Hilbert空间中闭的拟严格伪压缩映像的收缩投影方法_英文_

Hilbert空间中闭的拟严格伪压缩映像的收缩投影 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 _英文_ MA Le-rong, GAO Xing-hui ( College of Mathematics and Computer Science,Yan’an University,Yan’an 716000,Shaanxi) Abstract: The purpose of this paper is to propose a new shrinking projection method for quasi-strict pseudo- contractons and prove a strong convergence theorem for cosed and quasstrct pseudocontractons n a Hbert ili-i-iiilspace, The results of this paper improve and extend some recent relative results, Key words: shrinking projection methods; closed mapping; quasi-strict pseudo-contraction; fixed point 2000 MSC: 47H05; 47H09; 47H10 doi: 10, 3969 / , ssn, 1001 , 8395, 2011, 06, 002 ji 1 Introduction2 Preliminaries Let H be a Hilbert space with inner product It s we known that,n an nfntedmensonailliiii-iil 〈?,?〉,and C be a nonempty closed convex subset of Hilbert space,the normal Mann’s iterative algorithm H, Let T: C ? C be a self-mapping of C, We use has only weak convergence,in general,even for non- F( T) to denote the set of fixed points of T, expansive mappings, Consequently,in order to obtain Definition 2, 1 A mapping T: C?C is called a strong convergence, one has to modfy the normail quasi-strict pseudo-contraction if F( T) and there ?Mann’s iteration algorithm,the so called hybrid pro- exists a constant such that,0,1) κ? jection iteration method ( HPIA ) is such a modifica- 2 2 2 Tx ,p ,p + x ,Tx ,x ‖‖‖‖κ‖‖ ? ,1,tion, For 40 years,HPIA has received rapid devel- for a xC and pF( T) ,ll?? opments, For detas,the readers are referred to pail- Definition 2, 2 A mapping T: C?C is called a ,2 , 17, persand the references therein, strict pseudo-contraction if there exists a constant κ? Very recently,Aoyama,Kohsaka and Takahashi ,0,1) such that 2 2 studied the shrinking projection methods for firmly Tx ,Ty ,y +x ‖‖? ‖‖ 2 ( ,T) x ,( ,T) y,I I κ‖‖ nonexpansive mappings and nonexpansive for all x,yC, We also say that T is a -strict pseu- ?κmappings, The shrinking projection methods are do-contraction, smper than the hybrd proecton method, iliji Definition 2, 3 A mapping T: C?C is called a,3, In this paper,motivated by Marino and Xuandnonexpansve mappng if Tyx , yiiTx , ‖‖?‖‖ Aoyama, Kohsaka and Takahashi, we introduce a for all x,yC,? kind of new shrinking projection methods for quasi- Rema 2, 1 It s cear that the cass of strct rkilli strict pseudo-contractions and obtain a strong conver- pseudo-contractions contains properly the class of non- expansive mappings as a subclass, That is,T is nonex- gence theorem for closed quasi-strict pseudo-contrac- pansive if and only if T is a 0-strict