Abstract and Applied Analysis

Volume 2012 (2012), Article ID 536283, 20 pages

doi:10.1155/2012/536283

Research Article

An Iterative Shrinking Projection Method for Solving Fixed Point Problems of Closed and -Quasi-Strict Pseudocontractions along with Generalized Mixed Equilibrium Problems in Banach Spaces

Kasamsuk Ungchittrakool1,2

¹Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

²Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 22 April 2012; Revised 4 July 2012; Accepted 29 July 2012

Academic Editor: Simeon Reich

Copyright © 2012 Kasamsuk Ungchittrakool. 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 provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Moreover, we also find the significant inequality related to the mappings mentioned in the paper and the mappings defined from generalized mixed equilibrium problems on Banach spaces. We propose an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and -quasi-strict pseudo-contractions. Our results hold in reflexive, strictly convex, and smooth Banach spaces with the property (). The results of this paper improve and extend the corresponding results of Zhou and Gao (2010) and many others.

1. Introduction

It is well known that, in an infinite-dimensional Hilbert space, the normal Mann’s iterative algorithm [1] has only weak convergence, in general, even for nonexpansive mappings [2, 3]. Consequently, in order to obtain strong convergence, Nakajo and Takahashi [4] modified the normal Mann’s iteration algorithm, later well known as a hybrid projection iteration method. Since then, the hybrid method has undergone rapid developments. For the details, the readers are referred to papers [5–9] and the references therein.

Let be a smooth Banach space, and let be the dual of . The function is defined by

for all , which was studied by Alber [10], Kamimura and Takahashi [11], and Reich [12], where is the normalized duality mapping from to defined by

where denotes the duality paring. It is well known that if is smooth, then is single valued and if is strictly convex, then is injective (one to one). Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed points of . A point in is said to be an asymptotic fixed point of [12] if contains a sequence , which converges weakly to such that the strong . The set of asymptotic fixed points of   will be denoted by . A mapping from into itself is called nonexpansive if for all and relatively nonexpansive [13–16] if and for all and . The asymptotic behavior of relatively nonexpansive mapping was studied in [13–16].

In 2005, Matsushita and Takahashi [16] proposed the hybrid iteration method with generalized projection for relatively nonexpansive mapping in the framework of uniformly smooth and uniformly convex Banach spaces as follows:

where is the duality mapping on and is the generalized projection from onto a nonempty closed convex subset . Based on the guidelines of Matsushita and Takahashi [16], Plubtieng and Ungchittrakool [17, 18] studied and developed (1.3) to the case of two relatively nonexpansive mappings and finite family of relatively nonexpansive mappings, respectively.

On the other hand, for a real Banach space and the dual , let be a nonempty closed convex subset of . Let a bifunction, a real-valued function, and be a nonlinear mapping. The generalized mixed equilibrium problem is to find such that

The solution set of (1.4) is denoted by , that is,

If , the problem (1.4) reduces to the mixed equilibrium problem for , denoted by , which is to find such that

If , the problem (1.4) reduces to the mixed variational inequality of Browder type, denoted by , which is to find such that

If and , the problem (1.4) reduces to the equilibrium problem for (for short, , denoted by , which is to find such that

If and , the problem (1.4) reduces to the minimization problem for , denoted by , which is to find such that

The above formulation (1.7) was shown in [19] 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 optimization problem, which can also be written in the form of (1.8). However, (1.4) is very general, it covers the problems mentioned above as special cases.

In 2007, S. Takahashi and W. Takahashi [20] and Tada and Takahashi [21, 22] proved weak and strong convergence theorems for finding a common element of the set of solution of an equilibrium problem (1.8) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Takahashi et al. [9] 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. [9], Takahashi and Zembayashi [23] (see also [24]) introduced and proved a hybrid projection algorithm for solving equilibrium problems and fixed point problems of a relatively nonexpansive mapping in the framework of uniformly smooth and uniformly convex Banach space as follows:

where is the generalized projection from onto . Under some appropriate assumptions on , , and , they proved that the sequence converges strongly to .

