0 Fixed Point Theory and Applications

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Convergence of perturbed composite implicit iteration process for a finite family of asymptotically nonexpansive mappings

Xuewu Wang*

"Correspondence: wangxuewuxx@163.com Schoolof Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, Shandong 264005, China

Abstract

In this paper, we introduce a perturbed composite implicit iterative process with errors for a finite family of asymptotically nonexpansive mappings. Under Opial's condition, semicompact and lim infn^TO d(xn,F(T)) = 0 conditions, respectively, we prove that this iterative scheme converges weakly or strongly to a common fixed point of a finite family of asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize and improve the corresponding results of Sun (J. Math. Anal. Appl. 286:351-358,2003), Chang (J. Math. Anal. Appl. 313:273-283, 2006), Gu (J. Math. Anal. Appl. 329:766-776, 2007), Thakur (Appl. Math. Comput. 190:965-973,2007), Rafiq (Rostock. Math. Kolloqu. 62:21-39, 2007) and some others. MSC: 47H9; 47H10

Keywords: asymptotically nonexpansive mapping; uniformly convex Banach space; perturbation; composite implicit iterative process; weak and strong convergence; common fixed point

ft Spri

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1 Introduction

Let E be a real Banach space, K be a nonempty convex subset of E. Let (T1, T2,..., TN} be a finite family of mappings from K into itself, and F(Ti) be the set of fixed points of Ti (i e I — {1,2, ...,N}). F (T) denotes the set of common fixed points of (T^ T2,...,TN}.

Recently, Xu and Ori [1] have introduced an implicit iteration process for a finite family of nonexpansive mappings as follows:

Vn > 1, (1)

where Tn — Tn(mod N) (here the mod N function takes values in I), {an} be a real sequence in [0,1], x0 be an initial point in K.

Sun [2] have extended this iterative process defined by Xu and Ori to a new iterative process for a finite family of asymptotically nonexpansive mappings, which is defined as follows:

xn — anxn-1 + (1 - an) Tkxn, n > 1, (2)

where n = (k - 1)N + i, i e I.

© 2013 Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

Chang [3] have discussed the convergence of the implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings as follows:

Xn = anXn-i + (1 - an) T-^x-n + Un, n > 1, (3)

where n = (k(n) - 1)N + i(n), i(n) e I, and k(n) > 1 with k(n) ^to as n ^to. Under the hypotheses ^ \\un\\ < to and some appropriate conditions, they proved some results of weak and strong convergence for {xn} defined by (3). However, the condition ^^ \\un \\ < to is not too reasonable, because this implies that {un} are very small for n sufficiently big.

Gu [4] has extended the above implicit iteration processes. A composite implicit iteration process with random errors was introduced as follows:

Xn = (1 - an - Yn)Xn-1 + anTnJn + YnUn, n > 1;

yn = (1 - Pn - Sn)Xn + Pn TnXn + SnVn, n > 1,

where {an}, {pn}, {yn}, {Sn} are four real sequences in [0,1] satisfying an + yn < 1 and pn + Sn < 1 for all n > 1, {un}, {vn} are two sequences in K and X0 is an initial point. Some theorems were established on the strong convergence of the composite implicit iteration process defined by (4) for a finite family of mappings in real Banach spaces.

Thakur [5] has improved the composite implicit iteration process defined by (4) as follows:

Xn = (1 - an)Xn-1 + an T^yn, n > 1; ^

yn = (1 - Pn)Xn + PnT^Xn, n > 1.

Some theorems were proved on the weak and strong convergence of the composite implicit iteration process defined by (5) for a finite family of mappings in real uniformly convex Banach spaces.

Rafiq [6] have improved the implicit iterative process. The Mann type implicit iteration process was introduced in Hilbert spaces as follows:

Xn = anXn-i + (1 -an)Tvn, n > 1, (6)

where vn is a perturbation ofXn, and satisfy ^n>1 \\Xn - vn\\ < to. Moreover, Ciric [7] also did some work in this respect.

