In this paper, we study the existence of a common fixed point for uniformly continuous one parameter semigroups of nonlinear self-mappings on a closed convex subset $C$ of a real Banach space $X$ with uniformly normal structure such that the semigroup has a bounded orbit. This result applies, in particular, to the study of an asymptotic stability criterion for a class of semigroup of nonlinear uniformly continuous infinite-dimensional systems.