# Asymptotic Almost Periodic Functions with Range in a Topological Vector Space

Authors: Liaqat Ali Khan; Saud M. Alsulami
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

The notion of asymptotic almost periodicity wasfirst introduced by Fréchet in 1941 in the case of finite dimensional range spaces. Later, its extension to the case of Banach range spaces and locally convex range spaces has been considered by several authors. In this paper, we have generalized the concept of asymptotic almost periodicity to the case where the range space is a general topological vector space, not necessarily locally convex. Our results thus widen the scope of applications of asymptotic almost periodicity.


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## Body

## 1. Introduction

The theory of almost periodic functions was mainly created and published during 1924–1926 by the Danish mathematician Harold Bohr. In 1933, Bochner [1] published an important article devoted to extension of the theory of almost periodic functions on the real line ℝ with values in a Banach space E. His results were further developed by several mathematicians, see, for example [2–7].The concept of asymptotic almost periodicity was first considered by Fréchet [8, 9] in 1941 for functions f:ℝ+→E with the range E restricted to a finite dimensional space. The semigroup case of ℝ+ turns out to be significantly different from the group case of ℝ. If E is a Banach space or a locally convex case and ℝ+=[0,∞) replaced by Ja=[a,∞), with a∈ℝ, this notion has been extensively studied in recent years (see [10–14]). In this paper, we generalize the concept of asymptotic almost periodicity to the case of E, a general topological vector space.


## 2. Preliminaries

In this section, we give prerequisites on topological vector spaces and almost periodic functions for our main results of Section3.Throughout this paper,E denotes a nontrivial Hausdorff topological vector space (in short, a TVS) with a base 𝒲 of closed balanced shrinkable neighborhoods of 0. (A neighborhood W of 0 in E is called shrinkable [15] if rW¯⊆intW for 0≤r<1.) By [15, Theorem 4 and 5], every Hausdorff TVS has a base of shrinkable neighborhoods of 0, and also the Minkowski functional ρW of any such neighborhood W is continuous and satisfies
(1)W¯={x∈E:ρW(x)≤1},intW={x∈E:ρW(x)<1}.
We mention that, for any neighborhood W of 0 in E, ρW need not be absolutely homogeneous or subadditive; however, the following useful properties hold [15, 16].(a)
ρW is positively homogeneous; it is absolutely homogeneous if W is balanced.(b)
IfV is a balanced neighborhood of 0 in E with V+V⊆W, then
(2)ρW(x+y)≤ρV(x)+ρV(y)∀x,y∈E.A complete metrizable TVS is called anF-space.Notations. Let X be a completely regular Hausdorff space, and let C(X,E) be the set of all continuous functions E-valued functions on X. Furthermore, let
(3)Cb(X,E)={f∈C(X,E):f(X)isboundedinE},Cpc(X,E)={f∈C(X,E):f(X)isprecompactinE},C0(X,E)={f∈C(X,E):fvanishesatinfinityonX}.
Clearly, Cpc(X,E)⊆Cb(X,E)⊆C(X,E), C0(X,E)⊆Cb(X,E), and all these sets are vector spaces over 𝕂 with the pointwise operations of addition and scalar multiplication. The uniform topologyu on Cb(X,E) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form
(4)Nb(X,W)={f∈Cb(X,E):f(X)⊆W},
where W∈𝒲. The compact-open topologyk on C(X,E) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form
(5)N(K,W)={f∈C(X,E):f(K)⊆W},
where K⊆X is compact and W∈𝒲. Clearly, k≤u on Cb(X,E). (For details, see [16].)We state the following two general versions of the Arzelà-Ascoli theorem [16, 17] for reference purpose.Theorem 1.
LetX be a locally compact space. A subset 𝒜 of C0(X,E) is precompact if and only if the following conditions hold:(i)
𝒜 is equicontinuous:(ii)
𝒜(x)={f(x):f∈𝒜} is precompact in E for each x∈X;(iii)
𝒜 vanishes at infinity on X; that is, given W∈𝒲, there exists a compact set K⊆X such thatf(y)∈W for all f∈𝒜 and y∈X∖K;Before stating the next result, we introduce the following notation: for any𝒜⊆Cpc(X,E) and W∈𝒲, let
(6)Gx(𝒜,W)={y∈X:f(y)-f(x)∈W∀f∈𝒜},x∈X.Theorem 2.
