Mathematica Moravica Vol. 19-2 2015, 49 64 O Weak ad Strog Covergece Theorems for a Fiite Family of Noself I-asymptotically Noexpasive Mappigs Birol Güdüz ad Sezgi Akbulut Abstract. We prove the weak ad strog covergece of S iterative scheme to a commo fixed poit of a family of oself asymptotically I-oexpasive mappigs {T i} N i ad a family of oself asymptotically oexpasive mappigs {I i} N i, defied o a oempty closed covex subset of a Baach space. Our scheme coverges faster tha Ma ad Ishikawa iteratio for cotractios. Our weak covergece theorem is proved uder more geeral setup of space as differet from weak covergece theorems proved i previously. 1. Itroductio Let X be a real ormed liear space ad K be a oempty subset of X. Let T be a self-mappig of K. 1 T is said to be oexpasive if T x T y x y holds for all x, y K. 2 T is said to be asymptotically oexpasive if there exists a sequece {k } [0,, with lim k = 0, such that T x T y 1 + k x y for all x, y K ad 1. 3 T is called uiformly L-Lipschitzia if there exists costat L > 0 such that T x T y L x y for all x, y K ad positive iteger 1. Beig a importat geeralizatio of oexpasive mappig, the cocept of asymptotically oexpasive self-mappig was proposed by Goebel ad Kirk [1] i 1972. I [1], it was proved that if X is uiformly covex, ad K is a bouded closed ad covex subset of X, the every asymptotically oexpasive self mappig has a fixed poit. Iterative techiques for asymptotically oexpasive self-mappig i Baach spaces icludig Ma type ad Ishikawa type iteratios processes have bee studied extesively by various authors. However, if the domai of T, 2010 Mathematics Subject Classificatio. Primary: 47H10; Secodary: 54H25. Key words ad phrases. Noself asymptotically I-oexpasive mappigs; Iterative scheme; Kadec Klee property; Coditio B; Commo fixed poit. 49 c 2015 Mathematica Moravica
50 A Iterative Scheme for Commo Fixed Poits... DT, is a proper subset of E ad this is the case i several applicatios, ad T maps DT ito E, the the iteratio processes of Ma type ad Ishikawa type ad their modificatios itroduced may fail to be well defied. A subset K of X is said to be a retract of X if there exists a cotiuous map P : X K such that P x = x, for all x K. Every closed covex subset of a uiformly covex Baach space is a retract. A map P : X K is said to be a retractio if P 2 = P. It follows that, if a map P is a retractio, the P y = y for all y i the rage of P. I 2003, Chidume, Ofoedu ad Zegeye [15] further geeralized the cocept of asymptotically oexpasive self-mappig, ad proposed the cocept of oself asymptotically oexpasive mappig, which is defied as follows: Defiitio 1.1 [15]. Let K be a oempty subset of a real ormed space X ad P : X K be a oexpasive retractio of X oto K. 1 Mappig T : K X is called asymptotically oexpasive if there exists a sequece {k } [0, with k 0 as such that for ay positive iteger ad all x, y K T P T 1 x T P T 1 y 1 + k x y. 2 No-self mappig T : K X is said to be uiformly L-Lipschitzia if there exists a costat L > 0 such that for all x, y K T P T 1 x T P T 1 y L x y. Note that if P is a idetity mappig, the the above defiitios reduce to those of a self-mappig T. By usig the followig iterative algorithm: x +1 = P 1 α x + α T P T 1 x, x 1 K, 1, Chidume et al. [15] gave some strog ad weak covergece theorems for oself asymptotically oexpasive mappigs i uiformly covex Baach spaces. They also established a demiclosedess priciple. Recetly, Temir [4, 5] itroduced the followig defiitios: Defiitio 1.2. Let K be a oempty subset of real ormed liear space E. For self mappigs T, I : K K, 1 T is called be asymptotically I-oexpasive [4, 5] o K if there exists a sequece {v } [0, with lim v = 0 such that T x T y 1 + v I x I y for all x, y K ad 1. 2 T is called I-uiformly Lipschitz if there exists Γ > 0 such that T x T y Γ I x I y, x, y K ad 1.
