- Research Article
- Open access
- Published:
Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space
Fixed Point Theory and Applications volume 2009, Article number: 483497 (2009)
Abstract
We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover, we also apply our results to a class of relatively nonexpansive mappings, and hence, we immediately obtain the results announced by Qin and Su's result (2007), Nilsrakoo and Saejung's result (2008), Su et al.'s result (2008), and some known corresponding results in the literatures.
1. Introduction
Let be a nonempty closed convex subset of a real Banach space
. A mapping
is said to be nonexpansive if
for all
We denote by
the set of fixed points of
, that is
. A mapping
is said to be quasi-nonexpansive if
and
for all
and
. It is easy to see that if
is nonexpansive with
, then it is quasi-nonexpansive. Some iterative processes are often used to approximate a fixed point of a nonexpansive mapping. The Mann's iterative algorithm was introduced by Mann [1] in 1953. This iterative process is now known as Mann's iterative process, which is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ1_HTML.gif)
where the initial guess is taken in
arbitrarily and the sequence
is in the interval
.
In 1976, Halpern [2] first introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ2_HTML.gif)
see also Browder [3]. He pointed out that the conditions and
are necessary in the sence that, if the iteration (1.2) converges to a fixed point of
, then these conditions must be satisfied.
In 1974, Ishikawa [4] introduced a new iterative scheme, which is defined recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ3_HTML.gif)
where the initial guess is taken in
arbitrarily and the sequences
and
are in the interval
.
Concerning a family of nonexpansive mappings it has been considered by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings; see, for example, [5]. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [6]).
Zhang and Su [7] introduced the following implicit hybrid method for a finite family of nonexpansive mappings in a real Hilbert space:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ4_HTML.gif)
where ,
and
are sequences in
and
for some
and
for some
.
In 2008, Nakprasit et al. [8] established weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. In the same year, Cho et al. [9] introduced the normal Mann's iterative process and proved some strong convergence theorems for a finite family nonexpansive mapping in the framework Banach spaces.
To find a common fixed point of a family of nonexpansive mappings, Aoyama et al. [10] introduced the following iterative sequence. Let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ5_HTML.gif)
for all , where
is a nonempty closed convex subset of a Banach space,
is a sequence of
and
is a sequence of nonexpansive mappings. Then they proved that, under some suitable conditions, the sequence
defined by (1.5) converges strongly to a common fixed point of
.
In 2008, by using a (new) hybrid method, Takahashi et al. [11] proved the following theorem.
Theorem 1.1 (Takahashi et al. [11]).
Let be a Hilbert space and let
be a nonempty closed convex subset of
. Let
and
be families of nonexpansive mappings of
into itself such that
and let
. Suppose that
satisfies the NST-condition
with
. For
and
, define a sequence
of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ6_HTML.gif)
where for all
and
is said to satisfy the NST-condition
with
if for each bounded sequence
,
implies that
for all
. Then,
converges strongly to
.
Note that, recently, many authors try to extend the above result from Hilbert spaces to a Banach space setting.
Let be a real Banach space with dual
. Denote by
the duality product. The normalized duality mapping
from
to
is defined by
for all
. The function
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ7_HTML.gif)
A mapping is said to be hemi-relatively nonexpansive (see [12]) if
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ8_HTML.gif)
A point in
is said to be an asymptotic fixed point of
[13] if
contains a sequence
which converges weakly to
such that the strong
. The set of asymptotic fixed points of
will be denoted by
. A hemi-relatively nonexpansive mapping
from
into itself is called relatively nonexpansive if
; see [14–16]) for more details.
On the other hand, Matsushita and Takahashi [17] introduced the following iteration. A sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ9_HTML.gif)
where the initial guess element is arbitrary,
is a real sequence in
,
is a relatively nonexpansive mapping, and
denotes the generalized projection from
onto a closed convex subset
of
. Under some suitable conditions, they proved that the sequence
converges weakly to a fixed point of
.
Recently, Kohsaka and Takahashi [18] extended iteration (1.9) to obtain a weak convergence theorem for common fixed points of a finite family of relatively nonexpansive mappings by the following iteration:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ10_HTML.gif)
where and
with
, for all
. Moreover, Matsushita and Takahashi [14] proposed the following modification of iteration (1.9) in a Banach space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ11_HTML.gif)
and proved that the sequence converges strongly to
.