LetX be a k-space. Then, for any 𝒜⊆Cpc(X,E), the following are equivalent:(1)
𝒜 is precompact;(2)
consider the following:(i)𝒜 is equicontinuous;(ii)𝒜(x) is precompact in E for each x∈X;(iii) givenW∈𝒲, there exists a compact set K⊆X such that {Gx(𝒜,W):x∈K} covers X;(3)
(i)𝒜(X)={f(x):x∈X,f∈𝒜} is precompact in E;(ii) givenW∈𝒲, there exists a finite set F⊆X such that {Gx(𝒜,W):x∈F} covers X;(4)
(i)𝒜(x) is precompact in E for each x∈X;(ii) givenW∈𝒲, there exists a finite set F⊆X such that {Gx(𝒜,W):x∈F} covers X.Definition 3.
A subsetP of ℝ is called relatively dense in ℝ if there exists a number ℓ>0 such that every interval of length ℓ in ℝ contains at least one point of P.Definition 4.
A functionf:ℝ→E is called almost periodic if it is continuous and, for each W∈𝒲, there exists a number ℓ=ℓ(W,f)>0 such that each interval of length ℓ in ℝ contains a point τ=τ(W,f) such that
(7)f(t+τ)-f(t)∈W∀t∈ℝ.
A number τ∈ℝ for which (7) holds is called a (W,f)-translation numberof f. The above property says that, for each W∈𝒲, the function f has a set of (W,f)-translation numbers P=P(W,f) which is relatively dense in ℝ. The set of all almost periodic functions f:ℝ→E is denoted by AP(ℝ,E). For any f:ℝ→E and a fixed ω∈ℝ, the ω-translate of f is defined as the function fω:ℝ→E defined by
(8)fω(t)=f(t+ω),t∈ℝ.
We will denote H(f)={fω:ω∈ℝ}, the set of all translates of f.Theorem 5 (see [4, 7]).
LetE be a TVS. Let f∈AP(ℝ,E). Then,(a)
f has totally bounded range f(ℝ); hence f is bounded;(b)
f is uniformly continuous on ℝ.Remark 6.
Clearly, by the above theorem,AP(ℝ,E)⊆Cb(ℝ,E). We shall see below that AP(ℝ,E)is a vector space.Theorem 7 (see [4, 7]).
LetE be a TVS. If {fn} is a sequence in AP(ℝ,E) such that fn→uf, then f∈AP(ℝ,E).Theorem 8 (Bochner’s criterion [4, 7]).
LetE be a TVS and f:ℝ→E a continuous function. (a)
If the set of translatesH(f)={fω:ω∈ℝ} is u-sequentially compact in Cb(ℝ,E), then f is almost periodic.(b)
Conversely, iff is almost periodic and, in addition, E is an F-space, then the set of translates H(f)={fω:ω∈ℝ} is u-compact in Cb(ℝ,E).
Thus, forE being an F-space, a continuous function f:ℝ→E is almost periodic if and only if the set H(f)={fω:ω∈ℝ} is u-compact in Cb(ℝ,E).Theorem 9 (see [4, 7]).
LetE be a TVS. If f:ℝ→E is almost periodic, then the functions (i) λf(λ∈𝕂), (ii) f¯(t)≡f(-t), and (iii) fω(t)=f(t+ω)(ω∈ℝ) are also almost periodic.Theorem 10 (see[4, 7]).
LetE be an F-space. (a)
Iff,g∈AP(ℝ,E), then f+g∈AP(ℝ,E); hence AP(ℝ,E) is a vector space.(b)
(AP(ℝ,E),u) is complete.


## 3. Main Results

For a fixeda∈ℝ, let Ja=[0,∞)={t∈ℝ:t≧a}. A subset P of Ja is said to be relatively dense in Ja if there exists ℓ>0 such that, for each t∈Ja, the closed interval [t,t+ℓ] contains at least one member of P.We define the notion of asymptotic almost periodicity in the case where the range is in a general topological vector spaceE as follows.Definition 11.
