Abstract and Applied Analysis

Volume 2013 (2013), Article ID 107296, 13 pages

http://dx.doi.org/10.1155/2013/107296

Research Article

A New Hybrid Projection Algorithm for System of Equilibrium Problems and Variational Inequality Problems and Two Finite Families of Quasi--Nonexpansive Mappings

Pongrus Phuangphoo1,2 and Poom Kumam¹

¹Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thung Kru, Bangkok 10140, Thailand

²Department of Mathematics, Faculty of Education, Bansomdejchaopraya Rajabhat University (BSRU), Thonburi, Bangkok 10600, Thailand

Received 28 July 2012; Revised 24 September 2012; Accepted 28 September 2012

Academic Editor: Yongfu Su

Copyright © 2013 Pongrus Phuangphoo and Poom Kumam. 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 a modified Mann’s iterative procedure by using the hybrid projection method for solving the common solution of the system of equilibrium problems for a finite family of bifunctions satisfying certain condition, the common solution of fixed point problems for two finite families of quasi--nonexpansive mappings, and the common solution of variational inequality problems for a finite family of continuous monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by some mild conditions. Our result presented in this paper improves and generalizes some well-known results in the literature.

1. Introduction

Throughout this paper, we denote by and the set of all real numbers and the set of all positive numbers, respectively. We also assume that is a real Banach space and is the dual space of . Let be a nonempty, closed and convex subset of a real Banach space with the dual . We recall the following definitions.

A mapping is said to be nonexpansive if

A mapping is said to be monotone if for each , such that

A mapping is said to be -strongly monotone if there exists a constant such that

A mapping is said to be -inverse strongly monotone if there exists a constant such that

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

Clearly, the class of monotone mappings include the class of -inverse strongly monotone mappings.

Let be a monotone mapping. The variational inequality problem is to find a point such that

The set of the solution of the variational inequality problem is denoted by .

Let be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space , let be a mapping, and be the set of fixed points of .

A point is said to be a fixed point of if . The set of the solution of the fixed point of is denoted by .

A point is said to be an asymptotic fixed point of if there exists a sequence such that and . We denoted the set of all asymptotic fixed points of by .

A point is said to be a strong asymptotic fixed point of if there exists a sequence such that and . We denoted the set of all strong asymptotic fixed points of by .

Let be a bifunction. The equilibrium problem is to find a point such that

The set of the solution of equilibrium problem is denoted by . Numerous problems in sciences, mathematics, optimizations, and economics reduced to find a solution of equilibrium problems. The equilibrium problems include variational inequality problems and fixed point problem, and optimization problems as special cases (see, e.g., [1–3]). Recently, many authors have considered the problem for finding the common solution of fixed point problems, the common solution of equilibrium problems, and the common solution of variational inequality problems.

In 1953, Mann [4] introduced the iterative sequence which is defined by

where the initial element is arbitrary, is a nonexpansive mapping, and is the sequence in such that and . The sequence of (7) is generally referred to as the Mann iteration.

In 2009, Takahashi and Zembayashi [5] introduced the following iterative scheme by the shrinking projection method, and they proved that converges strongly to , under appropriate conditions.

Theorem TZ.  Let be a uniformly smooth and uniformly convex real Banach space, and let be a nonempty, closed and convex subset of . Let be a bifunction from to satisfying (A1)–(A4) and let be a relatively nonexpansive mapping from into itself such that . Let be a sequence generated by

for every , where is the duality mapping on , the sequence satisfies , and satisfies . Then, the sequence converges strongly to , where is the generalized projection of onto .

In 2009, Qin et al. [6] extended the iterative process (8) from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings. In 2011, Zegeye and Shahzad [7] introduced an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems, and common solutions of the finite family of variational inequality problems for monotone mappings in Banach spaces.

Theorem ZS.  Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space with the dual . Let , , be a finite family of bifunctions. Let , be a finite family relatively quasi-nonexpansive mappings and , be a finite family of continuous monotone mappings.

For , they define mappings by

where ( D), N), ( L) , and for some . Assume that . Let be a sequence generated by

where the real numbers such that . Then, converges strongly to an element of .

