Banach Journal of Mathematical Analysis

Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems

Yeol Je Cho, Mi Kwang Kang, and Zi-Ming Wang

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In this paper, we introduce a new hybrid projection algorithm based on the shrinking projection methods for two hemi-relatively nonexpansive mappings. Using the new algorithm, we prove some strong convergence theorems for finding a common element in the fixed points set of two hemi-relatively nonexpansive mappings, the solutions set of a variational inequality and the solutions set of an equilibrium problem in a uniformly convex and uniformly smooth Banach space. Furthermore, we apply our results to finding zeros of maximal monotone operators. Our results extend and improve the recent ones announced by Li [J. Math. Anal. Appl. 295 (2004) 115--126], Fan [J. Math. Anal. Appl. 337 (2008) 1041--1047], Liu [J. Glob. Optim. 46 (2010) 319--329], Kamraksa and Wangkeeree [J. Appl. Math. Comput. DOI: 10.1007/s12190-010-0427-2] and many others.

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Banach J. Math. Anal., Volume 6, Number 1 (2012), 11-34.

First available in Project Euclid: 14 May 2012

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Zentralblatt MATH identifier

Primary: 47H05: Monotone operators and generalizations
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Variational inequalities equilibrium problem hemi-relatively nonexpansive mappings shrinking projection method Banach space


Wang, Zi-Ming; Kang, Mi Kwang; Cho, Yeol Je. Convergence theorems based on the shrinking projection method for hemi-relatively nonexpansive mappings, variational inequalities and equilibrium problems. Banach J. Math. Anal. 6 (2012), no. 1, 11--34. doi:10.15352/bjma/1337014662.

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