The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-Ï•-nonexpansive mappings

Saewan Siwaporn, Kumam, Kumam Poom

Description

In this article, we introduce a new hybrid projection iterative scheme based on the shrinking projection method for finding a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points for a pair of asymptotically quasi- Ï• -nonexpansive mappings in Banach spaces and set of variational inequalities for an α -inverse strongly monotone mapping. The results obtained in this article improve and extend the recent ones announced by Matsushita and Takahashi (Fixed Point Theory Appl. 2004(1):37-47, 2004), Qin et al. (Appl. Math. Comput. 215:3874-3883, 2010), Chang et al. (Nonlinear Anal. 73:2260-2270, 2010), Kamraksa and Wangkeeree (J. Nonlinear Anal. Optim.: Theory Appl. 1(1):55-69, 2010) and many others. AMS Subject Classification : 47H05, 47H09, 47J25, 65J15.

Sujets

• Variational inequality
• Banach space

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 4
• Langue: English

Extrait

Saewan and KumamFixed Point Theory and Applications2011,2011:9 http://www.fixedpointtheoryandapplications.com/content/2011/1/9

R E S E A R C HOpen Access The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi jnonexpansive mappings * Siwaporn Saewan and Poom Kumam

* Correspondence: poom. [e-mail] Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Abstract In this article, we introduce a new hybrid projection iterative scheme based on the shrinking projection method for finding a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points for a pair of asymptotically quasijnonexpansive mappings in Banach spaces and set of variational inequalities for anainverse strongly monotone mapping. The results obtained in this article improve and extend the recent ones announced by Matsushita and Takahashi (Fixed Point Theory Appl. 2004(1):3747, 2004), Qin et al. (Appl. Math. Comput. 215:38743883, 2010), Chang et al. (Nonlinear Anal. 73:2260 2270, 2010), Kamraksa and Wangkeeree (J. Nonlinear Anal. Optim.: Theory Appl. 1 (1):5569, 2010) and many others. AMS Subject Classification: 47H05, 47H09, 47J25, 65J15. Keywords:Generalized mixed equilibrium problem, Asymptotically quasij?ϕ?nonex pansive mapping, Strong convergence theorem, Variational inequality, Banach spaces

1. Introduction LetEbe a Banach space with norm ||∙||,Cbe a nonempty closed convex subset ofE, and letE*denote the dual ofE. Letf:C × C®ℝbe a bifunction,:C®ℝbe a realvalued function, andB:C®E*be a mapping. Thegeneralized mixed equilibrium problem, is to findxÎCsuch that f x,y+Bx,y−x+ϕy−ϕx≥0,∀y∈C(1:1) The set of solutions to (1.1) is denoted by GMEP(f, B,), i.e., GMEPf,B,ϕ={x∈C:f x,y+Bx,y−x+ϕy−ϕx≥0,∀y∈C}(1:2) IfB≡0, then the problem (1.1) reduces into themixed equilibrium problem for f, denoted by MEP(f,), is to findxÎCsuch that f x,y+ϕy−ϕx≥0,∀y∈C (1:3) If≡0, then the problem (1.1) reduces into thegeneralized equilibrium problem, denoted by GEP(f,B), is to findxÎCsuch that

© 2011 Saewan and Kumam; 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.

Saewan and KumamFixed Point Theory and Applications2011,2011:9 http://www.fixedpointtheoryandapplications.com/content/2011/1/9

f x,y+Bx,y−x ≥0,∀y∈C(1:4) Iff≡0, then the problem (1.1) reduces into themixed variational inequalityof Browder type, denoted by MVI(B,C), is to findxÎCsuch that Bx,y−x+ϕy−ϕx≥0,∀y∈C (1:5) If≡0, then the problem (1.5) reduces into theclassical variational inequality, denoted by VI(B,C), which is to findxÎCsuch that Bx,−x ≥0,∀ ∈C (1:6) IfB≡0 and≡0, then the problem (1.1) reduces into theequilibrium problem for f, denoted by EP(f), which is to findxÎCsuch that f x,y≥0,∀y∈C (1:7) Iff≡0, then the problem (1.3) reduces into theminimize problem, denoted by Arg min (), which is to findxÎCsuch that ϕy−ϕx≥0,∀y∈C (1:8) The above formulation (1.6) was shown in [1] to cover monotone inclusion pro blems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other pro blems, for example, the complementarity problem, fixed point problem and optimiza tion problem, which can also be written in the form of an EP(f). In other words, the EP(f) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many articles have appeared in the literature on the existence of solutions of EP(f); see, for example [14] and refer ences therein. Some solution methods have been proposed to solve the EP(f) in Hilbert spaces and Banach spaces; see, for example [520] and references therein. x+y A Banach spaceEis said to bestrictly convexif <1 for allx,yÎEwith ||x|| = ||y|| = 1 andx≠y. LetU= {xÎE: ||x|| = 1} be the unit sphere ofE. Then, a ||x+ty|| − ||x|| Banach spaceEis said to besmoothif the limit lim exists for eachx,y t→0 ÎU. It is also said to beuniformly smoothif the limit exists uniformly inx,yÎU. Let Ebe a Banach space. Themodulus of convexityofEis the functionδ: [0, 2]®[0, 1] defined by x+y δ(ε) = inf{1− ||||:x,y∈E,||x||=||y||= 1,||x−y|| ≥ε}

A Banach spaceEisuniformly convexif and only ifδ(ε)>0 for allεÎ(0, 2]. Letp be a fixed real number withp≥2. A Banach spaceEis said to bepuniformly convex p if there exists a constantc >0 such thatδ(ε)≥cεfor allεÎ[0, 2]; see [21,22] for more details. Observe that everypuniformly convex is uniformly convex. One should note that no Banach space ispuniformly convex for 1< p <2. It is well known that a Hilbert space is2uniformly convex, uniformly smooth. For eachp >1, thegeneralized E* duality mapping J p :E®2 isdefined by

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