Advertisement

Fixed Point Theory and Applications

, 2009:369215 | Cite as

A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces

  • Rabian Wangkeeree
  • Uthai Kamraksa
Open Access
Research Article

Abstract

We introduce an iterative scheme for finding a common element of the set of common fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. As applications, at the end of the paper we utilize our results to study the problem of finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of Open image in new window -strictly pseudocontractive mappings. The results presented in the paper improve some recent results of Qin and Cho (2008).

Keywords

Hilbert Space Variational Inequality Monotone Mapping Nonexpansive Mapping Iterative Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Throughout this paper, we always assume that Open image in new window is a real Hilbert space with inner product Open image in new window and norm Open image in new window , respectively, Open image in new window is a nonempty closed convex subset of Open image in new window , and Open image in new window is the metric projection of Open image in new window onto Open image in new window . In the following, we denote by Open image in new window strong convergence and by Open image in new window weak convergence. Recall that a mapping Open image in new window is called nonexpansive if

We denote by Open image in new window the set of fixed points of Open image in new window . Recall that a mapping Open image in new window is said to be

(i)monotone if Open image in new window , for all Open image in new window ;

(ii) Open image in new window -Lipschitz if there exists a constant Open image in new window such that Open image in new window , for all Open image in new window ;

(iii) Open image in new window -inverse-strongly monotone [1, 2] if there exists a positive real number Open image in new window such that

Remark 1.1.

It is obvious that any Open image in new window -inverse-strongly monotone mapping Open image in new window is monotone and Open image in new window -Lipschitz continuous.

Let Open image in new window be a mapping. The classical variational inequality problem is to find a Open image in new window such that

The set of solutions of variational inequality (1.3) is denoted by Open image in new window . The variational inequality has been extensively studied in the literature; see, for example, [3, 4] and the references therein.

A self-mapping Open image in new window is a contraction if there exists a constant Open image in new window such that
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [5, 6, 7, 8] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space:
where Open image in new window is a linear bounded operator, Open image in new window is the fixed point set of a nonexpansive mapping Open image in new window , and Open image in new window is a given point in Open image in new window . Let Open image in new window be a real Hilbert space. Recall that a linear bounded operator Open image in new window is strongly positive if there is a constant Open image in new window with property
Recently, Marino and Xu [9] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [10]:
where Open image in new window is a strongly positive bounded linear operator on Open image in new window . They proved that if the sequence Open image in new window of parameters satisfies appropriate conditions, then the sequence Open image in new window generated by (1.7) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem

where Open image in new window is a potential function for Open image in new window (i.e., Open image in new window for Open image in new window ).

On the other hand, two classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Mann [11] and is defined as follows:

where the sequence Open image in new window is in the interval Open image in new window .

The second iteration process is referred to as Ishikawa's iteration process [12] which is defined recursively by
where Open image in new window and Open image in new window are sequences in the interval Open image in new window . However, both (1.16) and (1.11) have only weak convergence in general (see [13], e.g.). Very recently, Qin and Cho [14] introduced a composite iterative algorithm Open image in new window defined as follows:

where Open image in new window is a contraction, Open image in new window is a nonexpansive mapping, and Open image in new window is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters, Open image in new window defined by (1.12) converges strongly to a fixed point of Open image in new window , which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

On the other hand, for finding an element of Open image in new window , under the assumption that a set Open image in new window is nonempty, closed, and convex, a mapping Open image in new window is nonexpansive and a mapping Open image in new window is Open image in new window -inverse-strongly monotone, Takahashi and Toyoda [15] introduced the following iterative scheme:
where Open image in new window is a sequence in Open image in new window , and Open image in new window is a sequence in Open image in new window . They proved that if Open image in new window , then the sequence Open image in new window generated by (1.13) converges weakly to some Open image in new window . Recently, Iiduka and Takahashi [16] proposed another iterative scheme as follows

where Open image in new window is an Open image in new window -inverse strongly monotone mapping, Open image in new window and Open image in new window satisfy some parameters controlling conditions. They showed that if Open image in new window is nonempty, then the sequence Open image in new window generated by (1.14) converges strongly to some Open image in new window .

