- Research Article
- Open access
- Published:
On the Convergence of an Implicit Iterative Process for Generalized Asymptotically Quasi-Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 714860 (2010)
Abstract
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.
1. Introduction and Preliminaries
Let be a real Banach space and
.
is said to be uniformly convex if for any
there exists
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ2_HTML.gif)
  is said to bequasi-nonexpansive if
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ4_HTML.gif)
Then is quasi-nonexpansive but not nonexpansive.
is said to beasymptotically nonexpansive if there exists a positive sequence
with
as
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ6_HTML.gif)
is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ7_HTML.gif)
Putting , we see that
as
. Then (1.7) is reduced to the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ8_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ9_HTML.gif)
Putting , we see that
as
. Then (1.9) is reduced to the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ10_HTML.gif)
is said to be  generalized asymptotically nonexpansive if there exist two positive sequences
with
and
with
as
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ12_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ14_HTML.gif)
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 :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ15_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ16_HTML.gif)
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
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ17_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ18_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ19_HTML.gif)
If , where
denotes the identity mapping, for each
, then the implicit iterative process (1.18) is reduced to the following explicit iterative process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ20_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ22_HTML.gif)
for all and
such that
.
2. Main Results
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ23_HTML.gif)
Proof.
First, we show that the sequence generated in (1.18) is well defined. For each
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ24_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ26_HTML.gif)
Notice that . We see from the restrictions (a) and (b) that there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ27_HTML.gif)
where . It follows from (2.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ29_HTML.gif)
where and
are appropriate constants such that
and
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ30_HTML.gif)
In view of the restrictions (a) and (b), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ31_HTML.gif)
Since is a continuous, strictly increasing, and convex function with
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ32_HTML.gif)
Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ33_HTML.gif)
From Lemma 1.3, we also see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ34_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ35_HTML.gif)
In view of the restrictions (a) and (b), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ36_HTML.gif)
Since is a continuous, strictly increasing, and convex function with
, we obtain that (2.11) holds. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ37_HTML.gif)
In view of (2.10) and (2.11), we see from the restriction (b) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ38_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ39_HTML.gif)
Since for any positive integer , it can be written as
, where
, observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ40_HTML.gif)
Since for each ,
(mod
), on the other hand, we obtain from
that
. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ41_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ42_HTML.gif)
Substituting (2.20) into (2.18), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ43_HTML.gif)
In view of (2.11) and (2.17), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ44_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ45_HTML.gif)
It follows from (2.16) and (2.22) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ46_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ47_HTML.gif)
From (2.17) and (2.24), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ48_HTML.gif)
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ49_HTML.gif)
Letting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ50_HTML.gif)
In view of (2.10) and (2.16), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ51_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ52_HTML.gif)
In view of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ53_HTML.gif)
we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ54_HTML.gif)
In view of (2.10), (2.17), and (2.29), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ55_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ56_HTML.gif)
From (2.16) and (2.33), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ57_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ58_HTML.gif)
It follows from (2.17) and (2.35) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ59_HTML.gif)
Note that any subsequence of a convergent number sequence converges to the same limit. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ60_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ61_HTML.gif)
Since is Lipshcitz continuous, we obtain from (2.38) that
. Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ62_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ63_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ64_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ65_HTML.gif)
where . It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ66_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ67_HTML.gif)
For each , we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ68_HTML.gif)
Notice that . We see from the restrictions (a) and (b) that there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ69_HTML.gif)
where . Notice that the sequence
is bounded. It follows from (2.46) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ70_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F714860/MediaObjects/13663_2010_Article_1327_Equ71_HTML.gif)
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
.
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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
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DOI: https://doi.org/10.1155/2010/714860