Asymptotic Almost Periodic Functions with Range in a Topological Vector Space

Abstract

The notion of asymptotic almost periodicity was …first 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.

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 𝐽_𝑎=[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 Section 3.

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 𝜌_𝑊 of any such neighborhood 𝑊 is continuous and satisfies 𝑊=𝑥∈𝐸∶𝜌_𝑊(𝑥)≤1,int𝑊=𝑥∈𝐸∶𝜌_𝑊(𝑥)<1.(1) We mention that, for any neighborhood 𝑊 of 0 in 𝐸, 𝜌_𝑊 need not be absolutely homogeneous or subadditive; however, the following useful properties hold [15, 16].(a)𝜌_𝑊 is positively homogeneous; it is absolutely homogeneous if 𝑊 is balanced.(b)If 𝑉 is a balanced neighborhood of 0 in 𝐸 with 𝑉+𝑉⊆𝑊, then 𝜌_𝑊𝑥+𝑦≤𝜌_𝑉(𝑥)+𝜌_𝑉𝑦∀𝑥,𝑦∈𝐸.(2)

A complete metrizable TVS is called an 𝐹-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 𝐶_𝑏()=()(𝑋),𝐶𝑋,𝐸𝑓∈𝐶𝑋,𝐸∶𝑓isboundedin𝐸_𝑝𝑐()=()(𝑋),𝐶𝑋,𝐸𝑓∈𝐶𝑋,𝐸∶𝑓isprecompactin𝐸₀(𝑋,𝐸)=𝑓∈𝐶(𝑋,𝐸)∶𝑓vanishesatinfinityon𝑋.(3) Clearly, 𝐶_𝑝𝑐(𝑋,𝐸)⊆𝐶_𝑏(𝑋,𝐸)⊆𝐶(𝑋,𝐸), 𝐶₀(𝑋,𝐸)⊆𝐶_𝑏(𝑋,𝐸), and all these sets are vector spaces over 𝕂 with the pointwise operations of addition and scalar multiplication. The uniform topology 𝑢 on 𝐶_𝑏(𝑋,𝐸) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form 𝑁_𝑏(𝑋,𝑊)=𝑓∈𝐶_𝑏(𝑋,𝐸)∶𝑓(𝑋)⊆𝑊,(4) where 𝑊∈𝒲. The compact-open topology 𝑘 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 𝑁(𝐾,𝑊)=𝑓∈𝐶(𝑋,𝐸)∶𝑓(𝐾)⊆𝑊,(5) where 𝐾⊆𝑋 is compact and 𝑊∈𝒲. Clearly, 𝑘≤𝑢 on 𝐶_𝑏(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. Let 𝑋 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 any 𝒜⊆𝐶_𝑝𝑐(𝑋,𝐸) and 𝑊∈𝒲, let 𝐺_𝑥()=𝑦(𝑥)𝒜,𝑊𝑦∈𝑋∶𝑓−𝑓∈𝑊∀𝑓∈𝒜,𝑥∈𝑋.(6)

Theorem 2. Let 𝑋 be a 𝑘-space. Then, for any 𝒜⊆𝐶_𝑝𝑐(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) given 𝑊∈𝒲, there exists a compact set 𝐾⊆𝑋 such that {𝐺_𝑥(𝒜,𝑊)∶𝑥∈𝐾} covers 𝑋;(3) (i) 𝒜(𝑋)={𝑓(𝑥)∶𝑥∈𝑋,𝑓∈𝒜} is precompact in 𝐸; (ii) given W∈𝒲, there exists a finite set F⊆X such that {G_x(𝒜,W)∶x∈F} covers X;(4) (i) 𝒜(𝑥) is precompact in 𝐸 for each 𝑥∈𝑋; (ii) given 𝑊∈𝒲, there exists a finite set 𝐹⊆𝑋 such that {𝐺_𝑥(𝒜,𝑊)∶𝑥∈𝐹} covers 𝑋.

