General Variational Inequalities Research Articles

Recent Developments in General Quasi Variational Inequalities

In this paper, we present a number of new and known numerical techniques for solving general quasi variational inequalities, introduced by Noor [34] in 1988, using various techniques including projection, Wiener-Hopf equations, auxiliary principle, dynamical systems coupled with finite difference approach and sensitivity analysis. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis is also investigated. Some special cases are discussed as applications of the main results. Several open problems are suggested for future research.

New Auxiliary Principle Technique for General Harmonic Directional Variational Inequalities

This paper explores the utilisation of harmonic variational inequalities to establish the minimum value among two locally Lipschitz continuous harmonic convex functions. This investigation introduces novel classes of harmonic directed variational inequalities, particularly focusing on scenarios like harmonic complementarity and related optimization challenges. The study proposes and analyses various inertial iterative strategies for addressing harmonic directed variational inequalities through the auxiliary principle technique. It examines convergence criteria under specific weak conditions, emphasising the simplicity of the approach compared to other methodologies. The findings presented herein have broad applicability in the context of harmonic variational inequalities and optimization problems, though they are limited to theoretical exploration. Further research is required to implement these strategies numerically.

Error Bounds and Merit Functions for Exponentially General Variational Inequalities

In this paper, some new classes of classes of exponentially general variational inequalities are introduced. It is shown that the odd-order and nonsymmetric exponentially boundary value problems can be studied in the framework of exponentially general variational inequalities. We consider some classes of merit functions for exponentially general variational inequalities. Using these functions, we derive error bounds for the solution of exponentially general variational inequalities under some mild conditions. Since the exponentially general variational inequalities include general variational inequalities, quasi-variational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. Results obtained in this paper represent a refinement of previously known results for several classes of variational inequalities and related optimization problems.

New Classes of Exponentially General Nonconvex Variational Inequalities

In this paper, some new classes of exponentially general nonconvex variational inequalities are introduced and investigated. Several special cases are discussed as applications of these nonconvex variational inequalities. Projection technique is used to establish the equivalence between the non covex variational inequalities and fixed point problem. This equivalent formulation is used to discuss the existence of the solution. Several inertial type methods are suggested and analyzed for solving exponentially general nonconvex variational inequalities. using the technique of the projection operator and dynamical systems. Convergence analysis of the iterative methods is analyzed under suitable and appropriate weak conditions. In this sense, our result can be viewed as improvement and refinement of the previously known results. Our methods of proof are very simple as compared with other techniques.

A novel iterative approach for resolving generalized variational inequalities

<abstract><p>For figuring out general variational inequalities, we propose a novel and innovative iterative method. First, we demonstrate that the fixed point formulation and general vaiational inequality are equivalent. The fixed point formulation is used to formulate the explicit and implicit schemes. The general variational inequalities are the basis for the new algorithms. The newly developed algorithm is demonstrated numerically. For figuring out general variational inequalities, these new methods are innovative. Additionally, the convergence analysis is provided under certain favorable conditions.</p></abstract>

Iterative schemes for solving general variational inequalities

Iterative schemes for solving general variational inequalities

Iterative algorithms for general variational inequalities involving relaxed Lipschitz operators

In this paper, we establish and discuss the existence of a solution of the general variational inequality : Find u ∈ H, and ω ∈ T(u) such that h(u) ∈ K and This variational inequality was studied by Verma [3] in case s ≡ 0, and g ≡ I.

An inexact operator splitting method for general mixed variational inequalities

The present paper aims to deal with an inexact implicit method with a variable parameter for generalmixed variational inequalities in the setting of real Hilbert spaces. Under standard assumptions, the globalconvergence of the proposed method is proved. Numerical example is presented to illustrate the proposedmethod and convergence result. The results and method presented in this paper generalize, extend and unifysome known results in the literature.

A differential variational inequality in the study of contact problems with wear

We start with a mathematical model which describes the sliding contact of a viscoelastic body with a moving foundation. The contact is frictional and the wear of the contact surfaces is taken into account. We prove that this model leads to a differential variational inequality in which the unknowns are the displacement field and the wear function. Then, inspired by this model, we consider a general differential variational inequality in reflexive Banach spaces, governed by four parameters. We prove the unique solvability of the inequality as well as the continuous dependence of its solution with respect to the parameters. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact for which we deduce the existence of a unique solution as well as the existence of optimal control for an associate optimal control problem. We also present the corresponding mechanical interpretations.

A New Iterative Algorithm for General Variational Inequality Problem with Applications

This study aims at investigation of a generalized variational inequality problem. We initiate a new iterative algorithm and examine its convergence analysis. Using this newly proposed iterative method, we estimate the common solution of generalized variational inequality problem and fixed points of a nonexpansive mapping. A numerical example is illustrated to verify our existence result. Further, we demonstrate that the considered iterative algorithm converges with faster rate than normal S -iterative scheme. Furthermore, we apply our proposed iterative algorithm to estimate the solution of a convex minimization problem and a split feasibility problem.

