The paper concerns conditions under which there exist \(\bar{x}\in K\) and \(\bar{u}\in{\mathcal F}(\bar{x})\) such that \(W\cdot\Psi(\bar{u},\bar{x};y)\leq0\) for all \(y\in K\). Here, \(K\) is a convex subset of a topological vector space \(X\), \(\mathcal F\) is a multivalued map from \(K\) into a topological vector space \(Y\), \(\Psi\) is a map from \(Y\times K\times K\) into \(\mathbb{R}^l\), \(W\in \mathbb{R}^{l}\) is a weight vector, and “\(\cdot\)” stands for the inner product on \(\mathbb{R}^l\). Specifically, \(l=l_1+\dots+l_n\) so that \(W=(W_1,\dots,W_n)\) and \(\Psi=(\Psi_1,\dots,\Psi_n)\). Further, \(Y=Y_1^{l_1}\times\dots\times Y_n^{l_n}\) and \({\mathcal F}(x)=(F_{i_1}(x)\times\dots\times F_{i_{l_i}}(x))_{i=1,n}\). Finally, \(X=X_1\times\dots\times X_n\), \(K=K_1\times\dots\times K_n\), and \(\Psi(u,x;y)=(\Psi_i((u_{i_j})_{j=1,l_i},x_i;y_i))_{i=1,n}\). The developed theory encompasses the weighted variational inequalities from Q.H.Ansari, Z.Khan and A.H.Siddiqi [J. Optim.Theory Appl.127, No.2, 263–283 (2005; Zbl 1108.49004)].