The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of a countable family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Moreover, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. Our result improve and extend the corresponding results announced by Takahashi and Zembayashi (2008 and 2009), and many others.
1. Introduction
Let be a real Banach space and the dual space of . Let be a nonempty closed convex subset of and a bifunction from to , where denotes the set of real numbers. The equilibrium problem is to find such that
The set of solutions of (1.1) is denoted by . Given a mapping , let for all . Then, if and only if for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduced to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, Blum and Oettli [1], Combettes and Hirstoaga [2], and Moudafi [3].
Recall that a mapping is said to be nonexpansive if
We denote by the set of fixed points of . If a Banach space is uniformly convex, is bounded, closed and convex, and is a nonexpansive mapping of into itself, then is nonempty; see [4] for more details. Recently, many authors studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces, respectively; see, for instance, [5–13] and the references therein.
A popular method is the hybrid projection method developed by Nakajo and Takahashi [14], Kamimura and Takahashi [15], and MartinezYanes and Xu [16]; see also [5, 17–20] and references therein. Recently Takahashi et al. [21] introduced an alterative projection method, which is called the shrinking projection method, and they showed several strong convergence theorems for a family of nonexpansive mappings. In 2008, Takahashi and Zembayashi [12] introduced two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solution of an equilibrium problem in a Banach space. Then they prove strong and weak convergence of the sequences. Very recently, Takahashi and Zembayashi [13] proved a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using a new hybrid method.
On the other hand, motivated by Nakajo and Takahashi [14], Matsushita and Takahashi [17] reformulated the definition of the notion and obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Very recently, Aoyama et al. [22] introduce a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings. Let and
for all where is a nonempty closed convex subset of a Banach space; is a sequence in and is a sequence of nonexpansive mappings with some condition. They proved that defined by (1.3) converges strongly to a common fixed point of
Motivated and inspired by the research going on in this direction, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of relatively nonexpansive mappings in a Banach space by using the shrinking projection method. Further, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. The result obtained in this paper improves and extends the corresponding result of [13] and many others.
2. Preliminaries
Let be a real Banach space with norm and let be the dual of . For all and , we denote the value of at by . The normalized duality mapping from to is defined by
for . By HahnBanach theorem, is nonempty; see [4] for more details. We denote the strong convergence and the weak convergence of a sequence to in by and , respectively. We also denote the weak convergence of a sequence to in by A Banach space is said to be strictly convex if for all with and . It is also said to be uniformly convex if for each , there exists such that for with and . A uniformly convex Banach space has the KadecKlee property, that is, and imply . Let be the unit sphere of . Then the Banach space is said to be smooth provided that
exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . It is well know that if is smooth, strictly convex and reflexive, then the duality mapping is single valued, onetoone and onto.
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
It is obvious from the definition of the function that for all . Following Alber [23], the generalized projection from onto is defined by
If is a Hilbert space, then and is the metric projection of onto . We know the following lemmas for generalized projections.
Lemma 2.1 (Alber [23], Kamimura and Takahashi [15]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space . Then
Lemma 2.2 (Alber [23], Kamimura and Takahashi [15]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space let and let Then
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space ; let be a mapping from into itself. We denote by the set of fixed point of . A point is said to be an asymptotic fixed point of if there exists in which converges weakly to and . The set of asymptotic fixed points of will be denoted by . Following Matsushita and Takahashi [17], a mapping from into itself is said to be relatively nonexpansive if is nonempty, and .
The following lemma is according to Matsushita and Takahashi [17].
Lemma 2.3 (Matsushita and Takahashi [17]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , and let be a relatively nonexpansive mapping from into itself. Then is closed and convex.
We also know the following three lemmas.
Lemma 2.4 (Kamimura and Takahashi [15]).
Let be a uniformly convex and smooth Banach space, and let , be sequences in such that either or is bounded. If , then .
Lemma 2.5 (Xu [24], Zlinescu [25, 26]).
Let be a uniformly convex Banach space, and let . Then there exists a strictly increasing, continuous, and convex function such that and
for all and , where
Lemma 2.6 (Kamimura and Takahashi [15]).
Let be a smooth and uniformly convex Banach space and let Then there exists a strictly increasing, continuous, and convex function such that and
for all
For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:
for all ;
is monotone, that is, for all ;
for each ,
for each is convex and lower semicontinuous.
