# Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive MappingsAcademic research paper on "Mathematics" 0 0
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## Academic research paper on topic "Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings"

﻿Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 428241,11 pages doi:10.1155/2008/428241

Research Article

Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings

Chao Wang and Jinghao Zhu

Department of Applied Mathematics, Tongji University, Shanghai 200092, China Correspondence should be addressed to Chao Wang, wangchaoxj20002000@yahoo.com.cn Received 1 April 2008; Revised 12 June 2008; Accepted 19 July 2008 Recommended by Simeon Reich

We introduce a new three-step iterative scheme with errors. Several convergence theorems of this scheme are established for common fixed points of nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our theorems improve and generalize recent known results in the literature.

Copyright © 2008 C. Wang and J. Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let K be a nonempty closed convex subset of real normed linear space E. Recall that a mapping T : K ^ K is called asymptotically nonexpansive if there exists a sequence{rn} c [0,to), with limn^TOrn = 0 such that \\Tnx - Tny\\ < (1 + rn)\\x - y\\, for all x,y e K and n > 1. Moreover, it is uniformly L-Lipschitzian if there exists a constant L > 0 such that \\Tnx - Tny\\ < L\\x - y\\, for all x,y e K and each n > 1. Denote and define by F(T) = {x e K : Tx = x} the set of fixed points of T. Suppose F(T) = 0. A mapping T is called asymptotically quasi-non-expansive if there exists a sequence {rn} c [0,to), with limn^TO rn = 0 such that \\Tnx - p\\ < (1 + rn) \\x - p\\, for all x,y e K, p e F(T), and n > 1.

It is clear from the above definitions that an asymptotically nonexpansive mapping must be uniformly L-Lipschitzian as well as asymptotically quasi-non-expansive, but the converse does not hold. Iterative technique for asymptotically nonexpansive self-mapping in Hilbert spaces and Banach spaces including Mann-type and Ishikawa-type iteration processes has been studied extensively by many authors; see, for example, [1-6].

Recently, Chidume et al.  have introduced the concept of nonself asymptotically nonexpansive mappings, which is the generalization of asymptotically nonexpansive mappings. Similarly, the concept of nonself asymptotically quasi-non-expansive mappings

can also be defined as the generalization of asymptotically quasi-non-expansive mappings and nonself asymptotically nonexpansive mappings. These mappings are defined as follows.

Definition 1.1. Let K be a nonempty closed convex subset of real normed linear space E, let P : E ^ K be the nonexpansive retraction of E onto K, and let T : K ^ E be a nonself mapping.

(i) T is said to be a nonself asymptotically nonexpansive mapping if there exists a sequence {rn} c [0, to), with limn^TO rn = 0 such that

||T(PT)n-1x - T(PT)n-1y|| < (1 + rn)\\x - y\\, (1.1)

for all x,y e K and n > 1.

(ii) T is said to be a nonself uniformly L-Lipschitzian mapping if there exists a constant L> 0 such that

||T(PT)n-1x - T(PT)n-1y\\ < L\\x - y\\, (1.2)

for all x,y e K and n > 1.

(iii) T is said to be a nonself asymptotically quasi-non-expansive mapping if F(T) / 0 and there exists a sequence {rn} c [0, to), with limn^TO rn = 0 such that

||T (PT)n-1x - p\\ < (1 + rn )\\x - p\\, (1.3)

for all x,y e K, p e F(T), and n > 1.

By studying the following iteration process (Mann-type iteration):

x1 e K, xn+1 = P((1 - an)xn + anT(PT)n-1xn), Vn > 1, (1.4)

where {an} c [0,1], Chidume et al.  obtained many convergence theorems for the fixed points of nonself asymptotically nonexpansive mapping T. Later on, Wang  generalized the iteration process (1.4) as follows (Ishikawa-type iteration):

x1 e K,

xn+1 = P((1 - an)xn + anT\{PT1)n~lyn), (1.5)

yn = P((1 - pn)xn + PnT2(PT2)n-1xn), Vn > 1

where T\,T2 : K ^ E are nonself asymptotically nonexpansive mappings and {an}, [pn] c [0,1]. Also, he got several convergence theorems of the iterative scheme (1.5) under proper conditions.

