Iterative methods for solving variational inequalities in Euclidean space

作者：Gibali Aviv^*; Reich Simeon; Zalas Rafal

来源：Journal of Fixed Point Theory and Applications, 2015, 17(4): 775-811.

DOI：10.1007/s11784-015-0256-x

• Almost cyclic control
• common fixed point
• intermittent control
• iterative method
• metric projection
• nonexpansive operator
• quasi-nonexpansive operator
• subgradient projection
• variational inequality

In this paper, we investigate the convergence properties of an iterative method for solving variational inequalities in Euclidean space. We show that under certain assumptions the method can be applied to variational inequalities defined over the common fixed point set of a given infinite family of cutter operators. The main step of our method consists in the computation of the metric projection onto a certain superhalf- space, which is constructed using the input data defining the problem. Moreover, in the case where the common fixed point set is a finite intersection of the fixed point sets of cutters for which Opial's closedness principle holds, we show that the iterative method can be easily combined with either sequential, composition or convex combination type methods. We also discuss ways of applying our method to more general classes of operators such as quasi-nonexpansive and demi-contractive ones.

• 出版日期2015-12