Inspired and motivated by the above works, in this paper we will extend and improve the above iterative process to a perturbed composite implicit iterative process for a finite family of asymptotically nonexpansive mappings as follows:

Xn = (1 - an - Yn)Xn-1 + anT^yn + YnUn, n > 1; (7)

yn = (1 - Pn - Sn)Xn-1 + PnTk^~Xn + SnVn, n > 1,

where n = (k(n) - 1)N + i(n), i(n) e I, Tn = Tn(modn), {an}, {Pn}, {Yn}, {Sn} are four real sequences in [0,1] satisfying an + Yn < 1 and Pn + Sn < 1 for all n > 1, {un}, {vn} are two sequences in K and X0 is an initial point. {X n} be a sequence in K satisfying ^ n>1 \\Xn - X n\\ <

to, which implies that \\Xn -Xn \\ ^ 0 (n ^to). Therefore, Xn is known as the perturbation of Xn, and {Xn} is known as the perturbed sequence of {Xn}. This sequence {Xn} defined by (7) is said to be the perturbed composite implicit iterative sequence with random errors.

Especially, (I) in the iterative process defined by (7), when Pn = 0, Sn = 0 for all n > 1, we have

Xn = (1 - an - Yn)Xn-1 + an + YnUn, n > 1. (8)

At this time, the perturbed composite implicit iterative sequence generated by (7) becomes a Mann-type iterative sequence with random errors.

(II) In the iterative process defined by (7), when Pn = 1, Sn = 0 for all n > 1, we have

Xn = (1 - an - Yn)Xn-1 + an T^Xn + YnUn, n > 1. (9)

At this time, the perturbed composite implicit iterative sequence generated by (7) becomes a perturbed implicit iterative sequence with random errors.

(III) In the iterative process defined by (7), when Xn-1 = Xn for all n > 1, we have

Xn = (1 - an - Yn)Xn-1 + anT^yn + YnUn, n > 1; (10)

yn = (1 - Pn - Sn)Xn-1 + PnT^Xn-1 + SnVn, n > 1.

At this time, the perturbed composite implicit iterative sequence generated by (7) becomes an Ishikawa-type iterative sequence with random errors for a finite family of asymptotically nonexpansive mappings {Ti, i e I}.

From the above iterative processes defined by (1)-(6) and (8)-(10), we know that the iterative process (7) improves and extends some iterative process introduced by the recent literature. Moreover, we point out that the iterative process, defined by (7), in which it is not necessary to compute the value of the given operator at Xn, but compute an approximate point of Xn, are particularly useful in the numerical analysis. Therefore, the iterative sequence generated by (7) is better than some implicit iterative sequences at the existent aspect.

The main purpose of this paper is to study the convergence of the perturbed composite implicit iterative sequence {Xn} defined by (7) for a finite family of asymptotically nonexpansive mappings under Opial's condition, semicompact and liminfn^TO d(Xn,F(T)) = 0 conditions, respectively. The results presented in this paper generalized and improve the corresponding results of Sun [2], Chang [3], Gu [4], Thakur [5], Rafiq [6], and some others [1, 7-15].

2 Preliminaries

For the sake of convenience, we first recall some definitions and conclusions.

Definition 2.1 Let K be a closed subset of the real Banach space E and T: K ^ K be a mapping.

1. T is said to be semicompact, if for any bounded sequence {Xn} in K such that \\ TXn - Xn \\ ^ 0 (n ^ to), then there exists a subsequence {Xni} of {Xn} such that Xni ^ X e E,

2. T is said to be demiclosed at the origin, if for each sequence {xn} in K, the conditions xn ^ xo weakly and Txn ^ 0 strongly imply Tx0 — 0;

3. T is said to be asymptotically nonexpansive, if there exists a sequence hn e [1, with limn^TO hn — 1 such that

|| Tnx - Tny\< K\\x-y\\, Vx,y e K,n > 1. (11)

4. Let T is said to be uniformly L-Lipschitizian if there exists a constant L >0 such that

\Tnx - Tny\ < L\\x - y\\, Vx, y e E, n > 1.

Definition 2.2 [16] A Banach space X is said to satisfy Opial's condition if xn ^ x weakly as n ^^ and x — y imply that limsupn^TO \\xn - x\\ < limsupn^TO \\xn - y\\.

