Tap chi Khoa boc 2012-23b 32-41 Tnrdng Dai hpc Cdn Tha
SU* TON TAI NGHIEM CUA BAI TOAN BAO HAM TU'A BIEN PHAN VA LTNG DUNG
Ldm Quoc Anh' \d Phan Dgi Nhan"
ABSTRACT
We consider quasivariational inclusion problem in topological vector spaces. Sufficient conditions for the solution existence are established. Applications to somes special cases of quasivariational inclusion such as Ky Fan inequality, variational inequality and optimization problem.
Keywords: Quasivariational inclusion problems, Ky Fan inequality, variational inequality, optimization problem
Title: Existence of solutions to quasivariational inclusion problem and applications TOM TAT
Chung Idi xet bdi lodn bao hdm lua bien phdn trong khong gian vecto tdpd. Thiit lap cdc diiu kiin du cho sir ton tgi nghiem. Ap dung vdo mat so trudng hgp ddc biit cda bdi todn bao hdm tua biin phdn nhu. bdt ddng thuc Ky Fan. bdt ddng thuc biin phdn vd bdi todn toi uu
Tir khda: Bai todn bao ham tvra biin phan, bat dang thuc Ky Fan, bat ddng thirc bien phdn, bdi todn tii iru
1 M d D A U
Cho X la khdng gian vecto tdpd Hausdorff thyc, A^X la. tap hgp con ldi, ddng khac rdng ciia X, va I' la khdng gian vecta tdpd. Xet cac anh xa da tri S^: A-* 2\ S^: A-* 2'* CO gia tti khac rong, va F:AxA^2' .Ta xet bai toan bao ham tya bidn phan sau:
(QVIP): Tira xe5,(jc) sao cho,
0 e F(3t,y). vdi ragi ysS^(x).
Bai toan bao ham tya bien phan la dang tdng quat ciia nhieu bai toan quan trpng trong Iy thuyet tdi uu, sau day chiing ta xet mdt sd trudng hgp dac biet ciia bai toan nay de Iam thi du minh hpa.
Bdi todn lira cdn bdng vecta dgng I:
Cho 5 " : X - > 2 ' , G:XxX-^2' la cac ham da tri, va C e 7 la tap hgp ddng vdi phan ttong khac rong. Ta xet cac bai toan sau:
(QEP'):Tim xsc\S(x) sao cho.
G(^,.v)n(I'\-intC)?i0, vdi mpi yeS(x).
(SQEP'): Tun xeclS(x) sao cho,
Gix.y)Q(Y\-mtC), vdi mpi y^S(x).
Bdi todn cdn bdng dgng 2:
_ Khoa Su pham, Tnrang Dai hpc Can The - To Toan. Tmcmg Chujen L> Tir Trgng. TPCT
T^ chi Khoa hgc 2012:23b 32-41 Truang Dgi bpc Cdn Tha
Cho S:A^2^.V:A-*2^ !a cac ham d a t n . / : ^ x ^ - > r l a anh xa dan tri. Gia sir rang cac gia tri cua T la ddng vdi phan ttong khac rong va khac vdi Y. Xet bai toan can bdng:
(QEP-): Tim x e S^(x) sao cho,
fix, y) £ T(xX vdi mpi v e S{x).
Bdi todn bao hdm tua bien phdn:
Cho P,Q:Xx-X ->2^ la cac ham da tri. Bai toan bao hara tya bidn phan dugc xet ttong Hai va Khanh (2007) cd dang:
(QVIP'):T\m TeSj(Jc) sao cho.
P(x.y)^Qix.yX vdi mpi y^S-,(x).
Bai todn quan he bien phdn:
Cho R(x.y)\a he thuc lien kdt giua X.VBX. ta thay rang RCQ the ddng nhat vdi tap hgp con M = {(x,y)s^Xy.X :R(x,\) dugc thda man} cua khdng gian tich
X^X.
(QVRP): Tim 3ce5,(J) sao cho.
R(x.\) thda man. vdi mpi ye^S-^(x).
