Nonlinear Variational Inequalities Research Articles

Gauss-Seidel Type Iterative Algorithm for a Generalized System of Extended Non-linear Variational Inequalities

This study is focused on a new generalized system of extended non-linear variational inequalities (GSENVI, for short) involving 3k-distinct non-linear relaxed cocoercive operators. We give the equivalent formulation of GSENVI in a more convenient form by using auxiliary principal technique. Through the projection technique, we demonstrate that the non-linear projection equations are analogous to equivalent form of GSENVI. By the use of alternative fixed point formulations, we proposed the k-steps Gauss-Seidel type iterative algorithms to obtain an approximate solution of the GSENVI. Further, we discuss the convergence of proposed k-step Gauss Seidel type iterative algorithms. Several special cases of GSENVI are discussed for the reliability of our findings.

Fractional elliptic obstacle systems with multivalued terms and nonlocal operators

In this paper, we introduce and study a fractional elliptic obstacle system, which is composed of two elliptic inclusions with fractional (pi, qi)-Laplace operators, nonlocal functions, and multivalued terms. The weak solution of fractional elliptic obstacle system is formulated by a fully nonlinear coupled system driven by two nonlinear and nonmonotone variational inequalities with constraints. The nonemptiness and compactness of solution set in the weak sense are proved via employing a surjectivity theorem to the multivalued operators formulated by the sum of a multivalued pseudomonotone operator and a maximal monotone operator.

The Unified Description of Abstract Convexity Structures

The convexity of space is essential in nonlinear analysis, variational inequalities and optimization theory because it guarantees the existence and uniqueness of solutions to a certain extent. Because of its wide variety of applications, mathematicians have extensively promoted and researched convexity. This paper reviews some representative convexity structures and discusses their relations from their definitions, unifying them in the abstract convex structure. Moreover, applications of main convexity structures including KKM theory and fixed point theory will be reviewed.

A Full Approximation Scheme Multilevel Method for Nonlinear Variational Inequalities

A Full Approximation Scheme Multilevel Method for Nonlinear Variational Inequalities

Nonlinear variational inequalities with bilateral constraints coinciding on a set of positive measure

Nonlinear variational inequalities with bilateral constraints coinciding on a set of positive measure

Solutions for the Nonlinear Mixed Variational Inequality Problem in the System

Our paper proposes a system of nonlinear mixed variational inequality problems (SNMVIPs) on Banach spaces. Under suitable assumptions, using the K-Fan fixed point theorem and Minty techniques, we demonstrate that the solution set to the SNMVIP is nonempty, weakly compact, and unique. Additionally, we suggest a stability result for the SNMVIPs by perturbing the duality mappings. Furthermore, we present an optimal control problem that is governed by the SNMVIPs and show that it can be solved.

Existence of solutions for nonlinear elliptic variational inequalities with L 1 data

This article addresses the regularizing effect of the interaction between the coefficient of the zero-order term and the datum f in some elliptic variational inequalities. Even if the datum f only belongs to L 1 , the existence of bounded weak solutions to such problems are proven. This generalizes, to some extent, the previous results in the literature.

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

Portfolio-consumption choice problem with voluntary retirement and consumption constraints

We study the investment–consumption choice problem with voluntary retirement and upside/downside consumption constraints in infinite horizon, which can be formulated as a two-phase mixed stochastic control problem including stopping time coupled with controls. By verification theorem, we show the strong solution of a fully nonlinear differential variational inequality, satisfying some special properties, is the value function of this problem. Then we eliminate the nonlinearity through Legendre transformation, and obtain the closed-form formulas of the optimal investment–consumption feedback strategy, the optimal retirement strategy and the value function by solving the dual equation. Since the introduction of consumption constraints, these formulas are both segmented and semi-explicit, and bring certain difficulties to the analysis of the properties of the optimal strategies and the value function. The regularity of the feedback strategy and the value function are obtained, as well as their monotonicity or non-monotonicity with respect to various parameters are discussed by some stochastic analysis methods and differential equation techniques.

Nonlinear Variational Inequalities with Bilateral Constraints Coinciding on a Set of Positive Measure

Nonlinear Variational Inequalities with Bilateral Constraints Coinciding on a Set of Positive Measure

An upper bound for a condition number theorem of variational inequalities

Nonlinear variational inequalities in Banach spaces are considered. A notion of (absolute) condition number with respect to the right-hand side is introduced. A distance among variational inequalities is defined. We prove that the distance to suitably restricted ill-conditioned variational inequalities is bounded from above by a multiple of the reciprocal of the condition number. By using an analogous lower bound of the companion paper [14], we obtain a full condition number theorem for variational inequalities. The particular case of convex optimization problems is also considered. Known results dealing with optimization problems are thereby generalized.

Multigrid methods for non coercive variational inequalities

In this study, our examination centers around the numerical resolution of non-coercive issues using a multi-grid approach. Our particular emphasis is directed towards employing multi-grid methodologies to tackle non-linear variational inequalities. Our primary goal involves confirming the consistent convergence of the multi-grid algorithm. To attain this objective, we make use of fundamental sub-differential calculus and glean insights from the convergence principles of non-linear multi-grid techniques.

The Approximation of Fourth Order Nonlinear Variational Inequalities

In this paper we deal with existence and the approximation of the solution of the elliptic variational inequality from an abstract axiomatic point of view. We give an error estimate for the dierence of the two solutions in an appropriate norm. Also, we present some computational results.

Nonlinear Variational Inequalities in PDE-Constrained Optimization

Nonlinear Variational Inequalities (NVI) have emerged as a powerful mathematical framework for addressing complex optimization problems constrained by Partial Differential Equations (PDEs). This article explores the integration of NVI into PDE-constrained optimization, providing insights into the mathematical foundations, numerical techniques, and practical applications of this synergy. By understanding the role of NVI in PDE-constrained optimization, researchers and practitioners can tackle challenging optimization problems arising in fields such as engineering, physics, and computational science.

Inverse Problems with Nonlinear Variational Inequalities

Inverse problems are a class of challenging mathematical problems that involve estimating the properties or parameters of a system from observed data. In recent years, Nonlinear Variational Inequalities (NVI) have emerged as a powerful framework for solving inverse problems with complex constraints. This article explores the application of NVI to inverse problems, discussing their mathematical foundations, computational techniques, and real-world applications. By understanding the role of NVI in inverse problems, researchers and practitioners can tackle a wide range of problems in fields such as imaging, geophysics, and medical diagnostics.

Multigrid Methods for the Solution of Nonlinear Variational Inequalities

In this research, we investigate the numerical solution of second member problems that depend on the solution obtained through a multigrid method. Specifically, we focus on the application of multigrid techniques for solving nonlinear variational inequalities. The main objective is to establish the uniform convergence of the multigrid algorithm. To achieve this, we employ elementary subdifferential calculus and draw insights from the convergence theory of nonlinear multigrid methods.

Nonlinear Parabolic Variational Inequalities with Variable Time-Delay in Time Unbounded Domains

Nonlinear Parabolic Variational Inequalities with Variable Time-Delay in Time Unbounded Domains

Convergence of Solutions of Nonlinear Elliptic Variational Inequalities with Measurable Bilateral Constraints

Convergence of Solutions of Nonlinear Elliptic Variational Inequalities with Measurable Bilateral Constraints

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.

Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems

<abstract><p>We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.</p></abstract>