Devaney chaos of multiple mappings from a set-valued view

In this paper, we introduce the definitions of periodic point, transitivity, sensitivity and Devaney chaos of multiple mappings from a set-valued perspective. We study the relation between multiple mappings and its continuous self-maps and show that multiple mappings and its continuous self-maps do not imply each other in terms of periodic point or transitivity. While the sufficient condition for multiple mappings to have periodic points, to be transitive and sensitive is provided separately. And we show that transitivity plus dense periodic point set implies sensitivity. Also, a sufficient condition for multiple mappings to be Devaney chaotic is given.