Ba\ gid ta chi ra rang, vdi viec xay dyng ham muc tieu thich hgp, cac bai toan tten trd thanh cac trudng hgp dac biet cua bai toan (QVIP).
De chuydn (QEP^)ve mpt trudng hgp dac biet ciia (QVIP), ta dat S,(x)^ c\S(x).S^(x) = Six) va i^(.T..i)-G(3:.v)-(r\-intC). Khi do:
0 e F(x,y) <^ G(jc,>')n(r \ - i n t C ) * 0 .
Bai toan (SQEP') ciing la mgt trudng hgp dac biet ciia (QVIP) vdi.
5,(.T)= c\S(x),S,ix) = S(x) va F(x.y) = Y\(G(x,y)-i^m\C). Khi do:
0 e F(x, y)« G(x. y)^(Y\- intC).
Tuong ty. ddi vdi bai toan (QEP-). ta dat S^(x) = S2(x) s S(x). va F(x.} ) = fix.y)-r(x). Khi do:
0 e F(x,y) « /(:c..v) e r(x).
Dd chuyen bai toan (QVRP) ve trudng hgp dac biet ciia bai toan (QVIP), ta dat Y = XxX va F(x.y) = (x,y)-M. Khi dd:
R(x,y) thda man khi va chi khi 0 e F(x,y).
Trudc hdt ta thay vangiQVIP) la mpt trudng hgp dac biet ciia (QVIP^). vdi F(x. v) s Q(x, y) va P(x, y) = 0. Tuy nhidn. vdi viec xac dinh quan he R(x. y) thda man khi va chi khi P(x,y)^Q(x.y), thi bai toan (QVIP*) lai la trudng hpp ridng cua bai toan (QVIP).
Dinh nghia 1.1 (Fan, 1961) Ham da tri Hciia tap con A cua khdng gian vecta t d p d X v a o A'dugc gpi la anh xa KKM ttong A. ndu vdi mdi [x^.x,....,x„}^A ta cd:
conv {.Yi,x,,...,jc„} c I J " H(x,), a day conv{-} la ki hidu bao ldi ciia tap •••'".
Tgp chi Khoa hpc 2012.23b 32-41 Truang Dgi hpc Cdn Tha
Dinh ty I . l (Fan. 1961) Gid sirX Id khong gian vecta tdpd. A ^ X la tap loi khdc rdng va H: A-^2^ la mot dnh xg KKM vai gid tri dong. Niu A la compact thi
n,^ff(x)*0.
Dinh ly 1.2 (Yannelis, 1983) Cho A Id tap hap con compact, loi khdc rong cua khong gian vecta thuc Hausdorff, vd P:A^2* la hdm da tri thoa man dieu kien X«convP(x).vdi mgixeA. Niu vai moi y s A,P~'(y) = {xe. A:y EP(x)} la tap hap md trong A. thi tdn tgi x' sA sao cho P{x') = 0-
Dinh ly 1.3 (Park, 1992) Cho Xld khong gian vecta tdpd Hausdorff thuc, AcXld tap hgp con loi khdc rdng vd D cA Id tap hap compact khdc rong. cho S: A ->2^, L: A ->2^ Id cac hdm da tri. Gid sir rdng:
(a) Vai moi x sA, L(x) Id Wi va S(x) ^L(x):
(b) vai moi x e D, S(x) i^0;
(c) vdimgiy eA thi 5"'(.v) la md trong A;
(d) vai moi tap con hOu hgn NcuaA co mot tap con compact, loi L\ sao cho N ^Ls-cA vd vai moi x E L^.\ D, S(x) nL^ -^0.
Kill do L CO diem bdt dong.
Nhgn xet 1.1 Dieu kien birc (d) d Djnh Iy 1.3 cd thd thay thd bdi gia thidt birc sau:
(d') ton tgi mot lap compact loi K^ A sao cho. v&i moi x^A\D, Ion tgi y^K, di x€S~\}).
That vay. gia s u e d (d') va dat A'^e.^ la huu ban. Dat L^ =COTW(KKJN), thi vol rapixei^ \D, tdn tai j e A ' e i ^ , vdi x&S~\y). Vi thh.,y^S(x)r^K^S(x)c\L^, cd nghTa la (d) dugc thoa raan.
