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Mathematics
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Variational Inequality and Mathematical Analysis
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Variational Inequality and Mathematical Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 May 2024) | Viewed by 6401

Special Issue Editors

Dr. Godwin Chdid Ugwunnadi

E-Mail Website
Guest Editor
Department of Mathematics, University of Eswatini, Private Bag 4, Kwaluseni, Eswatini. Department of Mathematics and Applied Mathematics, Sefako Makgato Health Science University , P.O. Box 94, Pretoria 0204, South Africa.
Interests: Fixed point theory and application in equilibrium problem;variational inequality problem;optimization problems
Dr. Hammed Anuoluwapo Abass

E-Mail Website
Guest Editor
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa
Interests: fixed point theory
Dr. Maggie Aphane

E-Mail Website
Guest Editor
Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, P.O. Box 60, Pretoria 0204, South Africa
Interests: pseudomonotone; fixed point problem; extragradient method; subgradient method; equilibrium problem; common fixed point; strong convergence; topology

Special Issue Information

Dear Colleagues,

In the area of mathematical analysis, methods of solving variational inequalities and fixed point problems are among the most powerful and important techniques in the study of nonlinear occurrence. Numerous studies in pure and applied sciences have made extensive use of fixed-point methods, including physics, chemistry, biology, economics, engineering, computer science, image processing and game theory. Most problems do not always have an exact solution, so it is crucial to create a useful tool that can approximate the result. It is possible to formulate many situations in terms of fixed-point problems, including those involving nonlinear equations, calculus of variations, partial differential equations, nonlinear analysis, optimization problems, variational inequality problems, complementarity problems, equilibrium problems, split feasibility problems, differential equations, dynamical systems, mathematics of finance, and various engineering fields and inverse problems.

Due to the significance and active influence of fixed-point variational inequality problems in other real-life phenomena, this Special Issue aims is to compile significant contributions regarding existence theory and approximation techniques for solving problems related to variational inequality and/or inclusion, fixed point, equilibrium problems, optimization problems, problems with quasi-variational inequality, bilevel optimization problems, split feasibility problems, split common fixed point problems, split bilevel optimization problems, etc., along with some practical applications and numerical experiments.

Dr. Godwin Chdid Ugwunnadi
Dr. Hammed Anuoluwapo Abass
Dr. Maggie Aphane
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • variational inequality
  • fixed point problems
  • nonlinear occurrence
  • approximation techniques
  • nonlinear equations
  • calculus of variations
  • partial differential equations
  • nonlinear analysis
  • variational inequality problems
  • equilibrium problems

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Published Papers (6 papers)

