A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM, FIXED POINT#br#
PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS, AND QUASI-VARIATIONAL INCLUSION PROBLEM

doi:10.1016/S0252-9602(12)60033-2

Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (2): 499-519.doi: 10.1016/S0252-9602(12)60033-2

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A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM, FIXED POINT#br# PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS, AND QUASI-VARIATIONAL INCLUSION PROBLEM

LIU Min*, ZHANG Shi-Sheng

Department of Mathematics, Yibin university, Yibin 644007, China

Received:2009-01-13 Online:2012-03-20 Published:2012-03-20
Contact: Liu Min,liuminybsc@yahoo.com.cn E-mail:liuminybsc@yahoo.com.cn; changss@yahoo.cn
Supported by:
The project is supported by Scientific Research Fund of Sichuan Provincial Ed-ucation Department (09ZB102) and Scientific Research Fund of Science and Technology Deportment of Sichuan Provincial (2011JYZ011).

Abstract

Abstract:

In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone map-pings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Takahashi and Toyoda [9], Takahashi and Takahashi [10] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [11], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12].

Key words: k-strict pseudo-contractive mappings, generalized equilibrium problem, vis-cosity approximation method, variational inclusion problem, multi-valued

CLC Number:

47H09

LIU Min, ZHANG Shi-Sheng. A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM, FIXED POINT#br# PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS, AND QUASI-VARIATIONAL INCLUSION PROBLEM[J].Acta mathematica scientia,Series B, 2012, 32(2): 499-519.

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References

[1] Mann W R. Mean value methods in iteration. Proc Amer Math Soc, 1953, 4: 506–510

[2] Acedo G L, Xu H. Iterative methods for strict pseudo-contractions in Hilbert space. Nonlinear Anal, 2007, 67: 2258–2271

[3] Verma R U. Approximation solvability of a class of monotone set-valued variational inclusions involving (A, )-monotone mappings. J Math Anal Appl, 2008, 337: 969–975

[4] Zhang S. Set-valued variational inclusions in Banach space. J Math Anal Appl, 2000, 248: 438–454

[5] Zhang S. Existence and approximation of solutions of Set-valued variational inclusions in Banach space. Nonlinear Anal, 2001, 47(1): 583–594

[6] Demyanov V F, Stavroulakis G E, Polyakova L N, et al. Quasi-differentiability and nonsmooth modeling in mechanics, engineering and economics. Holland: Kluwer Academic, 1996

[7] Lions J L, Stampacchia G. Variational inequalities. Comm Pure Appl Math, 1967, 20: 493–517

[8] Chang S S, Joseph Lee J H, Chan C K. Algorithms of common solutions for quasi-variational inclusion and fixed point problems. Appl Math Mech, 2008, 29: 1–11

[9] Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mapping and monotone mapings. J Optim Theory Appl, 1998, 118: 417–428

[10] Takahashi S, Takahshi W. Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert space. J Math Anal Appl, 2007, 331: 506–515

[11] Chang S S, Joseph Lee H W, Chan C. A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Analysis, 2009, 70(9): 3307–3319

[12] Takahashi S, Takahashi W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Analysis, 2008, 69(3): 1025–1033

[13] Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Amer Math Soc, 1967, 73: 591–597

[14] Pascali Dan. NolinearMappings of Monotone Type. The Netherlands: Sijthoff and Noordhoff International Publishers, 1978

[15] Shimoji K, Takahashi W. Strong convergence to common fixed ponts of infinite nonexpansive mappings and applications. Taiwanese J Math, 2001, 5: 387–404

[16] Suzuki T. Strong convergence of Krasnoselskii and Mann type sequence for one parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl, 2005, 305: 227–239

[17] Zhang S. Some problems and results in the study of nonlinear analysis. Nonlinear Anal, 1997, 30: 4197–4208

[18] Blum E, Oettli W. From optimization and varitional inequalities to equilibrium problems. Math Student, 1994, 63: 123–145

[19] Combettes P L, Hirstoaga S A. Equilibrium problems in Hilbert space. J Nonlinear Convex Anal, 2005, 6: 117–136

A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM, FIXED POINT#br# PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS, AND QUASI-VARIATIONAL INCLUSION PROBLEM

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