Research Article | Open Access

Utith Inprasit, Weerayuth Nilsrakoo, "A Halpern-Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems", *Abstract and Applied Analysis*, vol. 2011, Article ID 632857, 22 pages, 2011. https://doi.org/10.1155/2011/632857

# A Halpern-Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems

**Academic Editor:**Norimichi Hirano

#### Abstract

A new modified Halpern-Mann type iterative method is constructed. Strong convergence of the scheme to a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. The results presented in this work improve on the corresponding ones announced by many others.

#### 1. Introduction

Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. Let be a Banach space, the dual space of , and a nonempty closed convex subset of . Let be a bifunction. The *equilibrium problem* is to find such that
The set of solutions of (1.1) is denoted by . The equilibrium problems include fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases. Some methods have been proposed to solve the equilibrium problems (see, e.g., [1, 2]). In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty, and they also proved a strong convergence theorem.

Let be a smooth Banach space and the normalized duality mapping from to . Alber [4] considered the following functional defined by
Using this functional, Matsushita and Takahashi [5, 6] studied and investigated the following mappings in Banach spaces. A mapping is *relatively nonexpansive* if the following properties are satisfied: (R1), (R2) for all and ,(R3),

where and denote the set of fixed points of and the set of asymptotic fixed points of , respectively. It is known that satisfies condition (R3) if and only if is demiclosed at zero, where is the identity mapping; that is, whenever a sequence in converges weakly to and converges strongly to 0, it follows that . In a Hilbert space , the duality mapping is an identity mapping and for all . Hence, if is nonexpansive (i.e., for all ), then it is relatively nonexpansive. Several articles have appeared providing methods for approximating fixed points of relatively nonexpansive mappings (see, e.g., [5–19] and the references therein). Matsushita and Takahashi [5] introduced the following iteration: a sequence defined by
where is arbitrary, is an appropriate sequence in , is a relatively nonexpansive mapping, and denotes the generalized projection from onto a closed convex subset of . They proved that the sequence converges *weakly* to a fixed point of . Moreover, Matsushita and Takahashi [6] proposed the following modification of iteration (1.3):
and proved that the sequence converges *strongly* to . The iteration (1.4) is called *the hybrid method.* To generate the iterative sequence, we use the generalized metric projection onto for . It always exists, because each is nonempty, closed, and convex. However, in a practical case, it is not easy to be calculated. In particular, as becomes larger, the shape of becomes more complicate, and the projection will take much more time to be calculated.

In order to overcome this difficulty, Nilsrakoo and Saejung [15] modified Halpern and Mann's iterations for finding a fixed point of a relatively nonexpansive mapping in a Banach space as follows: and where , , and are appropriate sequences in with , and they proved that converges strongly to .

Many authors studied the problems of finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [20–33] and the references therein). In a Hilbert space , S. Takahashi and W. Takahashi [34] introduced the iteration as follows: sequence generated by ,
where is an appropriate sequence in , is nonexpansive, and is an appropriate positive real sequence. They proved that converges strongly to an element in . In 2009, Takahashi and Zembayashi [30] proposed the iteration in a uniformly smooth and uniformly convex Banach space as follows: a sequence generated by ,
where is relatively nonexpansive, is an appropriate sequence in , and is an appropriate positive real sequence. They proved that if is weakly sequentially continuous, then converges *weakly* to an element in . Consequently, there are many results presented strong convergence theorems for finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems by using the hybrid method. However, Nilsrakoo [35] introduced the Halpern-Mann iteration guaranteeing the strong convergence as follows: and
and proved that and converge strongly to .

Motivated by Nilsrakoo and Saejung [15] and Nilsrakoo [35], we present a strong convergence theorem of a new modified Halpern-Mann iterative scheme to find a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth. The results in this work improve on the corresponding ones announced by many others.