Definition 3. A subset 𝑃 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 function 𝑓∶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 𝑓(𝑡+𝜏)−𝑓(𝑡)∈𝑊∀𝑡∈ℝ.(7) A number 𝜏∈ℝ for which (7) holds is called a (𝑊,𝑓)-translation number of 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 𝑓_𝜔∶ℝ→𝐸 defined by 𝑓_𝜔(𝑡)=𝑓(𝑡+𝜔),𝑡∈ℝ.(8) We will denote 𝐻(𝑓)={𝑓_𝜔∶𝜔∈ℝ}, the set of all translates of 𝑓.

Theorem 5 (see [4, 7]). Let 𝐸 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)⊆𝐶_𝑏(ℝ,𝐸). We shall see below that 𝐴𝑃(ℝ,𝐸) is a vector space.

Theorem 7 (see [4, 7]). Let 𝐸 be a TVS. If {𝑓_𝑛} is a sequence in 𝐴𝑃(ℝ,𝐸) such that 𝑓→−_𝑛^𝑢𝑓, then 𝑓∈𝐴𝑃(ℝ,𝐸).

Theorem 8 (Bochner’s criterion [4, 7]). Let 𝐸 be a TVS and 𝑓∶ℝ→𝐸 a continuous function. (a)If the set of translates 𝐻(𝑓)={𝑓_𝜔∶𝜔∈ℝ} is 𝑢-sequentially compact in 𝐶_𝑏(ℝ,𝐸), then 𝑓 is almost periodic. (b)Conversely, if 𝑓 is almost periodic and, in addition, 𝐸 is an 𝐹-space, then the set of translates 𝐻(𝑓)={𝑓_𝜔∶𝜔∈ℝ} is 𝑢-compact in 𝐶_𝑏(ℝ,𝐸).
Thus, for 𝐸 being an 𝐹-space, a continuous function 𝑓∶ℝ→𝐸 is almost periodic if and only if the set 𝐻(𝑓)={𝑓_𝜔∶𝜔∈ℝ} is 𝑢-compact in 𝐶_𝑏(ℝ,𝐸).

Theorem 9 (see [4, 7]). Let 𝐸 be a TVS. If 𝑓∶ℝ→𝐸 is almost periodic, then the functions (i) 𝜆𝑓 (𝜆∈𝕂), (ii) 𝑓(𝑡)≡𝑓(−𝑡), and (iii) 𝑓_𝜔(𝑡)=𝑓(𝑡+𝜔) (𝜔∈ℝ) are also almost periodic.

Theorem 10 (see[4, 7]). Let 𝐸 be an 𝐹-space. (a)If 𝑓,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 fixed 𝑎∈ℝ, let 𝐽_𝑎=[0,∞)={𝑡∈ℝ∶𝑡≧𝑎}. A subset 𝑃 of 𝐽_𝑎 is said to be relatively dense in 𝐽_𝑎 if there exists ℓ>0 such that, for each 𝑡∈𝐽_𝑎, 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 space 𝐸 as follows.

Definition 11. Consider a fixed 𝑎∈ℝ. A continuous function 𝑓∶𝐽_𝑎→𝐸 will be called asymptotically almost periodic if, given any 𝑊∈𝒲, there exists 𝑟=𝑟(𝑊,𝑓)≧𝑎 and a relatively dense set 𝑃=𝑃(𝑊,𝑓) in 𝐽_𝑟 such that, for each 𝜏∈𝑃 and every 𝑡∈𝐽_𝑟 with 𝑡+𝜏≧𝑟, 𝑓(𝑡+𝜏)−𝑓(𝑡)∈𝑊.(9)

In this section, we obtain extension of some results of [13] to our general setting.

