- Research Article
- Open access
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Approximation of Common Fixed Points of a Countable Family of Relatively Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 407651 (2009)
Abstract
We introduce two general iterative schemes for finding a common fixed point of a countable family of relatively nonexpansive mappings in a Banach space. Under suitable setting, we not only obtain several convergence theorems announced by many authors but also prove them under weaker assumptions. Applications to the problem of finding a common element of the fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem are also discussed.
1. Introduction and Preliminaries
Let be a nonempty subset of a Banach space
, and let
be a mapping from
into itself. When
is a sequence in
, we denote strong convergence of
to
by
and weak convergence by
. We also denote the weak
convergence of a sequence
to
in the dual
by
. A point
is an asymptotic fixed point of
if there exists
in
such that
and
. We denote
and
by the set of fixed points and of asymptotic fixed points of
, respectively. A Banach space
is said to be strictly convex if
for
and
. It is also said to be uniformly convex if for each
, there exists
such that
for
and
. The space
is said to be smooth if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ1_HTML.gif)
exists for all . It is also said to be uniformly smooth if the limit exists uniformly in
.
Many problems in nonlinear analysis can be formulated as a problem of finding a fixed point of a certain mapping or a common fixed point of a family of mappings. This paper deals with a class of nonlinear mappings, so-called relatively nonexpansive mappings introduced by Matsushita and Takahashi [1]. This type of mappings is closely related to the resolvent of maximal monotone operators (see [2–4]).
Let be a smooth, strictly convex and reflexive Banach space and let
be a nonempty closed convex subset of
. Throughout this paper, we denote by
the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ2_HTML.gif)
where is the normalized duality mapping from
to the dual space
given by the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ3_HTML.gif)
We know that if is smooth, strictly convex, and reflexive, then the duality mapping
is single-valued, one-to-one, and onto. The duality mapping
is said to be weakly sequentially continuous if
implies that
(see [5] for more details).
Following Matsushita and Takahashi [6], a mapping is said to be relatively nonexpansive if the following conditions are satisfied:
(R1) is nonempty;
(R2) for all
,
;
(R3).
If satisfies (R1) and (R2), then
is called relatively quasi-nonexpansive [7]. Obviously, relative nonexpansiveness implies relative quasi-nonexpansiveness but the converse is not true. Relatively quasi-nonexpansive mappings are sometimes called hemirelatively nonexpansive mappings. But we do prefer the former name because in a Hilbert space setting, relatively quasi-nonexpansive mappings are nothing but quasi-nonexpansive.
In [2], Alber introduced the generalized projection from
onto
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ4_HTML.gif)
If is a Hilbert space, then
and
becomes the metric projection of
onto
. Alber's generalized projection is an example of relatively nonexpansive mappings. For more example, see [1, 8].
In 2004, Masushita and Takahashi [1, 6] also proved weak and strong convergence theorems for finding a fixed point of a single relatively nonexpansive mapping. Several iterative methods, as a generalization of [1, 6], for finding a common fixed point of the family of relatively nonexpansive mappings have been further studied in [7, 9–14].
Recently, a problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping is studied by Takahashi and Zembayashi in [15, 16]. The purpose of this paper is to introduce a new iterative scheme which unifies several ones studied by many authors and to deduce the corresponding convergence theorems under the weaker assumptions. More precisely, many restrictions as were the case in other papers are dropped away.
First, we start with some preliminaries which will be used throughout the paper.
Lemma 1.1 (see [7, Lemma ]).
Let be a nonempty closed convex subset of a strictly convex and smooth Banach space
and let
be a relatively quasi-nonexpansive mapping from
into itself. Then
is closed and convex.
Lemma 1.2 (see [17, Proposition ]).
Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ5_HTML.gif)
for all and
.
Lemma 1.3 (see [17]).
Let be a smooth and uniformly convex Banach space and let
. Then there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ6_HTML.gif)
for all .
Lemma 1.4 (see [17, Proposition ]).
Let be a smooth and uniformly convex Banach space and let
and
be sequences of
such that either
or
is bounded. If
, then
.
Lemma 1.5 (see [2]).
Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space
, let
and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ7_HTML.gif)
Lemma 1.6 (see [18]).
Let be a uniformly convex Banach space and let
. Then there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ8_HTML.gif)
for all and
.
We next prove the following three lemmas which are very useful for our main results.
Lemma 1.7.
