Iterative roots of upper semicontinuous multifunctions

The square iterative roots for strictly monotonic and upper semicontinuous functions with one set-valued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203-220, 2009 ). As a continuation, we study both strictly monotonic and nonmonotonic multifunctions. We present sufficient and necessary conditions under which those multifunctions have n th iterative roots. This equivalent condition and the construction method of n th iterative roots extend the previous results.