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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 479438, 13 pages
http://dx.doi.org/10.1155/2012/479438
Research Article

Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 22 May 2012; Accepted 5 July 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Jingling Zhang et al. 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.

Abstract

The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.

1. Introduction and Preliminaries

Iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by some authors, see for instance [14]. Let be a nonempty closed convex subset of a smooth Banach space and a multivalued mapping such that is nonempty for all . In [4], Homaeipour and Razani have defined the relatively nonexpansive multivalued mapping and have proved some convergence theorems.

Theorem SA 1 (see [4]). Let be a uniformly convex and uniformly smooth Banach space, and a nonempty closed convex subset of . Suppose is a relatively nonexpansive multivalued mapping. Let be a sequence of real numbers such that and . For a given , let be the iterative sequence defined by If is weakly sequentially continuous, then converges weakly to a fixed point of .

Theorem SA 2 (see [4]). Let be a uniformly convex and uniformly smooth Banach space, and a nonempty closed convex subset of . Suppose is a relatively nonexpansive multivalued mapping. Let be a sequence of real numbers such that and . For a given , let be the iterative sequence defined by If the interior of is nonempty, then converges strongly to a fixed point of .

Let be a Banach space with dual . We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing. It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of .

As we all know that if is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently it is not available in more general Banach spaces. In this connection, Alber [5] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that is a smooth Banach space. Consider the functional defined as [5, 6] by Observe that, in a Hilbert space , (1.4) reduces to .

The generalized projection is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the following minimization problem: existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, eg., [57]). In Hilbert space, . It is obvious from the definition of function that

Remark 1.1. If is a reflexive strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if then . From (1.6), we have . This implies . From the definitions of , we have , that is, , see [8, 9] for more details.

Let be a nonempty closed convex subset of a smooth Banach space and a multivalued mapping such that is nonempty for all . A point is called a fixed point of , if . The set of fixed points of is represented by . A point is called an asymptotic fixed point of , if there exists a sequence in which converges weakly to and , where . The set of asymptotic fixed points of is represented by . Moreover, a multivalued mapping is called relatively nonexpansive multivalued mapping, if the following conditions are satisfied:(1),(2),  for  all   ,  for  all  ,  for  all ,(3).

In [4], authors also give an example which is a relatively nonexpansive multivalued mapping but not a nonexpansive multivalued mapping.

The purpose of this paper is to present the notion of weak relatively nonexpansive multivalued mapping and to prove the strong convergence theorems for the weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. We first give the definition of weak relatively nonexpansive multivalued mapping as follows.

Let be a nonempty closed convex subset of a smooth Banach space and a multivalued mapping such that is nonempty for all . A point is called an strong asymptotic fixed point of , if there exists a sequence in which converges strongly to and , where . The set of the strong asymptotic fixed points of is represented by . Moreover, a multivalued mapping is called weak relatively nonexpansive multivalued mapping, if the following conditions are satisfied: (I), (II),  for  all  ,  for  all  ,  for  all  , (III).

We need the following Lemmas for the proof of our main results.

Lemma 1.2 (see Kamimura and Takahashi [7]). Let be a uniformly convex and smooth Banach space and let , be two sequences of . If and either or is bounded, then .

Lemma 1.3 (see Alber [5]). Let be a nonempty closed convex subset of a smooth Banach space and . Then, if and only if

Lemma 1.4 (see Alber [5]). Let be a reflexive, strictly convex and smooth Banach space, let be a nonempty closed convex subset of , and let . Then

Lemma 1.5. Let be a uniformly convex and smooth Banach space, let be a closed convex subset of , and let be a weak relatively nonexpansive multivalued mapping. Then is closed and convex.

Proof. First, we show is closed. Let be a sequence in such that . Since is weak relatively nonexpansive multivalued mapping, we have Therefore By Lemma 1.2, we have , hence, , so . Therefore is closed. Next, we show is convex. Let , put for any . For , we have By Lemma 1.2, we have , so , that is . Therefore, is convex. This completes the proof.

Remark 1.6. Let be a strictly convex and smooth Banach space, and a nonempty closed convex subset of . Suppose is a weak relatively nonexpansive multivalued mapping. If , then .

2. An Example of Weak Relatively Nonexpansive Multivalued Mapping

Next, we give an example which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping.