Qin and Su [15] showed that the sequence , which is generated by relatively nonexpansive mappings
in a Banach space
, as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ12_HTML.gif)
converges strongly to
Moreover, they also showed that the sequence , which is generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ13_HTML.gif)
converges strongly to
In 2008, Nilsrakoo and Saejung [19] used the following Mann's iterative process:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ14_HTML.gif)
and showed that the sequence converges strongly to a common fixed point of a countable family of relatively nonexpansive mappings.
Recently, Su et al. [12] extended the results of Qin and Su [15], Matsushita and Takahashi [14] to a class of closed hemi-relatively nonexpansive mapping. Note that, since the hybrid iterative methods presented by Qin and Su [15] and Matsushita and Takahashi [14] cannot be used for hemi-relatively nonexpansive mappings. Thus, as we know, Su et al. [12] showed their results by using the method as a monotone (CQ) hybrid method.
In this paper, motivated by Qin and Su [15], Nilsrakoo and Saejung [19], we consider the modified Ishikawa iterative (1.12) and Halpern iterative processes (1.13), which is different from those of (1.12)–(1.14), for countable hemi-relatively nonexpansive mappings. By using the shrinking projection method, some strong convergence theorems in a uniformly convex and uniformly smooth Banach space are provided. Our results extend and improve the recent results by Nilsrakoo and Saejung's result [19], Qin and Su [15], Su et al. [12], Takahashi et al.'s theorem [11], and many others.
2. Preliminaries
In this section, we will recall some basic concepts and useful well-known results.
A Banach space is said to be strictly convex if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ15_HTML.gif)
for all with
and
. It is said to be uniformly convex if for any two sequences
in
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ16_HTML.gif)
holds.
Let be the unit sphere of
. Then the Banach space
is said to be smooth if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ17_HTML.gif)
exists for each It is said to be uniformly smooth if the limit is attained uniformly for
. In this case, the norm of
is said to be Gâteaux differentiable. The space
is said to have uniformly Gâteaux differentiable if for each
, the limit (2.3) is attained uniformly for
. The norm of
is said to be uniformly Fréchet differentiable (and
is said to be uniformly smooth) if the limit (2.3) is attained uniformly for
.
In our work, the concept duality mapping is very important. Here, we list some known facts, related to the duality mapping , as follows.
(a) (
, resp.) is uniformly convex if and only if
(
, resp.) is uniformly smooth.
(b) for each
.
-
(c)
If
is reflexive, then
is a mapping of
onto
.
-
(d)
If
is strictly convex, then
for all
.
-
(e)
If
is smooth, then
is single valued.
-
(f)
If
has a Fr
chet differentiable norm, then
is norm to norm continuous.
-
(g)
If
is uniformly smooth, then
is uniformly norm to norm continuous on each bounded subset of
.
-
(h)
If
is a Hilbert space, then
is the identity operator.
For more information, the readers may consult [20, 21].
If is a nonempty closed convex subset of a real Hilbert space
and
is the metric projection, then
is nonexpansive. Alber [22] has recently introduced a generalized projection operator
in a Banach space
which is an analogue representation of the metric projection in Hilbert spaces.
The generalized projection is a map that assigns to an arbitrary point
the minimum point of the functional
, that is,
, where
is the solution to the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ18_HTML.gif)
Notice that the existence and uniqueness of the operator is followed from the properties of the functional
and strict monotonicity of the mapping
, and moreover, in the Hilbert spaces setting we have
. It is obvious from the definition of the function
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ19_HTML.gif)
Remark 2.1.
If is a strictly convex and a smooth Banach space, then for all
,
if and only if
, see Matsushita and Takahashi [14].
To obtain our results, following lemmas are important.
Lemma 2.2 (Kamimura and Takahashi [23]).
Let be a uniformly convex and smooth Banach space and let
. Then there exists a continuous strictly increasing and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ20_HTML.gif)
for all
Lemma 2.3 (Kamimura and Takahashi [23]).
Let be a uniformly convex and smooth real Banach space and let
be two sequences of
. If
and either
or
is bounded, then
.
