Mixed quasi variational inequalities

Computers & Mathematics with Applications, 2011

In this paper, we introduce and consider a new class of mixed variational inequalities involving four operators, which are called extended general mixed variational inequalities. Using the resolvent operator technique, we establish the equivalence between the extended general mixed variational inequalities and fixed point problems as well as resolvent equations. We use this alternative equivalent formulation to suggest and analyze some iterative methods for solving general mixed variational inequalities. We study the convergence criteria for the suggested iterative methods under suitable conditions. Our methods of proof are very simple as compared with other techniques. The results proved in this paper may be viewed as refinements and important generalizations of the previous known results.

Optimization Letters, 2010

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

Korean Journal of Computational & Applied Mathematics, 2002

In this paper, we use the dynamical systems technique to suggest and investigate some inertial proximal methods for solving mixed variational inequalities and related optimization problems. It is proved that the convergence analysis of the proposed methods requires only the monotonicity. Some special cases are also considered. Our method of proof is very simple as compared with other techniques. Ideas and techniques of this paper may be extended for other classes of variational inequalities and equilibrium problems.

2016

In this paper, we introduce and study a new class of quasi variational inequalities, known as multivalued extended general quasi variational inequalities. It is shown that the multivalued extended general quasi variational inequalities are equivalent to the fixed point problems. We use this alternative equivalent formulation to suggest and analyze some iterative methods. We consider the convergence analysis of an iterative method under suitable conditions. We also introduce a new class of Wiener-Hopf equations, known as multivalued extended general implicit Wiener-Hopf equations. We establish the equivalence between the multivalued extended general quasi variational inequalities and multivalued extended general implicit Wiener-Hopf equations. Using this equivalence, we suggest and analyze some iterative methods. Several special cases are also discussed. The ideas and techniques of this paper may stimulate further research in this field.

International Journal of the Physical Sciences, 2011

In this paper, we use the auxiliary principle technique coupled with the principle of iterative regularization to suggest and analyze some new iterative algorithms for solving mixed quasi variational inequalities. We also study the convergence criteria of these algorithms under some suitable and mild conditions. Several special cases are also considered. Results obtained in this paper continue to hold for these special cases.

Journal of Mathematical Analysis and Applications, 2005

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.

Computers & Mathematics with Applications, 2000

An iterative method for solving general mixed variational inequalities is suggested by using the auxiliary principle technique. The convergence of the proposed method only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. As special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems. (~

Applied Mathematics and Computation, 2004

In this paper, we use the auxiliary principle technique to suggest and analyze some iterative methods for solving generalized mixed quasi-variational-like inequalities. We show that the convergence of the proposed method requires either pseudomonotonicity or partially relaxed strongly monotonicity under some mild condition. As special cases, one can obtain a number of new and known algorithms for solving variational inequalities and complementarity problems. Our results represent significant and important refinements of the previously known results.

In this paper, we consider a new class of quasi variational inequalities involving three operators, which is called the extended general quasi variational inequality. It is shown that the extended general quasi variational inequalities are equivalent to the fixed problems. This equivalence is used to suggest and analyze some iterative methods for solving the extended general quasi variational inequalities. Convergence analysis is also considered. We have also shown that the extended general quasi variational inequalities are equivalent to the extended general implicit Wiener-Hopf equations. This alternative formulation is used to suggest and analyze some iterative methods. The convergence analysis of these new methods under some suitable conditions is investigated. Several special cases are discussed. Since the extended general quasi variational inequalities include general variational inequalities, quasi variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. Results of this paper may stimulate further research in this fascinating area.

Computers & Mathematics with Applications, 2000

In this paper, we introduce and study a new class of quasi-variational inequalities, which is called the generalized nonlinear set-valued mixed quasi-variational inequality. Using the resolvent operator technique for maximal monotone mapping, we construct some new iterative algorithms for solving this class of generalized nonlinear set-valued mixed quasi-variational inequalities. We prove the existence of solution for this kind of generalized nonlinear set-valued mixed quasivariational inequalities without compactness and the convergence of iterative sequences generated by the algorithms. We also discuss the convergence and stability of perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities.