doi:.36/scieceasia53-874.6.3.67 ScieceAsia 3 (6: 67-7 Weak ad Strog Covergece Theorems of New Iteratios with Errors for Noexasive Noself-Maigs Sorsak Thiawa * ad Suthe Suatai ** Deartmet of Mathematics Faculty of Sciece Chiag Mai Uiversity Chiag Mai 5 Thailad. * ** Corresodig authors E-mails: sorsakt@u.ac.th * ad scmti5@chiagmai.ac.th ** Received Oct 5 Acceted 3 Ja 6 ABSTRAC I this aer a ew three-ste iterative scheme for oexasive oself- maigs i Baach saces is defied ad weak ad strog covergece theorems are established for the ew iterative scheme i a uiformly covex Baach sace. KEYWORDS: Noexasive oself-maig comletely cotiuous uiformly covex three-ste iteratio. INTRODUCTION Fixed-oit iteratio rocesses for oexasive oself-maig i Baach saces icludig Ma ad Ishikawa iteratio rocesses have bee studied extesively by may authors -3. I Noor 4 itroduced a three-ste iterative scheme ad studied the aroximate solutios of variatioal iclusio i Hilbert saces. I 998 Jug ad Kim 5 roved the existece of a fixed oit for a oexasive oselfmaig i a uiformly covex Baach sace with a uiformly Gateaux ˆ differetiable orm. Ta ad Xu itroduced a modified Ishikawa rocess to aroximate fixed oits of oexasive self-maigs defied o oemty closed covex bouded subsets of a uiformly covex Baach sace. Suatai 6 defied a ew three-ste iteratio which is a extesio of Noor iteratios ad gave some weak ad strog covergece theorems of such iteratios for asymtotically oexasive maigs i uiformly Baach saces. Recetly Shahzad 7 exteded Ta ad Xu s results (Theorem.35 to the case of oexasive oself-maig i a uiformly covex Baach sace. Motivatig these facts a ew class of three-ste iterative scheme is itroduced ad studied i this aer. The scheme is defied as follows. Let X be a ormed sace C a oemty covex subset of X P: X C a oexasive retractio of X oto C ad C X a give maig. The for a give x C comute the sequece { x } { y } ad { z } by the iterative scheme z = ( a b x atx bu y = P( ( c d z ctx v (. x = (( α β α β P y Tx w where { u} { v} { w } are bouded sequeces i C ad { a} { b} { c} { d } { α} { β } are. If C C ad a = b = c = d = β the the iterative scheme (. reduces to the usual Ma iterative scheme x = α ( α Tx x where { α } are aroriate sequeces i [ ]. The urose of this aer is to establish several strog covergece theorems for the three-ste scheme (. for comletely cotiuous oexasive oselfmaigs i a uiformly covex Baach sace ad weak covergece theorems for the scheme (. for oexasive oself-maigs i a uiformly covex Baach sace with Oial s coditio. Now we recall the well kow cocets ad results. Let X be ormed sace ad C a oemty subset of X. A maig C C is said to be oexasive o C if Tx Ty x y for all xy C. Recall that a Baach sace X is said to satisfy Oial s coditio 8 if x x weakly as ad x y imlyig that lim su x < lim su x y. aroriate sequeces i [ ] I the sequel the followig lemmas are eeded to rove our mai results. Lemma. Let {a }{b } ad {δ } be sequeces of oegative real umbers satisfyig the iequality ( a δ = a b. If δ < ad b < the = ( lima exists. =
