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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 636217, 13 pages
http://dx.doi.org/10.1155/2012/636217
Research Article

On Asymptotically Quasi--Nonexpansive Mappings in the Intermediate Sense

1Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China

Received 1 September 2012; Accepted 9 October 2012

Academic Editor: Yongfu Su

Copyright © 2012 Xiaolong Qin and Lin Wang. 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

A projection iterative process is investigated for the class of asymptotically quasi--nonexpansive mappings in the intermediate sense. Strong convergence theorems of common fixed points of a family of asymptotically quasi--nonexpansive mappings in the intermediate sense are established in the framework of Banach spaces.

1. Introduction

Fixed point theory as an important branch of nonlinear analysis theory has been applied in many disciplines, including economics, image recovery, mechanics, quantum physics, and control theory; see, for example, [14]. The theory itself is a beautiful mixture of analysis, topology, and geometry. During the four decades, many famous existence theorems of fixed points were established; see, for example, [513]. However, from the standpoint of real world applications it is not only to know the existence of fixed points of nonlinear mappings but also to be able to construct an iterative algorithm to approximate their fixed points. The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively; see [1416] for more details and the reference therein. Iterative methods play an important role in the computation of fixed points of nonlinear mappings. Indeed, many well-known problems can be studied by using algorithms which are iterative in their nature.

In this paper, we introduced a new class of nonlinear mappings: asymptotically quasi--nonexpansive mappings in the intermediate sense and considered the problem of approximating a common fixed point of a family of the mappings based on a projection iterative process.

The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, strong convergence of a projection iterative algorithm is obtained in a reflexive, strictly convex, and smooth Banach space such that both and   enjoy Kadec-Klee property. Some corollaries as the immediate results of main results are given.

2. Preliminaries

Let be a real Hilbert space, a nonempty subset of , and a mapping. The symbol stands for the fixed point set of . Recall the following. is said to be nonexpansive if and only if

is said to be quasi-nonexpansive if and only if , and

We remark here that a nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive: however, the inverse may be not true. See the following example [17].

Example 2.1. Let and define a mapping by by Then is quasi-nonexpansive but not nonexpansive.

is said to be asymptotically nonexpansive if and only if there exists a sequence with as such that It is easy to see that a nonexpansive mapping is an asymptotically nonexpansive mapping with the sequence . The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [7]. Since 1972, a host of authors have studied the convergence of iterative algorithms for such a class of mappings.

is said to be asymptotically quasi-nonexpansive if and only if , and there exists a sequence with as such that It is easy to see that a quasi-nonexpansive mapping is an asymptotically quasi-nonexpansive mapping with the sequence .

is said to be asymptotically nonexpansive in the intermediate sense if and only if it is continuous, and the following inequality holds:

is said to be asymptotically quasi-nonexpansive in the intermediate sense if and only if and the following inequality holds:

The class of mappings which are asymptotically nonexpansive in the intermediate sense was considered by Bruck et al. [18]. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense may not be Lipschitz continuous. However, asymptotically nonexpansive mappings are Lipschitz continuous.

In what follows, we always assume that is a Banach space with the dual space . Let be a nonempty, closed, and convex subset of . We use the symbol to stand for the normalized duality mapping from to   defined by where denotes the generalized duality pairing of elements between and . It is well known that if is strictly convex, then is single valued; if is reflexive and smooth, then is single valued and demicontinuous; see [19] for more details and the references therein.

It is also well known that if is a nonempty, closed, and convex subset of a Hilbert space , and is the metric projection from 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 [20] introduced a generalized projection operator in Banach spaces which is an analogue of the metric projection in Hilbert spaces.

Recall that a Banach space is said to be strictly convex if for all with , and . It is said to be uniformly convex if for any two sequences and in such that and . Let be the unit sphere of . Then the Banach space is said to be smooth provided exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for all .