In 2000, Noor  first introduced a three-step iterative sequence and studied the approximate solutions of variational inclusion in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle. Glowinski and Tallec  showed that the three-step iterative schemes perform better than the Mann-type and Ishikawa-type iterative schemes. On the other hand, Xu and Noor  introduced and studied a three-step scheme to approximate fixed points of asymptotically nonexpansive mappings in Banach spaces. Cho et al.  and Plubtieng et al.  extended the work of Xu and Noor to the three-step iterative scheme with errors, and gave weak and strong convergence theorems for asymptotically nonexpansive mappings in Banach spaces.

Inspired and motivated by these facts, a new class of three-step iterative schemes with errors, for three nonself asymptotically quasi-non-expansive mappings, is introduced and studied in this paper. This scheme can be viewed as an extension for (1.4), (1.5), and others. This scheme is defined as follows.

Let K be a nonempty convex subset of real normed linear space X,let P : E ^ K be the nonexpansive retraction of E onto K, and let T1 ,T2,T3 : K ^ E be three nonself asymptotically quasi-non-expansive mappings. Compute the sequences{xn}, {yn}, and {zn} by

x1 e K,

Xn+1 = P(anTl( PTi) n-1yn + finXn + JnWn), yn = P(a'nT2( PT2) n-1Zn + P'nXn + y'n Vn),

Zn = P(<T3(PT3)n-1Xn + ¡"nXn + Y>n), Vn > 1

where {an}, «}, («ii}, {¡¡n}, {¡¡n}, (\$!}, (in}, (in}, and {j^} are real sequences in [0,1] with

an + ¡¡n + jn = a'n + ¡¡'n + y'n = a'n + fin + J^ = 1, and {wn}, {vn}, and {wn} are bounded sequences

in K. n n n n n n

Remark 1.2. (i) If T1 = T2 = T3 := T, jn = y'n = = 0, and a'n = a'n = 0, then scheme (1.6) reduces to the Mann-type iteration (1.4).

(ii) If T2 = T3, jn = Yn = Yn = 0, and a'n = 0, then scheme (1.6) reduces to the Ishikawa-type iteration (1.5).

(iii) If T1, T2, and T3 are three self-asymptotically nonexpansive mappings, then scheme (1.6) reduces to the three-step iteration with errors defined by [12,13], and others.

The purpose of this paper is to study the iterative sequences (1.6) to converge to a common fixed point of three nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our results extend and improve the corresponding results in [5, 7, 8,11-13], and many others.

2. Preliminaries and lemmas

In this section, we first recall some well-known definitions.

A real Banach space E is said to be uniformly convex if the modulus of convexity of E:

6e(£) = inf {1 - : ||x|| = ||y|| = 1, ||x - y|| = e}> 0, (2.1)

for all 0 < £ < 2 (i.e., Se(e) is a function (0,2] ^ (0,1)).

A subset K of E is said to be a retract if there exists continuous mapping P : E ^ K such that Px = x, for all x e K, and every closed convex subset of a uniformly convex Banach space is a retract. A mapping P : E ^ E is said to be a retraction if P2 = P.

A mapping T : K ^ E with F(T) / 0 is said to satisfy condition (A) (see ) if there exists a nondecreasing function f : [0, to) ^ [0, to) with f (0) = 0, for all r e (0, to), such that

||x - Tx|| > f (d(x,F(T))), (2.2)

for all x e K, where d(x,F(T)) = inf{||x - x* || : x* e F(T)}.

We modify this condition for three mappings T1,T2,T3 : K ^ E as follows. Three mappings T1, T2, T3 : K ^ E, where K is a subset of E, are said to satisfy condition (B) if there

exist a real number a> 0 and a nondecreasing function f : [0, to) — [0, to) with f (0) = 0, for all r e (0, to), such that

||x - T1x| > af(d(x,F)) or ||x - T2x|| > af(d(x,F)) or ||x - T3x|| > af(d(x,F)),

for all x e K, where F = F(T1)nF(T2) nF(T3) / 0. Note that condition (B) reduces to condition (A) when T1 = T2 = T3 and a = 1.

A mapping T : K — E is said to be semicompact if, for any sequence {xn} in K such that \\xn-Txn\\ —> 0 (n — to), there exists subsequence [xnj} of {xn} such that [xnj} converges strongly to x* e K.

Next we state the following useful lemmas.

