Research Article | Open Access

# Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

**Academic Editor:**Yuriy Rogovchenko

#### Abstract

Let be a reflexive Banach space having a weakly sequentially continuous duality mapping with gauge function , a nonempty closed convex subset of , and a multivalued nonself-mapping such that is nonexpansive, where . Let be a contraction with constant . Suppose that, for each and , the contraction defined by has a fixed point . Let , and be three sequences in satisfying approximate conditions. Then, for arbitrary , the sequence generated by for all converges strongly to a fixed point of .

#### 1. Introduction

Let be a Banach space and a nonempty closed subset of . We will denote by the family of nonempty closed subsets of , by the family of nonempty closed bounded subsets of , by the family of nonempty compact subsets of , and by the family of nonempty compact convex subsets of . Let be the Hausdorff distance on ; that is,
for all , where is the distance from the point to the subset . Recall that a mapping is a * contraction* on if there exists a constant such that .

A multivalued mapping is said to be a * contraction* if there exists a constant such that
If (2) is valid when , the is called nonexpansive. A point is a fixed point for a multivalued mapping if . Banach's contraction principle was extended to a multivalued contraction by Nadler [1] in 1969. The set of fixed points of is denoted by .

Given a contraction with constant and , we can define a contraction by Then is a multivalued, and hence it has a (nonunique, in general) fixed point (see [1]); that is, If is single valued, we have which was studied by Moudafi [2] (see also Xu [3]). As a special case of (5), has been considered by Browder [4], Halpern [5], Jung and Kim [6, 7], Kim and Takahashi [8], Reich [9], Singh and Watson [10], Takahashi and Kim [11], Xu [12], and Xu and Yin [13] in a Hilbert space and Banach spaces.

In 2007, Jung [14] established the strong convergence of defined by for the multivalued nonexpansive nonself-mapping in a reflexive Banach space having a uniformly Gâteaux differentiable norm under the assumption .

In order to give a partial answer to Jung's open question [14] *Can the assumption ** be omitted?*, in 2008, Shahzad and Zegeye [15] considered a class of multivalued mapping under some mild conditions as follows.

Let be a closed convex subset of a Banach space . Let be a multivalued nonself-mapping and Then is multivalued, and is nonempty and compact for every . Instead of we consider, for , that

It is clear that , and if is nonexpansive and is weakly inward, then is weakly inward contraction. Theorem 1 of Lim [16] guarantees that has a fixed point, point ; that is, If is single valued, then (10) is reduced to (6).

Shahzad and Zegeye [15] also gave the strong convergence result of defined by (10) in a reflexive Banach space having a uniformly Gâteaux differentiable norm, which unified, extended, and complemented several known results including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], and Xu and Yin [13].

In 2009, motivated by the results of Rafiq [20] and Yao et al. [21], Ceng and Yao [22] considered the following iterative scheme.

Theorem CY (see [22, Theorem 3.1]). *Let be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, a nonempty closed convex subset of , and a multivalued nonself-mapping such that is nonexpansive. Suppose that is a nonexpansive retract of and that for each and the contraction defined by has a fixed point . Let , , and be three sequences in satisfying the following conditions:*(i)*;*(ii)* and .** For arbitrary initial value and a fixed element , let the sequence be generated by
**
Then converges strongly to a fixed point of . *

Theorem CY also improves, develops, and complements the corresponding results in Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Shahzad and Zegeye [15], and Xu and Yin [13] to the iterative scheme (11). For convergence of related iterative schemes for several nonlinear mappings, we can refer to [23–26] and the references therein.

In this paper, inspired and motivated by the above-mentioned results, we consider a viscosity iterative method for a multivalued nonself-mapping in a reflexive Banach space having a weakly sequentially continuous duality mapping and establish the strong convergence of the sequence generated by the proposed iterative method. Our results improve and develop the corresponding results of Ceng and Yao [22], as well as some known results in the earlier and recent literature, including those of Jung [14], Jung and Kim [6, 7], Kim and Jung [17], López Acedo and Xu [18], Sahu [19], Shahzad and Zegeye [15], Xu [12], and Xu and Yin [13], to the viscosity iterative scheme in different Banach space.

#### 2. Preliminaries

Let be a real Banach space with norm , and let be its dual. The value of at will be denoted by .

A Banach space is called * uniformly convex* if for every , where the modulus of convexity of is defined by
for every with . It is well known that if is uniformly convex, then is reflexive and strictly convex (cf. [27]).

By a gauge function we mean a continuous strictly increasing function defined on such that and . The mapping defined by
is called the *duality mapping* with gauge function . In particular, the duality mapping with gauge function denoted by is referred to as the * normalized duality mapping*. It is known that a Banach space is smooth if and only if the normalized duality mapping is single valued. The following property of duality mapping is also well known:
where is the set of all real numbers; in particular, for all ([28]).

We say that a Banach space has a * weakly sequentially continuous duality mapping* if there exists a gauge function such that the duality mapping is single valued and continuous from the weak topology to the topology; that is, for any with . For example, every space has a weakly continuous duality mapping with gauge function . It is well known that if is a Banach space having a weakly sequentially continuous duality mapping with gauge function , then has the * opial condition* [29]; this is, whenever a sequence in converges weakly to , then
A mapping is *-nonexpansive* [30] if, for all and with , there exists with such that
It is known that -nonexpansiveness is different from nonexpansiveness for multivalued mappings. There are some -nonexpansiveness multivalued mappings which are not nonexpansive and some nonexpansive multivalued mappings which are not -nonexpansive [31].

We introduce some terminology for boundary conditions for nonself-mappings. The * inward set* of at is defined by
Let with
for any . Note that, for a convex set , we have , the closure of . A multivalued mapping is said to satisfy the * weak inwardness condition* if for all .

Finally, the following lemma was given by Xu [32] (also see Xu [33]).

Lemma 1. *If is a closed bounded convex subset of a uniformly convex Banach space and is a nonexpansive mapping satisfying the weak inwardness condition, then has a fixed point.*

#### 3. Main Results

Now, we establish strong convergence of a viscosity iterative scheme for a multivalued nonself-mapping.

Theorem 2. *Let be a reflexive Banach space having a weakly sequentially continuous duality mapping with gauge function . Let be a nonempty closed convex subset of and a multivalued nonself-mapping such that and is nonexpansive. Let be a contraction with constant . Suppose that for each and , the contraction defined by has a fixed point . Let , , and be three sequences in satisfying the following conditions:*(i)*;*(ii)* and .** For arbitrary initial value , let the sequence be defined by
**
Then converges strongly to a fixed point of . *

* Proof. *First, observe that, for each ,
From , , it follows that . Also, notice that, for each and , the contraction defined by has a fixed point . Thus, for and , there exists such that
This shows that the sequence can be defined well via the following:
Therefore, for any , we can find some such that
Next, we divide the proof into several steps. *Step *1. We show that is bounded. Indeed, notice that whenever is a fixed point of . Let . Then and we have
It follows that
and so
By induction, we have
Hence is bounded and so are and . *Step *2. We show that . In fact, since for some , by conditions (i) and (ii), we have
*Step *3*.* We show that there exists . In fact, since and are bounded and is reflexive, there exists a subsequence of such that . For this , by compactness of , we can find , , such that
Now suppose that and . The sequence has a convergent subsequence, which is denoted again by with . Assume that . Since a Banach space having the weakly sequentially continuous duality mapping satisfies the opial condition [29], by Step 2, we have
which is a contradiction. Hence we have , and so . *Step *4. We show that for . Indeed, for , by (24), we have
So,
Thus it follows that
Hence, from condition (ii), we conclude that
*Step *5. We show that , where is defined as in Step 3. Indeed, for ,
Interchanging and in (35), we obtain
and so
Using the fact that is weakly sequentially continuous, Step 4, and condition (ii), we have
This implies that as . In fact, if , then , and by (38) . This means that , which is a contradiction. Thus we proved that there exists a subsequence of which converges strongly to a fixed point of . *Step *6. We show that the entire sequence converges strongly to . Suppose that there exists another subsequence of such that as . Since as , it follows that and so ; that is, . Notice that . Since is bounded and the duality mapping is single valued and weakly sequentially continuous from to , we have

Thus, from Steps 2 and 4, it follows that

By the same argument, we also have

Therefore, from (40) and (41), we obtain
and so . Thus . This completes the proof.

