## Advances on Multivalued Operators and Related Fixed Point Problems

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# Strong Convergence of a General Iterative Method for a Countable Family of Nonexpansive Mappings in Banach Spaces

**Academic Editor:**Chi-Ming Chen

#### Abstract

We introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Banach space. We prove strong convergence theorems for the sequences produced by the methods and approximate a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. It is important to state clearly that the contribution of this paper in relation with the previous works (Marino and Xu, 2006) is a technical method to prove strong convergence theorems of a general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces. Our results improve and generalize many known results in the current literature.

#### 1. Introduction

Viscosity approximation method for finding the fixed points of nonexpansive mappings was first proposed by Moudafi [1]. He proved the convergence of the sequence generated by the proposed method. In 2004, Xu [2] proved the strong convergence of the sequence generated by the viscosity approximation method to a unique solution of a certain variational inequality problem defined on the set of fixed points of a nonexpansive map (see also [3]). Marino and Xu [4] considered a general iterative method and proved that the sequence generated by the method converges strongly to a unique solution of a certain variational inequality problem which is the optimality condition for a particular minimization problem. Liu [5] and Qin et al. [6] also studied some applications of the iterative method considered in [4]. Yamada [7] introduced the so-called hybrid steepest-descent method for solving the variational inequality problem and also studied the convergence of the sequence generated by the proposed method. Recently, Tian [8] combined the iterative methods of [4, 7] in order to propose implicit and explicit schemes for constructing a fixed point of a nonexpansive mapping defined on a real Hilbert space. He also proved the strong convergence of these two schemes to a fixed point of under appropriate conditions. Related iterative methods for solving fixed point problems, variational inequalities, and optimization problems can be found in [9â€“14] and the references therein. By virtue of the projection, the authors in [13, 15] extended the implicit and explicit iterative schemes proposed in [8]. The approximation methods for common fixed points of a countable family of nonexpansive mappings have been recently studied by several authors; see, for example, [16, 17].

The purpose of this paper is to introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a Banach space. We prove strong convergence theorems for the sequences produced by the methods for a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. Our results improve and generalize many known results in the current literature; see, for example, [4, 7, 8, 13â€“15, 18â€“20].

#### 2. Preliminaries

Throughout this paper, we denote the set of real numbers and the set of positive integers by and , respectively. Let be a Banach space with the norm and the dual space . When is a sequence in , we denote the strong convergence of to by and the weak convergence by . For any sequence in , we denote the strong convergence of to by , the weak convergence by , and the weak-star convergence by . The normalized duality mapping is defined by
The modulus of convexity of is denoted by
for every with . A Banach space is said to be *uniformly convex* if for every . Let . The norm of is said to be *GÃ¢teaux differentiable* if for each , the limit
exists. In this case, is called *smooth*. If the limit (3) is attained uniformly in , then is called *uniformly smooth*. The Banach space is said to be *strictly convex* if whenever and . It is well known that is uniformly convex if and only if is uniformly smooth. It is also known that if is reflexive, then is strictly convex if and only if is smooth; for more details, see [21]. Now, we define a mapping , the modulus of smoothness of , as follows:
It is well known that is uniformly smooth if and only if . Let be such that . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all . If a Banach space admits a sequentially continuous duality mapping from weak topology to weak star topology, then is single valued and also is smooth; for more details, see [22]. In this case, the normalized duality mapping is said to be weakly sequentially continuous; that is, if is a sequence with , then [22]. A Banach space is said to satisfy the *Opial property* [23] if for any weakly convergent sequence in with weak limit ,
for all with . It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces () satisfy the Opial property; for example, see [22, 23]. It is also known that if admits a weakly sequentially continuous duality mapping, then is smooth and enjoys the Opial property; see for more details [22].

Let be a real Banach space and a nonempty subset of . Let be a mapping. We denote by the set of fixed points of ; that is, .

*Definition 1. *Let be a nonempty, closed, and convex subset of a real Banach space . An operator is said to be(i)*accretive* if there exists such that
(ii)*-strongly accretive* if, for some , there exists such that
(iii)*-Lipschitzian* if, for some ,
in particular, if , then is called a contraction;(iv) nonexpansive if
A linear bounded operator is said to be strongly positive if there exists such that

*Remark 2. *Let be a nonempty, closed, and convex subset of a real Banach space and let be a nonexpansive mapping. Then is an accretive operator, where is the identity mapping. Indeed, for any we have
which means that is accretive.

