On a variational-type inequality and its complementarity problem in Hausdorff topological vector spaces (Q1574270)

The authors have considered the problem: Given \(z\in K\), find \(x\in k\) such that \[ \Biggl\langle T\Biggl({(z+ x)\over 2}\Biggr),y- x\Biggr\rangle\geq 0,\quad\forall v\in K,\tag{1} \] which is called the variational like inequality. Note that for \(z= x\), Problem (1) is the classical variational inequality, which has been studied extensively. The authors have studied the existence of the solution of Problem (1) in the setting of the Hausdorff topological vector spaces using the KKM maps. The results are trivial generalizations of the known results and have no significant applications. In fact, problem (1) is itself artificial and has no applications.