In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative procedure for solving the common solution of system of equilibrium problems for a finite family of bifunctions satisfying certain conditions and the common solution of fixed point problems for two countable families of quasi--nonexpansive mappings and the common solution of variational inequality problems for a finite family of monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by the conditions. The results obtained in this paper extend and improve several recent results in this area.

2. Preliminaries

A Banach space is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in such that , and

Let = be the unit sphere of . Then the Banach space is said to be smooth if

exists for each . It is said to be uniformly smooth if the limit (12) is attained uniformly for all .

Let be a Banach space. Then a function is said to be the modulus of smoothness of if

The space is said to be smooth if , and is said to be uniformly smooth if and only if .

The modulus of convexity of is the function defined by

A Banach space is said to be uniformly convex if and only if for all .

We recall the following definitions.

Definition 1. Let be a nonempty set.

(1)A mapping is said to be closed if for each and imply .(2)A mapping is said to be quasi--nonexpansive (relatively quasi-nonexpansive) if , and

(3)A mapping is said to be relatively nonexpansive [8, 9] if ,   and

(4)A mapping is said to be weak relatively nonexpansive [10] if ,   and

Remark 2. We here the following basic properties.

(1)Each relatively nonexpansive mapping is closed.(2)The class of quasi--nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse may be not true.(3)The class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.(4)The class of quasi--nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.(5)If is a real uniformly smooth Banach space, then is uniformly continuous on each bounded subset of .(6)If is a strictly convex reflexive Banach space, then is hemicontinuous, that is, is norm-to- continuous.(7)If is a smooth and strictly convex reflexive Banach space, then is single-valued, one-to-one, and onto.(8)A Banach space is uniformly smooth if and only if is uniformly convex.(9)Each uniformly convex Banach space has the Kadec-Klee property, that is, for any sequence , if and , then .(10)A Banach space is strictly convex if and only if is strictly monotone, that is,

(11)Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.(12) is uniformly convex, then is uniformly norm-to-norm continuous on each bounded subset of .

Let be a real Banach space and be a sequence in . We denote by and the strong convergence and weak convergence of , respectively. The normalized duality mapping from to is defined by

where denotes the duality pairing. It is well known that if is smooth, then is single-valued and demicontinuous, and if is uniformly smooth, then is uniformly continuous on bounded subset of . Moreover, if is 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 to and so and (see [11, 12]). We note that in a Hilbert space , the mapping is the identity operator.

Now, let be a smooth and strictly convex reflexive Banach space. As Alber (see [13]) and Kamimura and Takahashi (see [14]) did, the Lyapunov functional is defined by

It follows from Kohsaka and Takahashi (see [15]) that if and only if and that

Further suppose that is nonempty, closed and convex subset of . The generalized, projection (Alber see [13]) is defined by for each ,

Remark 3. If is a real Hilbert space , then and (the metric projection of onto ).

Lemma 4 (Alber [13]). Let be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space , and let . Then

Lemma 5 (Kamimura and Takahashi [14]). Let be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space , and let and . Then,

Lemma 6 (Qin et al. [6] and Kohsaka and Takahashi [16]). Let be a smooth, strictly convex and reflexive Banach space, and is a continuous monotone mapping with . Then, it is proved in [16] that the resolvent , for is quasi--nonexpansive. Moreover, if is quasi--nonexpansive, then using the definition of one can show that is closed and convex (see [6]).

Lemma 7 (Kamimura and Takahashi [14]). Let be a uniformly convex and smooth real Banach space and let and be two sequences of . If and either or is bounded, then .

Lemma 8. Let be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let and be two sequences of , and . If and , then .

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

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

Lemma 9 (Blum and Oettli [1]). Let be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space and let be a bifunction satisfying the following conditions (A1)–(A4). Let be any given number and be any point. Then, there exists a point such that

Lemma 10 (Takahashi and Zembayashi [5]). Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space , and be a bifunction satisfying the following conditions (A1)–(A4). Let be any given number and be any point defined a mapping as follows. Then, there exists a point such that

Then, the following conclusions hold:

(1) is single-valued; (2) is a firmly nonexpansive type mapping, that is,

(3);(4) is a closed and convex subset of ; (5).

