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.     

Existence Theorems for Variational Inequalities in Banach Spaces

In this paper, by employing the notion of generalized projection operators and the well-known Fan’s lemma, we establish some existence results for the variational inequality problem and the quasivariational inequality problem in reflexive, strictly convex, and smooth Banach spaces. We propose also an iterative method for approximate solutions of the variational inequality problem and we establish some convergence results for this iterative method. L. C. Zeng, His research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation, Shanghai, China. J. C. Yao, His research was partially supported by the National Science Council of the Republic of China     

Generalized Vector Variational Inequalities

In this paper, we introduce a generalized vector variational inequality problem (GVVIP) which extends and unifies vector variational inequalities as well as classical variational inequalities in the literature. The concepts of generalized C-pseudomonotone and generalized hemicontinuous operators are introduced. Some existence results for GVVIP are obtained with the assumptions of generalized C-pseudomonotonicity and generalized hemicontinuity. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained.     

Iterative Method for Set-Valued Mixed Quasi-variational Inequalities in a Banach Space

This paper introduces an iterative method for finding approximate solutions of a set-valued mixed quasivariational inequality in the setting of a Banach space. Existence of a solution of this rather general problem and the convergence of the proposed iterative method to a solution are established.The first two authors were partially supported by the National Science Council of the Republic of China. The third author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in ShanghaiCommunicated by     

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 Quasimonotone Variational Inequalities

The purpose of this paper is to prove the existence of solutions of the Stampacchia variational inequality for a quasimonotone multivalued operator without any assumption on the existence of inner points. Moreover, the operator is not supposed to be bounded valued. The result strengthens a variety of other results in the literature.     

A Strongly Convergent Direct Method for Monotone Variational Inequalities in Hilbert Spaces

We introduce a two-step direct method, like Korpelevich's, for solving monotone variational inequalities. The advantage of our method over that one is that ours converges strongly in Hilbert spaces, whereas only weak convergence has been proved for Korpelevich's algorithm. Our method also has the following desirable property: the sequence converges to the solution of the problem that lies closest to the initial iterate.     

Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces

In this paper, we propose three different kinds of iteration schemes to compute the approximate solutions of variational inequalities in the setting of Banach spaces. First, we suggest Mann-type steepest-descent iterative algorithm, which is based on two well-known methods: Mann iterative method and steepest-descent method. Second, we introduce modified hybrid steepest-descent iterative algorithm. Third, we propose modified hybrid steepest-descent iterative algorithm by using the resolvent operator. For the first two cases, we prove the convergence of sequences generated by the proposed algorithms to a solution of a variational inequality in the setting of Banach spaces. For the third case, we prove the convergence of the iterative sequence generated by the proposed algorithm to a zero of an operator, which is also a solution of a variational inequality.     

Exceptional Families of Elements for Variational Inequalities in Banach Spaces

In keeping with very recent efforts to establish a useful concept of an exceptional family of elements for variational inequality problems rather than complementarity problems as in the past, we propose such a concept. It generalizes previous ones to multivalued variational inequalities in general normed spaces and allows us to obtain several existence results for variational inequalities corresponding to earlier ones for complementarity problems. Compared with the existing literature, we consider problems in more general spaces and under considerably weaker assumptions on the defining map.     

Hilbert空间中的广义松弛上强制变分不等式组

主要利用三步投影方法模型讨论了带误差估计的广义非线性上强制变分不等式组的逼近解及其收敛性,所得到结果推广和改进了一系列最新结果.     

Solution Methods for Pseudomonotone Variational Inequalities

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, [2006]) and Noor (J. Optim. Theory Appl. 115:447–452, [2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, [2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, [2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators. This research was supported in part by a grant from the National Sun Yat-Sen University, Kaohsiung, Taiwan. It has been carried out under the agreement between the National Sun Yat-Sen University, Kaohsiung, Taiwan and the University of Pisa, Pisa, Italy. The authors thank the anonymous referee for useful comments and suggestions.     

Optimal Control of Variational Inequalities with Delays in the Highest Order Spatial Derivatives

In this paper, an optimal control problem for parabolic variational inequalities with delays in the highest order spatial derivatives is investigated. The well-posedness of such kinds of variational inequalities is established. The existence of optimal controls under a Cesari-type condition is proved, and the necessary conditions of Pontryagin type for optimal controls is derived.     

<Emphasis Type="Italic">m</Emphasis>-Dissipativity of Some Gradient Systems with Measurable Potential

We consider the operator in L ²(B, ν) and in L ¹(B, ν) with Neumann boundary condition, where U is an unbounded function belonging to for some q ∈(1, ∞), B is the possibly unbounded convex open set in where U is finite and ν(dx) = C exp (−2U (x))dx is a probability measure, infinitesimally invariant for N ₀. We prove that the closure of N ₀ is a m-dissipative operator both in L ²(B, ν) and in L ¹(B, ν). Moreover we study the properties of ergodicity and strong mixing of the measure ν in the L ² case.      

Hilbert空间中的广义非线性变分不等式组的投影算法

利用投影技术讨论了Hilbert空间中一类含松弛伪上强制映射的广义非线性变分不等式组的逼近解及其收敛性,所得到结果推广和统一了系列最新结果.     

Variational Inequalities for Combined Control and Stopping Game

Abstract

We study a nonlinear elliptic variational inequality associated with the combined stochastic control problem. By using the dynamic programming principle and the method of penalization, we establish the existence of a unique viscosity solution of the variational inequality and describe it as the value function of the corresponding combined stochastic game problem.     

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

On Abstract Variational Inequalities in Viscoplasticity with Frictional Contact

In this paper, we study quasistatic abstract variational inequalities with time-dependent constraints. We prove existence results and present an approximation method valid for nonsmooth constraints. Then, we apply our results to the approximation of the quasistatic evolution of an elastic body in bilateral contact with a rigid foundation. The contact involves viscous friction of the Tresca or Coulomb type. We prove existence results for approximate problems and give a full asymptotic analysis, proving strong or weak convergence results. Our work is motivated by the numerical study in the paper [Delost, M.: Quasistatic Problem with Frictional Contact: Comparison between Numerical Methods and Asymptotic Analysis Related to Semi Discrete and Fully Discrete Approximations. University of Nice, Nice (2007, to appear)] and explains the choice of the approximation made in it.     

Stability for Equilibrium Problems: From Variational Inequalities to Dynamical Systems

We study the connections between solutions of variational inequalities and equilibrium points of a generalized dynamical system. Furthermore, we analyze some stability questions arising in this field.     

Hilbert空间中广义变分不等式和Wiener-Hopf方程(英文)

In this paper,we consider the solvability of generalized variational inequalities involving multi-valued relaxed monotone operators in the framework of Hilbert spaces.Our results mainly improve the corresponding results announced by Verma[R U Verma,Generalized variational inequalities involving multivalued relaxed monotone operators,Appl Math Lett,1997,10：107-109]and many others.     

Continuity Results for a Class of Variational Inequalities with Applications to Time-Dependent Network Problems

We consider a class of monotone variational inequalities in which both the operator and the convex set are parametrized by continuous functions. Under suitable assumptions, we prove the continuity of the solution with respect to the parameter. As an important application, we consider the case of finite dimensional variational inequalities on suitable polyhedra. We demonstrate the applicability of our results to time dependent traffic equilibrium problem.