Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Volume 009, Artcle ID 391839, 9 pages do:10.1155/009/391839 Research Artcle A New Iteratve Method for Commo Fxed Pots of a Fte Famly of Noexpasve Mappgs Suwcha Imag ad Suthep Suata Departmet of Mathematcs, Faculty of Scece, Chag Ma Uversty, Chag Ma, 5000, Thalad Correspodece should be addressed to Suthep Suata, scmt005@chagma.ac.th Receved 16 December 008; Accepted 9 Aprl 009 Recommeded by Je Xao Let X be a real uformly covex Baach space ad C a closed covex oempty subset of X. Let {T } r be a fte famly of oexpasve self-mappgs of C. For a gve x 1 C, let{x } ad {x }, 1,,...,r, be sequeces defed x 0 x, x 1 a 1 1 T 1x 0 1 a 1 1 x 0, x a a 1 T 1x 1 a a 1 x,..., x 1 a r 1 T 1x r a r x r 1 x, 1, where a j a r r T r x r 1 a r r 1 T r 1x r 0, 1 for all j {1,,...,r}, N r 1 a r ad 1,,...,j. I ths paper, weak ad strog covergece theorems of the sequece {x } to a commo fxed pot of a fte famly of oexpasve mappgs T 1,,...,r are establshed uder some certa cotrol codtos. Copyrght q 009 S. Imag ad S. Suata. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. 1. Itroducto Let X be a real Baach space, C a oempty closed covex subset of X, adt : C C a mappg. Recall that T s oexpasve f Tx Ty x y for all x, y C. Let T : C C, 1,,...,r,be oexpasve mappgs. Let Fx T deote the fxed pots set of T, that s, Fx T : {x C : T x x}, ad let F : r Fx T. For a gve x 1 C, ad a fxed r N N deote the set of all postve tegers, compute the teratve sequeces {x 0 }, {x 1 }, {x },...,{x r } by x 0 x, x 1 a 1 1 T 1x 0 1 a 1 1 x 0
Iteratoal Joural of Mathematcs ad Mathematcal Sceces x x 1 x r a a 1 T 1x 1 a a 1 x,. a r r T r x r 1 a r r 1 T r 1x r a r 1 T 1x r a r r 1 a r 1 x, 1, 1.1 where a j 0, 1 for all j {1,,...,r}, N ad 1,,...,j.Ifa j : 0, for all N, j {1,,...,r 1} ad 1,,...,j, the 1.1 reduces to the teratve scheme x 1 S x, 1, 1. where S : a r r T r a r r 1 T r 1 a r 1 T 1 1 a r r a r r 1 a r 1 I, a r 0, 1 for all 1,,...,r ad N. If a j : 0, for all N, j {1,,...,r 1}, 1,,...,j ad a r : α, for all N for all 1,,...,r, the 1.1 reduces to the teratve scheme defed by Lu et al. 1 x 1 Sx, 1, 1.3 where S : α r T r α r 1 T r 1 α 1 T 1 1 α r α r 1 α 1 I, α 0 for all, 3,...,r ad 1 α r α r 1 α 1 > 0. They showed that {x } defed by 1.3 coverges strogly to a commo fxed pot of T,,,...,r, Baach spaces, provded that T,,,...,r satsfy codto A. The result mproves the correspodg results of Krk, Mat ad Saha 3 ad Setor ad Dotso 4. If r ada 1 : 0 for all N, the 1.1 reduces to a geeralzato of Ma ad Ishkawa terato gve by Das ad Debata 5 ad Takahash ad Tamura 6. Ths scheme dealts wth two mappgs: x 1 x 1 x a 1 1 T 1x a 1 a 1 1 1 a x, x, 1, 1.4 where {a 1 1 }, {a } are approprate sequeces 0, 1. The purpose of ths paper s to establsh strog covergece theorems a uformly covex Baach space of the teratve sequece {x } defed by 1.1 to a commo fxed pot of T 1,,...,r uder some approprate cotrol codtos the case that oe of T 1,,...,r s completely cotuous or semcompact or {T } r satsfes codto B. Moreover, weak covergece theorem of the teratve scheme 1.1 to a commo fxed pot of T 1,,...,r s also establshed a uformly covex Baach spaces havg the Opal s codto.