Consider a fixeda∈ℝ. A continuous function f:Ja→E will be called asymptotically almost periodicif, given any W∈𝒲, there exists r=r(W,f)≧a and a relatively dense set P=P(W,f) in Jr such that, for each τ∈P and every t∈Jr with t+τ≧r,
(9)f(t+τ)-f(t)∈W.In this section, we obtain extension of some results of [13] to our general setting.Definition 12.
Consider a fixeda∈ℝ. A subset 𝒜 of C(Ja,E) is called(i)
equialmost periodic if, given W∈𝒲, there exists a relatively dense set P=P(W,𝒜) in ℝ such that, for each τ∈P and every t∈ℝ,
(10)f(t+τ)-f(t)∈W∀f∈𝒜;(ii)
equiasymptotically almost periodic if, given W∈𝒲, there exists r=r(W,𝒜)≧a and a relatively dense set P=P(W,𝒜) in Jr such that, for each τ∈P and every t∈Jr with t+τ≧r,
(11)f(t+τ)-f(t)∈W∀f∈𝒜.Lemma 13.
(a) For anya∈ℝ, let T=Ja, and let 𝒜 be a precompact subset of (C(T,E),k). If 𝒜 is also equi-asymptotically almost periodic, then 𝒜 is uniformly equicontinuous on T and 𝒜(T) is precompact in E.
(b) LetT=ℝ, and let 𝒜 be a precompact subset of (C(T,E),k). If 𝒜 is equi-almost periodic, then 𝒜 is uniformly equicontinuous on T and 𝒜(T) is precompact in E.Proof.
(a) SupposeT=Ja with 𝒜 being a precompact subset of (C(T,E),k) and also equi-asymptotically almost periodic.
First,𝒜 is uniformly equicontinuous on Ja as follows. Let W∈𝒲. Choose balanced V∈𝒲 such that V+V+V⊆W. There exist r≧a, ℓ>0, and a relatively dense set P in Jr such that, for each τ∈P and every t∈Jr with t+τ≧r,
(12)f(t+τ)-f(t)∈V∀f∈𝒜,
while [t,t+ℓ]∩P≠⌀ for any t∈Jr. Let d=max{r,ℓ}, and choose τk∈[kd,(k+1)d]∩P,k=1,2,… By Theorem 1, 𝒜 is equicontinuous on Ja, and hence 𝒜 is uniformly equi-continuous on the closed interval [a,5d]. Then there exists δ∈(0,d/2) such that
(13)f(t1)-f(t2)∈Vwheneverf∈𝒜,t1,t2∈[a,5d]with|t1-t2|<δ.
Now, let t1,t2>4d with |t1-t2|<δ. Choose k∈ℕ such that t1,t2∈[kd,(k+2)d] and put s1=t1-τk-2,  s2=t2-τk-2. Then it is easy to see that s1,s2∈[d,4d] and |s1-s2|<δ, and so by (12) and (13),
(14)f(t1)-f(t2)=(f(s1+τk-2)-f(s1))+(f(s1)-f(s2))+(f(s2+τk-2)-f(s2))∈V+V+V⊆W
for any f∈𝒜. This shows that 𝒜 is uniformly equicontinuous on Ja.
Next,𝒜(Ja) is precompact in E as follows. Let W,V,r,ℓ,P, and d be as above. By the equicontinuity of 𝒜, we can choose a finite (open) cover {Gi:1≤i≤n} of [a,3d] and ti∈Gi,i=1,…,n, such that
(15)f(t)-f(ti)∈Vfort∈Gi,i∈{1,…,n}andallf∈𝒜.
If t>3d, choose k∈ℕ such that t∈[kd,(k+1)d]. Putting s=t-tk-2, we have s∈[d,3d]. Then s∈Gi for some i∈{1,…,n}, and therefore, for any f∈𝒜, by (12) and (15),
(16)f(t)-f(ti)=(f(s+τk-2)-f(s))+(f(s)-f(ti))∈V+V⊆W.
