Convergence Theorem for Equilibrium Problems and Fixed Point Problems of Infinite Family of Nonexpansive Mappings

Yao, Yonghong; Liou, Yeong-Cheng; Yao, Jen‐Chih

doi:10.1155/2007/64363

Cited by 79 publications

(51 citation statements)

References 8 publications

(9 reference statements)

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“…Remark 2.7 ( [26], Remark 3.1). It can be known from Lemma 2.6 that if D is a nonempty bounded subset of C, then for > 0, there exists n 0 ≥ k such that for all n > n 0…”

Section: Lemma 26 ([20]mentioning

confidence: 97%

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Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Ceng¹,

Latif²,

Al-Mazrooei³

2016

J. Nonlinear Sci. Appl.

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In this paper, we introduce a multi-step iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inclusions for maximal monotone and inverse strong monotone mappings, the set of solutions of general system of variational inequalities and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. This multi-step iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, projection method, and strongly positive bounded linear operator and W -mapping approaches. We establish the strong convergence of the sequences generated by the proposed algorithm to a common element of above mentioned problems under appropriate assumptions, which also solves some optimization problem. The result presented in this paper improves and extends some corresponding ones in the earlier and recent literature.

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“…Remark 2.7 ( [26], Remark 3.1). It can be known from Lemma 2.6 that if D is a nonempty bounded subset of C, then for > 0, there exists n 0 ≥ k such that for all n > n 0…”

Section: Lemma 26 ([20]mentioning

confidence: 97%

“…During the last decade, many authors proposed different kinds of algorithms to find the common solutions of some problems mentioned above; See, for example, [13,15,23,25,26] and the references therein.…”

Section: System Of Variational Inclusion Problemsmentioning

confidence: 99%

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Ceng¹,

Latif²,

Al-Mazrooei³

2016

J. Nonlinear Sci. Appl.

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“…In particular, in order to study the EP (1.4) coupled with the fixed point problem, many authors have introduced iterative methods for finding a common element of the set of the solutions of the EP (1.4) and the set of fixed points of a countable family of nonexpansive mappings; see [6,8,13,23,24,27,28,32,37,38] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems

Jung¹

2017

J. Nonlinear Sci. Appl.

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In this paper, we introduce two modified hybrid iterative methods (one implicit method and one explicit method) for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality problem for a continuous monotone mapping and the set of fixed points of a continuous pseudocontractive mapping in Hilbert spaces, and show under suitable control conditions that the sequences generated by the proposed iterative methods converge strongly to a common element of three sets, which solves a certain variational inequality. As a direct consequence, we obtain the unique minimum-norm common point of three sets. The results in this paper substantially improve upon, develop and complement the previous well-known results in this area. c 2017 All rights reserved.Keywords: Hybrid iterative method, generalized mixed equilibrium problem, continuous monotone mapping, continuous pseudocontractive mapping, variational inequality, fixed point, ρ-Lipschitzian and η-strongly monotone mapping, metric projection. 2010 MSC: 49J30, 49J40, 47H09, 47H10, 47J20, 47J25, 47J05, 49M05.

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“…Using this result, in 2007, Yao et al [8], proposed the following explicit scheme with respect to W-mappings for an infinite family of nonexpansive mappings:…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, motivated and inspired by Yao et al [8,[10][11][12][13][14][15], Lau et al [16], Jitpeera et al [9], Kangtunyakarn [17] and Kim [18], Atsushiba and Takahashi [19], Saeidi [20], Piri [21][22][23] and Piri and Badali [24], we introduce the following iterative scheme for finding a common element of the set of solutions for a system of equilibrium problems…”

Section: Introductionmentioning

confidence: 99%

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

Piri

2012

Fixed Point Theory Appl

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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.

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Convergence Theorem for Equilibrium Problems and Fixed Point Problems of Infinite Family of Nonexpansive Mappings

Abstract: We introduce an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinite nonexpansive mappings in a Hilbert space. We prove a strong-convergence theorem under mild assumptions on parameters.

Cited by 79 publications

References 8 publications

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems

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

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