The following result has been proved in [24].

Lemma 3. *Let be a real -uniformly smooth Banach space. Then there exists a best uniformly smooth constant such that
**
for all .*

Let and be nonempty subsets of real Banach space with . A mapping is said to be sunny if for each and . A mapping is said to be a retraction if for each .

The following result has been proved in [25].

Lemma 4. *Let and be nonempty subsets of a real Banach space with and a retraction from into . Then is sunny and nonexpansive if and only if
**
for all and .*

Lemma 5 (demiclosedness principle [26]). *Let be a closed and convex subset of a real -uniformly smooth Banach space and let the normalized duality mapping be weakly sequentially continuous at zero. Suppose that is a nonexpansive mapping with . If is a sequence in that converges weakly to and if converges strongly to , then ; in particular, if , then .*

Lemma 6 (see [27]). *Let be a sequence of nonnegative real numbers satisfying the inequality
**
where and satisfy the conditions*(i)* and , or equivalently, ;*(ii)*, or*(ii)â€²*.**Then, .*

Lemma 7 (see [28]). *Let and be two sequences in a Banach space such that
**
where satisfies the following conditions: . If , then .*

Let be a subset of a real Banach space and a family of mappings of such that . Then is said to satisfy the -condition [29] if for each bounded subset of ,

Lemma 8 (see [29]). *Let be a subset of a real Banach space and a family of mappings of into itself which satisfies the -condition. Then, for each , converges strongly to a point in . Moreover, let the mapping be defined by
**
Then for each bounded subset of ,
**
In the sequel, one will write that satisfies the -condition if satisfies the -condition and is defined by Lemma 8 with .*

We end this section with the following simple examples of mappings satisfying the -condition (see also Lemma 19).

*Example 9. *(i) Let be a Banach space. For any , let a mapping be defined by
Then, is a nonexpansive mapping for each . It could easily be seen that satisfies the -condition, where for all .

(ii) Let be a smooth Banach space and let be any element of . For any , we define a mapping by
for all . We define also a mapping by
for all . It is easy to verify that satisfies the -condition.

(iii) Let , where
Let be a sequence defined by
where
for all . It is clear that the sequence converges weakly to . Indeed, for any , we have
as . It is also obvious that for any with sufficiently large. Thus, is not a Cauchy sequence. We define a countable family of mappings by
for all and . It is clear that for all . It is obvious that is a quasi-nonexpansive mapping for each . Thus is a countable family of quasi-nonexpansive mappings.

Let for all . It is easy to see that
Then, we obtain that is a quasi-nonexpansive mapping with . Let be a bounded subset of . Then there exists such that . On the other hand, for any , we have
Furthermore, we have
Therefore, satisfies the -condition.

#### 3. Fixed Point and Convergence Theorems

Let be a -uniformly smooth Banach space with the -uniform smooth constant and let be a closed and convex subset of . Let be a -Lipschitzian and -strongly accretive operator with constants , let be an -Lipschitzian mapping with constant , and let be a nonexpansive mapping with . Suppose that , . Define a mapping by From the definition of we deduce that Indeed, for any , in view of (31) we obtain On the other hand, it is easy to see that is continuous on compact interval . In fact, employing Lâ€™HÃ´pitalâ€™s Rule, we conclude that . Thus, Set and if . Then we have Assume now that satisfies . Then we get In this section, we introduce the following implicit scheme that generates a net in an implicit way: We prove the strong convergence of to a fixed point of which solves the variational inequality We first prove the following extension of Lemma 3.1 in [7] in a -uniformly smooth Banach space.

Lemma 10. *Let be a 2-uniformly smooth Banach space with the 2-uniform smooth constant and let be a closed and convex subset of . Let be a -Lipschitzian and -strongly accretive operator with , , and . In association with a nonexpansive mapping , define the mapping by
**
Then, is a contraction with contraction constant , where .*

*Proof. *In view of Lemma 3, we conclude that
for all . Put . Then by the assumptions and , we infer that
Let . Then we have
Therefore, is a contraction with contraction constant , which completes the proof.

*Remark 11. *Let be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant and a closed convex subset of . Let be a -Lipschitzian and -strongly accretive operator with constants and let be an -Lipschitzian mapping with constant . Assume is a nonexpansive mapping with . Let , , and , where satisfies (34). For any , let the mapping be defined by
Using Remark 11, it could easily be seen that
Thus in view of Banach contraction principle, the contraction mapping has a unique fixed point in , which uniquely solves the fixed point equation (37).

