Local Uniqueness of Solutions of General Variational Inequalities

We use the concept of Fréchet approximate Jacobian matrices to establish the local uniqueness of solutions to general variational inequalities which involve continuous, not necessarily locally Lipschitz continuous data.     

The Fréchet Approximate Jacobian and Local Uniqueness in Variational Inequalities

We introduce the concept of Fréchet approximate Jacobian matrices for continuous vector functions and use it to establish some sufficient criteria for the local uniqueness of solutions to a variational inequality problem involving continuous, not necessarily locally Lipschitz functions. Examples are also given to illustrate the usefulness of our approach.     

一类型变分不等式解的存在性唯一性问题及其对力学中的Signorini问题的应用

在本文中,我们研究了一类变分不等式解的存在性和唯一性问题,作为应用,讨论了力学中的著名的Signorini问题,改进了这一问题有解的条件.     

Existence and Uniqueness of Viscosity Solutions for Nonlinear Variational Inequalities Associated with Mixed Control

The author investigates the nonlinear parabolic variational inequality derived from the mixed stochastic control problem on finite horizon. Supposing that some sufficiently smooth conditions hold, by the dynamic programming principle, the author builds the Hamilton-Jacobi-Bellman(HJB for short) variational inequality for the value function.The author also proves that the value function is the unique viscosity solution of the HJB variational inequality and gives an application to the quasi-variat...     

Global Approximation of Solutions of Time-Dependent Variational Inequalities

We consider a class of parametric variational inequalities where both the operator and the convex set depend on time. This kind of variational inequalities are useful to model many time dependent equilibrium problems. We study the Lipschitz continuity of the solutions with respect to the time parameter and construct approximations for them which minimize the average worst case error. Some improved estimates of the Lipschitz constant for this class of problems are given. In order to illustrate our procedure, we study a classical network equilibrium problem.     

Applications of Generalized Variational and Quasivariational Inequalities with Operator Solutions in a TVS

In a recent paper, Domokos and Kolumbán introduced variational inequalities with operator solutions to provide a unified approach to several kinds of variational inequalities and vector variational inequalities in Banach spaces. Inspired by their work, in a former paper, we extended the scheme of vector variational inequalities with operator solutions from the single-valued case to the multivalued one and provided some applications to generalized vector variational inequalities and generalized quasivector variational inequalities in normed spaces. As a continuation of the former work, in this paper, we further extend those results to more general and tangible cases in the context of Hausdorff topological vector spaces or locally convex spaces. This work was supported by KOSEF Grant R01-2003-000-10825-0.     

On Two Applications of H-Differentiability to Optimization and Complementarity Problems

In a recent paper, Gowda and Ravindran (Algebraic univalence theorems for nonsmooth functions, Research Report, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD 21250, March 15, 1998) introduced the concepts of H-differentiability and H-differential for a function f : R ⁿ R ⁿ and showed that the Fréchet derivative of a Fréchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitzian function, the Bouligand subdifferential of a semismooth function, and the C-differential of a C-differentiable function are particular instances of H-differentials.In this paper, we consider two applications of H-differentiability. In the first application, we derive a necessary optimality condition for a local minimum of an H-differentiable function. In the second application, we consider a nonlinear complementarity problem corresponding to an H-differentiable function f and show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. These two applications were motivated by numerous studies carried out for C ¹, convex, locally Lipschitzian, and semismooth function by various researchers.     

On F-Implicit Generalized Vector Variational Inequalities

In this paper, we study the F-implicit generalized (weak) case for vector variational inequalities in real topological vector spaces. Both weak and strong solutions are considered. These two sets of solutions coincide whenever the mapping T is single-valued, but not set-valued. We use the Ferro minimax theorem to discuss the existence of strong solutions for F-implicit generalized vector variational inequalities.     

