Open Access

Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions

Fixed Point Theory and Applications20112011:941090

https://doi.org/10.1155/2011/941090

Received: 18 October 2010

Accepted: 27 December 2010

Published: 4 January 2011

Abstract

We introduce a new iterative algorithm for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common element in the solutions set of a system of generalized equilibrium problems and the common fixed points set of an infinitely countable family of strict pseudocontractions.

1. Introduction

Let be a real Hilbert space with the inner product and inducted norm . Let be a nonempty, closed, and convex subset of . Let be a family of bifunctions, and let be a family of nonlinear mappings, where is an arbitrary index set. The system of generalized equilibrium problems is to find such that
(1.1)
If is a singleton, then (1.1) reduces to find such that
(1.2)

The solutions set of (1.2) is denoted by . If , then the solutions set of (1.2) is denoted by , and if , then the solutions set of (1.2) is denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and the Nash equilibrium problem in noncooperative games; see also [1, 2]. Some methods have been constructed to solve the system of equilibrium problems (see, e.g., [37]). Recall that a mapping is said to be

(1)monotone if
(1.3)
(2)α-inverse-strongly monotone if there exists a constant such that
(1.4)

It is easy to see that if is α-inverse-strongly monotone, then is monotone and -Lipschitz.

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

(A1) for all ,

(A2) is monotone, that is, for all ,

(A3)for each , ,

(A4)for each , is convex and lower semicontinuous.

Throughout this paper, we denote the fixed points set of a nonlinear mapping by . Recall that is said to be a -strict pseudocontraction if there exists a constant such that
(1.5)
It is well known that (1.5) is equivalent to
(1.6)

It is worth mentioning that the class of strict pseudocontractions includes properly the class of nonexpansive mappings. It is also known that every -strict pseudocontraction is -Lipschitz; see [8].

In 1953, Mann [9] introduced the iteration as follows: a sequence defined by and
(1.7)

where . If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that , then the sequence defined by (1.7) converges weakly to a fixed point of (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [10]).

In 1967, Browder and Petryshyn [11] introduced the class of strict pseudocontractions and proved existence and weak convergence theorems in a real Hilbert setting by using Mann iterative algorithm (1.7) with a constant sequence for all . Recently, Marino and Xu [8] and Zhou [12] extended the results of Browder and Petryshyn [11] to Mann's iteration process (1.7). Since 1967, the construction of fixed points for pseudocontractions via the iterative process has been extensively investigated by many authors (see, e.g., [1322]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping, a bifunction, and let be an inverse-strongly monotone mapping.

In 2008, Moudafi [23] introduced an iterative method for approximating a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem as follows: a sequence defined by and
(1.8)

where and . He proved that the sequence generated by (1.8) converges weakly to an element in under suitable conditions.

Due to the weak convergence, recently, S. Takahashi and W. Takahashi [24] introduced another modification iterative method of (1.8) for finding a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem in the framework of a real Hilbert space. To be more precise, they proved the following theorem.

Theorem 1.1 (see [24]).

Let be a closed convex subset of a real Hilbert space , and let be a bifunction satisfying (A1)–(A4). Let be an α-inverse-strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that . Let and , and let and be sequences generated by
(1.9)

where , and satisfy

(i) and ,

(ii) ,

(iii) ,

(iv) .

Then, converges strongly to .

Recently, Yao et al. [25] introduced a new modified Mann iterative algorithm which is different from those in the literature for a nonexpansive mapping in a real Hilbert space. To be more precise, they proved the following theorem.

Theorem 1.2 (see [25]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping such that . Let , and let be two real sequences in . For given arbitrarily, let the sequence , , be generated iteratively by
(1.10)

Suppose that the following conditions are satisfied:

(i) and ,

(ii) ,

then, the sequence generated by (1.10) strongly converges to a fixed point of .

We know the following crucial lemmas concerning the equilibrium problem in Hilbert spaces.

Lemma 1.3 (see [1]).

Let be a nonempty, closed, and convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4). Let and . Then, there exists such that
(1.11)

Lemma 1.4 (see [26]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A4). For and , define the mapping as follows:
(1.12)

Then, the following statements hold:

(1) is single-valued,

(2) is firmly nonexpansive, that is, for any ,
(1.13)

(3) ,

(4) is closed and convex.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let for each . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let be a countable family of -strict pseudocontractions. For each , denote the mapping by , where is the mapping defined as in Lemma 1.4.

Motivated and inspired by Marino and Xu [8], Moudafi [23], S. Takahashi and W. Takahashi [24], and Yao et al. [25], we consider the following iteration: and
(1.14)

where , and .

