Summary: We consider a nonlinear Neumann eigenvalue problem driven by a possibly nonhomogeneous differential operator which incorporates as a special case the \(p\)-Laplacian. We assume that the right-hand side nonlinearity is \(( p - 1)\)-superlinear, but need not satisfy the Ambrosetti-Rabinowitz condition or to be monotone. We show that, for all values of the parameter \(\lambda\) in an upper half line, the problem has two positive and two negative solutions. Subsequently, for the case of the \(p\)-Laplacian, we also produce a nodal solution. Finally, for the semilinear case we show that the problem has two nodal solutions.