Let Let be a closed convex subset of a smooth Banach space
. Let
be a relatively quasi-nonexpansive mapping from
into
and let
be a family of relatively quasi-nonexpansive mappings from
into itself such that
. The mapping
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ9_HTML.gif)
for all and
,
such that
. If
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ10_HTML.gif)
Proof.
The proof of this lemma can be extracted from that of Lemma 1.8; so it is omitted.
If has a stronger assumption, we have the following lemma.
Lemma 1.8.
Let be a closed convex subset of a uniformly smooth Banach space
. Let
. Then, there exists a strictly increasing, continuous, and convex function
such that
and for each relatively quasi-nonexpansive mapping
and each finite family of relatively quasi-nonexpansive mappings
such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ11_HTML.gif)
for all and
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ12_HTML.gif)
and
,
such that
.
Proof.
Let . From Lemma 1.6 and
is uniformly convex, then there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ13_HTML.gif)
for all and
. Let
and
be relatively quasi-nonexpansive for all
such that
. For
and
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ14_HTML.gif)
and hence . Consequently, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ15_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ16_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ17_HTML.gif)
Lemma 1.9.
Let be a closed convex subset of a uniformly smooth and strictly convex Banach space
. Let
be a relatively quasi-nonexpansive mapping from
into
and let
be a family of relatively quasi-nonexpansive mappings from
into itself such that
. The mapping
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ18_HTML.gif)
for all and
,
such that
. Then, the following hold:
(1),
(2) is relatively quasi-nonexpansive.
Proof.
() Clearly,
. We want to show the reverse inclusion. Let
and
. Choose
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ19_HTML.gif)
From Lemma 1.8, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ20_HTML.gif)
From for all
and by the properties of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ21_HTML.gif)
for all . From
is one to one, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ22_HTML.gif)
for all . Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ23_HTML.gif)
Thus .
-
(2)
It follows directly from the above discussion.
2. Weak Convergence Theorem
Theorem 2.1.
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
be a family of relatively quasi-nonexpansive mappings and let
be a family of relatively quasi-nonexpansive mappings such that
. Let the sequence
be generated by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ24_HTML.gif)
for any ,
for all
,
such that
for all
. Then
converges strongly to
, where
is the generalized projection of
onto
.
Proof.
Let . Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ25_HTML.gif)
From Lemma 1.7, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ26_HTML.gif)
Therefore exists. This implies that
and
are bounded for all
.
Let . From (2.3) and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ27_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ28_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ29_HTML.gif)
This implies that exists. This together with the boundedness of
gives
. Using Lemma 1.3, there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ30_HTML.gif)
Since is a convergent sequence, it follows from the properties of
that
is a Cauchy sequence. Since
is closed, there exists
such that
.
We first establish weak convergence theorem for finding a common fixed point of a countable family of relatively quasi-nonexpansive mappings. Recall that, for a family of mappings with
, we say that
satisfies the NST-condition [19] if for each bounded sequence
in
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ31_HTML.gif)
where denotes the set of all weak subsequential limits of a sequence
.
Theorem 2.2.
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
be a family of relatively quasi-nonexpansive mappings satisfying NST-condition and let
be a family of relatively nonexpansive mappings such that
and suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ32_HTML.gif)
for all ,
and
. Let the sequence
be generated by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ33_HTML.gif)
for any ,
for all
,
such that
for all
,
for all
. If
is weakly sequentially continuous, then
converges weakly to
, where
.
Proof.
Let . From Theorem 2.1,
exists and hence
and
are bounded for all
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ34_HTML.gif)
By Lemma 1.8, there exists a strictly increasing, continuous, and convex function such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ35_HTML.gif)
In particular, for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ36_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ37_HTML.gif)
for all . Since
for all
and the properties of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ38_HTML.gif)
for all . Since
is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ39_HTML.gif)
for all . Since
is bounded, there exists a subsequence
of
such that
. Since
is relatively nonexpansive,
for all
.
We show that . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ40_HTML.gif)
We note from (2.15) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ41_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ42_HTML.gif)
Moreover, by (2.9) and the existence of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ43_HTML.gif)
It follows from Lemma 1.4 that . From (2.19) and
, we have
. Since
satisfies NST-condition, we have
. Hence
.
Let . From Lemma 1.5 and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ44_HTML.gif)
From Theorem 2.1, we know that . Since
is weakly sequentially continuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ45_HTML.gif)
Moreover, since is monotone,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ46_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ47_HTML.gif)
Since is strictly convex,
. This implies that
and hence
.
