General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions
© Wen and Chen; licensee Springer 2012
Received: 18 December 2011
Accepted: 11 July 2012
Published: 27 July 2012
In this paper, we modify the general iterative method to approximate a common element of the set of solutions of generalized equilibrium problems and the set of common fixed points of a finite family of k-strictly pseudo-contractive nonself mappings. Strong convergence theorems are established under some suitable conditions in a real Hilbert space, which also solves some variation inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.
MSC:47H05, 47H09, 47H10.
The generalized equilibrium problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, mini-max problems, the Nash equilibrium problem in noncooperative games and others (see, e.g., [1–3]).
T is said to be pseudo-contractive if , and is also said to be strongly pseudo-contractive if there exists a positive constant such that is pseudo-contractive. Clearly, the class of k-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions. We remark also that the class of strongly pseudo-contractive mappings is independent of the class of k-strict pseudo-contractions (see, e.g., [4, 5]).
Iterative methods for equilibrium problems and fixed point problems of nonexpansive mappings have been extensively investigated. However, iterative schemes for strict pseudo-contractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn  initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1.3) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudo-contraction. On the other hand, strict pseudo-contractions have more powerful applications than nonexpansive mappings do in solving inverse problems; see, e.g., [6–18, 20–27] and the references therein. Therefore it is interesting to develop the effective iterative methods for equilibrium problems and fixed point problems of strict pseudo-contractions.
and is also the optimality condition for some minimization problem.
Recently, Takahashi and Takahashi  considered the equilibrium problem and nonexpansive mapping by viscosity approximation methods. To be more precise, they proved the following theorem.
Then and converge strongly to , where .
In 2009, Ceng et al.  further studied the equilibrium problem and fixed point problems of strict pseudo-contraction mappings T by an iterative scheme for finding an element of . Very recently, by using the general iterative method Liu  proposed the implicit and explicit iterative processes for finding an element of and then obtained some strong convergence theorems, respectively. On the other hand, Takahashi and Takahashi  considered the generalized equilibrium problem and nonexpansive mapping in a Hilbert space. Moreover, they constructed an iterative scheme for finding an element of and then proved a strong convergence of the iterative sequence under some suitable conditions.
where constant , f is a contraction and A, B are two operators, is a finite family of -strict pseudo-contractions, is a finite sequence of positive numbers, , and are some sequences with certain conditions.
Our purpose is not only to modify the general iterative method to the case of a finite family of -strictly pseudo-contractive nonself mappings, but also to establish strong convergence theorems for a generalized equilibrium problem and -strict pseudo-contractions in a real Hilbert space, which also solves some variation inequality problems. Our theorems presented in this paper improve and extend the corresponding results of [12, 15, 16, 18, 20, 21, 25].
To study the generalized equilibrium problem (1.1), we may assume that the bi-function satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semi-continuous.
In order to prove our main results, we need the following lemmas and propositions.
is firmly nonexpansive, i.e, for all ;
is closed and convex.
Lemma 2.2 
, , .
Lemma 2.3 
Assume that B is a strongly positive linear bounded operator on the Hilbert space H with a coefficient and . Then .
Lemma 2.4 
If is a k-strict pseudo-contraction, then the fixed point set is closed convex so that the projection is well defined.
Let be a k-strict pseudo-contraction. For , define by for each . Then S is a nonexpansive mapping such that .
Lemma 2.6 
Proposition 2.1 (See, e.g., Acedo and Xu )
Let K be a nonempty closed convex subset of the Hilbert space H. Given an integer , assume that is a finite family of -strict pseudo-contractions. Suppose that is a positive sequence such that . Then is a k-strict pseudo-contraction with .
Proposition 2.2 (See, e.g., Acedo and Xu )
Let and be given as in Proposition 2.1 above. Then .
3 Main results
, and ;
, and ;
Proof Putting , we have is a k-strict pseudo-contraction and by Proposition 2.1 and 2.2, where .