Birol Güdüz ad Sezgi Akbulut 51 I 2007, Agarwal et al.[2] itroduced the followig iteratio scheme: x 1 = x K, x +1 = 1 α T x + α T y, y = 1 β x + β T x, 1. where {α } ad {β } are i 0, 1. They showed that this scheme coverges at a rate same as that of Picard iteratio. Also, Agarwal et al.[2] showed that this scheme coverges faster tha Ma ad Ishikawa iteratio for cotractios. Kha ad Abbas [3] geeralized correspodig theorems of Agarwal et al. [2] to the case of two mappigs. Recetly, Temir [4] studied the weak ad strog covergece of implicit iteratio process to a commo fixed poit for a fiite family of asymptotically I-oexpasive mappigs i Baach spaces. I [7], Gu proved some covergece theorems of o-implicit iteratio process with errors for a fiite families of I-asymptotically oexpasive mappigs i Baach spaces. I [6], Yag ad Xie first itroduced the class of I-asymptotically oexpasive oself-maps, the proved covergece theorems of a iteratio process to a commo fixed poits of a fiite family of I-asymptotically oexpasive oself-mappigs. Defiitio 1.3 [6]. Let K be a oempty subset of a real ormed space X ad P : X K be a oexpasive retractio of X oto K. Let T, I : K X be two mappigs. 1 T is called oself I-asymptotically oexpasive if there exists sequece {u } [0, with lim u = 0, such that T P T 1 x T P T 1 y 1 + u IP I 1 x IP I 1 y for all x, y K ad 1. 2 T is said to be uiformly Γ-Lipschitzia if there exists Γ > 0 such that for all x, y K ad all positive iteger T P T 1 x T P T 1 y Γ IP I 1 x IP I 1 y. Motivated ad ispired by the above works, i this paper, we itroduce a ew explicit iterative sequece {x } as follows: Let K be a oempty subset of a Baach space X. Let {T i } N i : K X be N oself I i -asymptotically oexpasive mappigs ad {I i } N i=1 : K X be N oself asymptotically oexpasive mappigs. Let {α } ad {β } be two real sequeces i [0, 1]. The the sequece {x } is geerated as follows: 1 x +1 = P 1 α T i P T i 1 x + α I i P I i 1 y y = P 1 β x + β T i P T i 1, 1, x
52 A Iterative Scheme for Commo Fixed Poits... where = k 1N + i, i = i J := {1, 2,..., N} is a positive iteger ad k as. Thus, 1 ca be expressed i the followig form: x +1 = P 1 α T i P T i k 1 x + α I i P I i k 1 y y = P 1 β x + β T i P T i k 1, 1. x I this paper, we get some weak ad strog covergece theorems of this iterative sequece for a family of oself asymptotically I-oexpasive mappigs {T i } N i ad a family of oself asymptotically oexpasive mappigs {I i } N i. Our theorems also are ew i case of self-mappigs. 2. Prelimiaries ad otatios I order to prove the mai results of this work, we eed the followig some cocepts ad results: A Baach space X is said to satisfy Opial s coditio if, for ay sequece {x } i X, x x implies that lim sup x x < lim sup x y for all y X with y x, where x x meas that {x } coverges weakly to x. A Baach space X is said to have a Fréchet differetiable orm [10] if for all x S X = {x X : x = 1} x + ty x lim t 0 t exists ad is attaied uiformly i y S X. A mappig T with domai DT ad rage RT i X is said to be demiclosed at p if wheever {x } is a sequece i DT such that x x DT ad T x p the T x = p. A mappig T : K K is said to be semicompact if, for ay bouded sequece {x } i K such that x T x 0 as, there exists