We next apply our result for finding a common element of a fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem. This problem is extensively studied in [11, 14–16]. Let be a subset of a Banach space
and let
be a bifunction. The equilibrium problem for a bifunction
is to find
such that
for all
. The set of solutions above is denoted by
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ48_HTML.gif)
To solve the equilibrium problem, we usually assume that a bifunction satisfies the following conditions (
is closed and convex):
(A1) for all
;
(A2) is monotone, that is,
;
(A3)for all ,
;
(A4)for all ,
is convex and lower semicontinuous.
The following lemma gives a characterization of a solution of an equilibrium problem.
Lemma 2.3.
Let be a nonempty closed convex subset of a Banach space
. Let
be a bifunction from
satisfying (A1)–(A4). Suppose that
. Then
if and only if
for all
.
Proof.
Let , then
for all
. From (A2), we get that
for all
.
Conversely, assume that for all
. For any
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ49_HTML.gif)
Then and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ50_HTML.gif)
So for all
. From (A3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ51_HTML.gif)
Hence .
Takahashi and Zembayashi proved the following important result.
Lemma 2.4 (see [15, Lemma ]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space
. Let
be a bifunction from
satisfying (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ52_HTML.gif)
for all . Then, the following hold:
(1) is single-valued;
(2) is a firmly nonexpansive-type mapping [20], that is, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ53_HTML.gif)
(3);
(4) is closed and convex.
We now deduce Takahashi and Zembayashi's recent result from Theorem 2.2.
Corollary 2.5 (see [15, Theorem ]).
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
be a bifunction from
to
satisfying (A1)–(A4) and let
be a relatively nonexpansive mapping from
into itself such that
. Let the sequence
be generated by
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ54_HTML.gif)
for every ,
satisfying
and
for some
. If
is weakly sequentially continuous, then
converges weakly to
, where
.
Proof.
Put where
is defined by Lemma 2.4. Then
. By reindexing the sequences
and
of this iteration, we can apply Theorem 2.2 by showing that the family
satisfies the condition (2.9) and NST-condition. It is proved in [15, Lemma
] that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ55_HTML.gif)
To see that satisfies NST-condition, let
be a bounded sequence in
such that
and
. Suppose that there exists a subsequence
of
such that
. Then
. Since
is uniformly continuous on bounded sets and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ56_HTML.gif)
From the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ57_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ58_HTML.gif)
and is lower semicontinuous and convex in the second variable, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ59_HTML.gif)
Thus for all
. From Lemma 2.3, we have
. Then
satisfies the NST-condition. From Theorem 2.2 where
,
converges weakly to
, where
.
Using the same proof as above, we have the following result.
Corollary 2.6 (see [11, Theorem ]).
Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space
. Let
be a bifunction from
to
satisfies (A1)–(A4) and let
be two relatively nonexpansive mappings such that
. Let the sequence
be generated by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ60_HTML.gif)
Assume that ,
and
are three sequences in
satisfying the following restrictions:
(a);
(b),
;
(c) for some
.
If is weakly sequentially continuous, then
converges weakly to
, where
.
The following result also follows from Theorem 2.2.
Corollary 2.7 (see [9, Theorem ]).
Let be a uniformly smooth and uniformly convex Banach space and let
be a nonempty closed convex subset of
. Let
be a finite family of relatively nonexpansive mappings from
into itself such that
is a nonempty and let
and
be sequences such that
and
for all
and
for all
. Let
be a sequence of mappings defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ61_HTML.gif)
for all and let the sequence
be generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ62_HTML.gif)
Then the following hold:
(1)the sequence is bounded and each weak subsequential limit of
belongs to
;
(2)if the duality mapping from
into
is weakly sequentially continuous, then
converges weakly to the strong limit of
.
Proof.
Since is relatively nonexpansive, the family
satisfies the NST-condition. Moreover,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ63_HTML.gif)
Thus the conclusions of this corollary follow.
3. Strong Convergence Theorem
In this section, we prove strong convergence of an iterative sequence generated by the hybrid method in mathematical programming. We start with the following useful common tools.
Lemma 3.1.
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
and
be families of relatively quasi-nonexpansive mappings such that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ64_HTML.gif)
for all ,
and
. Let
be such that
and
are bounded for all
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ65_HTML.gif)
where for all
and
satisfy
for all
,
for all
and
. Then the following statements hold:
(1) for all
,
(2),
(3),
(4)if , then
for all
,
(5)if , then
and
.
