The Space of Continuous Periodic Functions Is a Set of First Category in 𝐴𝑃(𝑋)

Abstract

We prove that the space of continuous periodic functions is a set of first category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of first category in the space of almost automorphic functions.

1. Introduction

Since the last century, the study on almost periodic type functions and their applications to evolution equations has been of great interest for many mathematicians. There is a large literature on this topic. Several books are especially devoted to almost periodic type functions and their applications to differential equations and dynamical systems. For example, let us indicate the books of Amerio and Prouse [1], Bezandry and Diagana [2], Bohr [3], Corduneanu [4], Diagana [5], Fink [6], Levitan and Zhikov [7], N’Guérékata [8, 9], Pankov [10], Shen and Yi [11], Zaidman [12], and Zhang [13].

Although almost periodic functions have a very wide range of applications now, it seems that giving an example of almost periodic (not periodic) functions is more difficult than giving an example of periodic functions. Also, there is a similar problem for almost automorphic functions. In this paper, we aim to compare the “amount” of almost periodic functions (not periodic) with the “amount” of continuous periodic functions, and we also discuss the related problems for almost automorphic functions.

2. Main Results

Throughout the rest of this paper, we denote by ℝ the set of real numbers, by 𝑋 a Banach space, and by 𝐶(ℝ,𝑋) the set of all continuous functions 𝑓∶ℝ→𝑋.

Definition 1 (see [4]). A function 𝑓∈𝐶(ℝ,𝑋) is called almost periodic if, for every 𝜀>0, there exists 𝑙(𝜀)>0 such that every interval of length 𝑙(𝜀) contains a number 𝜏 with the property that sup_𝑡∈ℝ𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀. (1) We denote the collection of all such functions by 𝐴𝑃(𝑋).

Recall that 𝐴𝑃(𝑋) is a Banach space under the supremum norm.

Definition 2. A function 𝑓∈𝐶(ℝ,𝑋) is called periodic if there exists 𝑙>0 such that 𝑓(𝑡+𝑙)=𝑓(𝑡),∀𝑡∈ℝ. (2) Here, 𝑙 is called a period of 𝑓. We denote the collection of all such functions by 𝑃(𝑋). For 𝑓∈𝑃(𝑋), we call 𝑙₀ the fundamental period if 𝑙₀ is the smallest period of 𝑓.

Remark 3. Similar to the proof in [4, page 1], it is not difficult to show that if 𝑓∈𝑃(𝑋) is not constant, and then 𝑓 has the fundamental period.

Definition 4 (see [8]). A function 𝑓∈𝐶(ℝ,𝑋) is called almost automorphic if, for every real sequence (𝑠_𝑚), there exists a subsequence (𝑠_𝑛) such that 𝑔(𝑡)=lim_𝑛→∞𝑓𝑡+𝑠_𝑛 (3) is well defined for each 𝑡∈ℝ and lim_𝑛→∞𝑔𝑡−𝑠_𝑛=𝑓(𝑡) (4) for each 𝑡∈ℝ. Denote by 𝐴𝐴(𝑋) the set of all such functions.

Recall that there exists an almost automorphic function which is not almost periodic, for instance, the following function: 𝑓(𝑡)1=sin2+cos𝑡+cos√2𝑡,𝑡∈ℝ. (5)

Before the proof of our main results, we need to recall the notion about the first category.

Definition 5 (see [14]). Let 𝑆 be a topological space. A set 𝐸⊂𝑆 is said to be nowhere dense if its closure has an empty interior. The sets of the first category in 𝑆 are those that are countable unions of nowhere dense sets. Any subset of 𝑆 that is not of the first category is said to be of the second category in 𝑆.

Theorem 6. 𝑃(𝑋) is a set of first category in 𝐴𝑃(𝑋).