*Remark 12. *Let be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant and a closed convex subset of . Let be a -Lipschitzian and -strongly accretive operator with constants and let be an -Lipschitzian mapping with constant . Assume is a nonexpansive mapping with . Let , , and , where satisfies (34). Then
That is, is strongly accretive with coefficient .

In the following result, we drive some important properties of the net which will be used in the sequel.

Proposition 13. *Let be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant and let be a closed and convex subset of . Let be a -Lipschitzian and -strongly accretive operator with constants and let be an -Lipschitzian mapping with constant . Assume is a nonexpansive mapping with . Let , , and , where satisfies (34). For each , let denote a unique solution of the fixed point equation (37). Then, the following properties hold for the net :*(1)* is bounded;*(2)*;*(3)* defines a continuous curve from ** into *.

*Proof. *(1) Let be taken arbitrarily. Then, in view of Lemma 10 we obtain
This implies that
This shows that is bounded.

(2) Since is bounded, we have that and are bounded too. In view of the definition of we conclude that
as .

(3) Take arbitrarily. Then, we have
This implies that
The boundedness of implies that defines a continuous curve from into .

Theorem 14. *Let be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant and let be a closed and convex subset of . Let be a -Lipschitzian and -strongly accretive operator with constants and let be an -Lipschitzian mapping with constant . Assume is a nonexpansive mapping with . Let , , and , where satisfies (34). For each , let denote a unique solution of the fixed point equation (37). Then the net converges strongly, as , to a fixed point of which solves the variational inequality (38), or equivalently, .*

*Proof. *In view of Remark 11 the variational inequality (38) has a unique solution, say . We show that as . To this end, let be given arbitrary. Set
Then we have and hence
Since is a nonexpansive mapping from onto , in view of Lemma 4, we conclude that
Exploiting Lemma 10, (37), and (52), we obtain
This implies that
Let be such that as . Letting , it follows from Proposition 13(2) that . The boundedness of implies that there exists such that as . In view of Lemma 5, we deduce that . Since as , it follows from (55) that . Thus we have well defined. Next, we show that solves the variational inequality (38). We first notice that
This, together with (52), implies that
Since is nonexpansive, in view of Remark 2, we conclude that is accretive. This implies that
Replacing by in (58), taking the limit , and noticing that is bounded for , we obtain
Thus, we have a solution of the variational inequality (38). Consequently, by uniqueness. Therefore, as . The variational inequality (38) can be written as
Thus, in view of Lemma 4, it is equivalent to the following fixed point equation:
This completes the proof.

Theorem 15. *Let be a uniformly convex and 2-uniformly smooth Banach space with the 2-uniform smooth constant and let be a nonempty, closed and convex subset of . Suppose that the normalized duality mapping is weakly sequentially continuous at zero. Let be a -Lipschitzian and -strongly accretive operator with constants and let be an -Lipschitzian mapping with constant . Let , , and , where satisfies (34). Assume is a sequence of nonexpansive mappings from into itself such that . Suppose in addition that is a nonexpansive mapping such that satisfies the -condition. For given arbitrarily, let the sequence be generated iteratively by
**
where is the sunny nonexpansive retraction from onto and and are two real sequences in satisfying the following control conditions:
**
Then, the sequence converges strongly to which solves the variational inequality
*

*Proof. *We divide the proof into several steps.*Step 1*. We claim that the sequence is bounded. Let be fixed. In view of (62)â€“(64) and Lemma 10, we obtain
Since is nonexpansive, for all , it follows from (62) and (65) that
By induction, we conclude that is bounded. This implies that the sequences , , , and are bounded too. Let and set . Then we have a bounded subset of and .*Step 2*. We claim that . For this purpose, we denote a sequence by . Then we have
This implies that
In view of Lemma 8 and (63)(a) we conclude that
Utilizing Lemma 7, we deduce that
It follows from (63)(b) and (70) that
Observe now that
This implies that
Utilizing Lemma 7 and taking into account , we deduce that
On the other hand, we have
Employing Lemma 8, we obtain
*Step 3*. We prove that there exists such that
where is as in Theorem 14. We first note that there exists a subsequence of such that
Since is bounded, without loss of generality, we may assume that as . In view of Lemma 6 and Step 2, we conclude that . This, together with (78), implies that
*Step 4*. We claim that .

For each , by Lemma 10 and (36) we obtain
This implies that
where
In view of (81), we conclude that