Iteratoal Joural of Mathematcs ad Mathematcal Sceces 3. Prelmares I ths secto, we recall the well-kow results ad gve a useful lemma that wll be used the ext secto. Recall that a Baach space X s sad to satsfy Opal s codto 7 f x x weakly as ad x / y mply that lm sup x x < lm sup x y. A fte famly of mappgs T : C C 1,,...,r wth F : r Fx T / s sad to satsfy codto B 8 f there s a odecreasg fucto f : 0, 0, wth f 0 0adf t > 0 for all t 0, such that max 1 r { x T x } f d x, F for all x C, where d x, F f{ x p : p F}. Lemma.1 see 9, Theorem. Let p>1, r>0 be two fxed umbers. The a Baach space X s uformly covex f ad oly f there exsts a cotuous, strctly creasg, ad covex fucto g : 0, 0,, g 0 0 such that λx 1 λ y p λ x p 1 λ y p wp λ g x y,.1 for all x, y B r {x X : x r},λ 0, 1, where w p λ λ 1 λ p λ p 1 λ.. Lemma. see 10, Lemma 1.6. Let X be a uformly covex Baach space, C a oempty closed covex subset of X, ad T : C C oexpasve mappg. The I T s demclosed at 0, that s, f x x weakly ad x Tx 0 strogly, the x Fx T. Lemma.3 see 11, Lemma.7. Let X be a Baach space whch satsfes Opal s codto ad let {x } be a sequece X. Letu, v X be such that lm x u ad lm x v exst. If {x k } ad {x mk } are subsequeces of {x } whch coverge weakly to u ad v, respectvely, the u v. Lemma.4. Let X be a uformly covex Baach space ad B r {x X : x r},r > 0. The for each N, there exsts a cotuous, strctly creasg, ad covex fucto g : 0, 0,, g 0 0 such that α x α x α 1 α g x 1 x,.3 for all x B r ad all α 0, 1 1,,..., wth α 1. Proof. Clearly.3 holds for 1,, by Lemma.1. Next, suppose that.3 s true whe k 1. Let x B r ad α 0, 1, 1,,...,kwth k α 1. The α k 1 / 1 k α x k 1 α k / 1 k α x k B r.bylemma.1, weobtathat α k 1 1 k α x k 1 α k 1 k α x k α k 1 1 k α x k 1 α k 1 k α x k..4
4 Iteratoal Joural of Mathematcs ad Mathematcal Sceces By the ductve hypothess, there exsts a cotuous, strctly creasg ad covex fucto g : 0, 0,, g 0 0 such that k 1 k 1 β y β y β1 β g y1 y.5 for all y B r ad all β 0, 1, 1,,...,k 1wth k 1 β 1. It follows that k k k αk 1 x k 1 α x α x 1 α 1 k α k k α x α k 1 x k 1 1 α 1 k α k α x k 1 α αk 1 x k 1 1 k α k α x α 1 α g x 1 x. α k x k 1 k α α k x k 1 k α α k x k 1 k α α 1 α g x 1 x α 1 α g x 1 x.6 Hece, we have the lemma. 3. Ma Results I ths secto, we prove weak ad strog covergece theorems of the teratve scheme 1.1 for a fte famly of oexpasve mappgs a uformly covex Baach space. I order to prove our ma results, the followg lemmas are eeded. The ext lemma s crucal for provg the ma theorems. Lemma 3.1. Let X be a Baach space ad C a oempty closed ad covex subset of X. Let{T } r be a fte famly of oexpasve self-mappgs of C. Leta j 0, 1 for all j {1,,...,r}, N ad 1,,...,j. For a gve x 1 C, let the sequece {x } be defed by 1.1. IfF /, the x 1 p x p for all N ad lm x p exsts for all p F. Proof. Let p F. For each 1, we ote that x 1 p a 1 1 T 1x 1 a 1 1 x p a 1 1 T1 x p 1 a 1 x 1 p a 1 1 x p 1 a 1 x 1 p 3.1 x p.