2 S i r TON TAI N G H I E M CUA BAI TOAN BAO H A M TU'A B I E N PHAN Dinh ly 2.1 Xet bdi todn (QVIP) gid sir cdc diiu sau duac nghiem diing:
(i) Vai moi tap con huu hgn {x^,x,,...,x^^} vav&imoi xe.conv{x^.x-„...,x^}
ton tgi JG{1,2... ,n} sao cho Q€^F{X,X\;
(ii) Sl Iddnhxgdong, convSJ^x)^S^{x) vd S-T'(y) Id md trong A, vdi mgi x,ye A;
(iii) vdi moi ye A tdp hap {xsA:0^ F(x,y)] la tap dong trong A;
(iv) A la tap compact.
Khi do ton tgi xeS^ (x) sao cho 0 e F(x,y), vdi mgiy e S,(x).
Chirng minh.
Vdi x.ysA. dat:
E = {xeA:xs S,(x)]. P(x) ^{yeA:0€ F(x,y)},
\S^(x)r\P(x) neu xsE, [S,{x) neu x€A\E, Q(y)^A\':>^\y).
Tgp chi Khoa boc 2012:23b 32-41 Truong Dai hpc Cdn Tha
Ta chung minh Q la anh xa KKM ttong A. That vay, gia sir co td hgp ldi
^ = X"=i'^j^' ti-ong v4sao cho x^yj^^Q(y^), nghia la xe<i>~\yj) hay y^^<^(x) vdi mpi j = 1, . . . , « .
* Ndu xeE ta CO y^ eP(x), nghJa la O^F(x.y^), vdi mpi j = 1, ..., «, didu ndy mau thudn vdi (i).
Ndu xsA\E thi y^E^(.i) = S.(x),j^\,...,„. Vay j ' ^ econv5.(x), suy ra
^ = 2"=i'^7->'y ^ conv5',(.v)c5',(.T), mau thudn. Do dd Q la anh xa KKM trong ^.
Kd tiep ta chiing minh tinh ddng cua Q (y). Vdi mgi y^A ta co, 0-' (y) = [ £ n 57' (y) n /""' (y)] w [(A \ E) n S:' (y)]
= [ ( ^ \ £ ) u / > - ' ( y ) ] n 5 ; ' ( y ) . fi(7) = ^ \ { [ ( ^ \ £ ) w / ' - ' ( y ) ] n 5 : ' ( y ) )
= {^\[(^\£)u/'-'(y)]}u(^\5:'(y))
= [Er^{A\F^\y))']u{A\S;J(y)).
Vi 5", dong ndn E ddng. Mat khac.
^\p-'(y) = {xe^:ye/'(j:)}
= {xeA:OeF(x,y)]
la tap ddng. Tir do ta suy ra Q(y) la ddng. Ap dung Dinh Iy 1.1 ta cd mpt didm x sao cho
xef]^^^Q(y) = A\\J^^^<I>''(y).
Vi the, X s <!>"' (y). vdi mgi y^A, nghia la 0 ( ? ) = 0 .
* Ndu xeA\E thi (D(x) - 5'-, ( J ) = 0 . mau thuan.
*Ndu xeE. t a c o 0 = <t»(J) = 5, ( j : ) n P ( x ) . Nhu thd, vdi mgi y e 5 , { j ) , y g / ' ( j ) , tuc la O e F ( 7 , y ) , vdi mpi y€5,(A'). Didu nay c6 nghia la, tdn tai jce5, (3c) sao cho 0 e F(lc.y), vdi mpi y^S, (.r).
Thdng thudng sy tdn tai nghiem ciia bai toan ludn Hen quan ddn tinh chat Hen tuc ciia ham muc tieu, do dd gia thidt (iii) ttong Djnh ]}> 2.1, yeu hon tinh chdt lien tuc, va nhu vay sy xuat hidn cua gia thidt nay ttong dinh Iy la didu tdt ydu. Tuy nhien, cac gia thiet con lai cd ve khdng lien quan den tinh lidn tuc, cac thi du sau day chi ra rang cac gia thidt ttdn la edt yeu.
