Convergence of a regularization algorithm for nonexpansive and monotone operators in Hilbert spaces
© Yuan and Zhang; licensee Springer. 2014
Received: 28 May 2014
Accepted: 15 August 2014
Published: 2 September 2014
Variational inequality, fixed point and generalized equilibrium problems are investigated via a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solution of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by many authors.
1 Introduction and preliminaries
In this paper, we always assume that H is a real Hilbert space with inner product and induced norm for . Let C be a nonempty, closed, and convex subset of H.
For such a case, A is also said to be α-inverse-strongly monotone.
In this paper, we always use to denote the solution set of (1.1) and use denote the metric projection from H onto C. It is well known that is a solution of (1.1) iff x is a fixed point of the mapping , where is a constant, I stands for the identity mapping. If A is strongly monotone and Lipschitz continuous, the existence and uniqueness of solutions of equilibrium (1.1) is guaranteed by the Banach contraction principle.
Recall that a set-valued mapping is said to be monotone iff, for all , , and imply . M is maximal iff the graph of R is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if, for any , , for all implies .
Let be a mapping. stands for the fixed point set of S; that is, .
For such a case, S is also said to be α-contractive. We know that the mapping enjoys a unique fixed point and Picard’s algorithm can be employed to approximate its unique fixed point.
If C is a closed, bounded and convex subset of H, then is not empty; see .
Such a mapping is nonexpansive from C to C and it is called a W-mapping generated by and ; see  and the references therein.
In this paper, the set of such an is denoted by .
If , then the problem (1.3) is reduced to the classical variational inequality (1.1).
To study equilibrium problems (1.3) and (1.4), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A4) for each , is convex and weakly lower semi-continuous.
Many important problems have reformulations which require finding solutions of equilibriums (1.3) and (1.4), for instance, image recovery, inverse problems, network allocation, transportation problems and optimization problems; see [3–11] and the references therein. For solving solutions of equilibriums (1.3) and (1.4), regularization methods recently have been extensively studied; see [11–28] and the references therein.
In this paper, motivated and inspired by the research going on in this direction, we study the variational inequality (1.1), and the fixed point and equilibrium problem (1.3) based on a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solutions of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Chang et al. , Takahashi and Takahashi  and Hao .
The following lemmas play an important role in our paper.
Lemma 1.1 
- (2)is firmly nonexpansive, i.e., for any ,
is closed and convex.
Lemma 1.2 
Lemma 1.3 
is nonexpansive and , for each ;
for each and for each positive integer k, the limit exists.
- (3)the mapping defined by(1.5)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 1.4 
Lemma 1.5 
Throughout this paper, we always assume that , .
Lemma 1.6 
Then D is maximal monotone and if and only if .
2 Main results
In view of Lemma 1.5, we obtain from (2.15) . It follows that . Thus one derives a contradiction. Thus, we have .
Since B is Lipschitz continuous, we see that . Notice that D is maximal monotone and hence . This shows that . In the same way, we find that .
It follows from (A3) that . This proves that .
From the restriction (d), we obtain from Lemma 1.2 . This completes the proof. □
Taking with and choosing a sequence of real numbers with , we obtain the desired result by Theorem 2.1. □
that is, . This completes the proof. □
The authors are grateful to the referees for useful suggestions which improved the contents of the paper.
- Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.Google Scholar
- Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 2001, 5: 387–404.MathSciNetGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
- He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv. Fixed Point Theory 2012, 2: 47–57.Google Scholar
- Kim KS, Kim JK, Lim WH: Convergence theorems for common solutions of various problems with nonlinear mapping. J. Inequal. Appl. 2014., 2014: Article ID 2Google Scholar
- Park S: A review of the KKM theory on -spaces or GFC-spaces. Adv. Fixed Point Theory 2013, 3: 355–382.Google Scholar
- Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618. 10.1016/S0252-9602(12)60127-1View ArticleGoogle Scholar
- Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.Google Scholar
- Al-Bayati AY, Al-Kawaz RZ: A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization. J. Math. Comput. Sci. 2012, 2: 937–966.MathSciNetGoogle Scholar
- Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5View ArticleMathSciNetGoogle Scholar
- Chang SS, Lee HWJ, 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.2008.04.035View ArticleMathSciNetGoogle Scholar
- Qin X, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017View 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
- Zhang L, Tong H: An iterative method for nonexpansive semigroups, variational inclusions and generalized equilibrium problems. Adv. Fixed Point Theory 2014, 4: 325–343.Google Scholar
- Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 2010, 48: 423–445. 10.1007/s10898-009-9498-8View ArticleMathSciNetGoogle Scholar
- Lv S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 19Google Scholar
- Li DF, Zhao J: On variational inequality, fixed point and generalized mixed equilibrium problems. J. Inequal. Appl. 2014., 2014: Article ID 203Google Scholar
- Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4Google Scholar
- Mahato NK, Nahak C: Equilibrium problem under various types of convexities in Banach space. J. Math. Comput. Sci. 2011, 1: 77–88.MathSciNetGoogle Scholar
- Qin X, Cho SY, Kang SM: Some results on variational inequalities and generalized equilibrium problems with applications. Comput. Appl. Math. 2010, 29: 393–421.MathSciNetGoogle Scholar
- Wu C, Liu A: Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl. 2012., 2012: Article ID 90Google Scholar
- Wang ZM, Zhang X: Shrinking projection methods for systems of mixed variational inequalities of Browder type, systems of mixed equilibrium problems and fixed point problems. J. Nonlinear Funct. Anal. 2014., 2014: Article ID 15Google Scholar
- Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008View ArticleMathSciNetGoogle Scholar
- Jeong JU: Strong convergence theorems for a generalized mixed equilibrium problem and variational inequality problems. Fixed Point Theory Appl. 2013., 2013: Article ID 65Google Scholar
- Cho SY: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013, 57: 1429–1446. 10.1007/s10898-012-0017-yView ArticleGoogle Scholar
- Rodjanadid B, Sompong S: A new iterative method for solving a system of generalized equilibrium problems, generalized mixed equilibrium problems and common fixed point problems in Hilbert spaces. Adv. Fixed Point Theory 2013, 3: 675–705.Google Scholar
- Wen DJ, Chen YA: General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 125Google Scholar
- Cho SY, Qin X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput. 2014, 235: 430–438.View ArticleMathSciNetGoogle Scholar
- Hao Y: On variational inclusion and common fixed point problems in Hilbert spaces with applications. Appl. Math. Comput. 2010, 217: 3000–3010. 10.1016/j.amc.2010.08.033View ArticleMathSciNetGoogle Scholar
- Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289View ArticleMathSciNetGoogle Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017View ArticleMathSciNetGoogle Scholar
- Opial Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0View ArticleMathSciNetGoogle Scholar
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