Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems

Piri

Description

We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. The results presented in this paper mainly extend and improved some well-known results in the literature. Mathematics Subject Classification (2000) : 47H09; 47H10; 47H20; 43A07; 47J25.

Sujets

• Strong convergence

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 6
• Langue: English

Extrait

PiriFixed Point Theory and Applications2012,2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99

R E S E A R C HOpen Access Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems Hossein Piri

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

Abstract We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. The results presented in this paper mainly extend and improved some wellknown results in the literature. Mathematics Subject Classification (2000): 47H09; 47H10; 47H20; 43A07; 47J25. Keywords:common fixed point, strong convergence, amenable semigroup, explicit iterative, system of equilibrium problem.

1. Introduction LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. LetA:C®Hbe a nonlinear mapping. The classical variational inequality problem is to finedxÎCsuch that Ax,y−x ≥0,∀y∈C. (1) The set of solution of (1) is denoted by VI(C,A), i.e., VI(C,A) ={x∈C:Ax,y−x ≥0,∀y∈C}. (2) Recall that the following definitions:

(1)Ais called monotone if Ax−Ay,x−y ≥0,∀x,y∈C.

(2)Ais calledastrongly monotone if there exists a positive constantasuch that 2  Ax−Ay,x−y ≥αx−y,∀x,y∈C.

© 2012 Piri; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PiriFixed Point Theory and Applications2012,2012:99 http://www.fixedpointtheoryandapplications.com/content/2012/1/99

(3)Ais calledμLipschitzian if there exist a positive constantμsuch that   Ax−Ay≤µx−y,∀x,y∈C.

(4)Ais calledainverse strongly monotone, if there exists a positive real numbera>0

such that 2  Ax−Ay,x−y ≥αAx−Ay,∀x,y∈C.

1 It is obvious that anyainverse strongly monotone mappingBis Lipschitzian. α

(5) A mappingT:C®Cis called nonexpansive if∥TxTy∥≤∥xy∥for allx,y ÎC. Next, we denote by Fix(T) the set of fixed point ofT. (6) A mappingf:C®Cis said to be contraction if there exists a coefficientaÎ (0, 1) such that   f(x)−f(y)≤αx−y,∀x,y∈C.

H (7) A setvalued mappingU:H®called monotone if for all2 isx,yÎH,fÎUx andgÎUyimply〈xy,fg〉≥0. H (8) A monotone mappingU:H®2 ismaximal if the graphG(U) ofUis not properly contained in the graph of any other monotone mapping.

It is known that a monotone mappingUis maximal if and only if for (x,f)ÎH×H, 〈xy,fg〉≤0 for every (y,g)ÎG(U) implies thatfÎUx. LetBbe a monotone map ping ofCintoHand letN C xbe the normal cone toCatxÎC, that is,N C x= {yÎH: 〈xz,y〉≤0,∀zÎC} and define  Bx+N C x,x∈C, Ux= ∅x∈C. ThenUis the maximal monotone and 0ÎUxif and only ifxÎVI(C,B); see [1]. LetFbe a bifunction ofC×Cintoℝ, whereℝis the set of real numbers. The equili brium problem forF:C × C®ℝis to determine its equilibrium points, i.e the set EP(F) ={x∈C:F(x,y)≥0,∀y∈C}. Let J ={F i } i∈I be a family of bifunctions fromC × Cintoℝ. The system of equili brium problems for J ={F i } i∈I is to determine common equilibrium points for J ={F i } i∈I , i.e the set EP( J ) ={x∈C:F i (x,y)≥0,∀y∈C,∀i∈I}. (3) Numerous problems in physics, optimization, and economics reduce into finding some element of EP(F). Some method have been proposed to solve the equilibrium problem; see, for instance [25]. The formulation (3), extend this formalism to systems of such problems, covering in particular various forms of feasibility problems [6,7].

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