Articles - Mathematics and its Applications

130 Mihail Megan, Codruța StoicaDefinition 5. The skew-evolution semiflow C = (Φ, ϕ) is exponentially dichotomicrelative to the projections family P : X → B(V ) (and we denoteP.e.d.) iff there exist a constant α > 0 and a nondecreasing mappingN : R + → [1, ∞) such that:(ed 1 ) e α(t−s) ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ N(s) ‖Φ P (s, t 0 , x 0 )v 0 ‖ ;(ed 2 ) e α(t−s) ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ N(t) ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y , where Q is the complementary of P .Remark 2. The skew-evolution semiflow C = (Φ, ϕ) is P.e.d. if and only ifthere exist a constant α > 0 and a nondecreasing mapping N : R + → [1, ∞)such that:(ed ′ 1 ) eα(t−s) ‖Φ P (t, s, x)v‖ ≤ N(s) ‖P (x)v‖ ;(ed ′ 2 ) eα(t−s) ‖Q(x)v‖ ≤ N(t) ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y .A particular case of P.e.d. is given byDefinition 6. The skew-evolution semiflow C = (ϕ, Φ) is called Barreira-Valls exponentially dichotomic relative to the projections family P : X →B(V ) (and we denote P.B.V.e.d.) iff there exist N ≥ 1, α > 0 and β ≥ 0such that:(BV ed 1 ) e α(t−s) ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ Ne βs ‖Φ P (s, t 0 , x 0 )v 0 ‖;(BV ed 2 ) e α(t−s) ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ Ne βt ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y .Remark 3. The skew-evolution semiflow C = (Φ, ϕ) is P.B.V.e.d. if andonly if there exist N ≥ 1, α > 0 and β ≥ 0 such that:(BV ed ′ 1 ) eα(t−s) ‖Φ P (t, s, x)v‖ ≤ Ne βs ‖P (x)v‖(BV ed ′ 2 ) eα(t−s) ‖Q(x)v‖ ≤ Ne βt ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y .Remark 4. It is obvious that if C is P.B.V.e.d., then it is P.e.d.The converse is not true, fact illustrated byExample 5. 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.