Summary: First the notion of generalized relaxed monotonicity is discussed and then based on a general auxiliary problem principle the approximation-solvability of a class of nonlinear variational inequalities is presented with applications. The nonlinear variational inequality problem considered here is as follows: find an element \(x^*\in K\) such that \[ \langle T(x^*),\eta(x, x^*)\rangle+ f(x)- f(x^*)\geq 0\quad\text{for all }x\in K, \] where \(T: K\to \mathbb{R}^n\) is a mapping from a nonempty closed invex subset \(K\) of \(\mathbb{R}^n\) into \(\mathbb{R}^n\), \(\eta: K\times K\to \mathbb{R}^n\) is a mapping, and \(f: K\to \mathbb{R}\) is a continuous invex functional on \(K\).