Abstract
In this paper, we provide a more general regularization method for seeking a solution to a class of monotone variational inequalities in a real Hilbert space, where the regularizer is a hemicontinuous and strongly monotone operator. As a discretization of the regularization method, we propose an iterative method. We then prove that the proposed iterative method converges in norm to a solution of the class of monotone variational inequalities. We also apply our results to the constrained minimization problem and the minimum-norm fixed point problem for a generalized Lipschitz continuous and pseudocontractive mapping. The results presented in the paper improve and extend recent ones in the literature.
Highlights
The present paper is devoted to presenting a more general regularization method for a class of monotone variational inequality problems (MVIPs) with a monotone and hemicontinuous operator F over a nonempty, closed, and convex subset C of a real Hilbert space H
Let C be a nonempty, closed, and convex subset of a real Hilbert space H
Let C be a nonempty, bounded, and closed convex subset of a real Hilbert space H
Summary
The present paper is devoted to presenting a more general regularization method for a class of monotone variational inequality problems (MVIPs) with a monotone and hemicontinuous operator F over a nonempty, closed, and convex subset C of a real Hilbert space H. Over the past five decades years or so, the researchers designed various iterative algorithms for solving MVIP An early and typical iterative algorithm for solving MVIP ) seems to be the projected gradient method (PGM), see, for instance, [ , ], which generates a sequence {xn} by the recursive proce-. It is well known that if F is k-Lipschitz continuous and η-strongly monotone, MVIP has a unique solution
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