Lemma 2.1 LetK be a nonempty subset ofE, T1, T2,...,TN : K ^ K be N asymptotically nonexpansive mappings. Then

(i) there exists a sequence {hn} c [1, with limn^TO hn — 1 such that

\T?x - Tfy|| < hn\\x - y\\, Vx, y e K, i e I, n > 1; (12)

(ii) {T1, T2,..., TN} is uniformly Lipschitzian, i.e., there exists a constant L such that

\T?x - Tfy|| < L\\x - y\\, Vx, y e K, i e I, n > 1. (13)

Proof Since T1, T2,...,TN : K ^ K are N asymptotically nonexpansive mappings, then for every i e I and n e N, there exists h^ e [1, with limn^TO h^ — 1 such that

\T?x - Tny\ < h^Wx - y\\, Vx, y e E.

Taking hn — max{hni),hn2,...,hnN)}, thenhn c [1, limn^TOhn — 1 and ( 12) holds.

An asymptotically nonexpansive mapping must is a uniformly Lipschitzian mapping. Hence, for every i e I and n e N, there exists Li such that

\T?x - T?y\ < Li\\x - y\\, Vx, y e E.

Taking L — max{L1,L2,...,LN}, it is obvious that (13) holds. □

Lemma 2.2 [17] LetE be a uniformly convex Banach space, K be a nonempty, closed and convex subset ofE and T: K ^ K be an asymptotically nonexpansive mapping. Then I - T is demi-closed at zero, i.e., for each sequence {xn} in K, if {xn} convergence weakly to q e E and {(I - T)xn} converges strongly to 0, then (I - T)q — 0.

Lemma 2.3 [18] LetE be a Banach space satisfying Opial's condition, {xn} be a sequence inE. Letu, v e Ebesuch thatlimn^m \\xn - u\\ and limn^TO \\xn - v\\ exist. If {xnk} and {xni} are two subsequences of {xn} which converge weakly to u and v, respectively, then u — v.

Lemma 2.4 [19] Let E be a uniformly convex Banach space, b, c be two constants with 0 < b < c <1. Suppose that {tn} is a sequence in [b, c] and {xn}, {yn} are two sequences in E. Then the conditions limn^TO \\tnxn + (1 - tn)yn\\ = d, limsupn^TO \\xn\\ < d, limsupn^TO \\yn\\ < d imply that limn^TO \\xn -yn\\ = 0, where d is a nonnegative constant.

Lemma 2.5 [20] Let {an}, {bn}, {Sn} are three sequences of nonnegative real numbers, if there exists n0 such that

an+1 < (1 + Sn)an + bn, Vn > n0,

where xto Sn < to and bn < to. Then

(i) limn^TO an exists;

(ii) limn^ to an = 0 whenever liminfn^m an = 0.

Lemma 2.6 Let E be a real Banach space and K be a nonempty closed convex subset of E. Let T1, T2,...,TN : K ^ K be N asymptotically nonexpansive mappings with F(T) = P|nl1 F(Ti) = 0. Let {un} and {vn} are two bounded sequences in K. If {an}, {ßn}, {yn}, {Sn} be four real sequences in [0,1] satisfying the following conditions:

(i) an + yn < 1 and ßn + Sn < 1 for all n > 1;

(ii) limsupn^TO an = a <1 or limsupn^TO ßn = ß <1;

(iii) eto=1 Yn < to, lto=1 Sn < to, eto=1(hn - 1) < to;

(iv) £\\xn - xn\\ < TO.

Let {xn} be the perturbed composite implicit iterative sequence defined by (7), then limn^TO \\xn - p\\ exists for all p e F(T).

Proof Takep e F(T), it follows from (7) and Lemma 2.1 that

\\xn -p\\ < || (1 - an - Yn)xn-1 + anT^yn + Ynun -p|

< (1- an - Yn )\\xn-1- p\\ + anhn\\yn - p\\ + Yn\Wn - p\\ (14)

\\yn -p\\ < |(1- ßn - Sn)xn-1+ ßnT^JCn + Snvn -p1

< (1-ßn - Sn)\xn-1-p\\ + ßnhn\\xn -p\\ + Sn\\vn -p\\

< (1-ßn - Sn)\\xn-1- p\\ + ßnhn\\xn - xn\\ + ßnhn\\xn - p\\ + SnPn - p\\. (15) Substituting (15) into (14) and simplifying, we obtain

(1 - anßnh2n)\\xn -p\\< [1 - an - Yn + anhn(1 - ßn - Sn)] \\xn-1 -p\\

+ anß n hn\\x n xn \\ + anSn hn\\vn -p\\ + Yn\\un -p\\. (16)

We notice the hypotheses on {an}, {ßn} and {hn}, by limsupn^TOan = a <1, there exists n0 e N such that

1- anßnh2n > 1- ank2n > ^(1- a)> 0, n > n0.

