Some generalizations of generalized bi-quasi-variational inequalities (Q1206542)

The authors introduce a class of new bi-quasi-variational inequalities and study the existence of solutions. Let \(\Omega\) be the real or the complex field, \(E\) a locally convex Hausdorff topological vector space over \(\Omega\), \(X\) a non-empty compact convex subset of \(E\), \(F\) and \(Z\) two topological vector spaces over \(\Omega\), \(\varphi:E\times E\times F\times Z\to\Omega\), \(S:X\to 2^ X\), \(M:X\to 2^ F\), \(T:X\to 2^ Z\). The main results of the paper are two theorems which establish (when \(\varphi\), \(S\), \(M\), \(T\) satisfy some conditions) existence results for the new bi-quasi-variational inequality: \(\hat y\in S(\hat y)\) and \(\inf_{w\in T(\hat y)}\text{Re} \varphi(x,\hat y,f,w)\leq 0\) for all \(x\in S(\hat y)\), \(f\in M(x)\).