#### 2. Preliminaries

We collect together some definitions and preliminaries which are needed in this paper. We say that a Banach space is *strictly convex* if the following implication holds for :
It is also said to be *uniformly convex* if for any , there exists such that
It is known that if is a uniformly convex Banach space, then is reflexive and strictly convex. We say that is *uniformly smooth* if the dual space of is uniformly convex. A Banach space is *smooth* if the limit exists for all norm one elements and in . It is not hard to show that if is reflexive, then is smooth if and only if is strictly convex.

Let be a smooth Banach space. The function (see [4]) is defined by
where the *duality mapping * is given by
It is obvious from the definition of the function that
for all . Moreover,
for all with and .

The following lemma is an analogue of Xu's inequality [36, Theorem 2] with respect to .

Lemma 2.1 (see [15, Lemma 2.2]). *Let be a uniformly smooth Banach space and . Then, there exists a continuous, strictly increasing, and convex function such that and
**
for all , and .*

It is also easy to see that if and are bounded sequences of a smooth Banach space , then implies that .

Lemma 2.2 (see [37, Proposition 2]). * Let be a uniformly convex and smooth Banach space, and let and be two sequences of such that or is bounded. If , then .*

*Remark 2.3. *For any bounded sequences and in a uniformly convex and uniformly smooth Banach space , we have

Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space . It is known that [4, 37] for any , there exists a unique point such that
Following Alber [4], we denote such an element by . The mapping is called the *generalized projection* from onto . It is easy to see that in a Hilbert space, the mapping coincides with the metric projection . Concerning the generalized projection, the followings are well known.

Lemma 2.4 (see [37, Propositions 4 and 5]). *Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space , and . Then, *(a)* if and only if for all ,*(b)* for all .*

*Remark 2.5. *The generalized projection mapping above is relatively nonexpansive and .

Let be a reflexive, strictly convex, and smooth Banach space. The duality mapping from onto coincides with the inverse of the duality mapping from onto ; that is, . We make use of the following mapping studied in Alber [4]: for all and . Obviously, for all and . We know the following lemma (see [4] and [38, Lemma 3.2]).

Lemma 2.6. *Let be a reflexive, strictly convex, and smooth Banach space, and let be as in (2.11). Then
**
for all and .*

Lemma 2.7 (see [39, Lemma 2.1]). *Let be a sequence of nonnegative real numbers. Suppose that
**
for all , where the sequences in and in satisfy conditions: , , and . Then .*

Lemma 2.8 (see [40, Lemma 3.1]). * Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then, there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
**
In fact, .*

For solving the equilibrium problem, we usually assume that a bifunction satisfies the following conditions (see, e.g., [1, 3, 30]): (A1) for all ,(A2) is monotone, that is, , for all , (A3)for all , , (A4)for all , is convex and lower semicontinuous.

The following lemma is a result which appeared in Blum and Oettli [1, Corollary 1].

Lemma 2.9 (see [1, Corollary 1]). *Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a bifunction satisfying conditions (A1)–(A4), and let and . Then, there exists such that
*

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.10 (see [30, Lemma 2.8]). *Let be a nonempty closed convex subset of a reflexive, strictly convex, and uniformly smooth Banach space . Let be a bifunction satisfying conditions (A1)–(A4). For , define a mapping so-called the resolvent of as follows:
**
for all . Then, the followings hold: *(i)* is single-valued, *(ii)* is a firmly nonexpansive-type mapping [11], that is, for all *(iii)*for all and , *(iv)*, *(v)* is closed and convex.*

*Remark 2.11. *Some well-known examples of resolvents of bifunctions satisfying conditions (A1)–(A4) are presented in [3, Lemma 2.15].

Lemma 2.12 2.12 (see [8, Lemma 2.3]). *Let be a nonempty closed convex subset of a Banach space , a bifunction from satisfying conditions (A1)–(A4), and . Then, if and only if for all .*

Lemma 2.13 2.13 (see [6], Proposition 2.4). *Let be a nonempty closed convex subset of a strictly convex and smooth Banach space and a relatively nonexpansive mapping. Then is closed and convex.*

#### 3. Main Results

In this section, we introduce a modified Halpern-Mann type iteration without *using the generalized metric projection* and prove a strong convergence theorem for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.

