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Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions
Fixed Point Theory and Applications volume 2011, Article number: 941090 (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
If is a singleton, then (1.1) reduces to find such that
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., [3–7]). Recall that a mapping is said to be
(1)monotone if
(2)α-inverse-strongly monotone if there exists a constant such that
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
It is well known that (1.5) is equivalent to
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
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., [13–22]).
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
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
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
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
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:
Then, the following statements hold:
(1) is single-valued,
(2) is firmly nonexpansive, that is, for any ,
(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
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,
for all , . In a real Hilbert space, we also know that
for all and .
In the sequel, we need the following lemmas.
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
where satisfies conditions: . If  , then as .
Lemma 2.3 (see [30]).
Assume that is a sequence of nonnegative real numbers such that
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
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 ,
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
Then, for each bounded subset of ,
The following results can be found in [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
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
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
where is the mapping defined as in Lemma 1.4. For each , we define the mapping as follows:
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,
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
It follows that
Hence, is bounded and so are and . Observe that
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
It follows that
where . So, we have
which implies that
for each . Since is firmly nonexpansive for each , we have for each and that
This implies that
where . This shows that
Hence,
From (3.10), we obtain
as . So, we can conclude that
for each . Observing
it follows by (3.16) that
From (3.6) and (3.18), we have
Hence,
as .
Next, we show that is relatively norm compact as . Let be a sequence such that as . Put . From (3.20), we obtain
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
From (A2), we have
Therefore,
For each and , put . Then, we have . From (3.24), we get that
We note that , as , and is a family of monotone mappings. It follows from (3.25) that
So, by (A1), (A4) and (3.26), we have for each and that
This implies that
Letting in (3.28), it follows from (A3) that
Hence ; consequently, . Further, we see that
So, we have
In particular,
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
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
Hence, by induction, is bounded and so are and .
Next, we show that
Since and ,
Set , . So, we have from (1.14) and (4.4) that
Since satisfies the -condition and , it follows that
So, by Lemma 2.2 and (ii), we obtain
Hence,
Observe that
as . Similar to the proof of Theorem 3.1, we obtain for each that
for some and . Then, from (4.10), we have
which implies that
So, from (4.8), (i), (ii) and for each , we have
for each . Similarly, from (4.11), we have
This implies that
From (i), (ii), (4.8), and (4.14), it follows that
for each .
Next, we show that
Observing
it follows, by (4.17), that
From (4.9) and (4.20), we have
We see that
So, by (4.7), (4.21), and Lemma 2.5, we have
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
Finally, we show that as . From (1.14), we have
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
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 .
References
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
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.
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.2009.01.115
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.
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.
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.
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.2009.05.028
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.2006.06.055
Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
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)90024-6
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)90085-6
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.2008.01.045
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.2006.08.036
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.
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.2008.10.062
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.2008.03.032
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.
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/s10898-009-9428-9
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.
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.2009.03.033
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.2007.09.009
Zhou HY: Convergence theorems for -strict pseudo-contractions in -uniformly smooth Banach spaces. Acta Mathematica Sinica 2010,26(4):743–758. 10.1007/s10114-010-7341-2
Moudafi A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. Journal of Nonlinear and Convex Analysis 2008,9(1):37–43.
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.2008.02.042
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.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.
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.54.4.1041
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.
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.2004.11.017
Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society 2002,66(1):240–256. 10.1112/S0024610702003332
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-z
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.2006.08.032
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.2009.09.007
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.2010.01.027
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.
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.
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Cholamjiak, P., Suantai, S. Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions. Fixed Point Theory Appl 2011, 941090 (2011). https://doi.org/10.1155/2011/941090
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DOI: https://doi.org/10.1155/2011/941090