STRONG CONVERGENCE OF MODIFIED MANN ITERATIONS FOR LIPSCHITZ PSEUDOCONTRACTIONS

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1 Joura of Mahemaica Scieces: Advaces ad Appicaios Voume, Number, 009, Pages STRONG CONVERGENCE OF MODIFIED MANN ITERATIONS FOR LIPSCHITZ PSEUDOCONTRACTIONS JING HAN ad YISHENG SONG Mahmaicas ad Sciece Educaio Deparme Jicheg Vocaioa ad Techica Coege Jicheg P. R. Chia Coege of Mahemaics ad Iformaio Sciece Hea Norma Uiversiy Xixiag P. R. Chia e-mai: Absrac I his paper, for a Lipschiz pseudocoracive mappig T, we sudy he srog covergece of he ieraive scheme geeraed by = ( α ) x + α (( β ) u + β Tx ), x + whe { β }, { α } saisfy ( i ) im α = 0; = ( ii) α = ; ( iii ) im β = Mahemaics Subjec Cassificaio: 49J30, 47H0, 47H7, 90C99, 47H06, 47J05, 47J5, 47H09. Keywords ad phrases: Lipschiz pseudocoracios, modified Ma ieraios, srog covergece, uiformy Gâeaux differeiabe orm. Received Ocober, Scieific Advaces Pubishers

2 48 JING HAN ad YISHENG SONG. Iroducio Le K be a oempy cosed covex subse of a smooh Baach space E. Le T : K K be a coiuous pseudocoracive mappig. The he mappig T = u + ( )T obviousy is a coiuous srogy pseudocoracive mappig from K o K for each fixed ( 0, ). Therefore, T has a uique fixed poi i K ([4, Coroary ]), i.e., for ay give ( 0, ), here exiss x K such ha x = u + ( ) Tx. (.) As 0, he srog covergece of he pah { x } has bee iroduced ad sudied by Browder [] for a oexpasive mappig T i Hiber space, by Reich [3] for a oexpasive sef-mappig T defied o a uiformy smooh Baach space, by Takahashi-Ueda [36] for a oexpasive sef-mappig T defied o a uiformy covex Baach space wih a uiformy Gâeaux differeiabe orm, by Xu [39] for a oexpasive sef-mappig T defied o a refexive Baach space which has a weaky coiuous duaiy mappig J ϕ, by Schu [7] for a coiuous pseudocoracive osef-mappig defied o a refexive Baach space havig weaky sequeiay coiuous duaiy mappig, by Moraes-Jug [] ad Udomee [37] for a coiuous pseudocoracive mappig T saisfyig he weaky iward codiio ad defied o a (refexive) Baach space wih a uiformy Gâeaux differeiabe orm. See aso Bruck [, 3], Reich [3, 5], Sog e a. [5, 8, 3, 33], Suzuki [35], ad ohers. O he oher had, Ma [] iroduced he foowig ieraio for T i a Hiber space: x = αx + ( α ) Tx, 0, (.) + where { α } is a sequece i [0, ]. Laery, Reich [4] sudied his ieraio i a uiformy covex Baach space wih a Fréche differeiabe orm ad obaied is weak covergece. I he as wey years of so, umerous papers have bee pubished o he ieraive approximaio of fixed pois of Lipschiz srogy pseudocoracive (ad correspodigy Lipschiz srogy accreive) mappigs usig he

3 LIPSCHITZ PSEUDOCONTRACTIONS MAPPINGS 49 Ma ieraio process. Resus which had bee kow oy for Hiber spaces ad oy for Lipschiz mappigs have bee exeded o more geera Baach spaces (see, e.g., [4, 6, 8-3, 5-7, 0, 3] ad ohers). This success, however, has o carried over o arbirary Lipschiz pseudocoracio T eve whe he domai of he operaor T is compac covex subse of a Hiber space. I fac, i was a og sadig ope quesio wheher or o he Ma ieraio process coverges uder his seig. I 999, Chidume-Moore [9] proposed he foowig probem i coecio wih he ieraive approximaio of fixed pois of pseudocoracios. Ope Probem. Does he Ma ieraio process aways coverge for coiuous pseudo-coracios, or for eve Lipschiz pseudocoracios? These quesios have recey bee resoved i he egaive by Chidume-Muagadura [8], who produced a exampe of a Lipschiz pseudo-coracive map defied o a compac covex subse of rea Hiber space wih a uique fixed poi for which o Ma sequece coverges. I order o ge a srog covergece resu, oe has o modify he orma Ma s ieraio agorihm. Some aemps have bee made ad severa impora resus have bee repored. Fox exampe, Kim-Xu [8], Chidume-Chidume [7], Suzuki [34] ad Sog-Che [9] dea wih srog covergece of he modified Ma ieraio (.3) for a oexpasive mappig T: for x, u, x 0 K + = αu + ( α ) ( βtx + ( β ) x ). (.3) Recey, Chidume-Ofoedu [6] ad Sog [30] ried o foud he srog covergece of he foowig ieraio { x } which idepede of he pah x ; = ( λ λ θ ) x + λ θ x + λ ( α Tx + ( α ) x ),. x + (.4) Very recey, Zhou [40] obaied he srog covergece heorem of he ieraive sequece (.5) for λ- sric pseudocoracio T i -uiformy smooh Baach space: for u, x0 E,