a subsequece say {x j } of {x } such that {x j } coverges strogly to some x i K. The mappig T : K K with F T is said to satisfy coditio A [11] if there is a odecreasig fuctio f : [0, [0, with f 0 = 0, f t > 0 for all t 0, such that x T x f d x, F T for all 1. Seter ad Dotso [11] poited out that every cotiuous ad semicompact mappig must satisfy Coditio A. Two mappigs T, S : K K are said to satisfy coditio A [12] if there is a odecreasig fuctio f : [0, [0, with f 0 = 0, f t > 0 for all t 0, such that 1 2 x T 1x + x T 2 x f d x, F
Birol Güdüz ad Sezgi Akbulut 53 for all x K, where d x, F = if { x p : p F := F T F S}. We modify these defiitios for our case as follows: A family {T i } N i : K X be N oself I i -asymptotically oexpasive mappigs ad a family {I i } N i : K X be N oself asymptotically oexpasive mappigs with F = N i=1 F T i F I i are said to satisfy coditio B if there is a odecreasig fuctio f : [0, [0, with f 0 = 0, f t > 0 for all t 0, such that max 1 1 i N 2 x T ix + x I i x f dx, F. Lemma 2.1 [8]. Let {a }, {b } ad {δ } be sequeces of oegative real umbers satisfyig the iequality a +1 1 + δ a + b, 1. If b < ad δ <, the lim a exists. Lemma 2.2 [9]. Suppose that X is a uiformly covex Baach space ad 0 < p t q < 1 for all 1. Also, suppose that {x } ad {y } are sequeces of X such that lim sup x r, lim sup y r ad hold for some r 0. The lim x y = 0. lim t x + 1 t y = r Lemma 2.3 [13]. Let X be a uiformly covex Baach space ad K a covex subset of X. The there exists a strictly icreasig cotiuous covex fuctio φ : [0, [0, with φ0 = 0 such that for each S : K K with Lipschitz costat L, αsx + 1 αsy S [αx + 1 αy] Lφ 1 x y + 1 L Sx Sy for all x, y K ad 0 < α < 1. Lemma 2.4 [15]. Let X be a real uiformly covex Baach space, K a oempty closed subset of E, ad let T : K X be a asymptotically oexpasive mappig with a sequece {k } [1, ad k 1 as, the E T is demiclosed at zero, where E is a idetity mappig. A Baach space X is said to have the Kadec Klee property if, for every sequece {x } i X, x x ad x x imply x x 0. Every locally uiformly covex space has the Kadec Klee property. I particular, L p spaces, 1 < p < have this property. Lemma 2.5 [13]. Let X be a real reflexive Baach space such that its dual X has Kadec Klee property. Let {x } be a bouded sequece i X ad q 1, q 2 ω w {x } weak limit set of {x }. Suppose lim αx + 1 αq 1 q 2 exists for all α [0, 1]. The q 1 = q 2.
54 A Iterative Scheme for Commo Fixed Poits... 3. Strog covergece theorems First, we prove the followig lemmas. Lemma 3.1. Let X be a real Baach space, K be a oempty closed covex subset of X which is also a oexpasive retract with retractio P. Let {T i } N i : K X be N oself I i -asymptotically oexpasive mappigs with sequeces l i [0, such that l i < ad {I i } N i : K X be N oself asymptotically oexpasive mappigs with sequeces k i [0, such that k i <. Suppose that for ay give x 1 K, the sequece {x } is geerated by 1 ad F = N F T i F I i. The i lim x q exists for all q F. ii lim d x, F exists, where d x, F = if p F x p. Proof. i Let q F. Settig { l = max l 1, l 2,..., l N Sice have 2 l i <, k i } <, so i=1 { }, k = max k 1, k 2,..., k N. l <, k <. Usig 1, we y q = P 1 β x + β T i P T i 1 x P q 1 β x + β T i P T i 1 x q Ti 1 β x q + β P T i 1 x q 1 β x q + β 1 + l I i P I i 1 x q 1 β x q + β 1 + l 1 + k x q = 1 + β l + β k + β l k x q 1 + l + k + l k x q = 1 + l 1 + k x q, ad so x +1 q = P 1 α T i P T i 1 x + α I i P I i 1 y P q 1 α T i P T i 1 x + α I i P I i 1 y q 1 α T i P T i 1 Ii x q + α P I i 1 y q 1 α 1 + l I i P I i 1 x q
