Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems

Yao, Yao Yonghong, Liou Yeong-Cheng, Wong Mu-Ming, Yao Jen-Chih

Description

In this paper, we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method. Mathematics Subject Classification (2000): 47H05; 47H09; 47H10; 47J05; 47J25.

Sujets

• Metric map
• Projection

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 16
• Langue: English

Extrait

Yaoet al.Fixed Point Theory and Applications2011,2011:53 http://www.fixedpointtheoryandapplications.com/content/2011/1/53

R E S E A R C HOpen Access Strong convergence of a hybrid method for monotone variational inequalities and fixed point problems 1 23* 4 Yonghong Yao , YeongCheng Liou , MuMing Wongand JenChih Yao

* Correspondence: [e-mail] 3 Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 32023, Taiwan Full list of author information is available at the end of the article

Abstract In this paper, we suggest a hybrid method for finding a common element of the set of solution of a monotone, Lipschitzcontinuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two wellknown methods: extragradient method andCQmethod. Under some mild conditions, we prove the strong convergence of the sequences generated by the proposed method. Mathematics Subject Classification (2000):47H05; 47H09; 47H10; 47J05; 47J25. Keywords:variational inequality problem, fixed point problems; monotone mapping, nonexpansive mapping, extragradient method,CQmethod, projection

1 Introduction LetHbe a real Hilbert space with inner product〈∙ , ∙〉and induced norm || ∙ ||. LetCbe a nonempty closed convex subset ofH. LetA:C®Hbe a nonlinear operator. It is well known that the variational inequality problem VI(C,A) is to finduÎCsuch that Au,v−u≥0,∀v∈C The set of solutions of the variational inequality is denoted byΩ. Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral and equilibrium problems, which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been developed for solving variational inequalities and related optimiza tion problems, see [1,125] and the references therein. Let us start with Korpelevich’s extragradient method which was introduced by Korpelevich [6] in 1976 and which generates a sequence {x n } via the recursion: y n =P C [x n −λAx n ], (1:1) x n+1 =P C [x n −λAy n ],n≥0 n whereP C is the metric projection fromRontoC,A:C®His a monotone opera tor andlis a constant. Korpelevich [6] proved that the sequence {x n } converges strongly to a solution ofV I(C,A). Note that the setting of the space is Euclid space n R .

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