Mathematics Volume : 4 Issue : 6 Jue 4 ISSN - 49-555X Data Depedece of New Iterative Schemes KEYWORDS CR Iteratio Data Depedece New Multistep Iteratio Quasi Cotractive * Aarti Kadia Assistat Professor Departmet of mathematics Deshbadhu College Delhi Uiversity New Delhi-9 * Correspodig Author ABSTRACT operators I this paper we prove data depedece of ew multistep iterative scheme for quasi cotractive operators that is by usig a approximate quasi -cotractive operator we approximate the fixed poit of the give INTRODUCTION Fixed poit theory is ivolved i various types of issues such as to fid the fixed poit existece ad uiqueess of fixed poit etc Data depedece of fixed poit is oe of these issues ad has become a subject of research iterest from some time ow The data depedece of various iterative schemes has bee studied by various authors like Rus ad Muresa [8] Rus et al [6 7] Beride[4] Espiola ad Petrusel [7] Marki [] Chifu ad Petrusel [4] Olatiwo [] Soltuz [9 ] Soltuz ad Grosa [] Chugh ad Kumar [5] Gursoy et al[5] ad several refereces thereof Our mai iterest i this paper is to show data depedece of ew multi step [] ad CR [6] iterative schemes usig quasi cotractive operators For the backgroud of our expositio we first metio some cotractive mappigs Zemfirescu [] established a ice geeralizatio of the Baach fixed poit theorem as follows: Let (X d be a complete metric space ad T: X X mappigs for which there exists real umbers αβγsatisfyig α < β < γ < respectively such that for each xy E at least oe of the followig is true: ( z d Ty αd( x y ( z d Ty β[ d( x Tx + d( y Ty] ( z d Ty γ[ d( x Ty + d( y Tx] 3 ( The the mappig T satisfyig ( is called Zamfirescu cotractio Remark Mappig which satisfy (z is called a Kaa mappig while the mappig satisfyig (z 3 is called Chatterjea operator The cotractive coditio ( implies Tx Ty δ x Tx + δ x y ad Tx Ty δ x Ty + δ x y β γ where δ max { α δ < β γ for all xy E Osilike ad Udomee [4] defied a ew geeral defiitio of quasi cotractive operator as follows: Tx Ty δ x y + L x Tx L δ [] (3 xy E ad some After that a more geeral defiitio was itroduced by Imoru ad Olatiwo [3] as follows: if there exists a costat δ < ad a mootoically icreasig ad cotiuous fuctio :[ [ such that for xy E Tx Ty δ x y + ϕ( x Tx ϕ with ϕ ( (4 Now we recapitulate some iterative schemes i the literature of fixed poit theory Let X be a Baach space ad E be a closed covex subset of X If T: X X a mappigs o E the F { p X : Tp p deotes the set of fixed poits of T For x { x is defied as x α x αty ( + + y ( β x + β Tx T E Ishikawa Iteratio [8] where { α ad { are real sequeces i [ ] satisfyig β α Observe that if β for each the the Ishikawa iteratio process (5 reduces to the Ma iteratio scheme For x E the Noor three step iterative scheme [3] { x is defied as x ( α x + + Ty y ( β x + Tz z ( γ x + Tx where { α { β ad { γ are real sequeces i [ ] satisfyig α If γ the Noor iteratio process (6 reduces to Ishikawa Iteratio scheme (5 I 7 Agarwal et al defied the Agarwal et al iterative scheme [] as x ( - α Tx + + α Ty y ( - β x + β Tx (7 INDIAN JOURNAL OF APPLIED RESEARCH X 335