In 2010, Zhou and Gao [25] introduced the definition of a quasi-strict pseudocontraction related to the function and proved a hybrid projection algorithm for finding a fixed point of a closed and quasi-strict pseudocontraction in more general framework than uniformly smooth and uniformly convex Banach spaces as follows:

where is the generalized projection from onto . They proved that the sequence converges strongly to .

Motivated and inspired by the above research work, in this paper, we provide some new type of mappings associated with pseudocontractions by introducing some actual examples in smooth and strictly convex Banach spaces. Furthermore, by employing the inequality that appeared in Lemma 2.10 together with (1.12) and some facts of Zhou and Gao [25], we create an iterative shrinking projection method for finding a common solution of generalized mixed equilibrium problems and fixed point problems of closed and -quasi-strict pseudocontractions in the framework of reflexive, strictly convex, and smooth Banach spaces with the property . The results of this research improve and extend the corresponding results of Zhou and Gao [25] and many others.

2. Preliminaries

In this paper, we denote by and a real Banach space and the dual space of , respectively. Let be a nonempty closed convex subset of . We denote by the normalized duality mapping from to defined by (1.1). Let be the unit sphere of . Then, a Banach space is said to be strictly convex for all and . It is also said to be uniformly convex if for any two sequences in such that . The Banach space is said to be smooth provided

exists for each . In this case, the norm of is said to be Gâteaux differentiable. The norm of is said to be Fréchet differentiable if for each , the limit (2.1) is attained uniformly for . The norm of is said to be uniformly Fréchet differentiable (and is said to be uniformly smooth) if the limit (2.1) is attained uniformly for .

A Banach space is said to have the property (K) (or Kadec-Klee property) if for any sequence such that weakly and , then . For more information concerning property the reader is referred to [26] and references cited there.

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

We also know the following properties (see [27–29] for details):

(i)if is smooth ( is strictly convex), then is single-valued,(ii)if is strictly convex ( is smooth), then is one-to-one (i.e., for all ),(iii)if is reflexive ( is reflexive), then is surjective,(iv)if is smooth and reflexive, then is single-valued and demicontinuous (i.e., if such that , then ),(v) is uniformly smooth if and only if is uniformly convex,(vi)if is uniformly convex, then

(a)it is strictly convex,(b)it is reflexive,(c)it satisfies the property ,

(vii)if is a Hilbert space, then is the identity operator.

It is also very 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 [10] recently introduced a generalized projection operator in a Banach space that is an analogue of the metric projection in Hilbert spaces.

Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , for any . 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 minimization problem

Existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [10, 11, 27, 29, 30]). In Hilbert spaces, and coincides with the metric projection .

It is obvious from the definition of function that

Remark 2.1. 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 (2.3), we have . This implies . From the definitions of , we have . That is, ; one may consult [27, 29] for the details.

is said to be a -quasi-strict pseudocontraction [25, page 230] if there exists a constant and such that for all and . In particular, is said to be -quasi-nonexpansive if and is said to be -quasi-pseudocontractive if .

Let be a nonempty closed convex subset of a real Banach space with the dual . A mapping is said to be monotone if, for each , the following inequality holds:

is said to be -inverse strongly monotone if there exists a positive real number such that

is said to be -quasi inverse strongly monotone if and there exists a positive real number such that for each , the following inequality holds:

Without loss of generality we can assume that the constant since if is -quasi inverse strongly monotone (or -inverse strongly monotone), then we can find such that and then (or ) for all (i.e., is -quasi inverse strongly monotone).

The following is the example of -quasi-strict pseudocontractions in the framework of a smooth and strictly convex Banach space .

Example 2.2. Let be a smooth and strictly convex Banach space with dual and a normalized duality mapping. Let , and define the mapping by

It follows that the mapping with the set of zero points is given as

Thus for each and if , we obtain

On the other hand, if , we have . Therefore, is -quasi inverse strongly monotone. So, we can define the following set:

where and (note that ). In the same way, let , and define

We claim that . Note that , and let us consider

Furthermore, it is also found that

This means that . So, we have the claim.

