  We consider Lurye (sometimes written Lur'e) systems whose nonlinear operator
is characterised by a possibly multivalued nonlinearity that is bounded above
and below by monotone functions. Stability can be established using a sub-class
of the Zames-Falb multipliers. The result generalises 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 the new mutlipliers with
two examples, one in continuous time and one 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 non-zero steady state exogenous
signals.
