270
where q* ::::: Ilqlloo [1 - XUnEz+[n+rt,n+rd')] and X denotes the characteristic function.
Note that
i + u(z)z4 - J-lq(') ::::: (+ Q.(4 - J-lq*(') ~ (+ U(O(4 - fl-q(.).
Moreover, we have ((t) ~ p :::::: (~lIqlloo f/4 V t ~ 0, because z(O) ::::: ((0) ~
g-l(fl-lIqlloo) ~ p in view of g(p) ~ Q.P4 ::::: fl-Ilqlloo' and fl-q(t) - Q.(4 < 0 V (t,O E
[0,00) x (p, 00). Now, observing that ( fulfils ( + Q.(4 ::::: 0 on [n + Tt, n + Tn we get
[
3
] 1/3
[
3
] 1/3
((t) ~ I+3~3(~-[t)-rn
for t E [n + Tt, n + Tn, since s f--> ~
solves
iJ + Q.TJ4 ::::: 0, TJ(O) ::::: p. We refer to [Diaz (1995)], where a very similar observation has
been made in the context of climate controllability.
7. S-shapedness of~.
As pointed out before, one can impose any growth restriction at infinity without touching climatological issues. The previous example suggests therefore to consider the case,
where g grows at most linearly at infinity. This leads to the following modifications for
(HI) and (H3):
(H'I) c,k E C 2 (M) positive; Q E C 2 (R+ xM), Q(·,x) I-periodic for x E M, q E
C 2 (R+) I-periodic, nonnegative, Jo1q(t)dt > 0, Q(t,x) ~ q(t) for x E M,t E
[0,1];
(H'3) 9 E C 2 (R+) ,g(O)::::: O,l(y) > 0 for y > 0, liminf g(y) > 0,
y-->oo
y
.
9 (y)
hmsup-- < 00, c*,y* E (O,OO),ygl(y) ~ c*g(y) for y E [y*, 00),
y-->oo
Y
lim 1(82a)(x,y)1 ::::: 0
y-->o+
gl (y)
uniformly for x EM.
Remarks 3.
(i) It is an interesting mathematical problem though to decide, whether we can relax
(H'3) as to allow for polynomial growth of g. The above example does not apply here
in view of the x-dependence of Q and a.
(ii) The function g is roughly speaking the minimum of the seasonally averaged
incoming solar radiation flux near the poles, which is positive at times corresponding
to the equinoxes.