Example 2.1. Let and It is obvious that converges weakly to . On the other hand, we have for any . Define a mapping as follows: It is also obvious that and Since is a Hilbert space, we have Next, we prove . In fact, that for any strong convergent sequence such that and as , then there exist sufficiently large nature number such that , for any . Then for , it follows from that and hence , this implies , so that . Then is a weak relatively multivalued nonexpansive mapping.
We second claim that is not relatively multivalued nonexpansive mapping. In fact, that and as hold, but .

3. Strong Convergence of Monotone Hybrid Algorithm

Theorem 3.1. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , and let be a weak relatively multivalued nonexpansive mapping such that . Assume that are three sequences in such that , and for some constant . Define a sequence in by the following algorithm: then converges to .

Proof. We first show that is closed and convex for all . From the definitions of , it is obvious that is closed for all . Next, we prove that is convex for all . Since is equivalent to It is easy to get that is convex for all .
Next, we show that for all . Indeed, for each , we have So, , which implies that for all .
Since and , then we get Therefore, is nondecreasing. On the other hand, by Lemma 1.4 we have for all and for all . Therefore, is also bounded. This together with (3.5) implies that the limit of exists. Put From Lemma 1.4, we have, for any positive integer , that for all . This together with (3.7) implies that uniformly for all , holds. By using Lemma 1.2, we get that uniformly for all , holds. Then is a Cauchy sequence, therefore there exists a point such that .
Since , from the definition of , we have This together with (3.9) and implies that Therefore, by using Lemma 1.2, we obtain Since is uniformly norm-to-norm continuous on bounded sets, then we have Noticing that which leads to From (3.14) and , we obtain Since is also uniformly norm-to-norm continuous on bounded sets, then we obtain Observe that It follows from (3.10) and (3.18) that Since we have proved that , which together with (3.21) and that, is weak relatively multivalued nonexpansive mapping, implies that .
Finally, we prove that . From Lemma 1.4, we have On the other hand, since and , for all . Also from Lemma 1.4, we have By the definition of , we know that Combining (3.22), (3.23), and (3.24), we know that . Therefore, it follows from the uniqueness of that . This completes the proof.

When in Theorem 3.1, we obtain the following result.

Theorem 3.2. Let be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of , and let be a weak relatively multivalued nonexpansive mapping such that . Assume that is a sequences in such that for some constant . Define a sequence in by the following algorithm: then converges to .

4. Applications for Maximal Monotone Operators

In this section, we apply the above results to prove some strong convergence theorem concerning maximal monotone operators in a Banach space .

Let be a multivalued operator from to with domain and range . An operator is said to be monotone if for each and . A monotone operator is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. We know that if is a maximal monotone operator, then is closed and convex. The following result is also well known.

Theorem 4.1 (see Rockafellar [10]). Let be a reflexive, strictly convex and smooth Banach space and let be a monotone operator from to . Then is maximal if and only if for all .

Let be a reflexive, strictly convex and smooth Banach space, and let be a maximal monotone operator from to . Using Theorem 4.1 and strict convexity of , we obtain that for every and , there exists a unique such that Then we can define a single valued mapping by and such a is called the resolvent of . We know that for all , see [9, 11] for more details. Using Theorem 3.1, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space. Such a problem has been also studied in [1, 7, 1020].

Theorem 4.2. Let be a uniformly convex and uniformly smooth Banach space, let be a maximal monotone operator from to with , and let be a multivalued mapping defined as follows: where is a set of real numbers such that . is called the multivalued resolvent of . Then is a weak relatively nonexpansive multivalued mapping.

Proof. Since so that . Next, we show From the monotonicity of , we have Finally, we show . Observe that is obvious. Next, we show . Let be a sequence such that and . There exist sequences in and such that Since is uniformly norm-to-norm continuous on bounded sets, we obtain It follows from and the monotonicity of that for all and . Letting , we have for all and . Therefore, from the maximality of , we obtain . Hence is a weak relatively nonexpansive multivalued mapping. This completes the proof.

By using Theorems 3.1 and 4.2, we directly obtain the following result.

Theorem 4.3. Let be a uniformly convex and uniformly smooth Banach space, let be a maximal monotone operator from to with , let be a multivalued resolvent of , where , and let be three sequences of real numbers such that and for some constant . Define a sequence of as follows: where is the duality mapping on . Then converges strongly to , where is the generalized projection from onto .

Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).

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