Birol Güdüz ad Sezgi Akbulut 55 3 + α 1 + k y q 1 α 1 + l 1 + k x q + α 1 + k 1 + l 1 + k x q = 1 + k 1 + l [1 α x q + α 1 + k x q ] 1 + k 1 + l 1 + α k x q 1 + l + 2k + 2l k + k 2 + l k 2 x q = 1 + δ x q, where δ = l + 2k + 2l k + k 2 + l k. 2 Sice l < ad k <, we obtai δ <. By Lemma 2.1 ad 3, we get lim x q exists. ii Takig ifimum over all q F i 3, we have 4 d x +1, F 1 + δ d x, F. It follows from 4 ad Lemma 2.1 that lim d x, F exists. Lemma 3.2. Let X be a real uiformly covex Baach space, K be a oempty closed covex subset of X which is also a oexpasive retract with retractio P. Let {T i } N i : K X be N oself I i -asymptotically oexpasive mappigs with sequeces [0, such that < ad l i k i {I i } N i : K X be N oself asymptotically oexpasive mappigs with sequeces [0, such that <. Let {α } ad {β } be sequeces i [a, 1 a] for some a 0, 1. Suppose that for ay give x 1 K, the sequece {x } is geerated by 1 ad F = N F T i F I i. The, k i i=1 lim x T i x = lim x I i x = 0 for all i J. Proof. By Lemma 3.1, lim x q exists for all q F. Call it c. Takig lim sup o both sides i the iequality 2, we get 5 lim sup y q lim sup x q = lim x q = c. Thus T i P T i 1 x q 1 + l I i P I i 1 x q for all 1 implies that 6 lim sup 1 + l 1 + k x q T i P T i 1 x q c. l i
56 A Iterative Scheme for Commo Fixed Poits... Next, I i P I i 1 y q 1 + k y q gives by 5 that 7 lim sup I i P I i 1 y q c. From 3, we have x +1 q 1 α T i P T i 1 x q + α I i P I i 1 y q 8 Sice 1 + δ x q. δ < ad lim x +1 q = c, lettig i the iequality 8, we have 9 lim 1 α T i P T i 1 x q + α I i P I i 1 y q = c. By usig 6, 7, 9 ad Lemma 2.2, we get 10 lim T i P T i 1 x I i P I i 1 y = 0. Usig 1, we have x +1 q 1 α T i P T i 1 x + α I i P I i 1 y q T i P T i 1 Ti x q + α P T i 1 x I i P I i 1 11 y Takig the limif o both sides i 11, by 10 ad lim x +1 q = c, we obtai 12 c lim if T i P T i 1 x q. This together with 6 implies that 13 lim T i P T i 1 x q = c. Further, Ti P Ti 1 x q T i P T i 1 x I i P I i 1 y + I i P I i 1 y q T i P T i 1 x I i P I i 1 y gives that + 1 + k y q 14 c lim if y q. By 5 ad 14, we obtai 15 lim y q = c.
Birol Güdüz ad Sezgi Akbulut 57 From 3, we have y q 1 β x q + β T i P T i 1 x q 1 + l 1 + k x q. Thus from 15, we get 16 lim 1 β x q + β T i P T i 1 x q = c Hece, usig lim x q = c, 13, 16 ad Lemma 2.2, we obtai 17 lim x T i P T i 1 x = 0. By y = P 1 β x + β T i P T i 1 x ad 17, sice P is oexpasive mappig, we have y x = P 1 β x + β T i P T i 1 x P x 1 β x + β T i P T i 1 x x 18 β Ti P T i 1 x x 0. Usig a similar method, together with 10 ad 17, we have x +1 x P 19 1 α T i P T i 1 x + α I i P I i 1 y P x 1 α T i P T i 1 x + α I i P I i 1 y x T i P T i 1 x x + α Ti P T i 1 x I i P I i 1 y 0 as. It follows from 18 ad 19 that 20 lim x +1 y = 0. Furthermore, from 19 ad x I i P I i 1 x y T i P T i 1 x + T i P T i 1 x I i P I i 1 y 0, we obtai 21 x +1 I i P I i 1 y x+1 x + x I i P I i 1 y 0.
