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In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operatorFover a closed and convex setC. We assume that the setCcan be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions onF. We provide several examples for the case whereCis the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a projection-type approximation method is introduced for solving a variational inequality problem. The proposed method involves only one projection per iteration and the underline operator is pseudo-monotone and L-Lipschitz-continuous. The strong convergence result of the iterative sequence generated by the proposed method is established, under mild conditions, in real Hilbert spaces. Sound computational experiments comparing our newly proposed method with the existing state of the art on multiple realistic test problems are given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new iterative method for solving a variational inequality over the set of common fixed points of a finite family of sequences of nearly non-expansive mappings in a real Hilbert space. And, using this result, we give some applications to the problem of finding a common fixed point of non-expansive mappings or non-expansive semigroups and the problem of finding a common null point of monotone operators. [ABSTRACT FROM AUTHOR]

In this paper, a new iterative method for finding the projection onto the intersection of two closed convex sets in the framework of Banach spaces is presented. It is a viscosity approximation method which produces a strongly convergent sequence. [ABSTRACT FROM AUTHOR]