Fixed Point Theory and Applications volume 2010, Article number: 714860 (2010)
The purpose of this paper is to introduce and consider a general implicit iterative process which includes Schu's explicit iterative processes and Sun's implicit iterative processes as special cases for a finite family of generalized asymptotically quasi-nonexpansive mappings. Strong convergence of the purposed iterative process is obtained in the framework of real Banach spaces.
Let 
be a real Banach space and 
. 
is said to be uniformly convex if for any 
there exists 
such that for any 
,
It is known that a uniformly convex Banach space is reflexive and strictly convex.
Let 
be a nonempty closed and convex subset of a Banach space 
. Let 
be a mapping. Denote by 
the fixed point set of 
.
Recall that 
is said to be nonexpansive if
is said to bequasi-nonexpansive if 
and
A nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive; however, the inverse may be not true. See the following example [1].
Example 1.1.
Let 
and define a mapping by 
by
Then 
is quasi-nonexpansive but not nonexpansive.
is said to beasymptotically nonexpansive if there exists a positive sequence 
with 
as 
such that
It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence 
. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972. It is known that if 
is a nonempty bounded closed convex subset of a uniformly convex Banach space 
, then every asymptotically nonexpansive mapping on 
has a fixed point. Further, the set 
of fixed points of 
is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.
is said to beasymptotically quasi-nonexpansive if 
, and there exists a positive sequence 
with 
as 
such that
is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
Putting 
, we see that 
as 
. Then (1.7) is reduced to the following:
The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk [3] (see also Bruck et al. [4]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if 
is a nonempty closed convex and bounded subset of a real Hilbert space, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point; see [5] more details.
is said to beasymptotically quasi-nonexpansive in the intermediate sense if it is continuous, 
, and the following inequality holds:
Putting 
, we see that 
as 
. Then (1.9) is reduced to the following:
is said to be generalized asymptotically nonexpansive if there exist two positive sequences 
with 
and 
with 
as 
such that
It is easy to see that the class of generalized asymptotically nonexpansive includes the class of asymptotically nonexpansive as a special case.
is said to begeneralized asymptotically quasi-nonexpansive if 
, and there exist two positive sequences 
with 
and 
with 
as 
such that
The class of generalized asymptotically quasi-nonexpansive was considered by Shahzad and Zegeye [6]; see [6, 7] for more details.
Recall that the modified Mann iteration which was introduced by Schu [8] generates a sequence 
in the following manner:
where 
is a sequence in the interval 
and 
: 
is an asymptotically nonexpansive mapping.
In 1991, Schu [8] obtained the following results.
Theorem Schu 1.
Let 
be a uniformly convex Banach space, 
closed bounded and convex, and 
asymptotically nonexpansive with sequence 
for which 
and 
is bounded away. Let 
be a sequence generated in (1.13). Then 
.
Theorem Schu 2.
Let 
be a uniformly convex Banach space, 
closed bounded and convex, and 
asymptotically nonexpansive with sequence 
for which 
and 
is bounded away. Let 
be a sequence generated in (1.13). Suppose that 
is compact for some positive integer 
. Then the sequence 
converges strongly to some fixed point of 
.
Theorem Schu 3.
Let 
be a uniformly convex Banach space, 
closed bounded and convex, and 
asymptotically nonexpansive with sequence 
for which 
and 
is bounded away. Let 
be a sequence generated in (1.13). Suppose that there exists a nonempty compact and convex subset 
of 
and 
such that
Then the sequence 
converges strongly to some fixed point of 
.
In 2007, Shahzad and Zegeye [6] considered the following implicit iterative process for a finite family of generalized asymptotically quasi-nonexpansive mappings 
:
where 
is the initial value and 
is a sequence 
. Since for each 
, it can be written as 
, where 
, 
is a positive integer, and 
as 
. Hence the above table can be rewritten in the following compact form:
We remark that the implicit iterative process (1.16) was first considered by Sun [9]; see [9] for more details.
Shahzad and Zegeye [6] obtained the following results.
Theorem SZ 1.