The following result is in Blum and Oettlli [1].
Lemma 2.7 (Blum and Oettlli [1]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , and let be a bifunction from satisfying , and let and . Then, there exists such that
We also know the following lemmas.
Lemma 2.8 (Takahashi and Zembayashi [12]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space , and let be a bifunction from satisfying . For and , define a mapping as follows:
for all , Then, the following holds:
(1) is singlevalued;
(2) is a firmly nonexpansivetype mapping, that is, for all
(3);
(4) is closed and convex.
Lemma 2.9 (Takahashi and Zembayashi [12]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , and let be a bifunction from satisfying . Then for , and ,
3. Main Results
Let be a nonempty closed convex subset of a Banach space let be a family of mappings of into itself with and denotes the set of all weak subsequential limits of a bounded sequence in . is said to satisfy the NSTcondition [27] if for every bounded sequence in ,
In this section, by using the NSTcondition, we prove two strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of relatively nonexpansive mappings in a Banach space.
Theorem 3.1.
Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying and let be a family of relatively nonexpansive mappings from into itself such that . Let be a sequence generated by and
for every , where is the duality mapping on , satisfies and for some Suppose that satisfy the NSTcondition. Then converges strongly to , where is the generalized projection of onto .
Proof.
Putting for all , we have from Lemma 2.9 that are relatively nonexpansive.
We first show that is closed and convex. It is obvious that is closed. Since
is convex. So, is a closed convex subset of for all .
Next, we show by induction that for all . From , we have . Suppose that for some . Let Since and are relatively nonexpansive, it follows that
Hence . This implies that for all . So, is well defined.
Next, we show that is bounded. From the definition of , we have
for all Thus is bounded and therefore and are also bounded.
From and , we have
This implies that is nondecreasing and so exists. Since
for all , it follows that From , we have
Therefore, we also have
Since and is uniformly convex and smooth, it follows from Lemma 2.4 that
Thus, we have
Since is uniformly normtonorm continuous on bounded sets, it follows by (3.11) that
Let . Since is a uniformly smooth Banach space, we note that is a uniformly convex Banach space. Therefore, by Lemma 2.5, there exists a continuous, strictly increasing, and convex function with such that
for and So, we have
for all Therefore, we have
Since
it follows that
From we have
Therefore, we note from the property of that
Since is uniformly normtonorm continuous on bounded sets, it follows that
Since satisfy the NSTcondition, we have . So, we assume that a subsequence of converges weakly to We shall show that . From and Lemma 2.9, we have
So, we note from (3.17) that
Since is uniformly convex and smooth and is bounded, it follows from Lemma 2.4 that
From , and , we have Since is uniformly normtonorm continuous on bounded sets and (3.23), it follows that
From we have
By the definition of , we have
Replacing by , we have from (A2) that
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting , we note from (3.25) and () that
For with and , let Since and , we have and hence So, from (), we have
This implies that
Letting from (A3), we have
Therefore, we obtain Finally, we will show that . Let . From and we note that
Since the norm is weakly lower semicontinuous, it follows that
From the definition of , we have . Hence Therefore, we obtain
Since has the KadecKlee property, it follows that Therefore, converges strongly to
As direct consequences of Theorem 3.1, we can obtain the following corollaries.
Corollary 3.2 (Takahashi and Zembayashi [13]).
Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying , and let be a relatively nonexpansive mapping from into itself such that . Let be a sequence generated by and
for every , where is the duality mapping on , satisfies and for some Then converges strongly to , where is the generalized projection of onto .
Proof.
Put . Let be a bounded sequence in with and let . Then there exists subsequence of such that . It follows directly from the definition of that . Hence satisfies NSTcondition, by Theorem 3.1; converges strongly to .
Corollary 3.3 (Takahashi and Zembayashi [12]).
Let be a uniformly convex and uniformly smooth Banach space; let be a nonempty closed convex subset of . Let be a bifunction from to satisfying . Let be a sequence generated by and
for every , where is the duality mapping on and for some Then, converges strongly to
Proof.
Putting in Theorem 3.1, we obtain Corollary 3.3.
Corollary 3.4.
Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a family of relatively nonexpansive mappings from into itself such that . Let be a sequence generated by and
for every , where is the duality mapping on , satisfies . Suppose that satisfy the NSTcondition. Then converges strongly to , where is the generalized projection of onto .