Lemma 11 (Zegeye and Shahzad [7]). Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space , and be a continuous monotone mapping. Let be any given number and be any point defined a mapping as follows:

Then, the following conclusions hold:

(1) is single valued; (2) is a firmly nonexpansive type mapping, that is,

(3);(4) is a closed and convex subset of ; (5).

Lemma 12 (Xu [17]). Let be a uniformly convex Banach space, let be a closed ball of , where . Then, there exists a continuous, strictly increasing, and convex function with such that

for all and all with .

3. Main Results

In this section, we prove a strong convergence theorem which solves the problem for finding a common solution of the system of equilibrium problems and variational inequality problems and fixed point problems in Banach spaces.

Theorem 13. Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that

(1) is a finite family of bifunctions satisfying conditions (A1)–(A4), where ;(2) and are two finite families of quasi--nonexpansive mappings from into , where ;(3) is a finite family of continuous monotone mappings, where ;(4)For , one defines the mappings by

Assume that is a nonempty and bounded in . Let be a sequence generated by

where , , , , for some and and are real numbers in such that .

Then, the sequence converges strongly to .

Proof. We will complete this proof by seven steps below.

Step  1. We will show that is closed and convex for each .

From the definition of , it is obvious that is closed. Moreover, since

It follows that is convex for each . Therefore, is closed and convex for each .

Step  2. We will show that for each .

From the assumption, we see that . Suppose that for some . Now, for , Since and are quasi--nonexpansive and by Lemmas 10 and 11, we compute

Therefore, . By the induction, this implies that for each .

Hence, the sequence is well defined.

Step  3. We will show that the sequence is bounded.

Let . From for each and is well defined. From the assumption of , we see that is closed and convex subset of .

Let , where is the unique element that satisfies . Now, we will show that as .

From the assumption of , we know that and since and . Using Lemma 4, we get

for all , where . Then, the sequence is bounded. Hence, the sequence is also bounded.

Step  4. We will show that there exists such that , as .

Since and , we have

Therefore, the sequence is nondecreasing. Hence exists. By the definition of , one has that and for any positive integer .

It follows that

Since exists, by taking in (37), we have .

From Lemma 7, it follows that as . Thus is a Cauchy sequence.

Without a loss of generalization, we can assume that . Since is bounded and is reflexive. Since is closed and convex, it follows that , . Moreover, by using the weak lower semicontinuous of the norm on and (35), we obtain

This implies that

By using Lemma 5, we have and hence, . By the definition of , we get

Therefore, . Since , by the Kadec-Klee property of , we obtain

From is uniformly continuous, we also have

Step  5. We will show that , and as .

Since , we have , as . Thus, from (21), we obtain

and so

This implies that is bounded. Note that reflexivity of implies reflexivity of . Thus, we assume that . Furthermore, the reflexivity of implies there exists such that . Then, it follows that

Taking the on both sides of (45) and using weak lower semicontinuous of norm to get that

Thus , and so . It follows that . Now, from (44) and the Kadec-Klee property of , we obtain

Thus the demicontinuity of implies that . Now, from (43) and the fact that has the Kadec-Klee property, we obtain

In the fact that and as , we get

Since , it follows from Lemma 11 that

From (21), we obtain

and so is bounded. Since is reflexive, we assume that . It follows that

Taking the on both sides of (52) and using the continuity of , we get that

This implies that and hence .

Now, from (51) and the Kadec-Klee property of , we obtain

In the fact that , and as , we get

Since , it follows from Lemma 10 that

From (21), we obtain

and so is bounded. Since is reflexive, we assume that . It follows that

Taking the on both sides of (58) and using the continuity of , we get that

This implies that and hence .

Now, from (57) and the Kadec-Klee property of , we obtain

Step  6. We will show that .

Substep 1. We will show that .

From the definition of of algorithm (32), we have

Let be such that , for all . Then from (61), we obtain

and that is

Now, we set , for all and .

