A Small-Gain Theorem for Monotone Systems with Multi-Valued Input-State   Characteristics
  We provide a new global small-gain theorem for feedback interconnections of monotone input-output systems with multi-valued input-state characteristics. This extends a recent small-gain theorem of Angeli and Sontag for monotone systems with singleton-valued characteristics. We prove our theorem using Thieme's convergence theory for asymptotically autonomous systems. An illustrative example is also provided. 
Michael Malisoff and Patrick de Leenheer