T h i d u 2.1 Cho X =Y = R.A=[0,+cc),S,(x)^S,(x)^A,F(x,y)=[y-x,-hm).
Ta th§y cac gia thiet ciia Dinh Iy 2.1 ddu dugc thda man, trir tinh compact ciia A.
Neu bai toan tdn tai nghiem thi tdn tai XE.,4 sao cho 0€F(x,y) vdi mpi ysA nghta la y<'x. vdi mpi ye A. didu nay khdng thd xay ra. Do do bai toan vo nghidm. ly do la (iii) bi vi pham.
T h i d u 2.2 Cho X = y = ^.A = [0,—lS,(x)^S,{\)^A,F(x,y) = [sm(x-y).\].
Tgp chi Khoa hpc 2012:23b 32-41 Tnrcmg Dgi hpc Cdn Tha
Khi do, cac gia thidt cua Djnh ly 2.1 dugc thda man trir (i).
Bdng each kidm tra tryc tidp ta thdy rdng, vdi mpi xe [Q,~], tdn tai y e [ 0 . ^ ] de s i n ( : f - i ) > 0 . Do do bai toan (QVIP) vd nghiem. Ly do la gia thidt (i) khdng nghidm diing. That vay.
Vdi J, =0..Y, = — tac6conv{A:,,x,} = [ 0 , — ] , ldy J : ^ — e [ 0 . — ] , x^-x = =>
sin(x, -x)>(i,x.-x=~=> sin(a:, - x ) > 0. Do do (i) la edt ydu.
" 6
T h i d u 2 . 3 C h o X - y - R , ^ = [0,2],5.(:<:)s[l,2],5-,(x) = [0.2],F(x,y) = [ x - y . 3 ] . Ta thdy cac gia thidt ciia Djnh Iy 2.1 ddu dugc thoa man, tru ti'nh chdt (ii). Dd thay bai toan la vd nghiem. Do do (ii) la khdng bd dugc.
He qua 2.2 Khdng dinh ciia Dinh ly 2.1 vdn dung khi diiu kien (iii) dugc thay bai dieu kiin sau:
(iii) Vaimoiy EA:X \-^ F(X, y) la dong trong A.
Chirng minh.
Ta chimg minh rang tir gia thiet (iii") suy ra gia thidt (iii). That vay, vdi mdi y e A, dat B = {xeA:OeF(x,y)}. Ta can chiing minh tap hgp B la ddng. Lay x^eB,x^^> X. do A ddng nen x e A Vi x^eB nen OeF(x^,y)., vdi mgi a. Do F(..y) ddng nen OeF(.\.v). nghia la xeB. Vay B la tap hgp ddng.
Dinh ly 2.3 Dinh ly 2.1 vdn diing niu ta thay gid thiet (i) bdng gid thiet sau:
(i') Vai mgi x eAtdp hgp {y sA: 0 g F(x,y)} la loi va 0 E F(x, x) Chung minh.
Dat:
E^{xeA:xeS,ix)},
P(x) = {yeAO^F(x,y)],
\S^(x)nP(x) ni'uxeE, [5,(J:) n d u j r e ^ \ £ .
Do gia thidt (i").i'(3:) la ldi vdi mgi xeA, hon niia vdi mpi xeA.x€ConvP(x).
That vay. do convP(x) = P(x) nen ndu cd xeconvP(x), thi ta suy ra xeP(x). Khi do 0 g F(x,x) ttai vdi gia thiet (i").
Kd tidp ta chung minh rdng x € conv 0 ( x ) . That vay,
+ Neu xeE tiii 0(x)aP(x), ndn conv-^(x)^P(x) vi xsP(x) suy ra j:econvO(.t);
+ Neu . T e ^ \ £ thi 0(x) = S._(x). Do do conv0(x)= convS,(x)^S,(x). Nhu vay, neu XG conv 0(.v) dan ddn xeS,(x), suy ra xeE, mau thudn vdi vide xeA\E.