Let 
be a real uniformly convex Banach space and 
be a nonempty closed convex subset of 
. Let 
, where 
, be 
uniformly Lipschitz, generalized asymptotically quasi-nonexpansive self-mappings of 
with 
, 
such that 
and 
for all 
. Suppose that 
and there exists one member 
in 
which is either semicompact or satisfies condition 
. Let 
for some 
. From arbitrary 
, define the sequence 
by (1.16). Then 
converges strongly to a common fixed point of the mappings 
.
Theorem SZ 2.
Let 
be a real uniformly convex Banach space and 
a nonemptyclosed convex subset of 
. Let 
, where 
, be 
generalized asymptotically quasi-nonexpansive self-mappings of 
with 
, 
such that 
and 
for all 
. Suppose that 
is closed. Let 
for some 
. From arbitrary 
, define the sequence 
by (1.16). Then 
converges strongly to a common fixed point of the mappings 
if and only if 
.
In this paper, motivated by the above results, we consider the following implicit iterative process for two finite families of generalized asymptotically quasi-nonexpansive mappings 
and 
:
where 
is the initial value, 
is a bounded sequence in 
, and 
, 
, 
, and 
are sequences 
such that 
for each 
. Since for each 
, it can be written as 
, where 
, 
is a positive integer and 
as 
. Hence the above table can be rewritten in the following compact form:
We remark that our implicit iterative process (1.18) which includes the explicit iterative process (1.13) and the implicit iterative process (1.16) as special cases is general.
If 
, where 
denotes the identity mapping, for each 
, then the implicit iterative process (1.18) is reduced to the following implicit iterative process:
If 
, where 
denotes the identity mapping, for each 
, then the implicit iterative process (1.18) is reduced to the following explicit iterative process:
The purpose of this paper is to study the convergence of the implicit iteration process (1.18) for two finite families of generalized asymptotically quasi-nonexpansive mappings. Strong convergence theorems are obtained in the framework of real Banach spaces. The results presented in this paper improve and extend the corresponding results in Shahzad and Zegeye [6], Sun [9], Chang et al. [10], Chidume and Shahzad [11], Guo and Cho [12], Kim et al. [13], Qin et al. [14], Thianwan and Suantai [15], Xu and Ori [16], and Zhou and Chang [17].
In order to prove our main results, we also need the following lemmas.
Lemma 1.2 (see [18]).
Let 
, 
, and 
be three nonnegative sequences satisfying the following condition:
where 
is some positive integer. If 
and 
, then 
exists.
Lemma 1.3 (see [19]).
Let 
be a real uniformly convex Banach space, 
a positive number, and 
a closed ball of 
. Then there exists a continuous, strictly increasing, and convex function 
with 
such that
for all 
and 
such that 
.
Lemma 2.1.
Let 
be a real uniformly convex Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
and 
a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
, where 
and 
and 
, where 
and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.18). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
, 
, and 
, where 
, for all 
;
(b)
.
Then
Proof.
First, we show that the sequence 
generated in (1.18) is well defined. For each 
, define a mapping 
as follows:
Notice that
From the restriction (a), we see that 
is a contraction for each 
. From Banach contraction mapping principle, we can prove that the sequence 
generated in (1.18) is well defined.
Fixing 
, we see that
Notice that 
. We see from the restrictions (a) and (b) that there exists a positive integer 
such that
where 
. It follows from (2.4) that
where 
is an appropriate constant such that 
. In view of the restrictions (a) and (b), we obtain from Lemma 1.2 that 
exists. It follows that the sequence 
is bounded. In view of Lemma 1.3, we see that
where 
and 
are appropriate constants such that 
and 
. This implies that
In view of the restrictions (a) and (b), we obtain that
Since 
is a continuous, strictly increasing, and convex function with 
, we obtain that
Next, we show that
From Lemma 1.3, we also see that
This implies that
In view of the restrictions (a) and (b), we obtain that
Since 
is a continuous, strictly increasing, and convex function with 
, we obtain that (2.11) holds. Notice that
In view of (2.10) and (2.11), we see from the restriction (b) that
which implies that
Since for any positive integer 
, it can be written as 
, where 
, observe that
Since for each 
, 
(mod 
), on the other hand, we obtain from 
that 
. That is,
Notice that
Substituting (2.20) into (2.18), we arrive at
In view of (2.11) and (2.17), we obtain that
Notice that
It follows from (2.16) and (2.22) that
Notice that
From (2.17) and (2.24), we arrive at
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that
Letting 
, we have
In view of (2.10) and (2.16), we see that
Observe that
In view of
we arrive at
In view of (2.10), (2.17), and (2.29), we obtain that
Notice that
From (2.16) and (2.33), we see that
On the other hand, we have
It follows from (2.17) and (2.35) that
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that
This completes the proof.
