Hindawi Publishing Corporaion Fixed Poin Theory and Applicaions Volume 011, Aricle ID 76859, 10 pages doi:101155/011/76859 Research Aricle An Implici Ieraion Mehod for Variaional Inequaliies over he Se of Common Fixed Poins for a Finie Family of onexpansive Mappings in Hilber Spaces guyen Buong 1 and guyen Thi Quynh Anh 1 Vienamese Academy of Science and Technology, Insiue of Informaion Technology, 18, Hoang Quoc Vie, Cau Giay, Ha oi 1100, Vienam Deparmen of Informaion Technology, Thai guyen aional Universiy, Thainguye 84803, Vienam Correspondence should be addressed o guyen Buong, nbuong@ioiacvn Received 17 December 010; Acceped 7 March 011 Academic Edior: Jong Kim Copyrigh q 011 Buong and T Quynh Anh This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied We inroduce a new implici ieraion mehod for finding a soluion for a variaional inequaliy involving Lipschiz coninuous and srongly monoone mapping over he se of common fixed poins for a finie family of nonexpansive mappings on Hilber spaces 1 Inroducion Le C be a nonempy closed and convex subse of a real Hilber space H wih inner produc, and norm,andlef : H H be a nonlinear mapping The variaional inequaliy problem is formulaed as finding a poin p C such ha F p,p p 0, p C 11 Variaional inequaliies were iniially sudied by Kinderlehrer and Sampacchia in 1 and ever since have been widely invesigaed, since hey cover as diverse disciplines as parial differenial equaions, opimal conrol, opimizaion, mahemaical programming, mechanics, and finance see 1 3
Fixed Poin Theory and Applicaions I is well known ha if F is an L-Lipschiz coninuous and η-srongly monoone, ha is, F saisfies he following condiions: F x F y L x y, F x F y,x y η x y, 1 where L and η are fixed posiive numbers, hen 11 has a unique soluion I is also known ha 11 is equivalen o he fixed-poin equaion p P C p μf p, 13 where P C denoes he meric projecion from x H ono C and μ is an arbirarily fixed posiive consan Le {T i } i 1 be a finie family of nonexpansive self-mappings of C For finding an elemen p i 1 Fix T i, Xu and Ori inroduced in 4 he following implici ieraion process For x 0 C and {β k } k 1 0, 1, he sequence {x k} is generaed as follows: x 1 β 1 x 0 1 β 1 T1 x 1, x β x 1 1 β T x, x β x 1 1 β T x, 14 x 1 β 1 x 1 β 1 T1 x 1, The compac expression of he mehod is he form x k β k x k 1 1 β k T k x k, k 1, 15 where T n T n mod, for ineger n 1, wih he mod funcion aking values in he se {1,,,} They proved he following resul Theorem 11 Le H be a real Hilber space and C a nonempy closed convex subse of HLe{T i } i 1 be nonexpansive self-maps of C such ha i 1 Fix T i /, wherefix T i {x C : T i x x} Le x 0 C and {β k } k 1 be a sequence in 0, 1 such ha lim k β k 0 Then, he sequence {x k } defined implicily by 15 converges weakly o a common fixed poin of he mappings {T i } i 1
