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, some properties of  pseudoinvex functions, defined by means of  limiting subdifferential, are discussed. Furthermore, the Minty Vector variational-like inequality,  the Stampacchia Vector variational-like inequality, and the  weak formulations of these two inequalities  defined by means of limiting subdifferential are studied. Moreover, some relationships  between the Vector variational-like inequalities and Vector optimization problems are established.

In this paper we introduce some new conditions of the solutions existence for variational-like inequalities with relaxed n-a pseudomonotone mappings in Banach spaces. The advantage of these new conditions is that they are easier to be verified than those that appear in some of the previous corresponding articles.

In this article, we examine a nonsmooth Vector optimization problem with locally Lipschitz approximately convex mappings in terms of the convexificator and provide some ideas for approximate effective solutions. Additionally, we define the relationship between the convexificator-based solutions of Stampacchia type Vector variational inequalities ($VVI$) and the approximate efficient approximation convex function of nonsmooth Vector optimization problems using the locally Lipschitz function. Furthermore, we provide a numerical example to demonstrate the veracity of our findings.

In this paper, we introduce two concepts of weakly relaxed {hg-ag} gÎG 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 {hg-ag} gÎG pseudomonotone and demipseudomonotone mappings in reflexive Banach spaces. Finally we show that our results improve and extend some corresponding results of Ref [6].

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.