4 50 JING HAN ad YISHENG SONG x = βu + γx + ( β γ )( αx + ( α ) Tx ), 0. (.5) + I is he purpose of his paper o prese he modified Ma s ieraio agorihm for a Lipschiz pseudocoracio T. Namey, a ew agorihm wi be proposed o fid a fixed poi of T. Our mehod is differe from he above severa ieraios where he mappig T ivovig wih he agorihm is averaged. Isead, our agorihm proposed beow works for every Lipschiz pseudocoracio T. More precisey, I wi repace he poi Tx i agorihm (.) wih ( β ) u + βtx a sep, for β ( 0, ). Tha is, my mehod produces a sequece { x } accordig o he ieraio process: x + = ( α ) x + α (( β ) u + βtx ), (.6) where { α } ad { β } are rea sequeces i ( 0, ) saisfyig he codiios: () i im α = 0; ( ii) α = ; ( iii) im β = 0. = I wi be proved ha { x } srogy coverges o some fixed poi of a Lipschiz pseudocoracio T. I paricuar, he parameers of our ieraive sequece are simper ad do deped o each oher.. Preimiaries Throughou his paper, a Baach space E wi aways be over he rea scaar fied. We deoe is orm by ad is dua space by E. The vaue of x E a y E is deoed by y, x, ad he ormaized E duaiy mappig from E io is deoed by J, ha is, J( x) = { f E : x, f = x f, x = f }, x E. Le F ( T ) = { x E : Tx = x}, he se of a fixed poi for a mappig T. Le S ( E) : = { x E; x = } deoe he ui sphere of a Baach space E. The space E is said o have (i) a uiformy Gâeaux differeiabe x + y x orm, if for each y i S ( E), he imi im is uiformy aaied 0 for x S( E); (ii) fixed poi propery for o-expasive sef-mappigs, if

5 LIPSCHITZ PSEUDOCONTRACTIONS MAPPINGS 5 each o-expasive sef-mappig defied o ay bouded cosed covex subse K of E has a eas a fixed poi. A Baach space E is said o be x + y sricy covex if x = y =, x y impies <. Reca ha a mappig T wih domai D ( T ) ad rage R ( T ) i Baach space E is caed srogy pseudo-coracive if, for a x, y D( T ), here exis k ( 0, ) ad j( x y) J( x y) such ha Tx Ty, j( x y) k x y. (.) Whie T is said o be pseudo-coracive if (.) hods for k =. T is said o be Lipschizia if, for a x, y D( T ), here exiss L > 0 such ha Tx Ty L x y. The mappig T is caed o-expasive if L = ad, furher, T is said o be coracive if L <. I is obvious ha (coracive) oexpasive mappig is a impora subcass of (Lipschiz srogy) pseudocoracive mappig, bu he coverse impicaio may be fase. This ca be see from he exisig exampes (see, e.g., [6, 8, 40]). If C ad D are oempy subses of a Baach space E such ha C is oempy cosed covex ad D C, he a mappig P : C D is caed a reracio from C o D if P is coiuous wih F ( P ) = D. A mappig P : C D is caed suy if P( Px + ( x Px )) = Px, x C wheever Px + ( x Px ) C ad > 0. A subse D of C is said o be a suy oexpasive rerac of C if here exiss a suy oexpasive reracio of C oo D. The erm suy oexpasive reracio was coied by Reich i [6]. For more deais, see [9, 5, 6]. Lemma. ([, 3, 33, 36]). Le E be a refexive Baach space which has boh fixed poi propery for o-expasive sef-mappigs ad a uiformy Gâeaux differeiabe orm or be a refexive ad sricy covex Baach space wih a uiformy Gâeaux differeiabe orm, ad K be a oempy cosed covex subse of E. Suppose ha T is a coiuous

