General Mixed Variational Inequalities Research Articles

An inexact operator splitting method for general mixed variational inequalities

The present paper aims to deal with an inexact implicit method with a variable parameter for generalmixed variational inequalities in the setting of real Hilbert spaces. Under standard assumptions, the globalconvergence of the proposed method is proved. Numerical example is presented to illustrate the proposedmethod and convergence result. The results and method presented in this paper generalize, extend and unifysome known results in the literature.

Strong and weak convergence theorems for general mixed equilibrium problems and general variational inequality problems and fixed point problems for two nonexpansive semigroups in Hilbert spaces

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of the general mixed equilibrium problem and the set of solutions of a general variational inequality for two cocoercive mappings and the set of common fixed points of two nonexpansive semigroups in Hilbert space. We obtain both strong and weak convergence theorems for the sequences generated by these iterative processes in Hilbert spaces. Our results improve and extend the results announced by many others.

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.

A Perturbed Iterative Algorithm for Split General Mixed Variational Inequality Problem

In this paper, we introduce a split general mixed variational inequality problem, which is a natural extension of a split variational inequality problem, split general quasi-variational inequality problem in Hilbert spaces. Using the resolvent operator technique, we propose a perturbed iterative algorithm for the split general mixed variational inequality problem. Further, we discuss the convergence criteria of the iterative algorithm. The results presented in this paper extend and improve many previously known results in this area.

Composite viscosity methods for common solutions of general mixed equilibrium problem, variational inequalities and common fixed points

In this paper, we introduce a new composite viscosity iterative algorithm and prove the strong convergence of the proposed algorithm to a common fixed point of one finite family of nonexpansive mappings and another infinite family of nonexpansive mappings, which also solves a general mixed equilibrium problem and a finite family of variational inequalities. An example is also provided in support of the main result. The main result presented in this paper improves and extends some corresponding ones in the earlier and recent literature.

A Resolvent Algorithm for System of General Mixed Variational Inequalities

In this paper, we suggest and analyze a new resolvent algorithm for finding the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point of a nonexpansive mapping in Hilbert spaces. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions. In this respect, our results present a refinement and improvement of the previously known results.

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

The original publication of the article contains errors which need to be amended as mentioned below: (1) In Section 2, “Formulations and basic facts”, lines 2 and 3, “Let T, g, h : H → H be three nonlinear single-valued operators” should be replaced by “Let T, g, h : H → H be three nonlinear single-valued operators such that g be an onto operator”. (2) In the problem (2.1), “ρφ(u)” must be changed to “ρφ(h(u))”. (3) In Section 6, part “(III)” must be edited as below: (III) Let the operators T and g be linear and suppose that the inverses of T and g, that is, T−1 and g−1 exist. Then (6.1) can be written as follows: T h−1 J φ z + ρ−1 Rφz = 0 ⇔ T (g−1(z − Rφz))+ ρ−1 Rφz = 0 ⇔ g−1(z − Rφz) = T−1(−ρ−1 Rφz) ⇔ z − Rφz = g(−ρ−1T−1 Rφz)

Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative algorithms for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of finite variational inequalities for inverse strongly monotone mappings and the set of common fixed points of an asymptotically k-strictly pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems by using the proposed algorithms under some suitable conditions. Our results improve and extend the corresponding results announced by 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.

Hadamard well-posedness of a general mixed variational inequality in Banach space

In this paper, we first introduce the concept of Hadamard well-posedness of a general mixed variational inequality in Banach space. Under some suitable conditions, relations between Levitin---Polyak well-posedness and Hadamard well-posedness of a general mixed variational inequality are studied. We also establish some characterizations of Hadamard well-posedness for a genaral mixed variational inequality. Finally, we derive some conditions under which a general mixed variational inequality is Hadamard well-posed.

Parametric Extended General Mixed Variational Inequalities

It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results.

SOME NEW RESOLVENT METHODS FOR SOLVING GENERAL MIXED VARIATIONAL INEQUALITIES

In this paper, we propose two methods for solving general mixed variational inequalities. In the first method, the new iterate is obtained by using a descent direction while the second method can be viewed as an extension of the first one by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. It is proved theoretically that the lower-bound of the progress obtained by the second method is greater than that by the first one. Under certain conditions, the global convergence of the both methods is proved. The comparison of these methods with other methods for solving the mixed genera variational inequalities is an open interesting problem.

Existence and iterative approximations of solutions for strongly nonlinear variational-like inequalities

Abstract. In this paper, we introduce and study a new class of stronglynonlinear variational-like inequalities. Under suitable conditions, we provethe existence of solutions for the class of strongly nonlinear variational-like inequalities. By making use of the auxiliary principle technique, wesuggest an iterative algorithm for the strongly nonlinear variational-likeinequality and give the convergence criteria of the sequences generatedby the iterative algorithm. 1. IntroductionIt is well known that there are lots of iterative type algorithms for ndingthe approximate solutions of various variational inequalities in Hilbert spaces[3] and [8-13]. Among the most e ective numerical technique is the projectionmethod and its variant forms. However, the standard projection technique canno longer be applied to suggest the iterative type algorithm for variational-likeinequalities. This fact motivated Gowinski, Lions and Tremoliers [7] to developthe auxiliary principle technique, which does not depend on the projection.Ding [4,5] and Ding and Tan [6] extended the auxiliary principle technique tosuggest several iterative algorithms to compute approximate solutions for someclasses of general nonlinear mixed variational inequalities and variational-likeinequalities.In this paper, we introduce and study a new class of strongly nonlinearvariational-like inequalities. By applying a result due to Chang [1], we provethe existence of solutions for the class of strongly nonlinear variational-likeinequalities. Using the auxiliary principle technique, we suggest and analyzea new three-step iterative algorithm for solving the class of strongly nonlinearvariational-like inequalities. The convergence criteria of the sequence generatedby the iterative algorithm are given.

Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces

AbstractIn this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semi-continuous function, φ and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear set-valued (mixed) variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of φ and g.2000 AMS Subject Classification: 47H05; 47H09; 47J25; 65J15.

Iterative methods for solving extended general mixed variational inequalities

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.

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.

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.

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.

New inexact implicit method for general mixed quasi variational inequalities

In this paper, we suggest and analyze a new self-adaptive inexact implicit method with a variable parameter for general mixed quasi variational inequalities, where the skew-symmetry of the nonlinear bifunction plays a crucial part in the convergence analysis of this method. We use a self-adaptive technique to adjust parameter ? at each iteration. The global convergence of the proposed method is proved under some mild conditions. Preliminary numerical results indicate that the self-adaptive adjustment rule is necessary in practice.

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.