluedcase.The probleu
F\-r,rther,
(1), (2) concerningsinglevaluedmappingsare studied by Ky Fan [6], Blum an
O"ttii ia, S], Oettli [10]and others. Our resultsare extensionsof their results. l
the existenceof equilibrium points concerni
[fO] Oettti provedsomeresultson
of scalarization. This method cannot t
method
the
vector functions by using
F of the form G + H, where G and '
mapping
the
applied to our casewith
weaken
the coerciveconditions (Conditic
We
also
,"[irfv different conditions.
(seeThmrem 3.3), using tl
conditions
by
other
Ag/ in Theorem 3.1 below)
weak lower C-continuo
The
space
X.
Banach
weak topology in a reflexive
(wEP,c)
in the casewhere
study
to
is
used
2.4
property of F in Theorem
cone'
polyhedral
has a cone C with C' a
Finally, we apply the obtainedresultsto showthe existenceof efHcientpoin
of subsetsin reflexiveBanachspacesand of solutionsof vectoroptimizationpro
'rariation
Iems. The resultscan be alsoappliedto the existenceof solutionsof
mappings.
multivalued
problem...
concerning
inequalities,Nash equilibrium
2. Preliminaries
Let x be atopological locally convexspace,D c X aconvex set. By R'
denotethe spaceofieal numberswith the usualtopology E: RU {+m}. Fir
of all we recall the followingdefinitions'
Definition 2.L. A function f : D + E tf calledo conaer (conu'ue)fuTcti
$@'+(1-a)v) 2 o/(')+(1-c)/(1
if f (ar+(l-c)y) S
" f @aIIr,A
) + ( 1 -€dom/:
a)/(d
{c e D I I@) < +oo} anda el0,
,Lrpl"tilr.i;9, iiia" for
I
Definitiol 2.2. Let {fola € I} be a fomitg of lunctionson D, where
equise
(lower)
upper
is
this
that
say
family
a nonernptyparameter set. We
in
continuousatrs e D il for euerye > 0 thereis aneighborhoodUolts
-€'
holds
respectively),
tor
suchthat f,(r) S f"(rs) +e (/"(") > h(to)
r€UfrD anda€1.
SufficientConditionsfor the Edstenen,
of EqltilibriumPoints
ZgT
F\rrther, let Y be another topologicallocally convex spacewith a cone C
and F be a multivaluedmappingfrom D to Y (denotedby F : D -r 2Y) which
meansthat F(r) is a set in Y for eachs e D.
We denoteby dom F the set of all s € D such that F(c) * 0. In this paper
without lossof generalitywe can assumedom F - D.
Definition 2.3. a) We say that F is upper (lower) C-untinuous at rs e D if
V of the origin in Y , therc,is a neighborhood
U of rs in
for eachneighborhood
x suchthat F(r) c F(cs) +v +C (F@s c r(c) +v - C, respectively)holds
forallr€UfldomF.
b) We soy that F is C-continuousof os if it is upper and lower C-continuous
at that point, and F is upper (respectiuely,
lower,... ) C-continuouson D if it
is upper (respectiuely,lower,.,. ) C-*ntinuou.s at euery point ol D.
c) We say that F is ueoklyupper(lower) C-wntinuow at os if the neighborhood
U of Es as aboueis in the wuk topologyof X.
Deftnitiort 2.4. F is said to beC-conueril F(aa + (1 - a)y) c aF(r) + (1 a)F(y) - C holdsfor all r,A € D ando € [0,1].
Let Yt denote the topological dual spaceof Y and
C' : {{ e Y' | (€,y) ) 0, for all y e C}.
Here by (€,y) we mean€(y).
This C' is called the polar coneof the cone C. For given F : D -r 2Y and
€ e C' we definethe functionGg : D +E UV Ge(") : supyeF(")(€,
Al, n e D.
Proposition 2.L. a) If F is C-conuea,then Gg is a cdnuexfunction.
b) If F is lowerC-continuousatag € domF, thenGq is alower semicontinuolls
at rg.
Proof. a) Let r; € D, i:
L,2, and a e [0,1]. Since
F(aq +(1 - a)rz)CaF(r1)+(t -o)F(x2)-C,
it follows that
Gq(aq + (1- a)rz):
r.r(o,il?r_.1,r,(€,v)
s
Y)
v r , , t r 1 o " , l i t l " l tG z ) - c ( € '
3
-',tr", I
yeo
F 1o,,ilf,
1a
(€'u)
sup (€,y) + (1- o) sup (€,y)
yeP(zrl
veF(c2)
a G s ( x 1 ) + ( 1- a ) G e @ z ) .
