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Proof. For Yo in P E, we denote by Yl and Y2 the solutions of (1.25) corresponding to
1/; = 1/;1 and 1/; = 1/;2, Then, as in (2.13), the variation of constants formula gives
Yi(i)=e-Atyo+ [e- A (t-S)PR(Yi(S)+1/;j(S)))dS, i=I,2.
(2.18)
Then
T1/;i(YO) = [°00 eAsQR(Yi(S) + 1/;i(Yi(S)))ds,
and setting Y = Yl - Y2, we find
T 1/;1 (yo) - T 1/;2 (yo )
= [°00 eAsQ [R(Yl(S) + 1/;l(Yl(S)) - R(Y2(S) + 1/;2(Y2(S)))) ds,
! T1/;l(YO) - T1/;2(YO) !E
::; [°00 leASQI.c(F,E) !R(Yl(S) + 1/;l(Yl(S)) - R(Y2(S) + 1/;2(Y2(S)))!F ds
::; (with (1.13))
here
::; k2 [°00 (!s!-" + A")eASr(s)ds;
res) = !R(Yl(S) + 1/;l(Yl(S))) - R(Y2(S) + 1/;2(Y2(S)))!F
::; (with (1.5) and 1/;1 E Fb,t)
::; M1(!Yl(S) - Y2(S)!E + !1/;2(Yl(S)) -1/;2(Y2(S))!E
(2.19)
+ !1/;l(Yl(S)) -1/;2(Yl(s))IE)
::; Ml [11/;1 -1/;21 00 + (1 +£)IY(s)IEJ,
with yes) = Yl(S) - Y2(S). Hence
IT1/;l(Yo) - T1/;2(yo)I E
::; Ml k2 [°00 (lsl-'" + A Ot)e As [11/;1 -1/;2!00 + (1 + €)IY( S )IE) ds
::; (with (2.4))
::; Mlk2(1 + )'Ot)A"'-1 11/;1 -1/;21 00
+Ml k2(1 + €) [°00 (IslOt
+ A Ot)e As Iy( S )IEds.
(2.20)
Now we ought to estimate ly(s)IE' We infer from (2.18), (1.4) and (1.12) that
yet) = it e-A(t-s) P [R(Yl(S) + 1/;1(YI(S))) - R(Y2(S) + 1/;2 (Y2(S)))) ds,
ly(t)IE ::; kIM1 ).'" 1° e-A(t-S)r(s)ds,