A semigroup {Vt, t ∈ R+, X} belongs to the class AK (or it is asymptotically compact) if it possesses the following property: for every B ∈ B such that γ+(B) ∈ B, each sequence of the form {Vtk(xk)}∞k=1, where xk ∈ B and tk +∞, is precompact.

Here we restrict ourselves to the case of continuous semigroups of class AK. We begin with two elementary propositions concerning continuous semigroups.

Proposition 3.1

For every compact set K and t ∈ R+the set γ+[0,t](K) is compact.

The proof is evident.

Proposition 3.2

If K is compact and γ+(K) is precompact then w(K) is a non-empty compact invariant set attracting K.

Proof In fact, this proposition was proved in Chapter 2. Denoting K1 := [γ+(K)]x we know this is compact and Vt(Ki) ⊂ K1. So we have a semigroup {Vt, t ∈ R+, K1) of continuous operators Vt acting on a metric space K1. Hence, this semigroup is of class K, and we may apply Theorem 2.1 to obtain the desired result.

Now we pass to the semigroups of class AK.

Proposition 3.3

Let {Vt, t ∈ R+, X} be a continuous semigroup of class AK. Suppose that K is a compact set such that the γ+ (K) is bounded. Then γ+(K) is precompact and thus the statement of Proposition 3.2 is true.

Proof Let ynn = 1,2,…, be an arbitrary sequence of points from γ+(K), i.e. yn = Vtn(xn) for some xn ∈ K and tn ∈ R+.