Lemma 2.1 (see ). Let {an}, {bn}, and {cn} be sequences of nonnegative real numbers satisfying the inequality

an+1 < (1 + cn)an + bn, Vn > 1. (2.4)

IfJhTO=1 cn< to and^TO=1 bn < to, then limn—TO an exists.

Lemma 2.2 (see ). Let E be a real uniformly convex Banach space and 0 < k < tn < q < 1, for all positive integer n > 1. Suppose that {xn} and {yn} are two sequences of E such that limsupn—TO\\xn\\ < r, limsupn—TO\\yn\\ < r, and limn—,x,\\tnxn + (1 - tn)yn\\ = r hold, for some r > 0; then limn—TO\\xn - yn\\ = 0.

3. Main results

In this section, we will prove the strong convergence of the iteration scheme (1.6) to a common fixed point of nonself asymptotically quasi-non-expansive mappings Ti,T2, and T3. We first prove the following lemmas.

Lemma 3.1. Let K be a nonempty closed convex subset of a real normed linear space E. Let T1, T2, T3 : K — E be nonself asymptotically quasi-non-expansive mappings with sequences {r^} such that xto=1 r« < to, for all i = 1,2,3. Suppose that {xn} is defined by (1.6) with £TO=1 Yn < to, £TO=1 Yn < to, and^TO=1 Yn < to. If F = F(T1) n F(T2) n F(T3) = 0, then limn—TO \\xn - p\\ exists, for all p e F.

Proof. Let p e F. Since {un}, {vn}, and {wn} are bounded sequences in K, therefore there exists M> 0 such that

M = max j sup11Un - psup^n - psup¡w« - p^. (3.1)

I n>1 n>1 n>1 J

Let rn = max{rni),r^2),r^3)} and kn = max{Yn,Y^,Yii'}. Then £TO=1 rn < to and £TO=1 kn < to. By (1.6), we have

l|xn+1 - pH = ||P[anT^PT1n-^yn + pnxn + YnWn] - P(p)H

< HanT1 (PTf-1)yn + pnxn + YnWn - (an + pn + Yn)pH (3 2)

< llanfT^PTf-1) yn - p] + pn(xn - p) + YnW - p) H

< an( 1 + rn)Hyn - pH + P«hX« - pH + knHWn - pH, ||yn - p|| = ||P [anT2(PT2n-1)zn + pn xn + Yn Vn] - P (p)||

< ||anT2(PT2n-1)zn + pnxn + Yn Vn - (a'n + p'n + Yn )p|| (3.3)

< a'n( 1 + rn) ¡zn - pH + p'nHX« - pH + kn ¡Vn - pH,

C. Wang and J. Zhu and similarly, we also have

\zn - p\\ < dn( 1 + Tn) \\xn - p\\ + ß'n\\xn - p\\ + kn\\un - p\\.

Substituting (3.4) into (3.3), we obtain

\\yn - p\\ < a'n(1 + r^[d'n (1 + Tn)\\xn - p\\ + ß'n,,\\xn - p\\ + kn\\un - p\\]

+ ß'n\\Xn - p\\ + kn\\Vn - p\\

< anan

(1 + rn) \\Xn - p\\ + a'nßln i1 + rn)\\xn - p\\ + ßn\\Xn - p\\ + a'nkn( 1 + Tn)\\un - p\\ + kn\\vn - p\\

< (1 - ß'n- rn X (1 + Tnf\\xn - p\\ + (1 - ß'„- rn Wn (1 + Tn) \ \xn - p\\ (3.5)

+ ßn\xn - p\\ + kn{ 1 + Tn)\\un - p\\ + kn\\vn - p\\

< (1 - ß'n- Yn )(< + ß'n )(1 + Tnf\\xn - p\\ + ß'n\\xn - p\\ + mn

< (1 - ß'n )(1 + Tn)2\\xn - p\\ + ß'n (1 + Tn)2\\x - p\ + mn

<1 + rn 2 \ xn - p\\ + mn,

where mn = kn(2+rn)M. Since ^TO=1 rn < to and ^TO=1 kn < to, then ^TO=1 mn < to. Substituting (3.5) into (3.2), we have

\\xn+1 - p\\ < an( 1 + rn) [(1 + r^\\xn - p\\ + mn] + ßn\\xn - p\\ + Yn\\Wn - p\\

