Journal of Applied Mathematics

Volume 2013 (2013), Article ID 761864, 21 pages

http://dx.doi.org/10.1155/2013/761864

Research Article

Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

Lu-Chuan Ceng¹ and Ching-Feng Wen²

¹Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

²Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 30 April 2013; Accepted 6 June 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Lu-Chuan Ceng and Ching-Feng Wen. 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 composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

1. Introduction

Let be a real Banach space whose dual space is denoted by . The normalized duality mapping is defined by

where denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that is nonempty for each . Let be a nonempty, closed, and convex subset of . A mapping is called nonexpansive if for every . The set of fixed points of is denoted by . We use the notation to indicate the weak convergence and the one to indicate the strong convergence. A mapping is said to be accretive if for each , there exists such that

It is said to be -strongly accretive if for each , there exists such that

for some . The mapping is called -inverse strongly-accretive if for each , there exists such that

for some and is said to be -strictly pseudocontractive if for each , there exists such that

for some .

Let denote the unite sphere of . A Banach space is said to be uniformly convex if for each , there exists such that for all ,

It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space is said to be smooth if the limit

exists for all ; in this case, is also said to have a Gâteaux differentiable norm. is said to have a uniformly Gâteaux differentiable norm if for each , the limit is attained uniformly for . Moreover, it is said to be uniformly smooth if this limit is attained uniformly for . The norm of is said to be the Fréchet differential if for each , this limit is attained uniformly for . In addition, we define a function called the modulus of smoothness of as follows:

It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then a Banach space is said to be -uniformly smooth if there exists a constant such that for all . As pointed out in [1], no Banach space is -uniformly smooth for . In addition, it is also known that is single-valued if and only if is smooth, whereas if is uniformly smooth, then the mapping is norm-to-norm uniformly continuous on bounded subsets of . If has a uniformly Gâteaux differentiable norm, then the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of .

Very recently, Cai and Bu [2] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space , which involves finding such that

where is a nonempty, closed, and convex subset of , , and are two nonlinear mappings, and and are two positive constants. Here the set of solutions of GSVI (9) is denoted by GSVI. In particular, if , a real Hilbert space, then GSVI (9) reduces to the following GSVI of finding such that

which and are two positive constants. The set of solutions of problem (10) is still denoted by GSVI. It is clear that the problem (10) covers as special case the classical variational inequality problem (VIP) of finding such that

The solution set of the VIP (11) is denoted by .

Recently, Ceng et al. [3] transformed problem (10) into a fixed point problem in the following way.

Lemma 1 (see [3]). For given is a solution of problem (10) if and only if is a fixed point of the mapping defined by

where and is the the projection of onto .

In particular, if the mappings is -inverse strongly monotone for , then the mapping is nonexpansive provided for .

Let be a nonempty, closed, and convex subset of a real smooth Banach space . Let be a sunny nonexpansive retraction from onto , and let be a contraction with coefficient . In this paper we introduce composite implicit and explicit iterative algorithms for solving GSVI (9) and the common fixed point problem of an infinite family of nonexpansive mappings of into itself. These composite iterative algorithms are based on Korpelevich’s extragradient method [4] and viscosity approximation method [5]. Let the mapping be defined by

We first propose a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space :

where is -inverse-strongly accretive with for and , , , and are the sequences in such that for all . It is proven that under appropriate conditions, converges strongly to , which solves the following VIP:

On the other hand, we also propose another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gateaux differentiable norm:

where is -strictly pseudocontractive and -strongly accretive with for and , , , and are the sequences in such that for all . It is proven that under mild conditions, also converges strongly to , which solves the VIP (15). The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 2 can be found in [6]. Lemma 3 is an immediate consequence of the subdifferential inequality of the function .

Lemma 2. Let be a sequence of nonnegative real numbers satisfying

where , , and satisfy the conditions:

(i) and ,(ii),(iii), , and .

Then .

Lemma 3. In a smooth Banach space , there holds the inequality

Lemma 4 (see [7]). Let and be bounded sequences in a Banach space , and let be a sequence in which satisfies the following condition:

Suppose that , , and . Then .

Let be a subset of , and let be a mapping of into . Then is said to be sunny if

whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for every where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . The following lemma concerns the sunny nonexpansive retraction.

Lemma 5 (see [8]). Let be a nonempty, closed, and convex subset of a real smooth Banach space . Let be a nonempty subset of . Let be a retraction of onto . Then the following are equivalent:

(i) is sunny and nonexpansive;(ii), ;(iii), , .