Proof.
Putting for all and in Theorem 3.1, we obtain Corollary 3.4.
Similarly as in the proof of Theorem 3.1, we can prove the following theorem.
Theorem 3.5.
Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of . Let be a bifunction from to satisfying and let be a family of relatively nonexpansive mappings from into itself such that . Let be a sequence generated by and
for every , where is the duality mapping on , satisfies and for some Suppose that satisfy the NSTcondition. Then converges strongly to , where is the generalized projection of onto .
Proof.
We first show that is closed and convex. It is obvious that is closed and is closed and convex. Since it follows that is convex. Hence is a closed and convex subset of for all Similarly as in proof of Theorem 3.1, we note that for all Next, we show by induction that for all . From , we note that Suppose that for some Then there exists such that From the definition of , we have
for all . Since , we have
and hence . So, we have and therefore, Hence for all . This implies that is well defined. By the same argument as in proof of Theorem 3.1, we can prove that the sequence converges strongly to .
Setting in Theorem 3.5, we have the following result.
Corollary 3.6 (Takahashi and Zembayashi [12, Theorem ]).
Let be a uniformly convex and uniformly smooth Banach space; let be a nonempty closed convex subset of . Let be a bifunction from to satisfying , and let be a relatively nonexpansive mapping from into itself such that . Let be a sequence generated by
for every , where is the duality mapping on , satisfies and for some Then, converges strongly to where is the generalized projection of onto .
4. Applications
In this section, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator in a Banach space by using the shrinking projection method.
Let be a real Banach space. An operator is said to be monotone if whenever . We denote the set by A monotone is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. If is maximal monotone, then the solution set is closed and convex.
Let be a smooth, strictly convex and reflexive Banach space, and let be a maximal monotone operator. Then for each and , there corresponds a unique element satisfying
see Barbu [28] or Takahashi [4]. We define the resolvent of by . In other words, for all . We know that is relatively nonexpansive and for all (see [4, 17]), where denotes the set of all fixed points of . We can also define, for each , the Yosida approximation of by We know that for all
We now consider the strong convergence theorem for finding a common element of the solution set of an equilibrium problem and the problem of finding a zero of a maximal monotone operator.
Theorem 4.1.
Let be a uniformly convex and uniformly smooth Banach space. Let be a maximal monotone operator, and let for all . Let be a nonempty closed convex subset of such that Let be a bifunction from to satisfying with . Let be a sequence generated by and
for every , where is the duality mapping on , , satisfy , and for some Then converges strongly to , where is the generalized projection of onto .
Proof.
Let be a bounded sequence in such that and let . Then, there exists a subsequence of such that . By the uniform smoothness of , we have
Since , we have
Let . Then it holds from the monotonicity of that
for all . Letting , we get . Then, the maximality of implies . Hence by Theorem 3.1, converges strongly to
In case . Putting for all and in Theorem 4.1, we obtain the following corollary.
Corollary 4.2.
Let be a uniformly convex and uniformly smooth Banach space. Let be a maximal monotone operator, and let for all , with . Let be a sequence generated by and
for every , where is the duality mapping on , , satisfy . Then converges strongly to , where is the generalized projection of onto .
Similarly as in the proof of Theorem 4.1, we can prove the following theorem.
Theorem 4.3.
Let be a uniformly convex and uniformly smooth Banach space. Let be a maximal monotone operator and let for all . Let be a nonempty closed convex subset of such that . Let be a bifunction from to satisfying with . Let be a sequence generated by and
for every , where is the duality mapping on , , satisfy , and for some Then converges strongly to , where is the generalized projection of onto .
Corollary 4.4.
Let be a uniformly convex and uniformly smooth Banach space. Let be a maximal monotone operator and let for all , with . Let be a sequence generated by and
for every , where is the duality mapping on , , satisfy . Then converges strongly to , where is the generalized projection of onto .
Acknowledgments
The first author thanks the National Research Council of Thailand to Naresuan University, 2009 for the financial support. Moreover, the second author would like to thank the National Centre of Excellence in Mathematics, PERDO, under the Commission on Higher Education, Ministry of Education, Thailand. This work is dedicated to Professor Wataru Takahashi on his retirement.
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