A power mapping and its fixed points

Abstract

In this paper, we show that for a nonempty finite set $U$ and a power mapping $T:U\longrightarrow \mathcal{P}^*(U)$, there exists a nonempty subset $F$ of $U$ such that $T'(F) = F$ where $T'(F)=\bigcup_{x\in F}T(x)$.  Also for a power mapping $T:U \longrightarrow \mathcal{P}(U)$, we get an equivalent condition for having a nonempty fixed points set. Finally, we present a method to obtain all of fixed point of $T$.