That is, 𝒜(Ja)⫅⋃i=1n(𝒜(ti)+W). By Theorem 1, for each i∈{1,…,n}, 𝒜(ti) is precompact in E. Thus 𝒜(Ja) is precompact in E.
(b) SupposeT=ℝ, and let 𝒜 be a precompact subset of (C(T,E),k) and also equi-almost periodic. Then, with minor changes in the above proof, it follows that 𝒜 is uniformly equicontinuous on T and 𝒜(T) is precompact in E.Theorem 14.
LetE be a TVS, and let T=Ja, where a∈ℝ. Then the following are equivalent for a subset 𝒜 of C(T,E).(1)
Consider the following:
(i)
𝒜 is precompact in (C(T,E),k);(ii)
𝒜 is equi-asymptotically almost periodic.(2)
The setH+(𝒜)={fω:f∈𝒜,ω∈ℝ+} of translates is a precompact subset of (Cb(T,E),u).Proof.
(
1
)
⇒
(
2
) First, we consider the case T=Ja and assume that 𝒜 is a k-precompact and equi-asymptotically almost periodic subset of C(Ja,E). By Lemma 13, 𝒜(Ja) is precompact in E. Since H+(𝒜) is contained in (Cpc(Ja,E),u), H+(𝒜)(t) is precompact in E for each t∈Ja. To show that H+(𝒜) is precompact in (Cpc(Ja,E),u), we need to verify that the finite covering condition 3(ii) of Arzelà-Ascoli Theorem 2 holds for H+(𝒜) in this setting. [Let W∈𝒲; choose a balanced V∈𝒲 such that V+V+V⊆W. As in Lemma 13, choose r≧a,l>0, and a relatively dense set P in Jr such that, for each τ∈P and every t∈Jr with t+τ≧r,
(17)f(t+τ)-f(t)∈V∀f∈𝒜,
while [t,t+l]∩P≠⌀ for any t∈Jr. Furthermore, we put d=max{r,l}, set τ0=0, and fix τk∈[kd,(k+1)d]∩P,k=1,2,…. By Lemma 13, 𝒜 is uniformly equicontinuous on Ja; H+(𝒜) is also clearly uniformly equicontinuous. In particular, we obtain a finite cover {Si:1≤i≤n} of [d,3d] by (relatively open) subsets of Jr and si∈Si,i=1,…,n, such that, for every f∈𝒜 and all ω∈ℝ+,(18)fω(s)-fω(si)∈Vwhenevers∈Si,i∈{1,…,n}.
Put Gi=⋃k=0∞(Si+τk), i=1,…,n. Now, taking t∈Gi for i∈{1,…,n}, choose s∈Si and k∈ℕ∪{0} so that t=s+τk. Using (17) and (18), we obtain
(19)fω(t)-fω(si)=(f(s+ω+τk)-f(s+ω))+(fω(s)-fω(si))∈V+V⊆W
for every f∈𝒜 and all ω∈ℝ+.
Next, for any𝒜⊆Cpc(Ja,E) and W∈𝒲, recall the notation (6):
(20)Gs(𝒜,W)={t∈Ja:f(t)-f(s)∈W∀f∈𝒜},s∈Ja.
It is easy to verify that Jd=[d,∞)⫅⋃i=1nGi, and so {Gsi(H+(𝒜),W):i=1,…,n} covers Jd. By the equicontinuity of H+(𝒜), it is possible to trivially cover [a,d] by finitely many sets of this same prescribed form, we see that 3(ii) of Arzelà-Ascoli Theorem 2 is satisfied. Thus H+(𝒜) is precompact in (Cb(Ja,E),u).
(
2
)
⇒
(
1
) Assume that (6) holds.
(1)(i) This is clearly satisfied.
