By David N. Cheban
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C(X, Y ), T, σ) is a dynamical system. 45. The dynamical system (C(X, Y ), T, σ) is called a dynamical system of shifts (dynamical system of translations or dynamical system of Bebutov) in the space of continuous functions C(X, Y ). Let us give some examples of dynamical systems of the type (C(X, Y ), T, σ) that are met in applications. 46. Assume X = T and by (X, T, π) denote a dynamical system on T, where π(x, t) = x + t. The dynamical system (C(T, Y ), T, σ) is called a dynamical system of Bebutov [86, 92, 93, 99, 100].
Let ε > 0 and g : R → B be a continuous function with the compact support such that 1/ p p |s|≤l+l0 g(s) − f (s) dμ(s) ≤ ε. 106) (because max|t|≤l |g(t + hn ) − g(t)| → 0 as hn → 0). Since ε is arbitrary, from the last relation we obtain 1/ p lim n→+∞ p |t |≤l f t + hn − f (t) dt = 0. 107), it follows the continuity of the mapping σ. The lemma is proved. 2. 69. A function ϕ ∈ Lloc (R; B; μ) is called S p almost periodic (almost periodic in the sense of Stepanoﬀ ), if the motion σ(·, ϕ) is almost periodic in the dynamical p system (Lloc (R; B; μ), R, σ).
Then the following conditions are equivalent: (1) (X, T1 , π), (Y , T2 , σ), h is a convergent nonautonomous dynamical system; (2) (a) limt→+∞ ρ(x1 t, x2 t) = 0 for every x1 , x2 ∈ X such that h(x1 ) = h(x2 ); (b) for every ε > 0 there exists δ > 0 such that from the inequality ρ(x1 , x2 ) < δ (h(x1 ) = h(x2 ) and x1 , x2 ∈ JX ) follows that ρ(x1 t, x2 t) < ε for all t ∈ T+ . Proof . Let us show that (1) implies (2). First of all, let us establish that (a) is true.