- Research Article
- Open access
- Published:
New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces
Fixed Point Theory and Applications volume 2011, Article number: 372975 (2011)
Abstract
We introduced a new iterative scheme for finding a common element in the set of common fixed points of a finite family of quasi-ϕ-nonexpansive mappings, the set of common solutions of a finite family of equilibrium problems, and the set of common solutions of a finite family of variational inequality problems in Banach spaces. The proof method for the main result is simplified under some new assumptions on the bifunctions.
1. Introduction
Throughout this paper, let denote the set of all real numbers. Let
be a smooth Banach space and
the dual space of
. The function
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ1_HTML.gif)
where is the normalized dual mapping from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ2_HTML.gif)
Let be a nonempty closed and convex subset of
. The generalized projection
is a mapping that assigns to an arbitrary point
the minimum point of the function
, that is,
, where
is the solution to the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ3_HTML.gif)
In Hilbert spaces, and
, where
is the metric projection. It is obvious from the definition of function
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ4_HTML.gif)
We remark that if is a reflexive, strictly convex and smooth Banach space, then for
,
if and only if
. For more details on
and
, the readers are referred to [1–4].
Let be a mapping from
into itself. We denote the set of fixed points of
by
.
is called to be nonexpansive if
for all
and quasi-nonexpansive if
and
for all
and
. A point
is called to be an asymptotic fixed point of
[5] if
contains a sequence
which converges weakly to
such that
. The set of asymptotic fixed points of
is denoted by
. The mapping
is said to be relatively nonexpansive [6–8] if
and
for all
and
. The mapping
is said to be
-nonexpansive if
for all
.
is called to be quasi-
-nonexpansive [9] if
and
for all
and
.
In 2005, Matsushita and Takahashi [10] introduced the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ5_HTML.gif)
where is the duality mapping on
,
is a relatively nonexpansive mapping from
into itself, and
is a sequence of real numbers such that
and
and proved that the sequence
generated by (1.5) converges strongly to
, where
is the generalized projection from
onto
.
Let be a bifunction from
to
. The equilibrium problem for
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ6_HTML.gif)
We use to denote the solution set of the equilibrium problem (1.6). That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ7_HTML.gif)
For studying the equilibrium problem, is usually assumed to satisfy the following conditions:
(A1) for all
;
(A2) is monotone, that is,
for all
;
(A3)for each ,
;
(A4)for each ,
is convex and lower semicontinuous.
Recently, many authors investigated the equilibrium problems in Hilbert spaces or Banach spaces; see, for example, [11–25]. In [20], Qin et al. considered the following iterative scheme by a hybrid method in a Banach space:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ8_HTML.gif)
where is a closed quasi-
-nonexpansive mapping for each
,
are real sequences in
satisfying
for each
and
for each
and
is a real sequence in
with
. Then the authors proved that
converges strongly to
, where
.
Very recently, Zegeye and Shahzad [25] introduced a new scheme for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions set of finite family of equilibrium problems, and common solutions set of finite family of variational inequality problems for monotone mappings in a Banach space. More precisely, let ,
, be a finite family of bifunctions,
,
, a finite family of relatively quasi-nonexpansive mappings, and
,
, a finite family of continuous monotone mappings. For
, define the mappings
,
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ9_HTML.gif)
where and
for some
. Zegeye and Shahzad [25] introduced the following scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ10_HTML.gif)
where ,
such that
. Further, they proved that
converges strongly to an element of
, where
.
In this paper, motivated and inspired by the iterations (1.8) and (1.10), we consider a new iterative process with a finite family of quasi--nonexpansive mappings for a finite family of equilibrium problems and a finite family of variational inequality problems in a Banach space. More precisely, let
be a family of quasi-
-nonexpansive mappings,
a finite family of bifunctions, and
a finite family of continuous monotone mappings such that
. Let
and
. Define the mappings
,
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ12_HTML.gif)
Consider the iteration
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ13_HTML.gif)
where are the real numbers in
satisfying
and for each
,
are the real numbers in
satisfying
. We will prove that the sequence
generated by (1.13) converges strongly to an element in
. In this paper, in order to simplify the proof, we will replace the condition (A3) with (A3'): for each fixed
,
is continuous.
