This subsection will be devoted to some applications of the spectral decomposition theorem to prove the existence of almost periodic solutions with specific spectral properties. In particular, we will revisit the classical result by Massera on the exis- tence of periodic solutions as well as its extensions. To this end, the following notion will play the key role.
Definition 2.14 Letσ(f) andσΓ(P) be defined as above. We say that the setσ(f) andσΓ(P) satisfythe spectral separation conditionif the setσΓ(P)\σ(f) is closed.
Corollary 2.11 Let f be almost periodic,σ(f)andσΓ(P)satisfy the spectral sep- aration condition. Moreover, letσ(f)be countable andX not contain any subspace which is isomorphic toc0. Then if there exists a bounded uniformly continuous so- lution uto Eq.(2.57), there exists an almost periodic solution w to Eq.(2.57) such thatσ(w) =σ(f).
Proof. We define in this caseS1:=σ(f),S2:=σΓ(P)\σ(f). Then, by Theorem 2.14 there exists a solutionwto Eq.(2.57) such thatσ(w)⊂σ(f). Using the estimate (2.67) we haveσ(w) =σ(f) . In particular, sinceσ(w) is countable andXdoes not containc0,wis almost periodic.
Remark 2.14 i) If σ(f) is finite, then sp(w) is discrete. Thus, the condition that Xdoes not contain any subspace isomorphic toc0can be dropped.
ii) In the case whereσΓ(P) is countable it is known that with additional ergodic conditions on u the solution u has ”similar spectral properties” as f (see Corollary 2.6). However, in many cases it is not expected that the solutionu itself has similar spectral properties asf as in the Massera type problem (see [147], [45], [206], [168] e.g.).
iii) In the case where P is compact (or merelyσΓ(P) is finite) the spectral sep- aration condition is always satisfied. Hence, we have a natural extension of a classical result for almost periodic solutions. In this case see also Corollary 2.12 below.
iv) We emphasize that the solution w in the statement of Corollary 2.11 is a
”σ(f)-spectral component” of the bounded solutionu. This will be helpful to find the Fourier coefficients ofwas part of those ofu.
v) In view of estimate (2.67) w may be seen as a ”minimal” solution in some sense.
Corollary 2.12 Let all assumptions of Corollary 2.11 be satisfied. Moreover, let σΓ(P)be countable. Then if there exists a bounded uniformly continuous solutionu to Eq.(2.57), it is almost periodic. Moreover, the following part of the Fourier series ofu
Σbλeiλt , bλ= lim
T→∞
1 2T
Z T
−T
e−iλξu(ξ)dξ, (2.78) whereeiλ∈σ(f), is again the Fourier series of another almost periodic solution to Eq.(2.57).
Proof. The assertion thatuis almost periodic is standard in view of (2.66) (see Chapter 1). It may be noted that in the case u is almost periodic, the spectral decomposition can be carried out in the function spaceAP(X) instead of the larger space BU C(R,X). Hence, we can decompose the solution uinto the sum of two almost periodic solutions with spectral properties described in Theorem 2.14. Using the definition of Fourier series of almost periodic functions we arrive at the next assertion of the corollary.
The next corollary will show the advantage of Theorem 2.15 which allows us to take into account the structure of sp(f) rather than that ofσ(f). To this end, we introduce the following terminology. A set of realsS is said to have an integer and finite basis if there is a finite subset T ⊂S such that any elements∈ S can be represented in the form s = n1b1+ã ã ã+nmbm, where nj ∈ Z, j = 1,ã ã ã, m, bj ∈T, j= 1,ã ã ã, m. Iff is quasi-periodic and the set of its Fourier-Bohr exponents is discrete (which coincides withsp(f) in this case), then the spectrum sp(f) has an integer and finite basis (see [137, p.48]). Conversely, iff is almost periodic and sp(f) has an integer and finite basis, then f is quasi-periodic. We refer the reader to [137, pp. 42-48] more information on the relation between quasi-periodicity and spectrum, Fourier-Bohr exponents of almost periodic functions.
Corollary 2.13 Let all assumptions of the second assertion of Theorem 2.15 be satisfied. Moreover, assume thatXdoes not containc0. Then ifsp(f)has an integer and finite basis, Eq.(2.72) has a quasi-periodic mild solutionwwithsp(w) =sp(f).
Proof. Under the corollary’s assumptions the spectrumsp(w) of the solutionw, as described in Theorem 2.15, is in particular countable. Hencewis almost periodic.
Sincesp(w) =sp(f),sp(w) has an integer and finite basis. Thuswis quasi-periodic.
Below we will consider some particular cases Example 2.10 Periodic solutions.
If σ(f) = {1} we are actually concerned with the existence of periodic solutions.