58 A Iterative Scheme for Commo Fixed Poits... Notice that x I i P I i 1 x x x +1 + x +1 I i P I i 1 y + I i P I i 1 y I i P I i 1 x x x +1 + x +1 I i P I i 1 y qquad + 1 + k y x. It follows from 18, 19 ad 21 that 22 lim x I i P I i 1 x = 0. Sice a asymptotically oexpasive mappig with respect to P must be uiformly Lipschitzia with respect to P, the we have x +1 I i x +1 x +1 I i P I i x +1 + I i P I i x +1 I i x +1 x +1 I i P I i x +1 + L I i P I i 1 x +1 x +1 x +1 I i P I i x +1 Ii + L P I i 1 x +1 I i P I i 1 y + I i P I i 1 y x +1 x +1 I i P I i x +1 + L 2 x +1 y + L I i P I i 1 y x +1. Takig lim o both sides i the above iequality, the from 20, 21 ad 22 we get lim x I i x = 0. Now we make use of the fact that every oself I asymptotically oexpasive mappig with respect to P must be I-uiformly Lipschitz with respect to P. The x +1 T i x +1 x +1 T i P T i x +1 + T i P T i x +1 T i P T i x + T i P T i x T i x +1 x +1 T i P T i x +1 + Γ I i P I i x +1 I i P I i x + Γ I i P I i x I i x +1 x +1 T i P T i x +1 + ΓL x +1 x + ΓL I i P I i 1 x x +1 x +1 T i P T i x +1 + ΓL x +1 x
yields by 17, 19 ad 22 Birol Güdüz ad Sezgi Akbulut 59 Ii + ΓL P I i 1 x x + x x +1 lim x T i x = 0. We ow prove strog covergece theorems of the iterative scheme 1 i Baach space ad uiformly covex Baach spaces. Theorem 3.1. Let X be a real Baach space, K be a oempty closed covex subset of X which is also a oexpasive retract with retractio P. Let {T i } N i : K X be N oself I i -asymptotically oexpasive mappigs with sequeces l i [0, such that l i < ad {I i } N i : K X be N oself asymptotically oexpasive mappigs with sequeces k i [0, such that k i <. Suppose that for ay give x 1 K, the sequece {x } is geerated by 1 ad F = N F T i F I i. The, {x } coverges strogly to a commo fixed poit of {T i } N i ad {I i } N i if ad oly if lim if d x, F = 0. Proof. The ecessity of Theorem is obvious. Let us proof the sufficiecy part of theorem. Let q F. The by Lemma 2.1 ii, lim d x, F exists ad by assumptio lim if d x, F = 0, we obtai lim d x, F = 0. Next, we show that {x } is a Cauchy sequece i K. Notice that from 3 for ay q K, we have x +m q exp δ x q < M x q for all m,, where M = exp δ +1 <. Sice lim d x, F = 0, for ay give ε > 0, there exists a positive iteger N 0 such that for all N 0, d x, F < ε 2M. There exists q 1 F such that x 0 q 1 < ε 2M. Hece, for all N 0 ad m 1, we have i=1 x +m x x +m q 1 + x q 1 M x 0 q 1 + M x 0 q 1 2M x 0 q 1 2M ε 2M = ε
60 A Iterative Scheme for Commo Fixed Poits... which implies that {x } is a Cauchy sequece i K. Thus, the completeess of X implies that {x } is coverget. Assume that {x } coverges to a poit q. The q K, because K is closed subset of X. Ad lim d x, F = 0 implies that lim d q, F = 0. F is closed, thus q F. This completes the proof. Applyig Theorem 3.1, we obtai a strog covergece theorem of the iterative scheme 1 uder the coditio B as follows. Theorem 3.2. Let X be a real uiformly covex Baach space, K be a oempty closed covex subset of X which is also a oexpasive retract with retractio P. Let {T i } N i : K X be N oself I i -asymptotically { oexpasive mappigs with sequeces k i l i } [0, such that l i < ad {I i } N i : K X be N oself asymptotically oexpasive mappigs with sequeces [0, such that k i <. Let {α } ad {β } be sequeces i [a, 1 a] for some a 0, 1. Suppose that for ay give x 1 K, the sequece {x } is geerated by 1 ad F = N F T i F I i. If {T i } N i ad {I i } N i satisfy the coditio B, the {x } coverges strogly to a commo fixed poit of {T i } N i ad {I i } N i. Proof. As is proved i Lemma 3.2 that 23 lim x T i x = lim x I i x = 0 for all i J. Because {T i } N i ad {I i } N i