- Open Access
Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings
© Thanh; licensee Springer. 2014
- Received: 7 May 2014
- Accepted: 10 September 2014
- Published: 25 September 2014
We give new hybrid variants of extragradient methods for finding a common solution of an equilibrium problem and a family of nonexpansive mappings. We present a scheme that combines the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this scheme is modified by projecting on a suitable convex set to get a better convergence property under certain assumptions in a real Hilbert space.
MSC:65K10, 65K15, 90C25, 90C33.
- equilibrium problems
- fixed point
- Lipschitz-type continuous
- extragradient method
- nonexpansive mappings
In the case , the mapping T is called nonexpansive on C. We denote by the set of fixed points of T.
where the function f and the mappings , , satisfy the following conditions:
(A1) for all and f is pseudomonotone on C,
(A2) f is Lipschitz-type continuous on C with constants and ,
(A3) f is upper semicontinuous on C,
(A4) For each , is convex and subdifferentiable on C,
Under certain conditions onto parameters and , he showed that the sequences , and weakly converge to the point in a real Hilbert space. At each main iteration n of the scheme, he only solved strongly convex problems on C, but the proof of convergence was still done under the assumptions that .
Under the restrictions of the control sequences , he showed that the sequence defined by (1.4) strongly converges to in a real Hilbert space ℋ, where .
In this paper, motivated by Ceng et al. [16, 17], Wang and Guo , Zhou , Nadezhkina and Takahashi , Cho et al. , Takahashi and Takahashi , Anh [6, 12] and Anh et al. [20, 21], we introduce several modified hybrid extragradient schemes to modify the iteration schemes (1.3) and (1.4) to obtain new strong convergence theorems for a family of nonexpansive mappings and the equilibrium problem in the framework of a real Hilbert space ℋ.
To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.
Lemma 1.1 (, Lemma 3.1)
Lemma 1.2 Let C be a closed convex subset of a real Hilbert space ℋ, and let be the metric projection from ℋ on to C (i.e., for , is the only point in C such that ). Given and . Then if only if there holds the relation for all .
for all .
for all and .
Now, we prove the main convergence theorem.
Then the sequences , and strongly converge to the same point .
Proof The proof of this theorem is divided into several steps.
Step 1. Claim that and are closed and convex for all .
Then is closed and convex. Thus, is closed and convex.
Step 2. Claim that for all .
First, we show that by induction on n. It suffices to show that .
By the definition of , we have , and so for all , which deduces that . This shows that for all .
Then we have and hence . This shows that , which yields that for all .
Step 3. Claim that the sequence is bounded and there exists the limit .
and hence . This together with the boundedness of implies that the limit exists.
Step 4. We claim that .
Passing the limit in (2.5) as , we get . Hence, is a Cauchy sequence in a real Hilbert space ℋ and so .
Step 5. We claim that , where .
It follows from Step 4 that . Hence .
Now we show that . By Step 5, we have as .
which implies that . Thus, the subsequences , , strongly converge to the same point . This completes the proof. □
From (2.10) and using the methods in Theorem 2.1, we can get the following convergence result.
Then the sequences , and converge strongly to the same point .
- Mastroeni G: Gap function for equilibrium problems. J. Glob. Optim. 2004, 27: 411–426.View ArticleMathSciNetGoogle Scholar
- Zeng LC, Ansari QH, Schaible S, Yao J-C: Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory 2011, 12(2):293–308.MathSciNetGoogle Scholar
- Peng JW: Iterative algorithms for mixed equilibrium problems, strict pseudocontractions and monotone mappings. J. Optim. Theory Appl. 2010, 144: 107–119. 10.1007/s10957-009-9585-5View ArticleMathSciNetGoogle Scholar
- Yao Y, Liou YC, Wu YJ: An extragradient method for mixed equilibrium problems and fixed point problems. Fixed Point Theory Appl. 2009., 2009: Article ID 632819Google Scholar
- Blum E, Oettli W: From optimization and variational inequality to equilibrium problems. Math. Stud. 1994, 63: 127–149.MathSciNetGoogle Scholar
- Anh PN: A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems. Bull. Malays. Math. Soc. 2013, 36: 107–116.MathSciNetGoogle Scholar
- Anh PN, Kim JK: Outer approximation algorithms for pseudomonotone equilibrium problems. Comput. Math. Appl. 2011, 61: 2588–2595. 10.1016/j.camwa.2011.02.052View ArticleMathSciNetGoogle Scholar
- Ceng LC, Ansari QH, Yao JC: Hybrid proximal-type and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31(7):763–797. 10.1080/01630563.2010.496697View ArticleMathSciNetGoogle Scholar
- Zeng LC, Al-Homidan S, Ansari QH: Hybrid proximal-type algorithms for generalized equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 973028Google Scholar
- Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2006, 128: 191–201. 10.1007/s10957-005-7564-zView ArticleMathSciNetGoogle Scholar
- Sun S: An alternative regularization method for equilibrium problems and fixed point of nonexpansive mappings. J. Appl. Math. 2012., 2012: Article ID 202860 10.1155/2012/202860Google Scholar
- Anh PN: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 2012, 154: 303–320. 10.1007/s10957-012-0005-xView ArticleMathSciNetGoogle 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.2006.08.036View ArticleMathSciNetGoogle Scholar
- Anh PN: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 2013, 62: 271–283. 10.1080/02331934.2011.607497View ArticleMathSciNetGoogle Scholar
- Zhou H: Strong convergence theorems for a family of Lipschitz quasi-pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2009, 71: 120–125. 10.1016/j.na.2008.10.059View ArticleMathSciNetGoogle Scholar
- Ceng LC, Ansari QH, Schaible S: Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems. J. Glob. Optim. 2012, 53(1):69–96. 10.1007/s10898-011-9703-4View ArticleMathSciNetGoogle Scholar
- Ceng LC, Ansari QH, Yao JC: Hybrid pseudoviscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 2010, 4(4):743–754. 10.1016/j.nahs.2010.05.001View ArticleMathSciNetGoogle Scholar
- Wang S, Guo B: New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces. J. Comput. Appl. Math. 2010, 233: 2620–2630. 10.1016/j.cam.2009.11.008View ArticleMathSciNetGoogle Scholar
- Cho YJ, Argyros IK, Petrot N: Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Comput. Math. Appl. 2010, 60: 2292–2301. 10.1016/j.camwa.2010.08.021View ArticleMathSciNetGoogle Scholar
- Anh PN, Kim JK, Nam JM: Strong convergence of an extragradient method for equilibrium problems and fixed point problems. J. Korean Math. Soc. 2012, 49: 187–200. 10.4134/JKMS.2012.49.1.187View ArticleMathSciNetGoogle Scholar
- Anh PN, Son DX: A new iterative scheme for pseudomonotone equilibrium problems and a finite family of pseudocontractions. J. Appl. Math. Inform. 2011, 29: 1179–1191.MathSciNetGoogle Scholar
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