It is known that, one of the important problem is to obtain sufficient conditions for the existence of interior global attractors for uniformly persistent dynamical systems. This is a nontrivial problem since the phase space \({\mathcal M}_0\) is an open subset of a complete metric space \(({\mathcal M},d)\). The main goal of this paper is to establish the existence of the interior global attractor (that is, the global attractor for \(T:({\mathcal M}_0, d)\to({\mathcal M}_0, d))\) and a fixed-point in \({\mathcal M}_0\). The authors provide four examples to show the existence of discrete – and continuous – time dynamical systems that admit global attractors, but no strong global attractors. Moreover, a simple periodic age-structured model is also studied to illustrate applicability in the case of convex \(k\)-contracting maps.