Now, we pick arbitrarily and claim that is a -quasi-strict pseudocontraction (i.e., s.t. for all and ).

Since , we have

Note that , and thus . It is not difficult to check that . Therefore, for each , we have

where . So, we obtain the desired result.

Next, we claim that the mapping as mentioned previously is closed. Let such that ; then we have

Moreover, . This shows that is a closed mapping.

Remark 2.3. A relatively nonexpansive mapping is a -quasi-strict pseudocontraction, but the converse may be not true.

Example 2.4. Let be a reflexive, strictly convex, and smooth Banach space. Let be a maximal monotone mapping such that is nonempty. Then, is a closed and -quasi-strict pseudocontraction from onto with constant and .

Example 2.5. Let be the generalized projection from a smooth, strictly convex, and reflexive Banach space onto a nonempty closed convex subset of . Then, is a closed and -quasi-strict pseudocontraction from onto with constant and .

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

(A1) for all ,(A2) is monotone, that is, for all ,(A3)for each ,

(A4)for each , is convex and lower semicontinuous.

Lemma 2.6 (Blum and Oettli [19]). Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , and let be a bifunction of into satisfying ()–(). Let    and . Then, there exists such that

Lemma 2.7 (Takahashi and Zembayashi [24]). Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , and let be a bifunction from to satisfying ()–(). 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

Lemma 2.8 (Takahashi and Zembayashi [24]). Let be a closed convex subset of a smooth, strictly convex and reflexive Banach space , let be a bifunction from to satisfying ()–(), and let . Then, for and ,

Lemma 2.9 (Zhang [31]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a continuous and monotone mapping, a lower semi-continuous and convex function, and a bifunction from to satisfying ()–(). For and , 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).

The following lemmas are crucial for the proofs of the main results in this paper.

Lemma 2.10. Let be a reflexive, strictly convex, and smooth Banach space. Assume that is a nonempty closed convex subset of . Let be a -quasi-strict pseudocontraction, and let be as in Lemma 2.9 such that . Then

for all and .

Proof. Let and . By the -quasi-strict pseudocontractility of and (2.4) we have

It follows from (2.4), Lemma 2.9 (5), and (2.27) that

and then

Lemma 2.11 (Alber and Reich [10]). Let be a nonempty closed convex subset of a smooth Banach space , , and . Then, if and only if

Lemma 2.12 (Alber [30]). Let be a reflexive, strictly convex, and smooth Banach space, let be a nonempty closed convex subset of , and let . Then

3. Main Result

Theorem 3.1. Let be a reflexive, strictly convex, and smooth Banach space such that and have the property . Assume that is a nonempty closed convex subset of . Let be a closed and -quasi-strict pseudocontraction, a bifunction from to satisfying ()–(), a lower semi-continuous and convex function, and 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. The proof is divided into seven steps.

Step  1. Show that is closed and convex.

From step 1 of Zhou and Gao [25], is closed and convex and by Lemma 2.9(4) is closed and convex. So, is closed and convex.

Step  2. Show that is closed and convex for all .

For , is closed and convex. Assume that is closed and convex for some . For , one obtains that

It is not hard to see that the continuity and linearity of and allow to be closed and convex. Then, for all , is closed and convex.

Step  3. Show that for all .

It is obvious that . Suppose that for some . For any , we have . Notice that , and then by Lemma 2.9 we obtain . So, it follows from Lemma 2.10 that

This means that . By mathematical induction, for all . Therefore, .

Step 4. Show that exists.

By and Lemma 2.12, we have

for each and for all . Therefore, the sequence is bounded.

On the other hand, noticing that and , for all . Therefore, is nondecreasing. It follows that the limit of exists.

Step  5. Show that as , where .

