Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings

In this paper, we consider the implicit quasi-variational inequality without continuity assumptions of data mappings. Our approach here is completely different from the one based on KKM theorem in the literature. Interesting applications to generalized quasi-variational inequalities for both discontinuous mappings and discontinuous fuzzy mappings are given.This paper was partially supported by the NSC grant no. 86-2115-M-110-004.     

Regularization of quasi-variational inequalities

An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Under suitable conditions, a sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the boundedness of regularized solutions, become sufficient solvability conditions. It turns out that the regularization theory is quite strong for quasi-variational inequalities with set-valued monotone maps but restrictive for generalized pseudo-monotone maps. The results are quite general and are applicable to ill-posed variational inequalities, hemi-variational inequalities, inverse problems, and split feasibility problem, among others.     

广义非线性含参隐拟变分包含解的灵敏性分析

引入并研究了一类新的广义非线性含参隐拟变分包含,用预解算子方法证明了其解的存在性,并在H ilbert空间中分析了这类变分包含解的灵敏性,这些结果改进和推广了近期文献的相关结果.     

Technical Note Discontinuous Implicit Quasivariational Inequalities in Normed Spaces

In this paper, we consider an implicit quasivariational inequality without continuity assumptions in normed spaces. The main result (Theorem 2.1) provides an infinite-dimensional version of Theorem 3.2 in Ref. 1. To achieve such a goal, we employ Theorem 3.2 in Ref. 1 and the technique of Cubiotti in Ref. 2. In particular, Theorem 3.1 covers a recent result of Cubiotti (Theorem 3.1 of Ref. 2) as a special case. Communicated by F. Giannessi This research was partially supported by the National Science Council of Taiwan, ROC.     

On the solution existence of implicit quasivariational inequalities with discontinuous multifunctions

In this article, we established some solution existence theorems for implicit quasivariational inequalities. We first established some results in finite-dimensional spaces and then a solution existence result in infinite-dimensional spaces was derived. Our theorems are proved for discontinuous mappings and sets which may be unbounded. The results presented in this article are improvements of results in Cubiotti, and Yao (Cubiotti, P. and Yao, J.C., 1997, Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings. Mathematical Methods of Operations Research, 46, 213–328; Cubiotti, P. and Yao, J.C., 2007, Discontinuous implicit generalized quasi-variational inequalities in Banach spaces. Journal of Global Optimization (To appear))     

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper,the perturbations of the Moore–Penrose metric generalized inverses of linear operators in Banach spaces are described.The Moore–Penrose metric generalized inverse is homogeneous and nonlinear in general,and the proofs of our results are different from linear generalized inverses.By using the quasi-additivity of Moore–Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition,we show some error estimates of perturbations for the singlevalued Moore–Penrose metric generalized inverses of bounded linear operators.Furthermore,by means of the continuity of the metric projection operator and the quasi-additivity of Moore–Penrose metric generalized inverse,an expression for Moore–Penrose metric generalized inverse is given.     

On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

We study the following generalized quasivariational inequality problem: given a closed convex set X in a normed space E with the dual E ^*, a multifunction and a multifunction Γ:X→2 ^X , find a point such that , . We prove some existence theorems in which Φ may be discontinuous, X may be unbounded, and Γ is not assumed to be Hausdorff lower semicontinuous. The authors express their sincere gratitude to the referees for helpful suggestions and comments. This research was partially supported by a grant from the National Science Council of Taiwan, ROC. B.T. Kien was on leave from National University of Civil Engineering, Hanoi, Vietnam.     

Orientor field equations in Banach spaces

This paper focuses on the interplay between seminormality requirements and convergence hypotheses on trajectories in the lower closure theorems for orientor field equations. With the use of a weak form of seminormality, called the intermediate property (Q*), we obtain lower closure theorems (and thereby closure theorems) for orientor field equations in Banach spaces, under the assumption of strong convergence of some coordinates of the trajectories, while only weak convergence is assumed in the others. In the Euclidean case, this requirement of property (Q*) is further reduced to mere Kuratowski property (K), under the usual growth-type conditions. Finally, in the appendix, property (Q*) is investigated in detail.This research has been partially supported by Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan.     