Theorem 3.1. *Let a uniformly convex and uniformly smooth Banach space, a nonempty closed convex subset of , be a finite family of a bifunction of into satisfying conditions (A1)–(A4), and a relatively nonexpansive mapping such that . Let be a finite family of the resolvents of with positive real sequences such that for all . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*,*(iv)*. ** Then, converges strongly to .*

*Proof. *For each , setting
We can see that . Since is nonempty, closed, and convex, we put . By Lemma 2.10(iii), we get
where . This together with (2.7) gives
By Lemma 2.6, we obtain
Next, we show that is bounded. We consider
By induction, we have
for all . This implies that is bounded, and so are , , , , and . Let be a function satisfying the properties of Lemma 2.1, where . It follows from (3.3) that

The rest of the proof will be divided into two cases.*Case 1. *Suppose that there exists such that is nonincreasing. In this situation, is then convergent. Then,
Notice that
From condition (ii),
It follows from (3.8) that
By the assumptions (i), (ii), and (iv),
By Remark 2.3, we get
From is continuous strictly increasing with , we have
Consequently,
This implies that
Since is bounded and is reflexive, we choose a subsequence of such that and
Let be fixed. Then, as . From and (3.14), we have
Then,
Replacing by , we have from (A2) that
Letting , we have from (3.19) and (A4) that
From Lemma 2.12, we have . Since satisfies condition (R3) and , we have . It follows that . By Lemma 2.4(a), we immediately obtain that
It follows from Lemma 2.7 and (3.5) that . Then, .*Case 2. *Suppose that there exists a subsequence of such that
for all . Then, by Lemma 2.8, there exists a nondecreasing sequence of positive integer numbers such that ,
for all sufficiently large numbers . We may assume without loss of generality that for all sufficiently large numbers . Since
we obtain
It follows from (3.8) that
Using the same proof of *Case 1*, we also obtain
From (3.5), we have
Since , we have
In particular, since , we get
It follows from (3.29) that . This together with (3.30) gives
But for all sufficiently large numbers , we conclude that .

From the two cases, we can conclude that converges strongly to and the proof is finished.

Setting , , and in Theorem 3.1, we have the following.

Corollary 3.2. *Let be a uniformly convex and uniformly smooth Banach space, a nonempty closed convex subset of , a bifunction of into satisfying conditions (A1)–(A4), and be a relatively nonexpansive mapping such that . Let be the resolvent of with a positive real sequence such that . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)*. ** Then, converges strongly to .*

Setting and in Corollary 3.2, we have the following result.

Corollary 3.3. *Let be a uniformly convex and uniformly smooth Banach space, a nonempty closed convex subset of , and a relatively nonexpansive mapping such that . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)*. ** Then converges strongly to .*

Next, we prove a strong convergence theorem for finding an element of the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.

Theorem 3.4. *Let be a uniformly convex and uniformly smooth Banach space, a nonempty closed convex subset of , a finite family of a bifunction of into satisfying conditions (A1)–(A4), and . Let be a finite family of the resolvents of with positive real sequences such that for all . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)* or . ** Then, converges strongly to .*

*Proof. *For each , setting
Since is nonempty, closed, and convex, we put . Using the same proof of Theorem 3.1 when is the identity operator, we can see that

The rest of the proof will be divided into two cases.*Case 1. *Suppose that there exists such that is non-increasing. In this situation, is then convergent. Then,
Notice that
From condition (ii),
It follows from (3.38) that
By the assumptions (i), (ii), and (iv),
By Remark 2.3, we get
Consequently,
This implies that
Since is bounded and is reflexive, we choose a subsequence of such that and
Let be fixed. Then, as . From and (3.14), we have
Then,
Replacing by , we have from (A2) that
Letting , we have from (3.49) and (A4) that
From Lemma 2.12, we have . By Lemma 2.4(a), we immediately obtain that
It follows from Lemma 2.7 and (3.39) that . Then, .*Case 2. *Suppose that there exists a subsequence of such that
for all . Using the same proof of *Case 2* in Theorem 3.1, we also conclude that .