Iteratoal Joural of Mathematcs ad Mathematcal Sceces 5 It follows from 3.1 that x p a a 1 T 1x 1 a a 1 x p a T x 1 p a 1 T1 x p 1 a a x 1 p a x 1 p a 1 x p 1 a a 1 x p 3. x p. By 3.1 ad 3., we have x 3 p a 3 3 T 3x a 3 a 3 1 T 1x a 3 T 3 x p a 3 T x 1 p a 3 T 1 x p 3 3 1 a 3 3 a 3 a 3 x 1 p a 3 x p a 3 x 1 p a 3 x p 1 1 a 3 3 a 3 a 3 x 1 p 1 a 3 3 a 3 a 3 1 1 x p 3.3 x p. By cotug the above argumet, we obta that x p x p 1,,...,r. 3.4 I partcular, we get x 1 p x p for all N, whch mples that lm x p exsts. Lemma 3.. Let X be a uformly covex Baach space ad C a oempty closed ad covex subset of X.Let{T } r be a fte famly of oexpasve self-mappgs of C wth F / ad a j 0, 1 for all j {1,,...,r}, N ad 1,,...,j such that j a j are 0, 1 for all j {1,,...,r} ad N. For a gve x 1 C,let{x } be defed by 1.1.If0 < lm f a r a r r 1 a r 1 < 1, the lm T x 1 x 0 for all 1,,...,r, lm T x x 0 for all 1,,...,r, lm x x 0 for all 1,,...,r. lm sup a r r Proof. Let p F, bylemma 3.1, sup x p <. Choose a umber s>0such that sup x p <s, t follows by 3.4 that {x p}, {T x 1 p} B s, for all {1,,...,r}.
6 Iteratoal Joural of Mathematcs ad Mathematcal Sceces By Lemma.4, there exsts a cotuous strctly creasg covex fucto g : 0, 0,, g 0 0 such that α x α x α 1 α g x 1 x, 3.5 for all x B s, 1,,...,r, α 0, 1 1,,..., wth α 1. By 3.4 ad 3.5, we have for x 1 p a r r T r x r 1 a r r 1 T r 1x r a r 1 T 1x r a r r 1 a r 1 x p a r r T r x r 1 p a r r 1 T r 1 x r p a r 1 T1 x p r a r r 1 a r x 1 p a r r T a r r 1 a r 1 g x 1 x a r x r 1 p a r x r p a r x p r r 1 r a r r 1 a r x 1 p a r r T a r r 1 a r 1 g x 1 x a r x p a r x p a r x p r r 1 r a r r 1 a r 1 x p a r r T a r r 1 a r 1 g x 1 x x p a r 1 r a r r 1 a r 1 1 g T x 1 x. 3.6 Therefore a r r T a r r 1 a r 1 g x 1 x x p x 1 p 3.7 for all 1,,...,r. Sce 0 < lm f a r t mples by Lemma 3.1 that lm g T x 1 cotuous at 0 wth g 0 0, t follows that lm T x 1 lm sup a r a r a r r r 1 1 < 1, x 0. Sce g s strctly creasg ad x 0 for all 1,,...,r.