This showsthat G6 is a convexfunction,
Nguyen Bo Minh ond Ngugen Xuan Tan
298
b) Let e > 0 be given. Since€ € C', there is a neighborhoodV of,the origin
in Y such that {(V) c (-e, e). As F is lower C-continuousat to € dom F, it
followsthat there existsa neighborhooilU of rs in X suchthat
F(ro) c F'(r) +V - C, forall c € U n D.
This implies
G6(cs) :
sup ({, y)
YeF(cs)
veF(rl
veV
ve-C
SGe(r)+e.
Hence
G c @ ) 2 G e ( " 0 )- e , f o r a l l s € U n D .
Therefore,Gg is lower semicontinuousat rs.
This completesthe proof of the proposition.
I
F\rrther, we recall that a barrel spaceis a topological locally convexspace,in
which any nonempty closed,symmetric, convexand absorbingset is a neighborhood of the origin (see,for example[1U). The followingtheoremis an extension
of the well-known Banach-Steinhaustheorem for a family of convexfunctions.
I is an indn set and to :
Theorem 2.2. Assurnethat X is o banel spaen,,
X + E, a € I, is conuer:and louer semicontinuornon sotneneighborhoodUo
of so € domfo lor ollo € I. In add,ition,supposethatforany r C X thereis a
constant7> 0 suchthatf"(*) <'f |or alla€ L Thenthefomilg{f"la € I}
is upper quisernienntinuousat ns.
Prcof. By setting /" (c) : lo(c + cs ) - /" ("0 ), if necessary,we may assumethat
ro:0 and /"(0) :0 for all a € ^I. For givene > 0 we put
A o : { r e X l / " ( " )S e } .
Since0 € /., we concludethat A" * 0.
Without lossof generality,we may assumethat Us is a closedconvexneighborhood of the origin. Flom the convexity and the lower semicontinuity of f"
on Us, it follows that Uq n A" is a nonempty closedconvex set for all c e .t.
F\rrther,putting U : fr.,el(Uon Aan(-A')), *" concludethat U is nonempty
closed,symmetricand convex. Now, we claim that U is absorbing.Indeed,let
r € X. By the hypothesesof the theorem there is a constant7 > 0 such that
( 7, for all o e /. Without loss of generalitSwe can
f"(x) ( 7 and f"(-*)
assume"y > e. Hence
: 9rf"{d
s r.
f"ef): f,{i, * (1- ;)0)=:,f"(")+ (r- 1)f"(0)
SinceUo is absorbing,there is a constant p ) 0 such thar exf p, -erlP € [/q. For
we concludethat (e/1)r e A, fl Uo. By a similar argument,
jo - *oh,p),
SufficientConditionsfor the Eristenceol Equi'libriumPoints
299
we obtain (-eln)" e Ao flUo, for all ae I, and then (r/lo)r eU.It means
that U is absorbing.SinceX is a barrel space,U is a neighborhoodof the origin
in X. For r € [/ we have
f"(r) S e :/"(0) * e, for all o e -I.
at the
This means that the family {1" I o e /} is upper equisemicontinuous
origin in X.
r
This completesthe proof of the theorem.
The following theoremsextend Theorem5.5 in [8].
2Y is a
Theorem 2.3. Supposethat Y is a Banach spaceand F : D
multiualuedrnappingwith F'(") - C conueafor all a e D. Then F is lower
U is lower
C-continuousat ns if and onlg if the familg {Ge lt e C', ll€ll
equisemicontinuousat tg.
Proof. We first assumethat F is lower C-continuousat ro. Let e > 0 be given.
theorem,the family {€ e C'lll€ll : 1} is equicontinuous.
By Banach-Steinhaus
Thereforethere is a neighborhoodV of,the origin in Y such that {(y) € (-e, e)
holds for all y e v and { € c" ll€ll : 1. Flom the lower c-continuity of F at
rs there exists a neighborhoodU of'xs in X such that
F(ro) c F(r) +V - C, for all r e U n D.
It follows that
Ge("0) :
suP (€,Y)
veF(zs)
veF(s)
YeV
ve-C
* e :G6(c)+ e
<
rilt,(€'Y)
h o l d s f o r a l lx € U F tD a n d € e C " l l € l l= 1 ' T h i s m e a n s t h a t t h e f a m i l y
at xs.
