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.     

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.     

Iterative Algorithm with Mixed Errors for Solving a New System of Generalized Nonlinear Variational-Like Inclusions and Fixed Point Problems in Banach Spaces

A new system of generalized nonlinear variational-like inclusions involving Amaximal m-relaxed η-accretive(so-called,(A, η)-accretive in [36]) mappings in q-uniformly smooth Banach spaces is introduced, and then, by using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings due to Lan et al., the existence and uniqueness of a solution to the aforementioned system is established. Applying two nearly uniformly Lipschitzian mappings S1 and S2 and using the resolvent operator technique associated with A-maximal m-relaxed η-accretive mappings, we shall construct a new perturbed N-step iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q =(S1, S2) which is the unique solution of the aforesaid system. We also prove the convergence and stability of the iterative sequence generated by the suggested perturbed iterative algorithm under some suitable conditions. The results presented in this paper extend and improve some known results in the literature.     

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.     

Extended general variational inequalities

In this work, we introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. Noor [M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.     

On a system of extended general variational inclusions

At the present paper, a new system of extended general nonlinear variational inclusions is introduced and using the resolvent operator technique, equivalence between the aforesaid system and the fixed point problem is verified. By using this alternative equivalent formulation, the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions is demonstrated and two new iterative schemes for solving this system of extended general nonlinear variational inclusions are suggested and analyzed. The convergence analysis of the proposed iterative methods under some suitable conditions is studied. Some errors in a recent article by Noor et al. (J Inequal Appl 2011:10, 2011) are found and the incorrectness of the results of the cited paper is proved. Also, the results of the aforementioned paper are revised and corrected.     

On a system of general mixed variational inequalities

In this paper, we introduce and consider a new system of general mixed variational inequalities involving three different operators. Using the resolvent operator technique, we establish the equivalence between the general mixed variational inequalities and the fixed point problems. We use this equivalent formulation to suggest and analyze some new explicit iterative methods for this system of general mixed 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 mixed variational inequalities involving two 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.     

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.     

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.     

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.     

Self-adaptive methods for mixed quasi-variational inequalities

It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed-point problems and the resolvent equations. In this paper, we use these alternative equivalent formulations to suggest and analyze some new self-adaptive iterative methods for solving mixed quasi-variational inequalities. Our results can be viewed as significant extensions of the previously known results for mixed quasi-variational inequalities. An example is given to illustrate the efficiency of the proposed method.     

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.     

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.     

Set-Valued Resolvent Equations and Mixed Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized set-valued mixed variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued variational inequalities, fixed point problems, and the generalized set-valued 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.     

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.     

Some iterative methods for nonconvex variational inequalities

It is well known that the nonconvex variational inequalities are equivalent to the fixed point problems. We use this equivalent alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational inequalities. We discuss the convergence of the iterative method under suitable conditions. We also introduce a new class of Wiener – Hopf equations. We establish the equivalence between the nonconvex variational inequalities and the Wiener – Hopf equations. This alternative equivalent formulation is used to suggest some iterative methods. We also consider the convergence analysis of these iterative methods. Our method of proofs is very simple compared to other techniques.     

Resolvent dynamical systems for mixed variational inequalities

In this paper, we suggest and analyze a class of resolvent dynamical systems associated with mixed variational inequalities. We study the globally asymptotic stability of the solution of the resolvent dynamical systems for the pseudomonotone operators. We also discuss some special cases, which can be obtained from our main results.     

A NEW SELF-ADAPTIVE ITERATIVE METHOD FOR GENERAL MIXED QUASI VARIATIONAL INEQUALITIES

The general mixed quasi variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method can not be applied to solve this problem due to the presence of nonlinear term. It is well known that the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. In this article, the authors use these alternative equivalent formulations to suggest and analyze a new self-adaptive iterative method for solving general mixed quasi variational inequalities. Global convergence of the new method is proved. An example is given to illustrate the efficiency of the proposed method.     

Split systems of general nonconvex variational inequalities and fixed point problems

The purpose of this paper is to introduce and study split systems of general nonconvex variational inequalities. Taking advantage of the projection technique over uniformly prox-regularity sets and utilizing two nonlinear operators, we propose and analyze an iterative scheme for solving the split systems of general nonconvex variational inequalities and fixed point problems. We prove that the sequence generated by the suggested iterative algorithm converges strongly to a common solution of the foregoing split problem and fixed point problem. The result presented in this paper extends and improves some well-known results in the literature. Numerical example illustrates the theoretical result.     

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.