Definition 12. Consider a fixed 𝑎∈ℝ. A subset 𝒜 of 𝐶(𝐽_𝑎,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, 𝑓(𝑡+𝜏)−𝑓(𝑡)∈𝑊∀𝑓∈𝒜;(10)(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 𝐽_𝑟 such that, for each 𝜏∈𝑃 and every 𝑡∈𝐽_𝑟 with 𝑡+𝜏≧𝑟, 𝑓(𝑡+𝜏)−𝑓(𝑡)∈𝑊∀𝑓∈𝒜.(11)

Lemma 13. (a) For any 𝑎∈ℝ, let 𝑇=𝐽_𝑎, 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) Let 𝑇=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) Suppose 𝑇=𝐽_𝑎 with 𝒜 being a precompact subset of (𝐶(𝑇,𝐸),𝑘) and also equi-asymptotically almost periodic.
First, 𝒜 is uniformly equicontinuous on 𝐽_𝑎 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 𝐽_𝑟 such that, for each 𝜏∈𝑃 and every 𝑡∈𝐽_𝑟 with 𝑡+𝜏≧𝑟, 𝑓(𝑡+𝜏)−𝑓(𝑡)∈𝑉∀𝑓∈𝒜,(12) while [𝑡,𝑡+ℓ]∩𝑃≠∅ for any 𝑡∈𝐽_𝑟. Let 𝑑=max{𝑟,ℓ}, and choose 𝜏_𝑘∈[𝑘𝑑,(𝑘+1)𝑑]∩𝑃,𝑘=1,2,… By Theorem 1, 𝒜 is equicontinuous on 𝐽_𝑎, and hence 𝒜 is uniformly equi-continuous on the closed interval [𝑎,5𝑑]. Then there exists 𝛿∈(0,𝑑/2) such that 𝑓𝑡₁−𝑓𝑡₂𝑡∈𝑉whenever𝑓∈𝒜,₁,𝑡₂∈[𝑎,5𝑑]with𝑡₁−𝑡₂<𝛿.(13) Now, let 𝑡₁,𝑡₂>4𝑑 with |𝑡₁−𝑡₂|<𝛿. Choose 𝑘∈ℕ such that 𝑡₁,𝑡₂∈[𝑘𝑑,(𝑘+2)𝑑] and put 𝑠₁=𝑡₁−𝜏_𝑘−2,  𝑠₂=𝑡₂−𝜏_𝑘−2. Then it is easy to see that 𝑠₁,𝑠₂∈[𝑑,4𝑑] and |𝑠₁−𝑠₂|<𝛿, and so by (12) and (13), 𝑓𝑡₁−𝑓𝑡₂=𝑓𝑠₁+𝜏_𝑘−2−𝑓𝑠₁+𝑓𝑠₁−𝑓𝑠₂+𝑓𝑠₂+𝜏_𝑘−2−𝑓𝑠₂∈𝑉+𝑉+𝑉⊆𝑊(14) for any 𝑓∈𝒜. This shows that 𝒜 is uniformly equicontinuous on 𝐽_𝑎.
Next, 𝒜(𝐽_𝑎) 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 {𝐺_𝑖∶1≤𝑖≤𝑛} of [𝑎,3𝑑] and 𝑡_𝑖∈𝐺_𝑖,𝑖=1,…,𝑛, such that 𝑓(𝑡)−𝑓𝑡_𝑖∈𝑉for𝑡∈𝐺_𝑖,𝑖∈{1,…,𝑛}andall𝑓∈𝒜.(15) If 𝑡>3𝑑, choose 𝑘∈ℕ such that 𝑡∈[𝑘𝑑,(𝑘+1)𝑑]. Putting 𝑠=𝑡−𝑡_𝑘−2, we have 𝑠∈[𝑑,3𝑑]. Then 𝑠∈𝐺_𝑖 for some 𝑖∈{1,…,𝑛}, and therefore, for any 𝑓∈𝒜, by (12) and (15), 𝑓(𝑡)−𝑓𝑡_𝑖=𝑓𝑠+𝜏_𝑘−2−𝑓(𝑠)+𝑓(𝑠)−𝑓𝑡_𝑖∈𝑉+𝑉⊆𝑊.(16) That is, 𝒜(𝐽_𝑎)⫅⋃^𝑛_𝑖=1(𝒜(𝑡_𝑖)+𝑊). By Theorem 1, for each 𝑖∈{1,…,𝑛}, 𝒜(𝑡_𝑖) is precompact in 𝐸. Thus 𝒜(𝐽_𝑎) is precompact in 𝐸.
(b) Suppose 𝑇=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. Let 𝐸 be a TVS, and let 𝑇=𝐽_𝑎, 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 set 𝐻⁺(𝒜)={𝑓_𝜔∶𝑓∈𝒜,𝜔∈ℝ⁺} of translates is a precompact subset of (𝐶_𝑏(𝑇,𝐸),𝑢).