satisfy Coditio B, therefore f dx, F max 1 1 i N 2 x T ix + x I i x. Thus it follows from 23 that lim f dx, F = 0. Sice f : [0, [0, is a odecreasig fuctio satisfyig f0 = 0, fr > 0 for all r 0,, therefore we have lim dx, F = 0. Now all the coditios of Theorem 3.1 are satisfied, therefore by its coclusio {x } coverges strogly to a poit of F. i=1 4. Weak covergece theorems Followig lemma is the key for our weak covergece result. Lemma 4.1. Let X be a real uiformly covex Baach space, K be a oempty closed covex subset of X which is also a oexpasive retract with retractio P. Let {T i } N i : K X be N oself I i -asymptotically oexpasive mappigs with sequeces [0, such that < ad l i {I i } N i : K X be N oself asymptotically oexpasive mappigs with l i
{ sequeces k i Birol Güdüz ad Sezgi Akbulut 61 } [0, such that k i <. Let {α } ad {β } be sequeces i [a, 1 a] for some a 0, 1. Suppose that for ay give x 1 K, the sequece {x } is geerated by 1 ad F = N F T i F I i. The lim tx + 1 t q 1 q 2 exists for all t [0, 1] ad q 1, q 2 F. Proof. By Lemma 3.1, lim x q = c for all q F. Sice {x } is bouded, there exists R > 0 such that {x } C : B R 0 K, where B R 0 = {x X : x R}. The C is a oempty closed covex bouded subset of X. Let a t = tx + 1 t p 1 p 2. The lim a 0 = p 1 p 2 ad from Lemma 3.1, lim a 1 = x p 2 exists. It ow remais to prove the lemma for t 0, 1. Defie U, W : C C by W x = P U x = P i=1 1 β x + β T i P T i 1 x 1 α T i P T i 1 x + α I i P I i 1 U x, 1 for all x K. The for all x, y K, we have P 1 β U x U y x + β T i P T i 1 x P 1 β y + β T i P T i 1 y 1 β x y + β T i P T i 1 x T i P T i y 1 1 β x y + β 1 + l I i P I i 1 x I i P I i 1 y 1 β x y + β 1 + l 1 + k x y 1 + l 1 + k x y ad so P 1 α T i P T i 1 x + α I i P I i 1 U x W x W y P 1 α T i P T i 1 y + α I i P I i 1 U y 1 α T i P T i 1 x T i P T i 1 y +α I i P I i 1 U x I i P I i 1 U y 1 α T i P T i 1 x T i P T i 1 y Ii + α P I i 1 U x I i P I i 1 U y 1 α 1 + l 1 + k x y + α 1 + k U x U y 1 α 1 + l 1 + k x y + α 1 + k 1 + l 1 + k x y
62 A Iterative Scheme for Commo Fixed Poits... 24 Settig 1 + l 1 + k [1 α + α 1 + k ] x y 1 + l 1 + k 1 + α k x y 1 + l 1 + k 2 x y. 25 R,m = W +m 1 W +m 2 W,, m 1, ad 26 b,m = R,m tx + 1 t q 1 tr,m x + 1 t R,m q 1. From 24 ad 25, we have 27 R,m x R,m y L x y for all x, y C, where L = +m 1 R,m q = q for all q F. Thus j= a t = tx + 1 t q 1 q 2 1 + l j 1 + k j 2 ad R,m x = x +m, b,m + R,m tx + 1 t q 1 q 2 b,m + L a t. It follows from 26, 27 ad Lemma 2.3 that b,m φ 1 x q 1 L 1 x +m q 1. By Lemma 3.1 ad lim L = 1, we have lim,m b,m = 0 ad so lim sup a t lim b,m + lim if L a t = lim if a t.,m This completes the proof. Fially, we give our weak covergece covergece theorem of the iterative scheme 1. But, we fistly wat to draw the attetio of the reader towards the followig remark. Remark 4.1. I [14], it is poit out that there exist uiformly covex Baach spaces which have either a Fréchet differetiable orm or the Opial property but their duals do have the Kadec Klee property. Ad the duals of reflexive Baach spaces with Fréchet differetiable orms or the Opial property have the Kadec Klee property. Theorem 4.1. Let X be a real uiformly covex Baach space such that its dual X has the Kadec Klee property ad K be a oempty closed covex subset of X which is also a oexpasive retract with retractio P. Let {T i } N i : K X be N oself I i -asymptotically oexpasive mappigs with sequeces l i [0, such that l i < ad {I i } N i : K X be N oself asymptotically oexpasive mappigs with sequeces k i
