Abstract
In this paper, we construct two multi-step inertial regularized methods for hierarchical inequality problems involving generalized Lipschitzian and hemicontinuous mappings in Hilbert spaces. Then we present two strong convergence theorems and some numerical experiments to show the effectiveness and feasibility of our new iterative methods.
Highlights
Throughout this paper, we let H be a real Hilbert space with the norm k · k and the inner product h·, ·i, respectively, C be a nonempty closed convex subset of H andA : H → H be a mapping
We focus on hierarchical variational inequality problems (HVIP) (4) when A and F satisfy the following conditions: (CD1) A is monotone on C and L-Lipschitzian on H
It is obvious that VI(C, A) = {0} and x ∗ = 0 is the unique solution of HVIP (4)
Summary
Throughout this paper, we let H be a real Hilbert space with the norm k · k and the inner product h·, ·i, respectively, C be a nonempty closed convex subset of H and. A is monotone and L-Lipschitzian (see, Definition 1), VI(C, A) 6= ∅, δ ∈ 0, L1 This algorithm has a weak convergence result. Both of these two methods have strong convergence results. In 2019, Dong et al [7] proposed the multi-step inertial Krasnosel’skiı-Mann algorithm for finding a fixed point of a nonexpansive mapping, as follows: yn = xn + ∑k∈Sn αk,n ( xn−k − xn−k−1 ), w = xn + ∑k∈Sn β k,n ( xn−k − xn−k−1 ),. In this paper, motivated by the results of [3,4,7], we construct a multi-step inertial regularized subgradient extragradient method and a multi-step inertial regularized Tseng’s extragradient method for solving HVIP (4) in a Hilbert space when F is a generalized.
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