Strongly nonlinear variational inequality problem and its alternative form and complementarity problem (Q1178961)

The paper deals with the problem: find \(u\in S=\{u\in K\mid\langle Tu,\nu - u\rangle\geq \langle Au,\nu -u\rangle\) for all \(\nu\in K\}\) where \(K\) is a closed convex set from a Hilbert space \(H\). \(T\) and \(A\) are monotone in the sense of Minty-Browder hemicontinuous mappings. For the case when \(K\) is a cone in \(H=R^ n\) an existence theorem is established. Note of the reviewer: The author gives a wrong proof for a wrong statement on the equivalence of the initial problem to its alternative form \[ u\in K: \langle T\nu ,\nu -u\rangle\geq \langle A\nu ,\nu - u\rangle\text{ for all }\nu\in K. \] Counterexample: \(K=H=R^ 1\), \(Tu=u\), \(Au=u^ 3\).