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Convergence of perturbed composite implicit iteration process for a finite family of asymptotically nonexpansive mappings
Fixed Point Theory and Applications volume 2013, Article number: 97 (2013)
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 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, 007), Rafiq (Rostock. Math. Kolloqu. 62:21-39, 2007) and some others.
MSC:47H9, 47H10.
1 Introduction
Let E be a real Banach space, K be a nonempty convex subset of E. Let be a finite family of mappings from K into itself, and be the set of fixed points of (). denotes the set of common fixed points of .
Recently, Xu and Ori [1] have introduced an implicit iteration process for a finite family of nonexpansive mappings as follows:
where (here the modN function takes values in I), be a real sequence in , 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:
where , .
Chang [3] have discussed the convergence of the implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings as follows:
where , , and with as . Under the hypotheses and some appropriate conditions, they proved some results of weak and strong convergence for defined by (3). However, the condition is not too reasonable, because this implies that 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:
where , , , are four real sequences in satisfying and for all , , are two sequences in K and 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:
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:
where is a perturbation of , and satisfy . 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:
where , , , , , , are four real sequences in satisfying and for all , , are two sequences in K and is an initial point. be a sequence in K satisfying , which implies that (). Therefore, is known as the perturbation of , and is known as the perturbed sequence of . This sequence 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 , for all , we have
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 , for all , we have
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 for all , we have
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 .
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 , but compute an approximate point of , 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 defined by (7) for a finite family of asymptotically nonexpansive mappings under Opial’s condition, semicompact and 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 be a mapping.
-
1.
T is said to be semicompact, if for any bounded sequence in K such that (), then there exists a subsequence of such that ;
-
2.
T is said to be demiclosed at the origin, if for each sequence in K, the conditions weakly and strongly imply ;
-
3.
T is said to be asymptotically nonexpansive, if there exists a sequence with such that
(11) -
4.
Let T is said to be uniformly L-Lipschitizian if there exists a constant such that
Definition 2.2 [16]
A Banach space X is said to satisfy Opial’s condition if weakly as and imply that .
Lemma 2.1 Let K be a nonempty subset of E, be N asymptotically nonexpansive mappings. Then
-
(i)
there exists a sequence with such that
(12) -
(ii)
is uniformly Lipschitzian, i.e., there exists a constant L such that
(13)
Proof Since are N asymptotically nonexpansive mappings, then for every and , there exists with such that
Taking , then , and (12) holds.
An asymptotically nonexpansive mapping must is a uniformly Lipschitzian mapping. Hence, for every and , there exists such that
Taking , it is obvious that (13) holds. □
Lemma 2.2 [17]
Let E be a uniformly convex Banach space, K be a nonempty, closed and convex subset of E and be an asymptotically nonexpansive mapping. Then is demi-closed at zero, i.e., for each sequence in K, if convergence weakly to and converges strongly to 0, then .
Lemma 2.3 [18]
Let E be a Banach space satisfying Opial’s condition, be a sequence in E. Let be such that and exist. If and are two subsequences of which converge weakly to u and v, respectively, then .
Lemma 2.4 [19]
Let E be a uniformly convex Banach space, b, c be two constants with . Suppose that is a sequence in and , are two sequences in E. Then the conditions , , imply that , where d is a nonnegative constant.
Lemma 2.5 [20]
Let , , are three sequences of nonnegative real numbers, if there exists such that
where and . Then
-
(i)
exists;
-
(ii)
whenever .
Lemma 2.6 Let E be a real Banach space and K be a nonempty closed convex subset of E. Let be N asymptotically nonexpansive mappings with . Let and are two bounded sequences in K. If , , , be four real sequences in satisfying the following conditions:
-
(i)
and for all ;
-
(ii)
or ;
-
(iii)
, , ;
-
(iv)
.
Let be the perturbed composite implicit iterative sequence defined by (7), then exists for all .
Proof Take , it follows from (7) and Lemma 2.1 that
and
Substituting (15) into (14) and simplifying, we obtain
We notice the hypotheses on , and , by , there exists such that
It follows from (16) that for
Hence, we have
where
and
From condition (iii), it is obvious that . In addition, since , are all bounded, we deduce that form (iii)-(iv). By virtue of (17) and Lemma 2.5, we obtain that exists. This completes the proof of Lemma 2.6. □
3 Main results and proofs
Theorem 3.1 Let E be a real Banach space and K be a nonempty, closed and convex subset of E. Let be N asymptotically nonexpansive mappings with . Let and are two bounded sequences in K. If , , , be four real sequences in satisfying the following conditions:
-
(i)
and for all ;
-
(ii)
or ;
-
(iii)
, , ;
-
(iv)
.
Then the perturbed composite implicit iterative sequence defined by (7) converges strongly to a common fixed point of if and only if .
