3.5.1. Almost Periodic Abstract Functional Differential Equations with Infinite Delay
In this section, we assume thatBis a fading memory space which is separable.
Now we shall consider the following functional differential equation du
dt =Au(t) +F(t, ut), (3.55)
whereAis the infinitesimal generator of a compact semigroup{T(t)}t≥0of bounded linear operators onXandF(t, φ)∈C(R× B;X).
We always impose the following conditions on (3.55) in addition to (H1)–(H2):
(H6) F(t, φ) is almost periodic int uniformly forφ∈ B, where F(t, φ) is said to be almost periodic intuniformly forφ∈ B, if for anyε >0 and any compact set W in B, there exists a positive numberl(ε, W) such that any interval of lengthl(ε, W) contains aτ for which
|F(t+τ, φ)−F(t, φ)|X≤ε for allt∈Rand allφ∈W.
A sequence {Fk} in C(R× B;X) is said to converge to G Bohr-uniformly on R× B ifFk converges toGuniformly onR×W for any compact set W in Bas k→ ∞. It is known (e.g. [121], [231, Theorems 2.2 and 2.3]) thatF(t, φ) is almost periodic in t uniformly for φ ∈ B if and only if for any sequence {tk} in R, the sequence{F(t+tk, φ)} contains a Bohr-uniformly convergent subsequence.
We denote by H(F) the set of all functions G(t, φ) such that {F(t+tk, φ)}
converges toG(t, φ) Bohr-uniformly for some sequence{tk}. In particular, Ω(F) is the subset ofH(F) for{tk} which tends to∞as k→ ∞. ClearlyG(t, φ) is almost periodic in t uniformly forφ∈ B ifG∈H(F). For ¯u(t) assured in (H4), we shall denote by Ω(¯u, F) the set of all (¯v, g)∈H(¯u, F) for which there exists a sequence {tk}, tk → ∞as k→ ∞, such that F(t+tk, φ)→G(t, φ)∈H(F) Bohr-uniformly and ¯u(t+tk)→v(t) compactly on¯ R. We can see that Ω(¯u, F) is nonempty and that ¯v(t) is the solution of
dv
dt =Av(t) +G(t, vt), (3.56)
whenever (¯v, G)∈Ω(¯u, F).
The following proposition follows from Theorem 3.4, immediately.
Proposition 3.3 Assume that conditions (H1), (H2), (H4) and (H6) hold. If the solutionu(t)¯ of(3.55)is an asymptotically almost periodic solution, then(3.55)has an almost periodic solution.
3.5.2. Existence Theorems of Almost Periodic Solutions for Nonlinear Systems
We shall discuss the existence of an almost periodic solution of an almost periodic system (3.55).
Theorem 3.17 Assume that conditions (H1), (H2), (H4) and (H6) hold. If the solution u(t)¯ of (3.55) is BC-TS, then it is asymptotically almost periodic in t.
Consequently,(3.55)has an almost periodic solution.
Proof. By Theorem 3.12, the solution ¯uof (3.55) isρ-TS w.r.t.Ufor any bounded setU inXsuch thatOu,R¯ +{¯u(t) :t∈R} ⊂Ui.
For any sequence {τk0} such that τk0 → ∞ as k → ∞, there is a subsequence {τk} of {τk0} and a (¯v, G) ∈ Ω(¯u, F) such that ¯u(t+τk) → v(t) compactly on¯ R andF(t+τk, φ) converges toG(t, φ) Bohr-uniformly onR× B. We shall show that
¯
u(t+τk) is convergent uniformly onR+.
