Global attractors and steady states for uniformly persistent dynamical systems

By appealing to the theory of global attractors on complete metric spaces, we obtain weaker sufficient conditions for the existence of interior global attractors for uniformly persistent dynamical systems, and hence generalize the earlier results on coexistence steady states. We also provide examples to show applicability of our interior fixed point theorem in the case of convex k-contracting maps, and to prove the existence of discrete- and continuous-time dynamical systems that admit global attractors, but no strong global attractors, which gives an affirmative answer to an open question presented by Sell and You [ Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002] in the case of continuous-time semiflows.