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [17, 18, 19, 20] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [21, 22]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [18, 23]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [23, 24]).

In this paper, we study the mapping Open image in new window defined by

where Open image in new window is a nonnegative real sequence with Open image in new window , for all Open image in new window , Open image in new window , form a family of infinitely nonexpansive mappings of Open image in new window into itself. Nonexpansivity of each Open image in new window ensures the nonexpansivity of Open image in new window . Such a Open image in new window is nonexpansive from Open image in new window to Open image in new window and it is called a Open image in new window -mapping generated by Open image in new window and Open image in new window .

In this paper, motivated and inspired by Su et al. [25], Marino and Xu [9], Takahashi and Toyoda [15], and Iiduka and Takahashi [16], we will introduce a new iterative scheme:

where Open image in new window is a mapping defined by (1.15), Open image in new window is a contraction, Open image in new window is strongly positive linear bounded self-adjoint operator, Open image in new window is a Open image in new window -inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences Open image in new window , Open image in new window , Open image in new window , and Open image in new window , the sequences Open image in new window defined by (1.16) converge strongly to a common element of the set of common fixed points of a family of Open image in new window and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

2. Preliminaries

Let Open image in new window be a real Hilbert space. It is well known that for any Open image in new window
Let Open image in new window be a nonempty closed convex subset of Open image in new window . For every point Open image in new window , there exists a unique nearest point in Open image in new window , denoted by Open image in new window , such that
Open image in new window is called the metric projection of Open image in new window onto Open image in new window . It is well known that Open image in new window is a nonexpansive mapping of Open image in new window onto Open image in new window and satisfies
for every Open image in new window Moreover, Open image in new window is characterized by the following properties: Open image in new window and
for all Open image in new window . It is easy to see that the following is true:
A Banach space Open image in new window is said to satisfy the Opial's condition if for each sequence Open image in new window in Open image in new window which converges weakly to a point Open image in new window we have

It is well known that each Hilbert space satisfies the Opial's condition.

A set-valued mapping Open image in new window is called monotone if for all Open image in new window , Open image in new window and Open image in new window imply Open image in new window . A monotone mapping Open image in new window is maximal if the graph of Open image in new window of Open image in new window is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping Open image in new window is maximal if and only if for Open image in new window , Open image in new window for every Open image in new window implies Open image in new window . Let Open image in new window be a monotone map of Open image in new window into Open image in new window and let Open image in new window be the normal cone to Open image in new window at Open image in new window , that is, Open image in new window and define

Then Open image in new window is the maximal monotone and Open image in new window if and only if Open image in new window ; see [26].

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1.

In a Hilbert space Open image in new window . Then the following inequality holds

Lemma 2.2 (see [27]).

Let Open image in new window and Open image in new window be bounded sequences in a Banach space Open image in new window and let Open image in new window be a sequence in Open image in new window with Open image in new window Suppose Open image in new window for all integers Open image in new window and Open image in new window Then, Open image in new window

Lemma 2.3 (see [28]).

Assume Open image in new window is a sequence of nonnegative real numbers such that

where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window such that

(1) Open image in new window

(2) Open image in new window or Open image in new window

Then Open image in new window

Lemma 2.4 (see [9]).

Assume that Open image in new window is a strongly positive linear bounded self-adjoint operator on a Hilbert space Open image in new window with coefficient Open image in new window and Open image in new window . Then Open image in new window .

Throughout this paper, we will assume that Open image in new window , for all Open image in new window . Concerning Open image in new window defined by (1.15), we have the following lemmas which are important to prove our main result.

Lemma 2.5 (see [29]).

