188
Hence
ITtf1(Yo)IE ::; k2MoA",-l(1 + '''')
and this is bounded by b, thanks to (1.7) and °(2.1), provided
Lemma 2.2. For tf1 in Fb,l, and for every YOl, Y02 in P E,
(2.6)
(2.7)
Proof: For i = 1,2, we consider the functions Y = Yi, which are solutions of (1.25) with
Yi(O) = YOi. From the definition (1.24) of T we then have
Ttf1(YoJ) - Ttf1(Y02) =
= [°00 eAsQ [R(Yl (s) + tf1(Yl (s ))) - R(Y2(S) + tf1(Y2(S )))] ds,
I Ttf1(YoJ) - Ttf1(Y02) IE::;
: : ; [°00 leAsQI C(F,E) IR(Yl (s) + tf1(Yl (s))) - R(Y2(S) + tf1(Y2 (s )))I F ds
::; (by (1.4), (1.13) and tf1 E Ft,b),
::; k2Ml [°00 (lsl-'" + A"')eAS(IYl(s) - Y2(s)IE + 1tf1(Yl(s) - tf1(Y2(s))IE)ds
::; k2Ml(1 +£) [°00 (Isl-'" + A"')ly(s)IEds,
(2.8)
where y(s) = Yl(S) - Y2(S). An estimate of ly(s)IE is given in Lemma 2.3 hereafter; this
yields (see (2.12))
where X = A + kl Ml (l + £)>''''. Hence, from the last inequality (2.8) we see that
I Ttf1(YOl) - Ttf1(Y02) IE::;
Due to (2.4),
::; 2kl k2Ml (1 + £) IYol - Y021 E [°00 (lsl-'" + A"')e(A-A)sds.
[°00 (Isl-'" + A"')e(A-A)sds = (A - X)",-l,,,, + A"'(A - X)-l
::; A"'(A - ~)-l(l + '''');