In this paper, we first prove a path convergence result for a nonexpansive mapping and a system of generalized equilibrium problems. Then, we prove a strong convergence theorem of the iteration process (1.14) for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in a real Hilbert space. Our results extend the main results obtained by Yao et al. [25] in several aspects.

2. Preliminaries

Let be a nonempty, closed, and convex subset of a real Hilbert space . For each , there exists a unique nearest point in , denoted by , such that . is called the metric projection of onto . It is also known that for and , is equivalent to for all . Furthermore,
(2.1)
for all , . In a real Hilbert space, we also know that
(2.2)

for all and .

In the sequel, we need the following lemmas.

Lemma 2.1 (see [27, 28]).

Let be a real uniformly convex Banach space, and let be a nonempty, closed, and convex subset of , and let be a nonexpansive mapping such that , then is demiclosed at zero.

Lemma 2.2 (see [29]).

Let and be two sequences in a Banach space such that
(2.3)

where satisfies conditions: . If   , then as .

Lemma 2.3 (see [30]).

Assume that is a sequence of nonnegative real numbers such that
(2.4)

where is a sequence in and is a sequence in such that 

(a)   ;  (b)   or .Then, .

Lemma 2.4 (see [31]).

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let the mapping be α-inverse-strongly monotone, and let be a constant. Then, we have
(2.5)

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

To deal with a family of mappings, the following conditions are introduced: let be a subset of a real Hilbert space , and let be a family of mappings of such that . Then, is said to satisfy the -condition [32] if for each bounded subset of ,
(2.6)

Lemma 2.5 (see [32]).

Let be a nonempty and closed subset of a Hilbert space , and let be a family of mappings of into itself which satisfies the -condition. Then, for each , converges strongly to a point in . Moreover, let the mapping be defined by
(2.7)
Then, for each bounded subset of ,
(2.8)

The following results can be found in [33, 34].

Lemma 2.6 (see [33, 34]).

Let be a closed, and convex subset of a Hilbert space . Suppose that is a family of -strictly pseudocontractive mappings from into with and is a real sequence in such that . Then, the following conclusions hold:

(1) is a -strictly pseudocontractive mapping,

(2) .

Lemma 2.7 (see [34]).

Let be a closed and convex subset of a Hilbert space . Suppose that is a countable family of -strictly pseudocontractive mappings of into itself with . For each , define by
(2.9)

where is a family of nonnegative numbers satisfying

(i) for all ,

(ii) for all ,

(iii) .

Then,

(1)Each is a -strictly pseudocontractive mapping.

(2) satisfies -condition.

(3)If is defined by
(2.10)

then and .

In the sequel, we will write satisfies the -condition if satisfies the -condition and is defined by Lemma 2.5 with .

3. Path Convergence Results

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping. Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let . For each , we denote the mapping by
(3.1)
where is the mapping defined as in Lemma 1.4. For each , we define the mapping as follows:
(3.2)
By Lemmas 1.4(2) and 2.4, we know that and are nonexpansive for each . So, the mapping is also nonexpansive for each . Moreover, we can check easily that is a contraction. Then, the Banach contraction principle ensures that there exists a unique fixed point of in , that is,
(3.3)

Theorem 3.1.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a nonexpansive mapping. Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let . For each , let the mapping be defined by (3.1). Assume that . For each , let the net be generated by (3.3). Then, as , the net converges strongly to an element in .

Proof.

First, we show that is bounded. For each , let and . From (3.3), we have for each that
(3.4)
It follows that
(3.5)
Hence, is bounded and so are and . Observe that
(3.6)

as since is bounded.

Next, we show that as . Denote for any and . We note that for each . From Lemma 2.4, we have for each and that
(3.7)
It follows that
(3.8)
where . So, we have
(3.9)
which implies that
(3.10)
for each . Since is firmly nonexpansive for each , we have for each and that
(3.11)
This implies that
(3.12)
where . This shows that
(3.13)
Hence,
(3.14)
From (3.10), we obtain
(3.15)
as . So, we can conclude that
(3.16)
for each . Observing
(3.17)
it follows by (3.16) that
(3.18)
From (3.6) and (3.18), we have
(3.19)
Hence,
(3.20)

as .

Next, we show that is relatively norm compact as . Let be a sequence such that as . Put . From (3.20), we obtain
(3.21)

Since is bounded, without loss of generality, we may assume that converges weakly to . Applying Lemma 2.1 to (3.21), we can conclude that .