Proof. (1)⇒(2) First, we consider the case 𝑇=𝐽_𝑎 and assume that 𝒜 is a 𝑘-precompact and equi-asymptotically almost periodic subset of 𝐶(𝐽_𝑎,𝐸). By Lemma 13, 𝒜(𝐽_𝑎) is precompact in 𝐸. Since 𝐻⁺(𝒜) is contained in (𝐶_𝑝𝑐(𝐽_𝑎,𝐸),𝑢), 𝐻⁺(𝒜)(𝑡) is precompact in 𝐸 for each 𝑡∈𝐽_𝑎. To show that 𝐻⁺(𝒜) is precompact in (𝐶_𝑝𝑐(𝐽_𝑎,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 𝐽_𝑟 such that, for each 𝜏∈𝑃 and every 𝑡∈𝐽_𝑟 with 𝑡+𝜏≧𝑟, 𝑓(𝑡+𝜏)−𝑓(𝑡)∈𝑉∀𝑓∈𝒜,(17) while [𝑡,𝑡+𝑙]∩𝑃≠∅ for any 𝑡∈𝐽_𝑟. Furthermore, we put 𝑑=max{𝑟,𝑙}, set 𝜏₀=0, and fix 𝜏_𝑘∈[𝑘𝑑,(𝑘+1)𝑑]∩𝑃,𝑘=1,2,…. By Lemma 13, 𝒜 is uniformly equicontinuous on 𝐽_𝑎; 𝐻⁺(𝒜) is also clearly uniformly equicontinuous. In particular, we obtain a finite cover {𝑆_𝑖∶1≤𝑖≤𝑛} of [𝑑,3𝑑] by (relatively open) subsets of 𝐽_𝑟 and 𝑠_𝑖∈𝑆_𝑖,𝑖=1,…,𝑛, such that, for every 𝑓∈𝒜 and all 𝜔∈ℝ⁺,𝑓_𝜔(𝑠)−𝑓_𝜔𝑠_𝑖∈𝑉whenever𝑠∈𝑆_𝑖,𝑖∈{1,…,𝑛}.(18) Put 𝐺_𝑖=⋃^∞_𝑘=0(𝑆_𝑖+𝜏_𝑘), 𝑖=1,…,𝑛. Now, taking 𝑡∈𝐺_𝑖 for 𝑖∈{1,…,𝑛}, choose 𝑠∈𝑆_𝑖 and 𝑘∈ℕ∪{0} so that 𝑡=𝑠+𝜏_𝑘. Using (17) and (18), we obtain 𝑓_𝜔(𝑡)−𝑓_𝜔𝑠_𝑖=𝑓𝑠+𝜔+𝜏_𝑘−𝑓(𝑠+𝜔)+𝑓_𝜔(𝑠)−𝑓_𝜔𝑠_𝑖∈𝑉+𝑉⊆𝑊(19) for every 𝑓∈𝒜 and all 𝜔∈ℝ⁺.
