Journal of Applied Mathematics

Volume 2012 (2012), Article ID 750732, 19 pages

doi:10.1155/2012/750732

Research Article

A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi--Asymptotically Nonexpansive Mapping in a Banach Space

Preedaporn Kanjanasamranwong,¹ Poom Kumam,² and Siwaporn Saewan¹

¹Department of Mathematics and Statistics, Faculty of Science, Thaksin University (TSU), Pa Phayom, Phatthalung 93110, Thailand

²Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 17 February 2012; Accepted 28 April 2012

Academic Editor: Yeol Je Cho

Copyright © 2012 Preedaporn Kanjanasamranwong 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 introduce the modified Halpern-type iterative method by the generalized f-projection operator for finding a common solution of fixed-point problem of a totally quasi--asymptotically nonexpansive mapping and a system of equilibrium problems in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Consequently, we prove the strong convergence for a common solution of above two sets. Our result presented in this paper generalize and improve the result of Chang et al., (2012), and some others.

1. Introduction

In 1953, Mann [1] introduced the following iteration process which is now known as Mann's iteration:

where is nonexpansive, the initial guess element is arbitrary, and is a sequence in . Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak conviergence (see [2, 3]).

Later, in 1967, Halpern [4] considered the following algorithm:

where is nonexpansive. He proved the strong convergence theorem of to a fixed point of under some control condition . Many authors improved and studied the result of Halpern [4] such as Qin et al. [5], Wang et al. [6], and reference therein.

In 2008-2009, Takahashi and Zembayashi [7, 8] studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of the Banach spaces.

On the other hand, Li et al. [9] introduced the following hybrid iterative scheme for approximation fixed points of relatively nonexpansive mapping using the generalized -projection operator in a uniformly smooth real Banach space which is also uniformly convex. They obtained strong convergence theorem for finding an element in the fixed point set of .

Recently, Ofoedu and Shehu [10] extended algorithm of Li et al. [9] to prove a strong convergence theorem for a common solution of a system of equilibrium problem and the set of common fixed points of a pair of relatively quasi-nonexpansive mappings in the Banach spaces by using generalized -projection operator. Chang et al. [11] extended and improved Qin and Su [12] to obtain a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space.

Very recently, Chang et al. [13] extended the results of Qin et al. [5] and Wang et al. [6] to consider a modification to the Halpern-type iteration algorithm for a total quasi--asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces.

The purpose of this paper is to be motivated and inspired by the works mentioned above, we introduce a modified Halpern-type iterative method by using the new hybrid projection algorithm of the generalized -projection operator for solving the common solution of fixed point for totally quasi--asymptoically nonexpansive mappings and the system of equilibrium problems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in this paper improve and extend the corresponding ones announced by many others.

2. Preliminaries and Definitions

Let be a real Banach space with dual , and let be a nonempty closed and convex subset of . Let be a bifunction, where is an arbitrary index set. The system of equilibrium problems is to find such that

If is a singleton, then problem (2.1) reduces to the equilibrium problem, which is to find such that

A mapping from into itself is said to be nonexpansive if

is said to be asymptotically nonexpansive if there exists a sequence with as such that

is said to be total asymptotically nonexpansive if there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that

A point is a fixed point of provided . Denote by the fixed point set of ; that is, . A point in is called an asymptotic fixed point of if contains a sequence which converges weakly to such that . The asymptotic fixed point set of is denoted by .

The normalized duality mapping is defined by . If is a Hilbert space, then , where is the identity mapping. Consider the functional defined by

where is the normalized duality mapping and denote the duality pairing of and .

If is a Hilbert space, then . It is obvious from the definition of that

A mapping from into itself is said to be -nonexpansive [14, 15] if

is said to be quasi--nonexpansive [14, 15] if and

is said to be asymptotically -nonexpansive [15] if there exists a sequence with as such that

is said to be quasi--asymptotically nonexpansive [15] if and there exists a sequence with as such that

is said to be totally quasi--asymptotically nonexpansive, if and there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that

A mapping from into itself is said to be closed if for any sequence such that and , then .

Alber [16] introduced the generalized projection is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution of the minimization problem:

The existence and uniqueness of the operator follows from the properties of the functional and the strict monotonicity of the mapping (see, e.g., [16–20]). If is a Hilbert space, then and becomes the metric projection . If is a nonempty, closed, and convex subset of a Hilbert space , then is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently, it is not available in more general Banach spaces. Later, Wu and Huang [21] introduced a new generalized -projection operator in the Banach space. They extended the definition of the generalized projection operators and proved some properties of the generalized -projection operator. Next, we recall the concept of the generalized -projection operator. Let be a functional defined by

where ,   is positive number, and is proper, convex, and lower semicontinuous. From the definition of , Wu and Huang [21] proved the following properties:

(1) is convex and continuous with respect to when is fixed; (2) is convex and lower semicontinuous with respect to when is fixed.