pseudo-contraction, tons n the framework of Hbert spaces, The resuts iiill Remark 2, 2 It is also clear that every strictimprove and extend the corresponding results of ,2 , pseudo-contraction with F( T) is quasi-strict pseu- ?3, and others, Recall that given a nonempty closed convex subFrom the definition of T,we get- 2 2 2 ,p ,p Tp p set C of a real Hilbert space H,for every point xH‖ ? ‖‖ ‖+ p ,Tp ,? κ‖‖ n n for each n, Taking the limit as n?yieldsthere exists a unique nearest point in C,denoted by ? 22 Px,such thatx , Pxx , yfor all yC,Tp ,p ,Tp ,p ‖‖?‖‖ ? ‖‖ ? κ‖‖ C C the pont Px s caed the metrc proecton of H onto iillijiSince 01,we have Tp = p and F( T) is closed, ， ?κ C ,3, We next show that F( T) is convex, For arbitrary C,L emma 2, 1Let C be a nonempty closed p,pF( T) ,t( 0,1) ,putting?? convex subset of a Hilbert space H, Given xH and z1 2 ? p= tp+ ( 1 ,t) p , t 1 2 C, Then,z = Px if 〈x , z,y , z〉for all yC,0 ? ? ?C we prove that Tp= p, Notcng ii,5, t t e 2, 2Let C be a nonempty cosed Lmmal p,p ,t) ,p = ( 1 p,‖ ‖‖‖ 1 t 1 2 convex subset of a Hilbert space H and P: H? C be C and the metric projection from H onto C, Then the follow - p,p = tp,p ,‖‖ ‖‖ 2 t 1 2 ing inequality holds: and usng Lemma 2, 3 ( ) ,we have iii2 2 2 y ,P x,P x,y ,+ x x ‖‖‖‖? ‖‖ C C 2 =p,T p‖‖ t t xH,yC, ?? 2 t( p,T p) + ( 1 ,t) ( p,T p) =‖‖ ,3, 1 t 2 t Lemma 2, 3Let H be a Hilbert space,2 2 + ( 1 ,t) p,Tp ,tp,T p ‖‖‖‖ 1 t t 2 There hod the foowng denttes: llliiii 222 2 2 t( 1 ,t) ,p t( ,p +pp‖‖‖‖ ? 1 2 1 t ( i) = x? 2〈x,y〉+ y,x ? y‖‖‖‖‖‖ 22p,T p,t) ( p,p +) + ( 1 κ‖‖ ‖‖ x,yH;2 t t t ? 2 2 2 2 2 ) ,t( 1 ,t) ,p = p p,T p‖‖κ‖‖ ( ii) tx + ( 1 , t) y= tx+ ( 1 , t) yt t ‖‖‖‖‖‖ ,1 2 2 2 2 2 ,t( 1 ,t) + ( 1 ,t) t ,t( 1 ,t) ,p ,p + ‖‖t( 1 , t) y,t0,1,,x,yH,x , ,‖‖ ?? 1 2 22p,T pp,T p,= κ‖‖κ‖‖ It is obvious from the definition of hat ?tt t t t ‖‖ 2 2 Snce 01,we have Tp= pand F ( T) s coni， i-? κ x ,y = x ,z +‖‖‖‖ t t 2 vex, By our assumption that F ( T ) ,one has ? z ,y + 2x ,z ,z ,y , ( 1)〈〉‖‖ Pxis well defined for every xH,? F( T) 0 0 for all x,y,zH,? Step 2 Show that Cis closed and convex for all n 3 Main results n1, It s obvous that C= C s cosed and iiil?1 convex, Assume that Cis closed and convex for some k Theorem 3, 1 Let C be a nonempty cosed conl- kN,For zC,one obtains that?? k + 1 vex subset of a Hbert space H and T: C ? C be a il , 1 κ 2,Tx 〈x,Tx ,x,z 〉,x‖? ‖k k k k k closed quasistrict pseudocontraction for some 0--，?κ 2 1 such that F( T) , Define a sequence { x} in ?n It is easy to see that Cis closed convex, Then,for C k + 1 by the following algorithm all n1,Cis closed convex, ?n xH, C= C, x= Px,? Step 3 Show that F( T) Cfor all n1, It is0 1 1 C0 ?,1 n , obvious that F( T) C = C, Suppose that F( T) C , 1 κ 1 k2, ,Tx xC= { z C:‖ ?