New Class Of Variational Inequalities Research Articles

MIXED QUASI VARIATIONAL INEQUALITIES INVOLVING FOUR NONLINEAR OPERATORS

In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the mixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

Regularized Nonconvex Mixed Variational Inequalities: Auxiliary Principle Technique and Iterative Methods

In this paper, we turn our attention to formulating and studying a new class of variational inequalities in a nonconvex setting, called regularized nonconvex mixed variational inequalities. By using the auxiliary principle technique, some new predictor corrector methods for solving such class of regularized nonconvex mixed variational inequalities are suggested and analyzed. The study of convergence analysis of the proposed iterative algorithms requires either pseudomonotonicity or partially mixed relaxed and strong monotonicity of the operator involved in regularized nonconvex mixed variational inequalities. As a consequence of our main results, we provide the correct versions of the algorithms and results presented in the literature.

Some remarks on regularized nonconvex variational inequalities

In this paper, we investigate and analyze the nonconvex variational inequalities introduced by Noor in (Optim. Lett. 3:411-418, 2009) and (Comput. Math. Model. 21:97-108, 2010) and prove that the algorithms and results in the above mentioned papers are not valid. To overcome the problems in the above cited papers, we introduce and consider a new class of variational inequalities, named regularized nonconvex variational inequalities, instead of the class of nonconvex variational inequalities introduced in the above mentioned papers. We also consider a class of nonconvex Wiener-Hopf equations and establish the equivalence between the regularized nonconvex variational inequalities and the fixed point problems as well as the nonconvex Wiener-Hopf equations. By using the obtained equivalence formulations, we prove the existence of a unique solution for the regularized nonconvex variational inequalities and propose some projection iterative schemes for solving the regularized nonconvex variational inequalities. We also study the convergence analysis of the suggested iterative schemes under some certain conditions. MSC:47H05, 47J20, 49J40, 90C33.

Penalization of history-dependent variational inequalities

The present paper represents a continuation of Sofonea and Matei's paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact. In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material's behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations. We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

Extended general bifunction variational inequalities

In this paper, we introduce and consider a new class of variational inequalities which is called the mixed extended general bifunction variational inequality. We suggest and analyze some proximal methods for solving the mixed extended general bifunction variational inequalities using the auxiliary principle technique. Convergence of these methods is considered under some mild suitable conditions. Several cases are also discussed. Results in this paper include some new and known results as special cases.

Some Aspects of Extended General Variational Inequalities

Noor (“Extended general variational inequalities,” 2009, “Auxiliary principle technique for extended general variational inequalities,” 2008, “Sensitivity analysis of extended general variational inequalities,” 2009, “Projection iterative methods for extended general variational inequalities,” 2010) introduced and studied a new class of variational inequalities, which is called the extended general variational inequality involving three different operators. This class of variational inequalities includes several classes of variational inequalities and optimization problems. The main motivation of this paper is to review some aspects of these variational inequalities including the iterative methods and sensitivity analysis. We expect that this paper may stimulate future research in this field along with novel applications.

Iterative Algorithms for General MultivaluedVariational Inequalities

We introduce and study some new classes of variational inequalities and the Wiener‐Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational in equalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed.

Iterative Schemes for a Class of Mixed Trifunction Variational Inequalities

We use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of variational inequalities, which is called the mixed trifunction variational inequality. The mixed trifunction variational inequality includes the trifunction variational inequalities and the classical variational inequalities as special cases. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of variational inequalities and its variant forms.

On nonconvex bifunction variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex bifunction variational inequality. We suggest and analyze some iterative methods for solving nonconvex bifunction variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only pseudomonotonicity, which is weaker condition than monotonicity. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.

Extragradient methods for solving nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which are called the nonconvex variational inequalities. Using the projection technique, we suggest and analyze an extragradient method for solving the nonconvex variational inequalities. We show that the extragradient method is equivalent to an implicit iterative method, the convergence of which requires only pseudo-monotonicity, a weaker condition than monotonicity. This clearly improves on the previously known result. Our method of proof is very simple as compared with other techniques.

