2.4. FIXED POINT THEOREMS AND FREDHOLM OPERATORS
2.4.5. Existence of Periodic Solutions : Compact Perturbations
We consider the existence of periodic solutions for Equation (2.155) by using The- orem 2.27. In particular, in order to check the conditionI−Tb(ω)∈Φ+(C),we will compute the α-measure of the operator Tb(t) as well as the radius re(T(t)) of theb essential spectrum ofT(t).b
For a subset D⊂E we denote byT(ã)D the family of functionsT(t)xdefined fort∈[0,∞) with a parameterx∈D, and byT(ã)D|[a, b] its restriction to [a, b]. We note thatα(Ω(0))≤α(Ω) for a bounded set Ω⊂ C, where Ω(0) ={φ(0) :φ∈Ω}.
PutδT := max{1, γT},where γT = lim supt→0kT(t)k.
Lemma 2.30 Let D⊂E be bounded. Then the following assertions hold true:
(1) If(T(t))t≥0 is aC0-semigroup onE, α(T(ã)D|[a, b])≤ sup
a≤s≤b
kT(s)kα(D), b > a≥0.
(2) If (T(t))t≥0 be a C0-semigroup on E such that T(t)x∈ D(A)for all x∈ E andt >0. If b > ε >0, then
(i) α(T(ã)D|[ε, b])≤supε≤s≤bα(T(s))α(D).
(ii) α(T(ã)D|[0, b])≤max{sup0≤s≤εkT(s)k,supε≤s≤bα(T(s))}α(D).
(3) Let(T(t))t≥0 be a compactC0-semigroup onE. If 0< ε < b, then (i) α(T(ã)D|[ε, b]) = 0.
(ii) α(T(ã)D|[0, b])≤δTα(D).
Proof. The assertion (1) is derived directly from the definition ofα(T(ã)D|[ε, b]).
SetH=T(ã)D|[ε, b].We claim that ω(t,H) = 0 fort >0 (see Section 4.2 for the deinition) ifT(t) has the property in (2) or (3). It is clear in the case (3). Consider the case (2). PutM := sup{|x|:x∈D}. Then M is finite, and|T(s)x−T(t)x| ≤ kT(s)−T(t)kM for s, t ≥0, x∈ D. Since T(t)x∈D(A), t > 0, x ∈ E, it follows
that T(t) is continuous fort >0 in the uniform operator topology, cf. [179, p.52].
This implies thatω(t,H) = 0 for t >0.
Now, from Lemma 4.2 it follows thatα(H) = supε≤s≤bα(H(s)). Sinceα(H(s))≤ α(T(s))α(D), we have the properties (2) (i) and (3) (i). Since
α(T(ã)D|[0, b])≤max{α(T(ã)D|[0, ε]), α(T(ã)D|[ε, b])}, we obtain the properties (2) (ii) and (3) (ii).
Lemma 2.31 The following assertions are valid:
(1) If(T(t))t≥0 be aC0-semigroup onE, then α(T(t))b ≤
sup0≤s≤tkT(s)k, r≥t≥0 supt−r≤s≤tkT(s)k, t > r.
(2) If (T(t))t≥0 be a C0-semigroup on E such that T(t)x∈ D(A)for all x∈ E andt >0, then
α(T(t))b ≤
max{δT,sup0<s≤tα(T(s))}, r≥t >0 supt−r≤s≤tα(T(s)), t > r.
(3) Let(T(t))t≥0 be a compactC0-semigroup fort > t0 on E. Then (i)If t0≥r,
α(Tb(t))≤
sup0≤s≤tkT(s)k, r > t≥0 supt−r≤s≤tkT(s)kδT, t0≥t≥r supt0−r≤s≤t0kT(s)kδT, t0+r≥t > t0
0, t > t0+r.
(ii)If r > t0,
α(T(t))b ≤
sup0≤s≤tkT(s)k, t0≥t≥0 sup0≤s≤t0kT(s)kδT, r≥t > t0
supt−r≤s≤t
0kT(s)kδT, t0+r≥t > r
0, t > t0+r.
