Nonlinear Mixed Variational-like Inequalities Research Articles

Solvability of the System of Extended Nonlinear Mixed Variational-Like Inequalities and Proximal Dynamical System

In this article, our goal is to study the system of extended nonlinear mixed variational-like inequalities (in short, SENMVLI) with a nonconvex functional in the setting of real Hilbert spaces and discuss the existence of solution of our considered problem. We propose a three-step iterative algorithm to calculate the approximate solutions of SENMVLI and investigate the convergence analysis as well as stability analysis of the proposed algorithm. Furthermore, we also study the proximal dynamical system for SENMVLI and prove that the trajectory of the solution of the extended proximal dynamical system converges globally exponentially to a unique solution of SENMVLI. Our suggested iterative algorithm and results have become the significant improvement, enhancement, and generalization of many previously known results in the literature.

Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces

We study a new class of mixed equilibrium problem, in short MEP, under weakly relaxed $ \alpha $-monotonicity in Banach spaces. This class of problems extends and generalizes some related fundamental results such as mixed variational-like inequalities, variational inequalities, and classical equilibrium problems as special cases. Existence and uniqueness of the solution to the problem is established. Auxiliary principle technique is used to obtain an iterative algorithm. Solvability of the auxiliary problem is established in the paper and finally the convergence of the iterates to the exact solution is proved. As applications of the approach developed in this paper, we study the existence and algorithmic approach for a general class of nonlinear mixed variational-like inequalities. The results obtained in this paper are interesting and improve considerably many existing results in literature.

Auxiliary Principle for a System of Generalized Set-Valued Nonlinear Mixed Variational-Like Inequalities in Banach Spaces

In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.

Generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map

In this paper, we introduce and study a new class of generalized nonlinear vector mixed quasi-variational-like inequalities governed by a multi-valued map in Hausdorff topological vector spaces which includes generalized vector mixed general quasi-variational-like inequalities, generalized nonlinear mixed variational-like inequalities, and so on. By using the fixed point theorem, we prove some existence theorems for the proposed inequality.

Generalized nonlinear variational-like inequalities in reflexive Banach spaces

In this paper, we introduce and study a new class of generalized nonlinear mixed variational-like inequalities in reflexive Banach spaces. By applying the auxiliary principle technique and the minimax inequality, we establish the existence and uniqueness theorems for solutions of generalized nonlinear mixed variational-like inequalities. Copyright © 2010 John Wiley & Sons, Ltd.

Existence and iterative algorithm of solutions for a system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems in Banach spaces

In this paper, we introduce and study a new system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Firstly, we prove the existence and uniqueness of solutions of the auxiliary problems for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Secondly, an iterative algorithm for solving the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems is constructed by using this existence and uniqueness result. Finally, we show the existence of solutions of the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and discuss the convergence analysis of this algorithm. These results improve, unify and generalize many corresponding known results given in literatures.

Auxiliary Principle for Generalized Strongly Nonlinear Mixed Variational-Like Inequalities

We introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities, which includes several classes of variational inequalities and variational-like inequalities as special cases. By applying the auxiliary principle technique and KKM theory, we suggest an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequality. The existence of solutions and convergence of sequence generated by the algorithm for the generalized strongly nonlinear mixed variational-like inequalities are obtained. The results presented in this paper extend and unify some known results.

Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces

In this paper, we introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities. First, the existence of solutions of the auxiliary problems for the generalized strongly nonlinear mixed variational-like inequalities is shown. Second, an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequalities is constructed by using this existence result. Finally, not only the existence of solutions of the generalized strongly nonlinear mixed variational-like inequalities is shown, but also the convergence of iterative sequences generated by the algorithm is also proven.

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle technique. The convergence of three-step iterative sequences generated by the algorithm is also proved.

Iterative algorithm for completely generalized set-valued strongly nonlinear mixed variational-like inequalities

A class of completely generalized set-valued strongly nonlinear mixed variational-likeinequalities is introduced. The auxiliary principle technique is extended to study this new class of mixed variational-like inequalities. The existence of a solution of the auxiliary problem for this new class of mixed variational-like inequalities is shown. The iterative algorithm for this new class of mixed variational-like inequalities is given by virtue of this existence result. Moreover, the existence of a solution of this new class of mixed variational-like inequalities and the strong convergence of iterative sequences generated by the algorithm are shown. The convergence criteria are different from some early and recent ones presented in the literature.

Iterative Algorithm for Generalized Set-Valued Strongly Nonlinear Mixed Variational-Like Inequalities

The auxiliary principle technique is extended to study the generalized strongly nonlinear mixed variational-like inequality problem for set-valued mappings without compact values. We establish first the existence of a solution of the related auxiliary problem. Then, the iterative algorithm for solving that problem is given by using this existence result. Moreover, the existence of a solution of the original problem and the convergence of iterative sequences generated by the algorithm are both derived.

Existence and algorithm of solutions for nonlinear mixed variational-like inequalities in Banach spaces

In this paper, we study a new class of nonlinear mixed variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality due to author, we first prove an existence uniqueness theorem of solutions for the nonlinear mixed variational-like inequalities. Next, we extend the auxiliary problem technique under reflexive Banach space setting to suggest and analyze an innovative iterative algorithm to compute the approximate solutions of the nonlinear mixed variational-like inequalities. The convergence criteria of the iterative algorithm is also given. These results improve and generalize many known results in recent literature.

General Nonlinear Mixed Variational-Like Inequality Problem Involving Relaxed Lipschitz Operators

In this article, we use the auxiliary principle technique to prove the existence, uniqueness theorem of solutions for the general nonlinear mixed variational-like inequality and special cases, which can be obtained from our main result are also discussed.

Auxiliary Principle and Iterative Algorithms for Generalized Set-Valued Strongly Nonlinear Mixed Variational-like Inequalities

In this paper, we extend the auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities. We prove the existence of a solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities, construct the iterative algorithm for the generalized set-valued strongly nonlinear mixed variational-like inequalities, and show the existence of a solution of the generalized set-valued strongly nonlinear mixed variational-like inequality by using the auxiliary principle technique. We also prove the convergence of iterative sequences generated by the algorithm.

General nonlinear mixed variational-link inequalities

In this paper. we introduce and study a new class of variational inequalities which is called the general strongly nonlinear mixed variational-like inequalities. This class is the most general and unifying one , The auxiliary principle technique is used to prove the unique existence of a solution of the variational-like inequality and to suggest a novel and innovative iterative algorithm. We also study the convergence analysis of the iterative algorithm. Several special cases. which can be obtained from our main results, are also discussed