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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 589282, 11 pages
http://dx.doi.org/10.1155/2013/589282
Research Article

An Iterative Shrinking Metric -Projection Method for Finding a Common Fixed Point of a Closed and Quasi-Strict-Pseudocontraction and a Countable Family of Firmly Nonexpansive Mappings and Applications in Hilbert Spaces

1Department of Mathematics, Faculty of Science, Naresuan University, 99 Moo 9, Phitsanulok-Nakhon Sawan Road, Tha Pho, Mueang, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 29 August 2013; Accepted 18 October 2013

Academic Editor: Shawn X. Wang

Copyright © 2013 Kasamsuk Ungchittrakool and Duangkamon Kumtaeng. 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

We create some new ideas of mappings called quasi-strict -pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric -projection, we propose an iterative shrinking metric -projection method for finding a common fixed point of a quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to find a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results.

1. Introduction

It is well known that the metric projection operators in Hilbert spaces and Banach spaces play an important role in various fields of mathematics such as functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problem (see, e.g., [1, 2]). In 1994, Alber [3] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by [3] and proved some properties of the generalized -projection operator. Fan et al. [5] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.

Let be a real Hilbert space; a mapping with domain and range in is called firmly nonexpansive if nonexpansive if Throughout this paper, stands for an identity mapping. The mapping is said to be a strict pseudocontraction if there exists a constant such that In this case, may be called a -strict pseudocontraction. We use to denote the set of fixed points of (i.e. . is said to be a quasi-strict pseudocontraction if the set of fixed point is nonempty and if there exists a constant such that

Construction of fixed points of nonexpansive mappings via Mann’s algorithm [6] has extensively been investigated in the literature; see, for example, [610] and references therein. However, we note that Mann’s iterations have only weak convergence even in a Hilbert space (see, e.g., [11]). Nakajo and Takahashi [12] modified the Mann iteration method so that strong convergence is guaranteed, later well known as a hybrid projection iteration method. Since then, the hybrid method has received rapid developments. For the details, the readers are referred to papers [1326] and the references therein.

On the other hand, for a real Banach space and the dual , let be a nonempty closed convex subset of . Let be a bifunction, let be a real-valued function, and let be a nonlinear mapping. The generalized mixed equilibrium problem is to find such that The solution set of (5) is denoted by ; that is, If , the problem (5) reduces to the mixed equilibrium problem for , denoted by , which is to find that such that If , the problem (5) reduces to the mixed variational inequality of Browder type, denoted by , which is to find that such that If and , the problem (5) reduces to the equilibrium problem for (for short, ), denoted by , which is to find that such that If and , the problem (5) reduces to the minimization problem for , denoted by , which is to find that such that

The previous formulation, (8), was shown in [27] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem, and the optimization problem, which can also be written in the form of (9). However, (5) is very general; it covers the problems mentioned above as special cases.

In 2007, S. T. Takahashi and W. T. Takahashi [28] and Tada and Takahashi [29, 30] proved weak and strong convergence theorems for finding a common element of the set of solutions of the equilibrium problem (9) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Takahashi et al. [22] studied a strong convergence theorem by the hybrid method for a family of nonexpansive mappings in Hilbert spaces as follows: , , and , and let where , for all , and is a sequence of nonexpansive mappings of into itself such that . They proved that if satisfies some appropriate conditions, then converges strongly to .

Motivated by Takahashi et al. [22], Takahashi and Zembayashi [31] (see also [32]) introduced and proved a hybrid projection algorithm for solving equilibrium problems and fixed point problems of a relatively nonexpansive mapping within the framework of a uniformly smooth and uniformly convex Banach space.

In 2011, Saewan and Kumam [33] introduced a new hybrid projection method based on the modified Mann iterative scheme by the generalized -projection operator for a countable family of relatively quasi-nonexpansive mappings and the solutions of the system of generalized mixed equilibrium problems. Later, they [34] also studied the new hybrid Ishikawa iteration process by the generalized -projection operator for finding a common element of the fixed point set for two countable families of weak relatively nonexpansive mappings and the set of solutions of the system of generalized Ky Fan inequalities in a uniformly convex and uniformly smooth Banach space.

Recently, Li et al. [35] have studied the following hybrid iterative scheme for a relatively nonexpansive mapping by using the generalized -projection operator in Banach spaces: Under some appropriate assumptions, they obtained strong convergence theorems in Banach spaces.

Motivated and inspired by the work mentioned above, in this paper, we are interested to study our theorems within the framework of a real Hibert space, and we create some new ideas of mappings called quasi-strict -pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric -projection, we propose an iterative shrinking metric -projection method for find a common fixed point of an quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to finding a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results.

