Journal of Applied Mathematics

Volume 2012 (2012), Article ID 516897, 19 pages

doi:10.1155/2012/516897

Research Article

Strong Convergence of a Hybrid Iteration Scheme for Equilibrium Problems, Variational Inequality Problems and Common Fixed Point Problems, of Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

Jing Zhao1,2

¹College of Science, Civil Aviation University of China, Tianjin 300300, China

²Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Received 3 April 2012; Accepted 5 June 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Jing Zhao. 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 introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.

1. Introduction and Preliminary

Let be a 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

If is -inverse strongly monotone, then it is Lipschitz continuous with constant , that is, , for all , and hence uniformly continuous.

Let be a nonempty closed convex subset of and a bifunction, where is the set of real numbers. The equilibrium problem for is to find such that

for all . The set of solutions of (1.3) is denoted by . Given a mapping , let for all . Then if and only if for all ; that is, is a solution of the variational inequality. Numerous problems in physics, optimization, engineering, and economics reduce to find a solution of (1.3). Some methods have been proposed to solve the equilibrium problem; see, for example, Blum and Oettli [1] and Moudafi [2]. For solving the equilibrium problem, let us assume that satisfies the following conditions:

(A1) for all ;(A2) is monotone, that is, for all ;(A3) for each , ;(A4) for each , the function is convex and lower semicontinuous.

Let be a Banach space with the dual . We denote by the normalized duality mapping from to defined by

where denotes the generalized duality pairing. Let dim , and the modulus of smoothness of is the function defined by

The space is said to be smooth if , for all and is called uniformly smooth if and only if . A Banach space is said to be strictly convex if for with and . The modulus of convexity of is the function defined by

is called uniformly convex if and only if for every . Let , then is said to be -uniformly convex if there exists a constant such that for all . Observe that every -uniformly convex space is uniformly convex. We know that if is uniformly smooth, strictly convex, and reflexive, then the normalized duality mapping is single-valued, one-to-one, onto and uniformly norm-to-norm continuous on each bounded subset of . Moreover, if is a reflexive and strictly convex Banach space with a strictly convex dual, then is single-valued, one-to-one, surjective, and it is the duality mapping from into and thus and (see, [3]). It is also well known that is uniformly smooth if and only if is uniformly convex.

Let be a nonempty closed convex subset of a Banach space and a mapping. A point is said to be a fixed point of provided . A point is said to be an asymptotic fixed point of provided contains a sequence which converges weakly to such that . In this paper, we use and to denote the fixed point set and the asymptotic fixed point set of and use to denote the strong convergence and weak convergence, respectively. Recall that a mapping is called nonexpansive if

A mapping is called asymptotically nonexpansive if there exists a sequence of real numbers with as such that

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [4] in 1972. They proved that if is a nonempty bounded closed convex subset of a uniformly convex Banach space , then every asymptotically nonexpansive self-mapping of has a fixed point. Further, the set is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [4–6] and the references therein).

It is 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 [7] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that is a smooth Banach space. Consider the functional defined by

Following Alber [7], 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:

It follows from the definition of the function that

If is a Hilbert space, then and is the metric projection of onto .

Remark 1.1 (see [8, 9]). If is a reflexive, strictly convex, and smooth Banach space, then for , if and only if .

Let be a nonempty, closed, and convex subset of a smooth Banach and a mapping from into itself. The mapping is said to be relatively nonexpansive if . The mapping is said to be -nonexpansive if . The mapping is said to be quasi--nonexpansive if . The mapping is said to be relatively asymptotically nonexpansive if there exists some real sequence with and as such that . The mapping is said to be -asymptotically nonexpansive if there exists some real sequence with and as such that . The mapping is said to be quasi--asymptotically nonexpansive if there exists some real sequence with and as such that . The mapping is said to be asymptotically regular on if, for any bounded subset of , . The mapping is said to be closed on if, for any sequence such that and , .

We remark that a -asymptotically nonexpansive mapping with a nonempty fixed point set is a quasi--asymptotically nonexpansive mapping, but the converse may be not true. The class of quasi--nonexpansive mappings and quasi--asymptotically nonexpansive mappings is more general than the class of relatively nonexpansive mappings and relatively asymptotically nonexpansive mappings, respectively.

Recently, many authors studied the problem of finding a common element of the set of fixed points of nonexpansive or relatively nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of variational inequalities in the frame work of Hilbert spaces and Banach spaces, respectively; see, for instance, [10–21] and the references therein.

In 2009, Takahashi and Zembayashi [22] introduced the following iterative process:

where is a bifunction satisfying (A1)–(A4), is the normalized duality mapping on and is a relatively nonexpansive mapping. They proved the sequence defined by (1.12) converges strongly to a common point of the set of solutions of the equilibrium problem (1.3) and the set of fixed points of provided the control sequences and satisfy appropriate conditions in Banach spaces.

