189
here we have used
which follows from (2.1), provided
(2.9)
Thus
I TtP(YOl) - TtP(Y02) IE::;
::; 2klk2Ml(1 +1!)(1 +,o)AO(A - X)-lIYOl - Y021 E ,
(2.10)
and this is bounded by I! IYOl - Y021E' thanks to (2.1) and provided
i.e., beside (2.10),
(2.11)
The proof of Lemma 2.2 will be complete after we prove Lemmas 2.3 and 2.4
hereafter.
Lemma 2.3. If Yl and Y2 are two solutions of (1.25) with Yi(O) = YOi E P E, i = 1,2
and tP E Fb,(, and if yet) = Yl(t) - Y2(t), then
ly(t)IE ::; 2kl IYOl - Y021 E e-~t, V t ::; 0,
where X = ), + kl M l (1 + I!),Q.
(2.12)
Proof: By the variation of constants formula, we write the solutions Yl,Y2 of (1.25) in
the form
Yi(t) = e-AtYoi + [e-A(t-.) PR(Yi(S) + tP(Yi(s)))ds.
(2.13)
Substracting these relations, we find
yet) = e- At (YOl - Y02) + 1t e-A(t-s) P [R(Yl(S) + tP(Yl(S))) - R(Y2(S) + tP(Y2(S)))] ds.
Hence upon using (1.4), (1.12) and tP E Fb,t :
ly(t)IE ::;le- At pl.c(E)IYOl-Y02IE+
+ 1
0 le-A(t-S) pi IR(Yl(S) + tP(Yl(S))) - R(Y2(S) + tP(Y2(s)))IF ds
t
C(F,E)
::; kllYOl - Y02I E e- M + klMl(l +I!)'O 1
0 e-A(t-S)ly(s)lds,