Projection iterative methods for solving some systems of general nonconvex variational inequalities

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.     

Projection algorithms for solving a system of general variational inequalities

In this paper, we introduce and consider a new system of general variational inequalities involving four different operators. Using the projection operator technique, we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving three operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.     

A parallel projection method for a system of nonlinear variational inequalities

In this paper, we introduce and consider a new system of nonlinear variational inequalities involving two different operators. Using the parallel projection technique, we suggest and analyze an iterative method for this system of variational inequalities. We establish a convergence result for the proposed method under certain conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

On General Mixed Variational Inequalities

In this paper, we introduce and consider a new class of mixed variational inequalities, which is called the general mixed variational inequality. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed-point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving the general mixed variational inequalities. We study the convergence criteria of the suggested iterative methods under suitable conditions. Using the resolvent operator technique, we also consider the resolvent dynamical systems associated with the general mixed variational inequalities. We show that the trajectory of the dynamical system converges globally exponentially to the unique solution of the general mixed variational inequalities. Our methods of proofs are very simple as compared with others’ techniques. Results proved in this paper may be viewed as a refinement and important generalizations of the previous known results.     

Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces

In this article, we introduce and consider a general system of variational inequalities. Using the projection technique, we suggest and analyse new iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving the single operator, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results improve and extend the recent ones announced by many others.     

The <Emphasis Type="Italic">p</Emphasis>-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with (<Emphasis Type="Italic">H</Emphasis>,η)-monotone operators

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with (H,η)-monotone operators which contains variational inequalities, variational inclusions, systems of variational inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (H,η)-monotone operators, we prove the existence of solutions and the convergence of some new p-step iterative algorithms for this system of generalized mixed quasi-variational inclusions and its special cases. The results in this paper unifies, extends and improves some known results in the literature.     

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.     

Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces

In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed iterative method under some suitable conditions.     

General variational inequalities and nonexpansive mappings

In this paper, we suggest and analyze some three-step iterative schemes for finding the common elements of the set of the solutions of the Noor variational inequalities involving two nonlinear operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of the suggested iterative schemes under some mild conditions. Since the Noor variational inequalities include variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.     

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.     

Resolvent Methods for Solving System of General Variational Inclusions

In this paper, we introduce and consider some new systems of extended general variational inclusions involving seven different operators. Using the resolvent operator technique, we show that the new systems of extended general variational inclusions are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some new iterative methods for this system of extended general variational inclusions. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also discussed. Results obtained in this paper can be viewed as pure mathematical contribution to variational analysis.     

Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators

In this paper, we introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities

Some new classes of extended general nonconvex set-valued variational inequalities and the extended general Wiener-Hopf inclusions are introduced. By the projection technique, equivalence between the extended general nonconvex set-valued variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener-Hopf inclusions is proved. Then by using this equivalent formulation, we discuss the existence of solutions of the extended general nonconvex set-valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set-valued variational inequalities. We also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set-valued variational inequalities. The results presented in this paper extend and improve some known results from the literature.     

Regularized mixed quasi equilibrium problems

In this paper, we introduce and study a new class of equilibrium problems, known as regularized mixed quasi equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for regularized equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving equilibrium problems and variational inequalities involving the convex sets.     

Mixed quasi nonconvex equilibrium problems

In this paper, we introduce and study a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for solving nonconvex equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier known results for solving equilibrium problems and variational inequalities involving the convex sets.     

一般多值混合隐拟变分不等式的解的存在性与算法

引入了实Hilbert空间中一类新的一般多值混合隐拟变分不等式.它概括了丁协平教授引入与研究过的熟知的广义混合隐拟变分不等式类成特例.运用辅助变分原理技巧来解这类一般多值混合隐拟变分不等式.首先,定义了具真凸下半连续的二元泛函的新的辅助变分不等式,并选取了一适当的泛函,使得其唯一的最小值点等价于此辅助变分不等式的解.其次,利用此辅助变分不等式,构造了用于计算一般多值混合隐拟变分不等式逼近解的新的迭代算法.在此,等价性保证了算法能够生成一列逼近解.最后,证明了一般多值混合隐拟变分不等式解的存在性与逼近解的收敛性.而且,给算法提供了新的收敛判据.因此,结果对M.A.Noor提出的公开问题给出了一个肯定答案,并推广和改进了关于各种变分不等式与补问题的早期与最近的结果,包括最近文献中涉及单值与集值映象的有关混合变分不等式、混合拟变不等式与拟补问题的相应结果.     

The existence and uniqueness of solution and the convergence of a multi-step iterative algorithm for a system of variational inclusions with (<Emphasis Type="Italic">A</Emphasis>, <Emphasis Type="Italic">η</Emphasis>, <Emphasis Type="Italic">m</Emphasis>)-accretive operators

In this paper, we introduce and study a new system of variational inclusions with (A, η, m)-accretive operators which contains variational inequalities, variational inclusions, systems of variational inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (A, η, m)-accretive operators, we prove the existence and uniqueness of solution and the convergence of a new multi-step iterative algorithm for this system of variational inclusions in real q-uniformly smooth Banach spaces. The results in this paper unifies, extends and improves some known results in the literature.      

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.     

Some Existence Results of Solutions for General Variational Inequalities

In this paper, we introduce a new class of operators. We present some fundamental properties of the operators belonging to this class and, as applications, we establish some existence results of the solutions for several general variational inequalities involving elements belonging to this class.