Articles - Mathematics and its Applications

138 Mihail Megan, Codruța Stoicaandt α‖Q(x)v‖ ≤ Nt α e −α(t−s) ‖Φ Q (t, s, x)v‖ ≤ Ns α ‖Φ Q (t, s, x)v‖ ,for all (t, s, x, v) ∈ ∆ × Y with s ≥ t 0 = 1.Finally, we obtain that C is P.u.p.d.Now, we give an example which shows that the converse of the precedingresult is not valid.Example 11. We consider the metric space (X, d), the Banach space Vand the evolution semiflow ϕ defined as in Example 3. Let us consider thecomplementary projections families P, Q : X → B(V ), P (x)v = (v 1 , 0),Q(x)v = (0, v 2 ), for all x ∈ X and all v = (v 1 , v 2 ) ∈ V , compatible with C.We consider the evolution cocycle Φ : ∆ × X → B(V ), defined by( s 2 + 1R )tΦ(t, s, x)v =t 2 + 1 e− s x(τ−s)dτ v 1 , t2 + 1Rs 2 + 1 e ts x(τ−s)dτ v 2 ,for (t, s, x) ∈ ∆ × X and v = (v 1 , v 2 ) ∈ V = R 2 . Using the inequalitieswe obtains 2 + 1t 2 + 1 ≤ s tesande t ≤ s , for t ≥ s ≥ 1,tt α ‖Φ P (t, s, x)v‖ ≤ tα (s 2 + 1)t 2 e −a(t−s) ‖P (x)v‖+ 1≤ tα · s( s) a= s α ‖P (x)v‖ ,t tfor all t ≥ s ≥ t 0 = 1 and (x, v) ∈ Y , where α = 1 + a.Similarly,t α ‖Q(x)v‖ = t · t a ≤ ts a e at e −as ‖Q(x)v‖ ≤ t s sα e a(t−s) ‖Q(x)v‖≤ sα (t 2 + 1)s 2 + 1 ea(t−s) ‖Q(x)v‖ ≤ s α ‖Φ Q (t, s, x)v‖ ,for all t ≥ s ≥ t 0 = 1 and (x, v) ∈ Y , with α = 1 + a. Thus, C is P.u.p.d.If we suppose that C is P.u.e.d., then there exist N ≥ 1, α > 0 and t 0 > 0such that(s 2 + 1)e α(t−s) ≤ N(t 2 + 1)e −a(t−s) ,for all t ≥ s ≥ t 0 . Then, for s = t 0 and t → ∞, we obtain a contradiction,which can be eliminated only if C is not P.u.e.d.