Iteratoal Joural of Mathematcs ad Mathematcal Sceces 7 For {1,,...,r}, we have T x x T x T x 1 T x 1 x x x 1 T x 1 x 1 j 1 a 1 j T j x j 1 x T x 1 x. 3.8 It follows from that T x x 0 as. 3.9 For {1,,...,r}, t follows from that x x a j j 1 T j x j 1 x 0 as. 3.10 Theorem 3.3. Let X be a uformly covex Baach space ad C a oempty closed ad covex subset of X. Let{T } r be a fte famly of oexpasve self-mappgs of C wth F /. Let the sequece {a j } 1 be as Lemma 3.. For a gve x 1 C, let sequeces {x } ad {x } 0, 1,...,r be defed by 1.1. If oe of {T } r s completely cotuous the {x } ad {x j } coverge strogly to a commo fxed pot of {T } r for all j 1,,...,r. Proof. Suppose that T 0 s completely cotuous where 0 {1,,...,r}. The there exsts a subsequece {x k } of {x } such that {T 0 x k } coverges. Let lm k T 0 x k q for some q C. ByLemma 3., lm T 0 x x 0. It follows that lm k x k q. Aga by Lemma 3., we have lm T x x 0for all 1,,...,r. It mples that lm k T x k q. By cotuty of T,wegetT q q, 1,,...,r.Soq F.ByLemma 3.1, lm x q exsts, t follows that lm x q 0. By Lemma 3., we have lm x j x 0 for each j {1,,...,r}. It follows that lm x j q for all j 1,,...,r. Theorem 3.4. Let X be a uformly covex Baach space ad C a oempty closed ad covex subset of X. Let{T } r be a fte famly of oexpasve self-mappgs of C wth F /. Let the sequece {a j } 1 be as Lemma 3.. For a gve x 1 C, let sequeces {x } ad {x } 0, 1,...,r be defed by 1.1. If the famly {T } r satsfes codto B the {x } ad {x j } coverge strogly to a commo fxed pot of {T } r for all j 1,,...,r. Proof. Let p F. The by Lemma 3.1, lm x p exsts ad x 1 p x p for all 1. Ths mples that d x 1,F d x,f for all 1, therefore, we get lm d x,f exsts. By Lemma 3., we have lm T x x 0 for each 1,,...,r. It follows, by the codto B that lm f d x,f 0. Sce f s odecreasg ad f 0 0, therefore, we get lm d x,f 0. Next we show that {x } s a Cauchy sequece. Sce
8 Iteratoal Joural of Mathematcs ad Mathematcal Sceces lm d x,f 0, gve ay ɛ>0, there exsts a atural umber 0 such that d x,f <ɛ/ for all 0. I partcular, d x 0,F <ɛ/. The there exsts q F such that x 0 q <ɛ/. For all 0 ad m 1, t follows by Lemma 3.1 that x m x x m q x q x0 q x0 q <ɛ. 3.11 Ths shows that {x } s a Cauchy sequece C, hece t must coverge to a pot of C. Let lm x p. Sce lm d x,f 0adF s closed, we obta p F. By Lemma 3., lm x j x 0 for each j {1,,...,r}. It follows that lm x j p for all j 1,,...,r. I Theorem 3.4,fa j : 0, for all N, j {1,,...,r 1} ad 1,,...,j,weobta the followg result. Corollary 3.5. Let X be a uformly covex Baach space ad C a oempty closed ad covex subset of X. Let{T } r be a fte famly of oexpasve self-mappgs of C wth F / ad a r 0, 1 for all 1,,...,r ad N such that r a r are 0, 1 for all N. For a gve x 1 C, let the sequece {x } be defed by 1.. If the famly {T } r satsfes codto B ad 0 < lm f a r lm sup a r a r a r r r 1 1 < 1, the the sequece {x } coverges strogly to a commo fxed pot of {T } r. Remark 3.6. I Corollary 3.5, fa r α, for all N ad for all 1,,...,r, the teratve scheme 1. reduces to the teratve scheme 1.3 defed by Lu et al. 1 ad we obta strog covergece of the sequece {x } defed by Lu et al. whe {T } r satsfes codto B whch s dfferet from the codto A defed by Lu et al. ad we ote that the result of Seter ad Dotso 4 s a specal case of Theorem 3.4 whe r 1. I the ext result, we prove weak covergece for the teratve scheme 1.1 for a fte famly of oexpasve mappgs a uformly covex Baach space satsfyg Opal s codto. Theorem 3.7. Let X be a uformly covex Baach space whch satsfes Opal s codto ad C a oempty closed ad covex subset of X. Let{T } r be a fte famly of oexpasve self-mappgs of C wth F /. For a gve x 1 C,let{x } be the sequece defed by 1.1. If the sequece {a j } 1 s as Lemma 3., the the sequece {x } coverges weakly to a commo fxed pot of {T } r. Proof. By Lemma 3., lm T x x 0 for all 1,,...,r. Sce X s uformly covex ad {x } s bouded, wthout loss of geeralty we may assume that x u weakly as for some u C.ByLemma., we have u F. Suppose that there are subsequeces {x k } ad {x mk } of {x } that coverge weakly to u ad v, respectvely. From Lemma., we have u, v F. ByLemma 3.1, lm x u ad lm x v exst. It follows from Lemma.3 that u v. Therefore {x } coverges weakly to a commo fxed pot of {T } r. For a j : 0, for all N, j {1,,...,r 1} ad 1,,...,j Theorem 3.7, we obta the followg result.