Lemma 2.4 (Alber [22]).
Let be a nonempty closed convex subset of a smooth real Banach space E and
. Then,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ21_HTML.gif)
Lemma 2.5 (Alber [22]).
Let be a reflexive strict convex and smooth real Banach space, let
be a nonempty closed convex subset of E and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ22_HTML.gif)
Lemma 2.6 (Matsushita and Takahashi [14]).
Let be a strictly convex and smooth real Banach space, let
be a closed convex subset of E, and let T be a hemi-relatively nonexpansive mapping from C into itself. Then F(T) is closed and convex.
Let be a subset of a Banach space
and let
be a family of mappings from
into
. For a subset
of
, one says that
(a) satisfies condition AKTT if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ23_HTML.gif)
(b) satisfies condition
AKTT if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ24_HTML.gif)
For more information, see Aoyama et al. [10].
Lemma 2.7 (Aoyama et al. [10]).
Let be a nonempty subset of a Banach space
and let
be a sequence of mappings from
into
. Let
be a subset of
with
satisfying condition AKTT, then there exists a mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ25_HTML.gif)
and
Inspired by Lemma 2.7, Nilsrakoo and Saejung [19] prove the following results.
Lemma 2.8 (Nilsrakoo and Saejung [19]).
Let be a reflexive and strictly convex Banach space whose norm is Fr
chet differentiable, let
be a nonempty subset of a Banach space
, and let
be a sequence of mappings from
into
. Let
be a subset of
with
satisfies condition
, then there exists a mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ26_HTML.gif)
and
Lemma 2.9 (Nilsrakoo and Saejung [19]).
Let be a reflexive and strictly convex Banach space whose norm is Fr
chet differentiable, let
be a nonempty subset of a Banach space
and let
be a sequence of mappings from
into
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Then there exists a mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ27_HTML.gif)
3. Modified Ishikawa Iterative Scheme
In this section, we establish the strong convergence theorems for finding common fixed points of a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. It is worth mentioning that our main theorem generalizes recent theorems by Su et al. [12] from relatively nonexpansive mappings to a more general concept. Moreover, our results also improve and extend the corresponding results of Nilsrakoo and Saejung [19]. In order to prove the main result, we recall a concept as follows. An operator in a Banach space is closed if
and
, then
.
Theorem 3.1.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of hemi-relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
and
are sequences in
such that
and
and let a sequence
in
by the following algorithm be:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ28_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. If
is uniformly continuous for all
, then
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
We first show that is closed and convex for each
. Obviously, from the definition of
, we see that
is closed for each
. Now we show that
is convex for any
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ29_HTML.gif)
this implies that is a convex set. Next, we show that
for all
. Indeed, let
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ31_HTML.gif)
Substituting (3.4) into (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ32_HTML.gif)
This means that, for all
. Consequently, the sequence
is well defined. Moreover, since
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ33_HTML.gif)
for all . Therefore,
is nondecreasing.
By the definition of and Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ34_HTML.gif)
for all . Thus,
is a bounded sequence. Moreover, by (2.5), we know that
is bounded. So,
exists. Again, by Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ35_HTML.gif)
for all . Thus,
as
.
Next, we show that is a Cauchy sequence. Using Lemma 2.2, for
such that
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ36_HTML.gif)
where is a continuous stricly increasing and convex function with
. Then the properties of the function
yield that
is a Cauchy sequence. Thus, we can say that
converges strongly to
for some point
in
. However, since
and
is bounded, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ37_HTML.gif)
Therefore as
.
Since , from the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ38_HTML.gif)
for all . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ39_HTML.gif)
By using Lemma 2.3, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ40_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ41_HTML.gif)
For each we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ42_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ43_HTML.gif)
By (3.14) and , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ44_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ45_HTML.gif)
By (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ46_HTML.gif)
Since is uniformly continuous, by (3.13) and (3.18), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ47_HTML.gif)
as , and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ48_HTML.gif)
Based on the hypothesis, we now consider the following two cases.
Case 1.
satisfies condition
. Applying Lemma 2.8 to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ49_HTML.gif)
Case 2.
satisfies condition AKTT. Apply Lemma 2.7 to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ50_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ51_HTML.gif)
Therefore, from the both two cases, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ52_HTML.gif)
Since is closed and
, we have
Moreover, by (3.7), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ53_HTML.gif)
for all . Therefore,
This completes the proof.
Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we obtain the following result for a countable family of relatively nonexpansive mappings of modified Ishikawa iterative process.
Corollary 3.2.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
and
are sequences in
such that
and
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ54_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. If
is uniformly continuous for all
, then
converges strongly to
, where
is the generalized projection from
onto
.
Theorem 3.3.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of hemi-relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ55_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
Proof.
In Theorem 3.1, if for all
then (3.1) reduced to (3.28).
Corollary 3.4.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ56_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
Notice that every uniformly continuous mapping must be a continuous and closed mapping. Then setting for all
, in Theorems 3.1 and 3.3, we immediately obtain the following results.
Corollary 3.5.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a closed hemi-relatively nonexpansive mapping such that
. Assume that
and
are sequences in
such that
and
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ57_HTML.gif)
for , where
is the single-valued duality mapping on
. If
is uniformly continuous, then
converges strongly to
.
Corollary 3.6.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a closed relatively nonexpansive mapping such that
. Assume that
and
are sequences in
such that
and
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ58_HTML.gif)
for , where
is the single-valued duality mapping on
. If
is uniformly continuous, then
converges strongly to
.
Proof.
Since a closed relatively nonexpansive mapping is a closed hemi-relatively one, Corollary 3.6 is implied by Corollary 3.5.
Corollary 3.7.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a closed hemi-relatively nonexpansive mapping from
into itself such that
. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ59_HTML.gif)
for , where
is the single-valued duality mapping on
. Then
converges strongly to
.
Corollary 3.8.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a closed relatively nonexpansive mapping from
into itself such that
. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ60_HTML.gif)
for , where
is the single-valued duality mapping on
. Then
converges strongly to
.
Similarly, as in the proof of Theorem 3.1, we obtain the following results.
Theorem 3.9.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of hemi-relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
and
are sequences in
such that
and
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ61_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. If
is uniformly continuous for all
, then
converges strongly to
, where
is the generalized projection from
onto
.
Corollary 3.10.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be closed hemi-relatively nonexpansive mappings from
into itself such that
. Assume that
and
are sequences in
such that
and
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ62_HTML.gif)
for , where
is the single-valued duality mapping on
. If
is uniformly continuous, then
converges strongly to
.
Theorem 3.11.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of relatively nonexpansive mappings from
into itself such that
is a nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ63_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
Proof.
Putting , for all
, in Theorem 3.9 we immediately obtain Theorem 3.11.
Corollary 3.12.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be closed hemi-relatively nonexpansive mappings from
into itself such that
. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ64_HTML.gif)
for , where
is the single-valued duality mapping on
. Then
converges strongly to
.
Remark 3.13.
Our results extend and improve the corresponding results in the following senses.
-
(i)
Corollary 3.10 improves Theorem 2.1 of Qin and Su [15] from relatively nonexpansive mappings to more general hemi-relatively nonexpansive mappings.
-
(ii)
Theorem 3.11 improves the algorithm in Theorem 3.1 of Nilsakoo and Saejung [19] from the Mann iteration process to modify Ishikawa iteration process and from countable relatively nonexpansive mappings to more general countable hemi-relatively nonexpansive mappings; that is, we relax the strong restriction
. From (i) and (ii), it means that we relax the strongly restriction as
from the assumption.
-
(iii)
Corollary 3.12 improves Theorem 3.1 of Matsushita and Takahashi [14] from relatively nonexpansive mappings to more general hemi-relatively nonexpansive mappings in a Banach space.
4. Halpern Iterative Scheme
In this section, we prove the strong convergence theorems for finding common fixed points of a countable family of hemi-relatively nonexpansive mappings, which can be viewed as a generalization of the recently result of [15, Theorem 2.2].
Theorem 4.1.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of hemi-relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ65_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
Proof.
As in the proof of Theorem 3.1, we have that is closed and convex for each
.
Next, we show that for all
. Indeed, let
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ66_HTML.gif)
This means that, for all
From Theorem 3.1, we obtain
and
exists. Since
and hence
, we also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ67_HTML.gif)
for all . Since
, thus,
as
.