Therefore, we get . From (63), it follows that

From the continuity of and (41) and (54), we have , , as , we obtain

Since is a monotone mapping, we also have .

Thus, it follows that

and hence

If , we obtain

This implies that .

Similarly, let be such that , for all .

Then, we have again that .

Continuing in the same way, we obtain that , where .

Hence, .

Substep 2. We will show that .

From the definition of of algorithm (32) and (A2), we have

Let be such that , for all . Then from (69), we obtain

From the continuity of and (41) and (60), we have , , as and we obtain

Therefore, .

Now, we set , for all and .

Consequently, we get . And so .

Therefore, from (A1), we obtain

Thus, . From (A3), if , then we get . This implies that .

Similarly, let be such that , for all . Then, we have again that .

Continuing in the same way, we obtain that , where . Hence, .

Substep 3. We will show that .

From algorithm (32) and Lemma 12, we compute

From (73), we have

From , , and , we obtain

Therefore,

It follows from the property of that

From (42), we have , as . Then,

and so

Moreover, the demicontinuity of implies that as . Thus, the Kadec-Klee property of , we obtain

Let be such that , for all .

Then, from (60), we have , as . It follows from (80) and the closedness of that

This implies that .

Similarly, let be such that , for all . Then, we have again that .

Continuing in the same way, we obtain that , where .

Hence, .

Substep 4. We will show that .

From (73), we have

From , , and , we obtain

Therefore,

It follows from the property of that

From (42), we have , as . Then,

and so

Moreover, the demicontinuity of implies that as . Thus, the Kadec-Klee property of , we obtain

Let be such that , for all .

Then, from (54), we have , as . It follows from (88) and the closedness of that

This implies that .

Similarly, let be such that , for all . Then, we have again that .

Continuing in the same way, we obtain that , where .

Hence, .

From Substeps (6.1)–(6.4), we can conclude that

Step  7. Finally, we will show that .

From , we have

Taking in (91), one has

Now, we have and by Lemma 5, we get

This completes the proof of Theorem 13.

If we set in Theorem 13, then we obtain the following result.

Corollary 14. Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that

(1) is a bifunction satisfying conditions (A1)–(A4);(2) and are two quasi--nonexpansive mappings from into ;(3) is a continuous monotone mapping.(4)For , one defines the mappings as follows:

Assume that is a nonempty and bounded in and let be a sequence generated by

where for some and , and , , are real numbers in such that .

Then, the sequence converges strongly to .

If we set , for any in Theorem 13, then we obtain the following result which extends and improves the result’s Zegeye and Shahzad [7].

Corollary 15 (Zegeye and Shahzad [7]). Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has Kadec-Klee property. Suppose that

(1) is a finite family of bifunctions satisfying conditions (A1)–(A4), where ;(2) is a finite family of quasi--nonexpansive mappings, where ;(3) is a finite family of continuous monotone mappings, where ;(4)For , one defines the mappings by

Assume that is a nonempty and bounded in and let be a sequence generated by

where , , , for some and , and , , are real numbers in (0,1) such that .

Then, the sequence converges strongly to .

4. Deduced Theorems

If we set , for any in Theorem 13, then we obtain the following result.

Corollary 16. Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that

(1) and are finite families of quasi--nonexpansive mappings from into , where ;(2) is a finite family of continuous monotone mappings, where ;(3)For , we define the mappings by

Assume that is a nonempty and bounded in and let be a sequence generated by

where , , , for some and , and , , are real numbers in (0,1) such that .

Then, the sequence converges strongly to .

If we set , for any in Theorem 13, then we obtain the following result.

Corollary 17. Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that

(1) is a finite family of bifunctions satisfying conditions (A1)–(A4), where ;(2) and are finite families of quasi--nonexpansive mappings from into , where ;(3)For , one defines the mappings by

Assume that is a nonempty and bounded in and let be a sequence generated by

where , , , for some and , and , , are real numbers in such that .

Then, the sequence converges strongly to .

If we set , for any and , for any in Theorem 13, then we obtain the following result.