Proof. For 𝑛=1,2,…, we denote 𝑃_𝑛=𝑓∈𝐶(ℝ,𝑋)∶thereexists𝑙∈[𝑛,𝑛+1]suchthat𝑓(𝑡+𝑙)=𝑓(𝑡)∀𝑡∈ℝ. (6)
Then, it is easy to see that 𝑃(𝑋)=^∞_𝑛=1𝑃_𝑛. (7) We divide the remaining proof into two steps.
Step??1. Every 𝑃_𝑛 is a closed subset of 𝐴𝑃(𝑋).
Let 𝑓∈𝐴𝑃(𝑋)⧵𝑃_𝑛. Then, for every 𝑙∈[𝑛,𝑛+1], there exists 𝑡_𝑙∈ℝ such that 𝑓(𝑡_𝑙+𝑙)≠𝑓(𝑡_𝑙). Denote 𝜀_𝑙1∶=4𝑓𝑡_𝑙+𝑙−𝑓𝑡_𝑙>0,𝑙∈[𝑛,𝑛+1]. (8) In addition, due to the continuity of 𝑓, for every 𝑙∈[𝑛,𝑛+1], there exists 𝛿_𝑙>0 such that 𝑓𝑡_𝑙+𝑠−𝑓𝑡_𝑙≥3𝜀_𝑙,∀𝑠∈𝑙−𝛿_𝑙,𝑙+𝛿_𝑙. (9) Obviously, we have [𝑛,𝑛+1]⊂_{[]𝑙∈𝑛,𝑛+1}𝑙−𝛿_𝑙,𝑙+𝛿_𝑙. (10) Then, by the Heine-Borel theorem, there exists 𝑙₁,…,𝑙_𝑘∈[𝑛,𝑛+1] such that [𝑛,𝑛+1]⊂^𝑘_𝑖=1𝑙_𝑖−𝛿_𝑙𝑖,𝑙_𝑖+𝛿_𝑙𝑖, (11) where 𝑘 is a fixed positive integer. Letting 𝜀=min_{1≤𝑖≤𝑘}{𝜀_𝑙𝑖}, and 𝑁𝑓,𝜀∶=𝑔∈𝐴𝑃(𝑋)∶𝑔−𝑓_{𝐴𝑃(𝑋)}<𝜀, (12) for every 𝑔∈𝑁(𝑓,𝜀), we claim that 𝑔∉𝑃_𝑛. In fact, for every 𝑙∈[𝑛,𝑛+1], there exists 𝑖∈{1,…,𝑘} such that 𝑙∈𝑙_𝑖−𝛿_𝑙𝑖,𝑙_𝑖+𝛿_𝑙𝑖. (13) Then, by (9), we have 𝑓𝑡_𝑙𝑖+𝑙−𝑓𝑡_𝑙𝑖≥3𝜀_𝑙𝑖≥3𝜀, (14) which yields that 𝑔𝑡_𝑙𝑖+𝑙−𝑔𝑡_𝑙𝑖≥𝑓𝑡_𝑙𝑖+𝑙−𝑓𝑡_𝑙𝑖−𝑓𝑡_𝑙𝑖+𝑙−𝑔𝑡_𝑙𝑖−𝑓𝑡+𝑙_𝑙𝑖−𝑔𝑡_𝑙𝑖≥3𝜀−𝜀−𝜀=𝜀>0, (15) where ‖𝑔−𝑓‖_{𝐴𝑃(ℝ)}<𝜀 was used. So, we know that 𝑁(𝑓,𝜀)⊂𝐴𝑃(𝑋)⧵𝑃_𝑛, which means that 𝑃_𝑛 is a closed subset of 𝐴𝑃(𝑋).
Step??2. Every 𝑃_𝑛 has an empty interior.
It suffices to prove that, for every 𝑓∈𝑃_𝑛 and 𝛿>0, 𝑁(𝑓,𝛿)⋂(𝐴𝑃(𝑋)⧵𝑃_𝑛)≠∅. Now let 𝑓∈𝑃_𝑛 and 𝛿>0. In the following, we discuss two cases.
Case I. 𝑓 is constant.
We denote 𝑓_𝛿(𝑡)=cos𝑡+cos√2𝑡3⋅𝛿⋅𝑥₀+𝑓(𝑡),𝑡∈ℝ, (16)
where 𝑥₀∈𝑋 is some constant with ‖𝑥₀‖=1. Then 𝑓_𝛿∈𝑁(𝑓,𝛿) and 𝑓_𝛿∉𝑃_𝑛 since 𝑓_𝛿 is not periodic.
Case II. 𝑓 is not constant.
By Remark 3, 𝑓 has a fundamental period 𝑙₀. We denote 𝑓_𝛿(𝑡)(𝑡)𝑡=𝑓+𝑓𝜋⋅𝛿𝑀_𝑓,𝑡∈ℝ, (17) where 𝑀_𝑓=sup_𝑡∈ℝ‖𝑓(𝑡)‖. Obviously, 𝑓_𝛿∈𝑁(𝑓,𝛿). Also, we claim that 𝑓_𝛿∉𝑃_𝑛. In fact, if this is not true, then there exists 𝑇∈[𝑛,𝑛+1] such that 𝑓_𝛿(𝑡+𝑇)=𝑓_𝛿(𝑡),𝑡∈ℝ, (18) that is, 𝑓(𝑡+𝑇)+𝑓𝑡+𝑇𝜋⋅𝛿𝑀_𝑓(𝑡)𝑡=𝑓+𝑓𝜋⋅𝛿𝑀_𝑓,𝑡∈ℝ. (19) Let 𝐹₁(𝑡)()(𝑡),𝐹=𝑓𝑡+𝑇−𝑓₂(𝑡)=𝛿𝑀_𝑓𝑓𝑡𝜋−𝑓𝑡+𝑇𝜋,𝑡∈ℝ. (20) Then 𝐹₁(𝑡)≡𝐹₂(𝑡). If 𝐹₁(𝑡)≡𝐹₂(𝑡)≡𝐶, where 𝐶 is a fixed constant, then 𝑓(𝑡+𝑇)=𝑓(𝑡)+𝐶,𝑡∈ℝ, (21) which yields 𝑓()(0)𝐶=𝑘𝑇−𝑓𝑘→0,𝑘→∞, (22) since 𝑓 is bounded. Thus, we have 𝑓()(𝑡)𝑡𝑡+𝑇=𝑓,𝑓𝜋=𝑓𝑡+𝑇𝜋,𝑡∈ℝ. (23) Noting that 𝑙₀ is the fundamental period of 𝑓 and 𝜋𝑙₀ is the fundamental period of 𝑓(⋅/𝜋), there exist two positive integers 𝑝and𝑞 such that 𝑝𝑙₀=𝑇=𝑞𝜋𝑙₀, (24) that is, 𝜋=𝑝/𝑞, which is a contradiction. If 𝐹₁=𝐹₂ is not constant, then, by Remark 3, we can assume that 𝑇₀ is the fundamental period of 𝐹₁ and 𝐹₂. Noting that 𝑙₀ is a period of 𝐹₁ and 𝜋𝑙₀ is a period of 𝐹₂, similar to the above proof, we can also show that 𝜋 is a rational number, which is a contradiction.
In conclusion, 𝑃(𝑋) is countable unions of closed subsets with empty interior. So 𝑃(𝑋) is a set of first category.

