Sharp Block–Sharkovsky Type Theorem for Multivalued Maps on the Circle and Its Application to Differential Equations and Inclusions

Abstract

This is a final part of the series of our papers devoted to a multivalued version of the (Sharkovsky type) Block cycle coexistence theorem. It improves our last general result in the sense that its part related to the usual ordering of positive integers becomes a full analogy of the standard single-valued case, while the alternative part related to the Sharkovsky ordering of positive integers is an analogy of the multivalued case for interval maps, provided there exists a fixed point. That is why we call the obtained theorem here as “sharp”. This theorem is still applied via the associated Poincaré translation operators to differential equations and inclusions on the circle. All the deterministic results are also randomized in an advantageous way.