Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities

Piri Hossein, Badali, Badali Ali

Description

In this paper, using strongly monotone and lipschitzian operator, we introduce a general iterative process for finding a common fixed point of a semigroup of nonexpansive mappings, with respect to strongly left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroups and the set of solutions of variational inequality for β -inverse strongly monotone mapping in a real Hilbert space. Under suitable conditions, we prove the strong convergence theorem for approximating a common element of the above two sets. Mathematics Subject Classification 2000: 47H09, 47H10, 43A07, 47J25

Sujets

• Projection
• Strong convergence
• Variational inequality

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 1 516
• Langue: English

Extrait

Piri and BadaliFixed Point Theory and Applications2011,2011:55 http://www.fixedpointtheoryandapplications.com/content/2011/1/55

R E S E A R C HOpen Access Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities * Hossein Piriand Ali Haji Badali

* Correspondence: h. [e-mail] Department of Mathematics, University of Bonab, Bonab 55517 61167, Iran

Abstract In this paper, using strongly monotone and lipschitzian operator, we introduce a general iterative process for finding a common fixed point of a semigroup of nonexpansive mappings, with respect to strongly left regular sequence of means defined on an appropriate space of bounded realvalued functions of the semigroups and the set of solutions of variational inequality forbinverse strongly monotone mapping in a real Hilbert space. Under suitable conditions, we prove the strong convergence theorem for approximating a common element of the above two sets. Mathematics Subject Classification 2000:47H09, 47H10, 43A07, 47J25 Keywords:projection, common fixed point, amenable semigroup, iterative process, strong convergence, variational inequality

1 Introduction Throughout this paper, we assume thatHis a real Hilbert space with inner product and norm are denoted by〈. , .〉and || . ||, respectively, and letCbe a nonempty closed convex subset ofH. A mappingTofCinto itself is called nonexpansive if ||TxTy ||≤||xy||, for allx, yÎH. ByFix(T), we denote the set of fixed points ofT(i.e.,Fix (T) = {xÎH:Tx=x}), it is well known thatFix(T) is closed and convex. Recall that a selfmappingf:C®Cis a contraction onCif there exists a constantaÎ[0, 1) such that ||f(x) f(y) ||≤a||xy|| for allx,yÎC. LetB:C®Hbe a mapping. The variational inequality problem, denoted byVI(C, B), is to finedxÎCsuch that Bx,y−x≥0 (1) for allyÎC. The variational inequality problem has been extensively studied in lit erature. See, for example, [1,2] and the references therein. Definition 1.1Let B:C®H be a mapping. Then B (1) is calledhstrongly monotone if there exists a positive constanthsuch that 2 Bx−By,x−y≥ηx−y,∀x,y∈C

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