It follows from (16) that for n > no

II* - »II < l~a" - Yn + anhn(1- ßn - Sn) 11* 1- »II

\\xn p\\ ^ 1 Ö 1,2 llxn-1 p\\

1 — anßnhn

(unßnh2n\\x n xn \\ + an^n hn\\Vn — p\\ + Yn\\Un — p\\)

1 — anßnh2n

anß n h 2n — an + anhn(1 — ßn)

1 — anßnh2n

\\Xn—1 — p\\

+ --(anpnh2n\\Xn -Xn\\ + an&nh„\\vn -p\\ + Yn\\Un -p\\)

< |l + n^a [anPnhn(hn - 1) + an(hn - 1)]| \\Xn-i -p\\ 2

+ --(anPnh2n\\Xn -Xn\\ + anSnhn\\Vn -p\\ + Yn\\Un -P^.

Hence, we have

\\Xn -p\<(1 + On)\\Xn-1-p\\ + Vn, n > no, (7)

On = --1anPnKh - 1) + an(hn - 1)1, n > no

1 - a L J

nn = --(anPnh2n\\Xn - Xn\ + an SnKW - p\\ + Yn\Un - p\\), n > no.

From condition (iii), it is obvious that ^'¡TO=1 On < to. In addition, since {\\un\\}, {\\vn\\} are all bounded, we deduce that ^nn < to form (iii)-(iv). By virtue of (17) and Lemma 2.5, we obtain that limn^TO \\xn -p \\ exists. This completes the proof of Lemma 2.6. □

3 Main results and proofs

Theorem 3.1 LetE be a realBanach space andK be a nonempty, closed and convex subset ofE. Let T1, T2,...,TN :K ^ K be N asymptotically nonexpansive mappings with F(T) = PlN=1 F(Ti) = 0. Let {un} and {vn} are two bounded sequences in K. If {an}, {fin}, {Yn}, {Sn} be four real sequences in [0,1] satisfying the following conditions:

(i) an + Yn < 1 and ¡3n + Sn < 1 for all n > 1;

(ii) limsupn^TO an < 1 or limsupn^TO < 1;

(iii) £TO=1 Yn < to, ETO=1 Sn < to, ETO=1(hn - 1) < to;

(iv) £ TO=1 \\Xn - Xn\\ < TO.

Then the perturbed composite implicit iterative sequence {xn} defined by (7) converges strongly to a common fixed point of {T1, T2,...,TN} if and only if liminfn^TO d(xn, F (T)) = 0.

Proof The necessity of Theorem 3.1 is obvious. Now we prove the sufficiency of Theorem 3.1.

For arbitraryp e F(T), it follows from (17) in Lemma 2.6 that

\\Xn -p\\ < (1 + 0h)II*„_! -p\\ + nn, Vn > «0,

where £TO=1 dn < to and £TO=1 n« < to. Hence, we have

d{xn,F(T)) < (1 + On)d(xn_i,F(T)) + nn, Vn > no. (18)

It follows from (18) and Lemma 2.5 that limit limn^TO d(xn, F(T)) exists. By the assumption, we have limn^TO d(xn, F(T)) = 0. Consequently, for any given e >0, there exists a positive integer N1 (N1 > n0) such that

d(xn,F(T)) < e, Ynk < e, t>k <1, Vn > N1, 88

k=n k=n

and there exists pi e F(T) such that \\xn _pi\\ < e/8, Vn > N1. By (18) and the inequality 1 + x < ex (x > 0), for any n > N1 and all m > 1, we have

\\xn+m _ xn\\ < exp{^n+m_1}\\xn+m_1 _ p1 \ + nn+m_1 + \\xn _ p1 \\

< exp{0n+m_1 +

n+m_2 - P1II + exp{0n+m_1}n n+m_2 + nn+m_1 + \\xn -P1W < ■ ■■

f n+m-1

i n+m-1 1 n+m-1

I m °k\ +1 - ^ill + expj Y^ M Y^

nk < e.