Proof. For a bounded set Ω⊂ Cand fort≥0, we have α(T(t)Ω)b ≤
max{α(T(ã)Ω(0)|[0, t]), α(Ω|[t−r,0])}, r≥t >0 α(T(ã)Ω(0)|[t−r, t]), t > r.
From this we can easily obtain the conclusion of the lemma. For example, we now show the relation
α(Tb(t))≤ sup
0≤s≤t0
kT(s)kδT for t∈(t0, r]
in the assertion 3)(ii). Using Lemma 2.30 we have that for anyε >0, α(T(t)Ω)b
≤ max{α(Ω|[t−r,0]), α(T(ã)Ω(0)|[0, t0+ε]), α(T(ã)Ω(0)|[t0+ε, t])}
≤ max{α(Ω), α(T(ã)Ω(0)|[0, t0+ε]}
≤ max{1, sup
0≤s≤t0+ε
kT(s)k}α(Ω)
≤ max{ sup
0≤s≤t0
kT(s)k, sup
t0≤s≤t0+ε
kT(s)k}α(Ω)
≤ max{ sup
0≤s≤t0
kT(s)k,kT(t0)k sup
0≤s≤ε
kT(s)k}α(Ω)
≤ max{ sup
0≤s≤t0
kT(s)k,kT(t0)kδT}α(Ω)
≤ sup
0≤s≤t0
kT(s)kδTα(Ω).
This implies the described inequality
Proposition 2.19 i) If(T(t))t≥0 is a compact C0-semigroup on E, thenTb(t) is a compact C0-semigroup fort > r onC,
ii) If (T(t))t≥0 is a compact C0-semigroup for t > t0 on E, then Tb(t) is a compact C0-semigroup fort > t0+ron C.
To show the existence of fixed points of T we will estimate the radius of the essential spectrum of the solution operator of Equation (2.155). Suppose that a function g : [0,∞) → [0,∞) is loacally bounded, and submultiplicative (that is, g(t+s)≤g(t)g(s) fort, s≥0). Then, it is well known that
t→∞lim t−1logg(t) = inf
t>0t−1logg(t),
which may be−∞, but not be +∞. This quantity is called thetype numberof the functiong(t). We denote respectively by
τ, τν,ˆτ ,τˆν, the type numbers of the functions
α(T(t)),kT(t)k, α(Tb(t)),kTb(t)k, provided that (T(t))t≥0 is aC0-semigroup on E.
In view of the Nussbaum formula we notice that re(Tb(t)) =eτ tˆ, t >0.
Thus, if ˆτ <0, thenI−Tb(ω)∈Φ+(C) (see Remark 2.21). Now we will give several conditions that ˆτ is negative.
Theorem 2.31 i) Let(T(t))t≥0be aC0-semigroup onE. Ifτνis negative, then ˆ
τν <0;and hence,ˆτ <0.
ii) Let (T(t))t≥0 be a C0-semigroup on E such that T(t)x∈ D(A)for all x∈E andt >0. If τ <0, thenτ <ˆ 0.
iii) If(T(t))t≥0 be a compactC0-semigroup on E or a compactC0-semigroup for t > t0 onE, thenτ <ˆ 0.
Proof. It is sufficient to prove only the assertion i). Sinceτν is negative, there is aàsuch that −τν> à >0 andkT(t)k ≤Màe−àt for allt≥0. Using the estimate (2.152), we see that
ˆ
τν = lim
t→∞
1
t logkTb(t)k
≤ lim
t→∞
1
t logMàe−à(t−r)
= −à <0,
from which we have the assertion i). The remainder follows from Lemma 2.31, cf.
[206, Theorem 4.8].
Proposition 2.20 Let (T(t))t≥0 be a compact C0-semigroup, or L(t,ã) be a com- pact operator for eacht∈R. Then
re(U(t, σ)) =re( ˆT(t−σ)) = exp(ˆτ(t−σ)), t > σ.
Proof. From the assumption and Proposition 2.9, it follows thatK(t, σ) is com- pact. Hence we have α(U(t, σ)n) = α(Tb(t−σ)n), n = 1,2,ã ã ã, which implies the formula in the proposition from the Nussbaum formula.