Mat khac. vdi mpi y e A,
<!)-'(y) = [ £ n 5;'(y) n P - ( y ) ] u [ ( ^ \ £) n 5;'(y)]
= [(^\£)wi'-'(y)]n5:'(y).
Tt^ chi Khoa hpc 2012 23b 32-41 Truang Dai hgc Can Tho
Do ^i la anh xa ddng nen E la tap hgp ddng. tiic la ^ \ £ la tap hgp md. Theo cac gia thidt (ii). ( i i i ) , S - \ y ) v a P " ' ( y ) ={xeA:yeP(x)} = {xeA:0^ F(x.y)} la cac tap hgp md. TTJ dd ta suy ra <^'''(y) la tap hgp md.
Ap dung Dinh ly 1.2. tdn tai x G A sao cho <J>(x) = 0 .
Ndu xeA\E thi ^(x) = S^(x)^0 ,v6ly.Vay 3 c e £ . T u c la t a c o . 'D( 3r ) = S, (x) ri P{x) = 0 .
Do do vdi mgi yeSi(xXyeP(x). nghia la O e F ( x . y ) . Noi each khac, tdn tai xeS,(Jc) sao cho. vdi mgi yeS^(jc,y),OeF(J,y).
Dinh 1^ 2.4 Dfnh ly 2.3 vdn diing neu ta thay gid thiit (iv) bdng gid thiit sau:
(iv ) Ton tgi mgt tap hgp con khdc rong compact D ^A sao cho, v&i mdi tap con hiru hgn N cua A. ton tgi mgt tap compact, loi L\ v&i N cLy ^ A.
thoa mdn cdc diiu kien sau:
(1) V&i mgi x EL^\D, Sytx) r-> Z,.^ ^ 0 ;
(2) v&i X E S}(x) n (L\ ' D), tdn tgi y e S:(x) n i y sao cho 0 0 Ffx. y).
Chirng minh.
Dat:
E = {xeA:xeS,(x)], P(x) = {y eA:O^F(x,y)],
lsJx)r\P(x) n€\i xeE, 0(x)= ,
152(A) neu A-£A\£,
|fconv5,(.v)lri/*(jc) ndu xeE.
Q(x) = V - '
[ conv5',(x) neu xeA\E.
Ap dyng Dinh ly 1.3 vdi Z ^ ^ va 5 = *. Ta chiing to rang cac gia thiet (a), (c). (d) cua Dinh Iy 1.3 dugc thda man, nhung Q khdng co diem bat ddng, va nhu the gia thidt (b) phai hi vi pham.
+ Ta CO Q(x) la Idi vdi mpi x. va 0(.T) e Q(x) bdi djnh nghia ciia Q, ndn (a) dugc nghidm diing.
+ Vdi mgi ye .4. ta cd:
(t>-\y)^[Er^S:\y)r^p-\y)'\'u[(A\E)-S:'(y)]
= \(A\E)^p-\y)']r,S:\y).
Suy ra
A\'^-\y) = {Er^(A\p-\y))]^{A\S{\y)). (2.1) Ta se chiing minh tap hgp nay la dong. Bang tinh dong cua Sj ttong gia thidt (ii). ta
thdy E la ddng. Theo gia thidt (H) ta suy ra.4\5,"'(y) ciing la ddng. Phan con lai ttong (2.1) la:
A\p-\y) = {x^A:y eP(x)]
Tap chi Khoa hoc 2012 23b 32-41 Tnwng Dai hpc Can Tha
la dong do (iii). Nhu tha A\<b;\\) Ja dong. tijc la (c) thoa man.
+ D6 kiem tra gia thi^t (d). ta xet D va L^ vai m6i JVxac dinh bcri gia thi^t (iv').
LSy x e i j \Z> tuy y. Neu j e ^ \ £ thi, ^(x)c^L„ = Ar\S.(x)r\L^ =S-(x)r^Ly^
0, do (iv'). Neu xeE thi xeS,(x)r^(Ly\D). Cung theo gia thiSt (iv") t6n tai ysS.(x)t^L... sao cho OgF(x.y). nghTa la y<=P(x). Nhu th^
yeS.(x)r^P(x) = ^x) va tl)(jc) n i^. ^e 0 . Do do (d) dugc nghiem dung.