< [«41 + rn)3 + ßn] \\xn - p\\ + an( 1 + rn)mn + Yn\\wn - p\\

< (an + ßn) (1 + rn)3\\xn - p\\ + (1 + rn)mn + kn\\wn - p\\ (3.6)

< (1 + rn)3\\xn - p\\ + (1 + r-n)mn + knM

< (1 + Cn) \\xn - p\\ + bn,

where cn = (1 + rn)3 - 1 and bn = (1 + rn)mn + knM. Since ^TO=i rn < to, ^TO=i kn < to, and XTO=1 mn < to, then ^TO=1 cn < to and ^TO=1 bn < to. It follows from Lemma 2.1 that limn—TO \\xn - p\\ exists. This completes the proof. □

Lemma 3.2. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T1, T2,T3: K — E be uniformly L-Lipschitzian nonself asymptotically quasi-non-expansive mappings with sequences {rn} such that £TO=1 rn < to, for all i = 1,2,3. Suppose that {xn} is defined by (1.6) with XTO=1 Yn < to,ZTO=1 Yn < to, and ^TO=1 Yn < to, where an, a'n, and a'n are three sequences in [e, 1 - e],for some e > 0. If F = F(T{) n FT) n F(T3) = 0, then

lim \\xn - T1xn\\ = lim \\xn - T2xn\\ = lim \\xn - T3xn\\ = 0. (3.7)

n—TO n—TO n—>TO

Proof. For any p e F, by Lemma 3.1, we see that limn—TO\\xn - p\\ exists. Assume limn—TO\\xn -p\\ = a, for some a > 0. For all n > 1, let rn = max{r^^T2^f3} and kn = max{yn,Y'n,yÜ}.

Then, Xrn < to and ^kn < to. From (3.5), we have

||yn - p|| < (1 + Tn)2\\xn - p|| + mn. (3.8)

Taking lim sup^^ on both sides in (3.8), since ^°°=1 rn< to and ^°°=1 mn < to, we obtain

limsup||yn - p|| < limsup|xn - p|| = lim ||xn - p|| = a (3.9)

so that

limsup|T1(PT1)n-1yn - p|| < limsup(1 + Tn) ||yn - p|| = limsup||yn - p|| < a. (3.10)

n—n—n—

Next consider

WT^ PT1) n-1yn - p + Jn(Wn - Xn) W < WT^ PT1) n-1yn - p! + knWWn - XnW. (3.11) Since limn—° kn = 0, we have

lim supPT1)n-1yn - p + jn(wn - Xn) W < a. (3.12)

Hxn - p + Jn(wn - Xn) || < Hxn - pH + kn||wn - Xn ||. (3.13)

lim sup ||xn - p + Yn(wn - Xn) || < a. (3.14)

This implies that

Further, observe that a = lim ||x„ - p||

n—OT 11 11

= lim HanTitPT\)n-1yn + pnXn + YnW - p|

n—>OT

= lim HanTi(PTi)n-iyn + (1 - an)xn - jnXn + YnW - (1 - a^p - a„pH

n—>OT

= lim HanTi (PTi)n-1yn - anp + anjnWn - anjnXn + (1 - a^Xn

n—>OT

- (i - an)p - YnXn + YnWn - anYnWn + anYnXnH = lim Han [Ti (PTi) n-iyn - p + Ynfan - Xn)] + (i - an) [Xn - p + Yn(Wn - Xn)] H-

n—>OT

By Lemma 2.2, (3.i2), (3.i4), and (3.i5), we have

lim||Ti(PTi)n-iyn - Xn || = 0- (3.i6)

n—OT

(3.15)

so that

Next consider

Thus, we have

limsup||X2(PX2)n 1zn - p + y'n (vn - Xn) || < a,

(3.17)

C. Wang and J. Zhu

Next we will prove that limn—x,||X2(PX2)n-1zn - xn || = 0. Since

||xn - p|| < ||Xi(PXi)n-1yn - Xn|| + ||Xi(PXi)n-1yn - p|| < ||X^PX1)n-1yn - Xn || + (1 + rn) ||yn - p|

and limn^^|X1(PX1)n-1yn - xnll = 0 = limn—(X) rn, we obtain

a = lim ||xn - p|| < liminf||yn - p||. (3.18)

Thus, it follows from (3.10) and (3.18) that

lim||yn - p|| = a. (3.19)

On the other hand, from (3.4), we have

||zn -p| < Vn(1 + rn) + P'n\ ||xn -p| + Mwn -p| (320)

< (1 + rn) ¡Xn - p| + kn^Un - py.