{Ge | € e C', ll€ll U is lowerequisemicontinuous
at rs, but F is not
Now, assumethat this family is lowerequisemicontinuous
lower C-continuousat os. This impliesthat there existsa convexneighborhood
- rs
V of the origin in Y suchthat one can find a net {t"} from X with lim ro
and
F@il I F(*,) +v - C, for all o.
Then, we can take go e F(rs) with
ao / F(r") +V - C.
Sincethe set cl (F(ro ) +V 1Z- C) is closedand convex,applying a separation
theorem one can find some{o from the topologicaldual spaceof Y with unit
norm such that
€o(y,) z €" (v)
for all g e F(ro) +VIZ- C. Assumethat thereis os suchthat eoo/ C'. Thus,
we can find ys € Cwith €"r(yo) < 0. Since'lyoeC for all 7 > 0, we have
-t€"0(yo)
€oo(yo)2 {", (roo) *€oo(u"o)
NgugenBa Minh and Nguyen Xuan Tan
300
for some zoo € F(r.,o)
us e V/2. Letting ? -r *oo, the right-hand side
"ttd
tends to *m and we have a contradiction. Therefore, we deduce that €o € C'
for all a.
For an arbitrary 6 > 0 there exist 9o € F(r"), Eo eV/2 and eo € C such
that
(4 o ,9 o ) > s u P (€ " , Y ) - *
J
yeF(t.
)
(€o,oo) > ,.rp(€.,r) - 9
, e + ' 3
- .
(€o,Eo)
r
f
"ru%(€",4
Hence,for zo - Ao + ta + eo e F(r") + VIZ - C, we have
€ o ( v o )2 € " ( " . )
yeF(x,)
uet
ce-C
: Ge.("") + suP(€",ol - 6.
u e V/ 2
at the origin, we can
Sincethe family {€" l€" e C', ll€ll - 1} is equicontinuous
assume
suP suP (€o,u) : 60> o'
v12
Consequently,
Ge"("0) > Gs("") + 6o- 6, for all {o € CP, ll€"ll : 1'
Taking e: (60- 6)12 for 6 ( 60,we obtain
Ge"("0) ) Ge,,(ro)+ e,
or
Ge,(ro) < Ge"("0) - e, for dl {o e C , ll€.ll : 1.
It contradictsthe lower equisemicontinuityof the family {Ge | € e C', ll(ll - 1}.
This completesthe proof of the theorem.
r
Next, we recall that a cone C c Y is said to be a polyhedral cone if. C by the @n\rexhull of n independent
cone(conv{yt , . . . ,An}), i.". a conespa^nned
v e c t o r {sA r r . . . , A n l .
Theorem 2.4, Let D, X,Y beos obweondletC Cy beaeonufficonewithCt
a polyhedrulcone. AssumethatF : D - 2Y is C -conva ond,lowq C -continuous
with F(x)+C convffifor ollr € D. ThenF is wdlg louuC-oontinuous.
Prcof. Assume that Ct - cone(convt€t,...'€"))- [t is clear that, for i :
1r2,. . . ,fl, GE, is a convex and lower semicontinuousfunction from D to E.
Therefore, it i; weakly lower semicontinuousfrom D tnE- Supposethat rs €
domF. We show that F is weakly locrer C-continuousat c6. Indeed, for any
given e > 0 and i : L,2,. .. ,fl, we Ganfind a neighborhoodU; of cs in the weak
topology of X such that
Sufficient Conditions for the Eristence of Equilibriurn Points
301
Ge,@), Gt,("0) - pse, for all r € U; i D,
whereBs : min ll DL, \Pjll, Dl=, ^r _ 1. Since{€i, i : 1,... ,n} is an
: 1, we haveps > 0. Putting U :
independentsystem
)l=rUr,
1nd.Dl=, lr
w e o b t a i nG e , @ )2 G e , ( " 0 ) - p s e , f o ra l l r € U n D a n d i - 1 , .. . , , T 1 , .
This showsthat the family {Ge, I i : I,. . . ,n} is weaklylowerequisemicontinuous at rg. Now, we claim that
G e @ ) > G 6 ( " 0 ) - e , f o r ar lel U a n d € e C ' , l l { l :l 1 .