Let Open image in new window be a nonempty closed convex subset of a Hilbert space Open image in new window , let Open image in new window be a family of infinitely nonexpansive mapping with Open image in new window , and let Open image in new window be a real sequence such that Open image in new window , for all Open image in new window . Then

(1) Open image in new window is nonexpansive and Open image in new window for each Open image in new window ;

(2)for each Open image in new window and for each positive integer Open image in new window , the limit Open image in new window exists;

(3)the mapping Open image in new window define by

is a nonexpansive mapping satisfying Open image in new window and it is called the Open image in new window -mapping generated by Open image in new window and Open image in new window

Lemma 2.6 (see [30]).

Let Open image in new window be a nonempty closed convex subset of a Hilbert space Open image in new window , let Open image in new window be a family of infinitely nonexpansive mappings with Open image in new window , and let Open image in new window be a real sequence such that Open image in new window , for all Open image in new window . If Open image in new window is any bounded subset of Open image in new window , then

3. Main Results

Now we are in a position to state and prove the main result in this paper.

Theorem 3.1.

Let Open image in new window be a closed convex subset of a real Hilbert space Open image in new window , let Open image in new window be a contraction of Open image in new window into itself, let Open image in new window be an Open image in new window -inverse strongly monotone mapping of Open image in new window into Open image in new window , and let Open image in new window be a family of infinitely nonexpansive mappings with Open image in new window . Let Open image in new window be a strongly positive linear bounded self-adjoint operator with the coefficient Open image in new window such that Open image in new window . Assume that Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be sequences in Open image in new window satisfying the following conditions:

(C1) Open image in new window

(C2) Open image in new window

(C3) Open image in new window

(C4) Open image in new window

(C5) Open image in new window .

Then the sequence Open image in new window defined by (1.16) converges strongly to Open image in new window , where Open image in new window which solves the following variational inequality:

Proof.

Since Open image in new window as Open image in new window by the condition (C1), we may assume, without loss of generality that Open image in new window for all Open image in new window . First, we will show that Open image in new window is nonexpansive. Indeed, for all Open image in new window and Open image in new window ,
which implies that Open image in new window is nonexpansive. Noticing that Open image in new window is a linear bounded self-adjoint operator, one has
Observing that
we obtain Open image in new window is positive. It follows that
Next, we observe that Open image in new window is bounded. Indeed, pick Open image in new window and notice that
It follows that
By simple induction, we have

which gives that the sequence Open image in new window is bounded, and so are Open image in new window and Open image in new window .

Next, we claim that

where Open image in new window is a constant such that Open image in new window . Similarly, there exists Open image in new window such that Open image in new window .

Observing that
we obtain that
It follows that
Noticing that
we obtain
It follows that
Substituting (3.11) into (3.14), we get
where Open image in new window is an appropriate constant such that

Putting Open image in new window , we get, Open image in new window .

Now, we compute Open image in new window . Observing that
It follows from (3.15) that
It follows that
Observing the conditions (C1) and (C4) and taking the superior limit as Open image in new window , we get
We can obtain Open image in new window easily by Lemma 2.2 since
one obtains that (3.7) holds. Setting Open image in new window , we have
Observing that
we arrive at
This implies
From (3.7) and (C1) we obtain that
This impies that
Since Open image in new window and from (3.7), we obtain
From (2.3), we have
so, we obtain
It follows that
which implies that
Applying (3.7), (3.30), and Open image in new window to the last inequality, we obtain that
It follows from (3.26) and (3.35) that
On the other hand, one has
which implies
From the conditions (C3), it follows that
Applying Lemma 2.6 and (3.39), we obtain that
It follows from (3.26) and (3.40) that
We observe that Open image in new window is a contraction. Indeed, for all Open image in new window , we have

Banach's Contraction Mapping Principle guarantees that Open image in new window has a unique fixed point, say Open image in new window . That is, Open image in new window .

Next, we claim that
Indeed, we choose a subsequence Open image in new window of Open image in new window such that
Since Open image in new window is bounded, there exists a subsequence Open image in new window of Open image in new window which converges weakly to Open image in new window . Without loss of generality, we can assume that Open image in new window . From Open image in new window we obtain Open image in new window Therefore, we have

Next we prove that Open image in new window .