where { α ad { β are sequeces of positive umbers i [ ] with α Quite recetly Phuegrattaa ad Suatai [5] itroduced SP iterative scheme as x ( α y + + αty y ( β z + β Tz z ( γ x + γ Tx (8 where { α { β ad { γ are sequeces of positive umbers i [ ] satisfyig α We shall use the followig iterative schemes: (a New multi step iterative scheme x+ ( α y + αty y ( β Tx + βty i k p k k y ( β x + β Tx (9 ad u+ ( α v + αtv v ( β Tu + βtv i k p k k v ( β u + β Tu where { α i { β i k k be the real sequeces of positive umbers i [ ] satisfyig α (b CR iterative scheme x E x+ ( α y + αty y ( β Tx + βtz z ( γ x + γtx ad u+ ( α v αtv + v ( β Tu + β Tw w ( γ u + γtu ( where{ α { β ad { γ are sequeces of positive umbers i [ ] satisfyig α due to Chugh ad Kumar [6] Now to prove our mai results we eed followig results i sequel Defiitio 3 [3] Let T T be two operators We say T is approximate operator of T if for all x X ad for a fixed TxTx ε ε > we have Volume : 4 Issue : 6 Jue 4 ISSN - 49-555X Lemma 4[] Let { be a oegative sequece for α which oe suppose there exists it satisfies the followig iequality: α ( λ α + λσ where λ ( N λ The lim supα lim sup σ I such that for all ad σ N Theorem 5 [7]: Let T: E E be a mappig satisfyig (4 ad { be defied by ( with real x sequece { α { β ad { γ i [ satisfyig α The the sequece { x coverges to uique fixed poit of T Theorem 6[9]: Let T: E E be a mappig satisfyig (4 ad { be defied by (9 with real x sequece { α i { β i k i [ satisfyig α The the sequece { x coverges to uique fixed poit of T Mai Results Theorem Let T : E E be a mappig satisfyig (4 Let T be a approximate operator of T as i Defiitioad{ x { u be two CR iterative schemes defied by ( associated to T ad T respectively where { α { β ad { γ are real ( i α( δ sequeces i [ satisfyig Let ( ii α p Tp ad q Tq the we have the followig estimate: 4ε pq δ Proof: Usig (4 ad ( we have the followig estimates: 336 X INDIAN JOURNAL OF APPLIED RESEARCH
x u ( α ( y v + α ( Ty Tv + + ( α y v + α Ty Tv ( α y v + α Ty Tv + Tv Tv { { ( ( ( α y v + α Ty Tv + Tv Tv ( α y v + α δ y v + ϕ T y y + ε ( α ( δ y v + αϕ Ty y + αε y v ( β Tu + β Tw ( β Tx Tu + β Tz Tw ( ( β Tx Tu + Tu Tu + β Tz Tw + Tw T w ( β { Tx Tu + Tu Tu + β{ Tz Tw + Tw Tw ( β { δ x u + ϕ x + ε + β{ δ z w + ϕ z + ε ( β δ x u + ( β ϕ x + ( β ε+ βδ z w + βϕ z + βε ( ad ( γ ( z w ( γ x u + Tx Tu ( γ x u + γ Tx Tu { ( ( ( γ x u + γ δ x u + ϕ Tx x + ε ( γ ( δ x u + γϕ Tx x + γε (3 Combiig ( ( ad (3 we have x ( β δ x u u ( α ( δ + ( β ϕ + ( β ε x α δ { βδ z w βϕ z βε ( Ty y αε + + + ( ( + + + αϕ + ( α ( δ( β δ x u + ( α ( δ β δ z w x z ( Ty y αε + ( α ( δ( β ϕ + ( α ( δ( β ε + ( α ( δ βϕ + ( α ( δ βε + αϕ + Volume : 4 Issue : 6 Jue 4 ISSN - 49-555X ( α( δ( β δ x u + ( α ( δ βδ ( γ ( δ + γϕ + γε { x u x x z ( Ty y + ( α( δ( β ϕ + ( α( δ( β ε + ( α ( δ βϕ + ( α ( δ βε+ αϕ + αε {( α( δ( β( δ( γ( δ x u δ( α( δ βγϕ x ( α( δ βδγ ε ( α( δ( β ϕ x ( α( δ( β ε z ( Ty y (4 + + + + + ( α ( δ βϕ + ( α ( δ βε+ αϕ + αε It may be oted that for { α { β { [ ad δ < the followig γ iequalities hold: ( α δ < ( α ( β( δ( γ( δ < αβδ < α It follows from assumptio (i that ( α( δ < α( δ α I Now usig (5 ad (6 i (4 we get ( α ( δ αϕ( z ( Ty y x u x u + Tx x + + + 4αε+ αϕ + αϕ which further implies x u ( α ( δ x u + + (5 (6 x + + z + ( Ty y {ϕ 4 ε ϕ ϕ + α( δ ( δ (7 Let us deote a x u r ( α δ ad x + + z + ( Ty y {ϕ 4 ε ϕ ϕ σ ( δ INDIAN JOURNAL OF APPLIED RESEARCH X 337
