Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of κi-strictly pseudocontractive mappings

Kangtunyakarn

Description

In this article, by using the S -mapping and hybrid method we prove a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of generalized equilibrium defined by Ceng et al., which is a solution of two sets of variational inequality problems. Moreover, by using our main result we have a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of solution of a finite family of generalized equilibrium defined by Ceng et al., which is a solution of a finite family of variational inequality problems.

Sujets

• Variational inequality

Informations

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

Extrait

KangtunyakarnFixed Point Theory and Applications2012,2012:30 http://www.fixedpointtheoryandapplications.com/content/2012/1/30

R E S E A R C HOpen Access Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of i strictly pseudocontractive mappings Atid Kangtunyakarn

Correspondence: [e-mail] Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Abstract In this article, by using theSmapping and hybrid method we prove a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of i strictly pseudocontractive mappings and the set of generalized equilibrium defined by Ceng et al., which is a solution of two sets of variational inequality problems. Moreover, by using our main result we have a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of i strictly pseudocontractive mappings and the set of solution of a finite family of generalized equilibrium defined by Ceng et al., which is a solution of a finite family of variational inequality problems. Keywords:κstrict pseudo contraction mapping,αinverse strongly monotone, gen eralized equilibrium problem, variational inequality, theSmapping

1 Introduction LetHbe a real Hilbert space and let C be a nonempty closed convex subset of H. A mappingTofHinto itself is callednonexpansiveif∥TxTy∥≤∥xy∥for allx, yÎH. We denote byF(T) the set of fixed points ofT(i.e.,F(T) = {xÎH:Tx=x}) Goebel and Kirk [1] showed thatF(T) is always closed convex, and also nonempty providedThas a bounded trajectory. Recall the mappingTis said to bestrict pseudocontrationif there existÎ[0, 1) such that 2 22   Tx−Ty≤x−y+κ(I−T)x−(I−T)y∀x,y∈D(T). (1:1) Note that the class ofstrict pseudocontractions strictly includes the class of nonex pansive mappings, that isTis nonexpansive if and only if T is 0strict pseudocontractive. If= 1,Tis said to bepseudocontraction mapping. Tisstrong pseudocontractionif there exists a positive constantlÎ(0, 1) such thatT+lIis pseudocontraction. In a real Hilbert spaceH(1.1) is equivalent to   2 1−κ 2  Tx−Ty,x−y≤x−y−(I−T)x−(I−T)y∀x,y∈D(T). (1:2) 2

© 2012 Kangtunyakarn; 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.

KangtunyakarnFixed Point Theory and Applications2012,2012:30 http://www.fixedpointtheoryandapplications.com/content/2012/1/30

Tis pseudocontraction if and only if   2  Tx−Ty,x−y≤x−y∀x,y∈D(T).

Tis strong pseudocontraction if there exists a positive constantlÎ(0, 1)   2  Tx−Ty,x−y≤(1−λ)x−y∀x,y∈D(T) The class ofstrict pseudocontractions falls into the one between classes of nonex pansive mappings and pseudocontraction mappings and class of strong pseudocon traction mappings is independent of the class ofstrict pseudocontraction. A mappingAofCintoHis calledinversestrongly monotone, see [2] if there exists a positive real numberasuch that   2  x−y,Ax−Ay≥αAx−Ay for allx, yÎC. The equilibrium problem forGis to determine its equilibrium points, i.e., the set EP(G) ={x∈G:G(x,y)≥0,∀y∈C}. (1:3) Given a mappingT:C®H, letG(x, y) =〈Tx, yx〉for allx, yÎC. Then,zÎEP (F) if and only if〈Tz, yz〉≥0 for allyÎC, i.e.,zis a solution of the variational inequality. LetA:C®Hbe a nonlinear mapping. The variational inequality problem is to find auÎCsuch that  v−u,Au  ≥ 0 (1:4) for allvÎC. The set of solutions of the variational inequality is denoted byVI(C, A). In 2005, Combettes and Hirstoaga [3] introduced some iterative schemes of finding the best approximation to the initial data when EP(G) is nonempty and proved strong convergence theorem. Also in [3] Combettes and Hiratoaga, following [4] defineS r :H®Cby 1   S r (x) ={z∈C:G(z,y) +y−z,z−x≥0∀y∈C}. (1:5) r hey proved that under suitable hypothesesG, S r is singlevalued and firmly nonex pansive withF(S r ) =EP(G). Numerous problems in physics, optimization, and economics reduce to find a ele ment ofEP(G) (see, e.g., [516]) LetCB(H) be the family of all nonempty closed bounded subsets ofHand H (., .) be the Hausdorff metric on CB(H) defined as   H (U,Vsup) = maxd(u,V), supd(U,v) ,∀U,V∈CB(H), u∈U v∈V whered(u, V) = inf vÎV d(u, v),d(U, v) = inf uÎU d(u, v), andd(u, v) =∥uv∥. LetCbe a nonempty closed convex subset ofH. Let:C®ℝbe a realvalued function,T:C®CB(H) a multivalued mapping andF:H×C×C®ℝan equili briumlike function, that is,F(w, u, v) +F(w, v, u) = 0 for all (w, u, v)ÎH×C×C which satisfies the following conditions with respect to the multivalued mapT:C® CB(H).

Page 2 of 21