It is well known that if a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto ; that is, . If is a nonempty, closed, and convex subset of a strictly convex and uniformly smooth Banach space and if is a nonexpansive mapping with the fixed point set , then the set is a sunny nonexpansive retract of .

Lemma 6 (see [9]). Given a number . A real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function , , such that

for all and such that and .

Lemma 7 (see [10]). Let be a nonempty, closed, and convex subset of a Banach space . Let , be a sequence of mappings of into itself. Suppose that . Then for each , converges strongly to some point of . Moreover, let be a mapping of into itself defined by for all . Then .

Let be a nonempty, closed, and convex subset of a Banach space , and let be a nonexpansive mapping with . As previous, let be the set of all contractions on . For and , let be the unique fixed point of the contraction on ; that is,

Lemma 8 (see [11, 12]). Let be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let be a nonempty, closed, and convex subset of , let be a nonexpansive mapping with , and . Then the net defined by converges strongly to a point in . If we define a mapping by , , then solves the VIP:

Lemma 9 (see [13]). Let be a nonempty, closed, and convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose that is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for is defined well, nonexpansive, and holds.

3. Implicit Iterative Schemes

In this section, we introduce our implicit iterative schemes and show the strong convergence theorems. We will use the following useful lemmas in the sequel.

Lemma 10 (see [2, Lemma 2.8]). Let be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space . Let the mapping be -inverse-strongly accretive. Then, one has

for , where . In particular, if , then is nonexpansive for .

Lemma 11 (see [2, Lemma 2.9]). Let be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive for . Let be the mapping defined by

If   for , then is nonexpensive.

Lemma 12 (see [2, Lemma 2.10]). Let be a nonempty, closed, and convex subset of a real 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be two nonlinear mappings. For given is a solution of GSVI (9) if and only if where .

Remark 13. By Lemma 12, we observe that

which implies that is a fixed point of the mapping .

We now state and prove our first result on the implicit iterative scheme.

Theorem 14. Let be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive for . Let be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping . For arbitrarily given , let be the sequence generated by

where for and , , , and are the sequences in such that , . Suppose that the following conditions hold:

(i) and ,(ii) and ,(iii).

Assume that for any bounded subset of , and let be a mapping of into itself defined by for all . Suppose that . Then converges strongly to , which solves the following VIP:

Proof. Take a fixed arbitrarily. Then by Lemma 12, we know that and for all . Moreover, by Lemma 11, we have

which hence implies that

Thus, from (27), we have

It immediately follows that is bounded, and so are the sequences , , and due to (30) and the nonexpansivity of .

Let us show that as . As a matter of fact, from (27), we have

Simple calculations show that

It follows that

which hence yields

Now, we write , , where . It follows that for all ,

This together with (35) implies that

where for some . So, from , condition (iii), and the assumption on , it immediately follows that

In terms of condition (ii) and Lemma 4, we get

Hence we obtain

Next we show that as .

For simplicity, put , , and . Then . From Lemma 10, we have

Substituting (41) into (42), we obtain

According to Lemma 3, we have from (27)

which hence yields

This together with (43) and the convexity of , we have

where for some . So, it follows that

Since for , from conditions (i), (ii), and (40), we obtain

Utilizing [14, Proposition 1] and Lemma 5, we have

which implies that

In the same way, we derive

which implies that

Substituting (50) into (52), we get

From (46) and (53), we have

which implies that

Utilizing conditions (i), (ii), from (40) and (48), we have

Utilizing the properties of and , we deduce that

From (57), we obtain

That is,

On the other hand, since and are bounded, by Lemma 6, there exists a continuous strictly increasing function , such that for

which together with (30) implies that

It immediately follows that

According to condition (ii), we get

Since , , and , we conclude that

Utilizing the property of , we have

We note that

So,

That is,

We observe that

Thus, from (59)–(68), we obtain that

By (70) and Lemma 7, we have

In terms of (59) and (71), we have

Define a mapping , where is a constant. Then by Lemma 9, we have that . We observe that

From (59) and (72), we obtain

Now, we claim that

where with being the fixed point of the contraction

Then solves the fixed point equation . Thus we have

By Lemma 3, we conclude that

where

It follows from (78) that

Letting in (80) and noticing (79), we derive

where is a constant such that for all and . Taking in (81), we have

On the other hand, we have

It follows that

Taking into account that as , we have from (82)

Since has a uniformly Fréchet differentiable norm, the duality mapping is norm-to-norm uniformly continuous on bounded subsets of . Consequently, the two limits are interchangeable, and hence (75) holds. From (68), we get . Noticing that is norm-to-norm uniformly continuous on bounded subsets of , we deduce from (75) that

Finally, let us show that as . We observe that

which implies that

By (27) and the convexity of , we get

which together with (88) leads to

Applying Lemma 2 to (88), we obtain that as . This completes the proof.

