197
Hence using again (1.4) and (1.13)
Iz(t) - cp(y(t))IE S k2e- At Izo - cp(Yo)I E +
+(kl£ + k2)M1 [((t -sr a + Aa)e-A(t-s)lz(s) - cp(Y(S))IEds,
e(At/2)lz(t) - cp(y(t))IE S k 2 e-(At/2) Izo - cp(Yo)I E
+(k1£ + k2)M1 1\(t -S)-a + A a)e- A (t-s)/2e At / 2 Iz(s) - cp(y(s »IEds.
We set g(s) = e As / 2 Iz(s) - cp(Y(S»IE and
G(t) = Sup g(s).
sElo,t]
We infer from (2.43) that for every t > 0,
get) S k21 zo - CP(YO)IE +
+ (kIf + k2)M 1 G(t) 1\(t -s)-a + A<»e- A (t-s)/2ds
S k21zo - cp(Yo)I E +
+ (kIf + k2)M1G(t) 1°00 (Isl-<> + A<»e- As / 2 ds
S (with (2.4»
S k21 zo - cp(yo)I E +
+ (kIf + k2)Ml «1/2),,-11'<> + 2)A <>-IG(t).
We deduce that if (2.2) is satisfied with
then
get) S G(t) S 2k21zo - cp(Yo)I E , V t > 0,
I.e.
(2.43)
(2.44)
(2.45)
This shows that the distance in E of u(t) to M is majorized by Iz(t) - cp(y(t))IE, i.e.
(2.46)