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New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces
Fixed Point Theory and Applicationsvolume 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 quasiϕnonexpansive 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
where is the normalized dual mapping from to defined by
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
In Hilbert spaces, and , where is the metric projection. It is obvious from the definition of function that
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 quasinonexpansive 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 quasinonexpansive [9] if and for all and .
In 2005, Matsushita and Takahashi [10] introduced the following algorithm:
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
We use to denote the solution set of the equilibrium problem (1.6). That is,
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:
where is a closed quasinonexpansive 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 quasinonexpansive 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 quasinonexpansive mappings, and , , a finite family of continuous monotone mappings. For , define the mappings , by
where and for some . Zegeye and Shahzad [25] introduced the following scheme:
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 quasinonexpansive 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 quasinonexpansive mappings, a finite family of bifunctions, and a finite family of continuous monotone mappings such that . Let and . Define the mappings , by
Consider the iteration
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
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 KadecKlee 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 KadecKlee property. For more details on the KadecKlee 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:
Let be a monotone mapping from into . The variational inequality problem on is formulated as follows:
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
Lemma 2.2 (see [1]).
Let be a reflexive, strictly convex and smooth Banach space, a nonempty closed and convex subset of , and . Then
Lemma 2.3 (see [20]).
Let be a strictly convex and smooth Banach space, a nonempty closed and convex subset of , and a quasinonexpansive 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

(b)
define a mapping by
Then the following conclusions hold:
(1) is singlevalued;
(2) is firmly nonexpansive, that is, for all ,
(3);
(4) is quasinonexpansive;
(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
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
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
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 T_{r} 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,
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
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 KadecKlee property. Let be a family of closed quasinonexpansive 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:
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 quasinonexpansive. Hence, for any , we have
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
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
It follows that
This implies that
Hence, we have as . In view of the KadecKlee property of , we get that
By the construction of , we have that and . It follows from Lemma 2.2 that
Letting , we obtain that . In view of , we have and hence
It follows that
From (1.4), we see that
Hence,
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
Take on both sides of (3.13) over and use weak lower semicontinuity of norm to get that
which implies that . Hence, . It follows that . Now, from (3.12) and KadecKlee property of , we obtain that as . Then the demicontinuity of implies that . Now, from (3.11) and the fact that has the KadecKlee property, we obtain that . Note that
It follows that
Since is uniformly normtonorm continuous on any bounded sets, we have
Since is uniformly smooth, we know that is uniformly convex. In view of Lemma 2.7, we see that, for any ,
It follows that
Note that
It follows from (3.16) and (3.17) that
By (3.19), (3.21), and , we have
It follows from the property of that
Since as and is demicontinuous, we obtain that . Note that
This implies that
Since enjoys the KadecKlee property, we see that
Note that
From (3.23) and (3.26), we arrive at
Note that is demicontinuous. It follows that . On the other hand, since
by (3.28) we conclude that as . Since enjoys the KadecKlee property, we obtain that
By repeating (3.18)–(3.30), we also can get
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
Since for each , we have
Letting in (3.35), we see that
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 KadecKlee property. Let be a closed quasinonexpansive 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:
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 quasinonexpansive 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:
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 ,
Since each is closed, we can conclude that , . Note that in a Hilbert space, a firmlynonexpansive 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 ,
or equivalently,
Let , where is a pseudocontraction. Then is a monotone mapping and . Moreover, . Indeed, it is easy to see that . Let . We have
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 quasinonexpansive 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:
where are defined by (1.11) with and is defined by
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 quasinonexpansive mapping, a bifunction satisfying the conditions (A1)–(A4), and a continuous pseudocontraction such that . Let . Define a sequence by the following manner:
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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Acknowledgment
This work was supported by the Natural Science Foundation of Hebei Province (A2010001482).
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Keywords
 Hilbert Space
 Banach Space
 Convex Subset
 Equilibrium Problem
 Nonexpansive Mapping