2. Preliminaries

In this section, some definitions are provided, and some relevant lemmas which are useful to prove in the following section are collected. Most of them are known, and others are not hard to find or understand their proofs. Throughout this paper, we will use the notation for weak convergence and for strong convergence.

Lemma 1 (see Takahashi [36]). Let be a sequence of real numbers. Then, if and only if, for any subsequence of , there exists a subsequence of such that .

Definition 2 (see [3740]). Let be a nonempty, closed, and convex subset of a Banach space , and let be a sequence of mappings of into itself such that . Then, is said to satisfy the -condition if, for each bounded sequence , implies that , where is the set of all weak cluster points of (i.e., ).

Let be a real Hilbert space, and let be nonempty, closed, and convex subset of . Let be a functional defined as follows (see Li et al. [35] (see also [4])): where , , is positive number, and is proper, convex, and lower semicontinuous. From the definitions of and , it is easy to see the following properties: (i) is convex and continuous with respect to when is fixed; (ii) is convex and lower semicontinuous with respect to when is fixed.

Definition 3 (see Li et al. [35] (see also [4])). Let be a real Hilbert space, and let be nonempty, closed, and convex subset of . We say that is a metric -projection operator if

Lemma 4 (see Li et al. [35, Lemma 3.1(ii)]). Let be a real Hilbert space, and let . Then, for every , if and only if

Lemma 5 (see Li et al. [35, Lemma 3.2]). Let be a real Hilbert space, let be a nonempty, closed, and convex subset of , and let , . Then,

Lemma 6 (see Deimling [41]). Let be a real Hilbert space, and be a lower semicontinuous convex functional. Then, there exist and such that

Due to the properties of , we have the motivation and ideas to create a new type of mappings which is general and covers a quasi-strict pseud-contraction as follows.

Definition 7. Let be a real Hilbert space, and a mapping with domain and range in is called quasi-strict -pseudocontraction if the set of fixed points is nonempty and if there exists a constant such that, for each ,

It is obvious from the previous definition that (19) is equivalent to

Definition 8. A mapping is said to be closed if, for any sequence with and , .

Example 9. Let be a mapping defined by , for all . Then, it is easy to see that . Moreover, it is found that for all , where , , and . Furthermore, if such that , then we have and . This means that is closed and quasi-strict -pseudocontraction.

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:(A1), for all ; (A2) is monotone; that is, ;(A3) for each , ;(A4) for each , is convex and lower semicontinuous.

Lemma 10 (see [27]). Let be a nonempty closed convex subset of , and let be a bifunction of into satisfying (A1)–(A4). Let 0 and . Then, there exists such that

Lemma 11 (see [32]). Let be a closed convex subset of and let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows: for all . Then, the following hold: (i) is single-valued; (ii) is firmly nonexpansive-type mapping; that is, for any , ;(iii); (iv) is closed and convex;(v), for all .

Lemma 12 (see [42]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Hilbert space . Let be a continuous and monotone mapping, let be a lower semicontinuous and convex function, and let be a bifunction from to satisfying (A1)–(A4). For and , then, there exists such that Define a mapping as follows: for all . Then, the following conclusions hold: (1) is single-valued; (2) is firmly nonexpansive type; that is, for all , ;(3); (4) is closed and convex;(5), for all .

Lemma 13. Let be a real Hilbert space, and let be a mapping. Then, the following are equivalent:(i) is firmly nonexpansive i.e.,, for all ;(ii), for all .

Proof. For each , we notice that The proof is complete.

The following lemma is important since it provides the significant inequality related to quasi-strict -pseudocontractions and firmly nonexpansive mappings within the framework of Hilbert spaces.

Lemma 14. Let be a nonempty, closed, convex subset of real Hilbert spaces . Let be a quasi-strict -pseudocontraction, and let be a firmly nonexpansive mapping such that . Then, for all and .

Proof. Let and . By the quasi-strict -pseudocontractility of , we have that It follows from Lemma 13 and (28) that and, then, This completes the proof.

3. Main Result

In this section, some available properties of a quasi-strict -pseudocontraction are used to prove that the set of fixed points is closed and convex. An iterative shrinking metric -projection method is provided in order to find a common fixed point of a quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings.

Lemma 15. Let be a nonempty, closed, convex subset of a real Hilbert space , and let be a quasi-strict -pseudocontraction. Then, the fixed point set of is closed and convex.

Proof. Let be a sequence in such that as . It follows from (28) that Taking on both sides of (31), so we have that This means that .
We next show that is convex. For arbitrary and , we let . By the definition of , we have that By (28), it is easy to see that (33) are equivalent to respectively. Multiplying by and on both sides of (34) and (35), respectively, and adding the two inequalities, we have that Hence, . This complete the proof.