By boundedness of the sequence {un} and by limn—OT rn = limn—OT kn = 0, we have

limsup||zn - p|| < limsup|xn -p|| = a (3.21)

n—<x> n—

limsup||X2(PX2)n 1zn - p|| < lim sup(1 + rn)||zn - p|| < a. (3.22)

||X2(PX2)n-1zn - p + rn (vn - Xn)|| < ||X2(PX2)n-1zn - p|| + kn||vn - Xn||. (3.23)

(3.24)

| Xn - p + Y'n (Vn - Xn) H < ^Xn - p| + kn^Vn - Xn^.

limsup|xn - p + y'n (vn - Xn) || < a. (3.25)

This implies that

Note that

a = lim |yn - p|

= ^JKXzC PX2) n-1zn + Xn + Yn Vn - p| (3.26)

= lim ||«n [X^( PX2) n-1zn - p + y'n (vn - X^] + (1 - a'n) [Xn - p + yn V - Xn)]||.

n—^^

It follows from Lemma 2.2, (3.24), and (3.25) that

(3.29)

limim^)" 1zn - Xn II = 0. (3.27)

n—>TO 11 N ' 11

Similarly, by using the same argument as in the proof above, we obtain

lim||T3(PT3)n-1Xn - Xn|| = 0. (3.28)

n—<X>" 11

Hence,

lim 1T^PTi)n-1 yn - Xn1 = lim ||T2(PT2)n-1Zn - Xn^ = lim ||Ts(PT3)n-1Xn - Xn^ = 0,

n—TO n—TO n—TO

and this implies that

||Xn+1 - Xn|| < an|T^PT^n-1yn - Xn|| + kn||ron - Xn|| —> 0 as n —>to. (3.30)

Since T1 is uniformly L-Lipschitzian mapping, then we have ||T1(PT1)n-1Xn - Xn ||

< ||T1(PT1)n-1Xn - T1(PT1)n-1yn|| + ||T1(PT1)n-1yn - Xn||

< L^Xn - yn! + HT1(PT1)n-1 yn - Xn!

< L^Xn - a'nT2( PT2) n-1Zn - fin Xn - in Vn! + HT1 (PT1) n-1yn - Xn!

< La'nHT2(PT2)n 1zn - Xn|| + LknHvn - Xn|| + ||T^PT^n 1yn - Xn|| —> 0 as n —> to,

(3.31)

||Xn T1Xn\\

< || Xn+1 -Xn || + ||Xn+1 - PT1) nXn+11| + PT1) nXn+1 - PT1) nXn| + |T^ PT1) nXn-T1Xn||

< W Xn+1 Xn y + W Xn+1 T1( PT1)r Xn+1 | + L| Xn+1 - Xn || + L|T^ PT1)n- Xn - Xn | .

(3.32)

It follows from (3.30), (3.31), and (3.32) that

lim || Xn - T^H = 0. (3.33)

n—TO

Next consider ||T2(PT2)n-1Xn - Xn y

< ||T2(PT2)n-1Xn - T2(PT2)n-1Zn|| + ||T2(PT2)n-1Zn - Xn||

< L|Xn - Zn || + ||T2(PT2)n-1Zn - Xn ||

< La'n||T3(PT3)n 1Xn - Xn|| + LknHun - Xn|| + ||T2(PT2)n 1Zn - Xn|| —> 0 as n —> to,

(3.34)

\Xn T2Xn\

< J]Xn+1 -Xn 11 + IIXn+1 -Tz(PT2)nXn+1 II + PT2)nXn+1 -Tz(PT2)nXn| + |T^(PT2)nXn-T2Xn|

< 1 Xn+1 Xn 1 + 1 Xn+1 T2 PT2 n II + L||T2( PT2)n-

(3.35)

It follows from (3.30), (3.34), and (3.35) that

lim \\xn - T2xnII = 0. (3.36)

n—>TO

Finally, we consider

\\xn - T3xn \\

< \\ Xn+1 -Xn\\ + \\xn+1 - T^ PT3) nXn+1 \\ + PT3) nXn+1 - T3( PT3) nXn\ + \T3( PT3) nXn-T3Xn\\

< \\ Xn+1 Xn \\ + \\ Xn+1 T^ PT3)n \\ + L\\T3( PT3)n-

(3.37)

It follows from (3.29), (3.30), and (3.37) that

lim \\xn - T3xn\\ = 0. (3.38)

n—TO

Therefore,

lim \\xn - T1xn\ = lim \\xn - T2xn\\ = lim \\xn - T3xn\\ = 0. (3.39)

n—TO n—>to n—TO

This completes the proof. □

Now, we give our main theorems of this paper.