It follows from the condition that the mapping is nonexpansive. From Lemma 2.1, we see that . Note that can be rewritten as and for each as .
which gives that sequence is bounded, and so are and .
which shows that is nonexpansive.
From , condition (i) and (3.23), we can arrive at the desired conclusion by Lemma 2.6. This completes the proof. □
, and ;
, and .
Proof Putting and , i.e., , the desired conclusion follows immediately from Theorem 3.1. This completes the proof. □
, and ;
, and ;
Proof Putting and , i.e., the generalized equilibrium problem (1.1) reduces to the normal equilibrium problem (1.2), the desired conclusion follows immediately from Theorem 3.1. This completes the proof. □
Remark 3.3 If and , then the algorithm (1.4) reduces to approximate the fixed point of k-strict pseudo-contractions, which includes the general iterative method of Marino and Xu  and the parallel algorithm of Acedo and Xu  as special cases.
Supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Changging (CSTC 2012jjA00039) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 110701).
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
- Moudafi A, Thera M: Proximal and dynamical approaches to equilibrium problems. Lecture Notes in Economics and Mathematical Systems 477. In Ill-posed Variational Problems and Regularization Techniques. Springer, Berlin; 1999:187–201.View ArticleGoogle Scholar
- Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
- Browder FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 1967, 24: 82–90.MathSciNetView ArticleGoogle Scholar
- Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleGoogle Scholar
- Scherzer O: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 1991, 194: 911–933.MathSciNetView ArticleGoogle Scholar
- Plubtieng S, Punpaeng R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 336: 455–469. 10.1016/j.jmaa.2007.02.044MathSciNetView ArticleGoogle Scholar
- Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleGoogle Scholar
- Ceng LC, Yao JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl. Math. Comput. 2008, 198: 729–741. 10.1016/j.amc.2007.09.011MathSciNetView ArticleGoogle Scholar
- Zhou H: Convergence theorems of fixed points for k -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2008, 69: 456–462. 10.1016/j.na.2007.05.032MathSciNetView ArticleGoogle Scholar
- Marino G, Xu HK: Weak and strong convergence theorems for k -strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336–349. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleGoogle Scholar
- Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleGoogle Scholar
- Plubtieng S, Punpaeng R: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Appl. Math. Comput. 2008, 197: 548–558. 10.1016/j.amc.2007.07.075MathSciNetView ArticleGoogle Scholar
- Chang SS, Joseph Lee HW, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleGoogle Scholar
- Ceng LC, Al-Homidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009, 223: 967–974. 10.1016/j.cam.2008.03.032MathSciNetView ArticleGoogle Scholar
- Liu Y: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2009, 71: 4852–4861. 10.1016/j.na.2009.03.060MathSciNetView ArticleGoogle Scholar
- Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleGoogle Scholar
- Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042MathSciNetView ArticleGoogle Scholar
- Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar
- Acedo GL, Xu HK: Iteration methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036MathSciNetView ArticleGoogle Scholar
- Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 72: 99–112. 10.1016/j.na.2009.06.042MathSciNetView ArticleGoogle Scholar
- Wen DJ: Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Anal. 2010, 73: 2292–2297. 10.1016/j.na.2010.06.010MathSciNetView ArticleGoogle Scholar
- Chang SS, Chan CK, Joseph Lee HW, Yang L: A system of mixed equilibrium problems, fixed point problems of strictly pseudo-contractive mappings and nonexpansive semi-groups. Appl. Math. Comput. 2010, 216: 51–60. 10.1016/j.amc.2009.12.060MathSciNetView ArticleGoogle Scholar
- Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010., 2010: Article ID 827082Google Scholar
- Wen DJ: Strong convergence theorems for equilibrium problems and k -strict pseudocontractions in Hilbert spaces. Abstr. Appl. Anal. 2011. doi:10.1155/2011/276874Google Scholar
- Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleGoogle Scholar
- Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.