Fixed Poin Theory and Applicaions 3 Furher, Zeng and Yao inroduced in 5 he following implici mehod For an arbirary iniial poin x 0 H, he sequence {x k } k 1 is generaed as follows: x 1 β 1 x 0 1 β 1 [ T1 x 1 λ 1 μf T 1 x 1 ], x β x 1 1 β [ T x λ μf T x ], x β x 1 1 β [ T x λ μf T x ], 16 x 1 β 1 x 1 β 1 [ T1 x 1 λ 1 μf T 1 x 1 ], The scheme is wrien in a compac form as x k β k x k 1 1 β k [ T k x k λ k μf T k x k ], k 1 17 They proved he following resul Theorem 1 Le H be a real Hilber space and F : H H a mapping such ha for some consans L, η > 0, F is L-Lipschiz coninuous and η-srongly monoone Le {T i } i 1 be nonexpansive selfmaps of H such ha C i 1 Fix T i / Leμ 0, η/l, and le x 0 H, wih {λ k } k 1 0, 1 and {β k } k 1 0, 1 saisfying he condiions: k 1 λ k < and α β k β, k 1, forsome α, β 0, 1 Then, he sequence {x k } defined by 17 converges weakly o a common fixed poin of he mappings {T i } i 1 The convergence is srong if and only if lim inf k d x k,c 0 Recenly, Ceng e al 6 exended he above resul o a finie family of asympoically self-maps Clearly, from k 1 λ k < we have ha λ k 0ask To obain srong convergence wihou he condiion k 1 λ k <, in his paper we propose he following implici algorihm: x T x, T : T 0 T T 1, 0, 1, 18 where T i are defined by T i x 1 β i x β i T ix, i 1,,, T 0 y I λ μf y, x, y H, 19 I denoes he ideniy mapping of H, and he parameers {λ }, {β i } 0, 1 for all 0, 1 saisfy he following condiions: λ 0as 0and0< lim inf 0 β i lim sup 0 βi < 1, i 1,,
4 Fixed Poin Theory and Applicaions Main Resul We formulae he following facs for he proof of our resuls Lemma 1 see 7 i x y x y, x y and for any fixed 0, 1, ii 1 x y 1 x y 1 x y, for all x, y H Pu T λ x Tx λμf Tx, x H, λ 0, 1 ; for any nonexpansive mapping T of H, we have he following lemma Lemma see 8 T λ x T λ y 1 λτ x y, for all x, y H and for a fixed number μ 0, η/l,whereτ 1 1 μ η μl 0, 1 Lemma 3 Demiclosedness Principle 9 Assume ha T is a nonexpansive self-mapping of a closed convex subse K of a Hiber space H IfT has a fixed poin, hen I T is demiclosed; ha is, whenever {x k } is a sequence in K weakly converging o some x K and he sequence { I T x k } srongly converges o some y, i follows ha I T x y ow, we are in a posiion o prove he following resul Theorem 4 Le H be a real Hilber space and F : H H a mapping such ha for some consans L, η > 0, F is L-Lipschiz coninuous and η-srongly monoone Le {T i } i 1 be nonexpansive selfmaps of H such ha C i 1 Fix T i / Leμ 0, η/l and le 0, 1, {λ }, {β i } 0, 1, such ha λ 0, as 0, 0 < lim inf 0 βi lim sup β i < 1, i 1,, 1 0 Then, he ne {x } defined by 18-19 converges srongly o he unique elemen p in 11 Proof By using Lemma wih T λ T 0,hais,T I, we have ha T x T y 1 λ τ T T 1 x T T 1 y 1 λ τ T i T 1 x T i T 1 y 1 λ τ T 1 x T 1 y 1 λ τ x y x, y H So, T is a conracion in H By Banach s Conracion Principle, here exiss a unique elemen x H such ha x T x for all 0, 1
Fixed Poin Theory and Applicaions 5 T i ex, we show ha {x } is bounded Indeed, for a fixed poin p C, we have ha p p for i 1,,, and hence x p T x p T x T T 1 p I λ μf T T 1 x I λ μf T T 1 p λ μf p 1 λ τ T T 1 x T T 1 p λ μ F p 1 λ τ T 1 T 1 x T 1 T 1 p λ μ F p 1 λ τ T i T 1 x T i T 1 p λ μ F p 1 λ τ T 1 x T 1 p λ μ F p 1 λ τ x p λ μ F p 3 Therefore, x p μ F p τ 4 ha implies he boundedness of {x } So, are he nes {F y Pu }, {y i }, i 1,, y 1 y 1 β 1 x β 1 T 1x, 1 β y 1 β T y 1, y i 1 β i y i 1 β i T iy i 1, 5 y 1 β y 1 β T y 1 Then, x I λ μf y 6