Proof.
() Since
and
is uniformly norm-to-norm continuous on bounded sets,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ66_HTML.gif)
We note here that is also bounded. For any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ67_HTML.gif)
 () Let
. Using (3.1) and the relative quasi-nonexpansiveness of each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ68_HTML.gif)
By Lemma 1.4 and the boundedness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ69_HTML.gif)
  () Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ70_HTML.gif)
we have .
() Assume that
. From
, we get that
. Since
is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ71_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ72_HTML.gif)
From (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ73_HTML.gif)
It follows from for all
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ74_HTML.gif)
for all . Since
is uniformly norm-to-norm continuous on bounded sets and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ75_HTML.gif)
for all , as desired.
() Assume that
. From the assumption and (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ76_HTML.gif)
Hence and
.
Lemma 3.2 (see [21, Lemma ]).
Let be a closed convex subset of a strictly convex, smooth and reflexive Banach space
satisfying Kadec-Klee property. Let
and
be a sequence in
such that
and
for all
. Then
.
Recall that a Banach space satisfies Kadec–Klee property if whenever
is a sequence in
with
and
, it follows that
.
3.1. The CQ-Method
Theorem 3.3.
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
be a family of relatively quasi-nonexpansive mappings satisfying NST-condition and let
be a family of relatively nonexpansive mappings such that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ77_HTML.gif)
for all ,
and
. Let the sequence
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ78_HTML.gif)
for every ,
for all
and
satisfying
for all
,
for all
. Then
converges strongly to
.
Proof.
The proof is broken into 4 steps.
Step 1 ( is well defined).
First, we show that is closed and convex. Clearly,
is closed and convex. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ79_HTML.gif)
then is closed and convex. Thus
is closed and convex.
We next show that . Let
. Then, from Lemma 1.7,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ80_HTML.gif)
Thus . Hence
for all
.
Next, we show by induction that for all
. Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ81_HTML.gif)
Suppose that for some
. From
and the definition of the generalized projection, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ82_HTML.gif)
for all . From
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ83_HTML.gif)
for all . Hence
, and we also have
. So, we have
for all
and hence the sequence
is well defined.
Step 2 ().
From the definition of , we have
. Using Lemma 1.2, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ84_HTML.gif)
for all . In particular, since
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ85_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ86_HTML.gif)
for all . This implies that
exists and
is bounded. Moreover, from (3.21) and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ87_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ88_HTML.gif)
It follows from that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ89_HTML.gif)
From (3.25), (3.26), and Lemma 1.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ90_HTML.gif)
So . Using Lemma 3.1(4), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ91_HTML.gif)
for all . Since each
is relatively nonexpansive,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ92_HTML.gif)
Step 3 ().
Let . From Lemma 3.1(2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ93_HTML.gif)
and . It follows from NST-condition that
.
Step 4 ().
From Steps 2 and 3, we have . The conclusion follows by Lemma 3.2 and (3.23).
We apply Theorem 3.3 and the proof of Corollary 2.5 and then obtain the following result.
Corollary 3.4.
Let ,
,
,
be as in Corollary 2.5. Let the sequence
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ94_HTML.gif)
for every ,
satisfying
and
for some
. Then,
converges strongly to
, where
is the generalized projection of
onto
.
Remark 3.5.
Corollary 3.4 improves the restriction on of [15, Theorem
]. In fact, it is assumed in [15, Theorem
] that
.
3.2. The Monotone CQ-Method
Let be a closed subset of a Banach space
. Recall that a mapping
is closed if for each
in
, if
and
, then
. A family of mappings
with
is said to satisfy the
-condition if for each bounded sequence
in
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ95_HTML.gif)
Remark 3.6.
-
(1)
If
satisfies NST-condition, then
satisfies
-condition.
-
(2)
If
and
is closed, then
satisfies
-condition.
Theorem 3.7.
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space
. Let
be a family of relatively quasi-nonexpansive mappings satisfying
-condition and let
be a family of closed relatively quasi-nonexpansive mappings such that
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ96_HTML.gif)
for all ,
and
. Let the sequence
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ97_HTML.gif)
for every ,
satisfying
and
for all
. Then
converges strongly to
.
Proof.
Step 1 ( is well defined).
This step is almost the same as Step 1 of the proof of Theorem 3.3, so it is omitted.
Step 2 ( is a Cauchy sequence in
).