*Remark 3. *(1) In Theorem 2, if , is a constant mapping, then the iterative scheme (19) is reduced to the iterative scheme (11) in Theorem CY of Ceng and Yao [22] in the Introduction section. Therefore Theorem 2 improves Theorem CY to the viscosity iterative scheme in different Banach space.

(2) In Theorem 2, we remove the assumption that is a nonexpansive retract of in Theorem CY.

(3) In Theorem 2, if for , then the iterative scheme (19) becomes the following scheme:
which is a viscosity iterative scheme for those in Shahzad and Zegeye [15]. Therefore Theorem 2 develops Theorem 3.1 of Shahzad and Zegeye [15], as well as Theorem 1 of Jung [14], to the viscosity iterative method in different Banach space.

(4) Theorem 2 also improves and complements the corresponding results of Kim and Jung [17] and Sahu [19] as well as Jung and Kim [6, 7], López Acedo and Xu [18], and Xu and Yin [13].

By definition of the Hausdorff metric, we obtain that if is -nonexpansive, then is nonexpansive. Hence, as a direct consequence of Theorem 2, we have the following result.

Corollary 4. *Let be a reflexive Banach space having a weakly sequentially continuous duality mapping with gauge function . Let be a nonempty closed convex subset of and a multivalued -nonexpansive nonself-mapping such that . Let be a contraction with constant . Suppose that, for each and , the contraction defined by has a fixed point . Let , , and be three sequences in satisfying the following conditions:*(i)*;*(ii)* and .** For arbitrary initial value , let the sequence be generated by (19). Then converges strongly to a fixed point of . *

It is well known that every nonempty closed convex subset of a strictly convex and reflexive Banach space is Chebyshev; that is, for any , there is a unique element such that . Thus, we have the following corollary.

Corollary 5. *Let be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping with gauge function . Let be a nonempty closed convex subset of and a multivalued nonself-mapping such that and is nonexpansive. Let be a contraction with constant . Suppose that, for each and , the contraction defined by has a fixed point . Let , , and be three sequences in satisfying the following conditions:*(i)*; *(ii)* and .** For arbitrary initial value , let the sequence be generated by (19). Then converges strongly to a fixed point of . *

* Proof. *In this case, is Chebyshev for each . So is a selector of and is single valued. Thus the result follows from Theorem 2.

Corollary 6. *Let be a strictly convex and reflexive Banach space having a weakly sequentially continuous duality mapping with gauge function . Let be a nonempty closed convex subset of and a multivalued -nonexpansive nonself-mapping such that . Let be a contraction with constant . Suppose that, for each and , the contraction defined by has a fixed point . Let , , and be three sequences in satisfying the following conditions:*(i)*;*(ii)* and .** For arbitrary initial value , let the sequence be generated by (19). Then converges strongly to a fixed point of . *

Corollary 7. *Let be a uniformly convex Banach space having a weakly sequentially continuous duality mapping with gauge function . Let be a nonempty closed convex subset of and a multivalued nonself-mapping satisfying the weak inwardness condition such that is nonexpansive. Let be a contraction with constant . Let , , and be three sequences in satisfying the following conditions:*(i)*;*(ii)* and .*

*Proof. *Define, for each and , the contraction by
As it is easily seen that also satisfies the weak inwardness condition: for all , it follows from Lemma 1 that has a fixed point denoted by . Thus the result follows from Theorem 2.

*Remark 8. *(1) As in [31], Shahzad and Zegeye [15] gave the following example of a multivalued such that is nonexpansive. Let , and let be defined by for . Then for . Also is -nonexpansive but not nonexpansive (see [31]).

(2) Corollaries 4–7 develop Corollaries 3.3–3.6 of Ceng and Yao [22] to the viscosity iterative method in different Banach spaces.

(3) By replacing the iterative scheme (11) in Theorem CY with the iterative scheme (19) in Theorem 2 and using the same proof lines as Theorem CY together with our method, we can also establish the viscosity iteration version of Theorem CY.

#### Acknowledgments

The author thanks the anonymous referees for their valuable comments and suggests, which improved the presentation of this paper and for providing some recent related papers. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012000895).

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#### Copyright

Copyright © 2013 Jong Soo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.