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.

Bifunction hemivariational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the bifunction hemivariational inequality. This new class includes several classes of variational inequalities as special cases. A number of iterative methods for solving bifunction hemivariational inequalities are suggested and analyzed by using the auxiliary principle technique. We also study the convergence analysis of these iterative methods under some mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique.

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. Variational inequalities theory, which was introduced by Stampacchia [35], provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences. During the last three decades, there has been considerable activity in the development of numerical techniques for solving variational inequalities. There is a substantial number of numerical methods, including projection method and its variant forms, Wiener‐Hopf equations, auxiliary principle, and descent framework for solving variational inequalities and complementarity problems; see [1‐35] and the references therein. It is worth mentioning that almost all the results regarding the existence of and iterative schemes for variational inequalities that have been investigated and considered are for the case where the underlying set is a convex set. This is because all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general when the sets are nonconvex. Noor [25, 28] has introduced and studied a new class of variational inequalities, which is called the nonconvex variational inequality in conjunction with the uniformly prox-regular sets, which are nonconvex and include the convex sets as a special case, see [7, 33]. Noor [25, 28] has shown that the projection technique can be extended for the nonconvex variational inequalities and has established the equivalence between the nonconvex variational inequalities and fixed point problems using essentially the projection technique. This equivalent alternative formulation is used to discuss the existence of a solution of the nonconvex variational inequalities, which is Theorem 3.1. We use this alternative equivalent formulation to suggest and analyze an implicit type iterative method for solving the nonconvex variational inequalities. In order to implement this new implicit method, we use the predictor-corrector technique to suggest a two-step method for solving the nonconvex variational inequalities, which is Algorithm 3.4. We also consider the convergence (Theorem 3.2) of the new iterative method under some suitable conditions. We have also suggested three-step iterative methods for solving nonconvex variational inequalities. Some special cases are also discussed. We also introduce and consider the problem of solving the Wiener‐Hopf equations. Using essentially the projection technique, we show that the nonconvex variational inequalities are equivalent to the Wiener‐Hopf equations. This alternative equivalent formulation is more general and flexible than the projection operator technique. This alternative equivalent formulation is used to suggest and analyze a number of iterative methods for solving the nonconvex variational inequalities. These iterative methods are the subject of Section 4. We also consider the convergence criteria of the proposed iterative methods under some suitable conditions. Several special cases are

Projection methods for nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex variational inequalities. We establish the equivalence between the nonconvex variational inequalities and the fixed-point problems using the projection technique. This equivalent formulation is used to discuss the existence of a solution of the nonconvex variational inequalities. We also use this equivalent alternative formulation to suggest and analyze a new iterative method for solving the nonconvex variational inequalities. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.

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.

Mixed quasi regularized variational inequalities

In this paper, we introduce and study a new class of variational inequalities, known as mixed quasi regularized variational inequality in the setting of nonconvexity. We use the auxiliary principle technique to suggest and analyze some iterative schemes for regularized variational inequalities. 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 general variational inequalities involving the convex sets. Mathematics subject classification (2000): 49J40, 90C30.

Hemivariational-like inequalities

In this paper, we introduce and consider a new class of variational inequalities, known as the hemivariational-like inequalities. It is shown that the hemivariational-like inequalities include hemi...

Hemivariational-like inequalities

In this paper, we introduce and consider a new class of variational inequalities, known as the hemivariational-like inequalities. It is shown that the hemivariational-like inequalities include hemivariational inequalities, variational-like inequalities and the classical variational inequalities as special cases. The auxiliary principle is used to suggest and analyze some iterative methods for solving hemivariational-like inequalities under mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique.

Projection Methods for Quasi-Variational Inequalities

In this paper, we consider a new class of variational inequalities which is called the general mixed multivalued nonlinear quasi-variational inequality. We define an iterative algorithm and prove the existence solution for our general mixed multivalued nonlinear quasi-variational inequality by using the updating projection method and convergence criteria for iterative sequences generated by algorithm 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.