6 5 JING HAN ad YISHENG SONG pseudocoracive mappig from K io K wih F ( T ) 0/. The as, x, defied by 0 x = u + ( ) Tx coverges srogy o a fixed poi P u of T, where P is he uique suy oexpasive rerac from K oo F ( T ). Lemma. (Liu [0] ad Xu [38]). Le { a } be a sequece of oegaive rea umbers saisfyig he propery: a ( ) a + b + c, 0, + where { }, { b }, { c } saisfy he resricios: (i) = ; ( ii) b < + ; ( iii) im sup c 0. = 0 = 0 The { a } coverges o zero as. 3. Mai Resus Theorem 3.. Le E be a refexive Baach space which has boh fixed poi propery for o-expasive sef-mappigs ad a uiformy Gâeaux differeiabe orm, ad K be a oempy cosed covex subse of E. Suppose T : K K is a Lipschizia pseudo-coracio wih a Lipschiz cosa L > 0 ad F ( T ) 0/, ad { x } is a sequece give by x+ = ( α ) x + α(( β ) u + βtx ). (3.) Assume ha { α } ad { β } are rea sequeces i (0, ) saisfyig he codiios: (ii) im α = 0; ( ii ) α = ; ( iii) im β = 0.. The, as, { } = coverges srogy o some fixed poi P u of T, where P is he uique suy oexpasive rerac from K oo F ( T ). x

7 LIPSCHITZ PSEUDOCONTRACTIONS MAPPINGS 53 Proof. The proof wi be spi io hree seps. Sep. { x } is bouded. Take p F ( T ). Choose M > 0 sufficiey arge such ha x p M, u p M L. We proceed by iducio o show ha x p M for a. Assume ha p M for some >. We show ha x x + p M. I fac, from (3.), we esimae as foows: x+ p α ( β ) u p + αβ Tx p + ( α ) x p α M ( β ) + αβlm + ( α ) M. L The whe L =, he resu is obvious. Beow e L >. We use he reducio o absurdiy. Suppose ha x + p >. We have M M < α M ( β ) + αβlm + ( α ) M. L The 0 < α ( β ) + αβl α, ad hece L 0 L < β + βl L, ha is, = < β. L + L This coradics o he codiios im β = 0. Therefore, x + p M for a. This proves he boudedess of he sequece { x }, which impies ha he sequece { Tx } is aso bouded. Sep. im sup u Pu, J( x Pu ) 0. + (3.) Le T be defied by T x : = ( α ) x + α Tx for each x K ad fixed α ( 0, ). The, we observe ha for each, T is a Lipschiz pseudocoracive mappig from K o isef wih Lipschiz cosa

8 54 JING HAN ad YISHENG SONG + α ( L ) > 0 ad F ( T ) = F ( T ). Moreover, he defiiio of T reduces o im T x x = im α Tx x = 0. (3.3) Seig T = u + ( ) T, he for each ( 0, ) ad ( 0, ), T obviousy is a coiuous srogy pseudocoracive mappig from K o K for each ( 0, ) ad each. Therefore, T has a uique fixed poi i K (see [4, Coroary ]), ha is, for each ( 0, ) ad each, α z = u + ( ) T z. The for each, i foows from Lemma. ha im z = Pu, is he 0 uique suy oexpasive reracio from K oo F ( T ) = F ( T ). The P P by he uiqueess of suy oexpasive reracio from K oo F ( T ), ad hece im z 0 = Pu for a. (3.4) Sice T is a pseudocoracive mappig for each, usig he equaiy z x = ( )( T z x ) + ( u x ), we have z x = ( ) T z x, J( z x ) + u x, J( z x ) = ( )( T z T x, J( z x ) + T x x, J( z x ) ) + J( z x ) u z, + z x ad hece, x z + Tx x J( z x ) + u z, J( z x ), Tx x u z, J( x z ) C,

9 LIPSCHITZ PSEUDOCONTRACTIONS MAPPINGS 55 for some cosa C > 0. Hece, oig (3.3), we obai im sup u z, J( x z ) 0. Therefore, for ay ε > 0, here exiss a posiive ieger N such ha for a N, ε u z, J( x z ) <. (3.5) O he oher had, sice J is orm opoogy o coiuous o bouded ses ad im Pu = 0, we have 0 N N u Pu, J( x Pu ) u z, J( x z ) z N weak opoogy uiformy N N N = u Pu, J( x Pu ) J ( x z ) + z Pu, J ( x z ) N N u Pu, J( x Pu ) J( x z ) + z Pu M 0, as 0. Hece, for he above ε > 0, δ > 0, such ha ( 0, δ), for a, we have u Pu, J ( x Pu ) N N ε u z, J ( x z ) +. By (3.5), we have ha ε ε im sup u Pu, J ( x Pu ) < + = ε. Sice ε is arbirary, (3.) is proved. Sep 3. im Pu = 0. x From (3.), we have x + Pu ( β )( u Pu ) + β α ( Tx Pu ) + ( α ) ( x Pu ), J( x Pu ) = α + ( β ) u Pu, J ( x Pu ) + β α Tx Pu x Pu α + +