Theorem 16. Let be a real Hilbert space, a nonempty, closed, be convex subset of , let be a closed and quasi-strict -pseudocontraction from into itself, and let be a countable family of firmly nonexpansive mappings from into itself which satisfies the -condition such that . Define a sequence in by the following algorithm: Then, the sequence converges strongly to .

Proof. The proof is divided into six steps.
Step 1. Show that is closed and convex, for all .
For , is closed and convex. Assume that is closed and convex for some . For , we have that It is not hard to see that the continuity and linearity of and together with the lower semicontinuity and convexity of allow to be closed and convex. Then, for all , is closed and convex.
Step 2. Show that .
It is obvious that . Suppose that for some . For any , we have , and by Lemma 14 we obtain that This means that . By mathematical induction, for all . Therefore, .
Step 3. Show that is bounded and that the exists.
Since is a convex and lower semicontinuous mapping, applying Lemma 6, we see that there exist and such that It follows that Since , it follows from (41) that for each . This implies that and are bounded. By the fact that and Lemma 5, we obtain that Since , is nondecreasing. Therefore, the limit of exists.
Step 4. Show that as , where .
Let . From the boundedness of , there exists such that Writing , it is easy to see that , where . Note that On the other hand, since , so , and, then, by weakly lower semicontinuity of and , we obtain that Combining (46) and (47), we obtain that It follows from (45) and (48), that and, then, Next, we consider that Combining (46) and (51), we obtain that and, then, Note that Then, we have that Combining (47) and (55), we obtain that and, then, By Lemma 1, this implies that On the other hand, we note that It follows from (44) and (53) that Therefore, as . This implies by Lemma 1 that Thus, It is easy to show that , for all . Hence, . Since , so, by Lemma 4, we have that Letting , so we obtain that which implies that .
Step 5. Show that .
Firstly, we prove that and are bounded. Indeed, taking and then by (28), we have that By a simple calculation, we get that where and . So we have that for all . Therefore, is bounded. Notice that, for each , for all . Therefore, is also bounded. Moreover, we note that Thus, by the fact that and by (58), we obtain that This means that For this reason, we have that Next, we have that That is, as . It follows from the closed mapping of , that ; thus, .
On the other hand, let us consider that It follows from the -condition of and (62), that . Therefore, .
Step 6. Show that .
Notice by Step 2 that ; so we have that , and then by Step 5 it yields that This shows that . It follows from the uniqueness . Then, converges strongly to . This completes the proof.

4. Deduced Theorems and Applications

In this section, some applications of the main theorem are provided in order to find some common solutions of problems in a Hilbert space.

If , then , for all , and , for all ; then, by Theorem 16, we obtain the following corollary.

Corollary 17. Let , , be the same as in Theorem 16, and be a closed and quasi-strict -pseudocontraction from into itself such that . Define a sequence in by the following algorithm: Then, the sequence converges strongly to .

If is a constant function, saying that , then , and it is not hard to see that coincides with a quasi-strict pseudocontraction. Thus, by Theorem 16, we obtain the following corollary.

Corollary 18. Let , , be the same as in Theorem 16, and let be a closed and quasi-strict pseudocontraction from into itself such that . Define a sequence in by the following algorithm: Then, the sequence converges strongly to .

Let be a firmly nonexpansive mapping and for all . Then, it is not hard to verify that a family satisfies -condition. Therefore, if , then Corollary 18 reduces to the following corollary.

Corollary 19. Let , , be the same as in Corollary 18, and let be a firmly nonexpansive mapping from into itself such that . Define a sequence in by the following algorithm: Then, the sequence converges strongly to .

Finally, we provide some applications of the main theorem to find a common solution of fixed point problems of a closed and quasi-strict -pseudocontraction and generalized mixed equilibrium problems via an iterative shrinking metric -projection method within the framework of Hilbert spaces.

Theorem 20. Let , , be the same as in Theorem 16, let be a bifunction from to satisfying (A1)–(A4), let be a lower semicontinuous and convex function, and let be a continuous and monotone mapping such that . Define a sequence in by the following algorithm: where and , for all , with . Then, converges strongly to .

Proof. By Lemma 12 it is not hard to see that . From the definition of , it is easy to see that , and by (62) we have that . Next, we will show that a countable family satisfies the -condition. It is sufficient to show that . It follows from (61) and (71) that as . Define by , for all . It is not hard to verify that satisfies conditions (A1)–(A4). By (A2), we have that By using (A4) and , we obtain, that , for all . For and , let . So, from (A1) and (A4), we have that Dividing by , we have that From (A3), we have that , for all , and, hence, , so . Therefore, satisfies -condition. Applying Theorem 16, we conclude that .

Acknowledgment

The authors would like to thank the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand, for supporting this paper.

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