Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

Sun Young Cho

doi:10.3390/sym11121502

Abstract

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

Highlights

In a real Hilbert space H, one employs h·, ·i and k · k to stand for its inner product and norm
Let us denote by Fix(S) the set of all fixed points of an operator S : C → H
Whenever ς = 0, S is called nonexpansive. This means that the class of nonexpansive mappings is a proper subclass of the one of strict pseudocontractions

Summary

Introduction

In a real Hilbert space H, one employs h·, ·i and k · k to stand for its inner product and norm. A number of authors have conducted various investigations on efficient iterative algorithms; for examples, see [2,3,4,5,6,7,8,9,10,11] Let both the operators A and B be inverse-strongly monotone from C to H and the self-mapping. Inspired by the above research works in [12,24,25,26,27], we are concerned with hybrid-adaptive step-sizes Tseng’s extragradient algorithms, that are more advantageous and more subtle than the above iterative algorithms because they involve solving the VIP with Lipschitzian, pseudomonotone operators, and the common fixed-point problem of a finite family of strict pseudocontractions in Hilbert spaces.

Preliminaries

Results

Applications