Qin et al. [8] introduced the following iterative scheme on the equilibrium problem (1.3) and a family of quasi--nonexpansive mapping:

Strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

Very recently, for finding a common element of Zegeye [23] proposed the following iterative algorithm:

where is closed quasi--nonexpansive mapping , is a bifunction satisfying (A1)–(A4) and is a -inverse strongly monotone mapping of into . Strong convergence theorems for iterative scheme (1.14) are obtained under some conditions on parameters in 2-uniformly convex and uniformly smooth real Banach space .

In this paper, inspired and motivated by the works mentioned above, we introduce an iterative process for finding a common element of the set of common fixed points of a finite family of closed quasi--asymptotically nonexpansive mappings, the solution set of equilibrium problem, and the solution set of the variational inequality problem for a -inverse strongly monotone mapping in Banach spaces. The results presented in this paper improve and generalize the corresponding results announced by many others.

In order to the main results of this paper, we need the following lemmas.

Lemma 1.2 (see [24]). Let be a 2-uniformly convex and smooth Banach space. Then, for all , one has

where is the normalized duality mapping of and is the 2-uniformly convex constant of .

Lemma 1.3 (see [7, 25]). Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space . Then

Lemma 1.4 (see [25]). Let be a smooth and uniformly convex Banach space and let and be sequences in such that either or is bounded. If , then .

Lemma 1.5 (see [7]). Let be a nonempty closed convex subset of a smooth Banach space , let and let . Then

We denote by the normal cone for at a point , that is, . We shall use the following lemma.

Lemma 1.6 (see [26]). Let be a nonempty closed convex subset of a Banach space and let be a monotone and hemicontinuous operator of into with . Let be an operator defined as follows:

Then is maximal monotone and .

We make use of the function defined by

for all and (see [7]). That is, for all and .

Lemma 1.7 (see [7]). Let be a reflexive, strictly convex, and smooth Banach space with as its dual. Then,

for all and .

Lemma 1.8 (see [1]). 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 1.9 (see [22]). Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space . Let be a bifunction from to satisfying (A1)–(A4). For and , define a mapping as follows:

for all . Then, the following hold:

(1) is single-valued;(2) is firmly nonexpansive, that is, for any ,

(3);(4) is closed and convex;(5).

Lemma 1.10 (see [8, 23]). Let be a uniformly convex Banach space, a positive number and a closed ball of . Then there exists a strictly increasing, continuous, and convex function with such that

for any , for all and such that .

Lemma 1.11 (see [27]). Let be a uniformly convex and uniformly smooth Banach space, a nonempty, closed, and convex subset of , and a closed quasi--asymptotically nonexpansive mapping from into itself. Then is a closed convex subset of .

2. Main Results

Theorem 2.1. Let be a nonempty, closed, and convex subset of a 2-uniformly convex and uniformly smooth real Banach space and a closed quasi--asymptotically nonexpansive mapping with sequence such that for each . Let be a bifunction from to satisfying (A1)–(A4). Let be a -inverse strongly monotone mapping of into with constant such that and is bounded. Assume that is asymptotically regular on C for each and for all and . Define a sequence in in the following manner:

for every , where is a real sequence in for some , is the normalized duality mapping on and . Assume that , , , are real sequences in such that and , for all . Let be a sequence in for some , where is the 2-uniformly convex constant of . Then the sequence converges strongly to .

Proof. We break the proof into nine steps.

Step 1. is well defined for .

By Lemma 1.11 we know that is a closed convex subset of for every . Hence is a nonempty closed convex subset of . Consequently, is well defined for .

Step 2. is closed and convex for each .

It is obvious that is closed and convex. Suppose that is closed and convex for some integer . Since the defining inequality in is equivalent to the inequality:

we have that is closed and convex. So is closed and convex for each . This in turn shows that is well defined.

Step 3. for all .

We do this by induction. For , we have . Suppose that for some . Let . Putting for all , we have that is quasi--nonexpansive from Lemma 1.9. Since is quasi--asymptotically nonexpansive, we have

Moreover, by Lemmas 1.3 and 1.7, we get that

Thus, since and is -inverse strongly monotone, we have from (2.4) that

Therefore, from (2.5), Lemma 1.2 and the fact that and for all and , we have

Substituting (2.6) into (2.3), we get

that is, . By induction, and the iteration algorithm generated by (2.1) is well defined.

Step 4. exists and is bounded.

Noticing that and Lemma 1.3, we have

for all and . This shows that the sequence is bounded. From and , we obtain that

which implies that is nondecreasing. Therefore, the limit of exists and is bounded.

Step 5. We have .