Cuoi ciing, gia sijr rang Q CO diem bat dong x e A. Nghia la J e Q(x).
Neu X GE thi X e i'(}r ), tiic la 0 i F(x. J), mau thuan vcri (i).
N6u Ic GA\E thi Jce c o n v ( 5 , ( J ) ) ^ 5, ( J ) CO nghTa la x e £", mau thuan.
Tir nhiJng dieu da chiing minh, ta suy ra gia thiet (b) cua Dinh h' 1.3 hi vi pham, tiiclatontai x e D^A saocho<I>(3c) = 0 .
Neu T GA\E thi S.{T) = ^{x)-0 mau thuan. Vi vay x&E va 0 = <1>(J) = S.(x) n P ( x ) , suyravCTimoi^'eS, (J),} s /*(?). Do do, QsF(x, r), vcri mpi ysS,(x).
3 MOT SO C N G D L T S G 3.1 Bat ding thirc Ky Fan
Cho X.A nhu a phin mo diu va f-.XxX-^H la m6t anh xa dem tri. Ta xet bit ding thurc Ky Fan sau:
(KF): Tim xeA sao cho
/(x.y) < 0, voi moi ye A.
Dinh nghia 3.1 Cho anh xa /z: A" -> R, va a e R.
(a) Tap a - m i l e tren ciia h. ky hi6u la lev^^h. dugc xac dinh boi:
lev^,h={xeX:h(x)>a}.
(b) Tap a -miic tren chat cua ciia h, ky hieu la /cv„A, dugc xac dinh bcri:
l<n;.>'^ixeX:h(x)>a}.
(c) Tap ff-miic duoi cua h, ky hieu la /ev„A, dugc xac dinh bcri:
l'i^s^h={xeX:h(x)<a}.
(d) Tap a - m u c duoi chat cua cua h, ky hieu la lev^^h, dugc xac djnh boi:
lev„h = ixeX:h(x)<a}.
Ket qua sau day dugc suy ra tir Dinh Iy 2.3.
He qua 3.1 Gia sir A la rap compaci, va cdc gia thiil sau day dirac ihoa man:
(i) Vcri moi x s A.lev.^f(x..) Idi va f(x.x) < 0;
(li/ v&i moi y e A.lev^^f(.,y) dong.
Khidolonlgi JeA di f(x.y)<Q, vcri moi yeA.
Chihig minb.
Dat F(.t,v) = [/(.T,j.).+oc). Khi do OeF(x,y) khi va chi khi /(x,y)<0.
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Ta kiem tra cac gia thidt cua Dinh ly 2.3 dugc thda man. Cac gia thiet (ii) va (iii) hidn nhien nghiem diing. Vi lev^f(x,.) ldi ndn { y e ^ : 0 ? F(-T.I )}Idi, va do
f(x.x)<Q n d n t a c d 0€F(j:,y). Do dd (i") thda man. Vdi (iii). vi lev^/{..y) dong ndn {jf e .4:0 e F(x. y)} la tap ddng. Ap dung Dinh ly 2.3 ta suy ra tdn tai 'xeA de
f(x.y)<0, vdi mpi yeA.
Dinh nghia 3.2 Cho h:X^3..
(a) h dugc ggi la nua lidn tyc ttdn (use) tai x^. ndu vdi mpi day {.r„} hdi tu ve .TQ thi ft(xo)>limsup/i(x„).
(b) h dugc gpi la nira lidn tuc dudi (la:) tai x^. ndu vdi mpi day {x„} hdi tu vd x^
thi h(x^)<\unmfh(x„).
D i n h n g h ^ 3.3 Cho A: A ' ^ ?_. va A la tap con khac rdng ciia X.
(a) h dugc ggi laidi ttong A. neu Vx,.x, e .•i.V/e[0,l], h(tx, + (1 - / ) x , ) < lh(x,) + (1 -t)h(x,).
(b) h dugc gpi la tya ldi trong A neu V.x, .x^eA.Vte [0,1].
h(tx,-i-(l~t)x.)<max[h(x^),hix.)).
