We continue our study of the existence of convergent iterations in the presence of computational errors for a nonexpansive setvalued mapping. In a recent paper we have shown that if for any initial point, there exists a trajectory of a nonexpansive set-valued mapping attracted by a given set, then this property is stable under small perturbations of the mapping. In particular, we have shown there that for any initial point, there exists a trajectory with a subsequence which is attracted by the attractor. In the present paper we show, under certain mild additional assumptions, that for any initial point, there exists a trajectory such that, for any given positive \(\epsilon\), almost all of its elements belong to an \(\epsilon\)-neighborhood of the attractor.