68 ScieceAsia 3 (6 ( lim a = wheever limif a Proof. See (Lemma. Lemma. Let X be a uiformly covex Baach sace ad Br = { x X: x r} r >. The there exists a cotiuous strictly icreasig ad covex g: g ( = such that fuctio [ [ for all r λβγ with λ β γ =. Proof. See 9 (Lemma.4. xyz B ad all [ ] Lemma.3 Let X be a uiformly covex Baach sace C a oemty closed covex subset of X ad C X be a oexasive maig. The I T is demi-closed at i.e. if x x weakly ad x Tx strogly the x F( T where FT ( is the set of fixed oit of T. Proof. See. Lemma.4 Let X be a Baach sace which satisfies Oial s coditio ad let { x } be a sequece i X. Let uv X be such that lim x u ad lim x v exist. If { x k } ad { x m k } are subsequeces of { x } which coverge weakly to u ad resectively the u = v. Proof. See 6 (Theorem.3. RESULTS I this sectio we rove weak ad strog covergece theorems for the three-ste iterative scheme (. for a oexasive oself-maig i a uiformly covex Baach sace. I order to rove our mai results the followig lemma is eeded. Lemma. Let X be a uiformly covex Baach sace C a oemty closed covex oexasive retract of X with P as a oexasive retractio ad C X a oexasive oself-maig with FT (. Suose that { α} { β} { a} { b} { c} ad { d } are real sequeces i [ ] such that c ad α β are i [ ] for all ad b < d < β <. For a give x C = = = let { x} { y } ad { z } be the sequeces defied as i (.. (i If is a fixed oit of T the lim x exists. (ii If < lim if α limsu( α β < the lim Tx y ( λx βy γz λ x β y γ z λβg x y < limsu < ad < lim if c lim su c < the lim Tx z < lim if α lim su α β < (iii If ( α β ( (iv If ( < lim if c limsu( c < ad limsua < the lim Tx Proof. Let F( T ad M = su { u : M = su v : } M3 = su { w : } M = max { Mi : i = 3 }. Usig (. we have (( ( (( a b x atx bu ( a b x a Tx T z = P a b x a Tx b u P b u ( a b x a x b u x Mb (( ( ( c d z c x Md ( c d( x Mb c x y = P c d z c Tx v P Md x Mb Md ad so x = P α β y α Tx β w P (( ( ( α β y α x β w ( α β( x Mb Md α x Mβ ( β x M b Hece the assertio (i follows from Lemma.. (ii By (i we kow that lim x exists for ay (. x Tx ad y are bouded. Also { u } { v } ad w are bouded by the assumtio. Now we set F T It follows that { } { } { } { } r = su { x : } r = su { Tx : } r3 = su { y : } r4 = su { z : } r5 = su { u : } r6 = su { v : }.
ScieceAsia 3 (6 69 By usig Lemma. we have ad so which leads to the followig: (. (( ( = z P a b x a Tx b u P ( a b ( x a ( Tx b ( u ( ( ( ( a b x a x b u a b x a Tx b u a a b g Tx x r b (( ( y = P c d z c Tx v P ( c d ( z c ( Tx ( v ( c d z c Tx ( ( ( v c c d g Tx z c d z c x ( ( v c c d g Tx z ( ( c d x r b c x ( ( r d c c d g Tx z ( c ( c d g( Tx z d x r b r d x r b r d (( α β α β ( = x P y Tx w P = ( α β α β ( α β ( y α ( Tx β ( w ( α β y α Tx β w α ( α β g( Tx y ( α β α β y Tx w y x w α ( α β g( Tx y ( ( α β x r b r d α x ( ( r β α α β g Tx y ( β x r b r d r β α ( α β g( Tx y { } { } r7 = su w : r = max r : i = 3 45 67. x r b r d r β ( g( Tx y ( β ( g( Tx y α α β i x r b d α α β ( g( Tx y α α β ( x r b β (. < lim if α limsu α β < the there If ( ad η η ( exist a ositive iteger such that < η < α ad α β < η < for all. It follows from (. that η η g Tx y = ( ( ( ( x r b β (.3 for all. Alyig (.3 for m we have m ( x m = g( Tx y m = η( η r ( b β d = x η( η m r ( b β = (.4 Lettig m i the iequality (.4 we get that g Tx y < ad therefore lim Tx y. Sice g is strictly icreasig ad cotiuous at with g( = it follows that lim Tx y (iii If < lim su( α β < ad < limif c limsu ( c < the by the same argumet as that give i (ii it ca be show that lim Tx z (iv If < lim if α lim su( α β < < lim if c limsu( c < ad limsu a < by (ii ad (iii we have lim Tx y = ad lim Tx z (.5 (( From y = P c d z ctx v we have = (( ( ( c d z ctx v = ( z c ( Tx z ( v z y x P c d z c Tx d v P x z c Tx z v z = P( ( a b x a Tx bu P( x c Tx z v z ( a b x a Tx bu c Tx z v z = a ( Tx b ( u c Tx z v z a Tx b u c Tx z v z a Tx c Tx z rb rd
7 ScieceAsia 3 (6 where r is defied by (.. Thus Tx Tx y y Tx y a Tx c Tx z rb rd ad so ( a Tx Tx y c Tx z rb rd Sice lim sua < ad lim b = lim d = it follows from (.5 that lim Tx Theorem. Let X be a uiformly covex Baach sace C a oemty closed covex oexasive retract of X with P as a oexasive retractio ad C X a comletely cotiuous oexasive oself-maig with FT (. Suose that { α} { β} { a} { b} { c} { d} are sequeces of real umbers i [ ] with c [ ] ad α β [ ] for all ad b < d < β < ad = = = < lim if limsu < ad (i α ( α β < limif c limsu c < ad limsua <. For { x} { y } ad { z } beig the sequeces defied by the three-ste iterative scheme (. we have { x} { y } ad { z } coverge strogly to a fixed oit of T. (ii ( Proof. By Lemma.(iv we have lim Tx (.6 Sice T is comletely cotiuous ad { x} C is bouded there exists a subsequece { } k such that { } k x of { } Tx coverges. Therefore from (.6 { x k } coverges. Let q = lim x. k k By the cotiuity of T ad (.6 we have that Tq = q so q is a fixed oit of T. By Lemma.(i lim x q exists. The lim x k k q Thus lim x q Sice y as ad z = P a b x atx bu P( x (( ( a b x atx bu x a Tx b ux as it follows that lim y = q ad lim z = q. x For a = b the Theorem. ca be reduced to the two-ste iteratio with errors. Corollary.3 Let X be a uiformly covex Baach sace C a oemty closed covex oexasive retract of X with P as a oexasive retractio ad C X a comletely cotiuous oexasive oself-maig with FT (. Suose that { c} { d } { α} { β} are real sequeces i [ ] satisfyig (i α ( α β < lim if lim su < ad (ii c ( c d < lim if limsu <. For a give x C defie y = P( ( c d x ctx v x = (( α β α β P y Tx w. The { x } ad { y } coverge strogly to a fixed oit of T. I the ext result we rove the weak covergece of the three-ste iterative scheme (. for oexasive oself-maigs i a uiformly covex Baach sace satisfyig Oial s coditio. Theorem.4 Let X be a uiformly covex Baach sace which satisfies Oial s coditio C a oemty closed covex oexasive retract of X with P as a oexasive retractio ad C X a oexasive oself-maig with FT (. Suose that { α} { β} { a} { b} { c} { d } are sequeces of real umbers i [ ] with c [ ] ad α β [ ] for all ad b < d < β < ad = = = (i α ( α β < lim if lim su < ad (ii ( < limif c limsu c < ad limsua <. Let { x } be the sequece defied by three-ste iterative scheme (.. The{ x } coverges weakly to a fixed oit of T. Proof. By usig the same roof as i Theorem. it ca be show that lim Tx Sice X is uiformly covex ad { x } is bouded we may assume that x u weakly as without loss of geerality. By Lemma.3 we have u F( T. Suose that subsequeces { x k } ad { x m k } of { x } coverge weakly to u ad v resectively. From Lemma.3