Nhan xet 3.1
(i) .\'eu /(..y) Iscthi lev^/(..y) dong.
(ii) \eu /(x.-) tua loi thi lev^f(x..) loi.
3.2 Bat dang thirc bien phan
Cho X la khdng gian dinh chuan. va A la tap con ldi khac rdng ciia X, va B:X-^X\ ttong do A"" la khdng gian doi nglu cdaX. Ta xet bai toan bat dang thuc bidn phan sau:
(VI): Tim xeA sao cho.
<S(x), \ - x ) >0, vdi mpi y eA.
Kdt qua sau day dugc suy ra tir He qua 3.1 \ a Nhan xet 3.1.
Hd q u a 3.2 Gid su .^ la tap compact vd vai mdi y e A dnh xg x\-^{B(x),x-y) nira lien tuc du&i trong A. Khi do tdn tgi xeA de (B(x). y-x)> 0. v&i moi y e A.
Chung minh.
Dat:
/ ( x , . 0 = ( 5 ( x ) . x - y > .
Ta kidm tta cac gia thidt ciia He qua 3.1 dupe thda man.
+ Ta chung minh vdi mdi x = .i. lev^f(x..) - I.v e A : <5(X).-Y-y> > 0} la tap ldi.
Ld} y^.y,ele\\J(x,.). tuc la (B(.v).x-y,> > 0 . ( 5 ( x ) , x - y , ) >0, va y, = n - , + ( 1 - 0 ; , • vdi re[0.1].
T a c o :
/ ( x . .V,) = (B(x). X - (n, ^ (1 - / ).i,))
= (B(x),a:+(l-r)x-(o-,+(l-O.V2)>
= <S(.T).r(x-y,)> + <S(.T).(l-rXA-y,))
= ^,5(x).x-l^^^(!-/KB(A•).-T-y.>>0.
Tep chi Khoa hpc 2012 23b32-4l Truong Dai hpc Can Tho
Suy ra y, e lev^(x..). Do do lev^Qf(x,.) Ia tap Idi.
+ Ta chung minh vdi radi ;• e A. lev^r,f(..y) ia tap dong.
Ldy x„e/eiv,/(..y).x„->x,suy ra <S(x„).x„-y><0. Do x\-^{B(x).x-y) Isc ndn {B(x),x-y) < lim inf<5(x„,x„ -y) < 0.
Nghia la. xe/ev,,/(..y). Vay /ev,o/(..y) dong. Mat khac /(-x,x) = <5(x).0> = 0. Do do cac gia thidt ciia He qua 3.1 nghiem diing. Ap dung He qua 3 . ! . ta suy ra tdn tai x e ^ dd
( 5 ( x ) . y - x ) > 0 , vdi mpi yeA.
3.3 Bai toan tdi ini
Cho X.A nhu phan md dau. va anh xa ipiX ->R. Ta xet bai toan tdi uu sau:
(OP): Tim min^(x). vdi x^A.
Dinh nghia 3.4 (Morgan va Scalzo. 2004. 2006) Cho X la khdng gian tdpd va / : AT -> R.
(a) / dugc gpi la tya nua Hen tuc ttdn tai .\„ G X neu,
[ / U ) > / ( x „ ) ] = > [ v d i m g i {.x„}-»-.To,/(.v)>limsup/(.r„)].
(b) / duoc ggi la tya nira lien tuc dudi tai x^ G AT neu,
[f(x) < /(A'O)] ^ [ vdi mpi [xJ -^ Xo./(x) < lim inf/(x„)].
(c) / dugc ggi la tya Hen tuc tai x\,eX, ndu / la tya nira lien tuc tren va tya nOra lien tuc dudi tai Xg.
Thi du sau day cho thdy khai niem tren la giam nhe that sy ciia khai nidm nira lien tuc ciia anh xa don tri.
Thi du 3.1 Xet / : P. ^ R dugc xac dinh bdi
{
x + 2 neu x > 0 , 0 neu x = 0, x - 2 neu x < 0 .Khi do / la tua lien tyc tai 0. nhung khdng niia lidn mc ttdn va nua Hen tuc dudi taiO.