I n d e e df ,o r { e C ' l l € - l :l 1 , w e c a n w r i t e€ : g D l = r ) i f i
f o r s o m ep > 0 .
havet: ll€ll: gllDf=, Ai{,11.Therefore,
't
1
0-#3;,
llfl'6'll
to
We
o rP s P < L .
Since
G g , ( c :)
s u p ( € i , y )>
yeF(z)
s u p ( € i , g ) - 0 0 e i,= 1 , . . . , n , r € U t D ,
yeF(cs)
multiplying both sidesof theseinequalitiesby gAr and taking the sum of them
we get
G s ( x ) :r g p . ( i g x , € , , a )
seF(x)
=
YeF(oe) =;
2Ge(ro) -e, for all c € U n D.
This showsthat the family tce | € € C',ll€ll = 1) is weaklylowerequisemicontinuous at rs. Applying Theorem 2.3, we concludethat F is weakly lower Ccontinuousat os.
This completesthe proof of the theorem.
r
Proposition 2.5. If F : D
2Y is lower C-continuouson D, then the
set A- {c e D I F(c) n intC : 0l is closed.
Proof. Without loss of generality,we assumethat int C # 0. Let rs e A,
the closureof .4. We claim that cs e A. Indeed,let xn € A and rn + ro.
We assumeon the contrar/ oe ( A, This meansthat F(ro) n int C t' 0 and
thereforewe can find a point yo e F(ro) *d a neighborhoodV of the origin in
Y suchthat ys +V CintC. SinceF is lowerC-continuousat rs, one can find
a neighborhoodU of.cs in X suchthat
F(ro) c F(r) +V - C, forall r € U n D,
Therefore,
Aoe F(r) +V - C, forall x €U frD,
302
Ngugen Ba Minh and Nguyen Xuan Tan
ort
0eF(r)-ao*V-C
cF(z)- intC-C
C F ( n ) - i n t C , f o r a l lr e U n D .
the fact that
This showsthat.F(r)n intC +0 forall x €U nD. It contradicts
rn €.4 and tn 4 rs. So rs e.A a"ndA is closed.
This completesthe proof of the proposition.
I
Next, we introduce the followingdefinitions:
Deffnition 2.5. We say that G : D x D -, 2Y is o rnonotonernappingif
G(*,y) + G(g,n) I -C holdsfor oll r,y € D.
Deffnition 2.6. Let K and D be nomemptyconuessubsetsin X ufith K C D.
Then coreDK, the eoreof K rcIotiaeto D, is defineilthrough@€ coreDK if and
o n l yi f a € . K a n dK f l ( o , U l + 0 f o r a l l y e D \ K , w h e r e ( a , y l :{ r € X l c aa * (1 - o)s for all a € [0,1)].
Finally, we recall the followingdefinition:
Deftnition 2.7. Assumethot B is o nonemptVset inY.
1. A point o € B is called an efficient or, Pareto-minimalpoint of B with
respectto C if , - A € C, for someA € B, then y - t € C.
The set of all the fficient pointsof B is denotel,bg Min (BIC)
2. In the casewhereint C is nonempty,r € B is elled o weoklyefficientpoint
of B uith respettto C if r € Min (B I {0} U int C).
The set of all efficient points of B is denotedby WMin (BlC).
3. The Main Results
Let X,Y be topological locally convexHausdorffspaces,D C X a nonempty
closedconvex set, C C Y a closedconvexpointed cone in Y. We generalize
Theorem3.1 in [12]to the caseconcerningmultivaluedmappingsby the following
theorem:
T h e o r e m 3 . 1 . L e t X , Y , D o n d C b e o s a b o u .Lee. t G : D x D - 2 Y
H : D x D +Y be rnappings sotisfyingthe folloaing unditions:
A1. 0 e G(r, r) lor all n € D,
A2. G is a rnonotonemappinguith G(r,y) rcmpoctfor oll r,y e D,
A3. For angfi&enr,U € D the moppingI : [0,Ll'2Y defind by
s ( t ) : G ( t Y+ ( 1 t ) t ' , Y )
and
is upper(-C)-continuousott:0,
A4. For-angfo"d r € D, themappingG(t,.), D - 2Y is C -unaer and lower
C-continuouson D,
SufficientConditionstor the Eristenceol EqltilibriumPoints
303
A5. H(r,r) :0 lor allr € D,
A6. For any fiied A e D, the mappingH(.,V) : D -> Y is (-C)-continuous on
D,
A7. For angfieenr € D, the mappingH(r,.), D +Y is C-conuea,
K D suchthatforanyn € D\
A 8 . T h e r ee d s t sa c o n u e r c o m p a c t s u b s e tC
cor€pK one canfind a point a € corzpK suchthat
G(r,a)+H(n,a)c-C.