Corollary 15. Let be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive for . Let be a contraction with coefficient . Let be a nonexpansive mapping of into itself such that , where is the fixed point set of the mapping . For arbitrarily given , let be the sequence generated by

where for and , , , and are the sequences in such that , . Suppose that the following conditions hold:

(i) and ,(ii) and ,(iii).

Then converges strongly to , which solves the following VIP:

Further, we illustrate Theorem 14 by virtue of an example, that is, the following corollary.

Corollary 16. Let be a nonempty, closed, and convex subset of a uniformly convex and 2-uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be a contraction with coefficient . Let be an -strictly pseudocontractive mapping of into itself, and let be a nonexpansive mapping of into itself such that . For arbitrarily given , let be the sequence generated by

where and , , , and are the sequences in such that , . Suppose that the following conditions hold:

(i) and ,(ii) and ,(iii).

Then converges strongly to , which solves the following VIP:

Proof. In Corollary 15, put , , , and . Since is an -strictly pseudocontractive mapping, it is clear that is an -inverse strongly accretive mapping. Hence, the GSVI (9) is equivalent to the following VIP of finding such that

which leads to . In the meantime, we have

In the same way, we get . In this case, it is easy to see that (91) reduces to (93). We claim that . As a matter of fact, we have, for ,

So, we conclude that . Therefore, the desired result follows from Corollary 15.

Remark 17. Theorem 14 improves, extends, supplements, and develops Cai and Bu [2, Theorem  3.1 and Corollary  3.2] and Jung [5, Theorem  3.1] in the following aspects.

(i)The problem of finding a point in Theorem 14 is more general and more subtle than the problem of finding a point in Jung [5, Theorem  3.1].(ii)The iterative scheme in [2, Theorem  3.1] is extended to develop the iterative scheme (27) of Theorem 14 by virtue of the iterative scheme of [5, Theorem  3.1]. The iterative scheme (27) of Theorem 14 is more advantageous and more flexible than the iterative scheme of [2, Theorem  3.1] because it involves several parameter sequences , , , and .(iii)The iterative scheme (27) in Theorem 14 is very different from everyone in both [2, Theorem  3.1] and [5, Theorem  3.1] because the mappings and in the iterative scheme of [2, Theorem  3.1] and the mapping in the iterative scheme of [5, Theorem  3.1] are replaced by the same composite mapping in the iterative scheme (27) of Theorem 14.(iv)The proof in [2, Theorem  3.1] depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces ([9]), and the inequality in smooth and uniform convex Banach spaces ([14, Proposition 1]). Because the composite mapping appears in the iterative scheme (27) of Theorem 14, the proof of Theorem 14 depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, the inequality in smooth and uniform convex Banach spaces, and the inequality in uniform convex Banach spaces (Lemma 6).(v)The iterative scheme in [2, Corollary 3.2] is extended to develop the new iterative scheme in Corollary 15 because the mappings and are replaced by the same composite mapping in Corollary 15.

4. Explicit Iterative Schemes

In this section, we introduce our explicit iterative schemes and show the strong convergence theorems. First, we give several useful lemmas.

Lemma 18. Let be a nonempty, closed, and convex subset of a smooth Banach space , and let the mapping be -strictly pseudocontractive and -strongly accretive with for . Then, for , we have

for . In particular, if , then is nonexpansive for .

Proof. Taking into account the -strict pseudocontractivity of , we derive for every

which implies that

Hence,

Utilizing the -strong accretivity and -strict pseudocontractivity of , we get

So, we have

Therefore, for , we have

Since , it follows immediately that

This implies that is nonexpansive for .

Lemma 19. Let be a nonempty, closed, and convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto , and let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let be the mapping defined by

If , then is nonexpansive.

Proof. According to Lemma 10, we know that is nonexpansive for . Hence, for all , we have

This shows that is nonexpansive. This completes the proof.