ScieceAsia 3 (6 7 u v F(T. By Lemma.(i lim x u ad lim x v exist. It follows from Lemma.4 that u = v. Therefore {x } coverges weakly to fixed oit of T. Whe a = b i Theorem.4 we obtai the weak covergece theorem of the two-ste iteratio with errors as follows: Corollary.5 Let X be a uiformly covex Baach sace which satisfies Oial s coditio C a oemty closed covex oexasive retract of X with P as a oexasive retractio ad T : C X a oexasive oself-maig with FT (. Suose that {c } {d } {α } {β } are sequeces of real umbers i [] such that (i α ( α β < lim if limsu < ad (ii c ( c d < lim if limsu <. For a give x C defie y = P( ( c d x ctx v x = ( α β α β P y Tx w. ( The {x } coverges weakly to a fixed oit of T. ACKNOWLEDGEMENTS 8. Rhoades BE (994 Fixed oit iteratios for certai oliear maigs. J. Math. Aal. Al. 83 8-. 9. Rhoades BE ad Soltuz SM (4 The equivalece betwee Ma-Ishikawa iteratios ad multiste iteratios. Noliear Aal. 58 9-8.. Seter HF ad Dotso WG (974 Aroximatig fixed oits of oexasive maigs. Proc. Amer. Math. Soc. 44 375-8.. Ta KK ad Xu HK (993 Aroximatig fixed oits of oexasive maig by the Ishikawa iteratio rocess. J. Math. Aal. Al. 78 3-8.. Xu HK (99 Iequality i Baach saces with alicatios. Noliear Aal. 6 7-38. 3. Zeg LC (998 A ote o aroximatig fixed oits of oexasive maigs by the Ishikawa iteratio rocess. J. Math. Aal. Al. 6 45-5. 4. Noor MA ( New aroximatio schemes for geeral variatioal iequalities. J. Math. Aal. Al. 5 7-9. 5. Jug JS ad Kim SS (998 Strog covergece theorems for oexasive oself-maigs i Baach saces. Noliear Aal. 33 3-9. 6. Suatai S (5 Weak ad strog covergece criteria of Noor iteratios for asymtotically oexasive maigs. J. Math. Aal. Al. 3 56-7. 7. Shahzad N (5 Aroximatig fixed oits of o-self oexasive maigs i Baach saces. Noliear Aal. 6 3-9. 8. Oial Z (976 Weak covergece of successive aroximatios for oexasive mais. Bull. Amer. Math. Soc. 73 59-7. 9. Cho YJ Zhou HY ad Guo G (4 Strog covergece theorems for oexasive oself-maigs i Baach saces. Noliear Aal. 47 77-7.. Browder FE (968 Semicotractive ad semiaccretive oliear maigs i Baach saces. Bull. Amer. Math. Soc. 74 66-5. The author would like to thak the Thailad Research Fud (RGJ Project for their fiacial suort durig the rearatio of this mauscrit. The first author was suorted by the Royal Golde Jubilee Grat PHD/6/547 ad the Graduate School Chiag Mai Uiversity Thailad. REFERENCES. Chidume CE ad Chika Moor (999 Fixed oit iteratio for seudocotractive mas. Proc. Amer. Math. Soc. 7 63-7.. Hicks T ad Kubicek J (977 O the Ma iteratio rocess i a Hilbert sace. J. Math. Aal. Al. 59 498-54. 3. Ishikawa S (976 Fixed oits ad iteratio of a oexasive maig i a Baach sace. Proc. Amer. Math. Soc. 59 65-7. 4. Ishikawa S (974 Fixed oit by a ew iteratio. Proc. Amer. Math. Soc. 44 47-5. 5. Kruel M (997 O a iequality for oexasive maigs i uiformly covex Baach saces. Rostock. Math. Kolloq. 5 5-3. 6. Ma WR (953 Mea value methods i iteratio. Proc. Amer. Math. Soc. 4 56-. 7. Reich S (979 Weak covergece theorems for oexasive maigs i Baach saces. J. Math. Aal. Al. 67 74-6.