Now from Theorem we have lim x p Also T satisfies coditio (4 ad Tp p F hece lim x Tx lim y Ty lim z Tz T Volume : 4 Issue : 6 Jue 4 ISSN - 49-555X u+ ( α v + αtv v Tu T v i k p p p v ( β u + β Tu ( β + β Because{ x { y { z poit of T coverges to fixed lim ϕ( x Tx lim ϕ( y Ty lim ϕ( z Tz Sice ϕ is cotiuous hece usig Lemma (7 yields 4ε pq ( δ Theorem Let T : E E be a mappig satisfyig (4 Let T be a approximate operator of T as i Defiitio ad { x { u be two iterative schemes defied by (9 associated to T ad T respectively where { α ad { β i i k are real sequeces i[ satisfyig i ( i o β α < i k Let p Tp ad q Tq ( ii α the we have the followig estimate kε pq δ Proof: For a give x E ad u E we cosider the followig iterative schemes for T ad T I this paper we use followig iterative scheme x+ ( α y + αty y ( β Tx + βty i k p k k y ( β x + β Tx (9 the usig (4 (8 ad (9 yield the followig estimates: x u ( α ( y v + α ( Ty Tv + + ( α y v + α Ty Tv ( α y v + α Ty Tv + Tv Tv ( α y v + α Ty Tv + α Tv Tv ( α ( δ y v + αϕ( y Ty + αε y v ( β Tu + β ( Ty Tv ( ( β Tx Tu + Tu Tu + β Ty Tv + Tv Tv { Tx Tu Tu Tu β { Ty Tv Tv Tv { x u x { y v ( Ty y ε ( β + + + ( β δ + ϕ + ε + β δ + ϕ + ( β δ + β ϕ ( x u ( Tx x ( + ( β ε+ βδ y v + βϕ Ty y + βε ad ( y v ( β ( x u + β ( Ty Tv 3 3 ( β δ x u + βδ y v + βϕ( y Ty 3 3 3 3 x + βε+ ( β ϕ + ( β ε Combiig ( ( ad ( we have ( x u ( α ( δ y v + αϕ( y Ty + αε + + α δ β δ x u + α δ β δ β x u ( ( ( ( ( ( + ( α ( δ βδ β y v + ( α ( δ β δβ ϕ( y Ty 3 3 3 3 + ( α( δ βδβε + ( α( δ βδ ( β ϕ x ( Ty y x + α δ β δ β ε + α δ β ϕ ( ( ( ( ( ( + ( α ( δ( β ε + ( α ( δ β ϕ + ( α ( δ βε+ αϕ( y Ty + αε Thus iductively we get (3 ad (8 338 X INDIAN JOURNAL OF APPLIED RESEARCH
( β δ x+ u+ ( α( δ x k 3 k 3 u + βδ ( β + + δ ββ ( β + ( α( δ δ ββ β y v k k k k 3 + [ αϕ ( y Ty + ( α( δ βϕ ( Ty y + ( α( δ( β ϕ x + ( α( δ βδ ( β ϕ x + ( α ( δ β δβ ϕ( y Ty ] + [ αε 3 3 ( ( ( ( ( + α δ β ε + α δ β ε ( α( δ βδ ( β ε ( α( δ βδβε ] + + + (4 Usig (4 ad (9 y v ( β ( x u + β Tu k k k k ( β x u + β Tx Tu k k ( β ( δ x u + β ϕ( x Tx + β ε k k k (5 Now by combiig (4 ad (5 x u x u ( β δ + + ( α( δ k 3 k 3 + βδ ( β + + δ ββ ( β x x ( + [ αϕ( y Ty + ( α ( δ βϕ Ty y ( α( δ( β ϕ + ( α( δ βδ ( β ϕ + 3 3 + ( α ( δ β δβ ϕ( y Ty k ( ( ( x Tx ] [ + α δ β δβ β ϕ + αε ( ( ( ( ( + α δ β ε + α δ β ε ( ( ( + α δ βδ β ε k ( α ( δ βδβε ( α ( δ βδβ β ε ] + + + Volume : 4 Issue : 6 Jue 4 ISSN - 49-555X x u [ ( α ( δ] x u + αϕ( y Ty + αϕ( y Ty + + 3 3 + αϕ ( y Ty + + αϕ ( x Tx + kε [ ( α ( δ] x u ϕ ( y Ty + ϕ( y Ty + + αϕ ( x Tx + kε + α( δ ( δ (9 Let us deote a x u r α ( δ ad { ϕ( y Ty + ϕ( y Ty + + ϕ( x Tx + kε σ : ( δ Now from Theorem we have lim x p Also T satisfies coditio (4 ad Tp p F usig the similar argumet as i Theorem 3 we get lim x Tx lim y Ty lim y Ty lim y Ty k k Sice ϕ is cotiuous we have lim ϕ( x Tx lim ϕ( y Ty lim ϕ( y Ty lim ϕ( y Ty k k Hece usig Lemma (9 yields kε pq δ T which further implies x u α δ β δ β δ x u k + + ( ( ( ( ( ( + αϕ( y Ty + δα β ϕ( y Ty + δαββϕ( y Ty + + δ α β β ϕ( x Tx 3 3 3 k k + αε+ δαβε+ + δ αβ β ε p k ( α δ < ( α < k ( β( δ( β ( δ k k δ αβ β < α It follows from assumptio (i that ( α( δ < α( δ α I (6 (7 (8 Hece usig (7 ad (8 i (6 we get INDIAN JOURNAL OF APPLIED RESEARCH X 339
