# An extragradient approximation method for variational inequality problem on fixed point problem of nonexpensive mappings and monotone mappings

## Alongkot Suvarnamani and Mongkol Tatong

Address: Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, (RMUTT) Thanyaburi, PathumThani, 12110, Thailand

E-mail:

kotmaster2@rmutt.ac.th

kotmaster2@hotmail.com

tatong_max1222@hotmail.com

Abstract: We introduce an iterative sequence for finding the common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for tree inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of paper we utilize our results to study the zeros of the maximal monotone and some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., [Math. Meth. Oper. Res., 67:375–390, 2008] and many others.

AMSclassification: primary 47J05; secondary 47J25, 47H09, 47H10.

Keywords: nonexpansive mapping, fixed point problems, Variational inequality, relaxed extragradient approximation method, maximal monotone.