In this paper, we introduce and study the system of general variational inequalities which is equivalent to the general variational inequality problem over the product of sets. The usual concept of monotonicity has been extended here. We establish existence results for the solution of general variational inequality problem over the product of sets in the setting of real Hausdroff topological ve...

where C ⊆ Rm is a closed convex and pointed cone with nonempty interior intC. (WVVI) was firstly introduced by Giannessi []. It has been shown to have many applications in vector optimization problems and traffic equilibrium problems (e.g., [, ]). Error bounds are to depict the distance from a feasible solution to the solution set, and have played an important role not only in sensitivity an...

Recently, Huang and Li [J. Li, N.J. Huang, Vector F-implicit complementarity problems in Banach spaces, Appl. Math. Lett. 19 (2006) 464–471] introduced and studied a new class of vector F-implicit complementarity problems and vector F-implicit variational inequality problems in Banach spaces. In this work, we study this class in topological vector spaces and drive some existence theorems for th...

Given an n n matrix M, a vector q in IR n , and a polyhedral convex set X = fxjAx b; Bx = dg, where A is an m n matrix and B is an p n matrix, the aane variational inequality problem is to nd x 2 X such that (Mx + q) T (y ? x) 0 for all y 2 X. If M is positive semi-deenite, the aane variational inequality can be transformed to a generalized complementarity problem, which can be solved in polyno...

We introduce and study a new class of generalized nonlinear variational-like inequalities. Under suitable conditions, we prove the existence of solutions for the class of generalized nonlinear variational-like inequalities. A new iterative algorithm for finding the approximate solutions of the generalized nonlinear variational-like inequality is given and the convergence of the algorithm is als...

In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E, a multifunction T : K → 2E∗ , and a vector b ∈ E, find x̄ ∈ K such that there exists ū ∈ T(x̄) satisfying 〈ū− b, y− x̄〉 ≥ 0, for all y ∈ K. By using generalized projection and the well-known Fan–KKM Theorem, we prove existence...

We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi 1995, 1996 for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is ...

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,...