Next, for any 𝒜⊆𝐶_𝑝𝑐(𝐽_𝑎,𝐸) and 𝑊∈𝒲, recall the notation (6): 𝐺_𝑠()=𝒜,𝑊𝑡∈𝐽_𝑎∶𝑓(𝑡)−𝑓(𝑠)∈𝑊∀𝑓∈𝒜,𝑠∈𝐽_𝑎.(20) It is easy to verify that 𝐽_𝑑=[𝑑,∞)⫅⋃^𝑛_𝑖=1𝐺_𝑖, and so {𝐺_𝑠𝑖(𝐻⁺(𝒜),𝑊)∶𝑖=1,…,𝑛} covers 𝐽_𝑑. 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 (𝐶_𝑏(𝐽_𝑎,𝐸),𝑢).
(2)⇒(1) Assume that (6) holds.
(1)(i) This is clearly satisfied.
(1)(ii) In view of the Arzelà-Ascoli Theorem 1, 𝐻⁺(𝒜) is a precompact subset of (𝐶_𝑏(𝐽_𝑎,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 {𝐺_𝑖∶1≤𝑖≤𝑛} of 𝐽_𝑏, where 𝑏=max{𝑎,1} and 𝐺_𝑖⫅𝐽_𝑏, 𝑖=1,…,𝑛, and 𝑡_𝑖∈𝐺_𝑖, 𝑖=1,…,𝑛, such that, for every 𝑓∈𝒜 and all 𝜔∈ℝ⁺,𝑓_𝜔(𝑡)−𝑓_𝜔𝑡_𝑖∈𝑊whenever𝑡∈𝐺_𝑖,𝑖∈{1,…,𝑛}.(21) Putting 𝑟=ℓ=max{𝑡₁,…,𝑡_𝑛}, we note that 𝑟≧𝑎 and ℓ>0. Now, let 𝑃=(⋃^𝑛_𝑖=1(𝐺_𝑖−𝑡_𝑖))∩𝐽_𝑟. Then 𝑃 is relatively dense in 𝐽_𝑟. (For any 𝑡∈𝐽_𝑟,𝑡+ℓ≧𝑏, we have 𝑡+ℓ∈𝐺_𝑖 for some 𝑖∈{1,…,𝑛}. Then 𝑡≦(𝑡+ℓ)−𝑡_𝑖≦𝑡+ℓ.) Now, given 𝑡∈𝐽_𝑟 and 𝜏∈𝑃, choose 𝑖∈{1,…,𝑛} and 𝑠∈𝐺_𝑖 such that 𝜏=𝑠−𝑡_𝑖. Since 𝑡−𝑡_𝑖≧0, we then have by (21) 𝑓(𝑡+𝜏)−𝑓(𝑡)=𝑓𝑡−𝑡_𝑖+𝑠−𝑓𝑡−𝑡_𝑖+𝑡_𝑖=𝑓_{𝑡−𝑡𝑖}(𝑠)−𝑓_{𝑡−𝑡𝑖}𝑡_𝑖∈𝑊(22) for all 𝑓∈𝒜. This proves that 𝒜 is equi-asymptotically almost periodic.