具有双障碍的退缩抛物变分不等式解的存在性和唯一性(英)

本文研究具有双障碍的退缩抛物变分不等式．我们利用罚技巧，有限逼近和先验估计方法，得到一类退缩抛物变分不等式弱解的存在性，并在一定条件之下，建立了弱解的唯一性．本文结论对广泛的一类抛物型变分不等式成立．     

四阶半线性椭圆变分不等式正解的存在性

本文通过Leray—Schauder度，给出四阶半线性椭圆变分不等式正解的存在性结果．     

Existence of Solutions and Star-shapedness in Minty Variational Inequalities

Minty Variational Inequalities (for short, Minty VI) have proved to characterize a kind of equilibrium more qualified than Stampacchia Variational Inequalities (for short, Stampacchia VI). This conclusion leads to argue that, when a Minty VI admits a solution and the operator F admits a primitive f (that is F= f′), then f has some regularity property, e.g. convexity or generalized convexity. In this paper we put in terms of the lower Dini directional derivative a problem, referred to as Minty VI(f′_,K), which can be considered a nonlinear extension of the Minty VI with F=f′ (K denotes a subset of ℝⁿ). We investigate, in the case that K is star-shaped, the existence of a solution of Minty VI(f’_,K) and increasing along rays starting at x^* property of (for short, F ɛIAR (K,x^*)). We prove that Minty VI(f’_,K) with a radially lower semicontinuous function fhas a solution x^* ɛker K if and only if FɛIAR(K, x^*). Furthermore we investigate, with regard to optimization problems, some properties of increasing along rays functions, which can be considered as extensions of analogous properties holding for convex functions. In particular we show that functions belonging to the class IAR(K,x^*) enjoy some well-posedness properties.     

Generalized Vector Variational and Quasi-Variational Inequalities with Operator Solutions

In a recent paper, Domokos and Kolumbán introduced variational inequalities with operator solutions to provide a suitable unified approach to several kinds of variational inequality and vector variational inequality in Banach spaces. Inspired by their work, in this paper, we further develop the new scheme of vector variational inequalities with operator solutions from the single-valued case into the multi-valued one. We prove the existence of solutions of generalized vector variational inequalities with operator solutions and generalized quasi-vector variational inequalities with operator solutions. Some applications to generalized vector variational inequalities and generalized quasi-vector variational inequalities in a normed space are also provided.     

微分方程解的存在与惟一性的一种证明

运用Banach压缩定理可证明微分方程解的存在与惟一性.     

摩擦约束弹性力学广义变分不等式解的存在性和唯一性

阐述了等价于摩擦约束弹性力学基本问题的广义变分不等式问题解的存在性和唯一性,进而提出广义变分不等式有限元近似及其离散解法。     

On the Index of Solvability for Variational Inequalities in Banach Spaces

We define the index of solvability, a topological characteristic, whose difference from zero provides the existence of a solution for variational inequalities of Stampacchia’s type with S ⁺-type and pseudo-monotone multimaps on reflexive separable Banach spaces. Some applications to a minimization problem and to a problem of economical dynamics are presented. The work is supported by the Russian FBR Grants 05-01-00100 and 07-01-00137 and by the NATO Grant ICS.NR.CLG 981757.     

带有单模糊映射的一般变分不等式解的存在性

众所周知，在各种变分不等式问题中，解的存在性是最基础也是最重要的问题之一．本文的目的是在Hilbert空间中研究带有单模糊映射的一般变分不等式解的存在性．此外，还讨论了几种特殊情况．     

Generalized Variational Inequalities with Pseudomonotone Operators Under Perturbations

Variational inequalities and generalized variational inequalities with perturbed operators and constraints are considered and convergence of solutions to such problems is proved under an assumption of pseudomonotonicity. The paper extends previous results given by the authors proved in the setting of monotone operators.     

On the Minimal Solutions of Variational Inequalities in Orlicz-Sobolev Spaces

In this paper, the author studies the existence of the minimal nonnegative solutions of some elliptic variational inequalities in Orlicz-Sobolev spaces on bounded or unbounded domains. She gets some comparison results between different solutions as tools to pass to the limit in the problems and to show the existence of the minimal solutions of the variational inequalities on bounded domains or unbounded domains. In both cases,coercive and noncoercive operators are handled. The sufficient and necessary conditions for the existence of the minimal nonnegative solution of the noncoercive variational inequality on bounded domains are established.     

Generalization of an Existence Theorem for Variational Inequalities

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.     

Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method

This paper deals with the convergence of the algorithm built on the auxiliary problem principle for solving pseudomonotone (in the sense of Karamardian) variational inequalities.