We are now in a position to give criteria for the existence of periodic solutions of Equation (2.155).
Theorem 2.32 Assume that at least one of the following conditions is satisfied:
i) (T(t))t≥0 is a compact C0-semigroup onE,
ii) L(t,ã)is a compact operator for eacht∈R andτν <0,
iii) (T(t))t≥0 is a compactC0-semigroup fort > t0 onE andL(t,ã)is a compact operator for eacht∈R.
iv) (T(t))t≥0 is aC0-semigroup onE such that T(t)x∈ D(A)for allx∈E and t >0,L(t,ã)is a compact operator for eacht∈R,andτ <0.
Then1 is a normal point ofU(ω,0), and hence, the following results hold true:
i) In the case where1 ∈ρ(U(ω,0)), Equation(2.155) has a unique ω-periodic solution.
ii) In the case where 1 is a normal eigenvalue of U(ω,0), if Equation (2.155) has an bounded solution, thenSL(ω)6=∅ anddimSL(ω)is finite.
Proof. From Theorem 2.31, Proposition 2.20 and Theorem 2.27 follows easily the conclusion of the theorem.
Finally, based on Theorem 2.23 we will give another proof of the assertion i) of the above theorem.
Lemma 2.32 Assume that(T(t))t≥0 is a compactC0-semigroup and that Equation (2.155) has a bounded solutionu(t) :=u(t, σ, φ, f). Then the setO:={u(t) :t≥σ}
is relatively compact in E, u(t) is uniformly continuous for t ≥ σ, and the set O:={ut:t≥σ} is relatively compact in C.
Proof. For the sake of simplicity of notations, we assumeσ= 0, and setF(t, φ) :=
L(t, φ) +f(t). Let h > 0. Since u(t,0, φ, f) =u(t, t−h, ut−h(0, φ, f), f) whenever t≥h, it follows that, fort≥h,
u(t) =T(h)u(t−h) + Z t
t−h
T(t−s)F(s, us)ds.
SinceT(h) is compact and{u(t−h) :t ≥h} is bounded, the set {T(h)u(t−h) : t ≥ h} is relatively compact. Since u(t) is bounded, there exists a B ≥ 0 such that |F(s, us)| ≤ B for s ≥ 0. Thus the norm of the above integral term is not greater than hγhB, where γh := sup{kT(t)k : 0 ≤ t ≤ h}. Hence we have that α(Oh)≤2hγhB, whereOh:={u(t) :t≥h}. Since the diameter of the setO\Oh
converges to zero ash→0, we have thatα(O) = 0; henceO is compact.
As computed above, we have that
|u(t)−u(t−h)| ≤ |T(h)u(t−h)−u(t−h)|+hγhB.
Since T(t) is uniformly continuos on compact sets,|u(t−h)−T(h)u(t−h)| → 0 ash→0 uniformly fort >0. Hence we have the second assertion in the lemma. In addition the third assertion follows from this and Lemma 4.2.
Theorem 2.33 Assume that (T(t))t≥0 is a compact C0-semigroup and that the equation (2.155) has a bounded solution. Then it has anω-periodic solution.
Proof. Letu(t, σ, φ0, f) be a bounded solution. Take an integerksuch thatkω >
σand definev(t) :=u(t+kω, σ, φ0, f) fort≥ −r. Since fort≥ −r,
u(t+kω, σ, φ0, f) =u(t+kω, kω, ukω(σ, φ0, f), f) =u(t,0, ukω(σ, φ0, f), f), v(t) is a bounded solution satisfying the equation fort ≥0. Hence we can assume thatσ= 0. DefineT φ=P φ+ψforφ∈ Cas before. SetD={Tnφ0:n= 0,1,2,ã ã ã}.
ThenTD ⊂ DandT is a continuous, linear affine map. NoticeTnφ0=unω(0, φ0, f) forn= 1,2,ã ã ã.Since{ut:t≥0} is relatively compact as proved above, Theorem 2.23 implies the conclusion of the theorem.