I k=n ) k=

Hence, {x«} is a Cauchy sequence in E. By the completeness of E, we can assume that xn ^ x e K. Next we prove that F(T) is a close subset of K. Let {pn} is a sequence in F(T) which converges strongly to some p, then we have for any i e I

\\p _ Tip\\<\\p -Pn\\ + \\Pn - Tip\ \ < (1 + L)\\p -Pn\\^0 (n ^ to).

Thus, p e F(T), and F(T) is closed. Since limn^TO d(xn,F(T)) = 0, then x* e F(T). Consequently, {xn} defined by (7) converges strongly to a common fixed point of {T1, T2,..., TN} in K. This completes the proof of Theorem 3.1. □

Theorem 3.2 LetE be a real uniformly convex Banach space satisfying Opial's condition and K be a nonempty closed convex subset ofE. Let T1, T2,...,TN : K ^ K be N asymptotically nonexpansive mappings with F(T) = P|N=1 F(Ti) = 0. Let {un} and {vn} are two bounded sequences in K. If {an}, {fin}, {yn}, {Sn} be four real sequences in [0,1] satisfying the following conditions:

(i) an + yn < 1 and ¡3n + Sn < 1 for all n > 1;

(ii) 0 < liminfn^TO an < limsupn^TO an < 1,limsupn^TO jin < 1;

(iii) £TO1 Yn < TO.Y.TO=1 Sn < to,£TO=1(hn -1) < to;

(iv) £^=1 \\x« -xn\\ < to.

Then the perturbed composite implicit iterative sequence {xn} defined by (7) converges weakly to a common fixed point of {T1, T2,..., TN} in K.

Proof First, we prove that limn—TO \\xn - TjXn \\ = 0 for all j e I.

For anyp e F(T), it follows from Lemma 2.6 that limn—TO \\xn -p\\ exists. Suppose that limn—TO \\xn -p\\ = d,we have from (7)

lim \\Xn -p\\ = lim 11(1-an)[xn-1 -p + Yn(un -Xn-1)]

n—>TO n—>TO L

+ an[T^yn -p + Yn(un -Xn-1^ || = d. (19)

Since limn—TO \\xn -p\\ = d, then {xn} be a bounded sequence. By virtue of the condition (iii) and the boundedness of sequences {xn} and {un}, we have

limsup||xn_i — p + Yn(un — xn—i)|

< lim sup \\xn—1 — p\\ + lim sup Yn\\un — xn—1\\ = d. (20)

It follows from£\\Xn -xn\\ < to that limniTO n -p\\ = limniTO \\xn -p\\ = d. We have

lim sup I r!(<n,J);yn — P + Yn(un — Xn—01

< lim sup hn\\yn - p\\ + lim sup Yn\\u - Xn-1\\

n—TO n—TO

< limsup[(1-Pn -Sn)\Xn-1-p\\ + Pnhn\^n -p\\ + Sn\\vn -p^ = d. (21)

n—TO

Therefore, by (19), (20), (21), (ii) and Lemma 2.4, we obtain that lim fT^yn - Xn-11 = 0.

n-—TO" i(n) "

Hence,

lim \\Xn - Xn-1 \\ < lim [an^^yn - Xn-1f + Yn\\un - Xn-1\\] = 0, (22)

M_l(Vl M_11 H"/ 'II -I

which implies that limniTO \\xn - xn+j\\ = 0 for all j e I. On the other hand, we also have

n^Wfcn - xn| < nli>ITOD\Xn - xn-l\ + Hxn-1 - r^nl + ¡T^yn - Tknxn ||]

< lim hn\\yn -xn\\ < lim \\yn -Xn-1\\ + lim \\xn -Xn-1\\

mœ mœ mœ

< nlim [fin1 T^Xn - Xn-11 + Sn\\Vn - Xn-1 \\]

< nlimro 1 Ti(n)Xn - Ti(n) Xn 1 + fin 1 Ti(n)Xn - Xn-11 ]