Obviously, the condition (A3') implies (A3). Under the condition (A3'), we will show that each (as well as
,
,
) is closed which is such that the proof for the main result of this paper is simplified.
2. Preliminaries
The modulus of smoothness of a Banach space is the function
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ14_HTML.gif)
The space is said to be smooth if
,
, and
is called uniformly smooth if and only if
.
A Banach space is said to be strictly convex if
for all
with
and
. It is said to be uniformly convex if
for any two sequences
and
in
such that
and
. It is known that if a Banach space
is uniformly smooth, then its dual space
is uniformly convex.
A Banach space is called to have the Kadec-Klee property if for any sequence
and
with
, where
denotes the weak convergence, and
, then
as
, where
denotes the strong convergence. It is well known that every uniformly convex Banach space has the Kadec-Klee property. For more details on the Kadec-Klee property, the reader is referred to [3, 4].
Let be a nonempty closed and convex subset of a Banach space
. A mapping
is said to be closed if for any sequence
such that
and
,
.
Let be a mapping.
is said to be monotone if for each
, the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ15_HTML.gif)
Let be a monotone mapping from
into
. The variational inequality problem on
is formulated as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ16_HTML.gif)
The solution set of the above variational inequality problem is denoted by .
Next we state some lemmas which will be used later.
Lemma 2.1 (see [1]).
Let be a nonempty closed and convex subset of a smooth Banach space
and
. Then,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ17_HTML.gif)
Lemma 2.2 (see [1]).
Let be a reflexive, strictly convex and smooth Banach space,
a nonempty closed and convex subset of
, and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ18_HTML.gif)
Lemma 2.3 (see [20]).
Let be a strictly convex and smooth Banach space,
a nonempty closed and convex subset of
, and
a quasi-
-nonexpansive mapping. Then
is a closed and convex subset of
.
Since the condition (A3') implies (A3), the following lemma is a natural result of [22, Lemmas 2.8 and 2.9].
Lemma 2.4.
Let be a closed and convex subset of a smooth, strictly convex and reflexive Banach space
. Let
be a bifunction from
satisfying (A1), (A2), (A3'), and (A4). Let
and
. Then
-
(a)
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ19_HTML.gif)
-
(b)
define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ20_HTML.gif)
Then the following conclusions hold:
(1) is single-valued;
(2) is firmly nonexpansive, that is, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ21_HTML.gif)
(3);
(4) is quasi-
-nonexpansive;
(5) is closed and convex;
(6).
Remark 2.5.
Let be a continuous monotone mapping and define
for all
. It is easy to see that
satisfies the conditions (A1), (A2), (A3'), and (A4) and
. Hence, for every real number
, if defining a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ22_HTML.gif)
then satisfies all the conclusions in Lemma 2.4. See [25, Lemma 2.4].
Lemma 2.6 (see [26]).
Let and
be two fixed real numbers. Then a Banach space
is uniformly convex if and only if there exists a continuous strictly increasing convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ23_HTML.gif)
for all and
, where
.
The following lemma can be obtained from Lemma 2.6 immediately; also see [20, Lemma 1.9].
Lemma 2.7 (see [20]).
Let be a uniformly convex Banach space,
a positive number, and
a closed ball of
. There exists a continuous, strictly increasing and convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ24_HTML.gif)
for all and
such that
.
Lemma 2.8.
Let be a closed and convex subset of a uniformly smooth and strictly convex Banach space
. Let
be a bifunction satisfying (A1), (A2), (A3'), and (A4). Let
and
be a mapping defined by (2.7). Then Tr is closed.
Proof.