Hence, Corollary 2.11 extends the classical result to a large class of evolution equa- tions which has 1 as an isolated point ofσΓ(P). Moreover, Corollary 2.12 provides a way to approximate the periodic solution. In particular, suppose thatσΓ(P) has finitely many elements, then we have the following:
Corollary 2.14 Let σΓ(P) have finitely many elements {à1,ã ã ã, àN} andu(ã) be a bounded uniformly continuous solution to Eq.(2.57). Then it is of the form
u(t) =u0(t) +
N
X
k=1
eiλktuk(t), (2.79) where u0 is a bounded uniformly continuous mild 1-periodic solution to the inho- mogeneous equation (2.57), uk, k = 1,ã ã ã, N,are 1-periodic solutions to Eq.(2.57) withf =−iλkuk , respectively, v(t) =PN
k=1eiλktuk(t)is a quasi periodic solution to the corresponding homogeneous equation of Eq.(2.57) and λ1,ã ã ã, λN are such that0< λ1,ã ã ã, < λN <2πandeiλj =àj, j= 1,ã ã ã, N.
Example 2.11 Anti-periodic solutions.
An anti-periodic (continuous) function f is defined to be a continuous one which satisfiesf(t+ω) =−f(t),∀t∈Rand hereω >0 is given. Thus,f is 2−ω-periodic.
It is known that, the space of anti-periodic functionsf with antiperiodω, which is denoted byAP(ω) , is a subspace ofBU C(R,X) with spectrum
sp(f)⊂ {2k+ 1
ω , k∈Z}.
Without loss of generality we can assume thatω= 1. Obviously,σ(f) ={−1},∀f ∈ AP(ω). In this case the spectral separation condition is nothing but the condition that{−1}is an isolated point ofσΓ(P).
Example 2.12
Letube a bounded uniformly continuous solution to Eq.(2.57) withf 2-periodic.
Let us define
F(t) = f(t)−f(t+ 1)
2 , G(t) = f(t) +f(t+ 1)
2 ,∀t∈R.
Then, it is seen that F is 1-anti-periodic and Gis 1-periodic. Applying Theorem 2.14 we see that there exist two solutions to Eq.(2.57) as two components ofuwhich are 1-antiperiodic and 1-periodic with forcing termsF, G, respectively. In particular, the sum of these solutions is a 2-periodic solution of Eq.(2.57) with forcing termf. Example 2.13
LetAbe a sectorial operator in a Banach spaceXand the mapt7→B(t)∈L(Xα,X) be H¨older continuous and 1-periodic. Then, as shown in [90, Theorem 7.1.3] the equation
dx
dt = (−A+B(t))x, (2.80)
generates an 1-periodic strongly continuous evolutionary process (U(t, s))t≥s. If, furthermore, A has compact resolvent, then the monodromy operator P of the process is compact. Hence, for every almost periodic functionf the sets σ(f) and σΓ(P) always satisfy spectral separation condition. In Section 1 we have shown that ifσΓ(P)∩σ(f) =, then there is a unique almost periodic solutionxf to the inhomogeneous equation
dx
dt = (−A+B(t))x+f(t) (2.81)
with property thatσ(xf)⊂σ(f) . Now suppose thatσΓ(P)∩σ(f)6=. By Corol- lary 2.11, if u is any bounded solution (the uniform continuity follows from the boundedness of such a solution to Eq.(2.81)), then there exists an almost periodic solutionwsuch that σ(w) =σ(f). We refer the reader to [90] and [179] for exam- ples from parabolic differential equations which can be included into the abstract equation (2.81).
Example 2.14
Consider the heat equation in materials
vt(t, x) = ∆v(t, x) +f(t, x), t∈R, x∈Ω v(t, x) = 0, t∈R, x∈∂Ω,
(2.82) where Ω ⊂ Rn denotes a bounded domain with smooth boundary ∂Ω. Let X = L2(Ω), A= ∆ withD(A) =W2,2(Ω)∩W01,2(Ω) . ThenAis selfadjoint and negative definite (see e.g. [179]). Hence σ(A)⊂(−∞,0). In particular σi(A) =. Eq.(22) now becomes
dv
dt =Av+f. (2.83)
We assume further thatf(t, x) =a(t)g(x) where ais a bounded uniformly contin- uous real function with sp(a) = Z∪πZ, g ∈ L2(Ω), g 6= 0. It may be seen that σ(f) =S1 andsp(f) has an integer and finite basis. Hence, Theorem 2.14 does not give any information on the existence of a solutionwwith specific spectral proper- ties. However, in this case Theorem 2.15 applies, namely, if Eq.(2.82) has a bounded solution, then it has a quasi periodic solution with the same spectrum asf. Example 2.15
We consider the case of Eq.(2.57) having exponential dichotomy. In this case, as is well known from Section 1, for every almost periodicf there exists a unique almost periodic solutionxf to Eq.(2.57). From the results above we see thatσ(xf) =σ(f).