Combining the above observation with Theorem 3.1 and Lemma 3.3, we get the following result onB-stabilities (cf. [101]). We emphasize that the additional condi- tion thatBis a uniform fading memory space cannot be removed because a fading memory spaceBmust be a uniform fading memory space whenever there is a func- tional differential equation onBwhich has aB-UAS solution ([107, Theorem 7.2.6], [159]).
Theorem 3.9 Let B be a uniform fading memory space which is separable, and suppose that the conditions(H1)–(H4) are satisfied. Then the following statements are equivalent:
i) The solutionu(t)¯ of(3.26) isB-UAS.
ii) The solutionu(t)¯ of(3.26) isB-US andB-attractive inΩ(F).
iii) The solutionu(t)¯ of(3.26) isB-UAS in Ω(F).
iv) The solutionu(t)¯ of(3.26) isB-WUAS in Ω(F).
Next we shall construct a quasi-process with X = BCρ associated with (3.28);
here and hereafter, BCρdenotes the space BC which is equipped with the metricρ defined by
ρ(φ, ψ) =
∞
X
n=0
1 2n
|φ−ψ|n
1 +|φ−ψ|n
, φ, ψ∈BCρ,
where|φ−ψ|n = sup−n≤θ≤0|φ(θ)−ψ(θ)|X. Then (BC, ρ) is a metric space. Fur- thermore, it is clear thatρ(φk, φ)→0 as k→ ∞if and only ifφk →φcompactly onR−.
We first provide an example which shows that a process on BCρ cannot be always constructed for functional differential equations with infinite delay.
Example 3.1 Consider a scalar delay equation
˙ x(t) =
∞
X
n=1
(1/n3)x(t−n), (3.31)
which is a special case of (3.26) with B = Cg0(R) (g(s) = s+ 1), A = 0 and F(t, φ) =P∞
n=1(1/n3)φ(−n).
It is clear that the conditions (H1)–(H3) are satisfied for this equation. Consider a sequence {φk} ⊂BC defined by φk(θ) = 0 if−k ≤θ ≤0, k4 if θ ≤ −k−1 and linear if −k−1 ≤ θ ≤ −k. Clearly φk → 0 in BCρ. Let denote by x(t, s, φ) the solution of (3.31) through (s, φ). Then
x(1,0, φk) = Z 1
0
∞
X
n=1
(1/n3)φk(s−n)ds
≥ 1/(k+ 2)3 Z 1
0
φk(s−k−2)ds
≥ k4/(k+ 2)3≥1
for k ≥ 10. Note that x(t,0,0) ≡ 0 and x1(0, φk) 6→ x1(0,0) in BCρ. Hence the associated mapping w : R+ ×R+ ×BCρ 7→ BCρ defined by w(t, s, φ) = xt+s(s, φ), (t, s, φ)∈R+×R+×BCρ, is not continuous onR+×R+×BCρ.
From the above example, we see that the concept of processes does not fit in with the study of theρ-stabilities in functional differential equations. In what follows, we
shall consider a subsetY of BCρand construct aY-quasi-process on BCρassociated with (3.26) to overcome the above difficulty.
Let U be a closed and bounded subset of X whose interior Ui contains the closure of the set{¯u(t) :t∈R}, where ¯uis the one in (H4). Set
BCUρ ={φ∈BCρ:φ(θ)∈U for all θ∈R−}.
It is clear that BCUρ is a nonempty closed subset of BCρ. With X = BCρ and Y = BCUρ, we shall construct the quasi-process associated with (3.28). Consider a functionwGρ :R+×R+×BCρ 7→BCρ defined by
wρG(t, s, φ) =ut+s(s, φ, G), (t, s, φ)∈R+×R+×BCρ, which is the restriction ofwGB toR+×R+×BCρ.
Lemma 3.6 wGρ is aBCUρ-quasi-process onBCρ.