Let converge to
and
converge to
. To end the conclusion, we need to prove that
. Indeed, for each
, Lemma 2.4 shows that there exists a unique
such that
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ25_HTML.gif)
Since is uniformly smooth,
is continuous on bounded set (note that
and
are both bounded). Taking the limit as
in (2.12), by using (A3'), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ26_HTML.gif)
which implies that . This completes the proof.
3. Main Results
Theorem 3.1.
Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space
which has the Kadec-Klee property. Let
be a family of closed quasi-
-nonexpansive mappings,
a finite family of bifunctions satisfying the conditions (A1), (A2), (A3'), and (A4), and
a finite family of continuous monotone mappings such that
. Let
. Let
be a sequence generated by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ27_HTML.gif)
where and
are defined by (1.11) and (1.12),
are the real numbers in
satisfying
and for each
,
are the real numbers in
satisfying
. Then the sequence
converges strongly to
, where
is the generalized projection from
onto
.
Proof.
First we prove that is closed and convex for each
. From the definition of
, it is obvious that
is closed. Moreover, since
is equivalent to
, it follows that
is convex for each
. By the definition of
, we can conclude that
is closed and convex for each
.
Next, we prove that for each
. From Lemma 2.4 and Remark 2.5, we see that each
(
) and
(
) are quasi-
-nonexpansive. Hence, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ28_HTML.gif)
which implies that for each
. So, it follows from the definition of
that
for each
. Therefore, the sequence
is well defined. Also, from Lemma 2.2 we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ29_HTML.gif)
for each . This shows that the sequence
is bounded. It follows from (1.4) that the sequence
is also bounded.
Since is reflexive, we may, without loss of generality, assume that
. Since
is closed and convex for each
, we can conclude that
for each
. By the definition of
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ30_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ31_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ32_HTML.gif)
Hence, we have as
. In view of the Kadec-Klee property of
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ33_HTML.gif)
By the construction of , we have that
and
. It follows from Lemma 2.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ34_HTML.gif)
Letting , we obtain that
. In view of
, we have
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ35_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ36_HTML.gif)
From (1.4), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ37_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ38_HTML.gif)
This implies that the sequence is bounded. Note that reflexivity of
implies reflexivity of
. Thus, we may assume that
. Furthermore, reflexivity of
implies that there exists
such that
. Then, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ39_HTML.gif)
Take on both sides of (3.13) over
and use weak lower semicontinuity of norm to get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ40_HTML.gif)
which implies that . Hence,
. It follows that
. Now, from (3.12) and Kadec-Klee property of
, we obtain that
as
. Then the demicontinuity of
implies that
. Now, from (3.11) and the fact that
has the Kadec-Klee property, we obtain that
. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ41_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ42_HTML.gif)
Since is uniformly norm-to-norm continuous on any bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ43_HTML.gif)
Since is uniformly smooth, we know that
is uniformly convex. In view of Lemma 2.7, we see that, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ44_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ45_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ46_HTML.gif)
It follows from (3.16) and (3.17) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ47_HTML.gif)
By (3.19), (3.21), and , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ48_HTML.gif)
It follows from the property of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ49_HTML.gif)
Since as
and
is demicontinuous, we obtain that
. Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ50_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ51_HTML.gif)
Since enjoys the Kadec-Klee property, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ52_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ53_HTML.gif)
From (3.23) and (3.26), we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ54_HTML.gif)
Note that is demicontinuous. It follows that
. On the other hand, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ55_HTML.gif)
by (3.28) we conclude that as
. Since
enjoys the Kadec-Klee property, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ56_HTML.gif)
By repeating (3.18)–(3.30), we also can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ57_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ58_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ59_HTML.gif)
Since each is closed, by (3.30) and (3.31) we conclude that
, that is,
,
. On the other hand, Lemma 2.4, Remark 2.5, and Lemma 2.8 show that
and
are closed. So, by (3.32) and (3.33) we have
and
. Now, it follows from Lemma 2.4 and Remark 2.5 that
(
) and
(
). Hence,
(
) and
(
). Therefore,
.