A solution differentiability result for evolutionary quasi-variational inequalities

We consider a class of evolutionary quasi-variational inequalities arising in the study of some network equilibrium problems. First we prove the existence and uniqueness of solutions and, subsequently, present a differentiability result based on projection arguments.     

First-order methods for certain quasi-variational inequalities in a Hilbert space

Sufficient conditions are obtained for quasi-variational inequalities of a special type with nonlinear operators in a Hilbert space to be uniquely solvable. A first-order continuous method and its discrete variant are constructed for inequalities of this kind. The strong convergence of these methods is proved.     

A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces

The purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem. Under suitable conditions, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces. The results presented in the paper improve and extend some recent results.     

Lipschitzian stability of parametric variational inequalities over generalized polyhedra in Banach spaces

This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis.     

Mean inequalities in certain Banach function spaces

Boundedness of the Hardy operator and its adjoint is characterized between Banach function spaces Xq and Lp. By applying a limiting procedure, corresponding boundedness of the geometric mean operator is also derived.     

Differential quasi-variational inequalities in finite dimensional spaces

In this paper, we introduce and study a new class of differential quasi-variational inequalities in finite dimensional Euclidean spaces. First, we prove existence theorems for Carathéodory weak solutions of the differential quasi-variational inequalities under various conditions. Furthermore, we establish a convergence result on Euler time-dependent procedure for solving the initial-value differential set-valued variational inequalities.     

Generalized quasi-variational inequalities: Duality under perturbations

The aim of this paper is to investigate, in the case of finite dimensional spaces, the stability of a duality scheme as well as of generalized Kuhn-Tucker conditions previously introduced by the authors for generalized quasi-variational inequalities with multifunction of the constraints described by a finite number of inequalities.     

An implicit function theorem for Banach spaces and some applications

We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach–Lie groups, as well as Weil’s local rigidity for homomorphisms from finitely generated groups to Banach–Lie groups.      

Projection algorithms for the system of generalized mixed variational inequalities in Banach spaces

Some projection algorithms are suggested for solving the system of generalized mixed variational inequalities, and the convergence of the proposed iterative methods are proved without any monotonicity assumption for the mappings in Banach spaces. Our theorems generalize some known results.     

Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces

In this paper, a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces is introduced and studied. We first generalize the notion of the Yosida approximation in Hilbert spaces introduced by Moudafi to reflexive Banach spaces. Further, by using the notion of the Yosida approximation, we consider a system of parametric generalized Wiener-Hopf equation problems and show its equivalence to the system of parametric generalized mixed implicit equilibrium problems. By using a fixed point formulation of the system of parametric generalized Wiener-Hopf equation problems, we study the behavior and sensitivity analysis of a solution set of the system of parametric generalized mixed implicit equilibrium problems. We prove that, under suitable assumptions, the solution set of the system of parametric generalized mixed implicit equilibrium problems is nonempty, closed and Lipschitz continuous with respect to the parameters. Our results are new, and improve and generalize some known results in this field.     

Solution behavior for parametric implicit complementarity problems

In this paper we study the solution behavior for a special class of quasi-variational inequalities, namely implicit complementarity problems. We derive conditions under which the perturbed solution of a parametric implicit complementarity problem is locally unique, continuous and Fréchet differentiable.     

Upper Semicontinuity of the Solution set to Parametric Vector Quasivariational Inequalities

We prove the upper semicontinuity (in term of the closedness) of the solution set with respect to parameters of vector quasivariational inequalities involving multifunctions in topological vector spaces under the semicontinuity of the data, avoiding monotonicity assumptions. In particular, a new quasivariational inequality problem is proposed. Applications to quasi-complementarity problems are considered This work was partially supported by the program “Optimisation et Mathématiques Appliquées” of C.I.U.F-C.U.D./C.U.I. of Belgium and by the National Basic Research Program in Natural Sciences of NCSR of Vietnam