广义混合隐拟h变分不等式的解的存在性与算法

本文在Hilbert空间中,引入了一类广义混合隐拟h变分不等式.运用变分原理,给出了广义混合隐拟h变分不等式逼近解的迭代算法,证明了这类变分不等式解的存在性定理,同时,得到迭代序列的收敛性.并改进和推广了[6~8]一些已知结果.     

关于混合拟似变分包含问题的解的迭代算法

本文对混合拟似变分包含问题提出新的辅助变分不等式,首先证明辅助变分不等式存在唯一解.然后,通过这一辅助形式建立混和拟似变分包含问题解的迭代算法.最后讨论在新的算法下迭代解的收敛性.     

广义非线性集值混合拟变分包含的扰动近似点算法

本文研究一类广义非线性集值混合拟变分包含,概括了尚明生等人引入与研究过的熟知的广义集值变分包含类成特例.运用预解算子的技巧,建立了广义非线性集值混合拟变分包含与不动点问题之间的等价性,其中,预解算子JρA(·,x)是具有常数1/(1+cρ)的Lipschitz连续算子.本文还建立了几个扰动迭代算法,并提供了由算法生成的逼近解的收敛判据,所得算法与结果改进与推广了尚明生等人的相应算法与结果.     

广义集值强非线性混合似变分不等式的辅助原理和三步迭代算法

使用辅助原理技巧研究了一类广义集值强非线性混合变分不等式．证明了此类集值强非线性混合变分不等式辅助问题解的存在性和唯一性;构建了一个新的三步迭代算法,通过辅助原理技巧,构建并计算此类非线性混合变分不等式的近似解,进一步证明非线性混合变分不等式解的存在性以及由算法产生的三个序列的收敛性．所得结论推广了近年来许多混合变分不等式和准变分不等式以及他们的有关结果．     

广义多值拟变分包含解的两步迭代

本文引入并研究了Hilbert空间中的一类广义多值拟变分包含问题.借助预解算子技巧构造了一个新的两步迭代算法来逼近广义多值拟变分包含的解,并且证明了其解的存在性以及迭代算法生成的迭代序列的收敛性.     

一般混合似变分不等式的隐式迭代算法

对一般混合似变分不等式的若干隐式迭代算法进行了研究;利用一般混合似变分不等式与不动点问题和预解方程的等价关系,采用分裂技巧和自适应迭代技巧结合,提出了一个求解一般混合似变分不等式的新的隐式迭代算法;并证明了该算法在算子T是g-单调连续的条件下收敛．     

求解广义混合隐拟平衡问题的预测修正算法

引入和研究了一类新的具有四元函数的广义混合隐拟平衡问题．这类平衡问题包含了很多已知的广义平衡问题和广义混合隐拟变分不等式问题作为很特殊的情形．利用辅助原理技巧建议和分析了求解广义混合隐拟平衡问题的预测修正迭代算法．所建议算法的收敛性仅需要映象的连续性和部分松弛强单调性．     

广义单调集值混合变分不等式的一类新算法

首先证明了广义单调集值混合变分不等式等价于一个新的不动点问题，在此基础上提出了解广义集值混合变分不等式及其相关优化问题的迭代算法，并给出了这类新算法的收敛性分析，我们的结果推广和综合了该领域的一些最新结论．     

广义混合拟变分不等式解的存在性及算法

应用辅助变分不等式技巧研究一类广义混合拟变分不等式解的存在性和迭代算法.所得到的结果回答了Noor提出的公开问题,改进和推广了一些较近的已知结果.     

集值混合变分不等式的一类自适应惯性投影次梯度算法

本文在实Hilbert空间上引入了一类求解集值混合变分不等式新的自适应惯性投影次梯度算法.在集值映射T为f-强伪单调或单调的条件下,我们证明了由该自适应惯性投影次梯度算法所产生的序列强收敛于集值混合变分不等式问题的的唯一解.     

Iterative methods for solving general quasi-variational inequalities

In this paper, we introduce a new class of variational inequalities, which is called the general quasi-variational inequality. We establish the equivalence among the general quasi variational inequality and implicit fixed point problems and the Wiener–Hopf equations. We use this equivalent formulation to discuss the existence of a solution of the general quasi-variational inequality. This equivalent formulation is used to suggest and analyze some iterative algorithms for solving the general quasi-variational inequality. We also discuss the convergence analysis of these iterative methods. Several special cases are also discussed.     

Set-valued mixed quasi-variational inequalities and implicit resolvent equations

In this paper, we introduce and study a new class of variational inequalities, which is called the set-valued mixed quasi-variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued mixed quasi-variational inequalities, fixed-point problems and the set-valued implicit resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.     

广义隐向量拟变分不等式解的存在性

本文研究一类广义隐式向量拟变分不等式问题,利用Fan-Kakutani不动点定理证明其解存在,推广了相关的文献中的结论.     

Merit functions for general mixed quasi-variational inequalities

In this paper, we present some merit functions for general mixed quasi-variational inequalities, and we obtain the equivalent optimization problems to general mixed quasi-variational inequalities. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, our results continue to hold for these problems. In this respect, results obtained in this paper represent an extension of previously known results.     

A Class of Generalized Evolutionary Problems Driven by Variational Inequalities and Fractional Operators

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators.

     

无限维Hillbert空间中单调变分不等式解的近似迭代算法

在无穷维Hillbert空间中研究了一类单调型变分不等式，把求单调型变分不等式解的问题转化为求强单调变分不等式的解，建立了一种新的迭代算法，并证明了由算法生成的迭代序列强收敛于单调变分不等式的解，从而推广了所列文献中的许多重要结果．     

Duality for implicit variational problems and numerical applications

The present paper deals with a dual characterization of the solutions of implicit variational problems. Some general results relating the solutions of the dual problem with those of the primal one are applied to variational and quasi-variational inequalities, Nash equilibria, saddle points and fixed points.

Finally a dual method for the numerical solutions of some quasi-variational inequalities is developed.