Lemma 20. Let be a nonempty, closed, and convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto , and let the mapping be -strictly pseudocontractive and -strongly accretive for . For given , is a solution of GSVI (9) if and only if where .

Proof. We can rewrite GSVI (9) as

which is obviously equivalent to

because of Lemma 5. This completes the proof.

Remark 21. By Lemma 20, we observe that

which implies that is a fixed point of the mapping . Throughout this paper, the set of fixed points of the mapping is denoted by .

We are now in a position to state and prove our result on the explicit iterative scheme.

Theorem 22. Let be a nonempty, closed, and convex subset of a uniformly convex Banach space which has a uniformly Gâteaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let be a contraction with coefficient . Let be an infinite family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping . For arbitrarily given , let be the sequence generated by

where for and , , , and are the sequences in such that , . Suppose that the following conditions hold:

(i),(ii) and ,(iii) or ,(iv) or ,(v) or ,(vi).

Assume that for any bounded subset of , and let be a mapping of into itself defined by for all . Suppose that . Then converges strongly to , which solves the following VIP:

Proof. Take a fixed arbitrarily. Then by Lemma 20, we know that and for all . Moreover, by Lemma 19, we have

From (113) we obtain

which implies that is bounded. By Lemma 19 we know from (113) that , , and are bounded.

Let us show that and as . As a matter of fact, from (113), we have

Simple calculations show that

It follows that

Now, we write , , where . It follows that, for all ,

This together with (117) implies that

Furthermore, we note that

Also, simple calculations show that

This together with (119) implies that

where for some . Utilizing Lemma 2, from conditions (ii)–(v) and the assumption on , we deduce that

Taking into account the boundedness of and , by Lemma 6, we know that there exists a continuous strictly increasing function , such that for

Since and are bounded, by Lemma 6, there exists a continuous strictly increasing function , such that for

which together with (124) implies that

It immediately follows that

According to condition (vi), we get

Since and , we conclude from conditions (i) and (vi) that

Utilizing the properties of and , we have

Note that

Thus, from (123), (130), and , it follows that

On the other hand, from (130), we get

This together with (132) implies that

By (130) and Lemma 7, we have

In terms of (134) and (135), we have

Define a mapping , where is a constant. Then by Lemma 9, we have that . We observe that

From (134) and (136), we obtain

Now, we claim that

where with being the fixed point of the contraction

Then solves the fixed point equation . Thus we have

By Lemma 3, we conclude that

where

It follows from (142) that

Letting in (144) and noticing (143), we derive

where is a constant such that for all and . Taking in (145), we have

On the other hand, we have

Hence it follows that

Taking into account that as , we have from (146)

Since has a uniformly Gâteaux differentiable norm, the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of . Consequently, the two limits are interchangeable, and hence (139) holds. From (123), we get . Noticing the norm-to-weak* uniform continuity of on bounded subsets of , we deduce from (139) that

Finally, let us show that as . We observe that

So, we have

Since and , , by Lemma 2, we conclude from (153) that as . This completes the proof.

Corollary 23. Let be a nonempty, closed, and convex subset of a uniformly convex Banach space which has a uniformly Gâteaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let the mapping be -strictly pseudocontractive and -strongly accretive with for . Let be a contraction with coefficient . Let be a nonexpansive mapping of into itself such that , where is the fixed point set of the mapping . For arbitrarily given , let be the sequence generated by

where for . Suppose that , , , and are the sequences in satisfying the following conditions:

(i),(ii) and ,(iii) or ,(iv) or ,(v) or ,(vi).

Then converges strongly to , which solves the following VIP:

Further, we illustrate Theorem 22 by virtue of an example, that is, the following corollary.

Corollary 24. Let be a nonempty, closed, and convex subset of a uniformly convex Banach space which has a uniformly Gâteaux differentiable norm. Let be a sunny nonexpansive retraction from onto . Let be a contraction with coefficient . Let be a self-mapping on such that is -strictly pseudocontractive and -strongly accretive with , and let be a nonexpansive mapping of into itself such that . For arbitrarily given , let be the sequence generated by

where . Suppose that , , , and are the sequences in satisfying the following conditions:

(i),(ii) and ,(iii) or ,(iv) or ,(v) or ,(vi).

Then converges strongly to , which solves the following VIP:

Proof. Utilizing the arguments similar to those in the proof of Corollary 16, we can obtain the desired result.

