On the Existence and Stability of an Infinite-Dimensional Invariant Torus

We consider an annular set of the form $K=B\times \mathbb{T}^{\infty}$, where $B$ is a closed ball of the Banach space $E$, $\mathbb{T}^{\infty}$ is the infinite-dimensional torus (the direct product of a countable number of circles with the topology of coordinatewise uniform convergence). For a certain class of smooth maps $\Pi\colon K\to K$, we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form
$$ A=\{(v, \varphi)\in K: v=h(\varphi)\in E,\,\varphi\in\mathbb{T}^{\infty}\}, $$
where $h(\varphi)$ is a continuous function of the argument $\varphi\in\mathbb{T}^{\infty}$. We also study the question of the $C^m$-smoothness of this manifold for any natural $m$.