Corollary 18. Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that and are finite families of quasi--nonexpansive mappings from into , where .

Assume that is a nonempty and bounded in and let be a sequence generated by

where , , for some and , and , , are real numbers in (0,1) such that .

Then, the sequence converges strongly to .

5. Some Applications

5.1. Application to Weak Relatively Nonexpansive Mappings

If we change the condition in Theorem 13 as follows: and are finite families of weak relatively nonexpansive mappings. From Remark 2, every weak relatively nonexpansive mappings is quasi--nonexpansive mappings. Then, we obtain the following result.

Corollary 19. Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that

(1) is a finite family of bifunctions satisfying conditions (A1)–(A4), where ;(2) and are finite families of weak relatively nonexpansive mappings from into , where ;(3) is a finite family of continuous monotone mappings, where ;(4)For , one defines the mappings by

Assume that is a nonempty and bounded in and let be a sequence generated by

where , , , , for some and , and , , are real numbers in (0,1) such that .

Then, the sequence converges strongly to .

5.2. Application to Relatively Nonexpansive Mappings

If we change the condition in Theorem 13 as follows: and are finite families of relatively nonexpansive mappings. From Remark 2 and every relatively nonexpansive mappings is weak relatively nonexpansive mappings and every weak relatively nonexpansive mappings is quasi--nonexpansive mappings. Then, we obtain the following result.

Corollary 20. Let be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space which has the Kadec-Klee property. Suppose that

(1) is a finite family of bifunctions satisfying conditions (A1)–(A4), where ;(2) and are finite families of relatively nonexpansive mappings from into , where ;(3) is a finite family of continuous monotone mappings, where ;(4)For , one defines the mappings by

Assume that is a nonempty and bounded in and let be a sequence generated by

where , , , , for some and , and , , are real numbers in (0,1) such that .

Then, the sequence converges strongly to .

5.3. Application to Hilbert Spaces

If , a real Hilbert space, then is uniformly smooth and strictly convex real Banach space. In this case, and . Then, we obtain the following result.

Corollary 21. Let be a nonempty, closed and convex subset of a real Hilbert space . Suppose that

(1) is a finite family of bifunctions satisfying conditions (A1)–(A4), where ;(2) and are finite families of nonexpansive mappings from into , where ;(3) is a finite family of continuous monotone mappings, where ;(4)For , one defines the mappings by

Assume that is a nonempty and bounded in and let be a sequence generated by

where , , , , for some and , and are real numbers in (0,1) such that .

Then, the sequence converges strongly to an element of .

Remark 22. Our theorem extends and improves the corresponding results in [5–7] in the following aspect.

(a)For the mapping, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, and weak relatively nonexpansive mappings to more general than quasi--nonexpansive mappings.(b)For the common solution, we extend the common solution of a single finite family of quasi--nonexpansive mappings to two finite families of quasi--nonexpansive mappings.

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. The first author would like to thank the Bansomdejchaopraya Rajabhat University (BSRU) for the financial support. Finally, the authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

References

E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

A. Moudafi, “A partial complement method for approximating solutions of a primal dual fixed-point problem,” Optimization Letters, vol. 4, no. 3, pp. 449–456, 2010.

P. M. Pardalos, T. M. Rassias, and A. A. Khan, Nonlinear Analysis and Variational Problems, vol. 35 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2010.

W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.

W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009.

X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.

H. Zegeye and N. Shahzad, “A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 1, pp. 263–272, 2011.

S.-y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

S. Plubtieng and K. Ungchittrakool, “Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, Article ID 583082, 19 pages, 2008.

Y. Su, H.-k. Xu, and X. Zhang, “Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 12, pp. 3890–3906, 2010.

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1990.

W. Takahashi, Nonlinear Functional Analysis, Yokohama, Yokohama, Japan, 2000.

Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996.

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.

F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 824–835, 2008.

F. Kohsaka and W. Takahashi, “Strong convergence of an iterative sequence for maximal monotone operators in a Banach space,” Abstract and Applied Analysis, vol. 2004, no. 3, pp. 239–249, 2004.

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