Riesz decomposition property implies asymptotic periodicity of positive and constrictive operators

Consider a linear and positive operator ${\mathbf{T}}$ acting on an ordered, $F$-normed linear space ${\mathbf{X}}$. Assume that there exists an open neighborhood ${\mathbf{U}} \ni {\mathbf{0}}$ such that the trajectory $\left\{ {{{\mathbf{T}}^n}({\mathbf{x}})} \right\}$ is attracted to a compact set ${{\mathbf{F}}_{\mathbf{U}}}$ whenever ${\mathbf{x}}$ is taken from ${\mathbf{U}}$ and that the positive cone ${{\mathbf{X}}_ + }$ is closed, proper, and reproducing. It is shown that if $({\mathbf{X}},{{\mathbf{X}}_ + })$ has the Riesz Decomposition Property then ${\mathbf{T}}$ has asymptotically periodic iterates.