IHCTHINGUYN TRNGI HCSP H M NGUYNSONGH XPXN G H I MCHOBT NGTHCBINPHN VIHV HNCCNHXK H NGGIN Ngnh TonGiitchMs 9460102 TMTT LUNNTINSTONHC THINGUYN 2018 Cngtrnhc honthnhti Trn g i hcSp h m i hcThiNguyn Ngi[.]
IHCTHINGUYN TRNGI HCSP H M NGUYNSONGH XPXN G H I MCHOBT NGTHCBINPHN VIHV H NCCN H XK H NGGIN Ngnh:TonGiitchMs : 9460102 TMTT LUNNTINST O NHC THINGUYN-2018 Cngtrnhc honthnhti: Trn g i hcSp h m-i hcThiNguyn Ngihn g dnkhoahc:GS.TS.NguynBn g Phnbin1: Phnbin2: Phnbin3: Lunn sc bo vt r cHin g chmlunn cpTrn g hpti:Trngi hcSphmi hcThiNguyn Voh i gi ngy thng nm2018 Ctht mhiulun n tithv i n: - ThvinQucgiaVitNam - TrungtmhcliuihcThiNguyn - Thv i ntrn g i hc Sphm-i hcThiNguyn Mu B i to n b t ng th c bi n ph n c xu t v o nh ng n m uc a th p ni n60thkX X , gnlinvinhngnghincucaLions,Stampacchiavc ngs( L i o n s vSta mpacchia, 1965, 1967; Hartman vStampacchia, 1966).Tn nay, b tngth c bi n ph n lu nlm tchn g h i n c u m a n g t n h t h i svthuhtcsquant mc anhi u nhk h o a h c t r o n g v n g o i nc Nhi u b i to n nh : b i to n c c tr ;bitoni mbtn g ; bitoncnbng;bitonb;phn g trnhvitontn iu;bitonbincd ngcaph n g tr nho hmri ng c t h q u y vm h nhbit o n b tn g th cb i np h nd ic cg i t h i tt h chh p.V t h b it o n n yl mtcngcm nhvthngnh ttrongnghincunhiumhnhbito nlthuytv ngdngthct VitNam,theonhiuconngtipcnkhcnhau,ccnhk h o a h ccn h ng ng g p quan tr ng cho b i to n n y cthknnhc c n h m n g h i n c u c a GS.TSKH Ph m KAnh (P.K Anh vtg, 2015, 2017); GS.TSKH Phan Qu c Kh nh(P.Q Kh nh vtg, 2005, 2006); GS.TSKH.inh ThL c ( T.L c vt g , 0 , 2014);GS.TSKH.LD ngMu(L.D.Muvt g , 2005,2012);GS.TSKH.Ph mHu S ch (P.H S ch vt g , 0 , 0 ) ; G S T S K H N g u y n X u n T n ( N X T n v t g , 2012,2013);GS.TSKH.Nguynn g Yn(N Ynvt g , 2005,2008);GS.TS.N guynBn g ( N B n g v t g , 1 , , 01 , ) ; P G S T S P h m N g c A n h ( P N A n h vt g , 0 , 0 , ) ; P G S T S N g u y n Quang Huy (N.Q Huy vtg, 2011) v PGS.TS.Ng uy nT h T h u Th y (N T.T Th y vt g , ,2 ) B nc nh, b t ngthcbinphnvm tsb i tonlinquancngva n g lt inghincucanhiutcgiltins g h i ncusinhtrongnc Mhnhbitonbtngthcbinphncincdng: Tmx∗∈Csaocho: ⟨F(x∗),x−x ∗⟩≥ , ∀x∈C , (0.1)trongC l tpconlingkhcrngcakhnggianHilbert H v F: H→ H l nhxxcnhtrnH Trongt r n g h pt pC c ab it o n( ) cc h o d idngn l t pi mb t ngchung c a m t hh u h n h a y v h n c c n h xkh ng gi n thb i t o n ( ) cli n hv i n h i u b i t o n t h c t i n n h b i t o n k h i p h ct n hi u, b i to n ph n ph ib ng th ng, ki