Theorem 3.3. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T1,T2,T3 : K — E be uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive mappings with sequences {r^} such that £^=1 r^ < to, for all i = 1,2,3, satisfying condition (B). Suppose that {xn} is defined by (1.6) with Yn < to, X^=1 Yn < to, and £^=1 Y^ < to, where an, a'n, and a'n are three sequences in [e, 1 - e],for some e > 0. If F = F(T1) n F(T2) n F(T3) / 0, then {xn} converges strongly to a common fixed point of T1, T2, and T3.

Proof. It follows from Lemma 3.2 that limn—TO||xn - T1xny = limn—TO||xn - T2xn || = limn—TO||xn -T3xn|| = 0. Since T1, T2, and T3 satisfy condition (B), we have limn—TO d(xn,F) = 0.

From Lemma 3.1 and the proof of Qihou , we can obtain that {xn} is a Cauchy sequence in K. Assume that limn—TO xn = p e K. Since limn—TO||xn - T1xn| = limn—TO||xn -T2xn|| = limn—TO||xn - T3xn|| = 0, by the continuity of T1, T2, and T3, we have p e F, that is, p is a common fixed point of T1, T2, and T3. This completes the proof. □

Corollary 3.4. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T1,T2,T3 : K — E be nonself asymptotically nonexpansive mappings with sequences {r^} such that XTO=1rn) < to, for all i = 1,2,3, satisfying condition (B). Suppose that {xn} is defined by (1.6) with XTO=1 Yn < to, £TO=1 Yn < to, and ^TO=1 YH < to, where an, a'n, and a'n are three sequences in [e, 1 - e], for some e > 0. If F = F(T1) n F(T2) n F(T3) / 0, then {xn} converges strongly to a common fixed point of T1, T2, and T3.

Proof. Since every nonself asymptotically nonexpansive mapping is uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive, the result can be deduced immediately from Theorem 3.3. This completes the proof. □

Theorem 3.5. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T\,T2,T3 : K ^ E be uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive mappings with sequences (r®) such that £°°=1 r^ < for all i = 1,2,3. Suppose that (xn) is defined by (1.6) with £°=1 jn < £°°=1j'n < and £°=1 yn < where an, a'n, and a'n are three sequences in [e, 1 - e\, for some e> 0. If F = F(T1) n F(T2) n F(T3) / 0 and one of T1, T2, and T3 is demicompact, then (xn) converges strongly to a common fixed point of T1, T2, and T3.

Proof. Without loss of generality, we may assume that T1 is demicompact. Since limn^°^xn -T1xny = 0, there exists a subsequence (xnj) c (xn) such that xnj ^ x* e K. Hence, from (3.39), we have

llx* — Tix*|| = limllxn, - Tixn,\\ = 0, i = 1,2,3. (3.40)

This implies that x* e F. By the arbitrariness of p e F, from Lemma 3.1, and taking p = x*, similarly we can prove that

lim \\xn — x*|| = d, (3.41)

where d > 0 is some nonnegative number. From xnj ^ x*, we know that d = 0, that is,

xn ^ x .

This completes the proof. □

Corollary 3.6. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T1,T2,T3 : K ^ E be nonself asymptotically nonexpansive mappings with sequences (r^) such that^°°=1 r() < for all i = 1,2,3. Suppose that (x

X°°=1 Yn < and X°°=1 jn < where an, a'n, and a'n are three sequences in [e, 1 — e\, for some e > 0. If F = F(T1) n F(T2) n F(T3) / 0 and one of T1, T2, and T3 is demicompact, then (xn) converges strongly to a common fixed point of T1, T2, and T3.

Acknowledgments

The authors would like to thank the referee and the editor for their careful reading of the manuscript and their many valuable comments and suggestions. This paper was supported by the National Natural Science Foundation of China (Grant no. 10671145).

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