In this paper, we introduce two concepts of weakly relaxed ${eta_gamma-alpha_gamma}_{gamma in Gamma}$ pseudomonotone and demipseudomonotone mappings in Banach spaces. Then we obtain some results of the solutions existence for a system of vector variational-like inequalities with weakly relaxed ${eta_gamma-alpha_gamma}_{gamma in Gamma}$ pseudomonotone and demipseudomonotone mappings in reflexive...

The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization problems (Refs. 1-3) and variational inequalities (Ref. 4). The key assumption to prove the global and strong convergence of such algorithms, as well as of most of the other algorithms...

In this paper, we consider a new class of generalized variational inequalities and a new class of generalized Wiener-Hopf equations involving set-valued and nonexpansive mappings in real Hilbert space. Using the projection technique, we show that the generalized variational inequalities are equivalent to the generalized Wiener-Hopf equations. Further, by using this alternative equivalence we su...

In this paper we propose an appealing inexact proximal alternating directions method (abbreviated as In-PADM) for solving a class of monotone variational inequalities with certain special structure, and this structure under consideration is common in practice. We prove convergence of In-PADM method while the inexact term is arbitrary but satisfied some suitable conditions. For solving the varia...

In this paper, we propose two new methods for solving mixed quasi variational inequalities by using the resolvent operator technique. Under certain conditions, the global convergence of the both methods is proved. The skew-symmetry of the nonlinear bifunction plays a crucial part in the convergence analysis of these new iterative methods. The comparison of these methods with other methods for s...

It is well known that the variational inequalities are equivalent to the fixed point problem. We use this alternative equivalent formulation to suggest and analyze some new proximal point methods for solving the variational inequalities. These new methods include the explicit, the implicit, and the extragradient methods as special cases. The convergence analysis of the new methods is considered...

In this paper, we suggest and analyze some iterative methods for solving the system of nonconvex variational inequalities with two nonlinear mappings by using the projection technique. We show that the strong convergence of these iterative methods to the solutions of the system of nonconvex variational inequalities. Consequently, the results presented in this paper can be viewed as an improveme...

Variational inequalities represent very useful and powerful tools for investigation and solution of many equilibrium type problems arising in Economics, Engineering, Operations Research and Mathematical Physics. The talk is devoted to several rather new approaches from [1]–[5] to construction of solution methods for variational inequalities with multi-valued cost mappings. We recall that the pr...

Parabolic variational inequalities of Allen-Cahn and CahnHilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active ...

In this paper, we introduce a new class of generalized implicit vector variational-like inequalities in Hausdorff topological vector spaces and Banach spaces which contain implicit vector equilibrium problems, implicit vector variational inequalities and implicit vector complementarity problems as special cases. We derive some new results by using the KKM–Fan theorem, under compact and noncompa...