Taiwanese Journal of Mathematics

Strong and Weak Convergence Theorems for Generalized Mixed Equilibrium Problem with Perturbation and Fixed Pointed Problem of Infinitely Many Nonexpansive Mappings

L. C. Ceng, Hui-Ying Hu, and M. M. Wong

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Abstract

Very recently, Plubtieng and Kumam [S. Plubtieng, P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009) 614-621] proposed an iterative algorithm for finding a common solution of a variational inequality problem for an inverse-strongly monotone mapping and a fixed point problem of a countable family of nonexpansive mappings, and obtained a weak convergence theorem. In this paper, based on Plubtieng-Kumam's iterative algorithm we introduce a new iterative algorithm for finding a common solution of a generalized mixed equilibrium problem with perturbation and a fixed point problem of a countable family of nonexpansive mappings in a Hilbert space. We first derive a strong convergence theorem for this new algorithm under appropriate assumptions and then consider a special case of this new algorithm. Moreover, we establish a weak convergence theorem for this special case under some weaker assumptions. Such a weak convergence theorem unifies, improves and extends Plubtieng-Kumam's weak convergence theorem. It is worth pointing out that the proof method of strong convergence theorem is very different from the one of weak convergence theorem.

Article information

Source
Taiwanese J. Math., Volume 15, Number 3 (2011), 1341-1367.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406303

Digital Object Identifier
doi:10.11650/twjm/1500406303

Mathematical Reviews number (MathSciNet)
MR2829915

Zentralblatt MATH identifier
1239.49005

Subjects
Primary: 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 47H17 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.

Keywords
generalized mixed equilibrium problem with perturbation fixed point problem variational inequality nonexpansive mapping demiclosedness principle inverse-strongly monotone mapping strong and weak convergence

Citation

Ceng, L. C.; Hu, Hui-Ying; Wong, M. M. Strong and Weak Convergence Theorems for Generalized Mixed Equilibrium Problem with Perturbation and Fixed Pointed Problem of Infinitely Many Nonexpansive Mappings. Taiwanese J. Math. 15 (2011), no. 3, 1341--1367. doi:10.11650/twjm/1500406303. https://projecteuclid.org/euclid.twjm/1500406303


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