Proof. From (H3) we easily see thatwGρ satisfies (p1) and (p2) withX = BCρ. We shall show that wGρ satisfies (p3) with X = BCρ and Y = BCUρ. Suppose the condition (p3) is not satisfied for wGρ. Then there exist a point (¯t,¯s,φ)¯ ∈ R+ ×R+ ×BCρ and sequences {tn} ⊂ R+, {sn} ⊂ R+ and {φn} ⊂ BCUρ such that (tn, sn, φn) → (¯t,¯s,φ) in¯ R+ × R+ ×BCρ as n → ∞ and that infnρ(utn+sn(sn, φn, G), ut+¯¯ s(¯s,φ, G))¯ >0. Then there exists an integerl >0 such that infn|utn+sn(sn, φn, G)−u¯t+¯s(¯s,φ, G)|¯ BC([−l,0];X)>0, and hence there exists a sequence{τn} ⊂[−l,0] such that
infn |u(tn+sn+τn, sn, φn, G)−u(¯t+ ¯s+τn,¯s,φ, G)|¯ X>0. (3.32) Sinceu(t,¯s,φ, G) is continuous in¯ t∈R, we get infn|u(tn+sn+τn, sn, φn, G)−u(tn+
¯
s+τn,s,¯ φ, G)|¯ X>0.Therefore it must hold thattn+τn ≥0 for all sufficiently large n, because of ρ(φn,φ)¯ → 0 asn → ∞. Thus we can assume that limn→∞τn = ¯τ for some ¯τ∈[−l,0] with ¯t+ ¯τ≥0. Since limn→∞|φn−φ|B= 0 by (A2), it follows from Proposition 3.1 that limn→∞|utn+sn+τn(sn, φn, G)−u¯t+¯s+¯τ(¯s,φ, G)|¯ B = 0;
which implies that limn→∞|u(tn+sn+τn, sn, φn, G)−u(¯t+ ¯s+ ¯τ ,s,¯ φ, G)|¯ X= 0 by (A1-iii). Therefore limn→∞|u(tn+sn+τn, sn, φn, G)−u(¯t+ ¯s+τn,s,¯ φ, G)|¯ X= 0, which is a contradiction to (3.32).
The mapping wρG constructed above is called the BCUρ-quasi-process on BCρ
generated by (3.28).
Now we consider the BCUρ-quasi-process wρF on BCρ generated by (3.26). By the same calculation as for wFB, we see that σ(τ)wFρ = wρFτ, Hσ(wρF) = {wGρ : G∈ H(F)} and Ωσ(wFρ) = {wGρ : G ∈Ω(F)}. Moreover, we see that Hσ(wFρ) is sequentially compact and the BCUρ- quasi-processwFB satisfies the condition (p4).
Fort∈R+, consider a mappingπρ(t) : BCρ×Hσ(wρF)7→BCρ×Hσ(wFρ) defined by
πρ(t)(φ, wρG) = (ut(0, φ, G), σ(t)wGρ), (φ, wρG)∈BCρ×Hσ(wFρ).
Notice that limn→∞|φn−φ|B = 0 whenever {φn} ⊂ BCUρ satisfies the condition limn→∞ρ(φn, φ) = 0. Therefore, repeating almost the same argument as in the proof of Lemma 3.6, one can see thatπρ(t) satisfies the condition (p5) withY = BCUρ, and henceπρ(t) is the skew product flow ofwρF.
Lemma 3.7 The skew product flowπρ(t)isBCUρ-strongly asymptotically smooth.
Proof. It suffices to show that for the set BCUρ ×Hσ(wρF) there exists a compact setJ ⊂BCUρ×Hσ(wFσ) with the property that{πρ(tn)(φn, wGρn)}has a subsequence which approaches toJ whenever sequences{tn} ⊂R+ and{(φn, wρGn)} ⊂BCUρ × Hσ(wFρ) satisfy limn→∞tn =∞and πρ(t)(φn, wρGn)⊂BCUρ ×Hσ(wFρ) for allt ∈ [0, tn].This can be done by the same arguments as in the proof of Lemma 3.3. Indeed, since the setU is bounded inX, puttingB = BCUρ ×Hσ(wρF) we can construct the setJB as in the proof of Lemma 3.3. Then the setJ := (JB∩BCUρ)×Hσ(wFρ) is a compact set in BCUρ×Hσ(wFρ).By virtue of (A2) and Ascoli-Arz´ela’s theorem,Jhas the desired property, because the functionxn(t) in the proof of Lemma 3.3 satisfies xn(t)∈OB and|xn(t)−xn(s)|X≤sup{|T(t−s)z−z|X:z∈OB}+C1L|t−s|for anys, twith 1≤s≤t≤tn and|t−s| ≤1.
For any functionξ:R7→Xsuch thatξ0∈BCUρ andξis continuous onR+, we define a continuous functionàξρ:R+7→BCρby
àξρ(t) =ξt, t∈R+.