Suppose that ¯u(t+τk) is not convergent uniformly onR+. Then, for someε >0 there are sequence{tj},{kj}and{mj} such that
kj→ ∞, mj→ ∞ as j→ ∞,
|¯u(τkj+tj)−u(τ¯ mj +tj)|X=ε (3.57) and
|¯u(τkj+t)−u(τ¯ mj +t)|X< ε on [0, tj). (3.58) Put vj(t) = ¯u(τkj +t) and wj(t) = ¯u(τmj +t). Since the sequences {vj(t)} and {wj(t)} converge to ¯v(t) uniformly on any compact interval inR, we can assume that
ρ(v0j, w0j) :=
∞
X
J=1
2−J|vj0−wj0|J/{1 +|vj0−wj0|J}< 1
j, j= 1,2,ã ã ã. (3.59) The set {vj0, w0j : j ∈ N} is relatively compact in B, because the set Xu,R¯ + is compact inBby Lemma 3.5 and vj0∈Xu,R¯ +. Moreover, the set{vj(t), wj(t) :j∈ N, t∈R+} is contained in the compact setOu,R¯ +. From these observations and Lemma 3.4 it follows that the set W := {vjt, wjt : j ∈ N, t∈ R+} is relatively compact inB. Consequently,
sup{|F(t+τk, φ)−G(t, φ)|X:t∈R, φ∈W} →0 as k→ ∞. (3.60) Define a continuous functionqj onR+ by
qj(t) =
( F(t+τkj, vjt)−F(t+τmj, wjt) 0≤t≤tj
qj,r(tj) tj< t.
By (3.60), we can choosej0=j0(ε)∈Nin such a way that sup{|qj(t)|X:j≥j0, t∈R+}< δ(ε/2)/2,
whereδ(ã) is the one for BC-TS of the solution ¯u(t) of (3.55). Since the functionvj is the solution of
dx
dt =Ax(t) +F(t+τmj, xt) +qj,r(t) fort∈[0, tj], and sincewj(t) is aBC-TSsolution of
dx
dt =Ax(t) +F(t+τmj, xt)
with the sameδ(ã) as the one for ¯u(t), from the fact that supt≥0|qj(t)|X < δ(ε/2) it follows that |vj(t)−wj(t)|X < ε/2 on [0, tj]. In particular, we have |vj(tj)− wj(tj)|X< ε, which contradicts (3.58).
The following theorem follows immediately from Theorems 3.11 and 3.17.
Theorem 3.18 Assume that conditions (H1)–(H4) and (H6) hold. If the solution u(t)isBC-UAS, then it is asymptotically almost periodic int. Consequently,(3.55) has an almost periodic solution.
Question. Assume that conditions (H1), (H2) and (H4) hold andF(t, φ) is periodic (we do not assume the regularity assumption on (3.55)).
i) Does there exist an almost periodic solution of (3.55) if ¯uis BC-US ? ii) Does there exist a harmonic solution of (3.55) if ¯uis BC-UAS ?
The B-stability implies the BC-stability. Therefore, the following results are direct consequences of Theorem 3.18.
Corollary 3.2 Assume that conditions (H1)–(H4) and (H6) hold. If the solution
¯
u(t)of (3.55) is B -TS or B -UAS, then it is asymptotically almost periodic in t.
Consequently,(3.55)has an almost periodic solution.
3.5.3. Existence Theorems of Almost Periodic Solutions for Linear Sys- tems
We consider the case where (3.55) is linear. First, we shall give the existence theorem of a bounded solution of
du
dt =Au(t) +F(t, ut) +f(t). (3.61) Proposition 3.4 Assume that conditions(H1), (H5)and(H6). If the null solution of (3.55)is BC-TS, then for any bounded and continuous function f(t)defined on R,(3.61)has a bounded solution defined onR+ which isBC-TS.
Proof. The solutionw(t), w(t) = 0 onR−,of dw
dt =Aw(t) +F(t, wt) +δ(1) 2Q f(t)
satisfies|w(t)|X <1 fort≥0, whereQ= supt∈R|f(t)|X andδ(1) is the one given for theBC-TS of the null solution of (3.61). Putting
¯
u(t) = 2Q δ(1)w(t),
¯
u(t) is a bounded solution of (3.61) and satisfies
|¯u(t)|X< 2Q δ(1) <∞ onR+.
This ¯u(t) also is BC-TS with same pair (ε, δ(ε)) as the one for BC-TS of the null solution of (3.55). Because, if|u¯σ−φ|BC < δ(ε) and h∈BC([σ,∞);X) with sup[0,∞)|h(t)|X< δ(ε), theny(t) :=u(t, σ, φ, F+f+h)−u(t) is a solution of (3.55)¯ through (σ, φưu¯σ) and |y(t)|X =|u(t, σ, φ, F+f+h)ưu(t)|¯ X < ε for all t≥σ, whereu(t, σ, φ, F +f+h) denotes the solution of
du
dt =Au(t) +F(t, ut) +f(t) +h(t), t≥σ, through (σ, φ).