(1)(ii) In view of the Arzelà-Ascoli Theorem1, H+(𝒜) is a precompact subset of (Cb(Ja,E),u). Furthermore, the Arzelà-Ascoli Theorem 2 can be used in showing that H+(𝒜) is equi-asymptotically almost periodic as follows. Fixing W∈𝒲, we use (3)(ii) of Arzelà-Ascoli Theorem 2 to obtain a finite cover {Gi:1≤i≤n} of Jb, where b=max{a,1} and Gi⫅Jb, i=1,…,n, and ti∈Gi, i=1,…,n, such that, for every f∈𝒜 and all ω∈ℝ+,(21)fω(t)-fω(ti)∈Wwhenevert∈Gi,i∈{1,…,n}.
Putting r=ℓ=max{t1,…,tn}, we note that r≧a and ℓ>0. Now, let P=(⋃i=1n(Gi-ti))∩Jr. Then P is relatively dense in Jr. (For any t∈Jr,t+ℓ≧b, we have t+ℓ∈Gi for some i∈{1,…,n}. Then t≦(t+ℓ)-ti≦t+ℓ.) Now, given t∈Jr and τ∈P, choose i∈{1,…,n} and s∈Gi such that τ=s-ti. Since t-ti≧0, we then have by (21)
(22)f(t+τ)-f(t)=f((t-ti)+s)-f((t-ti)+ti)=ft-ti(s)-ft-ti(ti)∈W
for all f∈𝒜. This proves that 𝒜 is equi-asymptotically almost periodic.We next obtain an analogue of the above result forT=ℝ (instead of T=Ja) and 𝒜 equi-almost periodic (instead of equi-asymptotically almost periodic) as follows.Theorem 15.
LetE be a TVS, and let T=ℝ. Then the following are equivalent for a subset 𝒜 of C(T,E).(1)
(i)𝒜 is precompact in (C(T,E),k);(ii)𝒜 is equi-almost periodic.(2)
The setH(𝒜)={fω:f∈𝒜,ω∈ℝ} of translates is a precompact subset of (Cb(T,E),u).Proof.
This follows from Theorem14 with minor changes, outlined as follows.
(
1
)
⇒
(
2
) Suppose 𝒜 is a k-precompact and equi-almost periodic subset of C(ℝ,E). For this part, we can easily adapt the arguments of Theorem 14 to show that H(𝒜) is a precompact subset of (Cpc(ℝ,E),u).
(
2
)
⇒
(
1
) In view of the Arzelà-Ascoli Theorem 1, 𝒜 is a precompact subset of (Cpc(ℝ,E),u). Further, 𝒜 is equi-almost periodic. In fact, fix W∈𝒲. Since H(𝒜) is a precompact, we use (3)(ii) of Arzelà-Ascoli Theorem 2 to obtain a finite cover {Gi:1≤i≤n} of ℝ and ti∈Gi,i=1,…,n, such that, for every f∈𝒜 and all ω∈ℝ,(23)fω(t)-fω(ti)∈Wwhenevert∈Gi,i∈{1,…,n}.
In this case, take ℓ>2max{|t1|,…,|tn|}, and put P=⋃i=1n(Gi-ti). Then P is relatively dense in ℝ. In fact, for any t∈ℝ, choose i∈{1,…,n} such that t+ℓ/2∈Gi. Then
(24)(t+ℓ2)-ti∈[t,t+ℓ]∩P.
Also, given t∈ℝ and any τ∈P, we can adapt the arguments of Theorem 14 till proving (22) and obtain
(25)f(t+τ)-f(t)∈W∀f∈𝒜.
Consequently, 𝒜 is equi-almost periodic.From Theorem14, we can deduce the following extension of Fréchet’s theorem.Theorem 16.
LetE be a TVS, and let a∈ℝ. For any f∈C(Ja,E), f is asymptotically almost periodic if and only if the set H+(f)={fω:ω≧0} is a precompact subset of (Cb(Ja,E),u).We next consider an alternate view of asymptotically almost periodic functions. Fixinga∈ℝ, if g:ℝ→E is a continuous almost periodic function and f∈C0(Ja,E), then the function f=g∣Ja+h is clearly asymptotically almost periodic on the interval Ja. On the other hand, it is also known that every asymptotically almost periodic function in C(Ja,E) can be so represented in case E is finite dimensional. We shall use Theorem 14 to study it in a more general situation. Recall that a TVSE is said to be quasicomplete if every bounded Cauchy net in E converges. Clearly completeness implies quasicompleteness.Theorem 17.
