We consider variational inequalities with different trial and test spaces and a possibly non-coercive bilinear form. Well-posedness is shown under general conditions that are e.g. valid for the space-time variational formulation of parabolic variational inequalities. Moreover, we prove an estimate for the error of a Petrov-Galerkin approximation in terms of the residual. For parabolic variation...

In this note we show that reversed variational inequalities cannot be studied in a general abstract framework as it happens for classical variational inequalities with Stampacchia's Lemma. Indeed, we provide two different situations for reversed variational inequalities which are of the same type from an abstract point of view, but which behave quite differently.

In this paper, we introduce and study a new system of generalized nonlinear mixed variational inequalities in Hilbert spaces. We prove the existence and uniqueness of solution for this system of generalized nonlinear mixed variational inequalities. We also give the convergence of the Mann iterative sequences with errors for this system of generalized nonlinear mixed variational inequalities in ...

In this paper, we suggest and analyze a new three-step iterative algorithm for solving the general variational inequalities. We also study the global convergence of the proposed method under some mild conditions. Since the general variational inequalities include variational inequalities, quasi variational inequalities and quasi complementarity problems as special cases, one can deduce similar ...

It is well known that the quasi variational inequalities are equivalent to the fixed point problems. We this equivalent alternative formulation to discuss the existence of a solution of quasi variational inequalities under some mild conditions. Since the quasi variational inequalities include variational inequalities, implicit complementarity problems and optimization problems as special cases,...

This paper deals with an application of duality theory in optimization to the construction of gap functions for quasi-variational inequalities. The same approach was investigated for variational inequalities and equilibrium problems in (Pac. J. Optim. 2(3): 667-678, 2006; Asia-Pac. J. Oper. Res. 24(3): 353-371, 2007), and the study shows that we can obtain some previous results for variational ...

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 variationallike inequality. The exi...

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 generali...

We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known threestep forward-backward splitting of Glowinski, Le Tallec, and M. A. Noor for so...

In this paper, we establish some results which exhibit an application for Michel–Penot subdifferential in nonsmooth vector optimization problems and vector variational-like inequalities. We formulate vector variational-like inequalities of Stampacchia and Minty type in terms of the Michel–Penot subdifferentials and use these variational-like inequalities as a tool to solve the vector optimizati...