Abstract and Applied AnalysisVolume 2011 (2011), Article ID 142626, 13 pagesdoi:10.1155/2011/142626

Research Article

Hybrid Projection Algorithms for Asymptotically Strict Quasi-ϕ-Pseudocontractions

Xiaolong Qin,1,2 Tianze Wang,¹ and Sun Young Cho³

¹School of Mathematics and Information Sciences, North China University of Water Conservancy and Electric Power, Zhengzhou 450011, China²Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China³Department of Mathematics, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 27 November 2010; Revised 5 March 2011; Accepted 31 March 2011

Academic Editor: Paul Eloe

Copyright © 2011 Xiaolong Qin 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.

A new nonlinear mapping is introduced. Hybrid projection algorithms are considered for the class of new nonlinear mappings. Strong convergence theorems are established in a real Banach space.

1. Introduction

Let be a real Banach space, a nonempty subset of , and a nonlinear mapping. Denote by the set of fixed points of . Recall that is said to be nonexpansive if We remark that the mapping is said to be quasinonexpansive if and (1.1) holds for all and . is said to be asymptotically nonexpansive if there exists a sequence with as such that We remark that the mapping is said to be asymptotically quasinonexpansive if and (1.2) holds for all and . The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that if is a nonempty bounded closed convex subset of a uniformly convex Banach space , then every asymptotically nonexpansive selfmapping has a fixed point in . Further, the set of fixed points of is closed and convex. Since 1972, many authors have studied the weak and strong convergence problems of iterative algorithms for the class of mappings.

Recall that is said to be a strict pseudocontraction if there exists a constant such that We remark that the mapping is said to be a strict quasipseudocontraction if and (1.3) holds for all and .

The class of strict pseudocontractions was introduced by Browder and Petryshyn [2]. In 2007, Marino and Xu [3] proved that the fixed point set of strict pseudocontractions is closed and convex. They also proved that is demiclosed at the origin in real Hilbert spaces. A strong convergence theorem of hybrid projection algorithms for strict pseudocontractions was established; see [3] for more details.

Recall that is said to be an asymptotically strict pseudocontraction if there exist a constant and a sequence with as such that We remark that the mapping is said to be an asymptotically strict quasipseudocontraction if and (1.4) holds for all and .

The class of asymptotically strict pseudocontractions was introduced by Qihou [4] in 1996. Kim and Xu [5] proved that the fixed-point set of asymptotically strict pseudocontractions is closed and convex. They also obtained a strong convergence theorem for the class of asymptotically strict pseudocontractions by hybrid projection algorithms. To be more precise, they proved the following theorem.

Theorem KX. Let be a closed convex subset of a Hilbert space , and let be an asymptotically -strict pseudocontraction for some . Assume that the fixed-point set of is nonempty and bounded. Let be the sequence generated by the following (CQ) algorithm: where Assume that the control sequence is chosen so that then converges strongly to .

It is well known that, in an infinite dimensional Hilbert space, the normal Mann iterative algorithm has only weak convergence, in general, even for nonexpansive mappings. Hybrid projection algorithms are popular tool to prove strong convergence of iterative sequences without compactness assumptions. Recently, hybrid projection algorithms have received rapid developments; see, for example, [3, 5–24]. In this paper, we will introduce a new mapping, asymptotically strict quasi--pseudocontractions, and give a strong convergence theorem by a simple hybrid projection algorithm in a real Banach space.

2. Preliminaries

Let be a Banach space with the dual space . We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing of elements between and ; see [25]. It is well known that if is strictly convex, then is single valued, and if is uniformly convex, then is uniformly continuous on bounded subsets of .

It is also well known 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 [26] recently introduced a generalized projection operator in a Banach space 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 in such that and . is said to have Kadec-Klee property if a sequence of satisfying that and , then . It is known that if is uniformly convex, then enjoys Kadec-Klee property; see [25, 27] for more details. Let be the unit sphere of then the Banach space is said to be smooth provided exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . It is well known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .

Let be a smooth Banach space. Consider the functional defined by Observe that, in a Hilbert space , (2.3) 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, [26–29]. In Hilbert spaces, . It is obvious from the definition of the function that

Remark 2.1. 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 (2.5), we have . This implies that . From the definition of , we see that . It follows that ; see [25, 27] for more details.

Now, we give some definitions for our main results in this paper.

Let be a closed convex subset of a real Banach space and a mapping.(1)A point in is said to be an asymptotic fixed point of [30] 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 [15, 31, 32] if The asymptotic behavior of a relatively nonexpansive mapping was studied in [30–32].(3) is said to be relatively asymptotically nonexpansive [6, 11] if where is a sequence such that as .(4) is said to be -nonexpansive [14, 16, 17] if (5) is said to be quasi--nonexpansive [14, 16, 17] if (6) is said to be asymptotically -nonexpansive [14] if there exists a real sequence with as such that (7) is said to be asymptotically quasi--nonexpansive [14] if there exists a real sequence with as such that (8) is said to be a strict quasi--pseudocontraction if , and there exists a constant such that We remark that is said to be a quasistrict pseudocontraction in [13]. (9) is said to be asymptotically regular on if, for any bounded subset of ,

Remark 2.2. The class of quasi-ϕ-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 mappings do not require , where denotes the asymptotic fixed-point set of .