Next, we show that . Note that for each . Hence, for each and , we obtain
(3.22)
From (A2), we have
(3.23)
Therefore,
(3.24)
For each and , put . Then, we have . From (3.24), we get that
(3.25)
We note that , as , and is a family of monotone mappings. It follows from (3.25) that
(3.26)
So, by (A1), (A4) and (3.26), we have for each and that
(3.27)
This implies that
(3.28)
Letting in (3.28), it follows from (A3) that
(3.29)
Hence ; consequently, . Further, we see that
(3.30)
So, we have
(3.31)
In particular,
(3.32)

Since , we have as . By using the same argument as in the proof of Theorem  3.1 of [25], we can show that as . This completes the proof.

4. Strong Convergence Results

Theorem 4.1.

Let be a nonempty, closed and convex subset of a real Hilbert space . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings and let be a countable family of -strict pseudocontractions for some such that . Assume that , , and for each satisfy the following conditions:

(i) and ,

(ii) .

Suppose that satisfies the -condition. Then, generated by (1.14) converges strongly to an element in .

Proof.

For each , define by , . Then, , since . Moreover, we know that satisfies the -condition, since satisfies the -condition. By Lemma 2.5, we can define the mapping by for . Hence, , , since for . Further, we know that is nonexpansive for each . Indeed, for each and , we have
(4.1)

Hence, is nonexpansive for each and so is .

Next, we show that is bounded. Denote for any and . We note that . From (1.14), we have for each that
(4.2)

Hence, by induction, is bounded and so are and .

Next, we show that
(4.3)
Since and ,
(4.4)
Set , . So, we have from (1.14) and (4.4) that
(4.5)
Since satisfies the -condition and , it follows that
(4.6)
So, by Lemma 2.2 and (ii), we obtain
(4.7)
Hence,
(4.8)
Observe that
(4.9)
as . Similar to the proof of Theorem 3.1, we obtain for each that
(4.10)
(4.11)
for some and . Then, from (4.10), we have
(4.12)
which implies that
(4.13)
So, from (4.8), (i), (ii) and for each , we have
(4.14)
for each . Similarly, from (4.11), we have
(4.15)
This implies that
(4.16)
From (i), (ii), (4.8), and (4.14), it follows that
(4.17)

for each .

Next, we show that
(4.18)
Observing
(4.19)
it follows, by (4.17), that
(4.20)
From (4.9) and (4.20), we have
(4.21)
We see that
(4.22)
So, by (4.7), (4.21), and Lemma 2.5, we have
(4.23)
Let the net be defined by (3.3). By Theorem 3.1, we have as . Moreover, by proving in the same manner as in Theorem  3.2 of [25], we can show that
(4.24)
Finally, we show that as . From (1.14), we have
(4.25)

By (i) and (4.24), it follows from Lemma 2.3 that . This completes the proof.

As a direct consequence of Lemmas 2.6 and 2.7 and Theorem 4.1, we obtain the following result.

Theorem 4.2.

Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a family of bifunctions, let be a family of -inverse-strongly monotone mappings, and let be a sequence of -strict pseudocontractions of into itself such that and . Assume that and for each . Define the sequence by and
(4.26)

where and are real sequences in which satisfy (i)-(ii) of Theorem 4.1 and is a real sequence which satisfies (i)–(iii) of Lemma 2.7. Then, converges strongly to an element in .

Remark 4.3.

Theorems 4.1 and 4.2 extend the main results in [25] from a nonexpansive mapping to an infinite family of strict pseudocontractions and a system of generalized equilibrium problems.

Remark 4.4.

If we take and for each , then Theorems 3.1, 4.1, and 4.2 can be applied to a system of equilibrium problems and to a system of variational inequality problems, respectively.

Remark 4.5.

Let be an infinite family of nonexpansive mappings of into itself, and let be real numbers such that for all . Moreover, let and be the -mappings [35] generated by and and and . Then, we know from [7, 35] that satisfies the -condition. Therefore, in Theorem 4.1, the mapping can be also replaced by .

Declarations

Acknowledgments

The authors would like to thank the referees for valuable suggestions. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, and the Thailand Research Fund. The first author is supported by the Royal Golden Jubilee Grant no. PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University
(2)
Centre of Excellence in Mathematics, CHE