We next obtain an analogue of the above result for 𝑇=ℝ (instead of 𝑇=𝐽_𝑎) and 𝒜 equi-almost periodic (instead of equi-asymptotically almost periodic) as follows.

Theorem 15. Let 𝐸 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 set 𝐻(𝒜)={𝑓_𝜔∶𝑓∈𝒜,𝜔∈ℝ} of translates is a precompact subset of (𝐶_𝑏(𝑇,𝐸),𝑢).

Proof. This follows from Theorem 14 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 (𝐶_𝑝𝑐(ℝ,𝐸),𝑢).
(2)⇒(1) In view of the Arzelà-Ascoli Theorem 1, 𝒜 is a precompact subset of (𝐶_𝑝𝑐(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 {𝐺_𝑖∶1≤𝑖≤𝑛} of ℝ and 𝑡_𝑖∈𝐺_𝑖,𝑖=1,…,𝑛, such that, for every 𝑓∈𝒜 and all 𝜔∈ℝ,𝑓_𝜔(𝑡)−𝑓_𝜔𝑡_𝑖∈𝑊whenever𝑡∈𝐺_𝑖,𝑖∈{1,…,𝑛}.(23) In this case, take ℓ>2max{|𝑡₁|,…,|𝑡_𝑛|}, and put 𝑃=⋃^𝑛_𝑖=1(𝐺_𝑖−𝑡_𝑖). Then 𝑃 is relatively dense in ℝ. In fact, for any 𝑡∈ℝ, choose 𝑖∈{1,…,𝑛} such that 𝑡+ℓ/2∈𝐺_𝑖. Then ℓ𝑡+2−𝑡_𝑖∈[𝑡,𝑡+ℓ]∩𝑃.(24) Also, given 𝑡∈ℝ and any 𝜏∈𝑃, we can adapt the arguments of Theorem 14 till proving (22) and obtain 𝑓(𝑡+𝜏)−𝑓(𝑡)∈𝑊∀𝑓∈𝒜.(25) Consequently, 𝒜 is equi-almost periodic.

From Theorem 14, we can deduce the following extension of Fréchet’s theorem.

Theorem 16. Let 𝐸 be a TVS, and let 𝑎∈ℝ. For any 𝑓∈𝐶(𝐽_𝑎,𝐸), 𝑓 is asymptotically almost periodic if and only if the set 𝐻⁺(𝑓)={𝑓_𝜔∶𝜔≧0} is a precompact subset of (𝐶_𝑏(𝐽_𝑎,𝐸),𝑢).

We next consider an alternate view of asymptotically almost periodic functions. Fixing 𝑎∈ℝ, if 𝑔∶ℝ→𝐸 is a continuous almost periodic function and 𝑓∈𝐶₀(𝐽_𝑎,𝐸), then the function 𝑓=𝑔∣𝐽_𝑎+ℎ is clearly asymptotically almost periodic on the interval 𝐽_𝑎. On the other hand, it is also known that every asymptotically almost periodic function in 𝐶(𝐽_𝑎,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 that 𝐸 is a quasi-complete TVS, and fix 𝑎∈ℝ. Then 𝑓∈𝐶(𝐽_𝑎,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(𝐽_𝑎,𝐸) such that 𝑓=𝑔∣𝐽_𝑎+ℎ.(26)