Assume thatE is a quasi-complete TVS, and fix a∈ℝ. Then f∈C(Ja,E) is asymptotically almost periodic if and only if there is a unique almost periodic function g∈C(ℝ,E) and a unique function h∈C0(Ja,E) such that
(26)f=g∣Ja+h.Proof.
SinceE is a quasi-complete TVS, it follows that both (Cpc(Ja,E),u) and (Cpc(ℝ,E),u) are quasi-complete [13, 16, 18]. Therefore, precompactness can be considered equivalent to relative compactness in either of these two spaces.
Consider an arbitrary asymptotically almost periodic functionf∈C(Ja,E). Then, for each pair (n,W)∈ℕ×𝒲, there exist r(n,W)≧max{a,n} and a relatively dense set P(n,W) in Jr(n,W) such that, for any τ∈P(n,W) and every t∈Jr(n,W),
(27)f(t+τ)-f(t)∈1nW.
Let us equip D=ℕ×𝒲 with the usual product order; that is, given (m,V),(n,W)∈D,(m,V)≦(n,W) if and only if m≦n and W⊆V (or equivalently ρV⊆ρW). Also, for each α=(n,W)∈D, we choose τα∈P(n,W). By Theorem 14, since {τα}α∈D is a net in ℝ+, there is a subnet {τα(λ)}λ∈D of {τα}α for which the net of translates {fτα(λ)} converges uniformly on Ja to some φ∈(Cpc(Ja,E),u).
We claim thatφ has an almost periodic extension g∈C(ℝ,E). For a fixed b∈ℝ∖Ja=(-∞,a), there exists λb∈D so that  b+τα(λ)≧a whenever λ∈D and λ≧λb; put Db={λ∈D:λ≧λb}. For each λ∈Db, let fτα(λ)*:Jb→X be an extension of fτα(λ) defined by
(28)fτα(λ)*(s)=f(s+τα(λ)),s∈Jb.
We can easily see that {fτα(λ)*:λ∈Db} is a bounded net in (Cpc(Ja,E),u). Now, let ε>0 and W∈𝒲 be given. Choose V∈𝒲 with V+V+V⊆W and choose n∈ℕ with n≧max{a-b,3/ε}, let τ∈P(n,V), and take λ0∈Db for which the following conditions are satisfied.(i)
Ifλ,μ∈Db and λ,μ≧λ0, then
(29)fτα(λ)(t)-fτα(μ)(t)∈ε3V∀t∈Ja,orρV[fτα(λ)(t)-fτα(μ)(t)]≤ε3∀t∈Ja.(ii)
In caseλ∈Db and λ≧λ0, b+τα(λ)≧r(n,V). For any s∈Jb, if λ,μ≧λ0 (as in case (i)), we then have by (2), (28), and (29) that
(30)ρW[fτα(λ)*(s)-fτα(λ)*(s)]≤ρV(f(s+τα(λ))-f(s+τα(λ)+τ))+ρV(fτα(λ)(s+τ)-fτα(μ)(s+τ))+ρV(f(s+τα(μ)+τ)-f(s+τα(μ)))≤ε3+ε3+ε3=ε.So {fτα(λ)}λ∈Db is a Cauchy net in (Cpc(Ja,E),u). Since it is also bounded and (Cpc(Ja,E),u) is quasi-complete, it converges uniformly on Jb to a function φb∈(Cpc(Ja,E),u). Clearly, we have that φb∣Ja=φ. If c∈ℝ with c≦b and if λc is any element in D for which c+τα(λ)≧a for all λ∈D such that λ≧λ0, then the corresponding net {fτα(λ)}λ≧λc of extensions from Ja to Jc will converge in Cpc(Jc,E) to a function φc. Clearly φc=φb on Jb. Define a function g:ℝ→E by
(31)g(t)={φ(t),t∈Jaφt(t),t∈ℝ∖Ja.
Then g is well defined and continuous on ℝ and g∣Ja=φ.
