A General Iterative Method for Variational Inequality Problems, Mixed Equilibrium Problems, and Fixed Point Problems of Strictly Pseudocontractive Mappings in Hilbert Spaces

• Main Authors: Wangkeeree Rattanaporn, Wangkeeree Rabian
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2009/519065

<p/> <p>We introduce an iterative scheme for finding a common element of the set of fixed points of a <inline-formula> <graphic file="1687-1812-2009-519065-i1.gif"/></inline-formula>-strictly pseudocontractive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping, and the set of solutions of the mixed equilibrium problem in a real Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we first apply our results to study the optimization problem and we next utilize our results to study the problem of finding a common element of the set of fixed points of two families of finitely <inline-formula> <graphic file="1687-1812-2009-519065-i2.gif"/></inline-formula>-strictly pseudocontractive mapping, the set of solutions of the variational inequality, and the set of solutions of the mixed equilibrium problem. The results presented in the paper improve some recent results of Kim and Xu (2005), Yao et al. (2008), Marino et al. (2009), Liu (2009), Plubtieng and Punpaeng (2007), and many others.</p>