We have the following theorem from Proposition 3.4 and Theorems 3.14 and 3.18, directly.
Theorem 3.19 Assume that conditions(H1), (H5)and(H6)hold. If the null solu- tion of(3.55) isBC-TS(equivalently,BC-UAS), then for any almost periodic func- tionf(t),(3.61)has an almost periodic solution.
Question. Assume that conditions (H1), (H4)–(H6) hold. Does there exist an al- most periodic solution of (3.61) if ¯uis BC-US ?
Under some additional conditions, we can derive the differentiability of the al- most periodic solution ensured in Theorem 3.19.
Theorem 3.20 Assume that conditions (H1), (H5) and (H6). Suppose that the semigroup{T(t)}t≥0 generated byAis analytic and compact, and thatF(t, φ)and f(t) are locally H¨older continuous in t uniformly for φ in bounded sets. Then the almost periodic solution ensured in Theorem 3.19 is continuously differentiable, and it satisfies the equation(3.61).
Proof. By consideringA−àIandF(t, φ)+àφ(0) with a sufficiently large constant à instead ofA and F(t, φ) respectively, it suffices to establish the theorem under the assumption that
kT(t)k ≤ce−λt, t≥0; (3.62)
here cand λare some positive constants. Let pbe the almost periodic solution of (3.61) ensured in Theorem 3.19, and setQ(t) =F(t, pt) +f(t).We claim that the function Q : R 7→ X is locally H¨older continuous. If the claim holds true, then the conclusion of the theorem follows from [179, Theorem 4.3.2]. Now, let r be a constant satisfyingr≥sup{|pt|B:t∈R}, and letϑ∈(0,1) be the H¨older exponent forF(t, φ) andf(t). Then there is a constantC such that
|F(t1, φ)−F(t2, φ)|X+|f(t1)−f(t2)|X≤C|t1−t2|ϑ (3.63) for (t1, φ), (t2, φ) ∈ R× B with |t1| ≤ r,|t2| ≤ r and |φ|B ≤ r. If s ≥ 1 and 0< h <1,then
|p(s+h)−p(s)|X ≤ |T(s+h)p(0)−T(s)p(0)|X
+|
Z s+h 0
T(s+h−τ)Q(τ)dτ− Z s
0
T(s−τ)Q(τ)dτ|X
= | Z s+h
s
AT(τ)p(0)dτ|X+ Z s+h
s
|T(s+h−τ)Q(τ)|Xdτ +|
Z s 0
Z h 0
(−A)1−ϑT(θ)(−A)ϑT(s−τ)Q(τ)dθdτ|X; here we used the relations
[T(h)−I]x= Z h
0
AT(τ)xdτ, x∈X, and
−AT(θ+τ)x= (−A)1−αT(θ)(−A)αT(τ)x,
where x ∈ X, θ > 0, τ > 0, 0 < α ≤ 1,(−A)α, the fractional power of −A, is well-defined in virtue of (3.62) (cf. [179, Section 2.6]). Using the inequality k(−A)αT(τ)k ≤cατ−αe−λτ forτ >0 (cf [179, Theorem 2.6.13 (c)]), one can easily deduce that
|p(s+h)−p(s)|X≤C1hϑ, s≥1, 0< h <1,
for some constantC1>0. Sincep(t) is almost periodic, the above inequality must hold for alls∈R.We thus get|pt+h−pt|BC≤C1hϑ fort∈Rand 0< h <1, and consequently
|pt+h−pt|B≤C2hϑ, t∈R, 0< h <1, (3.64) by (3.1), where C2 =C1J. Then, if |t| ≤ r and 0 < h <1, it follows from (3.63) and (3.64) that
|Q(t+h)−Q(t)|X ≤ |F(t+h, pt+h−pt)|X+|F(t+h, pt)−F(t, pt)|X
+|f(t+h)−f(t)|X
≤ {C2 sup
|τ|≤r+1
kF(τ,ã)k+C}hϑ, which shows the local H¨older continuity ofQ(t), as required.