Remark 25. As previous, we emphasize that our composite iterative algorithms (i.e., the iterative schemes (27) and (111)) are based on Korpelevich’s extragradient method and viscosity approximation method. It is well known that the so-called viscosity approximation method must contain a contraction on . In the meantime, it is worth pointing out that our proof of Theorems 14 and 22 must make use of Lemma 8 for implicit viscosity approximation method; that is, Lemma 8 plays a key role in our proof of Theorems 14 and 22. Therefore, there is no doubt that the contraction in Theorems 14 and 22 cannot be replaced by a general -Lipschitzian mapping with constant .

Remark 26. Theorem 22 improves, extends, supplements, and develops [2, Theorem  3.1 and Corollary  3.2] and [5, Theorems  3.1] in the following aspects.

(i)The problem of finding a point in Theorem 22 is more general and more subtle than the problem of finding a point in Jung [5, Theorem  3.1].(ii)The iterative scheme in [2, Theorem  3.1] is extended to develop the iterative scheme (111) of Theorem 22 by virtue of the iterative scheme of [5, Theorem  3.1]. The iterative scheme (111) in Theorem 22 is more advantageous and more flexible than the iterative scheme in [2, Theorem  3.1] because it involves several parameter sequences , , , and .(iii)The iterative scheme (111) in Theorem 22 is very different from everyone in both [2, Theorem  3.1] and [5, Theorem  3.1] because the mappings and in the iterative scheme of [2, Theorem  3.1] and the mapping in the iterative scheme of [5, Theorem  3.1] are replaced by the same composite mapping in the iterative scheme (111) of Theorem 22.(iv)The proof in [2, Theorem  3.1] depends on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. However, the proof of Theorem 22 does not depend on the argument techniques in [3], the inequality in 2-uniformly smooth Banach spaces, and the inequality in smooth and uniform convex Banach spaces. It depends on only the inequality in uniform convex Banach spaces.(v)The assumption of the uniformly convex and 2-uniformly smooth Banach space in [2, Theorem  3.1] is weakened to the one of the uniformly convex Banach space having a uniformly Gateaux differentiable norm in Theorem 22.(vi)The iterative scheme in [2, Corollary  3.2] is extended to develop the new iterative scheme in Corollary 15 because the mappings and are replaced by the same composite mapping in Corollary 23.

Finally, we observe that related results can be found in recent papers, for example, [15–24] and the references therein.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This research was partially supported by a Grant from NSC 101-2115-M-037-001.

References

Y. Takahashi, K. Hashimoto, and M. Kato, “On sharp uniform convexity, smoothness, and strong type, cotype inequalities,” Journal of Nonlinear and Convex Analysis, vol. 3, no. 2, pp. 267–281, 2002.

G. Cai and S. Q. Bu, “Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 538–546, 2012.

L. C. Ceng, C. Y. Wang, and J. C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375–390, 2008.

G. M. Korpelevič, “An extragradient method for finding saddle points and for other problems,” Èkonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.

J. S. Jung, “A new iteration method for nonexpansive mappings and monotone mappings in Hilbert spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 251761, 16 pages, 2010.

H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society 2, vol. 66, no. 1, pp. 240–256, 2002.

T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.

S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.

H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.

K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007.

J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509–520, 2005.

H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.

R. E. Bruck, “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,” Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973.

S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.

L. C. Ceng and J. C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007.

L. C. Zeng and J. C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006.

L. C. Ceng, Q. H. Ansari, and J. C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012.

L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010.

L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2116–2125, 2012.

L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Relaxed extragradient iterative methods for variational inequalities,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112–1123, 2011.

L. C. Ceng, Q. H. Ansari, N. C. Wong, and J. C. Yao, “An extragradient-like approximation method for variational inequalities and fixed point problems,” Fixed Point Theory and Applications, vol. 2011, article 22, 18 pages, 2011.

L. C. Ceng, S. M. Guu, and J. C. Yao, “Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method,” Fixed Point Theory and Applications, vol. 2011, Article ID 626159, 22 pages, 2011.

Y. Yao and J. C. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007.

L. C. Ceng, H. K. Xu, and J. C. Yao, “Strong convergence of an iterative method with perturbed mappings for nonexpansive and accretive operators,” Numerical Functional Analysis and Optimization, vol. 29, no. 3-4, pp. 324–345, 2008.