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.     

向量变分不等式的严格可行性问题

本篇文章首先定义了向量变分不等式的严格可行点概念,其次在假设了映射是强(D)-伪单调的情况下,证明了向量变分不等式解集非空有界与其严格可行点存在的等价性问题,推广了在数量变分不等式上得到的相应结果．     

General KKM Theorem with Applications to Minimax and Variational Inequalities

In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. By employing our general KKM theorem, we obtain a general minimax inequality which contains several existing ones as special cases. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general setting. We also establish several existence results for generalized variation inequalities and generalized quasi-variational inequalities.     

Generalized Variational Relation Problems with Applications

In this paper, we first obtain an existence theorem of the solutions for a variational relation problem. An existence theorem for a variational inclusion problem, a KKM theorem and an extension of the well know Ky Fan inequality will be established, as particular cases. Some applications concerning a saddle point problem with constraints, existence of a common fixed point for two mappings and an optimization problem with constraints, will be given in the last section of the paper.     

Generalized mixed variational-like inequality for random fuzzy mappings

In this paper, we introduce and study a new class of generalized mixed variational-like inequality for random fuzzy mappings(GMVLIP). An existence theorem for auxiliary problem of the GMVLIP is established. Further, by exploiting the theorem, we construct and analyze a new iterative algorithm for finding the solution of the GMVLIP. Furthermore, we prove the existence of a unique solution of the GMVLIP and discuss the convergence analysis of iterative sequence generated by the iterative algorithm.     

Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.     

Existence and uniqueness results for a class of quasi-hemivariational inequalities

In this paper we are concerned with the study of a nonstandard quasi-hemivariational inequality. Using a fixed point theorem for set-valued mappings the existence of at least one solution in bounded closed and convex subsets is established. We also provide sufficient conditions for which our inequality possesses solutions in the case of unbounded sets. Finally, the uniqueness and the stability of the solution are analyzed in a particular case.     

Multiple solutions for a class of elliptic hemivariational inequalities

The existence of multiple solutions to elliptic hemivariational inequality problems in bounded domains is investigated via a suitable nonsmooth version of a classical technique due to Struwe and a recent saddle point theorem for locally Lipschitz continuous functionals.     

New Approach to Solving a System of Variational Inequalities and Hierarchical Problems

This paper deals with a viscosity iterative method, in real Hilbert spaces, for solving a system of variational inequalities over the fixed-point sets of possibly discontinuous mappings. Under classical conditions, we prove a strong convergence theorem for our method. The proposed algorithm can be applied for instance to solving variational inequalities in some situations when the projection methods fail. Moreover, the techniques of analysis are novel and provide new tools in designing approximation schemes for combined and bilevel optimization problems.     

Duality and gap function for generalized multivalued ϵ-vector variational inequality

In this article, a generalized multivalued ?-vector variational inequality (GMVVI)_? is introduced. An existence theorem for (GMVVI)_? is established by means of the KKM theorem. Then, a dual ?-vector variational inequality (DVVI)_? for (GMVVI)_? is given, and an equivalence relation between (GMVVI)_? and (DVVI)_? is proved. Finally, gap functions are also proposed for these two problems, respectively.     

Coupled coincidence point theorems in ordered metric spaces

In this paper we extend the coupled contraction mapping theorem proved in partially ordered metric spaces by Gnana Bhaskar and Lakshmikantham (Nonlinear Anal. TMA 65:1379–1393, 2006) to a coupled coincidence point result for a pair of compatible mappings. A control function has been used in our theorem. The mappings are assumed to satisfy a weak contractive inequality. Our theorem improves the results of Harjani et al. (Nonlinear Anal. TMA 74:1749–1760, 2011). The result we have established is illustrated with an example which also shows that the improvement is actual.     

On a generalized sup-inf problem

In this paper, necessary and sufficient conditions for solvability of nonlinear inequality systems are given using certain generalized convexity concepts. Our results imply some theorems of Kirszbraun, Fan, Minty, Simons, Sebestyén, and Gwinner-Oettli.The authors are grateful to the referees for their helpful comments. They thank one of the referees, who emphasized the connection between the Wald minimax theorem and Theorem 2.1, and suggested an alternative proof of Theorem 2.1 in Section 4.     

An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces

ABSTRACT

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established under standard and mild conditions. Our scheme generalizes and extends some of the existing results in the literature, and moreover, its computational effort is less per each iteration compared with related works.     

集值变分不等式问题的例外簇

本文首先在Banach空间中证明了零调集值映射的一个Leray-Schauder型不动点定理,然后在Hilbert空间中定义了零调集值映射的变分不等式的例外簇,利用本文给出的不动点定理给出了无界集上的变分不等式问题存在解的一个充分条件．此条件弱于许多已知的关于变分不等式问题的解的存在性条件,并由此得到Hilbert空间中几个变分不等式约解的存在性定理．     

集值映射的对称向量拟均衡问题

研究向量集值映射的拟均衡问题的有效解,利用数值化方法与不动点定理,得到解的存在性定理.作为应用,得到广义向量鞍点,向量变分不等式与向量互补问题的存在性定理.     

Estimates on μ(z)-homeomorphisms of the unit disk

In the theory ofK-quasiconformal mappings, Mori's theorem shows thatK-quasiconformal mappings on the unit disk satisfy the Hölder condition, where the coefficient 16 is best possible. In this paper, we prove that self-μ(z)-homeomorphisms on the unit disk have an analogical result to Mori's theorem when the integral mean dilatations are controlled by log function. An unimprovable inequality is obtained.     

Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iteration process of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly-monotone mapping, and then obtain a weak convergence theorem. Using this result, we obtain a weak convergence theorem for a pair of a nonexpansive mapping and a strictly pseudocontractive mapping. Further, we consider the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping.     

The local homeomorphism property of spatial quasiregular mappings with distortion close to one

We give a quantitative proof for a theorem of Martio, Rickman and V?is?l? [MRV] on the rigidity of the local homeomorphism property of spatial quasiregular mappings with distortion close to one. The proof is based on a distortion theory established by using two main tools. First, we use a conformal invariant between sphere families and components of their preimages under quasiregular mappings. Secondly, we use Hall’s quantitative isoperimetric inequality result [H] to relate two different types of distortion. Received: April 2004 Revision: October 2004 Accepted: December 2004     

Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process     

一类积集上的广义向量变分不等式

研究一类积集上具某种权向量的广义向量变分不等式组及其广义向量变分不等式的有关问题,刻画它们之间解的相互关系.在映射的次连续性和关于某向量广义单调性的条件下,利用集值映射的不动点定理,对所讨论的几种类型的广义向量变分不等式给出解的存在性.