m so t n ng ln g c h o h t h n g m ng vi n th ng CDMA v k t h u t xltnhiubngtn cthn g d ngb it o nb tngth cb i np h nv oth ct i n,i hip h icnhngp hn g ph pgiis hiuqu chobito nny.Vl, mttrongnh ng h ng nghi n c u quan tr ng hi n d nhcsquan t m c a nhi u nhto n h ctrong v ngo inclvi cxu t c c ph ng ph p m i t m nghi m c a b i to n(0.1)h o c c i t i nhi uquc anhi uphngph pc Chonnayn g i t a thitl pcnhiukt h u tgiibtng thcbinphnd atrnphn g phpchiucaGoldstein(1964),Polyak(1966, 1967,1969),phngphpi m g n kc a Martinet(1970), Rokaffellar (1976), nguy n lb i t o n p h c a C o h e n ( ) , p h n g p h p h i u chnhdngBr owderT ikhonov (B r owder , 6 ; T ikhonov , 96 ), ph n g p h pi mgnkh i uc h nhc aL e h d i l i v M o u d a f i ( 9 ) , R y a z a n t s e v a ( 0 ) v p h n g p h pi mgnkquntn hdoAlvarezvAttouch(2001)xuthocdatrnmtskthuttm i m b tng nhphng ph p l p Krasnosel'skii-Mann (Mann, 1953; Krasnosel'skii,1955),phngph plpHalpern(1967)vphngphpxpxmm(Moudafi,2000) Ph ng php l pi n h nhgi i b i to n (0.1) lph ng ph p chi u gradient(Goldstein,1964;Zeidler,1990)cmt nhs a u : x0∈C, xk+1=PC(I−ρFF)(xk), k=0,1,2, (0.2) trongP Cl phpchi umtrict Hl nC ,Il nhxnvt r nH v ρFl mthngsdn g c nh Phn g ph p ( ) cc u tr cn gi n n n vi c v n d ng trongnhngtnhhungcthk h t h u ntin.Phn g php nylskthpgiavicsdngtrc tip dngngcaphpchiuP Cv p h n g phpkiun g dcnht Nhc nh ng ti n b ng k lthuy ti m b t ng c anh x kh ng gi n thk X X , phn g phplaighpng dcnhtcYamadavc ngs(Yamadav tg,1998, 1999) xu t nhl m t bi n thc a phn g p h p ng dc nh t t mcctiucamthml itrntpi mbtngchungcac cn h xkhnggin.c i m ch nh c a phn g p h p n y ld ng d ngngc a c cn h xkh ng gi n b t km t p i m b tngchung c a nlt p r ng bu c c a b i to n M t kh c, nhi ub i to n th ct ,ch ng h n b i to nxl t n hi u (Iiduka, 2010), ki m so t n ng lngchohthngmngvinthngCDMA(Iiduka,2012)hocphnphib ngthng(IidukavU c h i d a , 1 ) c t h a v b it o nt mn g h i mc ab tn g t h cb i np h ntrntpi m b t ngc a m t ho c m t hc c n h xkh ng gi n H n na,ch ng tabitrng,mitpconlin g uct h b i udindidnggiaom ccaccn a kh ng gian, dolgiaomc c a t p i m b tngc cn h xkh ng gi nlcctontchiulnnhngnakhnggianny.Vt h b itontmnghimcabt ngthcbinphn(0.1)trnmttpconlingct h q u y vv i ctmnghimbt ngth c bi n ph n tr n t pi m b tngchungc am thc cnhxkh ng gi n.Khi,mtvnt ralx cnhphn g phplpxpxn g h i mchobitonbt ngthcbinphn(0.1)nht h n onuchngtacd nghincaccn h xk h ng ginTi?