From the two cases, we can conclude that converges strongly to .

Finally, we give two explicit examples validating the assumptions in Theorem 3.1 as follows.

*Example 3.5 (Optimization). **Let ** be a uniformly convex and uniformly smooth Banach space, ** a nonempty bounded closed convex subset of **, and ** a lower semicontinuous and convex functional. For instance, let **, ** and ** be defined dy *
Then is lower semicontinuous and convex. For each , let be defined by for all . It is known [1, 11] that satisfies conditions (A1)–(A4), and . Let . Then, is relatively nonexpansive of into (see [5, 6]) and . Then, . Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to .

*Example 3.6 (The convex feasibility problem). **Let ** be a real Hilbert space, let ** be nonempty closed convex subsets of ** satisfying ** (e.g., **). Let ** be a finite family of bifunctions of ** into ** defined by *
where is a metric projection from onto . It is known [3, Lemma 2.15(iv)] that satisfies conditions (A1)–(A4) and . Let . Then, is relatively nonexpansive of into (see [5, 6]) and then . Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to .

#### 4. Deduced Theorems in Hilbert Spaces

In Hilbert spaces, if is quasi-nonexpansive such that is demiclosed at zero, then is relatively nonexpansive. We obtain the following result.

Theorem 4.1. *Let be a Hilbert space, a nonempty closed convex subset of , a finite family of a bifunction of into satisfying conditions (A1)–(A4), and a quasi-nonexpansive mapping such that is demiclosed at zero and . Let be a finite family of the resolvents of with real sequences such that for all . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)*. ** Then converges strongly to .*

Applying Theorem 4.1 and using the technique in [41], we have the following result.

Theorem 4.2. *Let be a Hilbert space, a nonempty closed convex subset of , a contraction of into itself (i.e., there is such that for all ), a finite family of a bifunction of into satisfying conditions (A1)–(A4), and be a nonexpansive mapping such that . Let be a finite family of the resolvents of with real sequences such that for all . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)*. ** Then, converges strongly to such that .*

*Proof. *We note that is contraction. By Banach contraction principle, let be the fixed point of and a sequence generated by and
Using Theorem 4.1, we have . Since and are nonexpansive,
Applying Lemma 2.7, and so .

Setting , , and in Theorem 4.1, we have the following.

Corollary 4.3. *Let be a Hilbert space, a nonempty closed convex subset of , a bifunction of into satisfying conditions (A1)–(A4), and a quasi-nonexpansive mapping such that is demiclosed at zero and . Let be the resolvent of with a positive real sequence such that . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)*. ** Then, converges strongly to .*

Corollary 4.4. *Let be a Hilbert space, a nonempty closed convex subset of , a contraction of into itself, a bifunction of into satisfying conditions (A1)–(A4), and a nonexpansive mapping such that . Let be the resolvent of with a positive real sequence such that . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)*. ** Then, converges strongly to such that .*

*Remark 4.5. *Corollary 4.4 improves and extends [42, Theorem 5]. More precisely, the conditions are removed.

Setting and in Corollary 4.3, we have the following.

Corollary 4.6. *Let be a Hilbert space, a nonempty closed convex subset of , and a quasi-nonexpansive mapping such that is demiclosed at zero and . Let be a sequence generated by and
**
where , , and are sequences in satisfying the following conditions: *(i)*, *(ii)*, *(iii)*, *(iv)*. ** Then, converges strongly to .*

Applying Theorem 3.4, we have the following result.

Theorem 4.7. *Let be a Hilbert space, a nonempty closed convex subset of , *