Then there existsa point E e D suchthat
G ( n , s ) + n @ , a ) -gi n t C
for all A € D, i.e. fr,is a solutionof the (WEP,C).
Rernark. If the set D is compact,then Assumption A8 is satisfiedvacuously
with K - D, sincethen D \ corepK :0. As in the proof of Theorem1 in [4],
we prove this theoremover three lemmata.
Lemma 8.2. Let D,K,G and H satisfythe assumptions
ol TheoremS.t.Then
there existsa pointi e K suchthat (C(y,t) - H(n,g)) n int0 :0 for all
a e K.
Proof. Indeed, for any y e K we set
S ( y ) : { r e D l ( G ( y ,t ) - H ( a , y ) ) n i n t C : 0 } .
SinceA e S(il, it followsthat S(y) #0for all y € D.By Proposition2.5,^9(g)
is a closedset in D.
Now, let {y; li e N} be a finite subset of,K and .I C N be an arbitrary
nonemptyfinite subset/:
{1,2,...,n}, whereN denotesthe set of natural
numbers.Take z € conv{ytll € /}, we have
+ ' . . r r \ n . - t n
withA;> 0, i . t,
,:E^,n,
I
Ai: 1.
We showthat z € Ul=, S(y,). Wu assumeon the contrary that z /Ui=rS(gr).
This implies
( G ( y r , z -) H ( r , y r ) ) n i n t C * 0 f o r a l l i :
L , 2 , . . ., r ' t .
Hence
;^,(G(v,,z)-H(r,yi))n
intC#0.
(B)
d=1
By AssumptionsA2 and A4, we have
n
n
I rrc(or,z) g |
i= 1
Hence
i,j=t
n
- C:
- C.
.rr.lrc(y;,y)
D r,r, (G(y,,y)* G(ai,,yi))
t,/:r-t
Nguyen Ba Minh and Nguyen Xuan Tan
304
n
n
rr.l,(G(y,,s)* G(yi,y))c -C.
rrC(yr,z) C I
I
(4)
i,j=L
d= t
F\rrther, it follows from AssumptionsAb and A7 that
o - H(z,4 ri\;H(z,vi) - c
i=1
tl
and then
-
I
t,
\;H(z,yr)e -C.
(b)
A combination of ( ) and (5) yields
-C.
5 ^,G(ar,z) H(r,yi))c
i=1
Togetherwith (3), this implies
n
IZ-r
d=1
f r ( C ( y i , z ) - H ( r , y d ) )n i n t C c - C o i n t C : 0 ,
andwe havea contradiction.Soz € UL, S(yr)andthen
c o n v{ v r l t - 1 , .. . , n } g U , S ( y i ) .
d=1
Applying the standard version of KKM's lemma (see,for example [], Theo
rem 24, Ch.6J), we deduc" fiLr S(ar) * 0. In other words,any finite subset
S(yr), Ai € K, has a nonemptyintersection.Thesesubsetsare nonemptyclosed
subsetsof the compact set K, so the entire family has nonempty intersection.
This meansthat
) s(il+s.
veK
Taking E € |'ex,S(y),
we can easilyverify that
( C ( v , i ) - H ( n 4 y ) )n i n t C : 0
holds for all A e K.
This completesthe proof of the lemma.
I
of Theorern3.1. Then
Lemma 3.3, Let D,K,G anil H satisfythe ossumptions
(i) implies(ii), where
(i) t € K, (G(y,fr)- H(n,y)) nint0 -0 holdsfor oIIy e K.
(ii) t € K, G(r,y) + H(t,v) g -intC holdsfor oIIy e K.
