In this paper, we consider different types of generalized vector Variational-like inequalities and study the relationships between their  olutions . We study the general forms of Stampacchia and Minty type vector Variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector Variational inequalities for the Clarke’s sub differential of non-differentiable locally Lipschitz functions and prove the existence of their solutions. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector Variational inequalities.

In this paper, we examine the perturbed absolute value Variational inequalities (PAVVI), a new class of Variational inequalities. For the (PAVVI), some new merit functions are established. We develop the error bounds for (PAVVI) using these merit functions. The results presented here recapture a number of previously established findings in the relevant fields because (PAVVI) include Variational inequalities, the absolute value complementarity problem, and systems of absolute value equations as special cases.

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, results proved in this paper continue to hold these problems. This shows that results proved in this paper can be viewed as an important and significant improvement and refinement of the previous results.

We introduce Variational inequality problems on Hilbert C-modules and we prove several existence results for Variational inequalities de ned on closed convex sets. Then relation between Variational inequalities, C-valued metric projection and  xed point theory on Hilbert C-modules is studied.

In this paper, we use the auxiliary principle technique to study the existence of a solution of the extended general Variational inequalities, which were introduced and studied by the author. Several special cases are also discussed.

In this paper, the concept of pseudoconvexity and quasiconvexity for continuous -time functions are studied and an equivalence condition for pseudoconvexity is obtained. Moreover, under pseudoconvexity assumptions, some relationships between Minty and Stampacchia vector Variational inequalities and continuous-time programming problems are presented. Finally, some characterizations of the solution sets of a single-valued continuous-time programming problem are obtained.

The purpose of this paper is to introduce some new concept and extend the usual ones are introduced for Variational inequality problems over arbitrary product sets. Our result can be viewed as an extension of the results obtained by Igor V. Konnov [Relatively monotone Variational inequalities over product sets, Operation research letters 28(2001), 21-26]. Moreover the results of this paper can be consider as a new version of the main results in [1].