Neutrosophic variational inequalities with applications in decision-making

In this paper, we introduced some new concepts of a neutrosophic set such as neutrosophic convex set, strongly neutrosophic convex set, neutrosophic convex function, strongly neutrosophic convex function, the minimum and maximum of a function f with respect to neutrosophic set, \(\min ,\) and \(\max \) neutrosophic variational inequality, neutrosophic general convex set, neutrosophic general convex function, and \(\min \), \(\max \) neutrosophic general variational inequality. We introduced some basic results on these new concepts. Moreover, we discussed the application of the neutrosophic set in optimization theory. We developed an algorithm using neutrosophic \(\min \) and \(\max \) variational inequalities and identified the maximum and minimum profit of the company.

Self-Adaptive Predictor-Corrector Approach for General Variational Inequalities Using a Fixed-Point Formulation

A literature review revealed that the general variational inequalities, fixed-point problems, and Winner–Hopf equations are equivalent. In this study, general variational inequality and fixed-point problem are considered. We introduced a new iterative method based on a self-adaptive predictor-corrector approach for finding a solution to the GVI. Adaptations in the fixed-point formulation and self-adaptive techniques have been used to predict a novel iterative approach. Convergence analyses of the suggested algorithm are demonstrated. Moreover, numerical analysis shows that we establish the new best method for solving general variational inequality which performs better than the previous one. Furthermore, it is known that GVI consisted of several classes including variational inequalities and related optimization problems, and results obtained in this study continue to hold for these problems.

New inertial approximation schemes for general quasi variational inclusions

In this article, we introduce and consider some new classes of general quasi variational inclusions, which provide us with unified, natural, novel and simple framework to consider a wide class of unrelated problems arising in pure and applied sciences. We prove that the general quasi variational inclusions are equivalent to the fixed point problems. This alternative formulation is used to discuss the existence of a solution as well as to propose some iterative methods. Convergence analysis is investigated under certain mild conditions. Since the general quasi variational inclusions include quasi variational inequalities, variational inequalities, and related optimization problems as special cases, our results continue to hold for these problems. It is an interesting problem to compare these methods with other technique for solving quasi variational inclusions for further research activities.

Penalty Method for Non-Stationary General Variational Like Inequalities

Penalty Method for Non-Stationary General Variational Like Inequalities

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Approximation-solvability of population biology systems based on p-Laplacian elliptic inequalities with demicontinuous strongly pseudo-contractive operators

The aim of this paper is to investigate existence, uniqueness and convergence of approximants of nonzero positive weak solutions for a class of population biology systems, which are models of one species based on p-Laplacian elliptic inequalities with demicontinuous strongly pseudo-contractive operators and involved logistic growth and harvesting rates. Toward this end, we foremost develop a kind of new general variational inequality principles with generalized duality mappings in reflexive Banach spaces. Then, we employ the new principle to obtain our main results for a general p-Laplacian elliptic inequality and the population biology systems. We note that the results are different from those relevant work of p-Laplacian elliptic inequalities in the literature, it is because a pseudo-contractive operator may not be an S-contractive operator and vice versa.

General Variational Inequalities: Existence of Solutions, Tikhonov-Type Regularization, and Well-posedness

General Variational Inequalities: Existence of Solutions, Tikhonov-Type Regularization, and Well-posedness

On the existence of solutions for the variational inequality problem $V I(A, psi, phi,g;K)$

In this paper, we are concerned with the existence of a solution u 2 K for the variational inequality problem $ V I(A; psi, phi, g;K)$. Furthermore, we propose some conditions that ensure the well-posedness of this problem. We study an operator type $g - ql$ which extends the linear problem. Finally, we investigate the existence of solutions for general vector variational inequalities in the inclusion form.

Iterative methods for extended general variational inequalities

Abstract In this paper, we suggest and analyze a new approximation schemes (3) to solve the extended general variational inequalities (2), which were introduced by Muhammad Aslam Noor (see[7, 9]). Using the projection operator technique, we establish the equivalence between the extended general variational inequalities and the fixed-point problem. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.

Generalized variational inequalities for maximal monotone operators

In this paper, we present some new results on the existence of solutions of generalized variational inequalities for set-valued mappings in reflexive Banach spaces with Fréchet differentiable norms. Moreover, the structure of the solution sets is investigated. The result obtained in this paper improves and extends the ones announced by Fang and Peterson [S. C. Fang and E. L. Peterson, Generalized Variational Inequalities, J. Optim. Theory Appl., 38 (1982), 363-383] and others.