It follows from (H4) that à¯uρ is an integral of the quasi-process wFρ on R+. Let η > 0 be chosen so that the interior of U contains the η-neighborhood of the set {¯u(t) :t∈R}. Then we easily see that (p6) is satisfied with δ1 :=η as Y = BCUρ, w = wρF and à = àuρ¯ because of the inequality |u(t+s, s, φ, F)−u(t¯ +s)|X ≤ ρ(wFρ(t, s, φ), àuρ¯(t)). Also, we getHσ(àuρ¯, wFρ) ={(à¯vρ, wρG) : (¯v, G)∈H(¯u, F)} and Ωσ(àuρ¯, wFρ) ={(à¯vρ, wρG) : (¯v, G)∈Ω(¯u, F)}.
The BCUρ-stabilities of the integral àuρ¯ of the quasi-process wρF yield the ρ- stabilities with respect toUfor the solution ¯u(t) of (3.26). For example, the solution
¯
u(t) of (3.26) isρ-uniformly stable with respect toU in Ω(F)(ρ-US with respect to U in Ω(F)), if for anyε >0 there exists aδ(ε)>0 such thatρ(¯ut(s, φ, G),¯vt)< ε for t ≥ s ≥ 0 whenever (¯v, G) ∈ Ω(¯u, F), ρ(φ,¯vs) < δ(ε) and φ(s) ∈ U for all s ∈ R−. The other ρ-stabilities with respect to U for ¯u(t) are given in a similar way; we omit the details.
The following result is a direct consequence of Theorem 3.9 and Lemmas 3.6 and 3.7.
Theorem 3.10 Let B be a fading memory space which is separable and suppose that the conditions (H1)–(H4) are satisfied. Also, let U be a closed and bounded subset ofXwhose interior contains the closure of the set {u(t) :¯ t∈R}. Then the following statements are equivalent:
i) The solutionu(t)¯ of(3.26) isρ-UAS with respect toU.
ii) The solutionu(t)¯ of (3.26) is ρ-US with respect to U and ρ-attractive with respect to U inΩ(F).
iii) The solutionu(t)¯ of(3.26) isρ-UAS with respect toU in Ω(F).
iv) The solutionu(t)¯ of(3.26) isρ-WUAS with respect toU inΩ(F).
In the above, if the termsρ(ut(σ, φ, F),u¯t) andρ(ut(σ, φ, G),¯vt) are replaced by
|u(t, σ, F)−u(t)|¯ Xand|u(t, σ, G)−¯v(t)|Xrespectively, then we have another concept of ρ-stability, which will be referred to as the (ρ,X)-stability. The equivalence of these two concepts ofρ-stabilities are given by the following proposition.
Proposition 3.2 Let U be a closed and bounded subset of X whose interior Ui contains the closure of the set {¯u(t) : t ∈R}. Then the solution u(t)¯ of (3.26) is ρ-USif and only if it is(ρ,X)-US.
Proof. The proof of the “only if” part is obvious. We shall establish the “if” parts.
Take any ε >0,(σ, φ)∈R+×BC with φ(s)∈ U, for all s∈U, ρ(φ,u¯σ)< δ(ε), where δ(ã) is the one for (ρ,X)-US of the solution ¯u(t) of (3.26). Then v(t) = v(t, σ, φ, F) satisfies
|v(t)−u(t)|¯ X< ε for t≥σ. (3.33) To estimate ρ(vt,u¯t), we first estimate |¯ut−vt|j. Let t ≥ σ, and denote by k the largest integer which does not exceed t−σ. If j ≤ k, then j ≤ t−σ; hence
|¯ut−vt|j = sup−j≤s≤0|¯u(t+s)−v(t+s)|X < εby (3.33). On the other hand, if j≥k+ 1, thenj≥t−σ, hence
|¯ut−vt|j = max{ sup
−j≤s≤σ−t
|¯u(t+s)−v(t+s)|X, sup
σ−t≤s≤0
|¯u(t+s)−v(t+s)|X}
≤max{ sup
−j≤θ≤0
|φ(θ)−u(σ¯ +θ)|X,sup
σ≤θ
|v(θ)−u(θ)|¯ X}
<|φưu(σ)|¯ j+ε by (3.33). Then
ρ(vt,u¯t) = (
k
X
j=1
+
∞
X
j=k+1
)2−j|vt−u¯t|j/[1 +|vt−u¯t|j] +ε
<
k
X
j=1
2−jε/(1 +ε) +
∞
X
j=k+1
2ưj[|φưu¯σ|j+ε]/[1 +|φưu¯σ|j+ε]
≤
k
X
j=1
2−jε/(1 +ε) +
∞
X
j=k+1
2ưj|φưu¯σ|j/[1 +|φưu¯σ|j]
< ε+δ(ε)≤2ε,
which shows that the solution ¯u(t) of (3.26) isρ-US withδ(ã/2).