Ket qua sau day dugc suy ra tir He qua 3.1.
He qua 3.3 Gid sir A la tap compact vd cdc gid thiit sau ddy nghiem dung:
(i) ip Id hdm lira loi trong A:
(ii) (p Id lira nita lien tuc dudi trong A.
Khi do bai todn (OP) co nghiem trong A.
Chung minb.
Vdi mdi x.yeA, dat
f{x.y) = (p(x)~(p(y).
Ta kiem tta cac gia thidt ciia He qua 3.1 nghidm diing ttong tmdng hgp nay.
+ Vdi moi x G A. xet
iev^Jix, ) = {y e ^ : q)(x)~(p(y) > Q}.
Tap chi Khoa hpc 2012-23b 32-41 Truang Dai hoc Cdn Tba
Gia sir y,,y2 e/ev^o/(x,.), va re[0.1], t a c o
^(x) -cp(ty\ +(\-t)y^) > ^ ( x ) - m a x {ip(y^), p ( y , ) } > 0,
vi yi,yi^iev.^f(x,.). Tir dd suy ra y, =ty^+(\~t)y2^lev^ofix,.). Do dd lev^f(x,.) la tap ldi.
+ V d i m d i yeA, ta se chi ra rdng /ei-,o/(.,y) = { x e ^ : ^ ( x ) - ^ ( y ) < 0 } la tap ddng.
Lay x„ elev.^^f(.,y),x„ -> x. Ta can chdng minh xG/ev^o/(.,y). Gia su ngugc lai, xg/ev^(,/(.,y), tuc la
(p(y)«p(x) (3.1) Theo tinh tua nua lidn tuc dudi cua 9>, tir (3.1) ta suy ra
f?(y) < lun inf tp(x„). (3.2) Mat khac, vi x„ e lev^^/(., y), ndn ta cd
(p(y)>ip(x„).
Dieu nay mau thuan vdi (3.2). Do do x e /eVgo/(.,y).
Ap dung He qua 3.1, ta suy ra tdn tai xeA dd.
/ ( ^ ^ y) = <P(^) - Hy) ^ 0> vdi mgi y^A.
Noi each khac, x e ^4 la nghidm cua bai toan (OP).
4 KET LUAN
Chiing tdi da sir dun^ cac djnh Iy vd diem bdt ddng dang KKM-Fan, dinh Iy vd phan tu tdi dai, dd thidt lap cac didu kien du cho sy tdn tai nghiem cua bai toan bao ham tya bien phan. Do bai toan bao ham tya bien phan chiia nhidu bai toan quan ttpng khac ttong ly thuyet tdi uu, nen cac kdt qua thu dugc ttong Muc 2 cd thd suy ra cac kdt qua tucmg iing cho cac trudng hgp dac biet ciia nd; ttong bai bao nay chiing tdi ap dung cho bai toan bat dang thiic Ky Fan, bat dang thuc bidn phan va bai toan tdi uu dd lam thi du minh hpa.
TAI LIEU THAM KHAO
Fan, K., 1961. A generalization of TychonofTs fixed point theroem. Math. Ann. 142:305-310.
Hai, N.X. and Khanh, P.Q., 2007. The solution existence of general variational inclusion problems. Journal of mathematical Analysis and Application, 328: 1268-1277.
Morgan, J, and Scalzo, V., 2006. Discontinuous but well-posed optimization problems.
SIAM J Optim. 17:861-870.
Morgan, J. and Scalzo, V., 2004. Pseudocontinuity in optimization and nozero sum games.
J. Optim- Theory Appl 120: 181-197.
Park,S., 1992. Some coincidence, theorem on acyclic multi functions and applications to KKM theory, fixed-point theory and application. Edied by K.K. Tan. Word Scientific, River Edge. New Jersey. 248-277.
Yannelis, Nicholas C a n d Prabhakar, N . D . , 1983. Existence of maximal elements and equilibria in linear topological spaces. Journal of Mathematical Economics 12: 233- 245. North-Holland.