Proof The necessity of Theorem 3.1 is obvious. Now we prove the sufficiency of Theorem 3.1.
For arbitrary , it follows from (17) in Lemma 2.6 that
where and . Hence, we have
It follows from (18) and Lemma 2.5 that limit exists. By the assumption, we have . Consequently, for any given , there exists a positive integer () such that
and there exists such that , . By (18) and the inequality (), for any and all , we have
Hence, is a Cauchy sequence in E. By the completeness of E, we can assume that . Next we prove that is a close subset of K. Let is a sequence in which converges strongly to some p, then we have for any
Thus, , and is closed. Since , then . Consequently, defined by (7) converges strongly to a common fixed point of in K. This completes the proof of Theorem 3.1. □
Theorem 3.2 Let E be a real uniformly convex Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of E. Let be N asymptotically nonexpansive mappings with . Let and are two bounded sequences in K. If , , , be four real sequences in satisfying the following conditions:
-
(i)
and for all ;
-
(ii)
;
-
(iii)
, , ;
-
(iv)
.
Then the perturbed composite implicit iterative sequence defined by (7) converges weakly to a common fixed point of in K.
Proof First, we prove that for all .
For any , it follows from Lemma 2.6 that exists. Suppose that , we have from (7)
Since , then be a bounded sequence. By virtue of the condition (iii) and the boundedness of sequences and , we have

It follows from that . We have

Therefore, by (19), (20), (21), (ii) and Lemma 2.4, we obtain that
Hence,
which implies that for all . On the other hand, we also have
It follows from (22), (23), conditions (ii) and (iv) that
Since for each , , , hence , i.e. and . Therefore, we have
and
In view of (25) and (26), we have
From (24) and (27), it is obviously that , which implies that
Consequently, we obtain that for all
By virtue of (28), we have for all .
Since E is uniformly convex, every bounded subset of E is weakly compact. Again since is a bounded subset in K, there exists a subsequence of such that converges weakly to q in K, and for all . By Lemma 2.2, we have that . Hence, for all . Therefore, .
Next, we prove that converges weakly to q. Suppose that contrary, then there exists a subsequence of such that converges weakly to and . Using the same method, we can prove that and limit exists. Without loss generality, we assume that , , where , are two nonnegative constants. By virtue of the Opial’s condition of E, we have
This is contradictory. Hence, , which implies that converges weakly to q. The proof of Theorem 3.2 is completed. □
Theorem 3.3 Let E be a real uniformly convex Banach space and K be a nonempty, closed and convex subset of E. Let be N asymptotically nonexpansive mappings with and at least there exists (), it is semicompact. Let and are two bounded sequences in K. If , , , be four real sequences in satisfying the following conditions:
-
(i)
and for all ;
-
(ii)
, ;
-
(iii)
, , ;
-
(iv)
.
Then the perturbed composite implicit iterative sequence defined by (7) converges strongly to a common fixed point of in K.
Proof Without loss of generality, we assume that is semicompact. By Theorem 3.2, we have . Hence, there exists a subsequence of such that as . Therefore, we have for all
It follows from (29) that for all . This implies that . Therefore, be a common fixed point of . By virtue of Lemma 2.6, exists. It follows from that . Hence, the perturbed composites implicit iterative sequence generated by (7) strongly converges to a common fixed point of . 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 of E. Let be N asymptotically nonexpansive mappings with and let is a bounded sequence in K. If , be two real sequences in satisfying the following conditions:
-
(i)
for all ;
-
(ii)
;
-
(iii)
, ;
-
(iv)
.
Then the perturbed implicit iterative sequence defined by (9) converges strongly to a common fixed point of if and only if .
Proof It is enough to take , for all 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 of E. Let be N asymptotically nonexpansive mappings with and let is a bounded sequence in K. If , be two real sequences in satisfying the following conditions:
-
(i)
for all ;
-
(ii)
;
-
(iii)
, .
Then the Mann type iterative sequence defined by (8) converges weakly to a common fixed point of in K.
Proof It is sufficient to take for all in Theorem 3.2. □
Corollary 3.6 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let be N asymptotically nonexpansive mappings with and at least there exists (), it is semicompact. Let is a bounded sequence in K. If , be two real sequences in satisfying the following conditions:
-
(i)
for all ;
-
(ii)
;
-
(iii)
, .
Then the Mann type iterative sequence defined by (8) converges strongly to a common fixed point of in K.
Proof It is enough to take for all in Theorem 3.3. □
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Wang, X. Convergence of perturbed composite implicit iteration process for a finite family of asymptotically nonexpansive mappings. Fixed Point Theory Appl 2013, 97 (2013). https://doi.org/10.1186/1687-1812-2013-97
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DOI: https://doi.org/10.1186/1687-1812-2013-97