10 56 JING HAN ad YISHENG SONG ( α ) x Pu, J ( x Pu ) + + ( x Pu x+ Pu α ) + + βα LM ( β ) u Pu, J ( x Pu ), + α + which impies ha x + Pu ( α ) x Pu + αθ, (3.6) θ + where = β LM + α ( β ) u Pu, J( x Pu ). Usig he codiio (iii) ad (3.), we have im sup θ 0. Hece, Appyig Lemma. o he iequaiy (3.6), we cocude ha im Pu = 0. This compees he proof. x We remark ha if E is sricy covex, he he propery ha E has he fixed poi propery for oexpasive sef-mappigs may be dropped. I fac, we have he foowig heorem. Theorem 3.. Le E be a refexive ad sricy covex Baach space wih a uiformy Gâeaux differeiabe orm. suppose K T, { x },, { α }, { β } are as Theorem 3.. The as, { x } coverges srogy o some fixed poi Pu of T, where P is he uique suy oexpasive rerac from K oo F ( T ). Proof. This foows from Lemma. ad he proof of Theorem 3. As a direc cosequece of Theorems 3. ad 3., we obai he foowig coroaries. Coroary 3.3. Le E be a refexive Baach space which has boh fixed poi propery for o-expasive sef-mappigs ad a uiformy Gâeaux differeiabe orm or be a refexive ad sricy covex Baach space wih a uiformy Gâeaux differeiabe orm. Suppose K, f, { x }, { α }, { β } are as Theorem 3., ad T : K K is a oexpasive mappig. The, as, { } coverges srogy o some fixed poi Pu of T, where P x is he uique suy oexpasive rerac from K oo F ( T ).

11 LIPSCHITZ PSEUDOCONTRACTIONS MAPPINGS 57 Remark. (i) There may Baach spaces which has fixed poi propery for o-expasive sef-mappigs. For exampe, compac Baach space, uiformy covex Baach space, uiformy smooh Baach space, refexive Baach space wih he Opia s propery, refexive Baach space wih orma srucure ad so o. (ii) I is easy o fid exampes of spaces which saisfy he fixed poi propery for o-expasive sef-mappigs, which are o sricy covex. O he oher had, i appears o be ukow wheher a refexive ad sricy covex Baach space saisfies he fixed poi propery for oexpasive sef-mappigs (see []). Refereces [] F. E. Browder, Fixed-poi heorems for ocompac mappigs i Hiber space, Proc. Na. Acad. Sci. USA 53 (965), [] R. E. Bruck, O he covex approximaio propery ad he asympoic behavior of oiear coracios i Baach spaces, Israe J. Mah. 38 (98), [3] R. E. Bruck, A simpe proof of he mea ergodic heorem for oiear coracios i Baach spaces, Israe J. Mah. 3 (979), [4] R. Che, Y. Sog ad H. Zhou, Covergece heorems for impici ieraio process for a fiie famiy of coiuous pseudocoracive mappigs, J. Mah. Aa. App. 34 (006), [5] R. Che, Y. Sog ad H. Zhou, Viscosiy approximaio mehods for coious pseudo-coracive mappigs, Aca Mahemaica Siica, Chiese series 49 (006), [6] C. E. Chidume ad E. U. Ofoedu, A ew ieraio process for geeraized Lipschiz pseudo-coracive ad geeraized Lipschiz accreive mappigs, Noiear Aa. 67 (007), [7] C. E. Chidume ad C. O. Chidume, Ieraive approximaio of fixed pois of oexpasive mappigs, J. Mah. Aa. App. 38 (006), [8] C. E. Chidume ad S. A. Muagadura, A exampe o he Ma ieraio mehod for Lipschiz pseudocoracios, Proc. Amer. Mah. Soc. 9(8) (00), [9] C. E. Chidume ad C. Moore, Fixed poi ieraio for pseudo-coracive maps, Proc. Amer. Mah. Soc. 7(4) (999), [0] C. E. Chidume, Goba ieraio schemes for srogy pseudo-coracive maps, Proc. Amer. Mah. Soc. 6(9) (998), [] C. E. Chidume, Approximaio of fixed pois of srogy pseudo-coracive mappigs, Proc. Amer. Mah. Soc. 0() (994),