‖? n +1 n n n ( 2) 2 , for some kN, For any p'F( T) ,one has p'C,???k , x,Tx ,x,z } ,〈〉 By using the definition of T,we have n n n , 22 2 ,x, Tx,p + x'xx= P‖‖ ? ‖,p 'κ‖ ,Tx , ‖ ‖ 0 k kkn +1 Ck n 1 + view of ( 1) ,we can obtain In Then xconverges strongy to p= Px,{ } l n 0 F( T) 0 2 2,x ,p '+Tx+ xProof We split the proof into six steps, ‖‖‖‖ k k k 2〈Tx,x ,x,p '〉Step 1 Show that F ( T) s cosed convex, We il?k k k 1 , Step 6 Show that p= Px, We prove firstκ 20 F( T) 0 Tx,x ‖‖ ? k k 2 that p= Tp , It follows from step 5 that x ?pC as? 0 0 n 0 〈x,Tx ,x,p '〉, n? , Hence , xas n ? , Sncex0 i? ‖ ‖ ? k k k n + 1 n whch mpes that pC, Ths prove that F( T) iili'i? xC,one has ?k + 1 n + 1 n + 1 C, Therefore,F( T) Cfor all n1, ?k + 1 n 1 , κ 2 ,Tx x‖‖ ? n n 2 Step 4 Show that lim x, x‖‖ n 0 exists, Since n? ?〈x,Tx ,x,x 〉?x= Px,by using Lemma 2, 2,one has n n n n +1 n C 0 n 2 2 ,Tx x,x x,?‖‖‖‖ n n n n +1 x,x ,x ,w ‖‖ ? ‖‖ n 0 0 which implies that , Txas x0 ‖‖2 2n n ,x ,w w ,x ? ‖‖‖‖0n n?, Since ? x?pas n?,one has Tx?pas n?,C C ??? ? n 0 n 0 for each wF( T) Cand for all n1, Therefore,?? n In vew of the coseness of T,we have p= Tp, We il0 0 the sequence { , x} is bounded,x‖‖n 0 next show that p= Px, From x= Px,one 0 F( T) 0 n C0 n On the other hand,noticing that x= Pxandn C 0 n sees 〈y , x,x, x〉0 for any yC, Since F( T) ?? n 0 n n x= PxCC,? n +1 C 0 n +1 n n +1 Cfor all n1,we arrive at 〈w , x,x, x〉0??n n 0 n one has , x, xfor a n xxll‖ ‖ ? ‖ ‖ n 0 n + 1 0 for any w F ( T ) , Take the limit as n ? yields? ? 1,? 〈w , p,x, p〉for any w F ( T ) , By using0 ? ?0 0 0 Therefore,the sequence { , x} is nondecreasx- ‖‖n 0 ng, It foows that m x, xexsts,illlii‖‖ n 0 n?? Lemma 2, 1,we have p= Px,0F( T)0 Step 5 Show that x?pas n?, By the con-? n 0 Remark 3, 1 Theorem 3, 1 extends Theorem struction of C,one has that CCand x= x n m n m C 0m 4, 1 of ,3, from strict pseudo-contraction to closed for any postve nteger mn, By Lemma 2, 2,Ciii?? n and quasistrict pseudocontraction, Our algorithm is --we have simpler than the algorithm of ,3,, Moreover, the 2 2 x,x ,P x= x‖‖‖ ?‖ m C 0 m n n proof method of Theorem 3, 1 is also different from the 2 2 x,x , x,x =P‖‖ ‖‖ m 0 C 0 0 n one used in ,3,, 2 2 x,x , ,x , x ‖‖‖‖ m 0 n 0 Remark 3, 2 The results of this paper also ex -Take the limit as m,n ? yeds x, xil? ‖ ‖ tend main results of ,2, and the others, m n 0, Hence { x} is Cauchy, So we can assume that x?p n n 0 References ,1, Haugazeau Y, Sur les inacuteequations variationnelles et la minimisation de fonctionnelles convexes,D,,Pa ris: Thacuteese Uni- versitacutee de,1968, ,2, Nakajo