Remark 2.3. In the framework of Hilbert spaces, quasi--nonexpansive mappings and asymptotically quasi--nonexpansive mappings are reduced to quasinonexpansive mappings and asymptotically quasinonexpansive mappings.

In this paper, we introduce a new nonlinear mapping: asymptotically strict quasi--pseudocontractions.

Definition 2.4. Recall that a mapping is said to be an asymptotically strict quasi--pseudocontraction if , and there exists a sequence with as and a constant such that

Remark 2.5. In the framework of Hilbert spaces, asymptotically strict quasi--pseudocontractions are asymptotically strict quasipseudocontractions.

Next, we give an example which is an asymptotically strict quasi--pseudocontraction.

Let , and let be the closed unit ball in . Define a mapping by where is a sequence of real numbers such that , , where , and . Then where is a fixed point of and is a real number. In view of , we see that is an asymptotically strict quasi--pseudocontraction.

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

Lemma 2.6 (see [29]). Let be a uniformly convex and smooth Banach space, and let , be two sequences of . If and either or is bounded, then .

Lemma 2.7 (see [26]). Let be a nonempty closed convex subset of a smooth Banach space and then if and only if

Lemma 2.8 (see [26]). Let be a reflexive, strictly convex, and smooth Banach space, a nonempty closed convex subset of , and then

3. Main Results

Theorem 3.1. Let be a nonempty closed and convex subset of a uniformly convex and smooth Banach space . Let be a closed and asymptotically strict quasi pseudocontraction with a sequence such that as . Assume that is uniformly asymptotically regular on and is nonempty and bounded. Let be a sequence generated in the following manner: where then the sequence converges strongly to .

Proof. The proof is split into five steps.Step 1. Show that is closed and convex.Let be a sequence in such that as . We see that . Indeed, we obtain from the definition of that In view of (2.6), we see that It follows that which implies that from which it follows that From Lemma 2.6, we see that as . This implies that as . From the closedness of , we obtain that . This proves the closedness of .Next, we show the convexness of . Let and , where . We see that . Indeed, we have from the definition of that By virtue of (2.6), we obtain that Multiplying and on both the sides of (3.7) and (3.8), respectively, yields that It follows that In view of Lemma 2.6, we see that as . This implies that as . From the closedness of , we obtain that . This proves that is convex. This completes Step 1.Step 2. Show that is closed and convex for each .It is not hard to see that is closed for each . Therefore, we only show that is convex for each . It is obvious that is convex. Suppose that is convex for some . Next, we show that is also convex for the same . Let and , where . It follows that where . From the above two inequalities, we can get that where . It follows that is closed and convex. This completes Step 2.Step 3. Show that for each .It is obvious that . Suppose that for some . For any , we see that On the other hand, we obtain from (2.6) that Combining (3.13) with (3.14), we arrive at which implies that . This shows that . This completes Step 3.Step 4. Show that the sequence is bounded.In view of , we see that In view of , we arrive at It follows from Lemma 2.8 that This implies that the sequence is bounded. It follows from (2.5) that the sequence is also bounded. This completes Step 4.Step 5. Show that , where , as .Since is bounded and the space is reflexive, we may assume that weakly. 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 .Now, we are in a position to show that . Notice that . On the other hand, we see from that from which it follows that as . In view of Lemma 2.6, we arrive at Note that as in view of It follows from (3.21) that On the other hand, we have It follows from the uniformly asymptotic regularity of and (3.23) that that is, . From the closedness of , we obtain that .Finally, we show that which completes the proof. Indeed, we obtain from that In particular, we have Taking the limit as in (3.27), we obtain that Hence, we obtain from Lemma 2.7 that . This completes the proof.

As applications of Theorem 3.1, we have the following.

Corollary 3.2. Let be a nonempty closed and convex subset of a uniformly convex and smooth Banach space . Let be a closed and asymptotically quasi snonexpansive mapping with a sequence such that as . Assume that is uniformly asymptotically regular on and is nonempty and bounded. Let be a sequence generated in the following manner: where then the sequence converges strongly to .

Proof. Putting in Theorem 3.1, we can conclude the desired conclusion easily.

Next, we give two theorems in the framework of real Hilbert spaces.

Theorem 3.3. Let a nonempty closed and convex subset of a real Hilbert space . Let be a closed and asymptotically strict quasipseudocontraction with a sequence such that as . Assume that is uniformly asymptotically regular on and is nonempty and bounded. Let be a sequence generated by the following manner: where then the sequence converges strongly to .

Theorem 3.4. Let be a nonempty closed and convex subset of a real Hilbert space . Let be a closed and asymptotically quasinonexpansive mapping with a sequence such that as . Assume that is uniformly asymptotically regular on and is nonempty and bounded. Let be a sequence generated by the following manner: where then the sequence converges strongly to .

Acknowledgments

The authors are grateful to the editor and the referees for their valuable comments and suggestions which improve the contents of the paper. The first author was partially supported by Natural Science Foundation of Zhejiang Province (Y6110270).

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