[0, such that Birol Güdüz ad Sezgi Akbulut 63 k i <. Let {α } ad {β } be sequeces i [a, 1 a] for some a 0, 1. Suppose that for ay give x 1 K, the sequece {x } is geerated by 1 ad F = N F T i F I i. The {x } coverges i=1 weakly to a commo fixed poit of {T i } N i ad {I i } N i. Proof. Sice {x } is bouded ad X is reflexive, there exists a subsequece { xj } of {x } which coverges weakly to some q K. Moreover, we have lim x T i x = lim x I i x = 0 for all i J by Lemma 3.2 ad so q F by Lemma 2.4. Now, we show that {x } coverges weakly to q. Suppose that {x k } is aother subsequece of {x } which coverges weakly to some q K. By the same method as above, we have q F ad q, q ω w {x }. By Lemma 4.1, lim tx + 1 t q q exists for all t [0, 1] ad so q = q by Lemma 2.5. Therefore, the sequece {x } coverges weakly to q. This completes the proof. Remark 4.2. 1 Our theorems ot oly improve ad geeralize importat related results of the previously kow results i this area, but also eve i the case of I i = E for all 1 i N, I i ad T i self-map of K for all 1 i N, α = 1 or β = 0 also all are ew. 2 If the error terms are added i 1 ad assumed to be bouded, the the results of this paper still hold. Refereces [1] K. Goebel, W.A. Kirk, A fixed poit theorem for asymptotically oexpasive mappigs, Proc. Amer. Math. Soc., 35 1972, 171 174. [2] R.P. Agarwal, Doal O Rega ad D.R. Sahu, Iterative costructio of fixed poits of early asymptotically oexpasive mappigs, J.Noliear Covex. Aal. 8 1 2007, 61 79. [3] S.H. Kha, M. Abbas, Approximatig commo fixed poits of early asymptotically oexpasive mappigs, Aal. Theory Appl., Vol. 27, No. 1 2011, 76 91. [4] S. Temir, O the covergece theorems of implicit iteratio process for a fiite family of I-asymptotically oexpasive mappigs, J. Comput. Appl. Math. 225 2009, 398 405. [5] S. Temir, O. Gul, Covergece theorem for I-asymptotically quasi-oexpasive mappig i Hilbert space, J. Math. Aal. Appl. 329 2007, 759 765. [6] L. Yag, X. Xie, Weak ad strog covergece theorems for a fiite family of I- asymptotically oexpasive mappigs, Appl. Math. Comput. 216 2010, 1057 1064. [7] F. Gu, Some covergece theorems of o-implicit iteratio process with errors for a fiite families of I-asymptotically oexpasive mappigs, Appl. Math. Comput. 216 2010, 161 172.
64 A Iterative Scheme for Commo Fixed Poits... [8] K.K. Ta ad H.K. Xu, Approximatig fixed poits of oexpasive mappigs by the Ishikawa iteratio process, Joural of Mathematical Aalysis ad Applicatios, vol. 178, o. 2 1993, pp. 301 308. [9] J. Schu, Weak ad strog covergece to fixed poits of asymptotically oexpasive mappigs, Bull. Austral. Math. Soc. 43 1991, 153-159. [10] M.O. Osilike, A. Udomee, Demiclosedess priciple ad covergece theorems for strictly pseudocotractive mappigs of Browder Petryshy type, J. Math. Aal. Appl. 256 2001, 431 445. [11] H.F. Seter, W.G. Dotso, Approximatig fixed poits of oexpasive mappigs, Proc. Amer. Math. Soc., 44 1974, 375-380. [12] S.H. Kha, H. Fukharuddi, Weak ad strog covergece of a scheme with errors for two oexpasive mappigs, Noliear Aal. 61 2005, 1295 1301. [13] J.G. Falset, W. Kaczor, T. Kuczumow, S. Reich, Weak covergece theorems for asymptotically oexpasive mappigs ad semigroups, Noliear Aal. 43 2001, 377 401. [14] W. Kaczor, Weak covergece of almost orbits of asymptotically oexpasive commutative semigroups, J. Math. Aal. Appl. 272 2002, 565 574. [15] C.E. Chidume, E.U. Ofoedu, H. Zegeye, Strog ad weak covergece theorems for asymptotically oexpasive mappigs, J. Math. Aal. Appl., 280 2003, 364-374. Birol Güdüz Departmet of Mathematics Faculty of Sciece ad Art Erzica Uiversity 24000, Erzica Turkey E-mail address: birolgdz@gmail.com Sezgi Akbulut Departmet of Mathematics Faculty of Sciece Ataturk Uiversity 25240, Erzurum Turkey E-mail address: sezgiakbulut@ataui.edu.tr