Abstract

Let \(X\) be a normed linear space with dual \(X^*\) and \(H\) be a mapping from \(X^{**}\) into \(X^{**}\). The mapping \(T\) is said to be \(H\)-monotone if \(\langle Tx-Ty, H(x-y) \rangle \geq 0\), \(\forall x,y \in X^{**}\). Let \(T: K \subset X^{**} \to 2^{X^*}\) be \(H\)-monotone. The author proves several existence theorems for the following nonlinear variational inequality problem: find \(x_0 \in K, w_0 \in Tx_0\) such that \(\langle w_0, H(x-x_0) \rangle \geq 0\) for all \(x \in K\).