New results on projection algorithms for solving systems of general variational inequalities

Projection algorithms for solving a system of general variational inequalities

Projection algorithms for the system of mixed 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.

We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces. Our main result extends a recent result acheived by Yao, Noor, Noor, Liou, and Yaqoob.

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.

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex 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 nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.

An explicit projection method for a system of nonlinear variational inequalities with different [formula omitted]-cocoercive mappings

In this paper, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem using the projection technique. This alternative equivalent formulation is used to suggest and analyze some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also considered. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

In this article, we introduce and consider a new system of mixed variational inequalities involving two different operators. Using the resolvent operator technique, we suggest and analyse some new explicit iterative methods for this system of 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 variational inequalities involving the single operator, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

The goal of this paper is to study a new system of a class of variational inequalities termed as absolute value variational inequalities. Absolute value variational inequalities present a rational, pragmatic, and novel framework for investigating a wide range of equilibrium problems that arise in a variety of disciplines. We first develop a system of absolute value auxiliary variational inequalities to calculate the approximate solution of the system of absolute variational inequalities, and then by employing the projection technique, we prove the existence of solutions of the system of absolute value auxiliary variational inequalities. By utilizing an auxiliary principle and the existence result, we propose several new iterative algorithms for solving the system of absolute value auxiliary variational inequalities in the frame of four different operators. Furthermore, the convergence of the proposed algorithms is investigated in a thorough manner. The efficiency and supremacy of the proposed schemes is exhibited through some special cases of the system of absolute value variational inequalities and an illustrative example. The results presented in this paper are more general and rehash a number of some previously published findings in this field.

In 1968, Brézis [Ann. Inst. Fourier (Grenoble), 18(1) (1968) 115‐175] initiated the study of the existence theory of a class of variational inequalities later known as variational inclusions, using proximal‐point mappings due to Moreau [Bull. Soc. Math. France, 93 (1965) 273‐299]. Variational inclusions include variational, quasi‐variational, variational‐like inequalities as special cases.In 1985, Pang [Math. Prog. 31 (1985) 206‐219] showed that a variety of equilibrium models can be uniformly modelled as a variational inequality defined on the product sets equivalent to a system of variational inequalities and discuss the convergence of method of decomposition for system of variational inequalities.Motivated by the recent research work in this directions, we consider some systems of variational (‐like) inequalities and inclusions; develop the iterative algorithms for finding the approximate solutions and discuss their convergence criteria. Further, we study the sensitivity analysis of solution of the system of variational inclusions. The techniques and results presented here improve the corresponding techniques and results for the variational inequalities and inclusions in the literature. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

This article deals with a system of quasilinear elliptic variational inequalities whose leading differential operator is a diagonal (p 1, p 2)-Laplace operator with 1 < p 1, p 2 < ∞, and whose lower order vector field f = (f 1, f 2) is a gradient field, which is not subject to any growth restriction, in particular, it may have supercritical growth, and thus coercivity is violated. The novelty of this article is to establish a variational approach of the, in general, noncoercive elliptic system of variational inequalities by introducing the concept of trapping region for such type of problems. The trapping region allows us to transform the given noncoercive system of variational inequalities into an associated ‘truncated’ system to which variational methods can be applied. We are going to prove that the ‘truncated’ system possesses solutions, which can be characterized as the critical points of a suitably constructed (nonsmooth) energy functional, and any critical point is shown to be a solution of the original problem within the trapping region. Moreover, applications to quasilinear elliptic systems under Dirichlet boundary conditions as well as to elliptic systems under obstacle constraints are treated by the theory developed in this article. †Dedicated to Professor R.P. Gilbert on the occasion of his 80th birthday.

Enclosure results for quasilinear systems of variational inequalities

In this article, we introduce and consider a new system of general nonconvex variational inequalities defined on uniformly prox-regular sets. We establish the equivalence between the new system of general nonconvex variational inequalities and the fixed point problems to analyze an explicit projection method for solving this system. We also consider the convergence of the projection method under some suitable conditions. Results presented in this article improve and extend the previously known results for the variational inequalities and related optimization problems.

We consider a new system for generalized variational inclusions in Hilbert spaces and define an iterative algorithm for finding the approximate solutions of this class of system of variational inclusions. We also establish that the approximate solutions obtained by our algorithm converge to the exact solution of new system of generalized variational inclusions. One can explore the role of our system of generalized variational inclusions for solving various known equilibrium problem and other related problems. References R. Ahmad and Q. H. Ansari, An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett. 13(5), 23--26, (2000). J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. X. P. Ding and C. L. Luo, Perturbed proximal point algorithms for general quasi-variational-like inclusions, J. Comput. Appl. Math. 113, 153--165, (2000). C. H. Lee, Q. H. Ansari and J. C. Yao, A perturbed algorithm for strongly nonlinear variational inclusions, Bull. Austral. Math. Soc. 62, 417--426, (2000). J. S. Pang, Asymmetric variational inequality problem over product of sets:Applications and iterative methods, Math Prog. 31, 206--219, (1985). R. U. Verma, Partially relaxed monotone mappings and a new system of nonlinear variational inequalities, Nonlinear Funct. Anal. Appl. 5(1), 65--72, (2000). R. U. Verma, Projection method, Algorithm and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41(7-8), 1025--1031, (2001). R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theory Appl. 121(1), 203--210, (2004). R. P. Agarwal, Y. J. Cho and N. J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13(6), 19--24, (2000). R. P. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J. Inequal. Appl. 7(6), 807--828, (2002). R. P. Agarwal, N. J. Huang and M. Y. Tan, Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions, Appl. Math. Lett. 17, 345--352, (2004). Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Austral. Math. Soc. 59(3), 433--442, (1999).