P r c o f .L e t E € K b e s u c h t h a t ( G ( y , i ) - H ( E , y ) ) n i n t C - 0 f o r " a l l y e K .
t € [0,lJ. It is clearthat rt € K
For fixeda € K, we set nt:ty+(l-t)fr,
f o r a l l t € [ 0 , 1 ]a n d t h e r e f o r(eG ( r r , t ) - H ( i , " r ) ) n i n t C : 0 . S i n c e
0 e G(x2,rt) c (1 - t)G(xt,fr)* tG(x1,U)- C,
hffwient Conditionsfor the Edstene,of EquilibriumPoints
it follows that
0 e (1 -t)(G(rt t) - H(n,"r)) + (1- t)fl(i,ri) *tG(r1,y) - C
c ( 1 - t ) ( G ( r 1 , t ) - H ( n , r t ) ) + ( L - t ) t f l ( r , y ) * t G ( n 1 , a . )- C .
If
then
-intC, for somet € [0,U
s - t)tff(n,y) *tG(r1,y) E
t1
trlt=\
rrr; ^ \\ ^ ;-+ r'r J a c,
r,.'(rt, t) - H(n,rr)) n int C * 0, for somet € [0,lJ,
a,ndwe have a contradiction.It implies
(L - t)tU(a,il * tG(r1,il g -int C.
Hence
(6)
(1 - t)H(E,y) + G(*r,y) g -int0, for t I 0.
We claim that (6) is also true for t - 0. Indeed, we define the mapping
f r [ 0 , l l - 2 Y by setting
F ( t ) - ( 1 - t ) H ( z , y ) * G ( * t , a ) , t € [ 0 ,1 ] .
Bt Asumption A3, F is upper (-C)-continuous at t - 0. Supposethat
t{O) c -intC. Sincetr'(O): H(i,y)+G(fr,y) is a compactset, there is a
-igbborhood V of the origin in Y such that tr'(O)+ V C -int C. The upper
of F at t : 0 impliesthat there exists 6 > 0 suchthat
(-C)ontinuity
F(t) c F(0) +V - C, for all t e (-6,6) n [0,1].
It iepties
.F(t)c -int0 - C --intC,
for all t € [0,6),
ttcn
11
-intC, for all t € [0,6),
1r - t)I/(i,y) +G(*r,y) c
1gd re have a contradiction.Thus we obtain F(0) E -int C. Therefore
H(n,il+G(n,y)Q-intC.
y e K is arbitrary, then we concludethat (ii) holds.
Thb completesthe proof of the lemma.
I
[
ma 3.4. Let D and K satisfy the assumptionsof Theorem3.1. Let Q :
D - 2Y be a multiualuedC-conuer mapping. Let rs € corepK be such that
+t"o) g -C, O(s) I -intC lor ally € K. ThenO(a)I -intC for alta e D.
pr1;61[.
Assumeon contrarythat thereexistsg € D\K suchthat O(y) C -int C.
lA z € ("0 ,A), z:' o,r1+ (1 - o)y for somea e [0,1). We have
- a)o
-c
(s)
',',
;'_,;";,1'|1;|,t'=T,ll?*
"
Tberefore
O(") e -intC, for all z € (rs,yl.
Ngugen Ba Minh and, Nguyen Xuan Tan
306
Sinceno € corcpK there existszo e (xo,y] nK. HencuiD(a) e -intC. This
contradictsthe fact that O(y) I -int C for all A € K, and the lemmais proved.
I
P'rvof of Theorern3.1. By Lemma3.2 there existsi e K with
( C ( z , s )- H ( i , y ) ) n i n t C : 0 ,
f o r a l ly e K .
Applying Lemma 3.3, we get
G(n,il + n@,y) g -int C, for all A e K.
(Z)
F\rrther, we definethe multivaluedmapping Q : D - 2v by
a(y)- G(n,il+ H(i,a), Ue D.
Assumptions44 and A7 showthat O is C-convexand it followsfrom (7) that
Afu) I -intC, for all y e K.
If t € cotepK, we set TO : u, otherwise, we set ug : @, where a is from
Assumption A8. Then, we always have A("0) g -C. Using Lemma 3.4, we
concludethat
-int
O(y) I
It followsthat
C, for all y e D.
G(r,a) + n@,y) g-intc,
This provesthe theorem.
for au y e D.
I
In the sequel, we weaken the coercivity requirement A8 for the case of
reflexive Banach space. We assumethat X is a reflexive Banach spacewith
the norm denotedUVll ' ll.
