Convergence Theorems Concerning Hybrid Methods for Strict Pseudocontractions and Systems of Equilibrium Problems
© Peichao Duan. 2010
Received: 23 May 2010
Accepted: 26 August 2010
Published: 30 August 2010
Let be strict pseudo-contractions defined on a closed and convex subset of a real Hilbert space . We consider the problem of finding a common element of fixed point set of these mappings and the solution set of a system of equilibrium problems by parallel and cyclic algorithms. In this paper, new iterative schemes are proposed for solving this problem. Furthermore, we prove that these schemes converge strongly by hybrid methods. The results presented in this paper improve and extend some well-known results in the literature.
for all ; see . We denote the fixed point set of by , that is, .
The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance, [1, 3, 4] and the references therein. Some methods have been proposed to solve the equilibrium problem (1.1); related work can also be found in [5–8].
Recently, Acedo and Xu  considered the problem of finding a common fixed point of a finite family of strict pseudo-contractive mappings by the parallel and cyclic algorithms. Very recently, Duan and Zhao  considered new hybrid methods for equilibrium problems and strict pseudocontractions. In this paper, motivated by [5, 8–12], applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the fixed point set of a finite family of strict pseudocontractions and the solution set of the system of equilibrium problems (1.1) by the hybrid methods.
We will use the following notations:
Lemma 2.2 (see ).
is convex (and closed).
Lemma 2.3 (see ).
Proposition 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Lemma 2.7 (see ).
3. Parallel Algorithm
In this section, we apply the hybrid methods to the parallel algorithm for finding a common element of the fixed point set of strict pseudocontractions and the solution set of the problem (1.1) in Hilbert spaces.
If , , and , we can obtain [14, Theorem ].
The proof of this theorem is similar to that of Theorem 3.1.
The sequence is well defined. We will show by induction that is closed and convex for all . For , we have which is closed and convex. Assume that for some is closed and convex; from Lemma 2.2, we have that is also closed and convex; The proof of is similar to the one in Step 1 of Theorem 3.1.
The proof of Step 2–Step 6 is similar to that of Theorem 3.1.
If , we can obtain the two corresponding theorems in .
4. Cyclic Algorithm
The proof of this theorem is similar to that of Theorem 3.1. The main points are the following.
The proof of this theorem is similar to that of Step 1 in Theorem 3.5 and Step 2–Step 6 in Theorem 4.1.
If , we can obtain the two corresponding theorems in .
This research is supported by Fundamental Research Funds for the Central Universities (GRANT:ZXH2009D021).
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005, 6(1):117–136.MathSciNetMATHGoogle Scholar
- 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)90085-6MathSciNetView ArticleMATHGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994, 63(1–4):123–145.MathSciNetMATHGoogle Scholar
- Colao V, Marino G, Xu HK: An iterative method for finding common solutions of equilibrium and fixed point problems. Journal of Mathematical Analysis and Applications 2008, 344(1):340–352. 10.1016/j.jmaa.2008.02.041MathSciNetView ArticleMATHGoogle Scholar
- Liu Y: A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis. Theory, Methods & Applications 2009, 71(10):4852–4861. 10.1016/j.na.2009.03.060MathSciNetView ArticleMATHGoogle Scholar
- Marino G, Xu HK: 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.2006.06.055MathSciNetView ArticleMATHGoogle Scholar
- Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003, 279(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar
- Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. Journal of Optimization Theory and Applications 2007, 133(3):359–370. 10.1007/s10957-007-9187-zMathSciNetView ArticleMATHGoogle Scholar
- 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.2006.08.036MathSciNetView ArticleMATHGoogle Scholar
- Duan PC, Zhao J: Strong convergence theorems by hybrid methods for strict pseudo-contractions and equilibrium problems. Fixed Point Theory and Applications 2010, 2010:-13.Google Scholar
- Kumam P: A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping. Nonlinear Analysis. Hybrid Systems 2008, 2(4):1245–1255. 10.1016/j.nahs.2008.09.017MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008, 341(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar
- Martinez-Yanes C, Xu HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis. Theory, Methods & Applications 2006, 64(11):2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleMATHGoogle Scholar
- Yao YH, Chen RD: Strong convergence theorems for strict pseudo-contractions in Hilbert spaces. Journal of Applied Mathematics and Computing 2010, 32(1):69–82. 10.1007/s12190-009-0233-xMathSciNetView ArticleMATHGoogle Scholar
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.