Recall that a mapping 
is said to be semicompact if for any bounded sequence 
in 
such that 
as 
, then there exists a subsequence 
such that 
.
Next, we give strong convergence theorems with the help of the semicompactness.
Theorem 2.2.
Let 
be a real uniformly convex Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
, and let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
, where 
and 
and 
, where 
and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.18). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
, 
, and 
, where 
, for all 
;
(b)
.
If one of 
or one of 
is semicompact, then the sequence 
converges strongly to some point in 
.
Proof.
Without loss of generality, we may assume that 
is semicompact. From (2.38), we see that there exits a subsequence 
of 
converging strongly to 
. For each 
, we get that
Since 
is Lipshcitz continuous, we obtain from (2.38) that 
. Notice that
Since 
is Lipshcitz continuous, we obtain from (2.27) that 
. This means that 
. In view of Lemma 2.1, we obtain that 
exists. Therefore, we can obtain the desired conclusion immediately.
If 
, where 
denotes the identity mapping, for each 
, then Theorem 2.2 is reduced to the following.
Corollary 2.3.
Let 
be a real uniformly convex Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
, and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.19). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
and 
, where 
, for all 
;
(b)
.
If one of 
is semicompact, then the sequence converges 
strongly to some point in 
.
If 
, where 
denotes the identity mapping, for each 
, then Theorem 2.2 is reduced to the following.
Corollary 2.4.
Let 
be a real uniformly convex Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.20). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
, 
, and 
, for all 
;
(b)
.
If one of 
is semicompact, then the sequence 
converges strongly to some point in 
.
In 2005, Chidume and Shahzad [11] introduced the following conception. Recall that a family 
with 
is said to satisfy Condition
on 
if there is a nondecreasing function 
with 
and 
for all 
such that for all 
Based on Condition 
, we introduced the following conception for two finite families of mappings. Recall that two families 
and 
with 
are said to satisfy Condition 
on 
if there is a nondecreasing function 
with 
and 
for all 
such that for all 
Next, we give strong convergence theorems with the help of Condition 
.
Theorem 2.5.
Let 
be a real uniformly convex Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
, and let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
, where 
and 
and 
, where 
and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.18). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
, 
, and 
, where 
, for all 
;
(b)
.
If 
and 
satisfy Condition 
, then the sequence converges strongly to some point in 
.
Proof.
In view of Condition 
, we obtain from (2.27) and (2.38) that 
, which implies 
. Next, we show that the sequence 
is Cauchy. In view of (2.6), for any positive integers 
, where 
, we see that
where 
. It follows that
It follows that 
is a Cauchy sequence in 
and so 
converges strongly to some 
. Since 
and 
are Lipschitz for each 
, we see that 
is closed. This in turn implies that 
. This completes the proof.
If 
, where 
denotes the identity mapping, for each 
, then Theorem 2.2 is reduced to the following.
Corollary 2.6.
Let 
be a real uniformly convex uniformly convex Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
and where 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.19). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
and 
, where 
, for all 
;
(b)
.
If 
satisfies Condition 
, then the sequence 
converges strongly to some point in 
.
If 
, where 
denotes the identity mapping, for each 
, then Theorem 2.2 is reduced to the following.
Corollary 2.7.
Let 
be a real uniformly convex Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
, and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.20). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
, 
and 
, for all 
;
(b)
.
If 
satisfies Condition 
, then the sequence 
converges strongly to some point in 
.
Finally, we give a strong convergence theorem criterion.
Theorem 2.8.
Let 
be a real Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
, and let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
, where 
and 
and 
, where 
and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.18). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
, 
, and 
, where 
, for all 
;
(b)
.
Then 
converges strongly to some point in 
if and only if 
.
Proof.