Iteratoal Joural of Mathematcs ad Mathematcal Sceces 9 Corollary 3.8. Let X be a uformly covex Baach space whch satsfes Opal s codto ad C a oempty closed ad covex subset of X. Let{T } r be a fte famly of oexpasve self-mappgs of C wth F / ad a r 0, 1 for all 1,,...,r ad N such that r a r are 0, 1 for all N. For a gve x 1 C, let{x } be the sequece defed by 1.. If0 < lm f a r lm sup a r a r a r r r 1 1 < 1, the the sequece {x } coverges weakly to a commo fxed pot of {T } r. Remark 3.9. I Corollary 3.8, fa r α, for all N ad for all 1,,...,r, the we obta weak covergece of the sequece {x } defed by Lu et al. 1. Ackowledgmets The authors would lke to thak the Commsso o Hgher Educato, the Thalad Research Fud, the Thaks Uversty, ad the Graduate School of Chag Ma Uversty, Thalad for ther facal support. Refereces 1 G. Lu, D. Le, ad S. L, Approxmatg fxed pots of oexpasve mappgs, Iteratoal Joural of Mathematcs ad Mathematcal Sceces, vol. 4, o. 3, pp. 173 177, 000. W. A. Krk, O successve approxmatos for oexpasve mappgs Baach spaces, Glasgow Mathematcal Joural, vol. 1, o. 1, pp. 6 9, 1971. 3 M. Mat ad B. Saha, Approxmatg fxed pots of oexpasve ad geeralzed oexpasve mappgs, Iteratoal Joural of Mathematcs ad Mathematcal Sceces, vol. 16, o. 1, pp. 81 86, 1993. 4 H. F. Seter ad W. G. Dotso Jr., Approxmatg fxed pots of oexpasve mappgs, Proceedgs of the Amerca Mathematcal Socety, vol. 44, o., pp. 375 380, 1974. 5 G. Das ad J. P. Debata, Fxed pots of quasoexpasve mappgs, Ida Joural of Pure ad Appled Mathematcs, vol. 17, o. 11, pp. 163 169, 1986. 6 W. Takahash ad T. Tamura, Covergece theorems for a par of oexpasve mappgs, Joural of Covex Aalyss, vol. 5, o. 1, pp. 45 56, 1998. 7 Z. Opal, Weak covergece of the sequece of successve approxmatos for oexpasve mappgs, Bullet of the Amerca Mathematcal Socety, vol. 73, o. 4, pp. 591 597, 1967. 8 C. E. Chdume ad N. Shahzad, Strog covergece of a mplct terato process for a fte famly of oexpasve mappgs, Nolear Aalyss, vol. 6, o. 6A, pp. 1149 1156, 005. 9 H. K. Xu, Iequaltes Baach spaces wth applcatos, Nolear Aalyss: Theory, Methods & Applcatos, vol. 16, o. 1, pp. 117 1138, 1991. 10 Y. J. Cho, H. Zhou, ad G. Guo, Weak ad strog covergece theorems for three-step teratos wth errors for asymptotcally oexpasve mappgs, Computers & Mathematcs wth Applcatos, vol. 47, o. 4-5, pp. 707 717, 004. 11 S. Suata, Weak ad strog covergece crtera of Noor teratos for asymptotcally oexpasve mappgs, Joural of Mathematcal Aalyss ad Applcatos, vol. 311, o., pp. 506 517, 005.