K,Takahashi W, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,J,, J Math Anal App,2003,279: 372 , 379, l ,3, Marino G,Xu H K, Weak and strong convergence theorems for strict pseudo , contractions in Hilbert spaces,J,, J Math Anal Appl,2007,329: 336 , 346, ,4, Zhou H Y, Convergence theorems of fixed points for Lipschitz pseudo , contractions in Hilbert spaces,J,, J Math Anal Appl, 2008,343 546 , 556, : ,5, Zhou H Y, Demiclosedness principle with applications for asymptotically pseudo , contractions in Hilbert spaces,J,, Nonlinear Anal,2009,70( 9) : 3140 , 3145, ,6, Zhou H Y,Su Y F, Strong convergence theorems for a famy of quas , asymptotc pseudo , contractons n Hbert spacesiliiiiil ,J,, Nonnear Ana,2009,70: 4047 , 4052, lil ,7, Su Y F,Wang D X,Shang M J, Strong convergence of monotone hybrid algorithm for hemi , relatively nonexpansive mappings,J,, Fixed Point Theory and Applications,2008,2008: 1 , 8, 9 Qin X L Su Y F, Strong convergence theorems for relatively nonexpansive mappings in a Banach space J , Nonlinear Anal 2007,67: 1958 , 1965, ,10, Carlos M Y,Xu H K, Strong convergence of the CQ method for fixed point iteration processes,J,,N onlinear Anal,2006,64: 2400 , 2411, ,11, Matsushita S,Takahashi W, A strong convergence theorem for relatively nonexpansive mappings in a Banach space,J ,, J Approximation Theory,2005,134: 257 , 266, ,12, 高兴慧,周海云( 拟 φ , 渐近非扩展映像族的公共不动点的迭代算法,J,, 系统科学与数学,2010,30( 4) : 486 , 492,,13, Zhou H Y,Gao X H, A strong convergence theorem for a family of quasi , , nonexpansive mappings in a Banach spaceφ ,J,, Fxed Pont Theory and Appcatons,2009,2009: 1 , 12, iilii ,14, ,,( Banach J,, ,2009,, ,高兴慧马乐荣周海云空间中拟 φ 渐近非扩展映像不动点的迭代算法数学的实践与认识 39( 9) : 220 , 224, ,15, Iemoto S,Takahashi W, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space,J,, Nonlinear Anal,2009,71: 2082 , 2089, ,16, ,,( Banach J,, : ,2010,35( 3) : 33 , 36,,马乐荣高兴慧周海云空间中增生算子的粘滞逼近问 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 西南师范大学学报自然科学版 ,17, ,,( ,J,, : ,高兴慧马乐荣周海云非扩张映像和非扩展映像公共不动点的强收敛定理西南师范大学学报自然科学版 2010,35( 3) : 29 , 32, ,18, ( ,J,, : ,2010,33( 3) : 317 , 322,程支明渐近非扩张映射族公共不动点的粘性逼近四川师范大学学报自然科学版 ,19, ( k , ,J,, : ,2011,34( 1) : 63 , 70,刘敏广义平衡问题与无限族 严格伪压缩映象的强收敛定理四川师范大学学报自然科学版 ,20, ( ,J,, : ,2010,33( 5) : 638 , 644,王勇严格伪压缩映象与广义拟平衡问题的强收敛定理四川师范大学学报自然科学版 Hilbert 空间中闭的拟严格伪压缩映像的 收缩投影方法 , 马乐荣高兴慧 ( ,716000)延安大学 数学与计算机科学学院陕西 延安 bert ,,( : Hil摘要在 空间中引入和研究了一种新的收缩投影迭代算法用以逼近拟严格伪压缩映像的不动点在适当的条件 ,,下利用所提出的收缩投影算法证明了闭的拟严格伪压缩映像的不动点的强收敛定理所得结果改进和推广了近期文献的相 (关结果 : ; ; ; 关键词收缩投影方法闭映像拟严格伪压缩映像不动点 : O177, 91 :A :1001 , 8395( 201106 ) , 0780 , 04中图分类号文献标识码文章编号 ( )编辑 李德华 : 2009 , 10 , 24收稿日期 : ( 10771050) 11JK0486) ( 基金项目国家自然科学基金和陕西省教育厅科研计划基金资助项目