In this paper we study pullback attractors of multi-valued dynamical systems that are asymptotically convergent. It is shown that, under certain conditions, the components of the pullback attractor of a dynamical system can converge in time to those of the pullback attractor of the limiting dynamical system. Particular examples are asymptotically autonomous and asymptotically periodic pullback attractors. Different criteria theorems requiring different conditions are established and their applicability and advantages are highlighted.
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