References

  1. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.MATHMathSciNetGoogle Scholar
  2. Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Mathematical Systems. Volume 477. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar
  3. Colao V, Acedo GL, Marino G: An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):2708–2715. 10.1016/j.na.2009.01.115MATHMathSciNetView ArticleGoogle Scholar
  4. Cholamjiak P, Suantai S: Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in Banach spaces. Abstract and Applied Analysis 2010, 2010:-17.Google Scholar
  5. Jaiboon C: The hybrid steepest descent method for addressing fixed point problems and system of equilibrium problems. Thai Journal of Mathematics 2010, 8: 275–292.MATHMathSciNetGoogle Scholar
  6. Jitpeera T, Kumam P: An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings. Journal of Nonlinear Analysis and Optimization 2010, 1: 71–91.MathSciNetGoogle Scholar
  7. Peng J-W, Yao J-C: A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):6001–6010. 10.1016/j.na.2009.05.028MATHMathSciNetView ArticleGoogle Scholar
  8. Marino G, Xu H-K: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,329(1):336–346. 10.1016/j.jmaa.2006.06.055MATHMathSciNetView ArticleGoogle Scholar
  9. Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MATHMathSciNetView ArticleGoogle Scholar
  10. Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 1979,67(2):274–276. 10.1016/0022-247X(79)90024-6MATHMathSciNetView ArticleGoogle Scholar
  11. 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)90085-6MATHMathSciNetView ArticleGoogle Scholar
  12. Zhou H: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,343(1):546–556. 10.1016/j.jmaa.2008.01.045MATHMathSciNetView ArticleGoogle Scholar
  13. Acedo GL, Xu HK: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2007,67(7):2258–2271. 10.1016/j.na.2006.08.036MATHMathSciNetView ArticleGoogle Scholar
  14. Ceng LC, Shyu DS, Yao JC: Relaxed composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive mappings. Fixed Point Theory and Applications 2009, 2009:-16.Google Scholar
  15. Ceng L-C, Petruşel A, Yao J-C: Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings. Applied Mathematics and Computation 2009,209(2):162–176. 10.1016/j.amc.2008.10.062MATHMathSciNetView ArticleGoogle Scholar
  16. Ceng L-C, Al-Homidan S, Ansari QH, Yao J-C: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. Journal of Computational and Applied Mathematics 2009,223(2):967–974. 10.1016/j.cam.2008.03.032MATHMathSciNetView ArticleGoogle Scholar
  17. Cholamjiak P, Suantai S: Weak convergence theorems for a countable family of strict pseudocontractions in banach spaces. Fixed Point Theory and Applications 2010, 2010:-16.Google Scholar
  18. Peng J-W, Yao J-C: Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping. Journal of Global Optimization 2010,46(3):331–345. 10.1007/s10898-009-9428-9MATHMathSciNetView ArticleGoogle Scholar
  19. Zhang Y, Guo Y: Weak convergence theorems of three iterative methods for strictly pseudocontractive mappings of Browder-Petryshyn type. Fixed Point Theory and Applications 2008, 2008:-13.Google Scholar
  20. Zhang H, Su Y: Convergence theorems for strict pseudo-contractions in -uniformly smooth Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(10):4572–4580. 10.1016/j.na.2009.03.033MATHMathSciNetView ArticleGoogle Scholar
  21. Zhou H: Convergence theorems for -strict pseudo-contractions in 2-uniformly smooth Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(9):3160–3173. 10.1016/j.na.2007.09.009MATHMathSciNetView ArticleGoogle Scholar
  22. Zhou HY: Convergence theorems for -strict pseudo-contractions in -uniformly smooth Banach spaces. Acta Mathematica Sinica 2010,26(4):743–758. 10.1007/s10114-010-7341-2MATHMathSciNetView ArticleGoogle Scholar
  23. Moudafi A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. Journal of Nonlinear and Convex Analysis 2008,9(1):37–43.MATHMathSciNetGoogle Scholar
  24. Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications 2008,69(3):1025–1033. 10.1016/j.na.2008.02.042MATHMathSciNetView ArticleGoogle Scholar
  25. Yao Y, Liou YC, Marino G: Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces. Fixed Point Theory and Applications 2009, 2009:-7.Google Scholar
  26. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.MATHMathSciNetGoogle Scholar
  27. Browder FE: Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041MATHMathSciNetView ArticleGoogle Scholar
  28. Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar
  29. 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.2004.11.017MATHMathSciNetView ArticleGoogle Scholar
  30. Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar
  31. Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2006,128(1):191–201. 10.1007/s10957-005-7564-zMATHMathSciNetView ArticleGoogle Scholar
  32. 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.2006.08.032MATHMathSciNetView ArticleGoogle Scholar
  33. Boonchari D, Saejung S: Weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings. Journal of Computational and Applied Mathematics 2009,233(4):1108–1116. 10.1016/j.cam.2009.09.007MATHMathSciNetView ArticleGoogle Scholar
  34. Boonchari D, Saejung S: Construction of common fixed points of a countable family of -demicontractive mappings in arbitrary Banach spaces. Applied Mathematics and Computation 2010,216(1):173–178. 10.1016/j.amc.2010.01.027MATHMathSciNetView ArticleGoogle Scholar
  35. 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.MATHMathSciNetGoogle Scholar

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© P. Cholamjiak and S. Suantai. 2011

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