A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces

Francis Akutsah; Akindele Adebayo Mebawondu; Hammed Anuoluwapo Abass; Ojen Kumar Narain

doi:10.3934/naco.2021046

Article Contents

2023, Volume 13, Issue 1: 117-138. Doi: 10.3934/naco.2021046

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A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces

1.
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
2.
DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)
3.
Mountain Top University, Prayer City, Ogun State, Nigeria

^* Corresponding author: Francis Akutsah
^* Corresponding author: Francis Akutsah

Received: February 2021

Revised: September 2021

Accepted: October 2021

Early access: November 2021

Published: March 2023

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Abstract

In this work, we propose a new inertial method for solving strongly monotone variational inequality problems over the solution set of a split variational inequality and composed fixed point problem in real Hilbert spaces. Our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm as well as the Lipschitz constant of the operator. In addition, we prove that the proposed method converges strongly to a minimum-norm solution of the problem without using the conventional two cases approach. Furthermore, we present some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature. The results obtained in this paper extend, generalize and improve results in this direction.

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Mathematics Subject Classification: Primary: 47H10; 49J20; Secondary: 49J40.

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Figure 1. Example 1, Top Left: Case Ⅰ; Top Right: Case Ⅱ; Bottom Left: case Ⅲ; Bottom Right: Case Ⅳ

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Figure 2. Example 2, Top Left: Case A; Top Right: Case B; Bottom Left: Case C; Bottom Right: Case D

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Figure 3. Example 3, Top Left: Case Ⅰ; Top Right: Case Ⅱ; Bottom Left: case Ⅲ; Bottom Right: Case Ⅳ

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