Page 56 - Textos de Matemática Vol. 36

46 Chapter 2. Global aspects
1
0 otherwise. If for
it holds that
c˜μ :=
c1μ if μ is such that : R(aμ) = R(ak) and I(aμ),I(ak) commensurable,
∆0(t) = eitI(aj)v˜(aj)D(aj,C˜1) j
|∆0(t)|≥δ>0 for t∈R,
where the summation j runs over all indices with R(aj) = R(ak) and I(aj), I(ak) commensurable, then W(·;WC˜1) is a periodic solution of equation RDER with
lim (W (t, WC1 ) − W (t, WC˜1 )) = 0 t→∞
and the convergence takes place at an exponential rate.
The next theorem is quite similar but here we deal with incommensurable imaginary parts of ζ(a).
Theorem 2.4.10. Let C ∈ Gap(ak) where
Gap(ak) := {C ∈ Vn(C(m+n)×n)|D(ak, C) ̸= 0 and
D(aj,C) = 0 if R(aj) ̸= R(ak)}.
If detQ(t,C) ̸≡ 0 and detQ0(C) ̸= 0, then W(·;WC) is an almost periodic solution of equation RDER which is bounded if det Q(t; C ) ̸= 0 for |It| ≤ ε > 0. The solution W(·;WC) satisfies
etζ(aj )vlα(aj )D(aj , C)
R(aj )=R(ak )
Notice that the functions of form (2.42) are periodic if the elements of the set
{ζ(aj) − ζ(ak)|R(aj) = R(ak)}
are commensurable imaginary numbers. Once again let C1 ∈ G˜ap(ak) where
G˜ap(ak) := {C ∈ Vn(C(m+n)×n)|D(ak,C) ̸= 0 and D(aj,C) = 0 if R(aj) > R(ak)}
R(aj )=R(ak ) wlα(t;WC)=
etζ(aj)v˜(aj)D(aj,C) . (2.42) Every almost periodic solution is of this form for some matrix C ∈ Gap(ak).