Recall that a Banach space enjoys Kadec-Klee property if for any sequence and with , and , then as . For more details on Kadec-Klee property, the readers can refer to [19, 21] and the references therein. It is well known that if is a uniformly convex Banach spaces, then enjoys Kadec-Klee property.

Let be a smooth Banach space. Consider the functional defined by Notice that, in a Hilbert space , (2.9) is reduced to for all . The generalized projection is a mapping that assigns to an arbitrary point , the minimum point of the functional ; that is, , where is the solution to the following minimization problem: The existence and uniqueness of the operator follow from the properties of the functional and the strict monotonicity of the mapping ; see, for example, [19, 20]. In Hilbert spaces, . It is obvious from the definition of the function that

Remark 2.2. If is a reflexive, strictly convex, and smooth Banach space, then, for all , if and only if . It is sufficient to show that if , then . From (2.11), we have . This implies that . From the definition of , we see that . It follows that ; see [20] for more details.

Next, we recall the following.

(1) A point in is said to be an asymptotic fixed point of [22] if and only if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by .

(2) is said to be relatively nonexpansive if and only if The asymptotic behavior of relatively nonexpansive mappings was studied in [23, 24].

(3) is said to be relatively asymptotically nonexpansive if and only if where is a sequence such that as .

Remark 2.3. The class of relatively asymptotically nonexpansive mappings was first considered in Su and Qin [25]; see also, Agarwal et al. [26], and Qin et al. [27].
(4) is said to be quasi--nonexpansive if and only if

(5) is said to be asymptotically quasi--nonexpansive if and only if there exists a sequence with as such that

Remark 2.4. The class of quasi--nonexpansive mappings and the class of asymptotically quasi--nonexpansive mappings were first considered in Zhou et al. [28]; see also Qin et al. [29], Qin, and Agarwal [30], Qin et al. [31], Qin et al. [32], and Qin et al. [33].

Remark 2.5. The class of quasi--nonexpansive mappings and the class of asymptotically quasi--nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi--nonexpansive mappings and asymptotically quasi--nonexpansive do not require .

Remark 2.6. The class of quasi--nonexpansive mappings and the class of asymptotically quasi--nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

In this paper, based on asymptotically (quasi-) nonexpansive mappings in the intermediate sense which was first considered by Bruck et al. [18], we introduce and consider the following new nonlinear mapping: asymptotically (quasi-) -nonexpansive mappings in the intermediate sense.

(6) is said to be an asymptotically -nonexpansive mapping in the intermediate sense if and only if

(7) is said to be an asymptotically quasi--nonexpansive mapping in the intermediate sense if and only if , and

Remark 2.7. The class of asymptotically (quasi-)  -nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically (quasi-) nonexpansive mappings in the intermediate sense in the framework of Banach spaces.

Let and . Define the following mapping by Then is an asymptotically -nonexpansive mapping in the intermediate sense with the fixed point set . We also have the following: where . Hence, we have (8) The mapping is said to be asymptotically regular on if and only if

In order to prove our main results, we also need the following lemmas.

Lemma 2.8   (see [20]). Let be a nonempty, closed, and convex subset of a smooth Banach space , and . Then if and only if

Lemma 2.9 (see [20]). Let be a reflexive, strictly, convex, and smooth Banach space, a nonempty, closed, and convex subset of , and . Then

3. Main Results

Theorem 3.1. Let be a reflexive, strictly convex, and smooth Banach space such that both and have Kadec-Klee property. Let be a nonempty, bounded closed, and convex subset of . Let be an index set, and a closed asymptotically quasi--nonexpansive mapping in the intermediate sense, for every . Assume that is nonempty, and is asymptotically regular, for every . Let be a sequence generated in the following manner: where . Then converges strongly to , where stands for the generalized projection from onto .

