Three-step iterations for nonexpansive mappings and inverse-strongly monotone mappings

Abstract

This paper introduces a three-step iteration for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inversestrongly monotone mapping by viscosity approximation methods in a Hilbert space. The authors show that the iterative sequence converges strongly to a common element of the two sets, which solves some variational inequality. Subsequently, the authors consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inversestrongly monotone mapping. The results obtained in this paper extend and improve the corresponding results announced by Nakajo, Takahashi, and Toyoda.

Variational inequalities and fixed point problems : a survey

“…Ax, x ≥ ∥x∥ 2 , x ∈ H. In 2009, motivated and inspired by above results, Meijuan Shang, Yongfu SU, Xiaolong Qin [41], introduced a general three-step iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping by viscosity approximation methods in a Hilbert space. They showed that the iterative sequence converges strongly to a common element of two sets, which solves some variational inequality.…”

A Modified Form of Inertial Viscosity Projection Methods for Variational Inequality and Fixed Point Problems

“…Chen et al [13] incorporated viscosity approximation methods for fnding the common elements to monotone and nonexpansive mappings. Numerous algorithms use viscosity approximation methods to fnd the common element of variational inequality problem and fxed point problem such as Ceng and Yao's [14] strong convergence result by combining the extragradient method and viscosity approximation method such that the two sequences generated by the algorithm converge strongly to the common element, a general three-step iterative process by Shang et al [15] in which two projections are calculated onto C in frst two steps, and in the third step, the third projection onto C is combined using viscosity approximation method, a generalized viscosity type extragradient method by Anh et al [16] which uses a strongly positive linear bounded operator to converge to the common element for variational inequality problem, fxed point problem and equilibrium problem, and two-step extragradient-viscosity method by Hieu et al [17] in which frst step calculates three projections onto C and second step combines the projections using viscosity approximation method.…”