By using the same argument as in Theorem 3.1, we obtain that is a Cauchy sequence, thus
converges strongly to
for some point
in
. By using Lemma 2.3, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ68_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ69_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ70_HTML.gif)
this gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ71_HTML.gif)
By (4.5) and , we obtain
Since
is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ72_HTML.gif)
It follows from (4.4) that as
, and since
is uniformly norm-to-norm continuous on bounded sets, we get
From the conditions
, AKTT, Lemmas 2.7 and 2.8, by using the same line as in the proof of Theorem 3.1, the both two cases, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ73_HTML.gif)
Finally, we prove that , where
. Let
be a subsequence of
such that
Replacing
, from
and
, we have
. On the other hand, from weakly lower semicontinuity of the norm, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ74_HTML.gif)
From the definition of , since
, we have
This implies
. Using the Kadec-Klee property ([24]) of the space
, we obtain that
converges strongly to
. Since
is an arbitrary weakly convergent sequence of
, we can conclude that
convergence strongly to
Corollary 4.2.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a closed hemi-relatively nonexpansive mapping from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ75_HTML.gif)
for , where
is the single-valued duality mapping on
. Then
converges strongly to
.
Proof.
By setting for all
, we immediately obtain the result.
Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we immediately obtain the following corollaries.
Corollary 4.3.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ76_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
Corollary 4.4.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a closed relatively nonexpansive mapping from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ77_HTML.gif)
for , where
is the single-valued duality mapping on
. Then
converges strongly to
.
Similarly, as in the proof of Theorem 4.1, we obtain the following result.
Theorem 4.5.
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of hemi-relatively nonexpansive mappings from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ78_HTML.gif)
for , where
is the single-valued duality mapping on
. Suppose that for each bounded subset
of
, the ordered pair
satisfies either condition AKTT or condition
. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
If , then Theorem 4.5 reduces to the following corollary.
Corollary 4.6 (see [15, Theorem 2.2]).
Let be a uniformly convex and uniformly smooth Banach space and let
be a nonempty bounded closed convex subset of
. Let
be a closed relatively nonexpansive mapping from
into itself such that
. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ79_HTML.gif)
for , where
is the single-valued duality mapping on
. Then
converges strongly to
.
5. Some Applications to Hilbert Spaces
It is well known that, in the Hilbert space setting, the concepts of hemi-relatively nonexpansive mappings and quasi-nonexpansive mappings are the equivalent. Thus, the following results can be obtained.
Theorem 5.1.
Let be a Hilbert space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of quasi-nonexpansive mappings from
into itself such that
is nonempty. Assume that
and
are sequences in
such that
and
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ80_HTML.gif)
for . Suppose that for each bounded subset
of
, the ordered pair
satisfies condition AKTT. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. If
is uniformly continuous for all
, then
converges strongly to
.
Proof.
Since is an identity operator, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ81_HTML.gif)
for every Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ82_HTML.gif)
for every and
Hence,
is quasi-nonexpansive if and only if
is hemi-relatively nonexpansive. Then, by Theorem 3.1, we obtain the result.
Theorem 5.2.
Let be a Hilbert space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of quasi-nonexpansive mappings from
into itself such that
is nonempty. Assume that
is sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ83_HTML.gif)
for . Suppose that for each bounded subset
of
, the ordered pair
satisfies condition AKTT. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
Proof.
In Theorem 5.1 setting for all
, then (5.1) reduces to (5.4).
Theorem 5.3.
Let be a Hilbert space and let
be a nonempty bounded closed convex subset of
. Let
be a sequence of quasi-nonexpansive mappings from
into itself such that
is nonempty. Assume that
is a sequence in
such that
and let a sequence
in
be defined by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F483497/MediaObjects/13663_2009_Article_1145_Equ84_HTML.gif)
for . Suppose that for each bounded subset
of
, the ordered pair
satisfies condition AKTT. Let
be the mapping from
into itself defined by
for all
and suppose that
is closed and
. Then
converges strongly to
.
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The authors would like to thank the referees for the valuable suggestions which helped to improve this manuscript. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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Petrot, N., Wattanawitoon, K. & Kumam, P. Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space. Fixed Point Theory Appl 2009, 483497 (2009). https://doi.org/10.1155/2009/483497
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DOI: https://doi.org/10.1155/2009/483497