Articles - Mathematics and its Applications

128 Mihail Megan, Codruța StoicaExample 2. Let us consider a skew-evolution semiflow C = (Φ, ϕ) and aparameter λ ∈ R. We define the mappingΦ λ : ∆ × X → B(V ), Φ λ (t, t 0 , x) = e λ(t−t 0) Φ(t, t 0 , x).One can remark that C λ = (Φ λ , ϕ) also satisfies the conditions of Definition3, being called λ-shifted skew-evolution semiflow on Y .Let us consider on the Banach space V the Cauchy problem{ ˙v(t) = Av(t), t > 0v(0) = v 0where A is an operator which generates a C 0 -semigroup S = {S(t)} t≥0 .Then Φ(t, s, x)v = S(t − s)v, where t ≥ s ≥ 0, (x, v) ∈ Y , defines anevolution cocycle. Moreover, the mapping defined by Φ λ : ∆ × X → B(V ),Φ λ (t, s, x)v = S λ (t−s)v, where S λ = {S λ (t)} t≥0 is generated by the operatorA − λI, is also an evolution cocycle.Example 3. Let f : R + → R ∗ + be a decreasing function with the propertythat there exists lim f(t) = a > 0. We denote by C = C(R + , R + ) the sett→∞of all continuous functions x : R + → R + , endowed with the topology ofuniform convergence on compact subsets of R + , metrizable by means of thedistanced(x, y) =∞∑n=11 d n (x, y)2 n 1 + d n (x, y) , where d n(x, y) = sup |x(t) − y(t)|.t∈[0,n]If x ∈ C, then, for all t ∈ R + , we denote x t (s) = x(t + s), x t ∈ C. LetX be the closure in C of the set {f t , t ∈ R + }. It follows that (X, d) is ametric space. The mapping ϕ : ∆ × X → X, ϕ(t, s, x) = x t−s is an evolutionsemiflow on X.We consider V = R 2 , with the norm ‖v‖ = |v 1 | + |v 2 |, v = (v 1 , v 2 ) ∈ V .If u : R + → R ∗ +, then the mapping Φ u : ∆ × X → B(V ) defined byΦ u (t, s, x)v =( u(s)u(t) e− R ts x(τ−s)dτ v 1 , u(t)u(s) e R ts x(τ−s)dτ v 2),is an evolution cocycle over ϕ and C = (Φ u , ϕ) is a skew-evolution semiflow.