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Research

16 pages, 15928 KiB  
Article
An Optimal ADMM for Unilateral Obstacle Problems
by Shougui Zhang, Xiyong Cui, Guihua Xiong and Ruisheng Ran
Mathematics 2024, 12(12), 1901; https://doi.org/10.3390/math12121901 - 19 Jun 2024
Viewed by 746
Abstract
We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an [...] Read more.
We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an ADMM is applied to the corresponding saddle-point problem. Through eliminating the primal and auxiliary unknowns, a pure dual algorithm is then used. The convergence of the proposed method is analyzed, and a simple strategy is presented for selecting the optimal parameter, with the largest and smallest eigenvalues of the iterative matrix. Several numerical experiments confirm the theoretical findings of this study. Full article
(This article belongs to the Special Issue Variational Inequality and Mathematical Analysis)
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30 pages, 365 KiB  
Article
A New Robust Iterative Scheme Applied in Solving a Fractional Diffusion Model for Oxygen Delivery via a Capillary of Tissues
by Godwin Amechi Okeke, Akanimo Victor Udo, Nadiyah Hussain Alharthi and Rubayyi T. Alqahtani
Mathematics 2024, 12(9), 1339; https://doi.org/10.3390/math12091339 - 28 Apr 2024
Viewed by 1307
Abstract
In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is [...] Read more.
In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green’s functions with an example. Full article
(This article belongs to the Special Issue Variational Inequality and Mathematical Analysis)
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18 pages, 503 KiB  
Article
An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation
by Pius W. M. Chin, Claude R. B. Moutsinga and Khadijo R. Adem
Mathematics 2024, 12(6), 867; https://doi.org/10.3390/math12060867 - 15 Mar 2024
Cited by 1 | Viewed by 891
Abstract
The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to [...] Read more.
The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to analyze because of the presence of the nonlinearity term, which is always very difficult to approximate. This paper aims to design a reliable scheme that consists of a combination of the nonstandard finite difference in time method, the Galerkin method and the compactness methods in space variables. This method is used to show that the solution of the problem exists uniquely. The a priori estimate from the existence process is applied to the scheme to show that the numerical solution from the scheme converges optimally in the L2 as well as the H1 norms. We proceed to show that the scheme preserves the decaying properties of the exact solution. Numerical experiments are introduced with a chosen example to validate the proposed theory. Full article
(This article belongs to the Special Issue Variational Inequality and Mathematical Analysis)
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14 pages, 267 KiB  
Article
Generalized Vector Quasi-Equilibrium Problems
by Adela Elisabeta Capătă
Mathematics 2024, 12(6), 809; https://doi.org/10.3390/math12060809 - 9 Mar 2024
Viewed by 815
Abstract
The aim of this paper is to present new existence results for solutions to a generalized quasi-equilibrium problem with set-valued mappings and moving cones. The key to this approach is a new Browder-type fixed point theorem, which permits working in a new direction [...] Read more.
The aim of this paper is to present new existence results for solutions to a generalized quasi-equilibrium problem with set-valued mappings and moving cones. The key to this approach is a new Browder-type fixed point theorem, which permits working in a new direction with the milder condition of transfer open-valued mapping and considering weaker assumptions on the coving cone. These results are applied to some generalized vector quasi-equilibrium problems with trifunctions and to a vector quasi-equilibrium problem with fuzzy mappings in a fuzzy environment. Full article
(This article belongs to the Special Issue Variational Inequality and Mathematical Analysis)
10 pages, 272 KiB  
Article
Proximal Analytic Center Cutting Plane Algorithms for Variational Inequalities and Nash Economic Equilibrium
by Renying Zeng
Mathematics 2024, 12(2), 177; https://doi.org/10.3390/math12020177 - 5 Jan 2024
Viewed by 941
Abstract
In this study, we proposed proximal analytic center cutting plane algorithms for solving variational inequalities whose domains are normal regions. Our algorithms stop with a solution of the variational inequality after a finite number of iterations, or we may find a sequence that [...] Read more.
In this study, we proposed proximal analytic center cutting plane algorithms for solving variational inequalities whose domains are normal regions. Our algorithms stop with a solution of the variational inequality after a finite number of iterations, or we may find a sequence that converges to the solution of the variational inequality. We introduced the definition of the Nash economic equilibrium solution over a normal region and proved a sufficient condition for our Nash economic solution. An example of Nash equilibrium over a normal region is also provided. Our proximal analytic center cutting plane algorithms are constructive proofs of our Nash equilibrium problems. Full article
(This article belongs to the Special Issue Variational Inequality and Mathematical Analysis)
28 pages, 2401 KiB  
Article
On Bilevel Monotone Inclusion and Variational Inequality Problems
by Austine Efut Ofem, Jacob Ashiwere Abuchu, Hossam A. Nabwey, Godwin Chidi Ugwunnadi and Ojen Kumar Narain
Mathematics 2023, 11(22), 4643; https://doi.org/10.3390/math11224643 - 14 Nov 2023
Cited by 2 | Viewed by 1125
Abstract
In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of [...] Read more.
In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted. Full article
(This article belongs to the Special Issue Variational Inequality and Mathematical Analysis)
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