VISCOSITY APPROXIMATION TO COMMON FIXED POINTS OF kn- LIPSCHITZIAN NONEXPANSIVE MAPPINGS IN BANACH SPACES

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1 Joral o Maheaical Scieces: Advaces ad Alicaios Vole Nber 9 Pages -35 VISCOSIY APPROXIMAION O COMMON FIXED POINS OF - LIPSCHIZIAN NONEXPANSIVE MAPPINGS IN BANACH SPACES HONGLIANG ZUO ad MIN YANG Deare o Maheaics Hea Noral Uiversiy Hea 4537 P. R. Chia e-ail: yag_i8@yahoo.c Absrac Le E be a real Baach sace wih iorly Gâea diereiable or ossessig ior oral srcre. K is a oey boded closed cove sbse o E ad { } ( ) is a seqece o Lischiia oeasive aigs ro K io isel sch ha li ad F ( ) / ad be a coracio o K. Uder siable codiios o seqece { } we show he seqece { } deied as ( ) eiss ad coverges srogly o a ied oi o a aig. Ad we aly i o rove he ieraive rocess deied by K ad ( ) coverges srogly o he sae oi. Maheaics Sbec Classiicaio: 47H5 47H 47H7. Keywords ad hrases: iorly Gâea diereiable or ior oral srcre viscosiy aroiaio ehods Lischiia aigs. Received Seeber Scieiic Advaces Pblishers

2 HONGLIANG ZUO ad MIN YANG. Irodcio Le E be a real Baach sace wih dal E ad K be a oey closed cove sbse o E. Le J : E deoe he oralied E daliy aig deied by J { E } E. deoes he geeralied daliy airig. I he seqel we shall deoe he sigle-valed daliy aig by. A aig : K K is called Lischiia i here eiss a osiive cosa L sch ha y L y y K. is also called L- Lischiia i is L- Lischiia ad L < L he is L- Lischiia. hrogho he aer we asse ha every Lischiia aig is L- Lischiia wih L. I L is ow as a oeasive aig ad is asyoically oeasive i wih li sch ha here eiss a seqece { } [ ) y y y K or all iegers. We deoe F ( ) { E; }. I [3] Modai roosed a viscosiy aroiaio ehod o selecig a ariclar ied oi o a give oeasive aig i Hilber saces. I H is a Hilber sace : K K a oeasive selaig o a oey closed cove sbse K o h ad : K K is a coracio he roved he ollowig heores. heore M [3 heore.]. he seqece { } geeraed by he schee ε ( ) ε ε coverges srogly o he iqe solio o he variaioal ieqaliy: sch ha ~ F ( )

3 VISCOSIY APPROXIMAION O COMMON FIXED 3 ( I ) ~ or all F ( ) where { ε } is a seqece o osiive bers edig o ero. heore M [3 heore.]. Wih a iiial seqece { } by K deied he ε ( ). ε ε Sosed ha li ε ε ; ad li. he ε ε { } coverges srogly o he iqe solio o he variaioal ieqaliy: sch ha ~ F ( ) ( I ) ~ or all F ( ). X [7] sdied he viscosiy aroiaio ehods roosed by Modai or a oeasive aig i a iorly sooh Baach sace. I deoes he se o all coracios o K he roved he K ollowig heores: heore HKX [7 heore 4.]. Le E be a iorly sooh Baach sace K a closed cove sbse o E ad : K K a oeasive aig wih F ( ) / ad. he he ah K { } deied by ( ) ( ) ( ) coverges srogly o a oi i F ( ). I we deie Q : F ( K ) by Q( ) li he Q ( ) solves he variaioal ieqaliy: ( I ) Q( ) ( Q( ) ) F ( ). K

4 4 HONGLIANG ZUO ad MIN YANG heore HKX [7 heore 4.]. Le E be a iorly sooh Baach sace K a closed cove sbse o E ad : K K a oeasive aig wih F ( ) / ad. Asse ha K ( ) saisies he ollowig codiios: (i) li ; (ii) ; (iii) eiher li or <. he he seqece { } geeraed by ( ) ( ) K coverges srogly o a oi i F ( ). Recely Shaad ad Udoee [5] obaied ied oi solios o variaioal ieqaliies or a asyoically oeasive aig deied o a real Baach sace wih iorly Gâea diereiable or ossessig ior oral srcre. Nilsraoo ad Saeg [4] esablished wea ad srog covergece heore or a coable aily o cerai Lischiia aigs ad i a real Hilber sace. hey roved he ollowig heore. heore W. N. S. S. [4 heore 5]. Le C be a oey closed cove sbse o a real Hilber sace H. Le { } ( ) is a seqece o L Lischiia oeasive aigs ro C io isel sch ha ( L ) < ad F ( ) /. Le { } be a seqece i C deied by C ad ( )