Definition 2.1. Let be a real Banach space with its dual . Let be a nonempty, closed, and convex subset of . We say that is a generalized -projection operator if

A Banach space with norm is called strictly convex if for all with and . Let be the unit sphere of . A Banach space is called smooth if the limit exists for each . It is also called uniformly smooth if the limit exists uniformly for all . The modulus of smoothness of is the function defined by . The modulus of convexity of (see [22]) is the function defined by . In this paper we denote the strong convergence and weak convergence of a sequence by and , respectively.

Remark 2.2. The basic properties of , , , and (see [18]) are as follows.

(i)If is an arbitrary Banach space, then is monotone and bounded. (ii)If is a strictly convex, then is strictly monotone.(iii)If is a smooth, then is single valued and semicontinuous. (iv)If is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . (v)If is reflexive smooth and strictly convex, then the normalized duality mapping is single valued, one-to-one, and onto. (vi)If is a reflexive strictly convex and smooth Banach space and is the duality mapping from into , then is also single valued, bijective, and is also the duality mapping from into , and thus and . (vii)If is uniformly smooth, then is smooth and reflexive.(viii) is uniformly smooth if and only if is uniformly convex.(ix)If is a reflexive and strictly convex Banach space, then is norm-weak*-continuous.

Remark 2.3. If is a reflexive, strictly convex, and smooth Banach space, then , if and only if . It is sufficient to show that if then . From (2.6), we have . This implies that . From the definition of , one has . Therefore, we have (see [18, 20, 23] for more details).

Recall that a Banach space has the Kadec-Klee property [18, 20, 24], if for any sequence and with and , then as . It is well known that if is a uniformly convex Banach space, then has the Kadec-Klee property.

We also need the following lemmas for the proof of our main results.

Lemma 2.4 (see Change et al. [25]). Let be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Let be a closed and total quasi--asymptotically nonexpansive mapping with nonnegative real sequence and with , as and a strictly increasing continuous function with . If , then the fixed point set is a closed convex subset of .

Lemma 2.5 (see Wu and Hung [21]). Let be a real reflexive Banach space with its dual and a nonempty, closed, and convex subset of . The following statement hold:

(1) is a nonempty, closed and convex subset of for all ; (2)if is smooth, then for all , if and only if

(3)if is strictly convex and is positive homogeneous (i.e., for all such that where ), then is single-valued mapping.

Lemma 2.6 (see Fan et al. [26]). Let be a real reflexive Banach space with its dual and be a nonempty, closed and convex subset of . If is strictly convex, then is single valued.

Recall that is single-valued mapping when is a smooth Banach space. There exists a unique element such that where . This substitution in (2.14) gives

Now we consider the second generalized projection operator in Banach space (see [9]).

Definition 2.7. Let be a real smooth Banach space, and let be a nonempty, closed, and convex subset of . We say that is generalized -projection operator if

Lemma 2.8 (see Deimling [27]). Let be a Banach space, and let be a lower semicontinuous convex function. Then there exist and such that

Lemma 2.9 (see Li et al. [9]). Let be a reflexive smooth Banach space, and let be a nonempty, closed, and convex subset of . The following statements hold:

(1) is nonempty, closed and convex subset of for all ; (2)for all , if and only if

(3)if is strictly convex, then is single-valued mapping.

Lemma 2.10 (see Li et al. [9]). Let be a real reflexive smooth Banach space, let be a nonempty, closed, and convex subset of , , and let . Then

Remark 2.11. Let be a uniformly convex and uniformly smooth Banach space and for all , then Lemma 2.10 reduces to the property of the generalized projection operator considered by Alber [16].

If for all and , then the definition of totally quasi--asymptotically nonexpansive is equivalent to if , and there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that

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.

For example, let be a continuous and monotone operator of into and define

Then, satisfies (A1)–(A4). The following result is in Blum and Oettli [28].

Lemma 2.12 (see Blum and Oettli [28]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let and . Then, there exists such that

Lemma 2.13 (see Takahashi and Zembayashi [8]). Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , and let be a bifunction from to satisfying conditions (A1)–(A4). For all and , define a mapping as follows:

Then the following hold:

(1) is single-valued; (2) is a firmly nonexpansive-type mapping [29]; that is, for all ,

(3); (4) is closed and convex.

Lemma 2.14 (see Takahashi and Zembayashi [8]). Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let . Then, for and ,

3. Main Result

Theorem 3.1. Let be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. For each , let be a bifunction from to which satisfies conditions (A1)–(A4). Let be a closed totally quasi--asymptotically nonexpansive mappings with nonnegative real sequences , with , as , and a strictly increasing continuous function with . Let be a convex and lower semicontinuous function with such that for all and . Assume that . For an initial point and , one define the sequence by

where is a sequence in , and for some . If , then converges strongly to .

Proof. We split the proof into four steps.

Step 1. First, we show that is closed and convex for all .

Clearly is closed and convex. Suppose that is closed and convex for all . For any , we know that is equivalent to

So, is closed and convex. Hence by induction is closed and convex for all .

Step 2. We will show that the sequence is well defined.

We will show by induction that for all . It is obvious that =. Suppose that for some . Let , put for all , , we have that

From (3.3) and which is a totally quasi- asymptotically nonexpansive mappings, it follows that

This shows that which implies that , and hence, for all . and the sequence is well defined. From , we see that

Since for each , we arrive at

Hence, the sequence is well defined.