We claim thatg is also almost periodic. Let ε>0 and W∈𝒲. Choose a balanced U∈𝒲 with U+U+U⊆W. Choose n∈ℕ such that n≧3/ε and the set P=P(n,W)∪{τ∈ℝ:-τ∈P(n,W)} is relatively dense in ℝ. Next, given t∈ℝ and τ∈P, first choose b∈ℝ∖Ja with b≦min{t,t+τ}, and then take λ∈D such that(a)
b+τα(λ)≧r(n,W)+|τ|,(b)
φb(s)-f(s+τα(λ))∈(ε/3)U for every s∈Jb.
Then, by (2), (31), and (b),
(32)ρW[g(t+τ)-g(t)]≤ρU(φb(t+τ)-f(t+τ+τα(λ)))+ρU(f(τ+τα(λ)+τ)-f(τ+τα(λ)))+ρU(f(τ+τα(λ))-φb(t))≤ε3+ε3+ε3=ε,
and so g is almost periodic.
It remains to show thath=f-g∣Ja vanishes at infinity on Ja. Fix W∈𝒲. Choose a balanced V∈𝒲 with V+V⊆W, and choose sufficiently large n∈ℕ such that there exists λ∈D such that α(λ)≧(n,V) and
(33)ftα(λ)(t)-φ(t)∈V∀t∈Ja;
we may assume that α(λ)=(m,V), where m∈ℕ. Thus, if t∈Jr(m,V), then
(34)h(t)=f(t)-φ(t)=(f(t)-f(t+τα(λ)))+(f(t+τα(λ))-φ(t))∈V+V⊆W.
Finally, we show that the functionsg and h in the representation (26) are unique. First observe that an almost periodic function θ∈C(ℝ,E) must be identically zero on ℝ if θ∣Ja∈C0(Ja,E). Therefore, given almost periodic functions g and φ in C(ℝ,E), we only need to verify that g+φ is also almost periodic. For this, choosing any net {ωα}α in ℝ, we apply Theorem 15 (twice) to obtain a subnet {ωα(λ)}λ such that the corresponding nets of translates {gωα(λ)} both converge in (Cpc(ℝ,E),u). Another application of Theorem 15 now gives us that g+φ is almost periodic.Scope of Applications. (1) The importance of such a work has been highlighted in ([12]; [13], p. 19-20). If, for a given Banach space E, a linear operator A:𝒟(A)⫅E→E is the infinitesimal generator of a C0-semigroup (T(t))t>0 of bounded linear operators from E to E, then, for any x0∈𝒟(A), the unique strong solution of the abstract Cauchy problem
(CP)dx(t)dt=A[x(t)],t>0,x(0)=x0∈E,
associated with A, is given by the motion x(·)=T(·)x0 of (T(t))t>0 through x0. See also [10].In the qualitative study of the solution, one of the problems is to determine its asymptotic behaviour ast→∞. In this regard, a useful concept is the so-called positive ω-limit set
(35)ω+(x0)={y∈E:∃0≤tn⟶∞suchthatT(tn)x0.⟶y}
of all possible limit points. The basic result is that: if the orbit γ+(x0)={T(t)x0:t≥0} of the motion is relatively compact, then ω+(x0) is nonempty, compact, connected, and invariant. A qualitative much stronger mode of asymptotic behaviour results if not only γ+(x0) is relatively compact, but also the set H+(T(·)x0)={Tω(·)x0:ω>0}, Tω(t)x0=T(t+ω)x0, of all translates of the motion is a relatively compact subset of the space (Cb(ℝ+,E),∥·∥∞). This observation raises the following problem.(*) Characterize those f∈Cb(ℝ+,E) for which H+(f) is relatively compact in (Cb(ℝ+,E),∥·∥∞).Clearly, our results contain a complete solution of problem(*) in the general setting.(2) In [3, 19], it has been obtained that in the (nonlocally convex) p-Fréchet space E, 0<p<1, (CP) has a unique solution x(t)=T(t)(x), with
(36)T(t)(x)=limn→∞(1-tnA)-1(x),
with the limit being taken in the p-norm of E.Our results thus widen the scope of applications of asymptotic almost periodicity to the nonlocally convex setting.



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