Banach空间中Noor型变分不等式的广义投影法

该文提出并分析了用广义投影方法解Noor型广义变分不等式问题.在较弱的条件下考虑了一个迭代格式的收敛性.由于广义变分不等式包含了许多变分不等式和相补问题为特例,因此该文得出的结果可以应用到这些问题中.这些结果是以前众多学者所做工作的完善和改进.     

Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers.     

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·     

Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces

The purpose of this paper is to suggest and analyze a number of iterative algorithms for solving the generalized set-valued variational inequalities in the sense of Noor in Hilbert spaces. Moreover, we show some relationships between the generalized set-valued variational inequality problem in the sense of Noor and the generalized set-valued Wiener-Hopf equations involving continuous operator. Consequently, by using the equivalence, we also establish some methods for finding the solutions of generalized set-valued Wiener-Hopf equations involving continuous operator. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory.     

Some iterative schemes for general mixed variational inequalities

In this paper, we consider a class of variational inequalities which is called the general mixed variational inequality. It is known that the general mixed variational inequalities are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some three-step iterative schemes for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the mixed variational inequalities. We also study the convergence criteria of three-step iterative method under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.     

Three-step iterations for variational inequalities and nonexpansive mappings

In this paper, we introduce a new three-step iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality using the technique of updating the solution. We show that the sequence converges strongly to a common element of two sets under some control conditions. Results proved in this paper may be viewed as an improvement and refinement of the recent results of Noor and Huang [M. Aslam Noor, Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput., in press] and Yao and Yao [Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput., in press].     

A self-adaptive method for solving general mixed variational inequalities

The general mixed variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method cannot be applied to solve this problem due to the presence of the nonlinear term. To overcome this disadvantage, Noor [M.A. Noor, Pseudomonotone general mixed variational inequalities, Appl. Math. Comput. 141 (2003) 529-540] used the resolvent equations technique to suggest and analyze an iterative method for solving general mixed variational inequalities. In this paper, we present a new self-adaptive iterative method which can be viewed as a refinement and improvement of the method of Noor. Global convergence of the new method is proved under the same assumptions as Noor's method. Some preliminary computational results are given.     

Generalized multivalued nonlinear quasivariational inclusions

In this paper, we introduce and study a few classes of generalized multivalued nonlinear quasivariational inclusions and generalized nonlinear quasivariational inequalities, which include many classes of variational inequalities, quasivariational inequalities and variational inclusions as special cases. Using the resolvent operator technique for maximal monotone mapping, we construct some new iterative algorithms for finding the approximate solutions of these classes of quasivariational inclusions and quasivariational inequalities. We establish the existence of solutions for this generalized nonlinear quasivariational inclusions involving both relaxed Lipschitz and strongly monotone and generalized pseudocontractive mappings and obtain the convergence of iterative sequences generated by the algorithms. Under certain conditions, we derive the existence of a unique solution for the generalized nonlinear quasivariational inequalities and obtain the convergence and stability results of the Noor type perturbed iterative algorithm. The results proved in this paper represent significant refinements and improvements of the previously known results in this area.     

A general iterative method for addressing mixed equilibrium problems and optimization problems

In this paper, we introduce a new general iterative method for finding a common element of the set of solutions of a mixed equilibrium problem (MEP), the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of variational inequalities for a ξ-inverse-strongly monotone mapping in Hilbert spaces. Furthermore, we establish the strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under some suitable conditions, which solves some optimization problems. Our results extend and improve the recent results of Yao et al. [Y. Yao, M.A. Noor, S. Zainab, Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl. 354 (2009) 319-329; Y. Yao, M. A. Noor, Y.C. Liou, On iterative methods for equilibrium problems, Nonlinear Anal. 70 (1) (2009) 479-509] and many others.     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.