Existence theorems of generalized variational inequalities and generalized complementarity problems are obtained in topological vector spaces for demi operators which are upper hemicontinuous along line segments in a convex set X . Fixed point theorems are also given in Hilbert spaces for set-valued operators T which are upper hemicontinuous along line segments in X such that I − T are demi ope...

We introduce an inequality which may be viewed as a generalization of both the Brascamp-Lieb inequality and its reverse (Barthe’s inequality), and prove its information-theoretic (i.e. entropic) formulation. This result leads to a unified approach to functional inequalities such as the variational formula of Rényi entropy, hypercontractivity and its reverse, strong data processing inequalities,...

In this paper, we study variational inequalities with multivalued mappings. By employing Fan's lemma, we establish the existence result for the dual formulation of variational inequalities with semistrictly quasimonotone mappings. We also show that similar results for quasimonotone variational inequalities do not hold.

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 ...

The equilibrium problem attracted much attention in recent years, since apart from its theoretical interest includes, as particular cases, scalar and vector optimization problems, saddlepoint (minimax) problems, variational inequalities, Nash equilibria problems, complementarity problems, fixed point problems, etc. However, a seemingly interesting special case, where the function involved is re...

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 ...