We can follow the proof of Theorem 3.3 and conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ98_HTML.gif)
exists. Moreover, as for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ99_HTML.gif)
Since is bounded, it follows from Lemma 1.3 that there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ100_HTML.gif)
Since exists, we have that
is a Cauchy sequence. Therefore,
for some
.
Step 3 ().
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ101_HTML.gif)
By Lemma 1.4 and the boundedness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ102_HTML.gif)
So, we have . Using Lemma 3.1(4), we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ103_HTML.gif)
for all . Since each
is closed,
.
Step 4 ().
Let . From Lemma 3.1(2), we have
and
. It follows from
-condition that
.
Step 5 ().
From Steps 3 and 4, we have . The conclusion follows by Lemma 3.2 and (3.23).
Letting identity and
yield the following result.
Corollary 3.8 (see [12, Theorem ]).
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space
. Let
be a closed relatively quasi-nonexpansive mapping such that
. Assume that
is a sequence in
such that
. Define a sequence
in
by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ104_HTML.gif)
Then converges strongly to
.
Letting identity and
yield the following result.
Corollary 3.9 (see [13, Theorem ]).
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space
. Let
be two closed relatively quasi-nonexpansive mappings from
into itself such that
. Define a sequence
in
be the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ105_HTML.gif)
with the conditions: with
and
(1);
(2);
(3) for some
.
Then converges strongly to
.
Remark 3.10.
Using Theorem 3.7, we can show that the conclusion of Corollary 3.9 remains true under the more general restrictions on and
:
(1) are arbitrary;
(2) and
.
3.3. The Shrinking Projection Method
Theorem 3.11.
Let be as in Theorem 3.7. Let the sequence
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ106_HTML.gif)
for every ,
for all
and
satisfies
for all
,
for all
. Then
converges strongly to
.
Proof.
The proof is almost the same as the proofs of Theorems 3.3 and 3.7; so it is omitted.
In particular, applying Theorem 3.11 gives the following result.
Corollary 3.12.
Let be as in Corollary 2.5. Let the sequence
be generated by
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ107_HTML.gif)
for every , where
is the duality mapping on
. Assume that
satisfies
and
for some
. Then
converges strongly to
, where
is the generalized projection of
onto
.
Remark 3.13.
Corollary 3.12 improves the restriction on of [16, Theorem
]. In fact, it is assumed in [16, Theorem
] that
.
Corollary 3.14 (see [11, Theorem ]).
Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space
. Let
be a bifunction from
to
satisfying (A1)–(A4) and let
be two closed relatively quasi-nonexpansive mappings such that
. Let the sequence
be generated by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ108_HTML.gif)
Assume that ,
and
are three sequences in
satisfying the restrictions:
(a);
(b),
;
(c) for some
.
Then converges strongly to
.
Remark 3.15.
The conclusion of Corollary 3.14 remains true under the more general assumption; that is, we can replace (b) by the following one:
(b′) and
.
We also deduce the following result.
Corollary 3.16 (see [14, Theorem ]).
Let be as in Corollary 3.14. Let the sequences
,
and
be generated by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ109_HTML.gif)
Assume that ,
and
are three sequences in
satisfying the following restrictions:
(a);
(b) for all
and
;
(c),
;
(d) for some
.
Then and
converge strongly to
.
Remark 3.17.
The conclusion of Corollary 3.16 remains true under the more general restrictions; that is, we replace (b) and (c) by the following one:
(b′) and
.
Corollary 3.18 (see [10, Theorem ]).
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
. Let
be a family of relatively nonexpansive mappings such that
and let
. For
and
, define a sequence
of
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F407651/MediaObjects/13663_2009_Article_1277_Equ110_HTML.gif)
where satisfies the following restrictions:
(i) for all
and
;
(ii) for all
,
for all
. If
(a)either for all
or
(b) and
for all
.
then the sequence converges strongly to
.
Remark 3.19.
The conclusion of Corollary 3.18 remains true under the more general restrictions on :
(1) are arbitrary.
(2) for all
.
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Acknowledgments
The authors would like to thank the referee for their comments on the manuscript. The first author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand, and the second author is supported by the Thailand Research Fund under Grant MRG5180146.
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Boonchari, D., Saejung, S. Approximation of Common Fixed Points of a Countable Family of Relatively Nonexpansive Mappings. Fixed Point Theory Appl 2010, 407651 (2009). https://doi.org/10.1155/2010/407651
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DOI: https://doi.org/10.1155/2010/407651