Finally, we prove that . From
, by Lemma 2.1, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ60_HTML.gif)
Since for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ61_HTML.gif)
Letting in (3.35), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ62_HTML.gif)
In view of Lemma 2.1, we can obtain that . This completes the proof.
Remark 3.2.
Obviously, the proof process of is simple since we replace the condition (A3) with (A3') which is such that
and
are closed. In fact, although the condition (A3') is stronger than (A3), it is not easier to verify the condition (A3) than verify the condition (A3'). Hence, from this point, the condition (A3') is acceptable. On the other hand, the definition of
is of some interest.
If for each
,
for each
and
for each
, then Theorem 3.1 reduces to the following result.
Corollary 3.3.
Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space
which has the Kadec-Klee property. Let
be a closed quasi-
-nonexpansive mapping,
a bifunction satisfying the conditions (A1), (A2), (A3'), and (A4) and
a continuous monotone mapping such that
. Let
. Let
be a sequence defined by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ63_HTML.gif)
where and
are defined by (1.11) and (1.12) with
and
,
are the real numbers in
satisfying
. Then the sequence
converges strongly to
, where
is the generalized projection from
onto
.
Corollary 3.4.
Let be a nonempty closed and convex subset of a Hilbert space
. Let
be a family of closed quasi-nonexpansive mappings,
a finite family of bifunctions satisfying the conditions (A1)–(A4), and
a finite family of continuous monotone mappings such that
. Let
. Define a sequence
by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ64_HTML.gif)
where and
are defined by (1.11) and (1.12) with
(
is the identity mapping),
are the real numbers in
satisfying
and for each
,
are the real numbers in
satisfying
. Then the sequence
converges strongly to
, where
is the projection from
onto
.
Proof.
By the proof of Theorem 3.1, we have as
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ65_HTML.gif)
Since each is closed, we can conclude that
,
. Note that in a Hilbert space, a firmly-nonexpansive mapping is also nonexpansive. Hence,
and
are nonexpansive for each
and
. By demiclosed principle, we can conclude that
and
for each
and
. That is,
. Then by the final part of proof of Theorem 3.1, we have
. This completes the proof.
Let be a Hilbert space and
a nonempty closed and convex subset of
. A mapping
is called a pseudocontraction if for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ66_HTML.gif)
or equivalently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ67_HTML.gif)
Let , where
is a pseudocontraction. Then
is a monotone mapping and
. Moreover,
. Indeed, it is easy to see that
. Let
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ68_HTML.gif)
for all . Take
. Then we have
. That is,
. This shows that
, which implies that
. So,
. Based this, we have following result.
Corollary 3.5.
Let be a nonempty closed and convex subset of a Hilbert space
. Let
be a family of closed quasi-nonexpansive mappings,
a finite family of bifunctions satisfying the conditions (A1)–(A4), and
a finite family of continuous pseudocontractions such that
. Let
. Define a sequence
by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ69_HTML.gif)
where are defined by (1.11) with
and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ70_HTML.gif)
are the real numbers in
satisfying
and for each
,
are the real numbers in
satisfying
. Then the sequence
converges strongly to
, where
is the projection from
onto
.
If ,
, and
for each
,
, and
, then Corollary 3.5 reduced the following result.
Corollary 3.6.
Let be a nonempty closed and convex subset of a Hilbert space
. Let
be a closed quasi-nonexpansive mapping,
a bifunction satisfying the conditions (A1)–(A4), and
a continuous pseudocontraction such that
. Let
. Define a sequence
by the following manner:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F372975/MediaObjects/13663_2010_Article_1402_Equ71_HTML.gif)
where is defined by (1.11) with
and
,
is defined by (3.44)
, and
are the real numbers in
satisfying
. Then the sequence
converges strongly to
, where
is the projection from
onto
.
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This work was supported by the Natural Science Foundation of Hebei Province (A2010001482).
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Wang, S., Zhou, C. New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces. Fixed Point Theory Appl 2011, 372975 (2011). https://doi.org/10.1155/2011/372975
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DOI: https://doi.org/10.1155/2011/372975