By Lemma 1.3, we have, for any positive integer , that

In view of Step  4 we deduce that as . It follows from Lemma 1.4 that as . Hence is a Cauchy sequence of . Since is a Banach space and is closed subset of , there exists a point such that .

Step 6. We have.

By taking in (2.10), we have

From Lemma 1.4, it follows that

Noticing that , we obtain

From (2.11), for any , and Lemma 1.4, we know

Notice that

for all . It follows from (2.12) and (2.14) that

which implies that as . Since is uniformly norm-to-norm continuous on bounded sets, from (2.16), we have

Let . Since is uniformly smooth Banach space, we know that is a uniformly convex Banach space. Therefore, from Lemma 1.10 we have, for any , that

Therefore, from (2.6) and (2.18), we have

It follows from that

On the other hand, we have

It follows from (2.16) and (2.17) that

Since and , from (2.20) and (2.22) we have

Therefore, from the property of , we obtain

Since is uniformly norm-to-norm continuous on bounded sets, we have

and hence as . Since , it follows from the asymptotic regularity of that

That is, as . From the closedness of , we get . Similarly, one can obtain that for . So, .

Moreover, from (2.19) we have that

which implies that

Now, Lemmas 1.3 and 1.7 imply that

In view of Lemma 1.2 and the fact that for all , , we have

From (2.28) and Lemma 1.4 we get

and hence as .

Step 7. We have .

Let be an operator as follows:

By Lemma 1.6, is maximal monotone and . Let . Since , we have . It follows from that

On the other hand, from and Lemma 1.5 we obtain that

and hence

Then, from (2.33) and (2.35), we have

Hence we have as , since the uniform continuity of and imply that the right side of (2.36) goes to as . Thus, since is maximal monotone, we have and hence .

Step 8. We have .

Let . From , (2.3), (2.6) and Lemma 1.9 we obtain that

It follows from (2.22) and that as . Now, by Lemma 1.4 we have that as . Consequently, we obtain that and from as . From the assumption , we get

Noting that , we obtain

From (A2), we have

Letting , we have from , (2.38) and (A4) that . For with and , let . Since and , we have and hence . Now, from (A1) and (A4) we have

and hence . Letting , from (A3), we have . This implies that . Therefore, in view of Steps  6, 7, and 8 we have .

Step 9. We have .

From , we get

Since for all , we arrive at

Letting , we have

and hence by Lemma 1.5. This completes the proof.

Strong convergence theorem for approximating a common element of the set of solutions of the equilibrium problem and the set of fixed points of a finite family of closed quasi--asymptotically nonexpansive mappings in Banach spaces may not require that be 2-uniformly convex. In fact, we have the following theorem.

Theorem 2.2. Let be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space and a closed quasi--asymptotically nonexpansive mapping with sequence such that for each . Let be a bifunction from to satisfying (A1)–(A4) such that and is bounded. Assume that is asymptotically regular on for each . Define a sequence in in the following manner:

for every , where is a real sequence in for some , is the normalized duality mapping on and . Assume that , , , are real sequences in such that and , for all . Then the sequence converges strongly to .

Proof. Put in Theorem 2.1. We have . Thus, the method of proof of Theorem 2.1 gives the required assertion without the requirement that is 2-uniformly convex.

As some corollaries of Theorems 2.1 and 2.2, we have the following results immediately.

Corollary 2.3. Let be a nonempty, closed, and convex subset of a Hilbert space and a closed quasi--asymptotically nonexpansive mapping with sequence such that for each . Let be a bifunction from to satisfying (A1)–(A4). Let be a -inverse strongly monotone mapping of into with constant such that and is bounded. Assume that is asymptotically regular on C for each and for all and . Define a sequence in in the following manner:

for every , where is a real sequence in for some and . Assume that , , , are real sequences in such that and , for all . Let be a sequence in for some . Then the sequence converges strongly to .

Corollary 2.4. Let be a nonempty, closed, and convex subset of a Hilbert space and a closed quasi--asymptotically nonexpansive mapping with sequence such that for each . Let be a bifunction from to satisfying (A1)–(A4) such that and is bounded. Assume that is asymptotically regular on C for each . Define a sequence in in the following manner:

for every , where is a real sequence in for some and . Assume that , , , are real sequences in such that and , for all . Let be a sequence in for some . Then the sequence converges strongly to .

Remark 2.5. Theorems 2.1 and 2.2 extend the main results of [23] from quasi--nonexpansive mappings to more general quasi--asymptotically nonexpansive mappings.

Acknowledgments

The research was supported by Fundamental Research Funds for the Central Universities (Program no. ZXH2012K001), supported in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing and it was also supported by the Science Research Foundation Program in Civil Aviation University of China (2012KYM04).

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