Articles - Mathematics and its Applications

132 Mihail Megan, Codruța StoicaRemark 5. The skew-evolution semiflow C = (Φ, ϕ) is P.u.e.d. if and onlyif there exist some constants N ≥ 1 and α > 0 such that:(ued ′ 1 ) eα(t−s) ‖Φ P (t, s, x)v‖ ≤ N ‖P (x)v‖ ;(ued ′ 2 ) eα(t−s) ‖Q(x)v‖ ≤ N ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y .Remark 6. It is obvious that if C is P.u.e.d., then it is P.B.V.e.d.The following example shows that the converse implication is not valid.Example 6. We consider the metric space (X, d), the Banach space V andthe 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.The mapping Φ : ∆ × X → B(V ), defined by=Φ(t, s, x)v =(v 1 e t sin t−s sin s−2(t−s)−R ts x(τ−s)dτ , v 2 e 3(t−s)−2t cos t+2s cos s+R ts x(τ−s)dτ )is an evolution cocycle over the evolution semiflow ϕ.We observe that for α = 1 + a we have thate α(t−s) ‖Φ P (t, s, x)v‖ ≤ e α(t−s) e −(1+a)t e (3+a)s ‖P (x)v‖ ≤ e 2s ‖P (x)v‖ ,for all (t, s, x, v) ∈ ∆ × Y. Similarly,e α(t−s) ‖Q(x)v‖ ≤ e α(t−s) e −3t+3s+2t cos t−2s cos s−R ts x(τ−s)dτ ‖Φ Q (t, s, x)v‖≤ ‖Φ Q (t, s, x)v‖ ,for all (t, s, x, v) ∈ ∆×Y. Thus, conditions (BV ed ′ 1 ) and (BV ed′ 2 ) are satisfiedforα = 1 + a, N = 1 and β = min{0, 2}.This shows that C = (Φ, ϕ) is P.B.V.e.d.If we suppose that C is P.u.e.d., then there exist N > 1 and α > 0 suchthate α(t−s) e t sin t−s sin s−2t+2s−R ts x(τ−s)dτ ≤ N,