First, we prove that Open image in new window .

Suppose the contrary, Open image in new window , that is, Open image in new window . Since Open image in new window , by the Opial's condition and (3.41), we have

This is a contradiction, which shows that Open image in new window .

Next, we prove Open image in new window . For this purpose, let Open image in new window be the maximal monotone mapping defined by (2.7):
On the other hand, from Open image in new window , we have
that is,
Therefore, we obtian
Noting that Open image in new window as Open image in new window and Open image in new window is Lipschitz continuous, hence from (3.18), we obtain

Since Open image in new window is maximal monotone, we have Open image in new window , and hence Open image in new window .

The conclusion Open image in new window is proved.

Hence by (3.45), we obtain
Since Open image in new window , it follows from (3.39), (3.41), and (3.53) that
Hence (3.43) holds. Using (3.26) and (3.54), we have
Now, from Lemma 2.1, it follows that
for all Open image in new window . It then follows that

Using (C1), (3.54), and (3.55), we get Open image in new window . Now applying Lemma 2.3 to (3.58), we conclude that Open image in new window . This completes the proof.

Remark 3.2.

Theorem 3.1 mainly improve the results of Qin and Cho [14] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.

4. Applications

In this section, we obtain two results by using a special case of the proposed method.

Theorem 4.1.

Let Open image in new window be a real Hilbert space, let Open image in new window be an Open image in new window -inverse strongly monotone mapping on Open image in new window , let Open image in new window be a family of infinitely nonexpansive mappings with Open image in new window . Let Open image in new window a contraction with coefficient Open image in new window , and let Open image in new window be a strongly positive bounded linear operator on Open image in new window with coefficient Open image in new window and Open image in new window . Suppose the sequences Open image in new window , Open image in new window , and Open image in new window be generated by

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window are sequences in Open image in new window satisfying the following conditions:

(C1) Open image in new window

(C2) Open image in new window

(C3) Open image in new window

(C4) Open image in new window

(C5) Open image in new window .

Then Open image in new window , Open image in new window , and Open image in new window converge strongly to Open image in new window which solves the variational inequality:

Proof.

We have Open image in new window and Open image in new window . Applying Theorem 3.1, we obtain the desired result.

Next, we will apply the main results to the problem for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of Open image in new window -strictly pseudocontractive mappings.

Definition 4.2.

A mappings Open image in new window is said to be a Open image in new window -strictly pseudocontractive mapping if there exists Open image in new window such that

The following lemmas can be obtained from [31, Proposition 2.6] by Acedo and Xu, easily.

Lemma 4.3.

Let Open image in new window be a Hilbert space, let Open image in new window be a closed convex subset of Open image in new window . For any integer Open image in new window , assume that, for each Open image in new window is a Open image in new window -strictly pseudocontractive mapping for some Open image in new window . Assume that Open image in new window is a positive sequence such that Open image in new window . Then Open image in new window is a Open image in new window -strictly pseudocontractive mapping with Open image in new window .

Lemma 4.4.

Let Open image in new window and Open image in new window be as in Lemma 4.3. Suppose that Open image in new window has a common fixed point in Open image in new window . Then Open image in new window .

Let Open image in new window be a Open image in new window -strictly pseudocontractive mapping for some Open image in new window . We define a mapping Open image in new window , where Open image in new window is a positive sequence such that Open image in new window , then Open image in new window is a Open image in new window -inverse-strongly monotone mapping with Open image in new window . In fact, from Lemma 4.3, we have
On the other hand
Hence, we have

This shows that Open image in new window is Open image in new window -inverse-strongly monotone.

Theorem 4.5.