Proof. Since 𝐸 is a quasi-complete TVS, it follows that both (𝐶_𝑝𝑐(𝐽_𝑎,𝐸),𝑢) and (𝐶_𝑝𝑐(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 function 𝑓∈𝐶(𝐽_𝑎,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 𝐽_{𝑟(𝑛,𝑊)} such that, for any 𝜏∈𝑃(𝑛,𝑊) and every 𝑡∈𝐽_{𝑟(𝑛,𝑊)}, 𝑓()(𝑡)∈1𝑡+𝜏−𝑓𝑛𝑊.(27) 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 𝜌_𝑉⊆𝜌_𝑊). Also, for each 𝛼=(𝑛,𝑊)∈𝐷, we choose 𝜏_𝛼∈𝑃(𝑛,𝑊). By Theorem 14, since {𝜏_𝛼}_𝛼∈𝐷 is a net in ℝ⁺, there is a subnet {𝜏_𝛼(𝜆)}_𝜆∈𝐷 of {𝜏_𝛼}_𝛼 for which the net of translates {𝑓_{𝜏𝛼(𝜆)}} converges uniformly on 𝐽_𝑎 to some 𝜑∈(𝐶_𝑝𝑐(𝐽_𝑎,𝐸),𝑢).
We claim that 𝜑 has an almost periodic extension 𝑔∈𝐶(ℝ,𝐸). For a fixed 𝑏∈ℝ⧵𝐽_𝑎=(−∞,𝑎), there exists 𝜆_𝑏∈𝐷 so that  𝑏+𝜏_𝛼(𝜆)≧𝑎 whenever 𝜆∈𝐷 and 𝜆≧𝜆_𝑏; put 𝐷_𝑏={𝜆∈𝐷∶𝜆≧𝜆_𝑏}. For each 𝜆∈𝐷_𝑏, let 𝑓^∗_{𝜏𝛼(𝜆)}∶𝐽_𝑏→𝑋 be an extension of 𝑓_{𝜏𝛼(𝜆)} defined by 𝑓^∗_{𝜏𝛼(𝜆)}(𝑠)=𝑓𝑠+𝜏_𝛼(𝜆),𝑠∈𝐽_𝑏.(28) We can easily see that {𝑓^∗_{𝜏𝛼(𝜆)}∶𝜆∈𝐷_𝑏} is a bounded net in (𝐶_𝑝𝑐(𝐽_𝑎,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∈𝐷_𝑏 for which the following conditions are satisfied.(i)If 𝜆,𝜇∈𝐷_𝑏 and 𝜆,𝜇≧𝜆₀, then 𝑓_{𝜏𝛼(𝜆)}(𝑡)−𝑓_{𝜏𝛼(𝜇)}(𝑡)∈𝜀3𝑉∀𝑡∈𝐽_𝑎,or𝜌_𝑉𝑓_{𝜏𝛼(𝜆)}(𝑡)−𝑓_{𝜏𝛼(𝜇)}(𝑡≤𝜀)3∀𝑡∈𝐽_𝑎.(29)(ii)In case 𝜆∈𝐷_𝑏 and 𝜆≧𝜆₀, 𝑏+𝜏_𝛼(𝜆)≧𝑟(𝑛,𝑉). For any 𝑠∈𝐽_𝑏, if 𝜆,𝜇≧𝜆₀ (as in case (i)), we then have by (2), (28), and (29) that 𝜌_𝑊𝑓^∗_{𝜏𝛼(𝜆)}(𝑠)−𝑓^∗_{𝜏𝛼(𝜆)}(𝑠)≤𝜌_𝑉𝑓𝑠+𝜏_𝛼(𝜆)−𝑓𝑠+𝜏_𝛼(𝜆)+𝜏+𝜌_𝑉𝑓_{𝜏𝛼(𝜆)}(𝑠+𝜏)−𝑓_{𝜏𝛼(𝜇)}(𝑠+𝜏)+𝜌_𝑉𝑓𝑠+𝜏_𝛼(𝜇)+𝜏−𝑓𝑠+𝜏_𝛼(𝜇)≤𝜀3+𝜀3+𝜀3=𝜀.(30)So {𝑓_{𝜏𝛼(𝜆)}}_{𝜆∈𝐷𝑏} is a Cauchy net in (𝐶_𝑝𝑐(𝐽_𝑎,𝐸),𝑢). Since it is also bounded and (𝐶_𝑝𝑐(𝐽_𝑎,𝐸),𝑢) is quasi-complete, it converges uniformly on 𝐽_𝑏 to a function 𝜑_𝑏∈(𝐶_𝑝𝑐(𝐽_𝑎,𝐸),𝑢). Clearly, we have that 𝜑_𝑏∣𝐽_𝑎=𝜑. If 𝑐∈ℝ with 𝑐≦𝑏 and if 𝜆_𝑐 is any element in 𝐷 for which 𝑐+𝜏_𝛼(𝜆)≧𝑎 for all 𝜆∈𝐷 such that 𝜆≧𝜆₀, then the corresponding net {𝑓_{𝜏𝛼(𝜆)}}_{𝜆≧𝜆𝑐} of extensions from 𝐽_𝑎 to 𝐽_𝑐 will converge in 𝐶_𝑝𝑐(𝐽_𝑐,𝐸) to a function 𝜑_𝑐. Clearly 𝜑_𝑐=𝜑_𝑏 on 𝐽_𝑏. Define a function 𝑔∶ℝ→𝐸 by 𝑔(𝑡)=𝜑(𝑡),𝑡∈𝐽_𝑎𝜑_𝑡(𝑡),𝑡∈ℝ⧵𝐽_𝑎.(31) Then 𝑔 is well defined and continuous on ℝ and 𝑔∣𝐽_𝑎=𝜑.
