We consider Lurye (sometimes written Lur’e) systems, whose nonlinear operator is characterized by a possibly multivalued nonlinearity that is bounded above and below by monotone functions. Stability can be established using a subclass of the Zames–Falb multipliers. The result generalizes similar approaches in the literature. Appropriate multipliers can be found using convex searches. Because the multipliers can be used for multivalued nonlinearities, they can be applied after loop transformation. We illustrate the power of new multipliers with two examples: one in continuous time and the other in discrete time: in the first, the approach is shown to outperform available stability tests in the literature; in the second, we focus on the special case for asymmetric saturation with important consequences for systems with nonzero steady-state exogenous signals