(i∈Iv iIl t pchs n o).Xutphtttn g ny,nm2001,Yamada xy d ng phn g p h p l a i g h p ngd c nh t mp h n g p h p n y h i tm nh vm t th nh ph n n m t pi m b t ng chung c a hh u h n c c n h xkh ngginn g thithamnlnghimcabitonbtn g thcbinphn(0.1) Ct h ,k h i C : = F i x (T)l t pi mb tn g c am tn h x k h ngg i n,Y a m a d a thitlpcnhlh itmnhsau nhl0 ChoF : H →H ln h x l i n t cL -Lipschitzv η -n i um nht r n H C h o T:H →H lnhx kh ng g int rnH v iFix (T)=/∅.Gi s ρF ∈ ( ,2η/LL2)vd yλk( ,1]thamncciukin : ∞ ∈ l i m λ =0, (L1) k (L2) k→∞ Σ λk=∞, (L3) lim(λk−λk+1)λ−2=0 k+1 k→∞ k=1 Khi,vii mbanu tyx 0∈H ,dylpxcnhbi xk+1= T (xk)−λ k+1ρFF(T(xk)), (0.3) k=0,1,2, hitmnhtinghimduynhtx ∗c abiton (0.1) Trongtrn g hpC lt pimbtngchungcamthh uhnccn h xk h ngginT i: H → H (i= ,2,3, ,N),d yl pxo a y v ngx px n g h i mch o bit o n( ) cYamadaxy dngcd ng xk+1= T [k+1](xk)−λk+1ρFF(T[k+1](xk)), (0.4) k=0,1,2, y,[ k]:=k modNlh mmodulolygit r t r o n g {1,2,3, ,N} nhl0 ChoF :H → H l nh x lint cL -Lipschitzv η -ni um nhtr nH Ch o \N Ti:H→Hl hh uhncc nhxk h nggintrnHv iC:= Fix(Ti)/=∅v i=1 C= Fix (T1T2 TN)= Fix (T2T3 TNT1)=···= Fix (TNT1 TN−1) Gis ρF ∈(0,2η/LL2)v dyλ k∈(0,1]t h amncci uki n: (L1)l i m λk=0, (L2) Σ ∞ Σ ∞ |λk−λk+N| 1.Gi \∞ s{Ti}lhvh nmc ccnhxkhnggintrnE v i C: = Fix(Ti) ∅ i=1 Lpbito nbtn g thcbinph n,kh i ulV I P (F,C),c phtbiunhs a u : ∗ Tmx∗∈Csaocho: ⟨F(x∗),j(x−x ∗)⟩≥ , trongjl nh xing uchun t c c aE lnghimcabitonVIP∗(F,C) ∀x∈C, (1.2) i mx∗∈Cth a m n(1.2)cg i 1.3.2 Phn g phplaighp n g dcnht Trong ph n n y, ch ng t i str nh b y chi ti t m t sn g h i n c u m r n g ho c c i binphn g phplaighpn g dcnhtx pxn g h i mchobitonbtn g thcbinphncdng(0.1)hoc(1.2) KhiClt pi m b tngchung c a m thh u h n c cn h xkh ng gi n trongkhnggianHilbertth c,nm 2003,XuvKimchngminhcktqutngt nhl0 h l0 khithayth( L ) v( L ) ∗t ngn g bicci ukin (L4)limλk/Lλk+1=1 v (L4)∗limλk/Lλk+N= k→∞ k→∞ Cththy r ng,i uki n (L4) yuh n th cs(L3), h n n uki n (L4) cho ph p tact h l a c h n v i d y t h a m s c h n h t c {1/Lk}trong khi( L ) k h n g t h a m n M ∗ ∗ t khc,khngkhk h nchr a rngi ukin(L3) suyrai ukin(L4) nugiihnlimλk/Lλk+Nt nti Nm2007,Zengvc ngsxutphn g phplpxoayvng k→∞ xk+1= T [k+1](xk)−λ k+1ρFk+1F(T[k+1](xk)), k=0,1,2, (1.3) vithamsρF k+1khngphilh ngsc n h nht r o n g (0.4)vi ukint lnccdythamsl pcngc c ibinm bosh it nhl1 ChoF :H → H l nh x lint cL -Lipschitzv η -ni um nhtr nH Ch o Ti:H→H l hhuhncc nhxkhnggintrn H v iC:= TN Fix(Ti)/=∅v i=1 C= Fix (T1T2 TN)= Fix (T2T3 TNT1)=···= Fix (TNT1 TN−1)