6 Fixed Poin Theory and Applicaions Moreover, x p I λ μf y y y 1 p p λ μ F y p λ μ F y 1 p λ μ F y x p λ μ F y y,y p λ μ F y,y p λ μ F y,y p λ μ F y,y p λ μ F y 7 Thus, η y p F p,y p λ μ F y 8 Furher, for he sake of simpliciy, we pu y 0 x and prove ha y i 1 0, 9 as 0fori 1,, Le { k } 0, 1 be an arbirary sequence converging o zero as k and x k : x k We have o prove ha y i 1 T k i y i 1 k 0, where yi k are defined by 5 wih k and y i k yi k Le{x l } be a subsequence of {x k } such ha lim sup y i 1 k k k y i 1 lim l l 10 l Le {x kj } be a subsequence of {x l } such ha lim sup xk p xkj lim p 11 k j
Fixed Poin Theory and Applicaions 7 From 6 and Lemma 1, i implies ha x kj p I λ kj μf y p y p λ kj μ F y,x kj kj p 1 β kj λ kj μ F y 1 1 β y 1 λ kj μ F p β T kj y 1 T p y kj,x kj p p β T y 1 T p y kj,x kj p y 1 p λ kj μ F y 1 p λ kj μ F y,x kj kj p y,x kj kj p x kj p λ kj μ F y,x kj kj p 1 Hence, y i lim x kj p lim j j p, i 1,, 13 By Lemma 1, y i p 1 β i y i 1 p β i Ti y i 1 p β i 1 β i y i 1 1 β i y i 1 p β i y i 1 β i 1 β i y i 1 y i 1 p β i 1 β i y i 1 y 0 p β i p 1 β i y i 1 x kj p β i 1 β i y i 1, i 1,, 14
8 Fixed Poin Theory and Applicaions Wihou loss of generaliy, we can assume ha α β i β for some α, β 0, 1 Then, we have α 1 β y i 1 xkj p y i p 15 This ogeher wih 13 implies ha lim y i 1 j 0, i 1,, 16 I means ha y i 1 T i y i 1 0as 0fori 1,, ex, we show ha x T i x 0as 0 In fac, in he case ha i 1 we have y 0 x So, x T 1 x 0as 0 Furher, since y 1 T 1x 1 β 1 x T 1 x, 17 and x T 1 x 0, we have ha y 1 T 1x 0 Therefore, from x y 1 x T 1 x T 1 x y 1, 18 i follows ha x y 1 0as 0 On he oher hand, since y T y 1 1 β y 1 T y 1 0, y x 1 β y 1 x β T y 1 x 1 β y 1 x β T y 1 y1 y 1 x, 19 we obain ha y x 0as 0 ow, from x T x x y y T y 1 T y 1 T x x y y T y 1 y 1 x, 0 and x y, y T y 1, y1 x 0, i follows ha x T x 0 Similarly, we obain ha x T i x 0, for i 1,, and y x 0as 0 Le {x k } be any sequence of {x } converging weakly o p as k Then, x k T i x k 0, for i 1,, and {y } also converges weakly o p ByLemma 3, we have k p C i 1 Fix T i and from 8, i follows ha F p,p p 0 p C 1
Fixed Poin Theory and Applicaions 9 Since p, p C, by replacing p by p 1 p in he las inequaliy, dividing by and aking 0 in he jus obained inequaliy, we obain F p,p p 0 p C The uniqueness of p in 11 guaranees ha p p Again, replacing p in 8 by p,we obain he srong convergence for {x } This complees he proof 3 Applicaion Recall ha a mapping S : H H is called a γ-sricly pseudoconracive if here exiss a consan γ 0, 1 such ha Sx Sy x y γ I S x I S y, x, y H 31 I is well known 10 ha a mapping T : H H by Tx αx 1 α Sx wih a fixed α γ,1 for all x H is a nonexpansive mapping and Fix T Fix S Using his fac, we can exend our resul o he case C i 1 Fix S i, where S i is γ i -sricly pseudoconracive as follows Le α i γ i, 1 be fixed numbers Then, C i 1 Fix T i wih T i y α i y 1 α i S i y,a nonexpansive mapping, for i 1,,, and hence T i y 1 β i y β i T i y 1 β i 1 α i y β i 1 α i S i y, i 1,, 3 So, we have he following resul Theorem 31 Le H be a real Hilber space and F : H H a mapping such ha for some consans L, η > 0, F is L-Lipschiz coninuous and η-srongly monoone Le {S i } i 1 be γ i-sricly pseudoconracive self-maps of H such ha