The necessity is obvious. We only show the sufficiency. Assume that
For each 
, we see that
Notice that 
. We see from the restrictions (a) and (b) that there exists a positive integer 
such that
where 
. Notice that the sequence 
is bounded. It follows from (2.46) that
where 
is an appropriate constant such that 
. In view of the restrictions (a) and (b), we obtain from Lemma 1.2 that 
exists. This implies that
In view of Theorem 2.5, we can conclude the desired conclusion easily.
If 
, where 
denotes the identity mapping, for each 
, then Theorem 2.2 is reduced to the following.
Corollary 2.9.
Let 
be a real Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
and where 
. Let 
, 
, 
and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.19). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
and 
, where 
, for all 
;
(b)
.
Then 
converges strongly to some point in 
if and only if 
.
If 
, where 
denotes the identity mapping, for each 
, then Theorem 2.2 is reduced to the following.
Corollary 2.10.
Let 
be a real Banach space and 
a nonempty closed convex subset of 
. Let 
be a uniformly 
-Lipschitz and generalized asymptotically quasi-nonexpansive mapping with sequences 
and 
such that 
and 
for each 
. Assume that 
is nonempty. Let 
be a bounded sequence in 
, 
, and 
. Let 
, 
, 
, and 
be sequences in 
such that 
for each 
. Let 
be a sequence generated in (1.20). Assume that the following restrictions are satisfied:
(a)there exist constants 
such that 
, 
, and 
, for all 
;
(b)
.
Then 
converges strongly to some point in 
if and only if 
.
Dotson, WG Jr.: Fixed points of quasi-nonexpansive mappings. Australian Mathematical Society A 1972, 13: 167–170. 10.1017/S144678870001123X
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3
Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Israel Journal of Mathematics 1974, 17: 339–346. 10.1007/BF02757136
Bruck R, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloquium Mathematicum 1993,65(2):169–179.
Xu HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Analysis: Theory, Methods & Applications 1991,16(12):1139–1146. 10.1016/0362-546X(91)90201-B
Shahzad N, Zegeye H: Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps. Applied Mathematics and Computation 2007,189(2):1058–1065. 10.1016/j.amc.2006.11.152
Saejung S, Suantai S, Yotkaew P: a note on common fixed point of multistep Noor iteration with errors for a finite family of generalized asymptotically quasi-nonexpansive mapping. Abstract and Applied Analysis 2009, 2009:-9.
Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society 1991,43(1):153–159. 10.1017/S0004972700028884
Sun Z: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. Journal of Mathematical Analysis and Applications 2003,286(1):351–358. 10.1016/S0022-247X(03)00537-7
Chang SS, Tan KK, Lee HWJ, Chan CK: On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications 2006,313(1):273–283. 10.1016/j.jmaa.2005.05.075
Chidume CE, Shahzad N: Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2005,62(6):1149–1156. 10.1016/j.na.2005.05.002
Guo W, Cho YJ: On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings. Applied Mathematics Letters 2008,21(10):1046–1052. 10.1016/j.aml.2007.07.034
Kim JK, Nam YM, Sim JY: Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonxpansive type mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e2839-e2848. 10.1016/j.na.2009.06.090
Qin X, Cho YJ, Shang M: Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings. Applied Mathematics and Computation 2009,210(2):542–550. 10.1016/j.amc.2009.01.018
Thianwan S, Suantai S: Weak and strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. Scientiae Mathematicae Japonicae 2007,66(1):73–81.
Xu H-K, Ori RG: An implicit iteration process for nonexpansive mappings. Numerical Functional Analysis and Optimization 2001,22(5–6):767–773. 10.1081/NFA-100105317
Zhou Y, Chang S-S: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numerical Functional Analysis and Optimization 2002,23(7–8):911–921. 10.1081/NFA-120016276
Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications 1993,178(2):301–308. 10.1006/jmaa.1993.1309
Hao Y, Cho SY, Qin X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory and Applications 2010, -11.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Qin, X., Kang, S. & Agarwal, R. On the Convergence of an Implicit Iterative Process for Generalized Asymptotically Quasi-Nonexpansive Mappings. Fixed Point Theory Appl 2010, 714860 (2010). https://doi.org/10.1155/2010/714860
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/714860