5 VISCOSIY APPROXIMAION O COMMON FIXED 5 or all N where { } is a seqece i [ ) wih ( ). Le s{ : B} < or ay boded sbse B o C ad be a aig o C io isel deied by li or all C ad sose ha F ( ) F ( ). he { } coverges wealy o ω F ( ). Moreover li P ω. F I his aer we sdied he coable aily o cerai Lischiia aigs i he real Baach sace ad roved he seqece coverges srogly o he iqe solio o soe variaioal ieqaliy i or ore geeral sace. Or heore also eeds heores 3. ad 3. o [5] o ore geeral class o oeasive aigs.. Preliiaries Le S : { E : } deoe he i shere o he Baach sace E he sace E is said o have a Gâea diereiable or i he lii y li ( ) eiss or each y S ad we call E sooh; ad E is said o have a iorly Gâea diereiable or i or each y S he lii () is aaied iorly or S. Frher E is said o be iorly sooh i he lii () eiss iorly or ( y) S S. I E is sooh he ay daliy aig o E is sigle-valed ad i E has a iorly Gâea diereiable or he he daliy aig is or-o- wea iorly coios o boded ses. A boded cove sbse K o a Baach sace E is said o have oral srcre i every cove sbse H o K ha coais ore ha oe oi coais a oi H sch ha s { y y H } < d( H ) where d( H ) s{ y y H } deoes he diaeer o H. A Baach sace E is said o have oral srcre i every boded cove

6 6 HONGLIANG ZUO ad MIN YANG sbse o E has oral srcre. E is said o have ior oral srcre i here eiss < c < sch ha or ay sbse K as above here eiss K sch ha s{ y y K } < c( d( K )). I is ow ha every sace wih a ior oral srcre is releive ad ha all iorly cove ad iorly sooh Baach saces have ior oral srcre. Sose a liear coios cioal µ o l sch ha µ µ (). he µ is called a ea o N i ad oly i he ieqaliies i { a ; N } µ ( a) s{ a ; N } hold or each a ( a a ) l we deoe by µ ( a ) isead o µ ( a). A ea is called a Baach lii i ( a ) µ ( a ). µ Lea. [6]. Le C be a oey closed cove sbse o a Baach sace E wih a iorly Gâea diereiable or le S be a ide se le { : S} be a boded se o E ad le µ be a ea o S. Le C he µ i µ y C y i ad oly i µ J( ) or all C where J is he daliy aig o E. Lea.. Le E be a arbirary real Baach sace. he y y ( y) () y E ad ( y) J( y). Lea.3 []. Le { a } { } { } b c be seqeces o oegaive bers saisyig where { ω } [ ]. I he li a. ( ω ) a b c a ω b o( ω ) ad c <

7 VISCOSIY APPROXIMAION O COMMON FIXED 7 Lea.4 []. Le K be a oey boded closed cove sbse o a releive Baach sace E. Ad sose ha K has oral srcre. I φ is a aig o K io isel which does o icrease disaces he φ has a ied oi. Lea.5 [7]. Le { a } be a seqece o oegaive real bers saisyig he roery ( γ ) a γ β a where { γ } ( ) ad { β } is real ber seqece sch ha (i) γ ; (ii) li β. he { a } coverges o ero as. 3. Mai Resls heore 3.. Le E be a real Baach sace wih iorly Gâea diereiable or ossessig ior oral srcre. Sose K is a oey boded closed cove sbse o E ad { } ( ) is a seqece o Lischiia oeasive aigs ro K io isel sch ha li ad F ( ) /. Le be a aig o K io isel deied by li or all K wih li s ad sose ha F ( ) F ( ). : K K K ( ) a coracio wih cosa [ ). Le { } ( ) be sch ha li ad li. he (i) For each ieger here is a iqe K sch ha

8 8 HONGLIANG ZUO ad MIN YANG (ii) he as ( ) ( ) ; (3.) coverges srogly o soe ied oi o sch ha is he iqe solio i F o he ollowig variaioal ieqaliy: ( I ) ( ) or all F. (3.) Proo. A irs we show ha here eiss iqe solio o he eqaliy (3.). I ac or each ieger by he codiios o he aig φ ( ) is a coracio. I ollows here eiss iqe K sch ha φ. Sice K is boded ad K { } { ( )} { } are boded. ( ) ( ). By he codiio li s K. Deie he aig v/ : K R by v/ : µ K. Sice E is releive v / as ad v/ is coios ad cove we have ha he se C : { y K : v/ ( y) i K v/ } /. C is closed ad boded. Fro v/ ( ) µ µ µ v/ hereore ( C) C. By Lea.4 has a ied oi i C. Le C F ( ) ad sig Lea. we have µ ( ) or all K. I ollows ha

