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

Abstract

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.