We claim that 𝑔 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⧵𝐽_𝑎 with 𝑏≦min{𝑡,𝑡+𝜏}, and then take 𝜆∈𝐷 such that(a)𝑏+𝜏_𝛼(𝜆)≧𝑟(𝑛,𝑊)+|𝜏|,(b)𝜑_𝑏(𝑠)−𝑓(𝑠+𝜏_𝛼(𝜆))∈(𝜀/3)𝑈 for every 𝑠∈𝐽_𝑏.
Then, by (2), (31), and (b), 𝜌_𝑊𝑔(𝑡+𝜏)−𝑔(𝑡)≤𝜌_𝑈𝜑_𝑏(𝑡+𝜏)−𝑓𝑡+𝜏+𝜏_𝛼(𝜆)+𝜌_𝑈𝑓𝜏+𝜏_𝛼(𝜆)+𝜏−𝑓𝜏+𝜏_𝛼(𝜆)+𝜌_𝑈𝑓𝜏+𝜏_𝛼(𝜆)−𝜑_𝑏(𝑡)≤𝜀3+𝜀3+𝜀3=𝜀,(32) and so 𝑔 is almost periodic.
It remains to show that ℎ=𝑓−𝑔∣𝐽_𝑎 vanishes at infinity on 𝐽_𝑎. 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 𝑓_{𝑡𝛼(𝜆)}(𝑡)−𝜑(𝑡)∈𝑉∀𝑡∈𝐽_𝑎;(33) we may assume that 𝛼(𝜆)=(𝑚,𝑉), where 𝑚∈ℕ. Thus, if 𝑡∈𝐽_{𝑟(𝑚,𝑉)}, then ℎ(𝑡)(𝑡)(𝑡)=𝑓(𝑡)=𝑓−𝜑−𝑓𝑡+𝜏_𝛼(𝜆)+𝑓𝑡+𝜏_𝛼(𝜆)−𝜑(𝑡)∈𝑉+𝑉⊆𝑊.(34)
Finally, we show that the functions 𝑔 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∣𝐽_𝑎∈𝐶₀(𝐽_𝑎,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 {𝜔_𝛼}_𝛼 in ℝ, we apply Theorem 15 (twice) to obtain a subnet {𝜔_𝛼(𝜆)}_𝜆 such that the corresponding nets of translates {𝑔_{𝜔𝛼(𝜆)}} both converge in (𝐶_𝑝𝑐(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 𝑑𝑥(𝑡)[𝑥(𝑡𝑥(0)𝑑𝑡=𝐴)],𝑡>0,=𝑥₀()∈𝐸,CP 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 as 𝑡→∞. In this regard, a useful concept is the so-called positive 𝜔-limit set 𝜔⁺𝑥₀=𝑦∈𝐸∶∃0≤𝑡_𝑛⟶∞suchthat𝑇𝑡_𝑛𝑥_0.⟶𝑦(35) 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)={𝑇_𝜔(⋅)𝑥₀∶𝜔>0}, 𝑇_𝜔(𝑡)𝑥₀=𝑇(𝑡+𝜔)𝑥₀, of all translates of the motion is a relatively compact subset of the space (𝐶_𝑏(ℝ⁺,𝐸),∥⋅∥_∞). This observation raises the following problem.  (∗) Characterize those 𝑓∈𝐶_𝑏(ℝ⁺,𝐸) for which 𝐻⁺(𝑓) is relatively compact in (𝐶_𝑏(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 𝑇(𝑡)(𝑥)=lim_𝑛→∞𝑡1−𝑛𝐴⁻¹(𝑥),(36) 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.

Acknowledgments

This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Project no. 97/130/1432. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are also grateful to the referee for several helpful comments.