9 VISCOSIY APPROXIMAION O COMMON FIXED 9 By he codiios o { } we have µ ( ). (a) ( ) ( ) F ( ). By he deiiio o he seqece { } we have ( ) ( ) F ( ). Sice K is boded i ollows ha li s ( ) ( ) F ( ). (b) Fro ( ) ( ) ( ) ( ). Ad ( a) ( b) µ. hs here eiss a sbseqece { } { } sch ha as. Asse ha here is aoher sbseqece { l } { } sch ha q as l. For l se q by ( b ) ( q). For q se by ( b ) q ( q) ( q ). l q ( q) ( q) q. as We s have q ad he iqeess is roved. hs ad F is iqe. Agai sig (b) we have ( I ) ( ) or all F. his cocldes he roo. Rear. For a asyoically oeasive aig se he we ca ge heore 3. o [5] as a corollary: Le E be a real Baach sace wih iorly Gâea diereiable or ossessig ior oral srcre K a oey closed cove ad boded sbse o E : K K a asyoically oeasive aig wih

10 3 seqece { } [ ) HONGLIANG ZUO ad MIN YANG ad : K K a coracio wih cosa ( ) [ ). Le { } ( ) be sch ha li ad li. he (i) For each ieger here is a iqe K sch ha (ii) he as ( ) ( ) ; coverges srogly o soe ied oi o sch ha is he iqe solio i F o he ollowig variaioal ieqaliy: ( I ) ( ) or all F. heore 3.. Le E be a real Baach sace wih iorly Gâea diereiable or ossessig ior oral srcre. Sose K is a oey closed cove sbse o E ad { } ( ) is a seqece o - Lischiia oeasive aigs ro K io isel sch ha li ad F ( ) /. Le { } ( { ( ) i }) be sch ha li ( ) s{ : B} < ad li. Ad or ay boded sbse B o K. : K K a coracio wih cosa [ ). Le { } be a seqece i K deied by K ad he (i) is boded ; ( ) ( ).

11 VISCOSIY APPROXIMAION O COMMON FIXED 3 (ii) Le C is a boded sbse o K coaiig { } be a aig o C io isel deied by li or all K ad sose ha. F F he { } coverges srogly o sch ha is he iqe solio i F o he ieqaliy (3.). Proo. Firsly we show ha { } is boded. ae. F I ollows ha ( ) [ ( )] { }. a By idcio { } a ad { } is boded which leads o he bodedess o { } ad { }. Usig he assio ha li we ge ha. he [ ] [ ]

12 HONGLIANG ZUO ad MIN YANG 3 [ ] [ ]. By Lea.3 Le [ ] ω hs. > ω > ω Le. b Sice { } ad { } are boded ad > ω he we have < ω M b as where { { } { } }. s > M Le c sig he assio { } < B : s or ay boded sbse B o C hs. < c Now. li li a he.

13 VISCOSIY APPROXIMAION O COMMON FIXED 33 Deie. Se le < we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). Sice C is boded or soe cosa > N i ollows ha N s li. li s N By heore 3. F which solve he variaioal ieqaliy (3.). Sice is or o wea coios o boded ses i he lii as we obai ha s li (c) here eiss a seqece { } ε ε or all sch ha

14 HONGLIANG ZUO ad MIN YANG 34 ε wih. ε so ha [ ] ( ). Le. : γ For soe γ > N as ad ( ) γ ( ) se λ he { } γ λ is a boded seqece. Le > M be a cosa sch ha. M γ λ he we have

15 VISCOSIY APPROXIMAION O COMMON FIXED 35 λ ( γ ) γε γ ( γ ) M γ ε. Usig Lea.5 we coclde ha. his colees he roo o he heore. Reereces [] W. A. Kir A ied oi heore or aigs which do o icrease disaces Aer. Mah. Mohly. 7 (965) 4-6. [] L. S. Li Ishiawa ad Ma ieraive rocess wih errors or oliear srogly accreive aigs i Baach Saces J. Mah. Aal. Al. 94 (995) 4-5. [3] A. Modai Viscosiy aroiaio ehods or ied-oi robles J. Mah. Aal. Al. 4 () [4] W. Nilsraoo ad S. Saeg Wea ad srog covergece heores or coable Lischiia aigs ad is alicaios Noliear Aalysis. I Press. [5] N. Shahad ad A. Udoee Fied oi solios o variaioal ieqaliies or asyoically oeasive aigs i Baach saces Noliear Aalysis 64 (6) [6] W. aahashi ad Y. Ueda O Reich s srog covergece heores or resolves o accreive oeraors J. Mah. Aal. Al. 4 (984) [7] Hog-K X Viscosiy aroiaio ehods or oeasive aigs J. Mah. Aal. Al. 98 (4) g

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