Since for any positive integer and by Lemma 2.12, we have

Letting in (3.5), one has as . Without loss of generality, we can assume that weakly as (passing to a subsequence if necessary). It is easy to show that for all . Hence . Noticing that , we have

which implies that as . Hence . By the property of , we have . From Lemma 2.11, we have

Hence

which implies that .

Step  6. Show that .

We prove first that and are bounded. Indeed, taking , we have

Then

where and . Thus

for all . Therefore is bounded. Note that for all . Therefore is also bounded. From , one has

By Step  5, we obtain that . Taking limit on both sides of (3.12), we obtain as . Noting that , , and , it is implied that

and consequently

This implies that and are bounded. Since is reflexive, is also reflexive. So we can assume that

weakly. On the other hand, in view of the reflexivity of , one has , which means that, for , there exists , such that . It follows that

Taking on both sides of the equality above, we have

Therefore and consequently , which implies that . Hence

weakly. Since and has the property , we have

Noting that is demicontinuous, we have

weakly. Since and has the property , we obtain that as . Similarly, it is not difficult to show that as . From and the closeness property of , we have .

Next, we want to show that . Define by for all . It is not hard to verify that satisfies conditions ()–(). It follows from and that

By using and , we obtain for all . For and , let . So, from and , we have

Dividing by , we have

From we have for all , and hence . So, .

Step  7. Show that .

It follows from Steps 5 and 6 that

which implies that . Hence, . Then converges strongly to . This completes the proof.

If is closed -quasi-nonexpansive, then Theorem 3.1 is reduced to the following corollary.

Corollary 3.2. Let be a reflexive, strictly convex, and smooth Banach space such that and have the property . Assume that is a nonempty closed convex subset of . Let be a -closed quasi-nonexpansive mapping, a bifunction from to satisfying ()–(), a lower semi-continuous and convex function, and a continuous and monotone mapping such that . Define a sequence in by the following algorithm:

where for all with . Then converges strongly to .

If is a Hilbert space, then Theorem 3.1 is reduced to the following corollary.

Corollary 3.3. Let be a Hilbert space. Assume that is a nonempty closed convex subset of . Let be a closed and -quasi-strict pseudocontraction, a bifunction from to satisfying (A 1)–(A 4), a lower semi-continuous and convex function, and a continuous and monotone mapping such that . Define a sequence in by the following algorithm:

where and for all with . Then converges strongly to .

If and , then we have the following corollary.

Corollary 3.4. Let be a reflexive, strictly convex, and smooth Banach space such that and have the property . Assume that is a nonempty closed convex subset of . Let be a closed and -quasi-strict pseudocontraction and a bifunction from to satisfying (A 1)–(A 4) such that . Define a sequence in by the following algorithm:

where and for all with . Then converges strongly to .

Corollary 3.5 ([25, Theorem 3.1]). Let be a reflexive, strictly convex and smooth Banach space such that and have the property . Assume that is a nonempty closed convex subset of . Let be a -closed quasi-strict pseudocontraction. Define a sequence in by the following algorithm:

where . Then converges strongly to .

Proof. Put , , , and for all in Theorem 3.1. Then, for all . So, for all (note that ). Since and then for all , we have for all . Thus and for all . For this reason, (1.12) is a special case of (3.1). Applying Theorem 3.1, we have the desired result.

Remark 3.6. It is well known that every uniformly convex and uniformly smooth Banach space satisfies all assumptions of Banach space in Theorem 3.1. On the other hand, in general, Musielak-Orlicz space [26] need not be uniformly convex or uniformly smooth, however, any strictly convex, reflexive and smooth Musielak-Orlicz space [26] satisfies all Banach-space assumptions of Theorem 3.1. It can be written as the following diagram:

For this reason, Theorem 3.1 can be viewed as a more general one and can be applied widely in both the fixed point problems and the equilibrium problems.

Acknowledgments

The author is supported by Thailand Research Fund under Grant no. MRG5380249 and the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand. The author would like to thank Simeon Reich and an anonymous referee for their valuable comments and suggestions, which were helpful in improving the paper.

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