Taiwanese Journal of Mathematics

STRONG CONVERGENCE THEOREMS BY A RELAXED EXTRAGRADIENT-LIKE

Lu-Chuan Ceng, Adrian Petrusel, and Mu-Ming Wong

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Abstract

Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis 69 (2008) 1025-1033] suggested and analyzed an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In this paper, we introduce a general system of generalized equilibria with inverse-strongly monotone mappings in a real Hilbert space. First, this system of generalized equilibria is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient-like method converges strongly to a solution of this system of generalized equilibria. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.

Article information

Source
Taiwanese J. Math., Volume 14, Number 4 (2010), 1689-1711.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405978

Digital Object Identifier
doi:10.11650/twjm/1500405978

Mathematical Reviews number (MathSciNet)
MR2663942

Zentralblatt MATH identifier
1225.49011

Subjects
Primary: 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49J40: Variational methods including variational inequalities [See also 47J20] 47J25: Iterative procedures [See also 65J15] 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.

Keywords
relaxed extragradient-like method nonexpansive mapping common fixed point demi-closedness principle inverse-strongly monotone mapping system of generalized equilibria

Citation

Ceng, Lu-Chuan; Petrusel, Adrian; Wong, Mu-Ming. STRONG CONVERGENCE THEOREMS BY A RELAXED EXTRAGRADIENT-LIKE. Taiwanese J. Math. 14 (2010), no. 4, 1689--1711. doi:10.11650/twjm/1500405978. https://projecteuclid.org/euclid.twjm/1500405978


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