< nlim [Pnhn\\Xn - Xn\\ + fin1 T^Xn - Xn1 + fin\\Xn-1-Xn\\\. (23) It follows from (22), (23), conditions (ii) and (iv) that

lim ¡T^X - Xn | =0. (24)

Since for each n > N, n = (n - N)(modN), n = (k(n) - 1)N + i(n), hence n - N = [(k(n) -1) - 1]N + i(n -N), i.e. k(n -N) = k(n) -1 and i(n - N) = i(n). Therefore, we have

|| ) Xn - Tk-N Xn-N | = \Tn' ) Xn - Tn( ) Xn-N | < LWXn - Xn-N W (25)

|| >j,k(n)-1 || _ || rpk(n-N) | /O^A

|| T n-N Xn-N - Xn-N || = || T n-N Xn-N - Xn-N 11. (6)

In view of (25) and (26), we have

llxn-1 - TnXn\\ < |xn-1 - Tk nxn | + | TnXn - Tn nxn |

< \\Xn - Xn-l\\ + ^n - T^X n| + L |xn Tn( ) Xn |

< \Xn - Xn-1 \ + |xn - Tn ( )Xn |

+ L( |Tn( ) Xn - Tn-N Xn-N | + W Tn-N Xn-N - Xn | )

< \Xn - Xn-1 \\ + \Xn - Tn ( )Xn |

+ (L2 + L)\\Xn - Xn-N H + L | Tn_nNNNXn-N - Xn-N |. (27)

From (24) and (27), it is obviously that limn—TO \\Xn-1 - TnXn \ = 0, which implies that lim \\Xn - TnXn\\ < lim (\Xn_1 - TnXn\\ + \\Xn -Xn-1\) = 0.

n—>to n—>TO '

Consequently, we obtain that for all i e I

\\Xn - Tn+iXn\\ < \\Xn - Xn+i \ + \Xn+i - Tn+iXn+i\\ + \\Tn+iXn+i - Tn+iXn\\

< (1+ L)\\Xn - Xn+M + \Xn+i - Tn+iXn+M — 0 (n —TO). (28)

By virtue of (28), we have limn—TO \\Xn - TiXn \ = 0 for all i e I.

Since E is uniformly convex, every bounded subset of E is weakly compact. Again since {Xn} is a bounded subset in K, there exists a subsequence {Xnk} of {Xn} such that {Xnk} converges weakly to q in K, and limnk—TO \\Xnk - TiXnk \ = 0 for all i e I .By Lemma 2.2, we have that (I - Ti)q = 0. Hence, q e F(Ti) for all i e I. Therefore, q e F(T).

Next, we prove that {Xn} converges weakly to q. Suppose that contrary, then there exists a subsequence {Xnj} of {Xn} such that {Xnj} converges weakly to qi e K and q = qi. Using the same method, we can prove that q1 e F(T) and limit limn—TO \\Xn - q1 \ exists. Without loss generality, we assume that limn—TO \\Xn - q\\ = d1, limn—TO \\Xn - q1 H = d2, where d1, d2 are two nonnegative constants. By virtue of the Opial's condition of E, we have

d1 = lim sup \\Xnk - q\\ < lim sup \\Xnk - q1H = lim sup \\Xn - q1H

nk—TO nk—TO n—TO

= limsup \\Xnj - q1 \ < limsup \\Xnj - q\\ = d1.

nj—TO nj—TO

This is contradictory. Hence, q = q1, which implies that {Xn} converges weakly to q. The proof of Theorem 3.2 is completed. □

Theorem 3.3 LetE be a real uniformly convex Banach space andK be a nonempty, closed and convex subset ofE. Let T1, T2,...,TN : K — K be N asymptotically nonexpansive mappings with F(T) = P|n=1 F(Ti) = 0 and at least there exists Ti (i e I), it is semicompact. Let {un} and {vn} are two bounded sequences inK. If {an}, {pn}, {Yn}, {Sn} be four real sequences in [0,1] satisfying the following conditions:

(i) an + Yn < 1 and pn + Sn < 1 for all n > 1;

(ii) 0 < liminfn—TO an < limsupn—TO an <1, limsupn—TO pn <1;

(iii) eto=1 Yn < to, £TO=1 Sn < to, eto=1(hn -1) < to;

(iv) ETO=1 \\xn -xn\\ < to.