Let Open image in new window be a closed convex subset of a real Hilbert space Open image in new window . For any integer Open image in new window , assume that, for each Open image in new window is a Open image in new window -strictly pseudocontractive mapping for some Open image in new window . Let Open image in new window be a family of infinitely nonexpansive mappings with Open image in new window . Let Open image in new window a contraction with coefficient Open image in new window and let Open image in new window be a strongly positive bounded linear operator with coefficient Open image in new window and Open image in new window . Let the sequences Open image in new window , Open image in new window , and Open image in new window be generated by

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window are the sequences in Open image in new window satisfying the following conditions:

(C1) Open image in new window

(C2) Open image in new window

(C3) Open image in new window

(C4) Open image in new window

(C5) Open image in new window

Then Open image in new window , Open image in new window , and Open image in new window converge strongly to Open image in new window which solves the variational inequality:

Proof.

Taking Open image in new window , we know that Open image in new window is Open image in new window -inverse strongly monotone with Open image in new window . Hence, Open image in new window is a monotone Open image in new window -Lipschitz continuous mapping with Open image in new window . From Lemma 4.4, we know that Open image in new window is a Open image in new window -strictly pseudocontractive mapping with Open image in new window and then Open image in new window by Chang [30, Proposition 1.3.5]. Observe that

The conclusion of Theorem 4.5 can be obtained from Theorem 3.1.

Remark 4.6.

Theorem 4.5 is a generalization and improvement of the theorems by Qin and Cho [14], Iiduka and Takahashi [16, Thorem 3.1], and Takahashi and Toyoda [15].

Notes

Acknowledgments

The authors would like to thank the referee for the comments which improve the manuscript. R. Wangkeeree was supported for CHE-PhD-THA-SUP/2551 from the Commission on Higher Education and the Thailand Research Fund under Grant TRG5280011.

References

  1. 1.
    Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis 1998,6(4):313–344. 10.1023/A:1008643727926MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Yao J-C, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications. Volume 76. Springer, New York, NY, USA; 2005:501–558.CrossRefGoogle Scholar
  4. 4.
    Zeng LC, Schaible S, Yao JC: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s10957-004-1182-zMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numerical Functional Analysis and Optimization 1998,19(1–2):33–56.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society. Second Series 2002,66(1):240–256. 10.1112/S0024610702003332MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications, Studies in Computational Mathematics. Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.Google Scholar
  9. 9.
    Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Qin X, Cho Y: Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Computational and Applied Mathematics. In pressGoogle Scholar
  15. 15.
    Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2005,61(3):341–350. 10.1016/j.na.2003.07.023MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(8):2350–2360. 10.1016/j.na.2006.08.032MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996,202(1):150–159. 10.1006/jmaa.1996.0308MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Shang M, Su Y, Qin X: Strong convergence theorems for a finite family of nonexpansive mappings. Fixed Point Theory and Applications 2007, Article ID 76971 2007:-9.Google Scholar
  20. 20.
    Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001,5(2):387–404.MathSciNetMATHGoogle Scholar
  21. 21.
    Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Review 1996,38(3):367–426. 10.1137/S0036144593251710MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Combettes PL: The foundations of set theoretic estimation. Proceedings of the IEEE 1993,81(2):182–208.CrossRefGoogle Scholar
  23. 23.
    Youla DC: Mathematical theory of image restoration by the method of convex projections. In Image Recovery: Theory and Application. Edited by: Star H. Academic Press, Orlando, Fla, USA; 1987:29–77.Google Scholar
  24. 24.
    Iusem AN, De Pierro AR: On the convergence of Han's method for convex programming with quadratic objective. Mathematical Programming Series B 1991,52(1–3):265–284.MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Su Y, Shang M, Qin X: A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings. Journal of Applied Mathematics and Computing 2008,28(1–2):283–294. 10.1007/s12190-008-0103-yMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970,149(1):75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Chang SS: Variational Inequalities and Related Problems. Chongqing Publishing House, Chongqing, China; 2007.Google Scholar
  31. 31.
    Acedo GL, Xu H-K: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2007,67(7):2258–2271. 10.1016/j.na.2006.08.036MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© R.Wangkeeree and U. Kamraksa. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand

Personalised recommendations