Step 3. We will show that .

Let is convex and lower semicontinuous function, follows from Lemma 2.8, there exist and such that

Since , it follows that

For and , we have

This shows that is bounded and so is . From the fact that and , it follows from Lemma 2.10 that

That is, is nondecreasing. Hence, we obtain that exists. Taking , we obtain

Since is reflexive, is bounded, and is closed and convex for all . Without loss of generality, we can assume that . From the fact that , we get that

Since is convex and lower semicontinuous, we have

By (3.12) and (3.13), we get

That is, ; this implies that ; by virtue of the Kadec-Klee property of , we obtain that

We also have

From (3.15), we get that

We show that .

Since and the definition of , we have

is equivalent to

From (3.11), (3.15), and (3.17), it follows that

From (2.7), we have

Since , we have

It follow that

That is, is bounded in and is reflexive; we assume that . In view of , there exists such that . It follows that

Taking on both sides of the equality above and is the weak lower semicontinuous, it yields that

That is, , which implies that . It follows that . From (3.23) and the Kadec-Klee property of we have as . Note that is norm-weak-continuous; that is, . From (3.22) and the Kadec-Klee property of , we have

For , by nonexpansiveness, we observe that

By Lemma 2.14, we have for

Since as , we get as , for . From (2.7), it follow that

Since , we also have

Since is bounded and is reflexive, without loss of generality we assume that . We know that is closed and convex for each it is obvious that . Again since

taking on the both sides of equality above, we have

That is, for all ; it follow that

from (3.30), (3.33), and the Kadec-Klee property, it follows that

By using triangle inequality, we have

Since as , we have

Again by using triangle inequality, we have

From (3.36), we also have

Since is uniformly norm-to-norm continuous, we obtain

From , we have as for all , and

By (A2), that

and as , we get , for all . For , define , then which imply that . From (A1), we obtain that

We have that . From (A3), we have , for all and . That is, , for all. This imply that .

(b) We show that .

Since and the definition of , we have

is equivalent to

Following (3.11), (3.15), and (3.17), we get that

From (2.7), we also have

It follows that

This implies that is bounded in . Since is reflexive and is also reflexive, we can assume that . In view of the reflexive of , we see that . There exists such that . It follows that

Taking on both sides of the equality above and in view of the weak lower semicontinuity of norm , it yields that

That is , which implies that . It follows that .From (3.47) and the Kadec-Klee property of we have as . Since is norm-weak-continuous, as . From (3.46) and the Kadec-Klee property of , we have

Since is bounded, then a mapping is also bounded. From the condition , we have that

From (3.47), we get

Since is norm-weak*-continuous,

On the other hand, we observe that

In view of (3.52), we obtain . Since has the Kadee-Klee property, we get

From , we get ; that is, . In view of closeness of , we have . This implies that . From (a) and (b), it follows that .

Step 4. We will show that .

Since is closed and convex set from Lemma 2.9, we have which is single valued, denoted by . By definition and , we also have

By the definition of and , we know that, for each given is convex and lower semicontinuous with respect to . So

From the definition of and since , we conclude that and as . The proof is completed.

Setting and in Theorem 3.1, then we have the following corollary.

Corollary 3.2. Let be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. For each , let be a bifunction from to which satisfies conditions (A1)–(A4). Let be a closed and quasi--asymptotically nonexpansive mappings, and let be a convex and lower semicontinuous function with such that for all and . Assume that . For an initial point and , we define the sequence by

where is a sequence in , , and for some . If , then converges strongly to .

Let be a real Banach space, and let be a nonempty closed convex subset of . Given a mapping , let for all . Then if and only if for all ; that is, is a solution of the classical variational inequality problem. The set of this solution is denoted by . For each and , we define the mapping by

Hence, we obtain the following corollary.

Corollary 3.3. Let be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. For each , let be a continuous monotone mapping of into . Let be a closed totally quasi--asymptotically nonexpansive mappings with nonnegative real sequences , with , as and a strictly increasing continuous function with , and let be a convex and lower semicontinuous function with such that for all and . Assume that . For an initial point and , one defines the sequence by

where , is a sequence in , and for some . If , then converges strongly to .

If for all , we have and . From Theorem 3.1, we obtain the following corollary.

Corollary 3.4. Let be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. For each , let be a bifunction from to which satisfies conditions (A1)–(A4). Let be a closed totally quasi--asymptotically nonexpansive mappings with nonnegative real sequences , with , as and a strictly increasing continuous function with . Assume that . For an initial point and , we define the sequence by

where is a sequence in , , and for some . If , then converges strongly to .

Remark 3.5. Our main result extends and improves the result of Chang et al. [13] in the following sense.

(i)From the algorithm we used new method replace by the generalized -projection method which is more general than generalized projection. (ii)For the problem, we extend the result to a common problem of fixed point problems and equilibrium problems.

Acknowledgments

The authors would like to thank The National Research Council of Thailand (NRCT) and Faculty of Science, King Mongkut's University of Technology Thonburi (Grant NRCT-2555). Furthermore, the authors would like to express their thanks to the referees for their helpful comments.

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