Remark 7. Since 𝐴𝑃(𝑋) is a set of second category, it follows from Theorem 6 that 𝐴𝑃(𝑋)⧵𝑃(𝑋) is a set of second category, which means that, to some extent, the “amount” of almost periodic functions (not periodic) is far more than the “amount” of continuous periodic functions.

Theorem 8. 𝐴𝑃(𝑋) is a set of first category in 𝐴𝐴(𝑋).

Proof. Firstly, 𝐴𝑃(𝑋) is a closed subset of 𝐴𝐴(𝑋). Secondly, 𝐴𝑃(𝑋) has an empty interior in 𝐴𝐴(𝑋). In fact, letting 𝜑(𝑡)1=sin2+cos𝑡+cos√2𝑡,𝑡∈ℝ, (25) for every 𝑓∈𝐴𝑃(𝑋) and 𝛿>0, we have 𝑓_𝛿∉𝐴𝑃(𝑋) and 𝑓_𝛿∈𝑁(𝑓,𝛿), where 𝑓_𝛿(𝑡)=𝑓(𝑡)+𝛿𝑥₀𝑀_𝜑𝜑(𝑡),𝑡∈ℝ,𝑀_𝜑=sup_𝑡∈ℝ𝜑(𝑡), (26) and 𝑥₀∈𝑋 is some constant with ‖𝑥₀‖=1. This completes the proof.

Remark 9. By Theorem 8, 𝐴𝐴(𝑋)⧵𝐴𝑃(𝑋) is a set of second category in 𝐴𝐴(𝑋), which means that, to some extent, the “amount” of almost automorphic functions (not almost periodic) is far more than the “amount” of almost periodic functions.

Acknowledgments

H.-S. Ding acknowledges support from the NSF of China (11101192), the Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), and the Jiangxi Provincial Education Department (GJJ12173).