Theorem 3.5. Let X bearufl,exiaeBanachspace,D.C,Y be as aboue.LetG
A1 - AT in TheoremS.Lwith the lowerC-continuity
and H satisfyAssumptions
A4 andA6 replaced
and (-C)-continuitg of G(*,.) andH(.,y) in Assumptions
by the weaklowerC-continuityand the weak(-C)-continuity, respectively,
In addition, assurnethat thereis a point a e D suchthat, for euerysequence
{*"} uithlim,.++m ll""ll : *F, one of the follouing conditionsholds:
(H1) Thereis no ) 0 suchthat G(r"o,a) * H(rno,a) g -C.
(H2) Thereareno ) 0 andy e D withlly-all < 11"",-all suchthatG(xno,A)*
H(*no,y)E -C.
(H3) Thereare no and g € D suchthat G(y,xr) - H(*n,V) g C, for aIIn 2
n0.
Then thereis E € D suchthat G(n,y)+ H(r,il g -intC.
P r o o f .L e t D n - { r € D I l l t - o l l S n } , f o r n : 1 , 2 , . . . S i n c e Xi s a r e f l e x i v e
Banach space,Drr, n : 7r2r. . . , are convexweakly compact subsetsin X.
Applying Theorem 3.1 with D - D,, and using the weak topology in X, we
concludethat thereexistfin € Dn, fr: Lr2r.. . , suchthat
Suficient Conditions tor the Eri,stenceol Eryitibrium Points
G(r",y) + H(r",g) g -intC for all A € Dn.
If llr" -all < n for somen, then tn € coreDr,. We definethe mapping
Fz D-2Y bysetting
(8)
F(r) -- G(rn,s) + H(rn,t), r e D.
It is clear that F is a C-convexmultivalued mapping and
G ( r " , c , , )* H ( * " , a n ) c - C
F'(") ( -int0, for all x € Dn.
F(r"):
Applying Lemma3.4, we deduce
F'(") f -int C, for all r e D.
This meansthat c' is a solutionof the problem(WEP,C). Therefore,it remains
to investigatethe ca.sellc,' - oll : n for all n ) 1.
that Condition (H1) holds. We showthat c,,o is a solution
First, we a^ssume
to the problem (WEP,C). Indeed,for any o € D\D,. there is a positivenumber
t € [0, 1] such that ta + (1 - t)x e D. Hence
G(r,o ,ta * (1 - t)r) * H (xno,ta * (1 - t)c)
e { G @ " 0 , o ) * H ( r n o , a ) )+ ( 1 - t X G ( " , , . ,x ) + H ( x n o , r ) ) - C .
SinceG(rno,a) + H(nno,a) C -C, it followsthat G(rns.ta + (1 - t)c) +
H ( r n o , t a * ( 1 - t ) c ) c ( 1 - t x c ( * n o , t ) + H ( r n o , s ) ) - C i f G ( r n o , x )*
H(rno,r) e -intC. Then
G(rno,ta* (1- t)r) * H(rno,ta* (1- t)r) e -intC,
a contradiction with the fact that c,,o is a solution of the problem (WEP,C)
on D,,. Consequently,
G(*no,n)+ H(*no'n) g -intC for all x € D, hencer,,o
is a solutionto the problem(WEP,C) on D.
Under Condition (H2), we observethat
G(rn* y) + H(rno,il C -C
and fly - oll < llr,,o - all : no. We definethe function F as in (8). Then
y € coreDDnoand F(y) C -C, F(r) E -int C for all c E Dno. Applying
Lemma 3'4 we have
F(") / -intc, for all a e D.
It follows that
G(rno,c) * H(*no,r) g -intC, for all n,e D
or, on, is a solutionto the problem(WEP,C).
Finally, if Condition (H3) is satisfied,we show that Condition (H2) also
holds. Indeed,since
G(*",y)+G(u,rn)c-C,
we concludethat
G ( r ^ , a ) + H ( * n , y ) C H ( r " , Y ) - G ( U , r n )- C c - C .
308
Nguyen Bo Minh and Ngugen Xuan Tan
< llt" - oll. Thus Condition (H2) is
For n sufficientlylarge,we have lly "ll
also satisfied. This completesthe proof of the theorem.
r
The aboveconditions(H1)- (H3) are similar to Conditions(H1)- (H3) in
4.5].