Then the perturbed composite implicit iterative sequence {xn} defined by (7) converges strongly to a common fixed point of {T1, T2,..., TN} in K.

Proof Without loss of generality, we assume that T1 is semicompact. By Theorem 3.2, we have limn—TO \\xn - T1xn\ = 0. Hence, there exists a subsequence {xnj} of {xn} such that {x„j} — x* as j —to. Therefore, we have for all i e I

fTix* - x*f < f Tix Tixnj f + \\ Tixnj xnj \\ + fxnj x *f. (29)

It follows from (29) that \\ Tix* -x* \\ = 0 for all i e I. This implies thatx* e F(T). Therefore, x* be a common fixed point of {Ti, i e I}. By virtue of Lemma 2.6, limn—TO \\xn - x* \\ exists. It follows from xnj ^ x e E that limn—«, \\xn - x*\\ = 0. Hence, the perturbed composites implicit iterative sequence {xn} generated by (7) strongly converges to a common fixed point of {Ti, i e I}. This completes the proof of Theorem 3.3. □

Corollary 3.4 Let E be a real Banach space and K be a nonempty closed convex subset ofE. Let T1, T2,...,TN : K — K be N asymptotically nonexpansive mappings with F (T) = Pl^ F(Ti) = 0 and let {un} is a bounded sequence in K. If {an}, {Yn} be two real sequences in [0,1] satisfying the following conditions:

(i) an + Yn < 1 for all n > 1;

(ii) limsupn—TO an <1;

(iii) EYn < TO, ETO=1(hn - 1) < TO;

(iv) E TO=1 Pn - xn\\ < TO.

Then the perturbed implicit iterative sequence {xn} defined by (9) converges strongly to a common fixed point of {T1, T2,...,TN} ifandonly if liminfn—TO d(xn, F(T)) = 0.

Proof It is enough to take pn = 1, Sn = 0 for all n e N in Theorem 3.1. □

Corollary 3.5 Let E be a real uniformly convex Banach space satisfying Opial's condition and K be a nonempty closed convex subset ofE. Let T1, T2,...,TN : K — K be N asymptotically nonexpansive mappings with F(T) = pN1 F(Ti) = 0 and let {un} is a bounded sequence in K. If {an}, {Yn} be two real sequences in [0,1] satisfying the following conditions:

(i) an + Yn < 1 for all n > 1;

(ii) 0 < liminfn—TO an < limsupn—TO an <1;

(iii) EYn < to, ETO=1(hn -1) < TO.

Then the Mann type iterative sequence {xn} defined by (8) converges weakly to a common fixed point of {T1, T2,..., TN} in K.

Proof It is sufficient to take = Sn = 0 for all n e N in Theorem 3.2. □

Corollary 3.6 LetE be a real uniformly conveX Banach space and K be a nonempty closed conveX subset ofE. Let T1, T2,...,TN : K — K be N asymptotically noneXpansive mappings with F(T) = f|; = F(Ti) = 0 and at least there eXists Ti (i e I), it is semicompact. Let {un} is a bounded sequence in K. If{an}, {yn} be two real sequences in [0,1] satisfying the following conditions:

(i) an + yn < 1 for all n > 1;

(ii) 0 < liminfn—TO an < limsupn—TO an <1;

(iii) eto1 Yn < to, eto=1(hn -1)< to.

Then the Mann type iterative sequence {Xn} defined by (8) converges strongly to a common fiXedpoint of {T1, T2,..., TN} in K.

Proof It is enough to take jin = Sn = 0 for all n e N in Theorem 3.3. □

Competing interests

The author did not provide this information.

Acknowledgements

The authors are gratefulto the anonymous referee forvaluable suggestions which helped to improve this manuscript.

Received: 4 July 2012 Accepted: 26 March 2013 Published: 12 April 2013

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doi:10.1186/1687-1812-2013-97

Cite this article as: Wang: Convergence of perturbed composite implicit iteration process for a finite family of asymptotically nonexpansive mappings. Fixed Point Theory and Applications 2013 2013:97.

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