Proof. The proof is split into the following 5 steps.
Step 1. It show that is closed and convex.
Since is closed, we can easily conclude that is closed. The proof that is closed. We only prove that is convex. Let and , where , for every . We see that . Indeed, we see from the definition of that In view of (2.12), we obtain (3.2) that Multiplying and on the both sides of (3.3), respectively, yields that . This implies that In light of (2.11), we arrive at It follows that Since   is reflexive, we may, without loss of generality, assume that . In view of the reflexivity of , we have . This shows that there exists an element such that . It follows that Taking on the both sides of the equality above, we obtain that This implies that , that is, . It follows that . In view of Kadec-Klee property of , we obtain from (3.6) that . Since is demicontinuous, we see that . By virtue of Kadec-Klee property of , we see from (3.5) that as . Hence , as . In view of the closedness of , we can obtain that , for every . This shows, for every , that is convex. This proves that is convex. This completes Step 1.
Step 2. It show that is closed and convex, .
It suffices to show, for any fixed but arbitrary , that is closed and convex, for every . This can be proved by induction on . It is obvious that is closed and convex. Assume that is closed and convex for some . We next prove that is closed and convex for the same . This completes the proof that is closed and convex. The closedness of is clear. We only prove the convexness. Indeed, , we see that , and In view of (3.9), we find that where , . It follows that is convex. This in turn implies that is closed, and convex. This completes Step 2.
Step 3. It show that , .
It is obvious that . Suppose that for some . For any , we see that On the other hand, we obtain from (2.12) that Combining (3.11) with (3.12), we arrive at which implies that . This proves that , . This completes Step 3.
Step  4. It show that , where , as .
Since is bounded and the space is reflexive, we may assume that . Since is closed and convex, we see that . On the other hand, we see from the weakly lower semicontinuity of the norm that which implies that as . Hence, as . In view of Kadec-Klee property of , we see that as . On the other hand, we see from that . It follows that from which it follows that as . In view of (2.11), we see that as . This in turn implies that as . Hence, as . This shows that is bounded. Since is reflexive, we see that is also reflexive. We may, without loss of generality, assume that . In view of the reflexivity of , we have . This shows that there exists an element such that . It follows that Taking on the both sides of the equality above, we obtain that This implies that , that is, . It follows that . In view of Kadec-Klee property of , we obtain that . Since is demicontinuous, we see that . In the light of Kadec-Klee property of , we see that , , as . On the other hand, we have It follows from the asymptotic regularity of that as . That is, . From the closedness of , we obtain that . This implies that .
Step 5. It show that .
In view of , we see from Lemma 2.8 that Since , we arrive at Letting in the above, we arrive at It follows from Lemma 2.8 that . This completes the proof of Theorem 3.1.

Remark 3.2. The space in Theorem 3.1 can be applicable to , . Since the class of asymptotically quasi--nonexpansive mappings includes the class of asymptotically quasi--nonexpansive mappings as a special case, we see that Theorem 3.1 still holds for the class of asymptotically quasi--nonexpansive mappings.

For a single mapping, we can easily conclude the following.

Corollary 3.3. Let be a reflexive, strictly convex, and smooth Banach space such that both and   have Kadec-Klee property. Let be a nonempty, bounded, closed, and convex subset of . Let be a closed asymptotically quasi--nonexpansive mapping in the intermediate sense. Assume that is nonempty, and is asymptotically regular. Let be a sequence generated in the following manner: where . Then converges strongly to , where stands for the generalized projection from onto .

In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following.

Corollary 3.4. Let be a nonempty, bounded, closed, and convex subset of a Hilbert space . Let be an index set, and a closed asymptotically quasi-nonexpansive mapping in the intermediate sense, for every . Assume that is nonempty, and is asymptotically regular, for every . Let be a sequence generated in the following manner: where . Then converges strongly to , where stands for the metric projection from onto .

For a single mapping, we can easily conclude the following.

Corollary 3.5. Let be a nonempty, bounded, closed, and convex subset of a Hilbert space . Let be a closed asymptotically quasi-nonexpansive mapping in the intermediate sense. Assume that is nonempty, and is asymptotically regular. Let be a sequence generated in the following manner: where . Then converges strongly to , where stands for the metric projection from onto .

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