Corollary
[9,
Banachspace,D C X be a closedconuex
Corollary 3.6. Let X be a refl,esiae
set. Let Y be a Banach spaceand C C X be a closedconuerpointedconewith
C' a polghedralcone. Let G and H satisfyAssurnptionsA1 -> AT in Theorem
3.L with the (-C)-continuitU of H(.,y) in AssumptionA6 replacedbg the weak
(-C)-continuity ol H(.,A). In addition,assurnethat G(r,y) + C is conuexfor
a l l ( r , y ) e D x D a n d t h e r e i sa p o i n t a € D s u c h t h a t t o re u e r ys e q u e n c{ re" l
withlimn-* ll",,ll : +m, one of Conditions(H1)-(H3) holds. Then thereis
n e D suchthat G(n,y) + H(i,y) g -int C.
Proof. Theorem 2.4 ensuresthat, for any fixed r € D, G(*,.) is C-convex
and weakly lower C-continuous.Therefore,the proof of this corollary follows
immediately from Theorem3.5.
Next, as in [12], the aboveresultscan be applied to differentproblems.
Herewe only showsomeapplicationsto vectoroptimizationproblems.
Corollary 3.7. Let D be a conuexclosedsubsetof a refl,edueBanachspace
X with a closedconuerpointed.coneC. Each of the tollowing conditionsis
sufficient/or WMin(D I C) * 0: Thereis a point a € D such that for euery
sequenee
{""} C D with lim,,-*- llt"ll : +m,
(Gl) thereis ns ) 0 suchthat rro € a * C,
(G2) therearen0 ) 0 andUe D suchthatlly-oll < ll",,o-all andtrn, € A*C,
( G 3 ) t h e r ea r a T L s ) 0a n d y e D w i t h t r e A * C , f o r a l l n ) n s .
Prcof. The corollary follows immediately from Corollary 3.6 with X : Y,
G(r,A):A-n, H=0.
Corollary 3.8. Let X be a topologicallocallgconuerHausdorffspace,D c X
& nonen'Lptyclosedconuexset, Let Y be a Banach spaceand C C Y be a
closedconaer pointed cone with C' polyhedralcone. Let f : D + Y be a
singleaalued
C-conuex,C-continuousmapping. In addition, assrrrnethat there
existsa nonernptyconuexweaklycompact subsetK C D such that for any
x € D \ corelrK one canfind a point o € corepK with f (r) e f (a) + C.
Then there eristsa point n €.D suchthat /(t) € WMin(/( D)lC).
r,A e
Proof.We definethe mappingG : DxD -t Y by G(*,il : I@)-/("),
D. UsingTheorem2.4,we concludethat for any fixedr e D, G(r,') : D -'
Y is C-convex and weakly C-continuous. Therefore, consideringthe weak
topologyin X, we apply Theorem3.1 with G and H:0.
This completesthe proof of the corollary.
I
Corollary 3.9. Let X, D,Y, C be as in Corollary3.6, Let f : D -> Y be a
Sufficient Conditionsfor the Ed,stenceof Equilibrium Points
309
singleaaluedC-conuer and C-continuousmapping. Then eachof the following
unditions is sufficientfor WMin (f (D)lC) # 0, Thereis a point a € D such
that for euerysequence
{*"} C D utithlim,,y6 ll""ll - *oo,
(IL) there is ns ) 0 suchthat f (r", ) e /(o) * C,
( I 2 ) t h e r ea r e n o ) 0 a n d A e D s a c h t h altl y - " l l < l l " , , o- a l l ' a n d , . f ( " " 0 )e
f(y)+c,
(I3) thereara rls ) 0 and y e D suchthat f(r") e f(il * C, for atl n ) no.
Proof.We definethe mappingG : DxD + Y by G(r,il - f fu)-f (*), r,A €
D, and apply Corollary 3.6 with G as aboveand I/ : 0 to concludethat there
existsa point i e D suchthat G(n,y) / -intC for all y € D. This implies
f (y) - f (fr) / -intO, for all u € D,
or
f @) €wMin (/(D) lc).
This completesthe proof of the corollary.
I
To concludethe paper we makethe following remark.
Remark, Instead of the lower C-continuity of G(r,.) for any fixed r € G and
the (-C)-contlnuity of H(.,il for any fixed v e D in A4 and AO we assume
that for any given U e D, the set
S(y) - {r e D | (c(y,x) - H(qy)) n(C\ {0}) :0}
is closed. Then, the conclusionsof Theorems3.1, 3.5, Corollary 8.6 remain
true for (EP,C). The proofsare exactly as the aboveone with intC replaced
by C \ t0) everywhere.Thereforethe conclusions
of Corollaries3.7-3.9 also
remaintrue for Min (D I C) and Min (/(D) lC).
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