C i 1 Fix S i / Leα i γ i, 1,μ 0, η/l and le 0, 1, {λ }, {β i } 0, 1, such ha λ 0, as 0, 0 < lim inf 0 βi Then, he ne {x } defined by lim sup β i < 1, i 1,, 33 0 x T x, T : T T 0 T 1, 0, 1, 34 where T i,fori 1,,, are defined by 3 and T 0 x I λ μf x, converges srongly o he unique elemen p in 11 I is known in 11 ha Fix S C where S i 1 ξ is i wih ξ i > 0and i 1 ξ i 1 for γ i -sricly pseudoconracions {S i } i 1 Moreover, S is γ-sricly pseudoconracive wih γ max{γ i :1 i } So, we also have he following resul
10 Fixed Poin Theory and Applicaions Theorem 3 Le H be a real Hilber space and F : H H a mapping such ha for some consans L, η > 0, F is L-Lipschiz coninuous and η-srongly monoone Le {S i } i 1 be γ i- sricly pseudoconracive self-maps of H such ha C i 1 Fix S i / Leα γ,1, where γ max{γ i :1 i }, μ 0, η/l, and le 0, 1, {λ }, {β } 0, 1, such ha λ 0, as 0, 0 < lim inf β lim sup β < 1 0 0 35 Then, he ne {x }, defined by 1 x T x, T : T0 β 1 α I β 1 α ξ i S i, 0, 1, 36 i 1 where T 0 I λ μf, ξ i > 0, and i 1 ξ i 1, converges srongly o he unique elemen p in 11 Acknowledgmen This work was suppored by he Vienamese aional Foundaion of Science and Technology Developmen References 1 D Kinderlehrer and G Sampacchia, An Inroducion o Variaional Inequaliies and Their Applicaions, vol 88 of Pure and Applied Mahemaics, Academic Press, ew York, Y, USA, 1980 R Glowinski, umerical Mehods for onlinear Variaional Problems, Springer Series in Compuaional Physics, Springer, ew York, Y, USA, 1984 3 E Zeidler, onlinear Funcional Analysis and Is Applicaions III, Springer, ew York, Y, USA, 1985 4 H-K Xu and R G Ori, An implici ieraion process for nonexpansive mappings, umerical Funcional Analysis and Opimizaion, vol, no 5-6, pp 767 773, 001 5 L-C Zeng and J-C Yao, Implici ieraion scheme wih perurbed mapping for common fixed poins of a finie family of nonexpansive mappings, onlinear Analysis Theory, Mehods & Applicaions, vol 64, no 11, pp 507 515, 006 6 L-C Ceng, -C Wong, and J-C Yao, Fixed poin soluions of variaional inequaliies for a finie family of asympoically nonexpansive mappings wihou common fixed poin assumpion, Compuers & Mahemaics wih Applicaions, vol 56, no 9, pp 31 3, 008 7 G Marino and H-K Xu, Weak and srong convergence heorems for sric pseudo-conracions in Hilber spaces, Journal of Mahemaical Analysis and Applicaions, vol 39, no 1, pp 336 346, 007 8 Y Yamada, The hybrid seepes-descen mehod for variaional inequaliies problems over he inesecionof he fixed poin ses of nonexpansive mappings, in Inhenly Parallel Algorihms in Feasibiliy and Opimizaion and Their Applicaions, D Bunariu, Y Censor, and S Reich, Eds, pp 473 504, orh-holland, Amserdam, Holland, 001 9 K Goebel and W A Kirk, Topics in Meric Fixed Poin Theory, vol 8 of Cambridge Sudies in Advanced Mahemaics, Cambridge Universiy Press, Cambridge, UK, 1990 10 H Zhou, Convergence heorems of fixed poins for κ-sric pseudo-conracions in Hilber spaces, onlinear Analysis Theory, Mehods & Applicaions, vol 69, no, pp 456 46, 008 11 G L Acedo and H-K Xu, Ieraive mehods for sric pseudo-conracions in Hilber spaces, onlinear Analysis Theory, Mehods & Applicaions, vol 67, no 7, pp 58 71, 007