Volume : 4 Issue : 6 Jue 4 ISSN - 49-555X REFERENCE [] Agarwal RP O Rega D ad Sahu DR: Iterative costructio of fixed poits of early asymptotically oexpasive mappigs Joural of Noliear ad Covex Aalysis 8( (7 6-79 [] BE Rhoades SM Soltuz The equivalece betwee Ma-Ishikawa iteratios ad multi-step iteratio Noliear Aalysis 58(4 9-8 [3] CO Imoru MO Olatiwo O the stability of Picard ad Ma iteratio processes Carpathia Joural of Mathematics 9(3 o 55-6 [4] Chifu G Petru sel Existece ad Data Depedece of Fixed Poits ad Strict Fixed Poits for Cotractive- Type Multivalued Operators Fixed Poit Theory ad Applicatios 7(7 Article ID 3448 8 pages [5] F Gursoy et al Data depedece results of ew multi-step ad s-iterative schemes for cotractive-like operators Fixed Poit Theory ad Applicatio 86/687-8-3-76 [6] IA Rus A Petru sel A Sˆıtamaria Data depedece of the fixed poits set of multivalued weakly Picard operators Stud Uiv Babes-Bolyai Math 46 ( ( - [7] IA Rus A Petru sel A Sˆıtamaria Data depedece of the fixed poit set of some multivalued weakly Picard operators Noliear Aalysis: Theory Methods & Applicatios 5 (3 947 959 [8] IA Rus S Muresa Data depedece of the fixed poits set of weakly Picard operators Stud Uiv Babes-Bolyai 43 (998 79-83 [9] JO Olaleru H Akewe O multistep iterative scheme for approximatig the commo fixed poits of cotractive-like operators It Joural of Mathematics ad mathematical scieces Volume Article ID 53964 pages [] JT Marki Cotiuous depedece of fixed poit sets Proc AMS 38 (973 545-547 [] M O Olatiwo O the cotiuous depedece of the fixed poits for ( -cotractive-type operators Kragujevac Joural of Mathematics 34( 9- [] M O Olatiwo Some results o the cotiuous depedece of the fixed poits i ormed liear space Fixed Poit Theory (9 o 5-57 [3] MA Noor New approximatio schemes for geeral variatioal iequalities Joural of Mathematical Aalysis ad Applicatios 5( o 7-9 [4] MO Osilike A Udomee Short proofs of stability results for fixed poit iteratio procedures for a class of cotractive-type mappigs Idia Joural of Pure ad Applied Mathematics 3(999 o 9-34 [5] R Chugh V Kumar Data depedece of Noor ad SP iterative schemes whe dealig with quasi-cotractive operators Iteratioal Joural of Computer Applicatios 3( o5 [6] R Chugh V Kumar S Kumar Strog covergece of a ew three step iterative scheme i Baach spaces America Joural of Computatioal Mathematics ( 345-357 [7] R Esp ıola A Petru sel Existece ad data depedece of fixed poits for multivalued operators o gauge spaces J Math Aal Appl 39 (5 4 43 [8] S Ishikawa Fixed poits by a ew iteratio method Proc Amer Math Soc 44(974 47-5 [9] SM Soltuz Data depedece for Ishikawa iteratio Lecturas Mathematicas 5(4 o 49-55 [] SM Soltuz Data depedece for Ma iteratio Octogo Math Magazie 9( 85-88 [] SM Soltuz T Grosa Data depedece for Ishikawa iteratio whe dealig with cotractive like operators Fixed Poit Theory ad Applicatios 8(8 Article ID 496 7 pages [] T Zamfirescu Fixed poit theorems i metric spaces Archiv der Mathematik 3(97 o 9-98 [3] V Beride Iterative Approximatio of Fixed Poits Spriger Berli (7 [4] V Beride O the approximatio of fixed poits of weak cotractive mappigs Carpathia J Math 9(3 o 7- [5] W Phuegrattaa S Suatai O the rate of covergece of Ma Ishikawa Noor ad SP iteratios for cotiuous fuctios o a arbitrary iterval Joural of Computatioal ad Applied Mathematics 35( 36-34 [6] W Takahashi Iterative methods for approximatio of fixed poits ad their applicatios Joural of the Operatios Research Society of Japa 43( o 87-8 [7] WR Ma Mea value methods i iteratios Proc Amer Math Soc 4(953 56-5 [8] Xu MA Noor Ishikawa ad Ma iteratio process with errors for oliear strogly accretive operator equatios J Math Aal Appl 4(998 9-34 X INDIAN JOURNAL OF APPLIED RESEARCH