12 58 JING HAN ad YISHENG SONG [] C. E. Chidume, Ieraive approximaio of Lipschiz sricy pseudo-coracive mappigs, Proc. Amer. Mah. Soc. 99() (987), [3] Y. J. Cho ad X. Qi, Viscosiy approximaio mehods for a famiy of m-accreive mappigs i refexive Baach spaces, Posiiviy (008), DOI 0.007/s [4] K. Deimig, Zero of accreive operaors, Mauscripa Mah. 3 (974), [5] L. Deg ad X. P. Dig, Ieraive approximaio of Lipschiz sricy pseudocoracive mappigs i uiformy smooh Baach spaces, Noiear Aa. 4(7) (995), [6] L. Deg, O chidumes ope probems, J. Mah. Aa. App. 74() (993), [7] T. L. Hicks ad J. R. Kubicek, O he Ma ieraio process i Hiber space, J. Mah. Aa. App. 59 (977), [8] T. H. Kim ad H. K. Xu, Srog covergece of modified Ma ieraios, Noiear Aa. 6 (005), [9] E. Kopecká ad S. Reich, Noexpasive reracs i Baach spaces, Baach Ceer Pubicaios 77 (007), [0] L. S. Liu, Ishikawa ad Ma ieraio process wih errors for oiear srogy accreive mappigs i Baach spaces, J. Mah. Aa. App. 94 (995), 4-5. [] W. R. Ma, Mea vaue mehods i ieraio, Proc. Amer. Mah. Soc. 4 (953), [] C. H. Moraes ad J. S. Jug, Covergece of pahs for pseudo-coracive mappigs i Baach spaces, Proc. Amer. Mah. Soc. 8 (000), [3] S. Reich, Srog covergece heorems for resoves of accreive operaors i Baach spaces, J. Mah. Aa. App. 75 (980), [4] S. Reich, Weak covergece heorems for oexpasive mappigs i Baach spaces, J. Mah. Aa. App. 67 (979), [5] S. Reich, Approximaig zeros of accreive operaors, Proc. Amer. Mah. Soc. 5 (975), [6] S. Reich, Asympoic behavior of coracios i Baach spaces, J. Mah. Aa. App. 44 (973), [7] J. Schu, Approximaig fixed pois of Lipschizia pseudocoracive mappigs, Houso J. Mah. 9 (993), [8] Y. Sog ad S. Xu, Srog covergece heorems for oexpasive semigroup i Baach spaces, J. Mah. Aa. App. 338 (008), 5-6. [9] Y. Sog ad R. Che, Srog covergece of a ieraive mehod for o-expasive mappigs, Mahemaische Nachriche 8(8) (008), [30] Y. Sog, A oe o he paper A ew ieraio process for geeraized Lipschiz pseudo-coracive ad geeraized Lipschiz accreive mappigs, Noiear Aa. 68 (008),

13 LIPSCHITZ PSEUDOCONTRACTIONS MAPPINGS 59 [3] Y. Sog, O a Ma ype impici ieraio process for coiuous pseudo-coracive mappigs, Noiear Aa. 67 (007), [3] Y. Sog ad R. Che, Covergece heorems of ieraive agorihms for coiuous pseudo-coracive mappigs, Noiear Aa. 67 (007), [33] Y. Sog ad R. Che, A approximaio mehod for coiuous pseudocoracive mappigs, J. Iequa. App. (006), -9. [34] T. Suzuki, Srog covergece of approximaed sequeces for oexpasive mappigs i Baach spaces, Proc. Amer. Mah. Soc. 35 (007), [35] T. Suzuki, Srog covergece heorems for ifiie famiies of oexpasive mappigs i geera Baach spaces, Fixed Poi Theory ad Appicaios 005() (005), [36] W. Takahashi ad Y. Ueda, O Reich s srog covergece for resoves of accreive operaors, J. Mah. Aa. App. 04 (984), [37] A. Udomee, Pah covergece, approximaio of fixed pois ad variaioa souios of Lipschiz pseudocoracios i Baach spaces, Noiear Aa. 67 (007), [38] H. K. Xu, Ieraive agorihms for oiear operaors, J. Lodo Mah. Soc. 66 (00), [39] H. K. Xu, Srog covergece of a ieraive mehod for oexpasive ad accreive operaors, J. Mah. Aa. App. 34 (006), [40] H. Zhou, Covergece heorems for λ- sric pseudo-coracios i -uiformy smooh Baach spaces, Noiear Aaysis (007), doi:0.06/ j.a g