The authors first briefly review and summarize representative existence results of a solution of the vector variational inequality problem, the vector variational-like inequality problem, the generalized vector variational inequality problem and the generalized vector variational-like inequality problem.
Let \(X\) and \(Z\) be Hausdorff topological vector spaces and let \((Y,C)\) be an ordered Hausdorff topological vector space with a positive cone \(C\) and \(\text{int }C\neq\emptyset\) with the following partial order relation, \(\forall x,y\in Y\): \[ y\ngeq_{\text{int }C}\Rightarrow y- x\not\in\text{int }C. \] Let \(K\subset X\) and \(E\subset Z\) be nonempty subsets. Let \(\eta: K\times K\to K\) be a vector-valued mapping, \(V: K\rightrightarrows E\) a set-valued mapping, and \(H: X\times Z\to L(X,Y)\) a continuous mapping. Then they consider the following more general generalized vector variational-like inequality problem:
Find \(y\in K\), \(z\in V(y)\) such that \[ \langle H(y, z),\eta(x,y)\rangle\ngeq_{-\text{int }C} 0,\quad\forall x\in K. \] By using the scalarization method and the Browder fixed-point theorem, the authors established an existence result for a solution of the above problem.
For the entire collection see [Zbl 0952.00009].